Latent class analysis variable selection

We propose a method for selecting variables in latent class analysis, which is the most common model-based clustering method for discrete data. The method assesses a variable's usefulness for clustering by comparing two models, given the clustering variables already selected. In one model the variable contributes information about cluster allocation beyond that contained in the already selected variables, and in the other model it does not. A headlong search algorithm is used to explore the model space and select clustering variables. In simulated datasets we found that the method selected the correct clustering variables, and also led to improvements in classification performance and in accuracy of the choice of the number of classes. In two real datasets, our method discovered the same group structure with fewer variables. In a dataset from the International HapMap Project consisting of 639 single nucleotide polymorphisms (SNPs) from 210 members of different groups, our method discovered the same group structure with a much smaller number of SNP

Levels of Distribution and the Affine Sieve

This article is an expanded version of the author's lecture in the Basic Notions Seminar at Harvard, September 2013. Our goal is a brief and introductory exposition of aspects of two topics in sieve theory which have received attention recently: (1) the spectacular work of Yitang Zhang, under the title "Level of Distribution," and (2) the so-called "Affine Sieve," introduced by Bourgain-Gamburd-Sarnak.Comment: 34 pages, 2 figure

Bear River Mutual Insurance Company v. David and Deanna Williams : Brief of Appellee

APPEAL FROM THE ORDER GRANTING PLAINTIFF BEAR RIVER MUTUAL INSURANCE COMPANY\u27S MOTION FOR SUMMARY JUDGMENT OF THE THIRD DISTRICT COURT IN AND FOR SALT LAKE COUNTY, STATE OF UTAH. SALT LAKE DEPARTMENT, THE HONORABLE JUDGE LEON A. DEVE

Improved bounds for the two-point logarithmic Chowla conjecture

Let $\lambda$ be the Liouville function, defined as $\lambda(n) := (-1)^{\Omega(n)}$ where $\Omega(n)$ is the number of prime factors of $n$ with multiplicity. In 2021, Helfgott and Radziwi{\l}{\l} proved that $$\sum_{n\leq x} \frac{1}{n} \lambda(n) \lambda(n+1) \ll \frac{\log x}{(\log \log x)^{1/2}},$$improving earlier results by Tao and Ter\"av\"ainen. We prove that $$\sum_{n\leq x} \frac{1}{n} \lambda(n) \lambda(n+1) \ll (\log x)^{1-c}$$for some absolute constant $c>0$. This appears to be best possible with current methods.Comment: 75 page