Symmetry

22 pages, 7870 KiB

Open AccessArticle

Effects of Support Friction on Mixed-Mode I/II Fracture Behavior of Compacted Clay Using Notched Deep Beam Specimens under Symmetric Fixed Support

by Shiyuan Huang, Xudong Li, Wenbing Yu, Xiaofeng Zhang and Hongbo Du

Symmetry 2023, 15(6), 1290; https://doi.org/10.3390/sym15061290 - 20 Jun 2023

Cited by 1 | Viewed by 1358

Abstract

This paper investigates the effects of support friction on mixed-mode I/II fracture behavior of compacted clay using notched deep beam (NDB) specimens under symmetric fixed support. Numerical models of 330 NDB specimens were established considering the crack inclination angle, crack length, support span, [...] Read more.

This paper investigates the effects of support friction on mixed-mode I/II fracture behavior of compacted clay using notched deep beam (NDB) specimens under symmetric fixed support. Numerical models of 330 NDB specimens were established considering the crack inclination angle, crack length, support span, and support friction coefficient, and the normalized fracture parameters (Y_I, Y_II, and T*) of NDB specimens were calibrated. The numerical results showed that the values of Y_I, Y_II, and T* decreased at different degrees after considering the support friction. Notably, the support friction coefficient could significantly change the loading pattern at the crack tip. To verify this phenomenon, 12 compacted clay NDB specimens were prepared, and a mixed-mode I/II fracture test was performed under fixed support conditions; the phenomenon of asymmetric crack propagation was studied. The test data were processed using the numerical calibration results of Y_I, Y_II, and T* with and without consideration of friction. Afterward, the test data were compared and analyzed by combining the generalized maximum tangential stress (GMTS) and the maximum tangential stress (MTS) criteria. The analysis indicated that the real fracture characteristics of compacted clay NDB specimens could not be reflected when conducting mixed-mode I/II fracture tests under symmetric fixed support conditions if the test results were analyzed by Y_I, Y_II, and T* without considering support friction, as in previous studies. Full article

(This article belongs to the Special Issue Symmetry, Asymmetry and Nonlinearity in Geomechanics)

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13 pages, 1043 KiB

Open AccessArticle

A New Two-Parameter Discrete Distribution for Overdispersed and Asymmetric Data: Its Properties, Estimation, Regression Model, and Applications

by Amani Alrumayh and Hazar A. Khogeer

Symmetry 2023, 15(6), 1289; https://doi.org/10.3390/sym15061289 - 20 Jun 2023

Cited by 5 | Viewed by 1871

Abstract

A novel discrete Poisson mixing probability distribution with two parameters has been developed by combining the Poisson distribution with the transmuted moment exponential distribution. It is possible to deduce several mathematical properties, such as the moment-generating function, ordinary moments, moments about the mean, [...] Read more.

A novel discrete Poisson mixing probability distribution with two parameters has been developed by combining the Poisson distribution with the transmuted moment exponential distribution. It is possible to deduce several mathematical properties, such as the moment-generating function, ordinary moments, moments about the mean, skewness, kurtosis, and the dispersion index. The maximum likelihood estimation method is utilized to estimate the model’s parameters. A thorough simulation study is utilized to determine the behavior of the generated estimators. Estimating model parameters using a Bayesian methodology is another primary topic of this research. The behavior of Bayesian estimates is evaluated by first charting the trace, then generating 1,005,000 iterations of the Markov chain Monte Carlo method. In addition to this, we suggest a new count regression model that uses Poisson and negative binomial models in an alternating fashion. In conclusion, asymmetric datasets derived from various research areas are utilized for practical applications. Full article

(This article belongs to the Special Issue Symmetrical and Asymmetrical Distributions in Statistics and Probability)

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14 pages, 4274 KiB

Open AccessArticle

Heat and Mass Transfer Gravity Driven Fluid Flow over a Symmetrically-Vertical Plane through Neural Networks

by Fuad A. Awwad, Emad A. A. Ismail and Taza Gul

Symmetry 2023, 15(6), 1288; https://doi.org/10.3390/sym15061288 - 20 Jun 2023

Cited by 8 | Viewed by 1623

Abstract

This paper explores the numerical optimization of heat and mass transfer in the buoyancy-driven Al₂O₃-water nanofluid flow containing electrified Al₂O₃-nanoparticles adjacent to a symmetrically-vertical plane wall. The proposed model becomes a set of nonlinear problems [...] Read more.

This paper explores the numerical optimization of heat and mass transfer in the buoyancy-driven Al₂O₃-water nanofluid flow containing electrified Al₂O₃-nanoparticles adjacent to a symmetrically-vertical plane wall. The proposed model becomes a set of nonlinear problems through similarity transformations. The nonlinear problem is solved using the bvp4c method. The results of the proposed model concerning heat and mass transfer with nanoparticle electrification and buoyancy parameters are depicted in the Figures and Tables. It was revealed that the electrification of nanoparticles enhances the heat and mass transfer capabilities of the Al₂O₃ water nanoliquid. As a result, the electrification of nanoparticles could be an important mechanism to improve the transmission of heat and mass in the flow of Al₂O₃-water nanofluids. Furthermore, the numerical solutions of the nanofluid model of heat/mass transfer using the deep neural network (DNN) along with the procedure of Bayesian regularization scheme (BRS), DNN-BRS, was carried out. The DNN process is provided by taking eight and ten neurons in the first and second hidden layers along with the log-sigmoid function. Full article

(This article belongs to the Special Issue Fluid Dynamics and Magnetogasdynamics)

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18 pages, 2210 KiB

Open AccessArticle

An Effective Framework for Intellectual Property Protection of NLG Models

by Mingjie Li, Zichi Wang and Xinpeng Zhang

Symmetry 2023, 15(6), 1287; https://doi.org/10.3390/sym15061287 - 20 Jun 2023

Cited by 1 | Viewed by 1900

Abstract

Natural language generation (NLG) models combined with increasingly mature and powerful deep learning techniques have been widely used in recent years. Deployed NLG models in practical applications may be stolen or used illegally, and watermarking has become an important tool to protect the [...] Read more.

