This book is devoted to the structure of the absolute Galois groups of certain algebraic extensions of the field of rational numbers. Its main result, a theorem proved by the authors and Florian Pop in 2012, describes the absolute Galois group of distinguished semi-local algebraic (and other) extensions of the rational numbers as free products of the free profinite group on countably many generators and local Galois groups. This is an instance of a positive answer to the generalized inverse problem of Galois theory. Adopting both an arithmetic and probabilistic approach, the book carefully sets out the preliminary material needed to prove the main theorem and its supporting results. In addition, it includes a description of Melnikov's construction of free products of profinite groups and, for the first time in book form, an account of a generalization of the theory of free products of profinite groups and their subgroups. The book will be of interest to researchers in field arithmetic, Galois theory and profinite groups.

Through three editions, Cryptography: Theory and Practice, has been embraced by instructors and students alike. It offers a comprehensive primer for the subject's fundamentals while presenting the most current advances in cryptography. The authors offer comprehensive, in-depth treatment of the methods and protocols that are vital to safeguarding the seemingly infinite and increasing amount of information circulating around the world. Key Features of the Fourth Edition: New chapter on the exciting, emerging new area of post-quantum cryptography (Chapter 9). New high-level, nontechnical overview of the goals and tools of cryptography (Chapter 1). New mathematical appendix that summarizes definitions and main results on number theory and algebra (Appendix A). An expanded treatment of stream ciphers, including common design techniques along with coverage of Trivium. Interesting attacks on cryptosystems, including: padding oracle attack correlation attacks and algebraic attacks on stream ciphers attack on the DUAL-EC random bit generator that makes use of a trapdoor. A treatment of the sponge construction for hash functions and its use in the new SHA-3 hash standard. Methods of key distribution in sensor networks. The basics of visual cryptography, allowing a secure method to split a secret visual message into pieces (shares) that can later be combined to reconstruct the secret. The fundamental techniques cryptocurrencies, as used in Bitcoin and blockchain. The basics of the new methods employed in messaging protocols such as Signal, including deniability and Diffie-Hellman key ratcheting.

Linear algebra is growing in importance. 3D entertainment, animations in movies and video games are developed using linear algebra. Animated characters are generated using equations straight out of this book. Linear algebra is used to extract knowledge from the massive amounts of data generated from modern technology. The Fourth Edition of this popular text introduces linear algebra in a comprehensive, geometric, and algorithmic way. The authors start with the fundamentals in 2D and 3D, then move on to higher dimensions, expanding on the fundamentals and introducing new topics, which are necessary for many real-life applications and the development of abstract thought. Applications are introduced to motivate topics. The subtitle, A Geometry Toolbox, hints at the book's geometric approach, which is supported by many sketches and figures. Furthermore, the book covers applications of triangles, polygons, conics, and curves. Examples demonstrate each topic in action. This practical approach to a linear algebra course, whether through classroom instruction or self-study, is unique to this book. New to the Fourth Edition: Ten new application sections. A new section on change of basis. This concept now appears in several places. Chapters 14-16 on higher dimensions are notably revised. A deeper look at polynomials in the gallery of spaces. Introduces the QR decomposition and its relevance to least squares. Similarity and diagonalization are given more attention, as are eigenfunctions. A longer thread on least squares, running from orthogonal projections to a solution via SVD and the pseudoinverse. More applications for PCA have been added. More examples, exercises, and more on the kernel and general linear spaces. A list of applications has been added in Appendix A. The book gives instructors the option of tailoring the course for the primary interests of their students: mathematics, engineering, science, computer graphics, and geometric modeling.

0.1. General remarks. For any algebraic system A, the set SubA of all subsystems of A partially ordered by inclusion forms a lattice. This is the subsystem lattice of A. (In certain cases, such as that of semigroups, in order to have the right always to say that SubA is a lattice, we have to treat the empty set as a subsystem.) The study of various inter-relationships between systems and their subsystem lattices is a rather large field of investigation developed over many years. This trend was formed first in group theory; basic relevant information up to the early seventies is contained in the book [Suz] and the surveys [K Pek St], [Sad 2], [Ar Sad], there is also a quite recent book [Schm 2]. As another inspiring source, one should point out a branch of mathematics to which the book [Baer] was devoted. One of the key objects of examination in this branch is the subspace lattice of a vector space over a skew field. A more general approach deals with modules and their submodule lattices. Examining subsystem lattices for the case of modules as well as for rings and algebras (both associative and non-associative, in particular, Lie algebras) began more than thirty years ago; there are results on this subject also for lattices, Boolean algebras and some other types of algebraic systems, both concrete and general. A lot of works including several surveys have been published here.

