Download or read book An Introduction to Central Simple Algebras and Their Applications to Wireless Communication written by Grégory Berhuy and published by American Mathematical Soc.. This book was released on 2013-07-05 with total page 288 pages. Available in PDF, EPUB and Kindle. Book excerpt: Central simple algebras arise naturally in many areas of mathematics. They are closely connected with ring theory, but are also important in representation theory, algebraic geometry and number theory. Recently, surprising applications of the theory of central simple algebras have arisen in the context of coding for wireless communication. The exposition in the book takes advantage of this serendipity, presenting an introduction to the theory of central simple algebras intertwined with its applications to coding theory. Many results or constructions from the standard theory are presented in classical form, but with a focus on explicit techniques and examples, often from coding theory. Topics covered include quaternion algebras, splitting fields, the Skolem-Noether Theorem, the Brauer group, crossed products, cyclic algebras and algebras with a unitary involution. Code constructions give the opportunity for many examples and explicit computations. This book provides an introduction to the theory of central algebras accessible to graduate students, while also presenting topics in coding theory for wireless communication for a mathematical audience. It is also suitable for coding theorists interested in learning how division algebras may be useful for coding in wireless communication.

Download or read book Number Theory Meets Wireless Communications written by Victor Beresnevich and published by Springer Nature. This book was released on 2021-01-08 with total page 281 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume explores the rich interplay between number theory and wireless communications, reviewing the surprisingly deep connections between these fields and presenting new research directions to inspire future research. The contributions of this volume stem from the Workshop on Interactions between Number Theory and Wireless Communication held at the University of York in 2016. The chapters, written by leading experts in their respective fields, provide direct overviews of highly exciting current research developments. The topics discussed include metric Diophantine approximation, geometry of numbers, homogeneous dynamics, algebraic lattices and codes, network and channel coding, and interference alignment. The book is edited by experts working in number theory and communication theory. It thus provides unique insight into key concepts, cutting-edge results, and modern techniques that play an essential role in contemporary research. Great effort has been made to present the material in a manner that is accessible to new researchers, including PhD students. The book will also be essential reading for established researchers working in number theory or wireless communications looking to broaden their outlook and contribute to this emerging interdisciplinary area.

Download or read book The Ricci Flow Techniques and Applications written by Bennett Chow and published by American Mathematical Soc.. This book was released on 2015-10-19 with total page 397 pages. Available in PDF, EPUB and Kindle. Book excerpt: Ricci flow is a powerful technique using a heat-type equation to deform Riemannian metrics on manifolds to better metrics in the search for geometric decompositions. With the fourth part of their volume on techniques and applications of the theory, the authors discuss long-time solutions of the Ricci flow and related topics. In dimension 3, Perelman completed Hamilton's program to prove Thurston's geometrization conjecture. In higher dimensions the Ricci flow has remarkable properties, which indicates its usefulness to understand relations between the geometry and topology of manifolds. This book discusses recent developments on gradient Ricci solitons, which model the singularities developing under the Ricci flow. In the shrinking case there is a surprising rigidity which suggests the likelihood of a well-developed structure theory. A broader class of solutions is ancient solutions; the authors discuss the beautiful classification in dimension 2. In higher dimensions they consider both ancient and singular Type I solutions, which must have shrinking gradient Ricci soliton models. Next, Hamilton's theory of 3-dimensional nonsingular solutions is presented, following his original work. Historically, this theory initially connected the Ricci flow to the geometrization conjecture. From a dynamical point of view, one is interested in the stability of the Ricci flow. The authors discuss what is known about this basic problem. Finally, they consider the degenerate neckpinch singularity from both the numerical and theoretical perspectives. This book makes advanced material accessible to researchers and graduate students who are interested in the Ricci flow and geometric evolution equations and who have a knowledge of the fundamentals of the Ricci flow.

Download or read book Galois Theories of Linear Difference Equations An Introduction written by Charlotte Hardouin and published by American Mathematical Soc.. This book was released on 2016-04-27 with total page 185 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is a collection of three introductory tutorials coming out of three courses given at the CIMPA Research School “Galois Theory of Difference Equations” in Santa Marta, Columbia, July 23–August 1, 2012. The aim of these tutorials is to introduce the reader to three Galois theories of linear difference equations and their interrelations. Each of the three articles addresses a different galoisian aspect of linear difference equations. The authors motivate and give elementary examples of the basic ideas and techniques, providing the reader with an entry to current research. In addition each article contains an extensive bibliography that includes recent papers; the authors have provided pointers to these articles allowing the interested reader to explore further.

