Type Of Symmetry Research Articles

(Co)homology of racks and multiple group racks for compact oriented surfaces in the 3-sphere

We introduce (co)homology theory for multiple group racks and construct cocycle invariants of compact oriented surfaces in the 3-sphere using their 2-cocycles, where a multiple group rack is a rack consisting of a disjoint union of groups. We provide a method to find new rack cocycles and multiple group rack cocycles from existing rack cocycles and quandle cocycles. Evaluating cocycle invariants, we determine several symmetry types, caused by chirality and invertibility, of compact oriented surfaces in the 3-sphere.

Symmetry classification and enumeration of square-tile sikku kolams

Pulli kolam is an ancient mathematical art form that is still practiced today in south India by over a quarter million people. ‘Pulli’ in the Tamizh language means ‘dots’. A specific type of pulli kolam is sikku kolam where a series of dots are placed using rice flour and lines are drawn around them following three simple rules. Here we focus on sikku kolams assembled from square tiles. Kolams are classified by their point group symmetry and formulae are derived for counting the number of kolams of each symmetry type. Some interesting constraint problems are posed that could help formulate new games based on this elegant art form.

Linear Programming with Unitary-Equivariant Constraints

Unitary equivariance is a natural symmetry that occurs in many contexts in physics and mathematics. Optimization problems with such symmetry can often be formulated as semidefinite programs for a dp+q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d^{p+q}$$\\end{document}-dimensional matrix variable that commutes with U⊗p⊗U¯⊗q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$U^{\\otimes p} \\otimes {\\bar{U}}^{\\otimes q}$$\\end{document}, for all U∈U(d)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$U \\in \ extrm{U}(d)$$\\end{document}. Solving such problems naively can be prohibitively expensive even if p+q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p+q$$\\end{document} is small but the local dimension d is large. We show that, under additional symmetry assumptions, this problem reduces to a linear program that can be solved in time that does not scale in d, and we provide a general framework to execute this reduction under different types of symmetries. The key ingredient of our method is a compact parametrization of the solution space by linear combinations of walled Brauer algebra diagrams. This parametrization requires the idempotents of a Gelfand–Tsetlin basis, which we obtain by adapting a general method inspired by the Okounkov–Vershik approach. To illustrate potential applications of our framework, we use several examples from quantum information: deciding the principal eigenvalue of a quantum state, quantum majority vote, asymmetric cloning and transformation of a black-box unitary. We also outline a possible route for extending our method to general unitary-equivariant semidefinite programs.

Fate of stringy noninvertible symmetries

Noninvertible symmetries in quantum field theory (QFT) generalize the familiar product rule of groups to a more general fusion rule. In many cases, gauged versions of these symmetries can be regarded as dual descriptions of invertible gauge symmetries. One may ask: are there any other types of noninvertible gauge symmetries? In theories with gravity we find a new form of noninvertible gauge symmetry that emerges in the limit of fundamental, tensionless strings. These stringy noninvertible gauge symmetries appear in standard examples such as non-Abelian orbifolds. Moving away from the tensionless limit always breaks these symmetries. We also find that both the conventional form of noninvertible gauge symmetries and these stringy generalizations are realized in AdS/CFT. Although generically broken, approximate noninvertible symmetries have implications for swampland constraints: in certain cases they can be used to prove the existence of towers of states related to the distance conjecture, and can sometimes explain the existence of slightly subextremal states which fill in the gaps in the sublattice weak gravity conjecture. Published by the American Physical Society 2024

Symmetry in Signals: A New Insight

Symmetry is a fundamental property of many natural systems, which is observable through signals. In most out-of-equilibrium complex dynamic systems, the observed signals are asymmetric. However, for certain operating modes, some systems have demonstrated a resurgence of symmetry in their signals. Research has naturally focused on examining time invariance to quantify this symmetry. Measures based on the statistical and harmonic properties of signals have been proposed, but most of them focused on harmonic distortion without explicitly measuring symmetry. This paper introduces a new mathematical framework based on group theory for analyzing signal symmetry beyond time invariance. It presents new indicators to evaluate different types of symmetry in non-stochastic symmetric signals. Both periodic and non-periodic symmetric signals are analyzed to formalize the problem. The study raises critical questions about the completeness of symmetry in signals and proposes a new classification for periodic and non-periodic signals that goes beyond the traditional classification based on Fourier coefficients. Furthermore, new measures such as “symmetrometry” and “distorsymmetry” are introduced to quantify symmetry. These measures outperform traditional indicators like Total Harmonic Distortion (THD) and provide a more accurate measurement of symmetry in complex signals from applications where duty cycle plays a major role.

