Action Of Group Research Articles

Isotropy group on some topological transformation group structures

This paper explores the topological properties of irresolute topological groups, their quotient maps, and the role of topology in normal subgroups. It provides a detailed analysis\linebreak using examples and counterexamples. The study focuses on the essential features of irresolute topological groups and their quotient groups, for understanding the topological aspects of isotropy groups. For a trans\-for\-ma\-tion group $(\mathsf{H}, \mathsf{Y}, \psi)$ and a point $y \in \mathsf{Y},$ the set \centerline{$\mathsf{H}_{y} = \{h \in \mathsf{H} \colon hy = y\}$} \noi consisting of elements of $\mathsf{H}$ that fix $y$, is called the isotropy group at $y$. The paper highlights the distinct topological characteristics of isotropy groups in transformation group structure. It demonstrates that if $(\mathsf{H}, \mathsf{Y}, \psi)$ is an Irr$^{*}$-topological transformation group, then $( \mathsf{H}/ \mathop{Ker} \psi, \mathsf{Y}, \overline{\psi})$ forms an effective Irr$^{*}$-topological transformation group. By investigating both irresolute topological groups and isotropy groups, the study provides a clear understanding of their topological features. This research improves our understanding of these groups by offering clear examples and counterexamples, leading to a thorough conclusion about their different topological features.

Jacobian varieties with group algebra decomposition not affordable by Prym varieties

The action of a finite group G on a compact Riemann surface X naturally induces another action of G on its Jacobian variety J(X). In many cases, each component of the group algebra decomposition of J(X) is isogenous to a Prym varieties of an intermediate covering of the Galois covering πG:X→X/G; in such a case, we say that the group algebra decomposition is affordable by Prym varieties. In this article, we present an infinite family of groups that act on Riemann surfaces in a manner that the group algebra decomposition of J(X) is not affordable by Prym varieties; namely, affine groups Aff(Fq) with some exceptions: q=2, q=9, q a Fermat prime, q=2n with 2n−1 a Mersenne prime and some particular cases when X/G has genus 0 or 1. In each one of this exceptional cases, we give the group algebra decomposition of J(X) by Prym varieties.

On the Construction of Congruences over Generalized Fuzzy G-Acts

Group action is defined to support Cayley’s claim that every group is isomorphic to a suitable subgroup of a symmetric group. Group actions have a wide range of applications, including the analysis of symmetries of geometric objects and algorithms and cryptographic systems. In 1965, Zadeh introduced the concept of fuzzy sets and provided a mathematical formulation for various concepts in this area. Since then, the concept of fuzziness has been integrated into several branches of mathematics to address the uncertainties of real-life scenarios. This article introduces the concept of group action in a fuzzy environment, termed fuzzy G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {G}$$\\end{document}-subacts. The study provide the concept of fuzzy G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {G}$$\\end{document}-orbits and fuzzy G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {G}$$\\end{document}-stabilizers and clearly outlines fuzzy permutation representations of G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {G}$$\\end{document} and fuzzy G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {G}$$\\end{document}-morphisms. The research findings significantly contributes to the understanding of fuzzy G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {G}$$\\end{document}-congruences and fuzzy quotient G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {G}$$\\end{document}-subacts with the help of fuzzy G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {G}$$\\end{document}-partitions. This approach not only refines the underlying theories but also opens up new possibilities for practical implementation. Thus, the study demonstrates how the more complex fuzzy theory can expand and enrich the mathematical structures of abstract algebra, making them highly applicable.

Minimal compact group actions on C∗-algebras with simple fixed point algebras

The notion of quasi-product actions of a compact group on a C[Formula: see text]-algebra was introduced by Bratteli et al. in their attempt to seek an equivariant analogue of Glimm’s characterization of non-type I C[Formula: see text]-algebras. We show that a faithful minimal action of a second countable compact group on a separable C[Formula: see text]-algebra is quasi-product whenever its fixed point algebra is simple. This was previously known only for compact abelian groups and for profinite groups. Our proof relies on a subfactor technique applied to finite index inclusions of simple C[Formula: see text]-algebras in the purely infinite case, and also uses ergodic actions of compact groups in the general case. As an application, we show that if moreover the fixed point algebra is a Kirchberg algebra, such an action is always isometrically shift-absorbing, and hence is classifiable by the equivariant KK-theory due to a recent result of Gabe-Szabó.

