Entropy
http://www.mdpi.com/journal/entropy
Latest open access articles published in Entropy at http://www.mdpi.com/journal/entropy<![CDATA[Entropy, Vol. 17, Pages 6801-6833: On the Calculation of System Entropy in Nonlinear Stochastic Biological Networks]]>
http://www.mdpi.com/1099-4300/17/10/6801
Biological networks are open systems that can utilize nutrients and energy from their environment for use in their metabolic processes, and produce metabolic products. System entropy is defined as the difference between input and output signal entropy, i.e., the net signal entropy of the biological system. System entropy is an important indicator for living or non-living biological systems, as biological systems can maintain or decrease their system entropy. In this study, system entropy is determined for the first time for stochastic biological networks, and a computation method is proposed to measure the system entropy of nonlinear stochastic biological networks that are subject to intrinsic random fluctuations and environmental disturbances. We find that intrinsic random fluctuations could increase the system entropy, and that the system entropy is inversely proportional to the robustness and stability of the biological networks. It is also determined that adding feedback loops to shift all eigenvalues to the farther left-hand plane of the complex s-domain could decrease the system entropy of a biological network.Entropy2015-10-081710Article10.3390/e17106801680168331099-43002015-10-08doi: 10.3390/e17106801Bor-Sen ChenShang-Wen WongCheng-Wei Li<![CDATA[Entropy, Vol. 17, Pages 6783-6800: New Patterns in Steady-State Chemical Kinetics: Intersections, Coincidences, Map of Events (Two-Step Mechanism)]]>
http://www.mdpi.com/1099-4300/17/10/6783
New patterns of steady-state chemical kinetics for continuously stirred-tank reactors (CSTR) have been found, i.e., intersections, maxima and coincidences, for two-step mechanism A↔B→C. There were found elegant analytical relationships for characteristics of these patterns (space times, values of concentrations and rates) allowing kinetic parameters to be easily determined. It was demonstrated that for the pair of species involved into the irreversible reaction (B and C), the space time of their corresponding concentration dependence intersection is invariant and does not depend on the initial conditions of the system. Maps of patterns are presented for visualization of their combinations and ranking in space time, and values of concentration and rates.Entropy2015-10-071710Article10.3390/e17106783678368001099-43002015-10-07doi: 10.3390/e17106783Daniel Branco-PintoGregory YablonskyGuy MarinDenis Constales<![CDATA[Entropy, Vol. 17, Pages 6765-6782: A Tale of Two Entangled Instabilities—The Dual Role of δ-O in HgBa2Can-1CunO2(n+1)+δ]]>
http://www.mdpi.com/1099-4300/17/10/6765
Low-energy instabilities in the hole-doped cuprates include, besides short range antiferromagnetic fluctuations and superconductivity, also ubiquitous translational and rotational symmetry breakings. The overwhelming majority of interpretations of these possibly related properties rely on mappings onto three bands spanned by the three atomic orbitals Cu3d(x2−y2)(σ), O2px(σ), and O2py(σ), these three local orbitals spanning the Zhang–Rice band (ZRB), the lower Hubbard bands (LHB) and the upper Hubbard bands (UHB), respectively. Here we demonstrate by means of supercell Density Functional Theory (DFT) (a) how oxygen intercalation affects the structures of the buffer layers, and (b) how the attenuated crystal field pulls two additional oxygen bands in the CuO2 plane to the Fermi level. The self-consistent changes in electronic structure reflected in the corresponding changes in external potential comprise formal properties of the Hohenberg–Kohn theorems. Validation of present days’ approximate exchange-correlation potentials to capture these qualitative effects by means of supercell DFT is made by comparing computed doping dependent structural shifts to corresponding experimentally observed correlations. The simplest generalization of Bardeen–Cooper–Schrieffer (BCS) theory is offered to articulate high-critical temperature superconductivity (HTS) from a normal state where crystal field causes states related to two non-hybridizing bands to coalesce at EF.Entropy2015-10-051710Article10.3390/e17106765676567821099-43002015-10-05doi: 10.3390/e17106765Itai Panas<![CDATA[Entropy, Vol. 17, Pages 6753-6764: Local Fractional Homotopy Perturbation Method for Solving Non-Homogeneous Heat Conduction Equations in Fractal Domains]]>
http://www.mdpi.com/1099-4300/17/10/6753
In this article, the local fractional Homotopy perturbation method is utilized to solve the non-homogeneous heat conduction equations. The operator is considered in the sense of the local fractional differential operator. Comparative results between non-homogeneous and homogeneous heat conduction equations are presented. The obtained result shows the non-differentiable behavior of heat conduction of the fractal temperature field in homogeneous media.Entropy2015-10-051710Article10.3390/e17106753675367641099-43002015-10-05doi: 10.3390/e17106753Yu ZhangCarlo CattaniXiao-Jun Yang<![CDATA[Entropy, Vol. 17, Pages 6743-6752: Quantum Secure Direct Communication Based on Dense Coding and Detecting Eavesdropping with Four-Particle Genuine Entangled State]]>
http://www.mdpi.com/1099-4300/17/10/6743
A novel quantum secure direct communication protocol based on four-particle genuine entangled state and quantum dense coding is proposed. In this protocol, the four-particle genuine entangled state is used to detect eavesdroppers, and quantum dense coding is used to encode the message. Finally, the security of the proposed protocol is discussed. During the security analysis, the method of entropy theory is introduced, and two detection strategies are compared quantitatively by comparing the relationship between the maximal information that the eavesdroppers (Eve) can obtain, and the probability of being detected. Through the analysis we can state that our scheme is feasible and secure.Entropy2015-09-301710Article10.3390/e17106743674367521099-43002015-09-30doi: 10.3390/e17106743Jian LiZeshi PanFengqi SunYanhua ChenZheng WangZuozhi Shi<![CDATA[Entropy, Vol. 17, Pages 6712-6742: Thermodynamic Analysis of Closed Steady or Cyclic Systems]]>
http://www.mdpi.com/1099-4300/17/10/6712
Closed, steady or cyclic thermodynamic systems, which have temperature variations over their boundaries, can represent an extremely large range of plants, devices or natural objects, such as combined heating, cooling and power plants, computers and data centres, and planets. Energy transfer rates can occur across the boundary, which are characterized as heat or work. We focus on the finite time thermodynamics aspects, on energy-based performance parameters, on rational efficiency and on the environmental reference temperature. To do this, we examine the net work rate of a closed, steady or cyclic system bounded by thermal resistances linked to isothermal reservoirs in terms of the first and second laws of thermodynamics. Citing relevant references from the literature, we propose a methodology that can improve the thermodynamic analysis of an energy-transforming or an exergy-destroying plant. Through the reflections and analysis presented, we have found an explanation of the second law that clarifies the link between the Clausius integral of heat over temperature and the reference temperature of the Gouy–Stodola theorem. With this insight and approach, the specification of the environmental reference temperature in exergy analysis becomes more solid. We have explained the relationship between the Curzon Ahlborn heat engine and an irreversible Carnot heat engine. We have outlined the nature of subsystem rational efficiencies and have found Rant’s anergy to play an important role. We postulate that heat transfer through thermal resistance is the sole basis of irreversibility.Entropy2015-09-291710Article10.3390/e17106712671267421099-43002015-09-29doi: 10.3390/e17106712Jim McGovern<![CDATA[Entropy, Vol. 17, Pages 6698-6711: Two-Dimensional Lattice Boltzmann for Reactive Rayleigh–Bénard and Bénard–Poiseuille Regimes]]>
http://www.mdpi.com/1099-4300/17/10/6698
We perform a computer simulation of the reaction-diffusion and convection that takes place in Rayleigh–Bénard and Bénard–Poiseuille regimes. The lattice Boltzmann equation (LBE) is used along with the Boussinesq approximation to solve the non-linear coupled differential equations that govern the systems’ thermo-hydrodynamics. Another LBE, is introduced to calculate the evolution concentration of the chemical species involved in the chemical reactions. The simulations are conducted at low Reynolds numbers and in terms of steady state between the first and second thermo-hydrodynamics instability. The results presented here (with no chemical reactions) are in good agreement with those reported in the scientific literature which gives us high expectations about the reliability of the chemical kinetics simulation. Some examples are provided.Entropy2015-09-291710Article10.3390/e17106698669867111099-43002015-09-29doi: 10.3390/e17106698Suemi Rodríguez-RomoOscar Ibañez-Orozco<![CDATA[Entropy, Vol. 17, Pages 6683-6697: Feature Extraction Method of Rolling Bearing Fault Signal Based on EEMD and Cloud Model Characteristic Entropy]]>
http://www.mdpi.com/1099-4300/17/10/6683
The randomness and fuzziness that exist in rolling bearings when faults occur result in uncertainty in acquisition signals and reduce the accuracy of signal feature extraction. To solve this problem, this study proposes a new method in which cloud model characteristic entropy (CMCE) is set as the signal characteristic eigenvalue. This approach can overcome the disadvantages of traditional entropy complexity in parameter selection when solving uncertainty problems. First, the acoustic emission signals under normal and damage rolling bearing states collected from the experiments are decomposed via ensemble empirical mode decomposition. The mutual information method is then used to select the sensitive intrinsic mode functions that can reflect signal characteristics to reconstruct the signal and eliminate noise interference. Subsequently, CMCE is set as the eigenvalue of the reconstructed signal. Finally, through the comparison of experiments between sample entropy, root mean square and CMCE, the results show that CMCE can better represent the characteristic information of the fault signal.Entropy2015-09-251710Article10.3390/e17106683668366971099-43002015-09-25doi: 10.3390/e17106683Long HanChengwei LiHongchen Liu<![CDATA[Entropy, Vol. 17, Pages 6663-6682: Identification of Green, Oolong and Black Teas in China via Wavelet Packet Entropy and Fuzzy Support Vector Machine]]>
http://www.mdpi.com/1099-4300/17/10/6663
To develop an automatic tea-category identification system with a high recall rate, we proposed a computer-vision and machine-learning based system, which did not require expensive signal acquiring devices and time-consuming procedures. We captured 300 tea images using a 3-CCD digital camera, and then extracted 64 color histogram features and 16 wavelet packet entropy (WPE) features to obtain color information and texture information, respectively. Principal component analysis was used to reduce features, which were fed into a fuzzy support vector machine (FSVM). Winner-take-all (WTA) was introduced to help the classifier deal with this 3-class problem. The 10 × 10-fold stratified cross-validation results show that the proposed FSVM + WTA method yields an overall recall rate of 97.77%, higher than 5 existing methods. In addition, the number of reduced features is only five, less than or equal to existing methods. The proposed method is effective for tea identification.Entropy2015-09-251710Article10.3390/e17106663666366821099-43002015-09-25doi: 10.3390/e17106663Shuihua WangXiaojun YangYudong ZhangPreetha PhillipsJianfei YangTi-Fei Yuan<![CDATA[Entropy, Vol. 17, Pages 6643-6662: Modified Gravity Models Admitting Second Order Equations of Motion]]>
http://www.mdpi.com/1099-4300/17/10/6643
The aim of this paper is to find higher order geometrical corrections to the Einstein–Hilbert action that can lead only to second order equations of motion. The metric formalism is used, and static spherically-symmetric and Friedmann–Lemaître space-times are considered, in four dimensions. The Fulling, King, Wybourne and Cummings (FKWC) basis is introduced in order to consider all of the possible invariant scalars, and both polynomial and non-polynomial gravities are investigated.Entropy2015-09-251710Article10.3390/e17106643664366621099-43002015-09-25doi: 10.3390/e17106643Aimeric ColléauxSergio Zerbini<![CDATA[Entropy, Vol. 17, Pages 6617-6642: Shannon Entropy-Based Wavelet Transform Method for Autonomous Coherent Structure Identification in Fluid Flow Field Data]]>
http://www.mdpi.com/1099-4300/17/10/6617
The coherent secondary flow structures (i.e., swirling motions) in a curved artery model possess a variety of spatio-temporal morphologies and can be encoded over an infinitely-wide range of wavelet scales. Wavelet analysis was applied to the following vorticity fields: (i) a numerically-generated system of Oseen-type vortices for which the theoretical solution is known, used for bench marking and evaluation of the technique; and (ii) experimental two-dimensional, particle image velocimetry data. The mother wavelet, a two-dimensional Ricker wavelet, can be dilated to infinitely large or infinitesimally small scales. We approached the problem of coherent structure detection by means of continuous wavelet transform (CWT) and decomposition (or Shannon) entropy. The main conclusion of this study is that the encoding of coherent secondary flow structures can be achieved by an optimal number of binary digits (or bits) corresponding to an optimal wavelet scale. The optimal wavelet-scale search was driven by a decomposition entropy-based algorithmic approach and led to a threshold-free coherent structure detection method. The method presented in this paper was successfully utilized in the detection of secondary flow structures in three clinically-relevant blood flow scenarios involving the curved artery model under a carotid artery-inspired, pulsatile inflow condition. These scenarios were: (i) a clean curved artery; (ii) stent-implanted curved artery; and (iii) an idealized Type IV stent fracture within the curved artery.Entropy2015-09-251710Article10.3390/e17106617661766421099-43002015-09-25doi: 10.3390/e17106617Kartik BulusuMichael Plesniak<![CDATA[Entropy, Vol. 17, Pages 6598-6616: Ultrasound Detection of Scatterer Concentration by Weighted Entropy]]>
http://www.mdpi.com/1099-4300/17/10/6598
Ultrasound backscattering signals depend on the microstructures of tissues. Some studies have applied Shannon entropy to analyze the uncertainty of raw radiofrequency (RF) data. However, we found that the sensitivity of entropy in detecting various scatterer concentrations is limited; thus, we propose a weighted entropy as a new information entropy-based approach to enhance the performance of scatterer characterization. A standard simulation model of ultrasound backscattering was used to generate backscattered RF signals with different number densities of scatterers. The RF signals were used to estimate the weighted entropy according to the proposed algorithmic scheme. The weighted entropy increased from 0.08 to 0.23 (representing a dynamic range of 0.15) when the number density of scatterers increased from 2 to 32 scatterers/mm2. In the same range of scatterer concentration, the conventional entropy increased from 0.16 to 0.19 (a dynamic range of 0.03). The results indicated that the weighted entropy enables achieving a more sensitive detection of the variation of scatterer concentrations by ultrasound.Entropy2015-09-251710Article10.3390/e17106598659866161099-43002015-09-25doi: 10.3390/e17106598Po-Hsiang Tsui<![CDATA[Entropy, Vol. 17, Pages 6576-6597: Bayesian Inference on the Memory Parameter for Gamma-Modulated Regression Models]]>
http://www.mdpi.com/1099-4300/17/10/6576
