The relation between the holonomy along a loop with the curvature form is a well-known fact, where the small square loop approximation of aholonomy Hγ,O is proportional to Rσ. In an attempt to generalize the relation for arbitrary loops, we encounter the following ambiguity. For a given loop γ embedded in a manifold M, Hγ,O is an element of a Lie group G; the curvature Rσ∈g is an element of the Lie algebra of G. However, it turns out that the curvature form Rσ obtained from the small loop approximation is ambiguous, as the information of γ and Hγ,O are insufficient for determining a specific plane σ responsible for Rσ. To resolve this ambiguity, it is necessary to specify the surface S enclosed by the loop γ; hence, σ is defined as the limit of S when γ shrinks to a point. In this article, we try to understand this problem more clearly. As a result, we obtain an exact relation between the holonomy along a loop with the integral of the curvature form over a surface that it encloses. The derivation of this result can be viewed as an alternative proof of the non-Abelian Stokes theorem in two dimensions with some generalizations.
{"title":"Alternative Derivation of the Non-Abelian Stokes Theorem in Two Dimensions","authors":"Seramika Ariwahjoedi, Freddy Permana Zen","doi":"10.3390/sym15112000","DOIUrl":"https://doi.org/10.3390/sym15112000","url":null,"abstract":"The relation between the holonomy along a loop with the curvature form is a well-known fact, where the small square loop approximation of aholonomy Hγ,O is proportional to Rσ. In an attempt to generalize the relation for arbitrary loops, we encounter the following ambiguity. For a given loop γ embedded in a manifold M, Hγ,O is an element of a Lie group G; the curvature Rσ∈g is an element of the Lie algebra of G. However, it turns out that the curvature form Rσ obtained from the small loop approximation is ambiguous, as the information of γ and Hγ,O are insufficient for determining a specific plane σ responsible for Rσ. To resolve this ambiguity, it is necessary to specify the surface S enclosed by the loop γ; hence, σ is defined as the limit of S when γ shrinks to a point. In this article, we try to understand this problem more clearly. As a result, we obtain an exact relation between the holonomy along a loop with the integral of the curvature form over a surface that it encloses. The derivation of this result can be viewed as an alternative proof of the non-Abelian Stokes theorem in two dimensions with some generalizations.","PeriodicalId":48874,"journal":{"name":"Symmetry-Basel","volume":"120 ","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136103530","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}
In this paper, we study the Oresme hybrationals that generalize Oresme hybrid numbers and Oresme rational functions. We give a reccurence relation and a generating function for Oresme hybrationals. Moreover, we give some of their properties, among others, Binet formulas and general bilinear index-reduction formulas, through which we can obtain Catalan-, Cassini-, Vajda-, and d’Ocagne-type identities.
{"title":"On Some Combinatorial Properties of Oresme Hybrationals","authors":"Iwona Włoch, Natalia Paja, Anetta Szynal-Liana","doi":"10.3390/sym15111996","DOIUrl":"https://doi.org/10.3390/sym15111996","url":null,"abstract":"In this paper, we study the Oresme hybrationals that generalize Oresme hybrid numbers and Oresme rational functions. We give a reccurence relation and a generating function for Oresme hybrationals. Moreover, we give some of their properties, among others, Binet formulas and general bilinear index-reduction formulas, through which we can obtain Catalan-, Cassini-, Vajda-, and d’Ocagne-type identities.","PeriodicalId":48874,"journal":{"name":"Symmetry-Basel","volume":"26 7","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136134606","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}
A self-organized geometric model is proposed for data dimension reduction to improve the robustness of manifold learning. In the model, a novel mechanism for dimension reduction is presented by the autonomous deforming of data manifolds. The autonomous deforming vector field is proposed to guide the deformation of the data manifold. The flattening of the data manifold is achieved as an emergent behavior under the virtual elastic and repulsive interaction between the data points. The manifold’s topological structure is preserved when it evolves to the shape of lower dimension. The soft neighborhood is proposed to overcome the uneven sampling and neighbor point misjudging problems. The simulation experiment results of data sets prove its effectiveness and also indicate that implicit features of data sets can be revealed. In the comparison experiments, the proposed method shows its advantage in robustness.
