Direct and inverse polynomial perturbations of hermitian linear functionals

AbstractThis paper is devoted to the study of direct and inverse (Laurent) polynomial modifications of moment functionals on the unit circle, i.e., associated with hermitian Toeplitz matrices. We present a new approach which allows us to study polynomial modifications of arbitrary degree.The main objective is the characterization of the quasi-definiteness of the functionals involved in the problem in terms of a difference equation relating the corresponding Schur parameters. The results are presented in the general framework of (non-necessarily quasi-definite) hermitian functionals, so that the maximum number of orthogonal polynomials is characterized by the number of consistent steps of an algorithm based on the referred recurrence for the Schur parameters.The non-uniqueness of the inverse problem makes it more interesting than the direct one. Due to this reason, special attention is paid to the inverse modification, showing that different approaches are possible depending on the data about the polynomial modification at hand. These different approaches are translated as different kinds of initial conditions for the related inverse algorithm.Some concrete applications to the study of orthogonal polynomials on the unit circle show the effectiveness of this new approach: an exhaustive and instructive analysis of the functionals coming from a general inverse polynomial perturbation of degree one for the Lebesgue measure; the classification of those pairs of orthogonal polynomials connected by a certain type of linear relation with constant polynomial coefficients; and the determination of those orthogonal polynomials whose associated ones are related to a degree one polynomial modification of the original orthogonality functional

Recurrence for discrete time unitary evolutions

We consider quantum dynamical systems specified by a unitary operator U and an initial state vector \phi. In each step the unitary is followed by a projective measurement checking whether the system has returned to the initial state. We call the system recurrent if this eventually happens with probability one. We show that recurrence is equivalent to the absence of an absolutely continuous part from the spectral measure of U with respect to \phi. We also show that in the recurrent case the expected first return time is an integer or infinite, for which we give a topological interpretation. A key role in our theory is played by the first arrival amplitudes, which turn out to be the (complex conjugated) Taylor coefficients of the Schur function of the spectral measure. On the one hand, this provides a direct dynamical interpretation of these coefficients; on the other hand it links our definition of first return times to a large body of mathematical literature.Comment: 27 pages, 5 figures, typos correcte

A CMV connection between orthogonal polynomials on the unit circle and the real line

M. Derevyagin, L. Vinet and A. Zhedanov introduced in Derevyagin et al. (2012) a new connection between orthogonal polynomials on the unit circle and the real line. It maps any real CMV matrix into a Jacobi one depending on a real parameter λ. In Derevyagin et al. (2012) the authors prove that this map yields a natural link between the Jacobi polynomials on the unit circle and the little and big −1 Jacobi polynomials on the real line. They also provide explicit expressions for the measure and orthogonal polynomials associated with the Jacobi matrix in terms of those related to the CMV matrix, but only for the value λ = 1 which simplifies the connection –basic DVZ connection–. However, similar explicit expressions for an arbitrary value of λ –(general) DVZ connection– are missing in Derevyagin et al. (2012). This is the main problem overcome in this paper. This work introduces a new approach to the DVZ connection which formulates it as a two-dimensional eigenproblem by using known properties of CMV matrices. This allows us to go further than Derevyagin et al. (2012), providing explicit relations between the measures and orthogonal polynomials for the general DVZ connection. It turns out that this connection maps a measure on the unit circle into a rational perturbation of an even measure supported on two symmetric intervals of the real line, which reduce to a single interval for the basic DVZ connection, while the perturbation becomes a degree one polynomial. Some instances of the DVZ connection are shown to give new one-parameter families of orthogonal polynomials on the real line.The work of the first, third and fourth authors has been supported in part by the research project MTM2017-89941-P from Ministerio de Economía, Industria y Competitividad of Spain and the European Regional Development Fund (ERDF), by project UAL18-FQM-B025-A (UAL/CECEU/FEDER) and by projects E26 17R and E48 20R of Diputación General de Aragón (Spain) and the ERDF 2014–2020 “Construyendo Europa desde Aragón”. The work of the second author has been partially supported by the research project PGC2018–096504-B-C33 supported by Agencia Estatal de Investigación of Spain

Fractional Moment Estimates for Random Unitary Operators

We consider unitary analogs of $d-$dimensional Anderson models on $l^2(\Z^d)$ defined by the product $U_\omega=D_\omega S$ where $S$ is a deterministic unitary and $D_\omega$ is a diagonal matrix of i.i.d. random phases. The operator $S$ is an absolutely continuous band matrix which depends on parameters controlling the size of its off-diagonal elements. We adapt the method of Aizenman-Molchanov to get exponential estimates on fractional moments of the matrix elements of $U_\omega(U_\omega -z)^{-1}$, provided the distribution of phases is absolutely continuous and the parameters correspond to small off-diagonal elements of $S$. Such estimates imply almost sure localization for $U_\omega$

