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Axioms
Volume 13
Issue 5
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Axioms, Volume 13, Issue 5 (May 2024) – 58 articles

Cover Story (view full-size image): Exceptional simple Jordan algebras play an important role in the structure theory of Jordan algebras, and increasingly also in certain areas of physics like particle physics and general relativity. The first Tits construction is a well-known tripling process to construct these algebras. Its ingredients are a central simple algebra of degree three and a nonzero scalar in the base field needed when defining the multiplication of the algebra. What happens when we employ an element in the central simple algebra instead of this scalar? In answering this question, we find a new family of highly nonassociative algebras, which obey surprisingly similar identities to their classical “cousins”. View this paper
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25 pages, 11760 KiB  
Article
Regular, Beating and Dilogarithmic Breathers in Biased Photorefractive Crystals
by Carlos Alberto Betancur-Silvera, Aurea Espinosa-Cerón, Boris A. Malomed and Jorge Fujioka
Axioms 2024, 13(5), 338; https://doi.org/10.3390/axioms13050338 - 20 May 2024
Viewed by 725
Abstract
The propagation of light beams in photovoltaic pyroelectric photorefractive crystals is modelled by a specific generalization of the nonlinear Schrödinger equation (GNLSE). We use a variational approximation (VA) to predict the propagation of solitary-wave inputs in the crystals, finding that the VA equations [...] Read more.
The propagation of light beams in photovoltaic pyroelectric photorefractive crystals is modelled by a specific generalization of the nonlinear Schrödinger equation (GNLSE). We use a variational approximation (VA) to predict the propagation of solitary-wave inputs in the crystals, finding that the VA equations involve a dilogarithm special function. The VA predicts that solitons and breathers exist, and the Vakhitov–Kolokolov criterion predicts that the solitons are stable solutions. Direct simulations of the underlying GNLSE corroborates the existence of such stable modes. The numerical solutions produce both regular breathers and ones featuring beats (long-period modulations of fast oscillations). In the latter case, the Fourier transform of amplitude oscillations reveals a nearly discrete spectrum characterizing the beats dynamics. Numerical solutions of another type demonstrate the spontaneous splitting of the input pulse in two or several secondary ones. Full article
(This article belongs to the Special Issue Nonlinear Schrödinger Equations)
►▼ Show Figures

Figure 1

Figure 1
<p>Values of the VA-predicted VK derivative <math display="inline"><semantics> <mrow> <mi>d</mi> <mi>P</mi> <mo>/</mo> <mi>d</mi> <mi>k</mi> </mrow> </semantics></math> for the family of stationary solitons based on ansatz (16) as produced by Equation (31) with the coefficients fixed as per Equation (32).</p> Full article ">Figure 2
<p><math display="inline"><semantics> <mrow> <mfenced open="|" close="|" separators="|"> <mrow> <mi>U</mi> <mo>(</mo> <mi>s</mi> <mo>,</mo> <mi>ζ</mi> <mo>)</mo> </mrow> </mfenced> </mrow> </semantics></math> as produced by ansatz (16), with the variational parameters obtained from the numerical solution of Equations (21)–(24) with coefficients (32) and an initial condition corresponding to input (41), which represents the fixed point of the equations. The values of coefficients (32) correspond to <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>E</mi> </mrow> <mrow> <mi>p</mi> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mn>12.7</mn> <mo> </mo> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">V</mi> <mo>/</mo> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">m</mi> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>E</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mn>3.99518</mn> <mo> </mo> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">V</mi> <mo>/</mo> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">m</mi> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>E</mi> </mrow> <mrow> <mi>p</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mn>2</mn> <mo> </mo> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">V</mi> <mo>/</mo> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">m</mi> </mrow> </semantics></math> and the physical parameters shown in Equation (47). The counterpart of this picture, produced by full simulations of the GNLSE (7), is presented below in <a href="#sec4dot3-axioms-13-00338" class="html-sec">Section 4.3</a>.</p> Full article ">Figure 3
<p>The VA-predicted breather for the coefficients chosen as per Equation (42) and the input corresponding to Equation (43). The values of coefficients (42) correspond to <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>E</mi> </mrow> <mrow> <mi>p</mi> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mn>2.31</mn> <mo> </mo> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">V</mi> <mo>/</mo> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">m</mi> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>E</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mn>4.00035</mn> <mo> </mo> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">V</mi> <mo>/</mo> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">m</mi> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>E</mi> </mrow> <mrow> <mi>p</mi> </mrow> </msub> <mo>=</mo> <mn>9.34</mn> <mo> </mo> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">v</mi> <mo>/</mo> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">m</mi> </mrow> </semantics></math> and the physical parameters shown in Equation (47). The counterpart of this dynamical picture, produced by full simulations of the GNLSE (7), is presented below in <a href="#sec4dot3-axioms-13-00338" class="html-sec">Section 4.3</a>.</p> Full article ">Figure 4
<p>Solutions of the Euler–Lagrange Equations (21)–(24) for variational parameters <math display="inline"><semantics> <mrow> <mfenced open="{" close="}" separators="|"> <mrow> <mi>A</mi> <mfenced separators="|"> <mrow> <mi>ζ</mi> </mrow> </mfenced> <mo>,</mo> <mo> </mo> <mi>a</mi> <mfenced separators="|"> <mrow> <mi>ζ</mi> </mrow> </mfenced> <mo>,</mo> <mo> </mo> <mi>b</mi> <mfenced separators="|"> <mrow> <mi>ζ</mi> </mrow> </mfenced> <mo> </mo> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">d</mi> <mo> </mo> <mi>c</mi> <mfenced separators="|"> <mrow> <mi>ζ</mi> </mrow> </mfenced> </mrow> </mfenced> </mrow> </semantics></math> for the same case as shown in <a href="#axioms-13-00338-f003" class="html-fig">Figure 3</a>.</p> Full article ">Figure 5
<p>The same as in <a href="#axioms-13-00338-f003" class="html-fig">Figure 3</a>, but for the parameters of (44) and an input corresponding to Equation (45). The values of the coefficients (44) correspond to <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>E</mi> </mrow> <mrow> <mi>p</mi> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mn>0.6</mn> <mo> </mo> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">V</mi> <mo>/</mo> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">m</mi> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>E</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mn>3.99518</mn> <mo> </mo> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">V</mi> <mo>/</mo> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">m</mi> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>E</mi> </mrow> <mrow> <mi>p</mi> </mrow> </msub> <mo>=</mo> <mn>9.34</mn> <mo> </mo> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">V</mi> <mo>/</mo> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">m</mi> </mrow> </semantics></math> and the other physical parameters shown in Equation (47). The counterpart of this dynamical picture, produced by full simulations of the GNLSE (7), is presented below in <a href="#sec4dot3-axioms-13-00338" class="html-sec">Section 4.3</a>.</p> Full article ">Figure 6
<p>Solutions of the Euler–Lagrange Equations (21)–(24) for variational parameters <math display="inline"><semantics> <mrow> <mfenced open="{" close="}" separators="|"> <mrow> <mi>A</mi> <mfenced separators="|"> <mrow> <mi>ζ</mi> </mrow> </mfenced> <mo>,</mo> <mo> </mo> <mi>a</mi> <mfenced separators="|"> <mrow> <mi>ζ</mi> </mrow> </mfenced> <mo>,</mo> <mo> </mo> <mi>b</mi> <mfenced separators="|"> <mrow> <mi>ζ</mi> </mrow> </mfenced> <mo> </mo> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">d</mi> <mo> </mo> <mi>c</mi> <mfenced separators="|"> <mrow> <mi>ζ</mi> </mrow> </mfenced> </mrow> </mfenced> </mrow> </semantics></math> for the same case as shown in <a href="#axioms-13-00338-f005" class="html-fig">Figure 5</a>.</p> Full article ">Figure 7
<p>The same as in <a href="#axioms-13-00338-f003" class="html-fig">Figure 3</a>, but for the parameters of (46) and an input corresponding to Equation (41). The values of coefficients (46) correspond to <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>E</mi> </mrow> <mrow> <mi>p</mi> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mn>8.32</mn> <mo> </mo> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">V</mi> <mo>/</mo> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">m</mi> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>E</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mn>3.99518</mn> <mo> </mo> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">V</mi> <mo>/</mo> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">m</mi> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>E</mi> </mrow> <mrow> <mi>p</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mn>2</mn> <mo> </mo> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">V</mi> <mo>/</mo> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">m</mi> </mrow> </semantics></math> and the other physical parameters shown in Equation (47).</p> Full article ">Figure 8
<p>Solutions of the Euler–Lagrange Equations (21)–(24) for the variational parameters <math display="inline"><semantics> <mrow> <mfenced open="{" close="}" separators="|"> <mrow> <mi>A</mi> <mfenced separators="|"> <mrow> <mi>ζ</mi> </mrow> </mfenced> <mo>,</mo> <mo> </mo> <mi>a</mi> <mfenced separators="|"> <mrow> <mi>ζ</mi> </mrow> </mfenced> <mo>,</mo> <mo> </mo> <mi>b</mi> <mfenced separators="|"> <mrow> <mi>ζ</mi> </mrow> </mfenced> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">d</mi> <mo> </mo> <mi>c</mi> <mfenced separators="|"> <mrow> <mi>ζ</mi> </mrow> </mfenced> </mrow> </mfenced> </mrow> </semantics></math> for the same case as shown in <a href="#axioms-13-00338-f007" class="html-fig">Figure 7</a>.</p> Full article ">Figure 9
<p>Soliton obtained with ansatz (33). In this case, the variational parameters evolve according to the Euler–Lagrange Equations (36)–(39) with the coefficients shown in Equation (49) and the initial condition given in Equation (48). The values of the coefficients in (49) correspond to <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>E</mi> </mrow> <mrow> <mi>p</mi> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mn>4.61</mn> <mo> </mo> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">V</mi> <mo>/</mo> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">m</mi> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>E</mi> </mrow> <mrow> <mi>p</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mn>2</mn> <mo> </mo> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">V</mi> <mo>/</mo> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">m</mi> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>E</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mn>3.99518</mn> <mo> </mo> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">V</mi> <mo>/</mo> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">m</mi> </mrow> </semantics></math> and the physical parameters shown in Equation (47).</p> Full article ">Figure 10
<p>VA-predicted breather obtained with the ansatz (33). It is plotted with the variational parameters evolving according to Euler–Lagrange Equations (36)–(39) with the coefficients chosen as per Equation (50) and the initial condition corresponding to Equation (48). The values of the coefficients of (50) correspond to <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>E</mi> </mrow> <mrow> <mi>p</mi> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mn>4.31</mn> <mo> </mo> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">V</mi> <mo>/</mo> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">m</mi> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>E</mi> </mrow> <mrow> <mi>p</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mn>2</mn> <mo> </mo> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">V</mi> <mo>/</mo> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">m</mi> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>E</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mn>3.99518</mn> <mo> </mo> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">V</mi> <mo>/</mo> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">m</mi> </mrow> </semantics></math> and the other physical parameters shown in Equation (47).</p> Full article ">Figure 11
