Entropy

509 KiB

Open AccessReview

by Alan Heavens

Entropy 2016, 18(6), 236; https://doi.org/10.3390/e18060236 - 22 Jun 2016

Cited by 8 | Viewed by 4503

Fisher matrices play an important role in experimental design and in data analysis. Their primary role is to make predictions for the inference of model parameters—both their errors and covariances. In this short review, I outline a number of extensions to the simple [...] Read more.

Fisher matrices play an important role in experimental design and in data analysis. Their primary role is to make predictions for the inference of model parameters—both their errors and covariances. In this short review, I outline a number of extensions to the simple Fisher matrix formalism, covering a number of recent developments in the field. These are: (a) situations where the data (in the form of (

x, y

) pairs) have errors in both x and y; (b) modifications to parameter inference in the presence of systematic errors, or through fixing the values of some model parameters; (c) Derivative Approximation for LIkelihoods (DALI) - higher-order expansions of the likelihood surface, going beyond the Gaussian shape approximation; (d) extensions of the Fisher-like formalism, to treat model selection problems with Bayesian evidence. Full article

(This article belongs to the Special Issue Applications of Fisher Information in Sciences)

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Open AccessArticle

A PUT-Based Approach to Automatically Extracting Quantities and Generating Final Answers for Numerical Attributes

by Yaqing Liu, Lidong Wang, Rong Chen, Yingjie Song and Yalin Cai

Entropy 2016, 18(6), 235; https://doi.org/10.3390/e18060235 - 22 Jun 2016

Cited by 5 | Viewed by 4652

Abstract

Automatically extracting quantities and generating final answers for numerical attributes is very useful in many occasions, including question answering, image processing, human-computer interaction, etc. A common approach is to learn linguistics templates or wrappers and employ some algorithm or model to generate a [...] Read more.

Automatically extracting quantities and generating final answers for numerical attributes is very useful in many occasions, including question answering, image processing, human-computer interaction, etc. A common approach is to learn linguistics templates or wrappers and employ some algorithm or model to generate a final answer. However, building linguistics templates or wrappers is a tough task for builders. In addition, linguistics templates or wrappers are domain-dependent. To make the builder escape from building linguistics templates or wrappers, we propose a new approach to final answer generation based on Predicates-Units Table (PUT), a mini domain-independent knowledge base. It is deserved to point out that, in the following cases, quantities are not represented well. Quantities are absent of units. Quantities are perhaps wrong for a given question. Even if all of them are represented well, their units are perhaps inconsistent. These cases have a strong impact on final answer solving. One thousand nine hundred twenty-six real queries are employed to test the proposed method, and the experimental results show that the average correctness ratio of our approach is 87.1%. Full article

(This article belongs to the Special Issue Machine Learning and Entropy: Discover Unknown Unknowns in Complex Data Sets)

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375 KiB

Open AccessFeature PaperArticle

Constant Slope Maps and the Vere-Jones Classification

by Jozef Bobok and Henk Bruin

Entropy 2016, 18(6), 234; https://doi.org/10.3390/e18060234 - 22 Jun 2016

Cited by 5 | Viewed by 4283

Abstract

We study continuous countably-piecewise monotone interval maps and formulate conditions under which these are conjugate to maps of constant slope, particularly when this slope is given by the topological entropy of the map. We confine our investigation to the Markov case and phrase [...] Read more.

We study continuous countably-piecewise monotone interval maps and formulate conditions under which these are conjugate to maps of constant slope, particularly when this slope is given by the topological entropy of the map. We confine our investigation to the Markov case and phrase our conditions in the terminology of the Vere-Jones classification of infinite matrices. Full article

(This article belongs to the Special Issue Entropic Properties of Dynamical Systems)

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845 KiB

Open AccessArticle

Entropic Measure of Time, and Gas Expansion in Vacuum

by Leonid M. Martyushev and Evgenii V. Shaiapin

Entropy 2016, 18(6), 233; https://doi.org/10.3390/e18060233 - 21 Jun 2016

Cited by 5 | Viewed by 5064

Abstract

The study considers advantages of the introduced measure of time based on the entropy change under irreversible processes (entropy production). Using the example of non-equilibrium expansion of an ideal gas in vacuum, such a measure is introduced. It is shown that, in the [...] Read more.

The study considers advantages of the introduced measure of time based on the entropy change under irreversible processes (entropy production). Using the example of non-equilibrium expansion of an ideal gas in vacuum, such a measure is introduced. It is shown that, in the general case, this measure of time proves to be nonlinearly related to the reference measure assumed uniform by convention. The connection between this result and the results of other authors investigating the measure of time in some biological and cosmological problems is noted. Full article

(This article belongs to the Special Issue Exploring the Second Law of Thermodynamics)

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9468 KiB

Open AccessArticle

3D Buoyancy-Induced Flow and Entropy Generation of Nanofluid-Filled Open Cavities Having Adiabatic Diamond Shaped Obstacles

by Lioua Kolsi, Omid Mahian, Hakan F. Öztop, Walid Aich, Mohamed Naceur Borjini, Nidal Abu-Hamdeh and Habib Ben Aissia

Entropy 2016, 18(6), 232; https://doi.org/10.3390/e18060232 - 21 Jun 2016

Cited by 52 | Viewed by 5736

Abstract

A three dimensional computational solution has been obtained to investigate the natural convection and entropy generation of nanofluid-filled open cavities with an adiabatic diamond shaped obstacle. In the model, the finite volume technique was used to solve the governing equations. Based on the [...] Read more.

