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Volume 23
Issue 3
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Entropy, Volume 23, Issue 3 (March 2021) – 120 articles

Cover Story (view full-size image): State changes of systems due to interventions of "intelligent beings'' that can lead to a decrease in entropy are widely discussed in the literature. We propose here to take them as examples of "conditional actions'' and to describe them mathematically as "instruments'' following the quantum theory of measurement. As a detailed case study, we calculate the imperfect erasure of a qubit, which can also be considered as a conditional action and is realized by coupling a spin to another small spin system in its ground state. This analysis casts a critical light on the widely accepted Landauer–Bennett principle. From our point of view, erasing memory content appears as an additional conditional action, but it is neither necessary nor sufficient to solve Maxwell's demon paradox. View this paper
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26 pages, 3947 KiB  
Article
Non Stationary Multi-Armed Bandit: Empirical Evaluation of a New Concept Drift-Aware Algorithm
by Emanuele Cavenaghi, Gabriele Sottocornola, Fabio Stella and Markus Zanker
Entropy 2021, 23(3), 380; https://doi.org/10.3390/e23030380 - 23 Mar 2021
Cited by 20 | Viewed by 5698
Abstract
The Multi-Armed Bandit (MAB) problem has been extensively studied in order to address real-world challenges related to sequential decision making. In this setting, an agent selects the best action to be performed at time-step t, based on the past rewards received by [...] Read more.
The Multi-Armed Bandit (MAB) problem has been extensively studied in order to address real-world challenges related to sequential decision making. In this setting, an agent selects the best action to be performed at time-step t, based on the past rewards received by the environment. This formulation implicitly assumes that the expected payoff for each action is kept stationary by the environment through time. Nevertheless, in many real-world applications this assumption does not hold and the agent has to face a non-stationary environment, that is, with a changing reward distribution. Thus, we present a new MAB algorithm, named f-Discounted-Sliding-Window Thompson Sampling (f-dsw TS), for non-stationary environments, that is, when the data streaming is affected by concept drift. The f-dsw TS algorithm is based on Thompson Sampling (TS) and exploits a discount factor on the reward history and an arm-related sliding window to contrast concept drift in non-stationary environments. We investigate how to combine these two sources of information, namely the discount factor and the sliding window, by means of an aggregation function f(.). In particular, we proposed a pessimistic (f=min), an optimistic (f=max), as well as an averaged (f=mean) version of the f-dsw TS algorithm. A rich set of numerical experiments is performed to evaluate the f-dsw TS algorithm compared to both stationary and non-stationary state-of-the-art TS baselines. We exploited synthetic environments (both randomly-generated and controlled) to test the MAB algorithms under different types of drift, that is, sudden/abrupt, incremental, gradual and increasing/decreasing drift. Furthermore, we adapt four real-world active learning tasks to our framework—a prediction task on crimes in the city of Baltimore, a classification task on insects species, a recommendation task on local web-news, and a time-series analysis on microbial organisms in the tropical air ecosystem. The f-dsw TS approach emerges as the best performing MAB algorithm. At least one of the versions of f-dsw TS performs better than the baselines in synthetic environments, proving the robustness of f-dsw TS under different concept drift types. Moreover, the pessimistic version (f=min) results as the most effective in all real-world tasks. Full article
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Figure 1
<p>Examples of the four types of concept drift affecting the probability distribution of Class 3.</p> Full article ">Figure 2
<p>Reward distributions for <span class="html-italic">custom-decreasing</span> (<b>a</b>), <span class="html-italic">custom-increasing</span> (<b>b</b>), and <span class="html-italic">custom-stationary</span> (<b>c</b>) environments. Every function represents the evolution over time <span class="html-italic">t</span> of an arm reward probability in the custom environment.</p> Full article ">Figure 3
<p>Monthly frequency of crimes in each district for the Baltimore Crime dataset. Every line represents the evolution of the relative frequency of crimes in each district, through the months.</p> Full article ">Figure 4
<p>Histogram of the session length in the Local News dataset.</p> Full article ">Figure 5
<p>Frequency of the weekly topic interactions for the Local News dataset.</p> Full article ">Figure 6
<p>Frequency of the classes of air microbial organisms for the Air Microbes dataset. Every line represents the evolution of the relative frequency of a microbe class every two hours for 20 days.</p> Full article ">Figure 7
<p>Results of the MAB algorithms in environments with abrupt changes. Line plots represent the performance of each algorithm in terms of <span class="html-italic">RCR</span>, when the probability of drift <span class="html-italic">d</span> varies.</p> Full article ">Figure 8
<p>Results of the MAB algorithms in random environments with incremental changes. Line plots represent the performance of each algorithm in terms of <span class="html-italic">RCR</span>, when the drift probability <span class="html-italic">d</span> varies.</p> Full article ">Figure 9
<p>Results of the MAB algorithms in the <span class="html-italic">custom-decreasing</span> environment. Line plots represent the regret (<b>a</b>) and the cumulative reward (<b>b</b>) through 1000 steps for each algorithms.</p> Full article ">Figure 10
<p>Results of the MAB algorithms in the <span class="html-italic">custom-increasing</span> environment. Line plots represent the regret (<b>a</b>) and the cumulative reward (<b>b</b>) through 1000 steps for each algorithms.</p> Full article ">Figure 11
<p>Results of the MAB algorithms in the <span class="html-italic">custom-stationary</span> environment. Line plots represent the regret (<b>a</b>) and the cumulative reward (<b>b</b>) through 1000 steps for each algorithms.</p> Full article ">Figure A1
<p>Accuracy results over 10 replications for the Baltimore Crime dataset.</p> Full article ">Figure A2
<p>Accuracy results over 10 replications for the Insects datasets collection.</p> Full article ">Figure A2 Cont.
<p>Accuracy results over 10 replications for the Insects datasets collection.</p> Full article ">Figure A3
<p>Accuracy results over 10 replications for the Local News dataset.</p> Full article ">Figure A4
<p>Relative cumulative reward (RCR) results over 10 replications for the Bacteria dataset.</p> Full article ">
26 pages, 383 KiB  
Article
Potential Well in Poincaré Recurrence
by Miguel Abadi, Vitor Amorim and Sandro Gallo
Entropy 2021, 23(3), 379; https://doi.org/10.3390/e23030379 - 23 Mar 2021
Cited by 2 | Viewed by 2343
Abstract
From a physical/dynamical system perspective, the potential well represents the proportional mass of points that escape the neighbourhood of a given point. In the last 20 years, several works have shown the importance of this quantity to obtain precise approximations for several recurrence [...] Read more.
From a physical/dynamical system perspective, the potential well represents the proportional mass of points that escape the neighbourhood of a given point. In the last 20 years, several works have shown the importance of this quantity to obtain precise approximations for several recurrence time distributions in mixing stochastic processes and dynamical systems. Besides providing a review of the different scaling factors used in the literature in recurrence times, the present work contributes two new results: (1) For ?-mixing and ?-mixing processes, we give a new exponential approximation for hitting and return times using the potential well as the scaling parameter. The error terms are explicit and sharp. (2) We analyse the uniform positivity of the potential well. Our results apply to processes on countable alphabets and do not assume a complete grammar. Full article
(This article belongs to the Special Issue Extreme Value Theory)
11 pages, 1362 KiB  
Article
Design of Optimal Rainfall Monitoring Network Using Radar and Road Networks
by Taeyong Kwon, Seongsim Yoon and Sanghoo Yoon
Entropy 2021, 23(3), 378; https://doi.org/10.3390/e23030378 - 23 Mar 2021
Viewed by 2185
Abstract
Uncertainty in the rainfall network can lead to mistakes in dam operation. Sudden increases in dam water levels due to rainfall uncertainty are a high disaster risk. In order to prevent these losses, it is necessary to configure an appropriate rainfall network that [...] Read more.
Uncertainty in the rainfall network can lead to mistakes in dam operation. Sudden increases in dam water levels due to rainfall uncertainty are a high disaster risk. In order to prevent these losses, it is necessary to configure an appropriate rainfall network that can effectively reflect the characteristics of the watershed. In this study, conditional entropy was used to calculate the uncertainty of the watershed using rainfall and radar data observed from 2018 to 2019 in the Goesan Dam and Hwacheon Dam watersheds. The results identified radar data suitable for the characteristics of the watershed and proposed a site for an additional rainfall gauge. It is also necessary to select the location of the additional rainfall gauged by limiting the points where smooth movement and installation, for example crossing national borders, are difficult. The proposed site emphasized accessibility and usability by leveraging road information and selecting a radar grid near the road. As a practice result, the uncertainty of precipitation in the Goesan and Hwacheon Dam watersheds could be decreased by 70.0% and 67.9%, respectively, when four and three additional gauge sites were installed without any restriction. When these were installed near to the road, with five and four additional gauge sites, the uncertainty in the Goesan Dam and Hwacheon Dam watersheds were reduced by up to 71.1%. Therefore, due to the high degree of uncertainty, it is necessary to measure precipitation. The operation of the rainfall gauge can provide a smooth site and configure an appropriate monitoring network. Full article
(This article belongs to the Special Issue Application of Information-Theoretic Concepts in Bio-, Environmental and Engineering Science Research)
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<p>The location of radar equipment and gauged stations. (<b>a</b>) Study area and the location of radar. (<b>b</b>) Gauged stations in Goesan Dam watershed. (<b>c</b>) Gauged stations in Hwacheon Dam watershed.</p> Full article ">Figure 2
<p>Potential grid nearby the road. (<b>a</b>) Goesan. (<b>b</b>) Hwacheon.</p> Full article ">Figure 3
<p>Result of conditional entropy using gauged data. (<b>a</b>) Goesan. (<b>b</b>) Hwacheon.</p> Full article ">Figure 4
<p>Result of Spearman correlation considering gauged network. (<b>a</b>) Goesan. (<b>b</b>) Hwacheon.</p> Full article ">Figure 5
<p>Results of the location of additional rainfall stations in the Goesan Dam watershed. (<b>a</b>) Proposed gauge sequence. (<b>b</b>) Proposed gauge sequence considering road.</p> Full article ">Figure 6
<p>Results of the location of additional rainfall stations in the Hwacheon Dam watershed. (<b>a</b>) Proposed gauge sequence. (<b>b</b>) Proposed gauge sequence considering road.</p> Full article ">Figure 7
<p>Comparison of ER values of additional gauging sites. (<b>a</b>) Goesan. (<b>b</b>) Hwacheon.</p> Full article ">
14 pages, 326 KiB  
Article
On the Classical Capacity of General Quantum Gaussian Measurement
by Alexander Holevo
Entropy 2021, 23(3), 377; https://doi.org/10.3390/e23030377 - 21 Mar 2021
Cited by 10 | Viewed by 3059
Abstract
In this paper, we consider the classical capacity problem for Gaussian measurement channels. We establish Gaussianity of the average state of the optimal ensemble in the general case and discuss the Hypothesis of Gaussian Maximizers concerning the structure of the ensemble. Then, we [...] Read more.
In this paper, we consider the classical capacity problem for Gaussian measurement channels. We establish Gaussianity of the average state of the optimal ensemble in the general case and discuss the Hypothesis of Gaussian Maximizers concerning the structure of the ensemble. Then, we consider the case of one mode in detail, including the dual problem of accessible information of a Gaussian ensemble. Our findings are relevant to practical situations in quantum communications where the receiver is Gaussian (say, a general-dyne detection) and concatenation of the Gaussian channel and the receiver can be considered as one Gaussian measurement channel. Our efforts in this and preceding papers are then aimed at establishing full Gaussianity of the optimal ensemble (usually taken as an assumption) in such schemes. Full article
(This article belongs to the Special Issue Quantum Communication, Quantum Radar, and Quantum Cipher)
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<p>(color online) The Gaussian classical capacity (<a href="#FD49-entropy-23-00377" class="html-disp-formula">A6</a>) and the upper bound (<a href="#FD33-entropy-23-00377" class="html-disp-formula">33</a>) (<math display="inline"><semantics> <mrow> <mi>β</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>).</p> Full article ">
18 pages, 1247 KiB  
Article
A Generative Adversarial Network for Infrared and Visible Image Fusion Based on Semantic Segmentation
by Jilei Hou, Dazhi Zhang, Wei Wu, Jiayi Ma and Huabing Zhou
Entropy 2021, 23(3), 376; https://doi.org/10.3390/e23030376 - 21 Mar 2021
Cited by 49 | Viewed by 5762
Abstract
This paper proposes a new generative adversarial network for infrared and visible image fusion based on semantic segmentation (SSGAN), which can consider not only the low-level features of infrared and visible images, but also the high-level semantic information. Source images can be divided [...] Read more.
This paper proposes a new generative adversarial network for infrared and visible image fusion based on semantic segmentation (SSGAN), which can consider not only the low-level features of infrared and visible images, but also the high-level semantic information. Source images can be divided into foregrounds and backgrounds by semantic masks. The generator with a dual-encoder-single-decoder framework is used to extract the feature of foregrounds and backgrounds by different encoder paths. Moreover, the discriminator’s input image is designed based on semantic segmentation, which is obtained by combining the foregrounds of the infrared images with the backgrounds of the visible images. Consequently, the prominence of thermal targets in the infrared images and texture details in the visible images can be preserved in the fused images simultaneously. Qualitative and quantitative experiments on publicly available datasets demonstrate that the proposed approach can significantly outperform the state-of-the-art methods. Full article
(This article belongs to the Special Issue Advances in Image Fusion)
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<p>The entire procedure of semantic segmentation GAN (SSGAN) for image fusion. <math display="inline"><semantics> <msub> <mi>I</mi> <mi>r</mi> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mi>I</mi> <mi>m</mi> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mi>I</mi> <mi>v</mi> </msub> </semantics></math>, and <math display="inline"><semantics> <msub> <mi>I</mi> <mi>f</mi> </msub> </semantics></math> denote the infrared image, mask, visible image, and fused image, respectively. <math display="inline"><semantics> <msub> <mi>I</mi> <mi>d</mi> </msub> </semantics></math> is used as the reference data to input into the discriminator, and its generation process is described later.</p> Full article ">Figure 2
<p>The overall architecture of the generator. <math display="inline"><semantics> <msub> <mi>I</mi> <mi>r</mi> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mi>I</mi> <mi>m</mi> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mi>I</mi> <mi>v</mi> </msub> </semantics></math>, and <math display="inline"><semantics> <msub> <mi>I</mi> <mi>f</mi> </msub> </semantics></math> denote the infrared image, mask, visible image, and fused image, respectively. Encoder-fg and Encoder-bg are used to extract the features of the foreground and background, respectively, and the decoder is used to reconstruct the fused image.</p> Full article ">Figure 3
<p>The overall architecture of the discriminator.</p> Full article ">Figure 4
<p>Qualitative fusion results on four typical infrared and visible image pairs. From left to right: infrared images, visible images, mask, results of GTF, DTCWT, wavelet, FusionGAN, DenseFuse, and the proposed SSGAN.</p> Full article ">Figure 5
<p>Quantitative comparisons of the four metrics, i.e., entropy (EN), SD, MI, and visual information fidelity (VIF), on the TNOdataset. The five state-of-the-art methods, DTCWT, GTF, wavelet, DenseFuse, and FusionGAN, are used for comparison.</p> Full article ">Figure 6
<p>Qualitative fusion results on the INOdataset. The first row is the infrared image, visible image, and mask. The rest two rows (from left to right, top to bottom) are the fusion results of GTF, DTCWT, wavelet, FusionGAN, DenseFuse, and SSGAN.</p> Full article ">Figure 7
<p>Quantitative comparisons of the four metrics, i.e., EN, SD, MI, and VIF, on the INO dataset. The five state-of-the-art methods DTCWT, GTF, wavelet, DenseFuse and FusionGAN are used for comparison.</p> Full article ">Figure 8
<p>Fused results of the experiment related to the mask. From left to right: infrared images, visible images, results of replacing <math display="inline"><semantics> <msub> <mi>I</mi> <mi>d</mi> </msub> </semantics></math> with <math display="inline"><semantics> <msub> <mi>I</mi> <mi>r</mi> </msub> </semantics></math>, results of replacing <math display="inline"><semantics> <msub> <mi>I</mi> <mi>d</mi> </msub> </semantics></math> with <math display="inline"><semantics> <msub> <mi>I</mi> <mi>v</mi> </msub> </semantics></math>, and results of the proposed SSGAN.</p> Full article ">Figure 9
<p>Fused results of the experiment related to the discriminator. From left to right: infrared images, visible images, results of the SSGAN without the discriminator, and results of the proposed SSGAN.</p> Full article ">
13 pages, 798 KiB  
Article
The Impact of Visual Input and Support Area Manipulation on Postural Control in Subjects after Osteoporotic Vertebral Fracture
by Michalina Błażkiewicz, Justyna Kędziorek and Anna Hadamus
Entropy 2021, 23(3), 375; https://doi.org/10.3390/e23030375 - 20 Mar 2021
Cited by 10 | Viewed by 2858
Abstract
Osteoporosis is a prevalent health concern among older adults and is associated with an increased risk of falls that may result in fracture, injury, or even death. Identifying the risk factors for falls and assessing the complexity of postural control within this population [...] Read more.