In this work, we propose a Bayesian methodology to make inferences for the memory parameter and other characteristics under non-standard assumptions for a class of stochastic processes. This class generalizes the Gamma-modulated process, with trajectories that exhibit long memory behavior, as well as decreasing variability as time increases. Different values of the memory parameter influence the speed of this decrease, making this heteroscedastic model very flexible. Its properties are used to implement an approximate Bayesian computation and MCMC scheme to obtain posterior estimates. We test and validate our method through simulations and real data from the big earthquake that occurred in 2010 in Chile.Entropy2015-09-251710Article10.3390/e17106576657665971099-43002015-09-25doi: 10.3390/e17106576Plinio AndradeLaura RifoSoledad TorresFrancisco Torres-Avilés<![CDATA[Entropy, Vol. 17, Pages 6560-6575: Expected Utility and Entropy-Based Decision-Making Model for Large Consumers in the Smart Grid]]>
http://www.mdpi.com/1099-4300/17/10/6560
In the smart grid, large consumers can procure electricity energy from various power sources to meet their load demands. To maximize its profit, each large consumer needs to decide their energy procurement strategy under risks such as price fluctuations from the spot market and power quality issues. In this paper, an electric energy procurement decision-making model is studied for large consumers who can obtain their electric energy from the spot market, generation companies under bilateral contracts, the options market and self-production facilities in the smart grid. Considering the effect of unqualified electric energy, the profit model of large consumers is formulated. In order to measure the risks from the price fluctuations and power quality, the expected utility and entropy is employed. Consequently, the expected utility and entropy decision-making model is presented, which helps large consumers to minimize their expected profit of electricity procurement while properly limiting the volatility of this cost. Finally, a case study verifies the feasibility and effectiveness of the proposed model.Entropy2015-09-251710Article10.3390/e17106560656065751099-43002015-09-25doi: 10.3390/e17106560Bingtuan GaoCheng WuYingjun WuYi Tang<![CDATA[Entropy, Vol. 17, Pages 6534-6559: A Bayesian Decision-Theoretic Approach to Logically-Consistent Hypothesis Testing]]>
http://www.mdpi.com/1099-4300/17/10/6534
This work addresses an important issue regarding the performance of simultaneous test procedures: the construction of multiple tests that at the same time are optimal from a statistical perspective and that also yield logically-consistent results that are easy to communicate to practitioners of statistical methods. For instance, if hypothesis A implies hypothesis B, is it possible to create optimal testing procedures that reject A whenever they reject B? Unfortunately, several standard testing procedures fail in having such logical consistency. Although this has been deeply investigated under a frequentist perspective, the literature lacks analyses under a Bayesian paradigm. In this work, we contribute to the discussion by investigating three rational relationships under a Bayesian decision-theoretic standpoint: coherence, invertibility and union consonance. We characterize and illustrate through simple examples optimal Bayes tests that fulfill each of these requisites separately. We also explore how far one can go by putting these requirements together. We show that although fairly intuitive tests satisfy both coherence and invertibility, no Bayesian testing scheme meets the desiderata as a whole, strengthening the understanding that logical consistency cannot be combined with statistical optimality in general. Finally, we associate Bayesian hypothesis testing with Bayes point estimation procedures. We prove the performance of logically-consistent hypothesis testing by means of a Bayes point estimator to be optimal only under very restrictive conditions.Entropy2015-09-241710Article10.3390/e17106534653465591099-43002015-09-24doi: 10.3390/e17106534Gustavo da SilvaLuis EstevesVictor FossaluzaRafael IzbickiSergio Wechsler<![CDATA[Entropy, Vol. 17, Pages 6519-6533: Approximate Analytical Solutions of Time Fractional Whitham–Broer–Kaup Equations by a Residual Power Series Method]]>
http://www.mdpi.com/1099-4300/17/9/6519
In this paper, a new analytic iterative technique, called the residual power series method (RPSM), is applied to time fractional Whitham–Broer–Kaup equations. The explicit approximate traveling solutions are obtained by using this method. The efficiency and accuracy of the present method is demonstrated by two aspects. One is analyzing the approximate solutions graphically. The other is comparing the results with those of the Adomian decomposition method (ADM), the variational iteration method (VIM) and the optimal homotopy asymptotic method (OHAM). Illustrative examples reveal that the present technique outperforms the aforementioned methods and can be used as an alternative for solving fractional equations.Entropy2015-09-23179Article10.3390/e17096519651965331099-43002015-09-23doi: 10.3390/e17096519Linjun WangXumei Chen<![CDATA[Entropy, Vol. 17, Pages 6503-6518: Thermodynamic Metrics and Black Hole Physics]]>
http://www.mdpi.com/1099-4300/17/9/6503
We give a brief survey of thermodynamic metrics, in particular the Hessian of the entropy function, and how they apply to black hole thermodynamics. We then provide a detailed discussion of the Gibbs surface of Kerr black holes. In particular, we analyze its global properties and extend it to take the entropy of the inner horizon into account. A brief discussion of Kerr–Newman black holes is included.Entropy2015-09-22179Article10.3390/e17096503650365181099-43002015-09-22doi: 10.3390/e17096503Jan ÅmanIngemar BengtssonNarit Pidokrajt<![CDATA[Entropy, Vol. 17, Pages 6481-6502: A Bayesian Predictive Discriminant Analysis with Screened Data]]>
http://www.mdpi.com/1099-4300/17/9/6481
In the application of discriminant analysis, a situation sometimes arises where individual measurements are screened by a multidimensional screening scheme. For this situation, a discriminant analysis with screened populations is considered from a Bayesian viewpoint, and an optimal predictive rule for the analysis is proposed. In order to establish a flexible method to incorporate the prior information of the screening mechanism, we propose a hierarchical screened scale mixture of normal (HSSMN) model, which makes provision for flexible modeling of the screened observations. An Markov chain Monte Carlo (MCMC) method using the Gibbs sampler and the Metropolis–Hastings algorithm within the Gibbs sampler is used to perform a Bayesian inference on the HSSMN models and to approximate the optimal predictive rule. A simulation study is given to demonstrate the performance of the proposed predictive discrimination procedure.Entropy2015-09-21179Article10.3390/e17096481648165021099-43002015-09-21doi: 10.3390/e17096481Hea-Jung Kim<![CDATA[Entropy, Vol. 17, Pages 6462-6480: Subspace Coding for Networks with Different Level Messages]]>
http://www.mdpi.com/1099-4300/17/9/6462
We study the asymptotically-achievable rate region of subspace codes for wireless network coding, where receivers have different link capacities due to the access ways or the faults of the intermediate links in the network. Firstly, an outer bound of the achievable rate region in a two-receiver network is derived from a combinatorial method. Subsequently, the achievability of the outer bound is proven by code construction, which is based on superposition coding. We show that the outer bound can be achieved asymptotically by using the code presented by Koetter and Kschischang, and the outer bound can be exactly attained in some points by using a q-analog Steiner structure. Finally, the asymptotically-achievable rate region is extended to the general case when the network has m receivers with different levels.Entropy2015-09-21179Article10.3390/e17096462646264801099-43002015-09-21doi: 10.3390/e17096462Feng CaiNing CaiWangmei Guo<![CDATA[Entropy, Vol. 17, Pages 6447-6461: Rolling Bearing Fault Diagnosis Based on Wavelet Packet Decomposition and Multi-Scale Permutation Entropy]]>
http://www.mdpi.com/1099-4300/17/9/6447
This paper presents a rolling bearing fault diagnosis approach by integrating wavelet packet decomposition (WPD) with multi-scale permutation entropy (MPE). The approach uses MPE values of the sub-frequency band signals to identify faults appearing in rolling bearings. Specifically, vibration signals measured from a rolling bearing test system with different defect conditions are decomposed into a set of sub-frequency band signals by means of the WPD method. Then, each sub-frequency band signal is divided into a series of subsequences, and MPEs of all subsequences in corresponding sub-frequency band signal are calculated. After that, the average MPE value of all subsequences about each sub-frequency band is calculated, and is considered as the fault feature of the corresponding sub-frequency band. Subsequently, MPE values of all sub-frequency bands are considered as input feature vectors, and the hidden Markov model (HMM) is used to identify the fault pattern of the rolling bearing. Experimental study on a data set from the Case Western Reserve University bearing data center has shown that the presented approach can accurately identify faults in rolling bearings.Entropy2015-09-21179Article10.3390/e17096447644764611099-43002015-09-21doi: 10.3390/e17096447Li-Ye ZhaoLei WangRu-Qiang Yan<![CDATA[Entropy, Vol. 17, Pages 6433-6446: Dynamical Systems Induced on Networks Constructed from Time Series]]>
http://www.mdpi.com/1099-4300/17/9/6433
Several methods exist to construct complex networks from time series. In general, these methods claim to construct complex networks that preserve certain properties of the underlying dynamical system, and hence, they mark new ways of accessing quantitative indicators based on that dynamics. In this paper, we test this assertion by developing an algorithm to realize dynamical systems from these complex networks in such a way that trajectories of these dynamical systems produce time series that preserve certain statistical properties of the original time series (and hence, also the underlying true dynamical system). Trajectories from these networks are constructed from only the information in the network and are shown to be statistically equivalent to the original time series. In the context of this algorithm, we are able to demonstrate that the so-called adaptive k-nearest neighbour algorithm for generating networks out-performs methods based on ε-ball recurrence plots. For such networks, and with a suitable choice of parameter values, which we provide, the time series generated by this method function as a new kind of nonlinear surrogate generation algorithm. With this approach, we are able to test whether the simulation dynamics built from a complex network capture the underlying structure of the original system; whether the complex network is an adequate model of the dynamics.Entropy2015-09-18179Article10.3390/e17096433643364461099-43002015-09-18doi: 10.3390/e17096433Lvlin HouMichael SmallSongyang Lao<![CDATA[Entropy, Vol. 17, Pages 6412-6432: Energy and Exergy Analyses of a Combined Power Cycle Using the Organic Rankine Cycle and the Cold Energy of Liquefied Natural Gas]]>
http://www.mdpi.com/1099-4300/17/9/6412
In this work, energy and exergy analyses are carried out for a combined cycle consisting of an organic Rankine cycle (ORC) and a liquefied natural gas (LNG) Rankine cycle for the recovery of low-grade heat sources and LNG cold energy. The effects of the turbine inlet pressure and the working fluid on the system performance are theoretically investigated. A modified temperature-enthalpy diagram is proposed, which can be useful to see the characteristics of the combined cycle, as well as the temperature distributions in the heat exchangers. Results show that the thermal efficiency increases with an increasing turbine inlet pressure and critical temperature of the working fluid. However, the exergy efficiency has a peak value with respect to the turbine inlet pressure, and the maximum exergy efficiency and the corresponding optimum turbine inlet pressure are significantly influenced by the selection of the working fluid. The exergy destruction at the condenser is generally the greatest among the exergy destruction components of the system.Entropy2015-09-18179Article10.3390/e17096412641264321099-43002015-09-18doi: 10.3390/e17096412Ho LeeKyoung Kim<![CDATA[Entropy, Vol. 17, Pages 6397-6411: Wavelet Entropy as a Measure of Ventricular Beat Suppression from the Electrocardiogram in Atrial Fibrillation]]>
http://www.mdpi.com/1099-4300/17/9/6397
A novel method of quantifying the effectiveness of the suppression of ventricular activity from electrocardiograms (ECGs) in atrial fibrillation is proposed. The temporal distribution of the energy of wavelet coefficients is quantified by wavelet entropy at each ventricular beat. More effective ventricular activity suppression yields increased entropies at scales dominated by the ventricular and atrial components of the ECG. Two studies are undertaken to demonstrate the efficacy of the method: first, using synthesised ECGs with controlled levels of residual ventricular activity, and second, using patient recordings with ventricular activity suppressed by an average beat template subtraction algorithm. In both cases wavelet entropy is shown to be a good measure of the effectiveness of ventricular beat suppression.Entropy2015-09-17179Article10.3390/e17096397639764111099-43002015-09-17doi: 10.3390/e17096397Philip Langley<![CDATA[Entropy, Vol. 17, Pages 6379-6396: A New Process Monitoring Method Based on Waveform Signal by Using Recurrence Plot]]>
http://www.mdpi.com/1099-4300/17/9/6379
Process monitoring is an important research problem in numerous areas. This paper proposes a novel process monitoring scheme by integrating the recurrence plot (RP) method and the control chart technique. Recently, the RP method has emerged as an effective tool to analyze waveform signals. However, unlike the existing RP methods that employ recurrence quantiﬁcation analysis (RQA) to quantify the recurrence plot by a few summary statistics; we propose new concepts of template recurrence plots and continuous-scale recurrence plots to characterize the waveform signals. A new feature extraction method is developed based on continuous-scale recurrence plot. Then, a monitoring statistic based on the top- approach is constructed from the continuous-scale recurrence plot. Finally, a bootstrap control chart is built to detect the signal changes based on the constructed monitoring statistics. The comprehensive simulation studies show that the proposed monitoring scheme outperforms other RQA-based control charts. In addition, a real case study of progressive stamping processes is implemented to further evaluate the performance of the proposed scheme for process monitoring.Entropy2015-09-16179Article10.3390/e17096379637963961099-43002015-09-16doi: 10.3390/e17096379Cheng ZhouWeidong Zhang<![CDATA[Entropy, Vol. 17, Pages 6329-6378: Entropies from Coarse-graining: Convex Polytopes vs. Ellipsoids]]>
http://www.mdpi.com/1099-4300/17/9/6329
We examine the Boltzmann/Gibbs/Shannon SBGS and the non-additive Havrda-Charvát/Daróczy/Cressie-Read/Tsallis Sq and the Kaniadakis κ-entropy Sκ from the viewpoint of coarse-graining, symplectic capacities and convexity. We argue that the functional form of such entropies can be ascribed to a discordance in phase-space coarse-graining between two generally different approaches: the Euclidean/Riemannian metric one that reflects independence and picks cubes as the fundamental cells in coarse-graining and the symplectic/canonical one that picks spheres/ellipsoids for this role. Our discussion is motivated by and confined to the behaviour of Hamiltonian systems of many degrees of freedom. We see that Dvoretzky’s theorem provides asymptotic estimates for the minimal dimension beyond which these two approaches are close to each other. We state and speculate about the role that dualities may play in this viewpoint.Entropy2015-09-15179Article10.3390/e17096329632963781099-43002015-09-15doi: 10.3390/e17096329Nikos Kalogeropoulos<![CDATA[Entropy, Vol. 17, Pages 6318-6328: Viscosity-Induced Crossing of the Phantom Barrier]]>
http://www.mdpi.com/1099-4300/17/9/6318