The renewed tin-weighting treatment as sustainable and durable flame-retardant approach for protein silk fabric

The facile development of a sustainable and durable flame-retardant approach for protein silk is of interest. Inspired by silk tin-weighting technology, this study developed a novel and sustainable in-situ deposition strategy based on biomass phytic acid to impart durable flame-retardant performance to silk fabrics. The chemical structure of insoluble chelating precipitation, and the surface morphology, thermal stability, combustion behavior, flame-retardant capacity, laundering resistance, and flame-retardant mode of action of the tin-weighting silk samples, were explored. The Sn-, P-, Si-containing insoluble chelating precipitation formed within the fiber interior and combined with silk fibers through electrostatic attraction and metal salt chelation. As a result, the tin-weighting silk displayed excellent self-extinguishing capacity, with the damaged length reduced to 9.2 cm and the LOI increased to 31.6 %; it also achieved self-extinguishing after 30 washing cycles, demonstrating high flame-retardant efficacy and laundering resistance. Moreover, the tin-weighting silk also showed the obvious suppression in smoke and heat generation by 55.6 % and 35.7 %, respectively. The synergistic charring action of phosphate groups, tin metal salts, and silicates was beneficial for enhancing the fire safety of silk. The tin-weighting treatment also displayed a minor impact on mechanical performance of silk fabrics.

Centralizers of Hamiltonian finite cyclic group actions on rational ruled surfaces

Let M = ( M , ω ) M=(M,\omega ) be either S 2 × S 2 S^2 \times S^2 or C P 2 # C P 2 ¯ \mathbb {C}P^2\# \overline {\mathbb {C}P^2} endowed with any symplectic form ω \omega . Suppose a finite cyclic group Z n \mathbb {Z}_n is acting effectively on ( M , ω ) (M,\omega ) through Hamiltonian diffeomorphisms, that is, there is an injective homomorphism Z n ↪ H a m ( M , ω ) \mathbb {Z}_n\hookrightarrow Ham(M,\omega ) . In this paper, we investigate the homotopy type of the group S y m p Z n ( M , ω ) Symp^{\mathbb {Z}_n}(M,\omega ) of equivariant symplectomorphisms. We prove that for some infinite families of Z n \mathbb {Z}_n actions satisfying certain inequalities involving the order n n and the symplectic cohomology class [ ω ] [\omega ] , the actions extend to either one or two toric actions, and accordingly, that the centralizers are homotopically equivalent to either a finite dimensional Lie group, or to the homotopy pushout of two tori along a circle. Our results rely on J J -holomorphic techniques, on Delzant’s classification of toric actions, on Karshon’s classification of Hamiltonian circle actions on 4 4 -manifolds, and on the Chen-Wilczyński classification of smooth Z n \mathbb {Z}_n -actions on Hirzebruch surfaces.

Differentials of the basis in Clifford Geometric Algebra

In this paper we discuss the dynamic effects of the varying frames. The differential of frame or basis vectors is always equivalent to a linear transformation of the frame, and the linear transformation is not the same in different contexts. In differential geometry, the linear transformation is the connection operator. While in quantum mechanics, the operator algebra corresponds to the differentials of matrices. Corresponding to the variation of the metric, the variation of the frame contains a unusual fourth-order tensor. We also derive the Lie differential of the frame corresponding to the Lorentz transformation group. The definition of differential of the frame is different, so the corresponding linear transformation is also different. In this paper, the unified point of view to deal with the variation of frame or basis vectors will bring great convenience to the research and application of Clifford algebras.

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

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

Fundamental regions for non-isometric group actions

We generalise results about isometric group actions on metric spaces and their fundamental regions to the context of merely continuous group actions. In particular, we obtain results that yield the relative compactness of a fundamental region for a cocompact group action. As a consequence, we obtain a criterion for a cocompact cyclic group of semi-Riemannian homotheties to be inessential.

