Table Of ContentDDeeffoorrmmaattiioonnss ooffMMaatthheemmaattiiccaall SSttrruuccttuurreess IIII
Deformations
of
Mathematical Structures II
Hurwitz-Type Structures and Applications
to Surface Physics
Selected Papers from the Seminar on Deformations,
t6di -Malinka, 1988/92
Edited by
JULIAN LAW RYNOWICZ
Institute of Mathematics of the Polish Academy of Sciences,
and Institute of Physics of the University of 1:..6di,
1:..6di, Poland
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
Library of Congress Cataloging-in-Publication Data
Seminar on Deformations (1988-1992 : tâdz. Poland and Malinka. Polandl
Deformations of mathematical structures II : Hurwitz-type
structures and applicatlons to surf ace phyS1CS : selected papers
from the Seminar on Deformations. todz-Malinka.1988/92 I edited by
Julian tawrynowicz.
il. cm.
Includes index.
ISBN 978-94-010-4838-5 ISBN 978-94-011-1896-5 (eBook)
DOI 10.1007/978-94-011-1896-5
1. Functions of complex variables--Congresses. 2. Geometry.
Algebraic--Congresses. 3. Global analysis (Mathematlcs)
-Congresses. 4. Surfaces (Physicsl--Congresses. r. tawrynowicz.
Ju 1 i an. 1939- II. Ti tIe.
OA331.7.S48 1992
515' .9--dc20 93-33466
ISBN 978-94-010-4838-5
Printed an acid-free paper
All Rights Reserved
~994 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1994
Softcover reprint of the hardcover 1s t edition 1994
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner.
Seminar Participants (Mexico, D.F., 1988)
Prof. JoseAdem(Mexico, D.F.),first from the right, with Profs.
Juan Jose Rivaud (Mexico, D.F.), first from the left, and Julian
Lawrynowicz (L6di)
Prof. Jose Adem with Norma Cuellar and Profs. Jakub Rem
bielillski and Julian Lawrynowicz (L6di)
Seminar Participants (Malinka, 1992)
Front row: Prof. Robert Redon (Villeurbanne), Prof. Leszek Wojtczak (t6dz), Dr.
Ilona Zasada (t6dz), Eng. Elzbieta iegalska, M.Sc. (tomza), Zofia
Fijarczyk, M.Sc. (£6dz), Prof. Piotr Modrak (Warszawa)
Back row: Prof. Julian £awrynowicz (L6dz), Prof. Claude Surry (Saint Etienne),
Prof. William M. Pezzaglia Jr. (San Francisco, CA), Prof. Guy
Bertholon (Saint Etienne), Dr. Jerzy Rutkowski (1.6dz), Mme Mad
laine Bertholon (Saint Etienne), Dr. Yves Robach (Lyon), Prof. Louis
Porte (Lyon)
vi
CONTENTS
Foreword ix
Part 1. HURWITZ-TYPE STRUCTURES 1
(ed. by J. Lawrynowicz, F. Succi, and O. Suzuki)
