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Symmetry
Volume 15
Issue 12
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Symmetry, Volume 15, Issue 12 (December 2023) – 120 articles

Cover Story (view full-size image): This study considers the problem of constructing chaotic maps where the statistical mean value can be appropriately controlled by tuning the map's parameters. Such maps can be used in chaos-based applications. To showcase this, these maps are used as sources of randomness in generating unpredictable trajectories for UAVs surveilling an area. The trajectory inherits the randomness of the chaotic map used as a seed, which results in chaotic motion patterns. View this paper
Generating Chaotic Trajectories for Patrolling Agents
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23 pages, 2000 KiB  
Article
Blend of Deep Features and Binary Tree Growth Algorithm for Skin Lesion Classification
by Sunil Kumar, Vijay Kumar Nath and Deepika Hazarika
Symmetry 2023, 15(12), 2213; https://doi.org/10.3390/sym15122213 - 18 Dec 2023
Cited by 1 | Viewed by 2958
Abstract
One of the most frequently identified cancers globally is skin cancer (SC). The computeraided categorization of numerous skin lesions via dermoscopic images is still a complicated problem. Early recognition is crucial since it considerably increases the survival chances. In this study, we introduce [...] Read more.
One of the most frequently identified cancers globally is skin cancer (SC). The computeraided categorization of numerous skin lesions via dermoscopic images is still a complicated problem. Early recognition is crucial since it considerably increases the survival chances. In this study, we introduce an approach for skin lesion categorization where, at first, a powerful hybrid deep-feature set is constructed, and then a binary tree growth (BTG)-based optimization procedure is implemented using a support vector machine (SVM) classifier with an intention to compute the categorizing error and build symmetry between categories, for selecting the most significant features which are finally fed to a multi-class SVM for classification. The hybrid deep-feature set is constructed by utilizing two pre-trained models, i.e., Densenet-201, and Inception-v3, that are fine-tuned on skin lesion data. These two deep-feature models have distinct architectures that characterize dissimilar feature abstraction strengths. This effective deep feature framework has been tested on two publicly available challenging datasets, i.e., ISIC2018 and ISIC2019. The proposed framework outperforms many existing approaches and achieves notable {accuracy, sensitivity, precision, specificity} values of {98.50%, 96.60%, 97.84%, 99.59%} and {96.60%, 94.21%, 96.38%, 99.39%} for the ISIC2018 and ISIC2019 datasets, respectively. The proposed implementation of the BTG-based optimization algorithm performs significantly better on the proposed feature blend for skin lesion classification. Full article
(This article belongs to the Special Issue Symmetry/Asymmetry in Computer Vision and Image Processing)
►▼ Show Figures

Figure 1

Figure 1
<p>Sample images from each class of ISIC2019.</p> Full article ">Figure 2
<p>Framework of the proposed methodology.</p> Full article ">Figure 3
<p>The structure of a Densenet framework.</p> Full article ">Figure 4
<p>Inception blocks (<b>a</b>) block-1; (<b>b</b>) block-2; (<b>c</b>) block-3.</p> Full article ">Figure 5
<p>The structure of an Incpetion-v3 framework.</p> Full article ">Figure 6
<p>The idea of transfer learning.</p> Full article ">Figure 7
<p>An example of masking procedure.</p> Full article ">Figure 8
<p>Confusion matrix obtained for proposed framework on ISIC2018. (The correct and incorrect observations are displayed in diagonal and off-diagonal cells respectively. The values in last column displays the corresponding precision and false discovery rates. The values in last row correspond to recall and false negative rates. The last diagonal cell in the bottom right displays the overall accuracy).</p> Full article ">Figure 9
<p>ROC curve corresponding to proposed framework for ISIC 2018 dataset (<b>a</b>) original and (<b>b</b>) partially zoomed form of (<b>a</b>).</p> Full article ">Figure 10
<p>Confusion matrix obtained for proposed framework on ISIC2019. (The correct and incorrect observations are displayed in diagonal and off-diagonal cells respectively. The values in last column displays the corresponding precision and false discovery rates. The values in last row correspond to recall and false negative rates. The last diagonal cell in the bottom right displays the overall accuracy).</p> Full article ">Figure 11
<p>ROC curve corresponding to proposed framework for ISIC 2019 dataset (<b>a</b>) original and (<b>b</b>) partially zoomed form of (<b>a</b>).</p> Full article ">
14 pages, 1201 KiB  
Article
TT¯ Deformation: A Lattice Approach
by Yunfeng Jiang
Symmetry 2023, 15(12), 2212; https://doi.org/10.3390/sym15122212 - 18 Dec 2023
Cited by 2 | Viewed by 1277
Abstract
Integrable quantum field theories can be regularized on the lattice while preserving integrability. The resulting theories on the lattice are integrable lattice models. A prototype of such a regularization is the correspondence between a sine-Gordon model and a six-vertex model on a light-cone [...] Read more.
Integrable quantum field theories can be regularized on the lattice while preserving integrability. The resulting theories on the lattice are integrable lattice models. A prototype of such a regularization is the correspondence between a sine-Gordon model and a six-vertex model on a light-cone lattice. We propose an integrable deformation of the light-cone lattice model such that in the continuum limit we obtain the TT¯-deformed sine-Gordon model. Under this deformation, the cut-off momentum becomes energy dependent and the underlying Yang–Baxter integrability is preserved. Therefore, this deformation is integrable but non-local: similar to the TT¯ deformation of quantum field theory. Full article
(This article belongs to the Special Issue Symmetry and Chaos in Quantum Mechanics)
►▼ Show Figures

Figure 1

Figure 1
<p>Light-cone lattice regularization of spacetime. The spacial direction is compact.</p> Full article ">Figure 2
<p>Plot of <math display="inline"><semantics> <mrow> <msub> <mi>f</mi> <mi>N</mi> </msub> <mrow> <mo>(</mo> <mi>v</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> with <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>R</mi> <mo>=</mo> <mn>0.05</mn> <mo>,</mo> <mi>t</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> at <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mn>10</mn> <mo>,</mo> <mn>100</mn> </mrow> </semantics></math>.</p> Full article ">Figure 3
<p>Plot of <math display="inline"><semantics> <mrow> <msub> <mi>f</mi> <mi>N</mi> </msub> <mrow> <mo>(</mo> <mi>v</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> with <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>R</mi> <mo>=</mo> <mn>0.05</mn> <mo>,</mo> <mi>t</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> at <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>3</mn> <mo>,</mo> <mn>11</mn> <mo>,</mo> <mn>101</mn> </mrow> </semantics></math>.</p> Full article ">Figure 4
<p>Deformed spectrum for <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math> with different values of <math display="inline"><semantics> <mi mathvariant="sans-serif">Θ</mi> </semantics></math>. The horizontal axis is the deformation parameter <span class="html-italic">t</span>, while the vertical axis is the deformed ground-state energy <math display="inline"><semantics> <msubsup> <mover accent="true"> <mi>E</mi> <mo stretchy="false">˜</mo> </mover> <mn>2</mn> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </msubsup> </semantics></math>. The blue and red dots denote the values for positive and negative values of <span class="html-italic">t</span>, respectively. (<b>a</b>) Deformed energy with <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi mathvariant="sans-serif">Θ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>. (<b>b</b>) Deformed energy with <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi mathvariant="sans-serif">Θ</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>.</p> Full article ">
20 pages, 12180 KiB  
Article
Clustering of Floating Tracers in a Random Velocity Field Modulated by an Ellipsoidal Vortex Flow
by Konstantin Koshel, Dmitry Stepanov, Nata Kuznetsova and Evgeny Ryzhov
Symmetry 2023, 15(12), 2211; https://doi.org/10.3390/sym15122211 - 18 Dec 2023
Viewed by 1058
Abstract
The influence of a background vortex flow on the clustering of floating tracers is addressed. The vortex flow considered is induced by an ellipsoidal vortex evolving in a deformation. The system exhibits various vortex motion regimes: (1) a steady state, (2) oscillation and [...] Read more.
The influence of a background vortex flow on the clustering of floating tracers is addressed. The vortex flow considered is induced by an ellipsoidal vortex evolving in a deformation. The system exhibits various vortex motion regimes: (1) a steady state, (2) oscillation and (3) rotation of the ellipsoidal vortex core. The latter two induce an unsteady velocity field for the tracer, thus leading to irregular (chaotic) tracer motion. Superimposing a stochastic divergent velocity field onto the deterministic vortex flow allows us to observe significantly different tracer evolution. An ellipsoidal vortex has ellipsoidal symmetry, and the tracer’s trajectories exhibit the same symmetry inside the vortex. Outside the vortex, the external deformation flow symmetry dominates. Diffusion scattering and chaotic advection give tracers the opportunity to leave the region of ellipsoidal symmetry and form a picture of shear flow symmetry. We use the method of characteristics to integrate the floating tracer density evolution equation and the Euler Ito scheme for obtaining the floating tracer trajectories with a random velocity field. The cluster area and cluster mass from the statistical topography are used as the quantitative diagnostics of a floating tracer’s clustering. For the case of a steady ellipsoidal vortex embedded into the deformation flow with a random velocity field component, we found that the clustering characteristics were weakened by the steady vortex. For the cases of an unsteady ellipsoidal vortex, we observed clustering in the floating tracer density field if the contribution of the divergent component was greater than or equal to that of the rotational (nondivergent) component. Even when the initial floating tracer patch was set on the boundary of the oscillating ellipsoidal vortex, we observed the formation of clusters. In the case of a rotating ellipsoidal vortex, we also observed pronounced clustering. Thus, we argue that unsteady ellipsoidal vortex regimes (oscillation and rotation), which induce chaotic motion of the nearby passive tracer’s trajectories, are still conducive to clustering of floating tracers observed in the density field, despite the intense deformation introduced by strain and shear. Full article
(This article belongs to the Special Issue Geophysical Fluid Dynamics and Symmetry)
►▼ Show Figures

Figure 1

Figure 1
<p>Motion regimes of the ellipsoidal vortex embedded in a deformation flow. (<b>a</b>) Phase portrait of the ellipsoidal vortex motion regimes (<math display="inline"><semantics> <mrow> <mi>ε</mi> <mo>,</mo> <mi>θ</mi> </mrow> </semantics></math>). Various color points denote the initial states of the ellipsoidal vortex for three cases: the blue point (<math display="inline"><semantics> <mrow> <mi>ε</mi> <mo>=</mo> <mn>2.28</mn> <mo>,</mo> <mi>θ</mi> <mo>=</mo> </mrow> </semantics></math>−<math display="inline"><semantics> <mrow> <mi>π</mi> <mo>/</mo> <mn>4</mn> </mrow> </semantics></math>) denotes a steady state, when the vortex is stationary, the green point (<math display="inline"><semantics> <mrow> <mi>ε</mi> <mo>=</mo> <mn>3.5</mn> <mo>,</mo> <mi>θ</mi> <mo>=</mo> <mo>−</mo> <mi>π</mi> <mo>/</mo> <mn>4</mn> </mrow> </semantics></math>) denotes an oscillation state, and the blue point (<math display="inline"><semantics> <mrow> <mi>ε</mi> <mo>=</mo> <mn>2.56</mn> <mo>,</mo> <mi>θ</mi> <mo>=</mo> <mo>−</mo> <mi>π</mi> <mo>/</mo> <mn>4</mn> </mrow> </semantics></math>) denotes a case of the vortex’s rotation. (<b>b</b>) Passive tracer trajectories inside the vortex (blue closed lines) and outside the vortex (green closed line) in a steady state. The red line denotes a separatrix, and the hyperbolic points are denoted by red points. (<b>c</b>) Positions of the vortex core boundary under oscillating of the ellipsoidal vortex. Blue points denote the initial state of the ellipsoidal vortex, and black points denote the positions of the vortex core boundary after 1/4, 1/2, and 3/4 of a period. Green points denote the position of the recirculation zone boundaries at the initial time moment. (<b>d</b>) Positions of the vortex core boundary for a rotating ellipsoidal vortex. Blue and green points denote the initial state of the ellipsoidal vortex and its recirculation zones, respectively. Red points denote the vortex core boundary after 1/4 and 3/4 of a period. Black points denote the vortex core boundary after 1/2 of a period.</p> Full article ">Figure 2
