Alladi The Legacy of Alladi Ramakrishnan in the Mathematical Sciences

1. Auflage 2010
ISBN: 978-1-4419-6263-8
Verlag: Springer-Verlag
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 575 Seiten

In the spirit of Alladi Ramakrishnan's profound interest and contributions to three fields of science - Mathematics, Statistics, and Physics - this volume contains invited surveys and research articles from prominent members of these communities who also knew Ramakrishnan personally and greatly respected his influence in these areas of science. Historical photos, telegrams, and biographical narratives of Alladi Ramakrishnan's illustrious career of special interest are included as well.

Krishnaswami Alladi is the editor in Chief of Springer's Ramanujan Journal, and the Development in Mathematics Book series. He was chair of the mathematics department of the University of Florida. In addition to his contributions of bibliographic material about his father, Krishnaswami Alladi has directed and organized the development of this tribute volume. C.R. Rao is inarguably the most prominent academic in the field of Statistics in current times. John Klauder has served on the Physics Advisory Panel of the NSF and been Editor of the Journal of Mathematical Physics, and as President of the International Association of Mathematical Physics.

Autoren/Hrsg.

Alladi, Krishnaswami

Inhaltsverzeichnis

1;The Legacy of Alladi Ramakrishnan in the Mathematical Sciences;1
1.1;Preface;7
1.2;Contents;15
1.3;Part I The Legacy of Alladi Ramakrishnan;19
1.3.1;Contributions of Alladi Ramakrishnan to the Mathematical Sciences;20
1.3.2;Alladi Ramakrishnan's Theoretical Physics Seminar;27
1.3.3;Telegrams Received for the MATSCIENCE Inauguration;41
1.3.4;The Miracle has Happened;82
1.3.5;Overseas Trips of Alladi Ramakrishnan;87
1.3.6;List of Publications of Alladi Ramakrishnan;95
1.3.7;List of PhD Students of Alladi Ramakrishnan;102
1.4;Part II Pure Mathematics;103
1.4.1;Inversion and Invariance of Characteristic Terms: Part I;104
1.4.1.1;1 Introduction;104
1.4.1.2;2 Notation;106
1.4.1.3;3 Remarks and Lemmas;109
1.4.1.4;4 Newtonian Expansion;134
1.4.1.5;5 Quadratic Transformations;145
1.4.1.6;6 Dicritical Divisors;154
1.4.1.7;7 Field Generators;167
1.4.1.8;8 Preview of Part II;169
1.4.1.9;9 Epilogue;170
1.4.1.9.1;9.1 Trigonometry;170
1.4.1.9.2;9.2 Taylor Expansion and Valuations;172
1.4.1.9.3;9.3 Discrete Valuation Rings or DVRs;173
1.4.1.9.4;9.4 Newton Expansion and Hamburger-Noether Expansion;176
1.4.1.9.5;9.5 Taylor Series with Remainder;177
1.4.1.9.6;9.6 Polynomials and Power Series;178
1.4.1.10;References;178
1.4.2;Partitions with Non-Repeating Odd Partsand Q-Hypergeometric Identities;180
1.4.2.1;1 Introduction;180
1.4.2.2;2 The Series Expansion;181
1.4.2.3;3 Sylvester's Identity;184
1.4.2.4;4 Lebesgue's Identity;185
1.4.2.5;5 Three Parameter Extension;185
1.4.2.6;6 The Rogers-Fine Identity;187
1.4.2.7;7 Partition Interpretations;190
1.4.2.8;References;192
1.4.3;q-Catalan Identities;194
