This book offers an introduction to the theory of lie groups and their representations. It covers the essentials of the subject. Starting from basic undergraduate mathematics, it proceeds through the fundamentals of Lie Theory up to topics in representation theory, such as the Peter-Weyl theorem, Weyl's character formula, and the Borel-Weil theorem, all in the context of linear groups.

In this book, ring-theoretical properties of skew Laurent series rings A((x; )) over a ring A, where A is an associative ring with non-zero identity element are described. In addition, we consider Laurent rings and Malcev-Neumann rings, which are proper extensions of skew Laurent series rings.

'I like the authorsaEURO (TM) taste in footnotes, what with their frequent emphasis on history, i.e. the minutiae of the lives of many mathematicians appearing in these pages. Their remarks add a particular dimension of fun and pleasure to what I think is a very good book. ItaEURO (TM)s pitched at the right level, it does a lot of serious stuff in preparation for what is coming the studentsaEURO (TM) way in the future, and it does it well.'MAA ReviewsThis comprehensive two-volume book deals with algebra, broadly conceived. Volume 1 (Chapters 1-6) comprises material for a first year graduate course in algebra, offering the instructor a number of options in designing such a course. Volume 1, provides as well all essential material that students need to prepare for the qualifying exam in algebra at most American and European universities. Volume 2 (Chapters 7-13) forms the basis for a second year graduate course in topics in algebra. As the table of contents shows, that volume provides ample material accommodating a variety of topics that may be included in a second year course. To facilitate matters for the reader, there is a chart showing the interdependence of the chapters.

'I like the authorsaEURO (TM) taste in footnotes, what with their frequent emphasis on history, i.e. the minutiae of the lives of many mathematicians appearing in these pages. Their remarks add a particular dimension of fun and pleasure to what I think is a very good book. ItaEURO (TM)s pitched at the right level, it does a lot of serious stuff in preparation for what is coming the studentsaEURO (TM) way in the future, and it does it well.'MAA ReviewsThis comprehensive two-volume book deals with algebra, broadly conceived. Volume 1 (Chapters 1-6) comprises material for a first year graduate course in algebra, offering the instructor a number of options in designing such a course. Volume 1, provides as well all essential material that students need to prepare for the qualifying exam in algebra at most American and European universities. Volume 2 (Chapters 7-13) forms the basis for a second year graduate course in topics in algebra. As the table of contents shows, that volume provides ample material accommodating a variety of topics that may be included in a second year course. To facilitate matters for the reader, there is a chart showing the interdependence of the chapters.

'I like the authorsaEURO (TM) taste in footnotes, what with their frequent emphasis on history, i.e. the minutiae of the lives of many mathematicians appearing in these pages. Their remarks add a particular dimension of fun and pleasure to what I think is a very good book. ItaEURO (TM)s pitched at the right level, it does a lot of serious stuff in preparation for what is coming the studentsaEURO (TM) way in the future, and it does it well.'MAA ReviewsThis comprehensive two-volume book deals with algebra, broadly conceived. Volume 1 (Chapters 1-6) comprises material for a first year graduate course in algebra, offering the instructor a number of options in designing such a course. Volume 1, provides as well all essential material that students need to prepare for the qualifying exam in algebra at most American and European universities. Volume 2 (Chapters 7-13) forms the basis for a second year graduate course in topics in algebra. As the table of contents shows, that volume provides ample material accommodating a variety of topics that may be included in a second year course. To facilitate matters for the reader, there is a chart showing the interdependence of the chapters.

'I like the authorsaEURO (TM) taste in footnotes, what with their frequent emphasis on history, i.e. the minutiae of the lives of many mathematicians appearing in these pages. Their remarks add a particular dimension of fun and pleasure to what I think is a very good book. ItaEURO (TM)s pitched at the right level, it does a lot of serious stuff in preparation for what is coming the studentsaEURO (TM) way in the future, and it does it well.'MAA ReviewsThis comprehensive two-volume book deals with algebra, broadly conceived. Volume 1 (Chapters 1-6) comprises material for a first year graduate course in algebra, offering the instructor a number of options in designing such a course. Volume 1, provides as well all essential material that students need to prepare for the qualifying exam in algebra at most American and European universities. Volume 2 (Chapters 7-13) forms the basis for a second year graduate course in topics in algebra. As the table of contents shows, that volume provides ample material accommodating a variety of topics that may be included in a second year course. To facilitate matters for the reader, there is a chart showing the interdependence of the chapters.

