Some might say that the eigenproblem is one of the examples people discovered by looking at the sky and wondering. Even though it was formulated to explain the movement of the planets, today it has become the ansatz of solving many linear and nonlinear problems. Formulation in the terms of the eigenproblem is one of the key tools to solve complex problems, especially in the area of molecular geometry. However, the basic concept is difficult without proper preparation. A review paper covering basic concepts and algorithms is very useful. This review covers the basics of the topic. Definitions are provided for defective, Hermitian, Hessenberg, modal, singular, spectral, symmetric, skew-symmetric, skew-Hermitian, triangular, and Wishart matrices. Then, concepts of characteristic polynomial, eigendecomposition, eigenpair, eigenproblem, eigenspace, eigenvalue, and eigenvector are subsequently introduced. Faddeev–LeVerrier, von Mises, Gauss–Jordan, Pohlhausen, Lanczos–Arnoldi, Rayleigh–Ritz, Jacobi–Davidson, and Gauss–Seidel fundamental algorithms are given, while others (Francis–Kublanovskaya, Gram–Schmidt, Householder, Givens, Broyden–Fletcher–Goldfarb–Shanno, Davidon–Fletcher–Powell, and Saad–Schultz) are merely discussed. The eigenproblem has thus found its use in many topics. The applications discussed include solving Bessel’s, Helmholtz’s, Laplace’s, Legendre’s, Poisson’s, and Schrödinger’s equations. The algorithm extracting the first principal component is also provided.
{"title":"Eigenproblem Basics and Algorithms","authors":"Lorentz Jäntschi","doi":"10.3390/sym15112046","DOIUrl":"https://doi.org/10.3390/sym15112046","url":null,"abstract":"Some might say that the eigenproblem is one of the examples people discovered by looking at the sky and wondering. Even though it was formulated to explain the movement of the planets, today it has become the ansatz of solving many linear and nonlinear problems. Formulation in the terms of the eigenproblem is one of the key tools to solve complex problems, especially in the area of molecular geometry. However, the basic concept is difficult without proper preparation. A review paper covering basic concepts and algorithms is very useful. This review covers the basics of the topic. Definitions are provided for defective, Hermitian, Hessenberg, modal, singular, spectral, symmetric, skew-symmetric, skew-Hermitian, triangular, and Wishart matrices. Then, concepts of characteristic polynomial, eigendecomposition, eigenpair, eigenproblem, eigenspace, eigenvalue, and eigenvector are subsequently introduced. Faddeev–LeVerrier, von Mises, Gauss–Jordan, Pohlhausen, Lanczos–Arnoldi, Rayleigh–Ritz, Jacobi–Davidson, and Gauss–Seidel fundamental algorithms are given, while others (Francis–Kublanovskaya, Gram–Schmidt, Householder, Givens, Broyden–Fletcher–Goldfarb–Shanno, Davidon–Fletcher–Powell, and Saad–Schultz) are merely discussed. The eigenproblem has thus found its use in many topics. The applications discussed include solving Bessel’s, Helmholtz’s, Laplace’s, Legendre’s, Poisson’s, and Schrödinger’s equations. The algorithm extracting the first principal component is also provided.","PeriodicalId":48874,"journal":{"name":"Symmetry-Basel","volume":" 14","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135186938","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The expectation that the physical expansion of space occurs smoothly may be expressed mathematically as a requirement for continuity in the time derivative of the metric scale factor of the Friedmann–Robertson–Walker cosmology. We explore the consequences of imposing such a smoothness requirement, examining the forms of possible interpolating functions between the end of inflation and subsequent radiation- or matter-dominated eras, using a straightforward geometric model of the interpolating behavior. We quantify the magnitude of the cusp found in a direct transition from the end of slow-roll inflation to the subsequent era, analyze the validity of several smooth interpolator candidates, and investigate equation-of-state and thermodynamic constraints. We find an order-of-magnitude increase in the size of the universe at the end of the transition to a single-component radiation or matter era. We also evaluate the interpolating functions in terms of the standard theory of preheating and determine the effect on the number of bosons produced.
