M. Cantero - Academia.edu (original) (raw)

Papers by M. Cantero

Research paper thumbnail of Spectral transformations of hermitian linear functionals

Contemporary Mathematics, 2012

Research paper thumbnail of Direct and inverse polynomial perturbations of hermitian linear functionals

Journal of Approximation Theory, 2011

This paper is devoted to the study of direct and inverse (Laurent) polynomial modifications of mo... more This 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. c

Research paper thumbnail of Hermitian modifications of Toeplitz linear functionals and orthogonal polynomials

… García de Galdeano …, 2004

In this paper the following situation is considered: A moment functional L associated with a quas... more In this paper the following situation is considered: A moment functional L associated with a quasi-definite infinite Hermitian Toeplitz matrix is modified by means of a complex polynomial P of degree one, obtaining a new linear functional ˜ ... L = (P(z) + P(z −1))L. A ...

Research paper thumbnail of An extension of the associated rational functions on the unit circle

Journal of Approximation Theory, 2011

Research paper thumbnail of Turbidity current with a roof: Success and failure of RANS modeling for turbidity currents under strongly stratified conditions

Journal of Geophysical Research: Earth Surface, 2013

Density underflows in general and turbidity currents in particular differ from rivers in that the... more Density underflows in general and turbidity currents in particular differ from rivers in that their governing equations do not allow a steady, streamwise uniform "normal" solution. This is due to the fact that density underflows entrain ambient fluid, thus creating a tendency for underflow discharge to increase downstream. Recently, however, a simplified configuration known as the "turbidity current with a roof" (TCR) has been proposed. The artifice of a roof allows for steady, uniform solutions for flows driven solely by gravity acting on suspended sediment. A recent application of direct numerical simulation (DNS) of the Navier-Stokes equations by Cantero et al. (2009) has revealed that increasing dimensionless sediment fall velocity increases flow stratification, resulting in a damping of the turbulence. When the dimensionless fall velocity is increased beyond a threshold value, near-bed turbulence collapses. Here we use the DNS results as a means of testing the ability of three Reynolds-averaged Navier-Stokes (RANS) models of turbulent flow to capture stratification effects in the TCR. Results showed that the Mellor-Yamada and quasi-equilibrium k-ε models are able to adequately capture the characteristics of the flow under conditions of relatively modest stratification, whereas the standard k-ε model is a relatively poor predictor of turbulence characteristics. As stratification strengthens, however, the deviation of all RANS models from the DNS results increases. All are incapable of predicting the collapse of near-bed turbulence predicted by DNS under conditions of strong stratification. This deficiency is likely due to the inability of RANS models to replace viscous dissipation of turbulent energy with transfer to internal waves under conditions of strong stratification. Within the limits of modest stratification, the quasi-equilibrium k-ε model is used to derive predictors of flow which can be incorporated into simpler, layer-averaged models of turbidity currents.

Research paper thumbnail of An extension of the associated rational functions on the unit circle

Journal of Approximation Theory, 2011

A special class of orthogonal rational functions (ORFs) is presented in this paper. Starting with... more A special class of orthogonal rational functions (ORFs) is presented in this paper. Starting with a sequence of ORFs and the corresponding rational functions of the second kind, we define a new sequence as a linear combination of the previous ones, the coefficients of this linear combination being self-reciprocal rational functions. We show that, under very general conditions on the self-reciprocal coefficients, this new sequence satisfies orthogonality conditions as well as a recurrence relation. Further, we identify the Carathéodory function of the corresponding orthogonality measure in terms of such self-reciprocal coefficients.

Research paper thumbnail of The CGMV method for quantum walks

Quantum Information Processing, 2012

ABSTRACT We review the main aspects of a recent approach to quantum walks, the CGMV method. This ... more ABSTRACT We review the main aspects of a recent approach to quantum walks, the CGMV method. This method proceeds by reducing the unitary evolution to canonical form, given by the so-called CMV matrices, which act as a link to the theory of orthogonal polynomials on the unit circle. This connection allows one to obtain results for quantum walks which are hard to tackle with other methods. Behind the above connections lies the discovery of a new quantum dynamical interpretation for well known mathematical tools in complex analysis. Among the standard examples which will illustrate the CGMV method are the famous Hadamard and Grover models, but we will go further showing that CGMV can deal even with non-translation invariant quantum walks. CGMV is not only a useful technique to study quantum walks, but also a method to construct quantum walks à la carte. Following this idea, a few more examples illustrate the versatility of the method. In particular, a quantum walk based on a construction of a measure on the unit circle due to F. Riesz will point out possible non-standard behaviours in quantum walks.

Research paper thumbnail of Matrix-valued Szegő polynomials and quantum random walks

Communications on Pure and Applied Mathematics, 2009

We consider quantum random walks (QRW) on the integers, a subject that has been considered in the... more We consider quantum random walks (QRW) on the integers, a subject that has been considered in the last few years in the framework of quantum computation.

Research paper thumbnail of ONE-DIMENSIONAL QUANTUM WALKS WITH ONE DEFECT

Reviews in Mathematical Physics, 2012

The CGMV method allows for the general discussion of localization properties for the states of a ... more The CGMV method allows for the general discussion of localization properties for the states of a one-dimensional quantum walk, both in the case of the integers and in the case of the non negative integers. Using this method we classify, according to such localization properties, all the quantum walks with one defect at the origin, providing explicit expressions for the asymptotic return probabilities to the origin.

