Emerging Topics on Differential Geometry and Graph Theory [1 ed.] 9781611220698, 9781607410119

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Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved. Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved. Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

MATHEMATICS RESEARCH DEVELOPMENTS SERIES

EMERGING TOPICS ON DIFFERENTIAL GEOMETRY AND GRAPH THEORY

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MATHEMATICS RESEARCH DEVELOPMENTS SERIES Boundary Properties and Applications of the Differentiated Poisson Integral for Different Domains Sergo Topuria 2009. ISBN: 978-1-60692-704-5 Quasi-Invariant and Pseudo-Differentiable Measures in Banach Spaces Sergey Ludkovsky 2009. ISBN: 978-1-60692-734-2 Operator Splittings and their Applications Istvan Farago and Agnes Havasiy 2009. ISBN: 978-1-60741-776-7 Measure of Non-Compactness for Integral Operators in Weighted Lebesgue Spaces Alexander Meskhi 2009. ISBN: 978-1-60692-886-8

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Mathematics and Mathematical Logic: New Research Peter Milosav and Irene Ercegovaca (Editors) 2009. ISBN: 978-1-60692-862-2 Role of Nonlinear Dynamics in Endocrine Feedback Chinmoy K. Bose 2009. ISBN: 978-1-60741-948-8 Geometric Properties and Problems of Thick Knots Yuanan Diao and Claus Ernst 2009. ISBN: 978-1-60741-070-6 Lie Groups: New Research Altos B. Canterra (Editors) 2009. ISBN: 978-1-60692-389-4 Emerging Topics on Differential Geometry and Graph Theory Lucas Bernard and Francois Roux (Editors) 2010. ISBN: 978-1-60741-011-9

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MATHEMATICS RESEARCH DEVELOPMENTS SERIES

EMERGING TOPICS ON DIFFERENTIAL GEOMETRY AND GRAPH THEORY

LUCAS BERNARD AND

FRANCOIS ROUX Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

EDITORS

Nova Science Publishers, Inc. New York

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Copyright © 2010 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Any parts of this book based on government reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of such works.

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Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA Emerging topics on differential geometry and graph theory / [edited by] Lucas Bernard and François Roux. p. cm. Includes index. ISBN 978-1-61122-069-8 (eBook) 1. Geometry, Differential. 2. Graph theory. I. Bernard, Lucas, 1962- II. Roux, François, 1960QA641.E636 2009 516.3'6--dc22 2009016576

Published by Nova Science Publishers, Inc. ; New York

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CONTENTS

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Preface

vii

Chapter 1

Applications of Graph Theory in Mechanism Analysis T.J. Li, W.Q. Cao and T.H. Yan

Chapter 2

A Categorical Perspective on Connections with Application in the Formulation of Functorial Physical Dynamics Elias Zafiris

35

Chapter 3

The Notion of CR Hamiltonian Flows and the Local Embedding Problem of CR Structures Takao Akahori

79

Chapter 4

Equivariant Methods in Combinatorial Geometry Pavle V.M. Blagojević

95

Chapter 5

Something New about Reconstruction Elena Konstantinova

135

Chapter 6

Coalescence of Graphs A.A. Lushnikov

187

Chapter 7

Graph Analysis with Application to Economics André A. Keller

223

Chapter 8

Application of Hilbert Spaces to the Stability Study of Flows on a Sphere Yuri N. Skiba

299

Chapter 9

A Functorial Approach to the Infinitesimal Theory of Groupoids Hirokazu Nishimura

345

Chapter 10

A Model of Higher-Order Concurrent Programs Based on Graph Rewriting Masaki Murakami

373

Chapter 11

Protecting the Vertices of a Graph William F. Klostermeyer

395

Index Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

1

411

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PREFACE Differential geometry is a mathematical discipline that uses the methods of differential and integral calculus to study problems in geometry. Graph theory is also a growing area in mathematical research. In mathematics and computer science, graph theory is the study of mathematical structures used to model pairwise relations between objects from a certain collection. This book presents various theories and applications in both of these mathematical fields. Included are the concepts of dominating sets, one of the most widely studied concepts in graph theory, some current developments of graph theory in the fields of planar linkage mechanisms and geared linkage mechanisms, lie algebras and the application of CR Hamiltonian flows to the deformation theory of CR structures. Graph theory is a very important tool to analyze the various mechanisms. Chapter 1 presents some current developments of graph theory in the fields of planar linkage mechanisms and geared linkage mechanisms. First the topological graphs including the simple graph, bicolored graph and tricolored graph are introduced to describe the topological relationship between the kinematic pairs and the links in the planar linkage mechanisms with simple and multiple joints and geared linkage mechanisms, the mathematic expressions such as incidence matrix and adjacency matrix are established to make the graph-based models of mechanisms easy operations. Then the key problems of mechanism analysis using graph theory are settled including the number of circuits, circuit identification, isomorphism detection, structural decomposition, etc. Finally, the general approaches are derived for the structural and kinematic analysis of mechanisms based on their structural topological characteristics. Some examples are used to illustrate the procedures using graph theory to analyze the mechanisms. The mechanism of differential calculus and differential geometry is based on the fundamental notion of connection on a module over a commutative unital ring of scalars defined together with the associated de Rham complex. From a categorical point of view Chapter 2 shows that the generation of this mechanism is a consequence of the existence of a monad-comonad pair arising from the adjunctive correspondence between the inverse algebraic processes of extending and restricting the scalars. The categorical remodeling of the notion of connection as well as of the associated de Rham complex in a functorial way permits the applicability of algebraic differential calculus in topoi independently of the existence of any background manifold. This is of particular importance for the formulation of dynamics in physical theories, where the adherence to such a substratum is problematic due to singularities or other topological defects, for example in quantum gravity. Thus, the

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Lucas Bernard and Francois Roux

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adaptation of the functorial mechanism of connections in appropriate topoi, instantiating generalized localization environments of physical observables, induces a consistent covariant framework of dynamics in the regime of these topoi. In studying the deformation theory of CR structures, the notion of the Hamiltonian vector field is found. In Chapter 3, the authors develop this notion. Namely, the notion of CRHamiltonian flows is introduced, and they apply this notion to the local embedding problem. Partition problems are classical problems of the combinatorial geometry whose solutions often rely on the methods of the equivariant topology. The k-fan partition problems introduced in [11] and first discussed by equivariant methods in [2], [3] have forced some hard concrete combinatorial calculations in equivariant cohomology [5], [4]. These problems can be reduced, by the beautiful scheme of Bárány andMatoušek, [2], to topological problems of the existence of Δ2n equivariant maps V2(Ρ3) → Wn− U Α(α) from a Stiefel manifold of all

orthonormal 2-frames in Ρ3 to complements of appropriate arrangements. In Chapter 4 the authors present a set of techniques, based on the equivariant obstruction theory, which can help in answering the question of the existence of a equivariant map to a complement of an arrangement. With the help of the target extension scheme, introduced in [5], the authors are able to deal with problems where the existence of the map depends on more then one obstruction. The introduced techniques, with an emphasis on computation, are applied on the known results of the fan partition problems. In Chapter 5 the authors review new recent results on the vertex reconstruction and graph reconstruction from all metric balls of a given radius in a graph. These two new problems are not related to the famous Ulam’s vertex reconstruction conjecture and differ in principle from traditional packing and covering problems. The graph reconstruction problem is considered for the metric balls of radius two, and necessary and sufficient conditions for exact reconstruction of a graph are shown. It is also shown that the reconstruction of a graph from all its metric balls of radius greater than two requires stronger restrictions. The authors also show what the connection is of this problem with the chemical problem of structure reconstruction of unknown compounds from NMR spectroscopy data. The vertex reconstruction problem comes from coding theory and concerns a local reconstruction of a vertex from all metric balls of a given radius of a graph. Initially, this problem was studied for the Hamming and Johnson graphs which are distance–regular, and the first one is a Cayley graph. This problem is much more complicated when a Cayley graph is not distance–regular. The results are presented for simple, regular and distance–regular graphs, and for some Cayley graphs on the symmetric and hyperoctahedral groups which are not distance–regular. The authors also show why these graphs and groups are of special interest in molecular biology, computer science and physics. Open questions and conjectures are also discussed. Chapter 6 considers an initially empty (no edges) graph of order M that is assumed to evolve by adding one edge at a time. This edge can connect either two linked components and form a new component of a larger order (coalescence of graphs) or increase (by one) the number of edges in a given linked component. Assuming the rate of appearance of an extra– edge in the graph to be known, the Master equation governing the time evolution of the probability W(Q, t) to find the random graph in a given state Q is formulated. The latter is given by a set of the population numbers of linked components of order g with υ edges. This equation is then reformulated in terms of the generating functional for the probability W(Q, t). This functional is shown to meet the evolution equation which is just a linear second–order

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Preface

ix

multidimensional (generally infinite dimensional) partial differential equation. The analogy between coagulation processes and the evolution of random graphs is tracked. The coalescence alone is shown to be responsible for the kinetics of growth of the tree–like population. The respective evolution equation for the generating functional is simpler than in theory of random graphs. The analogy with coagulating systems allows one to consider very large graphs in the thermodynamic limit. At the initial stage of the stochastic coalescence of graphs all graphs are trees (in the thermodynamic limit) and only after the critical time when the giant component emerges the cycles appear in it. The analogy with the sol–gel transition in coagulating systems is traced. The exact solution of the evolution equation for the generating functional of the evolving random graph is obtained. The spectrum (the average population numbers of linked components) is found. This approach is then applied to random bipartite graphs. The exact expression for the generating functional of evolving bipartite graph is derived. Two Appendices introduce and analyze two special families of polynomials closely associated with the uni–partite and bipartite graphs. These polynomials play extremely important role in theory of graphs. The structural properties of large size nonlinear macroeconometric models are essential to their comprehensive analysis, to their adequate simulations of economic policies, and even to their iterative resolution. In Chapter 7, the concepts and techniques of the graph theory are used to discover such essential properties. Basic elements of graph theory are first introduced, using small-size theoretical models and sub-models. Further advanced topics are developed thereafter, such as the perfect matchings, the connectivity of weighted graphs and the circuit enumeration. A large-size macroeconometric model such as the FAIR model for the USA is also analysed using directed graphs. Avertex typology is deduced from the properties of a giant strong connected component. Exhaustive lists of circuits illustrate a great interdependency of the variables. The temporal dimension of such econometric model with lags is also considered. One appendix is devoted to the performances of specialized packages for graphs of the software Mathematica V6.0. The reference Klein-Goldberger model is used for that demonstration. The main properties of spherical harmonics and associated Legendre functions are briefly surveyed. Geographical coordinate maps for the sphere are defined and the well-known theorem about the partition of unity is given which is an important tool in the theory of integration of functions on smooth compact manifolds. Derivatives Ds and Λ s of real degree s of smooth functions on the unit sphere are defined, and a family of Hilbert spaces Ηs of generalized functions having fractional derivatives on the sphere is introduced. Instead of the modulus of continuity [1-3], in Chapter 8, the Hilbert spaces of functions are defined by means of multiplier operators [4-7]. Orthogonal projectors on the subspace Ηn of homogeneous spherical polynomials of degree n and on the subspace ΠN of spherical polynomials of degree n ≤ Ν are defined. Some structural properties of Hilbert spaces Ηs including various embedding theorems are given, and the rate of convergence of FourierLaplace series of functions of Hs is estimated. As applications of theoretical results, both the global asymptotic stability and the normalmode stability of incompressible flows on a rotating sphere are considered. In particular, conditions for the global asymptotic stability of solutions to the barotropic vorticity equation are given, and the spectral approximation (convergence of eigenvalues and eigenvectors of

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Lucas Bernard and Francois Roux

discrete spectral problems) in the numerical normal-mode stability study of nondivergent viscous flows on a rotating sphere is examined. Lie algebroids are by nomeans natural as an infinitesimal counterpart of groupoids. In Chapter 9 the authors propose a functorial construction called Nishimura algebroids for an infinitesimal counterpart of groupoids. Nishimura algebroids, intended for differential geometry, are of the same vein as Lawvere’s functorial notion of algebraic theory and Ehresmann’s functorial notion of theory called sketches. The authors study totally intransitive Nishimura algebroids in detail. Finally, the authors show that Nishimura algebroids naturally give rise to Lie algebroids. Chapter 10 presents a formal model of concurrent systems with higher-order communication to transfer program codes as data between processes. A number of models [1,14,15] are reported as models of higher-order communication and most of them are based on process algebra. However, as the authors reported [7,8], it is not easy to represent the scopes of names of communication channels precisely using process algebra. The authors present a model of concurrent programs with higher order communication that can represent the scopes of names precisely. The model presented here is an extension of the model reported in [8] that is based on graph rewriting. The authors define the equivalence relation such that if the equivalence holds on two programs then they are equivalent not only on their behaviors but on the scopes of names also. This article also presents a compilation of higherorder concurrent programs into first-order name passing programs on the graph rewriting model. Dominating sets are one of the most widely studied concepts in graph theory. If one imagines that there is a guard located at each of the vertices in a dominating set, one can see that those guards can “protect” all the vertices of the graph. That is, we think of a guard as being able to protect the vertex it is on as well as neighboring vertices. Several recent variations of dominating sets have been proposed to protect the vertices of a graph against sequences of attacks at vertices, since a dominating set may only be effective against a single attack at one vertex. That is, after a guard moves to defend the attack, the resulting configuration of guards is not necessarily a dominating set, and thus may be unable to fully protect the graph against subsequent attacks. In Chapter 11, the authors review key results on domination and survey results on Roman domination, weak Roman domination, secure domination, and eternal domination, as well as variations on these types of domination and related problems such as the kserver problem. One of our main goals will be to compare the minimum number of guards needed in these various scenarios with other graph parameters such as the domination number and independence number. A number of open problems are stated.

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Chapter 1

APPLICATIONS OF GRAPH THEORY IN MECHANISM ANALYSIS T.J. Li1, W.Q. Cao 2 and T.H. Yan3 1

School of Electromechanical Engineering, Xidian University, Shaanxi Province, Xi’an 710071, China 2 Department of Mechanical Engineering, Xi'an University of Technology, Shaanxi Province, Xi'an 710048, China 3 R&D Center, Shanghai Micro-Electronics Equipment Co, Ltd, Shanghai 201203, China

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Abstract Graph theory is a very important tool to analyze the various mechanisms. This chapter presents some current developments of graph theory in the fields of planar linkage mechanisms and geared linkage mechanisms. First the topological graphs including the simple graph, bicolored graph and tricolored graph are introduced to describe the topological relationship between the kinematic pairs and the links in the planar linkage mechanisms with simple and multiple joints and geared linkage mechanisms, the mathematic expressions such as incidence matrix and adjacency matrix are established to make the graph-based models of mechanisms easy operations. Then the key problems of mechanism analysis using graph theory are settled including the number of circuits, circuit identification, isomorphism detection, structural decomposition, etc. Finally, the general approaches are derived for the structural and kinematic analysis of mechanisms based on their structural topological characteristics. Some examples are used to illustrate the procedures using graph theory to analyze the mechanisms.

1. Introduction A Kinematic Chain (KC) is a chain with mobility, without any fixed link. In the bibliography it is frequently referred to as Basic Kinematic Chain (BKC). When one or more links of a BKC are fixed, it becomes either a mechanism if at least two links retain mobility, or a

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structure if it does not have mobility. A mechanism is a mechanical system that has the purpose of transmitting motion and/or force from a source to an output body. The research of mechanisms focus on two aspects: analysis and synthesis [1,2]. The analysis of mechanisms includes structural analysis, kinematic analysis, static analysis and dynamic analysis etc, the structural analysis is the most critical and challenging step, which can reveal the internal rule of the mechanism composition, and can help us to understand the structure of a mechanism, to analyze the transmission route in the mechanism, and to improve our ability in mechanism design. Mechanism synthesis includes number, type and dimension synthesis. Number and type synthesis together are called structural synthesis, which consists in selecting the type of mechanism and the type and number of its component parts: it could be linkage, gear, cam, belt, etc. or a combination thereof. Dimensional synthesis aims to determine the proper dimensions of parts so that the mechanism can perform the given motion, such as path following, rigid-body guidance and function generation or more complex tasks. The dimensional synthesis consists in computing the links lengths and pivots positions. This task is also related to that of optimization of the mechanism. The dimensional synthesis phase and all the subsequent stages of detailing design are very costly, so it is essential to have good topology alternatives as output of the type synthesis stage in order to avoid repeated simulations and also not to leave some alternatives without being explored. In this work we are limited to planar mechanisms of the linkage type and the geared linkage type. A linkage consists of links (or bars), usually considered rigid, which are connected by joints such as pins (or revolutes) or prismatics, to form open or closed chains. Linkages can be designed to perform complex tasks, such as nonlinear motion and force transmission. The geared linkage mechanism (GLM) combines the advantages of linkages and gear mechanisms, and can produce many kinds of complex motion, the corresponding KC is called the Geared Linkage Kinematic Chain (GLKC). The purpose of structural analysis is to disconnect the Assur groups from the mechanism and to determine their types and assembly order. The concept of ‘Assur groups’ was developed by Leonid Assur (1878-1920), a professor at the Saint-Petersburg Polytechnical Institute. In 1914 he published a treatise entitled “Investigation of plane bar mechanisms with lower pairs from the viewpoint of their structure and classification.” Now Assur groups are reported in research papers for diverse applications such as: position analysis, kinematic analysis, force analysis of mechanisms and others [1,2]. Central to Assur’s method is the decomposition of complex linkages and GLMs into fundamental, minimal pieces whose analyses could then be merged to give an overall analysis. Many of these approaches for Assur groups were developed from a range of examples, analyzed geometrically and combinatorially, but never defined with mathematical rigor. In order to analyze an existing mechanism or design a new mechanism, it is helpful to draw a simple diagram to indicate the kinematic relationship between links. Such a diagram is called the kinematic diagram of the mechanism. Kinematic diagrams are widely used for mechanism design and are especially helpful for the understanding of complicated mechanism topologies. It seems natural and advantageous to represent kinematic diagrams by links and pairs. But the process of mathematical manipulation of the kinematic diagram is very complex and difficult. For the mechanism analysis and/or synthesis, the understanding of the structural topological relationship is the most important part for engineers. Graph theory is a very important tool to analyze the topological properties of various mechanisms. In mathematics

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Applications of Graph Theory in Mechanism Analysis

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and computer science, graph theory is the study of graphs: mathematical structures used to model pairwise relations between objects from a certain collection [3,4]. A graph refers to a collection of vertices or nodes and a collection of edges that connect pairs of vertices. A graph may be undirected, meaning that there is no distinction between the two vertices associated with each edge, or its edges may be directed from one vertex to another. A graph structure can be extended by assigning a weight to each edge of the graph. Graphs with weights, or weighted graphs, are used to represent structures in which pairwise connections have some numerical values. Graph theory is widely used in network, chemistry, physics, sociology and life science [5]. Graph Theory has been used for analysis and enumeration of mechanisms by some researchers [6,7,8].This work presents some current developments of graph theory in the fields of Planar Linkage Mechanisms (PLMs) and GLMs. The graph representation of mechanisms is shown in section 2. The detection of isomorphism among KCs is developed in Section 3. The loop properties of KCs is developed in Section 4. The structural decomposition and kinematic analysis of mechanisms is developed in Section 5. The topics are developed with detailed examples.

2 Graph Representation of Mechanisms 2.1. Topological Graph Representation of Kcs with Simple Joints

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The graph G(E, V ) of a KC with simple joints is obtained by representing each link by a vertex and each kinematic pair by an edge connecting the corresponding vertices. So the theorems in graph theory can be used. The graph of an N-link chain can be represented by a Nth order symmetric zero-one matrix

A = (aij ) n×n

(1)

in which a ii = 0 for all i and aij = 1 or 0 depending on whether vertex i is connected directly to veritex j by an edge or not. For example, the six-link Stephenson chain of Fig. 1(a) is represented by the graph of Fig. 1(b), which is represented by the matrix ⎡0 ⎢1 ⎢ ⎢0 A=⎢ ⎢1 ⎢0 ⎢ ⎣1

1 0 1 0 1⎤ 0 1 0 0 0 ⎥⎥ 1 0 1 1 0⎥ ⎥ 0 1 0 0 0⎥ 0 1 0 0 1⎥ ⎥ 0 0 0 1 0⎦

Matrix A is the adjacency matrix of graph shown in Fig. 1 (b), which can also be considered as the link-link adjacency matrix of the chain of Fig. 1(a).

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(a)

(b)

Figure 1. Stephenson chain and graph representation.

The topological graph defined above cannot represent the topological relationship of a KC with multiple joints.

2.2. Bicolored Graph Representation of Kcs with Multiple Joints

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The bicolored graph is an alternative way to represent a KC with multiple joints [9]. In the bicolored graph, we describe a link by a kind of colored vertex drawn as a filled circle, and use another kind of colored vertex drawn as a nonfilled circle to represent a lower pair, the edge connection of vertices corresponds to the connection between the link and the kinematic pair in a KCs. An example is given in Fig.2, in which Fig. 2(a) shows a 8-bar one degree of freedom (DOF) linkage (8 links and 8 joints) and Fig. 2(b) shows its bicolored graph representation (notice here that there are 8 filled circles and 8 non-filled circles).

(a)

(b)

Figure 2. Bicolored graph for a KC with multiple joints.

The topological relationship between links and joints is more clear, as both links and joints are represented as vertices in the bicolored graph representation. The bicolored graph is an unweighted graph and thus relatively easier to be handled with graph theory. The bicolored graph representation can well facilitate a KC with simple joints. For example, Fig. 3(a) shows a 9-bar truss and Fig. 3(b) shows its bicolored graph representation.

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Applications of Graph Theory in Mechanism Analysis

(a)

5

(b)

Figure 3. Bicolored graph for a KC with simple joints.

It is evident that there exists a one-to-one correspondence between a KC and a topological graph. An incidence matrix of the bicolored graph for KCs is defined as

B = (bij ) n×m

(2)

where bij = 1 if link i is connected to joint j, and otherwise bij = 0 ; n is the number of

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links, and m is the number of joints. The corresponding adjacency matrix of the bicolored graph can be defined as

A = (aij ) ( n + mp )×( n + mp )

(3)

where aij = 1 if vertex i is incident to vertex j ( i ≠ j ), otherwise aij = 0 ; n is the number of vertices representing links, mp is the number of vertices representing joints in a bicolored graph. It is clear that the adjacency matrix can be derived from the incidence matrix for kinematic chains. Therefore the information stored in the incidence matrix should be sufficient for structural analysis or type synthesis.

2.3. Tricolored Graph Representation of Glms Tricolored graph is a kind of topological graph used to express the topological relationship between joints and links in a GLM [10]. We describe the link by a kind of colored vertex (here we use a filled circle points “●”), use another kind of colored vertex to represent lower pair (here it is described by an unfilled circle point “O”), and use another different kind of colored vertex to denote the higher pair (here it is described by an unfilled triangle point “ ”). The links and the kinematic pairs on them are connected with edges, that is, in the tricolored graph the edge connection of vertices corresponds to the connection between the

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link and the kinematic pair in a GLM. So the only topological tricolored graph for a GLM can be obtained. For example, Fig. 4(a) is a GLM of 4-bars, 3-gears and 6 links with one degree of freedom (DOF). Fig. 4(b) is its tricolored graph.

(a)

(b)

Figure 4. A GLM and its tricolored graph. (a) Kinematic diagram, (b) tricolored graph.

We define the incidence matrix of the tricolored graph as follows

M a = (aij ) n×mp

(4)

where aij = 1 if link i is incident on the lower pair j, aij = 2 if link i is incident on the

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higher pair j, and otherwise aij = 0 ; n is the total number of links in a GLM, mp is the total number of vertexes which denotes the number of kinematic pairs (lower and higher) in a GLM. Referring to Fig. 4, we have

⎡1 ⎢0 M a = A = ⎢⎢0 1 ⎢0 ⎢⎣0

1 1 0 0 0 0

0 1 1 0 1 0

0 0 1 1 0 1

2 0 0 0 2 0

0⎤ 0⎥ 0⎥ 0⎥ 2⎥ 2⎥⎦

We define the adjacency matrix of the tricolored graph as follows

M b = (bij )(n + mp )×(n + mp )

(5)

where bij = 1 if the vertex of link i is adjacency to the vertex of a lower pair j, bij = 2 if the vertex of link i is adjacency to the vertex of a higher pair j, and otherwise bij = 0 . When numbering the tricolored graph, the following rules should be complied with

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[1] Numbering the links with code from 1 to n ; [2] Numbering the kinematic pairs with code from n + 1 to n + mp , firstly numbering the joints of the lower pairs and then the higher pairs. Hence

⎡ 0 Mb = ⎢ T ⎣A

A⎤ 0 ⎥⎦

(6)

T

where A is the transferred matrix of A. In order to express the relationship between the links and joints quantitatively, we need to assign the certain values to the edges of the tricolored graph. The corresponding adjacency matrix is defined as follows:

M c = (c ij )(n + mp )x (n + mp )

(7)

where cij = 0 if the vertex of an input lower pair i is adjacent to the vertex of link j, cij = 1 if the vertex of link i is adjacent to the vertex of the lower pair j, cij = 2 if the vertex of link i is adjacent to the vertex of the higher pair j, and otherwise cij = ∞ .

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Fig. 5 is the weighted tricolored graph of the GLM shown in Fig. 4(a), and its adjacency matrix will be

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ Mc = ⎢ 0 ⎢1 ⎢ ⎢∞ ⎢∞ ⎢2 ⎢⎣∞

∞

∞ 1 1 ∞ ∞ ∞

∞ ∞ 1 1 ∞ ∞

0 ∞ ∞ ∞ ∞ ∞ ∞ 1 ∞ 1 ∞ 1 ∞ 2 ∞ ∞ 2 2

0 ∞ ∞ 0 ∞ ∞

1 ∞ ∞ 2 ∞⎤ 1 1 ∞ ∞ ∞⎥ ∞ 1 1 ∞ ∞⎥ ⎥ ∞ ∞ 1 ∞ ∞⎥ ∞ 1 ∞ 2 2⎥ ∞ ∞ 1 ∞ 2⎥ ⎥ ⎥ ⎥ ⎥ ∞ ⎥ ⎥ ⎥⎦

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T.J. Li, W.Q. Cao and T.H. Yan

According to the mathematical expressions of the tricolored graph, all circuits inside the mechanism can be found by the computer, along with the length of each circuit, which is the sum of weighted values of all the edges in a circuit.

2.4. Combinatorial Graph Representation of Glkcs Seeing that there exist not only lower pairs, but gear pairs in a GLKC, as shown in Fig. 6, a modified weighted topological graph is proposed to describe a GLKC [11], i.e., the vertexes and edges with weights represent the members and joints of GLKC respectively. The weight of each edge, we corresponds to the type of the joint. The relationship between the weights and the types of joints is given as follows

corresponding to a lower pair, and is not labeled, ⎧1, we = ⎨ ⎩n × 2, corresponding to n gear pairs. The weight of each vertex, wv corresponds to the type of member, the relationship between the weights and the types of members is given as follows

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⎧ g , if the member is formed by g ( g ≥ 2) gears fixed, wv = ⎨ ⎩0, otherwise, and is not labeled.

Figure 6. A GLKC.

Figure 7. Combinatorial graph of GLKC.

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With these conventions clearly understood we could obtain a modified weighted topological graph, here named a combinatorial graph. For example the combinatorial graph corresponding to the GLKC shown in Fig. 6 is shown in Fig. 7. It is evident that there exists a one-to-one correspondence between a GLKC and a combinatorial graph, thus the representation is exact and sole. Based on the combinatorial graph representation of a GLKC, the matrixes, with adjacency and incidence matrix as the most typical ones, can be used to represent a graph, so the study of a graph can be transformed into the study of the correspondent matrix. Because the incidence matrix cannot describe the weights of the vertexes in a combinatorial graph and the adjacency matrix is a symmetric, square one, we suggested a modified adjacency matrix, the Combinatorial Matrix (CM), which can represent the topological relationship among vertexes clearly. The matrix is defined as follows

CM = (aij )n×n

(8)

where

⎧w, if the weight of the edge incident with the vertex i and j is w; ⎪ aij = ⎨ g , if i = j, and the weight of the vertex is g , ⎪0, otherwise. ⎩

n ——the number of the members in GLKC.

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Referring to the combinatorial graph of Fig. 7, its CM will be

⎡0 ⎢1 ⎢ ⎢0 ⎢ CM = ⎢0 ⎢1 ⎢ ⎢1 ⎢0 ⎣

1 0 0 1 1 0⎤ 0 1 0 0 1 1⎥⎥ 1 0 1 0 0 1⎥ ⎥ 0 1 0 1 0 0⎥ 0 0 1 0 2 0⎥ ⎥ 1 0 0 2 2 2⎥ 1 1 0 0 2 0⎥⎦

3. Detection of Isomorphism Among Kcs and Glkcs 3.1. Detection of Isomorphism Among Kcs In the structural analysis and synthesis of mechanisms, an important and difficult problem is the identification of the topological structure of KCs, and then overcome the problem of isomorphism among KCs. The identification of isomorphism of KCs is of momentous significance for the structural synthesis and optimum seeking types. If the nonisomorphic KCs are mistaken for isomorphic ones, the new, valuable KCs are probably lost; if the isomorphic KCs are regarded as nonisomorphic ones, it leads to type repetition and there is no point. Since 1960’s a lot of methods have been proposed by many specialists for this problem.

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T.J. Li, W.Q. Cao and T.H. Yan

Among these methods, the characteristic polynomial of matrix was once used most widely and researched most deeply and lost effectiveness due to the lack of accuracy [12]. The minimum code [13], adjacent degree code[14], degree code[15] and identification code[16] etc. have been suggested. Besides, many other methods, such as hamming number and numerical strings technique [17,18], link’s adjacent-chain table [19], distance matrix using the distance concept[20] and symmetry group using the comparison matrix [21] etc. have also been presented .All of the above methods are used to detect isomorphism by using some structural information of KCs. However, with regard to these methods there is either lack of uniqueness and the basis of graph theory, or it takes too much time for determining isomorphism of KCs and graphs and is difficult to handle for computer .With this end in view, this work evolves a method using the powers of the adjacency matrix to identify the isomorphism of KCs and graphs quickly, conveniently, correctly and uniquely.

(1) Theory to Detect Isomorphism among Kcs Two KCs are isomorphic, that is, the equality of structures in number of elements (links and joints), and the sequence of their contacts, namely that they have identical structural characteristics. While studying the structural properties of KCs, it is useful to represent the chain by its graph whose vertices correspond to links, edges correspond to joints of the chain and edge-connection between vertices correspond to the kinematic pairing between the links of the chain. In the graph, the degree of a vertex is the number of edges incident with the vertex. Appling concepts from graph theory, the kth power of the adjacency matrix is the

k [ A]n×n = [a ij(k ) ]n×n , which have the physical significance in multiplication of the matrix, i.e.

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a (k )

that an element ij denotes the total number of paths staring at link i and terminating at link j and having a length of k joints. Then in a graph with n vertexes, suppose the degree of a vertex is m , all of the total numbers of the paths having a length of k edges from this vertex to all the vertexes in the graph constitute a row vector, i.e. where

pi

a

s1 ≥ s2 ≥

, pi ,

, pn )

,

is the total number of paths having a length of k edges from this vertex to the ith

vertex. Furthermore, by arranging the elements of the get

SGDm( k ) = ( p1 , p2 ,

vector

≥ si ≥

defined

as:

(k ) m

ASGD

≥ sn . It can be seen that the

SGDm( k )

in the decreasing order, we

= ( s1 , s2 , (k ) m

ASGD

, si ,

, sn )

,

where

of each vertex in the graph is a

structural invariant for the KC. If the graph is said to be an isomorphism of the graph B , a (k )

same ASGDm in graph B can be found certainly, if not, they are not isomorphic. It is known from above that two KCs are isomorphic if and only if the correspondent two graphs are isomorphic, i.e., the related adjacency matrixes after exchanging the rows and columns are equal.

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The detection of isomorphism of topological structures depends on the structural invariants. Given an adjacent matrix A of a graph, the following structural invariants can be easily found. ① The number of vertexes, that is, the number of links in the correspondent KC, which is the number of the rows of matrix A. ② The number of edges, that is, the number of the joints in the KC, which is equal to the number of the zero elements in the upper right triangular part of matrix A. ③ The number of the vertexes with the same degree. The number of the vertexes with degree d is the number of the rows, each of which has d non-zero elements. (k )

④ The ASGDm referring to every vertex. It is a necessary condition for isomorphism of two graphs that the structural invariants mentioned above are equal. ⑤ The adjacent matrix. It is a sufficient and necessary condition for two isomorphic graphs that the correspondent adjacent matrixes after exchanging the rows and columns are equal.

(2) Method to Detect Isomorphism among Kcs Let md denote the maximum of the degrees of vertexes in a graph, nv denote the number of the vertexes with degree md . (k )

For a specified graph, there is no need to calculate the ASGDm

with k → ∞ and (k )

referring to every vertex. It is enough to calculate and compare the ASGD m ( m = md , Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

k ≤ md ) of the nv vertexes. To resume, we can elucidate the general steps of identification of the isomorphism of two KCs in the following ways: • • • •

•

Step 1: Given the two graphs corresponding to the two KCs, and find the related adjacency matrixes A1 and A2. Step 2: Find and compare the two numbers of the vertexes in the two graphs, if they are not equal, the two graphs are nonisomorphic, go to step 9; otherwise go to step 3. Step 3: Find and compare the two numbers of the edges in the two graphs, if they are not equal, the two graphs are nonisomorphic, go to step 9; otherwise go to step 4. Step 4: Find and compare the two numbers of the vertexes with the same degree in the two graphs respectively, if they are not equal, the two graphs are nonisomorphic, go to step 9; otherwise go to step 5. Step 5: Find the ASGDm(2) of the nv vertexes in the two graphs respectively, i.e. (2) ASGD1(2) m and ASGD 2 m , which are nv × n vectors. Referring to every row in

(2)

the ASGD1(2) m , if a row in the ASGD 2 m can be found to be equal to it, go to step 6; otherwise, the two graphs are nonisomorphic, go to step 9.

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T.J. Li, W.Q. Cao and T.H. Yan •

and ASGD 2(2) Step 6: Let ASGD1(2) m remain one row and become 1× n vector m respectively, and satisfy ASGD1m = ASGD 2m . Then find and compare the (2)

ASGD1(mk )

(2)

ASGD 2 (mk ) ( k = 3, 4,

and

, md )

respectively,

if

ASGD1(mk ) ≠ ASGD 2(mk ) , the two graphs are nonisomorphic, go to step 9; otherwise, go to step 7. •

( md )

Step 7: Adjust the elements of the SGD 2 m

( md )

to make SGD 2m

= SGD1(mmd ) ,

meanwhile, exchange the correspondent rows and columns of matrix A2. For ( md ) example, if exchange the pi and p j in SGD 2 m , correspondingly, in matrix A2

exchange the ith row and jth row, simultaneously exchange the ith column and the '

jth column. As such matrix A2 after exchanged becomes another matrix A2 , go to step 8. '

•

Step 8: Compare A1 and A2 , if they are equal, two graphs are isomorphic; otherwise,

•

they are not. Step 9: End.

The above method can detect isomorphism of not only KCs but also the general graphs.

(3) Illustrations Two examples are given below to illustrate the approach.

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Example 1. Determination of Isomorphism of Kcs. There are three graphs shown in Fig. 8(a), (b) and (c), which represent corresponding three KCs of 10-link with one freedom. Here we are to use the power of adjacent matrix to determine whether they are isomorphic.

(a)

(b)

(c)

Figure 8. Graphs of the KCs.

For the graphs in Fig. 8(a), (b) and (c), because the number of the vertexes, the number of the edges, and the numbers of the vertexes with same degree are identical respectively, and Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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13 (k )

there is only one vertex with the maximum degree 4 in graph, then the ASGDm

of these

three vertexes are used for identification firstly. Their adjacency matrixes are

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⎡0 ⎢1 ⎢ ⎢0 ⎢ ⎢0 ⎢0 A1 = ⎢ ⎢1 ⎢1 ⎢ ⎢0 ⎢0 ⎢ ⎣⎢0

1 0 0 0 1 1 0 0 0⎤ ⎡0 1 0 0 0 ⎢1 0 1 0 0 ⎥ 0 1 0 0 0 0 0 1 0⎥ ⎢ ⎢0 1 0 1 0 1 0 1 0 0 0 0 0 0⎥ ⎢ ⎥ 0 1 0 1 0 1 0 0 0⎥ ⎢0 0 1 0 1 ⎢0 0 0 1 0 0 0 1 0 1 0 0 0 0⎥ ⎥ A2 = ⎢ 0 0 0 1 0 0 0 0 0⎥ ⎢1 0 0 0 1 ⎢1 0 0 0 0 ⎥ 0 0 1 0 0 0 1 0 1 ⎢ ⎥ 0 0 0 0 0 1 0 1 0⎥ ⎢0 1 0 0 1 ⎢0 0 0 0 0 ⎥ 1 0 0 0 0 0 1 0 1 ⎢ ⎥ ⎢⎣0 0 0 1 0 0 0 0 0 0 1 0 1 0 ⎥⎦

⎡0 1 0 ⎢1 0 1 ⎢ ⎢0 1 0 ⎢ ⎢0 0 1 ⎢0 0 0 A3 = ⎢ ⎢1 0 0 ⎢0 0 0 ⎢ ⎢0 0 0 ⎢0 1 0 ⎢ ⎢⎣1 0 0 Then we have

1 1 0 0 0⎤ 0 0 1 0 0 ⎥⎥ 0 0 0 0 0⎥ ⎥ 0 0 0 0 1⎥ 1 0 1 0 0⎥ ⎥ 0 0 0 0 0⎥ 0 0 1 0 0⎥ ⎥ 0 1 0 1 0⎥ 0 0 1 0 1⎥ ⎥ 0 0 0 1 0 ⎦⎥

0 0 1 0 0 0 1⎤ 0 0 0 0 0 1 0 ⎥⎥ 1 0 0 0 0 0 0⎥ ⎥ 0 1 0 1 0 0 0⎥ 1 0 1 0 0 1 1⎥ ⎥ 0 1 0 0 0 0 0⎥ 1 0 0 0 1 0 0⎥ ⎥ 0 0 0 1 0 1 0⎥ 0 1 0 0 1 0 0⎥ ⎥ 0 1 0 0 0 0 0 ⎥⎦

ASGD1(42 ) = (4,2,1,1,1,1,0,0,0,0 ),

ASGD3(42 ) = (4,2,1,1,1,1,0,0,0,0 ),

ASGD 2 (42 ) = (4,2,1,1,1,1,0,0,0,0 ),

ASGD1(42 ) = ASGD2 (42 ) = ASGD3(42 ) , ASGD1(43 ) = (6,6,6,6,3,1,1,1,1,0 ),

ASGD3(43) = (6,6,6,6,3,1,1,1,1,0) ,

ASGD 2 (43) = (7,6,6,5,2,2,1,1,1,0),

ASGD1(43 ) = ASGD3(43) ≠ ASGD2 (43) , So the graphs in Fig. 8 (b), (a) and Fig. 8 (b), (c) are not isomorphic. Now continue to determine whether the graphs in Fig. 8 (a) and (c) are isomorphic. (4) ASGD1(4) 4 = (24,15,9,8, 7, 7, 4, 2,1,1) , ASGD34 = (24,15,9,8, 7, 7, 4, 2,1,1)

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14

T.J. Li, W.Q. Cao and T.H. Yan (4) ASGD1(4) 4 = ASGD 34

(4) SGD1(4) 4 = (4,8,9, 2, 7, 7, 24,1,15,1) , SGD34 = (15,8,9, 2, 24,1, 7,7, 4,1)

(4)

Then adjust the elements of SGD14

to make SGD14 = SGD34 , meanwhile (4)

(4)

exchange the correspondent rows and columns of matrix A1 . Finally we obtain a matrix as follows

⎡0 ⎢1 ⎢ ⎢0 ⎢ ⎢0 ⎢0 A1' = ⎢ ⎢1 ⎢0 ⎢ ⎢0 ⎢0 ⎢ ⎢⎣1

1 0 0 0 1 0 0 0 1⎤ 0 1 0 0 0 0 0 1 0 ⎥⎥ 1 0 1 0 0 0 0 0 0⎥ ⎥ 0 1 0 1 0 1 0 0 0⎥ 0 0 1 0 1 0 0 1 1⎥ ⎥ = A3 0 0 0 1 0 0 0 0 0⎥ 0 0 1 0 0 0 1 0 0⎥ ⎥ 0 0 0 0 0 1 0 1 0⎥ 1 0 0 1 0 0 1 0 0⎥ ⎥ 0 0 0 1 0 0 0 0 0 ⎥⎦

So the two graphs in Fig. 8 (a) and (c) are isomorphic.

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Example 2: Determination of Isomorphism of Graphs This example considers two of the most complicated cospectral graphs that we have found in the literature. This pair, which are shown in Fig. 9, are not the topological graphs of possible KCs as they contain several rigid embedded structures. They can, however, be used to show the power and efficiency of the isomorphism detection method given above.

(a)

(b) Figure 9. Two cospectral graphs.

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For the graphs in Fig. 9 (a) and (b), because the numbers of the vertexes, the numbers of the edges, and the numbers of the vertexes with same degree are all respectively equal in the two graphs, and there three vertexes with the maximum degree 6 in each graph, also these three vertexes are in symmetry and with the same ASGDm( k ) in each graph, we choose one of (k )

them arbitrarily in each graph. Then the ASGDm

of the two vertexes are used for

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identification firstly. Their adjacency matrixes A1 and A2 are

⎡0 ⎢1 ⎢ ⎢1 ⎢ ⎢1 ⎢0 ⎢ ⎢0 ⎢0 ⎢ A1 = ⎢0 ⎢1 ⎢ ⎢0 ⎢ ⎢0 ⎢0 ⎢ ⎢0 ⎢0 ⎢ ⎣⎢0

1 1 1 0 0 0 0 1 0 0 0 0 0 0⎤ 0 1 0 1 0 1 0 0 0 0 0 0 0 0⎥⎥ 1 0 0 0 1 0 1 0 0 0 0 0 0 0⎥ ⎥ 0 0 0 1 1 1 0 0 1 0 0 0 0 1⎥ 1 0 1 0 1 0 1 0 0 1 0 1 0 0⎥ ⎥ 0 1 1 1 0 0 0 1 0 0 1 0 1 0⎥ 1 0 1 0 0 0 0 0 0 0 0 0 0 0⎥ ⎥ 0 1 0 1 0 0 0 0 0 0 0 0 0 0⎥ 0 0 0 0 1 0 0 0 0 0 0 0 0 0⎥ ⎥ 0 0 1 0 0 0 0 0 0 1 1 1 0 0⎥ ⎥ 0 0 0 1 0 0 0 0 1 0 1 0 1 0⎥ 0 0 0 0 1 0 0 0 1 1 0 0 0 1⎥ ⎥ 0 0 0 1 0 0 0 0 1 0 0 0 0 0⎥ 0 0 0 0 1 0 0 0 0 1 0 0 0 0⎥ ⎥ 0 0 1 0 0 0 0 0 0 0 1 0 0 0⎦⎥

⎡0 ⎢1 ⎢ ⎢1 ⎢ ⎢1 ⎢0 ⎢ ⎢0 ⎢1 ⎢ A2 = ⎢0 ⎢0 ⎢ ⎢0 ⎢ ⎢0 ⎢0 ⎢ ⎢0 ⎢0 ⎢ ⎣⎢0

1 1 1 0 0 1 0 0 0 0 0 0 0 0⎤ 0 1 0 1 0 0 1 0 0 0 0 0 0 0⎥⎥ 1 0 0 0 1 0 0 1 0 0 0 0 0 0⎥ ⎥ 0 0 0 1 1 0 0 1 1 0 0 0 0 1⎥ 1 0 1 0 1 1 0 0 0 1 0 1 0 0⎥ ⎥ 0 1 1 1 0 0 1 0 0 0 1 0 1 0⎥ 0 0 0 1 0 0 0 0 0 0 0 0 0 0⎥ ⎥ 1 0 0 0 1 0 0 0 0 0 0 0 0 0⎥ 0 1 1 0 0 0 0 0 0 0 0 0 0 0⎥ ⎥ 0 0 1 0 0 0 0 0 0 1 1 1 0 0⎥ ⎥ 0 0 0 1 0 0 0 0 1 0 1 0 1 0⎥ 0 0 0 0 1 0 0 0 1 1 0 0 0 1⎥ ⎥ 0 0 0 1 0 0 0 0 1 0 0 0 0 0⎥ 0 0 0 0 1 0 0 0 0 1 0 0 0 0⎥ ⎥ 0 0 1 0 0 0 0 0 0 0 1 0 0 0⎦⎥

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T.J. Li, W.Q. Cao and T.H. Yan

when k = 2,3,

,6 , ASGD1(6k ) = ASGD2 (6k ) . When k = 6 , we have

ASGD16(6 ) = (1042,623,623,601,601,547,547,405,405,329,329,311,311,169,169)

ASGD 2 (66 ) = (1042,623,623,601,601,547,547,405,405,329,329,311,311,169,169) ASGD1(66 ) = ASGD 2 (66 ) ,

SGD1(66 ) = (311,623,547,1042,601,601,169,329,405,311,547,623,405,329,169) ,

SGD 2 6(6 ) = (311,547,623,1042,601,601,405,329,169,311,547,623,405,329,169) , ( 6)

Then adjust the elements of SGD 2 6

(6)

to make SGD 2 (66 ) = SGD16 , meanwhile

exchange the correspondent rows and columns of matrix A2 . Finally we obtain a matrix as

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follows ⎡0 ⎢1 ⎢ ⎢1 ⎢ ⎢1 ⎢0 ⎢ ⎢0 ⎢0 ⎢ ' A2 = ⎢0 ⎢1 ⎢ ⎢0 ⎢ ⎢0 ⎢0 ⎢ ⎢0 ⎢0 ⎢ ⎢⎣0

1 1 1 0 0 0 0 1 0 0 0 0 0 0⎤ 0 1 0 1 0 1 0 0 0 0 0 0 0 0⎥⎥ 1 0 0 0 1 0 1 0 0 0 0 0 0 0⎥ ⎥ 0 0 0 1 1 1 0 0 1 0 0 0 0 1⎥ 1 0 1 0 1 0 1 0 0 0 1 0 1 0⎥ ⎥ 0 1 1 1 0 0 0 1 0 1 0 1 0 0⎥ 1 0 1 0 0 0 0 0 0 0 0 0 0 0⎥ ⎥ 0 1 0 1 0 0 0 0 0 0 0 0 0 0⎥ ≠ A1 0 0 0 0 1 0 0 0 0 0 0 0 0 0⎥ ⎥ 0 0 1 0 0 0 0 0 0 1 1 1 0 0⎥ ⎥ 0 0 0 0 1 0 0 0 1 0 1 0 1 0⎥ 0 0 0 1 0 0 0 0 1 1 0 0 0 1⎥ ⎥ 0 0 0 0 1 0 0 0 1 0 0 0 0 0⎥ 0 0 0 1 0 0 0 0 0 1 0 0 0 0⎥ ⎥ 0 0 1 0 0 0 0 0 0 0 1 0 0 0⎥⎦

So these two graphs are not isomorphic.

3.2. Detection of Isomorphism among Glkcs Proceeded from the topological characteristics of GLM's structure, a fully new graph, combinatorial graph, which is easy to identify and calculate to describe the topological structure of GLKC, is proposed in Section 2.4. In the following, this work establishes a systematic procedure for detecting isomorphism among GLKCs using the powers of CM.

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(1) Theory to Detect Isomorphism Among Glkcs Two GLKCs are isomorphic, that is, the equality of structures in number of elements (links and joints), and the sequence of their contacts, namely that they have identical topological characteristics of structure.

[ ( )]

The kth power of the CM is the multiplication of the matrix, i.e. [CM ]n× n = a ij k

(k )

where aij

k

n× n

,

is defined as the Generalized Distance (GD) from the ith vertex to the jth vertex.

Then in a combinatorial graph with n vertexes, suppose the degree of a vertex is m , all the GDs from this vertex to all the vertexes in the graph constitute a row vector, i.e.

SGD m(k ) = ( p1 , p 2 ,

, pi ,

, p n ) , where p i is the GD from this vertex to the ith vertex. (k )

Furthermore, by arranging the members of the SGDm in decreasing order, we get a vector defined as: ASGD m = (s1 , s 2 , (k )

, si ,

, s n ) , where s1 ≥ s2 ≥

≥ si ≥

≥ sn . It can

(k )

be seen that the ASGD m of each vertex in the graph is a structural invariant for a GLKC. If (k )

the graph is said to be an isomorphism of the graph B , a same ASGDm in graph B can be

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found certainly, if not, they are not isomorphic. It is known from above that two GLKCs are isomorphic if and only if the correspondent two combinatorial graphs are isomorphic, i.e., the related two CMs after exchanging the rows and columns are equal. The detection of isomorphism of topological structures is depended on the structural invariants. Given a CM of a combinatorial graph, the following structural invariants can be easily found. ① The number of vertexes, that is, the number of the links in the correspondent GLKC, which is the number of the rows of the CM. ② The number of edges, that is, the number of the pairs formed by two members in GLKC, which is equal to the number of the non-zero elements in the upper right triangular part of the CM (not equal to the number of the joints in GLKC). ③ The number of the edges with same weight. The number of the edges with weight w is equal to the number of the elements whose value is w in the upper right triangular part of the CM. ④ The number of the vertexes with same weight. The number of the vertexes with weight g is equal to the number of the elements whose value is g in the principal diagonal of the CM. ⑤ The number of the vertexes with same degree. The number of the vertexes with degree d is the number of the rows, each of which has d non-zero elements (apart from the principal diagonal elements). (k )

⑥ The ASGDm referring to every vertex.

It is a necessary condition for isomorphism of two combinatorial graphs that the structural invariants mentioned above are equal. ⑦ The CM.

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T.J. Li, W.Q. Cao and T.H. Yan

It is a sufficient and necessary condition for two isomorphic graphs that the correspondent CMs after exchanging the rows and columns are equal.

(2) Method to Detect Isomorphism Among Kcs Let mg denote the maximum of the weights of vertexes in the combinatorial graph, md denote the maximum of the degrees of vertexes with weight mg, nv denote the number of the vertexes with degree md . (k ) Given a combinatorial graph, there is no need to calculate the ASGDm with k→ ∞ and (k )

referring to every vertex. It is enough to calculate and compare the ASGDm ( m = md ,

k ≤ md ) of the nv vertexes. To resume, we can elucidate the general steps of identification of the isomorphism of two combinatorial graphs in the following ways: • •

• •

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•

•

•

Step 1: Given the two combinatorial graphs corresponding to the two GLKCs, and find the related CM1 and CM2 . Step 2: Find and compare the two numbers of the vertexes in the two graphs, if they are not equal, the two graphs are nonisomorphic, go to step 11; otherwise go to step 3. Step 3: Find and compare the two numbers of the edges in the two graphs, if they are not equal, the two graphs are nonisomorphic, go to step 11; otherwise go to step 4. Step 4: Find and compare the two numbers of the edges with same weight in the two graphs respectively, if they are not equal, the two graphs are nonisomorphic, go to step 11; otherwise go to step 5. Step 5: Find and compare the two numbers of the vertexes with same weight in the two graphs respectively, if they are not equal, the two graphs are nonisomorphic, go to step 11; otherwise go to step 6. Step 6: Find and compare the two numbers of the vertexes with same degree in the two graphs respectively, if they are not equal, the two graphs are nonisomorphic, go to step 11; otherwise go to step 7. (2 )

Step 7: Find the ASGDm

of the nv vertexes in the two graphs respectively, i.e.

ASGD1(m2 ) and ASGD 2 (m2 ) , which are nv×n vectors. Referring to every row in the ASGD1(m2 ) , if a row in the ASGD 2 (m2 ) can be found to be equal to it, go to step 8; otherwise, the two graphs are nonisomorphic, go to step 11. •

(2 )

(2 )

Step 8: Let ASGD1m and ASGD 2 m remain one row and become 1×n vectors (2 )

(2 )

respectively, and satisfy ASGD1m = ASGD 2 m . Then find and compare the

ASGD1(mk )

and

ASGD2 (mk )

( k = 3, 4,

, md )

respectively,

if

ASGD1(mk ) ≠ ASGD 2 (mk ) , the two graphs are nonisomorphic, go to step 11; otherwise, go to step 9. •

( md )

Step 9: Adjust the elements of the SGD2 m

( md )

to make SGD 2 m

= SGD1(mmd ) ,

meanwhile, exchange the correspondent rows and columns of the CM2. For example, Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Applications of Graph Theory in Mechanism Analysis ( md )

if exchange the p i and p j in SGD2 m

• •

19

, correspondingly, in CM2 exchange the

ith row and the jth row, simultaneously exchange the ith column and the jth column. As such the CM2 after exchanged becomes a matrix CM2', go to step 10. Step 10: Compare CM1 and CM2', if they are equal, the two graphs are isomorphic; otherwise, they are not. Step 11: End.

(3) Illustrations Two examples are given below to illustrate the above approach. Example 1: Determination of Isomorphism of the Glkcs, as Shown in Fig. 10(A) and (B)

Their combinatorial graphs are shown in Figure 10(c) and (d) respectively. Because the numbers of the vertexes, the numbers of the edges, the numbers of the edges with same weight, the numbers of the vertexes with same weight, and the numbers of the vertexes with same degree in the two combinatorial graphs are identical respectively, and there is only one (k )

vertex with the maximum degree 5 in each graph. Therefore, the ASGDm

of these two

vertexes are used for identification firstly. 8 2

2

3

3 8

7 4

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1

6

7

4

1

5

5 6

(a)

(b)

7

8 (2)

(2)

7

6

2 1

(2)

2 1 (2)

(2)

5

3

6

(2)

8

5

3

4 (c)

4 (d)

Figure 10. GLKCs and combinatorial graphs.

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20

T.J. Li, W.Q. Cao and T.H. Yan Their combinatorial matrixes are

⎡0 ⎢1 ⎢ ⎢2 ⎢ 0 CM 1 = ⎢ ⎢1 ⎢ ⎢0 ⎢1 ⎢ ⎣⎢1

1 2 0 1 0 1 1⎤ 0 1 0 0 0 0 1 ⎥⎥ 1 0 1 0 0 0 0⎥ ⎥ 0 1 0 1 1 0 0⎥ 0 0 1 0 1 1 0⎥ ⎥ 0 0 1 1 0 2 0⎥ 0 0 0 1 2 0 2⎥ ⎥ 1 0 0 0 0 2 0⎦⎥

⎡0 ⎢1 ⎢ ⎢0 ⎢ 0 CM 2 = ⎢ ⎢1 ⎢ ⎢1 ⎢1 ⎢ ⎣⎢2

1 0 0 1 1 1 2⎤ 0 1 0 0 0 1 1 ⎥⎥ 1 0 1 0 0 0 1⎥ ⎥ 0 1 0 1 2 0 0⎥ 0 0 1 0 1 0 0⎥ ⎥ 0 0 2 1 0 2 0⎥ 1 0 0 0 2 0 0⎥ ⎥ 1 1 0 0 0 0 0⎦⎥

ASGD15(2 ) = (8,3,3,3,3,3,1,1) , ASGD 2 5(2 ) = (8,3,3,3,3,3,1,1) , ASGD15(2 ) = ASGD 2 5(2 ) ,

ASGD15(3) = (22,21,17,17,12,12,10,5) , ASGD2 5(3) = (22,21,17,15,14,12,10,7 ) ,

ASGD15(3 ) ≠ ASGD 2 5(3 ) , So the two combinatorial graphs are not isomorphic, i.e., the two GLKCs in Fig. 10(a) and (b) are not isomorphic.

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Example 2: Determination of isomorphism of the GLKCs, as shown in Fig. 11(a) and (b).

Their combinatorial graphs are shown in Fig. 11(c) and (d) respectively. Because the numbers of the vertexes, the numbers of the edges, the numbers of the edges with same weight, the numbers of the vertexes with same weight, and the numbers of the vertexes with same degree in the two combinatorial graphs are identical respectively, and there is only one vertex with (k )

the maximum degree 5 in each graph. Therefore, the ASGDm of these two vertexes are used for identification firstly. 8

8 3

2

2 5

7

7

4

1

5 6

4

1

3 6

(a)

(b) Figure 11. Continued on next page.

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7(2)

8

(2)

7(2)

(2)

(2) 6

21

(2)

(2) 6

2

8

(2)

2 1

1 5

5

3

3

4

4 (c)

(d) Figure 11. GLKCs and combinatorial graphs.

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Their combinatorial matrixes are

⎡0 ⎢1 ⎢ ⎢0 ⎢ 0 CM 1 = ⎢ ⎢1 ⎢ ⎢1 ⎢1 ⎢ ⎢⎣0

⎡0 1 0 0 1 1 1 0⎤ ⎢1 0 1 0 0 0 1 1 ⎥⎥ ⎢ ⎢1 1 0 1 0 0 2 1⎥ ⎢ ⎥ 0 0 1 0 1 0 0 0⎥ CM 2 = ⎢ ⎢0 0 0 1 0 1 0 0⎥ ⎢ ⎥ 0 0 0 1 0 2 0⎥ ⎢1 ⎢1 1 2 0 0 2 2 2⎥ ⎢ ⎥ 1 1 0 0 0 2 0⎥⎦ ⎢⎣0

1 1 0 0 1 1 0⎤ 0 0 0 1 0 1 1 ⎥⎥ 0 0 1 0 1 0 0⎥ ⎥ 0 1 0 1 0 0 0⎥ 1 0 1 0 0 2 1⎥ ⎥ 0 1 0 0 0 2 0⎥ 1 0 0 2 2 2 2⎥ ⎥ 1 0 0 1 0 2 0⎥⎦

when k=2,3,4,5, ASGD15(k ) = ASGD 2 5( k ) . Here when k=5,

ASGD15(5 ) = ASGD 2 5(5 ) = (2962,1614,1558,1334,1243,1059,460,356) ,

SGD15(5 ) = (1059,1243,1614,356,460,1334,2962,1558) ,

SGD 2 5(5 ) = (1059,1243,460,356,1614,1334,2962,1558) , (5 )

Then adjust the elements of SGD 2 5

(5 )

(5 )

to make SGD 2 5 = SGD15 , meanwhile

exchange the correspondent rows and columns of the CM2. Finally we obtain a matrix as follows:

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22

T.J. Li, W.Q. Cao and T.H. Yan

⎡0 ⎢1 ⎢ ⎢0 ⎢ 0 CM 2 = ⎢ ⎢1 ⎢ ⎢1 ⎢1 ⎢ ⎣⎢0

1 0 0 1 1 1 0⎤ 0 1 0 0 0 1 1 ⎥⎥ 1 0 1 0 0 2 1⎥ ⎥ 0 1 0 1 0 0 0⎥ =CM1 0 0 1 0 1 0 0⎥ ⎥ 0 0 0 1 0 2 0⎥ 1 2 0 0 2 2 2⎥ ⎥ 1 1 0 0 0 2 0⎦⎥

So the two graphs in Fig. 11(c) and (d) are isomorphic, i.e., the two GLKCs in Fig. 11(a) and (b) are isomorphic.

4. Topology-Loop Characteristics of Kcs The topology-loop structures play a key role in the structural decomposition of mechanisms. In the following, a uniform formula of the number of the independent loops in KCs is presented, and the corresponding proved courses are given, which provides a sound theoretical basis for the mechanical analysis. For a KC with simple joints, its topological graph is obtained by representing each link by a vertex and each kinematic pair by an edge connecting the corresponding vertices. So the theorems in graph theory can be used. The number of the independent circuits can be determined by Euler's formula, that is

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ll = ml − nl + 1

(9)

ml is the number of edges in the topological graph, that is the number of joints in n the KC with simple joints; l is the number of vertexes in the topological graph, that is the

in which,

number of links in the KC with simple joints.

(a)

(b)

Figure 12. A KC with simple joints and its topological graph.

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23

For example, Fig. 12 shows a KC with simple joints and its topological graph, the number of topological loops is

ll = ml − nl + 1 = 12 − 10 + 1 = 3 For the bicolored graph representation and the tricolored graph representation of KCs, we will prove the number of topological loops satisfies Eq. (9) in the following section.

4.1. The Number of Topological Loops of Bicolored Graph The bicolored graph can represent the topological structure of KCs not only with simple joints but also with multiple joints, as shown in Fig. 2 and Fig. 3. In the bicolored graph of KC with simple joints, it can be seen that the number of vertexes is

nv = ml + nl

(10)

and the number of edges is ne = 2ml . Thus the number of topological loops of bicolored graph is

ll = ne − nv + 1 = 2ml − ml − nl + 1 = ml − nl + 1

(11)

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Equation (11) indicates that the number of topological loops of bicolored graph can be determined by Eq. (9). Referring to the KC and its bicolored graph shown in Fig. 3, the number of topological loops is

ll = ml − nl + 1 = 12 − 9 + 1 = 4 For the KCs with multiple joints, a multiple joint is formed when there is a link with joint elements more than 2, and all the distances among these joint elements are shrunk to zero, and this link is eliminated. As shown in Fig. 13(a), there are three joint elements on link 4, they would be shrunk into one multiple joint. In Fig. 13(b), there are four joint elements on link 5, they would be shrunk into one multiple joint.

(a)

(b)

Figure 13. Composition of multiple joints.

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T.J. Li, W.Q. Cao and T.H. Yan In a simple-joint KC, assume that there are p links with more than two pair elements

shrunk into p multiple joints, the total number of kinematic pairs of p links is p1 . In the formed KC with multiple joints, let the number of kinematic pairs be m1 , and the number of links be n1 , so we have m1 = ml − p , n1 = nl − p . That is

m1 − n1 + 1 = ml − nl + 1 In the bicolored graph of the multiple-joint KC, the number of edges is and the number of vertexes is

(12)

ne = 2ml − p1 ,

nv = ml + nl − p1 , so the number of topological loops is

ll = ne − nv + 1 = 2ml − p1 − ml − nl + p1 + 1 = ml − nl + 1

(13)

From Eqs. (12) and (13), there is

ll = m1 − n1 + 1

(14)

Referring to the multiple-joint KC and its bicolored graph shown in Fig. 2, the number of topological loops is

ll = m1 − n1 + 1 = 10 − 8 + 1 = 3

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4.2. The Number of Topological Loops of Tricolored Graph A GLKC can be formed by adding basic gear arrangement modules on the planar linkage KC [22], this planar linkage KC is called the attachable KC. Both the higher and lower pairs exist within a GLKC, which also composes both simple and multiple joints. Therefore, a GLKC can be hardly expressed topologically by an ordinary graph or an bicolored graph, which should be represented by the tricolored graph, as shown in Fig. 4. Now we will prove the number of topological loops of the tricolored graph can be determined by the following formula l = m − n +1 (15) in which, l is the number of topological loops, m is the number of kinematic pairs, n is the number of links in a GLKC. Equation (15) can be proved as follows. Suppose l l is the number of topological loops in the attachable KC, ml is the number of kinematic pairs in the attachable KC, nl is the number of bars in the attachable KC, nG is the number of moving gears in the GLKC, m h is the number of gear pairs in the GLKC. Then the number of topological loops in the attachable KC is ll = ml − nl + 1 . Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Applications of Graph Theory in Mechanism Analysis

25

The kinematic pairs of GLKC include the kinematic pairs of the attachable KC and those introduced by the basic gear arrangement modules. So the number of kinematic pairs of GLKC is m = ml + mh + nG . The links of GLKC include the bars of the attachable KC and the moving gears, so the number of links of GLKC is n = nl + nG . We have

m − n + 1 = ml + mh + nG − nl − nG + 1 = ll + mh

(16)

In the tricolored graph of GLKC, the number of vertexes is nv = nl + mh + ml + nG , the number of edges is ne = 2ml + nG + 2mh . So the number of topological loops of tricolored graph can be

l = n e − n v + 1 = 2 ml + n G + 2 m h − n l − m h − ml − n G + 1 = ml − n l + 1 + m h = l l + m h

(17)

From Eqs. (16) and (17), we have

l = m − n +1 Referring to the GLKC and its tricolored graph shown in Fig. 14, the number of topological loops is

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l = m − n +1 = 8 − 6 +1 = 3

(a)

(b) Figure 14. A GLKC and its tricolored graph.

5. Structural Decomposition of Mechanisms The topological representations of mechanisms provide convenience for their structural analysis and synthesis. By means of Assur groups and the type transformation theory [23], it is quite convenient to study the kinematic analysis of Planar Linkage Mechanisms (PLMs) and GLMs. Based on the structural topological characteristics, a systematic method for decomposing a PLM or GLM into a series of sequential independent kinematic units, such as the simple links and the dyad link groups is proposed.

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26

T.J. Li, W.Q. Cao and T.H. Yan

5.1. Principle of Structural Decomposition If we have some second coordinates (which are unknown) of the links in a PLM or GLM transferred hypothetically into the first coordinates (which are known), the whole structure of the mechanism may be transferred and decomposed into a combination of the independent kinematic units, that is, a mechanism can be divided into the frame, the input unit(s), the hypothetical unit(s), dyad(s) and the constraint unit(s). Whenever the hypothetical parameters of the second coordinates of links lead to satisfaction of the identity conditions of the constraint units, the hypothetical values of the second coordinates will become true ones, and the structure of the mechanism is said to be transformed and decomposed. The number of hypothetical parameters is called the number of type transformation, denoted by m .

5.2. Calculation of Transformation Number In a PLM or GLM, the number of links is denoted by n , the number of total pairs is denoted by p , the number of circuits is denoted by L . Then we have

L = ( p − n) + 1

(18)

Let LB be the chosen circuit set, so

L B = {L B1 (φ B1 ), L B 2 (φ B 2 ),

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where

, L Bj (φ Bj ),

, L BL (φ BL )}

(19)

φ Bj is the set of the 2nd coordinates in circuit LBj .

Since all the 1st coordinates of the inputs are known, we only need to take the 2nd coordinates into account. Let

φ NBj = φ B1 ∪ φ B 2 ∪

∪ φ B ( j −1) ( j = 2,3,

,L)

(20)

Its physical meaning is: for the chosen circuit set LB , when calculating the kinematic parameters of circuit j ,

φ NBj will be the set of the second coordinates which have appeared

in the former j − 1 circuits. Obviously

φ NB1 = {φ }, is a empty set.

Let deg ree(φ j ) denote the number of elements in the set

φ j , the number of

hypothetical parameters which should be inputted to the jth circuit will be

m j = deg ree[φ Bj − (φ NBj ∩ φ Bj )] − c j where c j = 3 − 1 = 2 , for the planar mechanisms. The number of type transformation will be

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(21)

Applications of Graph Theory in Mechanism Analysis L

L

m = ∑ m j = ∑| m j | +

j =1

in which m j

+

−

27

(22)

j =1

means m j > 0 , it indicates that the jth circuit has m j DOF and m j

hypothetic units; m j

−

means m j < 0 , it indicates that the jth circuit has m j constrain units

and m j identity conditions.

5.3. Types of Kinematic Units For PLMs considering the all elements of lower pairs, there are 9 types of simple link units, 27 types of dyad units and 7 types of constraint units [23]. Referring to GLMs, higher pair elements are introduced. Table 1 shows some typical types of kinematic units with revolute pairs and gear pairs. Table 1. Some types of kinematic units Simple link units

Dyad units

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Constraint units

-1

-2

-1

5.4. Criteria of Choosing the Sequential Circuits In order to get the minimum value of m , the following procedure is suggested: •

Step 1: Determine the first circuit and the calculating reference frame as follows (a) According to the weighted tricolored graph and its adjacent matrix, find out the circuit sets which contain the shortest length; (b) Within the above circuit sets, choose the circuits with the maximum kinematic pairs; (c) According to the circuits obtained from above, choose the circuit that contains the frame as the first decomposed circuit, and choose the frame as the calculating reference link. If there is no frame in the circuits, choose one to be the first

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28

T.J. Li, W.Q. Cao and T.H. Yan circuit and a link which contains the maximum kinematic pairs as the calculating reference link. •

Step 2: Decompose the second circuit as follows: (a) When the first circuit has been chosen, the following circuits may be constituted by adding a single open chain [24] to the previous circuits. Then choose the second circuit set in which the length of the added chain is the shortest. (b) Choose one of circuit sets obtained from above, in which the number of the kinematic pairs belonging to the links of the added chain is maximum.

•

Step 3: Continue the above process until all circuits are decomposed.

5.5. Examples of Structural Decomposition Some examples are as follows, to illustrate the method of structural decomposition stated above.

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Example 1: Fig. 15(A) Shows A PLM with one DOF, and Fig. 15(B) the Weighted Graph and Fig. 15(C) the Decomposing Procedure.

(a)

(b)

(c) Figure 15. Structural decomposition of a PLM. (a) Kinematic diagram, (b) weighted bicolored graph, (c) decomposing procedure.

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The chosen sequential circuits are:

LB = {L123410 , L11049 , L194568 , L1867 } , φB1 = {φ3 , φ4 , φ10 }, φ B 2 = {φ10 , φ 4 , φ9 },

φB 3 = {φ9 , φ4 , φ5 , φ6 , φ8 } , φB 4 = {φ8 , φ6 , φ7 } , φ NB1 = {φ }, φ NB 2 = {φB1} = {φ3 , φ4 , φ10 } ,

φ NB 3 = φB1 ∪ φB 2 = {φ3 , φ4 , φ10 , φ9 }, φ NB 4 = φB1 ∪ φB 2 ∪ φB 3 = {φ3 , φ4 , φ10 , φ9 , φ5 , φ6 , φ8 } , m1 = degree[φB1 − (φ NB1 ∩ φB1 )] − 2 = 3 − 2 = 1 ,

m2 = degree[φ B 2 − (φ NB 2 ∩ φB 2 )] = 3 − 2 − 2 = −1 , m3 = degree[φ B 3 − (φ NB 3 ∩ φB 3 )] = 5 − 2 − 2 = 1 ,

m4 = degree[φ B 4 − (φ NB 4 ∩ φB 4 )] = 3 − 2 − 2 = −1 Therefore m = 2 , that is there are two Assur groups connected in series. As shown in Fig. 15(b), the constraint unit is in circuits L11049 and L1867 , along with the identity condition of link lengths

(x

− x k ) + ( y j − y k ) = L9 , 2

j

2

2

(x

− x g ) + ( y f − y g ) = L7 . 2

f

2

2

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Example 2: Fig. 16(A) Shows a Hydraulic Mechanism with one DOF, and Fig. 16(B) the Weighted Bicolored Graph and Fig. 16(C) the Decomposing Procedure.

(a)

(b)

(c) Figure 16. Structural decomposition of a hydraulic mechanism. (a) Kinematic diagram, (b) weighted bicolored graph, (c) decomposing procedure.

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T.J. Li, W.Q. Cao and T.H. Yan The matrix of the weighted bicolored graph is ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ Mc = ⎢1 ⎢ ⎢1 ⎢∞ ⎢ ⎢∞ ⎢ ⎢1 ⎢∞ ⎢ ⎣∞

1 ∞ ∞

1 1

∞ ∞ 1 ∞ ∞ ∞ ∞ ∞

∞ ∞

1

∞ ∞

∞ ∞ ∞ ∞ 1 ∞ ∞ ∞

1 1

∞ 1 1 ∞ ∞ ∞ ∞ ∞ 1 ∞ ∞ ∞ ∞ 0 0 ∞

∞ ∞

1

1

∞ ∞ 1 ∞ ∞⎤ 1 ∞ ∞ ∞ ∞ ⎥⎥ 1 1 ∞ ∞ 1⎥ ⎥ ∞ 1 ∞ ∞ ∞⎥ ∞ ∞ 1 0 ∞⎥ ⎥ ∞ ∞ ∞ 0 1⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ∞ ⎥ ⎥ ⎥ ⎦

Then we have

LB = {L1234 , L15632 }, φB1 = {φ1 , φ2 , φ3 }, φB 2 = {φ1 , φ5 , φ3 , φ2 } ,

φ NB 2 = {φB1} = {φ1 , φ2 , φ3 } ,

φ NB1 = {φ } m1 = degree[φB1 − (φ NB1 ∩ φB1 )] − 2 = 3 − 2 = 1 , m2 = degree[φ B 2 − (φ NB 2 ∩ φB 2 )] = 4 − 3 − 2 = −1

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So m = 1 , the identity condition of the constraint unit is

(x

− xe ) + ( y g − y e ) = Leg , 2

g

2

2

Leg = L5 + L6 + s f

Example 3: Fig. 17(A) Shows a GLM with one DOF, and Fig. 17(B) the Weighted Tricolored Graph and Fig. 17(C) the Decomposing Procedure.

(a)

(b) Figure 17. Continued on next page.

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(c) Figure 17. Structural decomposition of a GLM. (a) Kinematic diagram, (b) weighted tricolored graph, (c) decomposing procedure.

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The matrix M c will be

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ M c = ⎢∞ ⎢∞ ⎢ ⎢∞ ⎢1 ⎢ ⎢0 ⎢ ⎣∞

∞ ∞ ∞ 1 0 ∞⎤ ∞ ∞ 1 1 ∞ ∞ ⎥⎥ ∞ ∞ 1 1 ∞ ∞ 2⎥ ⎥ 1 1 ∞ ∞ ∞ ∞⎥ 1 ∞ ∞ ∞ 0 2⎥ ⎥ ∞ ∞ 1 1 ⎥ ⎥ ∞ 1 1 ∞ ⎥ 1 1 ∞ ∞ ⎥ ⎥ ∞ 1 ∞ ∞ ∞ ⎥ ⎥ ∞ ∞ ∞ 0 ⎥ ∞ 2 ∞ 2 ⎦

The chosen sequential circuits are

LB = {L12345 , L345 }, φB1 = {φ2 , φ3 , φ4 }, φB 2 = {φ3 , φ4 , se } ,

φ NB1 = {φ }, φ NB 2 = φB1 = {φ2 , φ3 , φ4 } Furthermore, we have

m1 = degree[φ B1 − (φ NB1 ∩ φ B1 )] − 2 = degree{φ 2 , φ 3 , φ 4 } − 2 = 3 − 2 = 1 ,

m2 = degree[φ B2 − (φ NB2 ∩ φ B2 )] − 2 = degree[{φ3 , φ 4 , s e } − {φ3 , φ 4 }] − 2 = −1

Therefore m = 1 . As shown in Fig. 17(b), the constraint unit is in circuit L345 , along with the identity condition

see3 = see5 , that is the arc length of gear 3 rolled is equal to that

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T.J. Li, W.Q. Cao and T.H. Yan

Example 4: Fig. 18 Shows a Complex GLM, its Tricolored Graph, and the Decomposing Procedure.

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(a)

(b)

(c) Figure 18. Structural decomposition of a GLM. (a) Kinematic diagram, (b) weighted tricolored graph, (c) decomposing procedure.

The chosen sequential circuits are

LB = {L516 , L12345 , L627 , L734 } , φB1 = {φ6 , sk } , φB 2 = {φ2 , φ3 , φ4 } ,

φB 3 = {φ7 , φ6 , φ2 , sh } , φB 4 = {φ7 , φ3 , φ4 , s f }, φ NB1 = {φ }, φ NB 2 = {φB1} = {φ6 , sk }, φ NB 3 = φB1 ∪ φB 2 = {φ2 , φ3 , φ4 , φ6 , sk },

φ NB 4 = φB1 ∪ φB 2 ∪ φB 3 = {φ3 , φ4 , φ2 , φ6 , φ7 , sh , sk }

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Furthermore, we have

m1 = degree[φ B1 − (φ NB1 ∩ φ B1 )] − 2 = 2 − 2 = 0

m2 = degree[φ B 2 − (φ NB 2 ∩ φ B 2 )] − 2 = 3 − 2 = 1

m3 = degree[φ B 3 − (φ NB 3 ∩ φ B 3 )] − 2 = 4 − 2 − 2 = 0

m4 = degree[φ B 4 − (φ NB 4 ∩ φ B 4 )] − 2 = 4 − 3 − 2 = −1 Therefore m = 1 . The constraint unit is within L347 , along with the identity condition

s ff 7 = s ff 4 .

6. Conclusion This work introduces several applications of graph theory in mechanism analysis. Firstly the types of graphs are given to represent the different KCs and mechanisms. Then the methods for isomorphism detection, topology-loop analysis and identification, and the structural decomposition are introduced based on the graph representations of mechanisms. These methods can be performed easily in computers, which provide a sound basis for the mechanism analysis. For example, the method of structural decomposition can realize to decompose the complex PLMs and GLMs into the simple kinematic units, and analyzing these pieces one at a time is an effective way to analyze the overall motion.

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References [1] [2] [3] [4] [5] [6]

[7] [8]

Erdman, A. G. and Sandor, G. N. (1991). Mechanism design: analysis and synthesis. New Jersey: Prentice Hall. Norton, R. L. (2001). Design of machinery: an introduction to the synthesis and analysis of mechanisms and machines (2ed). Columbus: McGraw-Hill. West D. B. (2001). Introduction to graph theory. New Jersey: Prentice Hall. Bondy J. A. and Murty U. S. R. (1976). Graph theory with applications. London: The MaCmillan Press ltd. Golumbic M. C., Hartman I. B. A. (2005). Graph Theory, combinatorics and algorithms: interdisciplinary applications. New York: Springer. Olson D. G., Erdman A. G., and Riley D. R. (1985). A systematic procedure for type synthesis of mechanisms with literature review. Mechanism and Machine Theory,20, 285-295. Yan H. S. and Hwang Y. W. (1991). The specialization of mechanisms. Mechanism and Machine Theory, 26, 541-551. Tsai L. W. (2001). Mechanism design: enumeration of kinematic structures according to function. Boca Raton: CRC Press.

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34 [9]

[10] [11]

[12]

[13] [14]

[15] [16]

[17]

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[18] [19]

[20]

[21] [22]

[23] [24]

T.J. Li, W.Q. Cao and T.H. Yan Chu J. K., Cao W. Q. and Yang T. L. (1998). Type synthesis of baranov truss with multiple joints and multiple-joint links. 1998 ASME Design Engineering Technical Conference, September 13-16, Atlanta, Georgia, DETC’98/MECH-5972. Li T. J., Cao W. Q. (2005). Kinematic Analysis of geared linkage mechanisms. Mechanism and Machine Theory, 40, 1394-1413. Li T. J., Cao W. Q. and Chu J. K. (1998). The topological representation and detection of isomorphism among geared linkage kinematic chains. 1998 ASME Design Engineering Technical Conference, September 13-16, Atlanta, Georgia, DETC’98/MECH-5812. Mruthyunjaya T. S. and Balasubramanzan H. R. (1987). In quest of a reliable and efficient computational test of isomorphism in kinematic chains. Mechanism and Machine Theory, 22, 131-139 Ambekar A. G. and Agrawal V. P. (1987). Canonical numbering of kinematic chains and isomorphism problem: min code. Mechanism and Machine Theory, 22, 1-8 Luo Y. F., Yang T. L., Cao W. Q. (1991). Identification of spatial kinematic chains using incident degree and incident degree code. Proc. of the 8th world congress on theory of Mech. and Mech., 4, 999-1002 Tang C. S. and Liu T. (1988). The degree code-a new mechanism identifier. Trends and Developments in Mechanism, Machines and Robotics, 1, 147-151 Ambekar A. G. and Agrawal V. P. (1985). Identification and classification of kinematic chains and mechanisms using identification codes. The 4th Inter Symposium on Linkages and Computer Aided Design Methods, 1, 545-552 Rao A. C., Raju D. V. (1991). Application of the hamming number technique to deduct isomorphism among kinematic chains and inversions. Mechanism and Machine theory, 26, 55–75. Srinath A., Rao A. C. (2006). Correlation to detect isomorphism, parallelism and type of freedom. Mechanism and Machine Theory, 41, 646-655 Chu J. K. and Cao W. Q. (1994). Identification of isomorphism among kinematic chains and inversions using link’s adjacent-chain -table. Mechanism and Machine Theory, 29, 53-58 Yadav J. N. Pratap C. R. and Agrawal V. P. (1996). Computer aided detection of isomorphism among kinematic chains and mechanisms using the concept of modified distance. Mechanism and Machine Theory, 31, 439-444 Tuttle E. R. (1996). Generation of planar kinematic chains. Mechanism and Machine Theory, 31, 729-748 Li T. J., Cao W. Q. and Chu J. K. (1997). The computer automatic generation for the type synthesis of geared linkage mechanisms. Proceedings of International Conference on Mechanical Transmission and Mechanisms (MTM’97), Tianjin, China, 111-114 Cao W. Q. (2002). Analysis and synthesis of linkage mechanisms. Beijing: Public House of Science. Yang T. L., Yao F. H. (1994). Topological characteristics and automatic generation of structural synthesis of planar mechanisms based on the ordered single-opened-chains. Proceedings of the 1994 ASME Design Technical Conferences, 67-74

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ISBN 978-1-60741-011-9 c 2010 Nova Science Publishers, Inc.

Chapter 2

A C ATEGORICAL P ERSPECTIVE ON C ONNECTIONS WITH A PPLICATION IN THE F ORMULATION OF F UNCTORIAL P HYSICAL DYNAMICS Elias Zafiris∗ University of Athens, Department of Mathematics, Panepistimioupolis, 15784 Athens, Greece

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Abstract The mechanism of differential calculus and differential geometry is based on the fundamental notion of connection on a module over a commutative unital ring of scalars defined together with the associated de Rham complex. From a categorical point of view we show that the generation of this mechanism is a consequence of the existence of a monad-comonad pair arising from the adjunctive correspondence between the inverse algebraic processes of extending and restricting the scalars. The categorical remodeling of the notion of connection as well as of the associated de Rham complex in a functorial way permits the applicability of algebraic differential calculus in topoi independently of the existence of any background manifold. This is of particular importance for the formulation of dynamics in physical theories, where the adherence to such a substratum is problematic due to singularities or other topological defects, for example in quantum gravity. Thus, the adaptation of the functorial mechanism of connections in appropriate topoi, instantiating generalized localization environments of physical observables, induces a consistent covariant framework of dynamics in the regime of these topoi.

1.

Introduction

The basic nucleus of ideas developed in this article, demonstrates that, the conceptual categorical skeleton of the generative mechanism of differential calculus and, in extenso, differential geometry is a consequence of the existence of a monad-comonad pair, expressing the inverse algebraic processes of extending and restricting the scalars. Classically, the differential mechanism is based on the fundamental notion of a connection on a module ∗

E-mail address: [email protected]

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over a commutative unital ring of scalars defined together with the associated de Rham complex [1-3]. A connection on a module induces a process of infinitesimal extension of the scalars of the underlying ring, which is interpreted geometrically as a process of firstorder parallel transport along infinitesimally variable paths in the spectrum-space of this ring [4]. The next stage of development of the differential mechanism involves the satisfaction of appropriate global requirements referring to the transition from the infinitesimal to the global level. These requirements are of a homological nature and characterize the integrability property of the variation process induced by a connection. Moreover, they are properly addressed by the construction of the De Rham complex associated to an integrable connection. The non-integrability of a connection is characterized by the notion of curvature bearing the semantics of observable disturbances to the process of cohomologically unobstructed variation induced by the corresponding connection. It is instructive to emphasize that the conceptualization of the classical differential mechanism along these lines does not presuppose the existence of an underlying differential manifold. This observation is particularly important because it provides the possibility of abstracting in functorial terms both the definition of a connection and the associated De Rham complex as well. The functorial recasting of the differential mechanism becomes possible by the appropriate qualification of the information encoded in the extension/restriction of scalars adjunction, or equivalently, in the corresponding monad-comonad pair. This information permits the definition of the universal object of differential 1-forms, and subsequently, the definition of a connection, together with the associated de Rham complex. The benefits we obtain from the functorial reformulation of the differential mechanism are multiple. Firstly, the differential mechanism becomes explicitly non-dependent on the existence of a smooth underlying manifold. Secondly, it can be applied effectively in a variety of topological localization environments, conceived as categories of sheaves, under the satisfaction of the appropriate homological requirements. Thirdly, the differential mechanism can be abstracted further via the categorical notion of a monadic connection on a module of a monad, and thus, can be applied to more general bidirectional processes of extension/restriction of scalars from one level of structure to another, described by means of an adjunctive correspondence between those levels. Finally, the functorial recasting of the mechanism of generation of differential calculus is of paramount importance for theoretical physics, because it can be properly used for the functorial formulation of dynamics taking into account the localization properties of the observables in the regime of each physical theory. In relation to the formulation of functorial physical dynamics we claim the validity of a network of basic physical principles, which secure the intelligibility of the categorical framework according to the operational and theoretical requirements of each physical theory, and moreover, permit the applicability of the mechanism of differentials in a variety of regimes of observable structure. This is of particular importance in the current debate concerning the construction of a tenable quantum theory of gravity, conceived as a unifying dynamical framework of both General Relativity and Quantum Mechanics [4-17]. It turns out that the problematics referring to the background manifold independence of the dynamics necessitated by a consistent theory of Quantum Gravity, should take at face value the fact that the fixed manifold construct in General Relativity is just the byproduct of absolutely fixing physical representabilty in terms of real numbers. Moreover, this absolute

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fixation is completely independent of the possibility of formulating dynamics, since the differential mechanism can be generated functorially in terms of an extension/restriction of observables adjunction, in accordance with the appropriate homological constraints, by means of monadic connections. Hence the usual analytic differential geometric framework of smooth manifolds, needed for the formulation of General Relativity, is just a special coordinatization of the universal functorial mechanism of infinitesimal scalars extension, and thus should be substituted appropriately, in case a merging with Quantum Theory is being sought. The substitution is guided by the principle of local relativization of physical representability with respect to a localization topos, and the effectuation of a field-theoretic dynamical mechanism by means of a monadic connection arising from an extension/restriction of observables monad-comonad pair. In this communication we adopt as a basic conceptual, methodological and technical tool the algebraic, categorical and topos-theoretic perspective of modern mathematics [1, 3, 18-25]. In Section 2, we reconstruct the classical mechanism of differential calculus and differential geometry from the information encoded in the extension/restriction of scalars adjunction obtained by considering a base morphism of commutative unital rings. We define functorially the universal object of differential 1-forms, the notion of a connection on a module, and finally we set-up the algebraic De Rham complex associated with a connection. Moreover, from the integrability property of a connection we derive the notion of curvature. In Section 3, we define the abstract equivalent notion of a connection on a module for a monad, where the latter is obtained from the extension/restriction of scalars adjunction. In Section 4, we derive the gravitational field dynamical mechanism of the General Theory of Relativity from the classical extension/restriction monad-comonad pair and discuss in detail its physical semantics. In Section 5, we lay the foundations for the formulation of functorial physical theories of field dynamics. For this purpose we provide a conceptual analysis of the basic principles underlying the intelligibility of functorial physical dynamics. Subsequently, we apply these principles for the construction of the following: [i]. Topological sheaf-theoretic dynamics according to the framework of Abstract Differential Geometry, and [ii]. Quantum functorial dynamics form the classical to quantum extension/restriction monad-cominad pair. Finally, in Section 6 we conclude that, the generation of a dynamical mechanism of differentials can be considered in a unifying perspective as the natural outcome of the adjunctive correspondence between the inverse processes of extension/restriction of scalars according to well-defined homological requirements. Furthermore, the endorsement of the principle of topos-theoretic local relativization of representability, necessitated by the distinctive properties of localization of the observables, permits the transfer of the dynamical mechanism, arising from the monad-comonad pair of the inverse processes of extension/restriction of scalars, in the regime of the corresponding localization topoi.

2. 2.1.

The Extension/Restriction of Scalars Categorical Adjunction The Adjoint Pair of Extension/Restriction Functors

In general, given a morphism of commutative and unital rings of scalars, ψ : A → B, considered as an extension of scalar coordinates, we obtain simultaneously two functors Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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between the corresponding categories of their modules [1], pointing into opposite directions as follows: I. The functor ψ ∗ from the category of B-modules to the category of A-modules ψ ∗ : M(B) → M(A) where, for any B-module M , ψ ∗ (M ) := ResB A (M ) is the underlying abelian group M , with the A-action given by: m · a := m · ψ(a) (m ∈ M , a ∈ A). The functor ψ ∗ := ResB A := R is called the restriction of scalars functor from the category of B-modules to the category of A-modules. II. The functor ψ! from the category of A-modules to the category of B-modules ψ! : M(A) → M(B) where, for any A-module N , ψ! (N ) := ExtA B (N ) = N the action a · b = ψ(a) · b (a ∈ A, b ∈ B). The functor ψ! := ExtA B = [−]

O

A

N

A B,

and B is an A-module via

B := L

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is called the extension of scalars functor from the category of A-modules to the category of B-modules. Furthermore, the algebraic functorial processes of restriction and extension of scalars are inverse, viz., given a morphism of commutative and unital rings of scalars, ψ : A → B, there exists a categorical adjunction ψ! : M(A)

qqqq qqqqqqqqq qqqq qqqqq qqqq qqq qqqqq

M(B) : ψ ∗

where, ψ! := L is left adjoint to ψ ∗ := R, such that, we have a bijection of morphisms HomB (ψ! (N ), M ) ∼ = HomA (N, ψ ∗ (M )) which, is natural for any N in M(A) , and any M in M(B) . Moreover, the established extension/restriction of scalars categorical adjunction, L : M(A)

qqqq qqqqqqqqq qqqq qqqqq qqqqq qq qqqqq

M(B) : R

is completely characterized in terms of the unit and counit natural transformations. More specifically, for each module N in M(A) , the unit is defined as: σN : N

qqqqq qqqq qqq qqqqq

RLN

On the other side, for each module M of M(B) the counit is defined as: ǫM : LR(M )

qqqqq qqqq qqq qqqqq

M

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2.2.

39

The Universal Object of Differential 1-Forms

The first basic objective of the categorical perspective on abstract differential calculus is to express the notions of modules and derivations of a commutative unital ring B in B intrinsically with respect to the information contained in the category B. This can be accomplished by using the method of categorical relativization, which is based on the passage to the comma category B/B [26]. More concretely, the basic problem has to do with the possibility of representing the information contained in an B-module, where B is a commutative unital ring in B, with a suitable object of the relativization of B with respect to B, viz., with an object of the comma category B/B. For this purpose, we L define the split extension of the commutative L ring B by an B-module M , denoted by B M , as follows: The underlying set of B M is the cartesian product B × M , where the group and ring theoretic operations are defined respectively as; (a, m) + (b, n) := (a + b, m + n) (a, m) • (b, n) := (ab, a · n + b · m) L Notice that the identity element of B M is (1B , 0M ), and also that, the split extension L B M contains an ideal 0B × M := hM i, that corresponds naturally to the B-module M . Thus, given a commutative L ring B in B, the information of an B-module M , consists of an object hM i (ideal in B M ), together with a split short exact sequence in B; M hM i ֒→ B M →B

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We infer that the ideal hM i is identified with the kernel of the epimorphism B viz., M hM i = Ker(B M → B)

L

M → B,

From now on we focus our attention to the comma category B/B, noticing that idB : B → B is the terminal object in this category. IfL we consider the split extension of the commutative ring B, by an B-module M , that is B M , then the morphism: M λ:B M →B (a, m) 7→ a is obviously an object of B/B. Moreover, it easy to show that it is actually an abelian group object in the comma category B/B. This equivalently means that for every object ξ in B/B the set of morphisms HomB/B (ξ, λ) is an abelian group in Sets. Moreover, the arrow γ : κ → λ is a morphism of abelian groups in B/B if and only if for every ξ in B/B the morphism; γˆξ : HomB/B (ξ, κ) → HomB/B (ξ, λ) is a morphism of abelian groups in Sets. We denote the category of abelian group objects in B/B by the suggestive symbol [B/B]Ab . Based on our previous remarks, it is straightforward to show that the category of abelian group objects in B/B is equivalent with the category of B-modules, viz.: [B/B]Ab ∼ = M(B)

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Thus, we have managed to characterize intrinsically B-modules as abelian group objects in the relativization of the category of commutative unital rings B with respect to B, and moreover, we have concretely identified them as kernels of split extensions of B. The characterization of B-modules as abelian group objects in the comma category B/B is particularly useful if we consider an B-module M as a codomain for derivations of objects of B/B. For this purpose, let us initially notice that if k : A → B is an arbitrary object in B/B, then any B-module M is also an A-module via the morphism k. We define a derivations functor from the comma category B/B to the category of abelian groups Ab: Der(−, M ) : B/B → Ab Then, if we evaluate the derivations functor at the commutative arithmetic A we obtain: M Der(A, M ) ∼ M) = HomB/B (A, B This means that, given an object k : A → B in B/B, then a derivation d : A → M is the same as the following morphism in B/B: B qqqqqqqqqq qqq q

qqqqqqqqqqq qqq

@ @

k

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A

@λ @

d˜

@

qqqqq qqqq qqq qqqqq

@

B

L

M

L Now we notice that the morphism: λ : B M → B is actually an object in [B/B]Ab . Hence, we consider it as an object of [B/B] via the action of an inclusion functor: ΥB : [B/B]Ab ֒→ [B/B] [λ : B

M

M → B] 7→ [ΥB (λ) : ΥB (M ) → B]

Thus we obtain the isomorphism: Der(A, M ) ∼ = HomB/B (A, ΥB (M )) The inclusion functor ΥB has a left adjoint functor; ΩB/(−) : [B/B] → [B/B]Ab Consequently, if we further take into account the equivalence of categories [B/B]Ab ∼ = M(B) the above isomorphism takes the following final form: Der(A, M ) ∼ = HomM(B) (ΩB/A (A), M )

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We conclude that the derivations functor Der(−, M ) : B/B → Ab is being represented by the abelianization functor ΩB/(−) : [B/B] → [B/B]Ab . Furthermore, the evaluation of the abelianization functor ΩB/(−) at an object k : A → B of B/B, viz. ΩB/A (A), is interpreted as the B-module of differentials on A. Finally, it is straightforward to see that, evaluating at the terminal object of B/B we arrive at the following conclusion: The covariant functor of B-modules valued derivations of B, denoted by DerB (−), is being representable by the free B-module of differential 1-forms of B over A, denoted by ΩB/A := Ω1 B/A in the category of B-modules, according to the isomorphism: DerB (M ) ∼ = HomM(B) (ΩB/A , M ) Hence, the object Ω1 B/A is characterized categorically as the universal object of differential 1-forms in M(B) and the derivation dB/A : B → Ω1 B/A as the universal derivation. More precisely, the universal derivation dB/A is an additive A-linear morphism: dB/A : B → Ω1 B/A f 7→ dB/A (f ) which, moreover, satisfies the Leibniz rule:

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dB/A (f · g) = f · dB/A (g) + g · dB/A (f ) In this setting, the formal symbols of differentials {dB/A f, f ∈ B}, generate the universal object of differential 1-forms Ω1 B/A , modulo the Leibniz constraint, where the scalars of A are treated as constants. Furthermore, according to our initial conclusions, the information of the free B-module 1 of differential L 1-forms of B over A, viz. ΩB/A := Ω B/A , consists of an object hΩB/A i (ideal in B ΩB/A ), together with a split short exact sequence in B; M hΩB/A i ֒→ B ΩB/A → B where, the ideal hΩB/A i constitutes the kernel of the epimorphism B M hΩB/A i = Ker(B ΩB/A → B)

L

ΩB/A → B, viz.,

In this context, the universal derivation dB/A , viz., the above additive A-linear morphism satisfying the Leibniz rule, is equivalent with the morphism of commutative unital rings: M dB/A : B → B ΩB/A expressed equivalently in the following suggestive notation: M dB/A : B → B ΩB/A · ǫ f 7→ f + dB/A (f ) · ǫ where dB/A (f ) =: dB/A f is considered as the infinitesimal part of the extended scalar, and ǫ the infinitesimal unit obeying ǫ2 = 0. The algebra of infinitesimally extended scalars

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B ⊕ ΩB/A · ǫ is called the algebra of dual numbers over B with coefficients in the universal B-module of 1-forms ΩB/A . It is immediate to see that the infinitesimally extended ring B ⊕ΩB/A ·ǫ, as an abelian group is just the direct sum B ⊕ΩB/A , whereas the multiplication is defined by: ′

′

′

′

′

(f + dB/A f · ǫ) • (f + dB/A f · ǫ) = f · f + (f · dB/A f + f · dB/A f ) · ǫ Note that, we further require that the composition of the augmentation B ⊕ ΩB/A · ǫ → A, with dB/A is the identity.

