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Register a new userNo embedding of the automorphisms of a topological space into a compact metric space endows them with a composition that passes to the limit
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Added by
Patrizio Frosini
Publication date
2010
fields
Informatics Engineering
and research's language is
English
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No Arabic abstract
The Hausdorff distance, the Gromov-Hausdorff, the Frechet and the natural pseudo-distances are instances of dissimilarity measures widely used in shape comparison. We show that they share the property of being defined as $inf_rho F(rho)$ where $F$ is a suitable functional and $rho$ varies in a set of correspondences containing the set of homeomorphisms. Our main result states that the set of homeomorphisms cannot be enlarged to a metric space $mathcal{K}$, in such a way that the composition in $mathcal{K}$ (extending the composition of homeomorphisms) passes to the limit and, at the same time, $mathcal{K}$ is compact.
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Poisson processes in the space of $(d-1)$-dimensional totally geodesic subspaces (hyperplanes) in a $d$-dimensional hyperbolic space of constant curvature $-1$ are studied. The $k$-dimensional Hausdorff measure of their $k$-skeleton is considered. Explicit formulas for first- and second-order quantities restricted to bounded observation windows are obtained. The central limit problem for the $k$-dimensional Hausdorff measure of the $k$-skeleton is approached in two different set-ups: (i) for a fixed window and growing intensities, and (ii) for fixed intensity and growing spherical windows. While in case (i) the central limit theorem is valid for all $dgeq 2$, it is shown that in case (ii) the central limit theorem holds for $din{2,3}$ and fails if $dgeq 4$ and $k=d-1$ or if $dgeq 7$ and for general $k$. Also rates of convergence are studied and multivariate central limit theorems are obtained. Moreover, the situation in which the intensity and the spherical window are growing simultaneously is discussed. In the background are the Malliavin-Stein method for normal approximation and the combinatorial moment structure of Poisson U-statistics as well as tools from hyperbolic integral geometry.
It is shown that there exists a dihedral acute triangulation of the three-dimensional cube. The method of constructing the acute triangulation is described, and symmetries of the triangulation are discussed.
The morphometric approach [HRC13,RHK06] writes the solvation free energy as a linear combination of weight
Representing an atom by a solid sphere in $3$-dimensional Euclidean space, we get the space-filling diagram of a molecule by taking the union. Molecular dynamics simulates its motion subject to bonds and other forces, including the solvation free energy. The morphometric approach [HRC13,RHK06] writes the latter as a linear combination of weight
It is known that on a closed manifold of dimension greater than one, every smooth weak Riemannian metric on the space of smooth positive densities that is invariant under the action of the diffeomorphism group, is of the form $$ G_mu(alpha,beta)=C_1(mu(M)) int_M frac{alpha}{mu}frac{beta}{mu},mu + C_2(mu(M)) int_Malpha cdot int_Mbeta $$ for some smooth functions $C_1,C_2$ of the total volume $mu(M)$. Here we determine the geodesics and the curvature of this metric and study geodesic and metric completeness.
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