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Entropy, Volume 25, Issue 7 (July 2023) – 149 articles

Cover Story (view full-size image): Gear shifting is best known from car driving. On a flat highway, the highest gear gives the highest speed. When driving uphill, shifting to a lower gear may increase the speed and prevent stalling; the optimal gear number decreases with the increasing slope of the hill. Can living organisms shift gears? The answer is yes. Cells can engage in alternative pathways that enable them to continue making energy molecules (‘ATP’) when these contain more Gibbs energy, even though this uses more nutrient energy per energy molecule. The continued synthesis of the energy molecules enables the cells to utilize these for growth and survival when faced with thermodynamic challenges. View this paper
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17 pages, 18119 KiB  
Article
Friction and Stiffness Dependent Dynamics of Accumulation Landslides with Delayed Failure
by Srđan Kostić, Kristina Todorović, Žarko Lazarević and Dragan Prekrat
Entropy 2023, 25(7), 1109; https://doi.org/10.3390/e25071109 - 24 Jul 2023
Cited by 1 | Viewed by 1165
Abstract
We propose a new model for landslide dynamics under the assumption of a delay failure mechanism. Delay failure is simulated as a delayed interaction between adjacent blocks, which mimics the relationship between the accumulation and feeder part of the accumulation slope. The conducted [...] Read more.
We propose a new model for landslide dynamics under the assumption of a delay failure mechanism. Delay failure is simulated as a delayed interaction between adjacent blocks, which mimics the relationship between the accumulation and feeder part of the accumulation slope. The conducted research consisted of three phases. Firstly, the real observed movements of the landslide were examined to exclude the existence or the statistically significant presence of background noise. Secondly, we propose a new mechanical model of an accumulation landslide dynamics, with introduced delay failure, and with variable friction law. Results obtained indicate the onset of a transition from an equilibrium state to an oscillatory regime if delayed failure is assumed for different cases of slope stiffness and state of homogeneity/heterogeneity of the slope. At the end, we examine the influence of different frictional properties (along the sliding surface) on the conditions for the onset of instability. Results obtained indicate that the increase of friction parameters leads to stabilization of sliding for homogeneous geological environment. Moreover, increase of certain friction parameters leads to the occurrence of irregular aperiodic behavior, which could be ascribed to the regime of fast irregular sliding along the slope. Full article
(This article belongs to the Section Complexity)
►▼ Show Figures

Figure 1

Figure 1
<p>Location of the landslide, with engineering-geological map and distribution of the monitoring equipment. Red lines denote positions of faults, spiky lines denote the loess cliffs, points 18–60 denote the position of geodetic benches, point IB-1, IB-4, and IB-5 stand for the position of inclinometers.</p> Full article ">Figure 2
<p>Typical engineering-geological cross-section 1-1’ in direction of landslide movement, as shown in <a href="#entropy-25-01109-f001" class="html-fig">Figure 1</a>, according to data in [<a href="#B16-entropy-25-01109" class="html-bibr">16</a>].</p> Full article ">Figure 3
<p>Time series of the superficial displacements recorded by geodetic benches in the period 2011–2020, at the location of landslide “Plavinac” in Smederevo [<a href="#B15-entropy-25-01109" class="html-bibr">15</a>].</p> Full article ">Figure 4
<p>Recorded time series of the displacements along the depth, inclinometers IB-1, IB-4, and IB-5 [<a href="#B15-entropy-25-01109" class="html-bibr">15</a>].</p> Full article ">Figure 5
<p>Results of mutual information method for some of the recorded time series. Qualitatively similar results are obtained for the rest of the examined series.</p> Full article ">Figure 6
<p>Bifurcation diagrams k-τ for models (4)–(6): (<b>a</b>) model (4)—both slides exhibit Coulomb-like friction force (<span class="html-italic">ε</span> = 10<sup>−4</sup>, <span class="html-italic">p</span> = 1, <span class="html-italic">α</span> = 0.2, <span class="html-italic">V</span><sub>0</sub> = 0.2; initial conditions: <span class="html-italic">U</span><sub>1</sub> = 0.001, <span class="html-italic">U</span><sub>2</sub> = 0.0001; <span class="html-italic">V</span><sub>1</sub> = <span class="html-italic">V</span><sub>2</sub> = 0.1), (<b>b</b>) model (5)—both slides exhibit cubic friction force (<span class="html-italic">a</span> = 3.2, <span class="html-italic">b</span> = 7.2, <span class="html-italic">c</span> = 4.8; initial conditions: <span class="html-italic">U</span><sub>1</sub> = 0.001, <span class="html-italic">U</span><sub>2</sub> = 0.0001; <span class="html-italic">V</span><sub>1</sub><span class="html-italic">= V</span><sub>2</sub> = 0.1), (<b>c</b>) model (6)—feeder slope exhibits Coulomb-like friction force, accumulation slope exhibits cubic friction force. Parameter values and initial conditions are the same as in (<b>a</b>,<b>b</b>). EQ stands for equilibrium state (steady movements of low intensity), while PM denotes the regime of regular periodic oscillations.</p> Full article ">Figure 7
<p>Characteristic time series for the (<span class="html-italic">k</span>,<span class="html-italic">τ</span>) points as marked in <a href="#entropy-25-01109-f006" class="html-fig">Figure 6</a>: (<b>a</b>) refers to points in <a href="#entropy-25-01109-f006" class="html-fig">Figure 6</a>a, (<b>b</b>) refers to points in <a href="#entropy-25-01109-f006" class="html-fig">Figure 6</a>b, (<b>c</b>) refers to points in <a href="#entropy-25-01109-f006" class="html-fig">Figure 6</a>c.</p> Full article ">Figure 8
<p>Bifurcation diagrams <span class="html-italic">τ-α</span> (<b>a</b>) and <span class="html-italic">τ-ε</span> (<b>b</b>) for models (5) and (7). While <span class="html-italic">τ</span>, <span class="html-italic">ε</span>, and <span class="html-italic">α</span> are varied, other parameters are being held constant: <span class="html-italic">ε</span> = 10<sup>−4</sup>, <span class="html-italic">p</span> = 1, <span class="html-italic">α</span> = 0.2, <span class="html-italic">V<sub>0</sub></span> = 0.2, <span class="html-italic">a</span> = 4.8, <span class="html-italic">b</span> = −7.2, <span class="html-italic">c</span> = 3.2, initial conditions: <span class="html-italic">U</span><sub>1</sub> = 0.001, <span class="html-italic">U</span><sub>2</sub> = 0.0001; <span class="html-italic">V</span><sub>1</sub><span class="html-italic">= V</span><sub>2</sub><span class="html-italic">=</span> 0.1. EQ denotes the steady state (no motion), SS marks the steady sliding, and PM denotes the occurrence of periodic oscillations.</p> Full article ">Figure 9
<p>Bifurcation diagrams <span class="html-italic">τ-a</span> for models (6), (<b>a</b>) and (7), (<b>b</b>). While <span class="html-italic">τ</span> and <span class="html-italic">a</span> are varied, other parameters are being held constant: <span class="html-italic">ε</span> = 10<sup>−4</sup>, <span class="html-italic">p</span> = 1, <span class="html-italic">α</span> = 0.2, <span class="html-italic">V</span><sub>0</sub> = 0.2, <span class="html-italic">b</span> = −7.2, <span class="html-italic">c</span> = 3.2, initial conditions: <span class="html-italic">U</span><sub>1</sub> = 0.001, <span class="html-italic">U</span><sub>2</sub> = 0.0001; <span class="html-italic">V</span><sub>1</sub><span class="html-italic">= V</span><sub>2</sub> = 0.1. EQ denotes the steady state (no motion), SS marks the steady sliding, PM denotes the occurrence of periodic oscillations, while IM stands for irregular oscillations.</p> Full article ">Figure 10
<p>Bifurcation diagrams τ-b for models (6), (<b>a</b>) and (7), (<b>b</b>). While <span class="html-italic">τ</span> and <span class="html-italic">b</span> are varied, other parameters are being held constant: <span class="html-italic">ε</span> = 10<sup>−4</sup>, <span class="html-italic">p</span> = 1, <span class="html-italic">α</span> = 0.2, V<sub>0</sub> = 0.2, <span class="html-italic">a</span> = 4.8, <span class="html-italic">c</span> = 3.2, initial conditions: <span class="html-italic">U</span><sub>1</sub> = 0.001, <span class="html-italic">U</span><sub>2</sub> = 0.0001; <span class="html-italic">V</span><sub>1</sub><span class="html-italic">= V</span><sub>2</sub> = 0.1. EQ denotes the steady state (no motion), SS marks the steady sliding, PM denotes the occurrence of periodic oscillations, while IM stands for irregular oscillations.</p> Full article ">Figure 11
<p>Bifurcation diagrams <span class="html-italic">τ-c</span> for models (6), (<b>a</b>) and (7), (<b>b</b>). While <span class="html-italic">τ</span> and <span class="html-italic">c</span> are varied, other parameters are being held constant: <span class="html-italic">ε</span> = 10<sup>−4</sup>, <span class="html-italic">p</span> = 1, <span class="html-italic">α</span> = 0.2, V<sub>0</sub> = 0.2, <span class="html-italic">a</span> = 4.8, <span class="html-italic">b</span> = −7.2, initial conditions: U<sub>1</sub> = 0.001, U<sub>2</sub> = 0.0001; V<sub>1</sub> = V<sub>2</sub> = 0.1. EQ denotes the steady state (no motion), SS marks the steady sliding, and PM denotes the occurrence of periodic oscillations.</p> Full article ">
33 pages, 804 KiB  
Article
Kolmogorov Entropy for Convergence Rate in Incomplete Functional Time Series: Application to Percentile and Cumulative Estimation in High Dimensional Data
by Ouahiba Litimein, Fatimah Alshahrani, Salim Bouzebda, Ali Laksaci and Boubaker Mechab
Entropy 2023, 25(7), 1108; https://doi.org/10.3390/e25071108 - 24 Jul 2023
Viewed by 1361
Abstract
The convergence rate for free-distribution functional data analyses is challenging. It requires some advanced pure mathematics functional analysis tools. This paper aims to bring several contributions to the existing functional data analysis literature. First, we prove in this work that Kolmogorov entropy is [...] Read more.
The convergence rate for free-distribution functional data analyses is challenging. It requires some advanced pure mathematics functional analysis tools. This paper aims to bring several contributions to the existing functional data analysis literature. First, we prove in this work that Kolmogorov entropy is a fundamental tool in characterizing the convergence rate of the local linear estimation. Precisely, we use this tool to derive the uniform convergence rate of the local linear estimation of the conditional cumulative distribution function and the local linear estimation conditional quantile function. Second, a central limit theorem for the proposed estimators is established. These results are proved under general assumptions, allowing for the incomplete functional time series case to be covered. Specifically, we model the correlation using the ergodic assumption and assume that the response variable is collected with missing at random. Finally, we conduct Monte Carlo simulations to assess the finite sample performance of the proposed estimators. Full article
(This article belongs to the Special Issue Stochastic Models and Statistical Inference: Analysis and Applications)
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Figure 1

Figure 1
<p>A sample of 100 curves.</p> Full article ">Figure 2
<p>Case <math display="inline"><semantics><mrow><mo>(</mo><mrow><mi mathvariant="italic">op</mi><mo>.</mo><mi mathvariant="italic">norms</mi></mrow><mo>,</mo><mi>γ</mi><mo>)</mo><mo>=</mo><mo>(</mo><mn>0.98</mn><mo>,</mo><mn>5</mn><mo>)</mo></mrow></semantics></math>.</p> Full article ">Figure 3
<p>Case <math display="inline"><semantics><mrow><mo>(</mo><mrow><mi mathvariant="italic">op</mi><mo>.</mo><mi mathvariant="italic">norms</mi></mrow><mo>,</mo><mi>γ</mi><mo>)</mo><mo>=</mo><mo>(</mo><mn>0.98</mn><mo>,</mo><mn>0.5</mn><mo>)</mo></mrow></semantics></math>.</p> Full article ">Figure 4
<p>Case <math display="inline"><semantics><mrow><mo>(</mo><mrow><mi mathvariant="italic">op</mi><mo>.</mo><mi mathvariant="italic">norms</mi></mrow><mo>,</mo><mi>γ</mi><mo>)</mo><mo>=</mo><mo>(</mo><mn>0.48</mn><mo>,</mo><mn>5</mn><mo>)</mo></mrow></semantics></math>.</p> Full article ">Figure 5
<p>Case <math display="inline"><semantics><mrow><mo>(</mo><mrow><mi mathvariant="italic">op</mi><mo>.</mo><mi mathvariant="italic">norms</mi></mrow><mo>,</mo><mi>γ</mi><mo>)</mo><mo>=</mo><mo>(</mo><mn>0.48</mn><mo>,</mo><mn>0.5</mn><mo>)</mo></mrow></semantics></math>.</p> Full article ">Figure 6
<p>Case <math display="inline"><semantics><mrow><mo>(</mo><mrow><mi mathvariant="italic">op</mi><mo>.</mo><mi mathvariant="italic">norms</mi></mrow><mo>,</mo><mi>γ</mi><mo>)</mo><mo>=</mo><mo>(</mo><mn>0.98</mn><mo>,</mo><mn>5</mn><mo>)</mo></mrow></semantics></math>.</p> Full article ">Figure 7
<p>Case <math display="inline"><semantics><mrow><mo>(</mo><mrow><mi mathvariant="italic">op</mi><mo>.</mo><mi mathvariant="italic">norms</mi></mrow><mo>,</mo><mi>γ</mi><mo>)</mo><mo>=</mo><mo>(</mo><mn>0.98</mn><mo>,</mo><mn>0.5</mn><mo>)</mo></mrow></semantics></math>.</p> Full article ">Figure 8
<p>Case <math display="inline"><semantics><mrow><mo>(</mo><mrow><mi mathvariant="italic">op</mi><mo>.</mo><mi mathvariant="italic">norms</mi></mrow><mo>,</mo><mi>γ</mi><mo>)</mo><mo>=</mo><mo>(</mo><mn>0.48</mn><mo>,</mo><mn>5</mn><mo>)</mo></mrow></semantics></math>.</p> Full article ">Figure 9
<p>Case <math display="inline"><semantics><mrow><mo>(</mo><mrow><mi mathvariant="italic">op</mi><mo>.</mo><mi mathvariant="italic">norms</mi></mrow><mo>,</mo><mi>γ</mi><mo>)</mo><mo>=</mo><mo>(</mo><mn>0.48</mn><mo>,</mo><mn>0.5</mn><mo>)</mo></mrow></semantics></math>.</p> Full article ">
13 pages, 1538 KiB  
Article
Kaniadakis’s Information Geometry of Compositional Data
by Giovanni Pistone and Muhammad Shoaib
Entropy 2023, 25(7), 1107; https://doi.org/10.3390/e25071107 - 24 Jul 2023
Cited by 1 | Viewed by 1248
Abstract
We propose to use a particular case of Kaniadakis’ logarithm for the exploratory analysis of compositional data following the Aitchison approach. The affine information geometry derived from Kaniadakis’ logarithm provides a consistent setup for the geometric analysis of compositional data. Moreover, the affine [...] Read more.
We propose to use a particular case of Kaniadakis’ logarithm for the exploratory analysis of compositional data following the Aitchison approach. The affine information geometry derived from Kaniadakis’ logarithm provides a consistent setup for the geometric analysis of compositional data. Moreover, the affine setup suggests a rationale for choosing a specific divergence, which we name the Kaniadakis divergence. Full article
(This article belongs to the Special Issue Twenty Years of Kaniadakis Entropy: Current Trends and Future Perspectives)
►▼ Show Figures

Figure 1

Figure 1
<p>Kaniadakis divergence on compositional data.</p> Full article ">Figure 2
<p>(<b>A</b>) Mixture displacement on compositional data by taking 2008 as a reference and (<b>B</b>) mixture displacement on compositional data by taking mean as reference.</p> Full article ">Figure 3
<p>(<b>A</b>) Exponential displacement on compositional data by taking 2008 as a reference and (<b>B</b>) exponential displacement on compositional data by taking mean as a reference.</p> Full article ">
39 pages, 5033 KiB  
Article
Reasoning and Logical Proofs of the Fundamental Laws: “No Hope” for the Challengers of the Second Law of Thermodynamics
by Milivoje Kostic
Entropy 2023, 25(7), 1106; https://doi.org/10.3390/e25071106 - 24 Jul 2023
Cited by 1 | Viewed by 3028
Abstract
This comprehensive treatise is written for the special occasion of the author’s 70th birthday. It presents his lifelong endeavors and reflections with original reasoning and re-interpretations of the most critical and sometimes misleading issues in thermodynamics—since now, we have the advantage to look [...] Read more.
