Professor Xihua Cao (1920-2005) was a leading scholar at East China Normal University (ECNU) and a famous algebraist in China. His contribution to the Chinese academic circle is particularly the formation of a world-renowned 'ECNU School' in algebra, covering research areas include algebraic groups, quantum groups, algebraic geometry, Lie algebra, algebraic number theory, representation theory and other hot fields. In January 2020, in order to commemorate Professor Xihua Cao's centenary birthday, East China Normal University held a three-day academic conference. Scholars at home and abroad gave dedications or delivered lectures in the conference. This volume originates from the memorial conference, collecting the dedications of scholars, reminiscences of family members, and 16 academic articles written based on the lectures in the conference, covering a wide range of research hot topics in algebra. The book shows not only scholars' respect and memory for Professor Xihua Cao, but also the research achievements of Chinese scholars at home and abroad.

Hyperidentities are important formulae of second-order logic, and research in hyperidentities paves way for the study of second-order logic and second-order model theory.This book illustrates many important current trends and perspectives for the field of hyperidentities and their applications, of interest to researchers in modern algebra and discrete mathematics. It covers a number of directions, including the characterizations of the Boolean algebra of n-ary Boolean functions and the distributive lattice of n-ary monotone Boolean functions; the classification of hyperidentities of the variety of lattices, the variety of distributive (modular) lattices, the variety of Boolean algebras, and the variety of De Morgan algebras; the characterization of algebras with aforementioned hyperidentities; the functional representations of finitely-generated free algebras of various varieties of lattices and bilattices via generalized Boolean functions (De Morgan functions, quasi-De Morgan functions, super-Boolean functions, super-De Morgan functions, etc); the structural results for De Morgan algebras, Boole-De Morgan algebras, super-Boolean algebras, bilattices, among others.While problems of Boolean functions theory are well known, the present book offers alternative, more general problems, involving the concepts of De Morgan functions, quasi-De Morgan functions, super-Boolean functions, and super-De Morgan functions, etc. In contrast to other generalized Boolean functions discovered and investigated so far, these functions have clearly normal forms. This quality is of crucial importance for their applications in pure and applied mathematics, especially in discrete mathematics, quantum computation, quantum information theory, quantum logic, and the theory of quantum computers.

Factorization Method for Boundary Value Problems by Invariant Embedding presents a new theory for linear elliptic boundary value problems. The authors provide a transformation of the problem in two initial value problems that are uncoupled, enabling you to solve these successively. This method appears similar to the Gauss block factorization of the matrix, obtained in finite dimension after discretization of the problem. This proposed method is comparable to the computation of optimal feedbacks for linear quadratic control problems.

Fixed Point Theory and Graph Theory provides an intersection between the theories of fixed point theorems that give the conditions under which maps (single or multivalued) have solutions and graph theory which uses mathematical structures to illustrate the relationship between ordered pairs of objects in terms of their vertices and directed edges. This edited reference work is perhaps the first to provide a link between the two theories, describing not only their foundational aspects, but also the most recent advances and the fascinating intersection of the domains. The authors provide solution methods for fixed points in different settings, with two chapters devoted to the solutions method for critically important non-linear problems in engineering, namely, variational inequalities, fixed point, split feasibility, and hierarchical variational inequality problems. The last two chapters are devoted to integrating fixed point theory in spaces with the graph and the use of retractions in the fixed point theory for ordered sets.

Lie superalgebras are a natural generalization of Lie algebras, having applications in geometry, number theory, gauge field theory, and string theory. Introduction to Finite and Infinite Dimensional Lie Algebras and Superalgebras introduces the theory of Lie superalgebras, their algebras, and their representations. The material covered ranges from basic definitions of Lie groups to the classification of finite-dimensional representations of semi-simple Lie algebras. While discussing all classes of finite and infinite dimensional Lie algebras and Lie superalgebras in terms of their different classes of root systems, the book focuses on Kac-Moody algebras. With numerous exercises and worked examples, it is ideal for graduate courses on Lie groups and Lie algebras.

This is the first monograph devoted to clean ring and matrix theory. It aims to study a theory of expressing an element in a ring as the sum of some special ones, such as idempotents, units, nilpotents, tripotents, involutions, etc. A matrix over such rings is thereby expressed as the sum of some special matrices. Also another topics on the behaviors of topological properties and *-properties of such rings are investigated.The book is based on the results of various published papers, particularly, by the authors'. It is accessible for students familiar with general abstract algebra, while the topics are interesting for researchers in the field of ring, matrix and operator theory.

This volume presents lectures given at the Wisła 20-21 Winter School and Workshop: Groups, Invariants, Integrals, and Mathematical Physics, organized by the Baltic Institute of Mathematics. The lectures were dedicated to differential invariants – with a focus on Lie groups, pseudogroups, and their orbit spaces – and Poisson structures in algebra and geometry and are included here as lecture notes comprising the first two chapters. Following this, chapters combine theoretical and applied perspectives to explore topics at the intersection of differential geometry, differential equations, and category theory. Specific topics covered include: The multisymplectic and variational nature of Monge-Ampère equations in dimension four Integrability of fifth-order equations admitting a Lie symmetry algebra Applications of the van Kampen theorem for groupoids to computation of homotopy types of striped surfaces A geometric framework to compare classical systems of PDEs in the category of smooth manifolds Groups, Invariants, Integrals, and Mathematical Physics is ideal for graduate students and researchers working in these areas. A basic understanding of differential geometry and category theory is assumed.

