Géométrie nodale et valeurs propres de l’opérateur de Laplace et du p-laplacien

La présente thèse porte sur différentes questions émanant de la géométrie spectrale. Ce domaine des mathématiques fondamentales a pour objet d'établir des liens entre la géométrie et le spectre d'une variété riemannienne. Le spectre d'une variété compacte fermée M munie d'une métrique riemannienne $g$ associée à l'opérateur de Laplace-Beltrami est une suite de nombres non négatifs croissante qui tend vers l’infini. La racine carrée de ces derniers représente une fréquence de vibration de la variété. Cette thèse présente quatre articles touchant divers aspects de la géométrie spectrale. Le premier article, présenté au Chapitre 1 et intitulé « Superlevel sets and nodal extrema of Laplace eigenfunctions », porte sur la géométrie nodale d'opérateurs elliptiques. L’objectif de mes travaux a été de généraliser un résultat de L. Polterovich et de M. Sodin qui établit une borne sur la distribution des extrema nodaux sur une surface riemannienne pour une assez vaste classe de fonctions, incluant, entre autres, les fonctions propres associées à l'opérateur de Laplace-Beltrami. La preuve fournie par ces auteurs n'étant valable que pour les surfaces riemanniennes, je prouve dans ce chapitre une approche indépendante pour les fonctions propres de l’opérateur de Laplace-Beltrami dans le cas des variétés riemanniennes de dimension arbitraire. Les deuxième et troisième articles traitent d'un autre opérateur elliptique, le p-laplacien. Sa particularité réside dans le fait qu'il est non linéaire. Au Chapitre 2, l'article « Principal frequency of the p-laplacian and the inradius of Euclidean domains » se penche sur l'étude de bornes inférieures sur la première valeur propre du problème de Dirichlet du p-laplacien en termes du rayon inscrit d’un domaine euclidien. Plus particulièrement, je prouve que, si p est supérieur à la dimension du domaine, il est possible d'établir une borne inférieure sans aucune hypothèse sur la topologie de ce dernier. L'étude de telles bornes a fait l'objet de nombreux articles par des chercheurs connus, tels que W. K. Haymann, E. Lieb, R. Banuelos et T. Carroll, principalement pour le cas de l'opérateur de Laplace. L'adaptation de ce type de bornes au cas du p-laplacien est abordée dans mon troisième article, « Bounds on the Principal Frequency of the p-Laplacian », présenté au Chapitre 3 de cet ouvrage. Mon quatrième article, « Wolf-Keller theorem for Neumann Eigenvalues », est le fruit d'une collaboration avec Guillaume Roy-Fortin. Le thème central de ce travail gravite autour de l'optimisation de formes dans le contexte du problème aux valeurs limites de Neumann. Le résultat principal de cet article est que les valeurs propres de Neumann ne sont pas toujours maximisées par l'union disjointe de disques arbitraires pour les domaines planaires d'aire fixée. Le tout est présenté au Chapitre 4 de cette thèse.

Lattice constellations and codes from quadratic number fields

We propose new classes of linear codes over integer rings of quadratic extensions of Q, the field of rational numbers. The codes are considered with respect to a Mannheim metric, which is a Manhattan metric modulo a two-dimensional (2-D) grid. In particular, codes over Gaussian integers and Eisenstein-Jacobi integers are extensively studied. Decoding algorithms are proposed for these codes when up to two coordinates of a transmitted code vector are affected by errors of arbitrary Mannheim weight. Moreover, we show that the proposed codes are maximum-distance separable (MDS), with respect to the Hamming distance. The practical interest in such Mannheim-metric codes is their use in coded modulation schemes based on quadrature amplitude modulation (QAM)-type constellations, for which neither the Hamming nor the Lee metric is appropriate.

ENERGY AND VOLUME OF VECTOR FIELDS ON SPHERICAL DOMAINS

We present a "boundary version" for theorems about minimality of volume and energy functionals on a spherical domain of an odd-dimensional Euclidean sphere.