Osteoporosis is a prevalent health concern among older adults and is associated with an increased risk of falls that may result in fracture, injury, or even death. Identifying the risk factors for falls and assessing the complexity of postural control within this population is essential for developing effective regimes for fall prevention. The aim of this study was to assess postural control in individuals recovering from osteoporotic vertebral fractures while performing various stability tasks. Seventeen individuals with type II osteoporosis and 17 healthy subjects participated in this study. The study involved maintaining balance while standing barefoot on both feet for 20 s on an Advanced Mechanical Technology Inc. (AMTI) plate, with eyes open, eyes closed, and eyes closed in conjunction with a dual-task. Another three trials lasting 10 s each were undertaken during a single-leg stance under the same conditions. Fall risk was assessed using the Biodex Balance platform. Nonlinear measures were used to assess center of pressure (CoP) dynamics in all trials. Reducing the support area or elimination of the visual control led to increased sample entropy and fractal dimension. Results of the nonlinear measurements indicate that individuals recovering from osteoporotic vertebral fractures are characterized by decreased irregularity, mainly in the medio-lateral direction and reduced complexity. Full article
(This article belongs to the Special Issue Information Theory in Biomedical Data Mining)
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<p>Path length values in subsequent trials for people with osteoporosis (OS) and the healthy group (C), where: * marks statistically significant differences between groups, the red line marks statistically significant differences between trials for OS group, and the blue line marks statistically significant differences between trials for the C group (<span class="html-italic">p</span> &lt; 0.05).</p> Full article ">Figure 2
<p>Coefficient values in subsequent trials for people with osteoporosis (OS) and healthy group (C): (<b>a</b>) sample entropy (SampEn); (<b>b</b>) fractal dimension (FD), where x—medio-lateral direction, y—anterior–posterior direction. * marks statistically significant differences between groups, the red line marks statistically significant differences between trials for the OS group, and the blue line marks statistically significant differences between trials for the C group (<span class="html-italic">p</span> &lt; 0.05).</p> Full article ">
14 pages, 3405 KiB  
Communication
Industry 4.0 Quantum Strategic Organizational Design Configurations. The Case of 3 Qubits: One Reports to Two
by Javier Villalba-Diez, Juan Carlos Losada, Rosa María Benito and Ana González-Marcos
Entropy 2021, 23(3), 374; https://doi.org/10.3390/e23030374 - 20 Mar 2021
Cited by 5 | Viewed by 2657
Abstract
In this work we explore how the relationship between one subordinate reporting to two leaders influences the alignment of the latter with the company’s strategic objectives in an Industry 4.0 environment. We do this through the implementation of quantum circuits that represent decision [...] Read more.
In this work we explore how the relationship between one subordinate reporting to two leaders influences the alignment of the latter with the company’s strategic objectives in an Industry 4.0 environment. We do this through the implementation of quantum circuits that represent decision networks. This is done for two cases: One in which the leaders do not communicate with each other, and one in which they do. Through the quantum simulation of strategic organizational design configurations (QSOD) through 500 quantum circuit simulations, we conclude that in the first case both leaders are not simultaneously in alignment, and in the second case that both reporting nodes need to have an alignment probability higher than 90% to support the leader node. Full article
(This article belongs to the Special Issue Quantum Approach to Game Theory and Social Science)
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<p>Quantum simulation of Strategic Organizational Design (QSOD). Case of three qubits configuration in which one node reports to two. I. Without communication between the leaders. II. With communication between the leaders.</p> Full article ">Figure 2
<p>Correlation between <math display="inline"><semantics> <mrow> <mi>P</mi> <mo>(</mo> <mi>B</mi> <mo>=</mo> <mrow> <mo>|</mo> <mn>0</mn> <mo>〉</mo> </mrow> <mo>)</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>P</mi> <mo>(</mo> <mi>C</mi> <mo>=</mo> <mrow> <mo>|</mo> <mn>0</mn> <mo>〉</mo> </mrow> <mo>)</mo> </mrow> </semantics></math> for different values of <math display="inline"><semantics> <mrow> <msub> <mi>z</mi> <mn>1</mn> </msub> <mo>=</mo> <mi>P</mi> <mrow> <mo>(</mo> <mi>A</mi> <mo>=</mo> <mrow> <mo>|</mo> <mn>1</mn> <mo>〉</mo> </mrow> <mo>)</mo> </mrow> <mo>∈</mo> <msub> <mi>ξ</mi> <mn>1</mn> </msub> </mrow> </semantics></math> for the case of no communication between <span class="html-italic">B</span> and <span class="html-italic">C</span>.</p> Full article ">Figure 3
<p>Correlation between <math display="inline"><semantics> <mrow> <mi>P</mi> <mo>(</mo> <mi>B</mi> <mo>=</mo> <mrow> <mo>|</mo> <mn>0</mn> <mo>〉</mo> </mrow> <mo>)</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>P</mi> <mo>(</mo> <mi>C</mi> <mo>=</mo> <mrow> <mo>|</mo> <mn>0</mn> <mo>〉</mo> </mrow> <mo>)</mo> </mrow> </semantics></math> for different values of <math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mn>1</mn> </msub> <mo>=</mo> <mi>P</mi> <mrow> <mo>(</mo> <mi>B</mi> <mo>=</mo> <mrow> <mo>|</mo> <mn>1</mn> <mo>〉</mo> </mrow> <mo>|</mo> <mi>A</mi> <mo>=</mo> <mrow> <mo>|</mo> <mn>1</mn> <mo>〉</mo> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>y</mi> <mn>2</mn> </msub> <mo>=</mo> <mi>P</mi> <mrow> <mo>(</mo> <mi>C</mi> <mo>=</mo> <mrow> <mo>|</mo> <mn>1</mn> <mo>〉</mo> </mrow> <mo>|</mo> <mi>A</mi> <mo>=</mo> <mrow> <mo>|</mo> <mn>1</mn> <mo>〉</mo> </mrow> <mo>)</mo> </mrow> <mo>∈</mo> <msub> <mi>ξ</mi> <mn>2</mn> </msub> </mrow> </semantics></math> for the case of no communication between <span class="html-italic">B</span> and <span class="html-italic">C</span>.</p> Full article ">Figure 4
<p>Correlation between <math display="inline"><semantics> <mrow> <mi>P</mi> <mo>(</mo> <mi>B</mi> <mo>=</mo> <mrow> <mo>|</mo> <mn>0</mn> <mo>〉</mo> </mrow> <mo>)</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>P</mi> <mo>(</mo> <mi>C</mi> <mo>=</mo> <mrow> <mo>|</mo> <mn>0</mn> <mo>〉</mo> </mrow> <mo>)</mo> </mrow> </semantics></math>.</p> Full article ">Figure 5
<p>Alignment Probabilities of <math display="inline"><semantics> <mrow> <mi>P</mi> <mo>(</mo> <mi>A</mi> <mo>=</mo> <mrow> <mo>|</mo> <mn>0</mn> <mo>〉</mo> </mrow> <mo>)</mo> <mo>,</mo> <mi>P</mi> <mo>(</mo> <mi>B</mi> <mo>=</mo> <mrow> <mo>|</mo> <mn>0</mn> <mo>〉</mo> </mrow> <mo>)</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>P</mi> <mo>(</mo> <mi>C</mi> <mo>=</mo> <mrow> <mo>|</mo> <mn>0</mn> <mo>〉</mo> </mrow> <mo>)</mo> </mrow> </semantics></math> with <math display="inline"><semantics> <mrow> <msub> <mi>z</mi> <mn>11</mn> </msub> <mo>=</mo> <mi>P</mi> <mrow> <mo>(</mo> <mi>A</mi> <mo>=</mo> <mrow> <mo>|</mo> <mn>1</mn> <mo>〉</mo> </mrow> <mo>)</mo> </mrow> <mo>∈</mo> <mrow> <mo>[</mo> <mn>0.01</mn> <mo>,</mo> <mn>0.1</mn> <mo>]</mo> </mrow> </mrow> </semantics></math> for different values of fixed <math display="inline"><semantics> <mrow> <msub> <mi>z</mi> <mn>21</mn> </msub> <mo>=</mo> <mi>P</mi> <mrow> <mo>(</mo> <mi>B</mi> <mo>=</mo> <mrow> <mo>|</mo> <mn>1</mn> <mo>〉</mo> </mrow> <mo>|</mo> <mi>A</mi> <mo>=</mo> <mrow> <mo>|</mo> <mn>1</mn> <mo>〉</mo> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>z</mi> <mn>22</mn> </msub> <mo>=</mo> <mi>P</mi> <mrow> <mo>(</mo> <mi>B</mi> <mo>=</mo> <mrow> <mo>|</mo> <mn>1</mn> <mo>〉</mo> </mrow> <mo>|</mo> <mi>A</mi> <mo>=</mo> <mrow> <mo>|</mo> <mn>0</mn> <mo>〉</mo> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math>, and combinations of <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>11</mn> </msub> <mo>=</mo> <msub> <mi>x</mi> <mn>21</mn> </msub> <mo>=</mo> <msub> <mi>x</mi> <mn>21</mn> </msub> <mo>=</mo> <msub> <mi>y</mi> <mn>21</mn> </msub> </mrow> </semantics></math>.</p> Full article ">Figure 5 Cont.
<p>Alignment Probabilities of <math display="inline"><semantics> <mrow> <mi>P</mi> <mo>(</mo> <mi>A</mi> <mo>=</mo> <mrow> <mo>|</mo> <mn>0</mn> <mo>〉</mo> </mrow> <mo>)</mo> <mo>,</mo> <mi>P</mi> <mo>(</mo> <mi>B</mi> <mo>=</mo> <mrow> <mo>|</mo> <mn>0</mn> <mo>〉</mo> </mrow> <mo>)</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>P</mi> <mo>(</mo> <mi>C</mi> <mo>=</mo> <mrow> <mo>|</mo> <mn>0</mn> <mo>〉</mo> </mrow> <mo>)</mo> </mrow> </semantics></math> with <math display="inline"><semantics> <mrow> <msub> <mi>z</mi> <mn>11</mn> </msub> <mo>=</mo> <mi>P</mi> <mrow> <mo>(</mo> <mi>A</mi> <mo>=</mo> <mrow> <mo>|</mo> <mn>1</mn> <mo>〉</mo> </mrow> <mo>)</mo> </mrow> <mo>∈</mo> <mrow> <mo>[</mo> <mn>0.01</mn> <mo>,</mo> <mn>0.1</mn> <mo>]</mo> </mrow> </mrow> </semantics></math> for different values of fixed <math display="inline"><semantics> <mrow> <msub> <mi>z</mi> <mn>21</mn> </msub> <mo>=</mo> <mi>P</mi> <mrow> <mo>(</mo> <mi>B</mi> <mo>=</mo> <mrow> <mo>|</mo> <mn>1</mn> <mo>〉</mo> </mrow> <mo>|</mo> <mi>A</mi> <mo>=</mo> <mrow> <mo>|</mo> <mn>1</mn> <mo>〉</mo> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>z</mi> <mn>22</mn> </msub> <mo>=</mo> <mi>P</mi> <mrow> <mo>(</mo> <mi>B</mi> <mo>=</mo> <mrow> <mo>|</mo> <mn>1</mn> <mo>〉</mo> </mrow> <mo>|</mo> <mi>A</mi> <mo>=</mo> <mrow> <mo>|</mo> <mn>0</mn> <mo>〉</mo> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math>, and combinations of <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>11</mn> </msub> <mo>=</mo> <msub> <mi>x</mi> <mn>21</mn> </msub> <mo>=</mo> <msub> <mi>x</mi> <mn>21</mn> </msub> <mo>=</mo> <msub> <mi>y</mi> <mn>21</mn> </msub> </mrow> </semantics></math>.</p> Full article ">
24 pages, 3104 KiB  
Article
Vector Arithmetic in the Triangular Grid
by Khaled Abuhmaidan, Monther Aldwairi and Benedek Nagy
Entropy 2021, 23(3), 373; https://doi.org/10.3390/e23030373 - 20 Mar 2021
Cited by 2 | Viewed by 3352
Abstract
Vector arithmetic is a base of (coordinate) geometry, physics and various other disciplines. The usual method is based on Cartesian coordinate-system which fits both to continuous plane/space and digital rectangular-grids. The triangular grid is also regular, but it is not a point lattice: [...] Read more.