We show explicitly, by using astrophysical data plus reasonable assumptions for the bulk viscosity in the cosmic fluid, how the magnitude of this viscosity may be high enough to drive the fluid from its position in the quintessence region at present time t = 0 across the barrier w = −1 into the phantom region in the late universe. The phantom barrier is accordingly not a sharp mathematical divide, but rather a fuzzy concept. We also calculate the limiting forms of various thermodynamical quantities, including the rate of entropy production, for a dark energy fluid near the future Big Rip singularity.Entropy2015-09-14179Article10.3390/e17096318631863281099-43002015-09-14doi: 10.3390/e17096318Iver Brevik<![CDATA[Entropy, Vol. 17, Pages 6304-6317: Metrics and Energy Landscapes in Irreversible Thermodynamics]]>
http://www.mdpi.com/1099-4300/17/9/6304
We describe how several metrics are possible in thermodynamic state space but that only one, Weinhold’s, has achieved widespread use. Lengths calculated based on this metric have been used to bound dissipation in finite-time (irreversible) processes be they continuous or discrete, and described in the energy picture or the entropy picture. Examples are provided from thermodynamics of heat conversion processes as well as chemical reactions. Even losses in economics can be bounded using a thermodynamic type metric. An essential foundation for the metric is a complete equation of state including all extensive variables of the system; examples are given. Finally, the second law of thermodynamics imposes convexity on any equation of state, be it analytical or empirical.Entropy2015-09-10179Article10.3390/e17096304630463171099-43002015-09-10doi: 10.3390/e17096304Bjarne Andresen<![CDATA[Entropy, Vol. 17, Pages 6289-6303: Modeling of a Mass-Spring-Damper System by Fractional Derivatives with and without a Singular Kernel]]>
http://www.mdpi.com/1099-4300/17/9/6289
In this paper, the fractional equations of the mass-spring-damper system with Caputo and Caputo–Fabrizio derivatives are presented. The physical units of the system are preserved by introducing an auxiliary parameter σ. The input of the resulting equations is a constant and periodic source; for the Caputo case, we obtain the analytical solution, and the resulting equations are given in terms of the Mittag–Leffler function; for the Caputo–Fabrizio approach, the numerical solutions are obtained by the numerical Laplace transform algorithm. Our results show that the mechanical components exhibit viscoelastic behaviors producing temporal fractality at different scales and demonstrate the existence of Entropy 2015, 17 6290 material heterogeneities in the mechanical components. The Markovian nature of the model is recovered when the order of the fractional derivatives is equal to one.Entropy2015-09-10179Article10.3390/e17096289628963031099-43002015-09-10doi: 10.3390/e17096289José Gómez-AguilarHuitzilin Yépez-MartínezCelia Calderón-RamónInes Cruz-OrduñaRicardo Escobar-JiménezVictor Olivares-Peregrino<![CDATA[Entropy, Vol. 17, Pages 6270-6288: Determination of Sample Entropy and Fuzzy Measure Entropy Parameters for Distinguishing Congestive Heart Failure from Normal Sinus Rhythm Subjects]]>
http://www.mdpi.com/1099-4300/17/9/6270
Entropy provides a valuable tool for quantifying the regularity of physiological time series and provides important insights for understanding the underlying mechanisms of the cardiovascular system. Before any entropy calculation, certain common parameters need to be initialized: embedding dimension m, tolerance threshold r and time series length N. However, no specific guideline exists on how to determine the appropriate parameter values for distinguishing congestive heart failure (CHF) from normal sinus rhythm (NSR) subjects in clinical application. In the present study, a thorough analysis on the selection of appropriate values of m, r and N for sample entropy (SampEn) and recently proposed fuzzy measure entropy (FuzzyMEn) is presented for distinguishing two group subjects. 44 long-term NRS and 29 long-term CHF RR interval recordings from http://www.physionet.org were used as the non-pathological and pathological data respectively. Extreme (&gt;2 s) and abnormal heartbeat RR intervals were firstly removed from each RR recording and then the recording was segmented with a non-overlapping segment length N of 300 and 1000, respectively. SampEn and FuzzyMEn were performed for each RR segment under different parameter combinations: m of 1, 2, 3 and 4, and r of 0.10, 0.15, 0.20 and 0.25 respectively. The statistical significance between NSR and CHF groups under each combination of m, r and N was observed. The results demonstrated that the selection of m, r and N plays a critical role in determining the SampEn and FuzzyMEn outputs. Compared with SampEn, FuzzyMEn shows a better regularity when selecting the parameters m and r. In addition, FuzzyMEn shows a better relative consistency for distinguishing the two groups, that is, the results of FuzzyMEn in the NSR group were consistently lower than those in the CHF group while SampEn were not. The selections of m of 2 and 3 and r of 0.10 and 0.15 for SampEn and the selections of m of 1 and 2 whenever r (herein, rL = rG = r) are for FuzzyMEn (in addition to setting nL = 3 and nG = 2) were recommended to yield the fine classification results for the NSR and CHF groups.Entropy2015-09-10179Article10.3390/e17096270627062881099-43002015-09-10doi: 10.3390/e17096270Lina ZhaoShoushui WeiChengqiu ZhangYatao ZhangXinge JiangFeng LiuChengyu Liu<![CDATA[Entropy, Vol. 17, Pages 6258-6269: A Hydrodynamical Model for Carriers and Phonons With Generation-Recombination, Including Auger Effect]]>
http://www.mdpi.com/1099-4300/17/9/6258
The asymptotic procedure proposed allows to derive closed hydrodynamical equations from the kinetic equations of carriers and phonons (treated as a partecipating species) in a photon background. The direct generation-recombination processes are accounted for. The fluid-dynamical equations constructed for the chemical potentials of carriers, temperature, and drift velocity, are related to the extended thermodynamical (ET) ones for the chemical potentials of carriers, temperature, and drift velocity. In the drift-diffusion approximation the constitutive laws are derived and the Onsager relation recovered.Entropy2015-09-09179Article10.3390/e17096258625862691099-43002015-09-09doi: 10.3390/e17096258Alberto Rossani<![CDATA[Entropy, Vol. 17, Pages 6239-6257: Using Generalized Entropies and OC-SVM with Mahalanobis Kernel for Detection and Classification of Anomalies in Network Traffic]]>
http://www.mdpi.com/1099-4300/17/9/6239
Network anomaly detection and classification is an important open issue in network security. Several approaches and systems based on different mathematical tools have been studied and developed, among them, the Anomaly-Network Intrusion Detection System (A-NIDS), which monitors network traffic and compares it against an established baseline of a “normal” traffic profile. Then, it is necessary to characterize the “normal” Internet traffic. This paper presents an approach for anomaly detection and classification based on Shannon, Rényi and Tsallis entropies of selected features, and the construction of regions from entropy data employing the Mahalanobis distance (MD), and One Class Support Vector Machine (OC-SVM) with different kernels (Radial Basis Function (RBF) and Mahalanobis Kernel (MK)) for “normal” and abnormal traffic. Regular and non-regular regions built from “normal” traffic profiles allow anomaly detection, while the classification is performed under the assumption that regions corresponding to the attack classes have been previously characterized. Although this approach allows the use of as many features as required, only four well-known significant features were selected in our case. In order to evaluate our approach, two different data sets were used: one set of real traffic obtained from an Academic Local Area Network (LAN), and the other a subset of the 1998 MIT-DARPA set. For these data sets, a True positive rate up to 99.35%, a True negative rate up to 99.83% and a False negative rate at about 0.16% were yielded. Experimental results show that certain q-values of the generalized entropies and the use of OC-SVM with RBF kernel improve the detection rate in the detection stage, while the novel inclusion of MK kernel in OC-SVM and k-temporal nearest neighbors improve accuracy in classification. In addition, the results show that using the Box-Cox transformation, the Mahalanobis distance yielded high detection rates with an efficient computation time, while OC-SVM achieved detection rates slightly higher, but is more computationally expensive.Entropy2015-09-08179Article10.3390/e17096239623962571099-43002015-09-08doi: 10.3390/e17096239Jayro Santiago-PazDeni Torres-RomanAngel Figueroa-YpiñaJesus Argaez-Xool<![CDATA[Entropy, Vol. 17, Pages 6238: Correction on Davidson, R.M.; Lauritzen, A.; Seneff, S. Biological Water Dynamics and Entropy: A Biophysical Origin of Cancer and Other Diseases. Entropy 2013, 15, 3822-3876]]>
http://www.mdpi.com/1099-4300/17/9/6238
The authors wish to make the following correction to their paper [1]. The correct reference 190 in the reference list should be: [...]Entropy2015-09-08179Correction10.3390/e17096238623862381099-43002015-09-08doi: 10.3390/e17096238Robert DavidsonAnn LauritzenStephanie Seneff<![CDATA[Entropy, Vol. 17, Pages 6229-6237: On the Exact Solution of Wave Equations on Cantor Sets]]>
http://www.mdpi.com/1099-4300/17/9/6229
The transfer of heat due to the emission of electromagnetic waves is called thermal radiations. In local fractional calculus, there are numerous contributions of scientists, like Mandelbrot, who described fractal geometry and its wide range of applications in many scientific fields. Christianto and Rahul gave the derivation of Proca equations on Cantor sets. Hao et al. investigated the Helmholtz and diffusion equations in Cantorian and Cantor-Type Cylindrical Coordinates. Carpinteri and Sapora studied diffusion problems in fractal media in Cantor sets. Zhang et al. studied local fractional wave equations under fixed entropy. In this paper, we are concerned with the exact solutions of wave equations by the help of local fractional Laplace variation iteration method (LFLVIM). We develop an iterative scheme for the exact solutions of local fractional wave equations (LFWEs). The efficiency of the scheme is examined by two illustrative examples.Entropy2015-09-08179Article10.3390/e17096229622962371099-43002015-09-08doi: 10.3390/e17096229Dumitru BaleanuHasib KhanHossien JafariRahmat Khan<![CDATA[Entropy, Vol. 17, Pages 6213-6228: Statistical Manifolds with almost Quaternionic Structures and Quaternionic Kähler-like Statistical Submersions]]>
http://www.mdpi.com/1099-4300/17/9/6213
In this paper we investigate statistical manifolds with almost quaternionic structures. We define the concept of quaternionic Kähler-like statistical manifold and derive the main properties of quaternionic Kähler-like statistical submersions, extending in a new setting some previous results obtained by K. Takano concerning statistical manifolds endowed with almost complex and almost contact structures. Finally, we give a nontrivial example and propose some open problems in the field for further research.Entropy2015-09-07179Article10.3390/e17096213621362281099-43002015-09-07doi: 10.3390/e17096213Alina-Daniela VîlcuGabriel-Eduard Vîlcu<![CDATA[Entropy, Vol. 17, Pages 6200-6212: Thixotropic Phenomena in Water: Quantitative Indicators of Casimir-Magnetic Transformations from Vacuum Oscillations (Virtual Particles)]]>
http://www.mdpi.com/1099-4300/17/9/6200
The ~1.5 × 10−20 J which is considered a universal quantity and is associated with the movement of protons in water also relates to the ratio of the magnetic moment of a proton divided by its unit charge, multiplied by viscosity and applied over the O-H distance. There is quantitative evidence that thixotropy, the “spontaneous” increased viscosity in water when undisturbed, originates from the transformation of virtual particles or vacuum oscillations to real states through conversion of Casimir-magnetic energies that involve the frequency of the neutral hydrogen line and the upper bound threshold value for intergalactic magnetic fields. The results indicate that ½ of a single electron orbit is real (particle) and the other ½ is virtual (wave). The matter equivalent per s for virtual-to-real states for electrons in 1 mL of water with a neutral pH is consistent with the numbers of protons (H+) and the measured range of molecules in the coherent domains for both width and duration of growth and is similar to widths of intergalactic dust grains from which planets and stars may condense. The de Broglie momentum for the lower boundary of the width of coherent domains multiplied by the fine structure velocity of an electron is concurrent with the quantum when one proton is being removed from another and when the upper boundary of the rest mass of a photon is transformed by the product of velocities for putative “entanglement” and light. Theoretical and experimental results indicate that components of thixotropy, such as specific domains of intercalated water molecules, could display excess correlations over very large distances. Because the energies of the universal quantity and water converge it may be a special conduit for discrete transformations from virtual to real states.Entropy2015-09-07179Concept Paper10.3390/e17096200620062121099-43002015-09-07doi: 10.3390/e17096200Michael Persinger<![CDATA[Entropy, Vol. 17, Pages 6179-6199: Wavelet Entropy Automatically Detects Episodes of Atrial Fibrillation from Single-Lead Electrocardiograms]]>
http://www.mdpi.com/1099-4300/17/9/6179
This work introduces for the first time the application of wavelet entropy (WE) to detect episodes of the most common cardiac arrhythmia, atrial fibrillation (AF), automatically from the electrocardiogram (ECG). Given that AF is often asymptomatic and usually presents very brief initial episodes, its early automatic detection is clinically relevant to improve AF treatment and prevent risks for the patients. After discarding noisy TQ intervals from the ECG, the WE has been computed over the median TQ segment obtained from the 10 previous noise-free beats under study. In this way, the P-waves or the fibrillatory waves present in the recording were highlighted or attenuated, respectively, thus enabling the patient’s rhythm identification (sinus rhythm or AF). Results provided a discriminant ability of about 95%, which is comparable to previous works. However, in contrast to most of them, which are mainly based on quantifying RR series variability, the proposed algorithm is able to deal with patients under rate-control therapy or with a reduced heart rate variability during AF. Additionally, it also presents interesting properties, such as the lowest delay in detecting AF or sinus rhythm, the ability to detect episodes as brief as five beats in length or its integration facilities under real-time beat-by-beat ECG monitoring systems. Consequently, this tool may help clinicians in the automatic detection of a wide variety of AF episodes, thus gaining further knowledge about the mechanisms initiating this arrhythmia.Entropy2015-09-07179Article10.3390/e17096179617961991099-43002015-09-07doi: 10.3390/e17096179Juan RódenasManuel GarcíaRaúl AlcarazJosé Rieta<![CDATA[Entropy, Vol. 17, Pages 6169-6178: Short-Lived Lattice Quasiparticles for Strongly Interacting Fluids]]>
http://www.mdpi.com/1099-4300/17/9/6169
It is shown that lattice kinetic theory based on short-lived quasiparticles proves very effective in simulating the complex dynamics of strongly interacting fluids (SIF). In particular, it is pointed out that the shear viscosity of lattice fluids is the sum of two contributions, one due to the usual interactions between particles (collision viscosity) and the other due to the interaction with the discrete lattice (propagation viscosity). Since the latter is negative, the sum may turn out to be orders of magnitude smaller than each of the two contributions separately, thus providing a mechanism to access SIF regimes at ordinary values of the collisional viscosity. This concept, as applied to quantum superfluids in one-dimensional optical lattices, is shown to reproduce shear viscosities consistent with the AdS-CFT holographic bound on the viscosity/entropy ratio. This shows that lattice kinetic theory continues to hold for strongly coupled hydrodynamic regimes where continuum kinetic theory may no longer be applicable.Entropy2015-09-03179Article10.3390/e17096169616961781099-43002015-09-03doi: 10.3390/e17096169Miller JimenezSauro Succi<![CDATA[Entropy, Vol. 17, Pages 6150-6168: Conformal Gauge Transformations in Thermodynamics]]>
