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Volume 9
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Mathematics, Volume 9, Issue 9 (May-1 2021) – 157 articles

Cover Story (view full-size image): Transcendental equations of the kind f(x) = g(x) can be coupled and solved simultaneously with other related functions. A system of transcendental equation including a Sine–Gordon wave equation was modelled and simulated with five input functions. This shows the output wave function over a range of wavelengths. The frequency, amplitude and wavelength of the output wave can be controlled by adjusting the parameters and variables of the system of transcendental equation. View this paper.
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13 pages, 839 KiB  
Article
Convergence Analysis and Cost Estimate of an MLMC-HDG Method for Elliptic PDEs with Random Coefficients
by Meng Li and Xianbing Luo
Mathematics 2021, 9(9), 1072; https://doi.org/10.3390/math9091072 - 10 May 2021
Cited by 2 | Viewed by 1920
Abstract
We considered an hybridizable discontinuous Galerkin (HDG) method for discrete elliptic PDEs with random coefficients. By an approach of projection, we obtained the error analysis under the assumption that a(ω,x) is uniformly bounded. Together with the HDG method, [...] Read more.
We considered an hybridizable discontinuous Galerkin (HDG) method for discrete elliptic PDEs with random coefficients. By an approach of projection, we obtained the error analysis under the assumption that a(ω,x) is uniformly bounded. Together with the HDG method, we applied a multilevel Monte Carlo (MLMC) method (MLMC-HDG method) to simulate the random elliptic PDEs. We derived the overall convergence rate and total computation cost estimate. Finally, some numerical experiments are presented to confirm the theoretical results. Full article
(This article belongs to the Section Computational and Applied Mathematics)
►▼ Show Figures

Figure 1

Figure 1
<p>(<b>Left</b>): Plot of <math display="inline"><semantics> <mrow> <mi mathvariant="double-struck">E</mi> <mo>[</mo> <msub> <mfenced separators="" open="&#x2225;" close="&#x2225;"> <mi>u</mi> <mo>−</mo> <msub> <mi>u</mi> <mi>h</mi> </msub> </mfenced> <mrow> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo>(</mo> <msub> <mi>D</mi> <mi>h</mi> </msub> <mo>)</mo> </mrow> </mrow> </msub> <mo>]</mo> </mrow> </semantics></math> versus <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>/</mo> <mi>h</mi> </mrow> </semantics></math>. The slope of the line is <math display="inline"><semantics> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </semantics></math>. (<b>Right</b>): Plot of <math display="inline"><semantics> <mrow> <mi mathvariant="double-struck">E</mi> <mo>[</mo> <msub> <mfenced separators="" open="&#x2225;" close="&#x2225;"> <mi mathvariant="bold-italic">q</mi> <mo>−</mo> <msub> <mi mathvariant="bold-italic">q</mi> <mi>h</mi> </msub> </mfenced> <mrow> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo>(</mo> <msub> <mi>D</mi> <mi>h</mi> </msub> <mo>)</mo> </mrow> </mrow> </msub> <mo>]</mo> </mrow> </semantics></math>. The slope of the line is <math display="inline"><semantics> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </semantics></math>.</p> Full article ">Figure 2
<p>Total CPU time for the MLMC-HDG approximation of <math display="inline"><semantics> <mrow> <msup> <mi>E</mi> <mi>L</mi> </msup> <mrow> <mo>(</mo> <mi>u</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msup> <mi>E</mi> <mi>L</mi> </msup> <mrow> <mo>(</mo> <mi mathvariant="bold-italic">q</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>.</p> Full article ">Figure 3
<p>CPU time for different levels for the MLMC-HDG approximation of <math display="inline"><semantics> <mrow> <msup> <mi>E</mi> <mi>L</mi> </msup> <mrow> <mo>(</mo> <mi>u</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msup> <mi>E</mi> <mi>L</mi> </msup> <mrow> <mo>(</mo> <mi mathvariant="bold-italic">q</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>.</p> Full article ">
15 pages, 3449 KiB  
Article
Motion of an Unbalanced Impact Body Colliding with a Moving Belt
by Marek Lampart and Jaroslav Zapoměl
Mathematics 2021, 9(9), 1071; https://doi.org/10.3390/math9091071 - 10 May 2021
Cited by 2 | Viewed by 2195
Abstract
In the field of mechanical engineering, conveyors and moving belts are frequently used machine parts. In many working regimes, they are subjected to sudden loading, which can be a source of irregular motion in the impacting bodies and undesirable behavior in the working [...] Read more.
In the field of mechanical engineering, conveyors and moving belts are frequently used machine parts. In many working regimes, they are subjected to sudden loading, which can be a source of irregular motion in the impacting bodies and undesirable behavior in the working machine. This paper deals with a mechanical model where colisions between an impact body and a moving belt take place. The impact body is constrained by a flexible rope, the upper end of which is excited by a slider in the vertical direction. The behavior of the system was investigated in terms of its dependence on the amplitude and frequency of excitation given by the movement of the slider, and the eccentricity of the center of gravity of the impact body. Outputs of the computations indicate that different combinations of the analyzed parameters lead to high complexity of the system’s movement. The bifurcation analysis shows multiple periodic areas changed by chaotic regions. The research carried out provides more details about the behavior and properties of strongly nonlinear mechanical systems resulting from impacts and dry friction. The obtained information will enable designers to propose parameters for industrial machines that make it possible to avoid their working at undesirable operating levels. Full article
(This article belongs to the Special Issue Theory and Application of Dynamical Systems in Mechanics)
►▼ Show Figures

Figure 1

Figure 1
<p>Scheme of the investigated system.</p> Full article ">Figure 2
<p>Decomposition of the force <math display="inline"><semantics> <msub> <mi>F</mi> <mi>l</mi> </msub> </semantics></math>.</p> Full article ">Figure 3
<p>The function <math display="inline"><semantics> <msub> <mi>y</mi> <mi>z</mi> </msub> </semantics></math> (<a href="#FD9-mathematics-09-01071" class="html-disp-formula">9</a>) for <math display="inline"><semantics> <mrow> <mi>ω</mi> <mo>=</mo> <mn>2.6</mn> </mrow> </semantics></math> rad s<math display="inline"><semantics> <msup> <mrow/> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>z</mi> <mi>a</mi> </msub> <mo>=</mo> <mn>7.5</mn> </mrow> </semantics></math> mm, and the parameters set in <a href="#mathematics-09-01071-t001" class="html-table">Table 1</a>.</p> Full article ">Figure 4
<p>Investigated system’s (<a href="#FD1-mathematics-09-01071" class="html-disp-formula">1</a>) time histories of <span class="html-italic">x</span>, <span class="html-italic">y</span>, and <math display="inline"><semantics> <mi>φ</mi> </semantics></math> for <math display="inline"><semantics> <mrow> <msub> <mi>e</mi> <mi>T</mi> </msub> <mo>=</mo> </mrow> </semantics></math> 0.015 m, <math display="inline"><semantics> <mrow> <mi>ω</mi> <mo>=</mo> </mrow> </semantics></math> 4 rad s<math display="inline"><semantics> <msup> <mrow/> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>, and <math display="inline"><semantics> <mrow> <msub> <mi>z</mi> <mi>a</mi> </msub> <mo>=</mo> </mrow> </semantics></math> 0.009 m.</p> Full article ">Figure 5
<p>Investigated system’s (<a href="#FD1-mathematics-09-01071" class="html-disp-formula">1</a>) phase diagram and FFT(<span class="html-italic">y</span>) for <math display="inline"><semantics> <mrow> <msub> <mi>e</mi> <mi>T</mi> </msub> <mo>=</mo> </mrow> </semantics></math> 0.015 m, <math display="inline"><semantics> <mrow> <mi>ω</mi> <mo>=</mo> </mrow> </semantics></math> 4 rad s<math display="inline"><semantics> <msup> <mrow/> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>, and <math display="inline"><semantics> <mrow> <msub> <mi>z</mi> <mi>a</mi> </msub> <mo>=</mo> </mrow> </semantics></math> 0.009 m.</p> Full article ">Figure 6
<p>Investigated system’s (<a href="#FD1-mathematics-09-01071" class="html-disp-formula">1</a>) forces <math display="inline"><semantics> <msub> <mi>F</mi> <mi>c</mi> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mi>F</mi> <mi>t</mi> </msub> </semantics></math>, and <math display="inline"><semantics> <msub> <mi>F</mi> <mi>r</mi> </msub> </semantics></math> for <math display="inline"><semantics> <mrow> <msub> <mi>e</mi> <mi>T</mi> </msub> <mo>=</mo> </mrow> </semantics></math> 0.015 m, <math display="inline"><semantics> <mrow> <mi>ω</mi> <mo>=</mo> </mrow> </semantics></math> 4 rad s<math display="inline"><semantics> <msup> <mrow/> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>, and <math display="inline"><semantics> <mrow> <msub> <mi>z</mi> <mi>a</mi> </msub> <mo>=</mo> </mrow> </semantics></math> 0.009 m.</p> Full article ">Figure 7
<p>Investigated system’s (<a href="#FD1-mathematics-09-01071" class="html-disp-formula">1</a>) time histories of <span class="html-italic">x</span>, <span class="html-italic">y</span>, and <math display="inline"><semantics> <mi>φ</mi> </semantics></math> for <math display="inline"><semantics> <mrow> <msub> <mi>e</mi> <mi>T</mi> </msub> <mo>=</mo> </mrow> </semantics></math> 0.02 m, <math display="inline"><semantics> <mrow> <mi>ω</mi> <mo>=</mo> </mrow> </semantics></math> 2.8 rad s<math display="inline"><semantics> <msup> <mrow/> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>, and <math display="inline"><semantics> <mrow> <msub> <mi>z</mi> <mi>a</mi> </msub> <mo>=</mo> </mrow> </semantics></math> 0.0056 m.</p> Full article ">Figure 8
<p>Investigated system’s (<a href="#FD1-mathematics-09-01071" class="html-disp-formula">1</a>) phase diagram and FFT(<span class="html-italic">y</span>) for <math display="inline"><semantics> <mrow> <msub> <mi>e</mi> <mi>T</mi> </msub> <mo>=</mo> </mrow> </semantics></math> 0.02 m, <math display="inline"><semantics> <mrow> <mi>ω</mi> <mo>=</mo> </mrow> </semantics></math> 2.8 rad s<math display="inline"><semantics> <msup> <mrow/> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>, and <math display="inline"><semantics> <mrow> <msub> <mi>z</mi> <mi>a</mi> </msub> <mo>=</mo> </mrow> </semantics></math> 0.0056 m.</p> Full article ">Figure 9
<p>Investigated system’s (<a href="#FD1-mathematics-09-01071" class="html-disp-formula">1</a>) forces <math display="inline"><semantics> <msub> <mi>F</mi> <mi>c</mi> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mi>F</mi> <mi>t</mi> </msub> </semantics></math>, and <math display="inline"><semantics> <msub> <mi>F</mi> <mi>r</mi> </msub> </semantics></math> for <math display="inline"><semantics> <mrow> <msub> <mi>e</mi> <mi>T</mi> </msub> <mo>=</mo> </mrow> </semantics></math> 0.02 m, <math display="inline"><semantics> <mrow> <mi>ω</mi> <mo>=</mo> </mrow> </semantics></math> 2.8 rad s<math display="inline"><semantics> <msup> <mrow/> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>, and <math display="inline"><semantics> <mrow> <msub> <mi>z</mi> <mi>a</mi> </msub> <mo>=</mo> </mrow> </semantics></math> 0.0056 m.</p> Full article ">Figure 10
<p>Investigatedsystem’s (<a href="#FD1-mathematics-09-01071" class="html-disp-formula">1</a>) time histories of <span class="html-italic">x</span>, <span class="html-italic">y</span>, and <math display="inline"><semantics> <mi>φ</mi> </semantics></math> for <math display="inline"><semantics> <mrow> <msub> <mi>e</mi> <mi>T</mi> </msub> <mo>=</mo> </mrow> </semantics></math> 0.015 m, <math display="inline"><semantics> <mrow> <mi>ω</mi> <mo>=</mo> </mrow> </semantics></math> 2.2 rad s<math display="inline"><semantics> <msup> <mrow/> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>, and <math display="inline"><semantics> <mrow> <msub> <mi>z</mi> <mi>a</mi> </msub> <mo>=</mo> </mrow> </semantics></math> 0.0028 m.</p> Full article ">Figure 11
<p>Investigated system’s (<a href="#FD1-mathematics-09-01071" class="html-disp-formula">1</a>) phase diagram and FFT(<span class="html-italic">y</span>) for <math display="inline"><semantics> <mrow> <msub> <mi>e</mi> <mi>T</mi> </msub> <mo>=</mo> </mrow> </semantics></math> 0.015 m, <math display="inline"><semantics> <mrow> <mi>ω</mi> <mo>=</mo> </mrow> </semantics></math> 2.2 rad s<math display="inline"><semantics> <msup> <mrow/> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>, and <math display="inline"><semantics> <mrow> <msub> <mi>z</mi> <mi>a</mi> </msub> <mo>=</mo> </mrow> </semantics></math> 0.0028 m.</p> Full article ">Figure 12
<p>Investigated system’s (<a href="#FD1-mathematics-09-01071" class="html-disp-formula">1</a>) forces <math display="inline"><semantics> <msub> <mi>F</mi> <mi>c</mi> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mi>F</mi> <mi>t</mi> </msub> </semantics></math>, and <math display="inline"><semantics> <msub> <mi>F</mi> <mi>r</mi> </msub> </semantics></math> for <math display="inline"><semantics> <mrow> <msub> <mi>e</mi> <mi>T</mi> </msub> <mo>=</mo> </mrow> </semantics></math> 0.015 m, <math display="inline"><semantics> <mrow> <mi>ω</mi> <mo>=</mo> </mrow> </semantics></math> 2.2 rad s<math display="inline"><semantics> <msup> <mrow/> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>, and <math display="inline"><semantics> <mrow> <msub> <mi>z</mi> <mi>a</mi> </msub> <mo>=</mo> </mrow> </semantics></math> 0.0028 m.</p> Full article ">Figure 13
<p>Outputs of the contact test (<b>a</b>), the 0–1 test for chaos <span class="html-italic">K</span> (<b>b</b>), and sample entropy <math display="inline"><semantics> <msub> <mi>E</mi> <mrow> <mi>S</mi> <mi>a</mi> <mi>m</mi> <mi>p</mi> </mrow> </msub> </semantics></math> (<b>c</b>) for eccentricity <math display="inline"><semantics> <msub> <mi>e</mi> <mi>T</mi> </msub> </semantics></math> of 0.01 m with respect to the excitation frequency <math display="inline"><semantics> <mi>ω</mi> </semantics></math> and amplitude <math display="inline"><semantics> <msub> <mi>z</mi> <mi>a</mi> </msub> </semantics></math>.</p> Full article ">Figure 14
<p>Outputs of the contact test (<b>a</b>), the 0–1 test for chaos <span class="html-italic">K</span> (<b>b</b>), and sample entropy <math display="inline"><semantics> <msub> <mi>E</mi> <mrow> <mi>S</mi> <mi>a</mi> <mi>m</mi> <mi>p</mi> </mrow> </msub> </semantics></math> (<b>c</b>) for eccentricity <math display="inline"><semantics> <msub> <mi>e</mi> <mi>T</mi> </msub> </semantics></math> of 0.015 m with respect to the excitation frequency <math display="inline"><semantics> <mi>ω</mi> </semantics></math> and amplitude <math display="inline"><semantics> <msub> <mi>z</mi> <mi>a</mi> </msub> </semantics></math>.</p> Full article ">Figure 15
<p>Outputs of the contact test (<b>a</b>), the 0–1 test for chaos <span class="html-italic">K</span> (<b>b</b>), and sample entropy <math display="inline"><semantics> <msub> <mi>E</mi> <mrow> <mi>S</mi> <mi>a</mi> <mi>m</mi> <mi>p</mi> </mrow> </msub> </semantics></math> (<b>c</b>) for eccentricity <math display="inline"><semantics> <msub> <mi>e</mi> <mi>T</mi> </msub> </semantics></math> of 0.02 m with respect to the excitation frequency <math display="inline"><semantics> <mi>ω</mi> </semantics></math> and amplitude <math display="inline"><semantics> <msub> <mi>z</mi> <mi>a</mi> </msub> </semantics></math>.</p> Full article ">Figure 16
<p>Bifurcation diagrams and outputs of the 0–1 test for chaos <span class="html-italic">K</span> and sample entropy <math display="inline"><semantics> <msub> <mi>E</mi> <mrow> <mi>S</mi> <mi>a</mi> <mi>m</mi> <mi>p</mi> </mrow> </msub> </semantics></math> of <span class="html-italic">x</span> for <math display="inline"><semantics> <mrow> <msub> <mi>z</mi> <mi>a</mi> </msub> <mo>=</mo> <mn>0.008</mn> </mrow> </semantics></math> m with respect to the excitation frequency <math display="inline"><semantics> <mi>ω</mi> </semantics></math> rad s<math display="inline"><semantics> <msup> <mrow/> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math> of the sliding body: the first-column cases correspond to <math display="inline"><semantics> <mrow> <mi>ω</mi> <mo>∈</mo> <mo>[</mo> <mn>2</mn> <mo>,</mo> <mn>7</mn> <mo>]</mo> </mrow> </semantics></math> rad s<math display="inline"><semantics> <msup> <mrow/> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math> while the second-column cases to their magnifications <math display="inline"><semantics> <mrow> <mi>ω</mi> <mo>∈</mo> <mo>[</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>]</mo> </mrow> </semantics></math> rad s<math display="inline"><semantics> <msup> <mrow/> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>, and “A”, “B”, “C” to eccentricity <math display="inline"><semantics> <msub> <mi>e</mi> <mi>T</mi> </msub> </semantics></math> of 0.01 m, 0.015 m, and 0.02 m, respectively.</p> Full article ">Figure 17
<p>Bifurcation diagrams and outputs of the 0–1 test for chaos <span class="html-italic">K</span> and sample entropy <math display="inline"><semantics> <msub> <mi>E</mi> <mrow> <mi>S</mi> <mi>a</mi> <mi>m</mi> <mi>p</mi> </mrow> </msub> </semantics></math> of <span class="html-italic">y</span> for <math display="inline"><semantics> <mrow> <msub> <mi>z</mi> <mi>a</mi> </msub> <mo>=</mo> <mn>0.008</mn> </mrow> </semantics></math> m with respect to the excitation frequency <math display="inline"><semantics> <mi>ω</mi> </semantics></math> rad s<math display="inline"><semantics> <msup> <mrow/> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math> of the sliding body: the first-column cases correspond to <math display="inline"><semantics> <mrow> <mi>ω</mi> <mo>∈</mo> <mo>[</mo> <mn>2</mn> <mo>,</mo> <mn>7</mn> <mo>]</mo> </mrow> </semantics></math> rad s<math display="inline"><semantics> <msup> <mrow/> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math> while the second-column cases to their magnifications <math display="inline"><semantics> <mrow> <mi>ω</mi> <mo>∈</mo> <mo>[</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>]</mo> </mrow> </semantics></math> rad s<math display="inline"><semantics> <msup> <mrow/> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics></math>, and “A”, “B”, “C” to eccentricity <math display="inline"><semantics> <msub> <mi>e</mi> <mi>T</mi> </msub> </semantics></math> of 0.01 m, 0.015 m, and 0.02 m, respectively.</p> Full article ">
18 pages, 343 KiB  
Article
Adomian Decomposition and Fractional Power Series Solution of a Class of Nonlinear Fractional Differential Equations
by Pshtiwan Othman Mohammed, José António Tenreiro Machado, Juan L. G. Guirao and Ravi P. Agarwal
Mathematics 2021, 9(9), 1070; https://doi.org/10.3390/math9091070 - 10 May 2021
Cited by 25 | Viewed by 2778
Abstract
Nonlinear fractional differential equations reflect the true nature of physical and biological models with non-locality and memory effects. This paper considers nonlinear fractional differential equations with unknown analytical solutions. The Adomian decomposition and the fractional power series methods are adopted to approximate the [...] Read more.
