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Symmetry
Volume 12
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Symmetry, Volume 12, Issue 12 (December 2020) – 188 articles

Cover Story (view full-size image): The sky is always a source of wonders, as in the case of this beautiful sunset. The picture was taken in Milano - Italy - in 2017. The clouds illuminated by the twilight create a texture which resembles the supposed quantum structure of spacetime. The sky thank to cosmic messengers can furnish probes to investigate this background nature. In a multimessenger approach ultrahigh energy cosmic rays can be one of these promising means to probe the possible violation of standard physics symmetries in a Lorentz invariance violation context, induced by quantum gravity effects as predicted in our theory. View this paper.
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2 pages, 190 KiB  
Correction
Correction: Cohl, H.S.; Costas-Santos, R.S.; Ge, L. Terminating Basic Hypergeometric Representations and Transformations for the Askey–Wilson Polynomials Symmetry 2020, 12, 1290
by Howard S. Cohl, Roberto S. Costas-Santos and Linus Ge
Symmetry 2020, 12(12), 2120; https://doi.org/10.3390/sym12122120 - 21 Dec 2020
Cited by 1 | Viewed by 1618
Abstract
The authors wish to make the following corrections to their paper [...] Full article
(This article belongs to the Special Issue Symmetry in Special Functions and Orthogonal Polynomials)
18 pages, 6064 KiB  
Article
ICONet: A Lightweight Network with Greater Environmental Adaptivity
by Wei He, Yanmei Huang, Zanhao Fu and Yingcheng Lin
Symmetry 2020, 12(12), 2119; https://doi.org/10.3390/sym12122119 - 21 Dec 2020
Cited by 3 | Viewed by 2723
Abstract
With the increasing popularity of artificial intelligence, deep learning has been applied to various fields, especially in computer vision. Since artificial intelligence is migrating from cloud to edge, deep learning nowadays should be edge-oriented and adaptive to complex environments. Aiming at these goals, [...] Read more.
With the increasing popularity of artificial intelligence, deep learning has been applied to various fields, especially in computer vision. Since artificial intelligence is migrating from cloud to edge, deep learning nowadays should be edge-oriented and adaptive to complex environments. Aiming at these goals, this paper proposes an ICONet (illumination condition optimized network). Based on OTSU segmentation algorithm and fuzzy c-means clustering algorithm, the illumination condition classification subnet increases the environmental adaptivity of our network. The reduced time complexity and optimized size of our convolutional neural network (CNN) model enables the implementation of ICONet on edge devices. In the field of fatigue driving, we test the performance of ICONet on YawDD and self-collected datasets. Our network achieves a general accuracy of 98.56% and our models are about 590 kilobytes. Compared to other proposed networks, the ICONet shows significant success and superiority. Applying ICONet to fatigue driving detection is helpful to solve the symmetry of the needs of edge-oriented detection under complex illumination condition environments and the scarcity of related approaches. Full article
(This article belongs to the Section Computer)
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<p>The schematic of the illumination condition optimized network (ICONet).</p> Full article ">Figure 2
<p>The structure of the proposed convolutional neural network.</p> Full article ">Figure 3
<p>Work of frequency queue.</p> Full article ">Figure 4
<p>The distribution of related parameters used for illumination condition classification. (<b>a</b>) Average gray value; (<b>b</b>) OTSU threshold value; (<b>c</b>) the ratio of OTSU threshold value and average gray value; (<b>d</b>) minimum of the fuzzy c-means clustering algorithm target function.</p> Full article ">Figure 5
<p>Face detection and extraction of the region of interest.</p> Full article ">Figure 6
<p>The test accuracies of all the involved convolutional neural network (CNN) models in ICONet.</p> Full article ">Figure 7
<p>The test loss of all the involved CNN models in ICONet.</p> Full article ">Figure 8
<p>The model test accuracy comparison on the CEW (Closed Eyes in the Wild) dataset.</p> Full article ">Figure 9
<p>Model accuracies comparison among ICONet and the other two approaches.</p> Full article ">Figure 10
<p>The comprehensive judgement of fatigue driving.</p> Full article ">Figure 11
<p>The changing eye and mouth results when a yawn occurs. (<b>a</b>) The changing eye result; (<b>b</b>) the changing mouth result.</p> Full article ">Figure 12
<p>Fatigue driving detection under normal daylight.</p> Full article ">
25 pages, 3374 KiB  
Article
Major Depression and Brain Asymmetry in a Decision-Making Task with Negative and Positive Feedback
by Almira Kustubayeva, Altyngul Kamzanova, Sandugash Kudaibergenova, Veronika Pivkina and Gerald Matthews
Symmetry 2020, 12(12), 2118; https://doi.org/10.3390/sym12122118 - 21 Dec 2020
Cited by 12 | Viewed by 4976
Abstract
Depressed patients are characterized by hypoactivity of the left and hyperactivity of the right frontal areas during the resting state. Depression is also associated with impaired decision-making, which reflects multiple cognitive, affective, and attentional processes, some of which may be lateralized. The aim [...] Read more.
Depressed patients are characterized by hypoactivity of the left and hyperactivity of the right frontal areas during the resting state. Depression is also associated with impaired decision-making, which reflects multiple cognitive, affective, and attentional processes, some of which may be lateralized. The aim of this study was to investigate brain asymmetry during a decision-making task performed in negative and positive feedback conditions in patients with Major Depressive Disorder (MDD) in comparison to healthy control participants. The electroencephalogram (EEG) was recorded from 60 MDD patients and 60 healthy participants while performing a multi-stage decision-making task. Frontal, central, and parietal alpha asymmetry were analyzed with EEGlab/ERPlab software. Evoked potential responses (ERPs) showed general lateralization suggestive of an initial right dominance developing into a more complex pattern of asymmetry across different scalp areas as information was processed. The MDD group showed impaired mood prior to performance, and decreased confidence during performance in comparison to the control group. The resting state frontal alpha asymmetry showed lateralization in the healthy group only. Task-induced alpha power and ERP P100 and P300 amplitudes were more informative biomarkers of depression during decision making. Asymmetry coefficients based on task alpha power and ERP amplitudes showed consistency in the dynamical changes during the decision-making stages. Depression was characterized by a lack of left dominance during the resting state and left hypoactivity during the task baseline and subsequent decision-making process. Findings add to understanding of the functional significance of lateralized brain processes in depression. Full article
(This article belongs to the Special Issue Biological Psychology: Brain Asymmetry and Behavioral Brain)
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<p>Decision-making task scheme and stimulus sequence in two feedback conditions. English statements above the final slides are translated from the original Russian.</p> Full article ">Figure 2
<p>DSSQ scores before experiment: BEA—Energetic Arousal, BTA—Tense Arousal; BHT—Hedonic Tone; BAF—Anger/Frustration. MDD group in blue, Hth group in red. * <span class="html-italic">p</span> &lt; 0.05.</p> Full article ">Figure 3
<p>(<b>a</b>) ERPs for P300 to Start at P<sub>3</sub> (black line) and P<sub>4</sub> (red line) electrodes. (<b>b</b>) 2-d map of the mean P<sub>300</sub> amplitude between 200–500 ms. *** <span class="html-italic">p &lt;</span> 0.001.</p> Full article ">Figure 4
<p>(<b>a</b>) ERPs for P<sub>100</sub> to Hazard (F<sub>3</sub> black line and F<sub>4</sub> red line), upper) and Benefit (C<sub>3</sub> black line and C<sub>4</sub> red line, lower). (<b>b</b>) 2-d maps of the mean P<sub>100</sub> amplitude between 50–150 ms. ** <span class="html-italic">p</span> &lt; 0.01.</p> Full article ">Figure 5
<p>(<b>a</b>) ERPs for P<sub>100</sub> and P<sub>300</sub> to Choice on F<sub>3</sub> (black line) and F<sub>4</sub> (red line) electrodes (left dominance) and for P<sub>100</sub> to Choice on P<sub>3</sub> (black line) and P<sub>4</sub> (red line) electrodes (right dominance), * <span class="html-italic">p &lt;</span> 0.05. Right upper: 2-d map of mean P<sub>100</sub> amplitude. (<b>b</b>) lower: 2-d map of mean P<sub>300</sub> amplitude.</p> Full article ">Figure 6
<p>(<b>a</b>) ERPs for P<sub>300</sub> on feedback on P<sub>3</sub> (black line) and P<sub>4</sub> (red line) electrodes. (<b>b</b>) 2-d map of mean P<sub>300</sub> amplitude, ** <span class="html-italic">p &lt;</span> 0.01, *** <span class="html-italic">p &lt;</span> 0.001.</p> Full article ">Figure 7
<p>2-d maps of for P<sub>300</sub> amplitude to Start and Hazard 300 ms after stimulus onset; and P<sub>100</sub> to Benefit and Choice in 100 ms after stimulus onset in both feedback conditions: negative and positive.</p> Full article ">Figure 8
<p>ERPs 2-d maps of the amplitude for P<sub>300</sub> to Start (500 ms) and Hazard (400 ms) in MDD and Healthy groups.</p> Full article ">Figure 9
<p>Alpha power spectrum to ‘feedback’ between 8–13 Hz in MDD and Healthy groups and difference between groups (Hth-MDD).</p> Full article ">
22 pages, 5784 KiB  
Article
Multi-Level P2P Traffic Classification Using Heuristic and Statistical-Based Techniques: A Hybrid Approach
by Max Bhatia, Vikrant Sharma, Parminder Singh and Mehedi Masud
Symmetry 2020, 12(12), 2117; https://doi.org/10.3390/sym12122117 - 20 Dec 2020
Cited by 29 | Viewed by 3119
Abstract
Peer-to-peer (P2P) applications have been popular among users for more than a decade. They consume a lot of network bandwidth, due to the fact that network administrators face several issues such as congestion, security, managing resources, etc. Hence, its accurate classification will allow [...] Read more.
Peer-to-peer (P2P) applications have been popular among users for more than a decade. They consume a lot of network bandwidth, due to the fact that network administrators face several issues such as congestion, security, managing resources, etc. Hence, its accurate classification will allow them to maintain a Quality of Service for various applications. Conventional classification techniques, i.e., port-based and payload-based techniques alone, have proved ineffective in accurately classifying P2P traffic as they possess significant limitations. As new P2P applications keep emerging and existing applications change their communication patterns, a single classification approach may not be sufficient to classify P2P traffic with high accuracy. Therefore, a multi-level P2P traffic classification technique is proposed in this paper, which utilizes the benefits of both heuristic and statistical-based techniques. By analyzing the behavior of various P2P applications, some heuristic rules have been proposed to classify P2P traffic. The traffic which remains unclassified as P2P undergoes further analysis, where statistical-features of traffic are used with the C4.5 decision tree for P2P classification. The proposed technique classifies P2P traffic with high accuracy (i.e., 98.30%), works with both TCP and UDP traffic, and is not affected even if the traffic is encrypted. Full article
(This article belongs to the Section Computer)
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<p>Controlling the quality of service.</p> Full article ">Figure 2
<p>Multi-level P2P traffic classification technique.</p> Full article ">Figure 3
<p>Calculation of the packet hash-key.</p> Full article ">Figure 4
<p>Packet-level classification process (first step).</p> Full article ">Figure 5
<p>Connection pattern of source peers with the destination P2P peer.</p> Full article ">Figure 6
<p>Connection pattern of source P2P peer with the destination peers.</p> Full article ">Figure 7
<p>Flow-level classification process (second step).</p> Full article ">Figure 8
<p>Classification performance of the proposed hybrid technique.</p> Full article ">Figure 9
<p>Accuracy comparison of various hybrid P2P traffic classification techniques.</p> Full article ">Figure 10
<p>Accuracy comparison of proposed hybrid technique with existing non-hybrid techniques.</p> Full article ">
22 pages, 16219 KiB  
Article
Hypersonic Imaging and Emission Spectroscopy of Hydrogen and Cyanide Following Laser-Induced Optical Breakdown
by Christian G. Parigger, Christopher M. Helstern and Ghaneshwar Gautam
Symmetry 2020, 12(12), 2116; https://doi.org/10.3390/sym12122116 - 19 Dec 2020
Cited by 3 | Viewed by 2703
Abstract
This work communicates the connection of measured shadowgraphs from optically induced air breakdown with emission spectroscopy in selected gas mixtures. Laser-induced optical breakdown is generated using 850 and 170 mJ, 6 ns pulses at a wavelength of 1064 nm, the shadowgraphs are recorded [...] Read more.
This work communicates the connection of measured shadowgraphs from optically induced air breakdown with emission spectroscopy in selected gas mixtures. Laser-induced optical breakdown is generated using 850 and 170 mJ, 6 ns pulses at a wavelength of 1064 nm, the shadowgraphs are recorded using time-delayed 5 ns pulses at a wavelength of 532 nm and a digital camera, and emission spectra are recorded for typically a dozen of discrete time-delays from optical breakdown by employing an intensified charge-coupled device. The symmetry of the breakdown event can be viewed as close-to spherical symmetry for time-delays of several 100 ns. Spectroscopic analysis explores well-above hypersonic expansion dynamics using primarily the diatomic molecule cyanide and atomic hydrogen emission spectroscopy. Analysis of the air breakdown and selected gas breakdown events permits the use of Abel inversion for inference of the expanding species distribution. Typically, species are prevalent at higher density near the hypersonically expanding shockwave, measured by tracing cyanide and a specific carbon atomic line. Overall, recorded air breakdown shadowgraphs are indicative of laser-plasma expansion in selected gas mixtures, and optical spectroscopy delivers analytical insight into plasma expansion phenomena. Full article
(This article belongs to the Special Issue Atomic Processes in Plasmas and Gases: Symmetries and Beyond)
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<p>Schematic of the apparatus used for the shadowgraph experiments.</p> Full article ">Figure 2
<p>Photograph of the experimental arrangement for air breakdown.</p> Full article ">Figure 3
<p>Schematic of the apparatus used for the laser-induced breakdown experiments.</p> Full article ">Figure 4
<p>Typical result of the peakfit.m script applied to measured line-of-sight Δv = 0 CN spectra. Individual fitted peaks and the background variation (in green) are added up for the overall fit (in red) to the experimental data (dotted, in blue).</p> Full article ">Figure 5
<p>Line-of-sight geometry and Abel inversion method.</p> Full article ">Figure 6
<p>Single-shot shadowgraphs of expanding laser-induced plasma initiated with 170 mJ, 6 ns, 1064 nm pulses, and imaged with 5 ns, 532 nm backlight, time-delayed by (<b>a</b>) 200 ns; (<b>b</b>) 1200; (<b>c</b>) 2200 ns; (<b>d</b>) 4200 ns.</p> Full article ">Figure 6 Cont.
<p>Single-shot shadowgraphs of expanding laser-induced plasma initiated with 170 mJ, 6 ns, 1064 nm pulses, and imaged with 5 ns, 532 nm backlight, time-delayed by (<b>a</b>) 200 ns; (<b>b</b>) 1200; (<b>c</b>) 2200 ns; (<b>d</b>) 4200 ns.</p> Full article ">Figure 7
<p>Shadowgraphs subsequent to laser-plasma generation with 850 mJ, 6 ns, 1064 nm pulses. Time delays: (<b>a</b>) 25 ns; (<b>b</b>) 50 ns; (<b>c</b>) 100 ns; (<b>d</b>) 200 ns; (<b>e</b>) 400 ns; (<b>f</b>) 600 ns.</p> Full article ">Figure 7 Cont.
<p>Shadowgraphs subsequent to laser-plasma generation with 850 mJ, 6 ns, 1064 nm pulses. Time delays: (<b>a</b>) 25 ns; (<b>b</b>) 50 ns; (<b>c</b>) 100 ns; (<b>d</b>) 200 ns; (<b>e</b>) 400 ns; (<b>f</b>) 600 ns.</p> Full article ">Figure 8
<p>Shadowgraphs captured after laser-plasma generation with 850 mJ, 6 ns, 1064 nm pulses. Time delays: (<b>a</b>) 800 ns; (<b>b</b>) 1000 ns; (<b>c</b>) 1500 ns; (<b>d</b>) 2000 ns; (<b>e</b>) 3000 ns; (<b>f</b>) 4000 ns.</p> Full article ">Figure 8 Cont.
<p>Shadowgraphs captured after laser-plasma generation with 850 mJ, 6 ns, 1064 nm pulses. Time delays: (<b>a</b>) 800 ns; (<b>b</b>) 1000 ns; (<b>c</b>) 1500 ns; (<b>d</b>) 2000 ns; (<b>e</b>) 3000 ns; (<b>f</b>) 4000 ns.</p> Full article ">Figure 9
<p>Optical breakdown CN spectra in a 1:1 molar CO<sub>2</sub>:N<sub>2</sub> gas mixture held at atmospheric pressure for time delays of (<b>a</b>) 200 ns, (<b>b</b>) 450 ns, (<b>c</b>) 700 ns, and (<b>d</b>) 950 ns. Spectrometer-detector gate width: 125 ns. <span class="html-italic">*</span>, second-order atomic carbon line [<a href="#B5-symmetry-12-02116" class="html-bibr">5</a>].</p> Full article ">Figure 9 Cont.