http://www.mdpi.com/1099-4300/17/9/6150
In this work, we show that the thermodynamic phase space is naturally endowed with a non-integrable connection, defined by all of those processes that annihilate the Gibbs one-form, i.e., reversible processes. We argue that such a connection is invariant under re-scalings of the connection one-form, whilst, as a consequence of the non-integrability of the connection, its curvature is not and, therefore, neither is the associated pseudo-Riemannian geometry. We claim that this is not surprising, since these two objects are associated with irreversible processes. Moreover, we provide the explicit form in which all of the elements of the geometric structure of the thermodynamic phase space change under a re-scaling of the connection one-form. We call this transformation of the geometric structure a conformal gauge transformation. As an example, we revisit the change of the thermodynamic representation and consider the resulting change between the two metrics on the thermodynamic phase space, which induce Weinhold’s energy metric and Ruppeiner’s entropy metric. As a by-product, we obtain a proof of the well-known conformal relation between Weinhold’s and Ruppeiner’s metrics along the equilibrium directions. Finally, we find interesting properties of the almost para-contact structure and of its eigenvectors, which may be of physical interest.Entropy2015-09-02179Article10.3390/e17096150615061681099-43002015-09-02doi: 10.3390/e17096150Alessandro BravettiCesar Lopez-MonsalvoFrancisco Nettel<![CDATA[Entropy, Vol. 17, Pages 6129-6149: Nonlinear Predictive Control of a Hydropower System Model]]>
http://www.mdpi.com/1099-4300/17/9/6129
A six-dimensional nonlinear hydropower system controlled by a nonlinear predictive control method is presented in this paper. In terms of the nonlinear predictive control method; the performance index with terminal penalty function is selected. A simple method to find an appropriate terminal penalty function is introduced and its effectiveness is proved. The input-to-state-stability of the controlled system is proved by using the Lyapunov function. Subsequently a six-dimensional model of the hydropower system is presented in the paper. Different with other hydropower system models; the above model includes the hydro-turbine system; the penstock system; the generator system; and the hydraulic servo system accurately describing the operational process of a hydropower plant. Furthermore, the numerical experiments show that the six-dimensional nonlinear hydropower system controlled by the method is stable. In addition, the numerical experiment also illustrates that the nonlinear predictive control method enjoys great advantages over a traditional control method in nonlinear systems. Finally, a strategy to combine the nonlinear predictive control method with other methods is proposed to further facilitate the application of the nonlinear predictive control method into practice.Entropy2015-09-01179Article10.3390/e17096129612961491099-43002015-09-01doi: 10.3390/e17096129Runfan ZhangDiyi ChenXiaoyi Ma<![CDATA[Entropy, Vol. 17, Pages 6110-6128: Entropic Dynamics]]>
http://www.mdpi.com/1099-4300/17/9/6110
Entropic Dynamics is a framework in which dynamical laws are derived as an application of entropic methods of inference. No underlying action principle is postulated. Instead, the dynamics is driven by entropy subject to the constraints appropriate to the problem at hand. In this paper we review three examples of entropic dynamics. First we tackle the simpler case of a standard diffusion process which allows us to address the central issue of the nature of time. Then we show that imposing the additional constraint that the dynamics be non-dissipative leads to Hamiltonian dynamics. Finally, considerations from information geometry naturally lead to the type of Hamiltonian that describes quantum theory.Entropy2015-09-01179Review10.3390/e17096110611061281099-43002015-09-01doi: 10.3390/e17096110Ariel Caticha<![CDATA[Entropy, Vol. 17, Pages 6093-6109: Optimal Base Wavelet Selection for ECG Noise Reduction Using a Comprehensive Entropy Criterion]]>
http://www.mdpi.com/1099-4300/17/9/6093
The selection of an appropriate wavelet is an essential issue that should be addressed in the wavelet-based filtering of electrocardiogram (ECG) signals. Since entropy can measure the features of uncertainty associated with the ECG signal, a novel comprehensive entropy criterion Ecom based on multiple criteria related to entropy and energy is proposed in this paper to search for an optimal base wavelet for a specific ECG signal. Taking account of the decomposition capability of wavelets and the similarity in information between the decomposed coefficients and the analyzed signal, the proposed Ecom criterion integrates eight criteria, i.e., energy, entropy, energy-to-entropy ratio, joint entropy, conditional entropy, mutual information, relative entropy, as well as comparison information entropy for optimal wavelet selection. The experimental validation is conducted on the basis of ECG signals of sixteen subjects selected from the MIT-BIH Arrhythmia Database. The Ecom is compared with each of these eight criteria through four filtering performance indexes, i.e., output signal to noise ratio (SNRo), root mean square error (RMSE), percent root mean-square difference (PRD) and correlation coefficients. The filtering results of ninety-six ECG signals contaminated by noise have verified that Ecom has outperformed the other eight criteria in the selection of best base wavelets for ECG signal filtering. The wavelet identified by the Ecom has achieved the best filtering performance than the other comparative criteria. A hypothesis test also validates that SNRo, RMSE, PRD and correlation coefficients of Ecom are significantly different from those of the shape-matched approach (α = 0.05 , two-sided t- test).Entropy2015-09-01179Article10.3390/e17096093609361091099-43002015-09-01doi: 10.3390/e17096093Hong HeYonghong TanYuexia Wang<![CDATA[Entropy, Vol. 17, Pages 6072-6092: Distributing Secret Keys with Quantum Continuous Variables: Principle, Security and Implementations]]>
http://www.mdpi.com/1099-4300/17/9/6072
The ability to distribute secret keys between two parties with information-theoretic security, that is regardless of the capacities of a malevolent eavesdropper, is one of the most celebrated results in the field of quantum information processing and communication. Indeed, quantum key distribution illustrates the power of encoding information on the quantum properties of light and has far-reaching implications in high-security applications. Today, quantum key distribution systems operate in real-world conditions and are commercially available. As with most quantum information protocols, quantum key distribution was first designed for qubits, the individual quanta of information. However, the use of quantum continuous variables for this task presents important advantages with respect to qubit-based protocols, in particular from a practical point of view, since it allows for simple implementations that require only standard telecommunication technology. In this review article, we describe the principle of continuous-variable quantum key distribution, focusing in particular on protocols based on coherent states. We discuss the security of these protocols and report on the state-of-the-art in experimental implementations, including the issue of side-channel attacks. We conclude with promising perspectives in this research field.Entropy2015-08-31179Review10.3390/e17096072607260921099-43002015-08-31doi: 10.3390/e17096072Eleni DiamantiAnthony Leverrier<![CDATA[Entropy, Vol. 17, Pages 6056-6071: Self-Similar Solutions of Rényi’s Entropy and the Concavity of Its Entropy Power]]>
http://www.mdpi.com/1099-4300/17/9/6056
We study the class of self-similar probability density functions with finite mean and variance, which maximize Rényi’s entropy. The investigation is restricted in the Schwartz space S(Rd) and in the space of l-differentiable compactly supported functions Clc (Rd). Interestingly, the solutions of this optimization problem do not coincide with the solutions of the usual porous medium equation with a Dirac point source, as occurs in the optimization of Shannon’s entropy. We also study the concavity of the entropy power in Rd with respect to time using two different methods. The first one takes advantage of the solutions determined earlier, while the second one is based on a setting that could be used for Riemannian manifolds.Entropy2015-08-31179Article10.3390/e17096056605660711099-43002015-08-31doi: 10.3390/e17096056Agapitos Hatzinikitas<![CDATA[Entropy, Vol. 17, Pages 6044-6055: The Effect of a Long-Range Correlated-Hopping Interaction on Bariev Spin Chains]]>
http://www.mdpi.com/1099-4300/17/9/6044
We introduce a long-range particle and spin interaction into the standard Bariev model and show that this interaction is equivalent to a phase shift in the kinetic term of the Hamiltonian. When the particles circle around the chain and across the boundary, the accumulated phase shift acts as a twist boundary condition with respect to the normal periodic boundary condition. This boundary phase term depends on the total number of particles in the system and also the number of particles in different spin states, which relates to the spin fluctuations in the system. The model is solved exactly via a unitary transformation by the coordinate Bethe ansatz. We calculate the Bethe equations and work out the energy spectrum with varying number of particles and spins.Entropy2015-08-28179Article10.3390/e17096044604460551099-43002015-08-28doi: 10.3390/e17096044Tao YangFa-Kai WenKun HaoLi-Ke CaoRui-Hong Yue<![CDATA[Entropy, Vol. 17, Pages 6025-6043: New Exact Solutions of the New Hamiltonian Amplitude-Equation and Fokas Lenells Equation]]>
http://www.mdpi.com/1099-4300/17/9/6025
In this paper, exact solutions of the new Hamiltonian amplitude equation and Fokas-Lenells equation are successfully obtained. The extended trial equation method (ETEM) and generalized Kudryashov method (GKM) are applied to find several exact solutions of the new Hamiltonian amplitude equation and Fokas-Lenells equation. Primarily, we seek some exact solutions of the new Hamiltonian amplitude equation and Fokas-Lenells equation by using ETEM. Then, we research dark soliton solutions of the new Hamiltonian amplitude equation and Fokas-Lenells equation by using GKM. Lastly, according to the values of some parameters, we draw two and three dimensional graphics of imaginary and real values of certain solutions found by utilizing both methods.Entropy2015-08-27179Article10.3390/e17096025602560431099-43002015-08-27doi: 10.3390/e17096025Seyma DemirayHasan Bulut<![CDATA[Entropy, Vol. 17, Pages 6007-6024: A Gloss Composition and Context Clustering Based Distributed Word Sense Representation Model]]>
http://www.mdpi.com/1099-4300/17/9/6007
In recent years, there has been an increasing interest in learning a distributed representation of word sense. Traditional context clustering based models usually require careful tuning of model parameters, and typically perform worse on infrequent word senses. This paper presents a novel approach which addresses these limitations by first initializing the word sense embeddings through learning sentence-level embeddings from WordNet glosses using a convolutional neural networks. The initialized word sense embeddings are used by a context clustering based model to generate the distributed representations of word senses. Our learned representations outperform the publicly available embeddings on half of the metrics in the word similarity task, 6 out of 13 sub tasks in the analogical reasoning task, and gives the best overall accuracy in the word sense effect classification task, which shows the effectiveness of our proposed distributed distribution learning model.Entropy2015-08-27179Article10.3390/e17096007600760241099-43002015-08-27doi: 10.3390/e17096007Tao ChenRuifeng XuYulan HeXuan Wang<![CDATA[Entropy, Vol. 17, Pages 5995-6006: Proportionate Minimum Error Entropy Algorithm for Sparse System Identification]]>
http://www.mdpi.com/1099-4300/17/9/5995
Sparse system identification has received a great deal of attention due to its broad applicability. The proportionate normalized least mean square (PNLMS) algorithm, as a popular tool, achieves excellent performance for sparse system identification. In previous studies, most of the cost functions used in proportionate-type sparse adaptive algorithms are based on the mean square error (MSE) criterion, which is optimal only when the measurement noise is Gaussian. However, this condition does not hold in most real-world environments. In this work, we use the minimum error entropy (MEE) criterion, an alternative to the conventional MSE criterion, to develop the proportionate minimum error entropy (PMEE) algorithm for sparse system identification, which may achieve much better performance than the MSE based methods especially in heavy-tailed non-Gaussian situations. Moreover, we analyze the convergence of the proposed algorithm and derive a sufficient condition that ensures the mean square convergence. Simulation results confirm the excellent performance of the new algorithm.Entropy2015-08-27179Letter10.3390/e17095995599560061099-43002015-08-27doi: 10.3390/e17095995Zongze WuSiyuan PengBadong ChenHaiquan ZhaoJose Principe<![CDATA[Entropy, Vol. 17, Pages 5980-5994: Numerical Investigation into Natural Convection and Entropy Generation in a Nanofluid-Filled U-Shaped Cavity]]>
http://www.mdpi.com/1099-4300/17/9/5980
This current work studies the heat transfer performance and entropy generation of natural convection in a nanofluid-filled U-shaped cavity. The flow behavior and heat transfer performance in the cavity are governed using the continuity equation, momentum equations, energy equation and Boussinesq approximation, and are solved numerically using the finite-volume method and SIMPLE C algorithm. The simulations examine the effects of the nanoparticle volume fraction, Rayleigh number and the geometry parameters of the U-shaped cavity on the mean Nusselt number and total entropy generation. It shows that the mean Nusselt number increases and the total entropy generation reduces as the volume fraction of nanoparticles increases. In addition, the results show that the mean Nusselt number and the total entropy generation are both increased as the Rayleigh number increases. Finally, it also shows that mean Nusselt number can be increased and the total entropy generation can be reduced by extending the length of the low temperature walls or widening the width of the low temperature walls.Entropy2015-08-26179Article10.3390/e17095980598059941099-43002015-08-26doi: 10.3390/e17095980Ching-Chang ChoHer-Terng YauChing-Huang ChiuKuo-Ching Chiu<![CDATA[Entropy, Vol. 17, Pages 5965-5979: Friction Signal Denoising Using Complete Ensemble EMD with Adaptive Noise and Mutual Information]]>
http://www.mdpi.com/1099-4300/17/9/5965
During the measurement of friction force, the measured signal generally contains noise. To remove the noise and preserve the important features of the signal, a hybrid filtering method is introduced that uses the mutual information and a new waveform. This new waveform is the difference between the original signal and the sum of intrinsic mode functions (IMFs), which are obtained by empirical mode decomposition (EMD) or its improved versions. To evaluate the filter performance for the friction signal, ensemble EMD (EEMD), complementary ensemble EMD (CEEMD), and complete ensemble EMD with adaptive noise (CEEMDAN) are employed in combination with the proposed filtering method. The combination is used to filter the synthesizing signals at first. For the filtering of the simulation signal, the filtering effect is compared under conditions of different ensemble number, sampling frequency, and the input signal-noise ratio, respectively. Results show that CEEMDAN outperforms other signal filtering methods. In particular, this method is successful in filtering the friction signal as evaluated by the de-trended fluctuation analysis (DFA) algorithm.Entropy2015-08-25179Article10.3390/e17095965596559791099-43002015-08-25doi: 10.3390/e17095965Chengwei LiLiwei ZhanLiqun Shen<![CDATA[Entropy, Vol. 17, Pages 5938-5964: Geometry of Multiscale Nonequilibrium Thermodynamics]]>
