Arithmetic of arithmetic Coxeter groups.

In the 1990s, J. H. Conway published a combinatorial-geometric method for analyzing integer-valued binary quadratic forms (BQFs). Using a visualization he named the "topograph," Conway revisited the reduction of BQFs and the solution of quadratic Diophantine equations such as Pell's equation. It appears that the crux of his method is the coincidence between the arithmetic group [Formula: see text] and the Coxeter group of type [Formula: see text] There are many arithmetic Coxeter groups, and each may have unforeseen applications to arithmetic. We introduce Conway's topograph and generalizations to other arithmetic Coxeter groups. This includes a study of "arithmetic flags" and variants of binary quadratic forms.

Clonally heritable gene expression imparts a layer of diversity within cell types.

Cell types can be classified according to shared patterns of transcription. Non-genetic variability among individual cells of the same type has been ascribed to stochastic transcriptional bursting and transient cell states. Using high-coverage single-cell RNA profiling, we asked whether long-term, heritable differences in gene expression can impart diversity within cells of the same type. Studying clonal human lymphocytes and mouse brain cells, we uncovered a vast diversity of heritable gene expression patterns among different clones of cells of the same type in vivo. We combined chromatin accessibility and RNA profiling on different lymphocyte clones to reveal thousands of regulatory regions exhibiting interclonal variation, which could be directly linked to interclonal variation in gene expression. Our findings identify a source of cellular diversity, which may have important implications for how cellular populations are shaped by selective processes in development, aging, and disease. A record of this paper's transparent peer review process is included in the supplemental information.