Charlemagne and his heritage: 1200 years of civilization and science in Europe = Karl der Grosse und sein Nachwirken : 1200 Jahre Kultur und Wissenschaft in Europa 2503506747, 9782503506746

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CHARLE MAGNE AND HIS HERITA GE

1200 YEARS OF CIVILIZATION AND SCIENC E IN EUROPE

KARL

DER GROSSE UND SEIN NACHW IRKEN

1200 JARRE KULTUR UND WISSEN SCHAFT IN EUROPA

Charlemagne and his Heritage 1200 Years of Civilization and Science in Europe Karl der Grosse und sein N achwirken 1200 Jahre Kultur und Wissenschaft in Europ a

Volume 2 Mathematical Arts

edited by

P.L.

BUTZER,

H.TH.

JONGEN,

W.

ÜBERSCHELP

in collaboration with J.

BEMELMANS,

W.

DAHMEN,

A. KRœo, W. PLESKEN

BREPOLS

Caver illustration - Umschlagsillustration: Representation of the 'Pegasus 'sign of the zodiac (Ms. Leiden, Univ. Bibl. Voss. lat. Q 79,f 32 v.,first half ofthe 9th century). Abbildung des 'Pegasus' Tierkreissternbildes (Ms. Leiden, Univ. Bibl. Voss. lat. Q 79,

f 32 v., erste Hdlfte des 9. Jahrhunderts).

© Brepols, Turnhout 1998 D/1998/0095/48 Ali rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior permission of the publisher. ISBN 2-503-50674-7

Table of Conten ts Preface and Introducti on IX Acknowled gements XIII Opening Addresses XV Jürgen Linden, Heinrich Mussinghoff, Johannes Flof3, Burkhardt Müller

History of Mathema tics during the Past 1200 Years Gordon Leff Alcuin of York (c. 730/35 - 804) David Singmaste r The history of some of Alcuin's Propositiones Harald Gropp Propositio de lupo et capra fasciculo cauli - On the history of river-cross ing problems Paul L. Butzer, assisted by Helga Butzer Felleisen Scholars of the mathemat ical sciences in the Aachen-L iège-Maas tricht region during the past 1200 years; An overview Bettina Klaus·& Hans Peters Maastrich t in science from Minckelers to Debye: Sorne historical notes François J ongmans Les mathémati ciens liégois du 19ième siècle Bruce C. Berndt An overview of Ramanuja n's notebooks Vladimir Tikhomiro v The phenomen on of the Moscow mathemat ical school Peter Ullrich The genesis of Hensel's p-adic numbers Friedrich Schreiber Emigratio n in Europe - The life and work of Félix Pollaczek

3 11

31

43 91 107 119 147 163 179

V

Table of Contents

------------------------~

Geometry

Walter Benz Lorentz-Minkowski geometry, De Sitter's world and Einstein's cylinder universe H eiko H arborth Integral distances in point sets A. V. Kuz'minykh On mappings preserving only some distances Francis Buekenhout The rise of incidence geometry and buildings in the 20th century Benno Klotzek Diskontinuierliche Bewegungsgruppen verschiedenen Grades in verschiedenen Geometrien David G. Glynn On non-spread representations of projective planes of order qi in PG(2i, q) Martin Salzwedel On characterizing injective lineations in Desarguesian affine spaces Peter BrajJ On lattice polyhedra and pseudocircle arrangements

197 213 225 235

257

273 287 297

Combinatorics

Adalbert K erber Anwendungsorientierte Theorie endlicher Strukturen Peter Kirschenhofer Average case analysis of algorithms - Sorne examples Annette Schelten & Uwe Schelten Pancyclicity in graphs with independent claw centers Dietmar Cieslik The Steiner ratio of finite-dimensional Lp-spaces Michael Hauss A Boole-type formula involving conjugate Euler polynomials

VI

305 331 345 353 361

- - - - - - - - - - - - - - - - - - - - - - - - - Table of Contents

Operatio ns Research

Ralf Borndorfe r & Martin Grotschel & Andreas Label Alcuin's transporta tion problems and integer programm ing Walter Oberschelp Alcuin's camel and the jeep problem D. Pallaschke & S. Rolewicz Penalty and augmented Lagrangia n in general optimisati on problems P. Jonker & M. Pouw & G. Still & F. Twilt On the partition of real skew-symm etric n x n-matrices Bettina Klaus & Hans Peters & Tom Storcken Strategy-p roof reallocatio n of an infinitely divisible good Martin Gugat Parametri c convex optimizati on O.J. Vrieze State-acti on frequencies in average Markov decision processes

379 411 423 439 455 471 485

Miscellan eous Topics

Henri Cohen Computin g in algebraic number fields Jürgen Elstrodt A quick proof of the prime number theorem for arithmetic progressio ns Andrew Khrenniko v The Kolmogoro v complexity of p-adic random sequences H. W. Broer & H.M. Osinga & G. Vegter Computin g a normally attracting invariant manifold of a Poincaré map Bernd Krauskopf Stability loss near 1:4 resonance

505 521 531 541 551

Appendi x Addresses of Authors

565

Table of Contents of Volume 1

569

VII

Preface and Introduct ion This volume, the second of two, had its origins in the international and interdisciplinary Colloquium Carolus Magnus: 1200 Years of Civilization and Science in Europe that was held in Aachen l\Iarch 19-26, 1995. It was conducted as a collaborative undertaking between six chairs of mathematics and those of medieval history, Romance languages, and art history, at the Aachen University of Technology, together with the Diocese of Aachen. Charlemagne's empire encompassed present-day France, the Low Countries and western Germany, as well as northern Italy and a part of Spain. Not surprisingly, the Romance languages, particularly Latin, played an important role, through the medium of the ancient manuscripts that not only contribute to art history but are vital for our increasing appreciation of the understanding of mathematics and astronomy at the time. The abject of the Colloquium was not only to elucidate the intellectual world under Emperor Charlemagne, who selected Aachen as his permanent winter residence in 794/95, but also to elucidate the influence of that seminal era 1200 years later for modern science, especially for those subfields of mathematics and their applications, the roots of which may go back to the Carolingian court school. Volume 1, consisting of 20 papers written by renowned scholars, addresses historical and cultural themes of the so-called Carolingian Renaissance, including scholarly activities, the artes liberales, the literary, juridical and political sphere, the development of the church and the evolution of monasticism, as well as the artes mechanicae, the construction principles of the Aachen cathedral, cosmological speculations, and astronomical and geographical conceptions of the early Middle Ages. Volume 2 first focuses on arithmetic and geometry in the time of Charlemagne and then turns to those mathematical fields which especially Alcuin of York introduced at the court school between 782 and 796, and which continue to play a primary role in contemporary mathematics. As Martin Gri:itschel et al. write in their paper (Part IV): "Alcuin's (four) river crossing problems [... ] already display all the characteristics of today's large-scale transportation problems. [... ] they could have been the starting point of combinatorics, optimization, and operations research". Our emphasis is on selected tapies in modern geometry, combinatorics, operations research, number theory and related fields. This volume of 35 papers includes invited colloquium papers, selected contributed papers (all refereed), as well as 10 papers solicited after the colloquium by the editors in order to round off the broad and interdisciplinar y subject matter. The solicited papers were also written by experts in their fields. In accord with the editors' conception, the papers have been grouped into five sections. More concretely, Part I, consisting of ten papers on special aspects of

IX

Preface and Introduction - - - - - - - - - - - - - - - - - - - - - - -

the history of mathematics during the past 1200 years, first deals with Alcuin of York, presented by York university's medievalist Gordon Leff, with the history of some of Alcuin's 53 Propositiones, perhaps the oldest collection of mathematical problems written in Latin. Several of these lie at the basis of modern research, including the camel and the jeep problems, or the river-crossing problems. Three papers follow on the scholars of mathematical sciences born in the Aachen-Düren, Maastricht and Liège regions. Further papers include a review of the remarkable notebooks of the great Indian mathematician Ramanujan that include over 3000 mathematical daims without proof; articles on the Moscow mathematical school from 1914 on, which occupied a leading position in the world of mathematics; the genesis of Kurt Hensel's p-adic numbers; and finally the problem of emigration in Europe, the life and work of Félix Pollaczek being an example. Part II, on geometry, treats the following: Lorentz-l\Iinkowski geometry in connection with Einstein's law of gravitation, specifically characterizations of motions and distance in these geometries; integral distances in point sets, such as Fibonacci triangles, sequences of pentagons and platonic solid graphs; mappings preserving only some distances in Euclidean and Lobatchevskii spaces; the rise of incidence geometry and the theory of buildings from 1955 on; discontinuous transformation and motion groups with respect to different geometries; the characterization under mild hypotheses of injective lineations in desarguesian affine spaces; and finally the maximum number of vertices of convex lattice polyhedra under pseudocircle arrangements. Part III, on combinatorics, is concerned with: the constructive theory of finite structures, examples being chemical isomers, unlabelled graphs, errorcorrecting, that demonstrate the applicability of the algebraic methods used; with applications of varions mathematical methods to the analysis of the average case behaviour of some selected algorithms, representing a border area between mathematics and computer science; nonspread representations of projective planes of order qi in the desarguesian plane PG(2i, q); pancyclicity in connected graphs with independent claw centers; the Steiner ratio and Steiner trees in d-dimensional linear spaces with p-norm; and finally the introduction of conjugate Euler polynomials, together with the counterpart of the Boole summation formula now involving these polynomials, together with an application for an Euler-type formula for Dirichlet L-series. Part IV, on operations research, deals with: Alcuin's transportation problem using current mathematical methods, which provides the reader with a leisurely written introduction into modern integer programming; Alcuin's problem 52, the first appearance of the jeep problem in operations research, which is solved under general assumptions and compared with modern versions of the problem; a simple analysis, without topological assumptions, of the value fonction of constrained optimization problems with smooth goal fonctions; the partition

X

- - - - - - - - - - - - - - - - - - - - - - - Preface and Introduction

of the set of all real skew symmetric n x n matrices, so that only very special multiplicity distributions of eigenvalues are generally possible for such matrices which are smooth parameter dependent; the problem of reallocating the total endowment of an infinitely divisible commodity between agents with singlepeaked preferences, covering Pareto optimality, strategy-proofn ess, and equaltreatments conditions; one-sided differentiability of the optimal value fonction under Mangasarian-F romovitz or Slater constraint qualifications; and a procedure in l\farkov decision processes that performs the translation of state-action frequencies into strategies. Part V, on miscellaneous topics, including number theory, comprises: a package for number theory called PARI, - developed and enhanced during the past dozen years at Bordeaux, is especially adapted for computing in algebraic number fields; a quick proof of the prime number theorem for arithmetic progressions via Newman's version of a weakened Wiener-Ikehara theorem; a study of stochastic models in p-adic probability theory where relative frequencies of events oscillate with respect to the ordinary real metric, but stabilize with respect to one of the p-adic metrics; an algorithm, based on the technique of graph transform, for the computation of a normally hyperbolic invariant manifold of a diffeomorphism ; and finally, the loss of stability of a closed orbit of a vector field in a 1:4 resonance, which is a classical codimension - two bifurcation problem in the theory of dynamical systems. vVe feel that this diverse menu seems to show how the humble seeds that were planted 1200 years ago have not only taken roots, but have yielded rich and abundant fruit.

Aachen, February 1998

P.L. Butzer

J. Bemelmans

H.Th. Jongen

W. Dahmen

W. Oberschelp

A. Krieg

W. Plesken

XI

Acknowle dgements The seven members of the editorial board would first of all like to thank several of their colleagues who helped select the participants, with the organization of the Colloquim, and in making this volume potentially known to a broader public, in particular Walter Benz (Hamburg), Bruce Berndt (Urbana, Ill.), Frank Twilt (Enschede, NL). Special thanks are due to the eminent medieval historian Gordon Leff (York) who allowed us to publish his lecture on Alcuin which he presented on the occasion of a Symposium on Fourier Analysis and Applications, organized by Jim Clunie and Maurice Dodson at York in 1993. The editorial board offers its special thanks to the authors of the thirty five papers for their presentations, to Dr. J. Linden, Lord Mayor of Aachen, Dr. H. Mussinghoff, Bishop of Aachen, Burckhard Müller, former General Secretary of the Deutsche Forschungsgem einschaft (DFG), and Prof. J. Flol:ll:l, Past Vice-Rector of our RWTH Aachen, for their words of welcome, as well as the many participants who attended the colloquium and helped make it the success it turned out to be. As in the case of Volume I, the editors extend their sincere thanks for financial support to the DFG, the "Freunde der Aachener Hochschule" (FAHO), the ALMA Maastricht, the Euregio Rhein-Maas, the Robert-Bosch-S tiftung, and the Friedrich-Wilhe lm-Stiftung Aachen. In this respect, Prof. K. Habetha, Past Rector of the RWTH, as well as Chancellor J. Kessler were particularly active in finding funding sources for the Colloquium as well as for the publication of the two volumes. Our appreciation is also due PD. Dr. D. Wynands of the Bischofliche Akademie des Bistums Aachen. In fact, the Colloquium was a joint venture of the RWTH, the Bishop's Academy as well as the Cathedral Chapter, in charge of Dompropst Dr. H. Müllejans. The colloquium was organized by faculty and collaborators from eight chairs; however, Hubertus Jongen and Walter Oberschelp were especially responsible for the mathematical part and Paul Butzer for the historical sections, in collaboration with l\fax Kerner. The editing of two volumes was largely left in the hands of Paul Butzer. The editors are deeply thankful to Marc Henke, a physicist who prepared the 20 papers of Volume I and the 35 papers of Volume II in such a ready-to-print postscript version so that the publishing firm could proceed rapidly with publication. In fact, Marc Henke devoted himself with great dedication and initative, meticulously converting the manuscripts by means of the best, current techniques with the assistance of René Schrader-Boels che, also a physicist. After negotiations with a local publisher failed it was Karl. W. Butzer (Austin, Texas) who, apart from other assistance, suggested Brepols NV, a Belgian publishing house celebrating its 200th birthday in 1995. Finally, Reinhard Oberschelp (Hannover) supported the editors in their selection of Brepols. Although

XIII

Brepols' fame lies in the production of medieval works, the editors are especially grateful that the firm also accepted this volume dealing with the mathematical arts. Last but not least the editors are particularly grateful to Mr. Christophe Lebbe of Brepols for his cooperation and the quality production.

Aachen, February 1998

P.L. Butzer

J. Bemelmans

XIV

H.Th. Jongen

W. Dahmen

W. Oberschelp

A. Krieg

W. Plesken

Oberbür germeiste r Dr. Jürgen Linden

G ruBwo rt zur Eroffn ung des "Collo quium Carolu s Magnu s" Meine sehr geehrten Damen und Herren, oft, wenn ich das Aachener Rathaus, meine Arbeitsste lle, betrete, geht mein Blick zuvor die Rathaus-F assade hinauf, nicht etwa, um den Fassadens chmuck auf seine Standfesti gkeit zu überprüfen , sondern um mich am Sinnbild- und Figurenen semble zu erfreuen. Dem geübten Blick entgeht dabei nicht, daf3 sich unter den Fenstern des Kronnungs saales die sieben freien Künste in ReliefAbbildung en dargestellt finden: Neben der Grammati k, der Rhetorik, der Dialektik, der Astronomi e und der Musik zeigen sich Symbole für die Arithmeti k und der Geometrie . Diese Relief-Dar stellungen sind keineswegs nur dekorative Füllsel, sondern eine Bezugnah me auf die Aachener Stadtgesch ichte. Das Rathaus, im 14. Jahrhunde rt erbaut auf den Resten der karolingisc hen Konigsaul a, ist ein Nachfolgebau eines Teils der Pfalzanlag e Karls des Grof3en. Hier war die schola palatina beheimate t, ein Gelehrtenk reis, den Karl der Grof3e aus den unterschiedliche n Landern seines Reiches nach Aachen berief, um von hier aus die geistige Bildung zu fordern und zu befruchten , auch um das Bildungsw esen des karolingisc hen Reiches zu organisiere n. Diese multikultu relle Gruppe wurde zum lmpulsgeb er für die erste europiiisch e Renaissanc e, die karolingisc he Renaissance. Mit ihr wurde der Grundstei n einer Bildungsb ewegung gelegt, die die historische Entwicklu ng unseres Kontinent s nachhaltig beeinfiuf3te. Sie schuf den ersten Aufschwun g europaisch en Geistes, durch den unser Denken bis in die Gegenwar t hinein gepragt wird. Der Rekurs auf antike Werte in Verbindun g mit christliche m Glaubensb ewuf3tsein lief3en die karolingische Renaissan ce ein Maf3, Ordnung und Klarheit stiftende Macht werden. Das freie Gesprach der Gelehrten -wennglei ch in den weltanscha ulichen Grenzen mittelalter licher Theologie - wurde an der Hofakadem ie Karls Alltagswir klichkeit. Wissensch aft und wissenscha ftlicher Diskurs haben also eine lange Tradition in unserer Stadt. Die Rheinisch- Westfalisc he Technische Hochschul e Aachen ist zwar erst im vergangen en Jahrhunde rt gegründet worden, kann also nicht als Nachfolge institution der karolingisc hen Hofschule gelten, sehr wohl aber mit Stolz auf die kulturellen Wurzeln, die von Aachen aus Europa gegeben wurden, hinweisen. Ich freue mich aufrichtig, daf3 sie dies mit dem Colloquium Carolus Magnus tut.

XV

Oberbürgermeister Linden - - - - - - - - - - - - - - - - - - - - - -

Welchen Stellenwert mathematisches Wissen bereits vor 1.200 Jahren innehatte, zeigen die von mir zu Beginn genannten Symbole der Arithmetik und der Geometrie. Seit jeher gehê:iren diese zwei Disziplinen zum Kernbestand europaischen Wissenschaftsverstandnisses. Ich bin den Veranstaltern des Colloquium Carolus Magnus deshalb sehr dankbar, daB sie 2 and k < 0

by W. Benz [6], [5]. Proofs for theorems 1 and 2 are in our book B 1, section 6.6 and sections 6.13, 6.14, 6.15. Theorem 1 does not hold true in the case n following

=

2. We proved in B 1 the

Theorem 3. Let Π> 1 be a fixed real number. Then there exist topological mappings T : IR.2 ---+ IR.2 satisfying

(ii)

T

is a causal automorphism of IR. 2 ,

(iii) the images of the lines

are the lines

respectively.

200

_ _ Lorentz-Minkowski Geometry, De Sitter's World and Einstein's Cyl. U.

(iv) T is not an affine mapping. Remarks. 1) A bijection g of IR.n (2 ::; n phism of IR.n in the case \/x,yEJ!l.n X::;

< oo) is said to be a causal automor-

y Ç} g (x)::; g (y).

(x::; y is defined by d (x, y) ::; 0::; Yn - Xn.) 2) A usual Lorentz transformation of the plane which is supposed to be orthochronous satisfies (i), (ii), (iii) and it is moreover an affine mapping ofIR.2 . 3) For 1 =1- k

> 0 define fk(x)

:= {

x

for

x>O

kx for all x E IR. For fixed a > 1 put

~(x2+x1,x2-x1),

Q(x1,x2)

·-

"( (x1, x2)

·- (a - l)(x1, x1) +(a+ l)(-x2, x2),

Œk (x1,x2)

·-

(fk(x1),fk(x2)).

Then Tk := "(Œk(! satisfies (i), (ii), (iii), (iv) and it is moreover a topological mapping of IR.2 . The following characterization theorems for Lorentz transformations of IR.2 hold true (B 1). Theorem 4 a. Let a satisfying

( 1) \/ a,bEJ!l.2 d (a, b)

> 1 be a fixed real number and let T be a bijection of IR. 2

=

0

Ç}

d (T (a), T (b))

=

0,

(2) T(TJ) Ç TJ-a andT(TJa) Ç T) withT) = {(O,x2) 1x2 E IR}, T/{3={(x1,/Jx1)1 X1 E IR},

(3) the li mit

201

Walter Benz - - - - - - - - - - - - - - - - - - - - - - - - - -

Then there is a Lorentz transformation À of positive determinant and a k > 0 such that T (x) = k · >.(x) holds true for all x E JR.2 . Theorem 4 b. Let o: > 1 be a fixed real number and let T be a causal automorphism of JR.2 satisfying (2), (3) of theorem 4 a. Then there exist an orthochronous Lorentz transformation À of positive determinant and a k > 0 such that T ( x) = k · À ( x) holds true for all x E JR.2 . Theorem 4 c. Let o: > 1 be a fixed real number and let T be a causal automorphism of JR.2 satisfying

(i) (ii)

'ix,yEIR2 T T

(TJ)

=

(x)

=Y=} T

(-y1, Y2)

=

(-x1, x2),

TJ-a,

(iii) the two one-sided limits lim

7/J(x,o:x) and lim

x-+O-

X

Tf then v and c are positive real numbers with av 1

X -

Vt

7/J(x,o:x)

x-+O+

1

X

= c,

t-

then

2x X

x= R a n d t = R hold true where we put

The following theorem was proved by A.D. Alexandrov (A.D. Alexandrov and V.V. Ovchinnikova [4], A.D. Alexandrov [3].)

Theorem 5. Every causal automorphism of JR.n (3 ::; n < oo )is product of an orthochronous Lorentz transformation of JR.n and a dilatation. For an easy proof of this theorem see B 1, 235 f.

3

Einstein's Cylinder Universe

Suppose that ~n- 1 (n 2': 2) is the unit hypersphere of JR.nand that direct product := ~n-1 X JR.1.

en

202

en

is the

_ _ Lorentz-M inkowski Geometry, De Sitter's World and Einstein's Cyl. U.

The typical element of en will be written in the form (xo, Xn+1), and it will be considered as a point of ~n+i as well . The group Mn of motions of en is the group of ail Lorentz transforma tions f of ~n+ 1 satisfying f( en) c en. The geometry (en, Mn) is called n-dimensi onal ECU. Every circular cylinder Z c en of radius 1 is called a plane of ECU. Suppose that Z is a plane of ECU. Then every euclidean line C Z and every euclidean circle c Z and every circular helix c Z will be called a line of (en, Mn). Every circular helix of a plane Z that intersects every euclidean line on Z under 45° is said to be a light ray or a null line. If we assign to

the point~( COS Xn+l, X 0 , sin Xn+l) of (n+l)-dim ensional projective space rrn+l, we get the mapping of June Lester,

which is surjective from en onto the Lie quadric of rrn+i. The classes

for

X E

en are countable point sets of en. Define g (x, y)

for

X,

:=

xy - cos(xn+l - Yn+1)

y E en and n

xy := Lxi Yi· i=l

The following result is due to June Lester ([13]). Theorem 6. All bijections

f : en---+ en(n?: 3) su ch that g (x, y) = 0 holds true if, and only if g (f (x), f (y)) = 0 is satisfied, are given as follows: let À : Ln-l ---+ Ln-l be an arbitrary Lie transforma tion of Ln-l and let f be an arbitrary bijection of en which maps the class [x] onto the class a- 1 (.\a(x)) for all XE en. Theorem 6 cannot be improved such that a mapping f there is acting on the set of equivalenc e classes [x] the same way as a suitable mapping of Mn. Obviously,

203

Walter Benz - - - - - - - - - - - - - - - - - - - - - - - - -

is a Lie transformation satisfying

À

of Ln- 1 . But as it turns out there is no

r

E

Mn

[r (x)] = a- 1 [Àa (x)] for all

X

E

en.

Also the following results hold true. Theorem 7. All bijections

f : en ---} en (n 2: 3) such that images and inverse images of light rays are light rays are given exactly by the mappings as desribed in Theorem 1. Let W

# 0 be a

set. We then would like to determine all

f : en

X

en ---} W

such that

f(x,y) = f(r(x),1(y)) holds true for all

X'

(1)

y E en and all 1 in Mn. Define

W 0 := [-1,+l] x {r E IR. r 2: 0}. J

Theorem 8. Suppose that g is a mapping from W 0 into vV. Then

f(x,y) := g(xy, Jxn+l -Yn-11) is a solution of the functional equation (1). If, on the other hand,

f : en

X

en ---} W

is a solution of (1), then there exists a function g: Wo ---} W satisfying (2). We proved (see R4) Theorem 9. Suppose that

d : en

X

en ---} {r

E IR_

satisfies (i), (ii), (iii), (iv). (i) d is a 2-point-invariant of (en, Mn). (ii) d is additive on admissible point triples.

204

J

r 2: Ü}

(2)

_ _ Lorentz-Minkowski Geometry, De Sitter's World and Einstein's Cyl. U.

(iii) d (a, b) .

.

= 0 for the points a= (x 0 , 0), b = (x 0 , ~) with xo = (1, 0, ... , 0).

d(p,x)

(iv) hm f:l(p, x)

= 1

for an existing p E en where

X

tends top along an arbi-

trary line which is not a light ray. Here we put

Th en d (x, y)

= JI (arccos xy) 2

-

(xn+l - Yn+i) 2 I

(3)

for all

with 1/3 - al

x

(a cos a+ bsina, Xn+1),

y

(acos/3 + bsin/3, Yn+1)

< n, arccos xy

E [O, n], a 2 = 1 = b2 , ab= O.

Assumption (iv) reflects a little bit the fact that space time manifolds are locally of Lorentz Minkowski type: d (x, y) should be close to ll (x, y) in the case that x and y are close together. A differentiable manifold M together with a tensor field g which is supposed to be covariant of the second order and of Lorentz Minkowski signature for every x E M, is said to be a space time manifold of dimension n (:::0: 2) if n is the dimension of M. Lorentz Minkowski Geometry represents such a manifold, but also (ECU) and (DSW). The geometries (LMG) and (DSW) are even solutions of

=

Rie

Àg,

where Rie is the Ricci tensor with respect to g and its canonical affine connection, and where À is a constant scalar. Solutions of this equation are called Einstein spaces. (ECU) is not an Einstein space, but it is a solution of Rie

=

Àg

+ r;,T

where T is the energy-impulse tensor and where is Einstein's law of gravitation.

K

is a scalar. The last equation

A 0-preserving bijection of en is a bijection satisfying d (x, y)

=

0

{=}

d

(f (x),

f(y))

=

0

for all x, y E en, where d denotes the distance fonction (3) with arccosxy E [O,n].

205

Walter Benz - - - - - - - - - - - - - - - - - - - - - - - - - -

The following result holds true (see B2 and R3). Theorem 10. Let f be a 0-preserving bijection of en (n ~ 3). Then l is a light ray if and only if f (l) is a light ray. There exist bijections which are not 0-preserving, but such that images and inverse images of light rays are light rays. Remarks 1) The following bijection is not 0-preserving, but it maps light rays in both directions onto light rays. Put

Pv

:=

((-lt,o, ... ,0,7T+v7T)

for v E Z and put

f (x) = { P_v x

2

for X =_Pv and v =1 otherw1se.

The mapping f leaves invariant every equivalence class and it permutes the points of a given class. It hence maps light rays in both directions onto light rays. in view of theorem 7. Observe moreover that

2) Put f (x, y) := (arccos xy) 2

-

(xn+l - Yn+1) 2

with arccos xy E [O, 7r] for x, y E en. We are then interested in the set of all mappings rp : en ---+ en satisfying the functional equation f (x,y)

=

f (rp(x),rp(y))

for all X' y E en. We proved in B 2 that the set of all solutions is given precisely by the group Mn of motions of (en, Mn).

3) Let S =f. 0 and W be sets and let d be a mapping from S x S into W. Then (S, W, d) is called a distance space. d (x, y) is called the distance of x, y E S (in this order). Suppose that R is a commutative ring with identity element 1 such that 1+1 is a unit element in R. Define

s TV d(x,y)

206

{ (x1, ... , Xn+1) E Rn+l xi+ ... + X~ 1

RxR

(xy, (xn+l

- Yn+1) 2 )

= 1},

_ _ Lorentz-Minkowski Geometry, De Sitter's World and Einstein's Cyl. U.

where the scalar product

is given by

xy =

X1 Yl

+ ... + Xn Yn.

Two distance spaces (Si, Wi, di, i = 1, 2, are called isomorphic if, and only if, there exists a bijection

such that

d (a, b) = d (x, y)

?

d

(f (a), f (b))

= d

(f (x), f (y))

holds true for all a, b, x, y E 8 1 . Isomorphic distance spaces have isomorphic groups of isometries. (The group of all bijective and distance preserving mappings of a distance space is called its group of isometries.) (S, W, d) represents in the case R :=JE. the classical situation of (ECU). For general R we get general Einstein Cylinder Geometries. The following result (B 2) is of interest in this connection. Let n -2: 2 be an integer and let R be a ring as described previously. Then every distancepreserving mapping of (S, vV, d) must be bijective and they are all given by the transformations

where A is an n x n-matrix over R with

and where c and a are elements of R with c 2 = l.

4

De Sitter's World

In this section xy denotes the pseudo-euclidean scalar product

xy

= X1 Y1 + ... + Xn Yn

- Xn+l Yn+l

207

Walter Benz - - - - - - - - - - - - - - - - - - - - - - - - - -

for points X = (x1, ... 'Xn+1) and y = (y1, ... 'Yn+1) of ~n+ 1 .The set sn of points of n-dimensional de Sitter's world is then given by sn

= {x

E ~n+l

x2

1

=

l}.

The group .6. n of motions of sn is the set of all Lorentz transformations f of ~n+i satisfying !(Sn) c sn. The geometry(Sn, .6.n) is then called ndimensional de Sitter's world DSW. The lines of DSW are the connected components of JrnSn where Jr is any 2-dimensional plane through the origin. There are three types of lines:the so-called closed ones which are ellipses, the open lines which are branches of hyperbolas and the light rays (the null-lines) which are the euclidean lines. The following result is due to June Lester ([14]). Theorem 11. A bijection of sn(n 2: 3) such that images and inverse images of light rays are light rays is a motion of (sn' .6. n).

E.M. Schri:ider ([26]) generalized this theorem to the field case. We presenteda proof of Theorem 11 which is based on the Fundamental Theorem of Lie geometry. It is possible to determine all bijections of S 2 such that images and inverse images of light rays are light rays. Not all these bijections are motions. We proved ([9], see also B 2)

Theorem 12. All bijections of the set of points of de Sitter's plane such that images and pre-images of light rays are light rays and such that images of open lines are open lines are motions.

A main step of the proof of theorem 12 is the proof of Theorem 13. Suppose that

admissible, M' is g(G)-isometrizable, and M' :J M(g). Since w is a limit point of the set œ' for every k E N (where N is thE set of positive integers), there exist a mapping hk E H(J0 ) and a finite set Mk C 'P(IEn), which have the following properties: 1) for every k the set Mk is hk(G)-isometrizable , Mk :J M(hk); 2) for every k restrictions of the mappings hk and h 1 onto the set {A 1, B 1, ···: En} coincide; 3) a set

is

n~ -admissible.

Thus, for every k the set 9'J1 is {h1(A1), hi(B1), ... , h 1 (Bn)} U {hk(A 2)}isometrizable. If a set S C IEn contains the points hi (A 1), h 1(B 1), ... ,hi (En) in a general position, any isometry µ : S --+ IEn is uniquely defined by its values at these points. Consequently, the set 9'J1 is OO

{hi (Ai), h1 (Bi), ... , hi (En)}

LJ {hk (A2)} -

isometrizable.

k=l

Obviously, all the points hk(A 2) (k EN) are distinct. Since the set {hk(A 2) k E N} is bounded, there exist such (distinct) numbers ki, k2 E N that d(hk,(A2),hk2(A2)) < E. Put a(n', E) ~ d(hk, (A2), hk 2 (A2)), Wl ~f Mk, U Mk 2 • Let X', Y' be such points that d(X',Y') = a(n',E). Let v: IEn--+ IEn be such an isometry that v(hk 1 (A2)) = X',v(hk 2 (A2)) =Y'. Put 9'Jî:(n',X',Y') ~f {{v(X),v(Y)} : {X, Y} E Wl}. Obviously, the set 9'Jî:(O', X', Y') is O'-admissible and {X', Y'}isometrizable. Lemma 1 is proved. Lemma 2 . Let a set n' c 1R+ be such that card O' > N0 . For any two points X 0 , Y 0 there exists at most countable set SJ1(X0 , Y 0 , n') C 'P(IEn) which is O' -admissible and possesses the following property: if f : IEn --+ IEn is such a mapping that it follows from the condition {X, Y} E SJ1(X0 , Y 0 , n') that d(f(X), f(Y)) = d(X, Y), then d(f(Xo), f(Yo)):::; d(Xo, Ya).

