Compact Group Research Articles

Compact and ordered swarms of unmanned aerial vehicles in cluttered environments.

The globally coordinated motion produced by the classical swarm model is typically generated by simple local interactions at the individual level. Despite the success of these models in interpretation, they cannot guarantee compact and ordered collective motion when applied to the cooperation of unmanned aerial vehicle (UAV) swarms in cluttered environments. Inspired by the behavioral characteristics of biological swarms, a distributed self-organized Reynolds (SOR) swarm model of UAVs is proposed. In this model, a social term is designed to keep the swarm in a collision-free, compact, and ordered collective motion, an obstacle avoidance term is introduced to make the UAV avoid obstacles with a smooth trajectory, and a migration term is added to make the UAV fly in a desired direction. All the behavioral rules for agent interactions are designed with as simple a potential function as possible. And the genetic algorithm is used to optimize the parameters of the model. To evaluate the collective performance, we introduce different metrics such as (a) order, (b) safety, (c) inter-agent distance error, (d) speed range. Through the comparative simulation with the current advanced bio-inspired compact and Vasarhelyi swarm models, the proposed approach can guide the UAV swarm to pass through the dense obstacle environment in a safe and ordered manner as a compact group, and has adaptability to different obstacle densities.

Quantitative version of Weyl’s law

We prove a general estimate for the Weyl remainder of an elliptic, semiclassical pseudodifferential operator in terms of volumes of recurrence sets for the Hamilton flow of its principal symbol. This quantifies earlier results of Volovoy (Comm Partial Differential Equations 15:1509–1563, 1990; Ann Global Anal Geom 8:127–136, 1990). Our result particularly improves Weyl remainder exponents for compact Lie groups and surfaces of revolution. And gives a quantitative estimate for Bérard’s Weyl remainder in terms of the maximal expansion rate and topological entropy of the geodesic flow.

Structure–activity relationship on selective antibacterial activity of nonionic surfactants

AbstractThe antibacterial activity of surfactants is an important factor to consider for improving skin conditions in cosmetic and skin care products. In this study, we evaluated the antibacterial activity of 16 nonionic surfactants containing ethylene oxide, ester, and ether functional groups against Staphylococcus aureus (S. aureus) and Staphylococcus epidermidis (S. epidermidis). 12:0 MG ester and 6–12 diol, which showed antibacterial activity against both Staphylococci, had compact hydrophilic groups with low polarity. In contrast, 12:0 PEG‐10 ester and 12:0 Sor ester, which showed antibacterial activity only against S. aureus, had large hydrophilic groups and high polarity. These results suggest that the hydrophilicity and polarity of the surfactant may be involved in its antibacterial behavior against Staphylococci. These findings will be useful in the design of surfactant molecules and in the development of cosmetics and body cleansers that exhibit antibacterial activity only against S. aureus.

How to perform the coherent measurement of a curved phase space by continuous isotropic measurement. I. Spin and the Kraus-operator geometry of SL(2,C)

The generalized Q-function of a spin system can be considered the outcome probability distribution of a state subjected to a measurement represented by the spin-coherent-state (SCS) positive-operator-valued measure (POVM). As fundamental as the SCS POVM is to the 2-sphere phase-space representation of spin systems, it has only recently been reported that the SCS POVM can be performed for any spin system by continuous isotropic measurement of the three total spin components [E. Shojaee, C. S. Jackson, C. A. Riofrio, A. Kalev, and I. H. Deutsch, Phys. Rev. Lett. 121, 130404 (2018)]. This article develops the theoretical details of the continuous isotropic measurement and places it within the general context of curved-phase-space correspondences for quantum systems. The analysis is in terms of the Kraus operators that develop over the course of a continuous isotropic measurement. The Kraus operators of any spin j are shown to represent elements of the Lie group SL(2,C)≅Spin(3,C), a complex version of the usual unitary operators that represent elements of SU(2)≅Spin(3,R). Consequently, the associated POVM elements represent points in the symmetric space SU(2)∖SL(2,C), which can be recognized as the 3-hyperboloid. Three equivalent stochastic techniques, (Wiener) path integral, (Fokker-Planck) diffusion equation, and stochastic differential equations, are applied to show that the continuous isotropic POVM quickly limits to the SCS POVM, placing spherical phase space at the boundary of the fundamental Lie group SL(2,C) in an operationally meaningful way. Two basic mathematical tools are used to analyze the evolving Kraus operators, the Maurer-Cartan form, modified for stochastic applications, and the Cartan, decomposition associated with the symmetric pair SU(2) ⊂ SL(2,C). Informed by these tools, the three schochastic techniques are applied directly to the Kraus operators in a representation-independent – and therefore geometric – way (independent of any spectral information about the spin components).The Kraus-operator-centric, geometric treatment applies not just to SU(2) ⊂ SL(2,C), but also to any compact semisimple Lie group and its complexification. The POVM associated with the continuous isotropic measurement of Lie-group generators thus corresponds to a type-IV globally Riemannian symmetric space and limits to the POVM of generalized coherent states. This generalization is the focus of a sequel to this article.

