Mathematics | An Open Access Journal from MDPI

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class="custom-accordion-for-small-screen-content"> <div class="expanding-div collapsed"> <div class="generic-item article-item no-border"> <div class="article-content"> <div class="label right label__btn"> <span style="font-size: 12px; color: #1a1a1a;"> 21 pages, 4740 KiB   </span> <a href="/2227-7390/12/23/3827/pdf?version=1733244755" class="UD_Listings_ArticlePDF" title="Article PDF" data-name="Multi-Scale Geometric Feature Extraction and Global Transformer for Real-World Indoor Point Cloud Analysis" data-journal="mathematics"> <i class="material-icons custom-download"></i> </a> </div> <div class="article-icons"><span class="label openaccess" data-dropdown="drop-article-label-openaccess" aria-expanded="false">Open Access</span><span class="label articletype">Article</span></div> <a class="title-link" href="/2227-7390/12/23/3827">Multi-Scale Geometric Feature Extraction and Global Transformer for Real-World Indoor Point Cloud Analysis</a> <div class="authors"> by <span class="inlineblock "><strong>Yisheng Chen</strong>, </span><span class="inlineblock "><strong>Yu Xiao</strong>, </span><span class="inlineblock "><strong>Hui Wu</strong>, </span><span class="inlineblock "><strong>Chongcheng Chen</strong> and </span><span class="inlineblock "><strong>Ding Lin</strong></span> </div> <div class="color-grey-dark"> <em>Mathematics</em> <b>2024</b>, <em>12</em>(23), 3827; https://doi.org/10.3390/math12233827 (registering DOI) - 3 Dec 2024 </div> <div class="abstract-div"> <a href="#" onclick="$(this).next('.abstract-cropped').toggleClass('inline').next('.abstract-full').toggleClass('inline'); return false;"> <strong>Abstract </strong> </a> <div class="abstract-cropped inline"> Indoor point clouds often present significant challenges due to the complexity and variety of structures and high object similarity. The local geometric structure helps the model learn the shape features of objects at the detail level, while the global context provides overall scene <a href="#" data-counterslink = "https://www.mdpi.com/2227-7390/12/23/3827/more" onclick="$(this).parents('.abstract-cropped').toggleClass('inline').next('.abstract-full').toggleClass('inline'); return false;"> [...] Read more.</a> </div> <div class="abstract-full "> Indoor point clouds often present significant challenges due to the complexity and variety of structures and high object similarity. The local geometric structure helps the model learn the shape features of objects at the detail level, while the global context provides overall scene semantics and spatial relationship information between objects. To address these challenges, we propose a novel network architecture, PointMSGT, which includes a multi-scale geometric feature extraction (MSGFE) module and a global Transformer (GT) module. The MSGFE module consists of a geometric feature extraction (GFE) module and a multi-scale attention (MSA) module. The GFE module reconstructs triangles through each point’s two neighbors and extracts detailed local geometric relationships by the triangle’s centroid, normal vector, and plane constant. The MSA module extracts features through multi-scale convolutions and adaptively aggregates features, focusing on both local geometric details and global semantic information at different scale levels, enhancing the understanding of complex scenes. The global Transformer employs a self-attention mechanism to capture long-range dependencies across the entire point cloud. The proposed method demonstrates competitive performance in real-world indoor scenarios, with a mIoU of 68.6% in semantic segmentation on S3DIS and OA of 86.4% in classification on ScanObjectNN. <a href="/2227-7390/12/23/3827">Full article</a> </div> </div> <div class="belongsTo" style="margin-bottom: 10px;"> (This article belongs to the Special Issue <a href=" /journal/mathematics/special_issues/Machine_Learning_Mathematical_Modeling_Applications ">Machine Learning Methods and Mathematical Modeling with Applications</a>)<br/> </div> <a href="#" class="abstract-figures-show" data-counterslink = "https://www.mdpi.com/2227-7390/12/23/3827/show" ><span >►</span><span style=" display: none;">▼</span> Show Figures </a><div class="abstract-image-preview "><div class="arrow left-arrow" id="prev1536112"><i class="fa fa-caret-left"></i></div><div class="arrow right-arrow" id="next1536112"><i class="fa fa-caret-right"></i></div><div class="absgraph cycle-slideshow manual" data-cycle-fx="scrollHorz" data-cycle-timeout="0" data-cycle-next="#next1536112" data-cycle-prev="#prev1536112" data-cycle-progressive="#images1536112" data-cycle-slides=">div" data-cycle-log="false"><div class='openpopupgallery cycle-slide' data-imgindex='0' data-target='article-1536112-popup'><span class="helper"></span><img src="data:image/gif;base64,R0lGODlhAQABAAD/ACwAAAAAAQABAAACADs=" data-src="https://pub.mdpi-res.com/mathematics/mathematics-12-03827/article_deploy/html/images/mathematics-12-03827-g001-550.jpg?1733244896" alt="" style="border: 0;"><p>Figure 1</p></div><script id="images1536112" type="text/cycle" data-cycle-split="---"><div class='openpopupgallery' data-imgindex='1' data-target='article-1536112-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03827/article_deploy/html/images/mathematics-12-03827-g002-550.jpg?1733244897'><p>Figure 2</p></div> --- <div class='openpopupgallery' data-imgindex='2' data-target='article-1536112-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03827/article_deploy/html/images/mathematics-12-03827-g003-550.jpg?1733244898'><p>Figure 3</p></div> --- <div class='openpopupgallery' data-imgindex='3' data-target='article-1536112-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03827/article_deploy/html/images/mathematics-12-03827-g004-550.jpg?1733244899'><p>Figure 4</p></div> --- <div class='openpopupgallery' data-imgindex='4' data-target='article-1536112-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03827/article_deploy/html/images/mathematics-12-03827-g005-550.jpg?1733244900'><p>Figure 5</p></div> --- <div class='openpopupgallery' data-imgindex='5' data-target='article-1536112-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03827/article_deploy/html/images/mathematics-12-03827-g006-550.jpg?1733244901'><p>Figure 6</p></div> --- <div class='openpopupgallery' data-imgindex='6' data-target='article-1536112-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03827/article_deploy/html/images/mathematics-12-03827-g007-550.jpg?1733244902'><p>Figure 7</p></div> --- <div class='openpopupgallery' data-imgindex='7' data-target='article-1536112-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03827/article_deploy/html/images/mathematics-12-03827-g008-550.jpg?1733244903'><p>Figure 8</p></div> --- <div class='openpopupgallery' data-imgindex='8' data-target='article-1536112-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03827/article_deploy/html/images/mathematics-12-03827-g009-550.jpg?1733244905'><p>Figure 9</p></div></script></div></div><div id="article-1536112-popup" class="popupgallery" style="display: inline; line-height: 200%"><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03827/article_deploy/html/images/mathematics-12-03827-g001-550.jpg?1733244896" title=" <strong>Figure 1</strong><br/> <p>Architecture of the proposed network.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3827'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03827/article_deploy/html/images/mathematics-12-03827-g002-550.jpg?1733244897" title=" <strong>Figure 2</strong><br/> <p>Multi-scale geometric feature extraction.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3827'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03827/article_deploy/html/images/mathematics-12-03827-g003-550.jpg?1733244898" title=" <strong>Figure 3</strong><br/> <p>Multi-scale attention module.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3827'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03827/article_deploy/html/images/mathematics-12-03827-g004-550.jpg?1733244899" title=" <strong>Figure 4</strong><br/> <p>Global Transformer.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3827'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03827/article_deploy/html/images/mathematics-12-03827-g005-550.jpg?1733244900" title=" <strong>Figure 5</strong><br/> <p>Visualization comparison of semantic segmentation results on S3DIS. The red rectangular boxes indicate the contrasting regions.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3827'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03827/article_deploy/html/images/mathematics-12-03827-g006-550.jpg?1733244901" title=" <strong>Figure 6</strong><br/> <p>Performance of different numbers of input points.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3827'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03827/article_deploy/html/images/mathematics-12-03827-g007-550.jpg?1733244902" title=" <strong>Figure 7</strong><br/> <p>Confusion matrix on S3DIS Area 5 with the comparison of the baseline (<b>a</b>) and our PointMSGT (<b>b</b>). The confusion matrix indicates that the overall performance of our proposed method is superior to that of the baseline.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3827'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03827/article_deploy/html/images/mathematics-12-03827-g008-550.jpg?1733244903" title=" <strong>Figure 8</strong><br/> <p>Visual comparison of baseline and our method on S3DIS. The yellow rectangular boxes indicate the contrasting regions.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3827'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03827/article_deploy/html/images/mathematics-12-03827-g009-550.jpg?1733244905" title=" <strong>Figure 9</strong><br/> <p>Ablation study of convolution kernels. The red rectangular boxes indicate the contrasting regions.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3827'>Full article</a></strong> "></a></div> </div> </div> </div> <div class="extending-content content-ready"> <div class="expanding-div collapsed"> <div class="generic-item article-item"> <div class="article-content"> <div class="label right label__btn"> <span style="font-size: 12px; color: #1a1a1a;"> 26 pages, 802 KiB   </span> <a href="/2227-7390/12/23/3826/pdf?version=1733226536" class="UD_Listings_ArticlePDF" title="Article PDF" data-name="Variable Dose-Constraints Method for Enhancing Intensity-Modulated Radiation Therapy Treatment Planning" data-journal="mathematics"> <i class="material-icons custom-download"></i> </a> </div> <div class="article-icons"><span class="label openaccess" data-dropdown="drop-article-label-openaccess" aria-expanded="false">Open Access</span><span class="label articletype">Article</span></div> <a class="title-link" href="/2227-7390/12/23/3826">Variable Dose-Constraints Method for Enhancing Intensity-Modulated Radiation Therapy Treatment Planning</a> <div class="authors"> by <span class="inlineblock "><strong>Norihisa Obata</strong>, </span><span class="inlineblock "><strong>Omar M. Abou Al-Ola</strong>, </span><span class="inlineblock "><strong>Ryosei Nakada</strong>, </span><span class="inlineblock "><strong>Takeshi Kojima</strong> and </span><span class="inlineblock "><strong>Tetsuya Yoshinaga</strong></span> </div> <div class="color-grey-dark"> <em>Mathematics</em> <b>2024</b>, <em>12</em>(23), 3826; <a href="https://doi.org/10.3390/math12233826">https://doi.org/10.3390/math12233826</a> - 3 Dec 2024 </div> <div class="abstract-div"> <a href="#" onclick="$(this).next('.abstract-cropped').toggleClass('inline').next('.abstract-full').toggleClass('inline'); return false;"> <strong>Abstract </strong> </a> <div class="abstract-cropped inline"> The conventional approach to intensity-modulated radiation therapy treatment planning involves two distinct strategies: optimizing an evaluation function while accounting for dose constraints, and solving feasibility problems using feasibility-seeking projection methods that incorporate inequality constraints. This paper introduces a novel iterative scheme within the <a href="#" data-counterslink = "https://www.mdpi.com/2227-7390/12/23/3826/more" onclick="$(this).parents('.abstract-cropped').toggleClass('inline').next('.abstract-full').toggleClass('inline'); return false;"> [...] Read more.</a> </div> <div class="abstract-full "> The conventional approach to intensity-modulated radiation therapy treatment planning involves two distinct strategies: optimizing an evaluation function while accounting for dose constraints, and solving feasibility problems using feasibility-seeking projection methods that incorporate inequality constraints. This paper introduces a novel iterative scheme within the framework of continuous dynamical systems, wherein constraint conditions dynamically evolve to enhance the optimization process. The validity of dynamically varying dose constraints is theoretically established through the foundation of continuous-time dynamical systems theory. In particular, we formalize a system of differential equations, with both beam coefficients and dose constraints modeled as state variables. The asymptotic stability of the system’s equilibrium is rigorously proven, ensuring convergence to a solution. In practical terms, we leverage a discretized iteration formula derived from the continuous-time system to achieve rapid computational speed. The mathematical structure of the proposed approach, which directly incorporates dose-volume constraints into the objective function, facilitates significant computational efficiency and solution refinement. The proposed method has an inherent dynamics that approaches more desirable solutions within the set of solutions when the solution to the optimization problem is not an isolated point. This property guarantees the identification of optimal solutions that respect the prescribed dose-volume constraints while enhancing accuracy when such constraints are feasible. By treating dose constraints as variables and concurrently solving the optimization problem with beam coefficients, we can achieve more accurate results when compared with using fixed values for prescribed dose conditions. <a href="/2227-7390/12/23/3826">Full article</a> </div> </div> <a href="#" class="abstract-figures-show" data-counterslink = "https://www.mdpi.com/2227-7390/12/23/3826/show" ><span >►</span><span style=" display: none;">▼</span> Show Figures </a><div class="abstract-image-preview "><div class="arrow left-arrow" id="prev1535858"><i class="fa fa-caret-left"></i></div><div class="arrow right-arrow" id="next1535858"><i class="fa fa-caret-right"></i></div><div class="absgraph cycle-slideshow manual" data-cycle-fx="scrollHorz" data-cycle-timeout="0" data-cycle-next="#next1535858" data-cycle-prev="#prev1535858" data-cycle-progressive="#images1535858" data-cycle-slides=">div" data-cycle-log="false"><div class='openpopupgallery cycle-slide' data-imgindex='0' 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src='https://pub.mdpi-res.com/mathematics/mathematics-12-03826/article_deploy/html/images/mathematics-12-03826-g007-550.jpg?1733226656'><p>Figure 7</p></div> --- <div class='openpopupgallery' data-imgindex='7' data-target='article-1535858-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03826/article_deploy/html/images/mathematics-12-03826-g008-550.jpg?1733226658'><p>Figure 8</p></div> --- <div class='openpopupgallery' data-imgindex='8' data-target='article-1535858-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03826/article_deploy/html/images/mathematics-12-03826-g009-550.jpg?1733226660'><p>Figure 9</p></div> --- <div class='openpopupgallery' data-imgindex='9' data-target='article-1535858-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03826/article_deploy/html/images/mathematics-12-03826-g010-550.jpg?1733226661'><p>Figure 10</p></div> --- <div class='openpopupgallery' data-imgindex='10' data-target='article-1535858-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03826/article_deploy/html/images/mathematics-12-03826-g011-550.jpg?1733226662'><p>Figure 11</p></div> --- <div class='openpopupgallery' data-imgindex='11' data-target='article-1535858-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03826/article_deploy/html/images/mathematics-12-03826-g012-550.jpg?1733226665'><p>Figure 12</p></div> --- <div class='openpopupgallery' data-imgindex='12' data-target='article-1535858-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03826/article_deploy/html/images/mathematics-12-03826-g013-550.jpg?1733226666'><p>Figure 13</p></div> --- <div class='openpopupgallery' data-imgindex='13' data-target='article-1535858-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03826/article_deploy/html/images/mathematics-12-03826-g014-550.jpg?1733226668'><p>Figure 14</p></div> --- <div class='openpopupgallery' data-imgindex='14' data-target='article-1535858-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03826/article_deploy/html/images/mathematics-12-03826-g015-550.jpg?1733226669'><p>Figure 15</p></div> --- <div class='openpopupgallery' data-imgindex='15' data-target='article-1535858-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03826/article_deploy/html/images/mathematics-12-03826-g016-550.jpg?1733226672'><p>Figure 16</p></div></script></div></div><div id="article-1535858-popup" class="popupgallery" style="display: inline; line-height: 200%"><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03826/article_deploy/html/images/mathematics-12-03826-g001-550.jpg?1733226648" title=" <strong>Figure 1</strong><br/> <p>Schematic diagram of the trajectory in dose plane.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3826'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03826/article_deploy/html/images/mathematics-12-03826-g002-550.jpg?1733226650" title=" <strong>Figure 2</strong><br/> <p>Schematic diagram of a projected subspace <math display="inline"><semantics> <mi mathvariant="script">X</mi> </semantics></math> that is close to equilibrium in the space of (<b>a</b>) beamlets and (<b>b</b>) corresponding doses.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3826'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03826/article_deploy/html/images/mathematics-12-03826-g003-550.jpg?1733226651" title=" <strong>Figure 3</strong><br/> <p>(<b>a</b>) Phantom image of the C-shape (the red region represents Target as PTV, while the blue and dark gray regions, respectively, represent Core and Body as OARs) and examples of the dose distributions for (<b>b</b>) Core, (<b>c</b>) Target, and (<b>d</b>) Body (the unit of values indicated on the color bar is Gy).</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3826'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03826/article_deploy/html/images/mathematics-12-03826-g004-550.jpg?1733226652" title=" <strong>Figure 4</strong><br/> <p>Time evolution of the extended evaluation function <math display="inline"><semantics> <mrow> <mi>E</mi> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> </semantics></math> in Equation (<a href="#FD34-mathematics-12-03826" class="html-disp-formula">34</a>) for consistent planning.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3826'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03826/article_deploy/html/images/mathematics-12-03826-g005-550.jpg?1733226653" title=" <strong>Figure 5</strong><br/> <p>Trajectories of <math display="inline"><semantics> <msub> <mrow> <mo>(</mo> <msub> <mi>K</mi> <mi mathvariant="normal">T</mi> </msub> <mspace width="0.166667em"/> <mi>z</mi> <mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> <mn>669</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mrow> <mo>(</mo> <msub> <mi>w</mi> <mi mathvariant="normal">T</mi> </msub> <mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> <mn>669</mn> </msub> </semantics></math> for consistent planning. The yellow-shaded region represents <math display="inline"><semantics> <mrow> <mo>{</mo> <msub> <mrow> <mo>(</mo> <msub> <mi>K</mi> <mi mathvariant="normal">T</mi> </msub> <mspace width="0.166667em"/> <mi>e</mi> <mo>)</mo> </mrow> <mn>669</mn> </msub> <mspace width="0.277778em"/> <mo>:</mo> <mspace width="0.277778em"/> <mi>e</mi> <mo>∈</mo> <msubsup> <mi mathvariant="script">E</mi> <mrow> <mi mathvariant="normal">T</mi> </mrow> <mi mathvariant="normal">U</mi> </msubsup> <mo>∩</mo> <msubsup> <mi mathvariant="script">E</mi> <mrow> <mi mathvariant="normal">T</mi> </mrow> <mi mathvariant="normal">L</mi> </msubsup> <mo>}</mo> </mrow> </semantics></math>.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3826'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03826/article_deploy/html/images/mathematics-12-03826-g006-550.jpg?1733226655" title=" <strong>Figure 6</strong><br/> <p>A dose-volume histogram obtained using the variable dose-constraints (VDC) method for consistent planning. Doses for Core and Target (solid lines), as well as variable dose-constraints (dashed lines) for both overlapped with each other.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3826'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03826/article_deploy/html/images/mathematics-12-03826-g007-550.jpg?1733226656" title=" <strong>Figure 7</strong><br/> <p>Time evolution of the evaluation function <math display="inline"><semantics> <mrow> <mi>E</mi> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> </semantics></math> in Equation (<a href="#FD37-mathematics-12-03826" class="html-disp-formula">37</a>) using the fixed dose-constraints method for acceptable planning.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3826'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03826/article_deploy/html/images/mathematics-12-03826-g008-550.jpg?1733226658" title=" <strong>Figure 8</strong><br/> <p>Time evolutions of (upper panel) <math display="inline"><semantics> <mrow> <msubsup> <mi>η</mi> <mrow> <mi mathvariant="normal">C</mi> </mrow> <mi mathvariant="normal">U</mi> </msubsup> <mrow> <mo>(</mo> <mi>z</mi> <mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msubsup> <mi>η</mi> <mrow> <mi mathvariant="normal">T</mi> </mrow> <mi mathvariant="normal">U</mi> </msubsup> <mrow> <mo>(</mo> <mi>z</mi> <mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msubsup> <mi>η</mi> <mrow> <mi mathvariant="normal">T</mi> </mrow> <mi mathvariant="normal">L</mi> </msubsup> <mrow> <mo>(</mo> <mi>z</mi> <mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <msubsup> <mi>η</mi> <mrow> <mi mathvariant="normal">B</mi> </mrow> <mi mathvariant="normal">U</mi> </msubsup> <mrow> <mo>(</mo> <mi>z</mi> <mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math> and (lower panel) <math display="inline"><semantics> <mrow> <msubsup> <mi>δ</mi> <mrow> <mi mathvariant="normal">C</mi> </mrow> <mi mathvariant="normal">U</mi> </msubsup> <mrow> <mo>(</mo> <mi>z</mi> <mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msubsup> <mi>δ</mi> <mrow> <mi mathvariant="normal">T</mi> </mrow> <mi mathvariant="normal">U</mi> </msubsup> <mrow> <mo>(</mo> <mi>z</mi> <mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msubsup> <mi>δ</mi> <mrow> <mi mathvariant="normal">T</mi> </mrow> <mi mathvariant="normal">L</mi> </msubsup> <mrow> <mo>(</mo> <mi>z</mi> <mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msubsup> <mi>δ</mi> <mrow> <mi mathvariant="normal">B</mi> </mrow> <mi mathvariant="normal">U</mi> </msubsup> <mrow> <mo>(</mo> <mi>z</mi> <mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mo>Δ</mo> <mo>(</mo> <mi>z</mi> <mo>(</mo> <mi>n</mi> <mo>)</mo> <mo>)</mo> </mrow> </semantics></math> from top to bottom, which were obtained using the fixed dose-constraints method for acceptable planning.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3826'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03826/article_deploy/html/images/mathematics-12-03826-g009-550.jpg?1733226660" title=" <strong>Figure 9</strong><br/> <p>The trajectories of <math display="inline"><semantics> <msub> <mrow> <mo>(</mo> <msub> <mi>K</mi> <mi mathvariant="normal">T</mi> </msub> <mspace width="0.166667em"/> <mi>z</mi> <mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> <mn>3870</mn> </msub> </semantics></math> obtained using the fixed dose-constraints method for acceptable planning. The yellow-shaded region represents <math display="inline"><semantics> <mrow> <mo>{</mo> <msub> <mrow> <mo>(</mo> <msub> <mi>K</mi> <mi mathvariant="normal">T</mi> </msub> <mspace width="0.166667em"/> <mi>e</mi> <mo>)</mo> </mrow> <mn>3870</mn> </msub> <mspace width="0.277778em"/> <mo>:</mo> <mspace width="0.277778em"/> <mi>e</mi> <mo>∈</mo> <msubsup> <mi mathvariant="script">A</mi> <mrow> <mi mathvariant="normal">T</mi> </mrow> <mi mathvariant="normal">U</mi> </msubsup> <mo>∩</mo> <msubsup> <mi mathvariant="script">A</mi> <mrow> <mi mathvariant="normal">T</mi> </mrow> <mi mathvariant="normal">L</mi> </msubsup> <mo>}</mo> </mrow> </semantics></math>.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3826'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03826/article_deploy/html/images/mathematics-12-03826-g010-550.jpg?1733226661" title=" <strong>Figure 10</strong><br/> <p>The dose-volume histogram obtained using fixed the dose-constraints method for acceptable planning.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3826'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03826/article_deploy/html/images/mathematics-12-03826-g011-550.jpg?1733226662" title=" <strong>Figure 11</strong><br/> <p>Time evolution of the evaluation function <math display="inline"><semantics> <mrow> <mi>E</mi> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> </semantics></math> in Equation (<a href="#FD38-mathematics-12-03826" class="html-disp-formula">38</a>) using the variable dose-constraints method for acceptable planning.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3826'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03826/article_deploy/html/images/mathematics-12-03826-g012-550.jpg?1733226665" title=" <strong>Figure 12</strong><br/> <p>Time evolutions of (upper panel) <math display="inline"><semantics> <mrow> <msubsup> <mi>η</mi> <mrow> <mi mathvariant="normal">C</mi> </mrow> <mi mathvariant="normal">U</mi> </msubsup> <mrow> <mo>(</mo> <mi>z</mi> <mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msubsup> <mi>η</mi> <mrow> <mi mathvariant="normal">T</mi> </mrow> <mi mathvariant="normal">U</mi> </msubsup> <mrow> <mo>(</mo> <mi>z</mi> <mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msubsup> <mi>η</mi> <mrow> <mi mathvariant="normal">T</mi> </mrow> <mi mathvariant="normal">L</mi> </msubsup> <mrow> <mo>(</mo> <mi>z</mi> <mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <msubsup> <mi>η</mi> <mrow> <mi mathvariant="normal">B</mi> </mrow> <mi mathvariant="normal">U</mi> </msubsup> <mrow> <mo>(</mo> <mi>z</mi> <mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math> and (lower panel) <math display="inline"><semantics> <mrow> <msubsup> <mi>δ</mi> <mrow> <mi mathvariant="normal">C</mi> </mrow> <mi mathvariant="normal">U</mi> </msubsup> <mrow> <mo>(</mo> <mi>z</mi> <mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msubsup> <mi>δ</mi> <mrow> <mi mathvariant="normal">T</mi> </mrow> <mi mathvariant="normal">U</mi> </msubsup> <mrow> <mo>(</mo> <mi>z</mi> <mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msubsup> <mi>δ</mi> <mrow> <mi mathvariant="normal">T</mi> </mrow> <mi mathvariant="normal">L</mi> </msubsup> <mrow> <mo>(</mo> <mi>z</mi> <mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msubsup> <mi>δ</mi> <mrow> <mi mathvariant="normal">B</mi> </mrow> <mi mathvariant="normal">U</mi> </msubsup> <mrow> <mo>(</mo> <mi>z</mi> <mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mo>Δ</mo> <mo>(</mo> <mi>z</mi> <mo>(</mo> <mi>n</mi> <mo>)</mo> <mo>)</mo> </mrow> </semantics></math> from top to bottom, which were obtained using the variable dose-constraints method for acceptable planning.