Role Of The Different Sexuality Domains On The Sexual Function Of Women With Premature Ovarian Failure.

Women with premature ovarian failure (POF) often manifest complaints involving different aspects of sexual function (SF), regardless of using hormone therapy. SF involves a complex interaction between physical, psychological, and sociocultural aspects. There are doubts about the impact of different complaints on the global context of SF of women with POF. To evaluate the percentage of influence of each of the sexuality domains on the SF in women with POF. Cross-sectional study with 80 women with POF, matched by age to 80 women with normal gonadal function. We evaluated SF through the Female Sexual Function Index (FSFI), a comparison between the POF and control groups using the Mann-Whitney test. Component exploratory factor analysis was used to assess the proportional influence of each domain on the composition of the overall SF for women in the POF group. SF was evaluated using FSFI. Exploratory Factor Analysis for components was used to evaluate the role of each domain on the SF of women with POF. The FSFI score was significantly worse for women with POF, with a decrease in arousal, lubrication, orgasm, satisfaction, and dyspareunia. Exploratory factor analysis of SF showed that the domain with greater influence in the SF was arousal, followed by desire, together accounting for 41% of the FSFI. The domains with less influence were dyspareunia and lubrication, which together accounted for 25% of the FSFI. Women with POF have impaired SF, determined mainly by changes in arousal and desire. Aspects related to lubrication and dyspareunia complaints have lower determination coefficient in SF. These results are important in adapting the approach of sexual disorders in this group of women. Benetti-Pinto CL, Soares PM, Giraldo HPD, and Yela DA. Role of the different sexuality domains on the sexual function of women with premature ovarian failure. J Sex Med 2015;12:685-689.

FBXO25-associated nuclear domains: A novel subnuclear structure

Skp1, Cul1, Rbx1, and the FBXO25 protein form a functional ubiquitin ligase complex. Here, we investigate the cellular distribution of FBXO25 and its colocalization with some nuclear proteins by using immunochemical and biochemical approaches. FBXO25 was monitored with affinity-purified antibodies raised against the recombinant fragment spanning residues 2-62 of the FBXO25 sequence. FBXO25 protein was expressed in all mouse tissues tested except striated muscle, as indicated by immunoblot analysis. Confocal analysis revealed that the endogenous FBXO25 was partially concentrated in a novel dot-like nuclear domain that is distinct from clastosomes and other well-characterized structures. These nuclear compartments contain a high concentration of ubiquitin conjugates and at least two other components of the ubiquitin-proteasome system: 20S proteasome and Skp1. We propose to name these compartments FBXO25-associated nuclear domains. Interestingly, inhibition of transcription by actinomycin D or heat-shock treatment drastically affected the nuclear organization of FBXO25-containing structures, indicating that they are dynamic compartments influenced by the transcriptional activity of the cell. Also, we present evidences that an FBXO25-dependent ubiquitin ligase activity prevents aggregation of recombinant polyglutamine-containing huntingtin protein in the nucleus of human embryonic kidney 293 cells, suggesting that this protein can be a target for the nuclear FBXO25 mediated ubiquitination.

Statistics of opinion domains of the majority-vote model on a square lattice

The existence of juxtaposed regions of distinct cultures in spite of the fact that people's beliefs have a tendency to become more similar to each other's as the individuals interact repeatedly is a puzzling phenomenon in the social sciences. Here we study an extreme version of the frequency-dependent bias model of social influence in which an individual adopts the opinion shared by the majority of the members of its extended neighborhood, which includes the individual itself. This is a variant of the majority-vote model in which the individual retains its opinion in case there is a tie among the neighbors' opinions. We assume that the individuals are fixed in the sites of a square lattice of linear size L and that they interact with their nearest neighbors only. Within a mean-field framework, we derive the equations of motion for the density of individuals adopting a particular opinion in the single-site and pair approximations. Although the single-site approximation predicts a single opinion domain that takes over the entire lattice, the pair approximation yields a qualitatively correct picture with the coexistence of different opinion domains and a strong dependence on the initial conditions. Extensive Monte Carlo simulations indicate the existence of a rich distribution of opinion domains or clusters, the number of which grows with L(2) whereas the size of the largest cluster grows with ln L(2). The analysis of the sizes of the opinion domains shows that they obey a power-law distribution for not too large sizes but that they are exponentially distributed in the limit of very large clusters. In addition, similarly to other well-known social influence model-Axelrod's model-we found that these opinion domains are unstable to the effect of a thermal-like noise.

