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Symmetry
Volume 13
Issue 12
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Symmetry, Volume 13, Issue 12 (December 2021) – 221 articles

Cover Story (view full-size image): Laws of large numbers (LLNs) are theorems of great importance for probability and chance theory. Within chance theory the notion of an uncertain random variable was defined to model phenomena where both uncertainty and randomness are present. This work considers uncertain random variables as functions of random variables and uncertain variables, introduced within uncertainty theory. Sequences of uncertain random variables are defined for a fixed continuous function, strictly monotone with respect to its second argument, as are sequences of pairwise independent or independent, identically distributed random variables and regular, independent, and identically distributed uncertain variables. Some LLNs for such defined sequences of uncertain random variables are formulated and proven. Two of their counterparts for uncertain variables are also discussed. View this paper
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13 pages, 1423 KiB  
Article
Modelling Artificial Immunization Processes to Counter Cyberthreats
by Dmitry Zegzhda, Evgeny Pavlenko and Elena Aleksandrova
Symmetry 2021, 13(12), 2453; https://doi.org/10.3390/sym13122453 - 20 Dec 2021
Cited by 9 | Viewed by 2594
Abstract
This paper looks at the problem of cybersecurity in modern cyber-physical and information systems and proposes an immune-like approach to the information security of modern complex systems. This approach is based on the mathematical modeling in information security—in particular, the use of immune [...] Read more.
This paper looks at the problem of cybersecurity in modern cyber-physical and information systems and proposes an immune-like approach to the information security of modern complex systems. This approach is based on the mathematical modeling in information security—in particular, the use of immune methods to protect several critical system nodes from a predetermined range of attacks, and to minimize the success of an attack on the system. The methodological approach is to systematize the tasks, means and modes of immunization to describe how modern systems can counter the spread of computer attacks. The main conclusions and recommendations are that using an immunization approach will not only improve the security of systems, but also define principles for building systems that are resistant to cyber attacks. The immunization approach enables a symmetrical response to an intruder in a protected system to be produced rapidly. This symmetry provides a step-by-step neutralization of all stages of a cyber attack, which, combined with the accumulation of knowledge of the attacker’s actions, allows a base of defensive responses to be generated for various cyber attack scenarios. The theoretical conclusions are supported by practical experiments describing real-world scenarios for the use of immunization tools to protect against cyber threats. Full article
(This article belongs to the Section Computer)
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Figure 1

Figure 1
<p>Systematization of immunization means.</p> Full article ">Figure 2
<p>A node’s lifecycle.</p> Full article ">Figure 3
<p>Transitions cycle.</p> Full article ">Figure 4
<p>Experiments for different immunization types.</p> Full article ">Figure 5
<p>System component state during immunization process.</p> Full article ">
17 pages, 308 KiB  
Article
Information Reliability Assessment and Filtering for Decision Making. Case Study of Lithuanian and China Students’ Life Goals Survey Data
by Aleksandras Krylovas, Natalja Kosareva and Stanislav Dadelo
Symmetry 2021, 13(12), 2452; https://doi.org/10.3390/sym13122452 - 20 Dec 2021
Viewed by 2113
Abstract
This article presents the methodology and tools to evaluate the reliability of quantitative sociological research data. The problem of filtering unreliable data is usually solved by statistical methods. This article proposes an improved method for filtering unreliable data. In this case, the statistical [...] Read more.
This article presents the methodology and tools to evaluate the reliability of quantitative sociological research data. The problem of filtering unreliable data is usually solved by statistical methods. This article proposes an improved method for filtering unreliable data. In this case, the statistical methods are not applied to the initial data but the value of the distance function between the two preferences. This allows for the disclosure of conflicting or erroneous data. Calculation of the distance between two preferences and prioritisation of life goals are based on binary relation theory, where the properties of symmetry (antisymmetry) are very important. The article presents a case study on 11 life goals evaluation and ranking by Lithuanian and China students. The study revealed that the China student data filtered at least twice as much as the Lithuanian student data, i.e., they are less reliable. The filtered data show that students of both countries ranked the most and the least important life goals in a very similar way with minimum deviations detected in the ranking results. Full article
(This article belongs to the Special Issue Symmetric and Asymmetric Data in Solution Models, Part II)
14 pages, 7933 KiB  
Article
Stability Characteristics and Mechanism of U-Shaped Metal Bellows under Symmetrical Cyclic Tension and Compression Process
by Zengliang Hao, Biao Yao, Yuhang Chen and Junting Luo
Symmetry 2021, 13(12), 2451; https://doi.org/10.3390/sym13122451 - 20 Dec 2021
Cited by 2 | Viewed by 3043
Abstract
The U-shaped metal bellows expansion joint compensates for the pipeline displacement by its own deformation. The compensation performance of the metal bellows in the initial stage of tension and compression deformation is unstable. In this paper, the symmetrical cyclic tension and compression (SCTC) [...] Read more.
The U-shaped metal bellows expansion joint compensates for the pipeline displacement by its own deformation. The compensation performance of the metal bellows in the initial stage of tension and compression deformation is unstable. In this paper, the symmetrical cyclic tension and compression (SCTC) process of metal bellows was simulated by ABAQUS software. Then, the SCTC process experiment of metal bellows was completed on the universal material testing machine. The distribution law of axial load with displacement and that of axial stiffness and yield load with cycles of metal bellows were obtained. Finally, the X-ray diffraction peak confirmed the deformation-induced martensite in the wave trough and proved that the plastic strain and hardness values of metal bellows increased with the displacement amplitude. The microstructure in the wave trough area was observed by a Zeiss microscope, and the stability characteristics mechanism of the metal bellows was revealed. The martensite in the wave trough increases the grain boundary area under SCTC loading. The forward movement of the slip band in the grain caused by large deformation reached an equilibrium state with the resistance at the grain boundary, which promotes the macroscopic mechanical properties of the metal bellows to be stable characteristics under SCTC loading. Full article
(This article belongs to the Special Issue Symmetry/Asymmetry in Metal Forming)
►▼ Show Figures

Figure 1

Figure 1
<p>Schematic of the tensile specimen.</p> Full article ">Figure 2
<p>The schematic diagram of the SCTC process of metal bellows: (<b>a</b>) metal bellows model; (<b>b</b>) SCTC process.</p> Full article ">Figure 3
<p>The finite element model of metal bellows: (<b>a</b>) boundary conditions; (<b>b</b>) meshed model.</p> Full article ">Figure 4
<p>The SCTC experiment device.</p> Full article ">Figure 5
<p>Mechanical properties of 316L stainless steel.</p> Full article ">Figure 6
<p>The equivalent plastic strain distribution nephogram of the metal bellows under the 30th SCTC loading: (<b>a</b>) ±6% <span class="html-italic">L</span>; (<b>b</b>) ±8% <span class="html-italic">L</span>; (<b>c</b>) ±10% <span class="html-italic">L</span>; (<b>d</b>) ±12% <span class="html-italic">L</span>.</p> Full article ">Figure 7
<p>The distribution law of the maximum equivalent stress and strain values of the wave trough node with cycles: (<b>a</b>) equivalent stress; (<b>b</b>) equivalent strain.</p> Full article ">Figure 8
<p>The distribution curve of the axial load with displacement of the metal bellows under single symmetrical tension and compression.</p> Full article ">Figure 9
<p>The hysteresis loop of metal bellows during the SCTC process: (<b>a</b>) ±6% <span class="html-italic">L</span><span class="html-italic">;</span> (<b>b</b>) ±8% <span class="html-italic">L</span>; (<b>c</b>) ±10% <span class="html-italic">L</span>.</p> Full article ">Figure 10
<p>The distribution curves of the hysteresis loop width of metal bellows with cycles.</p> Full article ">Figure 11
<p>The comparison diagram of load–displacement curves.</p> Full article ">Figure 12
<p>The distribution curves of the <span class="html-italic">K</span><sub>T</sub> and <span class="html-italic">K</span><sub>C</sub> of the metal bellows with cycles.</p> Full article ">Figure 13
<p>The distribution curves of axial yield load of metal bellows with cycles.</p> Full article ">Figure 14
<p>The hardness distribution curves of the waveform nodes under different displacement amplitudes after stable deformation.</p> Full article ">Figure 15
<p>The XRD diffraction peak distribution curves of the metal bellows.</p> Full article ">Figure 16
<p>The microstructure of the wave trough of the metal bellows after 30 cycles of SCTC under different displacement amplitudes: (<b>a</b>) ±6% <span class="html-italic">L</span>; (<b>b</b>) ±8% <span class="html-italic">L</span>; (<b>c</b>) ±10% <span class="html-italic">L</span>; (<b>d</b>) ±12% <span class="html-italic">L</span>.</p> Full article ">
19 pages, 1549 KiB  
Article
A Dynamically Adjusted Subspace Gradient Method and Its Application in Image Restoration
by Jun Huo, Yuping Wu, Guoen Xia and Shengwei Yao
Symmetry 2021, 13(12), 2450; https://doi.org/10.3390/sym13122450 - 20 Dec 2021
Viewed by 2128
Abstract
In this paper, a new subspace gradient method is proposed in which the search direction is determined by solving an approximate quadratic model in which a simple symmetric matrix is used to estimate the Hessian matrix in a three-dimensional subspace. The obtained algorithm [...] Read more.
In this paper, a new subspace gradient method is proposed in which the search direction is determined by solving an approximate quadratic model in which a simple symmetric matrix is used to estimate the Hessian matrix in a three-dimensional subspace. The obtained algorithm has the ability to automatically adjust the search direction according to the feedback from experiments. Under some mild assumptions, we use the generalized line search with non-monotonicity to obtain remarkable results, which not only establishes the global convergence of the algorithm for general functions, but also R-linear convergence for uniformly convex functions is further proved. The numerical performance for both the traditional test functions and image restoration problems show that the proposed algorithm is efficient. Full article
(This article belongs to the Special Issue PDE, Optimization Modeling and Symmetry in Multi-Dimensional Data and Low-Level Vision Tasks)
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Figure 1

Figure 1
<p>Performance profiles for the CPU.</p> Full article ">Figure 2
<p>Performance profiles for the NFG.</p> Full article ">Figure 3
<p>Performance profiles for the NI.</p> Full article ">Figure 4
<p>20% noise. (<b>a</b>) original image; (<b>b</b>) DSCG; (<b>c</b>) TTS; (<b>d</b>) CG_DESCENT.</p> Full article ">Figure 5
<p>50% noise. (<b>a</b>) original image; (<b>b</b>) DSCG; (<b>c</b>) TTS; (<b>d</b>) CG_DESCENT.</p> Full article ">
8 pages, 264 KiB  
Article
On Baryogenesis from a Complex Inflaton
by James Unwin
Symmetry 2021, 13(12), 2449; https://doi.org/10.3390/sym13122449 - 20 Dec 2021
Viewed by 1929
Abstract
We derive the particle asymmetry due to inflationary baryogenesis involving a complex inflaton, obtaining a different result to that in the literature. While asymmetries were found to be significantly smaller than previously calculated, in certain parameter regions, baryogenesis can still be achieved. Full article
(This article belongs to the Special Issue Current Issues in Particle Physics)
16 pages, 301 KiB  
Article
Radius of Star-Likeness for Certain Subclasses of Analytic Functions
by Caihuan Zhang, Mirajul Haq, Nazar Khan, Muhammad Arif, Khurshid Ahmad and Bilal Khan
Symmetry 2021, 13(12), 2448; https://doi.org/10.3390/sym13122448 - 19 Dec 2021
Cited by 2 | Viewed by 2782
Abstract
In this paper, we investigate a normalized analytic (symmetric under rotation) function, f, in an open unit disk that satisfies the condition ℜfzgz>0, for some analytic function, g, with [...] Read more.