2.3.

The Notion of Connection

A natural problem arising in the categorical setting of differential calculus refers to the representability of the universal object of differential 1-forms from the information encoded in the extension/restriction of scalars categorical adjunction. For this purpose, we remind that the extension/restriction of scalars categorical adjunction, L : M(A)

qqqq qqqqqqqqq qqqq qqqqq qqqq qqq qqqqq

M(B) : R

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is completely characterized in terms of the unit and counit natural transformations, where, for each module N in M(A) and for each module M of M(B) we obtain the morphisms:

σN

σN : N → RL(N ) O : N → ResB A (N B) A O n 7→ n 1

ǫM

ǫM : LR(M ) → M O : ResB A (M ) B→M A O m b 7→ m · b

The counit of the adjunction, defined by the composite endofunctor: G := LR : M(B) → M(B) constitutes the first step of a functorial free resolution of a module M in M(B) . Actually, by iterating the endofunctor G, we may extend ǫM to a free simplicial resolution of M . We notice that, if we take M = B, then, ǫM coincides with the multiplication morphism O µ:B B→B A

b1

O

b2 7→ b1 · b2

Then by taking the kernel of the multiplication morphism of rings µ, identified as the kernel of the counit ǫB of the extension/restriction of scalars adjunction, that is, the ideal: O O I = {f1 ⊗ f2 ∈ B B : µ(f1 ⊗ f2 ) = 0} ⊂ B B A

A

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we are goingto prove that the morphism of B-modules: Σ : ΩB/A →

I I2

dB/A f 7→ 1 ⊗ f − f ⊗ 1 is an isomorphism. We can prove the above isomorphism as follows: The fractional object module structure defined by:

I I2

has an B-

f · (f1 ⊗ f2 ) = (f · f1 ) ⊗ f2 = f1 ⊗ (f · f2 ) for f1 ⊗ f2 ∈ I, f ∈ B. We can check that the second equality is true by proving that the difference of (f · f1 ) ⊗ f2 and f1 ⊗ (f · f2 ) belonging to I, is actually an element of I2 , viz., the equality is true modulo I2 . So we have: (f · f1 ) ⊗ f2 − f1 ⊗ (f · f2 ) = (f1 ⊗ f2 ) · (f ⊗ 1 − 1 ⊗ f ) The first factor of the above product of elements belongs to I by assumption, whereas the second factor also belongs to I, since we have that: µ(f ⊗ 1 − 1 ⊗ f ) = 0

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Hence the product of elements above belongs to I · I = I2 . Consequently, we can define a morphism of B-modules: I Σ : ΩB/A → 2 I dB/A f 7→ 1 ⊗ f − f ⊗ 1 Now, we construct the inverse of that morphism as follows: The B-module ΩB/A can be made an ideal in the algebra of dual numbers over B, viz., B ⊕ ΩB/A · ǫ. Moreover, we can define the morphism of rings: B × B → B ⊕ ΩB/A · ǫ (f1 , f2 ) 7→ f1 · f2 + f1 · df2 ǫ This is an A-bilinear morphism of rings, and thus, it gives rise to a morphism of rings: Θ : B⊗A B → B ⊕ ΩB/A · ǫ Then, by definition we have that Θ(I) ⊂ ΩB/A , and also, Θ(I2 ) = 0. Hence, there is obviously induced a morphism of B-modules: ΩB/A ←

I I2

which is the inverse of Σ. Consequently, we conclude that: I ΩB/A ∼ = 2 I Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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Thus, the universal object of differential 1-forms ΩB/A can be isomorphically represented by the information encoded in the counit of the extension/restriction of scalars adjunction, applied to B, by means of taking the kernel of the counit morphism at B, viz. the kernel of ǫB . Equivalently, the universal object of differential 1-forms of B over A, viz. ΩB/A , is represented by means of the following split short exact sequence: 0 → ΩB/A ֒→ GB → B → 0 O 0 → ΩB/A ֒→ B B→B→0 A

M(B) ,

Analogously, for each module M of we represent the module of differential 1-forms corresponding to M , denoted by ΩM , by means of the following split short exact sequence: 0 → ΩM ֒→ GM → M → 0 According to the above, we obtain that:

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J ΩM ∼ = 2 J where J = Ker(ǫM ) denotes the kernel of the counit of the extension/restriction of scalars adjunction applied at M in M(B) . At a further stage of development of these ideas, if we tensor the exact sequence representing the universal object of differential 1-forms ΩB/A with M over B we obtain: O O O O 0→M ΩB/A ֒→ M (B B) → M B→0 B B A B N N N Notice that, the term M B (B A B) is naturally isomorphic to the term M A B, via the B-linear morphism F defined by: O O O F (m b c) = m · b c where, the inverse morphism is given by: O O O F −1 (m b) = m 1 b N ByN the above, we N conclude that, ΩM is isomorphic to M B ΩB/A , being the image of M B ΩB/A in M A B, via the B-linear isomorphism F . Moreover, since the B-module N ofN differential 1-forms of B over A, is generated by the set of elements dB/A b = 1 b − b 1, the B-module of differential M , according to the above, is generated by N 1-forms ΩN the set of elements of the form m b − m · b 1. The representation of the the B-module of differential 1-forms ΩM intrinsically in terms of the information encoded in the kernel of the counit of the extension/restriction of scalars adjunction provides the appropriate setting for the introduction of the notion of connection as follows: Initially, we notice that the functor R(M ) is left exact, because it is the right adjoint functor of the extension/restriction of scalars adjunction. Thus, it preserves the short exact sequence defining the object of differential 1-forms ΩM , in the following form: 0 → R(ΩM ) → R(G(M )) → R(M ) Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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Hence, we immediately obtain that: Z R(ΩM ) ∼ = 2 Z where Z = Ker(R(ǫM )). Then, the notion of a connection ∇R(M ) on a module M in M(B) , is defined as the following A-linear morphism: ∇R(M ) : R(M ) → R(ΩM ) such that, the following condition is satisfied: ∇R(M ) (m · b) = ∇R(M ) (m) · b + m

O

b−m·b

O

1

Obviously, by the preceding remarks, the connection ∇R(M ) on the module M in M(B) is equivalent to the following A-linear morphism: O ∇R(M ) : R(M ) → R(M ΩB/A ) B

such that, the following condition is satisfied (Leibniz rule): O ∇R(M ) (m · b) = ∇R(M ) (m) · b + m dB/A b

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2.4.

The Algebraic De Rham Complex and the Notion of Curvature

The next stage of development of the categorical schema of elucidating the mechanism of differential calculus involves the satisfaction of appropriate global constraints, which impose consistency requirements referring to the transition from the infinitesimal to the global level. For this purpose it is necessary to employ the methodology of homological algebra [1]. We start by reminding the algebraic construction, for each n ∈ N , n ≥ 2, of the n-fold exterior product as follows: ^n ( ΩB/A )n := Ωn (B) = Ω1 (B) where Ω(B) := Ω1 (B), B := Ω0 (B). We notice that there exists an A-linear morphism: dn : Ωn (B) → Ωn+1 (B) for all n ≥ 0, such that d0 = d. Let ω ∈ Ωn (B), then ω has the form: X ^ ^ ω= fi (dli1 . . . dlin ) with fi , lij , ∈ B for all integers i, j. Further, we define: X ^ ^ ^ dn (ω) = dfi dli1 . . . dlin Then, we can easily see that the resulting sequence of A-linear morphisms; B → Ω1 (B) → . . . → Ωn (B) → . . .

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46

Elias Zafiris

is a complex of A-modules, called the algebraic de Rham complex of B. The notion of complex means that the composition of two consequtive A-linear morphisms vanishes, that is dn+1 ◦ dn = 0, simplified symbolically as: d2 = 0 If we assume that ∇E is a connection defined on the B-module E, then ∇E induces a sequence of A-linear morphisms: O O E → Ω1 (B) E → . . . → Ωn (B) E → ... B

B

or equivalently: E → Ω1 (E) → . . . → Ωn (E) → . . . where the morphism: ∇n : Ωn (B)

O

B

E → Ωn+1 (B)

O

B

E

is given by the formula: ∇n (ω ⊗ v) = dn (ω) ⊗ v + (−1)n ω ∧ ∇(v) for all ω ∈ Ωn (B), v ∈ E. It is immediate to see that ∇0 = ∇E . Let us denote by: O R∇ : E → Ω2 (B) E = Ω2 (E)

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B

the composition ∇1 ◦ ∇0 . We see that R∇ is actually an B-linear morphism, that is Bcovariant, and is called the curvature of the connection ∇E . We notice that, the latter sequence of A-linear morphisms, is actually a complex of A-modules if and only if: R∇ = 0 We say that the connection ∇E is integrable if R∇ = 0, and we refer to the above complex as the de Rham complex of the integrable connection ∇E on E in that case. It is also usual to call a connection ∇E flat if R∇ = 0. A flat connection defines a maximally undisturbed process of differential variation. In this sense, a non-vanishing curvature signifies the existence of disturbances from the maximally symmetric state of that variation.

3. 3.1.

The Abstract Equivalent Monadic Notion of Connection The Extension/Restriction of Scalars Monad-Comonad Pair

Categorically from the extension/restriction of scalars adjunction, L : M(A)

qqqq qqqqqqqqq qqqq qqqqq qqqqq qq qqqqq

M(B) : R

we may construct a corresponding monad-comonad pair as follows: Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Connections and Functorial Physical Dynamics

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We have already seen that the extension/restriction of scalars adjunction is completely characterized in terms of the unit and counit natural transformations. For any module N in M(A) , the unit is defined as: RLN σN : N qqqqq qqqq qqq qqqqq

On the other side, for each module M of M(B) the counit is: ǫM : LR(M )

qqqqq qqqq qqq qqqqq

M

The composite endofunctor G := LR : M(B) → M(B) , together with the natural transformations δ : G → G ◦ G, called comultiplication, and also, ǫ : G → I, called counit, where I is the identity functor on M(B) , is defined as a comonad (G, δ, ǫ) on the category M(B) , provided that the diagrams below commute for each object M of M(B) ; δM

GM

qqqqq qqqq qqq qqqqq

δM

G2 M δGM

qqqqq qqqqq qqqqqqqq

qqqqq qqqqq qqqqqqqq

GδM

G2 M

qqqqq qqqqq qq qqqqq

G3 M

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GM

δM qqqqq qqqqq qqqqqqqq

GM

qqqq qqqqqqqqq qqqq

ǫGM

@ @ @ @ @ @ @ @ @ @ @ @

G2 M

GǫM

qqqqq qqqqq qq qqqqq

GM

Correspondingly, the composite endofunctor X := RL : M(A) → M(A) , together with the natural transformations ω : X ◦ X → X, called multiplication, and also, σ : I → X, called unit, where I is the identity functor on M(B) , is defined as a monad (X, ω, σ) on the category M(A) , provided that, the diagrams below commute for each object N of M(A) ; XN

qqqq qqqqqqqqq qqqq

ωN

qqqqqqq qqqqq qqqqq

qqqqqqq qqqqq qqqqq

ωN

X2 N

X2 N ωXN

qqqq qqqqqqqqq qqqq

XωN

X3 N

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Elias Zafiris

XN qqqqqqqqqqq qqq

qqqqqqqqqq qqq q

@ @

XσN

X2 N

ωN

qqqqq qqqq qqq qqqqq

XN

qqqq qqqqqqqqq qqqq

@ σXN @ @ @

ωN

X2 N

For a comonad (G, δ, ǫ) on M(B) , a G-coalgebra (comodule) is an object M of M(B) , being equipped with a structural map κ : M → GM , such that the following conditions are satisfied: 1M = ǫM ◦ κ : M → M Gκ ◦ κ = δM ◦ κ : M → G2 M With the above obvious notion of morphism, this gives a category M(B) G of all Gcoalgebras. Correspondingly, if (X, ω, σ) is a monad on the category M(A) , we define the category of M(B) X -algebras (modules) as follows: Its objects are pairs (N, µN ), where, N in M(A) , and, µN : X(N ) → N is a morphism in M(A) , such that, the following conditions are satisfied: 1N = µ ◦ σN : N → N

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µ ◦ Xµ = µ ◦ ωN : X2 N → N We easily notice that, the pair (N, ωN ), where, N in M(A) and ω is the multiplication morphism, is an M(A) X -algebra. It is also essential to notice that, h : (N, µN ) → (Z, µZ ) is a morphism in M(B) X , if and only if, h : N → Z is a morphism in M(A) , and moreover, we have: h ◦ µN = µZ ◦ X(h) We may recapitulate, by noticing that, for the extension/restriction of scalars adjunction we have: [i]. The composite endofunctor X := RL : M(A) → M(A) , together with the natural transformations ω : X ◦ X → X (multiplication), where, ω = RǫL , and also, σ : I → X (unit), where, I is the identity functor on M(A) , constitutes a monad (X, ω, σ) on the cat(A) egory M(A) . Furthermore, the pair (R(M ), R(ǫM )) constitutes an MX -module. Notice (A) that, the latter is an MX -module, if and only if, M is in M(B) . [ii]. The composite endofunctor G := LR : M(B) → M(B) , together with the natural transformations δ : G → G ◦ G (comultiplication), where, δ = LσR , and also, ǫ : G → I (counit), where, I is the identity functor on M(B) , constitutes a comonad (G, δ, ǫ) on the (B) category M(B) . Furthermore, the pair (L(N ), L(σN )) constitutes an MG -comodule. More precisely, the unit and counit natural transformations are defined as follows: For each module N in M(A) and for each module M of M(B) we have the morphisms: σN : N → RL(N ) Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Connections and Functorial Physical Dynamics O σN : N → ResB A (N B) A O n 7→ n 1

ǫM

3.2.

49

ǫM : LR(M ) → M O : ResB A (M ) B→M A O m b 7→ m · b

Categorical Monadic Reformulation of Connections

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We have noticed in the previous section that the the pair (R(M ), R(ǫM )) constitutes an (A) (A) MX -module. Moreover, the latter is an MX -module, if and only if, M is in M(B) . A natural question posing itself in the context of the monad-comonad pair arising from the extension/restriction of scalars adjunction is the following: Is it possible to abstract categorically the notion of a connection on a module M is in M(B) in terms of an appropriate reformulation involving only the functorial information incorporated in the extension/restriction monad-comonad pair? A hint towards such a reformulation has been already provided by (A) the observation that (R(M ), R(ǫM )) is an MX -module, if and only if, M is in M(B) . Our strategy will be the following: We introduce the notion of a monadic connection on (A) the MX -module (R(M ), R(ǫM )) and subsequently, we show that the monadic definition is equivalent with the definition of a connection on the module M in M(B) provided in Section 2.2. (A) We define a monadic connection on the MX -module (R(M ), R(ǫM )) as a morphism in M(A) : ∇ : R(M ) → RLR(M ) such that, the following conditions are satisfied: (C1 ) : ∇R(ǫM ) − RǫL(R(M )) RL(∇) = 1RL(R(M )) − σR(M ) R(ǫM ) (C2 ) : R(ǫM )∇ = 0 The above definition is obtained by means of Grothendieck’s approach to the notion of connection as an infinitesimal gluing datum by means of descent theory [27]. Next, we will show that the definition of a connection on the module M in M(B) , is equivalent (A) with the definition of a monadic connection on the MX -module (R(M ), R(ǫM )), stated according to the framework of the established monad-comonad pair, corresponding to the extension/restriction of scalars adjunction. For that purpose, we observe that condition (C2 ), viz., R(ǫM )∇ = 0 is satisfied, if and only if, there exists: ∇R(M ) : R(M ) → R(ΩM ) such that: ∇ = R(ιM )∇R(M ) Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

50

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where, ιM denotes the injection ΩM ֒→ GM . Moreover, in that case, the condition (C1 ) is equivalent to the condition (C´1 ): ∇R(M ) R(ǫM ) − R(ΩǫM )RL(∇R(M ) ) = ΛM where, ΛM is the unique natural morphism: ΛM : RLR(M ) → R(ΩM ) satisfying the relation: R(ιM )ΛM = 1RLR(M ) − σR(M ) R(ǫM ) Conversely, if we consider the condition (C´1 ), and moreover, replace ∇R(M ) by R(ιM )∇R(M ) , we obtain the condition (C1 ). Thus, we conclude that, the information of a connection on R(M ) consists of a morphism ∇R(M ) : R(M ) → R(ΩM )

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which, satisfies additionally the condition (C´1 ). The latter condition can be equivalently be written in the following form, in terms of elements: O O ∇R(M ) (m · b) = ∇R(M ) (m) · b + m b−m·b 1 Hence, it is equivalent with the Leibniz condition. Thus, we conclude that, the notion of a (A) monadic connection on the MX -module (R(M ), R(ǫM )), stated according to the framework of the established monad-comonad pair (corresponding to the extension/restriction of scalars adjunction) is tautosemous with the classical notion of a connection on the module M in M(B) , as an A-linear morphism, that satisfies the Leibniz rule. In a nutshell, we have managed to express the notion of a connection on a module (A) M in M(B) as a monadic connection on the MX -module (R(M ), R(ǫM )) using exclusively the functorial information incorporated into the extension/restriction of scalars monad-comonad pair. Since the notion of monadic connection does not depend on the particular specification of the adjoint pair of functors involved, it can be applied for the generation of a mechanism of differential calculus in every adjoint situation, interpreted as an extension/restriction of generalized scalars bidirectional process. Of course, as we have proved previously, the specification of the adjoint pair according to the classical extension/restriction of scalars adjunction reproduces fully and faithfully the classical mechanism of differential calculus, generated by the notion of a connection on a module M in M(B) together with the associated De Rham complex. The benefits we obtain from the categorical abstraction of the notion of connection monadically is of particular importance in the formulation of a functorial dynamical mechanism of non-classical physical theories, as we shall see in the sequel.

4.

General Theory of Relativity from the Classical Extension/Restriction Monad-Comonad Pair

The basic defining feature of General Relativity is the abolishment of any fixed preexisting kinematical framework by making the metric tensor a dynamically variable object. This Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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essentially means that, the geometrical relations defined on a four dimensional smooth manifold, endowing it with the structure of a spacetime manifold, become variable. Moreover, they are constituted dynamically by the gravitation field, as well as other fields from which matter can be derived, by means of Einstein’s field equations, through the imposition of a compatibility requirement relating the metric tensor, which represents the spacetime geometry, with the affine connection, which represents the gravitational field. The dynamic variability of the geometrical structure on the spacetime manifold constitutes the means of dynamicalization of geometry in the descriptive terms of General Relativity, formulated in terms of the differential geometric framework on smooth manifolds. The intelligibility of the framework is enriched by the imposition of the principle of general covariance of the field equations under arbitrary coordinate transformations of the points of the manifold preserving the differential structure, identified as the group of manifold diffeomorphisms. As an immediate consequence, the points of the manifold lose any intrinsic physical meaning, in the sense that, they are not dynamically localizable entities in the theory. Most importantly, manifold points assume an indirect reference as indicators of spacetime events only after the dynamical specification of geometrical relations among them, as particular solutions of the generally covariant field equations. From an algebraic viewpoint, a real differential manifold M can be recovered completely from the R-algebra C ∞ (M ) of smooth real-valued functions on it, and in particular, the points of M may be recovered from the algebra C ∞ (M ) as the algebra morphisms C ∞ (M ) → R [1, 4, 18-20]. In this sense, manifold points constitute the R-spectrum of C ∞ (M ), being isomorphic with the maximal ideals of that algebra. Notice that, the R-algebra C ∞ (M ) is a commutative algebra that contains the field of real numbers R as a distinguished subalgebra. This particular specification incorporates the physical assumption that our form of observation is being represented globally by evaluations in the field of real numbers. In the setting of General Relativity the form of observation is being coordinatized by means of a commutative unital algebra of scalar coefficients, called an algebra of observables, identified as the R-algebra of smooth real-valued functions C ∞ (M ). Hence, the background substratum of the theory remains fixed as the R-spectrum of the coefficient algebra of scalars of that theory, and consequently, the points of the manifold M , although not dynamically localizable degrees of freedom of General Relativity, are precisely the semantic information carriers of an absolute representability principle, formulated in terms of global evaluations of the algebra of scalars in the field of real numbers. Consequently, we conclude that, the classical monad-comonad pair arising from the inverse functorial processes of extension/restriction of scalars, in the case of general relativity theory, refers to the identifications A = R and B = C ∞ (M ). Notice that, at the level of the R-spectrum of C ∞ (M ), the only observables are the smooth functions evaluated over the points of M . In physical terminology, the introduction of new observables is conceived as the result of interactions caused by the presence of a physical field, identified with the gravitational field in the context of General Relativity. The basic idea of Riemann that has been incorporated in the context of General Relativity is that geometry should be built from the infinitesimal to the global. Geometry in this context is understood in terms of metric structures that can be defined on a differential manifold. If we adopt the algebraic viewpoint, geometry as a result of interactions, requires the extension of scalars of the algebra C ∞ (M ) by infinitesimal quantities, defined

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as a fibration: d∗ : C ∞ (M ) ֒→ C ∞ (M ) ⊕ V · ǫ f 7→ f + d(f ) · ǫ where d(f ) =: df is considered as the infinitesimal part of the extended scalar, and ǫ the infinitesimal unit obeying ǫ2 = 0. The algebra of infinitesimally extended scalars C ∞ (M )⊕ V · ǫ is called the algebra of dual numbers over C ∞ (M ) with coefficients in the C ∞ (M )module V . It is immediate to see that the algebra C ∞ (M ) ⊕ V · ǫ, as an abelian group is just the direct sum C ∞ (M ) ⊕ V , whereas the multiplication is defined by: ′

′

′

′

′

(f + df · ǫ) • (f + df · ǫ) = f · f + (f · df + f · df ) · ǫ Note that, we further require that the composition of the augmentation C ∞ (M ) ⊕ V · ǫ → C ∞ (M ), with d∗ is the identity. Equivalently, the above fibration, viz., the homomorphism of algebras d∗ : C ∞ (M ) ֒→ C ∞ (M ) ⊕ V · ǫ, can be formulated as a derivation, that is, in terms of an additive R-linear morphism: d : C ∞ (M ) → V f 7→ df that, moreover, satisfies the Leibniz rule:

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d(f · g) = f · dg + g · df Since the formal symbols of differentials {df, f ∈ C ∞ (M )}, are reserved for the universal derivation, the C ∞ (M )-module V is identified as the free C ∞ (M )-module Ω of 1-forms generated by these formal symbols, modulo the Leibniz constraint, where the scalars of the distinguished subalgebra R, that is the real numbers, are treated as constants. Moreover, the free C ∞ (M )-module Ω can be constructed explicitly from N the∞fundamental form of ∞ ∞ ∞ scalars extension of C (M ), that is ι : C (M ) ֒→ C (M ) R C (M ) by considering the morphism: O δ : C ∞ (M ) C ∞ (M ) → C ∞ (M ) R

f1 ⊗ f2 7→ f1 · f2 Then, by taking the kernel of this morphism of algebras, that is, the ideal: O O I = {f1 ⊗ f2 ∈ C ∞ (M ) C ∞ (M ) : δ(f1 ⊗ f2 ) = 0} ⊂ C ∞ (M ) C ∞ (M ) R

R

we can easily show that, the morphism of C ∞ (M )-modules: Σ:Ω→

I I2

df 7→ 1 ⊗ f − f ⊗ 1 is an isomorphism. Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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Thus the free C ∞ (M )-module Ω of 1-forms is isomorphic with the free C ∞ (M )-module I of K¨ahler differentials of the algebra of scalars C ∞ (M ) over R, conceived as a distinI2 guished ideal in the algebra of infinitesimally extended scalars C ∞ (M ) ⊕ Ω · ǫ, due to interaction, according to the following split short exact sequence: Ω ֒→ C ∞ (M ) ⊕ ·ǫ ։ A or equivalently formulated as: O 0 → Ω → C ∞ (M ) C ∞ (M ) → C ∞ (M ) R

By dualizing, we obtain the dual

C ∞ (M )-module

of Ω, that is:

Ξ := Hom(Ω, C ∞ (M )) Thus, we have at our disposal, expressed in terms of infinitesimal scalars extension of the algebra C ∞ (M ), semantically intertwined with the generation of geometry as a result of interaction, new types of observables related with the incorporation of differentials and their duals, called vectors. Let us now explain the functionality of geometry, as related with the infinitesimally extended rings of scalars defined above, in the context of General Relativity. As we have argued before, the absolute representability principle of this theory, necessitates that our form of observation is tautosemous with real numbers representability. This means that all types of observables should possess uniquely defined dual types of observables, such that their representability can be made possible my means of real numbers. This is exactly the role of a geometry induced by a metric. Concretely, a metric structure assigns a unique dual to each observable, by effectuating an isomorphism between the C ∞ (M )-modules Ω and Ξ = Hom(Ω, C ∞ (M )), that is: Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

g:Ω≃Ξ df 7→ vf := g(df ) Thus the functional role of a metric geometry forces the observation of extended scalars, by means of representability in the field of real numbers, and is reciprocally conceived as the result of interactions causing infinitesimal variations in the scalars of the R-algebra C ∞ (M ). We emphasize that the intelligibility of the algebraic schema is based on the conception that infinitesimal variations in the scalars of C ∞ (M ), are caused by interactions, meaning that they are being effectuated by the presence of a physical field, identified as the gravitational field in the context of General Relativity. Thus, it is necessary to establish a purely algebraic representation of the notion of physical field and explain the functional role it assumes for the interpretation of that theory. The key idea for this purpose amounts to expressing the process of scalars extension in functorial terms, and by anticipation, identify the functor of infinitesimal scalars extension due to interaction with the physical field that causes it [26]. The gravitational physical field as the causal agent of interactions in the regime of classical general relativity, admits a purely algebraic description as the functor of gravitational infinitesimal scalars extension, called a gravitational connection-inducing functor: b : M(C ∞ (M )) → M(C ∞ (M )⊕Ω1 (C ∞ (M ))·ǫ) ∇

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Elias Zafiris E 7→ [C ∞ (M ) ⊕ Ω1 (C ∞ (M )) · ǫ]

O

C ∞ (M )

E

Thus, the effect of the action of the gravitational physical field on the vectors of the left C ∞ (M )-module E can be expressed by means of the following comparison morphism of left C ∞ (M )-modules: O ∇⋆ E : E → E ⊕ [Ω1 (C ∞ (M )) E] · ǫ ∞ C

(M )

Equivalently, the irreducible amount of information incorporated in the comparison morphism can be now expressed as a connection on E. The latter is defined, according to the previous framework, as an R-linear morphism of C ∞ (M )-modules [3]: O O ∇E : E → Ω1 (C ∞ (M )) E = E Ω1 (C ∞ (M )) = Ω1 (E) ∞ ∞ C

(M )

C

(M )

such that the following Leibniz type constraint is satisfied: ∇E (f · v) = f · ∇E (v) + df ⊗ v

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for all f ∈ C ∞ (M ), v ∈ E. In the context of General Relativity, the absolute representability principle over the field of real numbers, necessitates as we have explained above the existence of uniquely defined duals of observables. Thus, the gravitational field is identified with a linear connection on the C ∞ (M )-module Ξ = Hom(Ω1 , C ∞ (M )), being isomorphic with Ω1 , by means of a metric: ∗ g : Ω1 ≃ Ξ = Ω1 Consequently, the gravitational field may be represented by the pair (Ξ, ∇Ξ ). The metric compatibility of the connection required by the theory is simply expressed as: ∇HomC∞ (M ) (Ξ,Ξ∗ ) (g) = 0 Furthermore, the sequence of R-linear morphisms; C ∞ (M ) → Ω1 (C ∞ (M )) → . . . → Ωn (C ∞ (M )) → . . . is a complex of R-vector spaces, called the differential de Rham complex of C ∞ (M ). If we assume that, (E, ∇E ) represents the gravitational field, defined by a connection ∇E on the C ∞ (M )-module E, then ∇E induces a sequence of R-linear morphisms: O O n ∞ E → Ω1 (C ∞ (M )) E → . . . → Ω (C (M )) E → ... ∞ ∞ C

(M )

C

(M )

or equivalently: E → Ω1 (E) → . . . → Ωn (E) → . . . where the morphism: O ∇n : Ωn (C ∞ (M ))

C ∞ (M )

O E → Ωn+1 (C ∞ (M ))

C ∞ (M )

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E

Connections and Functorial Physical Dynamics

55

is given by the formula: ∇n (ω ⊗ v) = dn (ω) ⊗ v + (−1)n ω ∧ ∇(v) for all ω ∈ Ωn (C ∞ (M )), v ∈ E. It is immediate to see that ∇0 = ∇E . Let us denote by: O R∇ : E → Ω2 (C ∞ (M )) E = Ω2 (E) ∞ C

(M )

the composition ∇1 ◦ ∇0 . We see that R∇ is actually an C ∞ (M )-linear morphism, that is C ∞ (M )-covariant, and is called the curvature of the connection ∇E , being tautosemous with the Riemannian curvature of the spacetime manifold. We notice that, the latter sequence of R-linear morphisms, is actually a complex of R-vector spaces if and only if: R∇ = 0 We say that the connection ∇E is integrable if R∇ = 0, and we refer to the above complex as the de Rham complex of the gravitational integrable connection ∇E on E in that case. It is also usual to call a connection ∇E flat if R∇ = 0. A flat connection defines a maximally undisturbed process of dynamical variation caused by the corresponding physical field. In this sense, a non-vanishing curvature signifies the existence of disturbances from the maximally symmetric state of that variation, associated with the presence of matter distributions, being incorporated in the specification of the energy-momentum tensor. Taking into account the requirement of absolute representability over the real numbers, and thus considering the relevant evaluation trace operator by means of the metric, we arrive at Einstein’s field equations, which in the absence of matter sources read:

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R(∇Ξ ) = 0 where, R(∇Ξ ) denotes the relevant Ricci scalar curvature. We conclude that, the generation of gravitational field dynamics, in the context of General Relativity theory, can be considered in a unifying perspective as the natural outcome of the adjunctive correspondence between the inverse processes of extension and restriction of scalars, implied by the defining requirements of that theory, according to which, our form of observation is tautosemous with real numbers representability. The above has been called above, the principle of absolute representability, that forces the topos of Sets, as the proper universe of discourse, where, the dynamical mechanism of the corresponding monad-comonad pair is being implemented.

5. 5.1.

Functorial Physical Theories of Dynamics Conceptual Analysis of Basic Principles

The formulation of functorial physical theories of dynamics is of particular importance in the current debate referring to the construction of a theory of Quantum Gravity, viz. the construction of a quantum gravitational dynamical mechanism. It is significant to notice that the central focus of the studies pertaining to the formulation of a tenable theory of Quantum Gravity resolves around the issue of background manifold independence. In the context

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of our preceding analysis, we have constructed a general functorial framework of generation of a differential calculus mechanism based on the existence of a monad-comonad pair arising from a bidirectional adjunctive correspondence of extension/restriction of scalars. Most importantly, the generation of this mechanism is based on the notion of connection as well as on the associated De Rham complex. In particular, we have applied this framework in the case of General Relativity recovering the classical gravitational dynamics. Thus, by extrapolating, we may assert that the field dynamical mechanism of a physical theory is being generated conceptually by expressing the inverse algebraic processes of extending and restricting the scalars in a functorial form, and more precisely, by forming an appropriate monad-comonad pair. Of course, this strategy can be successful if certain requirements of a physical nature can be appropriately addressed. More precisely, the notion of a scalar acquires the meaning of an observable, the notion of a connection models a physical field, as a causal agent of the induced interactions due to its presence, and the notion of curvature measures the observable disturbance from the maximally symmetric state of dynamical variation defined by the non-obstructed temporal evolution of the corresponding field. Furthermore, the possibility of using the generalized notion of a monadic connection as the representative of an appropriate physical field provides the technical means to generate a mechanism of differential calculus in a complex regime of observables from information collected in a structured variety of simple regimes, if we manage to form a corresponding adjoint pair of extension/restriction functors from the simple ones to the complex and inversely. This possibility finds a concrete application in the problem of transferring the dynamical mechanism of General Relativity into the quantum regime of observable structure as we will argue in the sequel. From a physical standpoint, the formulation of functorial physical theories of dynamics permits the conceptual and technical replacement of the axiomatic set-theoretic model of the physical “continuum”, being at the core of the background spacetime manifold idealization of classical physical theories, by a topos-theoretic model, taking into account, the realistic operational systemic procedures of localization processes for discerning and coordinatizing observables. In particular, the significance of functorial physical dynamics lies on the fact that, the coordinatization of the universal mechanism of encoding physical interactions in terms of observables, by means of causal agents, viz., physical fields effectuating infinitesimal scalars extension, should respect only the following: Firstly, the ring or algebra-theoretic structure of observables used for the coordinatization of events in the physical “continuum”. Secondly, the homological requirements underlying the construction of algebraic de Rham complexes, involving the satisfaction of appropriate global constraints, imposing consistency constraints in the transition process of the mechanism of differential calculus from the infinitesimal to the global level. Consequently, the realization of a physical dynamical mechanism is not dependent on the codomain of representability of the observables, although it is subordinate to the algebra-theoretic characterization of their structures. In particular, it not constrained at all by the absolute representability principle over the field of real numbers, imposed by classical General Relativity, a byproduct of which is the fixed background manifold construct of that theory. In this vein of ideas, the absolute representability principle of classical General Relativity, formulated strictly in terms of real numbers, may be relativized locally without affecting the functionality of the general dynamical mechanism, being generated by the existence of

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a monad-comonad pair, corresponding to the inverse processes of extension/restriction of scalars. Consequently, it is possible to describe the dynamics of gravitational interactions in generalized localization environments, instantiated by suitable topoi. In particular, according to this strategy, the problem of quantization of gravity is equivalent to forcing the algebraic general relativistic dynamical mechanism of the gravitational connection inside an appropriate topos, which, is capable of incorporating the localization properties of observables in the quantum regime. The only cost to be paid for this topos-theoretic local relativization of representability is the rejection of the fixed background manifold structure of the classical theory. This is actually not a cost at all, since it would permit the intelligibility of the field equations over geometric realizations that include manifold singularities and other pathologies, without affecting the algebraic mechanism of dynamics. After these introductory remarks, it is instructive to delineate clearly the physical principles underlying the conceptualization of functorial physical dynamics. The first principle, called observability principle, specifies a physical observable structure as a locally commutative unital ring of scalars. This principle is elucidated by the semantic content associated with a structure of observables, qualified as appropriate for the effectuation of a dynamical mechanism, interpreted as a process of temporal evolution induced by some physical field. The term appropriate is being qualified precisely by a twofold determination of observable structures. The first component concerns their algebraic nature, by stipulating the structure of a commutative unital ring or algebra A over the real numbers. The significance of this stipulation lies on the fact that the categorical dual of a commutative unital algebra is understood as a geometric state space, called the spectrum of the algebra, such that, the elements of the algebra, viz. the observables, can be considered as functions on the spectrum. This consideration can be properly actualized, by employing the second component of determination of appropriate observable structures, concerning their topological nature. The latter is responsible for the localization of the information contained in the algebras of observables, with respect to a category of reference loci, being amenable to an operational specification. The net effect of the algebraic and topological organization of information, realized by means of a vertical (on the fibers) and horizontal (on the reference base) conceptual dimension respectively, can be formalized by the notion of a presheaf or sheaf of locally commutative unital algebras of continuous real-valued observables. The second principle is called the principle of local relativization of physical representability with respect to some suitable category of local reference loci of observation, actualized in the mathematical framework of localization topoi. This principle is elucidated by the implementation and interpretation of contextual physical localization processes. In classical theories localization has been conceived by means of metrical properties on a pre-existing smooth set-theoretic spacetime manifold. We claim that general localization schemes, in agreement with realistic operational measurement procedures applied to the quantum regime as well, should be understood in terms of topological covering systems of the physical “continuum”. The disassociation of the physical meaning of a localization process from its restricted classical metrical reference requires, first of all, the abstraction of the constitutive properties of localization in appropriate categorical terms, and then, the effectuation of these properties for the definition of localization systems of global event structures. Regarding these objectives, the sought abstraction is being implemented by means of covering devices on the base category of reference contexts, called in categorical terminology

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covering sieves. The constitutive properties of localization being abstracted categorically in terms of sieves, being qualified as covering ones, satisfy the following basic requirements: [i]. The covering sieves are covariant under pullback operations, viz., they are stable under change of a base reference context. Most importantly, the stability conditions are functorial. [ii]. The covering sieves are transitive, such that, intuitively stated, covering sieves of subcontexts of a context in covering sieves of this context, are also covering sieves of the context themselves. Thus, the notion of topological covering systems elucidate the primary functionality of a localization process, being constituted by the properties of covariance with respect to pullback operations of covering sieves and transitivity. Furthermore, they invoke suitable criteria for collating local observables into global ones. Notice that, the notion of functional dependence introduced by localization schemes, is formalized exclusively in functorial algebraic terms of relational information content with respect to the category of base reference loci, without any supporting notion of a smooth metrical set-theoretic background manifold. In this sense, the resolution focus in the physical “continuum” is being shifted from point-set to topological localization models, that effectively, induce a transition in the semantics of observables from a set-theoretic to a sheaf-theoretic one. Subsequently, this semantic transition effectuates the conceptual replacement of the classical metrical ruler of localization on a smooth background manifold, with a multiplicity of sheaf-cohomological rulers of algebraic-topological localization in the respective spectra of the topos of all presheaves or sheaves of algebras of observables. In a nutshell, the novel conception of physical localization schemes according to the above, forcing this semantic transition, is an outcome of the principle of local relativization of physical representability with respect to appropriate localization topoi. The third principle is called the principle of covariance of dynamics with respect to cohomologically well-behaved algebras of physical observables. This principle is elucidated by the property of functoriality of the dynamical mechanism of physical information propagation, with respect to generalized algebras of observables, complying with the cohomological conditions for the formation of exact complexes. The significance of this covariance property, invoking the formulation of functorial dynamics in the physical “continuum”, lies on the fact that, algebra sheaves of smooth real-valued functions, together with, their associated, by measurement, manifold R-spectrums do not constitute unique coordinatizations of the universal physical mechanism of qualitative information propagation via observables. Thus, the smoothness assumption on the structure of observables is not necessary for the formulation of the dynamical mechanism. In this perspective, the assumed collapse of dynamics at singularities of a base manifold is actually only phenomenal, meaning that, it appears to break down exclusively due to the employment of inappropriate coefficients (smooth functions) for the coordinatization of these loci in the physical “continuum”. The fourth principle is called the principle of functorial dynamical connectivity. This principle is elucidated by the process of functorial modeling of the notion of a physical field in terms of a monadic connection, bearing the semantics of a causal interactions-inducing agent, implemented by the algebraic process of infinitesimal scalar extensions of the algebra sheaf of local observables. Thus, according to this conceptualization the dynamics of information propagation being caused by the presence of some interaction field, as a particular application of the general functorial dynamical mechanism, is being generated by the functioning of a monadic connection. The integrability property of a connection defines a

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maximally undisturbed process of dynamical variation caused by the field it models. The disturbances from the maximally symmetric state of dynamical variation, caused by the non-integrability of the connection, become observable geometrically, meaning that, they become covariantly represented with respect to the local sections of the algebra sheaf of observables, by means of a non-vanishing curvature. Thus, the curvature of a connection can be effectively endowed with the physical semantics of the corresponding field strength. Finally, these disturbances may be cohomologically identified as obstructions to topological deformation caused by physical sources. Finally, the fifth principle is called the principle of co-algebraic generation of physical dynamics. This principle is elucidated by the fact that the conceptual skeleton of the generative mechanism of dynamics is a consequence of the existence of a monad-comonad pair, expressing the inverse algebraic processes of extending and restricting the scalars (observables). Hence, the generation of field dynamics can be considered in a unifying perspective as the natural outcome of the adjunctive correspondence between the inverse processes of extension and restriction of scalars. Furthermore, the endorsement of the principle of topos-theoretic local relativization of physical representability, necessitated by the distinctive properties of localization of observables in the regime of each physical theory, permits the transfer of the dynamical mechanism, arising from the monad-comonad pair of the inverse processes of extension/restriction of scalars, in the regime of the corresponding localization topoi.

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5.2.

Topological Sheaf-Theoretic Field Dynamics and Abstract Differential Geometry

According to the principles underlying the formulation of functorial physical theories of field dynamics, the absolute representability principle of classical General Relativity in terms of real numbers, may be relativized locally with respect to a category of measurement loci without affecting the functionality of the physical dynamical mechanism. Consequently, it is possible to describe the dynamics of gravitational interactions in generalized localization environments, instantiated by suitable topoi. The latter are understood in the sense of categories of presheaves, defined over a base category of reference loci, or categories of sheaves with respect to some suitable topological covering system. From a physical viewpoint, the construction of a sheaf of algebras of observables constitutes the natural outcome of a complete localization process [28-30]. Generally speaking, a localization process is being implemented in terms of an action of some category of reference contexts on a set-theoretic global algebra of observables. The latter, is then partitioned into sorts parameterized by the objects of the category of contexts. In this way, the functioning of a localization process can be represented by means of a fibered construct, understood geometrically as a presheaf, or equivalently, as a variable set (algebra) over the base category of contexts. The fibers of this presheaf may be thought, in analogy to the case of the action of a group on a set of points, as the “generalized orbits” of the action of the category of contexts. The notion of functional dependence incorporated in this action, forces the global algebraic structure of observables to fiber over the base category of reference contexts, giving rise to a presheaf of algebras of observables. According to the principle of observability, for every reference context of the base category, the set of local observables

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defined over it constitutes a commutative unital algebra over the real numbers. The transition from a presheaf to a sheaf of algebras of observables requires the following: Firstly, the concrete specification of a topological covering system on the category of reference contexts, interpreted as a localization scheme of the global algebraic structure of observables. Secondly, the collation of local observable information into global ones effectuated via a compatible family of local sections of the presheaf over a localization system. If a locally compatible choice of observables induces a unique global choice, then the condition for being a sheaf is satisfied. We note that, in general, there will be more locally defined or partial choices than globally defined ones, since not all partial choices need be extendible to global ones, but a compatible family of partial choices uniquely extends to a global one, or in other words, any presheaf uniquely defines a sheaf. In the sequel, we consider the localization environment of the topos of sheaves of sets Shv(X) defined over the category of open sets O(X ) of an abstract topological space X, ordered by inclusion. We define a topological covering system in the environment of O(X ) as follows: A topological covering system on O(X ) is an operation J, which assigns to each open reference domain U in O(X ), a collection J(U ) of U -sieves, called topological covering U -sieves, such that the following three conditions are satisfied: [1]. For every open reference domain U in O(X ) the maximal sieve {g : cod(g) = U } belongs to J(U ) (maximality condition). [2]. If S belongs to J(U ) and h : V → U is a figure of U , then h∗ (S) = {f : V → U, (h ◦ f ) ∈ S} belongs to J(V ) (stability condition).

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[3]. If S belongs to J(U ), and if for each figure h : Vh → U in S there is a sieve Rh belonging to J(Vh ), then the set of all composites h ◦ g, with h ∈ S, and g ∈ Rh , belongs to J(U ) (transitivity condition). As a consequence of the conditions above, we can check that any two U -covering sieves have a common refinement, that is: if S, R belong to J(U ), then S ∩ R belongs to J(U ). If we consider the partially ordered set of open subsets of a topological measurement space X, viewed as the category of base reference loci O(X ), then we specify that S is a covering U -sieve if and only if U is contained in the union of open sets in S. The above specification fulfills the requirements of topological covering sieves posed above, and consequently, defines a topological covering system on O(X ). From a physical perspective, the consideration of covering sieves as generalized measures of localization of observables within a global observable structure, gives rise to localization systems of the latter. Furthermore, we can show that, if A is the contravariant presheaf functor that assigns to each open set U ⊂ X, the commutative unital algebra of all continuous observables on U , then A is actually a sheaf of sets. This is intuitively clear since the specification of a topology on a measurement space X (and hence, of a topological covering system on O(X ) as previously) is solely used for the definition of the continuous observables on X, and thus, the continuity of each observable can be determined locally. This means that continuity respects the operation of restriction to open sets, and moreover that, continuous observables can be collated in a unique manner, as it is required for the satisfaction of the sheaf condition. More precisely, the sheaf condition means that the following sequence of

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commutative unital R-algebras of local observables is left exact; Y Y 0 → A(U ) → A(Ua ) → A(Uab )

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a

a,b

Up to this stage, we conclude that we have successfully implemented the first two principles of functorial dynamics in the localization environment of the topos of sheaves Shv(X). More precisely, according to the principle of local relativization of physical representability with respect to some suitable category of local reference loci of observation, we have consider the local relativization of representability with respect to the category of open loci of X, viz. O(X ), and moreover, we have defined covering sieves as generalized measures of localization of observables within a global observable structure. Moreover, according to the observability principle, for every open locus U in X, A(U ) is a commutative, unital R-algebra of continuous local sections of the sheaf of R-algebras A. Let us now implement the third principle, viz., the principle of covariance of dynamics with respect to cohomologically well-behaved algebras of physical observables, interpreted in the environment of the localization topos Shv(X). It is instructive to remind that the algebraic cohomological framework of formulation of dynamics is based for its conceptualization and operative efficacy, neither, on the methodology of real Analysis, nor, on the restrictive assumption of smoothness of observables, but only, on the functorial expression of the inverse processes of infinitesimal scalars extension/restriction. Nevertheless, it is instructive, to apply this algebraic framework for the case of algebra sheaves of smooth observables, in order to reproduce the differential geometric mechanism of smooth manifolds geometric spectra, interpreted in the localization environment of the topos Shv(X). For this purpose, we consider that A stands for the sheaf of algebras of R-valued smooth functions on X, denoted by C ∞ , whereas, Ωn (A) stand in this context for the locally free sheaves of C ∞ -modules of differential n-forms on X. In this case, the algebraic de Rham complex of A, gives rise to the corresponding differential de Rham complex of C ∞ , as follows: C ∞ → Ω1 (C ∞ ) → . . . → Ωn (C ∞ ) → . . . The crucial mathematical observation concerning this complex, refers to the fact that, the augmented differential de Rham complex 0 → R → C ∞ → Ω1 (C ∞ ) → . . . → Ωn (C ∞ ) → . . . is actually exact. The exactness of the augmented differential de Rham complex, as above, constitutes an expression of the lemma of Poincar´e, according to which, every closed C ∞ form on X is exact at least locally in X. Thus, the well-definability of the differential geometric dynamical mechanism of smooth manifolds is precisely due to the exactness of the augmented differential de Rham complex. This mathematical observation for the case of algebra sheaves of smooth observable coefficients, raises the issue of enrichment of the general functorial mechanism of infinitesimal scalars extensions, by the requirement of exactness of the respective augmented algebraic de Rham complex, securing in this sense, the well-definability of the dynamical mechanism for the general case, and reproducing the corresponding differential geometric mechanism of smooth manifolds faithfully, as well.

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This requirement implements precisely the principle of covariance of dynamics with respect to cohomologically well-behaved algebras of physical observables, interpreted in the localization topos Shv(X). A complete settlement of this issue, addressing the principle of covariance as above, comes from the mathematical theory of Abstract Differential Geometry (ADG) [3, 31-32]. Actually, the axiomatic development of (ADG) in a fully-fledged mathematical theory, has been based on the exploitation of the consequences of the above-stated mathematical observation for the case of algebra sheaves of smooth observable coefficients. In this sense, the operational machinery of (ADG) is essentially implemented by the imposition of the exactness requirement of the following abstract de Rham complex, interpreted inside the topos of sheaves Shv(X):

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0 → R → A → Ω1 (A) → . . . → Ωn (A) → . . . (ADG)’s power of abstracting and generalizing the classical differential calculus on smooth manifolds basically lies on the possibility of assuming other more general coordinate sheaves A, while, at the same time retaining, via the exactness of the algebraic augmented de Rham complex, as above, the mechanism of differentials, instantiated paradigmatically, in the first place, in the case of classical differential geometry on smooth manifolds [3, 4, 16, 26, 30-40]. For our physical purposes, we conclude that any cohomologically appropriate sheaf of algebras A, characterized by the exactness property posed previously, can be legitimately regarded as a sheaf of local observables, capable of providing a well-defined dynamical mechanism, independently of any smooth manifold background, analogous, however, to the one supported by smooth manifolds. Conclusively, it is instructive to recapitulate and add some further remarks on the physical semantics associated with the preceding algebraic cohomological dynamical framework by invoking the sheaf-theoretic terminology explicitly. The basic mathematical objects involved in the development of that framework consists of a sheaf of commutative unital algebras A, identified as a sheaf of algebras of local observables, a sheaf of locally free Amodules E of rank n, as well as, the sheaf of locally free A-modules of universal 1-forms Ω of rank n. We assume that these sheaves have a common base space, over which they are localized, namely, an arbitrary topological measurement space X. A topological covering system of X is defined simply by an open covering U = {U ⊆ X : U open in X} of X such that, any locally free A-module sheaf N splits locally, by definition, that is, with respect to every U in U, into a finite n-fold Whitney sum An of A with itself as N |U = An |U . For this reason, a topological covering system U of X may be called a coordinatizing open cover of N. Hence, the local sections of the structure R-algebra sheaf A relative to the coordinatizing open cover U obtain the meaning of local coordinates, while A itself may be called ‘the coefficient’ or ‘continuously variable real number coordinate sheaf’ of N . At a further stage of development, the implementation of the principle of functorial dynamical connectivity requires the functorial modeling of the notion of a physical field in terms of a monadic connection expressing the algebraic process of infinitesimal scalar extensions of the algebra sheaf of local observables. Finally, according to the principle of co-algebraic generation of physical dynamics, the generative mechanism of dynamics is a consequence of the existence of a monad-comonad pair, expressing the inverse alge-

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braic processes of extending and restricting the scalars (observables). Thus, the notion of a monadic connection is obtained by the monad-comonad pair of extension/restriction of scalars from the sheaf R to the sheaf A, interpreted inside the localization topos Shv(X). Thus, we conclude that a pair (E, ∇E ), consisting of a left A-module sheaf E and a connection ∇E on E, represents a local causal agent of a variable interaction geometry, viz., a physical field acting locally and causing infinitesimal variations of coordinates, standing for local observables. In this sense, the local sections of A-module sheaf E, relative to the open cover U, coordinatize the states of the corresponding physical field. The connection ∇E on E, is given by an R-linear morphism of A-modules sheaves: O O ∇E : E → Ω1 (A) E=E Ω1 (A) := Ω1 (E) A

A

such that, the following Leibniz condition holds: ∇E (f · v) = f · ∇E (v) + df ⊗ v for all f ∈ A, v ∈ E. Notice that, by definition, the connection ∇E is only an R-linear morphism of A-modules sheaves. Hence, although it is R-covariant, it is not A-covariant as well. The connection ∇E on E contains the irreducible amount of information encoded in the process of infinitesimal scalars extension caused by local interactions, induced by the corresponding field. A significant observation has to do with the fact that if E = A, considered as an Amodule over itself, then, the R-linear morphism of sheaves of A-modules

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d : Ω0 (A) := A → Ω(A)1 := Ω(A) is a natural connection, which is also integrable, or flat, since, hΩ(A)i is actually a complex, namely the algebraic de Rham complex of A. If we consider a coordinatizing open cover eU ≡ {U ; (ei )0≤i≤n−1 } of the A-module sheaf E of rank n, every continuous local section s ∈ E(U ), where, U ∈ U, can be expressed uniquely as a superposition s=

n X

si ei

i=1

with coefficients si in A(U ). The action of ∇E on these sections of E is expressed as follows: ∇E (s) =

n X

(si ∇E (ei ) + ei ⊗ d(si ))

i=1

where, ∇E (ei ) =

n X

ei ⊗ ωij , 1 ≤ i, j ≤ n

i=1

where, ω = (ωij ) denotes an n × n matrix of sections of local 1-forms. Consequently we have; Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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∇E (s) =

n X

ei ⊗ (d(si ) +

n X

i=1

sj ωij ) ≡ (d + ω)(s)

j=1

Thus, every connection ∇E , where, E is a locally free finite rank-n sheaf of modules E on X, can be decomposed locally as follows: ∇E = d + ω In this context, ∇E is identified as a covariant derivative, being decomposed locally as a sum consisting of a flat part tautosemous with d, and a generally non-flat part ω, called the gauge potential (vector potential), signifying a measure of deviation from the maximally undisturbed process of dynamical variation (represented by the flat part), caused by the corresponding physical field. The behavior of the gauge potential part ω of ∇E under local gauge transformations constitutes the ‘transformation law of vector potentials’ and is established in the following manner: Let eU ≡ {U ; ei=1···n } and hV ≡ {V ; hi=1···n T } be two coordinatizing open covers of E over the open sets U and V of X, such that U V 6= ∅. Let us denote by g = (gij ) the following change of local gauge matrix: hj =

n X

gij ei

i=1

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Under such a local gauge transformation (gij ), the gauge potential part ω of ∇E transforms as follows: ′ ω = g −1 ωg + g −1 dg Furthermore, it is instructive to find the local form of the curvature R∇ of a connection ∇E , where, E is a locally free finite rank-n sheaf of modules E on X, defined by the following A-linear morphism of sheaves: R∇ := ∇1 ◦ ∇0 : E → Ω2 (A)

O

A

E := Ω2 (E)

Due to its property of A-covariance, a non-vanishing curvature represents in this context, the A-covariant, and thus, observable (by A-scalars) disturbance from the maximally symmetric state of the variation caused by the corresponding physical field. In this sense, it may be accurately characterized physically as ‘gauge field strength’. Moreover, since the curvature R∇ is an A-linear morphism of sheaves of A-modules, R∇ may be thought of as an element N of End(E) A Ω2 (A) := Ω2 (End(E)), that is: R∇ ∈ Ω2 (End(E)) Hence, the local form of the curvature R∇ of a connection ∇E , consists of local n × n matrices having for entries local 2-forms on X. The behavior of the curvature R∇ of a connection ∇E under local gauge transformations constitutes the ‘transformation law of gauge field strengths’. If we agree that g = (gij ) denotes the change of gauge matrix, we have previously considered in the discussion of the

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transformation law of gauge potentials, we deduce the following local transformation law of gauge field strengths: g

R∇ 7→ R∇ ′ = g −1 (R∇ )g According to the above dynamical framework, applications of ADG include the reformulation of Gauge theories in sheaf-theoretic terms [31, 32], as well as, the evasion of the problem of manifold singularities appearing in the context of General Relativity [33-39]. Related with the first issue, ADG has modeled Yang-Mills fields in terms of appropriate pairs (E, DE ), where E are vector sheaves whose sections have been identified with the states of the corresponding particles, and DE are connections that act on the corresponding states causing interactions by means of the respective fields they refer to. Related with the second issue, ADG has replaced the sheaf of R-algebras C ∞ (M ) of smooth real-valued functions on a differential manifold with a sheaf of R-algebras that incorporates the singularities of the manifold in terms of appropriate ideals, allowing the formulation of Einstein’s equations in a covariant form with respect to the generalized scalars of that sheaf of R-algebras. An overview of the didactics of topological sheafification of field dynamics according to ADG has been recently presented in [16], whereas, the general underlying philosophical framework of ADG, pointing towards the general conception and implementation of relational mathematics has been recently summarized in [40].

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5.3.

Quantum Functorial Dynamics from the Classical to Quantum Extension/Restriction Monad-Comonad Pair

The basic distinguishing feature of Quantum Theory according to the Bohrian interpretation [41-43], in comparison to all classical theories, is the realization that physical observables are not definitely or sharply valued as long as a measurement has not taken place, meaning both, the precise specification of a concrete experimental context, and also, the registration of a value of a measured quantity in that context by means of an apparatus. Furthermore, Heisenberg’s uncertainty relations, determine the limits for simultaneous measurements of certain pairs of complementary physical properties, like position and momentum. In a well-defined sense, the uncertainty relations may be interpreted as measures of the valuation vagueness associated with the simultaneous determination of all physical observables in the same experimental context. In all classical theories, the valuation algebra is fixed once and for all to be the division algebra of real numbers R, reflecting the fact that values admissible as measured results must be real numbers, irrespective of the measurement context and simultaneously for all physical observables. The resolution of valuation vagueness in Quantum Theory can be algebraically comprehended through the principle of local relativization of representability of the global valuation algebra with respect to a category of commutative algebraic loci corresponding to locally prepared measurement contexts [44, 45]. At a quantum logical level commutative loci of measurement correspond to the spectra of local Boolean algebras, identified as subalgebras of a quantum algebra of observables. In the general case, commutative algebraic loci are identified with the spectra of commutative K-algebras, where K = Z2 , R, C, understood as subalgebras of a globally non-commutative algebra of quantum observables, represented in an irreducible way as an algebra of operators on a Hilbert space of quantum states. The underlying idea is based

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on the possibility of approximating the information content of a quantum algebra of observables in terms of the information obtained by structured interlocking families of local commutative algebras of classical observables, the spectra of whom are indentified as loci of quantum measurement. The crucial observation making the construction possible is that a globally non-commutative or partially commutative algebra of observables determines an underlying categorical diagram of commutative subalgebras. Then, each commutative subalgebra can be locally identified with an algebra of classical observables in the context of some appropriate localization scheme [26]. According to the previous remarks, the strategy adopted for the generation of a viable dynamical mechanism in the quantum regime of observable structure is based on two cornerstones: [i]. The first cornerstone requires the local relativization of quantum representability with respect to a category of reference loci identified as the spectra of local commutative subalgebras of a quantum algebra of observables under the effectuation of an appropriate localization scheme. From this requirement we obtain two consequences: (a). The first consequence is the depiction of the topos of presheaves PShv(AC ) := op SetsAC or the topos of sheaves Shv(AC ), defined over the base category (AC ) of commutative reference loci as the appropriate localization topos for the generation of a quantum dynamical mechanism [26, 28, 46-48]. In this context, see also [17, 49]. A suitable localization scheme applicable to the quantum regime of observable structure is obtained by the notion of a topological-categorical covering system, viz. a Grothendieck topology, in terms of covers on the base category AC [26, 28, 46]. More concretely, a Grothendieck topology suitable for this purpose is defined by means of a covering system S of epimorphic families on the base category of commutative contexts, by requiring that the morphism a ′ GS : AC → AC ′ Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

{s:AC →AC }∈S

′

where, AC , AC in AC , is an epimorphism in AQ , where AQ denotes the category of quantum observables algebras. In this way, the Grothendieck topology defined above specifies functorially a physical localization scheme suited for probing the quantum regime of observable structure. In passing we mention that the interpretational aspects of the proposed topos-theoretic relativization of physical representability in relation to the truthvalues structures of quantum logics (via the notion of subobject classifier) have been discussed extensively in [50-51]. (b). The second consequence is the implementation of the observability principle in the quantum regime. More precisely, the observability principle in the environment of the localization topos of sheaves Shv(AC ) for the previously defined Grothendieck topology, is implemented by showing that, a quantum algebra of observables is being expressed locally as a quotient of a commutative algebra over a categorical ideal, that contains information about all other commutative algebras being compatible with it (pull-back compatibility) in a localization system of the former. The observability principle in the quantum regime provides a topos-theoretic embodiment of Bohr’s interpretation doctrine of quantum phenomena [26]. [ii]. The second cornerstone of the adopted strategy requires the generation of a viable quantum dynamical mechanism by inter-relating the variable-classical with the quantum

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level of observable structure, conceived as a bidirectional generalized process of extension/restriction from one level to the other. The underlying idea of this conception is based on the principle of co-algebraic generation of dynamics by means of a monadic connection obtained from the existence of a monad-comonad pair being formed from the adjunctive correspondence of those two levels of observable structure. In this vein of ideas, we can prove that the sought adjunction has the following form [26]: L : SetsAC

op

⇆ AQ : R

which says that the Grothendieck functor of points of a quantum observables algebra restricted to commutative subalgebras defined by: R(AQ ) : AC 7→HomAQ (V (AC ), AQ ) has a left adjoint: L : SetsAC

op

which is defined for each presheaf P in SetsAC

→ AQ

op

as the colimit:

L(P) = P⊗AC M where M is a coordinatization functor, viz.: M : AC → AQ which assigns to commutative observables algebras in AC the underlying quantum algebras from AQ . Equivalently, there exists a bijection, natural in P and AQ as follows:

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N at(P, R(AQ )) ∼ = HomAQ (LP, AQ ) A very important observation, pertaining to the generation process of a viable dynamical mechanism in the quantum regime, has to do with the fact that, the left adjoint functor L = −⊗AC M : PSh(AC ) → AQ can be considered as the variable-classical to quantum observables extension functor, whereas, the right adjoint functor R : AQ → PSh(AC ), can be considered as the quantum to variable-classical localizing restriction of observables functor. Thus, a quantum dynamical mechanism can be generated by expressing the inverse algebraic processes of extending and restricting the scalars between the variable-classical and the quantum regimes in a functorial form, and more precisely, by forming an appropriate monad-comonad pair, on the basis of which, it is possible to develop a co-algebraic dynamical mechanism. A quantum dynamical mechanism, according to the previous remarks, can be formulated, in the most general terms, by associating appropriately the notion of a monadic connection to the information incorporated in the adjunction L ⊣ R as follows: L : SetsAC

op

qqqq qqqqqqqqq qqqq qqqqq qqqq qqq qqqqq

AQ : R

The above adjunction is completely characterized in terms of the unit and counit natural op transformations. For any presheaf P ∈ SetsAC , the unit is defined as: σP : P

qqqqq qqqq qqq qqqqq

RLP

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On the other side, for each object AQ of AQ the counit is: ǫAQ : LR(AQ )

qqqqq qqqqq qq qqqqq

AQ

The composite endofunctor G := LR : AQ → AQ , together with the natural transformations δ : G → G ◦ G, called comultiplication, and also, ǫ : G → I, called counit, where I is the identity functor on AQ , is defined as a comonad (G, δ, ǫ) on the category of quantum observables algebras AQ , provided that the diagrams below commute for each object AQ of AQ ; δ AQ

GAQ

G2 AQ

qqqqq qqqq qqq qqqqq

δ AQ

δGAQ qqqqq qqqqq qqqqqqq

qqqqq qqqqq qqqqqqq

GδAQ

G2 AQ

G3 AQ

qqqqq qqqq qqq qqqqq

GAQ

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δ AQ qqqqq qqqqq qqqqqqqq

GAQ

qqqq qqqqqqqqq qqqq

ǫGAQ

@ @ @ @ @ @ @ @ @ @ @ @

G2 AQ

GǫAQ

qqqqq qqqq qqq qqqqq

GAQ op

op

Correspondingly, the composite endofunctor X := RL : SetsAC → SetsAC , together with the natural transformations ω : X ◦ X → X, called multiplication, and also, σ : I → op X, called unit, where I is the identity functor on SetsAC , is defined as a monad (X, ω, σ) op on the category of presheaves of commutative observables algebras SetsAC , provided op that, the diagrams below commute for each object P of SetsAC ; XP

qqqq qqqqqqqqq qqqq

ωP

qqqqqqq qqqqq qqqqq

qqqqqqq qqqqq qqqqq

ωP

X2 P

X2 P ωXP

qqqq qqqqqqqqq qqqq

XωP

X3 P

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XP qqqqqqqqqqq qqq

qqqqqqqqqq qqq q

@ @

XσP

X2 P

ωP

qqqqq qqqq qqq qqqqq

XP

qqqq qqqqqqqqq qqqq

@ σXP @ @ @

ωP

X2 P

For a comonad (G, δ, ǫ) on AQ , a G-coalgebra (comodule) is an object AQ of AQ , being equipped with a structural map κ : AQ → GAQ , such that the following conditions are satisfied: 1AQ = ǫAQ ◦ κ : AQ → AQ Gκ ◦ κ = δAQ ◦ κ : AQ → G2 AQ With the above obvious notion of morphism, this gives a category AQ G of all G-coalgebras. Correspondingly, if (X, ω, σ) is a monad on the category of presheaves of commutative op op observables algebras SetsAC , we define the category of SetsAC X -algebras (modules) op op as follows: The objects of SetsAC X are pairs (Y, µY ), where, Y in SetsAC , and, op µY : X(Y) → Y is a morphism in SetsAC , such that, the following conditions are satisfied: 1Y = µ ◦ σY : Y → Y µ ◦ Xµ = µ ◦ ωY : X2 Y → Y Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

op

We easily notice that, the pair (Y, ωY ), where, Y in SetsAC and ω is the multiplication op morphism, is an SetsAC X -algebra. It is also essential to notice that, h : (Y, µY ) → op (Z, µZ ) is a morphism in SetsAC X , if and only if, h : Y → Z is a morphism in op SetsAC , and moreover, we have: h ◦ µY = µZ ◦ X(h) We may recapitulate, by noticing that, for the adjunction L : SetsAC

op

qqqq qqqqqqqq qqqqq qqqqq qqqq qqq qqqqq

AQ : R

we have: op op [i]. The composite endofunctor X := RL : SetsAC → SetsAC , together with the natural transformations ω : X ◦ X → X (multiplication), where, ω = RǫL , and op also, σ : I → X (unit), where, I is the identity functor on SetsAC , constitutes a monad op (X, ω, σ) on the category of presheaves of commutative observables algebras SetsAC . op Furthermore, the pair (R(AQ ), R(ǫAQ )) constitutes an SetsAC X -module. [ii]. The composite endofunctor G := LR : AQ → AQ , together with the natural transformations δ : G → G ◦ G (comultiplication), where, δ = LσR , and also, ǫ : G → I (counit), where, I is the identity functor on AQ , constitutes a comonad (G, δ, ǫ) on

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the category of quantum observables algebras AQ . Furthermore, the pair (L(Y), L(σY )) constitutes an AQ G -comodule. Analogously to the model case presented in Section 3.2, we define the notion of a op monadic connection on the SetsAC X -module (R(AQ ), R(ǫAQ )) as the following natural op transformation in SetsAC : ∇ : R(AQ ) → RLR(AQ ) such that, the following conditions are satisfied: (C1 ) : ∇R(ǫAQ ) − RǫL(R(AQ )) RL(∇) = 1RL(R(AQ )) − σR(AQ ) R(ǫAQ ) (C2 ) : R(ǫAQ )∇ = 0 The counit of the adjunction: ǫAQ : LR(AQ ) → AQ defined by the composite endofunctor: G := LF : AQ → AQ constitutes the first step of a functorial free resolution of a quantum observables algebra AQ in AQ . Actually, by iterating the endofunctor G, we may extend ǫAQ to a free simplicial resolution of AQ , as we have already shown previously. In this setting, we define the object of quantum differential 1-forms, by means of the following split short exact sequence:

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0 → ΩAQ → GAQ → AQ or equivalently, 0 → ΩAQ → R(AQ )⊗AC M → AQ According to the above, we obtain that: ΩAQ =

J J2

where J = Ker(ǫAQ ) denotes the kernel of the counit of the adjunction. At a next stage, we notice that the functor of points of a quantum observables algebra restricted to commutative subalgebras R(AQ ) is left exact, because it is the right adjoint functor of the established adjunction. Thus, it preserves the short exact sequence defining the object of quantum differential 1-forms, in the following form: 0 → R(ΩAQ ) → R(G(AQ )) → R(AQ ) Hence, we immediately obtain that: R(ΩAQ ) =

Z Z2

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where Z = Ker(R(ǫAQ )). Then, the principle of dynamical connectivity in the topos op SetsAC is being implemented in terms of the notion of a quantum interactions field, defined by means of the following functorial pair:
R(AQ ) := HomAQ (M(−), AQ ), ∇R(AQ ) where the quantum connection ∇R(AQ ) is defined as the following natural transformation: ∇R(AQ ) : R(AQ ) → R(ΩAQ ) obeying the Leibniz rule. Next, we show that the definition above, is equivalent with the definition of a connecop tion on the SetsAC X -module (R(AQ ), R(ǫAQ )), stated previously. For that purpose, we op remind that, a connection on R(AQ ) is defined as a natural transformation in SetsAC : ∇ : R(AQ ) → RLR(AQ ) such that, the conditions (C1 ) and (C2 ) are satisfied. In particular, condition (C2 ), viz., R(ǫAQ )∇ = 0 is satisfied, if and only if, there exists: ∇R(AQ ) : R(AQ ) → R(ΩAQ ) such that:

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∇ = R(ιAQ )∇R(AQ ) where, ιAQ denotes the injection ΩAQ ֒→ GAQ . Moreover, in that case, the condition (C1 ) is equivalent to the condition: (C´1 ) : ∇R(AQ ) R(ǫAQ ) − R(ΩǫAQ )RL(∇R(AQ ) ) = ΛAQ where, ΛAQ is the unique natural morphism: ΛAQ : RLR(AQ ) → R(ΩAQ ) satisfying the relation: R(ιAQ )ΛAQ = 1RLR(AQ ) − σR(AQ ) R(ǫAQ ) Conversely, if we consider the condition (C´1 ), and moreover, replace ∇R(AQ ) by R(ιAQ )∇R(AQ ) , we obtain the condition (C1 ). Thus, we conclude that, the information of a connection on R(AQ ) consists of a functorial morphism ∇R(AQ ) : R(AQ ) → R(ΩAQ ) which, satisfies additionally the condition (C´1 ), considered as the functorial expression of the Leibniz condition. Hence, we can introduce the notion of a quantum field, by means of Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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the functorial pair R(AQ ) := HomAQ (M(−), AQ ), ∇R(AQ ) , where the quantum connection ∇R(AQ ) is defined as above. The quantum connection ∇R(AQ ) induces a sequence of functorial morphisms, or equivalently, natural transformations as follows: R(AQ ) → R(ΩAQ ) → . . . → R(Ωn AQ ) → . . . Let us denote by: R∇ : R(AQ ) → R(Ω2 AQ )

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the composition ∇1 ◦ ∇0 in the obvious notation, where ∇0 := ∇R(AQ ) , which we call the curvature of the quantum connection ∇R(AQ ) . The latter sequence of functorial morphisms, is actually a complex, if and only if: R∇ = 0 We say that the quantum connection ∇R(AQ ) is integrable or flat if R∇ = 0, referring to the above complex as the functorial de Rham complex of the integrable connection ∇R(AQ ) in that case. The vanishing of the curvature of the quantum connection, R∇ = 0, can be used as a means of transcription of Einstein’s equations in the quantum regime in the absence of cohomological obstructions. According to the principle of dynamical connectivity, if a connection on R(AQ ) is integrable, then, it corresponds to a maximally undisturbed process of dynamical variation. The non-integrability of a quantum connection, should be expressed by a non-vanishing curvature. A non-vanishing curvature of this process of parallel transport is understood as the denotator of disturbances with respect to the property of integrability of the corresponding quantum connection. In the physical state of affairs, the curvature of a non-integrable quantum connection is interpreted as the observable field strength of the corresponding quantum potential. Intuitively, the curvature of a quantum connection is the observable attribute of the global non-integrability of infinitesimal interlocking of commutative algebras, in some underlying localizing diagram of a quantum algebra of observables.

6.

Conclusion

The generation of the mechanism of differential calculus and differential geometry has been considered from a categorical perspective based on the information encoded in the adjunctive correspondence between the inverse processes of extension/restriction of scalars. The model case has been the classical extension/restriction of scalars adjunction relating the categories of modules defined with respect to a morphism of commutative unital rings. The established adjunction is completely characterized from the unit and counit natural transformations, giving rise to a monad-comonad pair. Initially, we have defined the universal object of differential 1-forms via the kernel of the counit of the adjunction. Subsequently, we have introduced the notion of a connection on a module as an appropriate natural transformation (satisfying the Leibniz condition) making use of the fact that the right adjoint functor of the adjunction (restriction functor) is left exact, and thus, it preserves the short exact sequence defining the universal object of differential 1-forms. From then on, we have set-up functorially the algebraic De Rham complex associated with an integrable connection. The failure of integrability of a connection is described by means of the curvature of this connection. Finally the definition of a connection has been abstracted from the model case of

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the classical extension/restriction of scalars adjunction. This has become possible via the notion of a monadic connection making it applicable to more general situations involving an adjunction interpreted as a bidirectional process of extension/restriction. Moreover, we have proved that the general notion of a monadic connection is equivalent to the classical notion when restricted to the classical extension/restriction of scalars adjunction. As an important physical application we derive the dynamical framework of the General Theory or Relativity by purely algebraic means. At a further stage of development, we have laid the foundation for the formulation of functorial physical dynamics. The basis of this foundation is the realization that the differential mechanism, and thus, dynamics interpreted in a physical sense, is the outcome of the existence of a pair of adjoint functors, considered as a bidirectional process of extension/restriction of observables. The foundation of functorial physical dynamics is described by means of five physical principles. The first principle, called the observability principle specifies a structure of observables as a commutative algebra localized sheaf-theoretically over a base category of reference contexts, determined by operational means. The second principle is called the principle of local relativization of physical representability with respect to appropriate localization topoi, which capture the particular localization properties of observables in the regime of each physical theory. The implementation of this principle is related with a generalized conceptualization of localization schemes of observables, defined by means of topological covering systems (Grothendieck topologies) on the base category of reference contexts of observation. The third principle is called the principle of covariance of dynamics with respect to cohomologically well-behaved algebras of observables. The crucial requirement refers to the successful formation of augmented De Rham complexes in analogy to the model case of an algebra sheaf of smooth functions defined on a differential manifold. The forth principle is called the principle of functorial dynamical connectivity. According to this, the notion of a connection is endowed with the semantics of a physical field (potential) effectuating a process of infinitesimal observables extension. An integrable connection is a manifestation of an undisturbed process of dynamical variation caused by the corresponding field. The existence of disturbances from this state of dynamical variation is due to cohomological obstructions. The non-integrability of a connection is described by means of the corresponding curvature, which is interpreted physically as the observable field strength of the potential. Finally, the fifth principle is called the principle of co-algebraic generation of functorial field dynamics. It states that a dynamical physical mechanism is being generated conceptually by expressing the inverse algebraic processes of extending and restricting the scalars in a functorial form, and more precisely, by forming an appropriate monad-comonad pair. Hence, the generation of field dynamics can be considered in a unifying perspective as the natural outcome of the adjunctive correspondence between the inverse processes of extension and restriction of scalars. We elaborate the application of these principles in the following cases: Firstly, we consider the case of topological sheaf-theoretic field dynamics described by the framework of Abstract Differential Geometry. Secondly, we consider the case of quantum dynamics generated by the monadcomonad pair of extension/restriction of scalars from the variable-classical (commutative) to the quantum (non-commutative or partially commutative) regime of observable structure. In the latter case, quantization is considered as a process of local relativization of representability with respect to a Grothendieck topos defined over a base category of com-

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mutative reference contexts equipped with the categorical topology of epimorphic families of covers. Then, the notion of monadic connection obtained by the established monadcomonad pair provides the appropriate means for the effectuation of a quantum dynamical mechanism in this localization topos. The foundation laid down by the above network of physical principles constitutes the unifying conceptual skeleton of generating a dynamical mechanism, and thus, endowing with a physical dynamical interpretation, the establishment of a monad-comonad pair referring to the inverse processes of extension and restriction of scalars. This skeleton acquires a concrete formulation in the regime of each physical theory, by taking into account the localization properties of observables in each case, and hence, qualifying the notion of scalar appropriately in each case. More concretely, the notion of scalar, can be qualified as an observable, which respects the corresponding localization properties in the regime of each physical theory, if and only if, it acquires a variable reference with respect to different topoi. The latter are being constructed as categories of sheaves (with respect to Grothendieck covering systems) over the localization domains of observables in the regime of each physical theory. In this sense, by considering a topos-theoretic local relativization of physical representability, necessitated by the distinctive properties of localization of observables in each case, it is possible to transfer the dynamical mechanism, arising from the monad-comonad pair of the inverse processes of extension/restriction of scalars, in the regime of the corresponding localization topoi.

References

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[1] Eisenbud D., Commutative algebra with a view toward algebraic geometry, Graduate Texts in Math., vol. 150, Springer-Verlag, Berlin and New York, 1995. [2] Connes A. Noncommutative geometry, Academic Press, 1994. [3] Mallios A., Geometry of Vector Sheaves: An Axiomatic Approach to Differential Geometry, Vols 1-2, Kluwer Academic Publishers, Dordrecht, 1998. [4] Mallios A., Zafiris E. Topos-Theoretic Relativization of Physical Representability and Quantum Gravity, gr-qc/0610113, Foundations of Physics (to appear) (2008). [5] Ashtekar A., and Lewandowski J., Background independent quantum gravity: a status report, pre-print (2004); gr-qc/0404018. [6] Butterfield J., and Isham C. J., Some Possible Roles for Topos Theory in Quantum Theory and Quantum Gravity, Foundations of Physics, 30, 1707 (2000). [7] Crane L., Clock and Category: Is Quantum Gravity Algebraic?, Journal of Mathematical Physics, 36, 6180 (1995). [8] Finkelstein D. R., Quantum Relativity: A Synthesis of the Ideas of Einstein and Heisenberg, Springer-Verlag, Berlin-Heidelberg-New York, 1996. [9] Selesnick S. A., Quanta, Logic and Spacetime, (2nd. ed), World Scientific, 2004. Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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[10] Isham C. J., Some Reflections on the Status of Conventional Quantum Theory when Applied to Quantum Gravity, in The Future of Theoretical Physics and Cosmology: Celebrating Stephen Hawking’s 60th Birthday, Gibbons, G. W., Shellard, E. P. S. and Rankin, S. J. (Eds.), Cambridge University Press, Cambridge (2003); quantph/0206090. [11] Isham C. J., A new approach to quantising space-time: I. Quantising on a general category, Advances in Theoretical and Mathematical Physics, 7, 331 (2003); grqc/0303060. [12] Penrose R., The problem of spacetime singularities: implications for quantum gravity?, in The Future of Theoretical Physics and Cosmology: Celebrating Stephen Hawking’s 60th Birthday, Gibbons, G. W., Shellard, E. P. S. and Rankin, S. J. (Eds.), Cambridge University Press, Cambridge (2003). [13] Smolin L., The case for background independence, gr-qc/0507235. [14] Sorkin R. D., A Specimen of Theory Construction from Quantum Gravity, in The Creation of Ideas in Physics, Leplin, J. (Ed.), Kluwer Academic Publishers, Dordrecht (1995); gr-qc/9511063. [15] Stachel J., Einstein and Quantum Mechanics, in Conceptual Problems of Quantum Gravity, Ashtekar, A. and Stachel, J. (Eds.), Birkh¨auser, Boston-Basel-Berlin, 1991.

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[16] Raptis I., A Dodecalogue of Basic Didactics from Applications of Abstract Differential Geometry to Quantum Gravity, International Journal of Theoretical Physics, 46, 12 (2007). [17] Mallios A., A-Invariance: An Axiomatic Approach to Quantum Relativity, International Journal of Theoretical Physics, doi: 10.1007/s10773-007-9637-2 (Online), (2008). [18] Atiyah M. and MacDonald I., Introduction to Commutative Algebra, AddisonWesley, Reading, Massachusetts, 1969. [19] MacLane S., Categories for the Working Mathematician, Springer-Verlag, New York, 1971. [20] Shafarevich I. R., Basic Notions of Algebra, Springer, Berlin, 1997. [21] Bell J. L., From Absolute to Local Mathematics, Synthese 69 (1986). [22] Bell J. L., Categories, Toposes and Sets, Synthese, 51, No.3 (1982). [23] MacLane S. and Moerdijk I., Sheaves in Geometry and Logic, Springer-Verlag, New York, 1992. [24] Bell J. L., Toposes and Local Set Theories, Oxford University Press, Oxford, 1988. [25] Artin M., Grothendieck A., and Verdier J. L., Theorie de topos et cohomologie etale des schemas, Springer LNM 269 and 270, Springer-Verlag, Berlin, 1972. Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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[26] Zafiris E., Quantum Observables Algebras and Abstract Differential Geometry: The Topos-Theoretic Dynamics of Diagrams of Commutative Algebraic Localizations, International Journal of Theoretical Physics, 46, (2) (2007). [27] Menini C. and Stefan D. Descent Theory and Amitsur Cohomology of Triples, J. of Algebra, 266, (2003). [28] Zafiris E., Generalized Topological Covering Systems on Quantum Events Structures, Journal of Physics A: Mathematical and General 39, (2006). [29] Zafiris E., ‘Sheaf-Theoretic Representation of Quantum Measure Algebras’, Journal of Mathematical Physics, 47, 092103 (2006). [30] Zafiris E., ‘ A Sheaf-Theoretic Topos Model of the Physical “Continuum” and its Cohomological Observable Dynamics’, International Journal of General Systems, DOI: 10.1080/03081070701819285 (2008). [31] Mallios A., Modern Differential Geometry in Gauge Theories: vol. 1. Maxwell Fields, Birkh¨auser, Boston, 2006. [32] Mallios A., Modern Differential Geometry in Gauge Theories: vol. 2. Yang-Mills Fields, (forthcoming by Birkh¨auser, Boston, 2008). [33] Mallios A., Remarks on “singularities”, Progress in Mathematical Physics, Columbus, F. (Ed.), Nova Science Publishers, Hauppauge, New York (2003) (invited paper); gr-qc/0202028.

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[34] Mallios A., Quantum gravity and “singularities”, Note di Matematica, 25, 57 (2005)/(2006) (invited paper); physics/0405111. [35] Mallios A., Geometry and physics today, International Journal of Theoretical Physics 45(8), 1557 (2006). [36] Mallios A., Rosinger E. E., Abstract Differential Geometry, differential algebras of generalized functions, and de Rham cohomology, Acta Appl. Math. 55, 231 (1999). [37] Mallios A., Rosinger E. E., Space-Time foam dense singularities and de Rham cohomology, Acta Appl. Math. 67, 59 (2001). ˇ [38] Mallios A., and Raptis I., Finitary Cech-de Rham Cohomology: much ado without ∞ C -smoothness, International Journal of Theoretical Physics, 41, 1857 (2002); grqc/0110033. [39] Mallios A., and Raptis I., C ∞ -Smooth Singularities Exposed: Chimeras of the Differential Spacetime Manifold; gr-qc/0411121. [40] Mallios A., Relational Mathematics, Publ. Ecole Normale Superieure, Takaddoum, Rabat, Morocco (to appear) (2008). [41] Bohr N., Atomic Physics and Human Knowledge, John Wiley, New York, 1958. Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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[42] Folse H. J., The Philosophy of Niels Bohr. The Framework of Complementarity, New York, North Holland, 1985. [43] Bub J., Interpreting the Quantum World, Cambridge University Press, Cambridge, 1997. [44] Zafiris E., Probing Quantum Structure Through Boolean Localization Systems, International Journal of Theoretical Physics 39, (12) (2000). [45] Zafiris E., Boolean Coverings of Quantum Observable Structure: A Setting for an Abstract Differential Geometric Mechanism , Journal of Geometry and Physics 50, 99 (2004). [46] Zafiris E., Quantum Event Structures from the perspective of Grothendieck Topoi, Foundations of Physics 34(7), (2004). [47] Zafiris E., Interpreting Observables in a Quantum World from the Categorial Standpoint, International Journal of Theoretical Physics 43, (1) (2004). [48] Zafiris E., Topos Theoretical Reference Frames on the Category of Quantum Observables, quant-ph/0202057. [49] Mallios A., On Algebra Spaces, Contemporary Mathematics 427, 263 (2007).

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[50] Zafiris E., Categorical Foundations of Quantum Logics and Their Truth Values Structures, quant-ph/0402174. [51] Zafiris E., Topos-Theoretic Classification of Quantum Events Structures in terms of Boolean Reference Frames, International Journal of Geometric Methods in Modern Physics, 3, (8) (2006).

Reviewed by Professor Anastasios Mallios.

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In: Emerging Topics on Differential Geometry... Editors: L. Bernard and F. Roux, pp. 79-94

ISBN 978-1-60741-011-9 c 2010 Nova Science Publishers, Inc.

Chapter 3

T HE N OTION OF CR H AMILTONIAN F LOWS AND THE L OCAL E MBEDDING P ROBLEM OF CR S TRUCTURES Takao Akahori Department of Mathematics, School of Science, University of Hyogo, Himeji, Hyogo Abstract

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In studying the deformation theory of CR structures, the notion of the Hamiltonian vector field is found. In this paper, we develop this notion. Namely, the notion of CR-Hamiltonian flows is introduced, and we apply this notion to the local embedding problem.

1.

Introduction

For compact Kaehler manifolds, even though the deformation theory of complex structures is insensitive to the deformation of Einstein Kaehler metrics, several relations between the deformation theory of complex structures and the deformation theory of Einstein-Kaehler metrics, have been proved. So, it seems natural to try to obtain similar results for open manifolds with compact smooth boundary. In this paper, in order to attack this subject, we propose the fundamental notion over the strongly pseudo convex boundary,i.e., the strongly pseudo convex CR manifold. Let (M, 0 T ′′ ) be a strongly pseudo convex CR manifold with dimR M = 2n − 1 ≥ 7. Then, by the local embedding theorem of CR structures(by Kuranishi,for the case dimR M = 2n − 1 ≥ 9, and by myself, for the case dimR M = 7, this theorem is proved), we have an n-dimensional complex manifold N satisfying: our (M, 0 T ′′ ) is embedded as a real hypersurface. It is important that; this complex manifold N is uniquely determined as a germ of neighborhoods of M . With this in mind, Miyajima constructs the versal family of isolated singularities by his CR method(see [Mi]). But, in this paper, we don’t follow his work. Rather, we rely on the method in [A3],[ALM]. We set a contact form θ on M . Recently, related to the string theory (moduli space of D-branes),a special kind of displacements of the real hypersurface with a real closed two

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Takao Akahori

form M in the complex manifold N has been paid attention. Our program is that; by the successful method in the deformation methory of CR structures, we would like to solve the Monge-Anpere equation, which is the center in proving the existence of the Einstein Kaehler metric. However, this is in progress. In this paper, we propose a new notion of displacements of real hypersurfaces in a complex manifold, which should be some preparations for our program. For this, we recall the deformation theory of CR structures, which is well established (see [A3], [AGL]). In the deformation theory, the notion of CR-Hamiltonian vectors is introduced and plays an important role in constructing a versal family of deformations of CR structures. We recall the notion of CR-Hamiltonian vector(in [AGL], the terminology of the CR-Hamiltonian vector is not used). (CR-Hamitonian vector fields)(see p 522 in [AGL]). Let (M, 0 T ′′ , θ) be a CR structure with a contact form θ. Let ζ be the real vector field on M , determined by ; θ(ζ) = 1, dθ(ζ, W ) = 0, for any W ∈ 0 T ′ + 0 T ′′ . For any C ∞ function g on M , we can determine an element Xg of 0 T ′ , satisfying: for Y ∈ 0 T ′′ , dθ(Xg , Y ) = −∂ b g(Y ). Set Zg = gζ + Xg , and call Zg a CR-Hamiltonian vector field, determined by g. With this in mind, the notion of the CR-Hamiltonian flows is introduced as follows. Assume taht; we are given a C ∞ function g(t), parametrized by a real number t ∈ (−ǫ, ǫ). For this g(t), a flow of M is introduced by: (CR-Hamiltonian flows) Let ft be an embedding of M as a real hypersurface into a complex manifold N , parametrized by t ∈ (−ǫ, ǫ), ft : M → N.

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Assume that this ft is expressed by using complex coordinates, ft = (z1 (t), ., zn (t)), by: ∂zi (t) = (g(t)ζ + Xg(t) )zi (t), 1 ≤ i ≤ n. ∂t Here Xg(t) is the CR-Hamiltonian vector field determined by g(t). We call this flow a CR Hamiltonian flow. About the existence of the CR-Hamiltonian flow, we have Theorem 1 Let (M, 0 T ′′ ) be a strongly pseudo convex CR manifold, embedded as a real hypersurface in a complex manifold N . Let θ be the contact form on M . For any complex valued C ∞ function g(t),which is parametrized by t ∈ (−ǫ, ǫ), there is a unique CR-Hamiltonian flow, if necessary, we must shrink ǫ sufficiently small. By using this notion, we prove a more precise statement on local embedding problem of CR structures. Theorem 2 Let (M, 0 T ′′ ) be a strongly pseudo convex CR structure with dimR M ≥ 7, embedded as a real hypersurface in N , with a contact form θ. Let (M, φ(t) T ′′ ) be any abstract deformations of the given CR structure, parametrized by the real number t, and φ(o) = 0. Then, there is a complex valued C ∞ function g(t) satisfying:this deformation is realized by the CR-Hamiltonian flows of the original CR structures with respect to g(t). Because of our local embedding theorem of CR structures, (M, φ(t) T ′′ ) can be locally embedded in N . While, Theorem 2 means that; geometrically, (M, φ(t) T ′′ ) can be obtained by the CR-Hamiltonian flow of (M, 0 T ′′ ).

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The Notion of CR Hamiltonian Flows and the Local Embedding Problem...

2.

81

CR Structures and Its Deformation Theory

Let (M, 0 T ′′ ) be a CR structure. This means that M is a real odd-dimensional C ∞ manifold. And 0 T ′′ is a complex subbundle of the complexfied tangent bundle C ⊗ T M , satisfying;

0

T ′′ ∩ 0 T ′′ = 0, dimC

C ⊗ TM

= 1, + 0 T ′′ (2) [Γ(M, 0 T ′′ ), Γ(M, 0 T ′′ )] ⊂ Γ(M, 0 T ′′ ) (1)

0 T ′′

Let (M, 0 T ′′ ) be a strongly pseudo convex CR manifold. We take a supplement vector field ζ over M . And set T ′ = 0 T ′′ + C ⊗ ζ, Now we set a real one form θ by θ(ζ) = 1

(2.1)

θ |0 T ′′ +0 T ′ = 0

(2.2)

Here 0 T ′ = 0 T ′′ . We assume that our real one form θ satisfies θ(ζ) = 1

(2.3)

dθ(ζ, ∗) = 0,

(2.4)

(because of strongly pseudo convexity, by changing ζ, this is always possible). Set Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

C ⊗ T M = T ′ + 0 T ′′ .

(2.5)

Suppose that we are given a CR structure (M, 0 T ′′ ). Now consider an almost CR structure (M, E) over this M . Definition 2..1. Let E is a complex subbundle of the complexfied tangent bundle C ⊗ T M . (M, E) is an almost CR structure if and only if our E satisfies (1) E ∩ E = 0, dimC

C ⊗ TM =1 E+E

And an almost CR structure (M, E) is called a finite distance from (M, 0 T ′′ ) if and only if Definition 2..2. The following composition map of the inclusion and projection of C ⊗ T M = T ′ + 0 T ′′ to 0 T ′′ is isomorphic. inclusion map

E −−−−−−−−−→ C ⊗ T M = T ′ + 0 T ′′   yprojection to 0 T ′′ 0 T ′′

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For an almost CR structure, which is of finite distance from (M, 0 T ′′ ), the following propositions hold( for the more precise explanation, see Sect.3 in [AGL]). Proposition 2.1. An almost CR structure, which is of finite distance from (M, 0 T ′′ ), is parametrized by an element of Γ(M, T ′ ⊗ (0 T ′′ )∗ . The correspondence is that: for φ ∈ Γ(M, T ′ ⊗ (0 T ′′ )∗ ), φ ′′ T = {X ′ = X + φ(X), X ∈ 0 T ′′ }. Proposition 2.2. An almost CR structure (M, φ T ′′ ) is actually a CR structure if and only if (1)

∂ T ′ φ + R2 (φ) + R3 (φ) = 0. Here, R2 (φ) is the element of Γ(M, T ′ ⊗ ∧2 (0 T ′′ )∗ ), which is of second order with respect to φ(and this R2 (φ) includes the first order derivatives of φ), and R3 (φ) is the element of Γ(M, T ′ ⊗ ∧2 (0 T ′′ )∗ ), which is of third order with respect to φ. We explain (1) ∂T ′ . We set Γ(M, T ′ ) = {u : u is a T ′ -valued C ∞ section}, and a T ′ -valued ∂ b -operator ∂ T ′ : Γ(M, T ′ ) → Γ(M, T ′ ⊗ (0 T ′′ )∗ ),

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by: for u ∈ Γ(M, T ′ ), for p ∈ M , ˜ p , u]T ′ , u → ∂ T ′ u(Xp ) = [X ˜ p is a C ∞ extension of Xp , and [X ˜ p , u]T ′ means the T ′ part of [X ˜ p , u] with respect where X to (2.5). Then, like as for scalar valued differential forms, we have a differential complex The standard deformation complex ∂

∂

′

(1) ′

T 0 −−−−→ Γ(M, T ′ ) −−−T−→ Γ(M, T ′ ⊗ (0 T ′′ )∗ ) −−− −→

∂

(2) ′

∂

(3) ′

T T Γ(M, T ′ ⊗ ∧2 (0 T ′′ )∗ ) −−− −→ Γ(M, T ′ ⊗ ∧3 (0 T ′′ )∗ ) −−− −→ ....

For compact complex manifolds, this differential complex is elliptic. However, for our CR case, i.e., boundaries of open manifolds, this is only sub-elliptic. This causes the difficuly in constructing the versal family of deformations eventhough H j (M, T ′ ) =

(j)

Ker∂ T ′

(j−1)

Im∂ T ′

is a

finite dimensionalvector space if 1 ≤ j ≤ n − 2. To overcome this difficulty, we introduce the new complex. Now we recall the new deformation complex, which is developed in our papers ([A3],[AGL]). Let ; for a nonnegative integer j, (j)

Ej = {u : u ∈ 0 T ′ ⊗ ∧j (0 T ′′ )∗ (∂ T ′ u)Cζ⊗∧j+1 (0 T ′′ )∗ = 0}. Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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(j)

We note that ; the map ; u ∈ 0 T ′ ⊗ ∧j (0 T ′′ )∗ → (∂ T ′ u)Cζ⊗∧j+1 (0 T ′′ )∗ is the linear map (j)

which does not include the derivatives with respect to u. Here (∂ T ′ u)Cζ⊗∧j+1 (0 T ′′ )∗ means (j)

the projection of ∂ T ′ u to Cζ ⊗ ∧j+1 (0 T ′′ )∗ according to the vector bundle decomposition T ′ ⊗ ∧j+1 (0 T ′′ )∗ =0 T ′ ⊗ ∧j+1 (0 T ′′ )∗ + Cζ ⊗ ∧j+1 (0 T ′′ )∗ . j+1

In fact, we extend u to a global section of Γ(M,0 T ′ ⊗ ∧j (0 T ′′ )), u ˜, and project ∂ T ′ u ˜ to Cζ ⊗ ∧j+1 (0 T ′′ )∗ according to this vector bundle decomposition. Because of taking the (j) projection, (∂ T ′ u ˜)Cζ⊗∧j+1 (0 T ′′ )∗ does not depend on the extension. Then, we have Proposition 2.3. (see Theorem 2.2 in [A3]). j ∂ T ′ Γ(M, Ej ) ⊂ Γ(M, Ej+1 ). Proposition 2.4. (see Theorem 2.4 in [A3]). For j ≥ 1, j Ker∂ T ′ ∩ Γ(M, Ej ) j−1 ∂ T ′ Γ(M, Ej−1 )

≃ H j (M, T ′ ).

And for j = 0, E0 = 0. So we have some part of deformation complex. ∂

(1) ′

∂

(2) ′

T T Γ(M, E1 ) −−− −→ Γ(M, E2 ) −−− −→ . . .

About, the correct version will be in Sect.3.

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3.

CR Hamiltonian Vector Fields

In Sect.2, we recall some part of the new deformation complex. Here, we recall the rest.(This part is proved in Sect.4 in [AGL]. While, in [AGL], we never use the terminology ”CR Hamilonian vector field”. The terminology ”CR Hamiltonian vector field” is used, for the first time, in this paper.). For g ∈ Γ(M, C), we set an element of Γ(M, E1 ), by Dg := ∂ T ′ Zg . We have to explain some notations. For the notation, we recall the ∂ b - operator. Γ(M, C) → Γ(M, (0 T ′′ )∗ ), by; for u ∈ Γ(M, C), we set ∂ b u(Y ) = Y u. Then, we have the ∂ b -Dolbeault complex ∂

∂

(1)

b 0 −−−−→ Γ(M, C) −−−b−→ Γ(M, (0 T ′′ )∗ ) −−− −→ Γ(M, ∧2 (0 T ′′ )∗ )...

