Entropy

229 KiB

Open AccessArticle

On the Existence and Uniqueness of Solutions for Local Fractional Differential Equations

by Hossein Jafari, Hassan Kamil Jassim, Maysaa Al Qurashi and Dumitru Baleanu

Entropy 2016, 18(11), 420; https://doi.org/10.3390/e18110420 - 23 Nov 2016

Cited by 27 | Viewed by 4374

In this manuscript, we prove the existence and uniqueness of solutions for local fractional differential equations (LFDEs) with local fractional derivative operators (LFDOs). By using the contracting mapping theorem (CMT) and increasing and decreasing theorem (IDT), existence and uniqueness results are obtained. Some [...] Read more.

In this manuscript, we prove the existence and uniqueness of solutions for local fractional differential equations (LFDEs) with local fractional derivative operators (LFDOs). By using the contracting mapping theorem (CMT) and increasing and decreasing theorem (IDT), existence and uniqueness results are obtained. Some examples are presented to illustrate the validity of our results. Full article

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

1079 KiB

Open AccessReview

Periodic Energy Transport and Entropy Production in Quantum Electronics

by María Florencia Ludovico, Liliana Arrachea, Michael Moskalets and David Sánchez

Entropy 2016, 18(11), 419; https://doi.org/10.3390/e18110419 - 23 Nov 2016

Cited by 47 | Viewed by 6018

Abstract

The problem of time-dependent particle transport in quantum conductors is nowadays a well established topic. In contrast, the way in which energy and heat flow in mesoscopic systems subjected to dynamical drivings is a relatively new subject that cross-fertilize both fundamental developments of [...] Read more.

The problem of time-dependent particle transport in quantum conductors is nowadays a well established topic. In contrast, the way in which energy and heat flow in mesoscopic systems subjected to dynamical drivings is a relatively new subject that cross-fertilize both fundamental developments of quantum thermodynamics and practical applications in nanoelectronics and quantum information. In this short review, we discuss from a thermodynamical perspective recent investigations on nonstationary heat and work generated in quantum systems, emphasizing open questions and unsolved issues. Full article

(This article belongs to the Special Issue Quantum Thermodynamics)

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3755 KiB

Open AccessArticle

Prediction of Bearing Fault Using Fractional Brownian Motion and Minimum Entropy Deconvolution

by Wanqing Song, Ming Li and Jian-Kai Liang

Entropy 2016, 18(11), 418; https://doi.org/10.3390/e18110418 - 23 Nov 2016

Cited by 13 | Viewed by 6083

Abstract

In this paper, we propose a novel framework for the diagnosis of incipient bearing faults and trend prediction of weak faults which result in gradual aggravation with the bearing vibration intensity as the characteristic parameter. For the weak fault diagnosis, the proposed framework [...] Read more.

In this paper, we propose a novel framework for the diagnosis of incipient bearing faults and trend prediction of weak faults which result in gradual aggravation with the bearing vibration intensity as the characteristic parameter. For the weak fault diagnosis, the proposed framework adopts the improved minimum entropy deconvolution (MED) theory to identify the weak fault characteristics of mechanical equipment. From a large number of actual data analysis, once a bearing shows a weak fault, the bearing vibration intensity not only has random non-stationary, but also long-range dependent (LRD) characteristics. Therefore, the stochastic model with LRD−fractional Brown motion (FBM) is proposed to evaluate and predict the condition of slowly varying bearing faults which is a gradual process from weak fault occurrence to severity. For the FBM stochastic model, we mainly implement the derivation and the parameter identification of the FBM model. This is the first study to slowly fault prediction with stochastic model FBM. Experimental results show that the proposed methods can obtain the best performance in incipient fault diagnosis and bearing condition trend prediction. Full article

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

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225 KiB

Open AccessArticle

Simple Harmonic Oscillator Canonical Ensemble Model for Tunneling Radiation of Black Hole

by Jinbo Yang, Tangmei He and Jingyi Zhang

Entropy 2016, 18(11), 415; https://doi.org/10.3390/e18110415 - 23 Nov 2016

Cited by 1 | Viewed by 4810

Abstract

A simple harmonic oscillator canonical ensemble model for Schwarzchild black hole quantum tunneling radiation is proposed in this paper. Firstly, the equivalence between canonical ensemble model and Parikh–Wilczek’s tunneling method is introduced. Then, radiated massless particles are considered as a collection of simple [...] Read more.

A simple harmonic oscillator canonical ensemble model for Schwarzchild black hole quantum tunneling radiation is proposed in this paper. Firstly, the equivalence between canonical ensemble model and Parikh–Wilczek’s tunneling method is introduced. Then, radiated massless particles are considered as a collection of simple harmonic oscillators. Based on this model, we treat the black hole as a heat bath to derive the energy flux of the radiation. Finally, we apply the result to estimate the lifespan of a black hole. Full article

(This article belongs to the Special Issue Black Hole Thermodynamics II)

5296 KiB

Open AccessArticle

Analysis of the Complexity Entropy and Chaos Control of the Bullwhip Effect Considering Price of Evolutionary Game between Two Retailers

by Junhai Ma, Xiaogang Ma and Wandong Lou

Entropy 2016, 18(11), 416; https://doi.org/10.3390/e18110416 - 19 Nov 2016

Cited by 9 | Viewed by 4401

Abstract

In this research, a model is established to represent a supply chain, which consists of one manufacturer and two retailers. The price-sensitive demand model is considered and the price game system is built according to the rule of bounded rationality as well as [...] Read more.