{"title":"Learning by Autonomous Manifold Deformation with an Intrinsic Deforming Field","authors":"Xiaodong Zhuang, Nikos Mastorakis","doi":"10.3390/sym15111995","DOIUrl":"https://doi.org/10.3390/sym15111995","url":null,"abstract":"A self-organized geometric model is proposed for data dimension reduction to improve the robustness of manifold learning. In the model, a novel mechanism for dimension reduction is presented by the autonomous deforming of data manifolds. The autonomous deforming vector field is proposed to guide the deformation of the data manifold. The flattening of the data manifold is achieved as an emergent behavior under the virtual elastic and repulsive interaction between the data points. The manifold’s topological structure is preserved when it evolves to the shape of lower dimension. The soft neighborhood is proposed to overcome the uneven sampling and neighbor point misjudging problems. The simulation experiment results of data sets prove its effectiveness and also indicate that implicit features of data sets can be revealed. In the comparison experiments, the proposed method shows its advantage in robustness.","PeriodicalId":48874,"journal":{"name":"Symmetry-Basel","volume":"30 5","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136134607","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}
Fahd Masood, Clemente Cesarano, Osama Moaaz, Sameh S. Askar, Ahmad M. Alshamrani, Hamdy El-Metwally
This paper delves into the analysis of oscillation characteristics within third-order quasilinear delay equations, focusing on the canonical case. Novel sufficient conditions are introduced, aimed at discerning the nature of solutions—whether they exhibit oscillatory behavior or converge to zero. By expanding the literature, this study enriches the existing knowledge landscape within this field. One of the foundations on which we rely in proving the results is the symmetry between the positive and negative solutions, so that we can, using this feature, obtain criteria that guarantee the oscillation of all solutions. The paper enhances comprehension through the provision of illustrative examples that effectively showcase the outcomes and implications of the established findings.
{"title":"Kneser-Type Oscillation Criteria for Half-Linear Delay Differential Equations of Third Order","authors":"Fahd Masood, Clemente Cesarano, Osama Moaaz, Sameh S. Askar, Ahmad M. Alshamrani, Hamdy El-Metwally","doi":"10.3390/sym15111994","DOIUrl":"https://doi.org/10.3390/sym15111994","url":null,"abstract":"This paper delves into the analysis of oscillation characteristics within third-order quasilinear delay equations, focusing on the canonical case. Novel sufficient conditions are introduced, aimed at discerning the nature of solutions—whether they exhibit oscillatory behavior or converge to zero. By expanding the literature, this study enriches the existing knowledge landscape within this field. One of the foundations on which we rely in proving the results is the symmetry between the positive and negative solutions, so that we can, using this feature, obtain criteria that guarantee the oscillation of all solutions. The paper enhances comprehension through the provision of illustrative examples that effectively showcase the outcomes and implications of the established findings.","PeriodicalId":48874,"journal":{"name":"Symmetry-Basel","volume":"68 5","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136135856","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 random output of renewable energy and the disorderly grid connection of electric vehicles (EV) will pose challenges to the safe and stable operation of the power system. In order to ensure the reliability and symmetry of the microgrid operation, this paper proposes a microgrid optimization scheduling strategy considering the access of EVs. Firstly, in order to reduce the impact of random access to EVs on power system operation, a schedulable model of an EV cluster is constructed based on the Minkowski sum. Then, based on the wavelet neural network (WNN), the renewable energy output is predicted to reduce the influence of its output fluctuation on the operation of the power system. Considering the operation constraints of each unit in the microgrid, the network active power loss and node voltage deviation are taken as the optimization objectives, and the established microgrid model is equivalently transformed via second-order cone relaxation to improve its solution efficiency. Based on network reconfiguration and flexible load participation in demand response, the economy and reliability of system operation are improved. Finally, the feasibility and effectiveness of the proposed method are verified based on the simulation examples.
{"title":"Research on Optimal Scheduling Strategy of Microgrid Considering Electric Vehicle Access","authors":"Zhimin Wu, Yang Zou, Feng Zheng, Ning Liang","doi":"10.3390/sym15111993","DOIUrl":"https://doi.org/10.3390/sym15111993","url":null,"abstract":"The random output of renewable energy and the disorderly grid connection of electric vehicles (EV) will pose challenges to the safe and stable operation of the power system. In order to ensure the reliability and symmetry of the microgrid operation, this paper proposes a microgrid optimization scheduling strategy considering the access of EVs. Firstly, in order to reduce the impact of random access to EVs on power system operation, a schedulable model of an EV cluster is constructed based on the Minkowski sum. Then, based on the wavelet neural network (WNN), the renewable energy output is predicted to reduce the influence of its output fluctuation on the operation of the power system. Considering the operation constraints of each unit in the microgrid, the network active power loss and node voltage deviation are taken as the optimization objectives, and the established microgrid model is equivalently transformed via second-order cone relaxation to improve its solution efficiency. Based on network reconfiguration and flexible load participation in demand response, the economy and reliability of system operation are improved. Finally, the feasibility and effectiveness of the proposed method are verified based on the simulation examples.","PeriodicalId":48874,"journal":{"name":"Symmetry-Basel","volume":"21 4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136232009","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}
Abdulrahman B. M. Alzahrani, Mohamed A. Abdoon, Mohamed Elbadri, Mohammed Berir, Diaa Eldin Elgezouli
This study aims to find a solution to the symmetry chaotic jerk system by using a new ABC-FD scheme and the NILM method. The findings of the supplied methods have been compared to Runge–Kutta’s fourth order (RK4). It was discovered that the suggested techniques gave results comparable to the RK4 method. Our primary goal is to develop effective methods for addressing symmetrical, chaotic systems. Using ABC-FD and NILM presents innovative approaches for comprehending and effectively handling intricate dynamics. The findings of this study have significant significance for addressing the occurrence of chaotic behavior in diverse scientific and engineering contexts. This research significantly contributes to fractional calculus and its various applications. The application of ABC-FD, which can identify chaotic behavior, makes our work stand out. This novel approach contributes to advancing research in nonlinear dynamics and fractional calculus. The present study not only offers a resolution to the problem of symmetric chaotic jerk systems but also presents a framework that may be applied to tackle analogous challenges in several domains. The techniques outlined in this paper facilitate the development and computational analysis of prospective fractional models, thereby contributing to the progress of scientific and engineering disciplines.
