Search

We provide a unified treatment of several results concerning full groups of ample groupoids and paradoxical decompositions attached to them. This includes a criterion for the full group of an ample groupoid being amenable as well as comparison of its orbit, Koopman and groupoid-left-regular representations. Besides that, we unify several recent results about paradoxicality in semigroups and groupoids, relating embeddings of Thompson's group V into full groups of ample étale groupoids. [ABSTRACT FROM AUTHOR]

Given a finite category T , we consider the functor category A T , 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 A T for a suitable category T. For a replete full subcategory S of T , we define F to be the full subcategory of A T 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 A T. This allows us to study (higher) central extensions from categorical Galois theory in A T with respect to F and generalized Hopf formulae for homology. [ABSTRACT FROM AUTHOR]

Given a regular representation of a finite group G and a positive integer number n , we construct a (finite) topological space X such that its group of homotopy classes of self-homotopy equivalences E (X) and its group of homeomorphisms A u t (X) are isomorphic to G , and the action of G on the n -th homology group H n (X) is the regular representation. We also discuss other representations. [ABSTRACT FROM AUTHOR]

Objective: In the freeway tunnel approach section, lane-changing behaviors and transitions in the driving environment exacerbate traffic flow disruptions, increase driving risks, and lead to a higher accident rate. To this end, this study presents a method to explore the risk evolution process of lane-changing in these sections and evaluate its impact on traffic flow operations surrounding lane-changing vehicles. Methods: First, a driving risk potential field model based on the field theory, which consists of a vehicle kinetic potential field and a tunnel illumination potential field, is proposed to evaluate the driving risk. Furthermore, a "vehicle group" risk graph was constructed based on graph theory, incorporating both a node-coupling driving risk model and a topological potential entropy model. Finally, trajectory datasets were collected through naturalistic driving tests to analyze the evolution of lane-changing risk and the stability of the vehicle group. Results: From the analysis of coupling driving risk evolution, we found that in the [100, 500) m, [500, 1000) m, and [1000, 1500) m freeway tunnel approach sections, the coupled driving risk of lane-changing vehicle (LCV) was higher than that of the other vehicles in the vehicle group. In different tunnel approach sections, LCVs that received the highest risk were from different vehicles in vehicle group. LCVs received the highest field strength from the risk potential fields of the lateral vehicle (LV), front lateral vehicle (FLV), and front vehicle (FV) in the [100, 500) m, [500, 1000) m, [1000, 1500) m tunnel approach sections, respectively. Based on the absolute value of the vehicle group topological potential entropy, we observed the resilience of the vehicle group system improved with increasing distance from the tunnel entrance. Traffic flow regained stability more quickly after lane-changing disturbances in sections farther from the tunnel entrance. Conclusions: This study highlights that the section closer to the freeway tunnel entrance significantly impact lane-changing risk, and it takes longer for the vehicle group to recover its stability after a lane-changing disturbance. The research results offer a theoretical and methodological foundation for enhancing traffic safety measures and developing microscopic driving behavior models for freeway tunnel approach sections. [ABSTRACT FROM AUTHOR]

Introduction: China's marine cities have reached a critical juncture after 40 years of rapid development. In this new stage, where internal circulation is the main focus, there is a need to enhance the internal circulation capabilities of these cities and unleash their full economic potential. This paper aims to explore the positioning and improvement path of marine cities in China's internal circulation network, and fully unleash the development potential of marine cities. Methods: Based on data from 284 prefecture-level cities in China, this paper constructs the social network of China's urban internal circulation with the help of the modified gravity model, and explores the conditional configuration of the improvement of the status of marine cities in internal circulation network by using the fuzzy-set qualitative comparative analysis (fsQCA) method. Results and discussion: (1) The development level of marine cities' internal circulation can be categorized into three tiers, led by Shanghai. The development gap between the 14 marine cities has gradually widened over recent years. (2) Chinese marine cities can be divided into three groups in the topological structure of China's urban internal circulation network: core, periphery, and edge, with Shanghai being the core "bridge" in the network. The traditional advantages of some northern economically strong cities in the construction of the internal circulation network have gradually been lost, and many marine cities have seen their leadership and control over the internal circulation network significantly weakened. (3) No single factor is a necessary condition for achieving a high-level status of marine cities in the internal circulation network. (4) The four conditional variables of demand side, supply side, industrial linkage and inter-regional integration have two sufficient condition configurations to enhance the status of marine cities in internal circulation network. Among them, the "industry-regional integration"-dominated configuration with the core of unblocking the bottlenecks of the internal circulation is the main path. [ABSTRACT FROM AUTHOR]

