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.     

On the sum of the L1 influences of bounded functions

Let f: {-1, 1}ⁿ → [-1, 1] have degree d as a multilinear polynomial. It is well-known that the total influence of f is at most d. Aaronson and Ambainis asked whether the total L₁ influence of f can also be bounded as a function of d. Ba?kurs and Bavarian answered this question in the affirmative, providing a bound of O(d³) for general functions and O(d²) for homogeneous functions. We improve on their results by providing a bound of d² for general functions and O(d log d) for homogeneous functions. In addition, we prove a bound of d/(2p) + o(d) for monotone functions, and provide a matching example.     

Synthesis, assembly, and optical properties of shape- and phase-controlled ZnSe nanostructures

Shape-, size-, and phase-controlled ZnSe nanostructures were synthesized by thermolysis of zinc acetate and selenourea using liganding solvents of octadecylamine (ODA) and trioctylphosphineoxide (TOPO) at different molar ratios. Materials synthesized in pure ODA resulted in uniform ultranarrow nanorods and nanowires of 1.3 nm in diameter. Morphological change from nanowire to spherical particle of larger diameter occurs with increasing TOPO/ODA ratio. Variation of the TOPO content in the mixed solvent also allows control of the crystallographic phase of ZnSe (wurtzite or zinc blende). The conditions and mechanisms of shape and phase control are discussed. Ultra-high-density networks of the ordered wires are achieved using the Langmuir-Blodgett technique in a single step as an essential stage on the route to ultra-high-density semiconductor nanocircuit fabrication.     

Convergence of Discrete Green Functions with Neumann Boundary Conditions

In this article we prove convergence of Green functions with Neumann boundary conditions for the random walk to their continuous counterparts. Our methods rely on local central limit theorems for convergence of random walks on discretizations of smooth domains to Reflected Brownian motion.     

Sharp superconductor-insulator transition in short wires

Recent experiments on short MoGe nanowires show a sharp superconducting-insulating transition at the universal resistance R(Q)=h/(4e(2)), contrary to the expectation of a smooth temperature dependence of the resistance for such Josephson-like systems. We present a self-consistent renormalization-group treatment of interacting quantum phase slips in short superconducting wires, which reproduces this sharp universal transition. Our method should also apply to other systems in the sine-Gordon universality class, in the previously inaccessible intermediate-coupling regime.     

Optical AND gate based on soliton interaction in a fiber Bragg grating

Using computer simulations, we demonstrate an optical cascadable AND gate based on soliton interaction in a fiber Bragg grating. A single soliton that is launched into the device is backreflected. When two solitons are launched, one of the solitons is transmitted while the other is backreflected. The time delay between the solitons may be few times longer than the duration of the solitons. We show that the interaction causes an increase in the frequency of one of the solitons that enables its transmission through the grating bandgap.