Dual boundary element formulation applied to analysis of multi-fractured domains

This paper deals with analysis of multiple random crack propagation in two-dimensional domains using the boundary element method (BEM). BEM is known to be a robust and accurate numerical technique for analysing this type of problem. The formulation adopted in this work is based on the dual BEM, for which singular and hyper-singular integral equations are used. We propose an iterative scheme to predict the crack growth path and the crack length increment at each time step. The proposed scheme able us to simulate localisation and coalescence phenomena, which is the main contribution of this paper. Considering the fracture mechanics analysis, the displacement correlation technique is applied to evaluate the stress intensity factors. The propagation angle and the equivalent stress intensity factor are calculated using the theory of maximum circumferential stress. Examples of simple and multi-fractured domains, loaded up to the rupture, are considered to illustrate the applicability of the proposed scheme. (C) 2010 Elsevier Ltd. All rights reserved.

Analyzing static three-dimensional elastic domains with a new infinite boundary element formulation

A new two-dimensionally mapped infinite boundary element (IBE) is presented. The formulation is based on a triangular boundary element (BE) with linear shape functions instead of the quadrilateral IBEs usually found in the literature. The infinite solids analyzed are assumed to be three-dimensional, linear-elastic and isotropic, and Kelvin fundamental solutions are employed. One advantage of the proposed formulation over quadratic or higher order elements is that no additional degrees of freedom are added to the original BE mesh by the presence of the IBEs. Thus, the IBEs allow the mesh to be reduced without compromising the accuracy of the result. Two examples are presented, in which the numerical results show good agreement with authors using quadrilateral IBEs and analytical solutions. (C) 2010 Elsevier Ltd. All rights reserved.

A Functional Verification Methodology Based on Parameter Domains for Efficient Input Stimuli Generation and Coverage Modeling

Modern Integrated Circuit (IC) design is characterized by a strong trend of Intellectual Property (IP) core integration into complex system-on-chip (SOC) architectures. These cores require thorough verification of their functionality to avoid erroneous behavior in the final device. Formal verification methods are capable of detecting any design bug. However, due to state explosion, their use remains limited to small circuits. Alternatively, simulation-based verification can explore hardware descriptions of any size, although the corresponding stimulus generation, as well as functional coverage definition, must be carefully planned to guarantee its efficacy. In general, static input space optimization methodologies have shown better efficiency and results than, for instance, Coverage Directed Verification (CDV) techniques, although they act on different facets of the monitored system and are not exclusive. This work presents a constrained-random simulation-based functional verification methodology where, on the basis of the Parameter Domains (PD) formalism, irrelevant and invalid test case scenarios are removed from the input space. To this purpose, a tool to automatically generate PD-based stimuli sources was developed. Additionally, we have developed a second tool to generate functional coverage models that fit exactly to the PD-based input space. Both the input stimuli and coverage model enhancements, resulted in a notable testbench efficiency increase, if compared to testbenches with traditional stimulation and coverage scenarios: 22% simulation time reduction when generating stimuli with our PD-based stimuli sources (still with a conventional coverage model), and 56% simulation time reduction when combining our stimuli sources with their corresponding, automatically generated, coverage models.