THz optical response of Ba(Fe[formula omitted]Ni[formula omitted])[formula omitted]As[formula omitted] films analyzed within the three-band Eliashberg s[formula omitted]-wave model

The uncertainty of the nature of the normal state and superconducting condensate of unconventional superconductors continues to stimulate considerable speculation about the mechanism of superconductivity in these materials. Of particular interest are the type of symmetry of the order parameter and the basic electronic characteristics of the superconducting and normal states. We report the derivation of temperature dependences of the superconducting condensate plasma frequency, superfluid density, and London penetration depth by measuring terahertz spectra of conductivity and dielectric permittivity of the Ba(Fe1−xNix)2As2 thin films with different Ni concentrations. A comprehensive analysis of the experimental data was performed in the framework of the simple three-band Eliashberg model under the assumption that the superconducting coupling mechanism is mediated by antiferromagnetic spin fluctuations. The results of independent experiments support the choice of model parameters. Based on calculations of the temperature dependences of superconducting gaps, we may conclude that the obtained results are compatible with the scenario, in which Ba(Fe1−xNix)2As2 is a multiband superconductor with s±-wave pairing symmetry.

Defining the boundaries of the meaning of ornamental compositions in the art of the Pazyryk culture

The article is devoted to the analysis of the semiotic field of the term "ornamental composition" in the context of definitions of ornament and composition in the art of the Pazyryk culture and its historiography. The subject of the study is ornamental compositions from elite burials of the Pazyryk culture of different periods of its development – the early Second Bashadar kurgan and the Tuekta burial ground and the late Pazyryk burial ground. In addition, the paper examines the specifics of the use of the terms "composition" and "ornament" in order to highlight the history of their meanings, as well as analyzes certain provisions of theoretical works on the study of the theory of composition and ornament. The purpose of the study is to identify the components of the semantic field of an ornamental composition related to the field of interaction of the composition with the viewer, and not to the field of meaning of specific figurative images. The article proposes using the method of formal analysis in combination with some provisions of the Gestalt approach described in the works of E. Gombrich and R. Arnheim to analyze a series of ornamental compositions with and without figurative images in the art of the Pazyryk culture. The scientific novelty of the study is predetermined by the absence in the Russian historiography of works describing the specifics of the meaning of ornamental composition in the context of visual perception in the art of the Pazyryk culture. The researchers note that the significance of ornaments in Pazyryk art could be related to the meaning of multi-figure compositions (when zoomorphic images are included in the ornament) and reflect the worldview of the bearers of the Pazyryk culture, or the types of symmetry used in the construction of ornaments could reflect the social structure of society. However, in both of these trends, the analysis of an ornamental composition without figurative images remains outside the scope as a structural and semantic, integral work of art interacting with the viewer. As a result of the research, the assumption is put forward that it is possible to designate a single semantic field for ornamental and multi-figure compositions in Pazyryk art. Clarifying and expanding the scope of the ornamental composition allows us to identify a single specificity of interaction with the viewer of both figurative and non-figurative images of Pazyryk art, to emphasize the game element in the nature of culture.

An efficient approach for searching three-body periodic orbits passing through Eulerian configuration

A new efficient approach for searching three-body periodic equal-mass collisionless orbits passing through Eulerian configuration is presented. The approach is based on a symmetry property of the solutions at the half period. Depending on two previously established symmetry types on the shape sphere, each solution is presented by one or two distinct initial conditions (one or two points in the search domain). A numerical search based on Newton’s method on a relatively coarse search grid for solutions with relatively small scale-invariant periods T∗<70 is conducted. The linear systems at each Newton’s iteration are computed by high order high precision Taylor series method. The search produced 12,431 initial conditions (i.c.s) corresponding to 6333 distinct solutions. In addition to passing through the Eulerian configuration, 35 of the solutions are also free-fall ones. Although most of the found solutions are new, all linearly stable solutions among them (only 7) are old ones. Particular attention is paid to the details of the high precision computations and the analysis of accuracy. All i.c.s are given with 100 correct digits.

