Entropy

907 KiB

Open AccessArticle

Approximate Analytical Solutions of Time Fractional Whitham–Broer–Kaup Equations by a Residual Power Series Method

by Linjun Wang and Xumei Chen

Entropy 2015, 17(9), 6519-6533; https://doi.org/10.3390/e17096519 - 23 Sep 2015

Cited by 64 | Viewed by 5581

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Complex and Fractional Dynamics)

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314 KiB

Open AccessArticle

Thermodynamic Metrics and Black Hole Physics

by Jan E. Åman, Ingemar Bengtsson and Narit Pidokrajt

Entropy 2015, 17(9), 6503-6518; https://doi.org/10.3390/e17096503 - 22 Sep 2015

Cited by 12 | Viewed by 6485

Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Geometry in Thermodynamics)

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438 KiB

Open AccessArticle

A Bayesian Predictive Discriminant Analysis with Screened Data

by Hea-Jung Kim

Entropy 2015, 17(9), 6481-6502; https://doi.org/10.3390/e17096481 - 21 Sep 2015

Viewed by 5216

Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Inductive Statistical Methods)

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309 KiB

Open AccessArticle

Subspace Coding for Networks with Different Level Messages

by Feng Cai, Ning Cai and Wangmei Guo

Entropy 2015, 17(9), 6462-6480; https://doi.org/10.3390/e17096462 - 21 Sep 2015

Viewed by 4103

Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Section Information Theory, Probability and Statistics)

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1122 KiB

Open AccessArticle

Rolling Bearing Fault Diagnosis Based on Wavelet Packet Decomposition and Multi-Scale Permutation Entropy

by Li-Ye Zhao, Lei Wang and Ru-Qiang Yan

Entropy 2015, 17(9), 6447-6461; https://doi.org/10.3390/e17096447 - 21 Sep 2015

Cited by 91 | Viewed by 7528

Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Section Complexity)

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523 KiB

Open AccessArticle

Dynamical Systems Induced on Networks Constructed from Time Series

by Lvlin Hou, Michael Small and Songyang Lao

Entropy 2015, 17(9), 6433-6446; https://doi.org/10.3390/e17096433 - 18 Sep 2015

Cited by 8 | Viewed by 4988

Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Recent Advances in Chaos Theory and Complex Networks)

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856 KiB

Open AccessArticle

Energy and Exergy Analyses of a Combined Power Cycle Using the Organic Rankine Cycle and the Cold Energy of Liquefied Natural Gas

by Ho Yong Lee and Kyoung Hoon Kim

Entropy 2015, 17(9), 6412-6432; https://doi.org/10.3390/e17096412 - 18 Sep 2015

Cited by 34 | Viewed by 8935

Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Section Thermodynamics)

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Open AccessArticle

Wavelet Entropy as a Measure of Ventricular Beat Suppression from the Electrocardiogram in Atrial Fibrillation

by Philip Langley

Entropy 2015, 17(9), 6397-6411; https://doi.org/10.3390/e17096397 - 17 Sep 2015

Cited by 14 | Viewed by 6739

Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Wavelet Entropy: Computation and Applications)

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4866 KiB

Open AccessArticle

A New Process Monitoring Method Based on Waveform Signal by Using Recurrence Plot

by Cheng Zhou and Weidong Zhang

Entropy 2015, 17(9), 6379-6396; https://doi.org/10.3390/e17096379 - 16 Sep 2015

Cited by 4 | Viewed by 6360

Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Information Theoretic Learning)

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Open AccessArticle

Entropies from Coarse-graining: Convex Polytopes vs. Ellipsoids

by Nikos Kalogeropoulos

Entropy 2015, 17(9), 6329-6378; https://doi.org/10.3390/e17096329 - 15 Sep 2015

Cited by 8 | Viewed by 5204

Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Geometry in Thermodynamics)

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Open AccessArticle

Viscosity-Induced Crossing of the Phantom Barrier

by Iver Brevik

Entropy 2015, 17(9), 6318-6328; https://doi.org/10.3390/e17096318 - 14 Sep 2015

Cited by 27 | Viewed by 4388

Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Entropy in Quantum Gravity and Quantum Cosmology)

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Open AccessArticle

Metrics and Energy Landscapes in Irreversible Thermodynamics

by Bjarne Andresen

Entropy 2015, 17(9), 6304-6317; https://doi.org/10.3390/e17096304 - 10 Sep 2015

Cited by 12 | Viewed by 5442

Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Geometry in Thermodynamics)

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Open AccessArticle

Modeling of a Mass-Spring-Damper System by Fractional Derivatives with and without a Singular Kernel

by José Francisco Gómez-Aguilar, Huitzilin Yépez-Martínez, Celia Calderón-Ramón, Ines Cruz-Orduña, Ricardo Fabricio Escobar-Jiménez and Victor Hugo Olivares-Peregrino

Entropy 2015, 17(9), 6289-6303; https://doi.org/10.3390/e17096289 - 10 Sep 2015

Cited by 142 | Viewed by 35851

Abstract

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; [...] Read more.