<p>Solutions of the Euler–Lagrange Equations (36)–(39) for the variational parameters <math display="inline"><semantics> <mrow> <mfenced open="{" close="}" separators="|"> <mrow> <mi>A</mi> <mfenced separators="|"> <mrow> <mi>ζ</mi> </mrow> </mfenced> <mo>,</mo> <mo> </mo> <mi>a</mi> <mfenced separators="|"> <mrow> <mi>ζ</mi> </mrow> </mfenced> <mo>,</mo> <mo> </mo> <mi>b</mi> <mfenced separators="|"> <mrow> <mi>ζ</mi> </mrow> </mfenced> <mo> </mo> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">d</mi> <mo> </mo> <mi>c</mi> <mfenced separators="|"> <mrow> <mi>ζ</mi> </mrow> </mfenced> </mrow> </mfenced> </mrow> </semantics></math>, for the same case as shown in <a href="#axioms-13-00338-f010" class="html-fig">Figure 10</a>.</p> Full article ">Figure 12
<p>The same as in <a href="#axioms-13-00338-f010" class="html-fig">Figure 10</a>, except <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>=</mo> <mn>6.6</mn> </mrow> </semantics></math>. This value, and coefficients <math display="inline"><semantics> <mrow> <mi>α</mi> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>θ</mi> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>γ</mi> </mrow> </semantics></math> given in Equation (50), correspond to <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>E</mi> </mrow> <mrow> <mi>p</mi> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mn>3.9</mn> <mo> </mo> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">V</mi> <mo>/</mo> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">m</mi> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>E</mi> </mrow> <mrow> <mi>p</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mn>2</mn> <mo> </mo> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">V</mi> <mo>/</mo> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">m</mi> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>E</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mn>3.99518</mn> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">V</mi> <mo>/</mo> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">m</mi> </mrow> </semantics></math> and the other physical parameters shown in Equation (47).</p> Full article ">Figure 13
<p>Solutions of the Euler–Lagrange Equations (36)–(39) for the variational parameters <math display="inline"><semantics> <mrow> <mfenced open="{" close="}" separators="|"> <mrow> <mi>A</mi> <mfenced separators="|"> <mrow> <mi>ζ</mi> </mrow> </mfenced> <mo>,</mo> <mo> </mo> <mi>a</mi> <mfenced separators="|"> <mrow> <mi>ζ</mi> </mrow> </mfenced> <mo>,</mo> <mo> </mo> <mi>b</mi> <mfenced separators="|"> <mrow> <mi>ζ</mi> </mrow> </mfenced> <mo> </mo> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">d</mi> <mi>c</mi> <mfenced separators="|"> <mrow> <mi>ζ</mi> </mrow> </mfenced> </mrow> </mfenced> </mrow> </semantics></math>, for the same case as shown in <a href="#axioms-13-00338-f012" class="html-fig">Figure 12</a>.</p> Full article ">Figure 14
<p>Breather produced by ansatz (33). In this case, the variational parameters evolve according to the Euler–Lagrange Equations (36)–(39) along with the coefficients shown in Equation (52) and the initial condition given in Equation (51). The values of the coefficients in (52) correspond to <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>E</mi> </mrow> <mrow> <mi>p</mi> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mn>0.71</mn> <mo> </mo> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">V</mi> <mo>/</mo> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">m</mi> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>E</mi> </mrow> <mrow> <mi>p</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mn>2</mn> <mo> </mo> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">V</mi> <mo>/</mo> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">m</mi> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>E</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mn>3.99518</mn> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">V</mi> <mo>/</mo> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">m</mi> </mrow> </semantics></math> and the other physical parameters shown in Equation (47).</p> Full article ">Figure 15
<p>Solutions of the Euler–Lagrange Equations (36)–(39) for the variational parameters <math display="inline"><semantics> <mrow> <mfenced open="{" close="}" separators="|"> <mrow> <mi>A</mi> <mfenced separators="|"> <mrow> <mi>ζ</mi> </mrow> </mfenced> <mo>,</mo> <mo> </mo> <mi>a</mi> <mfenced separators="|"> <mrow> <mi>ζ</mi> </mrow> </mfenced> <mo>,</mo> <mo> </mo> <mi>b</mi> <mfenced separators="|"> <mrow> <mi>ζ</mi> </mrow> </mfenced> <mo> </mo> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">d</mi> <mo> </mo> <mi>c</mi> <mfenced separators="|"> <mrow> <mi>ζ</mi> </mrow> </mfenced> </mrow> </mfenced> </mrow> </semantics></math>, for the same case as shown in <a href="#axioms-13-00338-f014" class="html-fig">Figure 14</a>.</p> Full article ">Figure 16
<p>Results of the numerical simulations of Equation (7) with the same coefficients as in Equation (32) and the same input as in Equation (41), which were used above to produce the VA solution displayed in <a href="#axioms-13-00338-f002" class="html-fig">Figure 2</a>.</p> Full article ">Figure 17
<p>Profiles <math display="inline"><semantics> <mrow> <mo>|</mo> <mi>U</mi> <mfenced separators="|"> <mrow> <mi>s</mi> <mo>,</mo> <mi>ζ</mi> <mo>=</mo> <mn>20</mn> </mrow> </mfenced> <mo>|</mo> </mrow> </semantics></math> obtained from the numerical solution of Equation (7) shown in <a href="#axioms-13-00338-f016" class="html-fig">Figure 16</a> (black squares) and the variational solution shown in <a href="#axioms-13-00338-f002" class="html-fig">Figure 2</a> (the thin line).</p> Full article ">Figure 18
<p>Results of the numerical simulations of Equation (7) with the same coefficients as Equation (42) and the same input as Equation (43), which were used above to produce the VA solution displayed in <a href="#axioms-13-00338-f003" class="html-fig">Figure 3</a>.</p> Full article ">Figure 19
<p>Results of the numerical simulations of Equation (7) with the same coefficients as Equation (44) and the same input as Equation (45), which were used above to produce the VA solution displayed in <a href="#axioms-13-00338-f005" class="html-fig">Figure 5</a>.</p> Full article ">Figure 20
<p>Profiles <math display="inline"><semantics> <mrow> <mo>|</mo> <mi>U</mi> <mfenced separators="|"> <mrow> <mi>s</mi> <mo>,</mo> <mi>ζ</mi> <mo>=</mo> <mn>1</mn> </mrow> </mfenced> <mo>|</mo> </mrow> </semantics></math> obtained from the numerical solution of Equation (7) shown in <a href="#axioms-13-00338-f019" class="html-fig">Figure 19</a> (black dots) and the variational solution shown in <a href="#axioms-13-00338-f005" class="html-fig">Figure 5</a> (thin line).</p> Full article ">Figure 21
<p>Numerical solution of Equation (7) for the coefficients shown in (46) and the initial condition in (53).</p> Full article ">Figure 22
<p>Optical power <span class="html-italic">P</span> vs. the propagation constant <span class="html-italic">k</span> of the soliton solutions of Equation (7) with the coefficients given in Equation (49).</p> Full article ">Figure 23
<p>Separation <math display="inline"><semantics> <mrow> <mi>λ</mi> </mrow> </semantics></math> between the breather’s peaks as a function of <math display="inline"><semantics> <mrow> <mi>β</mi> </mrow> </semantics></math>. Black squares represent direct numerical solutions of Equation (7) and stars represent the variational solution using the ansatz (16). The input is taken as per Equation (45) and the other coefficients in Equation (7) are <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>31.60</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>θ</mi> <mo>=</mo> <mo>−</mo> <mn>13.53</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>.</p> Full article ">Figure 24
<p>Beating breather, obtained as a numerical solution of Equation (7) with the parameters of (54) and the input of (55). The values of the coefficients of (54) correspond to the physical parameters given in Equation (47): <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>E</mi> </mrow> <mrow> <mi>p</mi> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mn>40</mn> <mo> </mo> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">V</mi> <mo>/</mo> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">m</mi> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>E</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mn>3.92098</mn> <mo> </mo> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">V</mi> <mo>/</mo> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">m</mi> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>E</mi> </mrow> <mrow> <mi>p</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mn>2</mn> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">V</mi> <mo>/</mo> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">m</mi> </mrow> </semantics></math>.</p> Full article ">Figure 25
<p>The power spectrum of the amplitude function <math display="inline"><semantics> <mrow> <mfenced open="|" close="|" separators="|"> <mrow> <mi>U</mi> <mo>(</mo> <mi>ζ</mi> <mo>,</mo> <mi>s</mi> <mo>=</mo> <mn>0</mn> <mo>)</mo> </mrow> </mfenced> </mrow> </semantics></math> corresponding to the breather shown in <a href="#axioms-13-00338-f024" class="html-fig">Figure 24</a>.</p> Full article ">Figure 26
<p>Numerical solution of Equation (7) with values of the parameters of (54) and the input of (56). These coefficients correspond to the physical parameters shown in Equation (47), where <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>E</mi> </mrow> <mrow> <mi>p</mi> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mn>40</mn> <mo> </mo> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">V</mi> <mo>/</mo> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">m</mi> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>E</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mn>3.92098</mn> <mo> </mo> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">V</mi> <mo>/</mo> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">m</mi> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>E</mi> </mrow> <mrow> <mi>p</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mn>2</mn> <mo> </mo> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">V</mi> <mo>/</mo> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">m</mi> </mrow> </semantics></math>.</p> Full article ">Figure 27
<p>The same as in <a href="#axioms-13-00338-f026" class="html-fig">Figure 26</a>, but observed from a different direction.</p> Full article ">Figure 28
<p>Splitting of the input pulse into five secondary ones, exhibited by the numerical solution of Equation (7) with the coefficients shown in Equation (54) and the initial condition of (57). These coefficients correspond to the physical parameters given in Equation (47), where <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>E</mi> </mrow> <mrow> <mi>p</mi> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mn>40</mn> <mo> </mo> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">V</mi> <mo>/</mo> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">m</mi> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>E</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mn>3.92098</mn> <mo> </mo> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">V</mi> <mo>/</mo> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">m</mi> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>E</mi> </mrow> <mrow> <mi>p</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mn>2</mn> <mo> </mo> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">V</mi> <mo>/</mo> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">m</mi> </mrow> </semantics></math>.</p> Full article ">
13 pages, 299 KiB  
Article
High Perturbations of a Fractional Kirchhoff Equation with Critical Nonlinearities
by Shengbin Yu, Lingmei Huang and Jiangbin Chen
Axioms 2024, 13(5), 337; https://doi.org/10.3390/axioms13050337 - 20 May 2024
Viewed by 623
Abstract
This paper concerns a fractional Kirchhoff equation with critical nonlinearities and a negative nonlocal term. In the case of high perturbations (large values of α, i.e., the parameter of a subcritical nonlinearity), existence results are obtained by the concentration compactness principle together [...] Read more.