Dedicated to the memory of Wolfgang Classical Intersection Theory (see for example Wei! [Wei]) treats the case of proper intersections, where geometrical objects (usually subvarieties of a non singular variety) intersect with the expected dimension. In 1984, two books appeared which surveyed and developed work by the individual authors, co workers and others on a refined version of Intersection Theory, treating the case of possibly improper intersections, where the intersection could have ex cess dimension. The first, by W. Fulton [Full] (recently revised in updated form), used a geometrical theory of deformation to the normal cone, more specifically, deformation to the normal bundle followed by moving the zero section to make the intersection proper; this theory was due to the author together with R. MacPherson and worked generally for intersections on algeb raic manifolds. It represents nowadays the standard approach to Intersection Theory. The second, by W. Vogel [Vogl], employed an algebraic approach to inter sections; although restricted to intersections in projective space it produced an intersection cycle by a simple and natural algorithm, thus leading to a Bezout theorem for improper intersections. It was developed together with J. Stiickrad and involved a refined version of the classical technique ofreduc tion to the diagonal: here one starts with the join variety and intersects with successive hyperplanes in general position, laying aside components which fall into the diagonal and intersecting the residual scheme with the next hyperplane; since all the hyperplanes intersect in the diagonal, the process terminates.

This book presents a guide to the extensive literature on the topic of semirings and includes a complete bibliography. It serves as a complement to the existing monographs and a point of reference to researchers and students on this topic. The literature on semirings has evolved over many years, in a variety of languages, by authors representing different schools of mathematics and working in various related fields. Recently, semiring theory has experienced rapid development, although publications are widely scattered. This survey also covers those newly emerged areas of semiring applications that have not received sufficient treatment in widely accessible monographs, as well as many lesser-known or forgotten' works.

The author has been collecting the bibliographic data for this book since 1985. Over the years, it has proved very useful for specialists. For example, J.S. Golan wrote he owed ... a special debt to Kazimierz Glazek, whose bibliography proved to be an invaluable guide to the bewildering maze of literature on semirings'. U. Hebisch and H.J. Weinert also mentioned his collection of literature had been of great assistance to them. Now updated to include publications up to the beginning of 2002, this work is available to a wide readership.

Audience: This volume is the first single reference that can guide the interested scholar or student to the relevant publications in semirings, semifields, algebraic theory of languages and automata, positive matrices and other generalisations, and ordered semigroups and groups.

This is the fourth in a series of proceedings of the Combinatorial and Additive Number Theory (CANT) conferences, based on talks from the 2019 and 2020 workshops at the City University of New York. The latter was held online due to the COVID-19 pandemic, and featured speakers from North and South America, Europe, and Asia. The 2020 Zoom conference was the largest CANT conference in terms of the number of both lectures and participants. These proceedings contain 25 peer-reviewed and edited papers on current topics in number theory. Held every year since 2003 at the CUNY Graduate Center, the workshop surveys state-of-the-art open problems in combinatorial and additive number theory and related parts of mathematics. Topics featured in this volume include sumsets, zero-sum sequences, minimal complements, analytic and prime number theory, Hausdorff dimension, combinatorial and discrete geometry, and Ramsey theory. This selection of articles will be of relevance to both researchers and graduate students interested in current progress in number theory.

This book gives an intuitive and hands-on introduction to Topological Data Analysis (TDA). Covering a wide range of topics at levels of sophistication varying from elementary (matrix algebra) to esoteric (Grothendieck spectral sequence), it offers a mirror of data science aimed at a general mathematical audience. The required algebraic background is developed in detail. The first third of the book reviews several core areas of mathematics, beginning with basic linear algebra and applications to data fitting and web search algorithms, followed by quick primers on algebra and topology. The middle third introduces algebraic topology, along with applications to sensor networks and voter ranking. The last third covers key contemporary tools in TDA: persistent and multiparameter persistent homology. Also included is a user's guide to derived functors and spectral sequences (useful but somewhat technical tools which have recently found applications in TDA), and an appendix illustrating a number of software packages used in the field. Based on a course given as part of a masters degree in statistics, the book is appropriate for graduate students.