This volume contains selected refereed papers based on lectures presented at the a ~Fifth International Fez Conference on Commutative Algebra and Applicationsa (TM) that was held in Fez, Morocco in June 2008. The volume represents new trends and areas of classical research within the field, with contributions from many different countries. In addition, the volume has as a special focus the research and influence of Alain Bouvier on commutative algebra over the past thirty years.

This book summarizes the application of linear algebra-based controllers (LABC) for trajectory tracking for practitioners and students across a range of engineering disciplines. It clarifies the necessary steps to apply this straight-forward technique to a non-linear multivariable system, dealing with continuous or discrete time models, and outlines the steps to implement such controllers. In this book, the authors present an approach of the trajectory tracking problem in systems with dead time and in the presence of additive uncertainties and environmental disturbances. Examples of applications of LABC to systems in real operating conditions (mobile robots, marine vessels, quadrotor and pvtol aircraft, chemical reactors and First Order Plus Dead Time systems) illustrate the controller design in such a way that the reader attains an understanding of LABC.

With roots in the nineteenth century, Lie theory has since found many and varied applications in mathematics and mathematical physics, to the point where it is now regarded as a classical branch of mathematics in its own right. This graduate text focuses on the study of semisimple Lie algebras, developing the necessary theory along the way. The material covered ranges from basic definitions of Lie groups to the classification of finite-dimensional representations of semisimple Lie algebras. Written in an informal style, this is a contemporary introduction to the subject which emphasizes the main concepts of the proofs and outlines the necessary technical details, allowing the material to be conveyed concisely. Based on a lecture course given by the author at the State University of New York, Stony Brook, the book includes numerous exercises and worked examples, and is ideal for graduate courses on Lie groups and Lie algebras.

Among all computer-generated mathematical images, Julia sets of rational maps occupy one of the most prominent positions. Their beauty and complexity can be fascinating. They also hold a deep mathematical content.

Computational hardness of Julia sets is the main subject of this book. By definition, a computable set in the plane can be visualized on a computer screen with an arbitrarily high magnification. There are countless programs to draw Julia sets. Yet, as the authors have discovered, it is possible to constructively produce examples of quadratic polynomials, whose Julia sets are not computable. This result is striking - it says that while a dynamical system can be described numerically with an arbitrary precision, the picture of the dynamics cannot be visualized.

The book summarizes the present knowledge (most of it from the authors' own work) about the computational properties of Julia sets in a self-contained way. It is accessible to experts and students with interest in theoretical computer science or dynamical systems.

"Presenting the proceedings of a conference held recently at Northwestern University, Evanston, Illinois, on the occasion of the retirement of noted mathematician Daniel Zelinsky, this novel reference provides up-to-date coverage of topics in commutative and noncommutative ring extensions, especially those involving issues of separability, Galois theory, and cohomology."

This book examines the notions of dimension and decomposition for module categories. It discusses some basic properties of quasidecomposition functions and the complete lattice of all quasidecomposition functions taking values in a fixed given lattice.

This book examines some aspects of homogeneous Banach algebras and related topics to illustrate various methods used in several classes of group algebras. It guides the reader toward some of the problems in harmonic analysis such as the problems of factorizations and closed subalgebras.

"Integrates and summarizes the most significant developments made by Chinese mathematicians in rings, groups, and algebras since the 1950s. Presents both survey articles and recent research results. Examines important topics in Hopf algebra, representation theory, semigroups, finite groups, homology algebra, module theory, valuation theory, and more."

This volume features selected, refereed papers on various aspects of statistics, matrix theory and its applications to statistics, as well as related numerical linear algebra topics and numerical solution methods, which are relevant for problems arising in statistics and in big data. The contributions were originally presented at the 25th International Workshop on Matrices and Statistics (IWMS 2016), held in Funchal (Madeira), Portugal on June 6-9, 2016. The IWMS workshop series brings together statisticians, computer scientists, data scientists and mathematicians, helping them better understand each other's tools, and fostering new collaborations at the interface of matrix theory and statistics.