Natural language generation (NLG) models combined with increasingly mature and powerful deep learning techniques have been widely used in recent years. Deployed NLG models in practical applications may be stolen or used illegally, and watermarking has become an important tool to protect the Intellectual Property (IP) of these deep models. Watermarking technique designs algorithms to embed watermark information and extracts watermark information for IP identification of NLG models can be seen as a symmetric signal processing problem. In terms of IP protection of NLG models, however, the existing watermarking approaches cannot provide reliable and timely model protection and prevent illegal users from utilizing the original performance of the stolen models. In addition, the quality of watermarked text sequences generated by some watermarking approaches is not high. In view of these, this paper proposes two embedding schemes to the hidden memory state of the RNN to protect the IP of NLG models for different tasks. Besides, we add a language model loss to the model decoder to improve the grammatical correctness of the output text sequences. During the experiments, it is proved that our approach does not compromise the performance of the original NLG models on the corresponding datasets and outputs high-quality text sequences, while forged secret keys will generate unusable NLG models, thus defeating the purpose of model infringement. Besides, we also conduct sufficient experiments to prove that the proposed model has strong robustness under different attacks. Full article

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14 pages, 440 KiB

Open AccessArticle

Wavelet Multiscale Granger Causality Analysis Based on State Space Models

by Qiyi Zhang, Chuanlin Zhang and Shuangqin Cheng

Symmetry 2023, 15(6), 1286; https://doi.org/10.3390/sym15061286 - 20 Jun 2023

Cited by 1 | Viewed by 2318

Abstract

Granger causality (GC) is a popular method in causal linkage recovery and has been applied to various fields, such as economics and neuroscience. While the conventional Granger causality model is capable of identifying symmetrical causal relationships among variables, it is the asymmetric Granger [...] Read more.

Granger causality (GC) is a popular method in causal linkage recovery and has been applied to various fields, such as economics and neuroscience. While the conventional Granger causality model is capable of identifying symmetrical causal relationships among variables, it is the asymmetric Granger causality that provides a more comprehensive perspective of the short- and long-term interactions between variables, which is of greater value for empirical study. Traditional vector autoregressive models lack the ability to explore multiscale information flow and are affected by the moving average component. Therefore, by combining the wavelet-based approach and state space model, we propose a new Granger causality analysis method to overcome the inherent limitation of vector autoregressive models and extend to multiscale causality exploration. Two simulations were conducted to compare the proposed approach to an existing wavelet-based method, and five evaluation indicators were utilized. The results indicate that the proposed method efficiently identifies the accurate asymmetric causalities at varying scales, while improving accuracy and reducing bias as compared to the current wavelet-based method. In conclusion, the combination of the wavelet approach and state space method enhances the multiscale causality detecting capability and can potentially contribute to multiscale Granger causality research. Full article

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30 pages, 495 KiB

Open AccessFeature PaperArticle

Geometric Numerical Methods for Lie Systems and Their Application in Optimal Control

by Luis Blanco Díaz, Cristina Sardón, Fernando Jiménez Alburquerque and Javier de Lucas

Symmetry 2023, 15(6), 1285; https://doi.org/10.3390/sym15061285 - 19 Jun 2023

Cited by 1 | Viewed by 2091

Abstract

A Lie system is a nonautonomous system of first-order ordinary differential equations whose general solution can be written via an autonomous function, the so-called (nonlinear) superposition rule of a finite number of particular solutions and some parameters to be related to initial conditions. [...] Read more.

A Lie system is a nonautonomous system of first-order ordinary differential equations whose general solution can be written via an autonomous function, the so-called (nonlinear) superposition rule of a finite number of particular solutions and some parameters to be related to initial conditions. This superposition rule can be obtained using the geometric features of the Lie system, its symmetries, and the symmetric properties of certain morphisms involved. Even if a superposition rule for a Lie system is known, the explicit analytic expression of its solutions frequently is not. This is why this article focuses on a novel geometric attempt to integrate Lie systems analytically and numerically. We focus on two families of methods based on Magnus expansions and on Runge–Kutta–Munthe–Kaas methods, which are here adapted, in a geometric manner, to Lie systems. To illustrate the accuracy of our techniques we analyze Lie systems related to Lie groups of the form SL

(n, R)

, which play a very relevant role in mechanics. In particular, we depict an optimal control problem for a vehicle with quadratic cost function. Particular numerical solutions of the studied examples are given. Full article

(This article belongs to the Special Issue Symmetry and Applications of Differential Geometry to the Differential Equations of Mathematical Physics)

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27 pages, 491 KiB

Open AccessArticle

Computational Techniques for Solving Mixed (1 + 1) Dimensional Integral Equations with Strongly Symmetric Singular Kernel

by Sharifah E. Alhazmi, Amr M. S. Mahdy, Mohamed A. Abdou and Doaa Sh. Mohamed

Symmetry 2023, 15(6), 1284; https://doi.org/10.3390/sym15061284 - 19 Jun 2023

Cited by 9 | Viewed by 1392

Abstract

This paper describes an effective strategy based on Lerch polynomial method for solving mixed integral equations (MIE) in position and time with a strongly symmetric singular kernel in the space [...] Read more.

This paper describes an effective strategy based on Lerch polynomial method for solving mixed integral equations (MIE) in position and time with a strongly symmetric singular kernel in the space

L_{2} (- 1, 1) \times C [0, T], (T < 1) .

The Quadratic numerical method (QNM) was applied to obtain a system of Fredholm integral equations (SFIE), then the Lerch polynomials method (LPM) was applied to transform SFIE into a system of linear algebraic equations (SLAE). The existence and uniqueness of the integral equation’s solution are discussed using Banach’s fixed point theory. Also, the convergence and stability of the solution and the stability of the error are discussed. Several examples are given to illustrate the applicability of the presented method. The Maple program obtains all the results. A numerical simulation is carried out to determine the efficacy of the methodology, and the results are given in symmetrical forms. From the numerical results, it is noted that there is a symmetry utterly identical to the kernel used when replacing each x with y. Full article

(This article belongs to the Special Issue Fixed Point Theory and Its Applications Dedicated to the Memory of Professor William Arthur Kirk)

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24 pages, 1399 KiB

Open AccessArticle

The Unit Alpha-Power Kum-Modified Size-Biased Lehmann Type II Distribution: Theory, Simulation, and Applications

by Rabab S. Gomaa, Alia M. Magar, Najwan Alsadat, Ehab M. Almetwally and Ahlam H. Tolba

Symmetry 2023, 15(6), 1283; https://doi.org/10.3390/sym15061283 - 19 Jun 2023

Cited by 2 | Viewed by 1307

Abstract

In order to represent the data with non-monotonic failure rates and produce a better fit, a novel distribution is created in this study using the alpha power family of distributions. This distribution is called the alpha-power Kum-modified size-biased Lehmann type II or, in [...] Read more.