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3826'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03826/article_deploy/html/images/mathematics-12-03826-g013-550.jpg?1733226666" title=" <strong>Figure 13</strong><br/> <p>Phase plane diagram the visualizing trajectories of <math display="inline"><semantics> <mrow> <msub> <mi>K</mi> <mi mathvariant="normal">T</mi> </msub> <mspace width="0.166667em"/> <mi>z</mi> <mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>w</mi> <mi mathvariant="normal">T</mi> </msub> <mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> from (<b>a</b>) the initial points to (<b>b</b>) the convergence to the set that was associated with the acceptable dose-distribution system (indicated by the yellow shaded region).</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3826'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03826/article_deploy/html/images/mathematics-12-03826-g014-550.jpg?1733226668" title=" <strong>Figure 14</strong><br/> <p>Trajectories of <math display="inline"><semantics> <msub> <mrow> <mo>(</mo> <msub> <mi>K</mi> <mi mathvariant="normal">T</mi> </msub> <mspace width="0.166667em"/> <mi>z</mi> <mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> <mn>854</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mrow> <mo>(</mo> <msub> <mi>w</mi> <mi mathvariant="normal">T</mi> </msub> <mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> <mn>854</mn> </msub> </semantics></math>, which were obtained using the variable dose-constraints method for acceptable planning. The yellow-shaded region represents <math display="inline"><semantics> <mrow> <mo>{</mo> <msub> <mrow> <mo>(</mo> <msub> <mi>K</mi> <mi mathvariant="normal">T</mi> </msub> <mspace width="0.166667em"/> <mi>e</mi> <mo>)</mo> </mrow> <mn>854</mn> </msub> <mspace width="0.277778em"/> <mo>:</mo> <mspace width="0.277778em"/> <mi>e</mi> <mo>∈</mo> <msubsup> <mi mathvariant="script">A</mi> <mrow> <mi mathvariant="normal">T</mi> </mrow> <mi mathvariant="normal">U</mi> </msubsup> <mo>∩</mo> <msubsup> <mi mathvariant="script">A</mi> <mrow> <mi mathvariant="normal">T</mi> </mrow> <mi mathvariant="normal">L</mi> </msubsup> <mo>}</mo> </mrow> </semantics></math>.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3826'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03826/article_deploy/html/images/mathematics-12-03826-g015-550.jpg?1733226669" title=" <strong>Figure 15</strong><br/> <p>Dose-volume histogram obtained using the variable dose-constraints method for acceptable planning. Doses for Core and Target are represented by solid lines, while the variable dose-constraints (VDCs) are indicated by dashed lines.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3826'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03826/article_deploy/html/images/mathematics-12-03826-g016-550.jpg?1733226672" title=" <strong>Figure 16</strong><br/> <p>Dose-volume histograms for acceptable planning. Left panel (<b>a</b>): Core region obtained using the fixed method (solid line) along with fixed dose constraints (dashed line). Right panel (<b>a</b>): Core region obtained using the variable method (solid line) and fixed method (dashed-and-dotted line) along with variable dose constraints (dashed line). Left panel (<b>b</b>): Target region obtained using the fixed method (solid line) along with fixed dose constraints (dashed line). Right panel (<b>b</b>): Target region obtained using the variable method (solid line) and fixed method (dashed-and-dotted line), along with variable dose constraints (dashed line).</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3826'>Full article</a></strong> "></a></div> </div> </div> </div> <div class="expanding-div collapsed"> <div class="generic-item article-item"> <div class="article-content"> <div class="label right label__btn"> <span style="font-size: 12px; color: #1a1a1a;"> 15 pages, 358 KiB   </span> <a href="/2227-7390/12/23/3825/pdf?version=1733221047" class="UD_Listings_ArticlePDF" title="Article PDF" data-name="Bounded Rational Players in a Symmetric Random Exchange Market" data-journal="mathematics"> <i class="material-icons custom-download"></i> </a> </div> <div class="article-icons"><span class="label openaccess" data-dropdown="drop-article-label-openaccess" aria-expanded="false">Open Access</span><span class="label articletype">Article</span></div> <a class="title-link" href="/2227-7390/12/23/3825">Bounded Rational Players in a Symmetric Random Exchange Market</a> <div class="authors"> by <span class="inlineblock "><strong>Aliyu Yusuf</strong>, </span><span class="inlineblock "><strong>Bruno Oliveira</strong>, </span><span class="inlineblock "><strong>Alberto Pinto</strong> and </span><span class="inlineblock "><strong>Athanasios N. Yannacopoulos</strong></span> </div> <div class="color-grey-dark"> <em>Mathematics</em> <b>2024</b>, <em>12</em>(23), 3825; <a href="https://doi.org/10.3390/math12233825">https://doi.org/10.3390/math12233825</a> - 3 Dec 2024 </div> <div class="abstract-div"> <a href="#" onclick="$(this).next('.abstract-cropped').toggleClass('inline').next('.abstract-full').toggleClass('inline'); return false;"> <strong>Abstract </strong> </a> <div class="abstract-cropped inline"> A model of Edgeworthian economies is studied, in which participants are randomly chosen at each period to exchange two goods to increase their utilities, as described by the Cobb–Douglas utility function. Participants can trade deviating from their bilateral equilibrium, provided that the market <a href="#" data-counterslink = "https://www.mdpi.com/2227-7390/12/23/3825/more" onclick="$(this).parents('.abstract-cropped').toggleClass('inline').next('.abstract-full').toggleClass('inline'); return false;"> [...] Read more.</a> </div> <div class="abstract-full "> A model of Edgeworthian economies is studied, in which participants are randomly chosen at each period to exchange two goods to increase their utilities, as described by the Cobb–Douglas utility function. Participants can trade deviating from their bilateral equilibrium, provided that the market and the trade follow appropriate symmetry conditions. The article aims to study the convergence to equilibrium in a situation where individuals or small groups of participants trade in a market, and prices are determined by interactions between the participants rather than by demand and supply alone. A dynamic matching and bargaining game is considered, with statistical duality imposed on the market game, ensuring that each participant has a counterpart with opposite preferences. This guaranties that there is sufficient incentive for trade. It is shown that, in each period, the expected logarithm of the trading price in the Edgeworthian economy equals the expected Walrasian price. This demonstrates that, under symmetry conditions, the trading price in the Edgeworthian economy is related to the Walrasian price, indicating convergence of the trading price in the Edgeworthian economy towards the Walrasian price. The study suggests that, under the right conditions, the decentralized trading model leads to price convergence similar to what would be expected in a more classical Walrasian economy, where prices balance demand and supply. <a href="/2227-7390/12/23/3825">Full article</a> </div> </div> <a href="#" class="abstract-figures-show" data-counterslink = "https://www.mdpi.com/2227-7390/12/23/3825/show" ><span >►</span><span style=" display: none;">▼</span> Show Figures </a><div class="abstract-image-preview "><div class="arrow left-arrow" id="prev1535759"><i class="fa fa-caret-left"></i></div><div class="arrow right-arrow" id="next1535759"><i class="fa fa-caret-right"></i></div><div class="absgraph cycle-slideshow manual" data-cycle-fx="scrollHorz" data-cycle-timeout="0" data-cycle-next="#next1535759" data-cycle-prev="#prev1535759" data-cycle-progressive="#images1535759" data-cycle-slides=">div" data-cycle-log="false"><div class='openpopupgallery cycle-slide' data-imgindex='0' data-target='article-1535759-popup'><span class="helper"></span><img src="data:image/gif;base64,R0lGODlhAQABAAD/ACwAAAAAAQABAAACADs=" data-src="https://pub.mdpi-res.com/mathematics/mathematics-12-03825/article_deploy/html/images/mathematics-12-03825-g001-550.jpg?1733221116" alt="" style="border: 0;"><p>Figure 1</p></div><script id="images1535759" type="text/cycle" data-cycle-split="---"><div class='openpopupgallery' data-imgindex='1' data-target='article-1535759-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03825/article_deploy/html/images/mathematics-12-03825-g002-550.jpg?1733221116'><p>Figure 2</p></div></script></div></div><div id="article-1535759-popup" class="popupgallery" style="display: inline; line-height: 200%"><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03825/article_deploy/html/images/mathematics-12-03825-g001-550.jpg?1733221116" title=" <strong>Figure 1</strong><br/> <p>The Edgeworth box showing bilateral trade between two participants. The eye-shaped region is the intersection between the area above the indifference curve of participant <math display="inline"><semantics> <msub> <mi>A</mi> <mi>i</mi> </msub> </semantics></math> in <a href="#mathematics-12-03825-f001" class="html-fig">Figure 1</a> and the area below the indifference curve of participant <math display="inline"><semantics> <msub> <mi>A</mi> <mi>j</mi> </msub> </semantics></math>.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3825'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03825/article_deploy/html/images/mathematics-12-03825-g002-550.jpg?1733221116" title=" <strong>Figure 2</strong><br/> <p>The Edgeworth box with bounded rationality showing bilateral trade between two participants where the participants have different bargaining skills.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3825'>Full article</a></strong> "></a></div> </div> </div> </div> <div class="expanding-div collapsed"> <div class="generic-item article-item"> <div class="article-content"> <div class="label right label__btn"> <span style="font-size: 12px; color: #1a1a1a;"> 13 pages, 998 KiB   </span> <a href="/2227-7390/12/23/3824/pdf?version=1733218961" class="UD_Listings_ArticlePDF" title="Article PDF" data-name="A Parametric Six-Step Method for Second-Order IVPs with Oscillating Solutions" data-journal="mathematics"> <i class="material-icons custom-download"></i> </a> </div> <div class="article-icons"><span class="label openaccess" data-dropdown="drop-article-label-openaccess" aria-expanded="false">Open Access</span><span class="label feature" data-dropdown="drop-article-label-feature" aria-expanded="false">Feature Paper</span><span class="label articletype">Article</span></div> <a class="title-link" href="/2227-7390/12/23/3824">A Parametric Six-Step Method for Second-Order IVPs with Oscillating Solutions</a> <div class="authors"> by <span class="inlineblock "><strong>Dimitris F. Papadopoulos</strong></span> </div> <div class="color-grey-dark"> <em>Mathematics</em> <b>2024</b>, <em>12</em>(23), 3824; <a href="https://doi.org/10.3390/math12233824">https://doi.org/10.3390/math12233824</a> - 3 Dec 2024 </div> <div class="abstract-div"> <a href="#" onclick="$(this).next('.abstract-cropped').toggleClass('inline').next('.abstract-full').toggleClass('inline'); return false;"> <strong>Abstract </strong> </a> <div class="abstract-cropped inline"> In this paper, we develop an explicit symmetric six-step method for the numerical solution of second-order initial value problems (IVPs) with oscillating solutions. The proposed method is phase-fitted and incorporates a free coefficient as a parameter to optimize its performance. By exploring a <a href="#" data-counterslink = "https://www.mdpi.com/2227-7390/12/23/3824/more" onclick="$(this).parents('.abstract-cropped').toggleClass('inline').next('.abstract-full').toggleClass('inline'); return false;"> [...] Read more.</a> </div> <div class="abstract-full "> In this paper, we develop an explicit symmetric six-step method for the numerical solution of second-order initial value problems (IVPs) with oscillating solutions. The proposed method is phase-fitted and incorporates a free coefficient as a parameter to optimize its performance. By exploring a wide range of values for this parameter, we computationally determine the periodicity interval. The objective of this procedure is to identify the range of the parameter’s values for which the method remains stable. Based on the output from the periodicity interval analysis, we then aim to define the optimal values for the parameter by numerically solving three initial value problems. The results guided us in identifying these optimal values and confirm the high efficiency of the new method. The method’s efficiency is further validated for the chosen optimal parameter value for specific oscillatory problems, where it is compared with well-known phase-fitted methods. <a href="/2227-7390/12/23/3824">Full article</a> </div> </div> <div class="belongsTo" style="margin-bottom: 10px;"> (This article belongs to the Special Issue <a href=" /journal/mathematics/special_issues/H71XYUCZGH ">Numerical Methods Applied to Mathematical Problems</a>)<br/> </div> <a href="#" class="abstract-figures-show" data-counterslink = "https://www.mdpi.com/2227-7390/12/23/3824/show" ><span >►</span><span style=" display: none;">▼</span> Show Figures </a><div class="abstract-image-preview "><div class="arrow left-arrow" id="prev1535694"><i class="fa fa-caret-left"></i></div><div class="arrow right-arrow" id="next1535694"><i class="fa fa-caret-right"></i></div><div class="absgraph cycle-slideshow manual" data-cycle-fx="scrollHorz" data-cycle-timeout="0" data-cycle-next="#next1535694" data-cycle-prev="#prev1535694" data-cycle-progressive="#images1535694" data-cycle-slides=">div" data-cycle-log="false"><div class='openpopupgallery cycle-slide' data-imgindex='0' data-target='article-1535694-popup'><span class="helper"></span><img src="data:image/gif;base64,R0lGODlhAQABAAD/ACwAAAAAAQABAAACADs=" data-src="https://pub.mdpi-res.com/mathematics/mathematics-12-03824/article_deploy/html/images/mathematics-12-03824-g001-550.jpg?1733219114" alt="" style="border: 0;"><p>Figure 1</p></div><script id="images1535694" type="text/cycle" data-cycle-split="---"><div class='openpopupgallery' data-imgindex='1' data-target='article-1535694-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03824/article_deploy/html/images/mathematics-12-03824-g002-550.jpg?1733219115'><p>Figure 2</p></div> --- <div class='openpopupgallery' data-imgindex='2' data-target='article-1535694-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03824/article_deploy/html/images/mathematics-12-03824-g003-550.jpg?1733219116'><p>Figure 3</p></div> --- <div class='openpopupgallery' data-imgindex='3' data-target='article-1535694-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03824/article_deploy/html/images/mathematics-12-03824-g004-550.jpg?1733219117'><p>Figure 4</p></div> --- <div class='openpopupgallery' data-imgindex='4' data-target='article-1535694-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03824/article_deploy/html/images/mathematics-12-03824-g005-550.jpg?1733219118'><p>Figure 5</p></div> --- <div class='openpopupgallery' data-imgindex='5' data-target='article-1535694-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03824/article_deploy/html/images/mathematics-12-03824-g006-550.jpg?1733219120'><p>Figure 6</p></div> --- <div class='openpopupgallery' data-imgindex='6' data-target='article-1535694-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03824/article_deploy/html/images/mathematics-12-03824-g007-550.jpg?1733219121'><p>Figure 7</p></div></script></div></div><div id="article-1535694-popup" class="popupgallery" style="display: inline; line-height: 200%"><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03824/article_deploy/html/images/mathematics-12-03824-g001-550.jpg?1733219114" title=" <strong>Figure 1</strong><br/> <p>Method’s periodicity interval with respect to the coefficient <math display="inline"><semantics> <msub> <mi>a</mi> <mn>2</mn> </msub> </semantics></math>.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3824'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03824/article_deploy/html/images/mathematics-12-03824-g002-550.jpg?1733219115" title=" <strong>Figure 2</strong><br/> <p>Efficiency of the new method for all values of <math display="inline"><semantics> <msub> <mi>a</mi> <mn>2</mn> </msub> </semantics></math>—Schrödinger equation.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3824'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03824/article_deploy/html/images/mathematics-12-03824-g003-550.jpg?1733219116" title=" <strong>Figure 3</strong><br/> <p>Efficiency of the new method for all values of <math display="inline"><semantics> <msub> <mi>a</mi> <mn>2</mn> </msub> </semantics></math>—Duffing equation.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3824'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03824/article_deploy/html/images/mathematics-12-03824-g004-550.jpg?1733219117" title=" <strong>Figure 4</strong><br/> <p>Efficiency of the new method for all values of <math display="inline"><semantics> <msub> <mi>a</mi> <mn>2</mn> </msub> </semantics></math>—Stiefel–Bettis problem.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3824'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03824/article_deploy/html/images/mathematics-12-03824-g005-550.jpg?1733219118" title=" <strong>Figure 5</strong><br/> <p>Comparison for the Schrödinger equation.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3824'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03824/article_deploy/html/images/mathematics-12-03824-g006-550.jpg?1733219120" title=" <strong>Figure 6</strong><br/> <p>Comparison for the Duffing equation.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3824'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03824/article_deploy/html/images/mathematics-12-03824-g007-550.jpg?1733219121" title=" <strong>Figure 7</strong><br/> <p>Comparison for the Stiefel–Bettis problem.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3824'>Full article</a></strong> "></a></div> </div> </div> </div> <div class="expanding-div collapsed"> <div class="generic-item article-item"> <div class="article-content"> <div class="label right label__btn"> <span style="font-size: 12px; color: #1a1a1a;"> 21 pages, 1227 KiB   </span> <a href="/2227-7390/12/23/3823/pdf?version=1733215695" class="UD_Listings_ArticlePDF" title="Article PDF" data-name="netQDA: Local Network-Guided High-Dimensioanl Quadratic Discriminant Analysis" data-journal="mathematics"> <i class="material-icons custom-download"></i> </a> </div> <div class="article-icons"><span class="label openaccess" data-dropdown="drop-article-label-openaccess" aria-expanded="false">Open Access</span><span class="label articletype">Article</span></div> <a class="title-link" href="/2227-7390/12/23/3823">netQDA: Local Network-Guided High-Dimensioanl Quadratic Discriminant Analysis</a> <div class="authors"> by <span class="inlineblock "><strong>Xueping Zhou</strong>, </span><span class="inlineblock "><strong>Wei Chen</strong> and </span><span class="inlineblock "><strong>Yanming Li</strong></span> </div> <div class="color-grey-dark"> <em>Mathematics</em> <b>2024</b>, <em>12</em>(23), 3823; <a href="https://doi.org/10.3390/math12233823">https://doi.org/10.3390/math12233823</a> - 3 Dec 2024 </div> <div class="abstract-div"> <a href="#" onclick="$(this).next('.abstract-cropped').toggleClass('inline').next('.abstract-full').toggleClass('inline'); return false;"> <strong>Abstract </strong> </a> <div class="abstract-cropped inline"> Quadratic Discriminant Analysis (QDA) is a well-known and flexible classification method that considers differences between groups based on both mean and covariance structures. However, the connection structures of high-dimensional predictors are usually not explicitly incorporated into modeling. In this work, we propose a <a href="#" data-counterslink = "https://www.mdpi.com/2227-7390/12/23/3823/more" onclick="$(this).parents('.abstract-cropped').toggleClass('inline').next('.abstract-full').toggleClass('inline'); return false;"> [...] Read more.</a> </div> <div class="abstract-full "> Quadratic Discriminant Analysis (QDA) is a well-known and flexible classification method that considers differences between groups based on both mean and covariance structures. However, the connection structures of high-dimensional predictors are usually not explicitly incorporated into modeling. In this work, we propose a local network-guided QDA method that integrates the local connection structures of high-dimensional predictors. In the context of gene expression research, our method can identify genes that show differential expression levels as well as gene networks that exhibit different connection patterns between various biological state groups, thereby enhancing our understanding of underlying biological mechanisms. Extensive simulations and real data applications demonstrate its superior performance in both feature selection and outcome classification compared to commonly used discriminant analysis methods. <a href="/2227-7390/12/23/3823">Full article</a> </div> </div> <div class="belongsTo" style="margin-bottom: 10px;"> (This article belongs to the Special Issue <a href=" /journal/mathematics/special_issues/60613810L9 ">Statistical Forecasting: Theories, Methods and Applications</a>)<br/> </div> <a href="#" class="abstract-figures-show" data-counterslink = "https://www.mdpi.com/2227-7390/12/23/3823/show" ><span >►</span><span style=" display: none;">▼</span> Show Figures </a><div class="abstract-image-preview "><div class="arrow left-arrow" id="prev1535590"><i class="fa fa-caret-left"></i></div><div class="arrow right-arrow" id="next1535590"><i class="fa fa-caret-right"></i></div><div class="absgraph cycle-slideshow manual" data-cycle-fx="scrollHorz" data-cycle-timeout="0" data-cycle-next="#next1535590" data-cycle-prev="#prev1535590" data-cycle-progressive="#images1535590" data-cycle-slides=">div" data-cycle-log="false"><div class='openpopupgallery cycle-slide' data-imgindex='0' data-target='article-1535590-popup'><span class="helper"></span><img src="data:image/gif;base64,R0lGODlhAQABAAD/ACwAAAAAAQABAAACADs=" data-src="https://pub.mdpi-res.com/mathematics/mathematics-12-03823/article_deploy/html/images/mathematics-12-03823-g001-550.jpg?1733215761" alt="" style="border: 0;"><p>Figure 1</p></div><script id="images1535590" type="text/cycle" data-cycle-split="---"><div class='openpopupgallery' data-imgindex='1' data-target='article-1535590-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03823/article_deploy/html/images/mathematics-12-03823-g002a-550.jpg?1733215762'><p>Figure 2</p></div> --- <div class='openpopupgallery' data-imgindex='2' data-target='article-1535590-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03823/article_deploy/html/images/mathematics-12-03823-g002b-550.jpg?1733215762'><p>Figure 2 Cont.</p></div> --- <div class='openpopupgallery' data-imgindex='3' data-target='article-1535590-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03823/article_deploy/html/images/mathematics-12-03823-g003-550.jpg?1733215763'><p>Figure 3</p></div> --- <div class='openpopupgallery' data-imgindex='4' data-target='article-1535590-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03823/article_deploy/html/images/mathematics-12-03823-g004-550.jpg?1733215765'><p>Figure 4</p></div> --- <div class='openpopupgallery' data-imgindex='5' data-target='article-1535590-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03823/article_deploy/html/images/mathematics-12-03823-g005-550.jpg?1733215766'><p>Figure 5</p></div> --- <div class='openpopupgallery' data-imgindex='6' data-target='article-1535590-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03823/article_deploy/html/images/mathematics-12-03823-g006-550.jpg?1733215769'><p>Figure 6</p></div> --- <div class='openpopupgallery' data-imgindex='7' data-target='article-1535590-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03823/article_deploy/html/images/mathematics-12-03823-g007-550.jpg?1733215771'><p>Figure 7</p></div> --- <div class='openpopupgallery' data-imgindex='8' data-target='article-1535590-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03823/article_deploy/html/images/mathematics-12-03823-g008-550.jpg?1733215774'><p>Figure 8</p></div></script></div></div><div id="article-1535590-popup" class="popupgallery" style="display: inline; line-height: 200%"><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03823/article_deploy/html/images/mathematics-12-03823-g001-550.jpg?1733215761" title=" <strong>Figure 1</strong><br/> <p>Strong (<span class="html-italic">CST1</span>) and weak (<span class="html-italic">CLCA1</span>) DE genes identified in the nasal epithelium between children with atopic asthma and non-atopic controls in a childhood asthma study.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3823'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03823/article_deploy/html/images/mathematics-12-03823-g002a-550.jpg?1733215762" title=" <strong>Figure 2</strong><br/> <p>Differentially connected gene networks in the nasal epithelium between children with atopic asthma and non-atopic controls in a childhood asthma study. The genes, adapted from [<a href="#B5-mathematics-12-03823" class="html-bibr">5</a>], exhibit different hub-based networks in both the atopic asthma group (<b>top</b>) and non-atopic control group (<b>bottom</b>). In this representation, hub genes are defined as genes having connections to at least four other genes in either group.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3823'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03823/article_deploy/html/images/mathematics-12-03823-g002b-550.jpg?1733215762" title=" <strong>Figure 2 Cont.</strong><br/> <p>Differentially connected gene networks in the nasal epithelium between children with atopic asthma and non-atopic controls in a childhood asthma study. The genes, adapted from [<a href="#B5-mathematics-12-03823" class="html-bibr">5</a>], exhibit different hub-based networks in both the atopic asthma group (<b>top</b>) and non-atopic control group (<b>bottom</b>). In this representation, hub genes are defined as genes having connections to at least four other genes in either group.