Quotient Map Research Articles

Some characterizations of ω-balanced topological groups with a q-point

In this paper, we study some characterizations of q-spaces, strict q-spaces and strong q-spaces under ω-balanced topological groups. First we characterize a topological group G to be ω-balanced and a q-space whenever for each open neighborhood O of the identity in G, there is a countably compact invariant subgroup H which is of countable character in G, such that H ⊆ O and the canonical quotient mapping p : G → G/H is quasi-perfect and the quotient group G/H is metrizable. Secondly, we characterize G to be ω-balanced and a strict q-space, replacing the condition of being a q-space with an appropriate condition which is designed for the so-called “strict q-spaces”. Finally, we characterize G to be ω-balanced and a strong q-space, illustrating an additional condition to be replaced to that of q-space, or strict q-space. As applications of our three main results, we found new characterizations of the ω-narrow topological groups.

Isotropy group on some topological transformation group structures

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

Asymptotic lifting for completely positive maps

Let A and B be C⁎-algebras with A separable, let I be an ideal in B, and let ψ:A→B/I be a completely positive contractive linear map. We show that there is a continuous family Θt:A→B, for t∈[1,∞), of lifts of ψ that are asymptotically linear, asymptotically completely positive and asymptotically contractive. If ψ is of order zero, then Θt can be chosen to have this property asymptotically. If A and B carry continuous actions of a second countable locally compact group G such that I is G-invariant and ψ is equivariant, we show that the family Θt can be chosen to be asymptotically equivariant. If a linear completely positive lift for ψ exists, we can arrange that Θt is linear and completely positive for all t∈[1,∞). In the equivariant setting, if A, B and ψ are unital, we show that asymptotically linear unital lifts are only guaranteed to exist if G is amenable. This leads to a new characterization of amenability in terms of the existence of asymptotically equivariant unital sections for quotient maps.

Quotient Permutation BF-Algebras and Quotient Maps

The permutation BF-algebras was given as a new class of BF-algebras and their basic features in this work were studied. We considered and discussed some new notions in permutation BF-algebras, like permutation BF1/BF2-algebras, permutation BF-ideals, permutation BF-subalgebras, normal permutation BF-subalgebras. We further looked at the homomorphism of BF-algebras, congruence relation, quotient permutation BF-algebras, and Quotient map.

On countably complete topological groups

In this paper, we give a characterization of countably complete topological groups and study when a countably complete subgroup of a topological group is C-embedded. We mainly show that (1) a topological group G is countably complete (the notion introduced by M. Tkachenko in 2012) iff G contains a closed r-pseudocompact subgroup H such that the quotient space G/H is completely metrizable and the canonical quotient mapping π:G→G/H satisfies that π−1(F) is r-pseudocompact in G for each r-pseudocompact set F in G/H; (2) every countably complete weakly Ψω-factorizable and ω-balanced subgroup H of a topological group G is C-embedded; (3) every countably complete subgroup H of a pointwise pseudocompact topological group G is C-embedded; (4) every uniformly strongly countably complete and weakly Ψω-factorizable subgroup H of a topological group G is C-embedded. Further, an ω-narrow locally compact subgroup H of a topological group G is C-embedded.

Continuous dependence of linear differential systems on polynomial matrices

Abstract Linear differential systems, as it is known, are parameterized by means of polynomial matrices. Introducing appropriate topologies, it is shown that this parametrization becomes a quotient map.

Optimal control of Boolean control networks with state-triggered impulses

In the real world, abrupt changes caused by external environment and system interaction can exhibit a pattern resembling an “impulse” response. A novel model, known as the hybrid-index model, has recently been introduced for Boolean control networks (BCNs) to characterize such impulsive responses. The optimal control problem is a fundamental issue in control theory, and many control problems can be classified as special cases of optimal control problems. Nevertheless, the optimal control problem for the hybrid-index model remains unresolved. This study delves into the optimal control problem of impulsive Boolean control networks (IBCNs) under the hybrid-index model framework. Forward completeness serves as a crucial condition to avert Zeno behavior in the hybrid-index model of IBCNs. In contrast to prior studies on the hybrid-index model of IBCNs, the study shows that the optimal control problem of IBCNs can be addressed even in cases where the system is not forward complete. Leveraging the theory of semi-tensor product, both finite-horizon and infinite-horizon optimal control problems of IBCNs are tackled using the quotient mapping method and the recursive method for optimal control of Boolean control networks. The study yields the optimal cost and optimal control law through state feedback. To illustrate the efficacy of the proposed approach, a mathematical example and a biological example known as the λ switch are presented.

