Nothing Special   »   [go: up one dir, main page]
Next Issue
Volume 18, April
Previous Issue
Volume 18, February
You seem to have javascript disabled. Please note that many of the page functionalities won't work as expected without javascript enabled.
 
 
Search for Articles:
Title / Keyword
Author / Affiliation / Email
Journal
Article Type
 
 
Advanced Search
 
Section
Special Issue
Volume
Issue
Number
Page
 
4.9
2.1
Journals
Entropy
Volume 18
Issue 3
entropy-logo

Journal Menu

► Journal Menu

Journal Browser

► Journal Browser
share announcement

Entropy, Volume 18, Issue 3 (March 2016) – 38 articles

  • Issues are regarded as officially published after their release is announced to the table of contents alert mailing list.
  • You may sign up for e-mail alerts to receive table of contents of newly released issues.
  • PDF is the official format for papers published in both, html and pdf forms. To view the papers in pdf format, click on the "PDF Full-text" link, and use the free Adobe Reader to open them.
Order results
Result details
Section
Show export options expand_more Show export options expand_less
Select all
Export citation of selected articles as:
1271 KiB  
Article
System Entropy Measurement of Stochastic Partial Differential Systems
by Bor-Sen Chen, Chao-Yi Hsieh and Shih-Ju Ho
Entropy 2016, 18(3), 99; https://doi.org/10.3390/e18030099 - 18 Mar 2016
Cited by 3 | Viewed by 4955
Abstract
System entropy describes the dispersal of a system’s energy and is an indication of the disorder of a physical system. Several system entropy measurement methods have been developed for dynamic systems. However, most real physical systems are always modeled using stochastic partial differential [...] Read more.
System entropy describes the dispersal of a system’s energy and is an indication of the disorder of a physical system. Several system entropy measurement methods have been developed for dynamic systems. However, most real physical systems are always modeled using stochastic partial differential dynamic equations in the spatio-temporal domain. No efficient method currently exists that can calculate the system entropy of stochastic partial differential systems (SPDSs) in consideration of the effects of intrinsic random fluctuation and compartment diffusion. In this study, a novel indirect measurement method is proposed for calculating of system entropy of SPDSs using a Hamilton–Jacobi integral inequality (HJII)-constrained optimization method. In other words, we solve a nonlinear HJII-constrained optimization problem for measuring the system entropy of nonlinear stochastic partial differential systems (NSPDSs). To simplify the system entropy measurement of NSPDSs, the global linearization technique and finite difference scheme were employed to approximate the nonlinear stochastic spatial state space system. This allows the nonlinear HJII-constrained optimization problem for the system entropy measurement to be transformed to an equivalent linear matrix inequalities (LMIs)-constrained optimization problem, which can be easily solved using the MATLAB LMI-toolbox (MATLAB R2014a, version 8.3). Finally, several examples are presented to illustrate the system entropy measurement of SPDSs. Full article
►▼ Show Figures

Figure 1

Figure 1
<p>Finite difference grids on the spatio-domain <math display="inline"> <semantics> <mi>U</mi> </semantics> </math>.</p> Full article ">Figure 2
<p>The temperature distribution <math display="inline"> <semantics> <mrow> <mi>y</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </semantics> </math> of the heat transfer system given in Equation (71) at <math display="inline"> <semantics> <mrow> <mi>t</mi> <mo>=</mo> </mrow> </semantics> </math> 1, 10, 30 and 50 s. Due to the diffusion term <math display="inline"> <semantics> <mrow> <mi>κ</mi> <msup> <mo>∇</mo> <mn>2</mn> </msup> <mi>y</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </semantics> </math>, the temperature of heat system will be uniformly distributed gradually to increase the system entropy.</p> Full article ">Figure 3
<p>The temperature distribution <math display="inline"> <semantics> <mrow> <mi>y</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </semantics> </math> of heat transfer system in Equation (72) at <math display="inline"> <semantics> <mrow> <mi>t</mi> <mo>=</mo> </mrow> </semantics> </math> 1, 10, 30 and 50 s. Obviously, the temperature distribution of stochastic heat transfer system in Equation (72) is with more random fluctuations and with more system entropy than the heat transfer system in Equation (71). The temperature distribution is also uniformly distributed gradually to increase the system entropy as time goes on. In general, the temperature in <a href="#entropy-18-00099-f003" class="html-fig">Figure 3</a> is more random than <a href="#entropy-18-00099-f002" class="html-fig">Figure 2</a>, <span class="html-italic">i.e.</span>, with more system randomness and entropy.</p> Full article ">Figure 4
<p>(<b>a</b>) Spatial-time profiles of the real biochemical system in Equation (73); (<b>b</b>) Spatial-time profiles of the approximated system in Equation (60) based on the finite difference scheme and global linearization technique; (<b>c</b>) The error between the real biochemical system in Equation (73) and the approximated system in Equation (60). Obviously, the approximated system based on finite difference scheme and global linearization method can approximate the biochemical enzyme system quite well.</p> Full article ">Figure 5
<p>(<b>a</b>) Spatial-time profiles of the real biochemical system in Equation (74); (<b>b</b>) Spatial-time profiles of the approximated system in Equation (66) based on the finite difference scheme and global linearization technique; (<b>c</b>) The error between the real biochemical system in Equation (74) and the approximated system in Equation (66). Obviously, the approximated system in Equation (66) could approximate the real system in Equation (74) quite well.</p> Full article ">
611 KiB  
Article
Assessment of Nociceptive Responsiveness Levels during Sedation-Analgesia by Entropy Analysis of EEG
by José F. Valencia, Umberto S. P. Melia, Montserrat Vallverdú, Xavier Borrat, Mathieu Jospin, Erik W. Jensen, Alberto Porta, Pedro L. Gambús and Pere Caminal
Entropy 2016, 18(3), 103; https://doi.org/10.3390/e18030103 - 18 Mar 2016
Cited by 10 | Viewed by 6187
Abstract
The level of sedation in patients undergoing medical procedures is decided to assure unconsciousness and prevent pain. The monitors of depth of anesthesia, based on the analysis of the electroencephalogram (EEG), have been progressively introduced into the daily practice to provide additional information [...] Read more.
The level of sedation in patients undergoing medical procedures is decided to assure unconsciousness and prevent pain. The monitors of depth of anesthesia, based on the analysis of the electroencephalogram (EEG), have been progressively introduced into the daily practice to provide additional information about the state of the patient. However, the quantification of analgesia still remains an open problem. The purpose of this work was to analyze the capability of prediction of nociceptive responses based on refined multiscale entropy (RMSE) and auto mutual information function (AMIF) applied to EEG signals recorded in 378 patients scheduled to undergo ultrasonographic endoscopy under sedation-analgesia. Two observed categorical responses after the application of painful stimulation were analyzed: the evaluation of the Ramsay Sedation Scale (RSS) after nail bed compression and the presence of gag reflex (GAG) during endoscopy tube insertion. In addition, bispectrum (BIS), heart rate (HR), predicted concentrations of propofol (CeProp) and remifentanil (CeRemi) were annotated with a resolution of 1 s. Results showed that functions based on RMSE, AMIF, HR and CeRemi permitted predicting different stimulation responses during sedation better than BIS. Full article
(This article belongs to the Special Issue Recent Advances in Information Theory Application to Physiological Signals)
►▼ Show Figures

Figure 1

Figure 1
<p>AMIF computed from an EEG trace and parameters extracted from it: <span class="html-italic">max</span> (first relative maximum) and <span class="html-italic">FD</span> (decay for τ = 1).</p> Full article ">Figure 2
<p>RMSE course (mean ± standard error) obtained from the EEG segments for different time scale factor <span class="html-italic">ts</span> in: responsive states 2 ≤ RSS ≤ 5 (blue line with square marker) and RSS = 5 (black line with cross marker); and unresponsive state RSS = 6 (red line with circle marker). Significant statistical differences with <span class="html-italic">p</span>-value &lt; 0.05 are marked with the symbols “+” for Trial 1 (2 ≤ RSS ≤ 5 <span class="html-italic">vs.</span> RSS = 6) and with “*” for Trial 2 (RSS = 5 <span class="html-italic">vs.</span> RSS = 6).</p> Full article ">
3390 KiB  
Article
A Complexity-Based Approach for the Detection of Weak Signals in Ocean Ambient Noise
by Shashidhar Siddagangaiah, Yaan Li, Xijing Guo, Xiao Chen, Qunfei Zhang, Kunde Yang and Yixin Yang
Entropy 2016, 18(3), 101; https://doi.org/10.3390/e18030101 - 18 Mar 2016
Cited by 54 | Viewed by 7601
Abstract
There are numerous studies showing that there is a constant increase in the ocean ambient noise level and the ever-growing demand for developing algorithms for detecting weak signals in ambient noise. In this study, we utilize dynamical and statistical complexity to detect the [...] Read more.
There are numerous studies showing that there is a constant increase in the ocean ambient noise level and the ever-growing demand for developing algorithms for detecting weak signals in ambient noise. In this study, we utilize dynamical and statistical complexity to detect the presence of weak ship noise embedded in ambient noise. The ambient noise and ship noise were recorded in the South China Sea. The multiscale entropy (MSE) method and the complexity-entropy causality plane (C-H plane) were used to quantify the dynamical and statistical complexity of the measured time series, respectively. We generated signals with varying signal-to-noise ratio (SNR) by varying the amplification of a ship signal. The simulation results indicate that the complexity is sensitive to change in the information in the ambient noise and the change in SNR, a finding that enables the detection of weak ship signals in strong background ambient noise. The simulation results also illustrate that complexity is better than the traditional spectrogram method, particularly effective for detecting low SNR signals in ambient noise. In addition, complexity-based MSE and C-H plane methods are simple, robust and do not assume any underlying dynamics in time series. Hence, complexity should be used in practical situations. Full article
(This article belongs to the Special Issue Computational Complexity)
►▼ Show Figures

Figure 1

Figure 1
<p>Recordings of (<b>a</b>) ocean ambient noise, and (<b>b</b>,<b>c</b>) the ship at 2.5 km and 1 km away from the hydrophone for a duration of 60 s and a sampling frequency of 16 kHz, respectively.</p> Full article ">Figure 2
<p>Spectrogram of ambient noise, the ship signal, and varying SNRs of the ship signal.</p> Full article ">Figure 3
<p>(<b>a</b>) Location of ambient noise, ship and varying SNRs of the ship on the <span class="html-italic">C-H</span> plane, the upper (lower) line represent the maximum (minimum) values of <span class="html-italic">C<sub>SCM</sub></span> as a function of <span class="html-italic">H<sub>SCM</sub></span> for <span class="html-italic">M</span> = 4; (<b>b</b>) Representation of variation of statistical complexity measure as a function of varying SNRs of the ship; (<b>c</b>) Representation of variation of statistical complexity measure and entropy as a function of varying SNRs of the ship.</p> Full article ">Figure 3 Cont.
<p>(<b>a</b>) Location of ambient noise, ship and varying SNRs of the ship on the <span class="html-italic">C-H</span> plane, the upper (lower) line represent the maximum (minimum) values of <span class="html-italic">C<sub>SCM</sub></span> as a function of <span class="html-italic">H<sub>SCM</sub></span> for <span class="html-italic">M</span> = 4; (<b>b</b>) Representation of variation of statistical complexity measure as a function of varying SNRs of the ship; (<b>c</b>) Representation of variation of statistical complexity measure and entropy as a function of varying SNRs of the ship.</p> Full article ">Figure 4
<p>Represents the location of ambient noise and ship signal with SNR −19.2 dB on the <span class="html-italic">C-H</span> plane, the upper (lower) line represent the maximum (minimum) values of <span class="html-italic">C<sub>SCM</sub></span> as a function of <span class="html-italic">H<sub>SCM</sub></span> for <span class="html-italic">M</span> = 4. The red line represent the threshold of detection between ship signal and ambient noise.</p> Full article ">Figure 5
<p>Location of various ambient noises on the <span class="html-italic">C-H</span> plane. The upper (lower) line represents the maximum (minimum) values of <span class="html-italic">C<sub>SCM</sub></span> as a function of <span class="html-italic">H<sub>SCM</sub></span> for <span class="html-italic">M</span> = 4.</p> Full article ">Figure 6
<p>(<b>a</b>) Location of ambient noise, ship and varying SNRs of the ship on the <span class="html-italic">C-H</span> plane. the upper (lower) line represents the maximum (minimum) values of <span class="html-italic">C<sub>SCM</sub></span> as a function of <span class="html-italic">H<sub>SCM</sub></span> for <span class="html-italic">M</span> = 4; (<b>b</b>) representation of variation of statistical complexity measure and entropy as a function of varying SNRs of the ship.</p> Full article ">Figure 7
<p>(<b>a</b>) MSE analysis of ship signal with varying SNRs evaluated for <span class="html-italic">m</span> = 4 and <span class="html-italic">r</span> = 0.15; (<b>b</b>) dynamical complexity measure (sum of sample entropy values for scale 1–20, inclusively) <span class="html-italic">vs</span>. varying SNRs of the ship signal.</p> Full article ">Figure 8
<p>Representation of normalized DCM and SCM as a function of varying SNRs of the ship signal and percentage of ship signal in ambient noise respectively.</p> Full article ">
6815 KiB  
Article
Development of a Refractory High Entropy Superalloy
by Oleg N. Senkov, Dieter Isheim, David N. Seidman and Adam L. Pilchak
Entropy 2016, 18(3), 102; https://doi.org/10.3390/e18030102 - 17 Mar 2016
Cited by 235 | Viewed by 16563
Abstract
Microstructure, phase composition and mechanical properties of a refractory high entropy superalloy, AlMo0.5NbTa0.5TiZr, are reported in this work. The alloy consists of a nano-scale mixture of two phases produced by the decomposition from a high temperature body-centered cubic (BCC) [...] Read more.
Microstructure, phase composition and mechanical properties of a refractory high entropy superalloy, AlMo0.5NbTa0.5TiZr, are reported in this work. The alloy consists of a nano-scale mixture of two phases produced by the decomposition from a high temperature body-centered cubic (BCC) phase. The first phase is present in the form of cuboidal-shaped nano-precipitates aligned in rows along <100>-type directions, has a disordered BCC crystal structure with the lattice parameter a1 = 326.9 ± 0.5 pm and is rich in Mo, Nb and Ta. The second phase is present in the form of channels between the cuboidal nano-precipitates, has an ordered B2 crystal structure with the lattice parameter a2 = 330.4 ± 0.5 pm and is rich in Al, Ti and Zr. Both phases are coherent and have the same crystallographic orientation within the former grains. The formation of this modulated nano-phase structure is discussed in the framework of nucleation-and-growth and spinodal decomposition mechanisms. The yield strength of this refractory high entropy superalloy is superior to the yield strength of Ni-based superalloys in the temperature range of 20 °C to 1200 °C. Full article
(This article belongs to the Special Issue High-Entropy Alloys and High-Entropy-Related Materials)
►▼ Show Figures