On the Dimension of Limit Sets on ℙ(ℝ3) via Stationary Measures: Variational Principles and Applications

Abstract This paper investigates the (semi)group action of $\textrm{SL}_{n}({\mathbb R})$ on ${\mathbb P}({\mathbb R}^{n})$, a primary example of non-conformal, non-linear, and non-strictly contracting action. We establish variational principles of the affinity exponent for two main examples: the Borel Anosov representations and the Rauzy gasket. In [ 32], they obtain a dimension formula for the stationary measures on ${\mathbb P}({\mathbb R}^{3})$. Combined with our result, it allows us to study the Hausdorff dimension of limit sets of Anosov representations in $\textrm{SL}_{3}({\mathbb R})$ and the Rauzy gasket. It yields the equality between the Hausdorff dimensions and the affinity exponents in both settings, generalizing the classical Patterson–Sullivan formula. In the appendix, we improve the numerical lower bound of the Hausdorff dimension of Rauzy gasket to $1.5$.

Twelve hot questions in the management of hypertension in patients aged 80+ years and their answers with the help of the 2023 European Society of Hypertension Guidelines.

Arterial hypertension is a major risk factor for cardiovascular morbidity and mortality, and highly prevalent in older age, underscoring the importance of its appropriate management. The population is ageing at an increasing rate, with those aged 80+ years being the fastest growing population characterized by high heterogeneity in terms of functionality and autonomy. The prevalence of hypertension rises with increasing age, due to a significant increase in SBP largely as a result of age-related stiffening of the aorta and other large arteries, affecting almost 80% of those aged 80+ years. Appropriate management of blood pressure in this population is a priority for clinicians. Frailty is a condition characterized by marked vulnerability to adverse health outcomes and is common among older adults including those with hypertension. Hypertension increases frailty level and at the same time, individuals with increasing frailty present with more drug-related adverse effects meaning they are less tolerant to blood pressure lowering by medication. Thus, frailty is a factor that should be integrated when treating hypertension in this population. The European Society of Hypertension 2023 Guidelines on the management of Hypertension are the first international guidelines to integrate the concept of adapting blood pressure management in older adults according to their frailty/functionality level, and to propose practical tools for the application of this concept in the daily practice of physicians and other healthcare professionals. The present article prepared by the European Society of Hypertension Working Group on Hypertension in Older Adults aims to further address some important aspects mentioned concisely in the 2023 European Society of Hypertension guidelines, in order to help physicians and other healthcare professionals including those practicing in primary care. To this end, this study discusses 12 'hot questions' which are answered with the help of the 2023 European Society of Hypertension Guidelines. We hope the present article and Working Group's actions will contribute to understanding and applying the ideal management of hypertension in this most vulnerable population.

Event-triggered lifted robust MPC stabilization control for space telescope on dual-rate actuated rigid spacecraft systems

This study addresses the pose stabilization control problem for enhancing a space telescope mounted on a spacecraft system under a dual-rate actuated setup. An input-lifting approach is utilized to manage the dual rates in the control components, specifically the orbital space telescope and its loading platform. To counteract external spatial perturbations and spacecraft system uncertainties, an integrated control scheme is proposed, combining real-time model parameter estimation, active compensation control, and event-triggered lifted robust model predictive control (ET-LRMPC). The event-triggered mechanism in the proposed algorithm minimizes computational resource consumption while maintaining effective spacecraft control. Numerical simulations demonstrate that the proposed control scheme achieves favorable results under persistent external perturbations and model uncertainties. In terms of the overshoot, the algorithm proposed reduces the control effect by up to 87.1% in the spatial displacement of the payload and 98.6% in the Euler attitude angle compared to the conventional control algorithm. For the control of the base, the amount of overshoots with respect to the conventional control in spatial displacement and Euler attitude angle is reduced by 49% and 97.1%.

