1,721,018 research outputs found
Some explicit examples of minimizers for the irrigation problem
No full textWe construct some examples of explicit solutions to the problem where the minimum is over all connected compact sets of prescribed one-dimensional Hausdorff measure. More precisely we show that, if is a curve of length with curvature bounded by , and , then is a solution to the above problem with being the -neighbourhood of . In particular, regularity is optimal for this proble
Compliance estimates for two-dimensional problems with Dirichlet region of prescribed length
No full textIn this paper we prove some lower bounds for the compliance functional, in terms of the 1-dimensional Hausdorff measure of the Dirichlet region and the number of its connected components. When the measure of the Dirichlet region is large, these estimates are asymptotically optimal and yield a proof of a conjecture by Buttazzo and Santambrogi
Convergence conditions of some methods for the simultaneous computation of polynomial zeros
No full text
Universal bounds on the convergence rate of extreme Toeplitz eigenvalues
No full textAbstractLet {Tn} denote the sequence of Toeplitz matrices associated with f, a non-negative integrable function such that inff=0 and supf>0. It is well known that Tn is ill conditioned since λmin(Tn), the smallest eigenvalue of Tn, tends to zero as n→∞. If f satisfies some smoothness conditions, then the convergence rate depends on the zeros of f. Here we prove that λmin(Tn) mimics the zeros of f only up to exponential convergence, i.e., λmin(Tn) is always bounded from below by exp(−cn), where c>0 depends on f, under no smoothness assumption on f. Furthermore, for multivariate f, an even stronger bound is valid. We also investigate Toeplitz matrices generated by positive measures, not necessarily absolutely continuous with respect to the Lebesgue measure, showing that in this case the convergence to zero of λmin(Tn) can be arbitrarily fast
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