An original motivation for algebraic geometry was to understand curves and surfaces in three dimensions. Recent theoretical and technological advances in areas such as robotics, computer vision, computer-aided geometric design and molecular biology, together with the increased availability of computational resources, have brought these original questions once more into the forefront of research. One particular challenge is to combine applicable methods from algebraic geometry with proven techniques from piecewise-linear computational geometry (such as Voronoi diagrams and hyperplane arrangements) to develop tools for treating curved objects. These research efforts may be summarized under the term nonlinear computational geometry. This volume grew out of an IMA workshop on Nonlinear Computational Geometry in May/June 2007 (organized by I.Z. Emiris, R. Goldman, F. Sottile, T. Theobald) which gathered leading experts in this emerging field. The research and expository articles in the volume are intended to provide an overview of nonlinear computational geometry. Since the topic involves computational geometry, algebraic geometry, and geometric modeling, the volume has contributions from all of these areas. By addressing a broad range of issues from purely theoretical and algorithmic problems, to implementation and practical applications this volume conveys the spirit of the IMA workshop.

When soliton theory, based on water waves, plasmas, fiber optics etc., was developing in the 1960-1970 era it seemed that perhaps KdV (and a few other equations) were really rather special in the set of all interesting partial differential equations. As it turns out, although integrable systems are still special, the mathematical interaction of integrable systems theory with virtually all branches of mathematics (and with many currently developing areas of theoretical physics) illustrates the importance of this area. This book concentrates on developing the theme of the tau function. KdV and KP equations are treated extensively, with material on NLS and AKNS systems, and in following the tau function theme one is led to conformal field theory, strings, and other topics in physics. The extensive list of references contains about 1000 entries.

This book provides a detailed and largely self-contained description of various classical and new results on solvability and unsolvability of equations in explicit form. In particular, it offers a complete exposition of the relatively new area of topological Galois theory, initiated by the author. Applications of Galois theory to solvability of algebraic equations by radicals, basics of Picard-Vessiot theory, and Liouville's results on the class of functions representable by quadratures are also discussed. A unique feature of this book is that recent results are presented in the same elementary manner as classical Galois theory, which will make the book useful and interesting to readers with varied backgrounds in mathematics, from undergraduate students to researchers. In this English-language edition, extra material has been added (Appendices A-D), the last two of which were written jointly with Yura Burda.

This book consists of invited survey articles and research papers in the scientific areas of the "International Workshop on Operator Algebras, Operator Theory and Applications," which was held in Lisbon in July 2016. Reflecting recent developments in the field of algebras of operators, operator theory and matrix theory, it particularly focuses on groupoid algebras and Fredholm conditions, algebras of approximation sequences, C* algebras of convolution type operators, index theorems, spectrum and numerical range of operators, extreme supercharacters of infinite groups, quantum dynamics and operator algebras, and inverse eigenvalue problems. Establishing bridges between the three related areas of operator algebras, operator theory, and matrix theory, the book is aimed at researchers and graduate students who use results from these areas.

This book gives a self-contained introduction to the theory of lambda-rings and closely related topics, including Witt vectors, integer-valued polynomials, and binomial rings. Many of the purely algebraic results about lambda-rings presented in this book have never appeared in book form before. This book concludes with a chapter on open problems related to lambda-rings.

This book is part of Algebra and Geometry, a subject within the SCIENCES collection published by ISTE and Wiley, and the second of three volumes specifically focusing on algebra and its applications. Algebra and Applications 2 centers on the increasing role played by combinatorial algebra and Hopf algebras, including an overview of the basic theories on non-associative algebras, operads and (combinatorial) Hopf algebras. The chapters are written by recognized experts in the field, providing insight into new trends, as well as a comprehensive introduction to the theory. The book incorporates self-contained surveys with the main results, applications and perspectives. The chapters in this volume cover a wide variety of algebraic structures and their related topics. Alongside the focal topic of combinatorial algebra and Hopf algebras, non-associative algebraic structures in iterated integrals, chronological calculus, differential equations, numerical methods, control theory, non-commutative symmetric functions, Lie series, descent algebras, Butcher groups, chronological algebras, Magnus expansions and Rota-Baxter algebras are explored. Algebra and Applications 2 is of great interest to graduate students and researchers. Each chapter combines some of the features of both a graduate level textbook and of research level surveys.