This paper concerns a fractional Kirchhoff equation with critical nonlinearities and a negative nonlocal term. In the case of high perturbations (large values of α, i.e., the parameter of a subcritical nonlinearity), existence results are obtained by the concentration compactness principle together with the mountain pass theorem and cut-off technique. The multiplicity of solutions are further considered with the help of the symmetric mountain pass theorem. Moreover, the nonexistence and asymptotic behavior of positive solutions are also investigated. Full article
(This article belongs to the Section Mathematical Analysis)
16 pages, 310 KiB  
Article
Blow-Up Analysis of L2-Norm Solutions for an Elliptic Equation with a Varying Nonlocal Term
by Xincai Zhu and Chunxia He
Axioms 2024, 13(5), 336; https://doi.org/10.3390/axioms13050336 - 20 May 2024
Viewed by 625
Abstract
This paper is devoted to studying a type of elliptic equation that contains a varying nonlocal term. We provide a detailed analysis of the existence, non-existence, and blow-up behavior of L2-norm solutions for the related equation when the potential function [...] Read more.
This paper is devoted to studying a type of elliptic equation that contains a varying nonlocal term. We provide a detailed analysis of the existence, non-existence, and blow-up behavior of L2-norm solutions for the related equation when the potential function V(x) fulfills an appropriate choice. Full article
(This article belongs to the Special Issue Advances in Differential Equations and Its Applications)
31 pages, 1619 KiB  
Article
Respiratory Condition Detection Using Audio Analysis and Convolutional Neural Networks Optimized by Modified Metaheuristics
by Nebojsa Bacanin, Luka Jovanovic, Ruxandra Stoean, Catalin Stoean, Miodrag Zivkovic, Milos Antonijevic and Milos Dobrojevic
Axioms 2024, 13(5), 335; https://doi.org/10.3390/axioms13050335 - 18 May 2024
Cited by 3 | Viewed by 912
Abstract
Respiratory conditions have been a focal point in recent medical studies. Early detection and timely treatment are crucial factors in improving patient outcomes for any medical condition. Traditionally, doctors diagnose respiratory conditions through an investigation process that involves listening to the patient’s lungs. [...] Read more.
Respiratory conditions have been a focal point in recent medical studies. Early detection and timely treatment are crucial factors in improving patient outcomes for any medical condition. Traditionally, doctors diagnose respiratory conditions through an investigation process that involves listening to the patient’s lungs. This study explores the potential of combining audio analysis with convolutional neural networks to detect respiratory conditions in patients. Given the significant impact of proper hyperparameter selection on network performance, contemporary optimizers are employed to enhance efficiency. Moreover, a modified algorithm is introduced that is tailored to the specific demands of this study. The proposed approach is validated using a real-world medical dataset and has demonstrated promising results. Two experiments are conducted: the first tasked models with respiratory condition detection when observing mel spectrograms of patients’ breathing patterns, while the second experiment considered the same data format for multiclass classification. Contemporary optimizers are employed to optimize the architecture selection and training parameters of models in both cases. Under identical test conditions, the best models are optimized by the introduced modified metaheuristic, with an accuracy of 0.93 demonstrated for condition detection, and a slightly reduced accuracy of 0.75 for specific condition identification. Full article
(This article belongs to the Special Issue Advances in Parameter-Tuning Techniques for Metaheuristic Algorithms)
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<p>Audio samples of healthy patient and patients with bronchitis and chronic obstructive pulmonary disease conversion to mel spectrograms.</p> Full article ">Figure 2
<p>Introduced experimental framework flowchart.</p> Full article ">Figure 3
<p>KDE plot illustrating the comparison between the distribution of the results for the used methods for the binary (<b>top</b>) and multiclass (<b>bottom</b>) classifications.</p> Full article ">Figure 4
<p>Binary respiratory condition detection outcome distribution plots.</p> Full article ">Figure 5
<p>Binary respiratory condition detection swarm plots.</p> Full article ">Figure 6
<p>Binary respiratory condition detection convergence diagrams.</p> Full article ">Figure 7
<p>Best-performing classifiers (CNN–AGbAEHO) model PR and confusion matrix charts.</p> Full article ">Figure 8
<p>Respiratory conduction identification outcome distribution plots.</p> Full article ">Figure 9
<p>Respiratory conduction identification swarm plots.</p> Full article ">Figure 10
<p>Respiratory conduction identification convergence diagrams.</p> Full article ">Figure 11
<p>Respiratory conduction identification best-performing classifiers (CNN–AGbAEHO) model PR and confusion matrix charts.</p> Full article ">
21 pages, 354 KiB  
Article
Exponential Stability of the Numerical Solution of a Hyperbolic System with Nonlocal Characteristic Velocities
by Rakhmatillo Djuraevich Aloev, Abdumauvlen Suleimanovich Berdyshev, Vasila Alimova and Kymbat Slamovna Bekenayeva
Axioms 2024, 13(5), 334; https://doi.org/10.3390/axioms13050334 - 17 May 2024
Cited by 1 | Viewed by 613
Abstract
In this paper, we investigate the problem of the exponential stability of a stationary solution for a hyperbolic system with nonlocal characteristic velocities and measurement error. The formulation of the initial boundary value problem of boundary control for the specified hyperbolic system is [...] Read more.
In this paper, we investigate the problem of the exponential stability of a stationary solution for a hyperbolic system with nonlocal characteristic velocities and measurement error. The formulation of the initial boundary value problem of boundary control for the specified hyperbolic system is given. A difference scheme is constructed for the numerical solution of the considered initial boundary value problem. The definition of the exponential stability of the numerical solution in ℓ2-norm with respect to a discrete perturbation of the equilibrium state of the initial boundary value difference problem is given. A discrete Lyapunov function for a numerical solution is constructed, and a theorem on the exponential stability of a stationary solution of the initial boundary value difference problem in ℓ2-norm with respect to a discrete perturbation is proved. Full article
(This article belongs to the Special Issue Difference, Functional, and Related Equations)
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<p>Numerical solution graph of the initial boundary value difference problem (<xref ref-type="disp-formula" rid="FD36-axioms-13-00334">36</xref>) in <inline-formula><mml:math id="mm367"><mml:semantics><mml:msup><mml:mi>l</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:semantics></mml:math></inline-formula>-norm.</p> Full article ">
38 pages, 522 KiB  
Article
Static Spherically Symmetric Perfect Fluid Solutions in Teleparallel F(T) Gravity
by Alexandre Landry
Axioms 2024, 13(5), 333; https://doi.org/10.3390/axioms13050333 - 17 May 2024
Cited by 1 | Viewed by 636
Abstract
In this paper, we investigate static spherically symmetric teleparallel F(T) gravity containing a perfect isotropic fluid. We first write the field equations and proceed to find new teleparallel F(T) solutions for perfect isotropic and linear fluids. By [...] Read more.
In this paper, we investigate static spherically symmetric teleparallel F(T) gravity containing a perfect isotropic fluid. We first write the field equations and proceed to find new teleparallel F(T) solutions for perfect isotropic and linear fluids. By using a power-law ansatz for the coframe components, we find several classes of new non-trivial teleparallel F(T) solutions. We also find a new class of teleparallel F(T) solutions for a matter dust fluid. After, we solve the field equations for a non-linear perfect fluid. Once again, there are several new exact teleparallel F(T) solutions and also some approximated teleparallel F(T) solutions. All these classes of new solutions may be relevant for future cosmological and astrophysical applications. Full article
(This article belongs to the Special Issue Advances in Differential Geometry and Mathematical Physics)
12 pages, 291 KiB  
Article
Eigenvalue of (p,q)-Biharmonic System along the Ricci Flow
by Lixu Yan, Yanlin Li, Apurba Saha, Abimbola Abolarinwa, Suraj Ghosh and Shyamal Kumar Hui
Axioms 2024, 13(5), 332; https://doi.org/10.3390/axioms13050332 - 17 May 2024
Viewed by 720
Abstract
In this paper, we determine the variation formula for the first eigenvalue of (p,q)-biharmonic system on a closed Riemannian manifold. Several monotonic quantities are also derived. Full article
(This article belongs to the Special Issue Advances in Differential Geometry and Singularity Theory)
16 pages, 281 KiB  
Article
Solving Nonlinear Second-Order ODEs via the Eisenhart Lift and Linearization
by Andronikos Paliathanasis
Axioms 2024, 13(5), 331; https://doi.org/10.3390/axioms13050331 - 16 May 2024
Cited by 1 | Viewed by 627
Abstract
The linearization of nonlinear differential equations represents a robust approach to solution derivation, typically achieved through Lie symmetry analysis. This study adopts a geometric methodology grounded in the Eisenhart lift, revealing transformative techniques that linearize a set of second-order ordinary differential equations. The [...] Read more.
