Group Invariance Research Articles

WDVV equations and invariant bi-Hamiltonian formalism

The purpose of the paper is to show that, in low dimensions, the WDVV equations are bi-Hamiltonian. The invariance of the bi-Hamiltonian formalism is proved for N = 3. More examples in higher dimensions show that the result might hold in general. The invariance group of the bi-Hamiltonian pairs that we find for WDVV equations is the group of projective transformations. The significance of projective invariance of WDVV equations is discussed in detail. The computer algebra programs that were used for calculations throughout the paper are provided in a GitHub repository.

Symmetries of the Schrödinger–Pauli equation for neutral particles

By using the algebraic approach, the Lie symmetries of Schrödinger equations with matrix potentials are classified. Thirty three inequivalent equations of such type together with the related symmetry groups are specified, and the admissible equivalence relations are clearly indicated. In particular, the Boyer results concerning kinematical invariance groups for arbitrary potentials [C. P. Boyer, Helv. Phys. Acta 47, 450–605 (1974)] are clarified and corrected.

A novel determination of non-perturbative contributions to Bjorken sum rule

Based on the operator product expansion, the perturbative and nonperturbative contributions to the polarized Bjorken sum rule (BSR) can be separated conveniently, and the nonperturbative one can be fitted via a proper comparison with the experimental data. In the paper, we first give a detailed study on the pQCD corrections to the leading-twist part of BSR. Basing on the accurate pQCD prediction of BSR, we then give a novel fit of the non-perturbative high-twist contributions by comparing with JLab data. Previous pQCD corrections to the leading-twist part derived under conventional scale-setting approach still show strong renormalization scale dependence. The principle of maximum conformality (PMC) provides a systematic and strict way to eliminate conventional renormalization scale-setting ambiguity by determining the accurate alpha _s-running behavior of the process with the help of renormalization group equation. Our calculation confirms the PMC prediction satisfies the standard renormalization group invariance, e.g. its fixed-order prediction does scheme-and-scale independent. In low Q^2-region, the effective momentum of the process is small and in order to derive a reliable prediction, we adopt four low-energy alpha _s models to do the analysis, i.e. the model based on the analytic perturbative theory (APT), the Webber model (WEB), the massive pQCD model (MPT) and the model under continuum QCD theory (CON). Our predictions show that even though the high-twist terms are generally power suppressed in high Q^2-region, they shall have sizable contributions in low and intermediate Q^2 domain. Based on the more accurate scheme-and-scale independent pQCD prediction, our newly fitted results for the high-twist corrections at Q^2=1;mathrm{GeV}^2 are, f_2^{p-n}|_{mathrm{APT}}=-0.120pm 0.013, f_2^{p-n}|_mathrm{WEB}=-0.081pm 0.013, f_2^{p-n}|_{mathrm{MPT}}=-0.128pm 0.013 and f_2^{p-n}|_{mathrm{CON}}=-0.139pm 0.013; mu _6|_mathrm{APT}=0.003pm 0.000, mu _6|_{mathrm{WEB}}=0.001pm 0.000, mu _6|_mathrm{MPT}=0.003pm 0.000 and mu _6|_{mathrm{CON}}=0.002pm 0.000, respectively, where the errors are squared averages of those from the statistical and systematic errors from the measured data.

Factorial validity and measurement invariance of the Psychosocial Uncertainty Scale

This study presents the development of the Psychosocial Uncertainty Scale (PS-US), which articulates the perception of uncertainty in the social context and its psychological experience. It was validated with a sample of 1596 students and active professionals (employed and unemployed). By randomly dividing this sample in three sub-samples, the following analyses were performed: exploratory factor analysis (sample one: N = 827); preliminary confirmatory factor analysis identifying the final version of the scale (sample two: N = 382); confirmatory factor analysis (sample three: N = 387). Multi-group analysis was used to assess measurement invariance, gender, sociocultural level, and group of origin invariance, by using samples two and three. Group differences were explored with the complete sample through Multiple Indicators and Multiple Causes (MIMIC) Models. Associations between this scale and the Uncertainty response Scale were explored through Structural Equation Modelling. Exploratory and confirmatory analyses’ results showed good internal consistency and overall good psychometric qualities. The scale reached full metric invariance across groups, gender, SCL level and group of origin. Results highlight the sensitivity of the scale towards social vulnerability, proving the existence of sociocultural levels’ effects on experiences of psychosocial uncertainty within working contexts, relationships and community living and self-defeating beliefs; and gender and students versus professionals’ effects on psychosocial uncertainty. Furthermore, the scale associated significantly with Uncertainty Response Scale’s dimensions, specifically with emotional uncertainty, which can be considered a self-defeating strategy. Results suggest that emotional coping strategies, are explained by psychosocial uncertainty by 57%, and so, may have social origins.