{"title":"Exiting Inflation with a Smooth Scale Factor","authors":"Harry Oslislo, Brett Altschul","doi":"10.3390/sym15112042","DOIUrl":"https://doi.org/10.3390/sym15112042","url":null,"abstract":"The expectation that the physical expansion of space occurs smoothly may be expressed mathematically as a requirement for continuity in the time derivative of the metric scale factor of the Friedmann–Robertson–Walker cosmology. We explore the consequences of imposing such a smoothness requirement, examining the forms of possible interpolating functions between the end of inflation and subsequent radiation- or matter-dominated eras, using a straightforward geometric model of the interpolating behavior. We quantify the magnitude of the cusp found in a direct transition from the end of slow-roll inflation to the subsequent era, analyze the validity of several smooth interpolator candidates, and investigate equation-of-state and thermodynamic constraints. We find an order-of-magnitude increase in the size of the universe at the end of the transition to a single-component radiation or matter era. We also evaluate the interpolating functions in terms of the standard theory of preheating and determine the effect on the number of bosons produced.","PeriodicalId":48874,"journal":{"name":"Symmetry-Basel","volume":" 6","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135186927","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Gudrun Albrecht, Esmeralda Mainar, Juan Manuel Peña, Beatriz Rubio
This paper proposes a new approach to define two frequency trigonometric spline curves with interesting shape preserving properties. This construction requires the normalized B-basis of the space U4(Iα)=span{1,cost,sint,cos2t,sin2t} defined on compact intervals Iα=[0,α], where α is a global shape parameter. It will be shown that the normalized B-basis can be regarded as the equivalent in the trigonometric space U4(Iα) to the Bernstein polynomial basis and shares its well-known symmetry properties. In fact, the normalized B-basis functions converge to the Bernstein polynomials as α→0. As a consequence, the convergence of the obtained piecewise trigonometric curves to uniform quartic B-Spline curves will be also shown. The proposed trigonometric spline curves can be used for CAM design, trajectory-generation, data fitting on the sphere and even to define new algebraic-trigonometric Pythagorean-Hodograph curves and their piecewise counterparts allowing the resolution of C(3 Hermite interpolation problems.
{"title":"A Shape Preserving Class of Two-Frequency Trigonometric B-Spline Curves","authors":"Gudrun Albrecht, Esmeralda Mainar, Juan Manuel Peña, Beatriz Rubio","doi":"10.3390/sym15112041","DOIUrl":"https://doi.org/10.3390/sym15112041","url":null,"abstract":"This paper proposes a new approach to define two frequency trigonometric spline curves with interesting shape preserving properties. This construction requires the normalized B-basis of the space U4(Iα)=span{1,cost,sint,cos2t,sin2t} defined on compact intervals Iα=[0,α], where α is a global shape parameter. It will be shown that the normalized B-basis can be regarded as the equivalent in the trigonometric space U4(Iα) to the Bernstein polynomial basis and shares its well-known symmetry properties. In fact, the normalized B-basis functions converge to the Bernstein polynomials as α→0. As a consequence, the convergence of the obtained piecewise trigonometric curves to uniform quartic B-Spline curves will be also shown. The proposed trigonometric spline curves can be used for CAM design, trajectory-generation, data fitting on the sphere and even to define new algebraic-trigonometric Pythagorean-Hodograph curves and their piecewise counterparts allowing the resolution of C(3 Hermite interpolation problems.","PeriodicalId":48874,"journal":{"name":"Symmetry-Basel","volume":"102 27","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135136531","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Aiming at the fuzzification of a decision environment and the challenge of determining the weights associated with the interaction among decision-makers, this study offers an original method for (p,q)-rung probabilistic hesitant orthopair fuzzy multi-objective group decision-making, which is founded on the weight optimization principle. Firstly, the notion of a probabilistic hesitant fuzzy set is expanded to a (p,q)-rung. Secondly, the determination of subjective and objective weights is accomplished through the utilization of the Analytic Network Process (ANP) and the Entropy Method. According to the degree of deviation and dispersion of each weight, an optimal objective function is constructed, and the neural network is used to iteratively solve for the best scheme of the comprehensive weight. Subsequently, the Elimination Et Choice Translating Reality (ELECTRE) approach was refined and applied to decision-making in the (p,q)-rung probabilistic hesitant orthopair fuzzy environment. Finally, comparative analysis was used to demonstrate the new method’s effectiveness and superiority.
