Estimación de los puntos de Fekete en la esfera unidad

Esta tesina se centra en el problema matemático de conseguir distribuir bien una cantidad cualquiera de puntos sobre la superficie de una esfera. Este problema admite multitud de variantes; nosotros estudiaremos tres de ellas: disponer n cargas eléctricas puntuales iguales y del mismo signo sobre la superficie de la esfera de manera que estén en equilibrio electrostático; disponer n partículas puntuales sobre la superficie de la esfera de manera que sea máximo el producto de todas sus distancias euclídeas dos a dos y disponer n partículas puntuales sobre la superficie de la esfera de manera que sea máxima la suma de todas sus distancias euclídeas dos a dos. El primero de estos problemas suele aparecer en la literatura como problema de J.J. Thomson y el segundo como problema de Fekete, y a las distribuciones de puntos que resuelven este último se les da el nombre de puntos de Fekete de orden n en la esfera. También es habitual encontrar que esos tres problemas y otros similares se engloben bajo el nombre común de problema de los puntos de Fekete. Haciendo uso de los principios de la Teoría del Potencial es posible plantear los tres problemas anteriores en términos de la minimización de un cierto funcional de energía potencial cuya expresión varía según el caso. Nosotros proponemos un nuevo algoritmo numérico de localización de mínimos de esos funcionales de energía restringidos a la esfera unidad S2

Estimación de los puntos de Fekete en la esfera unidad

Esta tesina se centra en el problema matemático de conseguir distribuir bien una cantidad cualquiera de puntos sobre la superficie de una esfera. Este problema admite multitud de variantes; nosotros estudiaremos tres de ellas: disponer n cargas eléctricas puntuales iguales y del mismo signo sobre la superficie de la esfera de manera que estén en equilibrio electrostático; disponer n partículas puntuales sobre la superficie de la esfera de manera que sea máximo el producto de todas sus distancias euclídeas dos a dos y disponer n partículas puntuales sobre la superficie de la esfera de manera que sea máxima la suma de todas sus distancias euclídeas dos a dos. El primero de estos problemas suele aparecer en la literatura como problema de J.J. Thomson y el segundo como problema de Fekete, y a las distribuciones de puntos que resuelven este último se les da el nombre de puntos de Fekete de orden n en la esfera. También es habitual encontrar que esos tres problemas y otros similares se engloben bajo el nombre común de problema de los puntos de Fekete. Haciendo uso de los principios de la Teoría del Potencial es posible plantear los tres problemas anteriores en términos de la minimización de un cierto funcional de energía potencial cuya expresión varía según el caso. Nosotros proponemos un nuevo algoritmo numérico de localización de mínimos de esos funcionales de energía restringidos a la esfera unidad S2

THMC modelling of jet grouting

A framework for the study of the jet-grouting hydration reaction and of the associated thermo-hydro-mechanical and chemical (THMC) interactions with the surrounding soils has been developed. In this work, we summarize the basic formulation that may be used for the simulation of such interactions, including references to the balance equations governing the problem, to the release of heat during the curing of the jet-grouted mass, to the TMC behaviour of such material and to the THM behaviour of the surrounding soil. The approach presented falls within the framework of plasticity for saturated soils, and it has been implemented within a FEM code for the study of the potential effects of the THMC couplings associated to jet-grouting treatments. The results obtained with this program validate it as a proper tool for the systematic analysis of a number of questions of interest in engineering practice, allowing the assessment, among others, of the following issues: magnitude and rate of production of the thermo-plastic settlements caused by the heat release associated to the installation of jet-grouted columns in the soil; effects of the release of the hydration heat on the hydraulic conditions in the surrounding soil; effects of the boundary conditions, the relative position of the jet-grouted zones and its sequence of installation on the rate of increase of the stiffness and strength associated to the curing of the jet-grouted zones; effects of those factors on the impact of the heat release on the surrounding soils