GENERAL THEORY
Editors' note 3
P. YIU (BocaRaton, FL.): Compositionofsumsofsquareswithinteger
coefficients 7
HURWITZ-TYPE STRUCTURES EQUIPPED WITH ADDITIONAL
GEOMETRIC AND ANALYTIC STRUCTURES
V. V. KRAVCHENKO (Odessa) and M.V. SHAPIRO (Odessaand Me
xico, D.F.): Helmholtz operator with a quaternionic wave number
and the associated function theory 101
W. M. PEZZAGLIA Jr. (SanFrancisco, CA): Cliffordalgebraderivation
ofthe characteristic hypersurfaces ofMaxwell's equations 129
1. FURUOYA, S. KANEMAKI (Tokyo), J. LAWRYNOWICZ (L6dz),
and O. SUZUKI (Tokyo): Hermitian Hurwitz pairs 135
S. MALINOWSKI, J. REMBIELINSKI, W. TYBOR (L6dz), and L.C.
PAPALOUCAS (Athens): Operatorrealizationsofquantum Heisen-
berg-Weyl and SU(2)q algebras 155
B. Z. ILIEV (Sofia): Some generalizations of the Jacobi identity with
application to the curvature- and torsion-dependent hamiltonians of
physical systems 161
SPECIAL HURWITZ-TYPE STRUCTURES
AND THE MANY-PARTICLE PROBLEM
J. LAWRYNOWICZ (L6dz), R.M. PORTER, E. RAMIREZ de ARE
LLANO (Mexico, D.F.), and J. REMBIELINSKI (L6dz): On duali-
ties generated by the generalised Hurwitz problem 189
J. LAWRYNOWICZ (L6dz)andO. SUZUKI (Tokyo): Thedualitythe
orem for the Hurwitz pairs of bidimension (8,5) and the Penrose
theory 209
J. LAWRYNOWICZ (L6dz) and L. WOJTCZAK (L6dz) in coopera
tion with S. KOSHI (Sapporo) and O. SUZUKI (Tokyo): Stochasti
cal mechanics of particle systems in Clifford-analytical formulation
related to Hurwitz pairs of bidimension (8,5) 213
vii
viii CONTENTS
Part II. SURFACE PHYSICS STRUCTURES
263
(ed. by J. Lawrynowicz, C. Surry, and Leszek Wojiczak)
BOUNDARY CONDITIONS: BROKEN SYMMETRY
AND SURAFCE DECORATIONS
B. GAVEAU (Paris), J. LAWRYNOWICZ(L6dz),andL. WOJTCZAK
(L6dz): Statistical mechanics ofa crystal surface 265
F. de L. CASTILLO ALVARADO, G. CONTRERAS PUENTE (Me
xico, D.F.), J. LAWRYNOWICZ and L. WOJTCZAK (L6dz): A
Hurwitz-pair approach to the pre-melting problem 289
H. PUSZKARSKI (Poznan) and H. DREYSSE (Nancy): On the exis-
tence conditionsfor surface states in tight-binding thin films 299
1. WOJTCZAK (L6dz) and H. GARTNER (Kassel): Spin wave
resonanceprofiles 307
Ch OPITZ, H. MULLER(Jena) and L. SKALA (Praha): Deformation
ofphysical and chemicalproperties ofsolids causedby the existence
ofa solid surface 317
A. URBANIAK-KUCHARCZYK (L6dz): Spin polarization transport
in thin magneticfilm 327
NONLINEARSOLUTIONS AND DYNAMICAL PROPERTIES:
A NEAR-SURFACE REGION
M. W. KALINOWSKI (Warszawa): Quasilinear quantum field theory
(a programofquantization) 341
L. VALENTA and J. FIALA (Praha): Astudyon torsionvibrations in
linear chains 399
R. REDON (Lyon), Rose-MarieCOURTADEet C. SURRY (Saint Eti
enne): Etude monodimensionnelle d'une barre vibrante avec con-
tants unilateraux 405
L. VALENTA (Praha): Ageneralization ofthe Landau-Lifschitz equa-
tion includingnon-magnetic phenomena 431
J. J. ZEBROWSKI and A. SUKIENNICKI (Warszawa): Chaotic do
main wall dynamics and coherent spatial structures in a magnetic
systemwith surfaces 435
INDEX 461
FOREWORD
These Proceedings contain selected original papers by the speakers invited to the
Seminar on Deformations, organized in 1988/92 by Julian Lawrynowicz (L6di),
whosemost fruitful partstook placein 1988in E6di, Paris and Mexico City(Profs.
J. Adem, F. de1. CastilloAlvarado, G. ContrerasPuente, R.M. Porter, E. Ramirez
de Arellano - Mexico, D.F.; Prof. B. Gaveau - Paris; Profs. J. Lawrynowicz, J.
Rembielinski, L. Wojtczak - Mdi et all.), in 1990 in -Mdi, Tokyo and Sapporo
(Profs. S. Koshi - Sapporo, O. Suzuki - Tokyo, J. Lawrynowicz - L6di et all.), in
1991in t6diand Rome(Profs. S.Marchiafava,F.Succi- Rome,J.Lawrynowicz,1.
Wojtczak - l.6di et all.), and in 1992 in E6di and Malinka - Mazurian Lakeland,
Poland (Profs. C. Surry - Saint Etienne, J. Lawrynowicz, J. Rembielinski, 1.
Wojtczak - L6di et all.). The meetings ofthe Seminar and the Proceedings were
supported by the Polish state Committee for Scientific Research (KBN) and the
-L6di Society ofSciences and Arts (LTN).
The collection containsonly new material- 20 papersconnected with deforma
tions ofmathematical structures in the context ofHurwitz-type structures and ap
plications to surface physics: general theory (editors' note and one leading paper),
Hurwitz-typestructuresequippedwithadditionalgeometricandanalyticstructures
(5 papers), special Hurwitz-type structures and the many-particle problem (3 pa
pers); boundary conditions: broken symmetry and surface decorations (6 papers),
nonlinearsolutionsand dynamicalproperties: a near-surface region(5 papers). All
the papers are in final form: no version ofthem will be submitted for publication
elsewhere. The volumefollows two previous volumes with similar titles:
[1] Lawrynowicz, J. (ed.) (1985) Seminar on Deformations, Proceedings, L6di
Warsaw1982/84(Lecture NotesinMath. 1165), Springer-Verlag, Berlin-Heidel
berg-New York-Tokyo, x+ 331 pp.;
[2] l.awrynowicz, J. (ed.) (1989) Deformations of Mathematical Structures. Com
plex Analysis with Physical Applications. Selected Papers from the Seminar
on Deformations, -L6di-Lublin 1985/87, Kluwer Academic Publ., Dordrecht
Boston-London, xii + 352 pp.
