Symmetric Representation Research Articles

Prioritization of Human-Centric and Sustainable City Criteria by Proportional Spherical Fuzzy Analytic Hierarchy Process

Human-centric and sustainable cities have become one of the most popular research areas today. They are based on a holistic approach that aims to both improve people’s quality of life and ensure environmental sustainability. The weighting of numerous tangible and intangible criteria used in the performance evaluation of these cities has emerged as an important problem. The symmetrical representation of vague and imprecise data using fuzzy set theory is an absolute necessity for a successful weighting process. In this study, symmetrical representation of especially intangible criteria is carried out with proportional spherical fuzzy sets. Proportional spherical fuzzy sets provide significant convenience to the expert in determining the membership, non-membership and hesitancy degrees and ensure that the assigned values are more accurate and consistent. In the study, five main human centric and sustainable city (HCSC) criteria and 26 HCSC sub-criteria determined from the literature were weighted. In addition, criteria weights were obtained by classical spherical fuzzy analytic hierarchy process (AHP) for comparison purpose. The first four most important HCSC criteria were determined as water management and conservation, employment/unemployment rate, carbon emission reduction strategies and economic sustainability, respectively.

Semiclassical quantization of M5 brane probes wrapped on AdS3× S3 and defect anomalies

We consider two supersymmetric M5 brane probe solutions in AdS7 × S4 and one in AdS4 × S7 that all have the AdS3 × S3 world-volume geometry. The values of the classical action of the first two M5 probes (with S3 in AdS7 or in S4) are related to the leading N2 parts in the anomaly b-coefficient in the (2,0) theory corresponding to a spherical surface defect in symmetric or antisymmetric SU(N) representations. We present a detailed computation of the corresponding one-loop M5 brane partition functions finding that they vanish (in a particular regularization). This implies the vanishing of the order N0 part in the b-anomaly coefficients, in agreement with earlier predictions for their exact values. It remains, however, a puzzle of how to reproduce the non-vanishing order N terms in these coefficients within the semiclassical M5-brane probe setup.

Modular representations of GL2(𝔽𝑞) using calculus

Abstract We show that certain modular induced representations of GL 2 ⁢ ( F q ) \mathrm{GL}_{2}(\mathbb{F}_{q}) can be written as cokernels of operators acting on symmetric power representations of GL 2 ⁢ ( F q ) \mathrm{GL}_{2}(\mathbb{F}_{q}) . When the induction is from the Borel subgroup or the anisotropic torus, the operators involve multiplication by newly defined twisted Dickson polynomials or twisted Serre operators, respectively. Our isomorphisms are explicitly defined using differential operators. As a corollary, we improve some periodicity results for quotients in the theta filtration.

Construction and analysis of symmetric Sprott B multi-attractors with electric implementation

Abstract In chaos theory, a number of systems and models which apparently contain simple ordinary differential equations (ODEs) turn out to show a dynamic characterized by complicated behaviors and complex trajectories. One of such systems is the Sprott B model. We construct some set of multi-attractors based on the Sprott B model where additional parameters and operators are considered. After summarizing important preliminaries relevant to simple chaotic differential systems, the model is firstly solved analytically and numerically, then graphical simulations are provided. The later show coexistence and evolution of two chaotic attractors in a symmetrical representation. Lastly, similar results and expected outcomes are recovered via an electrical circuit implementation, realized using the Field Programmable Gate Array (FPGA) board, the Digital-to-Analog Converter (DAC) and the Rigol Oscilloscope. They also show progressing sets of coexisting multi-attractors.

Symmetric Piecewise Developable Approximations

AbstractWe propose a novel method for generating symmetric piecewise developable approximations for shapes in approximately global reflectional or rotational symmetry. Given a shape and its symmetry constraint, the algorithm contains two crucial steps: (i) a symmetric deformation to achieve a nearly developable model and (ii) a symmetric segmentation aided by the deformed shape. The key to the deformation step is the use of the symmetric implicit neural representations of the shape and the deformation field. A new mesh extraction from the implicit function is introduced to construct a strictly symmetric mesh for the subsequent segmentation. The symmetry constraint is carefully integrated into the partition to achieve the symmetric piecewise developable approximation. We demonstrate the effectiveness of our algorithm over various meshes.