{"title":"A Comparative Numerical Study of the Symmetry Chaotic Jerk System with a Hyperbolic Sine Function via Two Different Methods","authors":"Abdulrahman B. M. Alzahrani, Mohamed A. Abdoon, Mohamed Elbadri, Mohammed Berir, Diaa Eldin Elgezouli","doi":"10.3390/sym15111991","DOIUrl":"https://doi.org/10.3390/sym15111991","url":null,"abstract":"This study aims to find a solution to the symmetry chaotic jerk system by using a new ABC-FD scheme and the NILM method. The findings of the supplied methods have been compared to Runge–Kutta’s fourth order (RK4). It was discovered that the suggested techniques gave results comparable to the RK4 method. Our primary goal is to develop effective methods for addressing symmetrical, chaotic systems. Using ABC-FD and NILM presents innovative approaches for comprehending and effectively handling intricate dynamics. The findings of this study have significant significance for addressing the occurrence of chaotic behavior in diverse scientific and engineering contexts. This research significantly contributes to fractional calculus and its various applications. The application of ABC-FD, which can identify chaotic behavior, makes our work stand out. This novel approach contributes to advancing research in nonlinear dynamics and fractional calculus. The present study not only offers a resolution to the problem of symmetric chaotic jerk systems but also presents a framework that may be applied to tackle analogous challenges in several domains. The techniques outlined in this paper facilitate the development and computational analysis of prospective fractional models, thereby contributing to the progress of scientific and engineering disciplines.","PeriodicalId":48874,"journal":{"name":"Symmetry-Basel","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136232138","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}
Consider G to be a finite group and p to be a prime divisor of the order |G| in the group G. The main aim of this paper is to prove that the outcome in a recent paper of A. Laradji is true in the case of a p-constrained group. We observe that the generalization of the concept of Navarro’s vertex for an irreducible character in a p-constrained group G is generally undefined. We illustrate this with a suitable example. Let ϕ∈Irr(G) have a positive height, and let there be an anchor group Aϕ. We prove that if the normalizer NG(Aϕ) is p-constrained, then Op´(NG(Aϕ))≠{1G}, where Op´(NG(Aϕ)) is the maximal normal p´ subgroup of NG(Aϕ). We use character theoretic methods. In particular, Clifford theory is the main tool used to accomplish the results.
{"title":"On Height-Zero Characters in p-Constrained Groups","authors":"Manal H. Algreagri, Ahmad M. Alghamdi","doi":"10.3390/sym15111990","DOIUrl":"https://doi.org/10.3390/sym15111990","url":null,"abstract":"Consider G to be a finite group and p to be a prime divisor of the order |G| in the group G. The main aim of this paper is to prove that the outcome in a recent paper of A. Laradji is true in the case of a p-constrained group. We observe that the generalization of the concept of Navarro’s vertex for an irreducible character in a p-constrained group G is generally undefined. We illustrate this with a suitable example. Let ϕ∈Irr(G) have a positive height, and let there be an anchor group Aϕ. We prove that if the normalizer NG(Aϕ) is p-constrained, then Op´(NG(Aϕ))≠{1G}, where Op´(NG(Aϕ)) is the maximal normal p´ subgroup of NG(Aϕ). We use character theoretic methods. In particular, Clifford theory is the main tool used to accomplish the results.","PeriodicalId":48874,"journal":{"name":"Symmetry-Basel","volume":"283 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136232258","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}
Fuzzy logic is becoming one of the most-influential fields of modern mathematics with applications that impact not only other sciences, but society in general. This newly found interest in fuzzy logic is in part due to the crucial role it plays in the development of artificial intelligence. As a result, new tools and practices for the development of the above-mentioned field are in high demand. This is one of the issues this paper was composed to address. To be more specific, a sizable part of fuzzy logic is the study of fuzzy connectives. However, the current method used to generalize them is restricted to the use of basic automorphisms, which hinders the creation of new fuzzy connectives. For this reason, in this paper, a new method of generalization is conceived of that aims to generalize the fuzzy connectives using polynomial automorphism functions instead. The creation of these automorphisms is achieved through numerical analysis, an endeavor that is supported with programming applications that, using mathematical modeling, validate and visualize the research. Furthermore, the automorphisms satisfy all the necessary criteria that have been established for use in the generalization process and, consequently, are used to successfully generalize fuzzy connectives. The result of the new generalization method is the creation of new usable and flexible fuzzy connectives, which is very promising for the future development of the field.