http://www.mdpi.com/1099-4300/17/9/5938
The time evolution of macroscopic systems can be experimentally observed and mathematically described on many different levels of description. It has been conjectured that the governing equations on all levels are particular realizations of a single abstract equation. We support this conjecture by interpreting the abstract equation as a geometrical formulation of general nonequilibrium thermodynamics.Entropy2015-08-25179Review10.3390/e17095938593859641099-43002015-08-25doi: 10.3390/e17095938Miroslav Grmela<![CDATA[Entropy, Vol. 17, Pages 5920-5937: Entropy Associated with Information Storage and Its Retrieval]]>
http://www.mdpi.com/1099-4300/17/8/5920
We provide an entropy analysis for light storage and light retrieval. In this analysis, entropy extraction and reduction in a typical light storage experiment are identified. The spatiotemporal behavior of entropy is presented for D1 transition in cold sodium atoms. The governing equations are the reduced Maxwell field equations and the Liouville–von Neumann equation for the density matrix of the dressed atom.Entropy2015-08-24178Article10.3390/e17085920592059371099-43002015-08-24doi: 10.3390/e17085920Abu Alhasan<![CDATA[Entropy, Vol. 17, Pages 5903-5919: Maximal Repetitions in Written Texts: Finite Energy Hypothesis vs. Strong Hilberg Conjecture]]>
http://www.mdpi.com/1099-4300/17/8/5903
The article discusses two mutually-incompatible hypotheses about the stochastic mechanism of the generation of texts in natural language, which could be related to entropy. The first hypothesis, the finite energy hypothesis, assumes that texts are generated by a process with exponentially-decaying probabilities. This hypothesis implies a logarithmic upper bound for maximal repetition, as a function of the text length. The second hypothesis, the strong Hilberg conjecture, assumes that the topological entropy grows as a power law. This hypothesis leads to a hyperlogarithmic lower bound for maximal repetition. By a study of 35 written texts in German, English and French, it is found that the hyperlogarithmic growth of maximal repetition holds for natural language. In this way, the finite energy hypothesis is rejected, and the strong Hilberg conjecture is partly corroborated.Entropy2015-08-21178Article10.3390/e17085903590359191099-43002015-08-21doi: 10.3390/e17085903Łukasz Dębowski<![CDATA[Entropy, Vol. 17, Pages 5888-5902: Generalised Complex Geometry in Thermodynamical Fluctuation Theory]]>
http://www.mdpi.com/1099-4300/17/8/5888
We present a brief overview of some key concepts in the theory of generalized complex manifolds. This new geometry interpolates, so to speak, between symplectic geometry and complex geometry. As such it provides an ideal framework to analyze thermodynamical fluctuation theory in the presence of gravitational fields. To illustrate the usefulness of generalized complex geometry, we examine a simplified version of the Unruh effect: the thermalising effect of gravitational fields on the Schroedinger wavefunction.Entropy2015-08-20178Article10.3390/e17085888588859021099-43002015-08-20doi: 10.3390/e17085888P. Fernández de CórdobaJ. Isidro<![CDATA[Entropy, Vol. 17, Pages 5868-5887: Detection of Causality between Process Variables Based on Industrial Alarm Data Using Transfer Entropy]]>
http://www.mdpi.com/1099-4300/17/8/5868
In modern industrial processes, it is easier and less expensive to configure alarms by software settings rather than by wiring, which causes the rapid growth of the number of alarms. Moreover, because there exist complex interactions, in particular the causal relationship among different parts in the process, a fault may propagate along propagation pathways once an abnormal situation occurs, which brings great difficulty to operators to identify its root cause immediately and to take proper actions correctly. Therefore, causality detection becomes a very important problem in the context of multivariate alarm analysis and design. Transfer entropy has become an effective and widely-used method to detect causality between different continuous process variables in both linear and nonlinear situations in recent years. However, such conventional methods to detect causality based on transfer entropy are computationally costly. Alternatively, using binary alarm series can be more computational-friendly and more direct because alarm data analysis is straightforward for alarm management in practice. The methodology and implementation issues are discussed in this paper. Illustrated by several case studies, including both numerical cases and simulated industrial cases, the proposed method is demonstrated to be suitable for industrial situations contaminated by noise.Entropy2015-08-20178Article10.3390/e17085868586858871099-43002015-08-20doi: 10.3390/e17085868Weijun YuFan Yang<![CDATA[Entropy, Vol. 17, Pages 5848-5867: A Model for Scale-Free Networks: Application to Twitter]]>
http://www.mdpi.com/1099-4300/17/8/5848
In the last few years, complex networks have become an increasingly relevant research topic due to the large number of fields of application. Particularly, complex networks are especially significant in the area of modern online social networks (OSNs). OSNs are actually a challenge for complex network analysis, as they present some characteristics that hinder topology processing. Concretely, social networks’ volume is exceedingly big, as they have a high number of nodes and links. One of the most popular and influential OSNs is Twitter. In this paper, we present a model to describe the growth of scale-free networks. This model is applied to Twitter after checking that it can be considered a “scale-free” complex network fulfilling the small world property. Checking this property involves the calculation of the shortest path between any two nodes of the network. Given the difficulty of this computation for large networks, a new heuristic method is also proposed to find the upper bounds of the path lengths instead of computing the exact length.Entropy2015-08-17178Article10.3390/e17085848584858671099-43002015-08-17doi: 10.3390/e17085848Sofía AparicioJavier Villazón-TerrazasGonzalo Álvarez<![CDATA[Entropy, Vol. 17, Pages 5829-5847: Parametric Analysis of a Two-Shaft Aeroderivate Gas Turbine of 11.86 MW]]>
http://www.mdpi.com/1099-4300/17/8/5829
The aeroderivate gas turbines are widely used for power generation in the oil and gas industry. In offshore marine platforms, the aeroderivative gas turbines provide the energy required to drive mechanically compressors, pumps and electric generators. Therefore, the study of the performance of aeroderivate gas turbines based on a parametric analysis is relevant to carry out a diagnostic of the engine, which can lead to operational as well as predictive and/or corrective maintenance actions. This work presents a methodology based on the exergetic analysis to estimate the irrevesibilities and exergetic efficiencies of the main components of a two-shaft aeroderivate gas turbine. The studied engine is the Solar Turbine Mars 100, which is rated to provide 11.86 MW. In this engine, the air is compressed in an axial compressor achieving a pressure ratio of 17.7 relative to ambient conditions and a high pressure turbine inlet temperature of 1220 °C. Even if the thermal efficiency associated to the pressure ratio of 17.7 is 1% lower than the maximum thermal efficiency, the irreversibilities related to this pressure ratio decrease approximately 1 GW with respect to irreversibilities of the optimal pressure ratio for the thermal efficiency. In addition, this paper contributes to develop a mathematical model to estimate the high turbine inlet temperature as well as the pressure ratio of the low and high pressure turbines.Entropy2015-08-14178Article10.3390/e17085829582958471099-43002015-08-14doi: 10.3390/e17085829R. Lugo-LeyteM. Salazar-PereyraH. MéndezI. Aguilar-AdayaJ. Ambriz-GarcíaJ. Vargas<![CDATA[Entropy, Vol. 17, Pages 5811-5828: Combined Power Quality Disturbances Recognition Using Wavelet Packet Entropies and S-Transform]]>
http://www.mdpi.com/1099-4300/17/8/5811
Aiming at the combined power quality +disturbance recognition, an automated recognition method based on wavelet packet entropy (WPE) and modified incomplete S-transform (MIST) is proposed in this paper. By combining wavelet packet Tsallis singular entropy, energy entropy and MIST, a 13-dimension vector of different power quality (PQ) disturbances including single disturbances and combined disturbances is extracted. Then, a ruled decision tree is designed to recognize the combined disturbances. The proposed method is tested and evaluated using a large number of simulated PQ disturbances and some real-life signals, which include voltage sag, swell, interruption, oscillation transient, impulsive transient, harmonics, voltage fluctuation and their combinations. In addition, the comparison of the proposed recognition approach with some existing techniques is made. The experimental results show that the proposed method can effectively recognize the single and combined PQ disturbances.Entropy2015-08-12178Article10.3390/e17085811581158281099-43002015-08-12doi: 10.3390/e17085811Zhigang LiuYan CuiWenhui Li<![CDATA[Entropy, Vol. 17, Pages 5799-5810: Entropy Bounds and Field Equations]]>
http://www.mdpi.com/1099-4300/17/8/5799
For general metric theories of gravity, we compare the approach that describes/derives the field equations of gravity as a thermodynamic identity with the one which looks at them from entropy bounds. The comparison is made through the consideration of the matter entropy flux across (Rindler) horizons, studied by making use of the notion of a limiting thermodynamic scale l* of matter, previously introduced in the context of entropy bounds. In doing this: (i) a bound for the entropy of any lump of matter with a given energy-momentum tensor Tab is considered, in terms of a quantity, which is independent of the theory of gravity that we use; this quantity is the variation of the Clausius entropy of a suitable horizon when the element of matter crosses it; (ii) by making use of the equations of motion of the theory, the same quantity is then expressed as the variation of Wald’s entropy of that horizon (and this leads to a generalized form of the generalized covariant entropy bound, applicable to general diffeomorphism-invariant theories of gravity); and (iii) a notion of l* for horizons, as well as an expression for it, is given.Entropy2015-08-12178Article10.3390/e17085799579958101099-43002015-08-12doi: 10.3390/e17085799Alessandro Pesci<![CDATA[Entropy, Vol. 17, Pages 5784-5798: Computing and Learning Year-Round Daily Patterns of Hourly Wind Speed and Direction and Their Global Associations with Meteorological Factors]]>
http://www.mdpi.com/1099-4300/17/8/5784
Daily wind patterns and their relational associations with other metocean (oceanographic and meteorological) variables were algorithmically computed and extracted from a year-long wind and weather dataset, which was collected hourly from an ocean buoy located in the Penghu archipelago of Taiwan. The computational algorithm is called data cloud geometry (DCG). This DCG algorithm is a clustering-based nonparametric learning approach that was constructed and developed implicitly based on various entropy concepts. Regarding the bivariate aspect of wind speed and wind direction, the resulting multiscale clustering hierarchy revealed well-known wind characteristics of year-round pattern cycles pertaining to the particular geographic location of the buoy. A wind pattern due to a set of extreme weather days was also identified. Moreover, in terms of the relational aspect of wind and other weather variables, causal patterns were revealed through applying the DCG algorithm alternatively on the row and column axes of a data matrix by iteratively adapting distance measures to computed DCG tree structures. This adaptation technically constructed and integrated a multiscale, two-sample testing into the distance measure. These computed wind patterns and pattern-based causal relationships are useful for both general sailing and competition planning.Entropy2015-08-11178Article10.3390/e17085784578457981099-43002015-08-11doi: 10.3390/e17085784Hsing-Ti WuHsieh FushingLaurence Chuang<![CDATA[Entropy, Vol. 17, Pages 5771-5783: Active Control of a Chaotic Fractional Order Economic System]]>
http://www.mdpi.com/1099-4300/17/8/5771
In this paper, a fractional order economic system is studied. An active control technique is applied to control chaos in this system. The stabilization of equilibria is obtained by both theoretical analysis and the simulation result. The numerical simulations, via the improved Adams–Bashforth algorithm, show the effectiveness of the proposed controller.Entropy2015-08-11178Article10.3390/e17085771577157831099-43002015-08-11doi: 10.3390/e17085771Haci BaskonusToufik MekkaouiZakia HammouchHasan Bulut<![CDATA[Entropy, Vol. 17, Pages 5752-5770: Consistency of Learning Bayesian Network Structures with Continuous Variables: An Information Theoretic Approach]]>
http://www.mdpi.com/1099-4300/17/8/5752
We consider the problem of learning a Bayesian network structure given n examples and the prior probability based on maximizing the posterior probability. We propose an algorithm that runs in O(n log n) time and that addresses continuous variables and discrete variables without assuming any class of distribution. We prove that the decision is strongly consistent, i.e., correct with probability one as n ! 1. To date, consistency has only been obtained for discrete variables for this class of problem, and many authors have attempted to prove consistency when continuous variables are present. Furthermore, we prove that the “log n” term that appears in the penalty term of the description length can be replaced by 2(1+ε) log log n to obtain strong consistency, where ε &gt; 0 is arbitrary, which implies that the Hannan–Quinn proposition holds.Entropy2015-08-10178Article10.3390/e17085752575257701099-43002015-08-10doi: 10.3390/e17085752Joe Suzuki<![CDATA[Entropy, Vol. 17, Pages 5729-5751: Deformed Algebras and Generalizations of Independence on Deformed Exponential Families]]>
http://www.mdpi.com/1099-4300/17/8/5729
A deformed exponential family is a generalization of exponential families. Since the useful classes of power law tailed distributions are described by the deformed exponential families, they are important objects in the theory of complex systems. Though the deformed exponential families are defined by deformed exponential functions, these functions do not satisfy the law of exponents in general. The deformed algebras have been introduced based on the deformed exponential functions. In this paper, after summarizing such deformed algebraic structures, it is clarified how deformed algebras work on deformed exponential families. In fact, deformed algebras cause generalization of expectations. The three kinds of expectations for random variables are introduced in this paper, and it is discussed why these generalized expectations are natural from the viewpoint of information geometry. In addition, deformed algebras cause generalization of independences. Whereas it is difficult to check the well-definedness of deformed independence in general, the κ-independence is always well-defined on κ-exponential families. This is one of advantages of κ-exponential families in complex systems. Consequently, we can well generalize the maximum likelihood method for the κ-exponential family from the viewpoint of information geometry.Entropy2015-08-10178Article10.3390/e17085729572957511099-43002015-08-10doi: 10.3390/e17085729Hiroshi MatsuzoeTatsuaki Wada<![CDATA[Entropy, Vol. 17, Pages 5711-5728: Fruit Classification by Wavelet-Entropy and Feedforward Neural Network Trained by Fitness-Scaled Chaotic ABC and Biogeography-Based Optimization]]>
http://www.mdpi.com/1099-4300/17/8/5711
