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.

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.

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 updated edition of this classic book is devoted to ordinary representation theory and is addressed to finite group theorists intending to study and apply character theory. It contains many exercises and examples, and the list of problems contains a number of open questions.

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."

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."

This book contains select chapters on support vector algorithms from different perspectives, including mathematical background, properties of various kernel functions, and several applications. The main focus of this book is on orthogonal kernel functions, and the properties of the classical kernel functions-Chebyshev, Legendre, Gegenbauer, and Jacobi-are reviewed in some chapters. Moreover, the fractional form of these kernel functions is introduced in the same chapters, and for ease of use for these kernel functions, a tutorial on a Python package named ORSVM is presented. The book also exhibits a variety of applications for support vector algorithms, and in addition to the classification, these algorithms along with the introduced kernel functions are utilized for solving ordinary, partial, integro, and fractional differential equations. On the other hand, nowadays, the real-time and big data applications of support vector algorithms are growing. Consequently, the Compute Unified Device Architecture (CUDA) parallelizing the procedure of support vector algorithms based on orthogonal kernel functions is presented. The book sheds light on how to use support vector algorithms based on orthogonal kernel functions in different situations and gives a significant perspective to all machine learning and scientific machine learning researchers all around the world to utilize fractional orthogonal kernel functions in their pattern recognition or scientific computing problems.

The aim of this work is to present several topics in time-frequency analysis as subjects in abelian group theory. The algebraic point of view pre dominates as questions of convergence are not considered. Our approach emphasizes the unifying role played by group structures on the development of theory and algorithms. This book consists of two main parts. The first treats Weyl-Heisenberg representations over finite abelian groups and the second deals with mul tirate filter structures over free abelian groups of finite rank. In both, the methods are dimensionless and coordinate-free and apply to one and multidimensional problems. The selection of topics is not motivated by mathematical necessity but rather by simplicity. We could have developed Weyl-Heisenberg theory over free abelian groups of finite rank or more generally developed both topics over locally compact abelian groups. However, except for having to dis cuss conditions for convergence, Haar measures, and other standard topics from analysis the underlying structures would essentially be the same. A re cent collection of papers 17] provides an excellent review of time-frequency analysis over locally compact abelian groups. A further reason for limiting the scope of generality is that our results can be immediately applied to the design of algorithms and codes for time frequency processing."

This book defines and studies a combinatorial object called the pedigree and develops the theory for optimising a linear function over the convex hull of pedigrees (the Pedigree polytope). A strongly polynomial algorithm implementing the framework given in the book for checking membership in the pedigree polytope is a major contribution. This book challenges the popularly held belief in computer science that a problem included in the NP-complete class may not have a polynomial algorithm to solve. By showing STSP has a polynomial algorithm, this book settles the P vs NP question. This book has illustrative examples, figures, and easily accessible proofs for showing this unexpected result. This book introduces novel constructions and ideas previously not used in the literature. Another interesting feature of this book is it uses basic max-flow and linear multicommodity flow algorithms and concepts in these proofs establishing efficient membership checking for the pedigree polytope. Chapters 3-7 can be adopted to give a course on Efficient Combinatorial Optimization. This book is the culmination of the author's research that started in 1982 through a presentation on a new formulation of STSP at the XIth International Symposium on Mathematical Programming at Bonn.

This comprehensive book covers both long-standing results in the theory of polynomials and recent developments. After initial chapters on the location and separation of roots and on irreducibility criteria, the book covers more specialized polynomials.

Each undergraduate course of algebra begins with basic notions and results concerning groups, rings, modules and linear algebra. That is, it begins with simple notions and simple results. Our intention was to provide a collection of exercises which cover only the easy part of ring theory, what we have named the "Basics of Ring Theory." This seems to be the part each student or beginner in ring theory (or even algebra) should know - but surely trying to solve as many of these exercises as possible independently. As difficult (or impossible) as this may seem, we have made every effort to avoid modules, lattices and field extensions in this collection and to remain in the ring area as much as possible. A brief look at the bibliography obviously shows that we don't claim much originality (one could name this the folklore of ring theory) for the statements of the exercises we have chosen (but this was a difficult task: indeed, the 28 titles contain approximatively 15.000 problems and our collection contains only 346). The real value of our book is the part which contains all the solutions of these exercises. We have tried to draw up these solutions as detailed as possible, so that each beginner can progress without skilled help. The book is divided in two parts each consisting of seventeen chapters, the first part containing the exercises and the second part the solutions.

