The unique solution of the Karcher equation and the self-adjointness of the Karcher mean

Lawson and Lim showed that the Karcher equation for positive invertible operators on a Hilbert space has a unique solution using the method of the implicit function theorem of a Banach space. In this paper, in the framework of the operator inequality, we show the equivalence of the unique solution of the Karcher equation and the self-adjointness of the Karcher mean. For this, we reform the notion of the operator power means of negative order by virtue of the Tsallis relative operator entropy of negative order.     

Some properties of generalized Fisher information in the context of nonextensive thermostatistics

We present two extended forms of Fisher information that fit well in the context of nonextensive thermostatistics. We show that there exists an interplay between these generalized Fisher information, the generalized q$q$-Gaussian distributions and the q$q$-entropies. The minimum of the generalized Fisher information among distributions with a fixed moment, or with a fixed q$q$-entropy is attained, in both cases, by a generalized q$q$-Gaussian distribution. This complements the fact that the q$q$-Gaussians maximize the q$q$-entropies subject to a moment constraint, and yields new variational characterizations of the generalizedq$q$-Gaussians. We show that the generalized Fisher information naturally pop up in the expression of the time derivative of the q$q$-entropies, for distributions satisfying a certain nonlinear heat equation. This result includes as a particular case the classical de Bruijn identity. Then we study further properties of the generalized Fisher information and of their minimization. We show that, though non additive, the generalized Fisher information of a combined system is upper bounded. In the case of mixing, we show that the generalized Fisher information is convex for q≥1$q\ge 1$. Finally, we show that the minimization of the generalized Fisher information subject to moment constraints satisfies a Legendre structure analog to the Legendre structure of thermodynamics.     

Randomness in denoised stock returns: The case of Moroccan family business companies

In this paper, we scrutinize entropy in family business stocks listed on Casablanca stock exchange and market index to assess randomness in their returns. For this purpose, we adopt a novel approach based on combination of stationary wavelet transform and Tsallis entropy for empirical analysis of the return series. The obtained empirical results show strong evidence that their respective entropy functions are characterized by opposite dynamics. Indeed, the information contents of their respective dynamics are statistically and significantly different. Obviously, information on regular events carried by family business returns is more certain, whilst that carried by market returns is uncertain. Such results are definitively useful to understand the nonlinear dynamics on returns on family business companies and those of the market. Without a doubt, they could be helpful for quantitative portfolio managers and investors.     

Holographic considerations on a Machian Universe

MOND theory explains the rotation curves of the galaxies. Verlinde’s ideas establish an entropic origin for gravitational forces and Tsallis principle generalizes the theory of Boltzmann–Gibbs. In this work we have promoted a connection between these recent approaches, that at first sight seemed to have few or no points in common, using the Mach’s principle as the background. In this way we have used Tsallis formalism to calculate the main parameters of the Machian Universe including the Hubble parameter and the age of the Universe. After that, we have also obtained a new value for the Tsallis parameter via Mach’s principle. Using Verlinde’s entropic gravity we have obtained new forms for MOND’s well established ingredients. Finally, based on the relations between particles and bits obtained here, we have discussed the idea of bits entanglement in the holographic screen.     

Entropic functionals of Laguerre polynomials and complexity properties of the half‐line Coulomb potential

The stationary states of the half‐line Coulomb potential are described by quantum‐mechanical wavefunctions, which are controlled by the Laguerre polynomials L(x). Here, we first calculate the qth‐order frequency or entropic moments of this quantum system, which is controlled by some entropic functionals of the Laguerre polynomials. These functionals are shown to be equal to a Lauricella function F(${1 \over q}$ ,…,,${1 \over q}$ ,1) by use of the Srivastava‐Niukkanen linearization relation of Laguerre polynomials. The resulting general expressions are applied to obtain the following information‐theoretic quantities of the half‐line Coulomb potential: disequilibrium, Renyi and Tsallis entropies. An alternative and simpler expression for the linear entropy is also found by means of a different method. Then, the Shannon entropy and the LMC shape complexity of the lowest and highest (Rydberg) energetic states are explicitly given; moreover, sharp information‐theoretic‐based upper bounds to these quantities are found for general physical states. These quantities are numerically discussed for the ground and various excited states. Finally, the uncertainty measures of the half‐line Coulomb potential given by the information‐theoretic lengths are discussed. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

Tsallis distribution and luminescence decays

Usually, the Kohlrausch (stretched exponential) function is employed to fit the luminescence decays. In this work we propose to use the Tsallis distribution as an alternative to describe them. We show that the curves of the luminescence decay obtained from the Tsallis distribution are close to those ones obtained from the stretched exponential. Further, we show that our result can fit well the data of porous silicon at low temperature and simulation result of the trapping controlled luminescence model.     

Rényi and Tsallis entropies of the Dirichlet and Neumann one-dimensional quantum wells

A comparative analysis of the Dirichlet and Neumann boundary conditions (BCs) of the one-dimensional (1D) quantum well extracts similarities and differences of the Rényi R(α) as well as Tsallis T(α) entropies between these two geometries. It is shown, in particular, that for either BC the dependences of the Rényi position components on the parameter α are the same for all orbitals but the lowest Neumann one, for which the corresponding functional R is not influenced by the variation of α. Lower limit α _TH of the semi-infinite range of the dimensionless Rényi/Tsallis coefficient where momentum entropies exist crucially depends on the position BC and is equal to 1/4 for the Dirichlet requirement and 1/2 for the Neumann one. At α approaching this critical value, the corresponding momentum functionals do diverge. The gap between the thresholds α _TH of the two BCs causes different behavior of the Rényi uncertainty relations as functions of α. For both configurations, the lowest energy level at α = 1/2 does saturate either type of the entropic inequality, thus confirming an earlier surmise about it. It is also conjectured that the threshold α _TH of ½ is characteristic of any 1D non-Dirichlet system. Other properties are discussed and analyzed from the mathematical and physical points of view.     

Quantum vibrational partition function in the non-extensive Tsallis framework

The quantum vibrational partition function has been obtained in the Tsallis statistics framework for the entropic index, q, between 1 and 2. The effect of non-extensivity on the population of states and thermodynamic properties have been studied and compared with their corresponding values obtained in the Boltzmann-Gibbs (BG) statistics. Our results show that the non-extensive partition function of harmonic oscillator at any temperature is larger than its corresponding values for an extensive system and that their differences increase with temperature and entropic index. Also, the number of accessible states increases with q but, compared to the BG statistics, the occupation number decreases for low energy levels while the population of the higher energy levels increases. The internal energy and heat capacity have also been obtained for the non-extensive harmonic oscillator system. Results indicate that the heat capacity is greater than its corresponding value in the extensive (BG) system at low temperatures but that this trend is reversed at higher temperatures.     

Approach on Tsallis statistical interpretation of hydrogen-atom by adopting the generalized radial distribution function

This paper revisits the statistical interpretation of the hydrogen atom within the framework of Tsallis Statistical Mechanics in the Canonical Ensemble. The convergence of the partition function does not exhibit for all the temperatures, while the well-known T → T′ transformation method of Tsallis Statistics fails, since non-monotonicity is observed between the ordinary temperature, T, and the auxiliary one, T′. Here we re-examine the inconsistency of T → T′ transformation method, in the case where the partition function converges for all the temperatures, by considering the generalized radial distribution function. We find that both the transformation method inconsistency and the partition function divergence can be recovered for all the temperatures, if the hydrogen atom is restricted within a critical radius R _c  ≤ 4.832 bohr, while Tsallis entropic index values are given by . An erratum to this article can be found at