Softcover ISBN: 978-1-4704-7295-5
Product Code: SURV/273
Mathematical Surveys and Monographs, Volume: 273
2023; 240 pp
MSC: Primary 60; 62; 43; Secondary 39;
It is well known that if two independent identically distributed random variables are Gaussian, then their sum and difference are also independent. It turns out that only Gaussian random variables have such property. This statement, known as the famous Kac-Bernstein theorem, is a typical example of a so-called characterization theorem. Characterization theorems in mathematical statistics are statements in which the description of possible distributions of random variables follows from properties of some functions of these random variables. The first results in this area are associated with famous 20th century mathematicians such as G. Polya, M. Kac, S. N. Bernstein, and Yu. V. Linnik.
By now, the corresponding theory on the real line has basically been constructed. The problem of extending the classical characterization theorems to various algebraic structures has been actively studied in recent decades. The purpose of this book is to provide a comprehensive and self-contained overview of the current state of the theory of characterization problems on locally compact Abelian groups. The book will be useful to everyone with some familiarity of abstract harmonic analysis who is interested in probability distributions and functional equations on groups.
Graduate students and researchers interested in probability distributions or functional equations on groups.
Chapters
Preliminaries
Independent random variables with independent sum and difference
Characterization of probability distributions through the independence of linear forms
Characterization of probability distributions through the symmetry of the conditional distribution of one linear form given another
Characterization theorems on the field of p
-adic numbers
Miscellaneous characterization theorems
Softcover ISBN: 978-1-4704-6968-9
Product Code: CONM/784
Contemporary Mathematics, Volume: 784
2023; 206 pp
MSC: Primary 49; 65; 78; 80; 86;
This volume contains the proceedings of two AMS Special Sessions gRecent Developments on Analysis and Computation for Inverse Problems for PDEs,h virtually held on March 13?14, 2021, and gRecent Advances in Inverse Problems for Partial Differential Equations,h virtually held on October 23?24, 2021.
The papers in this volume focus on new results on numerical methods for various inverse problems arising in electrical impedance tomography, inverse scattering in radar and optics problems, reconstruction of initial conditions, control of acoustic fields, and stock price forecasting. The authors studied iterative and non-iterative approaches such as optimization-based, globally convergent, sampling, and machine learning-based methods.
The volume provides an interesting source on advances in computational inverse problems for partial differential equations.
Readership
Graduate students and research mathematicians interested in partial differential equations and numerical methods for inverse problems.
Ugur G. Abdulla and Saleheh Seif - Discretization and convergence of the EIT optimal control problem in Sobolev spaces with dominating mixed smoothness
Thuy T. Le - Global reconstruction of initial conditions of nonlinear parabolic equations via the Carleman-contraction method
Isaac Harris - Regularization of the factorization method with applications to inverse scattering
Thu Le, Dinh-Liem Nguyen, Vu Nguyen and Trung Truong - Sampling type method combined with deep learning for inverse scattering with one incident wave
Dinh-Liem Nguyen and Trung Truong - Fast numerical solutions to direct and inverse scattering for bi-anisotropic periodic Maxwellfs equations
Loc H. Nguyen and Huong T.T. Vu - Reconstructing a space-dependent source term via the quasi-reversibility method
Quyen Tran - Convergence analysis of Nedelec finite element approximations for a stationary Maxwellfs system
Mikhail V. Klibanov, Kirill V. Golubnichiy and Andrey V. Nikitin - Quasi-reversibility method and neural network machine learning for forecasting of stock option prices
Vo Anh Khoa, Michael Victor Klibanov, William Grayson Powell and Loc Hoang Nguyen - Numerical reconstruction for 3D nonlinear SAR imaging via a version of the convexification method
Vo Anh Khoa, Mai Thanh Nhat Truong, Imhotep Hogan and Roselyn Williams - Initial state reconstruction on graphs
Lander Besabe and Daniel Onofrei - Active control of scalar Helmholtz fields in the presence of known impenetrable obstacles
Softcover ISBN: 978-1-4704-7151-4
Product Code: STML/101
Student Mathematical Library Volume,101
2023; 129 pp
MSC: Primary 55; 51; 20;
This book is an elementary introduction to knot theory. Unlike many other books on knot theory, this book has practically no prerequisites; it requires only basic plane and spatial Euclidean geometry but no knowledge of topology or group theory. It contains the first elementary proof of the existence of the Alexander polynomial of a knot or a link based on the Conway axioms, particularly the Conway skein relation. The book also contains an elementary exposition of the Jones polynomial, HOMFLY polynomial and Vassiliev knot invariants constructed using the Kontsevich integral. Additionally, there is a lecture introducing the braid group and shows its connection with knots and links.