This book features papers presented during a special session on algebra, functional analysis, complex analysis, and pluripotential theory. Research articles focus on topics such as slow convergence, spectral expansion, holomorphic extension, m-subharmonic functions, pseudo-Galilean group, involutive algebra, Log-integrable measurable functions, Gibbs measures, harmonic and analytic functions, local automorphisms, Lie algebras, and Leibniz algebras. Many of the papers address the theory of harmonic functions, and the book includes a number of extensive survey papers. Graduate and researchers interested in functional analysis, complex analysis, operator algebras and non-associative algebras will find this book relevant to their studies. The special session was part of the second USA-Uzbekistan Conference on Analysis and Mathematical Physics held on August 8-12, 2017 at Urgench State University (Uzbekistan). The conference encouraged communication and future collaboration among U.S. mathematicians and their counterparts in Uzbekistan and other countries. Main themes included algebra and functional analysis, dynamical systems, mathematical physics and partial differential equations, probability theory and mathematical statistics, and pluripotential theory. A number of significant, recently established results were disseminated at the conference's scheduled plenary talks, while invited talks presented a broad spectrum of findings in several sessions. Based on a different session from the conference, Differential Equations and Dynamical Systems is also published in the Springer Proceedings in Mathematics & Statistics Series.

This market-leading text continues to provide students and instructors with sound, consistently structured explanations of the mathematical concepts. Designed for a one-term course that prepares students for further study in mathematics, the new ninth edition retains the features that have always made COLLEGE ALGEBRA a complete solution for both students and instructors: interesting applications, pedagogically effective design, and innovative technology combined with an abundance of carefully developed examples and exercises.

Aside from distribution theory, projections and the singular value decomposition (SVD) are the two most important concepts for understanding the basic mechanism of multivariate analysis. The former underlies the least squares estimation in regression analysis, which is essentially a projection of one subspace onto another, and the latter underlies principal component analysis, which seeks to find a subspace that captures the largest variability in the original space. This book is about projections and SVD. A thorough discussion of generalized inverse (g-inverse) matrices is also given because it is closely related to the former. The book provides systematic and in-depth accounts of these concepts from a unified viewpoint of linear transformations finite dimensional vector spaces. More specially, it shows that projection matrices (projectors) and g-inverse matrices can be defined in various ways so that a vector space is decomposed into a direct-sum of (disjoint) subspaces. Projection Matrices, Generalized Inverse Matrices, and Singular Value Decomposition will be useful for researchers, practitioners, and students in applied mathematics, statistics, engineering, behaviormetrics, and other fields.

The book is complemented by biographical information. This volume is dedicated to Peter Lancaster, an outstanding expert in operator and matrix theory, numerical analysis and applications, on the occasion of his seventieth birthday. The book contains a selection of recent original research papers in linear algebra and analysis, areas in which Peter Lancaster was very active. The articles are complemented by biographical data and a list of publications. Contributed volume in honor of Peter Lancaster, an outstanding expert in operator theory, matrix theory and numerical analysis. The articles have been carefully selected and refereed and cover topics in linear algebra and analysis where Peter Lancaster was very active.

One of the beautiful results in the representation theory of the finite groups is McKay's theorem on a correspondence between representations of the binary polyhedral group of SU(2) and vertices of an extended simply-laced Dynkin diagram.

The Coxeter transformation is the main tool in the proof of the McKay correspondence, and is closely interrelated with the Cartan matrix and Poincare series. The Coxeter functors constructed by Bernstein, Gelfand and Ponomarev plays a distinguished role in the representation theory of quivers.

On these pages, the ideas and formulas due to J. N. Bernstein, I. M. Gelfand and V. A. Ponomarev, H.S.M. Coxeter, V. Dlab and C.M. Ringel, V. Kac, J. McKay, T.A. Springer, B. Kostant, P. Slodowy, R. Steinberg, W. Ebeling and several other authors, as well as the author and his colleagues from Subbotin's seminar, are presented in detail. Several proofs seem to be new. "

In 1917, Johann Radon published his fundamental work, where he introduced what is now called the Radon transform. Including important contributions by several experts, this book reports on ground-breaking developments related to the Radon transform throughout these years, and also discusses novel mathematical research topics and applications for the next century.