The linearization of nonlinear differential equations represents a robust approach to solution derivation, typically achieved through Lie symmetry analysis. This study adopts a geometric methodology grounded in the Eisenhart lift, revealing transformative techniques that linearize a set of second-order ordinary differential equations. The research underscores the effectiveness of this geometric approach in the linearization of a class of Newtonian systems that cannot be linearized through symmetry analysis. Full article
(This article belongs to the Special Issue Differential Equations and Its Application)
13 pages, 506 KiB  
Article
Extremal Bicyclic Graphs with Respect to Permanental Sums and Hosoya Indices
by Tingzeng Wu, Yinggang Bai and Shoujun Xu
Axioms 2024, 13(5), 330; https://doi.org/10.3390/axioms13050330 - 16 May 2024
Viewed by 625
Abstract
Graph polynomials is one of the important research directions in mathematical chemistry. The coefficients of some graph polynomials, such as matching polynomial and permanental polynomial, are related to structural properties of graphs. The Hosoya index of a graph is the sum of the [...] Read more.
Graph polynomials is one of the important research directions in mathematical chemistry. The coefficients of some graph polynomials, such as matching polynomial and permanental polynomial, are related to structural properties of graphs. The Hosoya index of a graph is the sum of the absolute value of all coefficients for the matching polynomial. And the permanental sum of a graph is the sum of the absolute value of all coefficients of the permanental polynomial. In this paper, we characterize the second to sixth minimal Hosoya indices of all bicyclic graphs. Furthermore, using the results, the second to sixth minimal permanental sums of all bicyclic graphs are also characterized. Full article
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<p>Bicyclic graphs <math display="inline"><semantics> <mrow> <msub> <mi>B</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>B</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo>,</mo> <mi>r</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>B</mi> <mn>3</mn> </msub> <mrow> <mo>(</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo>,</mo> <mi>r</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>.</p> Full article ">Figure 2
<p>Graphs <span class="html-italic">G</span> and <math display="inline"><semantics> <msup> <mi>G</mi> <mo>*</mo> </msup> </semantics></math>.</p> Full article ">Figure 3
<p>Graphs <span class="html-italic">G</span>, <math display="inline"><semantics> <msubsup> <mi>G</mi> <mrow> <mn>1</mn> </mrow> <mo>*</mo> </msubsup> </semantics></math> and <math display="inline"><semantics> <msubsup> <mi>G</mi> <mrow> <mn>2</mn> </mrow> <mo>*</mo> </msubsup> </semantics></math>.</p> Full article ">Figure 4
<p>Graphs <span class="html-italic">G</span> and <math display="inline"><semantics> <msup> <mi>G</mi> <mo>*</mo> </msup> </semantics></math>.</p> Full article ">Figure 5
<p>Graphs <span class="html-italic">G</span> and <math display="inline"><semantics> <msup> <mi>G</mi> <mo>*</mo> </msup> </semantics></math>.</p> Full article ">Figure 6
<p>Graphs <span class="html-italic">G</span> and <math display="inline"><semantics> <msup> <mi>G</mi> <mo>*</mo> </msup> </semantics></math>.</p> Full article ">Figure 7
<p>Graphs <span class="html-italic">G</span> and <math display="inline"><semantics> <msup> <mi>G</mi> <mo>*</mo> </msup> </semantics></math>.</p> Full article ">Figure 8
<p>Graphs <math display="inline"><semantics> <mrow> <msubsup> <mi>B</mi> <mrow> <mn>1</mn> </mrow> <mn>1</mn> </msubsup> <mrow> <mo>(</mo> <mn>3</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mi>n</mi> <mo>−</mo> <mn>5</mn> <mo>)</mo> </mrow> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msubsup> <mi>B</mi> <mrow> <mn>2</mn> </mrow> <mn>1</mn> </msubsup> <mrow> <mo>(</mo> <mn>3</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mi>n</mi> <mo>−</mo> <mn>6</mn> <mo>)</mo> </mrow> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msubsup> <mi>B</mi> <mrow> <mn>3</mn> </mrow> <mn>1</mn> </msubsup> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mi>n</mi> <mo>−</mo> <mn>4</mn> <mo>)</mo> </mrow> </mrow> </semantics></math>.</p> Full article ">Figure 9
<p>Graphs <math display="inline"><semantics> <mrow> <msubsup> <mi>M</mi> <mrow> <mn>1</mn> </mrow> <mn>1</mn> </msubsup> <mrow> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mi>n</mi> <mo>−</mo> <mn>5</mn> <mo>)</mo> </mrow> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msubsup> <mi>M</mi> <mrow> <mn>1</mn> </mrow> <mn>2</mn> </msubsup> <mrow> <mo>(</mo> <mi>s</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msubsup> <mi>M</mi> <mrow> <mn>1</mn> </mrow> <mn>3</mn> </msubsup> <mrow> <mo>(</mo> <mi>n</mi> <mo>−</mo> <mn>6</mn> <mo>)</mo> </mrow> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msubsup> <mi>M</mi> <mrow> <mn>1</mn> </mrow> <mn>4</mn> </msubsup> <mrow> <mo>(</mo> <mi>s</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>.</p> Full article ">Figure 10
<p>Graphs <math display="inline"><semantics> <mrow> <msub> <mi>H</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>n</mi> <mo>−</mo> <mn>5</mn> <mo>)</mo> </mrow> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>H</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>s</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>H</mi> <mn>3</mn> </msub> <mrow> <mo>(</mo> <mi>s</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>H</mi> <mn>4</mn> </msub> <mrow> <mo>(</mo> <mi>s</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>H</mi> <mn>5</mn> </msub> <mrow> <mo>(</mo> <mi>n</mi> <mo>−</mo> <mn>5</mn> <mo>)</mo> </mrow> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>H</mi> <mn>6</mn> </msub> <mrow> <mo>(</mo> <mi>n</mi> <mo>−</mo> <mn>4</mn> <mo>)</mo> </mrow> </mrow> </semantics></math>.</p> Full article ">Figure 11
<p>Graphs <math display="inline"><semantics> <mrow> <msubsup> <mi>H</mi> <mrow> <mn>1</mn> </mrow> <mn>1</mn> </msubsup> <mrow> <mo>(</mo> <mi>s</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msubsup> <mi>H</mi> <mrow> <mn>1</mn> </mrow> <mn>2</mn> </msubsup> <mrow> <mo>(</mo> <mi>s</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msubsup> <mi>H</mi> <mrow> <mn>1</mn> </mrow> <mn>3</mn> </msubsup> <mrow> <mo>(</mo> <mi>s</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msubsup> <mi>H</mi> <mrow> <mn>1</mn> </mrow> <mn>4</mn> </msubsup> <mrow> <mo>(</mo> <mi>n</mi> <mo>−</mo> <mn>6</mn> <mo>)</mo> </mrow> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msubsup> <mi>H</mi> <mrow> <mn>1</mn> </mrow> <mn>5</mn> </msubsup> <mrow> <mo>(</mo> <mi>n</mi> <mo>−</mo> <mn>5</mn> <mo>)</mo> </mrow> </mrow> </semantics></math>.</p> Full article ">Figure 12
<p>Graphs <math display="inline"><semantics> <mrow> <msubsup> <mi>H</mi> <mrow> <mn>4</mn> </mrow> <mn>1</mn> </msubsup> <mrow> <mo>(</mo> <mi>s</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msubsup> <mi>H</mi> <mrow> <mn>4</mn> </mrow> <mn>2</mn> </msubsup> <mrow> <mo>(</mo> <mi>s</mi> <mo>,</mo> <mi>t</mi> <mo>,</mo> <mi>u</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msubsup> <mi>H</mi> <mrow> <mn>4</mn> </mrow> <mn>3</mn> </msubsup> <mrow> <mo>(</mo> <mi>s</mi> <mo>,</mo> <mi>t</mi> <mo>,</mo> <mi>u</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msubsup> <mi>H</mi> <mrow> <mn>4</mn> </mrow> <mn>4</mn> </msubsup> <mrow> <mo>(</mo> <mi>s</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msubsup> <mi>H</mi> <mrow> <mn>4</mn> </mrow> <mn>5</mn> </msubsup> <mrow> <mo>(</mo> <mi>s</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>.</p> Full article ">Figure 13
<p>Graphs <math display="inline"><semantics> <mrow> <msubsup> <mi>H</mi> <mrow> <mn>6</mn> </mrow> <mn>1</mn> </msubsup> <mrow> <mo>(</mo> <mi>t</mi> <mo>,</mo> <mi>u</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msubsup> <mi>H</mi> <mrow> <mn>6</mn> </mrow> <mn>2</mn> </msubsup> <mrow> <mo>(</mo> <mi>t</mi> <mo>,</mo> <mi>u</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msubsup> <mi>H</mi> <mrow> <mn>6</mn> </mrow> <mn>3</mn> </msubsup> <mrow> <mo>(</mo> <mi>t</mi> <mo>,</mo> <mi>u</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msubsup> <mi>H</mi> <mrow> <mn>6</mn> </mrow> <mn>4</mn> </msubsup> <mrow> <mo>(</mo> <mi>n</mi> <mo>−</mo> <mn>5</mn> <mo>)</mo> </mrow> </mrow> </semantics></math>.</p> Full article ">
19 pages, 1128 KiB  
Article
Nonuniform Sampling in Lp-Subspaces Associated with the Multi-Dimensional Special Affine Fourier Transform
by Yingchun Jiang and Jing Yang
Axioms 2024, 13(5), 329; https://doi.org/10.3390/axioms13050329 - 15 May 2024
Viewed by 615
Abstract
In this paper, the sampling and reconstruction problems in function subspaces of Lp(Rn) associated with the multi-dimensional special affine Fourier transform (SAFT) are discussed. First, we give the definition of the multi-dimensional SAFT and study its properties including [...] Read more.
