Global well-posedness for KdV in Sobolev Spaces of negative index

The initial value problem for the Korteweg-deVries equation on the line is shown to be globally well-posed for rough data. In particular, we show global well-posedness for initial data in H^s({\mathbb{R}), -3/10<s.Comment: 5 pages. Electronic Journal of Differential equations (submitted

Combining All Pairs Shortest Paths and All Pairs Bottleneck Paths Problems

We introduce a new problem that combines the well known All Pairs Shortest Paths (APSP) problem and the All Pairs Bottleneck Paths (APBP) problem to compute the shortest paths for all pairs of vertices for all possible flow amounts. We call this new problem the All Pairs Shortest Paths for All Flows (APSP-AF) problem. We firstly solve the APSP-AF problem on directed graphs with unit edge costs and real edge capacities in $\tilde{O}(\sqrt{t}n^{(\omega+9)/4}) = \tilde{O}(\sqrt{t}n^{2.843})$ time, where $n$ is the number of vertices, $t$ is the number of distinct edge capacities (flow amounts) and $O(n^{\omega}) < O(n^{2.373})$ is the time taken to multiply two $n$-by-$n$ matrices over a ring. Secondly we extend the problem to graphs with positive integer edge costs and present an algorithm with $\tilde{O}(\sqrt{t}c^{(\omega+5)/4}n^{(\omega+9)/4}) = \tilde{O}(\sqrt{t}c^{1.843}n^{2.843})$ worst case time complexity, where $c$ is the upper bound on edge costs

Estradiol, Progesterone, and Transforming Growth Factor α Regulate Insulin-Like Growth Factor Binding Protein-3 (IGFBP3) Expression in Mouse Endometrial Cells

Insulin-like growth factor 1 (IGF1) Is Involved in the proliferation of mouse and rat endometrial cells in a paracrine or autocrine manner. Insulin-like growth factor binding protein-3 (IGFBP3) modulates actions of IGFs directly or indirectly. The present study aimed to determine whether IGFBP3 is Involved In the regulation of proliferation of mouse endometrial cells. Mouse endometrial epithelial cells and stromal cells were isolated, and cultured In a serum free medium. IGF1 stimulated DNA synthesis by endometrial epithelial and stromal cells, and IGFBP3 Inhibited IGF1-induced DNA synthesis. Estradiol-17 beta (E2) decreased the Igfbp3 mRNA level in endometrial stromal cells, whereas It Increased the Igf1 mRNA level. Transforming growth factor alpha (TGF alpha) significantly decreased IGFBP3 expression at both the mRNA and secreted protein levels in endometrial stromal cells. Progesterone (134) did not affect the E2-induced down-regulation of Igfbp3 mRNA expression in endometrial stromal cells, although P4 alone increased Igfbp3 mRNA levels. The present findings suggest that in mouse endometrial stromal cells E2 enhances IGF1 action through enhancement of IGF1 synthesis and reduction of IGFBP3 synthesis, and that TGF alpha affects IGF1 actions through modulation of IGFBP3 levels