Symmetry

15 pages, 2061 KiB

Open AccessFeature PaperArticle

Synthesis of α-Hydroxyethylphosphonates and α-Hydroxyethylphosphine Oxides: Role of Solvents During Optical Resolution

by Zsuzsanna Szalai, Anna Sára Kis, József Schindler, Konstantin Karaghiosoff and György Keglevich

Symmetry 2024, 16(11), 1557; https://doi.org/10.3390/sym16111557 - 20 Nov 2024

Viewed by 377

Five chiral, racemic dialkyl α-hydroxyethylphosphonates and α-hydroxyethyl-diarylphosphine oxides were prepared in a scalable manner. Possibilities for the optical resolution of the racemic hydroxyphosphine oxides were explored via diastereomeric complex-forming experiments. The acidic calcium salt of O,O′-dibenzoyl-(2R,3R)-tartaric [...] Read more.

Five chiral, racemic dialkyl α-hydroxyethylphosphonates and α-hydroxyethyl-diarylphosphine oxides were prepared in a scalable manner. Possibilities for the optical resolution of the racemic hydroxyphosphine oxides were explored via diastereomeric complex-forming experiments. The acidic calcium salt of O,O′-dibenzoyl-(2R,3R)-tartaric acid and O,O′-di-p-tolyl-(2R,3R)-tartaric acid were chosen as the resolving agents. The role of the solvent in the enantiomeric discrimination was investigated. The diastereomeric complex could be obtained in a crystalline form for α-hydroxyethyl-diphenylphosphine oxide and α-hydroxyethyl-bis(4-methylphenyl)phosphine oxide. However, in the third case, for α-hydroxyethyl-bis(3,5-dimethylphenyl)phosphine oxide, there was no chiral discrimination. Decomposition of the recrystallized diastereomeric complex followed by decomposition of the complex yielded the target compounds in 77/90% enantiomeric excess. The absolute configuration of the hydroxyethyl-diphenylphosphine oxide was determined by single-crystal X-ray diffraction measurements. The interactions stabilizing the supramolecular associate were evaluated. Full article

(This article belongs to the Special Issue Asymmetric Centers in Optically Active Molecules Containing a Heteroatom)

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Graphical abstract

11 pages, 277 KiB

Open AccessArticle

A New Method of Teaching Quality Evaluation Based on the α-Pessimistic Shapley Value

by Yin Gao

Symmetry 2024, 16(11), 1556; https://doi.org/10.3390/sym16111556 - 20 Nov 2024

Viewed by 322

Abstract

Coalitional game with uncertain payoffs driven by uncertain variables stands for uncertain coalitional game. A

α

-pessimistic Shapley value as the solution of uncertain coalitional game is proposed in this paper, which can be played an important role in solving the problem of [...] Read more.

Coalitional game with uncertain payoffs driven by uncertain variables stands for uncertain coalitional game. A

α

-pessimistic Shapley value as the solution of uncertain coalitional game is proposed in this paper, which can be played an important role in solving the problem of profit allocation. Moreover, the

α

-pessimistic Shapley value is proved to be the only value satisfying symmetry, dummy and additivity. Based on the

α

-pessimistic Shapley value, a new method of teaching quality evaluation in the institutions of higher learning is presented. In order to obtain the index of teaching quality evaluation in uncertain environments, an

α

-pessimistic multilinear extension is defined. Meanwhile, some properties of

α

-pessimistic multilinear extension are investigated successfully. Moreover, the problem of teaching quality evaluation are considered in this paper. Full article

(This article belongs to the Special Issue Symmetry in Optimal Control and Applications)

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28 pages, 5048 KiB

Open AccessArticle

Research on Runway Capacity Evaluation of General Aviation Airport Based on Runway Expansion System

by Zhiyuan Chen, Huachun Xiang, Bangcun Han, Yachen Shen, Ting Zhou and Feng Zhang

Symmetry 2024, 16(11), 1555; https://doi.org/10.3390/sym16111555 - 20 Nov 2024

Viewed by 318

Abstract

To enhance the operational management capabilities of general aviation airports, this paper proposes a method for evaluating the runway capacity of general aviation airports based on the runway expansion system. Firstly, it provides a brief introduction to the flight rules of general aviation [...] Read more.

To enhance the operational management capabilities of general aviation airports, this paper proposes a method for evaluating the runway capacity of general aviation airports based on the runway expansion system. Firstly, it provides a brief introduction to the flight rules of general aviation airports and arrival and departure flight procedures with symmetrical characteristics, which serve as a theoretical basis for establishing the runway expansion system. Subsequently, a runway expansion system that covers symmetrical flight activities such as departure and arrival under a visual flight rule and an instrument flight rule is proposed, providing a conceptual model for evaluating the runway capacity of general aviation airports. On this foundation, the classical space–time analysis model is improved to establish a single runway arrival, departure, and mixed operation capacity evaluation model for general aviation airports. Finally, the reliability and rationality of this method are verified through case evaluations and three sets of numerical experiments with symmetrical relationships. The experiments demonstrate that this method can better reflect the actual conditions of the runways at general aviation airports while ensuring flight safety, and it can provide a reference for related research. Full article

(This article belongs to the Section Engineering and Materials)

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12 pages, 2095 KiB

Open AccessArticle

Phase Portraits and Abundant Soliton Solutions of a Hirota Equation with Higher-Order Dispersion

by Fengxia Wu, Nauman Raza, Younes Chahlaoui, Asma Rashid Butt and Haci Mehmet Baskonus

Symmetry 2024, 16(11), 1554; https://doi.org/10.3390/sym16111554 - 20 Nov 2024

Viewed by 243

Abstract

The Hirota equation, an advanced variant of the nonlinear Schrödinger equation with cubic nonlinearity, incorporates time-delay adjustments and higher-order dispersion terms, offering an enhanced approximation for wave propagation in optical fibers and oceanic systems. By utilizing the traveling wave transformation generated from Lie [...] Read more.

