Knopfmacher [13] introduced the idea of an additive arithmetic semigroup as a general setting for an algebraic analogue of number theory. Within his framework, Zhang [19] showed that the asymptotic distribution of the values taken by additive functions closely resembles that found in classical number theory, in as much as there are direct analogues of the Erdős–Wintner and Kubilius Main Theorems. In this paper, we use probabilistic arguments to show that similar theorems, and their functional counterparts, can be proved in a much wider class of decomposable combinatorial structures.

We study some fine arithmetic properties of the components of solutions of a decomposable form equation. Lower growth rates for the greatest prime factor of a component are obtained for density 1 of the solutions. Also, high pure powers are shown to occur rarely. Computations illustrate the applicability of our results; for example, to the study of units in abelian group rings.

We prove that the order of any subloop of a finite Moufang loop is a factor of the order of the loop, thus obtaining an analog of Lagrange's theorem for finite Moufang loops.

We study two systems of tangle equations that arise when modeling the action of the Integrase family of proteins on DNA. These two systems—direct and inverted repeats—correspond to two different possibilities for the initial DNA sequence. We present one new class of solutions to the tangle equations. In the case of inverted repeats we prove that any solution not in these, or 2 previously known classes, would have to belong to one specific class. In the case of direct repeats we prove that the three classes are the only solutions possible.

In this paper we report on a surprising relation between the transfer operators for the congruence subgroups $\Gamma_{0}(nm),n,m\in \mathbb{N}$, and some kind of Hecke operators on the space of vector valued period functions for the groups $\Gamma_{0}(n)$. We study special eigenfunctions of the transfer operators for the groups $\Gamma_{0}(nm)$ with eigenvalues $\mp$1 which are also solutions of the Lewis equations for these groups and which are determined by eigenfunctions of the transfer operator for the congruence subgroup $\Gamma_{0}(n)$. In the language of the Atkin–Lehner theory of old and new forms one should hence call them old eigenfunctions or old solutions of the Lewis equation for $\Gamma_{0}(n)$. It turns out that certain linear combinations of the components of these old solutions for the group $\Gamma_{0}(nm)$ determine for any $m$ a solution of the Lewis equation for the group $\Gamma_{0}(n)$ and hence also an eigenfunction of the transfer operator for this group.

Our construction gives linear operators $\tilde{T}_{n}$ in the space of vector valued period functions for the group $\Gamma_{0}(n)$ which are rather similar to the Hecke operators. Indeed, in the case of the group $\Gamma_{0}(1)\,{=}\,\SL(2,\mathbb{Z})$ these operators are just the well-known Hecke operators on the space of period functions for the modular group, derived previously using the Eichler–Manin–Shimura correspondence between period polynomials and modular forms for this group, and its extension to Maass wave forms by Lewis and Zagier.

We study the dynamics of non-entire transcendental meromorphic functions with a finite asymptotic value mapped after some iterations onto a pole. This situation does not appear in the case of rational or entire functions. We consider the family of non-entire functions \[ f(z)=\frac{a\exp(z^p)+b\exp(-z^p)}{c\exp(z^p)+d\exp(-z^p)} \] with this property, i.e. there exists a finite asymptotic value $\xi$ and a positive natural number $q$ such that $f^q(\xi)\,{=}\,\infty$, where $p\,{\geq}\,1$, $a,b,c,d\,{\in}\,\mathbb{C}$. We show that the Hausdorff dimension of the set of points in the Julia set with bounded trajectory is strictly greater than one. We then impose two conditions on functions from this family. If the first condition holds, then the Lebesgue measure of the Julia set is zero, but the Hausdorff dimension is equal to two. On the other hand, if a function $f$ satisfies the second condition, then the Lebesgue measure of the Julia set is positive. Finally, we describe the measurable dynamics of these functions and prove that there does not exist an invariant measure on the Julia set absolutely continuous with respect to the Lebesgue measure and finite on compact subsets of $\mathbb{C}$.

Meromorphic solutions of the Schröder equation $f(sz)\,{=}\,R(f(z)),$ where $|s|\,{>}\,1$ and $R(w)$ is a rational function with $\deg[R]\,{\geq}\,2$, are studied. We will show that, if $\arg[s]\notin 2\pi {\mathbb Q}$, then $f(z)$ has any Borel direction, without exceptional values other than Picard values, which depend on $R(w)$. Further the case $\arg[s]\,{\in}\,2 \pi {\mathbb Q}$ is also considered. We investigate the relation between Julia directions of $f(z)$ and the Julia set of $R(w)$.

By investigating the relation between growth and value distribution A. Melas established a qualitative version of a Picard type theorem for universal Taylor series, that is: every universal Taylor series on the open unit disk $D$, assumes every complex value with at most one exception on infinite subsets of $D$ that approach the boundary of $D$ rather slowly. On the other hand, we show that there are universal Taylor series on $D$ such that the infinite subset of $D$ on which exactly one value is assumed, can approach the boundary of $D$ arbitrarily fast. Hence in view of Melas' work our result is the best possible. We also study the problem of interpolation by universal Taylor series.

The main aim of this paper is to give an overview of work that started in the early 90's and contained non-linearly in a number of papers and preprints. The style will be informal since the author wants this paper to be accessible to non-experts. Formal definitions, such as “$\Omega$ is a Lipschitz domain” will not be given. They will be well known to experts and “intuitively” clear to non-experts. Technical statements or proofs will not be given either. The reader who is not happy with this may consult the references given. The reader who wants to read this overview and meets unknown words is advised in the first instance to ignore them.

We introduce new concepts and properties of lightlike distributions and foliations (of dimension and co-dimension 1) in a space-time manifold of dimension $n$, from a purely geometric point of view. Given an observer and a lightlike distribution $\Omega $ of dimension or co-dimension 1, its lightlike direction is broken down into two vector fields: a timelike vector field $U$ representing the observer and a spacelike vector field $S$ representing the relative direction of propagation of $\Omega $ for this observer. A new distribution $\Omega_U^-$ is defined, with the opposite relative direction of propagation for the observer $U$. If both distributions $\Omega $ and $\Omega_U^-$ are integrable, the pair $\{\Omega,\Omega_U^-\}$ represents the wave fronts of a stationary wave for the observer $U$. However, we show in an example that the integrability of $\Omega $ does not imply the integrability of $\Omega_U^-$.