In this paper, we investigate a normalized analytic (symmetric under rotation) function, f, in an open unit disk that satisfies the condition ℜfzgz>0, for some analytic function, g, with ℜz+1−2nzgz>0,∀n∈N. We calculate the radius constants for different classes of analytic functions, including, for example, for the class of star-like functions connected with the exponential functions, i.e., the lemniscate of Bernoulli, the sine function, cardioid functions, the sine hyperbolic inverse function, the Nephroid function, cosine function and parabolic star-like functions. The results obtained are sharp. Full article
(This article belongs to the Special Issue Symmetry in Geometric Functions and Mathematical Analysis II)
12 pages, 312 KiB  
Article
Stabilization of Periodical Discrete Feedback Control for Markov Jumping Stochastic Systems
by Zhiyou Liu, Lichao Feng, Xinbin Li, Zhigang Lu and Xianhui Meng
Symmetry 2021, 13(12), 2447; https://doi.org/10.3390/sym13122447 - 19 Dec 2021
Cited by 2 | Viewed by 1969
Abstract
Motivated by the two strategies of intermittent control and discrete feedback control, this paper aims to introduce a periodically intermittent discrete feedback control in the drift part to stabilize an unstable Markov jumping stochastic differential system. It is illustrated that, by the approach [...] Read more.
Motivated by the two strategies of intermittent control and discrete feedback control, this paper aims to introduce a periodically intermittent discrete feedback control in the drift part to stabilize an unstable Markov jumping stochastic differential system. It is illustrated that, by the approach of comparison principle, this can be achieved in the sense of almost sure exponential stability. Further, the stabilization theory is applied to Markov jumping stochastic recurrent neural networks. Full article
12 pages, 6689 KiB  
Article
A Design of Multi-Purpose Image-Based QR Code
by Shang-Kuan Chen and Yen-Wu Ti
Symmetry 2021, 13(12), 2446; https://doi.org/10.3390/sym13122446 - 18 Dec 2021
Cited by 6 | Viewed by 4559
Abstract
A multi-purpose image-based QR code is designed in this paper. There are four purposes for the generated image-based QR code. In the first purpose, the basic image-based QR code with the look of a host image is with an ingenious layout to be [...] Read more.
A multi-purpose image-based QR code is designed in this paper. There are four purposes for the generated image-based QR code. In the first purpose, the basic image-based QR code with the look of a host image is with an ingenious layout to be identified easier. In the second one, a saliency region detection method is adopted for enhancing the quality of the image-based QR code. In the third one, the host image is embedded into the image-based QR code for further access to the host image; Finally, the visual cryptography-based watermarking method is applied to the host image embedded image-based QR code. In the case that the specific users need verification from the image-based QR code, the binary verified image can be retrieved when the public share is available. The experimental results demonstrate that the generated image-based QR code not only looked better than some previous works but also had high quality host image embedded and identification ability. Full article
(This article belongs to the Special Issue Symmetry in Image Processing and Visualization)
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Figure 1

Figure 1
<p>The basic structure of QR-code.</p> Full article ">Figure 2
<p>(<b>a</b>) the generated QR code (<b>b</b>) original image (<b>c</b>) image-based QR code (<b>d</b>) the recovered image.</p> Full article ">Figure 3
<p>(<b>a</b>) the encoding process (<b>b</b>) the decoding process (<b>c</b>) the embedding processes shown with generated images.</p> Full article ">Figure 4
<p>The generated image-based QR-code of Wakahara et al. [<a href="#B13-symmetry-13-02446" class="html-bibr">13</a>].</p> Full article ">Figure 5
<p>The generated image-based QR-code of Fang et al. [<a href="#B18-symmetry-13-02446" class="html-bibr">18</a>].</p> Full article ">Figure 6
<p>The generated art QR-code of You et al. [<a href="#B19-symmetry-13-02446" class="html-bibr">19</a>].</p> Full article ">Figure 7
<p>The generated QR code.</p> Full article ">Figure 8
<p>(<b>a</b>–<b>d</b>) Host images (<b>e</b>–<b>h</b>) Image-based QR codes.</p> Full article ">Figure 9
<p>(<b>a</b>–<b>d</b>) The image-based QR codes with embedding corresponding host images, (<b>e</b>–<b>h</b>) Corresponding recovered images.</p> Full article ">Figure 9 Cont.
<p>(<b>a</b>–<b>d</b>) The image-based QR codes with embedding corresponding host images, (<b>e</b>–<b>h</b>) Corresponding recovered images.</p> Full article ">Figure 10
<p>(<b>a</b>) a binary color image (<b>b</b>–<b>e</b>) the stego-images that embed the public shares, (<b>f</b>–<b>i</b>) the results taking picture from the printed images of <a href="#symmetry-13-02446-f009" class="html-fig">Figure 9</a>a–d, respectively, (<b>j</b>–<b>m</b>) the solved binary color images from the results (<b>f</b>–<b>i</b>).</p> Full article ">Figure 10 Cont.
<p>(<b>a</b>) a binary color image (<b>b</b>–<b>e</b>) the stego-images that embed the public shares, (<b>f</b>–<b>i</b>) the results taking picture from the printed images of <a href="#symmetry-13-02446-f009" class="html-fig">Figure 9</a>a–d, respectively, (<b>j</b>–<b>m</b>) the solved binary color images from the results (<b>f</b>–<b>i</b>).</p> Full article ">Figure 11
<p>The generated image-based QR code by (<b>a</b>) Wakahara et al. [<a href="#B13-symmetry-13-02446" class="html-bibr">13</a>] and (<b>b</b>) the proposed method.</p> Full article ">Figure 12
<p>The comparison between the generated image-based QR code of (<b>a</b>) Fang et al. [<a href="#B18-symmetry-13-02446" class="html-bibr">18</a>] and (<b>b</b>) the one of the proposed method.</p> Full article ">Figure 13
<p>The comparison between the generated image-based QR code of (<b>a</b>) You et al. [<a href="#B19-symmetry-13-02446" class="html-bibr">19</a>] and (<b>b</b>) the one of the proposed method. (<b>c</b>) the original image.</p> Full article ">
10 pages, 342 KiB  
Article
Coherence Trapping in Open Two-Qubit Dynamics
by Mariam Algarni, Kamal Berrada, Sayed Abdel-Khalek and Hichem Eleuch
Symmetry 2021, 13(12), 2445; https://doi.org/10.3390/sym13122445 - 17 Dec 2021
Cited by 3 | Viewed by 2191
Abstract
In this manuscript, we examine the dynamical behavior of the coherence in open quantum systems using the l1 norm. We consider a two-qubit system that evolves in the framework of Kossakowski-type quantum dynamical semigroups (KTQDSs) of completely positive maps (CPMs). We find [...] Read more.
In this manuscript, we examine the dynamical behavior of the coherence in open quantum systems using the l1 norm. We consider a two-qubit system that evolves in the framework of Kossakowski-type quantum dynamical semigroups (KTQDSs) of completely positive maps (CPMs). We find that the quantum coherence can be asymptotically maintained with respect to the values of the system parameters. Moreover, we show that the quantum coherence can resist the effect of the environment and preserve even in the regime of long times. The obtained results also show that the initially separable states can provide a finite value of the coherence during the time evolution. Because of such properties, several states in this type of environments are good candidates for incorporating quantum information and optics (QIO) schemes. Finally, we compare the dynamical behavior of the coherence with the entire quantum correlation. Full article
(This article belongs to the Special Issue Measurements in Quantum Mechanics)
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Figure 1

Figure 1
<p>Dynamics of the coherence for initially separable states is displayed versus the time <span class="html-italic">t</span> for different values of the parameter <span class="html-italic">m</span>. The dashed blue (<math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>) and dash-dotted red (<math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>) lines correspond to an initially separable pure state <math display="inline"><semantics> <mrow> <mi>W</mi> <mo>(</mo> <mn>0</mn> <mo>)</mo> <mo>=</mo> <mo>|</mo> <mn>0</mn> <mo>〉</mo> <mo>〈</mo> <mn>0</mn> <mo>|</mo> </mrow> </semantics></math>; the solid green (<math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>) and dotted black (<math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>) lines correspond to an initially separable mixed state, <math display="inline"><semantics> <mrow> <mi>W</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> <mo>=</mo> <mn>1</mn> <mo>/</mo> <mn>2</mn> <mfenced separators="" open="(" close=")"> <mo>|</mo> <mn>2</mn> <mo>〉</mo> <mo>〈</mo> <mn>2</mn> <mo>|</mo> <mo>+</mo> <mo>|</mo> <mn>3</mn> <mo>〉</mo> <mo>〈</mo> <mn>3</mn> <mo>|</mo> </mfenced> </mrow> </semantics></math>. The influence of the environment on the measure of coherence <math display="inline"><semantics> <msub> <mi>C</mi> <mi>L</mi> </msub> </semantics></math> is obtained to be similar, and that the dissipation can be able to enhance the amount of the quantum coherence during the dynamics; then, the function <math display="inline"><semantics> <msub> <mi>C</mi> <mi>L</mi> </msub> </semantics></math> reaches a constant value and is preserved even in the regime of long times. Furthermore, the revival rate of the function <math display="inline"><semantics> <msub> <mi>C</mi> <mi>L</mi> </msub> </semantics></math> depends on the choice of the parameter <span class="html-italic">m</span>.</p> Full article ">Figure 2
<p>Dynamics of the coherence for initially entangled quantum states is displayed versus the time <span class="html-italic">t</span> for different values of the parameter <span class="html-italic">m</span>. The dashed blue (<math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>) and dash-dotted red (<math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>) lines correspond to an initially maximally pure state, <math display="inline"><semantics> <mrow> <mrow> <mo>|</mo> <mi mathvariant="sans-serif">Ψ</mi> <mo>〉</mo> </mrow> <mo>=</mo> <mn>1</mn> <mo>/</mo> <msqrt> <mn>2</mn> </msqrt> <mfenced separators="" open="(" close=")"> <mo>|</mo> <mn>01</mn> <mo>〉</mo> <mo>+</mo> <mo>|</mo> <mn>10</mn> <mo>〉</mo> </mfenced> </mrow> </semantics></math>; the solid green (<math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>) and dotted black (<math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>) lines correspond to an initial Horodecki state, <math display="inline"><semantics> <mrow> <mi>W</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mi>a</mi> <mo>|</mo> <mi mathvariant="sans-serif">Ψ</mi> <mo>〉</mo> <mo>〈</mo> <mi mathvariant="sans-serif">Ψ</mi> <mo>|</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>−</mo> <mi>a</mi> <mo>)</mo> </mrow> <mfenced separators="" open="(" close=")"> <mo>|</mo> <mn>0</mn> <mo>〉</mo> <mo>〈</mo> <mn>0</mn> <mo>|</mo> </mfenced> </mrow> </semantics></math>, with <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>0.4</mn> </mrow> </semantics></math>. The influence of the environment on the function <math display="inline"><semantics> <msub> <mi>C</mi> <mi>L</mi> </msub> </semantics></math> is obtained to be similar for initially entangled states, so the function <math display="inline"><semantics> <msub> <mi>C</mi> <mi>L</mi> </msub> </semantics></math> tends to attain a constant value and is preserved even in the regime of long times. The dependence of the quantum coherence on the initial conditions clearly demonstrates that an appropriate choice of system parameters can lead to the enhancement and preservation of quantum coherence during the dynamics.</p> Full article ">Figure 3
