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Entropy, Volume 16, Issue 12 (December 2014), Pages 6195-6738

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Open AccessArticle Complex Modified Hybrid Projective Synchronization of Different Dimensional Fractional-Order Complex Chaos and Real Hyper-Chaos
Entropy 2014, 16(12), 6195-6211; doi:10.3390/e16126195
Received: 27 August 2014 / Revised: 16 November 2014 / Accepted: 17 November 2014 / Published: 27 November 2014
Cited by 9 | PDF Full-text (2785 KB) | HTML Full-text | XML Full-text
Abstract
This paper introduces a type of modified hybrid projective synchronization with complex transformationmatrix (CMHPS) for different dimensional fractional-order complex chaos and fractional-order real hyper-chaos. The transformationmatrix in this type of chaotic synchronization is a non-square matrix, and its elements are complex numbers. Based
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This paper introduces a type of modified hybrid projective synchronization with complex transformationmatrix (CMHPS) for different dimensional fractional-order complex chaos and fractional-order real hyper-chaos. The transformationmatrix in this type of chaotic synchronization is a non-square matrix, and its elements are complex numbers. Based on the stability theory of fractional-order systems, by employing the feedback control technique, necessary and sufficient criteria on CMHPS are derived. Furthermore, CMHPS between fractional-order real hyper-chaotic Rössler system and other two different dimensional fractional-order complex Lorenz-like chaotic systems is provided as two examples to discuss reduced order and increased order synchronization, respectively. Full article
(This article belongs to the Special Issue Complex Systems and Nonlinear Dynamics)
Open AccessArticle Ordinal Patterns, Entropy, and EEG
Entropy 2014, 16(12), 6212-6239; doi:10.3390/e16126212
Received: 9 October 2014 / Revised: 13 November 2014 / Accepted: 19 November 2014 / Published: 27 November 2014
Cited by 11 | PDF Full-text (882 KB) | HTML Full-text | XML Full-text
Abstract
In this paper we illustrate the potential of ordinal-patterns-based methods for analysis of real-world data and, especially, of electroencephalogram (EEG) data. We apply already known (empirical permutation entropy, ordinal pattern distributions) and new (empirical conditional entropy of ordinal patterns, robust to noise empirical
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In this paper we illustrate the potential of ordinal-patterns-based methods for analysis of real-world data and, especially, of electroencephalogram (EEG) data. We apply already known (empirical permutation entropy, ordinal pattern distributions) and new (empirical conditional entropy of ordinal patterns, robust to noise empirical permutation entropy) methods for measuring complexity, segmentation and classification of time series. Full article
(This article belongs to the Special Issue Entropy and Electroencephalography)
Open AccessArticle Generation and Nonlinear Dynamical Analyses of Fractional-Order Memristor-Based Lorenz Systems
Entropy 2014, 16(12), 6240-6253; doi:10.3390/e16126240
Received: 1 October 2014 / Revised: 9 November 2014 / Accepted: 21 November 2014 / Published: 28 November 2014
Cited by 8 | PDF Full-text (14220 KB) | HTML Full-text | XML Full-text
Abstract
In this paper, four fractional-order memristor-based Lorenz systems with the flux-controlled memristor characterized by a monotone-increasing piecewise linear function, a quadratic nonlinearity, a smooth continuous cubic nonlinearity and a quartic nonlinearity are presented, respectively. The nonlinear dynamics are analyzed by using numerical simulation
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In this paper, four fractional-order memristor-based Lorenz systems with the flux-controlled memristor characterized by a monotone-increasing piecewise linear function, a quadratic nonlinearity, a smooth continuous cubic nonlinearity and a quartic nonlinearity are presented, respectively. The nonlinear dynamics are analyzed by using numerical simulation methods, including phase portraits, bifurcation diagrams, the largest Lyapunov exponent and power spectrum diagrams. Some interesting phenomena, such as inverse period-doubling bifurcation and intermittent chaos, are found to exist in the proposed systems. Full article
(This article belongs to the Special Issue Recent Advances in Chaos Theory and Complex Networks)
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Open AccessArticle On a Local Fractional Wave Equation under Fixed Entropy Arising in Fractal Hydrodynamics
Entropy 2014, 16(12), 6254-6262; doi:10.3390/e16126254
Received: 20 October 2014 / Revised: 20 November 2014 / Accepted: 24 November 2014 / Published: 28 November 2014
Cited by 13 | PDF Full-text (206 KB) | HTML Full-text | XML Full-text
Abstract
In this paper, based on fixed entropy, the adiabatic equation of state in fractal flow is discussed. The local fractional wave equation for the velocity potential is also obtained by using the non-differential perturbations for the pressure and density of fractal hydrodynamics. Full article
(This article belongs to the Section Complexity)
Open AccessArticle A Thermodynamical Selection-Based Discrete Differential Evolution for the 0-1 Knapsack Problem
Entropy 2014, 16(12), 6263-6285; doi:10.3390/e16126263