<p>Optical breakdown CN spectra in a 1:1 molar CO<sub>2</sub>:N<sub>2</sub> gas mixture held at atmospheric pressure for time delays of (<b>a</b>) 200 ns, (<b>b</b>) 450 ns, (<b>c</b>) 700 ns, and (<b>d</b>) 950 ns. Spectrometer-detector gate width: 125 ns. <span class="html-italic">*</span>, second-order atomic carbon line [<a href="#B5-symmetry-12-02116" class="html-bibr">5</a>].</p> Full article ">Figure 10
<p>Log-log plot of shock wave expansion measured perpendicular to the laser-propagation direction when using 850 mJ, 6 ns, 1064 nm pulses for optical breakdown in laboratory air [<a href="#B4-symmetry-12-02116" class="html-bibr">4</a>].</p> Full article ">Figure 11
<p>Hydrogen alpha plasma spectra images at 10 ns (left) and 15 ns (right) time delays. The red arrow indicates the measured plasma width [<a href="#B4-symmetry-12-02116" class="html-bibr">4</a>].</p> Full article ">Figure 12
<p>Plasma expansion speeds. The indicated time-delay error bars are due to the gate width of 5 ns [<a href="#B4-symmetry-12-02116" class="html-bibr">4</a>].</p> Full article ">Figure 13
<p>Inferred widths and calculated electron densities of C I 193.09 nm atomic carbon line in 2nd order vs. slit height for 1:1 molar CO<sub>2</sub>:N<sub>2</sub> gas mixture held at atmospheric pressure. Time delays: (<b>a</b>,<b>b</b>) 450 ns, and (<b>c</b>,<b>d</b>) 950 ns.</p> Full article ">Figure 14
<p>Temperature vs. slit height for filtered line-of-sight CN spectra for fixed volume of 1:1 molar CO<sub>2</sub>:N<sub>2</sub> gaseous mixture held at atmospheric pressure. Time delays: (<b>a</b>) 450 ns; (<b>b</b>) 950 ns.</p> Full article ">Figure 15
<p>Abel inverted CN spectra 1:1 molar CO<sub>2</sub>:N<sub>2</sub> gaseous mixture held at atmospheric pressure. Time delay: (<b>a</b>) 200 ns; (<b>b</b>) 450 ns; (<b>c</b>) 950 ns.</p> Full article ">Figure 16
<p>Abel inverted CN spectra 1:1 molar CO<sub>2</sub>:N<sub>2</sub> gaseous mixture held at atmospheric pressure. Time delay: (<b>a</b>) 1200 ns; (<b>b</b>) 1700 ns; (<b>c</b>) 2200 ns.</p> Full article ">
14 pages, 5016 KiB  
Article
Construction Method and Performance Analysis of Chaotic S-Box Based on a Memorable Simulated Annealing Algorithm
by Juan Wang, Yangqing Zhu, Chao Zhou and Zhiming Qi
Symmetry 2020, 12(12), 2115; https://doi.org/10.3390/sym12122115 - 19 Dec 2020
Cited by 13 | Viewed by 2213
Abstract
The substitution box (S-box) is the only nonlinear components in the symmetric block cipher. Its performance directly determines the security strength of the block cipher. With the dynamic characteristics degradation and the local periodic phenomenon of digital chaos, and the security problems caused [...] Read more.
The substitution box (S-box) is the only nonlinear components in the symmetric block cipher. Its performance directly determines the security strength of the block cipher. With the dynamic characteristics degradation and the local periodic phenomenon of digital chaos, and the security problems caused by them becoming more and more prominent, how to efficiently generate an S-box with security guarantee based on chaos has gradually attracted the attention of cryptographers. In this paper, a chaotic S-box construction method is proposed based on a memorable simulated annealing algorithm (MSAA). The chaotic S-box set is produced by using the nonlinearity and randomness of the dynamic iteration of digital cascaded chaotic mapping. The composite objective function is constructed based on the analysis of the performance indexes of S-box. The MSAA is used to efficiently optimize the S-box set. The matrix segmentation and scrambling operations are carried out on the optimized S-box. The cryptographic performance of chaotic S-box is tested and analyzed, and compared with the mainstream chaotic S-box of the same kind. The results show that the S-box constructed in this paper can not only stably and efficiently generate chaotic S-box with better performance, but also make an effective exploration of the construction of chaotic S-boxes based on intelligent algorithms. Full article
(This article belongs to the Section Mathematics)
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<p>Flowchart of MSAA.</p> Full article ">Figure 2
<p>Construction of chaotic S-Box based on MSAA.</p> Full article ">Figure 3
<p>Matrix segmentation and scrambling operations.</p> Full article ">
26 pages, 7203 KiB  
Article
Modulated Viscosity-Dependent Parameters for MHD Blood Flow in Microvessels Containing Oxytactic Microorganisms and Nanoparticles
by M. A. Elogail and Kh. S. Mekheimer
Symmetry 2020, 12(12), 2114; https://doi.org/10.3390/sym12122114 - 19 Dec 2020
Cited by 20 | Viewed by 3010
Abstract
This work’s primary purpose is to implement a numerical study that simulates blood flow through a microvessel involving oxytactic microorganisms and nanoparticles. The oxytactic microorganisms exhibit negative chemotaxis to gradients of oxygen (oxygen repellents). These microorganisms are to batter infected hypoxic tumor cells [...] Read more.
This work’s primary purpose is to implement a numerical study that simulates blood flow through a microvessel involving oxytactic microorganisms and nanoparticles. The oxytactic microorganisms exhibit negative chemotaxis to gradients of oxygen (oxygen repellents). These microorganisms are to batter infected hypoxic tumor cells as drug-carriers. The viscosity of blood is to vary with temperature, shear-thinning, and nanoparticle concentration. We have formulated a mathematical model then simplified it under assumptions of long wavelength and low Reynold’s number. The resulting non-linear coupled differential equation system is solved numerically with the MATHEMATICA software aid using the built-in command (ParametricNDSolve). This study treated all non-dimensional parameters defined in terms of viscosity to be variables (VP-Model), unlike some previous literature attempts that have considered these parameters mentioned above as constants (CP-Model). The achieved results assured the reliability of the (VP-Model) over the (CP-Model). Our results reveal that temperature and microorganism density increase with the thermophoresis parameter. The impact of increasing the Brownian motion parameter is to increase temperature and lessen microorganism density. Outcomes also indicate an enhancement in the microorganism density towards the hypoxic tumor regions located aside the microvessel walls by boosting oxygen concentrations in the streamflow. The current study is believed to provide further opportunities to improve drug-carrier applications in hypoxic tumor regions by better recognizing the flow features, heat, and mass transfer in such zones. Full article
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<p>A schematic representation of blood in a microvessel containing microorganisms and nanoparticles.</p> Full article ">Figure 2
<p><math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>a</mi> </mstyle> <mo>)</mo> </mrow> </mrow> </semantics></math> Influence of <math display="inline"><semantics> <mi>β</mi> </semantics></math> on <math display="inline"><semantics> <mi>θ</mi> </semantics></math>. <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>b</mi> </mstyle> <mo>)</mo> </mrow> </mrow> </semantics></math> Influence of <math display="inline"><semantics> <mi>β</mi> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>θ</mi> <mo>.</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>c</mi> </mstyle> <mo>)</mo> </mrow> </mrow> </semantics></math> Influence of <math display="inline"><semantics> <mi>β</mi> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>φ</mi> <mo>.</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>d</mi> </mstyle> <mo>)</mo> </mrow> </mrow> </semantics></math> Influence of <math display="inline"><semantics> <mi>β</mi> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>φ</mi> <mo>.</mo> </mrow> </semantics></math></p> Full article ">Figure 3
<p><math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>a</mi> </mstyle> <mo>)</mo> </mrow> </mrow> </semantics></math> Influence of <math display="inline"><semantics> <mrow> <msub> <mi>ρ</mi> <mrow> <mi>e</mi> <mi>v</mi> </mrow> </msub> </mrow> </semantics></math> on <math display="inline"><semantics> <mi>Ω</mi> </semantics></math>. <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>b</mi> </mstyle> <mo>)</mo> </mrow> </mrow> </semantics></math> Influence of <math display="inline"><semantics> <mrow> <msub> <mi>σ</mi> <mrow> <mn>1</mn> <mo> </mo> </mrow> </msub> </mrow> </semantics></math>on <math display="inline"><semantics> <mrow> <mi>Ω</mi> <mo>.</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>c</mi> </mstyle> <mo>)</mo> </mrow> </mrow> </semantics></math> Influence of <math display="inline"><semantics> <mrow> <msub> <mi>λ</mi> <mn>1</mn> </msub> </mrow> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>Ω</mi> <mo>.</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>d</mi> </mstyle> <mo>)</mo> </mrow> </mrow> </semantics></math> Influence of <math display="inline"><semantics> <mrow> <msub> <mi>λ</mi> <mn>1</mn> </msub> </mrow> </semantics></math> on <math display="inline"><semantics> <mrow> <msup> <mi>Ω</mi> <mo>′</mo> </msup> <mo>.</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>e</mi> </mstyle> <mo>)</mo> </mrow> </mrow> </semantics></math> Influence of <math display="inline"><semantics> <mrow> <msub> <mi>λ</mi> <mn>2</mn> </msub> </mrow> </semantics></math> on <math display="inline"><semantics> <mi>Ω</mi> </semantics></math>. <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>f</mi> </mstyle> <mo>)</mo> </mrow> </mrow> </semantics></math> Influence of <math display="inline"><semantics> <mrow> <msub> <mi>λ</mi> <mn>2</mn> </msub> </mrow> </semantics></math> on <math display="inline"><semantics> <mrow> <msup> <mi>Ω</mi> <mo>′</mo> </msup> <mo>.</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>g</mi> </mstyle> <mo>)</mo> </mrow> </mrow> </semantics></math> Influence of <math display="inline"><semantics> <mrow> <msub> <mi>R</mi> <mrow> <mi>b</mi> <mi>v</mi> </mrow> </msub> </mrow> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>θ</mi> <mo>.</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>h</mi> </mstyle> <mo>)</mo> </mrow> </mrow> </semantics></math> Influence of<math display="inline"><semantics> <mrow> <mo> </mo> <msub> <mi>R</mi> <mrow> <mi>b</mi> <mi>v</mi> </mrow> </msub> </mrow> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>u</mi> <mo>.</mo> </mrow> </semantics></math></p> Full article ">Figure 4
<p><math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>a</mi> </mstyle> <mo>)</mo> </mrow> </mrow> </semantics></math> Influence of <math display="inline"><semantics> <mrow> <msub> <mi>M</mi> <mi>v</mi> </msub> </mrow> </semantics></math> on <math display="inline"><semantics> <mi>θ</mi> </semantics></math>. <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>b</mi> </mstyle> <mo>)</mo> </mrow> </mrow> </semantics></math> Influence of <math display="inline"><semantics> <mrow> <msub> <mi>M</mi> <mi>v</mi> </msub> </mrow> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>φ</mi> <mo>.</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>c</mi> </mstyle> <mo>)</mo> </mrow> </mrow> </semantics></math> Influence of <math display="inline"><semantics> <mi>m</mi> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>θ</mi> <mo>.</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>d</mi> </mstyle> <mo>)</mo> </mrow> </mrow> </semantics></math> Influence of <math display="inline"><semantics> <mrow> <mi>m</mi> <mo> </mo> </mrow> </semantics></math>on <math display="inline"><semantics> <mrow> <mi>φ</mi> <mo>.</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>e</mi> </mstyle> <mo>)</mo> </mrow> </mrow> </semantics></math> Influence of <math display="inline"><semantics> <mrow> <msub> <mi>M</mi> <mi>v</mi> </msub> </mrow> </semantics></math> on <math display="inline"><semantics> <mrow> <mfrac> <mrow> <mi>d</mi> <mi>p</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </semantics></math>. <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>f</mi> </mstyle> <mo>)</mo> </mrow> </mrow> </semantics></math> Influence of <math display="inline"><semantics> <mi>m</mi> </semantics></math> on <math display="inline"><semantics> <mrow> <mfrac> <mrow> <mi>d</mi> <mi>p</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> <mo>.</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>g</mi> </mstyle> <mo>)</mo> </mrow> </mrow> </semantics></math> Influence of <math display="inline"><semantics> <mrow> <msub> <mi>M</mi> <mi>v</mi> </msub> </mrow> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>Ω</mi> <mo>.</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>h</mi> </mstyle> <mo>)</mo> </mrow> </mrow> </semantics></math> Influence of <math display="inline"><semantics> <mi>m</mi> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>Ω</mi> <mo>.</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>i</mi> </mstyle> <mo>)</mo> </mrow> </mrow> </semantics></math> Influence of <math display="inline"><semantics> <mrow> <msub> <mi>M</mi> <mn>0</mn> </msub> </mrow> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>Ω</mi> <mo>.</mo> </mrow> </semantics></math></p> Full article ">Figure 4 Cont.
<p><math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>a</mi> </mstyle> <mo>)</mo> </mrow> </mrow> </semantics></math> Influence of <math display="inline"><semantics> <mrow> <msub> <mi>M</mi> <mi>v</mi> </msub> </mrow> </semantics></math> on <math display="inline"><semantics> <mi>θ</mi> </semantics></math>. <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>b</mi> </mstyle> <mo>)</mo> </mrow> </mrow> </semantics></math> Influence of <math display="inline"><semantics> <mrow> <msub> <mi>M</mi> <mi>v</mi> </msub> </mrow> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>φ</mi> <mo>.</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>c</mi> </mstyle> <mo>)</mo> </mrow> </mrow> </semantics></math> Influence of <math display="inline"><semantics> <mi>m</mi> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>θ</mi> <mo>.</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>d</mi> </mstyle> <mo>)</mo> </mrow> </mrow> </semantics></math> Influence of <math display="inline"><semantics> <mrow> <mi>m</mi> <mo> </mo> </mrow> </semantics></math>on <math display="inline"><semantics> <mrow> <mi>φ</mi> <mo>.</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>e</mi> </mstyle> <mo>)</mo> </mrow> </mrow> </semantics></math> Influence of <math display="inline"><semantics> <mrow> <msub> <mi>M</mi> <mi>v</mi> </msub> </mrow> </semantics></math> on <math display="inline"><semantics> <mrow> <mfrac> <mrow> <mi>d</mi> <mi>p</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </semantics></math>. <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>f</mi> </mstyle> <mo>)</mo> </mrow> </mrow> </semantics></math> Influence of <math display="inline"><semantics> <mi>m</mi> </semantics></math> on <math display="inline"><semantics> <mrow> <mfrac> <mrow> <mi>d</mi> <mi>p</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> <mo>.</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>g</mi> </mstyle> <mo>)</mo> </mrow> </mrow> </semantics></math> Influence of <math display="inline"><semantics> <mrow> <msub> <mi>M</mi> <mi>v</mi> </msub> </mrow> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>Ω</mi> <mo>.</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>h</mi> </mstyle> <mo>)</mo> </mrow> </mrow> </semantics></math> Influence of <math display="inline"><semantics> <mi>m</mi> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>Ω</mi> <mo>.</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>i</mi> </mstyle> <mo>)</mo> </mrow> </mrow> </semantics></math> Influence of <math display="inline"><semantics> <mrow> <msub> <mi>M</mi> <mn>0</mn> </msub> </mrow> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>Ω</mi> <mo>.</mo> </mrow> </semantics></math></p> Full article ">Figure 5
<p>(<b>a</b>) Influence of <math display="inline"><semantics> <mrow> <msub> <mi>M</mi> <mi>v</mi> </msub> </mrow> </semantics></math> on <math display="inline"><semantics> <mi>θ</mi> </semantics></math>. (<b>b</b>) Influence of <math display="inline"><semantics> <mi>m</mi> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>φ</mi> <mo>.</mo> </mrow> </semantics></math> (<b>c</b>) Influence of <math display="inline"><semantics> <mrow> <msub> <mi>M</mi> <mi>v</mi> </msub> <mo> </mo> </mrow> </semantics></math>on <math display="inline"><semantics> <mrow> <mi>θ</mi> <mo>.</mo> </mrow> </semantics></math> (<b>d</b>) Influence of <math display="inline"><semantics> <mi>m</mi> </semantics></math> on <math display="inline"><semantics> <mi>φ</mi> </semantics></math>.</p> Full article ">Figure 5 Cont.