This comprehensive treatise is written for the special occasion of the author’s 70th birthday. It presents his lifelong endeavors and reflections with original reasoning and re-interpretations of the most critical and sometimes misleading issues in thermodynamics—since now, we have the advantage to look at the historical developments more comprehensively and objectively than the pioneers. Starting from Carnot (grand-father of thermodynamics to become) to Kelvin and Clausius (fathers of thermodynamics), and other followers, the most relevant issues are critically examined and put in historical and contemporary perspective. From the original reasoning of generalized “energy forcing and displacement” to the logical proofs of several fundamental laws, to the ubiquity of thermal motion and heat, and the indestructibility of entropy, including the new concept of “thermal roughness” and “inevitability of dissipative irreversibility,” to dissecting “Carnot true reversible-equivalency” and the critical concept of “thermal-transformer,” limited by the newly generalized “Carnot-Clausius heat-work reversible-equivalency (CCHWRE),” regarding the inter-complementarity of heat and work, and to demonstrating “No Hope” for the “Challengers” of the Second Law of thermodynamics, among others, are offered. It is hoped that the novel contributions presented here will enlighten better comprehension and resolve some of the fundamental issues, as well as promote collaboration and future progress. Full article
(This article belongs to the Special Issue Exploring Fundamentals and Challenges of Heat, Entropy, and the Second Law of Thermodynamics: Honoring Professor Milivoje M. Kostic on the Occasion of His 70th Birthday)
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Figure 1

Figure 1
<p><span class="html-italic">Reasoning concepts of forcing and energy displacement</span>: Energy of a particle (or equivalent field particle) (<b>Left</b>) or bulk body (<b>Right</b>) will not change without forced interactions, i.e., <span class="html-italic">interactive forcing</span> (action–reaction) and <span class="html-italic">energy displacement</span> (energy transfer and conservation). A particle or bulk body in motion will uniformly move (<b>Left</b>) or freely expand (<b>Right</b>) unless interacting and exchanging energy with another particle or body. Elementary particle or ideal body interactions are reversible, but real, collective bulk-structure interactions of bounded collective-particles are irreversible due to dissipation of collective bulk, macro-energy within interacting micro-structures made up of interacting particles or equivalent field-particles.</p> Full article ">Figure 2
<p><span class="html-italic">Thermal roughness</span> and <span class="html-italic">thermal friction</span> are the underlying cause and source of unavoidable irreversibility (2LT) since absolute-0 K temperature is unfeasible (3LT), i.e., perpetual “smooth surface” is impossible. Real surface is always “<span class="html-italic">Dynamic-and-Rough</span>” (chaotic dotted-line) since it is impossible to have a “<span class="html-italic">Still-and-Smooth</span>” surface (plane solid-line) due to perpetual and unavoidable, dynamic “Thermal-Motion (ThM)” of “Thermal Particles (ThP)” always above unachievable, absolute-0 K temperature (3LT)).</p> Full article ">Figure 3
<p><span class="html-italic">Carnot Equality</span> (as named here), <span class="html-italic">Q/Q</span><sub>0</sub><span class="html-italic">= T/T</span><sub>0</sub><span class="html-italic">,</span> or <span class="html-italic">Q/T</span> = <span class="html-italic">constant</span>, for reversible cycles (different from Carnot Theorem), is much more important than what it appears at first. It is probably the most important correlation in Thermodynamics and among the most important equations in natural sciences. Carnot’s ingenious reasoning unlocked the way (for Kelvin, Clausius, and others) for generalization of “thermodynamic reversibility,” definition of absolute thermodynamic temperature and a new thermodynamic property “entropy” (<span class="html-italic">Clausius Equality</span> is generalization of <span class="html-italic">Carnot Equality</span>), as well as the Gibbs free energy, one of the most important thermodynamic functions for characterization of electro-chemical systems and their equilibriums, resulting in formulation of the universal and far-reaching Second Law of Thermodynamics (2LT) (as originally stated by this author in 2008 [<a href="#B16-entropy-25-01106" class="html-bibr">16</a>] and 2011 [<a href="#B17-entropy-25-01106" class="html-bibr">17</a>]).</p> Full article ">Figure 4
<p><span class="html-italic">Carnot</span> (<span class="html-italic">steam power</span>) <span class="html-italic">Cycle</span> (solid lines): heat <span class="html-italic">Q<b><sub>H</sub></b></span> at <span class="html-italic">T<b><sub>H</sub></b></span> is reversibly converted to work <span class="html-italic">W<b><sub>Max</sub></b> =W<b><sub>T</sub></b></span> − <span class="html-italic">W<b><sub>C</sub></b></span> and to <span class="html-italic">Q<b><sub>L</sub></b></span> at <span class="html-italic">T<b><sub>L</sub></b></span>; and <span class="html-italic">Carnot reverse cycle</span> (dashed lines with reversed directions): work <span class="html-italic">W</span><b><sub>C<span class="html-italic">R</span></sub></b> <span class="html-italic">= W</span><sub>|<b><span class="html-italic">C</span></b></sub> − <span class="html-italic">W</span><sub>|<b><span class="html-italic">T</span></b></sub> and heat <span class="html-italic">Q<b><sub>L</sub></b></span> at <span class="html-italic">T<b><sub>L</sub></b></span>, are converted to <span class="html-italic">Q<b><sub>H</sub></b></span> at <span class="html-italic">T<b><sub>H</sub></b></span>. Thermal reservoirs’ high <span class="html-italic">T<b><sub>H</sub></b></span> and low <span class="html-italic">T<b><sub>L</sub></b></span> temperatures (dotted lines). <span class="html-italic">T</span> = turbine, <span class="html-italic">C</span> = compressor, <span class="html-italic">|X =</span> reverse of any <span class="html-italic">X</span>-quantity. All quantities are positive magnitudes [<a href="#B16-entropy-25-01106" class="html-bibr">16</a>,<a href="#B17-entropy-25-01106" class="html-bibr">17</a>].</p> Full article ">Figure 5
<p><span class="html-italic">Converting heat and internal energy to work</span>: In any <span class="html-italic">steady-state process</span> or <span class="html-italic">quasi-steady-cyclic process</span>, entropy input <span class="html-italic">S<sub>H</sub></span>, with heat <span class="html-italic">Q<sub>H</sub></span> (and with mass <span class="html-italic">m<sub>H</sub></span> if any) at <span class="html-italic">T<sub>H</sub></span> &gt; <span class="html-italic">T<sub>L</sub></span>, and if any irreversible generated entropy <span class="html-italic">S<sub>gen</sub></span> within, must be discharged with heat <span class="html-italic">Q<sub>L</sub></span> (and with mass <span class="html-italic">m<sub>L</sub></span> if any), as entropy <span class="html-italic">S<sub>L</sub></span> at <span class="html-italic">T<sub>L</sub>.</span> For ideal reversible process, <math display="inline"><semantics><mrow><msub><mrow><mi>Q</mi></mrow><mrow><mi>L</mi><mo>,</mo><mi>R</mi></mrow></msub></mrow></semantics></math> is “<span class="html-italic">not a loss but necessity</span>”, reducing maximum efficiency below 100%, such as in Carnot cycles (<span class="html-italic">Carnot Equality</span>). For real processes, irreversible work loss, <math display="inline"><semantics><mrow><msub><mrow><mi>W</mi></mrow><mrow><mi>L</mi><mi>O</mi><mi>S</mi><mi>S</mi></mrow></msub></mrow></semantics></math> = <math display="inline"><semantics><mrow><msub><mrow><mi>Q</mi></mrow><mrow><mi>g</mi><mi>e</mi><mi>n</mi></mrow></msub></mrow></semantics></math> = <math display="inline"><semantics><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>L</mi></mrow></msub><msub><mrow><mi>S</mi></mrow><mrow><mi>g</mi><mi>e</mi><mi>n</mi></mrow></msub></mrow></semantics></math>, is due to dissipation of work to heat. For closed-mass and cyclic processes, <span class="html-italic">m<sub>L</sub> = m<sub>H</sub> =</span> 0, and for adiabatic turbine (<span class="html-italic">Q<sub>H|L</sub> =</span> 0), <span class="html-italic">W<sub>OUT</sub> = E<sub>mH</sub> − E<sub>mL</sub></span>.</p> Full article ">Figure 6
<p><span class="html-italic">Reversible equivalency:</span> Formation of “non-equilibrium state” requires “formation work-energy,” ideally all stored as “state work-potential (WP)” to be retrieved back in an ideal reversible process (Figure (<b>Center</b>), “<span class="html-italic">Reversible Equivalency</span>”: ideal formation work equal to work potential). Due to irreversible dissipation of work to heat (work loss), real formation work-in is bigger than stored WP, and retrieved useful work-out is smaller (<b>Left</b>). Formation of non-equilibrium state with less than its WP or obtaining more useful work than WP would require a “miracle Work-GAIN” without due WP source (violation of 2LT), being against natural forcing and existence of equilibrium, thus impossible (<b>Right</b>). Therefore, all reversible processes must be maximally and equally efficient [<a href="#B17-entropy-25-01106" class="html-bibr">17</a>,<a href="#B18-entropy-25-01106" class="html-bibr">18</a>].</p> Full article ">Figure 7
<p><span class="html-italic">Carnot–Clausius Heat–Work Reversible Equivalency (CCHWRE)</span> (as named here), established <span class="html-italic">interchangeability and related equivalency between “Heat-and-Work”</span>, based on the early work of <span class="html-italic">Carnot</span> (1824), that all reversible processes and cycles have equal and maximum efficiency, and among others, <span class="html-italic">Kelvin</span> and <span class="html-italic">Clausius’</span> meticulous work, around 1850s, that finalized the <span class="html-italic">thermodynamic temperature, reversible cycle efficiency, Carnot Equality, Clausius (In)Equality, Entropy,</span> and generalized the <span class="html-italic">Second Law of Thermodynamics (2LT)</span>.</p> Full article ">Figure 8
<p><span class="html-italic">Carnot cycle with ideal gas</span>: Isothermal expansion and compression’s works result in cycle net-work out, while adiabatic expansion and compression’s works cancel out, but they change temperatures required for reversible heat transfer.</p> Full article ">Figure 9
<p>“<span class="html-italic">Perpetual-motion-like</span>” watch with 5–10 years battery life, as if its battery lasts forever. We could mistakenly hypothesize (as if we have proved experimentally), that it works without using energy (PMM1, 1LT violation), or it consumes energy from the surrounding thermal reservoir alone (PMM2, 2LT violation).</p> Full article ">Figure 10
<p>In summary—the Second Law of Thermodynamics (2LT).</p> Full article ">Figure A1
<p>Thermal-Transformer and Temperature-=Oscillator, in adiabatic piston-cylinder system isentropically produces perpetual temperature oscillations or temperature difference. If the depicted piston, from isothermal center-position, is compressing an ideal gas in partition B, thus isentropically increasing the gas temperature, then gas will expand in partition A and isentropically decrease its temperature, without any heat transfer. If displaced piston within the cylinder with ideal, inertial mass and elastic spring system is left free, it will perpetually oscillate, but without any perpetual work generation, similarly to an ideal pendulum oscillations, thus demonstrating a thermal ‘dynamic quasi-equilibrium’ with perpetual temperature oscillations (but not entropy oscillations); or, at any locked, stationary piston position, a self-sustained ‘structural equilibrium’ will establish with perpetual temperature difference, without violating the Second law of thermodynamics (2LT).</p> Full article ">
11 pages, 1135 KiB  
Article
Kinetic Models of Wealth Distribution with Extreme Inequality: Numerical Study of Their Stability against Random Exchanges
by Asim Ghosh, Suchismita Banerjee, Sanchari Goswami, Manipushpak Mitra and Bikas K. Chakrabarti
Entropy 2023, 25(7), 1105; https://doi.org/10.3390/e25071105 - 24 Jul 2023
Cited by 4 | Viewed by 1708
Abstract
In view of some recent reports on global wealth inequality, where a small number (often a handful) of people own more wealth than 50% of the world’s population, we explored if kinetic exchange models of markets could ever capture features where a significant [...] Read more.
In view of some recent reports on global wealth inequality, where a small number (often a handful) of people own more wealth than 50% of the world’s population, we explored if kinetic exchange models of markets could ever capture features where a significant fraction of wealth can concentrate in the hands of a few as the market size N approaches infinity. One existing example of such a kinetic exchange model is the Chakraborti or Yard-Sale model; in the absence of tax redistribution, etc., all wealth ultimately condenses into the hands of a single individual (for any value of N), and the market dynamics stop. With tax redistribution, etc., steady-state dynamics are shown to have remarkable applicability in many cases in our extremely unequal world. We show that another kinetic exchange model (called the Banerjee model) has intriguing intrinsic dynamics, where only ten rich traders or agents possess about 99.98% of the total wealth in the steady state (without any tax, etc., like external manipulation) for any large N value. We will discuss the statistical features of this model using Monte Carlo simulations. We will also demonstrate that if each trader has a non-zero probability f of engaging in random exchanges, then these condensations of wealth (e.g., 100% in the hand of one agent in the Chakraborti model, or about 99.98% in the hands of ten agents in the Banerjee model) disappear in the large N limit. Moreover, due to the built-in possibility of random exchange dynamics in the earlier proposed Goswami–Sen model, where the exchange probability decreases with the inverse power of the wealth difference between trading pairs, one does not see any wealth condensation phenomena. In this paper, we explore these aspects of statistics of these intriguing models. Full article
(This article belongs to the Special Issue Statistical Physics and Its Applications in Economics and Social Sciences)
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Figure 1

Figure 1
<p>Distributions of the fraction of total wealth (<math display="inline"><semantics><mrow><mi>M</mi><mo>=</mo><mi>N</mi></mrow></semantics></math>) ending up in the hands of the richest three traders. The error estimation is based on 10 runs. The typical errors in the distribution grow with <span class="html-italic">N</span> near the most probable value of the wealth fraction and are indicated for <span class="html-italic">N</span> = 800 for all three traders. Far away from the most probable values, the errors are less than the data point symbol sizes.</p> Full article ">Figure 2
<p>Distribution of the total wealth fraction possessed by the ten richest (at any time in the steady state and for different <span class="html-italic">N</span> values). The inset shows that the average of the total wealth fraction of the ten richest (for any time and any value of <span class="html-italic">N</span>) in the steady state is very close to 0.9998. Although the wealth share fractions of the richest ten traders have considerable fluctuations (see <a href="#entropy-25-01105-f001" class="html-fig">Figure 1</a>), their wealth fraction totals hardly have any fluctuations (much less than the symbol size in the inset). The error estimation is based on 10 runs. The typical errors in the distribution of total wealth of the ten richest are more than the data point symbol sizes near the most probable values, where indicated.</p> Full article ">Figure 3
<p>Wealth distribution <math display="inline"><semantics><mrow><mi>P</mi><mo>(</mo><mi>m</mi><mo>)</mo></mrow></semantics></math> among all agents against the wealth <span class="html-italic">m</span> in the B model for different probabilities <span class="html-italic">f</span> of DY random exchanges. Note that the fluctuations appear to grow more for the lower values of the distribution of wealth due to the log scale used in the y-axis.</p> Full article ">Figure 4
<p>To obtain the limiting values (for large <span class="html-italic">N</span>) of the average fraction of total wealth (<math display="inline"><semantics><mrow><mi>M</mi><mo>=</mo><mi>N</mi></mrow></semantics></math>) possessed by the ten richest traders in the steady state, we plot the fraction against <math display="inline"><semantics><mrow><mn>1</mn><mo>/</mo><msup><mi>N</mi><mn>2</mn></msup></mrow></semantics></math> (as with DY-type trades, each N trader interacts with the <span class="html-italic">N</span> − 1 other trader. The extrapolated values all seem to approach zero for any non-zero value of <span class="html-italic">f</span> (but there remains a constant 0.9998 for <span class="html-italic">f</span> = 0, as in the pure B model). The error estimation is based on 10 runs. Typical sizes of error bars are indicated.</p> Full article ">Figure 5
<p>(<b>A</b>) The distribution of residence times (in units of <span class="html-italic">N</span>) of the 10 fortunate traders and (in the inset) the variation of the most probable and average values of the residence times. (<b>B</b>) The distribution of the return time to fortune (becoming one of the 10 richest, starting from the 20th rank) and (in the inset) the variation of the most probable and average values of the return times (in units of <span class="html-italic">N</span>). The error estimation is based on 10 runs. The typical errors in the distribution of both the residence and return times grow with <span class="html-italic">N</span> near the most probable values of the respective quantities, and are indicated for <span class="html-italic">N</span> = 400 here when they are bigger than the symbol sizes.</p> Full article ">Figure 6
<p>Wealth distribution <math display="inline"><semantics><mrow><mi>P</mi><mo>(</mo><mi>m</mi><mo>)</mo></mrow></semantics></math> among all the agents against the wealth <span class="html-italic">m</span> in the C model for different probabilities <span class="html-italic">f</span> of DY random exchanges. Note that the fluctuations appear to grow more for the lower values of the distribution due to the log scale used in the y-axis.</p> Full article ">Figure 7
<p>The limiting values (for large <span class="html-italic">N</span>) of the average fraction of total wealth (<math display="inline"><semantics><mrow><mi>M</mi><mo>=</mo><mi>N</mi></mrow></semantics></math>) possessed by the ten richest traders in the steady state of the C model with the <span class="html-italic">f</span> fraction of DY-like trades. For <math display="inline"><semantics><mrow><mi>f</mi><mo>=</mo><mn>0</mn></mrow></semantics></math>, the money goes to one agent and the other nine agents contribute nothing. When we plot the fraction against <math display="inline"><semantics><mrow><mn>1</mn><mo>/</mo><msup><mi>N</mi><mn>2</mn></msup></mrow></semantics></math> (as with DY-type trades, each N trader interacts with <math display="inline"><semantics><mrow><mo>(</mo><mi>N</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></semantics></math> other traders), the extrapolated values all seem to approach zero for any non-zero value of <span class="html-italic">f</span>. The error estimation is based on 10 runs. Typical sizes of error bars are indicated.</p> Full article ">Figure 8
<p>Wealth distribution <math display="inline"><semantics><mrow><mi>P</mi><mo>(</mo><mi>m</mi><mo>)</mo></mrow></semantics></math> among all the agents against the wealth <span class="html-italic">m</span> in the GS model for different values of <math display="inline"><semantics><mi>α</mi></semantics></math>. Note that the fluctuations appear to grow more for the lower values of the distribution due to the log scale used in the y-axis.</p> Full article ">Figure 9
<p>Plot of the fraction of total wealth (<math display="inline"><semantics><mrow><mi>M</mi><mo>=</mo><mi>N</mi></mrow></semantics></math>) against <math display="inline"><semantics><mrow><mn>1</mn><mo>/</mo><msup><mi>N</mi><mn>2</mn></msup></mrow></semantics></math> for different values of <math display="inline"><semantics><mi>α</mi></semantics></math> in the GS model. The extrapolated (with <span class="html-italic">N</span>) values of the fraction all seem to approach zero for any non-zero value of <math display="inline"><semantics><mi>α</mi></semantics></math>. The error estimation is based on 10 runs. Typical sizes of error bars are indicated.</p> Full article ">Figure 10
<p>The DY fraction <span class="html-italic">f</span> dependence of the bare residence time (in units of interactions or exchanges) at different <span class="html-italic">N</span> values for the B-model (<b>A</b>) and for the C-model (<b>B</b>) are shown. Their power law fits with <span class="html-italic">f</span> for <span class="html-italic">N</span> = 400 are shown (for other <span class="html-italic">N</span> vales, the respective pre-factors change linearly with <span class="html-italic">N</span>). The insets show the <span class="html-italic">f</span> dependence of the residence times <math display="inline"><semantics><mi>τ</mi></semantics></math> (in units of <span class="html-italic">N</span>). Note that the limiting values of <math display="inline"><semantics><mi>τ</mi></semantics></math> at <span class="html-italic">f</span> = 0 are about 66 for the B-model, while they go to infinity for the C-model.</p> Full article ">
23 pages, 35184 KiB  
Article
Multivariate Modeling for Spatio-Temporal Radon Flux Predictions
by Sandra De Iaco, Claudia Cappello, Antonella Congedi and Monica Palma
Entropy 2023, 25(7), 1104; https://doi.org/10.3390/e25071104 - 24 Jul 2023
Cited by 1 | Viewed by 1584
Abstract
Nowadays, various fields in environmental sciences require the availability of appropriate techniques to exploit the information given by multivariate spatial or spatio-temporal observations. In particular, radon flux data which are of high interest to monitor greenhouse gas emissions and to assess human exposure [...] Read more.
