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Entropy, Volume 14, Issue 7 (July 2012), Pages 1127-1305

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Research

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Open AccessArticle Fourth Order Diffusion Equations with Increasing Entropy
Entropy 2012, 14(7), 1127-1139; doi:10.3390/e14071127
Received: 11 April 2012 / Revised: 19 June 2012 / Accepted: 19 June 2012 / Published: 25 June 2012
Cited by 1 | PDF Full-text (130 KB) | HTML Full-text | XML Full-text
Abstract
The general quasi-linear autonomous fourth order diffusion equation ut = −[G(u)uxxx + h(u, ux, uxx)]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 [...] Read more.
The general quasi-linear autonomous fourth order diffusion equation ut = −[G(u)uxxx + h(u, ux, uxx)]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. Full article
(This article belongs to the Special Issue Concepts of Entropy and Their Applications)
Open AccessArticle Statistical Information Based Single Neuron Adaptive Control for Non-Gaussian Stochastic Systems
Entropy 2012, 14(7), 1154-1164; doi:10.3390/e14071154
Received: 3 May 2012 / Revised: 22 June 2012 / Accepted: 27 June 2012 / Published: 2 July 2012
Cited by 2 | PDF Full-text (498 KB) | HTML Full-text | XML Full-text
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 [...] Read more.
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. Full article
Open AccessArticle Information Theory Estimators for the First-Order Spatial Autoregressive Model
Entropy 2012, 14(7), 1165-1185; doi:10.3390/e14071165
Received: 25 May 2012 / Revised: 29 June 2012 / Accepted: 30 June 2012 / Published: 4 July 2012
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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 [...] Read more.
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. Full article
Open AccessArticle Multivariate Multi-Scale Permutation Entropy for Complexity Analysis of Alzheimer’s Disease EEG
Entropy 2012, 14(7), 1186-1202; doi:10.3390/e14071186
Received: 1 April 2012 / Revised: 21 June 2012 / Accepted: 26 June 2012 / Published: 4 July 2012
Cited by 41 | PDF Full-text (279 KB) | HTML Full-text | XML Full-text
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Concepts of Entropy and Their Applications)
Open AccessArticle The Dark Energy Properties of the Dirac–Born–Infeld Action
Entropy 2012, 14(7), 1203-1220; doi:10.3390/e14071203
Received: 1 May 2012 / Revised: 30 May 2012 / Accepted: 18 June 2012 / Published: 9 July 2012
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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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Modified Gravity: From Black Holes Entropy to Current Cosmology)
Open AccessArticle Nonparametric Estimation of Information-Based Measures of Statistical Dispersion
Entropy 2012, 14(7), 1221-1233; doi:10.3390/e14071221
Received: 29 March 2012 / Revised: 20 June 2012 / Accepted: 4 July 2012 / Published: 10 July 2012
Cited by 5 | PDF Full-text (289 KB) | HTML Full-text | XML Full-text
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Concepts of Entropy and Their Applications)
Open AccessArticle Association of Finite-Dimension Thermodynamics and a Bond-Graph Approach for Modeling an Irreversible Heat Engine
Entropy 2012, 14(7), 1234-1258; doi:10.3390/e14071234
Received: 23 April 2012 / Revised: 22 June 2012 / Accepted: 2 July 2012 / Published: 12 July 2012
Cited by 3 | PDF Full-text (1036 KB) | HTML Full-text | XML Full-text
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 Ths) and a heat sink (at temperature Tcs). We [...] Read more.
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 Ths) and a heat sink (at temperature Tcs). 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 (TcsTλ,φ cs, ThsTλ,φ hs) and apparent conductances (KhKλ h, KcKλ 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. Full article
(This article belongs to the Special Issue Advances in Applied Thermodynamics)
Open AccessArticle Quantum Dynamical Entropies and Gács Algorithmic Entropy
Entropy 2012, 14(7), 1259-1273; doi:10.3390/e14071259
Received: 13 April 2012 / Revised: 8 June 2012 / Accepted: 3 July 2012 / Published: 12 July 2012
Cited by 2 | PDF Full-text (123 KB) | HTML Full-text | XML Full-text
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Concepts of Entropy and Their Applications)
Open AccessArticle Discrete Wavelet Entropy Aided Detection of Abrupt Change: A Case Study in the Haihe River Basin, China
Entropy 2012, 14(7), 1274-1284; doi:10.3390/e14071274
Received: 10 May 2012 / Revised: 5 July 2012 / Accepted: 11 July 2012 / Published: 17 July 2012
Cited by 4 | PDF Full-text (445 KB) | HTML Full-text | XML Full-text
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 [...] Read more.
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. Full article
Open AccessArticle Socio-Thermodynamics—Evolutionary Potentials in a Population of Hawks and Doves
Entropy 2012, 14(7), 1285-1295; doi:10.3390/e14071285
Received: 3 May 2012 / Revised: 25 June 2012 / Accepted: 4 July 2012 / Published: 23 July 2012
PDF Full-text (1402 KB) | HTML Full-text | XML Full-text
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Concepts of Entropy and Their Applications)
Open AccessArticle A Model of Nonsingular Universe
Entropy 2012, 14(7), 1296-1305; doi:10.3390/e14071296
Received: 19 April 2012 / Revised: 4 June 2012 / Accepted: 13 June 2012 / Published: 23 July 2012
Cited by 9 | PDF Full-text (112 KB) | HTML Full-text | XML Full-text
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Modified Gravity: From Black Holes Entropy to Current Cosmology)

Review

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Open AccessReview Exact Solutions in Modified Gravity Models
Entropy 2012, 14(7), 1140-1153; doi:10.3390/e14071140
Received: 24 April 2012 / Revised: 21 May 2012 / Accepted: 21 May 2012 / Published: 25 June 2012
Cited by 3 | PDF Full-text (267 KB) | HTML Full-text | XML Full-text
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Modified Gravity: From Black Holes Entropy to Current Cosmology)

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