Nonlinear fractional differential equations reflect the true nature of physical and biological models with non-locality and memory effects. This paper considers nonlinear fractional differential equations with unknown analytical solutions. The Adomian decomposition and the fractional power series methods are adopted to approximate the solutions. The two approaches are illustrated and compared by means of four numerical examples. Full article
►▼ Show Figures

Figure 1

Figure 1
<p>The ADM (<math display="inline"><semantics> <msub> <mi>σ</mi> <mn>5</mn> </msub> </semantics></math>) and FPSM solutions of Equation (<a href="#FD32-mathematics-09-01070" class="html-disp-formula">32</a>), <math display="inline"><semantics> <mrow> <mo>Υ</mo> <mo stretchy="false">(</mo> <mi>η</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> vs <math display="inline"><semantics> <mi>η</mi> </semantics></math>, for <math display="inline"><semantics> <mrow> <mi>h</mi> <mo>=</mo> <mo>{</mo> <mn>0.1</mn> <mo>,</mo> <mn>0.01</mn> <mo>,</mo> <mn>0.001</mn> <mo>}</mo> </mrow> </semantics></math>.</p> Full article ">Figure 2
<p>The ADM (<math display="inline"><semantics> <msub> <mi>σ</mi> <mn>5</mn> </msub> </semantics></math>) and FPSM solutions of Equation (<a href="#FD33-mathematics-09-01070" class="html-disp-formula">33</a>), <math display="inline"><semantics> <mrow> <mo>Υ</mo> <mo stretchy="false">(</mo> <mi>η</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> vs η, for <math display="inline"><semantics> <mrow> <mi>h</mi> <mo>=</mo> <mo>{</mo> <mn>0.1</mn> <mo>,</mo> <mn>0.01</mn> <mo>,</mo> <mn>0.001</mn> <mo>}</mo> </mrow> </semantics></math>.</p> Full article ">Figure 3
<p>The ADM (<math display="inline"><semantics> <msub> <mi>σ</mi> <mn>5</mn> </msub> </semantics></math>) and FPSM solutions of Equation (<a href="#FD34-mathematics-09-01070" class="html-disp-formula">34</a>), <math display="inline"><semantics> <mrow> <mo>Υ</mo> <mo stretchy="false">(</mo> <mi>η</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> vs η, for <math display="inline"><semantics> <mrow> <mi>h</mi> <mo>=</mo> <mo>{</mo> <mn>0.1</mn> <mo>,</mo> <mn>0.01</mn> <mo>,</mo> <mn>0.001</mn> <mo>}</mo> </mrow> </semantics></math>.</p> Full article ">Figure 4
<p>The ADM (<math display="inline"><semantics> <msub> <mi>σ</mi> <mn>5</mn> </msub> </semantics></math>) and FPSM solutions of Equation (<a href="#FD35-mathematics-09-01070" class="html-disp-formula">35</a>), <math display="inline"><semantics> <mrow> <mo>Υ</mo> <mo stretchy="false">(</mo> <mi>η</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> vs η, for <math display="inline"><semantics> <mrow> <mi>h</mi> <mo>=</mo> <mo>{</mo> <mn>0.1</mn> <mo>,</mo> <mn>0.01</mn> <mo>,</mo> <mn>0.001</mn> <mo>}</mo> </mrow> </semantics></math>.</p> Full article ">
16 pages, 3622 KiB  
Article
DeepBlockShield: Blockchain Agent-Based Secured Clinical Data Management Model from the Deep Web Environment
by Junho Kim and Mucheol Kim
Mathematics 2021, 9(9), 1069; https://doi.org/10.3390/math9091069 - 10 May 2021
Cited by 4 | Viewed by 3035
Abstract
With the growth of artificial intelligence in healthcare and biomedical research, many researchers are interested in large amounts of data in hospitals and medical research centers. Then the need for remote medicine services and clinical data utilization are expanding. However, since the misuse [...] Read more.
With the growth of artificial intelligence in healthcare and biomedical research, many researchers are interested in large amounts of data in hospitals and medical research centers. Then the need for remote medicine services and clinical data utilization are expanding. However, since the misuse and abuse of clinical data causes serious problems, the scope of its use is bound to have a limited range physically and logically. Then a security-enhanced data distribution system for medical deep web environments. Therefore, in this paper, we propose a blockchain-based clinical data management model named DeepBlockshield to prevent information leakage between the deep web and the surface web. Blockchain supports data integrity and user validation to support data sharing in closed networks. Meanwhile, the agent performs integrity verification between the blockchain and the deep web and strengthens the security between the surface web and the deep web. DeepBlockShield verifies the user’s validity through the records of the deep web and blockchain. Furthermore, we wrap the results analyzed by the valid request into a web interface and provide information to the requester asynchronously. In the experiment, the block generation cycle and size on the delay time was analyzed for verifying the stability of the blockchain network. As a result, it showed that the proposed approach guarantees the integrity and availability of clinical data in the deep web environment. Full article
(This article belongs to the Special Issue Advances in Mathematical Methods for Machine Learning Algorithms for Computer Aided Diagnostic Systems)
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<p>The concept of surface, deep and dark web.</p> Full article ">Figure 2
<p>DeepBlockShield Overview.</p> Full article ">Figure 3
<p>Separated access control module for data security.</p> Full article ">Figure 4
<p>Agent-based security management system and separate blockchain network.</p> Full article ">Figure 5
<p>User’s node subscription and approval flow.</p> Full article ">Figure 6
<p>Analysis request management process diagram.</p> Full article ">Figure 7
<p>Smart contract scenario diagram.</p> Full article ">Figure 8
<p>Researcher’s interface page.</p> Full article ">Figure 9
<p>Block Creation Time According to Block Size and Number of Nodes.</p> Full article ">Figure 10
<p>Block generation time according to consensus algorithm.</p> Full article ">
16 pages, 2112 KiB  
Article
Spatio-Temporal Traffic Flow Prediction in Madrid: An Application of Residual Convolutional Neural Networks
by Daniel Vélez-Serrano, Alejandro Álvaro-Meca, Fernando Sebastián-Huerta and Jose Vélez-Serrano
Mathematics 2021, 9(9), 1068; https://doi.org/10.3390/math9091068 - 10 May 2021
Cited by 8 | Viewed by 2998
Abstract
Due to the need to predict traffic congestion during the morning or evening rush hours in large cities, a model that is capable of predicting traffic flow in the short term is needed. This model would enable transport authorities to better manage the [...] Read more.
Due to the need to predict traffic congestion during the morning or evening rush hours in large cities, a model that is capable of predicting traffic flow in the short term is needed. This model would enable transport authorities to better manage the situation during peak hours and would allow users to choose the best routes for reaching their destinations. The aim of this study was to perform a short-term prediction of traffic flow in Madrid, using different types of neural network architectures with a focus on convolutional residual neural networks, and it compared them with a classical time series analysis. The proposed convolutional residual neural network is superior in all of the metrics studied, and the predictions are adapted to various situations, such as holidays or possible sensor failures. Full article
(This article belongs to the Special Issue Spatial Statistics with Its Application)
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<p>Map of Madrid with 340 of the 3400 sensors used for traffic prediction.</p> Full article ">Figure 2
<p>UML activity diagram of the process used to train and test the ARIMA models.</p> Full article ">Figure 3
<p>Number of collisions for different matrix sizes and distances from the original position when the station was repositioned. (<b>a</b>) Sensors in a 60 × 60 matrix. (<b>b</b>) Sensors in a 120 × 120 matrix.</p> Full article ">Figure 4
<p>UML activity diagram of the process used to train and test the network models.</p> Full article ">Figure 5
<p>One possible input and expected output for the network.</p> Full article ">Figure 6
<p>Tested CNN architecture.</p> Full article ">Figure 7
<p>With three layers of <math display="inline"><semantics> <mrow> <mn>5</mn> <mo>×</mo> <mn>5</mn> </mrow> </semantics></math> convolutions, the predicted value for a sensor (blue) could be affected by input sensors that are in a <math display="inline"><semantics> <mrow> <mn>13</mn> <mo>×</mo> <mn>13</mn> </mrow> </semantics></math> neighborhood.</p> Full article ">Figure 8
<p>ResNet architecture: (<b>a</b>) linear block (ResNet) and (<b>b</b>) regression ResNet.</p> Full article ">Figure 9
<p>Examples of predicted vs. expected results in several situations: (<b>a</b>) random common days, (<b>b</b>) public holidays, and (<b>c</b>) unexpected events.</p> Full article ">Figure 9 Cont.
<p>Examples of predicted vs. expected results in several situations: (<b>a</b>) random common days, (<b>b</b>) public holidays, and (<b>c</b>) unexpected events.</p> Full article ">
25 pages, 683 KiB  
Article
Bayesian Uncertainty Quantification for Channelized Reservoirs via Reduced Dimensional Parameterization
by Anirban Mondal and Jia Wei
Mathematics 2021, 9(9), 1067; https://doi.org/10.3390/math9091067 - 10 May 2021
Cited by 2 | Viewed by 1580
Abstract
In this article, we study uncertainty quantification for flows in heterogeneous porous media. We use a Bayesian approach where the solution to the inverse problem is given by the posterior distribution of the permeability field given the flow and transport data. Permeability fields [...] Read more.