Iterative Splitting Methods for Differential Equations explains how to solve evolution equations via novel iterative-based splitting methods that efficiently use computational and memory resources. It focuses on systems of parabolic and hyperbolic equations, including convection-diffusion-reaction equations, heat equations, and wave equations. In the theoretical part of the book, the author discusses the main theorems and results of the stability and consistency analysis for ordinary differential equations. He then presents extensions of the iterative splitting methods to partial differential equations and spatial- and time-dependent differential equations. The practical part of the text applies the methods to benchmark and real-life problems, such as waste disposal, elastics wave propagation, and complex flow phenomena. The book also examines the benefits of equation decomposition. It concludes with a discussion on several useful software packages, including r3t and FIDOS. Covering a wide range of theoretical and practical issues in multiphysics and multiscale problems, this book explores the benefits of using iterative splitting schemes to solve physical problems. It illustrates how iterative operator splitting methods are excellent decomposition methods for obtaining higher-order accuracy.

Under intense scrutiny for the last few decades, Multiple Objective Decision Making (MODM) has been useful for dealing with the multiple-criteria decisions and planning problems associated with many important applications in fields including management science, engineering design, and transportation. Rough set theory has also proved to be an effective mathematical tool to counter the vague description of objects in fields such as artificial intelligence, expert systems, civil engineering, medical data analysis, data mining, pattern recognition, and decision theory. Rough Multiple Objective Decision Making is perhaps the first book to combine state-of-the-art application of rough set theory, rough approximation techniques, and MODM. It illustrates traditional techniques-and some that employ simulation-based intelligent algorithms-to solve a wide range of realistic problems. Application of rough theory can remedy two types of uncertainty (randomness and fuzziness) which present significant drawbacks to existing decision-making methods, so the authors illustrate the use of rough sets to approximate the feasible set, and they explore use of rough intervals to demonstrate relative coefficients and parameters involved in bi-level MODM. The book reviews relevant literature and introduces models for both random and fuzzy rough MODM, applying proposed models and algorithms to problem solutions. Given the broad range of uses for decision making, the authors offer background and guidance for rough approximation to real-world problems, with case studies that focus on engineering applications, including construction site layout planning, water resource allocation, and resource-constrained project scheduling. The text presents a general framework of rough MODM, including basic theory, models, and algorithms, as well as a proposed methodological system and discussion of future research.

This open access book focuses on processing, modeling, and visualization of anisotropy information, which are often addressed by employing sophisticated mathematical constructs such as tensors and other higher-order descriptors. It also discusses adaptations of such constructs to problems encountered in seemingly dissimilar areas of medical imaging, physical sciences, and engineering. Featuring original research contributions as well as insightful reviews for scientists interested in handling anisotropy information, it covers topics such as pertinent geometric and algebraic properties of tensors and tensor fields, challenges faced in processing and visualizing different types of data, statistical techniques for data processing, and specific applications like mapping white-matter fiber tracts in the brain. The book helps readers grasp the current challenges in the field and provides information on the techniques devised to address them. Further, it facilitates the transfer of knowledge between different disciplines in order to advance the research frontiers in these areas. This multidisciplinary book presents, in part, the outcomes of the seventh in a series of Dagstuhl seminars devoted to visualization and processing of tensor fields and higher-order descriptors, which was held in Dagstuhl, Germany, on October 28-November 2, 2018.

This unique book provides the first introduction to crystal base theory from the combinatorial point of view. Crystal base theory was developed by Kashiwara and Lusztig from the perspective of quantum groups. Its power comes from the fact that it addresses many questions in representation theory and mathematical physics by combinatorial means. This book approaches the subject directly from combinatorics, building crystals through local axioms (based on ideas by Stembridge) and virtual crystals. It also emphasizes parallels between the representation theory of the symmetric and general linear groups and phenomena in combinatorics. The combinatorial approach is linked to representation theory through the analysis of Demazure crystals. The relationship of crystals to tropical geometry is also explained.

