Sylvester's question and the Random Acceleration Process

Let n points be chosen randomly and independently in the unit disk. "Sylvester's question" concerns the probability p_n that they are the vertices of a convex n-sided polygon. Here we establish the link with another problem. We show that for large n this polygon, when suitably parametrized by a function r(phi) of the polar angle phi, satisfies the equation of the random acceleration process (RAP), d^2 r/d phi^2 = f(phi), where f is Gaussian noise. On the basis of this relation we derive the asymptotic expansion log p_n = -2n log n + n log(2 pi^2 e^2) - c_0 n^{1/5} + ..., of which the first two terms agree with a rigorous result due to Barany. The nonanalyticity in n of the third term is a new result. The value 1/5 of the exponent follows from recent work on the RAP due to Gyorgyi et al. [Phys. Rev. E 75, 021123 (2007)]. We show that the n-sided polygon is effectively contained in an annulus of width \sim n^{-4/5} along the edge of the disk. The distance delta_n of closest approach to the edge is exponentially distributed with average 1/(2n).Comment: 29 pages, 4 figures; references added and minor change

Seed vigor, aging, and osmopriming affect anion and sugar leakage during imbition of maize (Zea mays L.) caryopses

Conductivity was significantly increased by aging and decreased by osmopriming of maize (Zea mays L.) caryopses. Chloride, phosphate, and sulfate were the main anions that leaked out of maize seeds; their leakage was closely related to conductivity, increased by aging, and decreased by osmopriming. The anion leakage of isolated embryos correlated closely to seed vigor and was more sensitive to aging and priming than that of the whole seed. Anion leakage may be a more sensitive measure for seed vigor than bulk conductivity readings. Aging did not increase the sugar leakage of whole seeds but significantly increased the sugar leakage of isolated embryos. Sugar leakage was not closely related to total soluble sugar content of seeds. While priming decreased seed conductivity, the decreased anion and sugar leakage of the primed seeds was mainly caused by the washing effect during priming. The total anions or sugars left in the polyethylene glycol (PEG) solution after priming and in the conductivity solution of the primed seeds was almost the same as in the conductivity solution of the unprimed seeds alone

The perimeter of large planar Voronoi cells: a double-stranded random walk

Let $p\_n$ be the probability for a planar Poisson-Voronoi cell to have exactly $n$ sides. We construct the asymptotic expansion of $\log p\_n$ up to terms that vanish as $n\to\infty$. We show that {\it two independent biased random walks} executed by the polar angle determine the trajectory of the cell perimeter. We find the limit distribution of (i) the angle between two successive vertex vectors, and (ii) the one between two successive perimeter segments. We obtain the probability law for the perimeter's long wavelength deviations from circularity. We prove Lewis' law and show that it has coefficient 1/4.Comment: Slightly extended version; journal reference adde

Some Open Points in Nonextensive Statistical Mechanics

We present and discuss a list of some interesting points that are currently open in nonextensive statistical mechanics. Their analytical, numerical, experimental or observational advancement would naturally be very welcome.Comment: 30 pages including 6 figures. Invited paper to appear in the International Journal of Bifurcation and Chao

Preface to special issue (Fast reaction - slow diffusion scenarios: PDE approximations and free boundaries)

This issue is focussed on the modeling, analysis and simulation of fast reaction-slow transport scenarios as well as corresponding fast-reaction limits. Within this framework, internal sharp and thin reaction layers form and travel through the spatial domain often producing unexpected effects. Such situations appear in a variety of significant applications; for example flame propagation in combustion, segregation and aggregation of biological individuals, chemical attack on reactive porous materials (such as concrete or natural stone), dissolution and precipitation reactions in minerals, tumor growth, grain boundary motion, and temperature-induced phase transitions in shape-memory alloys represent typical cases in which the fast process is localized within a a priori unknown internal active layer

Equivalence of stationary state ensembles

We show that the contact process in an ensemble with conserved total particle number, as simulated recently by Tome and de Oliveira [Phys. Rev. Lett. 86 (2001) 5463], is equivalent to the ordinary contact process, in agreement with what the authors assumed and believed. Similar conserved ensembles and equivalence proofs are easily constructed for other models.Comment: 5 pages, no figure