Extended Abstracts 2021/2022 : Methusalem Lectures / / edited by Duván Cardona, Joel Restrepo, Michael Ruzhansky

• Pubblicazione: Cham : , : Springer Nature Switzerland : , : Imprint : Birkhäuser, , 2024
• Edizione: 1st ed. 2024.
• Descrizione fisica: 1 online resource (262 pages)
• Disciplina: 515.353
• Soggetto topico: Mathematical analysis; Geometry, Differential; Analysis; Differential Geometry
• Persona (resp. second.): CardonaDuván; RestrepoJoel; RuzhanskyM (Michael)
• Nota di bibliografia: Includes bibliographical references.
• Nota di contenuto: Intro -- Preface -- Contents -- Part I Geometric Analysis -- Analysis on Noncompact Manifolds and Index Theory: Fredholm Conditions and Pseudodifferential Operators -- 1 Introduction -- 2 Motivation and Some Classical Results -- 2.1 A Classical Case: Smooth, Compact Manifolds -- 2.2 Motivating Example: Cylindrical Ends -- 2.2.1 Polar Coordinates -- 2.2.2 The Black-Scholes Operator -- 2.2.3 Generalized Spherical Coordinates -- 2.2.4 Manifolds with Cylindrical Ends -- 3 Fredholm Operators on Manifolds with Nice Ends (Lie Manifolds) -- 3.1 Lie Manifolds -- 3.1.1 Definition of Lie Manifolds -- 3.1.2 Lie Manifolds and Geometry -- 3.1.3 Differential Operators -- 3.2 Formulation of the Main Result: Fredholm Conditions on ``Nice'' Manifolds -- 3.3 Examples and Applications -- 4 Pseudodifferential Operators and Problems -- 4.1 Problems -- References -- Singular Value Decomposition for the X-Ray Transforms on the Reduced Heisenberg Group, and a Two-Radius Theorem -- 1 Introduction -- 2 Definitions -- 2.1 Heisenberg Group -- 2.2 X-Ray Transform -- 3 Spectral Decomposition of Ir, Ir* and Ir* Ir -- 4 Intertwining Differential Operators -- 5 Ongoing Work -- Appendix -- Entry Functions on the Heisenberg Group -- Special Functions -- References -- Nonlocal Functionals with Non-standard Growth -- 1 Introduction -- 2 Nonlocal Functionals with Orlicz Growth -- 3 Nonlocal Functionals with Variable Exponent -- 4 Nonlocal Double Phase Functionals -- 5 Nonlocal Functionals with Orthotropic Structure -- References -- A Variational Approach to the Hot Spots Conjecture -- 1 Introduction -- 2 A Non-standard Variational Principle -- 3 Application to the Hot Spots Conjecture -- References -- Endpoint Sobolev Inequalities for Vector Fields and Cancelling Operators -- 1 Sobolev Inequalities for Vector Fields -- 2 Sobolev Estimates for Cancelling Operators. ; 3 Duality Estimates for Cocancelling Operators -- 4 Further Results -- References -- Scattering of Maxwell Potentials on Curved Spacetimes -- 1 Introduction -- 2 Background Spacetimes and Field Equations -- 2.1 CSCD Spacetimes -- 2.2 Field Equations -- 3 Main Results -- 4 Remarks -- 4.1 Conformal Scale -- 4.2 Spaces of Data -- 4.3 Goursat Problem -- 4.4 Role of Timelike Conformal Symmetry -- References -- Part II Analysis and PDEs -- Remark on the Ill-Posedness for KdV-Burgers Equation in Fourier Amalgam Spaces -- 1 Introduction -- 2 Proof of Theorem 1 -- References -- -Convergence for the Bi-Laplace-Beltrami Equationon Hypersurfaces -- 1 Introduction -- 2 Auxiliary Materials -- 3 Variational Formulation of Model Problem and Convergence -- References -- Bounded Weak Solutions with Orlicz Space Data: An Overview -- 1 Introduction: Uniformly Elliptic Operators -- 2 Degenerate Elliptic Operators -- 3 Further Remarks -- References -- Recent Progress on the Mathematical Theory of Wave Turbulence -- 1 Introduction -- 1.1 Description of the Theory -- 