Complete Symmetry Research Articles

Complete symmetry breaking constraints for the class of uniquely Hamiltonian graphs

Complete symmetry breaking constraints for the class of uniquely Hamiltonian graphs

Ladder Operators and Hidden Algebras for Shape Invariant Nonseparable and Nondiagonalizable Modelswith Quadratic Complex Interaction. II. Three-Dimensional Model

A shape invariant nonseparable and nondiagonalizable three-dimensional model with quadratic complex interaction was introduced by Bardavelidze, Cannata, Ioffe, and Nishnianidze. However, the complete hidden symmetry algebra and the description of the associated states that form Jordan blocks remained to be studied. We present a set of six operators $\{A^{\pm},B^{\pm},C^{\pm}\}$ that can be combined to build a ${\mathfrak{gl}}(3)$ hidden algebra. The latter can be embedded in an ${\mathfrak{sp}}(6)$ algebra, as well as in an ${\mathfrak{osp}}(1/6)$ superalgebra. The states associated with the eigenstates and making Jordan blocks are induced in different ways by combinations of operators acting on the ground state. We present the action of these operators and study the construction of an extended biorthogonal basis. These rely on establishing various nontrivial polynomial and commutator identities. We also make a connection between the hidden symmetry and the underlying superintegrability property of the model. Interestingly, the integrals generate a cubic algebra. This work demonstrates how various concepts that have been applied widely to Hermitian Hamiltonians, such as hidden symmetries, superintegrability, and ladder operators, extend to the pseudo-Hermitian case with many differences.

Applications of Solvable Lie Algebras to a Class of Third Order Equations

A family of third-order partial differential equations (PDEs) is analyzed. This family broadens out well-known PDEs such as the Korteweg-de Vries equation, the Gardner equation, and the Burgers equation, which model many real-world phenomena. Furthermore, several macroscopic models for semiconductors considering quantum effects—for example, models for the transmission of electrical lines and quantum hydrodynamic models—are governed by third-order PDEs of this family. For this family, all point symmetries have been derived. These symmetries are used to determine group-invariant solutions from three-dimensional solvable subgroups of the complete symmetry group, which allow us to reduce the given PDE to a first-order nonlinear ordinary differential equation (ODE). Finally, exact solutions are obtained by solving the first-order nonlinear ODEs or by taking into account the Type-II hidden symmetries that appear in the reduced second-order ODEs.

A Study of Drag Reduction on Cylinders with Different V-Groove Depths on the Surface

In this study, the drag reduction effect is studied for a cylinder with different V-groove depths on its surface using a k-ω/SST (Shear Stress Transport) turbulence model of computational fluid dynamics (CFD), while a particle image velocimetry (PIV) system is employed to analyze the wake characteristics for a smooth cylinder and a cylinder with different V-groove depths on its surface at different Reynolds numbers. The study focuses on the characteristics of the different V-groove depths on lift coefficient, drag coefficient, the velocity distribution of flow field, pressure coefficient, vortex shedding, and vortex structure. In comparison with a smooth cylinder, the lift coefficient and drag coefficient can be reduced for a cylinder with different V-groove depths on its surface, and the maximum reduction rates of lift coefficient and drag coefficient are about 34.4% and 16%, respectively. Otherwise, the vortex structure presents a complete symmetry for the smooth cylinder, however, the symmetry of the vortex structure becomes insignificant for the V-shaped groove structure with different depths. This is also an important reason for the drag reduction effect of a cylinder with a V-groove surface.

Research on the Supply Chain Performance Evaluation Mechanism Based on Emergencies

Some emergencies have produced enormous changes in market demand in recent years as the economy has progressed. The supply chain’s capacity to adapt to this market demand has a substantial impact on the company’s future growth. As the frequency of emergencies rises, emergency management is becoming increasingly critical for enterprises. This article examines the responsiveness and profit status of the supply chain under the three conditions of total complete information symmetry, incomplete information symmetry, and information asymmetry based on the information sharing degree of supply chain firms. The research offers a novel approach to emergency management. It makes recommendations for how the company may share information to improve its emergency management capabilities. Finally, the feasibility of the model is verified and analyzed by a case.