1.4.3.1;1 Introduction;194
1.4.3.2;2 q-Touchard's Identity;198
1.4.3.3;3 Koshy's Identity;198
1.4.3.4;4 Jonah's Identity;199
1.4.3.5;5 Conclusion;199
1.4.3.6;References;200
1.4.4;Completing Brahmagupta's Extension of Ptolemy's Theorem;202
1.4.4.1;1 Introduction;202
1.4.4.2;2 Brahmagupta's Refinements of Ptolemy's Theorem;203
1.4.4.3;3 Further Results of Brahmagupta;204
1.4.4.4;4 The Third Diagonal of a Cyclic Quadrilateral;207
1.4.4.5;References;208
1.4.5;A Transformation Formula Involving the Gamma and Riemann Zeta Functions in Ramanujan's Lost Notebook;209
1.4.5.1;1 Introduction;209
1.4.5.2;2 Preliminary Results;212
1.4.5.3;3 First Proof of Theorem 1.1;213
1.4.5.4;4 Second Proof of (1.3);216
1.4.5.5;References;220
1.4.6;Ternary Quadratic Forms, Modular Equations, and Certain Positivity Conjectures;221
1.4.6.1;1 Introduction;222
1.4.6.2;2 Ramanujan's Modular Equations of Degree 3 and Associated Identities for Ternary Quadratic Forms with Discriminant 144;225
1.4.6.3;3 Ramanujan's Modular Equations of Degree 5 and Associated Identities for Ternary Quadratic Forms with Discriminant 400;231
1.4.6.4;4 Ternary Forms with Discriminant 784;236
1.4.6.5;5 Ternary Forms with Discriminant 3600;244
1.4.6.6;6 S-Genus;248
1.4.6.7;References;251
1.4.7;How Often is n! a Sum of Three Squares?;252
1.4.7.1;1 Introduction;252
1.4.7.2;2 The Substitution;253
1.4.7.3;3 Proof of Theorem 2;257
1.4.7.4;References;260
1.4.8;Eulerian Polynomials: From Euler's Time to the Present;261
1.4.8.1;1 Introduction;261
1.4.8.2;2 Euler's Definition of the Eulerian Polynomials;263
1.4.8.3;3 A Formulary for the Eulerian Polynomials;270
1.4.8.4;4 A Relation with the Tangent Numbers;273
1.4.8.5;5 The Carlitz q-Eulerian Polynomials;275
1.4.8.6;6 A Detour to Combinatorics;278
1.4.8.7;References;281
1.4.9;Crystal Symmetry Viewed as Zeta Symmetry II;282
1.4.9.1;1 Introduction;282
1.4.9.2;2 Lattice Zeta-Function;283
1.4.9.3;3 Results on Lattice Zeta-Functions;285
1.4.9.4;4 Applications;291
1.4.9.5;References;298
1.4.10;Positive Homogeneous Minima for a System of Linear Forms;300
1.4.10.1;References;304
1.4.11;The Divisor Matrix, Dirichlet Series, and SL(2, Z);305
1.4.11.1;1 Introduction;305
1.4.11.2;2 Basic Definitions and Notation;307
1.4.11.3;3 An Action of SL(2,Z) on Dirichlet Series;309
1.4.11.4;4 A Jordan Form of the Divisor Matrix;309
1.4.11.5;5 Construction of Representations;315
1.4.11.5.1;5.1 Transforming into Jordan Form;317
1.4.11.6;6 Proof of Theorem 3.1;321
1.4.11.7;7 Extending Representations to GL(2,Z);322
1.4.11.8;8 Uniqueness of M;323
1.4.11.9;9 Dirichlet Series in the SL(2,Z)-Orbit of (s);325
1.4.11.10;10 The Cubic Equation Relating (s) and (s);326
1.4.11.10.1;10.1 Some Generalizations;331
1.4.11.11;11 A Functional Equation for (s);332
1.4.11.12;References;333
1.4.12;Proof of a Conjecture of Alladi Ramakrishnan on Circulants;334
1.4.12.1;References;336
1.5;Part III Probability and Statistics;340
1.5.1;Branching Random Walks;341
1.5.1.1;1 Introduction;341
1.5.1.2;2 Branching Random Walks;342