Plasmodium vivax apical membrane antigen-1: comparative recognition of different domains by antibodies induced during natural human infection

The Apical Membrane Antigen-1 (AMA-1) of Plasmodium sp. has been suggested as a vaccine candidate against malaria. This protein seems to be involved in merozoite invasion and its extra-cellular portion contains three distinct domains: DI, DII, and DIII. Previously, we described that Plasmodium vivax AMA-1 (PvAMA-1) ectodomain is highly immunogenic in natural human infections. Here, we expressed each domain, separately or in combination (DI-II or DII-III), as bacterial recombinant proteins to map immunodominant epitopes within the PvAMA-1 ectodomain. IgG recognition was assessed by ELISA using sera of P. vivax-infected individuals collected from endemic regions of Brazil or antibodies raised in immunized mice. The frequencies of responders to recombinant proteins containing the DII were higher than the others and similar to the ones observed against the PvAMA-1 ectodomain. Moreover, ELISA inhibition assays using the PvAMA-1 ectodomain as substrate revealed the presence of many common epitopes within DI-II that are recognized by human immune antibodies. Finally, immunization of mice with the PvAMA-1 ectodomain induced high levels of antibodies predominantly to DI-II. Together, our results indicate that DII is particularly immunogenic during natural human infections, thus indicating that this region could be used as part of an experimental sub-unit vaccine to prevent vivax malaria. (C) 2008 Elsevier Masson SAS. All rights reserved.

Enhanced skin penetration of P20 phosphopeptide using protein transduction domains

Protein transduction domains (PTDs) were recently demonstrated to increase the penetration of the model peptide P20 when the PTD and P20 were covalently attached. Here, we evaluated whether non-covalently linked PTDs were capable of increasing the skin penetration of P20. Two different PTDs were studied: YARA and WLR. Porcine ear skin mounted in a Franz diffusion cell was used to assess the penetration of P20 in the stratum corneum (SC) and viable skin (VS); VS consists of dermis and epidermis without SC. The transdermal delivery of P20 was also assessed. At 1 mM, YARA promoted a 2.33-fold increase in the retention of P20 in the SC but did not significantly increase the amount of P20 that reached VS. WLR significantly increased (2.88-fold) the penetration of P20 in VS. Compared to the non-attached form, the covalently linked WLR fragment was two times more effective in promoting the penetration of P20 into VS. None of the PTDs promoted transdermal delivery of P20 at 4 h post-application. It was concluded that selected non-covalently linked PTDs can be used as a penetration enhancer, but greater skin penetration efficiency can be achieved by covalently binding the PTD to the therapeutic agent. (c) 2007 Elsevier B.V. All rights reserved.

Generalized optimal lattice covering of finite-dimensional Euclidean space

A generalization of the classical problem of optimal lattice covering of R-n is considered. Solutions to this generalized problem are found in two specific classes of lattices. The global optimal solution of the generalization is found for R-2. (C) 1998 Elsevier Science Inc. All rights reserved.

Asymptotic limits of a self-dual Ginzburg-Landau functional in bounded domains

In the author's joint paper [HJS] with Jest and Struwe, we discuss asymtotic limits of a self-dual Ginzburg-Landau functional involving a section of a line bundle over a closed Riemann surface and a connection on this bundle. In this paper, the author generalizes the above results [HJS] to the case of bounded domains.

Seeing is believing: Recent trends in the measurement of Ca2+ in subcellular domains and intracellular organelles

The role of Ca2+ in the regulation of the cell cycle has been investigated mostly in studies assessing global cytosolic free Ca2+. Recent studies, however, have used unique techniques to assess Ca2+ in subcellular organelles, such as mitochondria, and in discrete regions of the cytoplasm. These studies have used advanced fluorescence digital imaging techniques and Ca2+-sensitive fluorescence probes, and/or targeting of Ca2+-sensitive proteins to intracellular organelles. The present review describes the results of some of these studies and the techniques used. The novel techniques used to measure Ca2+ in microdomains and intracellular organelles are likely to be of great use in future investigations assessing Ca2+ homeostasis during the cell cycle.