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. Full article

(This article belongs to the Special Issue Complex and Fractional Dynamics)

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326 KiB

Open AccessArticle

Determination of Sample Entropy and Fuzzy Measure Entropy Parameters for Distinguishing Congestive Heart Failure from Normal Sinus Rhythm Subjects

by Lina Zhao, Shoushui Wei, Chengqiu Zhang, Yatao Zhang, Xinge Jiang, Feng Liu and Chengyu Liu

Entropy 2015, 17(9), 6270-6288; https://doi.org/10.3390/e17096270 - 10 Sep 2015

Cited by 75 | Viewed by 8719

Abstract

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 [...] Read more.

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 (>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, r_L = r_G = r) are for FuzzyMEn (in addition to setting n_L = 3 and n_G = 2) were recommended to yield the fine classification results for the NSR and CHF groups. Full article

(This article belongs to the Special Issue Recent Advances in Information Theory Application to Physiological Signals)

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Open AccessArticle

A Hydrodynamical Model for Carriers and Phonons With Generation-Recombination, Including Auger Effect

by Alberto Rossani

Entropy 2015, 17(9), 6258-6269; https://doi.org/10.3390/e17096258 - 9 Sep 2015

Viewed by 3603

Abstract

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 [...] Read more.

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. Full article

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Open AccessArticle

Using Generalized Entropies and OC-SVM with Mahalanobis Kernel for Detection and Classification of Anomalies in Network Traffic

by Jayro Santiago-Paz, Deni Torres-Roman, Angel Figueroa-Ypiña and Jesus Argaez-Xool

Entropy 2015, 17(9), 6239-6257; https://doi.org/10.3390/e17096239 - 8 Sep 2015

Cited by 9 | Viewed by 7580

Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Selected Papers from 1st International Electronic Conference on Entropy and its Applications)

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604 KiB

Open AccessCorrection

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

by Robert M. Davidson, Ann Lauritzen and Stephanie Seneff

Entropy 2015, 17(9), 6238; https://doi.org/10.3390/e17096238 - 8 Sep 2015

Viewed by 3768

Abstract

The authors wish to make the following correction to their paper [1]. The correct reference 190 in the reference list should be: [...] Full article

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Open AccessArticle

On the Exact Solution of Wave Equations on Cantor Sets

by Dumitru Baleanu, Hasib Khan, Hossien Jafari and Rahmat Ali Khan

Entropy 2015, 17(9), 6229-6237; https://doi.org/10.3390/e17096229 - 8 Sep 2015

Cited by 28 | Viewed by 5403

Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Wavelets, Fractals and Information Theory I)

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Open AccessArticle

Statistical Manifolds with almost Quaternionic Structures and Quaternionic Kähler-like Statistical Submersions

by Alina-Daniela Vîlcu and Gabriel-Eduard Vîlcu

Entropy 2015, 17(9), 6213-6228; https://doi.org/10.3390/e17096213 - 7 Sep 2015

Cited by 51 | Viewed by 5103

Abstract

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 [...] Read more.

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. Full article

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Open AccessConcept Paper

Thixotropic Phenomena in Water: Quantitative Indicators of Casimir-Magnetic Transformations from Vacuum Oscillations (Virtual Particles)

by Michael A. Persinger

Entropy 2015, 17(9), 6200-6212; https://doi.org/10.3390/e17096200 - 7 Sep 2015

Cited by 5 | Viewed by 4650

Abstract

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 [...] Read more.

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. Full article

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Open AccessArticle

Wavelet Entropy Automatically Detects Episodes of Atrial Fibrillation from Single-Lead Electrocardiograms

by Juan Ródenas, Manuel García, Raúl Alcaraz and José J. Rieta

Entropy 2015, 17(9), 6179-6199; https://doi.org/10.3390/e17096179 - 7 Sep 2015

Cited by 56 | Viewed by 9571

Abstract

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, [...] Read more.

