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Books > Science & Mathematics > Mathematics > Algebra

Schaum's Outline of Linear Algebra, Sixth Edition (Paperback, 6th edition): Seymour Lipschutz, Marc Lipson Schaum's Outline of Linear Algebra, Sixth Edition (Paperback, 6th edition)
Seymour Lipschutz, Marc Lipson
R588 Discovery Miles 5 880 Ships in 9 - 15 working days

Tough Test Questions? Missed Lectures? Not Enough Time? Textbook too pricey? Fortunately, there's Schaum's. This all-in-one-package includes more than 600 fully-solved problems, examples, and practice exercises to sharpen your problem-solving skills. Plus, you will have access to 25 detailed videos featuring math instructors who explain how to solve the most commonly tested problems--it's just like having your own virtual tutor! You'll find everything you need to build confidence, skills, and knowledge for the highest score possible. More than 40 million students have trusted Schaum's to help them succeed in the classroom and on exams. Schaum's is the key to faster learning and higher grades in every subject. Each Outline presents all the essential course information in an easy-to-follow, topic-by-topic format. Helpful tables and illustrations increase your understanding of the subject at hand. Schaum's Outline of Linear Algebra, Sixth Edition features: * Updated content to match the latest curriculum * Over 600 problems with step-by-step solutions * An accessible outline format for quick and easy review * Clear explanations for all linear algebra concepts * Access to revised Schaums.com website and new app with access to 25 problem-solving videos, and more

Noncommutative Polynomial Algebras of Solvable Type and Their Modules - Basic Constructive-Computational Theory and Methods... Noncommutative Polynomial Algebras of Solvable Type and Their Modules - Basic Constructive-Computational Theory and Methods (Hardcover)
Huishi Li
R4,504 Discovery Miles 45 040 Ships in 12 - 17 working days

Noncommutative Polynomial Algebras of Solvable Type and Their Modules is the first book to systematically introduce the basic constructive-computational theory and methods developed for investigating solvable polynomial algebras and their modules. In doing so, this book covers: A constructive introduction to solvable polynomial algebras and Groebner basis theory for left ideals of solvable polynomial algebras and submodules of free modules The new filtered-graded techniques combined with the determination of the existence of graded monomial orderings The elimination theory and methods (for left ideals and submodules of free modules) combining the Groebner basis techniques with the use of Gelfand-Kirillov dimension, and the construction of different kinds of elimination orderings The computational construction of finite free resolutions (including computation of syzygies, construction of different kinds of finite minimal free resolutions based on computation of different kinds of minimal generating sets), etc. This book is perfectly suited to researchers and postgraduates researching noncommutative computational algebra and would also be an ideal resource for teaching an advanced lecture course.

Psychology and the Social Class Worldview - A Narrative-Based Introduction (Paperback): Anne  E. Noonan, William Ming Liu Psychology and the Social Class Worldview - A Narrative-Based Introduction (Paperback)
Anne E. Noonan, William Ming Liu
R1,205 Discovery Miles 12 050 Ships in 12 - 17 working days

Combines discussion of a theoretical model with essays about social class and classism experience, offering a unique approach that helps students learn theory and apply it to different contexts. Accompanied by an Instructors' Guide which offers more in-depth exploration and commentary, as well as discussion questions, writing prompts, and activity suggestions, making this the ideal resource for lecturers teaching courses on social class. Addresses the gap in psychological literature in terms of considering social class in relation to psychology and shows how the SCWM-R model can be applied to real life scenarios.

An Invitation to Abstract Algebra (Hardcover): Steven J. Rosenberg An Invitation to Abstract Algebra (Hardcover)
Steven J. Rosenberg
R2,610 Discovery Miles 26 100 Ships in 12 - 17 working days