On an Anisotropic Logistic Equation

We consider a nonlinear Dirichlet problem driven by the (p(z),q)-Laplacian and with a logistic reaction of the equidiffusive type. Under a nonlinearity condition on a quotient map, we show existence and uniqueness of positive solutions and the result is global in parameter λ. If the monotonicity condition on the quotient map is not true, we can no longer guarantee uniqueness, but we can show the existence of a minimal solution uλ* and establish the monotonicity of the map λ⟼uλ* and its asymptotic behaviour as the parameter λ decreases to the critical value λ^1(q)>0 (the principal eigenvalue of (−Δq,W01,q(Ω))).

Partial actions of groups on profinite spaces

We show that for a partial action η with closed domain of a compact group G on a profinite space X the space of orbits X/~G is profinite, this leads to the fact that when G is profinite the enveloping space XG is also profinite. Moreover, we provide conditions for the induced quotient map πG : X → X / ∼G of η to have a continuous section. Relations between continuous sections of πG and continuous sections of the quotient map induced by the enveloping action of η are also considered. At the end of this work, we prove that the category of actions on profinite spaces with countable number of clopen sets is reflective in the category of actions of compact Hausdorff spaces having countable number of clopen sets.

K3 surfaces with a symplectic automorphism of order 4

AbstractGiven , a K3 surface admitting a symplectic automorphism of order 4, we describe the isometry on . Having called and , respectively, the minimal resolutions of the quotient surfaces and , we also describe the maps induced in cohomology by the rational quotient maps and : With this knowledge, we are able to give a lattice‐theoretic characterization of , and find the relation between the Néron–Severi lattices of and in the projective case. We also produce three different projective models for and , each associated to a different polarization of degree 4 on .

ON TOPOLOGICAL J- QUOTIENT MAPS

In this paper, J- Quotient maps, Strongly J- Quotient maps, [J]Quotient maps and Strongly J- Open maps utilizing J- Closed sets are introduced. The newly defined Quotient maps are analysed with various existing Quotient maps. Interrelations between J- Quotient maps, Strongly J- Quotient maps, [J]Quotient maps and Strongly J- Open maps are investigated. Here the properties of those functions are presented.

Output controllability of impulsive Boolean control networks based on hybrid‐index model

AbstractIn biological systems, impulses are often used to model mutations in gene expression. In order to describe the instantaneity of the impulse, this paper employs a hybrid‐index model of impulsive Boolean networks with state‐triggered impulses. Based on this model, the time‐domain output controllability is investigated. Using the quotient mapping method, the hybrid‐domain solutions of this model are mapped to the time‐domain solutions, mathematically. Then a necessary and sufficient condition for output controllability in the time domain is presented. Finally, an example is given to demonstrate the correctness and effectiveness of the obtained results.

Corrigendum to “K-theoretic Characterization of C*-algebras with Approximately Inner Flip”

Abstract An error in the original paper is identified and corrected. The $\textrm {C}^{\ast }$-algebras with approximately inner flip, which satisfy the UCT, are identified (and turn out to be fewer than what is claimed in the original paper). The action of the flip map on K-theory turns out to be more subtle, involving a minus sign in certain components. To this end, we introduce new geometric resolutions for $\textrm {C}^{\ast }$-algebras, which do not involve index shifts in K-theory and thus allow for a more explicit description of the quotient map in the Künneth formula for tensor products.

Group C*-Algebras of Locally Compact Groups Acting on Trees

Abstract It was proved by Samei and Wiersma that for every non-compact, closed subgroup $G$ of the automorphism group $\textrm {Aut}(T)$ of a (semi-)homogeneous tree $T$ acting transitively on the boundary $\partial T$ and every $2 \leq q < p \leq \infty $, the quotient map $C^{\ast }_{L^{p+}}(G) \twoheadrightarrow C^{\ast }_{L^{q+}}(G)$ is not injective. We prove that whenever $G$, moreover, acts transitively on the vertices of $T$ and has Tits’s independence property, the group $C^{\ast }$-algebras $C^{\ast }_{L^{p+}}(G)$ are the only group $C^{\ast }$-algebras of $G$ coming from ideals of the Fourier–Stieltjes algebra. We also show that given such a group $G$, every group $C^{\ast }$-algebra $C^{\ast }_{\mu }(G)$ that is distinguishable from $C^{\ast }(G)$ and whose dual space $C^{\ast }_{\mu }(G)^{\ast }$ is an ideal in $B(G)$ is abstractly $^{\ast}$-isomorphic to $C^{\ast }_{r}(G)$.