In this article, we study uncertainty quantification for flows in heterogeneous porous media. We use a Bayesian approach where the solution to the inverse problem is given by the posterior distribution of the permeability field given the flow and transport data. Permeability fields within facies are assumed to be described by two-point correlation functions, while interfaces that separate facies are represented via smooth pseudo-velocity fields in a level set formulation to get reduced dimensional parameterization. The permeability fields within facies and pseudo-velocity fields representing interfaces can be described using Karhunen–Loève (K-L) expansion, where one can select dominant modes. We study the error of posterior distributions introduced in such truncations by estimating the difference in the expectation of a function with respect to full and truncated posteriors. The theoretical result shows that this error can be bounded by the tail of K-L eigenvalues with constants independent of the dimension of discretization. This result guarantees the feasibility of such truncations with respect to posterior distributions. To speed up Bayesian computations, we use an efficient two-stage Markov chain Monte Carlo (MCMC) method that utilizes mixed multiscale finite element method (MsFEM) to screen the proposals. The numerical results show the validity of the proposed parameterization to channel geometry and error estimations. Full article
(This article belongs to the Special Issue Advances on Uncertainty Quantification: Theory and Modelling)
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<p>Illustration of the permeability field with facies.</p> Full article ">Figure 2
<p>Interface evolution by moving initial interface with different vertical velocity fields.</p> Full article ">Figure 3
<p>Interface updates using velocity representation at some fixed points.</p> Full article ">Figure 4
<p>Ordered eigenvalues for <math display="inline"><semantics> <mrow> <msub> <mi>l</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>l</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.4</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <msup> <mi>σ</mi> <mn>2</mn> </msup> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>.</p> Full article ">Figure 5
<p>Linear fit of <math display="inline"><semantics> <mrow> <mo>{</mo> <msup> <mrow> <mo>(</mo> <msubsup> <mo>∑</mo> <mrow> <mi>i</mi> <mo>=</mo> <mi>M</mi> <mo>+</mo> <mn>1</mn> </mrow> <mi>N</mi> </msubsup> <msubsup> <mi>λ</mi> <mrow> <mi>i</mi> </mrow> <mrow> <mo>(</mo> <mi>θ</mi> <mo>)</mo> </mrow> </msubsup> <mo>)</mo> </mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </msup> <mo>,</mo> <mo>|</mo> <msub> <mi>E</mi> <mi>π</mi> </msub> <mi>F</mi> <mo>−</mo> <msub> <mi>E</mi> <mover accent="true"> <mi>π</mi> <mo stretchy="false">˜</mo> </mover> </msub> <mi>F</mi> <mo>|</mo> <mo>}</mo> </mrow> </semantics></math>. <b>Left</b>: <math display="inline"><semantics> <mrow> <msub> <mi>l</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>l</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.4</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msup> <mi>σ</mi> <mn>2</mn> </msup> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <msubsup> <mi>σ</mi> <mrow> <mi>f</mi> </mrow> <mn>2</mn> </msubsup> <mo>=</mo> <mn>0.001</mn> </mrow> </semantics></math>; <b>Right</b>: <math display="inline"><semantics> <mrow> <msub> <mi>l</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.2</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>l</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.2</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msup> <mi>σ</mi> <mn>2</mn> </msup> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <msubsup> <mi>σ</mi> <mrow> <mi>f</mi> </mrow> <mn>2</mn> </msubsup> <mo>=</mo> <mn>0.005</mn> </mrow> </semantics></math>.</p> Full article ">Figure 6
<p>Plots of <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mo movablelimits="true" form="prefix">max</mo> <mo>{</mo> </mrow> <msup> <mrow> <mo>(</mo> <msubsup> <mo>∑</mo> <mrow> <mi>j</mi> <mo>=</mo> <msub> <mi>M</mi> <mn>1</mn> </msub> <mo>+</mo> <mn>1</mn> </mrow> <msub> <mi>N</mi> <mn>1</mn> </msub> </msubsup> <msubsup> <mi>λ</mi> <mrow> <mn>1</mn> <mi>j</mi> </mrow> <mrow> <mo>(</mo> <mi>θ</mi> <mo>)</mo> </mrow> </msubsup> <mo>)</mo> </mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </msup> <mo>,</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <msup> <mrow> <mo>(</mo> <msubsup> <mo>∑</mo> <mrow> <mi>j</mi> <mo>=</mo> <msub> <mi>M</mi> <mn>2</mn> </msub> <mo>+</mo> <mn>1</mn> </mrow> <msub> <mi>N</mi> <mn>2</mn> </msub> </msubsup> <msubsup> <mi>λ</mi> <mrow> <mn>2</mn> <mi>j</mi> </mrow> <mrow> <mo>(</mo> <mi>θ</mi> <mo>)</mo> </mrow> </msubsup> <mo>)</mo> </mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </msup> <mrow> <mo>}</mo> <mo>,</mo> </mrow> <msup> <mrow> <mo>(</mo> <msubsup> <mo>∑</mo> <mrow> <mi>i</mi> <mo>=</mo> <mover accent="true"> <mi>M</mi> <mo stretchy="false">˜</mo> </mover> <mo>+</mo> <mn>1</mn> </mrow> <mover accent="true"> <mi>N</mi> <mo stretchy="false">˜</mo> </mover> </msubsup> <msubsup> <mi>λ</mi> <mrow> <mi>i</mi> </mrow> <mrow> <mo>(</mo> <mi>α</mi> <mo>)</mo> </mrow> </msubsup> <mo>)</mo> </mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </msup> <mrow> <mo>,</mo> <mo>|</mo> </mrow> <msub> <mi>E</mi> <mi>π</mi> </msub> <mi>F</mi> <mo>−</mo> <msub> <mi>E</mi> <mover accent="true"> <mi>π</mi> <mo stretchy="false">˜</mo> </mover> </msub> <mi>F</mi> <mrow> <mo>|</mo> <mo>)</mo> </mrow> </mrow> </semantics></math>.</p> Full article ">Figure 7
<p><b>Top left</b>: The true log-permeability field. <b>Top right</b>: Initial log-permeability field. <b>Bottom left</b>: One of the sampled log-permeability field. <b>Bottom Right</b>: The mean of the sampled log-permeability field from two-stage MCMC using 20 K-L terms.</p> Full article ">Figure 8
<p><b>Left</b>: Red line designates the fine-scale reference fractional flow of oil, the blue line designates the initial fractional flow of oil, and the green line designates fractional flow of oil corresponding to mean of the sampled permeability field from two-stage MCMC. <b>Right</b>: Fractional flow errors vs. accepted iterations when sampled from the posterior distribution retaining 20 terms in K-L expansion.</p> Full article ">Figure 9
<p><b>Top left</b>: The true log-permeability field. <b>Top right</b>: Initial log-permeability field. <b>Bottom left</b>: One of the sampled log-permeability field. <b>Bottom Right</b>: The mean of the sampled log-permeability field from two-stage MCMC using 25 K-L terms.</p> Full article ">Figure 10
<p><b>Left</b>: Red line designates the fine-scale reference fractional flow of oil, the blue line designates the initial fractional flow of oil, and the green line designates fractional flow of oil corresponding to mean of the sampled permeability field from two-stage MCMC. <b>Right</b>: Fractional flow errors vs. accepted iterations when sampled from the posterior distribution retaining 25 terms in K-L expansion.</p> Full article ">
19 pages, 7710 KiB  
Article
Optimal Parameter Estimation Methodology of Solid Oxide Fuel Cell Using Modern Optimization
by Hesham Alhumade, Ahmed Fathy, Abdulrahim Al-Zahrani, Muhyaddin Jamal Rawa and Hegazy Rezk
Mathematics 2021, 9(9), 1066; https://doi.org/10.3390/math9091066 - 10 May 2021
Cited by 15 | Viewed by 2685
Abstract
An optimal parameter estimation methodology of solid oxide fuel cell (SOFC) using modern optimization is proposed in this paper. An equilibrium optimizer (EO) has been used to identify the unidentified parameters of the SOFC equivalent circuit with the assistance of experimental results. This [...] Read more.
An optimal parameter estimation methodology of solid oxide fuel cell (SOFC) using modern optimization is proposed in this paper. An equilibrium optimizer (EO) has been used to identify the unidentified parameters of the SOFC equivalent circuit with the assistance of experimental results. This is presented via formulating the modeling process as an optimization problem considering the sum mean squared error (SMSE) between the observed and computed voltages as the target. Two modes of the SOFC-based model are investigated under variable operating conditions, namely, the steady-state and the dynamic-state based models. The proposed EO results are compared to those obtained via the Archimedes optimization algorithm (AOA), Heap-based optimizer (HBO), Seagull Optimization Algorithm (SOA), Student Psychology Based Optimization Algorithm (SPBO), Marine predator algorithm (MPA), Manta ray foraging optimization (MRFO), and comprehensive learning dynamic multi-swarm marine predators algorithm. The minimum fitness function at the steady-state model is obtained via the proposed EO with value of 1.5527 × 10−6 at 1173 K. In the dynamic based model, the minimum SMSE is 1.0406. The obtained results confirmed the reliability and superiority of the proposed EO in constructing a reliable model of SOFC. Full article
(This article belongs to the Special Issue Advanced Modeling, Control, and Optimization Methods in Power Hybrid Systems - 2021)
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<p>Optimization process of EO.</p> Full article ">Figure 2
<p>The proposed steps incorporating EO.</p> Full article ">Figure 3
<p>The measured and calculated polarization curves of SOFC operated at 1073 K obtained via EO at (<b>a</b>) current density-voltage, (<b>b</b>) current density-power.</p> Full article ">Figure 4
<p>(<b>a</b>) Current-voltage curve, (<b>b</b>) Current-power curve of SOFC operated at 1073 K obtained via other approaches.</p> Full article ">Figure 5
<p>The variation of fitness function during iterative process for all employed optimizers applied for steady-state SOFC model.</p> Full article ">Figure 6
<p>The measured and calculated (<b>a</b>) current density-voltage, (<b>b</b>) current density-power of SOFC operated at 1173 K, 1213 K, and 1273 K obtained via the proposed EO.</p> Full article ">Figure 7
<p>The measured and calculated polarization curves of SOFC dynamic-state model operated at 1273 K obtained via EO (<b>a</b>) current-voltage, (<b>b</b>) current-power.</p> Full article ">Figure 8
<p>The measured and calculated polarization curves of SOFC dynamic-state model operated at 1273 K obtained via MPA, HBO, SOA, and MRFO.</p> Full article ">Figure 9
<p>The variation of fitness function during iterative process for all employed optimizers applied for dynamic-state SOFC model.</p> Full article ">Figure 10
<p>First load disturbance applied on SOFC stack dynamic model.</p> Full article ">Figure 11
<p>Second load disturbance applied on SOFC stack dynamic model.</p> Full article ">
16 pages, 2771 KiB  
Article
Theoretical Identification of Coupling Effect and Performance Analysis of Single-Source Direct Sampling Method
by Won-Kwang Park
Mathematics 2021, 9(9), 1065; https://doi.org/10.3390/math9091065 - 10 May 2021
Cited by 4 | Viewed by 1687
Abstract
Although the direct sampling method (DSM) has demonstrated its feasibility in identifying small anomalies from measured scattering parameter data in microwave imaging, inaccurate imaging results that cannot be explained by conventional research approaches have often emerged. It has been heuristically identified that the [...] Read more.
Although the direct sampling method (DSM) has demonstrated its feasibility in identifying small anomalies from measured scattering parameter data in microwave imaging, inaccurate imaging results that cannot be explained by conventional research approaches have often emerged. It has been heuristically identified that the reason for this phenomenon is due to the coupling effect between the antenna and dipole antennas, but related mathematical theory has not been investigated satisfactorily yet. The main purpose of this contribution is to explain the theoretical elucidation of such a phenomenon and to design an improved DSM for successful application to microwave imaging. For this, we first survey traditional DSM and design an improved DSM, which is based on the fact that the measured scattering parameter is influenced by both the anomaly and the antennas. We then establish a new mathematical theory of both the traditional and the designed indicator functions of DSM by constructing a relationship between the antenna arrangement and an infinite series of Bessel functions of integer order of the first kind. On the basis of the theoretical results, we discover various factors that influence the imaging performance of traditional DSM and explain why the designed indicator function successfully improves the traditional one. Several numerical experiments with synthetic data support the established theoretical results and illustrate the pros and cons of traditional and designed DSMs. Full article
(This article belongs to the Section Computational and Applied Mathematics)
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<p>Illustration of the simulation configuration.</p> Full article ">Figure 2
<p>Maps of <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="fraktur">F</mi> <mo form="prefix">DSM</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">r</mi> <mo>,</mo> <mi>m</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="fraktur">F</mi> <mo form="prefix">DSE</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">r</mi> <mo>,</mo> <mi>m</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> at <math display="inline"><semantics> <mrow> <mi>f</mi> <mo>=</mo> <mn>0.925</mn> <mrow> <mi mathvariant="normal">G</mi> <mi>Hz</mi> </mrow> </mrow> </semantics></math>. White-colored dashed line describes the <math display="inline"><semantics> <mrow> <mo>∂</mo> <mi mathvariant="script">D</mi> </mrow> </semantics></math>.</p> Full article ">Figure 3
<p>Maps of <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="fraktur">F</mi> <mo form="prefix">DSM</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">r</mi> <mo>,</mo> <mi>m</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="fraktur">F</mi> <mo form="prefix">DSE</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">r</mi> <mo>,</mo> <mi>m</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> at <math display="inline"><semantics> <mrow> <mi>f</mi> <mo>=</mo> <mn>1.245</mn> <mrow> <mi mathvariant="normal">G</mi> <mi>Hz</mi> </mrow> </mrow> </semantics></math>. White-colored dashed line describes the <math display="inline"><semantics> <mrow> <mo>∂</mo> <mi mathvariant="script">D</mi> </mrow> </semantics></math>.</p> Full article ">Figure 4
<p>Maps of <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="fraktur">F</mi> <mo form="prefix">DSM</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">r</mi> <mo>,</mo> <mi>m</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="fraktur">F</mi> <mo form="prefix">DSE</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">r</mi> <mo>,</mo> <mi>m</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> at <math display="inline"><semantics> <mrow> <mi>f</mi> <mo>=</mo> <mn>0.750</mn> <mrow> <mi mathvariant="normal">G</mi> <mi>Hz</mi> </mrow> </mrow> </semantics></math>. White-colored dashed line describes the <math display="inline"><semantics> <mrow> <mo>∂</mo> <mi mathvariant="script">D</mi> </mrow> </semantics></math>.</p> Full article ">Figure 5
<p>Multi-frequency imaging of <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="fraktur">F</mi> <mo form="prefix">DSM</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">r</mi> <mo>,</mo> <mi>m</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="fraktur">F</mi> <mo form="prefix">DSE</mo> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">r</mi> <mo>,</mo> <mi>m</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>. White-colored dashed line describes the <math display="inline"><semantics> <mrow> <mo>∂</mo> <mi mathvariant="script">D</mi> </mrow> </semantics></math>.</p> Full article ">
23 pages, 1071 KiB  
Article
2DOF IMC and Smith-Predictor-Based Control for Stabilised Unstable First Order Time Delayed Plants
by Mikulas Huba, Pavol Bistak and Damir Vrancic
Mathematics 2021, 9(9), 1064; https://doi.org/10.3390/math9091064 - 10 May 2021
Cited by 7 | Viewed by 3116
Abstract
The article brings a brief revision of the two-degree-of-freedom (2-DoF) internal model control (IMC) and the 2-DoF Smith-Predictor-based (SP) control of unstable systems. It shows that the first important reason for distinguishing between these approaches is the limitations of the control action. However, [...] Read more.
