http://hdl.handle.net/2117/3741 Thu, 23 May 2013 12:29:47 GMT 2013-05-23T12:29:47Z webmaster.bupc@upc.edu Universitat Politècnica de Catalunya. Servei de Biblioteques i Documentació no http://hdl.handle.net/2117/8290 Title: Kirchhoff indexes of a network Authors: Bendito Pérez, Enrique; Carmona Mejías, Ángeles; Encinas Bachiller, Andrés Marcos; Gesto Beiroa, José Manuel; Mitjana Riera, Margarida Abstract: In this work we define the effective resistance between any pair of vertices with respect to a value λ ≥ 0 and a weight ω on the vertex set. This allows us to consider a generalization of the Kirchhoff Index of a finite network. It turns out that λ is the lowest eigenvalue of a suitable semi-definite positive Schrödinger operator and ω is the associated eigenfunction. We obtain the relation between the effective resistance, and hence between the Kirchhoff Index, with respect to λ and ω and the eigenvalues of the associated Schrödinger operator. However, our main aim in this work is to get explicit expressions of the above parameters in terms of equilibrium measures of the network. From these expressions, we derive a full generalization of Foster’s formulae that incorporate a positive probability of remaining in each vertex in every step of a random walk. Finally, we compute the effective resistances and the generalized Kirchhoff Index with respect to a λ and ω for some families of networks with symmetries, specifically for weighted wagon-wheels and circular ladders. Wed, 21 Jul 2010 08:33:04 GMT http://hdl.handle.net/2117/8290 2010-07-21T08:33:04Z Bendito Pérez, Enrique; Carmona Mejías, Ángeles; Encinas Bachiller, Andrés Marcos; Gesto Beiroa, José Manuel; Mitjana Riera, Margarida no In this work we define the effective resistance between any pair of vertices with respect to a value λ ≥ 0 and a weight ω on the vertex set. This allows us to consider a generalization of the Kirchhoff Index of a finite network. It turns out that λ is the lowest eigenvalue of a suitable semi-definite positive Schrödinger operator and ω is the associated eigenfunction. We obtain the relation between the effective resistance, and hence between the Kirchhoff Index, with respect to λ and ω and the eigenvalues of the associated Schrödinger operator. However, our main aim in this work is to get explicit expressions of the above parameters in terms of equilibrium measures of the network. From these expressions, we derive a full generalization of Foster’s formulae that incorporate a positive probability of remaining in each vertex in every step of a random walk. Finally, we compute the effective resistances and the generalized Kirchhoff Index with respect to a λ and ω for some families of networks with symmetries, specifically for weighted wagon-wheels and circular ladders. http://hdl.handle.net/2117/1247 Title: A formula for the Kirchhoff index Authors: Bendito Pérez, Enrique; Carmona Mejías, Ángeles; Encinas Bachiller, Andrés Marcos; Gesto Beiroa, José Manuel Abstract: 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. Wed, 17 Oct 2007 14:55:49 GMT http://hdl.handle.net/2117/1247 2007-10-17T14:55:49Z Bendito Pérez, Enrique; Carmona Mejías, Ángeles; Encinas Bachiller, Andrés Marcos; Gesto Beiroa, José Manuel no Kirchhoff index, effective resistance, equilibrium measure 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. http://hdl.handle.net/2117/1246 Title: Application of the forces' method in dynamic systems Authors: Bendito Pérez, Enrique; Carmona Mejías, Ángeles; Encinas Bachiller, Andrés Marcos; Gesto Beiroa, José Manuel Abstract: 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. Wed, 17 Oct 2007 14:33:10 GMT http://hdl.handle.net/2117/1246 2007-10-17T14:33:10Z Bendito Pérez, Enrique; Carmona Mejías, Ángeles; Encinas Bachiller, Andrés Marcos; Gesto Beiroa, José Manuel no dynamic systems, forces' method, choreographies 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. http://hdl.handle.net/2117/1241 Title: Computational cost of the Fekete problem Authors: Bendito Pérez, Enrique; Carmona Mejías, Ángeles; Encinas Bachiller, Andrés Marcos; Gesto Beiroa, José Manuel Abstract: 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}) Thu, 11 Oct 2007 16:42:31 GMT http://hdl.handle.net/2117/1241 2007-10-11T16:42:31Z Bendito Pérez, Enrique; Carmona Mejías, Ángeles; Encinas Bachiller, Andrés Marcos; Gesto Beiroa, José Manuel no Fekete points, Smale's 7th 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}) http://hdl.handle.net/2117/1168 Title: Characterization of symmetric M-matrices as resistive inverses Authors: Bendito Pérez, Enrique; Carmona Mejías, Ángeles; Encinas Bachiller, Andrés Marcos; Gesto Beiroa, José Manuel Abstract: 