Download or read book Brauer Groups Tamagawa Measures and Rational Points on Algebraic Varieties written by Jorg Jahnel and published by American Mathematical Soc.. This book was released on 2014-12-02 with total page 280 pages. Available in PDF, EPUB and Kindle. Book excerpt: The central theme of this book is the study of rational points on algebraic varieties of Fano and intermediate type--both in terms of when such points exist and, if they do, their quantitative density. The book consists of three parts. In the first part, the author discusses the concept of a height and formulates Manin's conjecture on the asymptotics of rational points on Fano varieties. The second part introduces the various versions of the Brauer group. The author explains why a Brauer class may serve as an obstruction to weak approximation or even to the Hasse principle. This part includes two sections devoted to explicit computations of the Brauer-Manin obstruction for particular types of cubic surfaces. The final part describes numerical experiments related to the Manin conjecture that were carried out by the author together with Andreas-Stephan Elsenhans. The book presents the state of the art in computational arithmetic geometry for higher-dimensional algebraic varieties and will be a valuable reference for researchers and graduate students interested in that area.

Download or read book Topics in Quaternion Linear Algebra written by Leiba Rodman and published by Princeton University Press. This book was released on 2014-08-24 with total page 378 pages. Available in PDF, EPUB and Kindle. Book excerpt: Quaternions are a number system that has become increasingly useful for representing the rotations of objects in three-dimensional space and has important applications in theoretical and applied mathematics, physics, computer science, and engineering. This is the first book to provide a systematic, accessible, and self-contained exposition of quaternion linear algebra. It features previously unpublished research results with complete proofs and many open problems at various levels, as well as more than 200 exercises to facilitate use by students and instructors. Applications presented in the book include numerical ranges, invariant semidefinite subspaces, differential equations with symmetries, and matrix equations. Designed for researchers and students across a variety of disciplines, the book can be read by anyone with a background in linear algebra, rudimentary complex analysis, and some multivariable calculus. Instructors will find it useful as a complementary text for undergraduate linear algebra courses or as a basis for a graduate course in linear algebra. The open problems can serve as research projects for undergraduates, topics for graduate students, or problems to be tackled by professional research mathematicians. The book is also an invaluable reference tool for researchers in fields where techniques based on quaternion analysis are used.

Download or read book The Octagonal PETs written by Richard Evan Schwartz and published by American Mathematical Soc.. This book was released on 2014-07-03 with total page 226 pages. Available in PDF, EPUB and Kindle. Book excerpt: A polytope exchange transformation is a (discontinuous) map from a polytope to itself that is a translation wherever it is defined. The 1-dimensional examples, interval exchange transformations, have been studied fruitfully for many years and have deep connections to other areas of mathematics, such as Teichmüller theory. This book introduces a general method for constructing polytope exchange transformations in higher dimensions and then studies the simplest example of the construction in detail. The simplest case is a 1-parameter family of polygon exchange transformations that turns out to be closely related to outer billiards on semi-regular octagons. The 1-parameter family admits a complete renormalization scheme, and this structure allows for a fairly complete analysis both of the system and of outer billiards on semi-regular octagons. The material in this book was discovered through computer experimentation. On the other hand, the proofs are traditional, except for a few rigorous computer-assisted calculations.

Download or read book Complex Multiplication and Lifting Problems written by Ching-Li Chai and published by American Mathematical Soc.. This book was released on 2013-12-19 with total page 402 pages. Available in PDF, EPUB and Kindle. Book excerpt: Abelian varieties with complex multiplication lie at the origins of class field theory, and they play a central role in the contemporary theory of Shimura varieties. They are special in characteristic 0 and ubiquitous over finite fields. This book explores the relationship between such abelian varieties over finite fields and over arithmetically interesting fields of characteristic 0 via the study of several natural CM lifting problems which had previously been solved only in special cases. In addition to giving complete solutions to such questions, the authors provide numerous examples to illustrate the general theory and present a detailed treatment of many fundamental results and concepts in the arithmetic of abelian varieties, such as the Main Theorem of Complex Multiplication and its generalizations, the finer aspects of Tate's work on abelian varieties over finite fields, and deformation theory. This book provides an ideal illustration of how modern techniques in arithmetic geometry (such as descent theory, crystalline methods, and group schemes) can be fruitfully combined with class field theory to answer concrete questions about abelian varieties. It will be a useful reference for researchers and advanced graduate students at the interface of number theory and algebraic geometry.

