Legacy of Alladi Ramakrishnan in the Mathematical Sciences (eBook)

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.

The Legacy of Alladi Ramakrishnan in the Mathematical Sciences 1
Preface 7
Contents 15
Part I The Legacy of Alladi Ramakrishnan 19
Contributions of Alladi Ramakrishnan to the Mathematical Sciences 20
Alladi Ramakrishnan's Theoretical Physics Seminar 27
Telegrams Received for the MATSCIENCE Inauguration 41
The Miracle has Happened 82
Overseas Trips of Alladi Ramakrishnan 87
List of Publications of Alladi Ramakrishnan 95
List of PhD Students of Alladi Ramakrishnan 102
Part II Pure Mathematics 103
Inversion and Invariance of Characteristic Terms: Part I 104
1 Introduction 104
2 Notation 106
3 Remarks and Lemmas 109
4 Newtonian Expansion 134
5 Quadratic Transformations 145
6 Dicritical Divisors 154
7 Field Generators 167
8 Preview of Part II 169
9 Epilogue 170
9.1 Trigonometry 170
9.2 Taylor Expansion and Valuations 172
9.3 Discrete Valuation Rings or DVRs 173
9.4 Newton Expansion and Hamburger-Noether Expansion 176
9.5 Taylor Series with Remainder 177
9.6 Polynomials and Power Series 178
References 178
Partitions with Non-Repeating Odd Partsand Q-Hypergeometric Identities 180
1 Introduction 180
2 The Series Expansion 181
3 Sylvester's Identity 184
4 Lebesgue's Identity 185
5 Three Parameter Extension 185
6 The Rogers-Fine Identity 187
7 Partition Interpretations 190
References 192
q-Catalan Identities 194
1 Introduction 194
2 q-Touchard's Identity 198
3 Koshy's Identity 198
4 Jonah's Identity 199
5 Conclusion 199
References 200
Completing Brahmagupta's Extension of Ptolemy's Theorem 202
1 Introduction 202
2 Brahmagupta's Refinements of Ptolemy's Theorem 203
3 Further Results of Brahmagupta 204
4 The Third Diagonal of a Cyclic Quadrilateral 207
References 208
A Transformation Formula Involving the Gamma and Riemann Zeta Functions in Ramanujan's Lost Notebook 209
1 Introduction 209
2 Preliminary Results 212
3 First Proof of Theorem 1.1 213
4 Second Proof of (1.3) 216
References 220
Ternary Quadratic Forms, Modular Equations, and Certain Positivity Conjectures 221
1 Introduction 222
2 Ramanujan's Modular Equations of Degree 3 and Associated Identities for Ternary Quadratic Forms with Discriminant 144 225
3 Ramanujan's Modular Equations of Degree 5 and Associated Identities for Ternary Quadratic Forms with Discriminant 400 231
4 Ternary Forms with Discriminant 784 236
5 Ternary Forms with Discriminant 3600 244
6 S-Genus 248
References 251
How Often is n! a Sum of Three Squares? 252
1 Introduction 252
2 The Substitution 253
3 Proof of Theorem 2 257
References 260
Eulerian Polynomials: From Euler's Time to the Present 261
1 Introduction 261
2 Euler's Definition of the Eulerian Polynomials 263
3 A Formulary for the Eulerian Polynomials 270
4 A Relation with the Tangent Numbers 273
5 The Carlitz q-Eulerian Polynomials 275
6 A Detour to Combinatorics 278
References 281
Crystal Symmetry Viewed as Zeta Symmetry II 282
1 Introduction 282
2 Lattice Zeta-Function 283
3 Results on Lattice Zeta-Functions 285
4 Applications 291
References 298
Positive Homogeneous Minima for a System of Linear Forms 300
References 304
The Divisor Matrix, Dirichlet Series, and SL(2, Z) 305
1 Introduction 305
2 Basic Definitions and Notation 307
3 An Action of SL(2,Z) on Dirichlet Series 309
4 A Jordan Form of the Divisor Matrix 309
5 Construction of Representations 315
5.1 Transforming into Jordan Form 317
6 Proof of Theorem 3.1 321
7 Extending Representations to GL(2,Z) 322
8 Uniqueness of M 323
9 Dirichlet Series in the SL(2,Z)-Orbit of (s) 325
10 The Cubic Equation Relating (s) and (s) 326
10.1 Some Generalizations 331
11 A Functional Equation for (s) 332
References 333
Proof of a Conjecture of Alladi Ramakrishnan on Circulants 334
References 336
Part III Probability and Statistics 340
Branching Random Walks 341
1 Introduction 341
2 Branching Random Walks 342
3 Results on Branching Processes 343
4 Branching Random Walks 346
5 Energy Cascades 349
6 Extensions and Open Problems 351
6.1 Non-Gaussian Limits 351
6.2 Continuous Time 351
6.3 Critical Case 352
6.4 Multitype Case 352
References 353
A Commentary on the Logistic Distribution 354
1 Introduction 354
2 The Main Results 356
3 Summary 359
References 359
Entropy and Cross Entropy: Characterizationsand Applications 361
1 Introduction 361
2 Entropy Functional 362
