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Presents the foundational work of David Hilbert in a sequence of thematically organized essays. They first trace the roots of Hilbert's work to the radical transformation of mathematics in the 19th century and bring out his pivotal role in creating mathematicallogic and proof theory. They then analyze techniques and results of "classical" proof theory as well as their dramatic expansion in modern proof theory. This intellectual experience finally opens horizons for reflection on the nature of mathematics in the 21st century: Sieg articulates his position of reductive structuralism and explores mathematical capacities via computational models.
Mathematical logic --- Philosophy of science --- Hilbert, David --- Mathematics --- Mathématiques --- Philosophy. --- Philosophie --- Hilbert, David, --- Philosophie. --- Logic of mathematics --- Mathematics, Logic of --- Gilʹbert, D., --- Hilbert, D. --- 希爾伯特, --- Mathématiques
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Differential equations, Partial --- Riemann-Hilbert problems. --- Asymptotic theory.
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Differential equations --- Differential equations, Partial --- Riemann-Hilbert problems --- Equations aux dérivées partielles --- Riemann-Hilbert, Problèmes de --- Asymptotic theory. --- Théorie asymptotique --- Riemann-Hilbert problems. --- 51 <082.1> --- Mathematics--Series --- Equations aux dérivées partielles --- Riemann-Hilbert, Problèmes de --- Théorie asymptotique --- Hilbert-Riemann problems --- Riemann problems --- Boundary value problems --- Asymptotic theory in partial differential equations --- Asymptotic expansions --- Asymptotic theory
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In a certain sense characteristic functions and correlation functions are the same, the common underlying concept is positive definiteness. Many results in probability theory, mathematical statistics and stochastic processes can be derived by using these functions. While there are books on characteristic functions of one variable, books devoting some sections to the multivariate case, and books treating the general case of locally compact groups, interestingly there is no book devoted entirely to the multidimensional case which is extremely important for applications. This book is intended to fill this gap at least partially. It makes the basic concepts and results on multivariate characteristic and correlation functions easily accessible to both students and researchers in a comprehensive manner. The first chapter presents basic results and should be read carefully since it is essential for the understanding of the subsequent chapters. The second chapter is devoted to correlation functions, their applications to stationary processes and some connections to harmonic analysis. In Chapter 3 we deal with several special properties, Chapter 4 is devoted to the extension problem while Chapter 5 contains a few applications. A relatively large appendix comprises topics like infinite products, functional equations, special functions or compact operators.
Characteristic functions. --- Correlation (Statistics) --- Variables (Mathematics) --- Multivariate analysis. --- Multivariate distributions --- Multivariate statistical analysis --- Statistical analysis, Multivariate --- Analysis of variance --- Mathematical statistics --- Matrices --- Mathematical constants --- Mathematics --- Least squares --- Probabilities --- Regression analysis --- Statistics --- Instrumental variables (Statistics) --- Characteristic formula of an ideal --- Characteristic Hilbert functions --- Functions, Characteristic --- Functions, Hilbert --- Hilbert characteristic functions --- Hilbert functions --- Hilbert's characteristic functions --- Hilbert's functions --- Postulation formula --- Graphic methods --- Characteristic Functions. --- Fourier Transform. --- Moment Problem. --- Probability Distribution.
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Bernard Helffer's graduate-level introduction to the basic tools in spectral analysis is illustrated by numerous examples from the Schrödinger operator theory and various branches of physics: statistical mechanics, superconductivity, fluid mechanics and kinetic theory. The later chapters also introduce non self-adjoint operator theory with an emphasis on the role of the pseudospectra. The author's focus on applications, along with exercises and examples, enables readers to connect theory with practice so that they develop a good understanding of how the abstract spectral theory can be applied. The final chapter provides various problems that have been the subject of active research in recent years and will challenge the reader's understanding of the material covered.
Spectral theory (Mathematics) --- Functional analysis --- Hilbert space --- Measure theory --- Transformations (Mathematics)
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This lively and accessible book describes the theory and applications of Hilbert spaces and also presents the history of the subject to reveal the ideas behind theorems and the human struggle that led to them. The authors begin by establishing the concept of 'countably infinite', which is central to the proper understanding of separable Hilbert spaces. Fundamental ideas such as convergence, completeness and dense sets are first demonstrated through simple familiar examples and then formalised. Having addressed fundamental topics in Hilbert spaces, the authors then go on to cover the theory of bounded, compact and integral operators at an advanced but accessible level. Finally, the theory is put into action, considering signal processing on the unit sphere, as well as reproducing kernel Hilbert spaces. The text is interspersed with historical comments about central figures in the development of the theory, which helps bring the subject to life.
