Theoretical Physics, Wavelets, Analysis, Genomics: An Indisciplinary Tribute to Alex Grossmann

ANHA Series Preface
En guise de préface
Contents
Contributors
The Making of a Physicist
Alex Grossmann, a Rinascimento Multidisciplinary Man
1 A Jubilee of Multifold Research
2 Quantum Physics and Related Fields
3 Wavelets
4 Genomics
5 How to Conclude Such a Short But Mind-Boggling Overview?
6 Postlude: A Geometric Existentialist Way of Thinking
Generalized Affine Signal Analysis with Time-Delay Thresholds
Introductory Note on the Draft Paper ``Generalized Affine Signal Analysis with Time-Delay Thresholds'', by Jan W. Dash, Alex Grossmann and Thierry Paul
1 Wavelets in the Mid-1980s
1.1 The Group-Theoretic View of Wavelets
1.2 Overcompleteness Can Be a Virtue
1.3 Comments on the First Sentence: ``A Class of Functions Recently Introduced by Dash and Paul …''
1.4 Phase
2 Specific Comments on the Text
3 Le Baron de Prony's Overcomplete Set of ``Wavelets à la Neanderthal''
4 Alex
5 Addendum/Corrigendum
Alex Grossmann's PhD Thesis (Harvard 1959): Covariant Functions of Quantum Fields - Table of Contents and Introduction
Part I Quantum Mechanics and Theoretical Physics
Alex Grossmann, from Nested Hilbert Spaces to Partial Inner Product Spaces and Wavelets
1 Introduction: Some History
2 Rigged Hilbert Spaces
3 Nested Hilbert Spaces
4 Towards Partial Inner Product Spaces
5 Operators on PIP-Spaces
5.1 General Definitions
5.2 Homomorphisms and Orthogonal Projections
5.3 Symmetric Operators and Self-Adjointness
6 Applications in Quantum Mechanics
6.1 General Formulation
6.2 Resonances, Analyticity Properties
6.3 Condensed Matter Physics
7 The Legacy: Operator Partial Algebras
8 Towards 2D Wavelets
9 Epilogue
References
Combining Quantum Mechanical Languages (A Tribute to Alex Grossmann)
1 Introduction
2 The kq-Representation
3 Bloch-Like Functions
4 Phase of the Bloch Function
5 kq-Space
6 Conclusion
References
Alex Grossmann, Scattering Amplitude, Fermi Pseudopotential, and Particle Physics