The article brings a brief revision of the two-degree-of-freedom (2-DoF) internal model control (IMC) and the 2-DoF Smith-Predictor-based (SP) control of unstable systems. It shows that the first important reason for distinguishing between these approaches is the limitations of the control action. However, it also reminds that, in addition to the seemingly lucrative dynamics of transients, the proposed approaches can conceal a tricky behavior with a structural instability, which may manifest itself only after a longer period of time. Instead, as one of possible reliable alternatives, two-step IMC and filtered Smith predictor (FSP) design are applied to unstable first-order time-delayed (UFOTD) systems. Firstly, the 2-DoF P controller yielding a double real dominant closed loop pole is applied. Only then the 2-DoF IMC or FSP controllers are designed, providing slightly slower, but more robust transients. These remain stable even in the long run, while also showing increased robustness. Full article
(This article belongs to the Special Issue Advances in Study of Time-Delay Systems and Their Applications)
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Figure 1
<p>2-DoF IMC design for the stabilised <span class="html-italic">j</span>-th order plant model.</p> Full article ">Figure 2
<p>2-DoF IMC control of the first-order plants (<a href="#FD1-mathematics-09-01064" class="html-disp-formula">1</a>) based on the model (<a href="#FD2-mathematics-09-01064" class="html-disp-formula">2</a>) with <math display="inline"><semantics> <mrow> <msub> <mi>T</mi> <mi>d</mi> </msub> <mo>=</mo> <msub> <mover> <mi>T</mi> <mo>¯</mo> </mover> <mi>d</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> and a setpoint feedforward generated either by a single transfer function <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mi>w</mi> </msub> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> (TF-IMC) or by the primary loop (PL-IMC) with 2-DoF P control of the plant model <math display="inline"><semantics> <mrow> <msub> <mover> <mi>S</mi> <mo>¯</mo> </mover> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>; both augmented by a disturbance feedforward <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mi>o</mi> </msub> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> considering <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> in (<a href="#FD9-mathematics-09-01064" class="html-disp-formula">9</a>).</p> Full article ">Figure 3
<p>Interpretation of the disturbance feedforward impact in 2-DoF IMC transformed to a feedforward control of an equivalent plant <math display="inline"><semantics> <mrow> <msub> <mi>S</mi> <mi>e</mi> </msub> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>. Modification of <a href="#mathematics-09-01064-f001" class="html-fig">Figure 1</a> by moving block <math display="inline"><semantics> <msub> <mi>C</mi> <mi>w</mi> </msub> </semantics></math> before summation point (<b>above</b>); replacement of internal feedback with blocks <math display="inline"><semantics> <mover> <mi>S</mi> <mo>¯</mo> </mover> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mi>o</mi> </msub> <msub> <mi>C</mi> <mi>w</mi> </msub> </mrow> </semantics></math> by controller <span class="html-italic">R</span> (<b>below</b>).</p> Full article ">Figure 4
<p>Cascade SL-IMC with the first order plant stabilisation by 2-DoF P control (<math display="inline"><semantics> <mrow> <msub> <mi>T</mi> <mi>d</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>).</p> Full article ">Figure 5
<p>FSP structure for analysis according to [<a href="#B19-mathematics-09-01064" class="html-bibr">19</a>] with PI controller in the setpoint feedforward loop, prefilter <span class="html-italic">F</span> and output disturbance feedforward filter <math display="inline"><semantics> <msub> <mi>F</mi> <mi>r</mi> </msub> </semantics></math> from the reconstructed output disturbance <math display="inline"><semantics> <msub> <mover> <mi>d</mi> <mo>¯</mo> </mover> <mi>o</mi> </msub> </semantics></math> for UFOTD.</p> Full article ">Figure 6
<p>Transient responses corresponding to the FSP controller from <a href="#mathematics-09-01064-f005" class="html-fig">Figure 5</a> according to [<a href="#B19-mathematics-09-01064" class="html-bibr">19</a>] and demonstrating internal instability due to unbounded reconstructed disturbance signals <math display="inline"><semantics> <mrow> <msub> <mi>d</mi> <mrow> <mi>o</mi> <mi>r</mi> <mi>e</mi> <mi>c</mi> </mrow> </msub> <mo>=</mo> <msub> <mover> <mi>d</mi> <mo>¯</mo> </mover> <mi>o</mi> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>d</mi> <mrow> <mi>o</mi> <mi>f</mi> </mrow> </msub> <mo>=</mo> <msub> <mover> <mi>d</mi> <mo>¯</mo> </mover> <mrow> <mi>o</mi> <mi>f</mi> </mrow> </msub> </mrow> </semantics></math>.</p> Full article ">Figure 7
<p>IMC-like structure of the internally unstable FSP controller scheme “for analysis” from <a href="#mathematics-09-01064-f005" class="html-fig">Figure 5</a>, with feedforward controller <math display="inline"><semantics> <mrow> <msup> <mrow/> <mn>1</mn> </msup> <msub> <mi>C</mi> <mi>w</mi> </msub> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> (<a href="#FD24-mathematics-09-01064" class="html-disp-formula">24</a>), prefilter <math display="inline"><semantics> <mrow> <mi>F</mi> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> </semantics></math> (<a href="#FD22-mathematics-09-01064" class="html-disp-formula">22</a>) and disturbance feedforward filter <math display="inline"><semantics> <mrow> <msub> <mi>F</mi> <mi>r</mi> </msub> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> (<a href="#FD23-mathematics-09-01064" class="html-disp-formula">23</a>) of the reconstructed output disturbance <math display="inline"><semantics> <msub> <mover> <mi>d</mi> <mo>¯</mo> </mover> <mi>o</mi> </msub> </semantics></math> for UFOTD according to [<a href="#B19-mathematics-09-01064" class="html-bibr">19</a>] (<b>above</b>) and the structure “for implementation” after eliminating the unbounded reconstructed disturbance <math display="inline"><semantics> <msub> <mover> <mi>d</mi> <mo>¯</mo> </mover> <mi>o</mi> </msub> </semantics></math> and introducing an equivalent controller <math display="inline"><semantics> <mrow> <msup> <mrow/> <mn>1</mn> </msup> <msub> <mi>C</mi> <mi>e</mi> </msub> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> (<a href="#FD26-mathematics-09-01064" class="html-disp-formula">26</a>) corresponding to the feedforward <math display="inline"><semantics> <mrow> <msup> <mrow/> <mn>1</mn> </msup> <msub> <mi>C</mi> <mi>w</mi> </msub> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> with the internal feedback blocks <math display="inline"><semantics> <mrow> <msub> <mi>F</mi> <mi>r</mi> </msub> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> and the nominal plant model <math display="inline"><semantics> <mrow> <msub> <mi>P</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> <msup> <mi>e</mi> <mrow> <mo>−</mo> <msub> <mi>L</mi> <mi>n</mi> </msub> <mi>s</mi> </mrow> </msup> </mrow> </semantics></math> (<a href="#FD20-mathematics-09-01064" class="html-disp-formula">20</a>) (<b>below</b>).</p> Full article ">Figure 8
<p>Stabilisation of UFOTD plant model by 2-DoF P controller.</p> Full article ">Figure 9
<p>Equivalent scheme of the 2-DoF IMC control for the stabilised <span class="html-italic">j</span>th order plant model.</p> Full article ">Figure 10
<p>2-DoF PL-IMC with PD controller (<a href="#FD49-mathematics-09-01064" class="html-disp-formula">49</a>) for 2nd order model of SL (<a href="#FD42-mathematics-09-01064" class="html-disp-formula">42</a>) (blue) approximating the UFOTD plant (<a href="#FD1-mathematics-09-01064" class="html-disp-formula">1</a>) stabilised by 2DoF P controller (white background) designed to yield a double real dominant pole; the disturbance reconstruction &amp; disturbance feedforward (<a href="#FD44-mathematics-09-01064" class="html-disp-formula">44</a>) designed for the 2nd order SL approximation (<a href="#FD42-mathematics-09-01064" class="html-disp-formula">42</a>) (green); setpoint prefilter with the time constant <math display="inline"><semantics> <msub> <mi>T</mi> <mrow> <mi>D</mi> <mi>f</mi> </mrow> </msub> </semantics></math> used to reduce the PD controller output kicks after setpoint step changes.</p> Full article ">Figure 11
<p>Transient responses of the plant (<a href="#FD20-mathematics-09-01064" class="html-disp-formula">20</a>) for a setpoint step <math display="inline"><semantics> <mrow> <mi>w</mi> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math> at <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>50</mn> </mrow> </semantics></math> and an input disturbance step <math display="inline"><semantics> <mrow> <msub> <mi>d</mi> <mi>i</mi> </msub> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math> at <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>400</mn> </mrow> </semantics></math>, nominal plant parameters <math display="inline"><semantics> <mrow> <msub> <mover> <mi>K</mi> <mo>¯</mo> </mover> <mi>s</mi> </msub> <mo>=</mo> <msub> <mi>K</mi> <mi>s</mi> </msub> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mover> <mi>a</mi> <mo>¯</mo> </mover> <mo>=</mo> <mi>a</mi> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mover> <mi>T</mi> <mo>¯</mo> </mover> <mi>d</mi> </msub> <mo>=</mo> <msub> <mi>T</mi> <mi>d</mi> </msub> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>T</mi> <mi>c</mi> </msub> <mo>=</mo> <msub> <mi>T</mi> <mi>f</mi> </msub> <mo>=</mo> <msub> <mover> <mi>T</mi> <mo>¯</mo> </mover> <mi>d</mi> </msub> </mrow> </semantics></math>: with the 2-DoF FSP according to [<a href="#B19-mathematics-09-01064" class="html-bibr">19</a>], 1-DoF and 2-DoF SL-IMC controllers (denoted as P-FSP1 and P-FSP2) for unstable plant stabilised by 2-DoF P control derived by a two-step design from <a href="#sec4dot2-mathematics-09-01064" class="html-sec">Section 4.2</a> and <a href="#sec4dot3-mathematics-09-01064" class="html-sec">Section 4.3</a> and the controller with setpoint and disturbance reference models and disturbance observer from [<a href="#B20-mathematics-09-01064" class="html-bibr">20</a>]; <math display="inline"><semantics> <mrow> <msub> <mi>d</mi> <mrow> <mi>o</mi> <mi>r</mi> <mi>e</mi> <mi>c</mi> </mrow> </msub> <mo>=</mo> <msub> <mover> <mi>d</mi> <mo>¯</mo> </mover> <mi>o</mi> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>d</mi> <mrow> <mi>o</mi> <mi>f</mi> </mrow> </msub> <mo>=</mo> <msub> <mover> <mi>d</mi> <mo>¯</mo> </mover> <mrow> <mi>o</mi> <mi>f</mi> </mrow> </msub> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>d</mi> <mrow> <mi>i</mi> <mi>r</mi> <mi>e</mi> <mi>c</mi> </mrow> </msub> <mo>=</mo> <msub> <mover> <mi>d</mi> <mo>¯</mo> </mover> <mi>i</mi> </msub> </mrow> </semantics></math>.</p> Full article ">Figure 12
<p><math display="inline"><semantics> <mrow> <mi>I</mi> <mi>A</mi> <mi>E</mi> </mrow> </semantics></math> values corresponding to transients in <a href="#mathematics-09-01064-f011" class="html-fig">Figure 11</a>.</p> Full article ">Figure 13
<p>Transient responses of the plant (<a href="#FD20-mathematics-09-01064" class="html-disp-formula">20</a>) for a setpoint step <math display="inline"><semantics> <mrow> <mi>w</mi> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math> at <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>50</mn> </mrow> </semantics></math> and an input disturbance step <math display="inline"><semantics> <mrow> <msub> <mi>d</mi> <mi>i</mi> </msub> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math> at <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>400</mn> </mrow> </semantics></math>, nominal plant parameters <math display="inline"><semantics> <mrow> <msub> <mover> <mi>K</mi> <mo>¯</mo> </mover> <mi>s</mi> </msub> <mo>=</mo> <msub> <mi>K</mi> <mi>s</mi> </msub> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mover> <mi>a</mi> <mo>¯</mo> </mover> <mo>=</mo> <mi>a</mi> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>T</mi> <mi>d</mi> </msub> <mo>=</mo> <msub> <mover> <mi>T</mi> <mo>¯</mo> </mover> <mi>d</mi> </msub> <mo>/</mo> <mn>1.3</mn> </mrow> </semantics></math> , <math display="inline"><semantics> <mrow> <msub> <mi>T</mi> <mi>c</mi> </msub> <mo>=</mo> <msub> <mi>T</mi> <mi>f</mi> </msub> <mo>=</mo> <msub> <mover> <mi>T</mi> <mo>¯</mo> </mover> <mi>d</mi> </msub> </mrow> </semantics></math>: with the 2-DoF FSP according to [<a href="#B19-mathematics-09-01064" class="html-bibr">19</a>], 1-DoF and 2-DoF SL-IMC controllers (denoted as P-FSP1 and P-FSP2) for unstable plant stabilised by 2-DoF P control derived by a two-step design from <a href="#sec4dot2-mathematics-09-01064" class="html-sec">Section 4.2</a> and <a href="#sec4dot3-mathematics-09-01064" class="html-sec">Section 4.3</a> and the controller with setpoint and disturbance reference models and disturbance observer from [<a href="#B20-mathematics-09-01064" class="html-bibr">20</a>] with the modified tuning <math display="inline"><semantics> <mrow> <msub> <mi>T</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>0.9</mn> <msub> <mi>L</mi> <mi>n</mi> </msub> </mrow> </semantics></math>; <math display="inline"><semantics> <mrow> <msub> <mi>d</mi> <mrow> <mi>o</mi> <mi>r</mi> <mi>e</mi> <mi>c</mi> </mrow> </msub> <mo>=</mo> <msub> <mover> <mi>d</mi> <mo>¯</mo> </mover> <mi>o</mi> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>d</mi> <mrow> <mi>o</mi> <mi>f</mi> </mrow> </msub> <mo>=</mo> <msub> <mover> <mi>d</mi> <mo>¯</mo> </mover> <mrow> <mi>o</mi> <mi>f</mi> </mrow> </msub> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>d</mi> <mrow> <mi>i</mi> <mi>r</mi> <mi>e</mi> <mi>c</mi> </mrow> </msub> <mo>=</mo> <msub> <mover> <mi>d</mi> <mo>¯</mo> </mover> <mi>i</mi> </msub> </mrow> </semantics></math>.</p> Full article ">Figure 14
<p><math display="inline"><semantics> <mrow> <mi>I</mi> <mi>A</mi> <mi>E</mi> </mrow> </semantics></math> values corresponding to transients in <a href="#mathematics-09-01064-f013" class="html-fig">Figure 13</a>.</p> Full article ">Figure 15
<p>The shape related <math display="inline"><semantics> <mrow> <mi>T</mi> <msub> <mi>V</mi> <mi>y</mi> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>T</mi> <msub> <mi>V</mi> <mi>u</mi> </msub> </mrow> </semantics></math> values corresponding to transients in <a href="#mathematics-09-01064-f013" class="html-fig">Figure 13</a>.</p> Full article ">
16 pages, 1124 KiB  
Article
Brain Signals Classification Based on Fuzzy Lattice Reasoning
by Eleni Vrochidou, Chris Lytridis, Christos Bazinas, George A. Papakostas, Hiroaki Wagatsuma and Vassilis G. Kaburlasos
Mathematics 2021, 9(9), 1063; https://doi.org/10.3390/math9091063 - 9 May 2021
Cited by 2 | Viewed by 2992
Abstract
Cyber-Physical System (CPS) applications including human-robot interaction call for automated reasoning for rational decision-making. In the latter context, typically, audio-visual signals are employed. Τhis work considers brain signals for emotion recognition towards an effective human-robot interaction. An ElectroEncephaloGraphy (EEG) signal here is represented [...] Read more.
Cyber-Physical System (CPS) applications including human-robot interaction call for automated reasoning for rational decision-making. In the latter context, typically, audio-visual signals are employed. Τhis work considers brain signals for emotion recognition towards an effective human-robot interaction. An ElectroEncephaloGraphy (EEG) signal here is represented by an Intervals’ Number (IN). An IN-based, optimizable parametric k Nearest Neighbor (kNN) classifier scheme for decision-making by fuzzy lattice reasoning (FLR) is proposed, where the conventional distance between two points is replaced by a fuzzy order function (σ) for reasoning-by-analogy. A main advantage of the employment of INs is that no ad hoc feature extraction is required since an IN may represent all-order data statistics, the latter are the features considered implicitly. Four different fuzzy order functions are employed in this work. Experimental results demonstrate comparably the good performance of the proposed techniques. Full article
(This article belongs to the Special Issue Numerical Analysis and Scientific Computing)
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Figure 1

Figure 1
<p>Computation of an IN by algorithm distrIN. All the EEG signal values are <span class="html-italic">n</span> + 1 (sorted) real numbers.</p> Full article ">Figure 2
<p>(<b>a</b>) An EEG signal; (<b>b</b>) the distribution function of EEG signal values; (<b>c</b>) membership-function representation computed by algorithm distrIN; (<b>d</b>) interval representation for 32 levels.</p> Full article ">Figure 3
<p>kNN classification scheme for testing.</p> Full article ">Figure 4
<p>Reduction in cost during optimization for 40 generations.</p> Full article ">
16 pages, 1512 KiB  
Article
Approximation-Based Quantized State Feedback Tracking of Uncertain Input-Saturated MIMO Nonlinear Systems with Application to 2-DOF Helicopter
by Byung Mo Kim and Sung Jin Yoo
Mathematics 2021, 9(9), 1062; https://doi.org/10.3390/math9091062 - 9 May 2021
Cited by 12 | Viewed by 2327
Abstract
This paper addresses an approximation-based quantized state feedback tracking problem of multiple-input multiple-output (MIMO) nonlinear systems with quantized input saturation. A uniform quantizer is adopted to quantize state variables and control inputs of MIMO nonlinear systems. The primary features in the current development [...] Read more.
This paper addresses an approximation-based quantized state feedback tracking problem of multiple-input multiple-output (MIMO) nonlinear systems with quantized input saturation. A uniform quantizer is adopted to quantize state variables and control inputs of MIMO nonlinear systems. The primary features in the current development are that (i) an adaptive neural network tracker using quantized states is developed for MIMO nonlinear systems and (ii) a compensation mechanism of quantized input saturation is designed by constructing an auxiliary system. An adaptive neural tracker design with the compensation of quantized input saturation is developed by deriving an augmented error surface using quantized states. It is shown that closed-loop stability analysis and tracking error convergence are conducted based on Lyapunov theory. Finally, we give simulation and experimental results of the 2-degrees-of-freedom (2-DOF) helicopter system for verifying to the validity of the proposed methodology where the tracking performance of pitch and yaw angles is measured with the mean squared errors of 0.1044 and 0.0435 for simulation results, and those of 0.0656 and 0.0523 for experimental results. Full article
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Figure 1

Figure 1
<p>Block diagram of the proposed quantized-states-based adaptive tracking system in the presence of quantized input saturation.</p> Full article ">Figure 2
<p>Quanser Aero 2-DOF Helicopter [<a href="#B37-mathematics-09-01062" class="html-bibr">37</a>].</p> Full article ">Figure 3
<p>Tracking results for simulation (<b>a</b>) <math display="inline"><semantics> <msub> <mi>x</mi> <mrow> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>x</mi> <mrow> <mn>1</mn> <mo>,</mo> <mi>d</mi> </mrow> </msub> </semantics></math> (<b>b</b>) <math display="inline"><semantics> <msub> <mi>x</mi> <mrow> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>x</mi> <mrow> <mn>2</mn> <mo>,</mo> <mi>d</mi> </mrow> </msub> </semantics></math>.</p> Full article ">Figure 4
<p>Control input voltages <math display="inline"><semantics> <msub> <mi>V</mi> <mi>p</mi> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>V</mi> <mi>y</mi> </msub> </semantics></math> for simulation.</p> Full article ">Figure 5
<p>RBFNNs outputs and adaptive parameters for simulation (<b>a</b>) <math display="inline"><semantics> <mrow> <msubsup> <mover accent="true"> <mi mathvariant="bold-italic">W</mi> <mo stretchy="false">^</mo> </mover> <mrow> <mn>1</mn> </mrow> <mo>⊤</mo> </msubsup> <msub> <mi mathvariant="bold-italic">Q</mi> <mn>1</mn> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msubsup> <mover accent="true"> <mi mathvariant="bold-italic">W</mi> <mo stretchy="false">^</mo> </mover> <mrow> <mn>2</mn> </mrow> <mo>⊤</mo> </msubsup> <msub> <mi mathvariant="bold-italic">Q</mi> <mn>2</mn> </msub> </mrow> </semantics></math> (<b>b</b>) <math display="inline"><semantics> <mover accent="true"> <mi>B</mi> <mo stretchy="false">^</mo> </mover> </semantics></math>.</p> Full article ">Figure 6
<p>Experiment setup.</p> Full article ">Figure 7
<p>Tracking results for experiment (<b>a</b>) <math display="inline"><semantics> <msub> <mi>x</mi> <mrow> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>x</mi> <mrow> <mn>1</mn> <mo>,</mo> <mi>d</mi> </mrow> </msub> </semantics></math> (<b>b</b>) <math display="inline"><semantics> <msub> <mi>x</mi> <mrow> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>x</mi> <mrow> <mn>2</mn> <mo>,</mo> <mi>d</mi> </mrow> </msub> </semantics></math>.</p> Full article ">Figure 8
<p>Control input voltage <math display="inline"><semantics> <msub> <mi>V</mi> <mi>p</mi> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>V</mi> <mi>y</mi> </msub> </semantics></math> for experiment.</p> Full article ">Figure 9
<p>RBFNNs outputs and adaptive parameters for experiment (<b>a</b>) <math display="inline"><semantics> <mrow> <msubsup> <mover accent="true"> <mi mathvariant="bold-italic">W</mi> <mo stretchy="false">^</mo> </mover> <mrow> <mn>1</mn> </mrow> <mo>⊤</mo> </msubsup> <msub> <mi mathvariant="bold-italic">Q</mi> <mn>1</mn> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msubsup> <mover accent="true"> <mi mathvariant="bold-italic">W</mi> <mo stretchy="false">^</mo> </mover> <mrow> <mn>2</mn> </mrow> <mo>⊤</mo> </msubsup> <msub> <mi mathvariant="bold-italic">Q</mi> <mn>2</mn> </msub> </mrow> </semantics></math> (<b>b</b>) <math display="inline"><semantics> <mover accent="true"> <mi>B</mi> <mo stretchy="false">^</mo> </mover> </semantics></math>.</p> Full article ">
18 pages, 656 KiB  
Article
Spatiotemporal Econometrics Models for Old Age Mortality in Europe
by Patricia Carracedo and Ana Debón
Mathematics 2021, 9(9), 1061; https://doi.org/10.3390/math9091061 - 9 May 2021
Cited by 1 | Viewed by 2294
Abstract
In the past decade, panel data models using time-series observations of several geographical units have become popular due to the availability of software able to implement them. The aim of this study is an updated comparison of estimation techniques between the implementations of [...] Read more.