<p>Baseline experiments, where floating tracer clustering was induced by the random velocity field only (Equation (<a href="#FD5-symmetry-15-02211" class="html-disp-formula">5</a>)). Instances of spatial distributions of the floating tracer density (Equation (<a href="#FD13-symmetry-15-02211" class="html-disp-formula">13</a>)) when (<b>a</b>) there was a divergent component only (<math display="inline"><semantics> <mrow> <msub> <mi>γ</mi> <mi>r</mi> </msub> <mo>=</mo> <mn>1.0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>10.94</mn> </mrow> </semantics></math>), (<b>b</b>) contributions of the divergent and nondivergent components were equal (<math display="inline"><semantics> <mrow> <msub> <mi>γ</mi> <mi>r</mi> </msub> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>34.48</mn> </mrow> </semantics></math>), (<b>c</b>) the contribution of the divergent component was small (<math display="inline"><semantics> <mrow> <msub> <mi>γ</mi> <mi>r</mi> </msub> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>34.48</mn> </mrow> </semantics></math>), and (<b>d</b>) there was a nondivergent component only (<math display="inline"><semantics> <mrow> <msub> <mi>γ</mi> <mi>r</mi> </msub> <mo>=</mo> <mn>0.0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>34.48</mn> </mrow> </semantics></math>). Colored lines at the bottom frame denote the evolution of the clustering areas <math display="inline"><semantics> <mrow> <mo>〈</mo> <msub> <mi>s</mi> <mi>hom</mi> </msub> <mfenced separators="" open="(" close=")"> <mi>t</mi> <mo>;</mo> <mover accent="true"> <mi>ρ</mi> <mo>¯</mo> </mover> <mo>&gt;</mo> <mn>1</mn> </mfenced> <mo>〉</mo> </mrow> </semantics></math> (lower curves) and the clustering mass <math display="inline"><semantics> <mrow> <mo>〈</mo> <msub> <mi>m</mi> <mi>hom</mi> </msub> <mfenced separators="" open="(" close=")"> <mi>t</mi> <mo>;</mo> <mover accent="true"> <mi>ρ</mi> <mo>¯</mo> </mover> <mo>&gt;</mo> <mn>1</mn> </mfenced> <mo>〉</mo> </mrow> </semantics></math> (upper curves), depending on various contributions of the divergent and nondivergent components. Red lines denote <math display="inline"><semantics> <mrow> <mo>〈</mo> <msub> <mi>s</mi> <mi>hom</mi> </msub> <mfenced separators="" open="(" close=")"> <mi>t</mi> <mo>;</mo> <mover accent="true"> <mi>ρ</mi> <mo>¯</mo> </mover> <mo>&gt;</mo> <mn>1</mn> </mfenced> <mo>〉</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mo>〈</mo> <msub> <mi>m</mi> <mi>hom</mi> </msub> <mfenced separators="" open="(" close=")"> <mi>t</mi> <mo>;</mo> <mover accent="true"> <mi>ρ</mi> <mo>¯</mo> </mover> <mo>&gt;</mo> <mn>1</mn> </mfenced> <mo>〉</mo> </mrow> </semantics></math> curves, corresponding to the spatial distribution of the floating tracer density (see <a href="#symmetry-15-02211-f002" class="html-fig">Figure 2</a>a). Purple lines denote <math display="inline"><semantics> <mrow> <mo>〈</mo> <msub> <mi>s</mi> <mi>hom</mi> </msub> <mfenced separators="" open="(" close=")"> <mi>t</mi> <mo>;</mo> <mover accent="true"> <mi>ρ</mi> <mo>¯</mo> </mover> <mo>&gt;</mo> <mn>1</mn> </mfenced> <mo>〉</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mo>〈</mo> <msub> <mi>m</mi> <mi>hom</mi> </msub> <mfenced separators="" open="(" close=")"> <mi>t</mi> <mo>;</mo> <mover accent="true"> <mi>ρ</mi> <mo>¯</mo> </mover> <mo>&gt;</mo> <mn>1</mn> </mfenced> <mo>〉</mo> </mrow> </semantics></math> curves, corresponding to the spatial distribution of the floating tracer density (see <a href="#symmetry-15-02211-f002" class="html-fig">Figure 2</a>b). Green lines denote <math display="inline"><semantics> <mrow> <mo>〈</mo> <msub> <mi>s</mi> <mi>hom</mi> </msub> <mfenced separators="" open="(" close=")"> <mi>t</mi> <mo>;</mo> <mover accent="true"> <mi>ρ</mi> <mo>¯</mo> </mover> <mo>&gt;</mo> <mn>1</mn> </mfenced> <mo>〉</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mo>〈</mo> <msub> <mi>m</mi> <mi>hom</mi> </msub> <mfenced separators="" open="(" close=")"> <mi>t</mi> <mo>;</mo> <mover accent="true"> <mi>ρ</mi> <mo>¯</mo> </mover> <mo>&gt;</mo> <mn>1</mn> </mfenced> <mo>〉</mo> </mrow> </semantics></math> curves, corresponding to the spatial distribution of the floating tracer density (see <a href="#symmetry-15-02211-f002" class="html-fig">Figure 2</a>c). Blue lines denote <math display="inline"><semantics> <mrow> <mo>〈</mo> <msub> <mi>s</mi> <mi>hom</mi> </msub> <mfenced separators="" open="(" close=")"> <mi>t</mi> <mo>;</mo> <mover accent="true"> <mi>ρ</mi> <mo>¯</mo> </mover> <mo>&gt;</mo> <mn>1</mn> </mfenced> <mo>〉</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mo>〈</mo> <msub> <mi>m</mi> <mi>hom</mi> </msub> <mfenced separators="" open="(" close=")"> <mi>t</mi> <mo>;</mo> <mover accent="true"> <mi>ρ</mi> <mo>¯</mo> </mover> <mo>&gt;</mo> <mn>1</mn> </mfenced> <mo>〉</mo> </mrow> </semantics></math> curves, corresponding to the spatial distribution of the floating tracer density (see <a href="#symmetry-15-02211-f002" class="html-fig">Figure 2</a>d). Finally, black lines denote <math display="inline"><semantics> <mrow> <mo>〈</mo> <msub> <mi>s</mi> <mi>hom</mi> </msub> <mfenced separators="" open="(" close=")"> <mi>t</mi> <mo>;</mo> <mover accent="true"> <mi>ρ</mi> <mo>¯</mo> </mover> <mo>&gt;</mo> <mn>1</mn> </mfenced> <mo>〉</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mo>〈</mo> <msub> <mi>m</mi> <mi>hom</mi> </msub> <mfenced separators="" open="(" close=")"> <mi>t</mi> <mo>;</mo> <mover accent="true"> <mi>ρ</mi> <mo>¯</mo> </mover> <mo>&gt;</mo> <mn>1</mn> </mfenced> <mo>〉</mo> </mrow> </semantics></math> curves, corresponding to their analytical estimates (Equation (<a href="#FD18-symmetry-15-02211" class="html-disp-formula">18</a>)), when there was a divergent component only (<math display="inline"><semantics> <mrow> <msub> <mi>γ</mi> <mi>r</mi> </msub> <mo>=</mo> <mn>1.0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>10.94</mn> </mrow> </semantics></math>) [<a href="#B17-symmetry-15-02211" class="html-bibr">17</a>].</p> Full article ">Figure 3
<p>The same as <a href="#symmetry-15-02211-f002" class="html-fig">Figure 2</a>, but with the deterministic velocity component (Equation (<a href="#FD4-symmetry-15-02211" class="html-disp-formula">4</a>)) induced by a steady ellipsoidal vortex. Frames (<b>a</b>–<b>d</b>) corresponds the same valies of <math display="inline"><semantics> <msub> <mi>γ</mi> <mi>r</mi> </msub> </semantics></math> end <math display="inline"><semantics> <mi>τ</mi> </semantics></math> as in <a href="#symmetry-15-02211-f002" class="html-fig">Figure 2</a>. For cases (<b>c</b>,<b>d</b>), see the insets (<b>e</b>,<b>f</b>), respectively, at the time <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>94.44</mn> </mrow> </semantics></math>. Colored lines denote the clustering area <math display="inline"><semantics> <mfenced separators="" open="&#x2329;" close="&#x232A;"> <mrow> <msub> <mi>s</mi> <mi>hom</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>;</mo> <mover accent="true"> <mi>ρ</mi> <mo>¯</mo> </mover> <mo>&gt;</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </mfenced> </semantics></math> (lower curves) and the clustering mass <math display="inline"><semantics> <mfenced separators="" open="&#x2329;" close="&#x232A;"> <mrow> <msub> <mi>m</mi> <mi>hom</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>;</mo> <mover accent="true"> <mi>ρ</mi> <mo>¯</mo> </mover> <mo>&gt;</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </mfenced> </semantics></math> (upper curves) given various contributions from the divergent and nondivergent components. Curves of <math display="inline"><semantics> <mfenced separators="" open="&#x2329;" close="&#x232A;"> <mrow> <msub> <mi>m</mi> <mi>hom</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>;</mo> <mover accent="true"> <mi>ρ</mi> <mo>¯</mo> </mover> <mo>&gt;</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </mfenced> </semantics></math> for the random velocity field only(Equation (<a href="#FD5-symmetry-15-02211" class="html-disp-formula">5</a>)) are denoted by the colored asterisks.</p> Full article ">Figure 4
<p>The same as <a href="#symmetry-15-02211-f003" class="html-fig">Figure 3</a>, but the initial tracer patch was placed near the boundary of the steady ellipsoidal vortex.</p> Full article ">Figure 5
<p>The same as <a href="#symmetry-15-02211-f003" class="html-fig">Figure 3</a>, but for the oscillating ellipsoidal vortex regime.</p> Full article ">Figure 6
<p>The same as <a href="#symmetry-15-02211-f005" class="html-fig">Figure 5</a>, but the initial tracer patch was placed near the boundary of the oscillating ellipsoidal vortex.</p> Full article ">Figure 7
<p>The same as <a href="#symmetry-15-02211-f003" class="html-fig">Figure 3</a> but for the rotating ellipsoidal vortex.</p> Full article ">Figure 8
<p>The same as <a href="#symmetry-15-02211-f007" class="html-fig">Figure 7</a>, but the initial tracer patch was placed near the boundary of the rotating ellipsoidal vortex.</p> Full article ">
41 pages, 769 KiB  
Review
Quantum-to-Classical Coexistence: Wavefunction Decay Kinetics, Photon Entanglement, and Q-Bits
by Piero Chiarelli
Symmetry 2023, 15(12), 2210; https://doi.org/10.3390/sym15122210 - 18 Dec 2023
Cited by 3 | Viewed by 1208
Abstract
By utilizing a generalized version of the Madelung quantum hydrodynamic framework that incorporates noise, we derive a solution using the path integral method to investigate how a quantum superposition of states evolves over time. This exploration seeks to comprehend the process through which [...] Read more.
By utilizing a generalized version of the Madelung quantum hydrodynamic framework that incorporates noise, we derive a solution using the path integral method to investigate how a quantum superposition of states evolves over time. This exploration seeks to comprehend the process through which a stable quantum state emerges when fluctuations induced by the noisy gravitational background are present. The model defines the conditions that give rise to a limited range of interactions for the quantum potential, allowing for the existence of coarse-grained classical descriptions at a macroscopic level. The theory uncovers the smallest attainable level of uncertainty in an open quantum system and examines its consistency with the localized behavior observed in large-scale classical systems. The research delves into connections and similarities alongside other theories such as decoherence and the Copenhagen foundation of quantum mechanics. Additionally, it assesses the potential consequences of wave function decay on the measurement of photon entanglement. To validate the proposed theory, an experiment involving entangled photons transmitted between detectors on the moon and Mars is discussed. Finally, the findings of the theory are applied to the creation of larger Q-bit systems at room temperatures. Full article
(This article belongs to the Special Issue Quantum Entanglement and Quantum Optics: Latest Advances and Prospects)
►▼ Show Figures

Figure 1

Figure 1
<p>Schematic illustration of the experimental setup.</p> Full article ">
14 pages, 344 KiB  
Article
Proving Rho Meson Is a Dynamical Gauge Boson of Hidden Local Symmetry
by Koichi Yamawaki
Symmetry 2023, 15(12), 2209; https://doi.org/10.3390/sym15122209 - 18 Dec 2023
Cited by 2 | Viewed by 1008
Abstract
The rho meson has long been successfully identified with a dynamical gauge boson of Hidden Local Symmetry (HLS) Hlocal in the non-linear sigma model G/H gauge equivalent to the model having the symmetry Gglobal×Hlocal, with [...] Read more.
The rho meson has long been successfully identified with a dynamical gauge boson of Hidden Local Symmetry (HLS) Hlocal in the non-linear sigma model G/H gauge equivalent to the model having the symmetry Gglobal×Hlocal, with G=[SU(2)L×SU(2)R]≃O(4),H=SU(2)V≃O(3). However, under a hitherto unproven assumption that its kinetic term is dynamically generated, together with an ad hoc choice of the auxiliary field parameter “a=2”, we prove this assumption, thereby solving the long-standing mystery. The rho meson kinetic term is generated simply by the large N limit of the Grassmannian model G/H=O(N)/[O(N−3)×O(3)] gauge equivalent to O(N)global×[O(N−3)×O(3)]local, extrapolated to N=4, O(4)global×O(3)local, with all the phenomenologically successful “a=2 results”, i.e., ρ-universality, KSRF relation, and the Vector Meson Dominance, realized independently of the parameter “a”. This in turn establishes validity of the large N dynamics at the quantitative level directly by the experiments. The relevant cutoff reads Λ≃4πFπ for N=4, which is regarded as a matching scale of the HLS as a “magnetic dual” to QCD. Skyrmion is stabilized by such a dynamically generated rho meson without recourse to the underlying QCD, a further signal of the duality. The unbroken phase with a massless rho meson may be realized as a novel chiral-restored hadronic phase in the hot/dense QCD. Full article
(This article belongs to the Special Issue Symmetries and Ultra Dense Matter of Compact Stars II: Emergence of Symmetries in Strong Nuclear Correlations)
12 pages, 1815 KiB  
Article
Certain Results on Subclasses of Analytic and Bi-Univalent Functions Associated with Coefficient Estimates and Quasi-Subordination
by Elaf Ibrahim Badiwi, Waggas Galib Atshan, Ameera N. Alkiffai and Alina Alb Lupas
Symmetry 2023, 15(12), 2208; https://doi.org/10.3390/sym15122208 - 17 Dec 2023
Cited by 1 | Viewed by 1334
Abstract
The purpose of the present paper is to introduce and investigate new subclasses of analytic function class of bi-univalent functions defined in open unit disks connected with a linear q-convolution operator, which are associated with quasi-subordination. We find coefficient estimates of [...] Read more.
