Mathematical Analysis, its Applications and Computation: ISAAC 2019, Aveiro, Portugal, July 29–August 2 (Springer Proceedings in Mathematics & Statistics, 385)

✍ Scribed by Paula Cerejeiras (editor), Michael Reissig (editor)

• Publisher: Springer
• Year: 2022
• Tongue: English
• Leaves: 150
• Edition: 1st ed. 2022
• Category: Library

✦ Synopsis

This volume includes the main contributions by the plenary speakers from the ISAAC congress held in Aveiro, Portugal, in 2019. It is the purpose of ISAAC to promote analysis, its applications, and its interaction with computation. Analysis is understood here in the broad sense of the word, including differential equations, integral equations, functional analysis, and function theory. With this objective, ISAAC organizes international Congresses for the presentation and discussion of research on analysis.
The plenary lectures in the present volume, authored by eminent specialists, are devoted to some exciting recent developments in topics such as science data, interpolating and sampling theory, inverse problems, and harmonic analysis.

✦ Table of Contents

Preface
Contents
Notes on Computational Hardness of Hypothesis Testing: Predictions Using the Low-Degree Likelihood Ratio
Overview
1 Towards a Computationally-Bounded Decision Theory
1.1 Statistical-to-Computational Gaps in Hypothesis Testing
1.2 Classical Asymptotic Decision Theory
1.2.1 Basic Notions
1.2.2 Likelihood Ratio Testing
1.2.3 Le Cam's Contiguity
1.3 Basics of the Low-Degree Method
2 The Additive Gaussian Noise Model
2.1 The Model
2.2 Computing the Classical Quantities
2.3 Computing the Low-Degree Quantities
2.3.1 Proof 1: Hermite Translation Identity
2.3.2 Proof 2: Gaussian Integration by Parts
2.3.3 Proof 3: Hermite Generating Function
3 Examples: Spiked Matrix and Tensor Models
3.1 The Spiked Tensor Model
3.1.1 Proof of Theorem 2: Upper Bound
3.1.2 Proof of Theorem 2: Lower Bound
3.2 The Spiked Wigner Matrix Model: Sharp Thresholds
3.2.1 The Canonical Distinguishing Algorithm: PCA
3.2.2 Low-Degree Analysis: Informally, with the ``Gaussian Heuristic''
3.2.3 Low-Degree Analysis: Formally, with Concentration Inequalities
4 More on the Low-Degree Method
4.1 The LDLR and Thresholding Polynomials
4.2 Algorithmic Implications of the LDLR
4.2.1 Robustness
4.2.2 Connection to Sum-of-Squares
4.2.3 Connection to Spectral Methods
4.2.4 Formal Conjecture
4.2.5 Empirical Evidence and Refined Conjecture
4.2.6 Extensions
Appendix 1: Omitted Proofs
Neyman-Pearson Lemma
Equivalence of Symmetric and Asymmetric Noise Models
Low-Degree Analysis of Spiked Wigner Above the PCA Threshold
Appendix 2: Omitted Probability Theory Background
Hermite Polynomials
Subgaussian Random Variables
Hypercontractivity
References
Totally Positive Functions in Sampling Theory and Time-Frequency Analysis
1 Introduction
2 Totally Positive Functions
3 Back to Sampling: Shift-Invariant Spaces
4 Totally Positive Generators of Gaussian Type
4.1 Sampling with Derivatives
4.2 Some Proof Ideas
5 Time-Frequency Analysis and Gabor Frames
6 Zero-Free Short-Time Fourier Transforms
7 Totally Positive Functions and the Riemann Hypothesis
8 Summary
References
Multidimensional Inverse Scattering for the Schrödinger Equation
1 Introduction
2 Potential Applications
3 Direct Scattering
4 The Main Objective of Problem 1.2a at Fixed and Sufficiently Large E
5 Old General Result on Problem 1.2a for d≥2
6 Results of N6,N7
7 Faddeev Functions
8 Results of N10,N11
9 Examples of Non-uniqueness for Problem 1.3a
10 Results of N15,NS2 on Modified Problem 1.3a for d≥2
11 Results of AHN
12 Formulas of N14,N17 Reducing Problem 1.3b to Problem 1.2a
References
A Survey of Hardy Type Inequalities on Homogeneous Groups
1 Introduction
2 Hardy Type Inequalities on Stratified Groups
3 Hardy Type Inequalities on Homogeneous Groups
References
Bogdan Bojarski in Complex and Real Worlds
1 Scientific Career
2 The Partial Indices of Matrix Function
3 Quasiconformal Mapping
4 Boundary Value Problems
5 Riemann-Hilbert Problem for a Multiply Connected Domain
6 Conclusion
References