Proceedings of the Ninth International Colloquium on Differential Equations: Plovdiv, Bulgaria, 18–23 August, 1998 9783112318973, 9783112307700

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Proceedings of the Ninth International Colloquium on Differential Equations

Also available f r o m V S P Proceedings of the Eighth International Colloquium on Differential Equations Plovdiv, Bulgaria, 1 8 - 2 3 August 1997 Proceedings of the Seventh International Colloquium on Differential Equations Plovdiv, Bulgaria, 1 8 - 2 3 August 1996 Proceedings of the Sixth International Colloquium on Differential Equations Plovdiv, Bulgaria, 1 8 - 2 3 August 1995 Proceedings of the Fifth International Colloquium on Differential Equations Plovdiv, Bulgaria, 1 8 - 2 3 August 1994 Proceedings of the Fourth International Colloquium on Differential Equations Plovdiv, Bulgaria, 1 8 - 2 2 August 1993 Proceedings of the Third International Colloquium on Differential Equations Plovdiv, Bulgaria, 1 8 - 2 3 August 1992 Proceedings of the Third International Colloquium on Numerical Analysis Plovdiv, Bulgaria, 13 - 17 August 1994 Proceedings of the Second International Colloquium on Numerical Analysis Plovdiv, Bulgaria, 1 3 - 1 7 August 1993 Proceedings of the First International Colloquium on Numerical Analysis Plovdiv, Bulgaria, 13 - 17 August 1992

Proceedings of the Ninth International Colloquium on Differential Equations Plovdiv, Bulgaria, 18-23 August, 1998

Editor: D. Bainov

///VSPIII

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VSP BV P.O. Box 346 3700 AH Zeist The Netherlands

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© V S P BV 1999 First published in 1999 ISBN 90-6764-296-7

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CONTENTS

Preface 65-th Anniversary of Prof. Drumi Bainov P. Popivanov

Construction of Systems of Partial Differential Equations and Their Solutions X. Agrafiotou Inverse Spectral Problem for the Differential Equation of the Second Order with Singularity R. Kh. Amirov and Y. Cakmak Inverse Problem for the Sturm-Liouville According to a Spectrum and Normalizin R. Kh. Amirov and S. Gulyaz On the Range and Inversion of Certain Unsymmetric Operators A. L. Barrenechea and C. C. Pena Impulsive Hamilton-Jacobi Equations for Hybrid Control Systems S. A. Belbas Numerical Solution of Multi-Time Dynamic Programming Equations S. A. Belbas Overdetermined Differential Systems in Spaces of (Small) Gevrey Functions C. Boiti and M. Nacinovich Applications of Stochastic Differential Equations to Population Growth C. A. Braumann Nonlocal Evolution Problems L. Byszewski Matrix Semigroups and Differential Equations: Ergodism and Asymptotic Behaviour of Solutions with or without Impulses, a Survey (2) J.-P. Caubet Schwarz Algorithms for Nonconforming and Mixed Methods for Second-Order Problems Z. J. Chen Local Mesh Refinement for Degenerate Two-Phase Incompressible Flow Problems Z. J. Chen and R. E. Ewing The Existence of Infinitely Many Solutions of a Nonlinear Wave Equation Q. H. Choi and T. Jung

Contents

T h e Approximation of Continuous Periodic Functions by Means of Extrapolation Techniques for the De La Valle'e Poussin Operator F. Costabile, M. I. Gualtieri and S. Serra

99

Variational Inequalities, Equivalent Optimization Problems and Associated Lagrangean Function P. Daniele

107

Studying the Numerical Properties of Solvers for Systems of Nonlinear Equations D. Dent, M. Paprzycki and, A. Kucaba-Pi§tal

113

Superlinear Convergence of Asynchronous Waveform Relaxation Methods for Nonlinear ODEs M. El-Kyal, J.-C. Miellou and J. Bahi

119

A Discretization Scheme and Error Estimate for First-Order Systems and Elliptic Problems R. E. Ewing and J. Shen

127

Stability Estimates for a Boundary Identification Problem in a Thin Domain D. Fasino and G. Inglese

133

Partial Holder Continuity Results for Solution of non Linear non Variational Elliptic Systems with Limit Controlled Growth L. Fattorusso and G. Idone

139

Applications of the Lvapunov Direct Method for Stability Investigations of Some Functional Differential Equations P. Fergola and V.B. Kolmanovskii

145

On Global Weak Solutions of a Nonlinear Evolution Equation in Noncylindrical Domain J. Ferreira and M.A. Rojas-Medar

