On the mixing time of a simple random walk on the super critical percolation cluster

 We study the robustness under perturbations of mixing times, by studying mixing times of random walks in percolation clusters inside boxes in Z ^d . We show that for d≥2 and p>p _c (Z ^d ), the mixing time of simple random walk on the largest cluster inside is Θ(n ² ) – thus the mixing time is robust up to a constant factor. The mixing time bound utilizes the Lovàsz-Kannan average conductance method. This is the first non-trivial application of this method which yields a tight result. Received: 16 December 2001 / Revised version: 13 August 2002 / Published online: 19 December 2002     

The unification of hypo-plastic and elasto-plastic theories

A unifying theory of plasticity is developed that allows deducing models with either hypo-plastic, elasto-plastic or hyper-plastic constitutive structures. Elasto-plasticity is shown to be a singular case of hypo-plasticity. It is further demonstrated that certain conditions can be imposed to generate a new hierarchy of thermodynamically consistent hypo-plastic models, with a hyper-plastic structure as a singular case. The unifying theory is powerfully bridging between the tools created specifically for advancing models under either of those previous formulations.     

On the support of harmonic measure for the random walk

We show that harmonic measure for the simple random walk on then ×…×n cube in thed-dimensional lattice is supported on o(n ^d ) vertices. This research was supported in part by NSF grant # DMS-9353149.     

Instability of Set Recurrence and Green’s Function on Groups with the Liouville Property

Let μ and ν be probability measures on a group Γ and let G _μ and G _ν denote Green’s function with respect to μ and ν. The group Γ is said to admit instability of Green’s function if there are symmetric, finitely supported measures μ and ν and a sequence {x _n } such that G _μ (e, x _n )/G _ν (e,x _n ) →0, and Γ admits instability of recurrence if there is a set S that is recurrent with respect to ν but transient with respect to μ. We give a number of examples of groups that have the Liouville property but have both types of instabilities. Previously known groups with these instabilities did not have the Liouville property.     

Subdivision Schemes for Positive Definite Matrices

The class of symmetric positive definite matrices is an important class both in theory and application. Although this class is well studied, little is known about how to efficiently interpolate such data within the class. We extend the 4-point interpolatory subdivision scheme, as a method of interpolation, to data consisting of symmetric positive definite matrices. This extension is based on an explicit formula for calculating a binary “geodetic average”. Our method generates a smooth curve of matrices, which retain many important properties of the interpolated matrices. Furthermore, the scheme is robust and easy to implement.     

AFM-induced amine deprotection: triggering localized bond cleavage by application of tip/substrate voltage bias for the surface self-assembly of nanosized dendritic objects

An alpha,alpha-dimethyl-3,5-dimethoxybenzyloxycarbonyl (DDZ)-protected amine monolayer can be selectively deprotected by the application of a voltage bias from a conducting AFM tip to afford localized nanoscale patterns that can be visualized by self-assembly of dendritic molecular objects with terminal carboxylic acid groups and different aspect ratios.     

Covalent formation of nanoscale fullerene and dendrimer patterns

Localized patterns of amine-terminated monolayers obtained via the surface modification of a monolayer with the biased probe of an atomic force microscope were used to covalently attach buckminsterfullerene or dendrimers to the surface, affording lines as narrow as 20 nm.     

How to Pack Trees

In a virtual memory system, the address space is partitioned into pages, and the main memory serves as a cache to the disk. In this setting, we address the following problem: Given a tree, find an allocation of its nodes to pages, so-called a packing, which optimizes the cache performance for some access pattern to the tree nodes. We investigate a model for tree access in which a node is accessed only via the path leading to it from the root. Two cost functions are considered: the total number of different pages visited in the search, and the number of page faults incurred. It is shown that both functions can be optimized simultaneously. An efficient dynamic programming algorithm to find an optimal packing is presented. The problem of finding an optimal packing which also uses the minimum number of pages is shown to be NP-complete. However, an efficient approximation algorithm is presented. This algorithm finds a packing that uses the minimum number of pages and requires at most one extra page fault per search. Finally, we study this problem in the context of dynamic trees which allow insertions and deletions.     

Optimum heat treatment conditions for masking the bitterness of the clarithromycin wax matrix

The effects of the contents of aminoalkyl methacrylate copolymer E (AMCE) in a wax matrix on the mechanism of polymorphic transformation of glyceryl monostearate (GM) were clarified by evaluating the enthalpy change defined as 1.51 (DeltaH(1)-DeltaH(2))/DeltaH(2), where DeltaH(1) and DeltaH(2) denote the enthalpies in the first and second thermal analyses, respectively. Using this value, K(1), the rate constant of transformation from alpha-form to beta'-form, and K(2), the rate constant of transformation from beta'-form to beta-form, could be obtained. As the ratio of AMCE increased, K(2) increased, but a minimum point existed for K(1). K(1) was always larger than K(2), but gradually approached K(2) as the ratio of AMCE increased. The optimum temperature for the transformation of GM was 50 degrees C, at which the enthalpy change was maximum. To prepare the wax matrix preparation of clarithromycin (CAM), we considered 40 degrees C the optimum treatment temperature for the transformation of GM in a CAM wax matrix compounded from CAM, GM and AMCE, since the matrices were mutually welded at above 45 degrees C during the spray congealing process. Although K(1) and K(2) were almost the same at 40 degrees C, the rate of transformation was accelerated by tumbling. By applying the tumbling that accelerated the transformation of GM in a CAM wax matrix, almost all of the alpha-form disappeared, and the release of CAM from the wax matrix diminished when the enthalpy change was more than 0.8.