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Let F = ind lim Fn be an infinite-dimensional LF-space with density dens F = r ( ≥ ℵo) such that some Fn is infinite-dimensional and dens Fn = r. It is proved that every open subset of F is homeomorphic to the product of an l2(r)-manifold and R∞ = ind lim Rn (hence the product of an open subset of l2(r) and R∞). As a consequence, any two open sets in F are homeomorphic if they have the same homotopy...
Concrete and construction industries have consumed an enormous amount of resources and energy and generated a large amount of wastes. We need to properly manage them from the environmental point of view. In this paper, environmental aspects of concrete and some systems for environmental management in the design of concrete structures are discussed.
Let ConvF(Rn) be the space of all non-empty closed convex sets in Euclidean space Rn endowed with the Fell topology. We prove that ConvF(lRn) ≈ Rn x Q for every n > 1 whereas ConvF(R) ≈ R x I.
By Fin(X) (resp. Fink(X)), we denote the hyperspace of all non-empty finite subsets of X (resp. consisting of at most k points) with the Vietoris topology. Let ℓ2(τ) be the Hilbert space with weight τ and ℓf2(τ) the linear span of the canonical orthonormal basis of ℓ2(τ). It is shown that if E = ℓf2(τ) or E is an absorbing set in ℓ2(τ) for one of the absolute Borel classes aα(τ) and Mα(τ) of weight...
The notion of C^1-stably positively expansive differentiable maps on closed C^infinity manifolds is introduced, and it is proved that a differentiable map f is C^1-stably positively expansive if and only if f is expanding. Furthermore, for such maps, the [epsilon]-time dependent stability is shown. As a result, every expanding map is e-time dependent stable.
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