The NAC gene family plays a crucial role in plant growth, development, and responses to biotic and abiotic stresses. Paspalum Vaginatum, a warm-season turfgrass with exceptional salt tolerance, can be irrigated with seawater. However, the NAC gene family in seashore paspalum remains poorly understood. In this study, genome-wide screening and identification were conducted based on the NAC (NAM) domain hidden Markov model in seashore paspalum, resulting in the identification of 168 PvNAC genes. A phylogenetic tree was constructed, and the genes were classified into 18 groups according to their topological structure. The physicochemical properties of the PvNAC gene family proteins, their conserved motifs and structural domains, cis-acting elements, intraspecific collinearity analysis, GO annotation analysis, and protein–protein interaction networks were analyzed. The results indicated that the majority of PvNAC proteins are hydrophilic and predominantly localized in the nucleus. The promoter regions of PvNACs are primarily enriched with light-responsive elements, ABRE motifs, MYB motifs, and others. Intraspecific collinearity analysis suggests that PvNACs may have experienced a large-scale gene duplication event. GO annotation indicated that PvNAC genes were essential for transcriptional regulation, organ development, and responses to environmental stimuli. Furthermore, the protein interaction network predicted that PvNAC73 interacts with proteins such as BZIP8 and DREB2A to form a major regulatory hub. The transcriptomic analysis investigates the expression patterns of NAC genes in both leaves and roots under varying durations of salt stress. The expression levels of 8 PvNACs in roots and leaves under salt stress were examined and increased to varying degrees under salt stress. The qRT-PCR results demonstrated that the expression levels of the selected genes were consistent with the FPKM value trends observed in the RNA-seq data. This study established a theoretical basis for understanding the molecular functions and regulatory mechanisms of the NAC gene family in seashore paspalum under salt stress. [ABSTRACT FROM AUTHOR]

We prove that if G is a totally bounded Abelian group such that its dual group \widehat {G}_p equipped with the finite-open topology is a Baire group, then every compact subset of G must be finite. This solves an open question by Chasco, Domínguez, and Tkachenko. Among other consequences, we obtain an example of a group that is g-dense in its completion but is not g-barreled. This solves a question proposed by Außenhofer and Dikranjan. [ABSTRACT FROM AUTHOR]

We show that pseudocharacter turns out to be discretely reflexive in Lindelöf Σ -groups but countable tightness is not discretely reflexive in hereditarily Lindelöf spaces. We also establish that it is independent of ZFC whether countable character, countable weight or countable network weight is discretely reflexive in spaces C p (X) . Furthermore, we prove that any hereditary topological property is discretely reflexive in spaces C p (X) with the Lindelöf Σ -property. If C p (X) is a Lindelöf Σ -space and LD is a k -space for any discrete subspace D C p (X) , then it is consistent with ZFC that C p (X) has the Fréchet–Urysohn property. Our results solve two published open questions. [ABSTRACT FROM AUTHOR]

This paper aims to stimulate debate around the development of a place‐based research review methodology. We present place‐based reviews as a potential source of support for wellbeing‐related local policymaking. Our introductory discussions highlight an ever‐growing need for insights about specific localities and a lack in resources—including time—for local policymakers to engage with research. Additionally, increasing demands for local insights have been driven by devolution shifts, which redistribute policymaking responsibilities to local authorities. Hence, we explore the challenges and opportunities that arise when places are considered in reviewing research relevant to wellbeing. We build a case study around two related places of different scale: Truro, a small cathedral city in the United Kingdom's Southwest; and Cornwall, the regional county that contains Truro. We use these places as search terms in combination with terms concerning health and social care (HSC) services. HSC services are included as a component of our case study, as the topic is a consistent concern for wellbeing‐related policies. In our findings, we report a lack of papers on our smaller scale of place (Truro). One might expect this outcome. Nonetheless, we reflect on current research practices and processes that might have further limited our ability to generate insights about Truro. Encouragingly, our findings on Cornwall demonstrate the potential of place‐based reviews in supporting local policymaking more broadly. We make initial judgements around knowledge gaps—including the exclusion of perspectives from certain groups and identities—and topological insights, that is, those that are relevant to Cornwall as a whole. Our discussions also consider how place‐based reviews can be enhanced via the retrieval and inclusion of non‐academic studies. Finally, key questions to induce debate on this subject are posed in the conclusion. We aim to induce debate around the development of a place‐based review methodology. Such reviews have the potential to be useful tools for local policymakers and the establishment of place‐based policies around wellbeing. [ABSTRACT FROM AUTHOR]