Nowadays, various fields in environmental sciences require the availability of appropriate techniques to exploit the information given by multivariate spatial or spatio-temporal observations. In particular, radon flux data which are of high interest to monitor greenhouse gas emissions and to assess human exposure to indoor radon are determined by the deposit of uranium and radio (precursor elements). Furthermore, they are also affected by various atmospheric variables, such as humidity, temperature, precipitation and evapotranspiration. To this aim, a significant role can be recognized to the tools of multivariate geostatistics which supports the modeling and prediction of variables under study. In this paper, the spatio-temporal distribution of radon flux densities over the Veneto Region (Italy) and its estimation at unsampled points in space and time are discussed. In particular, the spatio-temporal linear coregionalization model is identified on the basis of the joint diagonalization of the empirical covariance matrices evaluated at different spatio-temporal lags and is used to produce predicted radon flux maps for different months. Probability maps, that the radon flux density in the upcoming months is greater than three historical statistics, are then built. This might be of interest especially in summer months when the risk of radon exhalation is higher. Moreover, a comparison with respect to alternative models in the univariate and multivariate context is provided. Full article
(This article belongs to the Special Issue Spatiotemporal Prediction and Simulation Methods at the Nexus of Statistical Physics, Spatial Statistics and Machine Learning)
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<p>(<b>Left panel</b>): map of Italian regions (study area in orange). (<b>Middle panel</b>): Veneto Provinces. (<b>Right panel</b>): location map of meteorological and radon sample points over the study area.</p> Full article ">Figure 2
<p>Colour maps of Rn flux (in KBqm<math display="inline"><semantics><msup><mrow/><mrow><mo>−</mo><mn>2</mn></mrow></msup></semantics></math> s<math display="inline"><semantics><msup><mrow/><mrow><mo>−</mo><mn>1</mn></mrow></msup></semantics></math>), <math display="inline"><semantics><msub><mi>T</mi><mi>M</mi></msub></semantics></math> (in °C), <math display="inline"><semantics><msub><mi>H</mi><mi>m</mi></msub></semantics></math> (in %) and <math display="inline"><semantics><mrow><mi>E</mi><msub><mi>T</mi><mn>0</mn></msub></mrow></semantics></math> (in mm) monthly averages calculated for (<b>a</b>) January, (<b>b</b>) April, (<b>c</b>) July and (<b>d</b>) October.</p> Full article ">Figure 3
<p>Box and whisker plots showing (<b>a</b>) Rn flux (in KBqm<math display="inline"><semantics><msup><mrow/><mrow><mo>−</mo><mn>2</mn></mrow></msup></semantics></math> s<math display="inline"><semantics><msup><mrow/><mrow><mo>−</mo><mn>1</mn></mrow></msup></semantics></math>), (<b>b</b>) <math display="inline"><semantics><msub><mi>T</mi><mi>M</mi></msub></semantics></math> (in °C), (<b>c</b>) <math display="inline"><semantics><msub><mi>H</mi><mi>m</mi></msub></semantics></math> (in %) and (<b>d</b>) <math display="inline"><semantics><mrow><mi>E</mi><msub><mi>T</mi><mn>0</mn></msub></mrow></semantics></math> (in mm) and their corresponding residual values (<b>e</b>–<b>h</b>), grouped by month. The symbol ∘ indicates values which lie more than 1.5 times the interquartile range from the first and third quartile.</p> Full article ">Figure 4
<p>Sample space–time direct covariance surfaces for the residuals of (<b>a</b>) Rn flux, (<b>e</b>) <math display="inline"><semantics><msub><mi>T</mi><mi>M</mi></msub></semantics></math>, (<b>h</b>) <math display="inline"><semantics><msub><mi>H</mi><mi>m</mi></msub></semantics></math> and (<b>j</b>) <math display="inline"><semantics><mrow><mi>E</mi><msub><mi>T</mi><mn>0</mn></msub></mrow></semantics></math> together with the cross-covariance surfaces of (<b>b</b>) Rn flux vs. <math display="inline"><semantics><msub><mi>T</mi><mi>M</mi></msub></semantics></math>, (<b>c</b>) Rn flux vs. <math display="inline"><semantics><msub><mi>H</mi><mi>m</mi></msub></semantics></math>, (<b>d</b>) Rn flux vs. <math display="inline"><semantics><mrow><mi>E</mi><msub><mi>T</mi><mn>0</mn></msub></mrow></semantics></math>, (<b>f</b>) <math display="inline"><semantics><msub><mi>T</mi><mi>M</mi></msub></semantics></math> vs. <math display="inline"><semantics><msub><mi>H</mi><mi>m</mi></msub></semantics></math>, (<b>g</b>) <math display="inline"><semantics><msub><mi>T</mi><mi>M</mi></msub></semantics></math> vs. <math display="inline"><semantics><mrow><mi>E</mi><msub><mi>T</mi><mn>0</mn></msub></mrow></semantics></math>, (<b>h</b>) <math display="inline"><semantics><msub><mi>H</mi><mi>m</mi></msub></semantics></math> and (<b>i</b>) <math display="inline"><semantics><msub><mi>H</mi><mi>m</mi></msub></semantics></math> vs. <math display="inline"><semantics><mrow><mi>E</mi><msub><mi>T</mi><mn>0</mn></msub></mrow></semantics></math>, computed on the basis of the estimator in (<a href="#FD4-entropy-25-01104" class="html-disp-formula">4</a>).</p> Full article ">Figure 5
<p>Sample spatio-temporal covariance surfaces (on the left) and fitted models (on the right), for the basic components at (<b>a</b>) very small, (<b>b</b>) small, (<b>c</b>) medium and (<b>d</b>) large scale of spatio-temporal variability.</p> Full article ">Figure 6
<p>Box and whisker plots of sample non-separability ratios classified by spatial (on the left) and temporal (on the right) lags, computed for the basic components at (<b>a</b>) very small (<b>b</b>) small, (<b>c</b>) medium and (<b>d</b>) large scale of spatio-temporal variability. The symbol * indicates values which lie more than 1.5 times the interquartile range from the first and third quartile.</p> Full article ">Figure 7
<p>Prediction maps of Rn flux (in KBqm<math display="inline"><semantics><msup><mrow/><mrow><mo>−</mo><mn>2</mn></mrow></msup></semantics></math> s<math display="inline"><semantics><msup><mrow/><mrow><mo>−</mo><mn>1</mn></mrow></msup></semantics></math>) monthly averages for (<b>a</b>) May, (<b>b</b>) August and (<b>c</b>) November 2022.</p> Full article ">Figure 8
<p>Risk maps of the probability that Rn flux predicted in August 2022 exceeds (<b>a</b>) the 25th percentile (<math display="inline"><semantics><mrow><msub><mi>z</mi><mn>1</mn></msub><mo>=</mo><mn>19.326</mn></mrow></semantics></math> KBqm<math display="inline"><semantics><msup><mrow/><mn>2</mn></msup></semantics></math>s<math display="inline"><semantics><msup><mrow/><mrow><mo>−</mo><mn>1</mn></mrow></msup></semantics></math>), (<b>b</b>) the mean (<math display="inline"><semantics><mrow><msub><mi>z</mi><mn>2</mn></msub><mo>=</mo><mn>22.75</mn></mrow></semantics></math> KBqm<math display="inline"><semantics><msup><mrow/><mn>2</mn></msup></semantics></math>s<math display="inline"><semantics><msup><mrow/><mrow><mo>−</mo><mn>1</mn></mrow></msup></semantics></math>), (<b>c</b>) the median value (<math display="inline"><semantics><mrow><msub><mi>z</mi><mn>3</mn></msub><mo>=</mo><mn>23.994</mn></mrow></semantics></math> KBqm<math display="inline"><semantics><msup><mrow/><mn>2</mn></msup></semantics></math>s<math display="inline"><semantics><msup><mrow/><mrow><mo>−</mo><mn>1</mn></mrow></msup></semantics></math>) of the corresponding historical measurements.</p> Full article ">
18 pages, 961 KiB  
Article
Kernel-Free Quadratic Surface Regression for Multi-Class Classification
by Changlin Wang, Zhixia Yang, Junyou Ye and Xue Yang
Entropy 2023, 25(7), 1103; https://doi.org/10.3390/e25071103 - 24 Jul 2023
Viewed by 1502
Abstract
For multi-class classification problems, a new kernel-free nonlinear classifier is presented, called the hard quadratic surface least squares regression (HQSLSR). It combines the benefits of the least squares loss function and quadratic kernel-free trick. The optimization problem of HQSLSR is convex and unconstrained, [...] Read more.
For multi-class classification problems, a new kernel-free nonlinear classifier is presented, called the hard quadratic surface least squares regression (HQSLSR). It combines the benefits of the least squares loss function and quadratic kernel-free trick. The optimization problem of HQSLSR is convex and unconstrained, making it easy to solve. Further, to improve the generalization ability of HQSLSR, a softened version (SQSLSR) is proposed by introducing an ε-dragging technique, which can enlarge the between-class distance. The optimization problem of SQSLSR is solved by designing an alteration iteration algorithm. The convergence, interpretability and computational complexity of our methods are addressed in a theoretical analysis. The visualization results on five artificial datasets demonstrate that the obtained regression function in each category has geometric diversity and the advantage of the ε-dragging technique. Furthermore, experimental results on benchmark datasets show that our methods perform comparably to some state-of-the-art classifiers. Full article
(This article belongs to the Section Information Theory, Probability and Statistics)
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Figure 1

Figure 1
<p>Classification results of the artificial dataset I.</p> Full article ">Figure 2
<p>Classification results of the artificial dataset II.</p> Full article ">Figure 3
<p>Classification results of the artificial dataset III.</p> Full article ">Figure 4
<p>Classification results of the artificial dataset IV.</p> Full article ">Figure 5
<p>Training samples and the differences caused by <math display="inline"><semantics><mi>ε</mi></semantics></math>-dragging technique: (<b>a</b>) sixty training samples in three classes; (<b>b</b>) the first component of the difference <math display="inline"><semantics><mi mathvariant="bold-italic">D</mi></semantics></math>; (<b>c</b>) the second component of the difference <math display="inline"><semantics><mi mathvariant="bold-italic">D</mi></semantics></math>; and (<b>d</b>) the third component of the difference <math display="inline"><semantics><mi mathvariant="bold-italic">D</mi></semantics></math>.</p> Full article ">Figure 6
<p>Convergence of SQSLSR.</p> Full article ">Figure 7
<p>Friedman test and Nemenyi post hoc test of accuracy.</p> Full article ">Figure 8
<p>Friedman test and the Nemenyi post hoc test of computation time.</p> Full article ">
19 pages, 578 KiB  
Article
Weighted Sum Secrecy Rate Maximization for Joint ITS- and IRS-Empowered System
by Shaochuan Yang, Kaizhi Huang, Hehao Niu, Yi Wang and Zheng Chu
Entropy 2023, 25(7), 1102; https://doi.org/10.3390/e25071102 - 24 Jul 2023
Viewed by 1411
Abstract
In this work, we investigate a novel intelligent surface-assisted multiuser multiple-input single-output multiple-eavesdropper (MU-MISOME) secure communication network where an intelligent reflecting surface (IRS) is deployed to enhance the secrecy performance and an intelligent transmission surface (ITS)-based transmitter is utilized to perform energy-efficient beamforming. [...] Read more.
In this work, we investigate a novel intelligent surface-assisted multiuser multiple-input single-output multiple-eavesdropper (MU-MISOME) secure communication network where an intelligent reflecting surface (IRS) is deployed to enhance the secrecy performance and an intelligent transmission surface (ITS)-based transmitter is utilized to perform energy-efficient beamforming. A weighted sum secrecy rate (WSSR) maximization problem is developed by jointly optimizing transmit power allocation, ITS beamforming, and IRS phase shift. To solve this problem, we transform the objective function into an approximated concave form by using the successive convex approximation (SCA) technique. Then, we propose an efficient alternating optimization (AO) algorithm to solve the reformulated problem in an iterative way, where Karush–Kuhn–Tucker (KKT) conditions, the alternating direction method of the multiplier (ADMM), and majorization–minimization (MM) methods are adopted to derive the closed-form solution for each subproblem. Finally, simulation results are given to verify the convergence and secrecy performance of the proposed schemes. Full article
(This article belongs to the Special Issue Quantum and Classical Physical Cryptography)
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Figure 1
<p>System model.</p> Full article ">Figure 2
<p>Simulation setup.</p> Full article ">Figure 3
<p>The convergence performance of ADMM and MM algorithms.</p> Full article ">Figure 4
<p>The convergence behavior of OA algorithms.</p> Full article ">Figure 5
<p>The WSSR versus the transmit power.</p> Full article ">Figure 6
<p>The WSSR versus the reflective element number.</p> Full article ">Figure 7
<p>TheWSSR versus L.</p> Full article ">Figure 8
<p>The WSSR versus pathloss exponent of the IRS–Bobs link.</p> Full article ">Figure 9
<p>The WSSR versus pathloss exponent of the IRS–Eves link.</p> Full article ">
19 pages, 4201 KiB  
Article
Critic Learning-Based Safe Optimal Control for Nonlinear Systems with Asymmetric Input Constraints and Unmatched Disturbances
by Chunbin Qin, Kaijun Jiang, Jishi Zhang and Tianzeng Zhu
Entropy 2023, 25(7), 1101; https://doi.org/10.3390/e25071101 - 24 Jul 2023
Viewed by 1633
Abstract
In this paper, the safe optimal control method for continuous-time (CT) nonlinear safety-critical systems with asymmetric input constraints and unmatched disturbances based on the adaptive dynamic programming (ADP) is investigated. Initially, a new non-quadratic form function is implemented to effectively handle the asymmetric [...] Read more.
In this paper, the safe optimal control method for continuous-time (CT) nonlinear safety-critical systems with asymmetric input constraints and unmatched disturbances based on the adaptive dynamic programming (ADP) is investigated. Initially, a new non-quadratic form function is implemented to effectively handle the asymmetric input constraints. Subsequently, the safe optimal control problem is transformed into a two-player zero-sum game (ZSG) problem to suppress the influence of unmatched disturbances, and a new Hamilton–Jacobi–Isaacs (HJI) equation is introduced by integrating the control barrier function (CBF) with the cost function to penalize unsafe behavior. Moreover, a damping factor is embedded in the CBF to balance safety and optimality. To obtain a safe optimal controller, only one critic neural network (CNN) is utilized to tackle the complex HJI equation, leading to a decreased computational load in contrast to the utilization of the conventional actor–critic network. Then, the system state and the parameters of the CNN are uniformly ultimately bounded (UUB) through the application of the Lyapunov stability method. Lastly, two examples are presented to confirm the efficacy of the presented approach. Full article
(This article belongs to the Section Complexity)
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<p>Convergence of the CNN weights.</p> Full article ">Figure 2
<p>Convergence of system states <math display="inline"><semantics><msub><mi>x</mi><mn>1</mn></msub></semantics></math>, <math display="inline"><semantics><msub><mi>x</mi><mn>2</mn></msub></semantics></math>, and <math display="inline"><semantics><msub><mi>x</mi><mn>3</mn></msub></semantics></math>.</p> Full article ">Figure 3
<p>The comparison between the safe and unsafe states.</p> Full article ">Figure 4
<p>Control input in the system.</p> Full article ">Figure 5
<p>Disturbance input in the system.</p> Full article ">Figure 6
<p>The cost function of the system.</p> Full article ">Figure 7
<p>Control input without asymmetric input constraints.</p> Full article ">Figure 8
<p>Convergence of the CNN weights.</p> Full article ">Figure 9
<p>Convergence of system states <math display="inline"><semantics><msub><mi>x</mi><mn>1</mn></msub></semantics></math> and <math display="inline"><semantics><msub><mi>x</mi><mn>2</mn></msub></semantics></math>.</p> Full article ">Figure 10
<p>The comparison between the safe and unsafe states.</p> Full article ">Figure 11
<p>Control input in the system.</p> Full article ">Figure 12
<p>Disturbance input in the system.</p> Full article ">Figure 13
<p>The cost function of the system.</p> Full article ">Figure 14
<p>Control input without asymmetric input constraints.</p> Full article ">
17 pages, 1044 KiB  
Article
A Novel Trajectory Feature-Boosting Network for Trajectory Prediction
by Qingjian Ni, Wenqiang Peng, Yuntian Zhu and Ruotian Ye
Entropy 2023, 25(7), 1100; https://doi.org/10.3390/e25071100 - 23 Jul 2023
Viewed by 2183
Abstract
Trajectory prediction is an essential task in many applications, including autonomous driving, robotics, and surveillance systems. In this paper, we propose a novel trajectory prediction network, called TFBNet (trajectory feature-boosting network), that utilizes trajectory feature boosting to enhance prediction accuracy. TFBNet operates by [...] Read more.
Trajectory prediction is an essential task in many applications, including autonomous driving, robotics, and surveillance systems. In this paper, we propose a novel trajectory prediction network, called TFBNet (trajectory feature-boosting network), that utilizes trajectory feature boosting to enhance prediction accuracy. TFBNet operates by mapping the original trajectory data to a high-dimensional space, analyzing the change rules of the trajectory in this space, and finally aggregating the trajectory goals to generate the final trajectory. Our approach presents a new perspective on trajectory prediction. We evaluate TFBNet on five real-world datasets and compare it to state-of-the-art methods. Our results demonstrate that TFBNet achieves significant improvements in the ADE (average displacement error) and FDE (final displacement error) indicators, with increases of 46% and 52%, respectively. These results validate the effectiveness of our proposed approach and its potential to improve the performance of trajectory prediction models in various applications. Full article
(This article belongs to the Special Issue Spatiotemporal Prediction and Simulation Methods at the Nexus of Statistical Physics, Spatial Statistics and Machine Learning)
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<p>Model architecture of the proposed TFBNet.</p> Full article ">Figure 2
<p>The architecture of DAGConv.</p> Full article ">Figure 3
<p>TFBNet’s prediction results on ETH.</p> Full article ">Figure 4
<p>TFBNet’s prediction results on HOTEL.</p> Full article ">Figure 5
<p>TFBNet’s prediction results on UNIV.</p> Full article ">Figure 6
<p>TFBNet’s prediction results on ZARA1.</p> Full article ">Figure 7
<p>TFBNet’s prediction results on ZARA2.</p> Full article ">
19 pages, 3411 KiB  
Article
Double-Layer Detection Model of Malicious PDF Documents Based on Entropy Method with Multiple Features
by Enzhou Song, Tao Hu, Peng Yi and Wenbo Wang
Entropy 2023, 25(7), 1099; https://doi.org/10.3390/e25071099 - 23 Jul 2023
Cited by 1 | Viewed by 1287
Abstract
Traditional PDF document detection technology usually builds a rule or feature library for specific vulnerabilities and therefore is only fit for single detection targets and lacks anti-detection ability. To address these shortcomings, we build a double-layer detection model for malicious PDF documents based [...] Read more.
Traditional PDF document detection technology usually builds a rule or feature library for specific vulnerabilities and therefore is only fit for single detection targets and lacks anti-detection ability. To address these shortcomings, we build a double-layer detection model for malicious PDF documents based on an entropy method with multiple features. First, we address the single detection target problem with the fusion of 222 multiple features, including 130 basic features (such as objects, structure, content stream, metadata, etc.) and 82 dangerous features (such as suspicious and encoding function, etc.), which can effectively resist obfuscation and encryption. Second, we generate the best set of features (a total of 153) by creatively applying an entropy method based on RReliefF and MIC (EMBORAM) to PDF samples with 37 typical document vulnerabilities, which can effectively resist anti-detection methods, such as filling data and imitation attacks. Finally, we build a double-layer processing framework to detect samples efficiently through the AdaBoost-optimized random forest algorithm and the robustness-optimized support vector machine algorithm. Compared to the traditional static detection method, this model performs better for various evaluation criteria. The average time of document detection is 1.3 ms, while the accuracy rate reaches 95.9%. Full article
(This article belongs to the Section Information Theory, Probability and Statistics)
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<p>The basic structure of PDFs.</p> Full article ">Figure 2
<p>Model framework.</p> Full article ">Figure 3
<p>(<b>a</b>) Frequency of characters in malicious PDFs; (<b>b</b>) number of characters in each type.</p> Full article ">Figure 4
<p>Frame of EMBORAM module.</p> Full article ">Figure 5
<p>AdaBoost-optimized random forest classification model.</p> Full article ">Figure 6
<p>Robustness-optimized support vector machine classification model.</p> Full article ">Figure 7
<p>(<b>a</b>) Effect of the threshold K on accuracy; (<b>b</b>) effect of threshold K on the number of feature sets.</p> Full article ">Figure 8
<p>(<b>a</b>) Effect of threshold B on accuracy; (<b>b</b>) effect of threshold B on the number of samples in the boundary set.</p> Full article ">Figure 9
<p>(<b>a</b>) Comparison of detection results in malicious samples; (<b>b</b>) comparison of detection results in malicious-high samples.</p> Full article ">Figure 10
<p>(<b>a</b>) Comparison of TP values; (<b>b</b>) comparison of FP values; (<b>c</b>) comparison of FN values.</p> Full article ">
23 pages, 5510 KiB  
Article
Comparative Exergy and Environmental Assessment of the Residual Biomass Gasification Routes for Hydrogen and Ammonia Production
by Gabriel Gomes Vargas, Daniel Alexander Flórez-Orrego and Silvio de Oliveira Junior
Entropy 2023, 25(7), 1098; https://doi.org/10.3390/e25071098 - 22 Jul 2023
Cited by 5 | Viewed by 2633
Abstract
The need to reduce the dependency of chemicals on fossil fuels has recently motivated the adoption of renewable energies in those sectors. In addition, due to a growing population, the treatment and disposition of residual biomass from agricultural processes, such as sugar cane [...] Read more.
