General Symmetry Research Articles

Product manifolds as realizations of general linear symmetries

This paper considers the relationship between geometry, symmetry and fundamental interactions — gravity and those mediated by gauge fields. We explore product spacetimes which (a) have the necessary symmetries for gauge interactions and four-dimensional gravity and (b) reduce to an [Formula: see text]-dimensional isotropic universe in their flat space limit. The key technique is looking at orbits of the operator form of symmetric rank-two tensors under changes of coordinate system. Orbits containing diagonal matrices are seen to correspond to product manifolds. The [Formula: see text] symmetry of the decompactified universe acts nonlinearly on such a product spacetime. We explore the resulting Kaluza–Klein theories, in which the internal symmetries act indirectly on space of the extra dimensions, and give two examples: a six-dimensional model in which the gauge symmetry is [Formula: see text] and a seven-dimensional model in which it is [Formula: see text]. We identify constraints that can be placed on any rank-two symmetric tensor to obtain such spacetimes: relationships between polynomial invariants. The multiplicities of its eigenvalues determine the dimensionalities of the factor spaces and hence the gauge symmetries. If the tensor in question is the Ricci tensor, other than two-dimensional factor spaces all the factor spaces are Einstein manifolds. This situation represents the classical vacuum of the Kaluza–Klein theory.

Invariance analysis and some new exact analytic solutions of the time-fractional coupled Drinfeld-Sokolov-Wilson equations

In this work, the fractional Lie symmetry method is used to find the exact solutions of the time-fractional coupled Drinfeld-Sokolov-Wilson equations with the Riemann-Liouville fractional derivative. Time-fractional coupled Drinfeld-Sokolov-Wilson equations are obtained by replacing the first-order time derivative to the fractional derivatives (FD) of order $\alpha$ in the classical Drinfeld-Sokolov-Wilson (DSW) model. Using the fractional Lie symmetry method, the Lie symmetry generators are obtained. With the help of symmetry generators, FCDSW equations are reduced into fractional ordinary differential equations (FODEs) with Erd$\acute{e}$lyi-Kober fractional differential operator. Also, we have obtained the exact solution of FCDSW equations and shown the effects of non-integer order derivative value on the solutions graphically. The effect of fractional order $\alpha$ on the behavior of solutions is studied graphically. Finally, new conservation laws are constructed along with the formal Lagrangian and fractional generalization of Noether operators. It is quite interesting the exact analytic solutions are obtained in explicit form.

General symmetry operators of the asymmetric quantum Rabi model

The true level crossing in the asymmetric quantum Rabi model without any obvious symmetry can be exhibited in the energy spectrum if the qubit bias is a multiple of the cavity frequency, which should imply the existence of the hidden symmetry. In this work, within a Bogoliubov operator approach, we can readily derive the symmetry operators associated with the hidden symmetry hierarchically for arbitrary multiples. The symmetry operators for small multiples in the literature can be extremely easily reproduced in our general scheme. In addition, a general parity operator is defined through the symmetry operator, which naturally includes the well-known parity operator of the symmetric model. We believe that the present approach can be straightforwardly extended to other asymmetric Rabi models to find the relevant symmetry operators.

Squared eigenfunction symmetries of the constrained integrable hierarchies

In this paper, we mainly investigate the constrained Kadomtsev–Petviashvili (KP) and modified KP (mKP) hierarchies. Squared eigenfunction (SE) symmetry is an important symmetry defined by eigenfunctions and adjoint eigenfunctions, which can be viewed as the generator of the additional symmetries. Due to special constraints on the Lax operators, it is found that there are three different cases for the generating eigenfunctions and adjoint eigenfunctions of the SE symmetries. Then we discuss the changes of the SE symmetries under the Miura links between the KP and mKP hierarchies. Next, the exponential of the SE symmetries is shown to be the binary Darboux transformations. At last based upon the results above, new additional symmetries for the constrained KP and mKP hierarchies are constructed, different from the additional Virasoro symmetries defined by Aratyn.

Non‐classical Lie symmetries for nonlinear time‐fractional Heisenberg equations

The current work is based on performing the non‐classical and classical Lie group analysis for a system of nonlinear time‐fractional Heisenberg equation. Here, we apply analysis of the Lie group symmetry as one of the powerful tools dealing with the large class of fractional order differential equations in the Riemann–Liouville (RL) sense. Indeed, classical and non‐classical Lie symmetries group is used to similarity reduction of nonlinear time‐fractional Heisenberg equation. Especially, based on the obtained Lie symmetry generators, we construct conservation laws for the classical and non‐classical vector fields related to the Heisenberg time‐fraction equation.

