Symmetric Maps Research Articles

Symmetric Reverse n-Derivations on Ideals of Semiprime Rings

This paper focuses on examining a new type of n-additive map called the symmetric reverse n-derivation. As implied by its name, it combines the ideas of n-additive maps and reverse derivations, with a 1-reverse derivation being the ordinary reverse derivation. We explore several findings that expand our knowledge of these maps, particularly their presence in semiprime rings and the way rings respond to specific functional identities involving elements of ideals. Also, we provide examples to help clarify the concept of symmetric reverse n-derivations. This study aims to deepen our understanding of these symmetric maps and their properties within mathematical structures.

Controlling the polarization through opposite orthogonal transformation with coupling-optimized composite metasurface

The flexible manipulation of incident polarized waves enhances design freedom for integrated and miniaturized THz devices. This work proposes a method of orthogonal decomposition and transformation to manipulate the polarization of incident fields. It enables symmetric mapping with respect to the S2 axis (x = y) on the Poincaré sphere for any polarized waves. Consequently, arbitrary deflection can be achieved by adjusting the angle between the incident wave and the device. Additionally, the spin will be reversed for circularly and elliptically polarized waves. A specific metasurface is designed to implement this proposal, incorporating a precisely tailored coupling structure to optimize the interaction between cells, which is crucial for maintaining transmittance and bandwidth. All designs are thoroughly validated through numerical simulations. This adaptability and flexibility make it suitable for various applications, including waveplates, polarization convertors and detectors, antennas and radars, etc.

On New Symmetric Fractional Discrete-Time Systems: Chaos, Complexity, and Control

This paper introduces a new symmetric fractional-order discrete system. The dynamics and symmetry of the suggested model are studied under two initial conditions, mainly a comparison of the commensurate order and incommensurate order maps, which highlights their effect on symmetry-breaking bifurcations. In addition, a theoretical analysis examines the stability of the zero equilibrium point. It proves that the map generates typical nonlinear features, including chaos, which is confirmed numerically: phase attractors are plotted in a two-dimensional (2D) and three-dimensional (3D) space, bifurcation diagrams are drawn with variations in the derivative fractional values and in the system parameters, and we calculate the Maximum Lyapunov Exponents (MLEs) associated with the bifurcation diagram. Additionally, we use the C0 algorithm and entropy approach to measure the complexity of the chaotic symmetric fractional map. Finally, nonlinear 3D controllers are revealed to stabilize the symmetric fractional order map’s states in commensurate and incommensurate cases.

Two novel temperature-based topological indices with strong potential to predict physicochemical properties of polycyclic aromatic hydrocarbons with applications to silicon carbide nanotubes

For a vertex x ∈ V G , the temperature T x of x is defined as T x = d x /n − d x . A topological/graphical index G I is a map GI:∑→R , where ∑ (resp. R ) is the set of chemical graphs (resp. real numbers). Graphical indices are employed in structure-property & structure-activity modeling to predict physicochemical/thermodynamic/bilogical characteristics of a compound. A temperature-based graphical index of a chemical graph G is defined as GI T ≔ ∑edges f(T x , t y ), where f(T x , T y ) is a symmetric 2-variable map. In this paper, we introduce two new novel temperature-based indices known as the reduced reciprocal product-connectivity temperature (RRPT) index and the geometric-arithmetic temperature (GAT) index. The predictive potential of these indices have been investigated by employing them in structure-property modeling of physicochemical properties of polycyclic aromatic hydrocarbons (PAHs). The normal boiling point (bp) and the standard enthalpy of formation ΔHfo are selected as representatives of physicochemical characteristics. Intermolecular & van der Waals kind of interactions have been represented by bp, whereas, ΔHfo advocates for thermal characteristics of a compound. In order to validate the statistical inference, the lower 22 PAHs have been opted as test molecules as their experimental data is also publicly available. We propose a computational method to compute all temperature indices in literature and employ it to compute them for the lower 22 PAHs. Besides all the existing temperature indices, both RRPT & GAT are used in a quality testing to predict bp and ΔHfo for lower PAHs. Our statistical analysis asserts that both RRPT & GAT outperformed all the existing temperature indices for correlating bp and ΔHfo for lower PAHs. Most appropriate data-fitting regression models have been suggested to be linear. Since RRPT has the both of correlation coefficients >0.95, the study implicates its further employability in structure-property modeling. Importantly, our research contributes towards countering proliferation of graphical indices. Applications to well-performing temperature indices to correlate physicochemical characteristics of silicon carbide nanotubes are presented.