The purpose of the present paper is to introduce and investigate new subclasses of analytic function class of bi-univalent functions defined in open unit disks connected with a linear q-convolution operator, which are associated with quasi-subordination. We find coefficient estimates of h2, h3 for functions in these subclasses. Several known and new consequences of these results are also pointed out. There is symmetry between the results of the subclass fq,∑ μ(ζ,n,ρ,σ,ϑ,γ,δ,φ) and the results of the subclass ℵ∑q,δλ,ζ,n,ρ,σ,ϑ,φ. Full article
(This article belongs to the Special Issue Symmetry in Geometric Theory of Analytic Functions)
19 pages, 1700 KiB  
Article
Investigation of Special Type-Π Smarandache Ruled Surfaces Due to Rotation Minimizing Darboux Frame in E3
by Emad Solouma, Ibrahim Al-Dayel, Meraj Ali Khan and Mohamed Abdelkawy
Symmetry 2023, 15(12), 2207; https://doi.org/10.3390/sym15122207 - 17 Dec 2023
Cited by 2 | Viewed by 1268
Abstract
This study begins with the construction of type-Π Smarandache ruled surfaces, whose base curves are Smarandache curves derived by rotation-minimizing Darboux frame vectors of the curve in E3. The direction vectors of these surfaces are unit vectors that convert Smarandache [...] Read more.
This study begins with the construction of type-Π Smarandache ruled surfaces, whose base curves are Smarandache curves derived by rotation-minimizing Darboux frame vectors of the curve in E3. The direction vectors of these surfaces are unit vectors that convert Smarandache curves. The Gaussian and mean curvatures of the generated ruled surfaces are then separately calculated, and the surfaces are required to be minimal or developable. We report our main conclusions in terms of the angle between normal vectors and the relationship between normal curvature and geodesic curvature. For every surface, examples are provided, and the graphs of these surfaces are produced. Full article
(This article belongs to the Special Issue Symmetry and Its Application in Differential Geometry and Topology, 2nd Edition)
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<p>The curve <math display="inline"><semantics> <mrow> <mi>φ</mi> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>M</mi> <mo>(</mo> <mi>s</mi> <mo>,</mo> <mi>υ</mi> <mo>)</mo> </mrow> </semantics></math>.</p> Full article ">Figure 2
<p>The type-<math display="inline"><semantics> <mo>Π</mo> </semantics></math> Smarandache ruled surfaces <math display="inline"><semantics> <mo>Ψ</mo> </semantics></math>, <math display="inline"><semantics> <mo>Γ</mo> </semantics></math>, and <math display="inline"><semantics> <mo>Λ</mo> </semantics></math> along <math display="inline"><semantics> <msub> <mi>γ</mi> <mn>1</mn> </msub> </semantics></math>.</p> Full article ">Figure 3
<p>The type-<math display="inline"><semantics> <mo>Π</mo> </semantics></math> Smarandache ruled surfaces <math display="inline"><semantics> <mo>Θ</mo> </semantics></math>, <math display="inline"><semantics> <mi mathvariant="normal">Υ</mi> </semantics></math>, and <math display="inline"><semantics> <mo>Ω</mo> </semantics></math> along <math display="inline"><semantics> <msub> <mi>γ</mi> <mn>2</mn> </msub> </semantics></math>.</p> Full article ">Figure 4
<p>The type-<math display="inline"><semantics> <mo>Π</mo> </semantics></math> Smarandache ruled surfaces <math display="inline"><semantics> <mo>Ξ</mo> </semantics></math>, <math display="inline"><semantics> <mo>Σ</mo> </semantics></math>, and <math display="inline"><semantics> <mo>Δ</mo> </semantics></math> along <math display="inline"><semantics> <msub> <mi>γ</mi> <mn>3</mn> </msub> </semantics></math>.</p> Full article ">
21 pages, 7397 KiB  
Article
Fog Computing Task Scheduling of Smart Community Based on Hybrid Ant Lion Optimizer
by Fengqing Tian, Donghua Zhang, Ying Yuan, Guangchun Fu, Xiaomin Li and Guanghua Chen
Symmetry 2023, 15(12), 2206; https://doi.org/10.3390/sym15122206 - 17 Dec 2023
Viewed by 1133
Abstract
Due to the problem of large latency and energy consumption of fog computing in smart community applications, the fog computing task-scheduling method based on Hybrid Ant Lion Optimizer (HALO) is proposed in this paper. This method is based on the Ant Lion Optimizer [...] Read more.
Due to the problem of large latency and energy consumption of fog computing in smart community applications, the fog computing task-scheduling method based on Hybrid Ant Lion Optimizer (HALO) is proposed in this paper. This method is based on the Ant Lion Optimizer (ALO. Firstly, chaotic mapping is adopted to initialize the population, and the quality of the initial population is improved; secondly, the Adaptive Random Wandering (ARW) method is designed to improve the solution efficiency; finally, the improved Dynamic Opposite Learning Crossover (DOLC) strategy is embedded in the generation-hopping stage of the ALO to enrich the diversity of the population and improve the optimization-seeking ability of ALO. HALO is used to optimize the scheduling scheme of fog computing tasks. The simulation experiments are conducted under different data task volumes, compared with several other task scheduling algorithms such as the original algorithm of ALO, Genetic Algorithm (GA), Whale Optimizer Algorithm (WOA) and Salp Swarm Algorithm (SSA). HALO has good initial population quality, fast convergence speed, and high optimization-seeking accuracy. The scheduling scheme obtained by the proposed method in this paper can effectively reduce the latency of the system and reduce the energy consumption of the system. Full article
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<p>Fog computing architecture model for smart communities.</p> Full article ">Figure 2
<p>Fog computing task-scheduling process.</p> Full article ">Figure 3
<p>Scatter plot comparison: (<b>a</b>) Scatterplot of random distribution; (<b>b</b>) Circle chaos mapping scatter plot.</p> Full article ">Figure 4
<p>Wandering mode comparison: (<b>a</b>) Random wandering; (<b>b</b>) Adaptive random wandering.</p> Full article ">Figure 5
<p>DOL asymmetric search space. (<b>a</b>) The location distribution of <math display="inline"><semantics> <mrow> <msup> <mi>X</mi> <mrow> <mi>R</mi> <mi>O</mi> </mrow> </msup> </mrow> </semantics></math>; (<b>b</b>) The search space of <math display="inline"><semantics> <mrow> <msup> <mi>X</mi> <mrow> <mi>D</mi> <mi>O</mi> </mrow> </msup> </mrow> </semantics></math> when the value of <math display="inline"><semantics> <mrow> <msup> <mi>X</mi> <mrow> <mi>R</mi> <mi>O</mi> </mrow> </msup> </mrow> </semantics></math> is smaller than <math display="inline"><semantics> <mrow> <msup> <mi>X</mi> <mi>O</mi> </msup> </mrow> </semantics></math>; (<b>c</b>) The search space of <math display="inline"><semantics> <mrow> <msup> <mi>X</mi> <mrow> <mi>D</mi> <mi>O</mi> </mrow> </msup> </mrow> </semantics></math> when the value of <math display="inline"><semantics> <mrow> <msup> <mi>X</mi> <mrow> <mi>R</mi> <mi>O</mi> </mrow> </msup> </mrow> </semantics></math> is greater than <math display="inline"><semantics> <mrow> <msup> <mi>X</mi> <mi>O</mi> </msup> </mrow> </semantics></math>; (<b>d</b>) The search space of <math display="inline"><semantics> <mrow> <msup> <mi>X</mi> <mrow> <mi>D</mi> <mi>O</mi> </mrow> </msup> </mrow> </semantics></math> when the value of <math display="inline"><semantics> <mrow> <msup> <mi>X</mi> <mrow> <mi>R</mi> <mi>O</mi> </mrow> </msup> </mrow> </semantics></math> exceeds the boundary of the Ant Lion trap.</p> Full article ">Figure 6
<p><math display="inline"><semantics> <mrow> <mi>A</mi> <mi>n</mi> <msubsup> <mi>t</mi> <mi>k</mi> <mrow> <mi>I</mi> <mi>t</mi> <mi>e</mi> <mi>r</mi> </mrow> </msubsup> </mrow> </semantics></math> Example of coding.</p> Full article ">Figure 7
<p>HALO flow chart.</p> Full article ">Figure 8
<p><math display="inline"><semantics> <mrow> <mi>o</mi> <mi>b</mi> <mi>j</mi> <mn>1</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>o</mi> <mi>b</mi> <mi>j</mi> <mn>2</mn> </mrow> </semantics></math> under different <span class="html-italic">ω</span>.</p> Full article ">Figure 9
<p>Latency results of three algorithms with different Task Data.</p> Full article ">Figure 10
<p>Energy consumption results of three algorithms with different Task Data.</p> Full article ">Figure 11
<p>Comparison of different algorithms on Task 350.</p> Full article ">Figure 12
<p>Latency results of different algorithms with different Task Data.</p> Full article ">Figure 13
<p>Energy consumption results of different algorithms with different Task Data.</p> Full article ">Figure 14
<p>Comparison of different algorithms on Task Data.</p> Full article ">Figure 14 Cont.
<p>Comparison of different algorithms on Task Data.</p> Full article ">
23 pages, 330 KiB  
Article
Existence and Uniqueness of Solutions of Hammerstein-Type Functional Integral Equations
by Cemil Tunç, Fehaid Salem Alshammari and Fahir Talay Akyildiz
Symmetry 2023, 15(12), 2205; https://doi.org/10.3390/sym15122205 - 15 Dec 2023
Cited by 6 | Viewed by 1219
Abstract
The authors deal with nonlinear and general Hammerstein-type functional integral equations (HTFIEs). The first objective of this work is to apply and extend Burton’s method to general and nonlinear HTFIEs in a Banach space via the Chebyshev norm and complete metric. The second [...] Read more.
The authors deal with nonlinear and general Hammerstein-type functional integral equations (HTFIEs). The first objective of this work is to apply and extend Burton’s method to general and nonlinear HTFIEs in a Banach space via the Chebyshev norm and complete metric. The second objective of the paper is to extend and improve some earlier results to nonlinear HTFIEs. The authors prove two new theorems with regard to the existence and uniqueness of solutions (EUSs) of HTFIEs via a technique called progressive contractions, which belongs to T. A. Burton, and the Chebyshev norm and complete metric. Full article
(This article belongs to the Special Issue Elementary Fixed Point Theory and Common Fixed Points II)
11 pages, 1502 KiB  
Article
The Importance of Being Asymmetric for Geophysical Vortices
by Georgi G. Sutyrin
Symmetry 2023, 15(12), 2204; https://doi.org/10.3390/sym15122204 - 15 Dec 2023
Cited by 2 | Viewed by 1076
Abstract
Several types of spatial symmetry in vortex structures within rotating stratified fluids are examined by looking at self-propagating configurations in the quasigeostrophic model. The role of symmetry breaking in the dynamics of geophysical waves, vortices and instabilities is highlighted. In particular, the energy [...] Read more.
Several types of spatial symmetry in vortex structures within rotating stratified fluids are examined by looking at self-propagating configurations in the quasigeostrophic model. The role of symmetry breaking in the dynamics of geophysical waves, vortices and instabilities is highlighted. In particular, the energy exchange of the large-scale vertical shear with monopolar and dipolar vortices is analyzed. Various coupled vortex-wave structures are described in terms of wavy and evanescent modes. The Rossby wave radiation is shown to induce a zonal asymmetry, which is needed for the energy support and self-amplification of vortices in large-scale flow. The consequences for the evolution of the most long-lived vortices in the subtropical westward flows are discussed. Full article
(This article belongs to the Special Issue Geophysical Fluid Dynamics and Symmetry)
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<p>The geopotential, zonal velocity and meridional velocity in a circular cyclone (<b>a</b>,<b>d</b>,<b>g</b>); in a meridional A-dipole (<b>b</b>,<b>e</b>,<b>h</b>); and in a zonal S-dipole (<b>c</b>,<b>f</b>,<b>k</b>).</p> Full article ">Figure 2
<p>The branches of evanescent and wavy modes for different values of the vertical shear in the WB flow: BC (red) and BEQT (blue) modes according to (23) (<b>a</b>); the effects of the vertical shear on the branches originating from BC and BT modes (<b>b</b>); the reconnection of branches near the marginal stability (<b>c</b>); and weakly unstable flow (<b>d</b>).</p> Full article ">Figure 3
<p>The geopotential Equation (32) in the upper (<b>a</b>) and lower (<b>c</b>) layers; zonally even (<b>b</b>) and odd (<b>d</b>) parts of the geopotential in the upper layer plotted in (<b>a</b>).</p> Full article ">
12 pages, 296 KiB  
Article
Generalized χ and η Cross-Helicities in Non-Ideal Magnetohydrodynamics
by Prachi Sharma and Asher Yahalom
Symmetry 2023, 15(12), 2203; https://doi.org/10.3390/sym15122203 - 15 Dec 2023
Cited by 1 | Viewed by 1215
Abstract
We study the generalized χ and η cross-helicities for non-ideal non-barotropic magnetohydrodynamics (MHD). χ and η, the additional label translation symmetry group, are used to generalize cross-helicity in ideal flows. Both new helicities are additional topological invariants of ideal MHD. To study [...] Read more.
