Binomial coefficients arise naturally in combinatorics. Recently the speaker initiated the study of certain divisibility properties of binomial coefficients, and products or sums of binomial coefficients. In this talk we introduce the speaker’s related results and various conjectures. The materials come from the author’s two preprints: 1. Z. W. Sun, Products and sums divisible by central binomi...

Some properties of diagonal binomial coefficients were studied in respect to frequency of their units digits. An approach was formulated that led to use of difference tables to predict if certain units digits can appear in the values of binomial coefficients at quadratic terms of the binomial theorem. Frequency distributions of units digits of binomial coefficients contain gaps (zero frequency)...

BACKGROUND The negative binomial distribution is used commonly throughout biology as a model for overdispersed count data, with attention focused on the negative binomial dispersion parameter, k. A substantial literature exists on the estimation of k, but most attention has focused on datasets that are not highly overdispersed (i.e., those with k>or=1), and the accuracy of confidence intervals ...

Modeling a nonlinear pay o¤ generating instrument is a challenging work. The models that are commonly used for pricing derivative might divided into two main classes; analytical and iterative models. This paper compares the Black-Scholes and binomial tree models.

Empirical evidence from developed countries consistently shows that working wives remain responsible for most household duties. The aim of this paper is to study the unequal distribution of housework between working spouses in Spain. Housework time allocation is modelled through a bivariate negative binomial distribution, conditional on a set of observable characteristics, such as paid labor co...

The scattering number of a graph G, denoted sc(G), is defined by sc(G)=max{c(G−S)− |S| : S ⊆ V(G) and c(G−S)≠ 1} where c(G−S) denotes the number of components in G− S. It is one measure of graph vulnerability. In this paper, general results on the scattering number of a graph are considered. Firstly, some bounds on the scattering number are given. Further, scattering number of a binomial tree i...

This note contains two main results. (1) (Discrete time) Suppose S is a martingale whose marginal laws agree with a geometric simple random walk. (In financial terms, let S be a risk-neutral asset price and suppose the initial option prices agree with the Cox–Ross–Rubinstein binomial tree model.) Then S is a geometric simple random walk. (2) (Continuous time) Suppose S = S0eσX−σ 〈X〉/2 is a cont...

A wide range of graphs with regular structures are shown to be embeddable in an injured hypercube with faulty links. These include rings, linear paths, binomial trees, binary trees, meshes, tori, and many others. Unlike many existing algorithms which are capable of embedding only one type of graphs, our algorithm embeds the above graphs in a uniied way, all centered around a notion called edge ...

The binomial tree method, first proposed by Cox, Ross, and Rubinstein [Journal of Financial Economics, 7 (1979), pp. 229–263], is one of the most popular approaches to pricing options. By introducing an additional path-dependent variable, such methods can be readily extended to the valuation of path-dependent options. In this paper, using numerical analysis and the notion of viscosity solutions...

The pricing models of American-type path-dependent options are of degenerate parabolic obstacle problems. The binomial tree method is the most popular approach to pricing options. For some special cases, this method is modified in order to make it feasible. The main purpose of this paper is, using numerical analysis and the notion of viscosity solutions, to show the uniform convergence of the b...