<p>Dynamics of the total quantum correlation and coherence for initially separable states is displayed versus the time <span class="html-italic">t</span> with <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>. The dashed blue (total quantum correlation) and dash-dotted red (quantum coherence) lines correspond to an initially separable pure state <math display="inline"><semantics> <mrow> <mi>W</mi> <mo>(</mo> <mn>0</mn> <mo>)</mo> <mo>=</mo> <mo>|</mo> <mn>0</mn> <mo>〉</mo> <mo>〈</mo> <mn>0</mn> <mo>|</mo> </mrow> </semantics></math>; the solid green (total quantum correlation) and dotted black (quantum coherence) lines correspond to an initially separable mixed state, <math display="inline"><semantics> <mrow> <mi>W</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> <mo>=</mo> <mn>1</mn> <mo>/</mo> <mn>2</mn> <mfenced separators="" open="(" close=")"> <mo>|</mo> <mn>2</mn> <mo>〉</mo> <mo>〈</mo> <mn>2</mn> <mo>|</mo> <mo>+</mo> <mo>|</mo> <mn>3</mn> <mo>〉</mo> <mo>〈</mo> <mn>3</mn> <mo>|</mo> </mfenced> </mrow> </semantics></math>. The asymptotic behavior of the functions <math display="inline"><semantics> <msub> <mi>C</mi> <mi>L</mi> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>U</mi> <mi>T</mi> </msub> </semantics></math> is shown to be comparable according to the various parameters that are involved in the state of two qubits, and the quantifiers exhibit the same behavior and maintain their value even over extended periods of time.</p> Full article ">Figure 4
<p>Dynamics of the total quantum correlation and coherence for initially entangled states is displayed versus the time with <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>. The dashed blue (total quantum correlation) and dash-dotted red (quantum coherence) lines correspond to an initially maximally pure state, <math display="inline"><semantics> <mrow> <mrow> <mo>|</mo> <mi mathvariant="sans-serif">Ψ</mi> <mo>〉</mo> </mrow> <mo>=</mo> <mn>1</mn> <mo>/</mo> <msqrt> <mn>2</mn> </msqrt> <mfenced separators="" open="(" close=")"> <mo>|</mo> <mn>01</mn> <mo>〉</mo> <mo>+</mo> <mo>|</mo> <mn>10</mn> <mo>〉</mo> </mfenced> </mrow> </semantics></math>; the solid green (total quantum correlation) and dotted black (quantum coherence) lines correspond to an initial Horodecki state, <math display="inline"><semantics> <mrow> <mi>W</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mi>a</mi> <mo>|</mo> <mi mathvariant="sans-serif">Ψ</mi> <mo>〉</mo> <mo>〈</mo> <mi mathvariant="sans-serif">Ψ</mi> <mo>|</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>−</mo> <mi>a</mi> <mo>)</mo> </mrow> <mfenced separators="" open="(" close=")"> <mo>|</mo> <mn>0</mn> <mo>〉</mo> <mo>〈</mo> <mn>0</mn> <mo>|</mo> </mfenced> </mrow> </semantics></math>, with <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>0.4</mn> </mrow> </semantics></math>. The asymptotic behavior of the functions <math display="inline"><semantics> <msub> <mi>C</mi> <mi>L</mi> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>U</mi> <mi>T</mi> </msub> </semantics></math> is shown to be comparable according to the various parameters that are involved in the state of two qubits, and the quantifiers exhibit the same behavior and maintain their value even over extended periods of time.</p> Full article ">
18 pages, 671 KiB  
Article
A New Generalization of the Student’s t Distribution with an Application in Quantile Regression
by Jimmy Reyes, Mario A. Rojas and Jaime Arrué
Symmetry 2021, 13(12), 2444; https://doi.org/10.3390/sym13122444 - 17 Dec 2021
Cited by 3 | Viewed by 2136
Abstract
In this work, we present a new generalization of the student’s t distribution. The new distribution is obtained by the quotient of two independent random variables. This quotient consists of a standard Normal distribution divided by the power of a chi square distribution [...] Read more.
In this work, we present a new generalization of the student’s t distribution. The new distribution is obtained by the quotient of two independent random variables. This quotient consists of a standard Normal distribution divided by the power of a chi square distribution divided by its degrees of freedom. Thus, the new symmetric distribution has heavier tails than the student’s t distribution and extensions of the slash distribution. We develop a procedure to use quantile regression where the response variable or the residuals have high kurtosis. We give the density function expressed by an integral, we obtain some important properties and some useful procedures for making inference, such as moment and maximum likelihood estimators. By way of illustration, we carry out two applications using real data, in the first we provide maximum likelihood estimates for the parameters of the generalized student’s t distribution, student’s t, the extended slash distribution, the modified slash distribution, the slash distribution generalized student’s t test, and the double slash distribution, in the second we perform quantile regression to fit a model where the response variable presents a high kurtosis. Full article
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<p>Generalized student’s pdf with <math display="inline"><semantics> <mrow> <mi>q</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> (solid line), student’s for <math display="inline"><semantics> <mrow> <mi>ν</mi> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math> pdf (dotted line), Normal pdf (dashed line), <math display="inline"><semantics> <mrow> <mi>G</mi> <mi>S</mi> <mi>L</mi> <mi>T</mi> </mrow> </semantics></math> (dashed and dotted line) and <math display="inline"><semantics> <mrow> <mi>D</mi> <mi>S</mi> <mi>L</mi> </mrow> </semantics></math> (thick dashed line) (<b>left</b>), and tails comparison (<b>right</b>).</p> Full article ">Figure 2
<p>Quantile function of the generalized student’s t distribution compared to quantile function of the student’s t for <math display="inline"><semantics> <mrow> <mi>ν</mi> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math> for <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>=</mo> <mn>0.975</mn> </mrow> </semantics></math> (<b>left</b>) and <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>=</mo> <mn>0.95</mn> </mrow> </semantics></math> (<b>right</b>).</p> Full article ">Figure 3
<p>Densidad de <math display="inline"><semantics> <mrow> <mi>G</mi> <mi>T</mi> </mrow> </semantics></math> evaluate in quantile theoretical compared to quantile, proposition 2 (<b>upper</b>), and qqplot (<b>under</b>).</p> Full article ">Figure 4
<p>Kurtosis of the <math display="inline"><semantics> <mrow> <mi>G</mi> <mi>T</mi> </mrow> </semantics></math> distribution compared with <span class="html-italic">T</span> distribution for <math display="inline"><semantics> <mrow> <mi>ν</mi> <mo>=</mo> <mn>8</mn> </mrow> </semantics></math>.</p> Full article ">Figure 5
<p>Histogram (<b>left</b>) and Comparison the tails (<b>right</b>) for ants dataset. Overlaid on top is the generalized student’s t density with parameters estimated via ML (solid line), the modified slash density (dashed line), the extended slash density (dotted line), the student’s t density (dashed line).</p> Full article ">Figure 6
<p>Q-q plots: student’s t (<b>a</b>), modified slash (<b>b</b>), extended slash (<b>c</b>), generalized student’s t (<b>d</b>).</p> Full article ">Figure 7
<p>Empirical cdf with estimated <span class="html-italic">T</span> c.d.f. (yellow color),estimated <math display="inline"><semantics> <mrow> <mi>M</mi> <mi>S</mi> </mrow> </semantics></math> cdf (red color), estimated <math display="inline"><semantics> <mrow> <mi>E</mi> <mi>S</mi> </mrow> </semantics></math> c.d.f. (green color), and estimated <math display="inline"><semantics> <mrow> <mi>G</mi> <mi>T</mi> </mrow> </semantics></math> c.d.f. (blue color).</p> Full article ">Figure 8
<p>Histogram (<b>left</b>) and comparison the tails (<b>right</b>) for ants dataset. Overlaid on top is the generalized student’s t density with parameters estimated via ML (solid line), the modified slash density (dashed line), the extended slash density (dotted line),the student’s t density (dashed line).</p> Full article ">Figure 9
<p>Quantile regression for BMI and LBM data with student’s t distribution (<b>left</b>), slash logistic distribution (<b>center</b>) and generalized student’s t distribution (<b>right</b>).</p> Full article ">
16 pages, 433 KiB  
Article
A Novel Model for Distributed Denial of Service Attack Analysis and Interactivity
by Ashraf Ahmad, Yousef AbuHour and Firas Alghanim
Symmetry 2021, 13(12), 2443; https://doi.org/10.3390/sym13122443 - 17 Dec 2021
Cited by 2 | Viewed by 2677
Abstract
A Distributed Denial of Service (DDoS) attack is a type of cybercrime that renders a target service unavailable by overwhelming it with traffic from several sources (attack nodes). In this paper, we focus on DDoS attacks on a computer network by spreading bots [...] Read more.
A Distributed Denial of Service (DDoS) attack is a type of cybercrime that renders a target service unavailable by overwhelming it with traffic from several sources (attack nodes). In this paper, we focus on DDoS attacks on a computer network by spreading bots throughout the network. A mathematical differential equation model is proposed to represent the dynamism of nodes at different compartments of the model. The model considers two levels of security, with the assumption that the recovered nodes do not return to the same security level. In previous models, the recovered nodes are returned to be suspect on the same security level, which is an unrealistic assumption. Moreover, it is assumed that the attacker can use the infected target nodes to attack again. With such epidemic-like assumptions of infection, different cases are presented and discussed, and the stability of the model is analyzed as well; reversing the symmetry transformation of attacking nodes population is also proven. The proposed model has many parameters in order to precisely describe the infection movement and propagation. Numerical simulation methods are used to solve the developed system of equations using MATLAB, with the intention of finding the best counteraction to control DDoS spread throughout a network. Full article
(This article belongs to the Topic Dynamical Systems: Theory and Applications)
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<p>Schematic presentation of the target and attack populations of DDoS.</p> Full article ">Figure 2
<p>Simulations of the proposed model (<a href="#FD6-symmetry-13-02443" class="html-disp-formula">6</a>), showing the percentage of infected nodes as a function of time.</p> Full article ">Figure 3
<p>The solution of system (<a href="#FD6-symmetry-13-02443" class="html-disp-formula">6</a>) when the parameters are set, as in Example 1.</p> Full article ">Figure 4
<p>The infected solution of system (<a href="#FD6-symmetry-13-02443" class="html-disp-formula">6</a>), when the parameters are set as in Example 2.</p> Full article ">Figure 5
<p>Class 1: the solutions of system (<a href="#FD6-symmetry-13-02443" class="html-disp-formula">6</a>), when <math display="inline"><semantics> <mrow> <mi>ϵ</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>. Class 2: the solutions of system (<a href="#FD6-symmetry-13-02443" class="html-disp-formula">6</a>), when <math display="inline"><semantics> <mrow> <mi>ϵ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, in which the values of the parameters are set as in Example 3. Sub-figures (<b>1a</b>,<b>2a</b>) simulate an attack population, and sub-figures (<b>1b</b>,<b>2b</b>), (<b>1c</b>,<b>2c</b>) simulate the target population with both low-security level and high-security level, respectively.</p> Full article ">Figure 6
<p>Class 1: the solutions of system (<a href="#FD6-symmetry-13-02443" class="html-disp-formula">6</a>), when <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>. Class 2: the solutions of system (<a href="#FD6-symmetry-13-02443" class="html-disp-formula">6</a>), when <math display="inline"><semantics> <mrow> <mi>η</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, in which the values of the parameters are set as in Example 4. Sub-figures (<b>1a</b>,<b>2a</b>) simulate an attack population, and sub-figures (<b>1b</b>,<b>2b</b>), (<b>1c</b>,<b>2c</b>) simulate the target population with both low-security level and high-security level, respectively.</p> Full article ">
14 pages, 6103 KiB  
Article
Hybrid Sliding Mode Control of Full-Car Semi-Active Suspension Systems
by Ayman Aljarbouh, Muhammad Fayaz, Muhammad Shuaib Qureshi and Younes Boujoudar
Symmetry 2021, 13(12), 2442; https://doi.org/10.3390/sym13122442 - 17 Dec 2021
Cited by 13 | Viewed by 3512
Abstract
With the advance in technology in driving vehicles, there is currently more emphasis on developing advanced control systems for better road handling stability and ride comfort. However, one of the challenging problems in the design and implementation of intelligent suspension systems is that [...] Read more.
