134,439 research outputs found

    The Average-Case Area of Heilbronn-Type Triangles

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    From among (n3) {n \choose 3} triangles with vertices chosen from nn points in the unit square, let TT be the one with the smallest area, and let AA be the area of TT. Heilbronn's triangle problem asks for the maximum value assumed by AA over all choices of nn points. We consider the average-case: If the nn points are chosen independently and at random (with a uniform distribution), then there exist positive constants cc and CC such that c/n3<μn<C/n3c/n^3 < \mu_n < C/n^3 for all large enough values of nn, where μn\mu_n is the expectation of AA. Moreover, c/n3<A<C/n3c/n^3 < A < C/n^3, with probability close to one. Our proof uses the incompressibility method based on Kolmogorov complexity; it actually determines the area of the smallest triangle for an arrangement in ``general position.''Comment: 13 pages, LaTeX, 1 figure,Popular treatment in D. Mackenzie, On a roll, {\em New Scientist}, November 6, 1999, 44--4

    Pade-resummed high-order perturbation theory for nuclear structure calculations

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    We apply high-order many-body perturbation theory for the calculation of ground-state energies of closed-shell nuclei using realistic nuclear interactions. Using a simple recursive formulation, we compute the perturbative energy contributions up to 30th order and compare to exact no-core shell model calculations for the same model space and Hamiltonian. Generally, finite partial sums of this perturbation series do not show convergence with increasing order, but tend to diverge exponentially. Nevertheless, through a simple resummation via Pade approximants it is possible to extract rapidly converging and highly accurate results for the ground state energy.Comment: 6 pages, 2 figures, 1 tabl

    Going astray:Classics and the NSS

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    Bonnie and Clyde in Texas: The End of the Texas Outlaw Tradition

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    Spartan Salute, Vol. II

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    Spartan Soldier, Vol. I

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    Pinching of the First Eigenvalue of the Laplacian and almost-Einstein Hypersurfaces of the Euclidean Space

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    In this paper, we prove new pinching theorems for the first eigenvalue of the Laplacian on compact hypersurfaces of the Euclidean space. These pinching results are associated with the upper bound for the first eigenvalue in terms of higher order mean curvatures. We show that under a suitable pinching condition, the hypersurface is diffeomorpic and almost isometric to a standard sphere. Moreover, as a corollary, we show that a hypersurface of the Euclidean space which is almost Einstein is diffeomorpic and almost isometric to a standard sphere.Comment: 18 page
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