The need to reduce the dependency of chemicals on fossil fuels has recently motivated the adoption of renewable energies in those sectors. In addition, due to a growing population, the treatment and disposition of residual biomass from agricultural processes, such as sugar cane and orange bagasse, or even from human waste, such as sewage sludge, will be a challenge for the next generation. These residual biomasses can be an attractive alternative for the production of environmentally friendly fuels and make the economy more circular and efficient. However, these raw materials have been hitherto widely used as fuel for boilers or disposed of in sanitary landfills, losing their capacity to generate other by-products in addition to contributing to the emissions of gases that promote global warming. For this reason, this work analyzes and optimizes the biomass-based routes of biochemical production (namely, hydrogen and ammonia) using the gasification of residual biomasses. Moreover, the capture of biogenic CO2 aims to reduce the environmental burden, leading to negative emissions in the overall energy system. In this context, the chemical plants were designed, modeled, and simulated using Aspen plus™ software. The energy integration and optimization were performed using the OSMOSE Lua Platform. The exergy destruction, exergy efficiency, and general balance of the CO2 emissions were evaluated. As a result, the irreversibility generated by the gasification unit has a relevant influence on the exergy efficiency of the entire plant. On the other hand, an overall negative emission balance of −5.95 kgCO2/kgH2 in the hydrogen production route and −1.615 kgCO2/kgNH3 in the ammonia production route can be achieved, thus removing from the atmosphere 0.901 tCO2/tbiomass and 1.096 tCO2/tbiomass, respectively. Full article
(This article belongs to the Special Issue Thermodynamic Optimization of Industrial Energy Systems)
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<p>Flowsheet of the biomass pre-treatment and gasification unit. See <a href="#app1-entropy-25-01098" class="html-app">Supplementary Material</a> for numbered stream properties. Flow properties (1–8) can be found in the <a href="#app1-entropy-25-01098" class="html-app">Supplementary Material, Tables S1–S3</a>.</p> Full article ">Figure 2
<p>Flowsheet of the syngas conditioning unit for hydrogen production. Flow properties (1–4) can be found in the <a href="#app1-entropy-25-01098" class="html-app">Supplementary Material, Tables S4–S6</a>.</p> Full article ">Figure 3
<p>Flowsheet of the syngas treatment unit for ammonia production. Flow properties (1–8) can be found in the <a href="#app1-entropy-25-01098" class="html-app">Supplementary Material, Tables S7–S9</a>.</p> Full article ">Figure 4
<p>Flowsheet of the syngas purification unit for the ammonia production route. Flow properties (1–4) can be found in the <a href="#app1-entropy-25-01098" class="html-app">Supplementary Material, Tables S10–S12</a>.</p> Full article ">Figure 5
<p>Flowsheet of the syngas purification unit for hydrogen production. Flow properties (1–3) can be found in the <a href="#app1-entropy-25-01098" class="html-app">Supplementary Material, Tables S13–S15</a>.</p> Full article ">Figure 6
<p>Flowsheet of the pressure swing adsorption for hydrogen recovery and purge gas combustion. Flow properties (1–5) can be found in the <a href="#app1-entropy-25-01098" class="html-app">Supplementary Material, Tables S16–S18</a>.</p> Full article ">Figure 7
<p>Flowsheet of the ammonia synthesis loop. Flow properties (1–9) can be found in the <a href="#app1-entropy-25-01098" class="html-app">Supplementary Material, Tables S19–S21</a>.</p> Full article ">Figure 8
<p>Flowcharts of the hydrogen production route from residual biomass.</p> Full article ">Figure 9
<p>Flowcharts of the ammonia production route from residual biomass.</p> Full article ">Figure 10
<p>Comparison of the biomass gasification modeling results between the simulation in this work and the literature data reported by Marcantonio et al. [<a href="#B41-entropy-25-01098" class="html-bibr">41</a>] for walnut husk.</p> Full article ">Figure 11
<p>Cold gas efficiency (CGE) and carbon conversion efficiency (CCE) in the gasification process of biomass residues.</p> Full article ">Figure 12
<p>Comparison of the exergy efficiencies of (<b>a</b>) hydrogen and (<b>b</b>) ammonia production routes using different types of residual biomass.</p> Full article ">Figure 13
<p>Cold and hot composite curves for different waste biomass conversion processes exhibiting no need for external heating requirements, but showing the need for further cooling requirements: (<b>a</b>) sugarcane bagasse to hydrogen; (<b>b</b>) sugarcane bagasse to ammonia; (<b>c</b>) sewage sludge to hydrogen; (<b>d</b>) sewage sludge to ammonia; (<b>e</b>) orange bagasse to hydrogen; (<b>f</b>) orange bagasse to ammonia.</p> Full article ">Figure 14
<p>Grand composite curves for different waste biomass conversion processes exhibiting no need for external heating requirements, but showing the need for further cooling requirements: (<b>a</b>) sugarcane bagasse to hydrogen; (<b>b</b>) sugarcane bagasse to ammonia; (<b>c</b>) sewage sludge to hydrogen; (<b>d</b>) sewage sludge to ammonia; (<b>e</b>) orange bagasse to hydrogen; (<b>f</b>) orange bagasse to ammonia.</p> Full article ">Figure 15
<p>General and detailed emissions (biogenic and fossil, emitted directly, indirectly, and avoided) for the conversion process of (<b>a</b>) hydrogen and (<b>b</b>) ammonia for different types of selected biomass.</p> Full article ">
17 pages, 4931 KiB  
Article
Chinese Few-Shot Named Entity Recognition and Knowledge Graph Construction in Managed Pressure Drilling Domain
by Siqing Wei, Yanchun Liang, Xiaoran Li, Xiaohui Weng, Jiasheng Fu and Xiaosong Han
Entropy 2023, 25(7), 1097; https://doi.org/10.3390/e25071097 - 22 Jul 2023
Cited by 1 | Viewed by 1785
Abstract
Managed pressure drilling (MPD) is the most effective means to ensure drilling safety, and MPD is able to avoid further deterioration of complex working conditions through precise control of the wellhead back pressure. The key to the success of MPD is the well [...] Read more.
Managed pressure drilling (MPD) is the most effective means to ensure drilling safety, and MPD is able to avoid further deterioration of complex working conditions through precise control of the wellhead back pressure. The key to the success of MPD is the well control strategy, which currently relies heavily on manual experience, hindering the automation and intelligence process of well control. In response to this issue, an MPD knowledge graph is constructed in this paper that extracts knowledge from published papers and drilling reports to guide well control. In order to improve the performance of entity extraction in the knowledge graph, a few-shot Chinese entity recognition model CEntLM-KL is extended from the EntLM model, in which the KL entropy is built to improve the accuracy of entity recognition. Through experiments on benchmark datasets, it has been shown that the proposed model has a significant improvement compared to the state-of-the-art methods. On the few-shot drilling datasets, the F-1 score of entity recognition reaches 33%. Finally, the knowledge graph is stored in Neo4J and applied for knowledge inference. Full article
(This article belongs to the Special Issue Entropy in Machine Learning Applications)
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<p>Pipeline of our work.</p> Full article ">Figure 2
<p>Example of NER’s template-based prompt method.</p> Full article ">Figure 3
<p>Comparison of the flow of Chinese and English EntLM prediction labels.</p> Full article ">Figure 4
<p>Overall architecture of CEntLM-KL.</p> Full article ">Figure 5
<p>Various types of relationships between entities.</p> Full article ">Figure 6
<p>Fusion entity extraction process.</p> Full article ">Figure 7
<p>Trie tree example.</p> Full article ">Figure 8
<p>Example of BIO labeling.</p> Full article ">Figure 9
<p>F-1 values for different types of entities.</p> Full article ">Figure 10
<p>Comparison of Entity Extraction Results between CEntLM and CEntLM-KL Models.</p> Full article ">Figure 11
<p>Number of Relationships.</p> Full article ">Figure 12
<p>One-hop Neo4J query results.</p> Full article ">Figure 13
<p>Multi-hop Neo4J query results.</p> Full article ">
20 pages, 1325 KiB  
Article
Feature Fusion Based on Graph Convolution Network for Modulation Classification in Underwater Communication
by Xiaohui Yao, Honghui Yang and Meiping Sheng
Entropy 2023, 25(7), 1096; https://doi.org/10.3390/e25071096 - 21 Jul 2023
Cited by 3 | Viewed by 1628
Abstract
Automatic modulation classification (AMC) of underwater acoustic communication signals is of great significance in national defense and marine military. Accurate modulation classification methods can make great contributions to accurately grasping the parameters and characteristics of enemy communication systems. While a poor underwater acoustic [...] Read more.
Automatic modulation classification (AMC) of underwater acoustic communication signals is of great significance in national defense and marine military. Accurate modulation classification methods can make great contributions to accurately grasping the parameters and characteristics of enemy communication systems. While a poor underwater acoustic channel makes it difficult to classify the modulation types correctly. Feature extraction and deep learning methods have proven to be effective methods for the modulation classification of underwater acoustic communication signals, but their performance is still limited by the complex underwater communication environment. Graph convolution networks (GCN) can learn the graph structured information of the data, making it an effective method for processing structured data. To improve the stability and robustness of AMC in underwater channels, we combined the feature extraction and deep learning methods by fusing the multi-domain features and deep features using GCN. The proposed method takes the relationships among the different multi-domain features and deep features into account. Firstly, a feature graph was built using the properties of the features. Secondly, multi-domain features were extracted from the received signals and deep features were extracted from the signals using a deep neural network. Thirdly, we constructed the input of GCN using these features and the graph. Then, the multi-domain features and deep features were fused by the GCN. Finally, we classified the modulation types using the output of GCN by way of a softmax layer. We conducted the experiments on a simulated dataset and a real-world dataset, respectively. The results show that the AMC based on GCN can achieve a significant improvement in performance compared to the current state-of-the-art methods. Our approach is robust in underwater acoustic channels. Full article
(This article belongs to the Special Issue Entropy and Information Theory in Acoustics III)
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<p>Framework of the proposed AMC method based on GCN.</p> Full article ">Figure 2
<p>The graph of the multi-domain features and deep features. The graph is undirected, the black edges denote the relationships between different domain, the blue edges denote the relationships between the nodes belonging to the same domain.</p> Full article ">Figure 3
<p>The architecture of the DAE.</p> Full article ">Figure 4
<p>Sound velocity profile.</p> Full article ">Figure 5
<p>Diagram of underwater acoustic channel.</p> Full article ">Figure 6
<p>Time delays and amplitudes of the two multi-path fading channels.</p> Full article ">Figure 7
<p>Performance of proposed method in the two underwater multi-path channels.</p> Full article ">Figure 8
<p>Visualization of the features of the fully connected layer: (<b>a</b>) features are learned from the signals in Ch1 (SNR = 6 dB); (<b>b</b>) features are learned from the signals in Ch2 (SNR = 6 dB).</p> Full article ">Figure 9
<p>Performance comparison with and without deep features from the time domain; F-Y and F-N mean with and without such deep features, respectively.</p> Full article ">Figure 10
<p>Performance comparison with and without deep features from STFT; F-Y and F-N mean with and without such deep features, respectively.</p> Full article ">Figure 11
<p>Performance comparison with and without HOC features; F-Y and F-N mean with and without HOC features, respectively.</p> Full article ">Figure 12
<p>Performance comparison with and without CS features; F-Y and F-N mean with and without CS features, respectively.</p> Full article ">Figure 13
<p>Performance comparison with and without HOM features; F-Y and F-N mean with and without HOM features, respectively.</p> Full article ">Figure 14
<p>Performance comparison with and without edges inside HOM features; E-Y and E-N mean with and without such edges respectively.</p> Full article ">Figure 15
<p>Performance comparison with state-of-the-art AMC methods: (<b>a</b>) comparison result in Ch1, (<b>b</b>) comparison result in Ch2.</p> Full article ">Figure 15 Cont.
<p>Performance comparison with state-of-the-art AMC methods: (<b>a</b>) comparison result in Ch1, (<b>b</b>) comparison result in Ch2.</p> Full article ">Figure 16
<p>Computational cost comparison of different methods. GCN1 denotes the process of the first step without GPU acceleration, GCN2 denotes the process of the first step with GPU acceleration.</p> Full article ">
11 pages, 295 KiB  
Article
Gas of Particles Obeying the Monotone Statistics
by Francesco Fidaleo
Entropy 2023, 25(7), 1095; https://doi.org/10.3390/e25071095 - 21 Jul 2023
Viewed by 1051
Abstract
The present note is devoted to the detailed investigation of a concrete model satisfying the block-monotone statistics introduced in a previous paper (joint, with collaborators) of the author. The model under consideration indeed describes the free gas of massless particles in a one-dimensional [...] Read more.
The present note is devoted to the detailed investigation of a concrete model satisfying the block-monotone statistics introduced in a previous paper (joint, with collaborators) of the author. The model under consideration indeed describes the free gas of massless particles in a one-dimensional environment. This investigation can have consequences in two fundamental respects. The first one concerns the applicability of the (block-)monotone statistics to concrete physical models, yet completely unknown. Since the formula for the degeneracy of the energy-levels of the one-particle Hamiltonian of a free particle is very involved, the second aspect might be related to the, highly nontrivial, investigation of the expected thermodynamics of the free gas of particles obeying the block-monotone statistics in arbitrary spatial dimensions. A final section contains a comparison between the various (block, strict, and weak) monotone schemes with the Boltzmann statistics, which describes the gas of classical particles. It is seen that the block-monotone statistics, which takes into account the degeneracy of the energy-levels, seems the unique one having realistic physical applications. Full article
(This article belongs to the Special Issue Quantum Probability and Randomness IV)
8 pages, 303 KiB  
Article
The Time Evolution of Mutual Information between Disjoint Regions in the Universe
by Biswajit Pandey
Entropy 2023, 25(7), 1094; https://doi.org/10.3390/e25071094 - 21 Jul 2023
Cited by 1 | Viewed by 1273
Abstract
We study the time evolution of mutual information between mass distributions in spatially separated but casually connected regions in an expanding universe. The evolution of mutual information is primarily determined by the configuration entropy rate, which depends on the dynamics of the expansion [...] Read more.
We study the time evolution of mutual information between mass distributions in spatially separated but casually connected regions in an expanding universe. The evolution of mutual information is primarily determined by the configuration entropy rate, which depends on the dynamics of the expansion and growth of density perturbations. The joint entropy between distributions from the two regions plays a negligible role in such evolution. Mutual information decreases with time in a matter-dominated universe, whereas it stays constant in a Λ-dominated universe. The ΛCDM model and some other models of dark energy predict a minimum in mutual information beyond which dark energy dominates the dynamics of the universe. Mutual information may have deeper connections to the dark energy and accelerated expansion of the universe. Full article
(This article belongs to the Special Issue Entropy, Time and Evolution II)
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<p>This figure shows two large identical volumes, <span class="html-italic">A</span> and <span class="html-italic">B</span>, divided into an equal number of subvolumes. Here, <math display="inline"><semantics><msub><mi>ρ</mi><mrow><mi>A</mi></mrow></msub></semantics></math> and <math display="inline"><semantics><msub><mi>ρ</mi><mrow><mi>B</mi></mrow></msub></semantics></math> refer to the density within any two subvolumes at a given instant, <span class="html-italic">t</span>, and <math display="inline"><semantics><msub><mi>r</mi><mrow><mrow><mi>A</mi><mi>B</mi></mrow></mrow></msub></semantics></math> is the radial separation between the two subvolumes under consideration. We consider <span class="html-italic">A</span> and <span class="html-italic">B</span> to be causally connected.</p> Full article ">
12 pages, 283 KiB  
Article
The Necessary and Sufficient Conditions When Global and Local Fidelities Are Equal
by Seong-Kun Kim and Yonghae Lee
Entropy 2023, 25(7), 1093; https://doi.org/10.3390/e25071093 - 21 Jul 2023
Viewed by 1130
Abstract
In the field of quantum information theory, the concept of quantum fidelity is employed to quantify the similarity between two quantum states. It has been observed that the fidelity between two states describing a bipartite quantum system A⊗B is always less [...] Read more.
In the field of quantum information theory, the concept of quantum fidelity is employed to quantify the similarity between two quantum states. It has been observed that the fidelity between two states describing a bipartite quantum system A⊗B is always less than or equal to the quantum fidelity between the states in subsystem A alone. While this fidelity inequality is well understood, determining the conditions under which the inequality becomes an equality remains an open question. In this paper, we present the necessary and sufficient conditions for the equality of fidelities between a bipartite system A⊗B and subsystem A, considering pure quantum states. Moreover, we provide explicit representations of quantum states that satisfy the fidelity equality, based on our derived results. Full article
(This article belongs to the Special Issue Quantum Shannon Theory and Its Applications)
11 pages, 1562 KiB  
Article
Degree-Based Graph Entropy in Structure–Property Modeling
by Sourav Mondal and Kinkar Chandra Das
Entropy 2023, 25(7), 1092; https://doi.org/10.3390/e25071092 - 21 Jul 2023
Cited by 8 | Viewed by 2218
Abstract
Graph entropy plays an essential role in interpreting the structural information and complexity measure of a network. Let G be a graph of order n. Suppose dG(vi) is degree of the vertex vi for each [...] Read more.
Graph entropy plays an essential role in interpreting the structural information and complexity measure of a network. Let G be a graph of order n. Suppose dG(vi) is degree of the vertex vi for each i=1,2,…,n. Now, the k-th degree-based graph entropy for G is defined as Id,k(G)=−∑i=1ndG(vi)k∑j=1ndG(vj)klogdG(vi)k∑j=1ndG(vj)k, where k is real number. The first-degree-based entropy is generated for k=1, which has been well nurtured in last few years. As ∑j=1ndG(vj)k yields the well-known graph invariant first Zagreb index, the Id,k for k=2 is worthy of investigation. We call this graph entropy as the second-degree-based entropy. The present work aims to investigate the role of Id,2 in structure property modeling of molecules. Full article
(This article belongs to the Special Issue Spectral Graph Theory, Topological Indices of Graph, and Entropy)
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<p>Molecular graph representations of octanes.</p> Full article ">Figure 2
<p>Linear fitting of <math display="inline"><semantics><msub><mi>I</mi><mrow><mi>d</mi><mo>,</mo><mn>2</mn></mrow></msub></semantics></math> with entropy and <math display="inline"><semantics><mrow><mi>H</mi><mi>V</mi><mi>A</mi><mi>P</mi></mrow></semantics></math> for octanes.</p> Full article ">Figure 3
<p>Linear fitting of <math display="inline"><semantics><msub><mi>I</mi><mrow><mi>d</mi><mo>,</mo><mn>2</mn></mrow></msub></semantics></math> with <math display="inline"><semantics><mrow><mi>D</mi><mi>H</mi><mi>V</mi><mi>A</mi><mi>P</mi></mrow></semantics></math> and <math display="inline"><semantics><mrow><mi>A</mi><mi>F</mi></mrow></semantics></math> for octanes.</p> Full article ">Figure 4
<p>Experimental vs. predicted <math display="inline"><semantics><mrow><mi>A</mi><mi>F</mi></mrow></semantics></math> and residual plot.</p> Full article ">Figure 5
<p>Molecular graphs of benzenoid hydrocarbons.</p> Full article ">Figure 6
<p>Linear fitting of <math display="inline"><semantics><msub><mi>I</mi><mrow><mi>d</mi><mo>,</mo><mn>2</mn></mrow></msub></semantics></math> with <math display="inline"><semantics><msub><mi>E</mi><mi>π</mi></msub></semantics></math> and BP for benzenoid hydrocarbons.</p> Full article ">Figure 7
<p>Molecular graphs of some chemicals useful in drug preparation.</p> Full article ">Figure 8
<p>Linear fitting of <math display="inline"><semantics><msub><mi>I</mi><mrow><mi>d</mi><mo>,</mo><mn>2</mn></mrow></msub></semantics></math> with BP and MR for some structures displayed in <a href="#entropy-25-01092-f007" class="html-fig">Figure 7</a>.</p> Full article ">
50 pages, 3321 KiB  
Review
Non-Equilibrium Thermodynamics of Heat Transport in Superlattices, Graded Systems, and Thermal Metamaterials with Defects
by David Jou and Liliana Restuccia
Entropy 2023, 25(7), 1091; https://doi.org/10.3390/e25071091 - 20 Jul 2023
Cited by 3 | Viewed by 1636
Abstract
In this review, we discuss a nonequilibrium thermodynamic theory for heat transport in superlattices, graded systems, and thermal metamaterials with defects. The aim is to provide researchers in nonequilibrium thermodynamics as well as material scientists with a framework to consider in a systematic [...] Read more.
