Topological Group Research Articles

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.

Digital topological groups

In this article, we develop the basic theory of digital topological groups. The basic definitions directly lead to two separate types, based on the details of the continuity required of the group multiplication. We define NP1- and NP2-digital topological groups, and investigate their properties and algebraic structure. The NP2-type is very restrictive, and we give a complete classification of NP2-digital topological groups. We also give many examples of NP1-digital topological groups. We define digital topological group homomorphisms, and describe the digital counterpart of the first isomorphism theorem.

Geometries of topological groups

The paper provides an overarching framework for the study of some of the intrinsic geometries that a topological group may carry. An initial analysis is based on geometric nonlinear functional analysis, that is, the study of Banach spaces as metric spaces up to various notions of isomorphism, such as bi-Lipschitz equivalence, uniform homeomorphism, and coarse equivalence. This motivates the introduction of the various geometric categories applicable to all topological groups, namely, their uniform and coarse structure, along with those applicable to a more select class, that is, (local) Lipschitz and quasimetric structure. Our study touches on Lie theory, geometric group theory, and geometric nonlinear functional analysis and makes evident that these can all be seen as instances of a single coherent theory.

Equidistribution of Continuous Functions Along Monotone Compact Covers

Abstract We give a necessary and sufficient condition for equidistribution of continuous functions along monotone compact covers on locally compact spaces. We show the existence of equidistributed mappings along Bohr nets arising from group actions. Using almost periodic means, we give an analogue of Weyl’s equidistribution criterion for continuous functions with values in arbitrary topological groups. We prove van der Corput’s inequality on the lattice ℕm for vectors in Hilbert spaces, and use this inequality to extend Hlawka’s equidistribution theorem to functions on the lattice ℕm (m ≥ 1) with values in arbitrary topological groups.

Groups of convex bodies

In this paper we introduce and study a topological abelian group of convex bodies, analogous to the scissors congruence group and McMullen’s polytope algebra, with the universal property that continuous valuations on convex bodies correspond to continuous homomorphisms on the group of convex bodies. To study this group, we first obtain a version of McMullen polynomiality for valuations that take values not in fields or vector spaces, but in abelian groups. Using this, we are able to equip the group of convex bodies with a grading that consists of real vector spaces in all positive degrees, mirroring one of the main structural properties of the polytope algebra. It is hoped that this work can serve as the starting point for a K-theoretic interpretation of valuations on convex bodies.

Learning Topology-Specific Experts for Molecular Property Prediction

Recently, graph neural networks (GNNs) have been successfully applied to predicting molecular properties, which is one of the most classical cheminformatics tasks with various applications. Despite their effectiveness, we empirically observe that training a single GNN model for diverse molecules with distinct structural patterns limits its prediction performance. In this paper, motivated by this observation, we propose TopExpert to leverage topology-specific prediction models (referred to as experts), each of which is responsible for each molecular group sharing similar topological semantics. That is, each expert learns topology-specific discriminative features while being trained with its corresponding topological group. To tackle the key challenge of grouping molecules by their topological patterns, we introduce a clustering-based gating module that assigns an input molecule into one of the clusters and further optimizes the gating module with two different types of self-supervision: topological semantics induced by GNNs and molecular scaffolds, respectively. Extensive experiments demonstrate that TopExpert has boosted the performance for molecular property prediction and also achieved better generalization for new molecules with unseen scaffolds than baselines. The code is available at https://github.com/kimsu55/ToxExpert.

The company you keep: Effect of close social subgroup influence on STEM degree persistence at a small liberal arts college

The company you keep: Effect of close social subgroup influence on STEM degree persistence at a small liberal arts college

Topological Symmetry Groups of the Petersen Graphs

The topological symmetry group of an embedding Γ of an abstract graph γ in S3 is the group of automorphisms of γ that can be realized by homeomorphisms of the pair (S3,Γ). These groups are motivated by questions about the symmetries of molecules in space. The Petersen family of graphs is an important family of graphs for many problems in low-dimensional topology, so it is desirable to understand the possible groups of symmetries of their embeddings in space. In this paper, we find all the groups that can be realized as topological symmetry groups for each of the graphs in the Petersen family. Along the way, we also complete the classification of the realizable topological symmetry groups for K3,3.

The AH conjecture for Cantor minimal dihedral systems

The AH conjecture relates the low-dimensional homology groups of a groupoid with the abelianization of its topological full group. We show that transformation groupoids of minimal actions of the infinite dihedral group on the Cantor set satisfy this conjecture. The proof uses Kakutani–Rokhlin partitions adapted to such systems.

