We provide some new results on the weak convergence of sequences or nets lying in L,((T, & r), X) = L,(p, X), 1 <p < co, i.e., the space of equivalence classes of X-valued (X is a Banach space) Bochner integrable functions on the finite measure space ( r, z, I”). Our theorems generalize in several directions recent resuls on weak sequential convergence in L,(p, X) obtained by M. A. Khan and M. ...

LP is a recently introduced formalism that extends answer set programs by adopting the log-linear weight scheme of Markov Logic. This paper investigates the relationships between LP and two other extensions of answer set programs: weak constraints to express a quantitative preference among answer sets, and P-log to incorporate probabilistic uncertainty. We present a translation of LP into progr...

It is known that the free topological group over the Tychonoff space X, denoted F(X), is a P space if and only if X is a P -space. This article is concerned with the question of whether one can characterize when F(X) is a weak P -space, that is, a space where all countable subsets are closed. Our main result is that F(X) is a weak P -space if and only if X is a weak P -space and every countable...

Let (Ω,Σ, μ) be a purely non-atomic measure space, and let 1 < p < ∞. If L(Ω,Σ, μ) is isomorphic, as a Banach space, to L(Ω,Σ, μ) for some purely atomic measure space (Ω,Σ, μ), then there is a measurable partition Ω = Ω1 ∪Ω2 such that (Ω1,Σ ∩ Ω1, μ|Σ∩Ω1) is countably generated and σ-finite, and that μ(σ) = 0 or ∞ for every measurable σ ⊆ Ω2. In particular, L(Ω,Σ, μ) is isomorphic to l.

We consider the p-Laplacian in R perturbed by a weakly coupled potential. We calculate the asymptotic expansions of the lowest eigenvalue of such an operator in the weak coupling limit separately for p > d and p = d and discuss the connection with Sobolev interpolation inequalities. AMS Mathematics Subject Classification: 49R05, 35P30

Let (Q.,l,ß) be a probability space, X a Banach space, and L (fi, X) the Banach space of Bochner integrable functions f: Í2 —> X . Let W = {/ € Ll(p, X) : for a.e. w in Í2, ||/(w)|| < 1} . In this paper we characterize the rwc (relatively weakly compact) subsets of L (ß, X). The main results are as follows: Theorem A. A subset H of W is rwc iff given any sequence (/„) in H there exists a sequen...

We answer a question of Cummings and Magidor by proving that the P-ideal dichotomy of Todorčević refutes κ,ω for any uncountable κ. We also show that the P-ideal dichotomy implies the failure of κ, ω1.

Dual linear programs are very useful to obtain bounds on the optimal value of the primal linear programs using the following two duality theorems. Theorem 1.1. Weak duality: Let x be a feasible solution for (P ) and y be a feasible solution for (D). Then, cx ≥ by. Theorem 1.2. Strong duality: For the linear programs in Figure 1, exactly one of the following possibilities occurs: 1. Both (P ) an...