This volume is the proceedings of the conference on Automorphic Representations, L-functions and Applications: Progress and Prospects, held at the Department of Mathematics of The Ohio State University, March 27-30, 2003, in honor of the 60th birthday of Steve Rallis. The theory of automorphic representations, automorphic L-functions and their applications to arithmetic continues to be an area of vigorous and fruitful research. The contributed papers in this volume represent many of the most recent developments and directions, including Rankin-Selberg L-functions (Bump, Ginzburg-Jiang-Rallis, Lapid-Rallis) the relative trace formula (Jacquet, Mao-Rallis) automorphic representations (Gan-Gurevich, Ginzburg-Rallis-Soudry) representation theory of p-adic groups (Baruch, Kudla-Rallis, Moeglin, Cogdell-Piatetski-Shapiro-Shahidi) p-adic methods (Harris-Li-Skinner, Vigneras), and arithmetic applications (Chinta-Friedberg-Hoffstein). The survey articles by Bump, on the Rankin-Selberg method, and by Jacquet, on the relative trace formula, should be particularly useful as an introduction to the key ideas about these important topics. This volume should be of interest both to researchers and students in the area of automorphic representations, as well as to mathematicians in other areas interested in having an overview of current developments in this important field.

Through this book, upper undergraduate mathematics majors will master a challenging yet rewarding subject, and approach advanced studies in algebra, number theory and geometry with confidence. Groups, rings and fields are covered in depth with a strong emphasis on irreducible polynomials, a fresh approach to modules and linear algebra, a fresh take on Groebner theory, and a group theoretic treatment of Rejewski's deciphering of the Enigma machine. It includes a detailed treatment of the basics on finite groups, including Sylow theory and the structure of finite abelian groups. Galois theory and its applications to polynomial equations and geometric constructions are treated in depth. Those interested in computations will appreciate the novel treatment of division algorithms. This rigorous text 'gets to the point', focusing on concisely demonstrating the concept at hand, taking a 'definitions first, examples next' approach. Exercises reinforce the main ideas of the text and encourage students' creativity.

This book is based on the extensive experience of teaching for mathematics, physics and engineering students in Russia, USA, South Africa and Sweden. The author provides students and teachers with an easy to follow textbook spanning a variety of topics. The methods of local Lie groups discussed in the book provide universal and effective method for solving nonlinear differential equations analytically. Introduction to approximate transformation groups also contained in the book helps to develop skills in constructing approximate solutions for differential equations with a small parameter.

The concepts and theorems of advanced calculus combined with related computational methods are essential to understanding nearly all areas of quantitative science. Analysis in Vector Spaces presents the central results of this classic subject through rigorous arguments, discussions, and examples. The book aims to cultivate not only knowledge of the major theoretical results, but also the geometric intuition needed for both mathematical problem-solving and modeling in the formal sciences.

The authors begin with an outline of key concepts, terminology, and notation and also provide a basic introduction to set theory, the properties of real numbers, and a review of linear algebra. An elegant approach to eigenvector problems and the spectral theorem sets the stage for later results on volume and integration. Subsequent chapters present the major results of differential and integral calculus of several variables as well as the theory of manifolds. Additional topical coverage includes:

Sets and functions

Real numbers

Vector functions

Normed vector spaces

First- and higher-order derivatives

Diffeomorphisms and manifolds

Multiple integrals

Integration on manifolds

Stokes' theorem

Basic point set topology

Numerous examples and exercises are provided in each chapter to reinforce new concepts and to illustrate how results can be applied to additional problems. Furthermore, proofs and examples are presented in a clear style that emphasizes the underlying intuitive ideas. Counterexamples are provided throughout the book to warn against possible mistakes, and extensive appendices outline the construction of real numbers, include a fundamental result about dimension, and present general results about determinants.

Assuming only a fundamental understanding of linear algebra and single variable calculus, Analysis in Vector Spaces is an excellent book for a second course in analysis for mathematics, physics, computer science, and engineering majors at the undergraduate and graduate levels. It also serves as a valuable reference for further study in any discipline that requires a firm understanding of mathematical techniques and concepts.

