 # CURRENT ARTICLES: VOLUME 23

## Mixing Numbers and Unfriendly Colorings of Graphs » NB PDF

Published September 28, 2021

Mathematica has many built-in functions for doing research in graph theory. Formerly it was necessary to load the Combinatorica package to access these functions; most are now available within Mathematica itself. This article studies a problem concerning the vertex coloring of graphs using Mathematica by introducing some user-defined functions. Read More »

## Numerical Contour Integration » NB PDF

### Applied to One- and Two-Dimensional Integrals over Distributions

Published July 16, 2021

We present a straightforward implementation of contour integration by setting options for and , taking advantage of powerful results in complex analysis. As such, this article can be viewed as documentation to perform numerical contour integration with the existing built-in tools. We provide examples of how this method can be used when integrating analytically and numerically some commonly used distributions, such as Wightman functions in quantum field theory. We also provide an approximating technique when time-ordering is involved, a commonly encountered scenario in quantum field theory for computing second-order terms in Dyson series expansion and Feynman propagators. We believe our implementation will be useful for more general calculations involving advanced or retarded Greens functions, propagators, kernels and so on. Read More »

## Unconditional Applicability of Lehmer’s Measure to the Two-Term Machin-like Formula for π » NB PDF

Published May 24, 2021

Lehmer defined a measure where the may be either integers or rational numbers in a Machin-like formula for . When the are integers, Lehmers measure can be used to determine the computational efficiency of the given Machin-like formula for . However, because the computations are complicated, it is unclear if Lehmers measure applies when one or more of the are rational. In this article, we develop a new algorithm for a two-term Machin-like formula for as an example of the unconditional applicability of Lehmers measure. This approach does not involve any irrational numbers and may allow calculating rapidly by the NewtonRaphson iteration method for the tangent function. Read More »

## Coverage versus Confidence » NB PDF

### A Tutorial

Published March 3, 2021
This article is intended to help students understand the concept of a coverage probability involving confidence intervals. Mathematica is used as a language for describing an algorithm to compute the coverage probability for a simple confidence interval based on the binomial distribution. Then, higher-level functions are used to compute probabilities of expressions in order to obtain coverage probabilities. Several examples are presented: two confidence intervals for a population proportion based on the binomial distribution, an asymptotic confidence interval for the mean of the Poisson distribution, and an asymptotic confidence interval for a population proportion based on the negative binomial distribution. Read More »