Verwandte Artikel zu Ramified Integrals, Singularities and Lacunas: 315...
Inhaltsangabe
Many special functions occuring in physics and partial differential equations can be represented by integral transformatIons: the fundamental solutions of many PDE’s, Newton-Coulomb potentials, hypergeometric functions, Feynman integrals, initial data of (inverse) tomography problems, etc. The general picture of such transfor mations is as follows. There is an analytic fibre bundle E --+ T, a differential form w on E, whose restrictions on the fibres are closed, and a family of cycles in these fibres, parametrized by the points of T and depending continuously on these points. Then the integral of the form w along these cycles is a function on the base. The analytic properties of such functions depend on the monodromy action, i.e., on the natural action of the fundamental group of the base in the homology of the fibre: this action on the integration cycles defines the ramification of the analytic continuation of our function. The study of this action (which is a purely topological problem) can answer questions about the analytic behaviour of the integral function, for instance, is this function single-valued or at least algebraic, what are the singular points of this function, and what is its asymptotics close to these points. In this book, we study such analytic properties of three famous classes of func tions: the volume functions, which appear in the Archimedes-Newton problem on in tegrable bodies; the Newton-Coulomb potentials, and the Green functions of hyperbolic equations (studied, in particular, in the Hada mard-Petrovskii-Atiyah-Bott-Garding lacuna theory).
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Reseña del editor
Many special functions occuring in physics and partial differential equations can be represented by integral transformatIons: the fundamental solutions of many PDE's, Newton-Coulomb potentials, hypergeometric functions, Feynman integrals, initial data of (inverse) tomography problems, etc. The general picture of such transfor mations is as follows. There is an analytic fibre bundle E --+ T, a differential form w on E, whose restrictions on the fibres are closed, and a family of cycles in these fibres, parametrized by the points of T and depending continuously on these points. Then the integral of the form w along these cycles is a function on the base. The analytic properties of such functions depend on the monodromy action, i.e., on the natural action of the fundamental group of the base in the homology of the fibre: this action on the integration cycles defines the ramification of the analytic continuation of our function. The study of this action (which is a purely topological problem) can answer questions about the analytic behaviour of the integral function, for instance, is this function single-valued or at least algebraic, what are the singular points of this function, and what is its asymptotics close to these points. In this book, we study such analytic properties of three famous classes of func tions: the volume functions, which appear in the Archimedes-Newton problem on in tegrable bodies; the Newton-Coulomb potentials, and the Green functions of hyperbolic equations (studied, in particular, in the Hada mard-Petrovskii-Atiyah-Bott-Garding lacuna theory).
Reseña del editor
This volume contains an introduction to the Picard--Lefschetz theory, which controls the ramification and qualitative behaviour of many important functions of PDEs and integral geometry, and its foundations in singularity theory.
Solutions to many problems of these theories are treated. Subjects include the proof of multidimensional analogues of Newton's theorem on the nonintegrability of ovals; extension of the proofs for the theorems of Newton, Ivory, Arnold and Givental on potentials of algebraic surfaces. Also, it is discovered for which d and n the potentials of degree d hyperbolic surfaces in R n are algebraic outside the surfaces; the equivalence of local regularity (the so-called sharpness), of fundamental solutions of hyperbolic PDEs and the topological Petrovskii--Atiyah--Bott--Gårding condition is proved, and the geometrical characterization of domains of sharpness close to simple singularities of wave fronts is considered; a `stratified' version of the Picard--Lefschetz formula is proved, and an algorithm enumerating topologically distinct Morsifications of real function singularities is given.
This book will be valuable to those who are interested in integral transforms, operational calculus, algebraic geometry, PDEs, manifolds and cell complexes and potential theory.
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Ramified Integrals, Singularities and Lacunas (Mathematics and Its Applications, 315)
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Ramified Integrals, Singularities and Lacunas
Verlag:
Springer Netherlands, Springer Netherlands, 1994
ISBN 10: 0792331931
ISBN 13: 9780792331933
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Hardcover
Anbieter: AHA-BUCH GmbH, Einbeck, Deutschland
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Buch. Zustand: Neu. Druck auf Anfrage Neuware - Printed after ordering - Many special functions occuring in physics and partial differential equations can be represented by integral transformatIons: the fundamental solutions of many PDE's, Newton-Coulomb potentials, hypergeometric functions, Feynman integrals, initial data of (inverse) tomography problems, etc. The general picture of such transfor mations is as follows. There is an analytic fibre bundle E --+ T, a differential form w on E, whose restrictions on the fibres are closed, and a family of cycles in these fibres, parametrized by the points of T and depending continuously on these points. Then the integral of the form w along these cycles is a function on the base. The analytic properties of such functions depend on the monodromy action, i.e., on the natural action of the fundamental group of the base in the homology of the fibre: this action on the integration cycles defines the ramification of the analytic continuation of our function. The study of this action (which is a purely topological problem) can answer questions about the analytic behaviour of the integral function, for instance, is this function single-valued or at least algebraic, what are the singular points of this function, and what is its asymptotics close to these points. In this book, we study such analytic properties of three famous classes of func tions: the volume functions, which appear in the Archimedes-Newton problem on in tegrable bodies; the Newton-Coulomb potentials, and the Green functions of hyperbolic equations (studied, in particular, in the Hada mard-Petrovskii-Atiyah-Bott-Garding lacuna theory). Artikel-Nr. 9780792331933
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