In this research, a model is established to represent a supply chain, which consists of one manufacturer and two retailers. The price-sensitive demand model is considered and the price game system is built according to the rule of bounded rationality as well as the entropy theory. With the increase of the price adjustment speed, the game system may go into chaos from the stable and periodic state. The bullwhip effect and inventory variance ratio of different stages that the system falls in are compared in real time. We also employ the delayed feedback control method to control the system and succeed in mitigating the bullwhip effect of the system. On the whole, the bullwhip effect and inventory variance ratio in the stable state are smaller than those in period-doubling and chaos. In the stable state, there is an optimal price adjustment speed to obtain both the lowest bullwhip effect and inventory variance ratio. Full article

(This article belongs to the Section Complexity)

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3797 KiB

Open AccessArticle

Application of Sample Entropy Based LMD-TFPF De-Noising Algorithm for the Gear Transmission System

by Shaohui Ning, Zhennan Han, Zhijian Wang and Xuefeng Wu

Entropy 2016, 18(11), 414; https://doi.org/10.3390/e18110414 - 18 Nov 2016

Cited by 19 | Viewed by 4317

Abstract

This paper investigates an improved noise reduction method and its application on gearbox vibration signal de-noising. A hybrid de-noising algorithm based on local mean decomposition (LMD), sample entropy (SE), and time-frequency peak filtering (TFPF) is proposed. TFPF is a classical filter method in [...] Read more.

This paper investigates an improved noise reduction method and its application on gearbox vibration signal de-noising. A hybrid de-noising algorithm based on local mean decomposition (LMD), sample entropy (SE), and time-frequency peak filtering (TFPF) is proposed. TFPF is a classical filter method in the time-frequency domain. However, there is a contradiction in TFPF, i.e., a good preservation for signal amplitude, but poor random noise reduction results might be obtained by selecting a short window length, whereas a serious attenuation for signal amplitude, but effective random noise reduction might be obtained by selecting a long window length. In order to make a good tradeoff between valid signal amplitude preservation and random noise reduction, LMD and SE are adopted to improve TFPF. Firstly, the original signal is decomposed into PFs by LMD, and the SE value of each product function (PF) is calculated in order to classify the numerous PFs into the useful component, mixed component, and the noise component; then short-window TFPF is employed for the useful component, long-window TFPF is employed for the mixed component, and the noise component is removed; finally, the final signal is obtained after reconstruction. The gearbox vibration signals are employed to verify the proposed algorithm, and the comparison results show that the proposed SE-LMD-TFPF has the best de-noising results compared to traditional wavelet and TFPF method. Full article

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258 KiB

Open AccessArticle

Existence of Solutions to a Nonlinear Parabolic Equation of Fourth-Order in Variable Exponent Spaces

by Bo Liang, Xiting Peng and Chengyuan Qu

Entropy 2016, 18(11), 413; https://doi.org/10.3390/e18110413 - 18 Nov 2016

Cited by 7 | Viewed by 3915

Abstract

This paper is devoted to studying the existence and uniqueness of weak solutions for an initial boundary problem of a nonlinear fourth-order parabolic equation with variable exponent [...] Read more.

This paper is devoted to studying the existence and uniqueness of weak solutions for an initial boundary problem of a nonlinear fourth-order parabolic equation with variable exponent

v_{t} + {div (| \nabla ▵ v |}^{p (x) - 2} {\nabla ▵ v) - | ▵ v |}^{q (x) - 2} ▵ v = g (x, v)

. By applying Leray-Schauder’s fixed point theorem, the existence of weak solutions of the elliptic problem is given. Furthermore, the semi-discrete method yields the existence of weak solutions of the corresponding parabolic problem by constructing two approximate solutions. Full article

(This article belongs to the Special Issue Entropic Aspects of Nonlinear Partial Differential Equations: Classical and Quantum Mechanical Perspectives)

2844 KiB

Open AccessArticle

Symplectic Entropy as a Novel Measure for Complex Systems

by Min Lei, Guang Meng, Wenming Zhang, Joshua Wade and Nilanjan Sarkar

Entropy 2016, 18(11), 412; https://doi.org/10.3390/e18110412 - 17 Nov 2016

Cited by 15 | Viewed by 4674

Abstract

Real systems are often complex, nonlinear, and noisy in various fields, including mathematics, natural science, and social science. We present the symplectic entropy (SymEn) measure as well as an analysis method based on SymEn to estimate the nonlinearity of a complex system by [...] Read more.

Real systems are often complex, nonlinear, and noisy in various fields, including mathematics, natural science, and social science. We present the symplectic entropy (SymEn) measure as well as an analysis method based on SymEn to estimate the nonlinearity of a complex system by analyzing the given time series. The SymEn estimation is a kind of entropy based on symplectic principal component analysis (SPCA), which represents organized but unpredictable behaviors of systems. The key to SPCA is to preserve the global submanifold geometrical properties of the systems through a symplectic transform in the phase space, which is a kind of measure-preserving transform. The ability to preserve the global geometrical characteristics makes SymEn a test statistic for the detection of the nonlinear characteristics in several typical chaotic time series, and the stochastic characteristic in Gaussian white noise. The results are in agreement with findings in the approximate entropy (ApEn), the sample entropy (SampEn), and the fuzzy entropy (FuzzyEn). Moreover, the SymEn method is also used to analyze the nonlinearities of real signals (including the electroencephalogram (EEG) signals for Autism Spectrum Disorder (ASD) and healthy subjects, and the sound and vibration signals for mechanical systems). The results indicate that the SymEn estimation can be taken as a measure for the description of the nonlinear characteristics in the data collected from natural complex systems. Full article

(This article belongs to the Section Complexity)

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979 KiB

Open AccessArticle

Multivariate Generalized Multiscale Entropy Analysis

by Anne Humeau-Heurtier

Entropy 2016, 18(11), 411; https://doi.org/10.3390/e18110411 - 17 Nov 2016

Cited by 13 | Viewed by 6719

Abstract

Multiscale entropy (MSE) was introduced in the 2000s to quantify systems’ complexity. MSE relies on (i) a coarse-graining procedure to derive a set of time series representing the system dynamics on different time scales; (ii) the computation of the sample entropy for each [...] Read more.