Received: 2 October 2014 / Revised: 8 November 2014 / Accepted: 21 November 2014 / Published: 28 November 2014
Cited by 6 | PDF Full-text (266 KB) | HTML Full-text | XML Full-text
Abstract
Many problems in business and engineering can be modeled as 0-1 knapsack problems. However, the 0-1 knapsack problem is one of the classical NP-hard problems. Therefore, it is valuable to develop effective and efficient algorithms for solving 0-1 knapsack problems. Aiming at the
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Many problems in business and engineering can be modeled as 0-1 knapsack problems. However, the 0-1 knapsack problem is one of the classical NP-hard problems. Therefore, it is valuable to develop effective and efficient algorithms for solving 0-1 knapsack problems. Aiming at the drawbacks of the selection operator in the traditional differential evolution (DE), we present a novel discrete differential evolution (TDDE) for solving 0-1 knapsack problem. In TDDE, an enhanced selection operator inspired by the principle of the minimal free energy in thermodynamics is employed, trying to balance the conflict between the selective pressure and the diversity of population to some degree. An experimental study is conducted on twenty 0-1 knapsack test instances. The comparison results show that TDDE can gain competitive performance on the majority of the test instances. Full article
(This article belongs to the Special Issue Entropy in Bioinspired Intelligence)
Open AccessArticle Adaptive Synchronization of Fractional Neural Networks with Unknown Parameters and Time Delays
Entropy 2014, 16(12), 6286-6299; doi:10.3390/e16126286
Received: 11 October 2014 / Revised: 23 November 2014 / Accepted: 26 November 2014 / Published: 1 December 2014
Cited by 11 | PDF Full-text (1391 KB) | HTML Full-text | XML Full-text
Abstract
In this paper, the parameters identification and synchronization problem of fractional-order neural networks with time delays are investigated. Based on some analytical techniques and an adaptive control method, a simple adaptive synchronization controller and parameter update laws are designed to synchronize two uncertain
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In this paper, the parameters identification and synchronization problem of fractional-order neural networks with time delays are investigated. Based on some analytical techniques and an adaptive control method, a simple adaptive synchronization controller and parameter update laws are designed to synchronize two uncertain complex networks with time delays. Besides, the system parameters in the uncertain network can be identified in the process of synchronization. To demonstrate the validity of the proposed method, several illustrative examples are presented. Full article
(This article belongs to the Special Issue Complex Systems and Nonlinear Dynamics)
Open AccessArticle Ab intio Investigation of the Thermochemistry and Kinetics of the SO2 + O3 → SO3 + O2 Reaction in Aircraft Engines and the Environment
Entropy 2014, 16(12), 6300-6312; doi:10.3390/e16126300
Received: 30 September 2014 / Revised: 12 November 2014 / Accepted: 14 November 2014 / Published: 1 December 2014
PDF Full-text (1044 KB) | HTML Full-text | XML Full-text
Abstract
In the present work, the mechanisms, thermochemistry and kinetics of the reaction of SO2 with O3 have been studied using the CCSD(T)/6-31G(d) + CF method. It has been shown that there exist two possible pathways A and B of the
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In the present work, the mechanisms, thermochemistry and kinetics of the reaction of SO2 with O3 have been studied using the CCSD(T)/6-31G(d) + CF method. It has been shown that there exist two possible pathways A and B of the SO2 + O3 → SO3 + O2 reaction. The two pathways’ A and B barrier heights are 0.61 kcal mol−1 and 3.40 kcal mol−1, respectively, while the energy of the SO2 + O3 → SO3 + O2 reaction is −25.25 kcal mol−1. The canonical variational transition state theory with small-curvature tunneling (CVT/SCT) has been applied to study the reaction kinetics. The CVT/SCT study shows that the rate constants K for pathways A and B, KA = 1.11 × 10−12exp(−2526.13/T) and KB = 2.7 × 10−14exp(−1029.25/T), respectively, grow as the temperature increases and are much larger than those of the SO2 + O3 → SO3 + O2 reaction over the entire temperature range of 200–1500 K. This indicates that ionization of O3 and high temperatures are favorable for the SO2 oxidation via the reaction with ozone. The new data obtained in the present study can be utilized directly for the evaluation of experiments and model predictions concerning SO2 oxidation and kinetic modeling of gas-phase chemistry of pollutants/nucleation precursors formed in aircraft engines and the Earth’s atmosphere. Full article
(This article belongs to the Section Thermodynamics)
Open AccessArticle Landscape Analysis of Geographical Names in Hubei Province, China
Entropy 2014, 16(12), 6313-6337; doi:10.3390/e16126313
Received: 20 July 2014 / Revised: 31 October 2014 / Accepted: 26 November 2014 / Published: 1 December 2014
Cited by 2 | PDF Full-text (6384 KB) | HTML Full-text | XML Full-text
Abstract
Hubei Province is the hub of communications in central China, which directly determines its strategic position in the country’s development. Additionally, Hubei Province is well-known for its diverse landforms, including mountains, hills, mounds and plains. This area is called “The Province of Thousand