Studying abstract algebra can be an adventure of awe-inspiring discovery. The subject need not be watered down nor should it be presented as if all students will become mathematics instructors. This is a beautiful, profound, and useful field which is part of the shared language of many areas both within and outside of mathematics. To begin this journey of discovery, some experience with mathematical reasoning is beneficial. This text takes a fairly rigorous approach to its subject, and expects the reader to understand and create proofs as well as examples throughout. The book follows a single arc, starting from humble beginnings with arithmetic and high-school algebra, gradually introducing abstract structures and concepts, and culminating with Niels Henrik Abel and Evariste Galois' achievement in understanding how we can-and cannot-represent the roots of polynomials. The mathematically experienced reader may recognize a bias toward commutative algebra and fondness for number theory. The presentation includes the following features: Exercises are designed to support and extend the material in the chapter, as well as prepare for the succeeding chapters. The text can be used for a one, two, or three-term course. Each new topic is motivated with a question. A collection of projects appears in Chapter 23. Abstract algebra is indeed a deep subject; it can transform not only the way one thinks about mathematics, but the way that one thinks-period. This book is offered as a manual to a new way of thinking. The author's aim is to instill the desire to understand the material, to encourage more discovery, and to develop an appreciation of the subject for its own sake.

Social Cognition and the Second Person in Human Interaction (Paperback): Diana I. Perez, Antoni Gomila Social Cognition and the Second Person in Human Interaction (Paperback)
Diana I. Perez, Antoni Gomila
R1,173 Discovery Miles 11 730 Ships in 12 - 17 working days

* Fosters a new consensus by articulating a balanced and integrative view to show the relevance of the empirical study of intentional interaction * Connects expression with moral psychology to give a detailed and perspicuous presentation of the second person interaction * A unique new resource for academics and students of social cognition, social and cognitive neuroscience, the cognitive sciences, philosophy, and metaphysics

The Power of Geometric Algebra Computing - For Engineering and Quantum Computing (Hardcover): Dietmar Hildenbrand The Power of Geometric Algebra Computing - For Engineering and Quantum Computing (Hardcover)
Dietmar Hildenbrand
R2,535 Discovery Miles 25 350 Ships in 12 - 17 working days

Introduces a new web-based optimizer for Geometric algebra algorithms; Supports many programming languages as well as hardware; Covers the advantages of High-dimensional algebras; Includes geometrically intuitive support of quantum computing

Social Cognition and the Second Person in Human Interaction (Hardcover): Diana I. Perez, Antoni Gomila Social Cognition and the Second Person in Human Interaction (Hardcover)
Diana I. Perez, Antoni Gomila
R3,620 Discovery Miles 36 200 Ships in 12 - 17 working days

* Fosters a new consensus by articulating a balanced and integrative view to show the relevance of the empirical study of intentional interaction * Connects expression with moral psychology to give a detailed and perspicuous presentation of the second person interaction * A unique new resource for academics and students of social cognition, social and cognitive neuroscience, the cognitive sciences, philosophy, and metaphysics

The Center and Focus Problem - Algebraic Solutions and Hypotheses (Paperback): M.N. Popa, V.V. Pricop The Center and Focus Problem - Algebraic Solutions and Hypotheses (Paperback)
M.N. Popa, V.V. Pricop
R1,853 Discovery Miles 18 530 Ships in 12 - 17 working days

The Center and Focus Problem: Algebraic Solutions and Hypotheses, M. N. Popa and V.V. Pricop, ISBN: 978-1-032-01725-9 (Hardback) This book focuses on an old problem of the qualitative theory of differential equations, called the Center and Focus Problem. It is intended for mathematicians, researchers, professors and Ph.D. students working in the field of differential equations, as well as other specialists who are interested in the theory of Lie algebras, commutative graded algebras, the theory of generating functions and Hilbert series. The book reflects the results obtained by the authors in the last decades. A rather essential result is obtained in solving Poincare's problem. Namely, there are given the upper estimations of the number of Poincare-Lyapunov quantities, which are algebraically independent and participate in solving the Center and Focus Problem that have not been known so far. These estimations are equal to Krull dimensions of Sibirsky graded algebras of comitants and invariants of systems of differential equations.

The Center and Focus Problem - Algebraic Solutions and Hypotheses (Hardcover): M.N. Popa, V.V. Pricop The Center and Focus Problem - Algebraic Solutions and Hypotheses (Hardcover)
M.N. Popa, V.V. Pricop
R4,503 Discovery Miles 45 030 Ships in 12 - 17 working days

The Center and Focus Problem: Algebraic Solutions and Hypotheses, M. N. Popa and V.V. Pricop, ISBN: 978-1-032-01725-9 (Hardback) This book focuses on an old problem of the qualitative theory of differential equations, called the Center and Focus Problem. It is intended for mathematicians, researchers, professors and Ph.D. students working in the field of differential equations, as well as other specialists who are interested in the theory of Lie algebras, commutative graded algebras, the theory of generating functions and Hilbert series. The book reflects the results obtained by the authors in the last decades. A rather essential result is obtained in solving Poincare's problem. Namely, there are given the upper estimations of the number of Poincare-Lyapunov quantities, which are algebraically independent and participate in solving the Center and Focus Problem that have not been known so far. These estimations are equal to Krull dimensions of Sibirsky graded algebras of comitants and invariants of systems of differential equations.