The first systematic, self-contained presentation of a theory of arbitrary order ODEs with unbounded operator coefficients in a Hilbert or Banach space. Developed over the last 10 years by the authors, it deals with conditions of solvability, classes of uniqueness, estimates for solutions and asymptotic representations of solutions at infinity.

This book presents the latest findings on statistical inference in multivariate, multilinear and mixed linear models, providing a holistic presentation of the subject. It contains pioneering and carefully selected review contributions by experts in the field and guides the reader through topics related to estimation and testing of multivariate and mixed linear model parameters. Starting with the theory of multivariate distributions, covering identification and testing of covariance structures and means under various multivariate models, it goes on to discuss estimation in mixed linear models and their transformations. The results presented originate from the work of the research group Multivariate and Mixed Linear Models and their meetings held at the Mathematical Research and Conference Center in Bedlewo, Poland, over the last 10 years. Featuring an extensive bibliography of related publications, the book is intended for PhD students and researchers in modern statistical science who are interested in multivariate and mixed linear models.

Just suppose, for a moment, that all rings of integers in algebraic number fields were unique factorization domains, then it would be fairly easy to produce a proof of Fermat's Last Theorem, fitting, say, in the margin of this page. Unfortunately however, rings of integers are not that nice in general, so that, for centuries, math ematicians had to search for alternative proofs, a quest which culminated finally in Wiles' marvelous results - but this is history. The fact remains that modern algebraic number theory really started off with in vestigating the problem which rings of integers actually are unique factorization domains. The best approach to this question is, of course, through the general the ory of Dedekind rings, using the full power of their class group, whose vanishing is, by its very definition, equivalent to the unique factorization property. Using the fact that a Dedekind ring is essentially just a one-dimensional global version of discrete valuation rings, one easily verifies that the class group of a Dedekind ring coincides with its Picard group, thus making it into a nice, functorial invariant, which may be studied and calculated through algebraic, geometric and co homological methods. In view of the success of the use of the class group within the framework of Dedekind rings, one may wonder whether it may be applied in other contexts as well. However, for more general rings, even the definition of the class group itself causes problems."

This volume contains selected papers presented at the Second Workshop on Clifford Algebras and their Applications in Mathematical Physics. These papers range from various algebraic and analytic aspects of Clifford algebras to applications in, for example, gauge fields, relativity theory, supersymmetry and supergravity, and condensed phase physics. Included is a biography and list of publications of Mario Schenberg, who, next to Marcel Riesz, has made valuable contributions to these topics. This volume will be of interest to mathematicians working in the fields of algebra, geometry or special functions, to physicists working on quantum mechanics or supersymmetry, and to historians of mathematical physics. "

This volume offers English translations of three early works by Ernst Schroeder (1841-1902), a mathematician and logician whose philosophical ruminations and pathbreaking contributions to algebraic logic attracted the admiration and ire of figures such as Dedekind, Frege, Husserl, and C. S. Peirce. Today he still engages the sympathetic interest of logicians and philosophers. The works translated record Schroeder's journey out of algebra into algebraic logic and document his transformation of George Boole's opaque and unwieldy logical calculus into what we now recognize as Boolean algebra. Readers interested in algebraic logic and abstract algebra can look forward to a tour of the early history of those fields with a guide who was exceptionally thorough, unfailingly honest, and deeply reflective.

This volume began as the last part of a one-term graduate course given at the Fields Institute for Research in the Mathematical Sciences in the Autumn of 1993. The course was one of four associated with the 1993-94 Fields Institute programme, which I helped to organise, entitled "Artin L-functions". Published as [132]' the final chapter of the course introduced a manner in which to construct class-group valued invariants from Galois actions on the algebraic K-groups, in dimensions two and three, of number rings. These invariants were inspired by the analogous Chin burg invariants of [34], which correspond to dimensions zero and one. The classical Chinburg invariants measure the Galois structure of classical objects such as units in rings of algebraic integers. However, at the "Galois Module Structure" workshop in February 1994, discussions about my invariant (0,1 (L/ K, 3) in the notation of Chapter 5) after my lecture revealed that a number of other higher-dimensional co homological and motivic invariants of a similar nature were beginning to surface in the work of several authors. Encouraged by this trend and convinced that K-theory is the archetypical motivic cohomology theory, I gratefully took the opportunity of collaboration on computing and generalizing these K-theoretic invariants. These generalizations took several forms - local and global, for example - as I followed part of number theory and the prevalent trends in the "Galois Module Structure" arithmetic geometry.