Now Let (M, 0 T ′′ , θ) be a CR structure with a contact form θ. Let ζ be the real vector field on M , determined by (2.1), (2.2). For any C ∞ function g on M , we can determine an element Xg of 0 T ′ , satisfying: for Y ∈ 0 T ′′ , dθ(Xg , Y ) = −(∂ b g)(Y ).

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This Xg is uniquely determined. Actually, this equation comes from: (∂ T ′ (gζ))(Y ) + [Xg , Y ])ζ = 0, for Y ∈ 0 T ′′ , where ∂ T ′ (gζ)(Y ) + [Xg , Y ])ζ means the coefficient of ζ of ∂ T ′ (gζ)(Y ) + [Xg , Y ]. While by (2.4), (∂ T ′ (gζ)(Y ))ζ = (Y g)ζ. Now set Zg = gζ + Xg , and call Zg a CR Hamiltonian vector field. Suppose that M is a real hypersurface in Cn . Then, on M , there is a canonical bundle isomorphism map π: T ′ to T ′ Cn on M (this map is introduced by: just taking a (1, 0) part of vectors of T ′ ). So, there is an η of Γ(M, T ′ Cn |M ), π(ζ) = η. This means: ζ = η + η. So, we can set a (1, 0) vector on M by: for a C ∞ function g, gη + Xg . We use the same notation Zg for this vector. Set Dg = Zg for a C ∞ function g on M . Now we have the complete deformation complex for CR structures. ∂

D

(1) ′

∂

(2) ′

T T 0 −−−−→ Γ(M, C) −−−−→ Γ(M, E1 ) −−− −→ Γ(M, E2 ) −−− −→ . . .

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4.

CR Hamiltonian Flows

Let M be a real hypersurface with a real one form θ, satisfying (2.3), (2.4) in a complex manifold N with complex dimension n. The triple (M, 0 T ′′ , θ) is called a CR structure with a contact form. Let Mt be a family of real hypersurfaces of Cn , depending on t ∈ (−ǫ, ǫ). Let t T ′′ be the induced CR structure over Mt from N . Now a quartet (M, π, (−ǫ, ǫ), Θ) is called a CR Hamiltonian system if (1) M is a real submanifold of N × (−ǫ, ǫ), and π is a smooth map from M to (−ǫ, ǫ), satisfying : Mt = π −1 (t). At t = o, M0 = M , (2) Θ is a real one form on M satisfying; the restriction of Θ to Mt (we write this by θt ), determines a contact form on Mt , and θ0 = θ, (3) there is a C ∞ function g on M, and there is a family of C ∞ maps ft from M to Mt = π −1 (t) satisfying : ft∗ θt = θ, on M.

(4.1)

For a point p of N , we take a local complex coordinate (z1 , · · · , zn ) of p. For the definition of CR Hamiltonian flows, we use this complex coordinate.Let zi (p, t) = zi · ft and gt = g · ft . ∂zi (p, t) = (gt ζ + Xgt )zi (p, t), on M × (−ǫ, ǫ). ∂t where Xgt is the vector field on M , determined in the way as in Sect.2, for gt . The quartet (M, π, (−ǫ, ǫ), Θ) is a CR Hamiltonian sysyem.

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(4.2)

The Notion of CR Hamiltonian Flows and the Local Embedding Problem...

5.

85

On the Existence of the Solution

We show the existence of the solution(the existence of the CR Hamiltonian flow for a short time). Let (M, 0 T ′′ ) be a strongly pseudo convex CR manifold with dimR M = 2n − 1 ≥ 7. Let p be a reference point of M . In this case, our CR manifold is embeddable in a complex euclidean space(see [A5],[Ku]). We are focussing its behavior. Namely, we are given a family of CR structures (M, φ(t) T ′′ ), where t ∈ (−ǫ, ǫ). Then,the problem is that: how the local embedding map, ft , t ∈ (−ǫ, ǫ) behaves if t moves. Theorem 5.1. (MainTheorem) For any C ∞ function g(t), which is parametrized by t ∈ (−ǫ, ǫ), there is a unique CR Hamiltonian flow, ft , satisfying; ft is an embedding of M as a real hypersurface into a complex manifold N , parametrized by t ∈ (−ǫ, ǫ), ft : M → N, and this ft is expressed by using complex coordinates of N as follows. (z1 (t), ., zn (t)). Then,

Let ft =

∂zi (t) = (g(t)ζ + Xg(t) )zi (t), 1 ≤ i ≤ n. ∂t Here Xg(t) is the CR-Hamiltonian vector field determined by g(t). We call this flow a CR Hamiltonian flow.

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For this, we recall the following well-known existence theorem of the solutions of a single non-linear partial differential equation. Theorem 5.2. On a neighborhood of the origin in Rm , consider the following single nonlinear partial differential equation. F (x1 , · · · , xm , s, p1 , · · · , pm ) = 0, Where F is a real valued C ∞ function on a neighborhood in Rm × R × Rm , and s is an ∂s ∂F unknown function, pi = ∂x . Suppose ∂p 6= 0 for some i, 1 ≤ i ≤ m at the origin. Then, i i for any initial value, we have a unique solution. The proof relies on the existence theorem of solutions of ordinary differential equations. On Rm × R × Rm = (x1 , · · · , xm , s, p1 , · · · , pm ), we solve the ordinary differential equation; dx1 = ··· = P1 = Here Pi =

∂F ∂pi .

dxm ds = Pm p1 P1 + · · · + pm Pm −dp1 −dpm = · · · = ∂F ∂F ∂F ∂F ∂x1 + p1 ∂s ∂xm + pm ∂s

We rewrite (4.2) as follows. ∂zi (p, t) = (g(t)ζ + Xg(t) + X g(t) )zi (p, t). ∂t

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(5.1)

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Takao Akahori

Our (5.1) is equivalent to (4.2), because Xg(t) ∈ φ(t) T ′ and so X g(t) zi (p, t) = 0. Proposition 5.3. If zi (p, t) is a solution of ∂ zi (p, t) = (g(t)ζ + Wt )zi (p, t), 1 ≤ i ≤ n, ∂t where Wt is a real vector field on M , then this zi (p, t) satisfies ∂ (1,0) zi (p, t) = (g(t)ζ + Wt )zi (p, t), 1 ≤ i ≤, ∂t (1,0)

where Wt is the (1, 0) part of the real vector field Wt with respect to the CR structure (M, t T ′′ ), where t T ′′ is determined by; t

T ′′ = {X ′ ; X ′ ∈ C ⊗ T M, X ′ zi (p, t) = 0, 1 ≤ i ≤ n}.

Proof. If zi (p, t) satisfies ∂ (1,0) zi (p, t) = (g(t)ζ + Wt )zi (p, t), 1 ≤ i ≤ n, ∂t and

(0,1)

Wt

zi (p, t) = 0,

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then obviously, ∂ zi (p, t) = (g(t)ζ + Wt )zi (p, t), 1 ≤ i ≤ . ∂t Conversely, we assume ∂ zi (p, t) = (g(t)ζ + Wt )zi (p, t), 1 ≤ i ≤ n. ∂t (1,0)

Let Wt is the (1, 0) part of the real vector field Wt with respect to the CR structure t ′′ (M, T ), determined by the embedding : p ∈ M → (z1 (p, t), . . . , zn (p, t)), then

(0,1)

Wt

zi (p, t) = 0.

So, we have our proposition. By using this well-known theorem, we solve (5.1). While, contrary to our setting, (5.1) is still not a real valued partial differential equation. Our g(t) is a complex valued C ∞ function. In order to solve (5.1), we consider an extra dimension. Namely, consider M × (−ǫ′ , ǫ′ ) = {(p, q) : p ∈ M, q ∈ (−ǫ′ , ǫ′ )},

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87

and embeddings(local C ∞ isomorphism map) of M × (−ǫ′ , ǫ′ ), f˜t , satsifying: f˜t |q=0 = ft , M × (−ǫ′ , ǫ′ ) → N. We write f˜t = (z1 (p, q, t), · · · , zn (p, q, t)). By Jt , the complex structure over M ×(−ǫ′ , ǫ′ ) is induced from N .. By (1, 1) tensor, we denote Jt for this complex structure. ζ is a vector field on M . By the inclusion map i: M → M × (−ǫ, ǫ), p → (p, o), we have a vector field i∗ ζ on M × (−ǫ′ , ǫ′ ). Let √ Wt := −1(i∗ ζ) − Jt (i∗ ζ). Then, Jt Wt =

√ √

−1Jt (i∗ ζ) − Jt2 (i∗ ζ)

−1Jt (i∗ ζ) + i∗ ζ √ = − −1Wt . =

So, this means that Wt is of (0, 1) type vector with respect to Jt . Hence √ √

−1(i∗ ζ) = Jt (i∗ ζ) + Wt .

(5.2)

Let g(t) = h(t) + −1k(t), where h(t) is a real part and k(t) is an imaginary part of g(t). For real valued C ∞ functions g(t), h(t) on M , the following are equivalent.   ˜ √   ∂ ft ˜ ˜ ˜ = h(t)(i∗ ζ)ft + −1k(t)(i∗ ζ)ft + Xt ft on M × o × (−ǫ, ǫ) ∂t   ′˜ Y ft = 0 for Y ′ ∈ t T ′′ on M × o × (−ǫ, ǫ),

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and ∂ f˜t = h(t)(i∗ ζ)f˜t + k(t)(Jt (i∗ ζ))f˜t + (i∗ (Xt + X t ))f˜t = 0 on M × o × (−ǫ, ǫ). ∂t (As Wt is of type (0, 1) with respect to Jt , Wt f˜t = must hold.(because the CR structure Jt is determined by the C ∞ map f˜t .)) ˜ = h(t) · π, k(t) ˜ = k(t) · π, where π is the projection map of M × (−ǫ′ , ǫ′ ) to Set h(t) ′ ′ M , π : M × (−ǫ , ǫ ) → M ,(p, q) → p. Now we consider the following real valued partial differential equation on (p, q, t) - space. ∂ f˜t ˜ ˜ ˜ ˜ = h(t)(i∗ ζ)f˜t + k(t)J t (i∗ ζ)ft + (i∗ )(Xt + X t )ft = 0, ∂t on (p, q, t) ∈ M × (−ǫ′ , ǫ′ ) × (−ǫ, ǫ). ˜

This is a real valued partial diferential equation and the coefficient of ∂∂tft is 1(nonzero). So our theorem is applicable(to solve the ordinary differential equation with respect to t). Hence, we have a solution zi (p, q, t), 1 ≤ i ≤ n, p(t), q(t), satisfying ; p(o) = p, q(o) = q. So, let zi (t) =: zi (p(t), q(t), t). Then, we have a solution.

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6.

An Application to the Local Embedding Problem of CR Structures (Setting)

Let (M, 0 T ′′ ) be a strongly pseudo convex CR manifold with realdimension 2n − 1. Suppose that our CR manifold is embedded in a complex eucliean space Cn . For a complexvalued C ∞ function g(t), parametrized by a real parameter t ∈ (−ǫ, ǫ), we have a special family of C ∞ embedding map ft (CR-Hamiltonian flows). We use the notation ft (p) = (z1 (p, t), · · · , zn (p, t)), for p ∈ M, Here zj (p, t) = zj · ft , and {zj }1≤j≤n are complex coordinates of the euclidean space. Then, we have a deformation of CR structures of (M, 0 T ′′ ), (M, φ(t) T ′′ ), defined by: (Y + φ(t)(Y ))zi (p, t) = 0, i = 1, ..., n, Y ∈ 0 T ′′ . We consider the inverse problem. Suppose that we are given a deformation (M, φ(t) T ′′ ). Then, the problem is that;is this deformation realized by a CR Hamiltonian flow? In this paper, we give an affirmative answer. We note dimR M = 2n − 1 ≥ 7. Then, by the Kuranishi’s local embedding theorem of CR structures(and also my paper [A5]), we can assume that: there is a simultaneous CR embedding kt : (M, φ(t) T ′′ ) → Cn ,

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if necessary we must shrink ǫ. The problem is that; for this family of embeddings, can we adjust kt to a CR Hamiltonian flow ft : M → Cn , satisfying; the induced CR structure by ft , is really (M, φ(t) T ′′ ) ? And by the definition, this CR embedding is a CR Hamiltonian flow if there is a g(t) satisying ∂zi (p, t) = (g(t)ζ + Xg(t) )zi (p, t), i = 1, ..., n. ∂t

(6.1)

(Y + φ(t)(Y ))zi (p, t) = 0, i = 1, ..., n, Y ∈ 0 T ′′ .

(6.2)

And

We solve a system of partial differential equations, (6.1) and (6.2) over M (unknown function is gt ). For this, we differentiate (6.1) with respect to t, then (

∂ ∂ φ(t))(Y )zi (p, t) + (Y + φ(t)(Y )) zi (p, t) = 0, i = 1, .., n. ∂t ∂t

Instead of (6.1), we adopt this. By the definition of the CR Hamiltonian flow, this becomes to Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

(6.3)

The Notion of CR Hamiltonian Flows and the Local Embedding Problem...

(

∂ φ(t))(Y )zi (p, t) + (Y + φ(t)(Y ))Zg(t)) zi (p, t) = 0, i = 1, .., n. ∂t

89

(6.4)

While, because of (Y + φ(t)(Y ))zi (p, t) = 0, we have (

∂ φ(t))(Y )zi (p, t) + {(Y + φ(t)(Y ))Zg(t) − Zg(t) (Y + φ(t)(Y ))}zi (p, t) = 0 ∂t That is to say,

(

∂ φ(t))(Y )zi (p, t) + [Y + φ(t)(Y ), zg(t) ]zi (p, t) = 0. ∂t

(6.5)

We set a C ∞ vector bundle decomposition C ⊗ T M = φ(t) T ′′ + T ′ . With respect to this decomposition, this becomes ∂ φ(t)(Y ) + [Y + φ(t)(Y ), Zg(t) ]T ′ = 0. ∂t

(6.6)

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Here [Y + φ(t)(Y ), Zg(t) ]T ′ means the projection of [Y + φ(t)(Y ), Zg(t) ] to T ′ according to (6.6)(we note that; zi (p, t) is the complex coordinate with respect to φ(t) T ′′ and so Y ′ zi (p, t) = 0 for Y ′ ∈ φ(t) T ′′ for every t. In the next section, we will solve this equation (the unknown function is g(t)).

7.

The Solution

The equation (6.6) can be written as follows. ∂φ(t) φ(t) + ∂ T ′ Zg(t) = 0. ∂t

(7.1)

φ(t)

We mst explain the operator ∂ T ′ . φ Let (M, φ T ′′ ) be a deformation of (M, 0 T ′′ ). Then, we can introduce ∂ T ′ operator by: φ

u ∈ Γ(M, T ′ ) → ∂ T ′ u ∈ Γ(M, T ′ ⊗ (0 T ′′ )∗ ), by φ

∂ T ′ u(X) = [X + φ(X), u]T ′ − φ([X + φ(X), u]0 T ′′ ). Here [X + φ(X), u]T ′ means the projection of [X + φ(X), u]T ′ to 0 T ′′ according to the C ∞ vector bundle decomposition C ⊗ T M = 0 T ′′ + T ′ . And [X + φ(X), u]0 T ′′ means the projection of [X + φ(X), u] to 0 T ′′ according to this decomposition. This definition makes sense. In fact,

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Lemma 7.1. For a C ∞ function, f, for u ∈ Γ(M, T ′ ), and for X ∈ 0 T ′′ , φ

φ

∂ T ′ u(f X) = f ∂ T ′ u. Proof. In fact, φ

∂ T ′ u(f X) = = [f X + φ(f X), u]T ′ − φ([f X + φ(f X), u]0 T ′′ ) = −u(f )φ(X) + f [φ(X), u]T ′ + u(f )φ(X) − f φ([X + φ(X), u]0 T ′′ ) φ

= f ∂ T ′ u(X)

(i)

φ,(i)

As for like ∂ T ′ , we have ∂ T ′ ,i = 1, 2, .... Especially, Lemma 7.2. φ,(1)

∂ T ′ u(X, Y ) = [X + φ(X), u(Y )]T ′ − [Y + φ(Y ), u(X)]T ′ − u([X + φ(X), Y + φ(Y )]0 T ′′ , for all X, Y ∈ 0 T ′′ . And like the case for (M, 0 T ′′ ), Lemma 7.3.

φ,(i+1) φ,(i) ∂T ′

∂T ′

= 0.

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Like the case for (M, 0 T ′′ ), for a CR structure (M, φ(t) T ′′ ), we can introduce a relative version; for g ∈ Γ(M, C), we set an element of Γ(M, E1 ), by φ(t)

Dφ(t) g := ∂ T ′ Zg . Here Ej , j = 1, 2, · · · are introduced in [A3]. We recall Ej bundles. And we have a differential complex 0 −−−−→

C ∞ (M )

φ(t),(1)

∂T ′

Dφ(t)

−−−−→ Γ(M, E1 ) −−−−−→ Γ(M, E2 )

For a family of deformations of CR structures, (M, φ(t) T ′′ )t∈(−ǫ,ǫ) , the Kodaira-Spencer class is given as ∂φ(t) ∂t t=o

at the origin o. For t = to , the Kodaira-Spencer class is given as ∂φ(t) . ∂t t=to In fact, as φ(t) T ′′ is integrable, φ(t) satisfies

[X + φ(t)(X), Y + φ(t)(Y )]T ′ = φ(t)([X + φ(t)(X), Y + φ(t)(Y )]0 T ′′ ). Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

The Notion of CR Hamiltonian Flows and the Local Embedding Problem... for all X, Y ∈ Γ(M, 0 T ′′ ). We operate

∂ ∂t

91

on this. Then,

∂φ(t) ∂φ(t) (X), Y + φ(t)(Y )]T ′ + [X + φ(t)(X), (Y )]T ′ ∂t ∂t ∂φ(t) = ( )([X + φ(t)(X), Y + φ(t)(Y )]0 T ′′ ∂t ∂φ(t) (X), Y + φ(t)(Y )]0 T ′′ + φ(t)([ ∂t ∂φ(t) + φ(t)([X + φ(t)(X), (Y )]0 T ′′ ) ∂t [

This equality means Proposition 7.4. φ(t),(1)

∂T ′

(

∂φ(t) ) = 0. ∂t φ(t),(1)

By Proposition 7.4, we have that; ∂φ(t) ∂t is ∂ T ′ ∂φ(t) φ(t) -boundary. For this, we have to solve; ∂t is D

- closed. Furthermore, we see that;

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∂φ(t) = Dφ(t) g(t), ∂t where g(t) is an unknown C ∞ function on M × (−ǫ, ǫ). We note that our problem is purely local(we are considering the local embedding of M at the reference point po ). So, in the case complex manifolds, our equation corresponds to the kind of ”Dolbeault lemma ” for ∂. For this, we recall the Kuranishi’s local a-priori estimate for ∂ b on a CR manifold in his local embedding theorem of CR structures. Theorem 7.5. (Kuranishi’s local a priori estimate for ∂ b on a CR manifold) Let (M, 0 T ′′ ) be a strongly pseudo convex CR structure with dimR M ≥ 7, embedded as a real hypersurface in Cn . For a point p of M , there is a neighborhood of p, and on this neighborhood, we have an L2 estimate. We see this theorem more precisely. Let po ∈ M be a reference point. Let M ⊂ Cn . Let h be a holomorphic function on Cn satisfying; (1) 2Re h(po ) = 0, (2) 2Re h(p) is strictly pluri-subharmonic on a neighborhood of the reference point po of M . Let Uǫ (h) = {p : p ∈ M, 2Re h(p) < ǫ}. The Kuranishi’s local a priori estimate is that; over Uǫ (h), for u ∈ Γ(Uǫ (h), (0 T ′′ )∗ ) ∩ ∗ Dom ∂ b , ∗

c k u kUǫ (h) 1, then [x] does not depend on particular x. (C) If there is a subspace Lj such that codimLi (Li ∩ Lj ) = 1 and x1 , x2 belong to different connected components of Li \ ∪j6=i Lj , then [x1 ] 6= [x2].

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ˇ Proof. (A) By the Ziegler-Zivaljevi´ c formula [23], there is a wedge decomposition ˆ (A) ' U

_ V ∈P A

A ˆ1 ∨ . . . ∨ L ˆk ∨ )=L Vˆ ∗ ∆(P ih and πih+1 −1 < πih , The vectors β(π) and γ(π) are uniquely defined by the permutation π, and two permutations are distinct if at least one of these vectors differ. As an example, for the permutation π = [0, 1, 6, 5, 4, 7, 2, 3, 8, 9] we have the partition [0, 9] into intervals [0, 1], [2, 4], [5], [6, 7], [8, 9], b(P ) = 4, β(π) = (+, +, −, +), and γ(π) = (−, 0, +). Now let us estimate the change of the number of breakpoints as a result of applying a reversal ri,j to a permutation π. To formulate the following statements we define one additional function δ(x, y) of two integer variables x and y such that δ(x, y) = 1 if |x−y| ≥ 2 and δ(x, y) = 0 if |x − y| ≤ 1. Statement 13 Let π = [π0 , π1 , . . . , πn , πn+1 ] and ri,j is a reversal on the interval [i, j], 1 ≤ i < j ≤ n. Then b(πri,j ) − b(π) = δ(πi−1 , πj ) + δ(πi , πj+1 ) − δ(πi−1 , πi ) − δ(πj , πj+1 ). Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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Indeed, a reversal ri,j can only change the number of breakpoints between positions i − 1 and i, and between positions j and j + 1. Statement 14 Let π = [π0 , π1 , . . . , πn , πn+1 ], 1 ≤ i < j ≤ n, and |αi | = |αj+1 | = 1. Then b(πri,j ) = b(π) + 1 if and only if αi = −αj+1 and either πj = πi − 2αi or πj = πi + 2αi ; otherwise b(πri,j ) = b(π) + 2. It is clear that by (12) the condition |αi | = |αj+1 | = 1 is equivalent to the condition δ(πi−1 , πi ) = δ(πj , πj+1 ) = 0 and we also have πi = πi−1 + αi

and πj+1 = πj + αj+1

(13)

Note that πi − αi , πi , πi + αi and πj − αj+1 , πj , πj + αj+1 are three successive integers in increasing or decreasing order. Since all πi−1 , πi , πj , πj+1 are distinct, we have 1) either δ(πi , πj+1 ) = 1 or πj+1 = πi + αi , since πi−1 = πi − αi ; 2) either δ(πi−1 , πj ) = 1 or πi−1 = πj − αj+1 , since πj+1 = πj + αj+1 . Therefore, if b(πri,j ) = b(π) + 1 or b(πri,j ) = b(π), then πi−1 = πj − αj+1 or πj+1 = πi + αi , by Statement 13. In each of these cases we have αi = −αj+1 ; otherwise, we would get πj = πi , by (13). Moreover, for αi = −αj+1 in the case when πi−1 = πj − αj+1 by (13) we have πi = πi−1 + αi = πj + 2αi = πj+1 + 3αi (14)

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and hence δ(πi−1 , πj ) = 0, δ(πi , πj+1 ) = 1; and in the case when πj+1 = πi + αi , by (13) we have πj = πj+1 + αi = πi + 2αi = πi−1 + 3αi (15) and hence δ(πi−1 , πj ) = 1, δ(πi , πj+1 ) = 0. It follows that αi = −αj+1 and either πj = πi − 2αi or πj = πi + 2αi , if b(πri,j ) = b(π) + 1, and that the case b(πri,j ) = b(π) is not possible under the conditions of this statement. On the other hand, the conditions αi = −αj+1 and πj = πi − 2αi or πj = πi + 2αi are sufficient for b(πri,j ) = b(π) + 1, because in these cases (14) and (15) are respectively valid and we have b(πri,j ) = b(π) + 1 in each of them. Note that Statement 14 presents conditions when a given reversal increases the number of breakpoints of a permutation π. In constructing effective algorithms for sorting by reversals it is significant to find a reversal which decreases by two the number of breakpoints of a permutation π having a decreasing interval. In particular, as it was proved in [32], such a reversal exists, if every reversal that removes a breakpoint of π gives a permutation with no decreasing intervals. The reversal graph Symn (R) is defined on the group Symn and
generated by the reversals from the set R = {ri,j ∈ Symn , 1 ≤ i < j ≤ n}, |R| = n2 . The path distance in this graph corresponds to the reversal distance between two permutations. Hence, the diameter of this graph is (n − 1) as it is presented by (11), and the only permutations needing these many reversals are the Gollan permutation γn and its inverse, where the Gollan permutation, in one–line notation, is defined as follows  3, 1, 5, 2, 7, 4, . . . , n − 3, n − 5, n − 1, n − 4, n, n − 2, if n is even γn = 3, 1, 5, 2, 7, 4, . . . , n − 6, n − 2, n − 5, n, n − 3, n − 1, if n is odd. The following statement collects some properties of this graph [33, 35]. Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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Elena Konstantinova

Statement 15 The
reversal graph Symn (R), n ≥ 3, is (i) a connected n2 -regular graph of order n! and diameter n − 1; (ii) not edge–transitive, not distance–regular and hence not distance–transitive. To find the values (2) when r = 1, 2 for the reversal graph, we have to describe spheres S1 and S2 and connections between them. This is our main goal now. It is clear that S1 = R and by Statement 13 any reversal has two breakpoints. The next two technical statements give us the full description of permutations π ∈ S2 , having three or four breakpoints. For shortness, we shall write a coordinate of a vector γ(π) as + ∨ 0 or − ∨ 0, if this coordinate might take one of the values + or 0 and − or 0 respectively. Statement 16 Let π = rk,l ri,j ∈ S2 such that b(π) = 3 and rk,l 6= ri,j for fixed k and l, 1 ≤ k < l ≤ n, and any i and j, 1 ≤ i < j ≤ n. Then π belongs to one of the disjoint sets Uh (k, l), h = 1, ..., 8, defined as follows: 1. If π ∈ U1 (k, l) then i = k − 1 ≥ 1, j = l − 1 and γ(π) = (+ ∨ 0, +); 2. If π ∈ U2 (k, l) then i = k + 1, j = l + 1 ≤ n and γ(π) = (+, + ∨ 0); 3. If π ∈ U3 (k, l) then i = k, k + 1 ≤ j ≤ l − 1 and γ(π) = (+, − ∨ 0); 4. If π ∈ U4 (k, l) then i = k, l + 1 ≤ j ≤ n and γ(π) = (− ∨ 0, +); 5. If π ∈ U5 (k, l) then 1 ≤ i ≤ k − 1, j = l and γ(π) = (+, − ∨ 0); 6. If π ∈ U6 (k, l) then k + 1 ≤ i ≤ l − 1, j = l and γ(π) = (− ∨ 0, +);

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7. If π ∈ U7 (k, l) then i = l + 1, l + 2 ≤ j ≤ n and γ(π) = (−, −); 8. If π ∈ U8 (k, l) then 1 ≤ i ≤ k − 2, j = k − 1 and γ(π) = (−, −). To prove this statement, first we consider the case when the conditions of Statement 14 are satisfied. Then b(rk,l ri,j ) = 3 if and only if αi = −αj+1 = ±1 and either πj = πi +2αi or πj = πi − 2αi . Since γ(rk,l ) = (−) there are two possibilities: 1) αi = 1, where 1 ≤ i ≤ k − 1 and αj+1 = −1, where k ≤ j ≤ l − 1; 2) αi = −1, where k + 1 ≤ i ≤ l and αj+1 = 1, where l + 1 ≤ j ≤ n. In the first case we have πi = i and πj = i + 2 and get the set U1 (k, l), since i ≤ k − 1 and πj ≥ k + 1 imply i = k − 1, πj = k + 1, and j = l − 1. In the second case we have πj = j and πi = j − 2 and get the set U2 (k, l), since j ≥ l + 1 and πi ≤ l − 1 imply j = l + 1, πi = l − 1, and i = k + 1. If the conditions of Statement 14 are not satisfied we have i = k or i = l + 1 when |πi − πi−1 | ≥ 2, and have j = k − 1 or j = l when |πj+1 − πj | ≥ 2. Since i < j and k < l (for i = k and j = l we have π = rk,l ri,j = I), these four cases are incompatible. The case i = k gives rise to the set U3 (k, l) when k + 1 ≤ j ≤ l − 1, and gives rise to the set U4 (k, l) when l + 1 ≤ j ≤ n. Analogously, the case j = l gives rise to the set U5 (k, l) when 1 ≤ i ≤ k − 1, and gives rise to the set U6 (k, l) when k + 1 ≤ i ≤ l − 1. The case i = l + 1 is possible only if l + 2 ≤ j ≤ n and we obtain the set U7 (k, l). Analogously, the

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case j = k − 1 is possible only if 1 ≤ i ≤ k − 2 and we obtain the set U8 (k, l). It is easily seen that permutations π of the last six sets have also three breakpoints. The sets Uh (k, l), h = 1, ..., 8, are disjoint since the reversals ri,j are distinct for all cases. These sets consist of permutations π with β(π) = (+, −, +) in the first six cases and with β(π) = (+, +, +) in the last two cases. The statement is proved. Corollary 3 The reversal graph Symn (R) does not contain triangles. Note, that from Statement 14 and the proof of Statement 16 we have that any permutation π = rk,l ri,j where rk,l 6= ri,j has three or four breakpoints. This also implies that there are no edges between vertices of the sphere S1 which all have two breakpoints, and this implies Corollary 3. Since any permutation π = rk,l ri,j where rk,l 6= ri,j has three or four breakpoints, then one can consider all cases to arrange numbers i, j, 1 ≤ i < j ≤ n, with the fixed numbers k, l, 1 ≤ k < l ≤ n, then exclude the cases of Statement 16, when π has three breakpoints, and get the description of all permutations π ∈ S2 with four breakpoints as follows. Statement 17 Let π = rk,l ri,j ∈ S2 such that b(π) = 4 and rk,l 6= ri,j for fixed k and l, 1 ≤ k < l ≤ n, and any i and j, 1 ≤ i < j ≤ n. Then π belongs to one of the disjoint sets Wh (k, l), h = 1, ..., 4, defined as follows: 1. If π ∈ W1 (k, l) then k + 1 < l + 1 < i < j or i + 1 < j + 1 < k < l; β(π) = (+, +, +, +), γ(π) = (−, + ∨ 0, −);

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2. If π ∈ W2 (k, l) then k < i < j < l or i < k < l < j; β(π) = (+, −, −, +), γ(π) = (− ∨ 0, +, − ∨ 0); 3. If π ∈ W3 (k, l) then 1 ≤ i ≤ k − 1, k ≤ j ≤ l − 1 and j 6= l − 1, when i = k − 1; β(π) = (+, −, +, +), γ(π) = (+ ∨ 0, −, − ∨ 0) or γ(π) = (+ ∨ 0, 0, −); 4. If π ∈ W4 (k, l) then k + 1 ≤ i ≤ l, l + 1 ≤ j ≤ n and j 6= l + 1, when i = k + 1; β(π) = (+, +, −, +), γ(π) = (− ∨ 0, −, + ∨ 0) or γ(π) = (−, 0, + ∨ 0). So, Statements 16 and 17 determine all disjoint sets of permutations π ∈ S2 . Now let us consider connections between S1 and S2 , and find c2 (π) = c2 (π, I) for any π ∈ S2 . To formulate the next result we fix a permutation π ∈ S2 and consider the lexicographic ordering on permutations rk,l ∈ S1 , 1 ≤ k < l ≤ n, assuming that rk,l < rk′ ,l′ if k ≤ k ′ and l < l′ . Any pair of edges {rk,l , π} and {rk′ ,l′ , π} in the reversal graph Symn (R) implies the following expression for π : rk,l ri,j = rk′ ,l′ ri′ ,j ′

(16)

where ri,j = rk,l π and ri′ ,j ′ = rk′ ,l′ π. We say that (16) is a representation of π if rk,l < rk′ ,l′ , and it is a minimal representation of π if rk,l is minimal in the lexicographic order permutation of the set S2,1 (π) = {x ∈ S1 : d(x, π) = 1}. Note that if π has only one representation then this representation is minimal and c2 (π) = 2. In a general case, if π has h minimal representations with h = 0, 1, . . . , then c2 (π) = h + 1. We define the permutation πk = [0, 1, . . . , k − 1, k + 1, k + 2, k, k + 3, . . . , n + 1]

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Elena Konstantinova

for 1 ≤ k ≤ n − 2 and for which b(πk ) = 3, β(πk ) = (+, −, +), γ(πk ) = (+, 0). Its inverse is πk−1 = [0, 1, . . . , k − 1, k + 2, k, k + 1, k + 3, . . . , n + 1] such that b(πk−1 ) = 3, β(πk−1 ) = (+, −, +) and γ(πk−1 ) = (0, +). It is easily seen that πk = rk,k+1 rk+1,k+2 = rk,k+2 rk,k+1 = rk+1,k+2 rk,k+2 and as a consequence of (π1 π2 )−1 = π2−1 π1−1 we have πk−1 = rk,k+1 rk,k+2 = rk,k+2 rk+1,k+2 = rk+1,k+2 rk,k+1 . We also define τk = [0, 1, ..., k − 1, k + 2, k + 3, k, k + 1, k + 4, ..., n + 1] for 1 ≤ k ≤ n−3 and for which b(τk ) = 3, β(τk ) = (+, −, +), γ(τk ) = (+, +), τk−1 = τk and τk = rk,k+2 rk+1,k+3 = rk+1,k+3 rk,k+2 . Statement 18 Given a permutation π ∈ S2 let k, 1 ≤ k ≤ n − 1, be the minimal integer such that rk,s ∈ S2,1 (π) for some s, k < s ≤ n. Then 1. c2 (π) = 3, if π = πk or π = πk−1 for k ≤ n − 2; 2. c2 (π) = 2, if π has one of the following representations: rk,k+2 rk+1,k+3 = rk+1,k+3 rk,k+2 = τk ,

(17)

rk,l rk,j = rk+l−j,l rk,l , k + 1 ≤ j ≤ l − 1, l > k + 2,

(18)

l + 1 ≤ j ≤ n, j > k + 2,

(19)

rk,l ri,j = ri,j rk,l ,

k < l < i < j,

(20)

rk,l ri,j = rl−j+k,l−i+k rk,l ,

k < i < j < l;

(21)

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k ≤ n − 3,

3. c2 (π) = 1, in the remaining cases. We shall prove this statement assuming that c2 (π) > 1 and rk,s ∈ S2,1 (π) for some integer s. Then the permutation π has a representation (16) with the minimal integer k. Our goal is to describe all such representations of π, to determine which of them are minimal and then to find c2 (π). If π has a representation (16) then by Statements 16 and 17, we have π ∈ Uh (k, l) ∩ Um (k ′ , l′ ), if b(π) = 3, and π ∈ Wh (k, l) ∩ Wm (k ′ , l′ ), if b(π) = 4, for some h and m. First we consider representations of permutations with three breakpoints. We have γ(π) = (− ∨ 0, +) for π from U4 or U6 , and γ(π) = (+, − ∨ 0) for π from U3 or U5 . Therefore, since γ(π) = (+ ∨ 0, +) for π ∈ U1 and γ(π) = (+, + ∨ 0) for π ∈ U2 , we have
(U2 (k, l) ∪ U3 (k, l) ∪ U5 (k, l)) ∩ U4 (k ′ , l′ ) ∪ U6 (k ′ , l′ ) = ∅ and


(U3 (k, l) ∪ U5 (k, l)) ∩ U1 (k ′ , l′ ) ∪ U4 (k ′ , l′ ) ∪ U6 (k ′ , l′ ) = ∅.

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We have γ(π) = (−, −) for π from U7 or U8 , hence
(U7 (k, l) ∪ U8 (k, l)) ∩ ∪6m=1 Um (k ′ , l′ ) = ∅. Moreover, Uh (k, l) ∩ Uh (k ′ , l′ ) = ∅ for any h = 1, ..., 8, since k and l are uniquely defined by any element of π ∈ Uh (k, l). These arguments show that if π has a representation (16), b(π) = 3, and π ∈ Uh (k, l) ∩ Um (k ′ , l′ ), then (h, m) or (m, h) must belong to the set A = {(2, 3), (2, 5), (3, 5), (4, 6), (4, 1), (6, 1), (2, 1), (7, 8)}. We shall prove that if rk,l ∈ S2,1 (π) then there exists a unique rk′ ,l′ such that π ∈ Uh (k, l) ∩ Um (k ′ , l′ ) with (h, m) ∈ A and π has the representation (16) with rk,l < rk′ ,l′ . It is also shown that for (m, h) ∈ A we have the expression (16) but it is not a representation of π since in these cases rk,l < rk′ ,l′ does not hold. Case 1: (h, m) = (2, 3). For π ∈ U2 (k, l) ∩ U3 (k ′ , l′ ), we have i = k + 1, j = l + 1, i′ = k ′ , k ′ = k, l′ = l + 1, j ′ = l′ + k ′ − l = l′ − k ′ − k − 1 = k + 1 = l, and get the representation rk,k+1 rk+1,k+2 = rk,k+2 rk,k+1 of π. Case 2: (h, m) = (2, 5). For π ∈ U2 (k, l) ∩ U5 (k ′ , l′ ), we have i = k + 1, j = l + 1, i′ = k = l − 1, j ′ = l′ = l + 1, k ′ = l, and rk,k+1 rk+1,k+2 = rk+1,k+2 rk,k+2 is the representation of π.

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Case 3: (h, m) = (3, 5). For π ∈ U3 (k, l) ∩ U5 (k ′ , l′ ), we have i = k, k + 1 ≤ j ≤ l − 1, i′ = k, j ′ = l = l′ , k ′ = k + l − j and get the representation rk,l rk,j = rk+l−j,l rk,l of π. This gives (18) when l > k + 2. For l = k + 2 we have j = k + 1 and get the representation rk,k+2 rk,k+1 = rk+1,k+2 rk,k+2 of π. Case 4: (h, m) = (4, 6). For π ∈ U4 (k, l) ∩ U6 (k ′ , l′ ), we have i = k, l + 1 ≤ j ≤ n, j ′ = l′ = j, k ′ = k, i′ = k ′ + l′ − l = k + j − l and get the representation rk,l rk,j = rk,j rk+j−l,j of π. This gives (19) when j > k + 2. For j = k + 2 we get the representation rk,k+1 rk,k+2 = rk,k+2 rk+1,k+2 of π. Case 5: (h, m) = (4, 1). For π ∈ U4 (k, l) ∩ U1 (k ′ , l′ ), we have i = k, i′ = k ′ − 1, j ′ = l′ − 1, k ′ = k + 1 = j − 1, k ′ = l, l′ = j, and get the following representation rk,k+1 rk,k+2 = rk+1,k+2 rk,k+1 of π. Case 6: (h, m) = (6, 1). For π ∈ U6 (k, l) ∩ U1 (k ′ , l′ ), we have j = l, i′ = k ′ − 1, j ′ = l′ − 1, k ′ = k + 1 = l − 1, l′ = l, i = k + 1, and rk,k+2 rk+1,k+2 = rk+1,k+2 rk,k+1 is the representation of π. Case 7: (h, m) = (2, 1). For π ∈ U2 (k, l) ∩ U1 (k ′ , l′ ), we have i = k + 1, j = l + 1, i′ = k ′ − 1, j ′ = l′ − 1, k ′ = k + 1 = l − 1, l′ = l + 1, and get the representation (17) of the permutation π = τk . Case 8: (h, m) = (7, 8). For π ∈ U7 (k, l) ∩ U8 (k ′ , l′ ), we have i = l + 1, i′ = k, j ′ = k ′ − 1 = l, l′ = j, l + 2 ≤ j ≤ n, and get the representation rk,l rl+1,j = rl+1,j rk,l of π. This gives (20) for i = l + 1. Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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Thus, the considered cases give us the representations of all permutations with three breakpoints. The cases 1–3 give the representations of the permutation πk , and the cases 4– 6 give the representations of the permutation πk−1 . Permutations πk and πk−1 differ from remaining permutations with three breakpoints presented by (17)–(20) since their internal intervals consist of one and two numbers. All remaining permutations π are distinct because γ(π) = (+, +) for π = τk (see 17) and γ(π) = (+, −), γ(π) = (−, +), γ(π) = (−, −) for (18)–(20), respectively. Permutations πk and πk−1 have two minimal representations (the cases 1,2 and the cases 4,5, respectively), and (17)–(20) are the only and hence minimal representations of distinct permutations. This implies the statement for permutations π with three breakpoints. Now we consider the representations of all permutations π with four breakpoints and use Statements 13 and 17. If π ∈ W1 (k, l) or π ∈ W2 (k, l), then there exist two reversals, namely ri,j and rk,l , each of which decreases by two the number of breakpoints and transforms π to an element of S1 . This gives rise to representations (20)–(21) of such permutations π. If π ∈ W3 (k, l) or π ∈ W4 (k, l) then there exists only one reversal, namely ri,j , which decreases by two the number of breakpoints of π. Thus, for a permutation π ∈ S2 with four breakpoints we have c2 (π) = 2, if π is represented by (20)–(21) and c1 (π) = 1, otherwise. This completes the proof.

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Theorem 7 The reversal graph Symn (R), n ≥ 3, (i) does not contain a subgraph isomorphic to K2,4 ; (ii) each of its vertices belongs to (n − 2) subgraphs isomorphic to K3,3 , 1 (iii) and to 12 (n − 3)(n − 1)(n2 + 2n + 4), n ≥ 4, subgraphs isomorphic to K2,2 which are not subgraphs of K3,3 . To prove the first fact of the theorem let us assume that the reversal graph contains K2,4 . Without loss of generality one can assume that I belongs to the smaller part of K2,4 . Hence there exist four distinct vertices from S1 which are adjacent to I. Another vertex of the smaller part belongs to S2 and it is also adjacent to the same four vertices. This contradicts Statement 18 and hence there are no subgraphs isomorphic to K2,4 in the graph Symn . From Statement 18 it also follows that there exist exactly (n−2) subgraphs K3,3 having I as one of vertices. Such subgraphs have parts {I, πk , πk−1 } and {rk,k+1 , rk,k+2 , rk+1,k+2 } for any k, 1 ≤ k ≤ n − 2. Since this graph is vertex–transitive, hence each of its vertices belongs to (n − 2) subgraphs isomorphic to K3,3 . To prove the last fact we only need to consider the total number N2 of permutations π ∈ S2 such that c2 (π) = 2. By Statement 18, there are exactly (n − 3) permutations having representations (17). The number of representations (18) equals n−3 X

n X

(l − k − 1)

k=1 l=k+3

and the number of representations (19) equals n−3 X

n X

(j − k − 1).

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Therefore, the number of representations (18)–(19) equals 2

n−3 X n−k X

(i − 1) =

k=1 i=3

n−3 X

(n2 − n − 2 − k(2n − 1) + k 2 ) =

k=1

 1 1 = (n − 3) (n − 2)(n + 1) − (2n − 1)(n − 2) + (n − 2)(2n − 5) = 2 6 

1 = (n − 3)(n − 2)(n + 2). 3
The number of representations (20)–(21) equals 2 n4 . The summation of all these numbers gives the required number N2 =

1 (n − 3)(n − 1)(n2 + 2n + 4). 12

Corollary 4 The reversal graph Symn (R) is not distance–regular and hence not distance– transitive. It follows immediately from Statement 18 since in the distance–regular graphs the numbers bi (u) and ci (u) depend only on i but not on the choice of a vertex u.

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Corollary 5 |S2 | = 61 (n4 − 2n3 − n2 − 16n + 42), n ≥ 3. Indeed, since there are N3 = 2(n−2) permutations for which c2 (π) = 3 and N2 permutations for which c2 (π) = 2, and since this graph does not contain triangles by Corollary 3, hence |S2 | = |S1 |(|S1 | − 1) − 2N3 − N2 ,
where |S1 | = n2 , and this completes the proof. The exhaustive analysis of structural properties of the reversal graph give us the following results in the reconstruction of permutations distorted by at most one or two reversal errors [33, 35]. Theorem 8 For the reversal graph Symn (R), n ≥ 3, we have N (Symn (R), 1) = 3. It follows from Corollary 3 that λ = 0, and we also have µ = 3 since maxπ∈S2 c2 (π) = 3 by Statement 18 for the reversal graph Symn (R). Hence, we get N (Symn (R), 1) = 3 by (6). Corollary 6 Any permutation π is reconstructible from any four distinct permutations in B1 (π) of the reversal graph Symn (R). Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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We denote by ti the number of sets of i distinct permutations in B1 (π) from which π
n is reconstructible and we denote by N = |B1 (π)| = 2 + 1 the number of permutations ti being at the reversal distance at most one from the permutation π. Then pi = N is the (i) probability of the event that a permutation π is reconstructible from i distinct permutations in B1 (π) under the condition that these permutations are uniformly distributed. It is evident that p1 = 0 and p4 = 1, i.e., we never can reconstruct any permutation π from a single permutation in B1 (π) and we can always reconstruct an arbitrary permutation π from four distinct permutations in B1 (π), by Corollary 6. Theorem 9 p2 ∼

1 3

and p3 → 1 as n → ∞.

t2 , where t2 is the number of sets of two distinct (N2 ) permutations in B1 (π) from which π is reconstructible.
Since the reversal graph is vertex– transitive, one can consider I instead of π. There are n2 distinct pairs (I, τ ) where τ ∈ S1 , which do not allow to reconstruct I uniquely. Moreover, a pair of permutations from S1 which are adjacent to one and the same permutation τ ∈ S2 does not allow to reconstruct 1 I. By Theorem 7, there are exactly N2 = 12 (n − 3)(n − 1)(n2 + 2n + 4) such pairs of permutations from S1 which do not belong to S2,1 (τ ) and 3(n − 2) such pairs of permutations from S1 which do belong to S2,1 (τ ), when τ = πk , 1 ≤ k ≤ n − 2. Therefore,     N n 1 − t2 = + N2 + 3(n − 2) = (n4 − 2n3 + 5n2 + 20n − 60). 2 12 2
Since N2 = 81 (n4 − 2n3 + 3n2 + 2n), we have p2 = tN2 ∼ 13 as n → ∞. (2) By the definition, we also have p3 = tN3 where t3 is the number of sets of three (3) distinct permutations in B1 (π), from which π is reconstructible. We show that there are exactly (n − 2) sets of permutations {x1 , x2 , x3 } ∈ B1 (I) such that {x1 , x2 , x3 } ∈ B1 (τ ) for some τ 6= I. By Corollary 3, the reversal graph Symn (R) does not contain triangles, hence {x1 , x2 , x3 } ∈ S1 (I) and τ ∈ S2 (I). By Statement
18, it takes place if and only if τ = πk or τ = πk−1 for some 1 ≤ k ≤ n − 2. Hence, N3 − t3 = n − 2. Since

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Indeed, by the definition, p2 =

  N n6 as n → ∞, ∼ 48 3 we get 1 − p3 ∼ n485 and hence p3 → 1 as n → ∞. As the following shows, in the case of at most two reversal errors the reconstruction of a permutation π requires many more distinct permutations in the metric ball B2 (π). Theorem 10 For the reversal graph Symn (R), n ≥ 3, we have 3 N (Symn (R), 2) ≥ (n − 2)(n + 1). 2 To prove this theorem let us recall that by (2) we have N (Symn (R), 2) =

max

π,τ ∈V (Symn (R)), π6=τ

|B2 (π) ∩ B2 (τ )|.

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Let τ = I and π ∈ S2 such that c2 (π) = 3 and π has the minimal representation (16). By Statement 18 we have that π = πk or π = πk−1 for k = 1, ..., n − 2. Now we show that for rki ,li ∈ S1 , i = 1, 2, 3, we have 3 | ∪3i=1 B1 (rki ,li )| ≤ (n − 2)(n + 1). 2 Indeed, the metric balls B1 (rki ,li ), i = 1, 2, 3, belong to B2 (π) ∩ B2 (I) and have three joint vertices I, πk , πk−1 , since Symn (R) does not contain triangles nor subgraphs isomorphic to K2,4 , by Corollary 3 and Theorem 7.
Each

of the3metric balls has size n n 2 + 1. So the required statement follows from 3 2 + 1 − 6 = 2 (n − 2)(n + 1). The inequality in Theorem 10 is attained for the permutations πk and πk−1 where reversals can be considered as transpositions (compare also with equality (10)). For example, for π = π2 = [1342] or π = π2−1 = [1423], and for r2,4 = [1432], r2,3 = [1324], r3,4 = [1243] one can check that |B2 (π) ∩ B2 (I)| = |B1 (r2,4 ) ∪ B1 (r2,3 ) ∪ B1 (r3,4 )| = 15 (see Figure 4.) 4321 r

1432 r

r 2314

2341 r

r 4132

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3214 r

r 2431

4123 r

r 1342

1234 r

2134 r

r 2413

r 2143 r

4312

r 3412 r 1423

r

r

1324

4231

r

r

1243

3421 |

r 3142

r 3241 r 4213

S0

S1

{z

}

r 3124

S2

Figure 4. The reversal graph Sym4 (R). Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

S3

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Elena Konstantinova

Now let us present a simple reconstruction algorithm for the cases when four distinct permutations are considered to reconstruct an arbitrary permutation. This algorithm also allows to determine whether given four permutations can be obtained from one and the same permutation by at most one reversal or not. In other words, given permutations x0 , x1 , x2 , x3 we determine whether there exists a permutation π for which xi ∈ B1 (π), i = 0, . . . , 3, and find this permutation π if it exists. −1 Let yi = x−1 0 xi , i = 0, . . . , 3, in particular, y0 = x0 x0 = I. We define the number m = |{i : i ∈ {1, 2, 3}, d(yi , I) = 1}| and consider the following cases. Case 1. Let m = 0. For the sought permutation τ = x−1 0 π we have τ ∈ B1 (I) and τ ∈ B1 (yi ), i = 1, 2, 3, if it exists. All I, y1 , y2 , y3 are at the distance two from each other. If B1 (I) ∩ B1 (y1 ) ∩ B1 (y2 ) ∩ B1 (y3 ) 6= ∅ then there exists the sought permutation τ, and hence π = x0 τ such that xi ∈ B1 (π) for i = 1, 2, 3. The permutation τ is unique since otherwise there exists σ ∈ B1 (I) for which |B1 (τ ) ∩ B1 (σ)| ≥ 4 that contradicts Theorem 8. Case 2. Let m = 1 and suppose d(y1 , I) = 1. Then x0 or x1 = x0 y1 might be equal π. Overwise, x0 , x1 and π form a triangle that contradicts Corollary 3. However, π 6= x0 since d(y2 , I) = d(x2 , x0 ) = 2 and d(y3 , I) = d(x3 , x0 ) = 2. Hence, π = x1 if d(y1 , y2 ) = d(y1 , y3 ) = 1, and π does not exist otherwise. Case 3. Let m ≥ 2 and suppose that d(y1 , I) = d(y2 , I) = 1. Then x0 , x1 or x2 might be equal π. However, π 6= x1 and π 6= x2 since d(y1 , y2 ) = d(x1 , x2 ) = 2. Hence, π = x0 if d(y3 , I) = 1, and π does not exist otherwise. For example, given permutations x0 = [125436], x1 = [124356], x2 = [123546], x3 = [421536] we find y1 = [124536], y2 = [125346], y3 = [421356] where d(yi , I) = 2 for all cases, and hence m = 0. So we find B1 (I) ∩ B1 (y1 ) = {[123546], [125436], [124356]}, B1 (I) ∩ B1 (y2 ) = {[125436], [124356], [123546]} and B1 (I) ∩ B1 (y3 ) = {[124356]}. Hence, τ = [124356] and π = x0 τ = [124536]. Now we consider the (unburnt) pancake graph Symn (P R) defined on the symmetric group and generated by the prefix–reversals from the set P R = {r1,j ∈ Symn , 1 < i ≤ n}, |P R| = n − 1. There is a similarity between graphs Symn (st) and Symn (P R) such that S2 (st) = S2 (P R) = K2 and S3 (st) = S3 (P R) = C6 . The distance in the pancake graph is defined as the least number of the prefix–reversals transforming one permutation into another. In computer science this graph corresponds to the n-dimensional pancake network. Symmetries of this network were considered in [40] where it was shown that it is not edge–transitive, not distance–regular and hence not distance–transitive. Some combinatorial properties of Symn (P R) are collected in the following statement. Statement 19 The pancake graph Symn (P R), n ≥ 3, is a connected (n − 1)-regular graph of order n! without cycles of lengths 3,4,5. From this statement we have that λ(Symn (P R)) = 0 and µ(Symn (P R)) = 1 since there are no triangles and quadrangles as well. It is not difficult to observe that N4 (Symn (P R), 2) = 4 for n ≥ 6, N3 (Symn (P R), 2) = 4 for n ≥ 4, N2 (Symn (P R), 2) = n for n ≥ 4 and N1 (Symn (P R), 2) = 2(n − 1) for n ≥ 4. This graph is one more example for which the inequality (7) is attained since it is (n−1)–regular graph with µ = 1. So, by (5) and (6) we get the following theorem.

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Theorem 11 For the pancake graph Symn (P R), n ≥ 4, we have N (Symn (P R), 1) = 2

and N (Symn (P R), 2) = 2(n − 1).

As one can see we have one and the same result for the star graph, for the bubble–sort graph and for the pancake graph. Thus, for each of these graphs an arbitrary permutation π is reconstructible from any three of its distinct 1–neighbors and from any (2n − 1) of its distinct 2–neighbors.

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3.5.

Cayley Graphs on the Group Hyperoctahedral Group

In this section we consider Cayley graphs on the hyperoctahedral group Bn of all permutations π σ acting on the set {±1, . . . , ±n} such that π σ (−i) = −π σ (i) for all i ∈ {1, . . . , n}. An element of Bn is a signed permutation, i.e., a permutation with a sign attached to every entry and determined by two data: |π(|i|)|, which permutes {1, . . . , n}, and the sign of π σ (i) for 1 ≤ i ≤ n. This gives a bijection between Bn and the wreath product Z2 ≀Symn of the ”sign–change” cyclic group Z2 with the symmetric group Symn ; thus |Bn | = 2n n!. We use the compact one–line notation for a signed permutation π σ as [π1 , π 2 , . . . , π i , . . . , πn ], where a bar is written over each element with a negative sign. The order–preserving map between {±1, . . . , ±n} and {1, . . . , n} gives an embedding of Bn in Sym2n ; the defining relation of Bn becomes π(2n + 1 − x) = 2n + 1 − π(x). The group Bn with the set S = {s0 , s1 , . . . , sn−1 } of generators where s0 = (1, −1) and si = (i, i + 1)(−i, −i − 1) for every 1 ≤ i ≤ n − 1 is the classical Coxeter group of type B. The presented graphs are generated by the sign–change transpositions tσij , 1 ≤ i < j ≤ n, which switches two elements i and j and their signs, e.g., [. . . , πi , . . . , π j , . . .]tσij = [. . . , πj , . . . , π i , . . .], and the sign–change ”transpositions” tσii , 1 ≤ i ≤ n, which changes the sign of the i-th element, e.g., [. . . , πi , . . .]tσii = [. . . , π i , . . .]. By analogy with the symmetric group, we also consider Cayley graphs on Bn generated by the sign–change bubble–sort transpositions tσii+1 , 1 ≤ i < n, and by the sign–change prefix–transpositions tσ1i , 1 < i ≤ n. We also consider Cayley graphs generated by the sign–change reverσ flipping the signs of elements on the segments [i, j], 1 ≤ i ≤ j ≤ n, e.g., sals ri,j σ = [. . . , π , π [. . . , πi , π i+1 , . . . , πj−1 , πj , . . .]ri,j j j−1 , . . . , πi+1 , π i , . . .], or by the sign– σ change prefix–reversals r1,i , 1 ≤ i ≤ n. For all these Cayley graphs which are not distance–regular their combinatorial properties are shown and the main results in reconstructing signed permutations from their neighbors are given. It is also shown that for some of these graphs the bounds on regular graphs are attained. 3.5.1.

The Cayley Graphs on Bn Generated by Transpositions

The transposition Cayley graph Bn (T σ ) on Bn is S generated by the sign–change transposiσ σ tions from the set T = {tii ∈ Bn , 1 ≤ i ≤ n} {tσij ∈ Bn , 1 ≤ i < j ≤ n}, |T σ | =
n+1 . The distance in this graph is defined as the least number of the sign–change trans2 positions transforming one permutation into another. The order of this graph corresponds to the order of Bn that is 2n n!. The basic facts about Bn (T σ ) are presented as follows.

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Statement 20 The transposition graph Bn (T σ ), n ≥ 2,
(i) is a connected bipartite n+1 -regular graph of order 2n n!; 2 (ii) does not contain subgraphs isomorphic to K2,3 ; (iii) each of its vertices belongs to 12 (n3 + 9n2 − 58n + 90), n ≥ 3, subgraphs isomorphic to K2,2 . All these facts are based on the properties of the signed permutations belonging to the sphere S2 = S2 (I + ), where I + = [+1, +2, · · · , +n] is the positive identity permutation on n elements. There is no a signed permutation π σ ∈ S2 for which c2 (π σ ) = 3 and this means that there are no subgraphs isomorphic to K2,3 in Bn (T σ ). On the other hand, c2 (π σ ) = 2 for a signed permutation π σ ∈ S2 if and only if π σ has one of the following representations tσi,i tσj,j = tσj,j tσi,i ,

1 ≤ i < j ≤ n,

(22)

tσk,k tσi,j = tσi,j tσl,l ,

1 ≤ i < j ≤ n, k = i, l = j or k = j, l = i,

(23)

tσk,k tσi,j = tσi,j tσk,k ,

1 ≤ i < j ≤ n, k 6= i and k 6= j, 1 ≤ k ≤ n,

(24)

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tσi,j tσk,l = tσi,j tσk,l , for some i, j, k, l. (25)
So, there are exactly 3 n2 signed permutations having representations (22) and (23);


the number of representations (24) equals n n−1 and there are 3 n−2 + 6 n−3 signed 2 2 2 permutations for which (25) holds. The required number in Statement 20 is obtained by the summation of all these numbers. Thus, by this statement there are no triangles in Bn (T σ ) and we have λ(Bn (T σ )) = 0. Moreover, since there are no subgraphs isomorphic to K2,3 in Bn (T σ ), hence we have µ(Bn (T σ )) = 2 and by (6) the following result holds. Theorem 12 For the transposition graph Bn (T σ ), n ≥ 2, we have N (Bn (T σ ), 1) = 2. Corollary 7 Any signed permutation π σ is reconstructible from any three distinct signed permutations in B1 (π σ ) of the transposition graph Bn (T σ ). By direct analysis and counting Ni (Bn (T σ ), 2) for i = 1, . . . , 4, (see also (5)) it can be shown that N (Bn (T σ ), 2) = N2 (Bn (T σ ), 2) = n(n + 1) and we have the following result. Theorem 13 For the transposition graph Bn (T σ ), n ≥ 2, we have N (Bn (T σ ), 2) = n(n + 1). Thus, in the transposition graph Bn (T σ ) an arbitrary signed permutation π σ is reconstructible from any three of its distinct 1–neighbors and from any (n2 + n + 1) of its distinct 2–neighbors. Now we consider the bubble–sort graph Bn (tσ ) generated by the sign–change transpositions from the set tσ = {tσi,i ∈ Bn , 1 ≤ i ≤ n}∪{tσi,i+1 ∈ Bn , 1 ≤ i < n}, |tσ | = 2n−1. These sign–change transpositions from the set tσ determine the graph distance in Bn (tσ ) in the usual way and the graph has the following properties.

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Statement 21 The bubble–sort graph Bn (tσ ), n ≥ 2, (i) is a connected bipartite (2n − 1)-regular graph of order 2n n!; (ii) does not contain subgraphs isomorphic to K2,3 ; (iii) each of its vertices belongs to (2n2 − 4n + 3), n ≥ 3, subgraphs isomorphic to K2,2 . It is clear that λ(Bn (tσ )) = 0 since this graph is bipartite. There does not exist a signed permutation π σ ∈ S2 such that c2 (π σ ) = 3, hence there are no subgraphs isomorphic to K2,3 in Bn (tσ ). The number of signed permutations π σ ∈ S2 for which c2 (π σ ) = 2 in Bn (tσ ) is calculated by formulas (22)–(25) taking into account that j
= i + 1
in (23) and (24), and j = i+1, l = k+1, k 6= i+1 in (25). From this, there are n2 + n−2 signed 2 permutations represented by (22) and (25). The numbers of representations (23) and (24) equal 2(n − 1) + (n − 2)(n − 1). These gives the total number (2n2 − 4n + 3) of subgraphs isomorphic to K2,2 in Bn (tσ ) for n ≥ 3. From the above consideration, µ(Bn (tσ )) = 2. It can be also shown that N2 (Bn (tσ ), 2) = 2(2n − 1) is the maximum value among Ni (Bn (tσ ), 2) for any i = 1, . . . , 4. Thus, by (5) and (6) we get the following two results. Theorem 14 For the bubble–sort graph Bn (tσ ), n ≥ 2, we have

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N (Bn (tσ ), 1) = 2

and N (Bn (tσ ), 2) = 2(2n − 1).

Hence, in the bubble–sort graph Bn (tσ ) an arbitrary signed permutation π σ is reconstructible from any three of its distinct 1–neighbors and from any (4n − 1) of its distinct 2–neighbors. This graph is an example for which the inequality (8) is attained. Now let us consider the star graph Bn (stσ ) which is generated by the sign–change transpositions from the set stσ = {tσi,i ∈ Bn , 1 ≤ i ≤ n} ∪ {tσ1,i ∈ Bn , 1 < i ≤ n}, |stσ | = 2n − 1. The properties of this graph are similar to the properties of the bubble–sort graph and they are presented by the following statements. Statement 22 The star graph Bn (stσ ), n ≥ 2, (i) is a connected bipartite (2n − 1)-regular graph of order 2n n!; (ii) does not contain subgraphs isomorphic to K2,3 ; (iii) each of its vertices belongs to 23 n(n − 1), n ≥ 3, subgraphs isomorphic to K2,2 . We have that λ(Bn (stσ )) = 0 since this graph is bipartite. There are no subgraphs isomorphic to K2,3 in Bn (stσ ) since there does not exist a signed permutation π σ ∈ S2 such that c2 (π σ ) = 3. The number of signed permutations π σ ∈ S2 for which c2 (π σ ) = 2 in Bn (tσ ) is calculated by formulas (22)–(24) when i = 1, k = 1, l = j or i = 1, l = 1, k = j in (23), and i = 1,
k 6= 1, in (24); formula (25) doesn’t hold in this case. So, these gives the total number n2 + 2(n − 1) + (n − 2)(n − 1) = 23 n(n − 1) of subgraphs isomorphic to K2,2 in Bn (stσ ). Hence, µ(Bn (stσ )) = 2. Moreover, again we have that N2 (Bn (stσ ), 2) = 2(2n − 1) gives the least value among Ni (Bn (stσ ), 2) for any i = 1, . . . , 4. So, by (5) and (6) we get the following two results.

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Theorem 15 For the star graph Bn (stσ ), n ≥ 2, we have

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N (Bn (stσ ), 1) = 2

and N (Bn (stσ ), 2) = 2(2n − 1).

So, in the star graph Bn (stσ ) an arbitrary signed permutation π σ is reconstructible from any three of its distinct 1–neighbors and from any (4n − 1) of its distinct 2–neighbors. This graph is one more example for which the inequality (8) is attained. Now let us give some remarks concerning the situation when the Cayley graph on the group Bn is generated only by the sign–change transpositions tσi,j , when i 6= j (the sign– change ”transpositions” tσi,i , 1 ≤ i ≤ n are not considered). What are the properties of the corresponding Cayley graphs? These graphs are disconnected and the number of connected components for these graphs depends on their generating sets. For example, if all transpositions tσi,j , 1 ≤ i < j ≤ n, are considered as the set of generators, then the corresponding
Cayley graph has two connected bipartite n2 –regular components of order 2n−1 n!. These components are isomorphic to each other. Each of these connected components represents a subgroup of Bn and by symmetry these subgroups are isomorphic. The even–signed permutation group Dn (for details about groups see [9]) which is the normal subgroup of Bn of index 2 whose elements are signed permutations with even numbers of negative elements is one of these subgroups for this Cayley graph. In the case, when the sign–change bubble–sort transpositions tσi,i+1 , where 1 ≤ i < n, are considered as the set of generators, the corresponding Cayley graph has 2n connected bipartite (n − 1)–regular components of order n!. Each of these connected components represents a subgroup of Bn isomorphic to Symn and these components are isomorphic to the bubble–sort graph Symn (t) (see Section 3.4.1.). Thus, each of these components has the same properties as the graph Symn (t). The similar situation appears when the sign–change prefix–transpositions tσ1,i , where 1 ≤ i < n, are considered as the set of generators. In this case, the corresponding Cayley graph has 2n connected bipartite (n − 1)–regular components of order n! each of which is isomorphic to the star graph Symn (st) (see Section 3.4.1.) and has the same properties as Symn (st). It is easily seen, that in the last two cases three distinct 1–neighbors are required to reconstruct an arbitrary signed permutation since there is the correspondence between Cayley graphs on the group Symn and connected components of Cayley graphs on the group Bn . 3.5.2.

The Cayley Graphs on Bn Generated by Reversals

As it was shown at the beginning of Section 3.5., the order–preserving map between {1, . . . , n} and {±1, . . . , ±n} gives an embedding of Bn in Sym2n . We use this fact and define a one–to–one correspondence between all 2n n! signed permutations on n elements and some permutations on 2n elements which form a subgroup Symσ2n of the group Sym2n . Then we define the reversal Cayley graph Bn (Rσ ) on the group Bn which is genσ , 1 ≤ i ≤ j ≤ n, from a set Rσ . This graph is erated by the sign–change reversals ri,j isomorphic to the reversal Cayley graph Symσ2n on the group Symσ2n which is generated by so–called even reversals from the set ER. The combinatorial properties of the graph Bn (Rσ ) ∼ = Symσ2n (ER) are investigated to present the main result on the vertex reconstruction problem for the signed permutations distorted by sign–change reversals. We use this approach to show the connection between the symmetric group Sym2n and the group Bn of signed permutations.

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We start with some additional definitions and notations form [36]. For a signed permutation π σ = [π1 , π 2 , . . . , π i , . . . , πn ], where a bar is written over each element with a negative sign, we also will use in this section the notation π σ = [σ1 π1 , . . . , σn πn ], where π = [π1 , ..., πn ] ∈ Symn and σ = (σ1 , ..., σn ), σi ∈ {+, −}, i = 1, ..., n. For example, for the permutation π σ = [234165] we also have π σ = [−2, +3, +4, −1, +6, −5] where π = [2, 3, 4, 1, 6, 5] and σ = (−, +, +, −, +, −). Now let us consider the symmetric group Sym2n of permutations on 2n elements. For any fixed i, 1 ≤ i ≤ n, the elements of the set {2i − 1, 2i} are called the vicinal numbers. Let us denote by Symσ2n the subset of permutations P of Sym2n which transform any set of vicinal numbers to a set of vicinal numbers. This means that for any P ∈ Symσ2n and i, 1 ≤ i ≤ n, there exists h, 1 ≤ h ≤ n, such that P (2i − 1) = 2h − 1 and P (2i − 1) = 2h

and

P (2i) = 2h

or

P (2i) = 2h − 1

and these h are distinct for distinct i. We define a one-to-one correspondence between Bn and Symσ2n by the following way. For any signed permutation π σ ∈ Bn we consider a permutation P ∈ Sym2n such that for any i, 1 ≤ i ≤ n, we have

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P (2i − 1) = 2π(i) −

σi + 1 2

and

P (2i) = 2π(i) +

σi − 1 . 2

(26)

For any σi = ±1, 1 ≤ i ≤ n, the numbers 2π(i) − σi2+1 and 2π(i) + σi2−1 are vicinal numbers, hence P ∈ Symσ2n . Moreover, this mapping is a one-to-one mapping since (26) defines a pair of vectors π and σ in a unique way. Let T = P Q ∈ Sym2n be the product of the permutations P, Q ∈ Sym2n such that T (i) = P (Q(i)), 1 ≤ i ≤ 2n, and let us show that Symσ2n is a subgroup of the group Sym2n . Statement 23 P Q ∈ Symσ2n for any P, Q ∈ Symσ2n . To prove this statement we let T = P Q where P = (..., 2π(i) −

σi − 1 σi + 1 , 2π(i) + , ...) ∈ Symσ2n , 2 2

γj + 1 γj − 1 , 2̺(j) + , ...) ∈ Symσ2n . 2 2 Now we check that for any j, 1 ≤ j ≤ n, we have Q = (..., 2̺(j) −

T (i) = 2π(̺(j)) −

σ̺(j) γj + 1 , 2

when i = 2j − 1,

σ̺(j) γj − 1 , when i = 2j. 2 From these formulas it will immediately follow that T ∈ Symσ2n . We consider two cases. T (i) = 2π(̺(j)) +

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(27) (28)

170

Elena Konstantinova Let i = 2j − 1. Then by (26) we have T (i) = P (Q(i)) = P (Q(2j − 1)) = P (2̺(j) − If γj = +1 then Q(2j − 1) = 2̺(j) − 1 and

γj +1 2 ).

P (Q(2j − 1)) = P (2̺(j) − 1) = 2π(̺(j)) −

σ̺(j) + 1 , 2

that corresponds to (27). If γj = −1 then Q(2j − 1) = 2̺(j) and P (Q(2j − 1)) = P (2̺(j)) = 2π(̺(j)) +

σ̺(j) − 1 , 2

that also corresponds (27). Now let i = 2j. Then by (26) we have T (i) = P (Q(i)) = P (Q(2j)) = P (2̺(j) + γj −1 2 ). If γj = +1 then Q(2j) = 2̺(j) and P (Q(2j)) = P (2̺(j)) = 2π(̺(j)) +

σ̺(j) − 1 , 2

that corresponds to (28). If γj = −1 then Q(2j) = 2̺(j) − 1 and

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P (Q(2j)) = P (2̺(j) − 1) = 2π(̺(j)) −

σ̺(j) + 1 , 2

that also corresponds to (28). The considered cases complete the proof. It follows from this statement that Symσ2n is a subgroup of the group Sym2n since I2n ∈ Symσ2n , where I2n is the identity permutation on 2n elements. We denote by (P )−1 the inverse of P ∈ Symσ2n and have P (P )−1 = (P )−1 P = I2n . For any T, P, Q ∈ Symσ2n we also have (T P )Q = T (P Q) since Symσ2n forms a subgroup of the group Sym2n . An even reversal r2i−1,2j , 1 ≤ i ≤ j ≤ n, is the operation of reversing segments [2i − 1, 2j] of a permutation from Sym2n , e.g., [. . . , π2i−2 , π2i−1 , π2i , . . . , π2j−1 , π2j , π2j+1 , . . .]r2i−1,2j = = [. . . , π2i−2 , π2j , π2j−1 , . . . , π2i , π2i−1 , π2j+1 , . . .]. Note that r2i−1,2j ∈ Symσ2n , r2i−1,2j r2i−1,2j = I2n and (r2i−1,2j )−1 = r2i−1,2j . It is easy to check that an even reversal on a permutation P ∈ Symσ2n corresponds to a sign– σ on a permutation π σ ∈ B , where change reversal ri,j n σ = [. . . , π j , π j−1 , . . . , πi+1 , π i , . . .]. [. . . , πi , π i+1 , . . . , πj−1 , πj , . . .]ri,j

The reversal distance d(P, Q) between two permutations P, Q ∈ Symσ2n is the least number d of even reversals needed to transform P to Q, i.e., P r2i1 −1,2j1 . . . r2id −1,2jd = Q. It was shown by Knuth in Exercise 5.1.4-43 in [39] that at most (n + 1) sign–change reversals are needed to sort any signed permutation to the identity permutation, for all n > 3. Since there is a one–to–one correspondence between Bn and Symσ2n and a sign–change reversal on Bn corresponds to an even reversal on Symσ2n , we have max

P ∈Symσ 2n

d(P, I2n ) = n + 1

for n > 3.

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In the remainder of this section it is assumed that P = [π0 , π1 , π2 , ..., π2i−1 , π2i , ..., π2n−1 , π2n , π2n+1 ], where π0 = 0 and π2n+1 = 2n + 1. By analogy with the permutations from the group Symn , we say that a permutation P ∈ Symσ2n has a breakpoint between positions 2i and 2i + 1 for i ∈ {0, . . . , n}, if |αi (P )| ≥ 2 where αi = αi (P ) = π2i+1 − π2i ,

i = 0, ..., n.

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Denote by b(P ) the number of breakpoints of P . Note that b(P ) = 0 if and only if P = I, and b(P ) ≥ 2 otherwise. If a permutation P has b = b(P ) ≥ 2 breakpoints between positions 2ih and 2ih +1 respectively, 0 ≤ ih ≤ n, h = 1, . . . , b, then the interval [0, 2n+1] is partitioned into b − 1 internal monotonicity intervals [2ih + 1, 2ih+1 ], h = 1, . . . , b − 1, consisting of an even number of successive integers in decreasing or increasing order and two external monotonicity intervals [0, 2i1 ] and [2ib +1, 2n+1] consisting of an odd number of successive integers in increasing order (the external intervals have only one number when i1 = 0 or ib = n respectively). We define the vector β(P ) = (β1 , ..., βb )(the up– and down–sequence of monotonicity of breakpoints), where  + if π2ih < π2ih +1 , h = 1, ..., b. βh = − if π2ih > π2ih +1 . Note that 0 ≤ π2i1 < π2i1 +1 and π2ib < π2ib +1 ≤ 2n + 1, hence β1 = βb = +. We also define the vector γ(P ) = (γ1 , . . . , γb−1 ) (the up– and down–sequence of monotonicity of internal intervals), where  + if π2ih+1 > π2ih +1 , h = 1, ..., b − 1. γh = − if π2ih+1 < π2ih +1 . The vectors β(P ) and γ(P ) are uniquely defined by the permutation P , and two permutations are distinct if at least one of these vectors differ. As an example, for the permutation π σ = [+1, −3, −2, +4, −5] we have the permutation P = [0, 1, 2, 6, 5, 4, 3, 7, 8, 10, 9, 11] for which the interval [0, 11] is partitioned into 5 monotonicity intervals [0, 2], [3, 6], [7, 8], [9, 10], [11], and we have b(P ) = 4, β(P ) = (+, +, +, +), and γ(P ) = (−, +, −). To estimate the change of the number of breakpoints as a result of applying an even reversal r2i−1,2j to P ∈ Symσ2n we use the function δ(x, y) defined in Section 3.4.2. as a function of two integer variables x and y such that δ(x, y) = 1 if |x−y| ≥ 2 and δ(x, y) = 0 if |x − y| ≤ 1. Statement 24 Let P = [π0 , π1 , ..., π2n , π2n+1 ] ∈ Symσ2n and r2i−1,2j is an even reversal, 1 ≤ i ≤ j ≤ n. Then b(P r2i−1,2j ) − b(P ) = δ(π2i−2 , π2j ) + δ(π2i−1 , π2j+1 ) −δ(π2i−2 , π2i−1 ) − δ(π2j , π2j+1 ). Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

172

Elena Konstantinova

Indeed, an even reversal r2i−1,2j can only change the number of breakpoints between positions 2i − 2 and 2i − 1, and between positions 2j and 2j + 1. This completes the proof. Now let us find b(r2k−1,2l r2i−1,2j ), 1 ≤ i ≤ j ≤ n, 1 ≤ k ≤ l ≤ n, i 6= k, j 6= l, under the condition δ(π2i−2 , π2i−1 ) = δ(π2j , π2j+1 ) = 0, that is |αi−1 | = |αj | = 1. Statement 25 If |αi−1 | = |αj | = 1, where 1 ≤ i ≤ j ≤ n, then b(r2k−1,2l r2i−1,2j ) = 4. Let P = [π0 , π1 , ..., π2n , π2n+1 ] = r2k−1,2l and hence b(P ) = 2. The condition |αi−1 | = |αj | = 1, where 1 ≤ i ≤ j ≤ n, holds for the following cases within the monotonicity intervals. Since πh = h when 0 ≤ h ≤ 2k − 2 or 2l + 1 ≤ h ≤ 2n + 1 and πh = 2l + (2k − 1) − h when 2k − 1 ≤ h ≤ 2l, we have αi−1 = π2i−1 − π2i−2 = 1, if 1 ≤ i ≤ k − 1 or l + 2 ≤ i ≤ n, and αi−1 = −1, if k + 1 ≤ i ≤ l, and αk−1 = αl = 2l − 2k + 2 ≥ 2. Analogously, we have αj = π2j+1 − π2j = 1, if 1 ≤ j ≤ k − 2 or l + 1 ≤ j ≤ n, and αj = −1, if k ≤ j ≤ l − 1. Thus, from these the condition of statement doesn’t hold for i = k or j = l. Now show that for i 6= k and j 6= l we have b(r2k−1,2l r2i−1,2j ) = 4. As it was mentioned above, the condition |αi−1 | = |αj | = 1 means δ(π2i−2 , π2i−1 ) = δ(π2j , π2j+1 ) = 0 and by Statement 24 we have b(r2k−1,2l r2i−1,2j ) = b(r2k−1,2l ) + δ(π2i−2 , π2j ) + δ(π2i−1 , π2j+1 ) =

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= δ(π2i−2 , π2j ) + δ(π2i−1 , π2j+1 ) + 2. Moreover, we have each of the equalities |π2i−2 − π2j | = 1 or |π2i−1 − π2j+1 | = 1 if and only if i = k and j = l. Therefore, for all other cases we have |π2i−2 − π2j | ≥ 2 and |π2i−1 − π2j+1 | ≥ 2, and hence, δ(π2i−2 , π2j ) = 1 and δ(π2i−1 , π2j+1 ) = 1, this means that b(r2k−1,2l r2i−1,2j ) = 4. The reversal graph Bn (Rσ ) is defined on the group Bn and generated by the
sign– σ ∈ B , 1 ≤ i ≤ j ≤ n}, |Rσ | = n+1 . The change reversals from the set Rσ = {ri,j n 2 distance in this graph coincides with the reversal distance between two signed permutations. Hence, the diameter of this graph is (n + 1) (see [39]) and the following permutations, written in one–line notation, are at this maximum distance from the positive identity permutation I + :  +n, +(n − 1), . . . , +1, if n is even, σ π = +2, +1, +3, +n, +(n − 1), . . . , +4, if n > 3 is odd. The following statement collects some properties of this graph. Statement 26 The reversal graph Bn (Rσ ), n ≥ 2, is
n+1 (i) a connected 2 -regular graph of order 2n n! and diameter n + 1; (ii) not edge–transitive, not distance–regular and hence not distance–transitive. Figure 5 presents the reversal graph B3 (Rσ ). As before, we consider the spheres Si = Si (I + ) for i ∈ {0, . . . , d} where d is the diameter. The presented graph has the diameter 3 (Knuth’s result is true for n > 3 when n is odd) and there are no edges between vertices of the sphere S3 .

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123 q

173

q 123 q 213

123 q

123 q 213 q

q 132 q 321 q 213

132 q 123 q

321 q

q 132 q 321 q 231

123 q 213 q

q 312 q 312 q 231

123 q 132 q

q 213 q 132

123 q q

213

q

321

q 321

q

q 231

213

q 312

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q

132 q

132

q 312

q

q 231

321

q 231

q

231

q 312

q

q 312

312

q 231

q

321 q

312

q 321

q

q 132

231 S0

S1

S2

q 213

S3

Figure 5. The reversal graph B3 (Rσ ) (without edges between vertices of the sphere S3 ).

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174

Elena Konstantinova

We also present the graph Symσ2n (ER) on the group Symσ2n generated by the
even n+1 σ reversals from the set ER = {r2i−1,2j ∈ Sym2n , 1 ≤ i ≤ j ≤ n}, |ER| = 2 . As it was shown early there is one–to–one correspondence between Bn and Symσ2n , and between the sign–change reversals from the set Rσ and the even reversals from the set ER. So, we have Bn (Rσ ) ∼ = Symσ2n (ER) and one can investigate the properties only one of them. Let us consider the graph Symσ2n (ER) and let us describe its spheres S1 and S2 and connections between them. As it was shown in Section 3.4.2. for the reversal graph Symn (R) on the symmetric group, we have to know these properties to get the values (2) when r = 1, 2. So, the investigation of the structural properties of the graph Symσ2n (ER) is our main goal now. It is clear that S1 = ER and by Statement 24 any even reversal has two breakpoints. The next two technical statements give us the full description of permutation P ∈ S2 , having three or four breakpoints. Statement 27 Let P = r2k−1,2l r2i−1,2j ∈ S2 such that b(P ) = 3 and r2k−1,2l 6= r2i−1,2j for fixed k and l, 1 ≤ k ≤ l ≤ n, and any i and j, 1 ≤ i ≤ j ≤ n. Then P belongs to one of the following disjoint sets Uh (k, l), h = 1, . . . , 6, defined as follows: 1. If P ∈ U1 (k, l) then i = k, k ≤ j ≤ l − 1 and γ(P ) = (+, −); 2. If P ∈ U2 (k, l) then i = k, l + 1 ≤ j ≤ n and γ(P ) = (−, +); 3. If P ∈ U3 (k, l) then 1 ≤ i ≤ k − 1, j = l and γ(P ) = (+, −); 4. If P ∈ U4 (k, l) then k + 1 ≤ i ≤ l, j = l and γ(P ) = (−, +); 5. If P ∈ U5 (k, l) then i = l + 1, l + 1 ≤ j ≤ n and γ(P ) = (−, −);

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6. If P ∈ U6 (k, l) then 1 ≤ i ≤ k − 1, j = k − 1 and γ(P ) = (−, −); Indeed, if the condition |αi−1 | = |αj | = 1, where 1 ≤ i ≤ j ≤ n, of Statement 25 is not satisfied, then we have i = k or i = l + 1, when |αi−1 | = |π2i−1 − π2i−2 | ≥ 2, and we have j = k − 1 or j = l, when |αj | = |π2j+1 − π2j | ≥ 2. These four cases i = k, i = l + 1, j = k − 1, j = l are incompatible, excepting the cases i = k and j = l when we have P = r2k−1,2l r2i−1,2j = I2n . The case i = k gives rise to the set U1 (k, l), when k ≤ j ≤ l − 1, and gives rise to the set U2 (k, l), when l + 1 ≤ j ≤ n. Analogously, the case j = l gives rise to the set U3 (k, l), when 1 ≤ i ≤ k − 1, and gives rise to the set U4 (k, l), when k + 1 ≤ i ≤ l. The case i = l + 1 is possible only if l + 1 ≤ j ≤ n and we obtain the set U5 (k, l). Analogously, the case j = k − 1 is possible only if 1 ≤ i ≤ k − 1 and we obtain the set U6 (k, l). It is easily seen that permutations P of all six sets have three breakpoints. The sets Uh (k, l), h = 1, ..., 6, are disjoint since the even reversals r2i−1,2j are distinct for all cases. These sets consist of permutations P with β(P ) = (+, −, +) in the first four cases and with β(P ) = (+, +, +) in the last two cases. Note that, from Statement 25 and the proof of Statement 27 we have that any permutation P = r2k−1,2l r2i−1,2j where r2k−1,2l 6= r2i−1,2j has three or four breakpoints. This also means that there are no edges between vertices of the sphere S1 which all have two breakpoints, and this implies that the graph Symσ2n (ER) does not contain triangles. Moreover, since Bn (Rσ ) ∼ = Symσ2n (ER) we have the following statement.

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Corollary 8 The reversal graph Bn (Rσ ) does not contain triangles. Statement 28 Let P = r2k−1,2l r2i−1,2j ∈ S2 such that b(P ) = 4 and r2k−1,2l 6= r2i−1,2j for fixed k and l, 1 ≤ k ≤ l ≤ n, and any i and j, 1 ≤ i ≤ j ≤ n. Then P belongs to one of the following disjoint sets Wh (k, l), h = 1, . . . , 4, defined as follows: 1. If P ∈ W1 (k, l) then k + 1 ≤ l + 1 < i ≤ j or i + 1 ≤ j + 1 < k ≤ l; β(P ) = (+, +, +, +) and γ(P ) = (−, +, −); 2. If P ∈ W2 (k, l) then k < i ≤ j < l or i < k ≤ l < j; β(P ) = (+, −, −, +) and γ(P ) = (−, +, −); 3. If P ∈ W3 (k, l) then i ≤ k − 1 < j < l; β(P ) = (+, −, +, +) and γ(P ) = (+, −, −); 4. If P ∈ W4 (k, l) then k < i < l + 1 ≤ j; β(P ) = (+, +, −, +) and γ(P ) = (−, −, +).

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Since any permutation P = r2k−1,2l r2i−1,2j where r2k−1,2l 6= r2i−1,2j has three or four breakpoints, then these statements are proved by considering all cases to arrange numbers i, j, 1 ≤ i ≤ j ≤ n, in accordance to the fixed numbers k, l, 1 ≤ k ≤ l ≤ n, and by excluding the cases i = k, i = l + 1, j = k − 1, j = l of Statement 27. Thus, Statements 27 and 28 give us the sets of all permutations P belonging to the sphere S2 . Our next goal is to consider the connections between spheres S1 and S2 , and find c2 (P ) = c2 (P, I2n ) for any P ∈ S2 . To formulate the next result we fix a permutation P ∈ S2 and consider the lexicographic ordering on the permutations r2k−1,2l ∈ S1 , 1 ≤ k ≤ l ≤ n, assuming that r2k−1,2l < r2k′ −1,2l′ if k ≤ k ′ and l < l′ . Any pair of edges {r2k−1,2l , P } and {r2k′ −1,2l′ , P } of the graph Symσ2n (ER) implies the following expression for P : r2k−1,2l r2i−1,2j = r2k′ −1,2l′ r2i′ −1,2j ′

(29)

where r2i−1,2j = r2k−1,2l P and r2i′ −1,2j ′ = r2k′ −1,2l′ P . We say that (29) is a representation of P if r2k−1,2l < r2k′ −1,2l′ , and it is a minimal representation of P if r2k−1,2l is minimal in the lexicographic order permutation of the set S2,1 (P ) = {x ∈ S1 : d(x, P ) = 1}. Note that if P has only one representation then this representation is minimal and c2 (P ) = 2. In a general case, if P has h minimal representations with h = 0, 1, . . ., then c2 (P ) = h + 1. Statement 29 Given a permutation P ∈ S2 let k, 1 ≤ k ≤ n − 1, be the minimal integer such that r2k−1,2s ∈ S2,1 (P ) for some s, k < s ≤ n. Then 1. c2 (P ) = 2, if P has one of the following representations r2k−1,2l r2k−1,2j = r2(k+l−j)−1,2l r2k−1,2l ,

k ≤ j ≤ l − 1;

(30)

r2k−1,2l r2k−1,2j = r2k−1,2j r2(k+j−l)−1,2j ,

l + 1 ≤ j ≤ n;

(31)

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Elena Konstantinova k ≤ l < i ≤ j;

(32)

r2k−1,2l r2i−1,2j = r2(l−j+k)−1,2(l−i+k) r2k−1,2l , k < i ≤ j < l;

(33)

r2k−1,2l r2i−1,2j = r2i−1,2j r2k−1,2l ,

2. c2 (P ) = 1, in the remaining cases. Let us assume that c2 (P ) > 1 and r2k−1,2s ∈ S2,1 (P ) for some integer s. Then the permutation P has a representation (29) with the minimal integer k. To prove this statement we describe all such representations of P , determine which of them are minimal and then find c2 (P ). If P has a representation (29) then by Statements 27 and 28, we have P ∈ Uh (k, l) ∩ Um (k ′ , l′ ) when b(P ) = 3, and P ∈ Wh (k, l) ∩ Wm (k ′ , l′ ) when b(P ) = 4, for some h and m. First we consider representations of permutations with three breakpoints. Note that for P ∈ U1 and P ∈ U3 we have γ(P ) = (+, −), and for P ∈ U2 and P ∈ U4 we have γ(P ) = (−, +). Therefore,
(U1 (k, l) ∪ U3 (k, l)) ∩ U2 (k ′ , l′ ) ∪ U4 (k ′ , l′ ) = ∅. Since γ(P ) = (−, −), when P ∈ U5 ∪ U6 , by the same argument, we have
(U5 (k, l) ∪ U6 (k, l)) ∩ ∪4m=1 Um (k ′ , l′ ) = ∅. Moreover, Uh (k, l) ∩ Uh (k ′ , l′ ) = ∅ for any h = 1, ..., 6,

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since k and l are uniquely defined by any element of P ∈ Uh (k, l). These arguments show that if P with b(P ) = 3 has a representation (29) and P ∈ Uh (k, l)∩Um (k ′ , l′ ), then (h, m) or (m, h) must belong to the set A = {(1, 3), (2, 4), (5, 6)} We shall prove that if r2k−1,2l ∈ S2,1 (P ) then there exists a unique r2k′ −1,2l′ such that P ∈ Uh (k, l) ∩ Um (k ′ , l′ ) with (h, m) ∈ A, and P has the representation (29) with r2k−1,2l < r2k′ −1,2l′ . It will follow from our proof that for cases (m, h) ∈ A we also have an expression (29) but it is not a representation of P since in these cases R2k−1,2l < r2k′ −1,2l′ does not hold. Case 1: (h, m) = (1, 3). For P ∈ U1 (k, l) ∩ U3 (k ′ , l′ ), we have i = k, k ≤ j ≤ l − 1, i′ = k, j ′ = l = l′ , k ′ = l + k − j and get the representation (30). Case 2: (h, m) = (2, 4). For P ∈ U2 (k, l) ∩ U4 (k ′ , l′ ), we have i = k, l + 1 ≤ j ≤ n, j ′ = l′ = j, k ′ = k, i′ = k ′ + l′ − l = k + j − l and get the representation (31). Case 3: (h, m) = (5, 6). For P ∈ U5 (k, l) ∩ U6 (k ′ , l′ ), we have i = l + 1, i′ = k, j ′ = k ′ − 1 = l, l′ = j, i = l + 1 ≤ j ≤ n and get the representation (32). Thus, (30)–(32) give the representations of all permutations with three breakpoints. Moreover, (30)–(32) are unique and hence minimal representations of distinct permutations. This implies the statements for permutations P with three breakpoints. Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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Now we use Statements 24 and 28 to consider the representations of all permutations with four breakpoints. If P ∈ W1 (k, l) or P ∈ W2 (k, l), then there exist two even reversals, namely r2i−1,2j and r2k−1,2l , which decrease by two the number of breakpoints and transform P to an element of S1 . This gives rise to representations (32)–(33) of such permutations P . If P ∈ W3 (k, l) or P ∈ W4 (k, l) then there exists only one even reversal, namely r2i−1,2j , which decreases by two the number of breakpoints of P . Thus, for a permutation P ∈ S2 with four breakpoints we have c2 (P ) = 2, if P is represented by (32)–(33), and c2 (P ) = 1, otherwise. This completes the proof. This statement can be easily reformulated for the reversal graph Bnσ on signed permutations π σ ∈ S2 for which c2 (π σ ) = 2 as follows. Let us recall that S2,1 (π σ ) = {x ∈ S1 : σ ∈ S , 1 ≤ k ≤ l ≤ n. d(x, π σ ) = 1} and rk,l 1 Statement 30 Given a permutation π σ ∈ S2 let k, 1 ≤ k ≤ n − 1, be the minimal integer σ ∈ S (π σ ) for some s, k < s ≤ n. Then such that rk,s 2,1 1. c2 (π σ ) = 2, if π σ has one of the following representations σ rσ = rσ σ rk,l k,j k+l−j,l rk,l ,

k ≤ j ≤ l − 1;

σ rk,l σ rk,l σ rk,l

2. c2

σ = rσ rσ rk,j l + 1 ≤ j ≤ n; k,j k+j−l,j , σ σ σ ri,j = ri,j rk,l , k ≤ l < i ≤ j; σ σ σ ri,j = rl−j+k,l−i+k rk,l , k < i ≤ j < l; (π σ ) = 1, in the remaining cases.

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From this statement we have the following results. Theorem 16 The reversal graph Bn (Rσ ), n ≥ 3, (i) does not contain subgraphs isomorphic to K2,3 ; 1 (ii) each of its vertices belongs to 12 (n − 1)n(n + 1)(n + 4) subgraphs isomorphic to K2,2 . Let us assume that the reversal graph Bn (Rσ ) contains subgraphs isomorphic to K2,3 . Without loss of generality one can assume that I + belongs to the smaller part of K2,3 . Hence there exist three distinct vertices from S1 which are adjacent to I + . Another vertex of the smaller part belongs to S2 and it is also adjacent to the same three vertices. That contradicts to Statement 30 and proves the statement that there are no subgraphs isomorphic to K2,3 in Bn (Rσ ). To prove the second statement, we only need to find the total number N2 of the permutations π σ ∈ S2 for which c2 (π σ ) = 2. By Statements 29 and 30, this number coincides with the number of all representations (30)-(33). Thus, the number of representations (30) equals n−1 n n−1 X X X n−k X (l − k) = h k=1 l=k+1

k=1 h=1

and the number of representations (31) equals n−1 X n−1 X

n−1 X n−k X

k=1 l=k

k=1 h=1

(n − l) =

h.

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Therefore, the number of representations (30)–(31) of permutations π σ equals 2

n−1 X n−k X

h=

k=1 h=1

n−1 X

(n2 − 2nk + k 2 + n − k) =

k=1

n(n − 1)(n + 1) . 3



The number of representations (32)–(33) equals n2 when k = l < i = j; equals
n3 in each of three cases k = l < i < j, k < l < i = j, k < i = j < l, and equals 2 n4 in the cases when all i, j, k, l differ. Therefore, the number of representations (32)–(33) equals       n n n (n − 1)n2 (n + 1) +3 +2 = . 2 3 4 12 So, the total number N2 follows from N2 =

n(n − 1)(n + 1) (n − 1)n2 (n + 1) + = 3 12 =

(n − 1)n(n + 1)(n + 4) . 12

Corollary 9

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|S2 | =

(n − 1)n(n + 1)2 . 6

By Theorem 16 there are
N2 permutations π σ ∈ S2 for which c2 (π σ ) = 2. So, to obtain |S2 | we use that |S1 | = n+1 and the fact that the reversal graph Bn (Rσ ) does not contain 2 triangles, by Corollary 8, and hence we have |S2 | = |S1 |(|S1 | − 1) − N2 . For example, for n = 3 we have N2 = 14 and |S2 | = 16 (see Figure 5). From the structural properties of the reversal graph Bn (Rσ ) considered above we immediately get results in the reconstruction of signed permutations distorted by at most one or two signed reversals. Theorem 17 For the reversal graph Bn (Rσ ), n ≥ 3, we have N (Bn (Rσ ), 1) = 2. Indeed, we have λ(Bn (Rσ )) = 0 from Corollary 8, and by Statement 30 we also have µ(Bn (Rσ )) = 2 since maxπσ ∈S2 c2 (π σ ) = 2. So, finally by (6) we get N (Bn (Rσ ), 1) = 2, and this completes the proof. Corollary 10 Any signed permutation π σ is reconstructible from any three distinct permutations in B1 (π σ ) of the reversal graph Bn (Rσ ). Let us denote by ti the number of sets of i distinct signed permutations in B1 (π σ ) from which π σ is reconstructible. We also denote by   n+1 N = |B1 (π σ )| = +1 2

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the number of signed permutations at reversal distance at most one from the signed permuti is the probability of the event that a signed permutation π σ is tation π σ . Then pi = N (i) reconstructible from i distinct signed permutations in B1 (π σ ) under the condition that these permutations are uniformly distributed. It is evident that p1 = 0 and p3 = 1, i.e., we never can reconstruct an arbitrary signed permutation π σ from a single permutation in B1 (π σ ) and we can always reconstruct an arbitrary permutation π σ from three distinct permutations in B1 (π σ ). Theorem 18 p2 ∼

1 3

as n → ∞.

t2 , where t2 is the number of sets of two distinct per(N2 ) mutations in B1 (π σ ) from which π σ is reconstructible. Since Bn (Rσ ) is vertex–transitive,
n+1 + σ we consider I instead of π . There are 2 distinct pairs (I + , τ σ ), where τ σ ∈ S1 , which do not allow to reconstruct I + uniquely. Moreover, a pair of permutations from S1 that adjacent to one and the same permutation τ σ ∈ S2 do not allow to reconstruct I + . By Theorem 16 there are exactly

Indeed, by the definition, p2 =

N2 =

1 4 (n + 4n3 − n2 − 4n) 12

such pairs of permutations from S1 . Therefore,   1 N n(n + 1) + N2 = (n4 + 4n3 + 5n2 + 2n). − t2 = 2 12 2

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Since

  N 1 = (n4 + 2n3 + 3n2 + 2n), 8 2

we have p2 =

t2 1
∼ as n → ∞. N 3 2

Thus, any unknown signed permutation is uniquely reconstructible from three of its distinct 1-neighbors. It is also shown that a signed permutation is reconstructible from two of its distinct 1-neighbors with probability p2 ∼ 31 as n → ∞ under the conditions that these signed permutations are uniformly distributed. By using Theorems 17 and 18 it is easy to realize a reconstruction algorithm for the cases when three or two distinct signed permutations are sufficient to reconstruct an arbitrary signed permutation. Theorem 19 For the reversal graph Bn (Rσ ), n ≥ 2, we have N (Bn (Rσ ), 2) ≥ n(n + 1). This result is obtained by the same method as for the reversal Cayley graph Symn (R) in Section 3.4.2.. The inequality in Theorem 19 is attained for permutations π σ with c2 (π) = 2 having the representations (30)–(33) where the sign–change reversals correspond to the sign–change transpositions. For example, for n = 3 this inequality is atσ σ σ ) ∪ = 132, and r1,3 = 321. One can check that B2 (r2,3 tained for π σ = 231, r2,3

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σ ) = {123, 123, 123, 123, 213, 132, 321, 231, 231, 231, 231, 213}, and hence we B2 (r1,3 σ ) ∪ B (r σ )| = 12 (see Figure 5). have |B2 (π σ ) ∩ B2 (I)| = |B2 (r2,3 2 1,3 Now let us consider the burnt pancake Cayley graph Bn (P Rσ ), also called burnt prefix–reversal graph, which is defined on Bn and generated by the sign–change prefix– σ ∈ B , 1 ≤ i ≤ n}, |P Rσ | = 2n − 1. The distance in reversals from the set P Rσ = {r1,i n this graph is defined as the least number of the sign–change prefix–reversals transforming one signed permutation into another. The burnt pancake graph has the following properties.

