Green matrices associated with generalized linear polyominoes

Carmona, Á.; Encinas, A.M.; Riera, Margarida Mitjana

doi:10.1016/j.laa.2013.12.039

Cited by 8 publications

(3 citation statements)

References 15 publications

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“…Since the cycle is obtained from the path by adding a new edge, we can see it as a perturbation of the path. Specifically, we analyze here the perturbation of the signless Laplacian of the path, since the perturbation of the combinatorial Laplacian has been treated in other works, see [5]. Recall that the path is always bipartite, and hence its signless Laplacian is singular and elliptic, but the cycle is elliptic only when the number of vertices is even.…”

Section: Perturbation Of Elliptic Schrödinger-like Operatorsmentioning

confidence: 99%

Perturbations of discrete elliptic operators

Carmona

Encinas

Riera

2015

Linear Algebra and its Applications

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Abstract. Given V a finite set, a self-adjoint operator on C(V ), K, is called elliptic if it is positive semi-definite and its lowest eigenvalue is simple. Therefore, there exists a unique, up to sign, unitary function ω ∈ C(V ) satisfying K(ω) = λω and then K is named (λ, ω)-elliptic. Clearly, a (λ, ω)-elliptic operator is singular iff λ = 0. Examples of elliptic operators are the so-called Schrödinger operators on finite connected networks, as well as the signless Laplacian of connected bipartite graphs. If K is a (λ, ω)-elliptic operator, it defines an automorphism on ω ⊥ , whose inverse is called orthogonal Green operator of K. We aim here at studying the effect of a perturbation of K on its orthogonal Green operator. The perturbation here considered is performed by adding a self-adjoint and positive semi-definite operator to K. As particular cases we consider the effect of changing the conductances on semi-definite Schödinger operators on finite connected networks and on the signless Laplacian of connected bipartite graphs. The expression obtained for the perturbed networks is explicitly given in terms of the orthogonal Green function of the original network.

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Section: Perturbation Of Elliptic Schrödinger-like Operatorsmentioning

confidence: 99%

Perturbations of discrete elliptic operators

Carmona

Encinas

Riera

2015

Linear Algebra and its Applications

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“…In [23] we presented an algorithm for solving the forward problem of determining u, given η. Our approach was a perturbative one, making use of known Green's functions for the time-independent diffusion equation (or Schrödinger equation) [3,8,9,7,12,13,14,15,40,42], with η identically zero. The corresponding inverse problem, which we refer to as graph optical tomography, is to recover the potential η from measurements of u on the boundary of the graph.…”

Section: Introductionmentioning

confidence: 99%

Optical tomography on graphs

Chung¹,

Gilbert²,

Hoskins³

et al. 2017

Inverse Problems

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We present an algorithm for solving inverse problems on graphs analogous to those arising in diffuse optical tomography for continuous media. In particular, we formulate and analyze a discrete version of the inverse Born series, proving estimates characterizing the domain of convergence, approximation errors, and stability of our approach. We also present a modification which allows additional information on the structure of the potential to be incorporated, facilitating recovery for a broader class of problems.

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“…Then, we can consider c 1 , c 2 as the boundary conditions with coefficients σ 11 , σ 12 , σ 13 , σ 14 and σ 21 , σ 22 , σ 23 , σ 24 , respectively, and we denote with (c 1 , c 2 ) the pair of boundary conditions determined by the matrix C = With this all functional notation, (2) and (3) are equivalent to the boundary value problem on the path I L γ q (u) = f on • I, c 1 (u) = f (0) and c 2 (u) = f (n + 1). (5) In terms of the boundary value problem, the invertibility conditions for M are exactly the same conditions to ensure that the above boundary value problem is regular, that is, it has a unique solution for each given data. Therefore, the computation of the inverse of M can be reduced to the calculus of the solution of boundary value problems for suitable data.…”

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confidence: 99%

Boundary value problems for second order linear difference equations: application to the computation of the inverse of generalized Jacobi matrices

Encinas

Jiménez

2019

RACSAM

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We have named generalized Jacobi matrices to those that are practically tridiagonal, except for the two final entries and the two first entries of its first and its last row respectively. This class of matrices encompasses both standard Jacobi and periodic Jacobi matrices that appear in many contexts in pure and applied mathematics. Therefore, the study of the inverse of these matrices becomes of specific interest. However, explicit formulas for inverses are known only in a few cases, in particular when the coefficients of the diagonal entries are subjected to some restrictions. We will show that the inverse of generalized Jacobi matrices can be raised in terms of the resolution of a boundary value problem associated with a second order linear difference equation. In fact, recent advances in the study of linear difference equations, allow us to compute the solution of this kind of boundary value problems. So, the conditions that ensure the uniqueness of the solution of the boundary value problem leads to the invertibility conditions for the matrix, whereas that solutions for suitable problems provide explicitly the entries of the inverse matrix. Keywords Second order difference equation • Boundary value problem • Generalized Jacobi matrix 1 Introduction and Preliminaries Throughout the paper, R denotes the field of real numbers. Given a positive integer n, we denote by R n the real coordinate space of n dimensions. Moreover, the components of the vector v ∈ R n are denoted by v j , j = 1,. .. , n, i.e., v = (v 1 ,. .. , vn) .

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Green matrices associated with generalized linear polyominoes

Cited by 8 publications

References 15 publications

Perturbations of discrete elliptic operators

Perturbations of discrete elliptic operators

Optical tomography on graphs

Boundary value problems for second order linear difference equations: application to the computation of the inverse of generalized Jacobi matrices

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