Compact Group Research Articles

New mid‐Cretaceous macrothelids showing a similar living mode to extant Macrothelidae (Araneae: Mygalomorphae)

AbstractFive new spider fossils of the family Macrothelidae (Araneae: Mygalomorphae) are described from mid‐Cretaceous Kachin amber, Myanmar. A new genus Acanthomacrothele gen. nov. is established for three new species: Acanthomacrothele pauciverrucae sp. nov., Acanthomacrothele geminata sp. nov., and Acanthomacrothele longicaudata sp. nov. Both male and female specimens of A. pauciverrucae sp. nov. are described, representing a rare case of a fossil spider with both sexes known. We also describe an unidentified juvenile of Macrothelidae to document different developmental stages of Cretaceous species. To test the phylogenetic position of the new fossils, we undertook phylogenetic analyses using both topology‐unconstrained and topology‐constrained methods. Our analyses supported the placement of Acanthomacrothele gen. nov. in Macrothelidae, although its phylogenetic relationship to other genera of Macrothelidae differed among different analytical methods. The fossils documented here have elongated, widely spaced posterior lateral spinnerets, compact rectangular group of eyes on a tubercle, and more spines on male tibia of anterior legs than female, similar to extant macrothelids, suggesting that they probably have similar retreat construction behavior, habitat preference, and mating position to their living relatives.

Covariant projective representations of Hilbert–Lie groups

Abstract Hilbert–Lie groups are Lie groups whose Lie algebra is a real Hilbert space whose scalar product is invariant under the adjoint action. These infinite-dimensional Lie groups are the closest relatives to compact Lie groups. In this paper, we study unitary representations of these groups from various perspectives. First, we address norm-continuous, also called bounded, representations. These are well known for simple groups, but the general picture is more complicated. Our first main result is a characterization of the discrete decomposability of all bounded representations in terms of boundedness of the set of coroots. We also show that bounded representations of type II and III exist if the set of coroots is unbounded. Second, we use covariance with respect to a one-parameter group of automorphisms to implement some regularity. Here we develop some perturbation theory based on half-Lie groups that reduces matters to the case where a “maximal torus” is fixed, so that compatible weight decompositions can be studied. Third, we extend the context to projective representations which are covariant for a one-parameter group of automorphisms. Here important families of representations arise from “bounded extremal weights”, and for these, the corresponding central extensions can be determined explicitly, together with all one-parameter groups for which a covariant extension exists.

Abelian groups with solvable Lie endomorphism rings

A description has been obtained of all divisible, primary, as well as separable and algebraically compact torsion-free groups, the Lie endomorphism ring of which is solvable.

Continuity of the stabilizer map and irreducible extensions

Let G be a locally compact group. For every G -flow X , one can consider the stabilizer map x \mapsto G_{x} , from X to the space \mathrm{Sub}(G) of closed subgroups of G . This map is not continuous in general. We prove that if one passes from X to the universal irreducible extension of X , the stabilizer map becomes continuous. This result provides, in particular, a common generalization of a theorem of Frolík (that the set of fixed points of a homeomorphism of an extremally disconnected compact space is open) and a theorem of Veech (that the action of a locally compact group on its greatest ambit is free). It also allows to naturally associate to every G -flow X a stabilizer G -flow \mathrm{S}_{G}(X) in the space \mathrm{Sub}(G) , which generalizes the notion of stabilizer uniformly recurrent subgroup associated to a minimal G -flow introduced by Glasner and Weiss.

Complex Brain Morphology Discovered in the Shark Parasite Nybelinia surmenicola (Cestoda: Trypanorhyncha).

The ultrastructure of the nervous system has been studied in sexually mature Nybelinia surmenicola (Cestoda: Trypanorhyncha) from the intestine of a shark Lamna ditropis. The central nervous system (CNS) reveals a complex organization within cestodes and corresponds to the trypanorhynch pattern of brain architecture. The brain of N. surmenicola is differentiated into nine clearly defined lobes and semicircular, median, and X-shaped cruciate commissures. A specific feature is the presence of a powerful extracellular capsule that surrounds the brain lobes with the cortical glial cells. Moreover, the architecture of the anterior lobes clearly distinguishes the species of Tentacularioidea. The neurons of the anterior lobes form compact groups looking like frontal horns. There are approximately 120 neurons in the anterior lobes and a preliminary estimate of more than 300 perikarya in the brain. Several ultrastructural types of neurons have been identified, differing in the size and shape of the soma, the density of the cytoplasm, and the ultrastructure of synaptic vesicles. Numerous synapses involving clear and electron-dense vesicles have been observed in neuropils. Two types of glial cells have been found in the brain that participate in neuronal metabolism and wrap around the giant axons, brain lobes, neuropil compartments, and the main nerve cords. Such a powerful extracellular fibrillar brain capsule has not been observed in the brain of other studied cestodes and has been demonstrated in this study for the first time. The differentiation of the brain lobes reveals the important role of the rhyncheal system in the evolution of cestodes and correlates with their behavior. The anterior nerves arising from the anterior lobes innervate the radial muscles stabilizing the position of the tentacle sheaths and movements of the attachment organs. The nervous system anatomy and the brain architecture may reflect the morphofunctional aspects of the tapeworm evolution.

