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Lectures: M.F. Atiyah: Classical groups and classical differential operators on manifolds.- R. Bott: Some aspects of invariant theory in differential geometry.- E.M. Stein: Singular integral operators and nilpotent groups.- Seminars: P. Malliavin: Diffusion et géométrie différentielle globale.- S. Helgason: Solvability of invariant differential operators on homonogeneous manifolds.
Manifolds (Mathematics) --- Differential operators --- Operators, Differential --- Mathematics. --- Mathematics, general. --- Differential equations --- Operator theory --- Math --- Science
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Transmutation Theory and Applications
Differential operators. --- Transmutation operators. --- Operators, Transmutation --- Operator theory --- Operators, Differential --- Differential equations
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Spectral Theory of Differential Operators
Operator theory --- Differential operators --- Spectral theory (Mathematics) --- Operators, Differential --- Differential equations
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The main aim of this book is to introduce the reader to the concept of comparison algebra, defined as a type of C*-algebra of singular integral operators. The first part of the book develops the necessary elements of the spectral theory of differential operators as well as the basic properties of elliptic second order differential operators. The author then introduces comparison algebras and describes their theory in L2-spaces and L2-Soboler spaces, and in particular their importance in solving functional analytic problems involving differential operators. The book is based on lectures given in Sweden and the USA.
Differential operators. --- Linear operators. --- Linear maps --- Maps, Linear --- Operators, Linear --- Operator theory --- Operators, Differential --- Differential equations
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Chapter 1 presents theorems on differentiable functions often used in differential topology, such as the implicit function theorem, Sard's theorem and Whitney's approximation theorem.The next chapter is an introduction to real and complex manifolds. It contains an exposition of the theorem of Frobenius, the lemmata of Poincaré and Grothendieck with applications of Grothendieck's lemma to complex analysis, the imbedding theorem of Whitney and Thom's transversality theorem. Chapter 3 includes characterizations of linear differentiable operators, due to Peetre and Hormander. The ineq
Differentiable manifolds. --- Complex manifolds. --- Differential operators. --- Operators, Differential --- Differential equations --- Operator theory --- Analytic spaces --- Manifolds (Mathematics) --- Differential manifolds
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The theory of Lebesgue and Sobolev spaces with variable integrability is experiencing a steady expansion, and is the subject of much vigorous research by functional analysts, function-space analysts and specialists in nonlinear analysis. These spaces have attracted attention not only because of their intrinsic mathematical importance as natural, interesting examples of non-rearrangement-invariant function spaces but also in view of their applications, which include the mathematical modeling of electrorheological fluids and image restoration. The main focus of this book is to provide a solid fu
Function spaces. --- Sobolev spaces. --- Differential operators. --- Spaces, Sobolev --- Function spaces --- Operators, Differential --- Differential equations --- Operator theory --- Spaces, Function --- Functional analysis
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This book examines the conditions for the semi-boundedness of partial differential operators, which are interpreted in different ways. For example, today we know a great deal about L2-semibounded differential and pseudodifferential operators, although their complete characterization in analytic terms still poses difficulties, even for fairly simple operators. In contrast, until recently almost nothing was known about analytic characterizations of semi-boundedness for differential operators in other Hilbert function spaces and in Banach function spaces. This book works to address that gap. As such, various types of semi-boundedness are considered and a number of relevant conditions which are either necessary and sufficient or best possible in a certain sense are presented. The majority of the results reported on are the authors’ own contributions.
Differential operators. --- Operators, Differential --- Differential equations --- Operator theory --- Operator theory. --- Differential equations, partial. --- Operator Theory. --- Partial Differential Equations. --- Partial differential equations --- Functional analysis --- Partial differential equations.
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This short introduction to microlocal analysis is presented, in the spirit of Hörmander, in the classical framework of partial differential equations. This theory has important applications in areas such as harmonic and complex analysis, and also in theoretical physics. Here Grigis and Sjöstrand emphasise the basic tools, especially the method of stationary phase, and they discuss wavefront sets, elliptic operators, local symplectic geometry, and WKB-constructions. The contents of the book correspond to a graduate course given many times by the authors. It should prove to be useful to mathematicians and theoretical physicists, either to enrich their general knowledge of this area, or as preparation for the current research literature.
Differential operators. --- Microlocal analysis. --- Functional analysis --- Operators, Differential --- Differential equations --- Operator theory --- Pseudodifferential operators --- Opérateurs pseudo-différentiels --- Singularities (Mathematics) --- Singularités (mathématiques) --- Analyse microlocale --- Fourier, Opérateurs intégraux de
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The study of arithmetic differential operators is a novel and promising area of mathematics. This complete introduction to the subject starts with the basics: a discussion of p-adic numbers and some of the classical differential analysis on the field of p-adic numbers leading to the definition of arithmetic differential operators on this field. Buium's theory of arithmetic jet spaces is then developed succinctly in order to define arithmetic operators in general. Features of the book include a comparison of the behaviour of these operators over the p-adic integers and their behaviour over the unramified completion, and a discussion of the relationship between characteristic functions of p-adic discs and arithmetic differential operators that disappears as soon as a single root of unity is adjoined to the p-adic integers. This book is essential reading for researchers and graduate students who want a first introduction to arithmetic differential operators over the p-adic integers.
Differential operators. --- Arithmetic functions. --- p-adic numbers. --- Numbers, p-adic --- Number theory --- p-adic analysis --- Functions, Arithmetic --- Functions of complex variables --- Operators, Differential --- Differential equations --- Operator theory
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Periodic differential operators have a rich mathematical theory as well as important physical applications. They have been the subject of intensive development for over a century and remain a fertile research area. This book lays out the theoretical foundations and then moves on to give a coherent account of more recent results, relating in particular to the eigenvalue and spectral theory of the Hill and Dirac equations. The book will be valuable to advanced students and academics both for general reference and as an introduction to active research topics.
Mathematics. --- Operator algebras. --- Singularities (Mathematics). --- Differential operators --- Differential equations --- Operator theory --- Mathematics --- Physical Sciences & Mathematics --- Calculus --- Differential operators. --- Operators, Differential --- Operator theory. --- Differential equations. --- Ordinary Differential Equations. --- Operator Theory. --- 517.91 Differential equations --- Functional analysis --- Math --- Science --- Differential Equations.
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