Vector arithmetic is a base of (coordinate) geometry, physics and various other disciplines. The usual method is based on Cartesian coordinate-system which fits both to continuous plane/space and digital rectangular-grids. The triangular grid is also regular, but it is not a point lattice: it is not closed under vector-addition, which gives a challenge. The points of the triangular grid are represented by zero-sum and one-sum coordinate-triplets keeping the symmetry of the grid and reflecting the orientations of the triangles. This system is expanded to the plane using restrictions like, at least one of the coordinates is an integer and the sum of the three coordinates is in the interval [?1,1]. However, the vector arithmetic is still not straightforward; by purely adding two such vectors the result may not fulfill the above conditions. On the other hand, for various applications of digital grids, e.g., in image processing, cartography and physical simulations, one needs to do vector arithmetic. In this paper, we provide formulae that give the sum, difference and scalar product of vectors of the continuous coordinate system. Our work is essential for applications, e.g., to compute discrete rotations or interpolations of images on the triangular grid. Full article
(This article belongs to the Special Issue Exploring the NP-Complexity of Nature: Critical Phenomena, Chaos, Fractals, Graphs, Boson Sampling, Quantum Computing and More)
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Graphical abstract

Graphical abstract
Full article ">Figure 1
<p>The three regular grids: the square, the hexagonal and the triangular grids and their grid points (midpoints of the pixels).</p> Full article ">Figure 2
<p>(<b>a</b>) The discrete symmetric coordinate system for the hexagonal and (<b>b</b>) for the triangular grids. The coordinate values can also be considered to be assigned the midpoints of the corresponding pixels.</p> Full article ">Figure 3
<p>Any grid-vector of specific length and direction will lead to a grid-point in the square and the hexagonal grids but not in the triangular grid.</p> Full article ">Figure 4
<p>The symmetric coordinate system for the trihexagonal grid can also be used for the triangular grid and also for its dual, for the hexagonal grid, at the same time.</p> Full article ">Figure 5
<p>(<b>a</b>,<b>b</b>): Dividing each triangle to three areas—A, B and C. The letters assigned to the isosceles triangles are based on the orientation of sides. (<b>c</b>): The areas actually rhombuses in the plane.</p> Full article ">Figure 6
<p>A composition of the barycentric technique and discrete coordinate system to address points <span class="html-italic">p</span> and <span class="html-italic">q</span> in the triangular plane by coordinate triplets in (<b>a</b>,<b>b</b>), respectively.</p> Full article ">Figure 7
<p>Addressing the points in a hexagon containing one area of each type, where the midpoint (<span class="html-italic">m</span>) of the hexagon has coordinates (<span class="html-italic">i</span>, <span class="html-italic">j</span>, <span class="html-italic">k</span>) and 0 <span class="html-italic">≤ u ≤</span> 1 and 0 <span class="html-italic">≤ v ≤</span> 1 in each area.</p> Full article ">Figure 8
<p>The corresponding constant coordinate value for each area.</p> Full article ">Figure 9
<p>The plane is tessellated by three types of rhombuses; it can also be seen as the surface of a mesh of the cubic grid having three faces of each cube on the surface.</p> Full article ">Figure 10
<p>(<b>a</b>) Consider vectors <span class="html-italic">v</span><sub>1</sub> = (0.387, −1, 0.213) and <span class="html-italic">v</span><sub>2</sub> = (0.677, 0, −0.477); both are Type B. In this case, the direct-sum of vectors will be <span class="html-italic">s</span> = (1.064, −1, −0.264), which is Type B as well and hence is a result-vector for Ω. (<b>b</b>) Consider vectors <span class="html-italic">v</span><sub>1</sub> = (0.173, −0.813, 0) and <span class="html-italic">v</span><sub>2</sub> = (0.677, 0, −0.477) of Types C and B<span class="html-italic">,</span> respectively. In this case, the direct-sum of vectors will be <span class="html-italic">s</span> = (0.851, −0.813, −0.477), which is not compatible with Ω showing the nonlinearity of the system.</p> Full article ">Figure 11
<p>(<b>a</b>) The six regions of the triangular plane. (<b>b</b>) The signs of the coordinate triplet for each region of the triangular plane.</p> Full article ">Figure 12
<p>Examples for vector arithmetic: (<b>a</b>) Consider the scalar product of vector <span class="html-italic">v</span> = (0.5, 0, −0.25) with a positive integer multiplier <span class="html-italic">n</span> = 4 (that is to compute <span class="html-italic">v + v + v + v</span>) which yields to vector <span class="html-italic">r</span> = (2, 0, −1). (<b>b</b>) Consider the addition of vectors <span class="html-italic">v</span><sub>1</sub> = (0.75, 0, −0.25) and <span class="html-italic">v</span><sub>2</sub> = (−0.25, 0, 0.75) which results in vector <span class="html-italic">r</span> = (0, −0.5, 0). (<b>c</b>) The addition of the opposite vectors <span class="html-italic">v</span><sub>1</sub> = (1, 0, 0) and <span class="html-italic">v</span><sub>2</sub> = (−1, 0, 0) give vector <span class="html-italic">r</span> = (0, 0, 0).</p> Full article ">Figure 13
<p>Example of image translation by vector addition: the points represented by vectors <span class="html-italic">v</span><sub>1</sub>, …, <span class="html-italic">v</span><sub>8</sub> have been translated by vector <span class="html-italic">k</span> = (−0.848, −0.353, 1) to get the points represented by vectors <span class="html-italic">u</span><sub>1</sub>, …, <span class="html-italic">u</span><sub>8</sub>.</p> Full article ">Figure 14
<p>Example for zooming an image by vector arithmetic: the star defined by points represented by vectors <span class="html-italic">a</span><sub>1</sub>, …, <span class="html-italic">j</span><sub>1</sub> given in blue color are doubled and tripled, the resulted vectors and shapes are presented in red and green color.</p> Full article ">
16 pages, 483 KiB  
Article
Monitoring the Zero-Inflated Time Series Model of Counts with Random Coefficient
by Cong Li, Shuai Cui and Dehui Wang
Entropy 2021, 23(3), 372; https://doi.org/10.3390/e23030372 - 20 Mar 2021
Cited by 3 | Viewed by 3217
Abstract
In this research, we consider monitoring mean and correlation changes from zero-inflated autocorrelated count data based on the integer-valued time series model with random survival rate. A cumulative sum control chart is constructed due to its efficiency, the corresponding calculation methods of average [...] Read more.
In this research, we consider monitoring mean and correlation changes from zero-inflated autocorrelated count data based on the integer-valued time series model with random survival rate. A cumulative sum control chart is constructed due to its efficiency, the corresponding calculation methods of average run length and the standard deviation of the run length are given. Practical guidelines concerning the chart design are investigated. Extensive computations based on designs of experiments are conducted to illustrate the validity of the proposed method. Comparisons with the conventional control charting procedure are also provided. The analysis of the monthly number of drug crimes in the city of Pittsburgh is displayed to illustrate our current method of process monitoring. Full article
(This article belongs to the Special Issue Time Series Modelling)
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<p>Sample path of ZIGINAR<math display="inline"><semantics> <msub> <mrow/> <mi>RC</mi> </msub> </semantics></math>(1) processes for <math display="inline"><semantics> <mrow> <mi>θ</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>5</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>=</mo> <mn>0.2</mn> <mo>,</mo> <mn>0.5</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.5</mn> <mo>,</mo> <mn>0.8</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>=</mo> <mn>0.3</mn> <mo>,</mo> <mn>0.8</mn> </mrow> </semantics></math>.</p> Full article ">Figure 2
<p>The performance of the Shewhart chart and the CUSUM chart to detect an increase in process mean, (<b>a</b>) changes only in <math display="inline"><semantics> <mi>θ</mi> </semantics></math>, (<b>b</b>) changes only in <span class="html-italic">p</span>.</p> Full article ">Figure 3
<p>The performance of the Shewhart chart and the CUSUM chart to detect an increase in process first-order correlation, (<b>a</b>) changes only in <math display="inline"><semantics> <mi>α</mi> </semantics></math>, (<b>b</b>) changes only in <math display="inline"><semantics> <mi>β</mi> </semantics></math>.</p> Full article ">Figure 4
<p>The plots about the zero-inflated drug crime series, (<b>a</b>) the sample path, (<b>b</b>) the histogram with ZIG fit, Geometric fit and Poisson fit.</p> Full article ">Figure 5
<p>The plots about the zero-inflated drug crime series, (<b>a</b>) the ACF plot, (<b>b</b>) the PACF plot.</p> Full article ">Figure 6
<p>The control charts for the zero-inflated drug crime series, the CUSUM charts with designs (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>h</mi> <mo>=</mo> <mn>34</mn> <mo>,</mo> <mspace width="3.33333pt"/> <mi>k</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>, (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>h</mi> <mo>=</mo> <mn>15</mn> <mo>,</mo> <mspace width="3.33333pt"/> <mi>k</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math>, (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>h</mi> <mo>=</mo> <mn>12</mn> <mo>,</mo> <mspace width="3.33333pt"/> <mi>k</mi> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math>, and (<b>d</b>) the Shewhart chart with <math display="inline"><semantics> <mrow> <mi>U</mi> <mi>C</mi> <mi>L</mi> <mo>=</mo> <mn>13</mn> </mrow> </semantics></math>.</p> Full article ">
15 pages, 3549 KiB  
Article
Demonstration of Three True Random Number Generator Circuits Using Memristor Created Entropy and Commercial Off-the-Shelf Components
by Scott Stoller and Kristy A. Campbell
Entropy 2021, 23(3), 371; https://doi.org/10.3390/e23030371 - 20 Mar 2021
Cited by 10 | Viewed by 3945
Abstract
In this work, we build and test three memristor-based true random number generator (TRNG) circuits: two previously presented in the literature and one which is our own design. The functionality of each circuit is assessed using the National Institute of Standards and Technology [...] Read more.
In this work, we build and test three memristor-based true random number generator (TRNG) circuits: two previously presented in the literature and one which is our own design. The functionality of each circuit is assessed using the National Institute of Standards and Technology (NIST) Statistical Test Suite (STS). The TRNG circuits were built using commercially available off-the-shelf parts, including the memristor. The results of this work confirm the usefulness of memristors for successful implementation of TRNG circuits, as well as the ease with which a TRNG can be built using simple circuit designs and off-the-shelf breadboard circuit components. Full article
(This article belongs to the Special Issue Nonlinear Dynamics and Entropy of Complex Systems: Advances and Perspectives II)
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<p>Example of Von Neumann whitening.</p> Full article ">Figure 2
<p>Circuits tested. (<b>a</b>) Jiang’s circuit modified for testing [<a href="#B1-entropy-23-00371" class="html-bibr">1</a>]; (<b>b</b>) Rai’s circuit adapted for testing [<a href="#B2-entropy-23-00371" class="html-bibr">2</a>]; (<b>c</b>) Our design, student-true random number generator (S-TRNG).</p> Full article ">Figure 2 Cont.
<p>Circuits tested. (<b>a</b>) Jiang’s circuit modified for testing [<a href="#B1-entropy-23-00371" class="html-bibr">1</a>]; (<b>b</b>) Rai’s circuit adapted for testing [<a href="#B2-entropy-23-00371" class="html-bibr">2</a>]; (<b>c</b>) Our design, student-true random number generator (S-TRNG).</p> Full article ">Figure 3
<p>Timing diagram explanation of Jiang’s true random number generator circuit.</p> Full article ">Figure 4
<p>Implementation of Jiang’s design on a breadboard. The circuit is connected to the Digilent AD2 on the right.</p> Full article ">Figure 5
<p>Breadboard implementation of Rai’s true random number generator circuit. The circuit is connected to the Digilent AD2 on the right.</p> Full article ">Figure 6
<p>Histograms of the entropy captured as a slow clock sampling a fast clock in a dual oscillator type random number generator used in the S-TRNG circuit. Top graph: with a memristor. Bottom graph: with only a resistor. A total of 6000 clock periods were sampled for each oscillator type.</p> Full article ">Figure 7
<p>Example clock periods for memristor-based and resistor-based oscillators (variability emphasized in this example).</p> Full article ">Figure 8
<p>(<b>a</b>) Breadboard implementation; (<b>b</b>) Printed circuit board (PCB) implementation of the S-TRNG circuit.</p> Full article ">Figure 8 Cont.
<p>(<b>a</b>) Breadboard implementation; (<b>b</b>) Printed circuit board (PCB) implementation of the S-TRNG circuit.</p> Full article ">
27 pages, 409 KiB  
Article
The Quantum Regularization of Singular Black-Hole Solutions in Covariant Quantum Gravity
by Massimo Tessarotto and Claudio Cremaschini
Entropy 2021, 23(3), 370; https://doi.org/10.3390/e23030370 - 20 Mar 2021
Cited by 4 | Viewed by 2391
Abstract
An excruciating issue that arises in mathematical, theoretical and astro-physics concerns the possibility of regularizing classical singular black hole solutions of general relativity by means of quantum theory. The problem is posed here in the context of a manifestly covariant approach to quantum [...] Read more.
An excruciating issue that arises in mathematical, theoretical and astro-physics concerns the possibility of regularizing classical singular black hole solutions of general relativity by means of quantum theory. The problem is posed here in the context of a manifestly covariant approach to quantum gravity. Provided a non-vanishing quantum cosmological constant is present, here it is proved how a regular background space-time metric tensor can be obtained starting from a singular one. This is obtained by constructing suitable scale-transformed and conformal solutions for the metric tensor in which the conformal scale form factor is determined uniquely by the quantum Hamilton equations underlying the quantum gravitational field dynamics. Full article
(This article belongs to the Special Issue Quantum Regularization of Singular Black Hole Solutions)
11 pages, 1340 KiB  
Article
Spectral Ranking of Causal Influence in Complex Systems
by Errol Zalmijn, Tom Heskes and Tom Claassen
Entropy 2021, 23(3), 369; https://doi.org/10.3390/e23030369 - 20 Mar 2021
Cited by 1 | Viewed by 4486
Abstract
Similar to natural complex systems, such as the Earth’s climate or a living cell, semiconductor lithography systems are characterized by nonlinear dynamics across more than a dozen orders of magnitude in space and time. Thousands of sensors measure relevant process variables at appropriate [...] Read more.
Similar to natural complex systems, such as the Earth’s climate or a living cell, semiconductor lithography systems are characterized by nonlinear dynamics across more than a dozen orders of magnitude in space and time. Thousands of sensors measure relevant process variables at appropriate sampling rates, to provide time series as primary sources for system diagnostics. However, high-dimensionality, non-linearity and non-stationarity of the data are major challenges to efficiently, yet accurately, diagnose rare or new system issues by merely using model-based approaches. To reliably narrow down the causal search space, we validate a ranking algorithm that applies transfer entropy for bivariate interaction analysis of a system’s multivariate time series to obtain a weighted directed graph, and graph eigenvector centrality to identify the system’s most important sources of original information or causal influence. The results suggest that this approach robustly identifies the true drivers or causes of a complex system’s deviant behavior, even when its reconstructed information transfer network includes redundant edges. Full article
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<p>(<b>a</b>) Time series of system <math display="inline"><semantics> <msub> <mi>S</mi> <mrow> <mo>(</mo> <msub> <mi>L</mi> <mn>1</mn> </msub> <mo>⇄</mo> <msub> <mi>L</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> </msub> </semantics></math> composed of bidirectionally delay-coupled Lorenz systems <math display="inline"><semantics> <msub> <mi>L</mi> <mn>1</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>L</mi> <mn>2</mn> </msub> </semantics></math>, generated from Equations (5a)–(5f). (<b>b</b>) Heatmaps of detection count per cause–effect relation, for FaultMap (left) and PCMCI (right). Direct cause–effect relations are denoted by (*).</p> Full article ">Figure 2
<p>Information transfer network of two bidirectionally delay-coupled Lorenz systems. Edges indicate direct (⟶) or transitive indirect (⤏) information transfer. Edge annotations denote information transfer delay (sec). Node importance indicates a node’s global network influence. Edge-weight represents level of information transfer (<math display="inline"><semantics> <msub> <mi>w</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> </semantics></math>).</p> Full article ">Figure 3
<p>Dynamic information transfer via bidirectional delay-coupling <math display="inline"><semantics> <mrow> <msub> <mi>Y</mi> <mn>1</mn> </msub> <mo>⇄</mo> <msub> <mi>Y</mi> <mn>2</mn> </msub> </mrow> </semantics></math> of Lorenz systems <math display="inline"><semantics> <msub> <mi>L</mi> <mn>1</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>L</mi> <mn>2</mn> </msub> </semantics></math>, and dynamic importance of the Lorenz system state variables. (<b>a</b>) Dynamic information transfer (delay) via bidirectional delay-coupling <math display="inline"><semantics> <mrow> <msub> <mi>Y</mi> <mn>1</mn> </msub> <mo>⇄</mo> <msub> <mi>Y</mi> <mn>2</mn> </msub> </mrow> </semantics></math> between Lorenz systems <math display="inline"><semantics> <msub> <mi>L</mi> <mn>1</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>L</mi> <mn>2</mn> </msub> </semantics></math>. (<b>b</b>) Distribution of information transfer (delay) in 3a. (<b>c</b>) Dynamic importance of state variables in delay-coupled Lorenz systems <math display="inline"><semantics> <msub> <mi>L</mi> <mn>1</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>L</mi> <mn>2</mn> </msub> </semantics></math>. (<b>d</b>) Distributions of Lorenz system state variable importance in 3c.</p> Full article ">Figure 3 Cont.
