Symmetry Research Articles

On some extensions of \pi -regular rings

Some variations of \pi -regular and nil clean rings were recently introduced in the works of the first author: “A generalization of \pi -regular rings, Turkish J. Math. 43 (2), 702–711 (2019)”, “A symmetrization in \pi -regular rings, Trans. A. Razmadze Math. Inst. 174 (3), 271–275 (2020)”, “A symmetric generalization of \pi -regular rings, Ric. Mat. 73 (1), 179–190 (2024)”. In this paper, we examine the structure and relationships between these classes of rings. Specifically, we prove that (m, n)-regularly nil clean rings are left-right symmetric and also show that the inclusions (D- regularly nil clean) (regularly nil clean) ((m, n)-regularly nil clean) hold, as well as we answer Questions 1, 2 and 3 posed in the third of the above-listed works. Moreover, we also consider other similar questions concerning the symmetric properties of certain classes of rings. For example, it is proven that centrally Utumi rings are always strongly \pi -regular.

FLSAD: Defending Backdoor Attacks in Federated Learning via Self-Attention Distillation

Federated Learning (FL), as a distributed machine learning framework, can effectively learn symmetric and asymmetric patterns from large-scale participants. However, FL is susceptible to malicious backdoor attacks through attackers injecting triggers into the backdoored model, resulting in backdoor samples being misclassified as target classes. Due to the stealthy nature of backdoor attacks in FL, it is difficult for users to discover the symmetric and asymmetric backdoor properties. Currently, backdoor defense methods in FL cause model performance degradation while reducing backdoors. In addition, some methods will assume the existence of clean samples, which does not match the realistic scenarios. To address such issues, we propose FLSAD, an effective backdoor defense method in FL via self-attention distillation. FLSAD can recover the triggers using an entropy maximization estimator. Based on the recovered triggers, we leverage the self-attention distillation to eliminate the backdoor. Compared with the baseline backdoor defense methods, FLSAD can reduce the success rates of different state-of-the-art backdoor attacks to 2% on four real-world datasets through extensive evaluation.

Global existence of bounded smooth solutions for the compressible ideal MHD system with planar symmetry

Global existence of bounded smooth solutions for the compressible ideal MHD system with planar symmetry

Martínez–Kaabar Fractal–Fractional Laplace Transformation with Applications to Integral Equations

This paper addresses the extension of Martinez–Kaabar (MK) fractal–fractional calculus (for simplicity, in this research work, it is referred to as MK calculus) to the field of integral transformations, with applications to some solutions to integral equations. A new notion of Laplace transformation, named MK Laplace transformation, is proposed, which incorporates the MK α,γ-integral operator into classical Laplace transformation. Laplace transformation is very applicable in mathematical physics problems, especially symmetrical problems in physics, which are frequently seen in quantum mechanics. Symmetrical systems and properties can be helpful in applications of Laplace transformations, which can help in providing an effective computational tool for solving such problems. The main properties and results of this transformation are discussed. In addition, the MK Laplace transformation method is constructed and applied to the non-integer-order first- and second-kind Volterra integral equations, which exhibit a fractal effect. Finally, the MK Abel integral equation’s solution is also investigated via this technique.

An Old Babylonian Algorithm and Its Modern Applications

In this paper, an ancient Babylonian algorithm for calculating the square root of 2 is unveiled, and the potential link between this primitive technique and an ancient Chinese method is explored. The iteration process is a symmetrical property, whereby the approximate root converges to the exact one through harmonious interactions between two approximate roots. Subsequently, the algorithm is extended in an ingenious manner to solve algebraic equations. To demonstrate the effectiveness of the modified algorithm, a transcendental equation that arises in MEMS systems is considered. Furthermore, the established algorithm is adeptly adapted to handle differential equations and fractal-fractional differential equations. Two illustrative examples are presented for consideration: the first is a nonlinear first-order differential equation, and the second is the renowned Duffing equation. The results demonstrate that this age-old Babylonian approach offers a novel and highly effective method for addressing contemporary problems with remarkable ease, presenting a promising solution to a diverse range of modern challenges.

A New High-Order Fractional Parallel Iterative Scheme for Solving Nonlinear Equations

Solving fractional-order nonlinear equations is crucial in engineering, where precision and accuracy are essential. This study introduces a novel fractional parallel technique for solving nonlinear equations. To enhance convergence, we incorporate a simple root-finding method of order 3γ + 1 as a correction term in the parallel scheme. Theoretical analysis shows that the parallel scheme achieves a convergence order of 6γ + 3. Using a dynamical system approach, we identify optimal parameter values, and the symmetry in the dynamical planes for different fractional parameters demonstrates the method’s stability and consistency in handling nonlinear problems. These parameter values are applied to the parallel scheme, yielding highly consistent results. Several engineering problems are examined to assess the method’s efficiency, stability, and consistency compared to existing methods.

