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In this paper, we build up a scaled homology theory, lc-homology, for metric spaces such that every metric space can be visually regarded as "locally contractible" with this newly-built homology. We check that lc-homology satisfies all Eilenberg-Steenrod axioms except the exactness axiom whereas its corresponding lc-cohomology satisfies exactness axiom for cohomology. This homology can relax the smooth manifold restrictions on the compact metric space such that the entropy conjecture will hold for the first lc-homology group. [ABSTRACT FROM AUTHOR]

In this article, we focus on the left (translation) actions on noncommutative compact connected Lie groups S U (2) × T n . We define the rotation vectors of the left actions induced by the elements in the maximal tori of S U (2) × T n , and utilize rotation vectors to give the complete topologically conjugate classifications of left actions. Algebraic conjugacy and smooth conjugacy are also considered. [ABSTRACT FROM AUTHOR]

In this article, we focus on the left (translation) actions on noncommutative compact connected Lie groups S U (2) × T n . We define the rotation vectors of the left actions induced by the elements in the maximal tori of S U (2) × T n , and utilize rotation vectors to give the complete topologically conjugate classifications of left actions. Algebraic conjugacy and smooth conjugacy are also considered. [ABSTRACT FROM AUTHOR]

Let x = { x n } n = 1 ∞ $\begin{array}{} \displaystyle \{x_n\}_{n=1}^{\infty} \end{array}$ be a sequence of positive numbers, and 𝓙x be the collection of all subsets A ⊆ ℕ such that ∑ k ∈ A $\begin{array}{} \displaystyle \sum_{k\in A} \end{array}$ xk < +∞. The aim of this article is to study how large the summable subsequence could be. We define the upper density of summable subsequences of x as the supremum of the upper asymptotic densities over 𝓙x, SUD in brief, and we denote it by D*(x). Similarly, the lower density of summable subsequences of x is defined as the supremum of the lower asymptotic densities over 𝓙x, SLD in brief, and we denote it by D*(x). We study the properties of SUD and SLD, and also give some examples. One of our main results is that the SUD of a non-increasing sequence of positive numbers tending to zero is either 0 or 1. Furthermore, we obtain that for a non-increasing sequence, D*(x) = 1 if and only if lim inf k → ∞ n x n = 0 , $\begin{array}{} \displaystyle \liminf_{k\to\infty}nx_n=0, \end{array}$ which is an analogue of Cauchy condensation test. In particular, we prove that the SUD of the sequence of the reciprocals of all prime numbers is 1 and its SLD is 0. Moreover, we apply the results in this topic to improve some results for distributionally chaotic linear operators. [ABSTRACT FROM AUTHOR]

In this paper, we give a new description of the relative topological structure group of PL manifolds with boundary, and put the relative L -group into an exact sequence of groups. Then we define the relative index of PL manifolds with boundary, and show that this relative index will induce an additive map from the relative L -group to the K -theory of the relative Roe algebra. [ABSTRACT FROM AUTHOR]

Abstract: We discuss the relation between Devaney chaos in the base system and Devaney chaos in its induced hyperspace system. We show that the latter need not implies the former. We also argue that this implication is not true even in the strengthened condition. Additionally we give an equivalent condition for periodic density in the hyperspace system. [Copyright &y& Elsevier]