In this paper, the sampling and reconstruction problems in function subspaces of Lp(Rn) associated with the multi-dimensional special affine Fourier transform (SAFT) are discussed. First, we give the definition of the multi-dimensional SAFT and study its properties including the Parseval’s relation, the canonical convolution theorems and the chirp-modulation periodicity. Then, a kind of function spaces are defined by the canonical convolution in the multi-dimensional SAFT domain, the existence and the properties of the dual basis functions are demonstrated, and the Lp-stability of the basis functions is established. Finally, based on the nonuniform samples taken on a dense set, we propose an iterative reconstruction algorithm with exponential convergence to recover the signals in a Lp-subspace associated with the multi-dimensional SAFT, and the validity of the algorithm is demonstrated via simulations. Full article
(This article belongs to the Special Issue Advances in Mathematical Methods in Signal Processing and Its Applications)
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<p>The real and imaginary parts of <span class="html-italic">f</span>.</p> Full article ">Figure 2
<p>The real and imaginary parts of the SAFT of <span class="html-italic">f</span>.</p> Full article ">Figure 3
<p>The real and imaginary parts after sampling <span class="html-italic">f</span>.</p> Full article ">Figure 4
<p>The real and imaginary parts of the reconstructed signal.</p> Full article ">
20 pages, 338 KiB  
Article
Coercive and Noncoercive Mixed Generalized Complementarity Problems
by Ram N. Mohapatra, Bijaya K. Sahu and Gayatri Pany
Axioms 2024, 13(5), 328; https://doi.org/10.3390/axioms13050328 - 15 May 2024
Viewed by 707
Abstract
Impressed with the very recent developments of noncoercive complementarity problems and the use of recession sets in complementarity problems, here, we discuss mixed generalized complementarity problems in Hausdorff topological vector spaces. We used the Tikhonov regularization procedure, as well as arguments from the [...] Read more.
Impressed with the very recent developments of noncoercive complementarity problems and the use of recession sets in complementarity problems, here, we discuss mixed generalized complementarity problems in Hausdorff topological vector spaces. We used the Tikhonov regularization procedure, as well as arguments from the recession analysis, to establish the existence of solutions for mixed generalized complementarity problems without coercivity assumptions in Banach spaces. Full article
(This article belongs to the Special Issue Large-Scale Optimization and Variational Inequalities: Theory and Applications)
5 pages, 179 KiB  
Editorial
Mathematical Methods in Applied Sciences
by Nuno R. O. Bastos and Touria Karite
Axioms 2024, 13(5), 327; https://doi.org/10.3390/axioms13050327 - 15 May 2024
Viewed by 657
Abstract
In this editorial, we introduce “Mathematical Methods in Applied Sciences”, a Special Issue of Axioms comprising 17 articles [...] Full article
(This article belongs to the Special Issue Mathematical Methods in the Applied Sciences)
20 pages, 280 KiB  
Article
Tractability of Multivariate Approximation Problem on Euler and Wiener Integrated Processes
by Jie Zhang
Axioms 2024, 13(5), 326; https://doi.org/10.3390/axioms13050326 - 15 May 2024
Viewed by 572
Abstract
This paper examines the tractability of multivariate approximation problems under the normalized error criterion for a zero-mean Gaussian measure in an average-case setting. The Gaussian measure is associated with a covariance kernel, which is represented by the tensor product of one-dimensional kernels corresponding [...] Read more.
This paper examines the tractability of multivariate approximation problems under the normalized error criterion for a zero-mean Gaussian measure in an average-case setting. The Gaussian measure is associated with a covariance kernel, which is represented by the tensor product of one-dimensional kernels corresponding to Euler and Wiener integrated processes with non-negative and nondecreasing smoothness parameters {rd}d∈N. We give matching sufficient and necessary conditions for various concepts of tractability in terms of the asymptotic properties of the regularity parameters, except for (s, 0)-WT. Full article
(This article belongs to the Special Issue Numerical Computation, Approximation of Functions and Applied Mathematics, 2nd Edition)
20 pages, 319 KiB  
Article
Multiplicity of Solutions for the Noncooperative Kirchhoff-Type Variable Exponent Elliptic System with Nonlinear Boundary Conditions
by Yiying Mao and Yang Yang
Axioms 2024, 13(5), 325; https://doi.org/10.3390/axioms13050325 - 14 May 2024
Viewed by 594
Abstract
Considering the solutions of a class of noncooperative Kirchhoff-type p(x)-Laplacian elliptic systems with nonlinear boundary conditions, we derive a sequence of solutions utilizing both the variational method and limit index theory under certain underlying assumptions. The novelty of this [...] Read more.
Considering the solutions of a class of noncooperative Kirchhoff-type p(x)-Laplacian elliptic systems with nonlinear boundary conditions, we derive a sequence of solutions utilizing both the variational method and limit index theory under certain underlying assumptions. The novelty of this study is that we verify the (PS)c* condition using another method, diverging from the approaches cited in the previous literature. Full article
(This article belongs to the Special Issue Differential Equations and Related Topics)
20 pages, 1388 KiB  
Article
A Vector Representation of Multicomplex Numbers and Its Application to Radio Frequency Signals
by Daniele Borio
Axioms 2024, 13(5), 324; https://doi.org/10.3390/axioms13050324 - 14 May 2024
Viewed by 640
Abstract
Hypercomplex numbers, which are multi-dimensional extensions of complex numbers, have been proven beneficial in the development of advanced signal processing algorithms, including multi-dimensional filter design, linear regression and classification. We focus on multicomplex numbers, sets of hypercomplex numbers with commutative products, and introduce [...] Read more.
Hypercomplex numbers, which are multi-dimensional extensions of complex numbers, have been proven beneficial in the development of advanced signal processing algorithms, including multi-dimensional filter design, linear regression and classification. We focus on multicomplex numbers, sets of hypercomplex numbers with commutative products, and introduce a vector representation allowing one to isolate the hyperbolic real and imaginary parts of a multicomplex number. The orthogonal decomposition of a multicomplex number is also discussed, and its connection with Hadamard matrices is highlighted. Finally, a multicomplex polar representation is provided. These properties are used to extend the standard complex baseband signal representation to the multi-dimensional case. It is shown that a set of 2n Radio Frequency (RF) signals can be represented as the real part of a single multicomplex signal modulated by several frequencies. The signal RFs are related through a Hadamard matrix to the modulating frequencies adopted in the multicomplex baseband representation. Moreover, an orthogonal decomposition is provided for the obtained multicomplex baseband signal as a function of the complex baseband representations of the input RF signals. Full article
(This article belongs to the Special Issue Advances in Mathematical Methods in Signal Processing and Its Applications)
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<p>Schematic representation of the Galileo OS frequency plan (top part) and representation of the associated RF up-conversion process as a multicomplex modulation.</p> Full article ">Figure 2
<p>Scatter plots of four baseband QPSK signals affected by noise. The signals are used as orthogonal components to form a single multicomplex signal <math display="inline"><semantics> <mrow> <mo>∈</mo> <msub> <mi mathvariant="double-struck">C</mi> <mn>3</mn> </msub> </mrow> </semantics></math> according to (<a href="#FD93-axioms-13-00324" class="html-disp-formula">93</a>).</p> Full article ">Figure 3
<p>Scatter plots of the four complex components of <math display="inline"><semantics> <mrow> <msub> <mi>z</mi> <mi>n</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>. <math display="inline"><semantics> <mrow> <msub> <mi>z</mi> <mi>n</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> is expressed as <math display="inline"><semantics> <mrow> <msubsup> <mi mathvariant="bold">k</mi> <mn>3</mn> <mi>T</mi> </msubsup> <msub> <mi mathvariant="bold">z</mi> <mi>M</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> with <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="bold">z</mi> <mi>M</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> a four-dimensional complex vector. Each subplot represents a different component of <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="bold">z</mi> <mi>M</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>.</p> Full article ">Figure 4
<p>Hyperbolic modulus of the multicomplex signal obtained in <a href="#axioms-13-00324-f003" class="html-fig">Figure 3</a>. The four real components of the hyperbolic modulus are depicted separately as a function of time.</p> Full article ">
14 pages, 684 KiB  
Article
Quantum Vision Transformers for Quark–Gluon Classification
by Marçal Comajoan Cara, Gopal Ramesh Dahale, Zhongtian Dong, Roy T. Forestano, Sergei Gleyzer, Daniel Justice, Kyoungchul Kong, Tom Magorsch, Konstantin T. Matchev, Katia Matcheva and Eyup B. Unlu
Axioms 2024, 13(5), 323; https://doi.org/10.3390/axioms13050323 - 13 May 2024
Viewed by 1363
Abstract
We introduce a hybrid quantum-classical vision transformer architecture, notable for its integration of variational quantum circuits within both the attention mechanism and the multi-layer perceptrons. The research addresses the critical challenge of computational efficiency and resource constraints in analyzing data from the upcoming [...] Read more.