We study the generalized χ and η cross-helicities for non-ideal non-barotropic magnetohydrodynamics (MHD). χ and η, the additional label translation symmetry group, are used to generalize cross-helicity in ideal flows. Both new helicities are additional topological invariants of ideal MHD. To study there behavior in non-ideal MHD, we calculate the time derivative of both helicities using non-ideal MHD equations in which viscosity, finite resistivity, and heat conduction are taken into account. Physical variables are divided into ideal and non-ideal quantities separately during the mathematical analysis for simplification. The analytical results indicate that χ and η cross-helicities are not strict constants of motion in non-ideal MHD and show a rate of dissipation that is comparable to the dissipation of other topological constants of motion. Full article
(This article belongs to the Section Physics)
17 pages, 536 KiB  
Article
A Simple Method for Constructing Symmetric Subdivision Schemes with High-Degree Polynomial Reproduction
by Jun Shi, Jieqing Tan and Li Zhang
Symmetry 2023, 15(12), 2202; https://doi.org/10.3390/sym15122202 - 15 Dec 2023
Cited by 1 | Viewed by 1095
Abstract
In this paper, we present an efficient method for constructing symmetric subdivision schemes reproducing high-degree polynomials, without solving a system of linear equations. Original symmetric subdivision and its deduced subdivisions have similar increasing characteristics to the family of pseudo-splines from the polynomial reproduction [...] Read more.
In this paper, we present an efficient method for constructing symmetric subdivision schemes reproducing high-degree polynomials, without solving a system of linear equations. Original symmetric subdivision and its deduced subdivisions have similar increasing characteristics to the family of pseudo-splines from the polynomial reproduction point of view. Several examples are given to illustrate the efficiency of the method. Full article
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<p>Basic limit functions of <math display="inline"><semantics> <mrow> <msub> <mi>S</mi> <mn>8</mn> </msub> <mo>,</mo> <msub> <mi>S</mi> <mrow> <mn>8</mn> <mrow> <mo>_</mo> <mn>1</mn> </mrow> </mrow> </msub> <mo>,</mo> <msub> <mi>S</mi> <mrow> <mn>8</mn> <mrow> <mo>_</mo> <mn>2</mn> </mrow> </mrow> </msub> <mo>.</mo> </mrow> </semantics></math></p> Full article ">Figure 2
<p>Basic limit functions of <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>C</mi> <mrow> <mn>2</mn> <mrow> <mo>_</mo> <mn>1</mn> </mrow> </mrow> </msub> <mo>.</mo> </mrow> </semantics></math></p> Full article ">Figure 3
<p>Basic limit functions of <math display="inline"><semantics> <mrow> <msub> <mi>S</mi> <mn>7</mn> </msub> <mo>,</mo> <msub> <mi>S</mi> <mrow> <mn>7</mn> <mrow> <mo>_</mo> <mn>1</mn> </mrow> </mrow> </msub> <mo>.</mo> </mrow> </semantics></math></p> Full article ">Figure 4
<p>Basic limit functions of <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mn>3</mn> </msub> <mo>,</mo> <msub> <mi>C</mi> <mrow> <mn>3</mn> <mrow> <mo>_</mo> <mn>1</mn> </mrow> </mrow> </msub> <mo>.</mo> </mrow> </semantics></math></p> Full article ">Figure 5
<p>Polynomial reproduction capabilities (red) of <math display="inline"><semantics> <msub> <mi>S</mi> <mrow> <mn>8</mn> <mrow> <mo>_</mo> <mn>2</mn> </mrow> </mrow> </msub> </semantics></math> for Legendre polynomials (blue) from degree 2 to degree 7 with 1-step subdivision from initial points (yellow).</p> Full article ">
13 pages, 463 KiB  
Article
Is Inconsistency in the Association between Frontal Alpha Asymmetry and Depression a Function of Sex, Age, and Peripheral Inflammation?
by Christopher F. Sharpley, Ian D. Evans, Vicki Bitsika, Wayne M. Arnold, Emmanuel Jesulola and Linda L. Agnew
Symmetry 2023, 15(12), 2201; https://doi.org/10.3390/sym15122201 - 15 Dec 2023
Viewed by 1303
Abstract
Although alpha asymmetry has been found to correlate with depression, there is some inconsistency across the wider literature, suggesting the influence of other factors. Some of these may be the presence of peripheral inflammation, age, and sex of participants. To test the interaction [...] Read more.
Although alpha asymmetry has been found to correlate with depression, there is some inconsistency across the wider literature, suggesting the influence of other factors. Some of these may be the presence of peripheral inflammation, age, and sex of participants. To test the interaction of these factors in terms of the association between alpha asymmetry and depression in a community sample, in this study, data were collected on resting frontal alpha asymmetry (FAA) under eyes closed and eyes open conditions, serum C-reactive protein (CRP), age, and self-rated depression in a sample of 44 males and 56 females aged from 18 to 75 years (M = 32.5 yr, SD = 14.1 yr). Using regression models, the results indicated a complex set of associations. FAA values across the FP2-FP1 sites predicted depression in the eyes open condition, but not for any other pairing of sites. Increases in CRP concentration predicted increases in depression for women but not for men. CRP predicted FAA across two frontal sites (F8-F7) under the eyes open condition only. As CRP increased, FAA favoured the left hemisphere for that pair of frontal sites, a result found more strongly for males. Age did not influence these associations. By reflecting a complex, multi-factor interaction, these findings may tentatively provide some explanation for the inconsistency in the wider literature for the FAA–depression hypothesis. Full article
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<p>Sex-based differences in the relationship between CRP concentration and FAA at F8-F7 in the eyes open condition. Positive FAA values indicate stronger alpha band power in the right hemisphere, while negative FAA values indicate stronger alpha band power in the left hemisphere.</p> Full article ">
12 pages, 1642 KiB  
Article
Synchronized Cyclograms to Assess Inter-Limb Symmetry during Gait in Women with Anorexia and Bulimia: A Retrospective Study
by Massimiliano Pau, Serena Cerfoglio, Paolo Capodaglio, Flavia Marrone, Leonardo Mendolicchio, Micaela Porta, Bruno Leban, Manuela Galli and Veronica Cimolin
Symmetry 2023, 15(12), 2200; https://doi.org/10.3390/sym15122200 - 15 Dec 2023
Cited by 1 | Viewed by 2030
Abstract
Anorexia nervosa (AN) and bulimia nervosa (BN) are eating diseases characterized by extreme eating behaviours impacting both mental and physical health. Aberrant musculoskeletal adaptations due to malnutrition affect motor abilities such as postural control and gait. To date, limited data is available with [...] Read more.
Anorexia nervosa (AN) and bulimia nervosa (BN) are eating diseases characterized by extreme eating behaviours impacting both mental and physical health. Aberrant musculoskeletal adaptations due to malnutrition affect motor abilities such as postural control and gait. To date, limited data is available with regards to gait symmetry in AN and BN. The aim of this study was to characterize inter-limb asymmetry during gait in two cohorts affected by AN and BN, respectively, using the synchronized cyclograms and to compare it with a healthy weight group. A total of 14 AN, 17 BN, and 11 healthy-weight females were assessed via 3D gait analysis. Gait spatio-temporal parameters were computed together with angle–angle diagrams, which were characterized in terms of their geometric features. Individuals with AN and BN were characterized by reduced speed and cadence and an abnormal increase in the duration of the double support phase with respect to the healthy controls. With respect to inter-limb symmetry, asymmetries were detected in both groups, with individuals with BN exhibiting significantly larger cyclogram areas at the hip joint with respect to the other groups (323.43 degrees2 vs. 253.74 degrees2 vs. 136.37 degrees2) and significantly higher orientation angle and Trend Symmetry at both knee and ankle joint. The cyclogram analysis suggests the presence of an altered gait symmetry in individuals with BN. In the AN group, it is possible to observe a similar trend; however, this is not statistically significant. Overall, the findings of this study may provide a novel perspective on the motor control dysfunction linked to eating disorders and aid clinicians in selecting a suitable rehabilitation scheme targeted at enhancing motor stability and control. Full article
(This article belongs to the Special Issue Symmetry/Asymmetry in Biomedical Engineering)
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<p>Markers’ placement according to the Davis protocol [<a href="#B44-symmetry-15-02200" class="html-bibr">44</a>].</p> Full article ">Figure 2
<p>Graphical representation of a cyclogram and its main features considered for the present study.</p> Full article ">Figure 3
<p>Examples of hip–hip cyclograms. The unaffected individual (green curve) is characterized by a very small area with an inclination close to the 45° line (which indicates perfect inter-limb symmetry). The individual with anorexia nervosa is characterized by a larger cyclogram’s area but still relatively well-oriented. The individual with bulimia nervosa exhibits a very large cyclogram with a high orientation angle, thus indicating a very poor inter-limb symmetry.</p> Full article ">
12 pages, 269 KiB  
Article
Non-Classical Symmetry Analysis of a Class of Nonlinear Lattice Equations
by Wenting Li, Yueting Chen and Kun Jiang
Symmetry 2023, 15(12), 2199; https://doi.org/10.3390/sym15122199 - 14 Dec 2023
Cited by 2 | Viewed by 1126
Abstract
In this paper, a non-classical symmetry method for obtaining the symmetries of differential–difference equations is proposed. The non-classical symmetry method introduces an additional constraint known as the invariant surface condition, which is applied after the infinitesimal transformation. By solving the governing equations that [...] Read more.
In this paper, a non-classical symmetry method for obtaining the symmetries of differential–difference equations is proposed. The non-classical symmetry method introduces an additional constraint known as the invariant surface condition, which is applied after the infinitesimal transformation. By solving the governing equations that satisfy this condition, we can obtain the corresponding reduced equation. This allows us to determine the non-classical symmetry of the differential–difference equation. This method avoids the complicated calculation involved in extending the infinitesimal generator and allows for a wider range of symmetry forms. As a result, it enables the derivation of a greater number of differential–difference equations. In this paper, two kinds of (2+1)-dimensional Toda-like lattice equations are taken as examples, and their corresponding symmetric and reduced equations are obtained using the non-classical symmetry method. Full article
(This article belongs to the Special Issue Symmetries in Differential Equations and Application—2nd Edition)
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<p>The plot of <math display="inline"><semantics> <mrow> <mi>u</mi> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> </semantics></math> (<a href="#FD72-symmetry-15-02199" class="html-disp-formula">72</a>).</p> Full article ">
12 pages, 279 KiB  
Article
A New Method for Finding Lie Point Symmetries of First-Order Ordinary Differential Equations
by Winter Sinkala
Symmetry 2023, 15(12), 2198; https://doi.org/10.3390/sym15122198 - 14 Dec 2023
Viewed by 1538
Abstract
The traditional algorithm for finding Lie point symmetries of ordinary differential equations (ODEs) faces challenges when applied to first-order ODEs. This stems from the fact that for first-order ODEs, unlike higher-order ODEs, the determining equation lacks derivatives, rendering it impossible to decompose into [...] Read more.
The traditional algorithm for finding Lie point symmetries of ordinary differential equations (ODEs) faces challenges when applied to first-order ODEs. This stems from the fact that for first-order ODEs, unlike higher-order ODEs, the determining equation lacks derivatives, rendering it impossible to decompose into simpler PDEs to be solved for the infinitesimals. Consequently, a common technique for determining Lie point symmetries of first-order ODEs involves making speculative assumptions about the form of the infinitesimal generator. In this study, we propose a novel and more efficient approach for finding Lie point symmetries of first-order ODEs and systems of first-order ODEs. Our method leverages the inherent connection between first-order ODEs and their corresponding second-order counterparts derived through total differentiation. By exploiting this connection, we develop a systematic algorithm for determining Lie point symmetries of a wide range of first-order ODEs. We present the algorithm and provide illustrative examples to demonstrate its effectiveness. Full article
22 pages, 564 KiB  
Article
AAHEG: Automatic Advanced Heap Exploit Generation Based on Abstract Syntax Tree
by Yu Wang, Yipeng Zhang and Zhoujun Li
Symmetry 2023, 15(12), 2197; https://doi.org/10.3390/sym15122197 - 14 Dec 2023
Cited by 1 | Viewed by 2333
Abstract
Automatic Exploit Generation (AEG) involves automatically discovering paths in a program that trigger vulnerabilities, thereby generating exploits. While there is considerable research on heap-related vulnerability detection, such as detecting Heap Overflow and Use After Free (UAF) vulnerabilities, among contemporary heap-automated exploit techniques, only [...] Read more.
