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Eigenvectors of graph Laplacians have not, to date, been the subject of expository articles and thus they may seem a surprising topic for a book. The authors propose two motivations for this new LNM volume: (1) There are fascinating subtle differences between the properties of solutions of Schrödinger equations on manifolds on the one hand, and their discrete analogs on graphs. (2) "Geometric" properties of (cost) functions defined on the vertex sets of graphs are of practical interest for heuristic optimization algorithms. The observation that the cost functions of quite a few of the well-studied combinatorial optimization problems are eigenvectors of associated graph Laplacians has prompted the investigation of such eigenvectors. The volume investigates the structure of eigenvectors and looks at the number of their sign graphs ("nodal domains"), Perron components, graphs with extremal properties with respect to eigenvectors. The Rayleigh quotient and rearrangement of graphs form the main methodology.

Eigenvectors. --- Laplacian operator. --- Graph theory. --- Vecteurs --- Laplacien --- Théorie des graphes --- Eigenvectors --- Laplacian operator --- Graph theory --- Applied Physics --- Algebra --- Engineering & Applied Sciences --- Mathematics --- Physical Sciences & Mathematics --- Graphs, Theory of --- Theory of graphs --- Operator, Laplacian --- Extremal problems --- Mathematics. --- Algebra. --- Matrix theory. --- Combinatorics. --- Linear and Multilinear Algebras, Matrix Theory. --- Combinatorics --- Mathematical analysis --- Math --- Science --- Combinatorial analysis --- Topology --- Differential equations, Partial --- Matrices --- Vector spaces --- Eigenfactor

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The Lévy Laplacian is an infinite-dimensional generalization of the well-known classical Laplacian. The theory has become well developed in recent years and this book was the first systematic treatment of the Lévy-Laplace operator. The book describes the infinite-dimensional analogues of finite-dimensional results, and more especially those features which appear only in the generalized context. It develops a theory of operators generated by the Lévy Laplacian and the symmetrized Lévy Laplacian, as well as a theory of linear and nonlinear equations involving it. There are many problems leading to equations with Lévy Laplacians and to Lévy-Laplace operators, for example superconductivity theory, the theory of control systems, the Gauss random field theory, and the Yang-Mills equation. The book is complemented by an exhaustive bibliography. The result is a work that will be valued by those working in functional analysis, partial differential equations and probability theory.

Functional analysis --- Laplacian operator. --- Lévy processes. --- Harmonic functions. --- Functions, Harmonic --- Laplace's equations --- Bessel functions --- Differential equations, Partial --- Fourier series --- Harmonic analysis --- Lamé's functions --- Spherical harmonics --- Toroidal harmonics --- Random walks (Mathematics) --- Operator, Laplacian --- Levy processes.

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The existence, for every sub-Laplacian, of a homogeneous fundamental solution smooth out of the origin, plays a crucial role in the book. This makes it possible to develop an exhaustive Potential Theory, almost completely parallel to that of the classical Laplace operator. This book provides an extensive treatment of Potential Theory for sub-Laplacians on stratified Lie groups. In recent years, sub-Laplacian operators have received considerable attention due to their special role in the theory of linear second-order PDE's with semidefinite characteristic form. It also provides a largely self-contained presentation of stratified Lie groups, and of their Lie algebra of left-invariant vector fields. The presentation is accessible to graduate students and requires no specialized knowledge in algebra nor in differential geometry. It is thus addressed, besides PhD students, to junior and senior researchers in different areas such as: partial differential equations; geometric control theory; geometric measure theory and minimal surfaces in stratified Lie groups.

Potential theory (Mathematics) --- Harmonic functions. --- Laplacian operator. --- Lie groups. --- Differential equations, Partial. --- Groups, Lie --- Lie algebras --- Symmetric spaces --- Topological groups --- Partial differential equations --- Operator, Laplacian --- Differential equations, Partial --- Functions, Harmonic --- Laplace's equations --- Bessel functions --- Fourier series --- Harmonic analysis --- Lamé's functions --- Spherical harmonics --- Toroidal harmonics --- Green's operators --- Green's theorem --- Potential functions (Mathematics) --- Potential, Theory of --- Mathematical analysis --- Mechanics --- Algebra. --- Differential equations, partial. --- Potential theory (Mathematics). --- Topological Groups. --- Partial Differential Equations. --- Potential Theory. --- Topological Groups, Lie Groups. --- Groups, Topological --- Continuous groups --- Mathematics --- Partial differential equations. --- Topological groups.