The Hirota equation, an advanced variant of the nonlinear Schrödinger equation with cubic nonlinearity, incorporates time-delay adjustments and higher-order dispersion terms, offering an enhanced approximation for wave propagation in optical fibers and oceanic systems. By utilizing the traveling wave transformation generated from Lie point symmetry analysis with the combination of generalized exponential differential rational function and modified Bernoulli sub-ODE techniques, several traveling wave solutions, such as periodic, singular-periodic, and kink solitons, emerge. To examine the solutions visually, parametric values are adjusted to create 3D, contour, and 2D illustrations. Additionally, the dynamic properties of the model are explored through bifurcation analysis. The exact results demonstrate that both techniques are practical and robust. Full article

(This article belongs to the Special Issue New Trends on the Mathematical Models and Solitons Arising in Real-World Problems, 2nd Edition)

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13 pages, 336 KiB

Open AccessArticle

Generalized Galilean Rotations

by Harun Barış Çolakoğlu, İskender Öztürk, Oğuzhan Çelik and Mustafa Özdemir

Symmetry 2024, 16(11), 1553; https://doi.org/10.3390/sym16111553 - 20 Nov 2024

Viewed by 231

Abstract

In this article, we give rotational motions on any straight line or any parabola in a scalar product space. To achieve this goal, we first define the generalized Galilean scalar product and determine the generalized Galilean skew symmetric and orthogonal matrices. Then, using [...] Read more.

In this article, we give rotational motions on any straight line or any parabola in a scalar product space. To achieve this goal, we first define the generalized Galilean scalar product and determine the generalized Galilean skew symmetric and orthogonal matrices. Then, using the well-known Rodrigues, Cayley, and Householder maps, we produce the generalized Galilean rotation matrices. Finally, we show that these rotation matrices can also be used to determine parabolic rotational motion. Full article

(This article belongs to the Section Mathematics)

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24 pages, 1513 KiB

Open AccessArticle

Dynamical Complexity of Modified Leslie–Gower Predator–Prey Model Incorporating Double Allee Effect and Fear Effect

by Manoj Kumar Singh, Arushi Sharma and Luis M. Sánchez-Ruiz

Symmetry 2024, 16(11), 1552; https://doi.org/10.3390/sym16111552 - 19 Nov 2024

Viewed by 294

Abstract

This contribution concerns studying a realistic predator–prey interaction, which was achieved by virtue of formulating a modified Leslie–Gower predator–prey model under the influence of the double Allee effect and fear effect in the prey species. The initial theoretical work sheds light on the [...] Read more.

This contribution concerns studying a realistic predator–prey interaction, which was achieved by virtue of formulating a modified Leslie–Gower predator–prey model under the influence of the double Allee effect and fear effect in the prey species. The initial theoretical work sheds light on the relevant properties of the solution, presence, and local stability of the equilibria. Both analytic and numerical approaches were used to address the emergence of diverse bifurcations, like saddle-node, Hopf, and Bogdanov–Takens bifurcations. It is noteworthy that while making the assumption that the characteristic equation of the Jacobian matrix J has a pair of imaginary roots

C (ρ) \pm ι D (ρ)

, it is sufficient to consider only

C (ρ) + ι D (ρ)

due to symmetry. The impact of the fear effect on the proposed model is discussed. Numerical simulation results are provided to back up all the theoretical analysis. From the findings, it was established that the initial condition of the population, as well as the phenomena (fear effect) introduced, played a crucial role in determining the stability of the proposed model. Full article

(This article belongs to the Special Issue Symmetry/Asymmetry in Life Sciences: Feature Papers 2024)

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14 pages, 2915 KiB

Open AccessArticle

Missing Data Imputation Based on Causal Inference to Enhance Advanced Persistent Threat Attack Prediction

by Xiang Cheng, Miaomiao Kuang and Hongyu Yang

Symmetry 2024, 16(11), 1551; https://doi.org/10.3390/sym16111551 - 19 Nov 2024

Viewed by 292

Abstract

With the continuous development of network security situations, the types of attacks increase sharply, but can be divided into symmetric attacks and asymmetric attacks. Symmetric attacks such as phishing and DDoS attacks exploit fixed patterns, resulting in system crashes and data breaches that [...] Read more.

With the continuous development of network security situations, the types of attacks increase sharply, but can be divided into symmetric attacks and asymmetric attacks. Symmetric attacks such as phishing and DDoS attacks exploit fixed patterns, resulting in system crashes and data breaches that cause losses to businesses. Asymmetric attacks such as Advanced Persistent Threat (APT), a highly sophisticated and organized form of cyber attack, because of its concealment and complexity, realize data theft through long-term latency and pose a greater threat to organization security. In addition, there are challenges in the processing of missing data, especially in the application of symmetric and asymmetric data filling, the former is simple but not flexible, and the latter is complex and more suitable for highly complex attack scenarios. Since asymmetric attack research is particularly important, this paper proposes a method that combines causal discovery with graph autoencoder to solve missing data, classify potentially malicious nodes, and reveal causal relationships. The core is to use graphic autoencoders to learn the underlying causal structure of APT attacks, with a special focus on the complex causal relationships in asymmetric attacks. This causal knowledge is then applied to enhance the robustness of the model by compensating for data gaps. In the final phase, it also reveals causality, predicts and classifies potential APT attack nodes, and provides a comprehensive framework that not only predicts potential threats, but also provides insight into the logical sequence of the attacker’s actions. Full article

(This article belongs to the Special Issue Symmetry and Asymmetry in Cybersecurity)

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16 pages, 754 KiB

Open AccessArticle

Data-Sharing System with Attribute-Based Encryption in Blockchain and Privacy Computing

by Hao Wu, Yu Liu, Konglin Zhu and Lin Zhang

Symmetry 2024, 16(11), 1550; https://doi.org/10.3390/sym16111550 - 19 Nov 2024

Viewed by 316

Abstract

With the development of the data-sharing system in recent years, financial management systems and their privacy have sparked great interest. Existing financial data-sharing systems store metadata, which include a hash value and database index on the blockchain, and store high-capacity actual data in [...] Read more.

With the development of the data-sharing system in recent years, financial management systems and their privacy have sparked great interest. Existing financial data-sharing systems store metadata, which include a hash value and database index on the blockchain, and store high-capacity actual data in the center database. However, current data-sharing systems largely depend on centralized systems, which are susceptible to distributed denial-of-service (DDoS) attacks and present a centralized attack vector. Furthermore, storing data in a local center database has a high risk of information disclosure and tampering. In this paper, we propose the ChainMaker Privacy Computing (CPC) system, a new decentralized data-sharing system for secure financial data, to solve this problem. It provides a series of financial data information and a data structure rather than actual data on the blockchain to protect the privacy of data. We utilize a smart contract to establish a trusted platform for the local database to obtain encrypted data. We design a resource catalog to provide a trusted environment of data usage in the privacy computing system that is visible for members on the blockchain. Based on cipher-policy attribute-based encryption (CP-ABE), We design a CPC-CP-ABE algorithm to enable fine-grained access control through attribute based encryption. Finally, We propose an efficient scheme that allows authenticated data-sharing systems to perform Boolean searches on encrypted data information. The results of experiment show that the CPC system can finish trusted data sharing to all organizations on the blockchain. Full article

(This article belongs to the Section Computer)

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16 pages, 1533 KiB

Open AccessArticle

A General Mollification Regularization Method to Solve a Cauchy Problem for the Multi-Dimensional Modified Helmholtz Equation

by Huilin Xu, Baoxia Wang and Duanmei Zhou

Symmetry 2024, 16(11), 1549; https://doi.org/10.3390/sym16111549 - 19 Nov 2024

Viewed by 332

Abstract

This paper considers a Cauchy problem for the multi-dimensional modified Helmholtz equation with inhomogeneous Dirichlet and Neumann data. The Cauchy problem is severely ill-posed, and a general mollification method is introduced to solve the problem. Both the a priori and a posteriori choice [...] Read more.

