The sharp Hausdorff measure condition for length of projections

In a recent paper, Pertti Mattila asked which gauge functions have the property that for any Borel set with Hausdorff measure 0$">, the projection of to almost every line has positive length. We show that finiteness of , which is known to be sufficient for this property, is also necessary for regularly varying . Our proof is based on a random construction adapted to the gauge function.

     

Implementation of Statistical Process Control for Proteomic Experiments Via LC MS/MS

Statistical process control (SPC) is a robust set of tools that aids in the visualization, detection, and identification of assignable causes of variation in any process that creates products, services, or information. A tool has been developed termed Statistical Process Control in Proteomics (SProCoP) which implements aspects of SPC (e.g., control charts and Pareto analysis) into the Skyline proteomics software. It monitors five quality control metrics in a shotgun or targeted proteomic workflow. None of these metrics require peptide identification. The source code, written in the R statistical language, runs directly from the Skyline interface, which supports the use of raw data files from several of the mass spectrometry vendors. It provides real time evaluation of the chromatographic performance (e.g., retention time reproducibility, peak asymmetry, and resolution), and mass spectrometric performance (targeted peptide ion intensity and mass measurement accuracy for high resolving power instruments) via control charts. Thresholds are experiment- and instrument-specific and are determined empirically from user-defined quality control standards that enable the separation of random noise and systematic error. Finally, Pareto analysis provides a summary of performance metrics and guides the user to metrics with high variance. The utility of these charts to evaluate proteomic experiments is illustrated in two case studies.

Cusp Forms on GL(2n) with GL(n) × GL (n) Periods,and Simple Algebras

The notion of a period of a cusp form on GL(2,D(??)), with respect to the diagonal subgroup D(??)^X × D(??)^X, is defined. Here D is a simple algebra over a global field F with a ring ?? of adeles. For D^x = GL(1), the period is the value at 1/2 of the L-function of the cusp form on GL(2, ??). A cuspidal representation is called cyclic if it contains a cusp form with a non zero period. It is investigated whether the notion of cyclicity is preserved under the Deligne - Kazhdan correspondence, relating cuspidal representations on the group and its split form, where D is a matrix algebra. A local analogue is studied too, using the global technique. The method is based on a new bi-period summation formula. Local multiplicity one statements for spherical distributions, and non - vanishing properties of bi - characters, known only in a few cases, play a key role.     

The rotor-router shape is spherical

This column is a place for those bits of contagious mathematics that travel from person to person in the community, because they are so elegant, suprising, or appealing that one has an urge to pass them on. Contributions are most welcome.     

Magnetization of disordered ballistic quantum billards

We study the magnetic response of mesoscopic quantum dots in the ballistic regime where the mean free path l_e is larger that the size L of the sample, yet smaller than L(K_FL)^d?1. In this regime, disorder plays an important role. Employing a semiclassical picture we calculate the contribution of long tranjectories which are strongly affected by static disorder and which differ sharply from those of clean systems. In the case of a magnetic field, they give rise to a large linear paramagnetic susceptibility (which is disorder independent), whose magnitude is in agreement with recent experimental results. In the case of a Aharonov-Bohm flux, the susceptibility is disorder dependent and is proportional to the mean free path as in the diffusive regime. We also discuss the corresponding non-linear susceptibilities.     

Evolving sets, mixing and heat kernel bounds

We show that a new probabilistic technique, recently introduced by the first author, yields the sharpest bounds obtained to date on mixing times of Markov chains in terms of isoperimetric properties of the state space (also known as conductance bounds or Cheeger inequalities). We prove that the bounds for mixing time in total variation obtained by Lovász and Kannan, can be refined to apply to the maximum relative deviation |pⁿ(x,y)/π(y)−1| of the distribution at time n from the stationary distribution π. We then extend our results to Markov chains on infinite state spaces and to continuous-time chains. Our approach yields a direct link between isoperimetric inequalities and heat kernel bounds; previously, this link rested on analytic estimates known as Nash inequalities.Research supported in part by NSF Grants #DMS-0104073 and #DMS-0244479.     

A universal screening assay for glycosynthases: directed evolution of glycosynthase XynB2(E335G) suggests a general path to enhance activity

Glycosynthases are catalytic mutants of mainly retaining glycoside hydrolases that catalyze the synthesis of oligosaccharides from their corresponding glycosyl-fluoride donors and suitable acceptors. Here we describe the development of a general, high-throughput screening procedure for glycosynthase activity, which is based on the release of hydrofluoric acid, a by-product of all glycosynthase reactions. This assay is sensitive, does not require the synthesis of special chromophoric or modified substrates, and, most importantly, is applicable for all glycosynthases. We used this screening procedure on error-prone PCR libraries to isolate improved glycosynthase variants of XynB2(E335G) glycosynthase, a family 52 beta-xylosidase from Geobacillus stearothermophilus. The improved variants exhibited higher K(M) values toward the acceptor and the donor, suggesting that enzyme-product release is rate determining for k(cat).     

On the densities of cliques and independent sets in graphs

Let r; s≥2 be integers. Suppose that the number of blue r-cliques in a red/blue coloring of the edges of the complete graph K _n is known and fixed. What is the largest possible number of red s-cliques under this assumption? The well known Kruskal-Katona theorem answers this question for r = 2 or s = 2. Using the shifting technique from extremal set theory together with some analytical arguments, we resolve this problem in general and prove that in the extremal coloring either the blue edges or the red edges form a clique.