In order to represent the data with non-monotonic failure rates and produce a better fit, a novel distribution is created in this study using the alpha power family of distributions. This distribution is called the alpha-power Kum-modified size-biased Lehmann type II or, in short, the AP-Kum-MSBL-II distribution. This distribution is established for modeling bounded data in the interval

(0, 1)

. The proposed distribution’s moment-generating function, mode, quantiles, moments, and stress–strength reliability function are obtained, among other attributes. To estimate the parameters of the proposed distribution, estimation methods such as the maximum likelihood method and Bayesian method are employed to estimate the unknown parameters for the AP-Kum-MSBL-II distribution. Moreover, the confidence intervals, credible intervals, and coverage probability are calculated for all parameters. The symmetric and asymmetric loss functions are used to find the Bayesian estimators using the Markov chain Monte Carlo (MCMC) method. Furthermore, the proposed distribution’s usefulness is demonstrated using three real data sets. One of them is a medical data set dealing with COVID-19 patients’ mortality rate, the second is a trade share data set, and the third is from the engineering area, as well as extensive simulated data, which were applied to assess the performance of the estimators of the proposed distribution. Full article

(This article belongs to the Special Issue Asymmetric and Symmetric Study on Image Processing and Statistical Data Analysis)

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13 pages, 338 KiB

Open AccessFeature PaperArticle

Legendre Series Analysis and Computation via Composed Abel–Fourier Transform

by Enrico De Micheli

Symmetry 2023, 15(6), 1282; https://doi.org/10.3390/sym15061282 - 19 Jun 2023

Viewed by 1523

Abstract

Legendre coefficients of an integrable function

f (x)

are proved to coincide with the Fourier coefficients with a nonnegative index of a suitable Abel-type transform of the function itself. The numerical computation of N Legendre coefficients can thus be carried out [...] Read more.

Legendre coefficients of an integrable function

f (x)

are proved to coincide with the Fourier coefficients with a nonnegative index of a suitable Abel-type transform of the function itself. The numerical computation of N Legendre coefficients can thus be carried out efficiently in

O (N l o g N)

operations by means of a single fast Fourier transform of the Abel-type transform of

f (x)

. Symmetries associated with the Abel-type transform are exploited to further reduce the computational complexity. The dual problem of calculating the sum of Legendre expansions at a prescribed set of points is also considered. We prove that a Legendre series can be written as the Abel transform of a suitable Fourier series. This fact allows us to state an efficient algorithm for the evaluation of Legendre expansions. Finally, some numerical tests are illustrated to exemplify and confirm the theoretical results. Full article

(This article belongs to the Section Mathematics)

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15 pages, 1015 KiB

Open AccessArticle

Relaxation Functions Interpolating the Cole–Cole and Kohlrausch–Williams–Watts Dielectric Relaxation Models

by Lingjie Duan, Junsheng Duan and Ming Li

Symmetry 2023, 15(6), 1281; https://doi.org/10.3390/sym15061281 - 19 Jun 2023

Viewed by 2103

Abstract

To describe non-Debye relaxation phenomena observed in dielectric materials, the Cole–Cole (CC) relaxation model in the frequency domain and the Kohlrausch–Williams–Watts (KWW) relaxation model in the time domain were introduced in the physics of dielectrics. In this paper, we propose a new relaxation [...] Read more.

To describe non-Debye relaxation phenomena observed in dielectric materials, the Cole–Cole (CC) relaxation model in the frequency domain and the Kohlrausch–Williams–Watts (KWW) relaxation model in the time domain were introduced in the physics of dielectrics. In this paper, we propose a new relaxation model with two parameters besides a relaxation time by expressing the relaxation function in the time domain in terms of the Mittag–Leffler functions. The proposed model represents a group of non-Debye relaxation phenomena and shows a transition between the CC and the KWW models. The relaxation properties described by the new model are analyzed, including the response function, the normalized complex dielectric permittivity, dielectric storage and loss factors as well as the relaxation frequency and time spectral functions. The presented relaxation function has a concise form and is expected to be applied to more complex relaxation phenomena. Full article

(This article belongs to the Section Physics)

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14 pages, 3390 KiB

Open AccessArticle

Computer Network Redundancy Reduction Using Video Compression

by Shabana Habib, Waleed Albattah, Mohammed F. Alsharekh, Muhammad Islam, Mohammad Munawar Shees and Hammad I. Sherazi

Symmetry 2023, 15(6), 1280; https://doi.org/10.3390/sym15061280 - 19 Jun 2023

Cited by 1 | Viewed by 1886

Abstract

Due to the strong correlation between symmetric frames, video signals have a high degree of temporal redundancy. Motion estimation techniques are computationally expensive and time-consuming processes used in symmetric video compression to reduce temporal redundancy. The block-matching technique is, on the other hand, [...] Read more.

Due to the strong correlation between symmetric frames, video signals have a high degree of temporal redundancy. Motion estimation techniques are computationally expensive and time-consuming processes used in symmetric video compression to reduce temporal redundancy. The block-matching technique is, on the other hand, the most popular and efficient of the different motion estimation and compensation techniques. Motion compensation based on the block-matching technique generally uses the minimization of either the mean square error (MSE) or mean absolute difference (MAD) in order to find the appropriate motion vector. This paper proposes to remove the highly temporally redundant information contained in each block of the video signal using the removing temporal redundancy (RTR) technique in order to improve the data rate and efficiency of the video signal. A comparison between the PSNR values of this technique and those of the JPEG video compression standard is made. As a result of its moderate memory and computation requirements, the algorithm was found to be suitable for mobile networks and embedded devices. Based on a detailed set of testing scenarios and the obtained results, it is evident that the RTR compression technique allowed a compression ratio of 22.71 and 95% loss in bit rate reduction while maintaining sufficient intact signal quality with minimized information loss. Full article

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13 pages, 2868 KiB

Open AccessArticle

A New Pelican Optimization Algorithm for the Parameter Identification of Memristive Chaotic System

by Qi Xiong, Jincheng She and Jinkun Xiong

Symmetry 2023, 15(6), 1279; https://doi.org/10.3390/sym15061279 - 19 Jun 2023

Cited by 6 | Viewed by 1659

Abstract

A memristor is a kind of nonlinear electronic component. Parameter identification for memristive chaotic systems is a multi-dimensional variable optimization problem. It is one of the key issues in chaotic control and synchronization. To identify the unknown parameters accurately and quickly, we introduce, [...] Read more.