Multiscale entropy (MSE) was introduced in the 2000s to quantify systems’ complexity. MSE relies on (i) a coarse-graining procedure to derive a set of time series representing the system dynamics on different time scales; (ii) the computation of the sample entropy for each coarse-grained time series. A refined composite MSE (rcMSE)—based on the same steps as MSE—also exists. Compared to MSE, rcMSE increases the accuracy of entropy estimation and reduces the probability of inducing undefined entropy for short time series. The multivariate versions of MSE (MMSE) and rcMSE (MrcMSE) have also been introduced. In the coarse-graining step used in MSE, rcMSE, MMSE, and MrcMSE, the mean value is used to derive representations of the original data at different resolutions. A generalization of MSE was recently published, using the computation of different moments in the coarse-graining procedure. However, so far, this generalization only exists for univariate signals. We therefore herein propose an extension of this generalized MSE to multivariate data. The multivariate generalized algorithms of MMSE and MrcMSE presented herein (MGMSE and MGrcMSE, respectively) are first analyzed through the processing of synthetic signals. We reveal that MGrcMSE shows better performance than MGMSE for short multivariate data. We then study the performance of MGrcMSE on two sets of short multivariate electroencephalograms (EEG) available in the public domain. We report that MGrcMSE may show better performance than MrcMSE in distinguishing different types of multivariate EEG data. MGrcMSE could therefore supplement MMSE or MrcMSE in the processing of multivariate datasets. Full article

(This article belongs to the Special Issue Multivariate Entropy Measures and Their Applications)

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369 KiB

Open AccessArticle

Information-Theoretic Analysis of Memoryless Deterministic Systems

by Bernhard C. Geiger and Gernot Kubin

Entropy 2016, 18(11), 410; https://doi.org/10.3390/e18110410 - 17 Nov 2016

Cited by 1 | Viewed by 4099

Abstract

The information loss in deterministic, memoryless systems is investigated by evaluating the conditional entropy of the input random variable given the output random variable. It is shown that for a large class of systems the information loss is finite, even if the input [...] Read more.

The information loss in deterministic, memoryless systems is investigated by evaluating the conditional entropy of the input random variable given the output random variable. It is shown that for a large class of systems the information loss is finite, even if the input has a continuous distribution. For systems with infinite information loss, a relative measure is defined and shown to be related to Rényi information dimension. As deterministic signal processing can only destroy information, it is important to know how this information loss affects the solution of inverse problems. Hence, we connect the probability of perfectly reconstructing the input to the information lost in the system via Fano-type bounds. The theoretical results are illustrated by example systems commonly used in discrete-time, nonlinear signal processing and communications. Full article

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

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1154 KiB

Open AccessArticle

Entropy-Based Experimental Design for Optimal Model Discrimination in the Geosciences

by Wolfgang Nowak and Anneli Guthke

Entropy 2016, 18(11), 409; https://doi.org/10.3390/e18110409 - 17 Nov 2016

Cited by 27 | Viewed by 6272

Abstract

Choosing between competing models lies at the heart of scientific work, and is a frequent motivation for experimentation. Optimal experimental design (OD) methods maximize the benefit of experiments towards a specified goal. We advance and demonstrate an OD approach to maximize the information [...] Read more.

Choosing between competing models lies at the heart of scientific work, and is a frequent motivation for experimentation. Optimal experimental design (OD) methods maximize the benefit of experiments towards a specified goal. We advance and demonstrate an OD approach to maximize the information gained towards model selection. We make use of so-called model choice indicators, which are random variables with an expected value equal to Bayesian model weights. Their uncertainty can be measured with Shannon entropy. Since the experimental data are still random variables in the planning phase of an experiment, we use mutual information (the expected reduction in Shannon entropy) to quantify the information gained from a proposed experimental design. For implementation, we use the Preposterior Data Impact Assessor framework (PreDIA), because it is free of the lower-order approximations of mutual information often found in the geosciences. In comparison to other studies in statistics, our framework is not restricted to sequential design or to discrete-valued data, and it can handle measurement errors. As an application example, we optimize an experiment about the transport of contaminants in clay, featuring the problem of choosing between competing isotherms to describe sorption. We compare the results of optimizing towards maximum model discrimination with an alternative OD approach that minimizes the overall predictive uncertainty under model choice uncertainty. Full article

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

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284 KiB

Open AccessArticle

Geometry Induced by a Generalization of Rényi Divergence

by David C. De Souza, Rui F. Vigelis and Charles C. Cavalcante

Entropy 2016, 18(11), 407; https://doi.org/10.3390/e18110407 - 17 Nov 2016

Cited by 15 | Viewed by 4137

Abstract

In this paper, we propose a generalization of Rényi divergence, and then we investigate its induced geometry. This generalization is given in terms of a φ-function, the same function that is used in the definition of non-parametric φ-families. The properties of [...] Read more.

