Paperback
Format: Paperback / softback, 174 pages, height x width: 240x168 mm, weight: 328 g, 35 Tables, color;
2 Illustrations, color; 1 Illustrations, black and white; XII, 174 p. 3 illus., 2 illus. in color.,
Series: Synthesis Lectures on Mathematics & Statistics
Pub. Date: 29-Oct-2023
ISBN-13: 9783031170713
This textbook provides an introduction to methods for solving nonlinear partial differential equations (NLPDEs). After the introduction of several PDEs drawn from science and engineering, readers are introduced to techniques to obtain exact solutions of NLPDEs. The chapters include the following topics: Nonlinear PDEs are Everywhere; Differential Substitutions; Point and Contact Transformations; First Integrals; and Functional Separability. Readers are guided through these chapters and are provided with several detailed examples. Each chapter ends with a series of exercises illustrating the material presented in each chapter. This Second Edition includes a new method of generating contact transformations and focuses on a solution method (parametric Legendre transformations) to solve a particular class of two nonlinear PDEs.
Nonlinear PDEs are Everywhere.- Differential Substitutions.- Point and
Contact Transformations.- First Integrals.- Functional Separability.
Format: Paperback / softback, 233 pages, height x width: 235x155 mm, weight: 385 g,
1 Illustrations, black and white; XIV, 233 p. 1 illus.,
Series: Algebra and Applications 30
Pub. Date: 24-Oct-2023
ISBN-13: 9783031122903
This book offers an original introduction to the representation theory of algebras, suitable for beginning researchers in algebra. It includes many results and techniques not usually covered in introductory books, some of which appear here for the first time in book form.
The exposition employs methods from linear algebra (spectral methods and quadratic forms), as well as categorical and homological methods (module categories, Galois coverings, Hochschild cohomology) to present classical aspects of ring theory under new light. This includes topics such as rings with several objects, the Harada?Sai lemma, chain conditions, and Auslander?Reiten theory. Noteworthy and significant results covered in the book include the Brauer?Thrall conjectures, Drozdfs theorem, and criteria to distinguish tame from wild algebras.
This text may serve as the basis for a second graduate course in algebra or as an introduction to research in the field of representation theory of algebras. The originality of the exposition and the wealth of topics covered also make it a valuable resource for more established researchers.
1 Introduction and First Examples.- 2 A Categorical Approach.- 3 Constructive Methods.- 4 Spectral Methods in Representation Theory.- 5 Group Actions on Algebras and Module Categories.- 6 Reflections and Weyl Groups.- 7 Simply Connected Algebras.- 8 Degenerations of Algebras.- 9 Further Comments.
Paperback
Format: Paperback / softback, 272 pages, height x width: 235x155 mm, weight: 444 g,
1 Illustrations, black and white; XVI, 272 p. 1 illus
Pub. Date: 24-Oct-2023
ISBN-13: 9783031121067
This book emphasizes the use of stochastic orders as motivational tools for developing new statistical procedures. Stochastic orders have found useful applications in many disciplines, including reliability theory, survival analysis, risk theory, finance, nonparametric methods, economics and actuarial science. Written by a statistician, this volume clarifies the connection between stochastic orders and nonparametric methods.
The importance of order statistics and spacings is well recognized. Classically, they mainly focus on the case when the observations are independent and identically distributed, however, several new developments have extended the comparison of order statistics to the case of non-identically distributed or non-independent observations. In addition to giving a detailed discussion of various topics in the general area of stochastic orders, a substantial part of the book is devoted to recent research on stochastic comparisons of order statistics and spacings, including a long chapter on dependence among them.
The book will be useful for graduate students and researchers in statistics, economics, actuarial science and other related disciplines. In particular, with close to 300 references, it will be a valuable resource for reliability theorists, applied probabilists and statisticians. Readers are expected to have taken a first-year graduate level course in mathematical statistics or in applied probability.
Introduction and Preliminaries.- Magnitude Orders.- Variability Orders.- Skewness and Relative Aging Orders.- Dependence Orders.- Stochastic comparisons of sample spacings.- Dependence among order statistics and spacings.- Stochastic Comparisons of Weighted Sums of Random Variables.- Stochastic comparisons of mixtures of distributions.
Paperback
Format: Paperback / softback, 406 pages, height x width: 235x155 mm, weight: 755 g,
1 Illustrations, color; 5 Illustrations, black and white; XIV, 406 p. 6 illus., 1 illus. in color.
Series: Graduate Texts in Mathematics 295
Pub. Date: 31-Oct-2023
ISBN-13: 9783031142079
This textbook introduces readers to the fundamental notions of modern probability theory. The only prerequisite is a working knowledge in real analysis. Highlighting the connections between martingales and Markov chains on one hand, and Brownian motion and harmonic functions on the other, this book provides an introduction to the rich interplay between probability and other areas of analysis.
