In this paper, we introduce the dual notion of strongly top modules and study some of the basic properties of this class of modules.

1. Basic Assumptions. Assume that (i) {Tt} is a strongly-continuous semigroup of bounded linear operators on the Banach space B = C0(I) = { f continuous on I = [0, 1] : f(0) = f(1) = 0 } (ii) If f(x) ≥ 0 and f ∈ B, then Ttf(x) ≥ 0 and Ttf(x) ≤ maxy∈I f(y). In general, the infinitesimal generator of a strongly-continuuous semigroup of linear operators Tt on any Banach space B is defined by Af = ...

1. Basic Assumptions. Assume that (i) {Tt} is a strongly-continuous semigroup of bounded linear operators on the Banach space B = C0(I) = { f continuous on I = [0, 1] : f(0) = f(1) = 0 } (ii) If f(x) ≥ 0 and f ∈ B, then Ttf(x) ≥ 0 and Ttf(x) ≤ maxy∈I f(y). In general, the infinitesimal generator of a strongly-continuuous semigroup of linear operators Tt on any Banach space B is defined by Af = ...

Suppose that f(z) is a transcendental entire function and that the Fatou set F (f) 6= ∅. Set B1(f) := sup U supz∈U log(|z| + 3) infw∈U log(|w|+ 3) and B2(f) := sup U supz∈U log log(|z|+ 30) infw∈U log(|w| + 3) , where the supremum supU is taken over all components of F (f). If B1(f) < ∞ or B2(f) < ∞, then we say F (f) is strongly uniformly bounded or uniformly bounded respectively. In this arti...

In this article we introduce and study the notions of double -lacunary strongly summable, double  Cesàro strongly summable, double  statistically convergent and double  -lacunary statistically convergent sequence of fuzzy numbers. Consequently we construct the spaces   2 F L  ,   2 F   ,   2 F S  and   2 F S  respectively and investigate the relationship among these spaces. F...

An R-module M is called strongly noncosingular if it has no nonzero Rad-small (cosingular) homomorphic image in the sense of Harada. It is proven that (1) an R-module M is strongly noncosingular if and only if M is coatomic and noncosingular; (2) a right perfect ring R is Artinian hereditary serial if and only if the class of injective modules coincides with the class of (strongly) noncosingula...

This paper generalizes the idea of strongly associative hyperoperation introduced in [7]  to the class of hyperrings. We introduce and investigate hyperrings of type 1, type 2 and SDIS. Moreover, we study some examples of these hyperrings and give a new kind of hyperrings called  totally hyperrings. Totally hyperrings give us a characterization of Krasner hyperrings. Also, we investigate these ...

A function f defined on a subset E of a 2-normed space X is strongly lacunary ward continuous if it preserves strongly lacunary quasi-Cauchy sequences of points in E; that is, (f(x k)) is a strongly lacunary quasi-Cauchy sequence whenever (x k) is strongly lacunary quasi-Cauchy. In this paper, not only strongly lacunary ward continuity, but also some other kinds of continuities are investigated...

We show that, assuming GCH, if κ is a Ramsey or a strongly Ramsey cardinal and F is a class function on the regular cardinals having a closure point at κ and obeying the constraints of Easton’s theorem, namely, F (α) ≤ F (β) for α ≤ β and α < cf(F (α)), then there is a cofinality preserving forcing extension in which κ remains Ramsey or strongly Ramsey respectively and 2δ = F (δ) for every regu...

Let f be a computable function from finite sequences of 0’s and 1’s to real numbers. We prove that strong f -randomness implies strong f randomness relative to a PA-degree. We also prove: if X is strongly f -random and Turing reducible to Y where Y is Martin-Löf random relative to Z, then X is strongly f -random relative to Z. In addition, we prove analogous propagation results for other notion...