Correction to: Expansive actions of automorphisms of locally compact groups $$\varvec{G}$$ on Sub$$_{\varvec{G}}$$

Correction to: Expansive actions of automorphisms of locally compact groups $$\varvec{G}$$ on Sub$$_{\varvec{G}}$$

Bismut Ricci flat generalized metrics on compact homogeneous spaces

A generalized metric on a manifold M M , i.e., a pair ( g , H ) (g,H) , where g g is a Riemannian metric and H H a closed 3 3 -form, is a fixed point of the generalized Ricci flow if and only if ( g , H ) (g,H) is Bismut Ricci flat: H H is g g -harmonic and R c ( g ) = 1 4 H g 2 Rc(g)=\tfrac {1}{4}H_g^2 . On any homogeneous space M = G / K M=G/K , where G = G 1 × G 2 G=G_1\times G_2 is a compact semisimple Lie group with two simple factors, under some mild assumptions, we exhibit a Bismut Ricci flat G G -invariant generalized metric, which is proved to be unique among a 4 4 -parameter space of metrics in many cases, including when K K is neither abelian nor semisimple. On the other hand, if K K is simple and the standard metric is Einstein on both G 1 / π 1 ( K ) G_1/\pi _1(K) and G 2 / π 2 ( K ) G_2/\pi _2(K) , we give a one-parameter family of Bismut Ricci flat G G -invariant generalized metrics on G / K G/K and show that it is most likely pairwise non-homothetic by computing the ratio of Ricci eigenvalues. This is proved to be the case for every space of the form M = G × G / Δ K M=G\times G/\Delta K and for M 35 = S O ( 8 ) × S O ( 7 ) / G 2 M^{35}=SO(8)\times SO(7)/G_2 .

Invariant means and multipliers on convolution quantum group algebras

Let [Formula: see text] be a locally compact quantum group. Then the space [Formula: see text] of trace class operators on [Formula: see text] is a Banach algebra with the convolution induced by the right fundamental unitary of [Formula: see text]. We show that properties of [Formula: see text] such as amenability, triviality and compactness are equivalent to the existence of left or right invariant means on the convolution Banach algebra [Formula: see text]. We also investigate the relation between the existence of certain (weakly) compact right and left multipliers of [Formula: see text] and some properties of [Formula: see text].

Identifying the Regions of a Space with the Self-Parameterized Recursively Assessed Decomposition Algorithm (SPRADA)

This paper introduces a non-parametric methodology based on classical unsupervised clustering techniques to automatically identify the main regions of a space, without requiring the objective number of clusters, so as to identify the major regular states of unknown industrial systems. Indeed, useful knowledge on real industrial processes entails the identification of their regular states, and their historically encountered anomalies. Since both should form compact and salient groups of data, unsupervised clustering generally performs this task fairly accurately; however, this often requires the number of clusters upstream, knowledge which is rarely available. As such, the proposed algorithm operates a first partitioning of the space, then it estimates the integrity of the clusters, and splits them again and again until every cluster obtains an acceptable integrity; finally, a step of merging based on the clusters’ empirical distributions is performed to refine the partitioning. Applied to real industrial data obtained in the scope of a European project, this methodology proved able to automatically identify the main regular states of the system. Results show the robustness of the proposed approach in the fully-automatic and non-parametric identification of the main regions of a space, knowledge which is useful to industrial anomaly detection and behavioral modeling.

Pseudocompact and precompact topological subsemigroups of topological groups

It is known that every pseudocompact topological group is precompact, we extend this result to a class of subsemigroup of topological groups. Then we use this results to prove that cancellative locally compact countably compact topological semigroups with open shifts are topological groups and to give a sufficient condition under which a locally compact monothetic topological semigroup is a compact topological group.

On the homology of spaces of equivariant maps

A spectral sequence calculating the homology groups of some spaces of maps equivariant under compact group actions is described. For the main motivating example, we calculate the rational homology groups of spaces of even and odd maps [Formula: see text], [Formula: see text], or, which is the same, the stable homology groups of spaces of non-resultant homogeneous polynomial maps [Formula: see text] of increasing degree. Also, we calculate the homology groups of spaces of [Formula: see text]-equivariant maps of odd-dimensional spheres for any [Formula: see text]. In auxiliary calculations, we find the homology groups of configuration spaces of projective and lens spaces with coefficients in several local systems.