In the past decade, panel data models using time-series observations of several geographical units have become popular due to the availability of software able to implement them. The aim of this study is an updated comparison of estimation techniques between the implementations of spatiotemporal panel data models across MATLAB and R softwares in order to fit real mortality data. The case study used concerns the male and female mortality of the aged population of European countries. Mortality is quantified with the Comparative Mortality Figure, which is the most suitable statistic for comparing mortality by sex over space when detailed specific mortality is available for each studied population. The spatial dependence between the 26 European countries and their neighbors during 1995–2012 was confirmed through the Global Moran Index and the spatiotemporal panel data models. For this reason, it can be said that mortality in European population aging not only depends on differences in the health systems, which are subject to national discretion but also on supra-national developments. Finally, we conclude that although both programs seem similar, there are some differences in the estimation of parameters and goodness of fit measures being more reliable MATLAB. These differences have been justified by detailing the advantages and disadvantages of using each of them. Full article
(This article belongs to the Special Issue Spatial Statistics with Its Application)
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<p>CMF in Europe for 1995 and 2012 for males.</p> Full article ">Figure 2
<p>CMF in Europe for 1995 and 2012 for females.</p> Full article ">Figure 3
<p>CMF variability in Europe for males and females.</p> Full article ">Figure 4
<p>CMF’s variability in Europe for each country for males and females.</p> Full article ">Figure 5
<p>Normality of the SLMSFE model residuals for males and females.</p> Full article ">
10 pages, 309 KiB  
Article
On the Oscillatory Properties of Solutions of Second-Order Damped Delay Differential Equations
by Awatif A. Hendi, Osama Moaaz, Clemente Cesarano, Wedad R. Alharbi and Mohamed A. Abdou
Mathematics 2021, 9(9), 1060; https://doi.org/10.3390/math9091060 - 9 May 2021
Cited by 4 | Viewed by 2193
Abstract
In the work, a new oscillation condition was created for second-order damped delay differential equations with a non-canonical operator. The new criterion is of an iterative nature which helps to apply it even when the previous relevant results fail to apply. An example [...] Read more.
In the work, a new oscillation condition was created for second-order damped delay differential equations with a non-canonical operator. The new criterion is of an iterative nature which helps to apply it even when the previous relevant results fail to apply. An example is presented in order to illustrate the significance of the results. Full article
(This article belongs to the Special Issue Orthogonal Polynomials and Special Functions)
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<p>Schematic diagram for main results.</p> Full article ">
27 pages, 5390 KiB  
Article
Techniques to Improve B2B Data Governance Using FAIR Principles
by Cristina Georgiana Calancea and Lenuța Alboaie
Mathematics 2021, 9(9), 1059; https://doi.org/10.3390/math9091059 - 9 May 2021
Cited by 5 | Viewed by 3336
Abstract
Sharing data along the economic supply/demand chain represents a catalyst to improve the performance of a digitized business sector. In this context, designing automatic mechanisms for structured data exchange, that should also ensure the proper development of B2B processes in a regulated environment, [...] Read more.
Sharing data along the economic supply/demand chain represents a catalyst to improve the performance of a digitized business sector. In this context, designing automatic mechanisms for structured data exchange, that should also ensure the proper development of B2B processes in a regulated environment, becomes a necessity. Even though the data format used for sharing can be modeled using the open methodology, we propose the use of FAIR principles to additionally offer business entities a way to define commonly agreed upon supply, access and ownership procedures. As an approach to manage the FAIR modelled metadata, we propose a series of methodologies to follow. They were integrated in a data marketplace platform, which we developed to ensure they are properly applied. For its design, we modelled a decentralized architecture based on our own blockchain mechanisms. In our proposal, each business entity can host and structure its metadata in catalog, dataset and distribution assets. In order to offer businesses full control over the data supplied through our system, we designed and implemented a sharing mechanism based on access policies defined by the business entity directly in our data marketplace platform. In the proposed approach, metadata-based assets sharing can be done between two or multiple businesses, which will be able to manually access the data in the management interface and programmatically through an authorized data point. Business specific transactions proposed to modify the semantic model are validated using our own blockchain based technologies. As a result, security and integrity of the FAIR data in the collaboration process is ensured. From an architectural point of view, the lack of a central authority to manage the vehiculated data ensures businesses have full control of the terms and conditions under which their data is used. Full article
(This article belongs to the Special Issue Business and Economics Mathematics)
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<p>DataShareFair Architecture Overview.</p> Full article ">Figure 2
<p>Multichain Blockchain Logical Domains Architecture [<a href="#B39-mathematics-09-01059" class="html-bibr">39</a>].</p> Full article ">Figure 3
<p>Blockchain Application General Architecture [<a href="#B39-mathematics-09-01059" class="html-bibr">39</a>].</p> Full article ">Figure 4
<p>DataShareFair Platform Architecture.</p> Full article ">Figure 5
<p>Business Accounts and Groups Management View (First Architectural Layer).</p> Full article ">Figure 6
<p>FAIR Data Marketplace Dataset View (First Architectural Layer).</p> Full article ">Figure 7
<p>FAIR Distribution Creation View (First Architectural Layer).</p> Full article ">Figure 8
<p>FAIR Metadata Management View (First Architectural Layer).</p> Full article ">Figure 9
<p>Dataset Edit View (First Architectural Layer).</p> Full article ">Figure 10
<p>FAIR Data Point Distribution Metadata.</p> Full article ">
12 pages, 1056 KiB  
Article
A Model for the Optimal Investment Strategy in the Context of Pandemic Regional Lockdown
by Antoine Tonnoir, Ioana Ciotir, Adrian-Liviu Scutariu and Octavian Dospinescu
Mathematics 2021, 9(9), 1058; https://doi.org/10.3390/math9091058 - 8 May 2021
Cited by 6 | Viewed by 2979
Abstract
The Covid-19 pandemic has generated major changes in society, most of them having a negative impact on the quality of life and income obtained by the population and businesses. The negative consequences have been highlighted in the decrease of the GPD level for [...] Read more.
The Covid-19 pandemic has generated major changes in society, most of them having a negative impact on the quality of life and income obtained by the population and businesses. The negative consequences have been highlighted in the decrease of the GPD level for regions, countries and even continents. Returning to pre-pandemic levels is a considerable effort for both economic and political decision-makers. This article deals with the construction of a mathematical model for economic aspects in the context of variable productivity in time. Through this mathematical model, we propose to maximize revenues in pandemic conditions, in order to limit the economic consequences of the lockdown. One advantage of the proposed model consists in the fact that it is based on units that can be regions, economic branches, economic units or fields of investment. Another strength of the model is determined by the fact that it offers the possibility to choose between two different investment strategies, based on the specific options of the decision makers: the consistent increase of the state revenues or the amelioration of the disparity phenomenon. Furthermore, our model extends previous approaches from the literature by adding some generalization options and the proposed model can be applied in lockdown cases and seasonal situations. Full article
(This article belongs to the Special Issue Business and Economics Mathematics)
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<p>Representation of the function <math display="inline"><semantics> <mrow> <msub> <mi>c</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> versus time for the first example (on the <b>left</b>) and for the second example (on the <b>right</b>).</p> Full article ">Figure 2
<p>Evolution of the capital <math display="inline"><semantics> <mrow> <msub> <mi>k</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> over time (on the <b>left</b>) and optimal index <math display="inline"><semantics> <mrow> <msup> <mi>l</mi> <mo>*</mo> </msup> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> of the region over time (on the <b>right</b>) taking <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>.</p> Full article ">Figure 3
<p>Representation of the function <math display="inline"><semantics> <mrow> <msub> <mi>g</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> versus time.</p> Full article ">Figure 4
<p>Evolution of the capital <math display="inline"><semantics> <mrow> <msub> <mi>k</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> over time (on the <b>left</b>) and optimal index <math display="inline"><semantics> <mrow> <msup> <mi>l</mi> <mo>*</mo> </msup> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> of the region over time (on the <b>right</b>) taking <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.8</mn> </mrow> </semantics></math>.</p> Full article ">Figure 5
<p>Evolution of the capital <math display="inline"><semantics> <mrow> <msub> <mi>k</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> over time (on the <b>left</b>) and optimal index <math display="inline"><semantics> <mrow> <msup> <mi>l</mi> <mo>*</mo> </msup> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> of the region over time (on the <b>right</b>) taking <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>.</p> Full article ">Figure 6
<p>Evolution of the capital <math display="inline"><semantics> <mrow> <msub> <mi>k</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> over time (on the <b>left</b>) and optimal index <math display="inline"><semantics> <mrow> <msup> <mi>l</mi> <mo>*</mo> </msup> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> of the region over time (on the <b>right</b>).</p> Full article ">Figure 7
<p>Representation of the function <math display="inline"><semantics> <mrow> <msub> <mi>c</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> versus time (on the <b>left</b>) and the function <math display="inline"><semantics> <mrow> <msub> <mi>g</mi> <mi>j</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> (on the <b>right</b>).</p> Full article ">Figure 8
<p>Evolution of the capital <math display="inline"><semantics> <mrow> <msub> <mi>k</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> over time (on the <b>left</b>) and optimal index <math display="inline"><semantics> <mrow> <msup> <mi>l</mi> <mo>*</mo> </msup> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> of the region over time (on the <b>right</b>).</p> Full article ">Figure 9
<p>Evolution of the capital <math display="inline"><semantics> <mrow> <msub> <mi>k</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> over time (on the <b>left</b>) and index <math display="inline"><semantics> <mrow> <msup> <mi>l</mi> <mo>*</mo> </msup> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> over time (on the <b>right</b>). These results have been obtained using strategy (<a href="#FD12-mathematics-09-01058" class="html-disp-formula">12</a>).</p> Full article ">
19 pages, 13300 KiB  
Article
Robust Mode Analysis
by Gemunu H. Gunaratne and Sukesh Roy
Mathematics 2021, 9(9), 1057; https://doi.org/10.3390/math9091057 - 8 May 2021
Cited by 2 | Viewed by 2305
Abstract
In this paper, we introduce a model-free algorithm, robust mode analysis (RMA), to extract primary constituents in a fluid or reacting flow directly from high-frequency, high-resolution experimental data. It is expected to be particularly useful in studying strongly driven flows, where nonlinearities can [...] Read more.
