Bifurcações de pontos de equilíbrio

Pós-graduação em Matemática Universitária - IGCE

Problemas elípticos com potencial que pode tender a zero no in?nito

Pós-graduação em Matemática - IBILCE

Existence and multiplicity results for quasilinear elliptic equations

The goal of this paper is to study the multiplicity of positive solutions of a class of quasilinear elliptic equations. Based on the mountain pass theorems and sub-and supersolutions argument for p-Laplacian operators, under suitable conditions on nonlinearity f (x, s), we show the following problem: -Delta(p)u = lambda f(x,u) in Omega, u/(partial derivative Omega) = 0, where Omega is a bounded open subset of R-N, N >= 2, with smooth boundary, lambda is a positive parameter and Delta(p) is the p-Laplacian operator with p > 1, possesses at least two positive solutions for large lambda.

Asymptotic bifurcation results for quasilinear elliptic operators

We develop results for bifurcation from the principal eigenvalue for certain operators based on the p-Laplacian and containing a superlinear nonlinearity with a critical Sobolev exponent. The main result concerns an asymptotic estimate of the rate at which the solution branch departs from the eigenspace. The method can also be applied for nonpotential operators.

Nonlinear periodic problems with a jumping reaction

We consider a periodic problem driven by the scalar $p-$Laplacian and with a jumping (asymmetric) reaction. We prove two multiplicity theorems. The first concerns the nonlinear problem ($1

Geometric deep learning for polygonal mesh denoising

Il seguente lavoro si propone come analisi degli operatori convoluzionali che caratterizzano le graph neural networks. ln particolare, la trattazione si divide in due parti, una teorica e una sperimentale. Nella parte teorica vengono innanzitutto introdotte le nozioni preliminari di mesh e convoluzione su mesh. In seguito vengono riportati i concetti base del geometric deep learning, quali le definizioni degli operatori convoluzionali e di pooling e unpooling. Un'attenzione particolare è stata data all'architettura Graph U-Net. La parte sperimentare riguarda l'applicazione delle reti neurali e l'analisi degli operatori convoluzionali applicati al denoising di superfici perturbate a causa di misurazioni imperfette effettuate da scanner 3D.

Existence of solutions for p(x)-Laplacian problems on a bounded domain

In this paper we study the following p(x)-Laplacian problem: -div(a(x)&VERBAR;&DEL; u&VERBAR;(p(x)-2)&DEL; u)+b(x)&VERBAR; u&VERBAR;(p(x)-2)u = f(x, u), x ε &UOmega;, u = 0, on &PARTIAL; &UOmega;, where 1< p(1) &LE; p(x) &LE; p(2) < n, &UOmega; &SUB; R-n is a bounded domain and applying the mountain pass theorem we obtain the existence of solutions in W-0(1,p(x)) for the p(x)-Laplacian problems in the superlinear and sublinear cases. © 2004 Elsevier Inc. All rights reserved.

The relevance of the Laplacian of intracule and extracule density distributions for analyzing electron-electron interactions in molecules

A topological analysis of intracule and extracule densities and their Laplacians computed within the Hartree-Fock approximation is presented. The analysis of the density distributions reveals that among all possible electron-electron interactions in atoms and between atoms in molecules only very few are located rigorously as local maxima. In contrast, they are clearly identified as local minima in the topology of Laplacian maps. The conceptually different interpretation of intracule and extracule maps is also discussed in detail. An application example to the C2H2, C2H4, and C2H6 series of molecules is presented

On the Fucik spectrum of the Laplacian on a torus

We study the Fucik spectrum of the Laplacian on a two-dimensional torus T(2). Exploiting the invariance properties of the domain T(2) with respect to translations we obtain a good description of large parts of the spectrum. In particular, for each eigenvalue of the Laplacian we will find an explicit global curve in the Fucik spectrum which passes through this eigenvalue; these curves are ordered, and we will show that their asymptotic limits are positive. On the other hand, using a topological index based on the mentioned group invariance, we will obtain a variational characterization of global curves in the Fucik spectrum; also these curves emanate from the eigenvalues of the Laplacian, and we will show that they tend asymptotically to zero. Thus, we infer that the variational and the explicit curves cannot coincide globally, and that in fact many curve crossings must occur. We will give a bifurcation result which partially explains these phenomena. (C) 2008 Elsevier Inc. All rights reserved.

Analysis of the Kohn Laplacian on the Heisenberg Group and on Cauchy-Riemann Manifolds

The purpose of this study is to analyse the regularity of a differential operator, the Kohn Laplacian, in two settings: the Heisenberg group and the strongly pseudoconvex CR manifolds. The Heisenberg group is defined as a space of dimension 2n+1 with a product. It can be seen in two different ways: as a Lie group and as the boundary of the Siegel UpperHalf Space. On the Heisenberg group there exists the tangential CR complex. From this we define its adjoint and the Kohn-Laplacian. Then we obtain estimates for the Kohn-Laplacian and find its solvability and hypoellipticity. For stating L^p and Holder estimates, we talk about homogeneous distributions. In the second part we start working with a manifold M of real dimension 2n+1. We say that M is a CR manifold if some properties are satisfied. More, we say that a CR manifold M is strongly pseudoconvex if the Levi form defined on M is positive defined. Since we will show that the Heisenberg group is a model for the strongly pseudo-convex CR manifolds, we look for an osculating Heisenberg structure in a neighborhood of a point in M, and we want this structure to change smoothly from a point to another. For that, we define Normal Coordinates and we study their properties. We also examinate different Normal Coordinates in the case of a real hypersurface with an induced CR structure. Finally, we define again the CR complex, its adjoint and the Laplacian operator on M. We study these new operators showing subelliptic estimates. For that, we don't need M to be pseudo-complex but we ask less, that is, the Z(q) and the Y(q) conditions. This provides local regularity theorems for Laplacian and show its hypoellipticity on M.