Statement 31 The burnt pancake graph Bn (P Rσ ), n ≥ 2, (i) is a connected (2n − 1)-regular graph of order 2n n!; (ii) does not contain triangles nor subgraphs isomorphic to K2,3 ; (iii) each of its vertices belongs to 23 n(n − 1) subgraphs isomorphic to K2,2 . There are no subgraphs isomorphic to K2,3 since there is no signed permutation π σ ∈ S2 for which c2 (π σ ) = 3. The number of subgraphs isomorphic to K2,2 having I + as one of the vertices is calculated by formulas (30)-(32). So, the number of representations (30)
Pn−1 n and (31) equals 2 i = 2 , when k = 1; the number of representations (32) equals i=1 2
n 2 , when i = 1, k = 1, 1 ≤ l < j ≤ n; (33) does not hold in this case. From these, the required number in (iii) is obtained. From this statement, we immediately have that λ(Bn (P Rσ )) = 0 and µ(Bn (P Rσ )) = 2. This means that any signed permutation is uniquely reconstructible from three of its distinct 1–neighbors in this case.

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3.6.

Open Questions and Conjectures

Now let us present some open problems in the vertex reconstruction from metric balls. We have shown some results for simple, regular and Cayley graphs. For example, for simple graphs it was shown that N (Γ, r) = max1≤s≤2r Ns (Γ, r) where Ns (Γ, r) = maxx,y∈V ; d(x,y)=s |Br (x) ∩ Br (y)|. The natural question asked by Siemons arises here. Question 1 What is the condition on s and r so that N (Γ, r) = Ns (Γ, r)? There are also no general results for N (Γ, r) when Γ is a regular graph. Another question still unanswered for a general Cayley graph was stated by Siemons as follows. Question 2 What are bounds on N (Γ, r) for any Cayley graph Γ = Cay(G, S) of a group G with a set S of generators and for any r > 0? In a general case it is unknown how many distinct permutations in the metric ball Br (π) are required to reconstruct a permutation π. The following conjecture was made by Levenshtein for the transposition Cayley graph. Conjecture 2 For r ≥ 1 and n ≥ 2r + 1, we have N (Symn (T ), r) = N2 (Symn (T ), 2) = |Br (e)

\

Br (π)|

for each 3–cycle permutation π. We also nothing know about the value N (Γ, r), r ≥ 3, when Γ is the reversal Cayley graph on Symn or Bn . Cayley graphs on other groups and generators are still not investigated on the vertex reconstruction problem. For example, which kind of results one can obtain for free groups?

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181

Connections with Some Other Problems

The following open combinatorial problem well–known as the pancake flipping problem concerns the diameter of pancake graph. The original pancake flipping problem was posed in 1975 in the American Mathematical Monthly [16] by Jacob E. Goodman writing under the name ”Harry Dweighter” (or ”Harried Waiter”) and it is stated as follows:

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”The chef in our place is sloppy, and when he prepares a stack of pancakes they come out all different sizes. Therefore, when I deliver them to a customer, on the way to the table I rearrange them (so that the smallest winds up on top, and so on, down to the largest on the bottom) by grabbing several pancakes from the top and flips them over, repeating this (varying the number I flip) as many times as necessary. If there are n pancakes, what is the maximum number of flips (as a function of n) that I will ever have to use to rearrange them?” It is clear that a stack of these n pancakes can be represented by a permutation on n elements and the problem is to find the minimum number of prefix–reversals (flips) needed to transform a permutation into the identity permutation. Clearly, this number of flips corresponds to the pancake graph diameter, and the problem is to find it. In 1979, Gates and Papadimitriou [21] presented the upper and lower bounds for the diameter of the pancake graph. In 1997, Heydari and Sudborough [26] improved the lower bound and computed the diameter up to 13. Recently an improved upper bound was presented by Sudborough in cooperation with a team at University of Texas at Dallas [61]. It is also known that sorting by the prefix–reversals, i.e., finding a sequence of the prefix–reversals sorting a permutation to the identity permutation, is an NP–hard problem (for details see [53] and [56]). An interesting variant of the pancake problem, known as the burnt pancake flipping problem, concerns the diameter of the burnt pancake graph and deals with the pancakes that are two–sided (one side is burnt). Initially, the pancakes are arbitrary ordered and each pancake may have either side up. After sorting, the pancakes must not only be in size order, but must have their burnt sides face down. Two–sided pancakes can be represented by a signed permutation on n elements with some elements negated. The problem is to find the least number of burnt flips (sign–change prefix–reversals) needed to transform a signed permutation into the positive identity permutation. It is clear that this number of burnt flips corresponds to the diameter of the burnt pancake graph. Cohen and Blum [13] showed that the burnt prefix–reversal diameter is at most 3n/2 and at least 2n−2 where the upper bound holds for n ≥ 10. It is conjectured that the diameter in this case is achieved by the negative identity permutation I − = [−1, −2, . . . , −n]. Later Hyedari and Sudborough [26] showed that if the conjecture is true then the burnt prefix–reversal diameter is at least 3(n + 1)/2 since I − can be sorted in 3(n + 1)/2 steps for all n = 3 (mod 4) and n ≥ 23. In general, a pancake stack is an example of a data structure. In molecular biology and computer science the problems presented above are called sorting by prefix–reversals. The pancake graph (on burnt or unburnt pancakes) has also practical applications in parallel processing since it corresponds to the n-dimensional pancake network such that this network has processors labeled with each of the n! distinct permutations of length n. Two processors are connected when the label of one is obtained from anothe by some prefix–reversal. The diameter of this network corresponds to the worst communication delay for transmitting in-

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Elena Konstantinova

formation in a system. It is known that this network has sublogarithmic diameter and degree as a function of the number of processors. The pancake sorting can also provide an effective routing algorithm between processors. There is a very nice survey by Heydemann [25] about Cayley graphs as interconnection networks, which can be recommended for additional reading. The main advantage in using Cayley graphs as models for interconnection networks is their vertex–transitivity which makes it possible to implement the same routing and communication schemes at each vertex of the network they model. Furthermore, some of them have other advantages such as edge–transitivity (line symmetry), hierarchical structure (allowing recursive construction), high fault tolerance and so on [2, 25, 40, 59]. Recent advances in genome identification have also brought to light questions in molecular biology very similar to the pancake problems. Differences in genomes are usually explained by accumulated differences built up in the genetic material due to random mutation and random mating. In 1986 another mechanism of evolution was discovered by Palmer and Herbon [54]. Comparing two genomes one can often find that these two genomes contain the same set of genes. But the order of the genes is different in different genomes. For example, it was found that both human X chromosome and mouse X chromosome contain eight genes which are identical. In human, the genes are ordered as [4, 6, 1, 7, 2, 3, 5, 8] and in mouse, they are ordered as [1, 2, 3, 4, 5, 6, 7, 8]. It was also found that a set of genes in cabbage are ordered as [1, −5, 4, −3, 2] and in turnip, they are ordered as [1, 2, 3, 4, 5.] The comparison of two genomes is significant because it provides us some insight as to how far away genetically these species are. If two genomes are similar to each other, they are genetically close. This has inspired some molecular biologists to look at the mechanisms which might shuffle the order of the genetic material. One way of doing this is the prefix–reversals or just reversals. Analyzing the transformation from one species to another is analogous to the problem of finding the shortest series of reversals to transform one into another. Thus, the analysis of genomes evolving by reversals leads to the combinatorial problem of sorting by reversals which was mentioned in Section 3.4.2.. Reversal distance measures the amount of evolution that must have taken place at the chromosome level, assuming evolution proceeded by reversals. Mathematical analysis of the problem was initiated by Sankoff [55] in 1992, and then continued by other authors. There are two algorithmic subproblems. The first one is to find the reversal distance between two permutations. In 2001, it was also shown by Bader et al. [3] that the reversal distance could be calculated in linear time for signed permutations. The next subproblem here is how to reconstruct a sequence of reversals which realizes the distance. Its solutions are far from unique. In 1994 it was shown by Kececioglu and Sankoff [31] that the problem is NP–hard for the unsigned permutations. It is polynomial for the signed permutations as it was shown by Hannenhalli and Pevzner [23] in 1999. The 1.5–approximation algorithm for sorting unsigned permutations was presented by Christie [15] in 1998. One of the most effective algorithms that sort signed permutations by reversals was presented by Kaplan and Verbinin [27] in 2003. For more details see the recently published books by Pevzner [53], Sankoff and El–Mabrouk [56]. Interestingly, that the problem of sorting permutations by the fixed–length reversals (k– reversals) is implicit in the popular puzzle TOP-SPIN, manufactured by the Binary Arts Corporation. T OP − SP IN T M consists of a permutation of 20 numbered disks on an oval track, with a turnstile capable of reversing a string of 4 consecutive disks. The goal is to sort the disks to the identity permutation using reversals. The more general problem, with

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permutations of n disks and a turnstile of size k was considered and solved by Chen and Skiena [12]. Note that sometimes it is impossible to sort permutation using only reversals of certain length. For example, an odd length reversal does not change the parity of the position of any element it acts on, so odd length reversals cannot sort any permutation of the form π = [2, 1, . . . , n]. The authors gave a complete description for all n and k of how many permutations of length n can be sorted using only reversals of length k. In particular, it was shown that O(n3/2 ) k-reversals suffice to transform any permutation to the identity √ permutation when k ≈ n. The number of connected components on the corresponding Cayley graphs generated by the fixed–length reversals was also discussed there. For instance, in the trivial case of n-reversals, any permutation can be transformed only to its inverse permutation. Thus, there are n!/2 connected components for the symmetric group Symn with k = n > 2. Coming back to the genome analysis we have to say that it is also a powerful technique for phylogenetic inference. Current methods are based on distances between genomes (permutations), which are usually defined as the least number of such and such operations (for example, reversals or transpositions) needed to transform one genome into another. Distances are used directly as data for phylogenetic reconstruction, or in more qualitative attempts to reconstruct ancestral genomes as it was shown in 2002 by Bourque and Pevzner [8]. Reconstructing evolutionary trees is a fundamental research problem in biology, with applications to protein structure, function prediction, pathway detection, sequence alignment, drug design, etc. [20].

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4.

Conclusion

In this review paper we have presented the main results obtained on the vertex and graph reconstruction problems during the last ten years. The first problem comes from coding theory where the vertices in graphs correspond to sequences and the sets of generators are given by single errors on sequences. Important examples of considered graphs are Cayley graphs which arise in applied areas, for example in molecular biology when reconstructing an unknown genome in evolutionary trees from its close neighbors (1– or 2–neighbors) is considered. Usually, genomes are presented by permutations or signed permutations hence the problem of reconstructing permutations from its neighbors arises in this context. The corresponding Cayley graphs on the symmetric and hyperoctahedral groups are generated by reversals or prefix–reversals since these operations represent evolutionary events. Another example of a ”useful” operation on permutations is a transposition of two elements. Cayley graphs generated by transpositions are widely used in the representation of interconnection networks in computer science. Coxeter generators presented by the transpositions of two neighbor elements are fundamental to physics. So, we have confined our attention on Cayley graphs of two given groups Symn and Bn with mentioned sets of generators since their properties provide us with an understanding the problem we are interested in. The second presented problem that is the graph reconstruction problem was motivated by the structure reconstruction of chemical compounds from NMR spectroscopy data. It was generalized as the problem of reconstructing an unknown graph from all its metric balls of a given radius r. It was shown that there are necessary and sufficient conditions for such exact reconstruction. Two cases were considered. Firstly, an exact graph reconstruction

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from the metric balls of radius r = 2 was presented. It was shown that hat any connected graph of diameter at least 4 and of girth at least 7 (in particular, a tree) can be exactly reconstructed from all its metric balls of radius 2. Then the case when the metric balls of larger radius r ≥ 2 are used for the graph reconstruction was considered. In this case for exact graph reconstruction much more stronger restrictions are required on the girth of a graph and its structure. Such conditions were found for simple connected graphs Γ without terminal vertices. This problem is open for other graphs. Both these problems have interesting connections with other combinatorial, graph– theoretical and applied problems which were also discussed in this paper.

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[6] Bondy, J. A.; Hemminger, R. L. J. Graph Theory 1977, 1, 227–268. [7] Bondy, J. A. In Book Surveys in Combinatorics (Proceedings of the 13th British Combinatorial Conference, Guildford, UK, 1991), London Mathematical Society Lecture Note Series, 1991; 166, pp.221–252. [8] Bourque, G.; Pevzner, P. A. Genome Res. 2002, 12, 26–36. [9] Bj¨orner, A.; Brenti, F. Combinatorics of Coxeter groups; Springer Verlag, Heidelberg, New York, 2005. [10] Brouwer, A. E.; Cohen, A. M.; Neumaier, A. Distance–regular graphs, SpringerVerlag, Berlin, Heidelberg, 1989. [11] Cameron, P. Congressus Numerantium 1986, 113, 31–41. [12] Chen, T.; Skiena, S. S. Discrete Appl. Math. 1996, 71, 269–295. [13] Cohen, D. S.; Blum, M. Discrete Appl. Math. 1995, 61, 105–120. [14] Conway, J. H.; Sloane, N. J. A. Sphere packings, lattices andgroups, Springer–Verlag, New York, Berlin, 1988. [15] Christie, D. A.; In Book Proc. SODA 1998; ACM Press, 1998; pp.244–252. [16] Dweighter, H. Amer. Math. Monthly 1975, 82, 1010. Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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[30] Kerber, A.; Laue, R., Gr¨uner, T.; Meringer, M. MATCH 1998, 37, 205–208. [31] Kececioglu, J.; Sankoff, D. In Book Proc. CPM 1994; LNCS, 807, Springer Verlag, 1994; pp.307–325. [32] Kececioglu, J.; Sankoff, D. Algorithmica 1995, 13, 180–210. [33] Konstantinova, E. Bayreuth. Math. Schr. 2005, 73, 213–227. [34] Konstantinova, In Book E. Proc. VII Intern. Conf. Discrete Models in Control System Theory; Moscow, MGU, 2006; pp.172–178. [35] Konstantinova, E. Discrete Appl. Math. 2007, 155, 2426–2434. [36] Konstantinova, E. Discrete Math. 2008, 308, 974–984. [37] Konstantinova, E. Linear Algebra Appl. 2008, doi:10.1016/j.laa2008.05.010. [38] Konstantinova, E.; Levenshtein, V.I.; Siemons, J. http://arxiv.org/abs/math/0702191. [39] Knuth, D. E. Sorting and searching; Addison–Wesley, Reading, Massachusetts, second edition, 1998; p.72 and p.615. [40] Lakshmivarahan, S.; Jwo, J. S.; Dhall, S. K. Parallel Comput. 1993, 19, 361–407. Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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[54] Palmer, J. D.; Herbon, L. A. Nucleid Acids Research 1986, 14, 9755–9764. [55] Sankoff, D. In Book Proc. CPM 1992; LNCS, 644; Springer Verlag, 1992; pp.121– 135. [56] Sankoff, D.; El–Mabrouk, N. In Book Current topics in computational molecular biology; Jiang, T.; Smith, T.; Xu, Y.; Zhang, M. Q.; Eds.; MIT Press, 2002. [57] Setubal, J. C.; Meidanis, J. Introduction to omputational molecular biology; PWS Publishing Company, 1997. [58] Smolenski, E. A. J. vychisl. mat. i mat. phisiki 1962, 3, 371–372. [59] Schibell, S. T.; Stafford, S. M. Discrete Appl. Math. 1992, 40, 333–357. [60] Stockmeyer, P. K. Congressus Numerantium 1988, 63, 188-200. [61] Sudborough, H.; Chitturi, C.; Fahle, W.; Meng, A.; Morales, L.; Shields, C.; Voit, see: http://www.diku.dk/forskning/topps/neilfest/abstract sudborough.html [62] Ulam, S. M. A collection of mathematical problems, Wiley, New York, 1960, p.29. [63] Zaretski, K. A. Usp. mat. nauk 1965, 20, 94–96. Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

In: Emerging Topics on Differential Geometry... Editors: L. Bernard and F. Roux, pp. 187-222

ISBN 978-1-60741-011-9 c 2010 Nova Science Publishers, Inc.

Chapter 6

C OALESCENCE OF G RAPHS A.A. Lushnikov Karpov Institute of Physical Chemistry, 10, Vorontsovo Pole, 105064 Moscow, Russia

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Abstract

This Chapter considers an initially empty (no edges) graph of order M that is assumed to evolve by adding one edge at a time. This edge can connect either two linked components and form a new component of a larger order (coalescence of graphs) or increase (by one) the number of edges in a given linked component. Assuming the rate of appearance of an extra–edge in the graph to be known, the Master equation governing the time evolution of the probability W (Q, t) to find the random graph in a given state Q is formulated. The latter is given by a set of the population numbers of linked components of order g with ν edges. This equation is then reformulated in terms of the generating functional for the probability W (Q, t). This functional is shown to meet the evolution equation which is just a linear second–order multidimensional (generally infinite dimensional) partial differential equation. The analogy between coagulation processes and the evolution of random graphs is tracked. The coalescence alone is shown to be responsible for the kinetics of growth of the tree–like population. The respective evolution equation for the generating functional is simpler than in theory of random graphs. The analogy with coagulating systems allows one to consider very large graphs in the thermodynamic limit. At the initial stage of the stochastic coalescence of graphs all graphs are trees (in the thermodynamic limit) and only after the critical time when the giant component emerges the cycles appear in it. The analogy with the sol–gel transition in coagulating systems is traced. The exact solution of the evolution equation for the generating functional of the evolving random graph is obtained. The spectrum (the average population numbers of linked components) is found. This approach is then applied to random bipartite graphs. The exact expression for the generating functional of evolving bipartite graph is derived. Two Appendices introduce and analyze two special families of polynomials closely associated with the uni–partite and bipartite graphs. These polynomials play extremely important role in theory of graphs.

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188

1.

A.A. Lushnikov

Introduction

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Let us imagine a system comprising M functional units connected with K ≤ M (M − 1)/2 links allowing for an exchange of information between them. One easily recognizes a schematic model of a computer, the Internet, a living organism, or the human brain. Other (more physical examples) of such systems are polymers, disordered materials, random electric chains. The structures of all these systems can be modelled by a random graph, where K edges randomly distributed among M vertices form clusters (linked components). Any realization of the random graph can be given by {ng,ν }, a set of the numbers of linked components of order g with ν edges. The problem is to find the probability for the realization of a given partition, once K = ν1 + ν2 + . . . νg + . . . edges are randomly distributed among M vertices. In this statement the problem had been introduced by Erd¨os and Renyi [1] and then had been considered by many authors, (see [2, 3, 4, 5, 6, 7, 8, 9, 10] and references therein).

Figure 1. Evolution of a random graph. Initially empty graph (fig. 1a) begins to evolve. An edge linking two vertices appears at a time. An example of a graph formed after a time t is shown in fig 1b. This graph comprises two linked components with 3 and 5 vertices. Next appearing edge can either link these two components and thus produce one component of order 8 as shown in fig 1c (coalescence of graphs) or to add the edge to one of these graphs not changing the distribution of the graphs over their order (fig. 1d). Another formulation of this problem is possible. I illustrate the idea by considering a model of the life–to–death transition. A young living organism can be regarded as a system of M functional units connected with M (M − 1)/2 links. The merciless time either spoils or breaks the links by one at a time demolishing thus the organism. The organism is still alive while it contains linked ”macroscopic” fragments (containing g ∝ M linked

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Figure 2. Evolution of a random bipartite graph. Initially empty bipartite graph (fig. 2a) begins to evolve. An edge linking two vertices of different sorts (hearts and circles) appears at a time. Heart–to–heart and circle–to–circle edges are forbidden. An example of a graph formed after a time t is shown in fig 2b. This graph comprises two linked components with 4 and 3 vertices. Next appearing edge can either link these two components and thus produce one component of order (4,3) as shown in fig 2c (coalescence of graphs) or to add the edge to one of these graphs not changing the distribution of the graphs over their order (fig. 2d).

functional units). Time elapses, the fragments disintegrate becoming smaller and smaller, and a moment comes up when the fragments become so small that the system ceases to work. The organism dies. Such a scenario of life–to–death transition might seem to be flat, if it were not for one important circumstance: the transition from the state, where there is a macroscopic cluster of linked functional units (life) to the state, where there are not such macroscopic clusters (death) is sharp as M −→ ∞, i.e. the macroscopic linked cluster disappears instantly, the cluster lifetime being independent of M . Now this scenario of the life–to–death transition is in a more extent consistent with our intuitive perception of this process. We thus come to another recipe for constructing the random graph based on removing edges. However, instead of tearing off the links we can imagine that starting with the complete graph (all graph vertices are connected to each other) we randomly add to it white edges which make invisible the black ones. In this case we again return to the well–known statement of the problem: to determine the properties of the graph as a function of the number of white edges which are distributed at random among M vertices. But now we study the structure of the ”holes” on the initially complete graph. At the moment when a macroscopic ”hole” forms the organism dies. I thus silently assume that a macrohole and a

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macrocluster cannot coexist. Most commonly accepted approaches to the problem of the evolution of random graphs rely upon rather sophisticated combinatorial considerations. There exist another group of authors — supporters of a kinetic approach [11, 12, 13] who apply a balance kinetic equation allowing one to study the time dependence of changes to the random graph properties. Normally the kinetic approach implies the thermodynamic limit, i.e., the population numbers ng,ν are very large. Here I propose an alternative approach based on the formulation and the solution of an equation describing the time evolution of the generating functional for the probability to find a given set of occupation numbers of linked components having exactly g vertices and ν edges at time t [14, 15]. Figures 1 and 2 explain my intentions. I consider evolution of random uni– and bipartite graphs. The graphs change their structure when a new edge appears in the graph. This new appearing edge leads either to coalescence of two linked components or to change to the degree of filling of the component that the new edge hits. The idea of this approach is entirely adopted from my recent articles [16, 17, 18, 19, 20], where the sol–to–gel transition was investigated in finite coagulating systems. Being applied to the problem of the evolution of random graphs this approach gives very impressive and elegant results that I never met before. Although the analogy between the evolution of a coagulating system and a random graph is quite straightforward and had been noticed long ago [11, 12, 13], the approach proposed below allows one not only to trace the fate of a random graph in detail, but to use it for obtaining some new interesting results in theory of random graphs. The results presented below are exact and valid for any finite random graph. It is shown that the total number of all linked components of order g exactly repeats the fate of the particle mass spectrum of a finite coagulating system with the kernel proportional to the product of masses of coalescing particles. The remainder of this Chapter is divided as follows. Next Section describes the statement of the problem, introduces the process governing the time evolution of random graphs and formulates the Master equation for the probability W (Q, t) to find the random graph in a given state Q. The latter is given by a set of the population numbers of linked components of order g with ν edges. This equation is then reformulated in terms of the generating functional for the probability W (Q, t). This functional is shown to meet the evolution equation which is just a linear second–order multidimensional (generally infinite dimensional) partial differential equation. The analogy between coagulation processes and the evolution of random graphs is followed in Section III. Here I consider the evolution of forest and show that coalescence alone is responsible for the kinetics of growth of the tree–like population. I introduce the ideas of my rather ancient work [15], where the coagulating systems were described by applying the Markus–Lushnikov process [5]. Again, I formulate the evolution equation for the respective generating functional. In the case of coagulating systems this equation is simpler than in theory of random graphs. The analogy with coagulating systems allows one to consider very large graphs in the thermodynamic limit (Section IV). Here I show that at the initial stage of the stochastic coalescence of graphs all graphs are trees (in the thermodynamic limit) and only after the critical time when the giant component emerges the cycles appear in it. The analogy with the sol–gel transition in the coagulating systems is traced in Section V. The exact solution of the evolution equation for the generating functional of the evolving random graph is outlined in Section VI. Here the exact

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spectrum (the average population numbers of linked components) is found. In Section VII the approach is applied to random bipartite graphs. The exact expression for the generating functional of evolving bipartite graph is derived and used for calculating the average graph spectrum. The thermodynamic limit for large bipartite graphs is analyzed in Section VIII. Concluding Section IX summarizes the results. Two Appendices introduce and analyze two special families of polynomials closely related to the uni–partite and bipartite graphs. These polynomials play extremely important role in theory of random graphs.

2.

Basic Equation

Let there be a graph of order M comprising N linked components. Each linked component can be characterized with its order g, the number of vertices, and the degree of filling ν (the number of edges in the component). It is clear that g − 1 ≤ ν ≤ g(g − 1)/2. The minimal value of ν corresponds to a tree of order g and the maximal one is the number of edges in the complete graph of order g. A bare vertex is also regarded as a linked component of order 1.

2.1.

The Master Equation

Any state of the graph can be given by the set of population numbers

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Q = (n1,0, n2,1, n3,2, n3,3, . . . ng,ν , . . .) = {ng,ν },

(1)

where ng,ν is the number of linked components of order g having exactly ν edges (g, ν– component). Of course, the number of vertices is independent of the realization of the state Q, i.e., X gng,ν (Q) = M. (2) g,ν

Here the argument Q means that the distribution of the population numbers ng,ν (Q) is the realization of the state Q. Let us consider an initially empty graph of order M (simply M bare vertices) and begin to add to this graph the edges (one edge at a time) linking two valent vertices. This process gives rise to either a coalescence of two linked components (l, λ) + (m, µ) −→ (l + m, λ + µ + 1)

(3)

or to filling a given linked component with one extra–edge (g, ν) −→ (g, ν + 1).

(4)

The graph thus evolves due to changing the number of linked components, their order, and their degree of filling (see Figs 1 and 2). There are two types of states preceding to the state Q (see Fig 1c,d): the state Q− = {n1,0, n2,1, n3,2, n3,3, . . . nl,λ + 1 . . . nm,µ + 1 . . . ng,λ+µ − 1, . . .}

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192

A.A. Lushnikov

and the state ˜ − = {n1,0, n2,1, n3,2, n3,3, . . . ng,ν−1 + 1, ng,ν − 1, . . .}. Q

(6)

For labelled graphs the efficiency of the coalescence process is proportional to lm, the number of ways to connect two linked components by an extra–edge. The efficiency for filling a linked component is proportional to g(g − 1)/2 − ν, the number of vacant placements for the extra–edge. Now we introduce the probability W (Q, t) to find the graph in the state Q at time t and write down the Master equation governing the time evolution of W (Q, t): X dW (Q, t) X = A(Q, Q−)W (Q− , t) − A(Q+ , Q)W (Q, t) dt − + Q

+

X

Q

˜ − )W (Q ˜ − , t) − B(Q, Q

˜− Q

X

˜ + , Q)W (Q, t). B(Q

(7)

˜+ Q

˜ + )− = (Q ˜ − )+ = Q, i.e., the state Q precedes to the Here (Q+ )− = (Q− )+ = Q and (Q + + ˜ . The transitions rates A and B are expressed in terms of the occupation states Q and Q numbers n as follows: A(Q, Q−) =

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and

1 nl,λ (Q− )(nm,µ (Q− ) − δl,m δλ,µ ) 2T

˜ −) = 1 B(Q, Q T





g(g − 1) ˜ − ). − ν + 1 ng,ν−1 (Q 2

(8)

(9)

Here 1/T is the rate of appearance of an extra–edge and δl,m is the Kroneker delta. It always appears when two identical graphs coalesce. In this case the number of pairs is given by the factor nm,µ (Q− )[nm,µ (Q− ) − 1]/2.

2.2.

Generating Functional and the Evolution Equation

It is more convenient to deal with the generating functional for W (Q, t), Ψ(X, t) =

X Q

W (Q, t)

Y

n(g,ν|Q) xg,ν ,

(10)

g,ν

where n(g, ν|Q) stands for the occupation number ng,ν belonging to the given state Q and X = {xg,ν } is the set of independent formal variables. The equation for Ψ is readily derived from Eqs (7), (8), and (9). Indeed, noticing that xl,λ

∂ ∂xl,λ−1

X

˜ Q

=



∂ ˜− XQ = xl,λ ∂xl,λ xl+m,λ+µ+1



l(l − 1) ˜+ − λ + 1 nl,λ−1 (Q)X Q 2 



l(l − 1) ˜− − λ nl,λ (Q)X Q , 2

∂2 + X Q = (nl,λ (Q)nm,µ (Q) − δl,m δλ,µnm,µ (Q))X Q , ∂xl,λ∂xm,µ

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193

∂2 X Q = (nl,λ (Q)nm,µ(Q) − δl,m δλ,µnm,µ (Q))X Q ∂xl,λ∂xm,µ

we come to the Evolution equation, T

∂Ψ = (Lˆf + Lˆc )Ψ, ∂t

(13)

where the multiplier T defines the scale of time. The right–hand side of this equation contains two differential operators, Lˆf and Lˆc . The ˆ f is responsible for the evolution of the filling of a linked cluster, operator L Lˆf =

X  l(l − 1) l,λ

2



− λ + 1 xl,λ

∂ ∂xl,λ−1

−

X  l(l − 1) l,λ

2



− λ xl,λ

∂ . ∂xl,λ

(14)

The summation in the first sum on the right–hand side of this equation goes from λmin = g to λmax = [g(g−1)/2+1] whereas the respective limits in the second sum are λmin = g−1 to λmax = g(g − 1)/2. It is thus more convenient to rewrite Lˆf as follows: Lˆf =

X  l(l − 1) l,λ

2



− λ (xl,λ+1 − xl,λ)

∂ . ∂xl,λ

(15)

Now the summation over λ goes from λmin = g − 1 to λmax = g(g − 1)/2. The coalescence operator has the form,

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1 X ∂2 lm(xl+m,λ+µ+1 − xl,λ xm,µ ) . Lˆc = 2 l,λ;m,µ ∂xl,λ∂xm,µ

(16)

Equation (13) replaces the Master equation Eq. (7) governing the time evolution of the probability W (Q, t). Equation (13) should be supplemented with the initial condition Ψ(X, t = 0) = Ψ0 (X),

(17)

where Ψ0 (x) is a known functional. For example, Ψ0 (X) = xM 1,0

(18)

corresponds to initially empty graph comprising M vertices or X

Ψ0 (X) = exp[

n ¯ 0l,λ(xl,λ − 1)]

(19)

l,λ

describes the Poisson distributed graph. Here n ¯ 0l,λ are the initial average occupation numbers. Of course, Ψ(X = 1, t) = 1, (20) which corresponds to the normalization of W (Q, t) to unity, i.e.,

P

Q W (Q, t)

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A.A. Lushnikov

At this step we introduce the operators of occupation numbers, of the total number of linked components of order l, of the total number of linked components, and of the total order of graph. They are: n ˆ l,λ = xl,λ

∂ , ∂xl,λ

ˆl = N

X λ

n ˆl,λ ,

ˆ= N

X

n ˆl,λ

ˆ = M

l,λ

X

lˆ nl,λ.

(21)

l

Any average value of interest can be expressed in terms of Ψ. For example, the average P spectrum of graphs n ¯ g,ν (t) = Q ng,ν (Q)W (Q, t) is, ˆ g,ν Ψ(X, t)|X=1. n ¯ g,ν (t) = n

(22)

ˆ commutes with Lf and Lc and thus with the evolution It can be readily checked that M operator, ˆ M] ˆ = 0, [L, (23) which means the conservation of the graph order. The analogy with the second quantization is now clearly seen: the operator ∂x acts as an annihilation and x as a creation Bose operator. Their commutator ∂x x − x∂x = 1. Hence, the first term on the right–hand side of Eq.(16) replaces two components of orders l and m ˆ m,µ − n ˆl,λ δm,l δµ,ν by one with the order equal to l + m. The second term written as n ˆ l,λ n does not change the state Q. It just multiplies the probability W (Q, t) by the number of ways to remove the pair of linked components of order l, λ and m, µ from a given state of the evolving graph.

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3.

Coalescence of Trees and Coagulation

If the cycling in the evolving graph is artificially forbidden, then we come to the problem of coalescence of labelled trees, where the number of edges is always fixed and equal to g − 1 in the linked component of order g. Each component is now entirely characterized with its order g, the rate of coalescence being proportional to the product of the orders of two coalescing graphs. This process is described by the chain of irreversible reactions (g) + (l) −→ (g + l)

(24)

and is referred to as coagulation. I shall try to explain in short the statement of this problem. Details can be found in refs [17, 18, 19, 20]. Let there be M monomers in a volume V . The monomers move, coalesce, produce dimers, trimers etc along the scheme (24). Let then Q = (n1 , n2 . . . ng . . .) = {ng (Q)}

(25)

be a state of the system given by the set of integers ng , the numbers of particles of mass g, with g being the number of monomeric units in a g–mer.. A single coagulation act (the collision of two particles + their coalescence) changes a preceding state Q− = (n1, . . . nl + 1, . . . nm + 1, . . .ng − 1 . . .)

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Coalescence of Graphs

195

Q− = (n1 , . . . nl + 2, . . .. . . ng − 1 . . .)

(27)

if l 6= m and if l = m, 2l = g to the state Q by coalescing the particles with masses l and m to one particle with mass g. In its turn, a next coagulation act transfers the state Q to the state Q+ according to the scheme Q− −→ Q −→ Q+ ,

(Q+ )− = Q.

(28)

The probability per unit time for two particles to collide and to coalesce is K(l, m)/V , where K(l, m) is the coagulation kernel (the efficiency of the coagulation process). The rate of the process (24) is then A(Q, Q− ) =

K(l, m) nl (Q− )(nm (Q− ) − δl,m ). 2V

(29)

The combinatorial multiplier here is just the number of ways to get a successfully coalescing pair of l– and m–mers. Next, we introduce the probability W (Q, t) to find the system in the state Q at time t. We can write down the Master equation for the probability W (Q, t). It is X dW (Q, t) X = A(Q, Q−)W (Q− , t) − A(Q+ , Q)W (Q, t). dt − + Q

(30)

Q

As in the case of graphs, we prefer to deal with the generating functional Ψ(X, t) =

X

W (Q, t)X Q

(31)

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Q

rather than with the probability W (Q, t). Here X stands for the set x1, x2 . . . and X Q = n (Q) n (Q) x1 1 x2 2 . . .. Of course, Ψ(X = 1, t) = 1, (32) P

which corresponds to the normalization of W (Q, t) to unity, i.e., Q W (Q, t) = 1. The equation for Ψ is readily derived from Eq. (30). Indeed, noticing that xl+m

∂2 + X Q = (nl (Q)nm (Q) − δl,m nm (Q))X Q ∂xl∂xm

and

∂2 X Q = (nl (Q)nm (Q) − δl,m nm (Q))X Q ∂xl∂xm we find instead of Eq. (30), ∂Ψ ˆ = LΨ, V ∂t where the evolution operator L is defined as xl xm

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(33)

(34)

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A.A. Lushnikov

The similarity of this derivation and that done for the evolving graph is clearly seen if we substitute K(g, l) = κgl and choose T = V /κ. The most widespread approach to coagulation, however, relies upon the Smoluchovski equation. The latter operates with the particle ( g–mers) concentrations cg (t) defined as cg (t) = lim

n ¯ g (t) . V

(35)

The above limiting procedure (referred to as the thermodynamic limit in statistical physics) ¯ g (t) −→ ∞, with their ratio being finite. assumes V −→ ∞, n A simple balance equation (the Smoluchowski equation) for cg (t) can be immediately written down, g−1 ∞ X dcg 1X = K(g − l, l)cg−lcl − cg K(g, l)cl. (36) dt 2 l=1 l=1 Here the coagulation kernel K(g, l) is the transition rate for the process (g) + (l) −→ (g + l). The first term on the right–hand side of Eq. (36) describes the gain in the g–mer concentration due to coalescence of (g − l)– and l–mers while the second one is responsible for the losses of g–mers due to their sticking to all other particles. In introducing the procedure Eq. (35) (the thermodynamic limit) some losses are eventual: the groups of particles with average population numbers growing slower than V have zero concentrations as V goes to infinity and are thus excluded from the consideration of the coagulation process.

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4.

Thermodynamic Limit for Random Graphs

First of all, one sees that Eq. (36) strongly differs from Eq. (33) for the generating functional. And nevertheless the Smoluchowski equation can be derived from Eq. (33). The main objective of this Section is the derivation of the Smoluchowski–type equation describing the evolution of the average population numbers n ¯ g,ν (t) in the limit of large M and T. We begin by rearranging the last term in Lˆc (Eq. (16)) as follows: 



∂2 1 X 2 ∂ 1 X ˆ 2. lm(xl,λxm,µ ) =  l δl,m δλ,µ  − M − 2 l,λ;m,µ ∂xl,λ∂xm,µ 2 l,λ;m,µ ∂xl,λ

(37)

¯M ˆ , where M ¯ is defined below by Eq. (41). Then Lc is ˆ2 ≈ M Let us now approximate M replaced with its approximate expression Lˆc ≈

LˆA c

"

#

1 X ∂2 ∂ ¯M ˆ . (38) = lm xl+m,λ+µ+1 + xl,λ δl,m δλ,µ − M 2 l,λ;m,µ ∂xl,λ∂xm,µ ∂xl,λ

Equation (13) with LˆA = Lˆf + LˆA c and the initial condition given by Eq. (19) can be solved. The solution looks as follows: X

ΨA (X, t) = exp[

(xg,ν − 1)¯ ng,ν (t)]

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Coalescence of Graphs

197

On substituting this into Eq. (13) with the approximate evolution operator LˆA gives the set of equations for n ¯ g,ν (t), g−1 X 1 X ν+1 1 d¯ ng,ν ¯+ = (g − l)l¯ ng−l,ν−λ−1n ¯l,λ − g¯ ng,ν M dt 2T l=1 λ=l−1 T

1 T

(









)

g2 g(g − 1) g(g − 1) n ¯ g,ν + −ν+1 n ¯ g,ν−1 − −ν n ¯ g,ν , 2 2 2

(40)

where ¯ = M

∞ l(l−1)/2 X X

l¯ nl,λ

(41)

l=1 λ=l−1

is the average mass of the graph. Because we applied the initial condition Eq. (19)) rather than Eq. (18), the summation over l in Eq. (41) is extended up to ∞. ¯ and the time variable At this stage we introduce the concentrations cg,ν (t) = n ¯ g,ν (t)/M ¯ τ = tM /T . Then we come to the set of equations for cg,ν , g−1 ν+1 1X X dcg,ν = (g − l)lcg−l,ν−λ−1cl,λ − gcg,ν + dτ 2 l=1 λ=l−1

1 ¯ M

(









g2 g(g − 1) g(g − 1) cg,ν + − ν + 1 cg,ν−1 − − ν cg,ν 2 2 2

)

(42)

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with the normalization condition ∞ l(l−1)/2 X X

lcl,λ = 1.

(43)

l=1 λ=l−1

We encounter the absolutely remarkable fact: the linear in c term can be ignored unless very ¯ ) linked components with the concentrations of order 1/M contribute to the large (g ∝ M second line of the above equation. ¯ Let us first consider the initial stage of the graph evolution where the term of order 1/M can be ignored. In this case the kinetics of the process is described by the equation, g−1

X 1 X ν+1 dcg,ν = (g − l)lcg−l,ν−λ−1cl,λ − gcg,ν . dτ 2 l=1 λ=l−1

(44)

One sees immediately that if we start from the initially empty graph cg,ν (t = 0) = δg,1δν,0 ,

(45)

then the cycles can never form. Indeed, the first term of Eq. (45) joins two linked components l and m and adds only one edge. If these components are trees, the resulting linked component will also be a tree, because the number of edges in such linked graph is ν = l − 1 + m − 1 + 1 = l + m − 1 = g − 1.

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A.A. Lushnikov P

Let us introduce cg (t) = ν cg,ν Then, on summing both sides of Eq. (42) over ν yields the Smoluchowski–type equation for cg (t), g−1 dcg 1X g2 = (g − l)lcg−lcl + ¯ cg − gcg . dτ 2 l=1 2M

(46)

¯ −1 is retained, because it can contribute if a large particle with the The term containing M ¯ appears (it does!) in the system after a finite (independent of M ) time. mass g ∝ M

5.

Gelation Catastrophe and Giant Component

The exact solution to Eq. (44) for initially empty graph is known already more than half a century g g−2 −gτ g−1 e τ cg (τ ) = . (47) g! This solution exists for all nonnegative τ . If, however, we try to find the total mass concenP tration of all linked components m(t) = g gcg (t) we come to a very strange conclusion (details see in ref. [18]) that (48) m(τ ) = 1 − µc (τ ), where µc (τ ) is the root of the transcendent equation

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τ=

1 1 ln µc 1 − µc

or e−µc τ = 1 − µc

(49)

This equation has only one root µc = 0 at τ ≤ 1. At τ > 1 another positive root appears. This second positive root should be chosen at τ > 1. The point is that the spectrum Eq. (47) becomes the algebraic function cg (1) ∝ g −5/2 and then shrinks, so its mass cannot remain constant. The second moment φ2 of cg (τ ) diverges at τ = 1. It follows immediately from Eq. (47). On multiplying both sides of Eq. (44) by g 2 and summing over all g gives dφ2 = φ22 dτ

(50)

or φ2 = (1 − τ )−1 . This paradox is referred to as the gelation catastrophe in theory of coagulation. The deficiency in the mass concentration is associated to the formation of one giant particle with the mass comparable to M . This particle has zero concentration in the thermodynamic limit and thus does not appear in the Smoluchowski equation. But this particle exists and actively participates in the life of smaller particles. Equation (46) takes into account the existence of such particles. In theory of graphs this effect is associated to the formation of a giant linked component whose order is comparable to the order of the whole graph.

6.

Exact Solution of Evolution Equation

As the order of the initial graph is conserved M=

X

lnl,λ,

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(51)

Coalescence of Graphs

199

the solution to the evolution equation (13) can be found in the form: ΨM = M !Coef z z

−(M )

exp

X

ag,ν (t)xg,ν z

g

!

,

(52)

g,ν

where the multiplier M ! provides the correct normalization of the generating functional, Ψ(1, t) = 1. The operation Coef introduced in [21] replaces the contour integration when one deals with formal series [22]. By definition Coef z

X

bl z l = b−1.

The operation Coef displays many features of ordinary residues. ˆ , i.e., The functional ΨM given by Eq. (52) is an eigenfunctional of the operator M ˆ ΨM = M ΨM . M On substituting Eq. (52) into Eq. (13) gives the set of equations for ag,ν (t) 







1 1 dag,ν = g(g − 1) − ν + 1 ag,ν−1 − g(g − 1) − ν ag,ν T dt 2 2 +

1 X 1 g2 lmal,λam,µ δg,l+m δν,λ+µ+1 − M gag,ν + ag,ν . 2 l,λ;m,µ 2 2

(53)

The initial condition to Eq. (53) is chosen in the form:

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ag,ν (0) = δg,1 δν,0 .

(54)

It is easy to check that this initial condition corresponds to Ψ(X, 0) = xM 1,0 , i.e., to the initially empty graph comprising M bare vertices. Next, all ag,ν (t) = 0, once ν lies beyond the permitted interval (g − 1) ≤ ν ≤ g(g − 1)/2. Now let us try to solve Eq. (53). To this end we introduce the bivariate generating function for ag,ν (t), X G(z, ζ; t) = z g ζ ν ag,ν (t). (55) g,ν

The summation on the right–hand side of the above equation goes over all g and ν. The equation for G(z, ζ; t) immediately follows from Eq. (53), ζ ∂G = T ∂t 2

"

∂G z ∂z

2

#





∂ ∂G ∂G 1 ∂G M ∂G +z z − (ζ − 1) ζ + z − z . ∂z ∂z ∂ζ 2 ∂z 2 ∂z

(56)

This equation reduces to a linear one for the function D(z, ζ; t) = exp[G(zet/2T , ζ; t)], 



ζ ∂ ∂D ∂D 1 ∂D 1 ∂D = z z − (ζ − 1) ζ + z . T ∂t 2 ∂z ∂z ∂ζ 2 ∂z

(57)

The initial condition for this equation follows from Eq. (54), D(z, ζ; 0) = ez . Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

(58)

200

A.A. Lushnikov

Equation (57) is readily solved by separating variables. Let D(z, ζ; t) =

X

Θκ (t, )Zn,κ (ζ)z n .

n,κ

Then Θκ (t) = eκt/T and κZn,κ





ζ dZn,κ n = n2 Zn,κ + (1 − ζ) ζ + Zn,κ . 2 dζ 2

(59)

Here κ is a separation constant. The solution to this equation is, Zn,κ (ζ) = bn,κ (1 − ζ)n

2 /2−κ

ζ κ−n/2 .

(60)

The function D should be analytical at ζ = 0. Hence, κ−n/2 = s, where s is a nonnegative integer. Next, the coefficients bn,κ should be chosen from the initial condition Eq. (58). It is easy to see that ! 1 (n2 − n)/2 . bn,κ = n! s We then come to the result, D(z, ζ; t) =

∞ X zn n=0

n!

2 −n)/2

ent/2T [ζet/T + (1 − ζ)](n

.

(61)

In order to return to ag,ν (t) we use the Knuth identity [23], ln

∞ X

(1 + δ)n(n−1)/2

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n=1

∞ X zn zn = δ n−1 Pn−1 (δ) n! n! n=1

(62)

(see refs [23, 24, 25] and Appendix A for the definition of the polynomials Pg (δ)). P Equation (62) allows us to restore Ag (ζ, t) = ν ag,ν (t)ζ ν , Ag (ζ, t) =

1 −g(M −1)t/2T t/T (e − 1)g−1 ζ g−1 Pg−1 (ζet/T − ζ). e g!

(63)

Next, we use the definition of Pg (see Appendix A), g(g−1)/2

δ g−1 Pg−1 (δ) =

X

Cg,ν δ ν ,

(64)

ν=g−1

where Cg,ν is the number of labelled linked graphs of order g with ν edges. Equation (64) is readily applied for restoring ag,ν (t). The result is ag,ν (t) =

1 −g(M −1)t/2T t/T e (e − 1)ν Cg,ν . g!

(65)

Now we are ready to find the average number of linked components of order g with ν edges. From Eqs (22) and (52) we have, n ¯g,ν (t) = M !ag,ν (t)Coef z z −M +g D(z, 1; t).

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(66)

Coalescence of Graphs At ζ = 1 Eq. (61) gives, D(z, 1; t) =

∞ X zn n=0

n!

en

2 t/2T

201

.

(67)

We thus come to the result, n ¯ g,ν (t) =



M g



e(g

2 −2M g+g)t/2T

(et/T − 1)ν Cg,ν .

(68)

Because the function D(z, 1; T ) coincides with the D–function corresponding to the spectrum of coagulating particles in the system with the kernel K(g, l) = gl considered in [17, 18, 19], we can easily derive the expression for the spectrum of linked components P over their orders (numbers of vertices) , n ¯g = n ¯ g,ν . The result has the same form as ν

the average particle mass spectrum in the coagulating system with the coagulation kernel K(g, l) = gl, n ¯ g (t) =



M g



e(g

2 −2M g+g)t/2T

(et/T − 1)g−1 Pg−1 (et/T − 1)

(69)

P

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In coagulating systems the total mass of this spectrum M = g g¯ ng is known to be conserved. After the critical time t = tc = 1 a giant particle forms with the mass comparable to M . Details see in refs [17, 18, 19]. Of course, the spectrum Eq. (69) contains this giant component. However, the time dependence of the distributions given by Eq. (68) differs from that given by Eq. (69) and the question comes up what does it go on to the partial distributions? Does a giant component present in it? In order to answer these questions let us apply the result Eq. (28) to the simplest situation ν = g − 1, i.e., all linked components are trees. In this case Cg,g−1 = g g−2 . The distribution of the trees is then given by the formula n ¯ g,g−1 (t) =



M g



e(g

2 −2M g+g)t/2T

(et/T − 1)g−1g g−2 .

(70)

In the thermodynamic limit (M, T −→ ∞, their ratio being constant, M/T = 1) Eq. (70) gives, g g−2 g−1 −gt t e . (71) n ¯ sg,g−1 (t) = M g! This is exactly the mass spectrum derived from the Smoluchowski equation with the coagulation kernel K(g, l) = gl. This spectrum does not contain a giant component at t > tc and the total mass is not conserved. The problem is then how to reconcile this fact with the statement that the giant component should appear? The answer to this question is simple. We loose the giant tree in the thermodynamic limit, but it presents in the exact spectrum Eq. (70). However, the probability to find this tree is small as Cg,g−1/ P Cg,ν . Nevertheless it is possible to see this giant tree in the exact ν spectrum Eq. (70). The idea is simple. If the exact spectrum Eq. (70) contains a hump at g ∝ M we can detect it against the background of the approximate spectrum Eq. (71) which nsg and exponentiate it, is a smooth function at such large g. We thus consider the ratio n ¯g /¯ n ¯g = exp[M Ω(µ, t)], n ¯ sg

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(72)

202

A.A. Lushnikov

where µ = g/M and 1 Ω(µ, t) = −(1 − µ) ln(1 − µ) − µ + µ2 t. 2

(73)

Differentiating Ω over µ and putting the result equal to zero we find the position µc of the maximum, 1 1 t= ln . (74) µc 1 − µc This is exactly the location of the giant component of the random graph [1] as well as the gel particle in the respective coagulating system [15].

7.

Bipartite Graphs

Similar results can be derived for bipartite random graphs. We assume that two sorts of vertices are randomly connected with edges. The edges connecting the vertices of the same sort are forbidden. A bipartite linked component is characterized by three integers, the numbers of vertices of two sorts, 1 ≤ m, n < ∞, and the number of edges m + n − 1 ≤ ν ≤ mn. Each state of the bipartite random graph is characterized by the set of occupation numbers (75) Q = (n1,0; 0, n0,1; 0, n1,1; 1, n2,1; 2...) = {nm,n: ν }. Let us begin to add the edges to the initially empty bipartite graphs. This process gives rise either to a coalescence of two linked components

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(m, n; ν) + (k, l; λ) −→ (m + k, n + l; ν + λ + 1)

(76)

or to filling a given linked component with one extra–edge, (m, n, ν) −→ (m, n; ν + 1)

(77)

˜ − −→ The time evolution of the graph goes along the route Q− −→ Q −→ Q+ or Q + ˜ . The coalescence process changes three occupation numbers in the preceding Q −→ Q − state Q nm,n; ν (Q− ) = nm,n; ν (Q) + 1,

nk,l; ν (Q− ) = nk,l; ν (Q) + 1,

nm+k,n+l; ν+λ+1 (Q− ) = nm+k,n+l; ν+λ+1 (Q) − 1. If an edge is added to a linked component then only two its occupation numbers change, nm,n; ν−1 (Q− ) = nm,n; ν (Q) − 1,

nm,n; ν (Q− ) = nm,n; ν (Q) + 1.

We again introduce the time dependent probability W (Q, t) and write down Eq. (7) wherein the transition rates A and B differ from those given by Eqs (8) and (9), A(Q, Q− ) =

1 (ml + nk)nm,n; ν (Q− )[nk,l; λ(Q− ) − δm,k δn,l δν,λ ] 2T

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Coalescence of Graphs

203

and

1 ˜ − ). (mn − ν + 1)nm,n; ν−1 (Q (79) T Now we introduce the generating functional Ψ(X, t) exactly in the same way as has been done in the case of uni–partite graphs (see Eq. (10)), ˜ −) = B(Q, Q

X

Ψ(X, t) =

Y

W (Q, t)

n(m,n; ν|Q) xm,n; ν

(80)

m,n; ν

Q

The equation for Ψ has the form of Eq. (13) with Lˆf =

X

(mn − ν)(xm,n; ν+1 − xm,n; ν )

m,n;ν

∂ ∂xm,n; ν

(81)

and 1X ∂2 (ml + nk)(xm+k,n+l: ν+λ+1 − xm,n; ν xk,l; λ) . Lˆc = 2 ∂xm,n; ν ∂xk,l; λ

(82)

Here the summation goes over all indexes m, n; ν and k, l; λ. The operators ˆ = M

X m,n

mxm,n; ν

∂ ∂xm,n; ν

,

ˆ= N

X

nxm,n; ν

m,n

∂ ∂xm,n; ν

(83)

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commute with Lˆf and Lˆc which mean the conservation of the graph orders, i.e., the numbers of vertices of each sort do not change with time. We construct the solution in the form, Ψ = M !N !Coef u,v [u−M v −N exp(

X

xm,n; ν am,n; ν (t)um v n )],

(84)

m,n;ν

where the coefficients am,n; ν (t) will be defined later on (see Eq. (88)). Here the operation Coef (see ref. [21]) is introduced as follows: Coef u,v

"

X

m n

bm,n u v

#

= b−1,−1.

(85)

m,n

The spectrum of linked components of order m, n with ν edges n ¯ m,n;ν (t) can be expressed in terms of am,n;ν (t) as follows:

∂Ψ({xm,n; ν }, t) n ¯ m,n; ν (t) = ∂xm,n; ν {x

m,n; ν }={1}

= M !N !am,n; ν (t)Coef u,v {u−M +m v −N +n exp[G(u, v; 1|t)]}. Here G(u, v; ζ|t) =

X

am,n; ν (t)um v n ζ ν

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(86) (87)

204

A.A. Lushnikov

is the generating function for am,n; ν (t). On substituting Ψ in the form (84) into Eq. (13) with Lˆf and Lˆc defined by Eqs (81) and (82) yields the set of equations for am,n; ν (t), T

m.n X ν−1 X dam,n; ν = (m − l)kam−l,n−k; ν−µ−1 (t)al,k; µ (t) + mnam,n; ν (t) dt k,l=0 µ=0

1 − (M n + N m)am,n; ν (t) + (mn − ν + 1)am,n; ν−1 − (mn − ν)am,n; ν . 2 This set is subject to the condition corresponding to the initially empty graph,

(88)

am,n; ν (0) = (δm,1δn,0 + δm,0 δn,1 )δν, 0 .

(89)

N It is easy to check that the condition (89) corresponds to Ψ|t=0 = xM 1,0; 0 x0,1; 0 . The equation for the generating function G can be readily derived from Eqs (87) and (88),

"

#





∂G ∂G ∂ 2G ∂G 1 ∂G ∂G ∂G =ζ u v + uv − (ζ − 1)ζ − Nu + Mv . T ∂t ∂u ∂v ∂u∂v ∂ζ 2 ∂u ∂v

(90)

The initial condition for this equation is, G(u, v; ζ|0) = u + v.

(91)

D(u, v; ζ|t) = exp[G(ueN t/2T , veM t/2T ; ζ|t)].

(92)

Now let

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Then, instead of Eq. (90) we derive a linear equation for D, T

∂ 2D ∂D ∂D = ζuv − (ζ − 1)ζ . ∂t ∂u∂v ∂ζ

(93)

The initial condition for this equation follows from Eq. (91), D(u, v; ζ|0) = eu+v .

(94)

Equation (93) is readily solved by separating variables. Let D(u, v; ζ|t) =

X

Θκ (t, )Zm,n,κ (ζ)umv n .

m,n,κ

Then Θκ (t) = eκt/T and κZm,n; κ = ζmnZm,n; κ + ζ(1 − ζ)

dZm,n; κ , dζ

(95)

where κ is a separation constant. The solution to this equation is Zm,n; κ (ζ) = bm,n; κ (1 − ζ)mn−κ ζ κ . Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

(96)

Coalescence of Graphs

205

The function D should be analytical at ζ = 0. Hence, κ = s, where s is a nonnegative integer. Next, the coefficients bm,n; κ should be chosen from the initial condition Eq. (94). It is easy to see that   1 mn bm,n; κ = . m!n! s We then come to the result, ∞ X um v n

D(u, v; ζ|t) =

m!n!

m,n=0

(1 − ζ + ζet/T )mn .

(97)

In order to return to ag,ν (t) we use the identity (see Appendix B) ln

X um v n m,n

m!n!

mn

(1 + δ)

∞ X um v n m+n−1 δ = Pm−1,n−1 (δ) m,n=1

m!n!

(98)

with δ = ζ(et − 1). Then Eqs (94) and (98) allow us to restore Am,n (ζ, t) =

X

am,n; ν (t)ζ ν .

ν

We find 1 −(mN +nM )t/2T ) t/T e (e − 1)m+n−1 ζ m+n−1 Pm−1,n−1 [ζ(et/T − 1)], m!n! (99) where the polynomial Pm,n (δ) is defined as the generating function for the numbers Cm,n;ν of a linked labelled bipartite graph of order m, n (see Appendix B), Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

Am,n (ζ, t) =

mn X

δ m+n−1 Pm−1,n−1 (δ) =

Cm,n;ν δ ν ,

(100)

ν=m+n−1

Equation (100) is readily applied for restoring am,n; ν (t). The result is am,n; ν (t) =

1 −(mN +nM )t/2T ) t/T e (e − 1)ν Cm,n; ν . m!n!

(101)

Now we are ready to find the average number of linked components of order m, n with ν edges. From Eq. (86) we have, n ¯m,n; ν (t) = M !N !am,n; ν (t)Coef u,v u−M +m v −N +n D(u, v; 1|t).

(102)

At ζ = 1 Eq. (97) gives, D(u, v; 1|t) =

∞ X um v n mnt/T e . m,n=0

m!n!

(103)

Hence, Coefu,v u−M +m v −N +n D(u, v; 1|t) =

exp(−mN t/2 − nMt/2 + mnt/V ) . (M − m)!(N − n)!

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(104)

206

A.A. Lushnikov

We thus come to the result, n ¯ m,n; ν (t) =



M m



N n



e(mn−mN −nM )t/T (et/T − 1)ν Cm,n; ν .

(105)

Because the function D(u, v, 1|t) coincides with the D–function corresponding to the spectrum of coagulating particles in the system with the kernel K(m, n; k, l) = mk + nl considered in [20], we can easily derive the expression for the spectrum of linked components P ¯ m,n; ν . The result has the same form as over their orders (numbers of vertices) , n ¯ m,n = n ν

the average particle mass spectrum in the coagulating system with the coagulation kernel K(g, l) = mk + nl, n ¯ m,n (t) =

8.



M m



N n



e(mn−mN −nM )t/T (et/T − 1)m+n−1 Pm,n (et/T − 1).

(106)

Bipartite Graphs in the Thermodynamic Limit

The derivation of the equation governing the time evolution of the random bipartite graphs in the thermodynamic limit (T = M + N ) does not differ from that that we just applied for uni–partite graphs (see Eq. (42)). The equation for the concentrations ¯m,n; ν (t)/(M + N ) cm,n; ν (t) = n of the bipartite graph of order m, n with ν edges repeats Eq. (42)

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m.n X ν−1 X dcm,n;ν = (m − l)kcm−l,n−k; ν−µ−1 (t)cl,k;µ(t) − (Mn + N m)cm,n;ν (t) dt k,l=0 µ=0

1 [(mn − ν + 1)cm,n;ν−1 (t) + νcm,n;ν (t)]. (107) M +N Here we introduced M = M/(M + N ), and N = N/(M + N ). Once again, as in the case of uni–partite graphs, one sees that the last term on the right–hand side of Eq. (107) does not contribute to the kinetics of the graph unless a giant linked component of order m, n ∝ M, N appears in the evolving graph. We thus come to the conclusion that the initially empty bipartite graph evolves exactly along the same scenario as the empty uni– partite graph. At the initial stage (below the critical time) the appearing new edges form trees. After the critical time the giant linked cluster emerges. It evolves by forming cycles, whereas the smaller linked components remain to be trees. It is also possible to formulate the equation for the total numbers of linked components P irrespective of their degree of filling cm,n (t) = ν cm,n;ν (t), +

m,n

X mncm,n dcm,n = k(m − l)cm−l,n−k (t)cl,k + − (Mn + N m)cm,n . dt M +N k,l=0

(108)

This equation also describes evolution of a coagulating mixture. The exact solution to this equation was found three decades ago in ref. [26]. It looks as follows: cm,n (t) = Mm N n

mn−1 nm−1 m+n−1 −(mN +nM)t t e . m!n!

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(109)

Coalescence of Graphs

207

In the limit of large m, n we get cm,n (t) ≈

1 2πtm3/2n3/2

eFs (m,n;t) ,

(110)

where Fs (m, n; t) = m(1 + ln(Mt) − N t) + n(1 + ln(N t) − Mt) + (n − m) ln

m . (111) n

It is easy to check that at t = tc = (MN )−1/2 √ √ √ √ M− N Fs (m, n; τc) = √ (m N − n M) MN √ m N −(m − n) ln √ . (112) n M √ √ At t = tc and m N = n M the function Fs is zero together with its first derivatives with respect to m and n, The spectrum becomes algebraic, as should it be at the critical point, √ MN . (113) cm,n (τc ) ≈ 2πm3/2n3/2

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After the critical point it is easy to find the deficits of vertices. To this end we consider the ratio of the exact spectrum of trees n ¯m,n;m+n−1 (Eq. (105)) to its thermodynamic limit (Eq. (109). n ¯ m,n;m+n−1 ≈ exp[−(M − m) ln(1 − m/M ) − (N − n) ln(1 − n/N ) − m − n + mnt/T ]. T c(m, n) This ratio is expected to have a hump at the critical values of m and n (see similar consideration for uni–partite graphs). Hence, on differentiating the above ratio over m and n and putting the result equal to zero we find the composition of the giant component νg = 1 − e−Mµg t,

µg = 1 − e−N νg t ,

(114)

where µg and νg are the mass concentrations of vertices involved into the giant√component. This set of equations has the solution µg (t) = νg (t) √ = 0 at t < tc = 1/ MN . The positive nontrivial solution exists only at t > tc = 1/ MN [20].

9.

Summary

The central results of this paper are: • Theory of stochastic coalescence of random graphs has been formulated. In contrast to commonly accepted approaches the main idea of the present one relies upon the description of the evolution of the random graph in terms of the probability W to find a given set of occupation numbers ({ng,ν } or {nm,n; ν } in the case of uni–partite

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208

A.A. Lushnikov graph and bipartite graph respectively) at time t. The Master equation for this probability Eq. (7) is reformulated in terms of the generating functional Ψ, for which the Evolution equation is derived. It has the form (see Eq. (13)); ∂τ Ψ = (Lc + Lf )Ψ, where the linear differential operators Lc and Lf Eqs (15), (16), (81), and (82) are responsible for coalescence (subscript c) of two linked components and for filling of a linked component respectively.

• This Evolution equation is solved exactly for uni– and bipartite random graphs. The solution is applied for restoring the average mass spectra of the linked components of the random graph. The results are (Eq. (68)), n ¯g,ν (t) =



M g



e(g

2 −2M g+g)t/2T

(et/T − 1)ν Cg,ν .

for uni–partite graphs and n ¯ m,n; ν (t) =



M m



N n



e(mn−mN −nM )t/T (et/T − 1)ν Cm,n; ν

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for bipartite ones (Eq. (105)). The coefficients C are just the numbers of linked components of fixed order with ν edges. These coefficients serve as the basis for introduction of special families of polynomials Pg (δ) (Eq. (115)) and Pm,n (δ) (Eq. (150)). These polynomials are well recognized to be of great importance for solving a number of combinatorial problems [23, 24, 25]. • A special attention is given to the thermodynamic limit M, T −→ ∞, with their ratio being constant. In this very limit the emergency of the giant component (in theory of random graphs) and gelation (in theory of coagulation) manifest themselves as a second–order phase transition. The analogy between evolution of coagulating system and time evolution of random graphs allows one to determine the critical time. To this end it is enough to derive and to solve a differential equation for the second moment of the particle mass distribution in the coagulating system with the kernel proportional to the product of the masses of colliding particles. Hence, it is important to answer the question, what is going on to the spectrum of linked components n ¯ g (tc ). The Smoluchowski equation gives the algebraic spectrum, M . n ¯ g (tc ) ≈ √ 2πg 5/2 The finiteness of M modifies this result as follows [17]: M 3 2 e−g /8M . n ¯ g (tc ) ≈ √ 5/2 2πg The exponential cutoff factor appears. Next, if we try to find the asymptotic behavior of the critical spectrum of trees (Eq. (70)), we find 1 3 2 e−g /6M . n ¯ g (tc ) ≈ √ 5/2 2πg There is a difference that reveals itself at g ∝ M 2/3. Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Coalescence of Graphs

209

• In the thermodynamic limit all linked components except the giant one are trees. Indeed, kinetics of their growth obeys the Smoluchowski equation (Eq. (40)). The coalescence process alone never produces something else except for trees if the preceding state contained only trees. This fact allows us to find the average number of edges in the giant component. To this end we derive the equation for the probability w(ν, t) to find exactly ν edges in the graph at time t. This equation is, T

dw = dt





M (M − 1) − ν + 1 w(ν − 1, t) − 2





M (M − 1) − ν w(ν, t). 2

The solution to this equation with the initial condition w(0, 0) = 1 is w(ν, t) =

ν¯(t) −¯ν (t) e ν!

where

M (M − 1) (1 − e−t/T ) 2 The total number of edges is distributed between trees and the giant component. Hence, the average number of edges in the giant component ν¯c (t) is the difference between the total number of edges and the number of edges in the trees, ν¯(t) =

ν¯c (t) = ν¯(t) −

X

¯ (t) (g − 1)¯ ng,g−1(t) = ν¯(t) − M (1 − µc (t)) + N

The number concentration of trees is [18] M 2t ¯ [1 − µc (t)]2, N(t) = M (1 − µc (t)) − 2T

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where M µc (t) is the order of the giant component (see Eq. (49)). Hence, ν¯c (t) = ν¯(t) −

M 2t (1 − µc )2 . 2T

At small t  T and T = M we finely have ν¯c (t) =

Mt [1 − (1 − µc )2 ]. 2

10. Conclusion The approach described above has been shown to be well suited for studying the properties of finite random graphs. I cannot boast that this approach is very simple and transparent. It includes several very risky (for mathematicians) steps like operations with infinite numbers of formal variables, the polynomials whose generating functions are not well defined (respective series are divergent), and the asymptotic estimates. Still all these steps are very common for theorists in physics (I mean quantum many–body problem, first of all). It seems to me that the above algebraic approach has many advantages as compared to the combinatorial considerations and I tried to demonstrate them in this Chapter. I hope to find the supporters of my approach, especially because too many unresolved problems wait for their turn.

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Appendix A. Polynomials Pg (δ) Definition and Exponential Generating Function Here I give a short introduction to theory of polynomials Pg (δ) appearing in the expressions for the mass spectra of coagulating particles. By definition of ref. [27] the polynomials Pg−1 (δ) are introduced as the generating functions for the numbers Cg,ν of linked labelled graphs of order g , g(g−1)/2

δ g−1 Pg−1 (δ) =

X

Cg,ν δ ν .

(115)

ν=g−1

The limits of summation in this equation are just a minimal number of edges in a tree of order g (lower limit) and the maximal number of edges g(g − 1)/2 in the complete graph (upper limit). The bivariate exponential generating function for the number of linked graphs with given numbers of vertexes and edges is defined as follows: w(ξ, δ) =

∞ g g(g−1)/2 X X ξ g=1

g!

ν

Cg,ν δ =

ν=g−1

∞ g X ξ g=1

g!

δ g−1 Pg−1 (δ).

(116)

According to the Riddell theorem [28] w(ξ, δ) = ln W(ξ, δ),

(117)

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with W(ξ, δ) being the exponential generating function of all labelled graphs. The  latter is  . readily found. Indeed, the number of ways to connect g vertices by ν edges is g(g−1)/2 ν Hence, the polynomial (1 + δ)g(g−1)/2 generates the numbers of graphs of order g having exactly ν edges. The exponential generating function for these graphs is thus W(ξ, δ) =

∞ g X ξ g=0

g!

(1 + δ)g(g−1)/2.

(118)

We thus come to the result [23] (see also Eq. (16) in ref. [28]) ln

∞ g X ξ g=0

g!

(1 + δ)g(g−1)/2 =

∞ g X ξ g=1

g!

δ g−1 Pg−1 (δ).

(119)

Integral Equation for the Generating Function Let us derive an integral equation for the exponential generating function of the polynomials Pg (δ), ∞ g X ξ Pg (δ) (120) y(ξ, δ) = g! g=1 From Eq. (116) it is clear, that ∂ξ w(ξ, δ) = y(ξδ, δ),

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(121)

Coalescence of Graphs

211

This fact helps us to derive the equation for y(ξ, δ). Let us differentiate both sides of Eq. (117) over ξ and notice that ∂ξ W(ξ, δ) = W[(1 + δ)ξ, δ]. The latter equality immediately follows from Eq (118). The result of these operations is

ln y(ξδ, δ) = w[(1 + δ)ξ, δ] − w(ξ, δ) =

(1+δ)ξ Z

y(xδ, δ)dx.

(122)

ξ

On replacing ξδ = ζ and x = ζ(1 + uδ)/δ reduces Eq. (122) to [17] ln y(ζ, δ) = ζ

Z1

y[ζ(1 + uδ), δ]du.

(123)

0

Recurrences Equations (117) and (121) allow for a derivation of a linear set of recurrence relations for the polynomials Pg (δ). The derivation applies the theorem: if ln

∞ X

∞ X

ak xk =

k=0

bk xk

k=0

then kak =

k X

(k − s)as bk−s

(124)

s=0

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On applying this theorem to Eqs (117) and (118) yields after a simple algebra [18, 19], g   X g δ m Pm (δ)(1 + δ)(m+1)(m−2g)/2 = 1.

(125)

m

m=0

Another recurrence relation comes up if we apply this theorem to Eq. (122) [25], g





1 X g−1 [(1 + δ)m − 1]Pm−1(δ)Pg−m (δ). Pg (δ) = δ m=1 m − 1

(126)

Recurrences for Derivatives of y(ξ, δ) Let us now expand y(ξ, δ) in the powers of δ, y(ξ, δ) =

∞ X δ k (k) k=0

k!

yδ (ξ, 0)

(127) (k)

Equation (123) allows us to derive the recurrence for determining the derivatives yδ (ξ, 0)

n X ξ m+1 ∂ n ln y(ξ, δ) = ∂δ n m+1 δ=0 m=0



n m



∂ m ∂ n−m y(ξ, δ)|δ=0 ∂ξ m ∂δ n−m

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It is important to notice that if we know the n–th derivative over δ then it is possible to find the n+m–th mixed derivative over δ (n times) and ξ (m times). It is easy to find first several (k) derivatives yδ and to discover that they have singularity of the type (1 − ln y(ξ, 0))s and that the maximal power s = 3k − 1. Equation (128) allows for expressing all derivatives of y at δ = 0 in terms of the combinations ξ a yob yob lna yo A(a, b, c) = = , (129) (1 − ln yo )c yoa (1 − ln yo )c where yo ξ) = y(ξ, 0). The latter equality follows from Eq. (123): ξ = ln(yo )/yo . For example, 1 yδ0 = A(2, 3, 2) 2 yξ0 = A(0, 2, 1)

(130) (131)

00 = 2A(0, 3, 2) + A(0, 3, 3) yξ,ξ

3 y”ξ,δ = A(1, 3, 2) + A(2, 4, 3) + A(2, 4, 4) 2

(132)

5 1 3 5 y”δδ = A(3, 4, 3) + A(3, 4, 4) + A(4, 5, 4) + A(4, 5, 5) 3 3 2 4

(133)

In deriving the above results the relations were used, A(a, b, c)A(x, y, z) = A(a + x, b + y, c + z)

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A0ξ (a, b, c) = aA(a − 1, b, c) + bA(a, b + 1, c + 1) + cA(a, b + 1, c + 2) Let us introduce [A(a, b, c)]g =

g! 2πi

I

A(a, b, c)dξ , ξ g+1

(134) (135)

(136)

where the integration goes counterclockwise along the contour surrounding the origin of coordinates in the complex plane ξ. After some tedious but straightforward transformations one gets (see next subsection) Ag (a, b, c) =

g!(g + b − a)g−a+c−1 (g − a)!(c − 2)!

Z∞

e−(g+b−a)u uc−2 (1 + u)g−a du

(137)

0

For example, Pg0 (0) =

1 A(2, 3, 2)g = g! expg−2 (g + 1), 2

(138)

where the truncated exponent is introduced as expG (x) =

G X xn n=1

n!

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Coalescence of Graphs

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Calculation of Ag We evaluate Ag (a, b, c) =

g! 2πi

I

A(a, b, c)dξ g! = ξ g+1 2πi

I

y b dy ξ g−a+1 (1 −

ln y)c

(140)

We replace ξ −→ y, ξ = ln y/y, dy ξ = (1 − ln y)/y 2, and then ln y = t, dy = et dt. Ag =

g! 2πi

I

et(b+g−a) dt tg−a+1 (1 − t)c−1

(141)

Let now α = b + g − a, Then 1 2πi =

I



∞ X eαtdt s+γ = β+1 γ+1 t (1 − t) γ s=0

 β  X s+γ αβ−s s=0

β = g − a,

γ

(β − s)!

=

g−a X s=0

γ = c − 2. 

1 2πi

c−2+s c−2



I

eαt dt = tβ−s+1

(g + b − a)g−a−s (g − a − s)!

Finally we have Ag (a, b, c) = g!

g−a X

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r=0

=

c−2+g −a−r c−2

g−a 



(g + b − a)r r!



X g−a g! (g + c − 2 − r − a)!(g + b − a)r (g − a)!(c − 2)! r=0 r

g! = (g − a)!(c − 2)!

Z∞

e−x xc−2 (x + g + b − a)g dx

(142)

0

From here we find Eq. (137) The result is especially simple for a = b = 0, g!

 g  X g + c − 2 − r gr r=0

c−2

g

 

X g 1 = (g + c − 2 − r)!g r r! (c − 2)! r=0 r

1 = (c − 2)!

Z∞

e−x xc−2 (x + g)g dx

0

and g g+c−1 Ag (0, 0, c) = (c − 2)!

Z∞

e−gu uc−2 (u + 1)g du

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Asymptotic Behavior of Pg (δ) Equation (125) allows us to derive the asymptotic formula for Pg (δ) in the limit of large g and small δ, gδ being finite. The idea is very simple. The maximal value of each term in the sum on the rhs of Eq. (125) should be of order unity. If we represent each term in the exponential form e−gΦ , use the Stirling formula for the binomial coefficients, and introduce µ = m/g, α = gδ Ψ = ln[δ m Pm (δ)], we find that Φ = µ ln µ + (1 − µ) ln(1 − µ) − µΨ(µα) −

α µ(µ − 2) = 0 2

(144)

and Φ0µ = 0, ln µ − ln(1 − µ) − µαΨ0 (µα) − Ψ(µα) − α(µ − 1) = 0

(145)

We introduce the variable x = µα and two unknown functions, µ(x) and Ψ(x). We obtain two equations for determining these functions µ ln µ + (1 − µ) ln(1 − µ) − µΨ −

x (µ − 2) = 0 2

(146)

and ln µ − ln(1 − µ) − (xΨ)0 − x(1 − 1/µ) = 0.

(147)

It is easy to check that

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µ(x) = 1 − e−x ,

and

Ψ(x) = ln[2 sinh(x/2)]

(148)

is the solution to Eqs (145) and (146). It is also important to notice that if we return to the variables µ, t, then there are two solutions for µ, µ = 0 for all t and µ = 1 − e−2µt for t > tc . From the second Eq. (148) we finally find that at large g and small δ [17, 18, 19] Pg (δ) ∝

sinhg (gδ/2) (δ/2)g

(149)

Concluding Comments The first five P –polynomials are P0 (δ) = P1 (δ) = 1, and P2 (δ) = 3 + δ,

P3 (δ) = 16 + 15δ + 6δ 2 + δ 3

P4 (δ) = 125 + 222δ + 205δ 2 + 120δ 3 + 45δ 4 + 10δ 5 + δ 6 The polynomials Pg (δ) are commonly introduced via the Mallows–Riordan polynomials Fg (1 + δ) = Pg (δ) and Eq. (126) as the definition of these polynomials [23, 24, 25]. Then Eqs (115) and (119) appear as a consequence of (126).

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Appendix B. Polynomials Pm,n (δ) Definition and Exponential Generating Function The polynomials Pm−1,n−1 (δ) are introduced similarly. By definition δ m+n−1 Pm+n−1 (δ) is the generating functions for the numbers Cm,n;ν of linked labelled bipartite graphs of order m, n, mn X

δ m+n−1 Pm−1,n−1 (δ) =

Cm,n;ν δ ν .

(150)

ν=m+n−1

The limits of summation in this equation are just a minimal number of edges in a tree of order m, n (lower limit) and the maximal number of edges mn in the complete graph (upper limit). The exponential generating function for the number of linked bipartite graphs is defined as follows: ∞ X ξ m η n m+n−1 δ Pm−1.n−1 (δ).

w(ξ, η; δ) =

m,n=1

m!n!

(151)

Again, according to the Riddell theorem [28] w(ξ, η; δ) = ln W(ξ, η; δ),

(152)

with W(ξ, η; δ) being the exponential generating function of all labelled bipartite graphs. The latter is readily found. Indeed, the number of ways to connect m, n vertexes by ν edges mn
is ν . Hence, the polynomial (1 + δ)mn generates the numbers of graphs of order m, n having exactly ν edges. The exponential generating function for these graphs is thus

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W(ξ, η; δ) =

X ξmηn m,n

m!n!

(1 + δ)mn .

(153)

Here the summation goes over all nonnegative integers m, n. Hence, ln

X ξm ηn m,n

m!n!

(1 + δ)mn =

∞ X ξ m η n m+n−1 δ Pm−1,n−1 (δ). m,n=1

m!n!

(154)

Let us now introduce x = 1 + δ. Then Eq. (153) takes the form, W(ξ, η; x) =

X ηn n

n!

n

eξx = eξ

Hence, the function w(ξ, η) = ξ + ln

X ηn n

" k X ηn n=0

n!

n!

ξ(xn−1)

e

n −1)

eξ(x #

generates all cm,n (x) with n ≤ k, where cm,n (x) = (x − 1)m+n−1 Pm−1,n−1 (x − 1) For example, ∂η w(ξ, η)|η=0 = e(x−1)ξ

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(156)

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A.A. Lushnikov

and thus cm,1 = (x − 1)m . Next, 2 −1)ξ

2 ∂η,η w(ξ, η)|η=0 = e(x

− 2e2(x−1)ξ .

This function generates all cm,2 cm,2 = (x − 1)m [(x + 1)m − 2m ] It is not so difficult to continue in this spirit, cm,3 = (x − 1)m [(x2 + x + 1)m − 3(x + 2)m + 2 · 3m ]. In particular, we find, c2,2 = (x − 1)3(x + 3) c23 = (x − 1)4(x2 + 4x + 7), c33 = (x − 1)5(x4 + 5x3 + 15x2 + 29x + 31).

(157)

Sum Rules and Recurrences Equation (152) allows for deriving two sum rules for cm,n (x). It is evident that W∂ξ w = ∂ξ W

and

W∂η w = ∂η W.

(158)

On substituting here expansions (153) and (154) gives,

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m,n X



m p

 

m,n



X m n x−pn−qm+pq cp+1,q (x) = q p p,q=0

 

n x−pn−qm+pq cp,q+1 (x) = 1. q p,q=0 (159) These identities are of use in deriving the expression for the composition spectrum in the thermodynamic limit. Now let us derive the set of recurrence relations for cm,n (x) (Eq. (164), it is easy to see that ∂ξ W(ξ, η; x) = W(ξ, xη; x) and ∂η W(ξ, η; x) = W(xξ, η; x).

(160)

Hence ∂ξ w(ξ, η, x) = ew(ξ,xη;x)−w(ξ,η;x), and ∂η w(ξ, η, x) = ew(xξ,η;x)−w(ξ,η;x).

(161)

On combining equations (161) and (153) yields, X

"

#

X ξ mη n ξ mη n n = exp (x − 1) . cm+1,n (x) cm,n (x) m!n! m!n! m,n m,n Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

(162)

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217

Next, we apply equation (161) for finding ∂ 2 w/∂ξ∂η, X X ξm ηn X ξ α η β β+1 ξγ ηδ ∂ 2w = cm+1,n+1 = cα,β+1 (x − 1) cγ+1,δ . ∂ξ∂η m,n m!n! α!β! γ!δ! α,β γ,δ

(163)

The summations in two above equations go over all nonnegative integers. Equation (163) is equivalent to the set of recurrence relations, cm+1,n+1 =



m,n X p,q=0

m m−q





n cm−q,p+1 cq+1,n−p (xp+1 − 1). n−p

(164)

Polynomials Fm,n (x) At this stage we introduce the polynomials Fm,n (x). They are defined as cm+1,n+1 (x) = (x − 1)m+n+1 Fm,n (x),

(165)

or Pm,n (δ) = Fm,n (1 + δ) On substituting equation (165) into equation (164) gives the recurrence for the polynomials Fm,n (x), Fm+1,n+1 (x) =

m,n X



p,q=1

m+1 m+1−q



The lowest order polynomials are, F0,0



n+1 xp+1 − 1 Fm−q,p (x)Fq,n−p (x) . n+1−p x−1 (166) = 0, F0,1 = F1,0 = 1, and

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F1,1(x) = x+3, F1,2 (x) = F2,1(x) = x2 +4x+7, F2,2(x) = x4 +5x3+15x2 +29x+31, ... We finally come to the central identity allowing for deriving the exact mass spectrum, ln

X ξm ηn xmn m,n

m!n!

!

=

∞ X ξmηn m,n=1

m!n!

(x − 1)m+n−1 Fm−1,n−1 (x) + ξ + η.

(167)

Generating Function and Integral Equations The exponential generating function for the polynomials Fm,n (x) is introduced as Φ(ξ, η; x) =

∞ X

Fm−1.n−1 (x)

m,n=1

ξmηn + ξ + η. m!n!

(168)

On the other hand, using equation (165) we find, Φ[(x − 1)ξ, (x − 1)η; x] = (x − 1)w(ξ, η; x).

(169)

Equation (161) allows us to derive the set of equations for Φ(ξ, η; x). We introduce (x − 1)ξ = a and (x − 1)η = b. Then Eq. (169) can be rewritten as 



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On differentiating both sides of this equation with respect to a and b and using Eq. (161) yield the set of functional equations for Φ, 







Φ(a, xb; x) − Φ(a, b; x) ∂Φ(a, b; x) = exp , ∂a x−1 ∂Φ(a, b; x) Φ(xa, b; x) − Φ(a, b; x) = exp . ∂b x−1

(171)

Equations (171) can be rewritten in a more symmetric form, ln Xδ (a, b) = b

Z1

Yδ [a, (1 + uδ)b]du,

0

ln Yδ (a, b) = a

Z1

Xδ [(1 + uδ)a, b]du.

(172)

0

Here δ = x − 1 and Xδ (a, b) = ∂a Φ(a, b; 1 + δ),

Yδ (a, b) = ∂b Φ(a, b; 1 + δ).

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It is clear that Xδ and Yδ are the bivariate exponential generating functions for the polynomials Pm,n−1 (δ) = Fm,n−1 (1 + δ) and Pm−1,n (δ) = Fm−1,n (1 + δ), Xδ (ξ, η) =

X

Pm,n−1 (δ)

ξmηn , m!n!

Yδ (ξ, η) =

X

Pm−1,n (δ)

ξ mη n . m!n!

(173)

Pm,n (0) = (m + 1)n (n + 1)m.

(174)

It is easy to show that

Indeed, let us introduce u(ξ, η) = ln X(ξ, η; 0) and v(ξ, η) = ln Y (ξ, η; 0). From equation (172) we find, u = ηev ,

v = ξeu .

Then we can find Pm,n (0) from the following chain of equalities, Pm,n (0) = m!(n + 1)!Coef ξ,η

= m!(n + 1)!Coef u,v

X(ξ, η; 0) ξ m+1 η n+2

eu(m+2)+v(n+2) −(u+v) e (1 − uv). v m+1 un+2

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(175)

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Elementary algebraic transformations prove equation (174). The multiplier e−(u+v) (1−uv) in the last term is just the Jacobian appearing in replacing the variables ξ, η −→ u, v. It is easy to find the first several polynomials Pm,n (δ). From Eq. (157) we have, P1,2(δ) = P2,1 (δ) = δ 2 + 6δ + 12,

P1,1(δ) = δ + 4,

P2,2 (δ) = δ 4 + 9δ 3 + 36δ 2 + 78δ + 81, . . . The generating function Φ(ξ, η; 1) can be expressed in terms of X0 (ξ, η) and Y0 (ξ, η) as follows: (176) Φ(ξ, η; 1) = ln X0 + ln Y0 − ln X0 ln Y0 In order to prove this identity it is enough to differentiate Eq. (176) with respect to η. The left–hand side gives simply Φ0η (ξ, η; 1) = Y0 (ξ, η). We will show that the right–hand side of this equation reproduces the same result. Indeed, as it follows from Eq. (172) X0 and Y0 obey the set of transcendent equations, ln X0 = ηY0 ,

ln Y0 = ξX0

These equations allow us to find the derivatives, Y0 ∂X0 = , ∂η 1 − ln X0 ln Y0

∂Y0 Y0 ln Y0 = ∂η 1 − ln X0 ln Y0

Now we differentiate the right–hand side of Eq. (176),

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ln0η X0 + ln0η Y0 − ln0η X0 ln Y0 − ln X0 ln0η Y0 = ln0η X0 (1 − ln ηY0 ) + ln0η Y0 (1 − ln ηX0) =

Y0 (1 − ln Y0 ) + Y0 ln Y0 (1 − ln X0) = Y0 . 1 − ln X0 ln Y0

This result proves Eq. (176)

Asymptotic Behavior of Pm,n (δ) Now we use Eq. (159) for deriving the asymptotic formula for the polynomials Pm,n (δ) in the limit m, n −→ ∞, δ −→ 0, mδ, nδ < ∞. The answer can be easily guessed Pm,n (δ) ∝ mn nm hm (nδ)hn (mδ),

(177)

where h(x) =

sinh x/2 . x/2

(178)

In order to prove equation (177) we rewrite the first equation (159) in the exponential form, X 0 0 eΩ(m,n,p ,q ;δ) = 1, (179) p,q Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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where p0 = p/m, q 0 = q/n, and Ω(m, n, p0, q 0; δ) = −m[(1 − p0 ) ln(1 − p0 ) + p0 ln p0 ] − n[(1 − q 0) ln(1 − q 0) + q 0 ln q 0 ] −mn(p0 + q 0 − p0q 0 )δ + mp0 H(nq 0δ) + nq 0 H(mp0δ).

(180)

The idea is that the terms of the order of unity contribute to the right–hand side of equation (179). Next, these terms correspond to the maximum of eΩ . We assumed also that Cp,q ≈ epH(qδ)+qH(pδ), in accordance with equation (177) Let us analyze the condition for the maximum of Ω with respect to p0. It gives, −m[− ln(1 − p0) + ln p0] − mnδ(1 − q 0)+ mH(nq 0δ) + mnq 0 δH 0(mp0δ) = 0.

(181)

Now we introduce the variables x = mp0 δ and y = nq 0 δ and rewrite equation (181) in terms of these variables, ln(1 − p0) − ln p0 −

y + y + H(y) + yH 0(x) = 0. q0

(182)

x + x + H(x) + xH 0 (y) = 0. p0

(183)

Similar equation holds for q 0 , ln(1 − q 0 ) − ln q 0 − Together with the condition

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Ω(m, n, p0, q 0; δ) = 0.

(184)

we have three equations for determining p0 , q 0 and H as the functions of x and y. Equations (182) and (183) can be resolved under the condition that p0 (x, y) = f (y) and 0 q (x, y) = f (x). In this case a separation of variables is possible and both Eqs (182) and (183) give one and the same couple of equations for f (ζ) and H(ζ), 1 ln(1 − f ) − ln f + ζ + H(ζ) = − H 0(ζ) = a. ζ f

(185)

Here a is a separation constant. Its value determined from the condition (184) is a = 1/2. Then we find from Eq. (185), f (ζ) = 1 − e−ζ

and

H(ζ) = ln(2 sinh(ζ/2)).

(186)

It is easy to check that 0

p0 (x, y) = 1 − e−nq (x,y)δ ,

0

q 0(x, y) = 1 − e−mp (x,y)δ .

(187)

meet equation (187). At small ζ we find, H(ζ) ≈ ln ζ +

ζ2 . 24

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(188)

Coalescence of Graphs

221

References [1] Erd¨os P. and R´enyi A. A Magiar Tydomanyos Akatemia Matematikai Kutato Intezetenek Kzlemenyei 5 17–61 (1960). [2] Bollob´as B. Random graphs, Academic Press, London (1985). [3] Albert R. and Barab´asi A–L. Rev. Mod. Phys. 74 47–97 (2002). [4] Janson S. Luczak T., and Rucinski A. Random Graphs, Wiley, New York (2000). [5] Aldous D.J. Bernoulli 5 3–122 (1999). [6] Bollob´s B., Grimmet G., and Janson S. Probab. Theory Related Fields 104 283–317 (1996). [7] Han D. J. Phys. A: Math. Gen. 36 7485–7498 (2003). [8] Janson, S., Knuth, D.E., Luczak, T., and Pittel, B. Random Struct Algorithms 4, 233– 358 (1993). [9] Newman M.E.J., Strogatz S.H., and Watts D.J. Phys. Rev. E 64, 026118 (1–17) (2001). [10] Engel A., Monasson R, and Hartmann A.K. J. Stat. Phys. 117 387–426 (2004). [11] Spouge J. L. J. Stat. Phys. 38 573–588 (1985).

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[12] Ben–Naim E. and Krapivski P.L. Phys. Rev. E 71 026129 (1–11) (2005). [13] Buffet E. and J.V. Pul´e J.V. J. Stat. Phys. 64 87 – 110 (1991). [14] Marcus A.H. Technometrics 10 133–143 (1968). [15] Lushnikov A.A. J. Colloid Interface Sci. 65 276–285 (1978). [16] Lushnikov A.A. J. Phys. A: Math. Gen. 38 L777–L782 (2005). [17] Lushnikov A.A. Phys. Rev. Lett. 93 198302 (1–4) (2004). [18] Lushnikov A.A. Phys. Rev. E 71 046129 (1-11) (2005). [19] Lushnikov A.A. J. Phys. A: Math. Gen. 38 L35–L39 (2005). [20] Lushnikov A.A. J. Phys. A: Math. Gen. 38 L383–L387 (2005). [21] Egorychev G.P. Integral representation and counting the combinatorial sums (in Russian) Nauka, Moscow (1977). [22] Niven I. Amer. Math. Monthly 76 871–889 (1969). [23] Knuth D.E. Algorithmica 22 561–568 (1998). Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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[24] Flajolet P., .Poblete P., and Viola A. Algorithmica 22 490–515 (1998). [25] Kreweras G. Period. Math.Hungar. 11 309–320 (1980). [26] Domilovskii E.R.,, Lushnikov A.A., and Piskunov V.N. Dokl. Akad. Nauk SSSR (Sov. Phys. Doklady) 243 407–410 (1978). [27] Lushnikov A.A. Physica D 322 37–53 (2006).

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[28] Riddell R.J. and Ulenbeck G.E. J. Chem. Phys. 21 2056–2064 (1953).

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ISBN 978-1-60741-011-9 c 2010 Nova Science Publishers, Inc.

Chapter 7

G RAPH A NALYSIS WITH A PPLICATION TO E CONOMICS Andr´e A. Keller Universit´e de Haute Alsace, Campus de la Fonderie F68093 Mulhouse - France

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Abstract The structural properties of large size nonlinear macroeconometric models are essential to their comprehensive analysis, to their adequate simulations of economic policies, and even to their iterative resolution. In this study, the concepts and techniques of the graph theory are used to discover such essential properties. Basic elements of graph theory are first introduced, using small-size theoretical models and sub-models. Further advanced topics are developed thereafter, such as the perfect matchings, the connectivity of weighted graphs and the circuit enumeration. A large-size macroeconometric model such as the FAIR model for the USA is also analysed using directed graphs. Avertex typology is deduced from the properties of a giant strong connected component. Exhaustive lists of circuits illustrate a great interdependency of the variables. The temporal dimension of such econometric model with lags is also considered. One appendix is devoted to the performances of specialized packages for graphs of the software Mathematica V6.0. The reference Klein-Goldberger model is used for that demonstration.

1.

Introduction

This chapter explores the structure of macroeconomic models, using major concepts and algorithms of the graph theory (Gallo & Gilli [13], Gilli & Rossier [15], Greenberg & Maybee [17], Keller [26][27][29]). A vast literature introduces the definitions and theorems in handbooks (Gross & Yellen [19], Schrijver [44]), upon graphs and digraphs (Harary [21], Chartrand & Lesniak [6]), with applications in Gross & Yellen [20], and algorithms in Even [10], Kocay & Kreher [35],Sedgewick [45], Gondran & Minoux [16]. This contribution concerns the matching problem of economic models (Keller [28]), the construction of a more outstanding graph embedding by using a longest circuit, the enumeration of all the circuits (Keller [31]), the determination of the maximal list of edge-disjoint circuits and the introduction of a vertex typology based on the properties of all-pairs shortest

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paths matrices, the evaluation of the cohesion and vulnerability of weighted graphs when deleting some inessential arcs (Keller [30]), the extension to short and long run dynamic structures (Keller [32]). Different sizes of applications are considered. Small-size applications are those of a CS sub-model for the Netherlands, the Klein I reference model and a simplified walrasian model. The large-size application 1 considers the macro-econometric FAIR model for the USA by Fair [11] with 130 equations. The intermediate-size KleinGoldberger model (Klein & Goldberger [33][34]) with 26 equations in used in Appendix D. to evaluate the performances of specialized packages for graphs such as DecisionAnalysis‘Combinatorica (Pemmaraju & Skiena [42], Skiena [46]) and DiscreteMath‘GraphPlot. The computations are carried out using the software M AT HEM AT ICA V5.1 and V6.0 (Wolfram [54]). This chapter is divided into three main parts. In the first part, essential of graph theory is introduced with elementary examples and small-size economic models. The main subjects are the graph building and embeddings, the characteristics of a graph. In the second part, further topics of graph theory are introduced, such as elements of matching, connectivity and circuit theories. In the third part, graph theory is applied to the large-size FAIR macro-econometric model.

