Related papers: The heat flow conjecture for polynomials and rando…

We consider the passage of long polymers of length N through a hole in a membrane. If the process is slow, it is in principle possible to focus on the dynamics of the number of monomers s on one side of the membrane, assuming that the two…

We consider a system of two competing populations in two-dimensional heterogeneous environments. The populations are assumed to move horizontally and vertically with different probabilities, but are otherwise identical. We regard these…

Let $(G,\rho)$ be a stationary random graph, and use $B^G_{\rho}(r)$ to denote the ball of radius $r$ about $\rho$ in $G$. Suppose that $(G,\rho)$ has annealed polynomial growth, in the sense that $\mathbb{E}[|B^G_{\rho}(r)|] \leq O(r^k)$…

For univariate polynomials over arbitrary field the degree gives an upper bound on the number of roots (factor theorem) and as a related result for any finite point-set one can construct a polynomial of degree equal to the cardinality…

It is a common belief that in any environment where life is possible, life will be generated. Here it is suggested that the cause for a spontaneous generation of complex systems is probability driven processes. Based on equilibrium…

The aim of this paper is to investigate the distribution of a continuous homopolymer in the presence of an attractive finitely supported potential. The most intricate behavior can be observed if we simultaneously vary two parameters: the…

We introduce and study a conformal heat flow of harmonic maps defined by an evolution equation for a pair consisting of a map and a conformal factor of metric on the two-dimensional domain. This flow is designed to postpone finite time…

In this paper, we introduce the regularized conformal heat flow of $n$-harmonic maps, or simply regularized $n$-conformal heat flow from $n$-dimensional Riemannian manifold. This is a system of evolution equations combined with regularized…

On the long nuclear time scale of stellar main-sequence evolution, even weak mixing processes can become relevant for redistributing chemical species in a star. We investigate a process of "differential heating," which occurs when a…

We formulate conditions on a set of log-concave sequences, under which any linear combination of those sequences is log-concave, and further, of conditions under which linear combinations of log-concave sequences that have been transformed…

We discuss several conjectures about the real-rootedness of polynomials whose coefficients are determinants of coefficients of a real-rooted polynomial. We also consider some questions about matrices generalizing totally positive matrices,…

For one-dimensional growth processes we consider the distribution of the height above a given point of the substrate and study its scale invariance in the limit of large times. We argue that for self-similar growth from a single seed the…

In this paper, we study the evolution of a vortex filament in an incompressible ideal fluid. Under the assumption that the vorticity is concentrated along a smooth curve in $\mathbb{R}^3$, we prove that the curve evolves to leading order by…

Let $f$ be sampled uniformly at random from the set of degree $n$ polynomials whose coefficients lie in $\{ \pm 1\}$. A folklore conjecture, known to hold under GRH, states that the probability that $f$ is irreducible tends to $1$ as $n$…

We give a conditional proof of the Uniform Boundedness Conjecture of Morton and Silverman in the case of polynomials over number fields, assuming a standard conjecture in arithmetic geometry. Our technique simultaneously yields a dynamical…

We develop the complex-analytic viewpoint on the tree convolutions studied by the second author and Weihua Liu in "An operad of non-commutative independences defined by trees" (Dissertationes Mathematicae, 2020, doi:10.4064/dm797-6-2020),…

We study asymptotic clustering of zeros of random polynomials, and show that the expected discrepancy of roots of a polynomial of degree $n$, with not necessarily independent coefficients, decays like $\sqrt{\log n/n}$. Our proofs rely on…

If $p:\mathbb{C} \to \mathbb{C}$ is a non-constant polynomial, the Gauss--Lucas theorem asserts that its critical points are contained in the convex hull of its roots. We consider the case when $p$ is a random polynomial of degree $n$ with…

We discuss a relativistic diffusion in the proper time in an approach of Schay and Dudley. We derive (Langevin) stochastic differential equations in various coordinates.We show that in some coordinates the stochastic differential equations…

For a unimodular random graph $(G,\rho)$, we consider deformations of its intrinsic path metric by a (random) weighting of its vertices. This leads to the notion of the conformal growth exponent of $(G,\rho)$, which is the best asymptotic…