Arranged into three parts, the book begins with a rigorous treatment of measure theory, with applications to probability in mind. The second part of the book focuses on the basic concepts of probability theory such as random variables, independence, conditional expectation, and the different types of convergence of random variables. In the third part, in which all chapters can be read independently, the reader will encounter three important classes of stochastic processes: discrete-time martingales, countable state-space Markov chains, and Brownian motion. Each chapter ends with a selection of illuminating exercises of varying difficulty. Some basic facts from functional analysis, in particular on Hilbert and Banach spaces, are included in the appendix.
Measure Theory, Probability, and Stochastic Processes is an ideal text for readers seeking a thorough understanding of basic probability theory. Students interested in learning more about Brownian motion, and other continuous-time stochastic processes, may continue reading the authorfs more advanced textbook in the same series (GTM 274).
Part I. Measure Theory.
Chapter
1. Measurable Spaces.
Chapter
2.
Integration of Measurable Functions.
Chapter
3. Construction of Measures.-
Chapter
4. Lp Spaces.
Chapter
5. Product Measure.
Chapter
6. Signed
Measures.
Chapter
7. Change of Variables.- Part II. Probability Theory.-
Chapter
8. Foundations of Probability Theory.
Chapter
9. Independence.-
Chapter
10. Convergence of Random Variables.
Chapter
11. Conditioning.- Part
III. Stochastic Processes.
Chapter
12. Theory of Martingales.
Chapter
13.
Markov Chains.
Chapter
14. Brownian Motion.
Format: Paperback / softback, 327 pages, height x width: 235x155 mm, weight: 528 g,
1 Illustrations, black and white; XV, 327 p. 1 illus.,
Series: Studies in Universal Logic
Pub. Date: 03-Nov-2023
ISBN-13: 9783031148897
This book gives a comprehensive introduction to Universal Algebraic Logic. The three main themes are (i) universal logic and the question of what logic is, (ii) duality theories between the world of logics and the world of algebra, and (iii) Tarskian algebraic logic proper including algebras of relations of various ranks, cylindric algebras, relation algebras, polyadic algebras and other kinds of algebras of logic. One of the strengths of our approach is that it is directly applicable to a wide range of logics including not only propositional logics but also e.g. classical first order logic and other quantifier logics. Following the Tarskian tradition, besides the connections between logic and algebra, related logical connections with geometry and eventually spacetime geometry leading up to relativity are also part of the perspective of the book. Besides Tarskian algebraizations of logics, category theoretical perspectives are also touched upon.
This book, apart from being a monograph containing state of the art results in algebraic logic, can be used as the basis for a number of different courses intended for both novices and more experienced students of logic, mathematics, or philosophy. For instance, the first two chapters can be used in their own right as a crash course in Universal Algebra.
Preface.- Acknowledgement.- 1 Notation, Elementary Concepts.-1.1 Sets, classes, tuples, simple operations on sets.-1.2 Binary relations, equivalence relations, functions.- 1.3 Orderings, ordinals, cardinals.- 1.4 Sequences.- 1.5 Direct product of families of sets.- 1.6 Relations of higher ranks.- 1.7 Closure systems.- 1.8 First order logic (FOL).- 2 Basics from Universal Algebra.-2.1 Examples for algebras.- 2.2 Building new algebras from old ones (operations on algebras).- 2.2.1 Subalgebra.- 2.2.2 Homomorphic image.- 2.2.3 A distinguished example: Lattices.- 2.2.4 Congruence relation.- 2.2.5 Cartesian product, direct decomposition.- 2.2.6 Subdirect decomposition.- 2.2.7 Ultraproduct, reduced product.- 2.3 Categories.- 2.4 Variety characterization, quasi-variety characterization.- 2.5 Free algebras.- 2.6 Boolean Algebras.- 2.7 Discriminator varieties.- 2.8 Boas and BAOs.- 3 General framework and algebraization.- 3.1 Defining the framework for studying logics.- 3.2 Concrete logics in the new framework.- 3.3 Algebraization.- 3.3.1 Having connectives, formula algebra.- 3.3.2 Compositionality, tautological formula algebra.- 3.3.3 Algebraic counterparts of a logic.- 3.3.4 Substitution properties.- 3.3.5 Filter property.- 3.3.6 General Logics.- 3.4 Connections with Abstract Algebraic Logic, Abstract Model Theory and Institutions.