This paper considers a Cauchy problem for the multi-dimensional modified Helmholtz equation with inhomogeneous Dirichlet and Neumann data. The Cauchy problem is severely ill-posed, and a general mollification method is introduced to solve the problem. Both the a priori and a posteriori choice strategies of the regularization parameter are proposed, and error estimations of the corresponding regularization solutions are also presented. Finally, two numerical examples are introduced to show the effectiveness of the general mollification regularization method. Full article

(This article belongs to the Section Mathematics)

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19 pages, 3381 KiB

Open AccessReview

TIME REFRACTION and SPACETIME OPTICS

by José Tito Mendonça

Symmetry 2024, 16(11), 1548; https://doi.org/10.3390/sym16111548 - 19 Nov 2024

Viewed by 268

Abstract

A review of recent advances in spacetime optics is given, with special emphasis on time refraction. This is a basic optical process, occurring at a temporal discontinuity or temporal boundary, which is able to produce various different effects, such as frequency shifts, energy [...] Read more.

A review of recent advances in spacetime optics is given, with special emphasis on time refraction. This is a basic optical process, occurring at a temporal discontinuity or temporal boundary, which is able to produce various different effects, such as frequency shifts, energy amplification, time reflection, and photon emission. If, instead of a single discontinuity, we have two reverse temporal boundaries, we can form a temporal beam splitter, where temporal interferences can occur. It will also be shown that, in the presence of an axis of symmetry, such as a magnetic field, the temporal beam splitter can induce a rotation of the initial polarization state, similar to a Faraday rotation. Recent work on time crystals, superluminal fronts, and superfluid light will be reviewed. Time gates based on spacetime optical effects will be discussed. We also mention recent work on optical metamaterials. Finally, the quantum properties of time refraction, which imply the emission of photon from vacuum, are considered, while similar problems in high-energy QED associated with electron–positron pairs are briefly mentioned. Full article

(This article belongs to the Special Issue Symmetry/Asymmetry: Feature Review Papers 2024)

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22 pages, 1267 KiB

Open AccessArticle

A Hybrid Optimization Strategy for Minimizing Conversion Losses in Semi-Series-Resonant Dual-Active-Bridge Converter

by Shengzhi Zhou, Jianheng Huang, Jiahua Tang and Jihong Wang

Symmetry 2024, 16(11), 1547; https://doi.org/10.3390/sym16111547 - 19 Nov 2024

Viewed by 345

Abstract

To enhance the performance of resonant DC–DC converters, particularly under low-load conditions, a semi-series-resonant dual-active-bridge (Semi-SRDAB) converter with a hybrid optimization strategy is proposed. This strategy aims to reduce conduction-related losses and is designed for applications requiring a wide voltage range. The proposed [...] Read more.

To enhance the performance of resonant DC–DC converters, particularly under low-load conditions, a semi-series-resonant dual-active-bridge (Semi-SRDAB) converter with a hybrid optimization strategy is proposed. This strategy aims to reduce conduction-related losses and is designed for applications requiring a wide voltage range. The proposed Semi-SRDAB converter comprises a full-bridge inverter on the primary side and a hybrid-output bridge rectifier on the secondary side. It adopts phase-shift modulation combined with frequency modulation for power control. The hybrid optimization strategy for the Semi-SRDAB converter is investigated, beginning with the deduction of resonant current minimization using phasor analysis. Based on these analysis results, zero reactive power operation and soft-switching operation are achieved for both buck and boost modes. Successful validation has been demonstrated through experimental testing on a 300 W laboratory prototype. Enhanced conversion performance is confirmed by comparing the results with those from previous works. Full article

(This article belongs to the Section Engineering and Materials)

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20 pages, 713 KiB

Open AccessArticle

GRMD: A Two-Stage Design Space Exploration Strategy for Customized RNN Accelerators

by Qingpeng Li, Jian Xiao and Jizeng Wei

Symmetry 2024, 16(11), 1546; https://doi.org/10.3390/sym16111546 - 19 Nov 2024

Viewed by 278

Abstract

Recurrent neural networks (RNNs) have produced significant results in many fields, such as natural language processing and speech recognition. Owing to their computational complexity and sequence dependencies, RNNs need to be deployed on customized hardware accelerators to satisfy performance and energy-efficiency constraints. However, [...] Read more.

Recurrent neural networks (RNNs) have produced significant results in many fields, such as natural language processing and speech recognition. Owing to their computational complexity and sequence dependencies, RNNs need to be deployed on customized hardware accelerators to satisfy performance and energy-efficiency constraints. However, designing hardware accelerators for RNNs is challenged by the vast design space and the reliance on ineffective optimization. An efficient automated design space exploration (DSE) strategy that can balance conflicting objectives is wanted. To address the low efficiency and insufficient universality of the resource allocation process employed for hardware accelerators, we propose an automated two-stage design space exploration (DSE) strategy for customized RNN accelerators. The strategy combines a genetic algorithm (GA) and a reinforcement learning (RL) algorithm, and it utilizes symmetrical exploration and exploitation to find the optimal solutions. In the first stage, the area of the hardware accelerator is taken as the optimization objective, and the GA is used for partial exploration purposes to narrow the design space while maintaining diversity. Then, the latency and power of the hardware accelerator are taken as the optimization objectives, and the RL algorithm is used in the second stage to find the corresponding Pareto solutions. To verify the effectiveness of the developed strategy, it is compared with other algorithms. We use three different network models as benchmarks: a vanilla RNN, LSTM, and a GRU. The results demonstrate that the strategy proposed in this paper can provide better solutions and can achieve latency, power, and area reductions of 9.35%, 5.34%, and 11.95%, respectively. The HV of GRMD is reduced by averages of 6.33%, 6.32%, and 0.67%, and the runtime is reduced by averages of 18.11%, 14.94%, and 10.28%, respectively. Additionally, given different weights, it can make reasonable trade-offs between multiple objectives. Full article

(This article belongs to the Section Computer)

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14 pages, 724 KiB

Open AccessArticle

Mapping Properties of Associate Laguerre Polynomial in Symmetric Domains

by Sa’ud Al-Sa’di, Ayesha Siddiqa, Bushra Kanwal, Mohammed Ali Alamri, Saqib Hussain and Saima Noor

Symmetry 2024, 16(11), 1545; https://doi.org/10.3390/sym16111545 - 18 Nov 2024

Viewed by 346

Abstract

The significant characteristics of Associate Laguerre polynomials (ALPs) have noteworthy applications in the fields of complex analysis and mathematical physics. The present article mainly focuses on the inclusion relationships of ALPs and various analytic domains. Starting with the investigation of admissibility conditions of [...] Read more.