Automatic Exploit Generation (AEG) involves automatically discovering paths in a program that trigger vulnerabilities, thereby generating exploits. While there is considerable research on heap-related vulnerability detection, such as detecting Heap Overflow and Use After Free (UAF) vulnerabilities, among contemporary heap-automated exploit techniques, only certain automated exploit techniques can hijack program control flow to the shellcode. An important limitation of this approach is that it cannot effectively bypass Linux’s protection mechanisms. To solve this problem, we introduced Automatic Advanced Heap Exploit Generation (AAHEG). It first applies symbolic execution to analyze heap-related primitives in files and then detects potential heap-related vulnerabilities without a source code. After identifying these vulnerabilities, AAHEG builds an exploit abstract syntax tree (AST) to identify one or more successful exploit strategies, such as fast bin attack and Safe-unlink. AAHEG then selects exploitable methods via an abstract syntax tree (AST) and performs final testing to produce the final exploit. AAHEG chose to generate advanced heap-related exploits because the exploits can bypass Linux protections. Basically, AAHEG can automatically detect heap-related vulnerabilities in binaries without source code, build an exploit AST, choose from a variety of advanced heap exploit methods, bypass all Linux protection mechanisms, and generate final file-form exploit based on pwntools which can pass local and remote testing. Experimental results show that AAHEG successfully completed vulnerability detection and exploit generation for 20 Capture The Flag (CTF) binary files, 11 of which have all protection mechanisms enabled. Full article
(This article belongs to the Special Issue Advanced Studies of Symmetry/Asymmetry in Cybersecurity)
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<p>The heap structure in glibc.</p> Full article ">Figure 2
<p>Heap-related vulnerability.</p> Full article ">Figure 3
<p>Overview of AAHEG.</p> Full article ">Figure 4
<p>The function primitive to extract in binary files.</p> Full article ">Figure 5
<p>The Leak Libc AST through UAF vulnerability.</p> Full article ">Figure 6
<p>The Leak Libc AST through Heap Overflow vulnerability.</p> Full article ">Figure 7
<p>The Hijack Hooks AST through UAF vulnerability.</p> Full article ">Figure 8
<p>The Hijack Hooks AST through Heap Overflow vulnerability.</p> Full article ">Figure 9
<p>The Hijack Hooks AST through Off by Null vulnerability.</p> Full article ">Figure 10
<p>The Leak Libc AST through UAF vulnerability with bypassing tcache.</p> Full article ">Figure 11
<p>The flow chart of AAHEG.</p> Full article ">
21 pages, 905 KiB  
Article
Flexible Offloading and Task Scheduling for IoT Applications in Dynamic Multi-Access Edge Computing Environments
by Yang Sun, Yuwei Bian, Huixin Li, Fangqing Tan and Lihan Liu
Symmetry 2023, 15(12), 2196; https://doi.org/10.3390/sym15122196 - 14 Dec 2023
Viewed by 1538
Abstract
Nowadays, multi-access edge computing (MEC) has been widely recognized as a promising technology that can support a wide range of new applications for the Internet of Things (IoT). In dynamic MEC networks, the heterogeneous computation capacities of the edge servers and the diversified [...] Read more.
Nowadays, multi-access edge computing (MEC) has been widely recognized as a promising technology that can support a wide range of new applications for the Internet of Things (IoT). In dynamic MEC networks, the heterogeneous computation capacities of the edge servers and the diversified requirements of the IoT applications are both asymmetric, where and when to offload and schedule the time-dependent tasks of IoT applications remains a challenge. In this paper, we propose a flexible offloading and task scheduling scheme (FLOATS) to adaptively optimize the computation of offloading decisions and scheduling priority sequences for time-dependent tasks in dynamic networks. We model the dynamic optimization problem as a multi-objective combinatorial optimization problem in an infinite time horizon, which is intractable to solve. To address this, a rolling-horizon-based optimization mechanism is designed to decompose the dynamic optimization problem into a series of static sub-problems. A genetic algorithm (GA)-based computation offloading and task scheduling algorithm is proposed for each static sub-problem. This algorithm encodes feasible solutions into two-layer chromosomes, and the optimal solution can be obtained through chromosome selection, crossover and mutation operations. The simulation results demonstrate that the proposed scheme can effectively reduce network costs in comparison to other reference schemes. Full article
(This article belongs to the Special Issue Asymmetrical Network Control for Complex Dynamic Services)
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<p>System model.</p> Full article ">Figure 2
<p>Chromosome structure and encoding.</p> Full article ">Figure 3
<p>The Gantt chart corresponded to the given chromosome example.</p> Full article ">Figure 4
<p>Example of crossover operation.</p> Full article ">Figure 5
<p>Example of mutation operation.</p> Full article ">Figure 6
<p>Performance comparison between the proposed scheme and exhaustive search.</p> Full article ">Figure 7
<p>Performances of different application arrival rates <math display="inline"><semantics> <mi>λ</mi> </semantics></math> (app/s) (<math display="inline"><semantics> <mrow> <mover accent="true"> <mi>R</mi> <mo stretchy="false">¯</mo> </mover> <mo>=</mo> <mn>1000</mn> </mrow> </semantics></math> KB, <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>250</mn> </mrow> </semantics></math> ms). (<b>a</b>) Total network cost. (<b>b</b>) Total serving latency. (<b>c</b>) Total energy consumption. (<b>d</b>) Total number of tardy tasks.</p> Full article ">Figure 8
<p>Performances of under different mean input data size of tasks <math display="inline"><semantics> <mover accent="true"> <mi>R</mi> <mo stretchy="false">¯</mo> </mover> </semantics></math> (KB) (<math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>0.6</mn> </mrow> </semantics></math> app/s, <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>250</mn> </mrow> </semantics></math> ms). (<b>a</b>) Total network cost. (<b>b</b>) Total serving latency. (<b>c</b>) Total energy consumption. (<b>d</b>) Total number of tardy tasks.</p> Full article ">Figure 9
<p>Performances of different RHW time lengths <math display="inline"><semantics> <mi>τ</mi> </semantics></math> (ms) (<math display="inline"><semantics> <mrow> <mi>R</mi> <mo>=</mo> <mn>1000</mn> </mrow> </semantics></math> KB, <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>0.6</mn> </mrow> </semantics></math> app/s). (<b>a</b>) Total network cost. (<b>b</b>) Total serving latency. (<b>c</b>) Total energy consumption. (<b>d</b>) Total number of tardy tasks.</p> Full article ">
65 pages, 773 KiB  
Review
Generalized Equations in Quantum Mechanics and Brownian Theory
by Pierre-Henri Chavanis
Symmetry 2023, 15(12), 2195; https://doi.org/10.3390/sym15122195 - 13 Dec 2023
Cited by 2 | Viewed by 1288
Abstract
We discuss formal analogies between a nonlinear Schrödinger equation derived by the author from the theory of scale relativity and the equations of Brownian theory. By using the Madelung transformation, the nonlinear Schrödinger equation takes the form of hydrodynamic equations involving a friction [...] Read more.
We discuss formal analogies between a nonlinear Schrödinger equation derived by the author from the theory of scale relativity and the equations of Brownian theory. By using the Madelung transformation, the nonlinear Schrödinger equation takes the form of hydrodynamic equations involving a friction force, an effective thermal pressure, a pressure due to the self-interaction, and a quantum potential. These hydrodynamic equations have a form similar to the damped Euler equations obtained for self-interacting Brownian particles in the theory of simple liquids. In that case, the temperature is due to thermal motion and the pressure arises from spatial correlations between the particles. More generally, the correlations can be accounted for by using the dynamical density functional theory. We determine the excess free energy of Brownian particles that reproduces the standard quantum potential. We then consider a more general form of excess free energy functionals and propose a new class of generalized Schrödinger equations. For a certain form of excess free energy, we recover the generalized Schrödinger equation associated with the Tsallis entropy considered in a previous paper. Full article
(This article belongs to the Special Issue Hamiltonian and Overdamped Complex Systems, Symmetry of Phase-Space Occupancy)
9 pages, 260 KiB  
Article
A New Solution to the Strong CP Problem
by Sergey A. Larin
Symmetry 2023, 15(12), 2194; https://doi.org/10.3390/sym15122194 - 13 Dec 2023
Viewed by 2393
Abstract
We suggest a new solution to the strong CP problem. The solution is based on the proper use of the boundary conditions for the QCD-generating functional integral. We expand the perturbative boundary conditions to both perturbative and nonperturbative fields integrated into the QCD-generating [...] Read more.
We suggest a new solution to the strong CP problem. The solution is based on the proper use of the boundary conditions for the QCD-generating functional integral. We expand the perturbative boundary conditions to both perturbative and nonperturbative fields integrated into the QCD-generating functional integral. It allows us to nullify the CP odd term in the QCD Lagrangian and, thus, to solve the strong CP problem. The presently popular solution to the strong CP problem of introducing axions violates the principle of renormalizability of the Quantum Field Theory, which is very successful phenomenologically. Our solution obeys the principle of renormalizability of the Quantum Field Theory and does not involve new exotic particles like axions. Full article
(This article belongs to the Section Physics)
10 pages, 892 KiB  
Article
The Horizons in Circular Accelerated Motions and Its Consequences
by Jaume Giné
Symmetry 2023, 15(12), 2193; https://doi.org/10.3390/sym15122193 - 12 Dec 2023
Cited by 1 | Viewed by 1143
Abstract
In this work, we study the existence of horizons in circular accelerated motions and its consequences. One particular case is the existence of two horizons in any uniform circular motion. The radiation of the Poincaré invariant vacuum is related to the spontaneous breakdown [...] Read more.
In this work, we study the existence of horizons in circular accelerated motions and its consequences. One particular case is the existence of two horizons in any uniform circular motion. The radiation of the Poincaré invariant vacuum is related to the spontaneous breakdown of the conformal symmetry in Quantum Field Theory The main consequence of the existence of these horizons is the Unruh radiation coming from such horizons. This consequence allows us to study the possible experimental detection of the Unruh radiation in such motions. The radiation of the Poincaré invariant vacuum is related to the spontaneous breakdown of the conformal symmetry in Quantum Field Theory. This radiation is associated with an effective temperature that can be detected using an Unruh–DeWitt detector. In fact, this effective temperature at the relativistic limit depends linearly with respect to the proper acceleration. However, in general, this dependence is not linear, contrary of what happens in the classical Unruh effect. In the relativistic limit and high density case, the uniform circular motion becomes a rotating black hole. This allows for future studies of pre-black hole configurations. Full article
(This article belongs to the Section Physics)
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<p>The Rindler horizon to its left (at a distance <math display="inline"><semantics> <mrow> <msup> <mi>c</mi> <mn>2</mn> </msup> <mo>/</mo> <mi>a</mi> </mrow> </semantics></math> away).</p> Full article ">Figure 2
<p>Two hyperboloids and the infinite conus.</p> Full article ">Figure 3
<p>Rindler chart taking <math display="inline"><semantics> <mrow> <mover accent="true"> <mi>Y</mi> <mo stretchy="false">¯</mo> </mover> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> in <a href="#symmetry-15-02193-f001" class="html-fig">Figure 1</a>.</p> Full article ">
16 pages, 1629 KiB  
Article
Transformer-Based Recognition Model for Ground-Glass Nodules from the View of Global 3D Asymmetry Feature Representation
by Jun Miao, Maoxuan Zhang, Yiru Chang and Yuanhua Qiao
Symmetry 2023, 15(12), 2192; https://doi.org/10.3390/sym15122192 - 12 Dec 2023
Cited by 1 | Viewed by 1196
Abstract
Ground-glass nodules (GGN) are the main manifestation of early lung cancer, and accurate and efficient identification of ground-glass pulmonary nodules is of great significance for the treatment of lung diseases. In response to the problem of traditional machine learning requiring manual feature extraction, [...] Read more.
Ground-glass nodules (GGN) are the main manifestation of early lung cancer, and accurate and efficient identification of ground-glass pulmonary nodules is of great significance for the treatment of lung diseases. In response to the problem of traditional machine learning requiring manual feature extraction, and most deep learning models applied to 2D image classification, this paper proposes a Transformer-based recognition model for ground-glass nodules from the view of global 3D asymmetry feature representation. Firstly, a 3D convolutional neural network is used as the backbone to extract the features of the three-dimensional CT-image block of pulmonary nodules automatically; secondly, positional encoding information is added to the extracted feature map and input into the Transformer encoder layer for further extraction of global 3D asymmetry features, which can preserve more spatial information and obtain higher-order asymmetry feature representation; finally, the extracted asymmetry features are entered into a support vector machine or ELM-KNN model to further improve the recognition ability of the model. The experimental results show that the recognition accuracy of the proposed method reaches 95.89%, which is 4.79, 2.05, 4.11, and 2.74 percentage points higher than the common deep learning models of AlexNet, DenseNet121, GoogLeNet, and VGG19, respectively; compared with the latest models proposed in the field of pulmonary nodule classification, the accuracy has been improved by 2.05, 2.05, and 0.68 percentage points, respectively, which can effectively improve the recognition accuracy of ground-glass nodules. Full article
(This article belongs to the Special Issue Computer Vision, Pattern Recognition, Machine Learning, and Symmetry)
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<p>Model architecture.</p> Full article ">Figure 2
<p>Transformer Encoder Structure Diagram.</p> Full article ">Figure 3
<p>Examples of lung CT images.</p> Full article ">Figure 4
<p>Input pulmonary nodule images into 3D ResNet and Transformer Encoder for feature extraction.</p> Full article ">Figure 5
<p>Selection of Transformer Encoder Layers.</p> Full article ">
29 pages, 8262 KiB  
Article
Evaluation of the Prolate Spheroidal Wavefunctions via a Discrete-Time Fourier Transform Based Approach
by Natalie Baddour and Zuwen Sun
Symmetry 2023, 15(12), 2191; https://doi.org/10.3390/sym15122191 - 12 Dec 2023
Viewed by 1307
Abstract
Computation of prolate spheroidal wavefunctions (PSWFs) is notoriously difficult and time consuming. This paper applies operator theory to the discrete Fourier transform (DFT) to address the problem of computing PSWFs. The problem is turned into an infinite dimensional matrix operator eigenvalue problem, which [...] Read more.