A Novel Electromagnetic Positioning Prototype System With Simplified Receiver for Interventional Surgery Application

To meet the need of 3D positioning of surgical instru-ment in interventional surgery and the requirement of smaller size sensor for the narrow blood vessels, a new electromagnetic posi-tioning model is proposed with simplified receiver. Based on the electromagnetic theory and geometry principle, the electromag-netic field transmitter with groups of three orthogonal coils and the simplified receiver with smaller size than existing sensors are designed. Then, using Biot-Savart Law, the distance between re-ceiving end and geometric center of orthogonal coils is calculated, and the spatial coordinate of receiving end is computed with the spherical intersection formula. To further reduce the positioning error, a two-round accuracy improvement algorithm is designed for selecting the optimal topology of coil groups at the transmitter end and fitting correction of the calculated distance. We imple-ment the prototype system and perform both simulation experi-ments and actual experiments. The results show that our proposed new electromagnetic positioning method with simplified receiver has higher accuracy and stronger stability, compared with the tra-ditional positioning approach of three-coil transmitter and three-coil receiver.

On fuzzy paratopological group and decision making during robot crash

In this article, we introduce fuzzy paratopological group, fuzzy semitopological group and fuzzy quasitopological group with illustrative examples and properties. These new notions belong to fuzzy topological group. Here, we prove that each fuzzy regular paratopological group is completely regular by using fuzzy uniformities. Moreover, we prove some results related to fuzzy semitopological group and fuzzy quasitopological group. In addition, we provide an application in the area of decision making during robot crash by using our above stated notions and nano topology.

Basic theorems in local class field theory and some further exploration

The relation between the abelian extension of a field and the topological groups of the field itself can be constructed using class field theory. In this piece of writing, the author will introduce the fundamental theorems of local class field theory by conducting a method of literature study. These theorems are the Reciprocity Law and the Existence Theorem, respectively. In addition to that, the author will discuss several unresolved issues in class field theory and provide examples of their applications in number theory. In class field theory, the results can be shown in two different ways. Both can be considered broad strokes. The first step is to demonstrate that the local case is true. By employing the methodology of cohomology and the theory of Lubin and Tate, one can demonstrate the Local Reciprocity Law and the Local Existence Theorem. The fundamental theorems in the global case are going to be demonstrated by utilizing the local results in conjunction with cohomology. Directly demonstrating the Gocal Reciprocity Law is another viable option.

Propagation of quasi-diffraction-free lommel-airy vortex wavepackets in chiral media

Lommel beams have significant applications in optical communication and optical manipulation due to the changeable symmetry of transverse intensity distribution and constantly varying orbital angular momentum. In this paper, a novel quasi-diffraction-free Lommel-Airy vortex (LAiV) wavepackets is proposed, and spatiotemporal dynamics of the LAiV wavepackets propagating in chiral media are investigated. Results indicate that several variables such as topological charge group velocity asymmetry factor and chiral parameter could be used to regulate the profiles of the 3D spatiotemporal vortex rings. Particularly, the spatiotemporal propagation trajectories and intensity patterns of the LAiV wavepackets could be manipulated by altering the asymmetry factor and the chiral parameter These results would deepen the understanding of the spatiotemporal propagation properties of the LAiV wavepackets and are expected to promote the development of the novel optical communications in chiral media.

Topological groups with a compact open subgroup, relative hyperbolicity and coherence

The main objects of study in this article are pairs (G,H) where G is a topological group with a compact open subgroup, and H is a finite collection of open subgroups. We develop geometric techniques to study the notions of G being compactly generated and compactly presented relative to H. This includes topological characterizations in terms of discrete actions of G on complexes, quasi-isometry invariance of certain graphs associated to the pair (G,H) when G is compactly generated relative to H, and extensions of known results from the discrete case. For example, generalizing results of Osin for discrete groups, we show that in the case that G is compactly presented relative to H:•if G is compactly generated, then each subgroup H∈H is compactly generated;•if each subgroup H∈H is compactly presented, then G is compactly presented. The article also introduces an approach to relative hyperbolicity for pairs (G,H) based on Bowditch's work using discrete actions on hyperbolic fine graphs. For example, we prove that if G is hyperbolic relative to H then G is compactly presented relative to H.As an application of the results of the article we prove combination results for coherent topological groups with a compact open subgroup, and extend McCammond-Wise perimeter method to this general framework.