Simplicity of crossed products of the actions of totally disconnected locally compact groups on their boundaries

We prove that if a totally disconnected locally compact group admits a topologically free boundary, then the reduced crossed product of continuous functions on its Furstenberg boundary by the group is simple. We also prove a partial converse of this result.

Nuclear ribosomal transcription units in Asian Paragonimus species (Paragonimidae: Platyhelminthes): genetic characteristics, polymorphism, and implications for intersuperfamilial phylogeny.

The entire transcribed sequences (from the 5' terminus of 18S to the 3' terminus of 28S rRNA genes) of the ribosomal transcription units (rTU*) of five Asian Paragonimus species were obtained and characterized. The rTU* length was 7661bp for P. heterotremus (LC strain, Vietnam), 7422bp for P. iloktsuenensis (Amami strain, Japan), 6932bp for P. skrjabini miyazakii (OkuST1 strain, Japan), 7422bp for P. ohirai (Kino strain, Japan), and three strains of P. westermani: 8616bp (Megha strain, India), 7292bp (Bogil strain, South Korea), and 7052bp (QT2 strain, Vietnam) without intergenic spacer region (IGS). All seven Asian Paragonimus strains' genetic characteristics were described, including the length of individual genes/regions, repeat polymorphism, base composition, and skewness. To investigate the superfamilial relationships in the Xiphidiata, with a focus on the Troglotrematoidea and its associated superfamilies, we used the PhyML software package to create three comprehensive maximum-likelihood phylogenies. The datasets used were 83 concatenated 28S + 18S, 83 single complete 18S, and 157 single, partial 28S rDNA sequences, respectively, from entire rTUs and/or accessible ribosomal sequences of the same species from the suborders Xiphidiata, Echinostomata, and Haplosplanchnata, with a Schistosoma sequence as an outgroup. Three phylogenetic trees revealed that Echinostomata and Haplosplanchnata are monophyletic, while Xiphidiata is polyphyletic and contains the monophyletic Troglotrematoidea. The concatenated 18S + 28S and single 18S phylogenies revealed well-bootstrap supported seven superfamilies (Troglotrematoidea, Haploporoidea, Gorgoderoidea, Brachycladioidea, Microphalloidea, Plagiorchioidea, and Opecoeloidea) that are monophyletic in the Xiphidiata. The Haploporoidea was a basal superfamily nested close to the Gorgoderoidea and Troglotrematoidea and was not supported as a distinct suborder Haploporata. Six of seven xiphidiatan superfamilies were monophyletic in the partial 28S phylogeny, with the exception of Opecoeloidea, which was separated into two different subclades: Opecoelidae and Stenakridae/Zdzitowieckitrematidae. The monophyletic Haploporoidea/Haploporata was separated from the Gorgoderoidea associates and placed in a marginal group in Xiphidiata. There were two notable clusters in the Paragonimidae: mixed-Paragonimus, which included a fairly compact group of P. heterotremus strains, and P. westermani/siamensis, which was divided into geographical/country strain groups. In conclusion, combined ribosomal rDNA sequences were more effective than single rDNA markers in resolving interfamilial and familial relationships. The ribosomal datasets presented here will be useful for taxonomic reassessment, as well as evolutionary and population genetics research in the Troglotrematoidea and other superfamilies in the Xiphidiata and the class Trematoda.

A Cauchy–Davenport Theorem for Locally Compact Groups

Abstract We generalize the Cauchy–Davenport theorem to all locally compact groups.

Outcomes and prognostic factors of endodontically treated teeth filled with calcium silicate- or epoxy resin-based root canal sealers: A retrospective cohort study.