Moments of Artin–Schreier L-functions

Abstract We compute moments of L-functions associated to the polynomial family of Artin–Schreier covers over $\mathbb{F}_q$, where q is a power of a prime p &amp;gt; 2, when the size of the finite field is fixed and the genus of the family goes to infinity. More specifically, we compute the $k{\text{th}}$ moment for a large range of values of k, depending on the sizes of p and q. We also compute the second moment in absolute value of the polynomial family, obtaining an exact formula with a lower order term, and confirming the unitary symmetry type of the family.

Cosymplectic Geometry, Reductions, and Energy-Momentum Methods with Applications

Classical energy-momentum methods study the existence and stability properties of solutions of t-dependent Hamilton equations on symplectic manifolds whose evolution is given by their Hamiltonian Lie symmetries. The points of such solutions are called relative equilibrium points. This work devises a new cosymplectic energy-momentum method providing a new and more general framework to study t-dependent Hamilton equations. In fact, cosymplectic geometry allows for using more types of distinguished Lie symmetries (given by Hamiltonian, gradient, or evolution vector fields), relative equilibrium points, and reduction methods, than symplectic techniques. To make our work more self-contained and to fill some gaps in the literature, a review of the cosymplectic formalism and the cosymplectic Marsden–Weinstein reduction is included. Known and new types of relative equilibrium points are characterised and studied. Our methods remove technical conditions used in previous energy-momentum methods, like the Ad∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{Ad}^*$$\\end{document}-equivariance of momentum maps. Eigenfunctions of t-dependent Schrödinger equations are interpreted in terms of relative equilibrium points in cosymplectic manifolds. A new cosymplectic-to-symplectic reduction is developed and a new associated type of relative equilibrium points, the so-called gradient relative equilibrium points, are introduced and applied to study the Lagrange points and Hill spheres of a restricted circular three-body system by means of a not Hamiltonian Lie symmetry of the system.

On Symmetry in Physical Phenomena, Symmetry of an Electric Field and of a Magnetic Field

In this work, the classical concept of symmetry limited to geometric objects (figures and solids), which originated from ancient Greece, has been extended to allow for symmetry studies in other types of objects. By introducing the concepts of limiting point groups and kinematic elements characteristic for a studied object, it was determined what types of symmetries are exhibited by an electric field and a magnetic field. It was established that in order for a phenomenon to occur, a characteristic symmetry of a medium must be consistent with the characteristic symmetry of the phenomenon occurring in it. It was also determined that the symmetry elements of the causes must be found in the symmetry of their effects.

Anomaly inflow for dipole symmetry and higher form foliated field theories

In accordance with recent progress of fracton topological phases, unusual topological phases of matter hosting fractionalized quasiparticle excitations with mobility constraints, new type of symmetry is studied — multipole symmetry, associated with conservation of multipoles. Based on algebraic relation between dipole and global charges, we introduce a series of (d + 1)-dimensional BF theories with p-form gauge fields, which admit dipole of spatially extended excitations, and study their physical properties. We elucidate that gauge invariant loops have unusual form, containing linear function of the spatial coordinate, which leads to the position dependent braiding statistics and unusual ground state degeneracy dependence on the system size. We also show that the theories exhibit a mixed ’t Hooft anomaly between p-form and (d − p)-form dipole symmetries, which is canceled by an invertible theory defined in one dimensional higher via anomaly inflow mechanism.

Group-theoretic analysis of symmetry-preserving deployable structures and metamaterials.

Many deployable structures in nature, as well as human-made mechanisms, preserve symmetry as their configurations evolve. Examples in nature include blooming flowers, dilation of the iris within the human eye, viral capsid maturation and molecular and bacterial motors. Engineered examples include opening umbrellas, elongating scissor jacks, variable apertures in cameras, expanding Hoberman spheres and some kinds of morphing origami structures. In these cases, the structures either preserve a discrete symmetry group or are described as an evolution from one discrete symmetry group to another of the same type as the structure deploys. Likewise, elastic metamaterials built from lattice structures can also preserve symmetry type while passively deforming and changing lattice parameters. A mathematical formulation of such transitions/deployments is articulated here. It is shown that if [Formula: see text] is Euclidean space, [Formula: see text] is a continuous group of motions of Euclidean space and [Formula: see text] is the type of the discrete subgroup of [Formula: see text] describing the symmetries of the deploying structure, then the symmetry of the evolving structure can be described by time-dependent subgroups of [Formula: see text] of the form [Formula: see text], where [Formula: see text] is a time-dependent affine transformation. Then, instead of considering the whole structure in [Formula: see text], a 'sector' of it that lives in the orbit space [Formula: see text] can be considered at each instant in time, and instead of considering all motions in [Formula: see text], only representatives from right cosets in the space [Formula: see text] need to be considered. This article is part of the theme issue 'Current developments in elastic and acoustic metamaterials science (Part 1)'.