Families of Generalized Quasisymmetry Models: A ϕ-Divergence Approach

The quasisymmetry (QS) model for square contingency tables is revisited, highlighting properties and features on the basis of its alternative definitions. More parsimonious QS-type models, such as the ordinal QS model for ordinal classification variables and models based on association models (AMs) with homogeneous row and column scores, are discussed. All these models are linked to the local odds ratios (LOR). QS-type models and AMs were extended in the literature for generalized odds ratios other than LOR. Furthermore, in an information-theoretic context, they are expressed as distance models from a parsimonious reference model (the complete symmetry for QS and the independence for AMs), while they satisfy closeness properties with respect to Kullback–Leibler (KL) divergence. Replacing the KL by ϕ divergence, flexible classes of QS-type models for LOR, AMs for LOR, and AMs for generalized odds ratios were generated. However, special QS-type models that are based on homogeneous AMs for LOR have not been extended to ϕ-divergence-based classes so far, or the QS-type models for generalized odds ratios. In this work, we develop these missing extensions, and discuss QS-type models and their generalizations in depth. These flexible families enrich the modeling options, leading to models of better fit and sound interpretation, as illustrated by representative examples.

Holographic symmetry algebras for gauge theory and gravity

All 4D gauge and gravitational theories in asymptotically flat spacetimes contain an infinite number of non-trivial symmetries. They can be succinctly characterized by generalized 2D currents acting on the celestial sphere. A complete classification of these symmetries and their algebras is an open problem. Here we construct two towers of such 2D currents from positive-helicity photons, gluons, or gravitons with integer conformal weights. These generate the symmetries associated to an infinite tower of conformally soft theorems. The current algebra commutators are explicitly derived from the poles in the OPE coefficients, and found to comprise a rich closed subalgebra of the complete symmetry algebra.

Textile D-forms and D4d

D-forms were originally created from inflexible materials and have subsequently been considered as abstract mathematical objects. This paper describes a textile instance of a D-form, with ornamentation of the constituent surfaces as the highlighted feature. A set of 11 biscornu has been fashioned to provide a 3D sampler of the axial point group and its subgroups, using hitomezashi. Thus, this paper provides a link between the D-form literature and that of complete symmetry samplers in the fibre arts.

First-Principles Characterization of Surface Phonons of Halide Perovskite CsPbI3 and Their Role in Stabilization.

The stability of halide perovskites has been a long-standing issue for their real-world application. Approaches to improve stability include nanostructuring, dimensionality reduction, and strain engineering, where surfaces play an important role in the formation of a stable structure. To understand the mechanism we compute the lattice dynamics of the surface of CsPbI3 using density functional theory. We demonstrate, for the first time, that CsPbI3 crystals exhibit surface phonons that are localized on the outermost layers of the slabs, and we perform a complete symmetry characterization including an identification of the Raman/IR active modes. These surface phonons are present in the optically active cubic phase but are absent in the optically inactive "yellow" phase. Furthermore, we show that the surface suppresses bulk instabilities by hardening soft modes of the bulk cubic phase, resulting in phase stabilization and quenching of dynamical disorder. This study is fundamental for understanding the structural behavior of halide perovskite materials with high surface area-to-volume ratios, and for guiding stabilization strategies.

N-dimensional Smorodinsky–Winternitz model and related higher rank quadratic algebra

The N-dimensional Smorodinsky–Winternitz system is a maximally superintegrable and exactly solvable model, being subject of study from different approaches. The model has been demonstrated to be multiseparable with wavefunctions given by Laguerre and Jacobi polynomials. In this paper we present the complete symmetry algebra of the system, which it is a higher-rank quadratic one containing the recently discovered Racah algebra as subalgebra. The substructures of distinct quadratic algebras and their related Casimirs are also studied. In this way, from the constraints on the oscillator realizations of these substructures, the energy spectrum of the N-dimensional Smorodinsky–Winternitz system is obtained. We show that allows different set of substructures based on the Racah algebra which can be applied independently to algebraically derive the spectrum of the system.

Optimal system and classification of invariant solutions of nonlinear class of wave equations and their conservation laws

We study the classification of invariant solutions of a class of nonlinear wave equations using Lie symmetry analysis and the underlying optimal systems of subalgebras. We propose a classification of Lie generators via optimal systems for four cases that arise therein. These optimal systems are presented in a convenient tree leaf diagram. Corresponding to each class, complete symmetry reductions and the invariant solutions are presented. To the best of our knowledge, this classification of optimal systems is new and do not appear in the literature. Our results also lead to the establishment of the local conservation laws corresponding to each conserved vector via the multiplier approach.