155

Lifting Theory and its Application to Quantum Amplifier Processes Sh. Furuichi, K. Oshima and I. Hofuku

163

Tangential Oblique Derivative Problem for Nonlinear Elliptic Discontinuous Operators in the Plane S. Giuffre

169

Life Span of Solutions for Non Linear Cauchy Problem D. Gourdin and M. Mechab

177

Approximation of the McKendrick-Von Foester Equation for Population Growth D. Greenhalgh. I. Gyori and I. Kovacsvolgyi

185

Global Asymptotic Stability of FitzHugh-Nagumo System M. Hayashi

191

Contents

Oscillation Results for Semilinear Elliptic Equations V. B. Headley

197

A Simple Model of the Ocean with Eastern Boundary Slope Current P. F. Hodnett, Y. Yuan and J. Huthnance

205

New Algorithm of Curve Fitting for Non Decreasing Time Series D a t a I. Hofuku, K. Oshima, Sh. Furuichi and M. Ohya

213

T h e Asymptotic Distribution for the Eigenfunctions of the Sturm-Liouville Equation A. Jodayree Akbarfam and E. Pourreza

221

Existence of Group Invariant Solutions of a Certain Semilinear Elliptic Equation R. Kajikiya

229

Structured Programming Diagram PAD, P D E Solver, and Formation of Planetary System T. Kawai

235

Coefficients of an Asymptotic Expansion of logDet for Elliptic Operators with Parameter Y. Lee

241

Accurate Splatting of Octrees J- Y Liu and I. Gargantini

245

Singular Linear ODEs G. Lysik

251

Fine Regularity for Nonlinear Parabolic Systems M. Marino, A. Maugeri and J.Nonvariational Naumann Overlapping Multi-Subdomain Asynchronous Fixed Point Methods for Elliptic Boundary Value Problems J.-C. Miellou, M. Laaraj and M. J. Gander

257

261

Numerical Analysis of Natural Convection of a Viscoelastic Fluid, Using a Spectral Finite Difference Scheme Y. Mochimaru

267

Two-Step Variable Metric Conjugate Gradient Algorithms I. A. R. Moghrabi

273

Nonsmooth Variational Methods and Applications to Discontinuous Boundary Value Problems D. Motreanu and P. D. Panagiotopoulos

281

Mean Value T.Properties M. Nishio, Sugimoto for andtheN.Wave SuzukiEquation

289

viii

Contents

A Note on the Solutions of r " ( x ) r ( x ) = E. Omey

g(x)r'2(x) 293

Properties of General Elementary Matrices in Mathematical Science K. Oshima, I. Hofuku and S. Furuichi

299

Oblique Derivative Problem in Morrey Spaces D. Ii. Palagachev and M. A. Ragusa

307

Applying Probabilistic Neural Networks to the Multifont Recognition Problem M. Paprzycki

and S. Bowers

311

A Nested then Coupled Mediterranean Sea Layer-Level Model A. L. Perkins

319

One Phenomenon the Wave Equation in R 3 N.I. Popivanov andforTs.D. Hristov Blow Up of the Solutions for Quasilinear Hyperbolic Systems and Application to the Nonlinear Vibrating String Equation P. Popivanov and A. Slavova

331

Regularity of the First Derivatives of the Solutions of a Class of Linear Elliptic Equations of Second Order with Discontinuous Coefficients M. A. Ragusa

337

325

Fractal Geometry for Natural-Looking Tree Generation F. Raidan and I. A. R. Moghrabi

343

Exact Solutions of the Navier-Stokes Equations Ii. B. Ranger and R. A. Ross

353

Remarks on the N-Body Problem in Celestial Mechanics A. Rauh

361

Toeplitz-Theorem for Set-Valued Sequences in Banach-Spaces R. Rodler

369

Mixed Problem for a Damped Quasilinear L. P. San Gil Jutuca and M. Milla MirandaWave Equation Stability Z. Shao of Integral Manifolds of Neutral Functional Differential Equations

373 379

Numerical Accuracy of Dynamical System Integration K. Shida, Ii. Haraoka and T. Kawai

385

Approximations of a General Class of Quantum-Mechanical One-Dimensional Point Interactions by Local Self-Adjoint Interactions T. Shigehara, H. Mizoguchi, T. Mishima and T. Cheon

393

Contents

An Exponentially-Fitted Finite Difference Method for the Numerical Solution of Differential Equations with Engineering Applications T. E. Simos and G. Papakaliatakis

