Abstract:
background: bipolar disorder is a highly recurrent severe psychiatric disorder. the number of episodes has been found consistently associated with poor outcome. it has been suggested that bipolar patients with long duration of illness and highly recurrent course show great impairment of global functioning. objectives the aim of this study is to assess the clinical course and outcome of patients with bipolar disorder i and ii with a high number of mood episodes. methods: we compared a group of bipolar i and ii subjects whose number of episode was higher than ten (n = 167) with a similar-size representative sample of bipolar patients whose number of episodes was lower or equal than ten (n = 131). results: bipolar patients with more than 10 episodes have a more severe outcome of bipolar disorder. qualification and occupational status was clearly worse for the highly recurrent group which showed a predominance of depressive polarity. conclusions: these data suggest that bipolar patients with a highly recurrent course have significant functional impairment. with the passing of time, bipolar illness tends to be ruled by depressive features. treatment strategies may need to address this issue.

Abstract:
The macroscopic properties of a quantum system strongly depend on the spreading of the physical eigenfunctions (wavefunctions) of its Hamiltonian operador over its confined domain. The wavefunctions are often controlled by classical or hypergeometric-type orthogonal polynomials (Hermite, Laguerre and Jacobi). Here we discuss the spreading of these polynomials over its orthogonality interval by means of various information-theoretic quantities which grasp some facets of the polynomial distribution not yet analyzed. We consider the information-theoretic lengths closely related to the Fisher information and R\'enyi and Shannon entropies, which quantify the polynomial spreading far beyond the celebrated standard deviation.

Abstract:
The complexity measures of the Cr\'amer-Rao, Fisher-Shannon and LMC (L\'opez-Ruiz, Mancini and Calvet) types of the Rakhmanov probability density $\rho_n(x)=\omega(x) p_n^2(x)$ of the polynomials $p_n(x)$ orthogonal with respect to the weight function $\omega(x)$, $x\in (a,b)$, are used to quantify various two-fold facets of the spreading of the Hermite, Laguerre and Jacobi systems all over their corresponding orthogonality intervals in both analytical and computational ways. Their explicit (Cr\'amer-Rao) and asymptotical (Fisher-Shannon, LMC) values are given for the three systems of orthogonal polynomials. Then, these complexity-type mathematical quantities are numerically examined in terms of the polynomial's degree $n$ and the parameters which characterize the weight function. Finally, several open problems about the generalised hypergeometric functions of Lauricella and Srivastava-Daoust types, as well as on the asymptotics of weighted $L_q$-norms of Laguerre and Jacobi polynomials are pointed out.

Abstract:
The direct spreading measures of the Laguerre polynomials, which quantify the distribution of its Rakhmanov probability density along the positive real line in various complementary and qualitatively different ways, are investigated. These measures include the familiar root-mean-square or standard deviation and the information-theoretic lengths of Fisher, Renyi and Shannon types. The Fisher length is explicitly given. The Renyi length of order q (such that 2q is a natural number) is also found in terms of the polynomials parameters by means of two error-free computing approaches; one makes use of the Lauricella functions, which is based on the Srivastava-Niukkanen linearization relation of Laguerre polynomials, and another one which utilizes the multivariate Bell polynomials of Combinatorics. The Shannon length cannot be exactly calculated because of its logarithmic-functional form, but its asymptotics is provided and sharp bounds are obtained by use of an information-theoretic optimization procedure. Finally, all these spreading measures are mutually compared and computationally analyzed; in particular, it is found that the apparent quasi-linear relation between the Shannon length and the standard deviation becomes rigorously linear only asymptotically (i.e. for n>>1).

Abstract:
The measure of Jensen-Fisher divergence between probability distributions is introduced and its theoretical grounds set up. This quantity, in contrast to the remaining Jensen divergences, is very sensitive to the fluctuations of the probability distributions because it is controlled by the (local) Fisher information, which is a gradient functional of the distribution. So, it is appropriate and informative when studying the similarity of distributions, mainly for those having oscillatory character. The new Jensen-Fisher divergence shares with the Jensen-Shannon divergence the following properties: non-negativity, additivity when applied to an arbitrary number of probability densities, symmetry under exchange of these densities, vanishing if and only if all the densities are equal, and definiteness even when these densities present non-common zeros. Moreover, the Jensen-Fisher divergence is shown to be expressed in terms of the relative Fisher information as the Jensen-Shannon divergence does in terms of the Kullback-Leibler or relative Shannon entropy. Finally the Jensen-Shannon and Jensen-Fisher divergences are compared for the following three large, non-trivial and qualitatively different families of probability distributions: the sinusoidal, generalized gamma-like and Rakhmanov-Hermite distributions.

Abstract:
The asymptotics of the weighted $L_{p}$-norms of Hermite polynomials, which describes the R\'enyi entropy of order $p$ of the associated quantum oscillator probability density, is determined for $n\to\infty$ and $p>0$. Then, it is applied to the calculation of the R\'enyi entropy of the quantum-mechanical probability density of the highly-excited (Rydberg) states of the isotropic oscillator.

Abstract:
The information-theoretic representation of quantum systems, which complements the familiar energy description of the density-functional and wave-function-based theories, is here discussed. According to it, the internal disorder of the quantum-mechanical non-relativistic systems can be quantified by various single (Fisher information, Shannon entropy) and composite (e.g. Cramer-Rao, LMC shape and Fisher-Shannon complexity) functionals of the Schr\"odinger probability density. First, we examine these concepts and its application to quantum systems with central potentials. Then, we calculate these measures for hydrogenic systems, emphasizing their predictive power for various physical phenomena. Finally, some recent open problems are pointed out.

Abstract:
The uncertainty relation based on the Shannon entropies of the probability densities in position and momentum spaces is improved for quantum systems in arbitrary $D$-dimensional spherically symmetric potentials. To find it, we have used the $L^p$ -- $L^q$ norm inequality of L. De Carli and the logarithmic uncertainty relation for the Hankel transform of S. Omri. Applications to some relevant three-dimensional central potentials are shown.