Proof. Let sets

0~

(k

E

N) be such that

OO

1)

œ = u n~; k=i

2) for every k cardfl~ = cardfl'; 3) if k1 # k2, then n~, n n~2 = 0.

230

_ _ _ _ _ _ _ _ _ _ _ _ On l\1appings Preserving Only Sorne Distances

Let X 0 , Y0 be distinct points; let m EN be some number, which satisfies an inequality ~ < ~d(X0 , Y0 ) (we denote by m 0 the first of such numbers). Let M be su ch a set that card M = card n~ = card n'. Let Dç (Ç E M) be sets such that 1) n~ = LJ Dç; ÇEM

2) for every Ç card Dç = card n~; 3) if 6 =} Ç2 (6,6 E M), then Dù n Dç 2 = 0. By Lemma 1, for every Ç E M there exists a number o:(Dç, ~) such that 0 < o:(Dç, ~) O. Put b ~f inf { tk : k E N}. There exist a positive integer r :::0: 3 and (distinct) points T 1 , ... 1 Tri satisfying the following conditions: r~l d(X 0 , Y0 ) < ~; Xo = T 11 Y 0 = Tr; the points Tl, ... , Tr belong to the segment [Xo, Yo]; for every i (i = 2, ... , r - 1) d(Ti-1 1Ti) = d(Ti, Ti+1) = r~l d(Xo, Yo) (i.e., the points T 21 ... , Tr-l 3. More generally, he showed that there are no infinite n-sharply transitive groups at all for n > 3. He was advised by H. Hopf and on this occasion he came in touch with work on ends by Freudenthal of the 1930's and work of Hopf done in the 1940's. This was a first serious contact with topological groups. About H. Hopf, I want to refer to Hilton [82]. As a final remark here, I would like to observe that Tits' study of projective lines was good warming up for the Tits systems of rank 1 that he would discover later. Since history may repeat itself, let us also mention that buildings of rank one are trivial.

7

The Cremona plane

Another major question raised by Libois dealt with the Cremona group of all birational transformations of the complex plane. In this context, projective points are no longer invariant and, in agreement with the ideas explained earlier, he suggested that there is a "Cremona plane" which requires an axiomatization in terms of primitive objects (Cremona points) invariant under the Cremona group. One of the leading ideas was to use the concept of infinitely near points that has now again got credit in the rather different context of non-Standard mathematics. Libois and his student Pierre Defrise made significant contributions to this theme between 1939 and 1949 (see [92]) and, in 1950, young Tits solved it completely over any algebraically closed field, in a remarkable piece of work that has remained unpublished and of which few copies have been circulated. We may wonder indeed whether it is not in danger of being lost. I do have a partial copy, made hastily by hand in the 1960s. Tits made use, in particular, of his results on the one-dimensional projective groups. Again, this has something to do with the genesis of buildings because of the novel view that was required on projective spaces and because of the underlying "relativity of points" that would again be crucial in order to clarify building geometries or geometries over Coxeter diagrams. If he finds the leisure, Tits may explain us what an apartment is in the Cremona plane and once we grasp this, the entire matter of Cremona geometry will be under potential control for those who understand buildings. This does by the way explain a little of the simplifying power of buildings . By 1951, Tits was involved a lot more in the study of Lie groups. They were so much simpler than the Cremona group but he learned about them by himself, in the works of E. Cartan and this was not such an easy task, a comment once made also by H. Weyl (we quote from Tits

[112]).

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_ _ _ _ The Rise of Incidence Geometry and Buildings in the 20th Century

8

The Helmholtz-Lie space problem

This asks for a common characterization of Euclidean and non-Euclidean spaces on the basis of a "free mobility" principle common to their motion groups. We refer to Freudenthal [71] for a detailed account. In 1930, Kolmogorov gave a solution in terms of axioms for a group of homeomorphisms acting on a metrizable connected locally compact topological space. He developed a recurrent set of axioms (Mn), n a natural number and showed that his spaces were either Euclidean or non-Euclidean. In 1952, Tits classified all spaces satisfying (Ml) only (see [10], [14], [24]). From his list he checked that the union of (Ml) and (M2) implies (Mn) for all n with few exceptions made explicit. These results were meaningful for general relativity and they were therefore a particular matter of pride in the Séminaire de Mathématique Physique - to be distinguished from "Physique Mathématique" - that Libois and his colleagues Debever and Géhéniau were running in Ilrussels from 1950 to 1959 (see Bruffaerts [55]). The tools used by Tits had much to do with homogeneous spaces of Lie groups namely a pair consisting of a Lie group and some subroup which would behis basic starting point in order to create-discover buildings. In the words of Freudenthal [71] "the question had finally been settled by Tits". Freudenthal, instigated by [10], got the result under weaker conditions and by a different proof (see [70]) putting the original problem and its solution in a "final" form.

9

Exceptional Lie groups and Dynkin diagrams

In 1950, the simple complex Lie groups had been known for 60 years and they had been studied in many ways especially by Elie Cartan. We refer to [48],[61],[62]. The four infinite classes of simple Lie groups have a geometric interpretation in terms of automorphism groups of projective spaces and of their polarities that are symplectic, orthogonal or unitary. These are also called the classical groups after the famous book by H. Weyl. There are five exceptional groups called E 6 , E 7 , E 8 , F4 and G 2 represented in dimensions 14, 24, 27, 56 and 248. They had been discovered by Killing [85] in his classification work of spaces under the influence of Weierstrass (see Hawkins [78]). Cartan finalized their status and studied them further. These groups were not understood as symmetry groups of spaces. In his 1992 lecture [112] Tits explained the crucial importance of the exceptional groups. According to A. Borel he said, it is moral to study questions that apply also to the exceptional groups. Tits added that the exceptional groups made the theory of Lie groups live further till the present time. The history of the exceptional groups is related to the foundations of geometry as pointed out by Freudenthal [71]: " Though forshadowed by von Staudt's and Wiener's work, Hilbert's discovery of the relation between

245

Francis Buekenhout - - - - - - - - - - - - - - - - - - - - - - - -

geometric incidence theorems ( lock theorems ) and axioms for algebraic structures, is the most striking feature of his Grundlagen der Geometrie of 1899. To formulate these relations, we choose projective geometry instead of Euclidean, as did Hilbert. Adding Desargues theorem as a "lock incidence theorem" to the "trivial" incidence axioms, one gets a class of geometries which can be described algebraically as that of projective geometries over a ( non-necessar ily commutative ) field. Adding Pappus-Pasc al theorem algebraically means postulating commutativi ty. A momentous progress - the first one after Hilbert in the realm of ideas - was marked by 1foufang's discovery and analysis of harmonie geometries.T he harmonie lock incidence theorem says that a harmonie quadruple is uniquely determined by its first triple, so it does not depend on its particular construction . Moufang showed that affine coordinatiza tion is still possible by means of the harmonie theorem though the underlying algebraic structure may fail to be associative with respect to multiplicatio n. Associativity was replaced by a weaker law, called alternativity . . .. An example of a harmonie non-Desargu esian plane geometry was provided by the alternative field of the octaves .... in 1914, Elie Cartan mentioned without proof a representation of a real form of the exceptional group G 2 as the automorphis m group of the octaves ". In the early 1950's further relationships between the octaves and the exceptional groups G 2 , F 4 , E 6 were obtained by various authors such as Chevalley-Sc hafer on the basis of Cartan's work, A. Borel who got the plane of octaves, obtained somewhat earlier by G.Hirsch in a topological context and Freudenthal (see [71]). In 1953, Tits was engaged in this stream of work [12], [13] and he started a series of contacts with Freudenthal that would lead to a "magic square" explaining all exceptional groups in relationship with the real field, the complex field, the quaternions and the octaves. (On this matter see also Rosenfeld [102]. These efforts allowed at last to see the exceptional groups as automorphis m groups of spaces. However, a uniform description of geometries valid for all simple Lie groups was still lacking and Tits started looking for this. This is the point where the most important step on the road leading to buildings may have been made and where Coxeter-Dyn kin diagrams enter the scene. For the evolution and history of these pictures that have more content than any other I can think of, I refer to Coxeter [65]. I would like to mention that they are related to the 27 lines on a cubic surface whose automorphis m group is the exceptional Coxeter group of type E 6 , hence to configuratio ns and incidence and I refer again to Gropp [76], [77]. Thanks to Dynkin, also Witt (according to Tits [111]) the groups were represented by a simple drawing: the diagram. Therefore Tits thought that the geometric objects he was looking for had to be quite simple also. In the case of a diagram of type An, the nodes correspond to points, lines, planes, ... , hyperplanes as Tits observed and the diagram displays the duality of projective

246

_ _ _ _ The Rise of Incidence Geometry and Buildings in the 20th Century

geometry. In the diagram of type D 4 we can likewise read the principle of triality. Hence, the diagrams are among other things an "aide mémoire". This theme was started by Tits in [24]. Key results he used came from Morozov [95] and Karpelevic [83]. These authors had shown that in a simple Lie group, the conjugacy classes of those subgroups that are now called maximal parabolic subgroups, are in one-to-one correspondence with the nodes of the diagram. Tits, following the above interpretation of those subgroups in the case of the diagram An, choose a particular node, called this diagram a "Schliifli scheme", defined points as the subgroups corresponding to the particular node, subspaces as all maximal parabolic subgroups each of these being a set of points defined in an algorithmic way from the group and the subgroup. Later on Tits observed [26] that a geometry of type Es leads to a more symmetric set of axioms if points are no longer privileged and that 8 types of abjects are given together with an incidence relation. Hence, as in the Cremona plane, points bec:arne a relativistic concept. The impression may be that some geometric content was thus lost but according to Tits, in the lecture in honor of Libois, his choice would reveal a deeper geometric content that was not apparent earlier. From 1956 on, the following construction was available. Start from the group and its diagram. For any node i of the diagram, let the elements of type i of the geometry (once called subspaces of dimension i) be the maximal parabolic subgroups of type i and call two such elements incident if the intersection of the corresponding subgroups contains a Borel subgroup. Properties could be read from the diagram and more could be proved on that basis. Therefore, Tits started an axiomatic approach. It relies on the Coxeter-Dynki n diagram and geometries defined in some way over these. It involves an abstracted version of the classical concepts of subspace, quotient space and the vertex-figures of regular polyhedra and polytopes introduced one century before by Schliifli.

10

Chevalley groups and algebric Lie groups

After his constuction of a geometry over the exceptional groups, Tits realized that the axiomatization he produced allowed to find other properties exactly as in projective geometry. Thus he got a good intuition of his geometries and came to the idea that he might produce them over any ground field. This worked and he realized that program for E 6 and E1 thus producing, in [17], new finite simple groups. He was also on his way to extend his methods to Es when he became aware of Chevalley's construction [63]. His own project lost its interest as far as the construction of new groups was concerned. On the other hand, his former construction of geometries from the simple Lie groups carried over immediately to the Chevalley groups. This construction could be axiomatized and it would lead to the Tits systems in 1961. In [110], Tits spoke

247

Francis Buekenhout - - - - - - - - - - - - - - - - - - - - - - - -

of the "algebraization of the Lie-Cartan theory".

11

Generaliz ed polygons or buildings of rank 2

The classical groups can be associated to projective spaces, to their automorphism groups and to the centralizers of polarities (anti-automorph isms of order 2). It became clear to Tits that trialities of D 4 -geometries had to be classified and that their centralizers had to be analyzed. He did so and produced some of the classes of twisted Chevalley groups that were discovered independently by Steinberg (see [106]). While doing so, he was meeting new geometric objects whose properties inspired him the definition of generalized hexagons, hence generalized polygons. Buildings of rank 2 of spherical type were thus born and their apartments were rather explicit already as polygons or minimal circuits in the incidence graph.

12

The birth of buildings and Tits systems

By now, Tits had all pieces of the "building puzzle" in his hands. A uniform grouptheoretica l construction of diagram geometries from algebraic groups. A good grasp of their structure. A clear picture of the rank two buildings. The final touch could be given. The birth occurs in 1961 [47]. Tits refers to [24], [27] and [37] among other papers for a geometric interpretation of some groups, in particular a broad class of semi-simple algebraic groups over a not necessarily algebraically closed field. Here, he wants to give a general axiomatic frame to his results. He intoduces a set equipped with an incidence relation, a geometry and its residual geometries, examples including polygons, trees, polyhedra, projective geometries and coset geometries defined from the homogeneous spaces of a group provided with a family of 18 subgroups. Then he defines regular polyhedra, actually a generalization of the elementary regular polyhedra and polytopes; this new concept is now called a Coxeter complex. Here Tits refers to the work of Coxeter in which he got the finite Coxeter groups (1935), to his famous book Regular Polytopes and to [40], another major paper in this birth process of which few copies were circulated. It may be worth here to refer to Malkevich [93] for the history of polyhedra. Tits defines general diagrams that are now called Coxeter diagrams. They allow for any graph and any weight on each edge including an infinite weight. He recalls the main results of [40], in particular the geometric representation of a general Coxeter group that was taken over immediately in Bourbaki [53]. Now, he gets a polyhedral structure of type D where D is a Coxeter diagram and these are our buildings. Their structure is determined by skeletons now called apartments, that are isomorphic to the Coxeter complex. Basic theory is developed. Then the concept of a

248

_ _ _ _ The Rise of Incidence Geometry and Buildings in the 20th Century

Tits system is defined. Actually, Tits defines a relationshi p of admissibil ity of the action of a group on a building, he derives a Weyl structure from it in the group and starts proving properties that he would choose later on as axioms of a group with a BN-pair (see [45]) namely a Tits system as they were called by Bourbaki [53].

13

The theory of buildin gs

The theory of buildings (of spherical type) was well establishe d by 1974 but a great deal of it was anticipate d in a series of papers. We feel that this is another subject.

14

Remar ks

l. In order to put Tits' work in a broader mathemat ical context during the

20th century we refer especially to Pier [97] and Sasaki e.a. [103].

2. Tits' work, in particular his theory of buildings has to be understoo d in the context of a dialogue between the continuous and the discrete. 3. 1 am aware of the fact that I will have missed many interesting references for the present report and I would be grateful for additional material and comments . 4. The importanc e of buildings was understoo d right away by Bourbaki who included them already in one of his books, under the form of exercises, as early as 1968 [53]. On this occasion, Bourbaki created the real-estate terminolog y of buildings. Bourbaki acknowledges conversati ons with Tits. Therefore it is likely that Tits is one of the few persons who ever met Bourbaki. 5. Among the many reasons for which buildings do matter, one of the major theories is the classificati on of the finite simple groups for which we refer to Gorenstein [75]. 6. In the historical sketch leading to buildings we neglected another major branch of incidence geometry in the 20th century namely lattice theory. In this context Menger and other authors simplified the foundation s of projective geometry. A detailed account including references allowing to trace the historical evolution can be found in Birkhoff [52].

249

Francis Buekenhout _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ __

15

Tits' Bibliography from 1949 to 1962

I have been able to gather an almost complete set of Tits' papers and books with the kind help of Tits himself and with the help of Marcel Demeur who has collected them as well in the Service de Physique Nucléaire of the Université Libre de Bruxelles. The list provided here is copied fromTits [108]. except that his first four papers have been mentioned here as a unique item. At this time I am missing a copy of [11] thanks to the help of Prof. Ch.Binder who provided me with a copy of [9]. The manuscripts on the Cremona plane are from 1950 and have as title: - Caractérisation axiomatique de l'ensemble des points d'un plan crémonien. - Les courbes exceptionnelles. better than mine.

I would be grateful for a copy, possibly

These papers are quoted and discussed in Libois [92].

References 1. Généralisation des groupes projectifs I,11,III,IV. Acad.Roy.Belg., Bull.Cl.Sei.,

35, 197-208, 224-233,568-589, 756-773, 1949. 2. Groupes triplement transitifs et généralisations. Coloque d'Algèbre et de théorie des nombres du C.N.R.S., Paris, septembre1949, 207-208. 3. Généralisation d'un théorème de Kerékjarto. Ille Congrès Nat. Sei., Bruxelles, juillet 1950, 64-65. 4. Collinéations et transitivité. Ibidem, 66-67. 5. Les groupes projectifs: évolution et généralisations. Bull. Soc. Math. Belg., 3, 1-10 (1949-1950). 6. Sur les groupes triplement transitifs continus; généralisation d'un théorème de Kerekjarto. Compositio Mathematica, 9 , 85-96 (1951). 7. Généralisations des groupes projectifs basées sur leurs propriétés de transitivité. Mém. Acad. Roy. Belg.,27 (2), 115 pp. (1952). 8. Sur les groupes doublement transitifs continus. Comm. l\Iath. Helv., 26, 203224 (1952). 9. Caractérisation topologique de certains espaces métriques. Nachr. Oesterr. Math. Ges.,21/22 (Ber. III Oesterr. Mathematikerkongr., Salzburg, sept. 1952), p.51.

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_ _ _ _ The Rise of Incidence Geometry and Buildings in the 20th Century

10. Etude de certains espaces métriques. Bull. Soc. Math. Belg., 5, 40-52 (1952). 11. La notion d'homogénéité en géométrie. Sém. de synthèse sci.,Univ. Libre de Bruxelles, 1952, lp. 12. Le plan projectif des octaves et les groupes de Lie exceptionnels. Acad. Roy. Belg., Bull. Cl. Sci.,39, 309-329, (1953). 13. Le plan projectif des octaves et les groupes exceptionnels E5 et E1. Acad. Roy. Belg., Bull. CL Sei. 40, 29-40 (1954). 14. Sur l'article intitulé "Etude de certains espaces métriques". Bull. Soc. Math. Belg. 6, 126-127 (1953). 15. Espaces homogènes et groupes de Lie exceptionnels. Proc. Internat. Congr. Math., Amsterdam, sept. 1954, vol. I,495-496. 16. Etude géométrique d'une classe d'espaces homogènes. C. R. Acad. Sei. Paris, 239, 466-468 (1954). 17. Espaces homogènes et isotropes et espaces doublement homogènes, Ibid. 526527. 18. Sur les R-espaces. Ibid., 850-852. 19. Transitivité et groupes de mouvements. Schr. Forschungsinst. Mat., 1, (1957) (Ber. Riemann-Tagung, Berlin, oct. 1954), 98-111. 20. Groupes semi-simples complexes et géométrie projective. Séminaire Bourbaki, 112, llpp. (fév. 1955). 21. Sous-algèbres des algèbres de Lie complexes semi-simples. Séminaire Bourbaki, 119, 18 pp. (mai 1955). 22. Espaces homogènes et isotropes de la relativité. Helv. Phys. Acta, Suppl.IV, 1956 (Actes cinquantenaire théor. Relat., Berne, juillet 1955), 46-47. 23. Sur les groupes doublement transitifs continus: Corrections et compléments. Comment. Math. Helv., 30, 234-240 (1956). 24. Sur certaines classes d'espaces homogènes de groupes de Lie. Mém. Acad. Roy. Belg., 29 (3), 268 pp. (1955). 25. Sur la géométrie des R-espaces. Journ. Math. Pur. Appli., 36, 17-38 (1957). 26. Les groupes de Lie exceptionnels et leur interprétation géométrique. Bull. Soc. Math. Belg., 8, 48-81 (1956). 27. Sur les analogues algébriques des groupes semi-simples complexes. Colloque d'Algèbre Supérieure du C. B. R. M., Bruxelles, déc. 1956, 261-289.

251

Francis Buekenhout _________________________

28. Les "formes réelles" des groupes de type E5. Séminaire Bourbaki, 162, 15 pp. (fév. 1958). 29. Sur la trialité et les algèbres d'octaves. Acad. Roy. Belg., Bull. Cl. Sei., 44, 332-350 (1958). 30. Sur la trialité et certains groupes qui s'en déduisent. Pub!. l\fath. I. H. E. S. 2, 14-60 (1959). 31. Isotropie des espaces de Klein. Colloque de Géométrie différentielle globale du C. B. R. M., Bruxelles, déc. 1958, 153-161. 32. Les espaces isotropes de la relativité. Colloque sur la Théorie de la Relativité du C. B. R. M., Bruxelles, mai 1959, 107-119. 33. Sur la classification des groupes algébriques semi-simples. C. R.. Acad. Sei. Paris,249, 1438-1440 (1959). 34. Une remarque sur la structure des algèbres de Lie semi-simples complexes. Proc. Ned. Akad. Weten., A63 ( =lndag. l\.Iath.,22) 48-53 (1960). 35. Sur une classe de groupes de Lie résolubles. Bull. Soc. Math. Belg. 11 (2), 100-115 (1959). 36. Sur les groupes algébriques affins: théorèmes fondamentaux de structure; classification des groupes semi-simples et géométries asssociées. Centra Internazionale Matematico Estivo (C. 1. M. E. ), Saltino di Vallombrosa, sept. 1959; Rome, 1960, 111+74 pages. 37. Groupes algébriques semi-simples et géométrie associées. Proc. Coll. Algebraical and topological foundations of geometry, Utrecht, août 1959; Pergamon Press, Oxford, 1962, 175-192. 38. Les groupes simples de Suzuki et de Ree. Séminaire Bourbaki, 210, déc. 1960, 18 pages. 39. Sur les groupes d'affinités sans points fixe. Acad. Roy. Belg., Bull. Cl. Sei., 46, 954-956 (1960). 40. Groupes et géométries de Coxeter. Notes polycopiées, Inst. des Hautes Etudes Sei., Paris, juin 1961, 26 pp. 41. Sur une classe de groupes de Lie résolubles, corrections et additions. Bull. Soc. Math. Belg. 14 (2), 196-209 (1962). 42. Ovoïdes à translations. Rendiconti di Matematica, 21, 37-59 (1962). 43. Ovoïdes et groupes de Suzuki. Archiv der Math., 13, 187-198 (1962). 44. Espaces homogènes complexes compacts. Comment. Math. Helv., 7, 111-120 (1963).

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45. Théorème de Bruhat et sous-groupes paraboliques. C. R. Acad. Sei. Paris, 254, 2910-2912 (1962). 46. Une classe d'algèbres de Lie en relation avec les algèbres de Jordan. Proc. Ned. Akad. Wet., A65 (=Indag. Math., 24), 530-535 (1962). 47. Géométries polyédriques et groupes simples. Deuxième réunion du Groupement de mathématicie ns d'expression latine, Florence, sept. 1961, 66-88.

Other References 48. Akivis M. A. and B. Rosenfeld, Elie Cartan (1869-1951), translated by V. V. Goldberg, 1991. American Math. Society. Providence, RI. 1993. 49. Baer R. Linear Algebra and Projective Geometry, Academic Press, New-York. 1952. 50. Batten L. and A. Beutelspache r, The Theory of Finite Linear Spaces, Cambridge University Press, Cambridge, 1993. 51. Bensaude-Vin cent B. Langevin. Science et vigilance. Belin. Paris. 1987. 52. Birkhoff G. Lattice theory. American Math. Soc. Providence, R.I. 1967. 53. Bourbaki N. Groupes et algèbres de Lie. Herman, Paris. 1968. 54. Brouwer A., Cohen A.l\'1. and A. Neumaier. Distance regular graphs. SpringerVerlag. Berlin. 1989. 55. Bruffaerts X. Paul Libois : brouillon projet d'une biographie. licence. Université Libre de Bruxelles. 1994.

Mémoire de

56. Brown K. S. Buildings. Springer-Verl ag. Berlin. 1989. 57. Buekenhout F. (editor). Handbook of Incidence Geometry. Foundations. Elsevier. Amsterdam. 1995.

Buildings and

58. Buekenhout F. An Introduction to Incidence Geometry. In [58]. 1995. 59. Buekenhout F. A Belgian Mathematici an : Jacques TITS. Bull. Soc. Math. Belg. 42, 463-465, 1990. 60. Cartan E. Thèse. Paris. 1894. 61. Chern S. and C. Chevalley, Obituary: Elie Cartan and his mathematica l work. Bull. Amer. Math. Soc., vol 58, 1952, 217-250. 62. Chern S. S. Book Review of "Akivis M. A. and B. Rosenfeld, Elie Cartan (1869-1951), translated by V. V. Goldberg, 1991. American Math. Society, Providence, RI, 1993. Bulletin Amer. Math. Soc. 30, 95-96, 1994. 63. Chevalley C. Sur certains groupes simples. Tôhoku Math. J. (2) 7, 14- 66,1955.

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Francis Buekenhout - - - - - - - - - - - - - - - - - - - - - - - - -

64. Cohen A. M. Point-line spaces related to buildings. Chapter 11 in [57]. 1995. 65. Coxeter H. S. M. The Evolution of Coxeter-Dynkin Diagrams. Nieuw Archief Wiskunde, 4, 233-248, 1991. 66. Dahan-Dalmedico A. and J. Peiffer. Une histoire des mathématiques. Ed. du Seuil. Paris. 1986. 67. Dembowski P. Finite Geometries. Springer-Verlag. Berlin, 1968. 68. Dieudonné J. (editor) Abrégé d'histoire des mathématiques, 1700-1900, two volumes, Hermann, Paris, 1978. 69. Field J. V. and J. J. Gray. The Geometrical Work of Girard Desargues. Springer-Verlag. Berlin. 1987. 70. Freudenthal H. Neuere Fassungen des Riemannschen-Helmholtz-Lieschen Raumproblems, Math. Zeit. 63, 374-405, 1956. 71. Freudenthal, H. Lie Groups in the Foundations of Geometry. Math. 1,145-190 (1965).

Advances in

72. Freudenthal H. The impact of von Staudt's foundations of geometry. In Plaumann P. and K. Strambach (eds). Geometry-von Staudt's Point of View, 401425, Reidel. Dordrecht. 1981. 73. Guillaume M. Axiomatique et logique. Chapter 13 of [68], 315-430. 74. Guillaume l\1. La logique mathématique en sa jeunesse. In [98], 185-368, 1994. 75. Gorenstein D. Finite Simple Groups, An Introduction to their Classification, Plenum Press, New-York, 1982. 76. Gropp H. Enumeration of regular graphs 100 years ago. Discrete J\Iath. 101, 73-85, 1992. 77. Gropp H. On the history of configurations. Conference San Sebastian, 1990. 78. Hawkins T. Non-Euclidean geometry and Weierstrassian mathematics: the background to Killing's work on Lie algebras. Historia Math. 7, 289-342, 1980. 79. Hawkins T. The Erlanger Programm of Felix Klein: reflections on its place in the history of Mathematics, Historia Math. 11, 442-470, 1984. 80. Hermann R. Review of the book " The theory of spinors " by Elie Cartan. Amer. Math. Monthly. 90, 719- 720, 1983. 81. Hilbert D. Grundlagen der Geometrie. Teubner. Leipzig. 1899. 82. Hilton P. A brief, subjective history of homology and homotopy theory in this century. Math. Magaz. 61, 282-291, 1988.

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_ _ _ _ The Rise of Incidence Geometry and Buildings in the 20th Century

83. Karpelevic F. On the (non-semi-simple) maximal subalgebras of semi-simple Lie algebras (russian), Doklady Akad. Nauk. SSSR. 76, 775-778, 1951. 84. Karzel H. and H. J. Kroll. Geschichte der Geometrie seit Hilbert. Wissenschaftliche Buchgesellschaft. Darmstadt. 1988. 85. Killing W. Die Zusammensetzung der stetigen endliche Transformationsgruppen. l\Iath. Ann. 36, 161-189, 1890. 86. Kline M. Mathematics. A cultural approach. Addison-Wesley. Reading, MA. 1963. 87. Kline M. Mathematical Thought from Ancient to Modern Times. Oxford University Press. Oxford. 1972. 88. Libois P. De l'espace métrique à l'espace projectif. C. R. Ile Congrès National des Sciences. Bruxelles. 96-104. 1935. 89. Libois P. Synthèse des axiomatiques de l'algèbre et des géométries projective, affine et affine centrale. Congrès de l' Association française pour l'avancement des sciences, 77-82, Paris.1945. 90. Libois P. Espaces, géométries, groupes. C. R. IIIe Congrès National des Sciences. Bruxelles. 68-70; 1950. 91. Libois P. Les espaces. Thone. Liège. 1951. 92. Libois P. Elaboration d'une axiomatique du plan de Cremona. Atti Convegno Inter. Geom.Celeb. Centena. Nasci. Federigo Enriques. 115-123. Accad. Nazionale dei Lincei. Roma. 1973. 93. Malkevitch, J. Milestones in the History of Polyhedra. In Ed. Sénéchal M. and G. Fleck, Shaping space-A polyhedral approach. 80-92. Birkhauser. Basel. 1988 94. Mielants W. Can the best of all possible worlds be a random structure? To appear in Logique et Analyse. 1995. 95. l\Iorozov V. V. The non-semi-simple maximal subgroups of simple groups (Russian). Thesis, Kazan, 1943. 96. Pickert G. Pasch-oder Veblen-Axiom? Geo. Ded. 50, 81-86, 1994. 97. Pickert G. Projektive Ebenen. Springer-Verlag. Berlin. 1955. 98. Pier J.-P. (editor). Development ofmathematics 1900-1950. Birkhauser. Basel. 1994. 99. Reid, C. Hilbert. Springer-Verlag. Berlin. 1970. 100. Ronan M. Lectures on buildings. Perspectives in Math. 7 . Academic Press. London. 1989.

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Francis Buekenhout - - - - - - - - - - - - - - - - - - - - - - - - -

101. Ronan M. Buildings: main ideas and applications. Bull. London Math. Soc. 24 (1992) 1-51 and 97-126. 102. Rosenfeld B. A. A History ofNon-Euclidean Geometry-Evolution of the concept of space. Springer. Berlin. 1988. 103. Sasaki C., Sugiura M. and J. Dauben ( editors ). The intersection of History and Mathematics. Birkhiiuser Verlag. Basel. 1994. 104. Scharlau R. Buildings. Chapter 11 in [57]. 1995. 105. Segre B. Lectures on Modern Geometry, Cremonese, Roma, 1961. 105. Steinberg R. Variations on a theme of Chevalley. Pacifie J. :Math. 9, 875-891, 1959. 106. Stillwell J. :tviathematics and its history. Springer-Verlag. Berlin. 1989. 107. Tits J. Titres et travaux scientifiques de Jacques Tits. 1972. 108. Tits J. Buildings of Spherical type and Finite EN-Pairs, Lecture Notes in Math, vol 386, Springer, Berlin. 1974 109. Tits J. Leon inaugurale à la chaire de théorie des groupes, Collège de France, Paris, 1975. 110. Tits J. Buildings and Buekenhout geometries. In Ed. M. Collins, Finite Simple Groups II, Academic Press, New-York, 309-320, 1981. 111. Tits J. La théorie des groupes de Lie semi-simples: !'oeuvre d'Elie Cartan et Hermann Weyl (1900-1950). Lecture given at the conference "Development of Mathematics" (see [98]) in Bourglinster, Luxembourg, 1992. 112. Valette A. Quelques coups de projecteurs sur les travaux de Jacques TITS à l'occasion de la remise du prix Wolf. Gazette des mathématiciens, n 61, 61-79, 1994. 113. Veblen O. and W. H. Bussey. Math. soc. 7, 241-259, 1906.

Finite projective Geometries, Trans. Amer.

114. Veblen O. and J.W. Young. A set of assumptions for projective geometry. Amer. J. Math. 30, 347-380, 1908. 115. Veblen O. and J. W. Young. Projective Geometry, Blaisdell, New-York, vol 1, 1910 ; vol 2, 191 7. 116. Zassenhaus H. Über endliche Fastkürper. Abhand. Math. Sem. Hamburg. 11,187-220 (1935). 117. Zassenhaus H. Kennzeichnung endliche linearer Gruppen als Permutationsgruppen. Abhand. Math. Sem. Hamburg. 11, 17-40 (1935).