<p>(<b>a</b>) Influence of <math display="inline"><semantics> <mrow> <msub> <mi>M</mi> <mi>v</mi> </msub> </mrow> </semantics></math> on <math display="inline"><semantics> <mi>θ</mi> </semantics></math>. (<b>b</b>) Influence of <math display="inline"><semantics> <mi>m</mi> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>φ</mi> <mo>.</mo> </mrow> </semantics></math> (<b>c</b>) Influence of <math display="inline"><semantics> <mrow> <msub> <mi>M</mi> <mi>v</mi> </msub> <mo> </mo> </mrow> </semantics></math>on <math display="inline"><semantics> <mrow> <mi>θ</mi> <mo>.</mo> </mrow> </semantics></math> (<b>d</b>) Influence of <math display="inline"><semantics> <mi>m</mi> </semantics></math> on <math display="inline"><semantics> <mi>φ</mi> </semantics></math>.</p> Full article ">Figure 6
<p><math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>a</mi> </mstyle> <mo>)</mo> </mrow> </mrow> </semantics></math> Influence of <math display="inline"><semantics> <mrow> <msub> <mi>N</mi> <mrow> <mi>t</mi> <mi>v</mi> </mrow> </msub> </mrow> </semantics></math> on <math display="inline"><semantics> <mrow> <mfrac> <mrow> <mi>d</mi> <mi>p</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </semantics></math>. <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>b</mi> </mstyle> <mo>)</mo> </mrow> </mrow> </semantics></math> Influence of <math display="inline"><semantics> <mrow> <msub> <mi>N</mi> <mrow> <mi>t</mi> <mi>v</mi> </mrow> </msub> </mrow> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>θ</mi> <mo>.</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>c</mi> </mstyle> <mo>)</mo> </mrow> </mrow> </semantics></math> Influence of <math display="inline"><semantics> <mrow> <msub> <mi>N</mi> <mrow> <mi>t</mi> <mi>v</mi> </mrow> </msub> </mrow> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>Ω</mi> <mo>.</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>d</mi> </mstyle> <mo>)</mo> </mrow> </mrow> </semantics></math> Influence of <math display="inline"><semantics> <mrow> <msub> <mi>N</mi> <mrow> <mi>t</mi> <mi>v</mi> </mrow> </msub> <mo> </mo> </mrow> </semantics></math>on <math display="inline"><semantics> <mrow> <mi>φ</mi> <mo>.</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>e</mi> </mstyle> <mo>)</mo> </mrow> </mrow> </semantics></math> Influence of <math display="inline"><semantics> <mrow> <msub> <mi>N</mi> <mrow> <mi>b</mi> <mi>v</mi> </mrow> </msub> </mrow> </semantics></math> on <math display="inline"><semantics> <mrow> <mfrac> <mrow> <mi>d</mi> <mi>p</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </semantics></math>. <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>f</mi> </mstyle> <mo>)</mo> </mrow> </mrow> </semantics></math> Influence of<math display="inline"><semantics> <mrow> <mo> </mo> <msub> <mi>N</mi> <mrow> <mi>b</mi> <mi>v</mi> </mrow> </msub> </mrow> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>Ω</mi> <mo>.</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>g</mi> </mstyle> <mo>)</mo> </mrow> </mrow> </semantics></math> Influence of <math display="inline"><semantics> <mrow> <msub> <mi>N</mi> <mrow> <mi>b</mi> <mi>v</mi> </mrow> </msub> </mrow> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>θ</mi> <mo>.</mo> </mrow> </semantics></math> (h) Influence of <math display="inline"><semantics> <mrow> <msub> <mi>N</mi> <mrow> <mi>b</mi> <mi>v</mi> </mrow> </msub> </mrow> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>φ</mi> <mo>.</mo> </mrow> </semantics></math></p> Full article ">Figure 6 Cont.
<p><math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>a</mi> </mstyle> <mo>)</mo> </mrow> </mrow> </semantics></math> Influence of <math display="inline"><semantics> <mrow> <msub> <mi>N</mi> <mrow> <mi>t</mi> <mi>v</mi> </mrow> </msub> </mrow> </semantics></math> on <math display="inline"><semantics> <mrow> <mfrac> <mrow> <mi>d</mi> <mi>p</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </semantics></math>. <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>b</mi> </mstyle> <mo>)</mo> </mrow> </mrow> </semantics></math> Influence of <math display="inline"><semantics> <mrow> <msub> <mi>N</mi> <mrow> <mi>t</mi> <mi>v</mi> </mrow> </msub> </mrow> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>θ</mi> <mo>.</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>c</mi> </mstyle> <mo>)</mo> </mrow> </mrow> </semantics></math> Influence of <math display="inline"><semantics> <mrow> <msub> <mi>N</mi> <mrow> <mi>t</mi> <mi>v</mi> </mrow> </msub> </mrow> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>Ω</mi> <mo>.</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>d</mi> </mstyle> <mo>)</mo> </mrow> </mrow> </semantics></math> Influence of <math display="inline"><semantics> <mrow> <msub> <mi>N</mi> <mrow> <mi>t</mi> <mi>v</mi> </mrow> </msub> <mo> </mo> </mrow> </semantics></math>on <math display="inline"><semantics> <mrow> <mi>φ</mi> <mo>.</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>e</mi> </mstyle> <mo>)</mo> </mrow> </mrow> </semantics></math> Influence of <math display="inline"><semantics> <mrow> <msub> <mi>N</mi> <mrow> <mi>b</mi> <mi>v</mi> </mrow> </msub> </mrow> </semantics></math> on <math display="inline"><semantics> <mrow> <mfrac> <mrow> <mi>d</mi> <mi>p</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </semantics></math>. <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>f</mi> </mstyle> <mo>)</mo> </mrow> </mrow> </semantics></math> Influence of<math display="inline"><semantics> <mrow> <mo> </mo> <msub> <mi>N</mi> <mrow> <mi>b</mi> <mi>v</mi> </mrow> </msub> </mrow> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>Ω</mi> <mo>.</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>g</mi> </mstyle> <mo>)</mo> </mrow> </mrow> </semantics></math> Influence of <math display="inline"><semantics> <mrow> <msub> <mi>N</mi> <mrow> <mi>b</mi> <mi>v</mi> </mrow> </msub> </mrow> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>θ</mi> <mo>.</mo> </mrow> </semantics></math> (h) Influence of <math display="inline"><semantics> <mrow> <msub> <mi>N</mi> <mrow> <mi>b</mi> <mi>v</mi> </mrow> </msub> </mrow> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>φ</mi> <mo>.</mo> </mrow> </semantics></math></p> Full article ">Figure 7
<p><math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>a</mi> </mstyle> <mo>)</mo> </mrow> </mrow> </semantics></math> Influence of <math display="inline"><semantics> <mi>β</mi> </semantics></math> on <math display="inline"><semantics> <mi>ξ</mi> </semantics></math>. <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>b</mi> </mstyle> <mo>)</mo> </mrow> </mrow> </semantics></math> Influence of <math display="inline"><semantics> <mi>β</mi> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>ξ</mi> <mo>.</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>c</mi> </mstyle> <mo>)</mo> </mrow> </mrow> </semantics></math> Influence of <math display="inline"><semantics> <mi>β</mi> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>Ω</mi> <mo>.</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>d</mi> </mstyle> <mo>)</mo> </mrow> </mrow> </semantics></math> Influence of <math display="inline"><semantics> <mi>β</mi> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>φ</mi> <mo>.</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>e</mi> </mstyle> <mo>)</mo> </mrow> </mrow> </semantics></math> Influence of <math display="inline"><semantics> <mi mathvariant="sans-serif">α</mi> </semantics></math> on <math display="inline"><semantics> <mi>Ω</mi> </semantics></math>. <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>f</mi> </mstyle> <mo>)</mo> </mrow> </mrow> </semantics></math> Influence of <math display="inline"><semantics> <mi>n</mi> </semantics></math> on <math display="inline"><semantics> <mrow> <mfrac> <mrow> <mi>d</mi> <mi>p</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> <mo>.</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>g</mi> </mstyle> <mo>)</mo> </mrow> </mrow> </semantics></math> Influence of <math display="inline"><semantics> <mrow> <mi>W</mi> <mi>e</mi> </mrow> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>u</mi> <mo>.</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>h</mi> </mstyle> <mo>)</mo> </mrow> </mrow> </semantics></math> Influence of <math display="inline"><semantics> <mrow> <mi>W</mi> <mi>e</mi> </mrow> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>u</mi> <mo>.</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>i</mi> </mstyle> <mo>)</mo> </mrow> </mrow> </semantics></math> Influence of <math display="inline"><semantics> <mi>a</mi> </semantics></math> on <math display="inline"><semantics> <mrow> <mfrac> <mrow> <mi>d</mi> <mi>p</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> <mo>.</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>j</mi> </mstyle> <mo>)</mo> </mrow> </mrow> </semantics></math> Influence of <math display="inline"><semantics> <mi>a</mi> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>u</mi> <mo>.</mo> </mrow> </semantics></math></p> Full article ">Figure 7 Cont.
<p><math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>a</mi> </mstyle> <mo>)</mo> </mrow> </mrow> </semantics></math> Influence of <math display="inline"><semantics> <mi>β</mi> </semantics></math> on <math display="inline"><semantics> <mi>ξ</mi> </semantics></math>. <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>b</mi> </mstyle> <mo>)</mo> </mrow> </mrow> </semantics></math> Influence of <math display="inline"><semantics> <mi>β</mi> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>ξ</mi> <mo>.</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>c</mi> </mstyle> <mo>)</mo> </mrow> </mrow> </semantics></math> Influence of <math display="inline"><semantics> <mi>β</mi> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>Ω</mi> <mo>.</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>d</mi> </mstyle> <mo>)</mo> </mrow> </mrow> </semantics></math> Influence of <math display="inline"><semantics> <mi>β</mi> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>φ</mi> <mo>.</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>e</mi> </mstyle> <mo>)</mo> </mrow> </mrow> </semantics></math> Influence of <math display="inline"><semantics> <mi mathvariant="sans-serif">α</mi> </semantics></math> on <math display="inline"><semantics> <mi>Ω</mi> </semantics></math>. <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>f</mi> </mstyle> <mo>)</mo> </mrow> </mrow> </semantics></math> Influence of <math display="inline"><semantics> <mi>n</mi> </semantics></math> on <math display="inline"><semantics> <mrow> <mfrac> <mrow> <mi>d</mi> <mi>p</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> <mo>.</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>g</mi> </mstyle> <mo>)</mo> </mrow> </mrow> </semantics></math> Influence of <math display="inline"><semantics> <mrow> <mi>W</mi> <mi>e</mi> </mrow> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>u</mi> <mo>.</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>h</mi> </mstyle> <mo>)</mo> </mrow> </mrow> </semantics></math> Influence of <math display="inline"><semantics> <mrow> <mi>W</mi> <mi>e</mi> </mrow> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>u</mi> <mo>.</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>i</mi> </mstyle> <mo>)</mo> </mrow> </mrow> </semantics></math> Influence of <math display="inline"><semantics> <mi>a</mi> </semantics></math> on <math display="inline"><semantics> <mrow> <mfrac> <mrow> <mi>d</mi> <mi>p</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> <mo>.</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>j</mi> </mstyle> <mo>)</mo> </mrow> </mrow> </semantics></math> Influence of <math display="inline"><semantics> <mi>a</mi> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>u</mi> <mo>.</mo> </mrow> </semantics></math></p> Full article ">Figure 8
<p><math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>a</mi> </mstyle> <mo>)</mo> </mrow> </mrow> </semantics></math> Influence of<math display="inline"><semantics> <mrow> <mo> </mo> <msub> <mi>R</mi> <mrow> <mi>n</mi> <mi>v</mi> </mrow> </msub> </mrow> </semantics></math> on <math display="inline"><semantics> <mi>φ</mi> </semantics></math>. (<b>b</b>) Influence of<math display="inline"><semantics> <mrow> <mo> </mo> <msub> <mi>R</mi> <mrow> <mi>n</mi> <mi>v</mi> </mrow> </msub> </mrow> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>θ</mi> <mo>.</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>c</mi> </mstyle> <mo>)</mo> </mrow> </mrow> </semantics></math> Influence of<math display="inline"><semantics> <mrow> <mo> </mo> <msub> <mi>ϵ</mi> <mi>v</mi> </msub> <mo> </mo> </mrow> </semantics></math>on <math display="inline"><semantics> <mrow> <mi>φ</mi> <mo>.</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>d</mi> </mstyle> <mo>)</mo> </mrow> </mrow> </semantics></math> Influence of <math display="inline"><semantics> <mrow> <msub> <mi>ϵ</mi> <mi>v</mi> </msub> <mo> </mo> </mrow> </semantics></math>on <math display="inline"><semantics> <mrow> <mi>θ</mi> <mo>.</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>e</mi> </mstyle> <mo>)</mo> </mrow> </mrow> </semantics></math> Influence of<math display="inline"><semantics> <mrow> <mo> </mo> <msub> <mi>G</mi> <mrow> <mi>t</mi> <mi>v</mi> </mrow> </msub> </mrow> </semantics></math> on<math display="inline"><semantics> <mrow> <mo> </mo> <mfrac> <mrow> <mi>d</mi> <mi>p</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </semantics></math>. <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>f</mi> </mstyle> <mo>)</mo> </mrow> </mrow> </semantics></math> Influence of<math display="inline"><semantics> <mrow> <mo> </mo> <mo> </mo> <msub> <mi>G</mi> <mrow> <mi>c</mi> <mi>v</mi> </mrow> </msub> </mrow> </semantics></math> on <math display="inline"><semantics> <mrow> <mfrac> <mrow> <mi>d</mi> <mi>p</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> <mo>.</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>g</mi> </mstyle> <mo>)</mo> </mrow> </mrow> </semantics></math> Influence of<math display="inline"><semantics> <mrow> <mo> </mo> <msub> <mi>ϵ</mi> <mi>v</mi> </msub> </mrow> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>Ω</mi> <mo>.</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>h</mi> </mstyle> <mo>)</mo> </mrow> </mrow> </semantics></math> Influence of<math display="inline"><semantics> <mrow> <mo> </mo> <mi>q</mi> </mrow> </semantics></math> on <math display="inline"><semantics> <mrow> <mfrac> <mrow> <mi>d</mi> <mi>p</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> <mo>.</mo> </mrow> </semantics></math></p> Full article ">Figure 9
<p>Streamline patterns for different values of<math display="inline"><semantics> <mrow> <mo> </mo> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>a</mi> </mstyle> <mo>)</mo> </mrow> <mo> </mo> <mi>β</mi> </mrow> </semantics></math>. <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>b</mi> </mstyle> <mo>)</mo> </mrow> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>.</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>c</mi> </mstyle> <mo>)</mo> </mrow> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <msub> <mi>M</mi> <mi>v</mi> </msub> </mrow> </semantics></math>.</p> Full article ">Figure 9 Cont.
<p>Streamline patterns for different values of<math display="inline"><semantics> <mrow> <mo> </mo> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>a</mi> </mstyle> <mo>)</mo> </mrow> <mo> </mo> <mi>β</mi> </mrow> </semantics></math>. <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>b</mi> </mstyle> <mo>)</mo> </mrow> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>.</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>c</mi> </mstyle> <mo>)</mo> </mrow> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <msub> <mi>M</mi> <mi>v</mi> </msub> </mrow> </semantics></math>.</p> Full article ">
17 pages, 2465 KiB  
Article
The Cauchy Problem for the Generalized Hyperbolic Novikov–Veselov Equation via the Moutard Symmetries
by Alla A. Yurova, Artyom V. Yurov and Valerian A. Yurov
Symmetry 2020, 12(12), 2113; https://doi.org/10.3390/sym12122113 - 19 Dec 2020
Cited by 2 | Viewed by 2118
Abstract
We begin by introducing a new procedure for construction of the exact solutions to Cauchy problem of the real-valued (hyperbolic) Novikov–Veselov equation which is based on the Moutard symmetry. The procedure shown therein utilizes the well-known Airy function Ai(ξ) which [...] Read more.
We begin by introducing a new procedure for construction of the exact solutions to Cauchy problem of the real-valued (hyperbolic) Novikov–Veselov equation which is based on the Moutard symmetry. The procedure shown therein utilizes the well-known Airy function Ai(ξ) which in turn serves as a solution to the ordinary differential equation d2zdξ2=ξz. In the second part of the article we show that the aforementioned procedure can also work for the n-th order generalizations of the Novikov–Veselov equation, provided that one replaces the Airy function with the appropriate solution of the ordinary differential equation dn−1zdξn−1=ξz. Full article
(This article belongs to the Special Issue Symmetry: Feature Papers 2020)
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<p>The dromion-type Cauchy problem (<a href="#FD42-symmetry-12-02113" class="html-disp-formula">42</a>) for <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> with <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>0</mn> </msub> <mo>=</mo> <msub> <mi>y</mi> <mn>0</mn> </msub> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mrow> </semantics></math>. There are two equipotential lines at <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>−</mo> <msub> <mi>y</mi> <mn>0</mn> </msub> <mo>=</mo> <mo>±</mo> <mrow> <mo>(</mo> <mi>x</mi> <mo>−</mo> <msub> <mi>x</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> </mrow> </semantics></math>.</p> Full article ">Figure 2
<p>A sample of localized Cauchy problem (<a href="#FD43-symmetry-12-02113" class="html-disp-formula">43</a>) with <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>0</mn> </msub> <mo>=</mo> <msub> <mi>y</mi> <mn>0</mn> </msub> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mrow> </semantics></math>.</p> Full article ">
12 pages, 851 KiB  
Article
Preferring and Detecting Face Symmetry: Comparing Children and Adults Judging Human and Monkey Faces
by Anthony C. Little and Jack A. F. Griffey
Symmetry 2020, 12(12), 2112; https://doi.org/10.3390/sym12122112 - 19 Dec 2020
Cited by 1 | Viewed by 6153
Abstract
Background: Visual symmetry is often found attractive. Symmetry may be preferred either due to a bias in the visual system or due to evolutionary selection pressures related to partner preference. Simple perceptual bias views predict that symmetry preferences should be similar across types [...] Read more.
Background: Visual symmetry is often found attractive. Symmetry may be preferred either due to a bias in the visual system or due to evolutionary selection pressures related to partner preference. Simple perceptual bias views predict that symmetry preferences should be similar across types of stimuli and unlikely to be related to factors such as age. Methods: The current study examined preferences for symmetry across age groups (pre-puberty vs post-puberty) and stimuli type (human face vs monkey face). Pairs of images manipulated for symmetry were presented and participants asked to choose the image they preferred. Participants repeated the task and were asked to detect symmetry. Results: Both age of observer and stimuli type were associated with symmetry preferences. Older observers had higher preferences for symmetry but preferred it most in human vs monkey stimuli. Across both age groups, symmetry preferences and detection abilities were weakly related. Conclusions: The study supports some ideas from an evolutionary advantage view of symmetry preference, whereby symmetry is expected be higher for potential partners (here human faces) and higher post-puberty when partner choice becomes more relevant. Such potentially motivational based preferences challenge perceptual bias explanations as a sole explanation for symmetry preferences but may occur alongside them. Full article
(This article belongs to the Special Issue Empirical Aesthetics)
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<p>Examples of symmetric (left image of pair) and asymmetric (right image of pair) images for female humans (top left), male humans (top right), female monkeys (bottom left), and male monkeys (bottom right).</p> Full article ">Figure 2
<p>Preferences for symmetric vs. asymmetric images across different stimulus types by age group (+/− 1SEM). Scores greater than 50% indicated that symmetric versions were preferred over asymmetric versions.</p> Full article ">Figure 3
<p>Accurate detection for symmetric vs. asymmetric images across different stimulus types by age group (+/− 1SEM). Scores greater than 50% indicated that symmetric versions were correctly identified as more symmetric.</p> Full article ">
21 pages, 382 KiB  
Article
Recurrent Sequences Play for Survival Probability of Discrete Time Risk Model
by Andrius Grigutis and Jonas Šiaulys
Symmetry 2020, 12(12), 2111; https://doi.org/10.3390/sym12122111 - 18 Dec 2020
Cited by 4 | Viewed by 2331
Abstract
In this article we investigate a homogeneous discrete time risk model with a generalized premium income rate which can be any natural number. We derive theorems and give numerical examples for finite and ultimate time survival probability calculation for the mentioned model. Our [...] Read more.
