The Legacy of Alladi Ramakrishnan in the Mathematical Sciences

The Legacy of Alladi Ramakrishnan in the Mathematical Sciences

Preface

Contents

Part I The Legacy of Alladi Ramakrishnan

Contributions of Alladi Ramakrishnan to the Mathematical Sciences

Alladi Ramakrishnan's Theoretical Physics Seminar

Telegrams Received for the MATSCIENCE Inauguration

The Miracle has Happened

Overseas Trips of Alladi Ramakrishnan

List of Publications of Alladi Ramakrishnan

List of PhD Students of Alladi Ramakrishnan

Part II Pure Mathematics

Inversion and Invariance of Characteristic Terms: Part I

1 Introduction

2 Notation

3 Remarks and Lemmas

4 Newtonian Expansion

5 Quadratic Transformations

6 Dicritical Divisors

7 Field Generators

8 Preview of Part II

9 Epilogue

9.1 Trigonometry

9.2 Taylor Expansion and Valuations

9.3 Discrete Valuation Rings or DVRs

9.4 Newton Expansion and Hamburger-Noether Expansion

9.5 Taylor Series with Remainder

9.6 Polynomials and Power Series

References

Partitions with Non-Repeating Odd Partsand Q-Hypergeometric Identities

1 Introduction

2 The Series Expansion

3 Sylvester's Identity

4 Lebesgue's Identity

5 Three Parameter Extension

6 The Rogers-Fine Identity

7 Partition Interpretations

References

q-Catalan Identities

1 Introduction

2 q-Touchard's Identity

3 Koshy's Identity

4 Jonah's Identity

5 Conclusion

References

Completing Brahmagupta's Extension of Ptolemy's Theorem

1 Introduction

2 Brahmagupta's Refinements of Ptolemy's Theorem

3 Further Results of Brahmagupta

4 The Third Diagonal of a Cyclic Quadrilateral

References

A Transformation Formula Involving the Gamma and Riemann Zeta Functions in Ramanujan's Lost Notebook

1 Introduction

2 Preliminary Results

3 First Proof of Theorem 1.1

4 Second Proof of (1.3)

References

Ternary Quadratic Forms, Modular Equations, and Certain Positivity Conjectures

1 Introduction

2 Ramanujan's Modular Equations of Degree 3 and Associated Identities for Ternary Quadratic Forms with Discriminant 144

3 Ramanujan's Modular Equations of Degree 5 and Associated Identities for Ternary Quadratic Forms with Discriminant 400

4 Ternary Forms with Discriminant 784

5 Ternary Forms with Discriminant 3600

6 S-Genus

References

How Often is n! a Sum of Three Squares?

1 Introduction

2 The Substitution

3 Proof of Theorem 2

References

Eulerian Polynomials: From Euler's Time to the Present

1 Introduction

2 Euler's Definition of the Eulerian Polynomials

3 A Formulary for the Eulerian Polynomials

4 A Relation with the Tangent Numbers

5 The Carlitz q-Eulerian Polynomials

6 A Detour to Combinatorics

References

Crystal Symmetry Viewed as Zeta Symmetry II

1 Introduction

2 Lattice Zeta-Function

3 Results on Lattice Zeta-Functions

4 Applications

References

Positive Homogeneous Minima for a System of Linear Forms

References

The Divisor Matrix, Dirichlet Series, and SL(2, Z)