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Hubei Province is the hub of communications in central China, which directly determines its strategic position in the country’s development. Additionally, Hubei Province is well-known for its diverse landforms, including mountains, hills, mounds and plains. This area is called “The Province of Thousand Lakes” due to the abundance of water resources. Geographical names are exclusive names given to physical or anthropogenic geographic entities at specific spatial locations and are important signs by which humans understand natural and human activities. In this study, geographic information systems (GIS) technology is adopted to establish a geodatabase of geographical names with particular characteristics in Hubei Province and extract certain geomorphologic and environmental factors. We carry out landscape analysis of mountain-related geographical names and water-related geographical names respectively. In the end, we calculate the information entropy of geographical names of each county to describe the diversity and inhomogeneity of place names in Hubei province. Our study demonstrates that geographical names represent responses to the cultural landscape and physical environment. The geographical names are more interesting in specific landscapes, such as mountains and rivers. Full article
(This article belongs to the Special Issue Entropy and Space-Time Analysis in Environment and Health)
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Open AccessArticle The Information Geometry of Bregman Divergences and Some Applications in Multi-Expert Reasoning
Entropy 2014, 16(12), 6338-6381; doi:10.3390/e16126338
Received: 19 October 2014 / Revised: 24 November 2014 / Accepted: 25 November 2014 / Published: 1 December 2014
Cited by 8 | PDF Full-text (438 KB) | HTML Full-text | XML Full-text
Abstract
The aim of this paper is to develop a comprehensive study of the geometry involved in combining Bregman divergences with pooling operators over closed convex sets in a discrete probabilistic space. A particular connection we develop leads to an iterative procedure, which is
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The aim of this paper is to develop a comprehensive study of the geometry involved in combining Bregman divergences with pooling operators over closed convex sets in a discrete probabilistic space. A particular connection we develop leads to an iterative procedure, which is similar to the alternating projection procedure by Csiszár and Tusnády. Although such iterative procedures are well studied over much more general spaces than the one we consider, only a few authors have investigated combining projections with pooling operators. We aspire to achieve here a comprehensive study of such a combination. Besides, pooling operators combining the opinions of several rational experts allows us to discuss possible applications in multi-expert reasoning. Full article
(This article belongs to the Special Issue Maximum Entropy Applied to Inductive Logic and Reasoning)
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Open AccessArticle Entropy Analysis of RR and QT Interval Variability during Orthostatic and Mental Stress in Healthy Subjects
Entropy 2014, 16(12), 6384-6393; doi:10.3390/e16126384
Received: 17 October 2014 / Revised: 24 November 2014 / Accepted: 28 November 2014 / Published: 3 December 2014
Cited by 5 | PDF Full-text (1055 KB) | HTML Full-text | XML Full-text
Abstract
Autonomic activity affects beat-to-beat variability of heart rate and QT interval. The aim of this study was to explore whether entropy measures are suitable to detect changes in neural outflow to the heart elicited by two different stress paradigms. We recorded short-term ECG
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Autonomic activity affects beat-to-beat variability of heart rate and QT interval. The aim of this study was to explore whether entropy measures are suitable to detect changes in neural outflow to the heart elicited by two different stress paradigms. We recorded short-term ECG in 11 normal subjects during an experimental protocol that involved head-up tilt and mental arithmetic stress and computed sample entropy, cross-sample entropy and causal interactions based on conditional entropy from RR and QT interval time series. Head-up tilt resulted in a significant reduction in sample entropy of RR intervals and cross-sample entropy, while mental arithmetic stress resulted in a significant reduction in coupling directed from RR to QT. In conclusion, measures of entropy are suitable to detect changes in neural outflow to the heart and decoupling of repolarisation variability from heart rate variability elicited by orthostatic or mental arithmetic stress. Full article
(This article belongs to the Special Issue Entropy and Cardiac Physics)
Open AccessArticle Determining Common Weights in Data Envelopment Analysis with Shannon’s Entropy
Entropy 2014, 16(12), 6394-6414; doi:10.3390/e16126394
Received: 17 October 2014 / Revised: 29 November 2014 / Accepted: 2 December 2014 / Published: 4 December 2014
Cited by 7 | PDF Full-text (870 KB) | HTML Full-text | XML Full-text
Abstract
Data Envelopment Analysis (DEA) is a non-parametric method for evaluating the efficiency of Decision Making Units (DMUs) with multiple inputs and outputs. In the traditional DEA models, the DMU is allowed to use its most favorable multiplier weights to maximize its efficiency. There
[...] Read more.
