Differential Equations on Manifolds and Mathematical Physics: Dedicated to the Memory of Boris Sternin

Foreword
Boris Yu. Sternin (1939–2017)
Contents
Parametrix and Localized Solutions for Linearized Equations of Gas Dynamics
1. Introduction
2. Statement of the Problem
3. Decomposition of the Resolving Operator
3.1. Characteristics and Wave Fronts
3.2. Amplitudes of the Modes
3.3. Expansion of the Parametrix
3.4. Smooth Terms
3.5. Expansion of the Resolving Operator
4. Localized Solutions
Funding
References
Laplacians on Generalized Smooth Distributions as C*-Algebra Multipliers
1. Introduction
2. Distributions as Modules of Vector Fields
3. Horizontal Differential
4. Riemannian Metric on a Distribution
5. Horizontal Laplacian of a Distribution
6. Longitudinal Hypoellipticity
7. Horizontal Laplacian as a Multiplier
8. Leafwise Representations
9. Construction of a Parametrix
Funding
References
C*-Algebras Generated by Dynamical Systems and Nonlocal ψDO
1. Introduction
2.C*-Algebraic Preliminaries
2.1. C*-Algebras Generated by Dynamical Systems
2.2. Isomorphism Theorem
2.3. Finite Group. Isomorphism Theorem
3. Symbolic Calculus. Fredholm Property
3.1. Symbolic Calculus for FψDO
4. Symbol Invertibility Conditions
4.1. Finite Group G
4.2. Action of the Group Z
5. Index Formulas
5.1. Finite Group G
5.2. Group Z. Two-Term Operators
5.3. Multi-Term Operators. Reduction to Two-Term Operators
5.4. Multi-Term Operators. Lefschetz Numbers
5.5. Nontriviality of Lefschetz Numbers
Acknowledgements
References
Class of Fredholm Boundary Value Problems for the Wave Equation with Conditions on the Entire Boundary
1. Introduction
2. Statement of the Problem
3. Reduction to the Boundary
4. Fredholm Property of the Operators on the Boundary
5. Invertibility Conditions for Trajectory Symbols
6. Examples
Acknowledgements
Funding
References
Some Remarks on Equivariant Elliptic Operators and Their Invariants
1. Introduction
2. Equivariant Eta Invariant
Theorem 2.1 ([
Remark 2.2.
Remark 2.3.
Remark 2.4.
Theorem 2.5.
3. Equivariant Index
Theorem 3.1.
4. Equivariant APS Theorem
Theorem 4.1.
5. Examples
5.1. De Rham–Hodge Operator and the Equivariant Euler Characteristic
Example 5.1.
Example 5.2.
Example 5.3.
5.2. Dolbeault Operator on Complex Projective Space
5.3. Equivariant Eta Invariant of the Boundary Signature Operator
Funding
References
Homogenization of the Cauchy Problem for the Wave Equation with Rapidly Varying Coefficients and Initial Conditions
1. Introduction
2. 1D Wave Equation with Rapidly Oscillating Velocity
2.1. Model One-Dimensional Wave Equation
2.2. Fourier Transform Solution
2.3. Peierls Substitution
2.4. Mode Truncation
2.5. Polynomial Approximation to the Dispersion Relation and Simplification of the Asymptotics
2.6. Transition from Pseudodifferential Equations to the Limit Wave Equation and Other Reduced Equations
3. Generalizations: Inclusion of the Slow Dependence on x in the Velocity c and the Multidimensional Case
3.1. Slow Dependence on x and Distorted Fast Dependence
3.2. Extension of the Class of Rapidly Varying Initial Conditions
3.3. Generalization to the Multidimensional Wave Equation
3.4. Disadvantages of the Model Equation (3.16) and a Way around Difficulties in Its Practical Use
4. General Case: No Regular Dependence on the Fast Variables
4.1. General Asymptotic Cauchy Problem for the Wave Equation
4.2. Local Average
4.3. Homogenization Theorem in the Cauchy Problem for the Wave Equation
Acknowledgements
Funding
References
Resurgent Analysis of Singularly Perturbed Differential Systems: Exit Stokes, Enter Tes
1. Introduction. Model Problem
1.1. Model Problem
1.2. Multiple Resurgence
1.3. Normalizers ϴ±1
1.4. Elementary Multilinear Inputs: Biresurgent Monomials
2. Reminders on Resurgence, Moulds, and Hyperlogarithms
2.1. Reminders about Resurgence
2.2. Reminders about Moulds
2.3. Hyperlogarithmic Monomials and Monics
3. Weighted Products
3.1. Weighted Convolution weco
3.2. Weighted Multiplication wemu
3.3. Link with Biresurgent Monomials
4. Scramble Transform
4.1. Ordinary Scramble
4.2. υ-Augmented Scramble
4.3. Weighted Convolution with Polar or Hyperlogarithmic Inputs
5. Hyperlogarithmic Monomials under Alien Differentiation
5.1. Ordinary Monomials Sw(x)
5.2. υ-Augmented Monomials Sw(x)