This book introduces the study of algebra induced by combinatorial objects called directed graphs. These graphs are used as tools in the analysis of graph-theoretic problems and in the characterization and solution of analytic problems. The book presents recent research in operator algebra theory connected with discrete and combinatorial mathematical objects. It also covers tools and methods from a variety of mathematical areas, including algebra, operator theory, and combinatorics, and offers numerous applications of fractal theory, entropy theory, K-theory, and index theory.

This monograph is concerned with the mathematical analysis of patterns which are encountered in biological systems. It summarises, expands and relates results obtained in the field during the last fifteen years. It also links the results to biological applications and highlights their relevance to phenomena in nature. Of particular concern are large-amplitude patterns far from equilibrium in biologically relevant models. The approach adopted in the monograph is based on the following paradigms: * Examine the existence of spiky steady states in reaction-diffusion systems and select as observable patterns only the stable ones * Begin by exploring spatially homogeneous two-component activator-inhibitor systems in one or two space dimensions * Extend the studies by considering extra effects or related systems, each motivated by their specific roles in developmental biology, such as spatial inhomogeneities, large reaction rates, altered boundary conditions, saturation terms, convection, many-component systems. Mathematical Aspects of Pattern Formation in Biological Systems will be of interest to graduate students and researchers who are active in reaction-diffusion systems, pattern formation and mathematical biology.

For introductory sophomore-level courses in Linear Algebra or Matrix Theory. This text presents the basic ideas of linear algebra in a manner that offers students a fine balance between abstraction/theory and computational skills. The emphasis is on not just teaching how to read a proof but also on how to write a proof.

Groups that are the product of two subgroups are of particular interest to group theorists. In what way is the structure of the product related to that of its subgroups? This monograph gives the first detailed account of the most important results that have been found about groups of this form over the past 35 years. Although the emphasis is on infinite groups, some relevant theorems about finite products of groups are also proved. The material presented will be of interest for research students and specialists in group theory. In particular, it can be used in seminars or to supplement a general group theory course. A special chapter on conjugacy and splitting theorems obtained by means of the cohomology of groups has never appeared in book form and should be of independent interest.

This book discusses the application of independent continuous mapping method in predicting and the optimization of the mechanical performance of buckling with displacement, stress and static constrains. Each model is explained by mathematical theories and followed by simulation with frequently-used softwares. With abundant project data, the book is an essential reference for mechanical engineers, structural engineers and industrial designers.

We propose here a study of 'semiexact' and 'homological' categories as a basis for a generalised homological algebra. Our aim is to extend the homological notions to deeply non-abelian situations, where satellites and spectral sequences can still be studied.This is a sequel of a book on 'Homological Algebra, The interplay of homology with distributive lattices and orthodox semigroups', published by the same Editor, but can be read independently of the latter.The previous book develops homological algebra in p-exact categories, i.e. exact categories in the sense of Puppe and Mitchell - a moderate generalisation of abelian categories that is nevertheless crucial for a theory of 'coherence' and 'universal models' of (even abelian) homological algebra. The main motivation of the present, much wider extension is that the exact sequences or spectral sequences produced by unstable homotopy theory cannot be dealt with in the previous framework.According to the present definitions, a semiexact category is a category equipped with an ideal of 'null' morphisms and provided with kernels and cokernels with respect to this ideal. A homological category satisfies some further conditions that allow the construction of subquotients and induced morphisms, in particular the homology of a chain complex or the spectral sequence of an exact couple.Extending abelian categories, and also the p-exact ones, these notions include the usual domains of homology and homotopy theories, e.g. the category of 'pairs' of topological spaces or groups; they also include their codomains, since the sequences of homotopy 'objects' for a pair of pointed spaces or a fibration can be viewed as exact sequences in a homological category, whose objects are actions of groups on pointed sets.

This volume, first published in 2000, presents a classical approach to the foundations and development of the geometry of vector fields, describing vector fields in three-dimensional Euclidean space, triply-orthogonal systems and applications in mechanics. Topics covered include Pfaffian forms, systems in n-dimensional space, and foliations and their Godbillion-Vey invariant. There is much interest in the study of geometrical objects in n-dimensional Euclidean space and this volume provides a useful and comprehensive presentation.

