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Symmetry
Volume 13
Issue 5
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Symmetry, Volume 13, Issue 5 (May 2021) – 180 articles

Cover Story (view full-size image): In order to transport oxygen, the enzyme, Hemoglobin (Hb), transitions between two allosteric states, T and R. Oxygen binding reapportions rigidity balance in the allosteric transition between the T and R states. Similar functional rigidity changes, which are hallmarks of action at a distance, are found in other allosteric enzymes. View this paper.
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11 pages, 264 KiB  
Article
Cactus Graphs with Maximal Multiplicative Sum Zagreb Index
by Chunlei Xu, Batmend Horoldagva and Lkhagva Buyantogtokh
Symmetry 2021, 13(5), 913; https://doi.org/10.3390/sym13050913 - 20 May 2021
Cited by 5 | Viewed by 2130
Abstract
A connected graph G is said to be a cactus if any two cycles have at most one vertex in common. The multiplicative sum Zagreb index of a graph G is the product of the sum of the degrees of adjacent vertices in [...] Read more.
A connected graph G is said to be a cactus if any two cycles have at most one vertex in common. The multiplicative sum Zagreb index of a graph G is the product of the sum of the degrees of adjacent vertices in G. In this paper, we introduce several graph transformations that are useful tools for the study of the extremal properties of the multiplicative sum Zagreb index. Using these transformations and symmetric structural representations of some cactus graphs, we determine the graphs having maximal multiplicative sum Zagreb index for cactus graphs with the prescribed number of pendant vertices (cut edges). Furthermore, the graphs with maximal multiplicative sum Zagreb index are characterized among all cactus graphs of the given order. Full article
(This article belongs to the Special Issue Analytical and Computational Properties of Topological Indices)
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<p>Transformation A.</p> Full article ">Figure 2
<p>Transformations B and C.</p> Full article ">Figure 3
<p>Transformation D.</p> Full article ">Figure 4
<p>Transformation E.</p> Full article ">Figure 5
<p>Transformation F.</p> Full article ">Figure 6
<p>The graphs <math display="inline"><semantics> <msubsup> <mi>C</mi> <mrow> <mn>7</mn> <mo>,</mo> <mspace width="0.166667em"/> <mn>2</mn> </mrow> <mn>1</mn> </msubsup> </semantics></math> and <math display="inline"><semantics> <msubsup> <mi>C</mi> <mrow> <mn>8</mn> <mo>,</mo> <mspace width="0.166667em"/> <mn>2</mn> </mrow> <mn>2</mn> </msubsup> </semantics></math>.</p> Full article ">Figure 7
<p>The graph <math display="inline"><semantics> <mrow> <mi>G</mi> <mo>(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <msub> <mi>k</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>k</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> </semantics></math>.</p> Full article ">
15 pages, 2263 KiB  
Article
Comparison between Single-Phase Flow Simulation and Multiphase Flow Simulation of Patient-Specific Total Cavopulmonary Connection Structures Assisted by a Rotationally Symmetric Blood Pump
by Tong Chen, Xudong Liu, Biao Si, Yong Feng, Huifeng Zhang, Bing Jia and Shengzhang Wang
Symmetry 2021, 13(5), 912; https://doi.org/10.3390/sym13050912 - 20 May 2021
Cited by 5 | Viewed by 2701
Abstract
To accurately assess the hemolysis risk of the ventricular assist device, this paper proposed a cell destruction model and the corresponding evaluation parameters based on multiphase flow. The single-phase flow and multiphase flow in two patient-specific total cavopulmonary connection structures assisted by a [...] Read more.
To accurately assess the hemolysis risk of the ventricular assist device, this paper proposed a cell destruction model and the corresponding evaluation parameters based on multiphase flow. The single-phase flow and multiphase flow in two patient-specific total cavopulmonary connection structures assisted by a rotationally symmetric blood pump (pump-TCPC) were simulated. Then, single-phase and multiphase cell destruction models were used to evaluate the hemolysis risk. The results of both cell destruction models indicated that the hemolysis risk in the straight pump-TCPC model was lower than that in the curved pump-TCPC model. However, the average and maximum values of the multiphase flow blood damage index (mBDI) were smaller than those of the single-phase flow blood damage index (BDI), but the average and maximum values of the multiphase flow particle residence time (mPRT) were larger than those of the single-phase flow particle residence time (PRT). This study proved that the multiphase flow method can be used to simulate the mechanical behavior of red blood cells (RBCs) and white blood cells (WBCs) in a complex flow field and the multiphase flow cell destruction model had smaller estimates of the impact shear stress. Full article
(This article belongs to the Special Issue Biofluids in Medicine: Models, Computational Methods and Applications)
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<p>Structure of the axial blood pump.</p> Full article ">Figure 2
<p>Straight and curved TCPC models of patients and straight and curved pump-TCPC models: (<b>a</b>) straight TCPC model; (<b>b</b>) straight pump-TCPC model; (<b>c</b>) curved TCPC model; (<b>d</b>) curved pump-TCPC model. Abbreviations: LSVC, left superior vena cava; RSVC, right superior vena cava; IVC, inferior vena cava; LPA, left pulmonary artery; RPA, right pulmonary artery; HV1, HV2 and HV3, three hepatic veins.</p> Full article ">Figure 3
<p>Flow rate profile and pressure profile as the boundary conditions. (<b>a</b>) mass flow rate at SVC (solid line) and IVC (dot line); (<b>b</b>) static pressure at HV (solid line) and PA (dot line). Abbreviations: SVC, superior vena cava; IVC, inferior vena cava; PA, pulmonary artery; HV, hepatic veins</p> Full article ">Figure 4
<p>Distributions of particle BDI in the straight and curved pump-TCPC models at different moments (<b>a</b>–<b>d</b>: distributions of particle BDI at 0.2 s, 0.35 s, 0.6 s, and 0.8 s).</p> Full article ">Figure 5
<p>Distribution of particle mBDI in the straight and curved pump-TCPC models at different moments (<b>a</b>–<b>d</b>: distributions of particle mBDI at 0.2 s, 0.35 s, 0.6 s, and 0.8 s).</p> Full article ">Figure 6
<p>Normalized concentrations of blood cells on the wall (<b>a</b>,<b>c</b>: normalized concentrations of RBCs and WBCs on the wall of the blades; <b>b</b>,<b>d</b>: normalized concentrations of RBCs and WBCs on the inner wall of the housing).</p> Full article ">
13 pages, 524 KiB  
Article
The Superconducting Critical Temperature
by Mike Guidry, Yang Sun and Lian-Ao Wu
Symmetry 2021, 13(5), 911; https://doi.org/10.3390/sym13050911 - 20 May 2021
Cited by 2 | Viewed by 2515
Abstract
Two principles govern the critical temperature for superconducting transitions: (1) intrinsic strength of the pair coupling and (2) the effect of the many-body environments on the efficiency of that coupling. Most discussions take into account only the former, but we argue that the [...] Read more.
Two principles govern the critical temperature for superconducting transitions: (1) intrinsic strength of the pair coupling and (2) the effect of the many-body environments on the efficiency of that coupling. Most discussions take into account only the former, but we argue that the properties of unconventional superconductors are governed more often by the latter, through dynamical symmetry relating to normal and superconducting states. Differentiating these effects is essential to charting a path to the highest-temperature superconductors. Full article
(This article belongs to the Special Issue Hamiltonian and Overdamped Complex Systems, Symmetry of Phase-Space Occupancy)
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<p>(<b>a</b>) Emergent-symmetry truncation of Hilbert space to a collective subspace using principles of dynamical symmetry. (<b>b</b>) Comparison of matrix elements among different theories and data. Wavefunctions and operators are <span class="html-italic">not observables</span>. Only matrix elements are directly related to observables.</p> Full article ">Figure 2
<p>Schematic difference between bondwise <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>D</mi> <mo>,</mo> <mi>π</mi> <mo>)</mo> </mrow> </semantics></math> and onsite <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>S</mi> <mo>,</mo> <msup> <mi>S</mi> <mo>*</mo> </msup> <mo>)</mo> </mrow> </semantics></math> pair energies. If onsite repulsion is weak, the pairing states are nearly degenerated, yielding an SO(8) symmetry. If it is strong onsite pairs are pushed up in energy, reducing the symmetry to an effective SU(4) low-energy symmetry.</p> Full article ">Figure 3
<p>The relationship between SO(4), SO(8), and BCS SU(2) symmetry for conventional and unconventional superconductors.</p> Full article ">Figure 4
<p>SU(4) cuprate temperature <span class="html-italic">T</span> and doping <span class="html-italic">P</span> phase diagram compared with data taken from Refs. [<a href="#B34-symmetry-13-00911" class="html-bibr">34</a>,<a href="#B35-symmetry-13-00911" class="html-bibr">35</a>]. Strengths of the AF and singlet pairing correlations were determined in Ref. [<a href="#B27-symmetry-13-00911" class="html-bibr">27</a>] by global fits to cuprate data (inset plot). Pseudogap temperatures are indicated by <math display="inline"><semantics> <msup> <mi>T</mi> <mo>*</mo> </msup> </semantics></math>. The two PG curves correspond to whether momentum is resolved or not in the experiment. The inset shows the variation of the AF and pairing coupling with doping <span class="html-italic">P</span>.</p> Full article ">Figure 5
<p>Two fundamental SU(4) instabilities that govern the behavior of high temperature superconductors. The plots illustrate (<b>a</b>) the generalized Cooper instability and (<b>b</b>) The AF instability in terms of the values of the order parameters calculated within coherent state approximation.</p> Full article ">Figure 6
<p>(<b>a</b>–<b>c</b>) Coherent-state energy surfaces for symmetry limits of the SU(4) Hamiltonian [<a href="#B24-symmetry-13-00911" class="html-bibr">24</a>]. The horizontal axis measures AF order. Curves are labeled by lattice occupation fractions with the value 1 corresponding to half filling. The parameter <math display="inline"><semantics> <mi>σ</mi> </semantics></math> is the ratio of AF coupling to the sum of AF and pairing coupling strengths. (<b>d</b>–<b>f</b>) Effect of altering the ratio <math display="inline"><semantics> <mi>σ</mi> </semantics></math> for three values of doping in the cuprates. In (<b>d</b>,<b>f</b>) the system is in the stable minima associated with AF and SC, respectively, and changing <math display="inline"><semantics> <mi>σ</mi> </semantics></math> by 10% hardly alters the location of the energy minima, but in (<b>b</b>) the energy surface is critical and the perturbation can flip the nature of the ground state between SC and AF minima.</p> Full article ">Figure 7
<p>Universality of superconductivity and superfluidity. (<b>a</b>) Phase diagram for hole- and electron-doped cuprates [<a href="#B36-symmetry-13-00911" class="html-bibr">36</a>]. Superconducting (SC), antiferromagnetic (AF), and pseudogap (PG) regions are labeled, as are Néel (<math display="inline"><semantics> <msub> <mi>T</mi> <mi>N</mi> </msub> </semantics></math>), SC critical (<math display="inline"><semantics> <msub> <mi>T</mi> <mi>c</mi> </msub> </semantics></math>), and PG (<math display="inline"><semantics> <msup> <mi>T</mi> <mo>*</mo> </msup> </semantics></math>) temperatures. (<b>b</b>) Phase diagram for Fe-based SC [<a href="#B37-symmetry-13-00911" class="html-bibr">37</a>]. (<b>c</b>) Heavy-fermion phase diagram [<a href="#B38-symmetry-13-00911" class="html-bibr">38</a>]. (<b>d</b>) Phase diagram for an organic superconductor (SDW denotes spin density waves) [<a href="#B39-symmetry-13-00911" class="html-bibr">39</a>]. (<b>e</b>) Generic correlation-energy diagram for nuclear structure [<a href="#B40-symmetry-13-00911" class="html-bibr">40</a>].</p> Full article ">Figure 8
<p>Similarity in the dynamical symmetry chains and the ground coherent state energy surfaces for (<b>a</b>) dynamical symmetry in nuclear structure [<a href="#B16-symmetry-13-00911" class="html-bibr">16</a>], (<b>b</b>) high-temperature SC [<a href="#B23-symmetry-13-00911" class="html-bibr">23</a>,<a href="#B24-symmetry-13-00911" class="html-bibr">24</a>], and (<b>c</b>) monolayer graphene in a strong magnetic field [<a href="#B41-symmetry-13-00911" class="html-bibr">41</a>]. The plot contours show total energy as a function of an appropriate order parameter, with different curves corresponding to a particle number parameter.</p> Full article ">Figure 9
<p>(<b>a</b>,<b>b</b>) Formation of BCS superconductor by lowering the temperature of a Fermi liquid through <math display="inline"><semantics> <msub> <mi>T</mi> <mi>c</mi> </msub> </semantics></math>. Direction of vectors indicates relative strength of competing order (<span class="html-italic">x</span>) and SC (<span class="html-italic">y</span>); length indicates total SU(4) strength. The SC transition converts a high-entropy state (<b>a</b>) into a highly ordered one (<b>b</b>), implying a low <math display="inline"><semantics> <msub> <mi>T</mi> <mi>c</mi> </msub> </semantics></math>. (<b>c</b>,<b>d</b>) Formation of SC from a parent state having order that competes with SC but is related to SC by symmetry. This requires imposing SC order (<b>d</b>) on a state (<b>c</b>) already highly ordered, which can occur at a higher <math display="inline"><semantics> <msub> <mi>T</mi> <mi>c</mi> </msub> </semantics></math> because it is a collective rotation in the group space between two low-entropy states. (<b>e</b>) Collective rotation in SU(4) group space. (<b>f</b>) SU(4) Cooper instability.</p> Full article ">
22 pages, 4622 KiB  
Article
Towards Better Performance for Protected Iris Biometric System with Confidence Matrix
by Tong-Yuen Chai, Bok-Min Goi and Wun-She Yap
Symmetry 2021, 13(5), 910; https://doi.org/10.3390/sym13050910 - 20 May 2021
Cited by 12 | Viewed by 2301
Abstract
Biometric template protection (BTP) schemes are implemented to increase public confidence in biometric systems regarding data privacy and security in recent years. The introduction of BTP has naturally incurred loss of information for security, which leads to performance degradation at the matching stage. [...] Read more.
Biometric template protection (BTP) schemes are implemented to increase public confidence in biometric systems regarding data privacy and security in recent years. The introduction of BTP has naturally incurred loss of information for security, which leads to performance degradation at the matching stage. Although efforts are shown in the extended work of some iris BTP schemes to improve their recognition performance, there is still a lack of a generalized solution for this problem. In this paper, a trainable approach that requires no further modification on the protected iris biometric templates has been proposed. This approach consists of two strategies to generate a confidence matrix to reduce the performance degradation of iris BTP schemes. The proposed binary confidence matrix showed better performance in noisy iris data, whereas the probability confidence matrix showed better performance in iris databases with better image quality. In addition, our proposed scheme has also taken into consideration the potential effects in recognition performance, which are caused by the database-associated noise masks and the variation in biometric data types produced by different iris BTP schemes. The proposed scheme has reported remarkable improvement in our experiments with various publicly available iris research databases being tested. Full article
(This article belongs to the Section Computer)
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<p>The difference in the number of matching outcomes before/after binary-to-decimal transformation (<b>a</b>) and the loss of information through the product of binary codes (<b>b</b>).</p> Full article ">Figure 2
<p>Overview of the standard protected biometrics system and our proposed system.</p> Full article ">Figure 3
<p>Process of generating binary confidence matrix.</p> Full article ">Figure 4
<p>Process of generating probability confidence matrix.</p> Full article ">Figure 5
<p>Proposed matching strategy for binary confidence matrix.</p> Full article ">Figure 6
<p>Proposed matching strategy for probability confidence matrix.</p> Full article ">Figure 7
<p>Visualization of iris code and noise mask (bottom).</p> Full article ">Figure 8
<p>Overview of the methodology of Bloom filter.</p> Full article ">Figure 9
<p>Overview of the methodology of Bloom filter with the proposed noise-mask solution.</p> Full article ">Figure 10
<p>Genuine-imposter score distributions for (<b>a</b>) CASIAv1 (<b>b</b>) CASIAv4 (<b>c</b>) CASIAv3 (<b>d</b>) ND0405.</p> Full article ">Figure 10 Cont.