<p>Dynamic information transfer via bidirectional delay-coupling <math display="inline"><semantics> <mrow> <msub> <mi>Y</mi> <mn>1</mn> </msub> <mo>⇄</mo> <msub> <mi>Y</mi> <mn>2</mn> </msub> </mrow> </semantics></math> of Lorenz systems <math display="inline"><semantics> <msub> <mi>L</mi> <mn>1</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>L</mi> <mn>2</mn> </msub> </semantics></math>, and dynamic importance of the Lorenz system state variables. (<b>a</b>) Dynamic information transfer (delay) via bidirectional delay-coupling <math display="inline"><semantics> <mrow> <msub> <mi>Y</mi> <mn>1</mn> </msub> <mo>⇄</mo> <msub> <mi>Y</mi> <mn>2</mn> </msub> </mrow> </semantics></math> between Lorenz systems <math display="inline"><semantics> <msub> <mi>L</mi> <mn>1</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>L</mi> <mn>2</mn> </msub> </semantics></math>. (<b>b</b>) Distribution of information transfer (delay) in 3a. (<b>c</b>) Dynamic importance of state variables in delay-coupled Lorenz systems <math display="inline"><semantics> <msub> <mi>L</mi> <mn>1</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>L</mi> <mn>2</mn> </msub> </semantics></math>. (<b>d</b>) Distributions of Lorenz system state variable importance in 3c.</p> Full article ">Figure 4
<p>Top-ranked node <math display="inline"><semantics> <msub> <mi>P</mi> <mn>0</mn> </msub> </semantics></math> (root cause) transfers original information towards event <math display="inline"><semantics> <msub> <mi>P</mi> <mn>27</mn> </msub> </semantics></math> via a network of collateral effects <math display="inline"><semantics> <mrow> <mo>{</mo> <msub> <mi>P</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>P</mi> <mn>26</mn> </msub> <mo>}</mo> </mrow> </semantics></math> within an ASML subsystem. (<a href="#entropy-23-00369-f002" class="html-fig">Figure 2</a> legend applies here.)</p> Full article ">
18 pages, 3275 KiB  
Article
Stirling Refrigerating Machine Modeling Using Schmidt and Finite Physical Dimensions Thermodynamic Models: A Comparison with Experiments
by Cătălina Dobre, Lavinia Grosu, Alexandru Dobrovicescu, Georgiana Chişiu and Mihaela Constantin
Entropy 2021, 23(3), 368; https://doi.org/10.3390/e23030368 - 19 Mar 2021
Cited by 9 | Viewed by 3686
Abstract
The purpose of the study is to show that two simple models that take into account only the irreversibility due to temperature difference in the heat exchangers and imperfect regeneration are able to indicate refrigerating machine behavior. In the present paper, the finite [...] Read more.
The purpose of the study is to show that two simple models that take into account only the irreversibility due to temperature difference in the heat exchangers and imperfect regeneration are able to indicate refrigerating machine behavior. In the present paper, the finite physical dimensions thermodynamics (FPDT) method and 0-D modeling using the Schmidt model with imperfect regeneration were applied in the study of a ? type Stirling refrigeration machine.The 0-D modeling is improved by including the irreversibility caused by imperfect regeneration and the finite temperature difference between the gas and the heat exchangers wall. A flowchart of the Stirling refrigerator exergy balance is presented to show the internal and external irreversibilities. It is found that the irreversibility at the regenerator level is more important than that at the heat exchangers level. The energies exchanged by the working gas are expressed according to the practical parameters, necessary for the engineer during the entire project. The results of the two thermodynamic models are presented in comparison with the experimental results, which leads to validation of the proposed FPDT model for the functional and constructive parameters of the studied refrigerating machine. Full article
(This article belongs to the Special Issue Carnot Cycle and Heat Engine Fundamentals and Applications II)
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<p>Experimental device using a β-type Stirling refrigerating machine.</p> Full article ">Figure 2
<p>Representation of three volumes of machine and their boundaries.</p> Full article ">Figure 3
<p>Temperature gradient in refrigerator regenerator.</p> Full article ">Figure 4
<p>Exergy balance for β-type Stirling refrigerating machine.</p> Full article ">Figure 5
<p>Exergetic and entropic functional diagram of expansion volume.</p> Full article ">Figure 6
<p>Exergetic and entropic functional diagram of compression volume.</p> Full article ">Figure 7
<p>Exo-irreversible reversed Stirling cycle. (<b>A</b>) Logp-LogV diagram in the range limit of <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mrow> <mi>max</mi> </mrow> </msub> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>V</mi> <mrow> <mi>max</mi> </mrow> </msub> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>T</mi> <mi>l</mi> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>T</mi> <mi>l</mi> </msub> </mrow> </semantics></math>; (<b>B</b>) energy balance scheme.</p> Full article ">Figure 8
<p>Variation of overall heat transfer coefficient <math display="inline"><semantics> <mi>h</mi> </semantics></math> depending on rotational speed <math display="inline"><semantics> <mi>n</mi> </semantics></math>.</p> Full article ">Figure 9
<p>Flowchart of exergy balance equation.</p> Full article ">
25 pages, 412 KiB  
Article
Robust Estimation for Bivariate Poisson INGARCH Models
by Byungsoo Kim, Sangyeol Lee and Dongwon Kim
Entropy 2021, 23(3), 367; https://doi.org/10.3390/e23030367 - 19 Mar 2021
Cited by 8 | Viewed by 2935
Abstract
In the integer-valued generalized autoregressive conditional heteroscedastic (INGARCH) models, parameter estimation is conventionally based on the conditional maximum likelihood estimator (CMLE). However, because the CMLE is sensitive to outliers, we consider a robust estimation method for bivariate Poisson INGARCH models while using the [...] Read more.
In the integer-valued generalized autoregressive conditional heteroscedastic (INGARCH) models, parameter estimation is conventionally based on the conditional maximum likelihood estimator (CMLE). However, because the CMLE is sensitive to outliers, we consider a robust estimation method for bivariate Poisson INGARCH models while using the minimum density power divergence estimator. We demonstrate the proposed estimator is consistent and asymptotically normal under certain regularity conditions. Monte Carlo simulations are conducted to evaluate the performance of the estimator in the presence of outliers. Finally, a real data analysis using monthly count series of crimes in New South Wales and an artificial data example are provided as an illustration. Full article
(This article belongs to the Special Issue Time Series Modelling)
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<p>Monthly count series of liquor offences (LO) (<bold>left</bold>) and transport regulatory offences (TRO) (<bold>right</bold>) in Botany Bay.</p> Full article ">Figure 2
<p>Autocorrelation function (ACF) and partial autocorrelation function (PACF) of LO (<bold>top</bold>) and TRO (<bold>middle</bold>), and cross-correlation function (CCF) (<bold>bottom</bold>) between two series.</p> Full article ">
16 pages, 318 KiB  
Article
Classical and Quantum H-Theorem Revisited: Variational Entropy and Relaxation Processes
by Carlos Medel-Portugal, Juan Manuel Solano-Altamirano and José Luis E. Carrillo-Estrada
Entropy 2021, 23(3), 366; https://doi.org/10.3390/e23030366 - 19 Mar 2021
Cited by 2 | Viewed by 3159
Abstract
We propose a novel framework to describe the time-evolution of dilute classical and quantum gases, initially out of equilibrium and with spatial inhomogeneities, towards equilibrium. Briefly, we divide the system into small cells and consider the local equilibrium hypothesis. We subsequently define a [...] Read more.
We propose a novel framework to describe the time-evolution of dilute classical and quantum gases, initially out of equilibrium and with spatial inhomogeneities, towards equilibrium. Briefly, we divide the system into small cells and consider the local equilibrium hypothesis. We subsequently define a global functional that is the sum of cell H-functionals. Each cell functional recovers the corresponding Maxwell–Boltzmann, Fermi–Dirac, or Bose–Einstein distribution function, depending on the classical or quantum nature of the gas. The time-evolution of the system is described by the relationship dH/dt?0, and the equality condition occurs if the system is in the equilibrium state. Via the variational method, proof of the previous relationship, which might be an extension of the H-theorem for inhomogeneous systems, is presented for both classical and quantum gases. Furthermore, the H-functionals are in agreement with the correspondence principle. We discuss how the H-functionals can be identified with the system’s entropy and analyze the relaxation processes of out-of-equilibrium systems. Full article
(This article belongs to the Special Issue The Statistical Foundations of Entropy)
22 pages, 561 KiB  
Article
TSARM-UDP: An Efficient Time Series Association Rules Mining Algorithm Based on Up-to-Date Patterns
by Qiang Zhao, Qing Li, Deshui Yu and Yinghua Han
Entropy 2021, 23(3), 365; https://doi.org/10.3390/e23030365 - 19 Mar 2021
Cited by 6 | Viewed by 2926
Abstract
In many industrial domains, there is a significant interest in obtaining temporal relationships among multiple variables in time-series data, given that such relationships play an auxiliary role in decision making. However, when transactions occur frequently only for a period of time, it is [...] Read more.
In many industrial domains, there is a significant interest in obtaining temporal relationships among multiple variables in time-series data, given that such relationships play an auxiliary role in decision making. However, when transactions occur frequently only for a period of time, it is difficult for a traditional time-series association rules mining algorithm (TSARM) to identify this kind of relationship. In this paper, we propose a new TSARM framework and a novel algorithm named TSARM-UDP. A TSARM mining framework is used to mine time-series association rules (TSARs) and an up-to-date pattern (UDP) is applied to discover rare patterns that only appear in a period of time. Based on the up-to-date pattern mining, the proposed TSAR-UDP method could extract temporal relationship rules with better generality. The rules can be widely used in the process industry, the stock market, etc. Experiments are then performed on the public stock data and real blast furnace data to verify the effectiveness of the proposed algorithm. We compare our algorithm with three state-of-the-art algorithms, and the experimental results show that our algorithm can provide greater efficiency and interpretability in TSARs and that it has good prospects. Full article
(This article belongs to the Section Signal and Data Analysis)
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<p>The framework of TSARM-UDP.</p> Full article ">Figure 2
<p>The difference of calculating Support between Formula (2) and Formula (5).</p> Full article ">Figure 3
<p>The flowchart of the proposed TSARM-UDP.</p> Full article ">Figure 4
<p>Comparisons of mining results on the stock dataset (<math display="inline"><semantics> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> <mo>_</mo> <mi>c</mi> <mi>o</mi> <mi>n</mi> <mi>f</mi> <mo>=</mo> <mn>0.7</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>T</mi> <mo>=</mo> <mn>7</mn> </mrow> </semantics></math>).</p> Full article ">Figure 5
<p>Comparisons of the rule numbers and <math display="inline"><semantics> <msub> <mi>L</mi> <mi>k</mi> </msub> </semantics></math> on the stock dataset (<math display="inline"><semantics> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> <mo>_</mo> <mi>c</mi> <mi>o</mi> <mi>n</mi> <mi>f</mi> <mo>=</mo> <mn>0.8</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>T</mi> <mo>=</mo> <mn>7</mn> </mrow> </semantics></math>).</p> Full article ">Figure 6
<p><math display="inline"><semantics> <mrow> <msub> <mi>L</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>L</mi> <mn>2</mn> </msub> </mrow> </semantics></math>, <math display="inline"><semantics> <msub> <mi>L</mi> <mi>k</mi> </msub> </semantics></math>, and rule numbers comparison on the BF dataset (<math display="inline"><semantics> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> <mo>_</mo> <mi>c</mi> <mi>o</mi> <mi>n</mi> <mi>f</mi> <mo>=</mo> <mn>0.6</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>T</mi> <mo>=</mo> <mn>6</mn> </mrow> </semantics></math>).</p> Full article ">Figure 7
<p>Rule numbers and <math display="inline"><semantics> <msub> <mi>L</mi> <mi>k</mi> </msub> </semantics></math> comparison on the BF dataset (<math display="inline"><semantics> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> <mo>_</mo> <mi>c</mi> <mi>o</mi> <mi>n</mi> <mi>f</mi> <mo>=</mo> <mn>0.7</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>T</mi> <mo>=</mo> <mn>6</mn> </mrow> </semantics></math>).</p> Full article ">Figure 8
<p>Rule numbers and <math display="inline"><semantics> <msub> <mi>L</mi> <mi>k</mi> </msub> </semantics></math> comparison on the BF dataset (<math display="inline"><semantics> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> <mo>_</mo> <mi>c</mi> <mi>o</mi> <mi>n</mi> <mi>f</mi> <mo>=</mo> <mn>0.8</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>T</mi> <mo>=</mo> <mn>6</mn> </mrow> </semantics></math>).</p> Full article ">
32 pages, 459 KiB  
Review
Telegraphic Transport Processes and Their Fractional Generalization: A Review and Some Extensions
by Jaume Masoliver
Entropy 2021, 23(3), 364; https://doi.org/10.3390/e23030364 - 18 Mar 2021
Cited by 20 | Viewed by 2416
Abstract
We address the problem of telegraphic transport in several dimensions. We review the derivation of two and three dimensional telegrapher’s equations—as well as their fractional generalizations—from microscopic random walk models for transport (normal and anomalous). We also present new results on solutions of [...] Read more.
We address the problem of telegraphic transport in several dimensions. We review the derivation of two and three dimensional telegrapher’s equations—as well as their fractional generalizations—from microscopic random walk models for transport (normal and anomalous). We also present new results on solutions of the higher dimensional fractional equations. Full article
(This article belongs to the Special Issue Fractional Calculus and the Future of Science)
10 pages, 777 KiB  
Article
Discrete Versions of Jensen–Fisher, Fisher and Bayes–Fisher Information Measures of Finite Mixture Distributions
by Omid Kharazmi and Narayanaswamy Balakrishnan
Entropy 2021, 23(3), 363; https://doi.org/10.3390/e23030363 - 18 Mar 2021
Cited by 1 | Viewed by 2648
Abstract
In this work, we first consider the discrete version of Fisher information measure and then propose Jensen–Fisher information, to develop some associated results. Next, we consider Fisher information and Bayes–Fisher information measures for mixing parameter vector of a finite mixture probability mass function [...] Read more.
In this work, we first consider the discrete version of Fisher information measure and then propose Jensen–Fisher information, to develop some associated results. Next, we consider Fisher information and Bayes–Fisher information measures for mixing parameter vector of a finite mixture probability mass function and establish some results. We provide some connections between these measures with some known informational measures such as chi-square divergence, Shannon entropy, Kullback–Leibler, Jeffreys and Jensen–Shannon divergences. Full article
(This article belongs to the Special Issue Measures of Information)
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<p>3D-plot of the DJFI divergence between the PMFs <math display="inline"><semantics> <mrow> <mi mathvariant="bold-italic">P</mi> <mo>=</mo> <mo>(</mo> <mi>p</mi> <mo>,</mo> <mn>1</mn> <mo>−</mo> <mi>p</mi> <mo>)</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi mathvariant="bold-italic">Q</mi> <mo>=</mo> <mo>(</mo> <mi>q</mi> <mo>,</mo> <mn>1</mn> <mo>−</mo> <mi>q</mi> <mo>)</mo> </mrow> </semantics></math>.</p> Full article ">
29 pages, 1389 KiB  
Article
Mechanism Integrated Information
by Leonardo S. Barbosa, William Marshall, Larissa Albantakis and Giulio Tononi
Entropy 2021, 23(3), 362; https://doi.org/10.3390/e23030362 - 18 Mar 2021
Cited by 26 | Viewed by 6766
Abstract
The Integrated Information Theory (IIT) of consciousness starts from essential phenomenological properties, which are then translated into postulates that any physical system must satisfy in order to specify the physical substrate of consciousness. We recently introduced an information measure (Barbosa et al., 2020) [...] Read more.