- 4 Bridge between logic and algebra.- 4.1 Algebraic characterization of compactness properties.- 4.2 Algebraic characterizations of completeness properties.- 4.2.1 Hilbert-type inference systems.- 4.2.2 Completeness and soundness.- 4.3 Algebraic characterization of definability properties.- 4.3.1 Syntactical Beth definability property.- 4.3.2 Beth definability property.- 4.3.3 Local Beth definability property.-4.3.4 Weak Beth definability property.- 4.4 Algebraic characterization of interpolation properties.- 4.4.1 Interpolation properties.- 4.4.2 Amalgamation and interpolation properties.- 4.5 Decidability.- 4.6 Godel's incompleteness property.- 5 Applying the machinery: Examples.- 5.1 Classical propositional logic LC.- 5.2 Arrow logic L_{REL}.- 5.3 Finite-variable fragments of first-order logic, with substituted atomic formulas, L'_n.- 5.4 n-variable fragment L_n of rst-order logic, for n \le \omega.- 5.5 First-order logic with nonstandard semantics, L^{a}_{n}.- 5.6 Variable-dependent first-order logic, L^{vd}_{n}.- 5.7 First-order logic, ranked version, L^{ranked}_{FOL}.- 5.8 First-order logic, rank-free (or type-less) version, L^{rf}_{FOL}.- 6 Generalizations and new kinds of logics.- 6.1 Generalizations.- 6.2 New kinds of logics.- 7 Appendix: Algebras of relations.- 7.1 Algebras of binary relations.- 7.2 Algebras of unitary relations.- 7.3 All unitary relations together.- Bibliography.- Index.- Index of symbols.
Format: Paperback / softback, 388 pages, height x width: 235x155 mm, weight: 611 g, 15 Tables,
color; 15 Illustrations, color; 6 Illustrations, black and white; X, 388 p. 21 illus., 15 illus. in color.
Pub. Date: 02-Nov-2023
ISBN-13: 9783031053337
Analysis at Large is dedicated to Jean Bourgain whose research has deeply influenced the mathematics discipline, particularly in analysis and its interconnections with other fields. In this volume, the contributions made by renowned experts present both research and surveys on a wide spectrum of subjects, each of which pay tribute to a true mathematical pioneer. Examples of topics discussed in this book include Bourgains discretized sum-product theorem, his work in nonlinear dispersive equations, the slicing problem by Bourgain, harmonious sets, the joint spectral radius, equidistribution of affine random walks, Cartan covers and doubling Bernstein type inequalities, a weighted Prekopa-Leindler inequality and sumsets with quasicubes, the fractal uncertainty principle for the Walsh-Fourier transform, the continuous formulation of shallow neural networks as Wasserstein-type gradient flows, logarithmic quantum dynamical bounds for arithmetically defined ergodic Schrodinger operators, polynomial equations in subgroups, trace sets of restricted continued fraction semigroups, exponential sums, twisted multiplicativity and moments, the ternary Goldbach problem, as well as the multiplicative group generated by two primes in Z/QZ. It is hoped that this volume will inspire further research in the areas of analysis treated in this book and also provide direction and guidance for upcoming developments in this essential subject of mathematics.
On the joint spectral radius (E. Breuillard).- The failure of the
fractal uncertainty principle for the Walsh-Fourier transform (C. Demeter).-
The continuous formulation of shallow neural networks as Wasserstein-type
gradient flows (X. Fern?ndez-Real).- On the Origins, Nature and Impact of
Bourgain's Discretized Sum-Product Theorem (A. Gamburd).- Cartan Covers and
Doubling Bernstein Type Inequalities on Analytic Subsets of C2 (M.
Goldstein).- A Weighted Prekopa-Leindler inequality and sumsets with
quasicubes (B. Green).- Equidistribution of affine random walks on some
nilmanifolds (E. Lindenstrauss).- Logarithmic quantum dynamical bounds for
arithmetically defined ergodic Schrodinger operators with smooth potentials
(S. Jitomirskaya).- The slicing problem by Bourgain (B. Klartag).- On the
work of Jean Bourgain in nonlinear dispersive equations (E. Kenig).- On Trace
sets of restricted continued fraction semigroups (A. Kontorovich).-
Polynomial Equations in Subgroups and Applications (V. Konyagin).-
Exponential sums, twisted multiplicativity and moments (E. Kowalski).- The
ternary Goldbach problem with a missing digit and other primes of special
types (Th. Rassias).- A note on harmonious sets (Y. Franc cois Meyer).- On
the multiplicative group generated by two primes in Z/QZ (P. Varju).