2.

Graph Theory and Small Size Applications

2.1.

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2.1.1.

Graph Building and Embeddings Preliminaries and Notations

A digraph of order n and size m is a pair G = (V, E), where V is a finite set of vertices (with cardinality |V (G)| = n) and a set of ordered pairs from V (with cardinality |E(G)| = m). The oriented edges ek ∈ E, k = 1 . . . , m (k ∈ Nm ) are arcs. A forward arc is such as ek = (xk−1 , xk ) and ek = (xk , xk−1 ) denotes a backward arc. A chain c of G (or edgesequence) from vertex v0 to vk is an alternating sequence of vertices and arcs such that c = (v0 e1 v1 e2 . . . vk−1 ek vk ), tail(ei ) = vi−1 and head (ei ) = vi for i = 1, . . . , k (i ∈ Nk ). The chain is elementary if each vertex v appears only once. A directed graph G is the triple (V, E, f ) where f is a function to be defined. Precisely, a directed graph 2 is defined when f : E 7→ V ×V . For f (e) = (v1 , v2 ), the edge e is incident out of the vertex v1 and incident into the vertex v2 . The vertex v1 is said adjacent to v2 . Two adjacent edges have a common vertex, such as e1 = (v0 , v1 ) and e2 = (v1 , v2 ) since they share v1 . Simple graphs, without self-loops and parallel edges, will be considered in this chapter 3 . The out-degree denotes the number of edges incident out of some vertex v, whereas the in-degree is the number of edges incident into some vertex v. The directed distance d(u, v) is the length of the shortest 1

In scientific and engineering applications, large graphs generally involve millions of vertices. The Web graph contains more than one billion vertices, each representing a URL. Drawing and exploring such large graph have been soon examined by T. Munzner in: IEEE Computer and Applications, 18(4): 18-23. In this chapter, the concept of large graph has been much restricted to a hundred of equations, due to personal computation limitations. It could be easily extended to larger multi-country economic models with several thousands of equations. 2 The function f for an undirected graph is defined by f : E 7→ {{u, v}|u, v ∈ V }. 3 Self-loops are edges e such that for one vertex v, f (e) = (v, v). Parallel edges are such that e1 6= e2 and f (e1 ) = f (e2 ).

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Figure 1. Example of graph representations. path between the two vertices u and v 4 . The underlying undirected graph G0 = (V, E 0 ) of G = (V, E) is obtained by ignoring the direction of the arcs. The connectivity of such an underlying graph refers to the concept of weakly connectivity. Deleting an edge e yields G′ = (V, E ′ , f ′ ) where E ′ = E − {e}, f ′ being a restriction of f on E ′ (Prabhaker & Narsingh [43]). Deleting a vertex v renders G′ = (V ′ , E ′ , f ′ ), where V ′ = V − {v}, E ′ = {e ∈ E|f (e) 6= (v, u) or (u, v), for u ∈ V } and f ′ : E ′ 7→ V ′ × V ′ a restriction of f to E ′ . The n × n adjacency matrix A of G has entries ( 1 if (vi , vj ) ∈ E, aij = 0 otherwise. The adjacency list representation of an n-vertex graph consists of recording n lists, one list for each vertex containing all its adjacent vertices. Figure 1 shows a digraph and the 0-1 adjacency matrix. The dictionary of a graph needs less memory storage and is a more useful representation for computations. It consists of two lists of vertices : a list of addresses for each of the five vertices {1, 5, 8, 14, 17}, and a set of sub-lists, with one for each vertex {2,0,2,4, 1, 0, 3, 4, 0, 1, 2, 4, 5, 1, 0, 5, 0}. A sub-list contains successively the number of successors, a zero separator and the set of successors. It is reduced to a zero element if there are no successors. In this example, the vertex 1 has 2 successors 2 and 4 (in bold characters); the vertex 5 at address 17 has no successors and has a zero as sub-list. Different types of graphs will be considered in the applications with adequate simplified notations. Let V and E be the cardinal of the sets of vertices and edges, respectively, the notation G = (V, E) will be for graphs and digraphs, G0 = (V, E) for the underlying 4

The distance d(u, v) satisfies the three properties :(i) d(u, v) = ∞ if no path exists from u to v, (ii) d(u, v) = 0 ⇒ u = v, and (iii) the directed triangle inequality d(u, v) + d(v, w) ≤ d(u, w). Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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undirected graph, H = (V, E) for acyclic graphs, SCC = (V, E) for strong connected components, B = (V, E) for bipartite graphs, T = (V, E) for trees and N = (V, E) for a network. 2.1.2.

Application to the CS Sub-model

A small-size macro-economic model is the yearly Conjonctural-Structural (CS) industry sub-model (Van der Giessen [14]) for the Netherlands. Most of the variables are expressed in percentage change 5 . The set of variables consists of endogenous and exogenous variables. The variables are instantaneous, and may be delayed by several periods of year. The complete system is given by a set of econometric equations and definitions l˙ = 0.086 + 0.22 (p˙c + p˙c−1 + p˙c−2 ) + 0.79 h˙ − 1 − 0.09 w− 1 + l˙au , 2

˙ 1 + 0.25 pmgr ˙ + 0.072 pmc ˙ + pcau ˙ − ( p˙c = 0.006 + 0.5 (l˙ − h) − 2

n∗b−1 v−1

−

n∗b−2 v−2

y ab h˙ = ÷ − 1, y−1 ab −1 a˙b = c˙∗p − 0.4 ∆q + ∆abatv − 0.033 − 0.03

),

(2) (3)

−5 X i=0

(

1 + l˙ ) − 1, 1 − p˙i−wo

(4)

ab = ab−1 + ∆ab ,

(5)

al = al−1 + ∆ab − ∆az ,

(6)

∆Lb = ( Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

(1)

2

al (l˙ + 1) − 1) Lb−1 . al−1

(7)

The endogenous variables are: ab and a˙ b for the employment (with a dot for percentage changes), al for the dependent employment, h˙ for the labor productivity, l˙ for the wage level, Lb for the disposable wage income, and p˙c for the price level of private consumption. The exogenous variables are: abatv for the employment induced by change in working hours, az for the self-employment working population, c∗p for the productive capacity adjusted for autonomous influences, lau for the autonomous wage level, pcau for the autonomous price level of private consumption, pi−wo for the price level of imported goods (excluding housing), n ∗b /v for the ratio of stock (excluding live stock) to the total sales, pmc for the price level of imported goods, pmgr for the price level of imported raw material, and y for the gross industrial product. Equations (1), (2) and (4) are reaction functions, while the other equations are definitions. The CS sub-model is concentrated on the determination of wages, prices, labor productivity and employment in industries. Equation (1) is a Phillips-Lipsey function where wages rates are depending on prices (with lags over a period of two years), unemployment and 5

The variables on population and unemployment are in numbers (×100, 000). A dot above a symbol refers to percentage changes (divided by 100), such that l˙ = (l − l−1 )/l−1 , where l−1 is the value of l at the previous time period. The variable p˙ c−1 is delayed by one year, p˙ c−2 by two years. A variable h− 1 is (h + h−1 )/2, 2 and ∆q = q − q−1 . Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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labor productivity (with a lag of six months). Equation (2) is a price equation of usual type, where prices of expenditures are explained by wage costs, labor productivity and import prices. Prices are also influenced by stock positions. Equation (4) is for labor demand by industries. Labor demand depends on production capacity, real wage costs (with lags over a period of 5 years) and a technological trend. In this equation, real wages account for factor substitution. Thereafter, a static version of the CS sub-model is obtained as in van der Giessen [14] replacing all the predetermined variables by their observed values (yearly 1957 figures). A normalized system 6 is obtain when a variable in the LHS of each equation is calculated, once and only once, by the variables of the RHS. ˙ l˙ = 0.087 + 0.22 p˙c + 0.395 h, ˙ p˙c = 0.04 + 0.25 (l˙ − h), 37.26 h˙ = − 1, ab ∆ab = 0.297 − 0.1036 l,˙ ab = 36.02 + ∆ab , al = 26.69 + ∆ab , ˙ ∆Lb = −11.9 + 0.45 al (1 + l). Figure 2(a) shows the interactions between variables.

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2.1.3.

Circular Embeddings and Condensation

In a circular embedding, the vertices are placed computing evenly spaced points on the circumference of a unit circle. No three vertices will then be collinear. An illustration is given in Figure 2(a) for the directed graph G = (7, 10) with 7 vertices and 10 oriented edges (or arcs)7 . Definition 1 (Connected Components). Connected components of a graph are maximal connected subgraphs. Definition 2 (Strongly Connected Component). The strongly connected component of a graph (SCC) is a maximal strongly connected subgraph, where the vertex sets partition the set V and do not include all edges of G. Contracting a subset L of k vertices in G gives a graph H with n − k + 1 vertices. Each vertex v in L is mapped to vertex n − k + 1 in H. Every other vertex v in G is mapped to vertex v − i in H, if there are i vertices in L smaller than v. This mapping is constructed in O(n) time and creates the edges of H, in time proportional to the number of edges in G. In Figure 2(a), the maximal SCC is shown as a set of big gray points. The directed acyclic 6

The normalized system tells us which equation will calculate each endogenous variable of the model. This form is generally deduced from the economic theory and may be not unique. 7 This graph is simple since it has neither self-loops nor multi-arcs. It is a sparse graph since the cardinalities of the sets V and E are close together. Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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Figure 2. Graphs of the CS sub-model. graph (DAG) is achieved when contracting these sets. PROPOSITION: the condensation of a digraph is a directed acyclic digraph (DAG) which contains no circuits. This operation may produce multiple parallel edges. Contracting a pair of vertices v1 and v2 replaces them by one vertex v such that it is adjacent to any adjacent vertex of v1 and v2 8 . The contracted graph of the CS sub-model is shown in Figure 3.

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2.2.

Essential Graph Theory

Besides the contraction of a digraph to an acyclic DAG, other structural properties may be obtained when exploring the giant strong connected component (SCC). 2.2.1.

All-Pair Shortest Paths Matrix

The computation of the all-pairs shortest paths matrix is essential to judging the eccentricity of a giant SCC. This squared matrix collects all finite distances d(x, y) defined as the length of the shortest x − y paths using the Dijkistra’s algorithm or the Bellman-Ford algorithm (Pemmaraju & Skiena [42])9 . Several graph invariants are depending on this distance matrix. The all-pairs shortest paths matrix is shown in Figure 4(b) for the CS sub-model. 2.2.2.

Eccentricity, Central and Peripheral Vertices, Cut-points

The eccentricity for each vertex v is the maximum of the shortest paths starting from v. 8

Let G be an n-vertex graph and a subset S of k vertices to contract. The resulting graph H has n − k + 1 vertices. Each vertex in S is mapped to the n − k + 1 vertices of H. Contract runs in linear time O(|V |). 9 The all-pair shortest paths can be computed in O(|V |2 .|E|) time using the Bellman-Ford algorithm or in O(|V |3 ) time using the Dijkistra’s algorithm. Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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Figure 3. Contracted graph of the CS sub-model.

Definition 3 (Eccentricity). The eccentricity of a vertex v in graph G ecc(v) is the distance from v to any farthest vertex from it. Hence, we have ecc(v) = max {d(v, x)}. x∈V (G)

Definition 4 (Radius, Center, Diameter, Periphery). 1. The radius is the minimum eccentricity rad(G) = min {ecc(x)}. Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

x∈V (G)

The center of a graph G is the subgraph Z(G) induced on the set of central vertices. 2. The diameter is the maximum eccentricity diam(G) = max {ecc(x)}. x∈V (G)

The periphery of graph G is the subgraph per(G) induced on the set of peripheral vertices. In this application, ecc(v) = {3, 4, 3, 3, 3}, rad(SCC) = 3, diam(SCC) = 4. The ˙ ∆ab , ab }. A peripheral vertex is p˙c . The center is central vertices are Z(SCC) = {l,˙ h, shown with big gray points in Figure 4(a) . Definition 5 (Strongly Connected Digraph). A directed graph is strongly connected if there is an oriented path between every pair of vertices. A directed graph is weakly connected if there is a path between each pair of vertices in the underlying undirected graph. The articulation vertices (or cut-points) of the whole graph are vertices whose deletion disconnects the graph. In a k-connected graph, there does not exist a set of k−1 vertices whose deletion disconnects the graph. The cut-points are shown in light gray squares (Figure 4(a)). They are {l, ∆ab , al }.

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Figure 4. Graph properties of the CS sub-model. 2.2.3.

Rooted Embedding and Graph Traversals

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Rooted embeddings are used to represent hierarchies. One vertex is chosen as the root. The other vertices are ranked according to their distance (or depth) to the root. The rooted embedding of the graph is generally drawn choosing a central vertex as a root to have a more balanced embedding (Figure 5(a)).

Figure 5. Rooted embedding and disjoint circuits of the CS sub-model. The graph traversals are essential to explore all the vertices and edges of the graph G. Various graph properties are obtained. Two different approaches lead to linear time algorithms : the depth-first search (DFS) and the breadth-first search (BFS). The recursive function DFS starts with a vertex and scans its neighbors until the first unexplored vertex is found. The recursive function BFS starts with a vertex and explores all the adjacent vertices to the current vertex and then continues. In the DFS, the children of the first child of a vertex

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are visited before exploring its siblings. A DFS tree will then consist of edges connecting vertices to their parents. Any non-tree edge in the DFS is a back edge connecting a vertex to an ancestor. Each back edge (v, u) such makes a cycle from u to v in the DFS tree. The DFS will be preferred to find connected and bi-connected components (any subgraph with no articulation vertex). The BFS will be used for problems such as finding the shortest cycles in a graph10 . Graph traversals DFS and BFS of the CS sub-model, are shown in Figure 6 with the central vertex h˙ as a root.

Figure 6. Graph traversals of the CS sub-model.

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2.3. 2.3.1.

Vertex Typology of a Maximal SCC Vertex Typology

A typology may be based on the properties of an all-pairs shortest paths matrix. The eccentricity values are placed in a column vector to the right of the matrix. The central and peripheral vertices are deduced from this vector. Symmetrically, the maximum of each column is placed in a row vector below the matrix. This vector expresses the in-eccentricity from which the opposite concepts of in-radius and in-diameter are deduced. Both criteria are then crossed in a 3 × 3 table, where the columns state S successively for the peripherals P of the SCC, the center C and the remaining vertices P¯ C¯ with intermediate S ¯ properties. ¯ The rows state for similar in-values with the opposites ¬P , ¬C, ¬P ¬C. Four types of vertices are deduced. Type A shows an eccentric vertex which is weak perturbed and exerts low (or distant) influences. Type B∗ corresponds to vertices with low perturbations but strong (or fast) influences. Type C∗ states a strong integrated vertex which is close to all other interrelated vertices. Type D∗ shows a strong dominated vertex with a strong perturbation and a weak influence . 10

The BFS function returns three informative tables. The first table contains the BFS numbers of the vertices which are assigned in the order of the visit. The second table provides parent pointers where the parent of a vertex is the first visited neighboring vertex. The third table contains the distance of each vertex from the source of the search. Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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Figure 7. Vertex typology of the giant SCC in the CS sub-model.

2.3.2.

Structural Properties of the CS Sub-model

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The vertex typology of the giant SCC in the CS sub-model is shown in Figure 7. ˙ are integrated variables. The productivity h˙ The employment and wages {∆ab , ab , l} dominates and the price p˙c is a dominated variable.

3.

Further Graph Theory for Economic Models

Further important topics are considered with elementary matching, connectivity and circuits theories.

3.1. 3.1.1.

Matching Theory Introduction

A simplified walrasian model consists of eight variables: x, forthe quantity of good X, p for the price of good X, y for quantity of good Y ,q for the price of good Y, l for quantity of labor factor, k for the quantity of capital factor, s for the wage rates, r for the earnings on capital factor 11 . The equations are

11

x = x(p(−) , q(+) , s(+) , r(+) ),

(8)

y = y(p(+) , q(−) , s(+) , r(+) ),

(9)

l = l(p(−) , q(−) , s(−) , r(+) ),

(10)

The variables {x, y, l, k} are also stated as functions of the variables {p, q, s, r}.

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Figure 8. Assigning the equations of the Walras model.

k = k(p(−) , q(−) , s(+) , r(+) ),

(11)

l = a x + b y,

(12)

k = c x + d y,

(13)

p = a s + c r,

(14)

q = b s + d r = 1.

(15)

The indices of the variables {p, q , s , r} state the signs of the partial derivatives. All the parameters {a, b , c , d} are positive. Equations (8) and (9) are the demands of goods, equations (10) and (11) describe the allocation of factors, equations (12) and (13) are the linear production functions with fixed complementary factors and equations (14) and (15) are the cost functions with q = 1. The structure of the system is shown by the following boolean matrix M = (mij ), where the rows are the equations and the columns j the variables. If mij = 1, the variable j is present in the equation i. We have 

    M=     

1 0 0 0 1 1 0

0 1 0 0 1 1 0

1 1 1 1 0 0 1

0 0 1 0 1 0 0

0 0 0 1 0 1 0

1 1 1 1 0 0 1

1 1 1 1 0 0 1

         

At this stage the economist is confronted to an assignment problem (AP) when solving this model: how to calculate the 7 variables using the 7 equations? The assignment of the equations to the variables in Figure 8(b) is deduced from the economic theory. Since there is generally no unique solution to such problem, it is judicious to look for the other solutions. The concepts and algorithms of the matching theory will solve that problem.

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Andr´e A. Keller Perfect Bipartite Matching

Matching is a graph optimization problem (Lov´ asz & Plummer [38]). Some basic definitions are first needed. Definitions Definition 6 (Matching). A matching M in a graph G = (V, E) is a set M j E of pairwise nonadjacent edges. S Definition 7 (Bipartite Matching). A graph G = (U W, E) is bipartite if its set of vertices can be partitioned into two subsets U and W such that every edge in U has one endpoint in W . The sets U and W are the color classes of G and (U, W ) a bipartition of G.

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Definition 8 (Bipartite Matching of a Model). 1. A bipartite S matching M is a set of pairwise non-adjacent edges in a bipartite graph B = (U W, E) where U denotes the set of equations and W the set of variables. That is, M ⊆ E(G) such that e1 , e2 ∈ M, e1 = (i1 , j1 ), e2 = (i2 , j2 ) and i1 = i2 ⇔ j1 = j2 . S 2. A perfect matching p(M) of the bipartite graph G = (U W, E) is a pairing of the set U to the set W which uses each element of U and each element of W , once and only once. Such a matching covers all the vertices of the graph. The existence problem The Frobenius Theorem characterizes those bipartite graphs which have a perfect matching. Hall’s Theorem characterizes the bipartite graph which has a matching of U into W. K¨onig ’s Theorem gives a formula for the matching number ν(G) which is the size of the maximum matching, when a graph has no perfect matching. For this problem there is initially n! possible pairings (Hopcroft & Tarjan [23], Hopcroft & Karp [24]). Definition 9 (Complete Matching). A complete matching of a bipartite graph G = S (U W, E) is a matching M in which each vertex is incident on an edge of M. The matching M is an assignment from U to W . S Theorem 10 (Hall’s Theorem). Let G be a bipartite graph G = (U W, E). If X is any set in V (G), let |Γ(X)| be all vertices which are adjacent to at least one vertex of X. Then G has a complete matching of U into W , if and only if, |Γ(S)| ≥ |S| holds for every S ⊆ U . Corollary 1 (Frobenius’ Marriage Theorem). A bipartite graph G = (U, W ) has a perfect matching, if and only if, |U | = |W | and for each X ⊆ U , |X| ≤ |Γ(X)|. Let G be a graph and S a subgraph of G. The condition that the cardinality |S| exceeds the number of odd components c0 (G) for every subset of vertices is necessary and sufficient to have a 1-graph factor. Theorem 11 (Tutte’s Theorem). Let c0 (G) be the number of odd components of graph G. A graph G = (V, E) has a perfect matching, if and only if, c0 (G − S) ≤ |S| for all S ⊆ V (G). Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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Theorem 12 (Berge’s Theorem). Let M be a matching in a graph G. Then M is a maximum matching, if and only if, there exists no augmenting path in G relative to M. Definition 13 (Alternating Path). 1. A path P =< v1 , . . . , vm > is an alternating path with respect to the matching M if (vi , vi+1 ) ∈ M, if and only if, (vi+1 , vi+2 ) ∈ / M for 1 ≤ i ≤ m − 2. 2. An alternating path has edges that are alternately free such as e ∈ E − M and matched such as e ∈ M. An augmenting path starts at a free vertex and ends at another free vertex. Number of perfect matchings Definition 14 (Deficiency). Let G a bipartite graph with bipartition (U, W ), for X ⊆ U define the deficiency of X by def(X) = |X| − |Γ(X)|, where |Γ(X)| denotes all vertices which are adjacent to at least one element of X. Since def(∅) = 0, we have def(G) ≥ 0. As a consequence of the K¨onig or P. Hall Theorems, we have the following theorem Theorem 15 The matching number of a bipartite graph G is |U | − |def(G)|. Definition 16 (Bi-adjacency Matrix). Let G be a bipartite graph with bipartition sets (U, W ) of the same size |U | = |W | = n, where U = {u1 , . . . , un } and W = {w1 , ..., wn }. The bi-adjacency matrix A = (aij ) is defined by ( 1 if (ui wj ) ∈ E(G), aij = 0 otherwise.

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The entries aij ‘s denote the number of edges connecting ui to wj . Let A = (aij ) be an n × n matrix. Define the permanent of A by n X Y

per A =

ai,σ(i) ,

(16)

σ∈Sn i=1

where the sum is computed over all permutations σ of the numbers {1, . . . , n}. The only difference with the Leibniz formula of the determinant is that all terms have the same sign 12 . The formula (16) of per A contains n ! summands. 13 . If A is the bi-adjacency matrix, 12

The determinant is defined by detA =

X

sgn(σ).

σ∈Sn

n Y

ai,σ(i) ,

i=1

where sgn(σ) is -1 if σ is an odd number of inversions, and +1 otherwise. 13 We also have the Ryser formula Y X ai,j (−1)|s| per A = (−1)n s ⊆{1,...,n}

j ∈s

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each non-zero term corresponds to a perfect matching in the bipartite graph G. Then we have per A = Φ(G), where Φ(G) denotes the number of perfect matchings in G 14 . Theorem 17 Given a k-connected graph G containing at least one perfect matching. Then the number of perfect matchings in G is at least (k−2)/2

k !! =

Y

(k − 2 i).

i=0

A probabilistic estimate of the number of perfect matchings Φ(G) may be obtained. ~ be a random orientation of G, orienting each edge Theorem 18 Given a graph G. Let G ~ the skewindependently of the others with probability 1/2 in either direction. Let As (G) ~ symmetric adjacency matrix of G. Then, the expected value of det As (G) is Φ(G). Lemma 19 Let G be a simple bipartite graph with bipartition (U, W ) and assume that each vertex in U has degree at least k. Then if G has at least one perfect matching, it has k ! perfect matchings.

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Theorem 20 (K¨onig’s Minimax Theorem). The maximum size of a matching in a bipartite graph G is the minimum cardinality of a vertex cover in G. 3.1.3.

Maximal Matching Solutions S Given a bipartite graph G = (U W, E), a matching matches each vertex in U to one in W . Suppose that U states for a set of equations and W a set of variables. Hall’s marriage Theorem states that there is a matching in which every equation can be married, if and only if, every subset S of equations knows a subset of variables at least as large as |S|. A polynomial-time matching algorithm follows from Berge’s Theorem which states that a matching is maximum if and only if it contains no augmenting path (Hopcroft & Tarjan [23]). The algorithm starts with an arbitrary matching. This matching is the improved by finding, if any, S an M-augmenting path. Then M is replaced with the symmetric difference (M − P ) (P − M). The maximal matching will contain no augmenting path. The maximal matching of the Walras model is shown in Figure 9(a). The digraph of the model is deduced in Figure 9(b) with a longest circuit15 . 14

The following inequality may be useful for an estimation of the permanent per A ≤ (r1 !)1/r1 . . . (rn !)1/rn ,

where r1 . . . rn are the row sums. 15 The matching problem of the CS sub-model is detailed in Appendix B.. Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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Figure 9. Maximal matching of the Walras model.

3.1.4.

Multiple Matchings

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Equations of a growth model The macro-dynamic model of Vedel [51] is an attempt to conciliate fundamental elements of the macroeconomic theory : the standard IS - LM (Hicks-Hansen) model, the natural unemployment theory of Friedman and Phelps, the Cagan’s inflation theory, the Fisher’s equation and the Wicksell’s analysis. The equations have been rewritten in a more readable form. The variables are continuous functions of time and parameters are all positive, with 0 ≤ γ ≤ 1 and 0 ≤ µ ≤ 1. A doted variable X˙ states for a time derivation and log is the logarithm with base e. We have Y = Y γ Y˜ (1−γ) eα−δr + G,

(17)

M = k Y µ Y˜ (1−µ) e−ǫr , P

(18)

R=r+

P˙ e , Pe

(19)

P˙ P˙ e P¨e = λ ( − e ), P P P˙ e

(20)

Y P˙ P˙ e = ω log ¯ + e . P P Y

(21)

The endogenous variables are: P for the prices of goods, P e for expected price, R the nominal interest rate, r for the real interest rate, Y for the effective national revenue, Y¯ for the expected normal revenue. The exogenous variables are: G for the government expenditures and M for the nominal money supply. Equation (17) is the equilibrium condition on the market of goods : the private demand depends on an geometric average between the current and the expected revenue, as well as on real interest rates. In equation (18), the real money demand has the same explanatory variables (except the government expenditures).

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Equation (19) is the Fisher’s equation 16 , where the real and nominal rates are related. Equation (20) describes the formation of price expectation following Cagan 17 . Equation (21) expresses the Phillips curve introducing the natural unemployment of Friedman and Phelps 18 . Graph of the model This model may be rewritten (Vedel [51]) introducing the transitory component Y /Y˜ of the national product and the two parameters of budgetary policy z = 1 + G/Y γ Y˜ 1− γ × eα− δ r and b = log z 19 . We have the system (1 − γ) log v + δ r = α + b,

(22)

v˙ M˙ Y˜˙ P˙ + µ − ǫ ∆R = − , P v M Y˜

(23)

R = r + P˙ e ,

(24)

P˙ e P¨e P˙ = λ ( − e ), P P P˙ e

(25)

P˙ = ω log v + P˙ e . P

(26)

The presence of the variables {v, P, P e , R, r} in the equations (22) to (26) is shown in the following 0-1 matrix M = (mij ), where the rows are the equations and the columns the variables. If mij = 1, the variable j belongs to the equation i. We have

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

  M =  

1 1 0 0 1

0 1 0 1 1

0 0 1 1 1

0 1 1 0 0

1 0 1 0 0

     

Multiple perfect matchings Three theoretical interpretations are suitable for this model: a wicksellian, a friedmanian or an extreme monetarist interpretation (Vedel [51]) 20 . Figures 10 shows three perfect matchings which have one of these economic interpretations. Figures 11 are the corresponding graphs. 16

I. Fisher (1961), The theory of interest, in A.M. Kelley, Reprints of Economic Classics, New-York. P. Cagan (1956),The monetary dynamic of hyperinflation, in M. Friedman (editor), Studies in The Quantity Theory of Money, Chicago, Chicago University Press. 18 M. Friedman, (1968), The role of monetary policy, American Economic Review, 58: 193–194 ; E. S. Phelps and al.(1975), Microeconomic Foundations of Employment and Inflation, New-York: Norton. 19 The evolution of these parameters is comparable to ratio between public and private expenditures. 20 According to the wicksellian approach the economy fluctuates as the consequence of a gap between the real and the natural interest rates. According to the friedmanian approach, the level of activity is depending on the Phillips curve: the fluctuations around the equilibrium path are due to forecasting errors on inflation rate. It determines the nominal interest rate and the real interest rates adjust the market of goods. The Fisher’s equation relation calculates the nominal interest rate. According to the extreme monetarist interpretation the economic fluctuations are imputable to the money market. The real interest rate equilibrates the market of goods. 17

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Figure 10. Multiple perfect matchings of the growth model G.

Figure 11. Multiple graphs of the growth model G.

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3.1.5.

Finding All the Perfect Matchings

The algorithm of Fukuda and Matsui [12] uses the K-th best solution of assignment problems (AP) developed by Murty [41] and Chegireddy and Hamacher [7]. The computational time is O(n(n + m)) and it requires O(n + m) memory storage for each additional matching. Their recent algorithm requires O(e(n + m) + n5/2 ) computational time and O(nm) memory storage, where e is |Φ(G)|. First, the AP is solved by the Hungarian method 21 and each perfect matching is generated in a lexicographic order. The procedure is based on a binary partitioning where the enumeration problem can be partitioned into two subproblems. It generalizes the Murty’s algorithm for ranking the solutions of APs. Uno [50] proposes a new approach in two phases called trimming and balancing for developing fast algorithms.

3.2. 3.2.1.

Connectivity Theory Components, Independency, Cuts and Connectivity

Definition 21 (Directed Path). A directed path from v0 to vn is an alternating sequence P =< v0 , e1 , v1 , e2 , . . . , vn−1 , en , vn > of vertices and arcs such that tail(ei ) = vi−1 and head(ei ) = vi for i=1,. . . ,n. A directed path from x to y is an x-y directed path. 21

The Hungarian method for (AP) is due to H. Kuhn [36],[37] and been revised by J. Munkres [40]. The Hungarian algorithm solves the AP in polynomial time O(|V |3 ). Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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Definition 22 (Vertex-independency). A collection of paths between two vertices u and v of a graph G are independent if no two of them share another vertex. The maximum number of vertex-independent paths is denoted by λ(u, v). Similarly, a collection of paths will be independent if no two paths have an edge in common. The maximum cardinality of edge-independent paths between u and v is denoted by λ′ (u, v). Suppose that in a finite directed graph G = (V, E), every (u, v) have the capacity c(u, v) and that two vertices, the source s and the sink t have been distinguished. A cut is a split of the vertices into two sets S and T , such that s ∈ S and t ∈ T . The number of possible cuts is 2|V |−2 and the capacity of the cut (S, T ) is c(S, T ) =

X

c(u, v).

u∈S,v∈T |(u,v)∈E

Definition 23 (Vertex-cut). A vertex-cut in a strongly connected graph G = (V, E) is a vertex subset of V such that the vertex-deletion sub-digraph G−S is not strongly connected.

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3.2.2.

Menger’s Theorem

Menger’s Theorem 22 is a basic result about connectivity in an undirected graph. According to this theorem, if G is a connected graph, X and Y disjoint sets of vertices of G, then the minimum number of vertices whose deletion disconnect X and Y is equal to the maximum number of disjoint paths between the two sets. It has been proved for edge- and vertexconnectivity. Let G be a finite undirected graph and u and v two nonadjacent vertices. Then, the theorem states that the size of the minimum edge cut for u and v is equal to the maximum number of pairwise edge-independent paths from u to v.23 Theorem 24 (Directed Vertex-disjoint Version of the Menger’s Theorem). Let G = (V, E) be a digraph and s, t ∈ V . Then the maximum number of vertex-disjoint s−t paths is equal to the minimum size of an s − t cut. Proof(Schrijver [44]). Let k be the minimum of an s − t cut. Choose e = (u, v) ∈ E. If each s − t cut in G − e has size at least k, then inductively there exists k vertex-disjoint s − t paths in G − e, hence in G. Assume that G − e has an vertex-cut C of size ≤ k − 1. Then C ∪ {u} and C ∪ {v} are s − t vertex-cuts of G with size k. Now each s − (C ∪ {u}) vertex-cut B of G − e has size at least k, as it is s − t disconnecting in G. Indeed, each s − t path in G intersects C ∪ {u} and hence P contains an s − (C ∪ {u}) path in G − e. So P intersects B. Hence, by induction, G − e contains k disjoint s − (C ∪ {v}) − t paths. Any path in the collection intersects any path in the second collection only in C, since otherwise G − e contains an s − t path avoiding C. Hence, as |C| = k − 1, these paths can be pairwise concatenated to disjoint s − t paths, inserting arc between the path ending at u and starting at v.  22

Karl Menger, Zur allgemeinen Kurventheorie, Fundamenta Mathematicae, 10(1927): 96-115 . Similarly the vertex-connectivity version of the Menger’s Theorem states that the size of the minimum vertex cut is equal to the maximum number of pairwise vertex-independent paths from u to v. 23

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Corollary 2 (Directed Arc-disjoint Version of the Menger’s Theorem). Let G = (V, E) be a digraph and s, t ∈ V . Then the maximum number of arc-disjoint s-t paths is equal to the minimum size of an s − t cut. Definition 25 (Connectivity, Local Connectivity). The vertex-connectivity κ(G) is the cardinal of a smallest vertex cut. A graph is k-vertex-connected if κ(G) ≥ k. The local connectivity κ(u, v) is the cardinal of a smallest vertex cut separating u and v. Moreover, we have κ(G) = min κ(u, v) over all pairs u, v. All these concepts are valid for the edges of graph (edge-connectivity κ′ (G), k-edge connectivity with κ′ (G) ≥ k, local edge-connectivity κ′ (u, v)). Moreover, we have the property κ(G) ≤ κ′ (G). According to Menger’s Theorem, the equalities κ(u, v) = λ(u, v) and κ′ (u, v) = λ′ (u, v) are verified 24 .

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3.2.3.

Computational Methods

The Combinatorica functions in MATHEMATICA use traversals to determine whether a graph is connected, and to find the connected components. The traversals DFS and BFS are used to explore all the vertices and edges.The traversal DFS will be essential for algorithms that find SCC, bi-connected components and cycles (Sedgewick [45], Skiena [46]). According to Hopcroft and Tarjan [23] the DFS leads mainly to linear-time algorithms. The determination of some property of a graph often needs to explore all the edges and vertices in a particular way. The DFS is used to solve path problems. An application to the Petersen graph is shown in Figure 12. The DFS procedure starts from the vertex 6, visits recursively the adjacent vertices. Figure 12(b) is the DFS tree, with thick edges. Such spanning trees tend to have few but long branches. The thin non tree edges are the back edges. Each back edge (v, u) forms a cycle together with the path u-v in the DFS tree. Figure 12(c) shows a maximal list of 2 edge-disjoint cycles for this graph, (4, 1, 3, 5, 2, 4) and (10, 6, 7, 8, 9, 10) 25 . The BFS explores all the adjacent vertices to the current vertex. It is the basis for finding a shortest path and the all-pairs shortest paths connecting each pair of vertices. Figure 13(a) shows the BFS of the Petersen graph. The edges are shaded according to their distance from the root vertex 6 of the Petersen graph. The thick edges show the shortest paths found by the algorithm. The BFS tree in Figure 13(b) provides a representation of all the shortest paths from the root 6. Kocay & Kreher [35] show that the complexity 26 of the BFS is at most O(m). The Ford-Fulkerson algorithm for matching is illustrated in Appendix A. 24 The edge-connectivity version of the Menger’s Theorem was later on generalized by the max-flow min-cut Theorem, a statement optimization problem about maximal flows in flow networks (http://www.math.fau.edu/locke/Menger.htm). 25 The DFS of a graph, whose representation is an adjacency matrix requires a running-time of O(n). It requires O(n + m) for an adjacency-list representation. 26 The complexity of an algorithm is captured by the time or the number of steps it takes to complete a problem of size n. For example, when an algorithm has order of n2 time complexity the corresponding big O notation is O(n2 ) is of O(E.maxf low). The specialized algorithm of Edmonds-Karp finding paths with BFS is of O(V.E 2 ) time complexity.

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Figure 12. DFS search to the Petersen graph.

Figure 13. BFS search procedure with Petersen graph.

3.2.4.

Connectivity in Weighted Graphs

The Klein’s I small-size econometric model is used to build a weighted graph, where the intensity of the relationships between the variables are introduced.

Equation and graph Klein’s Model I [33] is a widely used econometric example of a simultaneous equation model (Greene [18], Maddala [39]). The seven equations model consist of three behavioral randomized equations (a consumption function, an investment Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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Figure 14. Graph representations of the Klein’s model I.

function, a demand for labor function) and four identities. The equations are ¯ g ) + α3 Pt−1 + εC Ct = α0 + α1 Pt + α2 (Wtp + W t , t It = β0 + β1 Pt + β2 Pt−1 + β3 Kt−1 + εIt , Wtp = γ0 + γ1 Xt + γ2 Xt−1 + γ3 t + εW t , ¯ ¯ Yt + Tt = Ct + It + Gt , ¯ g + Pt , Yt = W p + W Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

t

t

Kt = Kt−1 + It , ¯ g. Xt = Yt + T¯t − W t

¯ for the government spending (exogenous), I The variables are C for the consumption, G for the investment, K for the capital stock, P for theprivate profits, T¯ for the indirect taxes (exogenous), t¯ for the trend (exogenous), X for the demand, Y for the national income, W g for the government wage bill (exogenous), W p for the private wage bill. The coefficients αi′ s, βi′ s and γi′ s are the parameters to be estimated using yearly data from 1921 to 1941 27 . The residuals ǫ′ s are normally distributed RVs (random variables). Figures 14 show two isomorphic graphs. The vertices of G have been reordered in different sets of vertices : a set of vertices of a longest circuit, the remaining vertices of the giant SCC and the remaining vertices of the graph. 28

27 The data are listed by Greene [18] over this period. Updates are also given over the period from 1960 to 1984. 28 The reordered set of vertices is achieved applying P.A.P T , where A is the adjacency matrix to be permuted, and P the permutation matrix. Clearly, the product P.A permutes the lines of the matrix A, whereas the product A.P T permutes the columns of A.

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Weighting the arcs of the graph Numerous estimators may be calculated 29 . The comparison will be restricted to OLS (Ordinary Least Squares) and 2SLS (Two Stage Least Squares). Since the econometric equations are over-identified, the OLS method is inappropriate for this model. The standard errors (SE) we obtain for the OLS method are not correct, whereas they are at least asymptotically correct estimates for the 2SLS method. In general, SEs for the OLS estimates are lower than those of the 2SLS estimates. Table 1 presents the OLS-estimates and the 2SLS-estimates given by Maddala [39]. Table 1. OLS- and 2SLS estimates. Equation Consumption

Investment

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Wage bill

Variable Intercept P ¯g Wp + W P−1 Intercept P P−1 K−1 Intercept X X−1 t¯

OLS Estimate 16.237 0.193 0.796 0.089 10.125 0.479 0.333 -0.112 0.064 0.439 0.146 0.130

SE (1.203) (0.091) (0.040) (0.090) (5.465) (0.097) (0.100) (0.026) (1.151) (0.032) (0.037) (0.031)

2SLS Estimate 16.543 0.0190 0.810 0.214 20.284 0.149 0.616 -0.1578 0.065 0.438 0.146 0.130

SE (1.464) (0.013) (0.044) (0.118) (8.361) (0.191) (0.180) (0.040) (1.894) (0.065) (0.070) (0.053)

Definition 26 (Weighted Graph). A weighted graph associates a label or weight with every edge in the graph. In the behavioral equations of the model, the presence of explanatory variables is uncertain. The coefficients are random. The partial correlation is associated to each arc between one explanatory variable (on the RHS of the equation) and the explained variable (on the LHS). The partial correlation are t2j 2 ri.j = 2 , tj + dfi 2 is the squared partial correlation between variable i and variable j. The t2 denote where ri.j j the t-test squared statistics of Student and dfi the degree of freedom for the estimation. The results are shown in Table 2. Figure 15 shows the weighted graphs. Thereafter, the connectivity of the graph is tested while deleting inessential arcs due to a low explanatory capacity. The bar charts of Figures 16 show the effects of successive deletions of edges whose weight is low. The number of SCCs increases from 2 up to 7 and simultaneously the cardinality of the maximal SCC is decreasing from 6 to 1. 29

Greene [18] compares different methods: limited-information estimates (OLS, 2SLS, LIML) and fullinformation estimates (3SLS,FIML). The intuition suggests to prefer system methods (3SLS and FIML) rather single-equation methods (2SLS, OLS). Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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Table 2. Partial correlations using OLS- and 2SLS-estimates. Equation

Variable

Consumption

Intercept P ¯g Wp + W P−1 Intercept P P−1 K−1 Intercept X X−1 t¯

Investment

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Wage bill

Partial correlation (from OLS estimates) 0.9146 0.2092 0.9588 0.0539 0.1680 0.5892 0.3948 0.5219 0.0002 0.9172 0.4781 0.5085

Partial correlation (from 2SLS estimates) 0.8825 0.0013 0.9522 0.1621 0.2572 0.0346 0.4079 0.4754 0.00007 0.7276 0.2038 0.2614

Figure 15. Weighted graphs of the Klein’s model I.

3.3. 3.3.1.

Circuit Theory Definitions

A chain c is closed if its initial and terminal vertices coincide (v0 = vn ). A cycle is a closed simple chain. An elementary circuit is an elementary path with identical extremal vertices, such as (v1 , v2 ), (v2 , v3 ), . . . , (vn−1 , vn ), (vn , v1 ). A circuit is isometric, if for any two vertices u and v, it contains a shortest path from u to v and a shortest path from v to u. A short circuit has only one of the two last properties. A circuit is relevant, if and only if, it cannot be expressed as a linear combination of shorter circuits.

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Figure 16. Successive deletion of arcs in the Klein’s I model. 3.3.2.

Enumeration Problems

There are two orientations for enumerating problems on sets of objects. Indeed, one can either estimate how many objects are in the set or look for an exhaustive finding of the present objects. Both orientations will be briefly considered here. Counting problem In a complete directed graph Kn the number of elementary circuits is given by  n−1 X n (n − i)! n−i+1

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i=1

This number belongs to the open interval ((n−1)!, n!). For the complete graph K3 , we find   P2 n (4 − i)! = 5. Moreover, we have 3 circuits of length 2 {< 1, 2, 1 >, < i=1 4−i 2, 3, 2 >, < 3, 1, 3 >} and 2 circuits of length 3 {< 1, 2, 3, 1 >, < 1, 3, 2, 1 >}. Figure 17 shows the growth of the number of elementary circuits in the complete graphs K3 to K10 . These numbers grow faster than the exponential 2n . Estimates of the l-length circuits number Let us compute the number of circuits for each length l, 2 ≤ l ≤ n in a given complete graph G = (V, E)with cardinalities |V | = n and |E| = m 30 . The estimation of the number of circuits is given by n (n − 1)! k l ( ), l (n − l)! n − 1 where k denotes the integer part of m/n. For complete graphs K3 to K10 , the number of elementary circuits by length is shown in Table 3. . Proof. Let us consider an almost-complete digraph G = (V, E) with cardinalities |V | = n and |E| = m. The average number of arcs is the integer part of k = m/n. Starting from one vertex i, the total paths of length 1 is n−1 n−1 k, where the numerator states the number 30

For complete graphs Kn , the number of edges is given by the binomial



n 2



.

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Figure 17. Elementary circuits in complete graphs K3 to K10 .

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Table 3. Number of circuits by length in complete graphs. length 2 3 4 5 6 7 8 9 10 Total

K3 3 2 5

K4 6 8 6 20

K5 10 20 30 24 84

K6 15 40 90 144 120 409

K7 21 70 210 504 840 720 2365

K8 28 112 420 1344 3360 5760 5040 16064

K9 36 168 756 3024 10080 25920 45360 40320 125664

K10 45 240 1260 6048 25200 86400 226800 403200 362880 1112073

of extremal vertices other than i and the denominator the total number of possible extremal vertices. At an another vertex j, j 6= i, there are n − 2 extremal vertices left. Hence, the number of paths of length 1 starting from j is n−2 n−1 k. The number of paths of length 2 from vertex i is then n−2 (n − 1)(n − 2) 2 n−1 k× k = k . n−1 n−1 (n − 1)2 Hence, we have

(n−1)(n−2)...(n(l−1)) l−1 k (n−1)l−1

paths of length l − 1 starting from vertex i. The

average number of arcs joining a vertex other than i to i being length l is firstly estimated to

k n−1 ,

the total circuits of

k (n − 1)(n − 2) . . . (n(l − 1)) l−1 (n − 1)! k l k = ( ). n−1 (n − 1)l−1 (n − l)! n − 1 However for n vertices each circuit will be counted l times. Therefore the number of circuits

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Figure 18. Number of circuits in a digraph.

of length l may be estimated to n (n − 1)! k l ( ) . l (n − l)! n − 1

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Figure 18 shows an almost-complete digraph G = (6, 24). The discrepancies between the estimated and the true number of circuits have been computed and show close results. However, the errors become extremely high with larger non-complete graphs. Enumerating problem Following Prabhaker and Narsingh [43], every circuit enumeration algorithms to digraphs can be put into three classes of methods : search algorithms, algorithms using powers of the adjacency matrix, algorithms using line graphs (or edgedigraph). The search algorithms look for circuits in a set containing all the circuits 31 . In the integer powers p ≥ 1 to p = n of the adjacency matrix A, the nonzero elements apij are p-sequences from vertex vi to vj . The problem is to avoid non simple edge-sequences, where all vertices are not distinct, except for the extremities. The line graph L(G) of a graph G has a vertex associated to each edge of G and an edge of L(G) is drawn, if and only if, the two edges of G are adjacent 32 . The line graph of a p-circuit is also a p-circuit. There is a one-to-one correspondence between the circuits of G and L(G) (Bermond & Thomassen [4], Ehrenfeucht, Fosdick & Osterweil [9]). 3.3.3.

Tarjan’s Algorithm

The Tarjan’s algorithm for enumerating the elementary circuits of a digraph is given by Tarjan [47][48] in algol-like notation. The Tarjan’s algorithm is based on the Tiernan’s backtracking procedure [49] (Appendix C.) which explores the elementary paths and checks to 31

The problem of counting cycles is treated by Harary and Palmer [22]. line graph of a complete graph of order n and size m = n(n − 1)/2, contains m vertices and PThe n 2 i=1 di − m edges, where di are the degrees. 32

1 2

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verify if they are elementary circuits. A marking procedure avoids unnecessary searches. This algorithm has a running-time of O(n × m(c + 1)) where c denotes the number of elementary circuits and requires a O(n + m + S) space, where S is the sum of the lengths of all the elementary circuits 33 . A computer program 34 adopts these principles and rules. A adjacency-list representation is used to introduce the graph and the computer program produces a list of lexicographically ordered circuits. A main program is detecting the circuits, the subroutine SELECT is ordering each circuit from the smallest index, the subroutine CLASS is eliminating the repeated circuits by length and ordering the circuits lexicographically, the subroutine PRINT is editing the ordered circuits for each length 35 . Figure 19 shows a simplified diagram of the computer program.

Figure 19. Computer program for circuits enumeration.

33

A worst-case time bound is given by Johnson [25]. The digraph G = (3k + 3, 6k + 2) has exactly 3k elementary circuits with a maximum length of 2k + 2. The exploration from a given vertex takes O(k3 ) time. 34 Computer program BAOBAB (Fortran V,77) by A.A. Keller, PhD Paris 1977. Translated (f2c) to C++. 35 The Danˇ ut’s program [8] has been adapted for this chapter. This computer program is for enumerating paths or circuits of a digraph, having a specified starting vertex for all possible lengths. Thus for a complete graph K5 , the computer program finds 320 circuits, whose 236 of them are circular permutations of existing circuits and must be deleted. Hence, we have exactly 84 circuits in a K5 complete graph. This computer program may be used for looking for a longest circuit.

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Andr´e A. Keller Upper Bounds on Time and Space

Prabhaker & Narsingh [43] survey known algorithms for enumerating all circuits in the 1970’s. Table 4 shows the running-time and storage performances of some algorithms using the backtrack method. The Johnson’s algorithm [25] is the fastest one, since the running time between two consecutive circuits never exceeds the size of the graph (n + e). The elementary circuits are constructed from a root vertex u in the subgraph G − u and vertices larger than u36 . To avoid duplicate circuits, a vertex v is blocked when it is added to some elementary path beginning by u. This vertex stays blocked as long as every path from v to u intersects the current elementary path at a vertex other than u. Table 4. Complexity of backtrack algorithms to circuitsa . Author Time bound Johnson (1975) [25] (n + e)c Tarjan (1973) [47] n×e×c Tiernan (1970) [49] n(const)n Weinblatt (1972) [52] n(const)n a The graphs have n vertices, e edges and c circuits.

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3.3.5.

Space bound n+e n+e n+e n×c

Set of Circuits and Non-edge Disjoint Circuits

The graph of the CS sub-model consists of 3 circuits, such as C =< v1 , v2 , . . . , v1 >. The longest circuit is the oriented path < l, ∆ab , ab , h, pc , l >. Figure 20(b) shows an improved circular embedding with a longest circuit. The graph obtained after permutations of its vertices is isomorphic to G 37 . A maximal ˙ l,˙ ∆ab , h˙ > list of edge-disjoint cycles consists of two edge-independent circuits : < h, and < l,˙ p˙c , l˙ >.

3.4.

Dynamic Steady-State Graph

The delayed variables may be simply introduced into the graph. One method is consider a system close to a steady state situation. Additional arcs are introduced with one for each delay, whatever its length. A pseudograph is then obtained with self-loops and multiple edges (Figure 21(a)). A self-loop occurs when a variable is depending on itself in the case of hysteresis effects. Multiple edges appear when a variable observed at time t depends on another multiple delayed variable. Moreover the size of the longest circuit is increased to 5 vertices, and we have only one center with a lower radius rad(SCC) = 2. Since ecc(v) = {3, 4, 3, 3, 2} we have diam(SCC) = 4. The central vertex is the employment variable {ab } and the peripheral vertex is the price variable {p˙c }. Compared to the short term version, the set of variables price - productivity - wages are no longer central vertices.

36

According to the Tiernan’s principle [49]. There exists a bijection φ : V (G) 7→ V (H) such as for every pair u, v ∈ (G) one have an edge (uv) ∈ E(G) if and only if there is an edge (φ(u)φ(v)) ∈ E(H). 37

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Figure 20. Longest circuit of the CS sub-model.

Figure 21. Graphs of the dynamic CS sub-model.

4. 4.1.

Graph Theory and Large-Size Applications Graph Characteristics of a Large-Size Model

Large-size FAIR model The main characteristics of the FAIR econometric model for USA [11] are shown in Figure 2238 . The lists of variables are presented in Appendix E. Figure 23 represents a sub-matrix of the 130 × 130 adjacency matrix. Figure 24(a) is the bipartite graph of the model with equations on the LHS and variables on the RHS. Highlighted edges show that vertices 1, 34, 115, 120, 127 and 130 are present in equation 1. Figure 24(b) is the graph where the successors of vertex 1 are shown by highlighted arcs (1,34),(1,51),(1,52),(1,60),(1,65) and (1,116). Figures 23 and 24 illustrate the graph building. 38

The specification and estimates of the latest update of the US model are presented in Appendix A: The US model: July 31, 2008, available at http://fairmodel.econ.yale.edu. This site also offers the analytical power of a large-scale macro-econometric model for business forecasters and economic policy analysts. Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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Figure 22. Characteristics of the FAIR model.

Figure 23. Adjacency 20 × 20 sub-matrix of the FAIR model.

The circular embedding is drawn in Figure 25 with one of the longest circuit (the last of the 8 longest circuits) : big black points are those of the longest circuit, other black points denote the set of the remaining vertices of the giant SCC and the remaining vertices of the graph. The acyclic DAG in Figure 26 clarifies the whole structure and classifies the vertices according to their rank of influence from the top to the bottom of the acyclic graph. The resolution and the analysis of the model will take advantage from these properties. The reordered graph G = (130, 396) in Figure 25, has 130 vertices and 396 arcs . The acyclic

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Figure 24. Graph building and matching of the FAIR model.

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DAG H = (53, 170) in Figure 26 has been obtained by contracting each SCC. Structural Properties Several graph properties are derived from the all-pairs shortest paths matrix of the giant SCC. The graph of the FAIR model has a radius of 5 and a diameter of 11. The articulation vertices also play an important role. The properties of the FAIR model are shown with particular vertices in Figure 27. Figure 28 also shows the maximum clique of the graph in highlighted edges. Indeed, the largest clique induces a complete graph.

4.2. 4.2.1.

Connectivity of Large Graphs Connectivity Properties

Figures 29 and 30 show the ranked and the spring embeddings of the FAIR model. In a ranked embedding, vertices are placed on evenly spaced vertical lines. This embedding helps revealing features of the underlying (undirected) graph when the vertices are partitioned into subsets. Bi-connected components are maximal bi-connected subgraphs without cut-point (articulation vertex). The bipartite block-cutpoint graph B = (42, 39) shows the interaction between the 12 bi-connected components and 30 cut-points. The bi-connected components consist of one giant component of 119 vertices. The twelve bi-connected components are placed on the leftmost vertical line and the thirty cut-points on the rightmost vertical. The edges of B connects each cut-point to the bi-connected components that it belongs to. Any vertex, such as Y , that belongs to more than one bi-connected component is a cut-point. Eades (1984) 39 models the graph as a system of springs (Pemmaraju & Skiena [42]). 39

P. Eades (1984), A heuristic for graph drawing. Congressus Numerantium, 42: 149–160.

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Figure 25. Digraph of the FAIR model.

Hooke’s law 40 spaces the vertices. Hence, as in Figure 30, adjacent vertices attract each other with a force proportional to the logarithm of their separation and all non adjacent vertices repel each other with a force proportional to their distance. 4.2.2.

Depth - and Breadth-First Search

The depth- and breadth-first search have been computed taking a central vertex Y of the giant SCC as the root. Figure 31 shows the DFS tree and Figure 32 the BFS tree as being rooted at the source of search. The DFS consists of edges that connect vertices to their parents, the BFS shows all shortest paths from the root. It is not surprising that the DFS tree is much deeper with 28 levels than the BFS tree with 7 levels. 4.2.3.

Thresholds of Connectivity

A weak partial correlation (i.e. a low explanatory capacity) may justify the omission of such influence. The sensitivity of the structural properties to a progressive and cumulative 40

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Figure 26. DAG of the FAIR model.

deletion of coefficients will then be tested. The values of the partial correlations are first ordered increasingly. At each step of the procedure, the weakest coefficient is selected and deleted. Two connectivity properties of the graph are calculated at each step : the cardinality of the SCCs set and the cardinality of the giant SCC 41 . The procedure goes on until to an arbitrary (good) level of 90 per cent for partial correlations. Figure 33 shows the results in terms of the number of SCC(Figure 33(a)) and of the cardinality of the giant (the largest one) SCC (Figure 33(b)). The number of SCCs then increases and simultaneously the size of the giant SCC decreases. More interesting is to consider switching modes with jumps. These results may be considered in building procedure of such models. Moreover, The structure of the model seems to collapse at a percentage of 50 for the partial correlations (Figure 33).

41

The concept of tenacity to determine the strength of a network and its vulnerability is precisely based on our connectivity indicators. Indeed, the tenacity T (G) of a graph G = (V, E) is defined by T (G) = (G−A) }, where the minimum is taken over all vertex cutset A of G. The graph G − A is induced min{ |A|+τ ω(G−A) by V − A. The cardinality of the giant component of G − A is τ (G − A) and ω(G − A) states the cardinality of the set of components of G − A.

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Figure 27. Centers, peripherals and cut-points of the FAIR model.

4.2.4.

Vertex Typology

Table 5 shows the properties of the giant SCC. Vertices HO, JJS, Z(labour market) are low dominant with few perturbations and fast influences. The bank deposits MH is a rather dominated vertex. The vertices CS,CN,CD,IHH,Y,X for households expenditures and investment, production and sales are rather well integrated. Table 5. Vertex typology of the FAIR model.

A D∗ D

∗

B MH C∗ CD, CN , CS, IHH, X, Y C

HO, JJS, Z B C 67 variables

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Figure 28. Maximal clique of the FAIR model.

4.3. 4.3.1.

Circuit Enumeration of Large Graphs Enumeration of the Circuits

The variables of the FAIR model are highly interrelated with 7990 circuits, of which 8 include a maximum of 42 vertices. All the circuits are listed in a lexicographic order with increasing length. Figure 34 shows the number of circuits in the graph of the FAIR model and their distribution by length of the circuits 42 . The longest circuit of the static model is described by the path < CS, X, Y, HF, HF F, HO, HN, W A, LM, E, U, U B, Y D, IM, P D, RS, RM, RM A, CD, IBT G, IBT S, P H, M H, M B, BR, BO, AG, BG, IN T G, IN T ROW, CF, SF, AF, BF, IN T F, P IEF, T F S, T F G, DF, Y T, D1GM, RSA >. This longest is interpreted as the circular flows of goods and incomes in an open economy with the sequence : production and sales < X, Y > - labor market < HF, HF F, HO, HN, W A, LM, E, U, U B > - wage and price < Y D, P D > -taxes 42

For this application with G = (130, 396), the apparent computation time is about two hours on a personal computer which processor is AMD Athlon 3400+2.2 GHz. Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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Figure 29. Bipartite block-cutpoint graph of the FAIR model.

< IBT G, IBT S > - money market < M H, M B, BR, BO, AG, BG >, incomes and profits < CF, SF, AF, BF, P IEF, T F S, T F G, DF, Y T, D1GM, RSA > Figure 35(a) shows the longest circuit in the initial graph with black edges. The circuit has been reordered in Figure 35(b). 4.3.2.

Non-edge Disjoint Circuits

The maximal list of non-edge disjoint circuits for the FAIR model consists of 14 circuits whose maximal size is 15 vertices. The shortest circuits (Figure 36(a)) C1 ≡< CS, P H, CS > to C4 ≡< IHH, P H, IHH > are household expenditures - prices loops. The circuit C5 ≡< W F, P F, W F > is a wage price loop. The circuit C7 ≡< SG, AG, IN T G, Y T, T HG, SG > is a taxes loop. The circuits C6 ≡< IM, P D, P S, IN T F, Y D, IM > and C8 ≡< IM, X, P X, P EX, P D, P CS, P H, IM > are payment and sales loops of imports. The circuit C9 ≡< Y D, CS, IBT G, P H, L1, U, U B, Y D > is a price - wage - employment

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Figure 30. Spring embedding of the FAIR model.

loop. The circuit C10 ≡< IHH, X, M F, AF, BF, IN T F, IN T ROW, Y D, IHH >

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is a circuit for the finance of residential investments. The circuits C11 ≡< CD, X, Y, IKF, CCF, P IEF, T F G, DF, Y D, CD > and C12 ≡< RSA, CS, IBT S, IBT G, XX, P IEF, DF, Y T, D1GM, RSA > In Figure 37(a) are demand - production loops. The circuit C13 ≡< CF, SF, AF, IN T F, Y T, D1SM, RM A, CN, IBT S, XX, CF > is the circular flow of firms. The longest circuit C14 ≡< IBT G, IBT S, P H, M H, M B, BR, BD, BO, AG, BG, IN T G, IN T ROW, Y T, T HS, Y D, CN, IBT G > (Figure 37(b)) describes the whole circular flow of revenues in the economy. The Figure 38 shows the network of these circuits which are connected by an edge when the pair of circuits have at least one vertex in common. The maximum clique has been highlighted. It denotes a strong connection within the circuits.

4.4.

Dynamic Large Graphs

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Figure 31. DFS of the FAIR model.

Figure 32. BFS of the FAIR model.

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Figure 33. Structure sensitivity of the FAIR model.

Figure 34. Circuits distribution of the FAIR model.

term view of two or few periods, we have same graphs with one for each period are joint. In a long term perspective, when all dynamical effects have been exerted, a new graph is achieved when adding one edge for all the delays of a same variable. Short-run and long-run dynamic graphs for a model are then considered. 4.4.1.

Short-run Dynamic Graph Structures

A two-periods dynamically joined graph Due to simplicity, two-periods delays are only considered. The whole graph then consists of two joint same graphs, with one for each period as in Figure 39. A two-periods dynamically joined acyclic graph Figure 40 shows the two joint acyclic graphs. The reduction of the circuits is simply realized within each of the two periods, and the acyclic graphs are then joint without multiple edges.

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Figure 35. A longest circuit of the FAIR model. Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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Figure 37. Long edge-disjoint circuits of the FAIR model.

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Figure 38. Network of edge-disjoint circuits and maximum clique of the FAIR model.

Figure 39. Joint graphs of the dynamic FAIR model.

4.4.2.

Long-run Dynamic Graph Structures

Dynamic initial and acyclic graph Figure 41(a) shows an initial graph with added edges, with one oriented edge of each delay. Self-loops and multiple edges have been omitted. The additional edges introduce more and longest circuits. The acyclic graph in Figure 41(b) shows a more compact structure, compared to that of the static version ( Figure 26).

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Figure 40. Joint acyclic graphs of the dynamic FAIR model.

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Structure properties The structural properties of the giant SCC (with 93 vertices) are given in Figure 42, where the central vertices, the peripheral vertices and the cutpoints have been located. The center of the graph consists of the central vertices {CS, CN, CD, IHH, Y, IM, Y } (household expenditures, production, imports and sales) and then exert the fastest influence on the remaining vertices of maximal SCC. The articulation vertices of the SCC are incomes {W A, Y T }, production and profits {Y, P IEF } and financial assets {AF, AG}. However, peripheral vertices for job indicator, prices and deposits {LM, P G, P S, M B} exert a distant influence. The properties of the dynamic FAIR model are shown in Table 6. Table 6. Vertex typology of the dynamic FAIR model

A D∗ LM ,P G,P S,SG D

A.

B∗ IM C∗

CS,D1GM ,D1SM ,JJS,L1, T HG,T HS,W G,W M ,W S,Z B SH C 70 variables

C

Ford-Fulkerson Algorithm for Matching

S Let G(U W, E) be a graph with bipartition (U, W ) and let M be any matching in G. Suppose U1 and W1 are the sets of unmatched (or exposed) vertices. An M-augmenting path, if any, is to find connecting U1 to W1 . The set of vertices in U accessible from U1 on an M-alternating set is considered. The following algorithm (adapted from Gross & Yellen

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Figure 41. Graphs of the dynamic FAIR model.

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Figure 42. Central and peripheral vertices, cut-points of the dynamic FAIR model.

[19] p.1109) uses the graph form of G. In the following example, Figure 43 illustrates an arbitrary matching M = {(3, c), (4, d), (5, b)} in gray thick lines. At the next iteration, a set of free vertices is obtained F REE = 1, 2, a. Then \ \ SU = U F REE = {1, 2, 3, 4, 5} {1, 2, a} = {1, 2}. Since SEEN = ∅, it follows SW = {w|w ∈ / SEEN and(u, w, (c, 3), (3, a)}) ∈ E, u ∈ SU }, = {b, c}. Since the vertices b and c are both matched the algorithm continues \ \ SW F REE = {b, c} {1, 2, a} = ∅. S Then SEEN = ∅ SW = {∅, b, c}. We have SU = {u|(w, u) ∈ M, w ∈ SW = {3, 5},

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FORD- FULKERSON ALGORITHM S input: a bipartite graph G(U W, E) and an any arbitrary matching M output: a maximum matching M begin M := ∅ DONE := FALSE while NOT DONE let FREE be the T set of exposed vertices. set SU := U F REE SEEN := ∅ ST ILLOOKIN G := T RU E while ST ILLOOKIN G for an augmenting path set SW := / SEEN and(u, w) ∈ E, u ∈ §U } T {w |w ∈ if SW F REE 6= ∅ then an augmenting path exists constructing an augmenting path P L then M := M P ST ILLOOKIN G := F ALSE else continue lookingSfor an augmenting path SEEN := SEEN SW SU := {u|(w, u) ∈ M, w ∈ SW } if SU = ∅ ST ILLOOKIN G := F ALSE DON E := T RU E end

and according to (27), SU = {a, d}. The vertices a and d are free (or unmatched).Then SW

\

F REE = {a, d}

\ {1, 2, a} = {a} = 6 ∅.

An augmenting path exists. In Figure 43, the BFS tree has been rooted at the unmatched source 1. The augmenting path P is P =< (1, c), (3, a), (4, d), (5, b) > with two unmatched endpoints. The new matching is M P, M= M M (28) = < (3, c), (4, d), (5, b) > {(1, c)}, = < (1, c), (3, a), (4, d), (5, b) > . At the next iteration, F REE = {2} and SU = U

\

F REE = {1, 2, 3, 4, 5}

\ {2} = {2},

and following (28) SW = {b} is deduced and SW

\

F REE = {b}

\

{2} = ∅

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Figure 43. Example of the matching procedure.

such that there no augmenting path. Looking for a further augmenting path, SEEN is calculated [ [ SEEN = SEEN SW = ∅ {b} = {∅, b}. T Then SU = {5} and SW = {d}. Hence SW F REE = ∅, such that no augmenting path exists. We continue looking for an augmenting path with [ SEEN = {b} {d} = {b, d},

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and finally SW = ∅, SU = ∅. No further augmenting path has been found and the algorithm terminates with the maximum matching M = {(1, c), (3, a), (4, d)}. This maximum matching is illustrated in Figure 43. There are many other ways to solve this maximum flow problem. Another algorithms use the matrix form and consist in two phases, the labeling and the flipping phase.

B.

Matching Problem of the CS Sub-model

In the former normalization of the CS sub-model, the matching of the variables to the equations was imposed for economic theoretical reasons. Let us consider the matching problem in general. In the following 0-1 matrix M of the CS sub-model, the rows state for the equations and the columns for the variables. The entries mij are such that ( 1 if a variable ♯ in column j is present in equation ♯ in line i, mij = 0 otherwise. The matrix is shown in Figure 44. To solve the model, one has to assign each variable to one single equation. This problem is known as a matching problem. Given a graph G with bipartition V (G) = (U, W ). Let us denote ∇(v) the set of edges incident to v. A 0-1 vector

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Figure 44. Structure of the CS sub-model.

x in ℜE(G) is the incidence vector of a matching, if and only if, x(∇(v)) ≤ 1 for every vertex v ∈ V (G). The linear programming problem is maximize 1 .x subject to A .x ≤ 1

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x ≥ 0, where A = (ave ) is the incidence matrix of G with ( 1 if v is an end vertex of e, ave = 0 otherwise. The set of the inequations form a polytope M(G). The solutions are those which maximize the objective function 1.x The bi-adjacency matrix B = (bij ) is defined by ( 1 if (ui wj ) ∈ E(G), bij = 0 otherwise. The bi-adjacency matrix S of the CS sub-model is shown in Figure 45: for convenience, a bipartite graph G = (U W, E), where a black square states for one 1 and a blank for zero. A matching matches each vertex in U = {u1 ...un } to one in W = {w1 , . . . , wn }. Hall’s marriage Theorem states that there is a matching in which every equation can be married, if and only if, every subset S of equations knows a subset of variables at least as large as |S|. A polynomial-time matching algorithm follows from Berge’s Theorem, which states that a matching is maximum, if and only if, it contains no augmenting path. The algorithm starts with an arbitrary matching. This matching is the improved by finding, if any, an S M-augmenting path P . Then M is replaced with the symmetric difference (M − P ) (P − M). The matching is maximum, when it contains no augmenting path.

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Given a graph G with bipartition V (G) = (U, W ). Let us denote ∇(v) the set of edges incident to v. A 0-1 vector x in RE(G) is the incidence vector of a matching if and only if x(∇(v)) ≤ 1 for every vertex v ∈ V (G).

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Figure 45. Bi-adjacency matrix.

The linear programming problem is

maximize 1 .x subject to A .x ≤ 1 x ≥ 0,

where A = (ave ) is the incidence matrix of G with

ave

( 1 = 0

if the vertex v is incident on the edge e, otherwise.

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Figure 46. Incidence matrix.

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the objective function 1.e . The system of inequality constraints is e1 + e2 + e3

≤ 1 e1 ≥ 0

e4 + e5 + e6

≤ 1 e3 ≥ 0

e7 + e8

≤ 1 e4 ≥ 0

e9 + e10

≤ 1 e5 ≥ 0

e11 + e12

≤ 1 e6 ≥ 0

e13 + e14

≤ 1 e7 ≥ 0

e15 + e16 + e17

≤ 1 e8 ≥ 0

e1 + e4 + e9 + e15

≤ 1 e9 ≥ 0

e2 + e5

≤ 1 e10 ≥ 0

e3 + e6 + e7

≤ 1 e11 ≥ 0

e10 + e11 + e13

≤ 1 e12 ≥ 0

e8 + e12

≤ 1 e13 ≥ 0

e14 + e16

≤ 1 e14 ≥ 0

e17

≤ 1 e15 ≥ 0

e16 ≥ 0 e17 ≥ 0

The incidence matrix of the CS sub-model is shown in Figure 46. Among the optimal solutions of that system there will be a 0-1 vector. Theorem 27 (Lov´ asz & Plummer [38]) Let G be a bipartite graph. Then the vertices of the polytope {x ∈ RE(G) , x ≥ 0, A.x ≤ 1} are 0-1 vectors. In fact, they are exactly the incidence vectors of matchings. The computation renders a technical matching which is shown in Figure 47(a) . However, the economist would not accept this assignment of the variables to the equations.

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Figure 47. Maximum matchings.

Indeed, the econometric equations generally impose the causality issued from the explanatory to the explained variables. This will restrict the acceptable matchings. Let us impose that the econometric equations 1, 2 and 4 compute the adequate variable. Then the variable l˙ will be assigned to 1, variable p˙c to 2 and variable ∆ab to 4. The matching shown in Figure 47(b), will then be retained by the economists.

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C.

Backtracking Procedure

Suppose that someone is facing with the search of a True option. The options False and True are organized as with the tree of Figure 48. The procedure consists of starting at the root of the decision-tree, continuing until an option is found, and backtracking if necessary until the True option is found. In pseudo-code, the algorithm is described by the boolean function If solve(n) is True, vertex n is solvable. It is on a path from the root to some goal vertex and is a part of the solution. If solve(n) is False, then no path will include the vertex n to any goal vertex. A simple example illustrates this procedure. The visited nodes have been highlighted in Figure 48. The procedure may be 1: Start at root 1. The decisions are nodes 2 and 7. Choose node 2 2: At node 2, the decisions are nodes 3 and 4 3: Option 3 is False. Then go back to node 2 4: At node 2 choose node 4 (since node 3 was already tried) 5: At node 4, try option 5. It is False. Then go back to node 4 6: At node 4, try option 6. It is False. Then go back to node 4 7: Go back to node 2. It has been already visited. Then go back to root 1 8: At the root 1, choose node 7 9: At node 7, try option 8. It is True. Then stop.

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Algorithm : BACKTRACKING PROCEDURE input: an undirected graph G = (V, E) and any arbitrary vertex s. The tree consists of root s, leaves and a goal vertex output: a path to a goal vertex boolean solve(vertex s){ if s is a leaf vertex{ if the leaf is a goal vertex then{ print s return True } else return False } else{ for each vertex-child c of vertex s{ if solve(c) succeeds then{ print s return True } } return False } }

Figure 48. Example of backtracking.

D. D.1.

Graph Theory with Mathematica Introduction

For large macro-econometric models, where the equations are often non-linear, a block triangular boolean 0-1 matrix will tell how to organize the whole system of equations. This Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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will be helpful for recursive resolutions. Furthermore, the knowledge of the structural properties may show obvious but also unsuspected properties such that a dominant or dominated position, even within a set of interdependent variables. The degree of eccentricity of some component will then be considered. This problematic may be approached by means of oriented graphs and their algorithms, as it was proposed earlier by Keller [26] in the 1970’s. This analysis concerns performed embeddings for graphs using a longest circuit, the search after edge-disjoint cycles and a vertex typology. The application will be the dynamic KleinGoldberger model (Adelman & Adelman [3], Klein & Goldberger [34]) with 26 equations. The computations are done with M athematica 6.0 (Wolfram Research [54], Wolfram [53]) and specialized packages for graphs, such as DecisionAnalysis‘ Combinatorica (Pemmaraju & Skiena [42]) and DiscreteM ath‘GraphP lot. The documentation can be found at http : //library.wolf ram.com/inf ocenter/ Other Mathlink-compatible external programs written in Fortran F77L (Absoft [1][2]. The program Baobab (Keller [26][31]) returns an exhaustive list of lexicographically ordered circuits.

D.2.

Klein-Goldberger Model

The macro-econometric Klein-Goldberger model for the United States (henceforth KG model) has been the essential reference in earlier econometric modeling. The main purposes of such models are forecasting and simulation of economic policies as well (Bodkin [5]). Many analysis have been achieved with this small size keynesian model to discover its static and dynamic properties (Adelman & Adelman [3]).

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D.2.1.

Description of the Model

An economic model with n endogenous (Xi, i = 1, n) and m exogenous (Zk , k = 0, m−1) variables where Z0 is a constant, can be described by the following equations system Xi = fi (X1 , . . . , Xi−1 , Xi+1 , . . . , Xn , Z0 , Z1 , . . . , Zm ), i = 1, n. The KG model is a 26-equations macro-econometric model for USA over the annual period 1929-1952. The model has 26 endogenous variables and 14 exogenous variables: 21 endogenous are delayed by one year, one by 3 years, one by 5 years, and two exogenous variables are delayed by 1 year. Theoretically the KG model is a keynesian dynamic IS-LM model of an open economy, where the demand of goods and services plays an essential role in the production determination. The variables are defined in Tables 7 and 8. D.2.2.

Graph of the Model

To associate a graph to this model, the economist must tell how to read the model: there may be different ways to associate one endogenous variable to one equation in a model. A model may have different theoretical explanations. This AP problem is known as a maximal bipartite matching. A matching in the bipartite graph g = (X, Y, e) matches each vertex in X (the set of variables) to one in Y (the set of equations). The primitive

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Table 7. Numbered list of variables of the KG model. No 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

Symbol C I Sp Pc D W1 Nw w Fi A1 pa L1 L2 iL iS YTD P p K B T Tw Tc Tp Ta Y

Definition consumer expenditures (real) investment (real) corporate savings (deflated) corporate profits (deflated) capital consumption charges (real) private employee compensation (deflated) number of salaries worked hours imports farm income (deflated) agricultural prices liquid assets held by businesses (deflated) liquid assets held by persons (deflated) average yield on corporate bonds average yield on short term commercial paper Gross National Product nonwage nonfarm income (deflated) inflation stock of capital (real) corporate profits (deflated) net indirect taxes (deflated) net taxes on wage income (deflated) corporate income taxes (deflated) net taxes on nonwage (deflated) net taxes on farm income (deflated) domestic production

AddEdges[g, e] of the package Combinatorica returns a graph g(X,Y,e) . An adequate test for that type of graph is given by BipartiteQ[g]. The ranked embedded graph is rendered by RankedEmbedding[g, t], where parameter t numbers the vertices from 1 to 26 for the variables and from 27 to 52 for the equations. Finally, highlight edges show the retained matching. The complete set of primitives is given by g = AddEdges[EmptyGraph[26, T ype]], Directed], {listof edges}], where the term edges represents pairs like {x1 , y1 }, {x3 , y1 },{x2 , y2 },. . . such that equation y1 can determine variables x1 or x3 and where equation y2 determine x2 , ShowGraph[SetGraphOptions[RankedEmbedding[g, t], {matching}, options]], where matching represents the list of edges realizing the property of a matching in a bipartite graph. To load the package Combinatorica evaluate ≡< K, D, K > represents real capital accumulation in Figure 55(a) and the circuit < 9, 16, 6, 9 >≡< F i, Y T D, W1 , F i > illustrates the interactions between national product and importations. The circuit < 3, 10, 17, 4, 3 >≡< Sp , A1 , P, Pc , Sp > is

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Figure 54. Circuits of the KG model.

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the revenue allocation, and the circuit < 17, 1, 16, 5, 26, 17 >≡< P, C, Y T D, D, Y, P > is the circuit of the national revenue.

Figure 55. Maximal lists of edge-disjoint cycles in the KG model.

In the dynamic KG model (586 circuits), eight independent circuits have been found in Figure 55(b). The circuit < 17, 10, 17 >≡< P, A1 , P > illustrates how farm and non-farm incomes interact. The circuit < 18, 8, 18 >≡< p, w, p > is the Phillips curve. The circuit < 19, 2, 19 >≡< K, I, K > for investment. The circuit < 20, 3, 20 >≡< B, Sp , B > is between the corporate and savings. The consumption circuit (with thick black edges) is < 6, 1, 16, 6 >≡< W1 , C, Y T D, W1 >. The circuit with investment de-

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mand , inflation and revenues (with thick dashed edges) is < 10, 2, 16, 7, 18, 10 >≡< A1 , I, T T D, Nw , p, A1 >. The longest circuit of the revenue (with thick gray edges) is defined by < 4, 3, 10, 9, 16, 21, 26, 17, 4 >≡< PC , SP , A1 , F i, Y T D, T, Y, P, PC >

D.4.

Vertex Typology of the KG Model

The variable typology is based on the properties of the all-pairs shortest paths matrix. D.4.1.

All-Pairs Shortest Paths Matrix

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A computation of the all-pairs shortest paths matrix and a search for properties are carried out on the two versions of the KG model (Figures 56 and 57).

Figure 56. All-pairs shortest paths matrix of the static KG model.

D4.1.1. Static KG Model This computation of the largest all-pairs shortest paths matrix is essential to evaluate the eccentricity of different vertices. This matrix will be bordered by a column vector containEmerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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Figure 57. All-pairs shortest paths matrix of the dynamic KG model.

ing the eccentricity values. Each value represents the maximum of the shortest paths from one vertex to each every other vertices. The maximum eccentricity is the diameter. The minimum eccentricity is the radius of that graph. The all-pairs shortest paths are viewed by using by the primitive AllP airsShortestP ath[w]//T ableF orm. The largest all-pairs shortest paths matrix for the static version is given in Figure 56. A single-source shortest path consists of the shortest paths between one given source to all other vertices. The all-pairs shortest paths are then deduced, considering each vertex as a source successively (Figure 56). For example (C, W 1) = 2 signifies that the shortest path going from the source C and the the sink W 1 has two edges. We also have (w, P c) = 4 and (P c, w) = 5. The primitives of the package Combinatorica are used with SCC = DeteteV ertices[g, listof vertices]; AllP airsShortestP ath[SCC] //M atrixF orm. The command DeleteV ertices[g, {7, 17}] deletes the vertices 7 and 17 from the graph g, and all incident edges. To produce the table, the set of commands are of the type : F rameBox[StyleF orm[T ableF orm[s, T ableHeadings → {list, list}], options]//DisP layF orm

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Andr´e A. Keller The maxima of the shortest paths are given in a vector column {5, 5, 6, 5, 4, 6, 6, 5, 5, 6, 4, 5, 5, 6, 7, 5, 6, 5, 6, 6},

placed to the right to the matrix. The second element of that list indicates that the maximum distance from 2 : Sp is five edges. From this list, we deduce that the radius equals 4 and the diameter equals 7. These results are obtained by the following directives : Eccentricity[g] returns the eccentricity of each vertex of graph g, GraphCenter[g] gives the list of of the vertices of graph g with minimum eccentricity. Diameter[g] is the maximum length among all pairs of vertices in g and Radius[g] the minimum eccentricity of any vertex of g. The static KG model has two central vertices 6 : W1 (private compensation), 16 : GNP and one eccentric variable 21 : T (indirect taxes). In this model production and revenues play a central role. One taxes variable has a weak influence unless it is interdependent. From Figure 56, the perturbation (or in-eccentricity) evaluates the intensity of received influences from the other vertices of the SCC. The columns of the all-pair shortest paths matrix are explored. The maxima of shortest paths are given in a row vector below the matrix {4, 6, 5, 5, 6, 5, 6, 4, 4, 7, 5, 4, 6, 6, 4, 6, 5, 5, 7, 4} The in-central vertices are 1 : C (consumption), 9 : Fi (import) , 10 : A1 (farm income), 17 : P (non wga income) , 21 : T (indirect taxes) , 26 : Y (production). The in-eccentric vertices are : 11 : pa (prices in agriculture), 25 : Ta (farm income).

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D.4.2.

D4.1.2. Dynamic KG Model

The largest all-pairs shortest paths matrix in the dynamic version of the KG model is shown in Figure 57. In the dynamic version we have the same results for these examples : (C, W 1) = 2 , (w, P c) = 4 and (P c, w) = 5. The maxima of the shortest paths are given in a vector column {6, 6, 5, 6, 5, 4, 5, 5, 6, 5, 6, 7, 7, 5, 5, 4, 5, 6, 7, 7, 6, 5, 7, 6}. The radius is then equal to 4 and the diameter to 7. The dynamic KG model has two central vertices 6 : W1 (private compensation), 8 : w (worked hours) and five eccentric variables such as 12 : L1 (liquid assets held by persons), 13 : L2 (liquid assets held by businesses), 21 : T (indirect taxes), 22 : Tw (taxes on wage income), 25 : Ta (taxes on farm income). The variables of revenue and worked hours exert a greater influence on the rest of the model. The monetary variables (liquidities of persons and firms) are eccentric and exert less influences. In that sense , this is conformed to the theory. From Figure 57, the perturbation (or in-eccentricity) is deduced. The columns of the largest all-pair shortest paths matrix are explored. The maxima of shortest paths are given in a row vector {4, 6, 5, 5, 6, 5, 6, 4, 4, 7, 5, 4, 6, 6, 4, 6, 5, 5, 7, 4}. . The in-central vertices are 1 : C (consumption), 9 : Fi (Import) , 10 : A1 (farm income), 17 : P (non wage income) , 21 : T (indirect taxes) , 26 : Y (production). The in-eccentric vertices are : 11 : pa (prices in agriculture), 25 : Ta (taxes on farm income). Thus the total supply (production and import), the consumer demand and indirect taxes, non wage

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incomes are rather perturbed by the rest of the model. Agricultural prices and taxes are less influenced.

D.4.3.

Vertex Typology

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The vertex typology is based on the concepts of eccentricity and in-eccentricity. A vertex of the SCC may be a center or an eccentric vertex or else. It can be also an in-center or an in-eccentric vertex or or else. Crossing these properties, a 3 × 3 table is obtained. The columns (from left to right) represent each one a particular property : eccentric, central and else. The rows (from top to bottom) represent each of them one particular property : in − eccentric, in − central and else. Crossing these two features, different types for the vertices are deduced. The type A corresponds to an interrelated vertex which is eccentric and in − eccentric: this variable will then have little influence and will be less perturbed. The type B corresponds to a vertex which is weak dominant as a central vertex but an ineccentric vertex (type B∗ means strong dominant). The type C is a weak integrated vertex as it diffuses rapidly and is rapidly perturbed (type C∗ means strong integrated). The type D figures a weak dominated vertex since its influence is low and and its perturbation hight (type D∗ means strong dominated). The type E corresponds to vertices without remarkable features. According to Figure 58, the set of interrelated vertices is composed of weak integrated variables (C). According to that table, the weak dominant vertices correspond to agricultural prices pa and net taxes of farmers Ta . The strong dominated vertex is indirect taxes less transfers to agriculture T. Weak integrated vertices are the national revenue YTD and the production Y, the wages W1 and non-wages A1 , the consumption C and the imports Fi .

Figure 58. Vertex typology of the static KG model. Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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Andr´e A. Keller Figure 59 shows the typology of the dynamic KG model. The characteristics consist

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Figure 59. Vertex typology of dynamic KG model.

of weak integrated variables with gross national product YTD, consumption C, imports Fi , prices p and farm incomes A1 . The weak dominant variables are agricultural price pa and depreciation on real capital D. Liquidities L1 , L2 and taxes Tc are weak or strong dominated. Eccentric vertices correspond to net taxes T and Tw . The comparison between the two versions shows similarities for the weak integrated vertices, but new vertices enter the set of dominated vertices.

D.5.

Conclusion

Different structural properties of the KG model are deduced from the graph theory: 1) there may have different acceptable matchings, due to different economic theories , 2 ) the circular graph embedding may be improved by reordering the vertices of the giant SCC, using a longest circuit , 3) the enumeration of the circuits by lengths are useful computations to evaluate the interdependency in the model, 4) a vertex typology deduced from the structural properties.

E.

List of Variables of the FAIR Model

The symbol, the number and definition of the endogenous variables of the FAIR model are given in Table 9. The number of a variable refers to the equation that calculates this variable. The variables refer to levels, unless otherwise indicated. The deflated values are in billions of 1987 dollars. A numbered list of the symbols is also given in Table 10.

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Table 9. Alphabetical list of symbols in the FAIR model. Symbol

No

Definition

AA AB AF AG AH AR AS BF BG BO BR CCF CD CF CG CN CS CUR DF D1GM D1SM E EXPG EXPS GDP GDPD GDPR GNP GNPD GNPR HF HFF

89 73 70 77 66 75 79 55 56 22 56 21 3 68 25 2 1 26 18 90 91 85 106 113 82 84 83 129 80 130 14 100

total net wealth (deflated), households net financial assets, banks net financial assets, firms net financial assets, fed.government net financial assets, households net financial assets, foreign net financial assets, loc.government long term bond issues (estimated), firms long term bond issues (estimated), loc.government bank borrowing from the Fed total bank reserves capital consumption, firms consumer expenditures for durable goods (deflated) cash flow, firms capital gains/losses on corporate stocks, households consumer expenditures for nondurable goods (deflated) consumer expenditures for loc.government (deflated) currency held outside banks dividends paid marginal personal income tax rate (fed/government) marginal personal income tax rate (loc.government) total employment total expenditures, government total expenditures, loc.government gross domestic product GDP chain price index gross domestic product (deflated) gross national product GNP chain price index gross national product (deflated) number of hours paid per job, firms deviation of HF from its potential values

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Table 9. Continued Symbol

No

Definition

HN HO IBTG IBTS IHH IKF IM INTF INTG INTROW IVA IVF JF JHMIN JJ JJS KD KH KK KKMIN L1 L2 L3 LM M1 MB MF MH PCD PCGDPD PCGDPR PCM1 PCN

62 15 51 52 4 12 27 19 29 88 20 117 13 94 95 96 58 59 92 93 5 6 7 8 81 71 17 9 37 122 123 124 36

number of non overtime hours paid per job, firms number of overtime hours paid per job, firms indirect business taxes, fed.government indirect business taxes, loc.government residential investment (deflated), households nonresidential fixed investment (deflated), firms imports (deflated) net interest payments, firms net interest payments, fed.government net interest receipts, foreign inventory valuation adjustment inventory investment (deflated), firms number of jobs, firms number of required worker hours ratio worker hours to population 16 and over ratio of actual to potential JJ stock of durable goods (deflated) stock of housing (deflated), households capital stock (deflated) required capital (deflated) labor force of men 25-54 labor force of women 25-54 labor force of all others, 16+ number of jobs compared to employed people money supply net demand deposits and currency demand deposits and currency, firms demand deposits and currency, households price deflator for CD percentage change in GDPD percentage change in GDPR percentage change in M1 price deflator for nondurable goods consumption

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Table 9. Continued Symbol

No

Definition

PCS PD PEX PF PG PH PIEF PIH PIK PIV POP PROD PS PUG PUS PX RB RECG RECS RM RMA RS RSA SB SF SG SGP SH SHRPIE SIFG SIG SIHG SIS

35 33 32 10 40 34 67 38 39 42 120 118 41 104 110 31 23 105 112 24 128 30 127 72 69 76 107 65 121 54 103 53 109

price deflator for loc.government consumption price deflator for domestic sales price deflator for exports price deflator for EX, except farm products price deflator for CD price deflator for total expenses of households profits before tax, firms price deflator for residential investment price deflator for nonresidential fixed investment price deflator for inventory investment population 16+ labor productivity price deflator for purchases of goods purchases of goods and services, fed.government purchases of good and services, loc.government price deflator of total sales bond rate total receipts, fed.government total receipts, loc.government mortgage rate mortgage rate after tax three month Treasury bill rate bill rate after tax saving, banks saving, firms saving, government NIA balance, fed.government saving, households ratio profits to wage bill employer social insurance contribution total employer and employee social insurance employee social contribution total social insurance contributions to fed.government

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Table 9. Continued. Symbol

No

Definition

SR SRZ SS SSP TCG TCS TFG TFS THG THS TPG TRRSH U UB UBR UR V WA WF WG WH WM WR WS X XX Y YD YNL YS YT Z

74 116 78 114 102 108 49 50 47 48 101 111 86 28 125 87 63 126 16 44 43 45 119 46 60 61 11 115 99 98 64 97

saving, foreign saving rate, households saving, foreign NIA balance, services corporate profit taxes receipt,government corporate profit tax receipt, loc.government corporate profit taxes, firms to fed.government corporate profit taxes, firms to loc.government personal income taxes, households to fed.government personal income taxes, households to loc.government personal income tax receipts, fed.government total transfer payments, local to fed.government number of unemployed people unemployed insurance benefits unborrowed reserves saving, firms civilian unemployment rate stock of inventories by firms wage rate after tax average hourly earnings, excluding overtime in firms average hourly earnings of civilian workers average hourly earnings,excluding overtime of all average hourly earnings of military workers real wage rate of workers in firms average hourly earnings of workers in loc.government total sales (deflated), firms total sales, firms production, firms disposable income, households non labor income after tax, households potential output, firms taxable income, households labor constraint variable

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Table 10. Numbered list of symbols in the FAIR model. No 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33

Symbol CS CN CD IHH L1 L2 L3 LM MH PF Y IKF JF HF HO WF MF DF INTF IVA CCF BO RB RM CG CUR IM UB INTG RS PX PEX PD

No 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

Symbol PH PCS PCN PCD PIH PIK PG PS PIV WH WG WM WS THG THS TFG TFS IBTG IBTS SIH SIFG BF BG BR KD KH X XX HN V YT SH AH

No 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99

Symbol PIEF CF SF AF MB SB AB SR AR SG AG SS AS GNPD M1 GDP GDPR GDPD E U UR INTROW AA D1GM D1SM KK KKMIN JHMIN JJ JJS Z YS YNL

No 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130

Symbol HFF TPG TCG SIG PUG RECG EXPG SGP TCS SIS PUS TRRSH RECS EXPS SSP YD SRZ IVF PROD WR POP SHRPIE PCGDPD PCGDPR PCM1 UBR WA RSA RMA GNP GNPR

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References [1] Absoft (2000), ProFortran Windows : Fortran & C/C++ User Guide. Rochester Hills, USA : absoft, development tools and languages, http//www.absoft.com. [2] Absoft (2000), Fortran 77 : Reference Manual. Rochester Hills, USA : absoft, development tools and languages, http:www.absoft.com. [3] Adelman I. & Adelman, F.L. (1959), The dynamic properties of the Klein-Goldberger model, Econometrica 27(4): 596-625. [4] Bermond, J.C. & Thomassen, C. (1981), Cycles in digraphs – a survey, Journal of Graph Theory 5: 1–43. [5] Bodkin, R.G., Klein, L.R & Marwah,K. (1991), A History of Macroeconometric Model-Building, Brookfield, Vermont USA : Edward Elgar Pusblishing Co. [6] Chartrand, G. & Lesniak, L. (2004), Graphs & Digraphs, 4th edition, New York:Chapman & Hall/CRC. [7] Chegireddy, C.R. & Hamacher, H.W. (1987), Algorithms for finding k-best perfect matchings, Discrete Applied Mathematics 18: 155-165. [8] Danˇ ut, M. (1993), On finding the elementary paths and circuits of a digraph, Science Bulletin of Polytechnic University of Bucarest, Series D 55(3-4): 29-33.

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[9] Ehrenfeucht, A., Fosdick, L.D. & Osterweil, L.J. (1973), An algorithm for finding the elementary circuits of a directed graph, University of Colorado at Boulder, Department of Computer Science, Report ♯CU-CS-024-73. [10] Even, S. (1979), Graph Algorithms, Rockville, Massachussets: Computer Science Press. [11] Fair, R.C. (1994), Testing Macroeconometric Models, Cambridge, Massachusetts, Harvard University Press. [12] Fukuda, K. & Matsui, T. (1992), Finding all minimum-cost perfect matchings in bipartite graphs, Networks 22:461-468. [13] Gallo, G.M. & Gilli, M.H. (1990), How to strip a model to its essential elements, Computer Science in Economics and Management 3, Kluwer Academic Publishers, pp.199-214. [14] Giessen (Van der -), A. A. (1970), Solving non-linear systems by computer; a new method, Statistica Neerlandica 24/1:41-50. [15] Gilli, M. & Rossier, E. (1981), Understanding complex systems, Automatica 17 (4): 646-652. [16] Gondran, M. & Minoux, M.(1984),Graphs and Algorithms, New York: John Wiley. Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Graph Analysis with Application to Economics

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[17] Greenberg H.J.& Maybee, J.S. (Editors)(1981), Computer-Assisted Analysis and Model Simplification, New York: Academic Press. [18] Greene, W.H. (1991), Econometric Analysis, New York, MacMillan Publishing Co. [19] Gross, J.L. & Yellen, J. (editors)(2003), Handbook of Graph Theory, in Discrete Mathematics and Its Applications, series Editor Kenneth H. Rosen, New York:CRC Press. [20] Gross, J.L. & Yellen, J. (2006), Graph Theory and its Application, New York: Chapman & Hall. CRC Press. [21] Harary, F.(1994), Graph Theory, Boulder, CO: Westview Press. [22] Harary, F. & Palmer, E.M. (1973), Graphical Enumeration, New York: Academic Press. [23] Hopcroft J. & Tarjan, R.E. (1973), Algorithm 447 : efficient algorithms for graph manipulation, Communications ACM 16: 372–378. [24] Hopcroft, J. & Karp, R.M. (1973), An O(n5/2 ) algorithm for maximum matching in general graphs, SIAM Journal on Computing 2: 225–231. [25] Johnson, D.B. (1975), Finding all the elementary circuits of a directed graph, SIAM Journal on Computing 4(1): 77–84.

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[26] Keller, A.A. (1977), Essai sur les Structures Compar´ees des Mod`eles Macro´economiques de Pr´evision : Construction d’une Typologie par l’Etude des Graphes Associ´es et l’Analyse Factorielle, Th`ese pour le Doctorat d’Etat de Sciences Economiques, Universit´e de Paris I. [27] Keller, A.A. (1984), Semi-reduced forms of econometric models, in: Analysing the Stucture of Econometric Models, The Hague:Martinus Nijhoff, pp. 89 - 113. [28] Keller, A.A. (2006), Matching theory and economic building, Conference “Optimal Discrete Structures and Algorithms”, U niversit¨ at Rostock, Institut f u ¨r Mathematik, Institut f u ¨r Informatik, September, 04-6, 2006, Electronic Notes in Discrete Mathematics 27:57–58, unpublished. [29] Keller, A.A. (2007), Graph theory and economic models : from small to large size applications, Electronic Notes in Discrete Mathematics 28:469–476. [30] Keller, A.A. (2007), Connectivity threshold and vulnerability in large scale macroeconometric models, 21th British Combinatorial Conference, University of Reading (United Kingdom), July 8-13, 2007, unpublished. [31] Keller, A.A. (2008), Lexicographic all circuits enumeration in large scale macroeconometric models, in Advances in Computational Algorithms and Data Analysis, Ao, Sio-long, Rieger Burghard & Chen Su-Shing (editors) Series: lecture notes in electrical engineering, vol. 14, New York: Springer Verlag, pp.465-479.

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[32] Keller, A.A. (2007), Graph theory to the dynamic structure of macro-economic models, “9th Nordic Combinatorial Conference”, Bergen (Norway), November 23-24, 2007. unpublished. [33] Klein, L.R. (1950), Economic Fluctuations in the United States 1921-1941, Cowles Commission Monograph, No 11, New York:John Wiley & Sons, Inc. [34] Klein,L.R. & Goldberger, A.S. (1955) An econometric Model of the United States 1929-1952, Amsterdam : North-Holland. [35] Kocay, W. & Kreher, D.L. (2004), Graphs, Algorithms, and Optimization, New York:Chapman & Hall/CRC. [36] Kuhn, H. W. (1955), The Hungarian method for the assignment problem, Naval Research Logistics Quarterly 2, 83-97. [37] Kuhn, H. W. (1956), Variants of the Hungarian method for assignment problems, Naval Research Logistics Quarterly 3, 253-258. [38] Lov´asz L. & Plummer, M.D. (1986), Matching Theory, Annals of Discrete Mathematics 29, Mathematic Studies, New York:North-Holland. [39] Maddala, G.S. (1977), Econometrics, London, International Book Company. [40] Munkres, J. (1957), Algorithms for the assignment and transportation problems, Journal of the Society of Industrial and Applied Mathematics 5(1), 32-38.