In this paper, we propose a generalization of Rényi divergence, and then we investigate its induced geometry. This generalization is given in terms of a φ-function, the same function that is used in the definition of non-parametric φ-families. The properties of φ-functions proved to be crucial in the generalization of Rényi divergence. Assuming appropriate conditions, we verify that the generalized Rényi divergence reduces, in a limiting case, to the φ-divergence. In generalized statistical manifold, the φ-divergence induces a pair of dual connections

D^{(- 1)}

and

D^{(1)}

. We show that the family of connections

D^{(α)}

induced by the generalization of Rényi divergence satisfies the relation

D^{(α)} = \frac{1 - α}{2} D^{(- 1)} + \frac{1 + α}{2} D^{(1)}

, with

α \in [- 1, 1]

. Full article

(This article belongs to the Special Issue Differential Geometrical Theory of Statistics)

914 KiB

Open AccessArticle

Global Atmospheric Dynamics Investigated by Using Hilbert Frequency Analysis

by Dario A. Zappalà, Marcelo Barreiro and Cristina Masoller

Entropy 2016, 18(11), 408; https://doi.org/10.3390/e18110408 - 16 Nov 2016

Cited by 6 | Viewed by 4962

Abstract

The Hilbert transform is a well-known tool of time series analysis that has been widely used to investigate oscillatory signals that resemble a noisy periodic oscillation, because it allows instantaneous phase and frequency to be estimated, which in turn uncovers interesting properties of [...] Read more.

The Hilbert transform is a well-known tool of time series analysis that has been widely used to investigate oscillatory signals that resemble a noisy periodic oscillation, because it allows instantaneous phase and frequency to be estimated, which in turn uncovers interesting properties of the underlying process that generates the signal. Here we use this tool to analyze atmospheric data: we consider daily-averaged Surface Air Temperature (SAT) time series recorded over a regular grid of locations covering the Earth’s surface. From each SAT time series, we calculate the instantaneous frequency time series by considering the Hilbert analytic signal. The properties of the obtained frequency data set are investigated by plotting the map of the average frequency and the map of the standard deviation of the frequency fluctuations. The average frequency map reveals well-defined large-scale structures: in the extra-tropics, the average frequency in general corresponds to the expected one-year period of solar forcing, while in the tropics, a different behaviour is found, with particular regions having a faster average frequency. In the standard deviation map, large-scale structures are also found, which tend to be located over regions of strong annual precipitation. Our results demonstrate that Hilbert analysis of SAT time-series uncovers meaningful information, and is therefore a promising tool for the study of other climatological variables. Full article

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

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562 KiB

Open AccessArticle

Thermodynamics of Noncommutative Quantum Kerr Black Holes

by Lenin F. Escamilla-Herrera, Eri A. Mena-Barboza and José Torres-Arenas

Entropy 2016, 18(11), 406; https://doi.org/10.3390/e18110406 - 16 Nov 2016

Cited by 7 | Viewed by 3706

Abstract

The thermodynamic formalism for rotating black holes, characterized by noncommutative and quantum corrections, is constructed. From a fundamental thermodynamic relation, the equations of state and thermodynamic response functions are explicitly given, and the effect of noncommutativity and quantum correction is discussed. It is [...] Read more.

The thermodynamic formalism for rotating black holes, characterized by noncommutative and quantum corrections, is constructed. From a fundamental thermodynamic relation, the equations of state and thermodynamic response functions are explicitly given, and the effect of noncommutativity and quantum correction is discussed. It is shown that the well-known divergence exhibited in specific heat is not removed by any of these corrections. However, regions of thermodynamic stability are affected by noncommutativity, increasing the available states for which some thermodynamic stability conditions are satisfied. Full article

(This article belongs to the Special Issue Black Hole Thermodynamics II)

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787 KiB

Open AccessArticle

Efficient Multi-Label Feature Selection Using Entropy-Based Label Selection

by Jaesung Lee and Dae-Won Kim

Entropy 2016, 18(11), 405; https://doi.org/10.3390/e18110405 - 15 Nov 2016

Cited by 13 | Viewed by 6992

Abstract

Multi-label feature selection is designed to select a subset of features according to their importance to multiple labels. This task can be achieved by ranking the dependencies of features and selecting the features with the highest rankings. In a multi-label feature selection problem, [...] Read more.

Multi-label feature selection is designed to select a subset of features according to their importance to multiple labels. This task can be achieved by ranking the dependencies of features and selecting the features with the highest rankings. In a multi-label feature selection problem, the algorithm may be faced with a dataset containing a large number of labels. Because the computational cost of multi-label feature selection increases according to the number of labels, the algorithm may suffer from a degradation in performance when processing very large datasets. In this study, we propose an efficient multi-label feature selection method based on an information-theoretic label selection strategy. By identifying a subset of labels that significantly influence the importance of features, the proposed method efficiently outputs a feature subset. Experimental results demonstrate that the proposed method can identify a feature subset much faster than conventional multi-label feature selection methods for large multi-label datasets. Full article

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813 KiB

Open AccessArticle

Decision-Making Model under Risk Assessment Based on Entropy

by Xin Dong, Hao Lu, Yuanpu Xia and Ziming Xiong

Entropy 2016, 18(11), 404; https://doi.org/10.3390/e18110404 - 15 Nov 2016

Cited by 26 | Viewed by 6291

Abstract

Decision-making under risk assessment involves dealing with the matter of uncertainty, especially in projects such as tunnel construction. Risk control should include not only measures to reduce the possible consequence of incident, but also exploration measures (information collecting measures) to reduce the uncertainty [...] Read more.

