Rabinowitz / Zehnder Analysis, et Cetera

Research Papers Published in Honor of Jürgen Moser's 60th Birthday
1. Auflage 2014
ISBN: 978-1-4832-6886-6
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 706 Seiten, Web PDF

Analysis, et cetera: Research Papers Published in Honor of Jürgen Moser's 60th Birthday provides a collection of papers dedicated to Jürgen Moser on the occasion of his 60th birthday. This book covers a variety of topics, including Helmholtz equation, algebraic complex integrability, theory of Lie groups, and trigonometric polynomials. Organized into 31 chapters, this book begins with an overview of some basic consequences of the definition of algebraic complete integrability. This text then derives a representation theorem for solutions of the Helmholtz equation. Other chapters consider the integrable generalizations of the Volterra system and explain the dynamical system in the finite-dimensional case. This book discusses as well the global periodic solutions for the planar triple pendulum. The final chapter deals with the problem of deriving the macroscopic conservation laws, or the Euler equations, in accurate fashion from the microscopic equations of classical mechanics. This book is a valuable resource for mathematicians.

Autoren/Hrsg.

Rabinowitz, Paul H.

Zehnder, Eduard

Inhaltsverzeichnis

1;Front Cover;1
2;Analysis, et Cetera: Research Papers Published in Honor of
Jürgen Moser's 60th Birthday;4
3;Copyright Page;5
4;Table of Contents;6
5;Dedication;8
6;Acknowledgements;10
7;Contributors;12
8;Chapter 1. Painlevé Solutions and Algebraic Complex Integrability;14
8.1;§1. An elementary example of algebraic integrable systems;17
8.2;§2. Coherent Painlevé Solutions = A.C.I.;23
8.3;§3. Weight-homogeneous systems and their Laurent solutions;28
8.4;§4. Constructing the smooth embedding of the tori into
PN,quadratic differential equations and the special Painlevé solutions.;35
8.5;§5. The Kowalewski
Top;44
8.6;References;48
9;Chapter 2. A Representation Theorem for Solutions of the Helmholtz Equation and Resolvent Estimates for The Laplacian;52
9.1;1. Introduction;52
9.2;2. Preliminaries;54
9.3;3. The Estimates;56
9.4;4. The Representation Theorem;60
9.5;5. Resolvent Estimates in the Lower Half-Plane;70
9.6;APPENDIX;77
9.7;References;88
10;Chapter 3. Dynamics of Intersections;90
10.1;§1. Algebraic analysis;91
10.2;§2. Geometric analysis;92
10.3;§3. An example of super-exponential growth;94
10.4;§4. Remarks;96
11;Chapter 4. Laminations of 3-Tori by
Least Area Surfaces;98
11.1;Introduction;98
11.2;1. Notation and definitions;101
11.3;2. Estimates on the surface distance in terms of
the euclidean distance;103
11.4;3. The fundamental estimate;111
11.5;4. Comapactness properties of
F;115
11.6;5. Existence of laminations;120
11.7;REFERENCES;126
12;Chapter 5. Some Qualitative Properties of
Solutions of Semilinear Elliptic Equations in Cylindrical Domains;128
12.1;1. Introduction;128
12.2;2. Asymptotic behaviour of positive solutions of semilinear
elliptic equations in a cylinder;134
12.3;3. Generalized Eigenvalues of Some Associated Linear
Problems;138
12.4;4. Asymptotic behaviour near infinity;147
12.5;5. Monotonicity of travelling fronts;160
12.6;6. Symmetry of Solitary Waves;163
12.7;7. Travelling fronts-uniqueness and other properties;166
12.8;References;175