Data Envelopment Analysis (DEA) is a non-parametric method for evaluating the efficiency of Decision Making Units (DMUs) with multiple inputs and outputs. In the traditional DEA models, the DMU is allowed to use its most favorable multiplier weights to maximize its efficiency. There is usually more than one efficient DMU which cannot be further discriminated. Evaluating DMUs with different multiplier weights would also be somewhat irrational in practice. The common weights DEA model is an effective method for solving these problems. In this paper, we propose a methodology combining the common weights DEA with Shannon’s entropy. In our methodology, we propose a modified weight restricted DEA model for calculating non-zero optimal weights. Then these non-zero optimal weights would be aggregated to be the common weights using Shannon’s entropy. Compared with the traditional models, our proposed method is more powerful in discriminating DMUs, especially when the inputs and outputs are numerous. Our proposed method also keeps in accordance with the basic DEA method considering the evaluation of the most efficient and inefficient DMUs. Numerical examples are provided to examine the validity and effectiveness of our proposed methodology. Full article
Open AccessArticle Detecting Chaos from Agricultural Product Price Time Series
Entropy 2014, 16(12), 6415-6433; doi:10.3390/e16126415
Received: 13 July 2014 / Revised: 28 September 2014 / Accepted: 20 November 2014 / Published: 5 December 2014
Cited by 3 | PDF Full-text (1714 KB) | HTML Full-text | XML Full-text
Abstract
Analysis of the characteristics of agricultural product price volatility and trend forecasting are necessary to formulate and implement agricultural price control policies. Taking wholesale cabbage prices as an example, a multiple test methodology has been adopted to identify the nonlinearity, fractality, and chaos
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Analysis of the characteristics of agricultural product price volatility and trend forecasting are necessary to formulate and implement agricultural price control policies. Taking wholesale cabbage prices as an example, a multiple test methodology has been adopted to identify the nonlinearity, fractality, and chaos of the data. The approaches used include the R/S analysis, the BDS test, the power spectra, the recurrence plot, the largest Lyapunov exponent, the Kolmogorov entropy, and the correlation dimension. The results show that there is chaos in agricultural wholesale price data, which provides a good theoretical basis for selecting reasonable forecasting models as prediction techniques based on chaos theory can be applied to forecasting agricultural prices. Full article
(This article belongs to the Special Issue Complex Systems and Nonlinear Dynamics)
Open AccessArticle An Entropy-Based Damage Characterization
Entropy 2014, 16(12), 6434-6463; doi:10.3390/e16126434
Received: 21 November 2014 / Accepted: 2 December 2014 / Published: 5 December 2014
Cited by 10 | PDF Full-text (966 KB) | HTML Full-text | XML Full-text
Abstract
This paper presents a scientific basis for the description of the causes of damage within an irreversible thermodynamic framework and the effects of damage as observable variables that signify degradation of structural integrity. The approach relies on the fundamentals of irreversible thermodynamics and
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This paper presents a scientific basis for the description of the causes of damage within an irreversible thermodynamic framework and the effects of damage as observable variables that signify degradation of structural integrity. The approach relies on the fundamentals of irreversible thermodynamics and specifically the notion of entropy generation as a measure of degradation and damage. We first review the state-of-the-art advances in entropic treatment of damage followed by a discussion on generalization of the entropic concept to damage characterization that may offers a better definition of damage metric commonly used for structural integrity assessment. In general, this approach provides the opportunity to described reliability and risk of structures in terms of fundamental science concepts. Over the years, many studies have focused on materials damage assessment by determining physics-based cause and affect relationships, the goal of this paper is to put this work in perspective and encourage future work of materials damage based on the entropy concept. Full article
(This article belongs to the Section Thermodynamics)
Open AccessArticle A Memristive Diode Bridge-Based Canonical Chua’s Circuit
Entropy 2014, 16(12), 6464-6476; doi:10.3390/e16126464
Received: 30 September 2014 / Revised: 19 November 2014 / Accepted: 2 December 2014 / Published: 8 December 2014
Cited by 10 | PDF Full-text (1505 KB) | HTML Full-text | XML Full-text
Abstract
A novel memristor circuit is presented, which is generated from the canonical Chua’s circuit by replacing the Chua’s diode with a first order memristive diode bridge. The circuit dynamical characteristics with the variations of circuit parameters are investigated both theoretically and numerically. It
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A novel memristor circuit is presented, which is generated from the canonical Chua’s circuit by replacing the Chua’s diode with a first order memristive diode bridge. The circuit dynamical characteristics with the variations of circuit parameters are investigated both theoretically and numerically. It can be found that the circuit has three determined equilibrium points, including a zero saddle point and two nonzero saddle-foci with index 2. Specially, the circuit is non-dissipative in the neighborhood of the zero saddle point, and there exists complex nonlinear phenomena of coexisting bifurcation modes and coexisting chaotic attractors. Experimental observations are performed to verify the numerical simulation results. Full article
(This article belongs to the Special Issue Complex Systems and Nonlinear Dynamics)
Open AccessArticle An Evolutionary Algorithm for the Texture Analysis of Cubic System Materials Derived by the Maximum Entropy Principle