Download or read book Stochastic Resonance written by Samuel Herrmann and published by American Mathematical Soc.. This book was released on 2013-12-11 with total page 209 pages. Available in PDF, EPUB and Kindle. Book excerpt: Stochastic resonance is a phenomenon arising in a wide spectrum of areas in the sciences ranging from physics through neuroscience to chemistry and biology. This book presents a mathematical approach to stochastic resonance which is based on a large deviations principle (LDP) for randomly perturbed dynamical systems with a weak inhomogeneity given by an exogenous periodicity of small frequency. Resonance, the optimal tuning between period length and noise amplitude, is explained by optimizing the LDP's rate function. The authors show that not all physical measures of tuning quality are robust with respect to dimension reduction. They propose measures of tuning quality based on exponential transition rates explained by large deviations techniques and show that these measures are robust. The book sheds some light on the shortcomings and strengths of different concepts used in the theory and applications of stochastic resonance without attempting to give a comprehensive overview of the many facets of stochastic resonance in the various areas of sciences. It is intended for researchers and graduate students in mathematics and the sciences interested in stochastic dynamics who wish to understand the conceptual background of stochastic resonance.

Download or read book Geometry of Isotropic Convex Bodies written by Silouanos Brazitikos and published by American Mathematical Soc.. This book was released on 2014-04-24 with total page 618 pages. Available in PDF, EPUB and Kindle. Book excerpt: The study of high-dimensional convex bodies from a geometric and analytic point of view, with an emphasis on the dependence of various parameters on the dimension stands at the intersection of classical convex geometry and the local theory of Banach spaces. It is also closely linked to many other fields, such as probability theory, partial differential equations, Riemannian geometry, harmonic analysis and combinatorics. It is now understood that the convexity assumption forces most of the volume of a high-dimensional convex body to be concentrated in some canonical way and the main question is whether, under some natural normalization, the answer to many fundamental questions should be independent of the dimension. The aim of this book is to introduce a number of well-known questions regarding the distribution of volume in high-dimensional convex bodies, which are exactly of this nature: among them are the slicing problem, the thin shell conjecture and the Kannan-Lovász-Simonovits conjecture. This book provides a self-contained and up to date account of the progress that has been made in the last fifteen years.

Download or read book Coding Theory and Applications written by Raquel Pinto and published by Springer. This book was released on 2015-07-24 with total page 347 pages. Available in PDF, EPUB and Kindle. Book excerpt: The topics covered in this book, written by researchers at the forefront of their field, represent some of the most relevant research areas in modern coding theory: codes and combinatorial structures, algebraic geometric codes, group codes, quantum codes, convolutional codes, network coding and cryptography. The book includes a survey paper on the interconnections of coding theory with constrained systems, written by an invited speaker, as well as 37 cutting-edge research communications presented at the 4th International Castle Meeting on Coding Theory and Applications (4ICMCTA), held at the Castle of Palmela in September 2014. The event’s scientific program consisted of four invited talks and 39 regular talks by authors from 24 different countries. This conference provided an ideal opportunity for communicating new results, exchanging ideas, strengthening international cooperation, and introducing young researchers into the coding theory community.

Download or read book The Dynamical Mordell Lang Conjecture written by Jason P. Bell and published by American Mathematical Soc.. This book was released on 2016-04-20 with total page 297 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Dynamical Mordell-Lang Conjecture is an analogue of the classical Mordell-Lang conjecture in the context of arithmetic dynamics. It predicts the behavior of the orbit of a point x under the action of an endomorphism f of a quasiprojective complex variety X. More precisely, it claims that for any point x in X and any subvariety V of X, the set of indices n such that the n-th iterate of x under f lies in V is a finite union of arithmetic progressions. In this book the authors present all known results about the Dynamical Mordell-Lang Conjecture, focusing mainly on a p-adic approach which provides a parametrization of the orbit of a point under an endomorphism of a variety.

Download or read book Fokker Planck Kolmogorov Equations written by Vladimir I. Bogachev and published by American Mathematical Society. This book was released on 2022-02-10 with total page 495 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book gives an exposition of the principal concepts and results related to second order elliptic and parabolic equations for measures, the main examples of which are Fokker–Planck–Kolmogorov equations for stationary and transition probabilities of diffusion processes. Existence and uniqueness of solutions are studied along with existence and Sobolev regularity of their densities and upper and lower bounds for the latter. The target readership includes mathematicians and physicists whose research is related to diffusion processes as well as elliptic and parabolic equations.

