In this article we present the Lp, p≥1, approximation properties of activated singular integral operators over the real line. We establish their approximation to the unit operator with rates. The kernels here come from neural network activation functions and we employ the related density functions. The derived inequalities use the high order Lp modulus of smoothness.
{"title":"Degree of Lp Approximation Using Activated Singular Integrals","authors":"George A. Anastassiou","doi":"10.3390/sym16081022","DOIUrl":"https://doi.org/10.3390/sym16081022","url":null,"abstract":"In this article we present the Lp, p≥1, approximation properties of activated singular integral operators over the real line. We establish their approximation to the unit operator with rates. The kernels here come from neural network activation functions and we employ the related density functions. The derived inequalities use the high order Lp modulus of smoothness.","PeriodicalId":501198,"journal":{"name":"Symmetry","volume":"4 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141947550","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we investigate evolution systems in two components, characterized by higher-order spatial derivatives and the presence of two arbitrary functions. Our study begins with an analysis of a fourth-order system. We perform a detailed group classification and identify specific forms of the constitutive functions that allow the system to exhibit additional symmetries in addition to spatial and temporal translations. We extend these results to nth-order systems. Moreover, we derive invariant solutions for these systems. Finally, for each order n, we are able to find non-negative solutions.
在本文中，我们研究了由两个部分组成的进化系统，其特点是具有高阶空间导数和存在两个任意函数。我们的研究从分析一个四阶系统开始。我们进行了详细的分组分类，并确定了构成函数的特定形式，使系统除了空间和时间平移外，还表现出额外的对称性。我们将这些结果扩展到 n 阶系统。此外，我们还得出了这些系统的不变解。最后，对于每个 n 阶，我们都能找到非负解。
{"title":"Symmetries and Invariant Solutions of Higher-Order Evolution Systems","authors":"Rita Tracinà","doi":"10.3390/sym16081023","DOIUrl":"https://doi.org/10.3390/sym16081023","url":null,"abstract":"In this paper, we investigate evolution systems in two components, characterized by higher-order spatial derivatives and the presence of two arbitrary functions. Our study begins with an analysis of a fourth-order system. We perform a detailed group classification and identify specific forms of the constitutive functions that allow the system to exhibit additional symmetries in addition to spatial and temporal translations. We extend these results to nth-order systems. Moreover, we derive invariant solutions for these systems. Finally, for each order n, we are able to find non-negative solutions.","PeriodicalId":501198,"journal":{"name":"Symmetry","volume":"21 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141947456","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this study, a high-gain broadband circularly polarized crossed dipole antenna is designed. This antenna utilizes two pairs of cross dipoles and a pair of phase delay lines to form circularly polarized radiation. Open-circuit stubs are symmetrically loaded on the four arms of these dipole pairs to introduce new circularly polarized resonating frequencies. Additionally, a symmetrically positioned rectangular ring patch is introduced directly beneath the cross dipoles to generate the third circularly polarized resonating frequency, thereby enhancing the axial ratio bandwidth of the antenna symmetrically. Furthermore, metal posts are symmetrically loaded at the four right angles of the rectangular ring patch to augment the antenna gain, maintaining the overall symmetrical balance crucial for optimal circularly polarized radiation performance. This symmetric design ensures that the antenna achieves a 3dB axial ratio bandwidth of 29.2% (1.9–2.55 GHz) and sustains a gain of 7.5 dB within the passband, showcasing excellent circularly polarized radiation attributes.