The significant characteristics of Associate Laguerre polynomials (ALPs) have noteworthy applications in the fields of complex analysis and mathematical physics. The present article mainly focuses on the inclusion relationships of ALPs and various analytic domains. Starting with the investigation of admissibility conditions of the analytic functions belonging to these domains, we obtained the conditions on the parameters of ALPs under which an ALP maps an open unit disc inside such analytical domains. The graphical demonstration enhances the outcomes and also proves the validity of our obtained results. Full article

(This article belongs to the Special Issue Symmetry in Geometric Theory of Analytic Functions)

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24 pages, 767 KiB

Open AccessArticle

Symmetrical Convergence Rates and Asymptotic Properties of Estimators in a Semi-Parametric Errors-in-Variables Model with Strong Mixing Errors and Missing Responses

by Jingjing Zhang, Haiqin Yan and Tingting Hu

Symmetry 2024, 16(11), 1544; https://doi.org/10.3390/sym16111544 - 18 Nov 2024

Viewed by 314

Abstract

This paper considers a semi-parametric errors-in-variables (EV) model,

η_{i} = x_{i} β + g (τ_{i}) + ϵ_{i}

,

ξ_{i} = x_{i} + δ_{i}

,

1 ⩽ i ⩽ n

. The properties of estimators [...] Read more.

This paper considers a semi-parametric errors-in-variables (EV) model,

η_{i} = x_{i} β + g (τ_{i}) + ϵ_{i}

,

ξ_{i} = x_{i} + δ_{i}

,

1 ⩽ i ⩽ n

. The properties of estimators are investigated under conditions of missing data and strong mixing errors. Three approaches are used to handle missing data: direct deletion, imputation, and the regression surrogate. Furthermore, estimators for the coefficient

β

and the nonparametric function

g (\cdot)

are obtained. Notably, both estimators achieve strong consistency at a rate of

o (n^{- 1 / 4})

, exhibiting a symmetry in their convergence rates, and they also demonstrate asymptotic normality. Additionally, the validity of our theoretical results is supported by simulations demonstrating the finite sample behaviour of these estimators. Full article

(This article belongs to the Section Mathematics)

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28 pages, 949 KiB

Open AccessArticle

Application of a Fractional Order PI Controller for a Speed Servo Drive Control

by Pavol Bistak, Igor Bélai, Igor Bélai, Jr., Damir Vrancic and Mikulas Huba

Symmetry 2024, 16(11), 1543; https://doi.org/10.3390/sym16111543 - 18 Nov 2024

Viewed by 351

Abstract

This paper deals with the tuning of the parameters of a fractional-order PI controller for the speed control of an electric servo drive in which the torque is set by a torque generator. The controller parameters are tuned using the multiple dominant pole [...] Read more.

This paper deals with the tuning of the parameters of a fractional-order PI controller for the speed control of an electric servo drive in which the torque is set by a torque generator. The controller parameters are tuned using the multiple dominant pole method (MDPM), while the fractional order integrator is approximated by the Oustaloup method. The input parameters required for tuning the controller using MDPM are calculated using the optimization algorithm presented in this paper. This algorithm selects the optimal parameters from a set of points in three-dimensional space, based on the symmetry around a central point. The controller tuning is performed for the normalized control loop model. The obtained optimized normalized fractional order PI controller can then be applied to a real servo drive with specific parameters. The proposed tuning was also verified experimentally, comparing the obtained closed-loop responses with those of the integer-order PI controller. Both simulation and experimental results showed a significant reduction in the integral of the absolute error at the disturbance step compared to a control loop using an integer-order PI controller. This results in a faster output response to load torque steps and a smaller control error in a real servo drive. Full article

(This article belongs to the Special Issue Symmetry in Process Optimization)

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19 pages, 2132 KiB

Open AccessArticle

An Experimental Study on Deuterium Production from Titanium Hydride Powders Subjected to Thermal Cycles

by Luca Gamberale and Giovanni Modanese

Symmetry 2024, 16(11), 1542; https://doi.org/10.3390/sym16111542 - 18 Nov 2024

Viewed by 308

Abstract

An extensive multi-year experimental study was conducted to investigate the potential production of deuterium from titanium hydride TiH_x powders subjected to specific thermal cycles. Mass spectrometry was performed, focusing on the variation in signal intensities at

m / z

= 2, 3, [...] Read more.

An extensive multi-year experimental study was conducted to investigate the potential production of deuterium from titanium hydride TiH_x powders subjected to specific thermal cycles. Mass spectrometry was performed, focusing on the variation in signal intensities at

m / z

= 2, 3, 4, 18, 19, 20, and 21, corresponding to fragments primarily involving deuterium, during the degassing of titanium hydride powders as the sample temperature was raised from room temperature to approximately 1100 °C. The results reveal an anomaly in the deuterium-to-hydrogen ratios, with the analysis indicating an increase in deuterium concentration by a factor of approximately 280 compared to its natural concentration on Earth. Three independent methods confirmed the excess deuterium. Simultaneously, flow calorimetry was performed during the degassing process, which did not show any measurable excess heat produced in the configuration used. This study was motivated by our novel theoretical predictions, based on the standard electroweak theory with gauge symmetry, suggesting the generation of slow neutrons within metal hydrides when exposed to coherent excitations. Our findings align with direct measurements of neutron emission by TiH_x powders under cavitation in liquid water, as recently published by Fomitchev-Zamilov. Full article

(This article belongs to the Section Physics)

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9 pages, 1139 KiB

Open AccessArticle

The Test–Retest Reliability of a Belt-Fixated Dynamometer for Assessing Maximal Knee Strength, Inter-Limb Asymmetries, and Hamstring–Quadriceps Ratios

by Nenad Nedović, Stevan Jovanović, Danilo Vujičić, Chris Bishop and Žiga Kozinc

Symmetry 2024, 16(11), 1541; https://doi.org/10.3390/sym16111541 - 18 Nov 2024

Viewed by 506

Abstract

Evaluating maximal strength, inter-limb asymmetries, and the hamstring-to-quadriceps (HQ) ratio is essential for identifying strength deficits in athletes. This cross-sectional study assessed the test–retest (inter-visit) reliability of the EasyForce dynamometer for knee extension and flexion strength in 21 young healthy participants (11 women [...] Read more.