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. Full article

(This article belongs to the Special Issue Wavelet Entropy: Computation and Applications)

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Open AccessArticle

Short-Lived Lattice Quasiparticles for Strongly Interacting Fluids

by Miller Mendoza Jimenez and Sauro Succi

Entropy 2015, 17(9), 6169-6178; https://doi.org/10.3390/e17096169 - 3 Sep 2015

Cited by 2 | Viewed by 4742

Abstract

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, [...] Read more.

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. Full article

(This article belongs to the Special Issue Non-Linear Lattice)

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237 KiB

Open AccessArticle

Conformal Gauge Transformations in Thermodynamics

by Alessandro Bravetti, Cesar S Lopez-Monsalvo and Francisco Nettel

Entropy 2015, 17(9), 6150-6168; https://doi.org/10.3390/e17096150 - 2 Sep 2015

Cited by 14 | Viewed by 5287

Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Geometry in Thermodynamics)

796 KiB

Open AccessArticle

Nonlinear Predictive Control of a Hydropower System Model

by Runfan Zhang, Diyi Chen and Xiaoyi Ma

Entropy 2015, 17(9), 6129-6149; https://doi.org/10.3390/e17096129 - 1 Sep 2015

Cited by 22 | Viewed by 5064

Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Applications of Information Theory in the Geosciences)

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225 KiB

Open AccessReview

Entropic Dynamics

by Ariel Caticha

Entropy 2015, 17(9), 6110-6128; https://doi.org/10.3390/e17096110 - 1 Sep 2015

Cited by 35 | Viewed by 5752

Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Dynamical Equations and Causal Structures from Observations)

837 KiB

Open AccessArticle

Optimal Base Wavelet Selection for ECG Noise Reduction Using a Comprehensive Entropy Criterion

by Hong He, Yonghong Tan and Yuexia Wang

Entropy 2015, 17(9), 6093-6109; https://doi.org/10.3390/e17096093 - 1 Sep 2015

Cited by 49 | Viewed by 8638

Abstract

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 E_com based [...] Read more.

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 E_com 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 E_com 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 E_com is compared with each of these eight criteria through four filtering performance indexes, i.e., output signal to noise ratio (SNR_o), 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 E_com has outperformed the other eight criteria in the selection of best base wavelets for ECG signal filtering. The wavelet identified by the E_com has achieved the best filtering performance than the other comparative criteria. A hypothesis test also validates that SNR_o, RMSE, PRD and correlation coefficients of E_com are significantly different from those of the shape-matched approach (α = 0.05 , two-sided t- test). Full article

(This article belongs to the Special Issue Wavelet Entropy: Computation and Applications)

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370 KiB

Open AccessReview

Distributing Secret Keys with Quantum Continuous Variables: Principle, Security and Implementations

by Eleni Diamanti and Anthony Leverrier

Entropy 2015, 17(9), 6072-6092; https://doi.org/10.3390/e17096072 - 31 Aug 2015

Cited by 265 | Viewed by 14211

Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Quantum Cryptography)

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313 KiB

Open AccessArticle

Self-Similar Solutions of Rényi’s Entropy and the Concavity of Its Entropy Power

by Agapitos N. Hatzinikitas

Entropy 2015, 17(9), 6056-6071; https://doi.org/10.3390/e17096056 - 31 Aug 2015

Cited by 2 | Viewed by 5040

Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Quantum Computation and Information: Multi-Particle Aspects)

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332 KiB

Open AccessArticle

The Effect of a Long-Range Correlated-Hopping Interaction on Bariev Spin Chains

by Tao Yang, Fa-Kai Wen, Kun Hao, Li-Ke Cao and Rui-Hong Yue

Entropy 2015, 17(9), 6044-6055; https://doi.org/10.3390/e17096044 - 28 Aug 2015

Cited by 1 | Viewed by 5128

Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Non-Linear Lattice)

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5437 KiB

Open AccessArticle

New Exact Solutions of the New Hamiltonian Amplitude-Equation and Fokas Lenells Equation

by Seyma Tuluce Demiray and Hasan Bulut

Entropy 2015, 17(9), 6025-6043; https://doi.org/10.3390/e17096025 - 27 Aug 2015

Cited by 41 | Viewed by 4898

Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Section Complexity)

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Entropy, Volume 17, Issue 9 (September 2015) – 35 articles , Pages 5938-6533

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