The Integrated Information Theory (IIT) of consciousness starts from essential phenomenological properties, which are then translated into postulates that any physical system must satisfy in order to specify the physical substrate of consciousness. We recently introduced an information measure (Barbosa et al., 2020) that captures three postulates of IIT—existence, intrinsicality and information—and is unique. Here we show that the new measure also satisfies the remaining postulates of IIT—integration and exclusion—and create the framework that identifies maximally irreducible mechanisms. These mechanisms can then form maximally irreducible systems, which in turn will specify the physical substrate of conscious experience. Full article
(This article belongs to the Special Issue Integrated Information Theory and Consciousness)
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Graphical abstract

Graphical abstract
Full article ">Figure 1
<p>Theory. (<b>a</b>) System <span class="html-italic">S</span> with four random variables. (<b>b</b>) Example of a mechanism <math display="inline"><semantics> <mrow> <mi>M</mi> <mo>=</mo> <mo>{</mo> <mi>A</mi> <mo>,</mo> <mi>C</mi> <mo>}</mo> </mrow> </semantics></math> in state <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mo>{</mo> <mo>↑</mo> <mo>,</mo> <mo>↑</mo> <mo>}</mo> </mrow> </semantics></math> constraining a cause purview <math display="inline"><semantics> <mrow> <mi>Z</mi> <mo>=</mo> <mo>{</mo> <mi>B</mi> <mo>}</mo> </mrow> </semantics></math> and an effect purview <math display="inline"><semantics> <mrow> <mi>Z</mi> <mo>=</mo> <mo>{</mo> <mi>B</mi> <mo>,</mo> <mi>D</mi> <mo>}</mo> </mrow> </semantics></math>. Dashed lines show the partitions. The bar plots show the probability distributions, that is the cause repertoire (left) and effect repertoire (right). The black bars show the probabilities when the mechanism is constraining the purview, and the white bars show the probabilities after partitioning the mechanism.</p> Full article ">Figure 2
<p>Intrinsicality. (<b>a</b>) Activation functions without bias (<math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>) and different levels of constraint (<math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math>). (<b>b</b>) System <span class="html-italic">S</span> analyzed in this figure. The remaining panels show on top the causal graph of the mechanism <math display="inline"><semantics> <mrow> <mi>M</mi> <mo>=</mo> <mo>{</mo> <mi>A</mi> <mo>}</mo> </mrow> </semantics></math> at state <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mo>{</mo> <mn>1</mn> <mo>}</mo> </mrow> </semantics></math> constraining different output purviews and on the bottom the probability distributions of the purviews (effect repertoires). The black bars show the probabilities when the mechanism is constraining the purview, and the white bars show the unconstrained probabilities after the complete partition <math display="inline"><semantics> <msup> <mi>ψ</mi> <mn>0</mn> </msup> </semantics></math>. The “*” indicates the state selected by the maximum operation in the intrinsic difference (ID) function. (<b>c</b>) The mechanism fully constrains the unit <span class="html-italic">B</span> in the purview <math display="inline"><semantics> <mrow> <mi>Z</mi> <mo>=</mo> <mo>{</mo> <mi>B</mi> <mo>}</mo> </mrow> </semantics></math> (<math display="inline"><semantics> <mrow> <msub> <mi>τ</mi> <mi>B</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>), resulting in state <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>=</mo> <mo>{</mo> <mo>↑</mo> <mo>}</mo> </mrow> </semantics></math> defining the amount of intrinsic information in the mechanism as <math display="inline"><semantics> <mrow> <mi>φ</mi> <mrow> <mo>(</mo> <mi>m</mi> <mo>,</mo> <mi>Z</mi> <mo>,</mo> <msup> <mi>ψ</mi> <mn>0</mn> </msup> <mo>)</mo> </mrow> <mrow> <mo>=</mo> <mi>I</mi> <mi>D</mi> <mo>(</mo> </mrow> <msub> <mi>π</mi> <mi>e</mi> </msub> <mrow> <mo>(</mo> <mi>B</mi> <mo>|</mo> <mi>M</mi> </mrow> <mo>=</mo> <mrow> <mo>↑</mo> <mo>)</mo> </mrow> <mo>∣</mo> <msubsup> <mi>π</mi> <mi>e</mi> <msup> <mi>ψ</mi> <mn>0</mn> </msup> </msubsup> <mrow> <mo>(</mo> <mi>B</mi> <mo>|</mo> <mi>M</mi> </mrow> <mo>=</mo> <mrow> <mo>↑</mo> <mo>)</mo> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>π</mi> <mi>e</mi> </msub> <mrow> <mo>(</mo> <mi>B</mi> <mo>=</mo> <mo>↑</mo> <mo>|</mo> <mi>A</mi> <mo>=</mo> <mo>↑</mo> <mo>)</mo> <mo>·</mo> <mo>|</mo> <mo form="prefix">log</mo> <mo>(</mo> </mrow> <msub> <mi>π</mi> <mi>e</mi> </msub> <mrow> <mo>(</mo> <mi>B</mi> <mo>=</mo> <mo>↑</mo> <mo>|</mo> <mi>A</mi> <mo>=</mo> <mo>↑</mo> <mo>)</mo> </mrow> <mo>/</mo> <msubsup> <mi>π</mi> <mi>e</mi> <msup> <mi>ψ</mi> <mn>0</mn> </msup> </msubsup> <mrow> <mo>(</mo> <mi>B</mi> <mo>=</mo> <mo>↑</mo> <mo>|</mo> <mi>M</mi> <mo>=</mo> <mo>↑</mo> <mo>)</mo> <mo>)</mo> <mo>|</mo> <mo>=</mo> <mn>1</mn> <mo>·</mo> <mn>0.69</mn> <mo>=</mo> <mn>0.69</mn> </mrow> </mrow> </semantics></math>. (<b>d</b>) After adding a slightly undetermined unit (<math display="inline"><semantics> <mrow> <msub> <mi>τ</mi> <mi>C</mi> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>) to the purview (<math display="inline"><semantics> <mrow> <mi>Z</mi> <mo>=</mo> <mo>{</mo> <mi>B</mi> <mo>,</mo> <mi>C</mi> <mo>}</mo> </mrow> </semantics></math>), the intrinsic information increases to <math display="inline"><semantics> <mrow> <mn>1.11</mn> </mrow> </semantics></math>. The new maximum state (<math display="inline"><semantics> <mrow> <mi>z</mi> <mo>=</mo> <mo>{</mo> <mo>↑</mo> <mo>,</mo> <mo>↑</mo> <mo>}</mo> </mrow> </semantics></math>) has now much higher informativeness (<math display="inline"><semantics> <mrow> <mrow> <mo>|</mo> <mo form="prefix">log</mo> <mo>(</mo> </mrow> <msub> <mi>π</mi> <mi>e</mi> </msub> <mrow> <mo>(</mo> <mi>B</mi> <mi>C</mi> <mo>=</mo> <mo>↑</mo> <mo>↑</mo> <mo>|</mo> <mi>A</mi> <mo>=</mo> <mo>↑</mo> <mo>)</mo> </mrow> <mo>/</mo> <msubsup> <mi>π</mi> <mi>e</mi> <msup> <mi>ψ</mi> <mn>0</mn> </msup> </msubsup> <mrow> <mo>(</mo> <mi>B</mi> <mi>C</mi> <mo>=</mo> <mo>↑</mo> <mo>↑</mo> <mo>|</mo> <mi>A</mi> <mo>=</mo> <mo>↑</mo> <mo>)</mo> <mo>)</mo> <mo>|</mo> <mo>=</mo> <mn>1.26</mn> </mrow> </mrow> </semantics></math>) but only slightly lower selectivity (<math display="inline"><semantics> <mrow> <mi>π</mi> <mo>(</mo> <mi>B</mi> <mi>C</mi> <mo>=</mo> <mo>↑</mo> <mo>↑</mo> <mo>|</mo> <mi>A</mi> <mo>=</mo> <mo>↑</mo> <mo>)</mo> <mo>=</mo> <mn>0.89</mn> </mrow> </semantics></math>), resulting in expansion. (<b>e</b>) When instead of <span class="html-italic">C</span>, we add the very undetermined unit <span class="html-italic">D</span> to the purview (<math display="inline"><semantics> <mrow> <msub> <mi>τ</mi> <mi>D</mi> </msub> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math>), the new purview (<math display="inline"><semantics> <mrow> <mi>Z</mi> <mo>=</mo> <mo>{</mo> <mi>B</mi> <mo>,</mo> <mi>D</mi> <mo>}</mo> </mrow> </semantics></math>) has a new maximum state (<math display="inline"><semantics> <mrow> <mi>z</mi> <mo>=</mo> <mo>{</mo> <mo>↑</mo> <mo>,</mo> <mo>↑</mo> <mo>}</mo> </mrow> </semantics></math>) with marginally higher informativeness (<math display="inline"><semantics> <mrow> <mrow> <mo>|</mo> <mo form="prefix">log</mo> <mo>(</mo> </mrow> <msub> <mi>π</mi> <mi>e</mi> </msub> <mrow> <mo>(</mo> <mi>B</mi> <mi>C</mi> <mo>=</mo> <mo>↑</mo> <mo>↑</mo> <mo>|</mo> <mi>A</mi> <mo>=</mo> <mo>↑</mo> <mo>)</mo> </mrow> <mo>/</mo> <msubsup> <mi>π</mi> <mi>e</mi> <msup> <mi>ψ</mi> <mn>0</mn> </msup> </msubsup> <mrow> <mo>(</mo> <mi>B</mi> <mi>C</mi> <mo>=</mo> <mo>↑</mo> <mo>↑</mo> <mo>|</mo> <mi>A</mi> <mo>=</mo> <mo>↑</mo> <mo>)</mo> <mo>)</mo> <mo>|</mo> <mo>=</mo> <mn>0.79</mn> </mrow> </mrow> </semantics></math>) and very low selectivity (<math display="inline"><semantics> <mrow> <msub> <mi>π</mi> <mi>e</mi> </msub> <mrow> <mo>(</mo> <mi>B</mi> <mi>C</mi> <mo>=</mo> <mo>↑</mo> <mo>↑</mo> <mo>|</mo> <mi>A</mi> <mo>=</mo> <mo>↑</mo> <mo>)</mo> </mrow> <mo>=</mo> <mn>0.55</mn> </mrow> </semantics></math>), resulting in dilution.</p> Full article ">Figure 3
<p>Information. (<b>a</b>) System <span class="html-italic">S</span> analyzed in this figure. All units have <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mo>−</mo> <mn>3</mn> </mrow> </semantics></math> (partially deterministic AND gates). The remaining panels show on the left the time unfolded graph of the mechanism <math display="inline"><semantics> <mrow> <mi>M</mi> <mo>=</mo> <mo>{</mo> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo>,</mo> <mi>C</mi> <mo>,</mo> <mi>D</mi> <mo>}</mo> </mrow> </semantics></math> constraining different output purviews and on the right the probability distribution of the purview <math display="inline"><semantics> <mrow> <mi>Z</mi> <mo>=</mo> <mo>{</mo> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo>,</mo> <mi>C</mi> <mo>}</mo> </mrow> </semantics></math> (effect repertoires). The black bars show the probabilities when the mechanism is constraining the purview, and the white bars show the unconstrained probabilities after the complete partition. The “*” indicates the state selected by the maximum operation in the ID function. (<b>b</b>) The mechanism at state <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mo>{</mo> <mo>↓</mo> <mo>,</mo> <mo>↓</mo> <mo>,</mo> <mo>↓</mo> <mo>,</mo> <mo>↓</mo> <mo>}</mo> </mrow> </semantics></math>. The purview state <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>=</mo> <mo>{</mo> <mo>↓</mo> <mo>,</mo> <mo>↓</mo> <mo>,</mo> <mo>↓</mo> <mo>}</mo> </mrow> </semantics></math> is not only the most constrained by the mechanism (high informativeness) but also very dense (high selectivity). As a result, it has intrinsic information higher than all other states in the purview and defines the intrinsic information of the mechanism as 0.27. (<b>c</b>) If we change the mechanism state to <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mo>{</mo> <mo>↓</mo> <mo>,</mo> <mo>↑</mo> <mo>,</mo> <mo>↑</mo> <mo>,</mo> <mo>↑</mo> <mo>}</mo> </mrow> </semantics></math>, the probability of observing the purview state <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>=</mo> <mo>{</mo> <mo>↓</mo> <mo>,</mo> <mo>↓</mo> <mo>,</mo> <mo>↓</mo> <mo>}</mo> </mrow> </semantics></math> is now <span class="html-italic">smaller</span> than chance. However, this probability is still very <span class="html-italic">different</span> from chance and therefore very constrained by the mechanism (high informativeness). At the same time, the state is still very <span class="html-italic">dense</span>, meaning it has a probability of happening much higher than all other states (high selectivity). Together, they define the intrinsic information of the state, which is higher than the intrinsic information of all other states in the purview, defining the intrinsic information of the mechanism as 0.08.</p> Full article ">Figure 4
<p>Integration. (<b>a</b>) System <span class="html-italic">S</span> analysed in this figure and in <a href="#entropy-23-00362-f005" class="html-fig">Figure 5</a>. All units have <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> (partially deterministic MAJORITY gates). The remaining panels show on the top the time unfolded graph of different mechanisms constraining different output purviews and on the bottom the probability distributions (effect repertoires). The black bars show the probabilities when the mechanism is constraining the purview, and the white bars show the partitioned probabilities. The “*” indicates the state selected by the maximum operation in the ID function. (<b>b</b>) The mechanism <math display="inline"><semantics> <mrow> <mi>M</mi> <mo>=</mo> <mo>{</mo> <mi>A</mi> <mo>,</mo> <mi>E</mi> <mo>}</mo> </mrow> </semantics></math> in state <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mo>{</mo> <mo>↑</mo> <mo>,</mo> <mo>↓</mo> <mo>}</mo> </mrow> </semantics></math> constraining the purview <math display="inline"><semantics> <mrow> <mi>Z</mi> <mo>=</mo> <mo>{</mo> <mi>A</mi> <mo>,</mo> <mi>E</mi> <mo>}</mo> </mrow> </semantics></math>. While the complete partition has nonzero intrinsic information, the mechanism is clearly not integrated, as revealed by the MIP partition <math display="inline"><semantics> <mrow> <msup> <mi>ψ</mi> <mo>*</mo> </msup> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <mo>(</mo> <mrow> <mo>{</mo> <mi>A</mi> <mo>,</mo> <mo>}</mo> </mrow> <mo>,</mo> <mrow> <mo>{</mo> <mi>A</mi> <mo>}</mo> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mrow> <mo>(</mo> <mrow> <mo>{</mo> <mi>E</mi> <mo>,</mo> <mo>}</mo> </mrow> <mo>,</mo> <mrow> <mo>{</mo> <mi>E</mi> <mo>}</mo> </mrow> <mo>)</mo> </mrow> <mo>}</mo> </mrow> </mrow> </semantics></math>, resulting in zero <span class="html-italic">integrated</span> information. (<b>c</b>) The mechanism <math display="inline"><semantics> <mrow> <mi>M</mi> <mo>=</mo> <mo>{</mo> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo>}</mo> </mrow> </semantics></math> in state <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mo>{</mo> <mo>↑</mo> <mo>,</mo> <mo>↑</mo> <mo>}</mo> </mrow> </semantics></math> constraining the purview <math display="inline"><semantics> <mrow> <mi>Z</mi> <mo>=</mo> <mo>{</mo> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo>}</mo> </mrow> </semantics></math>. The partition <math display="inline"><semantics> <mrow> <msup> <mi>ψ</mi> <mo>*</mo> </msup> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <mo>(</mo> <mrow> <mo>{</mo> <mi>A</mi> <mo>,</mo> <mo>}</mo> </mrow> <mo>,</mo> <mrow> <mo>{</mo> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo>}</mo> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mrow> <mo>(</mo> <mrow> <mo>{</mo> <mi>B</mi> <mo>}</mo> </mrow> <mo>,</mo> <mrow> <mo>{</mo> <mo>∅</mo> <mo>}</mo> </mrow> <mo>)</mo> </mrow> <mo>}</mo> </mrow> </mrow> </semantics></math> has less intrinsic information than any other partition, i.e., it is the MIP of this mechanism, and it defines the integrated information as 0.36. (<b>d</b>) The mechanism <math display="inline"><semantics> <mrow> <mi>M</mi> <mo>=</mo> <mo>{</mo> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo>,</mo> <mi>D</mi> <mo>}</mo> </mrow> </semantics></math> in state <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mo>{</mo> <mo>↑</mo> <mo>,</mo> <mo>↑</mo> <mo>,</mo> <mo>↓</mo> <mo>}</mo> </mrow> </semantics></math> constraining the purview <math display="inline"><semantics> <mrow> <mi>Z</mi> <mo>=</mo> <mo>{</mo> <mi>E</mi> <mo>,</mo> <mi>F</mi> <mo>}</mo> </mrow> </semantics></math>. The tri-partition <math display="inline"><semantics> <mrow> <msup> <mi>ψ</mi> <mo>*</mo> </msup> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <mo>(</mo> <mrow> <mo>{</mo> <mi>A</mi> <mo>}</mo> </mrow> <mo>,</mo> <mrow> <mo>{</mo> <mo>∅</mo> <mo>}</mo> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mrow> <mo>(</mo> <mrow> <mo>{</mo> <mo>∅</mo> <mo>,</mo> <mo>}</mo> </mrow> <mo>,</mo> <mrow> <mo>{</mo> <mi>F</mi> <mo>}</mo> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mrow> <mo>(</mo> <mrow> <mo>{</mo> <mi>B</mi> <mo>,</mo> <mi>D</mi> <mo>}</mo> </mrow> <mo>,</mo> <mrow> <mo>{</mo> <mi>E</mi> <mo>}</mo> </mrow> <mo>)</mo> </mrow> <mo>}</mo> </mrow> </mrow> </semantics></math> is the MIP and it shows that the mechanism is not integrated, i.e, the mechanism has zero integrated information.</p> Full article ">Figure 5