Let G be a topological group, and let C(G) denote the algebra of continuous, complex valued functions on G. We determine the solutions f , g , h ∈ C (G) of the Levi-Civita equation g (x y) = g (x) g (y) + f (x) h (y) , x , y ∈ G , that extends the cosine addition law. As a corollary we obtain the solutions f , g ∈ C (G) of the cosine subtraction law g (x y ∗) = g (x) g (y) + f (x) f (y) , x , y ∈ G where x ↦ x ∗ is a continuous involution of G. That x ↦ x ∗ is an involution, means that (x y) ∗ = y ∗ x ∗ and x ∗ ∗ = x for all x , y ∈ G . [ABSTRACT FROM AUTHOR]

The space of degree d smooth projective hypersurfaces of |$\mathbb{C}P^n$| admits a scanning map to a certain space of sections. We compute a rational homotopy model of the action by conjugation of the group |$U(n+1)$| on this space of sections, from which we deduce that the scanning map induces a monomorphism on cohomology when d  > 2. Our main technique is the rational homotopy theory of Sullivan and, more specifically, a Sullivan model for the action by conjugation of a connected topological group on a space of sections. [ABSTRACT FROM AUTHOR]

Automatically segmenting the lesion of clinical data can aid doctors in diagnosis. The key issues with clinical brain tumor segmentation are partial volume effect, bias field, and noise interference. This paper proposes a novel segmentation algorithm based on pathology topological merging to solve the above problems. Here, an improved superpixel technique is used to group the pathology topological blocks, and the vector distance is refined to avoiding the problem of grouping pixels with small similarity near the tumor contour into the same region. Furthermore, meaningful pathology topological blocks are formed, and the entire brain tumor is segmented based on the pathology topological relationship and weight between pathology topological blocks. The proposed method is validated on the BraTS 2015 dataset and 123 patient images with brain tumors from a local hospital, and the mean Dice, Jaccard, Precision, and Recall values are 0.91, 0.92, 0.90, and 0.91, respectively, indicating that the proposed method can efficiently and accurately distinguish brain tumors from other tissues (such as edema). The method can overcome some defects (e.g. partial volume effect, bias field, and noise interference) in medical brain images while having practical clinical application prospects. [ABSTRACT FROM AUTHOR]

This paper's primary purpose is to seek properties of isocompact spaces by topological groups. In this work, we propose and say that a topological space X has the isoc property if each family of isocompact subsets in X is weakly hereditarily closure-preserving. Our first result shows that each T2 topological group with a locally compact subgroup, where the quotient group is isocompact, is isocompact if it has the isoc property. Our second result provides a necessary and sufficient condition for an ω narrow to be Lindelöf. [ABSTRACT FROM AUTHOR]

Mobile Edge Computing (MEC) is a new computing paradigm that has shown great potential. How to extract the cooperative topological relationship between MEC servers to realize jointly computing is the key problem to solve the bottleneck of MEC computational capability. In previous studies, multi-MEC servers are regarded as unit computing nodes with the same cooperation relationship to jointly schedule offloading tasks, without considering the hierarchical and clustered topology of the server collaborative work. As a result, in the scenario of unbalanced distribution of computing resources, it is difficult to obtain the optimal joint scheduling strategy for offloading tasks according to the cooperation relationship and resource differences among MEC servers. Therefore, this paper considers introducing the topological relationship of group cooperation among multi-MEC servers to optimize the joint scheduling strategy, and proposes a Multi-Agent Hierarchical Graph Attention Soft Actor-Critic algorithm (MHSAC). Firstly, based on the differences in their own resources and the demands of the tasks they undertake, MEC servers are divided into series clusters. Then, a Hierarchical Graph Attention Network (HGAT) is used to model each agent to extract the physical communication topology information of the MEC server and the group topology information of multi-edge cooperation. The multi-agent soft Actor-Critic algorithm is used to obtain the offloading scheduling decision of multi-edge cooperation. Experiments show that the MHSAC algorithm that considering the topological relationship of multi-edge group cooperation can optimize load distribution under low latency and resource-limited requirements, achieving optimal load balancing values and task drop rates. [ABSTRACT FROM AUTHOR]

We study topological group theoretic properties of algebraic groups over local fields. In particular, we find conditions under which such groups have closed images under arbitrary continuous homomorphisms into arbitrary topological groups. [ABSTRACT FROM AUTHOR]

AbstractIf f: G → Y is an irreducible closed continuous mapping defined on a regular weakly collectionwise normal first-countable meta-Lindelöf locally σ paratopological group G onto a T1-space Y which has a neighborhood ωω-base at a point y ∈ Y, then f−1(y) is σ-compact in G. We prove that if a Fréchet-Urysohn space X has strong α4 -property and a weakly countably complete base , then X is first-countable, where M is a separable and metrizable space and = {F: F is a non-empty compact subset of M } and with the Vietoris topology. By this result we can get the first-countability of certain paratopological groups. [ABSTRACT FROM AUTHOR]

