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The book constructs explicitly the fundamental solution of the sub-Laplacian operator for a family of model domains in Cn+1. This type of domain is a good point-wise model for a Cauchy-Rieman (CR) manifold with diagonalizable Levi form. Qualitative results for such operators have been studied extensively, but exact formulas are difficult to derive. Exact formulas are closely related to the underlying geometry and lead to equations of classical types such as hypergeometric equations and Whittaker’s equations.

Laplacian operator. --- Cauchy-Riemann equations.

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Potential theory (Mathematics) --- Laplacian operator --- Differential equations, Partial

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Mathematical physics --- 51 --- Mathematics --- Lagrange equations. --- Laplacian operator. --- Sound-waves --- Wave equation. --- Scattering. --- 51 Mathematics --- Equations aux derivees partielles --- Scattering

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"Fix d [greater than or equal to] 2, and s [epsilon] (d - 1, d). We characterize the non-negative locally finite non-atomic Borel measures [mu] in Rd for which the associated s-Riesz transform is bounded in L²([mu]) in terms of the Wolff energy. This extends the range of s in which the Mateu-Prat-Verdera characterization of measures with bounded s-Riesz transform is known. As an application, we give a metric characterization of the removable sets for locally Lipschitz continuous solutions of the fractional Laplacian operator (-[delta])[infinity]/2, [infinity] [epsilon] (1, 2), in terms of a well-known capacity from non-linear potential theory. This result contrasts sharply with removability results for Lipschitz harmonic functions"--

Harmonic analysis. --- Analyse harmonique (mathématiques) --- Singular integrals. --- Intégrales singulières. --- Euclidean algorithm. --- Euclide, Algorithme d'. --- Lipschitz spaces. --- Laplacian operator. --- Calderón-Zygmund operator. --- Potential theory (Mathematics)

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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