THMC modelling of jet grouting

A framework for the study of the jet-grouting hydration reaction and of the associated thermo-hydro-mechanical and chemical (THMC) interactions with the surrounding soils has been developed. In this work, we summarize the basic formulation that may be used for the simulation of such interactions, including references to the balance equations governing the problem, to the release of heat during the curing of the jet-grouted mass, to the TMC behaviour of such material and to the THM behaviour of the surrounding soil. The approach presented falls within the framework of plasticity for saturated soils, and it has been implemented within a FEM code for the study of the potential effects of the THMC couplings associated to jet-grouting treatments. The results obtained with this program validate it as a proper tool for the systematic analysis of a number of questions of interest in engineering practice, allowing the assessment, among others, of the following issues: magnitude and rate of production of the thermo-plastic settlements caused by the heat release associated to the installation of jet-grouted columns in the soil; effects of the release of the hydration heat on the hydraulic conditions in the surrounding soil; effects of the boundary conditions, the relative position of the jet-grouted zones and its sequence of installation on the rate of increase of the stiffness and strength associated to the curing of the jet-grouted zones; effects of those factors on the impact of the heat release on the surrounding soils.Postprint (published version

Computational cost of the Fekete problem

We present here strong numerical and statistical evidence of the fact that the Smale's 7th problem can be answered affirmatively. In particular, we show that a local minimum for the logarithmic potential energy in the 2-sphere satisfying the Smale's conditions can be identified with a computational cost of approximately O(N^10}

Potential Theory for boundary value problems on finite networks

We aim here at analyzing self-adjoint boundary value problems on finite networks associated with positive semi-definite Schrödinger operators. In addition, we study the existence and uniqueness of solutions and its variational formulation. Moreover, we will tackle a well-known problem in the framework of Potential Theory, the so-called condenser principle. Then, we generalize of the concept of effective resistance between two vertices of the network and we characterize the Green function of some BVP in terms of effective resistances

Regular boundary value problems on a path throughout Chebyshev Polynomials

In this work we study the different types of regular boundary value problems on a path associated with the Schrödinger operator. In particular, we obtain the Green function for each problem and we emphasize the case of Sturm-Liouville boundary conditions. In addition, we study the periodic boundary value problem that corresponds to the Poisson equation in a cycle. In any case, the Green functions are given in terms of Chebyshev polynomials since they verify a recurrence law similar to the one verified by the Schrödinger operator on a path

Application of the forces' method in dynamic systems

We present here some applications of the Forces's method in dynamic systems. In particular, we consider the problem of the approximation of the trajectories of a conservative system of point masses by means of the minimization of the action integral and the computation of planar central configurations

Characterization of symmetric M-matrices as resistive inverses

We aim here at characterizing those nonnegative matrices whose inverse is an irreducible Stieltjes matrix. Specifically, we prove that any irreducible Stieltjes matrix is a resistive inverse. To do that we consider the network defined by the off-diagonal entries of the matrix and we identify the matrix with a positive definite Schrödinger operator which ground state is determined by the its lowest eigenvalue and the corresponding positive eigenvector. We also analyze the case in which the operator is positive semidefinite which corresponds to the study of singular irreducible symmetric M-matrices. The key tool is the definition of the effective resistance with respect to a nonnegative value and a weight. We prove that these effective resistances verify similar properties to those satisfied by the standard effective resistances which leads us to carry out an exhaustive analysis of the generalized inverses of singular irreducible symmetric M-matrices. Moreover we pay special attention on those generalized inverses identified with Green operators, which in particular includes the analysis of the Moore-Penrose inverse

A formula for the Kirchhoff index

We show here that the Kirchhoff index of a network is the average of the Wiener capacities of its vertices. Moreover, we obtain a closed-form formula for the effective resistance between any pair of vertices when the considered network has some symmetries which allows us to give the corresponding formulas for the Kirchhoff index. In addition, we find the expression for the Foster's n-th Formula