In this review, we discuss a nonequilibrium thermodynamic theory for heat transport in superlattices, graded systems, and thermal metamaterials with defects. The aim is to provide researchers in nonequilibrium thermodynamics as well as material scientists with a framework to consider in a systematic way several nonequilibrium questions about current developments, which are fostering new aims in heat transport, and the techniques for achieving them, for instance, defect engineering, dislocation engineering, stress engineering, phonon engineering, and nanoengineering. We also suggest some new applications in the particular case of mobile defects. Full article
(This article belongs to the Special Issue Thermodynamic Constitutive Theory and Its Application)
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<p>A superlattice of alternating layers of two semiconductors A and B. Note that, because of the geometrical structure, the system is anisotropic and its properties in the axis across the interfaces are not the same as in the axis parallel to the interfaces.</p> Full article ">Figure 2
<p>A superlattice composed of a regular distribution of point defects (B atoms or B nanodots, for instance Ge) into an originally homogeneous matrix of A, for example Si.</p> Full article ">Figure 3
<p>A graded system formed by a succession of different semiconductors.</p> Full article ">Figure 4
<p>A graded alloy of semiconductors A and B with different compositions, <math display="inline"><semantics> <mrow> <msub> <mi>A</mi> <mi>x</mi> </msub> <msub> <mi>B</mi> <mrow> <mn>1</mn> <mo>−</mo> <mi>x</mi> </mrow> </msub> </mrow> </semantics></math>, with <span class="html-italic">x</span> changing along one axis of the system.</p> Full article ">Figure 5
<p>Stress-induced heat rectification. Direct situation.</p> Full article ">Figure 6
<p>Stress-induced heat rectification. Reverse situation.</p> Full article ">Figure 7
<p>A sketch of <span class="html-italic">q</span> as a function of <math display="inline"><semantics> <mrow> <msub> <mi>T</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>T</mi> <mn>2</mn> </msub> </mrow> </semantics></math>, showing a nonmonotonic behavior. The zone to the right of the maximum exhibits a negative differential thermal conductivity.</p> Full article ">Figure 8
<p>Thermal transistor. The configuration sketched in the figure will work as a thermal transistor if the variations of the outgoing flux, <math display="inline"><semantics> <msub> <mi>q</mi> <mi>B</mi> </msub> </semantics></math>, are higher than those of the control flux <math display="inline"><semantics> <msub> <mi>q</mi> <mi>C</mi> </msub> </semantics></math>, i.e., if <math display="inline"><semantics> <mrow> <mfrac> <mrow> <mo>∂</mo> <msub> <mi>q</mi> <mi>B</mi> </msub> </mrow> <mrow> <mo>∂</mo> <msub> <mi>q</mi> <mi>C</mi> </msub> </mrow> </mfrac> <mo>&gt;</mo> <mn>1</mn> </mrow> </semantics></math>.</p> Full article ">Figure 9
<p>Thermoelectric energy conversion: a fraction of the heat flowing from <math display="inline"><semantics> <msub> <mi>T</mi> <mi>H</mi> </msub> </semantics></math> to <math display="inline"><semantics> <msub> <mi>T</mi> <mi>C</mi> </msub> </semantics></math> is converted into electric power.</p> Full article ">Figure 10
<p>Single-material thermoelectric energy converter: local efficiency versus position. Since <math display="inline"><semantics> <mrow> <mi>Z</mi> <mi>T</mi> </mrow> </semantics></math> has a narrow maximum at a given temperature, and since temperature depends on the position, the maximumm efficiency will only be achieved in a narrow region.</p> Full article ">Figure 11
<p>Two-material thermoelectric energy converter: local efficiency versus position. For each system (A and B), <math display="inline"><semantics> <mrow> <mi>Z</mi> <mi>T</mi> </mrow> </semantics></math> reaches the maximum at a given temperature dependent on the material. If the system were only composed of material A or of material B, the maximum thermoelectric efficiency would be achieved only in a single narrow region.</p> Full article ">Figure 12
<p>Thermal cloak: A sketch of a cylindrical thermal cloak. When put between two parallel plates at temperatures <math display="inline"><semantics> <msub> <mi>T</mi> <mn>1</mn> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mi>T</mi> <mn>2</mn> </msub> </semantics></math><math display="inline"><semantics> <mrow> <mo>(</mo> <mo>&lt;</mo> <msub> <mi>T</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> </semantics></math>, the heat flows along the cylindrical walls in such a way that it does not penetrate in the inner cavity (thermal protection), and that the heat flux distribution at the plate at <math display="inline"><semantics> <msub> <mi>T</mi> <mn>2</mn> </msub> </semantics></math> is the same as if the material between the plates were perfectly homogeneous (thermal invisibility).</p> Full article ">Figure 13
<p>Thermal concentrator. In contrast to the thermal cloak in the previous figure, which implied zero heat flux in the central cavity of the cylinder, the thermal concentrator increases the heat flux in the center. This requires a different spatial distribution of the thermal conductivity than in a thermal cloak.</p> Full article ">Figure 14
<p>Thermal inverter. The device in the figure ensures that the heat flux in the central cavity has a direction opposite to the heat flow outside, as if the thermal conductivity were negative in this region. In reality, this is not so, but the temperature is distributed in such a way that inside the region, <math display="inline"><semantics> <mrow> <msubsup> <mi>T</mi> <mn>1</mn> <mo>′</mo> </msubsup> <mo>&lt;</mo> <msubsup> <mi>T</mi> <mn>2</mn> <mo>′</mo> </msubsup> </mrow> </semantics></math>, in contrast to the external one, where <math display="inline"><semantics> <mrow> <msub> <mi>T</mi> <mn>1</mn> </msub> <mo>&gt;</mo> <msub> <mi>T</mi> <mn>2</mn> </msub> </mrow> </semantics></math>.</p> Full article ">Figure 15
<p>Heat transistor system, based on mobile defects driven by a heat flux considered in this Section.</p> Full article ">Figure 16
<p>Homogeneous distribution of defects in the layer B when <math display="inline"><semantics> <mrow> <mi mathvariant="bold">q</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>.</p> Full article ">Figure 17
<p>Representation of defects’ concentration when <math display="inline"><semantics> <mrow> <mi mathvariant="bold">q</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>.</p> Full article ">Figure 18
<p>Inhomogeneous distribution of defects in the layer B when <math display="inline"><semantics> <mrow> <mi mathvariant="bold">q</mi> <mo>≠</mo> <mn>0</mn> </mrow> </semantics></math>.</p> Full article ">Figure 19
<p>Representation of defects’ concentration when <math display="inline"><semantics> <mrow> <mi mathvariant="bold">q</mi> <mo>≠</mo> <mn>0</mn> </mrow> </semantics></math>.</p> Full article ">Figure 20
<p>Constant concentration of defects when <math display="inline"><semantics> <mrow> <mi>q</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>.</p> Full article ">Figure 21
<p>Concentration of defects when <math display="inline"><semantics> <mrow> <mi>q</mi> <mo>≠</mo> <mn>0</mn> </mrow> </semantics></math>.</p> Full article ">Figure 22
<p>Thermal resistance as a function of <math display="inline"><semantics> <mi mathvariant="bold">q</mi> </semantics></math>.</p> Full article ">Figure 23
<p>Temperature wave.</p> Full article ">Figure 24
<p>Charged defect oscillations in the layer B submitted to an oscillating electric field will produce oscillations in the heat flux from <span class="html-italic">A</span> to <span class="html-italic">C</span>.</p> Full article ">Figure 25
<p>Dispersion of electronic-dislocation (ED) waves: (I)—the electronic branch, (II)—the dislocation branch, after [<a href="#B76-entropy-25-01091" class="html-bibr">76</a>].</p> Full article ">
11 pages, 812 KiB  
Article
Quantum Adversarial Transfer Learning
by Longhan Wang, Yifan Sun and Xiangdong Zhang
Entropy 2023, 25(7), 1090; https://doi.org/10.3390/e25071090 - 20 Jul 2023
Cited by 1 | Viewed by 1569
Abstract
Adversarial transfer learning is a machine learning method that employs an adversarial training process to learn the datasets of different domains. Recently, this method has attracted attention because it can efficiently decouple the requirements of tasks from insufficient target data. In this study, [...] Read more.
Adversarial transfer learning is a machine learning method that employs an adversarial training process to learn the datasets of different domains. Recently, this method has attracted attention because it can efficiently decouple the requirements of tasks from insufficient target data. In this study, we introduce the notion of quantum adversarial transfer learning, where data are completely encoded by quantum states. A measurement-based judgment of the data label and a quantum subroutine to compute the gradients are discussed in detail. We also prove that our proposal has an exponential advantage over its classical counterparts in terms of computing resources such as the gate number of the circuits and the size of the storage required for the generated data. Finally, numerical experiments demonstrate that our model can be successfully trained, achieving high accuracy on certain datasets. Full article
(This article belongs to the Topic Quantum Information and Quantum Computing)
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<p>(<b>a</b>) Model architecture of ATL. The generator <math display="inline"><semantics><mi>G</mi></semantics></math> produces a data sample <math display="inline"><semantics><mrow><msup><mi>X</mi><mi>G</mi></msup></mrow></semantics></math> on a source data sample <math display="inline"><semantics><mrow><msup><mi>X</mi><mrow><msub><mi>R</mi><mi>s</mi></msub></mrow></msup></mrow></semantics></math> and a noise vector <math display="inline"><semantics><mover accent="true"><mi>z</mi><mo stretchy="false">→</mo></mover></semantics></math>. The discriminator <math display="inline"><semantics><mi>D</mi></semantics></math> distinguishes <math display="inline"><semantics><mrow><msup><mi>X</mi><mi>G</mi></msup></mrow></semantics></math> from the target data sample <math display="inline"><semantics><mrow><msup><mi>X</mi><mrow><msub><mi>R</mi><mi>t</mi></msub></mrow></msup></mrow></semantics></math>, assigning the judgement as “real” or “fake”. The classifier <math display="inline"><semantics><mi>T</mi></semantics></math> assigns task-specific labels <math display="inline"><semantics><mrow><mfenced close="}" open="{"><mi>λ</mi></mfenced></mrow></semantics></math> to fake data sample <span class="html-italic">X<sup>G</sup></span>. Note that source data sample <math display="inline"><semantics><mrow><msup><mi>X</mi><mrow><msub><mi>R</mi><mi>s</mi></msub></mrow></msup><mo> </mo></mrow></semantics></math> is only accepted into the next step when the efficiency of the training is increased, as marked by the dashed arrow. (<b>b</b>) The corresponding QATL scheme. The data samples (<math display="inline"><semantics><mrow><msub><mrow><msup><mi>X</mi><mrow><msub><mi>R</mi><mi>s</mi></msub></mrow></msup></mrow><mi>s</mi></msub></mrow></semantics></math>, <math display="inline"><semantics><mrow><msup><mi>X</mi><mrow><msub><mi>R</mi><mi>t</mi></msub></mrow></msup></mrow></semantics></math>, and <math display="inline"><semantics><mrow><msup><mi>X</mi><mi>G</mi></msup></mrow></semantics></math> ) are encoded by the quantum state <math display="inline"><semantics><mrow><msup><mi>X</mi><mrow><msub><mi>R</mi><mi>s</mi></msub></mrow></msup></mrow></semantics></math> (<math display="inline"><semantics><mrow><msup><mi>ρ</mi><mrow><msub><mi>R</mi><mi>s</mi></msub></mrow></msup></mrow></semantics></math>, <math display="inline"><semantics><mrow><msup><mi>ρ</mi><mrow><msub><mi>R</mi><mi>t</mi></msub></mrow></msup></mrow></semantics></math>, and <math display="inline"><semantics><mrow><msup><mi>ρ</mi><mi>G</mi></msup></mrow></semantics></math> ), respectively. The functioning of <math display="inline"><semantics><mi>G</mi></semantics></math>, <math display="inline"><semantics><mi>D</mi></semantics></math>, and <math display="inline"><semantics><mi>T</mi></semantics></math> is implemented by the quantum operators <math display="inline"><semantics><mover accent="true"><mi>G</mi><mo>^</mo></mover></semantics></math>, <math display="inline"><semantics><mover accent="true"><mi>D</mi><mo>^</mo></mover></semantics></math>, and <math display="inline"><semantics><mover accent="true"><mi>T</mi><mo>^</mo></mover></semantics></math>, respectively. The judgement of <math display="inline"><semantics><mover accent="true"><mi>D</mi><mo>^</mo></mover></semantics></math> is given by a quantum state (<math display="inline"><semantics><mrow><mfenced close="&#x232A;" open="|"><mrow><mi>r</mi><mi>e</mi><mi>a</mi><mi>l</mi></mrow></mfenced><mo> </mo></mrow></semantics></math> or <math display="inline"><semantics><mrow><mfenced close="&#x232A;" open="|"><mrow><mi>f</mi><mi>a</mi><mi>k</mi><mi>e</mi></mrow></mfenced></mrow></semantics></math> ) and the label is also encoded by <math display="inline"><semantics><mrow><mfenced close="&#x232A;" open="|"><mi>λ</mi></mfenced></mrow></semantics></math>. (<b>c</b>) The quantum circuit of QATL. The qubit numbers of the registers <b>Bath D</b>, <b>Out G|R<sub>t|s</sub></b>, and <b>Bath T</b> are <span class="html-italic">d</span>, <span class="html-italic">n</span>, and <span class="html-italic">p</span>, respectively. The registers <b>Out D</b> and <b>Test</b> both contain one qubit. Their qubits are initialized as <math display="inline"><semantics><mrow><mfenced close="&#x232A;" open="|"><mn>0</mn></mfenced></mrow></semantics></math>. The register <b>Bath G</b> stores an <math display="inline"><semantics><mi>m</mi></semantics></math>-qubit random state <math display="inline"><semantics><mrow><mfenced close="&#x232A;" open="|"><mi>z</mi></mfenced></mrow></semantics></math> generated by the environmental coupling. The register Label has the initial state <math display="inline"><semantics><mrow><mfenced close="&#x232A;" open="|"><mi>λ</mi></mfenced></mrow></semantics></math>. <math display="inline"><semantics><mrow><msub><mover accent="true"><mi>R</mi><mo>^</mo></mover><mi>t</mi></msub></mrow></semantics></math>, <math display="inline"><semantics><mrow><msub><mover accent="true"><mi>R</mi><mo>^</mo></mover><mi>s</mi></msub></mrow></semantics></math>, <math display="inline"><semantics><mrow><mover accent="true"><mi>G</mi><mo>^</mo></mover><mfenced><mrow><msub><mover accent="true"><mi>θ</mi><mo stretchy="false">→</mo></mover><mi>G</mi></msub></mrow></mfenced></mrow></semantics></math>, <math display="inline"><semantics><mrow><mover accent="true"><mi>D</mi><mo>^</mo></mover><mfenced><mrow><msub><mover accent="true"><mi>θ</mi><mo stretchy="false">→</mo></mover><mi>D</mi></msub></mrow></mfenced></mrow></semantics></math>, and <math display="inline"><semantics><mrow><mover accent="true"><mi>T</mi><mo>^</mo></mover><mfenced><mrow><msub><mover accent="true"><mi>θ</mi><mo stretchy="false">→</mo></mover><mi>T</mi></msub></mrow></mfenced></mrow></semantics></math> represent the unitary operators. The dashed box marks an option of applying two types of setups. <math display="inline"><semantics><mi>H</mi></semantics></math> is the Hadamard gate. <math display="inline"><semantics><mrow><mo>〈</mo><msub><mi>Z</mi><mrow><mi>o</mi><mi>u</mi><mi>t</mi><mi>D</mi></mrow></msub><mo>〉</mo></mrow></semantics></math> is the expectation value of operator <math display="inline"><semantics><mi>Z</mi></semantics></math>, as defined in the main text.</p> Full article ">Figure 2
<p>The values of quantum cost functions as a function of the training step on the Iris dataset. <span class="html-italic">V</span> is the total cost function. <span class="html-italic">V<sup>ML</sup></span>, <span class="html-italic">V<sup>GAN</sup></span>, <span class="html-italic">V<sup>DG</sup></span>, and <span class="html-italic">V<sup>DR</sup></span> represent various components of the cost function and are indicated by blue, green, red, and orange lines, respectively.</p> Full article ">
17 pages, 4797 KiB  
Article
Unstable Points, Ergodicity and Born’s Rule in 2d Bohmian Systems
by Athanasios C. Tzemos and George Contopoulos
Entropy 2023, 25(7), 1089; https://doi.org/10.3390/e25071089 - 20 Jul 2023
Cited by 4 | Viewed by 1165
Abstract
We study the role of unstable points in the Bohmian flow of a 2d system composed of two non-interacting harmonic oscillators. In particular, we study the unstable points in the inertial frame of reference as well as in the frame of reference of [...] Read more.