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. Wed, 01 Aug 2007 19:30:23 GMT http://hdl.handle.net/2117/1168 2007-08-01T19:30:23Z Bendito Pérez, Enrique; Carmona Mejías, Ángeles; Encinas Bachiller, Andrés Marcos; Gesto Beiroa, José Manuel no Schrödinger operator, M-matrices, generalized inverses, Green kernels, Moore-Penrose inverse, effective resistance, Kirchhoff index 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. http://hdl.handle.net/2117/589 Title: Potential Theory for boundary value problems on finite networks Authors: Bendito Pérez, Enrique; Carmona Mejías, Ángeles; Encinas Bachiller, Andrés Marcos; Gesto Beiroa, José Manuel Abstract: 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. Fri, 01 Dec 2006 17:44:29 GMT http://hdl.handle.net/2117/589 2006-12-01T17:44:29Z Bendito Pérez, Enrique; Carmona Mejías, Ángeles; Encinas Bachiller, Andrés Marcos; Gesto Beiroa, José Manuel no Discrete Potential Theory, Combinatorial Laplacian, Schrödinger operator, Boundary value problems, Effective resistance, Green function 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. http://hdl.handle.net/2117/588 Title: Bounds on the first non-null eigenvalue for self-adjoint boundary value problems on networks Authors: Bendito Pérez, Enrique; Carmona Mejías, Ángeles; Encinas Bachiller, Andrés Marcos; Gesto Beiroa, José Manuel Abstract: We aim here at obtaining bounds on the first non-null eigenvalue for self-adjoint boundary value problems on a weighted network by means of equilibrium measures, that includes the study of Dirichlet, Neumann and Mixed problems. We also show the sharpness of these bounds throughout the analysis of some known examples. In particular, we emphasize the case of distance-regular graphs, and we show that the bounds obtained are better than the known until now. Fri, 01 Dec 2006 17:29:22 GMT http://hdl.handle.net/2117/588 2006-12-01T17:29:22Z Bendito Pérez, Enrique; Carmona Mejías, Ángeles; Encinas Bachiller, Andrés Marcos; Gesto Beiroa, José Manuel no Networks, Self-adjoint eigenvalue problems, Discrete laplacian, Equilibrium measures, Distance-regular graphs We aim here at obtaining bounds on the first non-null eigenvalue for self-adjoint boundary value problems on a weighted network by means of equilibrium measures, that includes the study of Dirichlet, Neumann and Mixed problems. We also show the sharpness of these bounds throughout the analysis of some known examples. In particular, we emphasize the case of distance-regular graphs, and we show that the bounds obtained are better than the known until now. http://hdl.handle.net/2117/587 Title: Regular boundary value problems on a path throughout Chebyshev Polynomials Authors: Bendito Pérez, Enrique; Carmona Mejías, Ángeles; Encinas Bachiller, Andrés Marcos; Gesto Beiroa, José Manuel Abstract: 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. Fri, 01 Dec 2006 17:06:32 GMT http://hdl.handle.net/2117/587 2006-12-01T17:06:32Z Bendito Pérez, Enrique; Carmona Mejías, Ángeles; Encinas Bachiller, Andrés Marcos; Gesto Beiroa, José Manuel no Discrete Schrödinger operator, boundary value problems, paths, Chebyshev polynomials, Green functions 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. http://hdl.handle.net/2117/494 Title: Vector Calculus on weighted networks Authors: Bendito Pérez, Enrique; Carmona Mejías, Ángeles; Encinas Bachiller, Andrés Marcos Abstract: In this work we study the different type of regular boundary value problems on a path associated with the Schr\"odinger operator. In particular, we get the Poisson kernel and the Green function for each problem and we emphasize the cases of Dirichlet, Neumann, Mixed and periodic problems. In any case, the Poisson kernel and the Green function are given in terms of second and third kind Chebyshev polynomials since they verify a recurrence law similar to the one verified by the Sch\"odinger operator on a path. Mon, 02 Oct 2006 17:59:53 GMT http://hdl.handle.net/2117/494 2006-10-02T17:59:53Z Bendito Pérez, Enrique; Carmona Mejías, Ángeles; Encinas Bachiller, Andrés Marcos no Discrete operators, Discrete green identity, Vector calculus, Network cohomology In this work we study the different type of regular boundary value problems on a path associated with the Schr\"odinger operator. In particular, we get the Poisson kernel and the Green function for each problem and we emphasize the cases of Dirichlet, Neumann, Mixed and periodic problems. In any case, the Poisson kernel and the Green function are given in terms of second and third kind Chebyshev polynomials since they verify a recurrence law similar to the one verified by the Sch\"odinger operator on a path. http://hdl.handle.net/2117/493 