Multiscale Partition of Unity

We introduce a new Partition of Unity Method for the numerical homogenization of elliptic partial differential equations with arbitrarily rough coefficients. We do not restrict to a particular ansatz space or the existence of a finite element mesh. The method modifies a given partition of unity such that optimal convergence is achieved independent of oscillation or discontinuities of the diffusion coefficient. The modification is based on an orthogonal decomposition of the solution space while preserving the partition of unity property. This precomputation involves the solution of independent problems on local subdomains of selectable size. We deduce quantitative error estimates for the method that account for the chosen amount of localization. Numerical experiments illustrate the high approximation properties even for 'cheap' parameter choices.Comment: Proceedings for Seventh International Workshop on Meshfree Methods for Partial Differential Equations, 18 pages, 3 figure

Announcement of the Summer School of Biology for 1929 July 6-August 16

Official Publication of Cornell University V.20 1928/2

An Anisotropic Ballistic Deposition Model with Links to the Ulam Problem and the Tracy-Widom Distribution

We compute exactly the asymptotic distribution of scaled height in a (1+1)--dimensional anisotropic ballistic deposition model by mapping it to the Ulam problem of finding the longest nondecreasing subsequence in a random sequence of integers. Using the known results for the Ulam problem, we show that the scaled height in our model has the Tracy-Widom distribution appearing in the theory of random matrices near the edges of the spectrum. Our result supports the hypothesis that various growth models in $(1+1)$ dimensions that belong to the Kardar-Parisi-Zhang universality class perhaps all share the same universal Tracy-Widom distribution for the suitably scaled height variables.Comment: 5 pages Revtex, 3 .eps figures included, new references adde

Structural Relaxation and Frequency Dependent Specific Heat in a Supercooled Liquid

We have studied the relation between the structural relaxation and the frequency dependent thermal response or the specific heat, $c_p(\omega)$, in a supercooled liquid. The Mode Coupling Theory (MCT) results are used to obtain $c_p(\omega)$ corresponding to different wavevectors. Due to the two-step relaxation process present in the MCT, an extra peak, in addition to the low frequency peak, is predicted in specific heat at high frequency.Comment: 14 pages, 13 Figure

REST represses a subset of the pancreatic endocrine differentiation program.

To contribute to devise successful beta-cell differentiation strategies for the cure of Type 1 diabetes we sought to uncover barriers that restrict endocrine fate acquisition by studying the role of the transcriptional repressor REST in the developing pancreas. Rest expression is prevented in neurons and in endocrine cells, which is necessary for their normal function. During development, REST represses a subset of genes in the neuronal differentiation program and Rest is down-regulated as neurons differentiate. Here, we investigate the role of REST in the differentiation of pancreatic endocrine cells, which are molecularly close to neurons. We show that Rest is widely expressed in pancreas progenitors and that it is down-regulated in differentiated endocrine cells. Sustained expression of REST in Pdx1(+) progenitors impairs the differentiation of endocrine-committed Neurog3(+) progenitors, decreases beta and alpha cell mass by E18.5, and triggers diabetes in adulthood. Conditional inactivation of Rest in Pdx1(+) progenitors is not sufficient to trigger endocrine differentiation but up-regulates a subset of differentiation genes. Our results show that the transcriptional repressor REST is active in pancreas progenitors where it gates the activation of part of the beta cell differentiation program

Coloring translates and homothets of a convex body

We obtain improved upper bounds and new lower bounds on the chromatic number as a linear function of the clique number, for the intersection graphs (and their complements) of finite families of translates and homothets of a convex body in \RR^n.Comment: 11 pages, 2 figure