A three dimensional computational solution has been obtained to investigate the natural convection and entropy generation of nanofluid-filled open cavities with an adiabatic diamond shaped obstacle. In the model, the finite volume technique was used to solve the governing equations. Based on the configuration, the cavity is heated from the left vertical wall and the diamond shape was chosen as adiabatic. Effects of nanoparticle volume fraction, Rayleigh number (10³ ≤ Ra ≤ 10⁶) and width of diamond shape were studied as governing parameters. It was found that the geometry of the partition is a control parameter for heat and fluid flow inside the open enclosure. Full article

(This article belongs to the Special Issue Entropy in Nanofluids)

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Open AccessArticle

Product Design Time Forecasting by Kernel-Based Regression with Gaussian Distribution Weights

by Zhi-Gen Shang and Hong-Sen Yan

Entropy 2016, 18(6), 231; https://doi.org/10.3390/e18060231 - 21 Jun 2016

Cited by 6 | Viewed by 4623

Abstract

There exist problems of small samples and heteroscedastic noise in design time forecasts. To solve them, a kernel-based regression with Gaussian distribution weights (GDW-KR) is proposed here. GDW-KR maintains a Gaussian distribution over weight vectors for the regression. It is applied to seek [...] Read more.

There exist problems of small samples and heteroscedastic noise in design time forecasts. To solve them, a kernel-based regression with Gaussian distribution weights (GDW-KR) is proposed here. GDW-KR maintains a Gaussian distribution over weight vectors for the regression. It is applied to seek the least informative distribution from those that keep the target value within the confidence interval of the forecast value. GDW-KR inherits the benefits of Gaussian margin machines. By assuming a Gaussian distribution over weight vectors, it could simultaneously offer a point forecast and its confidence interval, thus providing more information about product design time. Our experiments with real examples verify the effectiveness and flexibility of GDW-KR. Full article

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

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2298 KiB

Open AccessArticle

Investigating Aging-Related Changes in the Coordination of Agonist and Antagonist Muscles Using Fuzzy Entropy and Mutual Information

by Wenbo Sun, Jingtao Liang, Yuan Yang, Yuanyu Wu, Tiebin Yan and Rong Song

Entropy 2016, 18(6), 229; https://doi.org/10.3390/e18060229 - 20 Jun 2016

Cited by 13 | Viewed by 10003

Abstract

Aging alters muscular coordination patterns. This study aimed to investigate aging-related changes in the coordination of agonist and antagonist muscles from two aspects, the activities of individual muscles and the inter-muscular coupling. Eighteen young subjects and 10 elderly subjects were recruited to modulate [...] Read more.

Aging alters muscular coordination patterns. This study aimed to investigate aging-related changes in the coordination of agonist and antagonist muscles from two aspects, the activities of individual muscles and the inter-muscular coupling. Eighteen young subjects and 10 elderly subjects were recruited to modulate the agonist muscle activity to track a target during voluntary isometric elbow flexion and extension. Normalized muscle activation and fuzzy entropy (FuzzyEn) were applied to depict the activities of biceps and triceps. Mutual information (MI) was utilized to measure the inter-muscular coupling between biceps and triceps. The agonist activation decreased and the antagonist activation increased significantly during elbow flexion and extension with aging. FuzzyEn values of agonist electromyogram (EMG) were similar between the two age groups. FuzzyEn values of antagonist EMG increased significantly with aging during elbow extension. MI decreased significantly with aging during elbow extension. These results indicated increased antagonist co-activation and decreased inter-muscular coupling with aging during elbow extension, which might result from the reduced reciprocal inhibition and the recruitment of additional cortical-spinal pathways connected to biceps. Based on FuzzyEn and MI, this study provided a comprehensive understanding of the mechanisms underlying the aging-related changes in the coordination of agonist and antagonist muscles. Full article

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

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Open AccessArticle

Optimal Noise Enhanced Signal Detection in a Unified Framework

by Ting Yang, Shujun Liu, Mingchun Tang, Kui Zhang and Xinzheng Zhang

Entropy 2016, 18(6), 213; https://doi.org/10.3390/e18060213 - 17 Jun 2016

Cited by 2 | Viewed by 4452

Abstract

In this paper, a new framework for variable detectors is formulated in order to solve different noise enhanced signal detection optimal problems, where six different disjoint sets of detector and discrete vector pairs are defined according to the two inequality-constraints on detection and [...] Read more.