Let G be a topological group, and let C(G) denote the algebra of continuous, complex valued functions on G. We find the solutions f , g , h ∈ C (G) of the Levi-Civita equation f (x y) = f (x) h (y) + g (x) f (y) , x , y ∈ G , which is an extension of the sine addition law. Representations of G on C 2 play an important role. As a corollary we get the solutions f , g ∈ C (G) of the sine subtraction law f (x y ∗) = f (x) g (y) - g (x) f (y) , x , y ∈ G , in which x ↦ x ∗ is a continuous involution, meaning that (x y) ∗ = y ∗ x ∗ and x ∗ ∗ = x for all x , y ∈ G . [ABSTRACT FROM AUTHOR]

In this paper, we present a homotopical framework for studying invertible gapped phases of matter from the point of view of infinite spin lattice systems, using the framework of algebraic quantum mechanics. We define the notion of quantum state types. These are certain lax-monoidal functors from the category of finite-dimensional Hilbert spaces to the category of topological spaces. The universal example takes a finite-dimensional Hilbert space ℋ to the pure state space of the quasi-local algebra of the quantum spin system with Hilbert space ℋ at each site of a specified lattice. The lax-monoidal structure encodes the tensor product of states, which corresponds to stacking for quantum systems. We then explain how to formally extract parametrized phases of matter from quantum state types, and how they naturally give rise to ℰ ∞ -spaces for an operad we call the "multiplicative" linear isometry operad. We define the notion of invertible quantum state types and explain how the passage to phases for these is related to group completion. We also explain how invertible quantum state types give rise to loop-spectra. Our motivation is to provide a framework for constructing Kitaev's loop-spectrum of bosonic invertible gapped phases of matter. Finally, as a first step toward understanding the homotopy types of the loop-spectra associated to invertible quantum state types, we prove that the pure state space of any UHF algebra is simply connected. [ABSTRACT FROM AUTHOR]

We obtain partial affirmative answers to the question of whether the isomorphism of the unitary groups of two C*-algebras, either as topological groups or as discrete groups, implies the isomorphism of the C*-algebras as real C*-algebras. [ABSTRACT FROM AUTHOR]

Copyright of Palestine Journal of Mathematics is the property of Palestine Polytechnic University and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this paper, the definition of a soft topological group based on the concept of soft points is introduced. Let SC(X,Y ) denote the collection of all soft continuous functions between soft topological spaces X and Y. We investigate the structures of abstract groups, and soft groups within the context of the soft function space. Furthermore, the soft topological group structure of the group SC(X,Y ) under soft topologies is explored. [ABSTRACT FROM AUTHOR]

Background: Mutations occurring in nucleic acids or proteins may affect the binding affinities of protein-nucleic acid interactions. Although many efforts have been devoted to the impact of protein mutations, few computational studies have addressed the effect of nucleic acid mutations and explored whether the identical methodology could be applied to the prediction of binding affinity changes caused by these two mutation types. Results: Here, we developed a generalized algorithm named PNBACE for both DNA and protein mutations. We first demonstrated that DNA mutations could induce varying degrees of changes in binding affinity from multiple perspectives. We then designed a group of energy-based topological features based on different energy networks, which were combined with our previous partition-based energy features to construct individual prediction models through feature selections. Furthermore, we created an ensemble model by integrating the outputs of individual models using a differential evolution algorithm. In addition to predicting the impact of single-point mutations, PNBACE could predict the influence of multiple-point mutations and identify mutations significantly reducing binding affinities. Extensive comparisons indicated that PNBACE largely performed better than existing methods on both regression and classification tasks. Conclusions: PNBACE is an effective method for estimating the binding affinity changes of protein-nucleic acid complexes induced by DNA or protein mutations, therefore improving our understanding of the interactions between proteins and DNA/RNA. [ABSTRACT FROM AUTHOR]

Let $ (G, X) $ (G , X) be a G-system, which means that X is a compact Hausdorff space and G is an infinite topological group continuously acting on X, and let μ be a G-invariant measure of $ (G, X) $ (G , X). In this paper, we introduce the concepts of rigidity, uniform rigidity and μ-Ω-equicontinuity of $ (G,X) $ (G , X) with respect to an infinite sequence Ω of G and the notions of μ-Ω-equicontinuity and μ-Ω-mean-equicontinuity of a function $ f\in L^2(\mu) $ f ∈ L 2 (μ) with respect to an infinite sequence Ω of G. Then we give some equivalent conditions for $ f\in L^2(\mu) $ f ∈ L 2 (μ) and $ (G,X) $ (G , X) to be rigid, respectively. In addition, if G is commutative and X satisfies the first axiom of countability, we present some equivalent conditions for $ (G,X) $ (G , X) to be uniformly rigid. [ABSTRACT FROM AUTHOR]