Computation of prolate spheroidal wavefunctions (PSWFs) is notoriously difficult and time consuming. This paper applies operator theory to the discrete Fourier transform (DFT) to address the problem of computing PSWFs. The problem is turned into an infinite dimensional matrix operator eigenvalue problem, which we recognize as being the definition of the DPSSs. Truncation of the infinite matrix leads to a finite dimensional matrix eigenvalue problem which in turn yields what is known as the Slepian basis. These discrete-valued Slepian basis vectors can then be used as (approximately) discrete time evaluations of the PSWFs. Taking an inverse Fourier transform further demonstrates that continuous PSWFs can be reconstructed from the Slepian basis. The feasibility of this approach is shown via theoretical derivations followed by simulations to consider practical aspects. Simulations demonstrate that the level of errors between the reconstructed Slepian basis approach and true PSWFs are low when the orders of the eigenvectors are low but can become large when the orders of the eigenvectors are high. Accuracy can be increased by increasing the number of points used to generate the Slepian basis. Users need to balance accuracy with computational cost. For large time-bandwidth product PSWFs, the number of Slepian basis points required increases for a reconstruction to reach the same error as for low time-bandwidth products. However, when the time-bandwidth products increase and reach maximum concentration, the required number of points to achieve a given error level achieves steady state values. Furthermore, this method of reconstructing the PSWF from the Slepian basis can be more accurate when compared to the Shannon sampling approach and traditional quadrature approach for large time-bandwidth products. Finally, since the Slepian basis represents the (approximate) sampled values of PSWFs, when the number of points is sufficiently large, the reconstruction process can be omitted entirely so that the Slepian vectors can be used directly, without a reconstruction step. Full article
(This article belongs to the Section Engineering and Materials)
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<p>Comparison of PSWFs obtained through the Chebfun software and Flammer’s tabulated values. (<b>a</b>) <span class="html-italic">c</span> = 5; (<b>b</b>) <span class="html-italic">c</span> = 1.</p> Full article ">Figure 2
<p>Slepian bases with different number of points N, (<b>a</b>–<b>f</b>) <span class="html-italic">N</span> from 5 to 501.</p> Full article ">Figure 3
<p>Reconstructed Slepian bases with different number of points <span class="html-italic">N</span>, (<b>a</b>–<b>f</b>) <span class="html-italic">N</span> from 5 to 501.</p> Full article ">Figure 4
<p>Error between sinc-series reconstructed Slepian bases with different <span class="html-italic">N</span> and PSWFs of order zero, (<b>a</b>–<b>f</b>) <span class="html-italic">N</span> from 5 to 501.</p> Full article ">Figure 5
<p>Error between sinc-series reconstructed Slepian bases with different <span class="html-italic">N</span> and PSWFs of order 1, (<b>a</b>–<b>f</b>) <span class="html-italic">N</span> from 5 to 501.</p> Full article ">Figure 6
<p>Error between sinc-series reconstructed Slepian bases with different <span class="html-italic">N</span> and PSWFs of order 2, (<b>a</b>–<b>f</b>) <span class="html-italic">N</span> from 5 to 501.</p> Full article ">Figure 7
<p>Error between sinc-series reconstructed Slepian bases with different <span class="html-italic">N</span> and PSWFs of order 3, (<b>a</b>–<b>f</b>) <span class="html-italic">N</span> from 5 to 501.</p> Full article ">Figure 8
<p>Eigenvector error vs. Slepian bases number of points, (<b>a</b>–<b>g</b>) shows the relation for different orders.</p> Full article ">Figure 9
<p>Eigenvalue error relation with number of points, for eigenvectors of order 0 to 6: (<b>a</b>–<b>g</b>) shows the relation for different orders.</p> Full article ">Figure 10
<p>Error between reconstructed Slepian bases and PSWF of order zero with time-bandwidth product <span class="html-italic">c</span> = 2π: (<b>a</b>) error between eigenvectors; (<b>b</b>) linear curve fitting with vertical axis taken to be negative inverse of the error between eigenvectors; (<b>c</b>) parabolic curve fitting of the error between eigenvectors; (<b>d</b>) error between eigenvalues; (<b>e</b>) linear curve fitting with vertical axis taken to be negative inverse of the error between eigenvalues; (<b>f</b>) parabolic curve fitting of the error between eigenvalues.</p> Full article ">Figure 11
<p>Comparison of rate of error decrease for different orders of eigenvectors.</p> Full article ">Figure 12
<p>Error trend with orders of the eigenvectors: (<b>a</b>–<b>i</b>) <span class="html-italic">N</span> from 17 to 49.</p> Full article ">Figure 13
<p>(<b>a</b>) Error between Slepian basis and PSWFs of order zero. (<b>b</b>) Number of points required to achieve less than 2% error against time-bandwidth products. (<b>c</b>) Detailed plot of Slepian basis with minimum N and PSWFs sampled at the same time. (<b>d</b>) Detailed plot of Slepian basis with larger N and PSWFs sampled at the same time.</p> Full article ">Figure 14
<p>(<b>a</b>) Error between Slepian basis and PSWF of order one. (<b>b</b>) Number of points required to achieve less than 2% error versus time-bandwidth product. (<b>c</b>) Detailed plot of Slepian basis with minimum <span class="html-italic">N</span> and sampled PSWF values. (<b>d</b>) Detailed plot of Slepian basis with larger <span class="html-italic">N</span> and sampled PSWF values.</p> Full article ">Figure 15
<p>(<b>a</b>) Error between Slepian basis and PSWF of order two. (<b>b</b>) Number of points required to achieve less than 2% error against time-bandwidth products. (<b>c</b>) Detailed plot of Slepian basis with minimum <span class="html-italic">N</span> and sampled PSWF values. (<b>d</b>) Detailed plot of Slepian basis with larger <span class="html-italic">N</span> and sampled PSWF values.</p> Full article ">Figure 16
<p>Errors between PSWFs generated with other methods and the proposed reconstructed Slepian basis approach, (<b>a</b>–<b>f</b>) <span class="html-italic">c</span> from 0.628 to 62.8.</p> Full article ">Figure 17
<p>Run times of generating PSWFs using different approaches, (<b>a</b>–<b>f</b>) <span class="html-italic">c</span> from 0.628 to 62.8.</p> Full article ">Figure A1
<p>Error between reconstructed Slepian basis and PSWF of order zero with time-bandwidth product <span class="html-italic">c</span> = 1; (<b>a</b>) error between eigenvectors; (<b>b</b>) linear curve fitting with vertical axis taken to be negative inverse of the error between eigenvectors; (<b>c</b>) parabolic curve fitting of the error between eigenvectors; (<b>d</b>) error between eigenvalues; (<b>e</b>) linear curve fitting with vertical axis taken to be negative inverse of the error between eigenvalues; (<b>f</b>) parabolic curve fitting of the error between eigenvalues.</p> Full article ">Figure A2
<p>Error between reconstructed Slepian basis and PSWF of order one with time-bandwidth product <span class="html-italic">c</span> = 1; (<b>a</b>) error between eigenvectors; (<b>b</b>) linear curve fitting with vertical axis taken to be negative inverse of the error between eigenvectors; (<b>c</b>) parabolic curve fitting of the error between eigenvectors; (<b>d</b>) error between eigenvalues; (<b>e</b>) linear curve fitting with vertical axis taken to be negative inverse of the error between eigenvalues; (<b>f</b>) parabolic curve fitting of the error between eigenvalues.</p> Full article ">
21 pages, 6054 KiB  
Article
The Place of Descriptive Geometry in the Face of Industry 4.0 Challenges
by M. Carmen Ladrón-de-Guevara-Muñoz, María Alonso-García, Óscar D. de-Cózar-Macías and E. Beatriz Blázquez-Parra
Symmetry 2023, 15(12), 2190; https://doi.org/10.3390/sym15122190 - 12 Dec 2023
Cited by 1 | Viewed by 1301
Abstract
Industrial process automation has long been the main goal in production lines that seek to decrease human involvement. However, it is broadly agreed that a collaboration between humans and technologies must still exist as human capital is required to provide certain skills and [...] Read more.
Industrial process automation has long been the main goal in production lines that seek to decrease human involvement. However, it is broadly agreed that a collaboration between humans and technologies must still exist as human capital is required to provide certain skills and abilities that machines cannot offer yet. For instance, in the context of design and simulation, CAD, CAM, and CAE professionals must count not only on a deep knowledge of the technology employed but also on specific skills that make the human factor an integral piece of the transition. These abilities are considered fundamental to achieving sustainable development in the industrial sector. This work focuses on analyzing through four study cases where whether specular o bilateral symmetry, a.k.a. planar symmetry is present, the weaknesses found in the human factor related to CAD training of future industrial engineers. The most common mistakes found when developing the different symmetric pieces proposed are thoroughly examined in order to define their origin, which mainly lies in students’ lack of descriptive geometry (DG) understanding. This is aggravated in some cases by the lack of spatial visualization abilities. The unstoppable and fast advances in design and simulation tools and technologies require humans to update their capabilities almost in real time. However, results show that this should not threaten the need for the human mind to spatially understand the changes being made on the screen. Otherwise, humans are at risk of ending up at the service of machines and technologies instead of the opposite. Full article
(This article belongs to the Special Issue Challenges and Trends in the Application of Descriptive Geometry Field under the Current Paradigm of Computer – Assisted Design (CAD))
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<p>Drawings A1 and B1 correspond to raw CAD outcomes when going from 3D to 2D views. A2 and B2 refer to frequent mistakes made by students related to outer and inner intersection lines. A3 and B3 display the rightful form of the drawing depending on whether it relates to a semi-cut or total cut.</p> Full article ">Figure 2
<p>Perspectives showing the piece with a quarter cut or total cut, depending on the case. Red areas highlight the piece faces that generate the front lines the students tend to miss.</p> Full article ">Figure 3
<p>Reinforcement of rectangular cross section.</p> Full article ">Figure 4
<p>Misrepresentations of reinforcements. In (<b>A</b>), the intersection line shown in the front view between the cylinder and the nerve would produce a different top view, where the top surface of the cylinder enlarges to meet the nerve; in (<b>B</b>), the intersection line between the nerve and the cylinder is wrongly placed at the end of the horizontal diameter since it would originate a symmetric void space on each side of the horizontal axis that would change how the top view is displayed; in (<b>C</b>), the end of the line that represents the nerve is wrongly located inside the top circular surface, which would remove part of the cylinder material at that point, showing an incomplete circumference of the cylinder top surface.</p> Full article ">Figure 5
<p>Ellipse section produced in a cylinder by an inclined plane. Points A, B, and C belong to the ellipse.</p> Full article ">Figure 6
<p>Resulting projections when A and C belong to the lid perimeter (top base of the cylinder).</p> Full article ">Figure 7
<p>Graphical determination of <span class="html-italic">d</span> and <span class="html-italic">h</span>.</p> Full article ">Figure 8
<p>Resulting projections of the solution where B is located on the lid perimeter.</p> Full article ">Figure 9
<p>Correct representation of a nerve or reinforcement meeting a cylinder.</p> Full article ">Figure 10
<p>Fundamentals for obtaining curves on nut/bolt faces.</p> Full article ">Figure 11
<p>Front, top, and side views of the hyperbolic curves drawn on the six faces of a nut.</p> Full article ">Figure 12
<p>Error caused by taking the distance between the hexagon opposite vertices as the diameter of the directive circle.</p> Full article ">Figure 13
<p>Correct modeling by taking the distance between faces (e/c) as the diameter of the directive circumference.</p> Full article ">Figure 14
<p>Eccentric end part.</p> Full article ">Figure 15
<p>Detailed engineering drawing of an eccentric end, without machining indications.</p> Full article ">Figure 16
<p>Capture and verification of points 1, 2, 3, etc., that belong to the intersection curves A, B, and C.</p> Full article ">
21 pages, 3809 KiB  
Article
Probing a Hybrid Channel for the Dynamics of Non-Local Features
by Atta ur Rahman, Macheng Yang, Sultan Mahmood Zangi and Congfeng Qiao
Symmetry 2023, 15(12), 2189; https://doi.org/10.3390/sym15122189 - 12 Dec 2023
Cited by 3 | Viewed by 1031
Abstract
Effective information transmission is a central element in quantum information protocols, but the quest for optimal efficiency in channels with symmetrical characteristics remains a prominent challenge in quantum information science. In light of this challenge, we introduce a hybrid channel that encompasses thermal, [...] Read more.