In this paper, a new framework for variable detectors is formulated in order to solve different noise enhanced signal detection optimal problems, where six different disjoint sets of detector and discrete vector pairs are defined according to the two inequality-constraints on detection and false-alarm probabilities. Then theorems and algorithms constructed based on the new framework are presented to search the optimal noise enhanced solutions to maximize the relative improvements of the detection and the false-alarm probabilities, respectively. Further, the optimal noise enhanced solution of the maximum overall improvement is obtained based on the new framework and the relationship among the three maximums is presented. In addition, the sufficient conditions for improvability or non-improvability under the two certain constraints are given. Finally, numerous examples are presented to illustrate the theoretical results and the proofs of the main theorems are given in the Appendix. Full article

(This article belongs to the Special Issue Statistical Significance and the Logic of Hypothesis Testing)

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Open AccessArticle

On Extensions over Semigroups and Applications

by Wen Huang, Lei Jin and Xiangdong Ye

Entropy 2016, 18(6), 230; https://doi.org/10.3390/e18060230 - 15 Jun 2016

Cited by 3 | Viewed by 4413

Abstract

Applying a theorem according to Rhemtulla and Formanek, we partially solve an open problem raised by Hochman with an affirmative answer. Namely, we show that if G is a countable torsion-free locally nilpotent group that acts by homeomorphisms on X, and [...] Read more.

Applying a theorem according to Rhemtulla and Formanek, we partially solve an open problem raised by Hochman with an affirmative answer. Namely, we show that if G is a countable torsion-free locally nilpotent group that acts by homeomorphisms on X, and

S \subset G

is a subsemigroup not containing the unit of G such that

f \in 〈 1, s f : s \in S 〉

for every

f \in C (X)

, then

(X, G)

has zero topological entropy. Full article

(This article belongs to the Special Issue Entropic Properties of Dynamical Systems)

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Open AccessArticle

Discrete Time Dirac Quantum Walk in 3+1 Dimensions

by Giacomo Mauro D’Ariano, Nicola Mosco, Paolo Perinotti and Alessandro Tosini

Entropy 2016, 18(6), 228; https://doi.org/10.3390/e18060228 - 14 Jun 2016

Cited by 7 | Viewed by 5508

Abstract

In this paper we consider quantum walks whose evolution converges to the Dirac equation in the limit of small wave-vectors. We show exact Fast Fourier implementation of the Dirac quantum walks in one, two, and three space dimensions. The behaviour of particle states—defined [...] Read more.

In this paper we consider quantum walks whose evolution converges to the Dirac equation in the limit of small wave-vectors. We show exact Fast Fourier implementation of the Dirac quantum walks in one, two, and three space dimensions. The behaviour of particle states—defined as states smoothly peaked in some wave-vector eigenstate of the walk—is described by an approximated dispersive differential equation that for small wave-vectors gives the usual Dirac particle and antiparticle kinematics. The accuracy of the approximation is provided in terms of a lower bound on the fidelity between the exactly evolved state and the approximated one. The jittering of the position operator expectation value for states having both a particle and an antiparticle component is analytically derived and observed in the numerical implementations. Full article

(This article belongs to the Special Issue Quantum Information and Communication: From Foundations to Applications)

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Open AccessArticle

Fractional-Order Grey Prediction Method for Non-Equidistant Sequences

by Yue Shen, Bo He and Ping Qin

Entropy 2016, 18(6), 227; https://doi.org/10.3390/e18060227 - 14 Jun 2016

Cited by 14 | Viewed by 6012

Abstract

There are lots of non-equidistant sequences in actual applications due to random sampling, imperfect sensors, event-triggered phenomena, and so on. A new grey prediction method for non-equidistant sequences (r-NGM(1,1)) is proposed based on the basic grey model and the developed fractional-order [...] Read more.

There are lots of non-equidistant sequences in actual applications due to random sampling, imperfect sensors, event-triggered phenomena, and so on. A new grey prediction method for non-equidistant sequences (r-NGM(1,1)) is proposed based on the basic grey model and the developed fractional-order non-equidistant accumulated generating operation (r-NAGO), and the accumulated order is extended from the positive to the negative. The whole r-NAGO deletes the randomness of original sequences in the form of weighted accumulation and improves the exponential law of accumulated sequences. Furthermore, the Levenberg–Marquardt algorithm is used to optimize the fractional order. The optimal r-NGM(1,1) can enhance the predicting performance of the non-equidistant sequences. Results of three practical cases in engineering applications demonstrate that the proposed r-NGM(1,1) provides the significant predicting performance compared with the traditional grey model. Full article

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

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Open AccessArticle

Information and Selforganization: A Unifying Approach and Applications

by Hermann Haken and Juval Portugali

Entropy 2016, 18(6), 197; https://doi.org/10.3390/e18060197 - 14 Jun 2016

Cited by 35 | Viewed by 8837

Abstract

Selforganization is a process by which the interaction between the parts of a complex system gives rise to the spontaneous emergence of patterns, structures or functions. In this interaction the system elements exchange matter, energy and information. We focus our attention on the [...] Read more.

