Symmetry Research Articles

Physical consequences of Lindbladian invariance transformations

On its own, the invariance properties of Markovian master equations have mostly played a mathematical or computational role in the evaluation of quantum open system dynamics. Because all forms of the equation lead to the same time evolution for the state of the system, the fixation of a particular form has only gained physical meaning when correlated with additional information such as in the evolution of quantum trajectories or the study of decoherence-free subspaces. Here, we show that these symmetry transformations can be exploited, on their own, to optimize practical physical tasks. In particular, we present a general formulation showing how they can be used to change the measurable values of physical quantities regarding the exchange of energy and/or information with the environment. We also analyze examples of optimization in quantum thermodynamics and, finally, discuss practical implementations in terms of quantum trajectories. Published by the American Physical Society 2024

Rich Dynamics Caused by a Fractional Diffusion Operator in Nonchaotic Rulkov Maps

There are few works about Neimark–Sacker bifurcating analysis on discrete dynamical systems with linear diffusion and delayed coupling under periodic/Neumann-boundary conditions. In this paper, we build up the framework for Neimark–Sacker bifurcations caused by Turing instability on high-dimensional discrete-time dynamical systems with symmetrical property in the linearized system. The fractional diffusion operator in higher-dimensional discrete dynamical systems is introduced and regular/chaotic Turing patterns are discovered by the computation of the largest Lyapunov exponents.

Magnetically tunable bound states in the continuum with arbitrary polarization and intrinsic chirality

Bound states in the continuum (BICs), which are exotic localized eigenstates embedded in the continuum spectrum and exhibit topological polarization singularities in momentum space, have recently attracted great attention in both fundamental and applied physics. Here, based on a magneto-optical (MO) photonic crystal (PhC) slab placed in external magnetic fields with time-reversal symmetry (TRS) breaking, we theoretically propose magnetically tunable BICs with arbitrary polarization covering the entire Poincaré sphere and efficient off-Γ chiral emission of circularly polarized states (C point). More interestingly, by further breaking the in-plane inversion symmetry of the MO PhC slab to generate a pair of C points spawning from the eliminated BICs and tuning the external magnetic field strength to move one C point to the Γ point, an at-Γ intrinsic chiral BIC exhibits chiral characteristics on both sides of the PhC slab with near-unity circular dichroism exceeding 0.99 and a high-quality factor of 46,000 owing to the preserved out-of-plane mirror symmetry. Moreover, the chirality of the chiral BICs can be inverted by flipping the magnetic bias. Our work opens an unprecedented avenue to explore the unique topological photonics of BICs with broken TRS and promises multiple applications in chiral-optical effects, structured light, and tunable optical devices.

A simulation platform for slender, semiflexible, and inextensible fibers with Brownian hydrodynamics and steric repulsion

The last few years have witnessed an explosion of new numerical methods for filament hydrodynamics. Aside from their ubiquity in biology, physics, and engineering, filaments present unique challenges from an applied-mathematical point of view. Their slenderness, inextensibility, semiflexibility, and meso-scale nature all require numerical methods that can handle multiple lengthscales in the presence of constraints. Accounting for Brownian motion while keeping the dynamics in detailed balance and on the constraint is difficult, as is including a background solvent, which couples the dynamics of multiple filaments together in a suspension. In this paper, we present a simulation platform for deterministic and Brownian inextensible filament dynamics, which includes nonlocal fluid dynamics and steric repulsion. For nonlocal hydrodynamics, we define the mobility on a single filament using line integrals of Rotne–Prager–Yamakawa regularized singularities and numerically preserve the symmetric positive definite property by using a thicker regularization width for the nonlocal integrals than for the self-term. For steric repulsion, we introduce a soft local repulsive potential defined as a double integral over two filaments, then present a scheme to identify and evaluate the nonzero components of the integrand. Using a temporal integrator developed in previous work, we demonstrate that Langevin dynamics sample from the equilibrium distribution of free filament shapes and that the modeling error in using the thicker regularization is small. We conclude with two examples, sedimenting filaments and cross-linked fiber networks, in which nonlocal hydrodynamics does and does not generate long-range flow fields, respectively. In the latter case, we show that the effect of hydrodynamics can be accounted for through steric repulsion.