Evaluating maximal strength, inter-limb asymmetries, and the hamstring-to-quadriceps (HQ) ratio is essential for identifying strength deficits in athletes. This cross-sectional study assessed the test–retest (inter-visit) reliability of the EasyForce dynamometer for knee extension and flexion strength in 21 young healthy participants (11 women and 10 men; age = 19.4 ± 0.7 years). The dynamometer demonstrated excellent relative reliability, with ICC values of 0.99 for knee extension and 0.95–0.98 for knee flexion. Absolute reliability was also acceptable (typical error = 5.63–16.44 N; coefficient of variation = 3.94–6.80%). Reliability for inter-limb asymmetries (ICC = 0.90) and HQ ratios (ICC = 0.91–0.92) was good to excellent. Agreement for inter-limb asymmetry direction between visits was excellent for knee extension (κ = 0.90) and substantial for knee flexion (κ = 0.71). These findings suggest that EasyForce is reliable for assessing muscle strength, inter-limb asymmetries, and HQ ratios in physically active adults. Future research should explore the broader applicability of EasyForce in muscle strength assessment, particularly for professional athletes and during rehabilitation. Full article

(This article belongs to the Special Issue Application of Symmetry in Biomechanics)

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26 pages, 2013 KiB

Open AccessArticle

A Three-Party Dynamic Pricing Mechanism for Customized Data Products Based on the Stackelberg Game and Bargaining Model

by Yanfeng Chen, Minchao Liu, Jiayi Zhang, Aiping Tan and Yan Wang

Symmetry 2024, 16(11), 1540; https://doi.org/10.3390/sym16111540 - 18 Nov 2024

Viewed by 314

Abstract

In the era of big data, breaking down data silos to enable efficient data transactions has become essential, with the fairness and transparency of pricing mechanisms being paramount. This study addresses these challenges by introducing a novel tripartite pricing model for customized data [...] Read more.

In the era of big data, breaking down data silos to enable efficient data transactions has become essential, with the fairness and transparency of pricing mechanisms being paramount. This study addresses these challenges by introducing a novel tripartite pricing model for customized data products that integrates the Stackelberg and bargaining game frameworks. By designing distinct utility functions for buyers, sellers, and the platform, the model effectively aligns the varying objectives of each participant. A dynamic adjustment mechanism further enhances this model by adaptively recalibrating the guidance price and pricing range based on real-time updates to buyer budgets and seller offers, thus ensuring fairness and responsiveness throughout the negotiation process. Experimental simulations comprising 100 transaction rounds across diverse buyer–seller profiles validate the model’s effectiveness, achieving a transaction success rate of 92.70% with an average of 6.86 bargaining rounds. These findings underscore the model’s capacity to optimize transaction outcomes, promote pricing equity, and enhance bargaining efficiency. The proposed model has broad applications in sectors such as finance, healthcare, and e-commerce, where precise data pricing mechanisms are essential to maximize transactional value. Full article

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15 pages, 3078 KiB

Open AccessArticle

Bilateral Correlational Behavior of Pyroglutamate Aminopeptidase I Activity in Rat Photoneuroendocrine Locations During a Standard 12:12 h Light–Dark Cycle

by Manuel Ramírez-Sánchez, Isabel Prieto, Ana Belén Segarra, Inmaculada Banegas, Magdalena Martínez-Cañamero, Germán Domínguez-Vías, Raquel Durán and Francisco Vives

Symmetry 2024, 16(11), 1539; https://doi.org/10.3390/sym16111539 - 17 Nov 2024

Viewed by 342

Abstract

We previously described the circadian variation and bilateral distribution of pyroglutamate aminopeptidase I (pGluPI) activity levels in photoneuroendocrine locations of adult male rats during a standard 12:12 h light–dark cycle. However, the correlational analysis between such locations has not yet been studied. This [...] Read more.

We previously described the circadian variation and bilateral distribution of pyroglutamate aminopeptidase I (pGluPI) activity levels in photoneuroendocrine locations of adult male rats during a standard 12:12 h light–dark cycle. However, the correlational analysis between such locations has not yet been studied. This may provide new data about the unilateral and bilateral functional interaction between photoneuroendocrine locations under light and dark conditions. We analyzed the correlations between locations of a photoneuroendocrine circuit consisting of retina, anterior hypothalamus, superior cervical ganglion, and pineal gland, as well as other related photoneuroendocrine locations: posterior hypothalamus, anterior pituitary, posterior pituitary, occipital cortex, and serum. In particular, we analyzed the correlations between the left retina or the right retina versus the rest of the locations, as well as the correlations between the left and right sides of paired structures at the different time points selected from 12 h light and 12 h dark periods. Also, the profiles of correlational results were compared with the corresponding mean levels. The results demonstrate the complexity of asymmetrical brain behavior. The correlation profile did not always parallel the profile observed with the mean activity values. The diurnal behavior of correlations with the left or right retina differed from one location to another. Likewise, the diurnal variation of correlations between the left and right sides of the paired structures differed between them. Particularly, while most correlations between the left versus right sides of paired structures showed positive values, that of the posterior hypothalamus showed a negative value at 13 h of light period. In addition, except the posterior hypothalamus, most paired locations only correlated significantly with right retina at 07 h of the light period. The results demonstrate the dynamic complexity of brain asymmetry, which represents a challenge for understanding its functional meaning. Full article

(This article belongs to the Special Issue Symmetry/Asymmetry in Life Sciences: Feature Papers 2024)

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21 pages, 7291 KiB

Open AccessArticle

Implications of Asymptomatic Carriers for Tuberculosis Transmission and Control in Thailand: A Modelling Approach

by Md Abdul Kuddus, Sazia Khatun Tithi and Thitiya Theparod

Symmetry 2024, 16(11), 1538; https://doi.org/10.3390/sym16111538 - 17 Nov 2024

Viewed by 674

Abstract

Tuberculosis (TB) is a regional and global bacterial illness that has been expanding and affecting individuals in every generation. An unknown percentage of asymptomatic hosts have TB, and the infection can spread while exhibiting no symptoms. These asymptomatic TB carriers, who contribute to [...] Read more.