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3823'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03823/article_deploy/html/images/mathematics-12-03823-g003-550.jpg?1733215763" title=" <strong>Figure 3</strong><br/> <p>Precision matrix of a connected component and its interpretation as a graph for simulation scenario 1. In scenario <span class="html-italic">S</span>1, we focused on a specific setting where the informative features were limited to strong DE features and DC features. We examined two different structures for the precision matrices of the connected components. In (<b>a</b>), the precision matrix in group one followed a star structure, while in (<b>b</b>), the precision matrix followed a band structure. The left side of both (<b>a</b>,<b>b</b>) display the precision matrices specific to each connected component. On the right side, there are graphical interpretations of the precision matrices, where features are represented by nodes, and connections are represented by edges. The strong DE feature, denoted as <span class="html-italic">X</span><sub>1</sub>, is highlighted in different colors in two groups. In the sub-scenarios of <span class="html-italic">S</span>1, specific combinations of features were considered. In sub-scenario <span class="html-italic">S</span>1-1, there was one DC feature, <span class="html-italic">X</span><sub>2</sub>. In sub-scenario <span class="html-italic">S</span>1-2, there were two DC features, <span class="html-italic">X</span><sub>2</sub> and <span class="html-italic">X</span><sub>3</sub>. In sub-scenario S1-3, there were three DC features, <span class="html-italic">X</span><sub>2</sub>, <span class="html-italic">X</span><sub>3</sub>, and <span class="html-italic">X</span><sub>4</sub>. All the partial correlation coefficients were set to <span class="html-italic">ρ</span> = 0.5.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3823'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03823/article_deploy/html/images/mathematics-12-03823-g004-550.jpg?1733215765" title=" <strong>Figure 4</strong><br/> <p>Precision matrix of a connected component and its interpretation as a graph for simulation scenario 2. In scenario <math display="inline"><semantics> <mrow> <mi>S</mi> <mn>2</mn> </mrow> </semantics></math>, the informative features could be classified into three distinct categories: strong DE, weak DE, and DC features. These categories were mutually exclusive. We examined two different structures for the precision matrices of the connected components. Star structure in (<b>a</b>) and band structure in (<b>b</b>). The left side of both (<b>a</b>,<b>b</b>) displayed the precision matrices specific to each connected component. On the right side, there are graphical interpretations of the precision matrices, where features are represented by nodes, and connections are represented by edges. The strong DE feature, was highlighted in different colors in two groups. In the sub-scenarios of <math display="inline"><semantics> <mrow> <mi>S</mi> <mn>2</mn> </mrow> </semantics></math>, specific combinations of features were considered. The total number of DC features and DE weak features combined summed up to three in each component. We systematically varied the number of DC features from three to zero, while simultaneously increasing the number of weak DE features from zero to three. For example, for the sub-scenario one in (<b>a</b>), there is one strong DE feature <math display="inline"><semantics> <msub> <mi>X</mi> <mn>1</mn> </msub> </semantics></math> and three DC features <math display="inline"><semantics> <mrow> <msub> <mi>X</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>X</mi> <mn>3</mn> </msub> <mo>,</mo> <mi>a</mi> <mi>n</mi> <mi>d</mi> <mspace width="3.33333pt"/> <msub> <mi>X</mi> <mn>4</mn> </msub> </mrow> </semantics></math>. Sub-scenario two in (<b>a</b>) included one strong DE feature <math display="inline"><semantics> <msub> <mi>X</mi> <mn>1</mn> </msub> </semantics></math>, one weak DE feature <math display="inline"><semantics> <msub> <mi>X</mi> <mn>2</mn> </msub> </semantics></math>, and two DC features <math display="inline"><semantics> <msub> <mi>X</mi> <mn>2</mn> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mi>X</mi> <mn>3</mn> </msub> </semantics></math>, and <math display="inline"><semantics> <msub> <mi>X</mi> <mn>4</mn> </msub> </semantics></math>. All the partial correlation coefficients were set to <math display="inline"><semantics> <mrow> <mi>ρ</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3823'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03823/article_deploy/html/images/mathematics-12-03823-g005-550.jpg?1733215766" title=" <strong>Figure 5</strong><br/> <p>Precision matrix of a connected component and its interpretation as a graph for simulation scenario 3. In scenario <math display="inline"><semantics> <mrow> <mi>S</mi> <mn>3</mn> </mrow> </semantics></math>, the informative features could be classified into three categories: strong DE, weak DE, and DC features. The weak DE and DC features were allowed to overlap. We examined two different structures for the precision matrices of the connected components. Star structure in (<b>a</b>) and band structure (<b>b</b>). The left side of both (<b>a</b>,<b>b</b>) display the precision matrices specific to each connected component. On the right side, there are graphical interpretations of the precision matrices, where features are represented by nodes, and connections are represented by edges. The width of the edges corresponds to the strength of the connection or partial correlation coefficients between the features. Thicker edges indicate stronger connections <math display="inline"><semantics> <mrow> <mi>ρ</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>, while thinner edges represent weaker connections <math display="inline"><semantics> <mrow> <msup> <mi>ρ</mi> <mo>′</mo> </msup> <mo>=</mo> <mn>0.4</mn> </mrow> </semantics></math> in (<b>a</b>) and <math display="inline"><semantics> <mrow> <msup> <mi>ρ</mi> <mo>′</mo> </msup> <mo>=</mo> <mn>0.30</mn> </mrow> </semantics></math> in (<b>b</b>). The strong DE feature is highlighted in different colors in two groups. In the sub-scenarios of <math display="inline"><semantics> <mrow> <mi>S</mi> <mn>3</mn> </mrow> </semantics></math>, specific combinations of features were considered. Within each connected component, we ensured the presence of one strong DE feature <math display="inline"><semantics> <msub> <mi>X</mi> <mn>1</mn> </msub> </semantics></math> and three DC features <math display="inline"><semantics> <msub> <mi>X</mi> <mn>1</mn> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mi>X</mi> <mn>2</mn> </msub> </semantics></math>, and <math display="inline"><semantics> <msub> <mi>X</mi> <mn>3</mn> </msub> </semantics></math>. However, the number of overlapping weak DE and DC features varied across different configurations. We explored scenarios with one, two, and three overlapping weak DE and DC features, respectively.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3823'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03823/article_deploy/html/images/mathematics-12-03823-g006-550.jpg?1733215769" title=" <strong>Figure 6</strong><br/> <p>Boxplots for outcome classification accuracy for scenario 1. In Scenario 1, we focused on a specific setting where the informative features were limited to strong DE features and DC features. In each connected component, there was one strong DE feature with one, two, or three DC features. The corresponding outcomes were graphically represented in each column of the plot. The subplots in the lower row demonstrated the findings obtained from the star structure, while the upper row depicted the results associated with the band structure.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3823'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03823/article_deploy/html/images/mathematics-12-03823-g007-550.jpg?1733215771" title=" <strong>Figure 7</strong><br/> <p>Boxplots for outcome classification accuracy for scenario 2. In Scenario 2, the informative features could be classified into three distinct categories: strong DE, weak DE, and DC features. These categories were mutually exclusive. The total number of DC features and weak DE features combined summed up to three in each component. We systematically varied the number of DC features from three to zero, while simultaneously increasing the number of weak DE features from zero to three. The corresponding outcomes were graphically represented in each column of the plot. The lower row of subplots demonstrated the findings obtained from the star structure, while the upper row depicted the results associated with the band structure.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3823'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03823/article_deploy/html/images/mathematics-12-03823-g008-550.jpg?1733215774" title=" <strong>Figure 8</strong><br/> <p>Boxplots for outcome classification accuracy for scenario 3. In Scenario 3, the informative features could be classified into three distinct categories: strong DE, weak DE, and DC features. The weak DE and DC features were allowed to overlap. Within each connected component, we ensured the presence of one strong DE feature and three DC features. The number of overlapping weak DE and DC features is one, two or three. The corresponding outcomes were graphically represented in each column of the plot. The lower row of subplots demonstrated the findings obtained from the star structure, while the upper row depicted the results associated with the band structure.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3823'>Full article</a></strong> "></a></div> </div> </div> </div> <div class="expanding-div collapsed"> <div class="generic-item article-item"> <div class="article-content"> <div class="label right label__btn"> <span style="font-size: 12px; color: #1a1a1a;"> 17 pages, 6358 KiB   </span> <a href="/2227-7390/12/23/3822/pdf?version=1733209806" class="UD_Listings_ArticlePDF" title="Article PDF" data-name="Continuous Multi-Target Approaching Control of Hyper-Redundant Manipulators Based on Reinforcement Learning" data-journal="mathematics"> <i class="material-icons custom-download"></i> </a> </div> <div class="article-icons"><span class="label openaccess" data-dropdown="drop-article-label-openaccess" aria-expanded="false">Open Access</span><span class="label articletype">Article</span></div> <a class="title-link" href="/2227-7390/12/23/3822">Continuous Multi-Target Approaching Control of Hyper-Redundant Manipulators Based on Reinforcement Learning</a> <div class="authors"> by <span class="inlineblock "><strong>Han Xu</strong>, </span><span class="inlineblock "><strong>Chen Xue</strong>, </span><span class="inlineblock "><strong>Quan Chen</strong>, </span><span class="inlineblock "><strong>Jun Yang</strong> and </span><span class="inlineblock "><strong>Bin Liang</strong></span> </div> <div class="color-grey-dark"> <em>Mathematics</em> <b>2024</b>, <em>12</em>(23), 3822; <a href="https://doi.org/10.3390/math12233822">https://doi.org/10.3390/math12233822</a> - 3 Dec 2024 </div> <div class="abstract-div"> <a href="#" onclick="$(this).next('.abstract-cropped').toggleClass('inline').next('.abstract-full').toggleClass('inline'); return false;"> <strong>Abstract </strong> </a> <div class="abstract-cropped inline"> Hyper-redundant manipulators based on bionic structures offer superior dexterity due to their large number of degrees of freedom (DOFs) and slim bodies. However, controlling these manipulators is challenging because of infinite inverse kinematic solutions. In this paper, we present a novel reinforcement learning-based <a href="#" data-counterslink = "https://www.mdpi.com/2227-7390/12/23/3822/more" onclick="$(this).parents('.abstract-cropped').toggleClass('inline').next('.abstract-full').toggleClass('inline'); return false;"> [...] Read more.</a> </div> <div class="abstract-full "> Hyper-redundant manipulators based on bionic structures offer superior dexterity due to their large number of degrees of freedom (DOFs) and slim bodies. However, controlling these manipulators is challenging because of infinite inverse kinematic solutions. In this paper, we present a novel reinforcement learning-based control method for hyper-redundant manipulators, integrating path and configuration planning. First, we introduced a deep reinforcement learning-based control method for a multi-target approach, eliminating the need for complicated reward engineering. Then, we optimized the network structure and joint space target points sampling to implement precise control. Furthermore, we designed a variable-reset cycle technique for a continuous multi-target approach without resetting the manipulator, enabling it to complete end-effector trajectory tracking tasks. Finally, we verified the proposed control method in a dynamic simulation environment. The results demonstrate the effectiveness of our approach, achieving a success rate of 98.32% with a 134% improvement using the variable-reset cycle technique. <a href="/2227-7390/12/23/3822">Full article</a> </div> </div> <div class="belongsTo" style="margin-bottom: 10px;"> (This article belongs to the Special Issue <a href=" /journal/mathematics/special_issues/Intell_Control_Appl_Nonlinear_Dyn_Syst ">Intelligent Control and Applications of Nonlinear Dynamic System</a>)<br/> </div> <a href="#" class="abstract-figures-show" data-counterslink = "https://www.mdpi.com/2227-7390/12/23/3822/show" ><span >►</span><span style=" display: none;">▼</span> Show Figures </a><div class="abstract-image-preview "><div class="arrow left-arrow" id="prev1535472"><i class="fa fa-caret-left"></i></div><div class="arrow right-arrow" id="next1535472"><i class="fa fa-caret-right"></i></div><div class="absgraph cycle-slideshow manual" data-cycle-fx="scrollHorz" data-cycle-timeout="0" data-cycle-next="#next1535472" data-cycle-prev="#prev1535472" data-cycle-progressive="#images1535472" data-cycle-slides=">div" data-cycle-log="false"><div class='openpopupgallery cycle-slide' data-imgindex='0' data-target='article-1535472-popup'><span class="helper"></span><img src="data:image/gif;base64,R0lGODlhAQABAAD/ACwAAAAAAQABAAACADs=" data-src="https://pub.mdpi-res.com/mathematics/mathematics-12-03822/article_deploy/html/images/mathematics-12-03822-g001-550.jpg?1733209879" alt="" style="border: 0;"><p>Figure 1</p></div><script id="images1535472" type="text/cycle" data-cycle-split="---"><div class='openpopupgallery' data-imgindex='1' data-target='article-1535472-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03822/article_deploy/html/images/mathematics-12-03822-g002-550.jpg?1733209880'><p>Figure 2</p></div> --- <div class='openpopupgallery' data-imgindex='2' data-target='article-1535472-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03822/article_deploy/html/images/mathematics-12-03822-g003-550.jpg?1733209881'><p>Figure 3</p></div> --- <div class='openpopupgallery' data-imgindex='3' data-target='article-1535472-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03822/article_deploy/html/images/mathematics-12-03822-g004-550.jpg?1733209882'><p>Figure 4</p></div> --- <div class='openpopupgallery' data-imgindex='4' data-target='article-1535472-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03822/article_deploy/html/images/mathematics-12-03822-g005-550.jpg?1733209884'><p>Figure 5</p></div> --- <div class='openpopupgallery' data-imgindex='5' data-target='article-1535472-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03822/article_deploy/html/images/mathematics-12-03822-g006-550.jpg?1733209885'><p>Figure 6</p></div> --- <div class='openpopupgallery' data-imgindex='6' data-target='article-1535472-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03822/article_deploy/html/images/mathematics-12-03822-g007-550.jpg?1733209886'><p>Figure 7</p></div> --- <div class='openpopupgallery' data-imgindex='7' data-target='article-1535472-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03822/article_deploy/html/images/mathematics-12-03822-g008-550.jpg?1733209888'><p>Figure 8</p></div> --- <div class='openpopupgallery' data-imgindex='8' data-target='article-1535472-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03822/article_deploy/html/images/mathematics-12-03822-g009-550.jpg?1733209889'><p>Figure 9</p></div> --- <div class='openpopupgallery' data-imgindex='9' data-target='article-1535472-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03822/article_deploy/html/images/mathematics-12-03822-g010-550.jpg?1733209890'><p>Figure 10</p></div> --- <div class='openpopupgallery' data-imgindex='10' data-target='article-1535472-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03822/article_deploy/html/images/mathematics-12-03822-g011-550.jpg?1733209891'><p>Figure 11</p></div> --- <div class='openpopupgallery' data-imgindex='11' data-target='article-1535472-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03822/article_deploy/html/images/mathematics-12-03822-g012-550.jpg?1733209892'><p>Figure 12</p></div> --- <div class='openpopupgallery' data-imgindex='12' data-target='article-1535472-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03822/article_deploy/html/images/mathematics-12-03822-g013-550.jpg?1733209894'><p>Figure 13</p></div></script></div></div><div id="article-1535472-popup" class="popupgallery" style="display: inline; line-height: 200%"><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03822/article_deploy/html/images/mathematics-12-03822-g001-550.jpg?1733209879" title=" <strong>Figure 1</strong><br/> <p>Overall model of the hyper-redundant manipulator.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3822'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03822/article_deploy/html/images/mathematics-12-03822-g002-550.jpg?1733209880" title=" <strong>Figure 2</strong><br/> <p>D-H coordinate frame of the hyper-redundant manipulator.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3822'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03822/article_deploy/html/images/mathematics-12-03822-g003-550.jpg?1733209881" title=" <strong>Figure 3</strong><br/> <p>Diagram of reinforcement learning process of this paper.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3822'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03822/article_deploy/html/images/mathematics-12-03822-g004-550.jpg?1733209882" title=" <strong>Figure 4</strong><br/> <p>Structure of the critic networks and actor networks. (<b>a</b>) DenseConnect. (<b>b</b>) SimpleDenseConnect.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3822'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03822/article_deploy/html/images/mathematics-12-03822-g005-550.jpg?1733209884" title=" <strong>Figure 5</strong><br/> <p>Distribution of the target points mapped to a workspace by different sampling in the joint space of a 12-link, 24-DOF, and hyper-redundant manipulator. (<b>a</b>) Normal Distribution I sampling. (<b>b</b>) Normal Distribution II sampling. (<b>c</b>) Uniform distribution sampling. (<b>d</b>) U-shaped distribution sampling.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3822'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03822/article_deploy/html/images/mathematics-12-03822-g006-550.jpg?1733209885" title=" <strong>Figure 6</strong><br/> <p>Different control tasks of the hyper-redundant manipulator. (<b>a</b>) Conventional multi-target approach control. (<b>b</b>) Continuous multi-target approach control.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3822'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03822/article_deploy/html/images/mathematics-12-03822-g007-550.jpg?1733209886" title=" <strong>Figure 7</strong><br/> <p>Problems of the direct samples from joint space. (<b>a</b>) Initial state model penetration. (<b>b</b>) Knotting. (<b>c</b>) Collision. (<b>d</b>) Inconsistent with reality.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3822'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03822/article_deploy/html/images/mathematics-12-03822-g008-550.jpg?1733209888" title=" <strong>Figure 8</strong><br/> <p>Performance of the 12-link, 24-DOF, and hyper-redundant manipulator on continuous multi-target approaching tasks trained by our method. (<b>a</b>) Circle on <math display="inline"><semantics> <mrow> <mi>Y</mi> <mi>O</mi> <mi>Z</mi> </mrow> </semantics></math> plane. (<b>b</b>) Circle on <math display="inline"><semantics> <mrow> <mi>X</mi> <mi>O</mi> <mi>Y</mi> </mrow> </semantics></math> plane. (<b>c</b>) Custom shape on <math display="inline"><semantics> <mrow> <mi>Y</mi> <mi>O</mi> <mi>Z</mi> </mrow> </semantics></math> plane.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3822'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03822/article_deploy/html/images/mathematics-12-03822-g009-550.jpg?1733209889" title=" <strong>Figure 9</strong><br/> <p>Performance of the 12-link, 24-DOF, and hyper-redundant manipulator on continuous multi-target approaching tasks trained by each episode reset. (<b>a</b>) TD3. (<b>b</b>) PPO.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3822'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03822/article_deploy/html/images/mathematics-12-03822-g010-550.jpg?1733209890" title=" <strong>Figure 10</strong><br/> <p>Training results of the different network structures under different control accuracies.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3822'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03822/article_deploy/html/images/mathematics-12-03822-g011-550.jpg?1733209891" title=" <strong>Figure 11</strong><br/> <p>Performance of the 12-link, 24-DOF, and hyper-redundant manipulator on the continuous multi-target approaching tasks trained by each episode reset. (<b>a</b>) Circle on <math display="inline"><semantics> <mrow> <mi>Y</mi> <mi>O</mi> <mi>Z</mi> </mrow> </semantics></math> plane. (<b>b</b>) Circle on <math display="inline"><semantics> <mrow> <mi>X</mi> <mi>O</mi> <mi>Y</mi> </mrow> </semantics></math> plane.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3822'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03822/article_deploy/html/images/mathematics-12-03822-g012-550.jpg?1733209892" title=" <strong>Figure 12</strong><br/> <p>Training effect of the planar hyper-redundant manipulator with 2-to-12 DOFs that were trained by never resetting.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3822'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03822/article_deploy/html/images/mathematics-12-03822-g013-550.jpg?1733209894" title=" <strong>Figure 13</strong><br/> <p>Training effect of the hyper-redundant manipulator by fixed-cycle reset. (<b>a</b>) Planar manipulator with 10 DOFs. (<b>b</b>) 3D manipulators with 4-to-20 DOFs.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3822'>Full article</a></strong> "></a></div> </div> </div> </div> <div class="expanding-div collapsed"> <div class="generic-item article-item"> <div class="article-content"> <div class="label right label__btn"> <span style="font-size: 12px; color: #1a1a1a;"> 18 pages, 590 KiB   </span> <a href="/2227-7390/12/23/3821/pdf?version=1733219322" class="UD_Listings_ArticlePDF" title="Article PDF" data-name="Dynamic Asset Allocation and Retirement Decision with Consumption Ratcheting and Effort Choice" data-journal="mathematics"> <i class="material-icons custom-download"></i> </a> </div> <div class="article-icons"><span class="label openaccess" data-dropdown="drop-article-label-openaccess" aria-expanded="false">Open Access</span><span class="label feature" data-dropdown="drop-article-label-feature" aria-expanded="false">Feature Paper</span><span class="label articletype">Article</span></div> <a class="title-link" href="/2227-7390/12/23/3821">Dynamic Asset Allocation and Retirement Decision with Consumption Ratcheting and Effort Choice</a> <div class="authors"> by <span class="inlineblock "><strong>Geonwoo Kim</strong> and </span><span class="inlineblock "><strong>Junkee Jeon</strong></span> </div> <div class="color-grey-dark"> <em>Mathematics</em> <b>2024</b>, <em>12</em>(23), 3821; <a href="https://doi.org/10.3390/math12233821">https://doi.org/10.3390/math12233821</a> - 2 Dec 2024 </div> <div class="abstract-div"> <a href="#" onclick="$(this).next('.abstract-cropped').toggleClass('inline').next('.abstract-full').toggleClass('inline'); return false;"> <strong>Abstract </strong> </a> <div class="abstract-cropped inline"> This study examines the problem of choosing optimal consumption, investment, early retirement, and effort levels for an economic agent who is unwilling to reduce their consumption over time. The agent’s income depends on a fixed wage rate, and varies proportionally with their level <a href="#" data-counterslink = "https://www.mdpi.com/2227-7390/12/23/3821/more" onclick="$(this).parents('.abstract-cropped').toggleClass('inline').next('.abstract-full').toggleClass('inline'); return false;"> [...] Read more.</a> </div> <div class="abstract-full "> This study examines the problem of choosing optimal consumption, investment, early retirement, and effort levels for an economic agent who is unwilling to reduce their consumption over time. The agent’s income depends on a fixed wage rate, and varies proportionally with their level of effort, which incurs disutility due to labor. By applying the dual-martingale method, we reformulate the problem into its dual form, dividing it into two distinct sub-problems: one focuses on selecting a consumption path that does not decrease, while the other determines the optimal timing for early retirement. Using this method, we find optimal strategies in the form of explicit closed-form solutions. <a href="/2227-7390/12/23/3821">Full article</a> </div> </div> <div class="belongsTo" style="margin-bottom: 10px;"> (This article belongs to the Special Issue <a href=" /journal/mathematics/special_issues/2Q6721PSK5 ">Stochastic Control and Optimization in Mathematical Finance</a>)<br/> </div> <a href="#" class="abstract-figures-show" data-counterslink = "https://www.mdpi.com/2227-7390/12/23/3821/show" ><span >►</span><span style=" display: none;">▼</span> Show Figures </a><div class="abstract-image-preview "><div class="arrow left-arrow" id="prev1535312"><i class="fa fa-caret-left"></i></div><div class="arrow right-arrow" id="next1535312"><i class="fa fa-caret-right"></i></div><div class="absgraph cycle-slideshow manual" data-cycle-fx="scrollHorz" data-cycle-timeout="0" data-cycle-next="#next1535312" data-cycle-prev="#prev1535312" data-cycle-progressive="#images1535312" data-cycle-slides=">div" data-cycle-log="false"><div class='openpopupgallery cycle-slide' data-imgindex='0' data-target='article-1535312-popup'><span class="helper"></span><img src="data:image/gif;base64,R0lGODlhAQABAAD/ACwAAAAAAQABAAACADs=" data-src="https://pub.mdpi-res.com/mathematics/mathematics-12-03821/article_deploy/html/images/mathematics-12-03821-g001-550.jpg?1733219390" alt="" style="border: 0;"><p>Figure 1</p></div><script id="images1535312" type="text/cycle" data-cycle-split="---"><div class='openpopupgallery' data-imgindex='1' data-target='article-1535312-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03821/article_deploy/html/images/mathematics-12-03821-g002-550.jpg?1733219391'><p>Figure 2</p></div> --- <div class='openpopupgallery' data-imgindex='2' data-target='article-1535312-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03821/article_deploy/html/images/mathematics-12-03821-g003-550.jpg?1733219392'><p>Figure 3</p></div> --- <div class='openpopupgallery' data-imgindex='3' data-target='article-1535312-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03821/article_deploy/html/images/mathematics-12-03821-g004-550.jpg?1733219395'><p>Figure 4</p></div></script></div></div><div id="article-1535312-popup" class="popupgallery" style="display: inline; line-height: 200%"><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03821/article_deploy/html/images/mathematics-12-03821-g001-550.jpg?1733219390" title=" <strong>Figure 1</strong><br/> <p>The consumption adjustment.