Bound states in the continuum in a dielectric metamaterial with symmetry breaking in two dimensions

Abstract Bound states in the continuum (BICs), since their ultra-high quality (Q)-factor to extremely enhance light matter interactions, have attracted extensive interest very recently. As a typical category, symmetry-protected BICs are predictive and easily manipulated by using structure’s symmetry. However, most of the studies focus only on the structures with symmetry breaking in one dimension, in which one BIC will emerge and exhibit an inverse square relationship to asymmetry parameter. The structures with symmetry breaking in two dimensions remain rather unexplored. We here propose a dielectric metamaterial made of a square lattice of disks with a small hole. As moving the hole away from the center, the in-plane inversion symmetry can be broken either in one dimension or in two dimensions. As usual, a symmetry-protected BIC dominated by magnetic dipole (MD) occurs in the first case. In the second case, symmetry-protected dual BICs arise, consisting of the usual MD-dominated BIC and a new electric dipole (ED)-dominated BIC that is in cross-polarization to incident wave. The new BIC possesses an even higher Q-factor, which can also be continuously tuned via the position of the hole. Besides the structural modulation, we show the polarization angle of incident wave will act as another degree of freedom for designing symmetry breaking in two dimensions, where the similar symmetry-protected dual BICs are observed as well. Our work provides an alternative scheme for engineering multiple BICs and improving Q-factor, which may pave the way for practical device applications.

The Pyramidalization of sp2 Centers in 3/2‐Systems Is a “Structural Breathing” Independent of Energy

AbstractIn 3/2‐systems, such as the acetate anion H3CαC′OO−, an sp3 center and an sp2 center are connected by a covalent bond. The interaction of threefold and twofold symmetry results in the pyramidalization of the sp2 center during rotation about bond Esp3−Esp2. Rotation angles ψ=0°, ±60°, ±120°, and ±180° account for conformations with a symmetry plane containing the planar sp2 center CαC′OO. However, in all conformations with rotation angles ψ≠0°, ±60°, ±120°, and ±180° this symmetry plane is lost and pyramidalization must occur with maxima at rotation angles ψ=±30°, ±90°, and ±150°, because the two sides of the sp2 center CαC′OO are different. Inevitably, this leads to a pyramidalization/rotation profile θ/ψ with three maxima, three minima, and six zero‐crossings. Thus, myriads of 3/2‐compounds pyramidalize their sp2 centers each moment to the order of about 2° in a coupled pyramidalization/rotation molecular motion independent of energy.

HARNESSING GROUP THEORY FOR SIMPLIFIED ELECTRIC CIRCUIT ANALYSIS

Electric circuit analysis is fundamental to advancing modern technology, yet traditional methods often become complex and inefficient when applied to large-scale or intricate networks. Group theory, a branch of abstract algebra renowned for its capacity to exploit symmetry, has shown promise as a solution to this challenge. By applying group theoretical principles to circuit analysis, particularly through symmetry operations and transformation groups, engineers can simplify complex systems, reducing computational demands and revealing deeper insights into circuit behavior. Through systematic symmetry-based simplifications, group theory can decrease the number of equations required by up to 40% in certain configurations, enhancing both analytical efficiency and practical implementation. However, challenges include the mathematical complexity inherent to group theory, the limitations in analyzing highly irregular or nonlinear circuits, and the need for compatible computational tools. Recommendations include the integration of group theoretical approaches into standard circuit analysis software, focused training for electrical engineers, and further research to address the application of these methods in emerging technologies such as quantum and neuromorphic computing. Therefore, this review aims to elucidate the practical potential of group theory in transforming electric circuit analysis, presenting a streamlined approach that balances theoretical rigor with engineering feasibility.

Fractional Viscoelastic Analysis of Transversely Isotropic Surrounding Rock Along Shallow Buried Elliptical Tunnel

ABSTRACTThis study introduces the fractional order Merchant model to analytically solve the stress and displacement fields of the transversely isotropic viscoelastic surrounding rock along shallow elliptical tunnels. First, the stress and displacement solutions of fractional order viscoelastic and transversely isotropic half planes under arbitrary loads are obtained using the Laplace transform and the elastic‐viscoelastic correspondence principle. Second, based on the above half plane solution and the solution of the deep buried elliptical tunnel problem, the Schwarz alternating method is introduced to obtain the analytical solution of the shallow buried elliptical tunnel. A MATLAB program is developed, and the accuracy of the theory and program in this study is verified by comparing it with the results of ABAQUS. Finally, the effects of transversely isotropic parameters, tunnel burial depth, and viscoelastic parameters on the stress and displacement of tunnel surrounding rock are analyzed.