With the advance in technology in driving vehicles, there is currently more emphasis on developing advanced control systems for better road handling stability and ride comfort. However, one of the challenging problems in the design and implementation of intelligent suspension systems is that there is currently no solution supporting the export of generic suspension models and control components for integration into embedded Electronic Control Units (ECUs). This significantly limits the usage of embedded suspension components in automotive production code software as it requires very high efforts in implementation, manual testing, and fulfilling coding requirements. This paper introduces a new dynamic model of full-car suspension system with semi-active suspension behavior and provides a hybrid sliding mode approach for control of full-car suspension dynamics such that the road handling stability and ride comfort characteristics are ensured. The semi-active suspension model and the hybrid sliding mode controller are implemented as Functional Mock-Up Units (FMUs) conforming to the Functional Mock-Up Interface for embedded systems (eFMI) and are calibrated with a set experimental tests using a 1/5 Soben-car test bench. The methods and prototype implementation proposed in this paper allow both model and controller re-usability and provide a generic way of integrating models and control software into embedded ECUs. Full article
(This article belongs to the Topic Dynamical Systems: Theory and Applications)
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<p>Schematic of the 7 DOF full-car suspension system.</p> Full article ">Figure 2
<p>The test bench used for experimental validation and testing.</p> Full article ">Figure 3
<p>The road profile.</p> Full article ">Figure 4
<p>Model simulation versus experimental data: The vertical displacements of the front wheel-axle assemblies.</p> Full article ">Figure 5
<p>Model simulation versus experimental data: The vertical displacements of the rear wheel–axle assemblies.</p> Full article ">Figure 6
<p>Model simulation versus experimental data: The vertical displacements of the front quarter-car bodies.</p> Full article ">Figure 7
<p>Model simulation versus experimental data: The vertical displacements of the rear quarter-car bodies.</p> Full article ">Figure 8
<p>Model simulation versus experimental data: The vertical displacement of the full-car body centroid.</p> Full article ">Figure 9
<p>Passive and controlled displacements of the full-car body centroid: Full-car body centroid displacement when <math display="inline"><semantics> <mrow> <mi>κ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>.</p> Full article ">Figure 10
<p>Passive and controlled displacements of the full-car body centroid: Full-car body centroid displacement when <math display="inline"><semantics> <mrow> <mi>κ</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>.</p> Full article ">Figure 11
<p>Passive and controlled displacements of the full-car body centroid: Full-car body centroid displacement when <math display="inline"><semantics> <mrow> <mi>κ</mi> <mo>=</mo> <mn>0.6</mn> </mrow> </semantics></math>.</p> Full article ">Figure 12
<p>Passive and controlled displacements of the front wheel–axle assemblies: The vertical displacement of the front wheel–axle assemblies when <math display="inline"><semantics> <mrow> <mi>κ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>.</p> Full article ">Figure 13
<p>Passive and controlled displacements of the front wheel–axle assemblies: The vertical displacement of the front wheel–axle assemblies when <math display="inline"><semantics> <mrow> <mi>κ</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>.</p> Full article ">Figure 14
<p>Passive and controlled displacements of the front wheel–axle assemblies: The vertical displacement of the front wheel–axle assemblies when <math display="inline"><semantics> <mrow> <mi>κ</mi> <mo>=</mo> <mn>0.6</mn> </mrow> </semantics></math>.</p> Full article ">
17 pages, 663 KiB  
Article
On the Crossing Numbers of the Join Products of Six Graphs of Order Six with Paths and Cycles
by Michal Staš
Symmetry 2021, 13(12), 2441; https://doi.org/10.3390/sym13122441 - 17 Dec 2021
Cited by 9 | Viewed by 2467
Abstract
The crossing number of a graph G is the minimum number of edge crossings over all drawings of G in the plane. The main purpose of this paper is to determine the crossing numbers of the join products of six symmetric graphs on [...] Read more.
The crossing number of a graph G is the minimum number of edge crossings over all drawings of G in the plane. The main purpose of this paper is to determine the crossing numbers of the join products of six symmetric graphs on six vertices with paths and cycles on n vertices. The idea of configurations is generalized for the first time onto the family of subgraphs whose edges cross the edges of the considered graph at most once, and their lower bounds of necessary numbers of crossings are presented in the common symmetric table. Some proofs of the join products with cycles are done with the help of several well-known auxiliary statements, the idea of which is extended by a suitable classification of subgraphs that do not cross the edges of the examined graphs. Full article
(This article belongs to the Special Issue Advanced Symmetry Methods for Dynamics, Control, Optimization and Applications)
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<p>Two possible non-isomorphic drawings of the graph <math display="inline"><semantics> <msup> <mi>G</mi> <mo>*</mo> </msup> </semantics></math> for which <math display="inline"><semantics> <mrow> <mi>cr</mi> <mo>(</mo> <msubsup> <mi>C</mi> <mn>4</mn> <mo>⋆</mo> </msubsup> <mo>)</mo> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, and also with a possibility of obtaining a subgraph <math display="inline"><semantics> <msup> <mi>T</mi> <mi>i</mi> </msup> </semantics></math> whose edges do not cross the edges of <math display="inline"><semantics> <msup> <mi>G</mi> <mo>*</mo> </msup> </semantics></math>. (<b>a</b>): the planar drawing of <math display="inline"><semantics> <msup> <mi>G</mi> <mo>*</mo> </msup> </semantics></math>; (<b>b</b>): the drawing of <math display="inline"><semantics> <msup> <mi>G</mi> <mo>*</mo> </msup> </semantics></math> with <math display="inline"><semantics> <mrow> <msub> <mi>cr</mi> <mi>D</mi> </msub> <mrow> <mo>(</mo> <msup> <mi>G</mi> <mo>*</mo> </msup> <mo>)</mo> </mrow> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>.</p> Full article ">Figure 2
<p>Four possible drawings of <math display="inline"><semantics> <msup> <mi>F</mi> <mi>i</mi> </msup> </semantics></math> with a configuration from <math display="inline"><semantics> <mi mathvariant="script">M</mi> </semantics></math>.</p> Full article ">Figure 3
<p>Two possible drawings of <math display="inline"><semantics> <msup> <mi>F</mi> <mi>k</mi> </msup> </semantics></math> with a configuration from <math display="inline"><semantics> <mi mathvariant="script">N</mi> </semantics></math>.</p> Full article ">Figure 4
<p>The drawing of <math display="inline"><semantics> <mrow> <msup> <mi>G</mi> <mo>*</mo> </msup> <mo>+</mo> <msub> <mi>D</mi> <mi>n</mi> </msub> </mrow> </semantics></math> with <math display="inline"><semantics> <mrow> <mn>6</mn> <mfenced open="&#x230A;" close="&#x230B;"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mfenced> <mfenced open="&#x230A;" close="&#x230B;"> <mfrac> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mfenced> <mo>+</mo> <mn>2</mn> <mfenced open="&#x230A;" close="&#x230B;"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mfenced> </mrow> </semantics></math> crossings.</p> Full article ">Figure 5
<p>Two graphs <math display="inline"><semantics> <msub> <mi>G</mi> <mn>1</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>G</mi> <mn>2</mn> </msub> </semantics></math> by adding one new edge to the graph <math display="inline"><semantics> <msup> <mi>G</mi> <mo>*</mo> </msup> </semantics></math>.</p> Full article ">Figure 6
<p>The drawing of <math display="inline"><semantics> <mrow> <msup> <mi>G</mi> <mo>*</mo> </msup> <mo>+</mo> <msub> <mi>P</mi> <mi>n</mi> </msub> </mrow> </semantics></math> with <math display="inline"><semantics> <mrow> <mn>6</mn> <mfenced open="&#x230A;" close="&#x230B;"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mfenced> <mfenced open="&#x230A;" close="&#x230B;"> <mfrac> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mfenced> <mo>+</mo> <mn>2</mn> <mfenced open="&#x230A;" close="&#x230B;"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mfenced> </mrow> </semantics></math> crossings for <span class="html-italic">n</span> even.</p> Full article ">Figure 7
<p>The drawing of <math display="inline"><semantics> <mrow> <msup> <mi>G</mi> <mo>*</mo> </msup> <mo>+</mo> <msub> <mi>P</mi> <mi>n</mi> </msub> </mrow> </semantics></math> with <math display="inline"><semantics> <mrow> <mn>6</mn> <mfenced open="&#x230A;" close="&#x230B;"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mfenced> <mfenced open="&#x230A;" close="&#x230B;"> <mfrac> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mfenced> <mo>+</mo> <mn>2</mn> <mfenced open="&#x230A;" close="&#x230B;"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mfenced> </mrow> </semantics></math> crossings for <span class="html-italic">n</span> odd.</p> Full article ">Figure 8
<p>The planar drawing of the graph <math display="inline"><semantics> <msup> <mi>H</mi> <mo>*</mo> </msup> </semantics></math>.</p> Full article ">Figure 9
<p>The drawing of <math display="inline"><semantics> <mrow> <msup> <mi>H</mi> <mo>*</mo> </msup> <mo>+</mo> <msub> <mi>P</mi> <mi>n</mi> </msub> </mrow> </semantics></math> with <math display="inline"><semantics> <mrow> <mn>6</mn> <mfenced open="&#x230A;" close="&#x230B;"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mfenced> <mfenced open="&#x230A;" close="&#x230B;"> <mfrac> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mfenced> <mo>+</mo> <mn>2</mn> <mfenced open="&#x230A;" close="&#x230B;"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mfenced> </mrow> </semantics></math> crossings.</p> Full article ">Figure 10
<p>Two graphs <math display="inline"><semantics> <msub> <mi>H</mi> <mn>1</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>H</mi> <mn>2</mn> </msub> </semantics></math> by adding new edges to the graph <math display="inline"><semantics> <msup> <mi>H</mi> <mo>*</mo> </msup> </semantics></math>.</p> Full article ">Figure 11
<p>The drawing of <math display="inline"><semantics> <mrow> <msup> <mi>H</mi> <mo>*</mo> </msup> <mo>+</mo> <msub> <mi>C</mi> <mi>n</mi> </msub> </mrow> </semantics></math> with <math display="inline"><semantics> <mrow> <mn>6</mn> <mfenced open="&#x230A;" close="&#x230B;"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mfenced> <mfenced open="&#x230A;" close="&#x230B;"> <mfrac> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mfenced> <mo>+</mo> <mn>2</mn> <mfenced open="&#x230A;" close="&#x230B;"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mfenced> <mo>+</mo> <mn>2</mn> </mrow> </semantics></math> crossings.</p> Full article ">
25 pages, 37268 KiB  
Article
Numerical Study on Effects of Geometric Parameters on the Release Characteristics of Straight Sudden Expansion Gas Extinguishing Nozzles
by Quanwei Li, Xiaohua He, Yongbing Chen, Jiang Lin, Yi Zhang, Ruiyu Chen and Xia Zhou
Symmetry 2021, 13(12), 2440; https://doi.org/10.3390/sym13122440 - 17 Dec 2021
Cited by 4 | Viewed by 2113
Abstract
In order to guide the optimization design of the nozzle of the aircraft-fixed gas fire extinguishing system, we studied the influence of nozzle geometric parameters including outlet–inlet area ratio, length–diameter aspect ratio, and wall roughness on the distribution of pressure and velocity in [...] Read more.
