Group Invariance Research Articles

On the minimum number of generators for the module of derivations of the ring of invariants for some dihedral groups

In this paper, we study the generators of the module of derivations of the ring of invariants for some dihedral groups. Let [Formula: see text] be an algebraically closed field of characteristic [Formula: see text] and [Formula: see text] be a finite dihedral group. Let [Formula: see text] be the ring of invariants obtained by the linear action of [Formula: see text] on [Formula: see text]. In [ R. V. Gurjar and A. Patra, On minimum number of generators for some derivation modules, J. Pure Appl. Algebra [Formula: see text](11) (2022) 107105], Gurjar–Patra proved that [Formula: see text]. We will give a better bound on [Formula: see text] in this paper.

Shock wave propagation in a real gas with or without gravitational field in the presence of magnetic field and monochromatic radiation via group invariance method

PurposeThis article aims to find the similarity solutions for the one-dimensional motion of spherical symmetric shock wave in non-ideal gas influenced by the azimuthal magnetic field and monochromatic radiation in the presence or absence of gravitational field. This paper also aims to study the effects of physical parameters on the strength of shock wave, and on the flow variables in the flow-field region behind the shock front.Design/methodology/approachThe Roche model is used to describe the gravitational field effects due to a massive nucleus at the point of symmetry. To derive the similarity solutions, the Lie group symmetry method has been used. Also, the numerical solutions to the present problem are obtained by using Rung–Kutta method of the fourth order with the use of Mathematica software. The effects of variation in the parameter of non-idealness of the gas, the gravitation parameter, the strength of the ambient magnetic field and the adiabatic index of the gas on the shock wave, and on the flow variables is discussed. A comparative study between with and without gravitational field is also, made.FindingsFor different choices of the arbitrary constants that appeared in the solution of infinitesimal generators, we have obtained seven distinct cases of similarity solutions. In the absence of the gravitational field, the similarity solution exists to the power and exponential law shock paths, but in the presence of gravitational field, the similarity solution exists to the power law shock path case only. In the absence of gravitational field, the shock strength is enhanced in the exponential law shock path case in comparison to the power law shock path case. It is found that the shock wave decays with an increase in the value of the adiabatic exponent, the strength of magnetic field, non-idealness of the gas or gravitational parameter.Research limitations/implicationsThe consideration of medium under the influence of gravitational field due to a heavy nucleus at the center and presence of magnetic field decrease the shock strength. This result may be helpful in designing space vehicle and jet engine.Practical implicationsThe result of the present study may be used in the analysis of data from the measurements by space craft in the solar wind and in neighborhood of the Earth’s magnetosphere.Social implicationsThe obtained results may be used for mankind.Originality/valueThe study of spherical shock wave propagation influenced by monochromatic radiation and azimuthal magnetic field in a non-ideal gas with or without gravitational field has yet to be discussed by any authors by using the Lie group symmetry method. In this article, we have discussed all possible cases of similarity solutions using the Lie group symmetry method, which is not studied by anyone as known to us.

Integrability and renormalizability for the fully anisotropic SU(2) principal chiral field and its deformations

For the class of 1 + 1 dimensional field theories referred to as the non-linear sigma models, there is known to be a deep connection between classical integrability and one-loop renormalizability. In this work, the phenomenon is reviewed on the example of the so-called fully anisotropic SU(2) Principal Chiral Field (PCF). Along the way, we discover a new classically integrable four parameter family of sigma models, which is obtained from the fully anisotropic SU(2) PCF by means of the Poisson-Lie deformation. The theory turns out to be one-loop renormalizable and the system of ODEs describing the flow of the four couplings is derived. Also provided are explicit analytical expressions for the full set of functionally independent first integrals (renormalization group invariants).

Witt invariants of Weyl groups

We describe the Witt invariants of a Weyl group over a field k0 by giving generators for the W(k0)-module of Witt invariants, under the assumption that the characteristic of k0 does not divide the order of the group. For the Weyl groups of types Bn, Cn, Dn, and G2, we show that the Witt invariants are generated as a W(k0)-algebra by trace forms and their exterior powers, extending a result due to Serre in type An. Many of our computational methods are applicable to computing Witt invariants of any smooth linear algebraic group over k0, including a technique for lifting module generators from cohomological invariants to Witt invariants.