This unique book provides the first introduction to crystal base theory from the combinatorial point of view. Crystal base theory was developed by Kashiwara and Lusztig from the perspective of quantum groups. Its power comes from the fact that it addresses many questions in representation theory and mathematical physics by combinatorial means. This book approaches the subject directly from combinatorics, building crystals through local axioms (based on ideas by Stembridge) and virtual crystals. It also emphasizes parallels between the representation theory of the symmetric and general linear groups and phenomena in combinatorics. The combinatorial approach is linked to representation theory through the analysis of Demazure crystals. The relationship of crystals to tropical geometry is also explained.

This book presents an introduction to variational analysis, a field which unifies theories and techniques developed in calculus of variations, optimization, and control, and covers convex analysis, nonsmooth analysis, and set-valued analysis. It focuses on problems with constraints, the analysis of which involves set-valued mappings and functions that are not differentiable. Applications of variational analysis are interdisciplinary, ranging from financial planning to steering a flying object. The book is addressed to graduate students, researchers, and practitioners in mathematical sciences, engineering, economics, and finance. A typical reader of the book should be familiar with multivariable calculus and linear algebra. Some basic knowledge in optimization, control, and elementary functional analysis is desirable, but all necessary background material is included in the book.

This book casts the theory of periods of algebraic varieties in the natural setting of Madhav Nori's abelian category of mixed motives. It develops Nori's approach to mixed motives from scratch, thereby filling an important gap in the literature, and then explains the connection of mixed motives to periods, including a detailed account of the theory of period numbers in the sense of Kontsevich-Zagier and their structural properties. Period numbers are central to number theory and algebraic geometry, and also play an important role in other fields such as mathematical physics. There are long-standing conjectures about their transcendence properties, best understood in the language of cohomology of algebraic varieties or, more generally, motives. Readers of this book will discover that Nori's unconditional construction of an abelian category of motives (over fields embeddable into the complex numbers) is particularly well suited for this purpose. Notably, Kontsevich's formal period algebra represents a torsor under the motivic Galois group in Nori's sense, and the period conjecture of Kontsevich and Zagier can be recast in this setting. Periods and Nori Motives is highly informative and will appeal to graduate students interested in algebraic geometry and number theory as well as researchers working in related fields. Containing relevant background material on topics such as singular cohomology, algebraic de Rham cohomology, diagram categories and rigid tensor categories, as well as many interesting examples, the overall presentation of this book is self-contained.

The aim of the Expositions is to present new and important developments in pure and applied mathematics. Well established in the community over more than two decades, the series offers a large library of mathematical works, including several important classics. The volumes supply thorough and detailed expositions of the methods and ideas essential to the topics in question. In addition, they convey their relationships to other parts of mathematics. The series is addressed to advanced readers interested in a thorough study of the subject. Editorial Board Lev Birbrair, Universidade Federal do Ceara, Fortaleza, Brasil Walter D. Neumann, Columbia University, New York, USA Markus J. Pflaum, University of Colorado, Boulder, USA Dierk Schleicher, Jacobs University, Bremen, Germany Katrin Wendland, University of Freiburg, Germany Honorary Editor Victor P. Maslov, Russian Academy of Sciences, Moscow, Russia Titles in planning include Yuri A. Bahturin, Identical Relations in Lie Algebras (2019) Yakov G. Berkovich, Lev G. Kazarin, and Emmanuel M. Zhmud', Characters of Finite Groups, Volume 2 (2019) Jorge Herbert Soares de Lira, Variational Problems for Hypersurfaces in Riemannian Manifolds (2019) Volker Mayer, Mariusz Urbanski, and Anna Zdunik, Random and Conformal Dynamical Systems (2021) Ioannis Diamantis, Bostjan Gabrovsek, Sofia Lambropoulou, and Maciej Mroczkowski, Knot Theory of Lens Spaces (2021)

This book deals with the determinants of linear operators in Euclidean, Hilbert and Banach spaces. Determinants of operators give us an important tool for solving linear equations and invertibility conditions for linear operators, enable us to describe the spectra, to evaluate the multiplicities of eigenvalues, etc. We derive upper and lower bounds, and perturbation results for determinants, and discuss applications of our theoretical results to spectrum perturbations, matrix equations, two parameter eigenvalue problems, as well as to differential, difference and functional-differential equations.