The concept of a preopen set in a topological space was introduced by H. H. Corson and E. Michael in 1964 [3]. A subset A of a topological space (X, τ) is called preopen or locally dense or nearly open if A ⊆ Int(Cl(A)). A set A is called preclosed if its complement is preopen or equivalently if Cl(Int(A)) ⊆ A. The term, preopen, was used for the first time by A. S. Mashhour, M. E. Abd El-Monse...

Abstract. The metric Dα(q, q) on the set Q of particle locations of a homogeneous Poisson process on Rd, defined as the infimum of ( ∑ i |qi − qi+1| α)1/α over sequences in Q starting with q and ending with q (where | · | denotes Euclidean distance) has nontrivial geodesics when α > 1. The cases 1 < α < ∞ are the Euclidean first-passage percolation (FPP) models introduced earlier by the authors...

in this errata, we reconsider and modify two propositions and their corollaries which were written on epi-retractable and co-epi-retractable modules.

We consider a closed semi-algebraic set X ⊂ R and a C semi-algebraic function f : R → R such that f|X has a finite number of critical points. We relate the topology of X to the topology of the sets {f ∗ α}, where ∗ ∈ {≤,=,≥} and α ∈ R, and the indices of the critical points of f|X and −f|X . We also relate the topology of X to the topology of the links at infinity of the sets {f ∗ α} and the in...

In this paper, we construct a class of non-weight modules over the affine-Virasoro algebra type A1 by taking tensor products finite number irreducible M(λ,α,β,γ) with highest weight V(η,ϵ,θ). We obtain necessary and sufficient conditions for such product to be irreducible, determine two isomorphic. also compare these other known modules, showing that are new.

this formula defines a Schwartz function, and hence the solution u = uf ∈ S also, and the mapping f 7→ uf is continuous in the sense that if fn is a sequence of Schwartz functions such that ‖fn − f‖α,β → 0 for every Schwartz semi-norm ‖ · ‖α,β , then also ‖un − u‖α,β → 0 for every Schwartz semi-norm, where un = ufn , u = uf . In fact the formula above extends to define a distributional solution...

A semi-dualizing module over a commutative noetherian ringA is a finitely generated module C with RHomA(C,C) ≃ A in the derived category D(A). We show how each such module gives rise to three new homological dimensions which we call C–Gorenstein projective, C–Gorenstein injective, and C–Gorenstein flat dimension, and investigate the properties of these dimensions.