General Statistical Design of an Experimental Problem for Harmonics

Four years ago, the Michelin Tire Corporation proposed a problem on experimental design, to improve the manufacturing process for their tires. The idea is basically to determine the effects of placements for various layers built up in the construction of a tire, to allow the design of a smooth tire with a smooth ride. A highly success solution was developed, and it has been reported that this method introduced savings of over half a million dollars in their test processes. This year, Michelin returned to the workshop with an extension to the original problem, to address specific refinements in the testing method. This report summarizes the work completed in course of the five day workshop. It was clear early in the workshop that this problem could be handled quickly by reviewing the analysis which was done in 2000, and extending those ideas to the new problems at hand. We reviewed the required Fourier techniques to describe the harmonic problem, and statistical techniques to deal with the linear model that described how to accurately measure quantities that come from real experimental measurements. The “prime method” and “good lattice points method” were reviewed and re-analysed so we could understand (and prove) why they work so well. We then looked at extending these methods and successfully found solutions to problem 1) and 2) posed by Michelin. Matlab code was written to test and verify the algorithms developed. We have some ideas on problems 3) and 4), which are also described

On limit points of the sequence of normalized prime gaps

Let $p_n$ denote the $n$th smallest prime number, and let $\boldsymbol{L}$ denote the set of limit points of the sequence $\{(p_{n+1} - p_n)/\log p_n\}_{n = 1}^{\infty}$ of normalized differences between consecutive primes. We show that for $k = 9$ and for any sequence of $k$ nonnegative real numbers $\beta_1 \le \beta_2 \le ... \le \beta_k$, at least one of the numbers $\beta_j - \beta_i$ ($1 \le i < j \le k$) belongs to $\boldsymbol{L}$. It follows at least $12.5$ of all nonnegative real numbers belong to $\boldsymbol{L}$.Comment: Revised and improve