Metropolis / Orszag / Rota Surveys in Applied Mathematics

Essays Dedicated to S.M. Ulam
1. Auflage 2014
ISBN: 978-1-4832-5813-3
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 316 Seiten, Web PDF

Surveys in Applied Mathematics: Essays Dedicated to S.M. Ulam covers the proceedings of the First Los Alamos Symposium on Mathematics in the Natural Sciences. The book focuses on the processes, principles, methodologies, and applications of mathematics in the natural sciences. The selection first offers information on the role of applied mathematics, shape of a curve, and biased versus unbiased estimation. Discussions focus on the James-Stein estimator, automorphic forms and Poincaré series, Poincaré metrics, Schottky space and augmented Schottky space, and Schottky groups and Riemann surfaces. The text then examines algorithms, Whitney numbers of geometric lattices, and continued fraction expansion of algebraic numbers. The book takes a look at bifurcations in reaction-diffusion problems, survey of some finite element methods proposed for treating the Dirichlet problem, and mathematics of quantum fields. Topics include Dirichlet problem, chemical waves and reaction-diffusion equations, and bifurcation theorems. The text then ponders on almost periodic behavior of nonlinear waves, turbulence theory, and renormalization group methods. The selection is a valuable source of information for mathematicians and researchers interested in applied mathematics.

Autoren/Hrsg.

Metropolis, N.

Orszag, S.

Rota, G. -C.