1 Harvard and Scattering Amplitude
2 Marseille and Fermi Pseudopotential
3 Particle Physics
References
Sixty Years of Hadronic Vacuum Polarization
1 Introduction
2 What Is Hadronic Vacuum Polarization?
3 HVP and the Muon μ-Particle
4 Mellin-Barnes Representation of aμHVP
5 The Time Momentum Representation
6 Limitations of the LQCD Evaluations
7 Mellin-Barnes Approximants
7.1 Euler Beta-Function Approximants
8 Time Momentum Representation Approximants
8.1 Heaviside Spectral Function Approximants
9 Conclusion
References
Standard Model, and Its Standard Problems
1 Prologue
2 A Short Introduction to the Standard Model A.D. 2021
2.1 The Neutrino Mass Sector Sν
2.2 A Short Thermal History of the Universe, Seen Through the Standard Model
3 The Anomaly Cancellation and Fixing of Weak Hypercharge
3.1 Cancelling the Anomalies
4 The Gauging of the Poincaré Group
4.1 Gravity in the Standard Model and the Cosmological Constant Problem
5 Global Anomalies and Neutrino Sector
5.1 What Gauge Field Configurations Contribute to the Global Anomalies?
5.2 B-L Anomaly Through Gravitation
6 Conclusion: When All Else Fails, Just Tell the Truth regan
References
SU(3) Higher Roots and Their Lattices
1 Introduction
2 Lattices of Higher Roots
2.1 Analogies and Warnings
2.2 On Extended Fusion Matrices and Their Periods
2.3 The Quiver of Higher Roots (``the Ocneanu Ribbon'')
2.3.1 The Ribbon R
2.4 An Euclidean Structure on the Space of Higher Roots
2.5 Rank of the System
2.6 A Graphical Summary
2.7 Choice of a Basis
2.8 Higher Roots and Their Inner Products: Summary of the Procedure
3 Theta Functions for Lattices of Higher Roots
3.1 Lattices and Theta Functions (Reminders)
3.2 Lattice Properties: Tables
3.3 Properties of the Lattices
3.4 Remarks
References
Quantum Field Theory with Dynamical Boundary Conditions and the Casimir Effect
1 Introduction
2 The Classical Problem
3 Quantum Field Theory
3.1 The Hadamard Property, the Renormalized Local State Polarization and the Local Casimir Energy
4 The Casimir Effect
4.1 Dirichlet Boundary Condition at z = 0
4.1.1 The Renormalized Local State Polarization
4.1.2 The Local Casimir Energy
4.2 Robin Boundary Condition at z = 0
4.2.1 The Renormalized Local State Polarization
4.2.2 The Local Casimir Energy
5 The Casimir Effect at Positive Temperature
6 Numerical Examples
7 Final Remarks
8 Appendix 1: Useful Formulae
9 Appendix 2: Asymptotic Estimates for the Eigenvalues
9.1 Dirichlet Boundary Condition at z = 0
9.2 Robin Boundary Condition at z = 0
10 Appendix 3: Calculations with a Robin Boundary Condition at z = 0
10.1 Renormalized Local State Polarization in the Bulk
10.2 Renormalized Local State Polarization in the Boundary
10.3 Local Casimir Energy in the Bulk
10.4 Local Casimir Energy in the Boundary
References
Where Is a Photon in an Interferometer?
1 The Experiment
2 The History of Post-selected States
3 A Quantum Circuit Model
4 Three Interfering Paths
4.1 The Amplitudes of the Outgoing State
4.2 The Small Parameter: =α-β
5 The Quad-Detector
5.1 Calibrating the Detector
6 Concluding Remarks
References
About the Derivation of the Quasilinear Approximation in Plasma Physics
1 Introduction and Notation
1.1 Notation and Some Hypotheses
1.2 The Rescaled Liouville Equation
1.3 Obstruction to the Convergence to a Non-degenerate Diffusion Matrix
1.4 The First Iteration of the Duhamel Formula and the Diffusion: Reynolds Electric Stress Tensor
2 The Non-self-consistent Stochastic Approach
2.1 Properties of the Reynolds Electric Stress Tensor
2.2 Decorrelation
2.3 Weak Limits
2.4 The Basic Stochastic Theorem
3 Returning to the Vlasov–Poisson Equations
3.1 Classical Stability Results for Nonlinear and Linearized Vlasov–Poisson Equations
3.2 Spectral Properties of Linearized Vlasov–Poisson Equations
3.3 Remark About the Landau Damping, Comparison with the Behavior of the Vector Field in the Rescaled Equation
3.4 Roadmap for the Short-Time Quasilinear Approximation
4 Remarks and Conclusion
References
Analysing the Scattering of Electromagnetic Ultra-wideband Pulses from Large-Scale Objects by the Use of Wavelets
1 Introduction
2 Direct and Scattered Fields
3 Time-Reversal Operator
4 Evaluating the Sparsity of the Time-Reversal Operator
5 Discussion and Conclusion
Appendix 1: Antennas Diagrams
Appendix 2: Conductivity and Polarisability
Appendix 3: Daubechies Wavelets and Wjj Function
Appendix 4: Singularities for the Parallelipedic Case
