Bivariate Gonarov polynomials and integer sequences

Univariate Gonarov polynomials arose from the Gonarov interpolation problem in numerical analysis.They provide a natural basis of polynomials for working with u-parking functions,which are integer sequences whose order statistics are bounded by a given sequence u.In this paper,we study multivariate Gonarov polynomials,which form a basis of solutions for multivariate Gonarov interpolation problem.We present algebraic and analytic properties of multivariate Gonarov polynomials and establish a combinatorial relation with integer sequences.Explicitly,we prove that multivariate Gonarov polynomials enumerate k-tuples of integers sequences whose order statistics are bounded by certain weights along lattice paths in Nk.It leads to a higher-dimensional generalization of parking functions,for which many enumerative results can be derived from the theory of multivariate Gonarov polynomials.     

Asymptotic enumeration of sparse 0-1 matrices with irregular row and column sums

Let s=(s₁,…,s_m) and t=(t₁,…,t_n) be vectors of non-negative integer-valued functions with equal sum . Let N(s,t) be the number of m×n matrices with entries from {0,1} such that the ith row has row sum s_i and the jth column has column sum t_j. Equivalently, N(s,t) is the number of labelled bipartite graphs with degrees of the vertices in one side of the bipartition given by s and the degrees of the vertices in the other side given by t. We give an asymptotic formula for N(s,t) which holds when S→∞ with 1?st=o(S^2/3), where and . This extends a result of McKay and Wang [Linear Algebra Appl. 373 (2003) 273-288] for the semiregular case (when s_i=s for 1?i?m and t_j=t for 1?j?n). The previously strongest result for the non-semiregular case required 1?max{s,t}=o(S^1/4), due to McKay [Enumeration and Design, Academic Press, Canada, 1984, pp. 225-238].     

Global G-Manifold Reductions and Resolutions

The purpose of this note is to exhibit some simple and basic constructions for smooth compact transformation groups, and some of their most immediate applications to geometry.     

Let’s deflate that beach ball

The European Physical Journal E - This report presents the results of high-pressure and broadband dielectric spectroscopy studies in polyvinylidene difluoride (PVDF) and barium strontium titanate...     

Silica-supported l-proline organocatalysts for asymmetric aldolisation

New heterogenised silica-based organocatalysts have been prepared via the sol–gel process from two silylated derivatives of l-proline, featuring either a carbamate or an ether linker. Co-gelification with variable amounts of TEOS was performed with and without porogen to yield high surface area solids. These materials were evaluated as heterogeneous phase organocatalysts for the asymmetric aldol reaction of p-nitrobenzaldehyde with acetone.