A memristor is a kind of nonlinear electronic component. Parameter identification for memristive chaotic systems is a multi-dimensional variable optimization problem. It is one of the key issues in chaotic control and synchronization. To identify the unknown parameters accurately and quickly, we introduce, in this paper, a modified Pelican Optimization Algorithm (POA) called the fractional-order chaotic Pareto Pelican Optimization Algorithm (FPPOA). First, the pelican population’s diversity is augmented with the integration of a fractional chaotic sequence. Next, the utilization of the Pareto distribution is incorporated to alter the hunting strategy of pelicans in the POA. These measures are effective in hastening the speed of finding an optimal solution and circumventing local optimization issues. Thirdly, the FPPOA is used to determine the values of the parameters of the simplest memristive chaotic system, which has a property of conditional symmetry. The proposed algorithm was evaluated during simulations, where it was utilized to solve six objective functions of varying unimodal and multimodal types. The performance of the FPPOA exceeds three traditional swarm intelligence optimization algorithms. In the parameter identification experiment, the results for the parameters with the FPPOA had error rates all within a 1% range. Extensive testing shows that our new strategy has a faster rate of convergence and better optimization performance than some other traditional swarm algorithms. Full article

(This article belongs to the Special Issue Discrete and Continuous Memristive Nonlinear Systems and Symmetry II)

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25 pages, 3225 KiB

Open AccessArticle

The Discrete Exponentiated-Chen Model and Its Applications

by Refah Alotaibi, Hoda Rezk, Chanseok Park and Ahmed Elshahhat

Symmetry 2023, 15(6), 1278; https://doi.org/10.3390/sym15061278 - 18 Jun 2023

Cited by 3 | Viewed by 1484

Abstract

A novel discrete exponentiated Chen (DEC) distribution, which is a subset of the continuous exponentiated Chen distribution, is proposed. The offered model is more adaptable to analyzing a wide range of data than traditional and recently published models. Several important statistical and reliability [...] Read more.

A novel discrete exponentiated Chen (DEC) distribution, which is a subset of the continuous exponentiated Chen distribution, is proposed. The offered model is more adaptable to analyzing a wide range of data than traditional and recently published models. Several important statistical and reliability characteristics of the DEC model are introduced. In the presence of Type-II censored data, the maximum likelihood and asymptotic confidence interval estimators of the model parameters are acquired. Two various bootstrapping estimators of the DEC parameters are also obtained. To examine the efficacy of the adopted methods, several simulations are implemented. To further clarify the offered model in the life scenario, the two applications, based on the number of vehicle fatalities in South Carolina in 2012 and the final exam marks in 2004 at the Indian Institute of Technology at Kanpur, are analyzed. The analysis findings showed that the DEC model is the most effective model for fitting the supplied data sets compared to eleven well-known models in literature, including: Poisson, geometric, negative binomial, discrete-Weibull, discrete Burr Type XII, discrete generalized exponential, discrete-gamma, discrete Burr Hatke, discrete Nadarajah-Haghighi, discrete modified-Weibull, and exponentiated discrete-Weibull models. Ultimately, the new model is recommended to be applied in many fields of real practice. Full article

(This article belongs to the Special Issue Symmetric and Asymmetric Distributions: New Developments and Applications)

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16 pages, 1074 KiB

Open AccessReview

X-ray Tests of General Relativity with Black Holes

by Cosimo Bambi

Symmetry 2023, 15(6), 1277; https://doi.org/10.3390/sym15061277 - 18 Jun 2023

Cited by 3 | Viewed by 1839

Abstract

General relativity is one of the pillars of modern physics. For decades, the theory has been mainly tested in the weak-field regime with experiments in the solar system and radio observations of binary pulsars. Until 2015, the strong-field regime was almost completely unexplored. [...] Read more.

General relativity is one of the pillars of modern physics. For decades, the theory has been mainly tested in the weak-field regime with experiments in the solar system and radio observations of binary pulsars. Until 2015, the strong-field regime was almost completely unexplored. Thanks to new observational facilities, the situation has dramatically changed in the last few years. Today, we have gravitational wave data of the coalesce of stellar-mass compact objects from the LIGO-Virgo-KAGRA collaboration, images at mm wavelengths of the supermassive black holes in M87

^{*}

and Sgr A

^{*}

from the Event Horizon Telescope collaboration, and X-ray data of accreting compact objects from a number of X-ray missions. Gravitational wave tests and black hole imaging tests are certainly more popular and are discussed in other articles of this Special Issue. The aim of the present manuscript is to provide a pedagogical review on X-ray tests of general relativity with black holes and to compare these kinds of tests with those possible with gravitational wave data and black hole imaging. Full article

(This article belongs to the Special Issue Role of Black Holes in Testing Modified Theories of Gravity)

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14 pages, 324 KiB

Open AccessArticle

An Extension of Sylvester’s Theorem on Arithmetic Progressions

by Augustine O. Munagi and Francisco Javier de Vega

Symmetry 2023, 15(6), 1276; https://doi.org/10.3390/sym15061276 - 18 Jun 2023

Cited by 1 | Viewed by 1695

Abstract

Sylvester’s theorem states that every number can be decomposed into a sum of consecutive positive integers except powers of 2. In a way, this theorem characterizes the partitions of a number as a sum of consecutive integers. The first generalization we propose of [...] Read more.

Sylvester’s theorem states that every number can be decomposed into a sum of consecutive positive integers except powers of 2. In a way, this theorem characterizes the partitions of a number as a sum of consecutive integers. The first generalization we propose of the theorem characterizes the partitions of a number as a sum of arithmetic progressions with positive terms. In addition to synthesizing and rediscovering known results, the method we propose allows us to state a second generalization and characterize the partitions of a number into parts whose differences between consecutive parts form an arithmetic progression. To achieve this, we will analyze the set of divisors in arithmetics that modify the usual definition of the multiplication operation between two integers. As we will see, symmetries arise in the set of divisors based on two parameters:

t_{1}

, being even or odd, and

t_{2}

, congruent to 0, 1, or 2 (mod 3). This approach also leads to a unique representation result of the same nature as Sylvester’s theorem, i.e., a power of 3 cannot be represented as a sum of three or more terms of a positive integer sequence such that the differences between consecutive terms are consecutive integers. Full article

(This article belongs to the Special Issue Advances in Combinatorics and Graph Theory)

10 pages, 16122 KiB

Open AccessArticle

Solving Fractional Order Differential Equations by Using Fractional Radial Basis Function Neural Network

by Rana Javadi, Hamid Mesgarani, Omid Nikan and Zakieh Avazzadeh

Symmetry 2023, 15(6), 1275; https://doi.org/10.3390/sym15061275 - 17 Jun 2023

Cited by 4 | Viewed by 1872

Abstract

Fractional differential equations (FDEs) arising in engineering and other sciences describe nature sufficiently in terms of symmetry properties. This paper proposes a numerical technique to approximate ordinary fractional initial value problems by applying fractional radial basis function neural network. The fractional derivative used [...] Read more.