{"title":"Weight Optimization Decision Algorithm in (p,q)-Rung Probabilistic Hesitant Orthopair Fuzzy Environments","authors":"Jinyan Bao, Xiangzhi Kong","doi":"10.3390/sym15112043","DOIUrl":"https://doi.org/10.3390/sym15112043","url":null,"abstract":"Aiming at the fuzzification of a decision environment and the challenge of determining the weights associated with the interaction among decision-makers, this study offers an original method for (p,q)-rung probabilistic hesitant orthopair fuzzy multi-objective group decision-making, which is founded on the weight optimization principle. Firstly, the notion of a probabilistic hesitant fuzzy set is expanded to a (p,q)-rung. Secondly, the determination of subjective and objective weights is accomplished through the utilization of the Analytic Network Process (ANP) and the Entropy Method. According to the degree of deviation and dispersion of each weight, an optimal objective function is constructed, and the neural network is used to iteratively solve for the best scheme of the comprehensive weight. Subsequently, the Elimination Et Choice Translating Reality (ELECTRE) approach was refined and applied to decision-making in the (p,q)-rung probabilistic hesitant orthopair fuzzy environment. Finally, comparative analysis was used to demonstrate the new method’s effectiveness and superiority.","PeriodicalId":48874,"journal":{"name":"Symmetry-Basel","volume":" 15","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135186937","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Recently, the uncertainty aspects of record values have been increasingly studied in the literature. In this paper, we study the residual Tsallis entropy of upper record values coming from random samples. In the continuous case, we define the Tsallis entropy quantity for the residual lifetime of upper record values in general distributions as the residual Tsallis entropy of upper record values coming from a uniform distribution. We also obtain a lower bound on the residual Tsallis entropy of upper data set values originating from an arbitrary continuous probability distribution. We also discuss the monotonic property of the residual Tsallis entropy of upper data sets.
{"title":"Residual Tsallis Entropy and Record Values: Some New Insights","authors":"Mansour Shrahili, Mohamed Kayid","doi":"10.3390/sym15112040","DOIUrl":"https://doi.org/10.3390/sym15112040","url":null,"abstract":"Recently, the uncertainty aspects of record values have been increasingly studied in the literature. In this paper, we study the residual Tsallis entropy of upper record values coming from random samples. In the continuous case, we define the Tsallis entropy quantity for the residual lifetime of upper record values in general distributions as the residual Tsallis entropy of upper record values coming from a uniform distribution. We also obtain a lower bound on the residual Tsallis entropy of upper data set values originating from an arbitrary continuous probability distribution. We also discuss the monotonic property of the residual Tsallis entropy of upper data sets.","PeriodicalId":48874,"journal":{"name":"Symmetry-Basel","volume":"102 26","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135136532","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
One of the most difficult aspects of scheduling operations on virtual machines in a multi-cloud environment is determining a near-optimal permutation. This task requires assigning various computing jobs with competing objectives to a collection of virtual machines. A significant number of NP-hard problem optimization methods employ multi-objective algorithms. As a result, one of the most successful criteria for discovering the best Pareto solutions is Pareto dominance. In this study, the Pareto front is calculated using a novel multi-objective minimum weight approach. In particular, we use particle swarm optimization (PSO) to expand the FR-MOS multi-objective scheduling algorithm by using fuzzy resource management to maximize variety and obtain optimal Pareto convergence. The competing objectives include reliability, cost, utilization of resources, risk probability, and time makespan. Most of the previous studies provide numerous symmetry or equivalent solutions as trade-offs for different objectives, and selecting the optimum solution remains an issue. We propose a novel decision-making strategy named minimum weight optimization (MWO). Multi-objective algorithms use this method to select a set of permutations that provide the best trade-off between competing objectives. MWO is a suitable choice for attaining all optimal solutions, where both the needs of consumers and the interests of service providers are taken into consideration. (MWO) aims to find the best solution by comparing alternative weights, narrowing the search for an optimal solution through iterative refinement. We compare our proposed method to five distinct decision-making procedures using common scientific workflows with competing objectives: Pareto dominance, multi-criteria decision-making (MCDM), linear normalization I, linear normalization II, and weighted aggregated sum product assessment (WASPAS). MWO outperforms these strategies according to the results of this study.