6. Tessellation Coefficients
6.1. Ordinary Tessellation Coefficients tes•.
6.2. υ-Augmented Tesselation Coefficients vtes• and tes•.
7. Weighted Products under Alien Differentiation
7.1. Second Bridge Equation
7.2. Third Bridge Equation
8. Bridge Equations I, II, and III
8.1. Equational Resurgence. First Bridge Equation
8.2. Coequational Resurgence. From the Molecular to the Higher Levels
8.3. Coequational Resurgenge. Second and Third Bridge Equations
9. Equational–Coequational Link at the Monomial Level
9.1. Equational Resurgence and Its Entire Coefficients W•∗ (x)
9.2. Coequational Resurgence and Its Resurgent Coefficients T•∗ (x)
9.3. Equational–Coequational Link W•∗∗(x) ⇐⇒ T•∗∗(x)
10. Equational–Coequational Link at the Global Level
10.1. Time-Independent Schrödinger Equation with Polynomial Potential
10.2. Equational Resurgence
10.3. Coequational Resurgence
10.4. Isographic Invariance
10.5. Idempotence of the Rotator
10.6. Equational–Coequational Linkage
11. Isography and Autarky
11.1. Universality of Isography
11.2. Autark Functions
12. Conclusion
Acknowledgements
References
Large-Time Decay of Solutions of the Damped Kawahara Equation on the Half-Line
Funding
References
Flat Vector Bundles and Open Covers
1. Introduction
2. Exterior Algebra Version of the Mathai–Quillen Formalism
3. Counting Formula for the Euler Number of Flat Vector Bundles
3.1. Flat Vector Bundles and the Counting Formula
3.2. Superconnections and Flat Vector Bundles
3.3. Analysis outside of B+
3.4. Proof of Proposition 3.2
3.5. Proof of Theorem 3.4
Acknowledgements
Funding
References
Hochshild’s Method for Describing the Mackenzie Obstruction to Construction of a Transitive Lie Algebroid
Introduction
1. Definition of a Transitive Lie Algebroid
2. Extensions of Lie Algebras
3. Definitions and Hochschild’s Terminology
4. Hochshild’s Description of the Obstruction (for Transitive Lie Algebroids)
References
Dual Linear Programming Problem and One-Dimensional Gromov Minimal Fillings of Finite Metric Spaces
Introduction
1. Preliminaries
1.1. Minimal Fillings of Finite Metric Spaces
1.2. Linear Programming
2. Minimal Parametric Fillings and Linear Programming
3. Examples
3.1. Four-Point Spaces
3.2. Five-Point Spaces
3.3. Six-Point Spaces
Acknowledgments
References
Complete Semiclassical Spectral Asymptotics for Periodic and Almost Periodic Perturbations of Constant Operators
1. Introduction
1.1. Preliminary Remarks
1.2. Main Theorem
1.3. Plan of the Paper
2. Proof of the Main Theorem
2.1. Preliminary Analysis
2.2. Gauge Transformation
2.3. Nonresonant Zone
2.3.1. Gauge transformation.
2.3.2. Propagation.
2.4. Resonant Zone
2.4.1. Case of d = 2.
2.4.2. General case: gauge transform.
2.4.3. General case: propagation.
2.5. End of the Proof
3. Generalizations and Discussion
3.1. Matrix Operators
3.2. Perturbations
3.2.1. Decaying perturbations.
3.2.2. Hybrid perturbations.
3.2.3. Nonresonant zone.
3.2.4. Resonant zone.
3.3. Differentiability
Funding
References
Ellipticity of Operators Associated with Morse–Smale Diffeomorphisms
Introduction
1. Statement of the Problem
2. Diffeomorphisms of Morse–Smale Type
Asymptotics of weights.