{"title":"Generation of Polynomial Automorphisms Appropriate for the Generalization of Fuzzy Connectives","authors":"Eleftherios Makariadis, Stefanos Makariadis, Avrilia Konguetsof, Basil Papadopoulos","doi":"10.3390/sym15111992","DOIUrl":"https://doi.org/10.3390/sym15111992","url":null,"abstract":"Fuzzy logic is becoming one of the most-influential fields of modern mathematics with applications that impact not only other sciences, but society in general. This newly found interest in fuzzy logic is in part due to the crucial role it plays in the development of artificial intelligence. As a result, new tools and practices for the development of the above-mentioned field are in high demand. This is one of the issues this paper was composed to address. To be more specific, a sizable part of fuzzy logic is the study of fuzzy connectives. However, the current method used to generalize them is restricted to the use of basic automorphisms, which hinders the creation of new fuzzy connectives. For this reason, in this paper, a new method of generalization is conceived of that aims to generalize the fuzzy connectives using polynomial automorphism functions instead. The creation of these automorphisms is achieved through numerical analysis, an endeavor that is supported with programming applications that, using mathematical modeling, validate and visualize the research. Furthermore, the automorphisms satisfy all the necessary criteria that have been established for use in the generalization process and, consequently, are used to successfully generalize fuzzy connectives. The result of the new generalization method is the creation of new usable and flexible fuzzy connectives, which is very promising for the future development of the field.","PeriodicalId":48874,"journal":{"name":"Symmetry-Basel","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136232694","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}
We prove the modular convexity of the mixed norm Lp(ℓ2) on the Sobolev space W1,p(Ω) in a domain Ω⊂Rn under the sole assumption that the exponent p(x) is bounded away from 1, i.e., we include the case supx∈Ωp(x)=∞. In particular, the mixed Sobolev norm is uniformly convex if 1
{"title":"Uniform Convexity in Variable Exponent Sobolev Spaces","authors":"Mostafa Bachar, Mohamed A. Khamsi, Osvaldo Méndez","doi":"10.3390/sym15111988","DOIUrl":"https://doi.org/10.3390/sym15111988","url":null,"abstract":"We prove the modular convexity of the mixed norm Lp(ℓ2) on the Sobolev space W1,p(Ω) in a domain Ω⊂Rn under the sole assumption that the exponent p(x) is bounded away from 1, i.e., we include the case supx∈Ωp(x)=∞. In particular, the mixed Sobolev norm is uniformly convex if 1<infx∈Ωp(x)≤supx∈Ωp(x)<∞ and W01,p(Ω) is uniformly convex.","PeriodicalId":48874,"journal":{"name":"Symmetry-Basel","volume":"281 4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136235563","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 main outcome of this work is the construction of a surface pencil with a similarity to Bertrand curves in Euclidean 3-space E3. Then, by exploiting the Serret–Frenet frame, we deduce the sufficient and necessary conditions for a surface pencil with Bertrand curves as joint curvature lines. Consequently, the expansion to the ruled surface pencil is also designed. As demonstrations of our essential findings, we illustrate some models to emphasize the process.
{"title":"A Surface Pencil with Bertrand Curves as Joint Curvature Lines in Euclidean Three-Space","authors":"Sahar H. Nazra, Rashad A. Abdel-Baky","doi":"10.3390/sym15111986","DOIUrl":"https://doi.org/10.3390/sym15111986","url":null,"abstract":"The main outcome of this work is the construction of a surface pencil with a similarity to Bertrand curves in Euclidean 3-space E3. Then, by exploiting the Serret–Frenet frame, we deduce the sufficient and necessary conditions for a surface pencil with Bertrand curves as joint curvature lines. Consequently, the expansion to the ruled surface pencil is also designed. As demonstrations of our essential findings, we illustrate some models to emphasize the process.","PeriodicalId":48874,"journal":{"name":"Symmetry-Basel","volume":"25 3","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136263070","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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