LEADER 05999nam 22005775 450 001 9910568282803321 005 20251113210446.0 010 $a3-030-97127-9 024 7 $a10.1007/978-3-030-97127-4 035 $a(MiAaPQ)EBC6986458 035 $a(Au-PeEL)EBL6986458 035 $a(CKB)22371876800041 035 $a(PPN)269148922 035 $a(OCoLC)1319859436 035 $a(DE-He213)978-3-030-97127-4 035 $a(EXLCZ)9922371876800041 100 $a20220511d2022 u| 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aMathematical Analysis, its Applications and Computation $eISAAC 2019, Aveiro, Portugal, July 29?August 2 /$fedited by Paula Cerejeiras, Michael Reissig 205 $a1st ed. 2022. 210 1$aCham :$cSpringer International Publishing :$cImprint: Springer,$d2022. 215 $a1 online resource (150 pages) 225 1 $aSpringer Proceedings in Mathematics & Statistics,$x2194-1017 ;$v385 311 08$aPrint version: Cerejeiras, Paula Mathematical Analysis, Its Applications and Computation Cham : Springer International Publishing AG,c2022 9783030971267 327 $aIntro -- 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. 327 $a2 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. 330 $aThis volume includes the main contributions by the plenary speakers from the ISAAC congress held in Aveiro, Portugal, in 2019. It is the purpose of ISAAC to promote analysis, its applications, and its interaction with computation. Analysis is understood here in the broad sense of the word, including differential equations, integral equations, functional analysis, and function theory. With this objective, ISAAC organizes international Congresses for the presentation and discussion of research on analysis. The plenary lectures in the present volume, authored by eminent specialists, are devoted to some exciting recent developments in topics such as science data, interpolating and sampling theory, inverse problems, and harmonic analysis. 410 0$aSpringer Proceedings in Mathematics & Statistics,$x2194-1017 ;$v385 606 $aMathematics$xData processing 606 $aHarmonic analysis 606 $aComputational Mathematics and Numerical Analysis 606 $aAbstract Harmonic Analysis 615 0$aMathematics$xData processing. 615 0$aHarmonic analysis. 615 14$aComputational Mathematics and Numerical Analysis. 615 24$aAbstract Harmonic Analysis. 676 $a515 676 $a515 702 $aCerejeiras$b Paula 702 $aReissig$b Michael 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910568282803321 996 $aMathematical analysis, its applications and computation$92985084 997 $aUNINA