4,071 research outputs found
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
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
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
Fractional Moment Estimates for Random Unitary Operators
We consider unitary analogs of dimensional Anderson models on
defined by the product where is a deterministic
unitary and is a diagonal matrix of i.i.d. random phases. The
operator 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 , provided the
distribution of phases is absolutely continuous and the parameters correspond
to small off-diagonal elements of . Such estimates imply almost sure
localization for
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