Selforganization is a process by which the interaction between the parts of a complex system gives rise to the spontaneous emergence of patterns, structures or functions. In this interaction the system elements exchange matter, energy and information. We focus our attention on the relations between selforganization and information in general and the way they are linked to cognitive processes in particular. We do so from the analytical and mathematical perspective of the “second foundation of synergetics” and its “synergetic computer” and with reference to several forms of information: Shannon’s information that deals with the quantity of a message irrespective of its meaning, semantic and pragmatic forms of information that deal with the meaning conveyed by messages and information adaptation that refers to the interplay between Shannon’s information and semantic or pragmatic information. We first elucidate the relations between selforganization and information theoretically and mathematically and then by means of specific case studies. Full article

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

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Open AccessArticle

Information-Theoretic-Entropy Based Weight Aggregation Method in Multiple-Attribute Group Decision-Making

by Dayi He, Jiaqiang Xu and Xiaoling Chen

Entropy 2016, 18(6), 171; https://doi.org/10.3390/e18060171 - 14 Jun 2016

Cited by 48 | Viewed by 8154

Abstract

Weight aggregation is the key process to solve a multiple-attribute group decision-making (MAGDM) problem. This paper is trying to propose a possible approach to objectivize subjective information and to aggregate information from attribute values themselves and decision-makers’ judgment. An MAGDM problem without information [...] Read more.

Weight aggregation is the key process to solve a multiple-attribute group decision-making (MAGDM) problem. This paper is trying to propose a possible approach to objectivize subjective information and to aggregate information from attribute values themselves and decision-makers’ judgment. An MAGDM problem without information about decision-makers’ and attributes’ weight is considered. In order to define decision-makers’ subjective preference, their utility function is introduced. The attributes value matrix is converted into a subjective attributes value matrix based on their subjective judgment on attribute values. By utilizing the entropy weighting technique, decision-maker’s subjective weight on attributes and objective weight on attributes are determined individually based on the subjective attributes value matrix and attributes value matrix. Based on the principle of minimum cross-entropy, all decision-makers’ subjective weights are integrated into a single weight vector that is closest to all decision-makers’ judgment without any extra information added. Then, by applying the principle of minimum cross-entropy again, a weight aggregation method is proposed to combine the subjective and objective weight of attributes. Finally, an MAGDM example of project choosing is presented to illustrate the procedure of the proposed method. Full article

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Open AccessArticle

Nano-Crystallization of High-Entropy Amorphous NbTiAlSiW_xN_y Films Prepared by Magnetron Sputtering

by Wenjie Sheng, Xiao Yang, Cong Wang and Yong Zhang

Entropy 2016, 18(6), 226; https://doi.org/10.3390/e18060226 - 13 Jun 2016

Cited by 80 | Viewed by 10824

Abstract

High-entropy amorphous NbTiAlSiW_xN_y films (x = 0 or 1, i.e., NbTiAlSiN_y and NbTiAlSiWN_y) were prepared by magnetron sputtering method in the atmosphere of a mixture of N₂ + Ar (N₂ + Ar = 24 [...] Read more.

High-entropy amorphous NbTiAlSiW_xN_y films (x = 0 or 1, i.e., NbTiAlSiN_y and NbTiAlSiWN_y) were prepared by magnetron sputtering method in the atmosphere of a mixture of N₂ + Ar (N₂ + Ar = 24 standard cubic centimeter per minute (sccm)), where N₂ = 0, 4, and 8 sccm). All the as-deposited films present amorphous structures, which remain stable at 700 °C for over 24 h. After heat treatment at 1000 °C the films began to crystalize, and while the NbTiAlSiN_y films (N₂ = 4, 8 sccm) exhibit a face-centered cubic (FCC) structure, the NbTiAlSiW metallic films show a body-centered cubic (BCC) structure and then transit into a FCC structure composed of nanoscaled particles with increasing nitrogen flow rate. The hardness and modulus of the as-deposited NbTiAlSiN_y films reach maximum values of 20.5 GPa and 206.8 GPa, respectively. For the as-deposited NbTiAlSiWN_y films, both modulus and hardness increased to maximum values of 13.6 GPa and 154.4 GPa, respectively, and then decrease as the N₂ flow rate is increased. Both films could be potential candidates for protective coatings at high temperature. Full article

(This article belongs to the Special Issue High-Entropy Alloys and High-Entropy-Related Materials)

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Open AccessArticle

Extreme Learning Machine for Multi-Label Classification

by Xia Sun, Jingting Xu, Changmeng Jiang, Jun Feng, Su-Shing Chen and Feijuan He

Entropy 2016, 18(6), 225; https://doi.org/10.3390/e18060225 - 8 Jun 2016

Cited by 31 | Viewed by 9041

Abstract

Extreme learning machine (ELM) techniques have received considerable attention in the computational intelligence and machine learning communities because of the significantly low computational time required for training new classifiers. ELM provides solutions for regression, clustering, binary classification, multiclass classifications and so on, but [...] Read more.

Extreme learning machine (ELM) techniques have received considerable attention in the computational intelligence and machine learning communities because of the significantly low computational time required for training new classifiers. ELM provides solutions for regression, clustering, binary classification, multiclass classifications and so on, but not for multi-label learning. Multi-label learning deals with objects having multiple labels simultaneously, which widely exist in real-world applications. Therefore, a thresholding method-based ELM is proposed in this paper to adapt ELM to multi-label classification, called extreme learning machine for multi-label classification (ELM-ML). ELM-ML outperforms other multi-label classification methods in several standard data sets in most cases, especially for applications which only have a small labeled data set. Full article

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Open AccessArticle

Entropy Generation on MHD Eyring–Powell Nanofluid through a Permeable Stretching Surface

by Muhammad Mubashir Bhatti, Tehseen Abbas, Mohammad Mehdi Rashidi, Mohamed El-Sayed Ali and Zhigang Yang

Entropy 2016, 18(6), 224; https://doi.org/10.3390/e18060224 - 8 Jun 2016

Cited by 135 | Viewed by 8240

Abstract

In this article, entropy generation of an Eyring–Powell nanofluid through a permeable stretching surface has been investigated. The impact of magnetohydrodynamics (MHD) and nonlinear thermal radiation are also taken into account. The governing flow problem is modeled with the help of similarity transformation [...] Read more.