Figure 1

Figure 1
<p>SEM micrographs of Atom-probe tomography (APT) nanotip preparation from the AlMo<sub>0.5</sub>NbTa<sub>0.5</sub>TiZr alloy by focused-ion beam (FIB) microscopy-based lift-out, mounting and sharpening process. (<b>a</b>) lift-out volume mounted on a silicon micropost; (<b>b</b>) Final APT nanotip after FIB Ga<sup>+</sup> ion milling with annular patterns. The basket-weave pattern of the precipitates is visible during the entire FIB preparation process.</p> Full article ">Figure 2
<p>X-ray diffraction patterns of AlMo<sub>0.5</sub>NbTa<sub>0.5</sub>TiZr: (<b>a</b>) after annealing at 1400 °C and (<b>b</b>) after following 50% compression deformation at 1000 °C.</p> Full article ">Figure 3
<p>SEM/backscatter electron (BSE) image of equiaxed grain structure of AlMo<sub>0.5</sub>NbTa<sub>0.5</sub>TiZr after annealing at 1400 °C for 24 h. Second-phase precipitates (which are dark) are present at grain boundaries.</p> Full article ">Figure 4
<p>High magnification SEM/BSE images of: (<b>a</b>) a two-phase basket-weave lamellar structure inside grains and (<b>b</b>) coarsened particles of the two phases at grain boundaries in AlMo<sub>0.5</sub>NbTa<sub>0.5</sub>TiZr.</p> Full article ">Figure 5
<p>Scanning transmission electron microscopy (STEM) high-angle annular dark-field image and fast Fourier transforms (inside the red squares) recorded from a survey sample extracted from the inside-grain region ([<a href="#B29-entropy-18-00102" class="html-bibr">29</a>], reprinted by permission of Taylor &amp; Francis Ltd.).</p> Full article ">Figure 6
<p>(<b>a</b>) Dark-field TEM micrograph of the nano-phase structure of AlMo<sub>0.5</sub>NbTa<sub>0.5</sub>TiZr; and (<b>b</b>) respective selected area diffraction pattern. The dark field image was obtained with a (001) superlattice reflection placed inside the objective aperture (indicated by the circle) ([<a href="#B29-entropy-18-00102" class="html-bibr">29</a>], reprinted by permission of Taylor &amp; Francis Ltd.).</p> Full article ">Figure 7
<p>APT reconstruction of the precipitate microstructure in the AlMo<sub>0.5</sub>NbTa<sub>0.5</sub>TiZr alloy in a rectangular parallel piped volume, 93 × 93 × 301 nm<sup>3</sup> in size, containing 41,677,323 atoms, (<b>a</b>) front view and (<b>b</b>) top view of the reconstructed volume. An 18 at.% Ta isoconcentration surface is superposed in red to outline the Ta-rich cuboidal precipitates. For clarity, only 40% of the Al atoms are shown, and only 5% of the atoms of all other elements.</p> Full article ">Figure 8
<p>Concentration profiles of Nb, Ta, Mo, Al, Ti and Zr through a row of aligned cuboidal precipitates. The concentration profiles were taken along the cylinder positioned in the reconstruction box cutting through the precipitates displayed. The reconstruction volume is seen in the same projection direction as in <a href="#entropy-18-00102-f007" class="html-fig">Figure 7</a>a, with only the precipitates used for these concentration profiles displayed. The average compositions of cuboidal precipitates and the thin matrix channels between them are listed in <a href="#entropy-18-00102-t002" class="html-table">Table 2</a>.</p> Full article ">Figure 9
<p>SEM/BSE images of the basket-weave nano-phase structure (at two different magnifications) in AlMo<sub>0.5</sub>NbTa<sub>0.5</sub>TiZr after 50% compression deformation at 1000 °C.</p> Full article ">Figure 10
<p>Comparison of the temperature dependences of: (<b>a</b>) yield strength and (<b>b</b>) specific yield strength of the RHEA superalloy AlMo<sub>0.5</sub>NbTa<sub>0.5</sub>TiZr and three Ni-based superalloys: precipitation strengthened IN718 [<a href="#B38-entropy-18-00102" class="html-bibr">38</a>] and Mar-M247 [<a href="#B39-entropy-18-00102" class="html-bibr">39</a>], and solid solution-strengthened Haynes<sup>®</sup> 230 [<a href="#B40-entropy-18-00102" class="html-bibr">40</a>].</p> Full article ">
952 KiB  
Article
Riemannian Laplace Distribution on the Space of Symmetric Positive Definite Matrices
by Hatem Hajri, Ioana Ilea, Salem Said, Lionel Bombrun and Yannick Berthoumieu
Entropy 2016, 18(3), 98; https://doi.org/10.3390/e18030098 - 16 Mar 2016
Cited by 13 | Viewed by 5943
Abstract
The Riemannian geometry of the space Pm, of m × m symmetric positive definite matrices, has provided effective tools to the fields of medical imaging, computer vision and radar signal processing. Still, an open challenge remains, which consists of extending these [...] Read more.
The Riemannian geometry of the space Pm, of m × m symmetric positive definite matrices, has provided effective tools to the fields of medical imaging, computer vision and radar signal processing. Still, an open challenge remains, which consists of extending these tools to correctly handle the presence of outliers (or abnormal data), arising from excessive noise or faulty measurements. The present paper tackles this challenge by introducing new probability distributions, called Riemannian Laplace distributions on the space Pm. First, it shows that these distributions provide a statistical foundation for the concept of the Riemannian median, which offers improved robustness in dealing with outliers (in comparison to the more popular concept of the Riemannian center of mass). Second, it describes an original expectation-maximization algorithm, for estimating mixtures of Riemannian Laplace distributions. This algorithm is applied to the problem of texture classification, in computer vision, which is considered in the presence of outliers. It is shown to give significantly better performance with respect to other recently-proposed approaches. Full article
(This article belongs to the Special Issue Differential Geometrical Theory of Statistics)
►▼ Show Figures

Figure 1

Figure 1
<p>Example of a texture from the VisTex database (<b>a</b>), one of its patches (<b>b</b>) and the corresponding outlier (<b>c</b>).</p> Full article ">Figure 2
<p>Classification results.</p> Full article ">
442 KiB  
Article
Constrained Inference When the Sampled and Target Populations Differ
by Huijun Yi and Bhaskar Bhattacharya
Entropy 2016, 18(3), 97; https://doi.org/10.3390/e18030097 - 16 Mar 2016
Viewed by 3483
Abstract
In the analysis of contingency tables, often one faces two difficult criteria: sampled and target populations are not identical and prior information translates to the presence of general linear inequality restrictions. Under these situations, we present new models of estimating cell probabilities related [...] Read more.
In the analysis of contingency tables, often one faces two difficult criteria: sampled and target populations are not identical and prior information translates to the presence of general linear inequality restrictions. Under these situations, we present new models of estimating cell probabilities related to four well-known methods of estimation. We prove that each model yields maximum likelihood estimators under those restrictions. The performance ranking of these methods under equality restrictions is known. We compare these methods under inequality restrictions in a simulation study. It reveals that these methods may rank differently under inequality restriction than with equality. These four methods are also compared while US census data are analyzed. Full article
(This article belongs to the Section Information Theory, Probability and Statistics)
►▼ Show Figures

Figure 1

Figure 1
<p>RRMSEs for data generated under four models (<b>a</b>) RAKE, (<b>b</b>) LSQ, (<b>c</b>) MCSQ, (<b>d</b>) MLRS, when <math display="inline"> <mrow> <mi>n</mi> <mo>=</mo> <mn>30</mn> </mrow> </math>. The horizontal reference line at 0 RRMSE corresponds to ML estimates under the model.</p> Full article ">Figure 2
<p>RRMSEs for data generated under four models (<b>a</b>) RAKE, (<b>b</b>) LSQ, (<b>c</b>) MCSQ, (<b>d</b>) MLRS, when <math display="inline"> <mrow> <mi>n</mi> <mo>=</mo> <mn>100</mn> </mrow> </math>. The horizontal reference line at 0 RRMSE corresponds to ML estimates under the model.</p> Full article ">Figure 3
<p>RRMSEs for data generated under four models (<b>a</b>) RAKE, (<b>b</b>) LSQ, (<b>c</b>) MCSQ, (<b>d</b>) MLRS, when <math display="inline"> <mrow> <mi>n</mi> <mo>=</mo> <mn>1000</mn> </mrow> </math>. The horizontal reference line at 0 RRMSE corresponds to ML estimates under the model.</p> Full article ">
1994 KiB  
Article
Preference Inconsistence-Based Entropy
by Wei Pan, Kun She and Pengyuan Wei
Entropy 2016, 18(3), 96; https://doi.org/10.3390/e18030096 - 15 Mar 2016
Cited by 1 | Viewed by 4507
Abstract
Preference analysis is a class of important issues in ordinal decision making. As available information is usually obtained from different evaluation criteria or experts, the derived preference decisions may be inconsistent and uncertain. Shannon entropy is a suitable measurement of uncertainty. This work [...] Read more.
Preference analysis is a class of important issues in ordinal decision making. As available information is usually obtained from different evaluation criteria or experts, the derived preference decisions may be inconsistent and uncertain. Shannon entropy is a suitable measurement of uncertainty. This work proposes the concepts of preference inconsistence set and preference inconsistence degree. Then preference inconsistence entropy is introduced by combining preference inconsistence degree and Shannon entropy. A number of properties and theorems as well as two applications are discussed. Feature selection is used for attribute reduction and sample condensation aims to obtain a consistent preference system. Forward feature selection algorithm, backward feature selection algorithm and sample condensation algorithm are developed. The experimental results show that the proposed model represents an effective solution for preference analysis. Full article
►▼ Show Figures

Figure 1

Figure 1
<p>Preference inconsistence entropy and attribute significance of single features (Pasture production). (<b>a</b>) Inconsistent entropy of single features; (<b>b</b>) Significance of single features; (<b>c</b>) Relation of inconsistence entropy and attribute significance.</p> Full article ">Figure 2
<p>Distribution, histogram and pie of selected samples for upward preference (Pasture Production). (<b>a</b>) Distribution of selected samples; (<b>b</b>) Histogram of selected samples; (<b>c</b>) Pie of selected samples.</p> Full article ">Figure 2 Cont.
<p>Distribution, histogram and pie of selected samples for upward preference (Pasture Production). (<b>a</b>) Distribution of selected samples; (<b>b</b>) Histogram of selected samples; (<b>c</b>) Pie of selected samples.</p> Full article ">Figure 3
<p>Distribution, histogram and pie of selected samples for downward preference (Pasture Production). (<b>a</b>) Distribution of selected samples; (<b>b</b>) Histogram of selected samples; (<b>c</b>) Pie of selected samples.</p> Full article ">Figure 4
<p>Preference inconsistence entropy and attribute significance of single features (Squash Harvest). (<b>a</b>) Inconsistence entropy of single features; (<b>b</b>) Significance of single features; (<b>c</b>) Relation of attribute significance and inconsistence entropy.</p> Full article ">Figure 5
<p>Distribution, histogram and pie of selected samples for upward preference (Squash Harvest). (<b>a</b>) Distribution of selected samples; (<b>b</b>) Histogram of selected samples.</p> Full article ">Figure 5 Cont.
<p>Distribution, histogram and pie of selected samples for upward preference (Squash Harvest). (<b>a</b>) Distribution of selected samples; (<b>b</b>) Histogram of selected samples.</p> Full article ">Figure 6
<p>Distribution and histogram of selected samples for downward preference (Squash Harvest). (<b>a</b>) Distribution of selected samples; (<b>b</b>) Histogram of selected samples.</p> Full article ">
403 KiB  
Article
A Cross-Entropy-Based Admission Control Optimization Approach for Heterogeneous Virtual Machine Placement in Public Clouds
by Li Pan and Datao Wang
Entropy 2016, 18(3), 95; https://doi.org/10.3390/e18030095 - 15 Mar 2016
Cited by 5 | Viewed by 4730
Abstract
Virtualization technologies make it possible for cloud providers to consolidate multiple IaaS provisions into a single server in the form of virtual machines (VMs). Additionally, in order to fulfill the divergent service requirements from multiple users, a cloud provider needs to offer several [...] Read more.
Virtualization technologies make it possible for cloud providers to consolidate multiple IaaS provisions into a single server in the form of virtual machines (VMs). Additionally, in order to fulfill the divergent service requirements from multiple users, a cloud provider needs to offer several types of VM instances, which are associated with varying configurations and performance, as well as different prices. In such a heterogeneous virtual machine placement process, one significant problem faced by a cloud provider is how to optimally accept and place multiple VM service requests into its cloud data centers to achieve revenue maximization. To address this issue, in this paper, we first formulate such a revenue maximization problem during VM admission control as a multiple-dimensional knapsack problem, which is known to be NP-hard to solve. Then, we propose to use a cross-entropy-based optimization approach to address this revenue maximization problem, by obtaining a near-optimal eligible set for the provider to accept into its data centers, from the waiting VM service requests in the system. Finally, through extensive experiments and measurements in a simulated environment with the settings of VM instance classes derived from real-world cloud systems, we show that our proposed cross-entropy-based admission control optimization algorithm is efficient and effective in maximizing cloud providers’ revenue in a public cloud computing environment. Full article
►▼ Show Figures

Figure 1

Figure 1
<p>The architecture of the cloud IaaS platform.</p> Full article ">Figure 2
<p>Iterations required by cross-entropy-based VM admission control optimization algorithm (CEVMAC) for reaching convergence under different experimental settings.</p> Full article ">Figure 3
<p>Run-time of CEVMAC on a 3.20-GHz i5-3470 CPU from Intel for different experimental settings.</p> Full article ">Figure 4
<p>Normalized revenue of the data center with different algorithms, for different numbers of VM requests and load factors.</p> Full article ">
482 KiB  
Review
Dark Energy: The Shadowy Reflection of Dark Matter?
by Kostas Kleidis and Nikolaos K. Spyrou
Entropy 2016, 18(3), 94; https://doi.org/10.3390/e18030094 - 12 Mar 2016
Cited by 20 | Viewed by 6933
Abstract
In this article, we review a series of recent theoretical results regarding a conventional approach to the dark energy (DE) concept. This approach is distinguished among others for its simplicity and its physical relevance. By compromising General Relativity (GR) and Thermodynamics at cosmological [...] Read more.
In this article, we review a series of recent theoretical results regarding a conventional approach to the dark energy (DE) concept. This approach is distinguished among others for its simplicity and its physical relevance. By compromising General Relativity (GR) and Thermodynamics at cosmological scale, we end up with a model without DE. Instead, the Universe we are proposing is filled with a perfect fluid of self-interacting dark matter (DM), the volume elements of which perform hydrodynamic flows. To the best of our knowledge, it is the first time in a cosmological framework that the energy of the cosmic fluid internal motions is also taken into account as a source of the universal gravitational field. As we demonstrate, this form of energy may compensate for the DE needed to compromise spatial flatness, while, depending on the particular type of thermodynamic processes occurring in the interior of the DM fluid (isothermal or polytropic), the Universe depicts itself as either decelerating or accelerating (respectively). In both cases, there is no disagreement between observations and the theoretical prediction of the distant supernovae (SNe) Type Ia distribution. In fact, the cosmological model with matter content in the form of a thermodynamically-involved DM fluid not only interprets the observational data associated with the recent history of Universe expansion, but also confronts successfully with every major cosmological issue (such as the age and the coincidence problems). In this way, depending on the type of thermodynamic processes in it, such a model may serve either for a conventional DE cosmology or for a viable alternative one. Full article
(This article belongs to the Special Issue Selected Papers from 13th Joint European Thermodynamics Conference)
►▼ Show Figures