We introduce and motivate the definition of the virtual Rokhlin property for topological groups. We then classify the 2‐manifolds whose homeomorphism groups have the virtual Rokhlin property. We also establish the analogous result for mapping class groups of 2‐manifolds. [ABSTRACT FROM AUTHOR]

In this paper, we introduce proximal groups that are a generalization of topological groups. A straight-forward application of proximal groups is given in the concise representation of collections of overlapping Hilbert envelope lobes attached to the peak points on oscillatory motion waveforms that occur in sequences of video frames that track object movements. [ABSTRACT FROM AUTHOR]

Let \operatorname {Diff}_{\partial }(D^{n}) be the topological group of diffeomorphisms of D^{n} which agree with the identity near the boundary. In this short note, we compute the fundamental group \pi _1 \operatorname {Diff}_{\partial }(D^{4k}) for k\geq 3. [ABSTRACT FROM AUTHOR]

Let |$X$| be a connected scheme locally of finite type over |${\mathbb{C}}$|⁠ , and let |$X^{\textrm{an}}$| be its associated analytic space. In this paper, we define a comparison map from the topological fundamental group of |$X^{\textrm{an}}$| to the pro-étale fundamental group of |$X$|⁠. [ABSTRACT FROM AUTHOR]

In this paper, the notion of a soft topological transformation group is defined and studied. For a soft topological transformation group, it is proven that a map from a soft topological space onto itself is soft homeomorphism. The collection of all soft homeomorphisms of the given soft topological space onto itself constitutes soft topological group under composition. Subsequently, it is proved that there is a homomorphism between soft topological group and the group structure on the collection of all soft homeomorphisms of given topological space. Subsequently, it is shown that the mapping space Map(Y;Y) is soft Hausdorff and verified that any subspace of the mapping space is soft Hausdorff. Additionally, it is proved that the set of all soft homeomorphisms on Y forms a soft discrete space, soft extremally disconnected space, soft Moscow space and a soft Moscow topological group. Later, it is shown that the map from a soft topological group to a mapping space is soft continuous. Finally, it is proved that distinct group structure generates distinct collection of all soft homeomorphisms of the specified soft topological space onto itself is a soft isomorphism. [ABSTRACT FROM AUTHOR]

This work presents a pioneering photonic crystal (PC) heterostructure design exploiting tailored topological corner states and cavities to unleash a fascinating topological rainbow effect. This effect arises from the strategic integration of a nontrivial topological PC with sharp corners within a trivial PC matrix, resulting in a heterostructure rich in corner states and cavities. The critical innovation lies in manipulating the sector angle of circular columns, granting dynamic control over the rainbow effect and light localization. This manipulation induces distinct group velocities for different light frequencies, leading to their separation and localization at specific corner states. This remarkable "rainbow trapping" phenomenon manifests as highly confined light exhibiting exceptional resilience against disorder. These findings illuminate a pathway toward crafting next‐generation photonic devices boasting unparalleled functionalities. The reconfigurable rainbow trapping holds immense potential for applications in wavelength division multiplexing, optical sensing, and even venturing into quantum information processing. [ABSTRACT FROM AUTHOR]

One of the main challenges in molecular science is to model a chemical complex and predict its thermochemical properties. Various researchers have developed a number of hypothetical techniques in this regard, and one of them is focused on the topological indices. A topological index is a number associated with a molecule's structural graph that is considered to be capable of predicting certain chemical and physical properties of molecules. Degree-based topological indices are one of the groups of topological indices that are crucial in chemical graph theory. Shannon's entropy is a fundamental factor of a subclass of topological indices that measures the structural information of the graphs. In this study, we examine the entropy based on topological indices such as the first and second Zagreb indices, the general Randić index, the harmonic and sum-connectivity index of Hanoi graph. Moreover, we determine a number of multiplicative invariants, and different polynomials for Hanoi graphs with graphical illustrations. [ABSTRACT FROM AUTHOR]

This document explores the concept of Stone-Wales rotations and their application in manipulating the structure of fullerenes. The rotations involve rearranging carbon atoms to create new rings, specifically heptagons. The authors discuss the implications of these rotations in various fields, including chemistry, nanoscience, and mathematical chemistry. They also highlight the potential for studying quantum phenomena and the creation of new molecular structures. The document provides graphical tools and examples to aid in understanding the effects of these rotations on fullerene surfaces. [Extracted from the article]

