Minimal representations of unitary operators and orthogonal polynomials on the unit circle

In this paper we prove that the simplest band representations of unitary operators on a Hilbert space are five-diagonal. Orthogonal polynomials on the unit circle play an essential role in the development of this result, and also provide a parametrization of such five-diagonal representations which shows specially simple and interesting decomposition and factorization properties. As an application we get the reduction of the spectral problem of any unitary Hessenberg matrix to the spectral problem of a five-diagonal one. Two applications of these results to the study of orthogonal polynomials on the unit circle are presented: the first one concerns Krein's Theorem; the second one deals with the movement of mass points of the orthogonality measure under monoparametric perturbations of the Schur parameters.Comment: 31 page

Matrix orthogonal polynomials whose derivatives are also orthogonal

In this paper we prove some characterizations of the matrix orthogonal polynomials whose derivatives are also orthogonal, which generalize other known ones in the scalar case. In particular, we prove that the corresponding orthogonality matrix functional is characterized by a Pearson-type equation with two matrix polynomials of degree not greater than 2 and 1. The proofs are given for a general sequence of matrix orthogonal polynomials, not necessarily associated with an hermitian functional. However, we give several examples of non-diagonalizable positive definite weight matrices satisfying a Pearson-type equation, which show that the previous results are non-trivial even in the positive definite case. A detailed analysis is made for the class of matrix functionals which satisfy a Pearson-type equation whose polynomial of degree not greater than 2 is scalar. We characterize the Pearson-type equations of this kind that yield a sequence of matrix orthogonal polynomials, and we prove that these matrix orthogonal polynomials satisfy a second order differential equation even in the non-hermitian case. Finally, we prove and improve a conjecture of Duran and Grunbaum concerning the triviality of this class in the positive definite case, while some examples show the non-triviality for hermitian functionals which are not positive definite.Comment: 49 page

An extension of the associated rational functions on the unit circle

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 Caratheodory function of the corresponding orthogonality measure in terms of such self-reciprocal coefficients. The new class under study includes the associated rational functions as a particular case. As a consequence of the previous general analysis, we obtain explicit representations for the associated rational functions of arbitrary order, as well as for the related Caratheodory function. Such representations are used to find new properties of the associated rational functions.Comment: 27 page

Minimal ureagenesis is necessary for survival in the murine model of hyperargininemia treated by AAV-based gene therapy.

Hyperammonemia is less severe in arginase 1 deficiency compared with other urea cycle defects. Affected patients manifest hyperargininemia and infrequent episodes of hyperammonemia. Patients typically suffer from neurological impairment with cortical and pyramidal tract deterioration, spasticity, loss of ambulation, seizures and intellectual disability; death is less common than with other urea cycle disorders. In a mouse model of arginase I deficiency, the onset of symptoms begins with weight loss and gait instability, which progresses toward development of tail tremor with seizure-like activity; death typically occurs at about 2 weeks of life. Adeno-associated viral vector gene replacement strategies result in long-term survival of mice with this disorder. With neonatal administration of vector, the viral copy number in the liver greatly declines with hepatocyte proliferation in the first 5 weeks of life. Although the animals do survive, it is not known from a functional standpoint how well the urea cycle is functioning in the adult animals that receive adeno-associated virus. In these studies, we administered [1-13C] acetate to both littermate controls and adeno-associated virus-treated arginase 1 knockout animals and examined flux through the urea cycle. Circulating ammonia levels were mildly elevated in treated animals. Arginine and glutamine also had perturbations. Assessment 30 min after acetate administration demonstrated that ureagenesis was present in the treated knockout liver at levels as low at 3.3% of control animals. These studies demonstrate that only minimal levels of hepatic arginase activity are necessary for survival and ureagenesis in arginase-deficient mice and that this level of activity results in control of circulating ammonia. These results may have implications for potential therapy in humans with arginase deficiency

One-dimensional quantum walks with one defect

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 at the origin

Apropa't a la ciencia

Presentamos una propuesta didáctica que permite contextualizar la ciencia en la sociedad: el taller ¿Cómo será la ciencia del futuro? realizado durante la exposición “Apropa’t a la ciència”, realizada en el Palau Robert de Barcelona entre abril y junio del 2007. El taller promueve la creatividad y la reflexión sobre la actividad científica, así como el debate sobre la relación entre la ciencia y la sociedad, mediante la discusión de noticias que sean portada de periódico en el año 2060, creadas por grupos de estudiantes después de su visita a la exposición. Analizamos la percepción social de ciencia, la inclusión de aspectos sociales y los temas les resultan más atractivos a partir de sus producciones e ideas previas. Simultáneamente nos interesa indagar la existencia o no de diferencias atribuibles al género en las visiones del alumnado