Figure 1

Figure 1
<p>Overplotted on the Hubble diagram of the SNe Ia sample used by Davis <span class="html-italic">et al</span>. [<a href="#B181-entropy-18-00094" class="html-bibr">181</a>], are the theoretical curves of the distance modulus in the iDMF model, <math display="inline"> <mrow> <mi>μ</mi> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> </math>, for <math display="inline"> <mrow> <mi>w</mi> <msub> <mi mathvariant="sans-serif">Ω</mi> <mi>M</mi> </msub> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>10</mn> </mrow> </math> (red solid line), <math display="inline"> <mrow> <mi>w</mi> <msub> <mi mathvariant="sans-serif">Ω</mi> <mi>M</mi> </msub> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>16</mn> </mrow> </math> (green line), and <math display="inline"> <mrow> <mi>w</mi> <msub> <mi mathvariant="sans-serif">Ω</mi> <mi>M</mi> </msub> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>19</mn> </mrow> </math> (blue solid line). The dashed line represents the theoretical curve associated to the distance modulus in the context of the collisionless-DM approach.</p> Full article ">Figure 2
<p>The age of the DM fluid perform polytropic flows (pDMF) model, <math display="inline"> <msub> <mi>t</mi> <mn>0</mn> </msub> </math>, in units of <math display="inline"> <msub> <mi>t</mi> <mrow> <mi>E</mi> <mi>d</mi> <mi>S</mi> </mrow> </msub> </math>, as a function of the polytropic exponent <math display="inline"> <mrow> <mi mathvariant="sans-serif">Γ</mi> <mo>&lt;</mo> <mn>1</mn> </mrow> </math> (red solid line). For every <math display="inline"> <mrow> <mi mathvariant="sans-serif">Γ</mi> <mo>&lt;</mo> <mn>1</mn> </mrow> </math>, <math display="inline"> <mrow> <msub> <mi>t</mi> <mn>0</mn> </msub> <mo>&gt;</mo> <msub> <mi>t</mi> <mrow> <mi>E</mi> <mi>d</mi> <mi>S</mi> </mrow> </msub> </mrow> </math>, and <math display="inline"> <msub> <mi>t</mi> <mn>0</mn> </msub> </math> approaches <math display="inline"> <msub> <mi>t</mi> <mrow> <mi>E</mi> <mi>d</mi> <mi>S</mi> </mrow> </msub> </math> only in the isothermal <math display="inline"> <mrow> <mo>(</mo> <mi mathvariant="sans-serif">Γ</mi> <mo>→</mo> <mn>1</mn> <mo>)</mo> </mrow> </math> limit. The horizontal solid line denotes the age of the Universe <math display="inline"> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>.</mo> <mn>483</mn> <mspace width="0.277778em"/> <msub> <mi>t</mi> <mrow> <mi>E</mi> <mi>d</mi> <mi>S</mi> </mrow> </msub> <mo>)</mo> </mrow> </math> in the isobaric <math display="inline"> <mrow> <mo>(</mo> <mi mathvariant="sans-serif">Γ</mi> <mo>=</mo> <mn>0</mn> <mo>)</mo> </mrow> </math>ΛCDM limit of the pDMF model.</p> Full article ">Figure 3
<p>The scale factor, <span class="html-italic">S</span>, of the pDMF model with <math display="inline"> <mrow> <msub> <mi mathvariant="sans-serif">Ω</mi> <mi>M</mi> </msub> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>274</mn> </mrow> </math> (in units of its present-time value, <math display="inline"> <msub> <mi>S</mi> <mn>0</mn> </msub> </math>), as a function of the cosmic time <span class="html-italic">t</span> (in units of <math display="inline"> <msub> <mi>t</mi> <mn>0</mn> </msub> </math>), for <math display="inline"> <mrow> <mi mathvariant="sans-serif">Γ</mi> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>5</mn> </mrow> </math> (orange), <math display="inline"> <mrow> <mi mathvariant="sans-serif">Γ</mi> <mo>=</mo> <mn>0</mn> </mrow> </math> (dashed), <math display="inline"> <mrow> <mi mathvariant="sans-serif">Γ</mi> <mo>=</mo> <mo>-</mo> <mn>0</mn> <mo>.</mo> <mn>5</mn> </mrow> </math> (blue), <math display="inline"> <mrow> <mi mathvariant="sans-serif">Γ</mi> <mo>=</mo> <mo>-</mo> <mn>1</mn> </mrow> </math> (red), and <math display="inline"> <mrow> <mi mathvariant="sans-serif">Γ</mi> <mo>=</mo> <mo>-</mo> <mn>2</mn> </mrow> </math> (green). For each and every curve, there is a value of <math display="inline"> <mrow> <mi>t</mi> <mo>&lt;</mo> <msub> <mi>t</mi> <mn>0</mn> </msub> </mrow> </math>, above which <math display="inline"> <mrow> <mi>S</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </math> becomes concave, <span class="html-italic">i.e.</span>, the Universe accelerates its expansion.</p> Full article ">Figure 4
<p>The transition redshift, <math display="inline"> <msub> <mi>z</mi> <mrow> <mi>t</mi> <mi>r</mi> </mrow> </msub> </math>, in the pDMF model as a function of the polytropic exponent, Γ (blue solid curve). For <math display="inline"> <mrow> <mi mathvariant="sans-serif">Γ</mi> <mo>&lt;</mo> <mo>-</mo> <mn>0</mn> <mo>.</mo> <mn>377</mn> </mrow> </math>, the Universe resides to the phantom realm (red dashed curve).</p> Full article ">Figure 5
<p>Overplotted to the Hubble diagram of the Union 2.1 Compilation are the best-fit curves (too close to be distinguished) representing the function <math display="inline"> <mrow> <mi>μ</mi> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> </math> in the pDMF model, for <math display="inline"> <mrow> <mo>-</mo> <mn>0</mn> <mo>.</mo> <mn>09</mn> <mo>&lt;</mo> <mi mathvariant="sans-serif">Γ</mi> <mo>≤</mo> <mn>0</mn> </mrow> </math>.</p> Full article ">
7269 KiB  
Article
Selected Remarks about Computer Processing in Terms of Flow Control and Statistical Mechanics
by Dominik Strzałka
Entropy 2016, 18(3), 93; https://doi.org/10.3390/e18030093 - 12 Mar 2016
Cited by 3 | Viewed by 6314
Abstract
Despite the fact that much has been said about processing in computer science, it seems that there is still much to do. A classical approach assumes that the computations done by computers are a kind of mathematical operation (calculations of functions values) and [...] Read more.
Despite the fact that much has been said about processing in computer science, it seems that there is still much to do. A classical approach assumes that the computations done by computers are a kind of mathematical operation (calculations of functions values) and have no special relations to energy transformation and flow. However, there is a possibility to get a new view on selected topics, and as a special case, the sorting problem is presented; we know many different sorting algorithms, including those that have complexity equal to O(n lg(n)) , which means that this problem is algorithmically closed, but it is also possible to focus on the problem of sorting in terms of flow control, entropy and statistical mechanics. This is done in relation to the existing definitions of sorting, connections between sorting and ordering and some important aspects of computer processing understood as a flow that are not taken into account in many theoretical considerations in computer science. The proposed new view is an attempt to change the paradigm in the description of algorithms’ performance by computational complexity and processing, taking into account the existing references between the idea of Turing machines and their physical implementations. This proposal can be expressed as a physics of computer processing; a reference point to further analysis of algorithmic and interactive processing in computer systems. Full article
(This article belongs to the Special Issue Computational Complexity)
►▼ Show Figures

Figure 1

Figure 1
<p>Diagram flow for the insertion-sort algorithm.</p> Full article ">Figure 2
<p>Modified diagram flow for the insertion-sort algorithm with a parasite path. The solid (green) line stands for algorithm behavior in the optimistic case, path P1; the dotted (blue) line expresses the flow in pessimistic case, path P2; and the dashed (red) line shows the flow of sorted keys in other cases, path P3.</p> Full article ">Figure 3
<p>One possible example of sorting processing. For each sorted key from the input set (blue line), the exact number of dominant operations was recorded (black line). As can be seen, when there is a falling trend in the input set, the number of dominant operations rises; when there is a rising trend in the input set, the number of operations falls. The red line shows how the algorithm should behave if we follow asymptotic analysis for the worst case.</p> Full article ">Figure 4
<p>Entropy <math display="inline"> <msub> <mi>S</mi> <mi>i</mi> </msub> </math> production for successive keys <math display="inline"> <msub> <mi>n</mi> <mi>i</mi> </msub> </math> (solid line with open squares) with the approximation given by Equation (<a href="#FD7-entropy-18-00093" class="html-disp-formula">7</a>) (line with open circles).</p> Full article ">Figure 5
<p>Example of entropy production during sorting. The dark blue line denotes the number of dominant operations; light blue line, the amount of produced entropy according to Equation (<a href="#FD12-entropy-18-00093" class="html-disp-formula">12</a>); red line, maximal number of dominant operations (pessimistic case); black line, the number of dominant operations in the average case; green line, predicted amount of produced entropy according to Equation (<a href="#FD7-entropy-18-00093" class="html-disp-formula">7</a>).</p> Full article ">Figure 6
<p>Dependencies between one trajectory of fractional Brownian motion (fBm) <math display="inline"> <mrow> <mi>X</mi> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> </math> and the number of dominant operations <math display="inline"> <mrow> <mi>Y</mi> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> </math> required for its sorting. Red lines show falling (global and minor trends) in the input set, while green lines refer to local and global rising trends. The blue line shows the number of dominant operations; it is clear that each falling trend in the input set (inversions in the input set) denotes the growth of the number of dominant operations, whereas rising trends (runs in the input set) are related to a decrease of the dominant operations.</p> Full article ">Figure 7
<p>Dependencies between one trajectory of fBm <math display="inline"> <mrow> <mi>X</mi> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> </math> and the amount of produced entropy during its sorting. It is very visible (compared to <a href="#entropy-18-00093-f006" class="html-fig">Figure 6</a>) that the levels of produced entropy <math display="inline"> <msub> <mi>S</mi> <mi>i</mi> </msub> </math> depend not only on the values of sorted keys (indirectly, on the number of dominant operations), but also on the change of trends (inversions and runs) in the input set.</p> Full article ">Figure 8
<p>Direct relation between the values of sorted keys <math display="inline"> <mrow> <mi>X</mi> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> </math> and the number of dominant operations <math display="inline"> <mrow> <mi>Y</mi> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> </math> needed for their sorting. The whole graph shows bifurcations in the algorithm behavior in relation to existing feedback.</p> Full article ">Figure 9
<p>The direct relation between the values of sorted keys <math display="inline"> <mrow> <mi>X</mi> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> </math> and the amount of produced entropy during their sorting <math display="inline"> <msub> <mi>S</mi> <mi>i</mi> </msub> </math>. Produced entropy depends not only on key values, but also on existing feedback.</p> Full article ">
3634 KiB  
Article
Long-Range Electron Transport Donor-Acceptor in Nonlinear Lattices
by Alexander P. Chetverikov, Werner Ebeling and Manuel G. Velarde
Entropy 2016, 18(3), 92; https://doi.org/10.3390/e18030092 - 11 Mar 2016
Cited by 6 | Viewed by 5224
Abstract
We study here several simple models of the electron transfer (ET) in a one-dimensional nonlinear lattice between a donor and an acceptor and propose a new fast mechanism of electron surfing on soliton-like excitations along the lattice. The nonlinear lattice is modeled as [...] Read more.
We study here several simple models of the electron transfer (ET) in a one-dimensional nonlinear lattice between a donor and an acceptor and propose a new fast mechanism of electron surfing on soliton-like excitations along the lattice. The nonlinear lattice is modeled as a classical one-dimensional Morse chain and the dynamics of the electrons are considered in the tight-binding approximation. This model is applied to the processes along a covalent bridge connecting donors and acceptors. First, it is shown that the electron forms bound states with the solitonic excitations in the lattice. These so-called solectrons may move with supersonic speed. In a heated system, the electron transfer between a donor and an acceptor is modeled as a diffusion-like process. We study in detail the role of thermal factors on the electron transfer. Then, we develop a simple model based on the classical Smoluchowski–Chandrasekhar picture of diffusion-controlled reactions as stochastic processes with emitters and absorbers. Acceptors are modeled by an absorbing boundary. Finally, we compare the new ET mechanisms described here with known ET data. We conclude that electron surfing on solitons could be a special fast way for ET over quite long distances. Full article
(This article belongs to the Special Issue Non-Linear Lattice)
►▼ Show Figures

Figure 1

Figure 1
<p>Evolution of one electron in probability density starting at position 100 and a soliton starting at position 70. Then, 20–40 time units after the start the electron is catched by the soliton and forms with it a bound state (called solectron) which moves with slightly subsonic velocity.</p> Full article ">Figure 2
<p>Evolution of an electron starting at position 100 between two solitons emitted at positions 70 and 130, respectively. The electron probability density splits between the two solitons.</p> Full article ">Figure 3
<p>Evolution of the electron probability density of an electron released into a heated lattice (<math display="inline"> <mrow> <mi>T</mi> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>1</mn> </mrow> </math>, <math display="inline"> <mrow> <mi>τ</mi> <mo>=</mo> <mn>10</mn> </mrow> </math>, <math display="inline"> <mrow> <mi>α</mi> <mo>=</mo> <mn>1</mn> <mo>.</mo> <mn>75</mn> </mrow> </math>) gets concentrated at places of local soliton excitations (with a size up to 10 lattice units) and survives there for a finite time (may be a few picoseconds), and then it moves to another solitonic “hot spot”.</p> Full article ">Figure 4
<p>The diffusion function <math display="inline"> <mrow> <mi>d</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </math> over the time t for the temperature <math display="inline"> <mrow> <mi>T</mi> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>5</mn> </mrow> </math>. It shows this ET process as diffusion-like.</p> Full article ">Figure 5
<p>The effective diffusion coefficient, <math display="inline"> <msub> <mi>D</mi> <mrow> <mi>e</mi> <mi>f</mi> <mi>f</mi> </mrow> </msub> </math>, based on thermal electron hopping between solitons leads to two different outcomes. The fat points are based on quantum-mechanical hopping modeled in TBA whereas the thin line is obtained by solving quantum-statistical master equations and include quantum-mechanical and thermal hopping effects in the stochastic description.</p> Full article ">Figure 6
<p>Evolution of the probability density, <math display="inline"> <msub> <mi>p</mi> <mi>n</mi> </msub> </math>, of an electron approaching an absorbing boundary.</p> Full article ">Figure 7
<p>Total probability P(t) according to Equation (<a href="#FD15-entropy-18-00092" class="html-disp-formula">15</a>) to find the electron in the lattice system, the probability decreases due to escape through the absorbing boundary. The blue line shows the value 1/e which is used for defining the “life-time” of residence of the electron in the lattice before being absorbed by the boundary.</p> Full article ">Figure 8
<p>Heated lattice (top-down values: <math display="inline"> <mrow> <mi>T</mi> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>01</mn> </mrow> </math>, 0.1 and 0.5 in 2D units). (<b>Left panels</b>): evolution of the local electron probability density (<math display="inline"> <msub> <mi>p</mi> <mi>n</mi> </msub> </math>). (<b>Right panels</b>): total density (P) at different temperatures in the regime of thermal solitons.</p> Full article ">Figure 9
<p>The same problem as in <a href="#entropy-18-00092-f008" class="html-fig">Figure 8</a> but for a set of noise realizations (<math display="inline"> <mrow> <mi>T</mi> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>15</mn> </mrow> </math>, in 2D units).</p> Full article ">Figure 10
<p>Transition time <math display="inline"> <msub> <mi>t</mi> <mrow> <mi>t</mi> <mi>r</mi> </mrow> </msub> </math> for moving from the donor to an absorbing acceptor divided by the distance <math display="inline"> <mrow> <mi>δ</mi> <mi>n</mi> <mo>=</mo> <mn>40</mn> </mrow> </math> as a function of temperature. These results were obtained using TBA. (dotted green line with the portion for <math display="inline"> <mrow> <mi>T</mi> <mo>&lt;</mo> <mn>0</mn> <mo>.</mo> <mn>1</mn> </mrow> </math> being spurious as discussed in the main text) and a master equation approach (solid red line [<a href="#B13-entropy-18-00092" class="html-bibr">13</a>]).</p> Full article ">Figure 11
<p>The (dimensionless) flow through an absorbing boundary at, say, location <math display="inline"> <msub> <mi>x</mi> <mn>1</mn> </msub> </math> according to the model of Chandrasekhar as a function of (dimensionless) time.</p> Full article ">Figure 12
<p>Estimated time <span class="html-italic">t</span> (log-scale) which an electron needs to travel a distance <math display="inline"> <mrow> <mi>l</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </math> (on the abscissa in Å units) by using several alternative mechanisms. On the <span class="html-italic">y</span>-axis, we represent (corresponding to standard plots) the log of the reciprocal time in seconds. The green curve shows the estimate obtained from our diffusion mechanism. The blue curve corresponds to the surf on an externally excited soliton which moves with sound velocity. The red curve represents an average of data points (denoted by crosses) measured for azurin and other biomolecules [<a href="#B18-entropy-18-00092" class="html-bibr">18</a>].</p> Full article ">
2518 KiB  
Article
Analysis of Entropy Generation in the Flow of Peristaltic Nanofluids in Channels With Compliant Walls
by Munawwar Ali Abbas, Yanqin Bai, Mohammad Mehdi Rashidi and Muhammad Mubashir Bhatti
Entropy 2016, 18(3), 90; https://doi.org/10.3390/e18030090 - 11 Mar 2016
Cited by 79 | Viewed by 6115
Abstract
Entropy generation during peristaltic flow of nanofluids in a non-uniform two dimensional channel with compliant walls has been studied. The mathematical modelling of the governing flow problem is obtained under the approximation of long wavelength and zero Reynolds number (creeping flow regime). The [...] Read more.
Entropy generation during peristaltic flow of nanofluids in a non-uniform two dimensional channel with compliant walls has been studied. The mathematical modelling of the governing flow problem is obtained under the approximation of long wavelength and zero Reynolds number (creeping flow regime). The resulting non-linear partial differential equations are solved with the help of a perturbation method. The analytic and numerical results of different parameters are demonstrated mathematically and graphically. The present analysis provides a theoretical model to estimate the characteristics of several Newtonian and non-Newtonian fluid flows, such as peristaltic transport of blood. Full article
(This article belongs to the Special Issue Entropy in Nanofluids)
►▼ Show Figures