Effective information transmission is a central element in quantum information protocols, but the quest for optimal efficiency in channels with symmetrical characteristics remains a prominent challenge in quantum information science. In light of this challenge, we introduce a hybrid channel that encompasses thermal, magnetic, and local components, each simultaneously endowed with characteristics that enhance and diminish quantum correlations. To investigate the symmetry of this hybrid channel, we explored the quantum correlations of a simple two-qubit Heisenberg spin state, quantified using measures such as negativity, ℓ1-norm coherence, entropic uncertainty, and entropy functions. Our findings revealed that the hybrid channel can be adeptly tailored to preserve quantum correlations, surpassing the capabilities of its individual components. We also identified optimal parameterizations to attain maximum entanglement from mixed entangled/separable states, even in the presence of local dephasing. Notably, various parameters and quantum features, including non-Markovianity, exhibited distinct behaviors in the context of this hybrid channel. Ultimately, we discuss potential experimental applications of this configuration. Full article
(This article belongs to the Section Physics)
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<p>The physical model of the hybrid channel with thermal, magnetic, and classical dephasing parts controlled by static noise employed for the dynamics of the two-qubit Heisenberg spin state characterized by various parameters, such as spin–spin, DM, and KSEA interaction.</p> Full article ">Figure 2
<p>Dynamics of negativity (<b>a</b>), entropic uncertainty (<b>b</b>), <math display="inline"><semantics> <msub> <mo>ℓ</mo> <mn>1</mn> </msub> </semantics></math>-norm coherence (<b>c</b>), and linear entropy (<b>d</b>) as functions of static noise disorder parameter <math display="inline"><semantics> <msub> <mo>Δ</mo> <mi>Q</mi> </msub> </semantics></math> against time in a two-spin system influenced by an external TMCC. For all the plots, we set <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>K</mi> <mi>z</mi> </msub> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>J</mi> <mo>/</mo> <mi>T</mi> <mo>/</mo> <msub> <mi>D</mi> <mi>z</mi> </msub> <mo>/</mo> <msub> <mi>J</mi> <mi>z</mi> </msub> <mo>/</mo> <msub> <mo>Δ</mo> <mi>z</mi> </msub> <mo>/</mo> <mi>B</mi> <mo>/</mo> <mi>J</mi> <mo>/</mo> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>.</p> Full article ">Figure 3
<p>Dynamics of negativity (<b>a</b>), entropic uncertainty (<b>b</b>), <math display="inline"><semantics> <msub> <mo>ℓ</mo> <mn>1</mn> </msub> </semantics></math>-norm coherence (<b>c</b>), and linear entropy (<b>d</b>) as functions of classical field’s coupling strength <math display="inline"><semantics> <mi>λ</mi> </semantics></math> against time in a two-spin state influenced by an external TMCC. For all the plots, we set <math display="inline"><semantics> <mrow> <msub> <mo>Δ</mo> <mi>Q</mi> </msub> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>K</mi> <mi>z</mi> </msub> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>J</mi> <mo>/</mo> <mi>T</mi> <mo>/</mo> <msub> <mi>D</mi> <mi>z</mi> </msub> <mo>/</mo> <msub> <mi>J</mi> <mi>z</mi> </msub> <mo>/</mo> <msub> <mo>Δ</mo> <mi>z</mi> </msub> <mo>/</mo> <mi>B</mi> <mo>/</mo> <mi>J</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>.</p> Full article ">Figure 4
<p>Dynamics of negativity (<b>a</b>), entropic uncertainty (<b>b</b>), <math display="inline"><semantics> <msub> <mo>ℓ</mo> <mn>1</mn> </msub> </semantics></math>-norm coherence (<b>c</b>), and linear entropy (<b>d</b>) as functions of temperature <span class="html-italic">T</span> against time in a two-spin system influenced by an external TMCC. For all the plots, we set <math display="inline"><semantics> <mrow> <msub> <mo>Δ</mo> <mi>Q</mi> </msub> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>K</mi> <mi>z</mi> </msub> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>J</mi> <mo>/</mo> <msub> <mi>D</mi> <mi>z</mi> </msub> <mo>/</mo> <msub> <mi>J</mi> <mi>z</mi> </msub> <mo>/</mo> <msub> <mo>Δ</mo> <mi>z</mi> </msub> <mo>/</mo> <mi>B</mi> <mo>/</mo> <mi>J</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>.</p> Full article ">Figure 5
<p>Dynamics of negativity (<b>a</b>), entropic uncertainty (<b>b</b>), <math display="inline"><semantics> <msub> <mo>ℓ</mo> <mn>1</mn> </msub> </semantics></math>-norm coherence (<b>c</b>), and linear entropy (<b>d</b>) as functions of KSEA interaction along the <span class="html-italic">z</span>-axis <math display="inline"><semantics> <msub> <mi>K</mi> <mi>z</mi> </msub> </semantics></math> against time in a two-spin state influenced by an external TMCC. For all the plots, we set <math display="inline"><semantics> <mrow> <msub> <mo>Δ</mo> <mi>Q</mi> </msub> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>J</mi> <mo>/</mo> <msub> <mi>D</mi> <mi>z</mi> </msub> <mo>/</mo> <msub> <mi>J</mi> <mi>z</mi> </msub> <mo>/</mo> <msub> <mo>Δ</mo> <mi>z</mi> </msub> <mo>/</mo> <mi>B</mi> <mo>/</mo> <mi>J</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>.</p> Full article ">Figure 6
<p>Dynamics of negativity (<b>a</b>), entropic uncertainty (<b>b</b>), <math display="inline"><semantics> <msub> <mo>ℓ</mo> <mn>1</mn> </msub> </semantics></math>-norm coherence (<b>c</b>), and linear entropy (<b>d</b>) as functions of magnetic field strength <span class="html-italic">B</span> against time in a two-spin state influenced by an external TMCC. For all the plots, we set <math display="inline"><semantics> <mrow> <msub> <mo>Δ</mo> <mi>Q</mi> </msub> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>K</mi> <mi>z</mi> </msub> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>J</mi> <mo>/</mo> <msub> <mi>D</mi> <mi>z</mi> </msub> <mo>/</mo> <msub> <mi>J</mi> <mi>z</mi> </msub> <mo>/</mo> <msub> <mo>Δ</mo> <mi>z</mi> </msub> <mo>/</mo> <mi>J</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>.</p> Full article ">Figure 7
<p>Dynamics of negativity (<b>a</b>), entropic uncertainty (<b>b</b>), <math display="inline"><semantics> <msub> <mo>ℓ</mo> <mn>1</mn> </msub> </semantics></math>-norm coherence (<b>c</b>), and linear entropy (<b>d</b>) as functions of DM interaction strength along the <span class="html-italic">z</span>-axis <math display="inline"><semantics> <msub> <mi>D</mi> <mi>z</mi> </msub> </semantics></math> against time in a two-spin state influenced by an external TMCC. For all the plots, we set <math display="inline"><semantics> <mrow> <msub> <mo>Δ</mo> <mi>Q</mi> </msub> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>K</mi> <mi>z</mi> </msub> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>J</mi> <mo>/</mo> <msub> <mi>D</mi> <mi>z</mi> </msub> <mo>/</mo> <msub> <mo>Δ</mo> <mi>z</mi> </msub> <mo>/</mo> <mi>B</mi> <mo>/</mo> <mi>J</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>.</p> Full article ">Figure 8
<p>Dynamics of negativity (<b>a</b>), entropic uncertainty (<b>b</b>), <math display="inline"><semantics> <msub> <mo>ℓ</mo> <mn>1</mn> </msub> </semantics></math>-norm coherence (<b>c</b>), and linear entropy (<b>d</b>) as functions of the symmetric exchange spin-spin interaction strength in the <span class="html-italic">z</span>-direction along <math display="inline"><semantics> <msub> <mo>Δ</mo> <mi>z</mi> </msub> </semantics></math> against time in a two-spin state influenced by an external TMCC. For all the plots, we set <math display="inline"><semantics> <mrow> <msub> <mo>Δ</mo> <mi>Q</mi> </msub> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>K</mi> <mi>z</mi> </msub> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>J</mi> <mo>/</mo> <msub> <mi>D</mi> <mi>z</mi> </msub> <mo>/</mo> <mi>B</mi> <mo>/</mo> <mi>J</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>.</p> Full article ">Figure 9
<p>Dynamics of negativity (<b>a</b>), entropic uncertainty (<b>b</b>), <math display="inline"><semantics> <msub> <mo>ℓ</mo> <mn>1</mn> </msub> </semantics></math>-norm coherence (<b>c</b>), and linear entropy (<b>d</b>) as functions of the Heisenberg exchange interaction strength <span class="html-italic">J</span> against time in a two-spin state influenced by an external TMCC. For all the plots, we set <math display="inline"><semantics> <mrow> <msub> <mo>Δ</mo> <mi>Q</mi> </msub> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>K</mi> <mi>z</mi> </msub> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>T</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>J</mi> <mo>/</mo> <msub> <mi>D</mi> <mi>z</mi> </msub> <mo>/</mo> <mi>B</mi> <mo>/</mo> <mi>J</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>.</p> Full article ">Figure 10
<p>(<b>a</b>) Dynamics of fidelity between states <math display="inline"><semantics> <mrow> <msub> <mi>ρ</mi> <mrow> <mi>s</mi> <mi>t</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>,</mo> <mi>T</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>ρ</mi> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mi>T</mi> <mo>)</mo> </mrow> </semantics></math> for the two-qubit case when exposed to the hybrid channel against various strengths of classical dephasing while setting <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>K</mi> <mi>z</mi> </msub> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>T</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>J</mi> <mo>/</mo> <msub> <mi>D</mi> <mi>z</mi> </msub> <mo>/</mo> <mi>B</mi> <mo>/</mo> <mi>J</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>. (<b>b</b>) Same as (<b>a</b>), but for <math display="inline"><semantics> <mrow> <msub> <mi>ρ</mi> <mrow> <mi>s</mi> <mi>t</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>,</mo> <mi>T</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> and <math display="inline"><semantics> <msub> <mi>ρ</mi> <mn>0</mn> </msub> </semantics></math>.</p> Full article ">
19 pages, 2352 KiB  
Article
The Contribution of the Corpus Callosum to the Symmetrical Representation of Taste in the Human Brain: An fMRI Study of Callosotomized Patients
by Gabriele Polonara, Giulia Mascioli, Ugo Salvolini, Aldo Paggi, Tullio Manzoni and Mara Fabri
Symmetry 2023, 15(12), 2188; https://doi.org/10.3390/sym15122188 - 12 Dec 2023
Cited by 2 | Viewed by 1293
Abstract
The present study was designed to establish the contribution of the corpus callosum (CC) to the cortical representation of unilateral taste stimuli in the human primary gustatory area (GI). Unilateral taste stimulation of the tongue was applied to eight patients with partial or [...] Read more.