Constraints, Conserved Charges and Extended BRST Algebra for a 3D Field-Theoretic Example for Hodge Theory

We perform the constraint analysis of a three (2 + 1)-dimensional (3D) field-theoretic example for Hodge theory (i) at the classical level within the ambit of Lagrangian formulation, and (ii) at the quantum level within the framework of Becchi-Rouet-Stora-Tyutin (BRST) formalism. We derive the conserved charges corresponding to the six continuous symmetries of our present theory. These six continuous summery transformations are the nilpotent (anti-)BRST and (anti-)co-BRST symmetries, a unique bosonic symmetry and the ghost-scale symmetry. It turns out that the Noether conserved (anti-)BRST charges are found to be non-nilpotent even though they are derived from the off-shell nilpotent versions of the continuous and infinitesimal (anti-)BRST symmetry transformations. We obtain the nilpotent versions of the (anti-)BRST charges from the non-nilpotent Noether (anti-)BRST charges and discuss the physicality criteria w.r.t. the latter to demonstrate that the operator forms of the first-class constraints (of the classical gauge theory) annihilate the physical states at the quantum level. This observation is consistent with Dirac's quantization conditions for the systems that are endowed with the constraints. We lay emphasis on the existence of a single (anti-)BRST invariant Curci-Ferrari (CF) type restriction in our theory and derive it from various theoretical angles.

Bilateral symmetry-based augmentation method for improved tooth segmentation in panoramic X-rays

Panoramic X-rays are crucial in dental radiology, providing detailed images that are essential for diagnosing and planning treatment for various oral conditions. The advent of automated methods that learn from annotated data promises to significantly aid clinical experts in making accurate diagnoses. However, these methods often require large amounts of annotated data, making the generation of high-quality annotations for panoramic X-rays both challenging and time-consuming. This paper introduces a novel bilateral symmetry-based augmentation method specifically designed to enhance tooth segmentation in panoramic X-rays. By exploiting the inherent bilateral symmetry of these images, our proposed method systematically generates augmented data, leading to substantial improvements in the performance of tooth segmentation models. By increasing the training data size fourfold, our approach proportionately reduces the effort required to manually annotate extensive datasets. These findings highlight the potential of leveraging the symmetrical properties of medical images to enhance model performance and accuracy in dental radiology. The effectiveness of the proposed method is evaluated on three widely adopted deep learning models: U-Net, SE U-Net, and TransUNet. Significant improvements in segmentation accuracy are observed with the proposed augmentation method across all models. For example, the average Dice Similarity Coefficient (DSC) increases by over 8%, reaching 76.7% for TransUNet. Further, comparisons with existing augmentation methods, including rigid transform-based and elastic grid-based techniques, show that the proposed method consistently outperforms them with additional improvements up to 5% in terms of average DSC, with the exact improvement varying depending on the model and training dataset size. We have made the data augmentation codes and tools developed based on our method available at https://github.com/wathoresanket/bilateralsymmetrybasedaugmentation.

Efficient mode coupling/(de)multiplexing between a few-mode fiber and a silicon photonic chip

Mode-division multiplexing (MDM) has attracted much attention due to its ability to further increase the transmission capacity of optical interconnects. While further developments of MDM optical interconnects are hindered by the coupling of few-mode fibers (FMFs) and silicon photonic chips, a high-efficiency, broadband, and scalable multimode FMF-chip interface is still eagerly desired. To address this challenge, a novel scheme for efficient multimode coupling is proposed by introducing a silica planar lightwave circuit as an intermediate. The core idea is to couple and demultiplex higher-order modes by leveraging the superiorities of silica optical waveguides for manipulating LP modes, facilitated through tailoring the mode conversion related to different mode symmetric properties. The demultiplexed modes are consequently butt-coupled to the silicon photonic chip in single-mode manner, thus being available for fulfilling further data transmitting/receiving/routing directly. As a proof of concept, a six-channel FMF-chip coupler working with the LP01-x/y, LP11a-x/y, and LP11b-x/y modes is designed with low coupling losses of 0.77–1.39 dB and low intermode crosstalk of <−27.2 dB in a broad bandwidth (>150 nm). Minimum coupling losses of 1.36–2.48 dB are experimentally demonstrated. It is the first demonstration for the integrated multimode FMF-chip coupler enabling the simultaneous coupling of six mode-channels, to the best of our knowledge. We believe that this work has the great potential for developing energy-efficient and low-cost chip-to-chip MDM interconnections in the future.