We study the role of unstable points in the Bohmian flow of a 2d system composed of two non-interacting harmonic oscillators. In particular, we study the unstable points in the inertial frame of reference as well as in the frame of reference of the moving nodal points, in cases with 1, 2 and multiple nodal points. Then, we find the contributions of the ordered and chaotic trajectories in the Born distribution, and when the latter is accessible by an initial particle distribution which does not satisfy Born’s rule. Full article
(This article belongs to the Special Issue Quantum Probability and Randomness IV)
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<p>(<b>a</b>) A nodal point-X-point complex and the deviations of the trajectories approaching the X-point. (<b>b</b>) The total potential close to a nodal point and its corresponding X-point (red dot) and Y-point (black dot). Both figures are drawn in the system <math display="inline"><semantics><mrow><mo>(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>)</mo></mrow></semantics></math> of the moving nodal point.</p> Full article ">Figure 2
<p>(<b>a</b>) A snapshot of the time-dependent Bohmian flow and the invariant curves of the Y-point at <math display="inline"><semantics><mrow><mi>t</mi><mo>=</mo><mn>1</mn></mrow></semantics></math> along with various Bohmian trajectories integrated from <math display="inline"><semantics><mrow><mi>t</mi><mo>=</mo><mn>0</mn></mrow></semantics></math> (green dots) up to <math display="inline"><semantics><mrow><mi>t</mi><mo>=</mo><mn>3</mn></mrow></semantics></math> (red dots) in the case of Equation (<a href="#FD6-entropy-25-01089" class="html-disp-formula">6</a>) (<math display="inline"><semantics><mrow><mi>a</mi><mo>=</mo><mi>b</mi><mo>=</mo><mn>1</mn><mo>,</mo><mi>c</mi><mo>=</mo><msqrt><mn>2</mn></msqrt><mo>/</mo><mn>2</mn></mrow></semantics></math>). The black dots correspond to <math display="inline"><semantics><mrow><mi>t</mi><mo>=</mo><mn>1</mn></mrow></semantics></math>, i.e., to the flow snapshot. We note that the flow changes in <math display="inline"><semantics><mrow><mi>t</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>3</mn><mo>]</mo></mrow></semantics></math>, but we still understand the form of the trajectories by reading the coordinates of the nodal point (<math display="inline"><semantics><msub><mi>x</mi><mi>N</mi></msub></semantics></math> red) and the Y-point (<math display="inline"><semantics><msub><mi>x</mi><mi>Y</mi></msub></semantics></math> green and <math display="inline"><semantics><mrow><msub><mi>y</mi><mi>N</mi></msub><mo>=</mo><msub><mi>y</mi><mi>Y</mi></msub></mrow></semantics></math> blue) for <math display="inline"><semantics><mrow><mi>t</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>3</mn><mo>]</mo></mrow></semantics></math>, as shown in (<b>b</b>). There, we see that <math display="inline"><semantics><mrow><msub><mi>y</mi><mi>N</mi></msub><mo>=</mo><msub><mi>y</mi><mi>Y</mi></msub></mrow></semantics></math> passes from <math display="inline"><semantics><mrow><mo>−</mo><mo>∞</mo></mrow></semantics></math> to <span class="html-italic">∞</span> (at <math display="inline"><semantics><mrow><mi>t</mi><mo>≃</mo><mn>1.84</mn></mrow></semantics></math>), <math display="inline"><semantics><msub><mi>x</mi><mi>N</mi></msub></semantics></math> changes its sign from negative to positive. (<b>c</b>) The Bohmian flow and the invariant curves of the Y-point at <math display="inline"><semantics><mrow><mi>t</mi><mo>=</mo><mn>2.5</mn></mrow></semantics></math>, where <math display="inline"><semantics><mrow><msub><mi>x</mi><mi>N</mi></msub><mo>&gt;</mo><mn>0</mn></mrow></semantics></math>. The stable/unstable invariant curves are calculated in positive/negative time <span class="html-italic">s</span>.</p> Full article ">Figure 3
<p>The distance between chaotic Bohmian trajectory and the nodal point (blue curve of the upper part) and the Y-point (red curve of the upper part) and the corresponding stretching number <span class="html-italic">a</span> for <math display="inline"><semantics><mrow><mi>t</mi><mo>∈</mo><mo>[</mo><mn>300</mn><mo>,</mo><mn>400</mn><mo>]</mo></mrow></semantics></math>. We observe that most of the significant scattering events correspond to the close approaches to the nodal points (and their associated X-points).</p> Full article ">Figure 4
<p>The distribution of the points (at every <math display="inline"><semantics><mrow><mo>Δ</mo><mi>t</mi><mo>=</mo><mn>0.05</mn></mrow></semantics></math>) of 3000 trajectories when the initial distribution satisfies BR, up to <math display="inline"><semantics><mrow><mi>t</mi><mo>=</mo><mn>3000</mn></mrow></semantics></math>.</p> Full article ">Figure 5
<p>The colorplots of two locally ergodic–chaotic trajectories separately up to <math display="inline"><semantics><mrow><mi>t</mi><mo>=</mo><mn>2</mn><mo>×</mo><msup><mn>10</mn><mn>6</mn></msup></mrow></semantics></math> (<b>a</b>) on the left and (<b>b</b>) on the right of <span class="html-italic">y</span>-axis in the single node case.</p> Full article ">Figure 6
<p>5000 initial conditions, in the case of a single nodal point, distributed according to BR at <math display="inline"><semantics><mrow><mi>t</mi><mo>=</mo><mn>0</mn></mrow></semantics></math> (<b>a</b>) on the <math display="inline"><semantics><mrow><mi>x</mi><mo>−</mo><mi>y</mi></mrow></semantics></math> plane and (<b>b</b>) projected on <math display="inline"><semantics><mrow><msub><mi>P</mi><mn>0</mn></msub><mo>=</mo><msup><mrow><mo>|</mo><msub><mo>Ψ</mo><mn>0</mn></msub><mo>|</mo></mrow><mn>2</mn></msup></mrow></semantics></math>. Blue/red initial conditions produce chaotic/ordered Bohmian trajectories.</p> Full article ">Figure 7
<p>Colorplot of 5000 trajectories (in the case of a single nodal point) with <math display="inline"><semantics><mrow><msub><mi>P</mi><mn>0</mn></msub><mo>≠</mo><msup><mrow><mo>|</mo><msub><mo>Ψ</mo><mn>0</mn></msub><mo>|</mo></mrow><mn>2</mn></msup></mrow></semantics></math>, at <math display="inline"><semantics><mrow><mi>t</mi><mo>=</mo><mn>3000</mn></mrow></semantics></math>. It is very different from that of the BR distribution in <a href="#entropy-25-01089-f004" class="html-fig">Figure 4</a>.</p> Full article ">Figure 8
<p>The stable (blue) and unstable (red) asymptotic curves of the Y-point in the case of two nodal points for (<b>a</b>) t = 1.5 and (<b>b</b>) t = 1.8 in the case of Equation (<a href="#FD19-entropy-25-01089" class="html-disp-formula">19</a>) (<math display="inline"><semantics><mrow><mi>a</mi><mo>=</mo><mn>1.23</mn><mo>,</mo><mi>b</mi><mo>=</mo><mn>1</mn><mo>,</mo><mi>c</mi><mo>=</mo><msqrt><mn>2</mn></msqrt><mo>/</mo><mn>2</mn></mrow></semantics></math>). We observe the change in the behavior of the nodal points from attractors to repellers.</p> Full article ">Figure 9
<p>5000 initial conditions in the case of 2 nodal points distributed according to BR at <math display="inline"><semantics><mrow><mi>t</mi><mo>=</mo><mn>0</mn></mrow></semantics></math> (<b>a</b>) on the <math display="inline"><semantics><mrow><mi>x</mi><mo>−</mo><mi>y</mi></mrow></semantics></math> plane and (<b>b</b>) projected on <math display="inline"><semantics><mrow><msub><mi>P</mi><mn>0</mn></msub><mo>=</mo><msup><mrow><mo>|</mo><msub><mo>Ψ</mo><mn>0</mn></msub><mo>|</mo></mrow><mn>2</mn></msup></mrow></semantics></math>. Blue/red initial conditions produce chaotic/ordered Bohmian trajectories.</p> Full article ">Figure 10
<p>The colorplot of the points of 5000 trajectories (in the case of two nodal points) initially satisfying BR, up to <math display="inline"><semantics><mrow><mi>t</mi><mo>=</mo><mn>3000</mn></mrow></semantics></math>.</p> Full article ">Figure 11
<p>The colorplots of (<b>a</b>) the ordered and (<b>b</b>) of the chaotic trajectories in an initial BR distribution (in the case of two nodal points).</p> Full article ">Figure 12
<p>The colorplots of 5000 trajectories in two initial distributions with <math display="inline"><semantics><mrow><msub><mi>P</mi><mn>0</mn></msub><mo>≠</mo><msup><mrow><mo>|</mo><msub><mo>Ψ</mo><mn>0</mn></msub><mo>|</mo></mrow><mn>2</mn></msup></mrow></semantics></math>, up to <math display="inline"><semantics><mrow><mi>t</mi><mo>=</mo><mn>3000</mn></mrow></semantics></math>. The shape of (<b>a</b>) is different from that of (<b>b</b>) and they both are very different from that of the BR distribution (<a href="#entropy-25-01089-f010" class="html-fig">Figure 10</a>).</p> Full article ">Figure 13
<p>The Bohmian flow along with the stationary nodal points (red dots), the moving nodal points (black dots) and the Y-points (green dots) at <math display="inline"><semantics><mrow><mi>t</mi><mo>=</mo><mn>0.1</mn></mrow></semantics></math>.</p> Full article ">Figure 14
<p>Details of the collision between the moving nodal point 19 and the fixed nodal point 19 at <math display="inline"><semantics><mrow><mi>y</mi><mo>=</mo><mn>0</mn></mrow></semantics></math>. (<b>a</b>) Before the collision (<math display="inline"><semantics><mrow><mi>t</mi><mo>=</mo><mn>1.8</mn></mrow></semantics></math>), the nodal point 19 and a nearby green point <math display="inline"><semantics><msub><mi>Y</mi><mn>19</mn></msub></semantics></math> move toward the fixed point 16 together with the green point <math display="inline"><semantics><msub><mi>Y</mi><mn>19</mn></msub></semantics></math> on the left of 19 (see the arrows). (<b>b</b>) At <math display="inline"><semantics><mrow><mi>t</mi><mo>=</mo><msub><mi>t</mi><mrow><mi>c</mi><mi>o</mi><mi>l</mi></mrow></msub><mo>=</mo><mn>1.8403</mn></mrow></semantics></math> we observe the collision. (<b>c</b>) After the collision (<math display="inline"><semantics><mrow><mi>t</mi><mo>=</mo><mn>1.87</mn></mrow></semantics></math>), the moving nodal point 19 and <math display="inline"><semantics><msub><mi>Y</mi><mn>19</mn></msub></semantics></math> are above point 16 but <math display="inline"><semantics><msub><mi>Y</mi><mn>19</mn></msub></semantics></math> is now on the right of the nodal point 19 and the green point <math display="inline"><semantics><msub><mi>Y</mi><mn>16</mn></msub></semantics></math> has moved to the left of 16.</p> Full article ">Figure 15
<p>The asymptotic curves of some Y-points of the upper left corner of <a href="#entropy-25-01089-f013" class="html-fig">Figure 13</a>, stable (blue) and unstable (red).</p> Full article ">Figure 16
<p>Colorplots of two different chaotic trajectories in the case of multiple nodal points up to <math display="inline"><semantics><mrow><mi>t</mi><mo>=</mo><mn>5</mn><mo>×</mo><msup><mn>10</mn><mn>6</mn></msup></mrow></semantics></math>: (<b>a</b>) <math display="inline"><semantics><mrow><mi>x</mi><mo>(</mo><mn>0</mn><mo>)</mo><mo>=</mo><mn>2.8</mn><mo>,</mo><mi>y</mi><mo>(</mo><mn>0</mn><mo>)</mo><mo>=</mo><mn>0.8</mn></mrow></semantics></math> and (<b>b</b>) <math display="inline"><semantics><mrow><mi>x</mi><mo>(</mo><mn>0</mn><mo>)</mo><mo>=</mo><mo>−</mo><mn>3</mn><mo>,</mo><mi>y</mi><mo>(</mo><mn>0</mn><mo>)</mo><mo>=</mo><mo>−</mo><mn>1</mn></mrow></semantics></math>. They are very similar, i.e., they are approximately ergodic.</p> Full article ">Figure 17
<p>10,000 initial conditions in the case of multiple nodal points distributed according to Born’s distribution at <math display="inline"><semantics><mrow><mi>t</mi><mo>=</mo><mn>0</mn></mrow></semantics></math>, chaotic (blue) and ordered (red).</p> Full article ">
11 pages, 266 KiB  
Article
Upgrading the Fusion of Imprecise Classifiers
by Serafín Moral-García, María D. Benítez and Joaquín Abellán
Entropy 2023, 25(7), 1088; https://doi.org/10.3390/e25071088 - 19 Jul 2023
Viewed by 994
Abstract
Imprecise classification is a relatively new task within Machine Learning. The difference with standard classification is that not only is one state of the variable under study determined, a set of states that do not have enough information against them and cannot be [...] Read more.
Imprecise classification is a relatively new task within Machine Learning. The difference with standard classification is that not only is one state of the variable under study determined, a set of states that do not have enough information against them and cannot be ruled out is determined as well. For imprecise classification, a mode called an Imprecise Credal Decision Tree (ICDT) that uses imprecise probabilities and maximum of entropy as the information measure has been presented. A difficult and interesting task is to show how to combine this type of imprecise classifiers. A procedure based on the minimum level of dominance has been presented; though it represents a very strong method of combining, it has the drawback of an important risk of possible erroneous prediction. In this research, we use the second-best theory to argue that the aforementioned type of combination can be improved through a new procedure built by relaxing the constraints. The new procedure is compared with the original one in an experimental study on a large set of datasets, and shows improvement. Full article
(This article belongs to the Special Issue Selected Featured Papers from Entropy Editorial Board Members)
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<p>Summary of the Bagging-ICDT method.</p> Full article ">Figure 2
<p>New bagging scheme with ICDT: Bagging-ICDT2B.</p> Full article ">
14 pages, 4647 KiB  
Article
Low Noise Opto-Electro-Mechanical Modulator for RF-to-Optical Transduction in Quantum Communications
by Michele Bonaldi, Antonio Borrielli, Giovanni Di Giuseppe, Nicola Malossi, Bruno Morana, Riccardo Natali, Paolo Piergentili, Pasqualina Maria Sarro, Enrico Serra and David Vitali
Entropy 2023, 25(7), 1087; https://doi.org/10.3390/e25071087 - 19 Jul 2023
Cited by 3 | Viewed by 1736
Abstract
In this work, we present an Opto-Electro-Mechanical Modulator (OEMM) for RF-to-optical transduction realized via an ultra-coherent nanomembrane resonator capacitively coupled to an rf injection circuit made of a microfabricated read-out able to improve the electro-optomechanical interaction. This device configuration can be embedded in [...] Read more.
In this work, we present an Opto-Electro-Mechanical Modulator (OEMM) for RF-to-optical transduction realized via an ultra-coherent nanomembrane resonator capacitively coupled to an rf injection circuit made of a microfabricated read-out able to improve the electro-optomechanical interaction. This device configuration can be embedded in a Fabry–Perot cavity for electromagnetic cooling of the LC circuit in a dilution refrigerator exploiting the opto-electro-mechanical interaction. To this aim, an optically measured steady-state frequency shift of 380 Hz was seen with a polarization voltage of 30 V and a Q-factor of the assembled device above 106 at room temperature. The rf-sputtered titanium nitride layer can be made superconductive to develop efficient quantum transducers. Full article
(This article belongs to the Special Issue Lectures on Recent Experimental Achievements in Quantum-Enhanced Technologies)
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<p>Opto–Electro-Mechanical Modulator. (<b>a</b>) Schematic of the setup with a detailed view of the cross-section of the OEMM embedded in the high-finesse Fabry–Perot Cavity. The rf weak <math display="inline"><semantics><mrow><mi>δ</mi><mi>V</mi></mrow></semantics></math> signal is transferred to the cavity optical field after polarizing the coupling capacitor <math display="inline"><semantics><mrow><msub><mi>C</mi><mi>m</mi></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></semantics></math> with a DC voltage bias <math display="inline"><semantics><msub><mi>V</mi><mrow><mi>D</mi><mi>C</mi></mrow></msub></semantics></math>. (<b>b</b>,<b>c</b>) The two components of the OEMM device: the electrode connected to the LC circuit (<b>b</b>) and the floating electrode on the silicon nitride membrane (<b>c</b>). Materials are also specified, while the substrate is floating-zone crystalline silicon for both components.</p> Full article ">Figure 2
<p>Dissipation of the metalized membrane. (<b>a</b>) The <span class="html-italic">Q</span>-factor of the bilayer SiN/TiN membrane without edge losses <math display="inline"><semantics><msub><mi>Q</mi><mrow><mi>S</mi><mi>i</mi><mi>N</mi><mo>/</mo><mi>T</mi><mi>i</mi><mi>N</mi></mrow></msub></semantics></math> (red) and with edge loss of the SiN layer <math display="inline"><semantics><msubsup><mi>Q</mi><mrow><mi>S</mi><mi>i</mi><mi>N</mi><mo>/</mo><mi>T</mi><mi>i</mi><mi>N</mi></mrow><mrow><mi>T</mi><mi>o</mi><mi>t</mi></mrow></msubsup></semantics></math> (blue). Green stars are the experimental points of the measured <span class="html-italic">Q</span>-factor for the two first axisymmetric modes with index (0,1) and (0,2). (<b>b</b>) Optical image of the SiN stoichiometric nanomembrane (light blue) with the TiN layer (brown). The membrane is endowed with the on-chip shield for recoil losses (see Ref. [<a href="#B54-entropy-25-01087" class="html-bibr">54</a>]) (right). Detailed view of the TiN notch in the membrane electrode used for the mode identification and the component assembly.</p> Full article ">Figure 3
<p>Main steps of the microfabrication process flow-chart.</p> Full article ">Figure 4
<p>Detailed view of the interaction region in the OEMM assembled device (<b>left</b>). The OEMMs clamped to the OFHC copper block (<b>center</b>) and the diving PCB board used in the optical setup (<b>right</b>).</p> Full article ">Figure 5
<p>Experimental setup. Michelson interferometer with shot-noise limited homodyne detection.</p> Full article ">Figure 6
<p>Mechanical mode characterization. (<b>a</b>) Voltage spectral noise of the homodyne signal (blue) and the shot-noise contribution (red). Mechanical modes of the functionalized <math display="inline"><semantics><mrow><msub><mi mathvariant="normal">Si</mi><mn>3</mn></msub><msub><mi mathvariant="normal">N</mi><mn>4</mn></msub></mrow></semantics></math> membrane correspondence to the peaks of the spectrum. The first mode has frequency <math display="inline"><semantics><mrow><msub><mi>f</mi><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></msub><mo>=</mo><mn>260.65</mn></mrow></semantics></math> kHz. The green dashed lines are the calculated frequencies from the FEM simulation. (<b>b</b>) Calculated mode shapes via FEM simulation, corresponding to the calculated mode frequencies present in the spectrum. Axisymmetric modes and modes with two-fold degeneracy are classified according to the number of nodal and circumference indexes. Frequency increases from the top to the bottom and from left to right.</p> Full article ">Figure 7
<p>Measurement of the mechanical <span class="html-italic">Q</span>-factor for different modes after the final assembling of the device. Red points correspond to measurements at <math display="inline"><semantics><msub><mi>V</mi><mrow><mi>D</mi><mi>C</mi></mrow></msub></semantics></math> = 0 V, the blue point corresponds to the measurement at <math display="inline"><semantics><msub><mi>V</mi><mrow><mi>D</mi><mi>C</mi></mrow></msub></semantics></math> = 30 V. Inset: voltage spectral noise (VSN) density of the homodyne signal during the ring-down measurement of the fundamental mode (0,1) at <math display="inline"><semantics><msub><mi>V</mi><mrow><mi>D</mi><mi>C</mi></mrow></msub></semantics></math> = 0 V (dotted red line) and <math display="inline"><semantics><msub><mi>V</mi><mrow><mi>D</mi><mi>C</mi></mrow></msub></semantics></math> = 30 V (dotted blue line) and corresponding fit (continuous lines), with best fit values <math display="inline"><semantics><mrow><msub><mi>τ</mi><mrow><mn>0</mn><mi>V</mi></mrow></msub><mo>=</mo><mn>4.668</mn><mo>±</mo><mn>0.008</mn></mrow></semantics></math> s and <math display="inline"><semantics><mrow><msub><mi>τ</mi><mrow><mn>30</mn><mi>V</mi></mrow></msub><mo>=</mo><mn>2.476</mn><mo>±</mo><mn>0.004</mn></mrow></semantics></math> s, respectively.</p> Full article ">Figure 8
<p>Measurement of the frequency shift of the (1,1) mode as a function of the DC voltage bias <math display="inline"><semantics><msub><mi>V</mi><mrow><mi>D</mi><mi>C</mi></mrow></msub></semantics></math> (blue square). The red line is the fit of the data using Equation (<a href="#FD15-entropy-25-01087" class="html-disp-formula">15</a>), with best value of the average distance between the electrodes and the membrane of <math display="inline"><semantics><mrow><mi>d</mi><mo>=</mo><mo>(</mo><mn>5.12</mn><mo>±</mo><mn>0.14</mn><mo>)</mo></mrow></semantics></math><math display="inline"><semantics><mi mathvariant="sans-serif">μ</mi></semantics></math>m and using the following parameters: effective area <math display="inline"><semantics><mrow><msub><mi>A</mi><mrow><mi>e</mi><mi>f</mi><mi>f</mi></mrow></msub><mo>=</mo><mn>0.075</mn></mrow></semantics></math><math display="inline"><semantics><msup><mi mathvariant="normal">mm</mi><mn>2</mn></msup></semantics></math>; membrane mass <math display="inline"><semantics><mrow><msub><mi>m</mi><mrow><mi>e</mi><mi>f</mi><mi>f</mi></mrow></msub><mo>=</mo><mn>420</mn></mrow></semantics></math> ng; measured unperturbed mode frequency <math display="inline"><semantics><mrow><msub><mi>f</mi><mn>0</mn></msub><mo>=</mo><msub><mi>f</mi><mn>11</mn></msub><mo>=</mo></mrow></semantics></math> 399,587 Hz.</p> Full article ">
18 pages, 1826 KiB  
Article
Probing Intrinsic Neural Timescales in EEG with an Information-Theory Inspired Approach: Permutation Entropy Time Delay Estimation (PE-TD)
by Andrea Buccellato, Yasir Çatal, Patrizia Bisiacchi, Di Zang, Federico Zilio, Zhe Wang, Zengxin Qi, Ruizhe Zheng, Zeyu Xu, Xuehai Wu, Alessandra Del Felice, Ying Mao and Georg Northoff
Entropy 2023, 25(7), 1086; https://doi.org/10.3390/e25071086 - 19 Jul 2023
Cited by 1 | Viewed by 1679
Abstract
Time delays are a signature of many physical systems, including the brain, and considerably shape their dynamics; moreover, they play a key role in consciousness, as postulated by the temporo-spatial theory of consciousness (TTC). However, they are often not known a priori and [...] Read more.