Fruit classification is quite difficult because of the various categories and similar shapes and features of fruit. In this work, we proposed two novel machine-learning based classification methods. The developed system consists of wavelet entropy (WE), principal component analysis (PCA), feedforward neural network (FNN) trained by fitness-scaled chaotic artificial bee colony (FSCABC) and biogeography-based optimization (BBO), respectively. The K-fold stratified cross validation (SCV) was utilized for statistical analysis. The classification performance for 1653 fruit images from 18 categories showed that the proposed “WE + PCA + FSCABC-FNN” and “WE + PCA + BBO-FNN” methods achieve the same accuracy of 89.5%, higher than state-of-the-art approaches: “(CH + MP + US) + PCA + GA-FNN ” of 84.8%, “(CH + MP + US) + PCA + PSO-FNN” of 87.9%, “(CH + MP + US) + PCA + ABC-FNN” of 85.4%, “(CH + MP + US) + PCA + kSVM” of 88.2%, and “(CH + MP + US) + PCA + FSCABC-FNN” of 89.1%. Besides, our methods used only 12 features, less than the number of features used by other methods. Therefore, the proposed methods are effective for fruit classification.Entropy2015-08-07178Article10.3390/e17085711571157281099-43002015-08-07doi: 10.3390/e17085711Shuihua WangYudong ZhangGenlin JiJiquan YangJianguo WuLing Wei<![CDATA[Entropy, Vol. 17, Pages 5695-5710: New Region Planning in France? Better Order or More Disorder?]]>
http://www.mdpi.com/1099-4300/17/8/5695
This paper grounds the critique of the reduction of regions in a country , not only in its geographical and social context but also in its entropic space. The various recent plans leading to the reduction of the number of regions in metropolitan France are discussed, based on the mere distribution in the number of municipalities in the plans and analyzed according to various distribution laws. Each case, except the present distribution with 22 regions, on the mainland, does not seem to fit presently used theoretical models. In addition, the number of inhabitants is examined in each plan. The same conclusion holds. Therefore, a theoretical argument based on entropy considerations is proposed, thereby pointing to whether more order or less disorder is the key question—discounting political considerations.Entropy2015-08-06178Article10.3390/e17085695569557101099-43002015-08-06doi: 10.3390/e17085695Marcel Ausloos<![CDATA[Entropy, Vol. 17, Pages 5673-5694: Binary Classification with a Pseudo Exponential Model and Its Application for Multi-Task Learning]]>
http://www.mdpi.com/1099-4300/17/8/5673
In this paper, we investigate the basic properties of binary classification with a pseudo model based on the Itakura–Saito distance and reveal that the Itakura–Saito distance is a unique appropriate measure for estimation with the pseudo model in the framework of general Bregman divergence. Furthermore, we propose a novelmulti-task learning algorithm based on the pseudo model in the framework of the ensemble learning method. We focus on a specific setting of the multi-task learning for binary classification problems. The set of features is assumed to be common among all tasks, which are our targets of performance improvement. We consider a situation where the shared structures among the dataset are represented by divergence between underlying distributions associated with multiple tasks. We discuss statistical properties of the proposed method and investigate the validity of the proposed method with numerical experiments.Entropy2015-08-06178Article10.3390/e17085673567356941099-43002015-08-06doi: 10.3390/e17085673Takashi TakenouchiOsamu KomoriShinto Eguchi<![CDATA[Entropy, Vol. 17, Pages 5660-5672: Gaussian Network’s Dynamics Reflected into Geometric Entropy]]>
http://www.mdpi.com/1099-4300/17/8/5660
We consider a geometric entropy as a measure of complexity for Gaussian networks, namely networks having Gaussian random variables sitting on vertices and their correlations as weighted links. We then show how the network dynamics described by the well-known Ornstein–Uhlenbeck process reflects into such a measure. We unveil a crossing of the entropy time behaviors between switching on and off links. Moreover, depending on the number of links switched on or off, the entropy time behavior can be non-monotonic.Entropy2015-08-06178Article10.3390/e17085660566056721099-43002015-08-06doi: 10.3390/e17085660Domenico FeliceStefano Mancini<![CDATA[Entropy, Vol. 17, Pages 5635-5659: Unconditionally Secure Quantum Signatures]]>
http://www.mdpi.com/1099-4300/17/8/5635
Signature schemes, proposed in 1976 by Diffie and Hellman, have become ubiquitous across modern communications. They allow for the exchange of messages from one sender to multiple recipients, with the guarantees that messages cannot be forged or tampered with and that messages also can be forwarded from one recipient to another without compromising their validity. Signatures are different from, but no less important than encryption, which ensures the privacy of a message. Commonly used signature protocols—signatures based on the Rivest–Adleman–Shamir (RSA) algorithm, the digital signature algorithm (DSA), and the elliptic curve digital signature algorithm (ECDSA)—are only computationally secure, similar to public key encryption methods. In fact, since these rely on the difficulty of finding discrete logarithms or factoring large primes, it is known that they will become completely insecure with the emergence of quantum computers. We may therefore see a shift towards signature protocols that will remain secure even in a post-quantum world. Ideally, such schemes would provide unconditional or information-theoretic security. In this paper, we aim to provide an accessible and comprehensive review of existing unconditionally securesecure signature schemes for signing classical messages, with a focus on unconditionally secure quantum signature schemes.Entropy2015-08-04178Review10.3390/e17085635563556591099-43002015-08-04doi: 10.3390/e17085635Ryan AmiriErika Andersson<![CDATA[Entropy, Vol. 17, Pages 5611-5634: Conspiratorial Beliefs Observed through Entropy Principles]]>
http://www.mdpi.com/1099-4300/17/8/5611
We propose a novel approach framed in terms of information theory and entropy to tackle the issue of the propagation of conspiracy theories. We represent the initial report of an event (such as the 9/11 terroristic attack) as a series of strings of information, each string classified by a two-state variable Ei = ±1, i = 1, …, N. If the values of the Ei are set to −1 for all strings, a state of minimum entropy is achieved. Comments on the report, focusing repeatedly on several strings Ek, might alternate their meaning (from −1 to +1). The representation of the event is turned fuzzy with an increased entropy value. Beyond some threshold value of entropy, chosen by simplicity to its maximum value, meaning N/2 variables with Ei = 1, the chance is created that a conspiracy theory might be initiated/propagated. Therefore, the evolution of the associated entropy is a way to measure the degree of penetration of a conspiracy theory. Our general framework relies on online content made voluntarily available by crowds of people, in response to some news or blog articles published by official news agencies. We apply different aggregation levels (comment, person, discussion thread) and discuss the associated patterns of entropy change.Entropy2015-08-04178Article10.3390/e17085611561156341099-43002015-08-04doi: 10.3390/e17085611Nataša GoloSerge Galam<![CDATA[Entropy, Vol. 17, Pages 5593-5610: Entropy Minimization Design Approach of Supersonic Internal Passages]]>
http://www.mdpi.com/1099-4300/17/8/5593
Fluid machinery operating in the supersonic regime unveil avenues towards more compact technology. However, internal supersonic flows are associated with high aerodynamic and thermal penalties, which usually prevent their practical implementation. Indeed, both shock losses and the limited operational range represent particular challenges to aerodynamic designers that should be taken into account at the initial phase of the design process. This paper presents a design methodology for supersonic passages based on direct evaluations of the velocity field using the method of characteristics and computation of entropy generation across shock waves. This meshless function evaluation tool is then coupled to an optimization scheme, based on evolutionary algorithms that minimize the entropy generation across the supersonic passage. Finally, we assessed the results with 3D Reynolds Averaged Navier Stokes calculations.Entropy2015-08-03178Article10.3390/e17085593559356101099-43002015-08-03doi: 10.3390/e17085593Jorge SousaGuillermo Paniagua<![CDATA[Entropy, Vol. 17, Pages 5580-5592: Adaptive Fuzzy Control for Nonlinear Fractional-Order Uncertain Systems with Unknown Uncertainties and External Disturbance]]>
http://www.mdpi.com/1099-4300/17/8/5580
In this paper, the problem of robust control of nonlinear fractional-order systems in the presence of uncertainties and external disturbance is investigated. Fuzzy logic systems are used for estimating the unknown nonlinear functions. Based on the fractional Lyapunov direct method and some proposed Lemmas, an adaptive fuzzy controller is designed. The proposed method can guarantee all the signals in the closed-loop systems remain bounded and the tracking errors converge to an arbitrary small region of the origin. Lastly, an illustrative example is given to demonstrate the effectiveness of the proposed results.Entropy2015-08-03178Article10.3390/e17085580558055921099-43002015-08-03doi: 10.3390/e17085580Ling LiYeguo Sun<![CDATA[Entropy, Vol. 17, Pages 5561-5579: A New Chaotic System with Positive Topological Entropy]]>
http://www.mdpi.com/1099-4300/17/8/5561
This paper introduces a new simple system with a butterfly chaotic attractor. This system has rich and complex dynamics. With some typical parameters, its Lyapunov dimension is greater than other known three dimensional chaotic systems. It exhibits chaotic behavior over a large range of parameters, and the divergence of flow of this system is not a constant. The dynamics of this new system are analyzed via Lyapunov exponent spectrum, bifurcation diagrams, phase portraits and the Poincaré map. The compound structures of this new system are also analyzed. By means of topological horseshoe theory and numerical computation, the Poincaré map defined for the system is proved to be semi-conjugate to 3-shift map, and thus the system has positive topological entropy.Entropy2015-08-03178Article10.3390/e17085561556155791099-43002015-08-03doi: 10.3390/e17085561Zhonglin WangJian MaZengqiang ChenQing Zhang<![CDATA[Entropy, Vol. 17, Pages 5549-5560: Convergence of a Fixed-Point Minimum Error Entropy Algorithm]]>
http://www.mdpi.com/1099-4300/17/8/5549
The minimum error entropy (MEE) criterion is an important learning criterion in information theoretical learning (ITL). However, the MEE solution cannot be obtained in closed form even for a simple linear regression problem, and one has to search it, usually, in an iterative manner. The fixed-point iteration is an efficient way to solve the MEE solution. In this work, we study a fixed-point MEE algorithm for linear regression, and our focus is mainly on the convergence issue. We provide a sufficient condition (although a little loose) that guarantees the convergence of the fixed-point MEE algorithm. An illustrative example is also presented.Entropy2015-08-03178Article10.3390/e17085549554955601099-43002015-08-03doi: 10.3390/e17085549Yu ZhangBadong ChenXi LiuZejian YuanJose Principe<![CDATA[Entropy, Vol. 17, Pages 5522-5548: Life’s a Gas: A Thermodynamic Theory of Biological Evolution]]>
http://www.mdpi.com/1099-4300/17/8/5522
This paper outlines a thermodynamic theory of biological evolution. Beginning with a brief summary of the parallel histories of the modern evolutionary synthesis and thermodynamics, we use four physical laws and processes (the first and second laws of thermodynamics, diffusion and the maximum entropy production principle) to frame the theory. Given that open systems such as ecosystems will move towards maximizing dispersal of energy, we expect biological diversity to increase towards a level, Dmax, representing maximum entropic production (Smax). Based on this theory, we develop a mathematical model to predict diversity over the last 500 million years. This model combines diversification, post-extinction recovery and likelihood of discovery of the fossil record. We compare the output of this model with that of the observed fossil record. The model predicts that life diffuses into available energetic space (ecospace) towards a dynamic equilibrium, driven by increasing entropy within the genetic material. This dynamic equilibrium is punctured by extinction events, which are followed by restoration of Dmax through diffusion into available ecospace. Finally we compare and contrast our thermodynamic theory with the MES in relation to a number of important characteristics of evolution (progress, evolutionary tempo, form versus function, biosphere architecture, competition and fitness).Entropy2015-07-31178Article10.3390/e17085522552255481099-43002015-07-31doi: 10.3390/e17085522Keith Skene<![CDATA[Entropy, Vol. 17, Pages 5503-5521: Optimization of the Changing Phase Fluid in a Carnot Type Engine for the Recovery of a Given Waste Heat Source]]>
http://www.mdpi.com/1099-4300/17/8/5503
A Carnot type engine with a changing phase during the heating and the cooling is modeled with its thermal contact with the heat source. In a first optimization, the optimal high temperature of the cycle is determined to maximize the power output. The temperature and the mass flow rate of the heat source are given. This does not take into account the converter internal fluid and its mass flow rate. It is an exogenous optimization of the converter. In a second optimization, the endogenous optimization, the isothermal heating corresponds only to the vaporization of the selected fluid. The maximization of the power output gives the optimal vaporization temperature of the cycled fluid. Using these two optima allows connecting the temperature of the heat source to the working fluid used. For a given temperature level, mass flow rate and composition of the waste heat to recover, an optimal fluid and its temperature of vaporization are deduced. The optimal conditions size also the internal mass flow rate and the compression ratio (pump size). The optimum corresponds to the maximum of the power output and must be combined with the environmental fluid impact and the technological constraints.Entropy2015-07-31178Article10.3390/e17085503550355211099-43002015-07-31doi: 10.3390/e17085503Mathilde BlaiseMichel FeidtDenis Maillet<![CDATA[Entropy, Vol. 17, Pages 5472-5502: The Intrinsic Cause-Effect Power of Discrete Dynamical Systems—From Elementary Cellular Automata to Adapting Animats]]>
http://www.mdpi.com/1099-4300/17/8/5472
Current approaches to characterize the complexity of dynamical systems usually rely on state-space trajectories. In this article instead we focus on causal structure, treating discrete dynamical systems as directed causal graphs—systems of elements implementing local update functions. This allows us to characterize the system’s intrinsic cause-effect structure by applying the mathematical and conceptual tools developed within the framework of integrated information theory (IIT). In particular, we assess the number of irreducible mechanisms (concepts) and the total amount of integrated conceptual information Φ specified by a system. We analyze: (i) elementary cellular automata (ECA); and (ii) small, adaptive logic-gate networks (“animats”), similar to ECA in structure but evolving by interacting with an environment. We show that, in general, an integrated cause-effect structure with many concepts and high Φ is likely to have high dynamical complexity. Importantly, while a dynamical analysis describes what is “happening” in a system from the extrinsic perspective of an observer, the analysis of its cause-effect structure reveals what a system “is” from its own intrinsic perspective, exposing its dynamical and evolutionary potential under many different scenarios.Entropy2015-07-31178Article10.3390/e17085472547255021099-43002015-07-31doi: 10.3390/e17085472Larissa AlbantakisGiulio Tononi<![CDATA[Entropy, Vol. 17, Pages 5450-5471: Probabilistic Forecasts: Scoring Rules and Their Decomposition and Diagrammatic Representation via Bregman Divergences]]>