Switchable planar chirality and spin texture in highly ordered ferroelectric hybrid perovskite domains

Chiral organic-inorganic hybrid perovskites offer a promising platform for developing non-linear chiro-optical applications and chiral-induced spin selectivity. Here, we show that achiral hybrid perovskites that have highly ordered ferroelectric domains with orthogonal polarization exhibit planar chirality, as manifested by second harmonic generation with strong circular dichroism. Interestingly, the handedness of the second harmonic generation circular dichroism response can be alternatingly switched between orthogonally polarized domains and domain walls. To correlate the origin of planar chirality in these domains, atom-resolved atomic force microscopy is used to reveal distinct arrangements of atomic rows in these ferroelectric crystals that indicate planar symmetry breaking. Remarkably, a large SHG anisotropy factor of 0.41 is measured at the domain wall. Additionally, spatially resolved two-photon photoluminescence excitation measurement provide evidence of larger Rashba spin-splitting in these chiral ferroelectric domains compared to domain-free areas. Our discoveries of domain-dependent planar chirality and spin texture hold great promise for advancement of domain-specific chiro-optical applications.

Molecular Engineering of Covalent Organic Frameworks for Photocatalytic Hydrogen Evolution from Water.

Covalent organic frameworks (COFs) have demonstrated significant potential as photocatalysts for efficiently generating hydrogen through photocatalytic water splitting. However, the design of COFs with distinct organic unit blocks at a molecular level profoundly influences their photocatalytic performance. In this study, we synthesized a series of β-ketoamine COFs through molecular engineering of nitrogen sites, including phenyl-structured TpBD, phenylpyridine-structured TpPpy, phenylpyrimidine-structured TpPpm, and bipyridine-structured TpBpy. Advanced characterization techniques reveal that TpPpm and TpBpy with more nitrogen sites exhibit superior efficiencies in electron transfer and charge separation compared to TpBD and TpPpy, thereby endowing them with enhanced photocatalytic performance for hydrogen evolution from water. As a result, the photocatalytic hydrogen production rates of TpPpm (33.80 mmol g-1 h-1) and TpBpy (29.18 mmol g-1 h-1) surpass those of TpBD (20.82 mmol g-1 h-1) and TpPpy (27.49 mmol g-1 h-1). Additionally, due to the different plane symmetries between Ppm and Bpy resulting from the various positions of nitrogen sites, TpPpm displays superior photochemical properties and better photocatalytic performance compared to TpBpy. Moreover, theoretical calculation results further confirm the exceptional intramolecular charge transfer ability of TpPpm among all COFs. This work underscores the significance of precisely controlling N sites in COFs for designing high-performance photocatalysts.

Travelling Solitary Wave Solutions to Non-Gaussian χ-Wick-Type Stochastic Burgers’ Equation with Variable Coefficients

In this work, we obtain non-Gaussian (NG) stochastic solutions to χ-Wick-type stochastic (χ-Wk-TS) Burgers’ equations with variable coefficients. An Exp-function method, the connection between white noise theory and hypercomplex systems (HCSs), the χ-Wick product (χ-Wk-product) and an χ-Hermite transform (χ-Hr-transform) are proposed. We provide a new set of non-Gaussian solitary wave solutions (NG-SWSs) to Burgers’ equations with variable coefficients. NG white noise functional solutions (NG-WNFSs) to χ-Wk-TS Burgers’ equations with variable coefficients are shown. The symmetry coefficients of partial differential equations and the symmetrical properties of SPDEs are critical in determining the best solution.