Time delays are a signature of many physical systems, including the brain, and considerably shape their dynamics; moreover, they play a key role in consciousness, as postulated by the temporo-spatial theory of consciousness (TTC). However, they are often not known a priori and need to be estimated from time series. In this study, we propose the use of permutation entropy (PE) to estimate time delays from neural time series as a more robust alternative to the widely used autocorrelation window (ACW). In the first part, we demonstrate the validity of this approach on synthetic neural data, and we show its resistance to regimes of nonstationarity in time series. Mirroring yet another example of comparable behavior between different nonlinear systems, permutation entropy–time delay estimation (PE-TD) is also able to measure intrinsic neural timescales (INTs) (temporal windows of neural activity at rest) from hd-EEG human data; additionally, this replication extends to the abnormal prolongation of INT values in disorders of consciousness (DoCs). Surprisingly, the correlation between ACW-0 and PE-TD decreases in a state-dependent manner when consciousness is lost, hinting at potential different regimes of nonstationarity and nonlinearity in conscious/unconscious states, consistent with many current theoretical frameworks on consciousness. In summary, we demonstrate the validity of PE-TD as a tool to extract relevant time scales from neural data; furthermore, given the divergence between ACW and PE-TD specific to DoC subjects, we hint at its potential use for the characterization of conscious states. Full article
(This article belongs to the Special Issue Temporo-Spatial Theory of Consciousness (TTC))
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<p>Time delay estimation through permutation entropy (PE) in a neural environment. (<b>a</b>) PE values as a function of increasing time embedding tau for a single channel hd-EEG recording. Estimating time delays in a physical system through PE relies on the notion that, when the tau parameter matches the time delay of the system, PE values are expected to dip significantly. The distribution of permutation patterns obtained with a tau matching the system’s dominating temporal scale is narrower; therefore, the system becomes more “predictable”: thus the dip in PE values. (Red star indicates the minimum of the PE vs time-delay graph.) (<b>b</b>) An example of a stationary integrate and fire (IAF) neuron signal. (<b>c</b>) Concatenating multiple (8) IAF stationary segments to obtain a simple case of a synthetic nonstationary signal allowed us to investigate the effects of nonstationarity on PE-TD.</p> Full article ">Figure 2
<p>Differential behavior of autocorrelation window (ACW-0) and PE-TD in model parameters for a model nonlinear delay system to perform the interpretation of estimated time delays. (<b>a</b>) Estimation error (in seconds) as a function of parameter c (degree of nonlinearity) when using ACW-0 to estimate the time delay of a Mackey–Glass oscillator. Stable performance is reached at c = 16. (<b>b</b>) Same graph for PE-TD. PE-TD behaves similarly to ACW-0, as the performance reaches an optimal accuracy around c = 16. However, after this increase in accuracy, the performance of PE-TD decreases steadily as a function of c. (<b>c</b>) Third plot of the estimation error as a function of both a (feedback strength) and tau (time delay) when using ACW-0. (<b>d</b>) Same graph for PE-TD. PE-TD seems to perform stably at earlier timescales and with weaker feedback strength than ACW-0.</p> Full article ">Figure 3
<p>The distribution of INTs in healthy populations and their abnormal average prolongation in disorders of consciousness (DoCs), probed with PE-TD. (<b>a</b>) Topopolot depicting the average distribution of INT values (in seconds) probed with PE-TD in a healthy population (N = 44). (<b>b</b>) Topoplot depicting the same average distribution of INT values probed with ACW-0. The overall INT scalp distribution is clearly consistent between the two different measures, as confirmed by a very high channel-wise correlation between the two measures (R = 0.90, <span class="html-italic">p</span> &lt; 0.001, presented in <a href="#entropy-25-01086-f004" class="html-fig">Figure 4</a>). (<b>c</b>) Violin plots for the subject average length of PE-TD values in DoCs vs. healthy controls (HC). (HC mean = 0.19 s; UWS mean = 0.24 s; MCS mean = 0.27 s; HC vs. UWS: <span class="html-italic">p</span> &lt; 0.001; HC vs. MCS: <span class="html-italic">p</span> &lt; 0.001; UWS vs. MCS: <span class="html-italic">p</span> &gt; 0.05). (** represents <span class="html-italic">p</span> &lt; 0.01 and *** represents <span class="html-italic">p</span> &lt; 0.001. n.s., when shown, stands for “non-significant” (<span class="html-italic">p</span> &gt; 0.05)).</p> Full article ">Figure 4
<p>State-dependent decreased channel-wise correlation between PE-TD and ACW-0 in loss of consciousness (LOC). (<b>a</b>) Pearson’s correlation coefficient in healthy subjects (R = 0.90, <span class="html-italic">p</span> &lt; 0.001). (<b>b</b>) Pearson’s correlation coefficient in UWS (R = 0.55, <span class="html-italic">p</span> &lt; 0.001). The decrease in correlation observed in UWS is confirmed by a bootstrap distribution test (<span class="html-italic">p</span> &lt; 0.001) and is validated with a Fisher’s z transform test (<span class="html-italic">p</span> &lt; 0.001) (see <a href="#sec2-entropy-25-01086" class="html-sec">Section 2</a>—Methods).</p> Full article ">
15 pages, 4993 KiB  
Article
EnRDeA U-Net Deep Learning of Semantic Segmentation on Intricate Noise Roads
by Xiaodong Yu, Ta-Wen Kuan, Shih-Pang Tseng, Ying Chen, Shuo Chen, Jhing-Fa Wang, Yuhang Gu and Tuoli Chen
Entropy 2023, 25(7), 1085; https://doi.org/10.3390/e25071085 - 19 Jul 2023
Cited by 6 | Viewed by 1939
Abstract
Road segmentation is beneficial to build a vision-controllable mission-oriented self-driving bot, e.g., the Self-Driving Sweeping Bot, or SDSB, for working in restricted areas. Using road segmentation, the bot itself and physical facilities may be protected and the sweeping efficiency of the SDSB promoted. [...] Read more.
Road segmentation is beneficial to build a vision-controllable mission-oriented self-driving bot, e.g., the Self-Driving Sweeping Bot, or SDSB, for working in restricted areas. Using road segmentation, the bot itself and physical facilities may be protected and the sweeping efficiency of the SDSB promoted. However, roads in the real world are generally exposed to intricate noise conditions as a result of changing weather and climate effects; these include sunshine spots, shadowing caused by trees or physical facilities, traffic obstacles and signs, and cracks or sealing signs resulting from long-term road usage, as well as different types of road materials, such as cement or asphalt; all of these factors greatly influence the effectiveness of road segmentation. In this work, we investigate the extension of Primordial U-Net by the proposed EnRDeA U-Net, which uses an input channel applying a Residual U-Net block as an encoder and an attention gate in the output channel as a decoder, to validate a dataset of intricate road noises. In addition, we carry out a detailed analysis of the nets’ features and segmentation performance to validate the intricate noises dataset on three U-Net extensions, i.e., the Primordial U-Net, Residual U-Net, and EnRDeA U-Net. Finally, the nets’ structures, parameters, training losses, performance indexes, etc., are presented and discussed in the experimental results. Full article
(This article belongs to the Special Issue New Trends in Fault Diagnosis and Prognosis for Engineering Applications: From Signal Processing to Machine Learning and Deep Learning)
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<p>Training flowchart of Residual U-net embedded with attention block, wherein four layers of the residual blocks (black box) for the input original image are sequentially transposed into the corresponding attention gate (rosewood color) for the output segmentation.</p> Full article ">Figure 2
<p>The schematic diagram of primordial U-Net framework.</p> Full article ">Figure 3
<p>Two schematic residual blocks, the left block of the dotted-line block indicates that the learning of the residual mapping <span class="html-italic">g</span>(<span class="html-italic">x</span>) = <span class="html-italic">f</span>(<span class="html-italic">x</span>) − <span class="html-italic">x</span>, making the identity mapping <span class="html-italic">f</span>(<span class="html-italic">x</span>) = <span class="html-italic">x</span> easier to learn, and the right diagram indicates the input transformed into the desired shape for the addition operation by 1 × 1 convolution.</p> Full article ">Figure 4
<p>Attention block embedded in primordial U-Net output channel.</p> Full article ">Figure 5
<p>Examples of collected visual dataset, wherein the left image indicated the road with the squiggly black lines (red box) and 15 cm height of the retained wall (green boxes), whereas the right one presented the road surface with the random sunshine spots shadowed by facilities or giant trees (yellow box).</p> Full article ">Figure 6
<p>Examples of collected images dataset used for validation, in which the dataset is categorized as road with sunshine spot shadowed by facilities.</p> Full article ">Figure 7
<p>Examples of visual dataset which is categorized as road with sunshine spots shadowed by giant tree.</p> Full article ">Figure 8
<p>Examples of visual dataset which is categorized as roads under fully cloudy status, wherein two types of road materials, including the cement road and the asphalt road, respectively, are observed.</p> Full article ">Figure 9
<p>Examples of training images including, Camvid dataset in the first three columns’ images (<b>a1</b>–<b>c1</b>), and self-collection dataset in the last three columns’ ones (<b>d1</b>–<b>e1</b>), wherein the first row contains the raw images (<b>a1</b>–<b>f1</b>), whereas the second row contains the corresponding Ground-True images (<b>a2</b>–<b>f2</b>).</p> Full article ">Figure 10
<p>The visualized map of training evaluation in terms of Epoch and training loss.</p> Full article ">Figure 11
<p>The segmentation experiments on three versions of U-net, including original U-net, Residual U-net, and EnRDeA U-net, validated on four types of road conditions, wherein (<b>a</b>) is the cement road under fully cloudy weather, (<b>b</b>) is shown as the cement road with sunshine-shadowed spots under sunny weather, (<b>c</b>) is indicated as the asphalt road in a darker situation, whereas (<b>d</b>) exhibits the asphalt road in a brighter situation.</p> Full article ">Figure 12
<p>Three indexes of CPA (<b>a</b>), Recall (<b>b</b>), and IOU (<b>c</b>) with correspondence to Epoch number in terms of three U-nets, in which the EnRDeA U-net (blue line) significantly performed the superior efficiency compared to the other two (black line and gray line).</p> Full article ">
21 pages, 366 KiB  
Article
Thermodynamics of Composition Graded Thermoelastic Solids
by Vito Antonio Cimmelli
Entropy 2023, 25(7), 1084; https://doi.org/10.3390/e25071084 - 19 Jul 2023
Viewed by 1152
Abstract
We propose a thermodynamic model describing the thermoelastic behavior of composition graded materials. The compatibility of the model with the second law of thermodynamics is explored by applying a generalized Coleman–Noll procedure. For the material at hand, the specific entropy and the stress [...] Read more.
We propose a thermodynamic model describing the thermoelastic behavior of composition graded materials. The compatibility of the model with the second law of thermodynamics is explored by applying a generalized Coleman–Noll procedure. For the material at hand, the specific entropy and the stress tensor may depend on the gradient of the unknown fields, resulting in a very general theory. We calculate the speeds of coupled first- and second-sound pulses, propagating either trough nonequilibrium or equilibrium states. We characterize several different types of perturbations depending on the value of the material coefficients. Under the assumption that the deformation of the body can produce changes in its stoichiometry, altering locally the material composition, the possibility of propagation of pure stoichiometric waves is pointed out. Thermoelastic perturbations generated by the coupling of stoichiometric and thermal effects are analyzed as well. Full article
(This article belongs to the Special Issue Thermodynamic Constitutive Theory and Its Application)
17 pages, 2150 KiB  
Article
Synchronization Induced by Layer Mismatch in Multiplex Networks
by Md Sayeed Anwar, Sarbendu Rakshit, Jürgen Kurths and Dibakar Ghosh
Entropy 2023, 25(7), 1083; https://doi.org/10.3390/e25071083 - 19 Jul 2023
Cited by 3 | Viewed by 1371
Abstract
Heterogeneity among interacting units plays an important role in numerous biological and man-made complex systems. While the impacts of heterogeneity on synchronization, in terms of structural mismatch of the layers in multiplex networks, has been studied thoroughly, its influence on intralayer synchronization, in [...] Read more.
Heterogeneity among interacting units plays an important role in numerous biological and man-made complex systems. While the impacts of heterogeneity on synchronization, in terms of structural mismatch of the layers in multiplex networks, has been studied thoroughly, its influence on intralayer synchronization, in terms of parameter mismatch among the layers, has not been adequately investigated. Here, we study the intralayer synchrony in multiplex networks, where the layers are different from one other, due to parameter mismatch in their local dynamics. In such a multiplex network, the intralayer coupling strength for the emergence of intralayer synchronization decreases upon the introduction of impurity among the layers, which is caused by a parameter mismatch in their local dynamics. Furthermore, the area of occurrence of intralayer synchronization also widens with increasing mismatch. We analytically derive a condition under which the intralayer synchronous solution exists, and we even sustain its stability. We also prove that, in spite of the mismatch among the layers, all the layers of the multiplex network synchronize simultaneously. Our results indicate that a multiplex network with mismatched layers can induce synchrony more easily than a multiplex network with identical layers. Full article
(This article belongs to the Special Issue Synchronization in Complex Networks of Nonlinear Dynamical Systems)
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Figure 1
<p>Schematic representation of a <math display="inline"><semantics><mrow><mi>Q</mi><mo>=</mo><mn>2</mn></mrow></semantics></math> layered multiplex network with <math display="inline"><semantics><mrow><mi>N</mi><mo>=</mo><mn>10</mn></mrow></semantics></math> nodes in each layer. The nodes in each layer are identical to one other, while being different for different layers, by means of parameter mismatch in the local dynamics. To illustrate this, the nodes in layer-1 are colored in red, and the nodes in layer-2 are colored in blue. Solid black lines portray the intralayer connections, while dashed black lines show the replica-wise interlayer connections.</p> Full article ">Figure 2
<p>Synchronization errors <math display="inline"><semantics><msub><mi>E</mi><mn>1</mn></msub></semantics></math> and <math display="inline"><semantics><msub><mi>E</mi><mn>2</mn></msub></semantics></math>, corresponding to layer-1 and layer-2, are depicted by solid and dashed curves as a function of intralayer coupling strength <math display="inline"><semantics><msub><mi>σ</mi><mrow><mi>i</mi><mi>n</mi><mi>t</mi><mi>r</mi><mi>a</mi></mrow></msub></semantics></math> for three different values of interlayer coupling strengths: <math display="inline"><semantics><mrow><msub><mi>σ</mi><mrow><mi>i</mi><mi>n</mi><mi>t</mi><mi>e</mi><mi>r</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow></semantics></math> (in red); <math display="inline"><semantics><mrow><msub><mi>σ</mi><mrow><mi>i</mi><mi>n</mi><mi>t</mi><mi>e</mi><mi>r</mi></mrow></msub><mo>=</mo><mn>0.01</mn></mrow></semantics></math> (in blue); and <math display="inline"><semantics><mrow><msub><mi>σ</mi><mrow><mi>i</mi><mi>n</mi><mi>t</mi><mi>e</mi><mi>r</mi></mrow></msub><mo>=</mo><mn>0.1</mn></mrow></semantics></math> (in green). The mismatch parameter is fixed at <math display="inline"><semantics><mrow><mo>Δ</mo><mi>ω</mi><mo>=</mo><mn>0.1</mn></mrow></semantics></math>. The connectivity probability of both layers is fixed at <math display="inline"><semantics><mrow><msub><mi>p</mi><mrow><mi>r</mi><mi>a</mi><mi>n</mi><mi>d</mi></mrow></msub><mo>=</mo><mn>0.1</mn></mrow></semantics></math>. All the curves are obtained by taking 10 different network realizations and initial conditions, which are drawn from the phase space of isolated node dynamics.</p> Full article ">Figure 3
<p>(<b>a</b>) Variation of intralayer synchronization error as a function of <math display="inline"><semantics><msub><mi>σ</mi><mrow><mi>i</mi><mi>n</mi><mi>t</mi><mi>r</mi><mi>a</mi></mrow></msub></semantics></math> for three different values of mismatch parameter <math display="inline"><semantics><mrow><mo>Δ</mo><mi>ω</mi><mo>=</mo><mn>0</mn></mrow></semantics></math> (in red), <math display="inline"><semantics><mrow><mo>Δ</mo><mi>ω</mi><mo>=</mo><mn>0.02</mn></mrow></semantics></math> (in blue), and <math display="inline"><semantics><mrow><mo>Δ</mo><mi>ω</mi><mo>=</mo><mn>0.1</mn></mrow></semantics></math> (in magenta), respectively. The interlayer coupling strength is fixed at <math display="inline"><semantics><mrow><msub><mi>σ</mi><mrow><mi>i</mi><mi>n</mi><mi>t</mi><mi>e</mi><mi>r</mi></mrow></msub><mo>=</mo><mn>0.1</mn></mrow></semantics></math>; (<b>b</b>) the maximum Lyapunov exponent <math display="inline"><semantics><msub><mi mathvariant="sans-serif">Λ</mi><mrow><mi>i</mi><mi>n</mi><mi>t</mi><mi>r</mi><mi>a</mi></mrow></msub></semantics></math> as a function of <math display="inline"><semantics><msub><mi>σ</mi><mrow><mi>i</mi><mi>n</mi><mi>t</mi><mi>r</mi><mi>a</mi></mrow></msub></semantics></math> for the same set of mismatch parameter values taken in the upper panel. The dashed horizontal black line represents the 0 line. In both the upper and lower panel, the network connectivity probability is taken to be <math display="inline"><semantics><mrow><msub><mi>p</mi><mrow><mi>r</mi><mi>a</mi><mi>n</mi><mi>d</mi></mrow></msub><mo>=</mo><mn>0.1</mn></mrow></semantics></math>.</p> Full article ">Figure 4
<p>Variation of intralayer synchronization error <math display="inline"><semantics><msub><mi>E</mi><mrow><mi>i</mi><mi>n</mi><mi>t</mi><mi>r</mi><mi>a</mi></mrow></msub></semantics></math> as a function of <math display="inline"><semantics><msub><mi>σ</mi><mrow><mi>i</mi><mi>n</mi><mi>t</mi><mi>r</mi><mi>a</mi></mrow></msub></semantics></math> and <math display="inline"><semantics><msub><mi>σ</mi><mrow><mi>i</mi><mi>n</mi><mi>t</mi><mi>e</mi><mi>r</mi></mrow></msub></semantics></math> for two different values of mismatch parameter <math display="inline"><semantics><mrow><mo>Δ</mo><mi>ω</mi><mo>=</mo><mn>0</mn></mrow></semantics></math> (<b>a</b>) and <math display="inline"><semantics><mrow><mo>Δ</mo><mi>ω</mi><mo>=</mo><mn>0.1</mn></mrow></semantics></math> (<b>b</b>), respectively. For both the subfigures, in the white region the system becomes unbounded. The color bars indicate the variation of intralayer synchronization error <math display="inline"><semantics><msub><mi>E</mi><mrow><mi>i</mi><mi>n</mi><mi>t</mi><mi>r</mi><mi>a</mi></mrow></msub></semantics></math>, where the 0 value of <math display="inline"><semantics><msub><mi>E</mi><mrow><mi>i</mi><mi>n</mi><mi>t</mi><mi>r</mi><mi>a</mi></mrow></msub></semantics></math> represents the emergence of intralayer synchrony. The connectivity probability in both the layers is fixed at <math display="inline"><semantics><mrow><msub><mi>p</mi><mrow><mi>r</mi><mi>a</mi><mi>n</mi><mi>d</mi></mrow></msub><mo>=</mo><mn>0.1</mn></mrow></semantics></math>.</p> Full article ">Figure 5
<p>Variation of intralayer synchronization error <math display="inline"><semantics><msub><mi>E</mi><mrow><mi>i</mi><mi>n</mi><mi>t</mi><mi>r</mi><mi>a</mi></mrow></msub></semantics></math> as a function of intralayer coupling strength <math display="inline"><semantics><msub><mi>σ</mi><mrow><mi>i</mi><mi>n</mi><mi>t</mi><mi>r</mi><mi>a</mi></mrow></msub></semantics></math> and edge-generating probability <math display="inline"><semantics><msub><mi>p</mi><mrow><mi>r</mi><mi>a</mi><mi>n</mi><mi>d</mi></mrow></msub></semantics></math> for two different values of mismatch parameter <math display="inline"><semantics><mrow><mo>Δ</mo><mi>ω</mi><mo>=</mo><mn>0</mn></mrow></semantics></math> (<b>a</b>) and <math display="inline"><semantics><mrow><mo>Δ</mo><mi>ω</mi><mo>=</mo><mn>0.1</mn></mrow></semantics></math> (<b>b</b>), respectively. In both the subfigures, the region in white indicates the unbounded region. The color bars indicate the variation of intralayer synchronization error <math display="inline"><semantics><msub><mi>E</mi><mrow><mi>i</mi><mi>n</mi><mi>t</mi><mi>r</mi><mi>a</mi></mrow></msub></semantics></math>, where the 0 value of <math display="inline"><semantics><msub><mi>E</mi><mrow><mi>i</mi><mi>n</mi><mi>t</mi><mi>r</mi><mi>a</mi></mrow></msub></semantics></math> represents the emergence of intralayer synchrony. The interlayer coupling strength is fixed at <math display="inline"><semantics><mrow><msub><mi>σ</mi><mrow><mi>i</mi><mi>n</mi><mi>t</mi><mi>e</mi><mi>r</mi></mrow></msub><mo>=</mo><mn>0.1</mn></mrow></semantics></math>.</p> Full article ">Figure 6
<p>(<b>a</b>) Variation of intralayer synchronization error as a function of <math display="inline"><semantics><msub><mi>σ</mi><mrow><mi>i</mi><mi>n</mi><mi>t</mi><mi>r</mi><mi>a</mi></mrow></msub></semantics></math> for three different values of mismatch parameter <math display="inline"><semantics><mrow><mo>Δ</mo><mi>ω</mi><mo>=</mo><mn>0</mn></mrow></semantics></math> (in red), <math display="inline"><semantics><mrow><mo>Δ</mo><mi>ω</mi><mo>=</mo><mn>0.05</mn></mrow></semantics></math> (in blue), and <math display="inline"><semantics><mrow><mo>Δ</mo><mi>ω</mi><mo>=</mo><mn>0.1</mn></mrow></semantics></math> (in magenta), respectively. The interlayer coupling strength is fixed at <math display="inline"><semantics><mrow><msub><mi>σ</mi><mrow><mi>i</mi><mi>n</mi><mi>t</mi><mi>e</mi><mi>r</mi></mrow></msub><mo>=</mo><mn>0.1</mn></mrow></semantics></math>; (<b>b</b>) the maximum Lyapunov exponent <math display="inline"><semantics><msub><mi mathvariant="sans-serif">Λ</mi><mrow><mi>i</mi><mi>n</mi><mi>t</mi><mi>r</mi><mi>a</mi></mrow></msub></semantics></math> as a function of <math display="inline"><semantics><msub><mi>σ</mi><mrow><mi>i</mi><mi>n</mi><mi>t</mi><mi>r</mi><mi>a</mi></mrow></msub></semantics></math> for the same set of mismatch parameter values taken in the upper panel. The dashed horizontal black line represents the 0 line. In both the upper and lower panel, the network connectivity probability is taken to be <math display="inline"><semantics><mrow><msub><mi>p</mi><mrow><mi>r</mi><mi>a</mi><mi>n</mi><mi>d</mi></mrow></msub><mo>=</mo><mn>0.1</mn></mrow></semantics></math>.</p> Full article ">
9 pages, 278 KiB  
Article
Thermodynamic Entropy as a Noether Invariant from Contact Geometry
by Alessandro Bravetti, Miguel Ángel García-Ariza and Diego Tapias
Entropy 2023, 25(7), 1082; https://doi.org/10.3390/e25071082 - 19 Jul 2023
Cited by 6 | Viewed by 1741
Abstract
We use a formulation of Noether’s theorem for contact Hamiltonian systems to derive a relation between the thermodynamic entropy and the Noether invariant associated with time-translational symmetry. In the particular case of thermostatted systems at equilibrium, we show that the total entropy of [...] Read more.