This volume is dedicated to Bill Helton on the occasion of his sixty fifth birthday. It contains biographical material, a list of Bill's publications, a detailed survey of Bill's contributions to operator theory, optimization and control and 19 technical articles. Most of the technical articles are expository and should serve as useful introductions to many of the areas which Bill's highly original contributions have helped to shape over the last forty odd years. These include interpolation, Szegoe limit theorems, Nehari problems, trace formulas, systems and control theory, convexity, matrix completion problems, linear matrix inequalities and optimization. The book should be useful to graduate students in mathematics and engineering, as well as to faculty and individuals seeking entry level introductions and references to the indicated topics. It can also serve as a supplementary text to numerous courses in pure and applied mathematics and engineering, as well as a source book for seminars.

Services requiring parts has become a $1.5 trillion business annually worldwide, creating a tremendous incentive to manage the logistics of these parts efficiently by making planning and operational decisions in a rational and rigorous manner. This book provides a broad overview of modeling approaches and solution methodologies for addressing service parts inventory problems found in high-powered technology and aerospace applications. The focus in this work is on the management of high cost, low demand rate service parts found in multi-echelon settings.

This unique book, with its breadth of topics and mathematical treatment, begins by first demonstrating the optimality of an order-up-to policy [or (s-1, s)] in certain environments. This policy is used in the real world and studied throughout the text. The fundamental mathematical building blocks for modeling and solving applications of stochastic process and optimization techniques to service parts management problems are summarized extensively. A wide range of exact and approximate mathematical models of multi-echelon systems is developed and used in practice to estimate future inventory investment and part repair requirements.

The text may be used in a variety of courses for first-year graduate students or senior undergraduates, as well as for practitioners, requiring only a background in stochastic processes and optimization. It will serve as an excellent reference for key mathematical concepts and a guide to modeling a variety of multi-echelon service parts planning and operational problems.

The rapidly-evolving theory of vertex operator algebras provides deep insight into many important algebraic structures. Vertex operator algebras can be viewed as "complex analogues" of both Lie algebras and associative algebras. The monograph is written in a n accessible and self-contained manner, with detailed proofs and with many examples interwoven through the axiomatic treatment as motivation and applications. It will be useful for research mathematicians and theoretical physicists working the such fields as representation theory and algebraic structure sand will provide the basis for a number of graduate courses and seminars on these and related topics.

This book is about Lie group analysis of differential equations for physical and engineering problems. The topics include: -- Approximate symmetry in nonlinear physical problems -- Complex methods for Lie symmetry analysis -- Lie group classification, Symmetry analysis, and conservation laws -- Conservative difference schemes -- Hamiltonian structure and conservation laws of three-dimensional linear elasticity -- Involutive systems of partial differential equations This collection of works is written in memory of Professor Nail H. Ibragimov (1939-2018). It could be used as a reference book in differential equations in mathematics, mechanical, and electrical engineering.

This monograph is a comprehensive account of formal matrices, examining homological properties of modules over formal matrix rings and summarising the interplay between Morita contexts and K theory. While various special types of formal matrix rings have been studied for a long time from several points of view and appear in various textbooks, for instance to examine equivalences of module categories and to illustrate rings with one-sided non-symmetric properties, this particular class of rings has, so far, not been treated systematically. Exploring formal matrix rings of order 2 and introducing the notion of the determinant of a formal matrix over a commutative ring, this monograph further covers the Grothendieck and Whitehead groups of rings. Graduate students and researchers interested in ring theory, module theory and operator algebras will find this book particularly valuable. Containing numerous examples, Formal Matrices is a largely self-contained and accessible introduction to the topic, assuming a solid understanding of basic algebra.

This book is for junior/senior-level first courses in linear algebra and assumes calculus as a prerequisite. This thorough and accessible text, from one of the leading figures in the use of technology in linear algebra, gives students a challenging and broad understanding of the subject. The author infuses key concepts with their modern practical applications to offer students examples of how mathematics is used in the real world. Each chapter contains integrated worked examples and chapter tests. The book stresses the important roles geometry and visualisation play in understanding linear algebra.