New Results Regarding Positive Periodic Solutions of Generalized Leslie–Gower-Type Population Models

In this paper, we focus on the existence of positive periodic solutions of generalized Leslie–Gower-type population models. Using the topological degree, we provide sufficient conditions to demonstrate the existence of positive periodic solutions to the considered models. It is interesting that the positive periodic solutions in this paper are general positive functions, not e exponential functions, which generalizes and improves the existing results. We note that due to the symmetrical property of periodic solutions, the results of this paper provide a deeper understanding of the periodic behavior of biological populations. Two numerical examples show the effectiveness of our main results.

NEC violation in [formula omitted] gravity in the context of a non-canonical theory via modified Raychaudhuri equation

In this work, we develop the Raychaudhuri equation in f(R̄,T̄) gravity in the setting of a non-canonical theory, namely K-essence theory. We solve the modified Raychaudhuri equation for the additive form of f(R̄,T̄), which is f1(R̄)+f2(T̄). For this solution, we employ two different scale factors to give two types of f(R̄,T̄) solutions. The ongoing debate between Fisher et al. and Harko et al. in 2020 regarding the additive form of f(R̄,T̄) may provide a resolution within the modified f(R̄,T̄) gravity theory. By conducting a viability test and analyzing energy conditions, we have determined that in the first scenario, the null energy condition (NEC) is violated between two regions where the NEC is satisfied. Additionally, we have observed that this violation of the NEC exhibits a symmetric property during the phase transition. These observations indicate that bouncing events may occur as a result of the symmetrical violation of the NEC during the expansion of the universe. Moreover, this model indicates that resonant-type quantum tunneling may take place during the period when the NEC is violated. The findings of NEC violation through the power law of scale factor may have empirical relevance in contemporary observations. In the second scenario, our model indicates that the strong energy condition is violated, but the NEC and weak energy conditions are satisfied. The effective energy density decreases and is positive, while the effective pressure and equation of state parameters are negative. This suggests that the universe is expanding with acceleration and is dominated by dark energy.

P-numerical semigroups with p-symmetric properties, II

Recently, the concept of the [Formula: see text]-numerical semigroup with [Formula: see text]-symmetric properties has been introduced. When [Formula: see text], the classical numerical semigroup with symmetric properties is recovered. In this paper, we further study the [Formula: see text]-numerical semigroup with [Formula: see text]-almost symmetric properties. We also give [Formula: see text]-generalized formulas of Watanabe and Johnson, and introduce [Formula: see text]-Arf numerical semigroup and study its properties.

Matrix expressions of symmetric n-player games

The symmetric property in n-player games is explored in this paper. First of all, some algebraically verifiable conditions are provided respectively for the totally symmetric, weakly symmetric and anonymous games, which are based on the construction of permutation matrix, and some interesting properties are revealed. Then we investigate how network structures affect the symmetry in a networked game. Some sufficient conditions are proposed and several nice properties are preserved when traditional networked games are extended to generalized networked games, in which the resulting network structure is more complex than a classical network graph.

Intercalation-Induced Monolayer Behavior in Bulk NbSe2.

Superconductivity at the 2D limit is significant for advancing fundamental physics, leading to extensive research on monolayer two-dimensional materials. In particular, monolayer transition metal dichalcogenides such as NbSe2 exhibit Ising superconductivity due to broken in-plane inversion symmetry and strong spin-orbit coupling, which has garnered significant attention. In this letter, we adopted an organic cation intercalation technique to modulate the interlayer interaction of NbSe2 by expanding the interlayer distance, thereby making intercalated NbSe2 behave similarly to monolayer NbSe2. The interlayer distances of NbSe2 intercalated with THA+, CTA+, and TDA+ cations are almost double or triple that of pristine NbSe2. The superconducting transition temperature (Tc) of THA-NbSe2 is comparable to that of pristine NbSe2, while the charge density wave (CDW) transition temperature is higher. The Tc of intercalated NbSe2 decreases with a reduced hole concentration, and the enhanced CDW is ascribed to the dimensional reduction. Notably, the in-plane upper critical field of intercalated NbSe2 significantly exceeds the Pauli paramagnetic limit, which is similar to the Ising superconductivity observed in monolayer NbSe2. Our work demonstrates that the organic cations intercalated two-dimensional materials exhibit behavior similar to their monolayer counterparts, providing a convenient platform for exploring and modulating physical phenomena at the two-dimensional limit.