<p>Genuine-imposter score distributions for (<b>a</b>) CASIAv1 (<b>b</b>) CASIAv4 (<b>c</b>) CASIAv3 (<b>d</b>) ND0405.</p> Full article ">Figure 11
<p>Example of ROC plots for the implementation of (<b>a</b>) binary and (<b>b</b>) probability confidence against enhanced IFO.</p> Full article ">Figure 12
<p>Unlinkability analysis of the proposed binary (first row) and probability (second row) confidence matrices for databases (<b>a</b>) CASIAv1, (<b>b</b>) CASIAv4, (<b>c</b>) ND0405.</p> Full article ">
12 pages, 1150 KiB  
Article
Effect of Plastic Anisotropy on the Collapse of a Hollow Disk under Thermal and Mechanical Loading
by Elena Lyamina
Symmetry 2021, 13(5), 909; https://doi.org/10.3390/sym13050909 - 20 May 2021
Cited by 3 | Viewed by 1791
Abstract
Plastic anisotropy significantly affects the behavior of structures and machine parts. Given the many parameters that classify a structure made of anisotropic material, analytic and semi-analytic solutions are very useful for parametric analysis and preliminary design of such structures. The present paper is [...] Read more.
Plastic anisotropy significantly affects the behavior of structures and machine parts. Given the many parameters that classify a structure made of anisotropic material, analytic and semi-analytic solutions are very useful for parametric analysis and preliminary design of such structures. The present paper is devoted to describing the plastic collapse of a thin orthotropic hollow disk inserted into a rigid container. The disk is subject to a uniform temperature field and a uniform pressure is applied over its inner radius. The condition of axial symmetry in conjunction with the assumption of plane stress, permits an exact analytic solution. Two plastic collapse mechanisms exist. One of these mechanisms requires that the entire disk is plastic. According to the other mechanism, plastic deformation localizes at the inner radius of the disk. Additionally, two special solutions are possible. One of these solutions predicts that the entire disk becomes plastic at the initiation of plastic yielding (i.e., plastic yielding simultaneously initiates in the entire disk). The other special solution predicts that the plastic localization occurs at the inner radius of the disk with no plastic region of finite size. An essential difference between the orthotropic and isotropic disks is that plastic yielding might initiate at the outer radius of the orthotropic disk. Full article
(This article belongs to the Special Issue Analysis and Design of Structures Made of Plastically Anisotropic Materials 2020)
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<p>Geometry of the boundary value problem and the boundary conditions.</p> Full article ">Figure 2
<p>Generic geometric interpretation of the main features of the solution.</p> Full article ">Figure 3
<p>Geometric interpretation of Equations (14) and (15): (<b>a</b>) generic behavior of the curves if plastic yielding initiates at the inner radius and (<b>b</b>) behavior of the curves for specific materials.</p> Full article ">Figure 4
<p>Geometric interpretation of Equations (14) and (15) for AA3104-H19.</p> Full article ">Figure 5
<p>Geometric interpretation of the collapse mechanisms for several materials.</p> Full article ">Figure 6
<p>Radial distribution of the radial stress in a DC06 disk at plastic collapse for several values of <span class="html-italic">p</span>.</p> Full article ">Figure 7
<p>Radial distribution of the circumferential stress in a DC06 disk at plastic collapse for several values of <span class="html-italic">p</span>.</p> Full article ">
18 pages, 1383 KiB  
Article
Survival and Reliability Analysis with an Epsilon-Positive Family of Distributions with Applications
by Perla Celis, Rolando de la Cruz, Claudio Fuentes and Héctor W. Gómez
Symmetry 2021, 13(5), 908; https://doi.org/10.3390/sym13050908 - 20 May 2021
Cited by 2 | Viewed by 2133
Abstract
We introduce a new class of distributions called the epsilon–positive family, which can be viewed as generalization of the distributions with positive support. The construction of the epsilon–positive family is motivated by the ideas behind the generation of skew distributions using symmetric kernels. [...] Read more.
We introduce a new class of distributions called the epsilon–positive family, which can be viewed as generalization of the distributions with positive support. The construction of the epsilon–positive family is motivated by the ideas behind the generation of skew distributions using symmetric kernels. This new class of distributions has as special cases the exponential, Weibull, log–normal, log–logistic and gamma distributions, and it provides an alternative for analyzing reliability and survival data. An interesting feature of the epsilon–positive family is that it can viewed as a finite scale mixture of positive distributions, facilitating the derivation and implementation of EM–type algorithms to obtain maximum likelihood estimates (MLE) with (un)censored data. We illustrate the flexibility of this family to analyze censored and uncensored data using two real examples. One of them was previously discussed in the literature; the second one consists of a new application to model recidivism data of a group of inmates released from the Chilean prisons during 2007. The results show that this new family of distributions has a better performance fitting the data than some common alternatives such as the exponential distribution. Full article
(This article belongs to the Special Issue Symmetric and Asymmetric Distributions: Theoretical Developments and Applications II)
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<p>Examples of the probability density <math display="inline"><semantics> <mrow> <mi>f</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math>, survival <math display="inline"><semantics> <mrow> <mi>S</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math> and hazard <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math> functions of epsilon-exponential distribution, <math display="inline"><semantics> <mrow> <mi>E</mi> <mi>E</mi> <mo>(</mo> <mi>σ</mi> <mo>,</mo> <mi>ε</mi> <mo>)</mo> </mrow> </semantics></math>, and epsilon-Weibull distribution, <math display="inline"><semantics> <mrow> <mi>E</mi> <mi>W</mi> <mo>(</mo> <mi>α</mi> <mo>,</mo> <mi>σ</mi> <mo>,</mo> <mi>ε</mi> <mo>)</mo> </mrow> </semantics></math>. Please note that the exponential and Weibull distributions correspond to the case <math display="inline"><semantics> <mrow> <mi>ε</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>.</p> Full article ">Figure 2
<p>Examples of the probability density <math display="inline"><semantics> <mrow> <mi>f</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math>, survival <math display="inline"><semantics> <mrow> <mi>S</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math> and hazard <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math> functions of epsilon-log-logistic distribution, <math display="inline"><semantics> <mrow> <mi>E</mi> <mi>L</mi> <mi>L</mi> <mo>(</mo> <mi>σ</mi> <mo>,</mo> <mi>ε</mi> <mo>)</mo> </mrow> </semantics></math>, and epsilon-gamma distribution, <math display="inline"><semantics> <mrow> <mi>E</mi> <mi>G</mi> <mo>(</mo> <mi>α</mi> <mo>,</mo> <mi>σ</mi> <mo>,</mo> <mi>ε</mi> <mo>)</mo> </mrow> </semantics></math>. Please note that the log-logistic and gamma distributions correspond to the case <math display="inline"><semantics> <mrow> <mi>ε</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>.</p> Full article ">Figure 3
<p>Skewness (CS) and kurtosis (CK) coefficientes for <math display="inline"><semantics> <mrow> <mi>X</mi> <mo>∼</mo> <mi>E</mi> <mi>E</mi> <mo>(</mo> <mi>σ</mi> <mo>,</mo> <mi>ε</mi> <mo>)</mo> </mrow> </semantics></math>.</p> Full article ">Figure 4
<p>The density functions of the fitted epsilon exponential, exponential, Weibull and exponentiated exponential distributions.</p> Full article ">Figure 5
<p>Fit of the survival functions: Kaplan–Meier estimator (solid line), exponential (dashed line red) and epsilon–exponential (dashed line blue) distributions.</p> Full article ">
10 pages, 2256 KiB  
Article
Logic Gates Formed by Perturbations in an Asynchronous Game of Life
by Yoshihiko Ohzawa and Yukio-Pegio Gunji
Symmetry 2021, 13(5), 907; https://doi.org/10.3390/sym13050907 - 20 May 2021
Cited by 1 | Viewed by 2294
Abstract
The game of life (GL), a type of two-dimensional cellular automaton, has been the subject of many studies because of its simple mechanism and complex behavior. In particular, the construction of logic circuits using the GL has helped to extend the concept of [...] Read more.
The game of life (GL), a type of two-dimensional cellular automaton, has been the subject of many studies because of its simple mechanism and complex behavior. In particular, the construction of logic circuits using the GL has helped to extend the concept of computation. Conventional logic circuits assume deterministic transitions due to the synchronicity of the classic GL. However, they are fragile to noise and cannot maintain the expected behavior in an environment with noise. In this study, a probabilistic logic gate model was constructed using perturbations in an asynchronous game of life (AGL). Since our asynchronous automaton had no heterogeneity in either the horizontal or vertical directions, it was symmetrical with respect to spatial structure. On the other hand, the construction of the logical gate was implemented to contain heterogeneity in the horizontal or vertical directions, which could allow an AND gate and an OR gate in a single system. It was based on the phase transition between connected and unconnected phases, which is newly discovered in this study. In the model, perturbations symmetrically entail operations successful and unsuccessful, and this symmetrical double action is given not to interfere with established operations but to make operations possible. Therefore, this model had a different meaning from logic gates that exclude perturbations or use them externally. The idea of this perturbation is analogous to the inherent noise that destroys and generates structures in biological swarms. Full article
(This article belongs to the Section Computer)
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<p>AGL systems. The systems show different patterns at <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mrow> <mi>async</mi> </mrow> </msub> <mo>=</mo> <mn>0.30</mn> </mrow> </semantics></math> (<b>left column</b>) and <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mrow> <mi>async</mi> </mrow> </msub> <mo>=</mo> <mn>0.70</mn> </mrow> </semantics></math> (<b>right column</b>). These systems continue the transition as in rows 1 to 4.</p> Full article ">Figure 2
<p>An example of a system in a connected state. (<b>a</b>) GL system. (<b>b</b>) This system is determined to be in a connected state because the “path” shown in red runs from one edge to another edge of the system.</p> Full article ">Figure 3
<p>Phase transition between the connected phase and unconnected phase with respect to the asynchronous rate. The plots are data from experiments, and the curve is a sigmoid function fitted to the data.</p> Full article ">Figure 4
<p>Phase transitions at various perturbation rates. (<b>a</b>–<b>e</b>) Plots of the experimental data and curves of the fitting functions. (<b>f</b>) Fitting functions for all perturbation rates.</p> Full article ">Figure 5
<p>(<b>a</b>) Logic gate model. The arrangement of its input areas is the same as that of the conductors in (<b>b</b>) a series circuit when viewed vertically and that of (<b>c</b>) a parallel circuit when viewed horizontally.</p> Full article ">Figure 6
<p>Examples of outputs in series and parallel circuits.</p> Full article ">Figure 7
<p>Outputs for the input (1, 0) in the series and parallel circuits of our model.</p> Full article ">Figure 8
<p>Outputs of the logic gate model at <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mrow> <mi>async</mi> </mrow> </msub> <mo>=</mo> <mn>0.350</mn> </mrow> </semantics></math>. (<b>a</b>) The series circuit behaved as a probabilistic AND gate, and (<b>b</b>) the parallel circuit behaved as a probabilistic OR gate.</p> Full article ">
12 pages, 764 KiB  
Article
Lateralized Declarative-Like Memory for Conditional Spatial Information in Domestic Chicks (Gallus gallus)
by Maria Loconsole, Elena Mascalzoni, Jonathan Niall Daisley, Massimo De Agrò, Giorgio Vallortigara and Lucia Regolin
Symmetry 2021, 13(5), 906; https://doi.org/10.3390/sym13050906 - 20 May 2021
Cited by 3 | Viewed by 2430
Abstract
Declarative memory is an explicit, long-term memory system, used in generalization and categorization processes and to make inferences and to predict probable outcomes in novel situations. Animals have been proven to possess a similar declarative-like memory system. Here, we investigated declarative-like memory representations [...] Read more.
Declarative memory is an explicit, long-term memory system, used in generalization and categorization processes and to make inferences and to predict probable outcomes in novel situations. Animals have been proven to possess a similar declarative-like memory system. Here, we investigated declarative-like memory representations in young chicks, assessing the roles of the two hemispheres in memory recollection. Chicks were exposed for three consecutive days to two different arenas (blue/yellow), where they were presented with two panels, each depicting a different stimulus (cross/square). Only one of the two stimuli was rewarded, i.e., it hid a food reward. The position (left/right) of the rewarded stimulus remained constant within the same arena, but it differed between the two arenas (e.g., reward always on the left in the blue context and on the right in the yellow one). At test, both panels depicted the rewarded stimulus, thus chicks had to remember food position depending on the previously experienced contextual rule. Both binocular and right-eye monocularly-tested chicks correctly located the reward, whereas left-eye monocularly-tested chicks performed at the chance level. We showed that declarative-like memory of integrated information is available at early stages of development, and it is associated with a left hemisphere dominance. Full article
(This article belongs to the Special Issue Symmetry and Asymmetry: From Evolution to Neuroscience)
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<p>The two contexts (blue and yellow arenas) used for the exposure phase. Only one of the screens hid a jar containing food. Food would be consistently located depending on the color of the context (e.g., always behind the stimulus depicting a square, thus in the figure on the left in the blue arena and on the right in the yellow one). Jars looked identical and were not visible from the chick’s starting point (shown in the picture).</p> Full article ">Figure 2
<p>The testing apparatus. In this example the S+ corresponded to the square. The dotted lines delineate the three choice areas (colored for illustrative purposes) within the apparatus. The red and the green areas indicate the wrong-choice area and the correct-choice area, respectively. The front area (where the chick is positioned in the picture) corresponds to the non-choice area. As for the exposure phase, the jars looked identical and were not visible from the chick’s starting point (shown in the picture). At test both jars were empty.</p> Full article ">Figure 3
<p>Average time (sec.) spent at test in the correct area (green) and in the incorrect area (red) in each condition (BIN, RE and LE). Whereas binocular and right eye chicks significantly spent longer time in the correct area, Left eye chicks behaved at the chance level. Asterisks indicate statistically significant contrasts: ** <span class="html-italic">p</span> &lt; 0.001; *** <span class="html-italic">p</span> &lt; 0.0001.</p> Full article ">Figure 4
<p>Time (sec.) spent in the correct choice area (dark blue and dark yellow) or incorrect choice area (light blue and light yellow) for each experimental condition separately illustrated for correct position (left/right) and color of the arena (blue/yellow). Dark blue indicates time spent in the correct choice area in the blue context; light blue indicates time spent in the incorrect choice area in the blue context. Dark yellow indicates time within the correct area in the yellow arena; light yellow indicates time spent within the incorrect area in the yellow arena. Asterisks indicate statistically significant contrasts: * <span class="html-italic">p</span> &lt; 0.05; *** <span class="html-italic">p</span> &lt; 0.0001.</p> Full article ">
17 pages, 669 KiB  
Article
An Unsupervised Learning Method for Attributed Network Based on Non-Euclidean Geometry
by Wei Wu, Guangmin Hu and Fucai Yu
Symmetry 2021, 13(5), 905; https://doi.org/10.3390/sym13050905 - 19 May 2021
Cited by 31 | Viewed by 8075
Abstract
Many real-world networks can be modeled as attributed networks, where nodes are affiliated with attributes. When we implement attributed network embedding, we need to face two types of heterogeneous information, namely, structural information and attribute information. The structural information of undirected networks is [...] Read more.