Tuberculosis (TB) is a regional and global bacterial illness that has been expanding and affecting individuals in every generation. An unknown percentage of asymptomatic hosts have TB, and the infection can spread while exhibiting no symptoms. These asymptomatic TB carriers, who contribute to the spread of the illness yet go mostly undetected, may make it more difficult to prevent transmission. In this study, we utilized the concept of symmetry to construct a manageable disease modelling framework for TB transmission and control. We developed a TB model to investigate the potential influence of asymptomatic carriers, symptomatic infections, and the entirety of TB prevalence on different approaches to treatment and prevention in Thailand. Annual TB incidence data from Thailand from 2000 to 2022 were used to calibrate the model parameters. We assessed the potential for reaching conflicting results about the management and spread of tuberculosis in Thailand. Our results showed that some TB strategies that were thought to be effective in reducing transmission may have the opposite impact, or that an intervention’s effectiveness might be overestimated, making it seem unfeasible in certain scenarios. For example, the objective of TB treatment, which attempts to decrease the occurrence of symptomatic TB infections, is to decrease the TB infection and propagation rates if the relative carrier (η) is less than one. Nonetheless, our results indicate that this strategy may increase the frequency of asymptomatic TB patients, symptomatic TB viral infections, and overall TB prevalence if η has been sufficiently understated. We also found that reducing only the progression rate of symptomatic TB infections cannot stop asymptomatic TB carriers and total TB prevalence, even when the relative infection of carriers (

η

) is less than unity. Our research provides a better understanding of the role of asymptomatic patients in spreading TB and highlights the need to accurately include bearers in models that guide Thailand’s TB control strategy. Full article

(This article belongs to the Special Issue Mathematical Modeling of the Infectious Diseases and Their Controls)

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22 pages, 517 KiB

Open AccessArticle

LIRL: Latent Imagination-Based Reinforcement Learning for Efficient Coverage Path Planning

by Zhenglin Wei, Tiejiang Sun and Mengjie Zhou

Symmetry 2024, 16(11), 1537; https://doi.org/10.3390/sym16111537 - 17 Nov 2024

Viewed by 399

Abstract

Coverage Path Planning (CPP) in unknown environments presents unique challenges that often require the system to maintain a symmetry between exploration and exploitation in order to efficiently cover unknown areas. This paper introduces latent imagination-based reinforcement learning (LIRL), a novel framework that addresses [...] Read more.

Coverage Path Planning (CPP) in unknown environments presents unique challenges that often require the system to maintain a symmetry between exploration and exploitation in order to efficiently cover unknown areas. This paper introduces latent imagination-based reinforcement learning (LIRL), a novel framework that addresses these challenges by integrating three key components: memory-augmented experience replay (MAER), a latent imagination module (LIM), and multi-step prediction learning (MSPL) within a soft actor–critic architecture. MAER enhances sample efficiency by prioritizing experience retrieval, LIM facilitates long-term planning via simulated trajectories, and MSPL optimizes the trade-off between immediate rewards and future outcomes through adaptive n-step learning. MAER, LIM, and MSPL work within a soft actor–critic architecture, and LIRL creates a dynamic equilibrium that enables efficient, adaptive decision-making. We evaluate LIRL across diverse simulated environments, demonstrating substantial improvements over state-of-the-art methods. Through this method, the agent optimally balances short-term actions with long-term planning, maintaining symmetrical responses to varying environmental changes. The results highlight LIRL’s potential for advancing autonomous CPP in real-world applications such as search and rescue, agricultural robotics, and warehouse automation. Our work contributes to the broader fields of robotics and reinforcement learning, offering insights into integrating memory, imagination, and adaptive learning for complex sequential decision-making tasks. Full article

(This article belongs to the Section Computer)

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16 pages, 35376 KiB

Open AccessArticle

Numerical Simulation on Medium-Deep Hole Straight Cut Blasting Based on the Principle of Segmented Charging

by Xiantang Zhang, Fuzhi Wang, Zhiyu Bai, Bin Shao, Yuchao Wei, Qingqian Wu and Jingshuang Zhang

Symmetry 2024, 16(11), 1536; https://doi.org/10.3390/sym16111536 - 16 Nov 2024

Viewed by 363

Abstract

The efficiency of rock excavation depends on cut blasting. However, medium-deep hole cutting blasting faces the challenges of large clamping action and unsatisfactory blasting efficiency. The study proposes sectional charge cutting blasting technology and analyzes the mechanism of cavity formation by establishing a [...] Read more.

The efficiency of rock excavation depends on cut blasting. However, medium-deep hole cutting blasting faces the challenges of large clamping action and unsatisfactory blasting efficiency. The study proposes sectional charge cutting blasting technology and analyzes the mechanism of cavity formation by establishing a numerical model. The results demonstrated that sectional charge blasting in the hole can expand the range of stress waves, and the segment interaction is also optimized by introducing a delay time difference. These factors contribute to an increase in the rock-breaking volume and an improvement in the degree of rock breaking. Furthermore, the cutting effects of different segmented proportional models are quantified. When the upper and lower sections are symmetrically charged, the damage range caused by the upper section is greater. The reason is that the clamping force exerted on the rock mass increases with the depth of the hole. In addition, when the upper section ratio is 0.4, the model exhibits the most excellent cavity volume; this results from charging according to the symmetry principle for optimal energy distribution. Full article

(This article belongs to the Special Issue Symmetry in Civil Transportation Engineering)

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33 pages, 394 KiB

Open AccessArticle

On the Problem of the Uniqueness of Fixed Points and Solutions for Quadratic Fractional-Integral Equations on Banach Algebras

by Kinga Cichoń, Mieczysław Cichoń and Maciej Ciesielski

Symmetry 2024, 16(11), 1535; https://doi.org/10.3390/sym16111535 - 16 Nov 2024

Viewed by 340

Abstract

In this paper, we study the problem of the uniqueness of fixed points for operators defined on subspaces of the space of continuous functions

C [a, b]

equipped with norms stronger than the supremum norm. We are particularly interested in [...] Read more.

