This book introduces a new geometric vision of continued fractions. It covers several applications to questions related to such areas as Diophantine approximation, algebraic number theory, and toric geometry. The second edition now includes a geometric approach to Gauss Reduction Theory, classification of integer regular polygons and some further new subjects. Traditionally a subject of number theory, continued fractions appear in dynamical systems, algebraic geometry, topology, and even celestial mechanics. The rise of computational geometry has resulted in renewed interest in multidimensional generalizations of continued fractions. Numerous classical theorems have been extended to the multidimensional case, casting light on phenomena in diverse areas of mathematics. The reader will find an overview of current progress in the geometric theory of multidimensional continued fractions accompanied by currently open problems. Whenever possible, we illustrate geometric constructions with figures and examples. Each chapter has exercises useful for undergraduate or graduate courses.

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.

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.

Invitation to Linear Algebra is an informative, clearly written, flexible textbook for instructors and students. Based on over 30 years of experience as a mathematics professor, the author invites students to develop a more informed understanding of complex algebraic concepts using innovative, easy-to-follow methods. The book is organized into lessons rather than chapters. This limits the size of the mathematical morsels that students must digest, making it easier for instructors to budget class time. Each definition is carefully explained with detailed proofs of key theorems, including motivation for each step. This makes the book more flexible, allowing instructors to choose material that reflects their and their students' interests. A larger than normal amount of exercises illustrate how linear and nonlinear algebra apply in the students' areas of study. Features The book's unique lesson format enables students to better understand algebraic concepts Students will learn key elements of linear algebra in an enjoyable fashion Large number of exercises illustrate the applications of the course material Allows instructors to create a course around individual lessons Detailed solutions and hints are provided to selected exercises

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.

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 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.

Complex analysis is found in many areas of applied mathematics, from fluid mechanics, thermodynamics, signal processing, control theory, mechanical and electrical engineering to quantum mechanics, among others. And of course, it is a fundamental branch of pure mathematics. The coverage in this text includes advanced topics that are not always considered in more elementary texts. These topics include, a detailed treatment of univalent functions, harmonic functions, subharmonic and superharmonic functions, Nevanlinna theory, normal families, hyperbolic geometry, iteration of rational functions, and analytic number theory. As well, the text includes in depth discussions of the Dirichlet Problem, Green's function, Riemann Hypothesis, and the Laplace transform. Some beautiful color illustrations supplement the text of this most elegant subject.

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.

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.

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.

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."

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.

Considered a classic by many, A First Course in Abstract Algebra is an in-depth introduction to abstract algebra. Focused on groups, rings and fields, this text gives students a firm foundation for more specialized work by emphasizing an understanding of the nature of algebraic structures.

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.

Introduction to Mathematical Modeling helps students master the processes used by scientists and engineers to model real-world problems, including the challenges posed by space exploration, climate change, energy sustainability, chaotic dynamical systems and random processes. Primarily intended for students with a working knowledge of calculus but minimal training in computer programming in a first course on modeling, the more advanced topics in the book are also useful for advanced undergraduate and graduate students seeking to get to grips with the analytical, numerical, and visual aspects of mathematical modeling, as well as the approximations and abstractions needed for the creation of a viable model.

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.

Spaces of homogeneous type were introduced as a generalization to the Euclidean space and serve as a suffi cient setting in which one can generalize the classical isotropic Harmonic analysis and function space theory. This setting is sometimes too general, and the theory is limited. Here, we present a set of fl exible ellipsoid covers of n that replace the Euclidean balls and support a generalization of the theory with fewer limitations.

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 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.

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.