Let X be an F-space, and let Y be a subspace of X of dimension one, with X/Y = lp (0 p oo). Provided p ~ 1, X ~lp; however if p = 1, we construct an example to show that X need not be locally convex. More generally we show that Y is any closed subspace of X, then if Y is an r-Banach space (0 r:5 1) and XI Y is a p-Banach space with p r S 1 then X is a p-Banach space; if Y and XI Y are B-convex ...

The aim of this paper is to establish necessary and sufficient conditions for exponential stability of semigroups of linear operators and to show how this conditions can be applied in order to characterize the exponential dichotomy. First, we prove that an exponentially bounded semigroup is exponentially stable if and only if it is (l(N, X), l∞(N, X))-stable, where p ∈ (1,∞). After that this re...

this study develops and analyzes preconditioned krylov subspace methods to solve linear systemsarising from discretization of the time-independent space-fractional models. first, we apply shifted grunwald formulas to obtain a stable finite difference approximation to fractional advection-diffusion equations. then, we employee two preconditioned iterative methods, namely, the preconditioned gene...

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for an arbitrary ring $r$, the zero-divisor graph of $r$, denoted by $gamma (r)$, is an undirected simple graph that its vertices are all nonzero zero-divisors of $r$ in which any two vertices $x$ and $y$ are adjacent if and only if either $xy=0$ or $yx=0$. it is well-known that for any commutative ring $r$, $gamma (r) cong gamma (t(r))$ where $t(r)$ is the (total) quotient ring of $r$. in this...

The hierarchical mixture of experts architecture provides a flexible procedure for implementing classification algorithms. The classification is obtained by a recursive soft partition of the feature space in a data-driven fashion. Such a procedure enables local classification where several experts are used, each of which is assigned with the task of classification over some subspace of the feat...