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This book presents the analytic foundations to the theory of the hypoelliptic Laplacian. The hypoelliptic Laplacian, a second-order operator acting on the cotangent bundle of a compact manifold, is supposed to interpolate between the classical Laplacian and the geodesic flow. Jean-Michel Bismut and Gilles Lebeau establish the basic functional analytic properties of this operator, which is also studied from the perspective of local index theory and analytic torsion. The book shows that the hypoelliptic Laplacian provides a geometric version of the Fokker-Planck equations. The authors give the proper functional analytic setting in order to study this operator and develop a pseudodifferential calculus, which provides estimates on the hypoelliptic Laplacian's resolvent. When the deformation parameter tends to zero, the hypoelliptic Laplacian converges to the standard Hodge Laplacian of the base by a collapsing argument in which the fibers of the cotangent bundle collapse to a point. For the local index theory, small time asymptotics for the supertrace of the associated heat kernel are obtained. The Ray-Singer analytic torsion of the hypoelliptic Laplacian as well as the associated Ray-Singer metrics on the determinant of the cohomology are studied in an equivariant setting, resulting in a key comparison formula between the elliptic and hypoelliptic analytic torsions.

Differential equations, Hypoelliptic. --- Laplacian operator. --- Metric spaces. --- Spaces, Metric --- Operator, Laplacian --- Hypoelliptic differential equations --- Generalized spaces --- Set theory --- Topology --- Differential equations, Partial --- Alexander Grothendieck. --- Analytic function. --- Asymptote. --- Asymptotic expansion. --- Berezin integral. --- Bijection. --- Brownian dynamics. --- Brownian motion. --- Chaos theory. --- Chern class. --- Classical Wiener space. --- Clifford algebra. --- Cohomology. --- Combination. --- Commutator. --- Computation. --- Connection form. --- Coordinate system. --- Cotangent bundle. --- Covariance matrix. --- Curvature tensor. --- Curvature. --- De Rham cohomology. --- Derivative. --- Determinant. --- Differentiable manifold. --- Differential operator. --- Dirac operator. --- Direct proof. --- Eigenform. --- Eigenvalues and eigenvectors. --- Ellipse. --- Embedding. --- Equation. --- Estimation. --- Euclidean space. --- Explicit formula. --- Explicit formulae (L-function). --- Feynman–Kac formula. --- Fiber bundle. --- Fokker–Planck equation. --- Formal power series. --- Fourier series. --- Fourier transform. --- Fredholm determinant. --- Function space. --- Girsanov theorem. --- Ground state. --- Heat kernel. --- Hilbert space. --- Hodge theory. --- Holomorphic function. --- Holomorphic vector bundle. --- Hypoelliptic operator. --- Integration by parts. --- Invertible matrix. --- Logarithm. --- Malliavin calculus. --- Martingale (probability theory). --- Matrix calculus. --- Mellin transform. --- Morse theory. --- Notation. --- Parameter. --- Parametrix. --- Parity (mathematics). --- Polynomial. --- Principal bundle. --- Probabilistic method. --- Projection (linear algebra). --- Rectangle. --- Resolvent set. --- Ricci curvature. --- Riemann–Roch theorem. --- Scientific notation. --- Self-adjoint operator. --- Self-adjoint. --- Sign convention. --- Smoothness. --- Sobolev space. --- Spectral theory. --- Square root. --- Stochastic calculus. --- Stochastic process. --- Summation. --- Supertrace. --- Symmetric space. --- Tangent space. --- Taylor series. --- Theorem. --- Theory. --- Torus. --- Trace class. --- Translational symmetry. --- Transversality (mathematics). --- Uniform convergence. --- Variable (mathematics). --- Vector bundle. --- Vector space. --- Wave equation.