Figure 1

Figure 1
<p>Geometry of the problem.</p> Full article ">Figure 2
<p>Temperature distribution for various values of <math display="inline"> <semantics> <mrow> <msub> <mi>N</mi> <mi>t</mi> </msub> </mrow> </semantics> </math> and <math display="inline"> <semantics> <mrow> <msub> <mi>N</mi> <mi>b</mi> </msub> </mrow> </semantics> </math>. (<b>a</b>) red line: <math display="inline"> <semantics> <mrow> <msub> <mi>N</mi> <mi>b</mi> </msub> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics> </math>, green line: <math display="inline"> <semantics> <mrow> <msub> <mi>N</mi> <mi>b</mi> </msub> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics> </math>, blue line: <math display="inline"> <semantics> <mrow> <msub> <mi>N</mi> <mi>b</mi> </msub> <mo>=</mo> <mn>0.6</mn> </mrow> </semantics> </math>, black line: <math display="inline"> <semantics> <mrow> <msub> <mi>N</mi> <mi>b</mi> </msub> <mo>=</mo> <mn>0.9</mn> </mrow> </semantics> </math>; (<b>b</b>) red line: <math display="inline"> <semantics> <mrow> <msub> <mi>N</mi> <mi>t</mi> </msub> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics> </math>, green line: <math display="inline"> <semantics> <mrow> <msub> <mi>N</mi> <mi>t</mi> </msub> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics> </math>, blue line: <math display="inline"> <semantics> <mrow> <msub> <mi>N</mi> <mi>t</mi> </msub> <mo>=</mo> <mn>0.6</mn> </mrow> </semantics> </math>, black line: <math display="inline"> <semantics> <mrow> <msub> <mi>N</mi> <mi>t</mi> </msub> <mo>=</mo> <mn>0.9</mn> </mrow> </semantics> </math>.</p> Full article ">Figure 3
<p>Concentration distribution for various values of <math display="inline"> <semantics> <mrow> <msub> <mi>N</mi> <mi>b</mi> </msub> </mrow> </semantics> </math> and <math display="inline"> <semantics> <mrow> <msub> <mi>N</mi> <mi>t</mi> </msub> </mrow> </semantics> </math>. (<b>a</b>) red line: <math display="inline"> <semantics> <mrow> <msub> <mi>N</mi> <mi>b</mi> </msub> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics> </math>, green line:<math display="inline"> <semantics> <mrow> <mtext> </mtext> <msub> <mi>N</mi> <mi>b</mi> </msub> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics> </math>, blue line: <math display="inline"> <semantics> <mrow> <msub> <mi>N</mi> <mi>b</mi> </msub> <mo>=</mo> <mn>0.6</mn> </mrow> </semantics> </math>, black line: <math display="inline"> <semantics> <mrow> <msub> <mi>N</mi> <mi>b</mi> </msub> <mo>=</mo> <mn>0.9</mn> </mrow> </semantics> </math>; (<b>b</b>) red line: <math display="inline"> <semantics> <mrow> <msub> <mi>N</mi> <mi>t</mi> </msub> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics> </math>, green line: <math display="inline"> <semantics> <mrow> <msub> <mi>N</mi> <mi>t</mi> </msub> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics> </math>, blue line: <math display="inline"> <semantics> <mrow> <msub> <mi>N</mi> <mi>t</mi> </msub> <mo>=</mo> <mn>0.6</mn> </mrow> </semantics> </math>, black line: <math display="inline"> <semantics> <mrow> <msub> <mi>N</mi> <mi>t</mi> </msub> <mo>=</mo> <mn>0.9</mn> </mrow> </semantics> </math>.</p> Full article ">Figure 4
<p>Entropy generation for various values of <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>1</mn> </msub> </mrow> </semantics> </math> and <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>2</mn> </msub> </mrow> </semantics> </math>. (<b>a</b>) red line: <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics> </math>, green line: <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics> </math>, blue line: <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.6</mn> </mrow> </semantics> </math>, black line: <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.9</mn> </mrow> </semantics> </math>; (<b>b</b>) red line: <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics> </math>, green line: <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics> </math>, blue line: <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.6</mn> </mrow> </semantics> </math>, black line: <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.9</mn> </mrow> </semantics> </math>.</p> Full article ">Figure 5
<p>Entropy generation for various values of <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>3</mn> </msub> </mrow> </semantics> </math> and <math display="inline"> <semantics> <mrow> <mi>B</mi> <msup> <mi mathvariant="sans-serif">Ω</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>.</mo> </mrow> </semantics> </math> (<b>a</b>) red line: <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics> </math>, green line: <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics> </math>, blue line:<math display="inline"> <semantics> <mrow> <mtext> </mtext> <msub> <mi>E</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0.6</mn> </mrow> </semantics> </math>, black line: <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0.9</mn> </mrow> </semantics> </math>; (<b>b</b>) red line: <math display="inline"> <semantics> <mrow> <mi>B</mi> <msup> <mi mathvariant="sans-serif">Ω</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics> </math>, green line: <math display="inline"> <semantics> <mrow> <mi>B</mi> <msup> <mi mathvariant="sans-serif">Ω</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics> </math>, blue line: <math display="inline"> <semantics> <mrow> <mi>B</mi> <msup> <mi mathvariant="sans-serif">Ω</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mn>0.6</mn> </mrow> </semantics> </math>, black line: <math display="inline"> <semantics> <mrow> <mi>B</mi> <msup> <mi mathvariant="sans-serif">Ω</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mn>0.9</mn> </mrow> </semantics> </math>.</p> Full article ">Figure 6
<p>Entropy generation for various values of <math display="inline"> <semantics> <mrow> <msub> <mi>N</mi> <mi>t</mi> </msub> </mrow> </semantics> </math> and <math display="inline"> <semantics> <mrow> <msub> <mi>N</mi> <mi>b</mi> </msub> </mrow> </semantics> </math>. (<b>a</b>) red line: <math display="inline"> <semantics> <mrow> <msub> <mi>N</mi> <mi>t</mi> </msub> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics> </math>, green line: <math display="inline"> <semantics> <mrow> <msub> <mi>N</mi> <mi>t</mi> </msub> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics> </math>, blue line: <math display="inline"> <semantics> <mrow> <msub> <mi>N</mi> <mi>t</mi> </msub> <mo>=</mo> <mn>0.6</mn> </mrow> </semantics> </math>, black line: <math display="inline"> <semantics> <mrow> <msub> <mi>N</mi> <mi>t</mi> </msub> <mo>=</mo> <mn>0.9</mn> </mrow> </semantics> </math>; (<b>b</b>) red line: <math display="inline"> <semantics> <mrow> <msub> <mi>N</mi> <mi>b</mi> </msub> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics> </math>, green line: <math display="inline"> <semantics> <mrow> <msub> <mi>N</mi> <mi>b</mi> </msub> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics> </math>, blue line: <math display="inline"> <semantics> <mrow> <msub> <mi>N</mi> <mi>b</mi> </msub> <mo>=</mo> <mn>0.6</mn> </mrow> </semantics> </math>, black line: <math display="inline"> <semantics> <mrow> <msub> <mi>N</mi> <mi>b</mi> </msub> <mo>=</mo> <mn>0.9</mn> </mrow> </semantics> </math>.</p> Full article ">Figure 7
<p>Entropy generation for various values of <math display="inline"> <semantics> <mrow> <mi>G</mi> <msub> <mi>r</mi> <mi>T</mi> </msub> </mrow> </semantics> </math> and<math display="inline"> <semantics> <mrow> <mtext> </mtext> <mi>G</mi> <msub> <mi>r</mi> <mi>F</mi> </msub> </mrow> </semantics> </math>. (<b>a</b>) red line: <math display="inline"> <semantics> <mrow> <mi>G</mi> <msub> <mi>r</mi> <mi>T</mi> </msub> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics> </math>, green line: <math display="inline"> <semantics> <mrow> <mi>G</mi> <msub> <mi>r</mi> <mi>T</mi> </msub> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics> </math>, blue line: <math display="inline"> <semantics> <mrow> <mi>G</mi> <msub> <mi>r</mi> <mi>T</mi> </msub> <mo>=</mo> <mn>0.6</mn> </mrow> </semantics> </math>, black line: <math display="inline"> <semantics> <mrow> <mi>G</mi> <msub> <mi>r</mi> <mi>T</mi> </msub> <mo>=</mo> <mn>0.9</mn> </mrow> </semantics> </math>; (<b>b</b>) red line: <math display="inline"> <semantics> <mrow> <mi>G</mi> <msub> <mi>r</mi> <mi>F</mi> </msub> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics> </math>, green line: <math display="inline"> <semantics> <mrow> <mi>G</mi> <msub> <mi>r</mi> <mi>F</mi> </msub> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics> </math>, blue line: <math display="inline"> <semantics> <mrow> <mi>G</mi> <msub> <mi>r</mi> <mi>F</mi> </msub> <mo>=</mo> <mn>0.6</mn> </mrow> </semantics> </math>, black line: <math display="inline"> <semantics> <mrow> <mi>G</mi> <msub> <mi>r</mi> <mi>F</mi> </msub> <mo>=</mo> <mn>0.9</mn> </mrow> </semantics> </math>.</p> Full article ">Figure 8
<p>(<b>a</b>) Entropy generation for various values of <math display="inline"> <semantics> <mi>ℰ</mi> </semantics> </math>. Red line: <math display="inline"> <semantics> <mrow> <mi>ε</mi> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics> </math>, green line: <math display="inline"> <semantics> <mrow> <mi>ε</mi> <mo>=</mo> <mn>0.03</mn> </mrow> </semantics> </math>, blue line: <math display="inline"> <semantics> <mrow> <mi>ε</mi> <mo>=</mo> <mn>0.06</mn> </mrow> </semantics> </math>, black line: <math display="inline"> <semantics> <mrow> <mi>ε</mi> <mo>=</mo> <mn>0.09</mn> </mrow> </semantics> </math>; (<b>b</b>) Velocity profile for various values of <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>1</mn> </msub> </mrow> </semantics> </math>. Red line: <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics> </math>, green line: <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics> </math>, blue line: <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.6</mn> </mrow> </semantics> </math>, black line: <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.9</mn> </mrow> </semantics> </math>.</p> Full article ">Figure 9
<p>Velocity profile for various values of <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>2</mn> </msub> </mrow> </semantics> </math> and <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>3</mn> </msub> </mrow> </semantics> </math>. (<b>a</b>) red line: <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics> </math>, green line: <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics> </math>, blue line: <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.6</mn> </mrow> </semantics> </math>, black line: <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.9</mn> </mrow> </semantics> </math>; (<b>b</b>) red line: <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics> </math>, green line: <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics> </math>, blue line: <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0.6</mn> </mrow> </semantics> </math>, black line: <math display="inline"> <semantics> <mrow> <msub> <mi>E</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0.9</mn> </mrow> </semantics> </math>.</p> Full article ">
240 KiB  
Article
A Novel Weak Fuzzy Solution for Fuzzy Linear System
by Soheil Salahshour, Ali Ahmadian, Fudziah Ismail and Dumitru Baleanu
Entropy 2016, 18(3), 68; https://doi.org/10.3390/e18030068 - 11 Mar 2016
Cited by 8 | Viewed by 4583
Abstract
This article proposes a novel weak fuzzy solution for the fuzzy linear system. As a matter of fact, we define the right-hand side column of the fuzzy linear system as a piecewise fuzzy function to overcome the related shortcoming, which exists in the [...] Read more.
This article proposes a novel weak fuzzy solution for the fuzzy linear system. As a matter of fact, we define the right-hand side column of the fuzzy linear system as a piecewise fuzzy function to overcome the related shortcoming, which exists in the previous findings. The strong point of this proposal is that the weak fuzzy solution is always a fuzzy number vector. Two complex and non-complex linear systems under uncertainty are tested to validate the effectiveness and correctness of the presented method. Full article
(This article belongs to the Special Issue Complex and Fractional Dynamics)
►▼ Show Figures

Figure 1

Figure 1
<p>A comparison between our proposed solution with the solution in [<a href="#B17-entropy-18-00068" class="html-bibr">17</a>]. It can be noticed, and was proved in [<a href="#B19-entropy-18-00068" class="html-bibr">19</a>] that (<b>a</b>) the solution in [<a href="#B17-entropy-18-00068" class="html-bibr">17</a>] is not a fuzzy vector; while using our new definition (Definition 6), (<b>b</b>) the obtained solution is a fuzzy vector. (<span class="html-italic">y</span>-axis represents the <span class="html-italic">α</span>-cuts and <span class="html-italic">x</span>-axis is for the value of fuzzy solution (<math display="inline"> <msub> <mi mathvariant="bold">z</mi> <mn>1</mn> </msub> </math>)).</p> Full article ">
2051 KiB  
Article
Two Universality Properties Associated with the Monkey Model of Zipf’s Law
by Richard Perline and Ron Perline
Entropy 2016, 18(3), 89; https://doi.org/10.3390/e18030089 - 9 Mar 2016
Cited by 4 | Viewed by 5195
Abstract
The distribution of word probabilities in the monkey model of Zipf’s law is associated with two universality properties: (1) the exponent in the approximate power law approaches −1 as the alphabet size increases and the letter probabilities are specified as the spacings from [...] Read more.
The distribution of word probabilities in the monkey model of Zipf’s law is associated with two universality properties: (1) the exponent in the approximate power law approaches −1 as the alphabet size increases and the letter probabilities are specified as the spacings from a random division of the unit interval for any distribution with a bounded density function on [0,1] ; and (2), on a logarithmic scale the version of the model with a finite word length cutoff and unequal letter probabilities is approximately normally distributed in the part of the distribution away from the tails. The first property is proved using a remarkably general limit theorem from Shao and Hahn for the logarithm of sample spacings constructed on [0,1] and the second property follows from Anscombe’s central limit theorem for a random number of independent and identically distributed (i.i.d.) random variables. The finite word length model leads to a hybrid Zipf-lognormal mixture distribution closely related to work in other areas. Full article
(This article belongs to the Section Information Theory, Probability and Statistics)
►▼ Show Figures

Figure 1

Figure 1
<p>Log-log plots of relative word frequencies by rank for four authors writing in four different European languages in four different centuries. The approximate −1 slope in all the graphs is an iconic feature of Zipf’s word frequency law.</p> Full article ">Figure 2
<p>Log-log plots of monkey word probabilities by rank showing the asymptotic tendency towards a <math display="inline"> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </math> exponent (slope on the log-log scales) using four different distributions to generate letter probabilities: equal probabilities in (<b>a</b>) and a generalized broken stick process in (<b>b</b>–<b>d</b>). <math display="inline"> <mrow> <mi>K</mi> <mo>=</mo> <mn>26</mn> </mrow> </math> letters were used in all cases and the largest <math display="inline"> <mrow> <msubsup> <mo>∑</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mn>4</mn> </msubsup> <msup> <mn>26</mn> <mi>i</mi> </msup> <mo>=</mo> <mn>475</mn> <mo>,</mo> <mn>255</mn> </mrow> </math> word probabilities are displayed.</p> Full article ">Figure 3
<p><a href="#entropy-18-00089-f003" class="html-fig">Figure 3</a>a is a log-log plot of monkey word probabilities by rank for all words of length <math display="inline"> <mrow> <mo>≤</mo> <mn>4</mn> </mrow> </math> non-space characters using letter probabilities from uniform spacings. The linear upper tail coincides closely with the previous <a href="#entropy-18-00089-f002" class="html-fig">Figure 2</a>b, but the approximate power law clearly breaks down. <a href="#entropy-18-00089-f003" class="html-fig">Figure 3</a>b shows the same word probabilities in a standard normal quantile plot. Its rough linearity confirms an approximate Gaussian fit over the whole distribution even though the upper tail is an approximate power law as seen on the left in <a href="#entropy-18-00089-f003" class="html-fig">Figure 3</a>a. We refer to this distribution as a Zipf-lognormal hybrid, and it has many connections to other distributions discussed in the statistical literature.</p> Full article ">
357 KiB  
Article
Maximizing Diversity in Biology and Beyond
by Tom Leinster and Mark W. Meckes
Entropy 2016, 18(3), 88; https://doi.org/10.3390/e18030088 - 9 Mar 2016
Cited by 28 | Viewed by 10344
Abstract
Entropy, under a variety of names, has long been used as a measure of diversity in ecology, as well as in genetics, economics and other fields. There is a spectrum of viewpoints on diversity, indexed by a real parameter q giving greater or [...] Read more.
Entropy, under a variety of names, has long been used as a measure of diversity in ecology, as well as in genetics, economics and other fields. There is a spectrum of viewpoints on diversity, indexed by a real parameter q giving greater or lesser importance to rare species. Leinster and Cobbold (2012) proposed a one-parameter family of diversity measures taking into account both this variation and the varying similarities between species. Because of this latter feature, diversity is not maximized by the uniform distribution on species. So it is natural to ask: which distributions maximize diversity, and what is its maximum value? In principle, both answers depend on q, but our main theorem is that neither does. Thus, there is a single distribution that maximizes diversity from all viewpoints simultaneously, and any list of species has an unambiguous maximum diversity value. Furthermore, the maximizing distribution(s) can be computed in finite time, and any distribution maximizing diversity from some particular viewpoint q > 0 actually maximizes diversity for all q. Although we phrase our results in ecological terms, they apply very widely, with applications in graph theory and metric geometry. Full article
(This article belongs to the Special Issue Information and Entropy in Biological Systems)
►▼ Show Figures