Semihypergroup Theory is the first book devoted to the semihypergroup theory and it includes basic results concerning semigroup theory and algebraic hyperstructures, which represent the most general algebraic context in which reality can be modelled. Hyperstructures represent a natural extension of classical algebraic structures and they were introduced in 1934 by the French mathematician Marty. Since then, hundreds of papers have been published on this subject.

Introduction to Political Psychology explores the many psychological patterns that influence individual political behavior. The authors introduce readers to a broad range of theories, concepts, and case studies of political activity, arguing that individuals are driven or motivated to act in accordance with personality characteristics, values, beliefs, and attachments to groups. The book explains many aspects of political behavior-whether seemingly pathological actions or normal decision-making practices, which sometimes work optimally, and sometimes fail. Thoroughly updated throughout, the book examines patterns of political behavior in areas including leadership, group behavior, voting, race, nationalism, terrorism, and war. This edition features coverage of the 2016 election and profiles former U.S. President Donald Trump, while also including updated data on race relations and extremist groups in the United States. Global issues are also considered, with case studies focused on Myanmar and Syria, alongside coverage of social issues including Black Lives Matter and the #MeToo movement. Accessibly written and comprehensive in scope, it is an essential companion for all graduate and upper-level undergraduate students of psychology, political science, and political psychology. It will also be of interest to those in the policy-making community, especially those looking to learn more about the extent to which perceptions, personality, and group dynamics affect the policy-making arena. It is accompanied by a set of online instructor resources.

The Linear Algebra Survival Guide offers a concise introduction to the difficult core topics of linear algebra, guiding you through the powerful graphic displays and visualization of Mathematica that make the most abstract theories seem simple - allowing you to tackle realistic problems using simple mathematical manipulations. This resource is therefore a guide to learning the content of Mathematica in a practical way, enabling you to manipulate potential solutions/outcomes, and learn creatively. No starting knowledge of the Mathematica system is required to use the book. Desktop, laptop, web-based versions of Mathematica are available on all major platforms. Mathematica Online for tablet and smartphone systems are also under development and increases the reach of the guide as a general reference, teaching and learning tool.

This book is intended as a textbook for a one-term senior undergraduate (or graduate) course in Ring and Field Theory, or Galois theory. The book is ready for an instructor to pick up to teach without making any preparations.The book is written in a way that is easy to understand, simple and concise with simple historic remarks to show the beauty of algebraic results and algebraic methods. The book contains 240 carefully selected exercise questions of varying difficulty which will allow students to practice their own computational and proof-writing skills. Sample solutions to some exercise questions are provided, from which students can learn to approach and write their own solutions and proofs. Besides standard ones, some of the exercises are new and very interesting. The book contains several simple-to-use irreducibility criteria for rational polynomials which are not in any such textbook.This book can also serve as a reference for professional mathematicians. In particular, it will be a nice book for PhD students to prepare their qualification exams.

This book examines ultrametric Banach algebras in general. It begins with algebras of continuous functions, and looks for maximal and prime ideals in connections with ultrafilters on the set of definition. The multiplicative spectrum has shown to be indispensable in ultrametric analysis and is described in the general context and then, in various cases of Banach algebras.Applications are made to various kind of functions: uniformly continuous functions, Lipschitz functions, strictly differentiable functions, defined in a metric space. Analytic elements in an algebraically closed complete field (due to M Krasner) are recalled with most of their properties linked to T-filters and applications to their Banach algebras, and to the ultrametric holomorphic functional calculus, with applications to spectral properties. The multiplicative semi-norms of Krasner algebras are characterized by circular filters with a metric and an order that are examined.The definition of the theory of affinoid algebras due to J Tate is recalled with all the main algebraic properties (including Krasner-Tate algebras). The existence of idempotents associated to connected components of the multiplicative spectrum is described.

This book highlights new developments in the wide and growing field of partial differential equations (PDE)-constrained optimization. Optimization problems where the dynamics evolve according to a system of PDEs arise in science, engineering, and economic applications and they can take the form of inverse problems, optimal control problems or optimal design problems. This book covers new theoretical, computational as well as implementation aspects for PDE-constrained optimization problems under uncertainty, in shape optimization, and in feedback control, and it illustrates the new developments on representative problems from a variety of applications.

This book consists of the expanded notes from an upper level linear algebra course given some years ago by the author. Each section, or lecture, covers about a week's worth of material and includes a full set of exercises of interest. It should feel like a very readable series of lectures. The notes cover all the basics of linear algebra but from a mature point of view. The author starts by briefly discussing fields and uses those axioms to define and explain vector spaces. Then he carefully explores the relationship between linear transformations and matrices. Determinants are introduced as volume functions and as a way to determine whether vectors are linearly independent. Also included is a full chapter on bilinear forms and a brief chapter on infinite dimensional spaces.The book is very well written, with numerous examples and exercises. It includes proofs and techniques that the author has developed over the years to make the material easier to understand and to compute.