
Venancio TomeoComplutense University of Madrid | UCM · Department of Algebra
Venancio Tomeo
PhD Mathematics
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Introduction
Venancio Tomeo currently works at the Department of Algebra, Complutense University of Madrid. Venancio does research in Computing in Mathematics, Natural Science, Engineering and Medicine, Mathematics Education and Algebra. Their most recent publication is 'Inversion of infinite reduced Hessenberg matrices and operators'.
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Publications
Publications (34)
En este trabajo vamos a estudiar las medias estadísticas desde el punto de vista elemental, de forma que los conceptos puedan ser asimilados por alumnos de Enseñanza Secundaria y Bachillerato. Para ello se consideran problemas sencillos de la vida ordinaria relacionados con la geometría elemental, con la física y con la economía.
Por parte de la Ge...
En el presente trabajo se estudian las series geométricas y las series aritmético-geométricas, de primer y de segundo orden, su convergencia y su suma, para aplicar estas fórmulas al Cálculo de probabilidades. Se trata de hallar la probabilidad, la esperanza matemática y la varianza en casos de jugadores que compiten en el lanzamiento de dados o mo...
En este trabajo se estudian diez ejercicios de probabilidad. Se comienza con el problema de encontrar una estrategia para que la princesa elija al mejor de sus pretendientes, suponiendo varias condiciones. A continuación se estudian las estrategias para el juego de las siete y media y el juego del blackjack, por su semejanza con el problema de la p...
El presente trabajo parte del triángulo aritmético y los números combinatorios, para estudiar alguna de sus aplicaciones, generalizar el concepto de matriz de Pascal, permitiendo que la primera fila y la primera columna sean sucesiones cualesquiera de números reales o complejos, y estudiar la relación de las matrices de Pascal con las matrices de T...
We extend the concept of Pascal matrix to all matrices with the property that entries are the sum of the left and above entries. We have obtained an algebraic relation that allows us to obtain a Toeplitz matrix from a Pascal matrix, and vice versa. This procedure can be appiled to any Pascal matrices and any Toeplitz matrices, Hermitian matrices or...
Motived by current applications of the comrade matrices of large order, the infinite invertible comrade matrices are considered. These matrices are a generalization of the companion matrix and a particular case of the bordered tridiagonal matrices. A method based on a simple but suitable factorization is proposed for obtaining, in the finite-dimens...
En este trabajo se calcula la expresión de la resolvente de las secciones finitas
y se demuestra una fórmula que conecta los polinomios ortogonales con los polinomios asociados. Esta fórmula generaliza un teorema de Van Assche para los
polinomios ortogonales que satisfacen una relación de recurrencia a tres términos.
Como consecuencia se prueba una...
Las relaciones entre las medias estadísticas se presentan en numerosos textos de Estadística descriptiva aportando en general comprobaciones de ellas con datos concretos pero sin ninguna demostración rigurosa. En este trabajo se presentan demostraciones que pueden llamarse "elementales" a pesar de su complejidad, para el caso de N datos. A continua...
El presente trabajo está enfocado a la construcción de la matriz D, Hessenberg superior con subdiagonal positiva, correspondiente al operador multiplicación por z en la base de los polinomios ortogonales. Para ello se comienza con el estudio de la matriz de Gram relativa a un producto escalar y de las propiedades de mínimo de los polinomios mónicos...
En este trabajo se define y se recuerdan las principales propiedades del campo de valores de una matriz y se visualizan campos de valores de matrices sencillas mediante el método de Montecarlo.
Se estudia además la caracterización de Murnaghan y se utiliza para dibujar campos de valores de otras matrices sencillas mediante el método de la envolvent...
Inversion of infinite invertible unreduced Hessenberg matrices has been studied in [1], with applications to infinite invertible unreduced tridiagonal matrices. The case of infinite invertible reduced Hesenberg matrices and infinite reduced tridiagonal matrices, when the number of zeros on the subdiagonal is finite or infinite, is now studied for a...
To improve on the shortcomings observed in symbolic algorithms introduced recently for related matrices, a reliable numerical solver is proposed for computing the solution of the matrix linear equation . The matrix coefficient is a nonsingular bordered -tridiagonal matrix. The particular structure of is exploited through an incomplete or full Given...
The arrowhead matrices define a class of one-term Sylvester matrix (OTSM) operators on a finite-dimensional Hilbert space through an elementary UDL factorization. It enables us to consider the infinite invertible arrowhead matrices UDL factored properly for introducing, under suitable conditions, the arrowhead operators
and their associated class o...
Motived by current applications of the bordered tridiagonal matrices of large order, the infinite invertible bordered tridiagonal matrices are considered. These matrices are a generalization of the tridiagonal and the arrowhead matrices. A method based on a simple but suitable factorization is proposed for obtaining, in the finite-dimensional case,...
A comprehensive treatment on compact representations for the solutions of linear difference equations with variable coefficients, of both -th and unbounded order, is presented. The equivalence between their celebrated combinatorial and determinantal representations is considered. A corresponding representation is proposed using determined nested su...
A block matrix analysis is proposed to justify, and modify, a known algorithm for computing in O(n)O(n) time the determinant of a nonsingular n×nn×n pentadiagonal matrix (n≥6)(n≥6) having nonzero entries on its second subdiagonal. Also, we describe a procedure for computing the inverse matrix with acceptable accuracy in O(n2)O(n2) time. In the gene...