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3821'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03821/article_deploy/html/images/mathematics-12-03821-g002-550.jpg?1733219391" title=" <strong>Figure 2</strong><br/> <p>Two boundaries <math display="inline"><semantics> <mrow> <msub> <mi>X</mi> <mi>A</mi> </msub> <mrow> <mo>(</mo> <mi>c</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>X</mi> <mi>R</mi> </msub> <mrow> <mo>(</mo> <mi>c</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3821'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03821/article_deploy/html/images/mathematics-12-03821-g003-550.jpg?1733219392" title=" <strong>Figure 3</strong><br/> <p>The relationships between <math display="inline"><semantics> <msub> <mi>X</mi> <mi>t</mi> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>e</mi> <mi>t</mi> </msub> </semantics></math>, and between <math display="inline"><semantics> <msub> <mi>X</mi> <mi>t</mi> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>π</mi> <mi>t</mi> </msub> </semantics></math>.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3821'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03821/article_deploy/html/images/mathematics-12-03821-g004-550.jpg?1733219395" title=" <strong>Figure 4</strong><br/> <p>Simulation paths for optimal strategies.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3821'>Full article</a></strong> "></a></div> </div> </div> </div> <div class="expanding-div collapsed"> <div class="generic-item article-item"> <div class="article-content"> <div class="label right label__btn"> <span style="font-size: 12px; color: #1a1a1a;"> 17 pages, 2537 KiB   </span> <a href="/2227-7390/12/23/3820/pdf?version=1733212070" class="UD_Listings_ArticlePDF" title="Article PDF" data-name="Collaborative Optimization Strategy for Dependent Task Offloading in Vehicular Edge Computing" data-journal="mathematics"> <i class="material-icons custom-download"></i> </a> </div> <div class="article-icons"><span class="label openaccess" data-dropdown="drop-article-label-openaccess" aria-expanded="false">Open Access</span><span class="label articletype">Article</span></div> <a class="title-link" href="/2227-7390/12/23/3820">Collaborative Optimization Strategy for Dependent Task Offloading in Vehicular Edge Computing</a> <div class="authors"> by <span class="inlineblock "><strong>Xiting Peng</strong>, </span><span class="inlineblock "><strong>Yandi Zhang</strong>, </span><span class="inlineblock "><strong>Xiaoyu Zhang</strong>, </span><span class="inlineblock "><strong>Chaofeng Zhang</strong> and </span><span class="inlineblock "><strong>Wei Yang</strong></span> </div> <div class="color-grey-dark"> <em>Mathematics</em> <b>2024</b>, <em>12</em>(23), 3820; <a href="https://doi.org/10.3390/math12233820">https://doi.org/10.3390/math12233820</a> - 2 Dec 2024 </div> <div class="abstract-div"> <a href="#" onclick="$(this).next('.abstract-cropped').toggleClass('inline').next('.abstract-full').toggleClass('inline'); return false;"> <strong>Abstract </strong> </a> <div class="abstract-cropped inline"> The advancement of the Internet of Autonomous Vehicles has facilitated the development and deployment of numerous onboard applications. However, the delay-sensitive tasks generated by these applications present enormous challenges for vehicles with limited computing resources. Moreover, these tasks are often interdependent, preventing parallel <a href="#" data-counterslink = "https://www.mdpi.com/2227-7390/12/23/3820/more" onclick="$(this).parents('.abstract-cropped').toggleClass('inline').next('.abstract-full').toggleClass('inline'); return false;"> [...] Read more.</a> </div> <div class="abstract-full "> The advancement of the Internet of Autonomous Vehicles has facilitated the development and deployment of numerous onboard applications. However, the delay-sensitive tasks generated by these applications present enormous challenges for vehicles with limited computing resources. Moreover, these tasks are often interdependent, preventing parallel computation and severely prolonging completion times, which results in substantial energy consumption. Task-offloading technology offers an effective solution to mitigate these challenges. Traditional offloading strategies, however, fall short in the highly dynamic environment of the Internet of Vehicles. This paper proposes a task-offloading scheme based on deep reinforcement learning to optimize the strategy between vehicles and edge computing resources. The task-offloading problem is modeled as a Markov Decision Process, and an improved twin-delayed deep deterministic policy gradient algorithm, LT-TD3, is introduced to enhance the decision-making process. The integration of LSTM and a self-attention mechanism into the LT-TD3 network boosts its capability for feature extraction and representation. Additionally, considering task dependency, a topological sorting algorithm is employed to assign priorities to subtasks, thereby improving the efficiency of task offloading. Experimental results demonstrate that the proposed strategy significantly reduces task delays and energy consumption, offering an effective solution for efficient task processing and energy saving in autonomous vehicles. <a href="/2227-7390/12/23/3820">Full article</a> </div> </div> <div class="belongsTo" style="margin-bottom: 10px;"> (This article belongs to the Special Issue <a href=" /journal/mathematics/special_issues/10F703WE9S ">Advanced Computational Intelligence in Cloud/Edge Computing</a>)<br/> </div> <a href="#" class="abstract-figures-show" data-counterslink = "https://www.mdpi.com/2227-7390/12/23/3820/show" ><span >►</span><span style=" display: none;">▼</span> Show Figures </a><div class="abstract-image-preview "><div class="arrow left-arrow" id="prev1535299"><i class="fa fa-caret-left"></i></div><div class="arrow right-arrow" id="next1535299"><i class="fa fa-caret-right"></i></div><div class="absgraph cycle-slideshow manual" data-cycle-fx="scrollHorz" data-cycle-timeout="0" data-cycle-next="#next1535299" data-cycle-prev="#prev1535299" data-cycle-progressive="#images1535299" data-cycle-slides=">div" data-cycle-log="false"><div class='openpopupgallery cycle-slide' data-imgindex='0' data-target='article-1535299-popup'><span class="helper"></span><img src="data:image/gif;base64,R0lGODlhAQABAAD/ACwAAAAAAQABAAACADs=" data-src="https://pub.mdpi-res.com/mathematics/mathematics-12-03820/article_deploy/html/images/mathematics-12-03820-g001-550.jpg?1733212224" alt="" style="border: 0;"><p>Figure 1</p></div><script id="images1535299" type="text/cycle" data-cycle-split="---"><div class='openpopupgallery' data-imgindex='1' data-target='article-1535299-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03820/article_deploy/html/images/mathematics-12-03820-g002-550.jpg?1733212225'><p>Figure 2</p></div> --- <div class='openpopupgallery' data-imgindex='2' data-target='article-1535299-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03820/article_deploy/html/images/mathematics-12-03820-g003-550.jpg?1733212226'><p>Figure 3</p></div> --- <div class='openpopupgallery' data-imgindex='3' data-target='article-1535299-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03820/article_deploy/html/images/mathematics-12-03820-g004-550.jpg?1733212227'><p>Figure 4</p></div> --- <div class='openpopupgallery' data-imgindex='4' data-target='article-1535299-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03820/article_deploy/html/images/mathematics-12-03820-g005-550.jpg?1733212228'><p>Figure 5</p></div> --- <div class='openpopupgallery' data-imgindex='5' data-target='article-1535299-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03820/article_deploy/html/images/mathematics-12-03820-g006-550.jpg?1733212229'><p>Figure 6</p></div> --- <div class='openpopupgallery' data-imgindex='6' data-target='article-1535299-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03820/article_deploy/html/images/mathematics-12-03820-g007-550.jpg?1733212230'><p>Figure 7</p></div></script></div></div><div id="article-1535299-popup" class="popupgallery" style="display: inline; line-height: 200%"><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03820/article_deploy/html/images/mathematics-12-03820-g001-550.jpg?1733212224" title=" <strong>Figure 1</strong><br/> <p>Autonomous vehicle cloud offloading scenario.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3820'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03820/article_deploy/html/images/mathematics-12-03820-g002-550.jpg?1733212225" title=" <strong>Figure 2</strong><br/> <p>Subtask dependency. (<b>a</b>) Example AR application model. (<b>b</b>) DAG applies subtask dependencies.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3820'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03820/article_deploy/html/images/mathematics-12-03820-g003-550.jpg?1733212226" title=" <strong>Figure 3</strong><br/> <p>LT-TD3 Algorithm architecture diagram.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3820'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03820/article_deploy/html/images/mathematics-12-03820-g004-550.jpg?1733212227" title=" <strong>Figure 4</strong><br/> <p>Experimental results of the average task delay for varying numbers of vehicles in the vehicle cloud.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3820'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03820/article_deploy/html/images/mathematics-12-03820-g005-550.jpg?1733212228" title=" <strong>Figure 5</strong><br/> <p>Experimental results of the average task delay in the vehicle cloud by data size.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3820'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03820/article_deploy/html/images/mathematics-12-03820-g006-550.jpg?1733212229" title=" <strong>Figure 6</strong><br/> <p>Experimental results of the task completion rate under different acceptable delays in vehicle cloud.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3820'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03820/article_deploy/html/images/mathematics-12-03820-g007-550.jpg?1733212230" title=" <strong>Figure 7</strong><br/> <p>Experimental results of the average task delay under different edge device resources in the vehicle cloud.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3820'>Full article</a></strong> "></a></div> </div> </div> </div> <div class="expanding-div collapsed"> <div class="generic-item article-item"> <div class="article-content"> <div class="label right label__btn"> <span style="font-size: 12px; color: #1a1a1a;"> 19 pages, 3243 KiB   </span> <a href="/2227-7390/12/23/3819/pdf?version=1733218704" class="UD_Listings_ArticlePDF" title="Article PDF" data-name="The Effect of Leachate Recycling on the Dynamics of Two Competing Bacteria with an Obligate One-Way Beneficial Relationship in a Chemostat" data-journal="mathematics"> <i class="material-icons custom-download"></i> </a> </div> <div class="article-icons"><span class="label openaccess" data-dropdown="drop-article-label-openaccess" aria-expanded="false">Open Access</span><span class="label articletype">Article</span></div> <a class="title-link" href="/2227-7390/12/23/3819">The Effect of Leachate Recycling on the Dynamics of Two Competing Bacteria with an Obligate One-Way Beneficial Relationship in a Chemostat</a> <div class="authors"> by <span class="inlineblock "><strong>Hanan H. Almuashi</strong>, </span><span class="inlineblock "><strong>Nada A. Almuallem</strong> and </span><span class="inlineblock "><strong>Miled El Hajji</strong></span> </div> <div class="color-grey-dark"> <em>Mathematics</em> <b>2024</b>, <em>12</em>(23), 3819; <a href="https://doi.org/10.3390/math12233819">https://doi.org/10.3390/math12233819</a> - 2 Dec 2024 </div> <div class="abstract-div"> <a href="#" onclick="$(this).next('.abstract-cropped').toggleClass('inline').next('.abstract-full').toggleClass('inline'); return false;"> <strong>Abstract </strong> </a> <div class="abstract-cropped inline"> In the present work, we study a simple mathematical model that describes the competition of two bacterial species with an obligate one-way beneficial relationship for a limited substrate in a bioreactor associated with leachate recirculation. The substrate is present into two forms, insoluble <a href="#" data-counterslink = "https://www.mdpi.com/2227-7390/12/23/3819/more" onclick="$(this).parents('.abstract-cropped').toggleClass('inline').next('.abstract-full').toggleClass('inline'); return false;"> [...] Read more.</a> </div> <div class="abstract-full "> In the present work, we study a simple mathematical model that describes the competition of two bacterial species with an obligate one-way beneficial relationship for a limited substrate in a bioreactor associated with leachate recirculation. The substrate is present into two forms, insoluble and soluble substrates, where the latter is consumed by the two competing bacteria, which have two general nonlinear growth rates. The reduction of the model to a 2D one facilitates the study of the nature of the equilibrium points. The dynamic system admits multiple steady states. We provide necessary and sufficient conditions on the added insoluble and soluble substrates and the dilution rate to guarantee the existence, uniqueness, and local and global stability of such steady states. It is deduced that the coexistence of both bacteria is possible, which contradicts the competitive exclusion principle. In the second step, we propose an optimal control on the leachate recirculation rate that reduces the organic matter inside the reactor. Finally, we provide some numerical examples that corroborate and reinforce the theoretical findings. <a href="/2227-7390/12/23/3819">Full article</a> </div> </div> <div class="belongsTo" style="margin-bottom: 10px;"> (This article belongs to the Section <a href="/journal/mathematics/sections/dynamic_systems">Dynamical Systems</a>)<br/> </div> <a href="#" class="abstract-figures-show" data-counterslink = "https://www.mdpi.com/2227-7390/12/23/3819/show" ><span >►</span><span style=" display: none;">▼</span> Show Figures </a><div class="abstract-image-preview "><div class="arrow left-arrow" id="prev1535278"><i class="fa fa-caret-left"></i></div><div class="arrow right-arrow" id="next1535278"><i class="fa fa-caret-right"></i></div><div class="absgraph cycle-slideshow manual" data-cycle-fx="scrollHorz" data-cycle-timeout="0" data-cycle-next="#next1535278" data-cycle-prev="#prev1535278" data-cycle-progressive="#images1535278" data-cycle-slides=">div" data-cycle-log="false"><div class='openpopupgallery cycle-slide' data-imgindex='0' data-target='article-1535278-popup'><span class="helper"></span><img src="data:image/gif;base64,R0lGODlhAQABAAD/ACwAAAAAAQABAAACADs=" data-src="https://pub.mdpi-res.com/mathematics/mathematics-12-03819/article_deploy/html/images/mathematics-12-03819-g001-550.jpg?1733218814" alt="" style="border: 0;"><p>Figure 1</p></div><script id="images1535278" type="text/cycle" data-cycle-split="---"><div class='openpopupgallery' data-imgindex='1' data-target='article-1535278-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03819/article_deploy/html/images/mathematics-12-03819-g002-550.jpg?1733218815'><p>Figure 2</p></div> --- <div class='openpopupgallery' data-imgindex='2' data-target='article-1535278-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03819/article_deploy/html/images/mathematics-12-03819-g003-550.jpg?1733218816'><p>Figure 3</p></div> --- <div class='openpopupgallery' data-imgindex='3' data-target='article-1535278-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03819/article_deploy/html/images/mathematics-12-03819-g004-550.jpg?1733218818'><p>Figure 4</p></div> --- <div class='openpopupgallery' data-imgindex='4' data-target='article-1535278-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03819/article_deploy/html/images/mathematics-12-03819-g005-550.jpg?1733218819'><p>Figure 5</p></div> --- <div class='openpopupgallery' data-imgindex='5' data-target='article-1535278-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03819/article_deploy/html/images/mathematics-12-03819-g006-550.jpg?1733218821'><p>Figure 6</p></div> --- <div class='openpopupgallery' data-imgindex='6' data-target='article-1535278-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03819/article_deploy/html/images/mathematics-12-03819-g007-550.jpg?1733218822'><p>Figure 7</p></div> --- <div class='openpopupgallery' data-imgindex='7' data-target='article-1535278-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03819/article_deploy/html/images/mathematics-12-03819-g008-550.jpg?1733218824'><p>Figure 8</p></div> --- <div class='openpopupgallery' data-imgindex='8' data-target='article-1535278-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03819/article_deploy/html/images/mathematics-12-03819-g009-550.jpg?1733218825'><p>Figure 9</p></div> --- <div class='openpopupgallery' data-imgindex='9' data-target='article-1535278-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03819/article_deploy/html/images/mathematics-12-03819-g010-550.jpg?1733218827'><p>Figure 10</p></div></script></div></div><div id="article-1535278-popup" class="popupgallery" style="display: inline; line-height: 200%"><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03819/article_deploy/html/images/mathematics-12-03819-g001-550.jpg?1733218814" title=" <strong>Figure 1</strong><br/> <p>Interactions between two competing bacteria associated with an obligate one-way benefit and recirculation of the leachate [<a href="#B43-mathematics-12-03819" class="html-bibr">43</a>].</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3819'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03819/article_deploy/html/images/mathematics-12-03819-g002-550.jpg?1733218815" title=" <strong>Figure 2</strong><br/> <p>A classical bioreactor (Chemostat) associated with leachate recirculation [<a href="#B41-mathematics-12-03819" class="html-bibr">41</a>].</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3819'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03819/article_deploy/html/images/mathematics-12-03819-g003-550.jpg?1733218816" title=" <strong>Figure 3</strong><br/> <p>Comportment of system (<a href="#FD2-mathematics-12-03819" class="html-disp-formula">2</a>); the comportment of the components (<b>left</b>) and the <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> </mrow> </semantics></math> comportment (<b>right</b>) for <math display="inline"><semantics> <mrow> <mi>D</mi> <mo>=</mo> <mn>5.9</mn> </mrow> </semantics></math>. The red dot (<b>right</b>) is the equilibrium point in the plane (<math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> </mrow> </semantics></math>).</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3819'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03819/article_deploy/html/images/mathematics-12-03819-g004-550.jpg?1733218818" title=" <strong>Figure 4</strong><br/> <p>Comportment of system (<a href="#FD2-mathematics-12-03819" class="html-disp-formula">2</a>); the comportment of the components (<b>left</b>) and the <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> </mrow> </semantics></math> comportment (<b>right</b>) for <math display="inline"><semantics> <mrow> <mi>D</mi> <mo>=</mo> <mn>6.8</mn> </mrow> </semantics></math>. The red dot (<b>right</b>) is the equilibrium point in the plane (<math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> </mrow> </semantics></math>).</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3819'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03819/article_deploy/html/images/mathematics-12-03819-g005-550.jpg?1733218819" title=" <strong>Figure 5</strong><br/> <p>Comportment of system (<a href="#FD2-mathematics-12-03819" class="html-disp-formula">2</a>); the comportment of the components (<b>left</b>) and the <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> </mrow> </semantics></math> comportment (<b>right</b>) for <math display="inline"><semantics> <mrow> <mi>D</mi> <mo>=</mo> <mn>6.9</mn> </mrow> </semantics></math>. The red dot (<b>right</b>) is the equilibrium point in the plane (<math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> </mrow> </semantics></math>).</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3819'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03819/article_deploy/html/images/mathematics-12-03819-g006-550.jpg?1733218821" title=" <strong>Figure 6</strong><br/> <p>Comportment of system (<a href="#FD2-mathematics-12-03819" class="html-disp-formula">2</a>); the comportment of the components (<b>left</b>) and the <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> </mrow> </semantics></math> comportment (<b>right</b>) for <math display="inline"><semantics> <mrow> <mi>D</mi> <mo>=</mo> <mn>7</mn> </mrow> </semantics></math>. The red dot (<b>right</b>) is the equilibrium point in the plane (<math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> </mrow> </semantics></math>).</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3819'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03819/article_deploy/html/images/mathematics-12-03819-g007-550.jpg?1733218822" title=" <strong>Figure 7</strong><br/> <p>Comportment of system (<a href="#FD2-mathematics-12-03819" class="html-disp-formula">2</a>); the comportment of the components (<b>left</b>) and the <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> </mrow> </semantics></math> comportment (<b>right</b>) for <math display="inline"><semantics> <mrow> <mi>D</mi> <mo>=</mo> <mn>8</mn> </mrow> </semantics></math>. The red dot (<b>right</b>) is the equilibrium point in the plane (<math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> </mrow> </semantics></math>).</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3819'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03819/article_deploy/html/images/mathematics-12-03819-g008-550.jpg?1733218824" title=" <strong>Figure 8</strong><br/> <p>Influence of the optimal strategy on the dynamics for <math display="inline"><semantics> <mrow> <msub> <mi>κ</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>κ</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>κ</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>D</mi> <mo>=</mo> <mn>5.9</mn> </mrow> </semantics></math>.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3819'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03819/article_deploy/html/images/mathematics-12-03819-g009-550.jpg?1733218825" title=" <strong>Figure 9</strong><br/> <p>Influence of the optimal strategy on the dynamics for <math display="inline"><semantics> <mrow> <msub> <mi>κ</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>κ</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>κ</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>D</mi> <mo>=</mo> <mn>5.9</mn> </mrow> </semantics></math>.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3819'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03819/article_deploy/html/images/mathematics-12-03819-g010-550.jpg?1733218827" title=" <strong>Figure 10</strong><br/> <p>Influence of the optimal strategy on the dynamics for <math display="inline"><semantics> <mrow> <msub> <mi>κ</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>κ</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>κ</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>D</mi> <mo>=</mo> <mn>5.9</mn> </mrow> </semantics></math>.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3819'>Full article</a></strong> "></a></div> </div> </div> </div> <div class="expanding-div collapsed"> <div class="generic-item article-item"> <div class="article-content"> <div class="label right label__btn"> <span style="font-size: 12px; color: #1a1a1a;"> 19 pages, 3327 KiB   </span> <a href="/2227-7390/12/23/3818/pdf?version=1733220856" class="UD_Listings_ArticlePDF" title="Article PDF" data-name="Mixed Multi-Strategy Improved Aquila Optimizer and Its Application in Path Planning" data-journal="mathematics"> <i class="material-icons custom-download"></i> </a> </div> <div class="article-icons"><span class="label openaccess" data-dropdown="drop-article-label-openaccess" aria-expanded="false">Open Access</span><span class="label articletype">Article</span></div> <a class="title-link" href="/2227-7390/12/23/3818">Mixed Multi-Strategy Improved Aquila Optimizer and Its Application in Path Planning</a> <div class="authors"> by <span class="inlineblock "><strong>Tianyue Bao</strong>, </span><span class="inlineblock "><strong>Jiaxin Zhao</strong>, </span><span class="inlineblock "><strong>Yanchang Liu</strong>, </span><span class="inlineblock "><strong>Xusheng Guo</strong> and </span><span class="inlineblock "><strong>Tianshuo Chen</strong></span> </div> <div class="color-grey-dark"> <em>Mathematics</em> <b>2024</b>, <em>12</em>(23), 3818; <a href="https://doi.org/10.3390/math12233818">https://doi.org/10.3390/math12233818</a> - 2 Dec 2024 </div> <div class="abstract-div"> <a href="#" onclick="$(this).next('.abstract-cropped').toggleClass('inline').next('.abstract-full').toggleClass('inline'); return false;"> <strong>Abstract </strong> </a> <div class="abstract-cropped inline"> With the growing prevalence of drone technology across various sectors, efficient and safe path planning has emerged as a critical research priority. Traditional Aquila Optimizers, while effective, face limitations such as uneven population initialization, a tendency to get trapped in local optima, and <a href="#" data-counterslink = "https://www.mdpi.com/2227-7390/12/23/3818/more" onclick="$(this).parents('.abstract-cropped').toggleClass('inline').next('.abstract-full').toggleClass('inline'); return false;"> [...] Read more.