Galois Theory and Its Algebraic Background (Paperback, 2nd Revised edition): D.J.H. Garling Galois Theory and Its Algebraic Background (Paperback, 2nd Revised edition)
D.J.H. Garling
R1,332 R1,099 Discovery Miles 10 990 Save R233 (17%) Ships in 12 - 17 working days

Galois Theory, the theory of polynomial equations and their solutions, is one of the most fascinating and beautiful subjects of pure mathematics. Using group theory and field theory, it provides a complete answer to the problem of the solubility of polynomial equations by radicals: that is, determining when and how a polynomial equation can be solved by repeatedly extracting roots using elementary algebraic operations. This textbook contains a fully detailed account of Galois Theory and the algebra that it needs and is suitable both for those following a course of lectures and the independent reader (who is assumed to have no previous knowledge of Galois Theory). The second edition has been significantly revised and re-ordered; the first part develops the basic algebra that is needed, and the second a comprehensive account of Galois Theory. There are applications to ruler-and- compass constructions, and to the solution of classical mathematical problems of ancient times. There are new exercises throughout, and carefully-selected examples will help the reader develop a clear understanding of the mathematical theory.

The Event of Psychopoetics - Imagination and the Rupture of Psychology (Paperback): Raul Garcia The Event of Psychopoetics - Imagination and the Rupture of Psychology (Paperback)
Raul Garcia
R1,168 Discovery Miles 11 680 Ships in 12 - 17 working days

The Event of Psychopoetics overviews and investigates the notion of psychopoetics, a sociopsychological event that involves re-creative slips and that emerges under certain cultural conditions and power relations in the context of everyday interaction and through certain modes of dialoguing and conversing. This transdisciplinary text takes the reader through the thought processes of Deleuze, Guattari, Agamben, Maffesoli, Foucault, Butler, Haraway, and Braidotti, among others, addressing debates that are integral to the critique of psychology and its devices of subjectivization and normalization. Garcia takes a unique approach by reflecting on how psychopoetics contrasts institutionalized dialogues, while constantly emphasizing the generative and transformative potency of social worlds effectuated in the impetuous play of poetics. The book combines the rigor of academic research with the creative display of ideas that open diverse, suggestive lines of reflection on everyday interlocution and its possibilities of reinvention, modes of social existence, and the relation between subjectivity and the designs of power. A truly unique reading experience, this book is ideal for students, instructors, and researchers in the fields of philosophy, social psychology and sociological thought, discourse studies, literary theory, and cultural analysis.

The Psychology of Extremism - A Motivational Perspective (Hardcover): Catalina Kopetz, Ewa Szumowska, Arie Kruglanski The Psychology of Extremism - A Motivational Perspective (Hardcover)
Catalina Kopetz, Ewa Szumowska, Arie Kruglanski
R3,932 Discovery Miles 39 320 Ships in 12 - 17 working days

* Includes interdisciplinary contributions and brings together research on a range of extreme behaviors in one volume, by making theoretical links between different contexts * Explores the brain, hormones, and behaviour to offer insights into the mechanisms and processes that enable extremism to explain their occurrence and the conditions under which they may be likely to emerge * Ideal reading for high-level students taking courses on extremism, academics, and professionals dealing with extreme behavior

Spatializing Social Media - Social Networks Online and Offline (Hardcover): Marco Bastos Spatializing Social Media - Social Networks Online and Offline (Hardcover)
Marco Bastos
R3,921 Discovery Miles 39 210 Ships in 12 - 17 working days