13;Chapter 6. On Integrable Generalizations
of Volterra Systems;178
13.1;1. Introduction;178
13.2;2. Poisson structure;179
13.3;3. The Lax representation;179
13.4;4. First integrals;180
13.5;5. Integrable reductions;180
13.6;6. Continuous limit for p=const;183
13.7;7. Continuous limit for p
. .8;183
13.8;8. Lie-algebraic generalizations of the Volterra system;185
13.9;9. Integrable dynamical systems in associative algebras;186
13.10;References;188
14;Chapter 7. Forced Oscillations
for the Triple Pendulum;190
14.1;1. Introduction;190
14.2;2. Idea of proof;194
14.3;3. Abstract critical point
result;196
14.4;4. Application to the triple pendulum;210
14.5;5. Relative category and cohomology;217
14.6;References;220
15;Chapter 8. Historical Remarks
on Gauss-Bonnet;222
15.1;1. Review of Surface Theory;223
15.2;2. Contributions of Gauss, Bonnet,
etc;226
15.3;3. Finsler Surfaces and Landsberg
Surfaces;227
15.4;REFERENCES;230
16;Chapter 9. KAM Integrability;232
16.1;1. Model;235
16.2;2. Primary resonances;237
16.3;3.
High-order resonances;239
16.4;4. Numerical experiments;240
16.5;5. A weak
adiabaticity?;245
16.6;6. Concluding remarks;247
16.7;References;248
17;Chapter 10. Anderson Localization
and KAM-Theory;250
17.1;1. Introduction;250
17.2;2. Formulation of the problem and main results for the case
of two frequencies;253
17.3;3. Description of the inductive procedure;254
17.4;4. The limiting density of states and
A(x);260
17.5;REFERENCES;261
18;Chapter 11. Nodal Sets of Eigenfunctions: Riemannian Manifolds
With Boundary;264
18.1;1. Introduction;264
18.2;BIBLIOGRAPHY;275
19;Chapter 12. Peculiarities in the Development
of the Theory of Lie Groups;276
19.1;0.
Introduction;276
19.2;2.
Friedrich Schur;278
19.3;3. Killing;280
19.4;4.
Weyl;284
19.5;5. Pictures;285
19.6;References;294
20;Chapter 13. Generalization of an Estimate
of Small Divisors by Siegel;296
20.1;1. INTRODUCTION;296
20.2;2. RESONANCES;299
20.3;3. A N EQUIVALENCE RELATION ON
ad( . );302
20.4;4. THE BASIC COMPENSATIONS;306
20.5;5. PROOF OF THEOREM;310
20.6;REFERENCE;312
21;Chapter 14. A Quick Proof of Fay's
Secant Identities;314
21.1;1. INTRODUCTION;314
21.2;2. BA AND THETA;315
21.3;BIBLIOGRAPHY;320
22;Chapter 15. Combinatorics of the Free Lie Algebra
and the Symmetric Group;322
22.1;ABSTRACT;322
22.2;1. The Dynkin idempotent;326
22.3;2. Linearity and shuffles;330
22.4;3. Lyndon words;337
22.5;4. The Klyachko idempotent;343
22.6;5. The action of
Sn;350
22.7;6. The action of
GL(N);358
22.8;7. The Poincaré-Birkhoff-Witt Theorem and the Reutenauer
idempotents;364
22.9;8. Multiplicities of the Irreducibles in LIE[a1, a2, . . .
an];382
22.10;REFERENCES;394
23;Chapter 16. Minimizing Variational Integrals
Among Diffeomorphisms;396
23.1;1. Nonlinear hyperelasticity and weak diffeomorphisms;396
23.2;2. Cartesian currents, weak diffeomorphisms and degree;407
23.3;References;416
24;Chapter 17. A New Capacity for
Symplectic Manifolds;418
24.1;1. Axioms for a symplectic capacity and some consequences;418
24.2;2. Construction of a new symplectic capacity;423
24.3;3. Capacity and the Weinstein Conjecture;436
24.4;References;439
25;Chapter 18. The Nash-Moser Theorem and
Paradifferential Operators;442
25.1;1. Introduction;442
25.2;2. Paradifferential
calculus;443
25.3;3. The isometric embedding
problem;447
25.4;4. Scales of Banach
spaces;451