The objectives of this study were to compare the clinical outcomes of root canal treatment between calcium silicate-based sealer using the cold hydraulic condensation technique and epoxy resin-based sealer using warm vertical compaction and to identify the prognostic factors affecting the outcomes of both obturation techniques. Dental records and radiographs of the teeth that received root canal treatment with calcium silicate-based sealer using the cold hydraulic condensation technique or epoxy resin-based sealer using warm vertical compaction during 2015-2021 were selected using inclusion and exclusion criteria. The cases were matched between the two groups based on four factors: primary root canal treatment or retreatment, tooth type, periapical lesion and its size and extension of root canal filling. The success rates of root canal treatment between two sealers/obturation techniques were analysed. The incidence and resorption of extruded sealer were evaluated using the McNemar's test. Generalized estimated equations were used to evaluate any prognostic factors. The median recall period of this study was 15 months. A total of 234 teeth, comprising 117 cases in each sealer/obturation group, were included. No significant difference was found in the success rates between the calcium silicate-based sealer using the cold hydraulic condensation technique group (91.5%) and the epoxy resin-based sealer using the warm vertical compaction group (94.9%). The incidence of sealer extrusion for the calcium silicate-based sealer (27.6%) did not significantly differ from that of the epoxy resin-based sealer (36.2%), with both sealers typically demonstrating no resorption at recalls. No prognostic factors related to the success rate of either technique were detected. The study findings suggest no difference in the success rate between calcium silicate-based sealer and epoxy resin-based sealer. The study did not identify any prognostic factors that significantly influenced the outcomes of the endodontic treatments.

The First N,O-Chelated Diphenylboron-Based Fluorescent Probe for Peroxynitrite and Its Bioimaging Applications

Peroxynitrite (ONOO−) is a reactive oxygen species (ROS) that takes part in the oxidation-reduction homeostasis while at the same time being responsible for activating numerous pathological pathways. Accordingly, monitoring the dynamic changes in ONOO− concentration has attracted a great deal of attention, undoubtedly prompting the development of appropriate fluorescent chemosensors. Herein, we developed a novel N,O-chelated diphenylboron-based fluorescent probe (DPB) for ONOO− featuring high selectivity, a quick response time (2.0 min), and a low detection limit (55 nM). DPB incorporates tetra-coordinated boron in the center of the fluorogenic core and a three-coordinated boron from the pinacolphenylboronate fragment, which acts as the recognition site for ONOO−. As confirmed by HR-MS and 1H NMR, the interaction of DPB with ONOO− led to an oxidative cleavage of pinacolphenylboronate moiety to produce strongly emissive derivative DPB-OH. The fluorescence enhancement is likely a result of a substantial deactivation of non-radiative decay due to the replacement of the bulky pinacolphenylboronate moiety with a compact hydroxyl group. Importantly, DPB probe exhibits negligible cytotoxicity and favorable biocompatibility allowing for an efficient tracking of ONOO− in living cells and zebrafish. Overall, the current study does not only represents the first N,O-chelated diphenylboron-based fluorescent probe for a specific analyte, but also serves as a guideline for designing more potent fluorescent probes based on the chemistry of boron chelates.

Convolution Algebras of Double Groupoids and Strict 2-Groups

Double groupoids are a type of higher groupoid structure that can arise when one has two distinct groupoid products on the same set of arrows. A particularly important example of such structures is the irrational torus and, more generally, strict 2-groups. Groupoid structures give rise to convolution operations on the space of arrows. Therefore, a double groupoid comes equipped with two product operations on the space of functions. In this article we investigate in what sense these two convolution operations are compatible. We use the representation theory of compact Lie groups to get insight into a certain class of 2-groups.

Inclusions of simple C⁎-algebras arising from compact group actions

Inclusions of operator algebras have long been studied. In particular, inclusions arising from actions of compact groups on factors were studied by Izumi-Longo-Popa and others. The correspondence between intermediate subfactors and subgroups is called the Galois correspondence. Analogues for actions on C⁎-algebras have been studied by Izumi, Cameron-Smith, Peligrad, and others. In this article, we give examples of compact group actions on simple C⁎-algebras for which the Galois correspondence holds.

Estimates for Sums of Eigenfunctions of Elliptic Pseudo-differential Operators on Compact Lie Groups

We extend the estimates proved by Donnelly and Fefferman, and by Lebeau, Robbiano and Zuazua, for sums of eigenfunctions of the Laplacian (on a compact manifold) to estimates for sums of eigenfunctions of any positive and elliptic pseudo-differential operator of positive order on a compact Lie group. Our criteria are imposed in terms of the positivity of the corresponding matrix-valued symbol of the operator. As an application of these inequalities in control theory, we obtain the null-controllability for diffusion models for elliptic pseudo-differential operators on compact Lie groups.

