A note about the average number of real roots of a Bernstein polynomial system

We prove that the real roots of normal random homogeneous polynomial systems with n+1$n+1$ variables and given degrees are, in some sense, equidistributed in the projective space P(Rⁿ⁺¹)$\mathbb{P}\left({\mathbb{R}}^{n+1}\right)$. From this fact we compute the average number of real roots of normal random polynomial systems given in the Bernstein basis.     

Bisections of graphs

A bisection of a graph is a bipartition of its vertex set in which the number of vertices in the two parts differ by at most 1, and its size is the number of edges which go across the two parts. In this paper, motivated by several questions and conjectures of Bollobás and Scott, we study maximum bisections of graphs. First, we extend the classical Edwards bound on maximum cuts to bisections. A simple corollary of our result implies that every graph on n vertices and m   edges with no isolated vertices, and maximum degree at most n/3+1$n/3+1$, admits a bisection of size at least m/2+n/6$m/2+n/6$. Then using the tools that we developed to extend Edwards?s bound, we prove a judicious bisection result which states that graphs with large minimum degree have a bisection in which both parts span relatively few edges. A special case of this general theorem answers a conjecture of Bollobás and Scott, and shows that every graph on n vertices and m   edges of minimum degree at least 2 admits a bisection in which the number of edges in each part is at most (1/3+o(1))m$(1/3+o(1))m$. We also present several other results on bisections of graphs.     

Kernel method and linear recurrence system

Based on the kernel method, we present systematic methods to solve equation systems on generating functions of two variables. Using these methods, we get the generating functions for the number of permutations which avoid 1234 and 12k(k-1)…3$12k(k-1)\dots 3$ and permutations which avoid 1243 and 12…k$12\dots k$.     

Perturbations of May–Leonard system

In this paper we study the quadratic homogeneous perturbations of the 3-dimensional May–Leonard system with α+β=2$\alpha +\beta =2$. It is shown that there are perturbed systems having exactly one or two limit cycles bifurcated from the periodic orbits of May–Leonard system. This is proved by estimating the number of zeros of the first and the second order Melnikov functions.