Higher-Order Cellular Automata Generated Symmetry-Protected Topological Phases and Detection Through Multi Point Strange Correlators

In computer and system sciences, higher-order cellular automata (HOCA) are a type of cellular automata that evolve over multiple time steps and generate complex patterns, which have various applications, such as secret-sharing schemes, data compression, and image encryption. In this paper, we introduce HOCA to quantum many-body physics and construct a series of symmetry-protected topological (SPT) phases of matter, in which symmetries are supported on a great variety of subsystems embbeded in the SPT bulk. We call these phases HOCA-generated SPT (HGSPT) phases. Specifically, we show that HOCA can generate not only well-understood SPTs with symmetries supported on either regular (e.g., linelike subsystems in the two-dimensional cluster model) or fractal subsystems, but also a large class of unexplored SPTs with symmetries supported on more choices of subsystems. One example is that has either fractal and linelike subsystem symmetries simultaneously or two distinct types of fractal symmetries simultaneously. Another example is in which chaotic-looking symmetries are significantly different from and thus cannot reduce to fractal or regular subsystem symmetries. We also introduce a new notation system to characterize HGSPTs. We prove that all possible subsystem symmetries in a square lattice can be locally simulated by an HOCA-generated symmetry. As the usual two-point strange correlators are trivial in most HGSPTs, we find that the nontrivial SPT orders can be detected by what we call . We propose a universal procedure to design the spatial configuration of the multi point strange correlators for a given HGSPT phase. Specifically, we find deep connections between multi point strange correlators and the spurious topological entanglement entropy (STEE), both exhibiting long-range behavior in a short-range entangled state. Our HOCA approaches and multi point strange correlators pave the way for a unified paradigm to design, classify, and detect phases of matter with symmetries supported on a great variety of subsystems, and also provide potential useful perspective in surpassing the computational irreducibility of HOCA in a quantum mechanical way. Published by the American Physical Society 2024

TNSP: A framework supporting symmetry and fermion tensors for tensor network state methods

Recent advancements have established tensor network states (TNS) as formidable tools for exploring the complex realm of strongly-correlated many-particle systems in both one and two dimensions. To tackle the challenges presented by strongly-correlated fermion systems, various fermion tensor network states (f-TNS) methodologies have been developed. However, implementing f-TNS methods poses substantial challenges due to their particularly complex nature, making development efforts significantly difficult. This complexity is further exacerbated by the lack of underlying software packages that facilitate the development of f-TNS. Previously, we developed TNSPackage, a software package designed for TNS methods [1]. Initially, this package was only capable of handling spin and boson models. To confront the challenges presented by f-TNS, TNSPackage has undergone significant enhancements in its latest version, incorporating support for both symmetry and fermion tensors. This updated version provides a uniform interface for the consistent management of tensors across boson, fermion, and various symmetry types, maintaining its user-friendly and versatile nature. This greatly facilitates the development of programs based on f-TNS. The new TNSP framework consists of two principal components: a low-level tensor package named TAT, which supports sophisticated tensor operations, and a high-level interface package called tetragono that is built upon TAT. The tetragono package is designed to significantly simplify the development of complex physical models on square lattices. The TNSPackage framework enables users to implement a wide range of physical models with greater ease, without the need to pay close attention to the underlying implementation details.

Impact of Fe-impurity centers on phonon subsystem of wurtzite-type ZnO

The paper addresses the question of the structural and vibrational properties of Fe-doped ZnO in the wurtzite structure with a lattice without an inversion center. The ZnO crystals containing Fe impurities in different charged states are studied by means of density functional theory (DFT) and potential-based methods. A first-principles simulation is performed to determine the local atomic configurations near the considered defects. The DFT results are compared with those obtained using the shell model. Both approaches give a similar pattern of lattice distortion around the Fe-impurity centers occupying cation sites in the isoelectronic charge state Fe2＋ and single-positive-charge state Fe3＋. The phonon local symmetrized densities of states projected onto the displacement of ions surrounding the defects are calculated by a well-established recursion method. The frequencies of defect vibrations of various symmetry types induced by Fe impurities are determined. The calculations made it possible to interpret the structure of the phonon sideband accompanying the zero-phonon line in polarized emission spectra attributed to the Fe3＋ intracenter transitions, as well as to estimate the role of Fe ions in the formation of observed peaks.