本研究设计了一种高增益宽带圆极化交叉偶极子天线。该天线利用两对交叉偶极子和一对相位延迟线形成圆极化辐射。在这对偶极子的四个臂上对称加载开路存根，以引入新的圆极化谐振频率。此外，还在交叉偶极子的正下方引入了一个对称定位的矩形环形贴片，以产生第三个圆极化谐振频率，从而对称地增强了天线的轴向比带宽。此外，在矩形环形贴片的四个直角处对称加载金属柱，以提高天线增益，从而保持对最佳圆极化辐射性能至关重要的整体对称平衡。这种对称设计确保该天线实现 29.2% 的 3dB 轴向比带宽（1.9-2.55 GHz），并在通带内保持 7.5 dB 的增益，展现出卓越的圆极化辐射特性。
{"title":"The Aesthetics and Pragmatics of Symmetry in High-Gain and Wideband Circularly Polarized Antenna Design","authors":"Chunping Liao, Wenyong Liu, Xianjing Lin","doi":"10.3390/sym16081016","DOIUrl":"https://doi.org/10.3390/sym16081016","url":null,"abstract":"In this study, a high-gain broadband circularly polarized crossed dipole antenna is designed. This antenna utilizes two pairs of cross dipoles and a pair of phase delay lines to form circularly polarized radiation. Open-circuit stubs are symmetrically loaded on the four arms of these dipole pairs to introduce new circularly polarized resonating frequencies. Additionally, a symmetrically positioned rectangular ring patch is introduced directly beneath the cross dipoles to generate the third circularly polarized resonating frequency, thereby enhancing the axial ratio bandwidth of the antenna symmetrically. Furthermore, metal posts are symmetrically loaded at the four right angles of the rectangular ring patch to augment the antenna gain, maintaining the overall symmetrical balance crucial for optimal circularly polarized radiation performance. This symmetric design ensures that the antenna achieves a 3dB axial ratio bandwidth of 29.2% (1.9–2.55 GHz) and sustains a gain of 7.5 dB within the passband, showcasing excellent circularly polarized radiation attributes.","PeriodicalId":501198,"journal":{"name":"Symmetry","volume":"43 17","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141924499","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Yi-Wei Jiang, Wei-Han Tan, Hua-Xing Chen, Er-Liang Cui
We study the strong decays of the ϕ(2170), along with its possible partner X(2436), as two fully strange tetraquark states of JPC=1−−. These two states are assumed to contain two strange quarks and two anti-strange quarks, with the flavor symmetry 6ss⊗6¯s¯s¯. We consider seven decay channels: ϕη, ϕη′, ϕf0(980), ϕf1(1420), h1(1415)η, h1(1415)η′, and h1(1415)f1(1420). Some of these channels are kinematically possible, and we calculate their relative branching ratios through the Fierz rearrangement. Future experimental measurements on these ratios could be useful in determining the nature of the ϕ(2170) and X(2436). The ϕ(2170) has been observed in the ϕf0(980), ϕη, and ϕη′ channels, and we propose to further examine it in the h1(1415)η channel. Evidences of the X(2436) have been observed in the ϕf0(980) channel, and we propose to verify whether this structure exists or not in the ϕη, ϕη′, h1(1415)η, and h1(1415)η′ channels.
{"title":"Strong Decays of the ϕ(2170) as a Fully Strange Tetraquark State","authors":"Yi-Wei Jiang, Wei-Han Tan, Hua-Xing Chen, Er-Liang Cui","doi":"10.3390/sym16081021","DOIUrl":"https://doi.org/10.3390/sym16081021","url":null,"abstract":"We study the strong decays of the ϕ(2170), along with its possible partner X(2436), as two fully strange tetraquark states of JPC=1−−. These two states are assumed to contain two strange quarks and two anti-strange quarks, with the flavor symmetry 6ss⊗6¯s¯s¯. We consider seven decay channels: ϕη, ϕη′, ϕf0(980), ϕf1(1420), h1(1415)η, h1(1415)η′, and h1(1415)f1(1420). Some of these channels are kinematically possible, and we calculate their relative branching ratios through the Fierz rearrangement. Future experimental measurements on these ratios could be useful in determining the nature of the ϕ(2170) and X(2436). The ϕ(2170) has been observed in the ϕf0(980), ϕη, and ϕη′ channels, and we propose to further examine it in the h1(1415)η channel. Evidences of the X(2436) have been observed in the ϕf0(980) channel, and we propose to verify whether this structure exists or not in the ϕη, ϕη′, h1(1415)η, and h1(1415)η′ channels.","PeriodicalId":501198,"journal":{"name":"Symmetry","volume":"90 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141947561","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Zuhur Alqahtani, B. Qaraad, A. Almuneef, Faizah Alharbi
This paper focuses on studying certain oscillatory properties of a new class of half-linear second-order differential equations with an advanced argument in a non-canonical case. By employing some new relations between the solution and its higher derivatives and taking into account the symmetry of positive and negative solutions, we have introduced new criteria to test whether all solutions of the studied equation exhibit oscillatory behavior. Our study aims to expand and enhance previous results, helping to understand these properties in the specified context. The results obtained are confirmed and clarified through an example involving Euler-type equations.