In this article we investigate a homogeneous discrete time risk model with a generalized premium income rate which can be any natural number. We derive theorems and give numerical examples for finite and ultimate time survival probability calculation for the mentioned model. Our proved statements for ultimate time survival probability calculation, at some level, are similar to the previously known statements for non-homogeneous risk models, where required initial values of survival probability for some recurrent formulas are gathered by certain limit laws. We also give a simplified proof that a ruin is almost unavoidable with a neutral net profit condition and state several conjectures on a certain type of recurrent matrices non-singularity. All the research done can be interpreted as a possibility that symmetric or asymmetric random walk (r.w.) hits (or not) the line u+κt and that possibility is directly related to the expected value of r.w. generating random variable which might be equal, above or bellow κ. Full article
(This article belongs to the Section Mathematics)
22 pages, 2863 KiB  
Review
Implications of Gauge-Free Extended Electrodynamics
by Donald Reed and Lee M. Hively
Symmetry 2020, 12(12), 2110; https://doi.org/10.3390/sym12122110 - 18 Dec 2020
Cited by 7 | Viewed by 3727
Abstract
Recent tests measured an irrotational (curl-free) magnetic vector potential (A) that is contrary to classical electrodynamics (CED). A (irrotational) arises in extended electrodynamics (EED) that is derivable from the Stueckelberg Lagrangian. A (irrotational) implies an irrotational (gradient-driven) electrical current density, J [...] Read more.
Recent tests measured an irrotational (curl-free) magnetic vector potential (A) that is contrary to classical electrodynamics (CED). A (irrotational) arises in extended electrodynamics (EED) that is derivable from the Stueckelberg Lagrangian. A (irrotational) implies an irrotational (gradient-driven) electrical current density, J. Consequently, EED is gauge-free and provably unique. EED predicts a scalar field that equals the quantity usually set to zero as the Lorenz gauge, making A and the scalar potential () independent and physically-measureable fields. EED predicts a scalar-longitudinal wave (SLW) that has an electric field along the direction of propagation together with the scalar field, carrying both energy and momentum. EED also predicts the scalar wave (SW) that carries energy without momentum. EED predicts that the SLW and SW are unconstrained by the skin effect, because neither wave has a magnetic field that generates dissipative eddy currents in electrical conductors. The novel concept of a “gradient-driven” current is a key feature of US Patent 9,306,527 that disclosed antennas for SLW generation and reception. Preliminary experiments have validated the SLW’s no-skin-effect constraint as a potential harbinger of new technologies, a possible explanation for poorly understood laboratory and astrophysical phenomena, and a forerunner of paradigm revolutions. Full article
(This article belongs to the Special Issue Symmetry, Extended Maxwell Equations and Non-local Wavefunctions)
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<p>Circuits for the Maxwell-Lodge effect representation [<a href="#B34-symmetry-12-02110" class="html-bibr">34</a>].</p> Full article ">Figure 2
<p>VPT with secondary circuit coil configurations [<a href="#B36-symmetry-12-02110" class="html-bibr">36</a>].</p> Full article ">Figure 3
<p>Comparison between EED and CED.</p> Full article ">Figure 4
<p>Cross-section of SLW antenna (<b>left</b>) and bifilar-coil-type SLW antenna (<b>right</b>) [<a href="#B68-symmetry-12-02110" class="html-bibr">68</a>].</p> Full article ">Figure 5
<p>Possible registration of solar SLW radiation [<a href="#B100-symmetry-12-02110" class="html-bibr">100</a>].</p> Full article ">Figure 6
<p>Spectral output of radiation as a function of tape angle [<a href="#B103-symmetry-12-02110" class="html-bibr">103</a>].</p> Full article ">
8 pages, 551 KiB  
Article
Planetary Systems and the Hidden Symmetries of the Kepler Problem
by József Cseh
Symmetry 2020, 12(12), 2109; https://doi.org/10.3390/sym12122109 - 18 Dec 2020
Cited by 2 | Viewed by 2329
Abstract
The question of whether the solar distances of the planetary system follow a regular sequence was raised by Kepler more than 400 years ago. He could not prove his expectation, inasmuch as the planetary orbits are not transformed into each other by the [...] Read more.
The question of whether the solar distances of the planetary system follow a regular sequence was raised by Kepler more than 400 years ago. He could not prove his expectation, inasmuch as the planetary orbits are not transformed into each other by the regular polyhedra. In 1989, Barut proposed another relation, which was inspired by the hidden symmetry of the Kepler problem. It was found to be approximately valid for our Solar System. Here, we investigate if exoplanet systems follow this rule. We find that the symmetry-governed sequence is valid in several systems. It is very unlikely that the observed regularity is by chance; therefore, our findings give support to Kepler’s guess, although with a different transformation rule. Full article
(This article belongs to the Special Issue Astronomy and Symmetry)
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<p>The logarithm of the measured (circles, connected by solid lines) and calculated (dots, dashed lines) data of the TRAPPIST-1 [<a href="#B25-symmetry-12-02109" class="html-bibr">25</a>] (<math display="inline"><semantics> <mi>λ</mi> </semantics></math> = 0.139) and Kepler-444 [<a href="#B26-symmetry-12-02109" class="html-bibr">26</a>] (<math display="inline"><semantics> <mi>λ</mi> </semantics></math> = 0.084) planetary systems.</p> Full article ">Figure 2
<p>The same as in <a href="#symmetry-12-02109-f001" class="html-fig">Figure 1</a> for the systems Kepler-20 [<a href="#B27-symmetry-12-02109" class="html-bibr">27</a>] (<math display="inline"><semantics> <mi>λ</mi> </semantics></math> = 0.208) and Kepler-296 [<a href="#B28-symmetry-12-02109" class="html-bibr">28</a>] (<math display="inline"><semantics> <mi>λ</mi> </semantics></math> = 0.240).</p> Full article ">Figure 3
<p>The same as in <a href="#symmetry-12-02109-f001" class="html-fig">Figure 1</a> for the systems HD 10180 [<a href="#B29-symmetry-12-02109" class="html-bibr">29</a>] (<math display="inline"><semantics> <mi>λ</mi> </semantics></math> = 0.395) and HD 40307 [<a href="#B30-symmetry-12-02109" class="html-bibr">30</a>] (<math display="inline"><semantics> <mi>λ</mi> </semantics></math> = 0.311).</p> Full article ">
12 pages, 439 KiB  
Article
Distribution Function, Probability Generating Function and Archimedean Generator
by Weaam Alhadlaq and Abdulhamid Alzaid
Symmetry 2020, 12(12), 2108; https://doi.org/10.3390/sym12122108 - 18 Dec 2020
Cited by 8 | Viewed by 2495
Abstract
Archimedean copulas form a very wide subclass of symmetric copulas. Most of the popular copulas are members of the Archimedean copulas. These copulas are obtained using real functions known as Archimedean generators. In this paper, we observe that under certain conditions the cumulative [...] Read more.
Archimedean copulas form a very wide subclass of symmetric copulas. Most of the popular copulas are members of the Archimedean copulas. These copulas are obtained using real functions known as Archimedean generators. In this paper, we observe that under certain conditions the cumulative distribution functions on (0, 1) and probability generating functions can be used as Archimedean generators. It is shown that most of the well-known Archimedean copulas can be generated using such distributions. Further, we introduced new Archimedean copulas. Full article
(This article belongs to the Section Mathematics)
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<p>Copula 2 with parameters <math display="inline"><semantics> <mrow> <mi>θ</mi> <mo>=</mo> <mn>0.5</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>10</mn> </mrow> </semantics></math>.</p> Full article ">Figure 2
<p>The density of Copula 2 with parameters <math display="inline"><semantics> <mrow> <mi>θ</mi> <mo>=</mo> <mn>0.5</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </semantics></math>.</p> Full article ">Figure 3
<p>Copula 3 with parameter <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math>.</p> Full article ">Figure 4
<p>The density of Copula 3 with parameter <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math>.</p> Full article ">Figure 5
<p>Copula 4 with parameter <math display="inline"><semantics> <mrow> <mi>θ</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>.</p> Full article ">Figure 6
<p>The density of Copula 4 with parameter <math display="inline"><semantics> <mrow> <mi>θ</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>.</p> Full article ">Figure 7
<p>Copula 5 with parameter <math display="inline"><semantics> <mrow> <mi>θ</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>.</p> Full article ">Figure 8
<p>The density of Copula 5 with parameter <math display="inline"><semantics> <mrow> <mi>θ</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>.</p> Full article ">
16 pages, 1100 KiB  
Article
Method for Effectiveness Assessment of Electronic Warfare Systems in Cyberspace
by Seungcheol Choi, Oh-Jin Kwon, Haengrok Oh and Dongkyoo Shin
Symmetry 2020, 12(12), 2107; https://doi.org/10.3390/sym12122107 - 18 Dec 2020
Cited by 7 | Viewed by 9047
Abstract
Current electronic warfare (EW) systems, along with the rapid development of information and communication technology, are essential elements in the modern battlefield associated with cyberspace. In this study, an efficient evaluation framework is proposed to assess the effectiveness of various types of EW [...] Read more.
Current electronic warfare (EW) systems, along with the rapid development of information and communication technology, are essential elements in the modern battlefield associated with cyberspace. In this study, an efficient evaluation framework is proposed to assess the effectiveness of various types of EW systems that operate in cyberspace, which is recognized as an indispensable factor affecting modern military operations. The proposed method classifies EW systems into primary and sub-categories according to EWs’ types and identifies items for the measurement of the effectiveness of each EW system by considering the characteristics of cyberspace for evaluating the damage caused by cyberattacks. A scenario with an integrated EW system incorporating two or more different types of EW equipment is appropriately provided to confirm the effectiveness of the proposed framework in cyber electromagnetic warfare. The scenario explicates an example of assessing the effectiveness of EW systems under cyberattacks. Finally, the proposed method is demonstrated sufficiently by assessing the effectiveness of the EW systems using the scenario. Full article
(This article belongs to the Special Issue Symmetry in Artificial Visual Perception and Its Application)
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<p>Electronic warfare sub-divisions and applications [<a href="#B6-symmetry-12-02107" class="html-bibr">6</a>].</p> Full article ">Figure 2
<p>Overview of the electronic warfare support (ES) test and evaluation (T&amp;E) process [<a href="#B6-symmetry-12-02107" class="html-bibr">6</a>].</p> Full article ">Figure 3
<p>Definition of cyber and electromagnetic activities (CEMA) [<a href="#B8-symmetry-12-02107" class="html-bibr">8</a>].</p> Full article ">Figure 4
<p>Cyber and electromagnetic activities elements [<a href="#B17-symmetry-12-02107" class="html-bibr">17</a>].</p> Full article ">Figure 5
<p>Overview of the effectiveness assessment. (<b>a</b>) Cyber battle damage assessment framework, and (<b>b</b>) workflow of EW damage assessment, which is a sub-system of cyberspace battle damage assessment framework (CBDAF).</p> Full article ">
11 pages, 2453 KiB  
Article
Unified Visual Working Memory without the Anterior Corpus Callosum
by Yair Pinto, Edward H.F. de Haan, Maria-Chiara Villa, Sabrina Siliquini, Gabriele Polonara, Claudia Passamonti, Simona Lattanzi, Nicoletta Foschi and Mara Fabri
Symmetry 2020, 12(12), 2106; https://doi.org/10.3390/sym12122106 - 18 Dec 2020
Cited by 2 | Viewed by 2851
Abstract
One of the most fundamental, and most studied, human cognitive functions is working memory. Yet, it is currently unknown how working memory is unified. In other words, why does a healthy human brain have one integrated capacity of working memory, rather than one [...] Read more.
One of the most fundamental, and most studied, human cognitive functions is working memory. Yet, it is currently unknown how working memory is unified. In other words, why does a healthy human brain have one integrated capacity of working memory, rather than one capacity per visual hemifield, for instance. Thus, healthy subjects can memorize roughly as many items, regardless of whether all items are presented in one hemifield, rather than throughout two visual hemifields. In this current research, we investigated two patients in whom either most, or the entire, corpus callosum has been cut to alleviate otherwise untreatable epilepsy. Crucially, in both patients the anterior parts connecting the frontal and most of the parietal cortices, are entirely removed. This is essential, since it is often posited that working memory resides in these areas of the cortex. We found that despite the lack of direct connections between the frontal cortices in these patients, working memory capacity is similar regardless of whether stimuli are all presented in one visual hemifield or across two visual hemifields. This indicates that in the absence of the anterior parts of the corpus callosum working memory remains unified. Moreover, it is important to note that memory performance was not similar across visual fields. In fact, capacity was higher when items appeared in the left visual hemifield than when they appeared in the right visual hemifield. Visual information in the left hemifield is processed by the right hemisphere and vice versa. Therefore, this indicates that visual working memory is not symmetric, with the right hemisphere having a superior visual working memory. Nonetheless, a (subcortical) bottleneck apparently causes visual working memory to be integrated, such that capacity does not increase when items are presented in two, rather than one, visual hemifield. Full article
(This article belongs to the Special Issue Symmetry and Asymmetry in Brain Behavior and Perception)
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<p>Magnetic Resonance Imaging (MRI) scans of patient MC (<b>left</b>) and DDC (<b>right</b>).</p> Full article ">Figure 2
<p>From top to bottom, an overview of Experiments 1–5. In all of the experiments, visual working memory was tested. MC performed Experiments 1–3 and DDC performed Experiments 4 and 5. “Diverso” indicates different, meaning that the item at the cued location changed from memory to test display. “Uguale” indicates same, meaning that the item at the cued location was the same in both the memory and test display.</p> Full article ">Figure 2 Cont.
<p>From top to bottom, an overview of Experiments 1–5. In all of the experiments, visual working memory was tested. MC performed Experiments 1–3 and DDC performed Experiments 4 and 5. “Diverso” indicates different, meaning that the item at the cued location changed from memory to test display. “Uguale” indicates same, meaning that the item at the cued location was the same in both the memory and test display.</p> Full article ">Figure 3
<p>An overview of the results of Experiments 1–5. The top panel shows the results of Experiments 1–3, the bottom panel shows the results of Experiments 4 and 5. LVF indicates left visual field, RVF right visual field, Bilateral is the average of upper and lower field. Avg indicates the average of the unilateral conditions, Sum indicates the sum of the unilateral conditions. Capacity is depicted on the <span class="html-italic">y</span>-axis. The results clearly show that across experiments capacity in the bilateral condition is only slightly higher than the average capacity in the unilateral conditions.</p> Full article ">
16 pages, 2283 KiB  
Article
Matrix Method by Genocchi Polynomials for Solving Nonlinear Volterra Integral Equations with Weakly Singular Kernels
by Elham Hashemizadeh, Mohammad Ali Ebadi and Samad Noeiaghdam
Symmetry 2020, 12(12), 2105; https://doi.org/10.3390/sym12122105 - 17 Dec 2020
Cited by 13 | Viewed by 2438
Abstract
In this study, we present a spectral method for solving nonlinear Volterra integral equations with weakly singular kernels based on the Genocchi polynomials. Many other interesting results concerning nonlinear equations with discontinuous symmetric kernels with application of group symmetry have remained beyond this [...] Read more.
In this study, we present a spectral method for solving nonlinear Volterra integral equations with weakly singular kernels based on the Genocchi polynomials. Many other interesting results concerning nonlinear equations with discontinuous symmetric kernels with application of group symmetry have remained beyond this paper. In the proposed approach, relying on the useful properties of Genocchi polynomials, we produce an operational matrix and a related coefficient matrix to convert nonlinear Volterra integral equations with weakly singular kernels into a system of algebraic equations. This method is very fast and gives high-precision answers with good accuracy in a low number of repetitions compared to other methods that are available. The error boundaries for this method are also presented. Some illustrative examples are provided to demonstrate the capability of the proposed method. Also, the results derived from the new method are compared to Euler’s method to show the superiority of the proposed method. Full article
(This article belongs to the Special Issue Integral Equations: Theories, Approximations and Applications)
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<p>The plots of the Genocchi polynomials.</p> Full article ">Figure 2
<p>Plot of comparison between the exact and approximate solutions of Example 1 for <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math>.</p> Full article ">Figure 3
<p>Plot of the absolute error with <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>5</mn> <mo>​</mo> </mrow> </semantics></math> for Example 1.</p> Full article ">Figure 4
<p>Plot of the absolute error with <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math> for Example 1.</p> Full article ">Figure 5
<p>Plot of the absolute error with <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>15</mn> </mrow> </semantics></math> for Example 1.</p> Full article ">Figure 6
<p>Plot of the absolute error with <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>20</mn> </mrow> </semantics></math> for Example 1.</p> Full article ">Figure 7
<p>Plot of approximate solutions by our method (Genocchi polynomials) with different values of <math display="inline"><semantics> <mrow> <mi>N</mi> </mrow> </semantics></math> on the interval <math display="inline"><semantics> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>ε</mi> <mo stretchy="false">]</mo> </mrow> </semantics></math> with <math display="inline"><semantics> <mrow> <mi>ε</mi> <mo>=</mo> <mn>0.002</mn> </mrow> </semantics></math> for Example 1.</p> Full article ">
16 pages, 271 KiB  
Article
Quasi-Arithmetic Type Mean Generated by the Generalized Choquet Integral
by Sebastian Wójcik
Symmetry 2020, 12(12), 2104; https://doi.org/10.3390/sym12122104 - 17 Dec 2020
Viewed by 1943
Abstract
It is known that the quasi-arithmetic means can be characterized in various ways, with an essential role of a symmetry property. In the expected utility theory, the quasi-arithmetic mean is called the certainty equivalent and it is applied, e.g., in a utility-based insurance [...] Read more.