In this article, entropy generation of an Eyring–Powell nanofluid through a permeable stretching surface has been investigated. The impact of magnetohydrodynamics (MHD) and nonlinear thermal radiation are also taken into account. The governing flow problem is modeled with the help of similarity transformation variables. The resulting nonlinear ordinary differential equations are solved numerically with the combination of the Successive linearization method and Chebyshev spectral collocation method. The impact of all the emerging parameters such as Hartmann number, Prandtl number, radiation parameter, Lewis number, thermophoresis parameter, Brownian motion parameter, Reynolds number, fluid parameter, and Brinkmann number are discussed with the help of graphs and tables. It is observed that the influence of the magnetic field opposes the flow. Moreover, entropy generation profile behaves as an increasing function of all the physical parameters. Full article

(This article belongs to the Special Issue Entropy in Nanofluids)

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Open AccessArticle

Entropy Generation on Nanofluid Flow through a Horizontal Riga Plate

by Tehseen Abbas, Muhammad Ayub, Muhammad Mubashir Bhatti, Mohammad Mehdi Rashidi and Mohamed El-Sayed Ali

Entropy 2016, 18(6), 223; https://doi.org/10.3390/e18060223 - 8 Jun 2016

Cited by 79 | Viewed by 7121

Abstract

In this article, entropy generation on viscous nanofluid through a horizontal Riga plate has been examined. The present flow problem consists of continuity, linear momentum, thermal energy, and nanoparticle concentration equation which are simplified with the help of Oberbeck-Boussinesq approximation. The resulting highly [...] Read more.

In this article, entropy generation on viscous nanofluid through a horizontal Riga plate has been examined. The present flow problem consists of continuity, linear momentum, thermal energy, and nanoparticle concentration equation which are simplified with the help of Oberbeck-Boussinesq approximation. The resulting highly nonlinear coupled partial differential equations are solved numerically by means of the shooting method (SM). The expression of local Nusselt number and local Sherwood number are also taken into account and discussed with the help of table. The physical influence of all the emerging parameters such as Brownian motion parameter, thermophoresis parameter, Brinkmann number, Richardson number, nanoparticle flux parameter, Lewis number and suction parameter are demonstrated graphically. In particular, we conferred their influence on velocity profile, temperature profile, nanoparticle concentration profile and Entropy profile. Full article

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

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Open AccessArticle

Stimuli-Magnitude-Adaptive Sample Selection for Data-Driven Haptic Modeling

by Arsen Abdulali, Waseem Hassan and Seokhee Jeon

Entropy 2016, 18(6), 222; https://doi.org/10.3390/e18060222 - 7 Jun 2016

Cited by 9 | Viewed by 5666

Abstract

Data-driven haptic modeling is an emerging technique where contact dynamics are simulated and interpolated based on a generic input-output matching model identified by data sensed from interaction with target physical objects. In data-driven modeling, selecting representative samples from a large set of data [...] Read more.

Data-driven haptic modeling is an emerging technique where contact dynamics are simulated and interpolated based on a generic input-output matching model identified by data sensed from interaction with target physical objects. In data-driven modeling, selecting representative samples from a large set of data in a way that they can efficiently and accurately describe the whole dataset has been a long standing problem. This paper presents a new algorithm for the sample selection where the variances of output are observed for selecting representative input-output samples in order to ensure the quality of output prediction. The main idea is that representative pairs of input-output are chosen so that the ratio of the standard deviation to the mean of the corresponding output group does not exceed an application-dependent threshold. This output- and standard deviation-based sample selection is very effective in applications where the variance or relative error of the output should be kept within a certain threshold. This threshold is used for partitioning the input space using Binary Space Partitioning-tree (BSP-tree) and k-means algorithms. We apply the new approach to data-driven haptic modeling scenario where the relative error of the output prediction result should be less than a perceptual threshold. For evaluation, the proposed algorithm is compared to two state-of-the-art sample selection algorithms for regression tasks. Four kinds of haptic related behavior–force datasets are tested. The results showed that the proposed algorithm outperformed the others in terms of output-approximation quality and computational complexity. Full article

(This article belongs to the Special Issue Machine Learning and Entropy: Discover Unknown Unknowns in Complex Data Sets)

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Open AccessFeature PaperArticle

Zero Entropy Is Generic

by Lewis Bowen

Entropy 2016, 18(6), 220; https://doi.org/10.3390/e18060220 - 4 Jun 2016

Cited by 7 | Viewed by 5207

Abstract

Dan Rudolph showed that for an amenable group, Γ, the generic measure-preserving action of Γ on a Lebesgue space has zero entropy. Here, this is extended to nonamenable groups. In fact, the proof shows that every action is a factor of a zero [...] Read more.