Entropy 2014, 16(12), 6477-6496; doi:10.3390/e16126477
Received: 25 July 2014 / Revised: 15 October 2014 / Accepted: 12 November 2014 / Published: 9 December 2014
PDF Full-text (1551 KB) | HTML Full-text | XML Full-text
Abstract
Based on the principle of maximum entropy method (MEM) for quantitative texture analysis, the differential evolution (DE) algorithm was effectively introduced. Using a DE-optimized algorithm with a faster but more stable convergence rate of iteration reliable complete orientation distributions (C-ODF) have
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Based on the principle of maximum entropy method (MEM) for quantitative texture analysis, the differential evolution (DE) algorithm was effectively introduced. Using a DE-optimized algorithm with a faster but more stable convergence rate of iteration reliable complete orientation distributions (C-ODF) have been obtained for deep-drawn IF steel sheets and the recrystallized aluminum foils after cold-rolling, which are designated as showing a macroscopic cubic-orthogonal symmetry. With special reference to the data processing, no more other assumptions are required for DE-optimized MEM except that the system entropy approach the maximum. Full article
Open AccessArticle Statistical Power Law due to Reservoir Fluctuations and the Universal Thermostat Independence Principle
Entropy 2014, 16(12), 6497-6514; doi:10.3390/e16126497
Received: 3 November 2014 / Revised: 26 November 2014 / Accepted: 2 December 2014 / Published: 9 December 2014
Cited by 19 | PDF Full-text (224 KB) | HTML Full-text | XML Full-text
Abstract
Certain fluctuations in particle number, \(n\), at fixed total energy, \(E\), lead exactly to a cut-power law distribution in the one-particle energy, \(\omega\), via the induced fluctuations in the phase-space volume ratio, \(\Omega_n(E-\omega)/\Omega_n(E)=(1-\omega/E)^n\). The only parameters are \(1/T=\langle \beta \rangle=\langle n \rangle/E\) and
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Certain fluctuations in particle number, \(n\), at fixed total energy, \(E\), lead exactly to a cut-power law distribution in the one-particle energy, \(\omega\), via the induced fluctuations in the phase-space volume ratio, \(\Omega_n(E-\omega)/\Omega_n(E)=(1-\omega/E)^n\). The only parameters are \(1/T=\langle \beta \rangle=\langle n \rangle/E\) and \(q=1-1/\langle n \rangle + \Delta n^2/\langle n \rangle^2\). For the binomial distribution of \(n\) one obtains \(q=1-1/k\), for the negative binomial \(q=1+1/(k+1)\). These results also represent an approximation for general particle number distributions in the reservoir up to second order in the canonical expansion \(\omega \ll E\). For general systems the average phase-space volume ratio \(\langle e^{S(E-\omega)}/e^{S(E)}\rangle\) to second order delivers \(q=1-1/C+\Delta \beta^2/\langle \beta \rangle^2\) with \(\beta=S^{\prime}(E)\) and \(C=dE/dT\) heat capacity. However, \(q \ne 1\) leads to non-additivity of the Boltzmann–Gibbs entropy, \(S\). We demonstrate that a deformed entropy, \(K(S)\), can be constructed and used for demanding additivity, i.e., \(q_K=1\). This requirement leads to a second order differential equation for \(K(S)\). Finally, the generalized \(q\)-entropy formula, \(K(S)=\sum p_i K(-\ln p_i)\), contains the Tsallis, Rényi and Boltzmann–Gibbs–Shannon expressions as particular cases. For diverging variance, \(\Delta\beta^2\) we obtain a novel entropy formula. Full article
(This article belongs to the Special Issue Entropic Aspects in Statistical Physics of Complex Systems)
Open AccessArticle Geometric Thermodynamics: Black Holes and the Meaning of the Scalar Curvature
Entropy 2014, 16(12), 6515-6523; doi:10.3390/e16126515
Received: 9 October 2014 / Revised: 2 December 2014 / Accepted: 4 December 2014 / Published: 11 December 2014
Cited by 10 | PDF Full-text (199 KB) | HTML Full-text | XML Full-text
Abstract
In this paper we show that the vanishing of the scalar curvature of Ruppeiner-like metrics does not characterize the ideal gas. Furthermore, we claim through an example that flatness is not a sufficient condition to establish the absence of interactions in the underlying
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In this paper we show that the vanishing of the scalar curvature of Ruppeiner-like metrics does not characterize the ideal gas. Furthermore, we claim through an example that flatness is not a sufficient condition to establish the absence of interactions in the underlying microscopic model of a thermodynamic system, which poses a limitation on the usefulness of Ruppeiner’s metric and conjecture. Finally, we address the problem of the choice of coordinates in black hole thermodynamics. We propose an alternative energy representation for Kerr-Newman black holes that mimics fully Weinhold’s approach. The corresponding Ruppeiner’s metrics become degenerate only at absolute zero and have non-vanishing scalar curvatures. Full article
(This article belongs to the Special Issue Entropy and Spacetime)
Open AccessArticle Consensus of Discrete Multiagent System with Various Time Delays and Environmental Disturbances
Entropy 2014, 16(12), 6524-6538; doi:10.3390/e16126524
Received: 4 October 2014 / Revised: 9 December 2014 / Accepted: 9 December 2014 / Published: 11 December 2014
Cited by 2 | PDF Full-text (209 KB) | HTML Full-text | XML Full-text
Abstract
In this paper, the consensus problem of discrete multiagent systems with time varying sampling periods is studied. Firstly, with thorough analysis of various delays among agents, the control input of each agent is designed with consideration of sending delay and receiving delay. With
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In this paper, the consensus problem of discrete multiagent systems with time varying sampling periods is studied. Firstly, with thorough analysis of various delays among agents, the control input of each agent is designed with consideration of sending delay and receiving delay. With construction of discrete dynamics of state error vector, it is proved by applying Halanay inequality that consensus of the system can be reached. Further, the definition of bounded consensus is proposed in the situation where environmental disturbances exist. In order to handle this problem, the Halanay inequality is extended into a more general one with boundedness property. Based on the new Halanay inequality obtained, the boundedness of consensus error is guaranteed. At last, simulation examples are presented to demonstrate the theoretical conclusions. Full article