Many real-world networks can be modeled as attributed networks, where nodes are affiliated with attributes. When we implement attributed network embedding, we need to face two types of heterogeneous information, namely, structural information and attribute information. The structural information of undirected networks is usually expressed as a symmetric adjacency matrix. Network embedding learning is to utilize the above information to learn the vector representations of nodes in the network. How to integrate these two types of heterogeneous information to improve the performance of network embedding is a challenge. Most of the current approaches embed the networks in Euclidean spaces, but the networks themselves are non-Euclidean. As a consequence, the geometric differences between the embedded space and the underlying space of the network will affect the performance of the network embedding. According to the non-Euclidean geometry of networks, this paper proposes an attributed network embedding framework based on hyperbolic geometry and the Ricci curvature, namely, RHAE. Our method consists of two modules: (1) the first module is an autoencoder module in which each layer is provided with a network information aggregation layer based on the Ricci curvature and an embedding layer based on hyperbolic geometry; (2) the second module is a skip-gram module in which the random walk is based on the Ricci curvature. These two modules are based on non-Euclidean geometry, but they fuse the topology information and attribute information in the network from different angles. Experimental results on some benchmark datasets show that our approach outperforms the baselines. Full article
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<p>Attributed networks and Ricci curvature. (1) (<b>a</b>) is an undirected nework, which consists of 7 nodes, and the weights of the edges are 1; (<b>b</b>) show the same network as (<b>a</b>), except that the weights of the edges are replaced by the Ricci curvatures of the edges; The blue circles and the yellow squares at the nodes represent two different attributes in these two attributed networks. (2) (<b>c</b>) represents another network where all nodes have the same attributes.</p> Full article ">Figure 2
<p>Illustration of an autoencoder module.</p> Full article ">Figure 3
<p>Description of curvatude-based aggregation layers and hyperbolic embedding layers in autoencoder.</p> Full article ">Figure 4
<p>Description of curvatude-based aggregation layers and hyperbolic embedding layers in autoencoder.</p> Full article ">Figure 5
<p>The effect of embedding dimension. (<b>a</b>) shows that RHAE’s node classification performance in dimension 16, 32, 64, 128, 256 is better than that of ANRL in Cora dataset. The performance of ANRL peaked at dimension 128 and begin to decline, while the performance of RHAE don’t decline at the above dimensions. (<b>b</b>) shows that on the Wiki dataset, RHAE still performs better than ANRL in the above dimensions, especially in the lower dimensions. This shows that RHAE does indeed capture the true geometry of the network.</p> Full article ">Figure 6
<p>Training time comparison.</p> Full article ">
10 pages, 238 KiB  
Article
Univalence Conditions for Gaussian Hypergeometric Function Involving Differential Inequalities
by Georgia Irina Oros
Symmetry 2021, 13(5), 904; https://doi.org/10.3390/sym13050904 - 19 May 2021
Cited by 7 | Viewed by 1764
Abstract
In their paper published in 1990, Miller and Mocanu have investigated the special function Gaussian hypergeometric function in view of its relation to the theory of analytic functions, stating conditions for this function to be univalent using [...] Read more.
In their paper published in 1990, Miller and Mocanu have investigated the special function Gaussian hypergeometric function in view of its relation to the theory of analytic functions, stating conditions for this function to be univalent using a,b,c∈ℝ, c≠0,−1,−2,…. The study done in this paper extends the results on the univalence of the considered function taking a,b,c∈ℂ, with c≠0,−1,−2,… two criteria being stated in the corollaries of the proved theorems. An interpretation of the univalence results from the sets inclusion view is also given, underlining the geometrical properties of the outcomes. Examples showing how the univalence results can be applied are also included. Full article
(This article belongs to the Special Issue Functional Equations and Analytic Inequalities)
21 pages, 311 KiB  
Article
Optimized Factor Approximants and Critical Index
by Simon Gluzman
Symmetry 2021, 13(5), 903; https://doi.org/10.3390/sym13050903 - 19 May 2021
Cited by 6 | Viewed by 3263
Abstract
Based on expansions with only two coefficients and known critical points, we consider a minimal model of critical phenomena. The method of analysis is both based on and inspired with the symmetry properties of functional self-similarity relation between the consecutive functional approximations. Factor [...] Read more.
Based on expansions with only two coefficients and known critical points, we consider a minimal model of critical phenomena. The method of analysis is both based on and inspired with the symmetry properties of functional self-similarity relation between the consecutive functional approximations. Factor approximants are applied together with various natural optimization conditions of non-perturbative nature. The role of control parameter is played by the critical index by itself. The minimal derivative condition imposed on critical amplitude appears to bring the most reasonable, uniquely defined results. The minimal difference condition also imposed on amplitudes produces upper and lower bound on the critical index. While one of the bounds is close to the result from the minimal difference condition, the second bound is determined by the non-optimized factor approximant. One would expect that for the minimal derivative condition to work well, the bounds determined by the minimal difference condition should be not too wide. In this sense the technique of optimization presented above is self-consistent, since it automatically supplies the solution and the bounds. In the case of effective viscosity of passive suspensions the bounds could be found that are too wide to make any sense from either of the solutions. Other optimization conditions imposed on the factor approximants, lead to better estimates for the critical index for the effective viscosity. The optimization is based on equating two explicit expressions following from two different definitions of the critical index, while optimization parameter is introduced as the trial third-order coefficient in the expansion. Full article
(This article belongs to the Special Issue Dynamical Processes in Heterogeneous and Discrete Media)
15 pages, 1027 KiB  
Article
A Generalization of the Importance of Vertices for an Undirected Weighted Graph
by Ronald Manríquez, Camilo Guerrero-Nancuante, Felipe Martínez and Carla Taramasco
Symmetry 2021, 13(5), 902; https://doi.org/10.3390/sym13050902 - 19 May 2021
Cited by 7 | Viewed by 2999
Abstract
Establishing a node importance ranking is a problem that has attracted the attention of many researchers in recent decades. For unweighted networks where the edges do not have any attached weight, many proposals have been presented, considering local or global information of the [...] Read more.
Establishing a node importance ranking is a problem that has attracted the attention of many researchers in recent decades. For unweighted networks where the edges do not have any attached weight, many proposals have been presented, considering local or global information of the networks. On the contrary, it occurs in undirected edge-weighted networks, where the proposals to address this problem have been more scarce. In this paper, a ranking method of node importance for undirected and edge-weighted is provided, generalizing the measure of line importance (DIL) based on the centrality degree proposed by Opsahl. The experimentation was done on five real networks and the results illustrate the benefits of our proposal. Full article
(This article belongs to the Special Issue Symmetry in Engineering Sciences Ⅲ)
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<p>Simple network example.</p> Full article ">Figure 2
<p>The relation between decline rate of network efficiency and the ranking DIL-W<math display="inline"><semantics> <msup> <mrow/> <mn>1</mn> </msup> </semantics></math>, DIL-W<math display="inline"><semantics> <msup> <mrow/> <mrow> <mn>0.5</mn> </mrow> </msup> </semantics></math> and DIL-W on the example networks.</p> Full article ">Figure 3
<p>The decline rate of the network efficiency as a function of deleting the top 10% of the vertices ranked by DIL-W<math display="inline"><semantics> <msup> <mrow/> <mn>1</mn> </msup> </semantics></math>, DIL-W<math display="inline"><semantics> <msup> <mrow/> <mrow> <mn>0.5</mn> </mrow> </msup> </semantics></math> and DIL-W from five real networks (Zacharys karate club, wild birds, US air transport, brain functional coactivations and colony of ants).</p> Full article ">Figure 4
<p>The decline rate of the network efficiency as a function of deleting the top 10% of the vertices ranked by DIL-W<math display="inline"><semantics> <msup> <mrow/> <mrow> <mn>0.1</mn> </mrow> </msup> </semantics></math>, DIL-W<math display="inline"><semantics> <msup> <mrow/> <mrow> <mn>0.3</mn> </mrow> </msup> </semantics></math>, DIL-W<math display="inline"><semantics> <msup> <mrow/> <mrow> <mn>0.5</mn> </mrow> </msup> </semantics></math>, DIL-W<math display="inline"><semantics> <msup> <mrow/> <mrow> <mn>0.7</mn> </mrow> </msup> </semantics></math>, DIL-W<math display="inline"><semantics> <msup> <mrow/> <mrow> <mn>0.9</mn> </mrow> </msup> </semantics></math> and DIL-W<math display="inline"><semantics> <msup> <mrow/> <mn>1</mn> </msup> </semantics></math> from five real networks (Zacharys karate club, wild birds, US air transport, brain functional coactivations and colony of ants).</p> Full article ">Figure 5
<p>On the left the relationship between the DIL-W and DIL-W <sup><span class="html-italic">α</span></sup>rankings for different values of <span class="html-italic">α</span>. On the right the correlation coefficient for the different values of <span class="html-italic">α</span>.</p> Full article ">Figure 5 Cont.
<p>On the left the relationship between the DIL-W and DIL-W <sup><span class="html-italic">α</span></sup>rankings for different values of <span class="html-italic">α</span>. On the right the correlation coefficient for the different values of <span class="html-italic">α</span>.</p> Full article ">
11 pages, 11866 KiB  
Article
Multiframe Super-Resolution of Color Images Based on Cross Channel Prior
by Shen Shi, Bin Xiangli and Zengshan Yin
Symmetry 2021, 13(5), 901; https://doi.org/10.3390/sym13050901 - 19 May 2021
Cited by 4 | Viewed by 2383
Abstract
Color images have a wider range of applications than gray images. There are two ways to extend the traditional super-resolution reconstruction method to color images: Super resolution reconstructs each channel of the color image individually; Change the RGB color bands into YCrCb color [...] Read more.
Color images have a wider range of applications than gray images. There are two ways to extend the traditional super-resolution reconstruction method to color images: Super resolution reconstructs each channel of the color image individually; Change the RGB color bands into YCrCb color bands, then super-resolution reconstructs the luminance component and interpolates the chrominance components.These algorithms cannot effectively utilize the property that the edges and textures are similar in the RGB channels, and the results of those methods may lead to color artifacts. Aiming to solve these problems, we propose a new super-resolution method based on cross channel prior. First, a cross channel prior is proposed to describe the similarity of gradient in RGB channels. Then, a new super-resolution method is proposed for color images via combination of the cross channel prior and the traditional super-resolution methods. Finally, the proposed method reconstructs the color channels alternately. The experimental results show that the proposed method could effectively suppress the generation of color artifacts and improve the quality of the reconstructed images. Full article
(This article belongs to the Section Computer)
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<p>Four HR images that often used in SR experiments. (<b>a</b>) airplane, (<b>b</b>) parrot, (<b>c</b>) boat, (<b>d</b>) kid, (<b>e</b>) butterfly, (<b>f</b>) face, (<b>g</b>) child, (<b>h</b>) bird.</p> Full article ">Figure 2
<p>SR results of image airplane by different SR methods. (<b>a</b>) the first frame of the LR images, (<b>b</b>) traditional bicubic interpolation, (<b>c</b>) reconstruct the RGB color channels independently, (<b>d</b>) SR with YCbCr color space, (<b>e</b>) SR using chrominance regularization, (<b>f</b>) our proposed method.</p> Full article ">Figure 3
<p>SR results of image parrot by different SR methods. (<b>a</b>) the first frame of the LR images, (<b>b</b>) traditional bicubic interpolation, (<b>c</b>) reconstruct the RGB color channels independently, (<b>d</b>) SR with YCbCr color space, (<b>e</b>) SR using chrominance regularization, (<b>f</b>) our proposed method.</p> Full article ">Figure 4
<p>SR results of image boat by different SR methods. (<b>a</b>) the first frame of the LR images, (<b>b</b>) traditional bicubic interpolation, (<b>c</b>) reconstruct the RGB color channels independently, (<b>d</b>) SR with YCbCr color space, (<b>e</b>) SR using chrominance regularization, (<b>f</b>) our proposed method.</p> Full article ">Figure 5
<p>SR results of image kid by different SR methods. (<b>a</b>) the first frame of the LR images, (<b>b</b>) traditional bicubic interpolation, (<b>c</b>) reconstruct the RGB color channels independently, (<b>d</b>) SR with YCbCr color space, (<b>e</b>) SR using chrominance regularization, (<b>f</b>) our proposed method.</p> Full article ">Figure 6
<p>SR results of image butterfly by different SR methods. (<b>a</b>) the first frame of the LR images, (<b>b</b>) traditional bicubic interpolation, (<b>c</b>) reconstruct the RGB color channels independently, (<b>d</b>) SR with YCbCr color space, (<b>e</b>) SR using chrominance regularization, (<b>f</b>) our proposed method.</p> Full article ">Figure 7
<p>SR results of image face by different SR methods. (<b>a</b>) the first frame of the LR images, (<b>b</b>) traditional bicubic interpolation, (<b>c</b>) reconstruct the RGB color channels independently, (<b>d</b>) SR with YCbCr color space, (<b>e</b>) SR using chrominance regularization, (<b>f</b>) our proposed method.</p> Full article ">Figure 8
<p>SR results of image child by different SR methods. (<b>a</b>) the first frame of the LR images, (<b>b</b>) traditional bicubic interpolation, (<b>c</b>) reconstruct the RGB color channels independently, (<b>d</b>) SR with YCbCr color space, (<b>e</b>) SR using chrominance regularization, (<b>f</b>) our proposed method.</p> Full article ">Figure 9
<p>SR results of image bird by different SR methods. (<b>a</b>) the first frame of the LR images, (<b>b</b>) traditional bicubic interpolation, (<b>c</b>) reconstruct the RGB color channels independently, (<b>d</b>) SR with YCbCr color space, (<b>e</b>) SR using chrominance regularization, (<b>f</b>) our proposed method.</p> Full article ">Figure 10
<p>Comparison of the reconstruction results of real images. The images are reconstructed by the comparison algorithms and our proposed algorithm respectively. (<b>a</b>,<b>f</b>,<b>k</b>) traditional bicubic interpolation and it details, (<b>b</b>,<b>g</b>,<b>l</b>) reconstrut the RGB color channels independently and the details, (<b>c</b>,<b>h</b>,<b>m</b>) SR with YCbCr color space and the details, (<b>d</b>,<b>i</b>,<b>n</b>) SR using chrominance regularization and the details, (<b>e</b>,<b>j</b>,<b>o</b>) our proposed method and the details.</p> Full article ">
11 pages, 2485 KiB  
Article
A Simulation on Relation between Power Distribution of Low-Frequency Field Potentials and Conducting Direction of Rhythm Generator Flowing through 3D Asymmetrical Brain Tissue
by Hao Cheng, Manling Ge, Abdelkader Nasreddine Belkacem, Xiaoxuan Fu, Chong Xie, Zibo Song, Shenghua Chen and Chao Chen
Symmetry 2021, 13(5), 900; https://doi.org/10.3390/sym13050900 - 19 May 2021
Cited by 2 | Viewed by 2251
Abstract
Although the power of low-frequency oscillatory field potentials (FP) has been extensively applied previously, few studies have investigated the influence of conducting direction of deep-brain rhythm generator on the power distribution of low-frequency oscillatory FPs on the head surface. To address this issue, [...] Read more.