{"title":"Scheduling Scientific Workflow in Multi-Cloud: A Multi-Objective Minimum Weight Optimization Decision-Making Approach","authors":"Mazen Farid, Heng Siong Lim, Chin Poo Lee, Rohaya Latip","doi":"10.3390/sym15112047","DOIUrl":"https://doi.org/10.3390/sym15112047","url":null,"abstract":"One of the most difficult aspects of scheduling operations on virtual machines in a multi-cloud environment is determining a near-optimal permutation. This task requires assigning various computing jobs with competing objectives to a collection of virtual machines. A significant number of NP-hard problem optimization methods employ multi-objective algorithms. As a result, one of the most successful criteria for discovering the best Pareto solutions is Pareto dominance. In this study, the Pareto front is calculated using a novel multi-objective minimum weight approach. In particular, we use particle swarm optimization (PSO) to expand the FR-MOS multi-objective scheduling algorithm by using fuzzy resource management to maximize variety and obtain optimal Pareto convergence. The competing objectives include reliability, cost, utilization of resources, risk probability, and time makespan. Most of the previous studies provide numerous symmetry or equivalent solutions as trade-offs for different objectives, and selecting the optimum solution remains an issue. We propose a novel decision-making strategy named minimum weight optimization (MWO). Multi-objective algorithms use this method to select a set of permutations that provide the best trade-off between competing objectives. MWO is a suitable choice for attaining all optimal solutions, where both the needs of consumers and the interests of service providers are taken into consideration. (MWO) aims to find the best solution by comparing alternative weights, narrowing the search for an optimal solution through iterative refinement. We compare our proposed method to five distinct decision-making procedures using common scientific workflows with competing objectives: Pareto dominance, multi-criteria decision-making (MCDM), linear normalization I, linear normalization II, and weighted aggregated sum product assessment (WASPAS). MWO outperforms these strategies according to the results of this study.","PeriodicalId":48874,"journal":{"name":"Symmetry-Basel","volume":" 13","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135186939","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Artificial intelligence (AI) frameworks are essential for development since they offer pre-built tools and libraries that speed up and simplify the production of AI models, leveraging symmetry to save time and effort. They guarantee effective computing by modifying code for particular hardware, facilitating quicker testing and deployment. The identification of a suitable and optimal AI framework for development is a multi-criteria decision-making (MCDM) dilemma, where the considered AI frameworks for development are evaluated by considering various criteria and these criteria may have dual aspects (positive and negative). Thus, in this manuscript, we diagnosed a technique of MCDM within the bipolar fuzzy set (BFS) for identification and selection of optimal AI framework for development. In this regard, we diagnosed probability aggregation operators (AOs) within BFS, such as probability bipolar fuzzy weighted averaging (P-BFWA), probability bipolar fuzzy ordered weighted averaging (P-BFOWA), immediate probability bipolar fuzzy ordered weighted averaging (IP-BFOWA), probability bipolar fuzzy weighted geometric (P-BFWG), probability bipolar fuzzy ordered weighted geometric (P-BFOWH), and immediate probability bipolar fuzzy ordered weighted geometric (IP-BFOWG) operators. The diagnosed technique would be based on these invented probably AOs. Afterward, in this manuscript, we took a case study and obtained the optimal AI framework for development by employing the diagnosed technique of MCDM. We also investigated the comparison of the devised theory with certain prevailing theories to reveal the dominance and significance of the devised theory.