As in the case previous volumes, each paper has been sent to two referees.
The mathematical part (Part I) begins with a very general setting of composi
tion ofsumsofsquares given by the leading paper by P. Yiu (pp. 7-100) and then
concentrates on deformations ofthe related Hurwitz pair structures involving two
vector spaces, Riemannian or Hermitian manifolds, or fibre bundles, related via
the generalized Hurwitz condition for three quadratic or bilinear forms. Actually,
the concept of a Hurwitz pair had been introduced in [1], pp. 184-195, and then
extensively studied in [2], pp. 215-223, 225-233 and 331-337. Now it appears that
Hurwitz pairsgive rise to the realization ofa remarkablefamily ofpartial differen-
IX
FOREWORD
tialequationsin terms ofa suitableCliffordalgebra, thelinearizedequations being
a counterpart of the Cauchy-Riemann or Fueter equations, generating regular or
monogenicfunctions resembling holomorphic mappings. The geometrical structure
introduced enriches our understanding of the Clifford structure. It permits us to
obtain a number ofresultsfor the Dirac and Breit operators, inparticularfor their
null solutions. A special attention is paid to the Hurwitz-typestructuresequipped
with additional geometric and analytic structures as well as to the interrelations
betweenthespecialHurwitz-typestructuresandthe manyparticleproblem. More
over, a studyofthe relationshipofthe Cliffordanalysis and the Penrose transform
isextendedto the Penrose-type and Kaluia-Klein type structures, andtheirduality.
In such a way the book can be considerednot only as a natural continuationof[1]
and [2], but also as a natural application and extension ofthe monograph
[3] Delanghe, R., Sommen, F., and Soucek, V. (1992) Clifford Algebra and Spinor
Valued Functions. A Function Theory for the Dirac Operator (Series: Mathe
matics and Its Applications 53), Kluwer Academic Publishers, Dordrecht-Bos
ton-London, xviii + 485 pp.
The physical part (Part II) contains a study of surface physics structures, in
particular boundary conditions: broken symmetry and surface decorations, as
well as nonlinear solutions and dynamical properties: a near-surface region. Sur
face physics structures are considered as examples of deformations in connection
with nonlinearity and reduction of dimensions. Some nonlinear and dynamical
properties are shown by means ofthe models of two-dimensional surfaces or one
dimensional chains, which usually represent the surface as well. Non-linear be
haviouris always connectedwith solitons or dynamical properties. Relations to the
Hurwitz pairs consist in the property that the Hurwitz structure allows to reduce
the nonlinearity by means ofa proper choice ofthe Clifford algebra.
The bookis intendedfor research mathematicians, mathematical physicists and
physicists, postgraduatestudents, andundergraduatestudentsofthelast twoyears
ofstudies as a supplementary reading in connection with: 1) complex and hyper
complex analysis, 2) complex analytic geometry, 3) complex algebraic geometry,
4) theory of quadratic forms, 5) quantum mechanics of one- and many-electron
atoms, 6) solidstatephysics, 7) statisticalmechanics and the structureofsurfaces,
and 8) symmetry breaking and elementary particle physics.
The organizers of the Seminar express their gratitude to the Kluwer Academic
Publishersfor its kind consent to publishthe Proceedings. They alsowish to thank
warmly Mrs. Elibieta Galuszka and Miss Barbara Galwa, M.Sc., who took care of
the technical preparation ofthis volume.
:t.6di and DeinzejGent, May 1993 Julianl.awrynowicz
Part I
Hurwitz-Type Structures
edited by
JULIAN tAWRYNOWICZ
Institute ofMathematics ofthe Polish Academy ofSciences
and Institute ofPhysi~softhe University ofh6di
~6di, Poland
FRANCESCO SUCCI
Dipartimento di Matematica "Guido Caste/nuovo"
della Universita di Roma I "La Sapienza"
Roma, Italy
and
OSAMU SUZUKI
Department ofMathematics
College ofHumanities and Sciences
ofthe Nihon University, Tokyo, Japan
A. GENERAL THEORY, pp. 3-100
B. HURWITZ-TYPE STRUCTURES EQUIPPED WITH ADDITIONAL GEO
METRIC AND ANALYTIC STRUCTURES, pp. 101-188
C. SPECIAL HURWITZ-TYPE STRUCTURES AND THE MANY-PARTICLE
PROBLEM, pp. 189-262