Comparative analysis of temperature-based graphical indices for correlating the total π-electron energy of benzenoid hydrocarbons

In a graph [Formula: see text], the temperature [Formula: see text] of a vertex [Formula: see text] is defined as [Formula: see text], where n is the order of G and [Formula: see text] is the valency/degree of x. A topological/graphical index [Formula: see text] is a map [Formula: see text], where ∑ (respectively, [Formula: see text]) is the set of simple connected graphs (respectively, real numbers). Graphical indices are employed in quantitative structure-property relationship (QSPR) modeling to predict physicochemical/thermodynamic/biological characteristics of a compound. A temperature-based graphical index of a chemical graph G is defined as [Formula: see text], where [Formula: see text] is a symmetric 2-variable map. In this paper, we introduce two new novel temperature-based indices named as the reduced reciprocal product-connectivity temperature ([Formula: see text]) index and the geometric-arithmetic temperature ([Formula: see text]) index. The predictive potential of these indices has been investigated by employing them in structure-property modeling of the total [Formula: see text]-electronic energy [Formula: see text] of benzenoid hydrocarbons. In order to validate the statistical inference, the lower 30 BHs have been opted as test molecules as their experimental data for [Formula: see text] is also publicly available. First, we employ a computer-based computational method to compute temperature indices of 30 lower BHs. Certain QPSR models are proposed by utilizing the experimental data of [Formula: see text] for the BHs. Our statistical analysis suggests that the most efficient regression models are, in fact, linear. Our statistical analysis asserts that both [Formula: see text] and [Formula: see text] outperformed all the existing temperature indices for correlating [Formula: see text] for the BHs. The results suggest their further employability in QSPR modeling. Importantly, our research contributes toward countering proliferation of graphical indices.

Geometric Characterizations of Symmetric Maps in the Complex Plane

Geometric Characterizations of Symmetric Maps in the Complex Plane

On the Hessian of Cubic Hypersurfaces

Abstract In this paper, we analyze the Hessian locus associated to a general cubic hypersurface, by describing its singular locus and its desingularization for every dimension. The strategy is based on strong connections between the Hessian and the quadrics defined as partial derivatives of the cubic polynomial. In particular, we focus our attention on the singularities of the Hessian hypersurface associated to the general cubic four-fold. It turns out to be a minimal surface of general type: its analysis is developed by exploiting the nature of this surface as a degeneracy locus of a symmetric vector bundle map and by describing an unramified double cover, which is constructed in a more general setting.

Symmetric n-Additive Mappings Admitting Semiprime Ring

Symmetric n-Additive Mappings Admitting Semiprime Ring

On Symmetric Additive Mappings and Their Applications

The key motive of this paper is to study symmetric additive mappings and discuss their applications. The study of these symmetric mappings makes it possible to characterize symmetric n-derivations and describe the structure of the quotient ring S/P, where S is any ring and P is a prime ideal of S. The symmetricity of additive mappings allows us to transfer ring theory results to functional analyses, particularly to C∗-algebras. Precisely, we describe the structures of C∗-algebras via symmetric additive mappings.

