Pak-Leong Cheung | Searcharxiv

arXiv2014

Higher Order, Polar and Sz.-Nagy's Generalized Derivatives of Random Polynomials with Independent and Identically Distributed Zeros on the Unit Circle

For random polynomials with i.i.d. (independent and identically distribu-ted) zeros following any common probability distribution $μ$ with support contained in the unit circle, the empirical measures of the zeros of their first and higher order derivatives will be proved to converge weakly to $μ$ a.s. (almost sure(ly)). This, in particular, completes a recent work of Subramanian on the first order derivative case where $μ$ was assumed to be non-uniform. The same a.s. weak convergence will also be shown for polar and Sz.-Nagy's generalized derivatives, on some mild conditions.

math.CV↗

arXiv2014

Finiteness of Fixed Equilibrium Configurations of Point Vortices in the Plane with Background Flow

For a dynamic system consisting of $n$ point vortices in an ideal plane fluid with a steady, incompressible and} irrotational background flow, a more physically significant definition of a fixed equilibrium configuration is suggested. Under this new definition, if the complex polynomial $w$ that determines the aforesaid background flow is non-constant, we have found an attainable generic upper bound \smash{$\fr{(m+n-1)!}{(m-1)!\,n_1!\cdots n_{i_0}!}$} for the number of fixed equilibrium configurations. Here, $m=°w$, $i_0$ is the number of species, and each $n_i$ is the number of vortices in a species. We transform the rational function system arisen from} fixed equilibria into a polynomial system, whose form is good enough to apply the BKK theory (named after D. N. Bernshtein, A. G. Khovanskii and A. G. Kushnirenko) to show the finiteness of its number of solutions. Having this finiteness, the required bound follows from Bézout's theorem or the BKK root count by T. Y. Li and X.-S. Wang.}

math.CV↗