Fully biobased approach for sustainable flame retardancy, antibacterial and anti-UV modification of silk fabric

The versatile modification of silk fabric using a fully biobased approach is intriguing. In this study, phytic acid and vanillin, which are plant extracts, were utilized to synthesize reactive vanillin phytate ester for modifying silk fabric to achieve durable flame retardancy, antibacterial and anti-UV performance. The multifunctional properties, thermal resistance, smoke and heat suppression capacity, flame-retardant durability and mechanism of modified silk fabrics were investigated. The vanillin phytate ester imparted high antimicrobial and UV resistance to silk fabric, achieving a 97 % inhibition rate against Escherichia coli and reaching an “excellent” level of UV protection. The modified silk fabric also possessed the self-extinguishing capacity, with a limiting oxygen index value as high as 33.5 %. There was also a significant reduction of 71.7 % in peak heat release and 40 % in smoke release compared with those of the original silk. After undergoing 20 washing cycles, the modified silk fabrics also self-extinguished and passed the vertical burning test, and such good washing resistance was achieved by the Schiff base cross-linking of vanillin phytate ester on the silk fabric. An intumescent flame-retardant mechanism was also identified by char residue analyses, which was driven by the synergistic charring action of phosphate groups and aromatic structures.

Extended Schur functions and bases related by involutions

We introduce two new bases of QSym, the reverse extended Schur functions and the row-strict reverse extended Schur functions, as well as their duals in NSym, the reverse shin functions and row-strict reverse shin functions. These bases are the images of the extended Schur basis and shin basis under the involutions ρ and ω, which generalize the classical involution ω on the symmetric functions. In addition, we prove a Jacobi-Trudi rule for certain shin functions using creation operators. We define skew extended Schur functions and skew-II extended Schur functions based on left and right actions of NSym and QSym respectively. We then use the involutions ρ and ω to translate these and other known results to our reverse and row-strict reverse bases.

Uncountable group of continuous transformations of unit segment preserving tails of Q_2-representation of numbers

We consider two-base Q2-representation of numbers of segment [0;1] which is defined by two bases q0 ∈ (0;1), q1 = 1-q0 and alphabet A={0,1}, (αn) ∈ A × A × .... It is a generalization of classic binary representation q0=1/2. In the article we prove that the set of all continuous bijections of segment [0;1] preserving "tails" of Q2-representation of numbers forms an uncountable non-abelian group with respect to composition such that it is a subgroup of the group of continuous transformations preserving frequencies of digits of Q2-representation of numbers. Construction of such transformations (bijections) is based on the left and right shift operators for digits of Q2-representation of numbers.

Family of 2:1 resonant quasi-periodic distant retrograde orbits in cislunar space

Given the current enthusiasm for lunar exploration, the 2:1 resonant distant retrograde orbit (DRO) in Earth-Moon space is of interest. To gain an in-depth understanding of the complex dynamic environment in cislunar space and provide more options for parking orbits, this paper investigates the existence of quasi-periodic orbits near the 2:1 resonant DRO in the circular restricted three-body problem (CR3BP). Firstly, the numerical computation approach, continuation strategy, and stability analysis method of quasi-periodic orbits are introduced. Then, addressing the primary challenges in the continuation progress, we have developed an adaptive continuation algorithm with automatic adjustment of the step size and the number of discrete points that characterize the invariant torus. Finally, two types of 2D quasi-DROs and their linear stability properties are explored. Using Poincaré sections, we investigated the boundaries of the maximum extent attainable by both 2D quasi-DRO families in the CR3BP at a specific Jacobi energy, confirming that both types of quasi-periodic families have reached their respective boundaries. The algorithm described in this paper is beneficial for facilitating the computation of quasi-periodic families and aids in discovering additional potential dynamical structures.

Application of nano- and micro-particle-based approaches for selected bronchodilators in management of asthma.