401

¿""-Estimates for Strong Solutions to Quasilinear Elliptic Equations L.G. Softova

407

On Convolutional Decompositions of the Generalized Inverse Gaussian Distributions and the Student t-Distributions K. Takano

411

T h e Complete Ducks Not Satisfying S 1 K. Tchizawa

417

Oscillation B. Travis Criteria for Linear Differential Equations Connection Between Differential Equations and Differential Geometry Gr. Tsagas and G. Dimou

423

427

Errors in Iteration Points in Oscillatory State for Systems of Algebraic Equations K. Tsuji

431

Equicontinuity Domains and Runge Domains M. Tsuji

439

Riemann Conditions for Quasi-Abelian Varieties T. Umeno

443

An Approximation of the Functional of a Free Boundary Problem Y. Yamaura

451

A Mathematical Expression of Vertical Distribution of Annual Ring Widths of Trees Based on Height Growth and Inflow of Solute Carbohydrate from Roots T. Yoshida

459

Preface The Ninth International Colloquium on Differential Equations was organized by the Institute for Basic Science of Inha University, the International Federation of Nonlinear Analysts, the Mathematical Society of Japan, Pharmaceutical Faculty of the Medical University of Sofia, University of Catania and UNESCO, with the cooperation of the Association Suisse d'Informatique, the Canadian Mathematical Society, Kyushu University, the London Mathematical Society, Technical University of Plovdiv, and it was partially sponsored by the Bulgarian Ministry of Education and Science under Grant MM-702. The Colloquium was held on August 18-23, 1998, in Plovdiv, Bulgaria. The subsequent colloquia will take place each year from 18 to 23 of August in Plovdiv, Bulgaria. The publishing of this volume is fully supported by the Bulgarian Ministry of Education and Science under Grant MM-702. Organizing

Committee:

G. Anastassiou, D. Bainov (Chairman), Q. H. Choi, J.Diblik, S. Hristova, J. Kajiwara, N. Kitanov (Secretary), D. Kolev (Secretary), V. Lakshmikantham, M. Marino, A. Maugeri (Vice Chairman), L. Medeiros, E. Minchev (Secretary), N. Popivanov, P. Popivanov, M. Robnik, M. Sambandham, H.M. Srivastava, R.U. Verma. Scientific

Committee:

G. Anastassiou (USA), D. Bainov (Bulgaria), C. Dafermos (USA), G. Da Prato (Italy), H. Fujita (Japan), A. Kaneko (Japan), N. Kenmochi (Japan), V. Lakshmikantham (USA), A. Maugeri (Italy), L. Medeiros (Brazil), E. Minchev (Bulgaria), N. Pavel (USA), N. Popivanov (Bulgaria), P. Popivanov (Bulgaria), M. Robnik (Slovenia), M. Sambandham (USA), H.M. Srivastava (Canada), Fr. Stenger (USA), E. Titi (USA), M. Tsutsumi (Japan). R.U. Verma (USA). The address of the Organizing Committee is: Drumi Bainov, P.O.Box 45, 1504 Sofia, Bulgaria. The Editor

65th Anniversary of Professor Drumi Bainov* Peter Popivanov

Institute of Mathematics, Bulgarian Academy of Sciences, Sofia 1113, Bulgaria Dear Professor Bainov, Dear Drumi,

On behalf of all participants of this Colloquium on Differential Equations and of your colleagues, I would like to congratulate you on your 65th birthday. W e all wish you good health, a lot of energy and a long and fruitful working career in the field of differential equations. You have written around 1000 papers, all published in authoritative journals, and 16 monographs. You are very active in didactic and organizing work and your results for differential equations with delay and with impulses are very well known in our community. You have founded the impressive Bulgarian group on differential equations and as organizer you have perfectly managed to find and bring together many young and enthusiastic researchers, directing their interests, energy and efforts in the field of differential equations. You have helped organize many conferences and congresses, of which the Colloquium on Differential Equations in Plovdiv is most famous. Thanks to you and your efforts as heart and engine of the organization, this conference has now its own history, traditions, and always outstanding participants. Despite the hard times we are now going through in Bulgaria, you have succeeded in organizing this prestigious scientific conference. The exchange of ideas, results and new trends in DE is certainly one of the greatest achievements of this forum. Again, our warmest congratulations and best wishes for a very happy and long life. Prof. P. Kendorov, Member of the Bulgarian Academy of Sciences Prof. P. Popivanov, Corr. Member of the Bulgarian Academy of Sciences Prof. S. Nedev, Institute of Mathematics of the Bulgarian Academy of Sciences Prof. M. Kaschiev, Institute of Mathematics of the Bulgarian Academy of Sciences Assoc. Prof. V. Kiryakova, Institute of Mathematics of the Bulgarian Academy of Sciences Prof. N. Popivanov, Faculty of Mathematics, Sofia University.

