A strategy for non-strictly convex transport costs and the example of ||x-y||p in R2

This paper deals with the existence of optimal transport maps for some optimal transport problems with a convex but non strictly convex cost. We give a decomposition strategy to address this issue. As part of our strategy, we have to treat some transport problems, of independent interest, with a convex constraint on the displacement. As an illustration of our strategy, we prove existence of optimal transport maps in the case where the source measure is absolutely continuous with respect to the Lebesgue measure and the transportation cost is of the form h(||x-y||) with h strictly convex increasing and ||. || an arbitrary norm in \R2

The Monge problem with vanishing gradient penalization: Vortices and asymptotic profile

We investigate the approximation of the Monge problem (minimizing \int\_$\Omega$ |T (x) -- x| d$\mu$(x) among the vector-valued maps T with prescribed image measure T \# $\mu$) by adding a vanishing Dirichlet energy, namely $\epsilon$ \int\_$\Omega$ |DT |^2. We study the $\Gamma$-convergence as $\epsilon$ $\rightarrow$ 0, proving a density result for Sobolev (or Lipschitz) transport maps in the class of transport plans. In a certain two-dimensional framework that we analyze in details, when no optimal plan is induced by an H ^1 map, we study the selected limit map, which is a new "special" Monge transport, possibly different from the monotone one, and we find the precise asymptotics of the optimal cost depending on $\epsilon$, where the leading term is of order $\epsilon$| log $\epsilon$|

Méthodes unidimensionnelles et d'évolution pour le transport optimal

Sur une droite, le transport optimal ne pose pas de difficultés. Récemment, ce constat a été utilisé pour traiter des problèmes plus généraux. En effet, on a remarqué qu'une habile désintégration permet souvent de se ramener à la dimension un, ce qui permet d'utiliser les méthodes afférentes pour obtenir un premier résultat, que l'on fait ensuite évoluer pour gagner en précision.Je montre ici l'efficacité de cette approche, en revenant sur deux problèmes déjà résolus partiellement de cette manière, et en complétant la réponse qui en avait été donnée.Le premier problème concerne le calcul de l'application de Yann Brenier. En effet, Guillaume Carlier, Alfred Galichon et Filippo Santambrogio ont prouvé que celle-ci peut être obtenue grâce à une équation différentielle, pour laquelle une condition initiale est donnée par le réarrangement de Knothe-Rosenblatt (lui-même défini via une succession de transformations unidimensionnelles). Ils n'ont cependant traité que des mesures finales discrètes ; j'étends leur résultat aux cas continus. L'équation de Monge-Ampère, une fois dérivée, donne une EDP pour le potentiel de Kantorovitch; mais pour obtenir une condition initiale, il faut utiliser le théorème des fonctions implicites de Nash-Moser.Le chapitre 1 rappelle quelques résultats essentiels de la théorie du transport optimal, et le chapitre 2 est consacré au théorème de Nash-Moser. J'expose ensuite mes propres résultats dans le chapitre 3, et leur implémentation numérique dans le chapitre 4.Enfin, le dernier chapitre est consacré à l'algorithme IDT, développé par François Pitié, Anil C. Kokaram et Rozenn Dahyot. Celui-ci construit une application de transport suffisamment proche de celle de M. Brenier pour convenir à la plupart des applications. Une interprétation en est proposée en termes de flot de gradients dans l'espace des probabilités, avec pour fonctionnelle la distance de Wasserstein projetée. Je démontre aussi l'équivalence de celle-ci avec la distance usuelle de Wasserstein.In dimension one, optimal transportation is rather straightforward. The easiness with which a solution can be obtained in that setting has recently been used to tackle more general situations, each time thanks to the same method. First, disintegrate your problem to go back to the unidimensional case, and apply the available 1D methods to get a first result; then, improve it gradually using some evolution process.This dissertation explores that direction more thoroughly. Looking back at two problems only partially solved this way, I show how this viewpoint in fact allows to go even further.The first of these two problems concerns the computation of Yann Brenier's optimal map. Guillaume Carlier, Alfred Galichon, and Filippo Santambrogio found a new way to obtain it, thanks to an differential equation for which an initial condition is given by the Knothe-Rosenblatt rearrangement. (The latter is precisely defined by a series of unidimensional transformations.) However, they only dealt with discrete target measures; I~generalize their approach to a continuous setting. By differentiation, the Monge-Ampère equation readily gives a PDE satisfied by the Kantorovich potential; but to get a proper initial condition, it is necessary to use the Nash-Moser version of the implicit function theorem.The basics of optimal transport are recalled in the first chapter, and the Nash-Moser theory is exposed in chapter 2. My results are presented in chapter 3, and numerical experiments in chapter 4.The last chapter deals with the IDT algorithm, devised by François Pitié, Anil C. Kokaram, and Rozenn Dahyot. It builds a transport map that seems close enough to the optimal map for most applications. A complete mathematical understanding of the procedure is, however, still lacking. An interpretation as a gradient flow in the space of probability measures is proposed, with the sliced Wasserstein distance as the functional. I also prove the equivalence between the sliced and usual Wasserstein distances.PARIS11-SCD-Bib. électronique (914719901) / SudocSudocFranceF

