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In this text, the authors consider fully discretized p-Laplacian problems (evolution, boundary value and variational problems) on graphs. The motivation of nonlocal continuum limits comes from the quest of understanding collective dynamics in large ensembles of interacting particles, which is a fundamental problem in nonlinear science, with applications ranging from biology to physics, chemistry and computer science. Using the theory of graphons, the authors give a unified treatment of all the above problems and establish the continuum limit for each of them together with non-asymptotic convergence rates. They also describe an algorithmic framework based proximal splitting to solve these discrete problems on graphs.

Graph theory. --- Laplacian operator. --- Differential equations, Partial.

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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. --- Calderón-Zygmund operator. --- Laplacian operator. --- Lipschitz spaces. --- Potential theory (Mathematics)

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Laplacian operator. --- Geometry, Differential. --- Eigenvalues. --- Géométrie différentielle --- Laplacien --- Valeurs propres --- Géométrie différentielle. --- Laplacien. --- Valeurs propres. --- Géometrie différentielle

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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