Some Results On Convex Greedy Embedding Conjecture for 3-Connected Planar Graphs

A greedy embedding of a graph $G = (V,E)$ into a metric space $(X,d)$ is a function $x : V(G) \to X$ such that in the embedding for every pair of non-adjacent vertices $x(s), x(t)$ there exists another vertex $x(u)$ adjacent to $x(s)$ which is closer to $x(t)$ than $x(s)$. This notion of greedy embedding was defined by Papadimitriou and Ratajczak (Theor. Comput. Sci. 2005), where authors conjectured that every 3-connected planar graph has a greedy embedding (possibly planar and convex) in the Euclidean plane. Recently, greedy embedding conjecture has been proved by Leighton and Moitra (FOCS 2008). However, their algorithm do not result in a drawing that is planar and convex for all 3-connected planar graph in the Euclidean plane. In this work we consider the planar convex greedy embedding conjecture and make some progress. We derive a new characterization of planar convex greedy embedding that given a 3-connected planar graph $G = (V,E)$, an embedding x: V \to \bbbr^2 of $G$ is a planar convex greedy embedding if and only if, in the embedding $x$, weight of the maximum weight spanning tree ($T$) and weight of the minimum weight spanning tree (\func{MST}) satisfies \WT(T)/\WT(\func{MST}) \leq (\card{V}-1)^{1 - \delta}, for some $0 < \delta \leq 1$.Comment: 19 pages, A short version of this paper has been accepted for presentation in FCT 2009 - 17th International Symposium on Fundamentals of Computation Theor

Singular shell embedded into a cosmological model

We generalize Israel's formalism to cover singular shells embedded in a non-vacuum Universe. That is, we deduce the relativistic equation of motion for a thin shell embedded in a Schwarzschild/Friedmann-Lemaitre-Robertson-Walker spacetime. Also, we review the embedding of a Schwarzschild mass into a cosmological model using "curvature" coordinates and give solutions with (Sch/FLRW) and without the embedded mass (FLRW).Comment: 25 pages, 2 figure

Performance of PRP associated with porous chitosan as a composite scaffold for regenerative medicine

This study aimed to evaluate the in vitro performance of activated platelet-rich plasma associated with porous sponges of chitosan as a composite scaffold for proliferation and osteogenic differentiation of human adipose tissue-derived mesenchymal stem cells. The sponges were prepared by controlled freezing (−20, −80, or −196°C) and lyophilization of chitosan solutions (1, 2, or 3% w/v). The platelet-rich plasma was obtained from controlled centrifugation of whole blood and activated with calcium and autologous serum. The composite scaffolds were prepared by embedding the sponges with the activated platelet-rich plasma. The results showed the performance of the scaffolds was superior to that of activated platelet-rich plasma alone, in terms of delaying the release of growth factors and increased proliferation of the stem cells. The best preparation conditions of chitosan composite scaffolds that coordinated the physicochemical and mechanical properties and cell proliferation were 3% (w/v) chitosan and a −20°C freezing temperature, while −196°C favored osteogenic differentiation. Although the composite scaffolds are promising for regenerative medicine, the structures require stabilization to prevent the collapse observed after five days2015CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICO - CNPQsem informaçã

Performance Of Prp Associated With Porous Chitosan As A Composite Scaffold For Regenerative Medicine.

This study aimed to evaluate the in vitro performance of activated platelet-rich plasma associated with porous sponges of chitosan as a composite scaffold for proliferation and osteogenic differentiation of human adipose tissue-derived mesenchymal stem cells. The sponges were prepared by controlled freezing (-20, -80, or -196°C) and lyophilization of chitosan solutions (1, 2, or 3% w/v). The platelet-rich plasma was obtained from controlled centrifugation of whole blood and activated with calcium and autologous serum. The composite scaffolds were prepared by embedding the sponges with the activated platelet-rich plasma. The results showed the performance of the scaffolds was superior to that of activated platelet-rich plasma alone, in terms of delaying the release of growth factors and increased proliferation of the stem cells. The best preparation conditions of chitosan composite scaffolds that coordinated the physicochemical and mechanical properties and cell proliferation were 3% (w/v) chitosan and a -20°C freezing temperature, while -196°C favored osteogenic differentiation. Although the composite scaffolds are promising for regenerative medicine, the structures require stabilization to prevent the collapse observed after five days.201539613

On disjoint paths in acyclic planar graphs

We give an algorithm with complexity $O(f(R)^{k^2} k^3 n)$ for the integer multiflow problem on instances $(G,H,r,c)$ with $G$ an acyclic planar digraph and $r+c$ Eulerian. Here, $f$ is a polynomial function, $n = |V(G)|$, $k = |E(H)|$ and $R$ is the maximum request $\max_{h \in E(H)} r(h)$. When $k$ is fixed, this gives a polynomial algorithm for the arc-disjoint paths problem under the same hypothesis