Decision-making under risk assessment involves dealing with the matter of uncertainty, especially in projects such as tunnel construction. Risk control should include not only measures to reduce the possible consequence of incident, but also exploration measures (information collecting measures) to reduce the uncertainty of the incident. The classical risk assessment model in engineering is

R = P \times C

which only takes account of the assessment and decision-making of possible consequences. It cannot provide theoretical guidance for taking exploration measures. The paper presents an advanced methodology to assess the effectiveness of exploration measures in decision-making. The methodology classifies risk into two attributes: hazard (expected value) and uncertainty (entropy). On this basis, a generalized model of decision-making under risk assessment is proposed. This model extends the use of the classical assessment model to a more general case. The reason for taking exploration measures and assessment of such measures’ effectiveness could be explained well by this developed model. This model can also serve as a descriptive model for many risk problems and provide a decision-making basis for a variety of risk types. Moreover, the assessment process and calculation method are applied with some case studies. Full article

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1538 KiB

Open AccessArticle

A Concept Lattice for Semantic Integration of Geo-Ontologies Based on Weight of Inclusion Degree Importance and Information Entropy

by Jia Xiao and Zongyi He

Entropy 2016, 18(11), 399; https://doi.org/10.3390/e18110399 - 15 Nov 2016

Cited by 8 | Viewed by 4752

Abstract

Constructing a merged concept lattice with formal concept analysis (FCA) is an important research direction in the field of integrating multi-source geo-ontologies. Extracting essential geographical properties and reducing the concept lattice are two key points of previous research. A formal integration method is [...] Read more.

Constructing a merged concept lattice with formal concept analysis (FCA) is an important research direction in the field of integrating multi-source geo-ontologies. Extracting essential geographical properties and reducing the concept lattice are two key points of previous research. A formal integration method is proposed to address the challenges in these two areas. We first extract essential properties from multi-source geo-ontologies and use FCA to build a merged formal context. Second, the combined importance weight of each single attribute of the formal context is calculated by introducing the inclusion degree importance from rough set theory and information entropy; then a weighted formal context is built from the merged formal context. Third, a combined weighted concept lattice is established from the weighted formal context with FCA and the importance weight value of every concept is defined as the sum of weight of attributes belonging to the concept’s intent. Finally, semantic granularity of concept is defined by its importance weight; we, then gradually reduce the weighted concept lattice by setting up diminishing threshold of semantic granularity. Additionally, all of those reduced lattices are organized into a regular hierarchy structure based on the threshold of semantic granularity. A workflow is designed to demonstrate this procedure. A case study is conducted to show feasibility and validity of this method and the procedure to integrate multi-source geo-ontologies. Full article

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Open AccessReview

Increase in Complexity and Information through Molecular Evolution

by Peter Schuster

Entropy 2016, 18(11), 397; https://doi.org/10.3390/e18110397 - 14 Nov 2016

Cited by 11 | Viewed by 5976

Abstract

Biological evolution progresses by essentially three different mechanisms: (I) optimization of properties through natural selection in a population of competitors; (II) development of new capabilities through cooperation of competitors caused by catalyzed reproduction; and (III) variation of genetic information through mutation or recombination. [...] Read more.

Biological evolution progresses by essentially three different mechanisms: (I) optimization of properties through natural selection in a population of competitors; (II) development of new capabilities through cooperation of competitors caused by catalyzed reproduction; and (III) variation of genetic information through mutation or recombination. Simplified evolutionary processes combine two out of the three mechanisms: Darwinian evolution combines competition (I) and variation (III) and is represented by the quasispecies model, major transitions involve cooperation (II) of competitors (I), and the third combination, cooperation (II) and variation (III) provides new insights in the role of mutations in evolution. A minimal kinetic model based on simple molecular mechanisms for reproduction, catalyzed reproduction and mutation is introduced, cast into ordinary differential equations (ODEs), and analyzed mathematically in form of its implementation in a flow reactor. Stochastic aspects are investigated through computer simulation of trajectories of the corresponding chemical master equations. The competition-cooperation model, mechanisms (I) and (II), gives rise to selection at low levels of resources and leads to symbiontic cooperation in case the material required is abundant. Accordingly, it provides a kind of minimal system that can undergo a (major) transition. Stochastic effects leading to extinction of the population through self-enhancing oscillations destabilize symbioses of four or more partners. Mutations (III) are not only the basis of change in phenotypic properties but can also prevent extinction provided the mutation rates are sufficiently large. Threshold phenomena are observed for all three combinations: The quasispecies model leads to an error threshold, the competition-cooperation model allows for an identification of a resource-triggered bifurcation with the transition, and for the cooperation-mutation model a kind of stochastic threshold for survival through sufficiently high mutation rates is observed. The evolutionary processes in the model are accompanied by gains in information on the environment of the evolving populations. In order to provide a useful basis for comparison, two forms of information, syntactic or Shannon information and semantic information are introduced here. Both forms of information are defined for simple evolving systems at the molecular level. Selection leads primarily to an increase in semantic information in the sense that higher fitness allows for more efficient exploitation of the environment and provides the basis for more progeny whereas understanding transitions involves characteristic contributions from both Shannon information and semantic information. Full article

(This article belongs to the Special Issue Information and Self-Organization)

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Open AccessArticle

Fractional-Order Identification and Control of Heating Processes with Non-Continuous Materials

by Riccardo Caponetto, Francesca Sapuppo, Vincenzo Tomasello, Guido Maione and Paolo Lino

Entropy 2016, 18(11), 398; https://doi.org/10.3390/e18110398 - 12 Nov 2016

Cited by 12 | Viewed by 3827

Abstract

The paper presents a fractional order model of a heating process and a comparison of fractional and standard PI controllers in its closed loop system. Preliminarily, an enhanced fractional order model for the heating process on non-continuous materials has been identified through a [...] Read more.