{"title":"Oscillatory Properties of Second-Order Differential Equations with Advanced Arguments in the Noncanonical Case","authors":"Zuhur Alqahtani, B. Qaraad, A. Almuneef, Faizah Alharbi","doi":"10.3390/sym16081018","DOIUrl":"https://doi.org/10.3390/sym16081018","url":null,"abstract":"This paper focuses on studying certain oscillatory properties of a new class of half-linear second-order differential equations with an advanced argument in a non-canonical case. By employing some new relations between the solution and its higher derivatives and taking into account the symmetry of positive and negative solutions, we have introduced new criteria to test whether all solutions of the studied equation exhibit oscillatory behavior. Our study aims to expand and enhance previous results, helping to understand these properties in the specified context. The results obtained are confirmed and clarified through an example involving Euler-type equations.","PeriodicalId":501198,"journal":{"name":"Symmetry","volume":"30 45","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141924720","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper is devoted to new propagation and regularity results for a class of first-order Hamilton–Jacobi–Bellman-type problems in a separable infinite-dimensional Hilbert space. Specifically, the related Cauchy problem is investigated by employing the Faedo–Galerkin approximation method. Under some structural assumptions, the main result is obtained by using the probabilistic representation formula of the solution in order to define the weak continuity assumptions, by assuming the existence of a symmetric positive definite Hilbert–Schmidt operator and by employing modulus continuity arguments.
{"title":"Regularization and Propagation in a Hamilton–Jacobi–Bellman-Type Equation in Infinite-Dimensional Hilbert Space","authors":"Carlo Bianca, C. Dogbé","doi":"10.3390/sym16081017","DOIUrl":"https://doi.org/10.3390/sym16081017","url":null,"abstract":"This paper is devoted to new propagation and regularity results for a class of first-order Hamilton–Jacobi–Bellman-type problems in a separable infinite-dimensional Hilbert space. Specifically, the related Cauchy problem is investigated by employing the Faedo–Galerkin approximation method. Under some structural assumptions, the main result is obtained by using the probabilistic representation formula of the solution in order to define the weak continuity assumptions, by assuming the existence of a symmetric positive definite Hilbert–Schmidt operator and by employing modulus continuity arguments.","PeriodicalId":501198,"journal":{"name":"Symmetry","volume":"89 3","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141921838","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The whimsical Las Vegas/Monte Carlo cubic dice are generalized to construct the combinatorial problem of enumerating all n-dimensional hypercube dice and dice of other shapes that exhibit cubic, icosahedral, and higher symmetries. By utilizing powerful generating function techniques for various irreducible representations, we derive the combinatorial enumerations of all possible dice in n-dimensional space with hyperoctahedral symmetries. Likewise, a number of shapes that exhibit icosahedral symmetries such as a truncated dodecahedron and a truncated icosahedron are considered for the combinatorial problem of dice enumerations with the corresponding shapes. We consider several dice with cubic symmetries such as the truncated octahedron, dodecahedron, and Rubik’s cube shapes. It is shown that all enumerated dice are chiral, and we provide the counts of chiral pairs of dice in the n-dimensional space. During the combinatorial enumeration, it was discovered that two different shapes of dice exist with the same chiral pair count culminating to the novel concept of isochiral polyhedra. The combinatorial problem of dice enumeration is generalized to multi-coloring partitions. Applications to chirality in n-dimension, molecular clusters, zeolites, mesoporous materials, cryptography, and biology are also pointed out. Applications to the nonlinear n-dimensional hypercube and other dicey encryptions are exemplified with romantic, clandestine messages: “I love U” and “V Elope at 2”.