Topological group actions by group automorphisms and Banach representations

Abstract We study Banach representability for actions of topological groups on groups by automorphisms (in particular, an action of a group on itself by conjugations). Every such action is Banach representable on some Banach space. The natural question is to examine when we can find representations on low complexity Banach spaces. In contrast to the standard left action of a locally compact second countable group G on itself, the conjugation action need not be reflexively representable even for SL 2 ⁡ ( ℝ ) {\operatorname{SL}_{2}(\mathbb{R})} . The conjugation action of SL n ⁡ ( ℝ ) {\operatorname{SL}_{n}(\mathbb{R})} is not Asplund representable for every n ≥ 4 {n\geq 4} . The linear action of GL n ⁡ ( ℝ ) {\operatorname{GL}_{n}(\mathbb{R})} on ℝ n {{\mathbb{R}}^{n}} , for every n ≥ 2 {n\geq 2} , is not representable on Asplund Banach spaces. On the other hand, this action is representable on a Rosenthal Banach space (not containing an isomorphic copy of l 1 {l_{1}} ). The conjugation action of a locally compact group need not be Rosenthal representable (even for Lie groups). As a byproduct, we obtain some counterexamples about Banach representations of homogeneous G-actions G / H {G/H} .

The Polish topology of the isometry group of the infinite dimensional hyperbolic space

We consider the isometry group of the infinite-dimensional separable hyperbolic space with its Polish topology. This topology is given by pointwise convergence. For non-locally compact Polish groups, some striking phenomena like automatic continuity or extreme amenability may happen. Our leading idea is to compare this topological group with usual Lie groups on one side and with non-Archimedean infinite-dimensional groups like \mathcal{S}_\infty , the group of all permutations of a countable set on the other side. Our main results are All along the text, we lead a parallel study with the sibling group \mathrm{Isom}(\mathcal{H}) , where \mathcal{H} is a separable Hilbert space.

Isotropic Gels of Cellulose Nanocrystals Grafted with Dialkyl Groups: Influence of Surface Group Topology from Nonlinear Oscillatory Shear

Attractive (non-self-assembling) aqueous cellulose nanocrystal (CNC) suspensions were topologically tailored into isotropic gels through the surface grafting of dialkyl groups. We thus focus on the influence of CNC concentration, including for pristine CNC, surface linker branching, branching degree, and the influence of side group size and branch-on-branch surface-grafted groups. The resulting mobility and strength of interaction in particle-particle interaction mediated by the surface groups was investigated from a rheological point of view. The emphasis is on nonlinear material parameters from Fourier-transform rheology and stress decomposition analysis. The results show that nonlinear material parameters are more sensitive than linear viscoelastic parameters to the onset of weakly interconnected networks in pristine CNC isotropic suspensions. All surface-modified CNC suspensions resulted in isotropic gels. The nonlinear material parameters were found to be broadly sensitive to CNC concentration, branching, degree of branching and surface-grafted linkers' length. However, the length of the grafted chains and the degree of branching were the primary factors influencing the nonlinear material response. Furthermore, the results showed evidence of two strain amplitude ranges with distinct nonlinear signatures that could be attributed to the disruption of weak network connection points and to distortions of more dense (aggregate) network regions, respectively.

Geometric arcs and fundamental groups of rigid spaces

Abstract We develop the notion of a geometric covering of a rigid space 𝑋, which yields a larger class of covering spaces than that studied previously by de Jong. Geometric coverings are closed under disjoint unions and are étale local on 𝑋. If 𝑋 is connected, its geometric coverings form a tame infinite Galois category and hence are classified by a topological group. The definition is based on the property of lifting of “geometric arcs” and is meant to be an analogue of the notion developed for schemes by Bhatt and Scholze.

Countably compact group topologies on arbitrarily large free Abelian groups

We prove that if there are c incomparable selective ultrafilters then, for every infinite cardinal κ such that κω=κ, there exists a group topology on the free Abelian group of cardinality κ without nontrivial convergent sequences and such that every finite power is countably compact. In particular, there are arbitrarily large countably compact groups. This answers a 1992 question of D. Dikranjan and D. Shakhmatov.

Embedding theorems for function spaces

In this paper, we have proved results similar to Tychonoff's Theorem on embedding a space of functions with the topology of pointwise convergence into the Tychonoff product of topological spaces, but applied to the function space C(X,Y) of all continuous functions from a topological space X into a uniform space Y with the topology of uniform convergence on a family of subsets of X and with the (weak) set-open topology. We also investigated the following question: how the topological embedding of the space C(X,Y) is related to algebraic structures (such as topological groups, topological rings and topological vector spaces) on C(X,Y).