Fractional differential equations (FDEs) arising in engineering and other sciences describe nature sufficiently in terms of symmetry properties. This paper proposes a numerical technique to approximate ordinary fractional initial value problems by applying fractional radial basis function neural network. The fractional derivative used in the method is considered Riemann-Liouville type. This method is simple to implement and approximates the solution of any arbitrary point inside or outside the domain after training the ANN model. Finally, three examples are presented to show the validity and applicability of the method. Full article

(This article belongs to the Special Issue Symmetry in Functional Equations: Methods, Applications and Mathematical Models)

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20 pages, 6167 KiB

Open AccessArticle

An Adaptive Fatigue Detection System Based on 3D CNNs and Ensemble Models

by Ahmed Sedik, Mohamed Marey and Hala Mostafa

Symmetry 2023, 15(6), 1274; https://doi.org/10.3390/sym15061274 - 16 Jun 2023

Cited by 2 | Viewed by 2193

Abstract

Due to the widespread issue of road accidents, researchers have been drawn to investigate strategies to prevent them. One major contributing factor to these accidents is driver fatigue resulting from exhaustion. Various approaches have been explored to address this issue, with machine and [...] Read more.

Due to the widespread issue of road accidents, researchers have been drawn to investigate strategies to prevent them. One major contributing factor to these accidents is driver fatigue resulting from exhaustion. Various approaches have been explored to address this issue, with machine and deep learning proving to be effective in processing images and videos to detect asymmetric signs of fatigue, such as yawning, facial characteristics, and eye closure. This study proposes a multistage system utilizing machine and deep learning techniques. The first stage is designed to detect asymmetric states, including tiredness and non-vigilance as well as yawning. The second stage is focused on detecting eye closure. The machine learning approach employs several algorithms, including Support Vector Machine (SVM), k-Nearest Neighbor (KNN), Multi-layer Perceptron (MLP), Decision Tree (DT), Logistic Regression (LR), and Random Forest (RF). Meanwhile, the deep learning approach utilizes 2D and 3D Convolutional Neural Networks (CNNs). The architectures of proposed deep learning models are designed after several trials, and their parameters have been selected to achieve optimal performance. The effectiveness of the proposed methods is evaluated using video and image datasets, where the video dataset is classified into three states: alert, tired, and non-vigilant, while the image dataset is classified based on four facial symptoms, including open or closed eyes and yawning. A more robust system is achieved by combining the image and video datasets, resulting in multiple classes for detection. Simulation results demonstrate that the 3D CNN proposed in this study outperforms the other methods, with detection accuracies of 99 percent, 99 percent, and 98 percent for the image, video, and mixed datasets, respectively. Notably, this achievement surpasses the highest accuracy of 97 percent found in the literature, suggesting that the proposed methods for detecting drowsiness are indeed effective solutions. Full article

(This article belongs to the Special Issue Image Processing and Symmetry: Topics and Applications)

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22 pages, 10429 KiB

Open AccessArticle

An Invitation Model Protocol (IMP) for the Bitcoin Asymmetric Lightning Network

by Ali Abdullah and A. M. Mutawa

Symmetry 2023, 15(6), 1273; https://doi.org/10.3390/sym15061273 - 16 Jun 2023

Cited by 1 | Viewed by 1682

Abstract

The Lightning Network (LN), a second-layer protocol built atop Bitcoin, promises swift, low-cost transactions, thereby addressing blockchain scalability and enhancing user privacy. As the global financial technology landscape evolves, the LN’s importance in the future of fintech and the Fourth Industrial Revolution (4IR) [...] Read more.

The Lightning Network (LN), a second-layer protocol built atop Bitcoin, promises swift, low-cost transactions, thereby addressing blockchain scalability and enhancing user privacy. As the global financial technology landscape evolves, the LN’s importance in the future of fintech and the Fourth Industrial Revolution (4IR) becomes increasingly pivotal. The anticipated rise of blockchain-based payments and smart contracts in businesses demands a more agile and secure payment system. However, the LN’s early stage raises valid concerns about security and reliability, especially when implemented on a huge asymmetric network such as the Internet, potentially hindering its broader adoption. Malicious nodes could intentionally cause payment failures or initiate attacks, such as DDoS attacks, by overwhelming other nodes in the network with channel-opening requests. As a result, users will be discouraged from using the LN; hence, the technology will become obsolete as individuals will not waste the time and power investment required for using this technology. Addressing these issues, this paper proposes an innovative invitation model protocol (IMP) to reinforce the LN’s security and reliability. The IMP creates an exclusive ‘Club’ within the LN, admitting only nodes verified as honest, thereby bolstering network security and reliability. The protocol empowers Club Founders to expel members exhibiting malicious activities, thereby preserving the invested time, energy, and funds of the network’s users. The IMP was rigorously tested using Amazon Web Services Virtual Machines within the Bitcoin and Lightning Network’s Testnet environment, which is a highly asymmetric network. The results demonstrated the protocol’s efficacy in fulfilling its objectives, marking a significant step towards a safer and more efficient blockchain transaction network. As the blockchain continues to revolutionize the financial sector, implementing robust security measures such as the IMP becomes essential. This research paper introduces a novel approach to enhancing the reliability and security of a Lightning Network (LN), and thus distinguishes itself from the existing literature, by introducing an additional step before establishing or joining such a network. The research underscores the critical role of such protocols in realizing the potential of the LN in powering the next wave of fintech and industrial innovation. Full article

(This article belongs to the Special Issue Computer Science and Symmetry/Asymmetry: Feature Papers)

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28 pages, 4619 KiB

Open AccessArticle

From Netlist to Manufacturable Layout: An Auto-Layout Algorithm Optimized for Radio Frequency Integrated Circuits

by Yiding Wei, Jun Liu, Dengbao Sun, Guodong Su and Junchao Wang

Symmetry 2023, 15(6), 1272; https://doi.org/10.3390/sym15061272 - 16 Jun 2023

Cited by 1 | Viewed by 1690

Abstract

Layout stitching is a repetitive and tedious task of the radio frequency integrated circuit (RFIC) design process. While academic research on layout splicing algorithms mainly focuses on analog and digital circuits, there is still a lack of well-developed algorithms for RFICs. An RFIC [...] Read more.