(This article belongs to the Special Issue Complex Systems and Nonlinear Dynamics)
Open AccessArticle Chaos Control and Synchronization of a Hyperchaotic Zhou System by Integral Sliding Mode control
Entropy 2014, 16(12), 6539-6552; doi:10.3390/e16126539
Received: 25 September 2014 / Revised: 25 November 2014 / Accepted: 3 December 2014 / Published: 12 December 2014
Cited by 6 | PDF Full-text (306 KB) | HTML Full-text | XML Full-text
Abstract
In this paper, an adaptive integral sliding mode control scheme is proposed for synchronization of hyperchaotic Zhou systems. In the proposed scheme, an integral sliding mode control is designed to stabilize a hyperchaotic Zhou system with known parameters to its unstable equilibrium at
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In this paper, an adaptive integral sliding mode control scheme is proposed for synchronization of hyperchaotic Zhou systems. In the proposed scheme, an integral sliding mode control is designed to stabilize a hyperchaotic Zhou system with known parameters to its unstable equilibrium at the origin. The control is then applied to the synchronization of two identical systems, i.e., a slave and a master hyperchaotic Zhou system with unknown parameters. The adaptive control mechanism introduced synchronizes the systems by estimating the unknown parameters. Simulation results have shown that the proposed method has an excellent convergence from both speed and accuracy points of view, and it outperforms Vaidyanathan’s scheme, which is a well-recognized scheme in this area. Full article
(This article belongs to the Special Issue Complex Systems and Nonlinear Dynamics)
Open AccessArticle Enhanced Automatic Wavelet Independent Component Analysis for Electroencephalographic Artifact Removal
Entropy 2014, 16(12), 6553-6572; doi:10.3390/e16126553
Received: 31 July 2014 / Revised: 4 December 2014 / Accepted: 5 December 2014 / Published: 17 December 2014
Cited by 3 | PDF Full-text (18016 KB) | HTML Full-text | XML Full-text
Abstract
Electroencephalography (EEG) is a fundamental diagnostic instrument for many neurological disorders, and it is the main tool for the investigation of the cognitive or pathological activity of the brain through the bioelectromagnetic fields that it generates. The correct interpretation of the EEG is
[...] Read more.
Electroencephalography (EEG) is a fundamental diagnostic instrument for many neurological disorders, and it is the main tool for the investigation of the cognitive or pathological activity of the brain through the bioelectromagnetic fields that it generates. The correct interpretation of the EEG is misleading, both for clinicians’ visual evaluation and for automated procedures, because of artifacts. As a consequence, artifact rejection in EEG is a key preprocessing step, and the quest for reliable automatic processors has been quickly growing in the last few years. Recently, a promising automatic methodology, known as automatic wavelet-independent component analysis (AWICA), has been proposed. In this paper, a more efficient and sensitive version, called enhanced-AWICA (EAWICA), is proposed, and an extensive performance comparison is carried out by a step of tuning the different parameters that are involved in artifact detection. EAWICA is shown to minimize information loss and to outperform AWICA in artifact removal, both on simulated and real experimental EEG recordings. Full article
(This article belongs to the Special Issue Entropy and Electroencephalography)
Open AccessArticle Automatic Sleep Stages Classification Using EEG Entropy Features and Unsupervised Pattern Analysis Techniques
Entropy 2014, 16(12), 6573-6589; doi:10.3390/e16126573
Received: 18 July 2014 / Revised: 28 November 2014 / Accepted: 9 December 2014 / Published: 17 December 2014
Cited by 10 | PDF Full-text (1157 KB) | HTML Full-text | XML Full-text
Abstract
Sleep is a growing area of research interest in medicine and neuroscience. Actually, one major concern is to find a correlation between several physiologic variables and sleep stages. There is a scientific agreement on the characteristics of the five stages of human sleep,
[...] Read more.
Sleep is a growing area of research interest in medicine and neuroscience. Actually, one major concern is to find a correlation between several physiologic variables and sleep stages. There is a scientific agreement on the characteristics of the five stages of human sleep, based on EEG analysis. Nevertheless, manual stage classification is still the most widely used approach. This work proposes a new automatic sleep classification method based on unsupervised feature classification algorithms recently developed, and on EEG entropy measures. This scheme extracts entropy metrics from EEG records to obtain a feature vector. Then, these features are optimized in terms of relevance using the Q-α algorithm. Finally, the resulting set of features is entered into a clustering procedure to obtain a final segmentation of the sleep stages. The proposed method reached up to an average of 80% correctly classified stages for each patient separately while keeping the computational cost low. Full article
(This article belongs to the Special Issue Entropy and Electroencephalography)
Open AccessArticle Depth Image Coding Using Entropy-Based Adaptive Measurement Allocation
Entropy 2014, 16(12), 6590-6601; doi:10.3390/e16126590
Received: 20 October 2014 / Revised: 7 December 2014 / Accepted: 12 December 2014 / Published: 17 December 2014
PDF Full-text (1435 KB) | HTML Full-text | XML Full-text
Abstract
Differently from traditional two-dimensional texture images, the depth images of three-dimensional (3D) video systems have significant sparse characteristics under the certain transform basis, which make it possible for compressive sensing to represent depth information efficiently. Therefore, in this paper, a novel depth image
[...] Read more.
