We extend the flattening transformation, which turns nested into flat data parallelism, to the full higher-order case, including lambda abstractions and data parallel arrays of functions. Our central observa-ions and data parallel arrays of functions. Our central observation is that flattening needs to transform the closures used to represent functional values. Thus, we use closure conversion b...

This article presents an introduction to the generalization of the crossed rule approach to the detection of KnuthhBendix completion procedure divergence. It introduces the closure chains, which are special rule closures constructed by means of particular substitution operations and operators, as a suitable formalism for a progress in this direction. Supporting substitution algebra is developed...

We continue the study of the closures of GL(V )-orbits in the enhanced nilpotent cone V × N begun by the first two authors. We prove that each closure is an invariant-theoretic quotient of a suitably-defined enhanced quiver variety. We conjecture, and prove in special cases, that these enhanced quiver varieties are normal complete intersections, implying that the enhanced nilpotent orbit closur...

We study the distribution of S-integral points on SL2-orbit closures of binary forms and prove an asymptotic formula for the number of S-integral points. This extends a result of Duke, Rudnick, and Sarnak. The main ingredients of the proof are the method of mixing developed by Eskin-McMullen and Benoist-Oh, Chambert-Loir-Tschinkel’s study of asymptotic volume of height balls, and Hassett-Tschin...