Table of Contents

Abstract: The general quasi-linear autonomous fourth order diffusion equation u_t = −[G(u)u_xxx + h(u, u_x, u_xx)]_x with positive variable diffusivity G(u) and lower-order flux component h is considered on the real line. A direct algorithm produces a general class of equations for which the Shannon entropy density obeys a reaction-diffusion equation with a positive irreducible source term. Such equations may have any positive twice-differentiable diffusivity function G(u). The forms of such equations are the indicators of more general conservation equations whose entropy equation may be expressed in an alternative reaction-diffusion form whose source term, although reducible, is positive.

Abstract: We review the exact solutions in modified gravity. It is one of the main problems of mathematical physics for the gravity theory. One can obtain an exact solution if the field equations reduce to a system of ordinary differential equations. In this paper we consider a number of exact solutions obtained by the method of separation of variables. Some applications to Cosmology and BH entropy are briefly mentioned.

Abstract: Based on information theory, the single neuron adaptive control problem for stochastic systems with non-Gaussian noises is investigated in this paper. Here, the statistic information of the output within a receding window rather than the output value is used for the tracking problem. Firstly, the single neuron controller structure, which has the ability of self-learning and self-adaptation, is established. Then, an improved performance criterion is given to train the weights of the single neuron. Furthermore, the mean-square convergent condition of the proposed control algorithm is formulated. Finally, comparative simulation results are presented to show that the proposed algorithm is superior to the PID controller. The contributions of this work are twofold: (1) the optimal control algorithm is formulated in the data-driven framework, which needn’t the precise system model that is usually difficult to obtain; (2) the control problem of non-Gaussian systems can be effectively dealt with by the simple single neuron controller under improved minimum entropy criterion.

Abstract: Information theoretic estimators for the first-order spatial autoregressive model are introduced, small sample properties are investigated, and the estimator is applied empirically. Monte Carlo experiments are used to compare finite sample performance of more traditional spatial estimators to three different information theoretic estimators, including maximum empirical likelihood, maximum empirical exponential likelihood, and maximum log Euclidean likelihood. Information theoretic estimators are found to be robust to selected specifications of spatial autocorrelation and may dominate traditional estimators in the finite sample situations analyzed, except for the quasi-maximum likelihood estimator which competes reasonably well. The information theoretic estimators are illustrated via an application to hedonic housing pricing.

Abstract: An original multivariate multi-scale methodology for assessing the complexity of physiological signals is proposed. The technique is able to incorporate the simultaneous analysis of multi-channel data as a unique block within a multi-scale framework. The basic complexity measure is done by using Permutation Entropy, a methodology for time series processing based on ordinal analysis. Permutation Entropy is conceptually simple, structurally robust to noise and artifacts, computationally very fast, which is relevant for designing portable diagnostics. Since time series derived from biological systems show structures on multiple spatial-temporal scales, the proposed technique can be useful for other types of biomedical signal analysis. In this work, the possibility of distinguish among the brain states related to Alzheimer’s disease patients and Mild Cognitive Impaired subjects from normal healthy elderly is checked on a real, although quite limited, experimental database.

Abstract: Introducing a new potential, we deduce a general Lagrangian for Dirac–Born– Infeld (DBI) inflation, in which the determinant of the induced metric naturally includes the kinetic energy and the potential energy. In particular, the potential energy and kinetic energy can convert into each other at any same order, which is in agreement with the limit of classical physics. We also present a general sound speed in the evolutions of the universe, and the exact expressions of energy-momentum tensor, pressure and density. Furthermore, from the results we obtain the new equation of states. The analytic form of the action that is consistent with data turns out to be surprisingly simple and easy to categorize. Finally, we examine properties of the dark energy and introduce a novel mechanism for realizing either quintessence or phantom dark energy dominated phases within a string theoretical context.