T h i s congratulatory letter was presented at the opening ceremony of the Ninth International Colloquium on Differential Equations.

9thlnt. Coll. on Differential Equations, pp. 1-6 D. Bainov (Ed) © VSP 1999

Construction of Systems of Partial Differential Equations and Their Solutions X. A G R A F I O T O U School of Technology, Division Thessaloniki 540 06, Greece

of Mathematics,

Aristotle

University of

Thessaloniki,

A b s t r a c t : The aim of the present paper is to solve a given system of partial differential equations and determine their solutions Key Words and Phrases: Partial solution, General solution, System of PDE. 1991 Mathematics Subject Classification. 35F30.

1.

Introduction

Partial differential equations and systems of partial differential equations play an important role in Physics, Biology and Technology. This means that some problems of Physics, Biology or Technology can be represented by a partial differential equation or a system of partial differential equations. Therefore we need to find a solution of this equation or of this system. The purpose of this paper is to give a solution of a system of partial differential equations. This system comes from a nilpotent Lie algebra and it can be used as a model in Nuclear Physics. The whole paper contains four paragraphs. Each of them is analyzed as follows: The first paragraph is the introduction. The second paragraph contains some general theory for homogeneous systems of linear partial differential equations. A special system of linear partial differential equations is study in the third paragraph. T h e new form of this system under some special transformation is included in the last paragraph.

2.

General Theory

We consider the homogeneous system of linear partial differential equations:

9ml(xi, • • • >xn)g^

+ •• •+

• • • , xn)g~

= 'mfcl, • • • . xn)

X.

2

Agrafiotou

which has m equations, which must be satisfied by the unknown:

dx\

' ÔZ2'

'

dxn

This means that we need to determine a function: / = /( il

xn)

(3)

whose first derivatives satisfy this system. In the general, to find a solution of this system is a very difficult problem. Some methods to determine such solutions are given in ([1]). In the next paragraph we give a system of linear partial differential equations whose coefficients are polynomials of first degree of x i , . . . , xn with n=13. This system is obtained from a special nilpotent Lie algebra of dimension 13, which can be used as mathematical model in Nuclear Physics.

3.

A Special System of P D E

We consider the homogeneous system of linear partial differential equations: d f

-X5-

OX df

df

£5 q

dx

6

ox df

df

110-5 ox 9

XT-df h x i odf— + X 1 3 df —= 0 0X4 OX 6 0x9

(6)

df

df

17-5 1-110-5— + OX3 0X6 - x + 2c'm + 4 . , , g ' Kn+1/2)2 ^ (n+1/2)2-" ^ 2e 9 (n+1/2) ^ (n+1/2) ^ (n+1/2)5-3" ^ (n+1/2)''- !'

,

l\ _ _ A 2 ^ ln2(n+l/2) (n+l/2)4-2P 2ir (n+1/2)3-!'

ln(n+l/2) '2 (n+l/2)3"!>

(n+1/2)3-!1

_ ¿ 3 ln3(n+l/2) 12* (n+1/2)2 (2.2)

< ln2(n+l/2) ' ln(n+l/2) Ty, p , f 1 ) ^ ' 4 (n+1/2)2 ^ 15 (n+1/2)2 ^ (n+1/2)2 ^ ^ n 7 ^ ^ Then, by (2.1) and (2.2) 1 j41n7r 6 c 2 = - - H i + — — - + AJl/ 5 + 7T 2(1 - P)ttP' T

2

¿4 In 7r 2 ^

6 +

+

1 /•»

2(1 - p)7rP

Therefore Hx-H2

= tr(c2 - c 2 ).

(2.3)

By hypothesis, the numbers {An} and {/i n } are distinct spectra of one and the same equation. Hence it follows from the results of the previous section that the normalizing consants of the boundary value problem (2.1)-(2.2) are given by the formula (1.18) . In the rest of this section we shall find the asymptotic formula for the numbers a n in terms of the known asymptotic formulae for the numbers An and ¡i n . We shall carry out this derivation in several stages. We first consider the infinite product = 1 +

^

Clearly ln4-(A n ) = E g l / l n ( l +

For sufficiently large n and

k^n -— (in what follows C will denote a constant, not necessarily the AJ-A„ n+1/2 same one). Therefore

ln$(A„) = - f ;

£

(-l)r An

(2.4)

Inverse

Svectral

11

Problem.