In this paper, we survey some recent results on actions of finite groups on topological manifolds. Given an action of a finite group G on a manifold X , these results provide information on the restriction of the action to a subgroup of G of index bounded above by a number depending only on X. Some of these results refer to the algebraic structure of the group, such as being abelian or nilpotent or admitting a generating subset of controlled size; other results refer to the geometry of the action, e.g. to the existence of fixed points, to the collection of stabilizer subgroups or to the action on cohomology. [ABSTRACT FROM AUTHOR]

The paper examines various properties of D T - k -subgroup structures and addresses an open problem on the existence of topological group structures on the n -dimensional Khalimsky (K -, for brevity) topological space and Marcus–Wyse (M -, for short) topological plane. In particular, we obtain many types of totally k -disconnected or k -connected subgroups of a k -connected D T - k -group. Besides, we prove that each of the n -dimensional K -topological space and the M -topological plane cannot be a typical topological group. Unlike an existence of a D T - k -group structure of (S C k n , l , ∗) (see Proposition 4.7), we prove that neither of (S C K n , l , ∗) and (S C M l , ∗) is a topological group, where S C K n , l (respectively, S C M l ) is a simple closed K - (respectively, M -) topological curve with l elements in ℤ n (respectively, ℤ 2 ) and the operation "∗" is a special kind of binary operation for establishing a group structure of each of S C K n , l and S C M l . Finally, given a D T - k -group structure of (S C k 1 n 1 , l 1 × S C k 2 n 2 , l 2 , ∗) , we find several types of D T - k -subgroup structures of it (see Theorem 5.7). [ABSTRACT FROM AUTHOR]

A (non-commutative) Ulam quasimorphism is a map |$q$| from a group |$\Gamma $| to a topological group |$G$| such that |$q(xy)q(y)^{-1}q(x)^{-1}$| belongs to a fixed compact subset of |$G$|⁠. Generalizing the construction of Barge and Ghys, we build a family of quasimorphisms on a fundamental group of a closed manifold |$M$| of negative sectional curvature, taking values in an arbitrary Lie group. This construction, which generalizes the Barge-Ghys quasimorphisms, associates a quasimorphism to any principal |$G$| -bundle with connection on |$M$|⁠. Kapovich and Fujiwara have shown that all quasimorphisms taking values in a discrete group can be constructed from group homomorphisms and quasimorphisms taking values in a commutative group. We construct Barge-Ghys type quasimorphisms taking prescribed values on a given subset in |$\Gamma $|⁠ , producing counterexamples to the Kapovich and Fujiwara theorem for quasimorphisms taking values in a Lie group. Our construction also generalizes a result proven by D. Kazhdan in his paper "On |$\varepsilon $| -representations". Kazhdan has proved that for any |$\varepsilon>0$|⁠ , there exists an |$\varepsilon $| -representation of the fundamental group of a Riemann surface of genus 2 which cannot be |$1/10$| -approximated by a representation. We generalize his result by constructing an |$\varepsilon $| -representation of the fundamental group of a closed manifold of negative sectional curvature taking values in an arbitrary Lie group. [ABSTRACT FROM AUTHOR]

This document explores the concept of Stone-Wales rotations and their application in manipulating the structure of fullerenes. The rotations involve rearranging carbon atoms to create new rings, specifically heptagons. The authors discuss the implications of these rotations in various fields, including chemistry, nanoscience, and mathematical chemistry. They also highlight the potential for studying quantum phenomena and the creation of new molecular structures. The document provides graphical tools and examples to aid in understanding the effects of these rotations on fullerene surfaces. [Extracted from the article]

Cluster analysis has been widely studied due to its importance and several methods have been developed for this purpose. However, these methods are designed to process on centralized data. In this paper, we present an asynchronous approach based on topological relationship. The proposed approach unfolds on three steps: First, each site searches for clusters (models) of its local data. Secondly, a central site proceeds to the analysis and search for the partition of the whole (the global model). Finally, we proceed with the search for the right number of groups of the global model. We note that each local data has their own number of clusters and it can be different from the number of clusters in the entire data. The experiments have clearly demonstrated the effectiveness of the proposed approach to find the partition closest to that obtained on the data set from its subsets. [ABSTRACT FROM AUTHOR]

We provide a definition of a ≺-asymptotic (we suggest the pronunciation "prec-asymptotic") pair in a topological action (X , G) of a countable amenable group G , where ≺ is an order on G of type Z. In the case where for some G -invariant Borel probability measure μ on X , the measure-preserving system (X , μ , G) factors, via a map φ , onto a multiorder (O ˜ , ν , G) , we also introduce the notion of a φ -asymptotic pair. Then we prove that if μ has positive measure-theoretic conditional entropy with respect to the multiorder factor, then the set of points which belong to φ -asymptotic pairs has positive measure μ. This result is a generalization of the Blanchard-Host-Ruette Theorem for classical topological dynamical systems (actions of Z). As a strengthening of our theorem, we show that for any system (X , G) of positive topological entropy, any multiorder (O ˜ , ν , G) and ν -almost every ≺ ∈ O ˜ , there exist ≺-asymptotic pairs in X. Finally, we characterize systems (X , G) of topological entropy zero as factors of topologically multiordered systems (in which case φ is defined μ -almost everywhere for every G -invariant measure μ) with no φ -asymptotic pairs. [ABSTRACT FROM AUTHOR]