Figure 1

Figure 1
<p>Two bird communities. Heights of stacks indicate species abundances. In (<b>a</b>), there are four species, with the first dominant and the others relatively rare; in (<b>b</b>), the fourth species is absent but the community is otherwise evenly balanced.</p> Full article ">Figure 2
<p>Visualizations of the main theorem: (<b>a</b>) in terms of how different values of q rank the set of distributions; and (<b>b</b>) in terms of diversity profiles.</p> Full article ">Figure 3
<p>Hypothetical three-species system. Distances between species indicate degrees of dissimilarity between them (not to scale).</p> Full article ">Figure 4
<p>Hypothetical community consisting of one species of oak (▪) and ten species of pine (•), to which one further species of pine is then added (◦). Distances between species indicate degrees of dissimilarity (not to scale).</p> Full article ">
6193 KiB  
Article
Selecting Video Key Frames Based on Relative Entropy and the Extreme Studentized Deviate Test
by Yuejun Guo, Qing Xu, Shihua Sun, Xiaoxiao Luo and Mateu Sbert
Entropy 2016, 18(3), 73; https://doi.org/10.3390/e18030073 - 9 Mar 2016
Cited by 13 | Viewed by 9335
Abstract
This paper studies the relative entropy and its square root as distance measures of neighboring video frames for video key frame extraction. We develop a novel approach handling both common and wavelet video sequences, in which the extreme Studentized deviate test is exploited [...] Read more.
This paper studies the relative entropy and its square root as distance measures of neighboring video frames for video key frame extraction. We develop a novel approach handling both common and wavelet video sequences, in which the extreme Studentized deviate test is exploited to identify shot boundaries for segmenting a video sequence into shots. Then, video shots can be divided into different sub-shots, according to whether the video content change is large or not, and key frames are extracted from sub-shots. The proposed technique is general, effective and efficient to deal with video sequences of any kind. Our new approach can offer optional additional multiscale summarizations of video data, achieving a balance between having more details and maintaining less redundancy. Extensive experimental results show that the new scheme obtains very encouraging results in video key frame extraction, in terms of both objective evaluation metrics and subjective visual perception. Full article
►▼ Show Figures

Figure 1

Figure 1
<p>An overview of our proposed key frame extraction method. Here, the red dotted line, the blue dotted line and the bold black point indicate the shot boundary, sub-shot boundary and key frame, respectively. RE, relative entropy; SRRE, square root of RE; ESD, extreme Studentized deviate.</p> Full article ">Figure 2
<p>The square root function on RE “amplifies” the distance between Frames #492 and #493, where the content changes little.</p> Full article ">Figure 3
<p>An example of obtaining a range of adaptive thresholds used for the ESD-based shot detection. Due to the space limitation, the maximum and minimum ratio values <span class="html-italic">versus</span> window size for only three of all of the test videos are shown (black, green and blue colors correspond to the three videos). The “×” markers respectively on the dashed and solid lines shows the maximum and minimum distance ratios for the valid shot boundaries. The upper and lower bounds of the threshold, corresponding to the red dashed and solid lines, respectively, are also displayed.</p> Full article ">Figure 4
<p>The wrong rate (number of wrong outliers/number of outliers) increases consistently when the window size becomes bigger.</p> Full article ">Figure 5
<p>Multiscale key frames of Video Clip “7”. Scale 5–18 are skipped. The key frames with the same outlines in color and line style are from the same shot. (<b>a</b>) Scale 1; (<b>b</b>) Scale 2; (<b>c</b>) Scale 3; (<b>d</b>) Scale 4; (<b>e</b>) Scale 19.</p> Full article ">Figure 6
<p>Objective comparisons of different methods. (<b>a</b>) Video sampling error (VSE) results from Videos 1–23; (<b>b</b>) VSE results from Videos 24–46; (<b>c</b>) fidelity (FID) results from Videos 1–23; (<b>d</b>) FID results from Videos 24–46. JSD, Jensen–Shannon divergence; ED, entropy difference; MI, mutual information; GGD, generalized Gaussian density.</p> Full article ">Figure 6 Cont.
<p>Objective comparisons of different methods. (<b>a</b>) Video sampling error (VSE) results from Videos 1–23; (<b>b</b>) VSE results from Videos 24–46; (<b>c</b>) fidelity (FID) results from Videos 1–23; (<b>d</b>) FID results from Videos 24–46. JSD, Jensen–Shannon divergence; ED, entropy difference; MI, mutual information; GGD, generalized Gaussian density.</p> Full article ">Figure 6 Cont.
<p>Objective comparisons of different methods. (<b>a</b>) Video sampling error (VSE) results from Videos 1–23; (<b>b</b>) VSE results from Videos 24–46; (<b>c</b>) fidelity (FID) results from Videos 1–23; (<b>d</b>) FID results from Videos 24–46. JSD, Jensen–Shannon divergence; ED, entropy difference; MI, mutual information; GGD, generalized Gaussian density.</p> Full article ">Figure 7
<p>Subjective comparisons on different methods. (<b>a</b>) Subjective scores from Videos 1–23; (<b>b</b>) subjective scores from Videos 24–46.</p> Full article ">Figure 7 Cont.
<p>Subjective comparisons on different methods. (<b>a</b>) Subjective scores from Videos 1–23; (<b>b</b>) subjective scores from Videos 24–46.</p> Full article ">Figure 8
<p>Runtime consumptions by different methods. (<b>a</b>) Runtimes from Videos 1–32; (<b>b</b>) runtimes from Videos 33–46.</p> Full article ">Figure 9
<p>Memory usage consumptions by different methods. (<b>a</b>) Memory usage from Video 1–32; (<b>b</b>) memory usage from Video 33–46.</p> Full article ">Figure 10
<p>Comparison of different methods on Video “25”. (<b>a</b>) Uniformly down-sampled images; (<b>b</b>) RE; (<b>c</b>) SRRE; (<b>d</b>) JSD; (<b>e</b>) ED; (<b>f</b>) GGD; (<b>g</b>) MI; (<b>h</b>) RE for wavelet video; (<b>i</b>) SRRE for wavelet video.</p> Full article ">Figure 11
<p>Objective comparisons of RE and SRRE.</p> Full article ">Figure 11 Cont.
<p>Objective comparisons of RE and SRRE.</p> Full article ">Figure 12
<p>Key frames selected by our methods using RE and SRRE on common Video “15”. (<b>a</b>) Uniformly down-sampled images; (<b>b</b>) RE; (<b>c</b>) SRRE.</p> Full article ">Figure 13
<p>Key frames selected by our methods using RE and SRRE on wavelet Video “15”. (<b>a</b>) Uniformly down-sampled images; (<b>b</b>) RE; (<b>c</b>) SRRE.</p> Full article ">Figure 14
<p>Key frames selected by our methods using RE on wavelet Video “30”. (<b>a</b>) Uniformly down-sampled images; (<b>b</b>) RE for common video; (<b>c</b>) RE for wavelet video.</p> Full article ">Figure 15
<p>Key frames selected by our methods using SRRE on wavelet Video “30”. (<b>a</b>) Uniformly down-sampled images; (<b>b</b>) SRRE for common video; (<b>c</b>) SRRE for wavelet video.</p> Full article ">
762 KiB  
Article
Entropy Production in the Theory of Heat Conduction in Solids
by Federico Zullo
Entropy 2016, 18(3), 87; https://doi.org/10.3390/e18030087 - 8 Mar 2016
Cited by 14 | Viewed by 6579
Abstract
The evolution of the entropy production in solids due to heat transfer is usually associated with the Prigogine’s minimum entropy production principle. In this paper, we propose a critical review of the results of Prigogine and some comments on the succeeding literature. We [...] Read more.
The evolution of the entropy production in solids due to heat transfer is usually associated with the Prigogine’s minimum entropy production principle. In this paper, we propose a critical review of the results of Prigogine and some comments on the succeeding literature. We suggest a characterization of the evolution of the entropy production of the system through the generalized Fourier modes, showing that they are the only states with a time independent entropy production. The variational approach and a Lyapunov functional of the temperature, monotonically decreasing with time, are discussed. We describe the analytic properties of the entropy production as a function of time in terms of the generalized Fourier coefficients of the system. Analytical tools are used throughout the paper and numerical examples will support the statements. Full article
(This article belongs to the Special Issue Maximum versus Minimum Entropy Generation: Theoretical Developments and Applications)
►▼ Show Figures

Figure 1

Figure 1
<p>From left to right: plots of the initial profile of temperature, of the entropy production and its derivative corresponding to the temperature described by Equation (44). In the first three plots above, the parameters are set as <math display="inline"> <semantics> <mrow> <msub> <mi>T</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>10</mn> <msub> <mi>T</mi> <mn>0</mn> </msub> </mrow> </semantics> </math> and <math display="inline"> <semantics> <mrow> <mi>p</mi> <mo>=</mo> <mn>15</mn> <msub> <mi>T</mi> <mn>0</mn> </msub> </mrow> </semantics> </math> ; for the plots below, <math display="inline"> <semantics> <mrow> <msub> <mi>T</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>10</mn> <msub> <mi>T</mi> <mn>0</mn> </msub> </mrow> </semantics> </math> and <math display="inline"> <semantics> <mrow> <mi>p</mi> <mo>=</mo> <mo>−</mo> <mn>15</mn> <msub> <mi>T</mi> <mn>0</mn> </msub> </mrow> </semantics> </math>.</p> Full article ">Figure 2
<p>From left to right: plots of the initial profile of temperature, of the entropy production and its derivative corresponding to the temperature described by Equation (45). The parameters are set as <math display="inline"> <semantics> <mrow> <msub> <mi>T</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>10</mn> <msub> <mi>T</mi> <mn>0</mn> </msub> </mrow> </semantics> </math> and <math display="inline"> <semantics> <mrow> <mi>q</mi> <mo>=</mo> <mo>−</mo> <msub> <mi>T</mi> <mn>0</mn> </msub> </mrow> </semantics> </math>.</p> Full article ">
2714 KiB  
Article
Exergy and Thermoeconomic Analysis for an Underground Train Station Air-Conditioning Cooling System
by Ke Yang Liao and Yew Khoy Chuah
Entropy 2016, 18(3), 86; https://doi.org/10.3390/e18030086 - 7 Mar 2016
Cited by 2 | Viewed by 7635
Abstract
The necessity of air-conditioning causes the enormous energy use of underground train stations. Exergy and thermoeconomic analysis is applied to the annual operation of the air-conditioning system of a large underground train station in Taiwan. The current installation and the monitored data are [...] Read more.
The necessity of air-conditioning causes the enormous energy use of underground train stations. Exergy and thermoeconomic analysis is applied to the annual operation of the air-conditioning system of a large underground train station in Taiwan. The current installation and the monitored data are taken to be the base case, which is then compared to three different optimized designs. The total revenue requirement levelized cost rate and the total exergy destruction rate are used to evaluate the merits. The results show that the cost optimization objective would obtain a lower total revenue requirement levelized cost rate, but at the expense of a higher total exergy destruction rate. Optimization of thermodynamic efficiency, however, leads to a lower total exergy destruction rate, but would increase the total revenue requirement levelized cost rate significantly. It has been shown that multi-objective optimization would result in a small marginal increase in total revenue requirement levelized cost rate, but achieve a significantly lower total exergy destruction rate. Results in terms of the normalized total revenue requirement levelized cost rate and the normalized total exergy destruction rate are also presented. It has been shown by second law analysis when applied to underground train stations that lower annual energy use and lower CO2 emissions can be achieved. Full article
(This article belongs to the Special Issue Entropy and the Economy)
►▼ Show Figures

Figure 1

Figure 1
<p>Station floor plan and zones served by a general and a 24-h AC system. (<b>a</b>) The ground floor plan; (<b>b</b>) the underground U-1 floor plan; (<b>c</b>) the underground U-2 floor plan.</p> Full article ">Figure 2
<p>The schematics of the 24-h and the general air-conditioning systems. (<b>a</b>) The schematics of the 24-h air-conditioning system; (<b>b</b>) the schematics of the general air-conditioning system. AHU, air handling unit; FC, fan coils.</p> Full article ">Figure 3
<p>Hourly electricity consumption of the underground train station in 2012.</p> Full article ">Figure 4
<p>Monthly electricity cost of the underground train station in 2012.</p> Full article ">Figure 5
<p>Daily cooling load of the underground train station under study.</p> Full article ">Figure 6
<p>Daily refrigerating exergy of the underground train station under study in 2012.</p> Full article ">Figure 7
<p>The monthly average COP of the chillers. (<b>a</b>) The 24-h air-conditioning system; (<b>b</b>) the general air-conditioning system.</p> Full article ">Figure 8
<p>The monthly system performance factor and second law efficiency.</p> Full article ">Figure 9
<p>The normalized Pareto optimal frontier.</p> Full article ">Figure 10
<p>Exergy destruction of the air-conditioning system.</p> Full article ">Figure 11
<p>Levelized costs rate for the four cases.</p> Full article ">Figure 12
<p>Percentage comparison of the four cases.</p> Full article ">
2402 KiB  
Article
iDoRNA: An Interacting Domain-based Tool for Designing RNA-RNA Interaction Systems
by Jittrawan Thaiprasit, Boonserm Kaewkamnerdpong, Dujduan Waraho-Zhmayev, Supapon Cheevadhanarak and Asawin Meechai
Entropy 2016, 18(3), 83; https://doi.org/10.3390/e18030083 - 7 Mar 2016
Cited by 1 | Viewed by 5093
Abstract
RNA-RNA interactions play a crucial role in gene regulation in living organisms. They have gained increasing interest in the field of synthetic biology because of their potential applications in medicine and biotechnology. However, few novel regulators based on RNA-RNA interactions with desired structures [...] Read more.
RNA-RNA interactions play a crucial role in gene regulation in living organisms. They have gained increasing interest in the field of synthetic biology because of their potential applications in medicine and biotechnology. However, few novel regulators based on RNA-RNA interactions with desired structures and functions have been developed due to the challenges of developing design tools. Recently, we proposed a novel tool, called iDoDe, for designing RNA-RNA interacting sequences by first decomposing RNA structures into interacting domains and then designing each domain using a stochastic algorithm. However, iDoDe did not provide an optimal solution because it still lacks a mechanism to optimize the design. In this work, we have further developed the tool by incorporating a genetic algorithm (GA) to find an RNA solution with maximized structural similarity and minimized hybridized RNA energy, and renamed the tool iDoRNA. A set of suitable parameters for the genetic algorithm were determined and found to be a weighting factor of 0.7, a crossover rate of 0.9, a mutation rate of 0.1, and the number of individuals per population set to 8. We demonstrated the performance of iDoRNA in comparison with iDoDe by using six RNA-RNA interaction models. It was found that iDoRNA could efficiently generate all models of interacting RNAs with far more accuracy and required far less computational time than iDoDe. Moreover, we compared the design performance of our tool against existing design tools using forty-four RNA-RNA interaction models. The results showed that the performance of iDoRNA is better than RiboMaker when considering the ensemble defect, the fitness score and computation time usage. However, it appears that iDoRNA is outperformed by NUPACK and RNAiFold 2.0 when considering the ensemble defect. Nevertheless, iDoRNA can still be an useful alternative tool for designing novel RNA-RNA interactions in synthetic biology research. The source code of iDoRNA can be downloaded from the site http://synbio.sbi.kmutt.ac.th. Full article
(This article belongs to the Special Issue Entropy and RNA Structure, Folding and Mechanics)
►▼ Show Figures

