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.     

Adaptive versus Non-Adaptive Security of Multi-Party Protocols

Security analysis of multi-party cryptographic protocols distinguishes between two types of adversarial settings: In the non-adaptive setting the set of corrupted parties is chosen in advance, before the interaction begins. In the adaptive setting the adversary chooses who to corrupt during the course of the computation. We study the relations between adaptive security (i.e., security in the adaptive setting) and nonadaptive security, according to two definitions and in several models of computation.     

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.     

From the microscopic to the mesoscopic properties of lyotropic reverse hexagonal liquid crystals

In the present study we aimed to explore a correlation between the microstructural properties of the lyotropic reverse hexagonal phase (HII) of the GMO/tricaprylin/phosphatidylcholine/water system and its mesoscopic structure. The mesoscopic organization of discontinuous and anisotropic domains was examined, in the native state, using environmental scanning electron microscopy. The topography of the HII mesophases was imaged directly in their hydrated state, as a function of aqueous-phase concentration and composition, when a proline amino acid was solubilized into the systems as a kosmotropic (water-structure maker) guest molecule. The domain structures of several dozen micrometers in size, visualized in the environmental scanning electron microscopy, were found to possess fractal characteristics, indicating a discontinuous and disordered alignment of the corresponding internal water rods on the mesoscale. On the microstructural level, SAXS measurements revealed that as water content (Cw) increases the characteristic lattice parameter of the mesophases increases as well. Using the water concentration as the mass measure of the mixtures, a scaling relationship between the lattice parameter and the concentration was found to obey a power law whereby the derived fractal dimension was the relevant exponent, confirming the causal link between the microscopic and mesoscopic organizations. The topography of the HII mesophase was found to be affected by the microstructural parameters and the composition of the samples. Thermal analysis experiments involving these systems further confirmed that the behavior of water underpins both microscopical and mesoscopic features of the systems. It was shown that both the swelling of the lattice parameter and the mesoscopic domains is correlated to the bulk water concentration in the water rods.     

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.