Dan Rudolph showed that for an amenable group, Γ, the generic measure-preserving action of Γ on a Lebesgue space has zero entropy. Here, this is extended to nonamenable groups. In fact, the proof shows that every action is a factor of a zero entropy action! This uses the strange phenomena that in the presence of nonamenability, entropy can increase under a factor map. The proof uses Seward’s recent generalization of Sinai’s Factor Theorem, the Gaboriau–Lyons result and my theorem that for every nonabelian free group, all Bernoulli shifts factor onto each other. Full article

(This article belongs to the Special Issue Entropic Properties of Dynamical Systems)

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Open AccessArticle

Application of Entropy-Based Metrics to Identify Emotional Distress from Electroencephalographic Recordings

by Beatriz García-Martínez, Arturo Martínez-Rodrigo, Roberto Zangróniz Cantabrana, Jose Manuel Pastor García and Raúl Alcaraz

Entropy 2016, 18(6), 221; https://doi.org/10.3390/e18060221 - 3 Jun 2016

Cited by 60 | Viewed by 8113

Abstract

Recognition of emotions is still an unresolved challenge, which could be helpful to improve current human-machine interfaces. Recently, nonlinear analysis of some physiological signals has shown to play a more relevant role in this context than their traditional linear exploration. Thus, the present [...] Read more.

Recognition of emotions is still an unresolved challenge, which could be helpful to improve current human-machine interfaces. Recently, nonlinear analysis of some physiological signals has shown to play a more relevant role in this context than their traditional linear exploration. Thus, the present work introduces for the first time the application of three recent entropy-based metrics: sample entropy (SE), quadratic SE (QSE) and distribution entropy (DE) to discern between emotional states of calm and negative stress (also called distress). In the last few years, distress has received growing attention because it is a common negative factor in the modern lifestyle of people from developed countries and, moreover, it may lead to serious mental and physical health problems. Precisely, 279 segments of 32-channel electroencephalographic (EEG) recordings from 32 subjects elicited to be calm or negatively stressed have been analyzed. Results provide that QSE is the first single metric presented to date with the ability to identify negative stress. Indeed, this metric has reported a discriminant ability of around 70%, which is only slightly lower than the one obtained by some previous works. Nonetheless, discriminant models from dozens or even hundreds of features have been previously obtained by using advanced classifiers to yield diagnostic accuracies about 80%. Moreover, in agreement with previous neuroanatomy findings, QSE has also revealed notable differences for all the brain regions in the neural activation triggered by the two considered emotions. Consequently, given these results, as well as easy interpretation of QSE, this work opens a new standpoint in the detection of emotional distress, which may gain new insights about the brain’s behavior under this negative emotion. Full article

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

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Open AccessArticle

Single Neuron Stochastic Predictive PID Control Algorithm for Nonlinear and Non-Gaussian Systems Using the Survival Information Potential Criterion

by Mifeng Ren, Ting Cheng, Junghui Chen, Xinying Xu and Lan Cheng

Entropy 2016, 18(6), 218; https://doi.org/10.3390/e18060218 - 3 Jun 2016

Cited by 9 | Viewed by 4782

Abstract

This paper presents a novel stochastic predictive tracking control strategy for nonlinear and non-Gaussian stochastic systems based on the single neuron controller structure in the framework of information theory. Firstly, in order to characterize the randomness of the control system, survival information potential [...] Read more.

This paper presents a novel stochastic predictive tracking control strategy for nonlinear and non-Gaussian stochastic systems based on the single neuron controller structure in the framework of information theory. Firstly, in order to characterize the randomness of the control system, survival information potential (SIP), instead of entropy, is adopted to formulate the performance index, which is not shift-invariant, i.e., its value varies with the change of the distribution location. Then, the optimal weights of the single neuron controller can be obtained by minimizing the presented SIP based predictive control criterion. Furthermore, mean-square convergence of the proposed control algorithm is also analyzed from the energy conservation perspective. Finally, a numerical example is given to show the effectiveness of the proposed method. Full article

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

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Open AccessCorrection

Correction: Wolpert, D.H. The Free Energy Requirements of Biological Organisms; Implications for Evolution. Entropy 2016, 18, 138

by David H. Wolpert

Entropy 2016, 18(6), 219; https://doi.org/10.3390/e18060219 - 2 Jun 2016

Cited by 3 | Viewed by 4040

Abstract

The following corrections should be made to the published paper [1]: [...] Full article

(This article belongs to the Special Issue Information and Entropy in Biological Systems)

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Open AccessArticle

Empirical Laws and Foreseeing the Future of Technological Progress

by António M. Lopes, José A. Tenreiro Machado and Alexandra M. Galhano

Entropy 2016, 18(6), 217; https://doi.org/10.3390/e18060217 - 2 Jun 2016

Cited by 8 | Viewed by 6309

Abstract

The Moore’s law (ML) is one of many empirical expressions that is used to characterize natural and artificial phenomena. The ML addresses technological progress and is expected to predict future trends. Yet, the “art” of predicting is often confused with the accurate fitting [...] Read more.