</a> </div> <div class="abstract-full "> With the growing prevalence of drone technology across various sectors, efficient and safe path planning has emerged as a critical research priority. Traditional Aquila Optimizers, while effective, face limitations such as uneven population initialization, a tendency to get trapped in local optima, and slow convergence rates. This study presents a multi-strategy fusion of the improved Aquila Optimizer, aiming to enhance its performance by integrating diverse optimization techniques, particularly in the context of path planning. Key enhancements include the integration of Bernoulli chaotic mapping to improve initial population diversity, a spiral stepping strategy to boost search precision and diversity, and a “stealing” mechanism from the Dung Beetle Optimization algorithm to enhance global search capabilities and convergence. Additionally, a nonlinear balance factor is employed to dynamically manage the exploration–exploitation trade-off, thereby increasing the optimization of speed and accuracy. The effectiveness of the mixed multi-strategy improved Aquila Optimizer is validated through simulations on benchmark test functions, CEC2017 complex functions, and path planning scenarios. Comparative analysis with seven other optimization algorithms reveals that the proposed method significantly improves both convergence speed and optimization accuracy. These findings highlight the potential of mixed multi-strategy improved Aquila Optimizer in advancing drone path planning performance, offering enhanced safety and efficiency. <a href="/2227-7390/12/23/3818">Full article</a> </div> </div> <div class="belongsTo" style="margin-bottom: 10px;"> (This article belongs to the Special Issue <a href=" /journal/mathematics/special_issues/6V12W61V71 ">Optimization Models and Algorithms in Data Science</a>)<br/> </div> <a href="#" class="abstract-figures-show" data-counterslink = "https://www.mdpi.com/2227-7390/12/23/3818/show" ><span >►</span><span style=" display: none;">▼</span> Show Figures </a><div class="abstract-image-preview "><div class="arrow left-arrow" id="prev1535201"><i class="fa fa-caret-left"></i></div><div class="arrow right-arrow" id="next1535201"><i class="fa fa-caret-right"></i></div><div class="absgraph cycle-slideshow manual" data-cycle-fx="scrollHorz" data-cycle-timeout="0" data-cycle-next="#next1535201" data-cycle-prev="#prev1535201" data-cycle-progressive="#images1535201" data-cycle-slides=">div" data-cycle-log="false"><div class='openpopupgallery cycle-slide' data-imgindex='0' data-target='article-1535201-popup'><span class="helper"></span><img src="data:image/gif;base64,R0lGODlhAQABAAD/ACwAAAAAAQABAAACADs=" data-src="https://pub.mdpi-res.com/mathematics/mathematics-12-03818/article_deploy/html/images/mathematics-12-03818-g001-550.jpg?1733220943" alt="" style="border: 0;"><p>Figure 1</p></div><script id="images1535201" type="text/cycle" data-cycle-split="---"><div class='openpopupgallery' data-imgindex='1' data-target='article-1535201-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03818/article_deploy/html/images/mathematics-12-03818-g002-550.jpg?1733220945'><p>Figure 2</p></div> --- <div class='openpopupgallery' data-imgindex='2' data-target='article-1535201-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03818/article_deploy/html/images/mathematics-12-03818-g003-550.jpg?1733220948'><p>Figure 3</p></div> --- <div class='openpopupgallery' data-imgindex='3' data-target='article-1535201-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03818/article_deploy/html/images/mathematics-12-03818-g004-550.jpg?1733220949'><p>Figure 4</p></div> --- <div class='openpopupgallery' data-imgindex='4' data-target='article-1535201-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03818/article_deploy/html/images/mathematics-12-03818-g005a-550.jpg?1733220953'><p>Figure 5</p></div> --- <div class='openpopupgallery' data-imgindex='5' data-target='article-1535201-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03818/article_deploy/html/images/mathematics-12-03818-g005b-550.jpg?1733220957'><p>Figure 5 Cont.</p></div> --- <div class='openpopupgallery' data-imgindex='6' data-target='article-1535201-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03818/article_deploy/html/images/mathematics-12-03818-g006-550.jpg?1733220960'><p>Figure 6</p></div></script></div></div><div id="article-1535201-popup" class="popupgallery" style="display: inline; line-height: 200%"><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03818/article_deploy/html/images/mathematics-12-03818-g001-550.jpg?1733220943" title=" <strong>Figure 1</strong><br/> <p>Distribution of population initialization: (<b>a</b>) scattered distribution of random; (<b>b</b>) scattered distribution of Bernoulli; (<b>c</b>) distribution histogram of random; (<b>d</b>) distribution histogram of Bernoulli.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3818'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03818/article_deploy/html/images/mathematics-12-03818-g002-550.jpg?1733220945" title=" <strong>Figure 2</strong><br/> <p>Flowchart of MMSIAO.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3818'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03818/article_deploy/html/images/mathematics-12-03818-g003-550.jpg?1733220948" title=" <strong>Figure 3</strong><br/> <p>Adaptation graph for different swarm intelligence algorithms.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3818'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03818/article_deploy/html/images/mathematics-12-03818-g004-550.jpg?1733220949" title=" <strong>Figure 4</strong><br/> <p>Three-dimensional model.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3818'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03818/article_deploy/html/images/mathematics-12-03818-g005a-550.jpg?1733220953" title=" <strong>Figure 5</strong><br/> <p>The path planning model is shown in the following views: (<b>a</b>) in two dimensions of Environment 1; (<b>b</b>) in two dimensions of Environment 2; (<b>c</b>) in three dimensions of Environment 1; (<b>d</b>) in three dimensions of Environment 2; (<b>e</b>) in a separate comparison of eight algorithms of Environment 1; (<b>f</b>) in a separate comparison of eight algorithms of Environment 2.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3818'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03818/article_deploy/html/images/mathematics-12-03818-g005b-550.jpg?1733220957" title=" <strong>Figure 5 Cont.</strong><br/> <p>The path planning model is shown in the following views: (<b>a</b>) in two dimensions of Environment 1; (<b>b</b>) in two dimensions of Environment 2; (<b>c</b>) in three dimensions of Environment 1; (<b>d</b>) in three dimensions of Environment 2; (<b>e</b>) in a separate comparison of eight algorithms of Environment 1; (<b>f</b>) in a separate comparison of eight algorithms of Environment 2.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3818'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03818/article_deploy/html/images/mathematics-12-03818-g006-550.jpg?1733220960" title=" <strong>Figure 6</strong><br/> <p>The model for path planning diagram of convergence. (<b>a</b>) Convergence diagram of the cost function in Environment 1; (<b>b</b>) convergence diagram of the cost function in Environment 2.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3818'>Full article</a></strong> "></a></div> </div> </div> </div> <div class="expanding-div collapsed"> <div class="generic-item article-item"> <div class="article-content"> <div class="label right label__btn"> <span style="font-size: 12px; color: #1a1a1a;"> 30 pages, 1186 KiB   </span> <a href="/2227-7390/12/23/3817/pdf?version=1733149629" class="UD_Listings_ArticlePDF" title="Article PDF" data-name="Synergistic Impact of Active Case Detection and Early Hospitalization for Controlling the Spread of Yellow Fever Outbreak in Nigeria: An Epidemiological Modeling and Optimal Control Analysis" data-journal="mathematics"> <i class="material-icons custom-download"></i> </a> </div> <div class="article-icons"><span class="label openaccess" data-dropdown="drop-article-label-openaccess" aria-expanded="false">Open Access</span><span class="label articletype">Article</span></div> <a class="title-link" href="/2227-7390/12/23/3817">Synergistic Impact of Active Case Detection and Early Hospitalization for Controlling the Spread of Yellow Fever Outbreak in Nigeria: An Epidemiological Modeling and Optimal Control Analysis</a> <div class="authors"> by <span class="inlineblock "><strong>Nawaf L. Alsowait</strong>, </span><span class="inlineblock "><strong>Mohammed M. Al-Shomrani</strong>, </span><span class="inlineblock "><strong>Ismail Abdulrashid</strong> and </span><span class="inlineblock "><strong>Salihu S. Musa</strong></span> </div> <div class="color-grey-dark"> <em>Mathematics</em> <b>2024</b>, <em>12</em>(23), 3817; <a href="https://doi.org/10.3390/math12233817">https://doi.org/10.3390/math12233817</a> - 2 Dec 2024 </div> <div class="abstract-div"> <a href="#" onclick="$(this).next('.abstract-cropped').toggleClass('inline').next('.abstract-full').toggleClass('inline'); return false;"> <strong>Abstract </strong> </a> <div class="abstract-cropped inline"> Capturing the factors influencing yellow fever (YF) outbreaks is essential for effective public health interventions, especially in regions like Nigeria, where the disease poses significant health risks. This study explores the synergistic effects of active case detection (ACD) and early hospitalization on controlling <a href="#" data-counterslink = "https://www.mdpi.com/2227-7390/12/23/3817/more" onclick="$(this).parents('.abstract-cropped').toggleClass('inline').next('.abstract-full').toggleClass('inline'); return false;"> [...] Read more.</a> </div> <div class="abstract-full "> Capturing the factors influencing yellow fever (YF) outbreaks is essential for effective public health interventions, especially in regions like Nigeria, where the disease poses significant health risks. This study explores the synergistic effects of active case detection (ACD) and early hospitalization on controlling YF transmission dynamics. We develop a dynamic model that integrates vaccination, active case detection, and hospitalization to enhance our understanding of disease spread and inform prevention strategies. Our methodology encompasses mechanistic dynamic modeling, optimal control analysis, parameter estimation, model fitting, and sensitivity analyses to study YF transmission dynamics, ensuring the robustness of control measures. We employ advanced mathematical techniques, including next-generation matrix methods, to accurately compute the reproduction number and assess outbreak transmissibility. Rigorous qualitative analysis of the model reveals two equilibria: disease-free and endemic, demonstrating global asymptotic stability and its impact on overall YF transmission dynamics, significantly affecting control and prevention mechanisms. Furthermore, through sensitivity analysis, we identify crucial parameters of the model that require urgent attention for more effective YF control. Moreover, our results highlight the critical roles of ACD and early hospitalization in reducing YF transmission. These insights provide a foundation for informed decision making and resource allocation in epidemic control efforts, ultimately contributing to the enhancement of public health strategies aimed at mitigating the impact of YF outbreaks. <a href="/2227-7390/12/23/3817">Full article</a> </div> </div> <div class="belongsTo" style="margin-bottom: 10px;"> (This article belongs to the Special Issue <a href=" /journal/mathematics/special_issues/9J7G6DSH17 ">Mathematical Analysis of Infectious Disease</a>)<br/> </div> </div> </div> </div> <div class="expanding-div collapsed"> <div class="generic-item article-item"> <div class="article-content"> <div class="label right label__btn"> <span style="font-size: 12px; color: #1a1a1a;"> 22 pages, 3505 KiB   </span> <a href="/2227-7390/12/23/3816/pdf?version=1733211120" class="UD_Listings_ArticlePDF" title="Article PDF" data-name="Fixed-Time Command-Filtered Control for Nonlinear Systems with Mismatched Disturbances" data-journal="mathematics"> <i class="material-icons custom-download"></i> </a> </div> <div class="article-icons"><span class="label openaccess" data-dropdown="drop-article-label-openaccess" aria-expanded="false">Open Access</span><span class="label articletype">Article</span></div> <a class="title-link" href="/2227-7390/12/23/3816">Fixed-Time Command-Filtered Control for Nonlinear Systems with Mismatched Disturbances</a> <div class="authors"> by <span class="inlineblock "><strong>Zhiqiang Wu</strong>, </span><span class="inlineblock "><strong>Jian Zhang</strong>, </span><span class="inlineblock "><strong>Lei Xing</strong> and </span><span class="inlineblock "><strong>Liyang Sun</strong></span> </div> <div class="color-grey-dark"> <em>Mathematics</em> <b>2024</b>, <em>12</em>(23), 3816; <a href="https://doi.org/10.3390/math12233816">https://doi.org/10.3390/math12233816</a> - 2 Dec 2024 </div> <div class="abstract-div"> <a href="#" onclick="$(this).next('.abstract-cropped').toggleClass('inline').next('.abstract-full').toggleClass('inline'); return false;"> <strong>Abstract </strong> </a> <div class="abstract-cropped inline"> This article concerns the issue of adaptive fuzzy command-filtered fixed-time control in the context of a category of nonlinear systems characterized by mismatched disturbances and unknown nonlinear functions. The backstepping-based disturbance observers are created to alleviate the effects of mismatched disturbances and Fuzzy <a href="#" data-counterslink = "https://www.mdpi.com/2227-7390/12/23/3816/more" onclick="$(this).parents('.abstract-cropped').toggleClass('inline').next('.abstract-full').toggleClass('inline'); return false;"> [...] Read more.</a> </div> <div class="abstract-full "> This article concerns the issue of adaptive fuzzy command-filtered fixed-time control in the context of a category of nonlinear systems characterized by mismatched disturbances and unknown nonlinear functions. The backstepping-based disturbance observers are created to alleviate the effects of mismatched disturbances and Fuzzy logic systems are brought into play to model those terms that are unknown. To address the complexity explosion issue in traditional backstepping control, this paper utilizes fixed-time command filters (FTCFs) to design a novel control approach. Moreover, filtering error compensation mechanisms are developed to eliminate the errors introduced by the FTCFs. This paper derives a novel adaptive fixed-time control protocol that successfully conquers the difficulties posed by unknown nonlinear functions and mismatched disturbances. The protocol, implemented within a backstepping framework, guarantees the boundedness of all signals and tracking errors within fixed time. The efficacy of the derived control protocol is illustrated through simulation examples. <a href="/2227-7390/12/23/3816">Full article</a> </div> </div> <a href="#" class="abstract-figures-show" data-counterslink = "https://www.mdpi.com/2227-7390/12/23/3816/show" ><span >►</span><span style=" display: none;">▼</span> Show Figures </a><div class="abstract-image-preview "><div class="arrow left-arrow" id="prev1535151"><i class="fa fa-caret-left"></i></div><div class="arrow right-arrow" id="next1535151"><i class="fa fa-caret-right"></i></div><div class="absgraph cycle-slideshow manual" data-cycle-fx="scrollHorz" data-cycle-timeout="0" data-cycle-next="#next1535151" data-cycle-prev="#prev1535151" data-cycle-progressive="#images1535151" data-cycle-slides=">div" data-cycle-log="false"><div class='openpopupgallery cycle-slide' data-imgindex='0' data-target='article-1535151-popup'><span class="helper"></span><img src="data:image/gif;base64,R0lGODlhAQABAAD/ACwAAAAAAQABAAACADs=" 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src='https://pub.mdpi-res.com/mathematics/mathematics-12-03816/article_deploy/html/images/mathematics-12-03816-g004-550.jpg?1733211222'><p>Figure 4</p></div> --- <div class='openpopupgallery' data-imgindex='4' data-target='article-1535151-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03816/article_deploy/html/images/mathematics-12-03816-g005-550.jpg?1733211222'><p>Figure 5</p></div> --- <div class='openpopupgallery' data-imgindex='5' data-target='article-1535151-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03816/article_deploy/html/images/mathematics-12-03816-g006-550.jpg?1733211223'><p>Figure 6</p></div> --- <div class='openpopupgallery' data-imgindex='6' data-target='article-1535151-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03816/article_deploy/html/images/mathematics-12-03816-g007-550.jpg?1733211223'><p>Figure 7</p></div> --- <div class='openpopupgallery' data-imgindex='7' data-target='article-1535151-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03816/article_deploy/html/images/mathematics-12-03816-g008-550.jpg?1733211224'><p>Figure 8</p></div> --- <div class='openpopupgallery' data-imgindex='8' data-target='article-1535151-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03816/article_deploy/html/images/mathematics-12-03816-g009-550.jpg?1733211224'><p>Figure 9</p></div> --- <div class='openpopupgallery' data-imgindex='9' data-target='article-1535151-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03816/article_deploy/html/images/mathematics-12-03816-g010-550.jpg?1733211225'><p>Figure 10</p></div> --- <div class='openpopupgallery' data-imgindex='10' data-target='article-1535151-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03816/article_deploy/html/images/mathematics-12-03816-g011-550.jpg?1733211225'><p>Figure 11</p></div> --- <div class='openpopupgallery' data-imgindex='11' data-target='article-1535151-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03816/article_deploy/html/images/mathematics-12-03816-g012-550.jpg?1733211226'><p>Figure 12</p></div> --- <div class='openpopupgallery' data-imgindex='12' data-target='article-1535151-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03816/article_deploy/html/images/mathematics-12-03816-g013-550.jpg?1733211226'><p>Figure 13</p></div> --- <div class='openpopupgallery' data-imgindex='13' data-target='article-1535151-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03816/article_deploy/html/images/mathematics-12-03816-g014-550.jpg?1733211227'><p>Figure 14</p></div> --- <div class='openpopupgallery' data-imgindex='14' data-target='article-1535151-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03816/article_deploy/html/images/mathematics-12-03816-g015-550.jpg?1733211228'><p>Figure 15</p></div> --- <div class='openpopupgallery' data-imgindex='15' data-target='article-1535151-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03816/article_deploy/html/images/mathematics-12-03816-g016-550.jpg?1733211228'><p>Figure 16</p></div> --- <div class='openpopupgallery' data-imgindex='16' data-target='article-1535151-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03816/article_deploy/html/images/mathematics-12-03816-g017-550.jpg?1733211229'><p>Figure 17</p></div> --- <div class='openpopupgallery' data-imgindex='17' data-target='article-1535151-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03816/article_deploy/html/images/mathematics-12-03816-g018-550.jpg?1733211229'><p>Figure 18</p></div> --- <div class='openpopupgallery' data-imgindex='18' data-target='article-1535151-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03816/article_deploy/html/images/mathematics-12-03816-g019-550.jpg?1733211230'><p>Figure 19</p></div> --- <div class='openpopupgallery' data-imgindex='19' data-target='article-1535151-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03816/article_deploy/html/images/mathematics-12-03816-g020-550.jpg?1733211230'><p>Figure 20</p></div> --- <div class='openpopupgallery' data-imgindex='20' data-target='article-1535151-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03816/article_deploy/html/images/mathematics-12-03816-g021-550.jpg?1733211231'><p>Figure 21</p></div></script></div></div><div id="article-1535151-popup" class="popupgallery" style="display: inline; line-height: 200%"><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03816/article_deploy/html/images/mathematics-12-03816-g001-550.jpg?1733211220" title=" <strong>Figure 1</strong><br/> <p>Block diagram of the fixed-time control scheme.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3816'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03816/article_deploy/html/images/mathematics-12-03816-g002-550.jpg?1733211221" title=" <strong>Figure 2</strong><br/> <p>The trajectories of the states <math display="inline"><semantics> <msub> <mi>x</mi> <mn>1</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>x</mi> <mi>d</mi> </msub> </semantics></math>.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3816'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03816/article_deploy/html/images/mathematics-12-03816-g003-550.jpg?1733211221" title=" <strong>Figure 3</strong><br/> <p>The trajectory of the state <math display="inline"><semantics> <msub> <mi>x</mi> <mn>2</mn> </msub> </semantics></math>.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3816'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03816/article_deploy/html/images/mathematics-12-03816-g004-550.jpg?1733211222" title=" <strong>Figure 4</strong><br/> <p>The trajectory of tracking error <math display="inline"><semantics> <mrow> <mrow> <mo stretchy="false">|</mo> </mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>x</mi> <mi>d</mi> </msub> <mrow> <mo stretchy="false">|</mo> </mrow> </mrow> </semantics></math>.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3816'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03816/article_deploy/html/images/mathematics-12-03816-g005-550.jpg?1733211222" title=" <strong>Figure 5</strong><br/> <p>The trajectory of control signal.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3816'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03816/article_deploy/html/images/mathematics-12-03816-g006-550.jpg?1733211223" title=" <strong>Figure 6</strong><br/> <p>The trajectories of the states <math display="inline"><semantics> <msub> <mi>x</mi> <mn>1</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>x</mi> <mi>d</mi> </msub> </semantics></math>.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3816'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03816/article_deploy/html/images/mathematics-12-03816-g007-550.jpg?1733211223" title=" <strong>Figure 7</strong><br/> <p>The trajectory of the state <math display="inline"><semantics> <msub> <mi>x</mi> <mn>2</mn> </msub> </semantics></math>.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3816'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03816/article_deploy/html/images/mathematics-12-03816-g008-550.jpg?1733211224" title=" <strong>Figure 8</strong><br/> <p>The trajectory of tracking error <math display="inline"><semantics> <mrow> <mrow> <mo stretchy="false">|</mo> </mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>x</mi> <mi>d</mi> </msub> <mrow> <mo stretchy="false">|</mo> </mrow> </mrow> </semantics></math>.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3816'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03816/article_deploy/html/images/mathematics-12-03816-g009-550.jpg?1733211224" title=" <strong>Figure 9</strong><br/> <p>The trajectory of control signal.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3816'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03816/article_deploy/html/images/mathematics-12-03816-g010-550.jpg?1733211225" title=" <strong>Figure 10</strong><br/> <p>The curves of the states <math display="inline"><semantics> <msub> <mi>x</mi> <mn>1</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>y</mi> <mi>d</mi> </msub> </semantics></math> for case 1.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3816'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03816/article_deploy/html/images/mathematics-12-03816-g011-550.jpg?1733211225" title=" <strong>Figure 11</strong><br/> <p>The curve of the tracking error for case 1.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3816'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03816/article_deploy/html/images/mathematics-12-03816-g012-550.jpg?1733211226" title=" <strong>Figure 12</strong><br/> <p>The curves of the states <math display="inline"><semantics> <msub> <mi>x</mi> <mn>1</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>y</mi> <mi>d</mi> </msub> </semantics></math> for case 2.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3816'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03816/article_deploy/html/images/mathematics-12-03816-g013-550.jpg?1733211226" title=" <strong>Figure 13</strong><br/> <p>The curve of the tracking error for case 2.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3816'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03816/article_deploy/html/images/mathematics-12-03816-g014-550.jpg?1733211227" title=" <strong>Figure 14</strong><br/> <p>The curves of the states <math display="inline"><semantics> <msub> <mi>x</mi> <mn>1</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>y</mi> <mi>d</mi> </msub> </semantics></math> for case 3.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3816'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03816/article_deploy/html/images/mathematics-12-03816-g015-550.jpg?1733211228" title=" <strong>Figure 15</strong><br/> <p>The curve of the tracking error for case 3.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3816'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03816/article_deploy/html/images/mathematics-12-03816-g016-550.jpg?1733211228" title=" <strong>Figure 16</strong><br/> <p>The curves of the states <math display="inline"><semantics> <msub> <mi>x</mi> <mn>1</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>y</mi> <mi>d</mi> </msub> </semantics></math> for Ref. [<a href="#B49-mathematics-12-03816" class="html-bibr">49</a>].</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3816'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03816/article_deploy/html/images/mathematics-12-03816-g017-550.jpg?1733211229" title=" <strong>Figure 17</strong><br/> <p>The curve of the tracking error for Ref. [<a href="#B49-mathematics-12-03816" class="html-bibr">49</a>].</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3816'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03816/article_deploy/html/images/mathematics-12-03816-g018-550.jpg?1733211229" title=" <strong>Figure 18</strong><br/> <p>The curves of the states <math display="inline"><semantics> <msub> <mi>x</mi> <mn>1</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>y</mi> <mi>d</mi> </msub> </semantics></math> for fixed-time controller Equation (<a href="#FD52-mathematics-12-03816" class="html-disp-formula">52</a>).</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3816'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03816/article_deploy/html/images/mathematics-12-03816-g019-550.jpg?1733211230" title=" <strong>Figure 19</strong><br/> <p>The curve of the tracking error for fixed-time controller Equation (<a href="#FD52-mathematics-12-03816" class="html-disp-formula">52</a>).</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3816'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03816/article_deploy/html/images/mathematics-12-03816-g020-550.jpg?1733211230" title=" <strong>Figure 20</strong><br/> <p>The curves of the states <math display="inline"><semantics> <msub> <mi>x</mi> <mn>1</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>y</mi> <mi>d</mi> </msub> </semantics></math> for finite-time controller Equation (<a href="#FD74-mathematics-12-03816" class="html-disp-formula">74</a>).</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3816'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03816/article_deploy/html/images/mathematics-12-03816-g021-550.jpg?1733211231" title=" <strong>Figure 21</strong><br/> <p>The curve of the tracking error for finite-time controller Equation (<a href="#FD74-mathematics-12-03816" class="html-disp-formula">74</a>).