This proceedings volume presents selected, peer-reviewed contributions from the 26th National School on Algebra, which was held in Constanta, Romania, on August 26-September 1, 2018. The works cover three fields of mathematics: algebra, geometry and discrete mathematics, discussing the latest developments in the theory of monomial ideals, algebras of graphs and local positivity of line bundles. Whereas interactions between algebra and geometry go back at least to Hilbert, the ties to combinatorics are much more recent and are subject of immense interest at the forefront of contemporary mathematics research. Transplanting methods between different branches of mathematics has proved very fruitful in the past - for example, the application of fixed point theorems in topology to solving nonlinear differential equations in analysis. Similarly, combinatorial structures, e.g., Newton-Okounkov bodies, have led to significant advances in our understanding of the asymptotic properties of line bundles in geometry and multiplier ideals in algebra. This book is intended for advanced graduate students, young scientists and established researchers with an interest in the overlaps between different fields of mathematics. A volume for the 24th edition of this conference was previously published with Springer under the title "Multigraded Algebra and Applications" (ISBN 978-3-319-90493-1).

This manual contains completely worked-out solutions for all the odd-numbered exercises in the text.

This book describes the solution of electrodynamic boundary problems, which arose in the practical life of a designer. Only a few problems can be solved analytically and some of these solutions are given in the book, for example, the computation of a strip line in a rectangular or circular cylinder capacitance. Practical lines' configurations require computational work. As the authors' practice shows, the use of Green functions, leading to singular integral equations, is a powerful and pretty universal method to solve different boundary problems, including electrodynamic ones. The book presents the results of computations of microstrip lines on magnetized (longitudinally and transversally) ferrite and semiconductor substrates taking into account all the geometric sizes. The properties of gyrotropic media are described in the book for the reader's convenience. The geometrical shape may be practically any. The integral equations are exact and give the proper field near the edges. Actually, the use of singular integral equations reduces the experimental verification to minimum. The book will be useful for students, engineers, designers and researchers. It contains a lot of computed results, which are verified experimentally and can be used immediately.

Kasch Modules.- Compactness in Categories and Interpretations.- A Ring of Morita Context in Which Each Right Ideal is Weakly Self-injective.- Splitting Theorems and a Problem of Muller.- Decompositions of D1 Modules.- Right Cones in Groups.- On Extensions of Regular Rings of Finite Index by Central Elements.- Intersections of Modules.- Minimal Cogenerators Over Osofsky and Camillo Rings.- Uniform Modules Over Goldie Prime Serial Rings.- Co-Versus Contravariant Finiteness of Categories of Representations.- Monomials and the Lexicographic Order.- Rings Over Which Direct Sums of CS Modules Are CS.- Exchange Properties and the Total.- Local Bijective Gabriel Correspondence and Torsion Theoretic FBN Rings.- Normalizing Extensions and the Second Layer Condition.- Generators of Subgroups of Finite Index in GLm (?G).- Weak Relative Injective M-Subgenerated Modules.- Direct Product and Power Series Formations Over 2-Primal Rings.- Localization in Noetherian Rings.- Projective Dimension of Ideals in Von Neumann Regular Rings.- Homological Properties of Color Lie Superalgebras.- Indecomposable Modules Over Artinian Right Serial Rings.- Nonsingular Extending Modules.- Right Hereditary, Right Perfect Rings Are Semiprimary.- On the Endomorphism Ring of a Discrete Module: A Theorem of F. Kasch.- Nonsingular Rings with Finite Type Dimension.

This invaluable reference is the first to present the general theory of algebras of operators on a Hilbert space, and the modules over such algebras. The new theory of operator spaces is presented early on and the text assembles the basic concepts, theory and methodologies needed to equip a beginning researcher in this area. A major trend in modern mathematics, inspired largely by physics, is toward noncommutative' or quantized' phenomena. In functional analysis, this has appeared notably under the name of operator spaces', which is a variant of Banach spaces which is particularly appropriate for solving problems concerning spaces or algebras of operators on Hilbert space arising in 'noncommutative mathematics'. The category of operator spaces includes operator algebras, selfadjoint (that is, C*-algebras) or otherwise. Also, most of the important modules over operator algebras are operator spaces. A common treatment of the subjects of C*-algebras, Non-selfadjoint operator algebras, and modules over such algebras (such as Hilbert C*-modules), together under the umbrella of operator space theory, is the main topic of the book. A general theory of operator algebras, and their modules, naturally develops out of the operator space methodology. Indeed, operator space theory is a sensitive enough medium to reflect accurately many important non-commutative phenomena. Using recent advances in the field, the book shows how the underlying operator space structure captures, very precisely, the profound relations between the algebraic and the functional analytic structures involved. The rich interplay between spectral theory, operator theory, C*-algebra and von Neumann algebra techniques, and theinflux of important ideas from related disciplines, such as pure algebra, Banach space theory, Banach algebras, and abstract function theory is highlighted. Each chapter ends with a lengthy section of notes containing a wealth of additional information.