On the modular isomorphism problem for 2-generated groups with cyclic derived subgroup

We continue the analysis of the modular isomorphism problem for [Formula: see text]-generated [Formula: see text]-groups with cyclic derived subgroup, [Formula: see text], started in [D. García-Lucas, Á. del Río and M. Stanojkowski, On group invariants determined by modular group algebras: Even versus odd characteristic, Algebra Represent. Theory 26 (2022) 2683–2707, doi:10.1007/s10468-022-10182-x]. We show that if [Formula: see text] belongs to this class of groups, then the isomorphism type of the quotients [Formula: see text] and [Formula: see text] are determined by its modular group algebra. In fact, we obtain a more general but technical result, expressed in terms of the classification [O. Broche, D. García-Lucas and Á. del Río, A classification of the finite 2-generator cyclic-by-abelian groups of prime-power order, Int. J. Algebra Comput. 33(4) (2023) 641–686]. We also show that for groups in this class of order at most [Formula: see text], the modular isomorphism problem has positive answer. Finally, we describe some families of groups of order [Formula: see text] whose group algebras over the field with [Formula: see text] elements cannot be distinguished with the techniques available to us.

Two characterizations of the smallest non-solvable group

We introduce two invariants for finite groups, and use each to characterize A 5.

Hilbert series for covariants and their applications to minimal flavor violation

We elaborate how to apply the Hilbert series method to enumerating group covariants, which transform under any given representation, including but going beyond group invariants. Mathematically, group covariants form a module over the ring of the invariants. The number of independent covariants is given by the rank of the module, which can be computed by taking a ratio of two Hilbert series. In many cases, the rank equals the dimension of the group covariant representation. When this happens, we say that there is a rank saturation. We apply this technology to revisit the hypothesis of Minimal Flavor Violation in constructing Effective Field Theories beyond the Standard Model. We find that rank saturation is guaranteed in this case, leading to the important consequence that the MFV symmetry principle does not impose any restriction on the EFT, i.e. MFV SMEFT = SMEFT, in the absence of additional assumptions.

Scattering of solutions with group invariance for the fourth-order nonlinear Schrödinger equation

In this paper, we consider the focusing, L 2-supercritical and H˙2 -subcritical fourth-order nonlinear Schrödinger equations. We show the scattering of group-invariant solutions below the ground state threshold, under the hypothesis that the threshold for group-invariant solutions is less than a certain value.

A study of self-adjointness, Lie analysis, wave structures, and conservation laws of the completely generalized shallow water equation

This article explores the analysis of the completely generalized Hirota–Satsuma–Ito equation through Lie symmetry analysis. The equation under consideration represents a more comprehensive form of the (2+1)-dimensional HSI equation, encompassing four additional second-order derivative terms: Δ3Hϖϖ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta _{3}H_{\\varpi \\varpi }$$\\end{document}, Δ4Hϖι\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta _{4}H_{\\varpi \\iota }$$\\end{document}, Δ3Hϖϖ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta _{3}H_{\\varpi \\varpi }$$\\end{document}, Δ4Hϖι\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta _{4}H_{\\varpi \\iota }$$\\end{document}, and Δ6Hιι\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta _{6}H_{\\iota \\iota }$$\\end{document}, emerging from the inclusion of second-order dissipative-type elements. We calculate the infinitesimal generators and determine the symmetry group for each generator using the Lie group invariance condition. Employing the conjugacy classes of the Abelian algebra, we transform the considered equation into an ordinary differential equation through similarity reduction. Subsequently, we solve these ordinary differential equations to derive closed-form solutions for the completely generalized Hirota–Satsuma–Ito equation under certain conditions. For other scenarios, we utilize the extended direct algebraic method to obtain soliton solutions. Furthermore, we rigorously calculated the conserved quantities corresponding to each symmetry generator, the conservation laws of the model are established using the multiplier approach. Additionally, we present the graphical representation of selected solutions for specific values of the physical parameters of the equation under scrutiny.

Cosmological solutions in the Brans–Dicke theory via invariants of symmetry groups

We proceed to obtain an exact analytical solution of the Brans–Dicke (BD) equations for the spatially flat ([Formula: see text]) Friedmann–Lamaître–Robertson–Walker (FLRW) cosmological model in both cases of the absence and presence of the cosmological constant. The solution method that we use to solve the field equations of the BD equations is called the “invariants of symmetry groups method” (ISG method). This method is based on the extended Prelle–Singer (PS) method and it employs the Lie point symmetries, [Formula: see text]-symmetries, and Darboux polynomials (DPs). Indeed, the ISG method tries to provide two independent first-order invariants associated to the one-parameter Lie groups of transformations keeping the ordinary differential equations (ODEs) invariant, as solutions. It should be noted for integrable ODEs that the ISG method guarantees the extraction of these two invariants. In this work, for the BD equations in FLRW cosmological model, we find the Lie point symmetries, [Formula: see text]-symmetries, and DPs, and obtain the basic quantities of the extended PS method (which are the null forms and the integrating factors). By making use of the extended PS method we find two independent first-order invariants in such a way that appropriate cosmological solutions from solving these invariants as a system of algebraic equations are simultaneously obtained. These solutions are wealthy in that they include many known special solutions, such as the O’Hanlon–Tupper vacuum solutions, Nariai’s solutions, Brans–Dicke dust solutions, inflationary solutions, etc.