Here a known result on the structure of finite Hessenberg matrices is extended to infinite Hessenberg matrices. Its consequences for the example of infinite Hessenberg–Toeplitz matrices are described. The results are applied also to the inversion of infinite tridiagonal matrices via recurrence relations. Moreover, since there are available free par...
A method, based on recurrence relations, is proposed for evaluating classical inverses of infinite (unreduced as well as reduced) tridiagonal matrices using a known result on inversion of finite Hessenberg matrices, applicable also on tridiagonal matrices. The recurrence relations for the inverse way are also provided.
A modification of the Ikebe algorithm for computing the lower half of the inverse of an (unreduced) upper Hessenberg matrix, extended to compute the entries of the superdiagonal, is considered in this paper. It enables us to compute the inverse of a quasiseparable Hessenberg matrix in O(n2)O(n2) times. A new factorization expressing the inverse of...
A representation for the entries of the inverse of general tridiagonal matrices is based on the determinants of their principal submatrices. It enables us to introduce, through the linear recurrence relations satisfied by such determinants, a simple algorithm for the entries of the inverse of any tridiagonal nonsingular matrix, reduced as well as u...
Adequate conditions, using a known result on a class of finite Hessenberg matrices, are here proposed to make available finite, and infinite, matrices as a rank one perturbation of strictly upper triangular matrices UV+T. Some characterizations on such matrices for generating Orthogonal Polynomials sequences are also considered.
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ABSTRACT:
The nested sums applied to a general three-term recurrence relation permits us to give compact representations of orthogonal polynomial sequences, which satisfy such kind of linear recurrence. We illustrate this model on particular examples of classical as well as n...
Ikebe algorithm for computing the lower half of the inverse of any (unreduced) upper Hessenberg matrix is extended here to compute the entries of the superdiagonal. It gives rise to an algorithm of inversion based on the factorization H^-1=H_L.U^-1. The lower Hessenberg matrix H_L is a quasiseparable one and U^-1 is upper triangular, with diagonal...
The general representation for the elements of the inverse of any Hessenberg matrix fo finite order is here extended to reduced case with a new proof. Those entries are given with proper Hessenbergians from the original matrix. It justifies both the use of linear recurrences of unbounded order for such computations on matrices of intermediate order...
In this paper we prove two consequences of the subnormal character of the Hessenberg matrix D when the hermitian matrix M of an inner product is a moment matrix. If this inner product is defined by a measure supported on an algebraic curve in the complex plane, then D satisfies the equation of the curve in a noncommutative sense. We also prove an e...
After a brief introduction and justification of the method, a simple numerical algorithm for the inversion of tridiagonal nonsingular matrices (unreduced as well as reduced) is introduced. We maintain here the numerical approach, without invoking the symbolic computation, wich has produced recent advances in the inversion of such matrices.
A new proof of the general representation for the entries of the inverse of any unreduced Hessenberg matrix of finite order is found. Also this formulation is extended to the inverses of reduced Hessenberg matrices. Those matrices are given with proper Hessenbergians from the original matrix. It justifies both the use of linear recurrences for such...
En este trabajo se presentan unas propiedades de los determinantes que, por distintas causas, no se incluyen en las explicaciones que damos a nuestros alumnos en el ´ultimo curso de bachillerato ni en las de primer curso de cualquiera de las carreras de ciencias. Se ha procurado motivar a profesores y alumnos mediante nombres apropiados y ejemplos...
En el ámbito de la educación han aparecido en los últimos tiempos numerosos trabajos destacando la importancia de la enseñanza de la estadística y la probabilidad dentro de la enseñanza secundaria, en la práctica suelen ser las grandes olvidadas por los profesores de matemáticas.
En este trabajo se intenta poner de relieve la importancia que tiene...
An explicit representation for the determinant of any upper Hessenberg matrix in C^nxn, with non null diagonal terms, is achieved by means of a quasitriangular matrix with the same determinant value. It gives rise to a representation with nested functions on matrix elements h_i,j from de original Hessenberg matrix. Like an interesting application,...
In this paper we prove a formula that connect monic polynomials with associated polynomials. This formula generalizes a Van Assche's theorem for Orthogonal Polynomials satisfying a three term recurrence relation. As consecuence we prove a formula for the trace of the resolvent of the Hessenberg matrix's finite sections. This formula generalizes one...
In this work we prove that Hessenberg’s infinite matrix, associated with a Hermitian OPS that generalizes the Jacobi matrix, is normal under the assumption that the OPS is generated from a discrete infinite bounded distribution of non-aligned points in the complex plane with some geometrical restrictions. This matrix is also normal if we consider a...
In this paper we consider a class of three-term recurrence relations, whose associated tridiagonal matrices are subnormal operators. In this cases, there are measures associated to the polynomials given by such relations. We study the support of these measures.
We solve the Hermitian moment problem associated to a non-symmetric tridiagonal matrix depending on two parameters. The problem has a solution when the operator on l 2 defined by this matrix is subnormal. We analyze the spectrum of the operator and also the spectrum of its normal extension for several values of the parameters. We get the support of...