The Moore’s law (ML) is one of many empirical expressions that is used to characterize natural and artificial phenomena. The ML addresses technological progress and is expected to predict future trends. Yet, the “art” of predicting is often confused with the accurate fitting of trendlines to past events. Presently, data-series of multiple sources are available for scientific and computational processing. The data can be described by means of mathematical expressions that, in some cases, follow simple expressions and empirical laws. However, the extrapolation toward the future is considered with skepticism by the scientific community, particularly in the case of phenomena involving complex behavior. This paper addresses these issues in the light of entropy and pseudo-state space. The statistical and dynamical techniques lead to a more assertive perspective on the adoption of a given candidate law. Full article

(This article belongs to the Special Issue Computational Complexity)

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294 KiB

Open AccessArticle

On Two-Distillable Werner States

by Dragomir Ž. Đoković

Entropy 2016, 18(6), 216; https://doi.org/10.3390/e18060216 - 2 Jun 2016

Cited by 9 | Viewed by 4022

Abstract

We consider bipartite mixed states ρ in a

d \otimes d

quantum system. We say that ρ is PPT if its partial transpose

1 \otimes T (ρ)

is positive semidefinite, and otherwise ρ is NPT. The well-known Werner states are divided [...] Read more.

We consider bipartite mixed states ρ in a

d \otimes d

quantum system. We say that ρ is PPT if its partial transpose

1 \otimes T (ρ)

is positive semidefinite, and otherwise ρ is NPT. The well-known Werner states are divided into three types: (a) the separable states (the same as the PPT states); (b) the one-distillable states (necessarily NPT); and (c) the NPT states which are not one-distillable. We give several different formulations and provide further evidence for the validity of the conjecture that Werner states of type (c) are not two-distillable. Full article

(This article belongs to the Special Issue Quantum Information 2016)

346 KiB

Open AccessArticle

General Bulk-Viscous Solutions and Estimates of Bulk Viscosity in the Cosmic Fluid

by Ben David Normann and Iver Brevik

Entropy 2016, 18(6), 215; https://doi.org/10.3390/e18060215 - 2 Jun 2016

Cited by 41 | Viewed by 4566

Abstract

We derive a general formalism for bulk viscous solutions of the energy-conservation equation for

ρ (a, ζ)

, both for a single-component and a multicomponent fluid in the Friedmann universe. For our purposes, these general solutions become valuable in estimating [...] Read more.

We derive a general formalism for bulk viscous solutions of the energy-conservation equation for

ρ (a, ζ)

, both for a single-component and a multicomponent fluid in the Friedmann universe. For our purposes, these general solutions become valuable in estimating the order of magnitude of the phenomenological viscosity in the cosmic fluid at present.

H (z)

observations are found to put an upper limit on the magnitude of the modulus of the present-day bulk viscosity. It is found to be

ζ_{0} \sim 10^{6}

Pa·s, in agreement with previous works. We point out that this magnitude is acceptable from a hydrodynamic point of view. Finally, we bring new insight by using our estimates of ζ to analyze the fate of the future universe. Of special interest is the case

ζ \propto \sqrt{ρ}

for which the fluid, originally situated in the quintessence region, may slide through the phantom barrier and inevitably be driven into a big rip. Typical rip times are found to be a few hundred Gy. Full article

(This article belongs to the Section Astrophysics, Cosmology, and Black Holes)

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3508 KiB

Open AccessArticle

Experimental Study of Single Phase Flow in a Closed-Loop Cooling System with Integrated Mini-Channel Heat Sink

by Lei Ma, Xuxin Zhao, Hongyuan Sun, Qixing Wu and Wei Liu

Entropy 2016, 18(6), 128; https://doi.org/10.3390/e18060128 - 2 Jun 2016

Cited by 13 | Viewed by 6271

Abstract

The flow and heat transfer characteristics of a closed-loop cooling system with a mini-channel heat sink for thermal management of electronics is studied experimentally. The heat sink is designed with corrugated fins to improve its heat dissipation capability. The experiments are performed using [...] Read more.

The flow and heat transfer characteristics of a closed-loop cooling system with a mini-channel heat sink for thermal management of electronics is studied experimentally. The heat sink is designed with corrugated fins to improve its heat dissipation capability. The experiments are performed using variable coolant volumetric flow rates and input heating powers. The experimental results show a high and reliable thermal performance using the heat sink with corrugated fins. The heat transfer capability is improved up to 30 W/cm² when the base temperature is kept at a stable and acceptable level. Besides the heat transfer capability enhancement, the capability of the system to transfer heat for a long distance is also studied and a fast thermal response time to reach steady state is observed once the input heating power or the volume flow rate are varied. Under different input heat source powers and volumetric flow rates, our results suggest potential applications of the designed mini-channel heat sink in cooling microelectronics. Full article

(This article belongs to the Section Thermodynamics)

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2283 KiB

Open AccessArticle

Harmonic Source Localization Approach Based on Fast Kernel Entropy Optimization ICA and Minimum Conditional Entropy

by Tianlei Zang, Zhengyou He, Ling Fu, Jing Chen and Qingquan Qian

Entropy 2016, 18(6), 214; https://doi.org/10.3390/e18060214 - 1 Jun 2016

Cited by 10 | Viewed by 5347

Abstract

Based on the fast kernel entropy optimization independent component analysis and the minimum conditional entropy, this paper proposes a harmonic source localization method which aims at accurately estimating harmonic currents and identifying harmonic sources. The injected harmonic currents are estimated by the fast [...] Read more.