Although the power of low-frequency oscillatory field potentials (FP) has been extensively applied previously, few studies have investigated the influence of conducting direction of deep-brain rhythm generator on the power distribution of low-frequency oscillatory FPs on the head surface. To address this issue, a simulation was designed based on the principle of electroencephalogram (EEG) generation of equivalent dipole current in deep brain, where a single oscillatory dipole current represented the rhythm generator, the dipole moment for the rhythm generator’s conducting direction (which was orthogonal and rotating every 30 degrees and at pointing to or parallel to the frontal lobe surface) and the (an)isotropic conduction medium for the 3D (a)symmetrical brain tissue. Both the power above average (significant power value, SP value) and its space (SP area) of low-frequency oscillatory FPs were employed to respectively evaluate the strength and the space of the influence. The computation was conducted using the finite element method (FEM) and Hilbert transform. The finding was that either the SP value or the SP area could be reduced or extended, depending on the conducting direction of deep-brain rhythm generator flowing in the (an)isotropic medium, suggesting that the 3D (a)symmetrical brain tissue could decay or strengthen the spatial spread of a rhythm generator conducting in a different direction. Full article
(This article belongs to the Special Issue Cognitive and Neurophysiological Models of Brain Asymmetry)
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<p>Flow diagram of study.</p> Full article ">Figure 2
<p>Flow chart of reconstructing NUBS model.</p> Full article ">Figure 3
<p>FP computation on the quasi-real head model surface. (<b>a</b>) Finite element method; (<b>b</b>) contour lines when the dipole current was located inside the central frontal lobe, x, y, z: 8.2, 10.9, 5.8; (<b>c</b>) zero-potential surface (green part); (<b>d</b>) schematic of alternative positions of a single dipole current (1 cm displacement along the Z axis). The dipole current amplitude was 0.1 nA.</p> Full article ">Figure 4
<p>Influence of dipole current position on distribution of FP power, when a dipole current is flowing directly to the frontal lobe surface and is conducted in an isotropic medium (left panel) or an anisotropic medium (right panel). (<b>a</b>) Located far from the frontal lobe; (<b>b</b>) located inside the central frontal lobe; (<b>c</b>) located near to the frontal lobe. Each sub-picture contains an FP distribution figure with partially enlarged picture.</p> Full article ">Figure 4 Cont.
<p>Influence of dipole current position on distribution of FP power, when a dipole current is flowing directly to the frontal lobe surface and is conducted in an isotropic medium (left panel) or an anisotropic medium (right panel). (<b>a</b>) Located far from the frontal lobe; (<b>b</b>) located inside the central frontal lobe; (<b>c</b>) located near to the frontal lobe. Each sub-picture contains an FP distribution figure with partially enlarged picture.</p> Full article ">Figure 5
<p>Influence of dipole current’s position on distribution of FP power, when a dipole current is flowing parallel to the frontal lobe surface and is conducted in an isotropic medium (left panel) or an anisotropic medium (right panel). (<b>a</b>) Located inside the right frontal lobe; (<b>b</b>) located inside the central frontal lobe; (<b>c</b>) located inside the left frontal lobe.</p> Full article ">Figure 6
<p>Influence of dipole moment on the SP area. (<b>a</b>) Direct to the frontal lobe surface, departing from -X axis and arriving at backward via Y axis (90 degree); (<b>b</b>) along the frontal lobe surface, departing from Z axis and arriving at backward via Y axis (90 degree).</p> Full article ">
10 pages, 1244 KiB  
Article
iRG-4mC: Neural Network Based Tool for Identification of DNA 4mC Sites in Rosaceae Genome
by Dae Yeong Lim, Mobeen Ur Rehman and Kil To Chong
Symmetry 2021, 13(5), 899; https://doi.org/10.3390/sym13050899 - 19 May 2021
Cited by 11 | Viewed by 2440
Abstract
DNA N4-Methylcytosine is a genetic modification process which has an essential role in changing different biological processes such as DNA conformation, DNA replication, DNA stability, cell development and structural alteration in DNA. Due to its negative effects, it is important to identify the [...] Read more.
DNA N4-Methylcytosine is a genetic modification process which has an essential role in changing different biological processes such as DNA conformation, DNA replication, DNA stability, cell development and structural alteration in DNA. Due to its negative effects, it is important to identify the modified 4mC sites. Further, methylcytosine may develop anywhere at cytosine residue, however, clonal gene expression patterns are most likely transmitted just for cytosine residues in strand-symmetrical sequences. For this reason many different experiments are introduced but they proved not to be viable choice due to time limitation and high expenses. Therefore, to date there is still need for an efficient computational method to deal with 4mC sites identification. Keeping it in mind, in this research we have proposed an efficient model for Fragaria vesca (F. vesca) and Rosa chinensis (R. chinensis) genome. The proposed iRG-4mC tool is developed based on neural network architecture with two encoding schemes to identify the 4mC sites. The iRG-4mC predictor outperformed the existing state-of-the-art computational model by an accuracy difference of 9.95% on F. vesca (training dataset), 8.7% on R. chinesis (training dataset), 6.2% on F. vesca (independent dataset) and 10.6% on R. chinesis (independent dataset). We have also established a webserver which is freely accessible for the research community. Full article
(This article belongs to the Special Issue Bioinformatics and Computational Biology)
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<p>iRG-4mC Architecture for Identification of 4mC sites.</p> Full article ">Figure 2
<p>Visual performance comparison between state-of-the-art i4mC-ROSE and proposed iRG-4mC.</p> Full article ">Figure 3
<p>ROC curve.</p> Full article ">
21 pages, 285 KiB  
Article
Multiple Critical Points for Symmetric Functionals without upper Growth Condition on the Principal Part
by Marco Degiovanni and Marco Marzocchi
Symmetry 2021, 13(5), 898; https://doi.org/10.3390/sym13050898 - 18 May 2021
Cited by 3 | Viewed by 1593
Abstract
This paper is concerned with variational methods applied to functionals of the calculus of variations in a multi-dimensional case. We prove the existence of multiple critical points for a symmetric functional whose principal part is not subjected to any upper growth condition. For [...] Read more.
This paper is concerned with variational methods applied to functionals of the calculus of variations in a multi-dimensional case. We prove the existence of multiple critical points for a symmetric functional whose principal part is not subjected to any upper growth condition. For this purpose, nonsmooth variational methods are applied. Full article
(This article belongs to the Special Issue Recent Advance in Mathematical Physics)
29 pages, 81673 KiB  
Article
On the Application of a Design of Experiments along with an ANFIS and a Desirability Function to Model Response Variables
by Carmelo J. Luis Pérez
Symmetry 2021, 13(5), 897; https://doi.org/10.3390/sym13050897 - 18 May 2021
Cited by 3 | Viewed by 3409
Abstract
In manufacturing engineering, it is common to use both symmetrical and asymmetrical factorial designs along with regression techniques to model technological response variables, since the in-advance prediction of their behavior is of great importance to determine the levels of variation that lead to [...] Read more.
In manufacturing engineering, it is common to use both symmetrical and asymmetrical factorial designs along with regression techniques to model technological response variables, since the in-advance prediction of their behavior is of great importance to determine the levels of variation that lead to optimal response values to be obtained. For this purpose, regression techniques based on the response surface method combined with a desirability function for multi-objective optimization are commonly employed, since it is usual to find manufacturing processes that require simultaneous optimization of several variables, which exhibit in many cases an opposite behavior. However, these regression models are sometimes not accurate enough to predict the behavior of these response variables, especially when they have significant non-linearities. To deal with this drawback, soft computing techniques are very effective in overcoming the limitations of conventional regression models. This present study is focused on the employment of a symmetrical design of experiments along with a new desirability function, which is proposed in this study, and with soft computing techniques based on fuzzy logic. It will be shown that more accurate results than those obtained from regression techniques are obtained. Moreover, this new desirability function is analyzed in this study. Full article
(This article belongs to the Special Issue Symmetry in Manufacturing Systems Engineering — Concept and Theory to Improve Design, Process, Quality and Circularity as the Basis for Future Manufacturing Philosophy)
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<p>Desirability function results using a symmetrical design of experiments (DOE) and an ANFIS.</p> Full article ">Figure 2
<p>Influence of <math display="inline"><semantics> <mrow> <mfenced> <mrow> <msub> <mi>k</mi> <mrow> <msub> <mi>c</mi> <mi>M</mi> </msub> </mrow> </msub> <mo> </mo> <mi>a</mi> <mi>n</mi> <mi>d</mi> <mo> </mo> <msub> <mi>k</mi> <mrow> <msub> <mi>c</mi> <mi>m</mi> </msub> </mrow> </msub> </mrow> </mfenced> </mrow> </semantics></math> constants over the arctangent transformation (<b>a</b>) to maximize an output response and (<b>b</b>) to minimize an output response.</p> Full article ">Figure 3
<p>Influence of <math display="inline"><semantics> <mrow> <mfenced> <mrow> <msub> <mi>t</mi> <mrow> <msub> <mi>c</mi> <mi>M</mi> </msub> </mrow> </msub> <mo> </mo> <mi>a</mi> <mi>n</mi> <mi>d</mi> <mo> </mo> <msub> <mi>t</mi> <mrow> <msub> <mi>c</mi> <mi>m</mi> </msub> </mrow> </msub> </mrow> </mfenced> </mrow> </semantics></math> constants of the arctangent transformation (<b>a</b>) to maximize an output response, and (<b>b</b>) to minimize an output response.</p> Full article ">Figure 4
<p>Influence of <math display="inline"><semantics> <mrow> <mfenced> <mrow> <msub> <mi>k</mi> <mrow> <msub> <mi>H</mi> <mi>M</mi> </msub> </mrow> </msub> <mo> </mo> <mi>a</mi> <mi>n</mi> <mi>d</mi> <mo> </mo> <msub> <mi>k</mi> <mrow> <msub> <mi>H</mi> <mi>m</mi> </msub> </mrow> </msub> </mrow> </mfenced> </mrow> </semantics></math> constants of the arctangent transformation (<b>a</b>) to maximize an output response, and (<b>b</b>) to minimize an output response.</p> Full article ">Figure 5
<p>Arctangent transformation to (<b>a</b>) maximize a variable, (<b>b</b>) minimize a variable.</p> Full article ">Figure 6
<p>Arctangent transformation to keep a variable in a range (LB = Lower Bound, UB = Upper Bound).</p> Full article ">Figure 7
<p>Arctangent transformation to (<b>a</b>) maximize a variable, where the value of <math display="inline"><semantics> <mrow> <msub> <mi>t</mi> <mrow> <msub> <mi>c</mi> <mrow> <msub> <mi>M</mi> <mi>j</mi> </msub> </mrow> </msub> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> and to (<b>b</b>) minimize a variable, where the value <math display="inline"><semantics> <mrow> <msub> <mi>t</mi> <mrow> <msub> <mi>c</mi> <mrow> <msub> <mi>m</mi> <mi>k</mi> </msub> </mrow> </msub> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>.</p> Full article ">Figure 8
<p>Arctangent transformation to keep a variable in a range (LB = Lower Bound; UB = Upper Bound).</p> Full article ">Figure 9
<p>Arctangent transformation to (left) maximize a variable and (right) minimize a variable by selecting the levels of transition to “0” and “1”, where and <math display="inline"><semantics> <mrow> <msub> <mi>H</mi> <mrow> <mn>1</mn> <mi>M</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>k</mi> <mrow> <msub> <mi>H</mi> <mrow> <mn>1</mn> <mi>M</mi> </mrow> </msub> </mrow> </msub> <mfenced> <mrow> <msub> <mi>M</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>m</mi> <mn>1</mn> </msub> </mrow> </mfenced> </mrow> </semantics></math> <span class="html-italic">and</span> <math display="inline"><semantics> <mrow> <msub> <mi>H</mi> <mrow> <mn>2</mn> <mi>M</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>k</mi> <mrow> <msub> <mi>H</mi> <mrow> <mn>2</mn> <mi>M</mi> </mrow> </msub> </mrow> </msub> <mfenced> <mrow> <msub> <mi>M</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>m</mi> <mn>1</mn> </msub> </mrow> </mfenced> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>H</mi> <mrow> <mn>1</mn> <mi>m</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>k</mi> <mrow> <msub> <mi>H</mi> <mrow> <mn>1</mn> <mi>m</mi> </mrow> </msub> </mrow> </msub> <mfenced> <mrow> <msub> <mi>M</mi> <mn>2</mn> </msub> <mo>−</mo> <msub> <mi>m</mi> <mn>2</mn> </msub> </mrow> </mfenced> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>H</mi> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>k</mi> <mrow> <msub> <mi>H</mi> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </msub> </mrow> </msub> <mfenced> <mrow> <msub> <mi>M</mi> <mn>2</mn> </msub> <mo>−</mo> <msub> <mi>m</mi> <mn>2</mn> </msub> </mrow> </mfenced> </mrow> </semantics></math>. In the case shown in the figure the same values have been selected <math display="inline"><semantics> <mrow> <mfenced close="]" open="["> <mrow> <mi>m</mi> <mo>,</mo> <mo> </mo> <mi>M</mi> </mrow> </mfenced> <mo>=</mo> <mfenced close="]" open="["> <mrow> <mn>4</mn> <mo>,</mo> <mn>10</mn> </mrow> </mfenced> </mrow> </semantics></math><span class="html-italic">,</span> <math display="inline"><semantics> <mrow> <mfenced> <mrow> <msub> <mi>k</mi> <mrow> <msub> <mi>H</mi> <mn>1</mn> </msub> </mrow> </msub> <mo>=</mo> <mn>5</mn> <mo>,</mo> <mo> </mo> <msub> <mi>k</mi> <mrow> <msub> <mi>H</mi> <mn>2</mn> </msub> </mrow> </msub> <mo>=</mo> <mn>0.5</mn> </mrow> </mfenced> </mrow> </semantics></math>.</p> Full article ">Figure 10
<p>Actual shape of <math display="inline"><semantics> <mrow> <msub> <mi>f</mi> <mn>1</mn> </msub> </mrow> </semantics></math> function.</p> Full article ">Figure 11
<p>Actual shape of <math display="inline"><semantics> <mrow> <msub> <mi>f</mi> <mn>2</mn> </msub> </mrow> </semantics></math> function.</p> Full article ">Figure 12
<p>Actual shape of <math display="inline"><semantics> <mrow> <msub> <mi>f</mi> <mn>3</mn> </msub> </mrow> </semantics></math> function.</p> Full article ">Figure 13
<p>Response surface of <math display="inline"><semantics> <mrow> <msub> <mi>f</mi> <mn>1</mn> </msub> </mrow> </semantics></math> obtained with the regression model.</p> Full article ">Figure 14
<p>Response surface of <math display="inline"><semantics> <mrow> <msub> <mi>f</mi> <mn>2</mn> </msub> </mrow> </semantics></math> obtained with the regression model.</p> Full article ">Figure 15
<p>Response surface of <math display="inline"><semantics> <mrow> <msub> <mi>f</mi> <mn>3</mn> </msub> </mrow> </semantics></math> obtained with the regression model.</p> Full article ">Figure 16
<p>Adaptive neuro-fuzzy inference system (ANFIS).</p> Full article ">Figure 17
<p>Response surface of <math display="inline"><semantics> <mrow> <msub> <mi>f</mi> <mn>1</mn> </msub> </mrow> </semantics></math> obtained with the ANFIS model.</p> Full article ">Figure 18
<p>Response surface of <math display="inline"><semantics> <mrow> <msub> <mi>f</mi> <mn>2</mn> </msub> </mrow> </semantics></math> obtained with the ANFIS model.</p> Full article ">Figure 18 Cont.
