We study the maximal displacement and related population for a branching Brownian motion in Euclidean space in terms of the principal eigenvalue of an associated Schrödinger type operator. We first determine their growth rates on the survival event. We then establish the upper deviation for the maximal displacement under the possibility of extinction. Under the nonextinction condition, we further discuss the decay rate of the upper deviation probability and the population growth at the critical phase.

A simply connected domain $\Omega \subset \mathbb{C}$ is convex in the positive direction if for every $z\in \Omega$, the half-line $\{z+t:t\ge 0\}$ is contained in $\Omega$. We provide necessary and sufficient conditions for the existence of an angular derivative at $\infty$ for domains convex in the positive direction which are contained either in a horizontal half-plane or in a horizontal strip. This class of domains arises naturally in the theory of semigroups of holomorphic functions, and the existence of an angular derivative has interesting consequences for the semigroup.

Fan–Jarvis–Ruan–Witten theory is a formulation of physical Landau–Ginzburg models with a rich algebraic structure, rooted in enumerative geometry. As a consequence of a major physical conjecture, called the Landau–Ginzburg/Calabi–Yau correspondence, several birational morphisms of Calabi–Yau orbifolds should correspond to isomorphisms in Fan–Jarvis–Ruan–Witten theory. In this paper, we exhibit some of these isomorphisms that are related to Borcea–Voisin mirror symmetry. In particular, we develop a modified version of Berglund–Hübsch–Krawitz mirror symmetry for certain Landau–Ginzburg models. Using these isomorphisms, we prove several interesting consequences in the corresponding geometries.

We correct a mistake in a lemma in the paper cited in the title and show that it did not affect any of the other results of the paper. To this end, we prove results on linearly disjoint field extensions that do not seem to be commonly known. We give an example to show that a separability assumption in one of these results cannot be dropped (doing so had led to the mistake). Further, we discuss recent generalizations of the original classification of defect extensions.

In this paper, we study the Erdős distinct distances problem for Cartesian product sets in the setting of arbitrary finite fields. More precisely, let ${\mathbb{F}}_q$ be an arbitrary finite field and $A$ be a set in ${\mathbb{F}}_q$. Suppose $|A\cap (aG\left)\right|\le |G{|}^{1/2}$ for any subfield $G$ and $a\in {\mathbb{F}}_q^{\ast }$, then $$\left|{\Delta }_{{\mathbb{F}}_q}\right(A^2\left)\right|=\left|\right(A-A{)}^2+(A-A{)}^2|\gg |A{|}^{1+\frac{1}{21}}.$$ Using the same method, we also obtain some results on sum–product type problems.