In order to guide the optimization design of the nozzle of the aircraft-fixed gas fire extinguishing system, we studied the influence of nozzle geometric parameters including outlet–inlet area ratio, length–diameter aspect ratio, and wall roughness on the distribution of pressure and velocity in the nozzle on the basis of CFD simulations. Although the structure of the nozzle is axisymmetric, the spatial distribution of the pressure and velocity during the flow and release of gas extinguishing agent is not completely symmetric. It was found that both of the outlet–inlet area ratio (δ) and the length–diameter aspect ratio (ξ) had a significant impact on the distribution characteristics of the pressure and axial velocity in the nozzle. With the increase of δ, the average pressure at the outlet cross-section of the nozzle decreased monotonically, while the average axial velocity at the outlet increased approximately linearly. When ξ≥2, the uniformity of the pressure and velocity distribution at the nozzle outlet was significantly improved. Moreover, with the increase of ξ, the average pressure and the average axial velocity of the outlet both showed a non-monotonic change trend, and the optimal value of ξ should be about 3.0. Compared with δ and ξ, the influence of the nozzle wall roughness (εN) on the flow and release characteristics of the extinguishing agent was weak. With the increase of εN, the average pressure of the nozzle outlet increased slightly, while the average axial velocity at the nozzle outlet decreased slightly. Full article
(This article belongs to the Special Issue Asymmetry in Fire Dynamics and Modelling)
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<p>Structural diagram of the model: (<b>a</b>) nozzle and pipeline; (<b>b</b>) profile of nozzle.</p> Full article ">Figure 2
<p>Structural diagram of the whole simulation model.</p> Full article ">Figure 3
<p>Diagram of grids: (<b>a</b>) overall view; (<b>b</b>) grids of nozzle.</p> Full article ">Figure 4
<p>Mesh dependence study.</p> Full article ">Figure 5
<p>Section pressure distribution and 3D streamlines under different area ratios.</p> Full article ">Figure 6
<p>Variation of pressure under different area ratios: (<b>a</b>) central axis pressure curves; (<b>b</b>) minimum pressure and distance; (<b>c</b>) pressure variation with the radius; (<b>d</b>) average pressure on the nozzle outlet cross-section.</p> Full article ">Figure 6 Cont.
<p>Variation of pressure under different area ratios: (<b>a</b>) central axis pressure curves; (<b>b</b>) minimum pressure and distance; (<b>c</b>) pressure variation with the radius; (<b>d</b>) average pressure on the nozzle outlet cross-section.</p> Full article ">Figure 7
<p>Section velocity distribution and 3D streamlines under different area ratios.</p> Full article ">Figure 8
<p>Variation of axial velocity under different area ratios: (<b>a</b>) central axis velocity curves; (<b>b</b>) maximum velocity and distance; (<b>c</b>) velocity variation with the radius; (<b>d</b>) average velocity on the nozzle outlet cross-section.</p> Full article ">Figure 8 Cont.
<p>Variation of axial velocity under different area ratios: (<b>a</b>) central axis velocity curves; (<b>b</b>) maximum velocity and distance; (<b>c</b>) velocity variation with the radius; (<b>d</b>) average velocity on the nozzle outlet cross-section.</p> Full article ">Figure 9
<p>Section pressure distribution and 3D streamlines under different length–diameter ratios.</p> Full article ">Figure 10
<p>Variation of pressure under different length–diameter aspect ratios: (<b>a</b>) central axis pressure curves; (<b>b</b>) minimum pressure and distance; (<b>c</b>) pressure variation with the radius; (<b>d</b>) average pressure on the nozzle outlet cross-section.</p> Full article ">Figure 11
<p>Section velocity distribution and 3D streamlines under different length–diameter ratios.</p> Full article ">Figure 12
<p>Variation of axial velocity under different length–diameter aspect ratios: (<b>a</b>) central axis velocity curves; (<b>b</b>) maximum velocity and distance; (<b>c</b>) velocity variation with the radius; (<b>d</b>) average velocity on the nozzle outlet cross-section.</p> Full article ">Figure 13
<p>Variation of pressure under different wall roughness: (<b>a</b>) central axis pressure curves; (<b>b</b>) minimum pressure and distance; (<b>c</b>) pressure variation with the radius; (<b>d</b>) average pressure on the nozzle outlet cross-section.</p> Full article ">Figure 13 Cont.
<p>Variation of pressure under different wall roughness: (<b>a</b>) central axis pressure curves; (<b>b</b>) minimum pressure and distance; (<b>c</b>) pressure variation with the radius; (<b>d</b>) average pressure on the nozzle outlet cross-section.</p> Full article ">Figure 14
<p>Variation of velocity under different wall roughness: (<b>a</b>) central axis velocity curves; (<b>b</b>) maximum velocity and distance; (<b>c</b>) velocity variation with the radius; (<b>d</b>) average velocity on the nozzle outlet cross-section.</p> Full article ">
28 pages, 1226 KiB  
Review
A Review of Interpretable ML in Healthcare: Taxonomy, Applications, Challenges, and Future Directions
by Talal A. A. Abdullah, Mohd Soperi Mohd Zahid and Waleed Ali
Symmetry 2021, 13(12), 2439; https://doi.org/10.3390/sym13122439 - 17 Dec 2021
Cited by 44 | Viewed by 6262
Abstract
We have witnessed the impact of ML in disease diagnosis, image recognition and classification, and many more related fields. Healthcare is a sensitive field related to people’s lives in which decisions need to be carefully taken based on solid evidence. However, most ML [...] Read more.
We have witnessed the impact of ML in disease diagnosis, image recognition and classification, and many more related fields. Healthcare is a sensitive field related to people’s lives in which decisions need to be carefully taken based on solid evidence. However, most ML models are complex, i.e., black-box, meaning they do not provide insights into how the problems are solved or why such decisions are proposed. This lack of interpretability is the main reason why some ML models are not widely used yet in real environments such as healthcare. Therefore, it would be beneficial if ML models could provide explanations allowing physicians to make data-driven decisions that lead to higher quality service. Recently, several efforts have been made in proposing interpretable machine learning models to become more convenient and applicable in real environments. This paper aims to provide a comprehensive survey and symmetry phenomena of IML models and their applications in healthcare. The fundamental characteristics, theoretical underpinnings needed to develop IML, and taxonomy for IML are presented. Several examples of how they are applied in healthcare are investigated to encourage and facilitate the use of IML models in healthcare. Furthermore, current limitations, challenges, and future directions that might impact applying ML in healthcare are addressed. Full article
(This article belongs to the Special Issue Machine Learning and Symmetry Numerical Analysis in Biomedical Informatics)
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<p>The trade-off between accuracy and interpretability.</p> Full article ">Figure 2
<p>The number of articles related to IML per year in Web of Science.</p> Full article ">Figure 3
<p>The taxonomy of machine learning interpretability.</p> Full article ">Figure 4
<p>A sample outcome of Weight plot interpretation [<a href="#B7-symmetry-13-02439" class="html-bibr">7</a>].</p> Full article ">Figure 5
<p>A sample outcome of PDP explanation [<a href="#B7-symmetry-13-02439" class="html-bibr">7</a>].</p> Full article ">Figure 6
<p>A sample outcome of ICE explanation [<a href="#B7-symmetry-13-02439" class="html-bibr">7</a>].</p> Full article ">Figure 7
<p>A sample outcome of ALE explanation [<a href="#B45-symmetry-13-02439" class="html-bibr">45</a>].</p> Full article ">Figure 8
<p>A sample outcome of LIME explanation [<a href="#B50-symmetry-13-02439" class="html-bibr">50</a>].</p> Full article ">Figure 9
<p>A sample outcome of SHAP explanations [<a href="#B50-symmetry-13-02439" class="html-bibr">50</a>].</p> Full article ">Figure 10
<p>A sample outcome of Anchors explanations [<a href="#B52-symmetry-13-02439" class="html-bibr">52</a>].</p> Full article ">Figure 11
<p>A sample outcome of saliency map [<a href="#B61-symmetry-13-02439" class="html-bibr">61</a>].</p> Full article ">Figure 12
<p>A sample outcome of global interpretation from XGBoost [<a href="#B37-symmetry-13-02439" class="html-bibr">37</a>].</p> Full article ">
23 pages, 14625 KiB  
Article
Applying GMEI-GAN to Generate Meaningful Encrypted Images in Reversible Data Hiding Techniques
by Chwei-Shyong Tsai, Hsien-Chu Wu, Yu-Wen Li and Josh Jia-Ching Ying
Symmetry 2021, 13(12), 2438; https://doi.org/10.3390/sym13122438 - 16 Dec 2021
Cited by 2 | Viewed by 2467
Abstract
With the rapid development of information technology, the transmission of information has become convenient. In order to prevent the leakage of information, information security should be valued. Therefore, the data hiding technique has become a popular solution. The reversible data hiding technique (RDH) [...] Read more.
With the rapid development of information technology, the transmission of information has become convenient. In order to prevent the leakage of information, information security should be valued. Therefore, the data hiding technique has become a popular solution. The reversible data hiding technique (RDH) in particular uses symmetric encoding and decoding algorithms to embed the data into the cover carrier. Not only can the secret data be transmitted without being detected and retrieved completely, but the cover carrier also can be recovered without distortion. Moreover, the encryption technique can protect the carrier and the hidden data. However, the encrypted carrier is a form of ciphertext, which has a strong probability to attract the attention of potential attackers. Thus, this paper uses the generative adversarial networks (GAN) to generate meaningful encrypted images for RDH. A four-stage network architecture is designed for the experiment, including the hiding network, the encryption/decryption network, the extractor, and the recovery network. In the hiding network, the secret data are embedded into the cover image through residual learning. In the encryption/decryption network, the cover image is encrypted into a meaningful image, called the marked image, through GMEI-GAN, and then the marked image is restored to the decrypted image via the same architecture. In the extractor, 100% of the secret data are extracted through the residual learning framework, same as the hiding network. Lastly, in the recovery network, the cover image is reconstructed with the decrypted image and the retrieved secret data through the convolutional neural network. The experimental results show that using the PSNR/SSIM as the criteria, the stego image reaches 45.09 dB/0.9936 and the marked image achieves 38.57 dB/0.9654. The proposed method not only increases the embedding capacity but also maintains high image quality in the stego images and marked images. Full article
(This article belongs to the Section Computer)
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<p>Reversible data hiding framework.</p> Full article ">Figure 2
<p>(<b>a</b>) VRAE and (<b>b</b>) RRBE framework.</p> Full article ">Figure 3
<p>GRDH flowchart.</p> Full article ">Figure 4
<p>First phase: CycleGAN training.</p> Full article ">Figure 5
<p>Second phase: DCGAN training.</p> Full article ">Figure 6
<p>Third phase: extractor architecture.</p> Full article ">Figure 7
<p>The flowchart of [<a href="#B29-symmetry-13-02438" class="html-bibr">29</a>].</p> Full article ">Figure 8
<p>The flowchart of [<a href="#B30-symmetry-13-02438" class="html-bibr">30</a>].</p> Full article ">Figure 9
<p>Hiding network of [<a href="#B30-symmetry-13-02438" class="html-bibr">30</a>].</p> Full article ">Figure 10
<p>Extraction network of [<a href="#B30-symmetry-13-02438" class="html-bibr">30</a>].</p> Full article ">Figure 11
<p>Flowchart of the proposed method.</p> Full article ">Figure 12
<p>Hiding network architecture.</p> Full article ">Figure 13
<p>Calculation of residual block.</p> Full article ">Figure 14
<p>Deconvolutional operation.</p> Full article ">Figure 15
<p>GMEI-GAN architecture.</p> Full article ">Figure 16
<p>The calculation of Sigmoid function used in the discriminator.</p> Full article ">Figure 17
<p>Extractor architecture.</p> Full article ">Figure 18
<p>The calculation of Sigmoid function in secret message’.</p> Full article ">Figure 19
<p>Recovery network architecture.</p> Full article ">Figure 20
<p>Experimental images.</p> Full article ">Figure 21
<p>(<b>a</b>,<b>c</b>,<b>e</b>,<b>g</b>,<b>i</b>,<b>k</b>) are cover images and their histograms. (<b>b</b>,<b>d</b>,<b>f</b>,<b>h</b>,<b>j</b>,<b>l</b>) are stego images and their histograms.</p> Full article ">Figure 22
<p>Results of GMEI-GAN and DecryptGAN. (<b>a</b>) shows the results that uses the Barbara image as cover image and the Airplane image as a marked image. (<b>b</b>) shows the results that uses the Baboon image as cover image and the Boat image as a marked image. (<b>c</b>) shows the results that uses the Peppers image as cover image and the Lena image as a marked image.</p> Full article ">Figure 23
<p>The marked image of Zhang et al. [<a href="#B14-symmetry-13-02438" class="html-bibr">14</a>], Sahu et al. [<a href="#B23-symmetry-13-02438" class="html-bibr">23</a>], the proposed method, and the target image.</p> Full article ">
15 pages, 1528 KiB  
Article
SimBetaReg Web-Tool: The Easiest Way to Implement the Beta and Simplex Regression Models
by Emrah Altun and Mahmoud El-Morshedy
Symmetry 2021, 13(12), 2437; https://doi.org/10.3390/sym13122437 - 16 Dec 2021
Cited by 2 | Viewed by 2482
Abstract
When the response variable is defined on the (0,1) interval, the beta and simplex regression models are commonly used by researchers. However, there is no software support for these models to make their implementation easy for researchers. In this study, we developed a [...] Read more.
