The Elekes–Szabó Theorem in four dimensions

Let F ∈ C[x, y, s, t] be an irreducible constant-degree polynomial, and let A,B,C,D ? C be finite sets of size n. We show that F vanishes on at most O(n^8/3) points of the Cartesian product A × B × C × D, unless F has a special group-related form. A similar statement holds for A,B,C,D of unequal sizes, with a suitably modified bound on the number of zeros. This is a four-dimensional extension of our recent improved analysis of the original Elekes–Szabó theorem in three dimensions. We give three applications: an expansion bound for three-variable real polynomials that do not have a special form, a bound on the number of coplanar quadruples on a space curve that is neither planar nor quartic, and a bound on the number of four-point circles on a plane curve that has degree at least five.     

The Toda system and multiple-end solutions of autonomous planar elliptic problems

We prove the existence of a new class of entire, positive solutions for the classical elliptic problem Δu−u+up=0 in R², when p>2. The solutions we construct are obtained by perturbing the function

     

Hitting Half-spaces by Bessel-Brownian Diffusions

The purpose of the paper is to find explicit formulas describing the joint distributions of the first hitting time and place for half-spaces of codimension one for a diffusion in ℝ ^n + 1, composed of one-dimensional Bessel process and independent n-dimensional Brownian motion. The most important argument is carried out for the two-dimensional situation. We show that this amounts to computation of distributions of various integral functionals with respect to a two-dimensional process with independent Bessel components. As a result, we provide a formula for the Poisson kernel of a half-space or of a strip for the operator (I − Δ) ^α/2, 0 < α < 2. In the case of a half-space, this result was recently found, by different methods, in Byczkowski et al. (Trans Am Math Soc 361:4871–4900, 2009). As an application of our method we also compute various formulas for first hitting places for the isotropic stable Lévy process.     

The 2-center problem in three dimensions

Let P be a set of n points in ${\mathbb{R}}^3$. The 2-center problem for P is to find two congruent balls of minimum radius whose union covers P. We present a randomized algorithm for computing a 2-center of P that runs in $O(\beta (r^{?})n^2{\log }^{4}n\log \log n)$ expected time; here $\beta (r)=1/{(1?r/r_0)}^3$, $r^{?}$ is the radius of the 2-center balls of P, and $r_0$ is the radius of the smallest enclosing ball of P. The algorithm is near quadratic as long as $r^{?}$ is not too close to $r_0$, which is equivalent to the condition that the centers of the two covering balls be not too close to each other. This improves an earlier slightly super-cubic algorithm of Agarwal, Efrat, and Sharir (2000) [2] (at the cost of making the algorithm performance depend on the center separation of the covering balls).     

Helical twisting power of antiferroelectric chiral terphenyls in different matrices

To determine the influence of chemical structure on the Helical Twisting Power (HTP), we tested four optically active dopants having a terphenyl rigid core and the same chiral centre but differing in the length of nonchiral terminal chain and the substitution of benzene rings with fluorine atoms. The compounds were added to different achiral liquid crystalline matrices: nematic and smectic C. It was found that HTP as well as its temperature variation depends on the kind of used matrices. It gives a conclusion that information about HTP obtained in one matrix cannot be uncritically transferred to another one.