In this paper, we study the problem of the uniqueness of fixed points for operators defined on subspaces of the space of continuous functions

C [a, b]

equipped with norms stronger than the supremum norm. We are particularly interested in Hölder spaces since they are natural ranges of integral operators of fractional order. Our goal is to preserve the expected regularity of the fixed points (solutions of the equations) when investigating their uniqueness, without assuming a contraction condition on the space under study. We claim some symmetry between the case of the obtained results and the case of the classical Banach fixed-point theorem in such spaces, even for operators which are not necessarily contractions in the sense of the norm of these subspaces. This result is of particular interest for the study of quadratic integral equations, and as an application example we prove the uniqueness theorem for such a kind equations with general fractional order integral operators, which are not necessarily contractions, in a suitably constructed generalized Hölder space. Full article

(This article belongs to the Special Issue New Trends in Fixed Point Theory with Emphasis on Symmetry)

8 pages, 560 KiB

Open AccessArticle

Lack of Brain Asymmetry in the Alpha Band During the Observation of Object Grasping in Reality Versus on Screen

by Celia Andreu-Sánchez, Miguel Ángel Martín-Pascual, Agnès Gruart and José María Delgado-García

Symmetry 2024, 16(11), 1534; https://doi.org/10.3390/sym16111534 - 16 Nov 2024

Viewed by 365

Abstract

The way audiovisuals are perceived is not completely understood. Previous works have shown that attention increases when watching audiovisuals compared with looking at real events, but depending on the editing style, and the interpreter, the understanding of the content may be different. The [...] Read more.

The way audiovisuals are perceived is not completely understood. Previous works have shown that attention increases when watching audiovisuals compared with looking at real events, but depending on the editing style, and the interpreter, the understanding of the content may be different. The study of brain asymmetries in this context aims to identify potential lateralizations in audiovisual perception. Previous studies have proven that seeing others grasp objects has a contralateral impact on somatosensory areas (C3 and C4) in the alpha band (8–12 Hz). In this work, we investigated whether brain activity could be asymmetrical in that band when looking at real grasping compared with watching grasping on a screen, and whether media expertise would be a determinant in this regard and presented the same narrative content both through live performance and as a movie to 40 participants (half of them media professionals) while recording their electroencephalography (EEG) activity. We conclude that the asymmetry in the alpha band in the somatosensory cortex is not affected by the medium through which the grasping actions are presented, that is, in reality or on screen. We also conclude that media expertise does not impact this asymmetry. Full article

(This article belongs to the Special Issue Brain Asymmetry in Cognitive and Behavioral Perception)

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23 pages, 314 KiB

Open AccessArticle

Constructing and Analyzing BiHom-(Pre-)Poisson Conformal Algebras

by Sania Asif and Yao Wang

Symmetry 2024, 16(11), 1533; https://doi.org/10.3390/sym16111533 - 15 Nov 2024

Viewed by 470

Abstract

This study introduces the notions of BiHom-Poisson conformal algebra, BiHom-pre-Poisson conformal algebra, and their related structures. We show that many new BiHom-Poisson conformal algebras can be constructed from a BiHom-Poisson conformal algebra. In particular, the direct product of two BiHom-Poisson conformal algebras is [...] Read more.

This study introduces the notions of BiHom-Poisson conformal algebra, BiHom-pre-Poisson conformal algebra, and their related structures. We show that many new BiHom-Poisson conformal algebras can be constructed from a BiHom-Poisson conformal algebra. In particular, the direct product of two BiHom-Poisson conformal algebras is also a BiHom-Poisson conformal algebra. We further describe the conformal bimodule and representation theory of the BiHom-Poisson conformal algebra. In addition, we define BiHom-pre-Poisson conformal algebra as the combination of BiHom-pre-Lie conformal algebra and BiHom-dendriform conformal algebra under some compatibility conditions. We further demonstrate a way to construct BiHom-Poisson conformal algebra from BiHom-pre-Poisson conformal algebra and provide the representation theory for BiHom-pre-Poisson conformal algebra. Finally, a detailed description of

O

-operators and Rota–Baxter operators on BiHom-Poisson conformal algebra is provided. Full article

(This article belongs to the Section Mathematics)

23 pages, 3161 KiB

Open AccessArticle

Dynamic Characterization and Optimization of Heat Flux and Thermal Efficiency of a Penetrable Moving Hemispherical Fin Embedded in a Shape Optimized Fe₃O₄-Ni/C₆H₁₈OSi₂ Hybrid Nanofluid: L-IIIA Solution

by Ammembal Gopalkrishna Pai, Rekha G. Pai, Karthi Pradeep and Likith Raj

Symmetry 2024, 16(11), 1532; https://doi.org/10.3390/sym16111532 - 15 Nov 2024

Viewed by 571

Abstract

The present paper reports the theoretical results on the thermal performance of proposed Integrated Hybrid Nanofluid Hemi-Spherical Fin Model assuming a combination of Fe₃O₄-Ni/C₆H₁₈OSi₂ hybrid nanofluid. The model leverages the concept of symmetrical [...] Read more.

The present paper reports the theoretical results on the thermal performance of proposed Integrated Hybrid Nanofluid Hemi-Spherical Fin Model assuming a combination of Fe₃O₄-Ni/C₆H₁₈OSi₂ hybrid nanofluid. The model leverages the concept of symmetrical geometries and optimized nanoparticle shapes to enhance the heat flux, with a focus on symmetrical design applications in thermal engineering. The simulations are carried out by assuming a silicone oil as a base fluid, due to its exceptional stability in hot and humid conditions, enriched with superparamagnetic Fe₃O₄ and Ni nanoparticles to enhance the heat transfer capabilities, with the aim of contributing to the field of nanotechnology, electronics and thermal engineering, The focus of this work is to optimize the heat dissipation in systems that require high thermal efficiency and stability such as automotive cooling systems, aerospace components and power electronics. In addition, the study explores the influence of key parameters such as heat transfer coefficients and thermal conductivity that play an important role in improving the thermal performance of cooling systems. The overall thermal performance of the model is evaluated based on its heat flux and thermal efficiency. The study also examines the impact of the shape optimized nanoparticles in silicone oil by incorporating shape-factor in its modelling equations and proposes optimization of parameters to enhance the overall thermal performance of the system. Darcy’s flow model is used to analyse the key parameters in the system and study the thermal behaviour of the hybrid nanofluid within the fin by incorporating natural convection, temperature-dependent internal heat generation, and radiation effects. By using the similarity approach, the governing equations were reduced to non-linear ordinary differential equations and numerical solutions were obtained by using four-stage Lobatto-IIIA numerical technique due to its robust stability and convergence properties. This enables a systematic investigation of various influential parameters, including thermal conductivity, emissivity and heat transfer coefficients. Additionally, it stimulates interest among researchers in applying mathematical techniques to complex heat transfer systems, thereby contributing towards the development of highly efficient cooling system. Our findings indicate that there is a significant enhancement in the heat flux as well as improvement in the thermal efficiency due to the mixture of silicone oil and shape optimized nanoparticles, that was visualized through comprehensive graphical analysis. Quantitatively, the proposed model displays a maximum thermal efficiency of 57.5% for lamina shaped nanoparticles at