This volume summarizes recent developments in the topological and algebraic structures in fuzzy sets and may be rightly viewed as a continuation of the stan dardization of the mathematics of fuzzy sets established in the "Handbook," namely the Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory, Volume 3 of The Handbooks of Fuzzy Sets Series (Kluwer Academic Publish ers, 1999). Many of the topological chapters of the present work are not only based upon the foundations and notation for topology laid down in the Hand book, but also upon Handbook developments in convergence, uniform spaces, compactness, separation axioms, and canonical examples; and thus this work is, with respect to topology, a continuation of the standardization of the Hand book. At the same time, this work significantly complements the Handbook in regard to algebraic structures. Thus the present volume is an extension of the content and role of the Handbook as a reference work. On the other hand, this volume, even as the Handbook, is a culmination of mathematical developments motivated by the renowned International Sem inar on Fuzzy Set Theory, also known as the Linz Seminar, held annually in Linz, Austria. Much of the material of this volume is related to the Twenti eth Seminar held in February 1999, material for which the Seminar played a crucial and stimulating role, especially in providing feedback, connections, and the necessary screening of ideas."

Tensor Analysis and Nonlinear Tensor Functions embraces the basic fields of tensor calculus: tensor algebra, tensor analysis, tensor description of curves and surfaces, tensor integral calculus, the basis of tensor calculus in Riemannian spaces and affinely connected spaces, - which are used in mechanics and electrodynamics of continua, crystallophysics, quantum chemistry etc.

The book suggests a new approach to definition of a tensor in space R3, which allows us to show a geometric representation of a tensor and operations on tensors. Based on this approach, the author gives a mathematically rigorous definition of a tensor as an individual object in arbitrary linear, Riemannian and other spaces for the first time.

It is the first book to present a systematized theory of tensor invariants, a theory of nonlinear anisotropic tensor functions and a theory of indifferent tensors describing the physical properties of continua.

The book will be useful for students and postgraduates of mathematical, mechanical engineering and physical departments of universities and also for investigators and academic scientists working in continuum mechanics, solid physics, general relativity, crystallophysics, quantum chemistry of solids and material science.

This book is concerned with the role played by modules of infinite length when dealing with problems in the representation theory of groups and algebras, but also in topology and geometry; it shows the intriguing interplay between finite and infinite length modules.
The volume presents the invited lectures of a conference devoted to "Infinite Length Modules," held at Bielefeld in September 1998, which brought together experts from quite different schools in order to survey surprising relations between algebra, topology and geometry. Some additional reports have been included in order to establish a unified picture. The collection of articles, written by well-known experts from all parts of the world, is conceived as a sort of handbook which provides an easy access to the present state of knowledge and its aim is to stimulate further development.

We dedicate this volume to Professor Parimala on the occasion of her 60th birthday. It contains a variety of papers related to the themes of her research. Parimala's rst striking result was a counterexample to a quadratic analogue of Serre's conjecture (Bulletin of the American Mathematical Society, 1976). Her in uence has cont- ued through her tenure at the Tata Institute of Fundamental Research in Mumbai (1976-2006),and now her time at Emory University in Atlanta (2005-present). A conference was held from 30 December 2008 to 4 January 2009, at the U- versity of Hyderabad, India, to celebrate Parimala's 60th birthday (see the conf- ence's Web site at http://mathstat.uohyd.ernet.in/conf/quadforms2008). The or- nizing committee consisted of J.-L. Colliot-Thel ' en ' e, Skip Garibaldi, R. Sujatha, and V. Suresh. The present volume is an outcome of this event. We would like to thank all the participants of the conference, the authors who have contributed to this volume, and the referees who carefully examined the s- mitted papers. We would also like to thank Springer-Verlag for readily accepting to publish the volume. In addition, the other three editors of the volume would like to place on record their deep appreciation of Skip Garibaldi's untiring efforts toward the nal publication.

This book is designed to serve as a textbook for courses offered to undergraduate and postgraduate students enrolled in Mathematics. Using elementary row operations and Gram-Schmidt orthogonalization as basic tools the text develops characterization of equivalence and similarity, and various factorizations such as rank factorization, OR-factorization, Schurtriangularization, Diagonalization of normal matrices, Jordan decomposition, singular value decomposition, and polar decomposition. Along with Gauss-Jordan elimination for linear systems, it also discusses best approximations and least-squares solutions. The book includes norms on matrices as a means to deal with iterative solutions of linear systems and exponential of a matrix. The topics in the book are dealt with in a lively manner. Each section of the book has exercises to reinforce the concepts, and problems have been added at the end of each chapter. Most of these problems are theoretical, and they do not fit into the running text linearly. The detailed coverage and pedagogical tools make this an ideal textbook for students and researchers enrolled in senior undergraduate and beginning postgraduate mathematics courses.

Positivity is one of the most basic mathematical concepts. In many areas of mathematics (like analysis, real algebraic geometry, functional analysis, etc.) it shows up as positivity of a polynomial on a certain subset of R^n which itself is often given by polynomial inequalities. The main objective of the book is to give useful characterizations of such polynomials. It takes as starting point Hilbert's 17th Problem from 1900 and explains how E. Artin's solution of that problem eventually led to the development of real algebra towards the end of the 20th century. Beyond basic knowledge in algebra, only valuation theory as explained in the appendix is needed. Thus the monograph can also serve as the basis for a 2-semester course in real algebra.