References
Species of Spaces
Contents
1 Introduction: A Space Journey Through the Wonderful Landscape of Quantum and Classical Mechanics
1.1 From Planck to Heisenberg
1.2 From Symbols to Operators: A Quick Journey in Quantizland
1.3 From Operators to Symbols: Tell Me Which Operator You Are, I will Tell You What Is Your Symbol
1.4 From Symbols to Classical: Spaces as Possible Symbolic Calculi
2 Quantizing Complex Canonical Transformations
2.1 Off-diagonal Toeplitz Representation
2.2 Link with Weyl
2.3 Flows on Extended phase space
2.4 Examples
2.5 Real Phase Space Associated with Complex Linear Symplectomorphisms
2.6 Noncommutative Geometry Interpretation
2.6.1 The Canonical Groupoid
2.6.2 Symbols
2.6.3 On the (Formal) Composition of Symbols
2.7 Non-Canonical Transforms
3 Bose-Einstein-Fermi at the Classical Level
3.1 Husimi
3.2 Wigner
3.3 Toeplitz
3.4 On Wigner Again
3.5 Off-diagonal Toeplitz Representations
3.6 Link with the Complex (Anti)metaplectic Representation
3.7 A Classical Phase Space with Symmetries Inherited Form Quantum Statistics
4 Noncommutative Moduli Spaces Underlying Topological Quantum Field Theory in Large Colouring Asymptotics
4.1 The Standard Geometric Quantization of the Sphere
4.2 a-Toeplitz Quantization
4.3 Symbolic Calculus
4.4 Main Result
4.5 Classical Limit and Underlying ``phase space''
5 Long Time Semiclassical Evolution
5.1 Propagation
5.2 Noncommutative Geometry Interpretation
5.3 Another Groupoid Approach
5.4 Extended Semiclassical Measures
5.5 Noncommutative Phase Space Associated to Time Arrow
6 The Noncommutative phase space Underlying the Quotient by the Quantum Flow
6.1 The Non-resonant Harmonic Oscillator Spectrum and the Non commutative Torus
6.2 The Space of Frequencies and the Noncommutative Torus
6.3 Extensions to Chaotic Systems
6.3.1 Homoclinic Foliations Versus Invariant Tori Fibration
6.3.2 The Construction of the Noncommutative Structure
6.3.3 Bohr-Sommerfeld Conditions I
6.3.4 The ``Poincaré'' Section
6.3.5 Bohr-Sommerfeld Rules II
6.3.6 Creation, Annihilation, and All That
6.3.7 A New and Noncommutative Framework for Classical Dynamics
6.4 Miscellaneous
6.5 Conclusion: The Quotient of the Phase Space by the Flow
7 Indeterminism Versus Unpredictability (How Quantum Indeterminism Would Have Chocked Laplace But Not Poincaré)
7.1 Measurement in Quantum Mechanics
7.2 Critics of the Deterministic Reason of Classical Mechanics
7.3 Pushing Sensitivity to Initial Conditions to Its Extreme
7.4 Some Space for Merging the Two
7.5 A Classical Phase Space Incorporating the Point t=∞
References
Part II Wavelets and Mathematical Analysis
Curved Model Sets and Crystalline Measures
1 Introduction
2 Crystalline Measures Are Almost Periodic
3 Traces of Radon Measures
4 Sparse Crystalline Measures
5 Curved Model Sets
6 Ahern Measures and Inner Functions
7 The Set Constructed by Kurasov and Sarnak Is a Curved Model Set
Appendix
References
Diffusion Maps: Using the Semigroup Property for Parameter Tuning
1 Introduction
2 Diffusion Operator and Diffusion Maps: A Brief Recap
2.1 Laplace–Beltrami Operator
2.2 Diffusion Operator
2.3 Approximating the Diffusion Operator on a Discrete Dataset
3 Setting the Diffusion Time t
3.1 Sensitivity to the Choice of t
3.2 Semigroup Test
4 Conclusion
Some Personal Comments
References
Wavelet Phase Harmonics
1 Introduction
2 Maximum Entropy Models
2.1 Macrocanonical Models
2.2 Microcanonical Models
3 High-Order Moments and Phase Harmonics
3.1 Fourier Phase and High-Order Moments
3.2 Phase Harmonics
3.2.1 Phase Windowed Fourier Transform
3.2.2 Fourier Phase Harmonics Covariances
4 Wavelet Transforms
4.1 Analytic Wavelets for 1D Signals
4.2 Rotated Wavelet Frames
5 Maximum Entropy Models with Wavelet Phases
5.1 Foveal Wavelet Covariance Models
5.2 Wavelet Phase Harmonic Models
5.2.1 Wavelet Phase Harmonics
5.2.2 Rectified Neural Network Coefficients
5.2.3 Maximum Entropy Wavelet Phase Harmonic Foveal Model
6 Conclusion
References
Multiscale Decompositions of Hardy Spaces
1 Introduction
2 Preliminaries and Notation
3 Malmquist-Takenaka Bases on the Torus
4 The Upper Half Plane
4.1 Malmquist-Takenaka Bases
4.2 A Multiscale Wavelet Decomposition
4.3 An Orthonormal System
4.3.1 Multiscale Decomposition
5 Adapted Malmquist-Takenaka Bases: ``Phase Unwinding''
5.1 The Unwinding Series
5.2 The Fast Algorithm of Guido and Mary Weiss ww
6 Iteration of Blaschke Products