Graphical abstract

Graphical abstract
Full article ">Figure 1
<p>Workflows of (<b>a</b>) iDoDe and (<b>b</b>) iDoRNA. Numerical numbers indicate three main modules: 1, Initialization; 2, Evaluation; 3, Reproduction. (D, interacting domain; IR, individual RNA; HR, hybridized RNA; HD, hamming distance; MFE, minimal free energy of hybridized RNA; <span class="html-italic">f<sub>sim</sub></span>, Similarity score; <span class="html-italic">f<sub>sta</sub></span>, Stability score; <span class="html-italic">F</span>, Fitness score).</p> Full article ">Figure 2
<p>The representation of RNA individuals.</p> Full article ">Figure 3
<p>Schematic diagram of the reproduction module.</p> Full article ">Figure 4
<p>The evolution of RNA solution at different <span class="html-italic">ω<sub>sim</sub></span> (<span class="html-italic">ω<sub>sim</sub></span> = 0.3, 0.5, 0.7, and 1.0 depicted with a solid circle, a clear diamond, a solid square, and a solid triangle) presented on (<b>a</b>) initial population, (<b>b</b>) 5th generation, (<b>c</b>) 15th generation, and (<b>d</b>) 30th generation.</p> Full article ">Figure 5
<p>Effect of (<b>a</b>) crossover rate (<span class="html-italic">Cr</span>) and (<b>b</b>) mutation (<span class="html-italic">Mr</span>) on fitness score.</p> Full article ">
2692 KiB  
Article
Assessing the Robustness of Thermoeconomic Diagnosis of Fouled Evaporators: Sensitivity Analysis of the Exergetic Performance of Direct Expansion Coils
by Antonio Piacentino and Pietro Catrini
Entropy 2016, 18(3), 85; https://doi.org/10.3390/e18030085 - 5 Mar 2016
Cited by 12 | Viewed by 6077
Abstract
Thermoeconomic diagnosis of refrigeration systems is a pioneering approach to the diagnosis of malfunctions, which has been recently proven to achieve good performances for the detection of specific faults. Being an exergy-based diagnostic technique, its performance is influenced by the trends of exergy [...] Read more.
Thermoeconomic diagnosis of refrigeration systems is a pioneering approach to the diagnosis of malfunctions, which has been recently proven to achieve good performances for the detection of specific faults. Being an exergy-based diagnostic technique, its performance is influenced by the trends of exergy functions in the “design” and “abnormal” conditions. In this paper the sensitivity of performance of thermoeconomic diagnosis in detecting a fouled direct expansion coil and quantifying the additional consumption it induces is investigated; this fault is critical due to the simultaneous air cooling and dehumidification occurring in the coil, that induce variations in both the chemical and thermal fractions of air exergy. The examined parameters are the temperature and humidity of inlet air, the humidity of reference state and the sensible/latent heat ratio (varied by considering different coil depths). The exergy analysis reveals that due to the more intense dehumidification occurring in presence of fouling, the exergy efficiency of the evaporator coil eventually increases. Once the diagnostic technique is based only on the thermal fraction of air exergy, the results suggest that the performance of the technique increases when inlet air has a lower absolute humidity, as evident from the “optimal performance” regions identified on a psychrometric chart. Full article
(This article belongs to the Special Issue Thermoeconomics for Energy Efficiency)
►▼ Show Figures

Graphical abstract

Graphical abstract
Full article ">Figure 1
<p>Performance of clean coils with different geometries at a same air inlet condition, (<b>a</b>) 3 rows; (<b>b</b>) 5 rows; (<b>c</b>) 7 rows.</p> Full article ">Figure 1 Cont.
<p>Performance of clean coils with different geometries at a same air inlet condition, (<b>a</b>) 3 rows; (<b>b</b>) 5 rows; (<b>c</b>) 7 rows.</p> Full article ">Figure 2
<p>Reference simplified physical scheme of the system.</p> Full article ">Figure 3
<p>Exergetic efficiency of the 5-rows evaporator coil, assumed as clean, for different air inlet conditions.</p> Full article ">Figure 4
<p>Malfunction cost <math display="inline"> <semantics> <mrow> <msubsup> <mrow> <mtext>MF</mtext> </mrow> <mn>4</mn> <mo>*</mo> </msubsup> </mrow> </semantics> </math>, Fuel Impact ΔF<sub>T</sub> and performance indicator of the diagnostic technique Ψ<sup>fault 4</sup>, for all the scenarios and different coil geometries: (<b>a</b>) 3 rows; (<b>b</b>) 5 rows; (<b>c</b>) 7 rows.</p> Full article ">Figure 4 Cont.
<p>Malfunction cost <math display="inline"> <semantics> <mrow> <msubsup> <mrow> <mtext>MF</mtext> </mrow> <mn>4</mn> <mo>*</mo> </msubsup> </mrow> </semantics> </math>, Fuel Impact ΔF<sub>T</sub> and performance indicator of the diagnostic technique Ψ<sup>fault 4</sup>, for all the scenarios and different coil geometries: (<b>a</b>) 3 rows; (<b>b</b>) 5 rows; (<b>c</b>) 7 rows.</p> Full article ">Figure 5
<p>Representation of performance of the thermoeconomic diagnostic technique on the psychrometric chart, for different coil geometries: (<b>a</b>) 3 rows; (<b>b</b>) 5 rows; (<b>c</b>) 7 rows.</p> Full article ">Figure 5 Cont.
<p>Representation of performance of the thermoeconomic diagnostic technique on the psychrometric chart, for different coil geometries: (<b>a</b>) 3 rows; (<b>b</b>) 5 rows; (<b>c</b>) 7 rows.</p> Full article ">
906 KiB  
Article
Entropy and Fractal Antennas
by Emanuel Guariglia
Entropy 2016, 18(3), 84; https://doi.org/10.3390/e18030084 - 4 Mar 2016
Cited by 176 | Viewed by 14437
Abstract
The entropies of Shannon, Rényi and Kolmogorov are analyzed and compared together with their main properties. The entropy of some particular antennas with a pre-fractal shape, also called fractal antennas, is studied. In particular, their entropy is linked with the fractal geometrical shape [...] Read more.
The entropies of Shannon, Rényi and Kolmogorov are analyzed and compared together with their main properties. The entropy of some particular antennas with a pre-fractal shape, also called fractal antennas, is studied. In particular, their entropy is linked with the fractal geometrical shape and the physical performance. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory I)
►▼ Show Figures

Figure 1

Figure 1
<p>Rényi and Shannon entropies for a binomial distribution with N = 20: they converge for <math display="inline"> <mrow> <mi>α</mi> <mo>→</mo> <mn>1</mn> </mrow> </math>, in accord with Equation (<a href="#FD3-entropy-18-00084" class="html-disp-formula">3</a>). The computation of both entropies was done for <math display="inline"> <mrow> <mi>b</mi> <mo>=</mo> <mn>2</mn> </mrow> </math>.</p> Full article ">Figure 2
<p>Kolmogorov entropy for 1D-regular, chaotic-deterministic and random systems. The attractor is the classical Lorentz attractor [<a href="#B27-entropy-18-00084" class="html-bibr">27</a>] with <math display="inline"> <mrow> <mi>σ</mi> <mo>=</mo> <mn>10</mn> </mrow> </math>, <math display="inline"> <mrow> <mi>R</mi> <mo>=</mo> <mn>28</mn> </mrow> </math>, <math display="inline"> <mrow> <mi>b</mi> <mo>=</mo> <mn>8</mn> <mo>/</mo> <mn>3</mn> </mrow> </math> and initial values <math display="inline"> <mrow> <mi>x</mi> <mo>(</mo> <mn>0</mn> <mo>)</mo> <mo>=</mo> <mi>z</mi> <mo>(</mo> <mn>0</mn> <mo>)</mo> <mo>=</mo> <mn>0</mn> </mrow> </math>, <math display="inline"> <mrow> <mi>y</mi> <mo>(</mo> <mn>0</mn> <mo>)</mo> <mo>=</mo> <mn>1</mn> </mrow> </math>, while the random motion is given by a 2D-random walk from <math display="inline"> <mrow> <mo>-</mo> <mn>1</mn> <mo>-</mo> <mi>j</mi> </mrow> </math> to <math display="inline"> <mrow> <mn>1</mn> <mo>+</mo> <mi>j</mi> </mrow> </math> of 500 elements in which <span class="html-italic">j</span> is the imaginary unit.</p> Full article ">Figure 3
<p>Graph of the <math display="inline"> <mrow> <msub> <mi mathvariant="script">M</mi> <mi>s</mi> </msub> <mrow> <mo>(</mo> <mi>A</mi> <mo>)</mo> </mrow> </mrow> </math>, where <span class="html-italic">A</span> is a bounded subset of the Euclidean metric space <math display="inline"> <msup> <mrow> <mi>R</mi> </mrow> <mi>n</mi> </msup> </math>. It takes only two possible values, and the Hausdorff–Besicovitch dimension of <span class="html-italic">A</span> is given by the value of <span class="html-italic">s</span> in which there is the jump from <span class="html-italic">∞</span> to zero.</p> Full article ">Figure 4
<p>Here, the first steps of the box-counting procedure about England’s coastline are represented.</p> Full article ">Figure 5
<p>Here, the von Kock curve (on the left) and the middle third Cantor set (on the right) are shown: <math display="inline"> <msub> <mi>A</mi> <mn>0</mn> </msub> </math> is the initiator of length equal to one in both cases; in the generator <math display="inline"> <msub> <mi>A</mi> <mn>1</mn> </msub> </math> for the von Kock curve, the middle third of the unit interval is replaced by the other two sides of an equilateral triangle, while that of the middle third Cantor set is obtained removing the middle third of the interval.</p> Full article ">Figure 6
<p>Archimedean spiral antenna (on the left) and commercial log-periodic dipole antenna of 16 elements (on the right) [<a href="#B7-entropy-18-00084" class="html-bibr">7</a>].</p> Full article ">Figure 7
<p>A Sierpinski triangle (on the left) and a Hilbert curve (on the right) are shown: as in <a href="#entropy-18-00084-f005" class="html-fig">Figure 5</a>, <math display="inline"> <msub> <mi>A</mi> <mn>0</mn> </msub> </math> is the initiator, and <math display="inline"> <msub> <mi>A</mi> <mn>1</mn> </msub> </math> is the generator. The Sierpinski triangle is constructed using the iterated function system (IFS), while the other construction is that of David Hilbert. The two relative antennas are shown below with their feed points.</p> Full article ">Figure 8
<p>Examples of non-fractal antennas that offer similar performance over their fractal counterparts. Three different non-fractal antennas are presented above: they outperform their fractal counterparts, while the current distribution on the Sierpinski gasket antenna at the first three resonance frequencies is shown in the middle of the page. The modified Parany antenna (starting from the classical Sierpinski gasket antenna) is represented below.</p> Full article ">Figure 9
<p>The Rényi entropy <math display="inline"> <msub> <mi>H</mi> <mi>α</mi> </msub> </math> of a Sierpinski gasket (<a href="#entropy-18-00084-f007" class="html-fig">Figure 7</a>) with <math display="inline"> <mrow> <mi>α</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </math>: the plot shows us that Hartley entropy <math display="inline"> <msub> <mi>H</mi> <mn>0</mn> </msub> </math> is an upper to both Shannon entropy <math display="inline"> <msub> <mi>H</mi> <mn>1</mn> </msub> </math> and collision entropy <math display="inline"> <msub> <mi>H</mi> <mn>2</mn> </msub> </math>. The main limit of this procedure is clearly the precision of the triangulation.</p> Full article ">
1614 KiB  
Article
Hierarchical Decomposition Thermodynamic Approach for the Study of Solar Absorption Refrigerator Performance
by Emma Berrich Betouche, Ali Fellah, Ammar Ben Brahim, Fethi Aloui and Michel Feidt
Entropy 2016, 18(3), 82; https://doi.org/10.3390/e18030082 - 4 Mar 2016
Cited by 2 | Viewed by 4874
Abstract
A thermodynamic approach based on the hierarchical decomposition which is usually used in mechanical structure engineering is proposed. The methodology is applied to an absorption refrigeration cycle. Thus, a thermodynamic analysis of the performances on solar absorption refrigerators is presented. Under the hypothesis [...] Read more.
A thermodynamic approach based on the hierarchical decomposition which is usually used in mechanical structure engineering is proposed. The methodology is applied to an absorption refrigeration cycle. Thus, a thermodynamic analysis of the performances on solar absorption refrigerators is presented. Under the hypothesis of an endoreversible model, the effects of the generator, the solar concentrator and the solar converter temperatures, on the coefficient of performance (COP), are presented and discussed. In fact, the coefficient of performance variations, according to the ratio of the heat transfer areas of the high temperature part (the thermal engine 2) Ah and the heat transfer areas of the low temperature part (the thermal receptor) Ar variations, are studied in this paper. For low values of the heat-transfer areas of the high temperature part and relatively important values of heat-transfer areas of the low temperature part as for example Ah equal to 30% of Ar, the coefficient of performance is relatively important (approximately equal to 65%). For an equal-area distribution corresponding to an area ratio Ah/Ar of 50%, the COP is approximately equal to 35%. The originality of this deduction is that it allows a conceptual study of the solar absorption cycle. Full article
(This article belongs to the Special Issue Entropy Generation in Thermal Systems and Processes 2015)
►▼ Show Figures

Figure 1

Figure 1
<p>Equivalent model of the solar absorption refrigeration cycle.</p> Full article ">Figure 2
<p>Hierarchical decomposition of the solar absorption refrigeration cycle.</p> Full article ">Figure 3
<p>Refrigeration and control device.</p> Full article ">Figure 4
<p>Solar concentrator temperature effect on the entropy (<span class="html-italic">T<sub>si</sub></span> = 28 °C, <span class="html-italic">T<sub>sf</sub></span> = 10 °C).</p> Full article ">Figure 5
<p>Solar concentrator temperature effect on the Coefficient of Performance for (<span class="html-italic">T<sub>si</sub></span> = 28 °C, <span class="html-italic">T<sub>sf</sub></span> = 10 °C).</p> Full article ">Figure 6
<p>Effect of the solar converter temperature on the Coefficient of Performance.</p> Full article ">Figure 7
<p>Effect of the generator temperature on the Coefficient of Performance.</p> Full article ">Figure 8
<p>Effect of the areas distribution on the Coefficient of Performance.</p> Full article ">
495 KiB  
Article
Phase Transitions in Equilibrium and Non-Equilibrium Models on Some Topologies
by Francisco W. De Sousa Lima
Entropy 2016, 18(3), 81; https://doi.org/10.3390/e18030081 - 3 Mar 2016
Cited by 2 | Viewed by 5489
Abstract
On some regular and non-regular topologies, we studied the critical properties of models that present up-down symmetry, like the equilibrium Ising model and the nonequilibrium majority vote model. These are investigated on networks, like Apollonian (AN), Barabási–Albert (BA), small-worlds (SW), Voronoi–Delaunay (VD) and [...] Read more.
On some regular and non-regular topologies, we studied the critical properties of models that present up-down symmetry, like the equilibrium Ising model and the nonequilibrium majority vote model. These are investigated on networks, like Apollonian (AN), Barabási–Albert (BA), small-worlds (SW), Voronoi–Delaunay (VD) and Erdös–Rényi (ER) random graphs. The review here is on phase transitions, critical points, exponents and universality classes that are compared to the results obtained for these models on regular square lattices (SL). Full article
(This article belongs to the Special Issue Recent Advances in Non-Equilibrium Statistical Mechanics and Its Application)
►▼ Show Figures

