https://doi.org/10.1142/12823 | April 2022
Pages: 436
In this monograph, we develop the theory of one of the most fascinating topics in coding theory, namely, perfect codes and related structures. Perfect codes are considered to be the most beautiful structure in coding theory, at least from the mathematical side. These codes are the largest ones with their given parameters. The book develops the theory of these codes in various metrics ? Hamming, Johnson, Lee, Grassmann, as well as in other spaces and metrics. It also covers other related structures such as diameter perfect codes, quasi-perfect codes, mixed codes, tilings, combinatorial designs, and more. The goal is to give the aspects of all these codes, to derive bounds on their sizes, and present various constructions for these codes.
The intention is to offer a different perspective for the area of perfect codes. For example, in many chapters there is a section devoted to diameter perfect codes. In these codes, anticodes are used instead of balls and these anticodes are related to intersecting families, an area that is part of extremal combinatorics. This is one example that shows how we direct our exposition in this book to both researchers in coding theory and mathematicians interested in combinatorics and extremal combinatorics. New perspectives for MDS codes, different from the classic ones, which lead to new directions of research on these codes are another example of how this book may appeal to both researchers in coding theory and mathematicians.
The book can also be used as a textbook, either on basic course in combinatorial coding theory, or as an advance course in combinatorial coding theory.
Preface
Introduction
Definitions and Preliminaries
Combinatorial Designs and Bounds
Linear Perfect Codes
Nonlinear Perfect Codes
Density and Quasi-Perfect Codes
Codes with Mixed Alphabets
Binary Constant-Weight Codes
NonBinary Constant-Weight Codes
Codes Over Subspaces
The Lee and the Manhattan Metrics
Tiling with a Cluster of Unit Cubes
Codes in Other Metrics
Undergraduate and graduate students, researchers in coding theory and mathematicians interested in combinatorics and extremal combinatorics.
https://doi.org/10.1142/12776 | May 2022
Pages: 400
It is unlikely that today there is a specialist in theoretical physics who has not heard anything about the algebraic Bethe ansatz. Over the past few years, this method has been actively used in quantum statistical physics models, condensed matter physics, gauge field theories, and string theory.
This book presents the state-of-the-art research in the field of algebraic Bethe ansatz. Along with the results that have already become classic, the book also contains the results obtained in recent years. The reader will get acquainted with the solution of the spectral problem and more complex problems that are solved using this method. Various methods for calculating scalar products and form factors are described in detail. Special attention is paid to applying the algebraic Bethe ansatz to the calculation of the correlation functions of quantum integrable models. The book also elaborates on multiple integral representations for correlation functions and examples of calculating the long-distance asymptotics of correlations.
This text is intended for advanced undergraduate and postgraduate students, and specialists interested in the mathematical methods of studying physical systems that allow them to obtain exact results.
Quantum Integrable Systems
Algebraic Bethe Ansatz
Quantum Inverse Problem
Composite Model
Scalar Products of Off-Shell Bethe Vectors
Scalar Products with On-Shell Bethe Vectors
Alternative Methods to Compute Scalar Products
Form Factors of the Monodromy Matrix Elements
Form Factors of Local Operators
Thermodynamic Limit
Multiple Integral Representations for Correlation Functions
Asymptotics of Correlation Functions via Form Factor Expansion
Appendices:
The Psi-Function and the Barnes G-Function
Finite-Size Corrections to the Excitation Energy
Identities for Fredholm Determinants
Integrals with Vandermonde Determinant
Advanced undergraduate and graduate students, researchers in mathematical
physics and theoretical physics.
https://doi.org/10.1142/12819 | May 2022
Pages: 200
This book is intended as a textbook for a one-term senior undergraduate (or graduate) course in Ring and Field Theory, or Galois theory. The book is ready for an instructor to pick up to teach without making any preparations.
The book is written in a way that is easy to understand, simple and concise with simple historic remarks to show the beauty of algebraic results and algebraic methods. The book contains 240 carefully selected exercise questions of varying difficulty which will allow students to practice their own computational and proof-writing skills. Sample solutions to some exercise questions are provided, from which students can learn to approach and write their own solutions and proofs. Besides standard ones, some of the exercises are new and very interesting. The book contains several simple-to-use irreducibility criteria for rational polynomials which are not in any such textbook.
This book can also serve as a reference for professional mathematicians. In particular, it will be a nice book for PhD students to prepare their qualification exams.
Basic Theory on Rings
Unique Factorization Domains
Modules and Noetherian Rings
Fields and Extension Fields
Automorphisms of Fields
Galois Theory
Sample Solutions
Appendix A: Equivalence Relations and Kuratowski-Zorn Lemma
References
Index
Senior undergraduate and graduate students in abstract algebra.
https://doi.org/10.1142/12812 | August 2022
Pages: 180
Seduction is not just an end result, but a process ? and in mathematics, both the end results and the process by which those end results are achieved are often charming and elegant.
This helps to explain why so many people ? not just those for whom math plays a key role in their day-to-day lives ? have found mathematics so seductive. Math is unique among all subjects in that it contains end results of amazing insight and power, and lines of reasoning that are clever, charming, and elegant. This book is a collection of those results and lines of reasoning that make us say, "OMG, that's just amazing," ? because that's what mathematics is to those who love it. In addition, some of the stories about mathematical discoveries and the people who discovered them are every bit as fascinating as the discoveries themselves.