25.5;5. An abstract Nash-Moser theorem;454
25.6;REFERENCES;462
26;Chapter 19. Symmetry Breaking in
Semilinear Elliptic Problems;464
26.1;1. Introduction;464
26.2;2. Abstract Bifurcation and Continuation Results;466
26.3;3. Symmetry Breaking Bifurcation;468
26.4;4. Analytic Properties Of Branches Bifurcating From Infinity;469
26.5;5. Connectivity Properties;473
26.6;6. Degree Calculations;476
26.7;7. Symmetry Breaking Bifurcation;479
26.8;REFERENCES;482
27;Chapter 20. Dynamics of Discrete
Frenkel—Kontorova Models;484
27.1;1. Introduction;484
27.2;2. Traveling waves;487
27.3;3. Proof of the theorem;490
27.4;Appendix: Three lemmas on stopping and on starting;501
27.5;References;505
28;Chapter 21. A Note on the Moser-Hald
Variation of Newton's Method;508
28.1;REFERENCES;512
29;Chapter 22. Complex Analysis On Riemann Surfaces
Motivated By The Operatorial String Theory;514
29.1;§1. Elementary definitions;514
29.2;§2. Laurent and Fourier decompositions on the Riemann
surfaces;519
29.3;§3. Riemann analogs of Heisenberg and Virasoro algebras;522
29.4;§4. Realisation of the Fock spaces and Verma modules in
semi-infinite exterior forms;527
29.5;References;532
30;Chapter 23. Periodic Solutions for Some Forced
Singular Hamiltonian Systems;534
30.1;Introduction;534
30.2;§1 Subharmonic Solutions;536
30.3;§2. Sub harmonic s without
(V1);544
30.4;§3. Another Minimax Approach;547
30.5;§4. An Intersection Theorem;555
30.6;REFERENCES;556
31;Chapter 24. On an Inequality for Trigonometric Polynomials
In Several Variables;558
31.1;1. Introduction;558
31.2;2. Reduction of theorem 1 to theorem 2;562
31.3;3. An estimate for the differential operator
D;565
31.4;4. An estimate for the Laplacian
. in the plane;568
31.5;5. A lower bound for the best possible value of
C;573
31.6;REFERENCES;575
32;Chapter 25. Bifurcation for Semi-linear Elliptic Problems on an Infinite Strip Via the
Nash-Moser Technique;576
32.1;Introduction and Formulation;576
32.2;Bifurcation via the Nash-Moser technique;581
32.3;Concluding Remarks;583
32.4;References;584
33;Chapter 26. Floer Homology, the Maslov Index
And Periodic Orbits of Hamiltonian Equations;586
33.1;1. Introduction;586
33.2;2. The Variational Approach and Floer
Homology;588
33.3;3. The Maslov Index;594
33.4;4. The Fredholm Index;599
33.5;5. Morse Inequalities;605
33.6;Acknowledgement;611
33.7;REFERENCES;611
34;Chapter 27. Determinants of Laplacians;
Heights and Finiteness;614
34.1;Section 0. Introduction;614
34.2;Section 1. Heights;616
34.3;Section 2. The Isospectral Problem;623
34.4;Appendix;629
34.5;REFERENCES;633
35;Chapter 28. Ergodic Schrödinger Operators;636
35.1;References;648
36;Chapter 29. Multiple Solutions to the Dirichlet Problem for the Equation of Prescribed Mean Curvature
Michael Struwe;652
36.1;Abstract;652
36.2;1. Introduction;652
36.3;2. Auxiliary results;656
36.4;3. Existence results;669
36.5;4. Remarks on the Plateau problem;675
36.6;Appendix;676
36.7;References;678
37;Chapter 30. A Priori Bounds for Graphs
With Prescribed Curvature;680
37.1;REFERENCES;688
38;Chapter 31. On the Derivation of Conservation
Laws for Stochastic Dynamics;690
38.1;1. Introduction;690
38.2;2. The Model;692
38.3;3. The Problem;694
38.4;4. The Solution;695
38.5;5. The Method;697
38.6;7. The Ubiquitous Large Deviations;702
38.7;8. Other Models;704
38.8;9. Notes and Comments;706
38.9;Bibliography;707