In this paper, we introduce a model-free algorithm, robust mode analysis (RMA), to extract primary constituents in a fluid or reacting flow directly from high-frequency, high-resolution experimental data. It is expected to be particularly useful in studying strongly driven flows, where nonlinearities can induce chaotic and irregular dynamics. The lack of precise governing equations and the absence of symmetries or other simplifying constraints in realistic configurations preclude the derivation of analytical solutions for these systems; the presence of flow structures over a wide range of scales handicaps finding their numerical solutions. Thus, the need for direct analysis of experimental data is reinforced. RMA is predicated on the assumption that primary flow constituents are common in multiple, nominally identical realizations of an experiment. Their search relies on the identification of common dynamic modes in the experiments, the commonality established via proximity of the eigenvalues and eigenfunctions. Robust flow constituents are then constructed by combining common dynamic modes that flow at the same rate. We illustrate RMA using reacting flows behind a symmetric bluff body. Two robust constituents, whose signatures resemble symmetric and von Karman vortex shedding, are identified. It is shown how RMA can be implemented via extended dynamic mode decomposition in flow configurations interrogated with a small number of time-series. This approach may prove useful in analyzing changes in flow patterns in engines and propulsion systems equipped with sturdy arrays of pressure transducers or thermocouples. Finally, an analysis of high Reynolds number jet flows suggests that tests of statistical characterizations in turbulent flows may best be done using non-robust components of the flow. Full article
(This article belongs to the Special Issue Dynamical Systems and Operator Theory)
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<p>The number of robust modes found in the turbulent jet flow analyzed in <a href="#sec5-mathematics-09-01057" class="html-sec">Section 5</a> as a function of <math display="inline"><semantics> <msub> <mi>δ</mi> <mi>R</mi> </msub> </semantics></math> when the cutoff <math display="inline"><semantics> <mo>Δ</mo> </semantics></math> is set at 0.75.</p> Full article ">Figure 2
<p>(<b>a</b>) Schematic of the reacting flow behind a v-gutter bluff body (adapted from the Ref. [<a href="#B38-mathematics-09-01057" class="html-bibr">38</a>]). Schematics showing (<b>b</b>) symmetric and (<b>c</b>) von Karman vortex shedding behind a symmetric barrier.</p> Full article ">Figure 3
<p>(<b>a</b>) The DMD spectrum for the reacting flow for a “lean” mixture with equivalence ratio <math display="inline"><semantics> <mrow> <mi>ϕ</mi> <mo>=</mo> <mn>0.8</mn> </mrow> </semantics></math> (adapted from the Ref. [<a href="#B31-mathematics-09-01057" class="html-bibr">31</a>]). The flow exhibits symmetric vortex shedding, as well as von Karman shedding at this equivalence ratio. (<b>b</b>) The mean and standard deviation for the robust modes 4, 5, 38, 58, and 61.</p> Full article ">Figure 4
<p>(<b>a</b>) Real and (<b>b</b>) imaginary parts of <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="sans-serif">Φ</mi> <mn>4</mn> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">x</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>, the symmetric mode with the largest energy, for the reacting flow at <math display="inline"><semantics> <mrow> <mi>ϕ</mi> <mo>=</mo> <mn>0.8</mn> </mrow> </semantics></math>. (<b>c</b>) Real and (<b>d</b>) imaginary parts of <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="sans-serif">Φ</mi> <mn>38</mn> </msub> <mrow> <mo>(</mo> <mi mathvariant="bold">x</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>, the asymmetric mode with the highest energy. This figure was adapted from the Ref. [<a href="#B31-mathematics-09-01057" class="html-bibr">31</a>].</p> Full article ">Figure 5
<p>Phase dynamics of robust modes for the reacting flow. “Angular velocities” <math display="inline"><semantics> <mrow> <msub> <mi>ω</mi> <mn>4</mn> </msub> <mo>≈</mo> <mn>0.073</mn> </mrow> </semantics></math> radians/frame and <math display="inline"><semantics> <mrow> <msub> <mi>ω</mi> <mn>5</mn> </msub> <mo>≈</mo> <mn>0.078</mn> </mrow> </semantics></math> radians/frame are nearly identical, suggesting that they belong to the same flow constituent. (<math display="inline"><semantics> <msub> <mi>ω</mi> <mn>61</mn> </msub> </semantics></math> is slightly higher and is not used in the reconstruction.) Similarly, <math display="inline"><semantics> <mrow> <msub> <mi>ω</mi> <mn>38</mn> </msub> <mo>≈</mo> <mn>0.055</mn> </mrow> </semantics></math> radian/frame and <math display="inline"><semantics> <mrow> <msub> <mi>ω</mi> <mn>58</mn> </msub> <mo>≈</mo> <mn>0.051</mn> </mrow> </semantics></math> radians/frame are close, suggesting that they form a second constituent. Successive curves are shifted for clarity. This figure was adapted from the Ref. [<a href="#B31-mathematics-09-01057" class="html-bibr">31</a>].</p> Full article ">Figure 6
<p>Several snapshots of the reacting flow at equivalence ratio <math display="inline"><semantics> <mrow> <mi>ϕ</mi> <mo>=</mo> <mn>0.8</mn> </mrow> </semantics></math> (<b>top row</b>), reconstructions of the symmetric (<b>second row</b>), and asymmetric (<b>bottom row</b>) flow constituents. This figure was adapted from the Ref. [<a href="#B31-mathematics-09-01057" class="html-bibr">31</a>].</p> Full article ">Figure 7
<p>Schematic of the bluff body combustion chamber with locations of pressure transducers. The trailing edge of the symmetric bluff body is assigned to be at downstream locations being positive. Note that transducers 7, 8, and 9 are placed opposite to 4, 5, and 6, respectively. A detailed explanation of imaging optics and measurement techniques can be found in the Ref. [<a href="#B42-mathematics-09-01057" class="html-bibr">42</a>]. Here, we confine our analysis to measurements from transducers 1–9.</p> Full article ">Figure 8
<p>(<b>a</b>) Symmetric robust mode: Projections of the single robust mode at <math display="inline"><semantics> <mrow> <mi>ϕ</mi> <mo>=</mo> <mn>0.95</mn> </mrow> </semantics></math>. The <span class="html-italic">x</span>-axis is the transducer number, and the <span class="html-italic">y</span>-axis the magnitude of the robust mode at the site. Note that values of the mode on sites 4, 5, and 6 are nearly identical to those at sites 7, 8, and 9. Pairs of transducers (4, 7), (5, 8), and (6, 9) are at identical axis locations, thus indicating that the robust mode represents a symmetric flow constituent. (<b>b</b>) Asymmetric robust mode: One robust modes of the flow at <math display="inline"><semantics> <mrow> <mi>ϕ</mi> <mo>=</mo> <mn>0.85</mn> </mrow> </semantics></math>. Note that it does not have the symmetry between the values at transducer pairs (4, 7), (5, 8) and (6, 9) seen in the mode of (<b>a</b>).</p> Full article ">Figure 9
<p>(<b>a</b>) Axisymmetric jet flow is generated by <math display="inline"><semantics> <msub> <mi>N</mi> <mn>2</mn> </msub> </semantics></math> gas ejected from cylinder of radius <math display="inline"><semantics> <mrow> <mi>D</mi> <mo>=</mo> <mn>4.7</mn> </mrow> </semantics></math> mm. Rectangular region of size <math display="inline"><semantics> <mrow> <mn>13.4</mn> </mrow> </semantics></math> mm × <math display="inline"><semantics> <mrow> <mn>17.7</mn> </mrow> </semantics></math> mm where velocity measurements were made is approximately <math display="inline"><semantics> <mrow> <mn>13</mn> <mi>D</mi> </mrow> </semantics></math> downstream of inlet. (<b>b</b>) Snapshot of velocity field for flow of <math display="inline"><semantics> <mrow> <mi>R</mi> <mi>e</mi> <mo>≈</mo> </mrow> </semantics></math> 21,000.</p> Full article ">Figure 10
<p>The energy of the first 60 dynamic modes. Robust modes are shown in the dark color. Note that robust modes are not those with the highest energy.</p> Full article ">Figure 11
<p>Flow patterns in the pair of robust modes 6 and 7 whose eigenvalues are <math display="inline"><semantics> <mrow> <mo>−</mo> <mn>0.013</mn> <mo>±</mo> <mn>0.489</mn> <mi>i</mi> </mrow> </semantics></math> and whose period is <math display="inline"><semantics> <mrow> <mi>T</mi> <mo>=</mo> <mn>1.28</mn> </mrow> </semantics></math> ms.</p> Full article ">Figure 12
<p>Distributions of (<b>a</b>) radial, and (<b>b</b>) axial velocity fluctuations at a fixed location. The blue lines are the best Gaussian fit to the data and red lines are deviations from the Gaussian.</p> Full article ">Figure 13
<p>(<b>a</b>) The diamonds show <math display="inline"><semantics> <mrow> <msub> <mi>S</mi> <mi>ζ</mi> </msub> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> as a function of the moment <math display="inline"><semantics> <mi>ζ</mi> </semantics></math> for <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>→</mo> <mo>∞</mo> </mrow> </semantics></math> for the jet flow. The dashed line is that given in Equation (<a href="#FD12-mathematics-09-01057" class="html-disp-formula">12</a>). (<b>b</b>) The corresponding comparison for the non-robust flow.</p> Full article ">
78 pages, 660 KiB  
Article
Boolean-Valued Set-Theoretic Systems: General Formalism and Basic Technique
by Alexander Gutman
Mathematics 2021, 9(9), 1056; https://doi.org/10.3390/math9091056 - 8 May 2021
Cited by 2 | Viewed by 1693
Abstract
This article is devoted to the study of the Boolean-valued universe as an algebraic system. We start with the logical backgrounds of the notion and present the formalism of extending the syntax of Boolean truth values by the use of definable symbols, internal [...] Read more.
This article is devoted to the study of the Boolean-valued universe as an algebraic system. We start with the logical backgrounds of the notion and present the formalism of extending the syntax of Boolean truth values by the use of definable symbols, internal classes, outer terms and external Boolean-valued classes. Next, we enrich the collection of Boolean-valued research tools with the technique of partial elements and the corresponding joins, mixings and ascents. Passing on to the set-theoretic signature, we prove that bounded formulas are absolute for transitive Boolean-valued subsystems. We also introduce and study intensional, predicative, cyclic and regular Boolean-valued systems, examine the maximum principle, and analyze its relationship with the ascent and mixing principles. The main applications relate to the universe over an arbitrary extensional Boolean-valued system. A close interrelation is established between such a universe and the intensional hierarchy. We prove the existence and uniqueness of the Boolean-valued universe up to a unique isomorphism and show that the conditions in the corresponding axiomatic characterization are logically independent. We also describe the structure of the universe by means of several cumulative hierarchies. Another application, based on the quantifier hierarchy of formulas, improves the transfer principle for the canonical embedding in the Boolean-valued universe. Full article
(This article belongs to the Special Issue Boolean Valued Analysis with Applications)
28 pages, 1569 KiB  
Article
How Governance Paradigms and Other Drivers Affect Public Managers’ Use of Innovation Practices. A PLS-SEM Analysis and Model
by Alberto Peralta and Luis Rubalcaba
Mathematics 2021, 9(9), 1055; https://doi.org/10.3390/math9091055 - 7 May 2021
Cited by 15 | Viewed by 4006
Abstract
Using the Unified Theory of Acceptance and Use of Technology for Innovations in the Public Sector (UTAUT-IPS) model, this study examined the influences on using a specific innovation practice on public managers. We based our analysis on an end-of-2019 sample of 227 Spanish [...] Read more.
Using the Unified Theory of Acceptance and Use of Technology for Innovations in the Public Sector (UTAUT-IPS) model, this study examined the influences on using a specific innovation practice on public managers. We based our analysis on an end-of-2019 sample of 227 Spanish public managers, aiming to answer the question “Are public innovation and project managers driven only by a governance paradigm, influencing their intention and usage of an innovation practice?” Using the Partial Least Squares Structural Equation Modelling (PLS-SEM) algorithm, we singled out the effects of the governance paradigm, performance expectancy, and motivation, among seven other behavioral composite variables. The PLS-Prediction-Oriented Segmentation routine was used to segment our sample into three distinct groups of innovation managers: (i) those driven by nearly all influences; (ii) those driven by results and the governance paradigm; and (iii) those driven by governance and habits. The three groups highlight the different practical approaches to public innovation and co-creation initiatives, which clearly reflect the complex process of deciding which tool (or tools) should be used to implement these. Our UTAUT-IPS model helps visualize this complex decision-making process. Full article
(This article belongs to the Special Issue Partial Least Squares Structural Equation Modeling (PLS-SEM) Applications in Economics and Finance)
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<p>Research model.</p> Full article ">Figure 2
<p>Steps for the validation of the UTAUT-IPS model (based on [<a href="#B87-mathematics-09-01055" class="html-bibr">87</a>,<a href="#B88-mathematics-09-01055" class="html-bibr">88</a>,<a href="#B89-mathematics-09-01055" class="html-bibr">89</a>]).</p> Full article ">Figure 3
<p>Structural (inner) model for the pooled sample. Note: ** <span class="html-italic">p</span> &lt; 0.05; *** <span class="html-italic">p</span> &lt; 0.001.</p> Full article ">
18 pages, 5241 KiB  
Article
Complex Uncertainty of Surface Data Modeling via the Type-2 Fuzzy B-Spline Model
by Rozaimi Zakaria, Abd. Fatah Wahab, Isfarita Ismail and Mohammad Izat Emir Zulkifly
Mathematics 2021, 9(9), 1054; https://doi.org/10.3390/math9091054 - 7 May 2021
Cited by 14 | Viewed by 2130
Abstract
This paper discusses the construction of a type-2 fuzzy B-spline model to model complex uncertainty of surface data. To construct this model, the type-2 fuzzy set theory, which includes type-2 fuzzy number concepts and type-2 fuzzy relation, is used to define the complex [...] Read more.
This paper discusses the construction of a type-2 fuzzy B-spline model to model complex uncertainty of surface data. To construct this model, the type-2 fuzzy set theory, which includes type-2 fuzzy number concepts and type-2 fuzzy relation, is used to define the complex uncertainty of surface data in type-2 fuzzy data/control points. These type-2 fuzzy data/control points are blended with the B-spline surface function to produce the proposed model, which can be visualized and analyzed further. Various processes, namely fuzzification, type-reduction and defuzzification are defined to achieve a crisp, type-2 fuzzy B-spline surface, representing uncertainty complex surface data. This paper ends with a numerical example of terrain modeling, which shows the effectiveness of handling the uncertainty complex data. Full article
(This article belongs to the Special Issue Fuzzy Sets, Fuzzy Logic and Their Applications 2020)
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<p>Definition of an IT2FN.</p> Full article ">Figure 2
<p>T2FDP around eight.</p> Full article ">Figure 3
<p>The alpha-cut operation toward T2FDP.</p> Full article ">Figure 4
<p>The illustration of the correlation between alpha values and T2FDPs.</p> Full article ">Figure 5
<p>The processes of defining, fuzzification, type-reduction and defuzzification towards T2FDP.</p> Full article ">Figure 6
<p>The example of T2FIBsC model: (<b>a</b>) with T2FCP; (<b>b</b>) without T2FCP.</p> Full article ">Figure 7
<p>The example T2FIBsS model: with (<b>a</b>) and; without (<b>b</b>) type-2 fuzzy data net.</p> Full article ">Figure 8
<p>The example fuzzified T2FIBsS model with fuzzified type-2 fuzzy data net.</p> Full article ">Figure 9
<p>The example of type-reduced fuzzified T2FIBsS together with type-reduced fuzzified T2FDPs net.</p> Full article ">Figure 10
<p>The example of defuzzification-reduced T2FIBsS with defuzzification-reduced T2FDPs.</p> Full article ">Figure 11
<p>Illustration of procedure in taking depth data point (in meter) of the seabed, which consists of uncertainty complex data.</p> Full article ">Figure 12
<p>The T2FIBsS modeling through fuzzification until defuzzification processes along with the error plot.</p> Full article ">Figure 13
<p>The T2FIBsS modeling through fuzzification until defuzzification processes with <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math> along the error plot.</p> Full article ">Figure 14
<p>The T2FIBsS modeling through fuzzification until defuzzification processes with <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.8</mn> </mrow> </semantics></math> along the error plot.</p> Full article ">
20 pages, 5767 KiB  
Article
Image Region Prediction from Thermal Videos Based on Image Prediction Generative Adversarial Network
by Ganbayar Batchuluun, Ja Hyung Koo, Yu Hwan Kim and Kang Ryoung Park
Mathematics 2021, 9(9), 1053; https://doi.org/10.3390/math9091053 - 7 May 2021
Cited by 6 | Viewed by 2357
Abstract
Various studies have been conducted on object detection, tracking, and action recognition based on thermal images. However, errors occur during object detection, tracking, and action recognition when a moving object leaves the field of view (FOV) of a camera and part of the [...] Read more.
Various studies have been conducted on object detection, tracking, and action recognition based on thermal images. However, errors occur during object detection, tracking, and action recognition when a moving object leaves the field of view (FOV) of a camera and part of the object becomes invisible. However, no studies have examined this issue so far. Therefore, this article proposes a method for widening the FOV of the current image by predicting images outside the FOV of the camera using the current image and previous sequential images. In the proposed method, the original one-channel thermal image is converted into a three-channel thermal image to perform image prediction using an image prediction generative adversarial network. When image prediction and object detection experiments were conducted using the marathon sub-dataset of the Boston University-thermal infrared video (BU-TIV) benchmark open dataset, we confirmed that the proposed method showed the higher accuracies of image prediction (structural similarity index measure (SSIM) of 0.9839) and object detection (F1 score (F1) of 0.882, accuracy (ACC) of 0.983, and intersection over union (IoU) of 0.791) than the state-of-the-art methods. Full article
(This article belongs to the Special Issue Computer Graphics, Image Processing and Artificial Intelligence)
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<p>Example of thermal image prediction.</p> Full article ">Figure 2
<p>Overall flowchart of the proposed method.</p> Full article ">Figure 3
<p>Procedure of preprocessing.</p> Full article ">Figure 4
<p>Example of the structure of the proposed IPGAN.</p> Full article ">Figure 5
<p>Example of the postprocessing.</p> Full article ">Figure 6
<p>Examples of dataset preparation. In (<b>a</b>–<b>c</b>), on the left: from top to bottom, an original thermal image and an ROI image. In (<b>a</b>–<b>c</b>), on the right: from top to bottom, a ground-truth image and an input image.</p> Full article ">Figure 6 Cont.
<p>Examples of dataset preparation. In (<b>a</b>–<b>c</b>), on the left: from top to bottom, an original thermal image and an ROI image. In (<b>a</b>–<b>c</b>), on the right: from top to bottom, a ground-truth image and an input image.</p> Full article ">Figure 7
<p>Training loss curves of GAN.</p> Full article ">Figure 8
<p>Examples of result images obtained using Methods 1 and 2. From left to right, the input, output, and ground-truth images, respectively, obtained using (<b>a</b>) Method 1 and (<b>b</b>) Method 2. The size of the input, output, and ground-truth images is 80 × 170 pixels.</p> Full article ">Figure 9
<p>Examples of result images obtained using Methods 3 and 4. From left to right, the input, output, and ground-truth images, respectively, obtained using (<b>a</b>) Method 3 and (<b>b</b>) Method 4.</p> Full article ">Figure 10
<p>Examples of result images obtained using Methods 5–7. From left to right, the input, output, and ground-truth images, respectively, obtained using (<b>a</b>) Method 5, (<b>b</b>) Method 6, and (<b>c</b>) Method 7.</p> Full article ">Figure 11
<p>Examples of result images obtained using Methods 4 and 7. From left to right, the input, output, and ground-truth images, respectively, obtained using (<b>a</b>) Method 4 and (<b>b</b>) Method 7.</p> Full article ">Figure 12
<p>Examples of result images obtained using the proposed method. In (<b>a</b>–<b>d</b>), from left to right, the original, ground-truth, and predicted (output) images, respectively.</p> Full article ">Figure 13
<p>Examples of detection results before and after image prediction. In (<b>a</b>–<b>d</b>), from left to right, the original input images, results with original input images, ground-truth images, results with ground-truth images, images predicted using our method, and results with predicted images, respectively.</p> Full article ">Figure 14
<p>Comparisons of the original images, ground-truth images, and prediction results obtained using the state-of-the-art methods and our method: (<b>a</b>) original images; (<b>b</b>) ground-truth images. Images predicted using: (<b>c</b>) Haziq et al.’s method; (<b>d</b>) Liu et al.’s method; (<b>e</b>) Shin et al.’s method; (<b>f</b>) Nazeri et al.’s method; (<b>g</b>) the proposed method.</p> Full article ">Figure 15
<p>Comparisons of detection results using the original images, ground-truth images, and the predicted images obtained using the state-of-the-art methods and our method. (<b>a</b>) Original images. Detection results using the (<b>b</b>) original images, (<b>c</b>) ground-truth images, (<b>d</b>) images predicted using Haziq et al.’s method, (<b>e</b>) images predicted using Liu et al.’s method, (<b>f</b>) images predicted using Shin et al.’s method, (<b>g</b>) images predicted using Nazeri et al.’s method, and (<b>h</b>) images predicted using our method.</p> Full article ">
25 pages, 15629 KiB  
Article
Design of a Computer-Aided Location Expert System Based on a Mathematical Approach
by Martin Straka
Mathematics 2021, 9(9), 1052; https://doi.org/10.3390/math9091052 - 7 May 2021
Cited by 7 | Viewed by 2662
Abstract
This article discusses how to calculate the location of a point on a surface using a mathematical approach on two levels. The first level uses the traditional calculation procedure via Cooper’s iterative method through a spreadsheet editor and a classic result display map. [...] Read more.