It is known that the quasi-arithmetic means can be characterized in various ways, with an essential role of a symmetry property. In the expected utility theory, the quasi-arithmetic mean is called the certainty equivalent and it is applied, e.g., in a utility-based insurance contracts pricing. In this paper, we introduce and study the quasi-arithmetic type mean in a more general setting, namely with the expected value being replaced by the generalized Choquet integral. We show that a functional that is defined in this way is a mean. Furthermore, we characterize the equality, positive homogeneity, and translativity in this class of means. Full article
(This article belongs to the Special Issue Symmetry in Functional Equations and Inequalities)
24 pages, 5191 KiB  
Article
The Study for Longitudinal Deformation of Adjacent Shield Tunnel Due to Foundation Pit Excavation with Consideration of the Retaining Structure Deformation
by Xinhai Zhang, Gang Wei and Chengwu Jiang
Symmetry 2020, 12(12), 2103; https://doi.org/10.3390/sym12122103 - 17 Dec 2020
Cited by 26 | Viewed by 3115
Abstract
By selecting the ratio of the cumulative maximum deformation of the retaining structure to the excavation depth as the control parameter of the retaining structure deformation, this paper established a sidewall unloading model which can consider the deformation of the retaining structure and [...] Read more.
By selecting the ratio of the cumulative maximum deformation of the retaining structure to the excavation depth as the control parameter of the retaining structure deformation, this paper established a sidewall unloading model which can consider the deformation of the retaining structure and the spatial effect of foundation pit excavation. Meanwhile, the impact region of the sidewall was divided to calculate the distribution of additional stress caused by foundation pit excavation. On this basis, through introducing the collaborative deformation model for rotation and dislocation of a shield tunnel, this paper studied the longitudinal deformation of the adjacent shield tunnel due to foundation pit excavation. Moreover, several engineering cases were given to verify the reliability of the proposed method, and the influencing factors were analyzed. The following conclusions were obtained: the axial horizontal displacement of the shield tunnel by the side of the foundation pit was normally distributed, and the calculated value was in good agreement with the measured value; the longitudinal deformation of the shield tunnel was mainly induced by the unloading effect of the sidewall of the foundation pit, which was parallel and closed to the tunnel; the soil excavation in the vicinity of the buried depth of the tunnel would result in a significant increase in longitudinal deformation; with the increase in the retaining structure deformation of the foundation pit, the longitudinal deformation of the adjacent shield tunnel and its influence scope also increased; the longitudinal deformation of the shield tunnel decreased with the increase of clearances between the foundation pit and tunnel; and finally, the excavation of the foundation pit had a great influence on the shallowly buried shield tunnel nearby, and the effect of foundation pit excavation on the tunnel decreased with the increase of the burial depth of the shield tunnel. Full article
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<p>Schematic diagram of the foundation pit unloading model (<span class="html-italic">K</span><sub>0</sub> is the coefficient of static earth pressure, <span class="html-italic">γ</span> is the unit weight of soil for calculation, <span class="html-italic">z</span> is the depth of the calculation point and <span class="html-italic">β</span> is the stress loss rate).</p> Full article ">Figure 2
<p>Deformation diagram of the sidewall retaining structure.</p> Full article ">Figure 3
<p>Schematic diagram for the spatial effect of the foundation pit.</p> Full article ">Figure 4
<p>The schematic diagram of the position relationship and influence between the foundation pit and the adjacent shield tunnel.</p> Full article ">Figure 5
<p>The schematic diagram about the partition of the region affected by the unloading of the sidewall.</p> Full article ">Figure 6
<p>Schematic diagram for collaborative deformation mode of the shield tunnel.</p> Full article ">Figure 7
<p>Calculation model for the collaborative deformation of rotation and dislocation between the shield tunnel segment rings.</p> Full article ">Figure 8
<p>Comparison of the calculated values and measured values of the horizontal displacement of the shield tunnel adjacent to the foundation pit.</p> Full article ">Figure 9
<p>Horizontal displacement curve of the tunnel, caused by the unloading of each sidewall of the foundation pit.</p> Full article ">Figure 10
<p>Horizontal displacement curve of the tunnel during the process of excavation, layer by layer.</p> Full article ">Figure 11
<p>Longitudinal distribution of dislocation and rotation between the segment rings of the shield tunnel.</p> Full article ">Figure 12
<p>Longitudinal distribution of shear forces between the segment rings in the shield tunnel.</p> Full article ">Figure 13
<p>Comparison of the calculated value and measured value of the horizontal displacement of the tunnel beside the foundation pit.</p> Full article ">Figure 14
<p>The value of dislocation and rotation angle between the shield tunnel segments.</p> Full article ">Figure 15
<p>Longitudinal distribution of shear forces between the shield tunnel segments.</p> Full article ">Figure 16
<p>The variation curve of the maximum calculated value of the shield tunnel’s horizontal displacement with different values of <span class="html-italic">v</span><sub>max</sub>/<span class="html-italic">H</span><sub>e</sub>.</p> Full article ">Figure 17
<p>The longitudinal distribution curve of the adjacent tunnel’s horizontal displacement when <span class="html-italic">v</span><sub>max</sub>/<span class="html-italic">H</span><sub>e</sub> takes different values.</p> Full article ">Figure 18
<p>The calculated curve of the maximum horizontal displacement of the shield tunnel by the side of the foundation pit changing with <span class="html-italic">s</span>/<span class="html-italic">H</span><sub>e</sub>.</p> Full article ">Figure 19
<p>The longitudinal distribution curve of the horizontal displacement of the shield tunnel by the side of the foundation pit under different <span class="html-italic">s</span>/<span class="html-italic">H</span><sub>e</sub> conditions.</p> Full article ">Figure 20
<p>The changing curve of the maximum horizontal displacement (calculated) of the adjacent shield tunnel with different <span class="html-italic">h</span>/<span class="html-italic">H</span><sub>e</sub> values.</p> Full article ">Figure 21
<p>The longitudinal distribution curve of the horizontal displacement of the shield tunnel by the side of the foundation pit under different <span class="html-italic">h</span>/<span class="html-italic">H</span><sub>e</sub> conditions.</p> Full article ">
17 pages, 379 KiB  
Article
Priority Measurement of Patches for Program Repair Based on Semantic Distance
by Yukun Dong, Meng Wu, Li Zhang, Wenjing Yin, Mengying Wu and Haojie Li
Symmetry 2020, 12(12), 2102; https://doi.org/10.3390/sym12122102 - 17 Dec 2020
Cited by 3 | Viewed by 2352
Abstract
Automated program repair is an effective way to ensure software quality and improve software development efficiency. At present, there are many methods and tools of automated program reapir in real world, but most of them have low repair accuracy, resulting in a large [...] Read more.
Automated program repair is an effective way to ensure software quality and improve software development efficiency. At present, there are many methods and tools of automated program reapir in real world, but most of them have low repair accuracy, resulting in a large number of incorrect patches in the generated patches. To solve this problem, we propose a patch quality evaluation method based on semantic distance, which measures the semantic distance of patches by using features of interval distance, output coverage, and path matching. For each evaluation feature, we give a quantitative formula to obtain a specific distance value and use the distance to calculate the recommended patch value to measure the quality of the patch. Our quality evaluation method evaluated 279 patches from previous program repair tools, including Nopol, DynaMoth, ACS, jGenProg, and CapGen. This quality evaluation method successfully arranged the correct patches before the plausible but incorrect patches, and it recommended the higher-ranked patches to users first. On this basis, we compared our evaluation method with the existing evaluation methods and judged the evaluation ability of each feature. We showed that our proposed patch quality evaluation method can improve the repair accuracy of repair tools. Full article
(This article belongs to the Special Issue Special issue on Symmetry and Asymmetry in IoT and CPS Security and Privacy)
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Figure 1

Figure 1
<p>Patch for Closure 62 in Defects4J.</p> Full article ">Figure 2
<p>A buggy program with two possible repairs.</p> Full article ">Figure 3
<p>Patch for Math 63 in Defects4j generated by CapGen.</p> Full article ">Figure 4
<p>Patch for Math 53 in Defects4j generated by jGenProg.</p> Full article ">Figure 5
<p>Patch for Math 50 in Defects4j generated by jGenProg.</p> Full article ">Figure 6
<p>Evaluation effect of different features on patches.</p> Full article ">Figure 7
<p>Patch for Math 101 in Defects4j generated by human.</p> Full article ">
13 pages, 1608 KiB  
Article
An Evaluation of Symmetries in Ground Reaction Forces during Self-Paced Single- and Dual-Task Treadmill Walking in the Able-Bodied Men
by Rawan Al-Juaid and Mohammad Al-Amri
Symmetry 2020, 12(12), 2101; https://doi.org/10.3390/sym12122101 - 17 Dec 2020
Cited by 7 | Viewed by 3087
Abstract
Gait is a complex autonomous activity that has long been viewed as a symmetrical locomotion, even when it adapts to secondary concurrent attention-demanding tasks. This study aimed to evaluate the symmetry of the three ground reaction forces (GRFs) in able-bodied individuals during self-paced [...] Read more.
Gait is a complex autonomous activity that has long been viewed as a symmetrical locomotion, even when it adapts to secondary concurrent attention-demanding tasks. This study aimed to evaluate the symmetry of the three ground reaction forces (GRFs) in able-bodied individuals during self-paced treadmill walking with and without concurrent cognitive demands. Twenty-five male participants (age: 34.00 ± 4.44 years) completed two gait assessment sessions, each of whom were familiarized with the walking trials during their first session. Both sessions involved six-minute self-paced treadmill walking under three conditions: single-task walking and walking while concurrently responding to auditory 1-back and 2-back memory tasks. The symmetry of the GRFs was estimated using a nonlinear approach. Changes in the symmetry and walking speed across conditions in both sessions were assessed using inferential statistics. Results demonstrated that the three GRFs deviated from perfect symmetry by ≥10%. Engaging working memory during walking significantly reduced the symmetry of the vertical GRF (p = 0.003), and its detrimental effects on walking speed were significantly reduced in the second session with respect to the first session (p < 0.05). The findings indicate imperfect gait symmetry in able-bodied individuals, suggesting that common perceptions of gait symmetry should be reconsidered to reflect its objective importance in clinical settings. Full article
(This article belongs to the Special Issue Symmetry and Biomechanics)
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Figure 1

Figure 1
<p>Bland–Altman plots for outcome measurements; the solid line represents the mean difference between the two sessions, while the upper and lower dashed lines represent the 95% limits of agreement.</p> Full article ">Figure 2
<p>Estimated marginal means of outcome measurements across walking conditions in both sessions.</p> Full article ">Figure 2 Cont.
<p>Estimated marginal means of outcome measurements across walking conditions in both sessions.</p> Full article ">Figure 3
<p>Mean plot of the symmetry scores of the three GRFs altogether across walking conditions in Session 2.</p> Full article ">
19 pages, 5624 KiB  
Article
Extending the Fully Bayesian Unfolding with Regularization Using a Combined Sampling Method
by Petr Baroň and Jiří Kvita
Symmetry 2020, 12(12), 2100; https://doi.org/10.3390/sym12122100 - 17 Dec 2020
Cited by 2 | Viewed by 2195
Abstract
Regularization extensions to the Fully Bayesian Unfolding are implemented and studied with an algorithm of combined sampling to find, in a reasonable computational time, an optimal value of the regularization strength parameter in order to obtain an unfolded result of a desired property, [...] Read more.
Regularization extensions to the Fully Bayesian Unfolding are implemented and studied with an algorithm of combined sampling to find, in a reasonable computational time, an optimal value of the regularization strength parameter in order to obtain an unfolded result of a desired property, like smoothness. Three regularization conditions using the curvature, entropy and derivatives are applied, as a model example, to several simulated spectra of top-pair quark pairs that are produced in high energy pp collisions. The existence of a minimum of a χ2 between the unfolded and particle-level spectra is discussed, with recommendations on the checks and validity of the usage of the regularization feature in Fully Bayesian Unfolding (FBU). Full article
(This article belongs to the Special Issue Particle Physics and Symmetry)
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Figure 1

Figure 1
<p>Unfolding components of the spectrum <math display="inline"><semantics> <msup> <mi>η</mi> <mrow> <mi>t</mi> <mover accent="true"> <mi>t</mi> <mo>¯</mo> </mover> </mrow> </msup> </semantics></math>: (<b>a</b>) particle spectra (green), pseudo data (blue), unfolding result <math display="inline"><semantics> <mover accent="true"> <mi mathvariant="bold">T</mi> <mo stretchy="false">^</mo> </mover> </semantics></math> (red), (<b>b</b>) efficiency <math display="inline"><semantics> <mi>ϵ</mi> </semantics></math> and acceptance <math display="inline"><semantics> <mi>η</mi> </semantics></math> corrections of statistically independent sets A and B, and (<b>c</b>) normalized migration matrix <math display="inline"><semantics> <msub> <mi>M</mi> <mi>A</mi> </msub> </semantics></math>.</p> Full article ">Figure 2
<p>(<b>a</b>) View of a part of the 16-dimensional log-likelihood <math display="inline"><semantics> <mrow> <mo form="prefix">log</mo> <mi>L</mi> <mo>(</mo> <mi mathvariant="bold">T</mi> <mo>)</mo> </mrow> </semantics></math> function as function of the 6th and 9th bin (<b>b</b>) normalized maximal values of <math display="inline"><semantics> <mrow> <mo form="prefix">log</mo> <mi>L</mi> <mo>(</mo> <msub> <mi>T</mi> <mn>6</mn> </msub> <mo>,</mo> <msub> <mi>T</mi> <mn>9</mn> </msub> <mo>)</mo> </mrow> </semantics></math>, and (<b>c</b>) normalized maximal values of <math display="inline"><semantics> <mrow> <mi>L</mi> <mo>(</mo> <msub> <mi>T</mi> <mn>6</mn> </msub> <mo>,</mo> <msub> <mi>T</mi> <mn>9</mn> </msub> <mo>)</mo> </mrow> </semantics></math>.</p> Full article ">Figure 3
<p>Marginalized 1D posteriors in the 6th (<b>a</b>) and 9th (<b>b</b>) bin of the <math display="inline"><semantics> <msup> <mi>η</mi> <mrow> <mi>t</mi> <mover accent="true"> <mi>t</mi> <mo>¯</mo> </mover> </mrow> </msup> </semantics></math> spectrum without regularization applied.</p> Full article ">Figure 4
<p>Unfolding the double-peaked <math display="inline"><semantics> <msup> <mi>η</mi> <mrow> <mi>t</mi> <mover accent="true"> <mi>t</mi> <mo>¯</mo> </mover> </mrow> </msup> </semantics></math> over-binned spectrum for different values of the regularization strength parameter <math display="inline"><semantics> <mi>τ</mi> </semantics></math>. The parameter <math display="inline"><semantics> <mi>τ</mi> </semantics></math> is normalized, such that <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <msub> <mi>τ</mi> <mi>rel</mi> </msub> </mrow> </semantics></math>, see <a href="#sec3-symmetry-12-02100" class="html-sec">Section 3</a>.</p> Full article ">Figure 5
<p>Relative <math display="inline"><semantics> <mrow> <msup> <mi>χ</mi> <mn>2</mn> </msup> <mrow> <mo>(</mo> <mi>τ</mi> <mo>)</mo> </mrow> <mo>/</mo> <mi>ndf</mi> </mrow> </semantics></math> as function of the regularization strength parameter <math display="inline"><semantics> <mi>τ</mi> </semantics></math> and its minimum at <math display="inline"><semantics> <mrow> <msub> <mi>τ</mi> <mi>opt</mi> </msub> <mo>=</mo> <mn>2089</mn> </mrow> </semantics></math>. The parameter <math display="inline"><semantics> <mi>τ</mi> </semantics></math> is normalized, such that <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <msub> <mi>τ</mi> <mi>rel</mi> </msub> </mrow> </semantics></math>, see <a href="#sec3-symmetry-12-02100" class="html-sec">Section 3</a>. The vertical line represents the minimum of <math display="inline"><semantics> <mrow> <msup> <mi>χ</mi> <mn>2</mn> </msup> <mo>/</mo> <mi>ndf</mi> </mrow> </semantics></math>.</p> Full article ">Figure 6
<p>The envelope of normalized regularization functions <math display="inline"><semantics> <mrow> <mi>S</mi> <mo>(</mo> <mi mathvariant="bold">T</mi> <mo>)</mo> </mrow> </semantics></math> in the 6th and 9th bin. For sampling purposes, the gradient of <math display="inline"><semantics> <mrow> <mi>L</mi> <mo>(</mo> <mi mathvariant="bold">T</mi> <mo>)</mo> <mo>−</mo> <mi>S</mi> <mo>(</mo> <mi mathvariant="bold">T</mi> <mo>)</mo> </mrow> </semantics></math> was used.</p> Full article ">Figure 7
