Dynamical quasitilings of amenable group

We prove that for any compact zero-dimensional metric space $X$ on which an infinite countable amenable group $G$ acts freely by homeomorphisms, there exists a dynamical quasitiling with good covering, continuity, F{\o}lner and dynamical properties, i.e to every $x\in X$ we can assign a quasitiling $\mathcal{T}_x$ of $G$ (with all the $\mathcal{T}_x$ using the same, finite set of shapes) such that the tiles of $\mathcal{T}_x$ are disjoint, their union has arbitrarily high lower Banach Density, all the shapes of $\mathcal{T}_x$ are large subsets of an arbitrarily large F{\o}lner set, and if we consider $\mathcal{T}_x$ to be an element of a shift space over a certain finite alphabet, then the mapping $x\mapsto \mathcal{T}_x$ is a factor map

Half-Skyrmions and Spike-Vortex Solutions of Two-Component Nonlinear Schrodinger Systems

Recently, skyrmions with integer topological charges have been observed numerically but have not yet been shown rigorously on two-component systems of nonlinear Schrodinger equations (NLSE) describing a binary mixture of Bose-Einstein condensates. Besides, half-skyrmions characterized by half-integer topological charges can also be found in the nonlinear sigma model which is a model of the Bose-Einstein condensate of the Schwinger bosons. Here we prove rigorously the existence of half-skyrmions which may come from a new type of soliton solutions called spike-vortex solutions of two-component systems of NLSE on the entire plane. These spike-vortex solutions having spikes in one component and a vortex in the other component may form half-skyrmions. By Liapunov-Schmidt reduction process, we may find spike-vortex solutions of two-component systems of NLSE.Comment: to appear in J.Math.Phy

ChIP'ing the mammalian genome: technical advances and insights into functional elements

Characterization of the functional components in mammalian genomes depends on our ability to completely elucidate the genetic and epigenetic regulatory networks of chromatin states and nuclear architecture. Such endeavors demand the availability of robust and effective approaches to characterizing protein-DNA associations in their native chromatin environments. Consider able progress has been made through the applica tion of chromatin immunoprecipitation (ChIP) to study chromatin biology in cells. Coupled with genome-wide analyses, ChIP-based assays enable us to take a global, unbiased and comprehensive view of transcriptional control, epigenetic regulation and chromatin structures, with high precision and versatility. The integrated knowledge derived from these studies is used to decipher gene regulatory networks and define genome organization. In this review, we discuss this powerful approach and its current advances. We also explore the possible future developments of ChIP-based approaches to interrogating long-range chromatin interactions and their impact on the mechanisms regulating gene expression