1 Introduction

2 Basic Definitions and Notation

3 An Action of SL(2,Z) on Dirichlet Series

4 A Jordan Form of the Divisor Matrix

5 Construction of Representations

5.1 Transforming into Jordan Form

6 Proof of Theorem 3.1

7 Extending Representations to GL(2,Z)

8 Uniqueness of M

9 Dirichlet Series in the SL(2,Z)-Orbit of (s)

10 The Cubic Equation Relating (s) and (s)

10.1 Some Generalizations

11 A Functional Equation for (s)

References

Proof of a Conjecture of Alladi Ramakrishnan on Circulants

References

Part III Probability and Statistics

Branching Random Walks

1 Introduction

2 Branching Random Walks

3 Results on Branching Processes

4 Branching Random Walks

5 Energy Cascades

6 Extensions and Open Problems

6.1 Non-Gaussian Limits

6.2 Continuous Time

6.3 Critical Case

6.4 Multitype Case

References

A Commentary on the Logistic Distribution

1 Introduction

2 The Main Results

3 Summary

References

Entropy and Cross Entropy: Characterizationsand Applications

1 Introduction

2 Entropy Functional

2.1 Maximum Entropy Principle

3 Cross Entropy

3.1 Characterization

3.2 Decomposition of H()

3.3 Some Applications of Cross Entropy

References

Optimal Weights for a Class of Rank Tests for Censored Bivariate Data

1 Introduction

2 Efficacies of Statistics in C1 and C2

3 Estimating Optimal Weights

4 Simulations

5 Example

6 Conclusions

References

Connections Between Bernoulli Strings and Random Permutations

1 Introduction

2 Examples

3 Conditional Marked Poisson Process (CMPP)

4 The Sequence Bern(a,b)

5 The Sequence Bern1(a,b)

6 Dependent Bernoulli Sequences

7 Some Open Problems

References

Storage Models for a Class of Master Equations with Separable Kernels

1 Introduction

2 First Passage Time for Overflow Without Emptiness

3 First Passage Time for Overflow with Arbitrary Number of Emptiness

4 Expected Amount of Overflow in a Given Time

5 Expected Amount of Overflow Allowing Arbitrary Number of Emptiness

6 Diffusion Approximation

References

Part IV Theoretical Physics and Applied Mathematics

Inverse Consistent Deformable Image Registration

1 Introduction

2 Proposed Method

2.1 Motivation and Ideas of Proposed Method

2.2 Alternative Formulation of (4) Using Deformation Fields

2.3 MLE Based Derivation for dis(S,T)

2.4 Proposed Model

3 Existence of Solutions

4 Numerical Scheme

5 Experimental Results

References

A Statistical Model for the Quark Structure of the Nucleon

1 Introduction

2 Deep Inelastic Scattering of Leptons

3 The Statistical Model of the Nucleon

4 The Thermodynamical Bag Model

5 The Nucleon Spin

6 Conclusion

References

On Generalized Clifford Algebras and their Physical Applications

1 Introduction

2 Projective Representations of Finite Abelian Groups and GCAs

3 Representations of GCAs

4 The Clifford Algebra

5 Alladi Ramakrishnan's L-Matrix Theory and -Operation

6 Dirac's Positive-Energy Relativistic Wave Equation

7 GCAs with Ordered -Commutation Relations

8 Weyl-Schwinger Unitary Basis for Matrix Algebra and Alladi Ramakrishnan's Matrix Decomposition Theorem

9 Finite-Dimensional Wigner Function

10 Finite-Dimensional Quantum Canonical Transformations

11 Magnetic Bloch Functions

12 Finite-Dimensional Quantum Mechanics

13 GCAs and Quantum Groups

14 Conclusion

References

(p, q)-Rogers-Szego Polynomial and the (p, q)-Oscillator

1 Introduction

2 Harmonic Oscillator

3 q-Oscillator and the Rogers-Szegö Polynomial

4 (p,q)-Oscillator and the (p,q)-Rogers-Szegö Polynomial

5 Conclusion

References

Rethinking Renormalization

References

Magnetism, FeS Colloids, and Origins of Life

1 Introduction

2 Quantum Searches and the Origins of Life

2.1 Quantum Searches and Biology

2.2 Spin and Magnetic Systems for the Origin of Life

2.3 Ferrofluids; Field-Induced Structures

2.4 Structured Magnetic Phases; Life-Like Dynamics

3 ``The Importance of Being Magnetic''

3.1 Confinement, Connectivity, Frustration-Complexity

3.2 Nested Hierarchy, Cooperative Dynamics

3.3 Polar Cell-Organization and Structures

3.4 Reversible Gel-Sol Transitions

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; Optical Activity

3.9 The Potential for a Quantum-Leap to Life

4 Framboids and the Mineral Greigite

4.1 Framboids; Importance of Physical Properties

4.2 Framboidal Greigite

4.3 Magnetic Interactions

4.4 Dynamic Ordering; Phyllotaxis; Quasiperiodicity

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

5 Mound Scenario of Russell et al. and Greigite

5.1 Mound Scenario of Russell et al.

5.2 Greigite Formation from FeS

5.3 The FeS Gel Environment and Framboids

5.4 Field Estimate from W-B Model; Motor-Like Dynamics

5.5 Enzyme Clusters and Natural Violarite Phases

5.6 Coherence: Ferromagnetic–Ferroelectric Effects

5.7 Preliminary Experimental Requirements

6 Conclusions

References

The Ehrenfest Theorem in Quantum Field Theory

1 Quantum Mechanics

2 Abelian Field Theory

3 Non-Abelian Field Theory

4 Summary

References