This article discusses how to calculate the location of a point on a surface using a mathematical approach on two levels. The first level uses the traditional calculation procedure via Cooper’s iterative method through a spreadsheet editor and a classic result display map. The second level uses the author-created computer-aided location expert system on the principle of calculation using Cooper’s iterative method with the direct graphical display of results. The problem is related to designing a practical computer location expert system, which is based on a new idea of using the resolution of a computer map as an image to calculate location. The calculated results are validated by comparing them with each other, and the defined accuracy for a particular example was achieved at the 32nd iteration with the position optima DC[x(32);y(32)] = [288.8;82.7], with identical results. The location solution in the case study to the defined accuracy was achieved at the 6th iteration with the position optima DC[x(6);y(6)] = [274;220]. The calculations show that the expert system created achieves the required parameters and is a handy tool for determining the location of a point on a surface. Full article
(This article belongs to the Section Engineering Mathematics)
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Graphical abstract

Graphical abstract
Full article ">Figure 1
<p>Steps of the calculation procedure.</p> Full article ">Figure 2
<p>The location of a distribution and supply region in the coordinate system.</p> Full article ">Figure 3
<p>Calculation of the direct distance between places in the distribution region.</p> Full article ">Figure 4
<p>Location of a distribution and supply region in the coordinate system.</p> Full article ">Figure 5
<p>Graphical representation of the solution of the distribution centre’s location.</p> Full article ">Figure 6
<p>Graphical display of a task solution that matches the diagonal intersection.</p> Full article ">Figure 7
<p>Image resolution is the basis for the data input into the location calculation.</p> Full article ">Figure 8
<p>Recording input data about location positions used in the location calculation.</p> Full article ">Figure 9
<p>Setting the required location calculation parameters.</p> Full article ">Figure 10
<p>Results of the calculation of the location of the point on the area in the proposed expert system.</p> Full article ">Figure 11
<p>Input data and calculation settings for calculating the optimal point location on the surface based on <a href="#mathematics-09-01052-t001" class="html-table">Table 1</a>.</p> Full article ">Figure 12
<p>Graphical display of the results and procedure for calculating iterations via the computer-aided location expert system that the author created.</p> Full article ">Figure 13
<p>Graphical display of results of the objective function values calculating iterations from five different starting points.</p> Full article ">Figure 14
<p>Positions of the places of supply of GAMA, Inc. Secovce in eastern Slovakia.</p> Full article ">Figure 15
<p>Setting the required location calculation parameters, and the parameters of the raw material quantities and unit prices to be inserted into the expert system.</p> Full article ">Figure 16
<p>Graphical display of the solution of the case study location problem, obtained using the computer-aided location expert system.</p> Full article ">Figure 17
<p>Graphical display of results of the objective function values calculating iterations from five different starting points.</p> Full article ">
18 pages, 1719 KiB  
Article
Exploring the Effect of Status Quo, Innovativeness, and Involvement Tendencies on Luxury Fashion Innovations: The Mediation Role of Status Consumption
by Andreea-Ionela Puiu, Anca Monica Ardeleanu, Camelia Cojocaru and Anca Bratu
Mathematics 2021, 9(9), 1051; https://doi.org/10.3390/math9091051 - 7 May 2021
Cited by 3 | Viewed by 4480
Abstract
The article explores the mechanisms that affect consumers’ interest in luxury clothing innovations. The actual research aims to investigate the effect of status quo and clothing involvement on consumer brand loyalty. More, it was intended to quantify the influence of the level of [...] Read more.
The article explores the mechanisms that affect consumers’ interest in luxury clothing innovations. The actual research aims to investigate the effect of status quo and clothing involvement on consumer brand loyalty. More, it was intended to quantify the influence of the level of engagement concerning clothing acquisition and the status quo tendency on the consumers’ level of interest toward innovative luxury fashion products. The models were analyzed through the partial-least square-path modeling method. The results revealed that status quo bias and consumers’ involvement in fashion influence their loyalty to brands and level of innovativeness. The novelty of the present research comes from the analysis of the impact of the status quo manifest variables on consumers’ innovative tendencies. Moreover, it was found that status consumption fully mediates the relationships among the investigated predictors and considered outcome variables. The mediator manifests the highest effect size of all investigated predictors. The actual paper advances research in a direction that was not sufficiently addressed in the past, introducing the status quo construct as the main predictor of peoples’ inclination to be loyal to a brand or to manifest a tendency toward innovativeness. Moreover, the article emphasizes the essential role manifested by social status in foreseeing a behavioral response. Full article
(This article belongs to the Special Issue Partial Least Squares Structural Equation Modeling (PLS-SEM) Applications in Economics and Finance)
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<p>The research model.</p> Full article ">Figure A1
<p>The first structural equation model—clothing brand loyalty (CBL) is the outcome variable.</p> Full article ">Figure A2
<p>The second structural equation model—innovativeness interest (CIN<sub>a</sub>) is the outcome variable.</p> Full article ">Figure A3
<p>The third structural equation model—innovativeness awareness (CIN<sub>b</sub>) is the outcome variable.</p> Full article ">
9 pages, 264 KiB  
Article
Characterization of Frequency Domains of Bandlimited Frame Multiresolution Analysis
by Zhihua Zhang
Mathematics 2021, 9(9), 1050; https://doi.org/10.3390/math9091050 - 7 May 2021
Cited by 5 | Viewed by 1472
Abstract
Framelets have been widely used in narrowband signal processing, data analysis, and sampling theory, due to their resilience to background noise, stability of sparse reconstruction, and ability to capture local time-frequency information. The well-known approach to construct framelets with useful properties is through [...] Read more.
Framelets have been widely used in narrowband signal processing, data analysis, and sampling theory, due to their resilience to background noise, stability of sparse reconstruction, and ability to capture local time-frequency information. The well-known approach to construct framelets with useful properties is through frame multiresolution analysis (FMRA). In this article, we characterize the frequency domain of bandlimited FMRAs: there exists a bandlimited FMRA with the support of frequency domain G if and only if G satisfies G⊂2G, ⋃m2mG≅Rd, and G\G2⋂G2+2πν≅∅(ν∈Zd). Full article
(This article belongs to the Section Computational and Applied Mathematics)
20 pages, 48474 KiB  
Article
Optimal Design of High-Voltage Disconnecting Switch Drive System Based on ADAMS and Particle Swarm Optimization Algorithm
by Benxue Liu, Peng Yuan, Mengjian Wang, Cheng Bi, Chong Liu and Xia Li
Mathematics 2021, 9(9), 1049; https://doi.org/10.3390/math9091049 - 6 May 2021
Cited by 4 | Viewed by 3052
Abstract
This paper focuses on the analysis of the stability of the GW17 high-voltage disconnecting switch drive system. Firstly, the optimization model of the disconnector is established, and the simulation analysis is carried out by ADAMS (Automatic Dynamic Analysis of Mechanical Systems) and the [...] Read more.
This paper focuses on the analysis of the stability of the GW17 high-voltage disconnecting switch drive system. Firstly, the optimization model of the disconnector is established, and the simulation analysis is carried out by ADAMS (Automatic Dynamic Analysis of Mechanical Systems) and the simulation results are verified by experiments. Afterwards, ADAMS optimization design and particle swarm optimization algorithm (PSO) are used to optimize the drive system of the disconnector, and the results are verified on the experimental platform. After optimization, the space rod is reduced by 15 mm, the minimum corner angle of the lower conductive rod is reduced by 71.0%, the minimum folding arm angle is reduced by 88.7% and the maximum force of the ball pair is reduced by 35.7%, which realizes the lightweight of the rod, reduces the wear of the ball pair, and improves the stability of the equipment operation. Full article
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<p>Structure composition diagram of high-voltage isolating switch.</p> Full article ">Figure 2
<p>Structure composition diagram of high-voltage isolating switch.</p> Full article ">Figure 3
<p>The spatial four-bar linkage model of each position.</p> Full article ">Figure 4
<p>Schematic diagram of the four-bar mechanism in the closed state. (<b>a</b>) Main view; (<b>b</b>) Top view.</p> Full article ">Figure 5
<p>Schematic diagram of the four-bar mechanism in the open state. (<b>a</b>) Main view; (<b>b</b>) Top view.</p> Full article ">Figure 6
<p>Motion diagram of folding arm system.</p> Full article ">Figure 7
<p>Solid model of GW17 disconnector.</p> Full article ">Figure 8
<p>Simulation result. (<b>a</b>) Included angle of each rod; (<b>b</b>) Force change of ball pair.</p> Full article ">Figure 9
<p>Installation position of angular displacement sensor.</p> Full article ">Figure 10
<p>Simulation and test data. (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>θ</mi> <mtext> </mtext> </mrow> </semantics></math>; (<b>b</b>) <math display="inline"><semantics> <mi>ϖ</mi> </semantics></math>.</p> Full article ">Figure 11
<p>Schematic diagram of swing arm structure.</p> Full article ">Figure 12
<p>Change curves of each parameter. (<b>a</b>) Variation curve of <math display="inline"><semantics> <mi>θ</mi> </semantics></math> with AB′; (<b>b</b>) Variation curve of <math display="inline"><semantics> <mi>θ</mi> </semantics></math> with BB′; (<b>c</b>) Variation curve of <math display="inline"><semantics> <mrow> <msub> <mi>θ</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> <mtext> </mtext> </mrow> </msub> </mrow> </semantics></math> with AB′ and BB′.</p> Full article ">Figure 13
<p>Change curves of each parameter. (<b>a</b>) <math display="inline"><semantics> <mrow> <msub> <mi>θ</mi> <mi>c</mi> </msub> <mo> </mo> </mrow> </semantics></math>; (<b>b</b>) <math display="inline"><semantics> <mi>ϖ</mi> </semantics></math>; (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>ε</mi> <mo> </mo> </mrow> </semantics></math>; (<b>d</b>) <math display="inline"><semantics> <mrow> <msub> <mi>F</mi> <mi>B</mi> </msub> </mrow> </semantics></math>.</p> Full article ">Figure 14
<p>Flow chart of particle swarm optimization algorithm.</p> Full article ">Figure 15
<p>Change curves of each parameter. (<b>a</b>) <math display="inline"><semantics> <mrow> <msub> <mi>θ</mi> <mi>c</mi> </msub> <mo> </mo> </mrow> </semantics></math>; (<b>b</b>) <math display="inline"><semantics> <mi>ϖ</mi> </semantics></math>; (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>ε</mi> <mo> </mo> </mrow> </semantics></math>; (<b>d</b>) <math display="inline"><semantics> <mrow> <msub> <mi>F</mi> <mi>B</mi> </msub> <mo> </mo> </mrow> </semantics></math>.</p> Full article ">Figure 16
<p>Experimental platform.</p> Full article ">
11 pages, 3301 KiB  
Article
Dynamic Analysis of a Fiber-Reinforced Composite Beam under a Moving Load by the Ritz Method
by Şeref D. Akbaş, Hakan Ersoy, Bekir Akgöz and Ömer Civalek
Mathematics 2021, 9(9), 1048; https://doi.org/10.3390/math9091048 - 6 May 2021
Cited by 105 | Viewed by 5007
Abstract
This paper presents the dynamic responses of a fiber-reinforced composite beam under a moving load. The Timoshenko beam theory was employed to analyze the kinematics of the composite beam. The constitutive equations for motion were obtained by utilizing the Lagrange procedure. The Ritz [...] Read more.
This paper presents the dynamic responses of a fiber-reinforced composite beam under a moving load. The Timoshenko beam theory was employed to analyze the kinematics of the composite beam. The constitutive equations for motion were obtained by utilizing the Lagrange procedure. The Ritz method with polynomial functions was employed to solve the resulting equations in conjunction with the Newmark average acceleration method (NAAM). The influence of fiber orientation angle, volume fraction, and velocity of the moving load on the dynamic responses of the fiber-reinforced nonhomogeneous beam is presented and discussed. Full article
(This article belongs to the Special Issue Numerical Analysis and Scientific Computing)
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<p>A simply supported FRC beam under a moving load.</p> Full article ">Figure 2
<p>Convergence of maximum dimensionless dynamic displacements (<math display="inline"> <semantics> <mrow> <msub> <mover accent="true"> <mi>v</mi> <mo>¯</mo> </mover> <mi>m</mi> </msub> </mrow> </semantics> </math>) of the FRC beam under a moving load for <span class="html-italic">V<sub>f</sub></span> = 0.3, <span class="html-italic">θ</span> = 10°, <span class="html-italic">V<sub>Q</sub></span> = 10 m/s.</p> Full article ">Figure 3
<p>Dynamic lateral displacements at midspan (v<sub>m</sub>) of a homogeneous and isotropic beam under a moving load.</p> Full article ">Figure 4
<p>Time history of dimensionless dynamic displacements at midspan of the FRC beam under a moving load for different values of the fiber orientation angle (<math display="inline"> <semantics> <mi>θ</mi> </semantics> </math>).</p> Full article ">Figure 4 Cont.
<p>Time history of dimensionless dynamic displacements at midspan of the FRC beam under a moving load for different values of the fiber orientation angle (<math display="inline"> <semantics> <mi>θ</mi> </semantics> </math>).</p> Full article ">Figure 5
<p>Time history of dimensionless dynamic displacements at midspan of the FRC beam under a moving load for different values of the volume fraction of fiber (<span class="html-italic">V<sub>f</sub></span>).</p> Full article ">Figure 5 Cont.
<p>Time history of dimensionless dynamic displacements at midspan of the FRC beam under a moving load for different values of the volume fraction of fiber (<span class="html-italic">V<sub>f</sub></span>).</p> Full article ">
25 pages, 2864 KiB  
Article
Alternative Financial Methods for Improving the Investment in Renewable Energy Companies
by José Luis Miralles-Quirós and María Mar Miralles-Quirós
Mathematics 2021, 9(9), 1047; https://doi.org/10.3390/math9091047 - 6 May 2021
Cited by 2 | Viewed by 2647
Abstract
Renewable energies have increased in importance in recent years due to the harm caused to the environment by fossil fuels. As a result, renewable energy companies seem to be profitable investment opportunities given their likely substantial future earnings. However, previous empirical evidence has [...] Read more.
Renewable energies have increased in importance in recent years due to the harm caused to the environment by fossil fuels. As a result, renewable energy companies seem to be profitable investment opportunities given their likely substantial future earnings. However, previous empirical evidence has not always agreed about this likely profitability. In addition, the methodologies employed in the existing empirical literature are complicated and not feasible for most investors to use. Therefore, it is proposed an approach which combines the use of performance measures, screening rules, devolatized returns and portfolio strategies, all of which can be implemented by investors. This approach results in high cumulative returns of more than 200% and other positive ratios, even when transaction costs are considered. This should encourage people to invest in these renewable energies and contribute to improving the environment. Full article
(This article belongs to the Special Issue Mathematical and Statistical Methods Applications in Finance)
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<p>Daily closing prices graphs.</p> Full article ">Figure 1 Cont.
<p>Daily closing prices graphs.</p> Full article ">Figure 2
<p>Cumulative returns of strategies. This figure displays the cumulative returns over the out-of-sample period for the naïve strategy, the best strategy using logarithm returns, labeled as initial, and the best one using standardized returns, labeled as final.</p> Full article ">Figure 3
<p>Cumulative returns of strategies (5-year rolling window). This figure displays the 5-year rolling window cumulative returns over the out-of-sample period for the naïve strategy, the best strategy using logarithm returns, labeled as initial, and the best one using standardized returns, labeled as final.</p> Full article ">Figure 4
<p>Sharpe ratios of strategies (5-year rolling window). This figure displays the 5-year rolling window Sharpe ratios over the out-of-sample period for the naïve strategy, the best strategy using logarithm returns, labeled as initial, and the best one using standardized returns, labeled as final.</p> Full article ">Figure 5
<p>Sortino ratios of strategies (5-year rolling window). This figure displays the 5-year rolling window Sortino ratios over the out-of-sample period for the naïve strategy, the best strategy using logarithm returns, labeled as initial, and the best one using standardized returns, labeled as final.</p> Full article ">Figure 6
<p>Omega ratios of strategies (5-year rolling window). This figure displays the 5-year rolling window Omega ratios over the out-of-sample period for the naïve strategy, the best strategy using logarithm returns, labeled as initial, and the best one using standardized returns, labeled as final.</p> Full article ">
17 pages, 4413 KiB  
Article
Time Series Clustering with Topological and Geometric Mixed Distance
by Yunsheng Zhang, Qingzhang Shi, Jiawei Zhu, Jian Peng and Haifeng Li
Mathematics 2021, 9(9), 1046; https://doi.org/10.3390/math9091046 - 6 May 2021
Cited by 6 | Viewed by 4118
Abstract
Time series clustering is an essential ingredient of unsupervised learning techniques. It provides an understanding of the intrinsic properties of data upon exploiting similarity measures. Traditional similarity-based methods usually consider local geometric properties of raw time series or the global topological properties of [...] Read more.
