A systematic exposition of Sheffer polynomial sequences via determinantal form is given. A general linear interpolation problem related to Sheffer sequences is considered. It generalizes many known cases of linear interpolation. Numerical examples and conclusions close the paper. 1. The modern umbral calculus In the 1970s Rota and his collaborators [17,19,20] began to construct a completely rig...

The purpose of this note is to investigate the close relationship that exists between the Euler-Seidel matrix [3, 4, 5, 6, 10] of an integer sequence, and the Hankel matrix [9] of that sequence. We do so in the context of sequences that have integral moment representations, though many of the results are valid in a more general context. While partly expository in nature, the note assumes a cert...

Abstract Bollobás and Riordan, in their paper ‘Metrics for sparse graphs’, proposed a number of provocative conjectures extending central results quasirandom graphs graph limits to graphs. We refute these by exhibiting sequence with convergent normalized subgraph densities (and pseudorandom C 4 -counts), but no limit expressible as kernel.

Recently, Zhu [34] introduced a Stirling-Whitney-Riordan triangle [Tn,k]n,k?0 satisfying the recurrence Tn,k = (b1k + b2)Tn?1,k?1 [(2?b1 a1)k a2 ?(b1 b2)]Tn?1,k ?(a1 ?b1)(k 1)Tn?1,k+1, where initial conditions 0 unless ? k n and T0,0 1. Denote by Tn Pnk =0 Tn,k. In this paper, we show asymptotic normality of give an formula Tn. As applications, many famous combinatorial numbers, such as Stirlin...

In this paper we prove the following conjecture by Bollobás and Komlós: For every γ > 0 and positive integers r and ∆, there exists β > 0 with the following property. If G is a sufficiently large graph with n vertices and minimum degree at least ( r−1 r +γ)n and H is an r-chromatic graph with n vertices, bandwidth at most βn and maximum degree at most ∆, then G contains a copy of H.

We outline the basic theory of Belyi surfaces, up to Belyi’s theorem (1979, [1]), which characterizes these spaces as precisely those Riemann surfaces that are defined over Q. We then detail the Brooks-Makover construction of a Belyi surface from an oriented cubic graph from [3]. Consequently, we can apply the model of Bollobás in order to randomly pick a Belyi surface. Finally, we briefly expl...