Compensation of the Distorted WDM Signals by Symmetric Dispersion Map with Nonuniform Zero-Crossing Place of Accumulated Dispersion in Midway-OPC System

The nonlinear Kerr effect and chromatic dispersion are the fundamental causes of optical signal degradation in single-mode fiber (SMF) and erbium-doped fiber-amplification (EDFA)-based wavelength division multiplexing (WDM) transmission. Dispersion management combined with a midway optical phase conjugator among the technologies for compensating for such optical signal distortion is known to not be limited by the modulation format and multiplexing technology. Optimization of the dispersion map can partially alleviate the capacity and maximum transmission distance limitations of the SMF and EDFA system. In this paper, we propose various types of symmetric dispersion maps in which the position of zero-crossing place of the cumulative dispersion is not constant, and analyze the effect of each dispersion map configuration on 40 Gb/s × 24-channel WDM signal distortion compensation. When designed with the residual dispersion per span (RDPS) around 400 ps/nm, it is confirmed that most of the proposed dispersion maps are more effective in compensating the distorted WDM signal than conventional dispersion map. In particular, we confirm that, among the proposed dispersion maps, the dispersion map in which the RDPS is designed uniformly for all fiber spans can increase the power margin of WDM channel and expand the range of the total residual dispersion in the dispersion-managed link.

Learning-Based Intrinsic Reflectional Symmetry Detection.

Reflectional symmetry is a ubiquitous pattern in nature. Previous works usually solve this problem by voting or sampling, suffering from high computational cost and randomness. In this article, we propose a learning-based approach to intrinsic reflectional symmetry detection. Instead of directly finding symmetric point pairs, we parametrize this self-isometry using a functional map matrix, which can be easily computed given the signs of Laplacian eigenfunctions under the symmetric mapping. Therefore, we manually label the eigenfunction signs for a variety of shapes and train a novel neural network to predict the sign of each eigenfunction under symmetry. Our network aims at learning the global property of functions and consequently converts the problem defined on the manifold to the functional domain. By disentangling the prediction of the matrix into separated bases, our method generalizes well to new shapes and is invariant under perturbation of eigenfunctions. Through extensive experiments, we demonstrate the robustness of our method in challenging cases, including different topology and incomplete shapes with holes. By avoiding random sampling, our learning-based algorithm is over 20 times faster than state-of-the-art methods, and meanwhile, is more robust, achieving higher correspondence accuracy in commonly used metrics.

On meromorphic solutions of Malmquist type difference equations

Recently, the present authors used Nevanlinna theory to provide a classification for the Malmquist type difference equations of the form \(f(z+1)^n=R(z,f)\) \((\dagger)\) that have transcendental meromorphic solutions, where \(R(z,f)\) is rational in both arguments. In this paper, we first complete the classification for the case \(\deg_{f}(R(z,f))=n\) of \((\dagger)\) by identifying a new equation that was left out in our previous work. We will actually derive all the equations in this case based on some new observations on \((\dagger)\). Then, we study the relations between \((\dagger)\) and its differential counterpart \((f')^n=R(z,f)\). We show that most autonomous equations, singled out from \((\dagger)\) with \(n=2\), have a natural continuum limit to either the differential Riccati equation \(f'=a+f^2\) or the differential equation \((f')^2=a(f^2-\tau_1^2)(f^2-\tau_2^2)\), where \(a\neq 0\) and \(\tau_i\) are constants such that \(\tau_1^2\neq\tau_2^2\). The latter second degree differential equation and the symmetric QRT map are derived from each other using the bilinear method and the continuum limit method.