异想天开的拉斯维加斯/蒙特卡罗立方体骰子被概括为构建一个组合问题，即枚举所有 n 维超立方体骰子和其他形状的骰子，这些骰子表现出立方体、二十面体和更高的对称性。通过利用各种不可还原表示的强大生成函数技术，我们得出了 n 维空间中所有可能的超八面体对称骰子的组合枚举。同样，我们还考虑了一些具有二十面体对称性的形状，如截断十二面体和截断二十面体，以解决具有相应形状的骰子的组合枚举问题。我们考虑了几种具有立方对称性的骰子，如截顶八面体、十二面体和魔方。结果表明，所有枚举出的骰子都是手性的，我们还提供了 n 维空间中手性骰子对的计数。在组合枚举的过程中，我们发现两种不同形状的骰子具有相同的手性对数，从而提出了等手性多面体的新概念。骰子枚举的组合问题被推广到多色分区。还指出了 n 维手性、分子团簇、沸石、介孔材料、密码学和生物学的应用。非线性 n 维超立方和其他棘手加密的应用以浪漫的秘密信息为例："我爱 U "和 "V 在 2 点私奔"。
{"title":"Enumeration of n-Dimensional Hypercubes, Icosahedra, Rubik’s Cube Dice, Colorings, Chirality, and Encryptions Based on Their Symmetries","authors":"K. Balasubramanian","doi":"10.3390/sym16081020","DOIUrl":"https://doi.org/10.3390/sym16081020","url":null,"abstract":"The whimsical Las Vegas/Monte Carlo cubic dice are generalized to construct the combinatorial problem of enumerating all n-dimensional hypercube dice and dice of other shapes that exhibit cubic, icosahedral, and higher symmetries. By utilizing powerful generating function techniques for various irreducible representations, we derive the combinatorial enumerations of all possible dice in n-dimensional space with hyperoctahedral symmetries. Likewise, a number of shapes that exhibit icosahedral symmetries such as a truncated dodecahedron and a truncated icosahedron are considered for the combinatorial problem of dice enumerations with the corresponding shapes. We consider several dice with cubic symmetries such as the truncated octahedron, dodecahedron, and Rubik’s cube shapes. It is shown that all enumerated dice are chiral, and we provide the counts of chiral pairs of dice in the n-dimensional space. During the combinatorial enumeration, it was discovered that two different shapes of dice exist with the same chiral pair count culminating to the novel concept of isochiral polyhedra. The combinatorial problem of dice enumeration is generalized to multi-coloring partitions. Applications to chirality in n-dimension, molecular clusters, zeolites, mesoporous materials, cryptography, and biology are also pointed out. Applications to the nonlinear n-dimensional hypercube and other dicey encryptions are exemplified with romantic, clandestine messages: “I love U” and “V Elope at 2”.","PeriodicalId":501198,"journal":{"name":"Symmetry","volume":"14 10","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141924999","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The mapping relation between the emission angle of the photoelectron and its ionization time (i.e., the angle–time mapping) is important for the attoclock measurement. For a long time, the angle–time mapping was assumed to be angularly uniform. Recent investigations have demonstrated that the angle–time mapping is discontinuous for the low-energy electron at the angle for the minimum yield. However, the previous results were interpreted based on the assumption of s-electron initial states for noble-gas atoms, and the effect of the initial orbital symmetry on the angle–time mapping has been rarely investigated. In this work, we investigate the influence of the initial orbital symmetry on time–energy distribution using F− ions as a specific example. We demonstrate that the initial orbital symmetry significantly impacts the time–energy distribution. This behavior can be well explained by the saddle-point method. More interestingly, it is found that the angle–time mapping is strongly dependent on the initial orbital symmetry in the elliptically polarized laser field, especially for the low-energy electrons. Our work holds great significance for further developing the attoclock scheme.