Layout stitching is a repetitive and tedious task of the radio frequency integrated circuit (RFIC) design process. While academic research on layout splicing algorithms mainly focuses on analog and digital circuits, there is still a lack of well-developed algorithms for RFICs. An RFIC system usually has a symmetrical layout, such as transmitter and receiver components, low-noise amplifier (LNA), an SPDT switch, etc. This paper aims to address this gap by proposing an automated procedure for the layout of RFICs by relying on the basic device/PCell structure based on the interconnection among circuit topologies. This approach makes the in-series generation of layouts and automatic splicing based on circuit logic possible, resulting in superior stitching performance compared with related modules in Advanced Design System. To demonstrate the physical application possibilities, we implemented our algorithm on an LNA and a switch circuit. Full article

(This article belongs to the Section Computer)

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14 pages, 395 KiB

Open AccessFeature PaperArticle

Dense Baryonic Matter Predicted in “Pseudo-Conformal Model”

by Mannque Rho

Symmetry 2023, 15(6), 1271; https://doi.org/10.3390/sym15061271 - 16 Jun 2023

Cited by 4 | Viewed by 1049

Abstract

The World-Class University/Hanyang Project launched in Korea in 2008 led to what is now called the “pseudo-conformal model” that addresses dense compact star matter and is confronted in this short note with the presently available astrophysical observables, with focus on those from gravity [...] Read more.

The World-Class University/Hanyang Project launched in Korea in 2008 led to what is now called the “pseudo-conformal model” that addresses dense compact star matter and is confronted in this short note with the presently available astrophysical observables, with focus on those from gravity waves. The predictions made nearly free of parameters by the model involving “topology change” remain more or less intact “un-torpedoed” by the data. Full article

(This article belongs to the Special Issue The Nuclear Physics of Neutron Stars)

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18 pages, 342 KiB

Open AccessArticle

Aspects of Submanifolds on (α, β)-Type Almost Contact Manifolds with Quasi-Hemi-Slant Factor

by Ali H. Hakami, Mohd Danish Siddiqi, Oǧuzhan Bahadir and Toukeer Khan

Symmetry 2023, 15(6), 1270; https://doi.org/10.3390/sym15061270 - 16 Jun 2023

Cited by 4 | Viewed by 2128

Abstract

In this study, the authors focus on quasi-hemi-slant submanifolds (

q h s

-submanifolds) of

(α, β)

-type almost contact manifolds, also known as trans-Sasakian manifolds. Essentially, we give sufficient and necessary conditions for the integrability of distributions using the [...] Read more.

In this study, the authors focus on quasi-hemi-slant submanifolds (

q h s

-submanifolds) of

(α, β)

-type almost contact manifolds, also known as trans-Sasakian manifolds. Essentially, we give sufficient and necessary conditions for the integrability of distributions using the concept of quasi-hemi-slant submanifolds of trans-Sasakian manifolds. We also consider the geometry of foliations dictated by the distribution and the requirements for submanifolds of trans-Sasakian manifolds with quasi-hemi-slant factors to be totally geodesic. Lastly, we give an illustration of a submanifold with a quasi-hemi-slant factor and discuss its application to number theory. Full article

(This article belongs to the Section Mathematics)

17 pages, 1204 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Influence of Magnetic Field and Porous Medium on the Steady State and Flow Resistance of Second Grade Fluids over an Infinite Plate

by Constantin Fetecau and Costică Moroşanu

Symmetry 2023, 15(6), 1269; https://doi.org/10.3390/sym15061269 - 16 Jun 2023

Cited by 3 | Viewed by 1345

Abstract

The main purpose of this work is to completely solve two motion problems of some differential type fluids when velocity or shear stress is given on the boundary. In order to do that, isothermal MHD motions of incompressible second grade fluids over an [...] Read more.

The main purpose of this work is to completely solve two motion problems of some differential type fluids when velocity or shear stress is given on the boundary. In order to do that, isothermal MHD motions of incompressible second grade fluids over an infinite flat plate are analytically investigated when porous effects are taken into consideration. The fluid motion is due to the plate moving in its plane with an arbitrary time-dependent velocity or applying a time-dependent shear stress to the fluid. Closed-form expressions are established both for the dimensionless velocity and shear stress fields and the Darcy’s resistance corresponding to the first motion. The dimensionless shear stress corresponding to the second motion has been immediately obtained using a perfect symmetry between the governing equations of velocity and the non-trivial shear stress. Furthermore, the obtained results provide the first exact general solutions for MHD motions of second grade fluids through porous media. Finally, for illustration, as well as for their use in engineering applications, the starting and/or steady state solutions of some problems with technical relevance are provided, and the validation of the results is graphically proved. The influence of magnetic field and porous medium on the steady state and the flow resistance of fluid are graphically underlined and discussed. It was found that the flow resistance of the fluid declines or increases in the presence of a magnetic field or porous medium, respectively. In addition, the steady state is obtained earlier in the presence of a magnetic field or porous medium. Full article

(This article belongs to the Section Mathematics)

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19 pages, 857 KiB

Open AccessFeature PaperArticle

Finite Reservoirs Corrections to Hamiltonian Systems Statistics and Time Symmetry Breaking

by Matteo Colangeli, Antonio Di Francesco and Lamberto Rondoni

Symmetry 2023, 15(6), 1268; https://doi.org/10.3390/sym15061268 - 15 Jun 2023

Cited by 1 | Viewed by 1207

Abstract

We consider several Hamiltonian systems perturbed by external agents that preserve their Hamiltonian structure. We investigate the corrections to the canonical statistics resulting from coupling such systems with possibly large but finite reservoirs and from the onset of processes breaking the time-reversal symmetry. [...] Read more.