http://www.mdpi.com/1099-4300/17/8/5450
A scoring rule is a device for evaluation of forecasts that are given in terms of the probability of an event. In this article we will restrict our attention to binary forecasts. We may think of a scoring rule as a penalty attached to a forecast after the event has been observed. Thus a relatively small penalty will accrue if a high probability forecast that an event will occur is followed by occurrence of the event. On the other hand, a relatively large penalty will accrue if this forecast is followed by non-occurrence of the event. Meteorologists have been foremost in developing scoring rules for the evaluation of probabilistic forecasts. Here we use a published meteorological data set to illustrate diagrammatically the Brier score and the divergence score, and their statistical decompositions, as examples of Bregman divergences. In writing this article, we have in mind environmental scientists and modellers for whom meteorological factors are important drivers of biological, physical and chemical processes of interest. In this context, we briefly draw attention to the potential for probabilistic forecasting of the within-season component of nitrous oxide emissions from agricultural soils.Entropy2015-07-31178Article10.3390/e17085450545054711099-43002015-07-31doi: 10.3390/e17085450Gareth HughesCairistiona Topp<![CDATA[Entropy, Vol. 17, Pages 5437-5449: Reaction Kinetic Parameters and Surface Thermodynamic Properties of Cu2O Nanocubes]]>
http://www.mdpi.com/1099-4300/17/8/5437
Cuprous oxide (Cu2O) nanocubes were synthesized by reducing Cu(OH)2 in the presence of sodium citrate at room temperature. The samples were characterized in detail by field-emission scanning electron microscopy, transmission electron microscopy, high-resolution transmission electron microscopy, X-ray powder diffraction, and N2 absorption (BET specific surface area). The equations for acquiring reaction kinetic parameters and surface thermodynamic properties of Cu2O nanocubes were deduced by establishment of the relations between thermodynamic functions of Cu2O nanocubes and these of the bulk Cu2O. Combined with thermochemical cycle, transition state theory, basic theory of chemical thermodynamics, and in situ microcalorimetry, reaction kinetic parameters, specific surface enthalpy, specific surface Gibbs free energy, and specific surface entropy of Cu2O nanocubes were successfully determined. We also introduced a universal route for gaining reaction kinetic parameters and surface thermodynamic properties of nanomaterials.Entropy2015-07-30178Article10.3390/e17085437543754491099-43002015-07-30doi: 10.3390/e17085437Xingxing LiHuanfeng TangXianrui LuShi LinLili ShiZaiyin Huang<![CDATA[Entropy, Vol. 17, Pages 5422-5436: A Robust Planning Algorithm for Groups of Entities in Discrete Spaces]]>
http://www.mdpi.com/1099-4300/17/8/5422
Automated planning is a well-established field of artificial intelligence (AI), with applications in route finding, robotics and operational research, among others. The task of developing a plan is often solved by finding a path in a graph representing the search domain; a robust plan consists of numerous paths that can be chosen if the execution of the best (optimal) one fails. While robust planning for a single entity is rather simple, development of a robust plan for multiple entities in a common environment can lead to combinatorial explosion. This paper proposes a novel hybrid approach, joining heuristic search and the wavefront algorithm to provide a plan featuring robustness in areas where it is needed, while maintaining a low level of computational complexity.Entropy2015-07-30178Article10.3390/e17085422542254361099-43002015-07-30doi: 10.3390/e17085422Igor WojnickiSebastian ErnstWojciech Turek<![CDATA[Entropy, Vol. 17, Pages 5402-5421: Fractional State Space Analysis of Economic Systems]]>
http://www.mdpi.com/1099-4300/17/8/5402
This paper examines modern economic growth according to the multidimensional scaling (MDS) method and state space portrait (SSP) analysis. Electing GDP per capita as the main indicator for economic growth and prosperity, the long-run perspective from 1870 to 2010 identifies the main similarities among 34 world partners’ modern economic growth and exemplifies the historical waving mechanics of the largest world economy, the USA. MDS reveals two main clusters among the European countries and their old offshore territories, and SSP identifies the Great Depression as a mild challenge to the American global performance, when compared to the Second World War and the 2008 crisis.Entropy2015-07-29178Article10.3390/e17085402540254211099-43002015-07-29doi: 10.3390/e17085402J. MachadoMaria MataAntónio Lopes<![CDATA[Entropy, Vol. 17, Pages 5382-5401: Multifractal Dimensional Dependence Assessment Based on Tsallis Mutual Information]]>
http://www.mdpi.com/1099-4300/17/8/5382
Entropy-based tools are commonly used to describe the dynamics of complex systems. In the last few decades, non-extensive statistics, based on Tsallis entropy, and multifractal techniques have shown to be useful to characterize long-range interaction and scaling behavior. In this paper, an approach based on generalized Tsallis dimensions is used for the formulation of mutual-information-related dependence coefficients in the multifractal domain. Different versions according to the normalizing factor, as well as to the inclusion of the non-extensivity correction term are considered and discussed. An application to the assessment of dimensional interaction in the structural dynamics of a seismic real series is carried out to illustrate the usefulness and comparative performance of the measures introduced.Entropy2015-07-29178Article10.3390/e17085382538254011099-43002015-07-29doi: 10.3390/e17085382José AnguloFrancisco Esquivel<![CDATA[Entropy, Vol. 17, Pages 5353-5381: Approximate Methods for Maximum Likelihood Estimation of Multivariate Nonlinear Mixed-Effects Models]]>
http://www.mdpi.com/1099-4300/17/8/5353
Multivariate nonlinear mixed-effects models (MNLMM) have received increasing use due to their flexibility for analyzing multi-outcome longitudinal data following possibly nonlinear profiles. This paper presents and compares five different iterative algorithms for maximum likelihood estimation of the MNLMM. These algorithmic schemes include the penalized nonlinear least squares coupled to the multivariate linear mixed-effects (PNLS-MLME) procedure, Laplacian approximation, the pseudo-data expectation conditional maximization (ECM) algorithm, the Monte Carlo EM algorithm and the importance sampling EM algorithm. When fitting the MNLMM, it is rather difficult to exactly evaluate the observed log-likelihood function in a closed-form expression, because it involves complicated multiple integrals. To address this issue, the corresponding approximations of the observed log-likelihood function under the five algorithms are presented. An expected information matrix of parameters is also provided to calculate the standard errors of model parameters. A comparison of computational performances is investigated through simulation and a real data example from an AIDS clinical study.Entropy2015-07-29178Article10.3390/e17085353535353811099-43002015-07-29doi: 10.3390/e17085353Wan-Lun Wang<![CDATA[Entropy, Vol. 17, Pages 5333-5352: Statistical Evidence Measured on a Properly Calibrated Scale across Nested and Non-nested Hypothesis Comparisons]]>
http://www.mdpi.com/1099-4300/17/8/5333
Statistical modeling is often used to measure the strength of evidence for or against hypotheses about given data. We have previously proposed an information-dynamic framework in support of a properly calibrated measurement scale for statistical evidence, borrowing some mathematics from thermodynamics, and showing how an evidential analogue of the ideal gas equation of state could be used to measure evidence for a one-sided binomial hypothesis comparison (“coin is fair” vs. “coin is biased towards heads”). Here we take three important steps forward in generalizing the framework beyond this simple example, albeit still in the context of the binomial model. We: (1) extend the scope of application to other forms of hypothesis comparison; (2) show that doing so requires only the original ideal gas equation plus one simple extension, which has the form of the Van der Waals equation; (3) begin to develop the principles required to resolve a key constant, which enables us to calibrate the measurement scale across applications, and which we find to be related to the familiar statistical concept of degrees of freedom. This paper thus moves our information-dynamic theory substantially closer to the goal of producing a practical, properly calibrated measure of statistical evidence for use in general applications.Entropy2015-07-29178Article10.3390/e17085333533353521099-43002015-07-29doi: 10.3390/e17085333Veronica VielandSang-Cheol Seok<![CDATA[Entropy, Vol. 17, Pages 5304-5332: Entropy Generation through Deterministic Spiral Structures in a Corner Boundary-Layer Flow]]>
http://www.mdpi.com/1099-4300/17/8/5304
It is shown that nonlinear interactions between boundary layers on adjacent corner surfaces produce deterministic stream wise spiral structures. The synchronization properties of nonlinear spectral velocity equations of Lorenz form yield clearly defined deterministic spiral structures at several downstream stations. The computational procedure includes Burg’s method to obtain power spectral densities, yielding the available kinetic energy dissipation rates within the spiral structures. The singular value decomposition method is applied to the nonlinear time series solutions yielding empirical entropies, from which empirical entropic indices are then extracted. The intermittency exponents obtained from the entropic indices allow the computation of the entropy generation through the spiral structures to the final dissipation of the fluctuating kinetic energy into background thermal energy, resulting in an increase in the entropy. The entropy generation rates through the spiral structures are compared with the entropy generation rates within an empirical turbulent boundary layer at several stream wise stations.Entropy2015-07-27178Article10.3390/e17085304530453321099-43002015-07-27doi: 10.3390/e17085304LaVar Isaacson<![CDATA[Entropy, Vol. 17, Pages 5288-5303: Escalation with Overdose Control is More Efficient and Safer than Accelerated Titration for Dose Finding]]>
http://www.mdpi.com/1099-4300/17/8/5288
The standard 3 + 3 or “modified Fibonacci” up-and-down (MF-UD) method of dose escalation is by far the most used design in dose-finding cancer trials. However, MF-UD has always shown inferior performance when compared with its competitors regarding number of patients treated at optimal doses. A consequence of using less effective designs is that more patients are treated with doses outside the therapeutic window. In June 2012, the U S Food and Drug Administration (FDA) rejected the proposal to use Escalation with Overdose Control (EWOC), an established dose-finding method which has been extensively used in FDA-approved first in human trials and imposed a variation of the MF-UD, known as accelerated titration (AT) design. This event motivated us to perform an extensive simulation study comparing the operating characteristics of AT and EWOC. We show that the AT design has poor operating characteristics relative to three versions of EWOC under several practical scenarios. From the clinical investigator’s perspective, lower bias and mean square error make EWOC designs preferable than AT designs without compromising safety. From a patient’s perspective, uniformly higher proportion of patients receiving doses within an optimal range of the true MTD makes EWOC designs preferable than AT designs.Entropy2015-07-27178Article10.3390/e17085288528853031099-43002015-07-27doi: 10.3390/e17085288André RogatkoGalen Cook-WiensMourad TighiouartSteven Piantadosi<![CDATA[Entropy, Vol. 17, Pages 5274-5287: Prebiotic Competition between Information Variants, With Low Error Catastrophe Risks]]>
http://www.mdpi.com/1099-4300/17/8/5274
During competition for resources in primitive networks increased fitness of an information variant does not necessarily equate with successful elimination of its competitors. If variability is added fast to a system, speedy replacement of pre-existing and less-efficient forms of order is required as novel information variants arrive. Otherwise, the information capacity of the system fills up with information variants (an effect referred as “error catastrophe”). As the cost for managing the system’s exceeding complexity increases, the correlation between performance capabilities of information variants and their competitive success decreases, and evolution of such systems toward increased efficiency slows down. This impasse impedes the understanding of evolution in prebiotic networks. We used the simulation platform Biotic Abstract Dual Automata (BiADA) to analyze how information variants compete in a resource-limited space. We analyzed the effect of energy-related features (differences in autocatalytic efficiency, energy cost of order, energy availability, transformation rates and stability of order) on this competition. We discuss circumstances and controllers allowing primitive networks acquire novel information with minimal “error catastrophe” risks. We present a primitive mechanism for maximization of energy flux in dynamic networks. This work helps evaluate controllers of evolution in prebiotic networks and other systems where information variants compete.Entropy2015-07-27178Article10.3390/e17085274527452871099-43002015-07-27doi: 10.3390/e17085274Radu PopaVily Cimpoiasu<![CDATA[Entropy, Vol. 17, Pages 5257-5273: A Pilot Directional Protection for HVDC Transmission Line Based on Relative Entropy of Wavelet Energy]]>
http://www.mdpi.com/1099-4300/17/8/5257
On the basis of analyzing high-voltage direct current (HVDC) transmission system and its fault superimposed circuit, the direction of the fault components of the voltage and the current measured at one end of transmission line is certified to be different for internal faults and external faults. As an estimate of the differences between two signals, relative entropy is an effective parameter for recognizing transient signals in HVDC transmission lines. In this paper, the relative entropy of wavelet energy is applied to distinguish internal fault from external fault. For internal faults, the directions of fault components of voltage and current are opposite at the two ends of the transmission line, indicating a huge difference of wavelet energy relative entropy; for external faults, the directions are identical, indicating a small difference. The simulation results based on PSCAD/EMTDC show that the proposed pilot protection system acts accurately for faults under different conditions, and its performance is not affected by fault type, fault location, fault resistance and noise.Entropy2015-07-27178Article10.3390/e17085257525752731099-43002015-07-27doi: 10.3390/e17085257Sheng LinShan GaoZhengyou HeYujia Deng<![CDATA[Entropy, Vol. 17, Pages 5241-5256: Neural Network Reorganization Analysis During an Auditory Oddball Task in Schizophrenia Using Wavelet Entropy]]>
http://www.mdpi.com/1099-4300/17/8/5241
The aim of the present study was to characterize the neural network reorganization during a cognitive task in schizophrenia (SCH) by means of wavelet entropy (WE). Previous studies suggest that the cognitive impairment in patients with SCH could be related to the disrupted integrative functions of neural circuits. Nevertheless, further characterization of this effect is needed, especially in the time-frequency domain. This characterization is sensitive to fast neuronal dynamics and their synchronization that may be an important component of distributed neuronal interactions; especially in light of the disconnection hypothesis for SCH and its electrophysiological correlates. In this work, the irregularity dynamics elicited by an auditory oddball paradigm were analyzed through synchronized-averaging (SA) and single-trial (ST) analyses. They provide complementary information on the spatial patterns involved in the neural network reorganization. Our results from 20 healthy controls and 20 SCH patients showed a WE decrease from baseline to response both in controls and SCH subjects. These changes were significantly more pronounced for healthy controls after ST analysis, mainly in central and frontopolar areas. On the other hand, SA analysis showed more widespread spatial differences than ST results. These findings suggest that the activation response is weakly phase-locked to stimulus onset in SCH and related to the default mode and salience networks. Furthermore, the less pronounced changes in WE from baseline to response for SCH patients suggest an impaired ability to reorganize neural dynamics during an oddball task.Entropy2015-07-27178Article10.3390/e17085241524152561099-43002015-07-27doi: 10.3390/e17085241Javier Gomez-PilarJesús PozaAlejandro BachillerCarlos GómezVicente MolinaRoberto Hornero<![CDATA[Entropy, Vol. 17, Pages 5218-5240: An Integrated Index for the Identification of Focal Electroencephalogram Signals Using Discrete Wavelet Transform and Entropy Measures]]>