The paper presents a fractional order model of a heating process and a comparison of fractional and standard PI controllers in its closed loop system. Preliminarily, an enhanced fractional order model for the heating process on non-continuous materials has been identified through a fitting algorithm on experimental data. Experimentation has been carried out on a finite length beam filled with three non-continuous materials (air, styrofoam, metal buckshots) in order to identify a model in the frequency domain and to obtain a relationship between the fractional order of the heating process and the different materials’ properties. A comparison between the experimental model and the theoretical one has been performed, proving a significant enhancement of the fitting performances. Moreover the obtained modelling results confirm the fractional nature of the heating processes when diffusion occurs in non-continuous composite materials, and they show how the model’s fractional order can be used as a characteristic parameter for non-continuous materials with different composition and structure. Finally, three different kinds of controllers have been applied and compared in order to keep constant the beam temperature constant at a fixed length. Full article

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

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Open AccessArticle

Kernel Density Estimation on the Siegel Space with an Application to Radar Processing

by Emmanuel Chevallier, Thibault Forget, Frédéric Barbaresco and Jesus Angulo

Entropy 2016, 18(11), 396; https://doi.org/10.3390/e18110396 - 11 Nov 2016

Cited by 19 | Viewed by 10502

Abstract

This paper studies probability density estimation on the Siegel space. The Siegel space is a generalization of the hyperbolic space. Its Riemannian metric provides an interesting structure to the Toeplitz block Toeplitz matrices that appear in the covariance estimation of radar signals. The [...] Read more.

This paper studies probability density estimation on the Siegel space. The Siegel space is a generalization of the hyperbolic space. Its Riemannian metric provides an interesting structure to the Toeplitz block Toeplitz matrices that appear in the covariance estimation of radar signals. The main techniques of probability density estimation on Riemannian manifolds are reviewed. For computational reasons, we chose to focus on the kernel density estimation. The main result of the paper is the expression of Pelletier’s kernel density estimator. The computation of the kernels is made possible by the symmetric structure of the Siegel space. The method is applied to density estimation of reflection coefficients from radar observations. Full article

(This article belongs to the Special Issue Differential Geometrical Theory of Statistics)

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312 KiB

Open AccessArticle

Unextendible Mutually Unbiased Bases (after Mandayam, Bandyopadhyay, Grassl and Wootters)

by Koen Thas

Entropy 2016, 18(11), 395; https://doi.org/10.3390/e18110395 - 11 Nov 2016

Cited by 9 | Viewed by 3764

Abstract

We consider questions posed in a recent paper of Mandayam et al. (2014) on the nature of “unextendible mutually unbiased bases.” We describe a conceptual framework to study these questions, using a connection proved by the author in Thas (2009) between the set [...] Read more.

We consider questions posed in a recent paper of Mandayam et al. (2014) on the nature of “unextendible mutually unbiased bases.” We describe a conceptual framework to study these questions, using a connection proved by the author in Thas (2009) between the set of nonidentity generalized Pauli operators on the Hilbert space of N d-level quantum systems, d a prime, and the geometry of non-degenerate alternating bilinear forms of rank N over finite fields

F_{d}

. We then supply alternative and short proofs of results obtained in Mandayam et al. (2014), as well as new general bounds for the problems considered in loc. cit. In this setting, we also solve Conjecture 1 of Mandayam et al. (2014) and speculate on variations of this conjecture. Full article

(This article belongs to the Section Quantum Information)

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454 KiB

Open AccessArticle

Rectification and Non-Gaussian Diffusion in Heterogeneous Media

by Paolo Malgaretti, Ignacio Pagonabarraga and J. Miguel Rubi

Entropy 2016, 18(11), 394; https://doi.org/10.3390/e18110394 - 11 Nov 2016

Cited by 11 | Viewed by 4137

Abstract

We show that when Brownian motion takes place in a heterogeneous medium, the presence of local forces and transport coefficients leads to deviations from a Gaussian probability distribution that make that the ratio between forward and backward probabilities depend on the nature of [...] Read more.

We show that when Brownian motion takes place in a heterogeneous medium, the presence of local forces and transport coefficients leads to deviations from a Gaussian probability distribution that make that the ratio between forward and backward probabilities depend on the nature of the host medium, on local forces, and also on time. We have applied our results to two situations: diffusion in a disordered medium, and diffusion in a confined system. For such scenarios, we have shown that our theoretical predictions are in very good agreement with numerical results. Moreover, we have shown that the deviations from the Gaussian solution lead to the onset of rectification. Our predictions could be used to detect the presence of local forces and to characterize the intrinsic short-scale properties of the host medium—a problem of current interest in the study of micro- and nano-systems. Full article

(This article belongs to the Special Issue Nonequilibrium Phenomena in Confined Systems)

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2517 KiB

Open AccessArticle

Feature Extraction of Ship-Radiated Noise Based on Permutation Entropy of the Intrinsic Mode Function with the Highest Energy

by Yu-Xing Li, Ya-An Li, Zhe Chen and Xiao Chen

Entropy 2016, 18(11), 393; https://doi.org/10.3390/e18110393 - 11 Nov 2016

Cited by 55 | Viewed by 5229

Abstract

In order to solve the problem of feature extraction of underwater acoustic signals in complex ocean environment, a new method for feature extraction from ship-radiated noise is presented based on empirical mode decomposition theory and permutation entropy. It analyzes the separability for permutation [...] Read more.