256

Benno Klotzek

Diskontin uierliche Bewegung sgruppen verschied enen Grades in verschied enen Geometri en

Abstract Because of new applications, the research of discontinuous groups with respect to different geometries has made considerable progress. One can state that conditions which all characterize the same discontinuous motion groups in Euclidean spaces of the same fini te dimension define different classes of discontinuous transformation groups in other geometries. Moreover in the following section 1, sixteen sonditions are discribed which define only eight conditions in metric spaces (section 2). There are results about new classes of so-called Di - and D;-discontinuou s motion groups of the Euclidean geometry in section 3: For instance, there are new generalized rosette groups, and new generalized frize groups of all seven typs, but only nine typs of the seventeen typs of ornamental groups has generalizations, four of them with two different generalizations. The sixteen conditions of section 1 are each other distinct conditions in equiaffine planes (section 4). Finally, we give new results about the discontinuons motion groups of the real pseudo-Euclidea n plan with close connections with the number theorie.

AMS Subject Classification: 51 Key words and Phrases:: discontinuons transformation groups of Euclidean and non-Euclidean geometries, discontinuous motion groups and generalized discontinuous motion groups, Di- and D;-discontinuou s motion groups, equiaffine planes, real pseudo-Euclidea n planes and number theory

Diskontinuierlic he Bewegungsgrup pen sind als Symmetriegrup pen von Mustern M intiutiv zu erfassen (Abb. 1): Hier ist die ,,Wiederholung " in vertikaler und horizontaler Richtung zu erkennen, wenn man sich das Muster in die gesamte Ebene fortgesetzt denkt. Neben den eine ,,Wiederholung " charakterisieren den Verschiebungen sind Axial- und Zentralsymmet rien zu erkennen, aber M bleibt auch bei gewissen Schubspiegelun gen fest. (lm allgemeinen sind neben 2- (Zentralsymme trie) noch 3-,4-, 6-ziihlige Drehzentren moglich.) Diskontinuitiit einer Bewegungsgrup pe B wird hier wie in der Kristallographi e mittels

257

Benno Klotzek - - -- - - - -- - - - - -- - -- - - - - - -

Abb. 1 Punktorbits bzw. -bahnen P 8 := {PTIT E B} definiert; im folgenden werden wir B f:. {id} voraussetzen. lm Rahmen der euklidischcn Geometrie end licher Dimension findet man ais (untereinander aquivalente) charakterisierende Eigenschaften diskontinuierlicher Bcwegungsgruppcn beispiclsweise (LEO)

V

P,X,r>O

card (Ur(X)

n P 8 ) < oo

(lokale Endlichkeit der Orbits nach Hilbert und Cohn/Vossen gemiiB [5]) (IPO)

V 3

P r>O

card (Ur(P) n P 8

)

=1

(Isoliertheit der Punkte in ihrem Orbit nach L. Fejes T6th gemaJ3 [2]). Die Frage, ob diese Bedingungcn stets aquivalent sind, hat die spatere Einführung cines ganzen Systems von ahnlichen Bedingungen initiert. In bezug auf LEO/ IPO wurden Beispiele für Bewegungsgruppen mit isolierten Orbitpunkten (IPO) angegeben, deren Orbits nicht, lokal endlich (-.LEO) sind. Es ergab sich einerseits, da6 durch LEOund !PO in jedem unendlich dimcnsionalen Hilbertraum vcrschiedene Klassen von Gruppen beschrieben werden (vgl. [10]); andererseits sind LEO und !PO in jcdcm cndlich dimensionalen normierten Raum gleichwertig, ja sogar in den metrischen R.aumen, in denen jedc beschrankte Menge total beschrankt (priikompakt) ist (vgl. [9]) . In diesem Bericht sollen Bedingungen, die durch naheliegende logische Abwandlungen aus LEO und IPO hervorgehen, in metrischen, aquiaffinen und pseudoeuklidischen Raumen untersucht werden. Insbcsondcre gcht es um Àquivalenzen, Abhiingigkeiten und Unabhangigkeiten.

1

Diskretheitsbeding ungen

Spezieller als die lokale Endlichkeit ist Endlichkeit, etwa von Rosettengruppen bekannt; aile anderen Bedingungen entstehen durch formale Abschwachung von

258

_ Diskontinuie rliche Bewegungsg ruppen versch. Grades in versch. Geometrien

LEO;mittels der Abkürzunge n cardpxr := card (Ur(X) n PB) und F(B) ·{Y: yB ={Y}} lauten sie:

(ü) Endlichkeit der Orbits (EO) (1)

(2)

V P,X,r>O

V

P,r>O

cardpxr cardppr

<
O


O

cardpxr Sn

(5' )

V 3 3 Ptf.F(B) X r>O

cardpxr Sn

(6n) V 3 P r>O

cardppr Sn

(6')

3 3 Prf.F(B) r>O

cardppr Sn

(7n) 3 V 3 X P r>O

cardpxr Sn

(7')

3 3 Ptf.F(B) X,r>O

cardpxr Sn

(8n) V 3 P X,r>O

cardpxr Sn

(8~) = (7~)

n

n

n

formulieren, wobei über n E N noch geeignet zu verfügen ist. Hierbei stimmen (3') und (4') sowie (7~) und (8~) überein; weitere Einsichten (z.B. in bezug aufUnabhiin gigkeit oder Àquivalenz) sind erst zu erwarten, wenn wir die Sprache, der wir uns hier bedienen, vor der Orbit eines Punktes 1 in par(i::, P) dicht. - Es kann keine zu E nicht parallele Ebene T/ mit derselbei Eigenschajt geben.

f) Wenn Dg und Vg die Bedingung (ii) erfüllen, dann liegt bezüglich der Orbit eines Punktes P f:c g im Zylinder Z(g, P) um g durch P dicht - Es kann keine Gerade h i g mit derselben Eigenschajt geben.

Vg

g) Wenn Sg die Bedingung (ii) erfüllt, dann liegt bezüglich Sg der Orbit eine. Punktes P f:c g entweder in einer Schraubenlinie (Sg heiJ3e dann besonden Gruppe) oder im Zylinder Z(g,P) dicht. - Eine Gerade h of g mit derselber Eigenschajt kann es nicht geben. Auf der Grundlage dieser Aussagen konnen D 3 - bzw. D~-diskrete Bewegungs· gruppen gefunden werden, die IPO nicht erfüllen; zum Vorgehen im ebenen Fall im wesentlichen nur einen ersten Eindruck vor der Reichhaltigkeit des Zuwachses vermitteln und gewisse AusschuJ3bedingun· gen angeben; eine Übersicht über den gesamten Zuwachs bleibe einer spiiterer Veroffentlichung vorbehalten.

Fall 1. Für alle Punkte 0 sind alle Untergruppen B der Bewegungsgrupp1 {TE B: QT = O} stcts D 3 -diskrete Bewegungsgrup pen. Zu diesen Bewegungsgrup pen gehoren nicht nur die endlichen Punktgrupper (vgl. [6]), sondern auch Gruppen B, die -.IPO erfüllen. Man gewinnt eine Klassifizierung solcher Gruppen mit Bezug auf Einteilungen der endlichen Punktgruppen:

A. Reduzible Punktgruppen mit der Achse g 3 0 An die Stelle der zyklischen Untergruppe der Drehungen um g werde eine beliebige Untergruppe Dg der Gruppe aller Drehungen um g gesetzt (beispielsweise mit den Drehwinkeln 27rm/3n,m E Z,n = 1,2, ... ), so daJ3 Dg die Bedingung -.IPO erfüllt. Weiterhin seien sa, sh, s 6 die Spiegelungen am Punkt 0, an der Geraden h mit 0 E hl_g bzw. an der Ebene E mit g, h c E. Dann erfüllen neben Dg auch Dg U shDg, Dg U soDg, Dg U shDg U so(Dg U shDg), Dg U s 6 Dg, Dg u SsDg u sh(Dg u SsDg), Dg u SjSsDg mit 0 E f, SJSsSJSs E Dg die Bedingung -.IPO, d.h., dajJ alle Typen reduzibler Punktgruppen einer Verallgemeinerung fahig sind, wobei nur g eine oo-zahlige Drehachse ist. Es sei auch noch bemerkt, daJ3 einige von ihnen zwar Db-, aber nicht D 0 -diskret sind. B. Irreduzible Punktgruppen Soll eine irreduzible Punktgruppe analog ein Beispiel der gesuchten Art, wobei die Menge der Punkte, in denen oo-zi:ihlige Drehachse n eine Kugel K(O,P) treffen, in K(O,P) dicht liegt. - Es gilt stets V = E; zu kli:iren ist noch, welche wesentlich verschiede ne Gruppen B+ mit wenigstens zwei oo-zi:ihligen Drehachse n existieren, welche Anreicher ungen von B+ als gesuchte Verallgem einerungen in Frage kommen.

Fall 2. Für alle Geraden g sind die Untergruppen B der Bewegungsgruppe {TE lIB: g7 = g} stets D~-diskrete Bewegungsgruppen. Hierunter fallen offenbar die reduziblen Punktgrup pen mit der Achse g, aber nicht nur sie; einfachste Beispiele mit -ilPO sind: (1) Gruppen Vg, die die Bedingung -ilPO erfüllen, sowie Gruppen, die aus einem V g wie die verallgeme inerten Friesgruppen konstruier t werden konnen. (2) Besondere Gruppen Sg, die die Bedingung -ilPO erfüllen. Die Elemente solcher Gruppen (2) lassen sich durch Paare ,,Drehwink el, zugehëiriger Vortrieb li:ings g" beschreibe n. Für eine feste Zahl À E JR\ {O} bestimmt die Signatur

offenbar ein Beispiel von (2); die Gruppe ist wegen der Irrationalit i:it von 7r translation s- und wegen À -1- 0 rotationsfr ei. Nicht ganz so leicht ist eine nichtbesondere Gruppe Sg anzugeben :

(3) Die von den Schraubungen der Signatur

(n-n,e- 2 n),n EN zyklisch erzeugten Gruppen S~ erzeugen eine translations- und rotationsfreie Gruppe B, die die Bedingung -ilPO erfüllt. Die Eigenscha ften von B beruhen u.a. auf der Transzend enz von 7r und e. Allgemein kann eine beliebige Untergrup pe B+ der Gruppe aller Verschiebu ngen li:ings g sowie Drehungen und Schraubun gen mit der Achse g Grundlage einer Gruppenk onstruktio n sein, bei der mit Spiegelung en an Ebenen c mit g C c oder gl_s sowie Dreh- bzw. Schubspie gelungen mit der Achse g angereichert wird. - Insbesond ere gilt V= Vg.

Fall 3. Für alle Ebenen c sind alle Untergruppen B der Bewegungsgruppe 7 { T E lIB : c = E:} stets D~-diskrete Bewegungsgruppen. Hierunter fallen die reduziblen Punktgrup pen mit der Fixe bene E:, aber auch andere Bewegung sgruppen, die -ilPO erfüllen. Die einfachste n Fi:ille sind:

265

(4) Es sei g eine Gerade mit g1-c derart, daj] V g C V gilt und gung -.!PO erfüllt.

Dg

die Bedin-

Da E als Fixe bene vorausgesetzt wird, folgt hier zuniichst V 6 = V und V g = E; insbesondere existiert wegen V g C V ein T -=/= id in V 6 und damit eine weitere, zu g parallele Gerade h := g" derart, daf3 Dh die Bedingung -.IPO erfüllt. Falls es in B die Spiegelung an einer Ebene 77 oder Dreh- bzw. Schubspiegelungen bezüglich 77 gibt, muf3 77 = c oder 771-E gelten, denn sonst wiire (i) verletzt. Falls es in B noch Drehungen oder Schraubungen mit der Achse f gibt, muf3 f C E oder f 1-E gelten; im ersten Fall ist dann der Drehwinkel 7r. Offenbar sind die Gruppen < Dg, U >, wobei Dg die Bedingung -.!PO erfüllt und U eine beliebige Untergruppe der Gruppe aller Verschiebungen von c ist, Beispiele der gesuchten Art.

(5) In E seien g und h nicht parallele Geraden, für die V g oder V h die Bedingung -.!PO erfüllt, wenigstens jedoch V g -=/= E und V h -=/= E gilt. Dann ist < V g, V h > ein Beispiel der gesuchten Art. Auch dieses Beispiel lief3e sich variieren. Für eine Erweiterung B von < Vg, Vh > gilt, was unter (4) gesagt wurde.

Fall 4. Sonstige

D~-diskrete

Bewegungsgruppen.

Infolge der angegebenen Ausschluf3bedingungen sind in vielen Fiillen die Erweiterungsmoglichkeiten schon umrissen. Falls wir beispielsweise von (4) und (5) ausgehen, konnen wir mit Verschiebungen 1-c erweitern. Um (i) nicht zu verletzen, muf3 es dann unter allen Abstiinden p(E, c") für T E V\ V 6 einen kleinsten p geben, und alle anderen sind dann ganzzahlige Vielfache von p.

4

Aquiaffi.ne Ebenen

Für positive reelle Zahlen r tritt an die Stelle der Umgebungen U,.(X) eines metrischen Raumes hier eine formal ebenso gebildete Kreisscheibe

wobei v(XY) 2 durch eine feste (evtl. nur semidefinite) quadratische Form bestimmt werde. Das ergibt auch nicht konvexe Kreisscheiben (Abb. 4). Auf3erdem ist die Menge {Y : lv(XY) 2 = O} nur im Falle der positiv definiten quadratischen Form einelementig; sie kann ein Paar sich schneidender (isotroper) Geraden i,j, eine (isotrope) Parabel p oder eine (isotrope) Gerade i sein. Wie bereits bezüglich der pseudoeuklidischen Ebene mit Kreisscheiben gemiif3 Abb. 4b bemerkt wurde, kann aus einem endlichen Durchschnitt U,.(X) n pB nicht mehr allgemein auf die Existenz einer reellen Zahl r > 0 mit cardpx,. ~ 1 geschlossen werden (vgl. [8]). 1

266

_ Diskont inuierlic he Bewegu ngsgrup pen versch. Grades in versch. Geomet rien

X

a)

r X

c)

d) Abb. 4

Deshalb wurden die Bedingu ngen (5n) - (8n) und (5~) - (8~) mit Riicksic ht auf Gruppe n B aquiaffi ner Transfo rmation en formulie rt, obwohl in metrisch en Raumen stets die Schrank e n = 1 genomm en werden kann. Eine Gruppe B aquiaffi ner Transfo rmation en heif3t beziiglich einer Kreissch eibe K Bewegungsgruppe genau dann, wenn alle TE Beine Zerlegu ng T 1T 11 besitzen , wobei T 1 die K begrenz enden Kurven 2. Ordnun g invarian t faf3t und 11 T eine Transla tion ist, die im Fall der Parabel n zusatzli ch deren gemeins ame Achse auf sich abbildet .

Für die aquiaffine Ebene werden abgeschwachte Bedingu ngen bei verallge meinertem Bewegungsbegriff in Verbindung mit zugeordneten Kreisscheiben betrachtet, aber gegebenenfalls auch Transfo rmation sgruppe n, die keine Bewegu ngsgruppen sind. In der Abb. 5 wurden die logischen Abhang igkeiten dargeste llt: Aufgrun d der Kenntni sse über Inaquiv alenzen in metrisch en Raumen konnten solche Beziehu ngen mit einem 0 mitcardpxr::; 1. b) Wenn die K reisscheiben K nicht hyperbolisch sind und B eine Bewegungsgruppe bezüglich K mit cardpxr < oo ist. dann gibt es ein r > 0 mit cardp xr ::; 2.

...

~

~..

Abb. 5

c) Wenn B bezüglich einer Kreisscheibe K Bewegungsgruppe ist und cardpxr

oo gilt, dann existiert ein r > 0 mit cardpxr ::; 8; dabei kann 8 durch 4 ersetzt werden, falls B = B+ nur orientierungserhaltende Bewegungen enthalt (Abb. 6).


.j) genau dann (6 1 ), wenn >. schlecht approximierbar ist; dabei ist eine schlecht approximierbare Zahl >. E Ill folgendermajJ en charakterisiert: 3

V

c=c(À) p,qEZ;q>O

i>.- 12q 1>-%q

Eine Gruppe pseudoeuklid ischer Bewegungen ist offenbar nur dann diskret bezüglich der eigenen Kreisscheibe n, wenn sie bezüglich der gewéihnlichen Topologie diskret ist. Deshalb konnten auf der Grundlage von [1] und obiger Aussagen zuniichst (6~)-diskrete Bewegungsg ruppen gemiif3 der obigen Definition aufgeziihlt werden; inzwischen wurden in [17] alle (8~)-diskreten Bewegungsgruppen aufgelistet, da der mittels

(8')

3

3

PrfcF(B) X,r>O

cardpxr < oo

eingeführte Begriff der sogenannten schwachdisk reten Bewegungsg ruppe mit

2()9

dem Begriff der (80)-diskreten Bewegungsgruppe zusammenfallt: Unter Berücksichtigung der zwar weniger geometrisch als physikalisch motivierten Begriffe der Raum- bzw. Zeitartigkeit werden in [17] von Alpers 17 Typen von Rosetten-, 12 Typen von Fries-, 13 Typen von Wandmustergruppen ohne nichtidentische Drehungen und die so bezeichneten Serien G[N,k,l] (a) - aus hyperbolischen Drehungen und Verschiebungen bestehend - mit 47 abgeleiteten Unterserien aufgeführt. Angesichts der Ergebnisse über D3-diskrete Bewegungsgruppen der euklidischen Geometrie seien die Siitze 2.2.1 und 2.3.6 aus [17] über die Erzeugung der Translationsuntergruppe V der Fries- bzw. W andmustergruppen aus [17] hervorgehoben:

Sie werden von einer bzw. von zwei nichtparallelen Verschiebungen erzeugt. Bezüglich der anderen Diskretheitsbedingungen wird in [17] die Ermittlung der wesentlichen Bedingungen (vgl. den Abschnitt 2) vermif3t. Aus der facettenreichen Untersuchung der ermittelten (80)-diskreten Bewegungsgruppen in bezug auf das Erfülltsein der schiirferen Bedingungen aus 1 sollen hier nur noch folgende bemerkenswerte Siitze aus [17] mitgeteilt werden:

a) Die Wandmustergruppen W1, W 1\,, Wfr, îVfr, l-Vfz, Wfz, îVf W2, WJ, W~l, W1r, W1z, Wi sind (6D-diskret, wenn ihre Translationsuntergruppen von zwei 2

,

À-vertraglichen Translationen erzeugt werden. Andernfalls kann (6~) für keine natürliche Zahl n erfüllt werden. - b) Alle W andmustergruppen mit Drehungen sind (6~)-diskret. - 3.2.6, S. 77 Eine Wandmustergruppe ist (4')-diskret bzw. (4)-diskret genau dann, wenn sie nichtidentische Verschiebungen in beiden isotropen Richtungen enthalt. - 3.4.3, S. 82 Zu keiner natürlichen Zahl n gilt es (8n)-diskrete Wandmustergruppen mit nichtidentischen Drehungen. - 3.6.4, S. 92 W 1 ist (6 1 )-diskret, wenn die Gruppe von zwei À-vertraglichen Translationen erzeugt wird. Es gibt keine weiteren (6n)-diskrete Wandmustergruppen (n EN). - 3.8.1, S. 97

References [1] Baltag, I.A., i V.P. Garit: Dvumernye diskretnye affinnye gruppy. Izdat. Stiinca, Kisinev 1981. [2] Fejes T6th, L.: Regular Figures. Pergamon Press, Oxford 1964. - Reguliire Figuren. Akademiai Kiado Budapest/B.G. Teubner Verlagsgesellschaft Leipzig 1965.

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_ Diskontinu ierliche Bewegung sgruppen versch. Grades in versch. Geornetrie n

[3] Flachsmeye r, J.: Zwei Orbittheore me für Bewegungs gruppen. Beitr. zur Algebra und Geometrie 29 (1989), 233-240. [4] Flachsmeye r, J.: When Do Discrete Groups Act Discontinuo usly? Coll. math. soc. Janos Bolyai 55. Topology, Pécs 1989, 217-222. [5] Hilbert, D. und ST. Cohn-Vosse n: Anschaulic he Geometrie. Verlag von Julius Spinger, Berlin 1932. [6] Klemm, J\1.: Symmetrie n von Ornamente n und Kristallen. Springer Verlag, Berlin/Hei delberg/Ne w York 1982. [7] Klotzek, B.: Ebene iiquiaffine Spiegelung sgeometrie. Math.Nach r. 55 (1973), 89-131. [8] Klotzek, B.: Sorne Discontinu ous Groups in non-Euclid ean Geometries . Preprint PHPl/83. - Discontinu ous Groups in some Metric and Nonmetric Spaces. Beitriige zur Algebra und Geometrie 21 (1986), 57-66. [9] Klotzek, B.: Discontinu ous Groups in Normed Spaces. Coll. math. soc. Janos Bolyai 48. Intuitive Geometry, Si6fok 1985, 299-316. [10] Klotzek, B.: Diskrete Bewegungs gruppen in metrischen Riiumen. Wiss. Z. PHP 32 (1988), 172-173. [11] Klotzek, B.: Verschiede ne Diskretheit sbegriffe in metrischen Riiumen. Beitriige zur Algebra und Geometrie 29 (1989), 121-129. [12] Klotzek, B. und H. NIMZ: Verschieden e Diskretheit sbegriffe in der iiquiaffinen Ebene. Beitriige zur Algebra und Geometrie 30 (1990), 69-77. [13] Klotzek, B.: Über diskrete Bewegungs gruppen der pseudoeukl idischen Ebene. Wiss. Z. PHP 34 (1990), 93-97. [14] Klotzek, B.: Diskontinu ierliche Bewegungs gruppen verschieden en Grades in metrischen Riiumen. Journal of Geom. 52 (1995), 130-141. [15] Schülzke, J.: Zur Euklidizitii t von endlich-dim ensionalen Minkowski- Riiumen. Diss., Potsdam 1985. [16] Nimz, H.: Diskrete Gruppen iiquiaffiner Transforma tionen Berücksich tigung nichteuklid ischer Geometrien . Diss., Potsdam 1988.

unter

[17] Alpers, K.: Die diskreten Bewegungs gruppen der pseudoeukl idischen Ebene. Diss., Potsdam 1995.

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David G. Glynn *

On N on-spread Representations of Projective Planes of Ortler qi in PG(2i, q) Abstract Using the notion of special tangent it is shown that there are precisely two kinds of representation on a cone with planar vertex of a projective plane of order q 3 in PG(6, q). The Desarguesian plane PG(2, q3) admits both kinds of representation. Non-spread representations of PG(2, qi) in PG(2i, q) can be found for all i 2: 3, and we give three methods, one algebraic and two geometric, to construct these. Sorne indications of the applications of such representations are given, including the representation of subplanes and conics of PG(2, q3 ) in PG(6, q). These non-standard representations can be generalised to representations of PG(n, q), and are also valid for projective geometries over infinite fields.

AMS Subject Classification: 05 B 15, 05 B 25, 51A15, 51A30, 51A45, 51E15 Key Words and Phrases: orthogonal array, projective space, plane, cone-representation

1

Introduction

The representation of a structure on a cone was studied in [6], [7] and [8]. See also [9] for a more comprehensive survey. First, the types of cones considered were point-cones (1-cones, or cones with points as vertices), but now the theory can explain (i-1 )-cones. An (i-1 )-cone of a projective space is the set of points on a collection of i-dimensional subspaces through a fixed (i - 1)-dimensional subspace, that is called the vertex. Let us now summarize a way that these representations can be constructed. We start off with a kind of incidence structure with parallelism, and for a faithful representation we assume that the order of the structure is qi, where q is the order of some field G; that is, q is a prime-power if finite, or is infinite. In order to have the representation of the structure on an (i - 1)-cone in PG (n, q), the projective geometry of dimension nover G (= GF(q) if q is finite), we first couvert the structure into a combinatorial array, which is a k x m array with

*

This research was supported by the GNSAGA of CNR in Italy.

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set of elements E, where IEI = qi. This conversion is the same as that which produces an orthogonal array from a dual k-net, or from a projective plane with specified point. Thus, the structure has a kind of parallelism: ie. a set of k lines passing through a particular point V. On each line there are qi points other than V, and so the whole structure has kqi + 1 points. On each line through V we label the points other than V with the elements from E in a 1-1 fashion. Then the k rows of the array correspond to the lin es through V. The m columns of the array correspond to the m lines of the structure not through V: each intersects any line through V in a unique point. The label of that point is then put into the combinatorial array in the corresponding position. Given the array we can easily reconstruct the structure by reversing the argument. Now we can relabel the array (or equiYalently the points of the structure not equal to V) with members of the vector space Gi, which has the same size as E. This can be done independently for each row of the array, or equivalently independently for each line through V. This new labelling is then called a substitution. Once it has been chosen there is a unique cone-representa tion that can be constructed: it is faithful if the labelling on each row is a bijection. See [8] or [9] for further details. We denote a d-dimensional subspace of the projective space PG(n, q) by [d]. The cone-representa tion of the structure has points that are on an (i -1)-cone of PG(n, q), where n = i + r - 1, and i is the index, and ris the rank of the representation. This cone has vertex W, that is an [i -1], and has k generator [i] 's passing through W. (These spaces correspond to the lines of the structure through V.) Them lines not through V correspond tom secant [r - l]'s skew to W that eut the cone in sets of k points that are always isomorphic to the base curve. Later in this paper we show how to construct various representations , and so there is no need for an example at this stage. The following ideas were introduced (with slightly different notation) in [9]. It seems that they are crucial to a better understanding of cone-representa tions. Definition 1.1 (i, s )-fibration An (i, s )-fibration ( s . 0 , >. 1 ) of PG(l, q) be

c i-t ( Ào

À1 ) (

co/3

C1

ci Co+ C1a

) ,

\Je

E

GF(q 2 ).

We can check easily that the Bose-representation is obtained. See below for the general case for all n. It is interesting to note that if we fix >.0 , ..\ 1 for all rows x of the OA, then we actually obtain the spread-representation of PG(2,q 2 ) in PG(4,q), ofrank 3 and index 2. Thus in some sense the Bose-representation of rank 4 is an "amalgamation" of representations of one less rank. We can generalise the Bose-representation to representations of PG(2, qn) of rank 2n and index n. Let us briefly see how to do that via a simple substitution of the OA. The rows of the OA are labelled by z E GF(qn) U {oo}, and the columns by (a, b) E G F( qn )2 . Suppose that the element in the general position of the OA is za + b. In the oo row put a in the (a, b)'th position. Let the elements of GF(qn) be represented by n x n matrices over GF(q) using a matrix satisfying an irreducible equation of degree nover GF(q). As notation replace x E GF(qn) by a matrix X. For each row z of the OA choose a nonzero row vector Àz E GF(qr. Multiply each matrix in the z'th row of the new OA on the left by Àz· This is the substitution that will lead to the generalised Bose-representation for all n. Let us check that the substitution is of rank 2n: all we need is to find a basis for the column space of the (qn + 1) x q2 n x n matrix over GF(q) that we have constructed. A basis is the set of columns of the following pair of (qn + 1) x n matrices. which have rows indexed by z E GF(qn) U{ oo }. In the z'th row of the first matrix G we put the 1 x n vector Àz. In the z'th row of the second matrix H we put ÀzZ. If z = oo the rows contain 0 and \"° respectively. Then the i'th component in the (a, b) 'th position of the 3-dimensional matrix above has Àz(ZA + B)ei in the z'th row, where ei is the column unit vector of length n with 1 in the i'th position and O's elsewhere. Now we can write this (qn + 1) x 1 column vector (fixing i, A, and B) as H.Aei + G.Bei, since

This shows that the representation has rank 2n. Notice that the rows of the pair of matrices G and H are the coordinates for the base curve of the representation,

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and that every point of PG(n - 1, q) can be made a base point by a suitable choice of Àz and Z. This corresponds to the fact that in the base [n - 1] every point is contained in a unique special tangent, and so these special tangents form a spread. As we vary the choice of the vectors Àz 's the points of the cone-represe ntation move in orbits that are special tangents of dimension n - l. In fact, every point of PG(3n - 1, q) is contained in a unique special tangent (including the vertex [n - 1]), of which there are q 2n + qn + 1 subspaces. There are the same number of secant [2n - l]'s (including those through the vertex). Each of these spaces contains a spread of qn + 1 [n - l]'s that are the special tangents (including possibly the vertex of the representatio n). Indeed, the vertex can be chosen to be any of the special tangents. In a similar way, representatio ns of PG(m, qi) of index i and rank mi in PG((m + l)i - 1, q) may be constructed; see [11] and [9]. We shall consider these briefly later on.

3

N on-Spre ad Represe ntations

The spread-repre sentation of a translation plane of index i and rank i + 1 has a minimal rank given the index; see [8]. It is an interesting problem to discover other representatio ns of planes of the same rank and index. Then the problem arises of classifying all projective planes with such a representatio n: are they all translation planes or their duals? We started this program by analysing the cases of index less than 3; see [8, Theorem 2.2]. It was proved that there are only the spread-repre sentations for i = 1 or 2. However, for i = 3 there is the surprising fact that a new kind of representati on may exist. We establish the more important properties of this and show that the Desarguesia n plane PG(2, q3 ) has the second kind of representati on in PG(6, q). Theorem 3.1 For i = 1 or 2 there is only one kind of cone-represe ntation of a projective plane of order qi, index i and rank i + 1: the projective plane is a translation plane and the representatio n is via a spread of PG(2i - 1, q). For i = 3 there are two possibilities: the spread-representation of a translation

plane, and another. (Assume q

> 2.)

Proof. The first part has been shown in [8]. Let us consider the case i = 3. We note an important result due to Bruck; see [3]: a k-net of order q that is extendible to an affine plane (a net with q + 1 parallel classes) is uniquely extendible if (q - k) 2 < q. The case we need here is that a (q3 - q - 1)-net of order q3 that is extendible is uniquely extendible. (We need (q + 1) 2 < q3 which is true if q > 2.) The extension of a k-net is relevant here because we can always extend a k-net represented on a cone that has a point of the base [r -1] not on the base

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_ _ _ _ _ _ _ _ _ _ _ _ Non-spread Representation s of Projective Planes

curve with the property that it is on no tangent line of the base curve. Just adjoin that point to the base curve obtaining a larger cone. The property that two secant spaces never contain the same two points of the cone is preserved, and so a larger net is obtained. In our case, consider a (q 3 +1)-net representing an affine plane of order q 3 on a [2]-cone of PG(6, q). It cannot be extended in any fashion, and this implies that each point of the base [3] not on the base curve C is on at least one tangent line to C. Also, each point P of [3] not on C is on at most q + 2 tangent lines to C, because the number of points not on C is q3 +q 2 +q+1- (q 3 +1) = q 2 + q, and the number of points not on C on q + 3 tangent lin es is 1 + (q + 3) ( q - 1) = q2 + 2q - 2 > q2 + q. If we delete the points of C on the tangent lines through P we obtain a smaller net that is extendible by adjoining P to the new base curve. However, from Bruck's theorem above the net we obtain is the same as if we added back one of the deleted points of C. Thus we see that there must be a special tangent line joining P to a point of C. In fact, any point of [6] not in the cone will be on a special tangent to the cone. Let us assume that our representation is not that of a translation plane with a spread of [5]. Then the largest dimension of a special tangent is 1; see [8]. Since any special tangent is contained in a maximal special tangent we see that there are precisely q + 1 special tangent lines which are mutually skew in the base [3], and they intersect the base curve in q + 1 distinct points. Thus there are q + 1 generators of the cone of mst (maximal special tangent) dimension 1, and the remaining q3 - q have mst dimension O. The spread-represen tation has mst dimensions 2 for one generator, and 0 for the remaining q3 . This is the new kind of representation that we shall consider in more detail in the following investigation. D Theorem 3.2 A projective plane 7r of order q3 with the second kind of representation of rank 4 and index 3 in PG(6, q) has a special point V with a certain set L of q + 1 lines through V. There is a collection of q6 + q5 + q4 subplanes of order q that each contains V and every line of L. These subplanes form in a well-defined manner the structure of PG(6, q): they correspond ta almost all the hyperplanes of PG(6, q).