3. Operators with Constant Coefficients
4. Operators with Variable Coefficients
Appendix. Difference Operators with Coefficients Stabilizing at Infinity
Acknowledgements
Funding
References
On Solutions of Elliptic Equations with Variable Exponents and Measure Data in Rn
1. Introduction
2. Capacity, Measure, and Anisotropic Sobolev–Orlicz Spaces with Variable Exponents
3. Assumptions and Main Results
4. Preliminaries
5. Proofs of Theorems 3.3 and 3.4
Funding
References
Semiclassical Quantum Maps of Semi-Hyperbolic Type
1. Introduction
1.1. Main Hypotheses
1.2. Examples
1.3. Main Result on Resonances in the Semi-Hyperbolic Case
1.4. Bohr–Sommerfeld Quantization Rules
1.5. Remarks on Trace Formulas
2. Hint on the Proof of Theorem 1.1
2.1. (Absolute) Monodromy Operator
2.2. Intertwining M(z) with M(w)
2.3. Grushin Problem
3. “Approximate” Theory
3.1. Birkhoff Normal Form
3.2. Microlocalization in the Complex Domain
3.3. Poisson Operator, its Normalization, and the Monodromy Operator
Appendix. A Short Review on Complex Scaling
Acknowledgements
Funding
References
Derivations of Group Algebras and Hochschild Cohomology
1. Introduction
2. Hochschild Cohomology
2.1. Derivations
2.2. Hochschild Cohomology
2.3. Hochschild Homology
3. Classifying Space BG of the Groupoid G
3.1. Right Action: The Classifying Space BrG of the Groupoid rG
3.2. Trivial Action: The Classifying Space BG of the Group G
3.3. Adjoint action: The Classifying Space BG
Acknowledgements
Funding
References
On a New Type of Periodic Fronts in Burgers Type Equations with Modular Advection
1. Introduction. Statement of the Problem. Assumptions. Construction of Formal Asymptotics
1.1. Assumptions
1.2. Construction of Asymptotic Approximations to the Solution
2. Existence Results
2.1. Existence Theorem
2.2. Construction of Upper and Lower Solutions
3. Local Uniqueness and Stability of the Solution of the Periodic Problem
Acknowledgements
Funding
References
Mellin Operators in Weighted Corner Spaces
Introduction
1. Parameter-Dependent Edge Operators
1.1. Manifolds with Conical Singularities and Edges
1.2. Edge-Degenerate Operators
1.3. Mellin Symbols with Parameters
1.4. Weighted Cone Sobolev Spaces
1.5. Smoothing Mellin Plus Green Operators
1.6. Edge Sobolev Spaces
1.7. Elements of the Edge Calculus
2. Corner Operators
2.1. Kegel Spaces with Multiple Weights
2.2. Mellin Quantization with Respect to Corner Parameters
2.3. Calculus for Corner Singularities
References
Quantum Hall Effect and Noncommutative Geometry
1. Introduction
2. Classical Hall Effect
3. Quantum Hall Effect
4. Classical Bloch Theory
5. Magnetic Schr¨odinger Operator
6. Algebras of Observables
6.1. Algebra A(σ)
6.2. Von Neumann Group Algebras
6.3. Algebras C0(σ) and C0(σ)
6.4. Von Neumann algebras with Coefficients in a Hilbert Space
6.5. Algebras of Observables
7. Hochschild Cohomology
7.1. Group Cohomology
7.2. Hochschild Cohomology of Algebras
8. Interpretation of the Quantum Hall Effect
8.1. Integer Quantum Hall Effect
8.2. Fractional Quantum Hall Effect
Funding
References
Theory of the PROTO-SPHERA Experiment
1. Introduction
2. Governing Equations
3. Components in Cylindrical Geometry and Steady State
3.1. Components in Cylindrical Geometry and Steady State
4. Numerical Results and Discussion
5. Analysis of Axial 1D Equilibrium
6. Conclusions
References