<p>Posterior shifting and narrowing with increasing the regularization strength parameter <math display="inline"><semantics> <mi>τ</mi> </semantics></math> in a selected single bin: (<b>a</b>) no regularization applied <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>2089</mn> </mrow> </semantics></math> (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>≈</mo> <msup> <mn>10</mn> <mn>4</mn> </msup> </mrow> </semantics></math>.</p> Full article ">Figure 8
<p>Mostly decreasing (<b>a</b>) curvature, (<b>b</b>) entropy, and (<b>c</b>) derivatives of the unfolded spectrum <math display="inline"><semantics> <msup> <mi>η</mi> <mrow> <mi>t</mi> <mover accent="true"> <mi>t</mi> <mo>¯</mo> </mover> </mrow> </msup> </semantics></math> with respect to <math display="inline"><semantics> <mi>τ</mi> </semantics></math>. The uncertainty band is evaluated as a standard deviation over 20 independent unfolding runs initiated with different random seeds.</p> Full article ">Figure 9
<p>Cross-bin correlation matrix built from the correlation factor of likelihood, <math display="inline"><semantics> <mrow> <mi>L</mi> <mo>(</mo> <msub> <mi>T</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>T</mi> <mi>j</mi> </msub> <mo>)</mo> </mrow> </semantics></math> while using the curvature regularization for three different values of <math display="inline"><semantics> <mi>τ</mi> </semantics></math>.</p> Full article ">Figure 10
<p>The averaged cross bin correlations <math display="inline"><semantics> <msub> <mover accent="true"> <mi>C</mi> <mo>¯</mo> </mover> <mi>abs</mi> </msub> </semantics></math> (black) and <math display="inline"><semantics> <mover accent="true"> <mi>C</mi> <mo>¯</mo> </mover> </semantics></math> (pink) using (<b>a</b>) curvature, (<b>b</b>) entropy, and (<b>c</b>) derivative regularization for the <math display="inline"><semantics> <msup> <mi>η</mi> <mrow> <mi>t</mi> <mover accent="true"> <mi>t</mi> <mo>¯</mo> </mover> </mrow> </msup> </semantics></math> spectrum. The uncertainty band is evaluated as a standard deviation over 20 independent unfolding runs initiated with different random seeds.</p> Full article ">Figure 11
<p>Variable <math display="inline"><semantics> <mrow> <msub> <mi>χ</mi> <mi>num</mi> </msub> <mrow> <mo>(</mo> <mi>τ</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> of the spectrum <math display="inline"><semantics> <msup> <mi>η</mi> <mrow> <mi>t</mi> <mover accent="true"> <mi>t</mi> <mo>¯</mo> </mover> </mrow> </msup> </semantics></math> using (<b>a</b>) curvature (<b>b</b>) entropy and (<b>c</b>) derivative regularization illustrating the effect of spectra smoothing. The uncertainty band is evaluated as a standard deviation over 20 independent unfolding runs initiated with different random seeds. The red line indicates the minimal value of the <math display="inline"><semantics> <mrow> <msub> <mi>χ</mi> <mi>num</mi> </msub> <mrow> <mo>(</mo> <mi>τ</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>.</p> Full article ">Figure 12
<p>Variable <math display="inline"><semantics> <mrow> <msub> <mi>χ</mi> <mi>denom</mi> </msub> <mrow> <mo>(</mo> <mi>τ</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> of the spectrum <math display="inline"><semantics> <msup> <mi>η</mi> <mrow> <mi>t</mi> <mover accent="true"> <mi>t</mi> <mo>¯</mo> </mover> </mrow> </msup> </semantics></math> using (<b>a</b>) curvature, (<b>b</b>) entropy, and (<b>c</b>) derivative regularization illustrating the effect of narrowing the posteriors and decreasing the uncertainty. The uncertainty band is evaluated as a standard deviation over 20 independent unfolding runs initiated with different random seeds.</p> Full article ">Figure 13
<p>Variables <math display="inline"><semantics> <mrow> <msup> <mi>χ</mi> <mn>2</mn> </msup> <mo>/</mo> <mi>ndf</mi> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>χ</mi> <mi>num</mi> </msub> <mo>/</mo> <msub> <mi>χ</mi> <mi>denom</mi> </msub> </mrow> </semantics></math> of the <math display="inline"><semantics> <msup> <mi>η</mi> <mrow> <mi>t</mi> <mover accent="true"> <mi>t</mi> <mo>¯</mo> </mover> </mrow> </msup> </semantics></math> spectrum using (<b>a</b>) curvature, (<b>b</b>) entropy, and (<b>c</b>) derivative regularization showing good correspondence. The uncertainty band is evaluated as a standard deviation over 20 independent unfolding runs initiated with different random seeds.</p> Full article ">Figure 14
<p>Variable <math display="inline"><semantics> <mrow> <msup> <mi>χ</mi> <mn>2</mn> </msup> <mrow> <mo>(</mo> <mi>τ</mi> <mo>)</mo> </mrow> <mo>/</mo> <mi>ndf</mi> </mrow> </semantics></math> of the spectrum <math display="inline"><semantics> <msup> <mi>η</mi> <mrow> <mi>t</mi> <mover accent="true"> <mi>t</mi> <mo>¯</mo> </mover> </mrow> </msup> </semantics></math> using (<b>a</b>) curvature, (<b>b</b>) entropy, and (<b>c</b>) derivative regularization comparing combined (faster) sampling (blue) and full sampling (red). The uncertainty band is evaluated as a standard deviation over 20 independent unfolding runs initiated with different random seeds. The vertical dotted lines indicate positions of <math display="inline"><semantics> <mrow> <msup> <mi>χ</mi> <mn>2</mn> </msup> <mo>/</mo> <mi>ndf</mi> </mrow> </semantics></math> minima for each sampling case.</p> Full article ">Figure 15
<p>Variables (<b>a</b>) <math display="inline"><semantics> <mrow> <msub> <mi>χ</mi> <mi>num</mi> </msub> <mrow> <mo>(</mo> <mi>τ</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>, (<b>b</b>) <math display="inline"><semantics> <mrow> <msub> <mi>χ</mi> <mi>denom</mi> </msub> <mrow> <mo>(</mo> <mi>τ</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> for the full sampling method; and (<b>c</b>) <math display="inline"><semantics> <mrow> <msup> <mi>χ</mi> <mn>2</mn> </msup> <mrow> <mo>(</mo> <mi>τ</mi> <mo>)</mo> </mrow> <mo>/</mo> <mi>ndf</mi> </mrow> </semantics></math> using full (red) and combined (blue) sampling of the <math display="inline"><semantics> <msup> <mi>m</mi> <mrow> <mi>t</mi> <mover accent="true"> <mi>t</mi> <mo>¯</mo> </mover> </mrow> </msup> </semantics></math> spectrum with an accidental minimum at <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>≈</mo> <mn>7000</mn> </mrow> </semantics></math> (curvature regularization). The vertical dotted lines indicate positions of <math display="inline"><semantics> <mrow> <msup> <mi>χ</mi> <mn>2</mn> </msup> <mo>/</mo> <mi>ndf</mi> </mrow> </semantics></math> minima for each sampling case.</p> Full article ">Figure 16
<p>The result of unfolding (<b>a</b>) without regularization (<b>b</b>) with regularization and (<b>c</b>) with regularization applied only at second half of the spectrum <math display="inline"><semantics> <msup> <mi>m</mi> <mrow> <mi>t</mi> <mover accent="true"> <mi>t</mi> <mo>¯</mo> </mover> </mrow> </msup> </semantics></math> while using the curvature in the case of a accidental minimum in <math display="inline"><semantics> <mrow> <msup> <mi>χ</mi> <mn>2</mn> </msup> <mrow> <mo>(</mo> <mi>τ</mi> <mo>)</mo> </mrow> <mo>/</mo> <mi>ndf</mi> </mrow> </semantics></math> for one representative random seed.</p> Full article ">Figure 17
<p>Variables (<b>a</b>) <math display="inline"><semantics> <mrow> <msub> <mi>χ</mi> <mi>num</mi> </msub> <mrow> <mo>(</mo> <mi>τ</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>, (<b>b</b>) <math display="inline"><semantics> <mrow> <msub> <mi>χ</mi> <mi>denom</mi> </msub> <mrow> <mo>(</mo> <mi>τ</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> for the full sampling method; and, (<b>c</b>) <math display="inline"><semantics> <mrow> <msup> <mi>χ</mi> <mn>2</mn> </msup> <mrow> <mo>(</mo> <mi>τ</mi> <mo>)</mo> </mrow> <mo>/</mo> <mi>ndf</mi> </mrow> </semantics></math> using full (red) and combined (blue) sampling of the <math display="inline"><semantics> <msubsup> <mi>p</mi> <mrow> <mi>T</mi> </mrow> <mrow> <mi>t</mi> <mo>,</mo> <mi>had</mi> </mrow> </msubsup> </semantics></math> spectrum with a vanishing minimum (derivative regularization). The vertical dotted lines indicate positions of <math display="inline"><semantics> <mrow> <msup> <mi>χ</mi> <mn>2</mn> </msup> <mo>/</mo> <mi>ndf</mi> </mrow> </semantics></math> minima for each sampling case.</p> Full article ">Figure 18
<p>Result of unfolding (<b>a</b>) without regularization and (<b>b</b>) with regularization of the spectrum <math display="inline"><semantics> <msubsup> <mi>p</mi> <mrow> <mi>T</mi> </mrow> <mrow> <mi>t</mi> <mo>,</mo> <mi>had</mi> </mrow> </msubsup> </semantics></math> using the derivatives in case of a hidden minimum in <math display="inline"><semantics> <mrow> <msup> <mi>χ</mi> <mn>2</mn> </msup> <mrow> <mo>(</mo> <mi>τ</mi> <mo>)</mo> </mrow> <mo>/</mo> <mi>ndf</mi> </mrow> </semantics></math> for one representative random seed. Spectrum becomes smoother, but <math display="inline"><semantics> <mrow> <msup> <mi>χ</mi> <mn>2</mn> </msup> <mo>/</mo> <mi>ndf</mi> </mrow> </semantics></math> does not improve due to the narrowing of posteriors.</p> Full article ">Figure 19
<p>Variables (<b>a</b>) <math display="inline"><semantics> <mrow> <msub> <mi>χ</mi> <mi>num</mi> </msub> <mrow> <mo>(</mo> <mi>τ</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>, (<b>b</b>) <math display="inline"><semantics> <mrow> <msub> <mi>χ</mi> <mi>denom</mi> </msub> <mrow> <mo>(</mo> <mi>τ</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> for the full sampling method; and (<b>c</b>) <math display="inline"><semantics> <mrow> <msup> <mi>χ</mi> <mn>2</mn> </msup> <mrow> <mo>(</mo> <mi>τ</mi> <mo>)</mo> </mrow> <mo>/</mo> <mi>ndf</mi> </mrow> </semantics></math> using full (red) and combined (blue) sampling of the <math display="inline"><semantics> <msup> <mi>η</mi> <mrow> <mi>t</mi> <mover accent="true"> <mi>t</mi> <mo>¯</mo> </mover> </mrow> </msup> </semantics></math> spectrum with the real minimum at <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>≈</mo> <mn>900</mn> </mrow> </semantics></math> (curvature regularization). The vertical dotted lines indicate positions of <math display="inline"><semantics> <mrow> <msup> <mi>χ</mi> <mn>2</mn> </msup> <mo>/</mo> <mi>ndf</mi> </mrow> </semantics></math> minima for each sampling case.</p> Full article ">Figure 20
<p>The result of unfolding (<b>a</b>) without regularization and (<b>b</b>) with regularization of the spectrum <math display="inline"><semantics> <msup> <mi>η</mi> <mrow> <mi>t</mi> <mover accent="true"> <mi>t</mi> <mo>¯</mo> </mover> </mrow> </msup> </semantics></math> while using the curvature in case of a real minimum in <math display="inline"><semantics> <mrow> <msup> <mi>χ</mi> <mn>2</mn> </msup> <mo>/</mo> <mi>ndf</mi> </mrow> </semantics></math> for one representative random seed.</p> Full article ">Figure 21
<p>Result of unfolding (<b>a</b>) without regularization and (<b>c</b>) with regularization of the spectrum <math display="inline"><semantics> <msup> <mi>η</mi> <mrow> <mi>t</mi> <mo>,</mo> <mi>had</mi> </mrow> </msup> </semantics></math> using the curvature regularization for one representative random seed. Variable <math display="inline"><semantics> <mrow> <msup> <mi>χ</mi> <mn>2</mn> </msup> <mrow> <mo>(</mo> <mi>τ</mi> <mo>)</mo> </mrow> <mo>/</mo> <mi>ndf</mi> </mrow> </semantics></math> (<b>b</b>) while using full (red) and combined (blue) sampling of the <math display="inline"><semantics> <msup> <mi>η</mi> <mrow> <mi>t</mi> <mo>,</mo> <mi>had</mi> </mrow> </msup> </semantics></math> spectrum with minimum at <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>1698</mn> </mrow> </semantics></math> (curvature regularization). In this case, the regularization is not needed. The vertical dotted lines indicate positions of <math display="inline"><semantics> <mrow> <msup> <mi>χ</mi> <mn>2</mn> </msup> <mo>/</mo> <mi>ndf</mi> </mrow> </semantics></math> minima for each sampling case.</p> Full article ">Figure 22
<p>Result of unfolding (<b>a</b>) without regularization and (<b>c</b>) with regularization of the spectrum <math display="inline"><semantics> <msubsup> <mi>p</mi> <mi>T</mi> <mrow> <mi>t</mi> <mover accent="true"> <mi>t</mi> <mo>¯</mo> </mover> </mrow> </msubsup> </semantics></math> using entropy regularization for one representative random seed. Variable <math display="inline"><semantics> <mrow> <msup> <mi>χ</mi> <mn>2</mn> </msup> <mrow> <mo>(</mo> <mi>τ</mi> <mo>)</mo> </mrow> <mo>/</mo> <mi>ndf</mi> </mrow> </semantics></math> (<b>b</b>) using full (red) and combined (blue) sampling of the <math display="inline"><semantics> <msubsup> <mi>p</mi> <mi>T</mi> <mrow> <mi>t</mi> <mover accent="true"> <mi>t</mi> <mo>¯</mo> </mover> </mrow> </msubsup> </semantics></math> spectrum with minimum at <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>2089</mn> </mrow> </semantics></math> (entropy regularization). The vertical dotted lines indicate the positions of <math display="inline"><semantics> <mrow> <msup> <mi>χ</mi> <mn>2</mn> </msup> <mo>/</mo> <mi>ndf</mi> </mrow> </semantics></math> minima for each sampling case.</p> Full article ">
27 pages, 6119 KiB  
Article
Automated Fitting Process Using Robust Reliable Weighted Average on Near Infrared Spectral Data Analysis
by Divo Dharma Silalahi, Habshah Midi, Jayanthi Arasan, Mohd Shafie Mustafa and Jean-Pierre Caliman
Symmetry 2020, 12(12), 2099; https://doi.org/10.3390/sym12122099 - 17 Dec 2020
Cited by 2 | Viewed by 2100
Abstract
With the complexity of Near Infrared (NIR) spectral data, the selection of the optimal number of Partial Least Squares (PLS) components in the fitted Partial Least Squares Regression (PLSR) model is very important. Selecting a small number of PLS components leads to under [...] Read more.
With the complexity of Near Infrared (NIR) spectral data, the selection of the optimal number of Partial Least Squares (PLS) components in the fitted Partial Least Squares Regression (PLSR) model is very important. Selecting a small number of PLS components leads to under fitting, whereas selecting a large number of PLS components results in over fitting. Several methods exist in the selection procedure, and each yields a different result. However, so far no one has been able to determine the more superior method. In addition, the current methods are susceptible to the presence of outliers and High Leverage Points (HLP) in a dataset. In this study, a new automated fitting process method on PLSR model is introduced. The method is called the Robust Reliable Weighted Average—PLS (RRWA-PLS), and it is less sensitive to the optimum number of PLS components. The RRWA-PLS uses the weighted average strategy from multiple PLSR models generated by the different complexities of the PLS components. The method assigns robust procedures in the weighing schemes as an improvement to the existing Weighted Average—PLS (WA-PLS) method. The weighing schemes in the proposed method are resistant to outliers and HLP and thus, preserve the contribution of the most relevant variables in the fitted model. The evaluation was done by utilizing artificial data with the Monte Carlo simulation and NIR spectral data of oil palm (Elaeis guineensis Jacq.) fruit mesocarp. Based on the results, the method claims to have shown its superiority in the improvement of the weight and variable selection procedures in the WA-PLS. It is also resistant to the influence of outliers and HLP in the dataset. The RRWA-PLS method provides a promising robust solution for the automated fitting process in the PLSR model as unlike the classical PLS, it does not require the selection of an optimal number of PLS components. Full article
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<p>The RMSECV and RMSEP of the classical PLSR on the simulated data with no contamination of outlier and high leverage points.</p> Full article ">Figure 2
<p>The RMSECV and RMSEP of the classical PLSR on the simulated data with contamination of outlier and high leverage points.</p> Full article ">Figure 3
<p>SEP values in the RRWA-PLS using different approach on the simulated data with contamination of outlier and HLP.</p> Full article ">Figure 4
<p>The mean weights of the WA-PLS and RRWA-PLS on the simulated data with and without contamination of outlier and HLP.</p> Full article ">Figure 5
<p>The RMSEP values of the classical PLS, WA-PLS, and RRWA-PLS on the simulated data with and without contamination of outlier and HLP.</p> Full article ">Figure 6
<p>Predicted against actual values on the simulated data using PLS with opt., WA-PLS, MWA-PLS, and RRWA-PLS.</p> Full article ">Figure 7
<p>The RMSE of the fitted PLSR through cross validation and the prediction ability using %ODM dataset.</p> Full article ">Figure 8
<p>The mean weights of the fitted PLSR in WA-PLS and RRWA-PLS methods using %ODM dataset.</p> Full article ">Figure 9
<p>The RMSEP values of classical PLS, WA-PLS, RRWA-PLS method using %ODM dataset.</p> Full article ">Figure 10
<p>The RMSE of the fitted PLSR through cross validation and the prediction ability using %OWM dataset.</p> Full article ">Figure 11
<p>The mean weights of the fitted PLSR in WA-PLS and RRWA-PLS methods using %OWM dataset.</p> Full article ">Figure 12
<p>The RMSEP values of classical PLS, WA-PLS, RRWA-PLS method using %OWM dataset.</p> Full article ">Figure 13
<p>The RMSE of the fitted PLSR through cross validation and the prediction ability using %FFA dataset.</p> Full article ">Figure 14
<p>The mean weights of the fitted PLSR in WA-PLS and RRWA-PLS methods using %FFA dataset.</p> Full article ">Figure 15
<p>The RMSEP values of classical PLS, WA-PLS, RRWA-PLS method using %FFA dataset.</p> Full article ">Figure 16
<p>Reliability values using RRWA-PLS method on different datasets: (<b>a</b>) artificial data; NIR spectral dataset: (<b>b</b>) %ODM; (<b>c</b>) %OWM; (<b>d</b>) %FFA.</p> Full article ">Figure 16 Cont.