Algebraic solutions for pricing American put options under the constant elasticity of variance (CEV) model: Application of the Lie group approach

In this work, the algebraic properties of the American put option under the constant elasticity of variance (CEV) model of financial markets are studied with the implementation of the symmetry approach based on Lie’s group theory. The CEV model is described mathematically in terms of the parabolic time–space partial differential equation (PDE) along with the suitable choice of a terminal condition. The classical Lie symmetry analysis is carried out for the governing PDE. Based on the infinitesimal Lie symmetry generators, new classes of group invariant solutions are derived from the mathematical model and the general as well as the specific cases of the CEV model are thoroughly examined. Finally, the results are plotted against the various emergent parameters and their effect are analyzed and discussed.

YÜZ BÖLÜMLERİNİN ÖZELLİKLERİ İLE ORTODONTİK MALOKLUZYONLAR ARASINDA HERHANGİ BİR İLİŞKİ VAR MI? BÖLÜM I: FRONTAL GÖRÜNÜM

Background: The purpose of this study was to evaluate frontal facial features of individuals and analyze their relevance to orthodontic problems. Methods: Frontal photographs were taken and cephalometric data of 250 patient were collected. Photographs were used to evaluate facial dimensions, general symmetry, forehead, trichion anatomical landmark position, nostril visibility, nasal root breaking point, glabella position, cantus positions, eyeball symmetry, sclera visibility, nose symmetry and shape, nose widths, mouth width, lip vermillion areas and chin. Correlation between photographic data and malocclusion groups was statistically evaluated. Results: Forehead shape, nostril visibility, mouth width and chin symmetry differences were statistically significant due to type of skeletal malocclusion. Conversely, there are significant correlations between forehead shape, nose symmetry, upper lip vermillion, and chin symmetry and dental malocclusions. Chin asymmetry is mostly seen with skeletal Class III malocclusion, while nasal asymmetry with dental Class III malocclusion. Asymmetry is mostly seen in lower third of the face. In dental Class III malocclusion patients, forehead was mostly "wide". In front view, skeletal Class I and II individuals have increased nostril visibility and Class III decreased nostril visibility. Conclusions: The photographic analyzes add perspective to orthodontics. Skeletal class III patients are prone to asymmetries especially in lower third of the face.

Investigation of wave solutions and conservation laws of generalized Calogero–Bogoyavlenskii–Schiff equation by group theoretic method

This work is focused to analyze the generalized Calogero–Bogoyavlenskii–Schiff equation (GCBSE) by the Lie symmetry method. GCBS equation has been utilized to explain the wave profiles in soliton theory. GCBSE was constructed by Bogoyavlenskii and Schiff in different ways (explained in the introduction section). With the aid of Lie symmetry analysis, we have computed the symmetry generators of the GCBSE and commutation relation. We observed from the commutator table, translational symmetries make an Abelian algebra. Then by using the theory of Lie, we have discovered the similarity variables, which are used to convert the supposed nonlinear partial differential equation (NLPDE) into a nonlinear ordinary differential equation (NLODE). Using the new auxiliary method (NAM), we have to discover some new wave profiles of GCBSE in the type of few trigonometric functions. These exits some parameters which we give to some suitable values to attain the different diagrams of some obtained solutions. Further, the GCBSE is presented by non-linear self-adjointness, and conserved vectors are discovered corresponding to each generator.

Anomaly cascade in ( 2+1 )-dimensional fermionic topological phases

We develop a theory of anomalies of fermionic topological phases of matter in (2+1)D with a general fermionic symmetry group $G_f$. In general, $G_f$ can be a non-trivial central extension of the bosonic symmetry group $G_b$ by fermion parity $(-1)^F$. We encounter four layers of obstructions to gauging the $G_f$ symmetry, which we dub the anomaly cascade: (i) An $\mathcal{H}^1(G_b,\mathbb{Z}_{\bf T})$ obstruction to extending the symmetry permutations on the anyons to the fermion parity gauged theory, (ii) An $\mathcal{H}^2(G_b, \ker r)$ obstruction to extending the $G_b$ group structure of the symmetry permutations to the fermion parity gauged theory, where $r$ is a map that restricts symmetries of the fermion parity gauged theory to the anyon theory, (iii) An $\mathcal{H}^3(G_b, \mathbb{Z}_2)$ obstruction to extending the symmetry fractionalization class to the fermion parity gauged theory, and (iv) the well-known $\mathcal{H}^4(G_b, U(1))$ obstruction to developing a consistent theory of $G_b$ symmetry defects for the fermion parity gauged theory. We describe how the $\mathcal{H}^2$ obstruction can be canceled by anomaly inflow from a bulk (3+1)D symmetry-protected topological state (SPT) and also its relation to the Arf invariant of spin structures on a torus. If any anomaly in the above sequence is non-trivial, the subsequent ones become relative anomalies. A number of conjectures regarding symmetry actions on super-modular categories, guided by general expectations of anomalies in physics, are also presented.