The Fourier transform and the Laplace transform of a positive measure share, together with its moment sequence, a positive definiteness property which under certain regularity assumptions is characteristic for such expressions. This is formulated in exact terms in the famous theorems of Bochner, Bernstein-Widder and Hamburger. All three theorems can be viewed as special cases of a general theorem about functions qJ on abelian semigroups with involution (S, +, *) which are positive definite in the sense that the matrix (qJ(sJ + Sk" is positive definite for all finite choices of elements St, . . . , Sn from S. The three basic results mentioned above correspond to (~, +, x* = -x), ([0, 00[, +, x* = x) and (No, +, n* = n). The purpose of this book is to provide a treatment of these positive definite functions on abelian semigroups with involution. In doing so we also discuss related topics such as negative definite functions, completely mono tone functions and Hoeffding-type inequalities. We view these subjects as important ingredients of harmonic analysis on semigroups. It has been our aim, simultaneously, to write a book which can serve as a textbook for an advanced graduate course, because we feel that the notion of positive definiteness is an important and basic notion which occurs in mathematics as often as the notion of a Hilbert space.

The main part of the book is based on a one semester graduate course for students in mathematics. I have attempted to develop the theory of hyperbolic systems of differen tial equations in a systematic way, making as much use as possible ofgradient systems and their algebraic representation. However, despite the strong sim ilarities between the development of ideas here and that found in a Lie alge bras course this is not a book on Lie algebras. The order of presentation has been determined mainly by taking into account that algebraic representation and homomorphism correspondence with a full rank Lie algebra are the basic tools which require a detailed presentation. I am aware that the inclusion of the material on algebraic and homomorphism correspondence with a full rank Lie algebra is not standard in courses on the application of Lie algebras to hyperbolic equations. I think it should be. Moreover, the Lie algebraic structure plays an important role in integral representation for solutions of nonlinear control systems and stochastic differential equations yelding results that look quite different in their original setting. Finite-dimensional nonlin ear filters for stochastic differential equations and, say, decomposability of a nonlinear control system receive a common understanding in this framework."

This is a comprehensive introduction to the modular representation theory of finite groups, with an emphasis on block theory. The two volumes take into account classical results and concepts as well as some of the modern developments in the area. Volume 1 introduces the broader context, starting with general properties of finite group algebras over commutative rings, moving on to some basics in character theory and the structure theory of algebras over complete discrete valuation rings. In Volume 2, blocks of finite group algebras over complete p-local rings take centre stage, and many key results which have not appeared in a book before are treated in detail. In order to illustrate the wide range of techniques in block theory, the book concludes with chapters classifying the source algebras of blocks with cyclic and Klein four defect groups, and relating these classifications to the open conjectures that drive block theory.

Industrial Mathematics is a relatively recent discipline. It is concerned primarily with transforming technical, organizational and economic problems posed by indus try into mathematical problems; "solving" these problems byapproximative methods of analytical and/or numerical nature; and finally reinterpreting the results in terms of the original problems. In short, industrial mathematics is modelling and scientific computing of industrial problems. Industrial mathematicians are bridge-builders: they build bridges from the field of mathematics to the practical world; to do that they need to know about both sides, the problems from the companies and ideas and methods from mathematics. As mathematicians, they have to be generalists. If you enter the world of indus try, you never know which kind of problems you will encounter, and which kind of mathematical concepts and methods you will need to solve them. Hence, to be a good "industrial mathematician" you need to know a good deal of mathematics as well as ideas already common in engineering and modern mathematics with tremen dous potential for application. Mathematical concepts like wavelets, pseudorandom numbers, inverse problems, multigrid etc., introduced during the last 20 years have recently started entering the world of real applications. Industrial mathematics consists of modelling, discretization, analysis and visu alization. To make a good model, to transform the industrial problem into a math ematical one such that you can trust the prediction of the model is no easy task."

The three volumes of Interest Rate Modeling present a comprehensive and up-to-date treatment of techniques and models used in the pricing and risk management of fixed income securities. Written by two leading practitioners and seasoned industry veterans, this unique series combines finance theory, numerical methods, and approximation techniques to provide the reader with an integrated approach to the process of designing and implementing industrial-strength models for fixed income security valuation and hedging. Aiming to bridge the gap between advanced theoretical models and real-life trading applications, the pragmatic, yet rigorous, approach taken in this book will appeal to students, academics, and professionals working in quantitative finance. Volume I provides the theoretical and computational foundations for the series, emphasizing the construction of efficient grid- and simulation-based methods for contingent claims pricing. The second part of Volume I is dedicated to local-stochastic volatility modeling and to the construction of vanilla models for individual swap and Libor rates. Although the focus is eventually turned toward fixed income securities, much of the material in this volume applies to generic financial markets and will be of interest to anybody working in the general area of asset pricing.