Técnicas analíticas en problemas multilineales

Tesis inédita de la Universidad Complutense de Madrid, Facultad de Ciencias Matemáticas, Departamento de Análisis Matemático, leída el 16/09/2016This Ph.D. dissertation mainly focuses on three multilinear problems and itsaimistodescribe analytical and topological techniques that we found useful to tackle these problems. The first problem comes from Quantum Information theory, it is the so-called the Separability Problem, and the other two were proposed by Gurariy. Let Mk denote the set of complex matrices of order k and let Pk be the set of positive semidefinite Hermitian matrices of Mk. The aim of this problem is to find a deterministic criterion to distinguish the separable states from the entangled states. In this work we shall only deal with the bipartite finite dimensional case, therefore the states are elements in the tensor product space Mk ⊗Mm. We say that B ∈ Mk ⊗Mm is separable if B =Σin=1 Ci ⊗Di, where Ci ∈ Pk and Di ∈ Pm, for every i. If B is not separable then B is entangled. Denote by VMkV the set {V XV,X ∈ Mk}, where V ∈ Mk is an orthogonal projection. We say that a linear transformation T :VMkV →WMmW is a positive map, if T(Pk ∩VMkV )⊂ Pm ∩WMmW. We say that a non null positive map T : VMkV →VMkV is irreducible if V ′ MkV ′ ⊂ VMkV is such that T(V ′ MkV ′)⊂ V ′ MkV ′ then V ′ = V or V ′ = 0. Let us say that T : VMkV → VMkV is a completely reducible map, if it is a positive map and if there are orthogonal projections V1,...,Vs ∈ Mk such that ViVj = 0 (i ≠ j), ViV = Vi (1 ≤ i ≤ s), VMkV = V1MkV1 ⊕ ... ⊕ VsMkVs ⊕ R, R ⊥ V1MkV1 ⊕ ... ⊕ VsMkVs satisfying: T(ViMkVi)⊂ ViMkVi (1 ≤ i ≤ s), TSis irreducible (1 ≤ i ≤ s), TSR ≡ 0. Let A =Σni=1 Ai ⊗Bi ∈ Mk ⊗Mm. Define GA : Mk →Mm, as GA(X)=Σni=1 tr(AiX)Bi and FA : Mm → Mk, as FA(X)=Σin=1 tr(BiX)Ai. Our main results are the following: If A ∈ Mk ⊗Mm is positive under partial transposition (PPT) or symmetric with positive coefficients (SPC) or invariant under realignment then FA ○GA : Mk → Mk is completely reducible...Esta tesis doctoral se centra principalmente en tres problemas multilineales y su objetivo es describir las técnicas analíticas y topológicas útiles para atacar estos problemas. El primer problema tiene su origen en la Teoría de Información Cuántica, es el llamado problema de la separabilidad de los estados cuánticos, y los otros dos fueron propuestos por Vladimir I. Gurariy. Denotemos por Mk al conjunto de las matrices complejas de orden k y Pk será el conjunto de matrices Hermíticas semidefinidas positivasde Mk. El objetivo de nuestro primer problema es encontrar un criterio determinístico para distinguir los estados separables de los estados entrelazados. Aqui sólo trabajamos con el caso bipartito de dimensión finita, luego los estados son los elementos del producto tensorial Mk ⊗Mm. Decimos que B ∈ Mk ⊗Mm es separable si B =Σni=1 Ci ⊗Di, donde Ci ∈ Pk y Di ∈ Pm, para cada i. Si B no es separable entonces B está entrelazada. Sea VMkV el conjunto {V XV,X ∈ Mk}, donde V ∈ Mk es una proyección ortogonal. Se dice que una transformación lineal T : VMkV →WMmW es una aplicación positiva, si T(Pk ∩VMkV )⊂ Pm ∩WMmW. Se dice que una aplicación no nula positiva T : VMkV → VMkV es irreducible si V ′ MkV ′ ⊂ VMkV es tal que T(V ′ MkV ′)⊂ V ′ MkV ′ entonces V ′ = V o V ′ = 0. Digamos que T : VMkV → VMkV es una aplicación completamente reducible, si es positiva y si hay proyecciones ortogonales V1,...,Vs ∈ Mk tales que ViVj = 0 (i ≠ j), ViV = Vi (1 ≤ i ≤ s), VMkV = V1MkV1⊕...⊕VsMkVs⊕R, R ⊥V1MkV1⊕...⊕VsMkVs y que satisfacen: T(ViMkVi)⊂ ViMkVi (1 ≤ i ≤ s), TSViMkVi es irreducible (1 ≤ i ≤ s), TSR ≡ 0. Sea A =Σni=1 Ai ⊗Bi ∈ Mk ⊗Mm. Defina GA : Mk →Mm, GA(X)=1 tr(AiX)Bi y FA : Mm →Mk, FA(X)=Σin=1 tr(BiX)Ai. Nuestros resultados principales son los seguintes: Si A es positiva bajo transposición parcial (PPT) o simétrica con coeficientes positivos (SPC) o invariante bajo realineamiento luego FA ○GA :Mk →Mk es completamente reducible...Depto. de Análisis Matemático y Matemática AplicadaFac. de Ciencias MatemáticasTRUEunpu

Implementation of a New Electronic Trigger for Borex @ Lngs

To adapt signals between the new trigger board and the DAQ of borexino experiment it was necessary to implement an interface board to deal with a large number of signals at different logic levels

INFLUENCE OF CULTURAL STRENGTHS ON THE RELATIONSHIPS AMONG ACCULTURATIVE STRESS, RACISM, AND MENTAL AND PHYSICAL HEALTH IN LATINO IMMIGRANTS

The Latino immigrant population in the United States has grown rapidly, now standing at over 56 million people. Due to this increase in Latino immigrants, investigation of their mental and physical health is crucial. Few studies have investigated conjointly both physical and mental health in Latino immigrant adults. Daily discrimination and acculturative stress have been found to affect the mental and physical health of Latino immigrants. Cultural strengths including social support, religiosity, and enculturation have been linked to Latino immigrant health. In the minority stress model, cultural strengths have been theorized to moderate relationships between discrimination and health. The purpose of this study was to examine the relationships among acculturative stress, discrimination, and mental and physical health. A secondary aim was to examine whether direct and indirect effects among these series of variables are moderated by social support, religiosity, and enculturation. A community sample of 204 Latino immigrants were recruited. Generally, bivariate associations between variables were congruent with previous research. Anxiety was found to mediated the effects of both acculturative stress and discrimination on physical health. Depression was found to mediate the effect of discrimination on physical health. Social support was found to moderate indirect effect of discrimination on physical health through depression. Enculturation moderated the indirect effects of both acculturative stress and discrimination on physical health through anxiety. Results from this study indicate that minority stressors can impact physical health through mental health, and these relationships can be buffered by links to cultural strengths including social support and enculturation