Abstract: We address the problem of non-parametric estimation of the recently proposed measures of statistical dispersion of positive continuous random variables. The measures are based on the concepts of differential entropy and Fisher information and describe the “spread” or “variability” of the random variable from a different point of view than the ubiquitously used concept of standard deviation. The maximum penalized likelihood estimation of the probability density function proposed by Good and Gaskins is applied and a complete methodology of how to estimate the dispersion measures with a single algorithm is presented. We illustrate the approach on three standard statistical models describing neuronal activity.

Abstract: In recent decades, the approach known as Finite-Dimension Thermodynamics has provided a fruitful theoretical framework for the optimization of heat engines operating between a heat source (at temperature T_hs) and a heat sink (at temperature T_cs). We will show in this paper that the approach detailed in a previous paper [1] can be used to analytically model irreversible heat engines (with an additional assumption on the linearity of the heat transfer laws). By defining two dimensionless parameters, the intensity of internal dissipation and heat leakage within a heat engine were quantified. We then established the analogy between an endoreversible heat engine and an irreversible heat engine by using the apparent temperatures (T_cs → Tλ,φ cs, T_hs → Tλ,φ hs) and apparent conductances (K_h → Kλ h, K_c → Kλ c). We thus found the analytical expression of the maximum power of an irreversible heat engine. However, these apparent temperatures should not be used to calculate the conversion efficiency at the optimal operating point by analogy with the case of an endoreversible heat engine.

Abstract: Several quantum dynamical entropies have been proposed that extend the classical Kolmogorov–Sinai (dynamical) entropy. The same scenario appears in relation to the extension of algorithmic complexity theory to the quantum realm. A theorem of Brudno establishes that the complexity per unit time step along typical trajectories of a classical ergodic system equals the KS-entropy. In the following, we establish a similar relation between the Connes–Narnhofer–Thirring quantum dynamical entropy for the shift on quantum spin chains and the Gács algorithmic entropy. We further provide, for the same system, a weaker linkage between the latter algorithmic complexity and a different quantum dynamical entropy proposed by Alicki and Fannes.

Abstract: Detection of abrupt change is a key issue for understanding the facts and trends of climate change, but it is also a difficult task in practice. The Mann-Kendall (MK) test is commonly used for treating the issue, while the results are usually affected by the correlation and seasonal characters and sample size of series. This paper proposes a discrete wavelet entropy-aided approach for abrupt change detection, with the temperature analyses in the Haihe River Basin (HRB) as an example. The results show some obviously abrupt temperature changes in the study area in the 1960s–1990s. The MK test results do not reflect those abrupt temperature changes after the 1980s. Comparatively, the proposed approach can detect all main abrupt temperature changes in HRB, so it is more effective than the MK test. Differing from the MK test which only considers series’ value order or the conventional entropy which mainly considers series’ statistical random characters, the proposed approach is to describe the complexity and disorderliness of series using wavelet entropy theories, and it can fairly consider series’ composition and characteristics under different scales, so the results can more accurately reflect not only the abrupt changes, but also the complexity variation of a series over time. However, since it is based on the entropy theories, the series analyzed must have big sample size enough and the sampling rates being smaller than the concerned scale for the accurate computation of entropy values.

Abstract: The socio-thermodynamics of a population of two competing species exhibits strong analogies with the thermodynamics of solutions and alloys of two constituents. In particular we may construct strategy diagrams akin to the phase diagrams of chemical thermodynamics, complete with regions of homogeneous mixing and miscibility gaps.

Abstract: In the background of Friedmann–Robertson–Walker Universe, there exists Hawking radiation which comes from the cosmic apparent horizon due to quantum effect. Although the Hawking radiation on the late time evolution of the universe could be safely neglected, it plays an important role in the very early stage of the universe. In view of this point, we identify the temperature in the scalar field potential with the Hawking temperature of cosmic apparent horizon. Then we find a nonsingular universe sourced by the temperature-dependent scalar field. We find that the universe could be created from a de Sitter phase which has the Planck energy density. Thus the Big-Bang singularity is avoided.