Mono- and bis(steroids) containing a cyclooctane core: Synthesis, antiproliferative activity, and action on cell cytoskeleton microtubules.

Steroid dimers of natural and synthetic origin possess an unusual and complex molecular architecture that may lead to the realization of peculiar effects in biological systems, in particular in different cancer cell lines. In the present work, diastereoselective ring-opening of mono- and polyoxiranes, containing a cyclooctane core, by azide-anion was performed to yield a series of azidoalcohols with different types of symmetry. The products were involved in copper-catalyzed azyde-alkyne cycloaddition (CuAAC) reaction with ethinylestradiol and ethinyltestosterone, and the resulting steroids and steroid dimers with triazole linkers were screened for their antiproliferative activity via (3-(4,5-dimethylthiazol-2-yl)2,5-diphenyl tetrazolium bromide) assay. All the compounds revealed cytotoxicity toward several cancer cell lines. The effect of the most potent compound, containing two estradiol moieties, on the microtubules (MT) dynamics was investigated by immunofluorescent microscopy. The disruption of the majority of interphase cell cytoplasmic MT and mitotic event disturbances in the presence of the studied compound were observed. The latter effect caused the appearance of numerous multinucleated cells.

The role of task on the human brain's responses to, and representation of, visual regularity defined by reflection and rotation

Identifying and segmenting objects in an image is generally achieved effortlessly and is facilitated by the presence of symmetry: a principle of perceptual organisation used to interpret sensory inputs from the retina into meaningful representations. However, while imaging studies show evidence of symmetry selective responses across extrastriate visual areas in the human brain, whether symmetry is processed automatically is still under debate. We used functional Magnetic Resonance Imaging (fMRI) to study the response to and representation of two types of symmetry: reflection and rotation. Dot pattern stimuli were presented to 15 human participants (10 female) under stimulus-relevant (symmetry) and stimulus-irrelevant (luminance) task conditions. Our results show that symmetry-selective responses emerge from area V3 and extend throughout extrastriate visual areas. This response is largely maintained when participants engage in the stimulus irrelevant task, suggesting an automaticity to processing visual symmetry. Our multi-voxel pattern analysis (MVPA) results extend these findings by suggesting that not only spatial organisation of responses to symmetrical patterns can be distinguished from that of non-symmetrical (random) patterns, but also that representation of reflection and rotation symmetry can be differentiated in extrastriate and object-selective visual areas. Moreover, task demands did not affect the neural representation of the symmetry information. Intriguingly, our MVPA results show an interesting dissociation: representation of luminance (stimulus irrelevant feature) is maintained in visual cortex only when task relevant, while information of the spatial configuration of the stimuli is available across task conditions. This speaks in favour of the automaticity for processing perceptual organisation: extrastriate visual areas compute and represent global, spatial properties irrespective of the task at hand.

Critical velocities of a three-layer composite tube incorporating the rotary inertia and material anisotropy

Critical velocities of a three-layer composite tube subjected to a uniform internal pressure moving at a constant velocity are obtained in closed-form expressions. A Love–Kirchhoff thin shell model including the rotary inertia and material anisotropy effects is used in the formulation. The composite tube is made of three perfectly bonded cylindrical layers of dissimilar materials, each of which can be orthotropic, transversely isotropic, cubic or isotropic. Closed-form formulas for the critical velocities are first derived for the general case by incorporating the effects of material anisotropy, rotary inertia and radial stress. Specific formulas are then obtained for composite tubes without the rotary inertia effect and/or the radial stress effect and with various types of material symmetry for each layer as special cases. It is also shown that the current model for three-layer tubes can be reduced to those for single- and two-layer tubes. To illustrate the newly derived formulas, an example is provided for a composite tube consisting of an isotropic inner layer, an orthotropic core, and an isotropic outer layer. All four critical velocities of the composite tube are computed using the new closed-form formulas. Three values of the lowest critical velocity of the three-layer composite tube are analytically obtained from three sets of the new formulas, which agree well with the value computationally determined by others.

RECURRENCE SETS FOR PARTIAL INVERSE

Two types of recurrence sets are introduced for inverse semigroup partial actions in topological spaces. Some applications to the basic dynamical properties of the action are indicated, covering topics as topological transitivity, limit points and periodic points. We then explore the connections of these recurrence sets with similar notions for related types of imperfect symmetries (prefix inverse semigroup expansions, partial group and groupoid actions).