Let PR(X) denote the hyperspace of non-empty finite subsets a topological space X with Pixley-Roy topology. In this paper, we investigate quasi-Rothberger property in PR(X). We prove that for X, followings are equivalent: (1) is quasi-Rothberger; (2) satisfies S1(?rc f?h,?wrc f?h); (3) separable and each co-finite subset S1(?pc f?h,?wpc (4) PR(Y) Y X. also characterize quasi-Menger quasi-Hurewi...

Throughout the sequel, E denotes a reflexive real Banach space and E∗ its topological dual. We also assume that E is locally uniformly convex. This means that for each x ∈ E, with ‖x‖ = 1, and each > 0, there exists δ > 0 such that, for every y ∈ E satisfying ‖y‖ = 1 and ‖x− y‖ ≥ , one has ‖x + y‖ ≤ 2(1 − δ). Recall that any reflexive Banach space admits an equivalent norm with which it is loca...

Cyclic patterns of neuronal activity are ubiquitous in animal nervous systems, and partially responsible for generating and controlling rhythmic movements such as locomotion, respiration, swallowing and so on. Clarifying the role of the network connectivities for generating cyclic patterns is fundamental for understanding the generation of rhythmic movements. In this paper, the storage of binar...

Elementarily free groups are the finitely generated groups with the same elementary theory as free groups. We prove that elementarily free groups are subgroup separable, answering a question of Zlil Sela. Limit groups arise naturally in the study of the set of homomorphisms to free groups and, in the guise of fully residually free groups, have long been studied in connection with the first-orde...

Being motivated by the famous Kaplansky theorem we study various sequential properties of a Banach space E and its closed unit ball B, both endowed with the weak topology of E. We show that B has the Pytkeev property if and only if E in the norm topology contains no isomorphic copy of l1, while E has the Pytkeev property if and only if it is finite-dimensional. We extend Schlüchtermann and Whee...

We consider the bandwidth provisioning problem in service overlay networks (SON). We formulate the bandwidth provisioning problem in an optimization framework taking the stochastic processes involved into consideration. First, we provide the optimal solution that maximizes the expected net revenue in a single-link network under stationary traffic demand. Then, we show that the revenue maximizat...

Kingman’s (1978) representation theorem (J. Lond. Math. Soc. (2) 18 374–380) states that any exchangeable partition of ℕ can be represented as a paintbox based on random mass-partition. Similarly, composition (i.e., ordered ℕ) an interval-partition (Gnedin (1997) Ann. Probab. 25 1437–1450). Our first main result is coalescent process (not necessarily Markovian) nondecreasing valued in interval-...

We show that if the set of all bounded strongly continuous cosine families on a Banach space X is treated as a metric space under the metric of the uniform convergence associated with the operator norm on the space L (X) of all bounded linear operators on X, then the isolated points of this set are precisely the scalar cosine families. By definition, a scalar cosine family is a cosine family wh...

The general notion of topology does not allow to compare neighborhoods of different points. Such a comparison is quite natural in various geometric contexts. The general setting for such a comparison is that of a uniform structure. The most common and natural way for a uniform structure to appear is via a metric, which was already mentioned on several occasions in Chapter 1, so we will postpone...

We show that if X is a Σ 1 separable metrizable space which is not σ-compact then C p (X), the space of bounded real-valued continuous functions on X with the topology of pointwise convergence, is Borel-Π 1 -complete. Assuming projective determinacy we show that if X is projective not σ-compact and n is least such that X is Σ1n then Cp(X), the space of real-valued continuous functions on X with...