Tight wavelet frames on zero-dimensional groups. Construction and approximation

We present a method for constructing tight wavelet frames on locally compact zero-dimensional groups. When constructing frames, we do not use the principle of unitary extension. We also consider the approximate properties of the resulting frames for functions from the Sobolev space with logarithmic weight.

Proper Equivariant Stable Homotopy Theory

This monograph introduces a framework for genuine proper equivariant stable homotopy theory for Lie groups. The adjective ‘proper’ alludes to the feature that equivalences are tested on compact subgroups, and that the objects are built from equivariant cells with compact isotropy groups; the adjective ‘genuine’ indicates that the theory comes with appropriate transfers and Wirthmüller isomorphisms, and the resulting equivariant cohomology theories support the analog of an R O ( G ) R O(G) -grading. Our model for genuine proper G G -equivariant stable homotopy theory is the category of orthogonal G G -spectra; the equivalences are those morphisms that induce isomorphisms of equivariant stable homotopy groups for all compact subgroups of G G . This class of π ∗ \pi _* -isomorphisms is part of a symmetric monoidal stable model structure, and the associated tensor triangulated homotopy category is compactly generated. Consequently, every orthogonal G G -spectrum represents an equivariant cohomology theory on the category of G G -spaces. These represented cohomology theories are designed to only depend on the ‘proper G G -homotopy type’, tested by fixed points under all compact subgroups. An important special case of our theory are infinite discrete groups. For these, our genuine equivariant theory is related to finiteness properties in the sense of geometric group theory; for example, the G G -sphere spectrum is a compact object in our triangulated equivariant homotopy category if the universal space for proper G G -actions has a finite G G -CW-model. For discrete groups, the represented equivariant cohomology theories on finite proper G G -CW-complexes admit a more explicit description in terms of parameterized equivariant homotopy theory, suitably stabilized by G G -vector bundles. Via this description, we can identify the previously defined G G -cohomology theories of equivariant stable cohomotopy and equivariant K-theory as cohomology theories represented by specific orthogonal G G -spectra.

Character formulas from matrix factorizations

abstract: In this paper I present a new and unified method of proving character formulas for discrete series representations of connected Lie groups by applying a Chern character-type construction to the matrix factorizations of Freed-Hopkins-Teleman. In the case of a compact group I recover the Kirillov formula, thereby exhibiting the work of Freed-Teleman as a categorification of the Kirillov correspondence. In the case of a real semisimple group I recover the Rossman character formula with only a minimal amount of analysis. The appeal of this method is that it relies almost entirely on highest-weight theory, which is a far more ubiquitous phenomenon than the varied techniques that were previously used to prove such formulas.

Compact quantum group structures on type-I $\mathrm{C}^*$-algebras

We prove a number of results having to do with equipping type-I \mathrm{C}^* -algebras with compact quantum group structures, the two main ones being that such a compact quantum group is necessarily co-amenable, and that if the \mathrm{C}^* -algebra in question is an extension of a non-zero finite direct sum of elementary \mathrm{C}^* -algebras by a commutative unital \mathrm{C}^* -algebra then it must be finite-dimensional.

L_p–L_q Fourier Multipliers on Locally Compact Quantum Groups

Let mathbb {G} be a locally compact quantum group with dual widehat{mathbb {G}}. Suppose that the left Haar weight varphi and the dual left Haar weight widehat{varphi } are tracial, e.g. mathbb {G} is a unimodular Kac algebra. We prove that for 1<ple 2 le q<infty , the Fourier multiplier m_{x} is bounded from L_p(widehat{mathbb {G}},widehat{varphi }) to L_q(widehat{mathbb {G}},widehat{varphi }) whenever the symbol x lies in L_{r,infty }(mathbb {G},varphi ), where 1/r=1/p-1/q. Moreover, we have ‖mx:Lp(G^,φ^)→Lq(G^,φ^)‖≤cp,q‖x‖Lr,∞(G,φ),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\Vert m_{x}:L_p(\\widehat{\\mathbb {G}},\\widehat{\\varphi })\\rightarrow L_q(\\widehat{\\mathbb {G}},\\widehat{\\varphi })\\Vert \\le c_{p,q} \\Vert x\\Vert _{L_{r,\\infty }(\\mathbb {G},\\varphi )}, \\end{aligned}$$\\end{document}where c_{p,q} is a constant depending only on p and q. This was first proved by Hörmander (Acta Math 104:93–140, 1960) for mathbb {R}^n, and was recently extended to more general groups and quantum groups. Our work covers all these results and the proof is simpler. In particular, this also yields a family of L_p-Fourier multipliers over discrete group von Neumann algebras. A similar result for mathcal {S}_p-mathcal {S}_q Schur multipliers is also proved.