Searchable encryption over encrypted speech retrieval scheme in cloud storage

Aiming at the privacy and security issues of sensitive speech data in cloud storage, and how to achieve efficient retrieval with privacy-preserving for encrypted speech data in cloud storage, a searchable encryption over encrypted speech retrieval scheme in cloud storage was proposed. Firstly, the data owner encrypts the original speech data adopting symmetric encryption algorithm Lorenz chaotic mapping. Secondly, the encrypted speech data is uploaded to the cloud server for storage, and meanwhile the Mel frequency cepstrum coefficient (MFCC) features of original speech are extracted as the input of the convolutional neural network (CNN) to execute the deep semantic feature extraction, which is taken as the speech keywords. Finally, the extracted speech keywords are encrypted and stored in the cloud. When the authorized user sends a retrieval request, the trained CNN is utilized to extract the keywords of the speech to be retrieved, and the search trapdoor is generated and sent to the cloud server. In the process of retrieval, the Euclidean distance is used to match the encrypted keywords with the search trapdoor. The theoretical analysis and experimental results show that the proposed scheme has higher security and retrieval accuracy, and can be suitable for the encrypted storage of speech data and efficient secure retrieval.

Gerrymandering in the laboratory

AbstractGerrymandering, the drawing of electoral boundaries for political advantage, is a controversial political topic. Previous theoretical work has treated gerrymandering as a one‐stage strategic game against nature, but this paper treats drawing districts as the first stage in a two stage game where parties subsequently compete for undecided voters within districts. In a laboratory experiment, subjects overwhelmingly engage in gerrymandering, which maximizes their probability of winning a majority of the districts and maximizes their expected payoff. However, the subjects self‐report being opposed to the practice in politics regardless of their own political ideology. When forced to draw the electoral map behind a veil of ignorance, the subjects indicate a clear preference for symmetric maps although not the map with the highest theoretical expected payoff. Additionally, while subjects overspend in second stage competition, average spending is similar across electoral maps.

Object Detection of Flexible Objects with Arbitrary Orientation Based on Rotation-Adaptive YOLOv5

It is challenging to accurately detect flexible objects with arbitrary orientation from monitoring images in power grid maintenance and inspection sites. This is because these images exhibit a significant imbalance between the foreground and background, which can lead to low detection accuracy when using a horizontal bounding box (HBB) as the detector in general object detection algorithms. Existing multi-oriented detection algorithms that use irregular polygons as the detector can improve accuracy to some extent, but their accuracy is limited due to boundary problems during the training process. This paper proposes a rotation-adaptive YOLOv5 (R_YOLOv5) with a rotated bounding box (RBB) to detect flexible objects with arbitrary orientation, effectively addressing the above issues and achieving high accuracy. Firstly, a long-side representation method is used to add the degree of freedom (DOF) for bounding boxes, enabling accurate detection of flexible objects with large spans, deformable shapes, and small foreground-to-background ratios. Furthermore, the further boundary problem induced by the proposed bounding box strategy is overcome by using classification discretization and symmetric function mapping methods. Finally, the loss function is optimized to ensure training convergence for the new bounding box. To meet various practical requirements, we propose four models with different scales based on YOLOv5, namely R_YOLOv5s, R_YOLOv5m, R_YOLOv5l, and R_YOLOv5x. Experimental results demonstrate that these four models achieve mean average precision (mAP) values of 0.712, 0.731, 0.736, and 0.745 on the DOTA-v1.5 dataset and 0.579, 0.629, 0.689, and 0.713 on our self-built FO dataset, exhibiting higher recognition accuracy and a stronger generalization ability. Among them, R_YOLOv5x achieves a mAP that is about 6.84% higher than ReDet on the DOTAv-1.5 dataset and at least 2% higher than the original YOLOv5 model on the FO dataset.

Symmetric Volume Maps: Order-invariant Volumetric Mesh Correspondence with Free Boundary

Although shape correspondence is a central problem in geometry processing, most methods for this task apply only to two-dimensional surfaces. The neglected task of volumetric correspondence—a natural extension relevant to shapes extracted from simulation, medical imaging, and volume rendering—presents unique challenges that do not appear in the two-dimensional case. In this work, we propose a method for mapping between volumes represented as tetrahedral meshes. Our formulation minimizes a distortion energy designed to extract maps symmetrically, i.e., without dependence on the ordering of the source and target domains. We accompany our method with theoretical discussion describing the consequences of this symmetry assumption, leading us to select a symmetrized ARAP energy that favors isometric correspondences. Our final formulation optimizes for near-isometry while matching the boundary. We demonstrate our method on a diverse geometric dataset, producing low-distortion matchings that align closely to the boundary.

