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The aim of this book is to develop a new approach which we called the hyper geometric one to the theory of various integral transforms, convolutions, and their applications to solutions of integro-differential equations, operational calculus, and evaluation of integrals. We hope that this simple approach, which will be explained below, allows students, post graduates in mathematics, physicists and technicians, and serious mathematicians and researchers to find in this book new interesting results in the theory of integral transforms, special functions, and convolutions. The idea of this approach can be found in various papers of many authors, but systematic discussion and development is realized in this book for the first time. Let us explain briefly the basic points of this approach. As it is known, in the theory of special functions and its applications, the hypergeometric functions play the main role. Besides known elementary functions, this class includes the Gauss's, Bessel's, Kummer's, functions et c. In general case, the hypergeometric functions are defined as a linear combinations of the Mellin-Barnes integrals. These ques tions are extensively discussed in Chapter 1. Moreover, the Mellin-Barnes type integrals can be understood as an inversion Mellin transform from the quotient of products of Euler's gamma-functions. Thus we are led to the general construc tions like the Meijer's G-function and the Fox's H-function.
In 1991-1993 our three-volume book "Representation of Lie Groups and Spe cial Functions" was published. When we started to write that book (in 1983), editors of "Kluwer Academic Publishers" expressed their wish for the book to be of encyclopaedic type on the subject. Interrelations between representations of Lie groups and special functions are very wide. This width can be explained by existence of different types of Lie groups and by richness of the theory of their rep resentations. This is why the book, mentioned above, spread to three big volumes. Influence of representations of Lie groups and Lie algebras upon the theory of special functions is lasting. This theory is developing further and methods of the representation theory are of great importance in this development. When the book "Representation of Lie Groups and Special Functions" ,vol. 1-3, was under preparation, new directions of the theory of special functions, connected with group representations, appeared. New important results were discovered in the traditional directions. This impelled us to write a continuation of our three-volume book on relationship between representations and special functions. The result of our further work is the present book. The three-volume book, published before, was devoted mainly to studying classical special functions and orthogonal polynomials by means of matrix elements, Clebsch-Gordan and Racah coefficients of group representations and to generaliza tions of classical special functions that were dictated by matrix elements of repre sentations.
defined as elements of Grassmann algebra (an algebra with anticom muting generators). The derivatives of these elements with respect to anticommuting generators were defined according to algebraic laws, and nothing like Newton's analysis arose when Martin's approach was used. Later, during the next twenty years, the algebraic apparatus de veloped by Martin was used in all mathematical works. We must point out here the considerable contribution made by F. A. Berezin, G 1. Kac, D. A. Leites, B. Kostant. In their works, they constructed a new division of mathematics which can naturally be called an algebraic superanalysis. Following the example of physicists, researchers called the investigations carried out with the use of commuting and anticom muting coordinates supermathematics; all mathematical objects that appeared in supermathematics were called superobjects, although, of course, there is nothing "super" in supermathematics. However, despite the great achievements in algebraic superanaly sis, this formalism could not be regarded as a generalization to the case of commuting and anticommuting variables from the ordinary Newton analysis. What is more, Schwinger's formalism was still used in practically all physical works, on an intuitive level, and physicists regarded functions of anticommuting variables as "real functions" == maps of sets and not as elements of Grassmann algebras. In 1974, Salam and Strathdee proposed a very apt name for a set of super points. They called this set a superspace.
Boolean valued analysis is a technique for studying properties of an arbitrary mathematical object by comparing its representations in two different set-theoretic models whose construction utilises principally distinct Boolean algebras. The use of two models for studying a single object is a characteristic of the so-called non-standard methods of analysis. Application of Boolean valued models to problems of analysis rests ultimately on the procedures of ascending and descending, the two natural functors acting between a new Boolean valued universe and the von Neumann universe.
This is the first thorough examination of weakly nonlocal solitary waves, which are just as important in applications as their classical counterparts. The book describes a class of waves that radiate away from the core of the disturbance but are nevertheless very long-lived nonlinear disturbances.
A number of optimization problems of the mechanics of space flight and the motion of walking robots and manipulators, and of quantum physics, eco momics and biology, have an irregular structure: classical variational proce dures do not formally make it possible to find optimal controls that, as we explain, have an impulse character. This and other well-known facts lead to the necessity for constructing dynamical models using the concept of a gener alized function (Schwartz distribution). The problem ofthe systematization of such models is very important. In particular, the problem of the construction of the general form of linear and nonlinear operator equations in distributions is timely. Another problem is related to the proper determination of solutions of equations that have nonlinear operations over generalized functions in their description. It is well-known that "the value of a distribution at a point" has no meaning. As a result the problem to construct the concept of stability for generalized processes arises. Finally, optimization problems for dynamic systems in distributions need finding optimality conditions. This book contains results that we have obtained in the above-mentioned directions. The aim of the book is to provide for electrical and mechanical engineers or mathematicians working in applications, a general and systematic treat ment of dynamic systems based on up-to-date mathematical methods and to demonstrate the power of these methods in solving dynamics of systems and applied control problems.
The subject of the present book is sub differential calculus. The main source of this branch of functional analysis is the theory of extremal problems. For a start, we explicate the origin and statement of the principal problems of sub differential calculus. To this end, consider an abstract minimization problem formulated as follows: x E X, f(x) --+ inf. Here X is a vector space and f : X --+ iR is a numeric function taking possibly infinite values. In these circumstances, we are usually interested in the quantity inf f( x), the value of the problem, and in a solution or an optimum plan of the problem (i. e. , such an x that f(x) = inf f(X», if the latter exists. It is a rare occurrence to solve an arbitrary problem explicitly, i. e. to exhibit the value of the problem and one of its solutions. In this respect it becomes necessary to simplify the initial problem by reducing it to somewhat more manageable modifications formulated with the details of the structure of the objective function taken in due account. The conventional hypothesis presumed in attempts at theoretically approaching the reduction sought is as follows. Introducing an auxiliary function 1, one considers the next problem: x EX, f(x) -l(x) --+ inf. Furthermore, the new problem is assumed to be as complicated as the initial prob lem provided that 1 is a linear functional over X, i. e.
This book gives a comprehensive introduction to modern quantum mechanics, emphasising the underlying Hilbert space theory and generalised function theory. All the major modern techniques and approaches used in quantum mechanics are introduced, such as Berry phase, coherent and squeezed states, quantum computing, solitons and quantum mechanics. Audience: The book is suitable for graduate students in physics and mathematics.
The subject of this book is connected with a new direction in mathematics, which has been actively developed over the last few years, namely the field of polynomial computer algebra, which lies at the intersection point of algebra, mathematical analysis and programming. There were several incentives to write the book. First of all, there has lately been a considerable interest in applied nonlinear problems characterized by multiple sta tionary states. Practical needs have then in their turn led to the appearance of new theoretical results in the analysis of systems of nonlinear algebraic equations. And finally, the introduction of various computer packages for analytic manipulations has made it possible to use complicated elimination-theoretical algorithms in prac tical research. The structure of the book is accordingly represented by three main parts: Mathematical results driven to constructive algorithms, computer algebra realizations of these algorithms, and applications. Nonlinear systems of algebraic equations arise in diverse fields of science. In particular, for processes described by systems of differential equations with a poly nomial right hand side one is faced with the problem of determining the number (and location) of the stationary states in certain sets.