<p>Exclusion. Causal graphs of different mechanisms constraining different purviews. The system <span class="html-italic">S</span> used in these examples is the same as in <a href="#entropy-23-00362-f004" class="html-fig">Figure 4</a>a. Each line shows the mechanism <span class="html-italic">M</span> constraining different purviews <span class="html-italic">Z</span>. (<b>a</b>) The mechanism <math display="inline"><semantics> <mrow> <mi>M</mi> <mo>=</mo> <mo>{</mo> <mi>A</mi> <mo>}</mo> </mrow> </semantics></math> at state <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mo>{</mo> <mo>↑</mo> <mo>}</mo> </mrow> </semantics></math>. The bottom line shows the purview <math display="inline"><semantics> <mrow> <mi>Z</mi> <mo>∈</mo> <mi>S</mi> </mrow> </semantics></math> with maximum integrated effect information and the MIP is the complete partition. (<b>b</b>) The mechanism <math display="inline"><semantics> <mrow> <mi>M</mi> <mo>=</mo> <mo>{</mo> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo>,</mo> <mi>C</mi> <mo>,</mo> <mi>D</mi> <mo>}</mo> </mrow> </semantics></math> at state <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mo>{</mo> <mo>↑</mo> <mo>,</mo> <mo>↑</mo> <mo>,</mo> <mo>↑</mo> <mo>,</mo> <mo>↓</mo> <mo>}</mo> </mrow> </semantics></math>. The bottom line is the purview <math display="inline"><semantics> <mrow> <mi>Z</mi> <mo>∈</mo> <mi>S</mi> </mrow> </semantics></math> with maximum integrated effect information and the MIP is <math display="inline"><semantics> <mrow> <msup> <mi>ψ</mi> <mo>*</mo> </msup> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <mo>(</mo> <mrow> <mo>{</mo> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo>,</mo> <mi>C</mi> <mo>}</mo> </mrow> <mo>,</mo> <mrow> <mo>{</mo> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo>,</mo> <mi>C</mi> <mo>}</mo> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mrow> <mo>(</mo> <mrow> <mo>{</mo> <mi>D</mi> <mo>}</mo> </mrow> <mo>,</mo> <mrow> <mo>{</mo> <mo>∅</mo> <mo>}</mo> </mrow> <mo>)</mo> </mrow> <mo>}</mo> </mrow> </mrow> </semantics></math>.</p> Full article ">Figure 6
<p>Integrated cause information. (<b>a</b>) Causal graph of mechanism <math display="inline"><semantics> <mrow> <mi>M</mi> <mo>=</mo> <mo>{</mo> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo>,</mo> <mi>C</mi> <mo>,</mo> <mi>D</mi> <mo>}</mo> </mrow> </semantics></math> at state <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mo>{</mo> <mo>↑</mo> <mo>,</mo> <mo>↑</mo> <mo>,</mo> <mo>↑</mo> <mo>,</mo> <mo>↓</mo> <mo>}</mo> </mrow> </semantics></math> constraining the purview <math display="inline"><semantics> <mrow> <mi>Z</mi> <mo>=</mo> <mo>{</mo> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo>,</mo> <mi>F</mi> <mo>}</mo> </mrow> </semantics></math>, which has the maximum integrated information of all <math display="inline"><semantics> <mrow> <mi>Z</mi> <mo>⊆</mo> <mi>S</mi> </mrow> </semantics></math> and defines the mechanism integrated information. (<b>b</b>) The black bars show the probabilities when the mechanism is constraining the purview (cause repertoire), and the white bars show the probabilities after the partition (partitioned cause repertoire). The “*” indicates the state selected by the maximum operation in the ID function and defines <math display="inline"><semantics> <msubsup> <mi>Z</mi> <mi>c</mi> <mo>*</mo> </msubsup> </semantics></math>.</p> Full article ">Figure A1
<p>Comparison between earth mover’s distance (EMD) and ID. Using the same system <span class="html-italic">S</span> used in <a href="#entropy-23-00362-f004" class="html-fig">Figure 4</a>a, we find the cause purview with maximum integrated information for the mechanism <math display="inline"><semantics> <mrow> <mi>M</mi> <mo>=</mo> <mo>{</mo> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo>}</mo> </mrow> </semantics></math> in state <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mo>{</mo> <mo>↑</mo> <mo>,</mo> <mo>↑</mo> <mo>}</mo> </mrow> </semantics></math>, which is larger when using the EMD measure (<b>a</b>) when compared to the ID measure (<b>b</b>). The integrated information when using the EMD measure is also larger than the ID measure.</p> Full article ">
23 pages, 23313 KiB  
Article
Hyper-Chaotic Color Image Encryption Based on Transformed Zigzag Diffusion and RNA Operation
by Duzhong Zhang, Lexing Chen and Taiyong Li
Entropy 2021, 23(3), 361; https://doi.org/10.3390/e23030361 - 17 Mar 2021
Cited by 46 | Viewed by 3425
Abstract
With increasing utilization of digital multimedia and the Internet, protection on this digital information from cracks has become a hot topic in the communication field. As a path for protecting digital visual information, image encryption plays a crucial role in modern society. In [...] Read more.
With increasing utilization of digital multimedia and the Internet, protection on this digital information from cracks has become a hot topic in the communication field. As a path for protecting digital visual information, image encryption plays a crucial role in modern society. In this paper, a novel six-dimensional (6D) hyper-chaotic encryption scheme with three-dimensional (3D) transformed Zigzag diffusion and RNA operation (HCZRNA) is proposed for color images. For this HCZRNA scheme, four phases are included. First, three pseudo-random matrices are generated from the 6D hyper-chaotic system. Second, plaintext color image would be permuted by using the first pseudo-random matrix to convert to an initial cipher image. Third, the initial cipher image is placed on cube for 3D transformed Zigzag diffusion using the second pseudo-random matrix. Finally, the diffused image is converted to RNA codons array and updated through RNA codons tables, which are generated by codons and the third pseudo-random matrix. After four phases, a cipher image is obtained, and the experimental results show that HCZRNA has high resistance against well-known attacks and it is superior to other schemes. Full article
(This article belongs to the Special Issue Entropy in Image Analysis III)
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Figure 1

Figure 1
<p>The attractors of six-dimensional (6D) hyper-chaotic system.</p> Full article ">Figure 2
<p><math display="inline"><semantics> <mrow> <mn>4</mn> <mo>×</mo> <mn>4</mn> </mrow> </semantics></math> traditional Zigzag scramble.</p> Full article ">Figure 3
<p>Color image to cube.The first row is three channels of a color image. The second row is the triangles generated from image. Additionally, the third row is the placement of triangles on a cube.</p> Full article ">Figure 4
<p>Three-dimensional (3D) transformed Zigzag diffusion. (<b>a</b>) is the Zigzag diffusion process on the front side of cube. (<b>b</b>) is the Zigzag diffusion process on the back side of cube.</p> Full article ">Figure 5
<p>3D transformed Zigzag path. For all triangles on the cube, 3D transformed Zigzag diffusion is implemented through this order.</p> Full article ">Figure 6
<p>The process of encryption.</p> Full article ">Figure 7
<p>Reverse traversal.</p> Full article ">Figure 8
<p>The order of Zigzag in decryption.</p> Full article ">Figure 9
<p>Histogram. image (<b>a</b>,<b>c</b>,<b>e</b>) are the histograms of three channels of Baboon, and image (<b>b</b>,<b>d</b>,<b>f</b>) are the histograms of corresponding channels of encrypted Baboon.</p> Full article ">Figure 10
<p>Correlations.The first row is correlations of plaintext images, and the second row is correlations of cipher images.</p> Full article ">Figure 11
<p>Cropping attack tests. The first row is cipher images with <math display="inline"><semantics> <mrow> <mn>12.5</mn> <mo>%</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mn>25</mn> <mo>%</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mn>50</mn> <mo>%</mo> </mrow> </semantics></math> data loss, and the second row is decrypted images from the first row.</p> Full article ">Figure 12
<p>Noise attack tests. The first row is cipher images with <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>%</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mn>5</mn> <mo>%</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mn>10</mn> <mo>%</mo> </mrow> </semantics></math> salt and pepper noise, and the second row is decrypted images from the first row.</p> Full article ">
11 pages, 2020 KiB  
Article
Experimental and Theoretical Analysis of Metal Complex Diffusion through Cell Monolayer
by Katarzyna Gałczyńska, Jarosław Rachuna, Karol Ciepluch, Magdalena Kowalska, Sławomir Wąsik, Tadeusz Kosztołowicz, Katarzyna D. Lewandowska, Jacek Semaniak, Krystyna Kurdziel and Michał Arabski
Entropy 2021, 23(3), 360; https://doi.org/10.3390/e23030360 - 17 Mar 2021
Cited by 2 | Viewed by 2391
Abstract
The study of drugs diffusion through different biological membranes constitutes an essential step in the development of new pharmaceuticals. In this study, the method based on the monolayer cell culture of CHO-K1 cells has been developed in order to emulate the epithelial cells [...] Read more.
The study of drugs diffusion through different biological membranes constitutes an essential step in the development of new pharmaceuticals. In this study, the method based on the monolayer cell culture of CHO-K1 cells has been developed in order to emulate the epithelial cells barrier in permeability studies by laser interferometry. Laser interferometry was employed for the experimental analysis of nickel(II) and cobalt(II) complexes with 1-allylimidazole or their chlorides’ diffusion through eukaryotic cell monolayers. The amount (mol) of nickel(II) and cobalt(II) chlorides transported through the monolayer was greater than that of metals complexed with 1-allylimidazole by 4.34-fold and 1.45-fold, respectively, after 60 min. Thus, laser interferometry can be used for the quantitative analysis of the transport of compounds through eukaryotic cell monolayers, and the resulting parameters can be used to formulate a mathematical description of this process. Full article
(This article belongs to the Special Issue Thermodynamic Modelling in Membrane)
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Figure 1

Figure 1
<p>Chemical structures of the tested compounds.</p> Full article ">Figure 2
<p>Scheme showing the laser interferometer and the experimental system used in this work. The barrier consisted of a PET membrane with a monolayer of CHO-K1 cells formed for 48 h at 37 °C with 5% CO<sub>2</sub>.</p> Full article ">Figure 3
<p>The amount of [Ni(1-allim)<sub>6</sub>](NO<sub>3</sub>)<sub>2</sub>, [Co(1-allim)<sub>6</sub>](NO<sub>3</sub>)<sub>2</sub> and metal chlorides transported through a monolayer of CHO-K1 cells formed on a PET membrane after 48 h at 37 °C with 5% CO<sub>2</sub>. Symbols represent empirical data, and solid lines represent theoretical functions.</p> Full article ">Figure 4
<p>Microscopy images of the CHO-K1 cell monolayer. (<b>A</b>,<b>C</b>) show optical microscopy images (100× magnification) of the cell monolayer stained by Giemsa before and after the diffusion measurement of the [Co(1-allim)<sub>6</sub>](NO<sub>3</sub>)<sub>2</sub> complex (60 min), respectively. (<b>B</b>,<b>D</b>) show optical microscopy images (100× magnification) of the cell monolayer before and after the diffusion measurement of the [Co(1-allim)<sub>6</sub>](NO<sub>3</sub>)<sub>2</sub> complex (60 min), respectively.</p> Full article ">Figure 5
<p>The scheme of the system under consideration. A more detailed description is in the text.</p> Full article ">
11 pages, 1608 KiB  
Article
Are Gait and Balance Problems in Neurological Patients Interdependent? Enhanced Analysis Using Gait Indices, Cyclograms, Balance Parameters and Entropy
by Malgorzata Syczewska, Ewa Szczerbik, Malgorzata Kalinowska, Anna Swiecicka and Grazyna Graff
Entropy 2021, 23(3), 359; https://doi.org/10.3390/e23030359 - 17 Mar 2021
Cited by 3 | Viewed by 2313
Abstract
Background: Balance and locomotion are two main complex functions, which require intact and efficient neuromuscular and sensory systems, and their proper integration. In many studies the assumption of their dependence is present, and some rehabilitation approaches are based on it. Other papers undermine [...] Read more.