Planar decomposition of the HOMFLY polynomial for bipartite knots and links

The theory of the Kauffman bracket, which describes the Jones polynomial as a sum over closed circles formed by the planar resolution of vertices in a knot diagram, can be straightforwardly lifted from sl(2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak {sl}(2)$$\\end{document} to sl(N)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak {sl}(N)$$\\end{document} at arbitrary N – but for a special class of bipartite diagrams made entirely from the antiparallel lock tangle. Many amusing and important knots and links can be described in this way, from twist and double braid knots to the celebrated Kanenobu knots for even parameters – and for all of them the entire HOMFLY polynomials possess planar decomposition. This provides an approach to evaluation of HOMFLY polynomials, which is complementary to the arborescent calculus, and this opens a new direction to homological techniques, parallel to Khovanov–Rozansky generalizations of the Kauffman calculus. Moreover, this planar calculus is also applicable to other symmetric representations beyond the fundamental one, and to links which are not fully bipartite what is illustrated by examples of Kanenobu-like links.

ADHM wilson line defect indices

The Coulomb and Higgs indices of the 3d N = 4 U(N) ADHM theories can be decorated by line defect operators as the line defect correlators. We obtain exact closed-form expressions and various non-trivial algebraic relations for the correlators of the Wilson lines in the fundamental and (anti)symmetric representations by means of the Hall-Littlewood expansion, the Fermi-gas method and the residue calculation. From the large N limit of the correlators we obtain the single particle gravity indices which are expected to encode the spectra of fluctuation modes on the gravity duals of line operators in M2-brane SCFTs.

Unveiling Gender Portrayals in Indonesian EFL textbooks: A Critical Discourse Analysis

This study aims to describe the representation of genders in five Indonesian EFL textbooks published by Erlangga Publishing and unveil the ideology and attitudes behind the image of genders. The methods of critical discourse analysis (CDA) proposed by Van Dijk, which comprise discourse, social cognition, and discourse and society, are used to unveil the patterns of gender portrayal and the attitudes behind it. Data for this study are the human images in five Indonesian EFL textbooks published by Erlangga Publishing. This study reveals that in terms of frequency, women are more dominant than men; women’s images appear more frequently than men's. In dealing with other aspects under study, men and women are depicted symmetrically. The symmetrical representation between men and women suggests the attitude and ideology of gender equality the authors uphold.

On the Geometric Pattern Transformation (GPT) Properties of Unidimensional Signals

The Geometric Pattern Transformation (GPT) has several advantages of use concerning contemporary algorithms that have been duly studied in previous research. Regarding some of its properties, four different but complementary aspects of the GPT are presented in this work. After a brief review of the GPT concept, how tied data are manifested in data sets is shown, to obtain a symmetric representation of the GPT, a linear transformation is performed that regularizes the geometric representation of the GPT and the theoretical relationship between the GPT and the phase-state representation of 1D signals is analyzed and formalized, then the study of the forbidden pattern is easily revealed, obtaining a strong relationship with the stable and unstable fixed points of the logistic equation. Finally, the characterization of colored noises and the application in real world signals taken through experimental procedures is analyzed. With these results, in this work is proposed an advance in the potential applications of the GPT in an integral way in the processing and analysis of data series.

Leading large N giant graviton correction to Schur correlators in large representations

We consider 4d N=4U(N) SYM and the leading giant graviton correction at large N to the Schur defect 2-point functions of 12-BPS Wilson lines in rank-k symmetric and antisymmetric representations. We study in particular the large k limit for the symmetric case and the regime 1≪k≪N in the antisymmetric one. In both cases we present exact results for the correction. Wilson lines in symmetric/antisymmetric representations admit a description in terms of D3k and D5k brane probes representing a collection of k fundamental strings. In this picture, giant graviton corrections come from fluctuations of brane probes in the presence of a wrapped D3 brane giant graviton. In the antisymmetric case, our leading correction matches the half-index of the 4d N=4 Maxwell theory living on the disk which is a part of the giant graviton divided out by the D5k probe, as recently proposed in arXiv:2404.08302. For the symmetric case at large k, we derive an explicit exact residue formula for the leading large N correction.

Symmetry-Optimized Dynamical Analysis of Optical Soliton Patterns in the Flexibly Supported Euler–Bernoulli Beam Equation: A Semi-Analytical Solution Approach

This study employs spatial optimization principles to investigate the nonlinear vibration of a flexibly supported Euler–Bernoulli beam, a (1 + 1)-dimensional system subjected to axial loads. The modified Khater method, a crucial tool in mechanical engineering, is utilized to analyze analytical solutions, which include a symmetric spatial representation of the waveform as an integral part of each solution. Notably, periodic soliton solutions for the nonlinear model closely align with numerical and approximate analytical solutions, demonstrating the accuracy of our modeling approach. Density diagrams, contour diagrams, and Poincaré maps depicting the obtained analytical solutions are presented to elucidate their accuracy and provide visual confirmation of the optimized engineering model’s physical significance. The planar dynamical system is derived through the Galilean transformation by employing mathematical models and appropriate parameter values, thereby further refining problem understanding. Sensitivity analysis is conducted, and phase portraits with equilibrium points are illustrated by analyzing a special case of the investigated dynamical system, emphasizing its symmetrical properties. Lastly, we perform a global analysis to identify periodic, quasi-periodic, and chaotic behaviors, with an extra weak forcing term confirmed by Poincaré maps and a two-dimensional symmetric basin of the largest Lyapunov exponent.