We study \mathbb {R}^k \times \mathbb {Z}^\ell actions on arbitrary compact manifolds with a projectively dense set of Anosov elements and 1-dimensional coarse Lyapunov foliations. Such actions are called totally Cartan actions. We completely classify such actions as built from low-dimensional Anosov flows and diffeomorphisms and affine actions, verifying the Katok-Spatzier conjecture for this class. This is achieved by introducing a new tool, the action of a dynamically defined topological group describing paths in coarse Lyapunov foliations, and understanding its generators and relations. We obtain applications to the Zimmer program. [ABSTRACT FROM AUTHOR]

In this paper, we present several characterization theorems on approach groups, and ultra approach groups. In so doing, we first give necessary and sufficient conditions for an approach structure to be compatible with group structure. We show that every ultra approach group is ultrauniformizable. Secondly, starting with an approach space, and its natural neighborhood system on a group, we characterize the resulting neighborhood approach group. Finally, we show that the category of ultra approach-Cauchy group is a topological category, and more importantly, we show that the category of ultra approach-Cauchy groups and the category of strongly normal ultra approach-limit groups are isomorphic. [ABSTRACT FROM AUTHOR]

In this study, we aim to define a new chaotic dynamical system family on a discrete pentagon fractal, Pd, a totally disconnected fractal set. One of the ways to define dynamical systems on the discrete n-ake fractal is to use the elements of its symme-try group. Thus, with the help of the elements of the symmetry group of the equilateral pentagon D5 and the shift map (ε), we obtain difierent dynamical systems via code representations of the points on Pd. Moreover, we investigate Devaney's chaos conditions for this family of dynamical systems. [ABSTRACT FROM AUTHOR]

For any smooth proper rigid space $X$ over a complete algebraically closed extension $K$ of $\mathbb {Q}_p$ we give a geometrisation of the $p$ -adic Simpson correspondence of rank one in terms of analytic moduli spaces: the $p$ -adic character variety is canonically an étale twist of the moduli space of topologically torsion Higgs line bundles over the Hitchin base. This also eliminates the choice of an exponential. The key idea is to relate both sides to moduli spaces of $v$ -line bundles. As an application, we study a major open question in $p$ -adic non-abelian Hodge theory raised by Faltings, namely which Higgs bundles correspond to continuous representations under the $p$ -adic Simpson correspondence. We answer this question in rank one by describing the essential image of the continuous characters $\pi ^{{\mathrm {\acute {e}t}}}_1(X)\to K^\times$ in terms of moduli spaces: for projective $X$ over $K=\mathbb {C}_p$ , it is given by Higgs line bundles with vanishing Chern classes like in complex geometry. However, in general, the correct condition is the strictly stronger assumption that the underlying line bundle is a topologically torsion element in the topological group $\operatorname {Pic}(X)$. [ABSTRACT FROM AUTHOR]

Our main aim is introduced some concepts in dynamical system in rough theory. We give the definition of periodic points and critical points and investigate their properties in rough actions. Also, we illustrated the relation between them. [ABSTRACT FROM AUTHOR]

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 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 transformation group (H,Y, ψ) and a point y ∈ Y, the set Hy = {h ∈ H: hy = y} consisting of elements of 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 (H,Y, ψ) is an Irr∗-topological transformation group, then (H/Ker ψ,Y, ψ) 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. [ABSTRACT FROM AUTHOR]

Dimensionality reduction techniques map values from a high dimensional space to one with a lower dimension. The result is a space which requires less physical memory and has a faster distance calculation. These techniques are widely used where required properties of the reduced-dimension space give an acceptable accuracy with respect to the original space. Many such transforms have been described. They have been classified in two main groups: linear and topological. Linear methods such as Principal Component Analysis (PCA) and Random Projection (RP) define matrix-based transforms into a lower dimension of Euclidean space. Topological methods such as Multidimensional Scaling (MDS) attempt to preserve higher-level aspects such as the nearest-neighbour relation, and some may be applied to non-Euclidean spaces. Here, we introduce nSimplexZen, a novel topological method of reducing dimensionality. Like MDS, it relies only upon pairwise distances measured in the original space. The use of distances, rather than coordinates, allows the technique to be applied to both Euclidean and other Hilbert spaces, including those governed by Cosine, Jensen–Shannon and Quadratic Form distances. We show that in almost all cases, due to geometric properties of high-dimensional spaces, our new technique gives better properties than others, especially with reduction to very low dimensions. [ABSTRACT FROM AUTHOR]