Proof. Let the vertex [2] of the cone in [6] be W: it corresponds to the point V of 7r. The lines of L correspond in [6] to a collection M of q + 1 [4]'s passing through W, each [4] containing a (3, 1)-fibration of q3 mutually skew special lines partitioning the [4] \ [2]; see Definition 1.1. Any pair of special lines in different [4]'s of M generate a secant [3], that in turn contains q + 1 special lines in the q + 1 subspaces of M. Any pair of distinct secant [3] 's intersect either in a special line, or in a point of the cone that is not on a subspace of M. Using these properties it is immediate that any hyperplane, containing three special lines not all in a [3] or in subspace of M, is generated by those lines,

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and contains precisely q 2 + q special lines. These correspond to q2 + q points of 7r, and if we adjoin V we obtain a subplane of order q containing each line of L. Such a hyperplane does not contain the vertex W; otherwise it would contain a whole [4] of M. There are q6 + q5 + q4 hyperplanes of [6] not containing W, and we can check that each of them is obtained by the above method. In fact we can count triples ofspecial lines, each on one ofthree chosen [4]'s of M, not all on a secant [3], in two ways. Thus the number is q3.q3.(q3 - 1) = (q6

+ q5 + q4)q.q.(q -1),

which is the number of hyperplanes multiplied by the number of triples of special lines with the desired properties in each. Note that there are q 2 + q special lines of the representation contained in each hyperplane h not through W. If we adjoin the line h n W we obtain a set of q 2 + q + 1 lines of h that represent the subplane of order q, where the points of the subplane are the lines, and the lines of the subplane are the [3] 's that each contain precisely q + 1 of the above lines. Converting this knowledge to 7r we see that any three non-collinear points (not V), each on a line of L, are contained in subplane of order q containing each line of L. The number of these subplanes is q6 + q5 + q4 . We could spend more space showing how to reconstruct PG(6, q) from 7r given knowledge of these subplanes (since they correspond to almost every hyperplane of [6]) but we leave that to the interested reader. D Here are some more properties of the second kind of representation. Consider a fixed point P of 7r not on a line of L. The q3 + 1 lines through P correspond to the same number of secant [3]'s through a point Q of [6] not on a [4] of M. (We include (Q, W/ temporarily as a secant [3].) Every pair of these distinct secants intersect only at Q. Thus we have a dual spread at Q, which defines a dual translation plane of order q3 at Q. The points of this structure are the planes skew to Q contained in the secant [3]'s through Q. The lines of this structure are the hyperplanes of [6] not containing Q. The hyperplanes of [6] either correspond to subplanes of 7r or to other easily described sets containing V (if the hyperplane contains the vertex W). Also, general planes of [6] correspond to subplanes of 7r, although planes in (Q, W/, not through Q and not equal to W, correspond to subsets of points of size q 2 , not containing P or V, of the line (P, V/. Thus the construction of the dual translation plane is straightforward using the properties of the conerepresentation. However, when we convert the properties to the plane 7r we see that there are many cases to consider, and the transformation from 7r to the dual translation plane becomes quite complicated. Thus we have the following result, which is a strong condition that 7r should have the second kind of cone-representation of rank 4 and index 3. Theorem 3.3 It is possible to describe a dual translation plane of order q3 at

280

_____ _____ __ Non-sprea d Represent ations of Projective Planes

any point P of a plane 7r of order q3 having a representa tion of the second kind, where P does not lie on any of the special set L of q + 1 lines through V. The description involves certain geometrica l properties, mostly involving subplanes of order q, of 1T.

Let us note that there are many projective planes of order q3 which have the required number, q6 + q 5 + q4 , of subplanes through a certain set of q + 1 lines passing through a point. However, it is not clear whether such planes have the second kind of representa tion, because complicate d conditions have to be satisfied. Consider a translation plane of order q 3 that has a spread of q 3 + 1 planes of PG(5, q) containing a 2-regulus. This is a set of q + 1 planes forming a Segre variety such that there are q 2 + q + 1 transversa l lines, each of which intersects each plane in a point. If we use the spread-rep resentatio n of the translation plane in PG(6, q), then we can construct subplanes by considerin g the planes of [6] passing through the transversa l lines, and not contained in the hyperplan e holding the spread. Thus we obtain (q 2 + q + l)q 4 subplanes containing the q + 1 points of the translation line correspond ing to the Segre variety. Dualising, we see that every dual translation plane with this additional property has the required number of subplanes of order q. Examples of such planes are dual semi-field planes, dual André planes, or dual near-field planes of order q3 . We have not calculated the required properties for all these kinds of planes. However, it should be a fairly straightfor ward to verify if such a plane has the second kind of representa tion in PG(6, q). In the easiest and classical case we have the following (algebraic) constructi on. Theorem 3.4 The Desargues ian plane PG(2, q3 ) has the second kind of representatio n of rank 4 and index 3 in PG(6, q). The special lines of the base PG(3, q) form a regulus. Proof. We make a substitutio n of the orthogona l array correspond ing to the plane. First, let GF(q 3 ) = GF(q)[x], where x 3 =a+ f3x + "(X 2 is irreducible and Œ, /3, and'/ E GF(q). Thus x 4 = Œ"f +(a+ /3'!)x + (/3 + "( 2 )x 2 . The standard 0 A of PG ( 2, q3 ) has rows correspond ing to k E G F ( q3 ) U { oo}, and columns correspond ing to (a, b) E GF(q 3 ) 2 , with element a+ kb occuring in the correspond ing row and column, (with b occuring in the oo row). Now let us write GF(q 3 ) as a vector space of dimension 3 over GF(q) with basis 1, x, x 2. Writing a= ao + a 1x + a2x 2, b = bo + b1x + b2x 2, and k = ko + k1x + k2x 2 , where ai, bi, ki E GF(q), we could expand a+ kb in terms of 1, x, and x 2 obtaining a substitutio n making the elements of the OA elements of GF(q) 3 . The reader could check that the resulting representa tion is the first kind of representa tion with a spread of rank 4 and index 3. This substitutio n is just a GF(q)-hom omorphism taking GF(q 3 ) to GF(q) 3 that is the same for

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all rows of the array. However, to obtain the second kind of representation we have to first multiply each row of the OA by a certain element of GF(q 3 ), and then use the homomorphism. To find this element, for k €f_ GF(q) write k = ko + k1x + k2x 2 =(Bk+ tkx)/(1 + rkx), for unique rk, Bk, tk E GF(q). In fact, rk = -k2/(k1 + k21), Bk = ko + k2ark, and tk = kork + k1 + k2f3rk· Notice that this is a 1-1 mapping from GF(q 3 ) \ GF(q) to PG(3, q) missing out a hyperbolic quadric. Then we multiply the row corresponding to k of the OA by 1 + rkx, thus obtaining (1 + rkx )a+ (Bk+ tkx)b in the (a, b)'th column. Then we substitute a= a 0 +a 1x+a 2x 2 and b = b0 +b 1x+b 2x 2 and expand out by coefficients of 1, x, and x 2 obtaining the substitution. Indeed this operation replaces a+ kb by S := ao + rka2a + Bkbo + tkb2a + (rkao + a1 + rka2f3 + Bkb1 + tkbo + tkb2f3)x + (a2 + a1rk + rka21 + b1tk + Bkb2 + tkb21)x 2. Considering this as an element of GF(q 3 ) we obtain a basis for the column space (over GF(q)) of these q3 - q rows of the OA generated by the vector of all one's, and the 3 vectors with rk, Bk, and tk in the k'th positions. Thus the representation indeed has rank 4 and index 3. The extension to the entire OA by adjoining the remaining q + 1 rows is done later. This shows that the base curve in PG(3, q) of the representation of this part of the OA is the whole space minus a hyperbolic quadric. To find the secant [3] 's, from which we shall determine the special tangents, we write the expression S above as the product (1,x,x 2)Ya,b(l,rk,Bk,tk)t, where Ya,b is the 3 x 4 matrix over GF(q)

This is the matrix of a secant [3] skew to the vertex. Indeed, coordinatise PG(6, q) with homogeneous coordinates (c, d), where c E GF(q) 3 (horizontal vector) and d E GF(q) 4 (vertical vector). The equation for the vertex plane W is d = 0, and the equation for the secant [3] corresponding to the above matrix (or column (a, b) of the OA) is c = Ya,bd. Now we know that any pair of such secant [3] 's intersects either in a point or in a special tangent. If (a, b) = (0, 0) the secant [3] has equation c = 0, and so the secant [3] corresponding to (a, b) =/=- (0, 0) intersects this [3], which is the natural base space of the cone, in a [j] if and only if Ya,b has rank 3 - j. Hence we should find that the rank of this matrix is either 2 or 3. Now the 4 columns of Ya,b can be written (a, Xa, b, Xb), where

X:=

282

a)

0 0 1 0 (3 ( 0 1 î'

,

_____ _____ __ Non-sprea d Represent ations of Projective Planes

where a and b E GF(q) 3 (vertical vectors). Since Xis a matrix satisfying the same irreducible equation over GF(q) as x, the mapping ai-+ Xa is a bijection and fixed-poin t-free in PG(2,q); ie. if a cf. 0, then a and Xa are independe nt. In fact it is a Singer collineatio n of order q2 + q + l. Thus, if (a, b) cf. (0, 0), at least two columns are independe nt, and we see that Ya,b has rank at least 2. Also, we see that the matrix has rank 2 if and only if a and b are dependent over GF(q). Otherwise the matrix has rank 3. Suppose the matrix Ya,b has rank 2. That is, let b =ra, where r E GF(q). Then the null space of the matrix should give a special line contained in the natural base space of the cone with equation c = O. This line is given by d = (-r,-sr,l ,s), where s E GF(q) U {oo}. By varying r we see that there are q + 1 special lines in the base space and they form one of the two reguli contained in the hyperbolic quadric with equation d0 d 3 - d 1 d = 0, where 2 d =(do, di, d2, d3)t. Now we are in a position to find all the special lines of the representa tion. They all project from the vertex W to the above regulus. Indeed the secant [3] 's correspond ing to Ya,b and Yc,d intersect in a special line if and only if the vectors a - c and b - d are dependent . Otherwise , the secants intersect in a point. Partition Ya,b as (ZaZb), where Za and zb are 3 X 2 matrices. Then we can calculate that the [3] correspond ing to Ya,b is generated by the two (special) lines (Zat,t,0), and (Zbs,0,s), where t and s move over GF(q) 2 and 0 is the zero vector (all vertical). Considerin g the intersectio n of the pair of subspaces we have (Zat + Zbs, t, s) = (Zct' + Zds', t', s') if and only ifs= s', t = t', and Za-ct = Zd-bs, if and only if t = -rs and b - d = r(a - c) (r E GF(q)). Hence a general special tangent line is

This also checks that the difference of the two matrices Ya,b and Yc,d has rank 2 if and only if the correspond ing secant [3] 's intersect in such a special line. D The following shows that problems involving conics through a fixed point of the Desargues ian plane might be solved using the new representa tion that we are discussing. Theorern 3.5 Consider the Desarguesian plane PG(2, q3 ) and the second kind of representa tion of rank 4 and index 3 in PG(6, q). Each non-degenerate conic through V, su ch that the tangent at V is not one of the q + 1 lin es of L through V, is represented in [6] by a plane skew to the vertex W not contained in a secant [3]. Such a plane of [6] intersects exactly q + 1 special lines in the q + 1 points of a conic. The plane also corresponds ta a subplane of order q of the PG(2, q3 ) that intersects the q + 1 lines of L in a subconic of order q of the original conic.

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David G. Glynn _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ __

Proof. Consider the planes of [6] not intersecting the plane vertex W of the representation. They project from W to a plane of the base [3] that intersects the regulus of special lines in one of two ways: either it is a tangent plane to the hyperbolic quadric of the regulus, or it cuts the hyperbolic quadric in a non-degenerate conic. Counting the number of planes of the former type we obtain (q + 1) 2 q9 . Th ose that are contained in a secant [3] number q6 ( q + 1) 2 . The number of planes containing a special tangent line is (q + 1) q3 ( q 4 + q3 ), and so every such plane is contained in a unique secant [3]. In terms of PG(2, q3 ), the (q + 1)2(q9 - q6 ) planes that are of the former type, and that do not lie in a secant [3], are subplanes of order q of PG(2, q3 ) that intersect the q + 1 lines through V in a subline (not through V) of order q. Let us now consider planes of the latter type. The number of these is (q 3 q)q 9 , while those that are contained in a secant [3] number (q 3 - q)q 6 . Hence the number of those that in the representation of PG(2, q3 ) are subplanes number (q 3 -q)(q 9 -q 6 ). Each of these subplanes intersects the q+ 1 lines in a subconic of order q (by the isomorphism which is the projection from V in the representation to the base). Now the number of conics through Win PG(2, q3 ) is (q 15 -q 6 ) (q 3 +1)/(q 6 +q 3 +1) = q6 (q 6 -1). The number with a given tangent at W is this divided by q3 + 1. Hence the number with a tangent not in the q + 1 lines of L is q6 ( q6 - 1) - q6 ( q3 - 1) (q + 1) = q6 ( q3 - 1) (q3 - q). Each such conic intersects the q + 1 lines in a subconic of order q plus the point V. Also, the subconic determines the subplane uniquely (if q > 2 there are four points, no three collinear). Hence we deduce that these subconics and their subplanes are represented in a precise way by planes of [6] of the latter type. D Next we can give two more methods (both geometric) to give the second kind of representation of PG(2, q3 ).

Method 3.6 Slicing the Representation of PG(2. q3 ) in PG(8, q). This method was discussed briefly in [9], as was the representation of PG(2, q3 ) in PG(8, q). This latter representation is a special case (put m = 2 and i = 3) of the representation of PG( m, qi) in PG ( (m + 1)i - 1, q)), which is a straightforward generalisation of the Bose-representation of PG(2, q 2 ) in PG(5, q) discussed in §2. See also [11]. AU we have to know is that the points of PG(2, q3 ) are represented in [8] by a "2-spread" of q6 + q3 + 1 mutually skew [2] 's, while the lines of PG(2, q3 ) are represented by a collection of the same number of [5]'s of a "dual 5-spread". Incidence is the natural one. Every [5] of the dual 5-spread contains precisely q 3 + 1 [2] 's of the 2-spread. Now let us choose any particular [2] of the 2-spread and call it W. Next we choose any [6] of [8] passing through TF. This can be done in two essentially different ways. 1. If the [6] contains a [5] of the 5-spread then the larger representation induces on [6] a spread-representation.

284

_ _ _ _ _ _ _ _ _ _ _ _ Non-spread Representation s of Projective Planes

2. Otherwise, the larger representation induces on [6] the second kind of representation. Here is another way of producing the second kind of representation, and as with the other methods, it is not hard to generalise this to cone-representi ons of higher index.

Method 3. 7 Finding a Certain Line of PG(3, q 3 ) with respect to a PG(3, q). Suppose we had a cone-representa tion of PG(2, q3 ) in PG(6, q), of rank 4 and index 3. Then we canuse the operation of contraction to find a representation of the same rank but of index 1 over the field GF(q 3 ). See [8], and especially [9] where the operations of contraction and expansion (to yield new cone-representations from old one's) are defined and used. Now we have a cone-representation of PG(2,q 3 ) in PG(4,q 3 ). Dualising we can see that the picture can be attained by having a line manda PG(3, q) embedded in PG(3, q3 ), such that there is a collection H of q3 + 1 planes of PG(3, q), that intersect min q3 + 1 distinct points. The dual cone-representa tion is constructed by embedding the PG(3, q3 ) as a hyperplane in PG(4, q3 ), and considering the q6 points of any fixed plane through m (not contained in the PG(3, q 3 )) as the "secant points" of the dual cone. The dual cone has as its set of q3 + 1 dual generators the planes of the set H. Now there are two possibilities for such a line m and set H. 1. m intersects PG(3, q) in a point P, and H is the set of all q3 planes of PG(3, q) not through P together with one chosen plane of PG(3, q) through P: this gives the spread-represen tation after dualization and expansion. 2. mis not contained in any plane of PG(3, q), nor does it intersect PG(3, q) 2 in a point. Let mq and mq be the conjugates of m with respect to the automorphic collineation of PG(3, q 3 ) fixing PG(3, q) point-wise. Then 2 m, mq and mq are pair-wise skew and so are contained in a regulus (and a hyperbolic quadric) that belongs to PG(3, q). Hence there are q + 1 skew lines belonging to the opposite regulus (of PG(3, q)) that intersect the three lines in points. Let H be the set of all q3 - q planes of PG(3, q) that are not tangential to the hyperbolic quadric, together with q + 1 planes, one through each line of the opposite regulus. This choice of m and H will clearly give the cone-representa tion of PG(2, q3 ) of the second kind.

In conclusion, these constructions can be generalised to give non-spread representations of PG(2, qi), for all i ;::: 3, and also similar "non-spread" represen-

285

David G. Glynn - - - - - - - - - - - - - - - - - - - - - -

tations of PG(m,qi) can be constructed 2 .

References [1] J. André, Über nicht-Desarguesche Ebenen mit transitiver Translationsgruppe, Math. Zeitschr. 60 (1954), 156-186. [2] R.C. Bose, On a representation of the Baer subplanes of the Desarguesian plane PG(2, q 2 ) in a projective five dimensional space PG(5, q), Atti dei Convegni Lincei 17 (1976), 381-391. [3] R.H. Bruck, Finite Nets. II. Uniqueness and Embedding, Pacifie J. Math.13 (1963), 421-457. [4] R.H. Bruck and R.C. Bose, The construction of translation planes from projective spaces, J. Algebra 1 (1964), 85-102. [5] P. Dembowski, Finite Geometries, Springer, Berlin, Heidelberg, New York, (1968). [6] D.G. Glynn, A geometrical representation theory for orthogonal arrays, Bull. Austral. Math. Soc.49 (1994), 311-324. [7] _ _ _ , On cone representations of translation planes, J. Stat. Plan. Inf. 58 (1997), 33-42 [8] _ _ _ , A general theory of cone-representations, accepted, Boll. U.M.I. [9] _ _ _ , A survey of cone-representations, accepted for the Proc. R.C. Bose Memorial Conference, Fort Collins, Colorado, (1995), special volume of J. Stat. Plan. Inf.

[10] B. Segre, Teoria di Galois, fibrazioni proiettive e geometrie non desarguesiane, Ann. Mat. Pura Appl. 64 (1964), 1-76. [11] J.A. Thas, The m-dimensional projective space Sm(Mn(GF(q))) over the total matrix algebra Mn(GF(q)) of the n X n-matrices with elements in the Galois field GF(q), Rend. Math. (6) 4 (1971), 459-532.

2

286

Note added in proof: The cone-representations of projective planes of order q 3 in PG(6, q) have now been completely classified; see D. G. Glynn, Geometriae Dedicata 66 (1997), 343-355. The planes with the second kind of cone-representation in PG(6, q) are precisely the dual generalized Desarguesian planes of Jha and Johnson.

Martin Saltzwedel *

On Character izing Injective Lineation s in Desargue sian Affine Spaces Abstract Point fonctions that preserve collinearity are called lineations. In geometry and linear algebra collineations have been studied extensivly. However, there are only a few results concerning the characterization of injective lineations on subsets of affine spaces under mild hypotheses. DAVID S. CARTER and ANDREW VOGT proved a representation of all lineations defincd on a whole affine or projective Plane. Using these results, A. BREZULEANU and D.-C. RÀDULESCU characterized full lineations of desarguesian affine and projective spaces of finite dimension in projective spaces. In this paper a characterization of lineations, defined on three lines in general position which are injective at the intersection point of any two of the lines, is given, using the theorem of MENELAOS. Similar results in the case of projective planes have been proved by J. AczÉL and W. BENZ. (Cf. also the theorem of SCHAEFFER). It is shown, that except of a few special cases a much simpler representation of the mapping can be given, if the domain is enlarged by only one point. Finally the upper results are used to characterize lineations in desargesian affine spaces of arbitrary dimension ;::> 2. Special lineations are given, induced by an endomorphism of the underlying division ring and some constants. These lineations need not be defined on the whole affine space. A representation theorem is presented that leads to a characterization of all "basisinjective" lineations of a desarguesian affine space into itself, i.e. lineations that map a given basis onto an independent set of points and fulfil some special conditions of injectivity. The theorem also gives a representation for many lineations only defined on a subset of an desarguesian affine space and describes the maximum domain of induced injective lineations. AMS Subject Classification: 51A10 , 51A25 , 51A30 Key Words and Phrases: affine and projective geometry, foundations of geometry, lineations, collineations, desarguesian affine space, characterization under mild hypotheses

*

The following results are part of the author's doctoral dissertation, Universitat Hamburg 1995.

287

Martin Saltzwedel _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ __

1

Affine lineations on three lines

1.1

Definitions and Preconditions

Let K be a division ring with at least 3 elements. Let A 2 (K) be the desarguesian affine plane over K with pointset P. The line connecting different points P, Q is denoted as P, Q. Every line is equated with its set of points. For three points P, Q, R not on a line let

(P,Q,R) :=P,QUQ,RUR,P be the "triangle" with its "vertices" P, Q, and R. A mapping f : V -t W is called "injective at X E V", if

VY E V :

f (X)

=

f (Y)

=?X = Y.

For to get a symmetric description of a desarguesian affine plane, barycentric coordinates are used, i.e. every point of the plane is denoted in the form of

xA+yB+zC

with

X+ y+

Z =

1,

where A, B, C are given points not on a line. Let IN be the the set of natural numbers { 1, 2, 3, ... } .

1.2

Description of collinearity

The theorem of l\IENELAOS leads to a characterization of lineations on triangles, which are injective at the vertices, by using a functional equation: Theorem 1: Let A, B, C E P be points not on a line. Let a : (A, B, C) -t (A, B, C) be a mapping. Then the following statements are equivalent:

1) a preserves collinearity, is injective at A, B, C, and A, B, C are fixed by Œ.

2) There are functions at 0 and 1 with

f:

K -t K, g: K -t K, h: K -t K all being injective

a(aA + (1- a)B)=f(a) ·A+ (1- f(a)) · B, a(bB + (1 - b)C)=g(b) · B + (1 - g(b)) · C,

a(cC + (1 - c)A)=h(c) · C for any a, b, c

E

+ (1 -

h(c)) ·A

K, with the property that f (0)

= g(O) = h(O) = 0,

f(l) = g(l) = h(l) = 1 and (1-a- 1 )(1-b- 1 )(1-c- 1 ) = 1

=?

hold true for any a, b, c E K\ {O}.

288

(1-f(a)- 1 )(1-g(b)- 1 )(1-h(c)- 1 ) = 1

_ _ _ On Characterizing Injective Lineations in Desarguesian Affine Spaces

: K\{0}---+ K\{1}, x f---7 cj>(x) = l-x- 1 is a bijection. Hencethe functional equation can be transformed into

l;/x,y,z

E

K\{1}: xyz = 1=?f(x)g(y)h (z)=1

so that any solution (f, g, h) corresponds to a solution (f, g, h) of the new functional equation with = of o cj>- 1 etc. If JKI < 6 there are special cases. In the case of IKI ~ 6 it is possible to define f (1), g(l), h(l) (uniquely), so that the functional equation is also fulfilled with

J

lE{x,y,z}. The mapping k given by

k: K---+ K, k(x)

:=

f(x)f(1)- 1

restricted to K\ { 0} is an endomorphism of the multiplicative group of K. With k(O) := 0, a:= f(l), c := h(l)

f(x)=k(x)a, g(x)=a- 1 k(x)c- 1 , and h(x)=ck(x) can be verified for all x E K. The injectivity of a at the vertices of the triangle implies

a,c,ac

E

{x

E K

1

x rf_ im(k) V (x = 1 /\ k injective)}.

Remark 1: This property is a restriction to the endomorphism; e.g. in the case of K =IR with k(x) := Jxl it follows from r,s E IR\im(k) that r,s < 0 and therefore rs > 0, i.e. rs = Jrsl E im(k). Remark 2: For any given a, c und k with the properties mentioned above there is a solution of the functional equation including the special conditions.

1.3

Characteriz ation of affine lineations on triangles

Theorem 2: Let A, B, C E P be points not on a line. Let a : (A, B, C) ---+ (A, B, C) be a mapping. Then are equivalent:

1) a preserves collinearity, is injective at A, B, C, and A, B, C are fixed by Œ.

2) There are functions f : K---+ K, g : K ---+ K, h : K ---+ K all being injective at 0 and 1 with

a(xA + (1 - x)B)=f(x) ·A+ (1 - f(x)) · B, a(yB + (1 - y)C)=g(y) · B + (1 - g(y)) · C, a(zC + (1 - z)A)=h(z) · C + (1 - h(z)) ·A

289

Martin Saltzwedel - - - - - - - - - - - - - - - - - - - - - -

for any x, y, z E K with the properties f (ü) f(l) = g(l) = h(l) = 1, so that either

=

g(O)

h(O)

0 and

IKl=5 /\ f=g=h /\ f(3)=3,f(2)=f(4)E{2,4}

is true, or that there is an endomorphism k of the multiplicative group of K together with certain constants a, c E K\ { 0} with a,c,ac E {x E Klx so that for x

E

\t im(k)

V (x = 1 /\ k injective)}

K\{0, l} f(x)=(l - k(l - x- 1 )a)- 1 , g(x)=(l - a- 1 k(l - x- 1 )c- 1 )- 1 , h(x)=(l - ck(l - x- 1 ))- 1

hold true.

2

Affine lineations with a domain containing at least a triangle and one additional point

If the domain of a lineation includes a triangle enlarged by one point, a much simpler representation can be found. Except of a few special cases this representation is essentially given by an injective endomorphism of the underlying division ring. Theo rem 3: Let A, B, C E P be points not on a line. Let p, q, r E K\ { 0} be constants with p + q + r = l. Then obviously X := pA + qB + rC is a point not on (A, B, C). Suppose there is a lineation o: defined on a supers et of (A, B, C) U {X} that fixes A, B, C and that is (restricted to (A, B, C)) injective at A, Band C. Then either an injective endomorphism k of the division ring and constants a, c E K\ {O} can be found, so that for all xA + yB + zC in the domain of o:

k(x)

+ k(y)a + k(z)c- 1 -=/= 0

and o:(xA + yB hold true,

290

+ zC)

= [k(x)

+ k(y)a + k(z)c- 1 ]- 1 (k(x)A + k(y)aB + k(z)c- 1 C)

_ _ _ On Characterizi ng Injective Lineations in Desarguesia n Affine Spaces

ordom(o:) is a subset of (A,B,C)U {-A+B+C ,A-B+C,A +B-C} and there are constants a, c E K\ {0, 1} with ac -1- 1, so that for all x, y, z E K\ {0, 1}

o:(xA + (1 - x)B)=[l - ai- 1 A+ [1 - a- 1 i- 1 B o:(yB + (1 - y)C)=[l - a- 1 c- 1 ]- 1 B + [1 - ca]- 1 C o:(zC + (1 - z)A)=[l - c]- 1 C + [1 - c- 1 i- 1 A and one of the following two cases hold true: Case 1: 2 = 0 (especially dom(o:) = (A, B, C) U {A+ B + C} and p = q = r = l), where o:(A + B + C) =: pA + qB + rC is restricted by

p + qa- 1

+ rc=O and p+ q + f=l.

Case 2: 2 -1- 0, D n {-A+ B point, which fulfills

+ C, A -

B

+ C, A+ B

- C} contains only one

r

o:(-A + B + C)=[l - a- 1 c- 1 1 B + [1 - car 1 c, a(A-B+C )=[l-c]- 1 C +[1-c- 1 ]- 1 A, or

o:(A + B - C)=[l - ai- 1 A + [1 - a- 1 ]- 1 B respectivly. Remark: There are lineations with l{xA + yB + zC E P k(z)c- 1 = O}I = 1.

3

1

k(x) + k(y)a +

Charact erizing "basisin jective" lineatio ns of desargu esian affine spaces of dimensi on > 2

In the following text the results of the sections 1-3 are used to get statements concerning lineations defined on subsets of desarguesian affine spaces of dimension ?: 2. Especially a characteriza tion of injective lineations will be proved.

3.1

Induced injective lineations

Let A be a desarguesian affine space with dim(A) ?: 2 over a division ring K. With a set I => {O} let (Bi)iEI be a basis of A, i.e. A= {LiEI XiBil LiEI Xi= 1}. Let k be an injective endomorphi sm of K. Theorem and definition:

The mapping

291

Martin Saltzwedel - - - - - - - - - - - - - - - - - - - - - - -

is a lineation. It shall be denoted as the "injective lineation of Â, induced by k and ci (i E /) ". In the case c0 = 1 this induced injective lineation is called "normalized". Lemma 1: Any induced injective lineation maps non-collinear points to noncollinear points.

3.2

Areas of uniqueness

By using the following definition the comparison of two lineations becomes easier, if one of them maps points not on a line to points not on a line:

Definition: Let D be a pointset of a desarguesian affine space. Let Ç Define recursively

9o çi+l :=

C

D.

:= Q

Çiu{P ED l 3A,B,C,D E çi: A

cJ B

/\

c cJ D

/\ A,BnC,D

= P}.

Th en OO

Eg(D)

:=

LJ Çi i=O

is called "Q-Area of uniqueness of D". Especially: If a hyperplane as well as a line that meets the hyperplane in exactly one point are contained in Ç c D and if any line contains at least four points, then follows Eg(D) = D.

3.3

Representation theorem

Definition: A mapping a of points of a desarguesian affine space into that space is called "basisinjective", if there is an affine basis (Bi)iEI with 0 E /, so that (Bil) (a(Bi))iEJ is independent.

°'IB

(BI2) \:/i E I\{O}: B· is injective at B 0 and Bi· o, i Every product of an induced injective lineation and an injective affine-linear mapping is obviously basisinjective. Theorem 4: Let  be a desarguesian affine space with affine basis B := (Bi)iEI and I =:i {O, 1, 2} (i.e. dimA:::: 2). Let K be the underlying division ring with IKI :::: 4. Let D be a subset of Â, so that Q :=

LJ

iEJ\ {O}

292

Bo, Bi U

LJ

iEI\ {0,1}

Bi, Bic D.

_ _ _ On Characterizing Injective Lineations in Desarguesian Affine Spaces

Suppose there is a point pB0

+ qB1 + r B2

E V, which fulfills

Then, if a : V-+ A is a basisinjective lineation with regard ta B, alcç(D) is injective and there is an injective endomorphism k of the multiplicative group ofK and there are constants c0 := 1, Ci E K\{O} (i E I\{O}) with the property, that the following statements hold true for all L xiBi E Eç(V): iEI

1)

2:: k(xi)ci -1- 0 iEI

3) The endomorphism k and the constants Ci are uniquely determined by a. 4) If k is surjective, Ci = 1 is true for all i E I, i.e. for all LiEI xiBi E Eg(V) it can be proved 2 :

a(I:>iBi) = L k(xi)a(Bi) iEI

5) U := {

2:: xiBi iEI

If U -1-

EA

1

iEI

2:: k(xi)ci iEI

=

o}

is an affine subspace of A.

0 and if K is regarded as left vectorspace over im( k), 1 E spanleft (1 codimaffine(U)

ci)iEI,

= dim(spanleft(l -

Ci)iEI),

and

2 ::=; codimaffine(U) ::=; [K: im(k)]ieft

can be proved. Remark 1: In the representation theorem the preconditions of injectivity are as weak as possible: If B. is not injective at B 0 and Bj for one j E I\ {O}

a!Bo,

J

there are lineations of radial or axial type, described in [C-V80].

Remark 2: There are products of an induced injective lineation and an injective linear map, which do not fit in the preconditions of the representation theorem: E.g. U is an affine hyperplane of A if k = id and if, with an arbitrary 2

This statement has already been published. Cf. e.g. [Ben92] and [B-R84].