Соотношение между категориями «поле» и «система» в языке.

The reviewed scientific work reflects the author’s innovative attempt to avoid confusion of such important categories as field and system in linguistic analysis. To achieve this aim, having considered various points of view on the field theory, the author proposes to accept as a characteristic of the field the fact of interaction between certain elements and interpret it as an attraction, which will allow only a certain number of invariant and functional-invariant groups to be considered as fields in the language. Other groups and elements that are characterized by causal relationships and interconnections will then have the reason to be considered as systems.

Invariant Scalar Product and Associated Structures for Tachyonic Klein–Gordon Equation and Helmholtz Equation

Although describing very different physical systems, both the Klein–Gordon equation for tachyons (m2<0) and the Helmholtz equation share a remarkable property: a unitary and irreducible representation of the corresponding invariance group on a suitable subspace of solutions is only achieved if a non-local scalar product is defined. Then, a subset of oscillatory solutions of the Helmholtz equation supports a unirrep of the Euclidean group, and a subset of oscillatory solutions of the Klein–Gordon equation with m2<0 supports the scalar tachyonic representation of the Poincaré group. As a consequence, these systems also share similar structures, such as certain singularized solutions and projectors on the representation spaces, but they must be treated carefully in each case. We analyze differences and analogies, compare both equations with the conventional m2>0 Klein–Gordon equation, and provide a unified framework for the scalar products of the three equations.

Fuzzy Results for Finitely Supported Structures

We present a survey of some results published recently by the authors regarding the fuzzy aspects of finitely supported structures. Considering the notion of finite support, we introduce a new degree of membership association between a crisp set and a finitely supported function modelling a degree of membership for each element in the crisp set. We define and study the notions of invariant set, invariant complete lattices, invariant monoids and invariant strong inductive sets. The finitely supported (fuzzy) subgroups of an invariant group, as well as the L-fuzzy sets on an invariant set (with L being an invariant complete lattice) form invariant complete lattices. We present some fixed point results (particularly some extensions of the classical Tarski theorem, Bourbaki–Witt theorem or Tarski–Kantorovitch theorem) for finitely supported self-functions defined on invariant complete lattices and on invariant strong inductive sets; these results also provide new finiteness properties of infinite fuzzy sets. We show that apparently, large sets do not contain uniformly supported, infinite subsets, and so they are invariant strong inductive sets satisfying finiteness and fixed-point properties.

Symmetric polynomials of algebras related with 2 × 2 generic traceless matrices

Let [Formula: see text] and [Formula: see text] be two generic traceless matrices of size [Formula: see text] with entries from a commutative associative polynomial algebra over a field [Formula: see text] of characteristic zero. Consider the associative unitary algebra [Formula: see text], and its Lie subalgebra [Formula: see text] generated by [Formula: see text] and [Formula: see text] over the field [Formula: see text]. It is well known that the center [Formula: see text] of [Formula: see text] is the polynomial algebra generated by the algebraically independent commuting elements [Formula: see text], [Formula: see text], [Formula: see text]. We call a polynomial [Formula: see text] symmetric, if [Formula: see text]. The set of symmetric polynomials is equal to the algebra [Formula: see text] of invariants of symmetric group [Formula: see text]. Similarly, we define the Lie algebra [Formula: see text] of symmetric polynomials in the Lie algebra [Formula: see text]. We give the description of the algebras [Formula: see text] and [Formula: see text], and we provide finite sets of free generators for [Formula: see text], and [Formula: see text] as [Formula: see text]-modules.

Controlling consensus in networks with symmetries

We study networks with linear dynamics where the presence of symmetries of the pair , induces a partition of the network nodes in clusters and the matrix A is not restricted to be in Laplacian form. For these networks, an invariant group consensus subspace can be defined, in which the nodes in the same cluster evolve along the same trajectory in time. We prove that the network dynamics is uncontrollable in directions orthogonal to this subspace. Under the assumption that the dynamics parallel to this subspace is controllable, we design optimal controllers that drive the group consensus dynamics towards a desired state. Then, we consider the problem of selecting additional control inputs that stabilize the group consensus subspace and obtain bounds on the minimum number of additional inputs and driver nodes needed to this end. Altogether, our results indicate that it is possible to independently design the control actions along and transverse to the group consensus subspace.