</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3816'>Full article</a></strong> "></a></div> </div> </div> </div> <div class="expanding-div collapsed"> <div class="generic-item article-item"> <div class="article-content"> <div class="label right label__btn"> <span style="font-size: 12px; color: #1a1a1a;"> 16 pages, 1491 KiB   </span> <a href="/2227-7390/12/23/3815/pdf?version=1733217560" class="UD_Listings_ArticlePDF" title="Article PDF" data-name="An Effective Hybrid Metaheuristic Approach Based on the Genetic Algorithm" data-journal="mathematics"> <i class="material-icons custom-download"></i> </a> </div> <div class="article-icons"><span class="label openaccess" data-dropdown="drop-article-label-openaccess" aria-expanded="false">Open Access</span><span class="label articletype">Article</span></div> <a class="title-link" href="/2227-7390/12/23/3815">An Effective Hybrid Metaheuristic Approach Based on the Genetic Algorithm</a> <div class="authors"> by <span class="inlineblock "><strong>Olympia Roeva</strong>, </span><span class="inlineblock "><strong>Dafina Zoteva</strong>, </span><span class="inlineblock "><strong>Gergana Roeva</strong>, </span><span class="inlineblock "><strong>Maya Ignatova</strong> and </span><span class="inlineblock "><strong>Velislava Lyubenova</strong></span> </div> <div class="color-grey-dark"> <em>Mathematics</em> <b>2024</b>, <em>12</em>(23), 3815; <a href="https://doi.org/10.3390/math12233815">https://doi.org/10.3390/math12233815</a> - 2 Dec 2024 </div> <div class="abstract-div"> <a href="#" onclick="$(this).next('.abstract-cropped').toggleClass('inline').next('.abstract-full').toggleClass('inline'); return false;"> <strong>Abstract </strong> </a> <div class="abstract-cropped inline"> This paper presents an effective hybrid metaheuristic algorithm combining the genetic algorithm (GA) and a simple algorithm based on evolutionary computation. The evolutionary approach (EA) is applied to form the initial population of the GA, thus improving the algorithm’s performance, especially its convergence <a href="#" data-counterslink = "https://www.mdpi.com/2227-7390/12/23/3815/more" onclick="$(this).parents('.abstract-cropped').toggleClass('inline').next('.abstract-full').toggleClass('inline'); return false;"> [...] Read more.</a> </div> <div class="abstract-full "> This paper presents an effective hybrid metaheuristic algorithm combining the genetic algorithm (GA) and a simple algorithm based on evolutionary computation. The evolutionary approach (EA) is applied to form the initial population of the GA, thus improving the algorithm’s performance, especially its convergence speed. To assess its effectiveness, the proposed hybrid algorithm, the EAGA, is evaluated on selected benchmark functions, as well as on a real optimisation process. The EAGA is used to identify parameters in a nonlinear system of differential equations modelling an <i>E. coli</i> fed-batch fermentation process. The obtained results are compared against published results from hybrid metaheuristic algorithms applied to the selected optimisation problems. The EAGA hybrid outperforms the competing algorithms due to its effective initial population generation strategy. The risk of premature convergence is reduced. Better numerical outcomes are achieved. The investigations validate the potential of the proposed hybrid metaheuristic EAGA for solving real complex nonlinear optimisation tasks. <a href="/2227-7390/12/23/3815">Full article</a> </div> </div> <div class="belongsTo" style="margin-bottom: 10px;"> (This article belongs to the Section <a href="/journal/mathematics/sections/mathematical_biology">Mathematical Biology</a>)<br/> </div> <a href="#" class="abstract-figures-show" data-counterslink = "https://www.mdpi.com/2227-7390/12/23/3815/show" ><span >►</span><span style=" display: none;">▼</span> Show Figures </a><div class="abstract-image-preview "><div class="arrow left-arrow" id="prev1535148"><i class="fa fa-caret-left"></i></div><div class="arrow right-arrow" id="next1535148"><i class="fa fa-caret-right"></i></div><div class="absgraph cycle-slideshow manual" data-cycle-fx="scrollHorz" data-cycle-timeout="0" data-cycle-next="#next1535148" data-cycle-prev="#prev1535148" data-cycle-progressive="#images1535148" data-cycle-slides=">div" data-cycle-log="false"><div class='openpopupgallery cycle-slide' data-imgindex='0' data-target='article-1535148-popup'><span class="helper"></span><img src="data:image/gif;base64,R0lGODlhAQABAAD/ACwAAAAAAQABAAACADs=" data-src="https://pub.mdpi-res.com/mathematics/mathematics-12-03815/article_deploy/html/images/mathematics-12-03815-g001-550.jpg?1733217671" alt="" style="border: 0;"><p>Figure 1</p></div><script id="images1535148" type="text/cycle" data-cycle-split="---"><div class='openpopupgallery' data-imgindex='1' data-target='article-1535148-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03815/article_deploy/html/images/mathematics-12-03815-g002-550.jpg?1733217673'><p>Figure 2</p></div> --- <div class='openpopupgallery' data-imgindex='2' data-target='article-1535148-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03815/article_deploy/html/images/mathematics-12-03815-g003-550.jpg?1733217674'><p>Figure 3</p></div> --- <div class='openpopupgallery' data-imgindex='3' data-target='article-1535148-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03815/article_deploy/html/images/mathematics-12-03815-g004-550.jpg?1733217675'><p>Figure 4</p></div> --- <div class='openpopupgallery' data-imgindex='4' data-target='article-1535148-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03815/article_deploy/html/images/mathematics-12-03815-g005-550.jpg?1733217676'><p>Figure 5</p></div> --- <div class='openpopupgallery' data-imgindex='5' data-target='article-1535148-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03815/article_deploy/html/images/mathematics-12-03815-g006-550.jpg?1733217680'><p>Figure 6</p></div></script></div></div><div id="article-1535148-popup" class="popupgallery" style="display: inline; line-height: 200%"><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03815/article_deploy/html/images/mathematics-12-03815-g001-550.jpg?1733217671" title=" <strong>Figure 1</strong><br/> <p>Pseudo-code of the hybrid EAGA.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3815'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03815/article_deploy/html/images/mathematics-12-03815-g002-550.jpg?1733217673" title=" <strong>Figure 2</strong><br/> <p>Model predictions of the process variables (biomass and substrate)—time profiles.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3815'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03815/article_deploy/html/images/mathematics-12-03815-g003-550.jpg?1733217674" title=" <strong>Figure 3</strong><br/> <p>EAGA hybrid model results—scatter plot.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3815'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03815/article_deploy/html/images/mathematics-12-03815-g004-550.jpg?1733217675" title=" <strong>Figure 4</strong><br/> <p>EAGA hybrid—convergence curves.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3815'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03815/article_deploy/html/images/mathematics-12-03815-g005-550.jpg?1733217676" title=" <strong>Figure 5</strong><br/> <p>Comparison of hybrid algorithm performance—box plot of objective function evaluations over 30 runs.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3815'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03815/article_deploy/html/images/mathematics-12-03815-g006-550.jpg?1733217680" title=" <strong>Figure 6</strong><br/> <p>Comparison of hybrid algorithm performance—histograms of objective function evaluations over 30 runs.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3815'>Full article</a></strong> "></a></div> </div> </div> </div> <div class="expanding-div collapsed"> <div class="generic-item article-item"> <div class="article-content"> <div class="label right label__btn"> <span style="font-size: 12px; color: #1a1a1a;"> 12 pages, 796 KiB   </span> <a href="/2227-7390/12/23/3814/pdf?version=1733146335" class="UD_Listings_ArticlePDF" title="Article PDF" data-name="The Dynamical Behaviors of a Fractional-Order Malware Propagation Model in Information Networks" data-journal="mathematics"> <i class="material-icons custom-download"></i> </a> </div> <div class="article-icons"><span class="label openaccess" data-dropdown="drop-article-label-openaccess" aria-expanded="false">Open Access</span><span class="label articletype">Article</span></div> <a class="title-link" href="/2227-7390/12/23/3814">The Dynamical Behaviors of a Fractional-Order Malware Propagation Model in Information Networks</a> <div class="authors"> by <span class="inlineblock "><strong>Xueying Shi</strong>, </span><span class="inlineblock "><strong>An Luo</strong>, </span><span class="inlineblock "><strong>Xiaoping Chen</strong>, </span><span class="inlineblock "><strong>Ying Huang</strong>, </span><span class="inlineblock "><strong>Chengdai Huang</strong> and </span><span class="inlineblock "><strong>Xin Yin</strong></span> </div> <div class="color-grey-dark"> <em>Mathematics</em> <b>2024</b>, <em>12</em>(23), 3814; <a href="https://doi.org/10.3390/math12233814">https://doi.org/10.3390/math12233814</a> - 2 Dec 2024 </div> <div class="abstract-div"> <a href="#" onclick="$(this).next('.abstract-cropped').toggleClass('inline').next('.abstract-full').toggleClass('inline'); return false;"> <strong>Abstract </strong> </a> <div class="abstract-cropped inline"> With the rapid development of communication and information technology, information networks have become increasingly common in our work and daily lives. The aim of this paper is to study the process of introducing a new fractional-order malware propagation model into information networks. Based <a href="#" data-counterslink = "https://www.mdpi.com/2227-7390/12/23/3814/more" onclick="$(this).parents('.abstract-cropped').toggleClass('inline').next('.abstract-full').toggleClass('inline'); return false;"> [...] Read more.</a> </div> <div class="abstract-full "> With the rapid development of communication and information technology, information networks have become increasingly common in our work and daily lives. The aim of this paper is to study the process of introducing a new fractional-order malware propagation model into information networks. Based on the matrix theory of eigenvalues, the conditions of local stability for the above model are discussed. In addition, taking time delay as the bifurcation parameter, Hopf bifurcation is considered. The correctness of our theoretical results is verified through numerical experiments. <a href="/2227-7390/12/23/3814">Full article</a> </div> </div> <div class="belongsTo" style="margin-bottom: 10px;"> (This article belongs to the Section <a href="/journal/mathematics/sections/dynamic_systems">Dynamical Systems</a>)<br/> </div> <a href="#" class="abstract-figures-show" data-counterslink = "https://www.mdpi.com/2227-7390/12/23/3814/show" ><span >►</span><span style=" display: none;">▼</span> Show Figures </a><div class="abstract-image-preview "><div class="arrow left-arrow" id="prev1535128"><i class="fa fa-caret-left"></i></div><div class="arrow right-arrow" id="next1535128"><i class="fa fa-caret-right"></i></div><div class="absgraph cycle-slideshow manual" data-cycle-fx="scrollHorz" data-cycle-timeout="0" data-cycle-next="#next1535128" data-cycle-prev="#prev1535128" data-cycle-progressive="#images1535128" data-cycle-slides=">div" data-cycle-log="false"><div class='openpopupgallery cycle-slide' data-imgindex='0' data-target='article-1535128-popup'><span class="helper"></span><img src="data:image/gif;base64,R0lGODlhAQABAAD/ACwAAAAAAQABAAACADs=" data-src="https://pub.mdpi-res.com/mathematics/mathematics-12-03814/article_deploy/html/images/mathematics-12-03814-g001-550.jpg?1733146406" alt="" style="border: 0;"><p>Figure 1</p></div><script id="images1535128" type="text/cycle" data-cycle-split="---"><div class='openpopupgallery' data-imgindex='1' data-target='article-1535128-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03814/article_deploy/html/images/mathematics-12-03814-g002-550.jpg?1733146408'><p>Figure 2</p></div> --- <div class='openpopupgallery' data-imgindex='2' data-target='article-1535128-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03814/article_deploy/html/images/mathematics-12-03814-g003-550.jpg?1733146409'><p>Figure 3</p></div> --- <div class='openpopupgallery' data-imgindex='3' data-target='article-1535128-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03814/article_deploy/html/images/mathematics-12-03814-g004-550.jpg?1733146411'><p>Figure 4</p></div> --- <div class='openpopupgallery' data-imgindex='4' data-target='article-1535128-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03814/article_deploy/html/images/mathematics-12-03814-g005-550.jpg?1733146412'><p>Figure 5</p></div> --- <div class='openpopupgallery' data-imgindex='5' data-target='article-1535128-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03814/article_deploy/html/images/mathematics-12-03814-g006-550.jpg?1733146414'><p>Figure 6</p></div> --- <div class='openpopupgallery' data-imgindex='6' data-target='article-1535128-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03814/article_deploy/html/images/mathematics-12-03814-g007-550.jpg?1733146415'><p>Figure 7</p></div> --- <div class='openpopupgallery' data-imgindex='7' data-target='article-1535128-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03814/article_deploy/html/images/mathematics-12-03814-g008-550.jpg?1733146416'><p>Figure 8</p></div></script></div></div><div id="article-1535128-popup" class="popupgallery" style="display: inline; line-height: 200%"><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03814/article_deploy/html/images/mathematics-12-03814-g001-550.jpg?1733146406" title=" <strong>Figure 1</strong><br/> <p>Waveform plot of system (15) at <math display="inline"><semantics> <msup> <mi>E</mi> <mo>⋄</mo> </msup> </semantics></math> when <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>8</mn> <mo>&lt;</mo> <msub> <mi>τ</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>10.6719</mn> </mrow> </semantics></math>.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3814'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03814/article_deploy/html/images/mathematics-12-03814-g002-550.jpg?1733146408" title=" <strong>Figure 2</strong><br/> <p>Waveform plot of system (15) at <math display="inline"><semantics> <msup> <mi>E</mi> <mo>⋄</mo> </msup> </semantics></math> when <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>8</mn> <mo>&lt;</mo> <msub> <mi>τ</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>10.6719</mn> </mrow> </semantics></math>.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3814'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03814/article_deploy/html/images/mathematics-12-03814-g003-550.jpg?1733146409" title=" <strong>Figure 3</strong><br/> <p>Waveform plot of system (15) at <math display="inline"><semantics> <msup> <mi>E</mi> <mo>⋄</mo> </msup> </semantics></math> when <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>8</mn> <mo>&lt;</mo> <msub> <mi>τ</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>10.6719</mn> </mrow> </semantics></math>.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3814'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03814/article_deploy/html/images/mathematics-12-03814-g004-550.jpg?1733146411" title=" <strong>Figure 4</strong><br/> <p>Waveform plot of system (15) at <math display="inline"><semantics> <msup> <mi>E</mi> <mo>⋄</mo> </msup> </semantics></math> when <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>11</mn> <mo>&gt;</mo> <msub> <mi>τ</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>10.6719</mn> </mrow> </semantics></math>.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3814'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03814/article_deploy/html/images/mathematics-12-03814-g005-550.jpg?1733146412" title=" <strong>Figure 5</strong><br/> <p>Waveform plot of system (15) at <math display="inline"><semantics> <msup> <mi>E</mi> <mo>⋄</mo> </msup> </semantics></math> when <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>11</mn> <mo>&gt;</mo> <msub> <mi>τ</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>10.6719</mn> </mrow> </semantics></math>.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3814'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03814/article_deploy/html/images/mathematics-12-03814-g006-550.jpg?1733146414" title=" <strong>Figure 6</strong><br/> <p>Waveform plot of system (15) at <math display="inline"><semantics> <msup> <mi>E</mi> <mo>⋄</mo> </msup> </semantics></math> when <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>11</mn> <mo>&gt;</mo> <msub> <mi>τ</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>10.6719</mn> </mrow> </semantics></math>.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3814'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03814/article_deploy/html/images/mathematics-12-03814-g007-550.jpg?1733146415" title=" <strong>Figure 7</strong><br/> <p>Phase diagram of system (15) at <math display="inline"><semantics> <msup> <mi>E</mi> <mo>⋄</mo> </msup> </semantics></math> when <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>8</mn> <mo>&lt;</mo> <msub> <mi>τ</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>10.6719</mn> </mrow> </semantics></math>.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3814'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03814/article_deploy/html/images/mathematics-12-03814-g008-550.jpg?1733146416" title=" <strong>Figure 8</strong><br/> <p>Phase diagram of system (15) at <math display="inline"><semantics> <msup> <mi>E</mi> <mo>⋄</mo> </msup> </semantics></math> when <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>11</mn> <mo>&gt;</mo> <msub> <mi>τ</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>10.6719</mn> </mrow> </semantics></math>.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3814'>Full article</a></strong> "></a></div> </div> </div> </div> <div class="expanding-div collapsed"> <div class="generic-item article-item"> <div class="article-content"> <div class="label right label__btn"> <a data-dropdown="drop-supplementary-1535117" aria-controls="drop-supplementary-1535117" aria-expanded="false" title="Supplementary Material"> <i class="material-icons">attachment</i> </a> <div id="drop-supplementary-1535117" class="f-dropdown label__btn__dropdown label__btn__dropdown--wide" data-dropdown-content aria-hidden="true" tabindex="-1"> Supplementary material: <br/> <a href="/2227-7390/12/23/3813/s1?version=1733145316"> Supplementary File 1 (ZIP, 923 B) </a><br/> </div> </div> <div class="label right label__btn"> <span style="font-size: 12px; color: #1a1a1a;"> 28 pages, 6391 KiB   </span> <a href="/2227-7390/12/23/3813/pdf?version=1733145315" class="UD_Listings_ArticlePDF" title="Article PDF" data-name="Evolution and Motivation of the Value-Added Trade Pattern of Producer Services Based on a Complex Network" data-journal="mathematics"> <i class="material-icons custom-download"></i> </a> </div> <div class="article-icons"><span class="label openaccess" data-dropdown="drop-article-label-openaccess" aria-expanded="false">Open Access</span><span class="label articletype">Article</span></div> <a class="title-link" href="/2227-7390/12/23/3813">Evolution and Motivation of the Value-Added Trade Pattern of Producer Services Based on a Complex Network</a> <div class="authors"> by <span class="inlineblock "><strong>Yan Li</strong>, </span><span class="inlineblock "><strong>Xuehan Liang</strong>, </span><span class="inlineblock "><strong>Sizhe Guan</strong> and </span><span class="inlineblock "><strong>Qingbo Huang</strong></span> </div> <div class="color-grey-dark"> <em>Mathematics</em> <b>2024</b>, <em>12</em>(23), 3813; <a href="https://doi.org/10.3390/math12233813">https://doi.org/10.3390/math12233813</a> - 2 Dec 2024 </div> <div class="abstract-div"> <a href="#" onclick="$(this).next('.abstract-cropped').toggleClass('inline').next('.abstract-full').toggleClass('inline'); return false;"> <strong>Abstract </strong> </a> <div class="abstract-cropped inline"> With the acceleration of the digital revolution and free trade, the producer services industry has become a new driving force for countries to enhance their global value chain (GVC). The further integration of GVCs has promoted the formation of a global value network <a href="#" data-counterslink = "https://www.mdpi.com/2227-7390/12/23/3813/more" onclick="$(this).parents('.abstract-cropped').toggleClass('inline').next('.abstract-full').toggleClass('inline'); return false;"> [...] Read more.</a> </div> <div class="abstract-full "> With the acceleration of the digital revolution and free trade, the producer services industry has become a new driving force for countries to enhance their global value chain (GVC). The further integration of GVCs has promoted the formation of a global value network covering a wider range of regions and industries. The way of establishing strong core capabilities and competitive advantages for a country’s producer services industry in the vast value network is of great practical significance for achieving industrial structure transformation and upgrading in an open economy. Therefore, based on the Asian Development Bank Multi-Regional Input–Output Database (ADB-MRIO) and the bilateral export value-added accounting method (WWZ), this study constructs a value network of producer services and analyzes its structural evolution and driving factors. The results show the following: (1) The domestic value-added (DVA) network and foreign value-added (FVA) networks of producer services trade exhibit dense characteristics in terms of trade depth. The DVA network has a small-world topological structure, while the FVA network does not have this feature most years. (2) Western countries, represented by the USA, Germany, and the UK, are located at the hub of the global value network, while China’s network status is rising and gradually occupying a core position not only in the Asian region but also in the world. The value chain of producer services shows a pattern of being led by developed countries and extending to developing countries. (3) The broader tendency of modularity is decreasing, indicating that the development process of the DVA and FVA networks is becoming globalized. The members of the community can not only achieve trade interdependence due to geographical proximity but also break through regional limitations and complement each other. (4) The network evolution process is driven by a variety of factors. Industrial and technological distances are conducive to the expansion of the global value flow in producer services. Geographical and linguistic proximity are the main influencing factors, while the role of institutional proximity tends to be weakened. <a href="/2227-7390/12/23/3813">Full article</a> </div> </div> <div class="belongsTo" style="margin-bottom: 10px;"> (This article belongs to the Special Issue <a href=" /journal/mathematics/special_issues/1M62T0181W ">New Advancements of Economic Networks</a>)<br/> </div> <a href="#" class="abstract-figures-show" data-counterslink = "https://www.mdpi.com/2227-7390/12/23/3813/show" ><span >►</span><span style=" display: none;">▼</span> Show Figures </a><div class="abstract-image-preview "><div class="arrow left-arrow" id="prev1535117"><i class="fa fa-caret-left"></i></div><div class="arrow right-arrow" id="next1535117"><i class="fa fa-caret-right"></i></div><div class="absgraph cycle-slideshow manual" data-cycle-fx="scrollHorz" data-cycle-timeout="0" data-cycle-next="#next1535117" data-cycle-prev="#prev1535117" data-cycle-progressive="#images1535117" data-cycle-slides=">div" data-cycle-log="false"><div class='openpopupgallery cycle-slide' data-imgindex='0' data-target='article-1535117-popup'><span class="helper"></span><img src="data:image/gif;base64,R0lGODlhAQABAAD/ACwAAAAAAQABAAACADs=" data-src="https://pub.mdpi-res.com/mathematics/mathematics-12-03813/article_deploy/html/images/mathematics-12-03813-g001-550.jpg?1733145385" alt="" style="border: 0;"><p>Figure 1</p></div><script id="images1535117" type="text/cycle" data-cycle-split="---"><div class='openpopupgallery' data-imgindex='1' data-target='article-1535117-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03813/article_deploy/html/images/mathematics-12-03813-g002-550.jpg?1733145387'><p>Figure 2</p></div> --- <div class='openpopupgallery' data-imgindex='2' data-target='article-1535117-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03813/article_deploy/html/images/mathematics-12-03813-g003-550.jpg?1733145389'><p>Figure 3</p></div> --- <div class='openpopupgallery' data-imgindex='3' data-target='article-1535117-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03813/article_deploy/html/images/mathematics-12-03813-g004-550.jpg?1733145390'><p>Figure 4</p></div> --- <div class='openpopupgallery' data-imgindex='4' data-target='article-1535117-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03813/article_deploy/html/images/mathematics-12-03813-g005-550.jpg?1733145392'><p>Figure 5</p></div> --- <div class='openpopupgallery' data-imgindex='5' data-target='article-1535117-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03813/article_deploy/html/images/mathematics-12-03813-g006-550.jpg?1733145394'><p>Figure 6</p></div> --- <div class='openpopupgallery' data-imgindex='6' data-target='article-1535117-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03813/article_deploy/html/images/mathematics-12-03813-g007-550.jpg?1733145394'><p>Figure 7</p></div> --- <div class='openpopupgallery' data-imgindex='7' data-target='article-1535117-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03813/article_deploy/html/images/mathematics-12-03813-g008-550.jpg?1733145396'><p>Figure 8</p></div> --- <div class='openpopupgallery' data-imgindex='8' data-target='article-1535117-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03813/article_deploy/html/images/mathematics-12-03813-g009-550.jpg?1733145398'><p>Figure 9</p></div> --- <div class='openpopupgallery' data-imgindex='9' data-target='article-1535117-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03813/article_deploy/html/images/mathematics-12-03813-g0A1-550.jpg?1733145401'><p>Figure A1</p></div> --- <div class='openpopupgallery' data-imgindex='10' data-target='article-1535117-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03813/article_deploy/html/images/mathematics-12-03813-g0A2-550.jpg?1733145403'><p>Figure A2</p></div></script></div></div><div id="article-1535117-popup" class="popupgallery" style="display: inline; line-height: 200%"><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03813/article_deploy/html/images/mathematics-12-03813-g001-550.jpg?1733145385" title=" <strong>Figure 1</strong><br/> <p>Accounting results of value-added trade, (<b>a</b>) The trade scale of each sub-item of value-added, (<b>b</b>) The trade proportion of each sub-item of value-added. Source: the authors calculated and collated the data from the ADB-MRIO database.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3813'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03813/article_deploy/html/images/mathematics-12-03813-g002-550.jpg?1733145387" title=" <strong>Figure 2</strong><br/> <p>GVC forward participation and backward participation in producer services.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3813'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03813/article_deploy/html/images/mathematics-12-03813-g003-550.jpg?1733145389" title=" <strong>Figure 3</strong><br/> <p>The positions of node degree centrality.