Based on the fast kernel entropy optimization independent component analysis and the minimum conditional entropy, this paper proposes a harmonic source localization method which aims at accurately estimating harmonic currents and identifying harmonic sources. The injected harmonic currents are estimated by the fast kernel entropy optimization independent component analysis (FKEO-ICA) in the absence of prior knowledge of harmonic impedances. Then, the minimum conditional entropy is applied to locate the harmonic sources based on the estimated harmonic currents. The proposed harmonic source localization method is validated on the IEEE 34-bus system. By applying the correlation coefficient and three error evaluation indicators, comparison has been made among the performances of the FKEO-ICA and three other ICA algorithms. The results show that the FKEO-ICA algorithm could achieve a significantly better accuracy of harmonic current estimation, while the minimum conditional entropy could determine the locations of harmonic sources precisely. Full article

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267 KiB

Open AccessArticle

Extended First Law for Entanglement Entropy in Lovelock Gravity

by David Kastor, Sourya Ray and Jennie Traschen

Entropy 2016, 18(6), 212; https://doi.org/10.3390/e18060212 - 30 May 2016

Cited by 13 | Viewed by 4178

Abstract

The first law for the holographic entanglement entropy of spheres in a boundary CFT (Conformal Field Theory) with a bulk Lovelock dual is extended to include variations of the bulk Lovelock coupling constants. Such variations in the bulk correspond to perturbations within a [...] Read more.

The first law for the holographic entanglement entropy of spheres in a boundary CFT (Conformal Field Theory) with a bulk Lovelock dual is extended to include variations of the bulk Lovelock coupling constants. Such variations in the bulk correspond to perturbations within a family of boundary CFTs. The new contribution to the first law is found to be the product of the variation

δ a

of the “A”-type trace anomaly coefficient for even dimensional CFTs, or more generally its extension

δ a^{*}

to include odd dimensional boundaries, times the ratio

S / a^{*}

. Since

a^{*}

is a measure of the number of degrees of freedom N per unit volume of the boundary CFT, this new term has the form

μ δ N

, where the chemical potential μ is given by the entanglement entropy per degree of freedom. Full article

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

885 KiB

Open AccessReview

A Confidence Set Analysis for Observed Samples: A Fuzzy Set Approach

by José Alejandro González, Luis Mauricio Castro, Víctor Hugo Lachos and Alexandre Galvão Patriota

Entropy 2016, 18(6), 211; https://doi.org/10.3390/e18060211 - 30 May 2016

Cited by 4 | Viewed by 6810

Abstract

Confidence sets are generally interpreted in terms of replications of an experiment. However, this interpretation is only valid before observing the sample. After observing the sample, any confidence sets have probability zero or one to contain the parameter value. In this paper, we [...] Read more.

Confidence sets are generally interpreted in terms of replications of an experiment. However, this interpretation is only valid before observing the sample. After observing the sample, any confidence sets have probability zero or one to contain the parameter value. In this paper, we provide a confidence set analysis for an observed sample based on fuzzy set theory by using the concept of membership functions. We show that the traditional ad hoc thresholds (the confidence and significance levels) can be attained from a general membership function. The applicability of the newly proposed theory is demonstrated by using well-known examples from the statistical literature and an application in the context of contingency tables. Full article

(This article belongs to the Special Issue Statistical Significance and the Logic of Hypothesis Testing)

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337 KiB

Open AccessArticle

Unified Quantum Model of Work Generation in Thermoelectric Generators, Solar and Fuel Cells

by Robert Alicki

Entropy 2016, 18(6), 210; https://doi.org/10.3390/e18060210 - 28 May 2016

Cited by 12 | Viewed by 5912

Abstract

In the previous papers, the idea of “hidden oscillations” has been applied to explain work generation in semiconductor photovoltaic cells and thermoelectric generators. The aim of this paper is firstly to extend this approach to fuel cells and, secondly, to create a unified [...] Read more.

In the previous papers, the idea of “hidden oscillations” has been applied to explain work generation in semiconductor photovoltaic cells and thermoelectric generators. The aim of this paper is firstly to extend this approach to fuel cells and, secondly, to create a unified quantum model for all types of such devices. They are treated as electron pumps powered by heat or chemical engines. The working fluid is electron gas and the necessary oscillating element (“piston”) is provided by plasma oscillation. Those oscillations are localized around the junction that also serves as a diode rectifying fast electric charge oscillations and yielding a final output direct current (DC). The dynamics of the devices are governed by the Markovian master equations that can be derived in a rigorous way from the underlying Hamiltonian models and are consistent with the laws of thermodynamics. The new ingredient is the derivation of master equations for systems driven by chemical reactions. Full article

(This article belongs to the Special Issue Quantum Thermodynamics)

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