In order to solve the problem of feature extraction of underwater acoustic signals in complex ocean environment, a new method for feature extraction from ship-radiated noise is presented based on empirical mode decomposition theory and permutation entropy. It analyzes the separability for permutation entropies of the intrinsic mode functions of three types of ship-radiated noise signals, and discusses the permutation entropy of the intrinsic mode function with the highest energy. In this study, ship-radiated noise signals measured from three types of ships are decomposed into a set of intrinsic mode functions with empirical mode decomposition method. Then, the permutation entropies of all intrinsic mode functions are calculated with appropriate parameters. The permutation entropies are obviously different in the intrinsic mode functions with the highest energy, thus, the permutation entropy of the intrinsic mode function with the highest energy is regarded as a new characteristic parameter to extract the feature of ship-radiated noise. After that, the characteristic parameters—namely, the energy difference between high and low frequency, permutation entropy, and multi-scale permutation entropy—are compared with the permutation entropy of the intrinsic mode function with the highest energy. It is discovered that the four characteristic parameters are at the same level for similar ships, however, there are differences in the parameters for different types of ships. The results demonstrate that the permutation entropy of the intrinsic mode function with the highest energy is better in separability as the characteristic parameter than the other three parameters by comparing their fluctuation ranges and the average values of the four characteristic parameters. Hence, the feature of ship-radiated noise can be extracted efficiently with the method. Full article

(This article belongs to the Section Complexity)

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Open AccessArticle

Angular Spectral Density and Information Entropy for Eddy Current Distribution

by Guolong Chen and Weimin Zhang

Entropy 2016, 18(11), 392; https://doi.org/10.3390/e18110392 - 10 Nov 2016

Cited by 8 | Viewed by 4432

Abstract

Here, a new method is proposed to quantitatively evaluate the eddy current distribution induced by different exciting coils of an eddy current probe. Probability of energy allocation of a vector field is modeled via conservation of energy and imitating the wave function in [...] Read more.

Here, a new method is proposed to quantitatively evaluate the eddy current distribution induced by different exciting coils of an eddy current probe. Probability of energy allocation of a vector field is modeled via conservation of energy and imitating the wave function in quantum mechanics. The idea of quantization and the principle of circuit sampling is utilized to discretize the space of the vector field. Then, a method to calculate angular spectral density and Shannon information entropy is proposed. Eddy current induced by three different exciting coils is evaluated with this method, and the specific nature of eddy current testing is discussed. Full article

(This article belongs to the Special Issue Maximum Entropy and Its Application II)

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234 KiB

Open AccessArticle

On Thermodynamics Problems in the Single-Phase-Lagging Heat Conduction Model

by Shu-Nan Li and Bing-Yang Cao

Entropy 2016, 18(11), 391; https://doi.org/10.3390/e18110391 - 09 Nov 2016

Cited by 3 | Viewed by 4070

Abstract

Thermodynamics problems for the single-phase-lagging (SPL) model have not been much studied. In this paper, the violation of the second law of thermodynamics by the SPL model is studied from two perspectives, which are the negative entropy production rate and breaking equilibrium spontaneously. [...] Read more.

Thermodynamics problems for the single-phase-lagging (SPL) model have not been much studied. In this paper, the violation of the second law of thermodynamics by the SPL model is studied from two perspectives, which are the negative entropy production rate and breaking equilibrium spontaneously. The methods for the SPL model to avoid the negative entropy production rate are proposed, which are extended irreversible thermodynamics and the thermal relaxation time. Modifying the entropy production rate positive or zero is not enough to avoid the violation of the second law of thermodynamics for the SPL model, because the SPL model could cause breaking equilibrium spontaneously in some special circumstances. As comparison, it is shown that Fourier’s law and the CV model cannot break equilibrium spontaneously by analyzing mathematical energy integral. Full article

(This article belongs to the Special Issue Advances in Applied Thermodynamics II)

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Open AccessReview

Possibility of Using Entropy Method to Evaluate the Distracting Effect of Mobile Phones on Pedestrians

by Nurul Retno Nurwulan and Bernard C. Jiang

Entropy 2016, 18(11), 390; https://doi.org/10.3390/e18110390 - 04 Nov 2016

Cited by 14 | Viewed by 5142

Abstract

The number of mobile phone users keeps increasing every year and mobile phones have become a primary need for most people. Ordinarily, people are not aware of the risk from a common dual-task study, such as using a mobile phone while walking or [...] Read more.

The number of mobile phone users keeps increasing every year and mobile phones have become a primary need for most people. Ordinarily, people are not aware of the risk from a common dual-task study, such as using a mobile phone while walking or simply standing. This study reviewed the methodology in evaluating the distracting effect of mobile phones on pedestrians. A comprehensive review of literature revealed that the most common method in quantifying pedestrian performance is to evaluate postural task performance. Since using a mobile phone while crossing the road is a type of dual-task study, it would give more clarity to investigate it using entropy methods that have been proven more sensitive than the traditional center of pressure (COP) in discriminating the changes in human balance. The descriptions of commonly used entropy methods were also given in order to give a broad overview of the possibility in applying the methods to further clarify the distracting effect of mobile phones. Full article

(This article belongs to the Special Issue Multivariate Entropy Measures and Their Applications)

7800 KiB

Open AccessArticle

Geometric Theory of Heat from Souriau Lie Groups Thermodynamics and Koszul Hessian Geometry: Applications in Information Geometry for Exponential Families

by Frédéric Barbaresco

Entropy 2016, 18(11), 386; https://doi.org/10.3390/e18110386 - 04 Nov 2016

Cited by 26 | Viewed by 9953

Abstract

We introduce the symplectic structure of information geometry based on Souriau’s Lie group thermodynamics model, with a covariant definition of Gibbs equilibrium via invariances through co-adjoint action of a group on its moment space, defining physical observables like energy, heat, and moment as [...] Read more.

