Black Box Galois Representations

We develop methods to study $2$-dimensional $2$-adic Galois representations $\rho$ of the absolute Galois group of a number field $K$, unramified outside a known finite set of primes $S$ of $K$, which are presented as Black Box representations, where we only have access to the characteristic polynomials of Frobenius automorphisms at a finite set of primes. Using suitable finite test sets of primes, depending only on $K$ and $S$, we show how to determine the determinant $\det\rho$, whether or not $\rho$ is residually reducible, and further information about the size of the isogeny graph of $\rho$ whose vertices are homothety classes of stable lattices. The methods are illustrated with examples for $K=\mathbb{Q}$, and for $K$ imaginary quadratic, $\rho$ being the representation attached to a Bianchi modular form. These results form part of the first author's thesis.Comment: 40 pages, 3 figures. Numerous minor revisions following two referees' report

Solutions to quasi-relativistic multi-configurative Hartree-Fock equations in quantum chemistry

We establish existence of infinitely many distinct solutions to the multi-configurative Hartree-Fock type equations for N-electron Coulomb systems with quasi-relativistic kinetic energy for the n th electron. Finitely many of the solutions are interpreted as excited states of the molecule. Moreover, we prove existence of a ground state. The results are valid under the hypotheses that the total charge Z of K nuclei is greater than N-1 and that Z is smaller than a critical charge. The proofs are based on a new application of the Lions-Fang-Ghoussoub critical point approach to nonminimal solutions on a complete analytic Hilbert-Riemann manifold

Computational aspects of Galois representations

This thesis contains a series of studies about 2-adic integral Galois representations unramified outside a finite set of primes. There are two main focuses of research: the study of 2-adic integral Galois representations and the study on how to compare two 2-adic integral Galois representations. Firstly, when studying a representation, we develop methods to determine whether the residual image is reducible or irreducible: in the irreducible case the residual image is completely determined. On the other hand, when the residual image is reducible we are able to make a choice of a stable lattice to completely determine the residual image. Lastly, from the choice of lattice, we are able to extend our methods to determine whether the representation is trivial modulo 2k+1 assuming that is trivial modulo 2k. Secondly, when comparing two 2-adic integral Galois representations, we are able to determine whether the representations are isogenous that is, after conjugation if necessary, their residual representations are the same. In some cases, this process follows the approach given in [11] by Ron Livné. Finally, the idea behind these studies was the notion of what we call a "Black Box representation", i.e., a system that will provide the characteristic polynomial of the representation for any prime not in the set of primes

Detección de zonas de penumbra y visibilidad a los flujos electromagnéticos en campos abiertos

Este artículo exhibe un procedimiento computacional que permite explorar, sobre un mapa de elevación del terreno (DTM) de una región geográfica de interés, las zonas que están en la línea de incidencia directa de los flujos de energía generados por fuentes puntuales localizadas. En éste se exponen visualmente, las zonas geográficas de visibilidad y penumbras a las radiaciones de fuentes electromagnéticas puntuales. En el desarrollo del ejercicio se contemplan dos aspectos: el primero elabora un modelo geométrico en 3D que represente el perfil topográfico de regiones geográficas, a partir de las bases de datos disponibles como resultado del proyecto SRTM (NGA and NASA) [8]. El segundo, propone un modelo de propagación para una fuente de radiación ondulatoria en el espacio libre de cargas y fuentes en regiones homogéneas e isotrópicas, de acuerdo con la solución de Maxwell a la propagación de ondas electromagnéticas y en concordancia con la ley de Snelly las técnicas del Ray-Tracing (RT). This article exhibits a computational procedure that allows, on a Terrain Elevation Map (TEM) of a geographic region of interest, are as that are in line with direct impact of energy flows generated from point sources located. This presents visually, the geographical are as of visibility and penumbras of electromagnetic radiation of punctual sources. In developing of the exercise ha ve be en covered two aspects: the first produces a 3D geometric model that represents the topographic profile of geographical regions, based on the available databases as a result ofthe project SRTM (NGA and NASA)[8]. The second, proposed a model of propagation for a radiation so urce wave in the space free of charge and sources in homogeneous and isotropic regions, according to the solution of Maxwell to the propagation of electromagnetic waves and in accordance with the law and Snell and techniques of Ray-Tracing (RT)

New trans dichloro (triphenylphosphine)platinum(II) complexes containing N-(butyl),N-(arylmethyl)amino ligands: Synthesis, cytotoxicity and mechanism of action

Some new platinum(II) complexes have been prepared, of general formula trans-[PtCl2(PPh3)NH(Bu)CH2Ar], where the dimension of the Ar residue in the secondary amines has been varied from small phenyl to large pyrenyl group. The obtained complexes, tested in vitro towards a panel of human tumor cell lines showed an interesting antiproliferative effect on both cisplatin-sensitive and -resistant cells. For the most cytotoxic derivative 2a the investigation on the mechanism of action highlighted the ability to induce apoptosis on resistant cells and interestingly, to inhibit the catalytic activity of topoisomerase II

Distribución del potencial electrostático en una placa cuadrada utilizando el método de elementos finitos

Este artículo expone la solución al problema de la distribución del potencial electrostático en una placa cuadrada mediante el Método de Elementos Finitos (Finite Elements Method, FEM). En este proceso se implementa el método pesaje residual en la formulación débil de la ecuación diferencial de Laplace con condiciones de frontera de Dirichlet para el potencial electrostático. La solución encontrada se basa en la selección de funciones lineales sobre un dominio discretizado en un número finito de elementos geométricos. La estrategia de solución se basa en la implementación del método de Galerkin en la escogencia de la función solución de la ecuación de Laplace llevada a su representación discreta.This paper presents the solution to the problem of the distribution of the electrostatic potential in a square plate by means of the Finite Elements Method (FEM). In this process the method is implemented minimo remainder in the weak formulation of the equation differential of Laplace with conditions of border of Dirichlet for the electrostatic potential. The found solution is based on the selection of linear functions on a dominion discreetin a finite number of geometric elements. The solution strategy is based on the implementation of the method of Galerkin in thechoosing of the function solution of the equation of Laplace taken to its discreet representation

New insights in the structure-activity relationships of 2-phenylamino-substituted benzothiopyrano[4,3-d]pyrimidines as kinase inhibitors

Inhibition of angiogenesis via blocking vascular endothelial growth factor receptor (VEGFR) signaling pathway emerged as an established approach in anticancer therapy. So far, many monoclonal antibodies and ATP-competitive small molecule inhibitors have been clinically validated and approved. In this study, structure-activity relationships (SAR) within the 2-phenylamino-substituted benzothiopyrano[4,3-d]pyrimidine class of kinase inhibitors were further refined by the synthesis and biological evaluation of new compounds 1â\u80\u9321 featuring different substitution patterns on the pendant phenyl moiety, combined with H, OCH3, or Cl at 8-position. Most compounds showed a promising human kinase insert domain receptor (KDR) inhibition profile, with IC50values in the submicromolar/low nanomolar range, and promising antiproliferative activity on human umbilical vein endothelial cells (HUVECs) as well as on a panel of three human tumor cell lines. The angio-kinase selectivity profile was assessed for the most promising compound 16 against a set of six human kinases. Finally, computational studies allowed clarifying at molecular level the interaction pattern established by the compounds with KDR, highlighting key stable cation-Ï\u80 interactions, and thus providing the basis for further designing novel inhibitors