We have L e m m a : Let | Xk - X°k |< a ( ¿ = 1 , 2 , . . . ) . Then f ö

ln(n+l/2)

r
oo ar i ¿^r=3 (n+l/2)r - (n+1/2)3 (n+1/2)-"

—

( 1\ ^ V" 1 /'

This estimate and the formula (2.6) show that A*-A°

l ~ ' ( X

k

- X ° \

2

_ / l \

(2.7)

Now we consider the behaviour of the sums appearing in this formula. Using the asymptotic formula (2.1) and (2.2) we see that the first sum satisfies. If we use expression Xk and A® asymptotic, we get the expression v^oo 'At-a; Elb= +

_ rf2ci-6c?T c Q\ 1 ^(3^-4)/. ln(n+l/2) =[ r ^ J ( 2-C2)( n + i/2)i-P + 12* H ^ - ^ C n + l / V + ^

(3 +

, (71-7?) | + 22p-233-2p +

+

ln(3/2) /

lr« 2

+

| ^ i ( c 5 _ C0)

(U . (73-7?) 1 1 , fs ( - 72) + 2p-»3®-pJ (n+1/2)« + U

^aJ^ j

~

1 \

l aCi\ (n+1/2)5 UV'

4

(2.8)

C291 * ^

Therefore, the formulae (2.7), (2.8) and (2.9) imply that i n * (A„) = |-Ai-6c°1 = [ 3—1 (C2 +

-

+O

1 , A(3!T2-4)/. ln(n+l/2) 2/ (n+l/2)^~P + 12* (C2 ~ c 2) (¿+1/2/

-(h

[ M a + i s - S l (3 + u l i s ^ l + MM2Z?1) _ E f c l i l

+ § £ ( c 5 - eg) +

+ ^ ( 7 2

- 72°) +

pTfW + O

Hence •(A.)

+

+ [ m a + (3 + iW- 1 !^

I

feSi

(71-7?)

+ 2F^i(.C5 - C5) + 2ip-i3i-ip

+ ,

+

_ ln(3/2) f

2 >>-'3i-A72

Q-, . (73-7?) 1

1

~ 72) + v ^ - p ] (n+l/2)l

, ^ ( + U

1 ^ (2.10)

R.Kh.

12

Amirov and Y. Cakmak

Now we consider the behaviour of the infinity products 2 ' c

,0

JT C] — P C ,

(C2 _ c 2 ) (n+l/2)3-p +

Ak)k> 1 i £_7+1/no, no,

U^L 1 £_ 7+ i/ ni „ endowed with the of Gelfand and Shilov [3], i.e. a an element € £-, if and only if e £_ 7 + i/ n o , ^ and k -> in

—r+l/"o, "0 • R e m a r k 3 By Remark 2, if f) < 7 we have £p C £ 7 and the topology of £p is stronger than that induced on it by £ 7 . R e m a r k 4 D ( R + ) C £ 7 C £ ( R + ) , the topology of S) (R+) is stronger than that induced on it by £ 7 and the topology of £ 7 is stronger than that induced on it by £ ( R + ) , i.e. £ 7 becomes a testing function space. R e m a r k 5 The space D ( R + ) is not dense in £ 7 . In fact, let 6 be a positive number, T) > - 7 and i € £ ( R + ) such that ((x) = 1 if 0 < x < 6/2 and £(x) = 0 if x > 6. If 4>{x) = xv ex i ( x ) , x > 0, then € £^,6- On the other hand, for any testing function

1R e m a r k 6 Multiplication by x A , A 6 R, is an isomorphism of £ 7 onto £ 7 _ a , its inverse being multiplication by x~x.