The aim of our work is to develop a new type of games which are related to (D, WD, LD) compactness of topological groups. We used an infinite game that corresponds to our work. Also, we used an alternating game in which the response of the second player depends on the choice of the first one. Many results of winning and losing strategies have been studied, consistent with the nature of the topological groups. As well as, we presented some topological groups, which fail to have winning strategies and we give some illustrated examples. Finally, the effect of functions on the aforementioned compactness strategies was studied. [ABSTRACT FROM AUTHOR]

We establish a general framework for representability of a metric group on a (well-behaved) class of Banach spaces. More precisely, let \mathcal {G} be a topological group, and \mathcal {A} a unital symmetric C^*-subalgebra of \mathrm {UC}(\mathcal {G}), the algebra of bounded uniformly continuous functions on \mathcal {G}. Generalizing the notion of a stable metric, we study \mathcal {A}-metrics \delta, i.e., the function \delta (e, \cdot) belongs to \mathcal {A}; the case \mathcal {A}=W\hskip -0.7mm A\hskip -0.2mm P(\mathcal {G}), the algebra of weakly almost periodic functions on \mathcal {G}, recovers stability. If the topology of G is induced by a left invariant metric d, we prove that \mathcal {A} determines the topology of \mathcal {G} if and only if d is uniformly equivalent to a left invariant \mathcal {A}-metric. As an application, we show that the additive group of C[0,1] is not reflexively representable; this is a new proof of Megrelishvili [ Topological transformation groups: selected topics , Elsevier, 2007, Question 6.7] (the problem was already solved by Ferri and Galindo [Studia Math. 193 (2009), pp. 99–108] with different methods and later the results were generalized by Yaacov, Berenstein, and Ferri [Math. Z. 267 (2011), pp.129–138]). Let now \mathcal {G} be a metric group, and assume \mathcal {A}\subseteq \mathrm {LUC}(\mathcal {G}), the algebra of bounded left uniformly continuous functions on \mathcal {G}, is a unital C^*-algebra which is the uniform closure of coefficients of representations of \mathcal {G} on members of \mathscr {F}, where \mathscr {F} is a class of Banach spaces closed under \ell _2-direct sums. We prove that \mathcal {A} determines the topology of \mathcal {G} if and only if \mathcal {G} embeds into the isometry group of a member of \mathscr {F}, equipped with the weak operator topology. As applications, we obtain characterizations of unitary and reflexive representability. [ABSTRACT FROM AUTHOR]

This paper studies topological definitions of chain recurrence and shadowing for continuous endomorphisms of topological groups generalizing the relevant concepts for metric spaces. It is proved that in this case the sets of chain recurrent points and chain transitive component of the identity are topological subgroups. Furthermore, we show that some dynamical properties are induced by the original system on quotient spaces. These results link an algebraic property to a dynamical property. [ABSTRACT FROM AUTHOR]

We study simple, properly ordered nonexpansive Bratteli-Vershik ($ BV $) systems. Correcting a mistake in an earlier paper, we redefine the classes standard nonexpansive ($ SNE $) and strong standard nonexpansive ($ SSNE $). We define also the classes of very well timed and well timed systems, their opposing classes of untimed and very untimed systems (which feature, as subclasses of 'Case (2)', in the work of Downarowicz and Maass as well as Hoynes on expansiveness of $ BV $ systems of finite topological rank), and several related classes according to the existence of indistinguishable pairs (of some 'depth') and their synchronization ('common cuts'). We establish some properties of these types of systems and some relations among them. We provide several relevant examples, including a problematic one that is conjugate to a well timed system while also (vacuously) in the classes 'Case (2)'. We prove that the class of all simple, properly ordered nonexpansive $ BV $ systems is the disjoint union of the ones conjugate to well timed systems and those conjugate to untimed systems, thereby showing that nonexpansiveness in $ BV $ systems arises in one of two mutually exclusive ways. [ABSTRACT FROM AUTHOR]

We present a unified theory for the almost periodicity of functions with values in an arbitrary Banach space, measures and distributions via almost periodic elements for the action of a locally compact abelian group on a uniform topological space. We discuss the relation between Bohr- and Bochner-type almost periodicity, and similar conditions, and how the equivalence among such conditions relates to properties of the group action and the uniformity. We complete the paper by demonstrating how various examples considered earlier all fit in our framework. [ABSTRACT FROM AUTHOR]