Search results for "Regular polygon"
showing 10 items of 132 documents
The Bourgain property and convex hulls
2007
Let (Ω, Σ, μ) be a complete probability space and let X be a Banach space. We consider the following problem: Given a function f: Ω X for which there is a norming set B ⊂ BX * such that Zf,B = {x * ○ f: x * ∈ B } is uniformly integrable and has the Bourgain property, does it follow that f is Birkhoff integrable? It turns out that this question is equivalent to the following one: Given a pointwise bounded family ℋ ⊂ ℝΩ with the Bourgain property, does its convex hull co(ℋ) have the Bourgain property? With the help of an example of D. H. Fremlin, we make clear that both questions have negative answer in general. We prove that a function f: Ω X is scalarly measurable provided that there is a n…
Equidistribution of Common Perpendicular Arcs
2019
In this chapter, we prove the equidistribution of the initial and terminal vectors of common perpendiculars of convex subsets, at the universal covering space level, for Riemannian manifolds and for metric and simplicial trees.
An abstract inf-sup problem inspired by limit analysis in perfect plasticity and related applications
2021
This paper is concerned with an abstract inf-sup problem generated by a bilinear Lagrangian and convex constraints. We study the conditions that guarantee no gap between the inf-sup and related sup-inf problems. The key assumption introduced in the paper generalizes the well-known Babuška–Brezzi condition. It is based on an inf-sup condition defined for convex cones in function spaces. We also apply a regularization method convenient for solving the inf-sup problem and derive a computable majorant of the critical (inf-sup) value, which can be used in a posteriori error analysis of numerical results. Results obtained for the abstract problem are applied to continuum mechanics. In particular…
From the nearest neighbour rule to decision trees
1998
This paper proposes an algorithm to design a tree-like classifier whose result is equivalent to that achieved by the classical Nearest Neighbour rule. The procedure consists of a particular decomposition of a d-dimensional feature space into a set of convex regions with prototypes from just one class. Some experimental results over synthetic and real databases are provided in order to illustrate the applicability of the method.
From Wigner-Yanase-Dyson conjecture to Carlen-Frank-Lieb conjecture
2020
Abstract In this paper we study the joint convexity/concavity of the trace functions Ψ p , q , s ( A , B ) = Tr ( B q 2 K ⁎ A p K B q 2 ) s , p , q , s ∈ R , where A and B are positive definite matrices and K is any fixed invertible matrix. We will give full range of ( p , q , s ) ∈ R 3 for Ψ p , q , s to be jointly convex/concave for all K. As a consequence, we confirm a conjecture of Carlen, Frank and Lieb. In particular, we confirm a weaker conjecture of Audenaert and Datta and obtain the full range of ( α , z ) for α-z Renyi relative entropies to be monotone under completely positive trace preserving maps. We also give simpler proofs of many known results, including the concavity of Ψ p…
Adaptive mesh reconstruction for hyperbolic conservation laws with total variation bound
2012
We consider 3-point numerical schemes, that resolve scalar conservation laws, that are oscillatory either to their dispersive or anti-diffusive nature. The spatial discretization is performed over non-uniform adaptively redefined meshes. We provide a model for studying the evolution of the extremes of the oscillations. We prove that proper mesh reconstruction is able to control the oscillations; we provide bounds for the Total Variation (TV) of the numerical solution. We, moreover, prove under more strict assumptions that the increase of the TV, due to the oscillatory behavior of the numerical schemes, decreases with time; hence proving that the overall scheme is TV Increase-Decreasing (TVI…
Resonance of minimizers forn-level quantum systems with an arbitrary cost
2004
We consider an optimal control problem describing a laser-induced population transfer on a n-level quantum system. For a convex cost depending only on the moduli of controls ( i.e. the lasers intensities), we prove that there always exists a minimizer in resonance. This permits to justify some strategies used in experimental physics. It is also quite important because it permits to reduce remarkably the complexity of the problem (and extend some of our previous results for n=2 and n=3): instead of looking for minimizers on the sphere one is reduced to look just for minimizers on the sphere . Moreover, for the reduced problem, we investigate on the question of existence of strict abnormal mi…
A quantitative reverse Faber-Krahn inequality for the first Robin eigenvalue with negative boundary parameter
2021
The aim of this paper is to prove a quantitative form of a reverse Faber-Krahn type inequality for the first Robin Laplacian eigenvalueλβwith negative boundary parameter among convex sets of prescribed perimeter. In that framework, the ball is the only maximizer forλβand the distance from the optimal set is considered in terms of Hausdorff distance. The key point of our stategy is to prove a quantitative reverse Faber-Krahn inequality for the first eigenvalue of a Steklov-type problem related to the original Robin problem.
Locally Convex Quasi *-Algebras of Operators
2011
This note is mainly concerned with locally convex quasi C*-normed *-algebras which arise as completions of C*-algebras of operators under certain topologies. Their importance is made clear by the representation theory of abstract locally convex quasi C*-normed *-algebras, investigated in previous papers and whose basic aspects are also overviewed here.