Existing Hierarchically Semi-Separable (HSS) construction algorithms for dense n×n matrices require as much as O(n2) peak workspace memory, at a cost of O(n2) flops. An algorithm is presented which requires O(n1.5) peak worskpace memory in the worst case, while still requiring only O(n2) flops.

We present a unified framework for solving linear and convex quadratic programs via interior point methods. At each iteration, this method solves an indefinite system whose matrix is [_~-2 A v] instead of reducing to obtain the usual AD2A v system. This methodology affords two advantages: (1) it avoids the fill created by explicitly forming the product AD2A v when A has dense columns; and (2) i...

A space is a Bake space if the intersection of countably many dense open sets is dense. We show that if X is a non-separable completely met&able linear space (pathconnected abelian topological group) then X contains two linear subspaces (subgroups) E and F such that both E and F are Baire but E x F is not. If X is a completely met&able linear space of weight K, then X is the direct sum E@F of t...

We prove that \(L_2(\mathbb {R})\) contains a Schauder basis of non-negative functions. Similarly, for all \(1<p<\infty \), \(L_{p}(\mathbb basic sequence functions such \(L_p(\mathbb embeds into the closed span sequence. as well if X is separable Banach space with bounded approximation property, then any set in dense quasi-basis (Schauder frame) X.

1. Introduction and notation. In this paper the word " local " is used in at least three different meanings. Our aim is to study local Banach spaces of Fréchet or other locally convex spaces, and it turns out that it is convenient to use the local theory of Banach spaces for this purpose. Recall that given a locally convex space E and a continuous seminorm p on E the completion of the normed sp...

in this paper, we will give necessary and sufficient conditions for certain hnnextensions of subgroup separable groups with normal associated subgroup to be conjugacyseparable. in fact, we will show that these hnn extensions are conjugacy separable if and onlyif the normalizer of one of its associated subgroup is conjugacy separable.

We prove that every separable and metrizable space admits a metrizable compactification with a remainder that is both path connected and locally path connected. This result answers a question of P. Simon. Connectedness and compactness are two fundamental topological properties. A natural question is whether a given space admits a connected (Hausdorff) compactification. This question has been st...

Let F be a continuous function from an open subset D of a separable Banach space X into a Banach space Y. We show that if there is a dense G8 subset A of D and a Gs subset H of X whose closure has nonempty interior, such that for each a E A and each x E H the directional derivative DxF(a) of F at a in the direction x exists, then F is Giteaux differentiable on a dense G8 subset of D. If X is re...

Let (X, d) be a quasi-convex, complete and separable metric space with reference probability measure m. We prove that the set of of real valued Lipschitz function with non zero point-wise Lipschitz constant m-almost everywhere is residual, and hence dense, in the Banach space of Lipschitz and bounded functions. The result is the metric analogous of a result proved for real valued Lipschitz maps...

We build on Gruenhage, Natkaniec, and Piotrowski’s study of thin, very thin, and slim dense sets in products, and the related notions of (NC) and (GC) which they introduced. We find examples of separable spaces X such that X has a thin or slim dense set but no countable one. We characterize ordered spaces that satisfy (GC) and (NC), and we give an example of a separable space which satisfies (G...