A Smarandache quasigroup(loop) is shown to be universal if all its f, g-principal isotopes are Smarandache f, g-principal isotopes. Also, weak Smarandache loops of Bol-Moufang type such as Smarandache: left(right) Bol, Moufang and extra loops are shown to be universal if all their f, g-principal isotopes are Smarandache f, gprincipal isotopes. Conversely, it is shown that if these weak Smaranda...

We prove that in a certain class E of nonseparable Banach spaces the norm topology of the dual ball is definable in terms of its weak∗ topology. Thus, if X, Y ∈ E and f : BX∗ −→ BY ∗ is a weak ∗-to-weak∗ homeomorphism, then f is automatically norm-to-norm continuous.

We study classes of filters F on N such that weak and strong F -convergence of sequences in l1 coincide. We study also analogue of l1 weak sequential completeness theorem for filter convergence.

in this paper, we investigate the concept of topological stationary for locally compact semigroups. in [4], t. mitchell proved that a semigroup s is right stationary if and only if m(s) has a left invariant mean. in this case, the set of values ?(f) where ? runs over all left invariant means on m(s) coincides with the set of constants in the weak* closed convex hull of right translates of f. th...

Let N0 be the set of all non-negative integers and P(N0) be its the power set. An integer additive set-indexer (IASI) of a graph G is an injective function f : V (G) ! P(N0) such that the induced function f+ : E(G) ! P(N0) defined by f+(uv) = f(u) + f(v) is also injective, where f(u) + f(v) is the sum set of f(u) and f(v). An integer additive set-indexer f is said to be a weak integer additive ...

Let f be a diffeomorphism of a closed C∞ three-dimensional manifold. In this paper, we introduce the notion of C1-stably weak shadowing for a closed f -invariant set, and prove that C1-generically, for an aperiodic chain component Cf of f isolated in the chain recurrent set, if f|Cf is C 1-stably weak shadowing, then there are a C1-neighborhood U(f) of f and an open and dense subset V of U(f) s...

ion. LEQ-λ Γ, X :pκ ` F ≤ F ′ : κ′ Γ ` λXF ≤ λXF ′ : pκ→ κ′ Application. There are two kinds of congruence rules for application: one kind states that if functions F and F ′ are in the subtyping relation, so are their values F G and F ′G at a certain argument G. LEQ-FUN Γ ` F ≤ F ′ : pκ→ κ′ p−1Γ ` G : κ Γ ` F G ≤ F ′G : κ′ The other kind of rules concern the opposite case: If F is a function an...

Let f n (; !) be a sequence of stochastic processes which converge weakly to a limit process f 0 (; !). We show under some assumptions the weak inclusion of the solution sets n (!) = f : f n (; !) = 0g in the limiting solution set 0 (!) = f : f 0 (; !) = 0g. If the limiting solutions are a.s. singletons, then weak convergence holds. Results of his type are called Z-theorems (Zero-theorems). Mor...

Let $T$ be a bounded operator on the Banach space $X$ and $ph$ be an analytic self-map of the unit disk $Bbb{D}$‎. ‎We investigate some operator theoretic properties of‎ ‎bilateral composition operator $C_{ph‎, ‎T}‎: ‎f ri T circ f circ ph$ on the vector-valued Hardy space $H^p(X)$ for $1 leq p leq‎ ‎+infty$.‎ ‎Compactness and weak compactness of $C_{ph‎, ‎T}$ on $H^p(X)$‎ ‎are characterized an...

In this paper, we investigate the concept of topological stationary for locally compact semigroups. In [4], T. Mitchell proved that a semigroup S is right stationary if and only if m(S) has a left Invariant mean. In this case, the set of values ?(f) where ? runs over all left invariant means on m(S) coincides with the set of constants in the weak* closed convex hull of right translates of f. Th...