Other important features of the book are the large number of original illustrations, numerous exercises and the absence of any references in the first eleven lectures. The last two lectures differ from the first eleven: they comprise a sketch of non-elementary topics and a brief history of the subject, including many references.
Undergraduate and graduate students interested in knot theory.
Knots and links, Reidmeister moves
The Conway polynomial
The arithemtic of knots
Some simple knot invariants
The Kauffman bracket
The Jones polynomial
Braids
Discriminants and finite type invariants
Vassiliev invariants
Combinatorial description of Vassiliev invariants
The Kontsevich integrals
Other important topics
A brief history of knot theory
Softcover ISBN: 978-1-4704-5672-6
Product Code: SURV/272
Mathematical Surveys and Monographs, Volume: 272
2023; 154 pp
MSC: Primary 11; 13;
Iwasawa theory began in the late 1950s with a series of papers by Kenkichi Iwasawa on ideal class groups in the cyclotomic tower of number fields and their relation to p
-adic L
-functions. The theory was later generalized by putting it in the context of elliptic curves and modular forms. The main motivation for writing this book was the need for a total perspective of Iwasawa theory that includes the new trends of generalized Iwasawa theory. Another motivation of this book is an update of the classical theory for class groups taking into account the changed point of view on Iwasawa theory.
The goal of this first part of the two-part publication is to explain the theory of ideal class groups, including its algebraic aspect (the Iwasawa class number formula), its analytic aspect (Leopoldt?Kubota L
-functions), and the Iwasawa main conjecture, which is a bridge between the algebraic and the analytic aspects.
The second part of the book will be published as a separate volume in the same series, Mathematical Surveys and Monographs of the American Mathematical Society.
Graduate students and researchers interested in number theory and arithmetic geometry.
Motivation and utility of Iwasawa theory
Zp-extension and Iwasawa algebra
Cyclotomic Iwasawa theory for ideal class groups
Bookguide
Appendix A
Softcover ISBN: 978-1-4704-7256-6
Student Mathematical Library, Volume: 102
2023; 286 pp
MSC: Primary 00; 11;
One of the great charms of mathematics is uncovering unexpected connections. In Numbers and Figures, Giancarlo Travaglini provides six conversations that do exactly that by talking about several topics in elementary number theory and some of their connections to geometry, calculus, and real-life problems such as COVID-19 vaccines or fiscal frauds. Each conversation is in two parts?an introductory essay which provides a gentle introduction to the topic and a second section that delves deeper and requires study by the reader. The topics themselves are extremely appealing and include, for example, Pick's theorem, Simpson's paradox, Farey sequences, the Frobenius problem, and Benford's Law.
Numbers and Figures will be a useful resource for college faculty teaching Elementary Number Theory or Calculus. The chapters are largely independent and could make for nice course-ending projects or even lead-ins to high school or undergraduate research projects. The whole book would make for an enjoyable semester-long independent reading course. Faculty will find it entertaining bedtime reading and, last but not least, readers more generally will be interested in this book if they miss the accuracy and imagination found in their high school and college math courses.
Undergraduate students interested in number theory and discrete mathematics.