<p>Response surface of <math display="inline"><semantics> <mrow> <msub> <mi>f</mi> <mn>2</mn> </msub> </mrow> </semantics></math> obtained with the ANFIS model.</p> Full article ">Figure 19
<p>Response surface of <math display="inline"><semantics> <mrow> <msub> <mi>f</mi> <mn>3</mn> </msub> </mrow> </semantics></math> obtained with the ANFIS model.</p> Full article ">Figure 20
<p>Score of the response variables of the DOE using the arctangent transformation (sub index “t”) and that proposed by Derringer and Suich [<a href="#B18-symmetry-13-00897" class="html-bibr">18</a>] (sub index “d”).</p> Full article ">Figure 21
<p>Desirability functions versus the number of points (N) employed in the ANFIS.</p> Full article ">Figure 22
<p>Transformation of the values of the response functions obtained by using the ANFIS (<b>a<sub>1</sub></b>–<b>a<sub>3</sub></b>) Using the proposed desirability function and (<b>b<sub>1</sub></b>–<b>b<sub>3</sub></b>) using the Derringer and Suich [<a href="#B18-symmetry-13-00897" class="html-bibr">18</a>] desirability function.</p> Full article ">Figure 22 Cont.
<p>Transformation of the values of the response functions obtained by using the ANFIS (<b>a<sub>1</sub></b>–<b>a<sub>3</sub></b>) Using the proposed desirability function and (<b>b<sub>1</sub></b>–<b>b<sub>3</sub></b>) using the Derringer and Suich [<a href="#B18-symmetry-13-00897" class="html-bibr">18</a>] desirability function.</p> Full article ">Figure 23
<p>Derringer and Suich [<a href="#B18-symmetry-13-00897" class="html-bibr">18</a>] desirability function and response variables versus the independent variables, using the ANFIS (only values within 80–100% of the maximum desirability value obtained with the Derringer and Suich desirability function (<math display="inline"><semantics> <mrow> <mi>m</mi> <mi>a</mi> <msub> <mi>x</mi> <mrow> <msub> <mi>D</mi> <mrow> <mi>g</mi> <mi>d</mi> </mrow> </msub> </mrow> </msub> <mo>=</mo> <mn>0.1202</mn> </mrow> </semantics></math>) have been plotted).</p> Full article ">Figure 24
<p>Proposed desirability function and response variables versus the independent variables, using the ANFIS (only values within 80–100% of the maximum desirability value obtained with the proposed desirability function (<math display="inline"><semantics> <mrow> <mi>m</mi> <mi>a</mi> <msub> <mi>x</mi> <mrow> <msub> <mi>D</mi> <mrow> <mi>g</mi> <mi>t</mi> </mrow> </msub> </mrow> </msub> <mo>=</mo> <mn>0.2534</mn> </mrow> </semantics></math>) have been plotted).</p> Full article ">
16 pages, 283 KiB  
Article
Boundary Value Problems of Hadamard Fractional Differential Equations of Variable Order
by Snezhana Hristova, Amar Benkerrouche, Mohammed Said Souid and Ali Hakem
Symmetry 2021, 13(5), 896; https://doi.org/10.3390/sym13050896 - 18 May 2021
Cited by 26 | Viewed by 2613
Abstract
A boundary value problem for Hadamard fractional differential equations of variable order is studied. Note the symmetry of a transformation of a system of differential equations is connected with the locally solvability which is the same as the existence of solutions. It leads [...] Read more.
A boundary value problem for Hadamard fractional differential equations of variable order is studied. Note the symmetry of a transformation of a system of differential equations is connected with the locally solvability which is the same as the existence of solutions. It leads to the necessity of obtaining existence criteria for a boundary value problem for Hadamard fractional differential equations of variable order. Also, the stability in the sense of Ulam–Hyers–Rassias is investigated. The results are obtained based on the Kuratowski measure of noncompactness. An example illustrates the validity of the observed results. Full article
(This article belongs to the Special Issue Functional Equations and Analytic Inequalities)
14 pages, 3562 KiB  
Article
Three-Dimensional Investigation of Hydraulic Properties of Vertical Drop in the Presence of Step and Grid Dissipators
by Rasoul Daneshfaraz, Ehsan Aminvash, Amir Ghaderi, Alban Kuriqi and John Abraham
Symmetry 2021, 13(5), 895; https://doi.org/10.3390/sym13050895 - 18 May 2021
Cited by 18 | Viewed by 3163
Abstract
In irrigation and drainage channels, vertical drops are generally used to transfer water from a higher elevation to a lower level. Downstream of these structures, measures are taken to prevent the destruction of the channel bed by the flow and reduce its destructive [...] Read more.
In irrigation and drainage channels, vertical drops are generally used to transfer water from a higher elevation to a lower level. Downstream of these structures, measures are taken to prevent the destruction of the channel bed by the flow and reduce its destructive kinetic energy. In this study, the effect of use steps and grid dissipators on hydraulic characteristics regarding flow pattern, relative downstream depth, relative pool depth, and energy dissipation of a vertical drop was investigated by numerical simulation following the symmetry law. Two relative step heights and two grid dissipator cell sizes were used. The hydraulic model describes fully coupled three-dimensional flow with axial symmetry. For the simulation, critical depths ranging from 0.24 to 0.5 were considered. Values of low relative depth obtained from the numerical results are in satisfactory agreement with the laboratory data. The simultaneous use of step and grid dissipators increases the relative energy dissipation compared to a simple vertical drop and a vertical drop equipped with steps. By using the grid dissipators and the steps downstream of the vertical drop, the relative pool depth increases. Changing the pore size of the grid dissipators does not affect the relative depth of the pool. The simultaneous use of steps and grid dissipators reduces the downstream Froude number of the vertical drop from 3.83–5.20 to 1.46–2.00. Full article
(This article belongs to the Special Issue Turbulence and Multiphase Flows and Symmetry)
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<p>3D view of a vertical drop equipped with steps and grid structures: (<b>a</b>) h/H = 0.3, (<b>b</b>) h/H = 0.4, and (<b>c</b>) the grid structure.</p> Full article ">Figure 2
<p>The geometric profile, vertical drop, and boundary conditions.</p> Full article ">Figure 3
<p>Geometric and hydraulic parameters for a vertical drop equipped with step and grid structures.</p> Full article ">Figure 4
<p>Variation of the RMSE versus cell size.</p> Full article ">Figure 5
<p>Comparison of experimental and numerical values of the relative depth downstream of the vertical drop with a step of 7.5 cm.</p> Full article ">Figure 6
<p>Flow over a vertical drop with h/H = 0.3: (<b>a</b>) drop equipped with a step, (<b>b</b>) drop equipped with both a step and grid structures.</p> Full article ">Figure 7
<p>Relative downstream depth versus relative critical depth.</p> Full article ">Figure 8
<p>Relative energy dissipation versus relative critical depth.</p> Full article ">Figure 9
<p>Energy dissipation is presented as relative to a simple vertical drop case.</p> Full article ">Figure 10
<p>Relative pool depth versus relative critical depth.</p> Full article ">Figure 11
<p>Comparison of predictions and measured results: (<b>a</b>) relative downstream depth, (<b>b</b>) relative energy dissipation, (<b>c</b>) relative pool depth.</p> Full article ">Figure 11 Cont.
<p>Comparison of predictions and measured results: (<b>a</b>) relative downstream depth, (<b>b</b>) relative energy dissipation, (<b>c</b>) relative pool depth.</p> Full article ">
20 pages, 329 KiB  
Article
Assessment of Two Privacy Preserving Authentication Methods Using Secure Multiparty Computation Based on Secret Sharing
by Diana-Elena Fălămaş, Kinga Marton and Alin Suciu
Symmetry 2021, 13(5), 894; https://doi.org/10.3390/sym13050894 - 18 May 2021
Cited by 10 | Viewed by 3126
Abstract
Secure authentication is an essential mechanism required by the vast majority of computer systems and various applications in order to establish user identity. Credentials such as passwords and biometric data should be protected against theft, as user impersonation can have serious consequences. Some [...] Read more.
Secure authentication is an essential mechanism required by the vast majority of computer systems and various applications in order to establish user identity. Credentials such as passwords and biometric data should be protected against theft, as user impersonation can have serious consequences. Some practices widely used in order to make authentication more secure include storing password hashes in databases and processing biometric data under encryption. In this paper, we propose a system for both password-based and iris-based authentication that uses secure multiparty computation (SMPC) protocols and Shamir secret sharing. The system allows secure information storage in distributed databases and sensitive data is never revealed in plaintext during the authentication process. The communication between different components of the system is secured using both symmetric and asymmetric cryptographic primitives. The efficiency of the used protocols is evaluated along with two SMPC specific metrics: The number of communication rounds and the communication cost. According to our results, SMPC based on secret sharing can be successfully integrated in real-word authentication systems and the communication cost has an important impact on the performance of the SMPC protocols. Full article
(This article belongs to the Section Computer)
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<p>The architecture of the SMPC authentication system based on Shamir secret sharing.</p> Full article ">
15 pages, 1104 KiB  
Article
Symmetrical Antioxidant and Antibacterial Properties of Four Romanian Cruciferous Extracts
by Delia Muntean, Mariana N. Ştefănuţ, Adina Căta, Valentina Buda, Corina Danciu, Radu Bănică, Raluca Pop, Monica Licker and Ioana M. C. Ienaşcu
Symmetry 2021, 13(5), 893; https://doi.org/10.3390/sym13050893 - 18 May 2021
Cited by 7 | Viewed by 2899
Abstract
Four alcoholic extracts from Romanian Cruciferous species—cabbage, acclimatized broccoli, black radish and cauliflower—were obtained in a microwave field. The extracts showed good and symmetric antioxidant activity (0.97–1.13 mmol/L TE) and good phenolic content (1001–1632 mg GAE/L). For the HPLC method, the limit of [...] Read more.
Four alcoholic extracts from Romanian Cruciferous species—cabbage, acclimatized broccoli, black radish and cauliflower—were obtained in a microwave field. The extracts showed good and symmetric antioxidant activity (0.97–1.13 mmol/L TE) and good phenolic content (1001–1632 mg GAE/L). For the HPLC method, the limit of detection (LOD), limit of quantitation (LOQ) and recovery degree were established. The small values of LOD and LOQ indicated a great fit of data. The HPLC method achieved satisfactory quantitative recoveries in the range of 96%–122%, except for the lowest sinigrin concentration (8.774 µg/mL). The presence of metals in the studied extracts falls within the allowed limits. The four Cruciferous extracts showed good and slightly asymmetric antibacterial activities against some Gram-positive and Gram-negative bacteria, including strains with known resistance to antibiotics. Moreover, greater inhibitory effects were exhibited against Gram-negative bacteria. Asymmetrically, no inhibition was observed on the fungal strains. Therefore, the present results may suggest that some alcoholic extract formulas of cabbage and black radish (presenting good antibacterial activity) might be helpful in the antimicrobial fight and could be successfully used on selected cases and strains. Full article
(This article belongs to the Section Life Sciences)
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<p>The structures of the GLS compounds commonly found in <span class="html-italic">Brassicaceae</span> spp.</p> Full article ">Figure 2
<p>Crude and freeze-dried cruciferous plants.</p> Full article ">Figure 3
<p>Chromatographic profiles of cruciferous vegetables: (<b>a</b>) broccoli; (<b>b</b>) cabbage; (<b>c</b>) black radish; (<b>d</b>) cauliflower.</p> Full article ">
19 pages, 329 KiB  
Article
Material Geometry of Binary Composites
by Marcelo Epstein
Symmetry 2021, 13(5), 892; https://doi.org/10.3390/sym13050892 - 18 May 2021
Cited by 3 | Viewed by 2245
Abstract
The constitutive characterization of the uniformity and homogeneity of binary elastic composites is presented in terms of a combination of the material groupoids of the individual constituents. The incorporation of these two groupoids within a single double groupoid is proposed as a viable [...] Read more.
The constitutive characterization of the uniformity and homogeneity of binary elastic composites is presented in terms of a combination of the material groupoids of the individual constituents. The incorporation of these two groupoids within a single double groupoid is proposed as a viable mathematical framework for a unified formulation of this and similar kinds of problems in continuum mechanics. Full article
(This article belongs to the Special Issue Applications of Differential Geometry to Continuum Mechanics)
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<p>A groupoid <math display="inline"><semantics> <mrow> <mi mathvariant="script">Z</mi> <mo>⇉</mo> <mi mathvariant="script">B</mi> </mrow> </semantics></math> as a cloud of arrows hovering over a meadow <math display="inline"><semantics> <mi mathvariant="script">B</mi> </semantics></math>.</p> Full article ">Figure 2
<p>The object inclusion map <math display="inline"><semantics> <mrow> <mi>ϵ</mi> <mo>:</mo> <mi mathvariant="script">B</mi> <mo>→</mo> <mi mathvariant="script">Z</mi> </mrow> </semantics></math> and the identities.</p> Full article ">Figure 3
<p>Tip-to-tail composition.</p> Full article ">Figure 4
<p>Inverse and identities.</p> Full article ">Figure 5
<p>A group as a groupoid over a singleton.</p> Full article ">Figure 6
<p>The deformation gradient.</p> Full article ">Figure 7
<p>A material isomorphism as a transplant.</p> Full article ">Figure 8
<p>Two uniform and homogeneous plates.</p> Full article ">Figure 9
<p>Two uniform plates.</p> Full article ">Figure 10
<p>Loss of uniformity from two uniform plates.</p> Full article ">Figure 11
<p>Loss of stress-free configurations from two homogeneous plates.</p> Full article ">Figure 12
<p>A locally trivial composite.</p> Full article ">Figure 13
<p>Partial preservation of uniformity: a laminate.</p> Full article ">Figure 14
<p>Two isotropic plates with loss of isotropy.</p> Full article ">
21 pages, 413 KiB  
Article
An Extension TOPSIS Method Based on the Decision Maker’s Risk Attitude and the Adjusted Probabilistic Fuzzy Set
by Donghai Liu, An Huang, Yuanyuan Liu and Zaiming Liu
Symmetry 2021, 13(5), 891; https://doi.org/10.3390/sym13050891 - 17 May 2021
Cited by 6 | Viewed by 2261
Abstract
The paper studies an extension TOPSIS method with the adjusted probabilistic linguistic fuzzy set in which the decision maker’s behavior tendency is considered. Firstly, we propose a concept of probabilistic linguistic q-rung orthopair set (PLQROS) based on the probability linguistic fuzzy set (PLFS) [...] Read more.
