P\'olya number of continuous-time quantum walks

We propose a definition for the P\'olya number of continuous-time quantum walks to characterize their recurrence properties. The definition involves a series of measurements on the system, each carried out on a different member from an ensemble in order to minimize the disturbance caused by it. We examine various graphs, including the ring, the line, higher dimensional integer lattices and a number of other graphs and calculate their P\'olya number. For the timing of the measurements a Poisson process as well as regular timing are discussed. We find that the speed of decay for the probability at the origin is the key for recurrence.Comment: 8 pages, no figures. Accepted for publication in Physical Review

Diffusion Limited Aggregation on a Cylinder

We consider the DLA process on a cylinder G x N. It is shown that this process "grows arms", provided that the base graph G has small enough mixing time. Specifically, if the mixing time of G is at most (log|G|)^(2-\eps), the time it takes the cluster to reach the m-th layer of the cylinder is at most of order m |G|/loglog|G|. In particular we get examples of infinite Cayley graphs of degree 5, for which the DLA cluster on these graphs has arbitrarily small density. In addition, we provide an upper bound on the rate at which the "arms" grow. This bound is valid for a large class of base graphs G, including discrete tori of dimension at least 3. It is also shown that for any base graph G, the density of the DLA process on a G-cylinder is related to the rate at which the arms of the cluster grow. This implies, that for any vertex transitive G, the density of DLA on a G-cylinder is bounded by 2/3.Comment: 1 figur

Turán type oscillation inequalities in Lq norm on the boundary of convex domains

Some 77 years ago P. Turan was the first to establish lower estimations of the ratio of the maximum norm of the derivatives of polynomials and the maximum norm of the polynomials themselves on the interval I := [-1,1] and on the unit disk D := {z. C : vertical bar z vertical bar similar to 1} under the normalization condition that the zeroes of the polynomial p all lie in the interval or in the disk, respectively. He proved that with n := deg p tending to infinity, the precise growth order of the minimal possible ratio of the derivative norm and the norm is v n for I and n for D. J. Erod continued the work of Turan and extended his results to several other domains. The growth of the minimal possible ratio of the root-norm of the derivative and the polynomial itself was proved to be of order n for all compact convex domains a decade ago. Although Tur'an himself gave comments about the above oscillation question in L-q norms, till recently results were known only for D and I. Here we prove that in L-q norm the oscillation order is again n for a certain class of convex domains, including all smooth convex domains and also convex polygonal domains having no acute angles at their vertices

Electrospun nanofiber-based niflumic acid capsules with superior physicochemical properties

The aim of this study was to assess whether nanofibrous drug mats have potential as delivery systems for poorly water-soluble drugs. Amorphous nanofiber mats from a model poorly water-soluble active pharmaceutical ingredient (API), niflumic acid, together with the polymer excipient, polyvinyl pyrrolidine, were prepared by nozzle-free electrospinning. This technique offers a scalable way for drug formulation, and by increasing the surface area of the drug, the dissolution rate and therefore bioavailability of the API can be improved. In this study, both the amount of the dissolved active ingredient and the dissolution kinetics has been improved significantly when the nanofibrous mats were used in the drug formulation. A 15-fold increase in the dissolved amount of the produced amorphous niflumic acid nanofiber was observed compared to the dissolved amount of the raw drug within the first 15 minutes. Capsule formulation was made by mixing the electrospun nanofibers with a microcrystalline cellulose filler agent. When comparing the dissolution rate of the capsule formulation on the market with the nanofibrous capsules, a 14-fold increase was observed in the dissolved drug amount within the first 15 minutes

Stochastic integration based on simple, symmetric random walks

A new approach to stochastic integration is described, which is based on an a.s. pathwise approximation of the integrator by simple, symmetric random walks. Hopefully, this method is didactically more advantageous, more transparent, and technically less demanding than other existing ones. In a large part of the theory one has a.s. uniform convergence on compacts. In particular, it gives a.s. convergence for the stochastic integral of a finite variation function of the integrator, which is not c\`adl\`ag in general.Comment: 16 pages, some typos correcte

Turan type converse Markov inequalities in L-q on a generalized Erod class of convex domains

P.Turan was the first to derive lower estimations on the uniform norm of the derivatives of polynomials p of uniform norm 1 on the disk D := {z is an element of C : vertical bar z vertical bar <= 1} and the interval I := [-1, 1], under the normalization condition that the zeros of the polynomial p in question all lie in D or I, resp. Namely, in 1939 he proved that with n := deg p tending to infinity, the precise growth order of the minimal possible derivative norm is n for D and root n for II. Already the same year J.Erod considered the problem on other domains. In his most general formulation, he extended Turan's order n result on 101 to a certain general class of piecewise smooth convex domains. Finally, a decade ago the growth order of the minimal possible norm of the derivative was proved to be n for all compact convex domains. Turk himself gave comments about the above oscillation question in L-q norm on D. Nevertheless, till recently results were known only for 11),)1 and so-called R-circular domains. Continuing our recent work, also here we investigate the Turan-Erod problem on general classes of domains. (C) 2017 Elsevier Inc. All rights reserved

Self-intersection local time of planar Brownian motion based on a strong approximation by random walks

The main purpose of this work is to define planar self-intersection local time by an alternative approach which is based on an almost sure pathwise approximation of planar Brownian motion by simple, symmetric random walks. As a result, Brownian self-intersection local time is obtained as an almost sure limit of local averages of simple random walk self-intersection local times. An important tool is a discrete version of the Tanaka--Rosen--Yor formula; the continuous version of the formula is obtained as an almost sure limit of the discrete version. The author hopes that this approach to self-intersection local time is more transparent and elementary than other existing ones.Comment: 36 pages. A new part on renormalized self-intersection local time has been added and several inaccuracies have been corrected. To appear in Journal of Theoretical Probabilit