Quasi-bound states-driven terahertz molecular fingerprint retriever

Limited by the weak resonant intensities of ordinary modes, the sensitivity of terahertz sensing is often insufficient to retreive trace molecular fingerprint features. Here, a quasi-bound states in the continuum-induced terahertz metasurface with high field enhancement is proposed to achieve trace molecular fingerprint sensing. The ratio of radiative loss to intrinsic loss of proposed metasurface can be accurately tailored by controlling in-plane symmetry, enabling working in the under-coupled, critically coupled and over-coupled regions. Based on the proposed metasurface, a pixelated metasensor has been designed for molecular fingerprint reconstruction, where negative, near-zero, and positive signal modulations can be observed in three different coupling regions. Finally, we demonstrate that the designed pixelated metasensor can achieve quantitative sensing of mixtures, providing an alternative technique for detecting and identifying trace molecules.

Isoptic Point of the Non-Cyclic Quadrangle in the Isotropic Plane

We study the non-cyclic quadrangle ABCD in the isotropic plane and its isoptic point. This is a continuation of the research carried out in a few previous papers. There, we put the non-cyclic quadrangle in the standard position, which enables us to prove its properties using a simple analytical method. In the standard position, the special hyperbola xy=1 is circumscribed to the quadrangle. Hereby, we use the same method to obtain several results related to the isoptic point of the non-cyclic quadrangle. The isoptic point T is the inverse image of points A′, B′, C′, D′ with respect to circumcircles of BCD, ACD, ABD, ACD, respectively, where A′, B′, C′, D′ are isogonal points to vertices A, B, C, D with respect to triangles BCD, ACD, ABD, ACD. The circumircles are seen from T under the equal angles. Our analysis is motivated by the Euclidean results already published in the literature.

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

This paper considers a semi-parametric errors-in-variables (EV) model, ηi=xiβ+g(τi)+ϵi, ξi=xi+δi, 1⩽i⩽n. The properties of estimators are investigated under conditions of missing data and strong mixing errors. Three approaches are used to handle missing data: direct deletion, imputation, and the regression surrogate. Furthermore, estimators for the coefficient β and the nonparametric function g(·) are obtained. Notably, both estimators achieve strong consistency at a rate of o(n−1/4), exhibiting a symmetry in their convergence rates, and they also demonstrate asymptotic normality. Additionally, the validity of our theoretical results is supported by simulations demonstrating the finite sample behaviour of these estimators.

Structural and dynamic characteristics of horseshoe vortex systems in front of rigid vegetation inclined upstream under conditions of overland flow

Inclined angle significantly impacts horseshoe vortex (HV) system and subsequent flow events upstream of vegetation stems, which are crucial for understanding of erosion mechanisms and geodynamics. Flume experiments were conducted to investigate dynamic characteristics of horseshoe vortex (HV) system upstream of inclined rigid vegetation stems under shallow overland flow conditions. Five inclination angles (10°, 20°, 30°, 40°, 50°) were tested alongside a vertical column (0°) across four low Reynolds number conditions (ReD = 2995–4639). Using high-precision particle image velocimetry (PIV) system, we measured the flow field in upstream symmetry plane of the cylinder. Then time-averaged primary HV features were analyzed in terms of the location, radius, vorticity, swirling strength, flow direction and vertical velocity. The increasing inclination angle weakens the formation of the HV system. This is evident in the decrease of vorticity and swirling strength, as well as the gradual diffusion and eventual rupture of the HV system. In the horizontal direction, the primary HV gradually moves away from the cylinder, with minimal vertical impact. The radius of the main HV is positively correlated with the tilt angle, while vorticity and rotation intensity are negatively correlated. We also examined the time-resolved characteristics of the velocity components. The probability density functions (PDFs) of streamwise and vertical velocity components show asymmetric double peaks, indicating two high-frequency events: backflow and downwelling. Linear random estimation revealed that the backflow event is driven by a high-momentum counter current passing through the primary HV, which is mainly dominated by a backflow mode, while the downwelling event arises from low-momentum fluid that cannot penetrate the HV, dominated by a zero-flow mode. These two modes exhibited minimum and maximum time percentages within the current time range, highlighting erosion dynamics from the perspectives of flow event frequency and momentum theory.

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.