Asthma is a chronic inflammatory condition that affects the airways, posing a substantial health threat to a large number of people worldwide. Bronchodilators effectively alleviate symptoms of airway obstruction by inducing relaxation of the smooth muscles in the airways, thereby reducing breathlessness and enhancing overall quality of life. The drug targeting to lungs poses significant challenges; however, this issue can be resolved by employing nano- and micro-particles drug delivery systems. This review provides brief insights about underlying mechanisms of asthma, including the role of several inflammatory mediators that contribute to the development and progression of this disease. This article provides an overview of the physicochemical features, pharmacokinetics, and mechanism of action of particular groups of bronchodilators, including sympathomimetics, PDE-4 inhibitors (phosphodiesterase-4 inhibitors), methylxanthines, and anticholinergics. This study presents a detailed summary of the most recent developments in incorporation of bronchodilators in nano- and micro-particle-based delivery systems which include solid lipid nanoparticles, bilosomes, novasomes, liposomes, polymeric nano- and micro-particles. Specifically, it focuses on breakthroughs in the categories of sympathomimetics, methylxanthines, PDE-4 inhibitors, and anticholinergics. These medications have the ability to specifically target alveolar macrophages, leading to a higher concentration of pharmaceuticals in the lung tissues.

Gauss Equations for Local Action-Angle Orbital Elements in Cislunar Space

To track orbits in cislunar space, predict where they will naturally move over time, and identify where unbounded trajectories come from, one may consider local action-angle orbital elements—coordinates that relate a spacecraft state to specific trajectories and exist in approximately integrable regions of the Earth–moon circular restricted three-body problem (CR3BP). As local action-angle elements are a semi-analytical analog to two-body orbital elements, the theory is extended to allow for the study of arbitrary perturbations to the dynamics. Namely, we derive Gauss equations for an arbitrary perturbing acceleration—continuous or discrete—to the CR3BP. Examples for continuous thrust and impulsive ΔV are provided in cislunar space, i.e., around Earth–moon L1,2 in the CR3BP. Strategies for instantaneous maneuver design and transfers between quasi-periodic orbits and manifolds are developed.

The Distribution of Planet Radius in Kepler Multiplanet Systems Depends on Gap Complexity

The distribution of small planet radius (<4 R ⊕) is an indicator of the underlying processes governing planet formation and evolution. We investigate the correlation between the radius distribution of exoplanets in Kepler multiplanet systems and the system-level complexity in orbital period spacing. Utilizing a sample of 234 planetary systems with three or more candidate planets orbiting FGK main-sequence stars, we measure the gap complexity (C) to characterize the regularity of planetary spacing and compare it with other measures of period spacing and spacing uniformity. We find that systems with higher gap complexity exhibit a distinct radius distribution compared to systems with lower gap complexity. Specifically, we find that the radius valley, which separates super-Earths and sub-Neptunes, is more pronounced in systems with lower gap complexity (C < 0.165). Planets in high-complexity systems (C > 0.35) exhibit a lower frequency of sub-Earths (2.5 times less) and sub-Neptunes (1.3 times less) and a higher frequency of super-Earths (1.4 times more) than planets in low-complexity systems. This may suggest that planetary systems with more irregular spacings are more likely to undergo dynamic interactions that influence planet scattering, composition, and atmospheric retention. The gap complexity metric proves to be a valuable tool in linking the orbital configurations of planets to their physical characteristics.

Jacobian Spheroids, Shallow Encounters, and the Keyhole Map

The study of the dynamics of flybys is featured in several space-mission-related applications, spanning from the design of interplanetary orbits to the computation of impact probabilities with given celestial bodies in planetary protection and defense tasks. While patched-conics, two-body models suffice in accuracy for trajectory planning cases, collision probability assessment requires instead more insightful approaches. High-fidelity simulations have shown impact patterns to happen outside nominally colliding trajectories in terms of two-body analysis, all caused by distant or weak interactions between the given trajectory-celestial body pairs. This work proposes a novel concept of sphere of influence based on the Jacobian of the three-body dynamics, resulting in a wider definition of encounters that encompasses weak interactions up to an arbitrary threshold. The so-defined Jacobian spheroids are then used to construct the Keyhole map, a graphical tool that maps in the orbital elements space both nominal and off-nominal keyholes, i.e., collision paths accounting for three-body weak interactions. The proposed concepts arise from and are compared against the on-ground casualty risk assessment of the European Space Agency’s mission JUICE.