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3813'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03813/article_deploy/html/images/mathematics-12-03813-g004-550.jpg?1733145390" title=" <strong>Figure 4</strong><br/> <p>Closeness centrality and betweenness centrality.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3813'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03813/article_deploy/html/images/mathematics-12-03813-g005-550.jpg?1733145392" title=" <strong>Figure 5</strong><br/> <p>The positions of node strength.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3813'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03813/article_deploy/html/images/mathematics-12-03813-g006-550.jpg?1733145394" title=" <strong>Figure 6</strong><br/> <p>PageRank centrality of the nodes.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3813'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03813/article_deploy/html/images/mathematics-12-03813-g007-550.jpg?1733145394" title=" <strong>Figure 7</strong><br/> <p>Variations in community modularity.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3813'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03813/article_deploy/html/images/mathematics-12-03813-g008-550.jpg?1733145396" title=" <strong>Figure 8</strong><br/> <p>Evolution of the DVA network community for producer services.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3813'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03813/article_deploy/html/images/mathematics-12-03813-g009-550.jpg?1733145398" title=" <strong>Figure 9</strong><br/> <p>Evolution of the FVA network community for producer services.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3813'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03813/article_deploy/html/images/mathematics-12-03813-g0A1-550.jpg?1733145401" title=" <strong>Figure A1</strong><br/> <p>Diagram of the DVA network of producer services.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3813'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03813/article_deploy/html/images/mathematics-12-03813-g0A2-550.jpg?1733145403" title=" <strong>Figure A2</strong><br/> <p>Diagram of the FVA network of producer services.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3813'>Full article</a></strong> "></a></div> </div> </div> </div> <div class="expanding-div collapsed"> <div class="generic-item article-item"> <div class="article-content"> <div class="label right label__btn"> <span style="font-size: 12px; color: #1a1a1a;"> 17 pages, 1115 KiB   </span> <a href="/2227-7390/12/23/3812/pdf?version=1733143373" class="UD_Listings_ArticlePDF" title="Article PDF" data-name="Uncertainty Modeling and Stability Assessment of Minimum Spanning Trees in Network Design" data-journal="mathematics"> <i class="material-icons custom-download"></i> </a> </div> <div class="article-icons"><span class="label openaccess" data-dropdown="drop-article-label-openaccess" aria-expanded="false">Open Access</span><span class="label feature" data-dropdown="drop-article-label-feature" aria-expanded="false">Feature Paper</span><span class="label articletype">Article</span></div> <a class="title-link" href="/2227-7390/12/23/3812">Uncertainty Modeling and Stability Assessment of Minimum Spanning Trees in Network Design</a> <div class="authors"> by <span class="inlineblock "><strong>Ahmad Hosseini</strong></span> </div> <div class="color-grey-dark"> <em>Mathematics</em> <b>2024</b>, <em>12</em>(23), 3812; <a href="https://doi.org/10.3390/math12233812">https://doi.org/10.3390/math12233812</a> - 2 Dec 2024 </div> <div class="abstract-div"> <a href="#" onclick="$(this).next('.abstract-cropped').toggleClass('inline').next('.abstract-full').toggleClass('inline'); return false;"> <strong>Abstract </strong> </a> <div class="abstract-cropped inline"> The Minimum Spanning Tree (MST) problem in networks focuses on finding efficient routes, with applications in transportation, logistics, telecommunications, and more. However, catastrophes can make these networks uncertain, requiring robust computational models for decision-making. This paper introduces an uncertainty theory-based model to analyze <a href="#" data-counterslink = "https://www.mdpi.com/2227-7390/12/23/3812/more" onclick="$(this).parents('.abstract-cropped').toggleClass('inline').next('.abstract-full').toggleClass('inline'); return false;"> [...] Read more.</a> </div> <div class="abstract-full "> The Minimum Spanning Tree (MST) problem in networks focuses on finding efficient routes, with applications in transportation, logistics, telecommunications, and more. However, catastrophes can make these networks uncertain, requiring robust computational models for decision-making. This paper introduces an uncertainty theory-based model to analyze the stability of MSTs in uncertain networks. By incorporating reliability and risk variables, we assess the robustness of uncertain MSTs (UMSTs) and address the challenge of computing link tolerances, which define the range within which network links can vary without compromising MST optimality. This study proposes computational formulations to systematically calculate these tolerances, offering a more efficient alternative to traditional re-optimization methods. <a href="/2227-7390/12/23/3812">Full article</a> </div> </div> <a href="#" class="abstract-figures-show" data-counterslink = "https://www.mdpi.com/2227-7390/12/23/3812/show" ><span >►</span><span style=" display: none;">▼</span> Show Figures </a><div class="abstract-image-preview "><div class="arrow left-arrow" id="prev1535100"><i class="fa fa-caret-left"></i></div><div class="arrow right-arrow" id="next1535100"><i class="fa fa-caret-right"></i></div><div class="absgraph cycle-slideshow manual" data-cycle-fx="scrollHorz" data-cycle-timeout="0" data-cycle-next="#next1535100" data-cycle-prev="#prev1535100" data-cycle-progressive="#images1535100" data-cycle-slides=">div" data-cycle-log="false"><div class='openpopupgallery cycle-slide' data-imgindex='0' data-target='article-1535100-popup'><span class="helper"></span><img src="data:image/gif;base64,R0lGODlhAQABAAD/ACwAAAAAAQABAAACADs=" data-src="https://pub.mdpi-res.com/mathematics/mathematics-12-03812/article_deploy/html/images/mathematics-12-03812-g001-550.jpg?1733143489" alt="" style="border: 0;"><p>Figure 1</p></div><script id="images1535100" type="text/cycle" data-cycle-split="---"><div class='openpopupgallery' data-imgindex='1' data-target='article-1535100-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03812/article_deploy/html/images/mathematics-12-03812-g002-550.jpg?1733143490'><p>Figure 2</p></div></script></div></div><div id="article-1535100-popup" class="popupgallery" style="display: inline; line-height: 200%"><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03812/article_deploy/html/images/mathematics-12-03812-g001-550.jpg?1733143489" title=" <strong>Figure 1</strong><br/> <p>An uncertain test network and a designated planning zone from the network in a grid layout.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3812'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03812/article_deploy/html/images/mathematics-12-03812-g002-550.jpg?1733143490" title=" <strong>Figure 2</strong><br/> <p>The UMST distributions: Inverse Uncertainty Distribution <math display="inline"><semantics> <mrow> <msup> <mrow> <mi mathvariant="sans-serif">Ψ</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mfenced separators="|"> <mrow> <mi>α</mi> </mrow> </mfenced> </mrow> </semantics></math> vs. Uncertainty Distribution <math display="inline"><semantics> <mrow> <mi mathvariant="sans-serif">Ψ</mi> <mfenced separators="|"> <mrow> <mi>x</mi> </mrow> </mfenced> </mrow> </semantics></math>.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3812'>Full article</a></strong> "></a></div> </div> </div> </div> <div class="expanding-div collapsed"> <div class="generic-item article-item"> <div class="article-content"> <div class="label right label__btn"> <span style="font-size: 12px; color: #1a1a1a;"> 15 pages, 359 KiB   </span> <a href="/2227-7390/12/23/3811/pdf?version=1733132533" class="UD_Listings_ArticlePDF" title="Article PDF" data-name="Stability of the F∗ Algorithm on Strong Pseudocontractive Mapping and Its Application" data-journal="mathematics"> <i class="material-icons custom-download"></i> </a> </div> <div class="article-icons"><span class="label openaccess" data-dropdown="drop-article-label-openaccess" aria-expanded="false">Open Access</span><span class="label articletype">Article</span></div> <a class="title-link" href="/2227-7390/12/23/3811">Stability of the <i>F</i><sup>&lowast;</sup> Algorithm on Strong Pseudocontractive Mapping and Its Application</a> <div class="authors"> by <span class="inlineblock "><strong>Taiwo P. Fajusigbe</strong>, </span><span class="inlineblock "><strong>Francis Monday Nkwuda</strong>, </span><span class="inlineblock "><strong>Hussaini Joshua</strong>, </span><span class="inlineblock "><strong>Kayode Oshinubi</strong>, </span><span class="inlineblock "><strong>Felix D. Ajibade</strong> and </span><span class="inlineblock "><strong>Jamiu Aliyu</strong></span> </div> <div class="color-grey-dark"> <em>Mathematics</em> <b>2024</b>, <em>12</em>(23), 3811; <a href="https://doi.org/10.3390/math12233811">https://doi.org/10.3390/math12233811</a> - 2 Dec 2024 </div> <div class="abstract-div"> <a href="#" onclick="$(this).next('.abstract-cropped').toggleClass('inline').next('.abstract-full').toggleClass('inline'); return false;"> <strong>Abstract </strong> </a> <div class="abstract-cropped inline"> This paper investigates the stability of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>F</mi><mo>&lowast;</mo></msup></semantics></math></inline-formula> iterative algorithm applied to strongly pseudocontractive mappings within the context of uniformly convex Banach spaces. The study leverages both analytic and numerical methods to demonstrate the convergence and stability of the algorithm. In comparison <a href="#" data-counterslink = "https://www.mdpi.com/2227-7390/12/23/3811/more" onclick="$(this).parents('.abstract-cropped').toggleClass('inline').next('.abstract-full').toggleClass('inline'); return false;"> [...] Read more.</a> </div> <div class="abstract-full "> This paper investigates the stability of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>F</mi><mo>&lowast;</mo></msup></semantics></math></inline-formula> iterative algorithm applied to strongly pseudocontractive mappings within the context of uniformly convex Banach spaces. The study leverages both analytic and numerical methods to demonstrate the convergence and stability of the algorithm. In comparison to previous works, where weak-contraction mappings were utilized, the strongly pseudocontractive mappings used in this study preserve the convergence property, exhibit greater stability, and have broader applicability in optimization and fixed point theory. Additionally, this work shows that the type of mapping employed converges faster than those in earlier studies. The results are applied to a mixed-type Volterra–Fredholm nonlinear integral equation, and numerical examples are provided to validate the theoretical findings. Key contributions of this work include the following: (i) the use of strongly pseudocontractive mappings, which offer a more stable and efficient convergence rate compared to weak-contraction mappings; (ii) the application of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>F</mi><mo>&lowast;</mo></msup></semantics></math></inline-formula> algorithm to a wider range of problems; and (iii) the proposal of future directions for improving convergence rates and exploring the algorithm’s behavior in Hilbert and reflexive Banach spaces. <a href="/2227-7390/12/23/3811">Full article</a> </div> </div> <a href="#" class="abstract-figures-show" data-counterslink = "https://www.mdpi.com/2227-7390/12/23/3811/show" ><span >►</span><span style=" display: none;">▼</span> Show Figures </a><div class="abstract-image-preview "><div class="absgraph cycle-slideshow"><div class='openpopupgallery cycle-slide' data-imgindex='0' data-target='article-1534906-popup'><span class="helper"></span><img src="data:image/gif;base64,R0lGODlhAQABAAD/ACwAAAAAAQABAAACADs=" data-src="https://pub.mdpi-res.com/mathematics/mathematics-12-03811/article_deploy/html/images/mathematics-12-03811-g001-550.jpg?1733132661" alt="" style="border: 0;"><p>Figure 1</p></div></div></div><div id="article-1534906-popup" class="popupgallery" style="display: inline; line-height: 200%"><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03811/article_deploy/html/images/mathematics-12-03811-g001-550.jpg?1733132661" title=" <strong>Figure 1</strong><br/> <p>Convergence of the different iterative schemes.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3811'>Full article</a></strong> "></a></div> </div> </div> </div> <div class="expanding-div collapsed"> <div class="generic-item article-item"> <div class="article-content"> <div class="label right label__btn"> <span style="font-size: 12px; color: #1a1a1a;"> 16 pages, 1534 KiB   </span> <a href="/2227-7390/12/23/3810/pdf?version=1733131569" class="UD_Listings_ArticlePDF" title="Article PDF" data-name="On the Dynamics of Some Higher-Order Nonlinear Difference Equations" data-journal="mathematics"> <i class="material-icons custom-download"></i> </a> </div> <div class="article-icons"><span class="label openaccess" data-dropdown="drop-article-label-openaccess" aria-expanded="false">Open Access</span><span class="label articletype">Article</span></div> <a class="title-link" href="/2227-7390/12/23/3810">On the Dynamics of Some Higher-Order Nonlinear Difference Equations</a> <div class="authors"> by <span class="inlineblock "><strong>Turki D. Alharbi</strong> and </span><span class="inlineblock "><strong>Md Rifat Hasan</strong></span> </div> <div class="color-grey-dark"> <em>Mathematics</em> <b>2024</b>, <em>12</em>(23), 3810; <a href="https://doi.org/10.3390/math12233810">https://doi.org/10.3390/math12233810</a> - 2 Dec 2024 </div> <div class="abstract-div"> <a href="#" onclick="$(this).next('.abstract-cropped').toggleClass('inline').next('.abstract-full').toggleClass('inline'); return false;"> <strong>Abstract </strong> </a> <div class="abstract-cropped inline"> This research investigates the dynamics of higher-order nonlinear difference equations, specifically concentrating on seventh-order instances. Analytical solutions are obtained for particular equations, a formidable task owing to the absence of explicit mathematical techniques for their resolution. The qualitative characteristics of solutions, such as <a href="#" data-counterslink = "https://www.mdpi.com/2227-7390/12/23/3810/more" onclick="$(this).parents('.abstract-cropped').toggleClass('inline').next('.abstract-full').toggleClass('inline'); return false;"> [...] Read more.</a> </div> <div class="abstract-full "> This research investigates the dynamics of higher-order nonlinear difference equations, specifically concentrating on seventh-order instances. Analytical solutions are obtained for particular equations, a formidable task owing to the absence of explicit mathematical techniques for their resolution. The qualitative characteristics of solutions, such as their stability, boundedness, and periodicity, are analysed by theoretical methods and numerical simulations. The results indicate that equilibrium points frequently lack local asymptotic stability, leading to intricate phenomena such as unbounded solutions and periodic attractors. These findings augment our understanding of nonlinear difference equations, offering significant implications for their use across various scientific fields. <a href="/2227-7390/12/23/3810">Full article</a> </div> </div> <div class="belongsTo" style="margin-bottom: 10px;"> (This article belongs to the Special Issue <a href=" /journal/mathematics/special_issues/Advances_Abstract_Inequalities_Partial_Functional_Differential_Equations_Their_Applications ">Advances in Abstract Inequalities, Partial Functional Differential Equations and Their Applications</a>)<br/> </div> <a href="#" class="abstract-figures-show" data-counterslink = "https://www.mdpi.com/2227-7390/12/23/3810/show" ><span >►</span><span style=" display: none;">▼</span> Show Figures </a><div class="abstract-image-preview "><div class="arrow left-arrow" id="prev1534877"><i class="fa fa-caret-left"></i></div><div class="arrow right-arrow" id="next1534877"><i class="fa fa-caret-right"></i></div><div class="absgraph cycle-slideshow manual" data-cycle-fx="scrollHorz" data-cycle-timeout="0" data-cycle-next="#next1534877" data-cycle-prev="#prev1534877" data-cycle-progressive="#images1534877" data-cycle-slides=">div" data-cycle-log="false"><div class='openpopupgallery cycle-slide' data-imgindex='0' data-target='article-1534877-popup'><span class="helper"></span><img src="data:image/gif;base64,R0lGODlhAQABAAD/ACwAAAAAAQABAAACADs=" data-src="https://pub.mdpi-res.com/mathematics/mathematics-12-03810/article_deploy/html/images/mathematics-12-03810-g001-550.jpg?1733131714" alt="" style="border: 0;"><p>Figure 1</p></div><script id="images1534877" type="text/cycle" data-cycle-split="---"><div class='openpopupgallery' data-imgindex='1' data-target='article-1534877-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03810/article_deploy/html/images/mathematics-12-03810-g002-550.jpg?1733131715'><p>Figure 2</p></div> --- <div class='openpopupgallery' data-imgindex='2' data-target='article-1534877-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03810/article_deploy/html/images/mathematics-12-03810-g003-550.jpg?1733131717'><p>Figure 3</p></div> --- <div class='openpopupgallery' data-imgindex='3' data-target='article-1534877-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03810/article_deploy/html/images/mathematics-12-03810-g004-550.jpg?1733131719'><p>Figure 4</p></div> --- <div class='openpopupgallery' data-imgindex='4' data-target='article-1534877-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03810/article_deploy/html/images/mathematics-12-03810-g005-550.jpg?1733131721'><p>Figure 5</p></div> --- <div class='openpopupgallery' data-imgindex='5' data-target='article-1534877-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03810/article_deploy/html/images/mathematics-12-03810-g006-550.jpg?1733131722'><p>Figure 6</p></div> --- <div class='openpopupgallery' data-imgindex='6' data-target='article-1534877-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03810/article_deploy/html/images/mathematics-12-03810-g007-550.jpg?1733131724'><p>Figure 7</p></div></script></div></div><div id="article-1534877-popup" class="popupgallery" style="display: inline; line-height: 200%"><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03810/article_deploy/html/images/mathematics-12-03810-g001-550.jpg?1733131714" title=" <strong>Figure 1</strong><br/> <p>Dynamic behavior of Equation (<a href="#FD3-mathematics-12-03810" class="html-disp-formula">3</a>).</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3810'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03810/article_deploy/html/images/mathematics-12-03810-g002-550.jpg?1733131715" title=" <strong>Figure 2</strong><br/> <p>Dynamic behavior of Equation (<a href="#FD8-mathematics-12-03810" class="html-disp-formula">8</a>).</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3810'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03810/article_deploy/html/images/mathematics-12-03810-g003-550.jpg?1733131717" title=" <strong>Figure 3</strong><br/> <p>Dynamic behavior of Equation (<a href="#FD10-mathematics-12-03810" class="html-disp-formula">10</a>).</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3810'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03810/article_deploy/html/images/mathematics-12-03810-g004-550.jpg?1733131719" title=" <strong>Figure 4</strong><br/> <p>Dynamic behavior of Equation (<a href="#FD13-mathematics-12-03810" class="html-disp-formula">13</a>).</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3810'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03810/article_deploy/html/images/mathematics-12-03810-g005-550.jpg?1733131721" title=" <strong>Figure 5</strong><br/> <p>Dynamic behavior of Equation (<a href="#FD13-mathematics-12-03810" class="html-disp-formula">13</a>).</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3810'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03810/article_deploy/html/images/mathematics-12-03810-g006-550.jpg?1733131722" title=" <strong>Figure 6</strong><br/> <p>Dynamic behavior of Equation (<a href="#FD14-mathematics-12-03810" class="html-disp-formula">14</a>).</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3810'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03810/article_deploy/html/images/mathematics-12-03810-g007-550.jpg?1733131724" title=" <strong>Figure 7</strong><br/> <p>Dynamic behavior of Equation (<a href="#FD14-mathematics-12-03810" class="html-disp-formula">14</a>).</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3810'>Full article</a></strong> "></a></div> </div> </div> </div> <div class="expanding-div collapsed"> <div class="generic-item article-item"> <div class="article-content"> <div class="label right label__btn"> <span style="font-size: 12px; color: #1a1a1a;"> 18 pages, 3115 KiB   </span> <a href="/2227-7390/12/23/3809/pdf?version=1733209688" class="UD_Listings_ArticlePDF" title="Article PDF" data-name="A Comprehensive Study of Dynamical Behavior and Nonlinear Structures of the Modified α Equation" data-journal="mathematics"> <i class="material-icons custom-download"></i> </a> </div> <div class="article-icons"><span class="label openaccess" data-dropdown="drop-article-label-openaccess" aria-expanded="false">Open Access</span><span class="label articletype">Article</span></div> <a class="title-link" href="/2227-7390/12/23/3809">A Comprehensive Study of Dynamical Behavior and Nonlinear Structures of the Modified <i>α</i> Equation</a> <div class="authors"> by <span class="inlineblock "><strong>Hassan Almusawa</strong>, </span><span class="inlineblock "><strong>Musawa Yahya Almusawa</strong>, </span><span class="inlineblock "><strong>Adil Jhangeer</strong> and </span><span class="inlineblock "><strong>Zamir Hussain</strong></span> </div> <div class="color-grey-dark"> <em>Mathematics</em> <b>2024</b>, <em>12</em>(23), 3809; <a href="https://doi.org/10.3390/math12233809">https://doi.org/10.3390/math12233809</a> - 2 Dec 2024 </div> <div class="abstract-div"> <a href="#" onclick="$(this).next('.abstract-cropped').toggleClass('inline').next('.abstract-full').toggleClass('inline'); return false;"> <strong>Abstract </strong> </a> <div class="abstract-cropped inline"> In this article, the modified <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula> equation is solved using the direct algebraic approach. As a result, numerous new and more generalized exact solutions for such equations have been found, taking into account the wide range of travelling structures. The rational, trigonometric, hyperbolic, <a href="#" data-counterslink = "https://www.mdpi.com/2227-7390/12/23/3809/more" onclick="$(this).parents('.abstract-cropped').toggleClass('inline').next('.abstract-full').toggleClass('inline'); return false;"> [...] Read more.</a> </div> <div class="abstract-full "> In this article, the modified <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula> equation is solved using the direct algebraic approach. As a result, numerous new and more generalized exact solutions for such equations have been found, taking into account the wide range of travelling structures. The rational, trigonometric, hyperbolic, and exponential functions with a couple of licentious parameters are thus included in these exact answers. Analytical solutions feature a variety of physical structures, which are visually studied to demonstrate their dynamic behavior in 2D and 3D. Considering the parameters, all feasible phase portraits are shown. Furthermore, we used numerical approaches to determine the nonlinear periodic structures of the mentioned model, and the data are graphically displayed. Additionally, we employed numerical approaches to determine the nonlinear conditions that contribute to the presented model, and the data are graphically displayed. After evaluating the influence of frequency following the application of an external periodic factor, sensitivity exploration is used to study quasi-periodic and chaotic behavior for several starting value problems. Furthermore, the function of physical characteristics is investigated using an external periodic force. Quasi-periodic and quasi-periodic-chaotic patterns are described with the inclusion of a perturbation term. The direct algebraic methodology would be used to derive the soliton solution of modified <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula> equation, from which the Galilean transformation derives traveling wave solutions of the considered and a bifurcation behavior is reported. Analytical and numerical methods have been used to have the condition of the travelling wave phase transformation. The well-judged values of parameters are enhanced well with a graphically formal analysis of such specific solutions to illustrate their propagation. Then a planer dynamical system is introduced, and a bifurcation analysis is utilized to identify the bifurcation structures of the dynamical model’s nonlinear wave propagation solutions. Additionally, the periodic and quasi-periodic behavior of the discussed equation is analyzed using sensitivity analysis for a range of beginning values. To further comprehend the dynamical behaviors of the resultant solutions, a graphic analysis is conducted. <a href="/2227-7390/12/23/3809">Full article</a> </div> </div> <a href="#" class="abstract-figures-show" data-counterslink = "https://www.mdpi.com/2227-7390/12/23/3809/show" ><span >►</span><span style=" display: none;">▼</span> Show Figures </a><div class="abstract-image-preview "><div class="arrow left-arrow" id="prev1534712"><i class="fa fa-caret-left"></i></div><div class="arrow right-arrow" id="next1534712"><i class="fa fa-caret-right"></i></div><div class="absgraph cycle-slideshow manual" data-cycle-fx="scrollHorz" data-cycle-timeout="0" data-cycle-next="#next1534712" data-cycle-prev="#prev1534712" data-cycle-progressive="#images1534712" data-cycle-slides=">div" data-cycle-log="false"><div class='openpopupgallery cycle-slide' data-imgindex='0' data-target='article-1534712-popup'><span class="helper"></span><img src="data:image/gif;base64,R0lGODlhAQABAAD/ACwAAAAAAQABAAACADs=" data-src="https://pub.mdpi-res.com/mathematics/mathematics-12-03809/article_deploy/html/images/mathematics-12-03809-g001-550.jpg?1733209767" alt="" style="border: 0;"><p>Figure 1</p></div><script id="images1534712" type="text/cycle" data-cycle-split="---"><div class='openpopupgallery' data-imgindex='1' data-target='article-1534712-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03809/article_deploy/html/images/mathematics-12-03809-g002-550.jpg?1733209769'><p>Figure 2</p></div> --- <div class='openpopupgallery' data-imgindex='2' data-target='article-1534712-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03809/article_deploy/html/images/mathematics-12-03809-g003a-550.jpg?1733209770'><p>Figure 3</p></div> --- <div class='openpopupgallery' data-imgindex='3' data-target='article-1534712-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03809/article_deploy/html/images/mathematics-12-03809-g003b-550.jpg?1733209771'><p>Figure 3 Cont.