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[41] Murty, K.G. (1968), An algorithm for ranking all the assignments in order of increasing cost, Operations Research 16, 682-687. [42] Pemmaraju, S. & Skiena, S. (2003), Combinatorics, and Graph Theory with M athematicar , Cambridge, UK: Cambridge University Press. [43] Prabhaker M. & Narsingh, D. (1976), On algorithms for enumerating all circuits of a graph, SIAM Journal on Computing 5(1):90–99. [44] Schrijver, A. (2003), Combinatorial Optimization : Polyedra and Efficiency, Volume A (Paths, Flows, Matchings), New York:Springer-Verlag. [45] Sedgewick, R. (2002), Algorithms in C, Part 5 (Graph Algorithms), New York:Addison-Wesley. [46] Skiena, S. I. (1990), Implementing Discrete Mathematics : Combinatorics and Graph Theory with M athematicar , New York:Addison-Wesley Publishing Co. [47] Tarjan, R.E. (1972), Depth-first search and linear graph algorithms, SIAM Journal on Computing 1:146–160. [48] Tarjan, R.E. (1973), Enumeration of the elementary circuits of a directed graph, SIAM Journal on Computing 2(3):211–216. Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Graph Analysis with Application to Economics

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[49] Tiernan, J.C. (1970), An efficient search algorithm to find the elementary circuits of a directed graph, Communications ACM 13:722–726. [50] Uno, T. (1998), A new approach for speeding up enumeration algorithms, Lecture Note in Computer Science 1533, Springer-Verlag: 287–296. [51] Vedel, Cl. (1981), Politique mon´etaire et politique budg´etaire dans un mod`ele dynamique d’une economie en croissance, Institut international de finances publiques, Tokyo, unpublished. [52] Weinblatt, H. (1972), A new search algorithm for finding the simple cycles of a finite directed graph, Journal of the ACM 19:43–56. [53] Wolfram research (1999), MATHEMATICAr 4 : Standard-On Packages, Champain, Illimois:Wolfram Media Inc., with associated web site http://www.wolfram.com.

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[54] Wolfram, S. (2003), The MATHEMATICAr Book, 5th edition, Champain, Illimois:Wolfram Media Inc., 2003. With associated web site: http://www.wolfram.com.

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In: Emerging Topics on Differential Geometry... Editors: L. Bernard and F. Roux, pp. 299-344

ISBN 978-1-60741-011-9 c 2010 Nova Science Publishers, Inc.

Chapter 8

A PPLICATION OF H ILBERT S PACES TO THE S TABILITY S TUDY OF F LOWS ON A S PHERE Yuri N. Skiba∗ Centro de Ciencias de la Atm´osfera, Universidad Nacional Aut´onoma de M´exico, Av. Universidad # 3000, C. P. 04510, M´exico D. F., M´exico

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Abstract The main properties of spherical harmonics and associated Legendre functions are briefly surveyed. Geographical coordinate maps for the sphere are defined and the well-known theorem about the partition of unity is given which is an important tool in the theory of integration of functions on smooth compact manifolds. Derivatives Ds and Λs of real degree s of smooth functions on the unit sphere are defined, and a family of Hilbert spaces Hs of generalized functions having fractional derivatives on the sphere is introduced. Instead of the modulus of continuity [1-3], in this work, the Hilbert spaces of functions are defined by means of multiplier operators [4-7]. Orthogonal projectors on the subspace Hn of homogeneous spherical polynomials of degree n and on the subspace PN of spherical polynomials of degree n ≤ N are defined. Some structural properties of Hilbert spaces Hs including various embedding theorems are given, and the rate of convergence of Fourier-Laplace series of functions of Hs is estimated. As applications of theoretical results, both the global asymptotic stability and the normal-mode stability of incompressible flows on a rotating sphere are considered. In particular, conditions for the global asymptotic stability of solutions to the barotropic vorticity equation are given, and the spectral approximation (convergence of eigenvalues and eigenvectors of discrete spectral problems) in the numerical normal-mode stability study of nondivergent viscous flows on a rotating sphere is examined.

1. Introduction Noticeable activity in the study of different questions of the theory of Fourier-Laplace series on the sphere is observed in the recent decades [8]. This is connected mainly with the ∗

E-mail address: [email protected]; Fax: (+52-55) 5622-4090; Tel.: (+52-55) 5622-4247

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Yuri N. Skiba

application of the theory of Fourier-Laplace series for the numerical solution of different problems on the sphere in such applied sciences as meteorology, weather forecast, climate theory and so forth. In particular, such problems as the convergence and summation of series on the sphere are of great importance at use of the spectral method for the discretization of partial differential equations on the sphere. Section 2 surveys briefly the main properties of Legendre polynomials, associated Legendre functions and spherical harmonics. In section 3, we consider maps with geographical coordinates for the sphere and the well-known theorem about the partition of unity which is an important tool in the integration of functions on smooth compact manifolds [15-17]. In detail, different maps on the sphere are studied for instance in [1,15,18]. Spaces of scalar functions having fractional derivatives on the sphere are mainly defined by two ways: either with the modulus of continuity of functions [1-3, 9-11] or with certain multiplier operators using geometrical properties of the sphere [4-7, 12-14]. In section 4, the second more simple and constructive approach is used. As in essence the unique differential operator invariant to isometries is the Laplace operator, it is natural to describe differential properties of functions on the sphere in terms of this operator. Orthogonal projectors on subspaces Hn of homogeneous spherical polynomials of degree n and on subspaces PN of spherical polynomials of degree n ≤ N are defined in section 4. Basic properties of Hilbert spaces Hs are discussed in section 5. According to the well-known Gibbs phenomenon [19], the convergence of Fourier-Laplace series is deteriorated near to sharp changes of a continuous function. For the sphere, as a compact manifold, there is a ready instrument of approximation, determined by the expansion of L2 -space in the direct sum of finite-dimensional subspaces Hn invariant to the group SO(3) of sphere rotations. The rate of convergence of Fourier-Laplace series of a functions from Hs is estimated in section 5 as well. In more details, the approximation theory of Fourier-Laplace series on the sphere is analyzed in [8]. As applications of theoretical results, we study two problems related to the stability of barotropic incompressible viscous and forced flows on a rotating sphere: the global asymptotic stability of the barotropic vorticity equation (BVE) and normal-mode instability of a steady solution to the BVE. In section 6, two conditions for the global asymptotic stability of a BVE solution are given which differ by the solution smoothness. In section 7, the spectral approximation for closed operators is used to estimate the rate of convergence of eigenvalues and eigenvectors of discrete spectral problems (arising in the numerical normalmode stability study) to an isolated eigenvalue of finite algebraic multiplicity m and ascent p and the corresponding eigenfunction of original differential spectral problem.

2. Spherical Harmonics Let S = {x ∈ R3 : |x| = 1} be a unit sphere in the three-dimensional Euclidean space; we denote by C∞ (S) the set of infinitely differentiable functions on S and by

hf, gi =

Z

f (x)g(x)dS

S

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Application of Hilbert Spaces to the Stability Study of Flows on a Sphere

301

kf k = hf, f i1/2

(2)

the inner product and norm in C∞ (S), respectively. Here x = (λ, µ) is a point on the sphere, dS = dλdµ is an element of sphere surface, µ = sin φ ; µ ∈ [−1, 1], φ is the latitude, λ ∈ [0, 2π) is the longitude and g is the complex conjugate of function g. It is known [17], that spherical harmonics Ynm (λ, µ)



2n + 1 (n − m)! = 4π (n + m)!

1/2

Pnm (µ) eimλ , n ≥ 0, |m| ≤ n

(3)

form orthogonal basis in C∞ (S) [17, 20]: D E Ynm , Ylk = δ mk δ nl

(4)

 1 , if m = k 0 , if m = 6 k

(5)

m Qm n (µ) = cnm Pn (µ)

(6)

where δ mk = is the Kronecker delta, and

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where Pnm (µ)


m/2
n 1 − µ2 dn+m µ2 − 1 = n n+m 2 n! dµ

(7)

is the associated Legendre function of degree n and zonal number m (see [17]) normalized by the constant cnm



2n + 1 (n − m)! = 4π (n + m)!

1/2

(8)

so that 2π

Z

1 −1

m Qm n (µ) Ql (µ) dµ = δ ml

(9)

m If m is even then Qm n (µ) and Pn (µ) are algebraic polynomials in µ. Each spherical harmonic is the eigenfunction of spectral problem

−∆Ynm = χn Ynm ,

|m| ≤ n

for symmetric and positive definite Laplace operator on S : Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

(10)

302

Yuri N. Skiba 

∂ −∆ = − ∂µ


∂ 1 ∂2 1−µ − ∂µ 1 − µ2 ∂λ2 2

(11)

χn = n (n + 1)

(12)

where n and m are integer, and

is the eigenvalue corresponding to Ynm (λ, µ) . For each integer n ≥ 0, the eigenvalue χn has multiplicity 2n + 1, and the span of 2n + 1 spherical harmonics Ynm (λ, µ) (|m| ≤ n) form a generalized (2n + 1)-dimensional eigenspace Hn = {ψ : −∆ψ = χn ψ}

(13)

corresponding to the eigenvalue χn [16, 17]. On the plane of wavenumbers (m, n) shown in Fig.1, the wavenumbers of the spherical harmonics from different subspaces Hn are on the different parallel lines. Besides, the black (and grey) circles correspond to the wavenumbers of the spherical harmonics which are antisymmetric (symmetric) with respect to equator (µ = 0).

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Remark 1. The subspace Hn is invariant not only with respect to the Laplace operator but also to any transformations of the SO(3) group of rotations of sphere about its arbitrary axis [16, 17]. The last assertion means that a spherical harmonic Ynm (λ, µ) from Hn can ′ ′ be represented in a new coordinate system (λ , µ ) by Ynm



′

′

λ ,µ



n X

=

  ′ αk m, n, N Ynk (λ, µ)

(14)

k= − n

  ′ where complex numbers αk m, n, N depend on the wavenumbers n , m and the pole ′

N of new coordinate system. In order to simplify notation we will widely use in this book a multi-index α ≡ (m, n) ≡ (mα , nα ) for the wavenumber (m, n): α ≡ (−m, n) ≡ (−mα , nα ) , Yα ≡ Ynm , X α(k)

≡

Pα ≡ Pnm ,

∞ X n X n=k m= −n

and

χα ≡ χn = n(n + 1), N X α(k)

≡

N n X X

.

(15)

n=k m= −n

Besides, we will denote by N, Z and R the sets of natural, integer and real numbers, respectively, and use the following subsets: Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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303

Figure 1. Wave numbers of harmonics Ynm (λ, µ) in C∞ (S).

R+ = {x ∈ R : x > 0},

R0 = {x ∈ R : x ≥ 0}

Z0 = {n ∈ Z : n ≥ 0}

(16)

We now present the basic properties of associated Legendre functions when n, m ∈ N [17]:

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1. A function Pnm (µ) is the solution of the Legendre equation ∂ ∂µ




∂ m m2 Pn (µ) + [χn − ]P m (µ) = 0 1−µ ∂µ 1 − µ2 n 2

(17)

2. Pnm (µ) ≡ 0 if m > n . 3. Two functions with complex conjugate wavenumbers α and α ¯ are related as Pα = (−1)mα Pα ,

Yα = (−1)mα Y α

(18)

4. The function Pnm (µ) is equal to zero in (n − m) points of interval (−1, 1). 5. Since Pnm (−µ) = (−1)n−m Pnm (µ)

(19)

functions Pnm (µ) and Ynm (λ, µ) are symmetric (antisymmetric) with respect to equator µ = 0 if (n − m) is even (odd). At the sphere poles, (|µ| = 1), Pnm (±1) = 0 if m 6= 0. The Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

304

Yuri N. Skiba

nonzero values at the poles have only the Legendre polynomials Pn (µ) ≡ Pn0 (µ), besides, Pn (±1) = (±1)n (see Fig.2). 6. The following assertions hold for Pnm (µ): Z

1 −1

Pnm (µ) Plm (µ) dµ = Z

1

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−1

2 (n + m)! δ nl , 2n + 1 (n − m)!

[Pnm (µ]2 (n + m)! dµ = , 2 1−µ 2m (n − m)!

(n, l ≥ m ≥ 0) ,

(n ≥ 0, 1 ≤ m ≤ n)

(20)

(21)

Figure 2. Legendre polynomials Pn (µ). Example 1. As an example, we now prove the orthogonality of Pnm (µ) and Plm (µ) if n 6= l. Multiplying (10) by Plm (µ) we obtain an equation. By changing the indices n and l in this equation we obtain one more equation. Subtracting one equation from the other we get ∂ ∂µ



2

1−µ




{Plm (µ)

 ∂ m ∂ m m P (µ) − Pn (µ) P (µ)} = ∂µ n ∂µ l

= [χl − χn ]Pnm (µ) Plm (µ)

Integrating (22) from µ = −1 to µ = 1 and taking into account that n 6= l we get Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

(22)

Application of Hilbert Spaces to the Stability Study of Flows on a Sphere Z

1 −1

305

Pnm (µ) Plm (µ) dµ = 0

7. The Legendre polynomials Pn (µ) can be obtained by expanding the function (1 − 2µz + z 2 )−1/2 in the power series of z for |z| < 1 [21]: (1 − 2µz + z 2 )−1/2 = P0 (µ) + P1 (µ)z + P2 (µ)z 2 + ...

(23)

The generating formula (23) is often used in various proofs [22]. Example 2. Let us show that (20) holds for l = n and m = 0, that is, Z

1

[Pn (µ)]2 dµ =

−1

2 2n + 1

(24)

Indeed, if we square both parts of (23) and then integrate the result between µ = −1 and µ = 1 we obtain Z

1

2 −1

(1 − 2µz + z ) −1

∞ X ∞ Z X dµ = { n=0 j=0

1

Pn (µ)Pj (µ)dµ}z n+j

−1

Integrating the left side of this equality and using the orthogonality of the Legendre polynomials we get ∞ ∞ Z 1 X X 1 1+z 2 2n ln( )= ( )z = { [Pn (µ)]2 dµ}z 2n z 1−z 2n + 1 −1 n=0

n=0

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the comparison of the corresponding coefficients of the two last series proves (24). 8. Let n ∈ Z0 . The inequality |Pn (µ)| ≤ 1 is satisfied for every Legendre polynomial. This estimate immediately follows from the integral presentation of Legendre polynomials: 1 Pn (µ) = 2π where i =

√

Z 0

2π



n p µ + i 1 − µ2 sin λ dλ

(25)

−1 [16].

9. The formula

= (−1)m

(−n)m (n + 1)m m!

Pnm (µ) = "     # ∞ 1 − µ m/2 X (−n)k (n + 1)k 1 − µ k 1+µ (m + 1)k k! 2

(26)

k=0

holds for every n ∈ R and m ∈ Z, |m| ≤ n [23]. Here (a)0 = 1 and (a)k = a (a + 1) . . . (a + k − 1) for any a ∈ R and k ∈ N.

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Yuri N. Skiba

10. Let n and m be complex numbers. The following recurrence formulae are valid for the associated Legendre functions [23]: p m 1 − µ2 Pnm+1 (µ) = (n + m) Pn−1 (µ) − (n − m) µPnm (µ)

(28)

m m (n − m + 1) Pn+1 (µ) = (2n + 1) µPnm (µ) − (n + m) Pn−1 (µ)

(29)

m m (n + m) (n + m − 1) Pn−1 (µ) − (n − m) (n − m + 1) Pn+1 (µ) p = (2n + 1) 1 − µ2 Pnm+1 (µ)

(30)

p p 1 − µ2 Pnm+1 (µ) = 2mµPnm (µ) − (n + m) (n − m + 1) 1 − µ2 Pnm−1 (µ)
d m m P (µ) = −nµPnm (µ) + (n + m) Pn−1 (µ) dµ n p
d m Pn (µ) = mµPnm (µ) − (n + m) (n − m + 1) 1 − µ2 Pnm−1 (µ) 1 − µ2 dµ p
d m 1 − µ2 Pnm+1 (µ) = mµPnm (µ) + 1 − µ2 P (µ) dµ n 1 − µ2

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d [Pn+1 (µ) − Pn−1 (µ)] = (2n + 1) Pn (µ) dµ

(31)

(32) (33) (34) (35)

It is easily seen that (34) follows from (28) and (32), (31) follows from (33) and (34), while (35) is valid due to (30) and (34). In particular, for integer m and n when m ≥ 0, functions Pnm can be calculated with the recurrent formula (29) using
m/2 1 − µ2 m m = (2m)!; Pm+1 (µ) = µ (2m + 1) Pm (µ) 2m m! For m < 0, the relation (18) should be used. 11. In practice, the normalized Legendre functions (6) with integer m and n can be calculated with the following recurrence formula m Pm (µ)

m m Dmn Qm n (µ) = µQn−1 (µ) − Dm,n−1 Qn−2 (µ)

1 − µ2


d m m Q (µ) = (2n + 1) Dmn Qm n−1 − nµQn dµ n

(36) (37)

[ 24], where Dmn =



n2 − m2 4n2 − 1

1/2

.

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Application of Hilbert Spaces to the Stability Study of Flows on a Sphere

307

In conclusion, we give without proof the frequently used addition theorem for spherical harmonics. Theorem 1. ([17]). Let ω be an angle between two unit radius-vectors x~1 , x~2 corresponding to points x1 , x2 ∈ S. Then Pn (x~1 · x~2 ) =

n X 4π Y m (x1 ) Ynm (x2 ) 2n + 1 m= −n n

(39)

where x~1 · x~2 = cos ω is the scalar product of vectors x~1 and x~2 . In particular, if x1 = x2 = x then Pn (0) = 1 and n X

|Ynm (x)|2

m= −n

=

n X

2 [Qm n (µ)] =

m= −n

2n + 1 4π

(40)

Due to (40), the values of spherical harmonics and associated Legendre functions are limited:

|Ynm (λ, µ)| = |Qm n (µ)| ≤

1 2



2n + 1 4π

1/2

,

|Pnm (µ)| ≤



(n + m)! (n − m)!

1/2

(41)

Evidently, |Pn (µ)| ≤ 1. One more useful addition theorem is obtained if we set f = Ynm (λ, µ) and g = Ynm (λ, µ) in the identity

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∆(f g) = f ∆g + g∆f + 2∇f · ∇g and then sum the result over m from −n to n then (40) leads to n X

|∇Ynm (x)|2 = χn

m= −n



2n + 1 4π



(42)

with χn given by (12).

3. Geographical Coordinate Maps Let us introduce an inner metric on a sphere. The unit sphere S is transformed to a metric space if we define the distance between two arbitrary points x, y ∈ S by ρ (x, y) = arccos (~x · ~y ) = ω.

(43)

where we use the notations of Theorem 1. In metrics (43), the sphere S is a convex set ([18], Theorem 18.4.2). Note that there exists the only shortest way between two points x, y ∈ S if y 6= −x. In the case y = −x the shortest way is not unique because any semicircle connecting the points x and −x represent the shortest way between these points. Any open (in metrics (43)) set Ω on S having a local coordinate system we will call the coordinate domain Ω ⊂ S .

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Definition 1. A couple (Ω, f ) consisting of a coordinate domain Ω ⊂ S and injection f : Ω 7→ R2 , f (x) = (x1 , x2 ), is called the map on sphere S. Moreover, the mapping f : x 7→ f (x) is almost always a homeomorphism. We will denote the inverse mapping by τ : (x1 , x2 ) ∈ R2 7−→ x ∈ S ⊂ R3 . The geographical coordinate map is one of the most often used maps on a sphere. Let N and P be the northern and southern poles of sphere S in a coordinate system (λ, µ). Geographical coordinates are valid on the whole sphere except for a closed set (large semicircle of sphere) Γ = {x ∈ S : |λ(x)| = π, µ(x) ∈ [−1, 1]} called the Date line. The mapping f is not single-valued on Γ. Indeed, the longitude λ can have two values +π and −π on Γ if |µ| < 1. Besides, both poles (N and P ) have no certain value of λ from interval [−π, π]. Definition 2 [18]. Let S be an open set S/Γ. We say that the couple (Ω, f ) is a geographical coordinate map, if f : x 7−→ (λ(x), µ(x)) is isomorphism between spherical points as elements of R3 and points (λ(x), µ(x)) of R2 . The image f (Ω) is an open set (a rectangular) Π = (−π, π) × (−1, 1) ⊂ R2 . The inverse mapping is defined by  p p 1 − µ2 cos λ, 1 − µ2 sin λ, µ ∈ R3 x = τ (λ, µ) =

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A plane element dλdµ in R2 corresponds to a surface element dS in R3 . The Jacobian of transformation τ is equal to unity [18]. Since the geographic coordinates are defined almost everywhere on S, the integral over S can be written as Z

ψ (x) dS (x) = S

Z

1

−1

Z

π

ψ (τ (λ, µ)) dλdµ

(44)

−π

Let us consider now two geographic coordinate maps (Ω1 , f1 ) and (Ω2 , f2 ) corresponding to different northern poles N1 and N2 of sphere S, respectively. If we take pole N2 so that its coordinates in the geographic system (λ1 , µ1 ) are (π/2, 0) then the Date line Γ2 of system (λ2 , µ2 ) is a part of the equator of system (λ1 , µ1 ) (Fig.4), that is, the lines Γ1 and Γ2 do not intersect, and the union Ω1 ∩ Ω2 of open sets Ω1 and Ω2 is an open cover of sphere S. Since sphere S is a differentiable compact manifold, the partition of unity theorem is valid for S: Theorem 2 [15]. Let {Ωi } be an open cover of the sphere S by a finite number of coordinate domains Ωi . There exists a partition of unity {ϕi (x)} subordinate to the cover {Ωi }, that is, a finite number of infinitely differentiable functions ϕi (x) ∈ C∞ (S) such that 1. supp ϕi (x) ⊂ Ωi for each i, 2. P 0 ≤ ϕi (x) ≤ 1 for each i and any x ∈ S, 3. i ϕi (x) = 1 for any x ∈ S. Here supp f (x) means the support of f (x), i.e., the closure of all points x ∈ S such that f (x) 6= 0.

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Figure 3. Geographic coordinate map.

Figure 4. Open cover of the sphere by two coordinate domains. Due to Theorem 2, any continuous function f (x) can be represented on S as f (x) =

X

fi (x)

(45)

i

where fi (x) = f (x) · ϕi (x) and ϕi (x) is the function from the partition of unity. Note that the integral Z XZ fi (x) dS (x) (46) f (x) dS (x) = S

i

Ωi

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does not depend on the choice of a concrete cover {Ωi } of sphere S [16]. We now give useful formulas connecting the latitudes and longitudes of any point of sphere in two different systems of geographical coordinates (λ, µ) and (λ1 , µ1 ) [25]: q p (47) µ1 = µµ0 + 1 − µ2 1 − µ20 cos (λ − λ0 ), q q µ = µ1 µ0 − 1 − µ21 1 − µ20 cos λ1 , (48) q p 1 − µ21 sin λ1 = 1 − µ2 sin (λ − λ0 ). (49) Here (λ0 , µ0 ) are the coordinates of pole N1 of coordinate system (λ1 , µ1 ) in the coordinate system (λ, µ). Equation (47) determines µ1 in terms of λ, µ, λ0 and µ0 , while equation (48) realizes the inverse transformation. As soon as µ1 is calculated from (47), λ1 can be found from (48) and (49), since they determine both cos λ1 and sin λ1 .Conversely, once µ is calculated from (48) λ − λ0 can be determined from (47) and (49). Note that (47)-(49) can also be interpreted as the formulae of transformation of spherical harmonics Y1m (λ, µ) (−1 ≤ m ≤ 1) in the subspace H1 under a rotation of sphere (see Remark 1 and (14)).

4. Orthogonal Projectors and Fractional Derivatives We now introduce operators of projection and fractional differentiation (fractional derivatives) of functions on the sphere [4, 12, 26-28]. b = {ψ m } = {ψ } of Fourier Definition 3. We denote by Φ the space of sequences ψ n

α

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coefficients

m ψm n = hψ, Yn i

(50)

of functions ψ ∈ C∞ (S). Due to the Parseval identity [17], the mapping C∞ (S) 7→ Φ is isometric isomorphism, conserving the norm and inner product of elements. Therefore we will identify the spaces C∞ (S) and Φ. Since (−∆)k ψ ∈ C∞ (S) for every ψ ∈ C∞ (S) and any k ∈ N, then due to the formula (−∆)k ψ =

∞ X n X

m [n (n + 1)]k ψ m n Yn

(51)

n=0 m= −n

b the Fourier coefficients ψ m n of every sequence ψ ∈ Φ tend to zero as n → ∞ faster than k the sequence 1/n for any degree k. Definition 4. We define Φ∗ as a space dual to Φ and consisting of formal Fourier series P h ∼ α hα Yα such that the sequence b h = {hα } of Fourier coefficients hα satisfies the condition X

ψ α hα < ∞

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b ∈ Φ.Note that symbol ∼ asserts nothing about the convergence of series. for all ψ Thus, Φ∗ represents a space of continuous linear functionals D E X b b h (ψ) ≡ ψ, h = ψ α hα

(53)

α

on Φ, and, due to isomorphism, can be identified with the space of such sequences b h = m m {hn } whose elements hn increase not faster than some degree n [29]. Definition 5. The closure of C∞ (S) in the norm (2) is the Hilbert space L2 (S) = {ψ ∈ Φ∗ : kψk < ∞} .

(54)

of generalized functions on S with inner product (1). The space L2 (S) is the direct orthogonal sum of subspaces Hn defined by (13) [16]: L2 (S) = ⊕∞ n=0 Hn

(55)

Definition 6. A function z(~x · ~y ) depending only on the distance ρ(x, y) between two sphere points x, y is called the zonal function. The convolution of a function ψ ∈ L2 (S) and a zonal function Z(~x · ~y ) ∈ L2 (S) is defined by Z 1 ψ (y) Z (~x · ~y ) dS (y) (56) (ψ ∗ z) (x) = 4π S

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(see (43) and [16]). Definition 7. Let n ∈ Z0 . We introduce an orthogonal projector Yn : L2 (S) 7→ Hn of L2 (S) on the subspace Hn of homogeneous spherical polynomials of degree n by Yn (ψ; x) = (2n + 1) (ψ ∗ Pn ) (x)

(57)

For the sake of brevity we will sometimes write simply Yn (ψ) instead of Yn (ψ; x) . Let us show that (57) is really the projector. First, n X

Yn (ψ; x) =

m ψm n Yn (x)

(58)

m= −n

and hence, Yn (ψ) ∈ Hn . Indeed, due to definition 7, Theorem 1 and (50) we have 2n + 1 Yn (ψ; x) = 4π

=

Z

ψ (y) Pn (~x · ~y ) dS (y) =

Ynm (x) hψ (y) , Ynm (y)i =

m= −n

n X

ψ (y) S

S n X

Z

Ynm (x) Ynm (y)dS (y) =

m= −n n X

m ψm n Yn (x)

m= −n

Second, due to (58) and Theorem 1, we get Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

(59)

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Yuri N. Skiba

Z n 2n + 1 X m (2n + 1) (Yn (ψ) ∗ Pn ) (x) = ψ Y m (y) Pn (~x · ~y ) dS (y) = 4π m= −n n S n =

n X

ψm n

m= −n

n X

Ynk (x)

Z S

k= −n

Ynm (y) Ynk (y)dS (y) = Yn (ψ; x)

and hence, Yn (Yn (ψ)) = Yn (ψ) for all functions ψ ∈ L2 (S). Obviously, any function from subspace H0 is constant: Z 1 Y0 (ψ) = ψ (y) dS (y) = Const 4π S

(60)

Definition 8. We introduce finite dimensional subspaces PN and PN of spherical 0 polynomials of degree n ≤ N (N ∈ Z0 ) as direct orthogonal sums of subspaces Hn :  N N N PN = ⊕N n=0 Hn , P0 = ⊕n=1 Hn = ψ ∈ P : Y0 (ψ) = 0 .

(61)

Thus, Y0 (ψ) = 0 for any ψ ∈ PN 0 . Definition 9. We define an orthogonal projector TN : L2 (S) 7−→ PN as TN ψ (x) =

∞ X

Yn (ψ; x) = (ψ ∗ SN ) (x)

(62)

N X

(63)

n=0

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where SN (~x · ~y ) =

(2n + 1) Pn (~x · ~y )

n=0

is the convolution kernel [30]. Thus, if ψ(x) ∈ L2 (S) then TN ψ is the triangular truncation of Fourier series of ψ using the spherical harmonics as basic functions. Note that ParsevalSteklov’s identities kψk2 =

X

|ψ α |2 =

X α(0)

ψ α hα =

kYn (ψ)k2

(64)

hYn (ψ) , Yn (h)i

(65)

n=0

α(0)

hψ, hi =

∞ X

∞ X n=0

hold for any functions ψ, h ∈ L2 (S) [17]. Due to (64), each function ψ(x) ∈ L2 (S) is represented by its own Fourier-Laplace series ψ (x) =

∞ X n=0

Yn (ψ; x) ≡

∞ X n X

m ψm n Yn (x)

n=0 m= −n

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kψ − TN ψk → 0 as N → ∞

(67)

that is,

Definition 10. Let s ∈ R+ and ψ(x) ∈ C∞ (S). A spherical operator Λs = (−∆)s/2 of fractional differentiation of real order s is defined by means of equations s/2 Yn (Λs ψ) = χs/2 Yn (ψ) , n Yn (ψ) = [n(n + 1)]

n ∈ Z0

(68)

Thus, Λs is a multiplier operator which is completely defined by infinite set of multiplicators s/2 s {χn }∞ n=0 [5,12,31]. We will consider Λ as a derivative of real order s of functions on a sphere. In particular, if s = 1 then operator Λ can be interpreted as the square root of nonnegative and symmetric Laplace operator (11). Thus, Λs ψ (x) =

∞ X

χs/2 n Yn (ψ; x) ≡

n=1

X

χs/2 α ψ α Yα (x)

(69)

α(1)

Definition 11. Let r ∈ R and ψ(x) ∈ C∞ (S). We define a multiplier operator Dr ≡ (E − ∆)r/2 by the equations Yn (Dr ψ) = dr/2 n Yn (ψ) , n ∈ Z0

(70)

or Dr ψ (x) =

∞ X

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n=0

dr/2 n Yn (ψ; x) ≡

X

dr/2 α ψ α Yα (x)

(71)

α(0)

where dα ≡ dn = 1 + χn = 1 + n (n + 1)

(72)

Remark 2. Note that unlike operator Λs defined only for positive real degree s, operator is defined for any real degree r. However, if we consider only the functions which are orthogonal to subspace H0 on the sphere S, that is, Y0 (ψ; x) = 0 (see (60)) then Λs can also be defined for any real degree s, and we do not need to introduce the operators Dr . Dr

It is well-known that the main disadvantage of local derivatives ∂ n /∂λn and ∂ n /∂µn is that they depend on the choice of a coordinate system, i.e., on a sphere rotation. Both the new derivatives Λs and Dr and the projectors Yn and TN are invariant with respect to any element of the group SO(3) of sphere rotations [32], and hence are free from this disadvantage. These operators can be considered as a generalization of the operators (−∆)n and (E − ∆)n from integer degrees n to real degrees s. Their application allows to specify the degree of smoothness of functions on the sphere. In fact, the degree of smoothness of a function is completely determined by the rate of decrease of its Fourier coefficients ψ m n as n grows [33].

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5. Hilbert Spaces Hs on a Sphere In this section we introduce a family of Hilbert spaces Hs of generalized functions (distributions) on a sphere that depends on a real parameter s, besides, a function ψ ∈ Hs for some s if its sth fractional derivative belongs to the space L2 (S) [4,12,26,28,34,35]. Definition 12. We denote by C∞ 0 (S) the space of infinitely differentiable functions which are orthogonal to any constant on a sphere: ∞ C∞ 0 (S) = {ψ ∈ C (S) : Y0 (ψ) = 0}

Remark 3. Since each function of C∞ 0 (S) has no projection onto subspace H0 , the s operator Λ may be defined on functions C∞ 0 (S) by means of (69) for every real degree s (s ∈ R). Definition 13. For any s ∈ R, we introduce in C∞ (S) an inner product h·, ·is and a norm k·ks in the following way: hψ, his = hDs ψ, Ds hi = =

∞ X

dsn hYn (ψ) , Yn (h)i ≡

n=0

dsα ψ α hα ,

(73)

α(0)

kψks = kDs ψk = hψ, ψi1/2 = s 1/2 (∞ )1/2  X  X ≡ = dsn kYn (ψ)k2 dsα |ψ α |2   n=0

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X

(74)

α(0)

In the space C∞ 0 (S), the inner product h·, ·is and norm k·ks are defined as hψ, his = hΛs ψ, Λs hi =

X

χsα ψ α hα ,

kψks =

α(1)

 X 

χsα |ψ α |2

α(1)

1/2  

(75)

Definition 14. Let s ∈ R. The Hilbert spaces obtained by closing the spaces C∞ (S) s s and C∞ 0 (S) in the norms (74) and (75) we denote as H and H0 , respectively. In particular, H0 = L2 (S). For the sake of brevity we will keep the symbols h·, ·i and k·k for the inner product and norm in H0 and H00 (see (1) and (2)). It will be shown below (Lemma 1) that the inclusions Φ = C∞ (S) ⊂ Hr ⊂ Hs ⊂ H0 ⊂ H−s ⊂ H−r ⊂ Φ∗ ,

(76)

are continuous if 0 < s < r . Also, it will be proved that the dual space (Hs )∗ coincides with the space H−s for all s ∈ R0 (Lemma 5). Let s, r ∈ R. Operator Dr : C∞ (S) 7→ C∞ (S) is symmetric: hDr ψ, his = hψ, Dr his ,

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and therefore can be extended for elements from Hs . Definition 15. A function (distribution) z ∈ Hs is called the rth derivative Dr ψ of a function ψ ∈ Hs if equation hz, his = hz, Dr his holds for all h ∈ C∞ (S) where Dr h is defined by (71). Remark 4. Evidently that for any ψ ∈ C∞ (S), the derivative Dr ψ defined by (71) satisfies Definition 15. It is easy to show that the extension of operator Dr : Hs 7→ Hs is closed. Indeed, if there exists {ψ n } ⊂ C∞ (S) such that kψ n − ψks → 0 and kDr ψ n − zks → 0 then hz, his = lim hDr ψ n , his = lim hψ n , Dr his = hψ, Dr his , n→∞

n→∞

and by Definition 15, ψ ∈ D(Dr ) and z = Dr ψ. Here D(Dr ) is the domain of operator Dr : Hs 7→ Hs . The closed symmetric operators Λr : Hs 7→ Hs (where r > 0) and Λr : Hs0 7→ Hs0 (where r ∈ R) are introduced in the similar way for s ∈ R. The following assertions establish the estimates of embedding operators for families of spaces Hs and Hs0 (see (76)).

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Lemma 1 [28]. Let s ∈ R, r ∈ R+ . Then 1. ψ ∈ Hs for all ψ ∈ Hs+r and

kψks ≤ kψks+r

(77)

kψks+r = kDr ψks

(78)

kzks ≤ 3−r/2 kzks+r

(79)

2. for all z ∈ H0s+r

Proof. First let ψ ∈ C∞ (S) and h = Ds ψ. Then it is obvious that (78) is valid. By (74) we have X

2 kψk2s+r = Ds+r ψ = kDr hk2 = drα |hα |2 α(k)

≥ min {drα } khk2 = min {(1 + χn )r } kψk2s α

n≥k

Therefore, we obtain (77) at k = 0. If ψ = z ∈ C∞ 0 (S) then k = 1 and inequality (79) holds. Consider now the common case when ψ ∈ Hs+r . There exists Cauchy sequence {ψ n } of functions ψ n ∈ C∞ (S) such that kψ n − ψks+r → 0. Since {ψ n } and {Dr ψ n } are Cauchy sequences in Hs and (77) and (78) are valid for them, we have

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316

Yuri N. Skiba kψks+r = lim kψ n ks+r = lim kDr ψ n ks = kDr ψks n→∞

We used here the fact that operator (79) are proved similarly.

n→∞

Dr

:

Hs

7−→ Hs is closed. The inequalities (77) and

Remark 5. The equality (78) is valid for any r ∈ R. Corollary 1. Let r ∈ R+ and s ∈ R. The operator D−r : Hs 7−→ Hs is bounded: ≤ kψks . The equality is realized at ψ = const , that is, the spectral norm of operator D−r : Hs 7−→ Hs equals unity and does not depend on s and r. The spectral norm of operator D−r : Hs0 7−→ Hs0 is equal to 3−r/2 , and hence is independent of s. Due to (78) and Remark 5 we have kD−r ψks

Corollary 2. Let s, r ∈ R. The mapping Dr : Hs+r 7−→ Hs is isometry and isomorphism. In particular, at r = −2s , the operator D−2s : H−s 7−→ Hs is isometric isomorphism. Lemma 2 (Poincare inequality). For all ψ ∈ H10 ,

kψk ≤ a0 k∇ψk

√ where a0 = 1/ 2.

(80)

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Proof. Let ψ ∈ C∞ 0 (S) and let Ω1 ∪ Ω2 be a cover of sphere S by coordinate domains Ωi = S \ Γ (see Definition 2). Due to Theorem 2 there exist a partition of unity {ϕi } subordinate to the cover {Ωi } (i = 1, 2). Using geographical coordinate maps (see section 3) and Theorem 2, we obtain k∇ψk2 =

Z S

=

2 Z X i=1

=

2 Z X i=1

=

S

2 Z X i=1

2 Z X i=1

Ωi


∇ψ (x) · ∇ ϕi (x) ψ (x) dS (x) =

  ∇ψ (τ (λi , µi )) · ∇ ϕi (τ (λi , µi )) ψ (τ (λi , µi )) dλi dµi =

Π1

Ωi

∇ψ · ∇ψdS =

[−∆ψ (τ (λi , µi ))] ϕi (τ (λi , µi )) ψ (τ (λi , µi )) dλi dµi =

Z X 2 [−∆ψ (x)] ϕi (x) ψ (x) dS (x) = ( ϕi ) [−∆ψ] ψdS =

 = h−∆ψ, ψi = Λ2 ψ, ψ = kΛψk2 ≡

S i=1

X

χα |ψ α |2 ≥ 2 kψk2 .

α(1)

The standard proceeding to limit for ψ ∈ H10 concludes the proof. Thus, we also proved that for all ψ ∈ H10 ,



(81) k∇ψk = (−∆)1/2 ψ = kΛψk

In fact, a more general assertion than Lemma 2 is valid:

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Application of Hilbert Spaces to the Stability Study of Flows on a Sphere √ Lemma 3. Let r, s, t, ∈ R , r < t , a = 2. Then for all ψ ∈ Hs+t 0 ,

kΛr ψks ≤ ar−t Λt ψ s

317

(82)

The proof of lemma at s = 0 is similar to that of (77) and (79). Since the operators Λr and Ds are commutative, Lemma 3 is valid for arbitrary real number s. The following proposition asserts the equivalence of norms defined with operators Λr and Ds . p Lemma 4 [26]. Let s ∈ R+ and b = 3/2. Then the following assertions are valid: a) for all ψ ∈ Hs , 1 (kψk + kΛs ψk) ≤ kψks ≤ bs (kψk + kΛs ψk) ; 2 b) for all ψ ∈ Hs0 ,

(83)

kΛs ψk ≤ kψks ≤ bs kΛs ψk ;

(84)



b−s Λ−s ψ ≤ kψk−s ≤ Λ−s ψ .

(85)

c) for all ψ ∈ H−s 0 ,

Proof. We prove the case a). The cases b) and c) are proved in the same way. Let ψ ∈ Hs . Then the right part of inequality (83) follows from the estimate kψk2s =

X

dsα |ψ α |2 = |ψ 0 |2 +

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α(0)

X

χsα 1 + χ−s α


s

|ψ α |2 ≤

α(1)

 s 3 2 kΛs ψk2 ≤ b2s (kψk + kΛs ψk)2 ≤ kψk + 2 Now we show the validity of left part of (83): kψk2s = |ψ 0 |2 +

X

(1 + χα )s |ψ α |2 ≥

α(1)

> |ψ 0 |2 +

1X 1X s |ψ α |2 + χα |ψ α |2 ≥ 2 2 α(1)

1 2

α(1)



kψk2 + kΛs ψk2 ≥

1 (kψk + kΛs ψk)2 4

The lemma is proved. It is easy to prove some inequalities for functions from spaces Hs . Indeed, due to (73)   X hψ, his = d(s+r)/2 ψ α d(s−r)/2 hα α α α(0)

and therefore generalized Schwartz inequality Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

318

Yuri N. Skiba |hψ, his | ≤ kψks+r khks−r

(86)

kψk2s ≤ kψks+r kψks−r

(87)

and the estimate

holds for any ψ ∈ Hs+r and any h ∈ Hs−r , where s ∈ R and r ∈ R0 . In particular, we have |hψ, hi| ≤ kψks khk−s

(88)

for all ψ ∈ Hs and h ∈ H−s (s ≥ 0). We now obtain two interpolating inequalities well known in the theory of periodic functions [36-38]. Let s, r, t ∈ R ; t ≤ r < s ; a, ρ > 0 and let Ψ ∈ Hs . If we take a = ρ1/(s−r) dα in the inequality 1 ≤ as−r + at−r ( dα is the multiplicator (72)) then we get drα ≤ ρdsα + ρp/(p−1) dtα Here

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p=

r−t 0 for all j ) if

P

j

αj < ∞ ([17], § 12]).

Theorem 3 [26]. Let s ∈ R , r ∈ R+ . Then bounded operators D−r : Hs 7−→ Hs and Λ−r : Hs0 7−→ Hs0 are compact operators at r > 0 , Hilbert-Schmidt operators at r > 1, and k-operators at r > 2. −r/2

−r/2

Proof. The multipliers χα and dα in canonical expansions (69) and (71) of operators Λ−r and D−r tend to zero if index α increases. Due to the theorem from [17] (see § 12]) the operators D−r and Λ−r are compact. Further, both the series ∞ X X X

−r

Λ−r Yα 2 = χα =

α(1)

α(1)

n=1

2n + 1 [n (n + 1)]r

and X X X X

−r

D−r Yα 2 =

Λ−r Yα 2 d−r = (1 + χ ) = 1 + α α

α(0)

α(0)

α(0)

α(1)

diverge if r ≤ 1P and converge at r > 1. This follows immediately from comparison of them with the series n n−r [39]. Here we used the relation 1 1 2n + 1 r = r−1 r + r [n (n + 1)] n (n + 1) n (n + 1)r−1

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Yuri N. Skiba P P −r/2 −r/2 Hence, both series α(1) χα and converge at r > 2. As a result, the α(0) dα validity of the other statements of theorem follows from [17] (see Lemma in § 12]. Remark 7. We can define the maximal (Schmidt’s) norms  1/2 X 

−r −r

D = d α ∗   α(0)

and

 1/2 X 

−r −r

Λ = χ α ∗  

(93)

α(1)

at r > 1 and the traces of matrices
X −r/2 Spur D−r = dα ,


X −r/2 Spur Λ−r = χα

α(0)

(94)

α(1)

at r > 2 of the operators D−r : Hs 7−→ Hs and Λ−r : Hs0 7−→ Hs0 , respectively. Values (93) and (94) are independent of the choice of orthonormal basis in Hs and Hs0 [40]. The following statement is the corollary of Theorem 3. Lemma 6. Let s ∈ R and r ∈ R+ . Then each set M bounded in the norm of Hs+r is compact in the norm of Hs . Proof. Let us show that any weakly converging in Hs+r sequence {ψ n } ⊂ M is the Cauchy sequence in Hs . Indeed, due to (78), we have

kψ n − ψ m ks = D−r (ψ n − ψ m ) s+r

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But, by Theorem 3, the sequence {D−r ψ n } converges in Hs+r -norm. Therefore, {ψ n } is the Cauchy sequence in Hs . The lemma is proved. Remark 8. It is sufficient to prove that the embedding Hs in Ht is compact for a fixed number t < s. Then the estimate (91) implies that the embedding Hs in Hr is compact for all r ∈ (t, s). The lemma 6 is a particular case of Rellich’s lemma [41]. Lemma 7 ([34], Assertion 4.9). Let s > 1. Then the space Hs is a normed ring, i.e., if ψ, h ∈ Hs then f = ψh ∈ Hs and kf ks ≤ C kψks khks

(95)

where C is a positive constant depending only on s. Due to (67), each function ψ(x) ∈ L2 (S) is approximated by finite sums (spherical polynomials) (62) of subspace PN . However, the rate of convergence is not determined here. We now give a simple lemma about an order of approximation of a smooth function on a sphere by finite sums (62). Lemma 8 [26]. Let s ∈ R , r ∈ R+ and ψ ∈ Hs+r . Then the best approximation in Hs -norm is given by the following estimate: kψ − TN ψks ≤ N −r kψks+r

(96)

Proof. Indeed, since χN +1 = (N +1)(N +2) > N 2 and, by (72), dN +1 = 1+χN +1 ≥ N 2 , we have for all ψ ∈ Hs+r , Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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kψ − TN ψk2s =

321

dsα |ψ α |2 ≤

α(N +1)

X

≤ d−r N +1 ·

2 −2r ds+r kψk2s+r . α |ψ α | ≤ N

α(N +1)

The theory of best approximation of functions on a sphere in Banach spaces Lp (S) (1 ≤ p ≤ ∞) is developed in [5,11,12,42-44]. Remark 9. The Bernstein-Nikol’skiy-Favard inequalities kψks+r = kDr ψks ≤ 3r/2 N r kψks ,

kΛr zks ≤ N r kzks

(97)

hold for any spherical polynomials ψ ∈ PN and z ∈ PN 0 , and all s ∈ R and r ∈ R+ [45]). The estimates (77) and (97) demonstrate the equivalence of norms kψks and kψks+r in finite-dimensional spaces PN and PN 0 . The proof of estimate (97) is trivial. Let C(S) and C0 (S) = {ψ ∈ C(S) : Y0 (ψ) = 0} be Banach spaces of continuous functions on the unit sphere S equipped with the norm kψkC(S) = max |ψ (x)|

(98)

x∈S

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Theorem 4. Let r > 1, ψ(x) ∈ Hr and z(x) ∈ Hr0 . Then ψ(x) and z(x) can be changed only on a set of measure zero so that to belong to C(S) and C0 (S) , respectively. Moreover, we have kψkC(S) ≤ Kr kψkr

(99)

kzkC0 (S) ≤ Mr kzkr

(100)

and

where

√ Kr = D−r ∗ / 4π

,

√ Mr = Λ−r ∗ / 4π

(101)

and kD−r k∗ and kΛ−r k are the Schmidt norms (93) of Hilbert-Schmidt operators D−r and Λ−r . Proof. Let ψ ∈ Hr . Then N X

|Yn (ψ; x)| ≤

n=0

≤

 N X 

α(0)

N  X

  −r/2 dr/2 |ψ | d |Y (x)| α α α α

α(0)

drα |ψ α |2

1/2  N  X 



α(0)

2 d−r α |Yα (x)|

1/2  

≤ Kr kψkr

where Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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Kr2 =

∞ X n=0

d−r n

n X

|Ynm (x)|2

(102)

m= −n

Due to Theorem 1 (see (40)), series (102) can be represented by the numerical series Kr2 =

∞ 1 X 1 X −r (2n + 1) d−r dα ≡ n 4π 4π n=0

α(0)

It follows from Theorem 3 and Remark P 7 that series (102) is uniformly convergent on S at r > 1, and therefore the series N n=0 Yn (ψ; x) converges uniformly and absolutely. Besides, the estimate (99) is valid for the function ψ(x). The estimate (100) is established similarly. The theorem is proved. Note that estimate kψkC(S) ≤ C kψkr is given for arbitrary domain in [34] without specifying the constant C. As applications of the theory of functions on the sphere we analyze in following sections the two problems: (1) the global asymptotic stability of solutions to the barotropic vorticity equation describing the motion of a two-dimensional incompressible viscous and forced fluid on a rotating sphere, and (2) the spectral approximation in the numerical stability study of nondivergent viscous flows on a rotating sphere.

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6. Global Asymptotic Stability of Barotropic Flows on a Sphere In this section, we consider the motion of two-dimensional incompressible viscous and forced fluid on a rotating sphere, described by the barotropic vorticity equation (BVE). The equation takes into account the linear drag, turbulence and forcing. Like in [28], the turbulent terms is represented by a real power of the Laplace operator so as to show its role explicitly (subsection 6.1). In subsection 6.2, we introduce a norm for perturbations of a basic solution and derive an equation describing the evolution of these perturbations. Finally, in subsection 6.3, two sufficient conditions for the global asymptotic stability of a solution to the barotropic vorticity equation are obtained. These conditions guarantee that the basic solution is the only attractor of the problem and all other solutions will exponentially tend to it with time. In a bounded domain on the plane, in the absence of linear drag, a condition for the global asymptotic stability were earlier obtained by Sundstr¨em [46] for the basic flow whose stream function had continuous derivatives up to the third order inclusive. The first condition for the global asymptotic stability obtained here generalized his result to flows on a sphere when the linear drag is also taken into account. However, in the general case, the solvability theorems proved for the vorticity equation (see, for example, [28] and [47]) do not guarantee the existence of the solution whose third or higher derivatives are continuous. For example, if the basic solution is a modon by Tribbia’s [48], Verkley’s [49] or Neven’s [50], subjected to dissipation but supported by a certain forcing, then its third derivatives are not continuous on the whole sphere, and the Sundstr¨em condition cannot be used. The second theorem proved here gives the global asymptotic stability condition in which the requirement on the smoothness of basic solution is weakened and is in full accordance

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323

with the solvability theorems. Examples are given for the basic flows having the form of a Legendre polynomial (zonal flow), Rossby-Haurwitz wave and pure dipole modon.

6.1. Barotropic Vorticity Equation In the first approximation, the large-scale dynamics of non-divergent viscous and forced barotropic atmosphere can be described on a rotating unit sphere S by the non-dimensional non-dimensional equation ∂ ∆Ψ + J (Ψ, ∆Ψ + 2µ) = −σ∆Ψ + ν(−∆)s+1 Ψ + F ∂t or ∂ ∆Ψ + J (Ψ, ∆Ψ + 2µ) = −[σ + νΛ2s ]∆Ψ + F ∂t with initial condition ∆Ψ (t, λ, µ) = ∆Ψ0 (λ, µ)

(103)

where ∆Ψ (t, x) is the vorticity, Ψ is the streamfunction, ∆Ψ + 2µ is the absolute vorticity, F (t, λ, µ) is the forcing,

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J(ψ, h) =

∂ψ ∂h ∂ψ ∂h − − = (→ n × ∇ψ) · ∇h ∂λ ∂µ ∂µ ∂λ

(104)

− is the Jacobian, → n is the unit outer normal to S ,and the terms J (Ψ, 2µ) = 2∂Ψ/∂λ and σ∆Ψ represent the super-rotation and the linear drag, respectively [28,51,52]. The turbulent term is taken here as ν(−∆)s+1 Ψ = −νΛ2(s+1) Ψ where s is a real, s ≥ 1. The s = 1 correspond to the viscosity term in the 2D Navier-Stokes equations, while s = 2 was used in [53-55]. It is obvious that is Ψ is a solution of (103) then Ψ+const is also the solution. Therefore we assume that Y0 (Ψ) = 0 and Y0 (F ) = 0 (see (60)), that is, Z Z Ψ dS = 0 and F dS = 0 (105) S

S

We will need Lemma 9. Let s > 0, and ψ is a sufficiently smooth function on a sphere. Then hJ (ψ, µ) , Λs ψi = 0

(106)

Proof. First note that operator Λs is symmetric: hh, Λs ψi = hΛs h, ψi

(107)

Indeed, by (66) and (69), the left-hand side of (107) can be written as

hh, Λs ψi =

Z

h Λs ψ dS = S

∞ X n=1

χs/2 n

n X m= −n

ψm n

Z S

 X ∞ n X m h Ynm dS = χs/2 ψm n n hn n=1

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Yuri N. Skiba

It is easy to check that the same result is also obtained for the right-hand side of (107): s

hΛ h, ψi =

Z

s

Λ h ψ dS = S

=

∞ X

χs/2 n

n=1

n X m= −n

∞ X

Z S

ψYnm dS

=

=

∞ X n=1

∞ X n=1

Moreover, the operators ∂/∂λ and



Λs

χs/2 n

hm n

Z S

m= −n

n=1

hm n

n X

χs/2 n

n X

χs/2 n

n X

ψYnm dS

hm n

m= −n

Z S



=

ψYnm dS



=

m ψm n hn

m= −n

are commutative. Indeed, due to (68),

∂ s m ∂ m m Y = Λs [imYnm ] = im [Λs Ynm ] = imχs/2 Λ Yn (108) n Yn = ∂λ n ∂λ for every basic function (spherical harmonic) Ynm (λ, µ). Therefore, the use of (104), (107), (108) and periodical conditions along the latitudinal circle leads to  Z 1 Z 2π ∂ψ s s hJ (ψ, µ) , Λ ψi = (Λ ψ)dλ dµ = ∂λ −1 0  Z 1 Z 2π ∂ψ n s/2 s/2 o = Λ (Λ ψ) dλ dµ = ∂λ −1 0    Z 1 Z 2π  s/2 ∂ψ s/2 = Λ (Λ ψ)dλ dµ = ∂λ −1 0   Z 1 Z 2π  ∂  s/2  s/2 Λ ψ (Λ ψ)dλ dµ = = ∂λ −1 0  Z Z 2π 1 1 ∂ n s/2 o2 = dλ dµ = 0 Λ ψ 2 −1 0 ∂λ

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Λs

Lemma is proved.

6.2. Dynamics of Perturbations e λ, µ) of BVE (103) with initial condition Suppose we analyze the stability of a solution ψ(t, e b b λ, µ). ψ(0, λ, µ), and let ψ(t, λ, µ) be another solution of (103) with initial condition ψ(0, Then   ∂ e + J(ψ, e ∆ψ) + 2 ∂ψ + J (ψ, ∆ψ) = −[σ + νΛ2s ]∆ψ ∆ψ + J ψ, ∆ψ ∂t ∂λ

(109)

holds for the difference b λ, µ) − ψ(t, e λ, µ) ψ(t, λ, µ) = ψ(t, of two solutions, besides,

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(110)

Application of Hilbert Spaces to the Stability Study of Flows on a Sphere b λ, µ) − ψ(0, e λ, µ) ψ(0, λ, µ) = ψ(0,

325 (111)

e at the initial moment. The function (110) represents a perturbation of the basic solution ψ. Taking the inner product (1) of equation (109) successively with ψ and ∆ψ and using (106), we obtain two equations D  E ∂ e + 2σK(t) + ν kψk2 = 0 K(t) + J ψ, ∆ψ), ψ s+1 ∂t D  E ∂ e + 2ση(t) + ν kψk2 = 0 η(t) − J ψ, ∆ψ), ∆ψ s+2 ∂t for the perturbation energy K(t) =

(112) (113)

1 1 k▽ψk2 ≡ kψk21 2 2

(114)

1 1 k∆ψk2 ≡ kψk22 2 2

(115)

(see (81)) and perturbation enstrophy η(t) = We used here the norm

kψks+1 = Λs+1 ψ

(116)

(see (74), (75) for the functions orthogonal to any constant on the sphere) and the relation Z Z J(ψ, f )hdS = − J(ψ, h)f dS

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S

S

valid for any complex differentiable functions ψ, h and f on sphere S [56]. It follows from (112) and (113) that the first Jacobian in (109) may change the perturbation enstrophy η(t) but does not affect the behavior of perturbation energy K(t). On the contrary, the second Jacobian in (109) has no effect on the perturbation enstrophy η(t) but may change the perturbation energy K(t). Both the super-rotation term and the non-linear term (the last two terms in the left part of (109)) have no influence on the behavior of K(t) and η(t). e = 0 is globally asymptotically stable due to (112) and Evidently, the zero solution ψ (113). Indeed, in this case the perturbation energy and enstrophy will be constant for a non-dissipative fluid (σ = µ = 0) and exponentially decrease otherwise. Let now the e be non-zero. Let p and q be non-negative real numbers, not equal to zero basic solution ψ simultaneously. Then the functional kψkG = [G(p, q, ψ, t)]1/2

(117)

where 1 G(p, q, ψ, t) ≡ G(t) = pK(t) + qη(t) = (p kψk21 + q kψk22 ) 2 makes a norm in the space of perturbations on the sphereS.

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(118)

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Yuri N. Skiba

Multiplying (112) and (113) by p and q, respectively, and combining the results, we obtain ∂ G(t) = −2σG(t) − J(t) − νp kψk2s+1 − νq kψk2s+2 ∂t

(119)

D E e − q∆ψ e J(t) = J(ψ, ∆ψ), pψ

(120)

where

By using the inequality (82) one can obtain − kψk2s+1 ≤ −2s kψk21 ,

− kψk2s+2 ≤ −2s kψk22

(121)

Then the estimation of the last two terms in (119) leads to ∂ G(t) ≤ −2ρG(t) − J(t) ∂t

(122)

with

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ρ = σ + 2s ν e λ, µ) belongs to Example 1. The super-rotation basic flow. Let the basic solution ψ(t, the subspace H1 defined by (13). Thus it represents a super-rotation flow about some axis of the sphere. Then, as it is shown in [57], J(t) ≡ 0, and, due to (120), the super-rotation flow is Liapunov stable if ρ = 0 (σ = ν = 0), and is the global attractor (asymptotically Liapunov stable) if ρ > 0. In particular case of a super-rotation flow around the polar axis e of standard geographical system of coordinates, ψ(µ) = Cµ where C is a constant, and the result J(t) ≡ 0 follows from (106). Example 2. The basic flow in the form of a homogeneous spherical polynomial. Let e ψ(t, λ, µ) ∈ Hn for some n ≥ 2, that is, e λ, µ) ≡ Yn (ψ) e = ψ(t,

n X m=−n

e m (t)Y m (λ, µ) ψ n n

(see (57). In particular, it may have the form (25) of the Legendre polynomial of degree n e : ψ(µ) = CPn (µ) (zonal flow). Due to (109), any initial perturbation of the subspace Hn (ψ(0, λ, µ) ∈ Hn ) will never leave Hn , and hence, J(t) ≡ 0. It follows from (122) that G(p, q, ψ, t) ≤ G(p, q, ψ, 0) exp(−2ρt)

(123)

and any initial perturbation ψ(0, λ, µ) of Hn will exponentially tend to zero with time not leaving Hn . It means that set Hn belongs to to the basin of attraction of the solution e λ, µ). ψ(t, e λ, µ) is a linear Remark 10. The same result is valid in the case when basic flow ψ(t, e λ, µ) may combination of the flows considered in the examples 1 and 2. In particular, ψ(t, be a Rossby-Haurwitz wave (see for definition [57-59]).

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6.3. Sufficient Conditions for Global Asymptotic Stability We now obtain sufficient conditions for damping out any perturbation of a non-stationary solution to the BVE (103) on the sphere S (see below theorems 5 and 6). In the particular case when s = 1 and σ = 0, theorem 5 is analogous to the assertion proved by Sundstr¨em [46] for flows in a limited domain on the plane. e λ, µ) of equation (103) is rather smooth, such First, assume that the basic solution ψ(t, that two values e λ, µ) p = sup max ∇∆ψ(t, (λ,µ)∈S

t

e q = sup max ∇ψ(t, λ, µ)

and

t

(λ,µ)∈S

(124)

are finite. Let us estimate the inner product (120) by means of functional (118) with p and q defined by (124): D E e − q∆ψ, e ψ), ∆ψ ≤ 2pq k∇ψk k∆ψk = 2pq kψk kψk ≤ 2√pqG(t) |J(t)| = J(pψ 1 2 Substitution of this inequality in equation (122) leads to e λ, µ) of equation (103) satisfies the conTheorem 5. If a limited smooth solution ψ(t,

dition

σ + 2s ν ≥

√

pq

(125)

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e λ, µ) will exponentially dewith p and q defined by (124), then any perturbation of ψ(t, crease with time in the norm (117). Note that in theorem 5, as well as in the condition by Sundstr¨em asymptotic-stability e [46], the basic solution is so smooth that ∇∆ψ(t, λ, µ) is a continuous function in the whole domain. However, as it was mentioned earlier, the existence of BVE solutions is proved only in the classes of twice continuously differentiable functions. We now show that the restriction (124) on the smoothness of basic solution can be weakened so as to meet the requirements of the solvability theorem. e λ, µ) such that Indeed, let us consider a bounded solution ψ(t, e p = sup max ∆ψ(t, λ, µ) t

and

(λ,µ)∈S

e q = sup max ψ(t, λ, µ) t

(λ,µ)∈S

(126)

are finite values. Using the ε-inequality ab ≤ a2 ε2 +

b2 4ε2

where ε > 0 and after that (116) and (81) we get |J(t)| ≤ 2pq k∇ψk k∇∆ψk = 2pq kψk1 k∆ψk1 = pq √ √ = ( pq kψk1 ) (2 pq kψk3 ) ≤ 2qε2 G(t) + 2 kψk23 ε

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(127)

328

Yuri N. Skiba Further, it follows from (118), (119) and (121) that ∂ G(t) ≤ −2ρG(t) − J(t) − νp kψk2s+1 − νq kψk2s+2 ∂t ≤ −2ρG(t) − J(t) − νq kψk2s+2 ≤ −2ρG(t) − J(t) − 2s−1 νq kψk23

(128)

Let us combine (127) with (128) and put ε2 = p/(2s−1 ν) in order to eliminate the two terms containing kψk23 . The resulting inequality leads to e λ, µ) of equation (103) is limited for all t ≥ 0 so Theorem 6 [51]. If a solution ψ(t, that there exist finite numbers p and q defined by (126), besides, ν(σ + 2s ν) ≥ 21−s pq

(129)

e λ, µ) will exponentially decrease with time in the norm (117). then any perturbation of ψ(t, Remark 11. Evidently that in the case of a steady solution are defined as e p = max ∆ψ(t, λ, µ) and q = max

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(λ,µ)∈S

e µ), p and q in (129) ψ(λ,

(λ,µ)∈S

e ψ(t, λ, µ)

(130)

e λ, µ) must have continuous According to the conditions (126), the basic solution ψ(t, derivatives on a sphere not up to the third order as in (124), but only up to the second order. Therefore (129) can be applied to a wider class of solutions to equation (103). For example, e λ, µ) is a modon by Tribbia [48], Verkley [49] or Neven [50], subjected to dissipation if ψ(t, but supported by a certain forcing, then condition (129) is applicable, whereas (125) may e not be used because the function ∇∆ψ(t, λ, µ) is not continuous on the boundary between the inner and outer regions of modon. Note that theorem 2, in contrast to theorem 1, requires a non-zero coefficient of turbulent viscosity ν. Example 3. Stability condition for Rossby-Haurwitz waves. Let σ = 0 and s = 1 in e µ) be a homogeneous spherical polynoequation (103), and let the stationary solution ψ(λ, mial of degree n of Hn (n > 2): e µ) = ψ(λ,

n X m=−n

e m Y m (λ, µ) ψ n n

(131)

e 0 is the only non-zero In particular, it can be a zonal flow (Legendre polynomial) if ψ n coefficient in (131). Such type of flow is supported by a steady forcing whose Fourier coefficients are equal to em Fnm = (−νχ2n + i2m)ψ n √ where i = −1 and χn is defined by (12). Then, due to example 2, the subspace Hn is the e µ) for any ν. Let us now apply theorem 2 for obtaining basin of attraction of solution ψ(λ, a condition under which solution (131) is the global attractor of equation (103). In our case, (126) leads to p = χn q, and hence (129) is reduced to

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p

χn /2

329

(132)

In particular, the last inequality is satisfied if ν ≥ qn. Thus, the larger q and degree n of the basic flow (131), the larger must be the coefficient ν to provide the asymptotic stability of flow (131) to all the perturbations. Example 4. Stability condition for a pure dipole modon. Let us now specify condition (129) for the Verkley [25, 49] pure dipole stationary modon provided σ = 0 and s = 1 in equation (103). Such a modon has the form e ′ , µ′ ) = R(µ′ ) cos λ′ ψ(λ

(133)

in the local coordinate system where R(µ′ )pi Pα1 (µ′ ) + ω i P11 (µ′ ) in the inner region Si , and R(µ′ ) = Po Pσ1 (µ′ ) + ω o P11 (µ′ ) in the outer region So of the modon. It is easy to show that e = −χ ψ e ∆ψ α + (χα − 2)ω i µ

(134)

e = −χ ψ e ∆ψ σ + (χσ − 2)ω o µ

(135)

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in the region Si , and

in the region So , besides, (χα − 2)ω i = (χσ − 2)ω o = 2 for stationary modon. In the relations (133)-(135), µ is the sine of latitude in the geographical coordinates whose pole is on the rotational axis of the sphere, and χα and χσ are eigenvalues of the spherical harmonics used for constructing the modon in the regions Si and So (S = Si ∪ So ), besides, χα is a positive number and χσ is a real. We now can estimate p in (126) as p ≤ max{χα , |χσ |}q + 2

(136)

where q is defined by (126). As a result, the sufficient condition (129) for the global asymptotic stability of modon accept the form 1 ν 2 ≥ q [max{χα , |χσ |}q + 2] 2

(137)

Thus, the stability condition explicitly depends on the structure of modon, besides, the pure dipole modons with smaller values of χα and |χσ | are expected to be more stable. Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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7. Linear Stability of Nondivergent Viscous Flows on a Sphere In recent years, the method of normal modes has been widely used in the linear stability study of atmospheric and hydrodynamic flows. Under the assumption that initial perturbations are infinitesimal, the model equations are linearized about the basic flow, and the spectral problem for the linearized operator of model is numerically solved to determine the growth rate and the spatial and temporal structure of unstable normal modes. The numerical solution of spectral problem raises certain questions of spectral approximation. Indeed, how to estimate the accuracy to which the normal modes are calculated on the sphere? How the structure and growth rate of the most unstable normal modes depend on the approximation of basic flow and its disturbances and on the degree of Laplace operator in the diffusion term? With the aim to demonstrate the importance of the questions, it will suffice to mention that unavoidable numerical errors do not allow to analyze the algebraic growth of disturbances, because a Jordan block of a matrix may be destroyed by any infinitesimal perturbation [60]. The method of normal modes has been used in many works devoted to the numerical stability study of nondivergent flows on a sphere[53,54,56, 61-71]. Discrete spectral problems are obtained by approximating the basic flow and disturbances by spherical polynomials of finite degrees. As regards the stability of barotropic atmospheric flows, the spectral approximation problem was for the first time debated in [49, 55] and in more detail examined from computational point of view in [53, 64]. The qualitative results obtained in these works can be formulated as follows: 1. If the basic flow is a polynomial exactly described by a discrete spectral problem, then the eigenvalues and eigenvectors obtained numerically converge to those of the differential spectral problem as the truncation number of the disturbance series tends lo infinity; besides, the convergence rate is slower ideal fluids and faster for viscous fluids. 2. Computer-calculated eigenvalues and eigenvectors are rather sensitive to small variations of the basic flow regardless of what fluid is considered: ideal or viscous [53]. Algebraically, these facts are quite natural, and can easily be explained. Indeed, the last fact is due to a rather high sensitivity of the spectral problem solution to small variations in the matrix entries. As to the first fact, we note that, in a viscous fluid, the vorticity equation operator linearized about a smooth basic flow has a compact resolvent, and, hence, isolated eigenvalues of finite multiplicity [28,72]. Also note that the rate of convergence estimates known up to now have been obtained solely for isolated eigenvalues of finite multiplicity [73-75]. For an ideal fluid, on the other hand, the linearized operator may have a non-empty continuous spectrum [67,76]. Besides, approximation errors peculiar to each numerical method destroy the infinite number of the conservation laws that control the behavior of ideal fluid solutions (zonal flows, Rossby-Haurwitz waves and modons) and their perturbations [57,77-79]. Thus, it is not surprising that a slower convergence of numerical solutions of spectral problem in the case of an ideal fluid is something more than a mere coincidence. In this section, the accuracy of a reconstruction of the normal modes in the numerical linear stability study of stationary non-divergent viscous flows on a rotating sphere is analyzed [83]. Banach spaces of smooth functions on a sphere and some embedding theorems are considered in subsection 7.1. The problem is formulated in subsection 7.2. Discrete

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331

spectral problems are obtained by truncating Fourier series (of the spherical harmonics) for the basic flow and disturbances to finite spherical sums of degrees K and N , respectively. The spectral approximation theory for the closed operators [73] briefly described in subsection 7.3 (see theorem 7) is used in subsection 7.4 to estimate the rate of convergence of eigenvalues and eigenfunctions of truncated spectral problems to the corresponding eigenvalues and eigenfunctions of original differential spectral problem (theorem 8). The estimates specify the results obtained in [28,62] for a polynomial basic flow, and generalize them to any sufficiently smooth basic flow on a sphere. It is shown that the convergence takes place if the truncation numbers K and N tend to infinity keeping the ratio N/K fixed. The convergence rate increases with the smoothness of basic flow and the power s of the Laplace operator in the turbulent diffusion term of vorticity equation. At the same time, the dependence of convergence rate on the diffusion coefficient is weak and expressed only through the constant T of estimate (151).

7.1. Banach Spaces of Smooth Functions on a Sphere Let p be a real, 1 ≤ p < ∞. The closure of C∞ 0 (S) in the norm 

kψkLp = 

Z S

1/p

|ψ(x)|p dS 

(138)

is the Banach space of measurable functions on the sphere S denoted here as Lp (S). For any real r, the space

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Wpr (S) = {ψ(x) ∈ Lp (S) | Λr ψ ∈ Lp (S) }

(139)

consists only of such functions ψ(x) of Lp (S) that have a finite norm kΛr ψkLp . In particular, Wp0 (S) = Lp (S) and W2r (S) = Hr . In order to solve the problem of embedding of a space Wpr (S) into another space Wps (S) it is convenient to introduce on the plane R2 a set   1 − − G = x ∈ R2 | x = ( , r) + t1 → e 1 + t2 → e2 (140) p − − being the conic span of the vectors → e = (− 1 , −1) and → e = (1, 0) for all t ≥ 0 and 1

2

2

t2 ≥ 0. The point ( p1 , r) is the node of G. We will need the next assertion.

1

Lemma 10. Let 1 ≤ p, q < ∞, and let r and s be real numbers. Then the space Wpr (S) is embedded into Wps (S) if and only if the point ( p1 , s) belongs to the set G. Lemma 10 is a particular case of the assertion proved by V.A. Ivanov for arbitrary compact globally symmetric Riemannian space of rank 1 (theorem 2 in [80]). It follows 1/2 from Lemma 10 as a corollary that the space W2 (S) is embedded into L4 (S) ≡ W40 (S), and



(141) kψkL4 ≤ C0 Λ1/2 ψ p ≡ C0 kψk1/2 L

where C0 is a constant independent of ψ [80]. This embedding can also be expressed in terms of spherical gradient ∇ by means of the following assertion

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332

Yuri N. Skiba Lemma 11 [4]. Let ψ(x) ∈ Wp1 (S). Then C1 kΛψkLp ≤ k∇ψkLp ≤ C2 kΛψkLp

(142)

where C1 and C2 are some constants independent of ψ.

7.2. Method of Normal Modes for a Nondivergent Flow on a Sphere e Let x ≡ (λ, µ) be a point of sphere S and ψ(x) be a stationary solution to vorticity equae tion (103). We now consider the linear stability of flow ψ(x) on S. By linearizing (103) e e about the basic flow ψ(x) we obtain that an infinitesimal perturbation of ψ(x) satisfies the equation ∂ζ = Lζ ∂t for perturbation vorticity ζ = ∆ψ, where e ζ) − [σ + νΛ2s ]ζ e ∆−1 ζ) − J(ψ, Lζ = J(Ω,

(143)

(144)

is the linear operator, and e e Ω(x) = ∆ψ(x) + 2µ

(145)

is the absolute vorticity of basic flow. We search the solution of (143) in the form of a normal mode

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ζ(t, x) = G(x) exp(ωt)

(146)

where ω = ω r + iω i is the eigenvalue and G(x) is the corresponding eigenfunction of the spectral problem LG(x) = ωG(x)

(147)

for the operator (144). The mode (146) is unstable if ω r > 0, decaying if ω r < 0, neutral if ω r = 0, and stationary if ω i = 0. Obviously, various errors generated by any numerical method used to solve the spectral problem (147) result in some distortions of both the eigenvalue ω and the eigenfunction G(x). Our aim is to estimate the contribution to such e distortions of the errors made on approximation of the basic flow ψ(x) and disturbances ζ(t, x). The differential spectral problem (147) is discretized by truncating Fourier series (66) e e for the disturbance ζ(t, x) and basic state functions ψ(x) and Ω(x) to finite spherical sums ζ N (t, x) = TN ζ(t, x) e (x) = TK ψ(x) e ψ K e K (x) = TK Ω(x) e Ω

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(148)

Application of Hilbert Spaces to the Stability Study of Flows on a Sphere

333

of degrees N and K, respectively [see (62)]. Thus, ζ N is a spherical polynomial in PN 0 , K e e while ψ K and ΩK belong to P0 . Then (147) can be approximated by spectral problem LM G(M, x) = ω(M )G(M, x)

(149)

e , fN ) − [σ + νΛ2s ] f e K , ∆−1 fN ) − J(ψ LM f = J(Ω K

(150)

for the operator

which maps the subspace PM 0 of spherical polynomials of degree M = K + N − 1 into M e itself. Indeed, fN ≡ TN f (x) belongs to PN 0 for any f (x) ∈ P0 , and hence, J(ψ K , fN ) M belongs to P0 . The proof of the last assertion is given in [24] by using the formula ∂ m Y (λ, µ) = imYnm (λ, µ) ∂λ n to write the Jacobian J(Ynm , Ykl ) as

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J(Ynm , Ykl )

  d m d l l m = i mPk (µ) Pn (µ) − lPn (µ) Pk (µ) exp i(m + l)λ dµ dµ

It is a spherical polynomial whose degree is less than or equal to k+n−1. Then the required e and fN in J(ψ e , fN ) and assertion is obtained by substituting the finite spherical sums ψ K K m l using the polynomial structure of the Jacobians J(Yn , Yk ). Obviously, the eigenvalues ω(M ) and eigenfunctions G(M, x) depend on the truncation numbers N and K. The problem is to estimate the convergence of the solution {ω(M ), G(M, x)} of discrete spectral problem (149) to the solution {ω, G(x)} of differential spectral problem (147) as both N and K (and hence M ) tend to infinity. e It is shown in [28] that if ν > 0 and ∇Ω(x) is bounded on S then operator (144) has a compact resolvent [40,72,82], and hence only a discrete spectrum of isolated eigenvalues of finite multiplicity. Besides, for each constant C, the circle |ω| ≤ C contains at most a finite number of the eigenvalues. In the next subsection we will need the following assertion proved in [83]. e Lemma 12 [83]. Let s ≥ 1 and Ω(x) ∈ Hr , where r ≥ 23 , M = K + N − 1, and N/K = B0 = const. Then kf ks ≤ T (k f k + kLM f k)

(151)

for any f (x) ∈ H2s , where k·k is the norm (2), T = 2s/2 T0 and T0 is a constant that e depends on Ω(x), σ, ν, s but is independent of f (x), N and K (and hence M ).

7.3. Spectral Approximation Due to (33), the eigenfunction G(x) of operator L specifies the amplitude and phase of the normal mode on the sphere S, and the corresponding eigenvalue ω characterizes its period and exponential growth or decay [55]. Then the spectral approximation problem arises as to whether the eigenvalues and eigenfunctions of the discrete spectral problem (149) will

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Yuri N. Skiba

approximate those of continuous problem (147). And how the numerical solution accuracy depends on the truncation numbers N and K? Some rate of convergence estimates have been obtained previously for the case when e only the disturbances are truncated, and the basic state ψ(x) is a spherical polynomial [28,62]. In the present work, both the basic flow and the disturbances are truncated. Note that, in the case of a viscous fluid, the operator L has a compact resolvent, and the spectral approximation theory for compact operators can, in principle, be used for the inverse of L [41,74,84]. However, we apply here the theory developed in [73] for the isolated eigenvalues of operator L itself, instead of its inverse. In doing so, we will be able to study the role of various terms of L in the rate of convergence estimates. We first briefly give the main results of [73] and then formulate the basic theorem. Let P ≡ H0 . Hereafterk ·k is the norm (2). The operator (144) is a closed linear operator L : P → P with the dense domain D(L) = H2s ⊂ P. Let ω be one of its isolated eigenvalues of finite algebraic multiplicity m and ascent p. The linear operator LM : PM → PM [see (150)] is considered as a closed operator LM : P → P with the nondense domain D(LM ) = PM . The distance between an element ζ ∈ P and the space PM is defined as δ(ζ, PM ) = inf kζ − f k f ∈PM

(152)

The domains D(L) and D(LM ) with the graph norms k ζk + kL ζk and k f k + kLM f k respectively, are Banach spaces. The distance between closed operators LM and L is defined by the gap between the corresponding closed graphs G(LM ) and G(L) in the space P × P:

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δ(LM , L) = δ(G(LM ), G(L)) = =

sup f ∈ D(LM ) k f k + kLM f k = 1

inf {kζ − f k + kLζ − LM f k}

ζ∈D(L)

(153)

[40,73]. Let us introduce the following notations: E(P) is the m-dimensional generalized eigenspace corresponding to the eigenvalue ω, F (ϑ) is an arbitrary holomorphic function defined in a neighborhood of ω, L|E(P) is the restriction of operator L to E(P), and  γ M = min δ(L|E(P) , LM ), δ(LM , L)

(154)

Denote b δ(A, B) = max {δ(A, B), δ(B, A)} . We will use the following assertion proved in [73]: Theorem 7 [73]. Let δ(LM , L) → 0 as M → ∞

(155)

δ(ζ, PM ) → 0 as M → ∞ ∀ζ ∈ D(L)

(156)

and

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335

If M is large enough: M ≥ M0 , then a sufficiently small neighborhood of ω contains exactly m eigenvalues ω 1 (M ), ..., ω m (M ) of operator LM repeated according to their multiplicity. Besides, δ(EM (PM ), E(P)) ≤ C γ M

(157)

b δ(LM |EM (PM ) , L|E(P) ) ≤ C γ M

(158)

m X 1 F (ω k (M )) ≤ C γ M F (ω) − m

(159)

k=1

max |ω − ω k (M )|p ≤ C γ M

1≤k≤m

(160)

In (157)-(160), the constants C are different, EM (PM ) is the direct sum of the generalized eigenspaces of operator LM corresponding to the eigenvalues ω 1 (M ), ..., ω m (M ), and LM |EM (PM ) : EM (PM ) → EM (PM )

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is the restriction of operator LM to EM (PM ). Note that (155) means the convergence of closed operators LM to L, and by (156), any element of D(L) is approximated by elements of PM . Estimate (157) indicates the approximation of generalized eigenvectors. In the particular case when F (ω) ≡ ω, and p > 1, the inequalities (159), (160) imply that as a rule the arithmetic mean of eigenvalues ω k (M ) gives a better approximation to ω than each individual eigenvalue ω k (M ).

7.4. Rate of Convergence Estimates In this subsection, we prove the main result on the rate of convergence of eigenvalues and eigenfunctions. Throughout this subsection, k ·k is the norm (2). e Theorem 8. Let s ≥ 1 and Ω(x) ∈ Hr , where r ≥ 23 , M = K + N − 1, and N/K = B0 = const. Then the rate of convergence estimates (157)-(160) of theorem 7 hold with e1 K −(r−3/2) + C e2 N −s γM ≤ C

(161)

e1 and C e2 depend on B0 , the constant T from Lemma 12 and the smoothness of where C basic flow r. Proof. Since D(L) = H2s , it follows from definition (152) and Lemma 8 that δ(ζ, PM ) = inf kζ − f k ≤ kζ − TM ζk ≤ M −2s kζk2s f ∈PM

(162)

for any ζ ∈ H2s , and condition (156) is fulfilled. To use the estimates (157)-(160) of theorem 7, we must show that the condition (155) also satisfied. Besides, by (154), Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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Yuri N. Skiba

γ M ≤ δ(LM , L), and hence, the rate of convergence in (157)-(160) is no less than that of δ(LM , L). We now estimate δ(LM , L). Assuming ζ = f in (153) we get δ(LM , L) ≤

kLf − LM f k sup f ∈ D(LM ) k f k + kLM f k = 1

(163)

Taking into account (144) and (150) we have I ≡ kLf − LM f k ≤ I1 + I2 + I3 + I4

(164)

where



e e

e

−1 I1 ≡ J(Ω − ΩK , ∆−1 f ) , I2 ≡ J(Ω K , ∆ (f − fN ))



e e

e

I3 ≡ J(ψ − ψ K , fN ) , I4 ≡ J(ψ, f − fN )

We will repeatedly use the inequality

kJ(ψ, h)k = k∇ψkL4 k∇hkL4

(165)

that follows from the formula |J(ψ, h)| = |∇ψ| |∇h| [see (104)], and Schwartz inequality. Due to Definition 10, Λ∆−1 f = −ΛΛ−2 f = −Λ−1 f

(166)

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for any f ∈ PM . Then using (165), Lemma 11, (166), (141), Lemma 8 and (82) we obtain

e e I1 ≤ ∇(Ω − ΩK )



e −Ω e K )

∇∆−1 f 4 ≤ C22 Λ( Ω

L 4

L4

L



e e = C22 Λ(Ω − ΩK )

L4

−1

Λ f



e e = C3 Ω − ΩK

3/2

L4



e e ≤ C02 C22 Λ(Ω − ΩK )



Λ∆−1 f

1/2

L4

−1

Λ f 1/2

e −(r−3/2) kf k kf k−1/2 ≤ C4 (Ω)K

(167)

e = 2−1/4 C3 e where C3 = C02 C22 , and C4 (Ω)

Ω

. r Let us estimate I3 . Using (165), Lemma 11, (141), the equality

e−ψ e ) = −Λ∆−1 (Ω e −Ω e K ) = −Λ−1 (Ω e −Ω eK) Λ(ψ K [valid due to (145) and (166)], Bernstein-Nikol’skiy-Favard inequality (97) and Lemma 8 we get



e e

e e I3 ≤ ∇(ψ − ψ K ) 4 k∇fN kL4 ≤ C22 Λ(ψ − ψ K ) 4 kΛfN kL4 L L



e e

e e ≤ C3 Λ(ψ − ψ K ) kΛfN k1/2 ≤ C3 Ω − ΩK N 3/2 kfN k 1/2

−1/2

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e e −(r−1) kf k ≤ C3 (168)

Ω K −(r−1) kf k ≤ C5 (Ω)K K r

e = C3 B 3/2 e where C5 (Ω) 0 Ω . In a similar manner, using (165), Lemma 11, (166), (141), r (75), (82), Lemma 8 and Lemma 12, we obtain

e I2 ≤ ∇ΩK



e

∇∆−1 (f − fN ) 4 ≤ C22

ΛΩK L 4

L



e ≤ C3 ΛΩ K

1/2

−1

Λ (f − fN ) 4 L 4

L



−1

eK

Λ (f − fN ) = C3

Ω

1/2

3/2

kf − fN k−1/2



e −(s+1/2) e −(s+1/2) (k f k + kLM f k) ≤ C3 Ω kf ks ≤ C6 (Ω)N

N r

e = C3 T e where C6 (Ω)

Ω

. r To estimate the last term I4 we will need the following inequality:  

e e e + k2µkr−1 max ∇ψ(x) ≤ C(ψ) ≡ Mr Ω x∈S

r−1

(169)

(170)

To prove (170) let us use formulas (59) and (42) to get n m

  X

e e x

ψ n |∇Ynm (x)| ≤ Yn ψ;

m= −n

(

n X

|∇Ynm (x)|2

)1/2

m= −n

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    2n + 1 1/2

e ≤ Yn ψ; x χn 4π Therefore, for any integer k, k k n m     2n + 1 1/2 X X X

e m e

Yn ψ; x χn ψ n |∇Yn (x)| ≤ 4π m= −n n=1

n=1

=

1/2 k n k n m

  o  2n + 1  X X X

e m (1+r)/2 −r e |∇Y (x)| ≤ χ Y ψ; x ψ χ

n

n n n n 4π m= −n n=1

n=1

≤

(

k X

n=1



e = M r ψ

1+r

)1/2 k 1 X (2n + 1) χ−r n 4π n=1  

e e ≤ Mr Ω + k2µkr−1 ≡ C(ψ)

  2

e x χ1+r

Yn ψ;

n



e = Mr ∆ψ

r−1

)1/2 (

r−1

(171)

where Mr is defined by (101) [see also formula (93)]. Since (171) is valid for any k and x ∈ S (the r.h.s. of (171) is independent of k and x), we have ∞ ∞ X n n X X m X m e e m e e ∇Y (x) ≤ max max ∇ψ(x) = max ψ ψ n |∇Ynm (x)| ≤ C(ψ) n n x∈S x∈S x∈S n=1 m= −n

n=1 m= −n

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The inequality (170) is proved. We now can estimate the last term I4 using (170), Lemma 8, Lemma12:   e e −s kf k I4 ≤ max ∇ψ(x) kf − fN k ≤ C(ψ)N s x∈S

e N −s T (k f k + kLM f k) ≡ C7 (ψ) e N −s (k f k + kLM f k) ≤ C(ψ)

(172)

e1 K −(r−3/2) + C e2 N −s δ(LM , L) ≤ C

(173)

e = T C(ψ). e Finally, using (163), (164),(167)-(169) and the condition k f k + where C7 (ψ) kLM f k = 1, we obtain

e Thus, condition (155) of theorem e1 = C4 (Ω) e + C5 (Ω) e and C e2 = C6 (Ω) e + C7 (ψ). where C 7 is also satisfied, and hence, theorem 8 is proved.

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Conclusion The main properties of spherical harmonics and associated Legendre functions are considered in section 2. Geographical coordinate maps for the sphere are defined in section 3. The well-known theorem about the partition of unity is also given here. This theorem is an important tool in integration theory on smooth compact manifolds. In section 4, derivatives Ds and Λs of real degree s of a smooth function on the unit two-dimensional sphere are defined, and a family of Hilbert spaces Hs of generalized functions is introduced. Orthogonal projectors on the subspace Hn of homogeneous spherical polynomials of degree n and subspace PN of spherical polynomials of degree n ≤ N are also defined. Some structural properties of the family of Hilbert spaces Hs , various embedding theorems and convergence of Fourier-Laplace series are considered in section 5. As applications of theoretical results, two problems devoted to the stability of flows on a rotating sphere are analyzed in sections 6 and 7. Theorems 5 and 6 proved in section 6 give conditions for the global asymptotic stability of solutions to the barotropic vorticity equation on a sphere. The two results differ by the smoothness of basic solutions. The spectral approximation in the numerical normal-mode stability study of nondivergent viscous flows on a rotating sphere is examined in section 7. The main result here is formulated in theorem 8 that establishes the rate of convergence of the eigenvalues and eigenfunctions of discrete spectral problems to an isolated eigenvalue of finite algebraic multiplicity m and ascent p and the corresponding eigenfunction of the original differential spectral problem. One more application of Hilbert spaces of smooth functions on a sphere can be found in [85].

Acknowledgements This research was partially supported by the grants No. 14539 (Sistema Nacional de Investigadores, CONACyT, Mexico), and by the three projects: PAPIIT-UNAM IN105608 (Mexico), 46265 A-1 (CONACYT, Mexico) and FOSEMARNAT-CONACyT 2004-01-160 (Mexico).

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References [1] Butzer P.L. and Johnen H., Lipschitz Spaces on Compact Manifolds, J. Funct. Anal. 7 (1971) 242-266. [2] Dzapharov A.S., Spherical Modulus of Continuity and Best Approximation of Functions on the Sphere by Means of Spherical Sums, Izv. Akad. Nauk Azerbaidzan. SSR, Ser. Fiz.-Tehn. Mat. Nauki 5 (1968) 3–9 (in Russian). [3] Dzapharov A.S., The Spherical Analogues of the Classical Theorems of D. Jackson and S. N. Bernstein (Russian), Dokl. Akad. Nauk SSSR 203 (1972) 278–281. [4] Gadjiev A.D., On Differentiability Properties of the Symbol of a Multidimensional Singular Integral Operator, Mat. Sb. 114 (1981) 483–510. [5] Kamzolov A.I., On the Best Approximation of Classes of Functions Wpα (S n ) by Polynomials in Spherical Harmonics, Mathematical Notes 32 (1982) 622-626. [6] Kamzolov A.I., Bernstein’s Inequality for Fractional Derivatives of Polynomials in Spherical Harmonics, Russ. Math. Surv. 39 (1984) 163-164. [7] Kamzolov A.I., Approximation of Functions on the Sphere S n , Serdica 10 (1984) 3-10.

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[8] Topuriya S.B., Fourier-Laplace Series on a Sphere. Tbilisi University, Tbilisi, 1987 (in Russian). [9] Arnold V.I., Conditions for Nonlinear Stability of Stationary Plane Curvilinear Flows of an Ideal Fluid, Sov. Math. Dokl. 6 (1965) 773-777. [10] Berens H., Butzer P.L. and Pawelke S., Limitierungsverfahren von Reihen mehrdimensionale Kugelfunktionen und deren Saturationsverhalten, Publ. RIMS, Kyoto Univ., Ser.A 4 (1968) 201-268. [11] Rustamov Kh.P., On Direct and Inverse Theorems on Best Lp -approximation on the Sphere, Dokl. Akad. Nauk SSSR 294 (1987) 788-791 (in Russian). [12] Ivanov V.A., On the Bernstein-Nikolskiy and Favard Inequalities on Compact Homogeneous Spaces of Rank 1, Usp. Mat. Nauk 38 (1983) 179-180 (in Russian). [13] Ivanov V.A. and Lizorkin P.I., Estimates of Harmonic and Spherical Polynomials in Integral Norms, Dokl. Akad. Nauk SSSR 286 (1986) 23-27 (in Russian). [14] Kamzolov A.I., On Approximation of Smooth Functions on the Sphere S n by Fourier Method, Mathematical Notes 31 (1982) 428-431. [15] Dubrovin B.A., Novikov S.P. and Fomenko A.T., Modern Geometry. Nauka, Moscow, 1986 (in Russian); Geometria Contemporanea. Metodi e applicazioni. Vol. II. (Italian). Editori Riuniti, Rome; Edizioni Mir, Moscow, 1988. Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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[16] Helgason S., Groups and Geometric Analysis. Integral Geometry, Invariant Differential Operators and Spherical Functions. Academic Press, Orlando, 1984. [17] Richtmyer R.D., Principles of Advanced Mathematical Physics. Springer-Verlag, New York, 1978 (Vol. 1); 1981 (Vol. 2). [18] Berger M., Geometry II. Mir, Moscow, 1984 (in Russian); Springer-Verlag, Berlin, 1987. [19] Mandelstam L.I., Lectures on Oscillations Theory. Nauka, Moscow, 1972 (in Russian). [20] Sobolev S.L., Equations of Mathematical Physics. Nauka, Moscow, 1966 (in Russian). [21] Nikiforov A.F. and Uvarov V.B., Special Functions of Mathematical Physics. Nauka, Moscow, 1984 (in Russian); Birkh¨auser, Boston, 1988. [22] Spiegel M.R., Theory and Problems of Fourier Analysis with Applications to Boundary Value Problems. Schaum’s Outline Series in Mathematics, McGrow-Hill Book Company, New York, 1974. [23] Abramowitz M. and Stegun I.A., Handbook of Mathematical Functions. Dover, New York, 1965.

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[24] Machenhauer B., Spectral Methods. In: Numerical Methods Used in Atmospheric Models, Vol. II, F. Mesinger and A. Arakawa (eds.), GARP Publication Series, 17, WMO, Geneva, 1977, 124-275. [25] Verkley W.T.M., The Construction of Barotropic Modons on a Sphere, J. Atmos. Sci. 41 (1984) 2492-2504. [26] Skiba Yu.N., Unique Solvability of the Barotropic Vorticity Equation for a Viscous Fluid in Classes of Generalized Functions on the Sphere. VINITI, Depart. Numer. Mathematics, Akad. Nauk SSSR, Moscow, 1988 (in Russian). [27] Skiba Yu.N., The Dynamics of Perturbations of Solutions to the Vorticity Equation for an Ideal Fluid on the Sphere. VINITI, Depart. Numer. Mathematics, Akad. Nauk SSSR, Moscow, 1989. [28] Skiba Yu.N., Mathematical Problems of the Dynamics of Viscous Barotropic Fluid on a Rotating Sphere. VINITI, Moscow, 1989 (in Russian); Indian Inst. Tropical Meteorology, Pune, India, 1990 (in English). [29] Kolmogorov A.N. and Fomin S.V., Elements of the Function Theory and Functional Analysis. Nauka, Moscow, 1976 (in Russian). [30] Kushnirenko G.G., Approximation of Continuous Functions on the Unit Sphere by Finite Spherical Sums, Trudy Khar’k. Polytechn. Inst., Ser. Eng., Phys. 25 (1959), 3-22 (in Russian). Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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[31] Calderon A.P. and Zigmund A., Singular Integral Operators and Differential Equations, Amer. J. Math. 79 (1957) 901-921. [32] Strichartz R.S., Multipliers for Spherical Harmonics Expansions, Trans. Amer. Math. Soc. 167 (1972) 115-124. [33] Mikhlin S.G., Multidimensional Singular Integrals and Integral Equations, Fizmatgiz, Moscow, 1962 (in Russian); Pergamon, Oxford, 1965. [34] Agranovich M.S., Elliptic Singular Integro-Differential Operators. Russ. Math. Surv. 20 (1965) 1-121. [35] Slobodetskii L.N., Estimates in L2 of Solutions of Linear Elliptic and Parabolic Systems, Vestnik Leningrad University, Mat. Mekh. Astronom. 7 (1960) 28-47. [36] Bers L., John F. and Schecter M., Partial Differential Equations. Lectures in Appl. Math., Vol. III, Wiley, New York, 1964. [37] Korneichuk N.P., Exact Constants in Approximation Theory. Nauka, Moscow, 1987 (in Russian); Cambridge University Press, Cambridge, 1991. [38] Samoilenko A.M., Elements of Mathematical Theory of Multi-frequency Oscillations. Nauka, Moscow, 1987 (Russian); Kluwer, Dordrecht, 1991. [39] Vorobiev N.N., The Theory of Series. Fizmatgiz, Moscow, 1986 (in Russian).

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[40] Kato T., Perturbation Theory for Linear Operators. Springer-Verlag, Berlin, Heidelberg, New York, 1980. [41] Bramble J.H. and Osborn J.E., Rate of Convergence Estimates for Nonselfadjoint Eigenvalue Approximations, Math. Comp. 27 (1973) 525-549. [42] Lizorkin P.I. and Nikolskiy S.M., Approximation of Functions on the Unit Sphere in L2 -norm, Dokl. Akad. Nauk SSSR 271 (1983) 1059-1063. [43] Nikolskiy S.M. and Lizorkin P.I., On the Approximation Theory on the Sphere. In: Studies in the Theory of Functions of Several Real Variables and the Approximation of Functions, Trudy Mat. Inst. Steklov, Akad. Nauk SSSR, 1985, 272–279 (in Russian). [44] Nikolskiy S.M. and Lizorkin P.I., Approximation of Functions on the sphere, Math. USSR-Izv. 30 (1988) 599–614. [45] Pawelke S., Uber die Approximationsordnung bei Kugelfunktionen und Algebraische Polynomen, Tohoku Mathematical J. 24 (1972) 473-486. [46] Sundstr¨em A., Stability Theorems for the Barotropic Vorticity Equations, Mon. Wea. Rev. 97 (1969) 340-345. [47] Szeptycki P., Equations of Hydrodynamics on Manifold Diffeomorfic to the Sphere, Bull. L’acad. Pol. Sci., Seria: Sci. Math., Astr., Phys. 21 (1973) 341-344. Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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[48] Tribbia J.J., Modons in Spherical Geometry, Geophys. Astrophys. Fluid Dyn. 30 (1984) 131-168. [49] Verkley W.T.M., Stationary Barotropic Modons in Westerly Background Flows, J. Atmos. Sci. 144 (1987) 2383-2398. [50] Neven E.C., Quadrupole Modons on a Sphere, Geophys. Astrophys. Fluid Dynamics 65 (1992) 105-126. [51] Skiba Yu.N., On Sufficient Conditions for Global Asymptotic Stability of the Barotropic Fluid on a Sphere, Il Nuovo Cimento C 17 (1994) 359-368. [52] Skiba Yu.N., On Dimension of Attractive Sets of Viscous Fluids on a Sphere Under Quasi-Periodic Forcing, Geophys. Astrophys. Fluid Dyn. 85 (1997) 233-242. [53] Anderson J.L., The Robustness of Barotropic Unstable Modes in a Zonally Varying Atmosphere, J. Atmos. Sci. 48 (1991) 2393-2410. [54] Branstator G.W., Analysis of General Circulation Model Sea Surface Temperature Anomaly Simulations Using a Linear Model, Part II: Eigenanalysis, J. Atmos. Sci. 42 (1985) 2225-2241. [55] Simmons A.J., Wallace J.M. and Branstator G.W., Barotropic Wave Propagation and Instability, and Atmospheric Teleconnection Patterns, J. Atmos. Sci. 40 (1983) 13631392.