Differently from traditional two-dimensional texture images, the depth images of three-dimensional (3D) video systems have significant sparse characteristics under the certain transform basis, which make it possible for compressive sensing to represent depth information efficiently. Therefore, in this paper, a novel depth image coding scheme is proposed based on a block compressive sensing method. At the encoder, in view of the characteristics of depth images, the entropy of pixels in each block is employed to represent the sparsity of depth signals. Then according to the different sparsity in the pixel domain, the measurements can be adaptively allocated to each block for higher compression efficiency. At the decoder, the sparse transform can be combined to achieve the compressive sensing reconstruction. Experimental results have shown that at the same sampling rate, the proposed scheme can obtain higher PSNR values and better subjective quality of the rendered virtual views, compared with the method using a uniform sampling rate. Full article
Open AccessArticle Detection and Modeling of Cyber Attacks with Petri Nets
Entropy 2014, 16(12), 6602-6623; doi:10.3390/e16126602
Received: 30 October 2014 / Revised: 4 December 2014 / Accepted: 16 December 2014 / Published: 19 December 2014
Cited by 12 | PDF Full-text (1300 KB) | HTML Full-text | XML Full-text
Abstract
The aim of this article is to present an approach to develop and verify a method of formal modeling of cyber threats directed at computer systems. Moreover, the goal is to prove that the method enables one to create models resembling the behavior
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The aim of this article is to present an approach to develop and verify a method of formal modeling of cyber threats directed at computer systems. Moreover, the goal is to prove that the method enables one to create models resembling the behavior of malware that support the detection process of selected cyber attacks and facilitate the application of countermeasures. The most common cyber threats targeting end users and terminals are caused by malicious software, called malware. The malware detection process can be performed either by matching their digital signatures or analyzing their behavioral models. As the obfuscation techniques make the malware almost undetectable, the classic signature-based anti-virus tools must be supported with behavioral analysis. The proposed approach to modeling of malware behavior is based on colored Petri nets. This article is addressed to cyber defense researchers, security architects and developers solving up-to-date problems regarding the detection and prevention of advanced persistent threats. Full article
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Open AccessArticle The McMillan Theorem for Colored Branching Processes and Dimensions of Random Fractals
Entropy 2014, 16(12), 6624-6653; doi:10.3390/e16126624
Received: 16 September 2014 / Revised: 18 November 2014 / Accepted: 12 December 2014 / Published: 19 December 2014
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Abstract
For the simplest colored branching process, we prove an analog to the McMillan theorem and calculate the Hausdorff dimensions of random fractals defined in terms of the limit behavior of empirical measures generated by finite genetic lines. In this setting, the role of
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For the simplest colored branching process, we prove an analog to the McMillan theorem and calculate the Hausdorff dimensions of random fractals defined in terms of the limit behavior of empirical measures generated by finite genetic lines. In this setting, the role of Shannon’s entropy is played by the Kullback–Leibler divergence, and the Hausdorff dimensions are computed by means of the so-called Billingsley–Kullback entropy, defined in the paper. Full article
(This article belongs to the Section Information Theory)
Open AccessArticle The Effects of Spontaneous Random Activity on Information Transmission in an Auditory Brain Stem Neuron Model
Entropy 2014, 16(12), 6654-6666; doi:10.3390/e16126654
Received: 30 April 2014 / Revised: 8 December 2014 / Accepted: 15 December 2014 / Published: 19 December 2014
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Abstract
This paper presents the effects of spontaneous random activity on information transmission in an auditory brain stem neuron model. In computer simulations, the supra-threshold synaptic current stimuli ascending from auditory nerve fibers (ANFs) were modeled by a filtered inhomogeneous Poisson process modulated by
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This paper presents the effects of spontaneous random activity on information transmission in an auditory brain stem neuron model. In computer simulations, the supra-threshold synaptic current stimuli ascending from auditory nerve fibers (ANFs) were modeled by a filtered inhomogeneous Poisson process modulated by sinusoidal functions at a frequency of 220–3520 Hz with regard to the human speech spectrum. The stochastic sodium and stochastic high- and low-threshold potassium channels were incorporated into a single compartment model of the soma in spherical bushy neurons, so as to realize threshold fluctuations or a variation of spike firing times. The results show that the information rates estimated from the entropy of inter-spike intervals of spike trains tend toward a convex function of the spontaneous rates when the intensity of sinusoidal functions decreases. Furthermore, the results show that a convex function of the spontaneous rates tends to disappear as the frequency of the sinusoidal function increases, such that the phase-locked response can be unobserved. It is concluded that this sort of stochastic resonance (SR) phenomenon, which depends on the spontaneous rates with supra-threshold stimuli, can better enhance information transmission in a smaller intensity of sinusoidal functions within the human speech spectrum, like the situation in which the regular SR can enhance weak signals. Full article
(This article belongs to the Special Issue Entropy in Human Brain Networks)
Open AccessArticle Fast Rate Estimation for RDO Mode Decision in HEVC
Entropy 2014, 16(12), 6667-6685; doi:10.3390/e16126667
Received: 19 August 2014 / Revised: 3 December 2014 / Accepted: 17 December 2014 / Published: 19 December 2014
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Abstract
The latter-day H.265/HEVC video compression standard is able to provide two-times higher compression efficiency compared to the current industrial standard, H.264/AVC. However, coding complexity also increased. The main bottleneck of the compression process is the rate-distortion optimization (RDO) stage, as it involves numerous