Download or read book Topological Modular Forms written by Christopher L. Douglas and published by American Mathematical Soc.. This book was released on 2014-12-04 with total page 353 pages. Available in PDF, EPUB and Kindle. Book excerpt: The theory of topological modular forms is an intricate blend of classical algebraic modular forms and stable homotopy groups of spheres. The construction of this theory combines an algebro-geometric perspective on elliptic curves over finite fields with techniques from algebraic topology, particularly stable homotopy theory. It has applications to and connections with manifold topology, number theory, and string theory. This book provides a careful, accessible introduction to topological modular forms. After a brief history and an extended overview of the subject, the book proper commences with an exposition of classical aspects of elliptic cohomology, including background material on elliptic curves and modular forms, a description of the moduli stack of elliptic curves, an explanation of the exact functor theorem for constructing cohomology theories, and an exploration of sheaves in stable homotopy theory. There follows a treatment of more specialized topics, including localization of spectra, the deformation theory of formal groups, and Goerss-Hopkins obstruction theory for multiplicative structures on spectra. The book then proceeds to more advanced material, including discussions of the string orientation, the sheaf of spectra on the moduli stack of elliptic curves, the homotopy of topological modular forms, and an extensive account of the construction of the spectrum of topological modular forms. The book concludes with the three original, pioneering and enormously influential manuscripts on the subject, by Hopkins, Miller, and Mahowald.

Download or read book Foundations of Free Noncommutative Function Theory written by Dmitry S. Kaliuzhnyi-Verbovetskyi and published by American Mathematical Soc.. This book was released on 2014-11-19 with total page 194 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this book the authors develop a theory of free noncommutative functions, in both algebraic and analytic settings. Such functions are defined as mappings from square matrices of all sizes over a module (in particular, a vector space) to square matrices over another module, which respect the size, direct sums, and similarities of matrices. Examples include, but are not limited to, noncommutative polynomials, power series, and rational expressions. Motivation and inspiration for using the theory of free noncommutative functions often comes from free probability. An important application area is "dimensionless" matrix inequalities; these arise, e.g., in various optimization problems of system engineering. Among other related areas are those of polynomial identities in rings, formal languages and finite automata, quasideterminants, noncommutative symmetric functions, operator spaces and operator algebras, and quantum control.

Download or read book Persistence Theory From Quiver Representations to Data Analysis written by Steve Y. Oudot and published by American Mathematical Soc.. This book was released on 2017-05-17 with total page 229 pages. Available in PDF, EPUB and Kindle. Book excerpt: Persistence theory emerged in the early 2000s as a new theory in the area of applied and computational topology. This book provides a broad and modern view of the subject, including its algebraic, topological, and algorithmic aspects. It also elaborates on applications in data analysis. The level of detail of the exposition has been set so as to keep a survey style, while providing sufficient insights into the proofs so the reader can understand the mechanisms at work. The book is organized into three parts. The first part is dedicated to the foundations of persistence and emphasizes its connection to quiver representation theory. The second part focuses on its connection to applications through a few selected topics. The third part provides perspectives for both the theory and its applications. The book can be used as a text for a course on applied topology or data analysis.

Download or read book Grid Homology for Knots and Links written by Peter S. Ozsváth and published by American Mathematical Soc.. This book was released on 2015-12-04 with total page 423 pages. Available in PDF, EPUB and Kindle. Book excerpt: Knot theory is a classical area of low-dimensional topology, directly connected with the theory of three-manifolds and smooth four-manifold topology. In recent years, the subject has undergone transformative changes thanks to its connections with a number of other mathematical disciplines, including gauge theory; representation theory and categorification; contact geometry; and the theory of pseudo-holomorphic curves. Starting from the combinatorial point of view on knots using their grid diagrams, this book serves as an introduction to knot theory, specifically as it relates to some of the above developments. After a brief overview of the background material in the subject, the book gives a self-contained treatment of knot Floer homology from the point of view of grid diagrams. Applications include computations of the unknotting number and slice genus of torus knots (asked first in the 1960s and settled in the 1990s), and tools to study variants of knot theory in the presence of a contact structure. Additional topics are presented to prepare readers for further study in holomorphic methods in low-dimensional topology, especially Heegaard Floer homology. The book could serve as a textbook for an advanced undergraduate or part of a graduate course in knot theory. Standard background material is sketched in the text and the appendices.