The Legacy of Alladi Ramakrishnan in the Mathematical Sciences

The Legacy of Alladi Ramakrishnan in the Mathematical Sciences

von: Krishnaswami Alladi, John R. Klauder, Calyampudi R. Rao

Springer-Verlag, 2010

ISBN: 9781441962638 , 575 Seiten

Format: PDF

Kopierschutz: Wasserzeichen

Windows PC,Mac OSX Apple iPad, Android Tablet PC's

Preis: 171,19 EUR

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The Legacy of Alladi Ramakrishnan in the Mathematical Sciences


 

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

529

2.4 Structured Magnetic Phases; Life-Like Dynamics

529

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

532

3.6 Kinetic Barriers; Records of Constraints via Hysteresis

533

3.7 Self-Reproduction; Pre-Bio-Molecular Motors

534

3.8 Pre-RNA World; Transfer Reactions; Optical Activity

536

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

539

4.2 Framboidal Greigite

540

4.3 Magnetic Interactions

540

4.4 Dynamic Ordering; Phyllotaxis; Quasiperiodicity

541

4.5 Magnetic Assemblies in the Laboratory; Long-Range Order?

542

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

547

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