Inhaltsverzeichnis

1;Front Cover;1
2;Surveys in Applied Mathematics: Essays dedicated to S.M. Ulam;4
3;Copyright Page
;5
4;Table of Contents;8
5;CONTRIBUTORS;10
6;PREFACE;12
7;INTRODUCTION;14
8;Chapter 1. On the Role of Applied Mathematics;20
8.1;I. INTRODUCTION;20
8.2;II. WHAT IS APPLIED MATHEMATICS ?;22
8.3;III. EDUCATION;33
8.4;IV. RECOMMENDATIONS;37
8.5;V. APPENDIX;39
8.6;REFERENCES;40
9;Chapter 2. On the Shape of a Curve;42
9.1;REFERENCES;57
10;Chapter 3. Automorphic Forms for Schottky Groups;58
10.1;1. SCHOTTKY GROUPS;58
10.2;2. SCHOTTKY SPACE AND AUGMENTED SCHOTTKY SPACE;61
10.3;3. SCHOTTKY GROUPS AND RIEMANN SURFACES;64
10.4;4. POINCARE METRICS;67
10.5;5. AUTOMORPHIC FORMS AND POINCARÉ SERIES;70
10.6;6. PROOFS OF THEOREMS 2 AND 3;73
10.7;7. PROOF OF THEOREM 4;85
10.8;REFERENCES;86
11;Chapter 4. Biased Versus Unbiased Estimation;88
11.1;1. INTRODUCTION;88
11.2;2. THE JAMES–STEIN
ESTIMATOR;90
11.3;3. THREE EXAMPLES OF BIASED ESTIMATION IN PRACTICE;95
11.4;REFERENCES;105
12;Chapter 5. Algorithms;108
12.1;COMPUTATIONAL COMPLEXITY;108
12.2;SOME RECENT POSITIVE RESULTS;113
12.3;ADDITIONAL COMMENTS AND CONCLUSIONS;118
12.4;SUMMARY AND CONCLUSIONS;119
12.5;REFERENCES;119
13;Chapter 6. Whitney Numbers of Geometric Lattices;122
13.1;INTRODUCTION;122
13.2;1. PRELIMINARIES;123
13.3;2. THE SHEAVES
Z AND M;125
13.4;3. THE SHEAF
W;128
13.5;4. A SPECTRAL SEQUENCE;131
13.6;5. STANDARD RESOLUTIONS;133
13.7;REFERENCES;135
14;Chapter 7. Continued Fraction Expansion of Algebraic Numbers;136
14.1;REFERENCES;141
15;Chapter 8. Random Time Evolution of Infinite Particle Systems;142
15.1;REFERENCES;145
16;Chapter 9. Bifurcations in Reaction-Diffusion Problems;148
16.1;I. BIFURCATION THEOREMS;148
16.2;II. CHEMICAL WAVES AND
REACTION–DIFFUSION EQUATIONS;154
16.3;REFERENCES;160
17;Chapter 10. Singular Perturbation;162
17.1;REFERENCES;174
18;Chapter 11. A Survey of Some Finite Element Methods Proposed
for Treating the Dirichlet Problem;176
18.1;1. INTRODUCTION;176
18.2;2. THE NEUMANN PROBLEM;177
18.3;3. THE DIRICHLET PROBLEM;179
18.4;REFERENCES;184
19;Chapter 12. The Mathematics of Quantum Fields;186
19.1;REFERENCES;195
20;Chapter 13. Renormalization Group Methods;198
20.1;1. INTRODUCTION;198
20.2;2. MULTIPLE SCALES;200
20.3;3. THE RENORMALIZATION GROUP: TECHNIQUE;203
20.4;4. THE RENORMALIZATION GROUP: CONSEQUENCES;209
20.5;5. OUTLOOK;213
20.6;REFERENCES;214
21;Chapter 14. Remarks on Turbulence Theory;216
21.1;1. INTRODUCTION;216
21.2;2. THE DYNAMICAL EQUATIONS;218
21.3;3. ABSOLUTE EQUILIBRIUM DISTRIBUTIONS;220
21.4;4. NONEQUILIBRIUM: CASCADE PHENOMENA;222
21.5;5. EDDY VISCOSITY AND RENORMALIZATION;227
21.6;6. CONVECTION INVARIANCE AND LAGRANGIAN COORDINATES;232
21.7;7. WHAT CAN WE CALCULATE?;237
21.8;ACKNOWLEDGMENT;241
21.9;REFERENCES;241
22;Chapter 15. On an Explicitly Soluble System of Nonlinear Differential
Equations Related to Certain Toda Lattices;244
22.1;REFERENCES;253
23;Chapter 16. Three Integrable Hamiltonian Systems
Connected with Isospectral Deformations;254
23.1;1. INTRODUCTION;254
23.2;2. ISOSPECTRAL DEFORMATIONS;258
23.3;3. THE W-PARTICLE SYSTEM ON THE LINE WITH THE
INVERSE SQUARE POTENTIAL;260
23.4;4. ASYMPTOTIC BEHAVIOR, MARCHIORO'S CONJECTURE;263
23.5;5. THE PERIODIC CASE—SUTHERLAND'S EQUATION;266
23.6;6. RATIONAL CHARACTER OF THE SOLUTION OF (2.4);269
23.7;7. THE SCATTERING PROBLEM ASSOCIATED WITH THE EQUATION OF
KAC AND V. MOERBEKE;273
23.8;REFERENCES;276
24;Chapter 17. Almost Periodic Behavior of Nonlinear Waves;278
24.1;1. INTRODUCTION;278
24.2;2. A METHOD FOR CONSTRUCTING NONLINEAR SYSTEMS
WITH MANY INTEGRALS;279
24.3;3. THE TODA LATTICE;280
24.4;4. THE KdV EQUATION;284
24.5;5. THE SINE-GORDON EQUATION;285
24.6;ACKNOWLEDGMENTS;288
24.7;REFERENCES;288
25;Chapter 18. The Real Numbers as a Wreath Product;290
25.1;1. INTRODUCTION;290
25.2;2. SYNOPSIS;291
25.3;3. STRINGS;293
25.4;4. EQUIVALENCE OF STRINGS;294
25.5;5. CARRIES1;298
25.6;6. CLEARING;300
25.7;7. ADDITION AND MULTIPLICATION;304
25.8;8. DIVISION;305
25.9;9. THE REAL NUMBERS;307
25.10;10. DIGITAL REPRESENTATION OF FIELDS;308
25.11;11. THE
p-ADIC FIELDS;311
25.12;12. FURTHER WORK;312
25.13;APPENDIX;314
25.14;REFERENCES;316