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## Prescription of Gauss curvature using optimal mass transport

### Geometriae Dedicata (2016-08-01) 183: 81-99 , August 01, 2016

In this paper we give a new proof of a theorem by Alexandrov on the Gauss curvature prescription of Euclidean convex sets. This proof is based on the duality theory of convex sets and on optimal mass transport. A noteworthy property of this proof is that it does not rely neither on the theory of convex polyhedra nor on P.D.E. methods (which appeared in all the previous proofs of this result).

## Sard theorems for Lipschitz functions and applications in optimization

### Israel Journal of Mathematics (2016-05-01) 212: 757-790 , May 01, 2016

We establish a “preparatory Sard theorem” for smooth functions with a partially affine structure. By means of this result, we improve a previous result of Rifford [17, 19] concerning the generalized (Clarke) critical values of Lipschitz functions defined as minima of smooth functions. We also establish a nonsmooth Sard theorem for the class of Lipschitz functions from R^{d} to R^{p} that can be expressed as finite selections of *C*^{k} functions (more generally, continuous selections over a compact countable set). This recovers readily the classical Sard theorem and extends a previous result of Barbet–Daniilidis–Dambrine [1] to the case *p* > 1. Applications in semi-infinite and Pareto optimization are given.

## Singular Arcs in the Generalized Goddard’s Problem

### Journal of Optimization Theory and Applications (2008-11-01) 139: 439-461 , November 01, 2008

We investigate variants of Goddard’s problems for nonvertical trajectories. The control is the thrust force, and the objective is to maximize a certain final cost, typically, the final mass. In this article, performing an analysis based on the Pontryagin maximum principle, we prove that optimal trajectories may involve singular arcs (along which the norm of the thrust is neither zero nor maximal), that are computed and characterized. Numerical simulations are carried out, both with direct and indirect methods, demonstrating the relevance of taking into account singular arcs in the control strategy. The indirect method that we use is based on our previous theoretical analysis and consists in combining a shooting method with an homotopic method. The homotopic approach leads to a quadratic regularization of the problem and is a way to tackle the problem of nonsmoothness of the optimal control.

## Topological degree and the Sperner lemma

### Journal of Optimization Theory and Applications (1982-07-01) 37: 371-377 , July 01, 1982

In this paper, we give a new proof of Sperner's lemma, in its superstrong from, using the topological degree. Thus, we point out a relation between several methods for fixed-point theorems using either the topological degree, or the KKM lemma, or the Sperner lemma.

## A functional limit theorem for waves reflected by a random medium

### Applied Mathematics and Optimization (1994-11-01) 30: 307-334 , November 01, 1994

We introduce a class of distribution-valued stochastic processes that arise in the study of pulse reflection from random media and we analyze their asymptotic properties when they are scaled in a natural way.

## Nonexistence of travelling wave solutions to nonelliptic nonlinear schrödinger equations

### Journal of Nonlinear Science (1996-03-01) 6: 139-145 , March 01, 1996

### Summary

By deriving Pohojaev-type identities we prove that nonelliptic nonlinear Schrödinger equations do not admit localized travelling wave solutions. Similary, we prove that the Davey-Stewartson hyperbolic-elliptic systems do not support travelling wave solutions except for a specific range of the parameters that comprises the DS II focusing case (where the existence of lumps is well known).

## Category theory in Homological Algebra

### Tool and Object (2007-01-01) 32: 93-160 , January 01, 2007

Before around 1955, CT was almost exclusively used in algebraic topology and served there, at least up to Eilenberg and Steenrod, mainly as a conceptual (or linguistic) framework for the organization of a knowledge system. Arrows and arrow composition played an important role there, and the new framework emphasizing these aspects changed considerably the organization of topology as a whole (compare [Volkert 2002] chapter 6), but this change was rather a shift of emphasis from problem solving to conceptual clarification than direct progress in solving the problems formerly considered as central in the discipline (as, for instance, the classification of 3-manifolds). In the domain of algebraic topology, it was Kan who entered first a level of conceptual innovation on which CT came to serve also as a means of *deduction*. This means that results in the topological context have been obtained by the application of results established on the categorial level—results deeper than those available using solely the base concepts of category theory, *i.e*., results the proof (and already the formulation) of which used new, more involved concepts like adjoint functors and the general limit concept^{162}.

## A study on Markovian maximality, change of probability and regularity

### Potential Analysis (1994-12-01) 3: 391-422 , December 01, 1994

Let*E* be a rigid separable Banach space and*m* a bounded Borel measure on*E*. Let Ext denote the family of all gradient type Dirichlet forms on*L*^{2}(*E, m*) such that the domain of their extended generators (cf. Definition 1.1) contain the smooth functions. We prove three results. First, we prove the existence of the maximum element in Ext whenever Ext is not empty. Secondly, let ℰ be the maximum element in Ext (when Ext ≠ Ø) and let φ be a positive function in D(ℰ). We define a new measure μ=φ^{2}·*m* and we consider the family Ext_{μ} associated with the measure μ. We prove that if ℰ is associated with a diffusion process, Ext_{μ} is not empty and its maximum element is also associated with a diffusion process. Finally, when*m* is a centered Gaussian measure on*E*, we can prove that Ext_{μ} contains exactly one element.

## Properties of the solutions of the Ginzburg-Landau equation on the bifurcation branch

### Nonlinear Differential Equations and Applications (2003-11-01) 10: 375-397 , November 01, 2003

### Abstract.

Generalizing previous results of M. Comte and P. Mironescu, it
is shown that for degree *d* large enough
(such that
$$ 8d \varepsilon_{d}^{2} -1 > \sqrt{4d-1} $$
), there
is a bifurcation branch in the set of the solutions of the Ginzburg-Landau
equation, emanating from the branch of radial solutions at the critical value
ε_{d} of the parameter. Moreover, the solutions on the bifurcation branch admit
exactly *d* zeroes, and the energy on the bifurcation branch is strictly smaller
than the energy on the radial branch.

## Measure-valued solution for non-Newtonian compressible isothermal monopolar fluid

### Acta Applicandae Mathematicae (1994-11-01) 37: 109-128 , November 01, 1994

The global existence of measure-valued solutions of initial boundary-value problems in bounded domains to systems of partial differential equations for viscous non-Newtonian isothermal compressible monopolar fluid and the global existence of the weak solution for multipolar fluid is proved.