Topological Group Research Articles

Density estimation for mixed Euclidean and non-Euclidean data in the presence of measurement error

In this paper, we study density estimation for mixed Euclidean and non-Euclidean variables that are subject to measurement errors. This problem is largely unexplored in statistics. We develop a new deconvolution density estimator and derive its finite-sample properties. We also derive its asymptotic properties including the rate of convergence in various modes and the asymptotic distribution. For the derivation, we apply Fourier analysis on topological groups, which has not been well used in statistics. We provide full practical details on the implementation of the estimator as well as several simulation studies and real data analysis.

Extensions of some topological properties in topological gyrogroups

Topological gyrogroups, with a weaker algebraic structure than topological groups, has been studied lately. In this paper, it is shown that (1) if H is a closed L-subgyrogroup of a topological gyrogroup G, then G is connected provided H and G/H are connected; (2) if H is a closed L-subgyrogroup of a topological gyrogroup G, then G is totally disconnected provided H and G/H are totally disconnected; (3) if H is a locally compact metrizable connected and normal subgyrogroup of a topological gyrogroup G, then G is sequentially connected provided the quotient gyrogroup G/H is sequentially connected. The above results generalize the corresponding results in topological groups [3,18].

Size Control and Enhanced Stability of Silver Nanoparticles by Cyclic Poly(ethylene glycol).

Silver nanoparticles (AgNPs) are used in a wide range of applications, and the size control and stability of the nanoparticles are crucial aspects in their applications. In the present study, cyclized poly(ethylene glycol) (c-PEG) with various molecular weights, along with linear PEG with hydroxy chain ends (HO-PEG-OH) and methoxy chain ends (MeO-PEG-OMe) were applied for the Tollens' synthesis of AgNPs. The particle size was significantly affected by the topology and end groups of PEG. For example, the size determined by TEM was 40 ± 7 nm for HO-PEG5k-OH, 21 ± 4 nm for c-PEG5k, and 48 ± 9 nm for MeO-PEG5k-OMe when the molar ratio of PEG to AgNO3 (ω) was 44. The stability of AgNPs was also drastically improved by cyclization; the relative UV-Vis absorption intensity (A/A0 × 100%) at λmax to determine the proportion of persisting AgNPs in an aqueous NaCl solution (37.5 mM) was 58% for HO-PEG5k-OH, 80% for c-PEG5k, and 40% for MeO-PEG5k-OMe, despite the fact that AgNPs with c-PEG5k were much smaller than those with HO-PEG5k-OH and MeO-PEG5k-OMe.

On (para)topological groups with a countably (s-)complete sieve

Countably sieve-complete spaces were defined by M. Michael in 1972. In this article, we introduce a notion called countably sieve-s-complete spaces. Some properties of countably sieve-complete (countably sieve-s-complete, strongly countably complete) spaces are discussed. We get the following main results.A topological group is countably sieve-complete if and only if G contains a closed countably compact subgroup H such that the quotient space G/H is completely metrizable and the canonical quotient mapping π:G→G/H is closed. By the above conclusion and a known conclusion, we get that a topological group G is strongly countably complete if and only if G is countably sieve-complete. A topological group G is countably sieve-s-complete if and only if G contains a sequentially compact closed subgroup H with a countable base of open neighborhoods such that the quotient space G/H is a completely metrizable space and the canonical quotient mapping π:G→G/H is closed. An ω-balanced topological group G is countably sieve-complete if and only if G contains a closed countably compact invariant subgroup H such that the quotient space G/H is a completely metrizable topological group and the canonical quotient mapping π:G→G/H is closed. A topological group G is ω-narrow and countably sieve-s-complete if and only if G contains a sequentially compact closed invariant subgroup H with a countable base of open neighborhoods such that the quotient space G/H is a completely metrizable second-countable topological group and the canonical quotient mapping π:G→G/H is closed.We show that if G is a regular countably sieve-complete semitopological group with Sm(G)≤ω and satisfies property (⁎), then G is a topological group. Every regular totally ω-narrow countably sieve-complete paratopological group is a topological group.

Combinatoric topological string theories and group theory algorithms

A number of finite algorithms for constructing representation theoretic data from group multiplications in a finite group G have recently been shown to be related to amplitudes for combinatoric topological strings (G-CTST) based on Dijkgraaf-Witten theory of flat G-bundles on surfaces. We extend this result to projective representations of G using twisted Dijkgraaf-Witten theory. New algorithms for characters are described, based on handle creation operators and minimal multiplicative generating subspaces for the centers of group algebras and twisted group algebras. Such minimal generating subspaces are of interest in connection with information theoretic aspects of the AdS/CFT correspondence. For the untwisted case, we describe the integrality properties of certain character sums and character power sums which follow from these constructive G-CTST algorithms. These integer sums appear as residues of singularities in G-CTST generating functions. S-duality of the combinatoric topological strings motivates the definition of an inverse handle creation operator in the centers of group algebras and twisted group algebras.