Validation of the multidimensional WHOQOL-OLD in Ghana: A study among population-based healthy adults in three ethnically different districts.

ObjectivesStudy of well‐being of older adults, a rapidly growing demographic group in sub‐Saharan Africa, depends on well‐validated tools like the WHOQOL‐OLD. This scale has been tested on different populations with reasonable validity results but has limited application in Africa. The specific goal of this paper was to examine the factor structure of the WHOQOL‐OLD translated into three Ghanaian languages: Ga, Akan, and Kasem. We also tested group invariance for sex and for type of community (distinguished by ethnicity/language).MethodsWe interviewed 353 older adults aged 60 years and above, selected from three ethnically and linguistically different communities. Using a cross‐sectional design, we used purpose and convenience methods to select participants in three geographically and ethnically distinct communities. Each community was made up of selected rural, peri‐urban, and urban communities in Ghana. The questionnaire was translated into three languages and administered to each respondent.ResultsThe results showed moderate to high internal consistency coefficient and factorial validity for the scale. Using confirmatory factor analysis, we found that the results supported a multidimensional structure of the WHOQOL‐OLD and that it did not differ for males and females, neither did it differ for different ethnic/linguistic groups.ConclusionsWe conclude that the translated versions of the measure are adequate tools for evaluation of quality of life of older adults among the respective ethnic groups studied in Ghana. These results will also enable comparison of quality of life between older adults in Ghana and in other cultures.

Topological states between inversion symmetric atomic insulators

One of the hallmarks of topological insulators is the correspondence between the value of its bulk topological invariant and the number of topologically protected edge modes observed in a finite-sized sample. This bulk-boundary correspondence has been well-tested for strong topological invariants, and forms the basis for all proposed technological applications of topology. Here, we report that a group of weak topological invariants, which depend only on the symmetries of the atomic lattice, also induces a particular type of bulk-boundary correspondence. It predicts the presence or absence of states localised at the interface between two inversion-symmetric band insulators with trivial values for their strong invariants, based on the space group representation of the bands on either side of the junction. We show that this corresponds with symmetry-based classifications of topological materials. The interface modes are protected by the combination of band topology and symmetry of the interface, and may be used for topological transport and signal manipulation in heterojunction-based devices.

Next-to-soft corrections for Drell-Yan and Higgs boson rapidity distributions beyond N3LO

We present a formalism that sums up both soft-virtual (SV) and next to SV (NSV) contributions to all orders in perturbative QCD for the rapidity distribution of any colorless particle produced in hadron colliders. We have exploited the factorization properties and the renormalisation group (RG) invariance of the differential cross-section to achieve this. Using the state-of-the-art multiloop and multileg results, we determine the complete NSV contributions to third order in strong coupling constant for the rapidity distributions for a pair of leptons in Drell-Yan and also for Higgs boson in gluon fusion as well as bottom quark annihilation. Using the integral representation of our all order $z$ space result, we show how the NSV contributions can be resummed in two-dimensional Mellin space.

Spaces of commuting elements in the classical groups

Let G be the classical group, and let Hom(Zm,G) denote the space of commuting m-tuples in G. First, we refine the formula for the Poincaré series of Hom(Zm,G) due to Ramras and Stafa by assigning (signed) integer partitions to (signed) permutations. Using the refined formula, we determine the top term of the Poincaré series, and apply it to prove the dependence of the topology of Hom(Zm,G) on the parity of m and the rational hyperbolicity of Hom(Zm,G) for m≥2. Next, we give a minimal generating set of the cohomology of Hom(Zm,G) and determine the cohomology in low dimensions. We apply these results to prove homological stability for Hom(Zm,G) with the best possible stable range. Baird proved that the cohomology of Hom(Zm,G) is identified with a certain ring of invariants of the Weyl group of G, and our approach is a direct calculation of this ring of invariants.

On the Holographic Type Dynamics in Complexity Economics

Abstract Assimilating any complex economic system with a fractal, in the most general Mandelbrot’s sense, non – differentiable behaviors in its economic dynamics are analyzed. As such, economic dynamics in the form of Schrödinger – type various regimes imply “holographic implementations” of the economic processes through group invariance of SL(2R) – type. Then, by means of previous group invariance as synchronization group between any economic system entities, both the phases and the amplitudes of the entities are affected from a homographic perspective. The usual “synchronization” manifested through the delay of the amplitudes and phases of the entities of the economic system must represent here only a fully particular case. In a special case of synchronization of economic system entities, given by Riccati type gauge, period doubling, damping oscillations, self – modulation and chaotic regimes emerge as natural behaviors in the economic dynamics of the economic processes.