Renormalization group improvement for thermally resummed effective potential

We propose a novel method for renormalization group improvement of thermally resummed effective potential. In our method, β-functions are temperature dependent as a consequence of the divergence structure in resummed perturbation theory. In contrast to the ordinary MS¯ scheme, the renormalization group invariance of the resummed finite-temperature effective potential holds order by order, which significantly mitigates a notorious renormalization scale dependence of phase transition quantities such as a critical temperature even at the one-loop order. We also devise a tractable method that enables one to incorporate temperature-dependent higher-order corrections by fully exploiting the renormalization group invariance. Published by the American Physical Society 2024

Exact FLRW cosmological solutions via invariants of the symmetry groups

Until now, various methods have been demonstrated to solve the Friedmann-Lamaítre-Robertson-Walker (FLRW) equations in the spatially flat (k = 0) cosmological model. In this study, in order to solve the field equations of the spatially flat FLRW cosmological model in the presence of Λ, a new method based on the invariants of the symmetry groups which we called ISG-method, is presented. This method is based on the extended Prelle-Singer (PS) method and it uses the Lie point symmetry, λ-symmetry and Darboux polynomials (DPs). We employ this method to extract systematically the two independent first integrals (or invariants) such as I1(a,ȧ)=c1 and I2(t,a,ȧ)=c2 associated to the group of the Lie point transformations keeping the Friedmann-Einstein dynamical equation (DE), ä=ϕ(t,a,ȧ) , invariant. The obtained solutions from solving the DEs of the FLRW cosmological model by the ISG-method are explicitly written, so that they are suitable for cosmological applications. Finally, as an application of the solutions we look at the age of universe in the presence of cosmological constant where the dominant matter of the Universe is considered to be fluid with the state parameter w. In this regard, we calculate the age of universe when the dominant matter is dust.

Separating invariants for two-dimensional orthogonal groups over finite fields

We described a minimal separating set for the algebra of O2+(Fq)-invariant polynomial functions of m-tuples of two-dimensional vectors over a finite field Fq.

Factor structure, group invariance, and concurrent validity of scores from the college eating and drinking behavior scale among U.S. college students

Food and alcohol disturbance (FAD) refers to the intersection of alcohol- and eating-related motives and behaviors, such as restricting food intake before or during alcohol use to offset caloric intake or to enhance intoxication. Valid assessment is critical for advancing research on FAD. We tested the factor structure, group invariance, and concurrent validity of the College Eating and Drinking Behavior Scale (CEDBS) in a large college student sample (n = 2610; Mage = 20.95, SD = 4.65; 71.8% female; 77% White; 86% non-Hispanic). Participants completed measures assessing antecedents of alcohol use (i.e., protective behavioral strategies and drinking motives), negative alcohol-related consequences, alcohol use severity, and risk for eating disorder. The 3-factor model of the 21-item CEDBS provided an adequate fit to the data (e.g., CFI = 0.916). These factors include Alternative Methods (4 items; “Use laxative prior to drinking alcohol”), Offset Calories (7 items; “Restrict calories prior to drinking to help maintain your figure”), and Quicker Intoxication (10 items; “Not eating before drinking alcohol because it gives you the best buzz”). The CEDBS was scalar invariant across subgroups of participants based on age, sex, race/ethnicity, socioeconomic status, sexual orientation, and political orientation. Quicker Intoxication was most strongly related to risk factors and negative consequences for alcohol (r = 0.204–0.379, all ps < 0.01), and Offset Calories was most strongly related to risk for eating disorders (r = 0.349, p < .01). These findings further support the CEDBS to assess FAD among college students.

A conjecture of Mallows and Sloane with the universal denominator of Hilbert series

Abstract A conjecture of Mallows and Sloane conveys the dominance of Hilbert series for finding basic invariants of finite linear groups if the Hilbert series of the invariant ring is of a certain explicit canonical form. However, the conjecture does not hold in general by a well-known counterexample of Stanley. In this article, we give a constraint on lower bounds for the degrees of homogeneous system of parameters of rings of invariants of finite linear groups depending on the universal denominator of Hilbert series defined by Derksen. We consider the conjecture with the universal denominator on abelian groups and provide some criteria guaranteeing the existence of homogeneous system of parameters of certain degrees. In this case, Stanley’s counterexample could be avoided, and the homogeneous system of parameters is optimal.