When the response variable is defined on the (0,1) interval, the beta and simplex regression models are commonly used by researchers. However, there is no software support for these models to make their implementation easy for researchers. In this study, we developed a web-tool, named SimBetaReg, to help researchers who are not familiar with programming to implement the beta and simplex regression models. The developed application is free and works independently from the operating systems. Additionally, we model the incidence ratios of COVID-19 with educational and civic engagement indicators of the OECD countries using the SimBetaReg web-tool. Empirical findings show that when the educational attainment, years in education, and voter turnout increase, the incidence ratios of the countries decrease. Full article
(This article belongs to the Special Issue Probability, Statistics and Applied Mathematics)
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<p>The pdf plots of the beta distribution.</p> Full article ">Figure 2
<p>The pdf plots of the simplex distribution.</p> Full article ">Figure 3
<p>COVID-19 incidence ratio of the OECD countries.</p> Full article ">Figure 4
<p>Histograms of the covariates.</p> Full article ">Figure 5
<p>QQ plots of the randomized quantile residuals for the beta and simplex regression models.</p> Full article ">Figure 6
<p>Uploading data for SimBetaReg.</p> Full article ">Figure 7
<p>Displaying the estimated model parameters by SimBetaReg.</p> Full article ">Figure 8
<p>Displaying residual plots by SimBetaReg.</p> Full article ">Figure 9
<p>Displaying the KS normality test for residuals by SimBetaReg.</p> Full article ">Figure 10
<p>Displaying interpretation of the estimated parameters of the model by SimBetaReg.</p> Full article ">Figure 11
<p>Displaying pseudo R-squared and LR test results by SimBetaReg.</p> Full article ">Figure 12
<p>Displaying the AIC and BIC values by SimBetaReg.</p> Full article ">
15 pages, 3177 KiB  
Article
Research on Optimization of Urban Public Transport Network Based on Complex Network Theory
by Zhongyi Lin, Yang Cao, Huasheng Liu, Jin Li and Shuzhi Zhao
Symmetry 2021, 13(12), 2436; https://doi.org/10.3390/sym13122436 - 16 Dec 2021
Cited by 9 | Viewed by 3997
Abstract
The urban public transportation system is an important part of urban transportation, and the rationality of public transportation routes layout plays a vital role in the transportation of the city. Improving the efficiency of public transportation can have a positive impact on the [...] Read more.
The urban public transportation system is an important part of urban transportation, and the rationality of public transportation routes layout plays a vital role in the transportation of the city. Improving the efficiency of public transportation can have a positive impact on the operation of the public transportation system. This paper uses complex network theory and the symmetry of the up and down bus routes and stations to establish an urban public transit network model and calculates the probability of passengers choosing different routes in the public transit network according to passenger travel impedance. Based on passenger travel impedance, travel path probability and passenger travel demand, the links are weighed, and the network efficiency calculation method is improved. Finally, the public transit network optimization model was established with network efficiency as the objective function and solved by the ant colony algorithm. In order to verify the effectiveness of the model and the solution method, this paper selects areas in Nanguan District of Changchun City for example analysis. The result shows that the efficiency of the optimized network is 8.5% higher than that of the original network, which proves the feasibility of the optimized model and solution method. Full article
(This article belongs to the Special Issue Advanced Transportation Technologies and Symmetries in Intelligent Transportation Systems)
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<p>Schematic diagram of passenger route selection.</p> Full article ">Figure 2
<p>Ant colony algorithm solution.</p> Full article ">Figure 3
<p>(<b>a</b>) Theoretical road network and station; (<b>b</b>) the result of the original optimization method; (<b>c</b>) the result of the improved optimization method.</p> Full article ">Figure 4
<p>Road network and traffic zones division.</p> Full article ">Figure 5
<p>Bus stations and transit network.</p> Full article ">Figure 6
<p>Schematic diagram of transit routes.</p> Full article ">Figure 7
<p>Optimized transit network.</p> Full article ">Figure A1
<p>(<b>a</b>) Distribution of <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mi>a</mi> </msub> </mrow> </semantics></math> and <span class="html-italic">Z</span> on the <span class="html-italic">xOz</span> plane; (<b>b</b>) the probability density function of Z.</p> Full article ">
18 pages, 356 KiB  
Article
Continuity and Analyticity for the Generalized Benjamin–Ono Equation
by Xiaolin Pan, Bin Wang and Rong Chen
Symmetry 2021, 13(12), 2435; https://doi.org/10.3390/sym13122435 - 16 Dec 2021
Viewed by 1926
Abstract
This work mainly focuses on the continuity and analyticity for the generalized Benjamin–Ono (g-BO) equation. From the local well-posedness results for g-BO equation, we know that its solutions depend continuously on their initial data. In the present paper, we further show that such [...] Read more.
This work mainly focuses on the continuity and analyticity for the generalized Benjamin–Ono (g-BO) equation. From the local well-posedness results for g-BO equation, we know that its solutions depend continuously on their initial data. In the present paper, we further show that such dependence is not uniformly continuous in Sobolev spaces Hs(R) with s>3/2. We also provide more information about the stability of the data-solution map, i.e., the solution map for g-BO equation is Hölder continuous in Hr-topology for all 0≤r<s with exponent α depending on s and r. Finally, applying the generalized Ovsyannikov type theorem and the basic properties of Sobolev–Gevrey spaces, we prove the Gevrey regularity and analyticity for the g-BO equation. In addition, by the symmetry of the spatial variable, we obtain a lower bound of the lifespan and the continuity of the data-to-solution map. Full article
(This article belongs to the Topic Dynamical Systems: Theory and Applications)
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<p>The relationship of <math display="inline"><semantics> <mrow> <mi>r</mi> <mo>,</mo> <mi>s</mi> </mrow> </semantics></math> and <math display="inline"><semantics> <mi>α</mi> </semantics></math>.</p> Full article ">
12 pages, 262 KiB  
Article
Exact Solutions and Conservation Laws of the Time-Fractional Gardner Equation with Time-Dependent Coefficients
by Ruixin Li and Lianzhong Li
Symmetry 2021, 13(12), 2434; https://doi.org/10.3390/sym13122434 - 16 Dec 2021
Cited by 2 | Viewed by 1886
Abstract
In this paper, we employ the certain theory of Lie symmetry analysis to discuss the time-fractional Gardner equation with time-dependent coefficients. The Lie point symmetry is applied to realize the symmetry reduction of the equation, and then the power series solutions in some [...] Read more.
In this paper, we employ the certain theory of Lie symmetry analysis to discuss the time-fractional Gardner equation with time-dependent coefficients. The Lie point symmetry is applied to realize the symmetry reduction of the equation, and then the power series solutions in some specific cases are obtained. By virtue of the fractional conservation theorem, the conservation laws are constructed. Full article
11 pages, 1093 KiB  
Article
The FUor Star V2493 Cyg (HBC 722)—Eleven Years at Maximum Brightness
by Evgeni Semkov, Sunay Ibryamov and Stoyanka Peneva
Symmetry 2021, 13(12), 2433; https://doi.org/10.3390/sym13122433 - 16 Dec 2021
Cited by 8 | Viewed by 2182
Abstract
At the time of stellar evolution, young stellar objects go through processes of increased activity and instability. Star formation takes place in several stages during which the star accumulates enough mass to initiate thermonuclear reactions in the nucleus. A significant percentage of the [...] Read more.
At the time of stellar evolution, young stellar objects go through processes of increased activity and instability. Star formation takes place in several stages during which the star accumulates enough mass to initiate thermonuclear reactions in the nucleus. A significant percentage of the mass of Sun-like stars accumulates during periods of increased accretion known as FUor outbursts. Since we know only about two dozen stars of this type, the study of each new object is very important for our knowledge. In this paper, we present data from photometric monitoring on a FUor object V2493 Cyg discovered in 2010. Our data were obtained in the optical region with BVRI Johnson–Cousins set of filters during the period from November 2016 to February 2021. The results of our observations show that during this period no significant changes in the brightness of the star were registered. We only detect variations with a small amplitude around the maximum brightness value. Thus, since 2013 V2493 Cyg remains at its maximum brightness, without a decrease in brightness. Such photometric behavior is not typical of other stars from FUor type. Usually, the light curves of FUors are asymmetrical, with a very rapid rise and gradual decline of the brightness. V2493 Cyg remains unique in this respect with a very rapid rise in brightness and prolonged retention in maximum light. Our period analysis made for the interval February 2013–February 2021 reveals a well-defined period of 914 ± 10 days. Such periodicity can be explained by dust structures remaining from star formation in orbit around the star. Full article
(This article belongs to the Special Issue Observational Studies of Star Formation)
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<p>Color images of V2493 Cyg obtained with the 2-m RCC telescope of NAO Rozhen, (<b>Left</b>): before the outburst on 16 August 2007, (<b>Right</b>): after the outburst on 3 August 2013.</p> Full article ">Figure 2
<p><math display="inline"><semantics> <mrow> <mi>B</mi> <mi>V</mi> <mi>R</mi> <mi>I</mi> </mrow> </semantics></math> light curves of V2493 Cyg for the period from June 2008 till February 2021.</p> Full article ">Figure 3
<p><math display="inline"><semantics> <mrow> <mi>B</mi> <mi>V</mi> <mi>R</mi> <mi>I</mi> </mrow> </semantics></math> Color evolution of V2493 Cyg from June 2008 till February 2021.</p> Full article ">Figure 4
<p><math display="inline"><semantics> <mrow> <mi>V</mi> <mo>/</mo> <mi>V</mi> </mrow> </semantics></math>-<span class="html-italic">I</span> diagram from our <span class="html-italic">V</span> and <span class="html-italic">I</span> photometric data.</p> Full article ">Figure 5
<p>The phase-folded <span class="html-italic">V</span> light curve and periodogram of V2493 Cyg for the period February 2013–February 2021.</p> Full article ">
23 pages, 2009 KiB  
Article
Authenticated Encryption Based on Chaotic Neural Networks and Duplex Construction
by Nabil Abdoun, Safwan El Assad, Thang Manh Hoang, Olivier Deforges, Rima Assaf and Mohamad Khalil
Symmetry 2021, 13(12), 2432; https://doi.org/10.3390/sym13122432 - 16 Dec 2021
Cited by 4 | Viewed by 2626
Abstract
In this paper, we propose, implement and analyze an Authenticated Encryption with Associated Data Scheme (AEADS) based on the Modified Duplex Construction (MDC) that contains a chaotic compression function (CCF) based on our chaotic neural network revised (CNNR). Unlike the standard duplex construction [...] Read more.