Signal processing --- Hilbert space. --- Banach spaces --- Hyperspace --- Inner product spaces --- Mathematics.
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The study of measure-valued processes in random environments has seen some intensive research activities in recent years whereby interesting nonlinear stochastic partial differential equations (SPDEs) were derived. Due to the nonlinearity and the non-Lipschitz continuity of their coefficients, new techniques and concepts have recently been developed for the study of such SPDEs. These include the conditional Laplace transform technique, the conditional mild solution, and the bridge between SPDEs and some kind of backward stochastic differential equations. This volume provides an introduction to
Stochastic partial differential equations. --- Banach spaces, Stochastic differential equations in --- Hilbert spaces, Stochastic differential equations in --- SPDE (Differential equations) --- Stochastic differential equations in Banach spaces --- Stochastic differential equations in Hilbert spaces --- Differential equations, Partial --- Differential equations, Nonlinear.
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The notes in this volume correspond to advanced courses held at the Centre de Recerca Matemàtica as part of the research program in Arithmetic Geometry in the 2009-2010 academic year. The notes by Laurent Berger provide an introduction to p-adic Galois representations and Fontaine rings, which are especially useful for describing many local deformation rings at p that arise naturally in Galois deformation theory. The notes by Gebhard Böckle offer a comprehensive course on Galois deformation theory, starting from the foundational results of Mazur and discussing in detail the theory of pseudo-representations and their deformations, local deformations at places l ≠ p and local deformations at p which are flat. In the last section,the results of Böckle and Kisin on presentations of global deformation rings over local ones are discussed. The notes by Mladen Dimitrov present the basics of the arithmetic theory of Hilbert modular forms and varieties, with an emphasis on the study of the images of the attached Galois representations, on modularity lifting theorems over totally real number fields, and on the cohomology of Hilbert modular varieties with integral coefficients. The notes by Lassina Dembélé and John Voight describe methods for performing explicit computations in spaces of Hilbert modular forms. These methods depend on the Jacquet-Langlands correspondence and on computations in spaces of quaternionic modular forms, both for the case of definite and indefinite quaternion algebras. Several examples are given, and applications to modularity of Galois representations are discussed. The notes by Tim Dokchitser describe the proof, obtained by the author in a joint project with Vladimir Dokchitser, of the parity conjecture for elliptic curves over number fields under the assumption of finiteness of the Tate-Shafarevich group. The statement of the Birch and Swinnerton-Dyer conjecture is included, as well as a detailed study of local and global root numbers of elliptic curves and their classification.
Galois theory --- Curves, Elliptic --- Hilbert modular surfaces --- Mathematics --- Physical Sciences & Mathematics --- Algebra --- Modular surfaces, Hilbert --- Elliptic curves --- Mathematics. --- Algebra. --- Algebraic geometry. --- Number theory. --- Number Theory. --- Algebraic Geometry. --- Galois theory. --- Curves, Elliptic. --- Hilbert modular surfaces. --- Forms, Modular --- Surfaces --- Equations, Theory of --- Group theory --- Number theory --- Curves, Algebraic --- Geometry, algebraic. --- Mathematical analysis --- Algebraic geometry --- Geometry --- Number study --- Numbers, Theory of --- Geometry, Algebraic.
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Stochastic partial differential equations --- Stochastic partial differential equations. --- Banach spaces, Stochastic differential equations in --- Hilbert spaces, Stochastic differential equations in --- SPDE (Differential equations) --- Stochastic differential equations in Banach spaces --- Stochastic differential equations in Hilbert spaces --- Differential equations, Partial --- Mathematical Statistics --- Équations aux dérivées partielles stochastiques --- Anàlisi estocàstica --- Equacions en derivades parcials --- Anàlisi estocàstica. --- Equacions en derivades parcials.
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This edition includes new material on discontinuous Galerkin methods, non-tensorial nodal spectral element methods in simplex domains, and stabilisation and filtering techniques.
Fluid dynamics. --- Spectral theory (Mathematics) --- Finite element method. --- FEA (Numerical analysis) --- FEM (Numerical analysis) --- Finite element analysis --- Numerical analysis --- Isogeometric analysis --- Functional analysis --- Hilbert space --- Measure theory --- Transformations (Mathematics) --- Dynamics --- Fluid mechanics
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