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Luca Rizzi - One of the best experts on this subject based on the ideXlab platform.
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On the Essential Self-Adjointness of Singular Sub-Laplacians
Potential Analysis, 2019Co-Authors: Valentina Franceschi, Davide Prandi, Luca RizziAbstract:We prove a general essential self-adjointness criterion for sub-Laplacians on complete sub-Riemannian manifolds, defined with respect to singular measures. We also show that, in the compact case, this criterion implies discreteness of the sub-Laplacian spectrum even though the total volume of the manifold is infinite. As a consequence of our result, the intrinsic sub-Laplacian (i.e. defined w.r.t. Popp’s measure) is essentially self-adjoint on the equiregular connected components of a sub-Riemannian manifold. This settles a conjecture formulated by Boscain and Laurent (Ann. Inst. Fourier (Grenoble) 63 (5), 1739–1770, 2013 ), under mild regularity assumptions on the singular region, and when the latter does not contain characteristic points.
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Intrinsic random walks and sub-Laplacians in sub-Riemannian geometry
Advances in Mathematics, 2017Co-Authors: Ugo Boscain, Robert Neel, Luca RizziAbstract:On a sub-Riemannian manifold we define two type of Laplacians. The macroscopic Laplacian ∆ω, as the divergence of the horizontal gradient, once a volume ω is fixed, and the microscopic Laplacian, as the operator associated with a geodesic random walk. We consider a general class of random walks, where all sub-Riemannian geodesics are taken in account. This operator depends only on the choice of a complement c to the sub-Riemannian distribution, and is denoted L c. We address the problem of equivalence of the two operators. This problem is interesting since, on equiregular sub-Riemannian manifolds, there is always an intrinsic volume (e.g. Popp's one P) but not a canonical choice of complement. The result depends heavily on the type of structure under investigation: • On contact structures, for every volume ω, there exists a unique complement c such that ∆ω = L c. • On Carnot groups, if H is the Haar volume, then there always exists a complement c such that ∆H = L c. However this complement is not unique in general. • For quasi-contact structures, in general, ∆P = L c for any choice of c. In particular, L c is not symmetric w.r.t. Popp's measure. This is surprising especially in dimension 4 where, in a suitable sense, ∆P is the unique intrinsic macroscopic Laplacian. A crucial notion that we introduce here is the N-intrinsic volume, i.e. a volume that depends only on the set of parameters of the nilpotent approximation. When the nilpotent approximation does not depend on the point, a N-intrinsic volume is unique up to a scaling by a constant and the corresponding N-intrinsic sub-Laplacian is unique. This is what happens for dimension smaller or equal than 4, and in particular in the 4-dimensional quasi-contact structure mentioned above.
Robert S Strichartz - One of the best experts on this subject based on the ideXlab platform.
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polynomials on the sierpinski gasket with respect to different Laplacians which are symmetric and self similar
arXiv: Classical Analysis and ODEs, 2019Co-Authors: Christian Loring, Jacob W Ogden, Ely Sandine, Robert S StrichartzAbstract:We study the analogue of polynomials (solutions to $\Delta^{n+1} u =0$ for some $n$) on the Sierpinski gasket ($SG$) with respect to a family of symmetric, self-similar Laplacians constructed by Fang, King, Lee, and Strichartz, extending the work of Needleman, Strichartz, Teplyaev, and Yung on the polynomials with respect to the standard Kigami Laplacian. We define a basis for the space of polynomials, the monomials, characterized by the property that a certain "derivative" is 1 at one of the boundary points, while all other "derivatives" vanish, and we compute the values of the monomials at the boundary points of $SG$. We then present some data which suggest surprising relationships between the values of the monomials at the boundary and certain Neumann eigenvalues of the family of symmetric self-similar Laplacians. Surprisingly, the results for the general case are quite different from the results for the Kigami Laplacian.
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using peano curves to construct Laplacians on fractals
Fractals, 2015Co-Authors: Denali Molitor, Nadia Ott, Robert S StrichartzAbstract:We describe a new method to construct Laplacians on fractals using a Peano curve from the circle onto the fractal, extending an idea that has been used in the case of certain Julia sets. The Peano curve allows us to visualize eigenfunctions of the Laplacian by graphing the pullback to the circle. We study in detail three fractals: the pentagasket, the octagasket and the magic carpet. We also use the method for two nonfractal self-similar sets, the torus and the equilateral triangle, obtaining appealing new visualizations of eigenfunctions on the triangle. In contrast to the many familiar pictures of approximations to standard Peano curves, that do no show self-intersections, our descriptions of approximations to the Peano curves have self-intersections that play a vital role in constructing graph approximations to the fractal with explicit graph Laplacians that give the fractal Laplacian in the limit.