Background: Balance and locomotion are two main complex functions, which require intact and efficient neuromuscular and sensory systems, and their proper integration. In many studies the assumption of their dependence is present, and some rehabilitation approaches are based on it. Other papers undermine this assumption. Therefore the aim of this study was to examine the possible dependence between gait and balance in patients with neurological or sensory integration problems, which affected their balance. Methods: 75 patients (52 with neurological diseases, 23 with sensory integration problems) participated in the study. They underwent balance assessment on Kistler force plate in two conditions, six tests on a Balance Biodex System and instrumented gait analysis with VICON. The gait and balances parameters and indices, together with entropy and cyclograms were used for the analysis. Spearman correlation, multiple regression, cluster analysis, and discriminant analysis were used as analytical tools. Results: The analysis divided patients into 2 groups with 100% correctly classified cases. Some balance and gait measures are better in the first group, but some others in the second. Conclusions: This finding confirms the hypothesis that there is no direct link between gait and balance deficits. Full article
(This article belongs to the Special Issue Information Theory in Biomedical Data Mining)
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<p>Cyclograms: (<b>a</b>) for a healthy subject, (<b>b</b>) exemplary one from one of the patients. The numbers in squares are perimeters of each cyclogram. The upper left graph represents cyclogram of hip-pelvis in sagittal plane, upper right: knee-hip in sagittal plane, lower left: ankle-knee in sagittal plane, and lower right the cyclogram of hip-pelvis in frontal plane.</p> Full article ">Figure 2
<p>The hierarchical graph of the cluster analysis. X axis—cases, Y-axis—Euclidean distance on which order of clustering was done. The vertical line shows the division into two groups.</p> Full article ">
17 pages, 495 KiB  
Article
Higher Dimensional Rotating Black Hole Solutions in Quadratic f(R) Gravitational Theory and the Conserved Quantities
by Gamal G. L. Nashed and Kazuharu Bamba
Entropy 2021, 23(3), 358; https://doi.org/10.3390/e23030358 - 17 Mar 2021
Cited by 4 | Viewed by 2908
Abstract
We explore the quadratic form of the f(R)=R+bR2 gravitational theory to derive rotating N-dimensions black hole solutions with ai,i?1 rotation parameters. Here, R is the Ricci scalar and [...] Read more.
We explore the quadratic form of the f(R)=R+bR2 gravitational theory to derive rotating N-dimensions black hole solutions with ai,i?1 rotation parameters. Here, R is the Ricci scalar and b is the dimensional parameter. We assumed that the N-dimensional spacetime is static and it has flat horizons with a zero curvature boundary. We investigated the physics of black holes by calculating the relations of physical quantities such as the horizon radius and mass. We also demonstrate that, in the four-dimensional case, the higher-order curvature does not contribute to the black hole, i.e., black hole does not depend on the dimensional parameter b, whereas, in the case of N>4, it depends on parameter b, owing to the contribution of the correction R2 term. We analyze the conserved quantities, energy, and angular-momentum, of black hole solutions by applying the relocalization method. Additionally, we calculate the thermodynamic quantities, such as temperature and entropy, and examine the stability of black hole solutions locally and show that they have thermodynamic stability. Moreover, the calculations of entropy put a constraint on the parameter b to be b<116? to obtain a positive entropy. Full article
(This article belongs to the Special Issue Modified Gravity: From Black Holes Entropy to Current Cosmology III)
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<p>The function <span class="html-italic">h</span>(<span class="html-italic">r</span>) vs. the radial coordinate <span class="html-italic">r</span> for (<b>a</b>) <span class="html-italic">N</span> = 4, <math display="inline"><semantics> <mrow> <msub> <mi>c</mi> <mn>1</mn> </msub> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mrow> </semantics></math>. (<b>b</b>) <span class="html-italic">N</span> = 5, <math display="inline"><semantics> <mrow> <msub> <mi>c</mi> <mn>2</mn> </msub> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mrow> </semantics></math> (all of the figures are reproduced using the Maple software 16).</p> Full article ">Figure 2
<p>Horizon <math display="inline"><semantics> <msub> <mi>r</mi> <mi>h</mi> </msub> </semantics></math> vs. (<b>a</b>,<b>d</b>) Hawking temperature (<b>b</b>,<b>e</b>); entropy (<b>c</b>,<b>f</b>) heat capacity for the four-dimensional and five-dimensional cases, respectively. In these figures, we take <math display="inline"><semantics> <mrow> <mi>b</mi> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math> when <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>b</mi> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math> when <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math>.</p> Full article ">
14 pages, 325 KiB  
Article
Critical Comparison of MaxCal and Other Stochastic Modeling Approaches in Analysis of Gene Networks
by Taylor Firman, Jonathan Huihui, Austin R. Clark and Kingshuk Ghosh
Entropy 2021, 23(3), 357; https://doi.org/10.3390/e23030357 - 17 Mar 2021
Cited by 1 | Viewed by 2966
Abstract
Learning the underlying details of a gene network with feedback is critical in designing new synthetic circuits. Yet, quantitative characterization of these circuits remains limited. This is due to the fact that experiments can only measure partial information from which the details of [...] Read more.
Learning the underlying details of a gene network with feedback is critical in designing new synthetic circuits. Yet, quantitative characterization of these circuits remains limited. This is due to the fact that experiments can only measure partial information from which the details of the circuit must be inferred. One potentially useful avenue is to harness hidden information from single-cell stochastic gene expression time trajectories measured for long periods of time—recorded at frequent intervals—over multiple cells. This raises the feasibility vs. accuracy dilemma while deciding between different models of mining these stochastic trajectories. We demonstrate that inference based on the Maximum Caliber (MaxCal) principle is the method of choice by critically evaluating its computational efficiency and accuracy against two other typical modeling approaches: (i) a detailed model (DM) with explicit consideration of multiple molecules including protein-promoter interaction, and (ii) a coarse-grain model (CGM) using Hill type functions to model feedback. MaxCal provides a reasonably accurate model while being significantly more computationally efficient than DM and CGM. Furthermore, MaxCal requires minimal assumptions since it is a top-down approach and allows systematic model improvement by including constraints of higher order, in contrast to traditional bottom-up approaches that require more parameters or ad hoc assumptions. Thus, based on efficiency, accuracy, and ability to build minimal models, we propose MaxCal as a superior alternative to traditional approaches (DM, CGM) when inferring underlying details of gene circuits with feedback from limited data. Full article
(This article belongs to the Special Issue Information Theory and Biology: Seeking General Principles)
14 pages, 315 KiB  
Article
Results on Varextropy Measure of Random Variables
by Nastaran Marzban Vaselabadi, Saeid Tahmasebi, Mohammad Reza Kazemi and Francesco Buono
Entropy 2021, 23(3), 356; https://doi.org/10.3390/e23030356 - 17 Mar 2021
Cited by 12 | Viewed by 2708
Abstract
In 2015, Lad, Sanfilippo and Agrò proposed an alternative measure of uncertainty dual to the entropy known as extropy. This paper provides some results on a dispersion measure of extropy of random variables which is called varextropy and studies several properties of this [...] Read more.
In 2015, Lad, Sanfilippo and Agrò proposed an alternative measure of uncertainty dual to the entropy known as extropy. This paper provides some results on a dispersion measure of extropy of random variables which is called varextropy and studies several properties of this concept. Especially, the varextropy measure of residual and past lifetimes, order statistics, record values and proportional hazard rate models are discussed. Moreover, the conditional varextropy is considered and some properties of this measure are studied. Finally, a new stochastic comparison method, named varextropy ordering, is introduced and some of its properties are presented. Full article
(This article belongs to the Special Issue Measures of Information)
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<p>The values of <math display="inline"><semantics> <mrow> <mi>V</mi> <mi>J</mi> <mo>(</mo> <mi>Y</mi> <mo>)</mo> </mrow> </semantics></math> for Bernoulli distribution.</p> Full article ">
18 pages, 310 KiB  
Article
Order-Stability in Complex Biological, Social, and AI-Systems from Quantum Information Theory
by Andrei Khrennikov and Noboru Watanabe
Entropy 2021, 23(3), 355; https://doi.org/10.3390/e23030355 - 16 Mar 2021
Cited by 8 | Viewed by 2574
Abstract
This paper is our attempt, on the basis of physical theory, to bring more clarification on the question “What is life?” formulated in the well-known book of Schrödinger in 1944. According to Schrödinger, the main distinguishing feature of a biosystem’s functioning is the [...] Read more.
This paper is our attempt, on the basis of physical theory, to bring more clarification on the question “What is life?” formulated in the well-known book of Schrödinger in 1944. According to Schrödinger, the main distinguishing feature of a biosystem’s functioning is the ability to preserve its order structure or, in mathematical terms, to prevent increasing of entropy. However, Schrödinger’s analysis shows that the classical theory is not able to adequately describe the order-stability in a biosystem. Schrödinger also appealed to the ambiguous notion of negative entropy. We apply quantum theory. As is well-known, behaviour of the quantum von Neumann entropy crucially differs from behaviour of classical entropy. We consider a complex biosystem S composed of many subsystems, say proteins, cells, or neural networks in the brain, that is, S=(Si). We study the following problem: whether the compound system S can maintain “global order” in the situation of an increase of local disorder and if S can preserve the low entropy while other Si increase their entropies (may be essentially). We show that the entropy of a system as a whole can be constant, while the entropies of its parts rising. For classical systems, this is impossible, because the entropy of S cannot be less than the entropy of its subsystem Si. And if a subsystems’s entropy increases, then a system’s entropy should also increase, by at least the same amount. However, within the quantum information theory, the answer is positive. The significant role is played by the entanglement of a subsystems’ states. In the absence of entanglement, the increasing of local disorder implies an increasing disorder in the compound system S (as in the classical regime). In this note, we proceed within a quantum-like approach to mathematical modeling of information processing by biosystems—respecting the quantum laws need not be based on genuine quantum physical processes in biosystems. Recently, such modeling found numerous applications in molecular biology, genetics, evolution theory, cognition, psychology and decision making. The quantum-like model of order stability can be applied not only in biology, but also in social science and artificial intelligence. Full article
(This article belongs to the Special Issue Quantum Models of Cognition and Decision-Making)
16 pages, 5950 KiB  
Review
Beating Standard Quantum Limit with Weak Measurement
by Geng Chen, Peng Yin, Wen-Hao Zhang, Gong-Chu Li, Chuan-Feng Li and Guang-Can Guo
Entropy 2021, 23(3), 354; https://doi.org/10.3390/e23030354 - 16 Mar 2021
Cited by 14 | Viewed by 4029
Abstract
Weak measurements have been under intensive investigation in both experiment and theory. Numerous experiments have indicated that the amplified meter shift is produced by the post-selection, yielding an improved precision compared to conventional methods. However, this amplification effect comes at the cost of [...] Read more.
Weak measurements have been under intensive investigation in both experiment and theory. Numerous experiments have indicated that the amplified meter shift is produced by the post-selection, yielding an improved precision compared to conventional methods. However, this amplification effect comes at the cost of a reduced rate of acquiring data, which leads to an increasing uncertainty to determine the level of meter shift. From this point of view, a number of theoretical works have suggested that weak measurements cannot improve the precision, or even damage the metrology information due to the post-selection. In this review, we give a comprehensive analysis of the weak measurements to justify their positive effect on prompting measurement precision. As a further step, we introduce two modified weak measurement protocols to boost the precision beyond the standard quantum limit. Compared to previous works beating the standard quantum limit, these protocols are free of using entangled or squeezed states. The achieved precision outperforms that of the conventional method by two orders of magnitude and attains a practical Heisenberg scaling up to n=106 photons. Full article
(This article belongs to the Special Issue Lectures on Recent Experimental Achievements in Quantum-Enhanced Technologies)
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<p>Scheme for a standard weak measurement (WM) protocol. The quantum system (QS) is pre-selected into a superposition of <math display="inline"><semantics> <mrow> <mo>|</mo> <mo>+</mo> <mn>1</mn> <mo>〉</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mo>|</mo> <mo>−</mo> <mn>1</mn> <mo>〉</mo> </mrow> </semantics></math>, between which an extra phase is introduced after the coupling with the measurement apparatus (MA). After the post-selection, the two observables of the MA, namely <span class="html-italic">p</span> and <span class="html-italic">q</span>, are shifted in proportion to the real and imaginary parts of the weak value, as shown in (<b>a</b>,<b>b</b>), respectively. (Adapted from [<a href="#B12-entropy-23-00354" class="html-bibr">12</a>]).</p> Full article ">Figure 2
<p>The precision to estimate coupling strength <span class="html-italic">g</span> with different probing states. (<b>a</b>) Probing with coherent state <math display="inline"><semantics> <mfenced open="|" close="&#x232A;"> <mi>α</mi> </mfenced> </semantics></math>: the precision of measuring a phase <math display="inline"><semantics> <mrow> <mi>A</mi> <mi>r</mi> <mi>g</mi> <mo>(</mo> <mi>α</mi> <mo>)</mo> </mrow> </semantics></math> is limited by the inherent uncertainty <math display="inline"><semantics> <mrow> <mo>Δ</mo> <mi>Re</mi> <mo>[</mo> <mi>α</mi> <mo>]</mo> <mo>=</mo> <mo>Δ</mo> <mi>Im</mi> <mo>[</mo> <mi>α</mi> <mo>]</mo> <mo>=</mo> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </semantics></math>, hence it is standard-quantum-limited. (<b>b</b>) Probing with mixed state: the initial state is a statistical ensemble of <math display="inline"><semantics> <mfenced open="|" close="&#x232A;"> <mi>α</mi> </mfenced> </semantics></math>, each of which acquires a phase decided by <math display="inline"><semantics> <mrow> <mo>|</mo> <mi>α</mi> <mo>|</mo> </mrow> </semantics></math>, and interferes (with itself) with varying probability due to the post-selection. A shift in the radical direction is generated and its level is proportional to the length squared. For a mixed state, the length can be increased to <math display="inline"><semantics> <mrow> <mo>∼</mo> <mo>|</mo> <mi>α</mi> <mo>|</mo> </mrow> </semantics></math>, which leads to Heisenberg scaling precision. (Adapted from [<a href="#B21-entropy-23-00354" class="html-bibr">21</a>]).</p> Full article ">Figure 3
<p>Setup of estimating single photon Kerr effect with mixed probe state. Single photons of 785 nm interact with strong pulses of 800 nm in the photonic crystal fiber (PCF). The amplitude of the strong pulses is modulated to generate a mixture with different coherent states. Post-selecting the single photons and detecting the intensity of corresponding strong pulses with a full HD oscilloscope, the coupling strength <span class="html-italic">g</span> can be estimated from the mean photon number shift of the strong pulses. The achieved precision <math display="inline"><semantics> <mrow> <mo>Δ</mo> <mi>g</mi> </mrow> </semantics></math> is inversely proportional to the mean photon number <span class="html-italic">N</span>, which is the so-called Heisenberg scaling. (Adapted from [<a href="#B21-entropy-23-00354" class="html-bibr">21</a>]).</p> Full article ">Figure 4
<p>A practical Heisenberg scaling in experiment.The estimation precision of <span class="html-italic">g</span> against the mean photon number <span class="html-italic">N</span> of the mixed state is plotted. For <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>&lt;</mo> <msup> <mn>10</mn> <mn>5</mn> </msup> </mrow> </semantics></math>, the precision follows a Heisenberg scaling of <math display="inline"><semantics> <mrow> <mo>Δ</mo> <mi>g</mi> <mo>≃</mo> <mn>6.3</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </msup> <msup> <mi>N</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </semantics></math> rad, shown as a green line, obtained by fitting these points. The red line is a bound on the precision for mixed states, taking account of the QFI for each member in the ensemble, given by <math display="inline"><semantics> <mrow> <mo>Δ</mo> <msub> <mi>g</mi> <mi>min</mi> </msub> <mo>≃</mo> <mn>0.95</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </msup> <msup> <mi>N</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </semantics></math> rad. (Adapted from [<a href="#B21-entropy-23-00354" class="html-bibr">21</a>]).</p> Full article ">Figure 5
<p>The classical information <math display="inline"><semantics> <msub> <mi>F</mi> <mi>p</mi> </msub> </semantics></math> contained in the post-selection process. <math display="inline"><semantics> <msub> <mi>F</mi> <mi>p</mi> </msub> </semantics></math> is calculated for varying interaction strength <span class="html-italic">g</span> and post-selection parameter <math display="inline"><semantics> <mi>ε</mi> </semantics></math>, when the mean photon number of strong pulses is <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>5</mn> <mo>×</mo> <msup> <mn>10</mn> <mn>4</mn> </msup> </mrow> </semantics></math>. As <math display="inline"><semantics> <mrow> <mi>g</mi> <mo>→</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <msub> <mi>F</mi> <mi>p</mi> </msub> </semantics></math> becomes dominant in <math display="inline"><semantics> <msub> <mi>F</mi> <mrow> <mi>t</mi> <mi>o</mi> <mi>t</mi> </mrow> </msub> </semantics></math> and scales with <math display="inline"><semantics> <msup> <mi>n</mi> <mn>2</mn> </msup> </semantics></math>, which means a practical Heisenberg scaling precision is attainable by measuring the successful post-selection probability. (Adapted from [<a href="#B22-entropy-23-00354" class="html-bibr">22</a>]).</p> Full article ">Figure 6
<p>Setup for estimating single photon Kerr effect by measuring the post-selection probability. The 815 nm photons serve as triggers and the heralded 785 nm photons interact with strong pulses (800 nm) in an 8 m long photonic crystal fiber (PCF), centering in a polarization Sagnac interferometer (PSI). The interaction strength <span class="html-italic">g</span> is estimated from the distribution of successful and failed post-selection probabilities. (Adapted from [<a href="#B22-entropy-23-00354" class="html-bibr">22</a>]).</p> Full article ">Figure 7
<p>A practical Heisenberg scaling approaching the Heisenberg limit.(<b>a</b>) Experimental verification of Heisenberg-limited precision. The achieved precision shows good agreement with the <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>/</mo> <mi>n</mi> </mrow> </semantics></math> fitting line up to <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <msup> <mn>10</mn> <mn>6</mn> </msup> </mrow> </semantics></math> photons, and an ultimate precision of <math display="inline"><semantics> <mrow> <mo>∼</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>10</mn> </mrow> </msup> </mrow> </semantics></math> rad is obtained. (<b>b</b>) The amount of extracted Fisher information (FI) for different <span class="html-italic">n</span>. The <math display="inline"><semantics> <msup> <mi>n</mi> <mn>2</mn> </msup> </semantics></math> scaling also indicates that the Heisenberg limit is approached in this measurement. (Adapted from [<a href="#B22-entropy-23-00354" class="html-bibr">22</a>]).</p> Full article ">
13 pages, 497 KiB  
Article
Simulating Finite-Time Isothermal Processes with Superconducting Quantum Circuits
by Jin-Fu Chen, Ying Li and Hui Dong
Entropy 2021, 23(3), 353; https://doi.org/10.3390/e23030353 - 16 Mar 2021
Cited by 8 | Viewed by 2871
Abstract
Finite-time isothermal processes are ubiquitous in quantum-heat-engine cycles, yet complicated due to the coexistence of the changing Hamiltonian and the interaction with the thermal bath. Such complexity prevents classical thermodynamic measurements of a performed work. In this paper, the isothermal process is decomposed [...] Read more.