293

Martin Saltzwedel - - - - - - - - - - - - - - - - - - - - - - - - -

j E J, ci := 1 for al! i E I\ {j} and Cj := 2 are given. Therefore, the corresponding induced injective lineation cannot be handled with the representation theorem.

Remark 3: Let A, A' be desarguesian affine spaces of dimension ~ 2 over the same division ring K. Then the representation theorem can also be proved for a mapping a : D --t A' if D C A. Remark 4: The representation theorem becomes false in the case IKI = 3 even if it is assumed that the plane, spanned by B 0 , B 1 and B 2 is contained in the domain of a (instead of the condition that there is one point pBo+qB 1 +r B 2 , which cannot be met).

4

Conclusions of the representatio n theorem

This section shall show connections between the representation theorem and some other theorems.

4.1

The fundamental theorem of affine geometry

Theorem 5: Let A and A' be desarguesian affine spaces over the division ring ~ 2. Furthermore, IKI ~ 3 is assumed.

K of dimension

(1) Every lineation of A into A' which is basisinjective with regard to a given basis (Pi)iEiu{o} can be denoted uniquely as a product of a normalized induced injective lineation ( with regard to the given basis) and an injective affine-linear mapping . If K is regarded as a left vectorspace over the image of the corresponding injective endomorphism k, 1

tf- spanleft(l

- ci)iEI\{o}

can be proved.

(2) Conversely, every product of an induced injective lineation and an injective affine-linear mapping is an injective basisinjective lineation. The following theorem is a corollar of the representation theorem and leads to a characterization of collineations under mild hypotheses:

Theorem 6: Let A be a desarguesian affine space of dimension ~ 2 over K. Suppose IKI ~ 3. All basisinjective lineations from A into A are collineations if and only if A has jinite dimension and if every injective endomorphism of the division ring K is surjective.

294

____ On Characterizing Injective Lineations in Desarguesian Affine Spaces

4.2

Subspace preserving mappings

Let A and A' be desarguesian affine spaces of dimension ;::: 2 over the same division ring K. Suppose IKI ;::: 3. Let (Bi)iEJ be an affine basis of A.

Definition: A mapping a : A --+ A' is called "subspace preserving" if there is an n E IN with n < dim A, so that points of any choosen n-dimensional subspace are mapped into an n-dimensional subspace. Lineations of course are subspace preserving. Furthermore, it can be proved the following

Theorem 7: Every subspace preserving mapping, which maps a given affine basis (Bi)iEI to independent points, is subspace preserving for any n E IN, and is therefore a lineation.

Reference s [Acz66]

AczÉL, J., Collineations on Three and on Four Lines of Projective Planes over Fields, Mathematica (Cluj) 8 (31), 7-13 (1966).

[A-B69]

AczÉL, J. und BENZ, \iV., Kollineationen auf Drei- und Vierecken in der Desarguesschen projektiven Ebene und Aquivalenz der Dreiecksnomogramme und der Dreigewebe von Loops mit der Isotopie-Isomorp hieEigenschaft, Aequationes Math. 3, 86-92 (1969).

[A-M67]

AczÉL, J. and McKIERNAN, M. A., On the Characterization of Plane Projektive and Complex Moebius-Transfo rmations, Math. Nachr. 33, 315-337 (1967).

[Ben92]

BENZ, W., Geometrische Transformatione n unter besonderer Berücksichtigung der Lorentztransform ationen, Mannheim - Leipzig - Wien Zürich 1992.

[Ben94]

BENZ, W., Real geometries, Mannheim -Leipzig- Wien - Zürich 1994.

[B-R84]

BREZULEANU, A. and RÀDULESCU, D.-C., About Full or Injective Lineations, Journal of Geometry 23, 45-60 (1984).

[C-V80]

CARTER, D. S. and VOGT, A., Collinearity-pres erving fonctions between Desarguesian planes. Memoirs of the American Mathematical Society 27 (235), 1980.

[Hav70]

HAVEL, V., On Collineations on Three and on Four Lines in a Projective Plane, Aequationes Math. 4, 51-55 (1970).

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Martin Saltzwedel - - - - - - - - - - - - - - - - - - - - - - - - -

296

[Kli92]

KLINGENBERG, W., Lineare Algebra und Geometrie, 3. Auflage, Berlin - Heidelberg - New York - London etc. 1992.

[Orb71]

ORBAN, B., Extension of Collineations Defined on Certain Sets of a Desarguesian Projective Plane, Aequationes Math. 6, 59-65 (1971).

[Rad69]

RADO, F., Darstellung nicht-injektiver Kollineationen eines projektiven Raumes durch verallgemeinerte semilineare Abbildungen. Math. Z. 110, 153-170 (1969).

[Rad70]

RADÔ, F., Non-injective Collineations on Sorne Sets in Desarguesian Projective Planes and Extension of Non-Commutative Valuations, Aequationes Math. 4, 307-321 (1970).

[Rad71a]

RADÔ, F., Extension of Collineations Defined on Subsets of a Translation Plane, Journal of Geom. 1, 1-17 (1971).

[Rad71b]

RADÔ, F., Congruence-preserving Isomorphisms of the Translation Group Associated with a Translation Plane, Canad. Journal of Math. XXIII, 214-221 (1971).

[Rig68]

RrGBY, J. F., Collineations on Quadrilaterals in Projective Planes, l\Iathematica (Cluj) 10 (33), 369-383 (1968).

[Sal92]

SALTZWEDEL, M., Beckman-Quarles-Charakterisie rungen von Abbildungen affiner Riiume - Eine Erweiterung des Satzes von Schaeffer auf hi:ihere Dimensionen mit veriinderten Voraussetzungen im Falle der Dimension 2, Diplomarbeit bei PROF. DR. W. BENZ, 10. Dez. 1992, Hamburg.

[Sal95]

SALTZWEDEL, 1\1., Kennzeichnung von Lineationen Desarguesscher affiner Riiume, doctoral dissertation, Hamburg (1995).

[Scha80]

SCHAEFFER, H., Über eine Verallgemeinerung des Fundamentalsatzes in desarguesschen affinen Ebenen, Techn. Univ. München TUM-M8010, Beitriige zur Geometrie und Algebra Nr. 6, 36-41 (1980).

Peter Brass

On Convex Lattice Polyhed ra and Pseudoc ircle Arrange ments Abstract We study the maximum number of vertices fd(n) that a d-dimensiona l convex lattice polyhedron can have if the vertices have integer coordinates in the range {1, ... , n }. We determine in dimension two for each k the smallest n su ch that { 1, ... , n} 2 contains a convex 4k-gon, i.e. f2(n) = 4k, and construct upper and lower bounds for fd(n), d ?:: 3. The twodimension al case is used to construct a pseudocircle arrangement which shows that the bound of Clarkson e.a. on the number of incidences between points and pseudounitcir cles is best possible. We construct for each n a strictly convex norm and a set of n points in the plane with en 34 unit distances with respect to that norm.

AMS Subject Classificatio n: 52 C 05 , 52 C 10 Key Words and Phrases: Lattice polyhedra, pseudocircle arrangements, unit distances

1

Results on lattice polyhed ra

Let fd(n) denote the maximum number of vertices of a d-dimension al convex làttice polyhedron whose vertices have integer coordinates from {1, ... ,n}. A simple lower bound, which was used for a construction in [8], is fd(n)?:: jnd-z. This can be proved by ordering the nd integer lattice points {1, ... ,n}d according to their distance to the center of the cube [1, n]d. There are at most d 1) 2 j + 1 possible different distances, and points at the same distance to the center are cospherical and therefore vertices of a convex polyhedron. A simple upper bound fd(n) ::; 2nd-l follows from fd(n) ::; nfd-i(n) (again the pigeonhole principle) and fi(n) = 2. For fixed dimension d the correct order of fd(n) is cdnd- 2+ d~l; this follows from a bound of the number of lattice points on the surface of a strictly convex body in terms of its volume by Andrews [3]. Although this cospherical construction gives surprisingly large vertex numbers for small, fixed n and large d, it is far from the true order of magnitude for d small and n large. In the planar case there are only 0 ec1og1agn ~) (which

li(n -

(

297

Peter Brass - - - - - - - - - - - - - - - - - - - - - - - - - -

is O(nE) for all positive E) concircular points in this piece of the integer lattice, it is known that f2 (n) = 8 ( n ~) . This planar case has already been frequently studied, beginning as part of number theory, by Jarnik [10] and others, and more recently in the framework of digital geometry [1]. In the planar case it is possible to determine h(n) exactly for many n; Theorem 1 below is slightly stronger than the recent result of AcKETA and ZuNié [1], but uses essentially the same proof as JARNIK. For technical reasons it is simpler to determine the inverse of the fonction h, namely, for given v the smallest n such that h(n) 2 v, i.e. the smallest n such that {1, ... , n } 2 contains a convex v-gon. We can do this only for those v divisible by four; between these values the behaviour of h seems to be irregular as illustrated by the following exact values.

n

h

1 2 3 4 5 6 7 8 9 10 11 12 13 n 1 4 6 8 9 10 12 13 14 16 16 16 18

= 13 and f2(10) = 16.

The figure below illustrates the cases f2(8)

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0 0

0

0

0

0 0

0

0

Figure 1:

k

Theorem 1.1 If 4

1

v, v = : 4 + 4 I: rp(i) + 4l, l < rp(k + 1), then n i=2

k

2+

I: irp(i) +

(k + l)l is the smallest n such that h(n)

= v.

i=2

k

Using

I: rp(i) = ;

2

k 2 +O(k log k) and

i=2

again J ARNIK's result

298

k

I: irp( i) = ; i=2

2

k 3 +O(k 2 log k) we obtain

_ _ _ _ _ _ On Convex Lattice Polyhedra and Pseudocirc le Arrangem ents

Corollary 1.2 h(n) = 3 (

~) ~ n~ + O(n~ logn).

lt was already shown by Valtr ([14], an improved version in [13]) that a much more general type of set (n 2 points in the plane among which the ratio of the largest to the smallest distance is bounded by cm for some a ;::::

jfl3)

contains c( a )n ~ points which form a convex polygon. The integer lattice is therefore a simple example which shows that one cannot expect a better bound under these assumptio ns, supplemen ting the previous examples of Alon e.a. [2] and V altr [13].

2

Applic ation to pseudo circle arrang ements

A finite set of wnvex closed curves in the plane is called an arrangeme nt of pseudocirc les if any two curves intersect at most twice. If furthermo re for any two points there are at most two curves which go through both points, it is called an arrangeme nt of pseudouni tcircles. A related concept are the arrangeme nts and weak arrangeme nts of curves studied in [9]. Such arrangeme nts generate subdivisio ns of the plane which are (by the pairwise intersectio n property) of relatively small complexit y (few cells, curve segments bounding cells, and points of intersectio n) and therefore useful in computational geometry. Also they were used in an attempt to prove the famous ERDOS conjecture ([6],[7]) on the maximum number of unit distances among n points in the (euclidean) plane, which is believed to be 0 (nec 10 ~1'0 ~ A set of unit circles of a strictly convex norm on IR.2 ( e.g. the euclidean norm) always is an arrangeme nt of pseudouni tcircles [4]. Given a set of n points in the plane, one may count the number of unit distances in this set by taking the set of circles of radius one around these points and counting point-circl e incidences (i.e. pairs of a point which lies on a circle) in this arrangeme nt of n pseudouni tcircles; each unit distance is thus counted twice. Clarkson e.a. [5] proved that between m points and an arrangeme nt of n pseudouni tcircles there are at most O(m~n~ +m+n) incidences . This proves an O(ni) upper bound for the number of unit distances among n points in the plane. This is up to now the best bound even in the euclidean case, where it was previously obtained in a different way in [11], but this bound holds also for any strictly convex norm on IR.2 (for norms which are not strictly convex the exact maximum number of unit distances was determine d in [4]). Using Theorem 1 we can show that in this setting the bound is best possible.

n).

Theorem 2.1 There is a c > 0 such that for each n there is a set of n points and a strictly convex norm If ·fin on IR.2 such that there are en! unit distances among the n points with respect to this norm.

299

Peter Brass _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ __

So this bound for the number of point-pseudounitcircle incidences is essentially best possible, at least in the case m = n. It remains an open problem whether the order f2 ( n ~) in Theorem 2 can be reached by a single strictly convex norm (which is independent of n). In fact, contrary to the statement in [12] it seems unknown whether it is possible to get

n (n~)

lattice points on the boundary

of a fixed scaled strictly convex dise nC for infinitely many n.

3

Proof of Theorems 1 and 2

To prove Theorem 1, consider a convex v-gon P1, ... ,Pv,Pv+l =Pl (oriented clockwise) in { 1, ... , n } 2 C 'li}. By convexity each line intersects the polygon either in a segment or in at most two points. We want to count the number of intersections with the 2n horizontal and vertical lattice lines { (x, i) 1 x E IR}, i = 1, ... ,n and {(i,x) 1xEIR},i=1, ... ,n, where we count the intersection in a horizontal or vertical segment as two intersections in the endpoints of the segment. Then there are at most 4n such intersections, with equality iff the polygon has two points on each of the four extremal lin es { ( x, 1) 1 x E IR}, {(x, n) 1 x E IR}, {(1, x) 1 x E IR}, and {(n, x) 1 x E IR}. Each intersection is either in a vertex of the polygon or in the interior of an edge, so we count each intersection exactly once if we assign each edge PiPi+ 1 the number of intersections in the half-open segment ]pi, Pi+1] as weight. This weight is invariant under lattice translations, weight(pi, P;+1) = w(Pi+l - Pi) where w ( (x,

y)) :=

max(lxl, 1) + max(IYI, 1) for (x, y) E 'J'} \ {0}.

Since the v-gon is convex and each Pi is a proper vertex of the convex hull, we may describe it by the set of edge-vectors E : = {P;+l - Pi 1 i = 1, ... , v} C Z 2 \ {O}. Each vector occurs at most once as an edge, and there are no two vectors in E that are positive multiples of each other. Let = Z \LJ: iZ c Z {O} be the set of integer points with relatively prime coordinates. Given E c Z 2 \ {O} we construct a new set Ê c by replacing each (x, y) E E by gcdCx,y) (x, y), which has the same direction, but

z;P :

2

2 2

smaller weight if gcd(x, y) 4n :'.:".

L eEE

-=/:-

2\

z;P

1. Then IEI = IÊI = v and

w(e) :'.:".

L

w(e) :'.:".

inf

2 Fczrp

L

w(e).

IFl~v eEF

eEÊ

z;P

To estimate the last term, we note that does not contain an element e with w(e) < 2. There are exactly 8 elements e E with w(e) = 2 (the vectors (1, 0), (1, 1), (0, 1), ( -1, 1), ( -1, 0), (-1, -1), (0, -1), (1, -1)) and 4tp( i)

z;P k

elements e with w(e) = i for i :'.:". 3. Thus ifv = 4+4

2:= i=2

300

tp(i)+4l, l

< tp(k+l),

_ _ _ _ _ _ On Convex Lattice Polyhedra and Pseudocirc le Arrangem ents

then

This proves the lower bound for n. To prove that for these numbers v = 4 + 4

k

2: cp(i) + i=2

k

4l, n = 2 +

2: icp(i) + i=2

(k+ l)l, l < cp(k+ 1), it is possible to select a convex v-gon in {l, ... ,n}2, we first note that we can select a v-element set E c z;P with 2: w ( e) = 4n

eEE and an ~-rotational symmetry ((x,y) E E iff (y,-x) E E), since each of the level sets {e E z;P f w(e) = i} already has this symmetry. We order now the elements of this set E, starting with e 1 = (0, 1), according to the increasing angle to ei, and construct Pl ... Pv by Po = Pv : = (0, 0), Pi : = l:~=l ej. This is a closed (l:eEE e = 0 since E is centrally symmetric ) convex lattice polygon. It has a ~-rotational symmetry , so the smallest circumscri bed lattice rectangle is a square [a+ 1, a+ m] x [b + 1, b + m], a, b, m E Z. This polygon touches each side of the square in a segment, since (O,l),(0,- 1),(1,0),( -1,0) E E. So, counting again the intersectio ns with horizontal and vertical lines in this square, the weight sum must be 4m. Thus m = n, and the polygon is inscribed in a lattice translate of { 1, ... , n } 2 . This completes the proof of Theo rem 1. To prove Theorem 2 we have to construct a unit circle and a set of points. k

Let m

= 2 + (~ icp(i)) + (k + l)l and take the set of n : = 4m 2 points

H, ~' ... , 2;"} 2 , which contains the points {l, ... ,m}2.

To construct the unit circle, we interpolate the vertices of the v-gon constructe d above by a strictly convex curve with ~-rotational symmetry. The center of this curve is the point E { ~, ~, ... , 2;" } 2 . If we translate this curve to each point of 2 { ~, ..• , ;" } , then for each translate at least ~ of the curve is contained in the square [~, 2;" ]2, so each of these points has distance one (with respect to this norm) to at least ~h(m) = O(m~) = O(ni) other points of this n-point set. So this set contains D( n ~) unit distances among n points in the plane, with respect to a strictly convex norm, as claimed in Theorem 2.

(mtl ,mr)

301

References [1] D.M. Acketa and J.D. Zunié: On the maximal number of edges of convex digital polygons included into the m x m-grid, J. Combinatorial Theory Ser. A 69 (1995) 358-368 [2] N. Alon, M. Katchalski, and R. Pulleyblank: The maximum size of a convex polygon in a restricted set of points in the plane, Discrete & Computional Geometry 4 (1989) 245-251 [3] G.E. Andrews: A lower bound for the volume of strictly convex bodies with many boundary lattice points, Trans. Amer. Math. Soc. 106 (1963) 270-279 [4] P. Brass: Erdos distance problems in normed spaces, to appear in: Computational Geometry: Theory and Applications [5] K.L. Clarkson, H. Edelsbrunner, L.J. Guibas, M. Sharir and E. Welzl: Combinatorial complexity bounds for arrangements of curves and spheres, Discrete & Computational Geometry 5 (1990) 99-160 [6] P. Erdos: On sets of distances of n points, Amer. Math. Monthly 53 (1946) 248-250 [7] P. Erdos: Sorne of my favourite unsolved problems, in A Tribute to Paul Erdos (A. BAKER e.a., Eds.), Cambridge Univ. Press (1990), 467-478 [8] P. Erdos, Z. Füredi, J. Pach and I.Z. Ruzsa: The grid revisited, Discrete Math. 111 (1993) 189-196 [9] B. Grünbaum: Arrangements and spreads, Regional Conference Series in Mathematics Vol. 10, AMS 1972 [10] V. Jarnik: Über die Gitterpunkte auf konvexen Kurven, Mathematische Zeitschrift 24 (1926) 500-518 [11] J. Spencer, E. Szemerédi and W. Trotter: Unit distances in the euclidean plane, in Graph Theory and Combinatorics (B. Bollobas, Ed.) Academic Press, London (1984), 293-304 [12] R.P.F. Swinnerton-Dyer: The number of lattice points on a convex curve, J. Number Theory 6 (1974) 128-135 [13] P. Valtr: Planar point sets with bounded ratios of distances, PhD-thesis, FU Berlin 1994 [14] P. Valtr: Convex independent sets and 7-holes in restricted planar point sets, Discrete & Computational Geometry 7 (1992) 135-152

302

Comb inatories

Adalbert Kerber

Anwend ungsorientie rte Theorie endlicher Strukturen Abstract

The constructive theory of finite structures will be sketched. Severa! examples (chemical isomers, unlabelled graphs, error-correcting codes and t-(v, k, .>-)-designs) demonstrate the applicability of a few basic algebraic methods that can be used. The crucial point is that these methods are easy to formulate and effective in several seemingly quiLe ..)-Design (V,B) ist demnach eine Teilmatrix von M~k' die gerade aus den Spalten besteht, die zu den Bléicken des Designs gehéirer{. Dies wiederum kann man leicht in der Sprache der linearen Gleichungssysteme formulieren:

322

- - - - - - - - - Anwcndungsorientierte Theorie endlicher Strukturen

Folgerung: Die t - (v, k, >.)-Designs mit Punktemenge V entsprechen den 01-Losungen x des linearen Gleichungssystems

Die Blocke des Designs zu x sind die k-Teilmenge n von V, die durch die von Null verschiedenen K oordinaten von x bestimmt werden.

Bei der Suche nach 7-Designs hatte man schon seit vielen Jahren die Hoffnung, es konne 7 - (33, 8, >.)-Designs geben, >. war aber unbekannt, auch das wohl ein Grund dafür, daJ3 dieses Problem so lange offen blieb. Bei diesen Parametern erhiilt man eine Matrix mit etwa 6 · 10 13 Elementen, und es ist klar, dal3 die Suche nach 0-1-Losungen einer Gleichung mit einer derart grol3en Koeffizientenmatrix hoffnungslos ist. Die Strategie, die hier dennoch zu einer Losung füh1te ist die folgende:

Mn

• Strategie 3: .Man schranke die Betrachtung bei zu umfangreichen Problemen auf Strukturen mit vorgegebener Stabilisatorklasse ein. Dieses Verfahren ist natürlich ziemlich riskant, wenn es uro die Losung von Existenzproblemen geht, denn es kann natürlich sehr leicht sein, dal3 es zu vorgegebener Konjugiertenklasse von Untergruppe n keine Struktur von diesem Typ gibt. Aber es gibt auch andere Fii.lle, zu denen glücklicherweise gewisse 7-Designs geh6ren: Ein Element 7r der symmetrischen Gruppe Sv heiJ3t genau dann ein Automorphismus des Designs (V, 13), wenn gilt 1rl3 := {7rB := {7rb 1b CD} 1BE13} = 13.

Eine Gruppe aus solchen Automorphismen heillt eine Gruppe von Automorphismen des Designs, und die maximale Gruppe dieser Art, also

Aut(V, 13) := { 7r E Sv 1 7rl3 = B}, heillt die (volle} Automorphismengruppe von (V, B). Die Aufgabe ist, aile Designs mit vorgegebenen Parametern und vorgegebencr Gruppe A $ Aut(V, B) zu ermitteln. (Das naheliegende weiterführende Problem der Bestimmung aller Designs mit A= Aut(V, B) wollen wir hier nicht diskutieren, vgl. (17],[18].) Um dieses Problem in die Reichweite moderner Computertechnik zu bringen, betrachtet man anstelle von M~k die folgende wesentlich kleinere Matrix. Zu vorgegebener Untergruppe A $ Sv bildet man eine sogenannte K ramer/Mesner-Matrix Mt~k := (m!J.,K ).

323

Adalbert Kerber - - - - -- - - - - - - -- - - - - - - - - -

(Diese Matrix ist übrigens eine Teilmatrix der Matrix A" zum Verband der Teilmengen von V, vgl. [13],[9], Kapitel 3.) T durchliiuft dabei eine Transversale von A\\(~) und K eine Transversale von A~(~). Die Eintriige dieser Matrix werden wie folgt definiert: m~,I< := l{K' E A(K) 1 T Ç K'}I.

Das fondamentale Resultat ist Der Satz von Kramer und Mesner: Die Menge aller t - (v, k, >..)-Designs mit A als einer Gruppe von Automorphismen ergibt sich aus der Menge aller 0-1-Losungen x des linearen Gleichungssystems

Auch hier greift Strategie 2, denn sie liefert Bijektionen

Für die Parameter t = 7, v = 33, k = 8 und die Wahl von A := Pr L 2 (32) ergibt ü). Further, let F : lR --+ C be the p--antiperiodic extension of fl[x,x+p) on JR, i.e., F(x + p) = -F(x) for all x E R Again, the Hilbert transform of the 2p--periodic fonction Fis of basic importance, F~(x) := (H 2PF)(x), from which the representation F~(x) =

{P

E-+0+

+-l p

lO

1

-p/2

F(x - u)

2 [P/ ) F(x - u)-l cot -7rU du E 2 2p

(

PV

2 F(x - u)

(7rU + -7r)) du

- -1 cot 2 2p

-p/2

+:p1 Jo{P/ =-

1-E +

-1 (

lim

(

-

1

2 cot

2

(7rUp - 7r)) du } 2

2

lp/2 F(x - u) cosec -7rU du

p

p

-p/2

can be deduced. Now the simplest case of the counterpart of Boole's summation formula, namely the case m = 1, shall be given. With this formula it is possible to evaluate the Hilbert transform F~ ( x +y) in terms of the boundary values f (x) and f(x + p) and an error term K((x, y).

Theorem 1 (Hilbert-Boole formula) Let f E

cC 1 l [x, x + p]

with x E lR arbitmry and fixed (p > 0). If Fis the p-antiperiodic extension of fl[x,x+p) on JR, one has for all y E JR\pZ

F~(x +y)=_.!_ log

7r

1

cot ny (f(x) 2p 1

+ f(x + p)) + K((x, y),

(6)

363

Michael Hauss - - - - - - - - - - - - - - - - - - - - - - - -

where

R']'(x,y)

=

~ PV foP [0 (y;u)

J'(x+u)du.

Proof Let x E fil. be arbitrary and fixed and y E fil. with y =f. kp (k E Z). Since all integrals exists, because F is a piecewise continuous fonction and 1/ sin 1fPu = cosec ";u is continuous on [-~, -f] U [c, ~], for all E > 0 small enough, there holds true

{1-E 1P/ F(x+y-u)cosec-du -p/2 + ( lr {1-E + 1p/2} F(x+w-u)cosec -du, 2

-l

p

}

'TfU

p

E

=-p

'TfU

-p/2

p

E

where w E ( 0, p) satisfies w = y + mp ( m E Z). Because of the antiperiodicity of F one has F(x +y - u) = (-l)m F(x + w - u). Now cosec 1fu is also a p p-antiperiodic fonction, so that F(x + w - u) cosec 1fu is a p-periodic fonction. p Thus, one has

( l)m {lw-E + lw+p/2} F(x + u) ---cosec 7r (w -

w-p/2 w+E P = -1 {1w-E + lp } f(x + u) cosec 7r(y _ u) du. P w+c P

u ) du

P

Ü

For x E (0, 1) one obtains (see Proposition 3.2 of [4])

d

~

-d [ 0 (x) X

=

2 d 'TfX ---d logcot7f X 2

=

2cosec7rx.

Because of the periodicity, this equation even holds true for all x E fil.\Z, and by partial integration there follows

{ rW-E + lp } j(X + U) 7r(y - u) du Jo w+c: P = [-~f(x+u)E0 (y;u)] (J:-f; + J:+J-~ ( rw-E + lp ) J'(x + u) (-'!!_ Eü(y - u)) du P Jo w+E 2 P 1 y ( 1 (1W-E lp ) y = 2 E0 (-) f(x) + f(x + p) ) + 2 + du+ P w+c J'(x + u)E0 (--) p

~

COSeC

P

U

1(

+2

364

Ü

f(x+w+c)E 0 (

y-w-E p

)-j(x+w-E)E0 (

y-w+E) p

)

.

_ _ _ _ _ A Boole-Type Formula Involving Conjugate Euler Polynomials

Since by [4], Proposition 3.2, there holds true

one has the identity

~

(f(x

2 = ( -1

+ w + s)E0 (y -

r+

l

~ log 7r

1

COt

w p

JTC

2p

1

c) - f(x

+w

(f (X + W + c)

-

-

s)E0 (y - w p

f (X + W

-

+ c ))

c)) .

Because f E C(l) [x, x + p] and w E (O,p), there exists an xo,w, 0 E (x s, x + w + s) C (x, x + p), where sis small enough, satisfying f

'(x O,w,c

+w

-

) = f(x + w + s) - f(x + w - s). 2c

Hence, in view of L'Hospital's rule one has lim

c-+0+

(-l)m+l~ log

1

7r

cot 7rc (f(x + w + s) - f(x + w - s)) = 0, 2p

which completes the proof.

1

•

Theorem 1 can now be given in a more general version. The intermediate values F~(x+y) are described in terms of the derivatives f(ml(x) and f(ml(x+ p) and an error term R-;:(x, y). Comparing this with (4), where the intermediate values f(x + hy) are given by the boundary values f(k) (x) and f(k) (x +y), it is the structural counterpart of Boole's summation formula in the Hilbert case, the Euler polynomials now being replaced by the conjugate Euler polynomials.

Theorem 2 (Hilbert-Boole formula) Let f E C(k) [x, x + p] with k E N and x E ffi. arbitrary and fixed. Further, let F be the p-antiperiodic extension of fi[x,x+p)- Then one has for all y E ffi.\pZ and n E {1, .. ., k}

p~ (x + y) =

-

~ log

+~

n-l

L

1

cot

;~ f (x) + f (x + p)) + 1(

p~E,-;:;(!.)(!(m)(x)+f(m)(x+p))

m=l m.

where for n E N (in the case n value)

+R';;(x,y),

P

= 1 the

integral is to be understood as a principal

365

J\fichael Hauss - - - - - - - - - - - - - - - - - - - - - - -

Proof Let f E C(k) [x, x + p] be given and let 1 ::::; m < k. Then, by partial integration, one has

R;;,(x,y)

=

2(~~11)!{ +E..

r

m la

=

: E;:,(y;u) f(m)(x+u)I: +

E;:,(Y - u)f(m+ll(x

+ u) du}

P

;:! E;;, (~) (!(m) (x) + f(m) (x + p)) + R;;,+1 (x, y).

In the case m = 1 the integral is to be understood as a principal value (see Theo rem 1). A telescopic argument gives (n E { 1, ... , k}) n-1

R';;(x, y)

=

L (R;;,+ (x, y) -

R;;,(x, y))+ K{(x, y)

1

m=l

n-1

= -

L

m

;m!E;;,(~)(!Cml(x)

+ f(m)(x+p)) +R!(x,y),

m=l

which completes the proof in view of Theorem 1.

•

If FE C(k)(~) is a given p-antiperiodic fonction and f(u) := F(u) for all u E [x, x + p] with x E ~ arbitrary, one has with y E ~\p/Z the convolution identity

(7) For the Poisson summation formulae there also exist analoga in the setting of Euler polynomials. Theorem 3 (Poisson-Boole formulae) Let f E CC 2 l[x,x + N + 1] with x E ~ and NE No arbitrary and fixed. Then with h E (0, 1) one has a) the first Poisson-Boole formula N

oo

L(-lt f(x + h + v) 2L1

N+l

=

v=O

k=O

cos((2k + l)w(h- t))f(x + t) dt. (8)

0

b) the second Poisson-Boole formula N

N+l

L(-lt f(x + h + v) =Le 1 v=O

366

kEZ

0

exp[(2k + l)wi(h - t)]f(x + t) dt. (9)

_ _ _ _ _ A Boole-Type Formula Involving Conjugate Euler Polynomials

Proof To prove part a), by partial integration one obtains rN+l

Jo

=

cos((2k + l}rr(h - t))f(x + t) dt

sin~~~k:l~~nh) (f(x) +

(-l)N f(x + N + 1)) -

_ cos((2k + l)nh) (f'(x) + (-l)N J'(x + N + l)) _ ((2k + l)n) 2 1 {N+l - (( 2k + l)n) 2 cos((2k + l)n(h - t))J"(x + t) dt.

Jo

f

Since the series

k=O

cos((2k + l)n(h - t)) ((2k + l)n) 2

is absolutely and uniformly convergent one gets {N+l

(~cos((2k+l)n(h-t)))f"(

6

}0

=

((2k+l)n) 2

x+t

rN+l

= - ~ Jo

cos((2k + l)n(h - t))f(x + t) dt+

+(f(x) + (-l)N f(x + N + l)) -(!'( ) X

0 bserving (h

)d t

+

N

(-l)NJ'( X+

+

f

sin~(2k + ~)nh) 2k + 1 7r k=O

l))

_

~ cos((2k + l)nh) 6 ((2k + l)n)2 .

E ( 0, 1))

E (h) = 4 ~ sin((2k + l)nh) 0 L..., (2k + l)n

and

k=O

E (h) = _ 4 ~ cos((2k + l)nh) 1 L..., ((2k + l)n) 2 ' k=O

one finally proves (8) by applying Boole's summation formula. Part b) then easily follows observing 2 cos x = eix + e-ix. • The summation in (9), sometimes called Eisenstein summation (see [12], p. 6), is defined by

Leak kEZ

N

=

lim

N--+co

L ak. k=-N

Theorem 3 now has also a counterpart in the Hilbert case. Essentially, the cosine fonction is replaced by the sine fonction and the alternating sum on the left side of (8) is replaced by the Hilbert transform F~(x +y).