光电子的发射角与其电离时间之间的映射关系（即角-时间映射关系）对attoclock 测量非常重要。长期以来，角度-时间映射被假定为角度均匀的。最近的研究表明，低能电子在最小产率角上的角时映射是不连续的。然而，以前的研究结果是基于惰性气体原子的 s 电子初始态假设来解释的，很少有人研究初始轨道对称性对角时映射的影响。在这项研究中，我们以 F- 离子为例，研究了初始轨道对称性对时间能量分布的影响。我们证明了初始轨道对称性对时间能量分布的重大影响。鞍点法可以很好地解释这种行为。更有趣的是，我们发现在椭圆偏振激光场中，角度-时间映射与初始轨道对称性密切相关，尤其是对低能电子而言。我们的工作对进一步发展attoclock方案具有重要意义。
{"title":"Effect of Orbital Symmetry on Time–Energy Distributions of F− Ions in the Attoclock Scheme","authors":"Jian-Hong Chen, Song-Feng Zhao","doi":"10.3390/sym16081019","DOIUrl":"https://doi.org/10.3390/sym16081019","url":null,"abstract":"The mapping relation between the emission angle of the photoelectron and its ionization time (i.e., the angle–time mapping) is important for the attoclock measurement. For a long time, the angle–time mapping was assumed to be angularly uniform. Recent investigations have demonstrated that the angle–time mapping is discontinuous for the low-energy electron at the angle for the minimum yield. However, the previous results were interpreted based on the assumption of s-electron initial states for noble-gas atoms, and the effect of the initial orbital symmetry on the angle–time mapping has been rarely investigated. In this work, we investigate the influence of the initial orbital symmetry on time–energy distribution using F− ions as a specific example. We demonstrate that the initial orbital symmetry significantly impacts the time–energy distribution. This behavior can be well explained by the saddle-point method. More interestingly, it is found that the angle–time mapping is strongly dependent on the initial orbital symmetry in the elliptically polarized laser field, especially for the low-energy electrons. Our work holds great significance for further developing the attoclock scheme.","PeriodicalId":501198,"journal":{"name":"Symmetry","volume":"139 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141947560","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper proposes the complex dynamics of collective behavior through an interdisciplinary approach that integrates individual cognition with potential fields. Firstly, the interaction between individual cognition and external potential fields in complex social systems is explored, integrating perspectives from physics, cognitive psychology, and social science. Subsequently, a new modeling method for the multidimensional potential field mechanism is proposed, aiming to reduce individual behavioral errors and cognitive dissonance, thereby improving system efficiency and accuracy. The approach uses cooperative control and distributed learning algorithms to simulate collective behavior, allowing individuals to iteratively adapt based on local information and collective intelligence. Simulations highlight the impact of factors such as individual density, noise intensity, communication radius, and negative potential fields on collective dynamics. For instance, in a high-density environment with 180 individuals, increased social friction and competition for resources significantly decrease collective search efficiency. Validation is achieved by comparing simulation results with existing research, showing consistency and improvements over traditional models. In noisy environments, simulations maintain higher accuracy and group cohesion compared to standard methods. Additionally, without communication, the Mean Squared Error (MSE) initially drops rapidly as individuals adapt but stabilizes over time, emphasizing the importance of communication in maintaining collective efficiency. The study concludes that collective behavior emerges from complex nonlinear interactions between individual cognition and potential fields, rather than being merely the sum of individual actions. These insights enhance the understanding of complex system dynamics, providing a foundation for future applications in adaptive urban environments and the design of autonomous robots and AI systems.