</p></div> --- <div class='openpopupgallery' data-imgindex='4' data-target='article-1534712-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03809/article_deploy/html/images/mathematics-12-03809-g004-550.jpg?1733209772'><p>Figure 4</p></div> --- <div class='openpopupgallery' data-imgindex='5' data-target='article-1534712-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03809/article_deploy/html/images/mathematics-12-03809-g005-550.jpg?1733209774'><p>Figure 5</p></div> --- <div class='openpopupgallery' data-imgindex='6' data-target='article-1534712-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03809/article_deploy/html/images/mathematics-12-03809-g006-550.jpg?1733209775'><p>Figure 6</p></div> --- <div class='openpopupgallery' data-imgindex='7' data-target='article-1534712-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03809/article_deploy/html/images/mathematics-12-03809-g007-550.jpg?1733209775'><p>Figure 7</p></div> --- <div class='openpopupgallery' data-imgindex='8' data-target='article-1534712-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03809/article_deploy/html/images/mathematics-12-03809-g008-550.jpg?1733209776'><p>Figure 8</p></div> --- <div class='openpopupgallery' data-imgindex='9' data-target='article-1534712-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03809/article_deploy/html/images/mathematics-12-03809-g009-550.jpg?1733209777'><p>Figure 9</p></div> --- <div class='openpopupgallery' data-imgindex='10' data-target='article-1534712-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03809/article_deploy/html/images/mathematics-12-03809-g010-550.jpg?1733209778'><p>Figure 10</p></div> --- <div class='openpopupgallery' data-imgindex='11' data-target='article-1534712-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03809/article_deploy/html/images/mathematics-12-03809-g011-550.jpg?1733209781'><p>Figure 11</p></div> --- <div class='openpopupgallery' data-imgindex='12' data-target='article-1534712-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03809/article_deploy/html/images/mathematics-12-03809-g012-550.jpg?1733209784'><p>Figure 12</p></div> --- <div class='openpopupgallery' data-imgindex='13' data-target='article-1534712-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03809/article_deploy/html/images/mathematics-12-03809-g013-550.jpg?1733209785'><p>Figure 13</p></div> --- <div class='openpopupgallery' data-imgindex='14' data-target='article-1534712-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03809/article_deploy/html/images/mathematics-12-03809-g014-550.jpg?1733209785'><p>Figure 14</p></div></script></div></div><div id="article-1534712-popup" class="popupgallery" style="display: inline; line-height: 200%"><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03809/article_deploy/html/images/mathematics-12-03809-g001-550.jpg?1733209767" title=" <strong>Figure 1</strong><br/> <p><math display="inline"><semantics> <msub> <mi>Z</mi> <mrow> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> </semantics></math> is graphically represented in 3D and 2D when <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>b</mi> <mo>=</mo> <mo>−</mo> <mn>3</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>μ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>χ</mi> <mo>=</mo> <mn>0.0444</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>H</mi> <mo>=</mo> <mo>−</mo> <mn>0.43</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>K</mi> <mo>=</mo> <mo>−</mo> <mn>3</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>M</mi> <mo>=</mo> <mo>−</mo> <mn>1.5</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>ρ</mi> <mo>=</mo> <mi>e</mi> </mrow> </semantics></math>.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3809'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03809/article_deploy/html/images/mathematics-12-03809-g002-550.jpg?1733209769" title=" <strong>Figure 2</strong><br/> <p><math display="inline"><semantics> <msub> <mi>Z</mi> <mrow> <mn>1</mn> <mo>,</mo> <mn>4</mn> </mrow> </msub> </semantics></math> is graphically represented in 3D and 2D when <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>b</mi> <mo>=</mo> <mo>−</mo> <mn>3</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>μ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>χ</mi> <mo>=</mo> <mn>0.0444</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>H</mi> <mo>=</mo> <mo>−</mo> <mn>0.43</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>K</mi> <mo>=</mo> <mo>−</mo> <mn>3</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>M</mi> <mo>=</mo> <mo>−</mo> <mn>1.5</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>ρ</mi> <mo>=</mo> <mi>e</mi> </mrow> </semantics></math>.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3809'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03809/article_deploy/html/images/mathematics-12-03809-g003a-550.jpg?1733209770" title=" <strong>Figure 3</strong><br/> <p><math display="inline"><semantics> <msub> <mi>Z</mi> <mrow> <mn>1</mn> <mo>,</mo> <mn>8</mn> </mrow> </msub> </semantics></math> is graphically represented in 3D and 2D when <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mo>−</mo> <mn>5</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>b</mi> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>μ</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>χ</mi> <mo>=</mo> <mo>−</mo> <mn>0.75</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>H</mi> <mo>=</mo> <mn>0.75</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>K</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>M</mi> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>ρ</mi> <mo>=</mo> <mi>e</mi> </mrow> </semantics></math>.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3809'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03809/article_deploy/html/images/mathematics-12-03809-g003b-550.jpg?1733209771" title=" <strong>Figure 3 Cont.</strong><br/> <p><math display="inline"><semantics> <msub> <mi>Z</mi> <mrow> <mn>1</mn> <mo>,</mo> <mn>8</mn> </mrow> </msub> </semantics></math> is graphically represented in 3D and 2D when <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mo>−</mo> <mn>5</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>b</mi> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>μ</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>χ</mi> <mo>=</mo> <mo>−</mo> <mn>0.75</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>H</mi> <mo>=</mo> <mn>0.75</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>K</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>M</mi> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>ρ</mi> <mo>=</mo> <mi>e</mi> </mrow> </semantics></math>.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3809'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03809/article_deploy/html/images/mathematics-12-03809-g004-550.jpg?1733209772" title=" <strong>Figure 4</strong><br/> <p><math display="inline"><semantics> <msub> <mi>Z</mi> <mrow> <mn>2</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> </semantics></math> is graphically represented in 3D and 2D when <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mo>−</mo> <mn>6</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>b</mi> <mo>=</mo> <mn>6</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>μ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>χ</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>q</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>H</mi> <mo>=</mo> <mo>−</mo> <mn>0.32</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>K</mi> <mo>=</mo> <mo>−</mo> <mn>6</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>M</mi> <mo>=</mo> <mn>1.3</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>ρ</mi> <mo>=</mo> <mi>e</mi> </mrow> </semantics></math>.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3809'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03809/article_deploy/html/images/mathematics-12-03809-g005-550.jpg?1733209774" title=" <strong>Figure 5</strong><br/> <p><math display="inline"><semantics> <msub> <mi>Z</mi> <mrow> <mn>2</mn> <mo>,</mo> <mn>4</mn> </mrow> </msub> </semantics></math> is graphically represented in 3D and 2D when <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mo>−</mo> <mn>4</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>b</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>μ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>χ</mi> <mo>=</mo> <mn>0.0444</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>H</mi> <mo>=</mo> <mn>0.43</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>K</mi> <mo>=</mo> <mo>−</mo> <mn>3</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>M</mi> <mo>=</mo> <mo>−</mo> <mn>1.5</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>ρ</mi> <mo>=</mo> <mi>e</mi> </mrow> </semantics></math>.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3809'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03809/article_deploy/html/images/mathematics-12-03809-g006-550.jpg?1733209775" title=" <strong>Figure 6</strong><br/> <p>Alternative choices of unknown coefficients <math display="inline"><semantics> <msub> <mi>ς</mi> <mn>1</mn> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mi>ς</mi> <mn>2</mn> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mi>ς</mi> <mn>3</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>ς</mi> <mn>4</mn> </msub> </semantics></math> have different phase portraits.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3809'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03809/article_deploy/html/images/mathematics-12-03809-g007-550.jpg?1733209775" title=" <strong>Figure 7</strong><br/> <p>Phase pictures for a variety of unknown coefficient values <math display="inline"><semantics> <msub> <mi>ς</mi> <mn>1</mn> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mi>ς</mi> <mn>2</mn> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mi>ς</mi> <mn>3</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>ς</mi> <mn>4</mn> </msub> </semantics></math>.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3809'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03809/article_deploy/html/images/mathematics-12-03809-g008-550.jpg?1733209776" title=" <strong>Figure 8</strong><br/> <p>Phase portraits of super nonlinear periodic.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3809'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03809/article_deploy/html/images/mathematics-12-03809-g009-550.jpg?1733209777" title=" <strong>Figure 9</strong><br/> <p>Sensitive behavior of 2D graphics (<a href="#FD10-mathematics-12-03809" class="html-disp-formula">10</a>) for <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>Z</mi> <mo>,</mo> <mi>V</mi> <mo>)</mo> <mo>=</mo> <mo>(</mo> <mn>0.12</mn> <mo>,</mo> <mn>0</mn> <mo>)</mo> </mrow> </semantics></math> in navy blue color and <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>Z</mi> <mo>,</mo> <mi>V</mi> <mo>)</mo> <mo>=</mo> <mo>(</mo> <mn>0.13</mn> <mo>,</mo> <mn>0</mn> <mo>)</mo> </mrow> </semantics></math> in red color.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3809'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03809/article_deploy/html/images/mathematics-12-03809-g010-550.jpg?1733209778" title=" <strong>Figure 10</strong><br/> <p>Sensitive behavior of 2D graphics (<a href="#FD10-mathematics-12-03809" class="html-disp-formula">10</a>) for <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>Z</mi> <mo>,</mo> <mi>V</mi> <mo>)</mo> <mo>=</mo> <mo>(</mo> <mn>1.05</mn> <mo>,</mo> <mn>0</mn> <mo>)</mo> </mrow> </semantics></math> in navy blue color and <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>Z</mi> <mo>,</mo> <mi>V</mi> <mo>)</mo> <mo>=</mo> <mo>(</mo> <mn>1.07</mn> <mo>,</mo> <mn>0</mn> <mo>)</mo> </mrow> </semantics></math> in red color.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3809'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03809/article_deploy/html/images/mathematics-12-03809-g011-550.jpg?1733209781" title=" <strong>Figure 11</strong><br/> <p>For <math display="inline"><semantics> <mrow> <msub> <mi>ς</mi> <mn>1</mn> </msub> <mo>=</mo> <mo>−</mo> <mn>1.0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>ς</mi> <mn>2</mn> </msub> <mo>=</mo> <mo>−</mo> <mn>0.1</mn> </mrow> </semantics></math><math display="inline"><semantics> <mrow> <msub> <mi>ς</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>2.26</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>ς</mi> <mn>4</mn> </msub> <mo>=</mo> <mn>50.007</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>ω</mi> <mo>=</mo> <mn>0.65</mn> </mrow> </semantics></math>, for the initial circumstances = (2.345, 0.346) are represented a nonlinear dynamical system.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3809'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03809/article_deploy/html/images/mathematics-12-03809-g012-550.jpg?1733209784" title=" <strong>Figure 12</strong><br/> <p>For <math display="inline"><semantics> <mrow> <msub> <mi>ς</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.964</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>ς</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>4.98</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>ς</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>1.65</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>ς</mi> <mn>4</mn> </msub> <mo>=</mo> <mn>50.007</mn> </mrow> </semantics></math> and for the initial circumstances = (0.29, 0.43) are represented a nonlinear dynamical system.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3809'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03809/article_deploy/html/images/mathematics-12-03809-g013-550.jpg?1733209785" title=" <strong>Figure 13</strong><br/> <p>Lyapunov exponent for different values of parameters with same initial conditions.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3809'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03809/article_deploy/html/images/mathematics-12-03809-g014-550.jpg?1733209785" title=" <strong>Figure 14</strong><br/> <p>The dynamical system exhibits multi-stability for <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>Z</mi> <mo>,</mo> <mi>V</mi> <mo>)</mo> <mo>=</mo> <mo>(</mo> <mn>2.345</mn> <mo>,</mo> <mn>0.346</mn> <mo>)</mo> </mrow> </semantics></math> in green and <math display="inline"><semantics> <mrow> <mo>(</mo> <mn>1.123</mn> <mo>,</mo> <mn>0.124</mn> <mo>)</mo> </mrow> </semantics></math> in blue.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3809'>Full article</a></strong> "></a></div> </div> </div> </div> <div class="expanding-div collapsed"> <div class="generic-item article-item"> <div class="article-content"> <div class="label right label__btn"> <span style="font-size: 12px; color: #1a1a1a;"> 14 pages, 881 KiB   </span> <a href="/2227-7390/12/23/3808/pdf?version=1733121671" class="UD_Listings_ArticlePDF" title="Article PDF" data-name="Degree Distribution of Evolving Network with Node Preference Deletion" data-journal="mathematics"> <i class="material-icons custom-download"></i> </a> </div> <div class="article-icons"><span class="label openaccess" data-dropdown="drop-article-label-openaccess" aria-expanded="false">Open Access</span><span class="label articletype">Article</span></div> <a class="title-link" href="/2227-7390/12/23/3808">Degree Distribution of Evolving Network with Node Preference Deletion</a> <div class="authors"> by <span class="inlineblock "><strong>Yue Xiao</strong> and </span><span class="inlineblock "><strong>Xiaojun Zhang</strong></span> </div> <div class="color-grey-dark"> <em>Mathematics</em> <b>2024</b>, <em>12</em>(23), 3808; <a href="https://doi.org/10.3390/math12233808">https://doi.org/10.3390/math12233808</a> - 2 Dec 2024 </div> <div class="abstract-div"> <a href="#" onclick="$(this).next('.abstract-cropped').toggleClass('inline').next('.abstract-full').toggleClass('inline'); return false;"> <strong>Abstract </strong> </a> <div class="abstract-cropped inline"> Discussing evolutionary network models and corresponding degree distributions under different mechanisms is applied basic research in network science. This study proposes a new evolutionary network model, which integrates node preference deletion and edge reconnection mechanisms and is also an extension of the existing <a href="#" data-counterslink = "https://www.mdpi.com/2227-7390/12/23/3808/more" onclick="$(this).parents('.abstract-cropped').toggleClass('inline').next('.abstract-full').toggleClass('inline'); return false;"> [...] Read more.</a> </div> <div class="abstract-full "> Discussing evolutionary network models and corresponding degree distributions under different mechanisms is applied basic research in network science. This study proposes a new evolutionary network model, which integrates node preference deletion and edge reconnection mechanisms and is also an extension of the existing evolutionary network model. In order to analyze the key statistical property of the model, the steady-state distribution, we propose a Markov chain method based on the enhanced stochastic process rule (ESPR). The ESPR method makes the evolving network’s topological structure and statistical properties consistent with those observed in the natural evolution process, ensures the theoretical results of the degree distribution of the evolving network model, and overcomes the limitations of using empirical methods for approximate analysis. Finally, we verify the accuracy of the steady-state distribution and tail feature estimation of the model through Monte Carlo simulation. This work has laid a solid theoretical foundation for the future development of evolutionary network models and the study of more complex network statistical properties. <a href="/2227-7390/12/23/3808">Full article</a> </div> </div> <a href="#" class="abstract-figures-show" data-counterslink = "https://www.mdpi.com/2227-7390/12/23/3808/show" ><span >►</span><span style=" display: none;">▼</span> Show Figures </a><div class="abstract-image-preview "><div class="arrow left-arrow" id="prev1534656"><i class="fa fa-caret-left"></i></div><div class="arrow right-arrow" id="next1534656"><i class="fa fa-caret-right"></i></div><div class="absgraph cycle-slideshow manual" data-cycle-fx="scrollHorz" data-cycle-timeout="0" data-cycle-next="#next1534656" data-cycle-prev="#prev1534656" data-cycle-progressive="#images1534656" data-cycle-slides=">div" data-cycle-log="false"><div class='openpopupgallery cycle-slide' data-imgindex='0' data-target='article-1534656-popup'><span class="helper"></span><img src="data:image/gif;base64,R0lGODlhAQABAAD/ACwAAAAAAQABAAACADs=" data-src="https://pub.mdpi-res.com/mathematics/mathematics-12-03808/article_deploy/html/images/mathematics-12-03808-g001-550.jpg?1733121786" alt="" style="border: 0;"><p>Figure 1</p></div><script id="images1534656" type="text/cycle" data-cycle-split="---"><div class='openpopupgallery' data-imgindex='1' data-target='article-1534656-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03808/article_deploy/html/images/mathematics-12-03808-g002-550.jpg?1733121788'><p>Figure 2</p></div> --- <div class='openpopupgallery' data-imgindex='2' data-target='article-1534656-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03808/article_deploy/html/images/mathematics-12-03808-g003-550.jpg?1733121790'><p>Figure 3</p></div> --- <div class='openpopupgallery' data-imgindex='3' data-target='article-1534656-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03808/article_deploy/html/images/mathematics-12-03808-g004-550.jpg?1733121792'><p>Figure 4</p></div> --- <div class='openpopupgallery' data-imgindex='4' data-target='article-1534656-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03808/article_deploy/html/images/mathematics-12-03808-g005-550.jpg?1733121793'><p>Figure 5</p></div> --- <div class='openpopupgallery' data-imgindex='5' data-target='article-1534656-popup'><span class="helper"></span><img src='https://pub.mdpi-res.com/mathematics/mathematics-12-03808/article_deploy/html/images/mathematics-12-03808-g006-550.jpg?1733121795'><p>Figure 6</p></div></script></div></div><div id="article-1534656-popup" class="popupgallery" style="display: inline; line-height: 200%"><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03808/article_deploy/html/images/mathematics-12-03808-g001-550.jpg?1733121786" title=" <strong>Figure 1</strong><br/> <p>Schematic diagram of the natural network evolution process. The network <math display="inline"><semantics> <mrow> <mo stretchy="false">(</mo> <msup> <mi>b</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mrow> </semantics></math><math display="inline"><semantics> <mrow> <mo stretchy="false">(</mo> <msup> <mi>c</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mrow> </semantics></math> represents the operation of adding nodes to the network <math display="inline"><semantics> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> at time <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>−</mo> <mn>1</mn> </mrow> </semantics></math>, and the network <math display="inline"><semantics> <mrow> <mo stretchy="false">(</mo> <msup> <mi>d</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mrow> </semantics></math><math display="inline"><semantics> <mrow> <mo stretchy="false">(</mo> <msup> <mi>e</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mrow> </semantics></math><math display="inline"><semantics> <mrow> <mo stretchy="false">(</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mrow> </semantics></math> represents the operation of deleting nodes from the network <math display="inline"><semantics> <mrow> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math><math display="inline"><semantics> <mrow> <mo stretchy="false">(</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> and reconnecting edges at time <span class="html-italic">t</span>, finally obtaining a series of new networks <math display="inline"><semantics> <mrow> <mo stretchy="false">(</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math><math display="inline"><semantics> <mrow> <mo stretchy="false">(</mo> <mi>e</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math><math display="inline"><semantics> <mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> at time <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>+</mo> <mn>1</mn> </mrow> </semantics></math>. Although the generation probabilities of networks <math display="inline"><semantics> <mrow> <mo stretchy="false">(</mo> <mi>e</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> are different, their topological structures are the same.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3808'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03808/article_deploy/html/images/mathematics-12-03808-g002-550.jpg?1733121788" title=" <strong>Figure 2</strong><br/> <p>Schematic diagram of network evolution process in SPR. According to the size of the network <math display="inline"><semantics> <mrow> <msub> <mi>G</mi> <mn>5</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics></math>, the nodes in the network are numbered from 1 to 5. <math display="inline"><semantics> <mrow> <msub> <mi>A</mi> <mi>i</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics></math> represents a series of new networks with different structures and probabilities obtained after the network <math display="inline"><semantics> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> is preferentially deleted and the edges are reconnected; <math display="inline"><semantics> <mrow> <msub> <mi>B</mi> <mi>i</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics></math> represents the isolated nodes of different types and probabilities obtained after deleting nodes from the network <math display="inline"><semantics> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math>; <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mi>i</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics></math> represents the transformation of the probabilities of different types of isolated nodes in <math display="inline"><semantics> <mrow> <msub> <mi>B</mi> <mi>i</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics></math> so that networks with different topological structures and probabilities in <math display="inline"><semantics> <mrow> <msub> <mi>A</mi> <mi>i</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics></math> can be smoothly reconstructed. Networks <math display="inline"><semantics> <mrow> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math><math display="inline"><semantics> <mrow> <mo stretchy="false">(</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> are the possible results of the real evolution of the network. The network in <math display="inline"><semantics> <mrow> <msub> <mi>A</mi> <mi>i</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics></math> is consistent with the real result in terms of the topological structure and generation probability.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3808'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03808/article_deploy/html/images/mathematics-12-03808-g003-550.jpg?1733121790" title=" <strong>Figure 3</strong><br/> <p>Schematic diagram of one-step state transition process of a node from <math display="inline"><semantics> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> at <span class="html-italic">t</span> to all possible states of <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>+</mo> <mn>1</mn> </mrow> </semantics></math> in ESPR.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3808'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03808/article_deploy/html/images/mathematics-12-03808-g004-550.jpg?1733121792" title=" <strong>Figure 4</strong><br/> <p>Exact solution (ES) verse computer simulation (CS).</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3808'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03808/article_deploy/html/images/mathematics-12-03808-g005-550.jpg?1733121793" title=" <strong>Figure 5</strong><br/> <p>The tail of degree distribution.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3808'>Full article</a></strong> "></a><a href="https://pub.mdpi-res.com/mathematics/mathematics-12-03808/article_deploy/html/images/mathematics-12-03808-g006-550.jpg?1733121795" title=" <strong>Figure 6</strong><br/> <p>Estimate of the tail of the exponent.</p> <strong style='display: block; margin-top: 10px; font-size: 18px;'><a style='color: #fff' href='/2227-7390/12/23/3808'>Full article</a></strong> "></a></div> </div> </div> </div> </div> <div class="generic-item last-item"> <a class="bold" href="/search?q=&journal=mathematics&sort=pubdate&page_count=50">More Articles...</a> </div> </div> </div> </div> <div id="left-column" class="content__column large-3 large-pull-6 medium-3 medium-pull-6 small-12 columns"> <div id="js-large-main-top-container"> <div id="js-main-top-container" class="content__container"> <a href="/journal/mathematics"> <img src="https://pub.mdpi-res.com/img/journals/mathematics-logo.png?8600e93ff98dbf14" alt="mathematics-logo" title="Mathematics" style="max-height: 60px; margin: 0 0 0 0;"> </a> <div 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