Comparison between Cylindrical, Trigonal, and General Symmetry Models for the Analysis of Polarization-Dependent Second Harmonic Generation Measurements Acquired from Collagen-Rich Equine Pericardium Samples

Polarization-dependent second harmonic generation (PSHG) microscopy is used as an innovative, high-resolution, non-destructive, and label-free diagnostic imaging tool to elucidate biological issues with high significance. In the present study, information on the structure and directionality of collagen fibers in equine pericardium tissue was collected using PSHG imaging measurements. In an effort to acquire precise results, three different mathematical models (cylindrical, trigonal, and general) were applied to the analysis of the recorded PSHG datasets. A factor called the “ratio parameter” was calculated to provide quantitative information. The implementation of the trigonal symmetry model to the recorded data led to the extraction of improved results compared with the application of the widely used cylindrical symmetry model. The best outcome was achieved through the application of the general model that does not include any kind of symmetry for the data processing. Our findings suggest that the trigonal symmetry model is preferable for the analysis of the PSHG datasets acquired from the collagenous tissues compared with the cylindrical model approach although an increased computational time is required.

A Complex Network-Based Airspace Association Network Model and Its Characteristic Analysis

As airspace congestion becomes more and more serious, it not only affects the efficiency and quality of aviation activities, but also poses a greater safety hazard. For in-depth exploration of the airspace network, a new airspace association network that differs from the traditional aviation approach of organizing the airspace into a network of sectors based on aircraft traffic flow is proposed. By judging the relationship between the size and location of the airspace and other properties, the establishment of an association network provides new ideas for airspace conflict detection and other aerial tasks. The three key factors of airspace use conflict were extracted and quantified, then the Analytic Hierarchy Process (AHP) was used to assign weights to basic symmetric operators corresponding to three factors to calculate the final airspace correction degree, which reveals the magnitude of correlation degree between the airspace caused by the combined action of the above three factors. Finally, an airspace association network model was established according to the association degree, using basic symmetric operators to explore the general symmetry and network characteristics in real-world airspace association networks, and the feature indices, such as degree, strength, average correlation, clustering coefficient, and structure information entropy of the network, are proposed. The results show that the nodes with higher node degree and node association strength are often key nodes in the network, and the areas with higher average association degree and clustering coefficient are often more prone to airspace conflicts. At the same time, it was also verified that the safety distance between airspace has the greatest impact on the characteristics of the airspace association network.

Existence of wormholes in f(𝒢) gravity using symmetries

The current study examines the geometry of static wormholes with anisotropic matter distribution in the context of modified [Formula: see text] gravity. We consider the well-known Noether and conformal symmetries, which help in investigating wormholes in [Formula: see text] gravity. For this purpose, we develop symmetry generators associated with conserved quantities by taking into consideration the [Formula: see text] gravity model. Moreover, we use the conservation relationship gained from the classical Noether method and conformal Killing symmetries to develop the metric potential. These symmetries provide a strong mathematical background to investigate wormhole solutions by incorporating some suitable initial conditions. The obtained conserved quantity performs a significant role in defining the essential physical characteristics of the shape-function and energy conditions. Further, we also describe the stability of obtained wormholes solutions by employing the equilibrium condition in modified [Formula: see text] gravity. It is observed from graphical representation of obtained wormhole solutions that Noether and conformal Killing symmetries provide the results with physically accepted patterns.

Lie group analysis for a higher-order Boussinesq-Burgers system

Boussinesq-Burgers (BB)-type equations have been proposed to model the shallow water waves. Under investigation in this Letter is a higher-order BB system. We obtain the Lie point symmetry generators, Lie symmetry groups and symmetry reductions for that system via the Lie group method. Hyperbolic-function, trigonometric-function and rational solutions for that system are derived.