367

Michael Hauss _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ __

Theorem 4 (1. Hilbert-Poisson- Boole formula) Let f E C( 2 ) [x, x + p] with x E lR arbitmry. Further, let F be the p-antiperiodic extension of fl[x,x+p). Then one has for y E JR\p.:Z F~(x+y) = -2

00

L

1P sin((2k+l)7r--)f(x+u)du. y-u

p k=a a

(10)

p

Proof There holds

E

R';J,'(x, y)=

2 = -

{p E!(y - u)J"(x + u) du P

la 2 ~

1T

f(+ 1

k=a

2k

1

)2 rp sin((2k + l)1Ty - u)J"(x + u) du,

la

p

and by partial integration one deduces

1 P

a

=

y-u sin((2k + l)7r--)J"(x + u) du p

-sin((2k + l)7r~) (f'(x) + J'(x + p)) p

-

7r(2k + 1) y p cos((2k + l)7rp) ( f(x) + f(x + p) ) -

-

7r2(2k + 1)21P y - U sin((2k + 1)7r--)f(x + u) du, p2 a p

which finally gives

R~(

2 x,y

) = 4 ~ cos((2k + l)7ry/p- 7r/2) (f'( ) f'( )) 26 ((2k+1)7r)2 x + x+p +

r

k=a

~ ~ cos((2k + l)7ry/p) (f( ) f( )) + 4 2~ (2k+l)7r X+ x+p + 2

+-

y L 1p sin((2k + l)7r--)f(x + u) du U

OO

p k=a a

=

p

-~E1(~) (J'(x) + 2 +-

00

L

~E0 (~) (!(x) +

1P sin((2k + 1)7r--)f(x + u) du. y-u

p k=a a

In view of Theorem 2 (with n

368

J'(x + p)) -

p =

2) the proof is complete.

•

f(x + p)) +

----~

A Boole-Type Formula Involving Conjugate Euler Polynomials

It is this theorem, which plays the major role in obtaining the partial fraction expansion from which the Euler-type formula for .C(2m) will be deduced. The so-called second Hilbert-Poisson-Boole formula finally gives a slightly different representation of ( 10).

Theorem 5 (2. Hilbert-Poisson-Boole formula) Let f E C( 2 ) [x, x + p] with x E IR. arbitrary. Further, let F be the p-antiperiodic extension off [x,x+p). Then for all y E IR.\pZ'.. there holds true 1

F~(x

1"' 1 y +y)=-:-L.,e sgn(k + ) exp((2k + l)ni-) ·

ip

2

kEZ p

1

(11)

p

u exp(-(2k + l)ni- )f(x + u) du. p

0

Proof Noting 2i sin( (2k+ l)n y;u )=exp( (2k+ l)niy;u )-exp(-(2k+ l)niy;u) one has N

-L 2

p k=O

1 =-:-ip

1P

y-u sin((2k + l)n--)f(x + u) du

O

p

~ L.,

sgn(k +

k=-N-1

which proves Theorem 5.

3

1 2)

1P O

y-u exp((2k + l)ni--)f(x + u) du, p

•

Partial Fraction Expansions

As in the Bernoulli case (see [7]) the second Poisson-Boole formula and the first Hilbert-Poisson-Boole formula can be used to obtain two partial fraction expansions, namely (12) and (20). At least (20) seems to be new. In this paper the most important fact regarding these two expansions is the possibility to deduce the well-known Euler and the new Euler-type formulae - see Section 4. Proposition 2 (Partial fraction expansion) For w E C with w one has the partial fraction expansion

7r

sec nw =

(-l)k Le k +w +2

1 .

+

~

r/:

Z'..

(12)

kEZ

369

Michael Hauss - - - - - - - - - - - - - - - - - - - - - - - - -

Proof ln the second Poisson-Boole formula (9) let f(z) := Then with x = 0 one has 1

N

(

c

2

,,-zw

(w

E

q.

-2?Tw)N+l

~(-l)"e-2,,-(h+v)w = e-2?Twh_-_-_e_ _ __

1 + e-211:w

L

v=O

For h = ~ one finally obtains

which completes the proof by writing iw instead of w.

•

Proposition 2 gives a partial fraction expansion of the exponential generating fonction of the Euler numbers, namely

(13) This generating fonction has a similar appearance when the Euler numbers are replaced by the Bernoulli numbers En. In fact, there holds true 1fW - ~ (-l)k B2k ( )2k ( 2 k) 1. 1fW 2 cot 2 - L

1fW

(lwl < 2)

k=O

and in this case one has the partial fraction expansion, the counterpart of (12), 7f

cot 1fW =

1 Le -k-+w

(w

E

C\Z)

kEZ

from which the Euler formula

(m EN) can be deduced. To obtain similar partial fraction expansions in the Hilbert case one needs to introduce the omega fonction (see [4] and [7]).

370

______ A Boole-Type Formula lnvolving Conjugate Euler Polynomials

Definition 1 (Omega function) For w the incomplete D-function is defined by

E

IC and-~ :Sa< b :S ~ (a, b # 0)

If 0 E (a. b), the integral is to be understood as the Cauchy principal value at zero. In the case a = -~ and b = ~ we write D(w) ·- D(w; -~, ~) in abbreviation, called the (complete) D-function. Thus, by the definition of the Hilbert transform, the (complete) omega fonction D( w) is the 1-periodic Hilbert transform at 0 of the periodic extension of f(x) := e-xw, x E [-~, ~) (w E IC). In [7] it was shown that the (complete) omega fonction essentially is the exponential generating fonction of the conjugate Bernoulli numbers BI: (~), oo

B~(l)

~ _5__L wk

L_, k=O

k!

w/2

=

-~D(w)

(14)

eW - 1

ln this sense the omega fonction is the Hilbert analogue of the fonction cot x. In [7] it was shown that there holds true the partial fraction expansion

(w

E IC\Œ)

(15)

from which the Euler-type formula

(m EN) was deduced. In the Euler case, in [4] for fonction

lwl < 7r and x

E (0, 1) the exponential generating

~ ---;y-w E;;:(x) n = ew 2exw { - "( ) ( w + 1 H 2w + e + 1)H" ( 2w; - 21 , - 2X) +

L.., n=O

+(e-w

(16)

X 1 } + l)D(2w; -1--, 2)

2

was deduced. In the most important case x = ~ this is the exponential generating fonction of the conjugate Euler numbers. ln this case one has a certain fonctional equation of the omega fonction.

371

Lemma 1 (Functional equation) For w E C one has D(2w;- 1 , 1 ) - 1 n(w) = - 1{ - D(2w) + (e w + l)D(2w; - 1 , - 1 ) (17) 44 2

2

2 4

} +(e-w+l)D(2w ; 1 , 1 ).

42

Proof Noting the duplication formula 2cot(2x) = cotx + cot(x + obtains 1 1 -w 1 3 D(w) = D(2w; - , ) + e D(2w; 4' 4)-

~),

one

44

Combining this result with the equation e-wn(2w·

~ ~)

'4'4

- D(2w·

-~ ~)

) 4'4

+ l)D(2w· '

= (ew

+(e

-w

-~ -~) +

2' 4 1 1 1 1 + l)D(2w; , ) - D(2w; - , ),

42

which readily can be shown, the proof is complete.

22

•

Inserting the functional equation (17) into (16), one obtains the generating fonction

(18) =

1

1 2 exw 1 sinh wu -4-du ew + 1 o+ sin 7rU

(lwl < 7r),

which is the counterpart of (14). In Theorem 3.13 of [4] the numbers En(~) are given in terms of certain integrals. In fact, besides E2rn(~) = 0 one has, e.g.,

~E1 = E1(~) = D'(O) -40'(0;-~, ~) =

-2

f,

112

1/4

cot Jru du+ 8

f,

(19)

112

112

u cot 7rU du - 2 /

1/4

u cot 1fU du. -1/2

The following result will be basic for the derivation of the Euler-type formula of Section 4. In fact, it is the counterpart of (15) in the case of the conjugate Euler polynomials.

Proposition 3 (Partial fraction expansion) For w E {z E C; w ~), k E Z} one has the partial fraction expansion 1 1 1 D(47rw; --, -) - -D(27rw) = 4 4 2 =

1

1 2 /

o+

sinh 27rwu . du sm 7rU

4w cosh JrW ~ (-l)k 7r ~4w 2 +(2k+1)2" k=O

372

=f.

i(k

+

(20)

______ A Boole-Type Formula Involving Conjugate Euler Polynomials

Proof In Theorem 4 let f(z) := e 27rwz (w E x = 0 (p = 1) one has 1 F~(-) = 2

2 =

-i

L

11

k=O

0

OO

1

11

11

0

0

k=O

e-(2k+l)7riu+27rwu du_ e-(2k+l)'9

=

-i

=

4w (1 + e27rw) f)-l)k+l

oo

~

{

Together with y = ~ and

sin((2k + l)7r(- - u))e 2 7rwu du 2

L {e(2k+l)'9 OO

q.

e(2k+l)7riu+27rwu du}

(-l)ki (-1)2k+le27rW - 1 - -(-l)ki (-1)2k+le27rw - 1} 7l' 2w-(2k+l)i 7l' 2w+(2k+l)i

k=O

7l'

4w 2

1

+ (2k + 1) 2

.

On the other hand, there holds true

PV

/

1/2 F(-2 -u)cosec7rudu= 1

-1/2

=

lim

(!-é + 11/2)

c--+O+

-1/2

-2e7rw rl/2

la+ Observing Theorem 4 the proof is complete.

exp(27rw(~ - u))

c

. Sln 7l'U

du

sin~ 27!'WU du. sm 7l'U •

Altogether the fonction -wewl 2 D,(w)/(ew - 1) can be regarded as the counterpart of the cotangent fonction in the Hilbert case and the fonction

-4~{n(2w· -~4 ~) - ~n(w)} eW + 1 4 2 l

l

as the counterpart of the secant fonction.

4

Euler-type formula for .C(2m)

In [4] the Euler-type formula for L(2m) using the conjugate Euler polynomials was deduced. Now we are able to give a second proof of this formula. In fact, we finally can prove the Euler-type formula without the use of the Fourier representation of the conjugate Euler numbers. In a similar way the Euler formula (3) can be deduced from the partial fraction expansion (12). Theorem 6 (Euler-type formula) For m E N there holds true

373

Proof On the one hand side one has for oo

(-l)k

(-l)k

oo

lwl
0, where rn is a rational factor containing the Euler numbers. Theorem 6 now solves this problem also for even n in a structural sense. In fact, the Euler-type formula states that .C(2m) = s2m 7r 2m, where s2m is a factor containing the conjugate Euler numbers (see the integral representations (19)). Whether or not this factor s 2 m is rational is still unknown.

374

_ _ _ _ _ _ A Boole-Type Formula Involving Conjugate Euler Polynomials

References [1] B.C. Berndt. Character analogues of the Poisson and Euler-MacLaurin summation formulas with applications. J. Number Theory, 7 (1975), 413-445. [2] B.C. Berndt. Ramanujan's Notebooks, Part I. Springer Verlag, New York, 1985. [3] J.M. Borwein, P.B. Borwein, and K. Dilcher. Pi, Euler numbers, and asymptotic expansions. Amer. Math. Monthly, 96 (1989), 681-687. [4] P.L. Butzer, S. Flocke, and M. Hauss. Euler fonctions Ea(z) with complexa and applications. In Approximation, Probability, and Related Fields (Froc. Conference at Santa Barbara, May 1993; Eds.: G. Anastassiou and S. T. Rachev), Plenum Press, New York, 1994, 127-150. [5] P.L. Butzer and R.J. Nessel. Fourier Analysis and Approximation; Volume !. Birkhauser Verlag, Basel, 1971. [6] S. Flocke. Fraktionierte Leopoldsche Bernoulli Zahlen. Master's Thesis, RWTH Aachen (in preperation). [7] 1\1. Hauss. An Euler-Maclaurin-type formula involving Bernoulli polynomials and an application to ((2m + 1). Comm. Appl. Anal., 1 (1997), 15-32. [8] D. Kershaw. Sorne refiections on the Euler-Maclaurin formula. In Numerische Integration; Tagung im M athematischen Forschungsinstitut Oberwolfach; 1. 7.10.1978 {Ed.: G. Hi.immerlin), pp. 175-186, Birkhauser Verlag, Base!, 1979. [9] D. Kershaw. Sorne refiections on the Euler-Maclaurin sum formula. In Numerische Integration; Tagung im Mathematischen Forschungsinstitut Oberwolfach 1981; ISNM 57, Birkhauser Verlag, Base!, 1982, 154-163. [10] L. Lewin. Polylogarithms and Associated Functions. North-Bolland, New York, 1981. [11] N.E. Nürlund. Vorlesungen über Differenzenrechnung. Chelsea Pub!. Company, New York, 1954 (1. Edition: Springer, Berlin 1924). [12] A. Weil. Number Theory: An Approach through History from Hammurapi ta Legendre. Birkhauser Verlag, Boston, 1984.

375

Operations Research

Ralf Borndürfer & Martin Grotschel & Andreas Lobel

Alcuin's Transportati on Problems and Integer Programmin g Abstract The need to solve transportation problems was and still is one of the driving forces behind the development of the mathematical disciplines of graph theory, optimization, and operations research. Transportation problems seem to occur for the first time in the literature in the form of the four "River Crossing Problems" in the book Propositiones ad acuendos iuvenes. The Propositiones - the oldest collection of mathematical problems written in Latin - date back to the Sth century A.D. and are attributed to Alcuin of York, one of the leading scholars of his time, a royal advisor to Charlemagne at his Frankish court. Alcuin's river crossing problems had no impact on the development of mathematics. However, they already display all the characteristics of today's large-scale real transportation problems. From our point of view, they could have been the starting point of combinatorics, optimization, and operations research. We show the potential of Alcuin's problems in this respect by investigating his problem 18 about a wolf, a goat and a bunch of cabbages with current mathematical methods. This way, we also provide the reader with a leisurely introduction into the modern theory of integer programming.

AMS Subject Classification (1991): 90C10, 90-01, 90B06, 01A35 Key Words and Phrases: Transportation Problems, Integer Programming, Alcuin of York

1

Introduction

The book Propositiones ad acuendos iuvenes seems to be the oldest collection of mathematical problems written in Latin. It aims at teaching students some basic skills in logic thinking and problem solving. The collection was probably written at the end of the 8th century A.D. and is attributed to Alcuin, see [13], an Anglo-Sa.xon monk, born in York in 735, one of the leading scholars of his time, head of the Frankish court school at Aachen and a royal advisor of Charlemagne, see [12]. [18] contains an English translation of Alcuin's problems, [13] the Latin "original" together with a German translation. Both

379

Ralf Borndi:irfer & Martin Gri:itschel & Andreas Li:ibel _ _ _ _ _ _ _ __

papers also discuss the history of the problems tracing most of them back to ancient Chinese, Egyptian, Greek etc. sources. There is one notable exception. The four "River Crossing Problems" (No. 17-20) seem to appear for the first time in Alcuin's book. We quote here the Singmaster and Hadley translation:

17. Propositio de tribus fratribus singulas habentibus sorores - About three friends and their sisters. Three friends each with a sister needed to cross a river. Each one of them coveted the sister of another. At the river they found only a small boat, in which only two of them could cross at a time. How did they cross the river without any of the women being defiled by the men? 18. Propositio de lupo et capra et fasciculo cauli - About a wolf, a goat and a bunch of cabbages. A man had to take a wolf, a goat and a bunch of cabbages across the river. The only boat he could find could only take two of them at a time. But he had been ordered to transfer all of these to the other side in good condition. How could this be done? 19. Propositio de vira et muliere ponderantibus plaustrum - About a very heavy man and woman. A man and wornan, each the weight of a cartload, with two children who together weigh as much as a cartload, have to cross a river. They find a boat which can only take one cartload. Make the transfer if you can, without sinking the boat.

20. Propositio de ericiis - About Hedgehogs. The Latin text seems to be defective. It seems to say: "About a male and female hedgehog with two young, having weight, wanting to cross a river." One can obviously view these problems as transportation problems subject to side constraints. Such problems have - in the centuries to corne - played prominent roles in shaping the mathematical disciplines of Discrete Mathematics (Combinatorics and Graph Theory), Optimization (Linear and Integer Programrning), and Operations Research. The river crossing problems do not seem to have had any impact on the development of mathematics. They were merely conceived as recreational mathematics. From today's point of view they could have been the starting point of combinatorics and optimization - as we will point out in this paper. But history took a different line. The roots of combinatorics seem to lie in counting objects and arranging numbers such as the construction of magic squares. Counting formulas and magic squares can be found in Hindu and Chinese culture more than two thousand

380

_ _ _ _ _ _ _ Alcuin's Transportation Problems and Integer Programming

years ago. Surprisingly, almost no literature on combinatorics can be found in classical Western civilization. Substantial progress, in particular in the study of magic squares, was made by Chinese and Islamic scholars in the period from 900 to 1300 A.D. Although there was some transmission of this knowledge to the West it was not until the 17th and 18th century that European mathematicians took up the subject seriously. Pascal's work on "his" triangle, otivated by his and Fermat's famous correspondence on games of chance around 1654, Leibniz's Dissertatio de Arte Combinatoria in 1666 and Jakob lernoulli's paper Ars Conjectandi, published posthumous in 1713, mark the beginnings of modern combinatorics; Euler's paper on the bridges of Kêinigsberg 1736 gave birth to the area of graph theory. Note again that Euler's work was inspired by a (recreational) transportation problem. A detailed account of the history of combinatorics can be found in [3]. Also around the end of the 17th century the foundations of optimization were laid through the calculus of variations. A prominent example of this type is the discovery of the brachystochrone by Jakob 1 Bernoulli and Euler in 1697 1 . However, modern optimization only took off in the 20th century. Its rapid development was and still is closely connected to the progress in computing technology. The origins of linear programming, probably the most intensively used optimization technique and the most cost-saving mathematical tool, can again be traced back to transportation problems. Independent developments in the Soviet Union by [21] (mathematical methods for the organization and planning of production), for which he received the 1975 Nobel Prize in Economies, and in the United States (logistics for the war in the Pacifie in the early 1940s) resulted in a general mathematical theory and powerful algorithmic tools that are able to solve linear programs of extremely large scale (see [10] and [24] for the history). Transportation problems often have integrality constraints because of the indivisibility of commodities or transportation units and, therefore, they also give rise to the study of combinatorial or integer programs. Problems of this type are the famous travelling salesman problem (see [20] for its history), the assignment and transportation problem (discussed in [19] and [22]), network and multicommodity fiow and vehicle routing problems (see, for surveys, [1], [11]). An important early example is the case of the Berlin air lift in 1948/1949, which is extensively discussed in [26]. What has all this to do with Alcuin? The point is that Alcuin's river crossing problems could have been the beginning of all these developments. The aim of our paper is to show the potential of his problems in this respect and - in 1

The brachystochrone problem was also solved by Jakob's younger brother Johann I, who posed the problem to the "deeper thinking mathematicians" of his time in 1696, Euler, de !'Hôpital, and Leibniz, but Jakob Bernoulli's and Euler's methods led to the invention of the calculus of variations.

381

Ra.If Borndorfer & Martin Grotschel & Andreas Lobe! - - - -- - - - -

this way - to provide the rcadcr with a leisurely introduction into the modern theory of integer programming and its algorithmic techniques.

2

Alcuin's Solution of his Transportation Problems

Alcuin did not develop general solution procedures for his problems. He simply stated his solution. For problem 18, it reads: "/ would take the goat and leave the wolf and the cabbage. Then I would retum and take the wolf across. Having put the wolf on the other side I would take the goat back over. Having left that behind, I would take the cabbage across. I would then row across again, and having picked up the goat take it over once more. By this procedure there would be some healthy rowing, but no lacerating catastrophe." Figures 1 and 2a give a pictorial description of the solution.

Figure 1: "I would take the goat and leave the wolf and the cabbage."

We can only guess how Alcuin arrivcd at his solutions. But it is very likely that a combination of trial and error, enumeration, and the use of logic implications was used - just as schoolchildren confronted with such problems today would do it. From our modern point of view, no mathematical theory was developed. Problems were solved by ad-hoc methods. This approach was still in use just one hundred years ago, as is pointed out in [28], whcrc a detailed history of problem 17 is givcn with its generalization to more than three couples and the additional use of an island in the river. Authors dcscribing such variations and generalizations of Alcuin 's transportation problems during the last 1200 years often reported wrong solutions. The errors were (and still are, if ad-hoc enumerative reasoning is uscd) due to the fact that, in general, the number of possibilities grows extremely fast when additional parametcrs are introduced. Nowadays this phenomenon is called combinatorial explosion. There is no way

382

_ _ _ _ Alcuin's Transportation Problems and Integer Programming

m

l=l

l=l

[~]

[~]

[~]

t=l

[:]

[~]

t=2

[~]

[~]

t=3

[~]

[~]

t=4

i

[~] [;] i

[g]

t =O

[l]

m i m m i m m i m m

i

i

i

[~]

[;l

[~]

t=5

[~]

[:]

[~]

t=6

i

l=l

[~] a)

i

m

t=7

m m m m m m m m b)

m i m m i m m i m m i rn

Figure 2: "Then 1 would return and take the wolf across. Having put the wolf on the other sicle 1 would take the goat back over. Having left that behind, I would take the cabbage across. 1 would then row across again, and having picked up the goat take it over once more."

383

Ralf Borndè:irfer & Martin Grè:itschel & Andreas Lè:ibel _ _ _ _ _ _ _ __

to avoid it. But today we know reasonable ways to control or structure the "space of solutions".

3

The Modern Approach

Our aim in this paper is to describe a general technique - based on geometry (polyhedral theory) and linear programming - with which Alcuin's transportation problems and their variants and generalizations can be mathematically modelled and solved 2 . In fact, we describe the methodology currently in use to salve truly large-scale integer programs coming up in practice. We sketch a few of these applications in the final section of this paper. To show the technique we concentrate on problem 18 about a wolf, a goat, and a bunch of cabbages. It should be clear from our description how the other transportation problems can be handled. We are, of course, aware that problem 18 is quite simple and that we are using a steam hammer to crack a nut. But the technique we describe is very flexible and applies to all kinds of generalizations. The wolf-goat-cabbage problem is not only just a nice example where the modelling technique can be described easily. In fact, quite surprisingly, all the mathematical difficulties that arise in integer programming already appear here. Our starting point is the symbolic description of Alcuin's solution shown in Figures 1 and 2a. We want to transform this description into a representation that admits the use of algebraic and geometric techniques and is suitable for computation. We indicated the presence of a wolf, goat, or cabbage by writing W, G, or C and the absence by '-'. Instead of W, G, and C, we could also write 1 and instead of'-' write O. To remember whether 1 stands for vV, G, or C we (arbitrarily) fix an order of the symbols. We agree that symbols always appear in triples, the first symbol in the triple referring to the wolf, the second to the goat, and the third to the cabbage. For example, with this convention the "old" triple (W, -, C) now reads (1, 0, 1). Applying this procedure to Figure 2a yields Figure 2b. The advantage of this representation is that we can now "compute with wolves, goats, and cabbages" using standard algebra without introducing special rules for this particular case. The procedure that we have just applied is a special case of a more general method to bring vector spaces into play. This technique is the following: We assume that we have a (finite) set N of n elements. We (arbitrarily) number the 2

384

Our approach is by no means the only modern mathematical method suitable to attack Alcuin's transportation problems. For instance, [4] suggest in exercise 1.8.4 to solve problem 18 as a shortest path problem in a certain state-time graph. This approach is simpler than ours, but it illustrates other techniques than the ones we want to discuss and is not as general.

_ _ _ _ _ _ _ Alcuin's Transportation Problems and Integer Programming

elements 1, ... , n and define a vector x = (x 1 , ... , Xn) where the components are indexed by the elements of N. If Mis a subset of N we can represent M by an n-component vector XM =

where XM ·= i

.

{l, 0,

( X1M , ...

if i E M if i r/:. M

,xnM) i = 1, ... ,n.

The vector xM is called the charncteristic or incidence vector of the subset M of N. In our case of problem 18 the set N consists of the wolf, the goat, and the cabbage. The wolf is the first element and thus represented by the number 1, the goat gets number 2, and the cabbage 3. This way N = {1, 2, 3} is a representation of the wolf, the goat, and the cabbage and, therefore, every 0/1-vector xM with three components can be viewed as the incidence vector of a "wolf-goat-cabbage configuration" M. There are several ways (and it is not completely straightforward) how to model the full problem 18 using this technique. We describe one version that is somewhat redundant but conceptionally clear. Our approach is to view a solution of problem 18 as a sequence of states at different points in time. Let us begin at time t = 0 (again this is just a convention). We introduce three vectors

x(O)

y(O) z(O)

(x(0,1),x(0,2),x(0,3)) E {0,1} 3 (y(O, 1),y(0,2),y(0,3)) E {0,1} 3 (z(O, 1),z(0,2),z(0,3)) E {0,1} 3

with the following meaning: x(O) represents the incidence vector of the wolfgoat-cabbage configuration on one side (we call it the left-hand side) of the river, y(O) the incidence vector of the wolf-goat-cabbage configuration in the boat, and z(O) the incidence vector on the other side (the right-hand side) of the river, all at time t = O. The initial configuration (the state at time t = 0) in problem 18 is

x(O)

(1,1,1)

y(O) z(O)

(0, 0, 0) (0,0,0).

(1)

(Clearly, our approach could handle any other initial state.) Now we have to make some conventions to record the transformation of states when items are shipped across the river. Let us denote by x(t),y(t),z(t) E {O, 1} 3 ,

385

Ralf Borndorfer & Martin Grotschel & Andreas Label _ _ _ _ _ _ _ __

as above, the states on the left bank, the boat and the right bank, respectively, at time t = 0, 1, 2, .... The states x(t) and z(t) correspond to the wolf-goatcabbage configuration after the t-th shipment has been completed, while y(t) records the boat configuration for the t-th shipment. This convention implies that

x(t+ 1) z(t + 1) x(t+l) z(t+l)

= =

x (t) - y (t + 1) } z(t) + y(t + 1)

if t

~ 0 is even

(2)

x(t)+y(t+l) } z(t)-y(t+l)

ift~lisodd.

(3)

These equations are algebraic representations of the state transitions. Equation (2) describes the transition when an item is shipped from the right-hand sicle to the left-hand sicle, i.e., when t is even, while equation (3) models shipments from the left to the right. For instance, consider the transition from the initial state state at (even) time t = 0 with wolf-goat-cabba ge configuration x(O), z(O) to the next state with wolf-goat-cabba ge configuration x(l), z(l) by using the first shipment y(l) (shipping the goat from the left bank to the right bank). Equation (2) yields: x(l) z(l)

= =

x(O)-y(l) z(O) + y(l)

= =

(1,1,1)-(0,l,O ) (0, 0, 0) + (0, 1, 0)

= =

(1,0,1) (0, 1, 0).

The relations (2) and (3) are not sufficient because we have to guarantee feasibility of the states according to Alcuin's stipulations. Since the man rowing the boat can take only one item at a time, we have to introduce additional inequalities

y(t, 1) + y(t, 2)

+ y(t, 3)

:::: 1

t = 1, 2, ....

(4)

We interpret Alcuin's phrase " ... transfer ... in good condition" that certain wolf-goat-cabba ge configurations are not allowed. 'Wolf and goat' or 'goat and cabbage' are not permitted on the same sicle without the man. We can take this into account by adding so-called set constraints to the model. The constraint

x(t)

E

{OH~}(~}(~}

which is equivalent to stipulating

386

m} Hl~

0 ieodd,

(5)

_ _ _ _ _ _ _ Alcuin's Transportation Problems and Integer Programming

requires that whenever the man is not on the left bank, i.e., t is odd, a configuration consisting of all three items, the wolf and the goat, or the goat and the cabbage is not permitted. The modelling of the constraint for the right bank follows the same principle, but needs one more thought. Our aim is to ship all items from the left to the right, i.e., to reach a configuration z(t) = (1, 1, 1). Clearly, when z(t) = (1, 1, 1) is reached for the first time, say at time ta, ta must be odd because a last item has to be moved by the man from the left to the right. When this happens we consider our problem solved and don't want further shipments to occur, i.e., we wish the states to be x(t) = (0, 0, 0), y(t) = (0, 0, 0), and z(t) = (1, 1, 1) for all t > ta. Thus, we do not exclude the state z(t) = (1, 1, 1) from our state space for t even. Therefore, our set constraint for the right bank reads

At this point we have arrived at a proper mathematical model of Alcuin's problem 18 in the following sense. For every set of 0/1-vectors x(t), y(t), z(t) E {O, 1 }3, t = 0, 1, 2, ... , satisfying the constraints (1 )-(6) and z(ta) = (1, 1, 1) for some ta 2 0, there is a feasible shipment of the wolf, the goat, and the bunch of cabbages such that all three items arrive on the right bank of the river at time ta. Conversely, every solution of Alcuin's problem 18 can be encoded into a set of 0/1-vectors as above. The formulation above is not finite, just as problem 18 is not, since we could always add a few irrelevant additional shipments producing arbitrarily long sequences. As one can see from his solution, Alcuin was clearly interested in a minimum number of shipments. This brings up the optimization aspect and ways to make the problem finite. One possibility to introduce a finite time horizon is to count the number L of possible states x(t), t odd, on the left bank and observe that, whenever a certain state x(t), t odd, appears for a second time, say x(s) = x(t), where s < t and s and t are odd, then all shipments in the time interval between s and t were unnecessary. The set constraints (5) imply that there are at most L := 5 different feasible wolf-goat-cabbage configurations for t odd on the left bank, and thus, if a feasible solution exists, there must be one with at most T = 2L - 1 = 2 · 5 - 1 = 9 shipments. Using this or similar observations we can introduce a finite time horizon T, and therefore Alcuin's problem 18 can be viewed as a finite combinatorial problem. l\fore precisely, by adding the "final state constraint"

z(T)

=

(1, 1, 1)

(7)

we can conclude that the wolf, the goat, and the cabbage can be shipped from

387

Ralf Borndôrfer & Martin Grotschel & Andreas Lobel _ _ _ _ _ _ __

the left to the right-hand sicle if and only if the following system (1)- (8) has a feasible solution.

