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Optimal bounds for the colored Tverberg problem

Pavle V. M. Blagojević, Benjamin Matschke, Günter M. Ziegler (2015)

Journal of the European Mathematical Society

We prove a “Tverberg type” multiple intersection theorem. It strengthens the prime case of the original Tverberg theorem from 1966, as well as the topological Tverberg theorem of Bárány et al. (1980), by adding color constraints. It also provides an improved bound for the (topological) colored Tverberg problem of Bárány & Larman (1992) that is tight in the prime case and asymptotically optimal in the general case. The proof is based on relative equivariant obstruction theory.

Optimal isometries for a pair of compact convex subsets of ℝⁿ

Irmina Herburt, Maria Moszyńska (2009)

Banach Center Publications

In 1989 R. Arnold proved that for every pair (A,B) of compact convex subsets of ℝ there is an Euclidean isometry optimal with respect to L₂ metric and if f₀ is such an isometry, then the Steiner points of f₀(A) and B coincide. In the present paper we solve related problems for metrics topologically equivalent to the Hausdorff metric, in particular for L p metrics for all p ≥ 2 and the symmetric difference metric.

Optimal packings for filled rings of circles

Dinesh B. Ekanayake, Manjula Mahesh Ranpatidewage, Douglas J. LaFountain (2020)

Applications of Mathematics

General circle packings are arrangements of circles on a given surface such that no two circles overlap except at tangent points. In this paper, we examine the optimal arrangement of circles centered on concentric annuli, in what we term rings. Our motivation for this is two-fold: first, certain industrial applications of circle packing naturally allow for filled rings of circles; second, any packing of circles within a circle admits a ring structure if one allows for irregular spacing of circles...

Optimality conditions for maximizers of the information divergence from an exponential family

František Matúš (2007)

Kybernetika

The information divergence of a probability measure P from an exponential family ℰ over a finite set is defined as infimum of the divergences of P from Q subject to Q ∈ ℰ . All directional derivatives of the divergence from ℰ are explicitly found. To this end, behaviour of the conjugate of a log-Laplace transform on the boundary of its domain is analysed. The first order conditions for P to be a maximizer of the divergence from ℰ are presented, including new ones when P  is not projectable to ℰ .

Optimisation hybride par colonies de fourmis pour le problème de découpe à deux dimensions

Alice Yalaoui, Chengbin Chu (2009)

RAIRO - Operations Research

Nous nous intéressons dans cet article au problème de découpe guillotine en deux dimensions noté 2BP/O/G. Il s'agit de découper un certain nombre de pièces rectangulaires dans un ensemble de plaques de matière première, elles même rectangulaires et identiques. Celles-ci sont disponibles en quantité illimitée. L'objectif est de minimiser le nombre de plaques utilisées pour satisfaire la demande, en appliquant une succession de coupes, dites guillotines, allant de bout en bout. Nous proposons une approche...

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