We introduce a hybrid quantum-classical vision transformer architecture, notable for its integration of variational quantum circuits within both the attention mechanism and the multi-layer perceptrons. The research addresses the critical challenge of computational efficiency and resource constraints in analyzing data from the upcoming High Luminosity Large Hadron Collider, presenting the architecture as a potential solution. In particular, we evaluate our method by applying the model to multi-detector jet images from CMS Open Data. The goal is to distinguish quark-initiated from gluon-initiated jets. We successfully train the quantum model and evaluate it via numerical simulations. Using this approach, we achieve classification performance almost on par with the one obtained with the completely classical architecture, considering a similar number of parameters. Full article
(This article belongs to the Section Mathematical Analysis)
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<p>The CMS coordinates the system against the backdrop of the LHC, with the location of the four main experiments (CMS, ALICE, ATLAS, and LHCb). The <span class="html-italic">z</span> axis points to the Jura mountains, while the <span class="html-italic">y</span>-axis points toward the sky. In spherical coordinates, the components of a particle momentum <math display="inline"><semantics> <mover accent="true"> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </semantics></math> are its magnitude <math display="inline"><semantics> <mrow> <mrow> <mo>|</mo> </mrow> <mover accent="true"> <mi>p</mi> <mo stretchy="false">→</mo> </mover> <mrow> <mo>|</mo> </mrow> </mrow> </semantics></math>, the polar angle <math display="inline"><semantics> <mi>θ</mi> </semantics></math> (measured from the <span class="html-italic">z</span>-axis), and the azimuthal angle <math display="inline"><semantics> <mi>φ</mi> </semantics></math> (measured from the <span class="html-italic">x</span>-axis). The transverse momentum <math display="inline"><semantics> <msub> <mover accent="true"> <mi>p</mi> <mo stretchy="false">→</mo> </mover> <mi>T</mi> </msub> </semantics></math> is the projection of <math display="inline"><semantics> <mover accent="true"> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </semantics></math> on the transverse (<math display="inline"><semantics> <mrow> <mi>x</mi> <mi>y</mi> </mrow> </semantics></math>) plane. This figure was generated with TikZ code adapted from Ref. [<a href="#B58-axioms-13-00323" class="html-bibr">58</a>].</p> Full article ">Figure 2
<p>Representative images of jets for both quarks (<b>top</b>) and gluons (<b>bottom</b>). The columns show the distinct sub-detectors: Tracks, ECAL, HCAL, and a composite image combining all three. All images are in log scale. Note that the ECAL and HCAL were upscaled to match the Tracks resolution.</p> Full article ">Figure 3
<p>Average images of quarks (<b>top</b>) and gluons (<b>bottom</b>) across the entire dataset. The columns show the distinct sub-detectors: Tracks, ECAL, HCAL, and a composite image combining all three. All images are in log scale. Note the more dispersed nature of the gluon jets across channels.</p> Full article ">Figure 4
<p>Model overview. QMHA stands for quantum multi-head attention and QMLP for quantum multi-layer perceptron. The drawing style of the illustration was inspired by Dosovitskiy et al. [<a href="#B12-axioms-13-00323" class="html-bibr">12</a>], the major difference being that here we use a quantum transformer encoder as depicted in the right side of the figure.</p> Full article ">Figure 5
<p>Variational quantum circuits used in the proposed QViT.</p> Full article ">Figure 6
<p>Binary cross-entropy loss evolution during training, computed at the end of each epoch on the training (dashed lines) and validation (solid lines) sets for both the baseline classical ViT (orange lines) and our hybrid QViT (purple lines).</p> Full article ">Figure 7
<p>AUC score evolution during training computed at the end of each epoch on the training (dashed lines) and validation (solid lines) sets for both the baseline classical ViT (orange lines) and our hybrid QViT (purple lines).</p> Full article ">Figure 8
<p>Receiver Operating Characteristic (ROC) curves for the baseline classical ViT (orange line) and our hybrid QViT (purple line). The black dashed line represents the performance of a random classifier.</p> Full article ">
17 pages, 804 KiB  
Article
A C0 Nonconforming Virtual Element Method for the Kirchhoff Plate Obstacle Problem
by Bangmin Wu and Jiali Qiu
Axioms 2024, 13(5), 322; https://doi.org/10.3390/axioms13050322 - 13 May 2024
Viewed by 738
Abstract
This paper investigates a novel C0 nonconforming virtual element method (VEM) for solving the Kirchhoff plate obstacle problem, which is described by a fourth-order variational inequality (VI) of the first kind. In our study, we distinguish our approach by introducing new internal [...] Read more.
This paper investigates a novel C0 nonconforming virtual element method (VEM) for solving the Kirchhoff plate obstacle problem, which is described by a fourth-order variational inequality (VI) of the first kind. In our study, we distinguish our approach by introducing new internal degrees of freedom to the traditional lowest-order C0 nonconforming VEM, which originally lacked such degrees. This addition not only facilitates error estimation but also enhances its intuitiveness. Importantly, our novel C0 nonconforming VEM naturally satisfies the constraints of the obstacle problem. We then establish an a priori error estimate for our novel C0 nonconforming VEM, with the result indicating that the lowest order of our method achieves optimal convergence. Finally, we present a numerical example to validate the theoretical result. Full article
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<p>The obstacle problem <span class="html-small-caps">P</span>.</p> Full article ">Figure 2
<p>The DOFs of the lowest-order <math display="inline"><semantics> <msup> <mi>C</mi> <mn>0</mn> </msup> </semantics></math> nonconforming VE on <math display="inline"><semantics> <msubsup> <mi>V</mi> <mi>h</mi> <mi>T</mi> </msubsup> </semantics></math>.</p> Full article ">Figure 3
<p>Relative errors of rectangular mesh and polygon mesh.</p> Full article ">Figure 4
<p>The numerical solution and exact solution.</p> Full article ">Figure 5
<p>The numerical solution minus the value of the obstacle function <math display="inline"><semantics> <mi>ξ</mi> </semantics></math>.</p> Full article ">
16 pages, 1807 KiB  
Article
Quasi-Configurations Derived by Special Arrangements of Lines
by Stefano Innamorati
Axioms 2024, 13(5), 321; https://doi.org/10.3390/axioms13050321 - 11 May 2024
Viewed by 682
Abstract
A quasi-configuration is a point–line incidence structure in which each point is incident with at least three lines and each line is incident with at least three points. We investigate derived quasi-configurations that arise both by duality and intersecting lines of three special [...] Read more.
A quasi-configuration is a point–line incidence structure in which each point is incident with at least three lines and each line is incident with at least three points. We investigate derived quasi-configurations that arise both by duality and intersecting lines of three special arrangements of lines. Sets with few intersection numbers are provided. Full article
(This article belongs to the Special Issue Theory of Curves and Knots with Applications)
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<p>The Hesse (9<sub>4</sub>,12<sub>3</sub>) configuration.</p> Full article ">Figure 2
<p>The Hesse (9<sub>4</sub>,12<sub>3</sub>) configuration with the 12 intersection points of the parallel lines.</p> Full article ">Figure 3
<p>The 12 intersection points of the parallel lines are contained in the nine lines of the Ceva arrangement.</p> Full article ">Figure 4
<p>The projective plane of order four.</p> Full article ">
13 pages, 286 KiB  
Article
Stability of the Borell–Brascamp–Lieb Inequality for Multiple Power Concave Functions
by Meng Qin, Zhuohua Zhang, Rui Luo, Mengjie Ren and Denghui Wu
Axioms 2024, 13(5), 320; https://doi.org/10.3390/axioms13050320 - 11 May 2024
Viewed by 704
Abstract
In this paper, we prove the stability of the Brunn–Minkowski inequality for multiple convex bodies in terms of the concept of relative asymmetry. Using these stability results and the relationship of the compact support of functions, we establish the stability of the Borell–Brascamp–Lieb [...] Read more.
In this paper, we prove the stability of the Brunn–Minkowski inequality for multiple convex bodies in terms of the concept of relative asymmetry. Using these stability results and the relationship of the compact support of functions, we establish the stability of the Borell–Brascamp–Lieb inequality for multiple power concave functions via relative asymmetry. Full article
(This article belongs to the Special Issue Advances in Convex Geometry and Analysis)
16 pages, 573 KiB  
Article
Strategic Behavior and Optimal Inventory Level in a Make-to-Stock Queueing System with Retrial Customers
by Yuejiao Wang and Chenguang Cai
Axioms 2024, 13(5), 319; https://doi.org/10.3390/axioms13050319 - 11 May 2024
Viewed by 646
Abstract
In this article, we consider a make-to-stock queueing system with retrial customers. Upon their arrival, customers make a decision to either join the system or not based on a reward–cost function. If customers join the retrial queue, they become repeat customers. Each repeat [...] Read more.
In this article, we consider a make-to-stock queueing system with retrial customers. Upon their arrival, customers make a decision to either join the system or not based on a reward–cost function. If customers join the retrial queue, they become repeat customers. Each repeat customer repeats their demand after an exponential amount of time until they have been successfully served. We explore the equilibrium strategies of customers in both the almost observable and unobservable cases. Furthermore, we also analyze the expected costs of the entire system based on the customers’ behavior in these two cases. Additionally, we determine the optimal inventory levels in both cases through numerical experiments. Full article
(This article belongs to the Section Mathematical Analysis)
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<p>State transition diagram for the original model.</p> Full article ">Figure 2
<p>State transition diagram for the almost observable case.</p> Full article ">Figure 3
<p>State transition diagram for the unobservable case.</p> Full article ">Figure 4
<p>Above: the expected cost function of the entire system in the almost observable case vs. <math display="inline"><semantics> <mi>μ</mi> </semantics></math> for <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>8</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>q</mi> <mo>=</mo> <mn>0.8</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>. Below: the expected cost function of the entire system in the almost observable case vs. <math display="inline"><semantics> <mi>α</mi> </semantics></math> for <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>8</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>μ</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>q</mi> <mo>=</mo> <mn>0.8</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>.</p> Full article ">Figure 5
<p>Above: the expected cost function of the entire system in the almost observable case vs. <span class="html-italic">q</span> for <math display="inline"><semantics> <mrow> <mi>μ</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>8</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>. Below: the expected cost function of the entire system in the almost observable case vs. <math display="inline"><semantics> <msub> <mi>C</mi> <mn>1</mn> </msub> </semantics></math> for <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>8</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>μ</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>q</mi> <mo>=</mo> <mn>0.8</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>.</p> Full article ">Figure 5 Cont.
<p>Above: the expected cost function of the entire system in the almost observable case vs. <span class="html-italic">q</span> for <math display="inline"><semantics> <mrow> <mi>μ</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>8</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>. Below: the expected cost function of the entire system in the almost observable case vs. <math display="inline"><semantics> <msub> <mi>C</mi> <mn>1</mn> </msub> </semantics></math> for <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>8</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>μ</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>q</mi> <mo>=</mo> <mn>0.8</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>.</p> Full article ">Figure 6
<p>Above: the expected cost function of the entire system in the unobservable case vs. <math display="inline"><semantics> <mi>μ</mi> </semantics></math> for <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>8</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>q</mi> <mo>=</mo> <mn>0.8</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>. Below: the expected cost function of the entire system in the unobservable case vs. <math display="inline"><semantics> <mi>α</mi> </semantics></math> for <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>8</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>μ</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>q</mi> <mo>=</mo> <mn>0.8</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>.</p> Full article ">Figure 6 Cont.