Title: Fekete points in non-smooth surfaces Authors: Bendito Pérez, Enrique; Carmona Mejías, Ángeles; Encinas Bachiller, Andrés Marcos; Gesto Beiroa, José Manuel Abstract: In this paper we present a procedure for the estimation of the Fekete points on a wide variety of non-regular objects in $R^3$. We understand the problem of the Fekete points in terms of the identification of good equilibrium configurations for a potential energy that depends on the relative position of N particles. Although the procedure that we present here works well for different potential energies, the examples showed refer to the electrostatic potential energy, that plays an special role in Potential Theory and Physics. The objects for which our procedure has been designed can be described basically as the finite union of piecewise regular surfaces and curves. For the determination of a good starting configuration for the search of the Fekete points on such objects, a sequence of approximating regular surfaces must be constructed. The numerical experience carried out until now suggests that the total computational cost of the obtaining of a nearly optimal configuration with the procedure introduced here is less than $N^3$ independently of the object considered. Mon, 02 Oct 2006 17:34:51 GMT http://hdl.handle.net/2117/493 2006-10-02T17:34:51Z Bendito Pérez, Enrique; Carmona Mejías, Ángeles; Encinas Bachiller, Andrés Marcos; Gesto Beiroa, José Manuel no Fekete points, Optimization, Non-smooth surfaces, Riesz Kernels In this paper we present a procedure for the estimation of the Fekete points on a wide variety of non-regular objects in $R^3$. We understand the problem of the Fekete points in terms of the identification of good equilibrium configurations for a potential energy that depends on the relative position of N particles. Although the procedure that we present here works well for different potential energies, the examples showed refer to the electrostatic potential energy, that plays an special role in Potential Theory and Physics. The objects for which our procedure has been designed can be described basically as the finite union of piecewise regular surfaces and curves. For the determination of a good starting configuration for the search of the Fekete points on such objects, a sequence of approximating regular surfaces must be constructed. The numerical experience carried out until now suggests that the total computational cost of the obtaining of a nearly optimal configuration with the procedure introduced here is less than $N^3$ independently of the object considered. http://hdl.handle.net/2117/492 Title: Estimation of Fekete points Authors: Bendito Pérez, Enrique; Carmona Mejías, Ángeles; Encinas Bachiller, Andrés Marcos; Gesto Beiroa, José Manuel Abstract: In this paper we present a new method to estimate Fekete points on surfaces. Although our method works in a general setting, we concentrate on its application to the unit sphere because it is the prototype problem and in the unit cube because its singularity. The algorithm we present here is very simple and it is based in a physical interpretation of the behavior of a system of particles when they search for a minimal energy configuration. Moreover, the algorithm is efficient and robust independently of the surface and the kernel used to define the energy. The algorithm allows us to work with a great number of particles, for instance, in less than a day of calculation time, we have obtained a good configuration for 50000 particles in the unit sphere without using symmetry properties and with a conventional PC. Mon, 02 Oct 2006 17:20:08 GMT http://hdl.handle.net/2117/492 2006-10-02T17:20:08Z Bendito Pérez, Enrique; Carmona Mejías, Ángeles; Encinas Bachiller, Andrés Marcos; Gesto Beiroa, José Manuel no Fekete points, Optimization, Equilibrium distributions, Regular surfaces In this paper we present a new method to estimate Fekete points on surfaces. Although our method works in a general setting, we concentrate on its application to the unit sphere because it is the prototype problem and in the unit cube because its singularity. The algorithm we present here is very simple and it is based in a physical interpretation of the behavior of a system of particles when they search for a minimal energy configuration. Moreover, the algorithm is efficient and robust independently of the surface and the kernel used to define the energy. The algorithm allows us to work with a great number of particles, for instance, in less than a day of calculation time, we have obtained a good configuration for 50000 particles in the unit sphere without using symmetry properties and with a conventional PC.