On the Range and Inversion of Certain Unsymmetric Operators

25

R e m a r k 7 Differentiation D is a continuous injection of £ 7 into (but not onto) £,+2R e m a r k 8 The operators (1) - (3) are defined on £-, because their kernels are bounded as i —• x + if x > 0. R e m a r k 9 If 0 < u < 7 and W denotes the usual Weyl transform of order a [4], then W" is a continuous injection of £ 7 into (but in general not onto) £ 7 _ f f . P r o p o s i t i o n 6 Let if e £7, n, j be non negative integers such that j < n + l . Then there are uniquely real constants b^j's, 0 < h < j < n, such that

Pn+1J

K„((, o)

1/2, ( , 7 £ R such that £ + 7 > 0. Then KK(£,a) linear - integral transformation of £7 into itself.

is a

(5)

continuous

P r o o f . Let tp e £-,, no 6 N such that

0. In particular, 5upp[A", 0 and writing 6 = KX\/t - 1 we have |ff„(i,aM*)|< Since Ju+m(z)/z"

(Kx/2y~°

( t - l ) ^ - |Ja^(B)

is bounded if z > 0, u > - 1 / 2 (c.f. [5]), say by the constant C„, m , we ob-

| A ' K ( i , a ) ^ ( x ) | < 2 a _ 1 C c t _i,o Be{a,c)

tain

0. From the estimate | A ' . ( i + n + 1 - j - h, a + j - / ¡ ) P n + u ( x , D)p(x)\ < (

k x

/ 2

{ t

~

+ (

fl

1 l\ Q-f

+ l

|Ja+j-h-,(6)

2 l i


0, 0 < a < 1. Then 0 , = A',(O,a) c i " o

(8)

the equality being true on £-,. R e m a r k 1 0 Reasoning in a heuristic manner, from t h e representation of t h e kernel of the operator 0 * we get 0 „ = W 1 (1/2), ( j ! ) " 1 ( « ¥ ) ' , i.e. t h e Neumann generalized series representation of Q K . Therefore we m a y realize 0 S = W 1 (I — k2 W 2 ) - 1 / 2 , wherein I denotes t h e identity operator. So, it seems reasonable to write 0 " 1 = —(I — k2 W 2 ) 1 ' 2 Dx. Precisely, considering again t h e corresponding Neumann generalized series, we obtain evidence of what could be t h e inverse of 0 „ . Hence it seems natural to represent the inverse operator 0 " 1 as &:\(x)

= -v

\x) - K r

Jx

J l { / C (

'"

j ) }

Z — X

V{z)

dz, X > 0.

(9)

On the other h a n d , the work of Bakievich [6] on non convolution operators with Bessel functions in their kernels suggests t h a t = - v U)

- 1 J " V^M*Y/*{Y

Ms far as the authors know, representations uniform convergence on [a, 6] of the integrands

1

- *)}

1 then 0 k ( £ 7 ) = W

1

^) =

P r o p o s i t i o n 11 The operator 0'1 defined by (9) is continuous 7 > 1. P r o p o s i t i o n 12 The operator

between

and £,+2 if

is continuous between the spaces SL, andSL,-\

i f f > 1.

T h e o r e m 13 IfO < a < 1, 0 < then KK(0,a) is a 1 - 1 map between testing subspaces of £(, namely 7\~ 1 _ Q [£{] and A'^j[£{] 2 , with inverse A'fc(0,a) - 1 = xa 0 i ^ . i - a 0 0 " 1 , it being an automorphism if 1/2 < a < 1. P r o o f . By this time it's sufficient to observe that W ~ a o

, =

if a < 1 and

KJQ, a) [A'~i_ q [£ c ]] = Af0~i[£ Ext),(M,F(K2)

fl I ( A ' i , A' 2 ))

-> Ext),(M,F(K2))

-> Ext],{M,r{Ki))

->....

If A'i a n d A'2 are convex, by t h e Ehrenpreis f u n d a m e n t a l principle we obtain t h a t Ext?v{M,7{Ki))

= 0

V; > 1, i = 1,2

a n d hence t h e long exact sequence above reduces to: 0 -

Exq,{M,?{,K2)CiI{KuKi)) Ext°v{M,T{Kx))

Extp(M,Jr{I is that for compact ideal p € HQ2(Im

convex

subset

for

every

the pair (K\, Ki)

compact

convex

Q\ of A'i and a constant

the following

holds

9) < HQl ( I m 9) + M(LQ

to be hyperbolic

subset

Q2 of K2 there

L > 0 such that for

for

M

exist

every

a

prime

true:

+ L log(l + |r|) + L

Vfl = (r, ( ) € V{p)

;

b) a necessary and sufficient condition for the pair (A'i, A'2) to be hyperbolic for M. in the class is that for every compact convex subset Q2 of A'2 there exist a compact convex subset Q1 of K\ and a constant L > 0 such that for every prime ideal p G j4s.s(.M) the following holds true: HQ2(Im9)