Spatializing Social Media charts the theoretical and methodological challenges in analyzing and visualizing social media data mapped to geographic areas. It introduces the reader to concepts, theories, and methods that sit at the crossroads between spatial and social network analysis to unpack the conceptual differences between online and face-to-face social networks and the nonlinear effects triggered by social activity that overlaps online and offline. The book is divided into four sections, with the first accounting for the differences between space (the geometrical arrangements that structure and enable forms of interaction) and place (the mechanisms through which social meanings are attached to physical locations). The second section covers the rationale of social network analysis and the ontological differences, stating that relationships, more than individual and independent attributes, are key to understanding of social behavior. The third section covers a range of case studies that successfully mapped social media activity to geographically situated areas and considers the inflection of homophilous dependencies across online and offline social networks. The fourth and last section of the book explores a range of networks and discusses methods for and approaches to plotting a social network graph onto a map, including the purpose-built R package Spatial Social Media. The book takes a non-mathematical approach to social networks and spatial statistics suitable for postgraduate students in sociology, psychology and the social sciences.

Understanding Media Psychology (Hardcover): Gayle S. Stever, David C. Giles, J. David Cohen, Mary E. Myers Understanding Media Psychology (Hardcover)
Gayle S. Stever, David C. Giles, J. David Cohen, Mary E. Myers
R4,077 Discovery Miles 40 770 Ships in 12 - 17 working days

* Written and structured in relation to media psychology courses and curricula and includes a range of pedagogical features to aid teaching and student learning including vocabulary and key terms, discussion questions, and boxed examples * Offers an up to date perspective by engaging with digital media and cyberpsychology, as well as topical issues such as fake news, positive psychology, gaming, online dating, and social media * Written by a team of expert authors to offer a comprehensive overview of the topic are that is suitable for undergraduate and introductory master's course around the world

The Art and Science of Expert Witness Testimony - A Multidisciplinary Guide for Professionals (Paperback): Karen Postal The Art and Science of Expert Witness Testimony - A Multidisciplinary Guide for Professionals (Paperback)
Karen Postal
R1,501 Discovery Miles 15 010 Ships in 12 - 17 working days

* Solid research basis, drawing on findings from a 4-year research project with in-depth interviews with judges, attorneys, and seasoned forensic neuropsychologists and psychologists as well as further interviews with professionals in other fields such as engineering, physics and economics. * Provides focused attention on how experts interact with judges, attorneys, and juries * Challenges experts to avoid the traps of professional jargon and traditional manners of presenting information/knowledge/opinions. * Provides a step-by-step approach to orienting the new academic to expert witnessing

The Art and Science of Expert Witness Testimony - A Multidisciplinary Guide for Professionals (Hardcover): Karen Postal The Art and Science of Expert Witness Testimony - A Multidisciplinary Guide for Professionals (Hardcover)
Karen Postal
R4,063 Discovery Miles 40 630 Ships in 12 - 17 working days

* Solid research basis, drawing on findings from a 4-year research project with in-depth interviews with judges, attorneys, and seasoned forensic neuropsychologists and psychologists as well as further interviews with professionals in other fields such as engineering, physics and economics. * Provides focused attention on how experts interact with judges, attorneys, and juries * Challenges experts to avoid the traps of professional jargon and traditional manners of presenting information/knowledge/opinions. * Provides a step-by-step approach to orienting the new academic to expert witnessing

Advances in Linear Algebra Research (Hardcover): Ivan Kyrchei Advances in Linear Algebra Research (Hardcover)
Ivan Kyrchei
R5,237 Discovery Miles 52 370 Ships in 12 - 17 working days

This book presents original studies on the leading edge of linear algebra. Each chapter has been carefully selected in an attempt to present substantial research results across a broad spectrum. The main goal of Chapter One is to define and investigate the restricted generalized inverses corresponding to minimization of constrained quadratic form. As stated in Chapter Two, in systems and control theory, Linear Time Invariant (LTI) descriptor (Differential-Algebraic) systems are intimately related to the matrix pencil theory. A review of the most interesting properties of the Projective Equivalence and the Extended Hermite Equivalence classes is presented in the chapter. New determinantal representations of generalized inverse matrices based on their limit representations are introduced in Chapter Three. Using the obtained analogues of the adjoint matrix, Cramer's rules for the least squares solution with the minimum norm and for the Drazin inverse solution of singular linear systems have been obtained in the chapter. In Chapter Four, a very interesting application of linear algebra of commutative rings to systems theory, is explored. Chapter Five gives a comprehensive investigation to behaviors of a general Hermitian quadratic matrix-valued function by using ranks and inertias of matrices. In Chapter Six, the theory of triangular matrices (tables) is introduced. The main "characters" of the chapter are special triangular tables (which will be called triangular matrices) and their functions paradeterminants and parapermanents. The aim of Chapter Seven is to present the latest developments in iterative methods for solving linear matrix equations. The problems of existence of common eigenvectors and simultaneous triangularization of a pair of matrices over a principal ideal domain with quadratic minimal polynomials are investigated in Chapter Eight. Two approaches to define a noncommutative determinant (a determinant of a matrix with noncommutative elements) are considered in Chapter Nine. The last, Chapter 10, is an example of how the methods of linear algebra are used in natural sciences, particularly in chemistry. In this chapter, it is shown that in a First Order Chemical Kinetics Mechanisms matrix, all columns add to zero, all the diagonal elements are non-positive and all the other matrix entries are non-negative. As a result of this particular structure, the Gershgorin Circles Theorem can be applied to show that all the eigenvalues are negative or zero.