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The latter-day H.265/HEVC video compression standard is able to provide two-times higher compression efficiency compared to the current industrial standard, H.264/AVC. However, coding complexity also increased. The main bottleneck of the compression process is the rate-distortion optimization (RDO) stage, as it involves numerous sequential syntax-based binary arithmetic coding (SBAC) loops. In this paper, we present an entropy-based RDO estimation technique for H.265/HEVC compression, instead of the common approach based on the SBAC. Our RDO implementation reduces RDO complexity, providing an average bit rate overhead of 1.54%. At the same time, elimination of the SBAC from the RDO estimation reduces block interdependencies, thus providing an opportunity for the development of the compression system with parallel processing of multiple blocks of a video frame. Full article
(This article belongs to the Section Information Theory)
Open AccessArticle Effect of the Postural Challenge on the Dependence of the Cardiovascular Control Complexity on Age
Entropy 2014, 16(12), 6686-6704; doi:10.3390/e16126686
Received: 28 October 2014 / Revised: 8 December 2014 / Accepted: 18 December 2014 / Published: 22 December 2014
Cited by 9 | PDF Full-text (1699 KB) | HTML Full-text | XML Full-text
Abstract
Short-term complexity of heart period (HP) and systolic arterial pressure (SAP) was computed to detect age and gender influences over cardiovascular control in resting supine condition (REST) and during standing (STAND). Healthy subjects (n = 110, men = 55) were equally divided
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Short-term complexity of heart period (HP) and systolic arterial pressure (SAP) was computed to detect age and gender influences over cardiovascular control in resting supine condition (REST) and during standing (STAND). Healthy subjects (n = 110, men = 55) were equally divided into five groups (21–30; 31–40; 41–50; 51–60; and 61–70 years of age). HP and SAP series were recorded for 15 min at REST and during STAND. A normalized complexity index (NCI) based on conditional entropy was assessed. At REST we found that both NCIHP and NCISAP decreased with age in the overall population, but only women were responsible for this trend. During STAND we observed that both NCIHP and NCISAP were unrelated to age in the overall population, even when divided by gender. When the variation of NCI in response to STAND (ΔNCI = NCI at REST-NCI during STAND) was computed individually, we found that ΔNCIHP progressively decreased with age in the overall population, and women were again responsible for this trend. Conversely, ΔNCISAP was unrelated to age and gender. This study stresses that the complexity of cardiovascular control and its ability to respond to stressors are more importantly lost with age in women than in men. Full article
(This article belongs to the Special Issue Entropy and Cardiac Physics)
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Open AccessArticle A Representation of the Relative Entropy with Respect to a Diffusion Process in Terms of Its Infinitesimal Generator
Entropy 2014, 16(12), 6705-6721; doi:10.3390/e16126705
Received: 23 October 2014 / Revised: 17 December 2014 / Accepted: 18 December 2014 / Published: 22 December 2014
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Abstract
In this paper we derive an integral (with respect to time) representation of the relative entropy (or Kullback–Leibler Divergence) R(μ||P), where μ and P are measures on C([0,T];Rd). The underlying measure P is a weak solution to a martingale problem with
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In this paper we derive an integral (with respect to time) representation of the relative entropy (or Kullback–Leibler Divergence) R(μ||P), where μ and P are measures on C([0,T];Rd). The underlying measure P is a weak solution to a martingale problem with continuous coefficients. Our representation is in the form of an integral with respect to its infinitesimal generator. This representation is of use in statistical inference (particularly involving medical imaging). Since R(μ||P) governs the exponential rate of convergence of the empirical measure (according to Sanov’s theorem), this representation is also of use in the numerical and analytical investigation of finite-size effects in systems of interacting diffusions. Full article
(This article belongs to the Section Information Theory)
Open AccessArticle A Large Deviation Principle and an Expression of the Rate Function for a Discrete Stationary Gaussian Process
Entropy 2014, 16(12), 6722-6738; doi:10.3390/e16126722
Received: 3 November 2014 / Revised: 17 December 2014 / Accepted: 18 December 2014 / Published: 22 December 2014
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Abstract
We prove a large deviation principle for a stationary Gaussian process over Rb,indexed by Ζd (for some positive integers d and b), with positive definite spectral density, andprovide an expression of the corresponding rate function in terms of the
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We prove a large deviation principle for a stationary Gaussian process over Rb,indexed by Ζd (for some positive integers d and b), with positive definite spectral density, andprovide an expression of the corresponding rate function in terms of the mean of the processand its spectral density. This result is useful in applications where such an expression isneeded. Full article
(This article belongs to the Section Statistical Mechanics)

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Open AccessCorrection Correction on Iliyasu, A.M. et al. Hybrid Quantum-Classical Protocol for Storage and Retrieval of Discrete-Valued Information. Entropy, 2014, 16, 3537-3551
Entropy 2014, 16(12), 6382-6383; doi:10.3390/e16126382
Received: 18 November 2014 / Accepted: 26 November 2014 / Published: 2 December 2014
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Abstract
The authors wish to make the following corrections to this paper [1]: The correct name of the fourth author is: Ahmed S. Salama. In the Acknowledgment Section, we added the research Project No. 2014/01/2079. Below is the corrected version of the section.[...]
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The authors wish to make the following corrections to this paper [1]: The correct name of the fourth author is: Ahmed S. Salama. In the Acknowledgment Section, we added the research Project No. 2014/01/2079. Below is the corrected version of the section.[...] Full article

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