Research paper thumbnail of Spectral transformations of hermitian linear functionals

Contemporary Mathematics, 2012

Research paper thumbnail of Direct and inverse polynomial perturbations of hermitian linear functionals

Journal of Approximation Theory, 2011

This paper is devoted to the study of direct and inverse (Laurent) polynomial modifications of mo... more This 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. c

Research paper thumbnail of Hermitian modifications of Toeplitz linear functionals and orthogonal polynomials

… García de Galdeano …, 2004

In this paper the following situation is considered: A moment functional L associated with a quas... more In this paper the following situation is considered: A moment functional L associated with a quasi-definite infinite Hermitian Toeplitz matrix is modified by means of a complex polynomial P of degree one, obtaining a new linear functional ˜ ... L = (P(z) + P(z −1))L. A ...

Research paper thumbnail of An extension of the associated rational functions on the unit circle

Journal of Approximation Theory, 2011

Research paper thumbnail of Turbidity current with a roof: Success and failure of RANS modeling for turbidity currents under strongly stratified conditions

Journal of Geophysical Research: Earth Surface, 2013

Density underflows in general and turbidity currents in particular differ from rivers in that the... more Density underflows in general and turbidity currents in particular differ from rivers in that their governing equations do not allow a steady, streamwise uniform "normal" solution. This is due to the fact that density underflows entrain ambient fluid, thus creating a tendency for underflow discharge to increase downstream. Recently, however, a simplified configuration known as the "turbidity current with a roof" (TCR) has been proposed. The artifice of a roof allows for steady, uniform solutions for flows driven solely by gravity acting on suspended sediment. A recent application of direct numerical simulation (DNS) of the Navier-Stokes equations by Cantero et al. (2009) has revealed that increasing dimensionless sediment fall velocity increases flow stratification, resulting in a damping of the turbulence. When the dimensionless fall velocity is increased beyond a threshold value, near-bed turbulence collapses. Here we use the DNS results as a means of testing the ability of three Reynolds-averaged Navier-Stokes (RANS) models of turbulent flow to capture stratification effects in the TCR. Results showed that the Mellor-Yamada and quasi-equilibrium k-ε models are able to adequately capture the characteristics of the flow under conditions of relatively modest stratification, whereas the standard k-ε model is a relatively poor predictor of turbulence characteristics. As stratification strengthens, however, the deviation of all RANS models from the DNS results increases. All are incapable of predicting the collapse of near-bed turbulence predicted by DNS under conditions of strong stratification. This deficiency is likely due to the inability of RANS models to replace viscous dissipation of turbulent energy with transfer to internal waves under conditions of strong stratification. Within the limits of modest stratification, the quasi-equilibrium k-ε model is used to derive predictors of flow which can be incorporated into simpler, layer-averaged models of turbidity currents.

Research paper thumbnail of An extension of the associated rational functions on the unit circle

Journal of Approximation Theory, 2011

A special class of orthogonal rational functions (ORFs) is presented in this paper. Starting with... more A special class of orthogonal rational functions (ORFs) is presented in this paper. Starting with a sequence of ORFs and the corresponding rational functions of the second kind, we define a new sequence as a linear combination of the previous ones, the coefficients of this linear combination being self-reciprocal rational functions. We show that, under very general conditions on the self-reciprocal coefficients, this new sequence satisfies orthogonality conditions as well as a recurrence relation. Further, we identify the Carathéodory function of the corresponding orthogonality measure in terms of such self-reciprocal coefficients.

Research paper thumbnail of The CGMV method for quantum walks

Quantum Information Processing, 2012

ABSTRACT We review the main aspects of a recent approach to quantum walks, the CGMV method. This ... more ABSTRACT We review the main aspects of a recent approach to quantum walks, the CGMV method. This method proceeds by reducing the unitary evolution to canonical form, given by the so-called CMV matrices, which act as a link to the theory of orthogonal polynomials on the unit circle. This connection allows one to obtain results for quantum walks which are hard to tackle with other methods. Behind the above connections lies the discovery of a new quantum dynamical interpretation for well known mathematical tools in complex analysis. Among the standard examples which will illustrate the CGMV method are the famous Hadamard and Grover models, but we will go further showing that CGMV can deal even with non-translation invariant quantum walks. CGMV is not only a useful technique to study quantum walks, but also a method to construct quantum walks à la carte. Following this idea, a few more examples illustrate the versatility of the method. In particular, a quantum walk based on a construction of a measure on the unit circle due to F. Riesz will point out possible non-standard behaviours in quantum walks.

Research paper thumbnail of Matrix-valued Szegő polynomials and quantum random walks

Communications on Pure and Applied Mathematics, 2009

We consider quantum random walks (QRW) on the integers, a subject that has been considered in the... more We consider quantum random walks (QRW) on the integers, a subject that has been considered in the last few years in the framework of quantum computation.

Research paper thumbnail of ONE-DIMENSIONAL QUANTUM WALKS WITH ONE DEFECT

Reviews in Mathematical Physics, 2012

The CGMV method allows for the general discussion of localization properties for the states of a ... more The CGMV method allows for the general discussion of localization properties for the states of a one-dimensional quantum walk, both in the case of the integers and in the case of the non negative integers. Using this method we classify, according to such localization properties, all the quantum walks with one defect at the origin, providing explicit expressions for the asymptotic return probabilities to the origin.