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[56] Skiba Yu.N., Linear Instability of Ideal Flows on a Sphere, Mathematical Methods in Applied Science, 2008, in press. [57] Skiba Yu.N., Dynamics of Perturbations of the Rossby-Haurwitz Wave and the Verkley Modon, Atm´osfera 6 (1993) 87-125. [58] Haurwitz B., The Motion of Atmospheric Disturbances on the Spherical Earth, J. Marine Research 3 (1940) 254 -267. [59] Rossby C.-G., Relation Between Variations in the Intensity of the Zonal Circulation of the Atmosphere and the Displacements of the Semi-Permanent Centers of Action, J. Marine Res. 2 (1939) 38-55. [60] Wilkinson J.H., The Algebraic Eigenvalue Problem. Clarendon Press, Oxford, 1965. [61] Baines P.G., The Stability of Planetary Waves on a Sphere, J. Fluid Mech. 73 (1976) 193-213. [62] Dymnikov V.P. and Skiba Yu.N., Barotropic Instability of Zonally Asymmetric Atmospheric Flows. In: Computational Processes and Systems, Issue 4. Nauka, Moscow, 1986, 63-104 (in Russian). [63] Dymnikov V.P. and Skiba Yu.N., Barotropic Instability of Zonally Asymmetric Atmospheric Flows over Topography, Sov. J. Numer. Anal. Math. Modelling 2 (1987) 83–98. Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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[64] Neven E.C., Linear Stability of Modons on a Sphere, J. Atmos. Sci. 58 (2001) 22802305. [65] Haarsma R.J. and Opsteegh J.D., Barotropic Instability of Planetary-Scale Flows, J. Atmos. Sci. 45 (1988) 2789-2810. [66] Legras B. and Ghil M., Persistent Anomalies, Blocking and Variations in Atmospheric Predictability, J. Atm. Sci. 42 (1985) 433-471. [67] Skiba Yu.N., On the Spectral Problem in the Linear Stability Study of Flows on a Sphere, J. Math. Anal. Appl. 270 (2002) 165-180. [68] Skiba Yu.N., Nonlinear and Linear Instability of the Rossby-Haurwitz Wave, Journal of Mathematical Sciences 149 (2008) 1708-1725. [69] Skiba Yu.N. and Adem J., On the Linear Stability Study of Zonal Incompressible Flows on a Sphere, Numer. Meth. Part. Differ. Equations 14 (1998) 649-665. [70] Skiba Yu.N. and P´erez-Garc´ıa I., On the Structure and Growth Rate of Unstable Modes to the Rossby-Haurwitz Wave, Numer. Meth. Part. Differ. Equations 21 (2005) 368-386. [71] Skiba Yu.N. and Strelkov A.Y., On the Normal Mode Instability of Modons and WuVerkley Waves, Geophys. Astrophys. Fluid Dyn. 93 (2000) 39-54.

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[72] Ladyzhenskaya O.A., Boundary Value Problems of Mathematical Physics. Nauka, Moscow, 1973 (in Russian). [73] Descloux J., Luskin M. and Rappaz J., Approximation of the Spectrum of Closed Operators: The Determination of Normal Modes of a Rotating Basin, Math. Comput. 153 (1981) 137-154. [74] Osborn J.E., Spectral Approximation for Compact Operators, Math. Comp. 29 (1975) 712-725. [75] Mills W.H., Optimal Error Estimates for the Finite Element Spectral Approximation of Noncompact Operators, SIAM J. Numer. Analysis 16 (1979) 704-718. [76] Dikiy L.A., Hydrodynamic Stability and the Dynamics of Atmosphere. Gidrometeoizdat, Leningrad, 1976 (in Russian). [77] Skiba Yu.N., Lyapunov Instability of the Rossby-Haurwitz waves and Dipole Modons, Sov. J. Numer. Anal. Math. Modelling 6 (1991) 515-534. [78] Skiba Yu.N., Rossby-Haurwitz Wave Stability, Izv. Atmos. Ocean. Physics 28 (1992) 388–394. [79] Skiba Yu.N., Stability of Barotropic Modons on a Sphere, Izv. Atmos. Ocean. Physics 28 (1992) 765-773. Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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[80] Ivanov V.A. and Skiba Yu.N., Some Embedding Theorems on Compact Homogeneous Spaces and Their Application to the Study of the Stability of Barotropic Fluid on a Sphere. VINITI, Depart. Numer. Mathematics, Akad. Nauk SSSR, Moscow, 1990 (in Russian). [81] Di Prima R.C. and Habetler G.J., A Completeness Theorem for Non-Selfadjoint Eigenvalue Problems in Hydrodynamic Stability, Arch. Ration. Mech. Analys. 34 (1969) 218-227. [82] Keldysh M.V. and Lidskiy V.B., On Spectral Theory of Non-Selfadjoint Operators. In: Proceedings, IV All-USSR. Mathem. Congress, Vol.1, Akad. Nauk SSSR, Moscow, 1963, 101-120 (in Russian). [83] Skiba Yu.N., Spectral Approximation in the Numerical Stability Study of NonDivergent Viscous Flows on a Sphere, Numer. Meth. Part. Differ. Equations 14 (1998) 143-157. [84] Osborn J.E., Approximation of the Eigenvalues of Non-Selfadjoint Operators, SIAM J. Numer. Anal. 4 (1967) 45-54.

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[85] Skiba Yu.N., Instability of the Rossby-Haurwitz Wave in Invariant Sets of Perturbations, J. Math. Analys. Appl. 290 (2004) 686-701.

Reviewed by Dr. Denis M. Filatov, Centro de Investigacion en Computacion (CIC), Instituto Politecnico Nacional (IPN), C.P. 07738, Mexico, D.F., MEXICO Email: [email protected]

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In: Emerging Topics on Differential Geometry... Editors: L. Bernard and F. Roux, pp. 345-372

ISBN 978-1-60741-011-9 c 2010 Nova Science Publishers, Inc.

Chapter 9

A F UNCTORIAL A PPROACH TO THE I NFINITESIMAL T HEORY OF G ROUPOID Hirokazu Nishimura∗ Institute of Mathematics, University of Tsukuba Tsukuba, Ibaraki, 305-8571, Japan

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Abstract Lie algebroids are by no means natural as an infinitesimal counterpart of groupoids. In this paper we propose a functorial construction called Nishimura algebroids for an infinitesimal counterpart of groupoids. Nishimura algebroids, intended for differential geometry, are of the same vein as Lawvere’s functorial notion of algebraic theory and Ehresmann’s functorial notion of theory called sketches. We study totally intransitive Nishimura algebroids in detail. Finally we show that Nishimura algebroids naturally give rise to Lie algebroids.

1.

Introduction

Many mathematicians innocently believe that infinitesimalization is no other than linearization. We contend that infinitesimalization is more than linearization. It is true that Lie algebras are the linearization of Lie groups, but it is by no means true that Lie algebras are the infinitesimalization of Lie groups. The fortunate success of the theory of Lie algebras together with their correspondence with Lie groups unfortunately enhanced their wrong conviction and blurred what are to be really the infinitesimalization of groups and, more generally, groupoids. In this paper we propose, after the manners of Lawvere’s functorial construction of algebraic theory and Ehresmann’s functorial notion of theory called sketches, a functorial construction of Nishimura algebroids for the infinitesimalization of groupoids. After giving some preliminaries and fixing notation in the coming section, we will introduce our main notion of Nishimura algebroid in 6 steps. Then we will study totally intransitve ∗

E-mail address: [email protected]

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Hirokazu Nishimura

Nishimura algebroids, in which the main result is that the linear part of any totally intransitve Nishimura algebroid is a Lie algebra bundle. As our final investigation we will show that Nishimura algebroids naturally give rise to Lie algebroids.

2. 2.1.

Preliminaries Synthetic Differential Geometry

Our standard reference on synthetic differential geometry is Lavendhomme [5]. In synthetic differential geometry we generally work within a good topos. If the reader is willing to know how to get such a topos, he or she is referred to Kock [4] or Moerdijk and Reyes [9]. We denote by R the internal set of real numbers, which is endowed with a cornucopia of nilpotent infinitesimals pursuant to the general Kock-Lawvere axiom. The internal category Inf of infinitesimal spaces comes contravariantly from the external category of Weil algebras over the set of real numbers by taking SpecR . We should note that every infinitesimal space D has a distinguished point, namely, 0D (often written simply 0), and every morphism in Inf preserves distinguished points. An arbitrarily chosen microlinear space M shall be fixed throughout the rest of this paper.

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2.2.

Groupoids

Our standard reference on groupoids is [7]. Let D be an object in Inf . Given m ∈ M and a groupoid G over M with its object inclusion map id : M → G and its source and target projections α, β : G → M , we denote by AD m G the totality of mappings γ : D → G with γ(0D ) = idm and (α ◦ γ)(d) = m for any d ∈ D. We denote by AD G the set-theoretic D union of AD m G’s for all m ∈ M . The canonical projection π : A G → M is defined as is D D D expected. The anchor aG : A G → M is defined to be simply aD G (γ) = β ◦ γ for any γ ∈ AD G, where M D is the space of mappings of D into M . We note that if the groupoid G is the pair groupoid M × M , then AD (M × M ) can canonically be identified with M D . We write IG for the inner subgroupoid of G, for which the reader is referred to p.14 of [7].

2.3.

Simplicial Spaces

The notion of simplicial space was discussed by Nishimura [10] and [12], where simplicial spaces were called simplicial objects in the former paper, while they were called simplicial infinitesimal spaces in the latter paper. Simplicial spaces are spaces of the form Dm {S} = {(d1 , ..., dm ) ∈ Dm |di1 ...dik = 0 for any (i1 , ..., ik ) ∈ S}, where S is a finite set of sequences (i1 , ..., ik ) of natural numbers with 1 ≤ i1 < ... < ik ≤ m. By way of example, we have D(2) = D2 {(1, 2)} and D(3) = D3 {(1, 2), (1, 3), (2, 3)}.

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Given two simplicial spaces Dm {S} and Dn {T }, we define another simplicial space Dm {S} ⊕ Dn {T } to be Dm {S} ⊕ Dn {T } = {(d1 , ..., dm , e1 , ..., en ) ∈ Dm+n |di1 ...dik = 0 for any (i1 , ..., ik ) ∈ S, ej1 ...ejl = 0 for any (j1 , ..., jl ) ∈ T , di ej = 0 for any 1 ≤ i ≤ m and 1 ≤ j ≤ n} We denote by Simp the full subcategory of Inf whose objects are all simplicial spaces. Obviously the category Simp is closed under direct products. The category Simp has finite coproducts. In particular, it has the initial object 1, which is also the terminal object.

3.

Nishimura Algebroids

Let M be a microlinear space. We will introduce our main notion of Nishimura algebroid over M step by step, so that the text is divided into six subsections.

3.1.

Nishimura Algebroids1

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Definition 1 A Nishimura algebroid1 over M is simply a contravariant functor A from the category Simp of simplicial spaces to the category MLSM of microlinear spaces over M mapping finite coproducts in Simp to finite products in MLSM . Given a simplicial space D in Simp, we will usually write π : AD → M for A(D). n In particular, we will often write An in place of AD . We will simply write π for the projection to M in preference to such a more detailed notation as πA,D , which should not D cause any possible confusion. Given m ∈ M , we write AD m for {x ∈ A | π(x) = m}. ′ Given a morphism f : D → D′ in Simp, we will usually write Af : AD → AD for A(f ). Given m ∈ M , there is a unique element in A1m , which we denote by 01m . Given an object D D in Simp, we define 0D m ∈ Am to be D→1 D 0D (0m ) m =A

Example 2 By assigning the space M D of mappings from D into M to each object D in ′ Simp and assigning M f : M D → M D to each morphism f : D → D′ in Simp, we have a Nishimura algebroid1 over M to be called the standard Nishimura algebroid1 over M and to be denoted by SM or more simply by S. Example 3 Let G be a groupoid over M . By assigning AD G to each object D in Simp ′ and assigning Af G : AD G → AD G to each morphism f : D → D′ in Simp, we have a Nishimura algebroid1 over M to be denoted by AG. Each σ ∈ Sn induces a morphism σ : Dn → Dn in Simp such that σ(d1 , ..., dn ) = (dσ(1) , ..., dσ(n) ) for any (d1 , d2 ) ∈ D2 . Given x ∈ An , we will often write σ x for Aσ (x). It is easy to see that τσ x =τ (σ x)

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Hirokazu Nishimura

for any x ∈ An and any σ, τ ∈ Sn . Given x ∈ An and a ∈ R, we define a · x (1 ≤ i ≤ n) to be i

a·x=A

((d1 ,,,,,dn )∈Dn 7→(d1 ,...,di−1 ,adi ,di+1 ,...,dn )∈Dn )

i

3.2.

(x)

Nishimura Algebroids2

Definition 4 A Nishimura algebroid1 A over M is called a Nishimura algebroid2 over M if the application of A to any quasi-colimit diagram in Simp results in a limit diagram. Remark 5 The notion of Nishimura algebroid2 over M can be regarded as a partial algebrization of microlinearity. Example 6 The standard Nishimura algebroid1 SM over M is a Nishimura algebroid2 over M . This follows simply from our assumption that M is a microlinear space. Example 7 Let G be a groupoid over M . Then the Nishimura algebroid1 AG over M is a Nishimura algebroid2 over M . This follows simply from our assumption that M and G are microlinear spaces.

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Let A be a Nishimura algebroid2 over M . Let m ∈ M with x, y ∈ A1m . By using the quasi-colimit diagram (1) of small objects referred to in Proposition 6 (§2.2) of Lavendhomme [5], there exists a unique z ∈ AD⊕D with Ai1 (z) = x and Ai2 (z) = y, where ij : D → D ⊕ D is the canonical injection (j = 1, 2). We define x + y to be A△ (z), where △ : D → D ⊕ D assigns (d, d) ∈ D ⊕ D to each d ∈ D. Given a ∈ R, we define ax to be A(d∈D7→ad∈D) (x) ∈ A1m . With these operations we have Theorem 8 Given a Nishimura algebroid2 A over M , A1m is an R-module for any m ∈ M . 1 is an R-module, for which Proof. The proof is essentially a familiar proof that Sm the reader is referred, e.g., to Lavendhomme [5], §3.1, Proposition 1. What we should do is only to reformulate the familiar proof genuinely in terms of diagrams. The details can safely be left to the reader. Let A be a Nishimura algebroid2 over M with m ∈ M . Let x, y ∈ A2m with 2

A((d1 ,d2 )∈D⊕D7→(d1 ,d2 )∈D ) (x) 2

= A((d1 ,d2 )∈D⊕D7→(d1 ,d2 )∈D ) (y)

(1)

By using the quasi-colimit diagram of small objects at page 92 of Lavendhomme [5], we 2 ⊕D are sure that there exists a unique z ∈ AD with m A((d1 ,d2 )∈D and A((d1 ,d2 )∈D ·

2 7→(d ,d ,0)∈D 2 ⊕D) 1 2

(z) = x

2 7→(d ,d ,d d )∈D 2 ⊕D) 1 2 1 2

We define y − x ∈ A1m to be A(d∈D7→(0,0,d)∈D

2 ⊕D)

(z) = y

(z).

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(2) (3)

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349

Proposition 9 Let x, y ∈ A2 abide by (1). Then we have A((d1 ,d2 )∈D

2 7→(d ,d )∈D 2 ) 2 1

·

(y) − A((d1 ,d2 )∈D

2 7→(d ,d )∈D 2 ) 2 1

(x)

·

=y−x 2 ⊕D

Proof. Let z ∈ AD m A((d1 ,d2 )∈D

obedient to (2) and (3). Then we have

2 7→(d ,d ,0)∈D 2 ⊕D) 1 2

◦ A((d1 ,d2 ,d3 )∈D

=A

((d1 ,d2 )∈D2 7→(d2 ,d1 ,0)∈D2 ⊕D)

=A

((d1 ,d2 )∈D2 7→(d2 ,d1 )∈D2 )

=A

((d1 ,d2 )∈D2 7→(d2 ,d1 )∈D2 )

2 ⊕D7→(d ,d ,d )∈D 2 ⊕D) 2 1 3

(z)

(z)

◦ A((d1 ,d2 )∈D

2 7→(d ,d ,0)∈D 2 ⊕D) 1 2

(z)

(x)

while we have A((d1 ,d2 )∈D

2 7→(d ,d ,d d )∈D 2 ⊕D) 1 2 1 2

◦ A((d1 ,d2 ,d3 )∈D

=A

((d1 ,d2 )∈D2 7→(d2 ,d1 ,d1 d2 )∈D2 ⊕D)

=A

((d1 ,d2 )∈D2 7→(d2 ,d1 )∈D2 )

=A

((d1 ,d2 )∈D2 7→(d2 ,d1 )∈D2 )

◦A

2 ⊕D7→(d ,d ,d )∈D 2 ⊕D) 2 1 3

(z)

(z)

((d1 ,d2 )∈D2 7→(d1 ,d2 ,d1 d2 )∈D2 ⊕D)

(z)

(y)

Therefore we have

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A((d1 ,d2 )∈D

2 7→(d ,d )∈D 2 ) 2 1

·

(y) − A((d1 ,d2 )∈D

=A

(d∈D7→(0,0,d)∈D2 ⊕D)

=A

(d∈D7→(0,0,d)∈D2 ⊕D)

◦A

2 7→(d ,d )∈D 2 ) 2 1

(x)

((d1 ,d2 ,d3 )∈D2 ⊕D7→(d2 ,d1 ,d3 )∈D2 ⊕D)

(z)

(z)

·

=y−x This completes the proof. Proposition 10 Let x, y ∈ A2 abide by (1). Then we have ·

·

x − y = −(y − x) Proof. Let z ∈ AD

2 ⊕D

abide by the conditions (2) and (3). Let u ∈ AD

u = A((d1 ,d2 ,d3 )∈D

2 ⊕D7→(d ,d ,d d −d )∈D 2 ⊕D) 1 2 1 2 3

2 ⊕D

be

(z)

Then we have A((d1 ,d2 )∈D

2 7→(d ,d ,0)∈D 2 ⊕D) 1 2

(u)

=A

((d1 ,d2 )∈D2 7→(d1 ,d2 ,0)∈D2 ⊕D)

=A

((d1 ,d2 )∈D2 7→(d1 ,d2 ,d1 d2 )∈D2 ⊕D)

◦ A((d1 ,d2 ,d3 )∈D

2 ⊕D7→(d ,d ,d d −d )∈D 2 ⊕D) 1 2 1 2 3

(z)

=y Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

(z)

350

Hirokazu Nishimura

while we have A((d1 ,d2 )∈D

2 7→(d ,d ,d d )∈D 2 ⊕D) 1 2 1 2

(u)

=A

((d1 ,d2 )∈D2 7→(d1 ,d2 ,d1 d2 )∈D2 ⊕D)

=A

((d1 ,d2 )∈D2 7→(d1 ,d2 ,0)∈D2 ⊕D)

◦ A((d1 ,d2 ,d3 )∈D

2 ⊕D7→(d ,d ,d d −d )∈D 2 ⊕D) 1 2 1 2 3

(z)

(z)

=x Therefore we have ·

x−y = A(d∈D7→(0,0,d)∈D

2 ⊕D)

(u)

=A

(d∈D7→(0,0,d)∈D2 ⊕D)

=A

(d∈D7→(0,0,−d)∈D2 ⊕D)

◦ A((d1 ,d2 ,d3 )∈D

2 ⊕D7→(d ,d ,d d −d )∈D 2 ⊕D) 1 2 1 2 3

(z)

(z)

·

= −(y − x) This completes the proof. Proposition 11 Let x, y ∈ A2 abide by (1) with a ∈ R. Then we have ·

·

a · y − a · x = a(y − x) i

i

(i = 1, 2)

Proof. Here we deal only with the case i = 1, leaving the other case to the reader. Let 2 2 z ∈ AD ⊕D abide by the conditions (2) and (3). Let u ∈ AD ⊕D be Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

u = A((d1 ,d2 ,d3 )∈D

2 ⊕D7→(ad

1 ,d2 ,ad3 )∈D

2 ⊕D)

(z)

Then we have A((d1 ,d2 )∈D

2 7→(d ,d ,0)∈D 2 ⊕D) 1 2

(u)

=A

((d1 ,d2 )∈D2 7→(d1 ,d2 ,0)∈D2 ⊕D)

=A

((d1 ,d2 )∈D2 7→(ad1 ,d2 ,0)∈D2 ⊕D)

=A

((d1 ,d2 )∈D2 7→(ad1 ,d2 )∈D2 )

◦ A((d1 ,d2 ,d3 )∈D

◦A

2 ⊕D7→(ad

1 ,d2 ,ad3 )∈D

2 ⊕D)

(z)

(z)

((d1 ,d2 )∈D2 7→(d1 ,d2 ,0)∈D2 ⊕D)

(z)

=a·x 1

while we have A((d1 ,d2 )∈D =A

(u)

((d1 ,d2 )∈D2 7→(d1 ,d2 ,d1 d2 )∈D2 ⊕D)

= A((d1 ,d2 =A

2 7→(d ,d ,d d )∈D 2 ⊕D) 1 2 1 2

)∈D2 7→(ad

1 ,d2 ,ad1 d2

((d1 ,d2 )∈D2 7→(ad1 ,d2 )∈D2 )

◦ A((d1 ,d2 ,d3 )∈D

)∈D2 ⊕D)

◦A

2 ⊕D7→(ad

1 ,d2 ,ad3 )∈D

(z)

((d1 ,d2 )∈D2 7→(d1 ,d2 ,d1 d2 )∈D2 ⊕D)

(z)

=a·y 1

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2 ⊕D)

(z)

A Functorial Approach to the Infinitesimal Theory of Groupoids

351

Therefore we have ·

a·y−a·x i

=A

i (d∈D7→(0,0,d)∈D2 ⊕D)

(u)

=A

(d∈D7→(0,0,d)∈D2 ⊕D)

=A

(d∈D7→(0,0,ad)∈D2 ⊕D)

◦ A((d1 ,d2 ,d3 )∈D

2 ⊕D7→(ad

1 ,d2 ,ad3 )∈D

2 ⊕D)

(z)

(z)

= A(d∈D7→ad∈D) ◦ A(d∈D7→(0,0,d)∈D

2 ⊕D)

(z)

·

= a(y − x) This completes the proof. Lemma 12 The following diagram is a quasi-colimit diagram: i

D2

←

D⊕D

ϕ1

i

ր

i

ց

ϕ3

D2 ⊕ D ⊕ D

D⊕D ց

ց D2

←− ր

ր

i

D2

ϕ2

i

←

D⊕D

i

where i : D ⊕ D → D2 is the canonical injection, and ϕ1 , ϕ2 , ϕ3 : D2 → D2 ⊕ D ⊕ D are defined to be

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ϕ1 (d1 , d2 ) = (d1 , d2 , 0, 0) ϕ2 (d1 , d2 ) = (d1 , d2 , d1 d2 , 0) ϕ3 (d1 , d2 ) = (d1 , d2 , 0, d1 d2 ) Proposition 13 Let x, y, z ∈ A2 with 2

A((d1 ,d2 )∈D⊕D7→(d1 ,d2 )∈D ) (x) 2

= A((d1 ,d2 )∈D⊕D7→(d1 ,d2 )∈D ) (y) 2

= A((d1 ,d2 )∈D⊕D7→(d1 ,d2 )∈D ) (z) Then we have

·

·

·

(y − x) + (z − y) + (x − z) = 0 Proof. Let u ∈ AD

2 ⊕D⊕D

be the unique one such that

x = A((d1 ,d2 )∈D

2 7→(d ,d ,0,0)∈D 2 ⊕D⊕D) 1 2

(u)

y=A

((d1 ,d2 )∈D2 7→(d1 ,d2 ,d1 d2 ,0)∈D2 ⊕D⊕D)

(u)

z=A

((d1 ,d2 )∈D2 7→(d1 ,d2 ,0,d1 d2 )∈D2 ⊕D⊕D)

(u)

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The unique existence of such u ∈ AD have

2 ⊕D⊕D

is guaranteed by the above lemma. Since we

x = A((d1 ,d2 )∈D =A

2 7→(d ,d ,0,0)∈D 2 ⊕D⊕D) 1 2

((d1 ,d2 )∈D2 7→(d1 ,d2 ,0)∈D2 ⊕D)

◦A

(u)

((d1 ,d2 ,d3 )∈D2 ⊕D7→(d1 ,d2 ,d3 ,0)∈D2 ⊕D⊕D)

(u)

and y = A((d1 ,d2 )∈D =A

2 7→(d ,d ,d d ,0)∈D 2 ⊕D⊕D) 1 2 1 2

((d1 ,d2 )∈D2 7→(d1 ,d2 ,d1 d2 )∈D2 ⊕D)

(u)

◦ A((d1 ,d2 ,d3 )∈D

2 ⊕D7→(d ,d ,d ,0)∈D 2 ⊕D⊕D) 1 2 3

(u)

we have ·

y−x = A(d∈D7→(0,0,d)∈D =A

2 ⊕D)

◦ A((d1 ,d2 ,d3 )∈D

(d∈D7→(0,0,d,0)∈D2 ⊕D⊕D)

2 ⊕D7→(d ,d ,d ,0)∈D 2 ⊕D⊕D) 1 2 3

(u)

(u)

(4)

Since we have y = A((d1 ,d2 )∈D

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=A

2 7→(d ,d ,d d ,0)∈D 2 ⊕D⊕D) 1 2 1 2

((d1 ,d2 )∈D2 7→(d1 ,d2 ,0)∈D2 ⊕D)

◦A

(u)

((d1 ,d2 ,d3 )∈D2 ⊕D7→(d1 ,d2 ,d1 d2 −d3 ,d3 )∈D2 ⊕D⊕D)

(u)

and z = A((d1 ,d2 )∈D =A

2 7→(d ,d ,0,d d )∈D 2 ⊕D⊕D) 1 2 1 2

((d1 ,d2 )∈D2 7→(d1 ,d2 ,d1 d2 )∈D2 ⊕D)

(u)

◦ A((d1 ,d2 ,d3 )∈D

2 ⊕D7→(d ,d ,d d −d ,d )∈D 2 ⊕D⊕D) 1 2 1 2 3 3

(u)

we have ·

z−y = A(d∈D7→(0,0,d)∈D =A

2 ⊕D)

◦ A((d1 ,d2 ,d3 )∈D

(d∈D7→(0,0,−d,d)∈D2 ⊕D⊕D)

2 ⊕D7→(d ,d ,d d −d ,d )∈D 2 ⊕D⊕D) 1 2 1 2 3 3

(u)

(u)

(5)

Since we have z = A((d1 ,d2 )∈D =A

2 7→(d ,d ,0,d d )∈D 2 ⊕D⊕D) 1 2 1 2

((d1 ,d2 )∈D2 7→(d1 ,d2 ,0)∈D2 ⊕D)

(u)

◦ A((d1 ,d2 ,d3 )∈D

2 ⊕D7→(d ,d ,0,d d −d )∈D 2 ⊕D⊕D) 1 2 1 2 3

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and x = A((d1 ,d2 )∈D =A

2 7→(d ,d ,0,0)∈D 2 ⊕D⊕D) 1 2

((d1 ,d2 )∈D2 7→(d1 ,d2 ,d1 d2 )∈D2 ⊕D)

(u)

◦ A((d1 ,d2 ,d3 )∈D

2 ⊕D7→(d ,d ,0,d d −d )∈D 2 ⊕D⊕D) 1 2 1 2 3

(u)

we have ·

x−z = A(d∈D7→(0,0,d)∈D =A

2 ⊕D)

◦ A((d1 ,d2 ,d3 )∈D

(d∈D7→(0,0,0,−d)∈D2 ⊕D⊕D)

2 ⊕D7→(d ,d ,0,d d −d )∈D 2 ⊕D⊕D) 1 2 1 2 3

(u)

(u) (6)

Since we have ·

y−x = A(d∈D7→(0,0,d,0)∈D

2 ⊕D⊕D)

(u) [(4)]

= A(d∈D7→(d,0)∈D⊕D) ◦ A((d1 ,d2 )∈D⊕D7→(0,0,d1 −d2 ,d2 )∈D

2 ⊕D⊕D)

(u)

and ·

z−y = A(d∈D7→(0,0,−d,d)∈D

2 ⊕D⊕D)

(u)

[(5)]

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= A(d∈D7→(0,d)∈D⊕D) ◦ A((d1 ,d2 )∈D⊕D7→(0,0,d1 −d2 ,d2 )∈D

2 ⊕D⊕D)

(u)

we have ·

·

(y − x) + (z − y) = A(d∈D7→(d,d)∈D⊕D) ◦ A((d1 ,d2 )∈D⊕D7→(0,0,d1 −d2 ,d2 )∈D =A

(d∈D7→(0,0,0,d)∈D2 ⊕D⊕D)

2 ⊕D⊕D)

(u)

(u)

(7)

Since we have ·

·

(y − x) + (z − y) = A(d∈D7→(0,0,0,d)∈D =A

2 ⊕D⊕D)

(d∈D7→(d,0)∈D⊕D)

◦A

(u) [(7)]

((d1 ,d2 )∈D⊕D7→(0,0,0,d1 −d2 )∈D2 ⊕D⊕D)

(u)

and ·

x−z = A(d∈D7→(0,0,0,−d)∈D

2 ⊕D⊕D)

(u)

[(6)]

= A(d∈D7→(0,d)∈D⊕D) ◦ A((d1 ,d2 )∈D⊕D7→(0,0,0,d1 −d2 )∈D

2 ⊕D⊕D)

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we have ·

·

·

{(y − x) + (z − y)} + (x − z) = A(d∈D7→(d,d)∈D⊕D) ◦ A((d1 ,d2 )∈D⊕D7→(0,0,0,d1 −d2 )∈D =A

(d∈D7→(0,0,0,0)∈D2 ⊕D⊕D)

2 ⊕D⊕D)

(u)

(u)

=0 This completes the proof. Let x, y ∈ A3m with 3

A((d1 ,d2 ,d3 )∈D×(D⊕D)7→(d1 ,d2 ,d3 )∈D ) (x) 3

= A((d1 ,d2 ,d3 )∈D×(D⊕D)7→(d1 ,d2 ,d3 )∈D ) (y)

(8)

. By using the first quasi-colimit diagram of small objects in Lemma 2.1 of Nishimura [10], D4 {(2,4),(3,4)} we are sure that there exists a unique z ∈ Am with A((d1 ,d2 ,d3 )∈D and A((d1 ,d2 ,d3 )∈D ·

We define y − x ∈ A2m to be A

3 7→(d ,d ,d ,0)∈D 4 {(2,4),(3,4)}) 1 2 3

(z) = x

3 7→(d ,d ,d ,d d )∈D 4 {(2,4),(3,4)}) 1 2 3 2 3

(z) = y

((d1 ,d2 )∈D2 7→(d1 ,0,0,d2 )∈D4 {(2,4),(3,4)})

1

(z).

Let x, y ∈ A3m with A((d1 ,d2 ,d3 )∈D

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=A

3 {(1,3)}7→(d ,d ,d )∈D 3 ) 1 2 3

(x)

((d1 ,d2 ,d3 )∈D3 {(1,3)}7→(d1 ,d2 ,d3 )∈D3 )

(y)

(9)

By using the second quasi-colimit diagram of small objects in Lemma 2.1 of Nishimura D4 {(1,4),(3,4)} [10], we are sure that there exists a unique z ∈ Am with A((d1 ,d2 ,d3 )∈D and A((d1 ,d2 ,d3 )∈D

3 7→(d ,d ,d ,0)∈D 4 {(1,4),(3,4)}) 1 2 3

(z) = x

3 7→(d ,d ,d ,d d )∈D 4 {(1,4),(3,4)}) 1 2 3 1 3

·

We define y − x ∈ A2m to be A((d1 ,d2 )∈D

(z) = y

2 7→(0,d ,0,d )∈D 4 {(1,4),(3,4)}) 1 2

2

(z).

Let x, y ∈ A3m with 3

A((d1 ,d2 ,d3 )∈(D⊕D)×D7→(d1 ,d2 ,d3 )∈D ) (x) = A((d1 ,d2 ,d3 )∈(D⊕D)×D7→(d1 ,d2 ,d3

)∈D3 )

(10)

(y)

By using the third quasi-colimit diagram of small objects in Lemma 2.1 of Nishimura [10], D4 {(1,4),(2,4)} we are sure that there exists a unique z ∈ Am with A((d1 ,d2 ,d3 )∈D and A((d1 ,d2 ,d3 )∈D ·

We define y − x ∈ A2m to be A 3

3 7→(d ,d ,d ,0)∈D 4 {(1,4),(2,4)}) 1 2 3

(z) = x

3 7→(d ,d ,d ,d d )∈D 4 {(1,4),(2,4)}) 1 2 3 1 2

(z) = y

((d1 ,d2 )∈D2 7→(0,0,d1 ,d2 )∈D4 {(1,4),(2,4)})

(z).

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Proposition 14 Let x, y ∈ A3m . 1. If they satisfy (8), then we have ·

y−x 1

= A((d1 ,d2 ,d3 )∈D

3 7→(d ,d ,d )∈D 3 ) 2 1 3

= A((d1 ,d2 ,d3 )∈D

3 7→(d ,d ,d )∈D 3 ) 3 2 1

= A((d1 ,d2 ,d3 )∈D

3 7→(d ,d ,d )∈D 3 ) 1 3 2

·

(y) − A((d1 ,d2 ,d3 )∈D

3 7→(d ,d ,d )∈D 3 ) 2 1 3

(y) − A((d1 ,d2 ,d3 )∈D

3 7→(d ,d ,d )∈D 3 ) 3 2 1

(y) − A((d1 ,d2 ,d3 )∈D

3 7→(d ,d ,d )∈D 3 ) 1 3 2

2 ·

3 ·

1

(x) (x) (x)

2. If they satisfy (9), then we have ·

y−x 2

= A((d1 ,d2 ,d3 )∈D

3 7→(d ,d ,d )∈D 3 ) 2 1 3

= A((d1 ,d2 ,d3 )∈D

3 7→(d ,d ,d )∈D 3 ) 3 2 1

= A((d1 ,d2 ,d3 )∈D

3 7→(d ,d ,d )∈D 3 ) 1 3 2

·

(y) − A((d1 ,d2 ,d3 )∈D

3 7→(d ,d ,d )∈D 3 ) 2 1 3

(y) − A((d1 ,d2 ,d3 )∈D

3 7→(d ,d ,d )∈D 3 ) 3 2 1

(y) − A((d1 ,d2 ,d3 )∈D

3 7→(d ,d ,d )∈D 3 ) 1 3 2

1 ·

2 ·

3

(x) (x) (x)

3. If they satisfy (10), then we have ·

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3

= A((d1 ,d2 ,d3 )∈D

3 7→(d ,d ,d )∈D 3 ) 2 1 3

= A((d1 ,d2 ,d3 )∈D

3 7→(d ,d ,d )∈D 3 ) 3 2 1

= A((d1 ,d2 ,d3 )∈D

3 7→(d ,d ,d )∈D 3 ) 1 3 2

·

(y) − A((d1 ,d2 ,d3 )∈D

3 7→(d ,d ,d )∈D 3 ) 2 1 3

(y) − A((d1 ,d2 ,d3 )∈D

3 7→(d ,d ,d )∈D 3 ) 3 2 1

(y) − A((d1 ,d2 ,d3 )∈D

3 7→(d ,d ,d )∈D 3 ) 1 3 2

3 ·

1 ·

2

(x) (x) (x)

Proof. The proof is similar to that in Proposition 9. The details can safely be left to the reader. Now we have ·

·

·

·

1

2

3

Theorem 15 The four strong differences −, −, − and − satisfy the general Jacobi identity. I.e., given x123 , x132 , x213 , x231 , x312 , x321 ∈ A3 , as long as the following three expressions are well defined, they sum up only to vanish: ·

·

·

1 ·

·

1 ·

2 ·

·

2 ·

(x123 − x132 ) − (x231 − x321 ) (x231 − x213 ) − (x312 − x132 ) (x312 − x321 ) − (x123 − x213 ) 3

3

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Proof. The theorem was already proved in case of the standard Nishimura algebroid SM in Nishimura’s [11], §3. What we should do is only to reformulate the above proof genuinely in terms of diagrams. The details can safely be left to the reader.

3.3.

Nishimura algebroids3

Definition 16 A Nishimura algebroid2 A over M is called a Nishimura algebroid3 over M providing that it is endowed with a natural transformation a from A to the standard Nishimura algebroid2 SM to be called the anchor natural transformation. Example 17 The standard Nishimura algebroid2 SM over M is canonically a Nishimura algebroid3 over M endowed with the identity natural transformation of SM . Example 18 Let G be a groupoid over M . Then the Nishimura algebroid2 AG over M is a Nishimura algebroid3 over M endowed with the anchor natural transformation assigning D D aD G : A → M to each object D in Simp.

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3.4.

Nishimura Algebroids4

We denote by ⊗A , or more simply by ⊗, the contravariant functor which assigns D1 ⊗D2 = {(ζ, x) ∈ (AD2 )D1 ×AD1 | a(x) = π D1 (ζ)} to each object (D1 , D2 ) in Simp×Simp and ′ ′ which assigns f ⊗g = (ζ ∈ (AD2 )D1 7→ Af ◦ζ ◦Ag ∈ (AD2 )D1 , Ag ) : D1 ⊗D2 → D1′ ⊗D2′ to each morphism (f, g) : (D1′ , D2′ ) → (D1 , D2 ) in Simp×Simp, where (AD2 )D1 denotes the space of mappings from the infinitesimal space D1 to AD2 , and π D1 (ζ) assigns π(ζ(d)) e A , or more simply by ⊗ f, the contravariant functor which to each d ∈ D1 . We denote by ⊗ D ×D 1 2 e assigns D1 ⊗D2 = A to each object (D1 , D2 ) in Simp × Simp and which assigns e = Af ×g : AD1 ×D2 → AD1′ ×D2′ to each morphism (f, g) : (D1′ , D2′ ) → (D1 , D2 ) in f ⊗g Simp × Simp. Definition 19 A Nishimura algebroid3 A over M is called a Nishimura algebroid4 over M providing that it is endowed with a natural isomorphism ∗A (denoted more simply ∗ unless there is possible confusion) from the contravariant functor ⊗ to the contravariant functor e abiding by the following conditions: ⊗ 1. For any (ζ, x) ∈ D1 ⊗ D2 with (D1 , D2 ) in Simp × Simp, we have π(ζ ∗ x) = π(x) and a(ζ ∗ x) = aD1 (ζ) where aD1 (ζ) assigns a(ζ(d1 ))(d2 ) to each (d1 , d2 ) ∈ D1 × D2 . 2. Let ij : Dj → D1 × D2 be the canonical injection with pj : D1 × D2 → Dj the canonical projection (j = 1, 2). Then we have Ai1 (ζ ∗ x) = x

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and Ai2 (ζ ∗ x) = ζ(0D2 ) for any (ζ, x) ∈ D1 ⊗ D2 , while we have 2 Ap1 (y) = (d ∈ D1 7→ 0D (ay)(d) ) ∗ y

for any y ∈ AD1 and 1 Ap2 (z) = (d ∈ D1 7→ z) ∗ 0D π(z)

for any z ∈ AD2 . 3. Let f ∈ RD . For any (ζ, x) ∈ (D1 × ... × Dn ) ⊗ D, we have A((d,d1 ,...,dn )∈D×D1 ×...×Dn 7→(d,d1 ,...,di−1 ,f (d)di ,di+1 ,...,dn )∈D×D1 ×...×Dn ) (ζ ∗ x) = {d ∈ D 7→ A((d1 ,...,dn )∈D1 ×...×Dn 7→(d1 ,...,di−1 ,f (d)di ,di+1 ,...,dn )∈D1 ×...×Dn ) ζ(d) ∈ AD1 ×...×Dn } ∗ x (1 ≤ i ≤ n) 4. For any x ∈ AD1 , any ζ1 ∈ (AD2 )D1 and any ζ2 ∈ (AD3 )D1 ×D2 with a(x) = π D1 (ζ1 ) and aD1 (ζ1 ) = π D1 ×D2 (ζ2 ), we have ζ2 ∗ (ζ1 ∗ x) = (ζ2 ∗D ζ1 ) ∗ x

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where ζ2 ∗D1 ζ1 ∈ (AD1 ×D2 )D1 is defined to be (ζ2 ∗D1 ζ1 )(d) = ζ2 (d, ·) ∗ ζ1 (d) for any d ∈ D1 . Remark 20 What we require in our definition of Nishimura algebroid4 over M is that while multiplication seen in groupoids is no longer in view in Nishimura algebroids, the remnants of multiplication and its associativity are to be still in view. Multiplication seems completely lost in the traditional definition of Lie algebroid. Example 21 The standard Nishimura algebroid3 SM over M is canonically a Nishimura D1 ×D2 is defined to be algebroid4 over M provided that ζ ∗SM x ∈ SM (d1 , d2 ) ∈ D1 × D2 7→ ζ(d1 )(d2 ) ∈ M Example 22 Let G be a groupoid over M . The Nishimura algebroid3 AG over M is a Nishimura algebroid4 over M provided that ζ ∗AG x ∈ (AG)D1 ×D2 is defined to be (d1 , d2 ) ∈ D1 × D2 7→ ζ(d1 )(d2 )x(d1 ) ∈ G Now we give some results holding for any Nishimura algebroid4 A over M . Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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Proposition 23 There is a bijective correspondence between the mappings Φ : D → A1m 2 and the elements x ∈ A2m with A(d∈D7→(d,0)∈D ) (x) = 0D m. Proof. This follows simply from the first condition in the definition of Nishimura 2 algebroid4 over M , which claims that the assignment of Φ ∗ 0D m ∈ Am to each mapping Φ : D → A1m gives such a bijective correspondence. It is easy to see that Lemma 24 Let p1 : D1 × D2 → D1 be the canonical projection as in the second condition of Definition 19. Then we have D1 ×D2 1 Ap1 (0D m ) = 0m

As an easy consequence of the above proposition, we have Theorem 25 Given a Nishimura algebroid4 A over M with m ∈ M , the R-module A1m is Euclidean. Proof. We have already proved that A1m is naturally an R-module. Let ϕ : D → A1m be a mapping. We will consider another mapping Φ : D → A1m defined to be Φ(d) = ϕ(d) − ϕ(0) 2 (d∈D7→(d,0)) (x) = 0D , while for any d ∈ D. Let us consider x = Φ ∗ 0D m m ∈ Am . We have A (d∈D7 → (0,d)) D it is easy to see that A (x) = Φ(0) = 0m . Therefore there is a unique y ∈ A1m 2 2 with A((d1 ,d2 )∈D 7→d1 d2 ∈D) (y) = x. Let us consider A((d1 ,d2 )∈D 7→d2 ∈D) (y) = y ∗ 0D m ∈ 2 Am . Then it is easy to see that

(d ∈ D 7−→ dy) ∗ 0D m

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= A((d1 ,d2 )∈D =A

2 7−→(d ,d d )∈D 2 ) 1 1 2

((d1 ,d2 )∈D2 7→d1 d2 ∈D)

(A((d1 ,d2 )∈D

2 7→d ∈D) 2

(y))

(y)

=x D Therefore we have Φ ∗ 0D m = (d ∈ D 7−→ dy) ∗ 0m , which implies that

ϕ(d) − ϕ(0) = dy for any d ∈ D. To see the uniqueness of such y ∈ A1m , let us suppose that some z ∈ A1m satisfies dz = 0D m ((d1 ,d2 )∈D for any d ∈ D. Since z ∗ 0D m =A

2 7→d ∈D) 2

(z), we have

D (d ∈ D → 0D m ) ∗ 0m

= (d ∈ D 7−→ dz) ∗ 0D m = A((d1 ,d2 )∈D =A

2 7−→(d ,d d )∈D 2 ) 1 1 2

((d1 ,d2 )∈D2 7→d1 d2 ∈D)

(A((d1 ,d2 )∈D

2 7→d ∈D) 2

(z))

(z)

2

D D Since (d ∈ D → 0D m ) ∗ 0m = 0m by Lemma 24 and the second condition of Definition 19, the desired uniqueness follows from Proposition 1 (§2.2) of Lavendhomme [5]. Now we will discuss the relationship between ∗ and strong differences. Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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1. For any ζ1 , ζ2 ∈ (A2 )D and any x ∈ A1 with

Proposition 26

a(x) = π D (ζ1 ) = π D (ζ2 ) and 2

A((d1 ,d2 )∈D⊕D7→(d1 ,d2 )∈D ) (ζ1 (d)) 2

= A((d1 ,d2 )∈D⊕D7→(d1 ,d2 )∈D ) (ζ2 (d)) for any d ∈ D, we have ·

·

(ζ2 − ζ1 ) ∗ x = ζ2 ∗ x − ζ1 ∗ x 1

·

where ζ2 − ζ1 ∈ (A1 )D is defined to be ·

·

(ζ2 − ζ1 )(d) = ζ2 (d) − ζ1 (d) for any d ∈ D. 2. For any x, y ∈ A2 and any ζ ∈ (A1 )D

2 ⊕D

with

a(x) = (d1 , d2 ) ∈ D2 7→ π(ζ(d1 , d2 , 0))

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a(y) = (d1 , d2 ) ∈ D2 7→ π(x(d1 , d2 , d1 d2 )) and 2

A((d1 ,d2 )∈D⊕D7→(d1 ,d2 )∈D ) (x) 2

= A((d1 ,d2 )∈D⊕D7→(d1 ,d2 )∈D ) (y) we have A((d1 ,d2 )∈D⊕D7→(d1 ,d2 )∈D

2)

·

({ζ ◦ (d ∈ D 7→ (0, 0, d) ∈ D2 ⊕ D)} ∗ (y − x)) ·

= {ζ ◦ ((d1 , d2 ) ∈ D2 7→ (d1 , d2 , d1 d2 ) ∈ D2 ⊕ D)} ∗ y − 3

2

2

{ζ ◦ ((d1 , d2 ) ∈ D 7→ (d1 , d2 , 0) ∈ D ⊕ D)} ∗ x e Proof. It suffices to note that given an object D in Simp, the contravariant functor ⊗D e (resp. D⊗) and therefore the functor ⊗D (resp. D⊗) map every quasi-colimit diagram of small objects in Simp to a limit diagram. Therefore the proof is merely a reformulation of Proposition 2.6 of Nishimura [10]. The details can safely be left to the reader.

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3.5.

Nishimura Algebroids5

Definition 27 A Nishimura algebroid4 A over M is called a Nishimura algebroid5 over M providing that the anchor natural transformation a from A to the standard Nishimura algebroid4 SM is a homomorphism of Nishimura algebroids4 over M . In other words, a Nishimura algebroid4 A over M is a Nishimura algebroid5 over M providing that for any (ζ, x) ∈ D1 ⊗A D2 with (D1 , D2 ) in Simp × Simp, we have a(ζ ∗A x) = aD1 (ζ) ∗SM a(x) Example 28 It is trivial to see that the standard Nishimura algebroid4 SM over M is a Nishimura algebroid5 over M , since a is the identity transformation. Example 29 Let G be a groupoid over M . It is easy to see that the Nishimura algebroid4 AG over M is a Nishimura algebroid5 over M . It is also easy to see that a homomorphism ϕ : G → G′ of groupoids over M naturally gives rise to a homomorphism Aϕ : AG → AG′ of Nishimura algebroids5 over M . Thus we obtain a functor A from the category of groupoids over M to the category of Nishimura algebroids5 over M . The following proposition should be obvious. Proposition 30 Let ϕ : A → A′ be a homomorphism of Nishimura algebroids5 over M . Then its kernel at each m ∈ M , denoted by kerm ϕ, assigning (kerm ϕ)D = {x ∈ AD | ϕ(x) = 0D m } to each object D in Simp and assigning the restriction ′ ′ (kerm ϕ)f : (kerm ϕ)D → (kerm ϕ)D of Af : AD → AD to each morphism f : D → D′ in Simp is naturally a Nishimura algebroid5 over a single point.

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3.6.

Nishimura Algebroids6

Let A be a Nishimura algebroid5 over M . Since the anchor natural transformation aA : A → SM is really a homomorphism of Nishimura algebroids5 over M , its kernel kerm aA at each m ∈ M is a Nishimura algebroid5 over a single point by dint of the last proposition of the previous subsection. By collecting kerm aA over all m ∈ M , we obtain a bundle of Nishimura algebroids5 over a single point, which is called the inner subalgebroid of A and which is denoted by IA. The reader should note that the inner subalgebroid IA of A can naturally be reckoned as a Nishimura algebroid5 over M (as a subalgebroid of A in a natural sense). In the next definition we will consider the frame groupoid of Nishimura algebroids5 over a single point for IA, which is denoted by ΦN ishi5 (IA). Definition 31 A Nishimura algebroid5 A over M is called a Nishimura algebroid6 over M providing that it is endowed with a homomorphism adA (usually written simply ad) of Nishimura algebroids5 over M from A to A(ΦN ishi5 (IA)) abiding by the following condition: 1. We have ad(x)(d1 ) ◦ ad(y)(d2 ) = (ad((ad(x)(d1 ))(y)))(d2 ) ◦ ad(x)(d1 ) for any objects D1 , D2 in Simp, any d1 ∈ D1 , any d2 ∈ D2 , any x ∈ AD1 and any y ∈ (IA)D2 with π(x) = π(y).

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2. Given x, y ∈ (IA)1 with π(x) = π(y), we have (ad(x))(d)(y) − y = d[x, y] for any d ∈ D. Example 32 Since the inner subalgebroid ISM of the standard Nishimura algebroid5 SM is trivial, SM is trivially a Nishimura algebroid6 over M . Example 33 Let G be a groupoid over M . By assigning a mapping y ∈ (IG)αx 7→ xyx−1 ∈ (IG)βx to each x ∈ G, we get a homomorphism of groupoids over M from G to Φgrp (IG), which naturally gives rise to a homomorphism of groupoids over M from G to ΦN ishi5 (A(IG)). Since A(IG) and I(AG) can naturally be identified, we have a homomorphism of groupoids over M from G to ΦN ishi5 (I(AG)), to which we apply the functor A so as to get the desired adAG as a homomorphism of Nishimura algebroids5 over M from AG to A(ΦN ishi5 (I(AG))).

4.

Totally Intransitive Nishimura Algebroids

Definition 34 A Nishimura algebroid A over M is said to be totally intransitive providing that its anchor natural transformation aA is trivial, i.e.,

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aA (x) = 0D m for any m ∈ M , any object D in Simp and any x ∈ AD m. Remark 35 A totally intransitive Nishimura algebroid A over M can naturally be regarded as a bundle of Nishimura algebroids over a single point over M . In this section an arbitrarily chosen totally intransitive Nishimura algebroid A over M shall be fixed. Definition 36 Given x ∈ AD1 and y ∈ AD2 with π(x) = π(y), we define x ⊛ y ∈ AD1 ×D2 to be (d ∈ D2 7→ x) ∗ y Proposition 37 For any x ∈ AD1 , y ∈ AD2 and z ∈ AD3 with π(x) = π(y) = π(z), we have x ⊛ (y ⊛ z) = (x ⊛ y) ⊛ z Proof. This follows simply from the fourth condition in Definition 19. Remark 38 By this proposition we can omit parentheses in a combination by ⊛. Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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The following proposition is the Nishimura algebroid counterpart of Proposition 3 (§3.2) of Lavendhomme [5]. Proposition 39 Let x ∈ A1 . Then we have A((d1 ,d2 )∈D(2)7−→d1 +d2 ∈D) (x) 2

= A((d1 ,d2 )∈D(2)7−→(d1 ,d2 )∈D ) (x ⊛ x) 2

= A((d1 ,d2 )∈D(2)7−→(d2 ,d1 )∈D ) (x ⊛ x) 2

Proof. Let z = A((d1 ,d2 )∈D(2)7−→(d1 ,d2 )∈D ) (x ⊛ x). Then we have A(d∈D7−→(d,0)∈D(2)) (z) 2

= A(d∈D7−→(d,0)∈D ) (x ⊛ x) =x and A(d∈D7−→(0,d)∈D(2)) (z) 2

= A(d∈D7−→(0,d)∈D ) (x ⊛ x) = (d ∈ D 7−→ x)(0) =x

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Therefore the desired first equality follows at once from the quasi-colimit diagram in Proposition 6 (§2.2) of Lavendhomme [5]. The desired second equality can be dealt with similarly. The following proposition is the Nishimura algebroid counterpart of Proposition 6 (§3.2) of Lavendhomme [5]. Proposition 40 Let x, y ∈ A1 with π(x) = π(y). Then we have x+y 2

= A(d∈D7−→(d,d)∈D ) (y ⊛ x) 2

= A(d∈D7−→(d,d)∈D ) (x ⊛ y) 2

Proof. Let z = A(d1 ,d2 )∈D(2)7−→(d1 ,d2 )∈D ) (y ⊛ x). Then we have A(d∈D7−→(d,0)∈D(2)) (z) 2

= A(d∈D7−→(d,0)∈D ) (y ⊛ x) =x and A(d∈D7−→(0,d)∈D(2)) (z) 2

= A(d∈D7−→(0,d)∈D ) (y ⊛ x) =y Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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Therefore it follows from the quasi-colimit diagram in Proposition 6 (§ 2.2) of Lavendhomme [5] that x+y = A(d∈D7−→(d,d)∈D(2)) (z) 2

= A(d∈D7−→(d,d)∈D ) (y ⊛ x) which establishes the first desired equality. The second desired equality follows similarly.

Proposition 41 Given x, y ∈ A1 with π(x) = π(y), there exists a unique z ∈ A1 with π(x) = π(y) = π(z) such that A((d1 ,d2 )∈D =A

2 7−→d d ∈D) 1 2

(z)

((d1 ,d2 )∈D2 7−→(d1 ,d2 ,−d1 ,−d2 )∈D4 )

(y ⊛ x ⊛ y ⊛ x)

Proof. We will show that 2

2 7−→(d ,d ,−d ,−d )∈D 4 ) 1 2 1 2

2

2 7−→(d ,d ,−d ,−d )∈D 4 ) 1 2 1 2

A(d∈D7→(d,0)∈D ) ◦ A((d1 ,d2 )∈D

(y ⊛ x ⊛ y ⊛ x)

= 0D π(x) and

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A(d∈D7→(0,d)∈D ) ◦ A((d1 ,d2 )∈D

(y ⊛ x ⊛ y ⊛ x)

= 0D π(y) Then the desired result will follow from the quasi-colimit diagram in Proposition 7 (§2.2) of Lavendhomme [5]. Now we deal with the first desired identity. Since the composition of d ∈ D 7→ (d, 0) ∈ D2 and (d1 , d2 ) ∈ D2 7−→ (d1 , d2 , −d1 , −d2 ) ∈ D4 is equal to the composition of d ∈ D 7→ (d, d) ∈ D2 and (d1 , d2 ) ∈ D2 7−→ (d1 , 0, −d2 , 0) ∈ D4 , we have 2

A(d∈D7→(d,0)∈D ) ◦ A((d1 ,d2 )∈D =A

(d∈D7→(d,d)∈D2 )

=A

(d∈D7→(d,d)∈D2 )

◦A

2 7−→(d ,d ,−d ,−d )∈D 4 ) 1 2 1 2

((d1 ,d2 )∈D2 7−→(d1 ,0,−d2 ,0)∈D4 )

(A(d2 ∈D7−→(−d2

,0)∈D2 )

(y ⊛ x ⊛ y ⊛ x) (y ⊛ x ⊛ y ⊛ x) 2

(y ⊛ x) ⊛ A(d1 ∈D7−→(d1 ,0)∈D ) (y ⊛ x))

2

= A(d∈D7→(d,d)∈D ) ((−x) ⊛ x) =x−x =

[by Proposition 40]

0D π(x)

Now we turn to the second desired identity. Since the composition of d ∈ D 7→ (0, d) ∈ D2 and (d1 , d2 ) ∈ D2 7−→ (d1 , d2 , −d1 , −d2 ) ∈ D4 is equal to the composition of d ∈ D 7→ Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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(d, d) ∈ D2 and (d1 , d2 ) ∈ D2 7−→ (0, d1 , 0, −d2 ) ∈ D4 , we have 2

A(d∈D7→(0,d)∈D ) ◦ A((d1 ,d2 )∈D =A

(d∈D7→(d,d)∈D2 )

=A

(d∈D7→(d,d)∈D2 )

◦A

2 7−→(d ,d ,−d ,−d )∈D 4 ) 1 2 1 2

((d1 ,d2 )∈D2 7−→(0,d1 ,0,−d2 )∈D4 )

(A(d2 ∈D7−→(0,−d2

)∈D2 )

(y ⊛ x ⊛ y ⊛ x) (y ⊛ x ⊛ y ⊛ x) 2

(y ⊛ x) ⊛ A(d1 ∈D7−→(0,d1 )∈D ) (y ⊛ x))

2

= A(d∈D7→(d,d)∈D ) ((−y) ⊛ y) =y−y

[by Proposition 40]

0D π(y)

=

The proof is now complete. Notation 42 We will denote the above z by [x, y]. Proposition 43 Given x, y ∈ A1 with π(x) = π(y), we have [y, x] = −[x, y] Proof. Let m = π(x) = π(y). We have A((d1 ,d2 )∈D

2 7−→d d ∈D) 1 2

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= A((d1 ,d2 )∈D

2

◦ A(d∈D7→(d,d)∈D ) ([x, y] ⊛ [y, x])

2 7−→(d d ,d d )∈D 2 ) 1 2 1 2

([x, y] ⊛ [y, x])

=A

((d1 ,d2 )∈D2 7−→(d1 ,d2 ,d1 ,d2 )∈D4 )

=A

((d1 ,d2 )∈D2 7−→(d1 ,d2 ,d1 ,d2 )∈D4 )

⊛A =A A

((d1 ,d2 )∈D2 7−→(d2 ,d1 )∈D2 )

◦A

([x, y] ⊛ [y, x])

([x, y])

(A((d1 ,d2 )∈D

2 7−→d d ∈D) 1 2

(A

([x, y])⊛

([y, x]))

((d1 ,d2 )∈D2 7−→(d1 ,d2 ,−d1 ,−d2 )∈D4 )

((d1 ,d2 )∈D2 7−→(d1 ,d2 ,−d1 ,−d2 )∈D4 )

((d1 ,d2 )∈D2 7−→(d1 ,d2 ,d1 ,d2 )∈D4 )

((d1 ,d2 )∈D2 7−→(d2 ,d1 ,−d2 ,−d1 )∈D4 )

=A

((d1 ,d2 )∈D2 7−→d1 d2 ∈D)

((d1 ,d2 )∈D2 7−→d1 d2 ∈D)

((d1 ,d2 )∈D2 7−→(d1 ,d2 ,d1 ,d2 )∈D4 )

=A A

◦A

(A

4 7→(d d ,d d )∈D 2 ) 1 2 3 4

([y, x]))

((d1 ,d2 )∈D2 7−→(d1 ,d2 ,d1 ,d2 )∈D4 )

((d1 ,d2 )∈D2 7−→(d2 ,d1 )∈D2 )

=A A

((d1 ,d2 )∈D2 7−→d1 d2 ∈D)

◦ A((d1 ,d2 ,d3 ,d4 )∈D

(A

(y ⊛ x ⊛ y ⊛ x)⊛

(x ⊛ y ⊛ x ⊛ y))

((d1 ,d2 )∈D2 7−→(d1 ,d2 ,−d1 ,−d2 )∈D4 )

(y ⊛ x ⊛ y ⊛ x)⊛

(x ⊛ y ⊛ x ⊛ y))

((d1 ,d2 )∈D2 7−→(d1 ,d2 ,d1 ,d2 )∈D4 )

◦ A((d1 ,d2 ,d3 ,d4 )∈D

4 7→(d ,d ,−d ,−d ,d ,d ,−d ,−d )∈D 8 ) 2 1 2 1 3 4 3 4

(y ⊛ x ⊛ y ⊛ x ⊛ x ⊛ y ⊛ x ⊛ y) = A((d1 ,d2 )∈D =A

2 7−→(d ,d ,−d ,−d ,d ,d ,−d ,−d )∈D 8 ) 2 1 2 1 1 2 1 2

(y ⊛ x ⊛ y ⊛ x ⊛ x ⊛ y ⊛ x ⊛ y)

((d1 ,d2 )∈D2 7−→(d2 ,d1 ,−d2 ,d1 ,d2 ,−d1 ,−d2 )∈D7 )

◦ A((d1 ,d2 ,d3 ,d4 ,d5 ,d6 ,d7 )∈D

7 7→(d ,d ,d ,−d ,d ,d ,d ,d )∈D 8 ) 1 2 3 4 4 5 6 7

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(y ⊛ x ⊛ y ⊛ x ⊛ x ⊛ y ⊛ x ⊛ y) = A((d1 ,d2 )∈D

2 7−→(d ,d ,−d ,d ,d ,−d ,−d )∈D 7 ) 2 1 2 1 2 1 2 2

(y ⊛ x ⊛ y ⊛ A(d∈D7→(−d,d)∈D ) (x ⊛ x) ⊛ y ⊛ x ⊛ y) = A((d1 ,d2 )∈D =A

2 7−→(d ,d ,−d ,d ,−d ,−d )∈D 6 ) 2 1 2 2 1 2

((d1 ,d2 )∈D2 7−→(d2 ,d1 ,d2 ,−d1 ,−d2 )∈D5 )

(y ⊛ x ⊛ y ⊛ y ⊛ x ⊛ y)

◦ A((d1 ,d2 ,d3 ,d4 ,d5 )∈D

5 7−→(d ,d ,−d ,d ,d ,d )∈D 6 ) 1 2 3 3 4 5

(y ⊛ x ⊛ y ⊛ y ⊛ x ⊛ y) = A((d1 ,d2 )∈D

2 7−→(d ,d ,d ,−d ,−d )∈D 5 ) 2 1 2 1 2

= A((d1 ,d2 )∈D

2 7−→(d ,d ,−d ,−d )∈D 4 ) 2 1 1 2

=A

((d1 ,d2 )∈D2 7−→(d2 ,d1 ,−d2 )∈D3 )

=A

((d1 ,d2 )∈D2 7−→(d2 ,d1 ,−d2 )∈D3 )

=A

((d1 ,d2 )∈D2 7−→(d2 ,−d2 )∈D2 )

=A

((d1 ,d2 )∈D2 7−→d2 ∈D)

=

◦A

2

(y ⊛ x ⊛ A(d∈D7→(−d,d)∈D ) (y ⊛ y) ⊛ x ⊛ y)

(y ⊛ x ⊛ x ⊛ y)

◦ A((d1 ,d2 ,d3 )∈D (y ⊛ A

3 7−→(d ,d ,−d ,d )∈D 4 ) 1 2 2 3

(d∈D7→(d,−d)∈D2 )

(y ⊛ x ⊛ x ⊛ y)

(x ⊛ x) ⊛ y)

(y ⊛ y)

(d∈D7→(d,−d)∈D2 )

(y ⊛ y)

2 0D m

Proposition 44 Given x, y ∈ A1 with π(x) = π(y), we have 2

A((d1 ,d2 )∈D⊕D7→(d1 ,d2 )∈D ) (y ⊛ x) 2

= A((d1 ,d2 )∈D⊕D7→(d1 ,d2 )∈D ) ◦ A((d1 ,d2 )∈D Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

and

·

[x, y] = y ⊛ x − A((d1 ,d2 )∈D

2 7−→(d ,d )∈D 2 ) 2 1

2 7−→(d ,d )∈D 2 ) 2 1

(x ⊛ y)

(x ⊛ y)

(11)

(12)

Proof. Our proof is the proof of Proposition 8 (§3.4) of Lavendhomme [5] in disguise. In order to show the identity (11), it suffices, by dint of the quasi-colimit diagram in Proposition 6 (§2.2) of Lavendhomme [5], to show that 2

A(d∈D7−→(d,0)∈D⊕D) ◦ A((d1 ,d2 )∈D⊕D7→(d1 ,d2 )∈D ) (y ⊛ x) 2

= A(d∈D7−→(d,0)∈D⊕D) ◦ A((d1 ,d2 )∈D⊕D7→(d1 ,d2 )∈D ) ◦ A((d1 ,d2 )∈D

2 7−→(d ,d )∈D 2 ) 2 1

(x ⊛ y)

(13)

and 2

A(d∈D7−→(0,d)∈D⊕D) ◦ A((d1 ,d2 )∈D⊕D7→(d1 ,d2 )∈D ) (y ⊛ x) 2

= A(d∈D7−→(0,d)∈D⊕D) ◦ A((d1 ,d2 )∈D⊕D7→(d1 ,d2 )∈D ) ◦ A((d1 ,d2 )∈D

2 7−→(d ,d )∈D 2 ) 2 1

(x ⊛ y)

(14)

Since the composition of d ∈ D 7−→ (d, 0) ∈ D ⊕ D and (d1 , d2 ) ∈ D ⊕ D 7→ (d1 , d2 ) ∈ D2 is equal to d ∈ D 7−→ (d, 0) ∈ D2 , and since the composition of d ∈ D 7−→ (d, 0) ∈ Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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D ⊕ D, (d1 , d2 ) ∈ D ⊕ D 7→ (d1 , d2 ) ∈ D2 and (d1 , d2 ) ∈ D2 7−→ (d2 , d1 ) ∈ D2 is equal to d ∈ D 7−→ (0, d) ∈ D2 , it is easy to see that both sides of the identity (13) are equal to x by the second condition in Definition 19. The identity (14) can be established similarly. Let 2 3 z = A((d1 ,d2 ,d3 )∈D ⊕D7−→(d2 ,d3 ,d1 )∈D ) (x ⊛ [x, y] ⊛ y) Then we have A((d1 ,d2 )∈D

2 7−→(d ,d ,0)∈D 2 ⊕D) 1 2

(z)

=A

((d1 ,d2 )∈D2 7−→(d1 ,d2 ,0)∈D2 ⊕D)

=A

((d1 ,d2 )∈D2 7−→(d2 ,0,d1 )∈D3 )

=A

((d1 ,d2 )∈D2 7−→(d2 ,d1 )∈D2 )

=A

((d1 ,d2 )∈D2 7−→(d2 ,d1 )∈D2 )

(x ⊛ A

=A

((d1 ,d2 )∈D2 7−→(d2 ,d1 )∈D2 )

(x ⊛ y)

◦ A((d1 ,d2 ,d3 )∈D

2 ⊕D7−→(d ,d ,d )∈D 3 ) 2 3 1

(x ⊛ [x, y] ⊛ y)

(x ⊛ [x, y] ⊛ y)

◦ A((d1 ,d2 )∈D

2 7−→(d ,0,d )∈D 3 ) 1 2

(d∈D7−→(d,0)∈D2 )

(x ⊛ [x, y] ⊛ y)

([x, y] ⊛ y))

while we have A((d1 ,d2 )∈D

2 7−→(d ,d ,d d )∈D 2 ⊕D) 1 2 1 2

=A =A

((d1 ,d2 )∈D2 7−→(d2 ,d1 d2 ,d1 )∈D3 )

(x ⊛ [x, y] ⊛ y)

=A

((d1 ,d2 )∈D2 7−→(d2 ,d1 d2 ,d1 )∈D3 )

(x ⊛ A(d∈D7→−d∈D) ◦ A(d∈D7→−d∈D) ([x, y]) ⊛ y)

= A((d1 ,d2 )∈D = A((d1 ,d2 )∈D

◦ A((d1 ,d2 ,d3 )∈D

2 ⊕D7−→(d ,d ,d )∈D 3 ) 2 3 1

(x ⊛ [x, y] ⊛ y)

2 7−→(d ,d d ,d )∈D 3 ) 2 1 2 1

◦ A((d1 ,d2 ,d3 )∈D

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(z)

((d1 ,d2 )∈D2 7−→(d1 ,d2 ,d1 d2 )∈D2 ⊕D)

3 7→(d ,−d ,d )∈D 3 ) 1 2 3

2 7−→(d ,−d d ,d )∈D 3 ) 2 1 2 1

(x ⊛ A(d∈D7→−d∈D) ([x, y]) ⊛ y)

(x ⊛ [y, x] ⊛ y)

[By Proposition 43] = A((d1 ,d2 )∈D

2 7−→(d ,−d ,d ,d )∈D 4 ) 2 2 1 1

=A

((d1 ,d2 )∈D2 7−→(d2 ,−d2 ,d1 ,d1 )∈D4 )

=A

((d1 ,d2 )∈D2 7−→(d2 ,−d2 ,d1 ,d1 )∈D4 )

(x ⊛ A((d1 ,d2 )∈D =A

◦ A((d1 ,d2 ,d3 ,d4 )∈D (x ⊛ A

((d1 ,d2 )∈D2 7→d1 d2 ∈D)

2 7−→(d ,d ,−d ,−d )∈D 4 ) 1 2 1 2

((d1 ,d2 )∈D2 7−→(d2 ,−d2 ,d1 ,d1 )∈D4 )

4 7→(d ,d d ,d )∈D 3 ) 1 2 3 4

(x ⊛ [y, x] ⊛ y)

([y, x]) ⊛ y)

(x ⊛ y ⊛ x ⊛ y) ⊛ y)

◦ A((d1 ,d2 ,d3 ,d4 )∈D

4 7→(d ,d ,d ,−d ,−d ,d )∈D 6 1 2 3 2 3 4

)

(x ⊛ x ⊛ y ⊛ x ⊛ y ⊛ y) = A((d1 ,d2 )∈D =A

2 7−→(d ,−d ,d ,d ,−d ,d )∈D 6 ) 2 2 1 2 1 1

((d1 ,d2 )∈D2 7−→(d2 ,d1 ,d2 ,d1 )∈D4 )

(x ⊛ x ⊛ y ⊛ x ⊛ y ⊛ y)

◦ A((d1 ,d2 ,d3 ,d4 )∈D

4 7−→(d ,−d ,d ,d ,−d ,d )∈D 6 ) 1 1 2 3 4 4

(x ⊛ x ⊛ y ⊛ x ⊛ y ⊛ y) = A((d1 ,d2 )∈D

2 7−→(d ,d ,d ,d )∈D 4 ) 2 1 2 1

2

(A(d∈D7→(−d,d)∈D ) (x ⊛ x) ⊛ y ⊛ x⊛

2

A(d∈D7→(d,−d)∈D ) (y ⊛ y)) = A((d1 ,d2 )∈D

2 7−→(d ,d ,d ,d )∈D 4 ) 2 1 2 1

D (0D m ⊛ y ⊛ x ⊛ 0m )

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Therefore we have ·

y ⊛ x − A((d1 ,d2 )∈D

2 7−→(d ,d )∈D 2 ) 2 1

=A

(d∈D7→(0,0,d)∈D2 ⊕D)

=A

(d∈D7→(0,0,d)∈D2 ⊕D)

(x ⊛ y)

(z) ◦ A((d1 ,d2 ,d3 )∈D

2 ⊕D7−→(d ,d ,d )∈D 3 ) 2 3 1

(x ⊛ [x, y] ⊛ y)

= [x, y] This completes the proof. 1. Given x ∈ A1 and y, z ∈ A2 with π(x) = π(y) = π(z), if we have

Proposition 45

2

2

A((d1 ,d2 )∈D⊕D7→(d1 ,d2 )∈D ) (y) = A((d1 ,d2 )∈D⊕D7→(d1 ,d2 )∈D ) (z) then we have 3

A((d1 ,d2 ,d3 )∈D×(D⊕D)7→(d1 ,d2 ,d3 )∈D ) (y ⊛ x) = 3

A((d1 ,d2 ,d3 )∈D×(D⊕D)7→(d1 ,d2 ,d3 )∈D ) (z ⊛ x) and

·

·

z ⊛ x − y ⊛ x = (z − y) ⊛ x 1

2. Given x, y ∈ A2 and z ∈ A1 with π(x) = π(y) = π(z), if we have 2

2

A((d1 ,d2 )∈D⊕D7→(d1 ,d2 )∈D ) (x) = A((d1 ,d2 )∈D⊕D7→(d1 ,d2 )∈D ) (y)

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then we have 3

A((d1 ,d2 ,d3 )∈(D⊕D)×D7→(d1 ,d2 ,d3 )∈D ) (z ⊛ x) = 3

A((d1 ,d2 ,d3 )∈(D⊕D)×D7→(d1 ,d2 ,d3 )∈D ) (z ⊛ y) and A((d1 ,d2 )∈D

2 7→(d ,d )∈D 2 ) 2 1

·

·

(z ⊛ y − z ⊛ x) = z ⊛ (y − x) 3

Proof. This follows simply from Proposition 26. Proposition 46 Given x, y, z ∈ A1 with π(x) = π(y) = π(z), let it be the case that u123 = z ⊛ y ⊛ x u132 = A((d1 ,d2 ,d3 )∈D

3 7→(d ,d ,d )∈D 3 ) 1 3 2

(y ⊛ z ⊛ x)

u213 = A

((d1 ,d2 ,d3 )∈D3 7→(d2 ,d1 ,d3 )∈D3 )

(z ⊛ x ⊛ y)

u231 = A

((d1 ,d2 ,d3 )∈D3 7→(d2 ,d3 ,d1 )∈D3 )

(x ⊛ z ⊛ y)

u312 = A

((d1 ,d2 ,d3 )∈D3 7→(d3 ,d1 ,d2 )∈D3 )

(y ⊛ x ⊛ z)

u321 = A

((d1 ,d2 ,d3 )∈D3 7→(d3 ,d2 ,d1 )∈D3 )

(x ⊛ y ⊛ z)

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Then the right-hands of the following three identities are meaningful, and all the three identities hold: ·

·

·

1 ·

·

1 ·

2 ·

·

2 ·

[x, [y, z]] = (u123 − u132 ) − (u231 − u321 ) [y, [z, x]] = (u231 − u213 ) − (u312 − u132 ) [z, [x, y]] = (u312 − u321 ) − (u123 − u213 ) 3

3

Proof. Here we deal only with the first identity, leaving the other two identities to the reader. We have [x, [y, z]] ·

= [y, z] ⊛ x − A((d1 ,d2 )∈D ·

= {z ⊛ y − A ·

− A((d1 ,d2 )∈D

2 7−→(d ,d )∈D 2 ) 2 1

((d1 ,d2 )∈D2 7−→(d2 ,d1 )∈D2 )

2 7−→(d ,d )∈D 2 ) 2 1

·

= {z ⊛ y ⊛ x − A

(y ⊛ z)} ⊛ x ·

(x ⊛ {z ⊛ y − A((d1 ,d2 )∈D

((d1 ,d2 )∈D2 7−→(d2 ,d1 )∈D2 )

1 ·

·

(x ⊛ [y, z])

− {x ⊛ z ⊛ y − x ⊛ A((d1 ,d2 )∈D 3 ·

= {z ⊛ y ⊛ x − A((d1 ,d2 ,d3 )∈D ·

2 7−→(d ,d )∈D 2 ) 2 1

3 7→(d ,d ,d )∈D 3 ) 1 3 2

·

−A

3 7→(d ,d ,d )∈D 3 ) 2 3 1

((d1 ,d2 ,d3 )∈D3 7→(d2 ,d3 ,d1 )∈D3 )

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1

·

= {z ⊛ y ⊛ x − A((d1 ,d2 ,d3 )∈D ·

·

− A((d1 ,d2 ,d3 )∈D

3 7→(d ,d ,d )∈D 3 ) 1 3 2

3 7→(d ,d ,d )∈D 3 ) 2 3 1

·

= {z ⊛ y ⊛ x − A((d1 ,d2 ,d3 )∈D ·

·

(y ⊛ z))}

(x ⊛ z ⊛ y)

3 7→(d ,d ,d )∈D 3 ) 1 3 2

3 7→(d ,d ,d )∈D 3 ) 2 3 1

2 7−→(d ,d )∈D 2 ) 2 1

(y ⊛ z ⊛ x)}

◦ A((d1 ,d2 ,d3 )∈D

1

− {A((d1 ,d2 ,d3 )∈D

(y ⊛ z ⊛ x)}

(x ⊛ A((d1 ,d2 )∈D

3 7→(d ,d ,d )∈D 3 ) 2 3 1

1

(y ⊛ z)}

(x ⊛ z ⊛ y)

1

− {A((d1 ,d2 ,d3 )∈D

(y ⊛ z)})

(y ⊛ z) ⊛ x}

1

− {A((d1 ,d2 ,d3 )∈D

2 7−→(d ,d )∈D 2 ) 2 1

3 7−→(d ,d ,d )∈D 3 ) 2 1 3

(x ⊛ y ⊛ z)}

(y ⊛ z ⊛ x)}

·

(x ⊛ z ⊛ y) − A((d1 ,d2 ,d3 )∈D

3 7→(d ,d ,d )∈D 3 ) 3 2 1

1

·

(x ⊛ y ⊛ z)}

·

= (u123 − u132 ) − (u231 − u321 ) 1

1

Theorem 47 Given m ∈ M , the Jacobi identity holds for A1m with respect to the Lie bracket [·, ·]. I.e., we have [x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0 for any x, y, z ∈ A1m .

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369

From Nishimura Algebroids to Lie Algebroids

Let A be a Nishimura algebroid over M . It is very easy to see that Proposition 48 By assigning Γ(A)D =Γ(AD ) to each object D in Simp and assigning ′ Γ(A)f : γ ∈ Γ(AD ) 7→ Af ◦ γ ∈ Γ(AD ) to each morphism f : D → D′ in Simp, we have a Nishimura algebroid2 Γ(A) over a single point, where Γ(AD ) denotes the space of global sections of the bundle AD over M . Endowed with the trivial anchor natural transformation, it is a Nishimura algebroid3 over a single point. Definition 49 Given X ∈ Γ(AD1 ) and Y ∈ Γ(AD2 ), we define Y ⊚ X ∈ Γ(AD1 ×D2 ) to be (Y ⊚ X)m = (Y ◦ a(Xm )) ∗ Xm for any m ∈ M . Now we have Proposition 50 Given X ∈ Γ(AD1 ), Y ∈ Γ(AD2 ) and Z ∈ Γ(AD3 ), we have Z ⊚ (Y ⊚ X) = (Z ⊚ Y ) ⊚ X Proof. Let m ∈ M . We have (Z ⊚ (Y ⊚ X))m = (Z ◦ a((Y ⊚ X)m )) ∗ (Y ⊚ X)m = (Z ◦ (a(Y ) ⊚ a(X))m ) ∗ {(Y ◦ a(Xm )) ∗ Xm }

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= [{m′ ∈ M 7→ (Z ◦ a(Ym′ )) ∗ Ym′ } ◦ a(Xm )] ∗ Xm [By the fourth condition in Definition 19] = ((Z ⊚ Y ) ⊚ X)m

Remark 51 By this proposition we can omit parentheses in a combination by ⊚. Proposition 52 By adopting ⊚ as ∗Γ(A) , our Nishimura algebroid3 Γ(A) over a single point is a Nishimura algebroid4 over a single point. Proof. The fourth condition in Definition 19 follows from Proposition 50. The other three conditions follow trivially. Therefore all the discussions of the previous section hold. In particular, we have Theorem 53 Given X, Y ∈ Γ(A1 ), we can define [X, Y ] ∈ Γ(A1 ) to be the unique one satisfying A((d1 ,d2 )∈D =A

2 7−→d d ∈D) 1 2

◦ [X, Y ]

((d1 ,d2 )∈D2 7−→(d1 ,d2 ,−d1 ,−d2 )∈D4 )

◦ (Y ⊚ X ⊚ Y ⊚ X),

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Proposition 54 Given X, Y ∈ Γ(A1 ) and f ∈ RM , we have Y ⊚ f X = f · (Y ⊚ X) 1

Proof. Let m ∈ M . (Y ⊚ f X)m = (Y ◦ a(f (m)Xm )) ∗ (f (m)Xm ) = (Y ◦ (f (m)a(Xm ))) ∗ (f (m)Xm ) = f (m) · (Y ⊚ X)m 1

Proposition 55 Given X, Y ∈ Γ(A1 ) and f ∈ RM , we have ·

f Y ⊚ X − f · (Y ⊚ X) = X(f )Y 2

2 ⊕D

Proof. Let m ∈ M . We define µ ∈ AD m µ = A((d1 ,d2 ,d3 )∈D

to be

2 ⊕D7→(d ,d f (m)+d a(X )(f ))∈D 2 ) m 1 2 3

((Y ⊚ X)m )

where a(Xm )(f ) is the Lie derivative of f with respect a(Xm ). It is easy to see that A((d1 ,d2 )∈D Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

=A

2 7→(d ,d ,d d )∈D 2 ⊕D) 1 2 1 2

(µ)

((d1 ,d2 )∈D2 7→(d1 ,d2 ,d1 d2 )∈D2 ⊕D)

◦ A((d1 ,d2 ,d3 )∈D

2 ⊕D7→(d ,d f (m)+d a(X )(f ))∈D 2 ) m 1 2 3

((Y ⊚ X)m )

=A

((d1 ,d2 )∈D2 7→(d1 ,d2 f (m)+d1 d2 a(Xm )(f ))∈D2 )

((Y ⊚ X)m )

=A

((d1 ,d2 )∈D2 7→(d1 ,d2 f (m)+d1 d2 a(Xm )(f ))∈D2 )

((Y ⊚ X)m )

=A

((d1 ,d2 )∈D2 7→(d1 ,d2 f (a(Xm )(d1 )))∈D2 )

((Y ⊚ X)m )

= (f Y ⊚ X)m [By the third condition in Definition 19] It is also easy to see that A((d1 ,d2 )∈D =A

2 7→(d ,d ,0)∈D 2 ⊕D) 1 2

◦ A((d1 ,d2 ,d3 )∈D =A

(µ)

((d1 ,d2 )∈D2 7→(d1 ,d2 ,0)∈D2 ⊕D) 2 ⊕D7→(d ,d f (m)+d a(X )(f ))∈D 2 ) m 1 2 3

((d1 ,d2 )∈D2 7→(d1 ,d2 f (m))∈D2 )

((Y ⊚ X)m )

((Y ⊚ X)m )

= (f · (Y ⊚ X))m 2

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Therefore we have ·

(f Y ⊚ X)m − (f · (Y ⊚ X))m =A =A

2 (d∈D7→(0,0,d)∈D2 ⊕D)

(d∈D7→(0,0,d)∈D2 ⊕D)

= A(d∈D7→(0,da(Xm

(µ) ◦ A((d1 ,d2 ,d3 )∈D

)(f ))∈D2 )

2 ⊕D7→(d ,d f (m)+d a(X )(f ))∈D 2 ) m 1 2 3

((Y ⊚ X)m )

((Y ⊚ X)m )

= a(Xm )(f )Ym This completes the proof. Proposition 56 Given X, Y ∈ Γ(A1 ) and f ∈ RM , we have [X, f Y ] = f [X, Y ] + a(X)(f )Y

Proof. We have [X, f Y ] ·

= f Y ⊚ X − A((d1 ,d2 )∈D

2 7−→(d ,d )∈D 2 ) 2 1

(X ⊚ f Y )

·

= {f Y ⊚ X − f · (Y ⊚ X)}− 2

{A

((d1 ,d2 )∈D2 7−→(d2 ,d1 )∈D2 )

·

(f · (X ⊚ Y )) − f · (Y ⊚ X)} 1

2

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= a(X)(f )Y + f [X, Y ] This completes the proof. Theorem 57 Given a Nishimura algebroid A over M , A1 is a Lie algebroid over M . Proof. This follows from Theorem 53 and Proposition 56.

References [1] Borceux, F.:Handbook of Categorical Algebra, 3 vols., Cambridge University Press, Cambridge, 1984. [2] Grabowski, J. and Urba´nski:Algebroids—general differential calculi on vector bundles, Journal of Geometry and Physics, 31 (1999), 111-141. [3] Grabowski, J.:Quasi-derivations and QD-algebroids, Reports on Mathematical Physics, 52 (2003), 445-451. [4] Kock, A.: Synthetic Differential Geometry, London Mathematical Society Lecture Note Series, 51, Cambridge University Press, Cambridge, 1981.

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[5] Lavendhomme, R.: Basic Concepts of Synthetic Differential Geometry, Kluwer, Dordrecht, 1996. [6] Lawvere, W.:Functorial semantics of algebraic theories and some algebraic problems in the context of functorial semantics of algebraic theories, Repr. Theory Appl. Categ., 5 (2004), 1-121. [7] Mackenzie, K. C. H.:General Theory of Lie Groupoids and Lie Algebroids, London Mathematical Society Lecture Note Series, 213, Cambridge University Press, Cambridge, 2005. [8] Makkai, M. and Par´e, R.:Accessible Categories:The Foundations of Categorical Model Theory, Contemporary Mathematics, 104 (1989), American Mathematical Society. [9] Moerdijk, I. and Reyes, G. E.: Models for Smooth Infinitesimal Analysis, SpringerVerlag, New York, 1991. [10] Nishimura, H.:Theory of microcubes, International Journal of Theoretical Physics, 36 (1997), 1099-1131. [11] Nishimura, H.:General Jacobi identity revisited, International Journal of Theoretical Physics, 38 (1999), 2163-2174. [12] Nishimura, H.:Synthetic differential geometry of higher-order total differentials, Cahiers de Topologie et G´eom´etrie Differ´entielle Cat´egoriques, (2006), 207-232.

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[13] Nishimura, H.:The Lie algebra of the group of bisections, Far East Journal of Mathematical Sciences, 24 (2007), 329-342. [14] Schubert, H.:Categories, Springer-Verlag, Berlin and Heidelberg, 1972.

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In: Emerging Topics on Differential Geometry... Editors: L. Bernard and F. Roux, pp. 373-394

ISBN 978-1-60741-011-9 c 2010 Nova Science Publishers, Inc.

Chapter 10

A M ODEL OF H IGHER -O RDER C ONCURRENT P ROGRAMS BASED ON G RAPH R EWRITING Masaki Murakami Department of Computer Science Graduate School of Natural Science and Technology, Okayama University 3-1-1 Tsushima-Naka, Okayama, 700-0082, Japan

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Abstract This article presents a formal model of concurrent systems with higher-order communication to transfer program codes as data between processes. A number of models [1,14,15] are reported as models of higher-order communication and most of them are based on process algebra. However as we reported [7, 8], it is not easy to represent the scopes of names of communication channels precisely using process algebra. We present a model of concurrent programs with higher order communication that can represent the scopes of names precisely. The model presented here is an extension of the model reported in [8] that is based on graph rewriting. We define the equivalence relation such that if the equivalence holds on two programs then they are equivalent not only on their behaviors but on the scopes of names also. This article also presents a compilation of higher-order concurrent programs into first-order name passing programs on the graph rewriting model.

1.

Introduction

In recent years, the network technology makes common to transfer program codes between hosts connected with networks. The capability for sending and receiving of program codes during the computation is one of the key technology for distributed systems. A number of formal models with higher-order communication are reported [1, 14, 15]. LHOπ (Local Higher Order π-calculus) [15] is the one of the most well studied model in that area. It is a subcalculus of higher order π-calculus with asynchronous communication capability and it has the expressive power to represent many practically and/or theoretically interesting examples. Especially an interesting result is reported; any LHOπ process can be compiled into a first-order asynchronous π-calculus process removing higher-order communication.

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First-order asynchronous π-calculus has nice properties for example a number of interesting equivalence relations coincide and are congruence relations on the calculus. Then we can discuss properties of LHOπ processes easily with the compilation. On the other hand, as we reported in [7,8], it is difficult to represent the scopes of names of communication channels using models based on process algebra. As well known, it is important to describe “which process knows whose name” for the security of concurrent systems. In other words, it is essential to describe “which process is in the scope of which name in the system” to model distributed systems. In many such models based on process algebra, the scope of a name is represented using a binary operation as the ν-operation. Thus the scope of a name is a subterm of an expression representing a system. For example, in a π-calculus term: νa2 (νa1 (b1 |b2 )|b3 ), the scope of the name a2 is the subterm (νa1 (b1 |b2 )|b3 ) and the scope of the name a1 is the subterm (b1 |b2 ). However, this method has problems as followings. For example, consider a system S consisting of a server and two clients. A client b1 communicates with the server b2 using a channel a1 whose name is known only by b2 and oneself, and a client b3 communicates with b2 using a channel a2 that is known only by b2 and oneself. In this system a1 and a2 are local names. As b2 and b1 knows the name a1 but b3 does not, then the scope of a1 includes b1 and b2 and the scope of a2 includes b3 and b2 . Thus the scopes of a1 and a2 are not nested as shown in Fig. 1.

Figure 1. Scopes of names in S. The method denoting local names as bound names using ν-operator cannot represent the scopes of a1 and a2 precisely because the scope of a name is a subterm of a term and then the scopes of bound variable are nested (or disjoint) in any π-calculus term. In order to represent the situation above using ν-operator, the scopes must be ‘encoded’ into the nested scopes. For example, we must denote the above example as νa1 (b1 |νa2 (b2 |b3 )) or νa2 (b3 |νa1 (b1 |b2 )). Then we require a number of inferences to ‘decode’ the system using the structural congruence rules to see the scope of a name precisely. And we must regard νa1 (b1 |νa2 (b3 |b2 )) and νa2 (b3 |νa1 (b1 |b2 )) equivalent because both of them represents the same system S shown in Fig.1.. Then it is impossible to represent scopes of a1 and a2 with just one expression precisely. Furthermore, it is sometimes impossible to represent the scope even for one name precisely with ν-operator. Consider the example, νa(¯ v a.P ) | v(x).Q where x does not occur in Q. In this example, a is a local name and its scope is v¯a.P . The scope of a is extruded by communication with prefixes v¯a and v(x). Then the result of the action is νa(P |Q) and Q

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is included in the scope of a. However, as a does not occur in Q, it is equivalent to (νaP )|Q by the rules of structural congruence. We cannot see the fact that a is ‘leaked’ to Q from the resulting expression: (νaP )|Q. Thus we must keep the trace of communications for the analysis of scope extrusion. This makes difficult to analyze extrusions of scopes of names by executions. These problems also happen in LHOπ . We need a model with higher-order communication that can represent the scopes of names precisely. We presented a model that is based on graph rewriting instead of process algebra as a solution for the problem on representation of scopes of names for first-order case [8]. We defined an equivalence relation on processes on scopes of names of communication channels. We extend the model for higher-order communication in this article. We present a model of concurrent systems with higher-order communication based on graph rewriting that can represent the scopes of names of communication channels precisely. And we define an equivalence relation on processes such that it holds if two processes are equivalent not only on their behavior but also on the scopes of the names of communication channels. We show that the similar results to that we showed on the first-order model hold also on higher-order case. This article also presents a compilation of higher-order concurrent programs into firstorder name passing programs on the graph rewriting model.

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2.

Basic Idea

The model presented here is based on graph rewriting system such as [2–6, 13, 16, 17]. We represent a concurrent program consists of a number of processes (and messages on the way) using a bipartite directed acyclic graph. A bipartite graph is a graph whose nodes are decomposed into two disjoint sets: source nodes and sink nodes such that no two graph nodes within the same set are adjacent. Every edge is directed from a source node to a sink node. The system S shown in Fig.1. is represented with a graph as Fig.2.1 .

Figure 2. Bipartite Directed Acyclic Graph. Processes and messages on the way are represented with source nodes. We call source nodes as behaviors. In Fig. 2., b1 , b2 and b3 are behaviors. A behavior node has a nested structure in general. Namely, a behavior has a graph structure of a program inside as Fig. 3.. Then the structure of a program is a recursive graph. Scopes of names are represented edges of the graph. 1

Sink nodes in the system other than a1 , a2 are not depicted. In this article, we do not depict names that are not focused in the discussion in figures. Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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Figure 3. A Node with Recursive Structure.

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message: A behavior node that represents a message is a node labeled with a name of the recipient n (it is called the subject of the message) and the contents of the message o (Fig.4.a). The contents of the message is a name or a program code as we allow higher-order messages. As a program code to be sent has a structure of a concurrent program consists of a number of processes and/or messages, then the content of a message may have bipartite graph structure again (Fig.4.b). Thus the message node has a nested structure that has a graph structure inside of the node and then a program is a recursive graph in general.

Figure 4. A Message Node.

message receiving: A message is received by a receiver process. A process that receives a message execute an input action and then continue the execution. We denote such receiver process with a node consists of its epidermis that denotes the first input action and its content that denotes the continuation. For example, a receiver that executes an input action α and then become a program P (denoted as α.P in CCS term), it is denoted with the epidermis labeled with α and the content P (Fig.3. ). As the continuation P is a concurrent program, then it has a graph structure inside of the node. Thus the receiver process also has the nested structure.

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Figure 5. Message Receiving. Message receiving is represented as follows. Consider a message to send an object n to a recipient with name m and the receiver of the message(Fig.5.a). The execution of the input action is represented by “peeling” the epidermis of the receiver process node. When the message is received then it vanishes, the epidermis of the receiver is removed and the content is exposed (Fig.5.b). Now the continuation P is activated. The received object n is substituted to the name x in the content P .

Figure 6. Message Sending. message sending: In asynchronous π-calculus, message sending is represented in the same way to process activation. We adopt the similar idea. Consider an example that executes an input action α and send a message m (Fig.6. left). When the action α is executed, then the epidermis is peeled and the message m is exposed as Fig 6. right. Now the message m is transmitted and m can move to the receiver. And the execution of Q continues.

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Figure 7. Receiving a name.

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first-order and higher-order communications: Consider the case that the variable x occurs as the subject of a message like xhui in the content of a receiver(Fig.7. left and Fig.8.a). As mentioned above, the object n of a message is a name or a program code. If the object n received is a name then the behavior xhui becomes a message nhui to send object u to a receiver with name n(Fig.7. right).

Figure 8. Receiving a program code. If n is a program code, then nhui becomes a program to be activated. As LHOπ , a program code to transfer is in the form of an abstraction in the message. An abstraction denoted as (y)Q consists of a graph Q representing a program and its input argument y. When an abstraction (y)Q is sent to the receiver and substituted to x in in Fig.8.a, the behavior node (y)Qhui is exposed and ready to be activated(Fig.8.b). To activate (y)Qhui, u is substituted to y in Q (Fig.8.c). This action corresponds to the β-conversion in LHOπ . Then we have a program Q with input value u, it is activated. Scopes of names Names of communication channel in the system are sink nodes of the graph in this model. We represent the scopes of the local names using edges of the graph. Consider a

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system consists of three processes (behaviors) b1 , b2 and b3 and two names a1 and a2 as in Fig.1.. Let b1 and b2 be in the scope of a1 and b2 and b3 be in the scope of a2 . In other words, b1 and b2 can communicate using the name a1 but b3 cannot. Namely, if a1 is the name of input channel in b1 , only b2 can sent a message to b1 with the name a1 but b3 cannot. And if a2 is the name of input channel of b3 , then only b2 can send a message but b1 cannot. We represent such system with a directed acyclic graph (DAG) such as Fig.2.. A process (or a message) is inside of the scope of a name if and only if there is an edge to the name from the process in the DAG. In Fig.2., as a1 is reachable from b1 and b2 , then b1 and b2 are in the scope of a1 , and b2 and b3 are in the scope of a2 as the sink node a2 is reachable from b2 and b3 . It is possible to represent local names that their scope are not nested with the model. It is difficult to represent this situation precisely with restriction operation, which is commonly used in models based on process algebra.