Micro-Segregation of Ethnic Minorities in Rome: Highlighting Specificities of National Groups in Micro-Segregated Areas

This paper aims to study ethnic micro-segregation in Rome, namely, high residential concentrations of ethnic groups at the micro-area level within neighbourhoods with low concentrations of these groups, with a focus on specific situations of spatial inequality often overlooked in the debate. The Italian capital is one of the five most populous cities in the European Union and a multi-ethnic metropolis with relatively low levels of segregation. It is an urban context that has been little studied, partly due to the lack of reliable and granular data. This work is based on unpublished individual data from the 2020 population register, disaggregated into 155 neighbourhoods and 13,656 census tracts with average populations of about 18,000 and 200 residents, respectively. The five minority groups considered, Bangladeshis, Chinese, Filipinos, Romanians, and migrants from developed economy countries (DECs), add up to 55% of the total foreign residents and show different settlement patterns. The concept of micro-segregated area (MSA) is central to the scope of the analysis. An MSA is a census tract that shows a strong over-representation of a specific ethnic group despite being located within a neighbourhood where that group is under-represented. MSAs can be considered ‘interstitial’ micro-areas embedded in ethnically diverse neighbourhoods. Descriptive analysis based on location quotient (LQ) mapping and bivariate logistic models is developed to highlight (a) differences in the settlement patterns of minority ethnic groups; (b) differences in the micro-segregation of minority ethnic groups in terms of socio-demographic characteristics, settlement location, and socioeconomic status; and (c) the particular characteristics of minority ethnic groups underlying these differences. The findings indicate that differences in settlement patterns can be related to the interplay between real estate constraints and labour market specialisation. National specificities in micro-segregation are mainly linked to length of stay, but the models of the Asian groups do not offer any empirical support for the spatial assimilation hypothesis, unlike those of Romanians and DECs citizens. Further development of this research will aim to explore segregation patterns and motivations to move to MSAs using a mixed method approach.

Some generalized countably compact properties in topological groups

In this paper, firstly, we study strict q-spaces, strong q-spaces and point pseudocompact spaces in topological groups in terms of preimages of metrizable spaces. A continuous mapping f:X→Y is (strongly) sequential-perfect if f is closed and (f−1(F)) f−1(y) is a sequentially compact set for each (sequentially compact set F⊆Y) y∈Y. We give some characterizations as follows:(1)A topological group G is a strict q-space if and only if it is a sequential-perfect preimage of a metrizable space.(2)A topological group G is a strong q-space if and only if it is a strongly sequential-perfect preimage of a metrizable space.(3)A topological group G is a pointwise pseudocompact space if and only if there is a continuous open mapping f from G onto a metrizable space M such that f−1(F) is an r-pseudocompact set in G for each r-pseudocompact set F in M.Secondly, we study completeness of generalized countably compact topological groups in terms of preimages of completely metrizable spaces and obtain the following results:(4)A topological group G is strictly countably sieve complete if and only if G contains a closed sequentially compact subgroup H such that the quotient space G/H is completely metrizable and the canonical quotient mapping π:G→G/H is sequential-perfect.(5)A topological group G is countably sieve-s-complete if and only if G contains a closed sequentially compact subgroup H such that the quotient space G/H is completely metrizable and the canonical quotient mapping π:G→G/H is strongly sequential-perfect.(6)A topological group G is countably sieve-p-complete if and only if G contains a closed r-pseudocompact subgroup H such that the quotient space G/H is completely metrizable and the canonical quotient mapping π:G→G/H satisfies that π−1(F) is r-pseudocompact in G for each r-pseudocompact set F in G/H.