Statistics for Iwasawa invariants of elliptic curves

We study the average behaviour of the Iwasawa invariants for the Selmer groups of elliptic curves, setting out new directions in arithmetic statistics and Iwasawa theory.

Group Invariant Solutions for Flow and Heat Transfer of Power-Law Nanofluid in a Porous Medium

The present work covers the flow and heat transfer model for the power-law nanofluid in the presence of a porous medium over the penetrable plate. The flow is caused by the impulsive movement of the plate embedded in Darcy’s type porous medium. The flow and heat transfer model has been examined with the effect of linear thermal radiation and the internal heat source or sink in the flow regime. The Rosseland approximation is utilized for the optically thick nanofluid. To form the closed-form solutions for the governing partial differential equations of conservation of mass, momentum, and energy, the Lie symmetry analysis is used to get the reductions of governing equations and to find the group invariants. These invariants are then utilized to obtain the exact solution for all three cases, i.e., shear thinning fluid, Newtonian fluid, and shear thickening fluid. In the end, all solutions are plotted for thecu-water nanofluid and discussed briefly for the different emerging flow and heat transfer parameters.

Relations for a class of terminating 4 F 3(4) hypergeometric series

We derive relations for a certain class of terminating hypergeometric series with three free parameters. The invariance group composed of these relations is shown to be isomorphic to the symmetric group . We further study relations for terminating series that fall under two families. By using a series reversal, we examine the corresponding terminating and series relations. We additionally derive formulas for the sums of the first n + 1 terms of several nonterminating and series. We also show how certain known summation formulas for terminating and series follow as limiting cases of some of our relations.

Running’ under tight constraints in pionless effective field theory

The contents of renormalization group invariance and equations under tight constraints are explored and demonstrated with closed-form on-shell [Formula: see text] matrices of pionless effective field theory for nuclear forces right within the effective field theory philosophy. The ‘running’ couplings under such tight constraints are presented in [Formula: see text] and uncoupled [Formula: see text] channels up to truncation order [Formula: see text]. Some linear relations are exposed and in turn employed in the pursuit of nonperturbative ‘running’ solutions which serves as an alternative choice without resorting to special prescription and additional operations or treatments. The utility of such ‘running’ behaviors inherent in the closed-form [Formula: see text]-matrices of pionless effective field theory is remarked and a number of important issues related to effective field theory constructed with various truncations are interpreted or discussed from the underlying theory perspective.

Optimal system, symmetry reductions and group-invariant solutions of (2+1)-dimensional ZK-BBM equation

The present article intends to generate optimal system of one dimensional subalgebra and group–invariant solutions of ZK–BBM equation with the aid of Lie group theory. The ZK–BBM equation is long wave equation with large wavelength, which describes the water wave phenomena in nonlinear dispersive system. The infinitesimal vectors, commutative relations and invariant functions for optimal system of ZK–BBM equation are derived under invariance of Lie groups. The invariance property leads to the reduction of independent variable and leaves the system invariant. Based on the optimal system, ZK–BBM equation is transformed into ordinary differential equations by twice reductions. These ODEs are solved under parametric constraints and result into invariant solutions. The obtained solutions are analyzed physically based on their numerical simulation. Consequently, elastic multisoliton, dark and bright lumps, compacton and annihilation profiles of the solutions are well presented graphically.

Singlet-triplet interaction of the adiabatic terms [formula omitted] and [formula omitted] in the linear triatomic molecules

The paper introduces the 4x4 vibronic matrix that describes singlet–triplet interaction of adiabatic terms 1Σ+ and 3Σ+ via deformation π-modes of the linear triatomic system. Our analysis takes into account spin-orbital coupling in ab initio Breit-Pauli form in the framework of two-electronic model. It is shown that the symmetry operators of electronic Hamiltonian include both space operations (acting on electronic coordinates) and matrix operations (acting on electronic spins). Eigenvalues of vibronic matrix (i.e. potential energy surfaces) are invariants of the molecular symmetry group C∞ν. The 4x4 vibronic matrix includes 4 electrostatic parameters and 3 parameters caused by spin-orbital interaction.