Two-point resistances in Archimedean resistor networks

The rise of novel materials such as graphene compounds or carbon nanotubes recently revived the interest on the role of symmetries upon electrical properties. Among possible solids, symmetrical polytopes, widely realized in natural and artificial matter, deserve particular attention. When equal resistors are attached between neighbouring nodes of Perfect (Platonic) solids (PS) along the edges, the two-point resistance between any pair of nodes was established in the 1990s by van Steenwijk using an elegant and efficient method. Using van Steenwijk’s method, Moddy and Aravind subsequently derived the resistance values for two Archimedean and two Catalan solids. Here, with the same method, we derive the exact expression for the two-point resistance between any two nodes for resistor networks made of equal resistors placed along the edges of the thirteen Archimedean solids (AS). While the calculation remains elementary for the simplest AS, a dedicated method using computer assistance was developed for the Snub Cube, the Snub Dodecahedron and the three AS with three different rotation symmetries (the Great Rhombicuboctahedron, the Small Rhombicosidodecahedron and the Great Rhombicosidodecahedron). The largest resistance value (191/90) is obtained between the two opposite nodes of the Truncated Dodecahedron, and the second largest (42815/21114) is obtained between the two opposite nodes of the Great Rhombicosidodecahedron. The smallest resistance value (14137/38016) is obtained between some of the neighbouring nodes of the Snub Cube, whereas the resistance between neighbouring nodes of the Icosahedron (11/30) is slightly smaller. Some general symmetry relations between two-point resistances in AS networks are also derived, as well as some relations with two-point resistances in PS networks. The complete set of exact two-point resistance values for PS and AS networks can be used to check numerical codes or to evaluate the capacity of regular solids to provide appropriate models for given experimental situations.

Fermionic symmetry fractionalization in (2+1) dimensions

We develop a systematic theory of symmetry fractionalization for fermionic topological phases of matter in (2+1)D with a general fermionic symmetry group $G_f$. In general $G_f$ is a central extension of the bosonic symmetry group $G_b$ by fermion parity, $(-1)^F$, characterized by a non-trivial cohomology class $[\omega_2] \in \mathcal{H}^2(G_b, \mathbb{Z}_2)$. We show how the presence of local fermions places a number of constraints on the algebraic data that defines the action of the symmetry on the super-modular tensor category that characterizes the anyon content. We find two separate obstructions to defining symmetry fractionalization, which we refer to as the bosonic and fermionic symmetry localization obstructions. The former is valued in $\mathcal{H}^3(G_b, K(\mathcal{C}))$, while the latter is valued in either $\mathcal{H}^3(G_b,\mathcal{A}/\{1,\psi\})$ or $Z^2(G_b, \mathbb{Z}_2)$ depending on additional details of the theory. $K(\mathcal{C})$ is the Abelian group of functions from anyons to $\mathrm{U}(1)$ phases obeying the fusion rules, $\mathcal{A}$ is the Abelian group defined by fusion of Abelian anyons, and $\psi$ is the fermion. When these obstructions vanish, we show that distinct symmetry fractionalization patterns form a torsor over $\mathcal{H}^2(G_b, \mathcal{A}/\{1,\psi\})$. We study a number of examples in detail; in particular we provide a characterization of fermionic Kramers degeneracy arising in symmetry class DIII within this general framework, and we discuss fractional quantum Hall and $\mathbb{Z}_2$ quantum spin liquid states of electrons.

Lie symmetry analysis, power series solutions and conservation laws of the time-fractional breaking soliton equation

The main attention of this work focus on extending the Lie symmetry and conservation law theories to the fractional partial differential equations involving the mixed derivative of the Riemann-Liouville time-fractional and first-order x-derivatives. More specifically, we first present a new prolongation formula of the infinitesimal generators of Lie symmetries for the time-fractional breaking soliton equation since the equation involves the mixed derivative, then perform Lie symmetry analysis for the equation. Furthermore, we construct an optimal system of one-dimensional Lie subalgebras and use them to reduce the equation to lower-dimensional fractional partial differential equations involving the Erdélyi-Kober operator. In order to construct the power series solution of the equation, we introduce the Hadamard's finite-part integral to deal with the divergence of the integrals. The convergence and error estimate of the power series solution are proved. Finally, a new conservation law formula for the equation is given by means of the nonlinear self-adjointness method and nontrivial conservation laws are found.

Trudinger–Moser‐type inequality with logarithmic convolution potentials

We establish Moser–Trudinger-type inequalities in the presence of a logarithmic convolution potential when the domain is a ball or the entire space R 2 $\mathbb {R}^2$ . Moreover, we characterize critical nonlinear growth rates for these inequalities to hold and for the existence of corresponding extremal functions. In addition, we show that extremal functions satisfy corresponding Euler–Lagrange equations, and we derive general symmetry and uniqueness results for solutions of these equations.