= (1, 1, 1) y(O) = (0, 0, 0) z(O) = (0, 0, O) x(t + 1) = x(t) - y(t + 1) } z(t + 1) = z(t) + y(t + 1) x(t + 1) = x(t) + y(t + 1) } z(t + 1) = z(t) - y(t + 1) y(t, 1) + y(t, 2) + y(t, 3) :::; 1 x(O)

x(t) E

z(t) E {

mHn ' GH~H rn

m'm'm'm'G).( rn

z(T) = (1, 1, 1) x(t),y(t),z(t) E {O, 1} 3

(1)

if 0:::; t:::; T-1 is cven(2) if 0 :::; t :::; T-1 is odd (3) 0:::; t:::; T

(4)

if 0 :::; t :::; T is odd

(5)

if 0 :::; t :::; T is evcn

(6)

0:::; t:::; T

(8)

(7)

Let us introduce some abbreviations at this point. Combining the 3-dimensional vectors x(t), y(t), and z(t) for all T + 1 time periods t = 0, 1, ... , Tinto 3 · (T + 1)-dimensional vectors X

(x(O), x(l), ... , x(T)) E JR 3 ·(T+l),

-

(y(O), y(l), .. ., y(T)) E JR 3·(T+i>,

y z

(z(O),z(l), .. .,z(T)) E IR3·(T+i>,

-

and th ose into a 3 · 3 · (T + 1)-dimensional vector u

:=

(x,y,z)EJR33 '(T-t1),

Alcuin's problcm 18 is to decidc whether there is a solution u = (x, y, z) to the system (1)- (8). There are different solutions of Alcuin's problem. It is therefore reasonable to compare the "quality" of different solutions by the evaluation of certain criteria. In mathematical language t,his means that we transform the combinatorial problem into a combinatorial optimization problem by introducing an objective

388

_______ Alcuin's Transportation Problems and Integer Programming

function that evaluates the quality of solutions. There are, as usual, many options. The most natural option is to minimize the number of trips the man has to do. This is probably what Alcuin had in mind. One is therefore tempted to minimize the fonction T

3

(9)

L LY(t,i) t=O i=l

which, unfortunately, does not do the job. Namely, this fonction just counts the loaded trips since, whenever the man crosses the river with one item (the boat carries at most one) at time t, exactly one of the three variables y(t, 1), y(t, 2) and y(t, 3) is equal to one while the others are zero. But this fonction does not count the unloaded trips. Thus, unloaded trips can be "added" to any optimum solution without affecting optimality. A typical "trick" in optimization, with which such situations can be handled, is scaling. We replace the objectiYe fonction 9, which has only 0/1-coefficients, by T

3

L 3t LY(t, i). i=l t=O

(10)

For any feasible solution (x, y, z) of (1)-(8), set 3

tmax(x, y, z)

:= max{ti

LY(t, i) = 1}, i=l

i.e., tmax is the time of the last nonempty shipment. Now suppose that (x, y, z) and (x', y', z') are two feasible solutions of (1)-(8) with to = tmax(x, y, z) < t~ = tmax (x', y', z'). Then the choice of the scaling factors implies that T

T

3

3

L3tLy(t,i) < 3to+l S 3t~ S L3tLY'(t,i). t=O i=l t=O i=l Hence if (x, y, z) is a feasible solution of (1)-(8) achieving a minimum value with respect to the objective fonction (10) then tmax(x, y, z) is the minimum number of trips necessary to take the wolf, the goat, and the bunch of cabbages across. There are other "tricks" to count the number of trips. We could introduce auxiliary "counting variables" that are one as long as the river is crossed. In our case we would define 0/1-variables w(t) fort= 1, ... , T satisfying

w(t) w(t)

'L,;=

> 1 y(t,i) > w(t + 1)

t

= 1, .. . ,T,

t = 1, ... , T- l.

389

Ralf Borndorfer & Martin Grotschel & Andreas Lobel _ _ _ _ _ _ _ __

Then the objective fonction

(11) would count the number of river crossings, and any minimum solution with respect to this fonction would be best in Alcuin's sense. Although this modelling technique is qui te general (other cases can be handled similarly) optimizers do not like auxiliary 0/1-variables since they usually lead to forther difficulties that we cannot discuss here. Another possibility, in the particular case of Alcuin's problem 18, is to minimize the fonction T

3

L: L:x(t,i) t=O

(12)

i=l

that models the aim to remove all items from the left bank as soon as possible. It is not difficult to show that minimizing (12) is equivalent to minimizing (11). As we have indicated, there is no unique natural way to model Alcuin's problem 18. But we hope that it has become clear that the general approach to formulate problem 18 as a 0/1-optimizatio n problem is quite flexible. Using similar observations and "tricks", generalizations and variants of problem 18 can be modelled easily. Different objective fonctions and forther or other restrictions can be taken into account without difficulty. It should also be apparent how Alcuin's other transportation problems can be phrased in this way. Summarizing the discussion of this section, we have shown that Alcuin's problem 18 is equivalent to finding an optimum solution to the 0/1-0ptimization Problem (3) with T = 9. This is the problem that we are going to discuss forther in subsequent sections. The choice of one of the three (or more) possible objective fonctions is a matter of taste. We always try to avoid auxiliary variables. That is why we rule out (11). Objective fonction (10) has large coefficients (which sometimes results in numerical problems). thus we opt for (12).

4

The Integer Programm ing Approach

We would like to solve Alcuin's problem 18 by means of integer programming methodology. The current model (3), however, is not suitable for this approach because of the set constraints. Constraints of this type can (occasionally) be exploited by dynamic programming or other search or implicit enumeration techniques in case the state or search spaces can be efficiently reduced. We do not discuss these methods here.

390

- - -- - - Alcuin's Transportation Problems and Integer Programming

minimize subject to

= (1, 1, 1) y(O) = (0, 0, 0) x(O)

z(O) = (0, 0, 0) x(t)-y(t+l)} z(t) + y(t + 1)

x(t+l) z(t + 1) x(t + 1) z(t + 1)

=

x(t) + y(t + 1) } z(t) - y(t + 1)

if 0 :::; t :::; T is even if 0 :::; t :::; T is odd

y(t, 1) + y(t, 2) + y(t, 3) ::; 1

x(t)

z(t) E

E

mH~Hn ·Œ ·Œ}

{O)·(~) ·O)·(D·G) ·0)} z(T) = (1, 1, 1) x(t), y(t), z(t) E {O, 1} 3

if 0 :::; t :::; T is odd

if 0 :::; t

~

T is even

O:S:t:S:T

Optimization Problem 1: A Model of Alcuin's Problem 18.

Our approach is to reformulate the set constraints geometrically by means of linear inequalities (and integrality requirements). Let us look at the set constraints

We view the five 3-dimensional 0/1-vectors

x'

·~

m~ m~ m·~ m~ m' ,x'

,x'

,x'

,x'

describing the set of possible x-states, as points in JR 3 . The convex hull of these

391

Ralf Borndûrfer & Martin Grotschel & Andreas Lôbel - - -- - - - -

points

forms a polytope that we call the x-state polytope. By a general theorem due

Figure 3: x-State Polytope.

to Weyl and Minkowski, see [291, there is, for every finite set V of points in lRn, a finite set of inequalities Ax $ b whose solution set is equal to the convex hull of V, and vice versa., for every bounded set P that is the solution set of somc inequality system Ax ::; b, there is a finite set V whose convex hull is equal to P. So we know that there is a system Ax $bof incqualities with

Therc are general techniques to determine, given a finite set of points in nr, an inequality system describing the convex hull of these points (such as FourierMotzkin elimination or the double description method, see [26] or [30]) . These methods are inhcrently exponential. In other words, there are examples of a few points that need an enormous number of inequalities for the description of the convex hull.

392

Alcuin's 'Iransportation Problems and Integer Programming

In our case we are lucky and can easily compute

conv

füH~H!HD ·Œ}

=

l

X1 X2 X3

X1 +x2 X2 + X3

> > > ::;;
0

Zz > 0 Z3

Z1

=

Z3

-Z1 + Z2 + Z3 Z1 + Z2 -

Z3

> ::;; < <
(=) i:: s· rn-

rn 0,....,

Ç:j

5·

rt

2"

rn 0

(1l

,..,

(Jq

(1l

~·

2

0 ,....,

rt

Uî"

r

OO

(1l

i:: ,..,

i)q"

"rj

1)111000000101010010101000010 00110011001101010001000110101 00001010000101110000001110000 00111 2)111000000101010010101000010 00110011001101010001000110101 00001010100001010100001010000 10111 3)111000000101010010101000010 10000101111001000101010010101 00001010000101110000001110000 00111 4)111000000101010010101000010 10000101111001000101010010101 00001010100001010100001010000 10111 5)111000000101010010101000010 10100001010100001000110011001 10101000100011010100001010000 10111 6)111000000101010010101000010 10100001010100001010000101111 00100010101001010100001010000 10111 7)111000000101010010101000010 00110011001101010000101011001 10101000100011010100001010000 10111 ( 8)111000000101010010101000010 00110011001101010001000110101 00001010000101110100101010000 10111 ( 9)111000000101010010101000010 00110011001101010001000110101 10011000100011010100001010000 10111 (10)1110000001010100101010000 10001100110011010100010001101 11010000101010010101000010100 0010111 (11)1110000001010100101010000 10001100110101100010001100110 01101010001000110101000010100 0010111 (12)1110000001010100101010000 10001100110101100010100001011 11001000101010010101000010100 0010111 (13)1110000001010100101010000 10100001011101001010001100110 01101010001000110101000010100 0010111 (14)1110000001010100101010000 10100001011101001010100001011 11001000101010010101000010100 0010111 (15)1110000001010100101010000 10100001011110010001010100101 01000010100001011101001010100 0010111 (16)1110000001010100101010000 10100001011110010001010100101 01100110001000110101000010100 0010111 (17)1110000001010100101010000 10100001011110010001010100101 11010000101010010101000010100 0010111 (18)1110000001010100101010000 10100001011110010001100010011 11001000101010010101000010100 0010111 (19)1110000001010100101110100 00101010010101000010001100110 01101010001000110101000010100 0010111 (20)1110000001010100101110100 00101010010101000010100001011 11001000101010010101000010100 0010111

>

>-) = c + >-(C - c), where c 2: 0 is the fixed consumption for the empty vehicle. Under this assumption an arbitrary distribution of positive loads >- 1 , · · · , Àn over n vehicles always causes the same total consumption n

n

i=l

i=l

This balance equation for LC gives no reason to reload anything as long as all vehicles are necessary, i.e., as long as n - 1 < L Ài· We shall assume LC in the following, since a nonlinear behavior of C (À) might induce a permanent loading instability: If we take, e.g., C(>-) = c + >- 2 (C - c), then equal distributions on all vehicles are favored and thus vehicles might be maintained, if they could be abandoned otherwise. On the other hand, if we take, e.g., C(>-) = c+ J:\(C - c), then the trend is to concentrate loads to the full capacity of few vehicles permanently. Therefore, the problem of optimizing the yield under a nonlinear fonction C(>-) requires a very general theory, which we cannot develop here, not even for the case of a convex fonction C(À). Under ZCR and LC we now investigate the following strategy HSC of the harmonically shrinking convoy (caravan), which is a generalisation of Singmaster's policy: Definition: Strategy HSC: At any time, all of m active vehicles have equal loads À. If À becomes ~ m~l, then one vehicle is abandoned and the total load of m - 1 is distributed equally to each of the remaining m - 1 vehicles. In other words: One vehicle is abandoned as soon as the total remaining load admits to do so. Proposition: HSC is a strategy to obtain a maximal yield. Proof: If we follow HSC, at any time the remaining material is carried in the cheapest way as a concequence of the balance equation for LC. We can solve the problem by means of a convoy as a consequence of ZCR.

415

Walter Oberschelp - - - - - - - - - - - - - - - - - - - - - - -

3

The yield formula for constant consumption

Let us first assume constant consumption C(.À) = C, i.e., c = C. We assume Q ;:::: C, for otherwise the problem is insolvable, since the desert cannot be crossed at all. We have already assumed C ;:::: l. For if C < 1, the problem is solvable - if at all - trivially by one single trip without depot. Then the spending is S = C and the yield is

Q-C

C

Y=--=l-Q Q' Alcuin's example is the case C = 1 and Q = 3. He gets the yield Y = 3- 3.

~

3

-

~

=

~

= o.2.

9

The optimal yield following Singmaster and the strategy HSC is

y =

3- 3.

10 30

2.

15 30

_§__

30

3

5 = - = 0.27. 18

We shall give an estimation of the optimal yield: Theorem 3.1 The optimal yield under constant cons11.mption C is 1 Y= e -C - -(1e -C ) 2Q

+0

(

1 ) -Q2 .

Proof. We only prove the theorem under the assumption that Q is an integer (remember that L = 1). Then Q vehicles are starting the convoy and after the distance d 0 one unit L = 1 of supplies is spent. We have (L =)1 = Q · C ·do.

Now according to strategy HSC one vehicle is abandoned. Similarly, the next quantity L = 1 is spent after the distance d 1 , where

1 = (Q - l)Cd1, since only Q - 1 vehicles were under way, etc. We have to determine R such that (remember D = 1), do+ d1

+ · · · + dR-1:Sl(= D)

do+d1+···+dR>l.

This means do+ d1

416

+ · · · + dR-1 + pdR = 1 with 0 :s; p < 1 .

(1)

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ Alcuin's Carnel and the Jeep Problem

Therefore the spending is S

= R +p

and the yield is

(2) Relation (1) can be written, following the definition of the di, as

(3) Thus we have to determine the integer R such that

(4) Here HQ is the partial sum of the harmonie series Q 1

HQ =

2:-. n

n=l

We use the well known approximation ([14], p.75) H Q = log Q + 'Y +

where

Î =

2~ + 0 ( ~ 2 )

(5)

,

0.5772157 is Euler's constant.

From (4) und (5) we obtain

Q

C=logQ-R

1

+ 2Q-

1 - 2p 2(Q-R)

+o

( 1 ) Q2

Avery rough approximation now shows that lim

Q--+oo

+o

(

) 1 (Q-R)2 .

(c - log Q~R) =

(6)

0, since

obviously Q - R tends to infinity, as can be seen from (3), because C is fixed and the harmonie series is divergent. Therefore, in exponentiating, we get two constants A,B, near 1, such that 0 =/=A::; e 0 QQR ::; B. Hence we can identify

0 (

J2) and 0 (rQ_1R)2).

Therefore, we get from (6),

Expanding the exponential fonction, we get from (6)

Q-R=Qe

-C (

1 - 2p 1 l+ 2Q - 2(Q-R)

+o (

1 ) ) . Q2

(7)

417

Walter Oberschelp - - - - - - - - - - - - - - - - - - - - - -

Now, putting X := Q - R, we obtain the quadratic equation

X

2

-X·Qe -C

(

1 ( - 1 ) ) +--Qe 1 - 2p -C =0 l+-+O 2Q Q2 2

with the solution (taking the positive branch of the square root)

X=~ e-C + ~e-C + 0

(à)+

Q2 1 Q 1 - 2p -e-2C + -e-2C + -e-2C + 0(1) + 0 ( -1 ) - --Qe-C 4 16 4 Q 2 =

~e-c + 0

Q e-C + 2 4

(]:_) + Q e-C /1+]:_(1- 2(1- 2p)eC) + 0 Q 2 Q

y

Expanding the square root according to

Vf+x = 1 + ~ + O(x 2 )

(__!__). Q2

gives

Q -C + -e 1 -C + 0 ( -1 ) + -e Q -C ( 1 + - 1 (1- 2(1- 2p)e c) + 0 ( - 1 )~ X=-e 2 4 Q 2 2Q Q2

1 -C --(1-2p) 1 =Qe -C +-e +O ( -1) . 2 2 Q

ln other words, Q- R

=

Qe -C -

1

2 (1 -

e -C ) + p + 0

(

1)

Q

.

(8)

Using (2), we get the yield - with p disappearing in the equation -

Y(Q)=e -C and the theorem is proved.

1Q(l-e -C )+0

2

(

Q12 )

(9) D

It is easy to calculate the reach of the convoy: If we assume a desert which is wider than the unit desert with width D = 1, we have the reach

Of course, if we put C = HQ, then the convoy reaches exactly the rim of the unit desert. With the usual assumption C = 1 in jeep problems the reach is HQ. We try to control this result by applying Theorem 3.1:

418

- - - - - - - - - - - - - - - - Alcuin's Carnel and the Jeep Problem

If we consider the marginal yield, i.e., Y( Q) = 0, and if we neglect the term

0 (

J

2 )

in the theorem, we get Q =

~(1 -

e- 0

),

or

C = log(2Q + 1) : : : :; log Q +log 2 =log Q + 0.6931472. The error from this estimation against the exact result C = Hq = log Q + 0.5772157 + 0 ( ~) is an over-estimate of only about 0.1159315 for C, which becomes relatively small, if Q is big. Let us go back to Alcuin: Example 3.2 Consider the optimal yield for Q = 3, C = 1 which is Y= 0, 27. By Theorem 3.1 we deduce

e- 0

-

2~ (1 -

e- 0

)

= 0.262526.

If the camel eats the double quantity C = 2, he does not reach the goal, since the reach is then ~ H 3 = 0.916 < l. For large Q, however, the relative yield Y2 (Q) tends to e- 2 = 0.135335, i.e., not more than 13. 53 percent can then be brought through the desert. For Q = 1000 and C = 1 we have the yield 0.367563331. On the other hand, the theorem yields ~ - 20~ 0 ( 1 - ~) = 0.367563381. If Q = 1000 and C = 2, we get the yield 0.134902368 compared to 0.13490251 from the theorem. As we have seen, Alcuin's problem gives the motivation for a general theory of this version of the jeep problem. We are convinced that most of the variations on the jeep problem, which are mentioned in the references, can be solved in a similar technique.

4

Linear consumption

We finally take a look at the more general case, where the consumption C is a linear fonction of the load

C(À) = c +À· (C - c). We ask for the remaining load À(x) in the (possibly virtual ) distance x, if we start with À (O) = 1 (= L). Of course, if C (À) = C is constant, then we have À(x) = 1- Cx. We now assume 0 < c < C. From the obvious infinitesimal relation À(x+dx) = À(x) - (c + À(x)(C - c))dx we get a differential equation for À= À (x):

À1 =-c+À(C-c).

419

Walter Oberschelp - - - - - - - - - - - - - - - - - - - - - - -

This is a simple lst order linear differential equation with constant coefficients. The solution is À(x) =

_1_ (ce-(C-c)x -

C-c

c).

(10)

i) If c = 0, i.e., if there is no consumption for the empty vehicle, we get >.(x) = e-Cx; then the reach is infinite, since e-Cx never becomes zero. ii) If c

>

reach

0, then for >.(x) = 0 we have the condition e-(C-c)x =-§i.e., the without depot is

log ç_

x=-c c.

(11)

-c

We now try to analyze the optimal strategy HSK for the case of linear, but non-constan t consumption . Again we assume that Q is an integer; thus we start a convoy with Q vehicles. At which distance do can we abandon the first vehicle? This is the case, if for each of the Q starting vehicles we have

Q-1

d

Ào=--

(12)

Q .

Wc salve (10) for x and get 1

c

C-c

>.(C-c)+c

x= --log

.

(13)

Putting in Àdo from (12), we obtain

1 CQ 1 CQ do=-- log =--log----C-c (Q-l)(C-c )+cQ C-c CQ-(C-c ) 1 Q =--log---C-c Q- cc/

Putting

Œ

=

cc/' we get Q

1

do = - l o g - - . aC Q-a

(14)

The second vehicle can be abandoned after another distance di with Àd1 =

Q- 2. Q-1

Then we deduce in the same way di i 1 Q-2 t d 2 -- ac og Q-2-a' e c.

420

(15)

· furthermore we get ac log ~ Q-i-a'

_L

- - - - - - - - - - - - - - - - Alcuin's Carnel and the Jeep Problem

Now we can sum up in a similar way, as we did in (1). Then we get a sum of the type

~dµ=-l log

~

aC

Q(Q-1) ... (Q-r) . (Q - Œ) (Q - 1 - Œ) ... (Q - r - Œ)

Now it is possible to express the descending products with the help of the ffunction. Using asymptotic techniques for f(z), one can find approximations for the yield, which generalize Theorem 3.1. We leave the details to a future paper. Acknowledgements: I thank Paul Butzer, Johannes Faassen, Rolf Mi:ihring, Günter Rote, Ingo Schiermeyer, Andreas Schikarski, Peter Schmitt for giving me valuable hints concerning the literature.

References [1] Brauer,U., Brauer W.; A new approach to the jeep problem. Bulletin of the EATCS 38(1989), 145-154 [2] Butzer, P.L., Lohrmann, D. (eds.); Science in Western and Eastern Civilization in Carolingian Times. Birkhauser, Basel 1993 [3] Dewdney, A.K.; Computer Recreations, Sei.Amer., Vol. 256, Issue 6 (1987), pp. 106-109, Vol. 257, Issue 5 (1987), p. 122 [4] Fine, N.J.; The Jeep Problem. Am.Math.Monthly 54(1947), 24-31 [5] Folkerts, M.; Die alteste mathematische Aufgabensammlung in lateinischer Sprache: Die Alkuin zugeschriebenen Propositiones ad acuendos iuvenes. Überlieferung, Inhalt, kritische Edition. Ôsterreich. Akad. Wiss. Math.-Natur. Kl. Denkschr. 116, 2, (6. Abh.), pp. 13-80, Wien 1978 [6] Folkerts, M.; Die Alkuin zugeschriebenen Propositiones ad acuendos iuvenes, in: Butzer-Lohrmann [2], pp. 273-281 [7] Gale, D.; The Jeep once more or Jeeper by the Dozen. Am.Math.Monthly 77(1970)' 493-501 [8] Gale, D.; The return of the Jeep. The Math. Intelligencer 16(1994), 42-44 [9] Gericke, H., Folkerts, M; Die Alcuin zugeschriebenen Propositones. Lat. Text und deutsche Übersetzung. in: Butzer-Lohrmann [2], pp. 273-362 [10] Gerthsen, C., Vogel, H.; Physik, 17. Aufi., Springer Verlag, Berlin 1993

421

Walter Oberschelp - - - - - - - - - - - - - - - - - - - - - - -

[11] Hadley, J., Singmaster, D.; Problems to sharpen the young. The :Math. Gazette 76(1992), 102-126 [12] Hausrath, A., Jackson, B., Mitchem, J., Schmeichel, E.; Gale's Round-Trip Jeep Problem. Am.Math.Month. 102(1995), 299-309 [13] Jackson, B., Mitchem, J., Schmeichel, E.; A solution to Dewdney'sjeep problem. Froc. 7th Quad.Inf.Conf. on Theory and Appl. of Graphs I (Alavi, Y., Schwenk, A. eds.), Wiley, New York 1995 [14] Knuth, D.; The Art of Computer Programming 1, 3rd ed., Addison-Wesley, Reading, Mass. 1997 [15] Phipps, C.G.; The jeep problem: A more general solution Am.Math.Month. _)24(1947), 458-462 [16] Rote, G., Zhang,G., Optimal logistics for expeditions: The jeep problem with complete refilling. Tech. Report 71 KFU Graz and TU Graz, SFB F009, 1996

422

D. Pallaschke & S. Rolewicz

Penalty and augmented Lagrangian in general optimization problems Abstract In this paper a simple analysis of the value fonction of a constrained optimization problem is presented. This is done in a purely algebraic way and no topological assumptions are made. Moreover, we tried to present this material by using extremely simple proofs. A complete presentation of this subject can be found in [12].

AMS Subject Classification: 52 A 07, 26 A 27, 90 C 30 Key words: Generalized Convexity, Value Function, Lagrange Duality

1

Introduction

In the theory of optimization some special types of nondifferentiable fonctions occour in a quite natural way. A typical example of such a nondifferentiable fonction is the value fonction of a constrained optimization problem with a smooth goal fonction under smooth constraints. In [2] the following example for nonsmoothness was considered. Let U be an open subset of a real normed vector space (Z, li.li), and let f,gi: U---+ lR be Fréchet differentiable fonctions. We consider the following smooth optimization problem:

min f(x)

(Po)

un der

9i(x)

=

Œii E

{l, ... ,k}.

Varying the right hand side of the constraints gives the value function of the problem (P0 ), which is defined by V(a1, ... , Œk)

Now suppose that

= min{f(x) 1 9i(x) = Œi,

i E {1, .. , k} }.

x E U is a solution of the problem (P0 ). Then obviously

423

D. Pallaschke & S. Rolewicz - - - - - - - - - - - - - - - - - - - -

i.e. the fonction x r-+ f(x) - V(g1(x), ... ,gk(x)) has a local minimum at x EU. If we could differentiate this fonction then the necessary conditions for an extremum would lead to the Lagrange multiplier rule

with

8 Ài = 8 Üi v 1 . ~ (oq , ... ,Œk) A

In [2] F. H. Clarke points out, that this alternative proof of the well-known Lagrange multiplier rule is meaningless " because it has a fatal fiaw: the implicite assumption that V is differentiable ". In [9] and [10] S. Kurcyusz studied optimization problems in Banach spaces for which no Lagrange multipliers exist. In this paper, we will follow the presentation of S. Kurcyusz and will study a generalized duality theory for optimization problems We begin with a reformulation ofthis problem (Po) in a more general context. Let X, Y be nonempty sets denoted as the space of domain and the space of parnmeters and let r: Y--+ 2X be a set-valued mapping. Let r- 1(x) = {y E Y set-valued mapping.

1

x E r(y)} be the inverse

Using this notation, we shall write min f(x) un der

x E r(yo) , Yo E Y. for a minimization problem. For X r: Y--+

2X

with

(a1, ... , Œk)

= JRk f-7

and

{x EU 9i(x) S: 1

Œi,

i E {1, ... , k}}

the standard formulation of a minimization problem is transformed in the way used in this article. Throughout this paper we will use the symbol X if we deal in the space of domain and the symbol Y if we consider the space of parameters. Denote by lR =IR U { -oo, +oo} the extended reals. Then the value function of the above minimization problem is given by fr: Y--+ JR

424

with

fr(yo)

= inf{f(x)

1

XE r(yo) }.

_ _ Penalty and Augmented Lagrangian in General Optimization Problems

For very simple smooth minimization problems, for instance, for minimization of a polynomial f (x) of degree 3 in one variable under t he constraint x ~ a

the value fonction

fr is not differentiable and not convex as indicated in the

following figure:

f '\,

Figure 1 In order to study local properties of the value fonction, we will introduce now the notion of a reference system. Denote by F(Y) the linear space of all real-valued fonctions defined on Y.

Definition 1.1 A set \P Ç F(Y) is called a reference system if the following two conditions are satisfied:

{i) for all c E lR and all E \P holds + c {ii} for all

À ;:::

0

E

\P,

and all E \P holds A E )

=

lR the function

sup(ef>(y) - f(y)) yEY

is said ta be the -Fenchel conjugate off. The second -Fenchel conjugate is defined as f** : Y--+

lR with

f**(y)

=

sup(ef>(y) - f*(ef>)). Eif>

The following properties of the -Fenchel conjugation can be shown.

Proposition 2.2 Suppose that Y is a nonempty set, Ç F(Y) is a reference system, f, g : Y --+ JR, c E ~ and 1> E . Then the following holds: (i) g ?:: f implies f* ?:: g*;

(ii) (f*+c)

=

f*(ef>-c) =

f*-c;

(iii) f(y) + f*(ef>) ?:: ef>(y) for all y E Y and all Inequality).

1>

E

(Fenchel-Moreau

Proof. (i) From g ?:: f it follows - f ?:: -g. Thus for every 1> E we have 1> - f ?:: 1> - g, which implies f* ?:: g*. (ii) Clearly,

(! + c)*(ef>) = sup(ef>(y) - f(y) - c) = sup((ef>(y) - c) - f(y)) = f*(1> - c). yEY

yEY

On the other hand, we have

(! + c)*(ef>) = sup(ef>(y) - f(y) - c) = sup(ef>(y) - f(y) - c = f*(ef>) - c. yEY

yEY

(iii) By definition of the -Fenchel conjugate we have f*(ef>) = supyEY(ef>(y) f(y)). Hence for all y E Y and all 1> E we have f*(ef>) ?:: ef>(y) - f(y), i.e.

426

_ _ Penalty and Augmented Lagrangian in General Optimization Problems

f*(

1

,yo)=-(fr)*()+(Yo) xEX (2) sup inf L(x, , Yo) = (Jr) 0 (yo) xEX

434

_ _ Penalty and Augmented Lagrangian in General Optimization Problems

(3) inf sup L(x, q), Yo) = (JI')(Yo) xEX E'P

These formulae can be proved as follows. (1) We have: inf L(x,q),y0 )=inf(f(x)xEX

xEX

sup yEr-'(x)

= inf (f(x) =inf(f(x)+ xEX

inf

=inf

xEX yEr- 1 (x)

inf

yEr-'(x)

inf

(x,y)E{xEr(y)

(-q)(y))+q)(yo))

(f(x)-q)(y)+q)(yo))

=(x,y)E{yE~~f(x)

=inf

-(-(q)(y))) + q)(yo))

sup yEr-'(x)

xEX

=

q)(y)+q)(y0 ))

1

1

xEX} (f(x)

yEY}

- q)(y))

+ q)(yo)

(f(x) - q)(y)) + q)(yo)

inf (f(x)-q)(y)+q)(yo))

yEY xEr(y)

=inf( inf (f(x)-q)(y))+q)(yo) yEY xEr(y)

= inf (JI'(y) - q)(y)) + q)(yo) yEY

=- sup(q)(y)) - fI'(y)) + q)(yo) yEY

=-(fr)*(q)) + q)(yo) (2) Since by formula (1) we have inf L(x, q), Yo) = -(fr)*(q)) + q)(yo),

xEX

it follows from Theorem 2.3 that 0 sup inf L(x, q), Yo) = sup(q)(yo) - (JI')*(q))) = (fr)**(y 0 ) = (JI') (y 0 ).

E'P

E'P xEX

(3) Since by Theorem 3.5 Pr(yo)

= {p: X

---t

ffi. p(x) = -

sup

1

q)(y) + q)(yo) , q) E -subdifferentiable. Let us assume that the strong duality holds. Then there exist a with infxExL(x, 0 define the fonction Ç(t, x) E X such that for b E Bµ(t, x), Ç(t, x)(b) = 1 and for b E B"(t,x), Ç(t,x)(b) = -1. Then (i) of the MFC unif. -+ f+ is valid with E = 1. Since Œ > 0, Ç(t, x) can be chosen such that the derivatives Ç'(t,x) are uniformly bounded; then there exists ry E IR such that for ryÇ(t,x), condition (ii) is valid. Thus the MFC unif. -+ f+ is valid with E = ry.

477

l\fartin Gugat - - - - - - - - - - - - - - - - - - - - - - - -

Hence W A(O) holds. Thus even if for t = 0, the two obstacles have contact, i.e. there exist points b E (0, 1) with µ(O)(b) = v(O)(b), the right-sided derivative v~ (0) of the value fonction exists. Example 5 Consider the following decision problem. A consumer intends to purchase an utility-maximizing basket of n goods subject to a budget constraint. The corresponding optimization problem is

P(f3): maxu(x) subject to pT x:::; f3 and Xi 2: 0 i

E {1, · · · ,n},

where u : IRn ---+ IR is the concave differentiable utility fonction, p E IRn is the price vector (p > 0) and f3 is the budget. Consider the budget as a parameter. Let v(f3) denote the maximal utility that can be obtained by the consumer with budget f3. For f3 > 0, A(f3) is valid and bath one-sided derivatives of v exist. For f3 = 0, the consumer cannot buy anything, thus v(O) = O. Moreover, Slater's condition is violated for f3 =O. For the verification of the MFC unif. ---+ 0+, the following definition can be used, with / E (0, oo) chosen sucht that condition (ii) holds: If Xi= 0, let Çi(t,x) =/and if Xi> 0, let Çi(t,x) = -21'E,]=iPJ/Pi· Since W A(O) is valid, Theorem 3 implies that the one-sided derivative v~ (0) of the optimal value fonction exists.

6

Outline of the Proofs

Let w(t) be the value of problem D(t). Say that the set-valued maps S and 6. are closed at f [closed ---+ t+], if for every sequence {(tk, Xk, yk)}k=l with Xk E S(tk) and Yk E 6.(tk) [and tk > t] for each k and tk ---+ f, Xk ----" x, Yk ----"* y*, the relation (x, y*) E S(f) x 6.(f) is valid. The proofs of the Theorems are based on the following Lemmas. Lemma 1 Let A(f) be valid. Then there exists a neighbourhood U off such that (i)-(v) hold. (i) vlu = wlu (strong duality on U) (ii) For all t EU we have S(t) x 6.(t) -1- !f>. (iii) The set UtEuS(t) x 6.(t) is bounded. (iv) M oreover, the maps S and 6. are closed at f. ( v) For all t E U condition A (t) is valid. Proof Let 6 be as in A(f). There exists s E (0, J) such that for all t E (f-s, f+s) the inequality g(t, x) < 0 is valid and v(t) is finite. Then vf(l-c:,t+c:) = wf(l-c:,l+c:) (see [2], Chapter III). For all t E (f - s,f + s), .C(t, f(t, x)) is bounded; thus S(t) #- !/> for all t E (f-s, f+s). Slater's condition implies that 6.(t) -1- !/>for all t E (f-s, f+s).