<p>Reliability values using RRWA-PLS method on different datasets: (<b>a</b>) artificial data; NIR spectral dataset: (<b>b</b>) %ODM; (<b>c</b>) %OWM; (<b>d</b>) %FFA.</p> Full article ">
13 pages, 1566 KiB  
Article
Efficacy of Autologous Fat Grafting in Restoring Facial Symmetry in Linear Morphea-Associated Lesions
by Camilla Baserga, Annalisa Cappella, Daniele M. Gibelli, Raffaele Sacco, Claudia Dolci, Federico Cullati, Aldo Bruno Giannì and Chiarella Sforza
Symmetry 2020, 12(12), 2098; https://doi.org/10.3390/sym12122098 - 17 Dec 2020
Cited by 4 | Viewed by 3436
Abstract
Morphea is a rare sclerotic autoimmune disorder primary affecting the skin and subcutaneous tissues. The linear head variants involve the facial area, with asymmetries and deformities. Eighteen patients with hemifacial deformity (age range 14–75 years) were assessed before surgery (T0), and after one [...] Read more.
Morphea is a rare sclerotic autoimmune disorder primary affecting the skin and subcutaneous tissues. The linear head variants involve the facial area, with asymmetries and deformities. Eighteen patients with hemifacial deformity (age range 14–75 years) were assessed before surgery (T0), and after one (T1, 18 patients) or two (T2, six patients) surgical treatments of facial autologous fat grafting. A stereophotogrammetric reconstruction of the facial surface was obtained for each patient and a group of control subjects, and facial symmetry was quantified according to the root mean square distance between homologous areas of trigeminal innervation. Values obtained from the control subjects were used to calculate z-scores for patients. At T0, all facial thirds of the patients resulted significantly more asymmetrical than those of the control subjects (Mann–Whitney test, p < 0.05), while at T1, the symmetry of the middle facial third did not differ from that of control subjects (p = 0.263). At T2, the upper and the lower facial thirds also did not differ from the control values (p > 0.05). The faster result obtained in the facial middle third was in accord with clinical findings. In conclusion, autologous fat grafting significantly improved facial asymmetry after one (middle facial third) or two (lower and upper thirds) treatments; the outcomes were efficaciously quantified by stereophotogrammetry. Full article
(This article belongs to the Section Life Sciences)
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<p>Three-dimensional digital model of the face divided into its three thirds. The landmarks used to identify the thirds are listed in the text.</p> Full article ">Figure 2
<p>Colorimetric maps showing facial asymmetry of Patient 2 before treatment (T0), (<b>a</b>), 6 months after the first treatment (T1), (<b>b</b>) and 12 months after the second treatment (T2), (<b>c</b>). Asymmetric areas are colored in blue (left side superimposed on the right side), while symmetrical areas are in green.</p> Full article ">
13 pages, 298 KiB  
Article
Symmetric Properties of Eigenvalues and Eigenfunctions of Uniform Beams
by Daulet Nurakhmetov, Serik Jumabayev, Almir Aniyarov and Rinat Kussainov
Symmetry 2020, 12(12), 2097; https://doi.org/10.3390/sym12122097 - 17 Dec 2020
Cited by 6 | Viewed by 2599
Abstract
In this paper, the models of Euler–Bernoulli beams on the Winkler foundations are considered. The novelty of the research is in consideration of the models with an arbitrary variable coefficient of foundation. Qualitative results that influence the symmetry of the coefficient of foundation [...] Read more.
In this paper, the models of Euler–Bernoulli beams on the Winkler foundations are considered. The novelty of the research is in consideration of the models with an arbitrary variable coefficient of foundation. Qualitative results that influence the symmetry of the coefficient of foundation on the spectral properties of the corresponding problems are obtained, for which specific variable coefficients of foundation are tested using numerical calculations. Three types of fixing at the ends are studied: clamped-clamped, hinged-hinged and free-free. The conditions of the stiffness and types of beam fixing have been found for the set of eigenvalues of boundary value problems on a full segment and can be represented as two groups of the eigenvalues of certain problems on a half segment. Such qualitative spectral properties of a mechanical system can contribute to the creation of various algorithms for nondestructive testing, which are widely used in technical acoustics. Full article
(This article belongs to the Section Mathematics)
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<p>Clamped-clamped Euler–Bernoulli beam.</p> Full article ">Figure 2
<p>Hinged-hinged Euler–Bernoulli beam.</p> Full article ">Figure 3
<p>Hinged-hinged Euler–Bernoulli beam.</p> Full article ">
29 pages, 1343 KiB  
Article
Hybrid Hesitant Fuzzy Multi-Criteria Decision Making Method: A Symmetric Analysis of the Selection of the Best Water Distribution System
by Samayan Narayanamoorthy, Veerappan Annapoorani, Samayan Kalaiselvan and Daekook Kang
Symmetry 2020, 12(12), 2096; https://doi.org/10.3390/sym12122096 - 17 Dec 2020
Cited by 13 | Viewed by 3280
Abstract
Every country’s influence and livelihood is centered on that country’s water source. Therefore, many studies are being conducted worldwide to improve and sustain water resources. In this research paper, we have selected and researched the water scheme for groundwater recharge and drinking water [...] Read more.
Every country’s influence and livelihood is centered on that country’s water source. Therefore, many studies are being conducted worldwide to improve and sustain water resources. In this research paper, we have selected and researched the water scheme for groundwater recharge and drinking water supply of drought prone areas. The water project is aimed at connecting the drought prone areas of the three districts of Tamil Nadu to filling up the ponds in their respective villages and raising the ground water level and meeting the drinking water requirement. We have chosen a multi-criteria decision method to select the best alternative in a complex situation. When reviewing the implementation of this water project, many experts and people who will benefit from this project may have some hesitation and ambiguity in their suggestion on choosing the best water distribution system.We believe that the benefits of this project can be fully availed of if we choose a water distribution system. Our contribution in this article is to choose the best water distribution system for this project by use of our proposed multi-criteria decision making (MCDM) methods, hesitant fuzzy standard deviation with multi-objective optimization method by ratio analysis (HFSDV-MOORA), hesitant fuzzy standard deviation with technique, for order preference by similarity to an ideal solution (HFSDV-TOPSIS) and hesitant fuzzy standard deviation with VIsekriterijumsko Kompromisno Rangiranje (HFSDV-VIKOR), which will provide the best solution for improving the water resource for the drought-prone areas of three districts. Finally, we have identified and compared the correlation coefficient between proposed methods. As a result of the study, it has been found that the best water supply system is closed concrete pipes laid along agricultural land through the rural areas. Full article
(This article belongs to the Special Issue Multi-Criteria Decision Aid methods in fuzzy decision problems – part II)
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<p>Flowchart of proposed method.</p> Full article ">Figure 2
<p>Water distribution system of selected drought prone areas.</p> Full article ">Figure 3
<p>Description of alternatives.</p> Full article ">Figure 4
<p>The hierarchical structure of selected criteria.</p> Full article ">Figure 5
<p>Graphical representation of SDV weights of criteria.</p> Full article ">Figure 6
<p>The result of HFSDV-MOORA method.</p> Full article ">Figure 7
<p>Graphical representation of TOPSIS values.</p> Full article ">Figure 8
<p>Graphical representation of HFSDV-VIKOR result.</p> Full article ">Figure 9
<p>The ranking result of each alternative by using weighted entropy</p> Full article ">Figure 10
<p>Comparison ranking result of each alternative by using standard deviation weight.</p> Full article ">Figure 11
<p>Comparison results: (<b>a</b>)—Comparison results between HFSDV-MOORA and HFWEM-MOORA, (<b>b</b>)—Comparison results between HFSDV-TOPSIS and HFWEM-TOPSIS, (<b>c</b>)—Comparison results between HFSDV-VIKOR and HFWEM-VIKOR</p> Full article ">Figure 12
<p>Sensitivity analysis results: (<b>a</b>)—When <math display="inline"><semantics> <mrow> <msub> <mi>W</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.20</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>W</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.25</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>W</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0.15</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>W</mi> <mn>4</mn> </msub> <mo>=</mo> <mn>0.25</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>W</mi> <mn>5</mn> </msub> <mo>=</mo> <mn>0.15</mn> </mrow> </semantics></math>, (<b>b</b>)—When <math display="inline"><semantics> <mrow> <msub> <mi>W</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>W</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>W</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0.25</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>W</mi> <mn>4</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>W</mi> <mn>5</mn> </msub> <mo>=</mo> <mn>0.25</mn> </mrow> </semantics></math>.</p> Full article ">
19 pages, 1596 KiB  
Review
A Brief Review of Chiral Chemical Potential and Its Physical Effects
by Li-Kang Yang, Xiao-Feng Luo, Jorge Segovia and Hong-Shi Zong
Symmetry 2020, 12(12), 2095; https://doi.org/10.3390/sym12122095 - 16 Dec 2020
Cited by 7 | Viewed by 2782
Abstract
Nontrivial topological gluon configuration is one of the remarkable features of the Quantum Chromodynamics (QCD). Due to chiral anomaly, the chiral imbalance between right- and left-hand quarks can be induced by the transition of the nontrivial gluon configurations between different vacuums. In this [...] Read more.
Nontrivial topological gluon configuration is one of the remarkable features of the Quantum Chromodynamics (QCD). Due to chiral anomaly, the chiral imbalance between right- and left-hand quarks can be induced by the transition of the nontrivial gluon configurations between different vacuums. In this review, we will introduce the origin of the chiral chemical potential and its physical effects. These include: (1) the chiral imbalance in the presence of strong magnetic and related physical phenomena; (2) the influence of chiral chemical potential on the QCD phase structure; and (3) the effects of chiral chemical potential on quark stars. Moreover, we propose for the first time that quark stars are likely to be a natural laboratory for testing the destruction of strong interaction CP. Full article
(This article belongs to the Section Physics)
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<p>The illustration of the charge separation effect and chiral magnetic effect. The blue arrows and the red arrows respectively denote the spin and the momentum of quarks. (1) At the beginning, the <span class="html-italic">u</span> and <span class="html-italic">d</span> quarks are all in the lowest Landau level and can only move along the direction of the magnetic field. (2) The quarks interact with a nontrivial gauge configuration with <math display="inline"><semantics> <msub> <mi>q</mi> <mi>w</mi> </msub> </semantics></math>. Assuming <math display="inline"><semantics> <mrow> <msub> <mi>q</mi> <mi>w</mi> </msub> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mrow> </semantics></math>, this gauge configuration can convert the chiralities of quarks from left-hand to right-hand. It will lead to the chiral imbalance between the left- and right-hand quarks. (3) In the presence of a strong magnetic, the <span class="html-italic">u</span> quarks (or <span class="html-italic">d</span> quark) with different chiralities move in different directions. Due to the chiral imbalance, the total net charge moving along the direction of the magnetic is <math display="inline"><semantics> <mrow> <mi>Q</mi> <mo>=</mo> <mn>2</mn> <mi>e</mi> </mrow> </semantics></math>. In addition, it will result in a charge difference between two domain walls perpendicular to the magnetic field. Reprinted from [<a href="#B28-symmetry-12-02095" class="html-bibr">28</a>], with permission from Elsevier.</p> Full article ">Figure 2
<p>Charge separation effect—the regions inside the domain walls with <math display="inline"><semantics> <mrow> <mi>θ</mi> <mo>≠</mo> <mn>0</mn> </mrow> </semantics></math>, outside with <math display="inline"><semantics> <mrow> <mi>θ</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>. The domain walls is charged in the case of a strong magnetic field, with the surface charge density ∼<math display="inline"><semantics> <mfrac> <mrow> <msup> <mi>e</mi> <mn>2</mn> </msup> <mi>θ</mi> <mi>B</mi> </mrow> <mrow> <mn>2</mn> <msup> <mi>π</mi> <mn>2</mn> </msup> </mrow> </mfrac> </semantics></math>. Reprinted from [<a href="#B36-symmetry-12-02095" class="html-bibr">36</a>], with permission from Elsevier.</p> Full article ">Figure 3
<p>Chiral magnetic effect—in the case of a strong magnetic field, in the region with <math display="inline"><semantics> <mrow> <mover accent="true"> <mi>θ</mi> <mo>˙</mo> </mover> <mo>≠</mo> <mn>0</mn> </mrow> </semantics></math>, an electric current <math display="inline"><semantics> <mrow> <mi mathvariant="bold">J</mi> <mo>∼</mo> <mfrac> <mrow> <msup> <mi>e</mi> <mn>2</mn> </msup> <mover accent="true"> <mi>θ</mi> <mo>˙</mo> </mover> <mi mathvariant="bold">B</mi> </mrow> <mrow> <mn>2</mn> <msup> <mi>π</mi> <mn>2</mn> </msup> </mrow> </mfrac> </mrow> </semantics></math> is induced. Reprinted from [<a href="#B36-symmetry-12-02095" class="html-bibr">36</a>], with permission from Elsevier.</p> Full article ">Figure 4
<p>The mass–radius relation of quark star with different chiral chemical potential at <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.9</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <msup> <mi>B</mi> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </msup> </semantics></math> = 90 MeV.</p> Full article ">
17 pages, 10827 KiB  
Article
Lifelong Machine Learning for Regional-Based Image Classification in Open Datasets
by Hashem Alyami, Abdullah Alharbi and Irfan Uddin
Symmetry 2020, 12(12), 2094; https://doi.org/10.3390/sym12122094 - 16 Dec 2020
Cited by 2 | Viewed by 2334
Abstract
Deep Learning algorithms are becoming common in solving different supervised and unsupervised learning problems. Different deep learning algorithms were developed in last decade to solve different learning problems in different domains such as computer vision, speech recognition, machine translation, etc. In the research [...] Read more.