6.1 Multiscale Decomposition
7 Remarks on Iteration of Blaschke Products as a ``Deep Neural Net''
Appendix: An Example
References
A Generalization of Gleason's Frame Function for Quantum Measurement
1 Introduction
1.1 Background
1.2 The Role of Gleason's Theorem and Our Goal
1.3 Outline
2 Gleason's Theorem
2.1 Preliminaries
2.2 Gleason's Theorem
2.3 The Case d=2
3 Parseval Frames and POVMs
3.1 Properties of Frames
3.2 POVMs
4 Gleason Functions for Parseval Frames
4.1 Quadratic Forms Are Gleason Functions for Parseval Frames
4.2 Basic Properties
4.3 Gleason Functions and Quadratic Forms
5 Gleason Functions of Degree N
5.1 Inclusion Theorem and a Problem
5.2 Proof of Theorem 14
6 An Application of Gleason Functions
Appendix
References
Post-Fourier Frequencies: Variations and Paradoxes
1 From Sine to Sinc
1.1 Instantaneous: Gabor–Ville
1.2 Local: Teager–Kaiser
1.3 Local and Instantaneous 1: Hilbert–Huang
1.4 Local and Instantaneous 2: Time–Frequency
1.5 Which Lesson? Local vs. Instantaneous
2 Frequencies and Spectral Bandwidth
2.1 Two Spectral Lines
2.2 ``Supershift'' and ``Superoscillation''
3 To Conclude
References
The Unreasonable Effectiveness of Haar Frames
1 Introduction
2 The Haar Basis
3 Relevance of Frames and Haar Decompositions in Machine Learning
3.1 Toward a Mathematical Formalism of DL
3.2 Graph-Based Deep Learning
4 Haar Frame
4.1 Uniform Regularity
4.2 Pointwise Regularity: The Haar Basis
4.3 The Haar–Weierstrass Function
4.4 Pointwise Regularity: The Haar Frame
5 Concluding Remarks and Open Problems
References
Part III Genomics and Biology
Quantifying the Rationality of Rhythmic Signals
1 Introduction
2 Spectrum of Frequency Ratios: Formalism and Time–Frequency Generalisation
2.1 Correlation Functions for Signal Comparison
2.2 Spectrum of Frequency Ratios: A Frequency Ratio Distribution
2.3 Wavelet Transform Formalism
2.3.1 Time and Frequency Window for the Analysing Wavelet
2.3.2 Choice of the Analysing Wavelet: The Grossmann Wavelet
2.4 Extension of Frequency Ratio Distributions to Time–Frequency Ratio Distributions
2.4.1 Computation of the Log-Frequency Correlation Function
3 Computation of Log-Frequency Distributions from Numerical and Real Signals
3.1 Model Signals Constructed from Sine Functions
3.2 Physiological Signals: Voice Recordings
3.3 Tuning Voice Pitches via the Computation of Correlation Functions
3.3.1 Reference Frequency Distribution S0,j
3.3.2 Cross-Correlation R[S0,j,S(Q)j]
3.3.3 Application to Two Voice Signals from the VOICED Database
3.3.4 Cross-Correlation R[S(Q)i,S(Q)j] of Two Voice Signals
3.4 Frequency Distribution Matching and Sonance
4 Conclusion
References
Four Billion Years: The Story of an Ancient Protein Family
1 Introduction
2 Sequence Comparisons with Alignment
2.1 Pairwise Alignment of Two Sequences
2.2 Simultaneous Alignment of Several Sequences
3 Alignment-Free Sequence Comparison and Local Decoding
3.1 N-mers and N-Local Decoding
3.2 MS4
3.2.1 The Partition Tree
3.2.2 Classes Selection
3.2.3 The Dissimilarity Matrix
3.3 Variable Length Local Decoding of Sequences
4 Results: A Brief Look at the History of DNA Topoisomerases IA
4.1 The Evolutionary History of Topoisomerases IA
4.2 The Subfunctionalization of Reverse Gyrases
5 From Molecular Phylogenies to the Tree of Life
6 Conclusions
References
Pseudo-Rate Matrices, Beyond Dayhoff's Model
1 Introduction
2 Classical Approaches, Scoring Matrices
2.1 Dayhoff Evolution Model—The PAM (Point Accepted Mutation) Matrix
2.2 BLOSUM, Another General Purpose Substitution Matrix
2.3 Available Biological Material for the Estimation of Scoring Matrices
3 Rate Matrices, Beyond Dayhoff's Model
3.1 Definitions and Notations
3.2 From Pairwise Alignments to Rate Matrix
3.2.1 Observed Transition and Rate Matrices
3.2.2 The Symmetrized Case
3.3 Multivariate Analysis of Observed Rate Matrices
4 Biological Validation of Observed Rate Matrices
4.1 Rate Matrices and Subfunctionalization in Reverse Gyrases
4.2 Several Rate Matrices Within a Protein Family: The Case of Mitochondrial Proteins
5 The Influence of Dissimilarity Measures on Sequence Clustering and Phylogeny Reconstruction
5.1 An Iterative-Rank Based Clustering
5.2 Application to mtDNA-Encoded Proteins of Tetrapods
5.3 The Impact of Symmetry Assumptions
6 Discussion—Conclusion
References
Applied and Numerical Harmonic Analysis (105 volumes)
Applied and Numerical Harmonic Analysis