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Author: J.F. Colombeau
Publisher: Elsevier
ISBN: 0080872247
Category : Mathematics
Languages : en
Pages : 297
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Book Description
The author's previous book `New Generalized Functions and Multiplication of Distributions' (North-Holland, 1984) introduced `new generalized functions' in order to explain heuristic computations of Physics and to give a meaning to any finite product of distributions. The aim here is to present these functions in a more direct and elementary way. In Part I, the reader is assumed to be familiar only with the concepts of open and compact subsets of R&eegr;, of C∞ functions of several real variables and with some rudiments of integration theory. Part II defines tempered generalized functions, i.e. generalized functions which are, in some sense, increasing at infinity no faster than a polynomial (as well as all their partial derivatives). Part III shows that, in this setting, the partial differential equations have new solutions. The results obtained show that this setting is perfectly adapted to the study of nonlinear partial differential equations, and indicate some new perspectives in this field.
Author: J.F. Colombeau
Publisher: Elsevier
ISBN: 0080872247
Category : Mathematics
Languages : en
Pages : 297
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Book Description
The author's previous book `New Generalized Functions and Multiplication of Distributions' (North-Holland, 1984) introduced `new generalized functions' in order to explain heuristic computations of Physics and to give a meaning to any finite product of distributions. The aim here is to present these functions in a more direct and elementary way. In Part I, the reader is assumed to be familiar only with the concepts of open and compact subsets of R&eegr;, of C∞ functions of several real variables and with some rudiments of integration theory. Part II defines tempered generalized functions, i.e. generalized functions which are, in some sense, increasing at infinity no faster than a polynomial (as well as all their partial derivatives). Part III shows that, in this setting, the partial differential equations have new solutions. The results obtained show that this setting is perfectly adapted to the study of nonlinear partial differential equations, and indicate some new perspectives in this field.
Author: Jean François Colombeau
Publisher:
ISBN: 9780444877567
Category : Mathematics
Languages : en
Pages : 281
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Book Description
Author: I. M. Gel′fand
Publisher: American Mathematical Soc.
ISBN: 1470426587
Category : Mathematics
Languages : en
Pages : 450
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Book Description
he first systematic theory of generalized functions (also known as distributions) was created in the early 1950s, although some aspects were developed much earlier, most notably in the definition of the Green's function in mathematics and in the work of Paul Dirac on quantum electrodynamics in physics. The six-volume collection, Generalized Functions, written by I. M. Gel′fand and co-authors and published in Russian between 1958 and 1966, gives an introduction to generalized functions and presents various applications to analysis, PDE, stochastic processes, and representation theory. Volume 1 is devoted to basics of the theory of generalized functions. The first chapter contains main definitions and most important properties of generalized functions as functional on the space of smooth functions with compact support. The second chapter talks about the Fourier transform of generalized functions. In Chapter 3, definitions and properties of some important classes of generalized functions are discussed; in particular, generalized functions supported on submanifolds of lower dimension, generalized functions associated with quadratic forms, and homogeneous generalized functions are studied in detail. Many simple basic examples make this book an excellent place for a novice to get acquainted with the theory of generalized functions. A long appendix presents basics of generalized functions of complex variables.
Author: F. Farassat
Publisher:
ISBN:
Category : Aerodynamics
Languages : en
Pages : 60
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Book Description
Author: Ram P. Kanwal
Publisher: Springer Science & Business Media
ISBN: 9780817640064
Category : Mathematics
Languages : en
Pages : 478
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Book Description
This second edition of Generalized Functions has been strengthened in many ways. The already extensive set of examples has been expanded. Since the publication of the first edition, there has been tremendous growth in the subject and I have attempted to incorporate some of these new concepts. Accordingly, almost all the chapters have been revised. The bibliography has been enlarged considerably. Some of the material has been reorganized. For example, Chapters 12 and 13 of the first edition have been consolidated into Chapter 12 of this edition by a judicious process of elimination and addition of the subject matter. The new Chapter 13 explains the interplay between the theories of moments, asymptotics, and singular perturbations. Similarly, some sections of Chapter 15 have been revised and included in earlier chapters to improve the logical flow of ideas. However, two sections are retained. The section dealing with the application of the probability theory has been revised, and I am thankful to Professor Z.L. Crvenkovic for her help. The new material included in this chapter pertains to the modern topics of periodic distributions and microlocal theory. I have demonstrated through various examples that familiarity with the generalized functions is very helpful for students in physical sciences and technology. For instance, the reader will realize from Chapter 6 how the generalized functions have revolutionized the Fourier analysis which is being used extensively in many fields of scientific activity.
Author: Michael Oberguggenberger
Publisher: Routledge
ISBN: 1351428039
Category : Mathematics
Languages : en
Pages : 400
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Book Description
Questions regarding the interplay of nonlinearity and the creation and propagation of singularities arise in a variety of fields-including nonlinear partial differential equations, noise-driven stochastic partial differential equations, general relativity, and geometry with singularities. A workshop held at the Erwin-Schrödinger International Institute for Mathematical Physics in Vienna investigated these questions and culminated in this volume of invited papers from experts in the fields of nonlinear partial differential equations, structure theory of generalized functions, geometry and general relativity, stochastic partial differential equations, and nonstandard analysis. The authors provide the latest research relevant to work in partial differential equations, mathematical physics, and nonlinear analysis. With a focus on applications, this books provides a compilation of recent approaches to the problem of singularities in nonlinear models. The theory of differential algebras of generalized functions serves as the central theme of the project, along with its interrelations with classical methods.