Folding QQ-relations and transfer matrix eigenvalues: Towards a unified approach to Bethe ansatz for super spin chains

Extending the method proposed in [1], we derive QQ-relations (functional relations among Baxter Q-functions) and T-functions (eigenvalues of transfer matrices) for fusion vertex models associated with the twisted quantum affine superalgebras Uq(gl(2r+1|2s)(2)), Uq(gl(2r|2s+1)(2)), Uq(gl(2r|2s)(2)), Uq(osp(2r|2s)(2)) and the untwisted quantum affine orthosymplectic superalgebras Uq(osp(2r+1|2s)(1)) and Uq(osp(2r|2s)(1)) (and their Yangian counterparts, Y(osp(2r+1|2s)) and Y(osp(2r|2s))) as reductions (a kind of folding) of those associated with Uq(gl(M|N)(1)). In particular, we reproduce previously proposed generating functions (difference operators) of the T-functions for the symmetric or anti-symmetric representations, and tableau sum expressions for more general representations for orthosymplectic superalgebras [2,3], and obtain Wronskian-type expressions (analogues of Weyl-type character formulas) for them. T-functions for spinorial representations are related to reductions of those for asymptotic limits of typical representations of Uq(gl(M|N)(1)).

Triangular solutions to the reflection equation for Uq(sln^)

We study solutions of the reflection equation related to the quantum affine algebra Uq(sln^) . First, we explain how to construct a family of stochastic integrable vertex models with fixed boundary conditions. Then, we construct upper- and lower-triangular solutions of the reflection equation related to symmetric tensor representations of Uq(sln^) with arbitrary spin. We also prove the star–star relation for the Boltzmann weights of the Ising-type model, conjectured by Bazhanov and Sergeev, and use it to verify certain properties of the solutions obtained.

Dominant subspaces of high-fidelity polynomial structured parametric dynamical systems and model reduction

In this work, we investigate a model order reduction scheme for high-fidelity nonlinear structured parametric dynamical systems. More specifically, we consider a class of nonlinear dynamical systems whose nonlinear terms are polynomial functions, and the linear part corresponds to a linear structured model, such as second-order, time-delay, or fractional-order systems. Our approach relies on the Volterra series representation of these dynamical systems. Using this representation, we identify the kernels and, thus, the generalized multivariate transfer functions associated with these systems. Consequently, we present results allowing the construction of reduced-order models whose generalized transfer functions interpolate these of the original system at pre-defined frequency points. For efficient calculations, we also need the concept of a symmetric Kronecker product representation of a tensor and derive particular properties of them. Moreover, we propose an algorithm that extracts dominant subspaces from the prescribed interpolation conditions. This allows the construction of reduced-order models that preserve the structure. We also extend these results to parametric systems and a special case (delay in input/output). We demonstrate the efficiency of the proposed method by means of various numerical benchmarks.

Smile aesthetics in Pakistani population: dentist preferences and perceptions of anterior teeth proportion and harmony

BackgroundThis study aimed to evaluate dentist perceptions of attractive smiles in the Pakistani population, considering different dental proportions.MethodsMaxillary casts and digital images were used to create symmetrical representations of anterior teeth. dentists’ preferences for good and bad teeth proportions, width/height ratios, and various dental proportions (golden, recurring esthetic dental (RED), golden percentage, Preston, and local/observed) were assessed using one sample and paired t-test. The Chi-square test was used to determine the gender disparities and factors affecting smile attractiveness. A p-value of ≤ 0.05 was taken as significant.ResultsThe RED proportion emerged as the preferred choice for normal-sized teeth, with specialists and general dentists favoring it over the golden proportion. For tall teeth, the golden proportion was predominantly preferred. The golden percentage received limited preference for aesthetic smile construction.ConclusionsThe smiles created using the principles of RED proportion were opted as the most attractive by local dentists. Factors such as tooth arrangement, color, and midline were highlighted as essential considerations in aesthetic smile construction.

