Let N(t) denote the number of ways of writing t as a binomial coefficient. We show that N(t) = O (log t)(log log log t) (log log t)3 .

The so-called ζ-analogues of the Stirling numbers of the first and second kind are considered. These numbers cover ordinary binomial and Gaussian coefficients, p, qStirling numbers and other combinatorial numbers studied with the help of object selection, Ferrers diagrams and rook theory. Our generalization includes these and now also the p, q-binomial coefficients. This special subfamily of F ...

February 15, 2013 Type Package Title Multiple comparisons using normal approximation Version 1.1-14 Date 2012-01-11 Author Frank Schaarschmidt, Daniel Gerhard, Martin Sill Maintainer Frank Schaarschmidt Depends R (>= 2.6.0), mvtnorm, multcomp Description Multiple contrast tests and simultaneous confidence intervals based on normal approximation. With impl...

A binomial-type operator on a stationary Gaussian process is introduced in order to model long memory in the spatial context. Consistent estimators of model parameters are demonstrated. In particular , it is shown thatˆdN − d = OP ((Log N) 3 N), where d = (d1, d2) denotes the long memory parameter.

The security dynamics described by the Black-Scholes equation with price-dependent variance can be approximated as a damped discrete-time hopping process on a recombining binomial tree. In a previous working paper, such a nonuniform tree was explicitly constructed in terms of the continuous-time variance. The present note outlines how the previous procedure could be extended to multifactor Blac...

The binomial coefficient of two words u and v is the number of times v occurs as a subsequence of u. Based on this classical notion, we introduce the mbinomial equivalence of two words refining the abelian equivalence. Two words x and y are m-binomially equivalent, if, for all words v of length at most m, the binomial coefficients of x and v and respectively, y and v are equal. The m-binomial c...

Characterizing the information carried by neural populations in the brain requires accurate statistical models of neural spike responses. The negative-binomial distribution provides a convenient model for over-dispersed spike counts, that is, responses with greater-than-Poisson variability. Here we describe a powerful data-augmentation framework for fully Bayesian inference in neural models wit...

Given two words u and v, the binomial coefficient ( u v ) is the number of ways v appears as a subword (or subsequence) of u. The Thue-Morse sequence is the infinite word t = abbabaab · · · obtained by iteration of the morphism τ (a) = ab and τ (b) = ba. We show that, for every prime p, and every positive integer n, there exists an integer m = f(p, n), such that, for every non-empty word v of l...

1. INTRODUCTION. There is no field with only one element, yet there is a well-defined notion of what projective geometry over such a field means. This notion is familiar to experts and plays an interesting role behind the scenes in combinatorics and algebra, but it is rarely discussed as such. The purpose of this article is to bring it to the attention of a broader audience, as the solution to ...