Figure 1

Figure 1
<p>The magnetization <span class="html-italic">M</span> (<b>a</b>, <b>d</b>, <b>g</b> ), Binder’s cumulant <math display="inline"> <msub> <mi>U</mi> <mn>4</mn> </msub> </math> (<b>b</b>, <b>e</b>, <b>h</b> ) and susceptibility (<b>c</b>, <b>f</b>, <b>i</b> ) <span class="html-italic">vs.</span> noise parameter <span class="html-italic">q</span> for <math display="inline"> <mrow> <mi>N</mi> <mo>=</mo> <mn>367</mn> </mrow> </math>, 1096, 3283, 9844, 29,527, 88,576 and 265,723 sites and for <math display="inline"> <msub> <mi>g</mi> <mi>n</mi> </msub> </math> generations (<math display="inline"> <mrow> <mi>n</mi> <mo>=</mo> <mn>6</mn> </mrow> </math>, 7, 8, 9, 10, 11 and 12) on an Apollonian (AN) network. We use reconnection probability from <math display="inline"> <mrow> <mi>p</mi> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>0</mn> </mrow> </math> (undirected AN (UAN)) (<b>a</b>, <b>b</b>, <b>c</b> ), <math display="inline"> <mrow> <mi>p</mi> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>1</mn> </mrow> </math> (<b>d</b>, <b>e</b>, <b>f</b> ) and <math display="inline"> <mrow> <mi>p</mi> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>9</mn> </mrow> </math> (directed AN (DAN)) (<b>g</b>, <b>h</b>, <b>i</b> ).</p> Full article ">Figure 2
<p>Plot of the ln<math display="inline"> <mrow> <mi>M</mi> <mo>(</mo> <msub> <mi>q</mi> <mi>c</mi> </msub> <mo>)</mo> </mrow> </math><span class="html-italic">vs.</span> ln<span class="html-italic">N</span> for <span class="html-italic">p</span>-values from <math display="inline"> <mrow> <mi>p</mi> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>0</mn> </mrow> </math> to <math display="inline"> <mrow> <mn>0</mn> <mo>.</mo> <mn>9</mn> </mrow> </math>.</p> Full article ">Figure 3
<p>Plot of ln<math display="inline"> <mrow> <mi>χ</mi> <mo>(</mo> <mi>N</mi> <mo>)</mo> </mrow> </math><span class="html-italic">vs.</span> ln<span class="html-italic">N</span> for some values of the reconnection probability from <math display="inline"> <mrow> <mi>p</mi> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>0</mn> </mrow> </math> to <math display="inline"> <mrow> <mn>0</mn> <mo>.</mo> <mn>9</mn> </mrow> </math>.</p> Full article ">Figure 4
<p>The exponents <math display="inline"> <mrow> <mn>1</mn> <mo>/</mo> <mi>ν</mi> </mrow> </math> obtained from the relation Equation (9) for AN.</p> Full article ">
282 KiB  
Article
Minimal Length, Measurability and Gravity
by Alexander Shalyt-Margolin
Entropy 2016, 18(3), 80; https://doi.org/10.3390/e18030080 - 2 Mar 2016
Cited by 13 | Viewed by 4223
Abstract
The present work is a continuation of the previous papers written by the author on the subject. In terms of the measurability (or measurable quantities) notion introduced in a minimal length theory, first the consideration is given to a quantum theory in the [...] Read more.
The present work is a continuation of the previous papers written by the author on the subject. In terms of the measurability (or measurable quantities) notion introduced in a minimal length theory, first the consideration is given to a quantum theory in the momentum representation. The same terms are used to consider the Markov gravity model that here illustrates the general approach to studies of gravity in terms of measurable quantities. Full article
(This article belongs to the Section Astrophysics, Cosmology, and Black Holes)
437 KiB  
Article
Entanglement Entropy in a Triangular Billiard
by Sijo K. Joseph and Miguel A. F. Sanjuán
Entropy 2016, 18(3), 79; https://doi.org/10.3390/e18030079 - 1 Mar 2016
Cited by 3 | Viewed by 5285
Abstract
The Schrödinger equation for a quantum particle in a two-dimensional triangular billiard can be written as the Helmholtz equation with a Dirichlet boundary condition. We numerically explore the quantum entanglement of the eigenfunctions of the triangle billiard and its relation to the irrationality [...] Read more.
The Schrödinger equation for a quantum particle in a two-dimensional triangular billiard can be written as the Helmholtz equation with a Dirichlet boundary condition. We numerically explore the quantum entanglement of the eigenfunctions of the triangle billiard and its relation to the irrationality of the triangular geometry. We also study the entanglement dynamics of the coherent state with its center chosen at the centroid of the different triangle configuration. Using the von Neumann entropy of entanglement, we quantify the quantum entanglement appearing in the eigenfunction of the triangular domain. We see a clear correspondence between the irrationality of the triangle and the average entanglement of the eigenfunctions. The entanglement dynamics of the coherent state shows a dependence on the geometry of the triangle. The effect of quantum squeezing on the coherent state is analyzed and it can be utilize to enhance or decrease the entanglement entropy in a triangular billiard. Full article
(This article belongs to the Special Issue Entanglement Entropy)
►▼ Show Figures

Figure 1

Figure 1
<p>The ground state eigenfunctions of the different irrational triangles are shown. <a href="#entropy-18-00079-f001" class="html-fig">Figure 1</a>a–d show the irrational triangles with numbers <math display="inline"> <mrow> <mi>N</mi> <mo>=</mo> <mn>3</mn> </mrow> </math>, <math display="inline"> <mrow> <mi>N</mi> <mo>=</mo> <mn>13</mn> </mrow> </math>, <math display="inline"> <mrow> <mi>N</mi> <mo>=</mo> <mn>23</mn> </mrow> </math> and <math display="inline"> <mrow> <mi>N</mi> <mo>=</mo> <mn>33</mn> </mrow> </math>, respectively. It can be easily seen that, the first triangle shown in (<b>a</b>) is a right triangle and the triangle shown in (<b>d</b>) is approximate to an equilateral triangle, where (<b>b</b>) and (<b>c</b>) show intermediate geometries.</p> Full article ">Figure 2
<p>In (<b>a</b>) the von Neumann entropy of entanglement <math display="inline"> <msub> <mi>S</mi> <mrow> <mi>v</mi> <mi>n</mi> </mrow> </msub> </math> for the ground state eigenfunctions of the irrational triangles is plotted against the different triangle configurations. It can be easily seen that in both cases the entanglement entropy reduces as we increase <span class="html-italic">N</span> or as we approach the equilateral triangle. Red curve shows the exponential fit of data, where the equation of the curve is given by <math display="inline"> <mrow> <msub> <mi>S</mi> <mrow> <mi>v</mi> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi>A</mi> <msup> <mrow> <mi>e</mi> </mrow> <mrow> <mo>−</mo> <mi>α</mi> <mi>N</mi> </mrow> </msup> <mo>+</mo> <mi>β</mi> </mrow> </math>, where <math display="inline"> <mrow> <mi>A</mi> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>01123</mn> </mrow> </math>, <math display="inline"> <mrow> <mi>α</mi> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>08247</mn> </mrow> </math> and <math display="inline"> <mrow> <mi>β</mi> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>1227</mn> </mrow> </math>. In (<b>b</b>) the angles of the triangle <math display="inline"> <msub> <mi>θ</mi> <mn>1</mn> </msub> </math>, <math display="inline"> <msub> <mi>θ</mi> <mn>2</mn> </msub> </math> and <math display="inline"> <msub> <mi>θ</mi> <mn>3</mn> </msub> </math> are plotted for different triangle configurations and it can be seen that as N grows, the angles <math display="inline"> <mrow> <mo>(</mo> <mi>θ</mi> <mo>/</mo> <mi>π</mi> <mo>)</mo> </mrow> </math> approaches the rational value of <math display="inline"> <mrow> <mn>1</mn> <mo>/</mo> <mn>3</mn> </mrow> </math>.</p> Full article ">Figure 3
<p>The 500th excited eigenfunctions of the different irrational triangles are shown in the figure. It is the highest eigenfunction utilized to compute the average von Neumann entropy of entanglement. <a href="#entropy-18-00079-f003" class="html-fig">Figure 3</a>a–d show the irrational triangles with numbers <math display="inline"> <mrow> <mi>N</mi> <mo>=</mo> <mn>3</mn> </mrow> </math>, <math display="inline"> <mrow> <mi>N</mi> <mo>=</mo> <mn>13</mn> </mrow> </math>, <math display="inline"> <mrow> <mi>N</mi> <mo>=</mo> <mn>23</mn> </mrow> </math> and <math display="inline"> <mrow> <mi>N</mi> <mo>=</mo> <mn>33</mn> </mrow> </math>, respectively. It can be easily seen that, the first triangle shown in (<b>a</b>) is a right triangle and the triangle shown in (<b>d</b>) is approximately closer to an equilateral triangle, where (<b>b</b>) and (<b>c</b>) show intermediate ones.</p> Full article ">Figure 4
<p>The average von Neumann entropy of entanglement <math display="inline"> <msub> <mover accent="true"> <mi>S</mi> <mo stretchy="false">¯</mo> </mover> <mrow> <mi>v</mi> <mi>n</mi> </mrow> </msub> </math> is plotted against the different triangle configurations. It can be easily seen that the entanglement entropy reduces as we increase <span class="html-italic">N</span> or as we approach the equilateral triangle.</p> Full article ">Figure 5
<p>The time evolution of the entanglement entropy <math display="inline"> <mrow> <msub> <mi>S</mi> <mrow> <mi>v</mi> <mi>n</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </math> for the coherent state with a center at the centroid of the triangle is plotted for triangles with different <span class="html-italic">N</span> by fixing the Planck constant <math display="inline"> <mrow> <mi>ℏ</mi> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>025</mn> </mrow> </math>.</p> Full article ">Figure 6
<p>In (<b>a</b>) the maxima of von Neumann entropy of entanglement <math display="inline"> <msub> <mi>S</mi> <mi>M</mi> </msub> </math> for the squeezed coherent state chosen at the centroid of the irrational triangles is plotted against the different triangle configurations with <math display="inline"> <mrow> <mi>ℏ</mi> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>1</mn> </mrow> </math>. In (<b>b</b>) the maxima of von Neumann entropy of entanglement <math display="inline"> <msub> <mi>S</mi> <mi>M</mi> </msub> </math> for the squeezed coherent state chosen at the centroid of the irrational triangles is plotted against the different triangle configurations with <math display="inline"> <mrow> <mi>ℏ</mi> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>025</mn> </mrow> </math>. It can be easily seen that in both cases the entanglement entropy reduces as we increase <span class="html-italic">N</span> or as we approach the equilateral triangle.</p> Full article ">
4556 KiB  
Article
Wavelet Entropy-Based Traction Inverter Open Switch Fault Diagnosis in High-Speed Railways
by Keting Hu, Zhigang Liu and Shuangshuang Lin
Entropy 2016, 18(3), 78; https://doi.org/10.3390/e18030078 - 1 Mar 2016
Cited by 26 | Viewed by 6680
Abstract
In this paper, a diagnosis plan is proposed to settle the detection and isolation problem of open switch faults in high-speed railway traction system traction inverters. Five entropy forms are discussed and compared with the traditional fault detection methods, namely, discrete wavelet transform [...] Read more.
In this paper, a diagnosis plan is proposed to settle the detection and isolation problem of open switch faults in high-speed railway traction system traction inverters. Five entropy forms are discussed and compared with the traditional fault detection methods, namely, discrete wavelet transform and discrete wavelet packet transform. The traditional fault detection methods cannot efficiently detect the open switch faults in traction inverters because of the low resolution or the sudden change of the current. The performances of Wavelet Packet Energy Shannon Entropy (WPESE), Wavelet Packet Energy Tsallis Entropy (WPETE) with different non-extensive parameters, Wavelet Packet Energy Shannon Entropy with a specific sub-band (WPESE3,6), Empirical Mode Decomposition Shannon Entropy (EMDESE), and Empirical Mode Decomposition Tsallis Entropy (EMDETE) with non-extensive parameters in detecting the open switch fault are evaluated by the evaluation parameter. Comparison experiments are carried out to select the best entropy form for the traction inverter open switch fault detection. In addition, the DC component is adopted to isolate the failure Isolated Gate Bipolar Transistor (IGBT). The simulation experiments show that the proposed plan can diagnose single and simultaneous open switch faults correctly and timely. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory I)
►▼ Show Figures

Figure 1

Figure 1
<p>Schematic diagram of the vector controlled traction system.</p> Full article ">Figure 2
<p>(<b>a</b>) A-phase torque in normal state; (<b>b</b>) A-phase currents in normal state; (<b>c</b>) A-phase torque when <span class="html-italic">S</span>1 open switch fault occurs; (<b>d</b>) A-phase currents when <span class="html-italic">S</span>1 open switch fault occurs.</p> Full article ">Figure 3
<p>Wavelet decomposition.</p> Full article ">Figure 4
<p>(<b>a</b>) A-phase fault current with <span class="html-italic">S</span>1 open (50 kHz sampling frequency); (<b>b</b>) First level detail coefficient; (<b>c</b>) Second level detail coefficient; (<b>d</b>) Third level detail coefficient.</p> Full article ">Figure 5
<p>(<b>a</b>) spectrum of fault free current (50 kHz sampling frequency and 0.2 s acquisition time); (<b>b</b>) spectrum of A-phase fault current with <span class="html-italic">S</span>1 open (50 kHz sampling frequency and 0.2 s acquisition time).</p> Full article ">Figure 6
<p>Frequency of each sub-band after wavelet packet decomposition.</p> Full article ">Figure 7
<p>(<b>a</b>) A-phase fault current with <span class="html-italic">S</span>1 open; (<b>b</b>) Coefficients of the sixth node in the third level.</p> Full article ">Figure 8
<p>(<b>a</b>) WPESE of A-phase fault current with <span class="html-italic">S</span>1 open; (<b>b</b>) WPETE of A-phase fault current with <span class="html-italic">S</span>1 open when <span class="html-italic">q</span> = 0.1; (<b>c</b>) WPETE of A-phase fault current with <span class="html-italic">S</span>1 open when <span class="html-italic">q</span> = 0.5; (<b>d</b>) WPETE of A-phase fault current with <span class="html-italic">S</span>1 open when <span class="html-italic">q</span> = 0.9.</p> Full article ">Figure 9
<p>(<b>a</b>) WPESE of A-phase fault current with <span class="html-italic">S</span>1 open; (<b>b</b>) WPETE of A-phase fault current with <span class="html-italic">S</span>1 open when <span class="html-italic">q</span> = 2; (<b>c</b>) WPETE of A-phase fault current with <span class="html-italic">S</span>1 open when <span class="html-italic">q</span> = 5; (<b>d</b>) WPETE of A-phase fault current with <span class="html-italic">S</span>1 open when <span class="html-italic">q</span> = 9.</p> Full article ">Figure 10
<p>WPETE<sub>3,6</sub> of fault current.</p> Full article ">Figure 11
<p>EMD results of A-phase fault current with <span class="html-italic">S</span>1 open.</p> Full article ">Figure 12
<p>(<b>a</b>) EMDESE of A-phase fault current with <span class="html-italic">S</span>1 open; (<b>b</b>) EMDETE of A-phase fault current with <span class="html-italic">S</span>1 open when <span class="html-italic">q</span> = 0.1; (<b>c</b>) EMDETE of A-phase fault current with <span class="html-italic">S</span>1 open when <span class="html-italic">q</span> = 0.5; (<b>d</b>) EMDETE of A-phase fault current with <span class="html-italic">S</span>1 open when <span class="html-italic">q</span> = 0.9.</p> Full article ">Figure 13
<p>(<b>a</b>) EMDESE of A-phase fault current with <span class="html-italic">S</span>1 open; (<b>b</b>) EMDETE of A-phase fault current with <span class="html-italic">S</span>1 open when <span class="html-italic">q</span> = 2; (<b>c</b>) EMDETE of A-phase fault current with <span class="html-italic">S</span>1 open when <span class="html-italic">q</span> = 5; (<b>d</b>) EMDETE of A-phase fault current with <span class="html-italic">S</span>1 open when <span class="html-italic">q</span> = 9.</p> Full article ">Figure 14
<p>(<b>a</b>) WPETE of fault current when <span class="html-italic">q</span> is in the range of [0.1,0.9]; (<b>b</b>) WPETE of fault current when <span class="html-italic">q</span> is in the range of [2,9]; (<b>c</b>) EMDETE of fault current when <span class="html-italic">q</span> is in the range of [0.1,0.9]; (<b>d</b>) EMDETE of fault current when <span class="html-italic">q</span> is in the range of [2,9].</p> Full article ">Figure 15
<p>(<b>a</b>) Current flow of USF; Current wave when USF occurs in(<b>b</b>) negative current flow; (<b>c</b>) positive current flow; (<b>d</b>) Current flow of LSF; Current wave when LSF occurs in (<b>e</b>) positive current flow; (<b>f</b>) negative current flow.</p> Full article ">Figure 16
<p>Flow chart of the proposed method.</p> Full article ">Figure 17
<p>(<b>a</b>) A-phase current in normal condition; (<b>b</b>) A-phase WPESE<sub>3,6</sub>; (<b>c</b>) B-phase current in normal condition; (<b>d</b>) B-phase WPESE<sub>3,6</sub>; (<b>e</b>) C-phase current in normal condition; (<b>f</b>) C-phase WPESE<sub>3,6</sub>.</p> Full article ">Figure 18
<p>Normal condition (<b>a</b>) Torque; (<b>b</b>) Speed of the traction motor; (<b>c</b>) Three phase current.</p> Full article ">Figure 19
<p>Fault A condition (<b>a</b>) A-phase WPESE<sub>3,6</sub>; (<b>b</b>) B-phase WPESE<sub>3,6</sub>; (<b>c</b>) A-phase DC component; (<b>d</b>) C-phase WPESE<sub>3,6</sub>.</p> Full article ">Figure 20
<p>Fault B condition (<b>a</b>) A-phase WPESE<sub>3,6</sub>; (<b>b</b>) B-phase WPESE<sub>3,6</sub>; (<b>c</b>) A-phase DC component; (<b>d</b>) B-phase DC component; (<b>d</b>) C-phase WPESE<sub>3,6</sub>; (<b>f</b>) C-phase DC component.</p> Full article ">
3246 KiB  
Article
Tea Category Identification Using a Novel Fractional Fourier Entropy and Jaya Algorithm
by Yudong Zhang, Xiaojun Yang, Carlo Cattani, Ravipudi Venkata Rao, Shuihua Wang and Preetha Phillips
Entropy 2016, 18(3), 77; https://doi.org/10.3390/e18030077 - 27 Feb 2016
Cited by 99 | Viewed by 10039
Abstract
This work proposes a tea-category identification (TCI) system, which can automatically determine tea category from images captured by a 3 charge-coupled device (CCD) digital camera. Three-hundred tea images were acquired as the dataset. Apart from the 64 traditional color histogram features that were [...] Read more.
This work proposes a tea-category identification (TCI) system, which can automatically determine tea category from images captured by a 3 charge-coupled device (CCD) digital camera. Three-hundred tea images were acquired as the dataset. Apart from the 64 traditional color histogram features that were extracted, we also introduced a relatively new feature as fractional Fourier entropy (FRFE) and extracted 25 FRFE features from each tea image. Furthermore, the kernel principal component analysis (KPCA) was harnessed to reduce 64 + 25 = 89 features. The four reduced features were fed into a feedforward neural network (FNN). Its optimal weights were obtained by Jaya algorithm. The 10 × 10-fold stratified cross-validation (SCV) showed that our TCI system obtains an overall average sensitivity rate of 97.9%, which was higher than seven existing approaches. In addition, we used only four features less than or equal to state-of-the-art approaches. Our proposed system is efficient in terms of tea-category identification. Full article
(This article belongs to the Special Issue Computational Complexity)
►▼ Show Figures