http://www.mdpi.com/1099-4300/17/8/5218
The dynamics of brain area influenced by focal epilepsy can be studied using focal and non-focal electroencephalogram (EEG) signals. This paper presents a new method to detect focal and non-focal EEG signals based on an integrated index, termed the focal and non-focal index (FNFI), developed using discrete wavelet transform (DWT) and entropy features. The DWT decomposes the EEG signals up to six levels, and various entropy measures are computed from approximate and detail coefficients of sub-band signals. The computed entropy measures are average wavelet, permutation, fuzzy and phase entropies. The proposed FNFI developed using permutation, fuzzy and Shannon wavelet entropies is able to clearly discriminate focal and non-focal EEG signals using a single number. Furthermore, these entropy measures are ranked using different techniques, namely the Bhattacharyya space algorithm, Student’s t-test, the Wilcoxon test, the receiver operating characteristic (ROC) and entropy. These ranked features are fed to various classifiers, namely k-nearest neighbour (KNN), probabilistic neural network (PNN), fuzzy classifier and least squares support vector machine (LS-SVM), for automated classification of focal and non-focal EEG signals using the minimum number of features. The identification of the focal EEG signals can be helpful to locate the epileptogenic focus.Entropy2015-07-27178Article10.3390/e17085218521852401099-43002015-07-27doi: 10.3390/e17085218Rajeev SharmaRam PachoriU. Acharya<![CDATA[Entropy, Vol. 17, Pages 5199-5217: Generalized Combination Complex Synchronization for Fractional-Order Chaotic Complex Systems]]>
http://www.mdpi.com/1099-4300/17/8/5199
Based on two fractional-order chaotic complex drive systems and one fractional-order chaotic complex response system with different dimensions, we propose generalized combination complex synchronization. In this new synchronization scheme, there are two complex scaling matrices that are non-square matrices. On the basis of the stability theory of fractional-order linear systems, we design a general controller via active control. Additionally, by virtue of two complex scaling matrices, generalized combination complex synchronization between fractional-order chaotic complex systems and real systems is investigated. Finally, three typical examples are given to demonstrate the effectiveness and feasibility of the schemes.Entropy2015-07-24178Article10.3390/e17085199519952171099-43002015-07-24doi: 10.3390/e17085199Cuimei JiangShutang LiuDa Wang<![CDATA[Entropy, Vol. 17, Pages 5171-5198: Information-Theoretic Characterization and Undersampling Ratio Determination for Compressive Radar Imaging in a Simulated Environment]]>
http://www.mdpi.com/1099-4300/17/8/5171
Assuming sparsity or compressibility of the underlying signals, compressed sensing or compressive sampling (CS) exploits the informational efficiency of under-sampled measurements for increased efficiency yet acceptable accuracy in information gathering, transmission and processing, though it often incurs extra computational cost in signal reconstruction. Shannon information quantities and theorems, such as source rate-distortion, trans-information and rate distortion theorem concerning lossy data compression, provide a coherent framework, which is complementary to classic CS theory, for analyzing informational quantities and for determining the necessary number of measurements in CS. While there exists some information-theoretic research in the past on CS in general and compressive radar imaging in particular, systematic research is needed to handle issues related to scene description in cluttered environments and trans-information quantification in complex sparsity-clutter-sampling-noise settings. The novelty of this paper lies in furnishing a general strategy for information-theoretic analysis of scene compressibility, trans-information of radar echo data about the scene and the targets of interest, respectively, and limits to undersampling ratios necessary for scene reconstruction subject to distortion given sparsity-clutter-noise constraints. A computational experiment was performed to demonstrate informational analysis regarding the scene-sampling-reconstruction process and to generate phase transition diagrams showing relations between undersampling ratios and sparsity-clutter-noise-distortion constraints. The strategy proposed in this paper is valuable for information-theoretic analysis and undersampling theorem developments in compressive radar imaging and other computational imaging applications.Entropy2015-07-24178Article10.3390/e17085171517151981099-43002015-07-24doi: 10.3390/e17085171Jingxiong ZhangKe YangFengzhu LiuYing Zhang<![CDATA[Entropy, Vol. 17, Pages 5157-5170: Non-Fourier Heat Transfer with Phonons and Electrons in a Circular Thin Layer Surrounding a Hot Nanodevice]]>
http://www.mdpi.com/1099-4300/17/8/5157
A nonlocal model for heat transfer with phonons and electrons is applied to infer the steady-state radial temperature profile in a circular layer surrounding an inner hot component. Such a profile, following by the numerical solution of the heat equation, predicts that the temperature behaves in an anomalous way, since for radial distances from the heat source smaller than the mean-free path of phonons and electrons, it increases for increasing distances. The compatibility of this temperature behavior with the second law of thermodynamics is investigated by calculating numerically the local entropy production as a function of the radial distance. It turns out that such a production is positive and strictly decreasing with the radial distance.Entropy2015-07-24178Article10.3390/e17085157515751701099-43002015-07-24doi: 10.3390/e17085157Vito CimmelliIsabella CarlomagnoAntonio Sellitto<![CDATA[Entropy, Vol. 17, Pages 5145-5156: Scale-Invariant Rotating Black Holes in Quadratic Gravity]]>
http://www.mdpi.com/1099-4300/17/8/5145
Black hole solutions in pure quadratic theories of gravity are interesting since they allow the formulation of a set of scale-invariant thermodynamics laws. Recently, we have proven that static scale-invariant black holes have a well-defined entropy, which characterizes equivalent classes of solutions. In this paper, we generalize these results and explore the thermodynamics of rotating black holes in pure quadratic gravity.Entropy2015-07-23178Article10.3390/e17085145514551561099-43002015-07-23doi: 10.3390/e17085145Guido CognolaMassimiliano RinaldiLuciano Vanzo<![CDATA[Entropy, Vol. 17, Pages 5133-5144: Setting Diverging Colors for a Large-Scale Hypsometric Lunar Map Based on Entropy]]>
http://www.mdpi.com/1099-4300/17/7/5133
A hypsometric map is a type of map used to represent topographic characteristics by filling different map areas with diverging colors. The setting of appropriate diverging colors is essential for the map to reveal topographic details. When lunar real environmental exploration programs are performed, large-scale hypsometric maps with a high resolution and greater topographic detail are helpful. Compared to the situation on Earth, fewer lunar exploration objects are available, and the topographic waviness is smaller at a large scale, indicating that presenting the topographic details using traditional hypsometric map-making methods may be difficult. To solve this problem, we employed the Chang’E2 (CE2) topographic and imagery data with a resolution of 7 m and developed a new hypsometric map-making method by setting the diverging colors based on information entropy. The resulting map showed that this method is suitable for presenting the topographic details and might be useful for developing a better understanding of the environment of the lunar surface.Entropy2015-07-22177Article10.3390/e17075133513351441099-43002015-07-22doi: 10.3390/e17075133Xingguo ZengLingli MuJianjun LiuYiman Yang<![CDATA[Entropy, Vol. 17, Pages 5117-5132: An Entropy-Based Approach to Path Analysis of Structural Generalized Linear Models: A Basic Idea]]>
http://www.mdpi.com/1099-4300/17/7/5117
A path analysis method for causal systems based on generalized linear models is proposed by using entropy. A practical example is introduced, and a brief explanation of the entropy coefficient of determination is given. Direct and indirect effects of explanatory variables are discussed as log odds ratios, i.e., relative information, and a method for summarizing the effects is proposed. The example dataset is re-analyzed by using the method.Entropy2015-07-22177Article10.3390/e17075117511751321099-43002015-07-22doi: 10.3390/e17075117Nobuoki EshimaMinoru TabataClaudio BorroniYutaka Kano<![CDATA[Entropy, Vol. 17, Pages 5101-5116: Analytic Exact Upper Bound for the Lyapunov Dimension of the Shimizu–Morioka System]]>
http://www.mdpi.com/1099-4300/17/7/5101
In applied investigations, the invariance of the Lyapunov dimension under a diffeomorphism is often used. However, in the case of irregular linearization, this fact was not strictly considered in the classical works. In the present work, the invariance of the Lyapunov dimension under diffeomorphism is demonstrated in the general case. This fact is used to obtain the analytic exact upper bound of the Lyapunov dimension of an attractor of the Shimizu–Morioka system.Entropy2015-07-22177Article10.3390/e17075101510151161099-43002015-07-22doi: 10.3390/e17075101Gennady LeonovTatyana AlexeevaNikolay Kuznetsov<![CDATA[Entropy, Vol. 17, Pages 5085-5100: Averaged Extended Tree Augmented Naive Classifier]]>
http://www.mdpi.com/1099-4300/17/7/5085
This work presents a new general purpose classifier named Averaged Extended Tree Augmented Naive Bayes (AETAN), which is based on combining the advantageous characteristics of Extended Tree Augmented Naive Bayes (ETAN) and Averaged One-Dependence Estimator (AODE) classifiers. We describe the main properties of the approach and algorithms for learning it, along with an analysis of its computational time complexity. Empirical results with numerous data sets indicate that the new approach is superior to ETAN and AODE in terms of both zero-one classification accuracy and log loss. It also compares favourably against weighted AODE and hidden Naive Bayes. The learning phase of the new approach is slower than that of its competitors, while the time complexity for the testing phase is similar. Such characteristics suggest that the new classifier is ideal in scenarios where online learning is not required.Entropy2015-07-21177Article10.3390/e17075085508551001099-43002015-07-21doi: 10.3390/e17075085Aaron MeehanCassio de Campos<![CDATA[Entropy, Vol. 17, Pages 5063-5084: Minimal Rényi–Ingarden–Urbanik Entropy of Multipartite Quantum States]]>
http://www.mdpi.com/1099-4300/17/7/5063
We study the entanglement of a pure state of a composite quantum system consisting of several subsystems with d levels each. It can be described by the Rényi–Ingarden–Urbanik entropy Sq of a decomposition of the state in a product basis, minimized over all local unitary transformations. In the case q = 0, this quantity becomes a function of the rank of the tensor representing the state, while in the limit q → ∞, the entropy becomes related to the overlap with the closest separable state and the geometric measure of entanglement. For any bipartite system, the entropy S1 coincides with the standard entanglement entropy. We analyze the distribution of the minimal entropy for random states of three- and four-qubit systems. In the former case, the distribution of the three-tangle is studied and some of its moments are evaluated, while in the latter case, we analyze the distribution of the hyperdeterminant. The behavior of the maximum overlap of a three-qudit system with the closest separable state is also investigated in the asymptotic limit.Entropy2015-07-20177Article10.3390/e17075063506350841099-43002015-07-20doi: 10.3390/e17075063Marco EnríquezZbigniew PuchałaKarol Życzkowski<![CDATA[Entropy, Vol. 17, Pages 5047-5062: Evaluation of the Atmospheric Chemical Entropy Production of Mars]]>
http://www.mdpi.com/1099-4300/17/7/5047
Thermodynamic disequilibrium is a necessary situation in a system in which complex emergent structures are created and maintained. It is known that most of the chemical disequilibrium, a particular type of thermodynamic disequilibrium, in Earth’s atmosphere is a consequence of life. We have developed a thermochemical model for the Martian atmosphere to analyze the disequilibrium by chemical reactions calculating the entropy production. It follows from the comparison with the Earth atmosphere that the magnitude of the entropy produced by the recombination reaction forming O3 (O + O2 + CO2 ⥦ O3 + CO2) in the atmosphere of the Earth is larger than the entropy produced by the dominant set of chemical reactions considered for Mars, as a consequence of the low density and the poor variety of species of the Martian atmosphere. If disequilibrium is needed to create and maintain self-organizing structures in a system, we conclude that the current Martian atmosphere is unable to support large physico-chemical structures, such as those created on Earth.Entropy2015-07-20177Article10.3390/e17075047504750621099-43002015-07-20doi: 10.3390/e17075047Alfonso Delgado-BonalF. Martín-Torres<![CDATA[Entropy, Vol. 17, Pages 5043-5046: Reply to C. Tsallis’ “Conceptual Inadequacy of the Shore and Johnson Axioms for Wide Classes of Complex Systems”]]>
http://www.mdpi.com/1099-4300/17/7/5043
In a recent PRL (2013, 111, 180604), we invoked the Shore and Johnson axioms which demonstrate that the least-biased way to infer probability distributions fpig from data is to maximize the Boltzmann-Gibbs entropy. We then showed which biases are introduced in models obtained by maximizing nonadditive entropies. A rebuttal of our work appears in entropy (2015, 17, 2853) and argues that the Shore and Johnson axioms are inapplicable to a wide class of complex systems. Here we highlight the errors in this reasoning.Entropy2015-07-17177Reply10.3390/e17075043504350461099-43002015-07-17doi: 10.3390/e17075043Steve PresséKingshuk GhoshJulian LeeKen Dill<![CDATA[Entropy, Vol. 17, Pages 5022-5042: The Critical Point Entanglement and Chaos in the Dicke Model]]>
http://www.mdpi.com/1099-4300/17/7/5022
Ground state properties and level statistics of the Dicke model for a finite number of atoms are investigated based on a progressive diagonalization scheme (PDS). Particle number statistics, the entanglement measure and the Shannon information entropy at the resonance point in cases with a finite number of atoms as functions of the coupling parameter are calculated. It is shown that the entanglement measure defined in terms of the normalized von Neumann entropy of the reduced density matrix of the atoms reaches its maximum value at the critical point of the quantum phase transition where the system is most chaotic. Noticeable change in the Shannon information entropy near or at the critical point of the quantum phase transition is also observed. In addition, the quantum phase transition may be observed not only in the ground state mean photon number and the ground state atomic inversion as shown previously, but also in fluctuations of these two quantities in the ground state, especially in the atomic inversion fluctuation.Entropy2015-07-16177Article10.3390/e17075022502250421099-43002015-07-16doi: 10.3390/e17075022Lina BaoFeng PanJing LuJerry Draayer