This monograph is devoted to the study of Polygroup Theory. It begins with some basic results concerning group theory and algebraic hyperstructures, which represent the most general algebraic context, in which reality can be modeled. Most results on polygroups are collected in this book. Moreover, this monograph is the first book on this theory. The volume is highly recommended to theoreticians in pure and applied mathematics.

This monograph is intended to present the fundamentals of the theory of abstract parabolic evolution equations and to show how to apply to various nonlinear dif- sion equations and systems arising in science. The theory gives us a uni?ed and s- tematic treatment for concrete nonlinear diffusion models. Three main approaches are known to the abstract parabolic evolution equations, namely, the semigroup methods, the variational methods, and the methods of using operational equations. In order to keep the volume of the monograph in reasonable length, we will focus on the semigroup methods. For other two approaches, see the related references in Bibliography. The semigroup methods, which go back to the invention of the analytic se- groups in the middle of the last century, are characterized by precise formulas representing the solutions of the Cauchy problem for evolution equations. The ?tA analytic semigroup e generated by a linear operator ?A provides directly a fundamental solution to the Cauchy problem for an autonomous linear e- dU lution equation, +AU =F(t), 0

Learn how to use the modern techniques offered by Maple V, a powerful and popular computer algebra system. The Maple V Primer: Release 4 covers all the basic topics a reader needs to know to use Maple V in its major revision encompassed in Release 4 to do algebra and calculus, solve equations, graph 2- and 3-dimensional plots, perform simple programming tasks, and prepare mathematical documents.
Every common command and function is supported by a specific example, so you won't waste time struggling with the syntax. Graphs, plots, and other Maple output are provided along with the syntax, so the user knows what to expect when she or he uses a particular command. And all the examples come with a short discussion, answering questions you might have about applying the example to your own work. This is a painless - even fun - way to learn how to use Maple V.

A generalization of Conventional Matrix Product (CMP), called the Semi-Tensor Product (STP), is proposed. It extends the CMP to two arbitrary matrices and maintains all fundamental properties of CMP. In addition, it has a pseudo-commutative property, which makes it more superior to CMP. The STP was proposed by the authors to deal with higher-dimensional data as well as multilinear mappings. After over a decade of development, STP has been proven to be a powerful tool in dealing with nonlinear and logical calculations. This book is a comprehensive introduction to the theory of STP and its various applications, including logical function, fuzzy control, Boolean networks, analysis and control of nonlinear systems, amongst others.

On Convex Combinations of Unitary Operators in C*-Algebras.- Approximately Inner Derivations, Decompositions and Vector Fields of Simple C*-Algebras.- Derivations in Commutative C*-Algebras.- Representation of Quantum Groups.- Automorphism Groups and Covariant Irreducible Representations.- Proper Actions of Groups on C*-Algebras.- On the Baum-Connes Conjecture.- On Primitive Ideal Spaces of C*-Algebras over Certain Locally Compact Groupoids.- On Sequences of Jones' Projections.- The Powers' Binary Shifts on the Hyperfinite Factor of Type II1.- Index Theory for Type III Factors.- Relative Entropy of a Fixed Point Algebra.- Jones Index Theory for C*-Algebras.- Three Tensor Norms for Operator Spaces.- Extension Problems for Maps on Operator Systems.- Multivariable Toeplitz Operators and Index Theory.- On Maximality of Analytic Subalgebras Associated with Flow in von Neumann Algebras.- Reflections Relating a von Neumann Algebra and Its Commutant.- Normal AW*-Algebras.