The present study was designed to establish the contribution of the corpus callosum (CC) to the cortical representation of unilateral taste stimuli in the human primary gustatory area (GI). Unilateral taste stimulation of the tongue was applied to eight patients with partial or total callosal resection by placing a small cotton pad soaked in a salty solution on either side of the tongue. Functional images were acquired with a 1.5 Tesla machine. Diffusion tensor imaging and tractography were also performed. Unilateral taste stimuli evoked bilateral activation of the GI area in all patients, including those with total resection of the CC, with a prevalence in the ipsilateral hemisphere to the stimulated tongue side. Bilateral activation was also observed in the primary somatic sensory cortex (SI) in most patients, which was more intense in the contralateral SI. This report confirms previous functional studies carried out in control subjects and neuropsychological findings in callosotomized patients, showing that gustatory pathways from tongue to cortex are bilaterally distributed, with an ipsilateral predominance. It has been shown that the CC does play a role, although not an exclusive one, in the bilateral symmetrical representation of gustatory sensitivity in the GI area, at least for afferents from one side of the tongue. Full article
(This article belongs to the Special Issue Symmetry and Asymmetry in Brain Behavior and Perception II)
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<p>Midsagittal image showing the extent of callosal resection in the 8 patients, showing total callosal resection in 3 patients (P2, P3, P5), partial posterior resection in 1 patient (P7), partial anterior resection sparing the splenium only (P9), the splenium and the callosal body (P11, P12) and partial central resection (P18).</p> Full article ">Figure 2
<p>Gustatory stimulation protocols used in the present study. Both protocols were on–off block-designed stimulation paradigm lasting 5 min. The first version (Protocol 1, top track) was composed of an initial 60 s rest, followed by two 30 s stimulus-on periods (red tracks), each followed by 90 s rest periods. The second version (Protocol 2, bottom track) included an initial 30 s period of rest followed by five 15 s stimulus-on periods (red tracks) interleaved with four 45 s rest periods, with a final 15 s stimulus-off period to complete the 5 min run.</p> Full article ">Figure 3
<p>Bilateral activation of GI area. (<b>A</b>,<b>B</b>) Salty stimulation of the left tongue in patients P3 and P2, respectively, both with total callosotomy. (<b>C</b>) Salty stimulation of right tongue in patient P7, with partial posterior resection. (<b>D</b>) Salty stimulation of right tongue in patient P18, with central callosotomy. (<b>E</b>,<b>F</b>) Salty stimulation of right tongue in patients P12 and P11, respectively, both with partial anterior resection. Green arrows, ipsilateral GI activation; red arrows, contralateral GI activation. R, right. Left hemisphere is shown on the right, according to the radiological convention.</p> Full article ">Figure 4
<p>Bilateral activation of SI area. (<b>A</b>,<b>B</b>) Salty stimulation of the left tongue in patients P3 and P2, respectively, both with total callosotomy. (<b>C</b>) Salty stimulation of right tongue in patient P7, with partial posterior resection. (<b>D</b>) Salty stimulation of right tongue in patient P18, with central callosotomy. (<b>E</b>,<b>F</b>) Salty stimulation of right tongue in patients P12 and P11, respectively, both with partial anterior resection. Green arrows, contralateral SI activation; red arrows, ipsilateral SI activation. R, right. Left hemisphere is shown on the right, according to the radiological convention.</p> Full article ">Figure 5
<p>(<b>A</b>) Activation in the PBN on both sides in patient P2 after salty stimulation of the right hemitongue. (<b>B</b>) In the same patient (P2), fibers from right GI area connect to ipsilateral PBN (yellow arrow), cross the midline toward the contralateral PBN and then connect to contralateral GI. (<b>C</b>,<b>D</b>) Similar pathways connecting GI areas in the two hemispheres are observed in another totally callosomized patient, P5, and in a patient with anterior callosotomy, P12, in whom the callosal fibers between GI areas are interrupted. Green arrows indicate the activation in bilateral PBN; yellow arrows indicate the fiber bundle crossing the midline at pontine level. R, right. Left hemisphere is shown on the right, according to the radiological convention.</p> Full article ">Figure 6
<p>Model of possible pathways explaining the bilateral activation of GI area, as well as in absence of the CC: the crossing point has been hypothesized at pontine level (<b>A</b>, arrow) or at medulla level (<b>B</b>, arrow). GI, primary gustatory area; NTS, nucleus of solitary tract; PBN, parabrachialis nucleus; VPMpc, ventroposteromedial nucleus, parvicellular portion.</p> Full article ">
15 pages, 424 KiB  
Article
A New Cosine-Originated Probability Distribution with Symmetrical and Asymmetrical Behaviors: Repetitive Acceptance Sampling with Reliability Application
by Huda M. Alshanbari, Gadde Srinivasa Rao, Jin-Taek Seong, Sultan Salem and Saima K. Khosa
Symmetry 2023, 15(12), 2187; https://doi.org/10.3390/sym15122187 - 12 Dec 2023
Cited by 2 | Viewed by 1374
Abstract
Several new acceptance sampling plans using various probability distribution methods have been developed in the literature. However, there is no published work on the design of new sampling plans using trigonometric-based probability distributions. In order to cover this amazing and fascinating research gap, [...] Read more.
Several new acceptance sampling plans using various probability distribution methods have been developed in the literature. However, there is no published work on the design of new sampling plans using trigonometric-based probability distributions. In order to cover this amazing and fascinating research gap, we first introduce a novel probabilistic method called a new modified cosine-G method. A special member of the new modified cosine-G method, namely, a new modified cosine-Weibull distribution, is examined and implemented. The density function of the new model possesses symmetrical as well as asymmetrical behaviors. The usefulness and superior fitting power of the new modified cosine-Weibull distribution are demonstrated by analyzing an asymmetrical data set. Furthermore, based on the new modified cosine-Weibull distribution, we develop a new repetitive acceptance sampling strategy for attributes with specified shape parameters. Finally, a real-world application is presented to illustrate the proposed repetitive acceptance sampling strategy. Full article
(This article belongs to the Special Issue Skewed (Asymmetrical) Probability Distributions and Applications across Disciplines III)
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<p>The visual representations for (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>F</mi> <mfenced separators="" open="(" close=")"> <mi>x</mi> <mo>;</mo> <mi mathvariant="bold-italic">η</mi> </mfenced> </mrow> </semantics></math> and (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>S</mi> <mfenced separators="" open="(" close=")"> <mi>x</mi> <mo>;</mo> <mi mathvariant="bold-italic">η</mi> </mfenced> </mrow> </semantics></math> of the NMC-Weibull distribution for <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mfenced separators="" open="(" close=")"> <mn>1.2</mn> <mo>,</mo> <mn>3.6</mn> <mo>,</mo> <mn>4.5</mn> <mo>,</mo> <mn>6.4</mn> </mfenced> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mfenced separators="" open="(" close=")"> <mn>0.3</mn> <mo>,</mo> <mn>0.1</mn> <mo>,</mo> <mn>0.02</mn> <mo>,</mo> <mn>0.001</mn> </mfenced> </mrow> </semantics></math>.</p> Full article ">Figure 2
<p>The PDF <math display="inline"><semantics> <mrow> <mi>f</mi> <mfenced separators="" open="(" close=")"> <mi>x</mi> <mo>;</mo> <mi mathvariant="bold-italic">η</mi> </mfenced> </mrow> </semantics></math> plots for <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mfenced separators="" open="(" close=")"> <mn>0.5</mn> <mo>,</mo> <mn>3.6</mn> <mo>,</mo> <mn>4.5</mn> <mo>,</mo> <mn>6.4</mn> </mfenced> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mfenced separators="" open="(" close=")"> <mn>0.2</mn> <mo>,</mo> <mn>0.1</mn> <mo>,</mo> <mn>0.02</mn> <mo>,</mo> <mn>0.001</mn> </mfenced> </mrow> </semantics></math>.</p> Full article ">Figure 3
<p>The HF <math display="inline"><semantics> <mrow> <mi>h</mi> <mfenced separators="" open="(" close=")"> <mi>x</mi> <mo>;</mo> <mi mathvariant="bold-italic">η</mi> </mfenced> </mrow> </semantics></math> plots for <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mfenced separators="" open="(" close=")"> <mn>0.5</mn> <mo>,</mo> <mn>1.4</mn> <mo>,</mo> <mn>1.1</mn> </mfenced> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mfenced separators="" open="(" close=")"> <mn>0.2</mn> <mo>,</mo> <mn>0.4</mn> <mo>,</mo> <mn>0.7</mn> </mfenced> </mrow> </semantics></math>.</p> Full article ">Figure 4
<p>Some baseline plots for the secondary reactor pump data set.</p> Full article ">Figure 5
<p>The log-likelihood profiles of <math display="inline"><semantics> <msub> <mover accent="true"> <mi>α</mi> <mo stretchy="false">^</mo> </mover> <mrow> <mi>M</mi> <mi>L</mi> <mi>E</mi> </mrow> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mover accent="true"> <mi>τ</mi> <mo stretchy="false">^</mo> </mover> <mrow> <mi>M</mi> <mi>L</mi> <mi>E</mi> </mrow> </msub> </semantics></math> of the NMC-Weibull model for the secondary reactor pumps data.</p> Full article ">Figure 6
<p>The optimal fitting comparison of the rival distributions for the data from the secondary reactor pumps.</p> Full article ">
16 pages, 333 KiB  
Article
Binomial Series-Confluent Hypergeometric Distribution and Its Applications on Subclasses of Multivalent Functions
by Ibtisam Aldawish, Sheza M. El-Deeb and Gangadharan Murugusundaramoorthy
Symmetry 2023, 15(12), 2186; https://doi.org/10.3390/sym15122186 - 11 Dec 2023
Cited by 1 | Viewed by 1159
Abstract
Over the past ten years, analytical functions’ reputation in the literature and their application have grown. We study some practical issues pertaining to multivalent functions with bounded boundary rotation that associate with the combination of confluent hypergeometric functions and binomial series in this [...] Read more.
Over the past ten years, analytical functions’ reputation in the literature and their application have grown. We study some practical issues pertaining to multivalent functions with bounded boundary rotation that associate with the combination of confluent hypergeometric functions and binomial series in this research. A novel subset of multivalent functions is established through the use of convolution products and specific inclusion properties are examined through the application of second order differential inequalities in the complex plane. Furthermore, for multivalent functions, we examined inclusion findings using Bernardi integral operators. Moreover, we will demonstrate how the class proposed in this study, in conjunction with the acquired results, generalizes other well-known (or recently discovered) works that are called out as exceptions in the literature. Full article
(This article belongs to the Special Issue Symmetry in Geometric Theory of Analytic Functions)
18 pages, 2117 KiB  
Article
A Novel Approach for Data Feature Weighting Using Correlation Coefficients and Min–Max Normalization
by Mohammed Shantal, Zalinda Othman and Azuraliza Abu Bakar
Symmetry 2023, 15(12), 2185; https://doi.org/10.3390/sym15122185 - 11 Dec 2023
Cited by 21 | Viewed by 3749
Abstract
In the realm of data analysis and machine learning, achieving an optimal balance of feature importance, known as feature weighting, plays a pivotal role, especially when considering the nuanced interplay between the symmetry of data distribution and the need to assign differential weights [...] Read more.
In the realm of data analysis and machine learning, achieving an optimal balance of feature importance, known as feature weighting, plays a pivotal role, especially when considering the nuanced interplay between the symmetry of data distribution and the need to assign differential weights to individual features. Also, avoiding the dominance of large-scale traits is essential in data preparation. This step makes choosing an effective normalization approach one of the most challenging aspects of machine learning. In addition to normalization, feature weighting is another strategy to deal with the importance of the different features. One of the strategies to measure the dependency of features is the correlation coefficient. The correlation between features shows the relationship strength between the features. The integration of the normalization method with feature weighting in data transformation for classification has not been extensively studied. The goal is to improve the accuracy of classification methods by striking a balance between the normalization step and assigning greater importance to features with a strong relation to the class feature. To achieve this, we combine Min–Max normalization and weight the features by increasing their values based on their correlation coefficients with the class feature. This paper presents a proposed Correlation Coefficient with Min–Max Weighted (CCMMW) approach. The data being normalized depends on their correlation with the class feature. Logistic regression, support vector machine, k-nearest neighbor, neural network, and naive Bayesian classifiers were used to evaluate the proposed method. Twenty UCI Machine Learning Repository and Kaggle datasets with numerical values were also used in this study. The empirical results showed that the proposed CCMMW significantly improves the classification performance through support vector machine, logistic regression, and neural network classifiers in most datasets. Full article
(This article belongs to the Topic Decision-Making and Data Mining for Sustainable Computing)
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Figure 1
<p>The general CCMMW methodology.</p> Full article ">Figure 2
<p>The CCMMW performance on different classifiers. (<b>a</b>) LR. (<b>b</b>) SVM. (<b>c</b>) k-NN. (<b>d</b>) NN. (<b>e</b>) NB.</p> Full article ">Figure 2 Cont.
<p>The CCMMW performance on different classifiers. (<b>a</b>) LR. (<b>b</b>) SVM. (<b>c</b>) k-NN. (<b>d</b>) NN. (<b>e</b>) NB.</p> Full article ">Figure 3
<p>CCMMW performance: average accuracy (20 datasets) of five classification methods (classifiers).</p> Full article ">Figure 4
<p>The value of C. (<b>a</b>) LR. (<b>b</b>) SVM. (<b>c</b>) k-NN. (<b>d</b>) NN. (<b>e</b>) NB.</p> Full article ">Figure 5
<p>Value range of C. (<b>a</b>) LR. (<b>b</b>) SVM. (<b>c</b>) k-NN. (<b>d</b>) NN. (<b>e</b>) NB.</p> Full article ">
20 pages, 393 KiB  
Review
Dynamical Asymmetries, the Bayes’ Theorem, Entanglement, and Intentionality in the Brain Functional Activity
by David Bernal-Casas and Giuseppe Vitiello
Symmetry 2023, 15(12), 2184; https://doi.org/10.3390/sym15122184 - 11 Dec 2023
Cited by 1 | Viewed by 1692
Abstract
We discuss the asymmetries of dynamical origin that are relevant to functional brain activity. The brain is permanently open to its environment, and its dissipative dynamics is characterized indeed by the asymmetries under time translation transformations and time-reversal transformations, which manifest themselves in [...] Read more.
We discuss the asymmetries of dynamical origin that are relevant to functional brain activity. The brain is permanently open to its environment, and its dissipative dynamics is characterized indeed by the asymmetries under time translation transformations and time-reversal transformations, which manifest themselves in the irreversible “arrow of time”. Another asymmetry of dynamical origin arises from the breakdown of the rotational symmetry of molecular electric dipoles, triggered by incoming stimuli, which manifests in long-range dipole-dipole correlations favoring neuronal correlations. In the dissipative model, neurons, glial cells, and other biological components are classical structures. The dipole vibrational fields are quantum variables. We review the quantum field theory model of the brain proposed by Ricciardi and Umezawa and its subsequent extension to dissipative dynamics. We then show that Bayes’ theorem in probability theory is intrinsic to the structure of the brain states and discuss its strict relation with entanglement phenomena and free energy minimization. The brain estimates the action with a higher Bayes probability to be taken to produce the aimed effect. Bayes’ rule provides the formal basis of the intentionality in brain activity, which we also discuss in relation to mind and consciousness. Full article
(This article belongs to the Special Issue The Study of Brain Asymmetry)
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