<p>Above: the expected cost function of the entire system in the unobservable case vs. <math display="inline"><semantics> <mi>μ</mi> </semantics></math> for <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>8</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>q</mi> <mo>=</mo> <mn>0.8</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>. Below: the expected cost function of the entire system in the unobservable case vs. <math display="inline"><semantics> <mi>α</mi> </semantics></math> for <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>8</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>μ</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>q</mi> <mo>=</mo> <mn>0.8</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>.</p> Full article ">Figure 7
<p>Above: the expected cost function of the entire system in the unobservable case vs. <span class="html-italic">q</span> for <math display="inline"><semantics> <mrow> <mi>μ</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>8</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>. Below: the expected cost function of the entire system in the unobservable case vs. <math display="inline"><semantics> <msub> <mi>C</mi> <mn>1</mn> </msub> </semantics></math> for <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>8</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>μ</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>q</mi> <mo>=</mo> <mn>0.8</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>.</p> Full article ">Figure 8
<p>The expected cost function of the entire system in two cases vs. <span class="html-italic">N</span> for <math display="inline"><semantics> <mrow> <mi>μ</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>8</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>q</mi> <mo>=</mo> <mn>0.8</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>.</p> Full article ">
15 pages, 267 KiB  
Article
Quasi-Contraction Maps in Subordinate Semimetric Spaces
by Areej Alharbi, Hamed Alsulami and Maha Noorwali
Axioms 2024, 13(5), 318; https://doi.org/10.3390/axioms13050318 - 10 May 2024
Viewed by 650
Abstract
Throughout this study, we discuss the subordinate Pompeiu–Hausdorff metric (SPHM) in subordinate semimetric spaces. Moreover, we present a well-behaved quasi-contraction (WBQC) to solve quasi-contraction (QC) problems in subordinate semimetric spaces under some local constraints. Furthermore, we provide examples to support our conclusion. Full article
(This article belongs to the Special Issue Research on Fixed Point Theory and Application)
24 pages, 398 KiB  
Article
A New Closed-Form Formula of the Gauss Hypergeometric Function at Specific Arguments
by Yue-Wu Li and Feng Qi
Axioms 2024, 13(5), 317; https://doi.org/10.3390/axioms13050317 - 10 May 2024
Cited by 4 | Viewed by 1141
Abstract
In this paper, the authors briefly review some closed-form formulas of the Gauss hypergeometric function at specific arguments, alternatively prove four of these formulas, newly extend a closed-form formula of the Gauss hypergeometric function at some specific arguments, successfully apply a special case [...] Read more.
In this paper, the authors briefly review some closed-form formulas of the Gauss hypergeometric function at specific arguments, alternatively prove four of these formulas, newly extend a closed-form formula of the Gauss hypergeometric function at some specific arguments, successfully apply a special case of the newly extended closed-form formula to derive an alternative form for the Maclaurin power series expansion of the Wilf function, and discover two novel increasing rational approximations to a quarter of the circular constant. Full article
23 pages, 353 KiB  
Article
Convergence Results for History-Dependent Variational Inequalities
by Mircea Sofonea and Domingo A. Tarzia
Axioms 2024, 13(5), 316; https://doi.org/10.3390/axioms13050316 - 10 May 2024
Viewed by 688
Abstract
We consider a history-dependent variational inequality in a real Hilbert space, for which we recall an existence and uniqueness result. We associate this inequality with a gap function, together with two additional problems: a nonlinear equation and a minimization problem. Then, we prove [...] Read more.
We consider a history-dependent variational inequality in a real Hilbert space, for which we recall an existence and uniqueness result. We associate this inequality with a gap function, together with two additional problems: a nonlinear equation and a minimization problem. Then, we prove that solving these problems is equivalent to solving the original history-dependent variational inequality. Next, we state and prove a convergence criterion, i.e., we provide necessary and sufficient conditions which guarantee the convergence of a sequence of functions to the solution of the considered inequality. Based on the equivalence above, we deduce various consequences that present some interest on their own, and, moreover, we obtain convergence results for the two additional problems considered. Finally, we apply our abstract results to the study of an inequality problem in solid mechanics. It concerns the study of a viscoelastic constitutive law with long memory and unilateral constraints, for which we deduce a convergence result and provide the corresponding mechanical interpretations. Full article
(This article belongs to the Section Hilbert’s Sixth Problem)
20 pages, 334 KiB  
Article
Parameter Estimation in Spatial Autoregressive Models with Missing Data and Measurement Errors
by Tengjun Li, Zhikang Zhang and Yunquan Song
Axioms 2024, 13(5), 315; https://doi.org/10.3390/axioms13050315 - 10 May 2024
Viewed by 944
Abstract
This study addresses the problem of parameter estimation in spatial autoregressive models with missing data and measurement errors in covariates. Specifically, a corrected likelihood estimation approach is employed to rectify the bias in the log-maximum likelihood function induced by measurement errors. Additionally, a [...] Read more.
This study addresses the problem of parameter estimation in spatial autoregressive models with missing data and measurement errors in covariates. Specifically, a corrected likelihood estimation approach is employed to rectify the bias in the log-maximum likelihood function induced by measurement errors. Additionally, a combination of inverse probability weighting (IPW) and mean imputation is utilized to mitigate the bias caused by missing data. Under several mild conditions, it is demonstrated that the proposed estimators are consistent and possess oracle properties. The efficacy of the proposed parameter estimation process is assessed through Monte Carlo simulation studies. Finally, the applicability of the proposed method is further substantiated using the Boston Housing Dataset. Full article
(This article belongs to the Special Issue Mathematical and Statistical Methods and Their Applications)
15 pages, 266 KiB  
Article
Some Results on Zinbiel Algebras and Rota–Baxter Operators
by Jizhong Gao, Junna Ni and Jianhua Yu
Axioms 2024, 13(5), 314; https://doi.org/10.3390/axioms13050314 - 10 May 2024
Viewed by 672
Abstract
Rota–Baxter operators (RBOs) play a substantial role in many subfields of mathematics, especially in mathematical physics. In the article, RBOs on Zinbiel algebras (ZAs) and their sub-adjacent algebras are first investigated. Moreover, all the RBOs on two and three-dimensional ZAs are presented. Finally, [...] Read more.
Rota–Baxter operators (RBOs) play a substantial role in many subfields of mathematics, especially in mathematical physics. In the article, RBOs on Zinbiel algebras (ZAs) and their sub-adjacent algebras are first investigated. Moreover, all the RBOs on two and three-dimensional ZAs are presented. Finally, ZAs are also realized in low dimensions of the RBOs of commutative associative algebras. It was found that not all ZAs can be attained in this way. Full article
24 pages, 20089 KiB  
Article
Basic Computational Algorithms for Representing an Aircraft Flight (Calculation of 3D Displacement and Displaying)
by Adan Ramirez-Lopez
Axioms 2024, 13(5), 313; https://doi.org/10.3390/axioms13050313 - 10 May 2024
Cited by 1 | Viewed by 1112
Abstract
This manuscript describes the computational process to calculate an airplane path and display it in a 2D and 3D coordinate system on a computer screen. The airplane movement is calculated as a function of its dynamic’s conditions according to physical and logical theory. [...] Read more.
This manuscript describes the computational process to calculate an airplane path and display it in a 2D and 3D coordinate system on a computer screen. The airplane movement is calculated as a function of its dynamic’s conditions according to physical and logical theory. Here, the flight is divided into maneuvers and the aircraft conditions are defined as boundary conditions. Then the aircraft position is calculated using nested loops, which execute the calculation procedure at every step time (Δt). The calculation of the aircraft displacement is obtained as a function of the aircraft speed and heading angles. The simulator was created using the C++ programming language, and each part of the algorithm was compiled independently to reduce the source code, allow easy modification, and improve the programming efficiency. Aerial navigation involves very complex phenomena to be considered for an appropriate representation; moreover, in this manuscript, the influence of the mathematical approach to properly represent the aircraft flight is described in detail. The flight simulator was successfully tested by simulating some basic theoretical flights with different maneuvers, which include stationary position, running along the way, take off, and some movements in the airspace. The maximum aircraft speed tested was 120 km/h, the maximum maneuver time was 12 min, and the space for simulation was assumed to be without obstacles. Here, the geometrical description of path and speed is analyzed according to the symmetric and asymmetric results. Finally, an analysis was conducted to evaluate the approach of the numerical methods used; after that, it was possible to confirm that precision increased as the step time was reduced. According to this analysis, no more than 500 steps are required for a good approach in the calculation of the aircraft displacement. Full article
(This article belongs to the Special Issue Advancements in Applied Mathematics and Computational Physics)
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Figure 1

Figure 1
<p>Assumptions for the graphical representation of the surface and the heading angle (direction of flight) according with earth position.</p> Full article ">Figure 2
<p>General flowchart for the flight simulator developed.</p> Full article ">Figure 3
<p>Flowchart for the reading data routine for each maneuver.</p> Full article ">Figure 4
<p>Flowchart for selecting any option to display the results. (<b>a</b>) Option for display the aircraft path and (<b>b</b>) option for display graphics and performance of the aircraft.</p> Full article ">Figure 5
<p>Flowchart sentence to find the maximum and minimum values for the aircraft displacement along the (x) axis.</p> Full article ">Figure 6
<p>Computational representation of the aircraft path for flight (1) using different terrain angles. (<b>a</b>) Using 20° and (<b>b</b>) using 8°.</p> Full article ">Figure 7
<p>Aircraft performance graphics for flight (1) (<b>a</b>) Aircraft speed. (<b>b</b>) Aircraft displacement.</p> Full article ">Figure 8
<p>Computational representation of the aircraft path for flight (2) using different terrain angles. (<b>a</b>) Using 20° and (<b>b</b>) using 8°.</p> Full article ">Figure 9
<p>Aircraft performance graphics for flight (2). (<b>a</b>) Aircraft speed. (<b>b</b>) Aircraft displacement.</p> Full article ">Figure 9 Cont.