We consider several Hamiltonian systems perturbed by external agents that preserve their Hamiltonian structure. We investigate the corrections to the canonical statistics resulting from coupling such systems with possibly large but finite reservoirs and from the onset of processes breaking the time-reversal symmetry. We analyze exactly solvable oscillator systems and perform simulations of relatively more complex ones. This indicates that the standard statistical mechanical formalism needs to be adjusted in the ever more investigated nano-scale science and technology. In particular, the hypothesis that heat reservoirs be considered infinite and be described by the classical ensembles is found to be critical when exponential quantities are considered since the large size limit may not coincide with the infinite size canonical result. Furthermore, process-dependent emergent irreversibility affects ensemble averages, effectively frustrating, on a statistical level, the time reversal invariance of Hamiltonian dynamics that are used to obtain numerous results. Full article

(This article belongs to the Special Issue Symmetry in Hamiltonian Dynamical Systems)

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20 pages, 583 KiB

Open AccessArticle

Topological Symmetry Groups of the Petersen Graphs

by Deion Elzie, Samir Fridhi, Blake Mellor, Daniel Silva and Robin T. Wilson

Symmetry 2023, 15(6), 1267; https://doi.org/10.3390/sym15061267 - 15 Jun 2023

Viewed by 1801

Abstract

The topological symmetry group of an embedding

Γ

of an abstract graph

γ

in

S^{3}

is the group of automorphisms of

γ

that can be realized by homeomorphisms of the pair

(S^{3}, Γ)

. These groups are motivated [...] Read more.

The topological symmetry group of an embedding

Γ

of an abstract graph

γ

in

S^{3}

is the group of automorphisms of

γ

that can be realized by homeomorphisms of the pair

(S^{3}, Γ)

. These groups are motivated by questions about the symmetries of molecules in space. The Petersen family of graphs is an important family of graphs for many problems in low-dimensional topology, so it is desirable to understand the possible groups of symmetries of their embeddings in space. In this paper, we find all the groups that can be realized as topological symmetry groups for each of the graphs in the Petersen family. Along the way, we also complete the classification of the realizable topological symmetry groups for

K_{3, 3}

. Full article

(This article belongs to the Special Issue Advances in Graph Theory and Symmetry/Asymmetry)

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31 pages, 445 KiB

Open AccessArticle

Borel Transform and Scale-Invariant Fractional Derivatives United

by Simon Gluzman

Symmetry 2023, 15(6), 1266; https://doi.org/10.3390/sym15061266 - 15 Jun 2023

Cited by 2 | Viewed by 2443

Abstract

The method of Borel transformation for the summation of asymptotic expansions with the power-law asymptotic behavior at infinity is combined with elements of scale-invariant fractional analysis with the goal of calculating the critical amplitudes. The fractional order of specially designed scale-invariant fractional derivatives [...] Read more.

The method of Borel transformation for the summation of asymptotic expansions with the power-law asymptotic behavior at infinity is combined with elements of scale-invariant fractional analysis with the goal of calculating the critical amplitudes. The fractional order of specially designed scale-invariant fractional derivatives u is used as a control parameter to be defined uniquely from u-optimization. For resummation of the transformed expansions, we employed the self-similar iterated roots. We also consider a complementary optimization, called b-optimization with the number of iterations b as an alternative fractional control parameter. The method of scale-invariant Fractional Borel Summation consists of three constructive steps. The first step corresponds to u-optimization of the amplitudes with fixed parameter b. When the first step fails, the second step corresponds to b-optimization of the amplitudes with fixed parameter u. However, when the two steps fail, the third step corresponds to the simplified, Borel-light technique. The marginal amplitude should be found by means of the self-similar iterated roots constructed for the transformed series, optimized with either of the two above approaches and corrected with a diagonal Padé approximants. The examples are given when the complementary optimizations,“horses-for-courses” approach outperforms other analytical methods in calculation of critical amplitudes. Full article

(This article belongs to the Special Issue Nonlinear Science and Numerical Simulation with Symmetry)

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22 pages, 3926 KiB

Open AccessArticle

A Novel Bat Algorithm with Asymmetrical Weighed Variational Method in the Path Planning of UAVs

by Xin Cao, Chenyi Wang and Weiping Li

Symmetry 2023, 15(6), 1265; https://doi.org/10.3390/sym15061265 - 15 Jun 2023

Cited by 1 | Viewed by 1358

Abstract

In this paper, a novel bat algorithm with an asymmetrical weighed variational method (AWVM-BA) is proposed. The proposed algorithm employs the BA with a point-to-point modified asymmetrical variation above the three-dimensional flying region, which treats the space as sets of geodesics in a [...] Read more.

In this paper, a novel bat algorithm with an asymmetrical weighed variational method (AWVM-BA) is proposed. The proposed algorithm employs the BA with a point-to-point modified asymmetrical variation above the three-dimensional flying region, which treats the space as sets of geodesics in a second order Euclidean weighed warped space. Mutation and the local selection procedure can be avoided at the same time, which solves the problem of a local optimum in concave regions. As shown in the results, the proposed algorithm does not have much impact on the calculation complexity and time in convex regions. It can greatly reduce the calculation time and avoid local optimization in concave regions. The disadvantage of the proposed algorithm is that the iteration number increases comparatively faster with the increase in the deviation of the wind speed. Therefore, it requires a higher hardware calculation ability. Full article

(This article belongs to the Section Engineering and Materials)

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18 pages, 554 KiB

Open AccessArticle

AI-Enabled Consensus Algorithm in Human-Centric Collaborative Computing for Internet of Vehicle

by Chenxi Sun, Danyang Li, Beilei Wang and Jie Song

Symmetry 2023, 15(6), 1264; https://doi.org/10.3390/sym15061264 - 15 Jun 2023

Cited by 2 | Viewed by 1731

Abstract

With the enhanced interoperability of information among vehicles, the demand for collaborative sharing among vehicles increases. Based on blockchain, the classical consensus algorithms in collaborative IoV (Internet of Vehicle), such as PoW (Proof of Work), PoS (Proof of Stake), and DPoS (Delegated Proof [...] Read more.