Time series clustering is an essential ingredient of unsupervised learning techniques. It provides an understanding of the intrinsic properties of data upon exploiting similarity measures. Traditional similarity-based methods usually consider local geometric properties of raw time series or the global topological properties of time series in the phase space. In order to overcome their limitations, we put forward a time series clustering framework, referred to as time series clustering with Topological-Geometric Mixed Distance (TGMD), which jointly considers local geometric features and global topological characteristics of time series data. More specifically, persistent homology is employed to extract topological features of time series and to compute topological similarities among persistence diagrams. The geometric properties of raw time series are captured by using shape-based similarity measures such as Euclidean distance and dynamic time warping. The effectiveness of the proposed TGMD method is assessed by extensive experiments on synthetic noisy biological and real time series data. The results reveal that the proposed mixed distance-based similarity measure can lead to promising results and that it performs better than standard time series analysis techniques that consider only topological or geometrical similarity. Full article
(This article belongs to the Special Issue Data Mining for Temporal Data Analysis)
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<p>Schematic diagram of the proposed time series clustering method with Topological-Geometric Mixed Distance.</p> Full article ">Figure 2
<p>(<b>a</b>–<b>d</b>) Time series S1–S4 in the original space; (<b>e</b>–<b>h</b>) Time series S1–S4 in the phase space.</p> Full article ">Figure 3
<p>An example of VR complexes with resulting persistence diagram. We consider the 1-dimension hole (<math display="inline"><semantics> <mrow> <msub> <mi>H</mi> <mn>1</mn> </msub> </mrow> </semantics></math>) here. From (<b>a</b>–<b>f</b>), we let the radius <math display="inline"><semantics> <mi>ε</mi> </semantics></math> grow gradually. As <math display="inline"><semantics> <mi>ε</mi> </semantics></math> increases, a 1-dimension hole appears and disappears in the region. The first 1-dimensional hole (yellow loop) appears at (<b>c</b>), and the second (red loop) appears at (<b>d</b>). If we continue increasing <math display="inline"><semantics> <mi>ε</mi> </semantics></math>, then the yellow loop disappears at (<b>e</b>), and the red loop disappears at (<b>f</b>). (<b>g</b>) Persistence diagram corresponding to topological changes in the previous VR complex. There are two persistence points <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mrow> <mn>0.28</mn> <mo>,</mo> <mo> </mo> <mn>0.48</mn> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mrow> <mn>0.37</mn> <mo>,</mo> <mo> </mo> <mn>0.82</mn> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math>, which represent the birth and death of the yellow loop and the red loop, respectively. A persistence diagram is a collection of all persistence points in the filtration.</p> Full article ">Figure 4
<p>The Wasserstein distance is computed by matching all points. If no corresponding matching point is found, it will match to diagonal.</p> Full article ">Figure 5
<p>The adaptive tuning effect f(x).</p> Full article ">Figure 6
<p>(<b>a</b>,<b>d</b>) are two examples of non-oscillatory (<b>a</b>) and oscillatory (<b>d</b>) time series data generated by the Hes1 model. (<b>b</b>,<b>e</b>) are the visualization results of the time series data after delay embedding and projected to a two-dimensional plot by PCA. (<b>c</b>,<b>f</b>) are the corresponding one-dimensional persistence diagrams. (<b>c</b>) corresponds to the non-oscillatory data of (<b>a</b>). And (<b>f</b>) corresponds to the oscillatory time-series in (<b>d</b>).</p> Full article ">Figure 7
<p>Clustering results for synthetic single-cell data with different intervals. (<b>a</b>–<b>d</b>): the results with <math display="inline"><semantics> <mrow> <mi>v</mi> <mo>=</mo> <mn>5</mn> <mo>,</mo> <mo> </mo> <mn>10</mn> <mo>,</mo> <mo> </mo> <mn>15</mn> <mo>,</mo> <mo> </mo> <mn>20</mn> </mrow> </semantics></math>. Clustering results are evaluated by ARI (yellow) and silhouette coefficient (blue).</p> Full article ">Figure 8
<p>Clustering results for synthetic single-cell data with different noises.</p> Full article ">Figure 9
<p>The TGMD metric space with different adjustment parameter visualized using UMAP. (<b>a</b>–<b>f</b>): the results with <math display="inline"><semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo> </mo> <mn>2</mn> <mo>,</mo> <mo> </mo> <mn>3</mn> <mo>,</mo> <mo> </mo> <mn>4</mn> <mo>,</mo> <mo> </mo> <mn>5</mn> </mrow> </semantics></math>, different colors represent different labels of the data.</p> Full article ">Figure 10
<p>Clustering results by different adjustment parameter <span class="html-italic">k</span>.</p> Full article ">
24 pages, 2425 KiB  
Article
A Model for the Evaluation of Critical IT Systems Using Multicriteria Decision-Making with Elements for Risk Assessment
by Davor Maček, Ivan Magdalenić and Nina Begičević Ređep
Mathematics 2021, 9(9), 1045; https://doi.org/10.3390/math9091045 - 6 May 2021
Cited by 6 | Viewed by 3205
Abstract
One of the important objectives and concerns today is to find efficient means to manage the information security risks to which organizations are exposed. Due to a lack of necessary data and time and resource constraints, very often it is impossible to gather [...] Read more.
One of the important objectives and concerns today is to find efficient means to manage the information security risks to which organizations are exposed. Due to a lack of necessary data and time and resource constraints, very often it is impossible to gather and process all of the required information about an IT system in order to properly assess it within an acceptable timeframe. That puts the organization into a state of increased security risk. One of the means to solve such complex problems is the use of multicriteria decision-making methods that have a strong mathematical foundation. This paper presents a hybrid multicriteria model for the evaluation of critical IT systems where the elements for risk analysis and assessment are used as evaluation criteria. The iterative steps of the design science research (DSR) methodology for development of a new multicriteria model for the objectives of evaluation, ranking, and selection of critical information systems are delineated. The main advantage of the new model is its use of generic criteria for risk assessment instead of redefining inherent criteria and calculating related weights for each individual IT system. That is why more efficient evaluation, ranking, and decision-making between several possible IT solutions can be expected. The proposed model was validated in a case study of online banking transaction systems and could be used as a generic model for the evaluation of critical IT systems. Full article
(This article belongs to the Special Issue Recent Process on Strategic Planning and Decision Making)
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<p>Research process and scientific methods.</p> Full article ">Figure 2
<p>Experts’ attitudes on the ISRA criteria.</p> Full article ">Figure 3
<p>Matrix for estimating influences and dependencies between ISRA criteria.</p> Full article ">Figure 4
<p>High-level components used in development of a new multicriteria model.</p> Full article ">Figure 5
<p>Multicriteria model with ISRA elements for the evaluation of critical IT systems.</p> Full article ">Figure 6
<p>Matrix for estimating influences and dependencies between inherent critera for transaction systems.</p> Full article ">Figure 7
<p>Evaluation steps.</p> Full article ">Figure A1
<p>Research questionnaire for the first Delphi round. Five risk assessment attributes were presented for evaluation (closed questions), while one open question was included for IT security experts to consider any additional elements that could be part of the new hybrid multicriteria model.</p> Full article ">Figure A2
<p>Set of the individual questions for IT security experts necessary to obtain the level of competence of respondents in the domain of information security.</p> Full article ">Figure A3
<p>Research questionnaire for the second Delphi round, which presented statistics from the first Delphi round and also included an exploitability (E) element for rating as a potential additional criterion for the new multicriteria model.</p> Full article ">
12 pages, 347 KiB  
Article
Secure HIGHT Implementation on ARM Processors
by Hwajeong Seo, Hyunjun Kim, Kyungbae Jang, Hyeokdong Kwon, Minjoo Sim, Gyeongju Song, Siwoo Uhm and Hyunji Kim
Mathematics 2021, 9(9), 1044; https://doi.org/10.3390/math9091044 - 6 May 2021
Cited by 1 | Viewed by 2011
Abstract
Secure and compact designs of HIGHT block cipher on representative ARM microcontrollers are presented in this paper. We present several optimizations for implementations of the HIGHT block cipher, which exploit different parallel approaches, including task parallelism and data parallelism methods, for high-speed and [...] Read more.
Secure and compact designs of HIGHT block cipher on representative ARM microcontrollers are presented in this paper. We present several optimizations for implementations of the HIGHT block cipher, which exploit different parallel approaches, including task parallelism and data parallelism methods, for high-speed and high-throughput implementations. For the efficient parallel implementation of the HIGHT block cipher, the SIMD instructions of ARM architecture are fully utilized. These instructions support four-way 8-bit operations in the parallel way. The length of primitive operations in the HIGHT block cipher is 8-bit-wise in addition–rotation–exclusive-or operations. In the 32-bit word architecture (i.e., the 32-bit ARM architecture), four 8-bit operations are executed at once with the four-way SIMD instruction. By exploiting the SIMD instruction, three parallel HIGHT implementations are presented, including task-parallel, data-parallel, and task/data-parallel implementations. In terms of the secure implementation, we present a fault injection countermeasure for 32-bit ARM microcontrollers. The implementation ensures the fault detection through the representation of intra-instruction redundancy for the data format. In particular, we proposed two fault detection implementations by using parallel implementations. The two-way task/data-parallel based implementation is secure against fault injection models, including chosen bit pair, random bit, and random byte. The alternative four-way data-parallel-based implementation ensures all security features of the aforementioned secure implementations. Moreover, the instruction skip model is also prevented. The implementation of the HIGHT block cipher is further improved by using the constant value of the counter mode of operation. In particular, the 32-bit nonce value is pre-computed and the intermediate result is directly utilized. Finally, the optimized implementation achieved faster execution timing and security features toward the fault attack than previous works. Full article
(This article belongs to the Special Issue Recent Advances in Security, Privacy, and Applied Cryptography)
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<p>Encryption of the HIGHT algorithm; X and SK indicate plaintext and round key, respectively.</p> Full article ">Figure 2
<p>Counter mode of operation for HIGHT, (<b>left</b>) red colored routes represent the constant nonce part, (<b>right</b>) blue colored routes represent the optimized part.</p> Full article ">
19 pages, 707 KiB  
Article
Mechanical Models for Hermite Interpolation on the Unit Circle
by Elías Berriochoa, Alicia Cachafeiro, Héctor García Rábade and José Manuel García-Amor
Mathematics 2021, 9(9), 1043; https://doi.org/10.3390/math9091043 - 6 May 2021
Cited by 4 | Viewed by 2023
Abstract
In the present paper, we delve into the study of nodal systems on the unit circle that meet certain separation properties. Our aim was to study the Hermite interpolation process on the unit circle by using these nodal arrays. The target was to [...] Read more.
In the present paper, we delve into the study of nodal systems on the unit circle that meet certain separation properties. Our aim was to study the Hermite interpolation process on the unit circle by using these nodal arrays. The target was to develop the corresponding interpolation theory in order to make practical use of these nodal systems linked to certain mechanical models that fit these distributions. Full article
(This article belongs to the Special Issue Mathematical Methods, Modelling and Applications)
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Figure 1
<p>Left: representation of <math display="inline"><semantics> <mrow> <mi>f</mi> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mo>ℜ</mo> <mo>(</mo> <msub> <mi mathvariant="script">H</mi> <mrow> <mo>−</mo> <mi>n</mi> <mo>,</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>f</mi> <mo>,</mo> <mi>z</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </semantics></math>. Right: representation of <math display="inline"><semantics> <mrow> <mo>ℜ</mo> <mo>(</mo> <msub> <mi mathvariant="script">H</mi> <mrow> <mo>−</mo> <mi>n</mi> <mo>,</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>f</mi> <mo>,</mo> <mi>z</mi> <mo>)</mo> </mrow> <mo>)</mo> <mo>−</mo> <mi>f</mi> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>, with <math display="inline"><semantics> <mstyle scriptlevel="0" displaystyle="true"> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> <mo>=</mo> <mstyle displaystyle="true"> <munderover> <mo>∑</mo> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mo>∞</mo> </munderover> </mstyle> <mfrac> <mn>1</mn> <msup> <mi>k</mi> <mn>6</mn> </msup> </mfrac> <mrow> <mo>(</mo> <msup> <mi>z</mi> <mi>k</mi> </msup> <mo>+</mo> <msup> <mi>z</mi> <mrow> <mo>−</mo> <mi>k</mi> </mrow> </msup> <mo>)</mo> </mrow> </mrow> </mstyle> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>=</mo> <msup> <mi>e</mi> <mrow> <mi>i</mi> <mi>θ</mi> </mrow> </msup> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>θ</mi> <mo>∈</mo> <mo>[</mo> <mn>0</mn> <mo>,</mo> <mn>2</mn> <mi>π</mi> <mo>]</mo> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>40</mn> </mrow> </semantics></math>.</p> Full article ">Figure 2
<p><math display="inline"><semantics> <mrow> <mi>f</mi> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mo>ℜ</mo> <mo>(</mo> <msub> <mi mathvariant="script">HF</mi> <mrow> <mo>−</mo> <mi>n</mi> <mo>,</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>f</mi> <mo>,</mo> <mi>z</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </semantics></math> with the nodes marked on the lines, <math display="inline"><semantics> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mi>z</mi> <mo>+</mo> <mfrac> <mn>1</mn> <mi>z</mi> </mfrac> </mrow> <mn>2</mn> </mfrac> <mo form="prefix">sin</mo> <mfenced separators="" open="(" close=")"> <mfrac> <mn>2</mn> <mrow> <mi>z</mi> <mo>+</mo> <mfrac> <mn>1</mn> <mi>z</mi> </mfrac> </mrow> </mfrac> </mfenced> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>400</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>=</mo> <msup> <mi>e</mi> <mrow> <mi>i</mi> <mi>θ</mi> </mrow> </msup> </mrow> </semantics></math>. Left: <math display="inline"><semantics> <mrow> <mi>θ</mi> <mo>∈</mo> <mo>[</mo> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> <mo>,</mo> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>4</mn> </mfrac> <mo>]</mo> </mrow> </semantics></math>. Right: <math display="inline"><semantics> <mrow> <mi>θ</mi> <mo>∈</mo> <mo>[</mo> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> <mo>,</mo> <mfrac> <mrow> <mn>9</mn> <mi>π</mi> </mrow> <mn>16</mn> </mfrac> <mo>]</mo> </mrow> </semantics></math>.</p> Full article ">Figure 3
<p>Left: representation of <math display="inline"><semantics> <mrow> <mi>f</mi> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mo>ℜ</mo> <mo>(</mo> <msub> <mi mathvariant="script">H</mi> <mrow> <mo>−</mo> <mi>n</mi> <mo>,</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>f</mi> <mo>,</mo> <mi>z</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </semantics></math>. Right: representation of <math display="inline"><semantics> <mrow> <mo>ℜ</mo> <mo>(</mo> <msub> <mi mathvariant="script">H</mi> <mrow> <mo>−</mo> <mi>n</mi> <mo>,</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>f</mi> <mo>,</mo> <mi>z</mi> <mo>)</mo> </mrow> <mo>)</mo> <mo>−</mo> <mi>f</mi> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>, with <math display="inline"><semantics> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mrow> <mo>(</mo> <mi>z</mi> <mo>−</mo> <mn>1.02</mn> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mfrac> <mn>1</mn> <mi>z</mi> </mfrac> <mo>−</mo> <mn>1.02</mn> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mi>z</mi> <mo>+</mo> <mn>1.2</mn> <mi>ı</mi> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mfrac> <mn>1</mn> <mi>z</mi> </mfrac> <mo>−</mo> <mn>1.2</mn> <mi>ı</mi> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>=</mo> <msup> <mi>e</mi> <mrow> <mi>i</mi> <mi>θ</mi> </mrow> </msup> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>1000</mn> </mrow> </semantics></math>.</p> Full article ">
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