<p>Aircraft performance graphics for flight (2). (<b>a</b>) Aircraft speed. (<b>b</b>) Aircraft displacement.</p> Full article ">Figure 10
<p>Computational representation of the aircraft path for flight (3) using different terrain angles. (<b>a</b>) Using 20° and (<b>b</b>) using 5°.</p> Full article ">Figure 11
<p>Aircraft performance graphics for flight (3). (<b>a</b>) Aircraft speed. (<b>b</b>) Aircraft displacement.</p> Full article ">Figure 12
<p>Calculated approaching for flight (1) curves for axis (x) direction Sud-North. (<b>a</b>) considering displacement values. (<b>b</b>) considering the previous calculated value. (<b>c</b>) considering the division of the displacement between the number of steps. (<b>d</b>) considering the approaching to the final value.</p> Full article ">Figure 13
<p>Calculated approaching for flight (1) curves for axis (y) direction Est-West. (<b>a</b>) considering displacement values. (<b>b</b>) considering the previous calculated value. (<b>c</b>) considering the division of the displacement between the number of steps. (<b>d</b>) considering the approaching to the final value.</p> Full article ">Figure 14
<p>Calculated approaching for flight (1) curves for axis (z) this is for the altitude. (<b>a</b>) considering displacement values. (<b>b</b>) considering the previous calculated value. (<b>c</b>) considering the division of the displacement between the number of steps. (<b>d</b>) considering the approaching to the final value.</p> Full article ">Figure 15
<p>Calculated approaching for flight (1) curves for total distance measured for the original point assumed as the aircraft was in a stationary position at (0,0,0). (<b>a</b>) considering displacement values. (<b>b</b>) considering the previous calculated value. (<b>c</b>) considering the division of the displacement between the number of steps. (<b>d</b>) considering the approaching to the final value.</p> Full article ">Figure 16
<p>Reduction in the error calculated for flight (1) over every axis of displacement. Analysis for every displacement calculated (<b>a</b>) for Sud-North direction. (<b>b</b>) for Est-West direction. (<b>c</b>) for altitude. (<b>d</b>) for total displacement.</p> Full article ">Figure 16 Cont.
<p>Reduction in the error calculated for flight (1) over every axis of displacement. Analysis for every displacement calculated (<b>a</b>) for Sud-North direction. (<b>b</b>) for Est-West direction. (<b>c</b>) for altitude. (<b>d</b>) for total displacement.</p> Full article ">
20 pages, 347 KiB  
Article
Canonical Metrics on Twisted Quiver Bundles over a Class of Non-Compact Gauduchon Manifold
by Shi-Fan Cai, Sudhakar Kumar Chaubey, Xin Xu, Pan Zhang and Zhi-Heng Zhang
Axioms 2024, 13(5), 312; https://doi.org/10.3390/axioms13050312 - 9 May 2024
Viewed by 792
Abstract
The aim of this paper is to prove a theorem for holomorphic twisted quiver bundles over a special non-compact Gauduchon manifold, connecting the existence of (σ,τ)-Hermite–Yang–Mills metric in differential geometry and the analytic (σ,τ) [...] Read more.
The aim of this paper is to prove a theorem for holomorphic twisted quiver bundles over a special non-compact Gauduchon manifold, connecting the existence of (σ,τ)-Hermite–Yang–Mills metric in differential geometry and the analytic (σ,τ)-stability in algebraic geometry. The proof of the theorem relies on the flow method and the Uhlenbeck–Yau’s continuity method. Full article
(This article belongs to the Special Issue Advances in Differential Geometry and Mathematical Physics)
18 pages, 1228 KiB  
Article
Study of the Six-Compartment Nonlinear COVID-19 Model with the Homotopy Perturbation Method
by Muhammad Rafiullah, Muhammad Asif, Dure Jabeen and Mahmoud A. Ibrahim
Axioms 2024, 13(5), 311; https://doi.org/10.3390/axioms13050311 - 9 May 2024
Viewed by 1111
Abstract
The current study aims to utilize the homotopy perturbation method (HPM) to solve nonlinear dynamical models, with a particular focus on models related to predicting and controlling pandemics, such as the SIR model. Specifically, we apply this method to solve a six-compartment model [...] Read more.
The current study aims to utilize the homotopy perturbation method (HPM) to solve nonlinear dynamical models, with a particular focus on models related to predicting and controlling pandemics, such as the SIR model. Specifically, we apply this method to solve a six-compartment model for the novel coronavirus (COVID-19), which includes susceptible, exposed, asymptomatic infected, symptomatic infected, and recovered individuals, and the concentration of COVID-19 in the environment is indicated by S(t), E(t), A(t), I(t), R(t), and B(t), respectively. We present the series solution of this model by varying the controlling parameters and representing them graphically. Additionally, we verify the accuracy of the series solution (up to the (n−1)th-degree polynomial) that satisfies both the initial conditions and the model, with all coefficients correct at 18 decimal places. Furthermore, we have compared our results with the Runge–Kutta fourth-order method. Based on our findings, we conclude that the homotopy perturbation method is a promising approach to solve nonlinear dynamical models, particularly those associated with pandemics. This method provides valuable insight into how the control of various parameters can affect the model. We suggest that future studies can expand on our work by exploring additional models and assessing the applicability of other analytical methods. Full article
(This article belongs to the Special Issue Dynamical Systems: Theory and Applications in Mathematical Biology)
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Figure 1
<p>Total population and the concentration of the COVID-19 in the environment.</p> Full article ">Figure 2
<p>The impact of <math display="inline"><semantics> <msub> <mi>β</mi> <mn>1</mn> </msub> </semantics></math> on susceptible people.</p> Full article ">Figure 3
<p>The impact of <math display="inline"><semantics> <msub> <mi>β</mi> <mn>2</mn> </msub> </semantics></math> on susceptible people.</p> Full article ">Figure 4
<p>The impact of <math display="inline"><semantics> <msub> <mi>β</mi> <mn>3</mn> </msub> </semantics></math> on susceptible people.</p> Full article ">Figure 5
<p>The impact of <math display="inline"><semantics> <msub> <mi>β</mi> <mn>4</mn> </msub> </semantics></math> on susceptible people.</p> Full article ">Figure 6
<p>The impact of <math display="inline"><semantics> <msub> <mi>β</mi> <mn>1</mn> </msub> </semantics></math> exposed people.</p> Full article ">Figure 7
<p>The impact of <math display="inline"><semantics> <msub> <mi>β</mi> <mn>2</mn> </msub> </semantics></math> exposed people.</p> Full article ">Figure 8
<p>The impact of <math display="inline"><semantics> <msub> <mi>β</mi> <mn>3</mn> </msub> </semantics></math> exposed people.</p> Full article ">Figure 9
<p>The impact of <math display="inline"><semantics> <msub> <mi>β</mi> <mn>4</mn> </msub> </semantics></math> exposed people.</p> Full article ">Figure 10
<p>The impact of <math display="inline"><semantics> <msub> <mi>ψ</mi> <mn>1</mn> </msub> </semantics></math> on the concentration of COVID-19 in the environment.</p> Full article ">Figure 11
<p>The impact of <math display="inline"><semantics> <msub> <mi>ψ</mi> <mn>2</mn> </msub> </semantics></math> on the concentration of COVID-19 in the environment.</p> Full article ">Figure 12
<p>The impact of <math display="inline"><semantics> <msub> <mi>ψ</mi> <mn>3</mn> </msub> </semantics></math> on the concentration of COVID-19 in the environment.</p> Full article ">Figure 13
<p>The impact of <math display="inline"><semantics> <mi>ϕ</mi> </semantics></math> on the concentration of COVID-19 in the environment.</p> Full article ">
3 pages, 171 KiB  
Editorial
Recent Advances in Fractional Calculus
by Péter Kórus and Juan Eduardo Nápoles Valdés
Axioms 2024, 13(5), 310; https://doi.org/10.3390/axioms13050310 - 8 May 2024
Viewed by 752
Abstract
This Special Issue of the scientific journal Axioms, entitled “Recent Advances in Fractional Calculus”, is dedicated to one of the most dynamic areas of mathematical sciences today [...] Full article
(This article belongs to the Special Issue Recent Advances in Fractional Calculus)
17 pages, 966 KiB  
Article
Study on SEAI Model of COVID-19 Based on Asymptomatic Infection
by Lidong Huang, Yue Xia and Wenjie Qin
Axioms 2024, 13(5), 309; https://doi.org/10.3390/axioms13050309 - 8 May 2024
Cited by 1 | Viewed by 956
Abstract
In this paper, an SEAI epidemic model with asymptomatic infection is studied under the background of mass transmission of COVID-19. First, we use the next-generation matrix method to obtain the basic reproductive number R0 and calculate the equilibrium point. Secondly, when [...] Read more.
In this paper, an SEAI epidemic model with asymptomatic infection is studied under the background of mass transmission of COVID-19. First, we use the next-generation matrix method to obtain the basic reproductive number R0 and calculate the equilibrium point. Secondly, when R0<1, the local asymptotic stability of the disease-free equilibrium is proved by Hurwitz criterion, and the global asymptotic stability of the disease-free equilibrium is proved by constructing the Lyapunov function. When R0>1, the system has a unique endemic equilibrium point and is locally asymptotically stable, and it is also proved that the system is uniformly persistent. Then, the application of optimal control theory is carried out, and the expression of the optimal control solution is obtained. Finally, in order to verify the correctness of the theory, the stability of the equilibrium point is numerically simulated and the sensitivity of the parameters of R0 is analyzed. We also simulated the comparison of the number of asymptomatic infected people and symptomatic infected people before and after adopting the optimal control strategy. This shows that the infection of asymptomatic people cannot be underestimated in the spread of COVID-19 virus, and an isolation strategy should be adopted to control the spread speed of the disease. Full article
(This article belongs to the Special Issue Recent Advances in Mathematical Modeling of COVID-19 and Other Infectious Diseases)
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Figure 1

Figure 1
<p>Flow chart of SEAI transmission.</p> Full article ">Figure 2
<p>Time series of <math display="inline"><semantics> <mrow> <mi>S</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>I</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>A</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math>.</p> Full article ">Figure 3
<p>Time series of <math display="inline"><semantics> <mrow> <mi>S</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>I</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>A</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math>.</p> Full article ">Figure 4
<p>Correlation between <math display="inline"><semantics> <msub> <mi>R</mi> <mn>0</mn> </msub> </semantics></math> and parameters. The units of each parameter are shown in <a href="#axioms-13-00309-t001" class="html-table">Table 1</a>.</p> Full article ">Figure 5
<p>The functional relationship between the control variable <math display="inline"><semantics> <mrow> <mi>u</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> and time <span class="html-italic">t</span>.</p> Full article ">Figure 6
<p>Comparing the number of asymptomatic infectives before and after implementing the control measures.</p> Full article ">Figure 7
<p>Comparing the number of symptomatic infectives before and after implementing the control measures.</p> Full article ">
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