Figure 1

Figure 1
<p>Tea image acquiring system.</p> Full article ">Figure 2
<p>Fractional Fourier transform (FRFT) of tri(<span class="html-italic">t</span>).</p> Full article ">Figure 3
<p>Flowchart of feature processing.</p> Full article ">Figure 4
<p>Diagram of one-hidden-layer feedforward neural network (OHL-FNN).</p> Full article ">Figure 5
<p>Diagram of Jaya algorithm.</p> Full article ">Figure 6
<p>Color histogram of green, Oolong, and black tea [<a href="#B28-entropy-18-00077" class="html-bibr">28</a>].</p> Full article ">Figure 7
<p>2D-FRFT of an image of Oolong tea.</p> Full article ">Figure 8
<p>Comparing KPCA with PCA over 3D simulation data. (<b>a</b>) 3D Simulation Data; (<b>b</b>) PCA Result with 2 PCs; (<b>c</b>) POL-KPCA with 2 PCs; (<b>d</b>) RBF-KPCA with 2 PCs.</p> Full article ">Figure 9
<p>Curve of accumulated variance <span class="html-italic">versus</span> principal component number.</p> Full article ">Figure 10
<p>Comparison of different FNN training methods.</p> Full article ">
3876 KiB  
Article
Multi-Agent System Supporting Automated Large-Scale Photometric Computations
by Adam Sȩdziwy and Leszek Kotulski
Entropy 2016, 18(3), 76; https://doi.org/10.3390/e18030076 - 27 Feb 2016
Cited by 8 | Viewed by 4371
Abstract
The technologies related to green energy, smart cities and similar areas being dynamically developed in recent years, face frequently problems of a computational nature rather than a technological one. The example is the ability of accurately predicting the weather conditions for PV farms [...] Read more.
The technologies related to green energy, smart cities and similar areas being dynamically developed in recent years, face frequently problems of a computational nature rather than a technological one. The example is the ability of accurately predicting the weather conditions for PV farms or wind turbines. Another group of issues is related to the complexity of the computations required to obtain an optimal setup of a solution being designed. In this article, we present the case representing the latter group of problems, namely designing large-scale power-saving lighting installations. The term “large-scale” refers to an entire city area, containing tens of thousands of luminaires. Although a simple power reduction for a single street, giving limited savings, is relatively easy, it becomes infeasible for tasks covering thousands of luminaires described by precise coordinates (instead of simplified layouts). To overcome this critical issue, we propose introducing a formal representation of a computing problem and applying a multi-agent system to perform design-related computations in parallel. The important measure introduced in the article indicating optimization progress is entropy. It also allows for terminating optimization when the solution is satisfying. The article contains the results of real-life calculations being made with the help of the presented approach. Full article
(This article belongs to the Section Complexity)
►▼ Show Figures

Figure 1

Figure 1
<p>(<b>a</b>) Actual road layout with physical luminaires <math display="inline"> <mrow> <msub> <mi>L</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>L</mi> <mn>2</mn> </msub> <mi>…</mi> </mrow> </math>; and (<b>b</b>) its simplified form. <math display="inline"> <mrow> <msubsup> <mi>L</mi> <mn>1</mn> <mo>′</mo> </msubsup> <mo>,</mo> <msubsup> <mi>L</mi> <mn>2</mn> <mo>′</mo> </msubsup> <mo>,</mo> <mi>…</mi> </mrow> </math> denote “averaged” luminaire positions. In both cases, the sample calculation fields are filled with an oblique hatching.</p> Full article ">Figure 2
<p>The real-life case of the sample calculation point. The dashed line delimits the region containing luminaires that has to be taken to compute luminance and illuminance at the point.</p> Full article ">Figure 3
<p>The initial phase of a computing process. <math display="inline"> <msub> <mi>R</mi> <mn>0</mn> </msub> </math> is an initial row of luminaires, and <math display="inline"> <msub> <mi>A</mi> <mn>0</mn> </msub> </math> is an area grouping all neighboring communication routes.</p> Full article ">Figure 4
<p>Two special cases in a calculation process. For both figures, the bold arrows indicate processing order. White bulbs denote adjusted luminaires, and dark gray ones are currently being adjusted. Non-filled bulbs stand for luminaires waiting for processing. (<b>a</b>) The calculation process meets a processed and tagged luminaire (<span class="html-italic">T</span>); (<b>b</b>) the calculation process meets a <span class="html-italic">forking point</span>. The <span class="html-italic">N</span> symbol denotes the first luminaire in the new segment.</p> Full article ">Figure 5
<p>The solid hypergraph (S-hypergraph) representing a cuboid <math display="inline"> <msub> <mi>B</mi> <mi>i</mi> </msub> </math> shown in <a href="#entropy-18-00076-f006" class="html-fig">Figure 6</a>. Note that the shape related details are hidden in the attributes of nodes, edges and hyperedges.</p> Full article ">Figure 6
<p>The Pentagon (left) as a loop of adhering solids (<math display="inline"> <msub> <mi>B</mi> <mi>i</mi> </msub> </math>’s) with the internal area (<span class="html-italic">A</span>) and the corresponding aggregated hypergraph (A-hypergraph) (right).</p> Full article ">Figure 7
<p>Centralized graph (left; the dashed line indicates the decomposition border) and its slashed form (right).</p> Full article ">Figure 8
<p>Multi-agent system activity schema. <span class="html-italic">Master agent</span> (MA) life cycle (left) and <span class="html-italic">computing agent</span> (CA) life cycle.</p> Full article ">Figure 9
<p>The sample map with segments processed by agents.</p> Full article ">Figure 10
<p>Computing task from the global and local perspective. (<b>a</b>) Global view on a computing task form an MA’s perspective. The numbers in circles denote the numbers of luminaires in particular subdomains. The black rectangle delimits the area enlarged in (<b>b</b>); lighting installation layout from the perspective of a CA. Note that several CAs can operate in such a region.</p> Full article ">
457 KiB  
Article
The Impact of Entropy Production and Emission Mitigation on Economic Growth
by Reiner Kümmel
Entropy 2016, 18(3), 75; https://doi.org/10.3390/e18030075 - 27 Feb 2016
Cited by 8 | Viewed by 6371
Abstract
Entropy production in industrial economies involves heat currents, driven by gradients of temperature, and particle currents, driven by specific external forces and gradients of temperature and chemical potentials. Pollution functions are constructed for the associated emissions. They reduce the output elasticities of the [...] Read more.
Entropy production in industrial economies involves heat currents, driven by gradients of temperature, and particle currents, driven by specific external forces and gradients of temperature and chemical potentials. Pollution functions are constructed for the associated emissions. They reduce the output elasticities of the production factors capital, labor, and energy in the growth equation of the capital-labor-energy-creativity model, when the emissions approach their critical limits. These are drawn by, e.g., health hazards or threats to ecological and climate stability. By definition, the limits oblige the economic actors to dedicate shares of the available production factors to emission mitigation, or to adjustments to the emission-induced changes in the biosphere. Since these shares are missing for the production of the quantity of goods and services that would be available to consumers and investors without emission mitigation, the “conventional” output of the economy shrinks. The resulting losses of conventional output are estimated for two classes of scenarios: (1) energy conservation; and (2) nuclear exit and subsidies to photovoltaics. The data of the scenarios refer to Germany in the 1980s and after 11 March 2011. For the energy-conservation scenarios, a method of computing the reduction of output elasticities by emission abatement is proposed. Full article
(This article belongs to the Special Issue Entropy and the Economy)
►▼ Show Figures

Figure 1

Figure 1
<p>Optimized potentials of energy conservation by heat-exanger networks, heat pumps, and cogeneration in the Federal Republic of Germany as a function of the price <math display="inline"> <mrow> <mi>b</mi> <mo>(</mo> <mi>F</mi> <mo>)</mo> </mrow> </math> of one unit of fuel <span class="html-italic">F</span> [<a href="#B37-entropy-18-00075" class="html-bibr">37</a>]. <math display="inline"> <mrow> <mi>N</mi> <mo>(</mo> <mi>F</mi> <mo>)</mo> </mrow> </math> is the annual quantity of primary energy required to satisfy the fixed demand for process heat and electricity. <span class="html-italic">C</span> and <math display="inline"> <msub> <mi>C</mi> <mn>0</mn> </msub> </math> are the total annual costs of providing <math display="inline"> <mrow> <mi>N</mi> <mo>(</mo> <mi>F</mi> <mo>)</mo> </mrow> </math> with and without energy-saving technologies. Solid <math display="inline"> <mrow> <mi>N</mi> <mo>(</mo> <mi>F</mi> <mo>)</mo> </mrow> </math> curve: upper cost limit is <math display="inline"> <mrow> <mn>1</mn> <mo>.</mo> <mn>0</mn> <msub> <mi>C</mi> <mn>0</mn> </msub> </mrow> </math>; dashed curve: upper limit is <math display="inline"> <mrow> <mn>1</mn> <mo>.</mo> <mn>1</mn> <msub> <mi>C</mi> <mn>0</mn> </msub> </mrow> </math>. Note the suppressed zero point of the <math display="inline"> <mrow> <mi>N</mi> <mo>(</mo> <mi>F</mi> <mo>)</mo> </mrow> </math> ordinate. The figure is reproduced with kind permission of H.-M. Groscurth.</p> Full article ">Figure 2
<p>Growth of installed PV capacity (lower curve, left ordinate, GW) and of annual PV refunding (upper curve, right ordinate, billion Euros) in Germany. This figure is reproduced with kind permission of H. Wirth. It is part of a forthcoming English version of [<a href="#B51-entropy-18-00075" class="html-bibr">51</a>]. Solid curves: actual data, dashed curves: projections.</p> Full article ">
943 KiB  
Article
Self-Replicating Spots in the Brusselator Model and Extreme Events in the One-Dimensional Case with Delay
by Mustapha Tlidi, Yerali Gandica, Giorgio Sonnino, Etienne Averlant and Krassimir Panajotov
Entropy 2016, 18(3), 64; https://doi.org/10.3390/e18030064 - 27 Feb 2016
Cited by 26 | Viewed by 5975
Abstract
We consider the paradigmatic Brusselator model for the study of dissipative structures in far from equilibrium systems. In two dimensions, we show the occurrence of a self-replication phenomenon leading to the fragmentation of a single localized spot into four daughter spots. This instability [...] Read more.
We consider the paradigmatic Brusselator model for the study of dissipative structures in far from equilibrium systems. In two dimensions, we show the occurrence of a self-replication phenomenon leading to the fragmentation of a single localized spot into four daughter spots. This instability affects the new spots and leads to splitting behavior until the system reaches a hexagonal stationary pattern. This phenomenon occurs in the absence of delay feedback. In addition, we incorporate a time-delayed feedback loop in the Brusselator model. In one dimension, we show that the delay feedback induces extreme events in a chemical reaction diffusion system. We characterize their formation by computing the probability distribution of the pulse height. The long-tailed statistical distribution, which is often considered as a signature of the presence of rogue waves, appears for sufficiently strong feedback intensity. The generality of our analysis suggests that the feedback-induced instability leading to the spontaneous formation of rogue waves in a controllable way is a universal phenomenon. Full article
(This article belongs to the Special Issue Recent Advances in Non-Equilibrium Statistical Mechanics and Its Application)
►▼ Show Figures

Figure 1

Figure 1
<p>Examples of 2D stationary localized structures obtained from numerical simulations of the model Equation 1. (<b>a</b>) and (<b>c</b>) correspond to a single spot in the spatial profile of the chemical concentrations <span class="html-italic">X</span> and <span class="html-italic">Y</span>, respectively. (<b>b</b>) and (<b>d</b>) correspond to four spots in the spatial profile of the chemical concentrations <span class="html-italic">X</span> and <span class="html-italic">Y</span>, respectively. Parameters are <math display="inline"> <mrow> <mi>A</mi> <mo>=</mo> <mn>0.6</mn> </mrow> </math>, <math display="inline"> <mrow> <mi>B</mi> <mo>=</mo> <mn>0.65</mn> </mrow> </math>, and <math display="inline"> <mrow> <mi>D</mi> <mo>=</mo> <mn>150</mn> </mrow> </math>. The mesh integration is <math display="inline"> <mrow> <mn>256</mn> <mo>×</mo> <mn>256</mn> </mrow> </math> points.</p> Full article ">Figure 2
<p>Self-replicating spots obtained from numerical simulations of the model Equation 1. (<b>a</b>)–(<b>h</b>): time evolution of the chemical concentration <span class="html-italic">X</span>. (<b>i</b>)–(<b>p</b>): time evolution of the chemical concentration <span class="html-italic">Y</span>. Parameters are <math display="inline"> <mrow> <mi>A</mi> <mo>=</mo> <mn>0.6</mn> </mrow> </math>, <math display="inline"> <mrow> <mi>B</mi> <mo>=</mo> <mn>0.75</mn> </mrow> </math>, and <math display="inline"> <mrow> <mi>D</mi> <mo>=</mo> <mn>150</mn> </mrow> </math>. The mesh integration is <math display="inline"> <mrow> <mn>256</mn> <mo>×</mo> <mn>256</mn> </mrow> </math> points.</p> Full article ">Figure 3
<p>The threshold associated with the pattern formation instability as a function of the strength of the delayed feedback, with <math display="inline"> <mrow> <mi>A</mi> <mo>=</mo> <mn>0.6</mn> </mrow> </math>.</p> Full article ">Figure 4
<p>The wavelength of the Turing–Prigogine instability as a function of the strength of the delayed feedback, with <math display="inline"> <mrow> <mi>A</mi> <mo>=</mo> <mn>0.6</mn> </mrow> </math>.</p> Full article ">Figure 5
<p>(<b>a</b>) Space-time map showing the evolution of <math display="inline"> <mrow> <mi>X</mi> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </math> in the Brusselator reaction-diffusion model. Triangles indicate pulses with an intensity 5–10-times larger than the stationary localized structures without delayed feedback. The red square shows an extreme event with amplitude <math display="inline"> <mrow> <mi>X</mi> <mo>&gt;</mo> <mn>20</mn> </mrow> </math>. The Brusselator parameters are <math display="inline"> <mrow> <mi>D</mi> <mo>=</mo> <mn>150</mn> </mrow> </math>, <math display="inline"> <mrow> <mi>A</mi> <mo>=</mo> <mn>0.6</mn> </mrow> </math>, <math display="inline"> <mrow> <mi>B</mi> <mo>=</mo> <mn>0.75</mn> </mrow> </math>, and the feedback parameters are <math display="inline"> <mrow> <msub> <mi>η</mi> <mi>x</mi> </msub> <mo>=</mo> <mo>−</mo> <msub> <mi>η</mi> <mi>y</mi> </msub> <mo>=</mo> <mn>2.5</mn> </mrow> </math> and <math display="inline"> <mrow> <mi>τ</mi> <mo>=</mo> <mn>10</mn> </mrow> </math>. (<b>b</b>) A cross-section at t = 78 showing the spatial profile <math display="inline"> <mrow> <mi>X</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math> of the rogue wave.</p> Full article ">Figure 6
<p>The number of events as a function of the amplitude of the pulses in the semi-logarithmic scale. The parameters are the same as in <a href="#entropy-18-00064-f005" class="html-fig">Figure 5</a>. The SWH denotes the significant wave height. The dashed line indicates 2 × SWH.</p> Full article ">
Show export options expand_more Show export options expand_less
Select all
Export citation of selected articles as:
 
Displaying articles 1-38
Previous Issue
Next Issue
Back to TopTop