{"title":"Bipolar Fuzzy Multi-Criteria Decision-Making Technique Based on Probability Aggregation Operators for Selection of Optimal Artificial Intelligence Framework","authors":"Yanhua Chen, Ubaid ur Rehman, Tahir Mahmood","doi":"10.3390/sym15112045","DOIUrl":"https://doi.org/10.3390/sym15112045","url":null,"abstract":"Artificial intelligence (AI) frameworks are essential for development since they offer pre-built tools and libraries that speed up and simplify the production of AI models, leveraging symmetry to save time and effort. They guarantee effective computing by modifying code for particular hardware, facilitating quicker testing and deployment. The identification of a suitable and optimal AI framework for development is a multi-criteria decision-making (MCDM) dilemma, where the considered AI frameworks for development are evaluated by considering various criteria and these criteria may have dual aspects (positive and negative). Thus, in this manuscript, we diagnosed a technique of MCDM within the bipolar fuzzy set (BFS) for identification and selection of optimal AI framework for development. In this regard, we diagnosed probability aggregation operators (AOs) within BFS, such as probability bipolar fuzzy weighted averaging (P-BFWA), probability bipolar fuzzy ordered weighted averaging (P-BFOWA), immediate probability bipolar fuzzy ordered weighted averaging (IP-BFOWA), probability bipolar fuzzy weighted geometric (P-BFWG), probability bipolar fuzzy ordered weighted geometric (P-BFOWH), and immediate probability bipolar fuzzy ordered weighted geometric (IP-BFOWG) operators. The diagnosed technique would be based on these invented probably AOs. Afterward, in this manuscript, we took a case study and obtained the optimal AI framework for development by employing the diagnosed technique of MCDM. We also investigated the comparison of the devised theory with certain prevailing theories to reveal the dominance and significance of the devised theory.","PeriodicalId":48874,"journal":{"name":"Symmetry-Basel","volume":" 34","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135186952","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Covariant integral quantization with rotational SO(3) symmetry is established for quantum motion on this group manifold. It can also be applied to Gabor signal analysis on this group. The corresponding phase space takes the form of a discrete-continuous hypercylinder. The central tool for implementing this procedure is the Weyl–Gabor operator, a non-unitary operator that operates on the Hilbert space of square-integrable functions on SO(3). This operator serves as the counterpart to the unitary Weyl or displacement operator used in constructing standard Schrödinger–Glauber–Sudarshan coherent states. We unveil a diverse range of properties associated with the quantizations and their corresponding semi-classical phase-space portraits, which are derived from different weight functions on the considered discrete-continuous hypercylinder. Certain classes of these weight functions lead to families of coherent states. Moreover, our approach allows us to define a Wigner distribution, satisfying the standard marginality conditions, along with its related Wigner transform.