This book contains material capable of being appreciated by students in elementary school ? as well as some material that will probably be new to even the more mathematically sophisticated. Most of the book can be easily understood by those whose only math courses are algebra and geometry, and who may have missed the magic, enchantment, and wonder that is the special province of mathematics.
Seduced by Numbers
Seduced by Arithmetic
Seduced by Patterns
Seduced by Analytic Geometry
Seduced by Mathematical Induction (and How to Avoid It)
Seduced by Calculus I
Seduced by Complex Numbers
Seduced by Infinite Series
Seduced by Probability
Seduced by Infinity
Seduced by Computers
Seduced by a Few of My Favorite Things
General public, mathematics teachers at the elementary and secondary levels.
https://doi.org/10.1142/12644 | September 2022
Pages: 1131
Volume I contains a long article by Misha Gromov based on his many years of involvement in this subject. It came from lectures delivered in Spring 2019 at IHES. There is some background given. Many topics in the field are presented, and many open problems are discussed. One intriguing point here is the crucial role played by two seemingly unrelated analytic means: index theory of Dirac operators and geometric measure theory.
Very recently there have been some real breakthroughs in the field. Volume I has several survey articles written by people who were responsible for these results.
For Volume II, many people in areas of mathematics and physics, whose work is somehow related to scalar curvature, were asked to write about this in any way thay pleased. This gives rise to a wonderful collection of articles, some with very broad and historical views, others which discussed specific fascinating subjects.
These two books give a rich and powerful view of one of geometry's very appealing sides.
Volume 1:
Four Lectures on Scalar Curvature
Scalar Curvature and Generalized Callias Operators
Convergence and Regularity of Manifolds with Scalar Curvature and Entropy Lower Bounds
Level Set Methods in the Study of Scalar Curvature
The Secret Hyperbolic Life of Positive Scalar Curvature
On The Scalar Curvature of 4-Manifolds
Volume 2:
Classical Relations to Topology and The Dirac Operator:
Some Topological Implications of Positive Scalar Curvature and Generalizations
Complete Manifolds with Positive Scalar Curvature
Manifolds with Boundary and Spaces of Metrics with Positive Scalar Curvature and Mean Curvature
Minimal Varieties
Positive Mass and Positive Energy
Positive Scalar Curvature on Generalized Spaces:
Polyhedra and Positive Scalar Curvature on Metric Spaces
Distance Estimates
Families and Foliations
Professional mathematicians and physicists, and certainly graduate students, in differential geometry and related areas in mathematics, and in general relativity and related areas in physics.
The books could easily be used for advanced graduate courses in mathematics and physics.
https://doi.org/10.1142/12719 | March 2022
Pages: 212
This book is mainly intended for first-year University students who undertake a basic abstract algebra course, as well as instructors. It contains the basic notions of abstract algebra through solved exercises as well as a "True or False" section in each chapter. Each chapter also contains an essential background section, which makes the book easier to use.
Contents:
Introduction
Sets and Logic
Mappings
Binary Relations
Groups
Rings and Fields
Polynomials and Rational Fractions
Bibliography
Index
First and second year mathematics and computer science students interested in abstract algebra. Also good for instructors.
Part I: General Theory
Part II: Special Classes of Functions and Sets
https://doi.org/10.1142/12797 | September 2022
Pages: 1600
The monograph provides a detailed and comprehensive presentation of the rich and beautiful theory of unilateral variational analysis in infinite dimensions. It is divided into two volumes named Part I and Part II. Starting with the convergence of sets and the semilimits and semicontinuities of multimappings, the first volume develops the theories of tangent cones, of subdifferentials, of convexity and duality in locally convex spaces, of extended mean value inequalities in absence of differentiability, of metric regularity, of constrained optimization problems.
The second volume is devoted to special classes of non-smooth functions and sets. It expands the theory of subsmooth functions and sets, of semiconvex functions and multimappings, of primal lower regular functions, of singularities of non-smooth mappings, of prox-regular functions and sets in general spaces, of differentiability of projection mapping and others for prox-regular sets. Both volumes I and II contain, for each chapter, extensive comments covering related developments and historical comments.
Connected area fields of the material are: optimization, optimal control, variational inequalities, differential inclusions, mechanics, economics. The book is intended for PhD students, researchers, and practitioners using unilateral variational analysis tools.
Semilimits and Semicontinuity of Multimappings
Tangent Cones and Clarke Subdifferential
Convexity and Duality in Locally Convex Spaces
Mordukhovich Limiting Normal Cone and Subdifferential
Ioffe Approximate Subdifferential
Sequential Mean Value Inequalities
Metric Regularity
Subsmooth Functions and Sets
Subdifferential Determination
Semiconvex Functions
Primal Lower Regular Functions and Prox-Regular Functions
Singular Points of Nonsmooth Functions
Non-Differentiability Points of Functions on Separable Banach Spaces
Distance Function, Metric Projection and Moreau Envelope
Prox-Regularity of Sets In Hilbert Space
Compatible Parametrization and Vial Property of Prox-Regular Sets, Exterior Sphere Condition
Differentiability of Metric Projection Onto Prox-Regular Set
Prox-Regularity of Sets In Uniformly Convex Banach Space
The book is intended for researchers, PhD students, graduate students concerned with variational analysis, practitioners using variational analysis tools.