The paper studies an extension TOPSIS method with the adjusted probabilistic linguistic fuzzy set in which the decision maker’s behavior tendency is considered. Firstly, we propose a concept of probabilistic linguistic q-rung orthopair set (PLQROS) based on the probability linguistic fuzzy set (PLFS) and linguistic q-rung orthopair set (LQROS). The operational laws are introduced based on the transformed probabilistic linguistic q-rung orthopair sets (PLQROSs) which have the same probability. Through this adjustment method, the irrationality of the existing methods in the aggregation process is avoided. Furthermore, we propose a comparison rule of PLQROS and the aggregated operators. The distance measure of PLQROSs is also defined, which can deal with the symmetric information in multi-attribute decision making problems. Considering that the decision maker’s behavior has a very important impact on decision-making results, we propose a behavioral TOPSIS decision making method for PLQROS. Finally, we apply the practical problem of investment decision to demonstrate the validity of the extension TOPSIS method, and the merits of the behavior decision method is testified by comparing with the classic TOPSIS method. The sensitivity analysis results of decision-maker’s behavior are also given. Full article
(This article belongs to the Special Issue Research on Fuzzy Logic and Mathematics with Applications)
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<p>The adjust process of PLQRONs <inline-formula><mml:math id="mm476" display="block"><mml:semantics><mml:mrow><mml:mi>p</mml:mi><mml:msubsup><mml:mi>l</mml:mi><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mn>1</mml:mn></mml:msubsup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula> and <inline-formula><mml:math id="mm477" display="block"><mml:semantics><mml:mrow><mml:mi>p</mml:mi><mml:msubsup><mml:mi>l</mml:mi><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msubsup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>.</p> Full article ">Figure 2
<p>The adjust process of PLQRONs <inline-formula><mml:math id="mm478" display="block"><mml:semantics><mml:mrow><mml:mi>p</mml:mi><mml:msubsup><mml:mi>l</mml:mi><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mn>1</mml:mn></mml:msubsup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula> and <inline-formula><mml:math id="mm479" display="block"><mml:semantics><mml:mrow><mml:mi>p</mml:mi><mml:msubsup><mml:mi>l</mml:mi><mml:mrow><mml:mi>s</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msubsup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>.</p> Full article ">Figure 3
<p>The results of change <italic>λ</italic> in behavioral TOPSIS method.</p> Full article ">Figure 4
<p>The results of change the parameter <italic>λ</italic> in traditional TOPSIS method.</p> Full article ">Figure 5
<p>The results of changed loss aversion parameter <italic>γ</italic>.</p> Full article ">Figure 6
<p>The results of change the risk preference parameter <italic>α</italic>.</p> Full article ">Figure 7
<p>The results of change the risk preference parameter <italic>β</italic>.</p> Full article ">
21 pages, 2641 KiB  
Article
A Control Based Mathematical Model for the Evaluation of Intervention Lines in COVID-19 Epidemic Spread: The Italian Case Study
by Paolo Di Giamberardino, Rita Caldarella and Daniela Iacoviello
Symmetry 2021, 13(5), 890; https://doi.org/10.3390/sym13050890 - 17 May 2021
Cited by 4 | Viewed by 2729
Abstract
This paper addresses the problem of describing the spread of COVID-19 by a mathematical model introducing all the possible control actions as prevention (informative campaign, use of masks, social distancing, vaccination) and medication. The model adopted is similar to SEIQR, with the infected [...] Read more.
This paper addresses the problem of describing the spread of COVID-19 by a mathematical model introducing all the possible control actions as prevention (informative campaign, use of masks, social distancing, vaccination) and medication. The model adopted is similar to SEIQR, with the infected patients split into groups of asymptomatic subjects and isolated ones. This distinction is particularly important in the current pandemic, due to the fundamental the role of asymptomatic subjects in the virus diffusion. The influence of the control actions is considered in analysing the model, from the calculus of the equilibrium points to the determination of the reproduction number. This choice is motivated by the fact that the available organised data have been collected since from the end of February 2020, and almost simultaneously containment measures, increasing in typology and effectiveness, have been applied. The characteristics of COVID-19, not fully understood yet, suggest an asymmetric diffusion among countries and among categories of subjects. Referring to the Italian situation, the containment measures, as applied by the population, have been identified, showing their relation with the government’s decisions; this allows the study of possible scenarios, comparing the impact of different possible choices. Full article
(This article belongs to the Special Issue Mathematical Modeling of Biological Systems: Geometry, Symmetry and Conservation Laws)
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Figure 1

Figure 1
<p>Infected patients <math display="inline"><semantics> <mrow> <msub> <mi>I</mi> <mi>d</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> estimated from the model versus the corresponding real data.</p> Full article ">Figure 2
<p>Recovered subjects <math display="inline"><semantics> <mrow> <msub> <mi>R</mi> <msub> <mi>I</mi> <mi>d</mi> </msub> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> estimated from the model versus the corresponding real data.</p> Full article ">Figure 3
<p>Deceased patients <math display="inline"><semantics> <mrow> <mi>D</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> estimated from the model versus the corresponding real data.</p> Full article ">Figure 4
<p>Reconstructed evolution of <math display="inline"><semantics> <mrow> <mover accent="true"> <mi>β</mi> <mo>¯</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>β</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>−</mo> <msub> <mi>u</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math>.</p> Full article ">Figure 5
<p>Reconstructed evolution of <math display="inline"><semantics> <mover accent="true"> <mi>a</mi> <mo>¯</mo> </mover> </semantics></math> related to the control action <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>.</p> Full article ">Figure 6
<p>Reconstructed evolution of <math display="inline"><semantics> <mrow> <mover accent="true"> <mi>c</mi> <mo>¯</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> corresponding to the control action <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mn>5</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>.</p> Full article ">Figure 7
<p>Reconstructed evolution of <math display="inline"><semantics> <mover accent="true"> <mi>γ</mi> <mo>¯</mo> </mover> </semantics></math> regarding the control action <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mn>3</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mn>4</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> according to the simplification in (<a href="#FD35-symmetry-13-00890" class="html-disp-formula">35</a>).</p> Full article ">Figure 8
<p>Reconstructed evolution of <math display="inline"><semantics> <mover accent="true"> <mi>v</mi> <mo>¯</mo> </mover> </semantics></math> corresponding to the control action <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mn>6</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>.</p> Full article ">Figure 9
<p>Error 1 <math display="inline"><semantics> <mrow> <mo>=</mo> <mfrac> <mrow> <mrow> <mo>|</mo> </mrow> <mrow> <msub> <mi>I</mi> <mrow> <mi>r</mi> <mi>e</mi> <mi>a</mi> <mi>l</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>−</mo> <msub> <mi>I</mi> <mi>d</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mo>|</mo> </mrow> </mrow> <mrow> <msub> <mi>I</mi> <mrow> <mi>r</mi> <mi>e</mi> <mi>a</mi> <mi>l</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> </semantics></math>.</p> Full article ">Figure 10
<p>Error 2 <math display="inline"><semantics> <mrow> <mo>=</mo> <mfrac> <mrow> <mrow> <mo>|</mo> </mrow> <mrow> <msub> <mi>R</mi> <mrow> <mi>r</mi> <mi>e</mi> <mi>a</mi> <mi>l</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>−</mo> <msub> <mi>R</mi> <msub> <mi>I</mi> <mi>d</mi> </msub> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mo>|</mo> </mrow> </mrow> <mrow> <msub> <mi>R</mi> <mrow> <mi>r</mi> <mi>e</mi> <mi>a</mi> <mi>l</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> </semantics></math>.</p> Full article ">Figure 11
<p>Error 3 <math display="inline"><semantics> <mrow> <mo>=</mo> <mfrac> <mrow> <mrow> <mo>|</mo> </mrow> <mrow> <msub> <mi>D</mi> <mrow> <mi>r</mi> <mi>e</mi> <mi>a</mi> <mi>l</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>−</mo> <mi>D</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mo>|</mo> </mrow> </mrow> <mrow> <msub> <mi>D</mi> <mrow> <mi>r</mi> <mi>e</mi> <mi>a</mi> <mi>l</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> </semantics></math>.</p> Full article ">Figure 12
<p>Scenario 1: evolution of <math display="inline"><semantics> <mrow> <msub> <mi>I</mi> <mi>d</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> versus the corresponding real data.</p> Full article ">Figure 13
<p>Scenario 1: evolution of the <math display="inline"><semantics> <mrow> <msub> <mi>R</mi> <msub> <mi>I</mi> <mi>d</mi> </msub> </msub> <mrow> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math> versus the corresponding real data.</p> Full article ">Figure 14
<p>Scenario 1: evolution of the <math display="inline"><semantics> <mrow> <mi>D</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> <mo>)</mo> </mrow> </semantics></math> versus the corresponding real data.</p> Full article ">Figure 15
<p>Scenarios 2 and 3: evolution of <math display="inline"><semantics> <mrow> <msub> <mi>I</mi> <mi>d</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> in the two scenarios versus the corresponding real data.</p> Full article ">Figure 16
<p>Scenarios 2 and 3: evolution of the <math display="inline"><semantics> <mrow> <msub> <mi>R</mi> <msub> <mi>I</mi> <mi>d</mi> </msub> </msub> <mrow> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math> in the two scenarios versus the corresponding real data.</p> Full article ">Figure 17
<p>Scenarios 2 and 3: evolution of the <math display="inline"><semantics> <mrow> <mi>D</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> <mo>)</mo> </mrow> </semantics></math> in the two scenarios versus the corresponding real data.</p> Full article ">Figure 18
<p>Scenarios 4 and 5: evolution of <math display="inline"><semantics> <mrow> <msub> <mi>I</mi> <mi>d</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> in the two scenarios versus the corresponding real data.</p> Full article ">Figure 19
<p>Scenarios 4 and 5: evolution of the <math display="inline"><semantics> <mrow> <msub> <mi>R</mi> <msub> <mi>I</mi> <mi>d</mi> </msub> </msub> <mrow> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math> in the two scenarios versus the corresponding real data.</p> Full article ">Figure 20
<p>Scenarios 4 and 5: evolution of the <math display="inline"><semantics> <mrow> <mi>D</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> in the two scenarios versus the corresponding real data.</p> Full article ">
52 pages, 4723 KiB  
Review
Phosphorus Compounds of Natural Origin: Prebiotic, Stereochemistry, Application
by Oleg I. Kolodiazhnyi
Symmetry 2021, 13(5), 889; https://doi.org/10.3390/sym13050889 - 17 May 2021
Cited by 42 | Viewed by 12792
Abstract
Organophosphorus compounds play a vital role as nucleic acids, nucleotide coenzymes, metabolic intermediates and are involved in many biochemical processes. They are part of DNA, RNA, ATP and a number of important biological elements of living organisms. Synthetic compounds of this class have [...] Read more.
Organophosphorus compounds play a vital role as nucleic acids, nucleotide coenzymes, metabolic intermediates and are involved in many biochemical processes. They are part of DNA, RNA, ATP and a number of important biological elements of living organisms. Synthetic compounds of this class have found practical application as agrochemicals, pharmaceuticals, bioregulators, and othrs. In recent years, a large number of phosphorus compounds containing P-O, P-N, P-C bonds have been isolated from natural sources. Many of them have shown interesting biological properties and have become the objects of intensive scientific research. Most of these compounds contain asymmetric centers, the absolute configurations of which have a significant effect on the biological properties of the products of their transformations. This area of research on natural phosphorus compounds is still little-studied, that prompted us to analyze and discuss it in our review. Moreover natural organophosphorus compounds represent interesting models for the development of new biologically active compounds, and a number of promising drugs and agrochemicals have already been obtained on their basis. The review also discusses the history of the development of ideas about the role of organophosphorus compounds and stereochemistry in the origin of life on Earth, starting from the prebiotic period, that allows us in a new way to consider this most important problem of fundamental science. Full article
(This article belongs to the Collection Feature Papers in Chemistry)
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Graphical abstract

Graphical abstract
Full article ">Figure 1
<p>Natural organophosphotus compounds.</p> Full article ">Figure 2
<p>Biologically important phosphates.</p> Full article ">Figure 3
<p>An example of the substrate-level phosphorylation.</p> Full article ">Figure 4
<p>Structure of nucleic acids.</p> Full article ">Figure 5
<p><b>A</b>-, <b>B</b> and <b>Z</b>-forms of DNA.</p> Full article ">Figure 6
<p>Nucleopeptides.</p> Full article ">Figure 7
<p>Cross-shaped structure of DNA: (<b>a</b>)—Schematic representation; (<b>b</b>)—Drawing from a photograph obtained using an electron micrograph of a four-way DNA junction (permission on reproduction of Cambridge University Press and Copyright Clearance Center, Order Number: 5065840479765 from 11 May 2021) [<a href="#B29-symmetry-13-00889" class="html-bibr">29</a>].</p> Full article ">Figure 8
<p>Schematic representation of DNA-based asymmetric catalysis using the supramolecular anchoring strategy.</p> Full article ">Figure 9
<p>L- and D-Enantiomers of DNA:<span class="html-small-caps"> L</span>-DNA (<b>left</b>) and natural <span class="html-small-caps">D</span>-DNA (<b>right</b>).</p> Full article ">Figure 10
<p>The asymmetric Michael addition reaction catalyzed by complexes formed between copper(II) ions and achiral ligands in the presence of DNA.</p> Full article ">Figure 11
<p>Some chiral compounds obtained by asymmetric DNA catalysis.</p> Full article ">Figure 12
<p>Some of the most important phosphagens.</p> Full article ">Figure 13
<p>Mechanisms of phosphoryl group transfer in substitution reactions at phosphorus atom. (*) is a well-known designation for chirality, the absolute configuration of which is not defined.</p> Full article ">Figure 14
<p>ATP hydrolysis by ATPase with formation of ADP.</p> Full article ">Figure 15
<p>(<b>a</b>) Mechanism of solvolysis and racemization of dinitrophenyl phosphorous acid dianion; (<b>b</b>) The metaphosphate-anion in a crystal lattice of fructoso-1,6-bisphosphatase found by X-ray crystallographic analysis.</p> Full article ">Figure 16
<p>Interconversions of phosphocreatine/creatine kinase: the system of ATP + creatine is in equilibrium with ADP + phosphocreatine.</p> Full article ">Figure 17
<p>Examples of <span class="html-italic">N</span>-phosphamides.</p> Full article ">Figure 18
<p>Examples of Phosphaguanidines.</p> Full article ">Figure 19
<p>Analogs of N-phosphocreatinine.</p> Full article ">Figure 20
<p>Regioselective intramolecular phosphorylation of glycolaldehyde (R = H) and D-glyceraldehyde (R = CH<sub>2</sub>OH) by amidotriphosphate.</p> Full article ">Figure 21
<p>Intramolecular phosphorylation by transient tethering in the <span class="html-italic">arabino</span> series.</p> Full article ">Figure 22