1.5.1.3;3 Results on Branching Processes;343
1.5.1.4;4 Branching Random Walks;346
1.5.1.5;5 Energy Cascades;349
1.5.1.6;6 Extensions and Open Problems;351
1.5.1.6.1;6.1 Non-Gaussian Limits;351
1.5.1.6.2;6.2 Continuous Time;351
1.5.1.6.3;6.3 Critical Case;352
1.5.1.6.4;6.4 Multitype Case;352
1.5.1.7;References;353
1.5.2;A Commentary on the Logistic Distribution;354
1.5.2.1;1 Introduction;354
1.5.2.2;2 The Main Results;356
1.5.2.3;3 Summary;359
1.5.2.4;References;359
1.5.3;Entropy and Cross Entropy: Characterizationsand Applications;361
1.5.3.1;1 Introduction;361
1.5.3.2;2 Entropy Functional;362
1.5.3.2.1;2.1 Maximum Entropy Principle;364
1.5.3.3;3 Cross Entropy;365
1.5.3.3.1;3.1 Characterization;365
1.5.3.3.2;3.2 Decomposition of H();366
1.5.3.3.3;3.3 Some Applications of Cross Entropy;367
1.5.3.4;References;368
1.5.4;Optimal Weights for a Class of Rank Tests for Censored Bivariate Data;370
1.5.4.1;1 Introduction;371
1.5.4.2;2 Efficacies of Statistics in C1 and C2;373
1.5.4.3;3 Estimating Optimal Weights;374
1.5.4.4;4 Simulations;377
1.5.4.5;5 Example;384
1.5.4.6;6 Conclusions;385
1.5.4.7;References;385
1.5.5;Connections Between Bernoulli Strings and Random Permutations;390
1.5.5.1;1 Introduction;391
1.5.5.2;2 Examples;392
1.5.5.3;3 Conditional Marked Poisson Process (CMPP);394
1.5.5.4;4 The Sequence Bern(a,b);395
1.5.5.5;5 The Sequence Bern1(a,b);397
1.5.5.6;6 Dependent Bernoulli Sequences;398
1.5.5.7;7 Some Open Problems;399
1.5.5.8;References;399
1.5.6;Storage Models for a Class of Master Equations with Separable Kernels;401
1.5.6.1;1 Introduction;401
1.5.6.2;2 First Passage Time for Overflow Without Emptiness;403
1.5.6.3;3 First Passage Time for Overflow with Arbitrary Number of Emptiness;408
1.5.6.4;4 Expected Amount of Overflow in a Given Time;409
1.5.6.5;5 Expected Amount of Overflow Allowing Arbitrary Number of Emptiness;411
1.5.6.6;6 Diffusion Approximation;412
1.5.6.7;References;414
1.6;Part IV Theoretical Physics and Applied Mathematics;416
1.6.1;Inverse Consistent Deformable Image Registration;417
1.6.1.1;1 Introduction;417
1.6.1.2;2 Proposed Method;420
1.6.1.2.1;2.1 Motivation and Ideas of Proposed Method;421
1.6.1.2.2;2.2 Alternative Formulation of (4) Using Deformation Fields;422
1.6.1.2.3;2.3 MLE Based Derivation for dis(S,T);423
1.6.1.2.4;2.4 Proposed Model;424
1.6.1.3;3 Existence of Solutions;426
1.6.1.4;4 Numerical Scheme;428
1.6.1.5;5 Experimental Results;430
1.6.1.6;References;438
1.6.2;A Statistical Model for the Quark Structure of the Nucleon;439
1.6.2.1;1 Introduction;439
1.6.2.2;2 Deep Inelastic Scattering of Leptons;442
1.6.2.3;3 The Statistical Model of the Nucleon;446
1.6.2.4;4 The Thermodynamical Bag Model;453
1.6.2.5;5 The Nucleon Spin;455
1.6.2.6;6 Conclusion;460
1.6.2.7;References;461
1.6.3;On Generalized Clifford Algebras and their Physical Applications;462
1.6.3.1;1 Introduction;463
1.6.3.2;2 Projective Representations of Finite Abelian Groups and GCAs;464
1.6.3.3;3 Representations of GCAs;465
1.6.3.4;4 The Clifford Algebra;467