478

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ Parametric Convex Optimization

Condition (ii) in A(t) implies that UtE(f- 0 ,f+ 0 iS(t) is bounded and Slater's condition implies the corresponding statement for the dual solutions. Let {tk}k=l be a sequence converging to f. For each k, let Xk E S(tk) and Yi. E ~(tk) such that Xk ----'" x weakly in X and Yi. ----'"* y* weakly* in Y*. The lower semicontinuity of g implies that g(f, x) S O. The lower semicontinuity of f yields the inequality

v(t) s f(f, x)slim inf f(tki xk) k-+oo

= lim inf v(tk) k-+oo

slimsupw(tk) = limsup inf L(tk, x, Yi.) k-+oo

k-+oo

xEX

S inf L(f, x, y*) S w(t) = v(t) xEX

since for each fixed t, the fonction H(y*) = infx L(t, x, y*) is upper semicontinuous (as the infimum of upper semicontinuous fonctions). Hence x E S(t) and y* E ~(t). Thus the maps Sand~ are closed at f. With U = (f - E, f + s), assertion (v) is obvious. D Lemma 2 Let W A(t) be valid. Then there exists ta > f such that (i)-(v) hold. (i) vl[t,t 0 ] = wl[t,t 0 ]· (ii) For all t E [f, ta] we have S(t) x ~(t) -/= · f"' 1lill - l"lill Ill Ut L(t-k' Xk' y-*) k--+oo tk

- t

k--+oo

· f"' = l"lID m Ut L(-t' Xk' y-*) . k--+ex;

Since X is reflexive, the last equality follows from the boundedness of {xk}k=l and the assumed continuity of OtL. Now we show (8). Let x E S(f). Then

v(tk)-:-- v(f)= inf L(tk,x,yk) - f(t,x) xEX

:SL(tk, x, Yk) - f(t, x) :SL(tk, x, Yk) - L(t, x, Yk) =(tk - f)8tL(tk, x, Yk), for some ik E (f, tk) due to the mean value theorem. Thus

. v(tk) - v(f) :S hmsuputL . "' ('tk,x,yk _ *) = hmsuputL . "' (-t,x,yk _ *) . hmsup k--+oo

tk - t

k--+=

k--+oo

The last equality follows from the boundedness of {yk}k=l and the assumed continuity of OtL (using Alaoglu's Theorem). D

480

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ Parametric Convex Optimization

Lemma 4 Assume that there exists t 0 > f such that (i)-(iv) from Lemma 2 hold. Then v~ (f) exists. For every sequence { (tk, Xk, y,4'.) }k"=i such that tk --+ f and for each k: tk > f, Xk E S(tk) and Y'k E b..(tk), we have v~(f)

=

lim OtL(tk, Xk, y'k)

k-+oo

=

inf

sup OtL(t, x, y*).

(9)

xES(f) y* E!J.(f)

Pro of Choose a sequence { (tk, x k, Y'k)} k=i as in the Lemma. Let Li = liminfk-+oo(v(tk) - v(f))/(tk - f). Extract a subsequence kj such that limj-+oo(v(tkj) - v(f))/(tkj - f) = L1 and Xkj --'xi E S(f). Then for all y* E b..(f): liminfj-+ooàtL(f,xkj,y*) = OtL(f,xi,y*). Thus (7) implies L1 ;::::

sup

àtL(f, xi, y*).

y*E!J.(f)

Let L2 = limsupk-+oo(v(tk) - v(f))/(tk - f). Extract a subsequence kz such that L2 = liml-+oo(v(tkJ - v(f))/(tk 1 - f) and Y'k 1 ----'* yj' E b..(f). Then for all x E S(f): limsupz-+oo OtL(l, x, ykJ = OtL(f, x, yî). Thus (8) implies inf àtL(l,x,y~).

L2 :S:

xES(f)

Hence we obtain the inequality

àtL(f,xi,y*)

Li::::: sup f}*E!J.(f)

sup àtL(l, x, y*)

;:::: inf

xES(f) y* E!J.(f)

;:::: inf 8tL(f, x, yr) ;:::: L2 ;:::: L1. xES(f)

Since the sequence {tk} was arbitrary, the existence of v~(f) =

inf

v~ (f)

follows; in fact,

sup OtL(f, x, y*).

xES(f) y* E!J.(f)

Let L3 = liminfk-+oo OtL(tk,xk,yk). Extract a subsequence ki such that L3 = limi-+oo OtL(tk" Xki, Y'kJ and Xki --' x2 E S(f). Then (8) applied to {k;}~i yields L3 = limsup8tL(f,x2,Y'kJ;:::: v~(f). i-+oo

Let L 4 = limsupk-+oo OtL(tk, Xk, Y'k)· In analogous manner, (7) yields L4 :S: v~(f). Thus L3 = L4 = v~(f). D The combination of Lemma 1 and Lemma 4 yields the proof of Theorem 2. The combination of Lemma 2 and Lemma 4 yields the proof of Theorem 3.

481

7

Acknowledge ment

I want to thank H. Th. Jongen, R. Hettich and R. Tichatschke for their advice. Moreover, I thank the anonymous referees for their helpful suggestions.

References [1] A. Auslender and R. Cominetti. First and Second Order Sensitivity Analysis of Nonlinear Programs under Directional Constraint Qualification Conditions. Optimization, 21: 351-363, 1990. [2] I. Ekeland and R. Temam. Convex Analysis and Variational Problems. NorthHolland, 1976. [3] J. Gauvin. A Necessary and Sufficient Regularity Condition to have Bounded Multipliers in l\fathematical Programming. Math. Progr., 12: 136-138, 1977. [4] J. Gauvin and F. Dubeau. Differential Properties of the Marginal Function in Mathematical Programming. Math. prog. study, 19: 101-119, 1982. [5] J. Gau vin and R. Janin. Directional Behaviour of Optimal Solutions in N onlinear Mathematical Programming. Math. Oper. Res., 13: 629-649, 1988. [6] M. Gugat. A Fast Algorithm for a Class of Genera!iile 1

P(7r)ij = P1(St = j \ St~l = i) = L

1riaPiaj,

i,j E E.

(1)

aEA(i)

The definitions above are completely standard and can be found in most references on MDP's.

Definition 2.1 A policy f is called ultimately stationary if there exists a stationary policy 7r, called the tail off, and a stopping time T, called the switching time of f, such that 1. ft = 7r for any t ;::::

T,

D

Remark 2.1 As defined above, generally the stopping time T may have a random nature. As an example, T could prescribe a switch at t, when ht E Îit C Ht, where Îit are switch demanding histories; evidently such a switch is stochastic by nature by the set up of the MDP. D

Let the random variable Rt denote the immediate reward at time t. Then for any policy f and initial state s, the expectation of Rt is given by

E1(Rt, s) =

L

P1(St = j, At= a\S1 = s)rja·

L

(2)

jEE aEA(j)

Let 0, then 7r is average-opti mal. If, in addition, (x , y ) 0 be to taken be can is an optimal extreme point solution of LP(5), then Jr Ey), U (Ex \ E in deterministi c by accordingly defining it on the states and x(7r 0 ; (3) = x,

where x(7r 0 ; (3) was defined in (4).

The next result establishes some of the properties of the policy 7r(x, y) induced by a feasible solution (x, y) via (6). Theorem 2.2 (Kallenber g) For (x, y) E F let 7r = 7r(x, y). Then 1. Let x be the [E[-dimensi onal vector with components Xs = 2=aEA(s) Xsou s E E. Then x is a stationary probability distribution of P( 7r), i.e. X~'P(

7r ) = X~1.

2. s E Ex implies that s is a recu'r'rent state in P( 7r), and the set Ex is closed in the Markov Chain P(7r). 3. If (3 > 0, then E = Ex U Ey, Ex is the set of all recurrent states of P( 7r), and Ey \Ex is the set of all tmnsient states.

491

O.J. Vrieze - - - - - - - - - - - - - - - - - - - - - - - - - -

4. Ifs tic

Ex U Ey, then for any j such that f31

> 0 and for any t

~

1,

The linear-progr amming approach to Average-Rew ard l'vIDPs consists of first finding an optimal extreme point solution ( x 0 , y 0 ) of F and then constructing a stationary policy, via (6), whose state-action frequency vector is equal to x 0 . A natural question is whether the same construction could be applied to any feasible solution (x, y) E F? That is, is it true that for any ( x, y) E F, x(K(x,y);/3 )

=

x?

(7)

(i.e., is K(x, y) always a solution to the TFP?). ln general, the answer is no However, the next result provides sufficient conditions under which such a construction is valid.

Theorem 2.3 (Kallenber g) Let (x, y) E F. 1. Let ( x, y) be an extreme point of F. Th en equation (7) holds. 2. Suppose that the Markov Chain P(K(x,y)) has exactly one recurrent class. Then equation (7) holds. 3. Suppose that Ysa

Xsa

LaEA(s) Ysa

LaEA(s) Xsa

for every s E Ex n Ey, a E A(s).

(8)

Then equation (7) holds. Observe that the condition in part 3 of the theorem is satisfied whenever Ex n Ey

=r/J.

Remark 2.2 If the MDP is unichain, then the equation (7) always holds since part 2 of Theorem 2.3 applies. 0

Now we define

The Target Frequency Problem (TFP): Given (x*, y*) E F - a feasible solution of LP (5) - construct a policy f E C 1 such that x(f; /3) = x*. We will refer to x* as the target frequency, and will say that policy f achieves the target frequency x* .

492

- - - - - - - - - - - - - - - - - - - - - - State-Action Frequencies

As discussed in the Introduction, the TFP arises naturally in MDPs with constraints and/or non-standard payoff criteria. Generally, the construction of policy f that solves the TFP is quite problematic, unless one of the conditions of theorem 2.4 applies (in those cases, the optimal policy is constructed via (6)). However, examples can be readily constructed where no stationary policy will suffice. Example 1 Consider the MDP presented m figure l. We have E = 1, 2, A(l) = {1, 2}, A(2) = {1 }, p 111 = p 122 = P212 = 1 (with all other transition 1 probabilities equal to 0), and (3 = (1, 0). The rewards rsa can be chosen arbitrarily. Here C(D) = {7î 1 , 7î 2 } where 7Ti takes action i in state 1, i = 1, 2. Clearly, and

Thus the set of possible state-action frequencies equal { x : x 11 E [O, 1], x 12 = O,x 21 =1-x 11 }. Let the target vector x* = (,\0, 1-À)' for some À E (0, 1). And take a suitable y* so that (x*, y*) E F (e.g., take y= (0, 1 - À, O)'). We first show that no stationary policy can achieve x*. Let 7T be any stationary policy. Note that if 7î 12 = 0 then 7T = 7T1, and x(7T;(3) = x(7î 1 ;(3). If 7î 12 > 0, then x(7î) = x(7î 2 ) since the Markov chain induced by 7T will eventually enter state 2 and will be absorbed there. Thus, the only two state-action frequencies that can be reached by stationary strategies are (1, 0, 0) and (0, 0, 1). It is straightforward to construct an ultimately-stationary policy that will achieve x*. Define the decision rule f 1 by

fi1

=

À,

li2 = 1 -

À,

fi2 = 1,

and let f = (f1, 7T 1 ), that is the policy f uses the decision rule f 1 at time 1 and uses the deterministic policy 7î 1 thereafter. It is easy to verify that x(f; (3) = x*. Note that the stationary policy 7î(x*, y*), constructed via equation (6), is equal to 7î 1 (for any choice of y* vector), and thus, intuitively, the ultimatelystationary policy constructed above 'plays a lottery' (throws a coin) in the initial period and then switches to the stationary policy 7î(x*,y*). It will be shown in the remainder of this paper that, a 'lottery-based' ultimatelystationary policy, whose tail consists of 7î(x*, y*), can be constructed for any given target vector. D

In the remainder of this section we present important preliminary results that indicate the special role played by the stationary policy 7î(x*, y*) in solving the TFP.

493

Q

.Q

1

2

Figure 1: MDP for example 1. Single arrows correspond to actions number 1 and double arrows to actions number 2. All actions are deterministic.

Remark 2.3 To simplify the notation in the remainder of the paper we introduce the following notational convention: for a vector z with components Z 8 a, s E E. a E A(s), we define Zs :=

L

Zsa

and

z'

:=

(z1, z2, ...

'ZN).

aE:A(s)

Also, if E'

c E

is a set of states, then ZE' :=

L L

Zsa·

sE:E' aE:A(s)

For example, the equation "' 6sE:Ex "' 6aEA(s) x sa XEx = 1.

1 could be rewritten as D

Theorem 2.4 For (x*,y*) E Flet 7r* = 7r(x*,y*). Let R 1 •... ,Rd be the recurrent classes of the Markov Chain P(7r*), and let T be the set of transien1 states. Choose some k E { 1, ... , d} su ch that Rk C E?r i.e. the recurrent cl as' Rk will be reached with positive probabil'ity for the initial distribution {3. Lei {3 1 be any initial distribution such that

L P,,.(St E RklS1 = s)/3; > 0

for some t ::>: 1.

sEE

i.e., the process has a positive probability of reaching Rk (in particular, {3 1 = {3 could be chosen). Then

for any s E Rk·

(9)

Proof: Since Rk is a recurrent class under 7r*, it follows that for any s E Rk and t ::>: 1,

P1'* (St+m E RklSt

494

=

s)

= 1,

m

= 1, 2, ....

- - - - - - - - - - - - - - - - - - - - - - State-Action Frequencies

It then follows from the definition of state-action frequencies that

x(7r*: {3

1

)Rk

=

L

PIT*

(Rk is ever reached IS1 =

s)/3; > 0,

(10)

sEE 1 where the last inequality holds by choice of {3 . Note also that (by hypothesis) Rk C Ex*, which implies that x'Rk > 0, and thus the ratios in (9) are welldefined. For each Rj, j E { 1, ... , d} there exists a unique stationary probability vector pJ of P(7r*), where we define ~ = 0 for all s (/':. Rj· It is well-known that the vectors pl, j E {1, ... , d} span the space of stationary probability vectors of P(7r*) (e.g., see [10], p. 26). It is easy to show from (4) that the vector x 1 with components X8 = x(7r*;f3 )s, s E Eisa stationary probability vector of P(7r*). It follows that

d

d

x = L >.]r),

where

j=l

L

ÀJ =

1,

ÀJ

~ 0 'ï/j E {1, ... , d}.

(11)

j=l

Summing (11) component-wise over Rk, we get x(7r*,(3

1

=

)Rk

L

Xs =

Àk > 0,

sERk

where the last inequality follows by (10). Thus, 1

x(7r*,(3 )s x(7r*,(3

i )Rk

=

ÀkP~

---V- =

k

Ps 'ï/ s E Rk.

(12)

k

Since by Theorem 2.2(1) it follows that the vector x* with components x; = x;, s E E, also, is a stationary probability distribution of P(7r*), and since x'Rk > 0 by choice of k, equation (12) also applies to x*, and we have

x;

-*-

k

= ps

'ï/ s E Rk.

XRk

D

Note that by Theorems 2.1, 2.2 the set Ex* is non-empty and is recurrent under 7r*. Thus, for each j E {1, ... , d}, either Rj c Ex· or Rj n Ex* = 0. Therefore, Rk satisfying the hypothesis of Theorem 2.4 can always be found. Theorem 2.4 implies that even though the policy 7r(x*, y*) might not achieve the target frequency x*, it spends the 'right' fraction of time in each recurrent state that might be visited under the initial distribution {3. This result can be easily extended to state-action pairs.

495

Corollary 2.1 Under the hypothesis of Theorem 2.4,

x(1T*,,6 1)sa x(1T*,,61)Rk

x;a

-*-, for any s E Rki a E A(s). XRk

Proof: By equation (4),

x(1T*,,6 1)sa

=

x(1T*,,6 1) 8 1T;a for all s E E,a E A(s).

Also, by definition of 7r* in (6),

Therefore, 1

x(1T*' /3 )sa X(1T*,,61)Rk

for any s E Rk, a E A(s).

D

The next result shows that a target frequency can always be achieved, provided the initial distribution is 'suitably altered'. Corollary 2.2 As in Theorem 2.4, for (x*,y*) E Flet 1T* = 7r(x*,y*). Let R 1 , ... , Rd be the recurrent classes of the Markov Chain P(7r*) induced by 1T* and let T be the set of transient states. Suppose ,6 1 is an initial distribution which satisfies

L 13;

=

x'Rk whenever Rk

C Ex*,

k E {1, ... , d}.

sERk

Then x(7r*;,6 1 ) = x*. Proof: First note that

and thus ,6 1 satisfying the hypothesis of the corollary exists. Moreover, J3kx* 1, so ,6 1 puts only positive weight on initial states that are recurrent w.r.t. P(7r*). Since for each k E {1, ... , d}, Rk is a recurrent class of P(7r*), we must have x(1T*,/3 1)Rk = /31k for all k E {1, ... ,d}. (13) The result follows from Corollary 2.1. D

496

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ State-Action Frequencies

1 Of course, replacing the original distribution f3 by some other (3 is not practical. However, as shown in the following corollary, essentially the same effect could be achieved by finding an ultimately stationary policy f such that the probability distribution over the states at the time of the switch to the 1 stationary part corresponds to (3 . The proof of theorem 2.5 is straightforw ard

and is omitted. Theorem 2.5 Let f be an ultimately-st ationary policy with tail

f3{ =

L

P1(ST

= i\S1 =

s)(3 8 , i

7r*,

and let

= {1, ... , N},

sEE

where f3 is the initial distribution and T is the switching time off. Suppose that {3 1 satisfies the hypothesis of Corollary 2.2. Then

x(f, (3)

=

x*.

Corollary 2.2 has thus reduced the TFP to the problem of finding the "front" of an ultimately stationary policy wich has tail 7r* and which satisfies the hypothesis of the corollary. The computation of such a policy is addressed in the sequel.

3

Achievi ng a Target Frequen cy with Lottery Based Policies

We are now in the position to define the lottery-base d policy f* that will later be shown to be a solution to the TFP. As in the preceding section, (x, y) will denote a particular feasible solution of the Linear Program(5); x represent the target frequency. For the description off* we need two stationary strategies, denoted by 7r(x, y) and 7r(y,x) respectively. Policy 7r(x,y) was already defined in (6) but for the sake of clarity we repeat the definition here 7r(X, Y)sa

7r(y, x)sa

Xsa/Xs Ysa/Ys arbitrary

if s E Ex , a E A (s) ifs E Ey \Ex , a E A(s) otherwise

(14)

Ysa/Ys Xsa/Xs arbitrary

ifsEEy, aEA(s) ifsEEx\E y, aEA(s) otherwise

(15)

{ {

Recall, by Theorem 2.2 part 4, that neither 7r( x, y) nor 7r(y, x) can leave the set of states Ex U Ey and that (3j = 0 for j t/:. (Ex U Ey).

In words,

f*

is defined as follows:

497

O.J. Vrieze - - - - - - - - - - - - - - - - - - - - - - - - - - - -

1. Start playing 7r(y,x) 2. Any time the process is in astate s E Ex, and when no switch has yet occured, perform a two-sided lottery with probabiliti es _Jf_+s and . x+s , Xs Ys Xs Ys respectivel y. 3. If the event associated with probability _Jf_+s , occurs, then go on with Xs Ys playing 7r(y, x). 4. Otherwise switch to 7r(x, y) and keep playing 7r(x, y) for the rest of the process. Lemma 3.1 1. The switching process in the definition off* can be interpreted as a stopping tirne.

2. With probability 1 there exists a finite tirne T for which the strategy 7r(y, x) has switched to 7r (x, y). Proof: 1. By definition of f* part 1 of then lemma holds. 2. First observe that 7r(y, x) and 7r(x, y) are the same in the states s E Ey \Ex. Further, under 7r(x, y) the states Ey \Ex are transient with respect to Ex (cf. part 3 of Theo rem 2. 2). Hence, starting from a state s E Ey \ Ex for both 7r(x, y) and 7r(y, x), the set of states Ex is reached with probability 1. In state s E Ex the one-stage switching probabilit y equals --"'-+s > 0, for any s. Ys Hence with probability 1 such a switch will occur in the Xsinfinite horizon process. (Notice that it could happen that under 7r(y, x) there is a positive probability of traveling back to Ey \Ex from Ex. However this does not affect the ab ove reasoning, since once again the system moves back to Ex with probability 1, etc.) D

Lemma 3.1, part 2, states that f* can be viewed as an ultimately stationary strategy. The following theorem is crucial for our assertion that f* salves the TFP. It states that the probabiliti es with which the switch occurs in the different states of Ex have exactly the right proportion s relative to one another. Let T be the (random) switching moment defined. Theorem 3.1

498

- - - - - - - - - - - - - - - - - - - - - State-Ac tion Frequenc ies Proof: Since the set of states Ex U Ey is closed with respect to bath 7r( x, y) and 7r(y, x), and since {3 only leads to positive frequenci es in this set, we may assume, for notationa l convenie nce, without loss of generalit y that Ex U Ey = E. From the definition of 7r(y,x) and the second equality of constrain t (5) we derive 1 (16) x' +y' - y' P(7r(y, x)) = {3

Let ~be a diagonal matrix of dimensio n jEj with ôii = xiY_;.Yi if i E Ex, Ôii = 1 if i E Ey \Ex and ÔiJ = 0 otherwise . Then, provided that the switch will not occur at a certain decision moment, the one step transitio ns are given by the matrix product

(17) On the other hand, switches at the different states are given by the diagonal elements of I - ~' where I is the identity matrix. Conseque ntly, the total long-run probabili ties of a switch at the different states is given by the expressio n {3 (I - ~) 1

+ {3 ~P(7r(y, x))(I 1

~)

+ {3

1

2 (~P(7r(y, x))) (J - ~)

+...

(18)

OO

1 1 = f3'2~)~P(7r(y,x)))t(J - ~) = {3 (I - ~P(7r(y,x)))- (J - ~).

t=O

(Here the s - th compone nt refers to a switch at state s) Now observe that y'= (x' +y')~. Hence (21) can be written as

x' +y' - (x' or

+ y')~P(7r(y, x))

=

{3

1

1 1 x' +y'= {3 (I - ~P(7r(y, x)))-

Hence, expressio n (23) can be reduced to (x' + y')(J - ~) = that, under f*, PJ*(ST = sj{3) = Xs for any s E Ex.

x', which shows

D

The main theorem of our paper has now become a combina tion of already proved facts. Theorem 3.2 The TFP is solved by f*.

499

Proof: Theorem 3.2 shows that f* is ultimatel y stationar y with the distribut ion at switching tirne equal to x. Obviousl y this switching time distribut ion x satisfies the condition s of Corollary 2.2, Theorem 2.6 now implies that the state action frequenc y distribut ion off* equals x*. D

4

Concl usion

We notice that our solution of the TFP is a computa tionally attractiv e one. Since to determin e the policy f* we only need to calculate the stationar y policies n(x,y) and n(y,x) from (17) and (18) and the lotteries, all ofwhich are easily compute d from the initial vector (x, y) in linear time. In addition the policy f* has a "turnpike " structure : it starts out following one stationar y strategy, n(y, x), and then at a random time, it switches to another policy, n(x, y). Thus, the impleme ntation off* is scarcely harder than the impleme ntation of a stationar y strategy.

500

_______________________ State-Action Frequencies

References [1] E. Altman and A. Shwartz, "Markov Decision Problems With State-Action Frequencies", SIAM J. Contrai Opt., 29, 1991, pp. 786-809. [2] M. Baykal-Gursoy and K.W. Ross, "Variability-Sennsitive Markov Decision Processes", Math. of Oper. Res. 17, 1992, pp. 558-571 [3] E.V. Denardo, "On Linear Programming in Markov Decision Processes", Management Science. 16, 1970, pp. 281-288 [4] C. Derman, Finite State Markovian Decision Processes. Academic Press, New York, 1970 [5] J.A. Filar, L.C.M. Kallenberg and H.-M. Lee, "Variance- Penalized Markov Decision Processes", Math. Oper. Res., 1, 1989, pp. 147-161. [6] J.A. Filar and D. Krass, "Hamiltonian Cycles and Markov Chains", Math. Oper. Res., 19, 1994, pp. 223-235. [7] A. Hordijk and L.C.M. Kallenberg, "Linear Programming and Markov Decision Chains", Management Science, 25, pp. 352-362, 1979. [8] A. Hordijk and L.C.M. Kallenberg, "Constrained Undiscounted Stochastic Dynamic Programming", Math. of Oper. Res.,2, pp. 159-217, 1984. [9] A. Hordijk, O.J. Vrieze and G.L. Wanrooij, "Semi- Markov strategies in Stochastic Garnes", !nt. J. of Game Theory, 12, pp. 81-89, 1983. [10] L.C.M. Kallenberg, Linear Programming and Finite Markovian Contrai Problems. Volume 148, Mathematical Center Tracts, Amsterdam, 1983 [11] M.L. Puterman, Markov Decision Processes, Wiley, New York, 1994. [12] D.J. White, "Mean, Variance and Probabilistic Criteria in Finte Markov Decision Processes", Journal of Optimization Theory and Applications, 56, 1988, pp.1-29.

501

Misce llaneo us Tapie s

Henri Cohen

Comput ing in Aige braie N umber Fields

Abstract

In this paper, we explain how one can now make efficient computations in algebraic number fields, using the Pari system developed by the author's group in Bordeaux. Recent advances such as the solution to the principal ideal problem, finding explicit Galois conjugates, computing in relative extensions, or computing ray class group of number fields and associated computations are also described. The complete system (sources and manual) is available by anonymous ftp. AMS Subject Classificatio n: 11 Y 40 , 11- 04 , 11R29 Key Words and Phrases: Number field, class group, unit group, computational number theory

During the past 10 years, with the help of a number of colleagues and students, we have developped a package for number theory called PARI. In the past 3 years, this package has been considerably enhanced by the addition of fonctions for computing in algebraic number fields, some of them being very sophisticated. The aim of this talk is to explain how to use this part of the PARI package. A complete description can be found in the user manual, as well as in [Coh]. Note that there is basically only one other package capable of performing the same kind of algebraic number theory computation s, the KANT package written under the supervision of M. Pohst now in Berlin. There are quite a number of differences between these packages. I will not talk about KANT, but mention important places where the functionaliti es of the two packages differ. Both packages are available without cost by anonymous ftp. Another important system should be mentioned, the MAGMA system, written under the supervision of J. Cannon in Sidney. This package, which is not free, now incorporates essentially all of KANT, and should incorporate most of the PARI functionaliti es by the end of next year. The advantage of using MAGMA over KANT or PARI is that it is a much more complete package which includes tools for working in many different parts of mathematics , it has a very nice mathematic al philosophy and user interface. The only problem is that you have to pay for these advantages. Apart from that, I recommend it highly.

505

1 1.1

Number Fields, Elements, ldeals Sorne Algebraic Number Theory

Recall that a number field is a finite extension K of the field Q of rational numbers. By the primitive element theorem, there exists B E K such that K = Q(B). If T(X) E Q[X] is the monic minimal polynomial of B over Q, T is irreducible in Q[X] and K is isomorphic to Q[X]/T(X)Q[X] by sending B to the class of X. An element of K can th us be represented in a unique way as a polynomial in B with coefficient sin Q of degree less than or equal to n - 1, where n = [K: Q] = deg(T) is the degree of the number field K. We will denote by '!LK the ring of integers of K. It is a free 'IL-module of rank n. A '!L-basis w1, ... , Wn of ZK will be called an integral basis of K. The determinant of the matrix Tr(wiwj) is independent of the choice of integral basis and will be called the discriminant d(K) of the number field K. If we denote by d(T) the discriminant of the polynomial T and f = [ZK : Z[B]] (calledthe index of B in K), we have the fondamental equality d(T) = d(K)j2. We will always assume that all this data has been computed before performing any operations in K (we will see how to do this later). Finally recall that '!LK is a Dedekind domain. The crucial properties that we will use are the existence and uniqueness of decomposition into prime ideals, and the approximation theorem. The fondamental theorem of arithmetic in '!LK states that if I is any (non-zero) fractional ideal of '!LK then I can be written in a unique way (up to permutation of the factors) as

I

=

Il pvP

(I)

with

Vp

(I) E 'IL

jJ

where the product is over a (finite) set of prime ideals of '!LK. In particular, if p is a prime number of 'IL, then g

p'!LK

= Il P~i

with

ei

>0

i=l

and it can easily be shown that the ideals Pi are all the prime ideals p of '!LK such that p n Z = p'!L, and that if [ZK /Pi: Z/p'!L] =fi then :Z:::::ir is the time T-map of the autonomous system (4), and T is the period of the limit cycle.) The initial data is shown in Figure 2 (top left).

546

_ _

Computing a normally attracting invariant manifold of a Poincaré map

We compute a branch of invariant circlcs with an accuracy of 10-4 . The initial incremcnt of the continuation parametcr f3 is set to 0.02, but is adjustcd when necessary. The algorithm automatically updatcs the mesh after each continuation step. y

y

X

X

Figure 2: The continuation of the forced Van der Pol oscillator: a 0.4, f3 runs from 0 to 0.360938. Top left: the initial circle with 50 mesh points. Top right: the last circle has 356 mesh points; the tangent and normal directions are draum for each fifth mesh point. Bottom: all circles that have been calculated in the continuation process.

w = 0.9;

547

H.W. Broer & H.M. Osinga & G. Vegter - - - -- - - - - - - - -

Figure 3: The invariant torus of the forced Van der Pol oscillator; a = 0.4, = 0.9 and {3 = 0.360938. We identified t = 0 with l = 2n and embedded the torus in JR3 .

w

Results The continuation process leads to a branch of invariant circles from /3 = 0 up to /3 = 0.360938. Figure 2 (top right) shows the last invariant cirde we were able to compute. Due to the automatic rnesh adaption it consists of 356 points. The continuation steps are shown in Figure 2 (bottom). For {3 ~ 0.3634 the invariant circle disappears in a normal saddle-node bifurcation. As {3 ranges from 0 to this final value, the dynamics on the invariant circle undergoes bifurcations , but apparcntly thesc do not affect our algorithm. In Figure 3 we integrated 51 of the 356 mcsh points on the circlc for /3 = 0.360938 in 50 steps along the vector field. This gives 50 circles as an approximati on of the invariant torus.

548

___ Computi ng a normally attractin g invariant manifold of a Poincaré map

Refer ences [1] A. Back, J. Guckenhe imer, l'vI.R. l\Iyers, F.J. vVicklin, and P.A. Worfolk. "DsTool: Compute r assisted exploratio n of dynamica l systems". Notices Amer. Math. Soc., 39(4):303 -309, 1992.

[2] H.W. Broer, H.M. Osinga, and G. Vegter. "Computi ng a normally hyperboli c invariant manifold of saddle type". ln G.S. Ladde and l\I. Samband ham (Eds.). Proceedin gs of Dynamic Systems & Applicati ons, 2: 83-90, Dynamic Publisher s, Inc., 1996.

[3] H.W. Broer, H.M. Osinga, and G. Vegter. "Algorith ms for computin g normally hyperboli c invariant manifolds ". z. angew. Math. Phys., 48: 480-524, 1997. [4] M.vV. Hirsch, C. Pugh, and M. Shub. Invariant manifolds . Springer Lect. Notes in Math. 583, 1977. [5] l\I.W. Hirsch and S. Smale. Différenti ai equations, dynamica l systems, and linear algebra. Academic Press, 1974. [6] A.J. Homburg , H.l\I. Osinga, and G. Vegter. "On the computat ion of invariant manifolds of fixed points". Z. angew. Math. Phys., 46:171-18 7, 1995. [7] H.l\I. Osinga. "Computi ng inyariant manifolds : Variation s on the graph transform. Ph.D. Thesis, University of Groninge n, 1996.

549

Bernd Krausk opf

Stabi lity Loss near 1 :4 Reso nanc e Abstrac t What happens when a closed orbit of a vector field !oses its stability in a 1:4 resonanc e? This classical codimen sion-two bifurcati on problem from the theory of dynamic al systems is discusse d using the example of a periodic ally forced oscillato r. The main techniqu e is to reduce the problem to the study of Z 4 -equivar iant planar vector fields. A conjectu re by Arnol'd states that the informat ion on ail possible unfoldin gs of such vector fields is containe d in the system i = e i°' z + e i