Nonverbal Communication (Hardcover, 2nd edition): Valerie Manusov, Laura K. Guerrero, Judee K. Burgoon Nonverbal Communication (Hardcover, 2nd edition)
Valerie Manusov, Laura K. Guerrero, Judee K. Burgoon
R4,546 Discovery Miles 45 460 Ships in 12 - 17 working days

--Core textbook featuring accessible but advanced coverage of theory, research, and applications in nonverbal communication from renowned scholars --Usable for undergraduate and graduate courses in communication and psychology departments --Includes a new chapter on identity and impression management, as well as fully updated research coverage throughout --Online resources include an extensive instructor's manual and test bank

Abstract Calculus - A Categorical Approach (Hardcover): Francisco Javier Garcia-Pacheco Abstract Calculus - A Categorical Approach (Hardcover)
Francisco Javier Garcia-Pacheco
R4,674 Discovery Miles 46 740 Ships in 12 - 17 working days

Features: Suitable for PhD candidates and researchers. Requires prerequisites in set theory, general topology and abstract algebra, but is otherwise self-contained.

Linear Algebra - Pearson New International Edition (Paperback, 4th edition): Stephen Friedberg, Arnold Insel, Lawrence Spence Linear Algebra - Pearson New International Edition (Paperback, 4th edition)
Stephen Friedberg, Arnold Insel, Lawrence Spence
R2,308 Discovery Miles 23 080 Ships in 9 - 15 working days

For courses in Advanced Linear Algebra. This top-selling, theorem-proof text presents a careful treatment of the principle topics of linear algebra, and illustrates the power of the subject through a variety of applications. It emphasizes the symbiotic relationship between linear transformations and matrices, but states theorems in the more general infinite-dimensional case where appropriate.

Introduction to Lattice Algebra - With Applications in AI, Pattern Recognition, Image Analysis, and Biomimetic Neural Networks... Introduction to Lattice Algebra - With Applications in AI, Pattern Recognition, Image Analysis, and Biomimetic Neural Networks (Hardcover)
Gerhard X. Ritter, Gonzalo Urcid
R3,193 Discovery Miles 31 930 Ships in 12 - 17 working days

Features Filled with instructive examples and exercises to help build understanding Suitable for researchers, professionals and students, both in mathematics and computer science Every chapter consists of exercises with solution provided online at www.Routledge.com/9780367720292

Algebra & Geometry - An Introduction to University Mathematics (Hardcover, 2nd edition): Mark V. Lawson Algebra & Geometry - An Introduction to University Mathematics (Hardcover, 2nd edition)
Mark V. Lawson
R4,366 Discovery Miles 43 660 Ships in 12 - 17 working days

Algebra & Geometry: An Introduction to University Mathematics, Second Edition provides a bridge between high school and undergraduate mathematics courses on algebra and geometry. The author shows students how mathematics is more than a collection of methods by presenting important ideas and their historical origins throughout the text. He incorporates a hands-on approach to proofs and connects algebra and geometry to various applications. The text focuses on linear equations, polynomial equations, and quadratic forms. The first few chapters cover foundational topics, including the importance of proofs and a discussion of the properties commonly encountered when studying algebra. The remaining chapters form the mathematical core of the book. These chapters explain the solutions of different kinds of algebraic equations, the nature of the solutions, and the interplay between geometry and algebra. New to the second edition Several updated chapters, plus an all-new chapter discussing the construction of the real numbers by means of approximations by rational numbers Includes fifteen short 'essays' that are accessible to undergraduate readers, but which direct interested students to more advanced developments of the material Expanded references Contains chapter exercises with solutions provided online at www.routledge.com/9780367563035