Relación entre la mortalidad materna y la gobernanza de los países iberoamericanos

Antecedentes/Objetivos: Pese al aumento de los recursos financieros y ser a menudo el 1er objetivo de los gobiernos, muchos países no están en camino de alcanzar el objetivo de Desarrollo del Milenio relacionado con la reducción de la mortalidad materna (MM). La gobernanza de los países es uno de los determinantes de la MM sobre el que hay disparidad de criterios y escasa literatura sobre el impacto de la misma en la MM. Objetivo: analizar la relación entre los factores de la gobernanza y la mortalidad materna en Iberoamérica, en 2012. Métodos: Estudio transversal ecológico para 2012, que utiliza el país como unidad de análisis sobre relación entre la gobernanza y la MM. Fuentes información: Estadísticas de Naciones Unidas, Banco Mundial, OMS/OPS. Se realizó un modelo de regresión lineal simple, y ajustado por riqueza (PIB). Resultados: Controlado por la riqueza del país, se detectan asociaciones significativas entre la MM con las variables de la gobernanza: Control de la corrupción R2 = 73,2% (p = 0,001), Estado de derecho R2 = 73% (p = 0,001), calidad regulatoria R2 = 70,9% (p = 0,002), efectividad gubernamental R2 = 69,5% (p = 0,003), voz y rendición de cuentas R2 = 68,3% (p = 0,004) y transparencia R2 = 66,8% (p = 0,003). Todas estas variables se asociación inversamente. Conclusiones: La mortalidad materna en 2012 está fuertemente relacionada con la capacidad y calidad gubernamental de los 19 países Iberoamericanos. En concreto, la MM se asocia con 6 de las 7 categorías componentes de la gobernanza. Afrontar la multicausalidad de la MM es un reto, entre los que se debe considerar y analizar el impulso político para la reducción de la corrupción, el desarrollo del estado de derecho, la calidad regulatoria, la efectividad gubernamental, el facilitar la voz y rendición de cuentas y la transparencia.Proyecto Prometeo. SENESCYT, Ecuador

Localization of the Grover walks on spidernets and free Meixner laws

A spidernet is a graph obtained by adding large cycles to an almost regular tree and considered as an example having intermediate properties of lattices and trees in the study of discrete-time quantum walks on graphs. We introduce the Grover walk on a spidernet and its one-dimensional reduction. We derive an integral representation of the $n$-step transition amplitude in terms of the free Meixner law which appears as the spectral distribution. As an application we determine the class of spidernets which exhibit localization. Our method is based on quantum probabilistic spectral analysis of graphs.Comment: 32 page

A connection between orthogonal polynomials on the unit circle and matrix orthogonal polynomials on the real line

Szego's procedure to connect orthogonal polynomials on the unit circle and orthogonal polynomials on [-1,1] is generalized to nonsymmetric measures. It generates the so-called semi-orthogonal functions on the linear space of Laurent polynomials L, and leads to a new orthogonality structure in the module LxL. This structure can be interpreted in terms of a 2x2 matrix measure on [-1,1], and semi-orthogonal functions provide the corresponding sequence of orthogonal matrix polynomials. This gives a connection between orthogonal polynomials on the unit circle and certain classes of matrix orthogonal polynomials on [-1,1]. As an application, the strong asymptotics of these matrix orthogonal polynomials is derived, obtaining an explicit expression for the corresponding Szego's matrix function.Comment: 28 page

Comparison between different multidimensional analytical systems for protein identification

Comunicaciones a congreso

Localization for Random Unitary Operators

We consider unitary analogs of $1-$dimensional Anderson models on $l^2(\Z)$ defined by the product $U_\omega=D_\omega S$ where $S$ is a deterministic unitary and $D_\omega$ is a diagonal matrix of i.i.d. random phases. The operator $S$ is an absolutely continuous band matrix which depends on a parameter controlling the size of its off-diagonal elements. We prove that the spectrum of $U_\omega$ is pure point almost surely for all values of the parameter of $S$. We provide similar results for unitary operators defined on $l^2(\N)$ together with an application to orthogonal polynomials on the unit circle. We get almost sure localization for polynomials characterized by Verblunski coefficients of constant modulus and correlated random phases