This book addresses selected topics in the theory of generalized inverses. Following a discussion of the "reverse order law" problem and certain problems involving completions of operator matrices, it subsequently presents a specific approach to solving the problem of the reverse order law for {1} -generalized inverses. Particular emphasis is placed on the existence of Drazin invertible completions of an upper triangular operator matrix; on the invertibility and different types of generalized invertibility of a linear combination of operators on Hilbert spaces and Banach algebra elements; on the problem of finding representations of the Drazin inverse of a 2x2 block matrix; and on selected additive results and algebraic properties for the Drazin inverse. In addition to the clarity of its content, the book discusses the relevant open problems for each topic discussed. Comments on the latest references on generalized inverses are also included. Accordingly, the book will be useful for graduate students, PhD students and researchers, but also for a broader readership interested in these topics.

ELEMENTARY LINEAR ALGEBRA, 8E, INTERNATIONAL METRIC EDITION's clear, careful, and concise presentation of material helps you fully understand how mathematics works. The author balances theory with examples, applications, and geometric intuition for a complete, step-by-step learning system. To engage you in the material, a new design highlights the relevance of the mathematics and makes the book easier to read. Data and applications reflect current statistics and examples, demonstrating the link between theory and practice. The companion website LarsonLinearAlgebra.com offers free access to multiple study tools and resources. CalcChat.com offers free step-by-step solutions to the odd-numbered exercises in the text.

Over the last decade, Computational Fluid Dynamics (CFD) has become a - ture technology for the development of new products in aeronautical industry. Aerodynamic design engineers have progressively taken advantage of the pos- bilities o?ered by the numericalsolutionof the Reynolds averagedNavier-Stokes (RANS) equations. Signi?cant improvements in physical modeling and solution algorithms as well as the enormous increase of computer power enable hi- ?delity numerical simulations in all stages of aircraft development. In Germany, the national CFD project MEGAFLOW furthered the dev- opment and availability of RANS solvers for the prediction of complex ?ow problemssigni?cantly. MEGAFLOWwasinitiated by the?rstaviationresearch programoftheFederalGovernmentin1995undertheleadershipoftheDLR(see Kroll, N. , Fassbender, J. K. (Eds). : MEGAFLOW - Numerical Flow Simulation for Aircraft Design; Notes on Numerical Fluid Mechanics and Multidisciplinary Design, Volume 89, Springer, 2005). A network from aircraft industry, DLR and several universities was created with the goal to focus and direct development activities for numerical ?ow simulation towards a common aerodynamic si- lation system providing both a block-structured (FLOWer-Code) and a hybrid (TAU-Code) parallel ? ow prediction capability. Today, both codes have reached a high level of maturity and reliability. They are routinely used at DLR and German aeronautic industry for a wide range of aerodynamic applications. For many universities the MEGAFLOW software represents a platform for the - provementofphysicalmodelsandfortheinvestigationofcomplex?owproblems. The network was established as an e?cient group of very closely co-operating partners with supplementing expertises and experience.

Identity Based Encryption (IBE) is a type of public key encryption and has been intensely researched in the past decade. Identity-Based Encryption summarizes the available research for IBE and the main ideas that would enable users to pursue further work in this area. This book will also cover a brief background on Elliptic Curves and Pairings, security against chosen Cipher text Attacks, standards and more. Advanced-level students in computer science and mathematics who specialize in cryptology, and the general community of researchers in the area of cryptology and data security will find Identity-Based Encryption a useful book. Practitioners and engineers who work with real-world IBE schemes and need a proper understanding of the basic IBE techniques, will also find this book a valuable asset.

This edited volume features a curated selection of research in algebraic combinatorics that explores the boundaries of current knowledge in the field. Focusing on topics experiencing broad interest and rapid growth, invited contributors offer survey articles on representation theory, symmetric functions, invariant theory, and the combinatorics of Young tableaux. The volume also addresses subjects at the intersection of algebra, combinatorics, and geometry, including the study of polytopes, lattice points, hyperplane arrangements, crystal graphs, and Grassmannians. All surveys are written at an introductory level that emphasizes recent developments and open problems. An interactive tutorial on Schubert Calculus emphasizes the geometric and topological aspects of the topic and is suitable for combinatorialists as well as geometrically minded researchers seeking to gain familiarity with relevant combinatorial tools. Featured authors include prominent women in the field known for their exceptional writing of deep mathematics in an accessible manner. Each article in this volume was reviewed independently by two referees. The volume is suitable for graduate students and researchers interested in algebraic combinatorics.