Galaxy evolution in compact groups – I. Revealing a transitional galaxy population through a multiwavelength approach

ABSTRACT Compact groups (CGs) of galaxies show members with morphological disturbances, mainly products of galaxy–galaxy interactions, thus making them ideal systems to study galaxy evolution, in high-density environment. To understand how this environment affects the properties of galaxies, we select a sample of 340 CGs in the Stripe 82 region, for a total of 1083 galaxies, and a sample of 2281 field galaxies as a control sample. By performing a multiwavelength morphological fitting process using Southern Photometric Local Universe Survey data, we divide our sample into early-type galaxies (ETGs), late-type galaxies (LTGs), and transition galaxies using the r-band Sérsic index and the colour (u − r). We find a bimodal distribution in the plane of the effective radius – Sérsic index, where a secondary ‘peculiar’ galaxy population of smaller and more compact galaxies is found in CGs, which is not observed in the control sample. This indicates that galaxies are undergoing a morphological transformation in CGs. In addition, we find significant statistical differences in the distribution of specific star-formation rate (sSFR) when we compare both environments for LTGs and ETGs. We also find a higher fraction of quenched galaxies and a lower median sSFR in CGs than in the control sample, suggesting the existence of environmental effects favouring the cessation of star formation, regardless of galaxy type. Our results support the notion that CGs promote morphological and physical transformations, highlighting their potential as ideal systems for galaxy pre-processing.

ABELIAN GROUPS DEFINABLE IN p-ADICALLY CLOSED FIELDS

Abstract Recall that a group G has finitely satisfiable generics (fsg) or definable f-generics (dfg) if there is a global type p on G and a small model $M_0$ such that every left translate of p is finitely satisfiable in $M_0$ or definable over $M_0$ , respectively. We show that any abelian group definable in a p-adically closed field is an extension of a definably compact fsg definable group by a dfg definable group. We discuss an approach which might prove a similar statement for interpretable abelian groups. In the case where G is an abelian group definable in the standard model $\mathbb {Q}_p$ , we show that $G^0 = G^{00}$ , and that G is an open subgroup of an algebraic group, up to finite factors. This latter result can be seen as a rough classification of abelian definable groups in $\mathbb {Q}_p$ .

Convergence of Fourier truncations for compact quantum groups and finitely generated groups

We generalize the Fejér-Riesz operator systems defined for the circle group by Connes and van Suijlekom to the setting of compact matrix quantum groups and their ergodic actions on C*-algebras. These truncations form filtrations of the containing C*-algebra. We show that when they and the containing C*-algebra are equipped with suitable quantum metrics, then under suitable conditions they converge to the containing C*-algebra for quantum Gromov-Hausdorff distance. Among other examples, our results are applicable to the quantum groups SUq(2) and their homogeneous spaces Sq2.

Harmonic analysis of little [formula omitted]-Legendre polynomials

Many classes of orthogonal polynomials satisfy a specific linearization property giving rise to a polynomial hypergroup structure, which offers an elegant and fruitful link to Fourier analysis, harmonic analysis and functional analysis. From the opposite point of view, this allows regarding certain Banach algebras as L1-algebras, associated with underlying orthogonal polynomials. The individual behavior strongly depends on these underlying polynomials. We study the little q-Legendre polynomials, which are orthogonal with respect to a discrete measure. We will show that their L1-algebras have the property that every element can be approximated by linear combinations of idempotents. This particularly implies that these L1-algebras are weakly amenable (i.e., every bounded derivation into the dual module is an inner derivation), which is known to be shared by any L1-algebra of a locally compact group; in the polynomial hypergroup context, weak amenability is rarely satisfied and of particular interest because it corresponds to a certain property of the derivatives of the underlying polynomials and their (Fourier) expansions w.r.t. the polynomial basis. To our knowledge, the little q-Legendre polynomials yield the first example of a polynomial hypergroup whose L1-algebra is weakly amenable and right character amenable but fails to be amenable. As a crucial tool, we establish certain uniform boundedness properties of the characters. Our strategy relies on the Fourier transformation on hypergroups, the Plancherel isomorphism, continued fractions, character estimations and asymptotic behavior.

Transference and restriction of Fourier multipliers on Orlicz spaces

AbstractLet G be a locally compact abelian group with Haar measure and be Young functions. A bounded measurable function m on G is called a Fourier ‐multiplier if defined for functions in such that , extends to a bounded operator from to . We write for the space of ‐multipliers on G and study some properties of this class. We give necessary and sufficient conditions for m to be a ‐multiplier on various groups such as , and . In particular, we prove that regulated ‐multipliers defined on coincide with ‐multipliers defined on the real line with the discrete topology D, under certain assumptions involving the norm of the dilation operator acting on Orlicz spaces. Also, several transference and restriction results on multipliers acting on and are achieved.