Galois theory and homology in quasi-abelian functor categories

Given a finite category T, we consider the functor category AT, where A can be any quasi-abelian category. Examples of quasi-abelian categories are given by any abelian category but also by non-exact additive categories as the categories of torsion(-free) abelian groups, topological abelian groups, locally compact abelian groups, Banach spaces and Fréchet spaces. In this situation, the categories of various internal categorical structures in A, such as the categories of internal n-fold groupoids, are equivalent to functor categories AT for a suitable category T. For a replete full subcategory S of T, we define F to be the full subcategory of AT whose objects are given by the functors F:T→A with F(T)=0 for all T∉S. We prove that F is a torsion-free Birkhoff subcategory of AT. This allows us to study (higher) central extensions from categorical Galois theory in AT with respect to F and generalized Hopf formulae for homology.

Lebesgue-Fourier algebras on coset spaces and approximate amenability

Let K be a compact subgroup of a locally compact group G. In this work, we initiate an investigation of the Lebesgue-Fourier algebra S 1 A(G/K) on the coset space G/K with pointwise multiplication; after some general results on the Banach algebra S 1 A(G/K), as our main result, we give some necessary conditions for that S 1 A(G/K) be approximately amenable in terms of algebraic and topological properties of G and K .

Amenability of finite energy path and loop groups

It is shown that the groups of finite energy (i.e., Sobolev class H^{1} ) paths and loops with values in a compact Lie group are amenable in the sense of Pierre de la Harpe, that is, every continuous action of such a group on a compact space admits an invariant regular Borel probability measure. To our knowledge, the strongest previously known result concerned the amenability of groups of continuous paths and loops (Malliavin and Malliavin 1992).

The space of locally elliptic subgroups of a locally compact group

Let G be a locally compact group. We denote by Sub ( G ) the space of closed subgroups of G equipped with the Chabauty topology. A closed subgroup H of G is called locally elliptic if every compact subset of H is contained in a compact subgroup. In this paper we establish that the subset Su b L E ( G ) of closed locally elliptic subgroups of G is Chabauty-closed if and only if the set com p n ( G ) consisting of the n-tuples of elements of G contained in a common compact subgroup, is closed in the Cartesian product G n , for every nonnegative integer n. Moreover, we prove that the subspace Su b L E ( G ) is Chabauty-closed in the case when the group G contains an open normal locally elliptic subgroup.

All Hartman–Mycielski groups are minimally almost periodic

For each topological group G, it is shown that the Hartman–Mycielski group G• over G is minimally almost periodic, that is, all continuous homomorphisms of G• to compact Hausdorff topological groups are trivial. We also show that the group G•, over a compactly generated topological group G, is algebraically generated by a compact connected subset. Consequently, a wealth of minimally almost periodic, connected, compactly generated topological Abelian groups exist.

Quantifying noncovariance of quantum channels with respect to groups

A quantum channel is covariant with respect to a group if it commutes with the action of the group. In general, a quantum channel may not be covariant with respect to a given group. The degree of noncovariance can vary between different channels, and it is desirable to have a quantitative characterization for the degree of channel noncovariance. In this work, we propose a measure based on the Hilbert-Schmidt norm to quantify noncovariance of quantum channels with respect to a group and demonstrate that it satisfies several desirable properties. Compared with the existing measures of channel noncovariance, our measure applies to not only compact Lie groups but also finite groups, and it is easy to evaluate. Using this measure and its modified version together with two existing measures, we evaluate and analyze channel noncovariance through an example, finding that these measures of channel noncovariance are closely related but differ from each other. They capture different perspectives of noncovariance of quantum channels. As applications, we provide a relation between channel noncovariance and approximate quantum error correction using our measures of channel noncovariance.

Quasi-invariant states

In this paper, we develop the theory of quasi-invariant (respectively, strongly quasi-invariant) states under the action of a group [Formula: see text] of normal ∗-automorphisms of a ∗-algebra (or von Neumann algebra) [Formula: see text]. We prove that these states are naturally associated to left-[Formula: see text]-[Formula: see text]-cocycles. If [Formula: see text] is compact, the structure of strongly [Formula: see text]-quasi-invariant states is determined. For any [Formula: see text]-strongly quasi-invariant state [Formula: see text], we construct a unitary representation associated to the triple [Formula: see text]. We prove, under some conditions, that any quantum Markov chain with commuting, invertible and Hermitian conditional density amplitudes on a countable tensor product of type I factors is strongly quasi-invariant with respect to the natural action of the group [Formula: see text] of local permutations and we give the explicit form of the associated cocycle. This provides a family of nontrivial examples of strongly quasi-invariant states for locally compact groups obtained as inductive limit of an increasing sequence of compact groups.