1.6.3.5;5 Alladi Ramakrishnan's L-Matrix Theory and -Operation;471
1.6.3.6;6 Dirac's Positive-Energy Relativistic Wave Equation;473
1.6.3.7;7 GCAs with Ordered -Commutation Relations;474
1.6.3.8;8 Weyl-Schwinger Unitary Basis for Matrix Algebra and Alladi Ramakrishnan's Matrix Decomposition Theorem;477
1.6.3.9;9 Finite-Dimensional Wigner Function;479
1.6.3.10;10 Finite-Dimensional Quantum Canonical Transformations;481
1.6.3.11;11 Magnetic Bloch Functions;482
1.6.3.12;12 Finite-Dimensional Quantum Mechanics;483
1.6.3.13;13 GCAs and Quantum Groups;484
1.6.3.14;14 Conclusion;484
1.6.3.15;References;485
1.6.4;(p, q)-Rogers-Szego Polynomial and the (p, q)-Oscillator;487
1.6.4.1;1 Introduction;487
1.6.4.2;2 Harmonic Oscillator;489
1.6.4.3;3 q-Oscillator and the Rogers-Szegö Polynomial;490
1.6.4.4;4 (p,q)-Oscillator and the (p,q)-Rogers-Szegö Polynomial;492
1.6.4.5;5 Conclusion;495
1.6.4.6;References;496
1.6.5;Rethinking Renormalization;498
1.6.5.1;References;523
1.6.6;Magnetism, FeS Colloids, and Origins of Life;524
1.6.6.1;1 Introduction;525
1.6.6.2;2 Quantum Searches and the Origins of Life;527
1.6.6.2.1;2.1 Quantum Searches and Biology;527
1.6.6.2.2;2.2 Spin and Magnetic Systems for the Origin of Life;528
1.6.6.2.3;2.3 Ferrofluids; Field-Induced Structures;529
1.6.6.2.4;2.4 Structured Magnetic Phases; Life-Like Dynamics;529
1.6.6.3;3 ``The Importance of Being Magnetic'';530
1.6.6.3.1;3.1 Confinement, Connectivity, Frustration-Complexity;530
1.6.6.3.2;3.2 Nested Hierarchy, Cooperative Dynamics;531
1.6.6.3.3;3.3 Polar Cell-Organization and Structures;531
1.6.6.3.4;3.4 Reversible Gel-Sol Transitions;532
1.6.6.3.5;3.5 Reversible Interactions; Weak Bonds;532
1.6.6.3.6;3.6 Kinetic Barriers; Records of Constraints via Hysteresis;533
1.6.6.3.7;3.7 Self-Reproduction; Pre-Bio-Molecular Motors;534
1.6.6.3.8;3.8 Pre-RNA World; Transfer Reactions; Optical Activity;536
1.6.6.3.9;3.9 The Potential for a Quantum-Leap to Life;537
1.6.6.4;4 Framboids and the Mineral Greigite;539
1.6.6.4.1;4.1 Framboids; Importance of Physical Properties;539
1.6.6.4.2;4.2 Framboidal Greigite;540
1.6.6.4.3;4.3 Magnetic Interactions;540
1.6.6.4.4;4.4 Dynamic Ordering; Phyllotaxis; Quasiperiodicity;541
1.6.6.4.5;4.5 Magnetic Assemblies in the Laboratory; Long-Range Order?;542
1.6.6.5;5 Mound Scenario of Russell et al. and Greigite;543
1.6.6.5.1;5.1 Mound Scenario of Russell et al.;543
1.6.6.5.2;5.2 Greigite Formation from FeS;545
1.6.6.5.3;5.3 The FeS Gel Environment and Framboids;546
1.6.6.5.4;5.4 Field Estimate from W-B Model; Motor-Like Dynamics;547
1.6.6.5.5;5.5 Enzyme Clusters and Natural Violarite Phases;549
1.6.6.5.6;5.6 Coherence: Ferromagnetic–Ferroelectric Effects;549
1.6.6.5.7;5.7 Preliminary Experimental Requirements;550
1.6.6.6;6 Conclusions;551
1.6.6.7;References;552
1.6.7;The Ehrenfest Theorem in Quantum Field Theory;560
1.6.7.1;1 Quantum Mechanics;560
1.6.7.2;2 Abelian Field Theory;563
1.6.7.3;3 Non-Abelian Field Theory;565
1.6.7.4;4 Summary;570
1.6.7.5;References;570