2.1 Maximum Entropy Principle 364
3 Cross Entropy 365
3.1 Characterization 365
3.2 Decomposition of H() 366
3.3 Some Applications of Cross Entropy 367
References 368
Optimal Weights for a Class of Rank Tests for Censored Bivariate Data 370
1 Introduction 371
2 Efficacies of Statistics in C1 and C2 373
3 Estimating Optimal Weights 374
4 Simulations 377
5 Example 384
6 Conclusions 385
References 385
Connections Between Bernoulli Strings and Random Permutations 390
1 Introduction 391
2 Examples 392
3 Conditional Marked Poisson Process (CMPP) 394
4 The Sequence Bern(a,b) 395
5 The Sequence Bern1(a,b) 397
6 Dependent Bernoulli Sequences 398
7 Some Open Problems 399
References 399
Storage Models for a Class of Master Equations with Separable Kernels 401
1 Introduction 401
2 First Passage Time for Overflow Without Emptiness 403
3 First Passage Time for Overflow with Arbitrary Number of Emptiness 408
4 Expected Amount of Overflow in a Given Time 409
5 Expected Amount of Overflow Allowing Arbitrary Number of Emptiness 411
6 Diffusion Approximation 412
References 414
Part IV Theoretical Physics and Applied Mathematics 416
Inverse Consistent Deformable Image Registration 417
1 Introduction 417
2 Proposed Method 420
2.1 Motivation and Ideas of Proposed Method 421
2.2 Alternative Formulation of (4) Using Deformation Fields 422
2.3 MLE Based Derivation for dis(S,T) 423
2.4 Proposed Model 424
3 Existence of Solutions 426
4 Numerical Scheme 428
5 Experimental Results 430
References 438
A Statistical Model for the Quark Structure of the Nucleon 439
1 Introduction 439
2 Deep Inelastic Scattering of Leptons 442
3 The Statistical Model of the Nucleon 446
4 The Thermodynamical Bag Model 453
5 The Nucleon Spin 455
6 Conclusion 460
References 461
On Generalized Clifford Algebras and their Physical Applications 462
1 Introduction 463
2 Projective Representations of Finite Abelian Groups and GCAs 464
3 Representations of GCAs 465
4 The Clifford Algebra 467
5 Alladi Ramakrishnan's L-Matrix Theory and -Operation 471
6 Dirac's Positive-Energy Relativistic Wave Equation 473
7 GCAs with Ordered -Commutation Relations 474
8 Weyl-Schwinger Unitary Basis for Matrix Algebra and Alladi Ramakrishnan's Matrix Decomposition Theorem 477
9 Finite-Dimensional Wigner Function 479
10 Finite-Dimensional Quantum Canonical Transformations 481
11 Magnetic Bloch Functions 482
12 Finite-Dimensional Quantum Mechanics 483
13 GCAs and Quantum Groups 484
14 Conclusion 484
References 485
(p, q)-Rogers-Szego Polynomial and the (p, q)-Oscillator 487
1 Introduction 487
2 Harmonic Oscillator 489
3 q-Oscillator and the Rogers-Szegö Polynomial 490
4 (p,q)-Oscillator and the (p,q)-Rogers-Szegö Polynomial 492
5 Conclusion 495
References 496
Rethinking Renormalization 498
References 523
Magnetism, FeS Colloids, and Origins of Life 524
1 Introduction 525
2 Quantum Searches and the Origins of Life 527
2.1 Quantum Searches and Biology 527
2.2 Spin and Magnetic Systems for the Origin of Life 528
2.3 Ferrofluids Field-Induced Structures
2.4 Structured Magnetic Phases Life-Like Dynamics
3 ``The Importance of Being Magnetic'' 530
3.1 Confinement, Connectivity, Frustration-Complexity 530
3.2 Nested Hierarchy, Cooperative Dynamics 531
3.3 Polar Cell-Organization and Structures 531
3.4 Reversible Gel-Sol Transitions 532
3.5 Reversible Interactions Weak Bonds
3.6 Kinetic Barriers Records of Constraints via Hysteresis
3.7 Self-Reproduction Pre-Bio-Molecular Motors
3.8 Pre-RNA World Transfer Reactions
3.9 The Potential for a Quantum-Leap to Life 537
4 Framboids and the Mineral Greigite 539
4.1 Framboids Importance of Physical Properties
4.2 Framboidal Greigite 540
4.3 Magnetic Interactions 540
4.4 Dynamic Ordering Phyllotaxis
4.5 Magnetic Assemblies in the Laboratory Long-Range Order?
5 Mound Scenario of Russell et al. and Greigite 543
5.1 Mound Scenario of Russell et al. 543
5.2 Greigite Formation from FeS 545
5.3 The FeS Gel Environment and Framboids 546
5.4 Field Estimate from W-B Model Motor-Like Dynamics
5.5 Enzyme Clusters and Natural Violarite Phases 549
5.6 Coherence: Ferromagnetic–Ferroelectric Effects 549
5.7 Preliminary Experimental Requirements 550
6 Conclusions 551
References 552
The Ehrenfest Theorem in Quantum Field Theory 560
1 Quantum Mechanics 560
2 Abelian Field Theory 563
3 Non-Abelian Field Theory 565
4 Summary 570
References 570