Sobolev spaces were firstly defined by the Russian mathematician, Sergei L. Sobolev (1908-1989) in the 1930s. Several properties of these spaces have been studied by mathematicians until today. Functions that account for existence and uniqueness, asymptotic behavior, blow up, stability and instability of the solution of many differential equations that occur in applied and in engineering sciences are carried out with the help of Sobolev spaces and embedding theorems in these spaces. An Introduction to Sobolev Spaces provides a brief introduction to Sobolev spaces at a simple level with illustrated examples. Readers will learn about the properties of these types of vector spaces and gain an understanding of advanced differential calculus and partial difference equations that are related to this topic. The contents of the book are suitable for undergraduate and graduate students, mathematicians, and engineers who have an interest in getting a quick, but carefully presented, mathematically sound, basic knowledge about Sobolev Spaces.

This book represents a novel approach for the trigonometry and an original scientific work in this field, by using the ensemble structure composed of the real analysis and the axiomatic fundaments of geometry. Throughout this e – book one presents, in a proper manner, definitions, properties, formulae and applications more specific of the subject title and its immediate connections. The book is recommended not only as a pertinent introduction for the high school students, being also very useful for the university students, mathematics teachers and anyone who is interested in the major elements regarding the essence of real mathematics.

All developments in mathematics and computer science facilitate development of industrial applications. This e-book approaches the subject in a profoundly interdisciplinary manner. The spectrum of subjects covered in this e-book includes mathematics, computer science, materials science and industrial applications. Specifically, the e-book elaborates on mathematical models of phase-field transitions. Nonlinear parabolic systems represent the core of this book. Mathematically speaking, it presents methods related to existence, uniqueness and regularity of solution, fractional steps, analysis of some boundary optimal control problems governed by phase-field transition system, conceptual algorithms to compute the approximate solution and boundary control.

Applied Computational Mathematics in Social Sciences adopts a modern scientific approach that combines knowledge from mathematical modeling with various aspects of social science. Special algorithms can be created to simulate an artificial society and a detailed analysis can subsequently be used to project social realities. This Ebook specifically deals with computations using the NetLogo platform, and is intended for researchers interested in advanced human geography and mathematical modeling studies.

Understanding Calculus as a Logical Extension of Arithmetic re-examines the calculus paradigm by expanding the set of ‘indeterminate forms’ espoused by l’Hôpital 320 years ago. Starting from the 54 possible binary combinations of the foundational numbers (zero, one and infinity), a replacement for the function theory formulated earlier by Newton and Leibniz, is presented. A logical extension of the three concepts of differentiation, integration and the Naperian base number e follows this introduction, which is interpreted as ‘zero divided by zero’, ‘infinity times zero’ and ‘one raised to the infinite power’, respectively. The concept that a number postulated as representing ‘nothing’ is reinforced a useful connection to calculus theory. This treatise proposes a ‘similar’ number to denote and quantify ‘all’, in order to understand the concept of infinity. Other topics covered in this text include analytical geometry, infinite sequences and infinite series. Understanding Calculus as a Logical Extension of Arithmetic is a useful reference on advanced calculus theory for mathematics students and researchers.

Complex Analysis for Science and Technology is a textbook for undergraduate and postgraduate students undertaking science, technology, engineering and mathematics (STEM) courses. The book begins with an introduction to basic complex numbers, followed by chapters covering complex functions, integrals, transformations and conformal mapping. Topics such as complex series and residue theory are also covered. Key features of this textbook include: -simple, easy-to-understand explanations of relevant concepts -a wide range of simple and complex examples -several figures where appropriate

1. Since the introduction of the information measure widely known as Shannon entropy, quantifiers based on information theory and concepts such as entropic forms and statistical complexities have proven to be useful in diverse scientific research fields. This book contains introductory tutorials suitable for the general reader, together with chapters dedicated to the basic concepts of the most frequently employed information measures or quantifiers and their recent applications to different areas, including physics, biology, medicine, economics, communication and social sciences. As these quantifiers are powerful tools for the study of general time and data series independently of their sources, this book will be useful to all those doing research connected with information analysis. The tutorials in this volume are written at a broadly accessible level and readers will have the opportunity to acquire the knowledge necessary to use the information theory tools in their field of interest.

Convergence Theorems for Lattice Group-valued Measures explains limit and boundedness theorems for measures taking values in abstract structures. The book begins with a historical survey about these topics since the beginning of the last century, moving on to basic notions and preliminaries on filters/ideals, lattice groups, measures and tools which are featured in the rest of this text. Readers will also find a survey on recent classical results about limit, boundedness and extension theorems for lattice group-valued measures followed by information about recent developments on these kinds of theorems and several results in the setting of filter/ideal convergence. In addition, each chapter has a general description of the topics and an appendix on random variables, concepts and lattices is also provided. Thus readers will benefit from this book through an easy-to-read historical survey about all the problems on convergence and boundedness theorems, and the techniques and tools which are used to prove the main results. The book serves as a primer for undergraduate, postgraduate and Ph. D. students on mathematical lattice and topological groups and filters, and a treatise for expert researchers who aim to extend their knowledge base.

Discrete Hilbert-type inequalities including Hilbert's inequality are important in mathematical analysis and its applications. In 1998, the author presented an extension of Hilbert's integral inequality with an independent parameter. In 2004, some new extensions of Hilbert's inequality were presented by introducing two pairs of conjugate exponents and additional independent parameters. Since then, a number of new discrete Hilbert-type inequalities have arisen. In this book, the author explains how to use the way of weight coefficients and introduce specific parameters to build new discrete Hilbert-type inequalities and consider some applications. The book is intended to augment the reader's understanding of Hilbert-type inequalities.

This e-book presents several research areas of elliptical problems solved by differential equations. The mathematical models explained in this e-book have been contributed by experts in the field and can be applied to a wide range of real life examples. Methods explained include pre-conditioned conjugate grid models, higher order discretization and finite element mesh generation. This e-book should be a useful reference for scholars and professionals interested in elliptical partial differential equations.