Finite-time isothermal processes are ubiquitous in quantum-heat-engine cycles, yet complicated due to the coexistence of the changing Hamiltonian and the interaction with the thermal bath. Such complexity prevents classical thermodynamic measurements of a performed work. In this paper, the isothermal process is decomposed into piecewise adiabatic and isochoric processes to measure the performed work as the internal energy change in adiabatic processes. The piecewise control scheme allows the direct simulation of the whole process on a universal quantum computer, which provides a new experimental platform to study quantum thermodynamics. We implement the simulation on ibmqx2 to show the 1/? scaling of the extra work in finite-time isothermal processes. Full article
(This article belongs to the Special Issue Carnot Cycle and Heat Engine Fundamentals and Applications II)
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<p>Simulation of the isothermal process on the superconducting quantum computer. The finite-time isothermal process is divided into series of piecewise adiabatic and isochoric processes. In the adiabatic process, the energy of the two-level system is tuned with the switched-off interaction between the system and the thermal bath. In the isochoric process, the interaction is switched on with the unchanged energy spacing <math display="inline"><semantics> <msub> <mi>ω</mi> <mi>j</mi> </msub> </semantics></math>. One qubit represents the simulated two-level system, and the ancillary qubits play the role of the thermal bath at the temperature <span class="html-italic">T</span>. After implementing the quantum circuit, the system qubit is measured to obtain the internal energy.</p> Full article ">Figure 2
<p>The quantum circuits in one elementary process. (<b>a</b>) The amplitude damping (pumping) channel <math display="inline"><semantics> <msubsup> <mi mathvariant="script">E</mi> <mrow> <mo>↓</mo> </mrow> <mrow> <mo>(</mo> <mi>j</mi> <mo>)</mo> </mrow> </msubsup> </semantics></math> (<math display="inline"><semantics> <msubsup> <mi mathvariant="script">E</mi> <mrow> <mo>↑</mo> </mrow> <mrow> <mo>(</mo> <mi>j</mi> <mo>)</mo> </mrow> </msubsup> </semantics></math>) in the hybrid simulation. (<b>b</b>) One elementary process in the hybrid simulation. The selection of the two sub-channels is realized by the classical random number generator. (<b>c</b>) One elementary process in the fully quantum simulation. The selection of the two sub-channels is assisted by another ancillary qubit. (<b>d</b>) Instruction of gates in the current simulation.</p> Full article ">Figure 3
<p>The circuit of the two-step isothermal process on ibmqx2. (<b>a</b>) Excited state population-energy (<math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mi>e</mi> </msub> <mo>−</mo> <mi>E</mi> </mrow> </semantics></math>) diagram. (<b>b</b>) The circuit for the hybrid simulation. In each elementary process, the X gate is (or not) implemented for the sub-channel selected as the amplitude pumping (damping) channel according to the classical random number. Each elementary process requires another ancillary qubit. (<b>c</b>) The circuit for the fully quantum simulation. Each elementary requires two ancillary qubits.</p> Full article ">Figure 4
<p><math display="inline"><semantics> <mrow> <mn>1</mn> <mo>/</mo> <mi>N</mi> </mrow> </semantics></math> scaling of the extra work for the discrete isothermal process. The operation time of each isochoric process is set as <math display="inline"><semantics> <mrow> <mi>δ</mi> <mi>τ</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math> (blue dashed curve) or 10 (red solid curve). The ibmqx2 simulation results for <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> </mrow> </semantics></math> and 4 are plotted. The empty squares present the results by the hybrid simulations, and the pentagrams for the fully quantum simulation. The <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>/</mo> <mi>N</mi> </mrow> </semantics></math> scaling is shown by the solid black curve.</p> Full article ">Figure 5
<p>Comparison of the ibmqx2 simulation and the numerical results. (<b>a</b>,<b>b</b>) show the microscopic work in the hybrid simulation with the step number <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mspace width="0.166667em"/> <mn>3</mn> </mrow> </semantics></math> and 4. The ibmqx2 simulation result (blue solid line) is compared with the numerical result (gray dashed line). (<b>c</b>,<b>d</b>) show the excited state population <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mi>e</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> at each step in the fully quantum simulation of the two-step isothermal process. The ibmqx2 simulation results (blue bar) are compared to the numerical results (gray bar).</p> Full article ">
19 pages, 4805 KiB  
Article
Network Analysis of Cross-Correlations on Forex Market during Crises. Globalisation on Forex Market
by Janusz Miśkiewicz
Entropy 2021, 23(3), 352; https://doi.org/10.3390/e23030352 - 15 Mar 2021
Cited by 12 | Viewed by 3143
Abstract
Within the paper, the problem of globalisation during financial crises is analysed. The research is based on the Forex exchange rates. In the analysis, the power law classification scheme (PLCS) is used. The study shows that during crises cross-correlations increase resulting in significant [...] Read more.
Within the paper, the problem of globalisation during financial crises is analysed. The research is based on the Forex exchange rates. In the analysis, the power law classification scheme (PLCS) is used. The study shows that during crises cross-correlations increase resulting in significant growth of cliques, and also the ranks of nodes on the converging time series network are growing. This suggests that the crises expose the globalisation processes, which can be verified by the proposed analysis. Full article
(This article belongs to the Special Issue Three Risky Decades: A Time for Econophysics?)
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<p>The mean value of the exchange rates return of the considered time series.</p> Full article ">Figure 2
<p>The frequency of connection presented in descending order. The time window <math display="inline"><semantics> <mrow> <msub> <mi>T</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>20</mn> </mrow> </semantics></math> days. The blue line denotes a group of currencies of similar frequency of being connected on the network.</p> Full article ">Figure 3
<p>The biggest clique size evolution. Time window size <math display="inline"><semantics> <mrow> <msub> <mi>T</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>20</mn> </mrow> </semantics></math> days.</p> Full article ">Figure 4
<p>Evolution of the communities number. Time window size <math display="inline"><semantics> <mrow> <msub> <mi>T</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>20</mn> </mrow> </semantics></math> days.</p> Full article ">Figure 5
<p>Evolution of the rank nodes histogram for converging network. The time window size <math display="inline"><semantics> <mrow> <msub> <mi>T</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>20</mn> </mrow> </semantics></math> days. The counts denote how many times the node of given rank (number of links) was observed on the network.</p> Full article ">Figure 6
<p>Evolution of the rank nodes histogram for diverging network. The time window size <math display="inline"><semantics> <mrow> <msub> <mi>T</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>20</mn> </mrow> </semantics></math> days. Counts denotes how many times the node of given rank (number of links) was observed on the network.</p> Full article ">Figure 7
<p>Evolution of the rank node entropy for diverging and converging networks. The time window size <math display="inline"><semantics> <mrow> <msub> <mi>T</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>20</mn> </mrow> </semantics></math> days. The blue circles and green squares denote the entropy of diverging and convergent network, respectively.</p> Full article ">Figure 8
<p>The frequency of connection presented in descending order. The time window <math display="inline"><semantics> <mrow> <msub> <mi>T</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>60</mn> </mrow> </semantics></math> days. The blue line denotes group of currencies of similar frequency of being connected on the network.</p> Full article ">Figure 9
<p>The biggest clique size evolution. Time window size <math display="inline"><semantics> <mrow> <msub> <mi>T</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>60</mn> </mrow> </semantics></math> days.</p> Full article ">Figure 10
<p>Evolution of the community number. The time window <math display="inline"><semantics> <mrow> <msub> <mi>T</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>60</mn> </mrow> </semantics></math> days.</p> Full article ">Figure 11
<p>Evolution of the rank nodes histogram for converging network. The time window size <math display="inline"><semantics> <mrow> <msub> <mi>T</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>60</mn> </mrow> </semantics></math> days. Counts denote how many times the node of given rank (number of links) was observed on the network.</p> Full article ">Figure 12
<p>Evolution of the rank nodes histogram for diverging network. The time window size <math display="inline"><semantics> <mrow> <msub> <mi>T</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>60</mn> </mrow> </semantics></math> days. Counts denote how many times the node of given rank (number of links) was observed on the network.</p> Full article ">Figure 13
<p>Evolution of the rank node entropy for diverging and converging networks. The time window size <math display="inline"><semantics> <mrow> <msub> <mi>T</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>60</mn> </mrow> </semantics></math> days. The blue circles and green squares denote the entropy of diverging and convergent network respectively.</p> Full article ">Figure 14
<p>The frequency of connection presented in descending order. The time window <math display="inline"><semantics> <mrow> <msub> <mi>T</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>120</mn> </mrow> </semantics></math> days. The blue line denotes group of currencies of similar frequency of being connected on the network.</p> Full article ">Figure 15
<p>The biggest clique size evolution. Time window size <math display="inline"><semantics> <mrow> <msub> <mi>T</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>120</mn> </mrow> </semantics></math> days.</p> Full article ">Figure 16
<p>Evolution of the community number. The time window <math display="inline"><semantics> <mrow> <msub> <mi>T</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>120</mn> </mrow> </semantics></math> days.</p> Full article ">Figure 17
<p>Evolution of the rank nodes histogram for converging network. The time window size <math display="inline"><semantics> <mrow> <msub> <mi>T</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>120</mn> </mrow> </semantics></math> days. Counts denotes how many times the node of given rank (number of links) was observed on the network.</p> Full article ">Figure 18
<p>Evolution of the rank nodes histogram for diverging network. The time window size <math display="inline"><semantics> <mrow> <msub> <mi>T</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>120</mn> </mrow> </semantics></math> days. Counts denote how many times the node of given rank (number of links) was observed on the network.</p> Full article ">Figure 19
<p>Evolution of the rank node entropy for diverging and converging networks. The time window size <math display="inline"><semantics> <mrow> <msub> <mi>T</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>120</mn> </mrow> </semantics></math> days. The blue circles and green squares denote the entropy of diverging and convergent network respectively.</p> Full article ">
14 pages, 716 KiB  
Article
Interstage Pressures of a Multistage Compressor with Intercooling
by Helen Lugo-Méndez, Teresa Lopez-Arenas, Alejandro Torres-Aldaco, Edgar Vicente Torres-González, Mauricio Sales-Cruz and Raúl Lugo-Leyte
Entropy 2021, 23(3), 351; https://doi.org/10.3390/e23030351 - 15 Mar 2021
Cited by 7 | Viewed by 3983
Abstract
This paper considers the criterion of minimum compression work to derive an expression for the interstage pressure of a multistage compressor with intercooling that includes the gas properties, pressure drops in the intercoolers, different suction gas temperatures, and isentropic efficiencies in each compression [...] Read more.
This paper considers the criterion of minimum compression work to derive an expression for the interstage pressure of a multistage compressor with intercooling that includes the gas properties, pressure drops in the intercoolers, different suction gas temperatures, and isentropic efficiencies in each compression stage. The analytical expression for the interstage pressures is applied to estimate the number of compression stages and to evaluate its applicability in order to estimate interstage pressures in the operation of multistage compressors, which can be especially useful when their measurements are not available. Full article
(This article belongs to the Section Thermodynamics)
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<p>Schematic diagram of an <math display="inline"><semantics> <msub> <mi>N</mi> <mi>c</mi> </msub> </semantics></math>-multistage compressor with intercooling.</p> Full article ">Figure 2
<p>Temperature–entropy diagram of an <math display="inline"><semantics> <msub> <mi>N</mi> <mi>c</mi> </msub> </semantics></math>-multistage compression process with intercooling.</p> Full article ">Figure 3
<p>Number of compression stages and number of intercooling stages as a function: (<b>a</b>) individual compressor pressure ratio and (<b>b</b>) overall pressure ratio.</p> Full article ">Figure 4
<p>Natural gas two-stage centrifugal compressor: site design, and actual operating conditions.</p> Full article ">Figure 5
<p>Percentage deviations in interstage pressure models under design and actual operating conditions.</p> Full article ">
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