1.1.1 Scope of the Theory -- 1.1.2 Applications -- 1.2 Mathematical Literature -- 2 Main Results -- 2.1 Discussions -- 2.2 Related Works -- 2.3 Main Ideas in the Proof -- References -- Laplace-Beltrami Equation on Lipschitz Hypersurfaces in the Generic Bessel Potential Spaces -- 1 Introduction and Formulation of the Main Results -- Introduction and formulation of the problems -- 2 Generic Bessel Potential Spaces and Mellin Convolutions -- 3 Localization and the Model Problems, Proof of the Main Result -- References -- On the Convergence Fourier Series and Greedy Algorithm by Multiplicative System -- 1 Introduction -- 2 New Results: (Theorems) -- References -- Asymptotics of Harmonic Functions in the Absence of Monotonicity Formulas -- 1 Introduction. ; 2 Alternative for Motonicity Formula: Convergence of Doubling Index -- 3 Uniqueness of Blowup and Expansion Formula -- References -- Semiregular Non-commutative Harmonic Oscillators: Some Spectral Asymptotic Properties -- 1 Introduction -- 2 Non-commutative Harmonic Oscillators (NCHOs) -- 3 Jaynes-Cumming Model Hamiltonian Analytical Study -- 3.1 The JC-Model by Semiregular NCHOs -- 3.2 The JC-Model for One Atom with N-Level and One Cavity-Mode in the -Configuration -- 3.3 The JC-Model for an N-Level Atom and n=N-1 Cavity-Modes in the -Configuration -- 3.4 The JC-Model for an N-Level Atom and n=N-1 Cavity-Modes in the -Configuration -- 3.5 The JC-Model for an N-Level Atom and n=N-1 Cavity-Modes in the So-Called -Configuration -- 4 The Asymptotics of the Eigenvalues Counting Function -- 4.1 The Semiregular Metric Globally Elliptic System Class (SMGES Class) -- 4.2 The Decoupling Theorem -- 4.3 The Weyl Law -- References -- Global Compactness, Subcritical Approximation of the Sobolev Quotient, and a Related Concentration Result in the HeisenbergGroup -- 1 Critical Sobolev Embeddings in the Heisenberg Group -- 2 Subcritical Approximation of the Sobolev Quotient -- 3 Struwe's Global Compactness in the Heisenberg Group -- 4 Asymptotics of the Optimal Functions -- 5 Localization of the Energy Concentration -- References -- A Note on Carleson-Hunt Type Theorems for Vilenkin-Fourier Series -- 1 Introduction -- 2 Preliminaries -- 3 On Martingale Inequalities -- 4 a.e. Convergence of Vilenkin-Fourier Series -- 5 Almost Everywhere Divergence of Vilenkin-Fourier Series -- References -- Self-Similar Gravitational Collapse for Polytropic Stars -- 1 Introduction -- 2 Strategy of Proof -- References -- Control of Parabolic Equations with Inverse Square Infinite Potential Wells -- 1 Critically Singular Heat Equations -- 2 Existing Results -- 3 The Main Result. ; 4 The Hilbert Uniqueness Method -- 5 The Global Carleman Estimate -- 6 Additional Remarks -- References -- On Geometric Estimates for Some Problems Arising from Modeling Pull-in Voltage in MEMS -- 1 Introduction -- 2 Preliminaries -- 3 The Pull-in Voltage for the Newtonian Potential -- References -- A Note on Fractional Powers of the Hermite Operator -- 1 Introduction -- 2 Fractional Powers H-s of the Hermite Operator -- 3 More on Fractional Powers of the Hermite Operator -- References -- Non-Standard Version of Egorov Algebra of Generalized Functions -- 1 Introduction -- 2 Notations and Set-Theoretical Framework -- 3 Non-Standard Version of Egorov Algebra *E(Ω) -- 4 Non-Standard Delta-Function -- 5 Embedding of Distributions Into *E(Ω) -- 6 Regular Algebra -- References -- Density Conditions for Coherent State Subsystems of