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fractal differential equations on the sierpinski gasket
Journal of Fourier Analysis and Applications, 1999Co-Authors: Kyallee Dalrymple, Robert S Strichartz, Jade P VinsonAbstract:Let Δ denote the symmetric Laplacian on the Sierpinski gasket SG defined by Kigami [11] as a renormalized limit of graph Laplacians on the sequence of pregaskets Gm whose limit is SG. We study the analogs of some of the classical partial differential equations with Δ playing the role of the usual Laplacian. For harmonic functions, biharmonic functions, and Dirichlet eigenfunctions of Δ, we give efficient algorithms to compute the solutions exactly, we display the results of implementing these algorithms, and we prove various properties of the solutions that are suggested by the data. Completing the work of Fukushima and Shima [8] who computed the Dirichlet eigenvalues and their multiplicities, we show how to construct a basis (but not orthonormal) for the eigenspaces, so that we have the analog of Fourier sine series on the unit interval. We also show that certain eigenfunctions have the property that they are a nonzero constant along certain lines contained in SG. For the analogs of the heat and wave equation, we give algorithms for approximating the solution, and display the results of implementing these algorithms. We give strong evidence that the analog of finite propagation for the wave equation does not hold because of inconsistent scaling behavior in space and time.
Elena Panteley - One of the best experts on this subject based on the ideXlab platform.
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On the influence of noise in randomized consensus algorithms
IEEE Control Systems Letters, 2021Co-Authors: Renato Vizuete, Paolo Frasca, Elena PanteleyAbstract:In this paper we study the influence of additive noise in randomized consensus algorithms. Assuming that the update matrices are symmetric, we derive a closed form expression for the mean square error induced by the noise, together with upper and lower bounds that are simpler to evaluate. Motivated by the study of Open Multi-Agent Systems, we concentrate on Randomly Induced Discretized Laplacians, a family of update matrices that are generated by sampling subgraphs of a large undirected graph. For these matrices, we express the bounds by using the eigenvalues of the Laplacian matrix of the underlying graph or the graph's average effective resistance, thereby proving their tightness. Finally, we derive expressions for the bounds on some examples of graphs and numerically evaluate them.
Valentina Franceschi - One of the best experts on this subject based on the ideXlab platform.
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On the Essential Self-Adjointness of Singular Sub-Laplacians
Potential Analysis, 2019Co-Authors: Valentina Franceschi, Davide Prandi, Luca RizziAbstract:We prove a general essential self-adjointness criterion for sub-Laplacians on complete sub-Riemannian manifolds, defined with respect to singular measures. We also show that, in the compact case, this criterion implies discreteness of the sub-Laplacian spectrum even though the total volume of the manifold is infinite. As a consequence of our result, the intrinsic sub-Laplacian (i.e. defined w.r.t. Popp’s measure) is essentially self-adjoint on the equiregular connected components of a sub-Riemannian manifold. This settles a conjecture formulated by Boscain and Laurent (Ann. Inst. Fourier (Grenoble) 63 (5), 1739–1770, 2013 ), under mild regularity assumptions on the singular region, and when the latter does not contain characteristic points.
U Lindstrom - One of the best experts on this subject based on the ideXlab platform.
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super Laplacians and their symmetries
Journal of High Energy Physics, 2017Co-Authors: P S Howe, U LindstromAbstract:A super-Laplacian is a set of differential operators in superspace whose highest-dimensional component is given by the spacetime Laplacian. Symmetries of super-Laplacians are given by linear differ ...
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super Laplacians and their symmetries
arXiv: High Energy Physics - Theory, 2016Co-Authors: P S Howe, U LindstromAbstract:A super-Laplacian is a set of differential operators in superspace whose highest-dimensional component is given by the spacetime Laplacian. Symmetries of super-Laplacians are given by linear differential operators of arbitrary finite degree and are determined by superconformal Killing tensors. We investigate these operators and their symmetries in flat superspaces. The differential operators form an algebra which can be identified in many cases with the tensor algebra of the relevant superconformal Lie algebra modulo a certain ideal, and which have applications to Higher Spin theories.
