Algorithmic and Hardness Results for the Colorful Components Problems

In this paper we investigate the colorful components framework, motivated by applications emerging from comparative genomics. The general goal is to remove a collection of edges from an undirected vertex-colored graph $G$ such that in the resulting graph $G'$ all the connected components are colorful (i.e., any two vertices of the same color belong to different connected components). We want $G'$ to optimize an objective function, the selection of this function being specific to each problem in the framework. We analyze three objective functions, and thus, three different problems, which are believed to be relevant for the biological applications: minimizing the number of singleton vertices, maximizing the number of edges in the transitive closure, and minimizing the number of connected components. Our main result is a polynomial time algorithm for the first problem. This result disproves the conjecture of Zheng et al. that the problem is $NP$-hard (assuming $P \neq NP$). Then, we show that the second problem is $APX$-hard, thus proving and strengthening the conjecture of Zheng et al. that the problem is $NP$-hard. Finally, we show that the third problem does not admit polynomial time approximation within a factor of $|V|^{1/14 - \epsilon}$ for any $\epsilon > 0$, assuming $P \neq NP$ (or within a factor of $|V|^{1/2 - \epsilon}$, assuming $ZPP \neq NP$).Comment: 18 pages, 3 figure

New Supporting Evidence for the Overdensity of Galaxies around the Radio-Loud Quasar SDSS J0836+0054 at z =5.8

Recently, Zheng et al. (2005) found evidence for an overdensity of galaxies around a radio-loud quasar, SDSS J0836+0054, at z=5.8 (a five arcmin$^2$ region). We have examined our deep optical imaging data (B, V, r', i', z', and NB816) taken with the Suprime-Cam on the Subaru Telescope. The NB816 narrow-band filter (lambda_c = 815 nm and $\Delta\lambda = 12$ nm) is suitable for searching for Ly$\alpha$ emitters at $z\approx 5.7$. We have found a new strong Ly$\alpha$ emitter at $z \approx 5.7$ close to object B identified by Zheng et al. Further, the non detection of the nine objects selected by Zheng et al. (2005) in our B, V, and r' images provides supporting evidence that they are high-z objects.Comment: 5 pages, 1 figure, accepted for PAS

Comment on ``Deterministic equations of motion and phase ordering dynamics''

Zheng [Phys. Rev. E {\bf 61}, 153 (2000), cond-mat/9909324] claims that phase ordering dynamics in the microcanonical $\phi^4$ model displays unusual scaling laws. We show here, performing more careful numerical investigations, that Zheng only observed transient dynamics mostly due to the corrections to scaling introduced by lattice effects, and that Ising-like (model A) phase ordering actually takes place at late times. Moreover, we argue that energy conservation manifests itself in different corrections to scaling.Comment: 5 pages, 4 figure

Conservativeness of non-symmetric diffusion processes generated by perturbed divergence forms

Let E be an unbounded open (or closed) domain in Euclidean space of dimension greater or equal to two. We present conservativeness criteria for (possibly reflected) diffusions with state space E that are associated to fairly general perturbed divergence form operators. Our main tool is a recently extended forward and backward martingale decomposition, which reduces to the well-known Lyons-Zheng decomposition in the symmetric case.Comment: Corrected typos, minor modification

Regulation of G protein-coupled receptors by palmitoylation and cholesterol

Due to their membrane location, G protein-coupled receptors (GPCRs) are subject to regulation by soluble and integral membrane proteins as well as membrane components, including lipids and sterols. GPCRs also undergo a variety of post-translational modifications, including palmitoylation. A recent article by Zheng et al. in BMC Cell Biology demonstrates cooperative roles for receptor palmitoylation and cholesterol binding in GPCR dimerization and G protein coupling, underlining the complex regulation of these receptors

Stochastic Optimization Theory of Backward Stochastic Differential Equations Driven by G-Brownian Motion

In this paper, we consider the stochastic optimal control problems under G-expectation. Based on the theory of backward stochastic differential equations driven by G-Brownian motion, which was introduced in [10.11], we can investigate the more general stochastic optimal control problems under G-expectation than that were constructed in [28]. Then we obtain a generalized dynamic programming principle and the value function is proved to be a viscosity solution of a fully nonlinear second-order partial differential equation.Comment: 25 page