The planar limit of the N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 E-theory: numerical calculations and the large λ expansion

We study correlation functions of local operators and Wilson loop expectation values in the planar limit of a 4d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 superconformal SU(N) YM theory with hypermultiplets in the symmetric and antisymmetric representations of the gauge group. This so called E theory is closely related to N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 SYM and has a holographic description in terms of a ℤ2 orientifold of AdS5 × S5. Using recent matrix model results based on supersymmetric localization we develop efficient numerical methods to calculate two- and three-point functions of certain single trace operators as well as 1/2-BPS Wilson loop expectation values as a function of the ’t Hooft coupling λ. We use our numerical results to arrive at simple analytic expressions for these correlators valid up to sixth order in the λ−1/2 strong coupling expansion. These results provide explicit field theory predictions for the α′ corrections to the supergravity approximation of type IIB string theory on the AdS5 × S5/ℤ2 orientifold.

Completeness of atomic structure representations

In this paper, we address the challenge of obtaining a comprehensive and symmetric representation of point particle groups, such as atoms in a molecule, which is crucial in physics and theoretical chemistry. The problem has become even more important with the widespread adoption of machine-learning techniques in science, as it underpins the capacity of models to accurately reproduce physical relationships while being consistent with fundamental symmetries and conservation laws. However, some of the descriptors that are commonly used to represent point clouds— notably those based on discretized correlations of the neighbor density that power most of the existing ML models of matter at the atomic scale—are unable to distinguish between special arrangements of particles in three dimensions. This makes it impossible to machine learn their properties. Atom-density correlations are provably complete in the limit in which they simultaneously describe the mutual relationship between all atoms, which is impractical. We present a novel approach to construct descriptors of finite correlations based on the relative arrangement of particle triplets, which can be employed to create symmetry-adapted models with universal approximation capabilities, and have the resolution of the neighbor discretization as the sole convergence parameter. Our strategy is demonstrated on a class of atomic arrangements that are specifically built to defy a broad class of conventional symmetric descriptors, showing its potential for addressing their limitations.

A Weighted Symmetric Graph Embedding Approach for Link Prediction in Undirected Graphs.

Link prediction is an important task in social network analysis and mining because of its various applications. A large number of link prediction methods have been proposed. Among them, the deep learning-based embedding methods exhibit excellent performance, which encodes each node and edge as an embedding vector, enabling easy integration with traditional machine learning algorithms. However, there still remain some unsolved problems for this kind of methods, especially in the steps of node embedding and edge embedding. First, they either share exactly the same weight among all neighbors or assign a completely different weight to each node to obtain the node embedding. Second, they can hardly keep the symmetry of edge embeddings obtained from node representations by direct concatenation or other binary operations such as averaging and Hadamard product. In order to solve these problems, we propose a weighted symmetric graph embedding approach for link prediction. In node embedding, the proposed approach aggregates neighbors in different orders with different aggregating weights. In edge embedding, the proposed approach bidirectionally concatenates node pairs both forwardly and backwardly to guarantee the symmetry of edge representations while preserving local structural information. The experimental results show that our proposed approach can better predict network links, outperforming the state-of-the-art methods. The appropriate aggregating weight assignment and the bidirectional concatenation enable us to learn more accurate and symmetric edge representations for link prediction.

A matrix-model approach to integrated correlators in a N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 SYM theory

In a N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 superconformal gauge theory with matter hypermultiplets transforming in the symmetric and anti-symmetric representations of SU(N), we study the integrated correlators of two Coulomb-branch operators and two moment-map operators using localization. In the corresponding matrix model we identify the operator associated with the integrated insertions of moment-map operators and provide for it an exact expression valid for all values of the coupling constant in the planar limit. This allows us to study the integrated correlators at strong-coupling where we show that they behave as the 2-point functions of the Coulomb-branch operators, up to an overall constant dependent only on the conformal dimensions of the latter. The strong-coupling relation between integrated correlators and 2-point functions turns out to be the same as in N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 SYM at large N, despite the reduced amount of supersymmetry in our theory.

Large N and large representations of Schur line defect correlators

We study the large N and large representation limits of the Schur line defect correlators of the Wilson line operators transforming in the (anti)symmetric, hook and rectangular representations for \U0001d4a9 = 4 U(N) super Yang-Mills theory. By means of the factorization property, the large N correlators of the Wilson line operators in arbitrary representations can be exactly calculated in principle. In the large representation limit they turn out to be expressible in terms of certain infinite series such as Ramanujan’s general theta functions and the q-analogues of multiple zeta values (q-MZVs). Several generating functions for combinatorial objects, including partitions with non-negative cranks and conjugacy classes of general linear groups over finite fields, emerge from the large N correlators. Also we find conjectured properties of the automorphy and the hook-length expansion satisfied by the large N correlators.