This e-book presents several basic methods and results of order theory that are currently used in various branches of mathematics. It presents topics that require a broad explanation in a concise and attractive way. Readers will easily identify problems/techniques/solutions without the frustration of long searches through previous notations, definitions and results. It begins with a sketch of axiomatic set theory and of the theory of categories. The fundamentals of the order relation are presented, including several properties equivalent to the axiom of choice, and culminating with well-ordered sets and transfinite numbers. This e-book also explains: (1) The complete existential theory of the various distinguished elements which a partially ordered set may possess, (2) The proof of the equivalence with the axiom of choice of 14 frequently used properties,(3) The fact that certain properties of ordinals are revealed as being just properties of well-ordered sets, and (4) The presentation of ordinals and cardinals both in axiomatic setting and following von Neumann’s construction. Proofs are given in detail along with numerous examples and exercises illustrating the theory and demonstrating applications. Several classes of lattices, closure operators and Galois connections include tools which serve as prerequisites to universal algebra. Other topics include embeddings and topological duality for distributive lattices. The e-book serves both as a textbook for undergraduate and graduate students and as a reference book for mathematicians working in fields different from set theory or algebra, to whom it provides a quick access to basic facts from order theory.

Lately there is an increasing interest in partial difference equations demonstrated by the enormous amount of research papers devoted to them. The initial reason for this increasing interest was the development of computers and the area of numerical analysis, where partial difference equations arise naturally when discretizing a partial differential equation. The aim of this e-book is to provide some recent advances in the field of partial difference equations. Applications of partial difference equations in numerical analysis and systems theory are also presented. This e-book will be of use to high level undergraduate students, graduate students and researchers working in the area of partial difference equations.

This book explains different types of subharmonic and harmonic functions. The book brings 12 chapters explaining general and specific types of subharmonic functions (eg. quasinearly subharmonic functions and other separate functions), related partial differential equations, mathematical proofs and extension results. The methods covered in the book also attempt to explain different mathematical analyses such as elliptical equations, domination conditions, weighted boundary behavior. The book serves as a reference work for scholars interested in potential theory and complex analysis.

The theory of summability has many uses throughout analysis and applied mathematics. Engineers and physicists working with Fourier series or analytic continuation will also find the concepts of summability theory valuable to their research. The concepts of summability have been extended to the sequences of fuzzy numbers and also to the theorems of ergodic theory. This ebook explains various aspects of summability and demonstrates applications in a coherent manner. The content can readily serve as a useful series of lecture notes on the subject. This ebook comprises of 8 chapters starting from classical sequence spaces and covering matrix transformations and fuzzy numbers. An accompanying bibliography with extensive references makes this a valuable source of information for readers interested in summability theory as well as other branches of science.

In The Generalized Relative Gol‘dberg Order and Type: Some Remarks on Functions of Complex Variables, the authors have discussed the generalized comparative growth analysis of entire functions of several complex variables at length. The discussion covers the important branch of complex analysis specially the theory of analytic functions of several variables. The book contains eight chapters. Chapter 1 presents introductory aspects of the topic and some preliminary definitions. In the proceeding chapters (2-8), the authors derive some results about the generalized Gol'dberg order (α,β) and generalized Gol'dberg type (α,β) of entire functions of several complex variables which progressively explores different variations of these variables, (upper and lower, functions, inequalities, growth measures). The book culminates in an explanation of the sum and product theorems depending on the generalized relative Gol'dberg order (α,β). This monograph is intended for mathematical researchers and enthusiasts who wish to expand their understanding about complex variables of several functions and complex and related aspects of complex analysis.

The Stokes phenomenon refers to the emergence of jump discontinuities in asymptotic expansions at specific rays in the complex plane. This book presents a radical theory for the phenomenon by introducing the concept of regularization. Two methods of regularization, Borel summation and Mellin-Barnes regularization, are used to derive general expressions for the regularized values of asymptotic expansions throughout the complex plane. Though different, both yield identical values, which, where possible, agree with the original functions. Consequently, asymptotics has been elevated to a true discipline yielding precise solutions. All researchers, who seek asymptotic solutions to problems, will find this a most valuable book.

This reference explains hybrid-Trefftz finite element method (FEM). Readers are introduced to the basic concepts and general element formulations of the method. This is followed by topics on non-homogeneous parabolic problems, thermal analysis of composites, and heat conduction in nonlinear functionally graded materials. A brief summary of the fundamental solution based-FEM is also presented followed by a discussion on axisymmetric potential problems and the rotordynamic response of tapered composites. The book is rounded by chapters that cover the n-sided polygonal hybrid finite elements and analysis of piezoelectric materials. Key Features - Systematic presentation of 9 topics - Covers FEMs in two sections: 1) hybrid-Trefftz method and 2) fundamental FEM solutions - Bibliographic references - Includes solutions to problems in the numerical analysis of different material types - Includes solutions to some problems encountered in civil engineering (seepage, heat transfer, etc). This reference is suitable for scholars involved in advanced courses in mathematics and engineering (civil engineering/materials engineering). Professionals involved in developing analytical tools for materials and construction testing can also benefit from the methods presented in the book.

Mathematical transformations have applications in many everyday artistic (computer graphics and design), industrial (manufacturing) and scientific (informatics) processes. Transformations: A Mathematical Approach covers both the mathematical basics of transformations and technical applications. Readers will find information on the mathematical operators for linear, nonlinear and affine transformations. Key Features -introduces readers to affine transformations, their properties and definitions -explains different linear and nonlinear transformations -covers the application of transformations in acoustics, actuary, bioinformatics, calculus, cybernetics, epidemiology, genetics, optics, physics, probability and vector analysis -includes carefully selected examples for easy understanding The combination of an easy-to understand text with information on a broad range of basic and applied topics related to transformations makes this textbook a handy resource for students of mathematics and allied disciplines, at all levels.

High air pollution levels pose a significant threat to plants, animals and human beings. Efforts by researchers are directed towards keeping air pollution levels below well defined ‘critical‘ levels in order to maintain a sustainable atmosphere and environmental system. The application of advanced mathematical models is important for researchers to achieve this goal as efficiently as possible. Mathematical models can be used to predict answers to many important questions about the environment. This application comes with several complex theoretical and practical obstacles which need to be resolved. A successfully applicable mathematical model needs to enable researchers to • Mathematically describe all important physical and chemical processes. • Apply fast and sufficiently accurate numerical methods. • Ensure that the model runs efficiently on modern high speed computers. • Use high quality input data, both meteorological data and emission inventories, in the runs. • Verify the model results by comparing them with reliable measurements taken in different parts of the spatial domain of the model. • Carry out long series of sensitivity experiments to check the response of the model to changes of different key parameters. • Visualize and animate the output results in order to make them easily understandable even to non-specialists. This monograph thoroughly describes mathematical methods useful for various situations in environmental modeling - including finite difference methods, splitting methods, parallel computation, etc. - and provides a framework for resolving problems posed in relation to the points listed above. Chapters are written by well-known specialists making this book a handy reference for researchers, university teachers and students working and studying in the areas of air pollution, meteorology, applied mathematics and computer science.