In this paper, we propose, implement and analyze an Authenticated Encryption with Associated Data Scheme (AEADS) based on the Modified Duplex Construction (MDC) that contains a chaotic compression function (CCF) based on our chaotic neural network revised (CNNR). Unlike the standard duplex construction (SDC), in the MDC there are two phases: the initialization phase and the duplexing phase, each contain a CNNR formed by a neural network with single layer, and followed by a set of non-linear functions. The MDC is implemented with two variants of width, i.e., 512 and 1024 bits. We tested our proposed scheme against the different cryptanalytic attacks. In fact, we evaluated the key and the message sensitivity, the collision resistance analysis and the diffusion effect. Additionally, we tested our proposed AEADS using the different statistical tests such as NIST, Histogram, chi-square, entropy, and correlation analysis. The experimental results obtained on the security performance of the proposed AEADS system are notable and the proposed system can then be used to protect data and authenticate their sources. Full article
(This article belongs to the Special Issue Chaotic Systems and Nonlinear Dynamics)
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<p>The <span class="html-italic">Duplex</span> construction.</p> Full article ">Figure 2
<p>Structure of the proposed <span class="html-italic">CNN-Duplex</span> for <span class="html-italic">AEADS</span>—Encryption process.</p> Full article ">Figure 3
<p>Structure of the proposed <span class="html-italic">CNN-Duplex</span> for <span class="html-italic">AEADS</span>—Decryption process.</p> Full article ">Figure 4
<p>Detailed structure of the <span class="html-italic">i</span>th <span class="html-italic">Chaotic compression function</span> of the proposed <span class="html-italic">CNN-Duplex</span> based on a one-layered <span class="html-italic">NL</span> for <span class="html-italic">AEADS</span>.</p> Full article ">Figure 5
<p>Detailed structure of the <span class="html-italic">k</span>th input neuron for the first choice of the proposed <span class="html-italic">CNN-Duplex</span>.</p> Full article ">Figure 6
<p>Detailed structure of the <span class="html-italic">k</span>th input neuron for the second choice of the proposed <span class="html-italic">CNN-Duplex</span>.</p> Full article ">Figure 7
<p>Detailed structure of <span class="html-italic">NL</span> Functions block.</p> Full article ">Figure 8
<p><span class="html-italic">NIST</span> test for one of ciphertext results.</p> Full article ">Figure 9
<p>Results of Lena image. (<b>a</b>) Lena image, (<b>b</b>) Ciphered Lena, (<b>c</b>) Histogram of Lena image, and (<b>d</b>) Histogram of ciphered Lena.</p> Full article ">Figure 10
<p>Results of Boat image. (<b>a</b>) Boat image, (<b>b</b>) Ciphered Boat, (<b>c</b>) Histogram of Boat image, and (<b>d</b>) Histogram of ciphered Boat.</p> Full article ">Figure 11
<p>Results of Camera man image. (<b>a</b>) Camera man image, (<b>b</b>) Ciphered Camera man, (<b>c</b>) Histogram of Camera man image, and (<b>d</b>) Histogram of ciphered Camera man.</p> Full article ">
20 pages, 817 KiB  
Article
Some New Reverse Hilbert’s Inequalities on Time Scales
by Ghada AlNemer, Ahmed I. Saied, Mohammed Zakarya, Hoda A. Abd El-Hamid, Omar Bazighifan and Haytham M. Rezk
Symmetry 2021, 13(12), 2431; https://doi.org/10.3390/sym13122431 - 15 Dec 2021
Cited by 13 | Viewed by 2082
Abstract
This paper is interested in establishing some new reverse Hilbert-type inequalities, by using chain rule on time scales, reverse Jensen’s, and reverse Hölder’s with Specht’s ratio and mean inequalities. To get the results, we used the Specht’s ratio function and its applications for [...] Read more.
This paper is interested in establishing some new reverse Hilbert-type inequalities, by using chain rule on time scales, reverse Jensen’s, and reverse Hölder’s with Specht’s ratio and mean inequalities. To get the results, we used the Specht’s ratio function and its applications for reverse inequalities of Hilbert-type. Symmetrical properties play an essential role in determining the correct methods to solve inequalities. The new inequalities in special cases yield some recent relevance, which also provide new estimates on inequalities of these type. Full article
(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials)
36 pages, 2195 KiB  
Article
A New Integrated FUCOM–CODAS Framework with Fermatean Fuzzy Information for Multi-Criteria Group Decision-Making
by Sanjib Biswas, Dragan Pamucar, Samarjit Kar and Shib Sankar Sana
Symmetry 2021, 13(12), 2430; https://doi.org/10.3390/sym13122430 - 15 Dec 2021
Cited by 37 | Viewed by 4387
Abstract
Smartphones have become an inevitable part of every facet of modern society. The selection of a particular smartphone brand from multiple options that are available is a complex and dynamic decision-making problem, involving multiple conflicting criteria that are associated with imprecise asymmetric information [...] Read more.
Smartphones have become an inevitable part of every facet of modern society. The selection of a particular smartphone brand from multiple options that are available is a complex and dynamic decision-making problem, involving multiple conflicting criteria that are associated with imprecise asymmetric information imposed by the uncertainty of the consumers. In this paper, we propose a novel hybrid full consistency method (FUCOM) and a combinative distance based assessment (CODAS) based on the multi-criteria group decision-making (MAGDM) framework in the Fermatean fuzzy (FF) domain for smartphone brand selection. We derive the criteria using the UTAUT2 (unified theory of acceptance and ese of technology) model. A group of 15 decision makers (DMs) participated in our study. We compare 14 leading smartphone brands in India and find that the brands having superior features of a good quality and selling a brand image at a affordable price outperform other smartphones. To check the validity of our framework, we compare the results using extant multi-criteria decision-making (MCDM) models. We observe our model provides a consistent solution. Furthermore, we carry out a sensitivity analysis for ascertaining the robustness and stability of the results generated by our model. The results of the sensitivity analysis show that our proposed framework delivers a stable and robust solution. Full article
(This article belongs to the Special Issue Multi-Criteria Decision Aid Methods in Fuzzy Decision Problems – Part III)
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Graphical abstract

Graphical abstract
Full article ">Figure 1
<p>Research framework (source: authors).</p> Full article ">Figure 2
<p>Difference among IFS, PyFS, and FFS based on spaces related to MD.</p> Full article ">Figure 3
<p>Comparison of ranking results.</p> Full article ">Figure 4
<p>Presentation of the results of the sensitivity analysis: (<b>a</b>) scheme (i) and (<b>b</b>) scheme (ii).</p> Full article ">
8 pages, 230 KiB  
Article
A New Formula for Calculating Uncertainty Distribution of Function of Uncertain Variables
by Yuxing Jia, Yuer Lv and Zhigang Wang
Symmetry 2021, 13(12), 2429; https://doi.org/10.3390/sym13122429 - 15 Dec 2021
Viewed by 1854
Abstract
As a mathematical tool to rationally handle degrees of belief in human beings, uncertainty theory has been widely applied in the research and development of various domains, including science and engineering. As a fundamental part of uncertainty theory, uncertainty distribution is the key [...] Read more.
As a mathematical tool to rationally handle degrees of belief in human beings, uncertainty theory has been widely applied in the research and development of various domains, including science and engineering. As a fundamental part of uncertainty theory, uncertainty distribution is the key approach in the characterization of an uncertain variable. This paper shows a new formula to calculate the uncertainty distribution of strictly monotone function of uncertain variables, which breaks the habitual thinking that only the former formula can be used. In particular, the new formula is symmetrical to the former formula, which shows that when it is too intricate to deal with a problem using the former formula, the problem can be observed from another perspective by using the new formula. New ideas may be obtained from the combination of uncertainty theory and symmetry. Full article
(This article belongs to the Special Issue Uncertainty Theory: Symmetry and Applications)
17 pages, 2739 KiB  
Article
Numerical Solutions of the Mathematical Models on the Digestive System and COVID-19 Pandemic by Hermite Wavelet Technique
by Kumbinarasaiah Srinivasa, Haci Mehmet Baskonus and Yolanda Guerrero Sánchez
Symmetry 2021, 13(12), 2428; https://doi.org/10.3390/sym13122428 - 15 Dec 2021
Cited by 27 | Viewed by 2684
Abstract
This article developed a functional integration matrix via the Hermite wavelets and proposed a novel technique called the Hermite wavelet collocation method (HWM). Here, we studied two models: the coupled system of an ordinary differential equation (ODE) is modeled on the digestive system [...] Read more.
