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Generalized gradient flow and singularities of the Riemannian distance function

Piermarco Cannarsa (2012/2013)

Séminaire Laurent Schwartz — EDP et applications

Significant information about the topology of a bounded domain Ω of a Riemannian manifold M is encoded into the properties of the distance, d ∂ Ω , from the boundary of Ω . We discuss recent results showing the invariance of the singular set of the distance function with respect to the generalized gradient flow of d ∂ Ω , as well as applications to homotopy equivalence.

Generalized midconvexity

Jacek Tabor, Józef Tabor, Krzysztof Misztal (2013)

Banach Center Publications

There are many types of midconvexities, for example Jensen convexity, t-convexity, (s,t)-convexity. We provide a uniform framework for all the above mentioned midconvexities by considering a generalized middle-point map on an abstract space X. We show that we can define and study the basic convexity properties in this setting.

Generalized versions of Ilmanen lemma: Insertion of C 1 , ω or C loc 1 , ω functions

Václav Kryštof (2018)

Commentationes Mathematicae Universitatis Carolinae

We prove that for a normed linear space X , if f 1 : X → ℝ is continuous and semiconvex with modulus ω , f 2 : X → ℝ is continuous and semiconcave with modulus ω and f 1 ≤ f 2 , then there exists f ∈ C 1 , ω ( X ) such that f 1 ≤ f ≤ f 2 . Using this result we prove a generalization of Ilmanen lemma (which deals with the case ω ( t ) = t ) to the case of an arbitrary nontrivial modulus ω . This generalization (where a C l o c 1 , ω function is inserted) gives a positive answer to a problem formulated by A. Fathi and M. Zavidovique in 2010.

Hidden structures in the class of convex functions and a new duality transform

Shiri Artstein-Avidan, Vitali Milman (2011)

Journal of the European Mathematical Society

Our main intention in this paper is to demonstrate how some seemingly purely geometric notions can be presented and understood in an analytic language of inequalities and then, with this understanding, can be defined for classes of functions and reveal new and hidden structures in these classes. One main example which we discovered is a new duality transform for convex non-negative functions on ℝ n attaining the value 0 at the origin (which we call “geometric convex functions”). This transform, together...

Interior sphere property of attainable sets and time optimal control problems

Piermarco Cannarsa, Hélène Frankowska (2006)

ESAIM: Control, Optimisation and Calculus of Variations

This paper studies the attainable set at time T>0 for the control system y ˙ ( t ) = f ( y ( t ) , u ( t ) ) u ( t ) ∈ U showing that, under suitable assumptions on f, such a set satisfies a uniform interior sphere condition. The interior sphere property is then applied to recover a semiconcavity result for the value function of time optimal control problems with a general target, and to deduce C1,1-regularity for boundaries of attainable sets.

Les effets de l’exposant de la fonction barrière multiplicative dans les méthodes de points intérieurs

Adama Coulibaly, Jean-Pierre Crouzeix (2003)

RAIRO - Operations Research - Recherche Opérationnelle

Les méthodes de points intérieurs en programmation linéaire connaissent un grand succès depuis l’introduction de l’algorithme de Karmarkar. La convergence de l’algorithme repose sur une fonction potentielle qui, sous sa forme multiplicative, fait apparaître un exposant p . Cet exposant est, de façon générale, choisi supérieur au nombre de variables n du problème. Nous montrons dans cet article que l’on peut utiliser des valeurs de p plus petites que n . Ceci permet d’améliorer le conditionnement de...

Les effets de l'exposant de la fonction barrière multiplicative dans les méthodes de points intérieurs

Adama Coulibaly, Jean-Pierre Crouzeix (2010)

RAIRO - Operations Research

Les méthodes de points intérieurs en programmation linéaire connaissent un grand succès depuis l'introduction de l'algorithme de Karmarkar. La convergence de l'algorithme repose sur une fonction potentielle qui, sous sa forme multiplicative, fait apparaître un exposant p. Cet exposant est, de façon générale, choisi supérieur au nombre de variables n du problème. Nous montrons dans cet article que l'on peut utiliser des valeurs de p plus petites que n. Ceci permet d'améliorer le conditionnement...

Currently displaying 61 – 80 of 178