Nilpotent Lie Groups -- 1 Introduction -- 2 Coherent State Systems for Nilpotent Lie Groups -- 3 Density Conditions for Lattice Subgroups -- 4 Necessary Density Conditions for Irregular Point Sets -- 4.1 Approximate Lattices -- 4.2 General Point Sets -- 5 Open Problems and Questions -- References -- Space-Time Mixed Norm Estimates in Riemannian Symmetric Spaces of Non-Compact Type -- 1 Introduction -- 2 Strichartz Inequality -- 3 Smoothing Property -- 4 Conclusion -- References -- Analysis on Compact Symmetric Spaces: Eigenfunctions and Nonlinear Schrödinger Equations -- 1 Compact Globally Symmetric Spaces -- 2 Eigenfunction Bounds and Spherical Functions -- 3 NLS on CGSSs -- 4 Conclusion -- References -- Part III Applied Mathematics -- On Empirical Bayes Approach to Inverse Problems -- 1 Introduction -- 2 Compact G* G, Inverse Signal-in-White-Noise Model -- 2.1 Constructing an Empirical Bayes Posterior -- 2.2 Main Result -- 3 Further Discussion -- 3.1 Uncertainty Quantification (UQ) Problem -- 3.2 The Case of Non-Linear G. ; 3.3 Examples of Inverse Problems Originated from PDEs -- 3.3.1 Heat Equation: Recovery of Initial Condition -- 3.3.2 Elliptic PDE: Recovery of a Functional Coefficient -- 3.3.3 Parabolic PDE: Recovery of a Functional Coefficient -- References -- The Interferon Influence on the Infection Wave Propagation -- 1 Introduction -- 2 IFN Action in Infected Cells -- 3 IFN Action in Uninfected Cells -- 4 IFN Redistribution -- 5 Conclusions and Perspectives -- References -- Machine Learning-Based Analysis of Human Motions for Parkinson's Disease Diagnostics -- 1 Introduction -- 2 Problem Statement -- 3 Proposed Solution -- 3.1 Motion Mass -- 3.2 Machine Learning Workflow -- 4 Discussion -- 5 Conclusions -- References.
• Sommario/riassunto: This volume presents modern developments in analysis, PDEs and geometric analysis by some of the leading worldwide experts, prominent junior and senior researchers who were invited to be part of the Ghent Analysis & PDE Center Methusalem Seminars from 2021 to 2022. The contributions are from the speakers of the Methusalem Colloquium, Methusalem Junior Seminar and Geometric Analysis Seminar. The volume has two main topics: 1. Analysis and PDEs. The volume presents recent results in fundamental problems for solving partial integro-differential equations in different settings such as Euclidean spaces, manifolds, Banach spaces, and many others. Discussions about the global and local solvability using micro-local and harmonic analysis methods, studies of new techniques and approaches arising from a physical perspective or the mathematical point of view have also been included. Several connected branches arising in this regard are shown. 2. Geometric analysis. The volume presents studies of modern techniques for elliptic and subelliptic PDEs that in recent times have been used to establish new results in differential geometry and differential topology. These topics involve the intrinsic research in microlocal analysis, geometric analysis, and harmonic analysis abroad. Different problems having relevant geometric information for different applications in mathematical physics and other problems of classification have been considered.
• Titolo autorizzato: Extended Abstracts 2021
• ISBN: 3-031-48579-3
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 9910842294303321
• Lo trovi qui: Univ. Federico II
• Serie: Research Perspectives Ghent Analysis and PDE Center, . 2948-1732 ; ; 3