Connectivity of Semiring Valued Graphs

Graph connectivity theory is important in network implementations, transportation, network routing and network tolerance, among other things. Separation edges and vertices refer to single points of failure in a network, and so they are often sought-after. Chandramouleeswaran et al. introduced the principle of semiring valued graphs, also known as S-valued symmetry graphs, in 2015. Since then, works on S-valued symmetry graphs such as vertex dominating set, edge dominating set, regularity, etc. have been done. However, the connectivity of S-valued graphs has not been studied. Motivated by this, in this paper, the concept of connectivity in S-valued graphs has been studied. We have introduced the term vertex S-connectivity and edge S-connectivity and arrived some results for connectivity of a complete S-valued symmetry graph, S-path and S-star. Unlike the graph theory, we have observed that the inequality for connectivity κ(G)≤κ′(G)≤δ(G) holds in the case of S-valued graphs only when there is a symmetry of the graph as seen in Examples 3–5.

A reaction–diffusion system with cross-diffusion: Lie symmetry, exact solutions and their applications in the pandemic modelling

A non-linear reaction–diffusion system with cross-diffusion describing the COVID-19 outbreak is studied using the Lie symmetry method. A complete Lie symmetry classification is derived and it is shown that the system with correctly specified parameters admits highly non-trivial Lie symmetry operators, which do not occur for all known reaction–diffusion systems. The symmetries obtained are also applied for finding exact solutions of the system in the most interesting case from applicability point of view. It is shown that the exact solutions derived possess typical properties for describing the pandemic spread under 1D approximation in space and lead to the distributions, which qualitatively correspond to the measured data of the COVID-19 spread in Ukraine.

On the ineffectiveness of constant rotation in the primitive equations and their symmetry analysis

Modern weather and climate prediction models are based on a system of nonlinear partial differential equations called the primitive equations. Lie symmetries of the primitive equations with zero external heating rate are computed and the structure of its maximal Lie invariance algebra, which is infinite-dimensional, is studied. The maximal Lie invariance algebra for the case of a nonzero constant Coriolis parameter is mapped to the case of vanishing Coriolis force. The same mapping allows one to transform the constantly rotating primitive equations to the equations in a resting reference frame. This mapping is used to obtain exact solutions for the rotating case from exact solutions for the nonrotating equations. Another important result of the paper is the computation of the complete point symmetry group of the primitive equations using the algebraic method.

Inverse Derivative Operator and Umbral Methods for the Harmonic Numbers and Telescopic Series Study

The formalism of differ-integral calculus, initially developed to treat differential operators of fractional order, realizes a complete symmetry between differential and integral operators. This possibility has opened new and interesting scenarios, once extended to positive and negative order derivatives. The associated rules offer an elegant, yet powerful, tool to deal with integral operators, viewed as derivatives of order-1. Although it is well known that the integration is the inverse of the derivative operation, the aforementioned rules offer a new mean to obtain either an explicit iteration of the integration by parts or a general formula to obtain the primitive of any infinitely differentiable function. We show that the method provides an unexpected link with generalized telescoping series, yields new useful tools for the relevant treatment, and allows a practically unexhausted tool to derive identities involving harmonic numbers and the associated generalized forms. It is eventually shown that embedding the differ-integral point of view with techniques of umbral algebraic nature offers a new insight into, and the possibility of, establishing a new and more powerful formalism.

The novel Ideal Symmetry Genetic Code table – Common purine-pyrimidine symmetry net for all RNA and DNA species

Ever since Nirenberg’s discovery in 1961 in which codons code individual amino acids, numerous scientists searched for symmetries within the genetic code. The standard genetic code (SGC) table is an alphabetic artificial construct based on the U-C-A-G ordering of nucleotides without natural symmetries. Up to the present, complete symmetry in the genetic code has not been found, leaving doubt as to whether the symmetrical nature as the protector of order even exists. Our novel Ideal Symmetry Genetic Code (ISyGC) table reflects a unique fundamental physicochemical purine-pyrimidine symmetry net for all more than thirty known variations of nuclear and mitochondrial genetic codes. The nuclear genetic code for RNA and DNA viruses also contains the same purine-pyrimidine symmetry net. We show that the ISyGC table leads to automatic transformation into a DNA sequence akin to the 5′3 codon and 3′5 anticodon patterns. As a result of purine-pyrimidine symmetries between codons in the ISyGC table, algorithms of the first two bases as well of the third base of codons show how tRNA cognate anticodons can recognize synonymous codons during mRNA decoding. We show that the ISyGC purine-pyrimidine net with its physicochemical properties represents an evolutionary common “frozen accident” at the onset of each genetic code creation and RNA to DNA evolution. As such, during all of evolution the unique fundamental purine-pyrimidine symmetry net of all genetic codes remains unchangeable. In this way, evolution is a road paved with symmetries.

