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This volume presents the general theory of generalized functions,
including the Fourier, Laplace, Mellin, Hilbert, Cauchy-Bochner and
Poisson integral transforms and operational calculus, with the
traditional material augmented by the theory of Fourier series,
abelian theorems, and boundary values of helomorphic functions for
one and several variables. The author addresses several facets in
depth, including convolution theory, convolution algebras and
convolution equations in them, homogenous generalized functions,
and multiplication of generalized functions. This book will meet
the needs of researchers, engineers, and students of applied
mathematics, control theory, and the engineering sciences.
The volume is based on the Sobolev-Schwartz concept of Generalized
Functions. It presents general theory including the Fourier,
Laplace, Mellin, Hilbert, Cauchy-Bochner, Poisson integral
transforms and operational calculus. Traditional material is
supplemented by the theory of Fourier series, abelian theorems,
boundary values of helomorphic functions for one and several
variables. There is detailed study of convolution theory,
convolution algebras and convolution equations in them, homogenous
generalized functions and multiplication of generalized functions
and some trends in these problems. Methods of the theory of
generalized functions are applied to some problems in mathematical
physics, for example: fundamental solutions of partial differential
equations and Cauchy problems. This volume also includes numerous
problems, exercises, examples and figures.
Plurisubharmonic functions playa major role in the theory of
functions of several complex variables. The extensiveness of
plurisubharmonic functions, the simplicity of their definition
together with the richness of their properties and. most
importantly, their close connection with holomorphic functions have
assured plurisubharmonic functions a lasting place in
multidimensional complex analysis. (Pluri)subharmonic functions
first made their appearance in the works of Hartogs at the
beginning of the century. They figure in an essential way, for
example, in the proof of the famous theorem of Hartogs (1906) on
joint holomorphicity. Defined at first on the complex plane IC, the
class of subharmonic functions became thereafter one of the most
fundamental tools in the investigation of analytic functions of one
or several variables. The theory of subharmonic functions was
developed and generalized in various directions: subharmonic
functions in Euclidean space IRn, plurisubharmonic functions in
complex space en and others. Subharmonic functions and the
foundations ofthe associated classical poten tial theory are
sufficiently well exposed in the literature, and so we introduce
here only a few fundamental results which we require. More detailed
expositions can be found in the monographs of Privalov (1937),
Brelot (1961), and Landkof (1966). See also Brelot (1972), where a
history of the development of the theory of subharmonic functions
is given."
Approach your problems from the right end It isn't that they can't
see the solution. It is and begin with the answers. Then one day,
that they can't see the problem. perhaps you will find the final
question. The Scandal of Father G. K. Chesterton. 'The Hermit Clad
in Crane Feathers' in R. Brown 'The point of a Pin'. van Gulik's
The Chinese Maze Murders. Growing specialization and
diversification have brought a host of monographs and textbooks on
increasingly specialized topics. However, the "tree" of knowledge
of mathematics and related fields does not grow only by putting
forth new branches. It also happens, quite often in fact, that
branches which were thought to be completely disparate are suddenly
seen to be related. Further, the kind and level of sophistication
of mathematics applied in various sciences has changed drastically
in recent years: measure theory is used (non trivially) in regional
and theoretical economics; algebraic geometry interacts with
physics; the Minkowsky lemma, coding theory and the structure of
water meet one another in packing and covering theory; quantum
fields, crystal defects and mathematical programming profit from
homotopy theory; Lie algebras are relevant to filtering; and
prediction and electrical engineering can use Stein spaces. And in
addition to this there are such new emerging subdisciplines as
"experimental mathematics," "CFD," "completely integrable systems,"
"chaos, synergetics and large-scale order," which are almost
impossible to fit into the existing classification schemes. They
draw upon widely different sections of mathematics."
Approach your problems from the right end It isn't that they can't
see the solution. It is and begin with the answers. Then one day,
that they can't see the problem. perhaps you will find the final
question. The Scandal of Father G. K. Chesterton. 'The Hermit Clad
in Crane Feathers' in R. Brown 'The point of a Pin'. van Gulik's
The Chinese Maze Murders. Growing specialization and
diversification have brought a host of monographs and textbooks on
increasingly specialized topics. However, the "tree" of knowledge
of mathematics and related fields does not grow only by putting
forth new branches. It also happens, quite often in fact, that
branches which were thought to be completely disparate are suddenly
seen to be related. Further, the kind and level of sophistication
of mathematics applied in various sciences has changed drastically
in recent years: measure theory is used (non trivially) in regional
and theoretical economics; algebraic geometry interacts with
physics; the Minkowsky lemma, coding theory and the structure of
water meet one another in packing and covering theory; quantum
fields, crystal defects and mathematical programming profit from
homotopy theory; Lie algebras are relevant to filtering; and
prediction and electrical engineering can use Stein spaces. And in
addition to this there are such new emerging subdisciplines as
"experimental mathematics," "CFD," "completely integrable systems,"
"chaos, synergetics and large-scale order," which are almost
impossible to fit into the existing classification schemes. They
draw upon widely different sections of mathematics."
This systematic exposition outlines the fundamentals of the theory
of single sheeted domains of holomorphy. It further illustrates
applications to quantum field theory, the theory of functions, and
differential equations with constant coefficients. Students of
quantum field theory will find this text of particular value. The
text begins with an introduction that defines the basic concepts
and elementary propositions, along with the more salient facts from
the theory of functions of real variables and the theory of
generalized functions. Subsequent chapters address the theory of
plurisubharmonic functions and pseudoconvex domains, along with
characteristics of domains of holomorphy. These explorations are
further examined in terms of four types of domains:
multiple-circular, tubular, semitubular, and Hartogs' domains.
Surveys of integral representations focus on the
Martinelli-Bochner, Bergman-Weil, and Bochner representations. The
final chapter is devoted to applications, particularly those
involved in field theory. It employs the theory of generalized
functions, along with the theory of functions of several complex
variables.
p-adic numbers play a very important role in modern number theory,
algebraic geometry and representation theory. Lately p-adic numbers
have attracted a great deal of attention in modern theoretical
physics as a promising new approach for describing the
non-Archimedean geometry of space-time at small distances.This is
the first book to deal with applications of p-adic numbers in
theoretical and mathematical physics. It gives an elementary and
thoroughly written introduction to p-adic numbers and p-adic
analysis with great numbers of examples as well as applications of
p-adic numbers in classical mechanics, dynamical systems, quantum
mechanics, statistical physics, quantum field theory and string
theory.
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