Monge's transport problem in the Heisenberg group

We prove the existence of solutions to Monge transport problem between two compactly supported Borel probability measures in the Heisenberg group equipped with its Carnot-Caratheodory distance assuming that the initial measure is absolutely continuous with respect to the Haar measure of the group

Solitary fibrous tumor of the pleura presenting with syncope episodes when coughing

<p>Abstract</p> <p>Background</p> <p>Solitary fibrous tumor of the pleura is a rarely encountered clinical entity which may have different clinical pictures. Although the majority of these neoplasms have a benign course, the malignant form has also been reported.</p> <p>Case presentation</p> <p>We herein describe a case of 72 year-old man with head, facial, and thoracic traumas caused by neurally-mediated situational syncope when coughing. The diagnostic work-up including chest x-ray, CT and PET, revealed a large solitary mass of the left hemithorax. Radical surgical resection of the mass was performed through a left lateral thoracotomy and completed with a wedge resection of the lingula. Hystological examination of the surgical specimen showed an encapsulated mass measuring 12 × 11.5 × 6 cm consistent with a solitary fibrous tumor of the pleura. It's surgical removal definitively resolved the neurologic manifestations. The patient had no postoperative complications. At two years follow-up the patient is free from recurrence and without clinical manifestations.</p> <p>Conclusion</p> <p>In our case its resection definitively resolved the episodes of situational syncope due, in our opinion, to the large thoracic mass compressing the phrenic nerve</p

Alveolar adenoma of the lung: unusual diagnosis of a lesion positive on PET scan. A case report

The authors report a clinical case of alveolar adenoma presenting as a solitary pulmonary nodule which was positive to PET and deeply located in the lung. Few cases of alveolar adenomas have been reported in literature; these lesions are considered pulmonary neoplasms with benign behaviour, usually presenting as a peripheral or subpleural coin lesion; the PET activities of such neoplasms were unknown

α-Synuclein is a Novel Microtubule Dynamase.

α-Synuclein is a presynaptic protein associated to Parkinson's disease, which is unstructured when free in the cytoplasm and adopts α helical conformation when bound to vesicles. After decades of intense studies, α-Synuclein physiology is still difficult to clear up due to its interaction with multiple partners and its involvement in a pletora of neuronal functions. Here, we looked at the remarkably neglected interplay between α-Synuclein and microtubules, which potentially impacts on synaptic functionality. In order to identify the mechanisms underlying these actions, we investigated the interaction between purified α-Synuclein and tubulin. We demonstrated that α-Synuclein binds to microtubules and tubulin α2β2 tetramer; the latter interaction inducing the formation of helical segment(s) in the α-Synuclein polypeptide. This structural change seems to enable α-Synuclein to promote microtubule nucleation and to enhance microtubule growth rate and catastrophe frequency, both in vitro and in cell. We also showed that Parkinson's disease-linked α-Synuclein variants do not undergo tubulin-induced folding and cause tubulin aggregation rather than polymerization. Our data enable us to propose α-Synuclein as a novel, foldable, microtubule-dynamase, which influences microtubule organisation through its binding to tubulin and its regulating effects on microtubule nucleation and dynamics

Human Lung Tissue Transcriptome:Influence of Sex and Age

Background Sex and age strongly influence the pathophysiology of human lungs, but scarce information is available about their effects on pulmonary gene expression. Methods We followed a discovery-validation strategy to identify sex-and age-related transcriptional differences in lung. Results We identified transcriptional profiles significantly associated with sex (215 genes; FDR <0.05) and age at surgery (217 genes) in non-involved lung tissue resected from 284 lung adenocarcinoma patients. When these profiles were tested in three independent series of non-tumor lung tissue from an additional 1,111 patients, we validated the association with sex and age for 25 and 22 genes, respectively. Among the 17 sex-biased genes mapping on chromosome X, 16 have been reported to escape X-chromosome inactivation in other tissues or cells, suggesting that this mechanism influences lung transcription too. Our 22 age-related genes partially overlap with genes modulated by age in other tissues, suggesting that the aging process has similar consequences on gene expression in different organs. Finally, seven genes whose expression was modulated by sex in non-tumor lung tissue, but no age-related gene, were also validated using publicly available data from 990 lung adenocarcinoma samples, suggesting that the physiological regulatory mechanisms are only partially active in neoplastic tissue. Conclusions Gene expression in non-tumor lung tissue is modulated by both sex and age. These findings represent a validated starting point for research on the molecular mechanisms underlying the observed differences in the course of lung diseases among men and women of different ages