Archivo Digital UPM: No conditions. Results ordered -Date Deposited. 2021-12-07T19:28:23Z EPrints https://oa.upm.es/style/images/logo-archivo-digital.png https://oa.upm.es/ 2014-05-14T11:14:17Z 2016-04-20T12:29:49Z https://oa.upm.es/id/eprint/2856 This item is in the repository with the URL: https://oa.upm.es/id/eprint/2856 2014-05-14T11:14:17Z Constant solutions in second osOrder linear homogeneous difference equations with variable coefficients The constant solutions of second order linear and homogeneous difference equation with variable coefficients are analyzed. Thus, a sufficient condition for the existence of the constant solution, with any initial solution, is provided. Also, fixed points and invariant subsets of the solutions are considered. Finally, a necessary and sufficient condition to maintain the existence of non-trivial fixed points in the product of transfer matrices of two dimensions, is given. Jesús Carmelo Abderramán Marrero 2011-10-13T07:45:17Z 2016-04-20T17:46:12Z https://oa.upm.es/id/eprint/9285 This item is in the repository with the URL: https://oa.upm.es/id/eprint/9285 2011-10-13T07:45:17Z Inverses of regular Hessenberg matrices A new proof of the general representation for the entries of the inverse of any unreduced Hessenberg matrix of nite order is found. Also this formulation is extended to the inverses of reduced Hessenberg matrices. Those entries are given with proper Hessenbergians from the original matrix. It justies both the use of linear recurrences for such computations and some elementary properties of the inverse matrix. As an application of current interest in the theory of orthogonal polynomials on the complex plane, the resolvent matrix associated to a nite Hes- senberg matrix in standard form is calculated. The results are illustrated with two examples on the unit disk. Jesús Carmelo Abderramán Marrero Venancio Tomeo Perucha 2010-09-27T07:49:06Z 2016-04-20T13:36:04Z https://oa.upm.es/id/eprint/4315 This item is in the repository with the URL: https://oa.upm.es/id/eprint/4315 2010-09-27T07:49:06Z General solution of linear homogeneous difference equations with variable coefficients. A constructive theory for the general solution of kth-order difference equation is given as in a forthcoming paper of the author. As complement of the analytical theory [George D. Birkhoff, General theory of linear difference equations, Transactions of the American Mathematical Society, volume 12, number 2, pages 243–284, 1911], this constructive approach permits us an explicit and nonrecurrent representation of the general solution, for any initial conditions, and any sequences of complex numbers . If k = 1, then the solution is straightforward. Jesús Carmelo Abderramán Marrero 2010-04-13T10:11:17Z 2016-04-20T12:29:51Z https://oa.upm.es/id/eprint/2857 This item is in the repository with the URL: https://oa.upm.es/id/eprint/2857 2010-04-13T10:11:17Z Chebyshev expansion for the component functions of the Almost-Mathieu Operator The component functions {Ψn(∈)} (n ∈ Z+) from difference Schrödinger operators, can be formulated in a second order linear difference equation. Then the Harper equation, associated to almost-Mathieu operator, is a prototypical example. Its spectral behavior is amazing. Here, due the cosine coefficient in Harper equation, the component functions are expanded in a Chebyshev series of first kind, Tn(cos2πθ). It permits us a particular method for the θ variable separation. Thus, component functions can be expressed as an inner product, Ψn(, λ, θ) = _T [ n(n−1) 2 ] (cos2πθ) • _A [ n(n−1) 2 ] (_, λ). A matrix block transference method is applied for the calculation of the vector _A [ n(n−1) 2 ] (_, λ). When θ is integer, Ψn(_) is the sum of component from _A [ n(n−1) 2 ]. The complete family of Chebyshev Polynomials can be generated, with fit initial conditions. The continuous spectrum is one band with Lebesgue measure equal to 4. When θ is not integer the inner product Ψn can be seen as a perturbation of vector _T [ n(n−1) 2 ] on the sum of components from the vector _A [ n(n−1) 2 ]. When θ = p q , with p and q coprime, periodic perturbation appears, the connected band from the integer case degenerates in q sub-bands. When θ is irrational, ergodic perturbation produces that one band spectrum from integer case degenerates to a Cantor set. Lebesgue measure is Lσ = 4(1 − |λ|), 0 < |λ| ≤ 1. In this situation, the series solution becomes critical. Jesús Carmelo Abderramán Marrero