We use a formulation of Noether’s theorem for contact Hamiltonian systems to derive a relation between the thermodynamic entropy and the Noether invariant associated with time-translational symmetry. In the particular case of thermostatted systems at equilibrium, we show that the total entropy of the system plus the reservoir are conserved as a consequence thereof. Our results contribute to understanding thermodynamic entropy from a geometric point of view. Full article
(This article belongs to the Special Issue Geometric Structure of Thermodynamics: Theory and Applications)
28 pages, 1596 KiB  
Article
Decision Support System for Prioritization of Offshore Wind Farm Site by Utilizing Picture Fuzzy Combined Compromise Solution Group Decision Method
by Yuan Rong and Liying Yu
Entropy 2023, 25(7), 1081; https://doi.org/10.3390/e25071081 - 18 Jul 2023
Cited by 9 | Viewed by 1404
Abstract
The selection of offshore wind farm site (OWFS) has important strategic significance for vigorously developing offshore new energy and is deemed as a complicated uncertain multicriteria decision-making (MCDM) process. To further promote offshore wind power energy planning and provide decision support, this paper [...] Read more.
The selection of offshore wind farm site (OWFS) has important strategic significance for vigorously developing offshore new energy and is deemed as a complicated uncertain multicriteria decision-making (MCDM) process. To further promote offshore wind power energy planning and provide decision support, this paper proposes a hybrid picture fuzzy (PF) combined compromise solution (CoCoSo) technique for prioritization of OWFSs. To begin with, a fresh PF similarity measure is proffered to estimate the importance of experts. Next, the novel operational rules for PF numbers based upon the generalized Dombi norms are defined, and four novel generalized Dombi operators are propounded. Afterward, the PF preference selection index (PSI) method and PF stepwise weights assessment ratio analysis (SWARA) model are propounded to identify the objective and subjective weight of criteria, separately. In addition, the enhanced CoCoSo method is proffered via the similarity measure and new operators for ranking OWFSs with PF information. Lastly, the applicability and feasibility of the propounded PF-PSI-SWARA-CoCoSo method are adopted to ascertain the optimal OWFS. The comparison and sensibility investigations are also carried out to validate the robustness and superiority of our methodology. Results manifest that the developed methodology can offer powerful decision support for departments and managers to evaluate and choose the satisfying OWFSs. Full article
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Figure 1

Figure 1
<p>Propounded hybrid PF-PSI-SWARA-CoCoSo approach for selecting OWFSs.</p> Full article ">Figure 2
<p>Decision results attained by diverse parameter <inline-formula><mml:math id="mm380"><mml:semantics><mml:mi>τ</mml:mi></mml:semantics></mml:math></inline-formula>.</p> Full article ">Figure 3
<p>Decision results of diverse types of weight for OWFSs.</p> Full article ">Figure 4
<p>Ranking results of the diverse PF method for OWFSs.</p> Full article ">
24 pages, 2685 KiB  
Article
Wavelet-Based Multiscale Intermittency Analysis: The Effect of Deformation
by José M. Angulo and Ana E. Madrid
Entropy 2023, 25(7), 1080; https://doi.org/10.3390/e25071080 - 18 Jul 2023
Cited by 1 | Viewed by 1157
Abstract
Intermittency represents a certain form of heterogeneous behavior that has interest in diverse fields of application, particularly regarding the characterization of system dynamics and for risk assessment. Given its intrinsic location-scale-dependent nature, wavelets constitute a useful functional tool for technical analysis of intermittency. [...] Read more.
Intermittency represents a certain form of heterogeneous behavior that has interest in diverse fields of application, particularly regarding the characterization of system dynamics and for risk assessment. Given its intrinsic location-scale-dependent nature, wavelets constitute a useful functional tool for technical analysis of intermittency. Deformation of the support may induce complex structural changes in a signal. In this paper, we study the effect of deformation on intermittency. Specifically, we analyze the interscale transfer of energy and its implications on different wavelet-based intermittency indicators, depending on whether the signal corresponds to a ‘level’- or a ‘flow’-type physical magnitude. Further, we evaluate the effect of deformation on the interscale distribution of energy in terms of generalized entropy and complexity measures. For illustration, various contrasting scenarios are considered based on simulation, as well as two segments corresponding to different regimes in a real seismic series before and after a significant earthquake. Full article
(This article belongs to the Section Signal and Data Analysis)
►▼ Show Figures

Figure 1

Figure 1
<p>From top to bottom: (<b>a</b>,<b>b</b>) simulated realizations of model (9), with (<b>a</b>) Cauchy and (<b>b</b>) Gaussian white noise; correspondingly in left and right columns, using Haar wavelet, (<b>c</b>,<b>d</b>) scalogram <math display="inline"><semantics><mrow><msubsup><mi>W</mi><mi>x</mi><mn>2</mn></msubsup><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow></mrow></semantics></math>, (<b>e</b>,<b>f</b>) <math display="inline"><semantics><mrow><mi>L</mi><mi>I</mi><msub><mi>M</mi><mi>x</mi></msub></mrow></semantics></math> map, (<b>g</b>,<b>h</b>) threshold exceedance set for <math display="inline"><semantics><mrow><mi>L</mi><mi>I</mi><msubsup><mi>M</mi><mi>x</mi><mn>2</mn></msubsup><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow><mo>&gt;</mo><mn>3</mn></mrow></semantics></math>, and (<b>i</b>,<b>j</b>) <math display="inline"><semantics><msub><mi>F</mi><mi>x</mi></msub></semantics></math> curve.</p> Full article ">Figure 2
<p>From top to bottom: (<b>a</b>,<b>b</b>) simulated realizations of model (9), with (<b>a</b>) Cauchy and (<b>b</b>) Gaussian white noise; correspondingly in left and right columns, using Morlet wavelet, (<b>c</b>,<b>d</b>) scalogram <math display="inline"><semantics><mrow><msubsup><mi>W</mi><mi>x</mi><mn>2</mn></msubsup><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow></mrow></semantics></math>, (<b>e</b>,<b>f</b>) <math display="inline"><semantics><mrow><mi>L</mi><mi>I</mi><msub><mi>M</mi><mi>x</mi></msub></mrow></semantics></math> map, (<b>g</b>,<b>h</b>) threshold exceedance set for <math display="inline"><semantics><mrow><mi>L</mi><mi>I</mi><msubsup><mi>M</mi><mi>x</mi><mn>2</mn></msubsup><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow><mo>&gt;</mo><mn>3</mn></mrow></semantics></math>, and (<b>i</b>,<b>j</b>) <math display="inline"><semantics><msub><mi>F</mi><mi>x</mi></msub></semantics></math> curve.</p> Full article ">Figure 2 Cont.
<p>From top to bottom: (<b>a</b>,<b>b</b>) simulated realizations of model (9), with (<b>a</b>) Cauchy and (<b>b</b>) Gaussian white noise; correspondingly in left and right columns, using Morlet wavelet, (<b>c</b>,<b>d</b>) scalogram <math display="inline"><semantics><mrow><msubsup><mi>W</mi><mi>x</mi><mn>2</mn></msubsup><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow></mrow></semantics></math>, (<b>e</b>,<b>f</b>) <math display="inline"><semantics><mrow><mi>L</mi><mi>I</mi><msub><mi>M</mi><mi>x</mi></msub></mrow></semantics></math> map, (<b>g</b>,<b>h</b>) threshold exceedance set for <math display="inline"><semantics><mrow><mi>L</mi><mi>I</mi><msubsup><mi>M</mi><mi>x</mi><mn>2</mn></msubsup><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow><mo>&gt;</mo><mn>3</mn></mrow></semantics></math>, and (<b>i</b>,<b>j</b>) <math display="inline"><semantics><msub><mi>F</mi><mi>x</mi></msub></semantics></math> curve.</p> Full article ">Figure 3
<p>Deformation <math display="inline"><semantics><mo>Φ</mo></semantics></math> given by (10).</p> Full article ">Figure 4
<p>From top to bottom: (<b>a</b>) simulated signal realization <span class="html-italic">x</span> generated from model (9) with Cauchy white noise, and (<b>b</b>) its (‘level’-type) deformation <math display="inline"><semantics><mrow><mi>x</mi><mo>[</mo><mo>Φ</mo><mo>]</mo></mrow></semantics></math>; correspondingly in left and right columns, (<b>c</b>,<b>d</b>) scalogram <math display="inline"><semantics><mrow><msup><mi>W</mi><mn>2</mn></msup><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow></mrow></semantics></math>, (<b>e</b>,<b>f</b>) <math display="inline"><semantics><mrow><mi>L</mi><mi>I</mi><mi>M</mi></mrow></semantics></math> map, (<b>g</b>,<b>h</b>) threshold exceedance set for <math display="inline"><semantics><mrow><mi>L</mi><mi>I</mi><msup><mi>M</mi><mn>2</mn></msup><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow><mo>&gt;</mo><mn>3</mn></mrow></semantics></math>, and (<b>i</b>,<b>j</b>) <span class="html-italic">F</span> curve.</p> Full article ">Figure 5
<p>From top to bottom: (<b>a</b>) simulated signal realization <span class="html-italic">x</span> generated from model (9) with Gaussian white noise, and (<b>b</b>) its (‘level’-type) deformation <math display="inline"><semantics><mrow><mi>x</mi><mo>[</mo><mo>Φ</mo><mo>]</mo></mrow></semantics></math>; correspondingly in left and right columns, (<b>c</b>,<b>d</b>) scalogram <math display="inline"><semantics><mrow><msup><mi>W</mi><mn>2</mn></msup><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow></mrow></semantics></math>, (<b>e</b>,<b>f</b>) <math display="inline"><semantics><mrow><mi>L</mi><mi>I</mi><mi>M</mi></mrow></semantics></math> map, (<b>g</b>,<b>h</b>) threshold exceedance set for <math display="inline"><semantics><mrow><mi>L</mi><mi>I</mi><msup><mi>M</mi><mn>2</mn></msup><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow><mo>&gt;</mo><mn>3</mn></mrow></semantics></math>, and (<b>i</b>,<b>j</b>) <span class="html-italic">F</span> curve.</p> Full article ">Figure 6
<p>From top to bottom: (<b>a</b>) simulated signal realization <span class="html-italic">x</span> generated from the ARIMA(1,2,1) model obtained by double integration of model (9) with Cauchy noise, and (<b>b</b>) its (‘flow’-type) deformation <math display="inline"><semantics><mrow><mi>x</mi><mo>[</mo><mover accent="true"><mo>Φ</mo><mo>˜</mo></mover><mo>]</mo></mrow></semantics></math>; correspondingly in left and right columns, (<b>c</b>,<b>d</b>) scalogram <math display="inline"><semantics><mrow><msup><mi>W</mi><mn>2</mn></msup><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow></mrow></semantics></math>, (<b>e</b>,<b>f</b>) <math display="inline"><semantics><mrow><mi>L</mi><mi>I</mi><mi>M</mi></mrow></semantics></math> map, (<b>g</b>,<b>h</b>) threshold exceedance set for <math display="inline"><semantics><mrow><mi>L</mi><mi>I</mi><msup><mi>M</mi><mn>2</mn></msup><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow><mo>&gt;</mo><mn>3</mn></mrow></semantics></math>, and (<b>i</b>,<b>j</b>) <span class="html-italic">F</span> curve.</p> Full article ">Figure 7
<p>From top to bottom: (<b>a</b>) simulated signal realization <span class="html-italic">x</span> generated from the ARIMA(1,2,1) model obtained by double integration of model (9) with Gaussian noise, and (<b>b</b>) its (‘flow’-type) deformation <math display="inline"><semantics><mrow><mi>x</mi><mo>[</mo><mover accent="true"><mo>Φ</mo><mo>˜</mo></mover><mo>]</mo></mrow></semantics></math>; correspondingly in left and right columns, (<b>c</b>,<b>d</b>) scalogram <math display="inline"><semantics><mrow><msup><mi>W</mi><mn>2</mn></msup><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow></mrow></semantics></math>, (<b>e</b>,<b>f</b>) <math display="inline"><semantics><mrow><mi>L</mi><mi>I</mi><mi>M</mi></mrow></semantics></math> map, (<b>g</b>,<b>h</b>) threshold exceedance set for <math display="inline"><semantics><mrow><mi>L</mi><mi>I</mi><msup><mi>M</mi><mn>2</mn></msup><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow><mo>&gt;</mo><mn>3</mn></mrow></semantics></math>, and (<b>i</b>,<b>j</b>) <span class="html-italic">F</span> curve.</p> Full article ">Figure 8
<p>(<b>a</b>,<b>b</b>) Shannon entropy, (<b>c</b>,<b>d</b>) Rényi entropy of order <math display="inline"><semantics><mrow><mi>q</mi><mo>=</mo><mn>3</mn></mrow></semantics></math>, and (<b>e</b>,<b>f</b>) Tsallis entropy of order <math display="inline"><semantics><mrow><mi>q</mi><mo>=</mo><mn>3</mn></mrow></semantics></math>, displayed in blue color for original signal generated from model (9) with (<b>a</b>,<b>c</b>,<b>e</b>) Cauchy and (<b>b</b>,<b>d</b>,<b>f</b>) Gaussian white noise, and in red color for the corresponding (‘level’-type) deformation.</p> Full article ">Figure 9
<p>(<b>a</b>,<b>b</b>) Shannon entropy, (<b>c</b>,<b>d</b>) Rényi entropy of order <math display="inline"><semantics><mrow><mi>q</mi><mo>=</mo><mn>3</mn></mrow></semantics></math>, and (<b>e</b>,<b>f</b>) Tsallis entropy of order <math display="inline"><semantics><mrow><mi>q</mi><mo>=</mo><mn>3</mn></mrow></semantics></math>, displayed in blue color for original signal generated from the ARIMA(1,2,1) model obtained by double integration of model (9) with (<b>a</b>,<b>c</b>,<b>e</b>) Cauchy and (<b>b</b>,<b>d</b>,<b>f</b>) Gaussian white noise, and in red color for the corresponding (‘flow’-type) deformation.</p> Full article ">Figure 10
<p>Seismic signal of L’Aquila earthquake (6 April 2009).</p> Full article ">Figure 11
<p>From top to bottom: (<b>a</b>) segment 1, (<b>b</b>) segment 2 of seismic signal; correspondingly in left and right columns, (<b>c</b>,<b>d</b>) scalogram <math display="inline"><semantics><mrow><msubsup><mi>W</mi><mi>x</mi><mn>2</mn></msubsup><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow></mrow></semantics></math>, (<b>e</b>,<b>f</b>) <math display="inline"><semantics><mrow><mi>L</mi><mi>I</mi><msub><mi>M</mi><mi>x</mi></msub></mrow></semantics></math> map, (<b>g</b>,<b>h</b>) threshold exceedance set for <math display="inline"><semantics><mrow><mi>L</mi><mi>I</mi><msubsup><mi>M</mi><mi>x</mi><mn>2</mn></msubsup><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow><mo>&gt;</mo><mn>3</mn></mrow></semantics></math>, and (<b>i</b>,<b>j</b>) <math display="inline"><semantics><msub><mi>F</mi><mi>x</mi></msub></semantics></math> curve.</p> Full article ">Figure 12
<p>From top to bottom: (<b>a</b>) segment 1 of seismic signal, and (<b>b</b>) its (‘level’-type) deformation; correspondingly in left and right columns, (<b>c</b>,<b>d</b>) scalogram <math display="inline"><semantics><mrow><msubsup><mi>W</mi><mi>x</mi><mn>2</mn></msubsup><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow></mrow></semantics></math>, (<b>e</b>,<b>f</b>) <math display="inline"><semantics><mrow><mi>L</mi><mi>I</mi><msub><mi>M</mi><mi>x</mi></msub></mrow></semantics></math> map, (<b>g</b>,<b>h</b>) threshold exceedance set for <math display="inline"><semantics><mrow><mi>L</mi><mi>I</mi><msubsup><mi>M</mi><mi>x</mi><mn>2</mn></msubsup><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow><mo>&gt;</mo><mn>3</mn></mrow></semantics></math>, and (<b>i</b>,<b>j</b>) <math display="inline"><semantics><msub><mi>F</mi><mi>x</mi></msub></semantics></math> curve.</p> Full article ">Figure 13
<p>From top to bottom: (<b>a</b>) segment 2 of seismic signal, and (<b>b</b>) its (‘level’-type) deformation; correspondingly in left and right columns, (<b>c</b>,<b>d</b>) scalogram <math display="inline"><semantics><mrow><msubsup><mi>W</mi><mi>x</mi><mn>2</mn></msubsup><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow></mrow></semantics></math>, (<b>e</b>,<b>f</b>) <math display="inline"><semantics><mrow><mi>L</mi><mi>I</mi><msub><mi>M</mi><mi>x</mi></msub></mrow></semantics></math> map, (<b>g</b>,<b>h</b>) threshold exceedance set for <math display="inline"><semantics><mrow><mi>L</mi><mi>I</mi><msubsup><mi>M</mi><mi>x</mi><mn>2</mn></msubsup><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow><mo>&gt;</mo><mn>3</mn></mrow></semantics></math>, and (<b>i</b>,<b>j</b>) <math display="inline"><semantics><msub><mi>F</mi><mi>x</mi></msub></semantics></math> curve.</p> Full article ">Figure 14
<p>(<b>a</b>,<b>b</b>) Shannon entropy, (<b>c</b>,<b>d</b>) Rényi entropy of order <math display="inline"><semantics><mrow><mi>q</mi><mo>=</mo><mn>3</mn></mrow></semantics></math>, and (<b>e</b>,<b>f</b>) Tsallis entropy of order <math display="inline"><semantics><mrow><mi>q</mi><mo>=</mo><mn>3</mn></mrow></semantics></math>, displayed in blue color for original seismic signal with (<b>a</b>,<b>c</b>,<b>e</b>) segment 1 and (<b>b</b>,<b>d</b>,<b>f</b>) segment 2, and in red color for the corresponding (‘level’-type) deformation.</p> Full article ">Figure 15
<p>Rényi-based generalized complexity under selected <math display="inline"><semantics><mrow><mo>(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>)</mo></mrow></semantics></math> values, for (<b>a</b>,<b>b</b>) original seismic signal <span class="html-italic">x</span>, (<b>a</b>) segment 1 and (<b>b</b>) segment 2, and (<b>c</b>,<b>d</b>) for the corresponding (‘level’-type) deformation <math display="inline"><semantics><mrow><mi>x</mi><mo>[</mo><mo>Φ</mo><mo>]</mo></mrow></semantics></math>.</p> Full article ">Figure 15 Cont.
<p>Rényi-based generalized complexity under selected <math display="inline"><semantics><mrow><mo>(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>)</mo></mrow></semantics></math> values, for (<b>a</b>,<b>b</b>) original seismic signal <span class="html-italic">x</span>, (<b>a</b>) segment 1 and (<b>b</b>) segment 2, and (<b>c</b>,<b>d</b>) for the corresponding (‘level’-type) deformation <math display="inline"><semantics><mrow><mi>x</mi><mo>[</mo><mo>Φ</mo><mo>]</mo></mrow></semantics></math>.</p> Full article ">
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