{"title":"Covariant Integral Quantization of the Semi-Discrete SO(3)-Hypercylinder","authors":"Jean-Pierre Gazeau, Romain Murenzi","doi":"10.3390/sym15112044","DOIUrl":"https://doi.org/10.3390/sym15112044","url":null,"abstract":"Covariant integral quantization with rotational SO(3) symmetry is established for quantum motion on this group manifold. It can also be applied to Gabor signal analysis on this group. The corresponding phase space takes the form of a discrete-continuous hypercylinder. The central tool for implementing this procedure is the Weyl–Gabor operator, a non-unitary operator that operates on the Hilbert space of square-integrable functions on SO(3). This operator serves as the counterpart to the unitary Weyl or displacement operator used in constructing standard Schrödinger–Glauber–Sudarshan coherent states. We unveil a diverse range of properties associated with the quantizations and their corresponding semi-classical phase-space portraits, which are derived from different weight functions on the considered discrete-continuous hypercylinder. Certain classes of these weight functions lead to families of coherent states. Moreover, our approach allows us to define a Wigner distribution, satisfying the standard marginality conditions, along with its related Wigner transform.","PeriodicalId":48874,"journal":{"name":"Symmetry-Basel","volume":" 6","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135187101","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Ayman Alahmade, Zeeshan Mujahid, Ferdous M. O. Tawfiq, Bilal Khan, Nazar Khan, Fairouz Tchier
In this research, we define a few subclasses of analytic functions which are connected to sine functions and the cardioid domain in the unit disk. We investigate initial coefficient bounds, the Fekete–Szego problem and second and third Hankel determinants for the functions f belonging to these newly defined classes. We also define the class of m-fold symmetric functions related with the sine function and then investigate the bounds of the third Hankel determinant for twofold symmetric and threefold symmetric functions.
{"title":"Third Hankel Determinant for Subclasses of Analytic and m-Fold Symmetric Functions Involving Cardioid Domain and Sine Function","authors":"Ayman Alahmade, Zeeshan Mujahid, Ferdous M. O. Tawfiq, Bilal Khan, Nazar Khan, Fairouz Tchier","doi":"10.3390/sym15112039","DOIUrl":"https://doi.org/10.3390/sym15112039","url":null,"abstract":"In this research, we define a few subclasses of analytic functions which are connected to sine functions and the cardioid domain in the unit disk. We investigate initial coefficient bounds, the Fekete–Szego problem and second and third Hankel determinants for the functions f belonging to these newly defined classes. We also define the class of m-fold symmetric functions related with the sine function and then investigate the bounds of the third Hankel determinant for twofold symmetric and threefold symmetric functions.","PeriodicalId":48874,"journal":{"name":"Symmetry-Basel","volume":"113 18","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135138114","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Rajesh Kumar, Lalnunenga Colney, Mohammad Nazrul Islam Khan
The main aim of the proposed paper is to investigate the lifts of Kenmotsu manifolds that admit NSNMC in the tangent bundle. We investigate several properties of the lifts of the curvature tensor, the conformal curvature tensor, and the conharmonic curvature tensor of Kenmotsu manifolds that admit NSNMC in the tangent bundle. We also study and discover that the lift of the Kenmotsu manifold that admit NSNMC is regular in the tangent bundle. Additionally, we find that the data provided by the lift of Ricci soliton on the lift of Ricci semi-symmetric Kenmotsu manifold that admits NSNMC in the tangent bundle are expanding.
{"title":"Proposed Theorems on the Lifts of Kenmotsu Manifolds Admitting a Non-Symmetric Non-Metric Connection (NSNMC) in the Tangent Bundle","authors":"Rajesh Kumar, Lalnunenga Colney, Mohammad Nazrul Islam Khan","doi":"10.3390/sym15112037","DOIUrl":"https://doi.org/10.3390/sym15112037","url":null,"abstract":"The main aim of the proposed paper is to investigate the lifts of Kenmotsu manifolds that admit NSNMC in the tangent bundle. We investigate several properties of the lifts of the curvature tensor, the conformal curvature tensor, and the conharmonic curvature tensor of Kenmotsu manifolds that admit NSNMC in the tangent bundle. We also study and discover that the lift of the Kenmotsu manifold that admit NSNMC is regular in the tangent bundle. Additionally, we find that the data provided by the lift of Ricci soliton on the lift of Ricci semi-symmetric Kenmotsu manifold that admits NSNMC in the tangent bundle are expanding.","PeriodicalId":48874,"journal":{"name":"Symmetry-Basel","volume":" 13","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135240767","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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