ISBN: 978-981-126-619-5 (hardcover)
The book aims at giving a monographic presentation of the abstract harmonic analysis of hypergroups, while combining it with applied topics of spectral analysis, approximation by orthogonal expansions and stochastic sequences. Hypergroups are locally compact Hausdorff spaces equipped with a convolution, an involution and a unit element. Related algebraic structures had already been studied by Frobenius around 1900. Their axiomatic characterisation in harmonic analysis was later developed in the 1970s. Hypergoups naturally emerge in seemingly different application areas as time series analysis, probability theory and theoretical physics.
The book presents harmonic analysis on commutative and polynomial hypergroups as well as weakly stationary random fields and sequences thereon. For polynomial hypergroups also difference equations and stationary sequences are considered. At greater extent than in the existing literature, the book compiles a rather comprehensive list of hypergroups, in particular of polynomial hypergroups. With an eye on readers at advanced undergraduate and graduate level, the proofs are generally worked out in careful detail. The bibliography is extensive.
Hypergroups
Basics of Harmonic Analysis of Hypergroups
Harmonic Analysis on Commutative Hypergroups
Fourier Analysis on Polynomial Hypergroups
Weakly Stationary Random Fields on a Commutative Hypergroup
Weakly Stationary Random Sequences on a Polynomial Hypergroup
Difference Equations and Stationary Sequences on Polynomial Hypergroups
Further Hypergroup Examples
Readership: Advanced undergraduate and graduate students, researchers in harmonic analysis and fields of application.
Pages: 358
Classical Deformation Theory is used for determining the completions of the local rings of an eventual moduli space. When a moduli variety exists, a main result in the book is that the local ring in a closed point can be explicitly computed as an algebraization of the pro-representing hull (therefore, called the local formal moduli) of the deformation functor for the corresponding closed point.
The book gives explicit computational methods and includes the most necessary prerequisites. It focuses on the meaning and the place of deformation theory, resulting in a complete theory applicable to moduli theory. It answers the question "why moduli theory" and it give examples in mathematical physics by looking at the universe as a moduli of molecules. Thereby giving a meaning to most noncommutative theories.
The book contains the first explicit definition of a noncommutative scheme, covered by not necessarily commutative rings. This definition does not contradict any of the previous abstract definitions of noncommutative algebraic geometry, but rather gives interesting relations to other theories which is left for further investigation.
Basic Introduction to Associative Modul
Associative Algebra
Associative Varieties I
Noncommutative Deformation Theory
Associative Varieties II
Computational Examples of Associative Varieties
Algebraic Invariant Theory
Pre-dynamic GIT
Dynamical Algebraic Structures
Readership: The target readership is graduate mathematicians, and it can be used as a textbook for graduate courses in algebra/algebraic geometry. The examples are good foundations for master and PhD theses. Can also be adopted to differential geometry and algebraic topology / K-theory and can be adopted to the recommended reading lists in such courses.
Pages: 200
In modern theoretical and applied mechanics, tensors and differential geometry are two almost essential tools. Unfortunately, in university courses for engineering and mechanics students, these topics are often poorly treated or even completely ignored. At the same time, many existing, very complete texts on tensors or differential geometry are so advanced and written in abstract language that discourage young readers looking for an introduction to these topics specifically oriented to engineering applications.
This textbook, mainly addressed to graduate students and young researchers in mechanics, is an attempt to fill the gap. Its aim is to introduce the reader to the modern mathematical tools and language of tensors, with special applications to the differential geometry of curves and surfaces in the Euclidean space. The exposition of the matter is sober, directly oriented to problems that are ordinarily found in mechanics and engineering. Also, the language and symbols are tailored to those usually employed in modern texts of continuum mechanics.
Though not exhaustive, as any primer textbook, this volume constitutes a coherent, self-contained introduction to the mathematical tools and results necessary in modern continuum mechanics, concerning vectors, 2nd- and 4th-rank tensors, curves, fields, curvilinear coordinates, and surfaces in the Euclidean space. More than 100 exercises are proposed to the reader, many of them complete the theoretical part through additional results and proofs. To accompany the reader in learning, all the exercises are entirely developed and solved at the end of the book.
Points and Vectors
Second Rank Tensors
Fourth Rank Tensors
Tensor Analysis: Curves
Tensor Analysis: Fields
Curvilinear Coordinates
Surfaces in the Euclidean Space
Readership: Graduate or PhD students and young researchers in mechanics, engineering, and applied mathematics. Applicable to courses in differential geometry or tensor algebra, and advanced applied or theoretical continuum mechanics courses for engineering students.
Pages: 150
ISBN: 978-981-126-709-3 (hardcover)
Hilbert-type inequalities, including Hilbert's inequalities proved in 1908, Hardy-Hilbert-type inequalities proved in 1934, and Yang-Hilbert-type inequalities first proved around 1998, play an important role in analysis and its applications. These inequalities are mainly divided in three classes: integral, discrete and half-discrete. During the last twenty years, there have been many research advances on Hilbert-type inequalities, and especially on Yang-Hilbert-type inequalities.
In the present monograph, applying weight functions, the idea of parametrization as well as techniques of real analysis and functional analysis, we prove some new Hilbert-type integral inequalities as well as their reverses with parameters. These inequalities constitute extensions of the well-known Hardy-Hilbert integral inequality. The equivalent forms and some equivalent statements of the best possible constant factors associated with several parameters are considered. Furthermore, we also obtain the operator expressions with the norm and some particular inequalities involving the Riemann-zeta function and the Hurwitz-zeta function. In the form of applications, by means of the beta function and the gamma function, we use the extended Hardy-Hilbert integral inequalities to consider several Hilbert-type integral inequalities involving derivative functions and upper limit functions. In the last chapter, we consider the case of Hardy-type integral inequalities. The lemmas and theorems within provide an extensive account of these kinds of integral inequalities and operators.
Efforts have been made for this monograph hopefully to be useful, especially to graduate students of mathematics, physics and engineering, as well as researchers in these domains.
Introduction
Equivalent Properties of a New Hilbert-Type Integral Inequality with Parameters
A New Hilbert-type Integral Inequality Involving One Derivative Function
A New Hilbert-type Integral Inequality Involving Two Derivative Functions and Others
New Hilbert-Type Integral Inequalities Involving Upper Limit Functions
Equivalent Properties of Two Kinds of Hardy-Type Integral Inequalities
References
Readership: Advanced undergraduate students, graduate students, established research mathematicians, researchers working in applied mathematical analysis.
Pages: 750
This volume features an extensive account of both research and expository papers in a wide area of engineering and mathematics and its various applications.
Topics treated within this book include optimization of control points, game theory, equilibrium points, algorithms, Cartan matrices, integral inequalities, Volterra integro-differential equations, Caristi-Kirk theorems, Laplace type integral operators, etc.
This useful reference text benefits graduate students, beginning research engineers and mathematicians as well as established researchers in these domains.
Readership: Researchers, professionals, academics, and graduate students computer engineering, computer maths and science and mathematical computation.
**********************************************************