Computable topological abelian groups

We study the algorithmic content of Pontryagin - van Kampen duality. We prove that the process of dualization is computable in the important cases of compact and locally compact totally disconnected Polish abelian groups. The applications of our main results include solutions to questions of Kihara and Ng about presentations of connected Polish spaces, and an unexpected arithmetical characterisation of direct products of solenoid groups among all Polish groups.

Formula omitted]-factorizable topological groups

In this paper, we define a new class of PM-factorizable topological groups. A topological group G is called PM-factorizable if, for every continuous real-valued function f on G, one can find a perfect homomorphism p:G→L onto a metrizable topological group L and a continuous real-valued function h on L such that f=h∘p. The relations between PR-factorizable, PM-factorizable and M-factorizable topological groups are studied. Also, some new characterizations of PR-factorizable, PM-factorizable and M-factorizable topological groups are obtained.

Is the Free Locally Convex Space L(X) Nuclear?

Given a class $${{\mathcal {P}}}$$ of Banach spaces, a locally convex space (LCS) E is called multi- $${{\mathcal {P}}}$$ if E can be isomorphically embedded into a product of spaces that belong to $${{\mathcal {P}}}$$ . We investigate the question whether the free locally convex space L(X) is strongly nuclear, nuclear, Schwartz, multi-Hilbert or multi-reflexive. If X is a Tychonoff space containing an infinite compact subset then, as it follows from the results of Außenhofer (Topol Appl 134:90–102, 2007), L(X) is not nuclear. We prove that for such X the free LCS L(X) has the stronger property of not being multi-Hilbert. We deduce that if X is a k-space, then the following properties are equivalent: (1) L(X) is strongly nuclear; (2) L(X) is nuclear; (3) L(X) is multi-Hilbert; (4) X is countable and discrete. On the other hand, we show that L(X) is strongly nuclear for every projectively countable P-space (in particular, for every Lindelöf P-space) X. We observe that every Schwartz LCS is multi-reflexive. It is known that if X is a $$k_\omega $$ -space, then L(X) is a Schwartz LCS (Außenhofer et al. in Stud Math 181(3):199–210, 2007), hence L(X) is multi-reflexive. We show that for every first-countable paracompact (in particular, for every metrizable) space X the converse is true, so L(X) is multi-reflexive if and only if X is a $$k_\omega $$ -space, equivalently if X is a locally compact and $$\sigma $$ -compact space. Similarly, we show that for any first-countable paracompact space X the free abelian topological group A(X) is a Schwartz group if and only if X is a locally compact space such that the set $$X^{(1)}$$ of all non-isolated points of X is $$\sigma $$ -compact.

Basic orders in descriptive combinatorics

Basic orders were originally introduced in order to illuminate Tukey theory of directed sets and in particular explain the phenomenon that Tukey reductions between definable directed sets tend to be definable themselves. We show here that they can be also used to calibrate between different properties of topological spaces and topological groups.

On the cardinality and structure of extremally disconnected groups with linear topology

A group topology is said to be linear if open subgroups form a base of neighborhoods of the identity element. Most of the known (consistent) examples of extremally disconnected groups have linear topology. It is proved that the existence of a nondiscrete extremally disconnected group of Ulam nonmeasurable cardinality with linear topology implies that of such a group of cardinality at most 2ω. Results on the structure of such groups are also presented.

On separation axioms of topological rough groups

On separation axioms of topological rough groups

Minimality of topological matrix groups and Fermat primes

Our aim is to study topological minimality of some natural matrix groups. We show that the special upper triangular group ST+(n,F) is minimal for every local field F of characteristic ≠2. This result is new even for the field R of reals and it leads to some important consequences. We prove criteria for the minimality and total minimality of the special linear group SL(n,F), where F is a subfield of a local field. This extends some known results of Remus–Stoyanov (1991) and Bader–Gelander (2017).One of our main applications is a characterization of Fermat primes, which asserts that for an odd prime p the following conditions are equivalent:(1)p is a Fermat prime;(2)SL(p−1,Q) is minimal, where Q is the field of rationals equipped with the p-adic topology;(3)SL(p−1,Q(i)) is minimal, where Q(i)⊂C is the Gaussian rational field.

Savage surfaces

Let G be the topological fundamental group of a given nonsingular complex projective surface. We prove that the Chern slopes c_1^2(S)/c_2(S) of minimal nonsingular surfaces of general type S with \pi_1(S) \simeq G are dense in the interval [1,3] .

Orderable groups and semigroup compactifications

Our aim is to find some new links between linear (circular) orderability of groups and topological dynamics. We suggest natural analogs of the concept of algebraic orderability for topological groups involving order-preserving actions on compact spaces and the corresponding enveloping semigroups in the sense of R. Ellis. This approach leads to several natural questions. Some of them might be useful also for discrete (countable) orderable groups.