3.

Formal Definitions

In this section, we present formal definitions of the model presented informally in previous section.

3.1.

Programs

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First, a countably-infinite set of names is presupposed as other formal models based on process algebra. Definition 3.1 (program, behaviour) (i) Let a1 , . . . , ak are distinct names. A program is a a bipartite directed acyclic graph with source nodes b1 , . . . , bm and sink nodes a1 , . . . , ak such that • Each source node bi (1 ≤ i ≤ m) is a behaviour. Duplicated occurrences of the same behavior are possible. • Each sink node is a name aj (1 ≤ j ≤ k). All aj ’s are distinct. • Each edge is directed from a source node to a sink node. Namely, an edge is an ordered pair (bi , aj ) of a source node and a name. For any source node bi and a name aj there is at most one edge from bi to ai . For a program P , we denote the multiset of all source nodes of P as src(P ), the set of all sink nodes as snk(P ) and the set of all edges as edge(P ). Note that the empty graph (P such that src(P ) = snk(P ) = edge(P ) = ∅) is a program. (ii) A behavior is an application, a message or a node consists of the epidermis and the content defined as follows. In the following of this definition, we assume that any element of snk(P ) nor x does not occur in anywhere else in the program. 1. A node labeled with a tuple of a name: n (called the subject of the message) and an object: o is a message and denoted as nhoi. 2. A tuple of a variable x and a program P is an abstraction and denoted as (x)P . Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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3. An object is a name or an abstraction. 4. A node labeled with a tuple of an abstraction and an object is an application. We denote an application as Ahoi where A is an abstraction and o is an object. 5. A node whose epidermis is labeled with ! and the content is a program P is a replication, and denoted as !P . 6. An input prefix is a node (denoted as a(x).P ) that the epidermis is labeled with a tuple of a name a and a variable x and the content is a program P . 7. A τ -prefix is a node (denoted as τ.P ) that the epidermis is labeled with a silent action τ and the content is a program P . The assumption for the elements of snk(P ) and x is from the idea that these names are “private” names on P and they can be renamed if necessary.

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Definition 3.2 (locality condition) A program P is local if for any input prefix c(x).Q that occurs in P , x does not occur in the epidermis of any input prefix in Q. The locality condition says that “anyone cannot use the name given from other one as the name of itself to receive messages”. Though this condition affects the expressive power of the model, in many practical examples, transfer of receiving capability is implemented with transfer of sending capability. We do not consider that the damage to the expressive power by this restriction is significant. So in this paper, we consider local and normal programs only. Theoretical motivations of this restriction are discussed in [15]. Definition 3.3 (composition) Let P and Q be programs and src(P ) ∩ src(Q) = ∅. The composition P kQ of P and Q is the program such that: • src(P kQ) = src(P ) ∪ src(Q). • snk(P kQ) = snk(P ) ∪ snk(Q). • edge(P kQ) = edge(P ) ∪ edge(Q). Intuitively, P kQ is the parallel composition of P and Q. Note that we do not assume snk(P ) ∩ snk(Q) = ∅. Obviously P kQ = QkP and ((P kQ)kR) = (P k(QkR)) for any P, Q and R from the definition. The empty graph 0 is the unit of “k”. Note that src(P ) ∪ src(Q) and edge(P ) ∪ edge(Q) denote the multiset unions while snk(P ) ∪ snk(Q) denotes the set union. Definition 3.4 (N -closure) For a normal program P and a set of names N such that N ∩ bn(P ) = ∅, the N -closure νN (P ) is the program such that: src(νN (P )) = src(P ) snk(νN (P )) = snk(P ) ∪ N edge(νN (P )) = edge(P ) ∪ {(b, n)|b ∈ src(P ), n ∈ N }

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Definition 3.5 (deleting a behaviour) For a normal program P and b ∈ src(P ), P \ b is a program that is obtained by deleting a node b and edges that are connected with b from P . Namely, src(P \ b) = src(P ) \ {b} snk(P \ b) = snk(P ) edge(P \ b) = edge(P ) \ {(b, n)|(b, n) ∈ edge(P )} Note that src(P ) \ {b} and edge(P ) \ {(b, n)|(b, n) ∈ edge(P )} denote the multiset subtractions.

3.2.

Operational Semantics

We define the operational semantics with a labeled transition system. Definition 3.6 (free/bound name) 1. For a behavior or an object p, the set of free names of p : fn(p) is defined as follows. • for a name a, fn(a) = {a}, • fn(ahoi) = fn(o) ∪ {a}, • fn((x)P ) = fn(P ) \ {x}, • fn(!P ) = fn(P ), • fn(τ.P ) = fn(P ), • fn(a(x).P ) = (fn(P ) \ {x}) ∪ {a} and

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• fn(o1 ho2 i) = fn(o1 ) ∪ fn(o2 ) 2. For a program P where src(P ) = {b1 , . . . , bm }, fn(P ) =

S

i fn(bi )

\ snk(P ).

The set of bound names of P (denoted as bn(P )) is the set of all names that occur in P (including elements of snk(P ) even if they do not occur in any element of src(P )) but not in fn(P ). Definition 3.7 (normal program) A program P is normal if for any b ∈ src(P ) and for any n ∈ fn(b) ∩ snk(P ), (b, n) ∈ edge(P ) and any programs occur in b is also normal. It is quite natural to assume the normality for programs, because anyone must know the name to use it. Definition 3.8 (substitution) The substitution of an object to a program, to a behaviour or to an object is defined recursively as follows. Let p be a behavior, an object or a program. For a name a, we assume that a ∈ fn(p). The substitution of an object o to name a in p is denoted as p[o/a] and defined as follows. • for a name c, c[o/a] =

(

o if c = a c otherwise

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• (o1 ho2 i)[o/a] = o1 [o/a]ho2 [o/a]i, • (!P )[o/a] =!(P [o/a]), • (c(x).P )[o/a] = c(x).(P [o/a]), • (τ.P )[o/a] = τ.(P [o/a]) and • for a program P and a ∈ fn(P ), P [o/a] = P ′ where P ′ is a program such that src(P ′ ) = {b[o/a]|b ∈ src(P )}, snk(P ′ ) = snk(P ) and edge(P ′ ) = {(b[o/a], n)|(b, n) ∈ edge(P )}. For the cases of abstraction and input prefix, note that we can assume x 6= a because a ∈ fn((x)P ) or ∈ fn(c(x).P ) without losing generality. (We can rename x if necessary.) We can also assume a 6= c from the locality condition as substitutions are applied only for the contents of an input prefix. Definition 3.9 (action) An action is a silent action τ , an output action or an input action. For a name a and an object o, an input action is a tuple a(o) and an output action is a tuple ahoi. α

Definition 3.10 (labeled transition) For an action α, → is the least binary relation on normal programs that satisfies the following rules.

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input : if b ∈ src(P ) and b = a(x).Q, then a(o)

P → (P \ b)kν{n|(b, n) ∈ edge(P )} νoQ[o/x] β-conversion : if b ∈ src(P ) and (y)Qhoi, then τ

P → (P \ b)kν{n|(b, n) ∈ edge(P )}Q[o/y] τ -action : if b ∈ src(P ) and b = τ.Q, then τ

P → (P \ b)kν{n|(b, n) ∈ edge(P )}Q α

α

replication 1 : P → P ′ if !Q ∈ src(P ), and P kν{n|(!Q, n) ∈ edge(P )}Q′ → P ′ , where Q′ is a program obtained from Q by renaming all names in snk(R) to distinct fresh names that do not occur in anywhere else, for all R’s where each R is a program that occur in Q (including Q itself).

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replication 2 : P → P ′ if !Q ∈ src(P ) and P kν{n|(!Q, n) ∈ edge(P )}(Q′1 kQ′2 ) → P ′ , where each Q′i (i = 1, 2) is a program obtained from Q by renaming all names in snk(R) to distinct fresh names that do not occur in anywhere else, for all R’s where each R is a program that occur in Q (including Q itself). ahvi

output : if b ∈ src(P ), b = ahvi then, P → P \ b communication : if b1 , b2 ∈ src(P ), b1 = ahoi, b2 = a(x).Q then, τ

P → ((P \ b1 ) \ b2 )kν{n|(b2 , n) ∈ edge(P )} νoQ[o/x] In all rules above except replication, the behavior that makes an action is removed from src(P ) to obtain src(P ′ ). Then the edges from the removed behaviors are no longer in edge(P ′ ). The following propositions are straightforward from the definition. Proposition 3.1 α For any program P, P ′ and any action α such that P → P ′ , 1. If P is local then P ′ is local. 2. If P is normal then P ′ is normal. Example 3.1 Consider the following (asynchronous) π-calculus processes:

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P1 = m(x).Q P2 = νn(m(u).(na|νc(cu)) | n(x).Q). Assume that x nor n do not occur in Q, then P1 and P2 are weakly bisimilar. Consider the case that a message mo is received by Pi (i = 1, 2). In P1 , the object o received by the execution of m(x) is not stored in Q. On the other hand, o remains in νc(co) in the case of P2 . If o is confidential data and that should not be kept by any in unauthorized processes (including P1 and P2 ), then P2 is an illegal process but P1 is not (because it discards o immediately). Thus P1 and P2 should be distinguished but usual behavioural equivalence in π-calculus cannot. One may say that stronger equivalence relations such as syntactic equivalence or structural congruence work. Of course, syntactic equivalence can distinguish these two cases, but it is not convenient. How about structural congruence? In fact, P1′ and P2′ are m(o)

not structural congruent where Pi′ is a process such that Pi ⇒ Pi′ (i = 1, 2). But consider the case that as o is strictly confidential data then just receiving o is not allowed for an unauthorized process and processes of the form α.Q are unauthorized. P1′ is Q and P2′ is νn(νc(co))) | Q. In this case, o occurs in P2′ but unauthorized n(x).Q does not receive o. Thus P2 is not illegal. On the other hand, though o does not occur in P1′ that is Q, o is received by P1 of the form α.Q. We cannot find that from P1′ without the history of communication. Thus it is not easy to find illegal case using structural congruence.

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Figure 9.a. Graph representation of P1 and mo.

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Figure 9.b. Graph representation of P2 and mo.

We can use bipartite directed acyclic graph model presented here to model this example. The example that corresponds to the system consists of P1 and the message mo is given by the graphs in Fig. 9.a. left 2 . These graphs evolve to the graph in Fig. 9. a. right (in the case that o is a name) that corresponds to the π-calculus process Q. This graph explicitly denotes that Q is in the scope of the newly imported name o. On the other hand the example of P2 with mo is Fig. 9.b. left. After receiving the message carrying o, that remains as the message node choi as Fig. 9.b. right. In this case, the result explicitly shows that Q is not in the scope of o.

3.3.

Behavioral Equivalence

The equivalence relation introduced in the next section is defined using the usual behavioral equivalence. Thus we define the weak bisimulation equivalence here. As usual, we denote α τ τ α τ τ τ α ˆ α P ⇒ P ′ if P → · · · →→→ · · · →→ P ′ . And we denote P ⇒ P ′ if P ⇒ P ′ when α 6= τ τˆ τ τ and P ⇒ P ′ if P → · · · → P ′ . For a sequence α of actions α1 , . . . αk , α

⇒

=

α ˆ

α ˆ

⇒1 · · · ⇒k

Weak bisimulation relation is defined as usual. 2

The sink nodes corresponding n are not depicted in the following examples.

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Definition 3.11 (weak bisimulation) A binary relation R on programs is a weak bisimuα lation if: for any (P, Q) ∈ R, for any α and P ′ if P → P ′ then there exists Q′ such that α ˆ Q ⇒ Q′ and (P ′ , Q′ ) ∈ R and vice verse. Definition 3.12 (weak bisimilarity) Weak bisimulation equivalence is defined as follows. ≃=

[

{R|R is a weak bisimulation}

It is easy to show ≃ is an equivalence relation. We can show that congruence property of ≃ wrt composition. Proposition 3.2 For any program R, if P ≃ Q then P kR ≃ QkR. proof:(outline) We can show that the following relation is a weak bisimulation from the definitions of “k” and bisimilarity. {(P kR, QkR)|P ≃ Q} We can also show that ≃ is a congruence relation on the set of first-order programs wrt any first-order context [10].

4.

Scope Equivalence

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This section presents an equivalence relation on programs which ensures that two systems are equivalent in their behavior and in the scopes of names [11]. We introduce preliminary notions first. Definition 4.1 For a process graph P and a name n such that n, P/n is the program defined as follows: src(P/n) = {b|b ∈ src(P ), (b, n) ∈ edge(P )} snk(P/n) = snk(P ) \ {n} edge(P/n) = {(b, a)|b ∈ src(P/n), a ∈ snk(P/n), (b, a) ∈ edge(P )} Intuitively P/n is the subsystem of P that consists of behaviors which are in the scope of n. Let P be an example of Fig.2., P/a1 is a subgraph of Fig.2.. obtained by removing the node of b3 (and the edge from b3 to a2 ) and a1 (and the edges to a1 ) as shown in Fig.10.. It consists of process nodes b1 and b2 and a name node a2 .

Figure 10. The graph P/a1 . Proposition 4.1 For any P, Q and n ∈ snk(P ) ∪ snk(Q), (P kQ)/n = P/nkQ/n. proof: We show the followings three equations.

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Masaki Murakami i) src((P kQ)/n) = src(P/nkQ/n) ii) snk((P kQ)/n) = snk(P/nkQ/n) iii) edge((P kQ)/n) = edge(P/nkQ/n)

i) For any b ∈ src((P kQ)/n), b is an element of src(P ) or src(Q), so we assume b ∈ src(P ) without losing generality. As b ∈ src((P kQ)/n), there is an edge from b to n in P kQ. There is an edge from b to n in P , because any edge in P kQ from b in src(P ) is an edge in P . Thus b ∈ src(P/n) from the definition. As src(P/nkQ/n) = src(P/n) ∪ src(Q/n), then b ∈ src(P/nkQ/n). Conversely, for any b ∈ src(P/nkQ/n) = src(P/n) ∪ src(Q/n), we assume b ∈ src(P/n) without losing generality. From the definition, b ∈ src(P ) then b ∈ src(P kQ). And from the definition, there is an edge from b to n in P , so there is an edge from b to n in P kQ also. Thus b ∈ src((P kQ)/n).

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ii) For any m ∈ snk((P kQ)/n), we assume m ∈ snk(P ) without losing generality and m 6= n. So we have m ∈ snk(P/n) from the definition. Thus m ∈ snk(P/n)∪snk(Q/n) = snk(P/nkQ/n). Conversely, for any m ∈ snk(P/nkQ/n), we assume m ∈ snk(P/n) without losing generality. From the definition, m ∈ snk(P ) and m 6= n. Thus m ∈ snk(P kQ) and Thus m ∈ snk((P kQ)/n). iii) For any (b, m) ∈ edge((P kQ)/n), (b, m) ∈ edge(P kQ) and n 6= m from the definition. Again we assume (b, m) ∈ edge(P ) without losing generality. As n 6= m, (b, m) ∈ edge(P/n). Then (b, m) ∈ edge((P/nkQ/n)) from the definition. On the other hand, for any (b, m) ∈ edge(P/nkQ/n), we assume (b, m) ∈ edge(P/n) without losing generality again. From the definition, (b, m) ∈ edge(P ). Thus (b, m) ∈ edge(P kQ). From (b, m) ∈ edge(P/n), m ∈ snk(P ) and m 6= n, so m ∈ snk(P kQ) and m ∈ snk((P kQ)/n). From the definition, we have (b, m) ∈ edge((P kQ)/n). Definition 4.2 (scope bisimulation) A binary relation R on programs is a scope bisimulation if for any (P, Q) ∈ R, • P/n is an empty graph iff Q/n is an empty for any n ∈ snk(P ) ∩ snk(Q), • P/n ≃ Q/n for any n ∈ snk(P ) ∩ snk(Q) and • R is a weak bisimulation. It is easy to show that the union of all scope bisimulations is a scope bisimulation and it is the unique largest scope bisimulation. Definition 4.3 (scope bisimulation) The largest scope bisimulation is scope equivalence and denoted as ⊥. It is obvious from the definition that ⊥ is an equivalence relation. It is easy to show that ⊥ is a congruence relation wrt k. Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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Proposition 4.2 If P ⊥ Q then P kR ⊥ QkR if snk(P ) ∩ snk(R) = snk(Q) ∩ snk(R). proof: We show that the following relation R is a scope bisimulation if snk(P )∩snk(R) = snk(Q) ∩ snk(R). R = {(P kR, QkR)|P ⊥ Q}

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i) Assume that (P kR)/n is an empty graph for some n ∈ snk(P kR) ∩ snk(QkR). First, consider the case such that n ∈ snk(P ) ∩ snk(Q) (otherwise n 6∈ snk(P ) ∪ snk(Q) from snk(P ) ∩ snk(R) = snk(Q) ∩ snk(R)). As (P kR)/n is an empty graph, there is no edge from any behavior in src(R) to n, so R/n is an empty graph. If P/n is not an empty graph, then there is an edge for some b ∈ src(P ) to n. As b ∈ src(P kR) and (b, n) ∈ edge(P kR), then (P kR)/n must be a non-empty. Thus P/n is an empty graph. From P ⊥ Q, if P/m is an empty graph then Q/m is an empty graph also for any m ∈ snk(P ) ∩ snk(Q). Then Q/n is an empty graph for n. From the definition, Q/nkR/n is an empty graph from both of Q/n and R/n are empty. Thus (QkR)/n is empty because it is equal to Q/nkR/n from Proposition 4.1. For the case such that n 6∈ snk(P ) ∪ snk(Q), obviously there is no edge from any b ∈ src(Q) to n in Q. And as (P kR)/n is an empty graph, there is no edge from any b ∈ src(R) to n in R. Then (QkR)/n is an empty graph from the definition. ii) We show that (P kR)/n ≃ (QkR)/n for any n ∈ snk(P kR) ∩ snk(QkR) if P ⊥ Q. As n ∈ snk(P kR) ∩ snk(QkR), n ∈ snk(P kR) and hence n ∈ snk(P ) ∪ snk(R). From Proposition 4.1, (P kR)/n = P/nkR/n for n ∈ snk(P ) ∪ snk(R). Consider the case such that n ∈ snk(P ) ∩ snk(Q). From P ⊥ Q, P/n ≃ Q/n for n ∈ snk(P ) ∩ snk(Q). By Proposition 3.2, we have P/nkR/n ≃ Q/nkR/n. And from Proposition 4.1 again, Q/nkR/n = (QkR)/n. Thus from the transitivity of ≃, (P kR)/n ≃ (QkR)/n. If n 6∈ snk(P ) ∩ snk(Q), n 6∈ snk(P ) and n 6∈ snk(Q) from snk(P ) ∩ snk(R) = snk(Q) ∩ snk(R). Thus there is no edge from any b ∈ src(P ) to n. Thus (P kR)/n is equal to R/n. Similarly, (QkR)/n is also equal to R/n. Thus (P kR)/n is equal to (QkR)/n, so (P kR)/n ≃ (QkR)/n from the reflexivity of ≃. α

iii) From Definition 3.10, we can show that any P1 such that P kR → P1 has the form of α α P ′ kR for some P ′ such that P → P ′ , P kR′ for some R′ such that R → R′ or P ′ kR′ for β′

β

some P ′ such that P → P ′ and R′ such that R → R′ where β and β ′ are ahvi and a(v) respectively or vice verse (and then α = τ ). For the first case, as ⊥ is the largest scope bisimulation, that is a weak bisimulation. α α Thus for any P ⊥ Q, for any P ′ such that P → P ′ , there exists Q′ such that Q → α α Q′ and P ′ ⊥ Q′ . From Q → Q′ , we have QkR → Q′ kR. From the definition of R, (P ′ kR, Q′ kR) ∈ R as P ′ ⊥ Q′ . α For the second case, obviously QkR → QkR′ . From the definition of R, (P kR′ , QkR′ ) ∈ R. The last case is the combination of the above two case. Namely, for P ′ such that β

β

P → P ′ , from P ⊥ Q and ⊥ is a weak bisimulation, so there exists Q′ such that Q ⇒ Q′ τ

β

β′

and P ′ ⊥ Q′ . On the other hand, if P kR → P ′ kR′ and P → P ′ , then R → R′ where β τ and β ′ are ahvi and a(v) respectively or vice verse. Thus we have QkR ⇒ Q′ kR′ . From Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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the definition of R, (P ′ kR′ , Q′ kR′ ) ∈ R as P ′ ⊥ Q′ . We can also show that ⊥ is a congruence relation on the set of first-order programs wrt any first-order context [12]. Definition 4.4 For a program P such that P = P1 kQ where P1 is not an empty graph and a name n ∈ snk(P ), if (b, n) ∈ edge(P ) for some b ∈ src(P1 ) (in other words, (b, n) ∈ edge(P1 )), then P1 ⊢ n in P . Intuitively, P1 ⊢ n in P means that at least one behaviour in the component P1 of P is in the scope of the name n. Definition 4.5 For a program P such that P = P1 kQ where P1 is not an empty graph and a name n ∈ snk(P ), if there exists a program P1′ kQ′ such that α

• P1 ⇒ P1′ for some α, β

• Q ⇒ Q′ for some β and • P1′ ⊢ n in P1′ kQ′ , then we denote P1 |= n (in P ).

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Intuitively for a subsystem P1 of P , P1 |= n means that at least one behaviour in P1 will eventually be in the scope of n (or is in the scope of n already). Theorem 4.1 Assume that P1 ⊥ P2 . Then for any name n ∈ snk(Q), P1 |= n in QkP1 if and only if P2 |= n in QkP2 . α

proof: We prove the following by the induction on the length k of α for P1 ⇒ P1′ . If there exists a program P1′ kQ′ such that α

• P1 ⇒ P1′ for some α, β

• Q ⇒ Q′ for some β and • P1′ ⊢ n in P1′ kQ′ , then there exists a program P2′ kQ′ such that α

• P2 ⇒ P2′ for the same α, • P2′ ⊢ n in P2′ kQ′ for the same Q′ . If k = 0, it means that P1 is P1′ . From P1 |= n, P1 ⊢ n in P1 kQ′ . Namely for some b ∈ src(P1 ), (b, n) ∈ edgeP1 kQ′ . Thus b ∈ src(P1 ), (b, n) ∈ edgeP1 . As P1 ⊥ P2 ,

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b′ ∈ src(P2 ), (b′ , n) ∈ edgeP2 otherwise P2 /n is empty and contradicts P1 ⊥ P2 . Thus P2 ⊢ n in P2 kQ′ . Then P2 |= n in QkP2 . α α′

0 Consider the case that the length of α is k + 1. Let α be α0 α′ . As P1 ⇒ P1′ , there

α

α′

exists P2′′ such that P1 →0 P1′′ and P1′′ ⇒ P1′ . From P1 ⊥ P2 , there exists P1′′ such that α ˆ

P2 ⇒0 P2′′ and P1′′ ⊥ P2′′ because ⊥ is a weak bisimulation. From the induction hypothesis, α′

α α′

0 there exists P2′′ ⇒ P2′ and P2′ ⊢ n in P2′ kQ′ . And we have P2 ⇒ P2′ .

This theorem says that if P1 ⊥ P2 and n will be received by P1 from Q eventually when they are executed concurrently, then P2 can receive n also from Q if Q is executed with P2 . Example 4.1 Let P1 be the (asynchronous) π-calculus processes defined in Example 3.1, and P3 be as following: P3 = νn(m(u).na | n(x).Q)

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that is similar to P2 in Example 3.1 but it does not keep the received object (a 6= u is assumed). We assume that x nor n do not occur in Q and then P1 is weakly bisimilar to P3 . We cannot distinguish P1 from P3 by behavioral equivalence of π-calculus. As we show in Example 3.1, the graph representation of P1 explicitly denotes that the process after the communication is in the scope of o, then P1 |= o in mhoikP1 . On the other hand the graph representation that corresponds P3 is Fig. 11 left. The process after the communication on m and n is Fig. 11 right in which Q is not in the scope of o. Then P3 6|= o in mhoikP3 . By Theorem 4.1, P1 6⊥ P3 from these results. In fact, P1′ /o 6≃ P3′ /o where P3′ is the process obtained from P3 by the communication with mhoi. Note that P1′ ≃ P3′ then ⊥ is strictly stronger than ≃.

Figure 11. The graph of P3 and mo. On the other hand, if a = u then Q is in the scope of o. In this case, P1′ /o ≃ P3′ /o.

5.

Compilation of Higher-Order Programs into First-Order Programs

This section presents the formal definition of the compilation procedure from higher-order process into first-order process. The basic idea for the compilation is “to translate a message sending a program code into one sending the location of the program code instead Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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Figure 12. An example with higher-order communication.

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of the program code itself”. The location of the program code is denoted with the name of communication channel name to access the program code. The essential step of the compilation is translate a higher-order message mh(y)Qi to carry a program code (y)Q into the template of the program code with location “c” where c is a fresh name and the message to send c (see Fig. 13.). We present “how the compilation mapping works” using an example informally. Consider an example that consists of a higher-order message mh(y)Qi to carry a program code (y)Q and two processes (Fig. 12(a)). (In Fig. 12-14., we depict behavior nodes only.) One process is a receiver process (Fig. 12(a) center) which invoke the received program with input a and forward the program to another receiver (Fig. 12(a) right) with message nhxi. The forwarded program code is received by the second receiver and activated with input value u. The execution of this example based on the operational semantics defined in Definition 3.10 is as follows. The program code (y)Q is received by the first receiver and an application (that is a pair of program code and an input value) (y)Qhai is invoked and the message nh(y)Qi to forward (y)Q is transmitted (Fig. 12(b)). The input value a is substituted to y the first instance of (y)Q is activated (Fig. 12(c) left) and the forward message is received by the second receiver (Fig. 12(c) right). Finally, the input value is substituted to y and the second instance of (y)Q is activated. As the result, two instances Q[a/y] and Q[u/y] is activated in this example (Fig. 12(d).). The example of Fig. 12(a). is translated to the first-order program as Fig. 14(a). using the compilation procedure in Fig. 13, there is no longer higher-order communication after the compilation. Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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Figure 13. Translation of a higher-order message. This example consist of the template of Q and the first-order message mhci (Fig. 14(a) left) with two receiver processes which are same to the case of before compilation. The message mhci is received by the first receiver and forwarded to the second receiver (Fig. 14(b)). The second receiver receives the forwarded message, now two first-order messages chvi and chui is transmitted (Fig. 14(c)). These two messages are received by the instances of (y)Q invoked by the template (Fig. 14(d)). Then two instances Q[a/y] and Q[u/y] is activated in this example as before compilation (Fig. 14(e)). Though the template of Q still remains, it no longer affects the computation. It is because even if a new instance is created, it is impossible to activate it. As there are no process in the scope of the name c elsewhere, the input prefix c(y) cannot be executed. The rest of this section presents the formal definition of the compilation procedure [9]. Definition 5.1 (compilation mapping) For a program P , hm(P ) is the multiset of all higherorder messages occur in src(P ). The mapping into the first-order program [[P ]] is defined as follows. • snk([[P ]]) = snk(P ) ∪ new(P ) where new(P ) = {nm |m ∈ hm(P ), nm is a fresh name that does not occur anywhere else.} • src([[P ]]) = {[[b]] | b ∈ src(P ), b 6∈ hm(P )} ∪

S

m∈hm(P ) [[m]].

• For a behavior b, [[b]] is defines as follows – [[!Q]] =![[Q]], – [[τ.Q]] = τ.[[Q]], – [[a(x).Q]] = a(x).[[Q]],

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Figure 14. The execution of the compiled example.

– [[ahni]] = ahni for a name n, – [[(z)Qhoi]] = [[Q[o/z]]] and – [[ah(y)Qi]] = {ahnm i, !(nm (z).[[Q]])} where nm ∈ new(P ) and m = ah(y)Qi. • edge([[P ]]) = {([[b]], n) | (b, n) ∈ edge(P ), b 6∈ hm(P )}∪

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{(ahnm i, nm ) | ahnm i ∈ src([[P ]]), nm ∈ new(P )}∪ {(!(nm (z).[[Q]]), nm ) | !(nm (z).[[Q]]) ∈ src([[P ]]), nm ∈ new(P )}. Note that there may be a case that the compilation procedure above does not terminate because of infinite applications of [[(z)Qhoi]] = [[Q[o/z]]]. However such program has an infinite sequence of τ -actions by β-conversions as an execution sequence of the program. Thus such program is an unusual one with infinite loops, then we do not consider such cases. We consider only programs that the compilation procedure terminates for the program.

6.

Conclusion

We presented a model of concurrent system with higher-order communication based on graph rewriting. The model can represent the scopes of channel names of programs precisely. We defined the equivalence relation for the scopes of names and we showed the relation is congruence on composition operation of programs. This article also presented a compilation method of higher-order communication into the first-order model based on graph rewriting. The idea adopted here is basically a translation of the compile method from LHOπ to the first-order π calculus [15] into graph rewriting model.

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References [1] Baldamus, M. and Dingel, J., Modal Characterization of Weak Bisimulation for Higher-order Processes, Proceedings of the Seventh International Joint Conference on the Theory and Practice of Software Development (TAPSOFT ’97) (1997) [2] Ehrig, H. and B. K¨onig, Deriving Bisimulation Congruences in the DPO Approach to Graph Rewriting with Borrowed Contexts, Mathematical Structures in Computer Science, vol.16, no.6, pp. 1133-1163, (2006) [3] Gadducci, F., Term Graph rewriting for the π-calculus, Proc. of APLAS ’03 (Programming Languages and Systems), LNCS 2895, pp. 37-54, (2003) [4] K¨onig, B., A Graph Rewriting Semantics for the Polyadic π-Calculus, Proc. of GTVMT ’00 (Workshop on Graph Transformation and Visual Modeling Techniques), pp. 451-458 (2000) [5] Lafont, Y., Interaction Nets. Proc. of POPL’90, ACM, pp. 95-108, (1990) [6] Milner, R., Bigraphical Reactive Systems, Proc. of CONCUR’01, LNCS 2154, Springer, pp. 16-35 (2001) [7] Murakami, M., A Model of Runtime Transformation for Distributed Systems Based on Directed Acyclic Graph Model, Journal of System Architectures, Elsevier, Vol. 50, pp. 417-425 (2004) Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

394

Masaki Murakami

[8] Murakami, M., A Formal Model of Concurrent Systems Based on Bipartite Directed Acyclic Graph, Science of Computer Programming, Elsevier, 61 pp. 38-47 (2006) [9] Murakami, M., Compilation of Higher-Order Concurrent Programs into First-Order Programs on Recursive Graph Rewriting Model, Proc. of FCS ’07, CSREA, pp. 367375 (2007) [10] Murakami, M., Congruence Results of Behavioral Equivalence for A Graph Rewriting Model of Concurrent Programs, Proc. of ICITA 2008, pp. 636-641 (2008) [11] Murakami, M., A Graph Rewriting Model of Concurrent Programs with Higher-Order Communication, Proc. of TMFCS-08, ISRST, pp. 80-87 (2008) [12] Murakami, M., Congruence Results of Scope Equivalence for a Graph Rewriting Model of Concurrent Programs, to appear in Proc. of ICTAC 2008 [13] Odersky, M. Functional Nets, European Symposium on Programming 2000, Lecture Notes in Computer Science 1782, Springer Verlag, (2000) [14] Sangiorgi, D. , π-calculus, internal mobility, and agent-passing calculi, Theoretical Computer Science 167(2), (1996), pp. 235-274 [15] Sangiorgi, D. Asynchronous Process Calculi: The First- and Higher-order Paradigms, Theoretical Computer Science, 253, pp. 311-350 (2001)

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[16] Sassone, V. and P. Soboci´nski, Reactive systems over cospans, Proc. of LICS ’05 IEEE, pp. 311-320, (2005) [17] Ueda, K. and N. Kato, Programming with Logical Links: Design of the LMNtal language, Proc. of PPL’03, JSSST, pp. 20-31 (2003)

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ISBN 978-1-60741-011-9 c 2010 Nova Science Publishers, Inc.

Chapter 11

P ROTECTING THE V ERTICES OF A G RAPH William F. Klostermeyer∗ School of Computing University of North Florida Jacksonville, FL 32224-2669

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Abstract Dominating sets are one of the most widely studied concepts in graph theory. If one imagines that there is a guard located at each of the vertices in a dominating set, one can see that those guards can “protect” all the vertices of the graph. That is, we think of a guard as being able to protect the vertex it is on as well as neighboring vertices. Several recent variations of dominating sets have been proposed to protect the vertices of a graph against sequences of attacks at vertices, since a dominating set may only be effective against a single attack at one vertex. That is, after a guard moves to defend the attack, the resulting configuration of guards is not necessarily a dominating set, and thus may be unable to fully protect the graph against subsequent attacks. In this chapter, we review key results on domination and survey results on Roman domination, weak Roman domination, secure domination, and eternal domination, as well as variations on these types of domination and related problems such as the kserver problem. One of our main goals will be to compare the minimum number of guards needed in these various scenarios with other graph parameters such as the domination number and independence number. A number of open problems are stated.

1.

Introduction

One of the most famous problems in combinatorics dealing with security guards is the art gallery problem, which asks how many guards are needed to view every point in an art gallery. The gallery is represented by a polygon with n vertices and Chv´atal proved in 1973 that at most ⌊ n3 ⌋ guards are needed [10]. The decision problem associated with finding the minimum problem, however, is N P -complete. In this paper, we will be interested in guarding, or defending, the vertices of a graph, rather than the space defined by a polygon. The general problem is as follows. Given ∗

E-mail address: [email protected]

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a simple, finite, undirected graph G = (V, E), we wish to defend the graph against a sequence of one or more attacks. Guards are located at (some) vertices and can defend an attack at the vertex at which they are located or can move to another vertex to either defend an attack there, or in some versions of the problem, to better configure their location to defend future attacks. The versions of the problem we consider have several variations depending on (1) the number of attacks, (2) how far a guard can move to defend an attack, (3) whether guards that are not defending the current attack are allowed to move in order to re-configure their location, or (4) the number of guards allowed or required at each vertex. We will detail each version of the problem within the subsequent sections of the paper. The bibliography contains a list of papers on the subject, some of which are not explicitly cited in the text. Throughout the paper, we shall let n = |V | and use V (G) and E(G) to denote the vertex set and edge set of G when necessary. We use R = r1 , r2 , . . . with ri ∈ V to denote the sequence of attacks. It will always be the case that attacks occur, and must be defended, one at a time.

2.

k-server Problem

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The k-server problem is an algorithmic problem, sometimes couched in the more general framework of metric spaces, but oftentimes focused on graphs. It was defined in [31] as follows. There are k mobile servers (a.k.a. guards) located at vertices in the graph. In response to an attack ri ∈ V , if there is no server located at ri , a server must be sent to ri . The objective is to minimize the total distance traveled by all the servers over the sequence of attacks. There are three main variations of the problem, which will define a general categorization we will use throughout the paper. (1) Offline problem: the entire sequence of attacks, r1 , r2 , . . . , rm , is known in advance. (2) Adaptive Online problem: the sequence of attacks is revealed one by one by an adversary (sometimes called an adaptive adversary). (3) Oblivious Online problem: the sequence of attacks is constructed in advance by an adversary, but revealed one by one. The adversary in this case is called an oblivious adversary. A simple polynomial time algorithm using dynamic programming can compute the optimal solution for the offline problem [31]. A faster algorithm using maximum flow techniques that runs in O(km2 ) time is given in [9]. Koutsoupias and Papadimitriou proved that a simple algorithm known as the workfunction algorithm is 2k − 1 competitive in [30]. In other words, the distance the servers travel using work function algorithm is at most 2k − 1 times the distance they would travel using any other algorithm, including an optimal algorithm that knew the attack sequence in advance, over all attack sequences. It is a well-known conjecture that the work function algorithm is k-competitive and that this would be best possible. A key difference between problems (2) and (3) is that a randomized algorithm can be effective in problem (3). Since an oblivious adversary cannot adapt the attack sequence to

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the moves of the algorithm, by using randomization, an algorithm may be able to effectively prevent an adversary from constructing a costly attack sequence. A famous result from [32] is an Hk competitive algorithm for the problem of k servers on a complete graph with k + 1 vertices, where Hk is the k th Harmonic number. This result is known to be optimal.

3.

Dominating Sets

Denote the open and closed neighborhoods of a vertex x ∈ V by N (x) and N [x], respectively. That is, N (x) = {v|xv ∈ E} and N [x] = N (x) ∪ {x}. A dominating set of G is a set D ⊆ V with the property that for each u ∈ V − D, there exists x ∈ D adjacent to u. The minimum cardinality amongst all dominating sets is the domination number γ(G). An excellent survey of domination is given in [20]. In terms of protecting the vertices of a graph, we can imagine that a dominating set can protect a graph against a single attack. That is, if we place guards at the vertices of a dominating set, then each vertex is distance at most one from a guard. Thus an attack can be defended by a guard either at the attacked vertex or at a neighboring vertex. It is easy to see that any maximal independent set is a dominating set. Hence the domination number of a graph is less than or equal to its independence number. We state two well-known results on dominating sets. Theorem 1 (Ore’s Theorem) Let G be a graph with no isolated vertices. Then γ(G) ≤ n2 . A simple probabilistic argument gives the following, originally proved by Arnautov and also Payan.

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. Theorem 2 Let G be a graph with minimum degree δ. Then γ ≤ n 1+ln(δ+1) δ+1 Better bounds exist in special cases when δ = 2 ( 2n 5 with seven exceptions) and δ = 3 if G is connected) [20]. Vizing’s conjecture is perhaps the most famous problem in domination theory, asserting that γ(G × H) ≥ γ(G)γ(H) [20]. G × H is defined to be the graph with vertex set V (G) × V (H), so vertices in G × H are denoted < u1 , v1 > and edges (< u1 , v1 >, < u2 , v2 >) where either u1 = u2 and (v1 v2 ) ∈ E(H) or v1 = v2 and (u1 , u2 ) ∈ E(G). The m×n grid graph is Pm ×Pn . The exact value of γ(Pm ×Pn ) is known only for small values 3n+4 of m. For example, γ(P2 × Pn ) = ⌊ n+2 2 ⌋ and γ(P3 × Pn ) = ⌊ 4 ⌋ [20]. γ(Pn × Pn ) is 2 2 known to be roughly between n −n−3 and n +4n−16 , cf. [20]. These results and problems 5 5 are motivation for studying other forms of domination in grids and other cartesian products of graphs. A total dominating set of G is a set D ⊆ V with the property that for each u ∈ V , there exists x ∈ D adjacent to u. The size of a minimum cardinality total dominating set in G is denoted γt (G). Note that the total domination number is only defined for graphs without isolated vertices. ( 3n 8

Theorem 3 (Cockayne, Dawes, and Hedetniemi) cf.[20] If G is a connected graph with at least three vertices, then γt ≤ 2n 3 . Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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A perfect code is a set of vertices D such that |N [v] ∩ D| = 1 for all vertices v ∈ V . It is easy to prove that if G has a perfect code, then each perfect code of G contains γ(G) vertices, cf. [20]. So a perfect code protects each vertex of a graph with exactly one guard. C4 is a simple example of a graph that does not contain a perfect code. A 2-packing is a set of vertices X ⊆ V such that for all vertices u, v ∈ X, N [u] ∩ N [v] = ∅. So a 2-packing is like a partial perfect code.

4.

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4.1.

Eternal Domination Eternal Dominating Sets

In the eternal dominating set problem, the sequence of attacks is infinitely long and the configuration of guards must induce a dominating set before and after each attack has been defended. At most one guard is located at each vertex. A guard can protect the vertex at which its located and can move to a neighboring vertex to defend an attack there and at most one guard is allowed to move for each attack (the version in which all guards are allowed to move is considered in the next subsection). Eternal dominating sets have been considered in a number of recent papers such as [1, 6, 17, 18, 25, 27] and others that we mention below. They are sometimes called eternal secure sets. An eternal dominating set (EDS) of G is a set D such for each sequence of attacks R = r1 , r2 , . . . with ri ∈ V there exists a sequence D = D1 , D2 . . . of dominating sets and a sequence of vertices s1 , s2 , . . ., where si ∈ Di ∩N [ri ], such that Di+1 = (Di −{si })∪{ri }. Note that si = ri is possible. The set Di+1 is the set of locations of the guards after the attack at ri is defended. If si 6= ri , we will say that the guard at si has moved to ri . The minimum cardinality among all eternal dominating sets is the eternal domination number γ ∞ (G). As pointed out in [1, 18, 36], it is more general to model this problem as a two-player game: the defender chooses D1 and the vertices s1 , s2 , . . . while the (adaptive) adversary chooses the vertices r1 , r2 , . . . (the defender chooses si to defend the attack the adversary makes at ri .) In other words, the location of an attack can be chosen by the adversary depending on the location of the guards. Hence this is an adaptive online problem, whereas the original definition from [6] used in the previous paragraph can be thought of as an oblivious online problem. As far as we know, all proofs in the literature can be applied to the more general adaptive online problem. Hence we shall assume that model from this point forward. The clique covering number θ(G) is the minimum number, k, of sets in a partition V = V1 ∪ · · · ∪ Vk of V such that the subgraph of G induced by each Vi is complete, i.e., θ(G) is equal to the chromatic number of the complement G of G. We denote the independence number of G by α(G). Burger et al. [6] and Goddard et al. [17] showed the following fundamental bounds. Fact 4 For all graphs G, α(G) ≤ γ ∞ (G) ≤ θ(G). Proof: To see the leftmost inequality, imagine a sequence of consecutive attacks at independent vertices. To see the rightmost inequality, observe that a single guard can defend all vertices of a clique. 2

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Goddard, Hedetniemi, and Hedetniemi [17] asked whether the eternal domination number can be bounded by a constant times the independence number and gave a proof for the case when α = 2. One of the main results on eternal domination is the following upper bound, due to Klostermeyer and MacGillivray [25]. Theorem 5 [25] For any graph G, γ ∞ (G) ≤

!

α(G) + 1 . 2

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Proof: Assume |V | > α+1 2 . The proof is essentially algorithmic. Consider disjoint independent sets Sα , Sα−1 , . . . , S1 , where Sα is a maximum independent set of G and, for t = α − 1, α − 2, . . . 1, the set St is either empty or an independent set of size t. Other than Sα , each St is not necessarily a maximal independent set. Among all collections of such
sets, we choose one such that | ∪αt=1 St | is maximum. Since |V | > α+1 , the set S 6= ∅. 1 2 Let Di−1 = ∪αt=1 St , for some i ≥ 1, where Di is as defined in the definition of eternal domination above. Suppose the ith attack ri is at vertex v. If there is a guard at v, then the attack is defended by the guard located at v. Otherwise, a guard from the set Sj with the smallest subscript among those with a vertex adjacent to v moves to v. Note that such a set Sj exists because Sα is a dominating set (as it is a maximum independent set). The key technical bit of the proof is to show that Di = (Di−1 \ {v}) ∪ {ri } can be partitioned into disjoint independent sets. If v ∈ Di−1 then Di = Di−1 and the statement is true in this case. If v 6∈ Di−1 , then a guard at g ∈ Sj moves to v. There are two cases. If (Sj − {g}) ∪ {v} is an independent set, then replacing Sj by (Sj − {g}) ∪ {v} yields another collection of disjoint independent sets as desired. Otherwise, v is adjacent to at least two vertices in Sj , j > 1. Let k be the greatest integer less than j such that Sk is not empty. It must be that k = j − 1; otherwise the fact that Sk ∪ {v} is independent (by the definition of j, no vertex in Sk is adjacent to v) contradicts the maximality of |∪αk=1 Sk |. Replacing Sj by Sj−1 ∪{v} and Sj−1 by Sj −{g} gives another collection of independent sets with the desired same maximality properties. 2 It was shown in [18] that this bound is sharp for certain graphs. Specifically, let G(n, k) be the graph with vertex set equal to the set of all k-subsets of an n-set and where two vertices are adjacent if and only if their intersection is nonempty (so G(n, k) is the complement of a Kneser graph). Theorem 6 [18] For each positive integer t, if
k is sufficiently large, then the graph G(kt+ k − 1, k) has eternal domination number t+1 2 . Subsequently, Regan [36] showed another graph for which the bound is sharp: the circulant graph C22 [1, 2, 4, 5, 9, 11]. A main problem is to demonstrate classes of graphs for which either γ ∞ (G) = α(G) or γ ∞ (G) = θ(G). Of course, it is the case that γ ∞ (G) = α(G) = θ(G) for all perfect graphs, since the clique-size equals the chromatic number for all perfect graphs and because the weak Perfect Graph Theorem shows that the complement of a perfect graph is perfect. Before proceeding, we should point out that the independence number, eternal domination number, and clique-covering number can vary widely.

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Proposition 7 [27] Let c and d be positive integers. Then there exists a connected graph G with α(G) + c < γ ∞ (G) and γ ∞ (G) + d < θ(G). Let Cnk be the k th power of the cycle on n vertices. We assume that 2k + 1 < n, when referring to Cnk or its complement. Theorem 8 Let G be a graph in one of the following classes. Then γ ∞ (G) = θ(G). (1) [27] Cnk , for all k ≥ 1, n ≥ 3 (2) [27] Cnk , for all k ≥ 1, n ≥ 3. (3) [36] Circular-arc graphs. (4) [1] K4 -minor free graphs. (5) Perfect graphs. (6) [1] Cm × Cn ; Pm × Cn . Proof Sketch of (3): Let G be a minimum counterexample. Note that G cannot contain vertices, u, v such that N [u] ⊆ N [v], else one could find a smaller counterexample. Considering the possible edge sets of G, it is either than case that G = Cnk , or there must exist u, v such that N [u] ⊆ N [v]. Proof Sketch of (4): Suppose G is K4 -minor-free and γ ∞ (G) < θ(G). One can show that G must be 2-connected. One can further show that G contains a cycle C with exactly two vertices, u, v of degree greater than two. The proof proceeds by cases depending on the lengths of the two paths between u and v on the cycle. The analysis of the cases in technical and we refer the reader to [1]. 2

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The following is one of the main questions. Question 1 Is γ ∞ (G) = θ(G) for all planar graphs G? One might ask the weaker question: for which k ≥ 2 do all k-outerplanar graphs have γ ∞ (G) = θ(G)? A k-outerplanar graph is a planar graph such that, given a planar embedding of the graph, deletion of the vertices on the exterior face leaves a (k − 1)outerplanar graph. Note that γ ∞ (G) = θ(G) for all outerplanar graphs G (equal to 1outerplanar graphs), since outerplanar graphs are series-parallel. We can characterize which graphs have eternal domination number equal to domination number. Theorem 9 [27] γ(G) = γ ∞ (G) if and only if γ(G) = θ(G). It remains to characterize which graphs have eternal domination number equal to their independence number. It was shown in [5] that α(G) = γ ∞ (G) in complete multi-partite graphs. The complexity of the eternal security problem was studied in [24], where it is shown to be N P -complete to decide if a set of vertices is an eternal dominating set in the offline version on the problem (that is, there are m requests that are known in advance) and the online oblivious version of the problem is shown to be complete for co − N P N P .

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Klostermeyer and Mynhardt give a number of results on eternal connected domination and eternal total domination [29]. Eternal connected and eternal total dominating sets are defined in the obvious manner. For example, in the case of eternal total dominating sets, one must maintain a total dominating set eternally while moving at most one guard to a neighboring vertex in defense of each attack. For instance, they show that the eternal total domination number is greater that the eternal domination number for all graphs. An interesting question from that paper asks whether the smallest eternal connected domination set must always be larger than the clique-covering number of a graph.

4.2.

Eternal m-dominating Sets

An eternal m-dominating set is defined similarly to an eternal dominating set, except that when an attack occurs, each guard is allowed to move to a neighboring vertex to either defend the attack there or better position themselves for the future. We will refer to this sometimes as the “all guards move” model (though of course, it is not required that each guard move upon each attack). The size of a smallest eternal m-dominating set in G is the ∞ (G) or simply γ ∞ . eternal m-domination number, and is denoted by γm m It is an open question whether, in this model, there is any advantage in allowing two guards to occupy the same vertex (there is no advantage allowing multiple guards to occupy a single vertex in the “one guard moves” model [6]). All known results apply to the restricted case when only one guard is allowed to occupy each vertex. ∞ (G) exactly for complete graphs, paths, cycles, and Goddard et al. [17] determine γm ∞ (G) = γ(G) for all Cayley graphs G. complete bipartite graphs. Further, they show that γm They also proved the following fundamental bound.

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∞ (G) ≤ α(G). Theorem 10 [17] For all graphs G, γ(G) ≤ γm

Proof: We include the proof of Burger, Hedetniemi, and Hedetniemi, as it is quite simple and elegant. The left inequality is obvious. If there is vertex v not contained in an maximum independent set, then delete the closed neighborhood of v and proceed by induction. Otherwise, let Mv denote a maximum independent set containing v. Defend an attack on v occurs by moving all guards to Mv from Mu (the previous configuration of guards). This is possible since Hall’s Marriage Theorem ensures there is a matching between Mv and Mu . 2 The upper bound in the previous theorem is not very tight in general. For example, K1,m has independence number m and can be defended with just two guards in this model. ∞ (G), γ ∞ (G) = γ ∞ (G), or Problem 1 Describe classes of graphs having γ(G) = γm m ∞ (G) = α(G). γm

In order to get a better upper bound, we define a neo-colonization to be a partition {V1 , V2 , . . . , Vt } of G such that each Vi induces a connected graph. A part Vi is assigned weight one if it induces a clique and 1 + γc (G[Vi ]), otherwise, where γc (G[Vi ]) is the size of the smallest connected dominating set in the subgraph induced by Vi . [A connected dominating set of G is a set D such that D is a dominating set and G[D] is connected]. Then θc (G) is the minimum weight of any neo-colonization of G. Goddard et al. [17]

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proved that γm (G) ≤ θc (G) ≤ γc (G) + 1. Klostermeyer and MacGillirvray proved the following and also gave a linear-time algorithm for the eternal m-domination number of a tree [27]. ∞ (T ). Theorem 11 Let T be a tree. Then θc (T ) = γm

It remains to characterize which graphs have eternal m-domination number equal to independence number. It also seems that the eternal m-domination number could be studied for various classes of graphs. ∞ (P × P ). Problem 2 Determine the value of γm n m

A different upper bound was proved in [8]. ∞ (G) ≤ ⌈ n ⌉. Theorem 12 [8] Let G be a graph with no isolated vertices. Then γm 2

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Proof: By induction on n. Let k = ⌈ n2 ⌉ and let v be a vertex. If degree(v) ≤ 2, the induction is easy to see. Otherwise, apply the inductive hypothesis to G − N (v), which can be defended with k − 1 guards. We use a single guard to defend the neighbors of v, pulling an additional guard from the rest of the graph to use on v if necessary. That is, initially locate a guard on v and k − 1 guards in G − N (v) − v. If an attack occurs on w ∈ N (v), move the guard from v to w and a guard from G − N (v) − v to v. An attack at a different neighbor of v is handled by moving the guard from w to v and the guard from v to the attack. An attack at a vertex not in N (v) causes us to move guards to that there is one guard on v and none on the neighbors of v. 2 The bound in Theorem 12 can be improved to ⌈ n2 ⌉ − 1 when δ(G) ≥ 2 (with four small exceptions) [8]. Nordaus-Gaddum results were also shown in [8], for example the following. ∞ (G) + γ ∞ (G) ≤ n + 1. Theorem 13 [8] γm m

Chambers et al. [8] also characterize the special graphs for which equality holds in Theorem 13. Klostermeyer and Mynhardt give a number of results on eternal connected domination and eternal total domination [29] in the “all-guards move” model.

5.

Other Types of Eternal Domination

Klostermeyer and Mynhardt prove a number of results on eternal connected domination and eternal total domination [29]. Note in the definition of eternal domination, the decision of which guard to send to defend an attack may require omnipotence! That is, the definition says “there exists” a guard to send to defend the attack such that the configuration of guards can defend all subsequent attacks. It is likely difficult in practice to decide, or compute, which guard to send to defend an attack. Burger et al. [6] consider the “foolproof” variation on eternal domination in which the resulting configuration of guards must be able to defend all subsequent attacks if

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a guard from any vertex adjacent to the attacked vertex is sent to defend the attack (assuming no guard is located at the attacked vertex). That is, no matter which guard is sent, the resulting configuration will be able to defend all future attacks (i.e., the resulting configurations will always be dominating sets). They proved that n − δ(G) guards are necessary and sufficient for all graphs G, where δ(G) is the minimum vertex degree in the graph. To see this, note that any set of n − δ(G) vertices form a dominating set. On the other hand, if we have fewer guards than n − δ(G) in a graph, then by a series of attacks, an adversary can force the closed neighborhood of a vertex to contain to guards. For example, consider C6 , and observe that γ ∞ (C6 ) = 3. Now suppose we could defend the graph with three guards in the foolproof model. Since we must be able to defend an attack with any neighboring guard, an adversary can force the three guards to migrate to three consecutive vertices, thereby leaving a vertex undefended. The previous problem has not been studied in the “all guards move” model.

6.

Secure Domination

A secure dominating set (SDS) must be able to defend a graph against two attacks. That is, D is a secure dominating set if for each r1 ∈ V there exists a dominating set D1 and a vertex s1 ∈ D ∩ N [r1 ], such that D1 = (D − {s1 }) ∪ {r1 }. The minimum cardinality amongst all SDSs is the secure domination number γs (G) of G. This concept was first defined in [15] the following basic inequality proved.

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Fact 14 [15] For all graphs G, γ(G) ≤ γs (G) ≤ 2γ(G). P2 and P3 are examples where the left and right inequalities, respectively, are sharp. It was shown in [13] that if G is claw-free, then γs (G) ≤ 3α(G)/2, and if, in addition, G is C5 -free, then γs (G) ≤ α(G). Klostermeyer and Mynhardt proved that γs (G) ≤ 2α(G), but posed the following question [28]. Question 2 Does there exist a graph G with γs (G) = 2α(G)? 7mk Cockayne et al. [15] showed that γs (Pm × Pk ) ≤ ⌈ mk 3 ⌉ + 2 and that 23 ≤ γs (Cm × Ck ) ≤ ⌈ mk 3 ⌉. They also proved that for all triangle-free graphs with maximum degree ∆, γs ≥ n(2∆ − 1)/(∆2 + 2∆ − 1) and that there exist graphs attaining the bound for each ∆. Cockayne improved this lower bound for trees with ∆ ≥ 3 to (∆n + ∆ − 1)/(3∆ − 1) and showed it is sharp for some trees [11].

Question 3 [17] Is γs (Cm × Cn ) = ⌈ mn 4 ⌉ for all m, n ≥ 4? The notion of a secure domination set was generalized in [5] to allow k > 1 attacks, where k is finite. This was called higher order domination and is also studied in [2]. Critical secure dominating sets are studied in [19]. Secure total dominating sets have been studied in [3, 28]. A secure total dominating set is a total dominating set such that if any vertex is attacked, we can move a guard to the attacked vertex and the resulting configuration of guards induces a total dominating set. It was proved in [28] that for all graphs G, the size of a smallest secure total dominating set is at most 3α(G) − 1 and they make the following interesting conjecture.

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Conjecture 4 [28] For all graphs G, the size of a smallest secure total dominating set is at most 2α(G).

7.

Roman Domination

The term Roman domination stems from the problem’s ancient origins in Emperor Constantine’s efforts to defend the Roman Empire from attackers [22, 38]. There are two variations of Roman domination that we consider in the next two subsections.

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7.1.

Roman Domination

From [12] “A Roman dominating function on a graph G=(V, E) is a function f : V → {0, 1, 2} satisfying the condition that every vertex u for which f (u) = 0 is adjacent to at least one vertex v for which f (v) = 2. The weight of a Roman dominating function is the P value f (V ) = u∈V f (u). The minimum weight of a Roman dominating function on a graph G is called the Roman domination number of G.” We denote the Roman domination number of G by γR (G). We think of f as specifying the number of guards located at each vertex. The concept was initially defined by Cockayne, Dreyer, and Hedetniemi [12] and also less formally in [37, 38]. It is easy to prove that for all graphs G, γ(G) ≤ γR (G) ≤ 2γ(G) cf. [15]. One can also show that γR (G) ≥ 2n/(∆(G) + 1), if ∆(G) ≥ 1 and this bound sharp for K1,n [15]. It is not hard to prove that γR (G) = γ(G) if and only if G = Kn [12]. To see this, observe that γR (G) = γ(G) = n. For the other direction, since the domination and Roman domination numbers are equal, it must be that f (v) = 1 for exactly γ(G) vertices. This means that no vertex v has f (v) = 2 and thus there can be no vertices have weight zero. It follows that G = Kn [12]. Cockayne et al. also characterize graphs with γR (G) = γ(G)+1 and γR (G) = γ(G)+ 2 [12]. For example, for connected graph G, γR (G) = γ(G) + 1 if and only if G contains a vertex of degree n − γ(G). On the other hand, there is not yet a complete characterization of graphs G having γR (G) = 2γ(G), so called Roman graphs. Henning characterized trees with this property [21] and Cockayne et al. give a partial characterization. They also show that such graphs must contain no vertices v with f (v) = 1. Theorem 15 [12] γR (G) = min{2γ(G − S) + |S| : S is a 2-packing}. It is tempting to conjecture that any graph G containing a perfect code is a Roman graph. This is because we can place two guards on each vertex of a perfect code. However, the conjecture is not true as the following graph illustrates. Take a K1,5 , let w be the vertex of degree five and let v be a vertex of degree one. To v attach a vertex x and to x attach a vertex y. Now add edges between w and each of the other four vertices adjacent to w. Note that {w, y} is a perfect code. However, letting f (x) = 2 and f (w) = 1 produces a Roman dominating set of weight three. Problem 3 Characterize which graphs having perfect codes are Roman graphs.

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Cockayne et al. [12] suggest an a line of future research considering the restriction of Roman domination in which the vertices v having f (v) ≥ 1 must induce an independent set. Prince [35] gives an interesting characterization of the Roman domination number in random graphs; showing in one probability model (generating a type of sparse graph) that the Roman domination number of a generated graph G is (1 + o(1))γ(G) with high probability, whereas in another probability model (generating a type of dense graph) the Roman domination number of a generated graph G is (2 − o(1))γ(G) with high probability. Chambers et al. proved the following bounds [7]. Theorem 16 [7] Let G be a connected graph. Then (1) γR (G) ≤ 4n/5. (2) γR (G) ≤ 8n/11 if δ(G) ≥ 2 and n ≥ 9. (3) γR (G)γR ((G) ≤ 16n/5. Chambers et al. also characterize the graphs having γR (G) = 4n/5 and show examples where (2) is sharp. The latter graphs are either C5 or n/5 P5 ’s connected via a path joining all the “centers” of the P5 ’s. They go on to make the following very appealing conjecture. Conjecture 5 [7] If G is a 2-connected graph, then γR (G) ≤ ⌈ 2n 3 ⌉. Cn is an example where the conjectured bound is sharp. Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

We next consider the Roman domination number in cartesian products of graphs. n n Theorem 17 [15] (1) γR (Pn × Pm ) ≤ 2(⌈ nm 5 ⌉ + ⌈ 5 ⌉ + ⌈ 5 ⌉) 2nm nm n n (2) ⌈ 5 ⌉ ≤ γR (Cn × Cm ) ≤ 2(⌈ 5 ⌉ + ⌈ 5 ⌉ + ⌈ 5 ⌉) (3) If 2 ≤ m ≤ n, γR (Kn × Km ) = 2m − 1 if m = n and γR (Kn × Km ) = 2m if m < n.

Precise values for γR (Pn × Pm ) for some values of m, n are given in [16]. Cockayne et al. [12] use the probabilistic method to prove a result similar to the one stated in Section 3 for dominating sets. . Theorem 18 Let G be a graph with minimum degree δ. Then γR (G) ≤ n 2+ln((δ+1)/2) δ+1 Some complexity results, including hardness of approximation results, on Roman domination and some variations of Roman domination can be found in [16, 34]. For example, a polynomial-time approximation algorithm for Roman domination is known with performance ration (2 + ln n) but none is possible with performance ratio less than or equal to c log n for any constant c unless P = N P . Better approximation results are shown in [34] for planar graphs.

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7.2.

William F. Klostermeyer

Weak Roman Domination

A weak Roman dominating function on a graph G=(V, E) is a function f : V → {0, 1, 2} (specifying the number of guards located at each vertex) satisfying the following. For each vertex u such that f (u) = 0, there exists a vertex v adjacent to u such that a guard at v can move to u and the resulting configuration of guards is “safe.” By safe, we mean that each vertex has at least one guard in its closed neighborhood. The weight of a weak Roman P dominating function is the value f (V ) = u∈V f (u). We denote the minimum value of a weak Roman domination number of G by γr (G). Weak Roman domination was originally defined by Cockayne et al. [15]. They proved the following fundamental bounds. Fact 19 [15] [23] For all graphs G, γ(G) ≤ γr (G) ≤ γs (G) ≤ 2γ(G).

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Fact 20 [15] [23] For all graphs G, γ(G) ≤ γr (G) ≤ γR (G) ≤ 2γ(G). Hence upper bounds given above for secure dominating sets are also upper bounds for weak Roman dominating sets. Though they have the same value for many graphs, we can separate γr and γs : γr (K1,n ) = 2 and γs (K1,n ) = n. Likewise, γr (Kn ) = 1 < γR (Kn ) = 2n 2 and γr (Cn ) = γs (Cn ) = ⌈ 3n 7 ⌉ < γR (Cn ) = ⌈ 3 ⌉. Henning and Hedetniemi [23] characterized the graphs G having γr (G) = γ(G) and the forests G having γr (G) = 2γ(G). They also showed it is N P -complete to decide if the weak Roman domination number of a bipartite or a chordal graph is at most k. Whether it is N P -complete for chordal bipartite graphs seems open. They further showed that the weak Roman domination number can be computed in polynomial time in trees. We repeat the question of Goddard, Hedetniemi, and Hedetniemi from above, but in the context of weak Roman domination. Question 6 [17] Is γr (Cm × Cn ) = ⌈ mn 4 ⌉ for all m, n ≥ 4? Finally, it would be of interest to further separate the parameter γr from γR and γs . Are there other nice classes of graphs for which the parameters are equal? Are there other classes of graphs for which the parameters are not equal? Which graphs have γr = 2γ?

8.

Conclusion

We have given an overview of several types of protection schemes for graphs, including stating a number of important open problems and conjectures. As many of these problems are fairly new to the literature, we have attempted to give a survey of the types of results that might be obtained by researchers in the future. It is clear however that there are also many other variations on these problems that could be considered. Two such examples include the problem of maintaining an independent set eternally, a problem defined in [36], and restricted versions of the eternal m-dominating set problem in which only a constant number of guards can move per request.

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References [1] M. Anderson, C. Barrientos, R. Brigham, J. Carrington, R. Vitray, and J. Yellen, Maximum demand graphs for eternal security, J. Combin. Math. Combin. Comput. vol. 61 (2007), 111-128. [2] S. Benecke, Higher Order Domination of Graphs, Master’s Thesis, University of Stellenbosch, 2004. [3] S. Benecke, E.J. Cockayne and C.M. Mynhardt, Secure total domination in graphs, Utilitas Math., to appear. [4] S. Benecke, P.J.P. Grobler and J.H. van Vuuren, Protection of complete multipartite graphs, Utilitas Math. 71 (2006), 161-168 [5] A.P. Burger, E.J. Cockayne, W.R. Gr¨undlingh, C.M. Mynhardt, J.H. van Vuuren and W. Winterbach, Finite order domination in graphs, J. Combin. Math. Combin. Comput., vol. 49 (2004), 159-175. [6] A.P. Burger, E.J. Cockayne, W.R. Gr¨undlingh, C.M. Mynhardt, J.H. van Vuuren and W. Winterbach, Infinite order domination in graphs, J. Combin. Math. Combin. Comput., vol. 50 (2004), 179-194. [7] E. Chambers, W. Kinnersly, N. Prince, and D. West, Extremal problems for Roman domination, manuscript (2008).

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[8] E. Chambers, W. Kinnersly, and N. Prince, Mobile eternal security in graphs, manuscript (2008). [9] M. Chrobak, H. Karloff, T. Payne, and S. Vishwanathan, New results on server problems, SIAM Journal on Discrete Mathematics, vol. 4 (1991), pp. 172-181 [10] V. Chv´atal, A combinatorial theorem in plane geometry, Journal of Combinatorial Theory, vol. 18 (1975), pp. 39-41 [11] E.J. Cockayne, Irredundance, secure domination and maximum degree in trees, Discrete Math., vol. 307 (2007), 12-17. [12] E.J. Cockayne, P.A. Dreyer, S.M. Hedetniemi and S.T. Hedetniemi, Roman domination in graphs, Discrete Math., vol. 278 (2004), 11-12. [13] E.J. Cockayne, O. Favaron and C.M. Mynhardt, Secure domination, weak Roman domination and forbidden subgraphs, Bull. Inst. Combin. Appl. vol. 39 (2003), 87100. [14] E.J. Cockayne, O. Favaron and C.M. Mynhardt, Total domination in Kr -covered graphs, Ars Combin. , vol. 71 (2004), 289-303. [15] E.J. Cockayne, P.J.P. Grobler, W.R. Gr¨undlingh, J. Munganga and J.H. van Vuuren, Protection of a graph, Utilitas Math., vol. 67 (2005), 19-32. Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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[16] P. Dreyer, Applications and variations of domination in graphs, PhD thesis, Rutgers University, 2000 [17] W. Goddard, S.M. Hedetniemi and S.T. Hedetniemi, Eternal security in graphs, J. Combin. Math. Combin. Comput., vol. 52 (2005), 169–180. [18] J. Goldwasser and W.F. Klostermeyer, Tight bounds for eternal dominating sets in graphs, Discrete Math., vol 308 (2008), pp. 2589-2593. [19] P.J.P. Grobler and C.M. Mynhardt, Secure domination critical graphs, manuscript, 2008. [20] T. W. Haynes, S. T. Hedetniemi and P. J. Slater, Fundamentals of Domination in Graphs. Marcel Dekker, New York, 1998. [21] M.A. Henning, A characterization of Roman trees, Discussiones Math. Graph Theory, vol. 22 (2002), 225-234. [22] M.A. Henning, Defending the Roman Empire from multiple attacks, Discrete Math., vol. 271 (2003), 101-115. [23] M.A. Henning and S.T. Hedetniemi, Defending the Roman Empire – A new strategy, Discrete Math., vol. 266 (2003), 239-251. [24] W. F. Klostermeyer, Complexity of Eternal Security, J. Comb. Math. Comb. Comput., vol. 61 (2007), pp. 135-141

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[25] W.F. Klostermeyer and G. MacGillivray, Eternal security in graphs of fixed independence number, J. Combin. Math. Combin. Comput., vol. 63 (2007), 97-101. [26] W. F. Klostermeyer and G. MacGillivray, Eternally Secure Sets, Independence Sets, and Cliques, AKCE International Journal of Graphs and Combinatorics, vol. 2 (2005), pp.119-122 [27] W.F. Klostermeyer and G. MacGillivray, Eternal dominating sets in graphs, J. Combin. Math. Combin. Comput., to appear, 2008. [28] W.F. Klostermeyer and C.M. Mynhardt, Secure domination and secure total domination in graphs, to appear in Discussiones Mathematicae Graph Theory (2008). [29] W.F. Klostermeyer and C.M. Mynhardt, Eternal total domination in graphs, submitted. [30] E. Koutsoupias and C. Papadimitriou, On the k-server conjecture, J. ACM, vol. 42 (1995), pp. 971-983. [31] M. Manasse, L. McGeoch, and D. Sleator, Competitive algorithms for server problems, Journal of Algorithms, vol. 11 (1990), pp. 208-230. [32] L. McGeoch and D. Sleator, A strongly competitive randomized paging algorithm for server, Algorithmica, vol. 6 (1991), pp. 816-825. Emerging Topics on Differential Geometry and Graph Theory, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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[33] C.M. Mynhardt, H.C. Swart and E. Ungerer, Excellent trees and secure domination, Utilitas Math., vol. 67 (2005), 255-267. [34] A. Pagourtzis, P. Penna, K. Schlude, K. Steinh¨oofel, D. Taylor, P. Widmayer, Server placements, Roman domination and other dominating set Variants, Proceedings 2002 IFIP Conference on Theoretical Computer Science, Montreal, Canada, pp. 280-291. [35] N. Prince, Thresholds for Roman domination, manuscript, 2008. [36] F. Regan, Dynamic variants of domination and independence in graphs, graduate thesis, Rheinischen Friedrich-Wilhlems University, Bonn, 2007. [37] C. S. ReVelle and K.E. Rosing, Defendens imperium romanum: a classical problem in military strategy, Amer. Math. Monthly, vol. 107 (2000), pp. 585-594

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[38] I. Stewart, Defend the Roman Empire! Scientific American, December 1999, 136-138.

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INDEX A

C

accuracy, 10, 330, 334 ACM, 184, 185, 295, 297, 393, 408 agent, 53, 56, 58, 63 agents, 56 algebraic geometry, 74 algorithm, 136, 137, 139, 141, 142, 144, 145, 146, 147, 164, 179, 182, 228, 236, 239, 241, 248, 249, 250, 266, 268, 270, 271, 274, 294, 295, 296, 297, 396, 397, 402, 405, 408 amplitude, 333 application, vii, 56, 58, 73, 94, 119, 123, 128, 224, 229, 241, 257, 276, 300, 313, 338, 348, 379, 380, 390 arithmetic, 40, 335 articulation, 229, 231, 253, 266 ASD, 142 assets, 266, 286, 289 assignment, 233, 234, 239, 273, 296, 358 asymptotic, x, 208, 209, 214, 219, 299, 300, 322, 329, 338 asymptotically, 244, 325, 326 asynchronous, 373, 374, 377, 383, 389 asynchronous communication, 373 atoms, 144, 145

C++, 249, 294 calculus, vii, 35, 36, 37, 39, 42, 45, 50, 56, 62, 72, 374, 393 capacity, 226, 227, 240, 244, 254 capital accumulation, 282 capital consumption, 277, 289 capital gains, 289 cash flow, 289 categorization, 396 causality, 274 cell, 97, 98, 100, 102, 103, 112, 113, 114, 115, 120, 129 channels, x, 137, 373, 374, 375 circular flow, 257, 259 classes, 97, 105, 106, 107, 116, 121, 126, 127, 130, 234, 248, 327, 399, 400, 401, 402, 406 classical, viii, 36, 37, 50, 51, 53, 56, 57, 58, 59, 62, 65, 66, 72, 73, 95, 96, 135, 137, 165, 409 closure, 132, 308, 311, 331 clusters, 188, 189 coagulation, ix, 187, 190, 194, 195, 196, 198, 201, 206, 208 coagulation process, ix, 187, 190, 195, 196 codes, x, 34, 373, 404 coding, viii, 135, 137, 145, 183 combinatorics, 33, 96, 133, 395 communication, x, 37, 181, 182, 373, 374, 375, 378, 383, 389, 390, 393 commutativity, 99 compatibility, 51, 54, 66 compensation, 277, 286 compilation, x, 373, 374, 375, 389, 390, 391, 393 complement, viii, 95, 104, 105, 106, 108, 111, 112, 117, 122, 125, 126, 130, 398, 399, 400 complex numbers, 302, 306 complex systems, 294 complexity, 142, 241, 400, 405 components, viii, ix, 105, 168, 183, 187, 188, 189, 190, 191, 192, 194, 197, 198, 200, 201, 202, 203, 205, 206, 208, 209, 226, 227, 231, 234, 241, 253, 255, 279 composites, 60

B Banach spaces, 321, 330, 334 benefits, 36, 50, 280, 281, 292 bisimulations, 386 Bohr, 66, 76, 77 bonds, 144, 145, 277 Boolean algebras, 65 borrowing, 289 Bose, 194 Boston, 76, 340 bounded solution, 327 bounds, 148, 149, 165, 180, 181, 397, 398, 405, 406, 408

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Index

composition, 2, 42, 46, 52, 55, 72, 81, 98, 100, 103, 104, 113, 207, 216, 363, 365, 380, 385, 393 compounds, viii, 135, 136, 137, 144, 145, 183 computation, viii, 92, 95, 96, 112, 113, 116, 127, 224, 228, 257, 273, 284, 373, 391 computer science, vii, viii, 3, 145, 152, 164, 181, 183 computing, 2, 112, 127, 129, 227 conceptualization, 36, 57, 58, 61, 73 configuration, x, 95, 96, 108, 109, 395, 398, 401, 402, 403, 406 congruence, 118, 374, 375, 383, 385, 386, 388, 393 conjecture, viii, 135, 136, 143, 180, 181, 396, 397, 403, 404, 405, 408 connectivity, ix, 58, 62, 71, 72, 73, 223, 224, 225, 232, 240, 241, 244, 255 constraints, 37, 45, 56, 273 construction, x, 36, 37, 45, 55, 56, 59, 66, 94, 96, 97, 98, 108, 182, 223, 345 consumer expenditure, 277, 289 consumption, 226, 242, 243, 277, 283, 286, 287, 288, 289, 290, 291 consumption function, 242 convergence, ix, x, 299, 300, 311, 320, 330, 331, 333, 334, 335, 336, 338 convex, 79, 80, 81, 85, 88, 91, 94, 132, 307 correlation, 244, 245, 2545, 255 CRC, 33, 133, 294, 295, 296 critical value, 207 currency, 289, 290 cycles, ix, 138, 140, 142, 153, 164, 187, 190, 197, 206, 231, 241, 248, 250, 276, 283, 297, 401 cycling, 194

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D data structure, 181 D-branes, 79 decomposition, vii, 1, 2, 3, 22, 28, 29, 31, 32, 33, 83, 89, 105, 113, 117, 118, 122, 123, 125, 127 defects, vii, 35 defense, 401 deficiency, 198, 235 deficits, 207 deformation, vii, viii, 59, 79, 80, 82, 83, 84, 88, 89, 92, 93, 94 degrees of freedom, 51 demand, 227, 237, 243, 259, 276, 286, 290, 407 density, 281 deposits, 256, 266, 290 depreciation, 288 derivatives, ix, 82, 83, 207, 211, 212, 219, 233, 299, 300, 310, 313, 322, 328, 338 detection, vii, 1, 3, 11, 14, 17, 33, 34, 183 deviation, 289 differentiation, 310, 313 diffusion, 330, 331 dipole, 323, 329 discretization, 300 disposable income, 292

distortions, 332 distribution, 188, 189, 191, 201, 208, 257, 261, 281, 315 dividends, 289 division, 65 DPO, 393 drug design, 183 durable goods, 289, 290

E earnings, 232, 292 economic policy, 251 economic theory, 227, 233 employee compensation, 277 employment, 58, 226, 232, 250, 289 encoding, 56, 137 energy-momentum, 55 environment, 60, 61, 66 equality, 10, 17, 43, 91, 116, 117, 118, 119, 125, 126, 163, 211, 212, 305, 316, 336, 362, 363, 402 equilibrium, 237, 238 estimating, 148 estimators, 244 Euclidean space, 104, 300 evolution, viii, ix, 56, 57, 182, 187, 190, 192, 193, 194, 195, 196, 197, 199, 202, 206, 207, 208, 238, 322 execution, 376, 377, 383, 390, 392, 393 exercise, 96, 103, 106, 123, 124 expansions, 216, 319 expenditures, 227, 238, 256, 258, 266, 289 exports, 291 external magnetic fields, 144 extrusion, 375

F fiber, 57, 59, 123, 124, 130 finance, 259 firms, 259, 280, 286, 289, 290, 291, 292 fixation, 37 flow, 80, 85, 88, 94, 241, 259, 270, 289, 322, 323, 326, 328, 329, 330, 331, 332, 334, 335, 396 fluctuations, 238 fluid, 322, 325, 330, 334 forecasting, 238, 276 Fortran, 249, 276, 294 Fourier, ix, 310, 312, 313, 328, 331, 332, 339, 340 fractional differentiation, 310, 313

G gauge, 64, 65 gelation, 198, 208 General Relativity, 36, 37, 50, 51, 53, 54, 55, 56, 59, 65 generation, vii, 2, 34, 35, 36, 37, 50, 53, 55, 56, 59, 62, 66, 67, 72, 73 generators, 118, 137, 146, 148, 152, 165, 168, 180, 183

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genes, 182 genome, 182, 183 genomes, 182, 183 girth, 138, 139, 143, 144, 184 goals, x, 395 goods and services, 276, 291 government, 237, 243, 289, 290, 291, 292 government expenditure, 237 gravitational field, 37, 51, 53, 54, 55 gravity, vii, 35, 36, 57, 74, 75, 76 grids, 397 gross domestic product, 289, 293 gross national product, 286, 288, 289, 293 groups, viii, 2, 25, 29, 39, 40, 96, 97, 132, 135, 137, 145, 148, 168, 180, 183, 184, 196, 345 growth, ix, 187, 190, 209, 237, 239, 246, 330, 333 growth rate, 330

independence, x, 36, 55, 75, 395, 397, 398, 399, 400, 401, 402, 408, 409 indicators, 51, 255 indices, 233, 304 induction, 240, 388, 389, 401, 402 industrial, 226 industry, 226 inequality, 123, 124, 152, 153, 163, 164, 167, 168, 179, 225, 236, 273, 305, 315, 316, 317, 318, 326, 327, 328, 329, 336, 337, 338, 398, 401, 403 inferences, 374 infinite, ix, 187, 190, 209, 313, 330, 393 inflation, 237, 238, 277, 284 inherited, 104, 117 injection, 71, 308, 348, 351, 356 insight, 182 instability, 300 insurance, 291, 292 H integration, ix, 199, 212, 299, 300, 338 intensity, 242, 286 Hamiltonian, vii, viii, 79, 80, 81, 83, 84, 85, 87, 88, 89, intentions, 190 91, 93, 94 interaction, 53, 58, 63, 253 harmonics, ix, 299, 300, 301, 302, 303, 307, 310, 312, interactions, 51, 53, 56, 57, 59, 63, 65, 71, 227, 282 329, 331, 338 interconnection networks, 152, 182, 183 Heisenberg, 65, 74 interdisciplinary, 33, 144 heuristic, 253 interest rates, 237, 238, 280 Hilbert Space, ix, 65, 299, 300, 301, 303, 305, 307, Internet, 188 309, 311, 313, 314, 315, 317, 319, 321, 323, 325, interpretation, 53, 57, 65, 66, 74, 238 327, 329, 331, 333, 335, 337, 338, 339, 341, 343 interval, 149, 154, 155, 171, 199, 246, 303, 308 homology, 95, 97, 105, 117, 123, 125, 126, 127, 129, intrinsic, 51 131 intuition, 244 homomorphism, 52, 99, 101, 104, 110, 360, 361 invariants, 11, 17, 228 House, 34 inventories, 292 household, 258, 266 inversions, 34, 235 households, 256, 289, 290, 291, 292 investment, 242, 243, 256, 277, 280, 281, 283, 290, housing, 226, 290 291 H-space, 104 ions, 75, 96, 291, 393 human, 182, 188 IS-LM, 276 human brain, 188 isomorphism, vii, 1, 3, 9, 10, 11, 12, 14, 16, 17, 18, 20, Hungarian, 239, 296 33, 34, 40, 41, 43, 44, 52, 53, 84, 87, 97, 100, 105, hydrodynamic, 330 106, 110, 112, 113, 117, 118, 126, 127, 135, 136, hydrogen, 144 138, 144, 308, 310, 311, 316, 356 hydrogen atoms, 144 iteration, 268, 269 hyperinflation, 238 IVF, 290, 293 hypothesis, 389, 402 J hysteresis, 250

I IKF, 259, 290, 293 images, 113, 116, 120, 127, 129 implementation, 57, 62, 65, 66, 73 imports, 258, 266, 277, 287, 288, 290 incidence, vii, 1, 5, 6, 9, 271, 272, 273 inclusion, 40, 60, 81, 87, 97, 105, 106, 111, 346 income, 226, 243, 277, 286, 289, 292 income tax, 277, 289, 292 incomes, 257, 258, 266, 283, 287, 288 incompressible, x, 299, 300, 322

Jacobian, 219, 308, 323, 325, 333 joints, vii, 1, 2, 3, 4, 5, 7, 8, 10, 11, 17, 22, 23, 24, 34

K kernel, 39, 41, 42, 44, 52, 70, 72, 98, 190, 195, 196, 201, 206, 208, 312, 360 kinetics, ix, 187, 190, 197, 206, 209 Kolmogorov, 340

L labor, 226, 227, 232, 243, 290, 291, 292

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labor force, 290 labor productivity, 226, 227, 291 labour market, 256 language, 97, 107, 132, 394 large-scale, 251, 323 lattice, 146, 148, 184 law, 64, 65, 254, 330 lead, 26, 230 Leibniz, 41, 45, 50, 52, 54, 63, 71, 72, 235 Lie algebra, 345, 346, 369, 372 Lie group, 345 line graph, 248 linear, ix, 54, 83, 104, 105, 106, 107, 109, 117, 128, 130, 182, 187, 190, 197, 199, 204, 208, 211, 228, 230, 233, 245, 271, 272, 296, 311, 318, 319, 322, 323, 326, 330, 332, 334, 346 linear function, 311, 318, 319 linear programming, 271, 272 linkage, vii, 1, 2, 4, 24, 34 links, vii, 1, 2, 4, 5, 6, 7, 10, 11, 17, 22, 24, 25, 26, 28, 34, 188, 189 liquid assets, 277, 280, 286 localization, viii, 35, 36, 37, 56, 57, 58, 59, 60, 61, 62, 63, 66, 73, 74 location, 202, 389, 390, 396, 398 locus, 61 London, 33, 184, 186, 221, 296, 371, 372 losses, 196, 289 Lyapunov, 343

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M machinery, 33, 62 macroeconomic, 223, 237 macroeconomic models, 223 magnetic, 144, 186 magnetic field, 144 manifold, vii, viii, ix, 35, 36, 37, 51, 55, 56, 57, 58, 61, 62, 65, 73, 79, 80, 81, 82, 84, 85, 88, 91, 92, 94, 95, 108, 109, 299, 300, 308, 338 manifolds, ix, 37, 51, 61, 62, 79, 82, 91, 94, 95, 299, 300, 338 manipulation, 2, 295 manners, 345 mapping, 141, 169, 227, 308, 310, 316, 347, 358, 361, 390, 391 market, 237, 238, 256, 257, 258 marriage, 236, 271 Massachusetts, 75, 185, 294 Mathematical Methods, 342 mathematicians, 209, 345 mathematics, vii, 2, 37, 65, 137 matrix, vii, 1, 3, 5, 6, 7, 9, 10, 11, 12, 14, 16, 17, 19, 21, 27, 30, 31, 63, 64, 136, 146, 147, 225, 228, 231, 233, 235, 236, 238, 241, 243, 248, 251, 253, 270, 271, 272, 273, 275, 278, 284, 285, 286, 330 measurement, 57, 58, 59, 60, 62, 65, 66 measures, 56, 60, 61, 65, 96, 107, 108, 109, 110, 128, 131, 132, 133, 182

messages, 375, 376, 380, 391 metric, viii, 50, 51, 53, 54, 55, 80, 135, 136, 137, 138, 139, 141, 142, 143, 144, 145, 147, 151, 162, 163, 180, 183, 184, 307, 396 metric geometry, 53 metric spaces, 396 Mexico, 338, 344 military, 292, 409 MMP, 269 mobility, 1, 2, 394 modeling, 58, 62, 276 models, vii, ix, x, 1, 56, 58, 59, 182, 223, 224, 255, 259, 275, 276, 295, 296, 373, 374, 379 modules, 24, 25, 38, 39, 48, 62, 64, 69, 72, 113, 114, 117 modulus, ix, 299, 300 molecular biology, viii, 135, 145, 154, 181, 182, 183, 186 molecules, 144 momentum, 65, 144 monetary policy, 238 money, 237, 238, 290 money supply, 237, 290 monomeric, 194 monomers, 194 monotone, 123 mortgage, 291 motion, 2, 33, 322 motivation, 397 mouse, 182 multidimensional, ix, 187, 190 multiplication, 10, 17, 42, 47, 48, 52, 68, 69, 103, 146, 357 multiplicity, 58, 300, 302, 330, 333, 334, 335, 338 multiplier, ix, 193, 195, 199, 219, 299, 300, 313 mutation, 182 mutations, 179

N national, 237, 238, 243, 282, 283, 287, 288, 289 national income, 243 national product, 238, 282 natural, x, 2, 37, 38, 42, 47, 48, 49, 50, 55, 59, 63, 67, 68, 69, 70, 71, 72, 73, 79, 96, 111, 117, 145, 180, 237, 238, 300, 302, 330, 345, 346, 356, 360, 361, 369, 381 Navier-Stokes equation, 323 Netherlands, 224, 226 network, 3, 36, 74, 164, 181, 182, 226, 255, 259, 373 Niels Bohr, 77 nodes, 3, 274, 375, 378, 379, 384, 385, 390 noise, 137 nonlinear, ix, 2, 85, 223, 275, 294, 325 normal, x, 94, 104, 168, 237, 300, 323, 330, 332, 333, 380, 381, 382, 383 normalization, 193, 195, 197, 199, 270 normal-mode, x, 299, 300, 338 norms, 314, 317, 320, 321, 334

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nuclear magnetic resonance, viii, 135, 144, 145, 183, 186 nuclei, 144 nucleus, 35

415

physical interaction, 56 physical properties, 65 physics, viii, 3, 36, 76, 135, 145, 152, 183, 196, 209, 254 planar, vii, 1, 2, 24, 26, 34, 400, 405 O Poisson, 193 polymers, 188 observations, 147 polynomials, ix, 187, 191, 200, 208, 209, 210, 211, obstruction, viii, 95, 96, 97, 98, 99, 100, 101, 102, 103, 214, 215, 217, 218, 219, 299, 300, 301, 304, 305, 107, 108, 111, 112, 113, 115, 116, 119, 120, 127, 311, 312, 320, 321, 330, 333, 338 130, 132 population, viii, ix, 187, 190, 191, 196, 226, 290, 291 oil, 127 potential output, 292 omission, 254 power, 10, 12, 14, 17, 62, 212, 251, 305, 322, 331, one-to-one mapping, 169 373, 380, 400 online, 398, 400 powers, 10, 16, 211, 248 open economy, 276 prediction, 183 operator, 55, 89, 92, 193, 194, 195, 197, 199, 300, 301, pre-existing, 57 302, 313, 314, 315, 316, 319, 322, 323, 330, 331, preference, 347 332, 333, 334, 335 price deflator, 290, 291 optimization, 2, 234, 241 price index, 289 orbit, 115, 116, 129 prices, 226, 227, 237, 266, 277, 286, 287, 288 ordinary differential equations, 85 primitives, 277, 278, 279, 281, 285 organic, 144 private, 226, 237, 238, 243, 277, 286, 380 organic compounds, 144 probability, viii, 96, 107, 109, 162, 179, 187, 188, 190, organism, 188, 189 192, 193, 194, 195, 201, 202, 207, 208, 209, 236, organization, 57 405 orientation, 104, 113, 117, 125, 129, 236 production, 227, 233, 256, 257, 266, 276, 277, 286, orthogonality, 304, 305 287, 292 oscillations, 144 production function, 233 overtime, 290, 292 productivity, 226, 227, 232, 291 profit, 281, 292 P profits, 243, 258, 266, 277, 291 program, x, 80, 249, 276, 279, 281, 373, 375, 376, 378, pairing, 10, 234 379, 380, 381, 382, 383, 385, 388, 389, 390, 391, paradox, 198 393 parallel processing, 181 programming, 396 parallelism, 34 projector, 311, 312 parameter, 88, 102, 277, 314, 406 property, 36, 37, 58, 62, 64, 72, 101, 102, 106, 107, parents, 231, 254 121, 142, 241, 277, 287, 385, 397, 404 partial differential equations, 88, 300 proposition, 86, 99, 101, 102, 103, 107, 109, 110, 113, particle mass, 190, 201, 206, 208 122, 127, 130, 317, 358, 360, 361, 362, 369 particles, 65, 190, 194, 195, 196, 198, 201, 206, 208, protein structure, 183 210 partition, viii, ix, 95, 96, 107, 108, 110, 111, 122, 132, pseudo, 79, 80, 81, 85, 88, 91, 94 154, 188, 227, 299, 300, 308, 309, 316, 338, 398, Q 401 PCS, 258, 291, 293 quantization, 57, 73, 194 perception, 189 quantum, vii, 35, 36, 37, 55, 56, 57, 65, 66, 67, 68, 70, performance, 405 71, 72, 73, 74, 75, 209 periodic, 132, 318 quantum dynamics, 73 Peripheral, 228 quantum gravity, vii, 35, 74, 75 permit, 36, 57 quantum phenomena, 66 personal, 224, 257, 289, 292 quantum state, 65 perturbation, 231, 286, 287, 325, 326, 327, 328, 330, quantum theory, 36 332 R Phillips curve, 238, 283 philosophical, 65 radius, viii, 135, 136, 137, 138, 139, 141, 143, 144, phylogenetic, 183 145, 183, 184, 229, 250, 253, 285, 286 physical fields, 56

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random, viii, ix, 182, 187, 188, 189, 190, 191, 202, 206, 207, 208, 209, 236, 243, 244, 405 random mating, 182 real numbers, 36, 51, 52, 53, 54, 55, 56, 57, 59, 60, 65, 302, 325, 331, 346 real wage, 227, 292 recall, 80, 82, 83, 85, 90, 91, 103, 119, 129, 153, 162 reconcile, 201 reconstruction, viii, 135, 136, 137, 138, 139, 141, 142, 143, 144, 145, 147, 148, 151, 153, 161, 162, 164, 168, 178, 179, 180, 183, 184, 186, 330 recurrence, 211, 216, 217, 306 reduction, 261 redundancy, 137 reference frame, 27 reflexivity, 387 regular, viii, 113, 119, 135, 137, 145, 146, 147, 148, 149, 153, 165, 180 rejection, 57 relationship, vii, 1, 2, 4, 5, 7, 8, 9, 254, 358 relationships, 242 relativity, 51, 53 remodeling, vii, 35 repetitions, 281 replication, 380, 382, 383 research, vii, 2, 135, 144, 183, 297, 338, 405 researchers, 3, 144, 406 reserves, 289, 292 residential, 259, 290, 291 residuals, 243 residues, 199 resolution, ix, 42, 58, 65, 70, 114, 223, 252 returns, 231, 276, 277, 286 revenue, 237, 283, 284, 286, 287 rings, 37, 38, 40, 41, 42, 43, 53, 72 Robotics, 34 Roman Empire, 404, 408, 409 rotations, 300, 302, 313 routing, 182 Russian, 135, 144, 221, 339, 340, 341, 342, 343, 344 Russian Academy of Sciences, 135, 144

S sales, 226, 256, 257, 258, 266, 291, 292 satisfaction, 26, 36, 45, 56, 60 saving rate, 292 savings, 277, 283 scalar, 37, 41, 51, 52, 55, 56, 58, 62, 74, 82, 93, 300, 307 schema, 45, 53, 75 SDS, 403 search, 102, 144, 230, 231, 242, 248, 254, 274, 276, 279, 284, 296, 297, 332 searching, 148, 185 security, 374, 395, 400, 407, 408 selecting, 2 self-employment, 226 semantic, 51, 57, 58

semantic content, 57 semantics, 36, 37, 58, 59, 62, 73, 372, 381, 390 semicircle, 307, 308 sensitivity, 254, 261, 330 separation, 200, 204, 220, 254 series, ix, 25, 29, 182, 199, 209, 295, 299, 300, 305, 310, 311, 312, 319, 320, 322, 330, 331, 332, 338, 403 seta, 89 SGD, 10, 12, 14, 21 SGP, 291, 293 Shanghai, 1 shape, 95 siblings, 231 sign, 99, 106, 165, 168, 169, 170, 172, 235 signs, 106, 107, 165, 233 similarity, 164, 196 simulation, 276 simulations, ix, 2, 223 sine, 329 singular, 112, 113, 119 singularities, vii, 35, 57, 58, 65, 75, 76, 79, 94 SIS, 291, 293 skeleton, 35, 59, 74, 97, 101, 102 smoothness, 58, 61, 300, 313, 322, 327, 331, 335, 338 sociology, 3 software, ix, 223, 224 solutions, viii, x, 51, 85, 95, 111, 182, 214, 233, 239, 271, 272, 273, 299, 322, 324, 327, 328, 330, 338 sorting, 154, 155, 181, 182 spacetime, 51, 55, 56, 57, 75 spatial, 34, 330 specialization, 33 species, 182 spectroscopy, viii, 135, 136, 144, 183 spectrum, ix, 57, 144, 187, 190, 191, 194, 198, 201, 203, 206, 207, 208, 216, 217, 330, 333 spheres, 118, 148, 156, 172, 174, 175 spin, 144 springs, 253 stability, x, 58, 60, 299, 300, 322, 324, 329, 330, 332, 338 standard error, 244 statistics, 244 steady state, 250 Stephen Hawking, 75 stochastic, ix, 187, 190, 207 stock, 226, 227, 243, 277, 290, 292 strength, 59, 64, 72, 73, 255 string theory, 79 subgroups, 168 substitution, 37, 227, 381 superposition, 63 symbols, 41, 52, 278, 288, 289, 293, 314 symmetry, 10, 15, 110, 168, 182, 185 syntactic, 383 synthesis, 2, 5, 9, 25, 33, 34 systems, ix, x, 57, 58, 60, 73, 74, 119, 145, 187, 188, 190, 201, 294, 310, 373, 374, 375, 385, 394

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T tax receipt, 292 taxes, 243, 258, 277, 286, 287, 288, 290, 292 technology, 373 temporal, ix, 56, 57, 223, 330 textbooks, 137 TFG, 257, 258, 259, 292, 293 thermodynamic, ix, 187, 190, 191, 196, 198, 201, 206, 207, 208, 209, 216 third order, 82, 322, 328 three-dimensional, 300 threshold, 147, 295 topological, vii, viii, 1, 2, 4, 5, 6, 8, 9, 11, 14, 16, 17, 22, 23, 24, 25, 34, 35, 36, 57, 58, 59, 60, 62, 65, 73, 95, 96 topological structures, 11, 17 topology, viii, 2, 60, 66, 74, 95, 107, 108, 110, 111, 133 Topos, 74, 76, 77 total employment, 289 total expenditures, 289 tradition, 106 transcription, 72 transfer, x, 37, 59, 74, 103, 292, 373, 378, 380 transfer payments, 292 transformation, 25, 26, 64, 65, 70, 71, 72, 182, 308, 310, 356, 360, 361, 369 transformations, 38, 42, 47, 48, 51, 64, 67, 68, 69, 72, 212, 219, 302 transition, ix, 36, 45, 56, 58, 60, 187, 188, 189, 190, 196, 202, 208, 381, 382 transition rate, 196, 202 transmission, 2, 137 transparent, 115, 209 transport, 36, 72 turbulent, 322, 323, 328, 331 two-dimensional, 322, 338 typology, ix, 223, 231, 232, 256, 266, 276, 284, 287, 288

U unemployment, 226, 237, 238, 292 unions, 105, 380 United Kingdom, 184, 295, 296 United States, 276, 296

V validity, 36, 317, 320 values, 3, 7, 8, 26, 65, 66, 142, 149, 152, 156, 174, 207, 227, 231, 255, 285, 288, 289, 304, 307, 308, 327, 329, 397, 405 variability, 51 variation, 36, 46, 55, 56, 59, 64, 72, 73, 402 vector, viii, 10, 12, 17, 65, 79, 80, 81, 83, 84, 85, 86, 87, 89, 92, 93, 106, 107, 109, 110, 133, 144, 146, 147, 154, 156, 171, 231, 270, 271, 272, 273, 284, 286, 371 vein, x, 56, 67, 345 viscosity, 323, 328 vulnerability, 224, 255, 295

W wage level, 226 wage rate, 232, 292 wages, 226, 232, 287 wealth, 289 web, 297 women, 290 workers, 292 working hours, 226 working population, 226

Y Yang-Mills, 65, 76 yield, 218, 277

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