<p>Formation of adenosine-5′-phosphoramide from ammonia and adenosine-5′-phosphosulfate.</p> Full article ">Figure 23
<p>Structures of the nucleotidyl derivatives Hint 2.</p> Full article ">Figure 24
<p>Nucleotides that bind protein of the histidine triad.</p> Full article ">Figure 25
<p>Natural products containing N-acylphosphoramide bonds.</p> Full article ">Figure 26
<p>Synthesis of phosmidosine diasteremers. <b>Reagents and conditions</b>: (I) MeOH-Et<sub>3</sub>N (9:1, <span class="html-italic">v</span>/<span class="html-italic">v</span>), rt, 1 h; (II) diisopropylammonium tetrazolide, CH<sub>2</sub>Cl<sub>2</sub>, rt, 1 h; (III) 17, 5-(3,5-dinitrophenyl)-1H-tetrazole, CH<sub>2</sub>Cl<sub>2</sub>-MeCN (1:1, <span class="html-italic">v</span>/<span class="html-italic">v</span>), rt, 10 min; tert-BuOOH, MeCN, rt, 5 min; I2, pyridine-H<sub>2</sub>O (9:1, <span class="html-italic">v</span>/<span class="html-italic">v</span>), rt, 30 min; (IV) 80% HCOOH, rt, 12 h. <span class="html-italic">(*) is a well-known designation for chirality, the absolute configuration of which is not defined</span>.</p> Full article ">Figure 27
<p>The synthesis of nucleotide antibiotic phosmidosine B.</p> Full article ">Figure 28
<p>Agrocine 84 and Biotin-Phosmidosine.</p> Full article ">Figure 29
<p>Metabolites of Agrocin 84.</p> Full article ">Figure 30
<p>Conversion of agrocin 84 into a toxic fragment of TM84.</p> Full article ">Figure 31
<p>Antibiotics of the McC family.</p> Full article ">Figure 32
<p>McC derivatives.</p> Full article ">Figure 33
<p>Examples of naturally occurring aminoacyladenylate-based antibiotics.</p> Full article ">Figure 34
<p>Phosphoramide antibiotics.</p> Full article ">Figure 35
<p>Protein-DNA intermediates.</p> Full article ">Figure 36
<p>Examples of N-phosphohistidine.</p> Full article ">Figure 37
<p>Natural phosphorus-sulfur bond containing phosphates.</p> Full article ">Figure 38
<p>Phosphorothioate modificated oligonucleotides.</p> Full article ">Figure 39
<p>The most famous natural phosphonates.</p> Full article ">Figure 40
<p>Overview of P-C phosphonate biosynthesis.</p> Full article ">Figure 41
<p>Synthesis of (2<span class="html-italic">S</span>,1<span class="html-italic">R</span>)-Fosfomycin starting from phosphonoenolpyruvate.</p> Full article ">Figure 42
<p>Formation of herbicide Bialaphos.</p> Full article ">Figure 43
<p>Biosynthesis of (2-amino-I-hydroxyethyl) phosphinic acid.</p> Full article ">Figure 44
<p>Biocatalytic epoxidation of (<span class="html-italic">S</span>)-2-hydroxyalkylphosphonic acids to epoxides.</p> Full article ">Figure 45
<p>Biocatalytic conversion of hydroxyphosphonate to fosfomycin.</p> Full article ">Figure 46
<p>Synthesis of a deuterated analog of (<span class="html-italic">S</span>,<span class="html-italic">R</span>)-phosphomycin.</p> Full article ">Figure 47
<p>Biologically active phosphonates.</p> Full article ">Figure 48
<p>Enzymatic synthesis of D- and L-stereomers of phosphinotricin.</p> Full article ">Figure 49
<p>The enzymatic synthesis of Phosphonothrixin.</p> Full article ">Figure 50
<p>Synthesis of phosphonothrixin.</p> Full article ">Figure 51
<p>Asymmetric synthesis of phosphonothrixin.</p> Full article ">Figure 52
<p>Determination of (<span class="html-italic">S</span>)-phosphonoth rixin configuration by chemical correlation.</p> Full article ">Figure 53
<p>Phosphorus derivatives of Plakotilen A and B.</p> Full article ">Figure 54
<p>Reaction system for ciliatocholic acid synthesis.</p> Full article ">Figure 55
<p>Nucleotide antibiotics—phosphadecin and phosphocytocin.</p> Full article ">Figure 56
<p>Representatives of phosphonolipids.</p> Full article ">Figure 57
<p>Hydrolytic treatment of alaphosphin peptides to obtain an alanine analog.</p> Full article ">Figure 58
<p>Hydrolytic treatment of dehydrophos peptides to release pyruvate analog.</p> Full article ">Figure 59
<p>Antibiotic phosphonopeptides.</p> Full article ">Figure 60
<p>Seven step synthesis of (<span class="html-italic">S</span>,<span class="html-italic">Z</span>)-APPA starting from (<span class="html-italic">R</span>)-Garner’s aldehyde. Reagents and conditions: (<b>a</b>) (CF<sub>3</sub>CH<sub>2</sub>O)<sub>2</sub>P(O)CH<sub>2</sub>CO<sub>2</sub>Me, 18-Crown-6, NaH, THF, 78 °C; (<b>b</b>) DIBAL-H 1 M sol. in hexanes, toluene, 78 °C; (<b>c</b>) CBr<sub>4</sub>, PPh<sub>3</sub>, CH<sub>2</sub>Cl<sub>2</sub>, 0 °C; (<b>d</b>) P(OMe)<sub>3</sub>, 80–100 °C; (<b>e</b>) Dowex 50WX4 H<sup>+</sup>, MeOH/H<sub>2</sub>O:(9/1); (<b>f</b>) (i) H<sub>5</sub>IO<sub>6</sub>/CrO<sub>3</sub>, CH<sub>3</sub>CN/H<sub>2</sub>O:(99/1), 0 °C; (ii) CH<sub>3</sub>I, K<sub>2</sub>CO<sub>3</sub>, CH<sub>3</sub>CN; (<b>g</b>) (i) 10% (CH<sub>3</sub>)<sub>3</sub>SiBr in CH<sub>2</sub>Cl<sub>2</sub>, and then MeOH/H<sub>2</sub>O:(9/1); (ii) 1 N LiOH, THF/H<sub>2</sub>O:(4/1).</p> Full article ">Figure 61
<p>Phosphonopeptides containing a nitrogen-nitrogen (NN) bond.</p> Full article ">Figure 62
<p>Determination of absolute configuration of phosphazinomycin by chemical correlation. (*)—center of chirality with an undefined absolute configuration.</p> Full article ">Figure 63
<p>Some representatives of phosphonopeptides.</p> Full article ">Figure 64
<p>Representatives of peptide antibiotics.</p> Full article ">Figure 65
<p>Stereocontrolled synthesis of the tyrosine hydroxyphosphonate analog.</p> Full article ">Figure 66
<p>Stereoselective synthesis of tripeptide renin inhibitors.</p> Full article ">Figure 67
<p>Phosphonic tripeptides and tetrapeptide.</p> Full article ">Figure 68
<p>Mechanism of Trialaphos and Phosalacine transformation inside of cell.</p> Full article ">
16 pages, 2937 KiB  
Article
A Topology Optimization Method Based on Non-Uniform Rational Basis Spline Hyper-Surfaces for Heat Conduction Problems
by Marco Montemurro and Khalil Refai
Symmetry 2021, 13(5), 888; https://doi.org/10.3390/sym13050888 - 17 May 2021
Cited by 19 | Viewed by 3758
Abstract
This work deals with heat conduction problems formulation in the framework of a CAD-compatible topology optimization method based on a pseudo-density field as a topology descriptor. In particular, the proposed strategy relies, on the one hand, on the use of CAD-compatible Non-Uniform Rational [...] Read more.
This work deals with heat conduction problems formulation in the framework of a CAD-compatible topology optimization method based on a pseudo-density field as a topology descriptor. In particular, the proposed strategy relies, on the one hand, on the use of CAD-compatible Non-Uniform Rational Basis Spline (NURBS) hyper-surfaces to represent the pseudo-density field and, on the other hand, on the well-known Solid Isotropic Material with Penalization (SIMP) approach. The resulting method is then referred to as NURBS-based SIMP method. In this background, heat conduction problems have been reformulated by taking advantage of the properties of the NURBS entities. The influence of the integer parameters, involved in the definition of the NURBS hyper-surface, on the optimized topology is investigated. Furthermore, symmetry constraints, as well as a manufacturing requirement related to the minimum allowable size, are also integrated into the problem formulation without introducing explicit constraint functions, thanks to the NURBS blending functions properties. Finally, since the topological variable is represented by means of a NURBS entity, the geometrical representation of the boundary of the topology is available at each iteration of the optimization process and its reconstruction becomes a straightforward task. The effectiveness of the NURBS-based SIMP method is shown on 2D and 3D benchmark problems taken from the literature. Full article
(This article belongs to the Special Issue Mathematical Theory, Methods, and Its Applications for Industry)
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Figure 1

Figure 1
<p>Geometry and boundary conditions of benchmark problems (<b>a</b>) BK1-2D and (<b>b</b>) BK2-2D.</p> Full article ">Figure 2
<p>Benchmark problem BK1-2D: sensitivity of the optimized topology to CP number and basis functions degrees, B-spline solutions of problem (<a href="#FD19-symmetry-13-00888" class="html-disp-formula">19</a>); the grey-scale bar refers to the pseudo-density field of Equation (<a href="#FD16-symmetry-13-00888" class="html-disp-formula">16</a>).</p> Full article ">Figure 2 Cont.
<p>Benchmark problem BK1-2D: sensitivity of the optimized topology to CP number and basis functions degrees, B-spline solutions of problem (<a href="#FD19-symmetry-13-00888" class="html-disp-formula">19</a>); the grey-scale bar refers to the pseudo-density field of Equation (<a href="#FD16-symmetry-13-00888" class="html-disp-formula">16</a>).</p> Full article ">Figure 3
<p>Benchmark problem BK1-2D: sensitivity of the optimized topology to CP number and basis functions degrees, NURBS solutions of problem (<a href="#FD19-symmetry-13-00888" class="html-disp-formula">19</a>); the grey-scale bar refers to the pseudo-density field of Equation (<a href="#FD16-symmetry-13-00888" class="html-disp-formula">16</a>).</p> Full article ">Figure 3 Cont.
<p>Benchmark problem BK1-2D: sensitivity of the optimized topology to CP number and basis functions degrees, NURBS solutions of problem (<a href="#FD19-symmetry-13-00888" class="html-disp-formula">19</a>); the grey-scale bar refers to the pseudo-density field of Equation (<a href="#FD16-symmetry-13-00888" class="html-disp-formula">16</a>).</p> Full article ">Figure 4
<p>Benchmark problem BK1-2D: dimensionless thermal compliance versus CPs number and degrees for B-spline and NURBS solutions</p> Full article ">Figure 5
<p>Benchmark problem BK2-2D: (<b>a</b>) B-spline and (<b>b</b>) NURBS solutions of problem (<a href="#FD19-symmetry-13-00888" class="html-disp-formula">19</a>) with <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>n</mi> <mi>CP</mi> </msub> <mo>=</mo> <mn>68</mn> <mo>×</mo> <mn>68</mn> </mrow> </semantics></math>; the grey-scale bar refers to the pseudo-density field of Equation (<a href="#FD16-symmetry-13-00888" class="html-disp-formula">16</a>).</p> Full article ">Figure 6
<p>Geometry and boundary conditions of benchmark problem BK1-3D.</p> Full article ">Figure 7
<p>Benchmark problem BK1-3D: (<b>a</b>) B-spline and (<b>b</b>) NURBS solutions of problem (<a href="#FD19-symmetry-13-00888" class="html-disp-formula">19</a>) with <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>n</mi> <mi>CP</mi> </msub> <mo>=</mo> <mn>30</mn> <mo>×</mo> <mn>30</mn> <mo>×</mo> <mn>30</mn> </mrow> </semantics></math>.</p> Full article ">
20 pages, 1432 KiB  
Article
Exact Likelihood Inference for a Competing Risks Model with Generalized Type II Progressive Hybrid Censored Exponential Data
by Subin Cho and Kyeongjun Lee
Symmetry 2021, 13(5), 887; https://doi.org/10.3390/sym13050887 - 17 May 2021
Cited by 7 | Viewed by 1918
Abstract
In many situations of survival and reliability test, the withdrawal of units from the test is pre-planned in order to to free up testing facilities for other tests, or to save cost and time. It is known that several risk factors (RiFs) compete [...] Read more.
In many situations of survival and reliability test, the withdrawal of units from the test is pre-planned in order to to free up testing facilities for other tests, or to save cost and time. It is known that several risk factors (RiFs) compete for the immediate failure cause of items. In this paper, we derive an inference for a competing risks model (CompRiM) with a generalized type II progressive hybrid censoring scheme (GeTy2PrHCS). We derive the conditional moment generating functions (CondMgfs), distributions and confidence interval (ConfI) of the scale parameters of exponential distribution (ExDist) under GeTy2PrHCS with CompRiM. A real data set is analysed to illustrate the validity of the method developed here. From the data, it can be seen that the conditional PDFs of MLEs is almost symmetrical. Full article
(This article belongs to the Section Mathematics)
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Figure 1

Figure 1
<p>GeTy2PrHCS. <b>Case</b> (<b>a</b>): <math display="inline"><semantics> <mrow> <mo>{</mo> <msub> <mi>X</mi> <mrow> <mn>1</mn> <mo>:</mo> <mi>m</mi> <mo>:</mo> <mi>n</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow> <mn>2</mn> <mo>:</mo> <mi>m</mi> <mo>:</mo> <mi>n</mi> </mrow> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow> <mi>m</mi> <mo>:</mo> <mi>m</mi> <mo>:</mo> <mi>n</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow> <mi>m</mi> <mo>+</mo> <mn>1</mn> <mo>:</mo> <mi>n</mi> </mrow> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow> <msub> <mi>d</mi> <mn>1</mn> </msub> <mo>:</mo> <mi>n</mi> </mrow> </msub> <mo>}</mo> </mrow> </semantics></math>, if <math display="inline"><semantics> <mrow> <msub> <mi>X</mi> <mrow> <mi>m</mi> <mo>:</mo> <mi>m</mi> <mo>:</mo> <mi>n</mi> </mrow> </msub> <mo>&lt;</mo> <msub> <mi mathvariant="script">T</mi> <mn>1</mn> </msub> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mo>ℜ</mo> <mi>m</mi> </msub> <mo>=</mo> <msub> <mo>ℜ</mo> <mrow> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo>⋯</mo> <mo>=</mo> <msub> <mo>ℜ</mo> <msub> <mi>d</mi> <mn>1</mn> </msub> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>. <b>Case</b> (<b>b</b>): <math display="inline"><semantics> <mrow> <mo>{</mo> <msub> <mi>X</mi> <mrow> <mn>1</mn> <mo>:</mo> <mi>m</mi> <mo>:</mo> <mi>n</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow> <mn>2</mn> <mo>:</mo> <mi>m</mi> <mo>:</mo> <mi>n</mi> </mrow> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow> <msub> <mi>d</mi> <mn>1</mn> </msub> <mo>:</mo> <mi>m</mi> <mo>:</mo> <mi>n</mi> </mrow> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow> <mi>m</mi> <mo>:</mo> <mi>m</mi> <mo>:</mo> <mi>n</mi> </mrow> </msub> <mo>}</mo> </mrow> </semantics></math>, if <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="script">T</mi> <mn>1</mn> </msub> <mo>&lt;</mo> <msub> <mi>X</mi> <mrow> <mi>m</mi> <mo>:</mo> <mi>m</mi> <mo>:</mo> <mi>n</mi> </mrow> </msub> <mo>&lt;</mo> <msub> <mi mathvariant="script">T</mi> <mn>2</mn> </msub> </mrow> </semantics></math>. <b>Case</b> (<b>c</b>): <math display="inline"><semantics> <mrow> <mo>{</mo> <msub> <mi>X</mi> <mrow> <mn>1</mn> <mo>:</mo> <mi>m</mi> <mo>:</mo> <mi>n</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mrow> <mn>2</mn> <mo>:</mo> <mi>m</mi> <mo>:</mo> <mi>n</mi> </mrow> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow> <msub> <mi>d</mi> <mn>1</mn> </msub> <mo>:</mo> <mi>m</mi> <mo>:</mo> <mi>n</mi> </mrow> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow> <msub> <mi>d</mi> <mn>2</mn> </msub> <mo>:</mo> <mi>m</mi> <mo>:</mo> <mi>n</mi> </mrow> </msub> <mo>}</mo> </mrow> </semantics></math>, if <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="script">T</mi> <mn>2</mn> </msub> <mo>&lt;</mo> <msub> <mi>X</mi> <mrow> <mi>m</mi> <mo>:</mo> <mi>m</mi> <mo>:</mo> <mi>n</mi> </mrow> </msub> </mrow> </semantics></math>.</p> Full article ">Figure 2
<p>The CondPDFs of <math display="inline"><semantics> <msub> <mover accent="true"> <mi>λ</mi> <mo>^</mo> </mover> <mn>1</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mover accent="true"> <mi>λ</mi> <mo>^</mo> </mover> <mn>2</mn> </msub> </semantics></math> for example.</p> Full article ">Figure 3
<p>Relative rMSEs for <math display="inline"><semantics> <msub> <mover accent="true"> <mi>λ</mi> <mo>^</mo> </mover> <mn>1</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mover accent="true"> <mi>λ</mi> <mo>^</mo> </mover> <mn>2</mn> </msub> </semantics></math></p> Full article ">Figure 4
<p>Relative ConfL for <math display="inline"><semantics> <msub> <mover accent="true"> <mi>λ</mi> <mo>^</mo> </mover> <mn>1</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mover accent="true"> <mi>λ</mi> <mo>^</mo> </mover> <mn>2</mn> </msub> </semantics></math>.</p> Full article ">
20 pages, 1733 KiB  
Article
A New Analysis of Fractional-Order Equal-Width Equations via Novel Techniques
by Muhammad Naeem, Ahmed M. Zidan, Kamsing Nonlaopon, Muhammad I. Syam, Zeyad Al-Zhour and Rasool Shah
Symmetry 2021, 13(5), 886; https://doi.org/10.3390/sym13050886 - 17 May 2021
Cited by 35 | Viewed by 3629
Abstract
In this paper, the new iterative transform method and the homotopy perturbation transform method was used to solve fractional-order Equal-Width equations with the help of Caputo-Fabrizio. This method combines the Laplace transform with the new iterative transform method and the homotopy perturbation method. [...] Read more.
