We characterize Hermitian cones among the surfaces of degree $q+1$ of $\text{PG}(3,q^{2})$ by their intersection numbers with planes. We then use this result and provide a characterization of nonsingular Hermitian varieties of $\text{PG}(4,q^{2})$ among quasi-Hermitian ones.

In a recent study of how the output voltage of a Hall plate is affected by the shape of the plate and the size of its contacts, U. Ausserlechner has come up with a remarkable double integral that can be viewed as a generalisation of the classical elliptic ‘arithmetic–geometric mean (AGM)’ integral. Here we discuss transformation properties of the integral, which were experimentally observed by Ausserlechner, as well as its analytical and arithmetic features including connections with modular forms.

Wreath products of nondiscrete locally compact groups are usually not locally compact groups, nor even topological groups. As a substitute introduce a natural extension of the wreath product construction to the setting of locally compact groups. Applying this construction, we disprove a conjecture of Trofimov, constructing compactly generated locally compact groups of intermediate growth without any open compact normal subgroup.

In 1991, Andrews and Hickerson established a new Bailey pair and combined it with the constant term method to prove some results related to sixth-order mock theta functions. In this paper, we study how this pair gives rise to new mock theta functions in terms of Appell–Lerch sums. Furthermore, we establish some relations between these new mock theta functions and some second-order mock theta functions. Meanwhile, we obtain an identity between a second-order and a sixth-order mock theta functions. In addition, we provide the mock theta conjectures for these new mock theta functions. Finally, we discuss the dual nature between the new mock theta functions and partial theta functions.

A new class of functions with a unique identification minor is introduced: functions determined by content and singletons. Relationships between this class and other known classes of functions with a unique identification minor are investigated. Some properties of functions determined by content and singletons are established, especially concerning invariance groups and similarity.

In this paper, we study the mixed Littlewood conjecture with pseudo-absolute values. For any pseudo-absolute-value sequence ${\mathcal{D}}$, we obtain a sharp criterion such that for almost every $\unicode[STIX]{x1D6FC}$ the inequality

$$\begin{eqnarray}|n|_{{\mathcal{D}}}|n\unicode[STIX]{x1D6FC}-p|\leq \unicode[STIX]{x1D713}(n)\end{eqnarray}$$

$(n,p)\in \mathbb{N}\times \mathbb{Z}$

$\unicode[STIX]{x1D713}$

${\mathcal{D}}_{1},{\mathcal{D}}_{2},\ldots ,{\mathcal{D}}_{k}$

$\unicode[STIX]{x1D713}(n)$

$\unicode[STIX]{x1D6FC}$

$$\begin{eqnarray}|n|_{{\mathcal{D}}_{1}}|n|_{{\mathcal{D}}_{2}}\cdots |n|_{{\mathcal{D}}_{k}}|n\unicode[STIX]{x1D6FC}-p|\leq \unicode[STIX]{x1D713}(n)\end{eqnarray}$$

$(n,p)\in \mathbb{N}\times \mathbb{Z}$

Rosay and Rudin introduced the notion of ‘tameness’ for discrete subsets of $\mathbf{C}^{n}$. We generalize the notion of tameness for discrete sets to arbitrary Stein manifolds, with special emphasis on complex Lie groups.

Denote by $\mathbb{P}$ the set of all prime numbers and by $P(n)$ the largest prime factor of positive integer $n\geq 1$ with the convention $P(1)=1$. In this paper, we prove that, for each $\unicode[STIX]{x1D702}\in (\frac{32}{17},2.1426\cdots \,)$, there is a constant $c(\unicode[STIX]{x1D702})>1$ such that, for every fixed nonzero integer $a\in \mathbb{Z}^{\ast }$, the set

$$\begin{eqnarray}\{p\in \mathbb{P}:p=P(q-a)\text{ for some prime }q\text{ with }p^{\unicode[STIX]{x1D702}}<q\leq c(\unicode[STIX]{x1D702})p^{\unicode[STIX]{x1D702}}\}\end{eqnarray}$$

$\mathbb{P}$

$\unicode[STIX]{x1D702}\in (\frac{32}{17},2.0606\cdots \,)$

$\unicode[STIX]{x1D702}\in (\frac{32}{17},2.1426\cdots \,)$