Deep Learning algorithms are becoming common in solving different supervised and unsupervised learning problems. Different deep learning algorithms were developed in last decade to solve different learning problems in different domains such as computer vision, speech recognition, machine translation, etc. In the research field of computer vision, it is observed that deep learning has become overwhelmingly popular. In solving computer vision related problems, we first take a CNN (Convolutional Neural Network) which is trained from scratch or some times a pre-trained model is taken and further fine-tuned based on the dataset that is available. The problem of training the model from scratch on new datasets suffers from catastrophic forgetting. Which means that when a new dataset is used to train the model, it forgets the knowledge it has obtained from an existing dataset. In other words different datasets does not help the model to increase its knowledge. The problem with the pre-trained models is that mostly CNN models are trained on open datasets, where the data set contains instances from specific regions. This results into predicting disturbing labels when the same model is used for instances of datasets collected in a different region. Therefore, there is a need to find a solution on how to reduce the gap of Geo-diversity in different computer vision problems in developing world. In this paper, we explore the problems of models that were trained from scratch along with models which are pre-trained on a large dataset, using a dataset specifically developed to understand the geo-diversity issues in open datasets. The dataset contains images of different wedding scenarios in South Asian countries. We developed a Lifelong CNN that can incrementally increase knowledge i.e., the CNN learns labels from the new dataset but includes the existing knowledge of open data sets. The proposed model demonstrates highest accuracy compared to models trained from scratch or pre-trained model. Full article
(This article belongs to the Section Computer)
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<p>Fraction of images in Open Images and ImageNet datasets collected in different regions. In both datasets, majority of the images are collected from US, UK, Canada and Australia. In the image the two-letter code represents country name (Source [<a href="#B1-symmetry-12-02094" class="html-bibr">1</a>]).</p> Full article ">Figure 2
<p>Photos of <span class="html-italic">bridegrooms</span> from different countries aligned by the log-likelihood that the classifier trained on Open Images assigns to the <span class="html-italic">bridegroom</span> class. Images from Ethiopia and Pakistan are not classified as accurately as images from the regions such as US, UK, Canada, Australia, where the model is trained on (Source [<a href="#B1-symmetry-12-02094" class="html-bibr">1</a>]).</p> Full article ">Figure 3
<p>A sample collection of images from the Wedding dataset.</p> Full article ">Figure 4
<p>The architecture of LeNet-5 (source: [<a href="#B24-symmetry-12-02094" class="html-bibr">24</a>]).</p> Full article ">Figure 5
<p>(<b>a</b>) ResNet architecture (Source: [<a href="#B25-symmetry-12-02094" class="html-bibr">25</a>]). (<b>b</b>) Inception architecture (Source: [<a href="#B2-symmetry-12-02094" class="html-bibr">2</a>]).</p> Full article ">Figure 6
<p>An example demonstration of how Tree-CNN learns new classes from the Wedding data set and adds new classes along with the classes already learned in ImageNet. (<b>a</b>) Shows the Tree-CNN on ImageNet (<b>b</b>) Shows how the Tree-CNN learns new features from Wedding data set and adds with the labels already learned from ImageNet data set.</p> Full article ">Figure 7
<p>Train and Test accuracy by LeNet, ResNet, Inception and Tree-CNN in classification of Wedding dataset.</p> Full article ">Figure 8
<p>Accuracy, Precision, Recall and F1-Score computed by LeNet, ResNet, Inception and Tree-CNN in making predictions inWedding dataset. Figure (<b>a</b>) shows the train and test accuracy in CNNs and Figure (<b>b</b>) shows the Precision, Recall, F1-Score by different CNNs.</p> Full article ">Figure 9
<p>Confusion Matrix computed by LeNet, ResNet, Inception and Tree-CNN in the prediction of Wedding dataset.</p> Full article ">Figure 10
<p>ROC curve obtained by LeNet, ResNet, Inception and Tree-CNN computed by making prediction in Wedding dataset.</p> Full article ">Figure 11
<p>Images of <span class="html-italic">bridegrooms</span> and <span class="html-italic">bride</span> taken from different regions and represented as a log-likelihood using model pre-trained with ImageNet and then learn new classes from the Wedding dataset.</p> Full article ">
13 pages, 2964 KiB  
Article
The Influence of Symmetrical Boundary Conditions on the Structural Behaviour of Sandwich Panels Subjected to Torsion
by Zbigniew Pozorski and Szymon Wojciechowski
Symmetry 2020, 12(12), 2093; https://doi.org/10.3390/sym12122093 - 16 Dec 2020
Cited by 8 | Viewed by 2317
Abstract
The paper discusses the influence of load and support conditions on the behaviour of sandwich panels subjected to torsion. 3-D numerical models are presented, in which various boundary conditions have been defined. The case of the load causing the concentrated torque in the [...] Read more.
The paper discusses the influence of load and support conditions on the behaviour of sandwich panels subjected to torsion. 3-D numerical models are presented, in which various boundary conditions have been defined. The case of the load causing the concentrated torque in the span is analyzed, and the load definition affects the structural response. The numerical results were compared with the results obtained for the analytical beam model, which included both free torsion and secondary warping torsion. The conditions under which the models achieve a high agreement between the results were determined, but the significant sensitivity of the solution was also indicated. In each case of the considered load and boundary conditions, the structural response shows appropriate symmetry. Full article
(This article belongs to the Section Computer)
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<p>Geometry of the analyzed sandwich panel (dimensions in mm).</p> Full article ">Figure 2
<p>Load models used to generate torsional moment <span class="html-italic">M</span>: (<b>a</b>) horizontal forces, (<b>b</b>) vertical forces.</p> Full article ">Figure 3
<p>Boundary conditions defined in reference points (RP) of the supporting plates: (<b>a</b>) model 1—displacement in the <span class="html-italic">y</span>-direction is impossible, (<b>b</b>) model 2 with a flexibility of the support in the <span class="html-italic">y</span>-direction, (<b>c</b>) model 3—support without the upper supporting plate.</p> Full article ">Figure 4
<p>Results obtained from HFP models in the longitudinal cross-section <span class="html-italic">y</span> = 250 mm: (<b>a</b>) shear stresses in the bottom facing, (<b>b</b>) shear stresses in the core.</p> Full article ">Figure 5
<p>Results obtained from VFP models in the longitudinal cross-section <span class="html-italic">y</span> = 250 mm: (<b>a</b>) shear stresses in the bottom facing, (<b>b</b>) shear stresses in the core.</p> Full article ">Figure 6
<p>Results obtained from HFP models in the longitudinal cross-section <span class="html-italic">y</span> = 250 mm: (<b>a</b>) normal stresses in the bottom facing, (<b>b</b>) vertical displacements of the bottom facing.</p> Full article ">Figure 7
<p>Results obtained from VFP models in the longitudinal cross-section <span class="html-italic">y</span> = 250 mm: (<b>a</b>) normal stresses in the bottom facing, (<b>b</b>) vertical displacements of the bottom facing.</p> Full article ">Figure 8
<p>Results obtained from BC1-VFP model in the longitudinal cross-section <span class="html-italic">y</span> = 500 mm: (<b>a</b>) normal stresses in the bottom facing, (<b>b</b>) visualization of stresses corresponding to different phenomena (torsion and local bending).</p> Full article ">
21 pages, 5570 KiB  
Article
Intermediate Gas Feed in Bi- or Triphasic Gas–Liquid(–Liquid) Segmented Slug Flow Capillary Reactors
by Niclas von Vietinghoff, David Hellmann, Jan Priebe and David W. Agar
Symmetry 2020, 12(12), 2092; https://doi.org/10.3390/sym12122092 - 16 Dec 2020
Cited by 3 | Viewed by 2247
Abstract
Segmented slug flow systems in capillaries have already shown good potential for process intensification, due to their symmetry in the characteristic flow pattern. However, several challenges remain in this technology. For instance, in gas-consuming reactions, like Aliq + Bgas→Cliq [...] Read more.
Segmented slug flow systems in capillaries have already shown good potential for process intensification, due to their symmetry in the characteristic flow pattern. However, several challenges remain in this technology. For instance, in gas-consuming reactions, like Aliq + Bgas→Cliq, the gas droplets shrink and may even disappear, limiting the conversions and throughputs of capillary reactor systems. To overcome such shortcomings, an intermediate gas feed was developed. In order to maintain the well-defined slug flow characteristics, it is necessary to introduce the gas rapidly and precisely, in small aliquots of <10 µL. This allows us to preserve the well-defined alternating triphasic slug flow. A miniaturized electrolysis cell, together with a flow-observing system, was thus devised and implemented successfully as an intermediate gas feed. Feeding a new gas droplet into an existing liquid–liquid segmented flow had a success rate of up to 99%, whereas refilling an existing gas droplet is often limited by a lack of coalescence. Here, only at low volumetric flows, 70% of the gas bubbles were refilled by coalescence. Full article
(This article belongs to the Special Issue Symmetry and Complexity of Catalysis in Flow Chemistry)
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<p>Biphasic slug flow with a continuous slug and a dispersed bubble, which can be either gas or liquid. Internal circulations in each phase indicated.</p> Full article ">Figure 2
<p>Schematic illustration of an intermediate gas feeding into an existing liquid/liquid two-phase segmented slug flow. (1) Observation and characterization of segmented slug flow before injection, (2) gas injection, (3) control unit and (4) observation and characterization of segmented slug flow after injection.</p> Full article ">Figure 3
<p><b>Left</b>: Schematic setup of the intermediate gas feeding with a solenoid valve and control communication between the transmissive sensor before the gas injection and solenoid valve. <b>Right</b>: Experimental setup of T-junction with included solenoid valve.</p> Full article ">Figure 4
<p>Schematic illustration and photographs of the electrolysis cells. <b>Top</b>: Cell 1 with a large free gas volume and the option of using a membrane that is able to separate oxygen and hydrogen. <b>Bottom</b>: Cell 2 with very small free gas volume. This cell does not use a membrane and therefore produces a mixture on hydrogen and oxygen, which is introduced in the capillary with an internal T-junction.</p> Full article ">Figure 5
<p>Schematic setup of the intermediate gas feeding via an electrolysis cell and control communication between the transmissive sensor before the gas injection and the electrolysis cell.</p> Full article ">Figure 6
<p>Illustration of the necessary steps to determine the operating parameters required for the control variables and the feeding after time delay (t<sub>delay</sub>). (1) Starting measuring slug velocity, v<sub>slug</sub>, and unit cell length, L<sub>unit</sub>. (2) Calculation of new distance to location of gas feeding. (3) Gas injection to unit cell.</p> Full article ">Figure 7
<p>Gas feeding into a single-phase (water) flow via a solenoid valve, under variation of the pressure difference. Feeding-frequency: 1 Hz.</p> Full article ">Figure 8
<p>Gas volumes produced by electrolysis as a function of time with different amperages. <b>Left</b>: Cell 1. <b>Right</b>: Cell 2.</p> Full article ">Figure 9
<p>Use of the electrolysis cells as gas source to generate a slug flow in a T-junction and comparison of the experimentally determined bubble lengths with theoretical values of the model of Garstecki and Fuerstmann [<a href="#B20-symmetry-12-02092" class="html-bibr">20</a>].</p> Full article ">Figure 10
<p>Measured volumetric gas flows after switching off the power supply for different free gas volumes in electrolysis Cell 1, with an initial phase ratio of 1 and a total volumetric flow of 2 mL/min in the capillary.</p> Full article ">Figure 11
<p>Dependence of bubble length with varying volumetric flow of the continuous phase. For volumetric flows from 0.25 to 2 mL/min, at t<sub>active</sub> = 40 ms, t<sub>off</sub> = 300 ms and I = 150 mA.</p> Full article ">Figure 12
<p>Comparison of the approximation methods with an actual bubble. Length of the actual bubble: 3.5 mm. Red: approximation with cylindrical volume and flat caps. Green: approximation of the bubble caps with perfect hemispheres.</p> Full article ">Figure 13
<p>Dependence of the bubble length with variation of the t<sub>active</sub>. For a period of gas production from 10 to 120 ms; t<sub>active</sub> = 40 ms; V<sub>cont</sub> = 0.5 mL/min; I = 150 mA.</p> Full article ">Figure 14
<p>Dependence of the bubble length with varying amperages. For amperages from 25 to 350 mA; t<sub>active</sub> = 40 ms; V<sub>cont</sub> = 0.5 mL/min.</p> Full article ">Figure 15
<p>Two release mechanisms for the gas injection. <b>Left</b>: release in the continuous phase. <b>Right</b>: the fed gas bubble is attached at the T-junction outlet, until the dispersed liquid bubble entrains it.</p> Full article ">Figure 16
<p>Resulting three-phase flow after successful intermediate gas feeding. V<sub>tot</sub> = 0.5 mL/min. Length of the liquid bubble: 3.1 mm. Length of the continuous slug: 3.5 mm. Length of the injected gas bubbles: 2 mm.</p> Full article ">Figure 17
<p>Schematic illustration of intermediate gas feeding into the water bubble with subsequent coalescence of the divided water bubbles.</p> Full article ">Figure 18
<p>Three-phase slug flow after intermediate gas feeding with long gas bubbles (7 mm). Length of liquid bubble: 2.9 mm. Length of continuous slug: 2.7 mm. V<sub>tot</sub> = 0.5 mL/min.</p> Full article ">Figure 19
<p>Three possible locations for injecting gas bubbles with the aim of refilling them. <b>Left</b>: Injection in the middle of the continuous phase. <b>Middle</b>: Injection in the middle of the gas phase. <b>Right</b>: Injection in the front cap of the gas phase.</p> Full article ">Figure 20
<p>Four cases in which a control of the manipulated variables for intermediate gas feeding is beneficial. (1) The gas injection was performed too late. (2) The injection was performed too early. (3) The injected gas bubble is too small. (4) The injected gas bubble is too large.</p> Full article ">Figure 21
<p>Time course of the controlled gas-bubble length injected with a P-controller. The amperage was controlled, and one measuring point corresponds to one gas bubble. Proportional gain for P-controller: KP = 0.75, and t<sub>active</sub> = 80 ms.</p> Full article ">
23 pages, 10583 KiB  
Article
Seismic Behavior of Moment-Resisting Frames with Conventional and Innovative Connections
by Sabatino Di Benedetto, Antonella Bianca Francavilla, Massimo Latour, Giovanni Ferrante Cavallaro, Vincenzo Piluso and Gianvittorio Rizzano
Symmetry 2020, 12(12), 2091; https://doi.org/10.3390/sym12122091 - 16 Dec 2020
Cited by 10 | Viewed by 4505
Abstract
In the last few decades, increasing efforts have been devoted to the development of beam-to-column connections able to accommodate the local ductility demand dissipating, contemporaneously, the seismic input energy. Among the typologies proposed, the so-called RBS (Reduced Beam Section) has gained wide acceptance [...] Read more.
In the last few decades, increasing efforts have been devoted to the development of beam-to-column connections able to accommodate the local ductility demand dissipating, contemporaneously, the seismic input energy. Among the typologies proposed, the so-called RBS (Reduced Beam Section) has gained wide acceptance in the construction market, leading to easy-to-construct and cost-effective solutions. As an alternative, new proposals based on the inclusion of friction devices in beam-to-column joints have recently been made. Such a practice has the merit, in case of destructive events, of exhibiting wide and stable hysteretic cycles concentrating damage in elements that undergo only minor yielding. Both RBS and friction joints have been widely studied, carrying out experimental tests on sub-assemblies investigating their cyclic rotational response. Nevertheless, the available experimental results on full-scale structures equipped with these connections are still quite limited. This is the reason why two experimental campaigns aimed at performing pseudo-dynamic testing of a full-scale two-storey steel building equipped with RBS and friction connections have been planned at the STRENGTH (STRuctural ENGineering Test Hall) Laboratory of the University of Salerno. The first experimental campaign with the structure equipped with RBSs has already been performed; the connections showed higher resistance than expected, and exhibited brittle fracture due to cyclic fatigue. The second campaign has not yet been carried out, but in this paper the blind analysis of the supposed behavior is reported. It is expected that the friction joints allow to dissipate the seismic input energy without any structural damage in the members, but only through the friction pads of the devices, which can be easily replaced at the end of a severe seismic event. Full article
(This article belongs to the Special Issue Recent Advances in Computational and Structural Engineering)
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Figure 1

Figure 1
<p>Building mock-up and its reference structure. (<b>a</b>) Lateral view of the mock-up (Moment Resisting Frames-Reduced Beam Section—MRF-RBS); (<b>b</b>) 3-D view of the reference structural scheme; and (<b>c</b>) Plan view and individuation of the tested frame.</p> Full article ">Figure 2
<p>Geometrical detail of the tested RBS connection. (<b>a</b>) Lateral view; (<b>b</b>) Front view; and (<b>c</b>) Plan view.</p> Full article ">Figure 3
<p>Friction joint.</p> Full article ">Figure 4
<p>(<b>a</b>) 3D numerical model of the structure with RBS connections. (<b>b</b>) Simplified representation of the finite element (FE) model of the connection.</p> Full article ">Figure 5
<p>Model calibration of the smooth link element.</p> Full article ">Figure 6
<p>Displacements histories. (<b>a</b>) Test 1; and (<b>b</b>) Test 2.</p> Full article ">Figure 7
<p>RBS hysteretic curves. (<b>a</b>) Test 1; and (<b>b</b>) Test 2.</p> Full article ">Figure 8
<p>Test 2. (<b>a</b>) Roof displacements; and (<b>b</b>) Moment-rotation curves (connection 1B).</p> Full article ">Figure 9
<p>Test 3. (<b>a</b>) Roof displacements; and (<b>b</b>) Moment-rotation curves (connection 1B).</p> Full article ">Figure 10
<p>Test 5. (<b>a</b>) Roof displacements; and (<b>b</b>) Moment-rotation curves (connection 1B).</p> Full article ">Figure 11
<p>Test 4. (<b>a</b>) Roof displacements; and (<b>b</b>) Moment-rotation curves (connection 1B).</p> Full article ">Figure 12
<p>Test 6. (<b>a</b>) Roof displacements; and (<b>b</b>) Moment-rotation curves (connection 1B).</p> Full article ">Figure 13
<p>(<b>a</b>) Hysteretic curves of the last applied accelerogram; and (<b>b</b>) failure in the welding of connection 1A.</p> Full article ">Figure 14
<p>Test 7. (<b>a</b>) Roof displacements; and (<b>b</b>) Moment-rotation curves (connection 1B).</p> Full article ">Figure 15
<p>FE model of the structure.</p> Full article ">Figure 16
<p>Test 1—Imperial Valley 1.10 g. (<b>a</b>) Base shear; and (<b>b</b>) Roof displacements.</p> Full article ">Figure 17
<p>Test 1—Imperial Valley 1.10 g: moment-rotation hysteretic curves.</p> Full article ">Figure 18
<p>Test 2—Spitak 0.80 g. (<b>a</b>) Base shear; and (<b>b</b>) Roof displacements.</p> Full article ">Figure 19
<p>Test 2—Spitak 0.80 g: moment-rotation hysteretic curves.</p> Full article ">Figure 20
<p>Global and local results of Test 3 (Artificial 0.50 g) and Test 4 (Santa Barbara 0.80 g). (<b>a</b>,<b>b</b>) Base shear; (<b>c</b>,<b>d</b>) Roof displacements; and (<b>e</b>,<b>f</b>) Hysteretic curves.</p> Full article ">Figure 21
<p>Test 5—Coalinga 0.80 g. (<b>a</b>) Base shear; and (<b>b</b>) Roof displacements.</p> Full article ">Figure 22
<p>Test 5—Coalinga 0.80 g: moment-rotation hysteretic curves.</p> Full article ">Figure 23
<p>Test 1. (<b>a</b>) Roof displacements; (<b>b</b>) Base shear.</p> Full article ">Figure 24
<p>Connection 1A (Test 1—Imperial Valley): comparison of the hysteretic curves.</p> Full article ">
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