Dual-spherical-wave optical scanning holographic microscopy with resolution enhancement and an extended depth of field

Conventional optical scanning holographic microscopy (OSHM) is realized by using a plane wave and a spherical wave as the illumination. The resolution of the conventional OSHM is limited by the numerical aperture of the spherical wave and cannot exceed the Rayleigh limit. In this paper, the OSHM by using dual spherical waves as the illumination is studied. The model of a Gaussian wave, together with the constraints on the signal strength and the visibility, are considered in the analysis of the dual-spherical-wave (DS) OSHM. For an object located at the symmetry plane of the DS-OSHM, the resolution can slightly exceed the Rayleigh limit, and the depth of field (DoF) is extended significantly. For a transparent object, the noise in the DS-OSHM is less than that of the conventional OSHM, thanks to the highly divergent scanning beam. If the object is off the symmetry plane, the resolution is still enhanced, but the extension of the DoF is limited. In addition, a clear reconstructed image will be observed not only at the original object plane, but also at its mirror plane. To the best of our knowledge, this phenomenon is reported for the first time.

Unconventional pairing in Ising superconductors: application to monolayer NbSe2

Abstract The presence of a non-centrosymmetric crystal structure and in-plane mirror symmetry allows an Ising spin–orbit coupling to form in some two-dimensional materials. Examples include transition metal dichalcogenide superconductors like monolayer NbSe2, MoS2, TaS2, and PbTe2, where a nontrivial nature of the superconducting state is currently being explored. In this study, we develop a microscopic formalism for Ising superconductors that captures the superconducting instability arising from a momentum-dependent spin- and charge-fluctuation-mediated pairing interaction. We apply our pairing model to the electronic structure of monolayer NbSe2, where first-principles calculations reveal the presence of strong paramagnetic fluctuations. Our calculations provide a quantitative measure of the mixing between the even- and odd-parity superconducting states and its variation with Coulomb interaction. Further, numerical analysis in the presence of an external Zeeman field reveals the role of Ising spin–orbit coupling and mixing of odd-parity superconducting state in influencing the low-temperature enhancement of the critical magnetic field.

Quaternion and Biquaternion Representations of Proper and Improper Transformations in Non-Cartesian Reference Systems

Quaternion and biquaternion symmetry transformations have been applied to non-Cartesian reference systems of direct and reciprocal crystal lattices. The transformations performed directly in the sets of crystal reference axes simplify the calculations, eliminate the need for orthogonalization, permit the use of crystallographic vectors for defining the directions of rotations and perform the computations directly in the crystal coordinates. The applications of the general quaternion transformations are envisioned for physical, chemical, crystallographic and engineering applications. The general quaternion multiplication rules for any symmetry-unrestricted lattices have been derived for the triclinic crystallographic system and have been applied to the biquaternion representations of all point-group symmetry elements, including the crystallographic hexagonal system. Cayley multiplication matrices for point-groups, based on the biquaternion symbols of proper and improper symmetry elements, have been exemplified.

Phenomenological models of ferroelastics with a fully symmetric order parameter: classification by methods of equivariant catastrophe theory

Using the methods of the catastrophe theory, which take into account the symmetry of the thermodynamic system, phenomenological models of phase transitions in proper and pseudo-proper ferroelastics have been constructed for interacting order parameters, one of which is fully symmetric, invariant with respect to all symmetry transformations of the crystal. A classification of models based on the number of control parameters, which depend on external thermodynamic conditions, has been carried out. A phase diagram of one of the models has been constructed, and the theoretical temperature dependence of the heat capacity has been calculated. The comparison of theoretical and experimental data showed satisfactory qualitative agreement.

Tunable mechanical properties of the 3D anticircular-curve transversal-isotropic auxetic structure

This work studies a three-dimensional anticircular-curve transversal-isotropic auxetic structure (3D-ATAS) with tunable Poisson’s ratio (υ) and tunable Young’s modulus (E) on the basis of hexagonal symmetry in the transverse plane using the anti-deformation method by the design of inclined rods as circular-curve rods in the opposite direction of the deformation (under compressive loading). By using the energy method, expressions of υ and E of 3D-ATAS are acquired, and based on the periodic boundary conditions, υ and E of 3D-ATAS are parametrically researched through numerical simulation and uniaxial compression experiments. The effects of anticircular-curve rod thickness t, anticircular-curve rod width b, and anticircular-curve cross-section angle θ to υ and E of 3D-ATAS are investigated. The tunable ranges of υ and E of 3D-ATAS are predicted, and the wide range of E is attained. Compared with the 3D circular-curve transversal-isotropic auxetic structure, E of 3D-ATAS is significantly enhanced in different directions, and the maximum enhanced up to 10 times. In the same way, both υ and E of 3D-ATAS are transversal-isotropic.