Precise determination of the top-quark on-shell mass via its scale- invariant perturbative relation to the top-quark mass * *Supported in part by the National Natural Science Foundation of China (12247129, 12175025, 12347101) and the Graduate Research and Innovation Foundation of Chongqing, China (ydstd1912)

The principle of maximum conformality (PMC) provides a systematic approach to solve the conventional renormalization scheme and scale ambiguities. Scale-fixed predictions of physical observables using the PMC are independent of the choice of renormalization scheme – a key requirement for renormalization group invariance. In this paper, we derive new degeneracy relations based on the renormalization group equations that involve both the usual β-function and the quark mass anomalous dimension -function. These new degeneracy relations enable improved PMC scale-setting procedures for correct magnitudes of the strong coupling constant and -running quark mass to be determined simultaneously. By using these improved PMC scale-setting procedures, the renormalization scale dependence of the -on-shell quark mass relation can be eliminated systematically. Consequently, the top-quark on-shell (or ) mass can be determined without conventional renormalization scale ambiguity. Taking the top-quark mass GeV as the input, we obtain GeV. Here, the uncertainties arise from errors combined with those from and the approximate uncertainty resulting from the uncalculated five-loop terms predicted through the Padé approximation approach.

Statistics for iwasawa invariants of elliptic curves, II

Statistics for iwasawa invariants of elliptic curves, II

Approximate N5LO Higgs Boson Decay Width Γ(H→γγ)

The precision and predictive power of perturbative QCD (pQCD) prediction depends on both a precise, convergent, fixed-order series and a reliable way of estimating the contributions of unknown higher-order (UHO) terms. It has been shown that by applying the principle of maximum conformality (PMC), which applies the renormalization group equation recursively to set the effective magnitude of αs of the process, the remaining conformal coefficients will be well matched with the corresponding αs at each order, leading to a scheme-and-scale invariant and more convergent perturbative series. The PMC series, being satisfied with the standard renormalization group invariance, has a rigorous foundation. Thus it not only can be widely applied to virtually all high-energy hadronic processes, but also can be a reliable platform for estimating UHO contributions. In this paper, by using the total decay width Γ(H→γγ) which has been calculated up to N4LO QCD corrections, we first derive its PMC series by using the PMC single-scale setting approach and then estimate its unknown N5LO contributions by using a Bayesian analysis. The newly suggested Bayesian-based approach estimates the magnitude of the UHO contributions based on an optimized analysis of the probability density distribution, and the predicted UHO contribution becomes more accurate when more loop terms have been known to tame the probability density function. Using the top-quark pole mass Mt = 172.69 GeV and the Higgs mass MH = 125.25 GeV as inputs, we obtain Γ(H→γγ)=9.56504keV, and the estimated N5LO contribution to the total decay width is ΔΓH=±1.65×10−4keV for the smallest credible interval of 95.5% degree of belief.

Discrepancy of coordinate system selection in backscattering Mueller matrix polarimetry: exploring photon coordinate system transformation invariants.

In biomedical studies, Mueller matrix polarimetry is gaining increasing attention because it can comprehensively characterize polarization-related vectorial properties of the sample, which are crucial for microstructural identification and evaluation. For backscattering Mueller matrix polarimetry, there are two photon coordinate selection conventions, which can affect the following Mueller matrix parameters calculation and information acquisition quantitatively. In this study, we systematically analyze the influence of photon coordinate system selection on the backscattering Mueller matrix polarimetry. We compare the Mueller matrix elements in the right-handed-nonunitary and non-right-handed-unitary coordinate systems, and specifically deduce the changes of Mueller matrix polar decomposition, Mueller matrix Cloude decomposition and Mueller matrix transformation parameters widely used in backscattering Mueller matrix imaging as the photon coordinate system varied. Based on the theoretical analysis and phantom experiments, we provide a group of photon coordinate system transformation invariants for backscattering Mueller matrix polarimetry. The findings presented in this study give a crucial criterion of parameters selection for backscattering Mueller matrix imaging under different photon coordinate systems.

QCD factorization from light-ray OPE

The energy-energy correlator (EEC) in Quantum Chromodynamics (QCD) serves as an important event shape for probing the substructure of jets in high-energy collisions. A significant progress has been made in understanding the collinear limit, where the angle between two detectors approaches zero, from the factorization formula in QCD and the light-ray Operator Product Expansion (OPE) in Conformal Field Theory. Building upon prior research on the renormalization of light-ray operators, we take an innovative step to extend the light-ray OPE into non-conformal contexts, with a specific emphasis on perturbative QCD. Our proposed form of the light-ray OPE is constrained by three fundamental properties: Lorentz symmetry, renormalization group invariance, and constraints from physical observables. This extension allows us to derive a factorization formula for the collinear limit of EEC, facilitating the future exploration and understanding on subleading power corrections in collinear limit.