Even symmetric chaotic and skewed maps as a technique in video encryption

The massive growth and use of digital multimedia through computer networks, including video and images, have increased the demand for protecting this digital data. To secure digital video, video encryption is frequently utilized. In this paper, a brand-new video scrambling technique based on two chaotic linearly symmetric maps and one chaotic tent map that has been twisted is suggested. The permutation procedure moves every frame pixel's position using a P-box created by permuting a linearly symmetric chaotic sequence. The diffusion technique employs both linearly symmetric chaos maps and distorted tent maps to create key streams. The keystream closely resembles simple frames because the pixels in the permuted frame indicate which of the two even symmetric chaos maps is replicated each time for the following byte. The information entropy, histogram, neighboring pixel correlation and sensitivity analysis, number of pixels changing regions (NPCR), and unified mean change intensity are used to thoroughly evaluate the recommended method's capacity to improve performance and security (UACI). Comparatively to other methods, the suggested algorithm is resistant to clipping, salt and pepper noise, speckle noise rotation assaults, and clipping. This positive outcome indicates that the plan can be successfully implemented for secure video communication applications.

Multiple Alternatives of Offset Boosting in a Symmetric Hyperchaotic Map

The offset as the average value of a variable plays an important role in signal processing and system design. Offset boosting can be realized by a non-bifurcation parameter or an initial condition. In this work, symmetric coexisting attractors with opposite polarity and a 2D hyperchaotic map with multiple modes of offset boosting are proposed, where the offset can be controlled both by the initial condition and system parameter, and as a result, multiple alternatives of offset boosting and offset competition show up. Consequently, the final offset is determined eventually by the balance of two factors. The theoretical findings are verified through the hardware experiment based on the STM32. Finally, a pseudo-random number generator (PRNG) is constructed based on the newly proposed hyperchaotic map, demonstrating its high performance in engineering applications.

Self-Dual Maps II: Links and Symmetry

In this paper, we investigate representations of links that are either centrally symmetric in or antipodally symmetric in . By using the notions of antipodally self-dual and antipodally symmetric maps, introduced and studied by the authors in [L. Montejano, J. L. Ramírez Alfonsín, and I. Rasskin, SIAM J. Discrete Math., 36 (2022), pp. 1551–1566], we are able to present sufficient combinatorial conditions for a link to admit such representations. The latter naturally provide sufficient conditions for to be amphichiral. We also introduce another (closely related) method yielding again sufficient conditions for to be amphichiral. We finally prove that a link , associated to a map , is amphichiral if the self-dual pairing of is not one of 6 specific cases among the classification of the 24 self-dual pairing .

Equivariant Relative Submajorization

We study a generalization of relative submajorization that compares pairs of positive operators on representation spaces of some fixed group. A pair equivariantly relatively submajorizes another if there is an equivariant subnormalized channel that takes the components of the first pair to a pair satisfying similar positivity constraints as in the definition of relative submajorization. In the context of the resource theory approach to thermodynamics, this generalization allows one to study transformations by Gibbs-preserving maps that are in addition time-translation symmetric. We find a sufficient condition for the existence of catalytic transformations and a characterization of an asymptotic relaxation of the relation. For classical and certain quantum pairs the characterization is in terms of explicit monotone quantities related to the sandwiched quantum Rényi divergences. In the general quantum case the relevant quantities are given only implicitly. Nevertheless, we find a large collection of monotones that provide necessary conditions for asymptotic or catalytic transformations. When applied to time-translation symmetric maps, these give rise to second laws that constrain state transformations allowed by thermal operations even in the presence of catalysts.