Author: R.S. Pathak
Publisher: Springer Science & Business Media
ISBN: 1489915915
Category : Social Science
Languages : en
Pages : 298
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Book Description
The International Symposium on Generalized Functions and Their Applications was organized by the Department of Mathematics, Banaras Hindu University, and held December 23-26, 1991, on the occasion of the Platinum Jubilee Celebration of the university. More than a hundred mathematicians from ten countries participated in the deliberations of the symposium. Thirty lectures were delivered on a variety of topics within the area. The contributions to the proceedings of the symposium are, with a few exceptions, expanded versions of the lectures delivered by the invited speakers. The survey papers by Komatsu and Hoskins and Sousa Pinto provide an up-to-date account of the theory of hyperfunctions, ultradistributions and microfunctions, and the nonstandard theory of new generalized functions, respectively; those by Stankovic and Kanwal deal with structures and asymptotics. Choquet-Bruhat's work studies generalized functions on manifold and gives applications to shocks and discrete models. The other contributions relate to contemporary problems and achievements in theory and applications, especially in the theory of partial differential equations, differential geometry, mechanics, mathematical physics, and systems science. The proceedings give a very clear impression of the present state of the art in this field and contain many challenges, ideas, and open problems. The volume is very helpful for a broad spectrum of readers: graduate students to mathematical researchers.
Author: Stevan Pilipović
Publisher: Birkhäuser
ISBN: 3319146181
Category : Mathematics
Languages : en
Pages : 288
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Book Description
This book gathers peer-reviewed contributions representing modern trends in the theory of generalized functions and pseudo-differential operators. It is dedicated to Professor Michael Oberguggenberger (Innsbruck University, Austria) in honour of his 60th birthday. The topics covered were suggested by the ISAAC Group in Generalized Functions (GF) and the ISAAC Group in Pseudo-Differential Operators (IGPDO), which met at the 9th ISAAC congress in Krakow, Poland in August 2013. Topics include Columbeau algebras, ultra-distributions, partial differential equations, micro-local analysis, harmonic analysis, global analysis, geometry, quantization, mathematical physics, and time-frequency analysis. Featuring both essays and research articles, the book will be of great interest to graduate students and researchers working in analysis, PDE and mathematical physics, while also offering a valuable complement to the volumes on this topic previously published in the OT series.
Author: Hebe de Azevedo Biagioni
Publisher: Springer
ISBN: 3540469818
Category : Mathematics
Languages : en
Pages : 226
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Book Description
This book provides a simple introduction to a nonlinear theory of generalized functions introduced by J.F. Colombeau, which gives a meaning to any multiplication of distributions. This theory extends from pure mathematics (it presents a faithful generalization of the classical theory of C? functions and provides a synthesis of most existing multiplications of distributions) to physics (it permits the resolution of ambiguities that appear in products of distributions), passing through the theory of partial differential equations both from the theoretical viewpoint (it furnishes a concept of weak solution of pde's leading to existence-uniqueness results in many cases where no distributional solution exists) and the numerical viewpoint (it introduces new and efficient methods developed recently in elastoplasticity, hydrodynamics and acoustics). This text presents basic concepts and results which until now were only published in article form. It is in- tended for mathematicians but, since the theory and applications are not dissociated it may also be useful for physicists and engineers. The needed prerequisites for its reading are essentially reduced to the classical notions of differential calculus and the theory of integration over n-dimensional euclidean spaces.
Author: Bogoljub Stankovic
Publisher: Springer Science & Business Media
ISBN: 1461310555
Category : Mathematics
Languages : en
Pages : 432
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Book Description
This Proceedings consists of a collection of papers presented at the International Conference "Generalized functions, convergence structures and their applications" held from June 23-27, 1987 in Dubrovnik, Yugoslavia (GFCA-87): 71 participants from 21 countr~es from allover the world took part in the Conference. Proceedings reflects the work of the Conference. Plenary lectures of J. Burzyk, J. F. Colombeau, W. Gahler, H. Keiter, H. Komatsu, B. Stankovic, H. G. Tillman, V. S. Vladimirov provide an up-to-date account of the cur rent state of the subject. All these lectures, except H. G. Tillman's, are published in this volume. The published communications give the contemporary problems and achievements in the theory of generalized functions, in the theory of convergence structures and in their applications, specially in the theory of partial differential equations and in the mathematical physics. New approaches to the theory of generalized functions are presented, moti vated by concrete problems of applications. The presence of articles of experts in mathematical physics contributed to this aim. At the end of the volume one can find presented open problems which also point to further course of development in the theory of generalized functions and convergence structures. We are very grateful to Mr. Milan Manojlovic who typed these Proce edings with extreme skill and diligence and with inexhaustible patience.