Weighting the relative importance of behaviors affecting gait score.

This research communication evaluates experts' opinions on the importance and weights of six gait aspects. In 2016, a Qualtrics (Qualtrics LLC., Provo, Utah) survey was distributed to lameness experts. Six gait aspects - general symmetry, tracking, spine curvature, head bobbing, speed and abduction as well as adduction were included. Respondents were asked to rank the gait aspects from 1 (most important) to 6 (least important), and to indicate which weight each gait aspect should receive when assessing lameness. For each gait aspect, frequency (percentage %) was used to describe the distribution of rank, and medians as well as 25th and 75th percentiles were used to summarize assigned weights. Thirty-nine percent of respondents ranked general symmetry first, followed by 32% for tracking, and 19% ranked spine curvature third. Head bobbing ranked fourth with 10% whereas, speed, abduction and adduction were not ranked. The median, 25th and 75th percentiles weight for each gait aspect were: general symmetry (25, 15, and 30), tracking (20, 10, and 30), spine curvature (20, 10, and 21), head bobbing (15, 10, and 20), speed (10, 5, and 20), and abduction and adduction (10, 5, and 10). General symmetry and tracking were deemed the most important gait aspects. A composite gait score can be calculated based on weighted importance of different gait aspects to indicate possible lameness.

BHLS_2 upgrade: tau spectra, muon HVP and the [pi ^0,~eta ,~{eta ^prime }] system

The generic hidden local symmetry (HLS) model has recently given rise to its hbox {BHLS}_2 variant, defined by introducing symmetry breaking mostly in the vector meson sector; the central mechanism is a modification of the covariant derivative at the root of the HLS approach. However, the description of the tau dipion spectra, especially the Belle one, is not fully satisfactory, whereas the simultaneous dealing with its annihilation sector (e^+ e^- rightarrow pi ^+ pi ^-/pi ^+ pi ^-pi ^0/ pi ^0 gamma /eta gamma /K^+ K^-/K_L K_S) is optimum. We show that this issue is solved by means of an additional breaking term which also allows us to consistently include the mixing properties of the [pi ^0,eta ,{eta ^prime }] system within this extended hbox {BHLS}_2 (hbox {EBHLS}_2) scope. This mechanism, an extension of the usual ’t Hooft determinant term, only affects the kinetic energy part of the hbox {BHLS}_2 Lagrangian. One thus obtains a fair account for the tau dipion spectra which complements the fair account of the annihilation channels already reached. The Belle dipion spectrum is found to provide evidence in favor of a violation of the conserved vector current (CVC) in the tau lepton decay; this evidence is enforced by imposing the conditions <0|J_mu ^q |[q^prime overline{q^prime }](p)>=ip_mu f_q delta _{q q^prime }, { [q {overline{q}}], q=u,d,s} on hbox {EBHLS}_2 axial current matrix elements. hbox {EBHLS}_2 is found to recover the usual (completed) formulae for the [pi ^0,~eta ,~{eta ^prime }] mixing parameters, and the global fits return mixing parameter values in agreement with expectations and better uncertainties. Updating the muon hadronic vacuum polarization (HVP), one also argues that the strong tension between the KLOE and BaBar pion form factors imposes to provide two solutions, namely a_mu ^{HVP-LO}(mathrm{KLOE})=687.48 pm 2.93 and a_mu ^{HVP-LO}(mathrm{BaBar})=692.53 pm 2.95, in units of 10^{-10}, rather than some combination of these. Taking into account common systematics, their differences from the experimental BNL-FNAL average value exhibit significance > 5.4sigma (KLOE) and > 4.1sigma (BaBar), with fit probabilities favoring the former.

A Family of Generalized Legendre-Based Apostol-Type Polynomials

Numerous polynomials, their extensions, and variations have been thoroughly explored, owing to their potential applications in a wide variety of research fields. The purpose of this work is to provide a unified family of Legendre-based generalized Apostol-Bernoulli, Apostol-Euler, and Apostol-Genocchi polynomials, with appropriate constraints for the Maclaurin series. Then we look at the formulae and identities that are involved, including an integral formula, differential formulas, addition formulas, implicit summation formulas, and general symmetry identities. We also provide an explicit representation for these new polynomials. Due to the generality of the findings given here, various formulae and identities for relatively simple polynomials and numbers, such as generalized Bernoulli, Euler, and Genocchi numbers and polynomials, are indicated to be deducible. Furthermore, we employ the umbral calculus theory to offer some additional formulae for these new polynomials.