A note on Pontryagin duality and continuous logic

We exhibit Pontryagin duality as a special case of Stone duality in a continuous logic setting. More specifically, given an abelian topological group A, and F the family (group) of continuous homomorphisms from A to the circle group T, then, viewing (A,+) equipped with the collection F as a continuous logic structure M, we show that the local type space SF(M) is precisely the Pontryagin dual of the group F where the latter is considered as a discrete group.We conclude, using Pontryagin duality (between compact and discrete abelian groups), that SF(M) is the Bohr compactification of the topological group A.

On closed subgroups of precompact groups

Abstract It is a theorem of W. W. Comfort and K. A. Ross that if 𝐺 is a subgroup of a compact Abelian group and 𝑆 denotes the continuous homomorphisms from 𝐺 to the one-dimensional torus, then the topology on 𝐺 is the initial topology given by 𝑆. Assume that 𝐻 is a subgroup of 𝐺. We study how the choice of 𝑆 affects the topological placement and properties of 𝐻 in 𝐺. Among other results, we have made significant progress toward the solution of the following specific questions. How many totally bounded group topologies does 𝐺 admit such that 𝐻 is a closed (dense) subgroup? If C S C_{S} denotes the poset of all subgroups of 𝐺 that are 𝑆-closed, ordered by inclusion, does C S C_{S} have a greatest (resp. smallest) element? We say that a totally bounded (topological, resp.) group is an SC group (topologically simple, resp.) if all its subgroups are closed (if 𝐺 and { e } \{e\} are its only possible closed normal subgroups, resp.) In addition, we investigate the following questions. How many SC-(topologically simple totally bounded, resp.) group topologies does an arbitrary Abelian group 𝐺 admit?

Analysis on topological alterations of functional brain networks after acute alcohol intake using resting-state functional magnetic resonance imaging and graph theory.

AimsAlcohol consumption could lead to a series of health problems and social issues. In the current study, we investigated the resting-state functional brain networks of healthy volunteers before and after drinking through graph-theory analysis, aiming to ascertain the effects of acute alcohol intake on topology and information processing mode of the functional brain networks.Materials and methodsThirty-three healthy volunteers were enrolled in this experiment. Each volunteer accepted alcohol breathalyzer tests followed by resting-state magnetic resonance imaging at three time points: before drinking, 0.5 h after drinking, and 1 h after drinking. The data obtained were grouped based on scanning time into control group, 0.5-h group and 1-h group, and post-drinking data were regrouped according to breath alcohol concentration (BrAC) into relative low BrAC group (A group; 0.5-h data, n = 17; 1-h data, n = 16) and relative high BrAC group (B group; 0.5-h data, n = 16; 1-h data, n = 17). The graph-theory approach was adopted to construct whole-brain functional networks and identify the differences of network topological properties among all the groups.ResultsThe network topology of most groups was altered after drinking, with the B group presenting the most alterations. For global network measures, B group exhibited increased global efficiency, Synchronization, and decreased local efficiency, clustering coefficient, normalized clustering coefficient, characteristic path length, normalized characteristic path length, as compared to control group. Regarding nodal network measures, nodal clustering coefficient and nodal local efficiency of some nodes were lower in B group than control group. These changes suggested that the network integration ability and synchrony improved, while the segregation ability diminished.ConclusionThis study revealed the effects of acute alcohol intake on the topology and information processing mode of resting-state functional brain networks, providing new perceptions and insights into the effects of alcohol on the brain.

Finite dimensional varieties over the Heisenberg group

Spectral analysis and synthesis studies translation invariant function spaces, so-called varieties over topological groups. The basic building blocks are the finite dimensional varieties. In the commutative case finite dimensional varieties are spanned by exponential polynomials. In non-commutative situations no relevant results exist. In this paper we consider finite dimensional left translation invariant linear spaces of continuous complex valued functions over the Heisenberg group. Using basic knowledge about Lie algebra we describe all left varieties of this type. In particular, it turns out that those function spaces are spanned by exponential polynomials as well.

A dichotomy for topological full groups

AbstractGiven a minimal action$\alpha $of a countable group on the Cantor set, we show that the alternating full group$\mathsf {A}(\alpha )$is non-amenable if and only if the topological full group$\mathsf {F}(\alpha )$is$C^*$-simple. This implies, for instance, that the Elek–Monod example of non-amenable topological full group coming from a Cantor minimal$\mathbb {Z}^2$-system is$C^*$-simple.

Closed subsets in Bishop topological groups

We introduce the notion of a Bishop topological group i.e., a group X equipped with a Bishop topology of functions F such that the group operations of X are Bishop morphisms with respect to F. A closed subset in the neighborhood structure of X induced by its Bishop topology F is defined in a positive way i.e., not as the complement of an open subset in X. The corresponding closure operator, although it is not topological, in the classical sense, does not involve sequences. As countable choice (CC) is avoided, and in agreement with Richman's critique on the use of CC in constructive mathematics, the fundamental facts on closed subsets in Bishop topological groups shown here have a clear algorithmic content. We work within Bishop's informal system of constructive mathematics BISH, without countable choice, equipped with inductive definitions with rules of countably many premises.