Extensions on two-component BKP and D type Drinfeld–Sokolov hierarchies

In this article, we define a two-component coupled BKP hierarchy defined by four pseudo-differential Lax operators and construct its additional symmetries. These additional symmetries have a structure of a coupled $$w^B_{\infty }\times w^B_{\infty } $$ Lie algebra. Further we construct a new coupled D type Drinfeld–Sokolov hierarchy, which is a reduction of the two-component coupled BKP hierarchy, and the hierarchy possesses a complete Virasoro symmetry.

Autologous Free Dermal-Fat-Fascial Graft for Parotidectomy Defects: A Case Series.

Background:Frey’s syndrome and facial asymmetry from loss of parotid tissue are long-term sequelae of parotid surgeries causing significant morbidity. Various techniques have been used to fill the parotidectomy defect, preserve facial contour symmetry, and prevent Frey’s syndrome. Free dermal-fat-fascial graft (DFFG) is one such technique; however, its use is largely undocumented in the literature. In this case series, we investigate the efficacy of free DFFG in reconstructing parotidectomy defects at 2 tertiary care centers.Materials and methods:Medical records of 54 patients who underwent primary parotidectomy and immediate reconstruction with autologous abdominal free DFFG by 2 surgeons in George Washington University Hospital and McGill University Health Centre between 2007 and 2019 were collected prospectively. Patients responded to 2 questionnaires addressing postoperative outcomes.Results:Fifty-four patients were included; 32 superficial parotidectomies and 22 total parotidectomies were performed for 39 benign and 15 malignant tumors. Thirty-seven patients could be reached. Out of 37 patients who responded to the first questionnaire, 59% (22) reported complete facial symmetry, 27% (10) reported mild hollowness, and 14% (5) reported mild fullness. None declared noticeable hollowness or fullness. While 81% (30) did not experience Frey’s syndrome, 5.4% (2) experienced mild symptoms without disability, and 13.5% (5) experienced debilitating symptoms. Out of 37 patients, 8 patients responded to a second questionnaire addressing the outcome of the abdominal graft donor site. In regard to the donor site, 87.5% (7) were satisfied or very satisfied from its cosmetic appearance, 75% (6) were not bothered by its cosmetic appearance, and 87.5% (7) had no discomfort at the graft donor site. Patients did not report any other symptom at the graft donor site.Conclusion:In this large series of total parotidectomies including malignant pathologies, autologous abdominal free DFFG effectively prevented Frey’s syndrome and preserved facial cosmesis in most patients.

Generalized stokes’ flow and radiative heat transfer model of a non-Newtonian fluid in a darcy porous medium subject to Navier’s slip conditions on the penetrable porous boundary: Group theoretical and compatibility analysis

In this work, the generalized Stokes’ model of a non-Newtonian fluid is discussed. The incompressible time-dependent flow and radiative heat transfer model of a non-Newtonian third grade fluid over a moving porous boundary is considered. The fluid flows into the porous region in the half space geometry. The flux disturbance caused by the impulsive movement of the porous boundary is studied. The energy equation is studied under the influence of internal heat source and thermal radiation. The flow and heat transfer model meets the Navier’s slip conditions at the boundary which allows the extremes of no-slipping at the penetrable surface. Lie symmetry analysis is carried out to calculate the complete Lie symmetry algebra associated with the modelled nonlinear partial differential equations. Symmetry operators are used to find self-similar transformations which help to perform reduction of the governing nonlinear partial differential equations to different classes of nonlinear ordinary differential equations. Reduced ordinary differential equations, as a result of the implication of symmetry reductions, are solved by employing the general compatibility approach. Furthermore, we have discussed the non-classical symmetry algebra, associated with the model, to classify all possible closed-form exact solutions of the flow model. The effect of various pertinent parameters on the flow and heat transfer model is studied in detail.

Complete symmetry reductions, conservation laws and exact solutions for a (2 + 1)-dimensional nonlinear Schrödinger equation

This paper deal with a (2 + 1)-dimensional nonlinear Schrödinger (NLS) equation by the symmetry analysis, the multiplier approach and Ibragimov's conservation law method. Based on the optimal system of one dimensional subalgebras, complete similarity reduction, two types of further similarity reduction and three kinds of group-invariant solutions are obtained. Secondly, the different relationship between the NLS equation and sine-Gordon is derived. At last, some conservation laws are constructed for the NLS equation using both the multiplier method and Ibragimov's one.