With the enhanced interoperability of information among vehicles, the demand for collaborative sharing among vehicles increases. Based on blockchain, the classical consensus algorithms in collaborative IoV (Internet of Vehicle), such as PoW (Proof of Work), PoS (Proof of Stake), and DPoS (Delegated Proof of Stake), only consider the node features, which is hard to adapt to the immediacy and flexibility of vehicles. On the other hand, classical consensus algorithms often require mass computing, which undoubtedly increases the communication overhead, resulting in the inability to achieve collaborative IoV under asymmetric networks. Therefore, proposing a low failure rate consensus algorithm that takes into account running time and energy consumption becomes a major challenge in IoV applications. This paper proposes an AI-enabled consensus algorithm with vehicle features, combining vehicle-based metrics and neural networks. First, we introduce vehicle-based metrics such as vehicle online time, performance, and behavior. Then, we propose an integral model and a hierarchical classification method, which combine with a BP neural network to obtain the optimal solution for interconnection. Among them, we also use Informer to predict the future online duration of vehicles, which effectively solves the situation that the primary node vehicle drops off in collaborative IoV. Finally, the experimentations show that the vehicle-based metrics eliminate the problem of the primary node vehicle being offline, which realizes the collaborative IoV considering vehicle features. Meanwhile, it reduces the vehicle network system delay and energy consumption. Full article

(This article belongs to the Special Issue Symmetry and Asymmetry in AI-Enabled Human-Centric Collaborative Computing)

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22 pages, 2681 KiB

Open AccessArticle

Analytical and Numerical Methods for Solving Second-Order Two-Dimensional Symmetric Sequential Fractional Integro-Differential Equations

by Sondos M. Syam, Z. Siri, Sami H. Altoum and R. Md. Kasmani

Symmetry 2023, 15(6), 1263; https://doi.org/10.3390/sym15061263 - 15 Jun 2023

Cited by 8 | Viewed by 1394

Abstract

In this paper, we investigate the solution to a class of symmetric non-homogeneous two-dimensional fractional integro-differential equations using both analytical and numerical methods. We first show the differences between the Caputo derivative and the symmetric sequential fractional derivative and how they help facilitate [...] Read more.

In this paper, we investigate the solution to a class of symmetric non-homogeneous two-dimensional fractional integro-differential equations using both analytical and numerical methods. We first show the differences between the Caputo derivative and the symmetric sequential fractional derivative and how they help facilitate the implementation of numerical and analytical approaches. Then, we propose a numerical approach based on the operational matrix method, which involves deriving operational matrices for the differential and integral terms of the equation and combining them to generate a single algebraic system. This method allows for the efficient and accurate approximation of the solution without the need for projection. Our findings demonstrate the effectiveness of the operational matrix method for solving non-homogeneous fractional integro-differential equations. We then provide examples to test our numerical method. The results demonstrate the accuracy and efficiency of the approach, with the graph of exact and approximate solutions showing almost complete overlap, and the approximate solution to the fractional problem converges to the solution of the integer problem as the order of the fractional derivative approaches one. We use various methods to measure the error in the approximation, such as absolute and

L_{2}

errors. Additionally, we explore the effect of the derivative order. The results show that the absolute error is on the order of

10^{- 14}

, while the

L_{2}

error is on the order of

10^{- 13}

. Next, we apply the Laplace transform to find an analytical solution to a class of fractional integro-differential equations and extend the approach to the two-dimensional case. We consider all homogeneous cases. Through our examples, we achieve two purposes. First, we show how the obtained results are implemented, especially the exact solution for some 1D and 2D classes. We then demonstrate that the exact fractional solution converges to the exact solution of the ordinary derivative as the order of the fractional derivative approaches one. Full article

(This article belongs to the Special Issue Advanced Analytical and Numerical Methods for Fractional Initial and Boundary Value Problems with Symmetry/Asymmetry)

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21 pages, 1597 KiB

Open AccessArticle

Generalized Support Vector Regression and Symmetry Functional Regression Approaches to Model the High-Dimensional Data

by Mahdi Roozbeh, Arta Rouhi, Nur Anisah Mohamed and Fatemeh Jahadi

Symmetry 2023, 15(6), 1262; https://doi.org/10.3390/sym15061262 - 15 Jun 2023

Cited by 2 | Viewed by 1433

Abstract

The analysis of the high-dimensional dataset when the number of explanatory variables is greater than the observations using classical regression approaches is not applicable and the results may be misleading. In this research, we proposed to analyze such data by introducing modern and [...] Read more.

The analysis of the high-dimensional dataset when the number of explanatory variables is greater than the observations using classical regression approaches is not applicable and the results may be misleading. In this research, we proposed to analyze such data by introducing modern and up-to-date techniques such as support vector regression, symmetry functional regression, ridge, and lasso regression methods. In this study, we developed the support vector regression approach called generalized support vector regression to provide more efficient shrinkage estimation and variable selection in high-dimensional datasets. The generalized support vector regression can improve the performance of the support vector regression by employing an accurate algorithm for obtaining the optimum value of the penalty parameter using a cross-validation score, which is an asymptotically unbiased feasible estimator of the risk function. In this regard, using the proposed methods to analyze two real high-dimensional datasets (yeast gene data and riboflavin data) and a simulated dataset, the most efficient model is determined based on three criteria (correlation squared, mean squared error, and mean absolute error percentage deviation) according to the type of datasets. On the basis of the above criteria, the efficiency of the proposed estimators is evaluated. Full article

(This article belongs to the Special Issue Symmetry in Multivariate Analysis)

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27 pages, 11599 KiB

Open AccessFeature PaperReview

Atropselective Organocatalytic Synthesis of Chiral Compounds Containing Nitrogen along the Axis of Chirality

by Ana Maria Faisca Phillips and Armando J. L. Pombeiro

Symmetry 2023, 15(6), 1261; https://doi.org/10.3390/sym15061261 - 15 Jun 2023

Cited by 5 | Viewed by 2813

Abstract

Atropisomers, i.e., stereoisomers that are distinct because their free rotation about a single bond is hindered by steric interactions between nearby bulky groups or by electrostatics, may interact with their surroundings in different ways, and may also exhibit different properties. They may be [...] Read more.

Atropisomers, i.e., stereoisomers that are distinct because their free rotation about a single bond is hindered by steric interactions between nearby bulky groups or by electrostatics, may interact with their surroundings in different ways, and may also exhibit different properties. They may be found as natural products, as pharmaceutical or agricultural active ingredients, as chiral ligands and organocatalysts, and in functional materials. Our ability to synthesize them stereoselectively and in a sustainable way, using achiral materials and simply with the aid of an organocatalyst and mild conditions, has become a hot topic in research. This review provides an overview of recent achievements in the synthesis of atropisomers containing C-N and N-N axes of chirality. Full article

(This article belongs to the Special Issue New Advances in Asymmetric Organocatalysis)

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