Quotient and transversal mappings for topological quasigroups

This article is devoted to studying the structure of topological left (or right) quasigroups, which play a great role in noncommutative geometry. Quotient and transversal mappings are important in the theory of differentiable manifolds and topological manifolds. Their transversal and quotient mappings are investigated. Necessary and sufficient conditions for their continuity are scrutinized. Examples are given. Homogeneous spaces are investigated related to topological quasigroups and their subquasigroups. For this purpose, the products of special types of topological left (or right) quasigroups, which are called smashed, are investigated. They are used to describe an extensive family of topological nondiscrete left (or right) quasigroups for which transversal mappings are continuous.

K-Theory of the maximal and reduced Roe algebras of metric spaces with A-by-CE coarse fibrations

Let [Formula: see text] be a discrete metric space with bounded geometry. In this paper, we show that if [Formula: see text] admits an “A-by-CE” coarse fibration, then the canonical quotient map [Formula: see text] from the maximal Roe algebra to the Roe algebra of [Formula: see text], and the canonical quotient map [Formula: see text] from the maximal uniform Roe algebra to the uniform Roe algebra of [Formula: see text], induce isomorphisms on [Formula: see text]-theory. A typical example of such a space arises from a sequence of group extensions [Formula: see text] such that the sequence [Formula: see text] has Yu’s property A, and the sequence [Formula: see text] admits a coarse embedding into Hilbert space. This extends an early result of Špakula and Willett [Maximal and reduced Roe algebras of coarsely embeddable spaces, J. Reine Angew. Math. 678 (2013) 35–68] to the case of metric spaces which may not admit a coarse embedding into Hilbert space. Moreover, it implies that the maximal coarse Baum–Connes conjecture holds for a large class of metric spaces which may not admit a fibered coarse embedding into Hilbert space.

Approximations of delocalized eta invariants by their finite analogues

For a given self-adjoint first order elliptic differential operator on a closed smooth manifold, we prove a list of results on when the delocalized eta invariant associated to a regular covering space can be approximated by the delocalized eta invariants associated to finite-sheeted covering spaces. One of our main results is the following. Suppose M is a closed smooth spin manifold and \widetilde{M} is a \Gamma -regular covering space of M . Let \langle \alpha \rangle be the conjugacy class of a non-identity element \alpha\in \Gamma . Suppose \{\Gamma_i\} is a sequence of finite-index normal subgroups of \Gamma that distinguishes \langle \alpha \rangle . Let \pi_{\Gamma_i} be the quotient map from \Gamma to \Gamma/\Gamma_i and \langle \pi_{\Gamma_i}(\alpha) \rangle the conjugacy class of \pi_{\Gamma_i}(\alpha) in \Gamma/\Gamma_i . If the scalar curvature on M is everywhere bounded below by a sufficiently large positive number, then the delocalized eta invariant for the Dirac operator of \widetilde{M} at the conjugacy class \langle \alpha \rangle is equal to the limit of the delocalized eta invariants for the Dirac operators of M_{\Gamma_i} at the conjugacy class \langle \pi_{\Gamma_i}(\alpha) \rangle , where M_{\Gamma_i}= \widetilde{M}/\Gamma_i is the finite-sheeted covering space of M determined by \Gamma_i . In another main result of the paper, we prove that the limit of the delocalized eta invariants for the Dirac operators of M_{\Gamma_i} at the conjugacy class \langle \pi_{\Gamma_i}(\alpha) \rangle converges, under the assumption that the rational maximal Baum–Connes conjecture holds for \Gamma .

MSSM: A Macaulay2 Package for the Vacuum Moduli Space

The Standard Model of particle physics has been amazingly successful at explaining interactions between three of the four fundamental forces of physics: the strong force, the weak force, and electro-magnetism. But it is an incomplete theory, failing to incorporate gravity. The Minimal Supersymmetric Standard Model (MSSM) is an approach to including gravity and supersymmetry in the standard model, by adding new particle states and interactions to the standard model. For example, supersymmetry pairs fermions (such as electrons) with bosons (such as photons). A starting point in the study of the MSSM is the Vacuum Moduli Space, which is a highly complicated algebraic variety introduced by Witten: it is the image of an affine variety X ⊆ C 49 under a symplectic quotient map to C 973 . We describe the Macaulay2 software package MSSM, which facilitates computational investigation of the Vacuum Moduli Space.