Let $E\subseteq \mathbb{R}$. Suppose there is an $s>0$ such that $$\left|\right\{k\in \mathbb{Z},-m\le k\le m-1:[k,k+1]\cap E\ne \varnothing \left\}\right|\ge m^s$$ for all sufficiently large $m\in \mathbb{N}$. Then there is an $n\in \mathbb{N}$ and a linear $T:{\mathbb{R}}^n\to \mathbb{R}$ such that $T\left(E^n\right)$ is dense. As a corollary, we show that if $E$ is in addition nowhere dense, then $(\mathbb{R},<,+,0,(x\mapsto \lambda x{)}_{\lambda \in \mathbb{R}},E)$ defines every bounded Borel subset of every ${\mathbb{R}}^n$.

We study the boundary asymptotics of the Carathéodory and Kobayashi–Eisenman volume elements on smoothly bounded convex finite type domains and Levi corank one domains.

Motivated by a recent result of Yoshino and the work of Bergh on reducible complexity, we introduce reducing versions of invariants of finitely generated modules over local rings. Our main result considers modules which have finite reducing Gorenstein dimension and determines a criterion for such modules to be totally reflexive in terms of the vanishing of Ext. Along the way, we give examples and applications, and in particular, prove that a Cohen–Macaulay local ring with canonical module is Gorenstein if and only if the canonical module has finite reducing Gorenstein dimension.

Let $C^{\left(\mathbf{n}\right)}=C^{\left(\mathbf{n}\right)}(\mathbb{D}\times \mathbb{D})$ be a Banach space of complex valued functions $f(x,y)$ that are continuous on the closed bidisc $\overline{\mathbb{D}\times \mathbb{D}}$, where $\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}$ is the unit disc in the complex plane $\mathbb{C}$ and has $n$th partial derivatives in $\mathbb{D}\times \mathbb{D}$ which can be extended to functions continuous on $\overline{\mathbb{D}\times \mathbb{D}}$. The Duhamel product is defined on $C^{\left(\mathbf{n}\right)}$ by the formula $$\left(f⊛g\right)(z,w)=\frac{{\partial }^2}{\partial z\partial w}{\int }_0^z{\int }_0^{w}f(z-u,w-v)g(u,v)dvdu.$$ In the present paper we prove that $C^{\left(\mathbf{n}\right)}(\mathbb{D}\times \mathbb{D})$ is a Banach algebra with respect to the Duhamel product $⊛$. This result extends some known results. We also investigate the structure of the set of all extended eigenvalues and extended eigenvectors of some double integration operator $W_{zw}$. In particular, the commutant of the double integration operator $W_{zw}$ is also described.

Given a triangular array with $m_n$ random variables in the $n$th row and a growth rate $\{k_n{\}}_{n=1}^{\infty }$ with ${lim\hspace{0.17em}sup\hspace{0.17em}}_{n\to \infty }(k_n/m_n)<1$, if the empirical distributions converge for any subarrays with the same growth rate, then the triangular array is asymptotically independent. In other words, if the empirical distribution of any $k_n$ random variables in the $n$th row of the triangular array is asymptotically close in probability to the law of a randomly selected random variable among these $k_n$ random variables, then two randomly selected random variables from the $n$th row of the triangular array are asymptotically close to being independent. This provides a converse law of large numbers by deriving asymptotic independence from a sample stability condition. It follows that a triangular array of random variables is asymptotically independent if and only if the empirical distributions converge for any subarrays with a given asymptotic density in $(0,1)$. Our proof is based on nonstandard analysis, a general method arisen from mathematical logic, and Loeb measure spaces in particular.

In the general framework of ${\mathbb{R}}^d$ equipped with Lebesgue measure and a critical radius function, we introduce several Hardy–Littlewood type maximal operators and related classes of weights. We prove appropriate two weighted inequalities for such operators as well as a version of Lerner’s inequality for a product of weights. With these tools we are able to prove factored weight inequalities for certain operators associated to the critical radius function. As it is known, the harmonic analysis arising from the Schrödinger operator $L=-\Delta +V$, as introduced by Shen, is based on the use of a related critical radius function. When our previous result is applied to this case, it allows to show some inequalities with factored weights for all first and second order Schrödinger–Riesz transforms.

A number of results are proved concerning the existence of nonreal zeros of derivatives of strictly nonreal meromorphic functions in the plane.