Algebra & Geometry - An Introduction to University Mathematics (Paperback, 2nd edition): Mark V. Lawson Algebra & Geometry - An Introduction to University Mathematics (Paperback, 2nd edition)
Mark V. Lawson
R1,675 Discovery Miles 16 750 Ships in 12 - 17 working days

Algebra & Geometry: An Introduction to University Mathematics, Second Edition provides a bridge between high school and undergraduate mathematics courses on algebra and geometry. The author shows students how mathematics is more than a collection of methods by presenting important ideas and their historical origins throughout the text. He incorporates a hands-on approach to proofs and connects algebra and geometry to various applications. The text focuses on linear equations, polynomial equations, and quadratic forms. The first few chapters cover foundational topics, including the importance of proofs and a discussion of the properties commonly encountered when studying algebra. The remaining chapters form the mathematical core of the book. These chapters explain the solutions of different kinds of algebraic equations, the nature of the solutions, and the interplay between geometry and algebra. New to the second edition Several updated chapters, plus an all-new chapter discussing the construction of the real numbers by means of approximations by rational numbers Includes fifteen short 'essays' that are accessible to undergraduate readers, but which direct interested students to more advanced developments of the material Expanded references Contains chapter exercises with solutions provided online at www.routledge.com/9780367563035

Algebraic Number Theory - A Brief Introduction (Paperback): J.S. Chahal Algebraic Number Theory - A Brief Introduction (Paperback)
J.S. Chahal
R1,689 Discovery Miles 16 890 Ships in 12 - 17 working days

This book offers the basics of algebraic number theory for students and others who need an introduction and do not have the time to wade through the voluminous textbooks available. It is suitable for an independent study or as a textbook for a first course on the topic. The author presents the topic here by first offering a brief introduction to number theory and a review of the prerequisite material, then presents the basic theory of algebraic numbers. The treatment of the subject is classical but the newer approach discussed at the end provides a broader theory to include the arithmetic of algebraic curves over finite fields, and even suggests a theory for studying higher dimensional varieties over finite fields. It leads naturally to the Weil conjecture and some delicate questions in algebraic geometry. About the Author Dr. J. S. Chahal is a professor of mathematics at Brigham Young University. He received his Ph.D. from Johns Hopkins University and after spending a couple of years at the University of Wisconsin as a post doc, he joined Brigham Young University as an assistant professor and has been there ever since. He specializes and has published several papers in number theory. For hobbies, he likes to travel and hike. His book, Fundamentals of Linear Algebra, is also published by CRC Press.

Nonlinear Second Order Parabolic Equations (Paperback): Mingxin Wang Nonlinear Second Order Parabolic Equations (Paperback)
Mingxin Wang
R1,435 Discovery Miles 14 350 Ships in 12 - 17 working days

The parabolic partial differential equations model one of the most important processes in the real-world: diffusion. Whether it is the diffusion of energy in space-time, the diffusion of species in ecology, the diffusion of chemicals in biochemical processes, or the diffusion of information in social networks, diffusion processes are ubiquitous and crucial in the physical and natural world as well as our everyday lives. This book is self-contained and covers key topics such as the Lp theory and Schauder theory, maximum principle, comparison principle, regularity and uniform estimates, initial-boundary value problems of semilinear parabolic scalar equations and weakly coupled parabolic systems, the upper and lower solutions method, monotone properties and long-time behaviours of solutions, convergence of solutions and stability of equilibrium solutions, global solutions and finite time blowup. It also touches on periodic boundary value problems, free boundary problems, and semigroup theory. The book covers major theories and methods of the field, including topics that are useful but hard to find elsewhere. This book is based on tried and tested teaching materials used at the Harbin Institute of Technology over the past ten years. Special care was taken to make the book suitable for classroom teaching as well as for self-study among graduate students. About the Author: Mingxin Wang is Professor of Mathematics at Harbin Institute of Technology, China. He has published ten monographs and textbooks and 260 papers. He is also a supervisor of 30 PhD students.

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