{"title":"Utilizing Potential Field Mechanisms and Distributed Learning to Discover Collective Behavior on Complex Social Systems","authors":"Junqiao Zhang, Qiang Qu, X. Chen","doi":"10.3390/sym16081014","DOIUrl":"https://doi.org/10.3390/sym16081014","url":null,"abstract":"This paper proposes the complex dynamics of collective behavior through an interdisciplinary approach that integrates individual cognition with potential fields. Firstly, the interaction between individual cognition and external potential fields in complex social systems is explored, integrating perspectives from physics, cognitive psychology, and social science. Subsequently, a new modeling method for the multidimensional potential field mechanism is proposed, aiming to reduce individual behavioral errors and cognitive dissonance, thereby improving system efficiency and accuracy. The approach uses cooperative control and distributed learning algorithms to simulate collective behavior, allowing individuals to iteratively adapt based on local information and collective intelligence. Simulations highlight the impact of factors such as individual density, noise intensity, communication radius, and negative potential fields on collective dynamics. For instance, in a high-density environment with 180 individuals, increased social friction and competition for resources significantly decrease collective search efficiency. Validation is achieved by comparing simulation results with existing research, showing consistency and improvements over traditional models. In noisy environments, simulations maintain higher accuracy and group cohesion compared to standard methods. Additionally, without communication, the Mean Squared Error (MSE) initially drops rapidly as individuals adapt but stabilizes over time, emphasizing the importance of communication in maintaining collective efficiency. The study concludes that collective behavior emerges from complex nonlinear interactions between individual cognition and potential fields, rather than being merely the sum of individual actions. These insights enhance the understanding of complex system dynamics, providing a foundation for future applications in adaptive urban environments and the design of autonomous robots and AI systems.","PeriodicalId":501198,"journal":{"name":"Symmetry","volume":"14 5","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141926306","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The effects of the helical trajectories of the perturbing electrons in magnetized plasmas on the dynamical Stark width of hydrogen or deuterium spectral lines have been studied analytically in our previous two papers—specifically in the situation where the magnetic field B is so strong that the dynamical Stark width of these lines reduces to the so-called adiabatic Stark width because the so-called nonadiabatic Stark width is completely suppressed. This situation corresponds, for example, to DA and DBA white dwarfs. We obtained those analytical results by using the formalism of the so-called conventional (or standard) theory of the impact Stark broadening: namely, by performing calculations in the second order of the Dyson perturbation expansion. The primary outcome was that the dynamical Stark broadening was found to not depend on the magnetic field B (for sufficiently strong B). In the present paper, we use the O4 symmetry of hydrogen atoms for performing the corresponding non-perturbative analytical calculations equivalent to accounting for all orders of the Dyson perturbation expansion. The results, obtained by using the O4 symmetry of hydrogen atoms, differ from our previous ones not only quantitatively, but—most importantly—qualitatively. Namely, the dynamical Stark broadening does depend on the magnetic field B, even for strong B. These results should be important for revising the interpretation of the hydrogen Balmer lines observed in DA and DBA white dwarfs. We also address confusion in the literature on this subject.
我们在前两篇论文中分析研究了磁化等离子体中扰动电子的螺旋轨迹对氢或氘光谱线动态斯塔克宽度的影响--特别是在磁场 B 非常强的情况下，由于所谓的非绝热斯塔克宽度被完全抑制，这些光谱线的动态斯塔克宽度会减小到所谓的绝热斯塔克宽度。例如，DA 和 DBA 白矮星就属于这种情况。我们是通过使用所谓的传统（或标准）撞击斯塔克展宽理论的形式来获得这些分析结果的：即通过戴森微扰展开的二阶计算。主要结果是发现动态斯塔克展宽与磁场 B 无关（对于足够强的磁场 B）。在本文中，我们利用氢原子的 O4 对称性进行了相应的非微扰分析计算，相当于考虑了戴森微扰展开的所有阶次。利用氢原子的 O4 对称性得到的结果不仅在数量上，而且最重要的是在质量上与我们以前的结果不同。也就是说，动态斯塔克展宽确实取决于磁场 B，甚至对于强磁场 B 也是如此。这些结果对于修正在 DA 和 DBA 白矮星中观测到的氢巴尔默线的解释非常重要。我们还讨论了文献中关于这个问题的混淆之处。
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