N_{c}

= 0.5,

N_{r}

= 0.2,

N_{g}

= 0.2 and

Θ_{a}

= 0.4. The maximum enhancement in the heat flux occurs when

N_{c}

doubles from 5 to 10 for

m_{2}

= 0.2 and

N_{r}

= 0.1. Optimal thermal performance is found for

N_{c}

,

N_{r}

and

m_{2}

values in the range 5 to 10, 0.2 to 0.4 and 0.4 to 0.8 respectively. Full article

(This article belongs to the Section Physics)

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17 pages, 5051 KiB

Open AccessArticle

Negative Solvatochromism of the Intramolecular Charge Transfer Band in Two Structurally Related Pyridazinium—Ylids

by Mihaela Iuliana Avădănei, Antonina Griţco-Todiraşcu and Dana Ortansa Dorohoi

Symmetry 2024, 16(11), 1531; https://doi.org/10.3390/sym16111531 - 15 Nov 2024

Viewed by 478

Abstract

Two charge transfer compounds based on pyridazinium ylids were studied by electronic absorption spectroscopy in binary and ternary solutions, with the purpose of evaluating their descriptors of the first singlet excited state and to estimate the strength of the intermolecular interactions in protic [...] Read more.

Two charge transfer compounds based on pyridazinium ylids were studied by electronic absorption spectroscopy in binary and ternary solutions, with the purpose of evaluating their descriptors of the first singlet excited state and to estimate the strength of the intermolecular interactions in protic solvents. The molecular descriptors of the excited state were comparatively estimated using the variational method and the Abe model of diluted binary solutions. Analysis of electronic properties using density functional theory was performed for several key solvents, in order to understand the solvatochromic behavior. The DFT calculations revealed that, in the polar and strongly interacting solvents, the carbanion and the terminal group become a stronger electron acceptor. The bathochromic shift of the ICT band was confirmed using DFT calculus. The ability of the two ylids to recognize and discriminate the solvents was analyzed with principal component analysis and with cluster analysis. Although the study was performed in 24 solvents, the results showed that the ylids were most sensitive to alcohols, so they can be a useful tool to identify and classify different types of low-alcoholic solvents. Full article

(This article belongs to the Collection Feature Papers in Chemistry)

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21 pages, 9459 KiB

Open AccessArticle

Development and Space Experiment Verification of Annular Liquid Flow Payload for China Space Station

by Jia Wang, Yi Chen, Di Wu, Liang Hu, Zhijun Ye, Li Duan, Xilin Zhao, Yongli Yin and Qi Kang

Symmetry 2024, 16(11), 1530; https://doi.org/10.3390/sym16111530 - 15 Nov 2024

Viewed by 315

Abstract

The annular flow payload is among the first batch of space science experimental projects carried out on the Fluid Physics Rack of the China Space Station. This paper provides a detailed introduction to the development of the payload, ground validation, and in orbit [...] Read more.

The annular flow payload is among the first batch of space science experimental projects carried out on the Fluid Physics Rack of the China Space Station. This paper provides a detailed introduction to the development of the payload, ground validation, and in orbit experiments. The payload, sized 320 mm × 200 mm × 220 mm, includes an annular flow model and supports supply (24 V, 12 V, and 5 V), communication, and data transmission. A multi-functional heating column in the annular flow model was designed, allowing for the column to operate in fixed, rotating, and lifting scenarios. In the first round, 96 sets of space experiments covering volume ratio ranges from 0.45 to 1.06 were carried out. The annular flow liquid surface state, temperature oscillation, and infrared temperature field evolution were obtained. Mode decomposition shows the oscillatory convection of the m = 4 travelling wave, and contains m = 3, m = 6, and m = 8 waves. Full article

(This article belongs to the Section Physics)

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11 pages, 2794 KiB

Open AccessArticle

Multilinear Variable Separation Approach in (4+1)-Dimensional Boiti–Leon–Manna–Pempinelli Equation

by Jia-Rong Zhu and Bo Ren

Symmetry 2024, 16(11), 1529; https://doi.org/10.3390/sym16111529 - 15 Nov 2024

Viewed by 455

Abstract

In this paper, we use the multilinear variable separation approach involving two arbitrary variable separation functions to construct a new variable separation solution of the (4+1)-dimensional Boiti–Leon–Manna–Pempinelli equation. Through considering different parameters, three types of local excitations including dromions, lumps, and ring solitons [...] Read more.

In this paper, we use the multilinear variable separation approach involving two arbitrary variable separation functions to construct a new variable separation solution of the (4+1)-dimensional Boiti–Leon–Manna–Pempinelli equation. Through considering different parameters, three types of local excitations including dromions, lumps, and ring solitons are constructed. Dromion molecules, lump molecules, ring soliton molecules, and their interactions are analyzed through the velocity resonance mechanism. In addition, the results reveal the elastic and inelastic interactions between solitons. We discuss some dynamical properties of these solitons and soliton molecules obtained analytically. Three-dimensional diagrams and contour plots of the solution are given to help understand the physical mechanism of the solutions. Full article

(This article belongs to the Special Issue Symmetry in Nonlinear Schrödinger Equations)

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25 pages, 1691 KiB

Open AccessArticle

The Application of Generalized Viscosity Implicit Midpoint Rule for Nonexpansive Mappings

by Huancheng Zhang

Symmetry 2024, 16(11), 1528; https://doi.org/10.3390/sym16111528 - 14 Nov 2024

Viewed by 364

Abstract

This paper proposes new iterative algorithms by using the generalized viscosity implicit midpoint rule in Banach space, which is also a symmetric space. Then, this paper obtains strong convergence conclusions. Moreover, the results generalize the related conclusions of some researchers. Finally, this paper [...] Read more.

This paper proposes new iterative algorithms by using the generalized viscosity implicit midpoint rule in Banach space, which is also a symmetric space. Then, this paper obtains strong convergence conclusions. Moreover, the results generalize the related conclusions of some researchers. Finally, this paper provides some examples to verify these conclusions. These conclusions further extend and enrich the relevant theory of symmetric space. Full article

(This article belongs to the Section Mathematics)

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