In this paper, the new iterative transform method and the homotopy perturbation transform method was used to solve fractional-order Equal-Width equations with the help of Caputo-Fabrizio. This method combines the Laplace transform with the new iterative transform method and the homotopy perturbation method. The approximate results are calculated in the series form with easily computable components. The fractional Equal-Width equations play an essential role in describe hydromagnetic waves in cold plasma. Our object is to study the nonlinear behaviour of the plasma system and highlight the critical points. The techniques are very reliable, effective, and efficient, which can solve a wide range of problems arising in engineering and sciences. Full article
(This article belongs to the Special Issue Methods on Discrete Dynamical Systems, Networks, and Optimization for Signal Modelling)
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<p>The actual and HPTM solution graphs at <math display="inline"><semantics> <mrow> <mi>ϱ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> of Example 1.</p> Full article ">Figure 2
<p>The HPTM solution of different fractional-order ϱ graph of Example 1.</p> Full article ">Figure 3
<p>The actual and HPTM solution graphs at <math display="inline"><semantics> <mrow> <mi>ϱ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> of Example 2.</p> Full article ">Figure 4
<p>The HPTM solution of different fractional-order ϱ graph of Example 2.</p> Full article ">Figure 5
<p>The actual and HPTM solution graphs at <math display="inline"><semantics> <mrow> <mi>ϱ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> of Example 3.</p> Full article ">Figure 6
<p>The HPTM solution of different fractional-order ϱ graph of Example 3.</p> Full article ">Figure 7
<p>The actual and ITM solution graphs at <math display="inline"><semantics> <mrow> <mi>ϱ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> with respect to ϕ and ℑ of Example 4.</p> Full article ">Figure 8
<p>The ITM solution of different fractional-order ϱ graphs with respect to ϕ and ℑ of Example 4.</p> Full article ">Figure 9
<p>The actual and ITM solution graphs at <math display="inline"><semantics> <mrow> <mi>ϱ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> with respect to ϕ and ℑ of Example 5.</p> Full article ">Figure 10
<p>The ITM solution of different fractional-order ϱ graphs with respect to ϕ and ℑ of Example 5.</p> Full article ">Figure 11
<p>The actual and ITM solution graphs at <math display="inline"><semantics> <mrow> <mi>ϱ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> with respect to ϕ and ℑ of Example 6.</p> Full article ">Figure 12
<p>The ITM solution of different fractional-order ϱ graphs with respect to ϕ and ℑ of Example 6.</p> Full article ">
12 pages, 638 KiB  
Article
Enhancing Ant-Based Algorithms for Medical Image Edge Detection by Admissible Perturbations of Demicontractive Mappings
by Vasile Berinde and Cristina Ţicală
Symmetry 2021, 13(5), 885; https://doi.org/10.3390/sym13050885 - 17 May 2021
Cited by 8 | Viewed by 2716
Abstract
The aim of this paper is to show analytically and empirically how ant-based algorithms for medical image edge detection can be enhanced by using an admissible perturbation of demicontractive operators. We thus complement the results reported in a recent paper by the second [...] Read more.
The aim of this paper is to show analytically and empirically how ant-based algorithms for medical image edge detection can be enhanced by using an admissible perturbation of demicontractive operators. We thus complement the results reported in a recent paper by the second author and her collaborators, where they used admissible perturbations of demicontractive mappings as test functions. To illustrate this fact, we first consider some typical properties of demicontractive mappings and of their admissible perturbations and then present some appropriate numerical tests to illustrate the improvement brought by the admissible perturbations of demicontractive mappings when they are taken as test functions in ant-based algorithms for medical image edge detection. The edge detection process reported in our study considers both symmetric (Head CT and Brain CT) and asymmetric (Hand X-ray) medical images. The performance of the algorithm was tested visually with various images and empirically with evaluation of parameters. Full article
(This article belongs to the Special Issue Fixed Point Theory and Computational Analysis with Applications)
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<p>Graph for the successive approximations points corresponding to <span class="html-italic">T</span>, <math display="inline"><semantics> <msub> <mi>T</mi> <mrow> <mn>0.1</mn> </mrow> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mi>T</mi> <mrow> <mn>0.2</mn> </mrow> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>T</mi> <mrow> <mn>0.5</mn> </mrow> </msub> </semantics></math>.</p> Full article ">Figure 2
<p>Distribution of iterations <math display="inline"><semantics> <msub> <mi>x</mi> <mi>n</mi> </msub> </semantics></math> with respect to the corresponding graphs of operators <span class="html-italic">T</span>, <math display="inline"><semantics> <msub> <mi>T</mi> <mrow> <mn>0.5</mn> </mrow> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>T</mi> <mrow> <mn>0.1</mn> </mrow> </msub> </semantics></math>.</p> Full article ">Figure 3
<p>Test-data: medical images (<math display="inline"><semantics> <mrow> <mi>H</mi> <mi>e</mi> <mi>a</mi> <mi>d</mi> <mo>_</mo> <mi>C</mi> <mi>T</mi> </mrow> </semantics></math> (<b>a</b>) [<a href="#B23-symmetry-13-00885" class="html-bibr">23</a>], <math display="inline"><semantics> <mrow> <mi mathvariant="italic">Brain</mi> <mo>_</mo> <mi mathvariant="italic">CT</mi> </mrow> </semantics></math> (<b>b</b>) and <math display="inline"><semantics> <mrow> <mi mathvariant="italic">Hand</mi> <mo>_</mo> <mi>X</mi> <mspace width="-0.166667em"/> <mo>-</mo> <mspace width="-0.166667em"/> <mi mathvariant="italic">ray</mi> </mrow> </semantics></math> [<a href="#B24-symmetry-13-00885" class="html-bibr">24</a>] (<b>c</b>)).</p> Full article ">Figure 4
<p>Detected edges for <span class="html-italic">Head CT</span> with test function: (<b>a</b>) <span class="html-italic">T</span>; (<b>b</b>) <math display="inline"><semantics> <msub> <mi>T</mi> <mrow> <mn>0.9</mn> </mrow> </msub> </semantics></math>; (<b>c</b>) <math display="inline"><semantics> <msub> <mi>T</mi> <mrow> <mn>0.5</mn> </mrow> </msub> </semantics></math>; and (<b>d</b>) <math display="inline"><semantics> <msub> <mi>T</mi> <mfrac> <mn>1</mn> <mn>15</mn> </mfrac> </msub> </semantics></math>.</p> Full article ">Figure 5
<p>Detected edges for <span class="html-italic">Brain CT</span> with test function: (<b>a</b>) <span class="html-italic">T</span>; (<b>b</b>) <math display="inline"><semantics> <msub> <mi>T</mi> <mrow> <mn>0.9</mn> </mrow> </msub> </semantics></math>; (<b>c</b>) <math display="inline"><semantics> <msub> <mi>T</mi> <mrow> <mn>0.5</mn> </mrow> </msub> </semantics></math>; and (<b>d</b>) <math display="inline"><semantics> <msub> <mi>T</mi> <mfrac> <mn>1</mn> <mn>15</mn> </mfrac> </msub> </semantics></math>.</p> Full article ">Figure 6
<p>Detected edges for Hand X-ray with test function: (<b>a</b>) <span class="html-italic">T</span>; (<b>b</b>) <math display="inline"><semantics> <msub> <mi>T</mi> <mrow> <mn>0.9</mn> </mrow> </msub> </semantics></math>; (<b>c</b>) <math display="inline"><semantics> <msub> <mi>T</mi> <mrow> <mn>0.5</mn> </mrow> </msub> </semantics></math>; and (<b>d</b>) <math display="inline"><semantics> <msub> <mi>T</mi> <mfrac> <mn>1</mn> <mn>15</mn> </mfrac> </msub> </semantics></math>.</p> Full article ">Figure 7
<p>Extracted edges for <span class="html-italic">Hand X-Ray</span> with different edge extraction methods: (<b>a</b>) <math display="inline"><semantics> <mrow> <mi mathvariant="italic">Prewitt</mi> </mrow> </semantics></math>; (<b>b</b>) <math display="inline"><semantics> <mrow> <mi mathvariant="italic">Sobel</mi> </mrow> </semantics></math>; (<b>c</b>) <math display="inline"><semantics> <mrow> <mi mathvariant="italic">Roberts</mi> </mrow> </semantics></math>; and (<b>d</b>) ACO with test function <math display="inline"><semantics> <msub> <mi>T</mi> <mfrac> <mn>1</mn> <mn>15</mn> </mfrac> </msub> </semantics></math>.</p> Full article ">
17 pages, 1943 KiB  
Article
A General Optimal Iterative Scheme with Arbitrary Order of Convergence
by Alicia Cordero, Juan R. Torregrosa and Paula Triguero-Navarro
Symmetry 2021, 13(5), 884; https://doi.org/10.3390/sym13050884 - 16 May 2021
Cited by 9 | Viewed by 2287
Abstract
A general optimal iterative method, for approximating the solution of nonlinear equations, of (n+1) steps with 2n+1 order of convergence is presented. Cases n=0 and n=1 correspond to Newton’s and Ostrowski’s schemes, [...] Read more.
A general optimal iterative method, for approximating the solution of nonlinear equations, of (n+1) steps with 2n+1 order of convergence is presented. Cases n=0 and n=1 correspond to Newton’s and Ostrowski’s schemes, respectively. The basins of attraction of the proposed schemes on different test functions are analyzed and compared with the corresponding to other known methods. The dynamical planes showing the different symmetries of the basins of attraction of new and known methods are presented. The performance of different methods on some test functions is shown. Full article
(This article belongs to the Special Issue Recent Advances and Application of Iterative Methods)
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Figure 1

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<p>Dynamical planes of <math display="inline"><semantics> <mrow> <mo form="prefix">cos</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> for the new and known methods: (<b>a</b>) Newton, (<b>b</b>) M4 (<a href="#FD4-symmetry-13-00884" class="html-disp-formula">4</a>), (<b>c</b>) K4 (<a href="#FD18-symmetry-13-00884" class="html-disp-formula">18</a>), (<b>d</b>) J4 (<a href="#FD18-symmetry-13-00884" class="html-disp-formula">18</a>), (<b>e</b>) M8 (<a href="#FD5-symmetry-13-00884" class="html-disp-formula">5</a>), (<b>f</b>) K8 (<a href="#FD20-symmetry-13-00884" class="html-disp-formula">20</a>) and (<b>g</b>) J8 (<a href="#FD19-symmetry-13-00884" class="html-disp-formula">19</a>).</p> Full article ">Figure 2
<p>Dynamical planes of <math display="inline"><semantics> <mrow> <msup> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mn>6</mn> </msup> <mo>−</mo> <mn>1</mn> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> for the new and known methods: (<b>a</b>) Newton, (<b>b</b>) M4 (<a href="#FD4-symmetry-13-00884" class="html-disp-formula">4</a>), (<b>c</b>) K4 (<a href="#FD18-symmetry-13-00884" class="html-disp-formula">18</a>), (<b>d</b>) J4 (<a href="#FD18-symmetry-13-00884" class="html-disp-formula">18</a>), (<b>e</b>) M8 (<a href="#FD5-symmetry-13-00884" class="html-disp-formula">5</a>), (<b>f</b>) K8 (<a href="#FD20-symmetry-13-00884" class="html-disp-formula">20</a>) and (<b>g</b>) J8 (<a href="#FD19-symmetry-13-00884" class="html-disp-formula">19</a>).</p> Full article ">Figure 3
<p>Dynamical planes of <math display="inline"><semantics> <mrow> <mo form="prefix">arctan</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> for the new and known methods: (<b>a</b>) Newton, (<b>b</b>) M4 (<a href="#FD4-symmetry-13-00884" class="html-disp-formula">4</a>), (<b>c</b>) K4 (<a href="#FD18-symmetry-13-00884" class="html-disp-formula">18</a>), (<b>d</b>) J4 (<a href="#FD18-symmetry-13-00884" class="html-disp-formula">18</a>), (<b>e</b>) M8 (<a href="#FD5-symmetry-13-00884" class="html-disp-formula">5</a>), (<b>f</b>) K8 (<a href="#FD20-symmetry-13-00884" class="html-disp-formula">20</a>) and (<b>g</b>) J8 (<a href="#FD19-symmetry-13-00884" class="html-disp-formula">19</a>).</p> Full article ">Figure 4
<p>Dynamical planes of <math display="inline"><semantics> <mrow> <mo form="prefix">arctan</mo> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>−</mo> <mfrac> <mrow> <mn>2</mn> <mi>x</mi> </mrow> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> for the new and known methods: (<b>a</b>) Newton, (<b>b</b>) M4 (<a href="#FD4-symmetry-13-00884" class="html-disp-formula">4</a>), (<b>c</b>) K4 (<a href="#FD18-symmetry-13-00884" class="html-disp-formula">18</a>), (<b>d</b>) J4 (<a href="#FD18-symmetry-13-00884" class="html-disp-formula">18</a>), (<b>e</b>) M8 (<a href="#FD5-symmetry-13-00884" class="html-disp-formula">5</a>), (<b>f</b>) K8 (<a href="#FD20-symmetry-13-00884" class="html-disp-formula">20</a>) and (<b>g</b>) J8 (<a href="#FD19-symmetry-13-00884" class="html-disp-formula">19</a>).</p> Full article ">
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