This article developed a functional integration matrix via the Hermite wavelets and proposed a novel technique called the Hermite wavelet collocation method (HWM). Here, we studied two models: the coupled system of an ordinary differential equation (ODE) is modeled on the digestive system by considering different parameters such as sleep factor, tension, food rate, death rate, and medicine. Here, we discussed how these parameters influence the digestive system and showed them through figures and tables. Another fractional model is used on the COVID-19 pandemic. This model is defined by a system of fractional-ODEs including five variables, called S (susceptible), E (exposed), I (infected), Q (quarantined), and R (recovered). The proposed wavelet technique investigates these two models. Here, we express the modeled equation in terms of the Hermite wavelets along with the collocation scheme. Then, using the properties of wavelets, we convert the modeled equation into a system of algebraic equations. We use the Newton–Raphson method to solve these nonlinear algebraic equations. The obtained results are compared with numerical solutions and the Runge–Kutta method (R–K method), which is expressed through tables and graphs. The HWM computational time (consumes less time) is better than that of the R–K method. Full article
(This article belongs to the Special Issue New Trends on the Mathematical Models and Solitons Arising in Real-World Problems)
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<p>Influence of <math display="inline"><semantics> <mi>α</mi> </semantics></math> on tension when <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>=</mo> <mn>0.2</mn> <mo>,</mo> <mi>δ</mi> <mo>=</mo> <mn>0.3</mn> <mo>,</mo> <mi>γ</mi> <mo>=</mo> <mn>0.4</mn> <mo>,</mo> <mi>d</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>.</p> Full article ">Figure 2
<p>Influence of <math display="inline"><semantics> <mi>α</mi> </semantics></math> on medicine when <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>=</mo> <mn>0.2</mn> <mo>,</mo> <mi>δ</mi> <mo>=</mo> <mn>0.3</mn> <mo>,</mo> <mi>γ</mi> <mo>=</mo> <mn>0.4</mn> <mo>,</mo> <mi>d</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>.</p> Full article ">Figure 3
<p>Influence of <math display="inline"><semantics> <mi>α</mi> </semantics></math> on food when <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>=</mo> <mn>0.2</mn> <mo>,</mo> <mi>δ</mi> <mo>=</mo> <mn>0.3</mn> <mo>,</mo> <mi>γ</mi> <mo>=</mo> <mn>0.4</mn> <mo>,</mo> <mi>d</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>.</p> Full article ">Figure 4
<p>Influence of <math display="inline"><semantics> <mi>β</mi> </semantics></math> on tension when <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mi>δ</mi> <mo>=</mo> <mn>0.3</mn> <mo>,</mo> <mi>γ</mi> <mo>=</mo> <mn>0.4</mn> <mo>,</mo> <mi>d</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>.</p> Full article ">Figure 5
<p>Influence of <math display="inline"><semantics> <mi>δ</mi> </semantics></math> on tension when <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mi>β</mi> <mo>=</mo> <mn>0.2</mn> <mo>,</mo> <mi>γ</mi> <mo>=</mo> <mn>0.4</mn> <mo>,</mo> <mi>d</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>.</p> Full article ">Figure 6
<p>Influence of <math display="inline"><semantics> <mi>δ</mi> </semantics></math> on medicine when <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mi>β</mi> <mo>=</mo> <mn>0.2</mn> <mo>,</mo> <mi>γ</mi> <mo>=</mo> <mn>0.4</mn> <mo>,</mo> <mi>d</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>.</p> Full article ">Figure 7
<p>Influence of <math display="inline"><semantics> <mi>γ</mi> </semantics></math> on medicine when <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mi>β</mi> <mo>=</mo> <mn>0.2</mn> <mo>,</mo> <mo> </mo> <mi>δ</mi> <mo>=</mo> <mn>0.2</mn> <mo>,</mo> <mi>d</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>.</p> Full article ">Figure 8
<p>Influence of <math display="inline"><semantics> <mi>d</mi> </semantics></math> on medicine when <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mi>β</mi> <mo>=</mo> <mn>0.2</mn> <mo>,</mo> <mo> </mo> <mi>δ</mi> <mo>=</mo> <mn>0.2</mn> <mo>,</mo> <mi>γ</mi> <mo>=</mo> <mn>0.4</mn> </mrow> </semantics></math>.</p> Full article ">Figure 9
<p>Graphical representation of the susceptible population at different values of <math display="inline"><semantics> <mi>δ</mi> </semantics></math> by HWM and the R–K method.</p> Full article ">Figure 10
<p>Graphical representation of infected population at different values of <math display="inline"><semantics> <mi>δ</mi> </semantics></math> by HWM and the R–K method.</p> Full article ">Figure 11
<p>Graphical representation of the exposed population at different values of <math display="inline"><semantics> <mi>δ</mi> </semantics></math> by HWM and the R–K method.</p> Full article ">Figure 12
<p>Graphical representation of the isolated population at different values of <math display="inline"><semantics> <mi>δ</mi> </semantics></math> by HWM and the R–K method.</p> Full article ">Figure 13
<p>Graphical representation of the recovered population at different values of <math display="inline"><semantics> <mi>δ</mi> </semantics></math> by HWM and the R–K method.</p> Full article ">
23 pages, 1199 KiB  
Article
The Multi-Compartment SI(RD) Model with Regime Switching: An Application to COVID-19 Pandemic
by Manuel L. Esquível, Nadezhda P. Krasii, Gracinda R. Guerreiro and Paula Patrício
Symmetry 2021, 13(12), 2427; https://doi.org/10.3390/sym13122427 - 15 Dec 2021
Cited by 6 | Viewed by 2424
Abstract
We study—with existence and unicity results—a variant of the SIR model for an infectious disease incorporating both the possibility of a death outcome—in a short period of time—and a regime switch that can account for the mitigation measures used to control the spreading [...] Read more.
We study—with existence and unicity results—a variant of the SIR model for an infectious disease incorporating both the possibility of a death outcome—in a short period of time—and a regime switch that can account for the mitigation measures used to control the spreading of the infections, such as a total lockdown. This model is parametrised by three parameters: the basic reproduction number, the mortality rate of the infected, and the duration of the disease. We discuss a particular example of application to Portuguese COVID-19 data in two short periods just after the start of the epidemic in 4 March 2020, with the first two cases dated that day. We propose a simple and effective method for the estimation of the main parameters of the disease, namely, the basic reproduction number and the mortality rate of the infected. We correct these estimated values to take into account the asymptomatic non-diagnosed members of the population. We compare the outcome of the model in the cases of the existence, or not, of a regime switch, and under three different scenarios, with a remarkable agreement between model and data deaths in the case of our basis scenario. In a final short remark, we deal with the existence of symmetries for the proposed model. Full article
(This article belongs to the Special Issue Probability, Statistics and Applied Mathematics)
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<p>Regime switching drive function (red) and its complement to one (blue).</p> Full article ">Figure 2
<p>Daily infected and death numbers and 12-day moving averages in Portugal.</p> Full article ">Figure 3
<p>Clades and strains of SARS-COVID-19 in Portugal by Instituto Ricardo Jorge.</p> Full article ">Figure 4
<p>Solutions of the SI(RD) ODE: constant parameters, R-RS and S-RS; <math display="inline"><semantics> <mrow> <mi>ζ</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>.</p> Full article ">Figure 5
<p>Comparing constant parameters with (R) and (S) regime switching; <math display="inline"><semantics> <mrow> <mi>ζ</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>.</p> Full article ">
15 pages, 1089 KiB  
Article
Rotational Cryptanalysis of MORUS
by Iftekhar Salam
Symmetry 2021, 13(12), 2426; https://doi.org/10.3390/sym13122426 - 15 Dec 2021
Viewed by 2041
Abstract
MORUS is one of the finalists of the CAESAR competition. This is an ARX construction that required investigation against rotational cryptanalysis. We investigated the power of rotational cryptanalysis against MORUS. We show that all the operations in the state update function of MORUS [...] Read more.
MORUS is one of the finalists of the CAESAR competition. This is an ARX construction that required investigation against rotational cryptanalysis. We investigated the power of rotational cryptanalysis against MORUS. We show that all the operations in the state update function of MORUS maintain the rotational pairs when the rotation distance is set to a multiple of the sub-word size. Our investigation also confirms that the rotational pairs can be used as distinguishers for the full version of MORUS if the constants used in MORUS are rotational-invariant. However, the actual constants used in MORUS are not rotational-invariant. The introduction of such constants in the state update function breaks the symmetry of the rotational pairs. Experimental results show that rotational pairs can be used as distinguishers for only one step of the initialization phase of MORUS. For more than one step, there are not enough known differences in the rotational pairs of MORUS to provide an effective distinguisher. This is due to the XOR-ing of the constants that are not rotational-invariant. Therefore, it is unlikely for an adversary to construct a distinguisher for the full version of MORUS by observing the rotational pairs. Full article
(This article belongs to the Special Issue Information Security and Cryptology: Advanced Technologies in Symmetry/Asymmetry)
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<p>State update function of MORUS.</p> Full article ">
16 pages, 2215 KiB  
Article
Key Concepts of Systemological Approach to CPS Adaptive Information Security Monitoring
by Maria Poltavtseva, Alexander Shelupanov, Dmitriy Bragin, Dmitry Zegzhda and Elena Alexandrova
Symmetry 2021, 13(12), 2425; https://doi.org/10.3390/sym13122425 - 15 Dec 2021
Cited by 9 | Viewed by 2668
Abstract
Modern cyber-physical systems (CPS) use digital control of physical processes. This allows attackers to conduct various cyberattacks on these systems. According to the current trends, an information security monitoring system (ISMS) becomes part of a security management system of CPS. It provides information [...] Read more.
Modern cyber-physical systems (CPS) use digital control of physical processes. This allows attackers to conduct various cyberattacks on these systems. According to the current trends, an information security monitoring system (ISMS) becomes part of a security management system of CPS. It provides information to make a decision and generate a response. A large number of new methods are aimed at CPS security, including security assessment, intrusion detection, and ensuring sustainability. However, as a cyber-physical system operates over time, its structure and requirements may change. The datasets available for the protection object (CPS) and the security requirements have become dynamic. This dynamic effect causes asymmetry between the monitoring data collection and processing subsystem and the presented security tasks. The problem herein is the choice of the most appropriate set of methods in order to solve the security problems of a particular CPS configuration from a particular bank of the available methods. To solve this problem, the authors present a method for the management of an adaptive information security monitoring system. The method consists of solving a multicriteria discrete optimization problem under Pareto-optimality conditions when the available data, methods or external requirements change. The experimental study was performed on an example of smart home intrusion detection. In the study, the introduction of a constraint (a change in requirements) led to the revision of the monitoring scheme and a different recommendation of the monitoring method. As a result, the information security monitoring system gains the property of adaptability to changes in tasks and the available data. An important result from the study is the fact that the monitoring scheme obtained using the proposed management method has a proven optimality under the given conditions. Therefore, the asymmetry between the information security monitoring data collection and processing subsystem and the set of security requirements in cyber-physical systems can be overcome. Full article
(This article belongs to the Topic Multi-Criteria Decision Making)
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<p>CPS data analysis in the system of information security monitoring (*** display the quantity of methods).</p> Full article ">Figure 2
<p>CPS adaptive ISMS scheme, highlighting the principles of the systematic approach.</p> Full article ">Figure 3
<p>Schematic of the experimental stand.</p> Full article ">Figure 4
<p>Estimation of the boundary condition by the time of the methods operation.</p> Full article ">Figure 5
<p>Evaluation of the boundary <span class="html-italic">Accuracy</span> condition.</p> Full article ">
11 pages, 1333 KiB  
Article
Investigation on the Prediction of Cardiovascular Events Based on Multi-Scale Time Irreversibility Analysis
by Xiaochuan Wu, Qianru Yang, Jin Li and Fengzhen Hou
Symmetry 2021, 13(12), 2424; https://doi.org/10.3390/sym13122424 - 15 Dec 2021
Cited by 2 | Viewed by 1896
Abstract
Investigation of the risk factors associated with cardiovascular disease (CVD) plays an important part in the prevention and treatment of CVD. This study investigated whether alteration in the multi-scale time irreversibility of sleeping heart rate variability (HRV) was a risk factor for cardiovascular [...] Read more.
Investigation of the risk factors associated with cardiovascular disease (CVD) plays an important part in the prevention and treatment of CVD. This study investigated whether alteration in the multi-scale time irreversibility of sleeping heart rate variability (HRV) was a risk factor for cardiovascular events. The D-value, based on analysis of multi-scale increments in HRV series, was used as the measurement of time irreversibility. Eighty-four subjects from an open-access database (i.e., the Sleep Heart Health Study) were included in this study. None of them had any CVD history at baseline; 42 subjects had cardiovascular events within 1 year after baseline polysomnography and were classed as the CVD group, and the other 42 subjects in the non-CVD group were age matched with those in the CVD group and had no cardiovascular events during the 15-year follow-up period. We compared D-values of sleeping HRV between the CVD and non-CVD groups and found that the D-values of the CVD group were significantly lower than those of the non-CVD group on all 10 scales, even after adjusting for gender and body mass index. Moreover, we investigated the performance of a machine learning model to classify CVD and non-CVD subjects. The model, which was fed with a feature space based on the D-values on 10 scales and trained by a random forest algorithm, achieved an accuracy of 80.8% and a positive prediction rate of 86.7%. These results suggest that the decreased time irreversibility of sleeping HRV is an independent predictor of cardiovascular events that could be used to assist the intelligent prediction of cardiovascular events. Full article
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<p>An example of a decision-making process based on a decision tree. The decision “included” or “excluded” was made according to whether the conditions (in the rectangles) were met.</p> Full article ">Figure 2
<p>Schematic diagram of the bagging ensemble learning algorithm. Bagging extracts m samples from the original sample set but ensures different content each time by bootstrapping and carries out T rounds of extraction to obtain T independent training sets.</p> Full article ">Figure 3
<p>Comparison of D-values (mean ± standard error) between non-CVD group and CVD group on 1–10 scale. * Significant difference in D-value between non-CVD group and CVD group (paired <span class="html-italic">t</span>-test, <span class="html-italic">p</span> &lt; 0.05) on the corresponding scale.</p> Full article ">Figure 4
<p>ROC curves of the HRV indices used to discriminate between CVD and non-CVD participants. The AUC of the ROC curves of these HRV indices reflects their sensitivity in capturing the pathological dynamics of CVD.</p> Full article ">
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