We establish asymptotic formulas for the number of integral points of bounded height on partial equivariant compactifications of vector groups.

We consider the 1-dimensional semilinear wave equation with power nonlinearity. We consider an arbitrary blowup solution $u(x,t)$, the graph $x\mapsto T\left(x\right)$ of its blowup points, and $\mathcal{S}\subset \mathbb{R}$ the set of all characteristic points. We show that $\mathcal{S}$ is locally finite.

Let $M^3$ be a closed Cauchy–Riemann (CR) 3-manifold. In this article, we derive a Bochner formula for the Kohn Laplacian in which the pseudo-Hermitian torsion does not play any role. By means of this formula we show that the nonzero eigenvalues of the Kohn Laplacian have a positive lower bound, provided that the CR Paneitz operator is nonnegative and the Webster curvature is positive. This means that $M^3$ is embeddable when the CR Yamabe constant is positive and the CR Paneitz operator is nonnegative. Our lower bound estimate is sharp. In addition, we show that the embedding is stable in the sense of Burns and Epstein.

We fix a nonzero integer $a$ and consider arithmetic progressions $amodq$, with $q$ varying over a given range. We show that, for certain specific values of $a$, the arithmetic progressions $amodq$ contain, on average, significantly fewer primes than expected. We improve on results of Fouvry, Bombieri, Friedlander, Iwaniec, Granville, Hildebrandt, and Maier.

We give a geometric proof that any minimal length element in a (twisted) conjugacy class of a finite Coxeter group $W$ has remarkable properties with respect to conjugation, taking powers in the associated braid monoid and taking the centralizer in $W$.