Spectra and Laplacian spectra of arbitrary powers of lexicographic products of graphs

Consider two graphs G and H. Let H^k[G] be the lexicographic product of H^k and G, where H^k is the lexicographic product of the graph H by itself k times. In this paper, we determine the spectrum of H^k[G]H and H^k when G and H are regular and the Laplacian spectrum of H^k[G] and H^k for G and H arbitrary. Particular emphasis is given to the least eigenvalue of the adjacency matrix in the case of lexicographic powers of regular graphs, and to the algebraic connectivity and the largest Laplacian eigenvalues in the case of lexicographic powers of arbitrary graphs. This approach allows the determination of the spectrum (in case of regular graphs) and Laplacian spectrum (for arbitrary graphs) of huge graphs. As an example, the spectrum of the lexicographic power of the Petersen graph with the googol number (that is, 10^100 ) of vertices is determined. The paper finishes with the extension of some well known spectral and combinatorial invariant properties of graphs to its lexicographic powers.

Pyrimidine-5'-nucleotidase Campinas, A New Mutation (p.r56g) In The Nt5c3 Gene Associated With Pyrimidine-5'-nucleotidase Type I Deficiency And Influence Of Gilbert's Syndrome On Clinical Expression.

Pyrimidine-5'-nucleotidase type I (P5'NI) deficiency is an autosomal recessive condition that causes nonspherocytic hemolytic anemia, characterized by marked basophilic stippling and pyrimidine nucleotide accumulation in erythrocytes. We herein present two African descendant patients, father and daughter, with P5'N deficiency, both born from first cousins. Investigation of the promoter polymorphism of the uridine diphospho glucuronosyl transferase 1A (UGT1A) gene revealed that the father was homozygous for the allele (TA7) and the daughter heterozygous (TA6/TA7). P5'NI gene (NT5C3) gene sequencing revealed a further change in homozygosity at amino acid position 56 (p.R56G), located in a highly conserved region. Both patients developed gallstones; however the father, who had undergone surgery for the removal of stones, had extremely severe intrahepatic cholestasis and, liver biopsy revealed fibrosis and siderosis grade III, leading us to believe that the homozygosity of the UGT1A polymorphism was responsible for the more severe clinical features in the father. Moreover, our results show how the clinical expression of hemolytic anemia is influenced by epistatic factors and we describe a new mutation in the P5'N gene associated with enzyme deficiency, iron overload, and severe gallstone formation. To our knowledge, this is the first description of P5'N deficiency in South Americans.

Preserved Fertility In A Patient With Gynecomastia Associated With The P.pro695ser Mutation In The Androgen Receptor.

The androgen insensitivity syndrome (AIS) is described as a dysfunction of the androgen receptor (AR) in 46,XY individuals, which can be associated with mutations in the AR gene or can be due to unknown mechanisms. Different mutations in AIS generally cause variable phenotypes that range from a complete hormone resistance to a mild form usually associated with male infertility. The purpose of this study was to search for mutations in the AR gene in a fertile man with gynecomastia and to evaluate the influence of the mutation on the AR transactivation ability. Sequencing of the AR gene revealed the p.Pro695Ser mutation. It is located within the AR ligand-binding domain. Bioinformatics analysis indicated a deleterious role, which was verified after testing transactivation activity and N-/C-terminal (N/C) interaction by in vitro expression of a reporter gene and 2-hybrid assays. p.Pro695Ser showed low levels of both transactivation activity and N/C interaction at low dihydrotestosterone (DHT) conditions. As the ligand concentration increased, both transactivation activity and N/C interaction also increased and reached normal levels. Therefore, this study provides functional insights for the p.Pro695Ser mutation described here for the first time in a patient with mild AIS. The expression profile of p.Pro695Ser not only correlates to the patient's phenotype, but also suggests that a high-dose DHT therapy may overcome the functional deficit of the mutant AR.

Glicoproteína-P, resistência a múltiplas drogas (MDR) e relação estrutura-atividade de moduladores

Multidrug resistance, MDR is a major obstacle for cancer chemotherapy. MDR can be reversed by drugs that vary in their chemical structure and main biological activity. Many efforts have been done to overcome MDR based on studies of structure-activity relationships and in this review we summarize some aspects of MDR mediated by P-glycoprotein (P-gp), as the most experimentally and clinically tested form of drug resistance. The most significant MDR mechanisms revealed until now are shortly discussed. Physicochemical and structural properties of MDR modulators, measures of the MDR reversal, and QSAR studies are included.