This monograph is devoted to the creation of a comprehensive formalism for quantitative description of polarized modes' linear interaction in modern single-mode optic fibers. The theory of random connections between polarized modes, developed in the monograph, allows calculations of the zero shift deviations for a fiber ring interferometer. The monograph addresses also the Sagnac effect and the Thomas precession. Devices such as gyroscopes, used in navigation and flight control, work based on this technology. Given the ever increasing market for navigation and air traffic, researchers and practitioners in research and industry need a fundamental and sound understanding of the principles. This work presents the underlying physical foundations.

The study of noncommutative rings is a major area in modern algebra. The structure theory of noncommutative rings was originally concerned with three parts: The study of semi-simple rings; the study of radical rings; and the construction of rings with given radical and semi-simple factor rings. Recently, this has extended to many new parts: The zero-divisor theory, containing the study of coefficients of zero-dividing polynomials and the study of annihilators over noncommutative rings, that is related to the Koethe's conjecture; the study of nil rings and Jacobson rings; the study of applying ring-theoretic properties to modules; representation theory; the study of relations between algebraic and concepts of other branches (for example, analytic and topological), etc. Thus, noncommutative rings are ubiquitous in mathematics, and occur in numerous sciences.This volume consists of a collection of original articles refereed by world experts that was presented at the Sixth China-Japan-Korea International Conference on Ring Theory. These articles exhibit new ideas, tools and techniques needed for successful research and investigation in noncommutative ring theory, and show the trend of current research. It is a useful resource book for beginners and advanced experts in ring theory.

In this book we want to explore aspects of coherence in homological algebra, that already appear in the classical situation of abelian groups or abelian categories. Lattices of subobjects are shown to play an important role in the study of homological systems, from simple chain complexes to all the structures that give rise to spectral sequences. A parallel role is played by semigroups of endorelations.These links rest on the fact that many such systems, but not all of them, live in distributive sublattices of the modular lattices of subobjects of the system.The property of distributivity allows one to work with induced morphisms in an automatically consistent way, as we prove in a 'Coherence Theorem for homological algebra'. (On the contrary, a 'non-distributive' homological structure like the bifiltered chain complex can easily lead to inconsistency, if one explores the interaction of its two spectral sequences farther than it is normally done.)The same property of distributivity also permits representations of homological structures by means of sets and lattices of subsets, yielding a precise foundation for the heuristic tool of Zeeman diagrams as universal models of spectral sequences.We thus establish an effective method of working with spectral sequences, called 'crossword chasing', that can often replace the usual complicated algebraic tools and be of much help to readers that want to apply spectral sequences in any field.

Outliers play an important, though underestimated, role in control engineering. Traditionally they are unseen and neglected. In opposition, industrial practice gives frequent examples of their existence and their mostly negative impacts on the control quality. The origin of outliers is never fully known. Some of them are generated externally to the process (exogenous), like for instance erroneous observations, data corrupted by control systems or the effect of human intervention. Such outliers appear occasionally with some unknow probability shifting real value often to some strange and nonsense value. They are frequently called deviants, anomalies or contaminants. In most cases we are interested in their detection and removal. However, there exists the second kind of outliers. Quite often strange looking data observations are not artificial data occurrences. They may be just representatives of the underlying generation mechanism being inseparable internal part of the process (endogenous outliers). In such a case they are not wrong and should be treated with cautiousness, as they may include important information about the dynamic nature of the process. As such they cannot be neglected nor simply removed. The Outlier should be detected, labelled and suitably treated. These activities cannot be performed without proper analytical tools and modeling approaches. There are dozens of methods proposed by scientists, starting from Gaussian-based statistical scoring up to data mining artificial intelligence tools. The research presented in this book presents novel approach incorporating non-Gaussian statistical tools and fractional calculus approach revealing new data analytics applied to this important and challenging task. The proposed book includes a collection of contributions addressing different yet cohesive subjects, like dynamic modelling, classical control, advanced control, fractional calculus, statistical analytics focused on an ultimate goal: robust and outlier-proof analysis. All studied problems show that outliers play an important role and classical methods, in which outlier are not taken into account, do not give good results. Applications from different engineering areas are considered such as semiconductor process control and monitoring, MIMO peltier temperature control and health monitoring, networked control systems, and etc.