A complete study on an important class of linear dynamical systems-positive linear systems

One of the most often-encountered systems in nearly all areas of science and technology, positive linear systems is a specific but remarkable and fascinating class. Renowned scientists Lorenzo Farina and Sergio Rinaldi introduce readers to the world of positive linear systems in their rigorous but highly accessible book, rich in applications, examples, and figures.

This professional reference is divided into three main parts: The first part contains the definitions and basic properties of positive linear systems. The second part, following the theoretical exposition, reports the main conceptual results, considering applicable examples taken from a number of widely used models. The third part is devoted to the study of some classes of positive linear systems of particular relevance in applications (such as the Leontief model, the Leslie model, the Markov chains, the compartmental systems, and the queueing systems). Readers familiar with linear algebra and linear systems theory will appreciate the way arguments are treated and presented.

Extraordinarily comprehensive, Positive Linear Systems features:

• Applications from a variety of backgrounds including modeling, control engineering, computer science, demography, economics, bioengineering, chemistry, and ecology
• References and annotated bibliographies throughout the book
• Two appendices concerning linear algebra and linear systems theory for readers unfamiliar with the mathematics used

Farina and Rinaldi make no effort to hide their enthusiasm for the topics presented, making Positive Linear Systems: Theory andApplications an indispensable resource for researchers and professionals in a broad range of fields.

Thisbookisintendedasanintroductiontoallthe?nitesimplegroups.During themonumentalstruggletoclassifythe?nitesimplegroups(andindeedsince), a huge amount of information about these groups has been accumulated. Conveyingthisinformationtothenextgenerationofstudentsandresearchers, not to mention those who might wish to apply this knowledge, has become a major challenge. With the publication of the two volumes by Aschbacher and Smith [12, 13] in 2004 we can reasonably regard the proof of the Classi?cation Theorem for Finite Simple Groups (usually abbreviated CFSG) as complete. Thus it is timely to attempt an overview of all the (non-abelian) ?nite simple groups in one volume. For expository purposes it is convenient to divide them into four basic types, namely the alternating, classical, exceptional and sporadic groups. The study of alternating groups soon develops into the theory of per- tation groups, which is well served by the classic text of Wielandt [170]and more modern treatments such as the comprehensive introduction by Dixon and Mortimer [53] and more specialised texts such as that of Cameron [19].

This volume is a collection of chapters covering recent advances in stochastic optimal control theory and algebraic systems theory. The book will be a useful reference for researchers and graduate students in systems and control, algebraic systems theory, and applied mathematics. Requiring only knowledge of undergraduate-level control and systems theory, the work may be used as a supplementary textbook in a graduate course on optimal control or algebraic systems theory.

Traditionally, Lie Theory is a tool to build mathematical models for physical systems. Recently, the trend is towards geometrisation of the mathematical description of physical systems and objects. A geometric approach to a system yields in general some notion of symmetry which is very helpful in understanding its structure. Geometrisation and symmetries are meant in their broadest sense, i.e., classical geometry, differential geometry, groups and quantum groups, infinite-dimensional (super-)algebras, and their representations. Furthermore, we include the necessary tools from functional analysis and number theory. This is a large interdisciplinary and interrelated field.
Samples of these new trends are presented in this volume, based on contributions from the Workshop Lie Theory and Its Applications in Physics held near Varna, Bulgaria, in June 2011.
This book is suitable for an extensive audience of mathematicians, mathematical physicists, theoretical physicists, and researchers in the field of Lie Theory."

This book provides an extensive treatment of Potential Theory for sub-Laplacians on stratified Lie groups. It also provides a largely self-contained presentation of stratified Lie groups, and of their Lie algebra of left-invariant vector fields. The presentation is accessible to graduate students and requires no specialized knowledge in algebra or differential geometry.