We introduce the symplectic structure of information geometry based on Souriau’s Lie group thermodynamics model, with a covariant definition of Gibbs equilibrium via invariances through co-adjoint action of a group on its moment space, defining physical observables like energy, heat, and moment as pure geometrical objects. Using geometric Planck temperature of Souriau model and symplectic cocycle notion, the Fisher metric is identified as a Souriau geometric heat capacity. The Souriau model is based on affine representation of Lie group and Lie algebra that we compare with Koszul works on G/K homogeneous space and bijective correspondence between the set of G-invariant flat connections on G/K and the set of affine representations of the Lie algebra of G. In the framework of Lie group thermodynamics, an Euler-Poincaré equation is elaborated with respect to thermodynamic variables, and a new variational principal for thermodynamics is built through an invariant Poincaré-Cartan-Souriau integral. The Souriau-Fisher metric is linked to KKS (Kostant–Kirillov–Souriau) 2-form that associates a canonical homogeneous symplectic manifold to the co-adjoint orbits. We apply this model in the framework of information geometry for the action of an affine group for exponential families, and provide some illustrations of use cases for multivariate gaussian densities. Information geometry is presented in the context of the seminal work of Fréchet and his Clairaut-Legendre equation. The Souriau model of statistical physics is validated as compatible with the Balian gauge model of thermodynamics. We recall the precursor work of Casalis on affine group invariance for natural exponential families. Full article

(This article belongs to the Special Issue Differential Geometrical Theory of Statistics)

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6281 KiB

Open AccessArticle

Texture Segmentation Using Laplace Distribution-Based Wavelet-Domain Hidden Markov Tree Models

by Yulong Qiao and Ganchao Zhao

Entropy 2016, 18(11), 384; https://doi.org/10.3390/e18110384 - 04 Nov 2016

Cited by 4 | Viewed by 6052

Abstract

Multiresolution models such as the wavelet-domain hidden Markov tree (HMT) model provide a powerful approach for image modeling and processing because it captures the key features of the wavelet coefficients of real-world data. It is observed that the Laplace distribution is peakier in [...] Read more.

Multiresolution models such as the wavelet-domain hidden Markov tree (HMT) model provide a powerful approach for image modeling and processing because it captures the key features of the wavelet coefficients of real-world data. It is observed that the Laplace distribution is peakier in the center and has heavier tails compared with the Gaussian distribution. Thus we propose a new HMT model based on the two-state, zero-mean Laplace mixture model (LMM), the LMM-HMT, which provides significantly potential for characterizing real-world textures. By using the HMT segmentation framework, we develop LMM-HMT based segmentation methods for image textures and dynamic textures. The experimental results demonstrate the effectiveness of the introduced model and segmentation methods. Full article

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

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278 KiB

Open AccessDiscussion

A Possible Ethical Imperative Based on the Entropy Law

by Mehrdad Massoudi

Entropy 2016, 18(11), 389; https://doi.org/10.3390/e18110389 - 03 Nov 2016

Cited by 2 | Viewed by 7665

Abstract

Lindsay in an article titled, “Entropy consumption and values in physical science,” (Am. Sci. 1959, 47, 678–696) proposed a Thermodynamic Imperative similar to Kant’s Ethical Categorical Imperative. In this paper, after describing the concept of ethical imperative as [...] Read more.

Lindsay in an article titled, “Entropy consumption and values in physical science,” (Am. Sci. 1959, 47, 678–696) proposed a Thermodynamic Imperative similar to Kant’s Ethical Categorical Imperative. In this paper, after describing the concept of ethical imperative as elaborated by Kant, we provide a brief discussion of the role of science and its relationship to the classical thermodynamics and the physical implications of the first and the second laws of thermodynamics. We finally attempt to extend and supplement Lindsay’s Thermodynamic Imperative (TI), by another Imperative suggesting simplicity, conservation, and harmony. Full article

1435 KiB

Open AccessArticle

Entropy Analysis of a Railway Network’s Complexity

by Duarte Valério, António M. Lopes and José A. Tenreiro Machado

Entropy 2016, 18(11), 388; https://doi.org/10.3390/e18110388 - 31 Oct 2016

Cited by 17 | Viewed by 4639

Abstract

Railway networks are among the many physical systems that reveal a fractal structure. This paper studies the Portuguese railway system, and analyzes how it evolved over time, namely what concerns the structure of its different levels, and its distribution over the territory. Different [...] Read more.

Railway networks are among the many physical systems that reveal a fractal structure. This paper studies the Portuguese railway system, and analyzes how it evolved over time, namely what concerns the structure of its different levels, and its distribution over the territory. Different mathematical tools are adopted, such as fractal dimension, entropy and state space portrait. The results are consistent with the historical evolution of the network. Full article

(This article belongs to the Section Complexity)

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Entropy, Volume 18, Issue 11 (November 2016) – 36 articles

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