In some recent papers, the authors considered regular continued fractions of the form \[ \bigg[a_{0};\underbrace{a,\ldots, a}_{m}, \underbrace{a^{2},\ldots, a^{2}}_{m}, \underbrace{a^{3},\ldots, a^{3}}_{m}, \ldots \bigg], \] where $a_{0} \,{\geq}\, 0, a \,{\geq}\, 2$ and $m \,{\geq}\, 1$ are integers. The limits of such continued fractions, for general $a$ and in the cases $m\,{=}\,1$ and $m\,{=}\,2$, were given as ratios of certain infinite series.

However, these formulae can be derived from known facts about two continued fractions of Ramanujan. Motivated by these observations, we give alternative proofs of the results of the previous authors for the cases $m\,{=}\,1$ and $m\,{=}\,2$ and also use known results about other $q$-continued fractions investigated by Ramanujan to derive the limits of other infinite families of regular continued fractions.

Lejeune–Jalabert and Reguera computed the geometric Poincaré series $P_{geom}(T)$ for toric surface singularities. They raised the question whether this series equals the arithmetic Poincaré series. We prove this equality for a class of toric varieties including the surfaces, and construct a counterexample in the general case. We also compute the motivic Igusa Poincaré series $Q_{geom}(T)$ for toric surface singularities, using the change of variables formula for motivic integrals, thus answering a second question of Lejeune–Jalabert and Reguera's. The series $Q_{geom}(T)$ contains more information than the geometric series, since it determines the dimension of the tangent space at the singularity. In some sense, this is the only difference between $Q_{geom}(T)$ and $P_{geom}(T)$.

Two spatial embeddings of a graph are said to be delta edge-homotopic if they are transformed into each other by self delta moves and ambient isotopies. In this paper we classify theta curves up to delta edge-homotopy in terms of the third coefficient of the Conway polynomial of an associated 2-component link. In particular, we show that every boundary theta curve is delta edge-homotopically trivial, and two cobordant theta curves are delta edge-homotopic.

J. Dean introduced the notion of primitive/Seifert-fibered knots and constructions to explain how Dehn surgeries on knots produce small Seifert fibered manifolds. We determine non-hyperbolic, primitive/Seifert-fibered knots, and show that for such knots any integral, small Seifert fibered surgery arises from a primitive/Seifert-fibered construction. We also show that any 2-orbifold $S^2(n_1, n_2, n_3)$ with a pair of indices coprime becomes the base orbifold of a Seifert fibered manifold obtained by Dehn surgery on a hyperbolic, primitive/Seifert-fibered knot in $S^3$.

We study eta-invariants of links and show that in many cases they form link concordance invariants, in particular that many eta-invariants vanish for slice links. This result contains and generalizes previous invariants by Smolinsky and Cha–Ko. We give a formula for the eta-invariant for boundary links. In several interesting cases this allows us to show that a given link is not slice. We show that even more eta-invariants have to vanish for boundary slice links.

We give an elegant simple proof of a result concerning the first normal bundle of ideal Lagrangian immersions. This enables us to simplify and clarify several of the results in an earlier paper.

A point in Baire space is found for which the first derived $\omega$-limit set is not Borel, whilst the second is empty. A second point is found for which the sequence of derived $\omega$-limit sets does not stabilise until the first uncountable ordinal. The two points are recursive.

It is proved that $fg$ is Henstock–Kurzweil integrable on a compact interval ${\mathop{\prod}_{i=1}^{m}}[a_i, b_i]$ in ${\mathbb R}^m$ for each Henstock–Kurzweil integrable function $f$ if and only if there exists a finite signed Borel measure $\nu$ on ${\mathop{\prod}_{i=1}^{m}}[a_i, b_i)$ such that $g$ is equivalent to $\nu({\mathop{\prod}_{i=1}^{m}}[a_i, \,{\cdot}\,))$ on ${\mathop{\prod}_{i=1}^{m}}[a_i, b_i]$.

We compare conical density properties and spherical density properties for general Borel measures on $\mathbb{R}^n$. As a consequence, we obtain results for packing and Hausdorff measures $\mathcal{P}_h$ and $\mathcal{H}_h$ provided that the gauge function $h$ satisfies certain conditions.

One consequence of our general results is the following: let $m, n\,{\in}\,\mathbb{N}, 0\,{\lt}\,s\,{\lt}\,m\,{\leq}\,n$, $0\,{\lt}\,\eta\,{\lt}\,1$, and suppose that $V$ is an $m$-dimensional linear subspace of $\mathbb{R}^n$. Let $\mu$ be either the $s$-dimensional Hausdorff measure or the $s$-dimensional packing measure restricted to a set $A$ with $\mu(A)\,{\lt}\,\infty$. Then for $\mu$-almost every $x\,{\in}\,\mathbb{R}^n$, there is $\theta\,{\in}\,V\,{\cap}\, S^{n-1}$ such that \[ \liminf\limits_{r\downarrow 0}r^{-s}\mu(B(x, r)\cap H(x,\theta,\eta))=0,\] where $H\left(x,\theta,\eta\right)\,{=}\,\{{y\in\mathbb{R}^n}{\ip{\br{y-x}}{\theta}\,{\gt}\,\eta|{y-x}|\}$.

Let $P$ be the self-similar probability corresponding to an iterated function system of contracting similitudes $S_1,\ldots,S_N$ with probabilities $p_1,\ldots,p_N$ and contraction constants $s_1,\ldots, s_N$. Let $D_r$ be the quantization dimension of $P$ of order $r\in(0,+\infty)$, and let $P_r$ be the self-similar probability corresponding to $S_1,\ldots,S_N$ with probabilities $(p_1s_1^r)^{D_r/(r+D_r)},\ldots,(p_Ns_N^r)^{D_r/(r+D_r)}$. It is shown that the quanitzation coefficient of $P$ of order $r$ exists and that the empirical measures on an arbitrary sequence of asymptotically optimal quantizing sets (of order $r$) weakly converges to $P_r$ provided $\log(p_1s_1^r),\ldots,\log(p_Ns_N^r)$ is not arithmetic and $S_1,\ldots,S_N$ satisfies the open set condition.

Classical solutions of the spherically symmetric Nordström–Vlasov system are shown to exist globally in time. The main motivation for investigating the mathematical properties of the Nordström–Vlasov system is its relation to the Einstein–Vlasov system. The former is not a physically correct model, but it is expected to capture some of the typical features of the latter, which constitutes a physically satisfactory, relativistic model but is mathematically much more complex. We show that classical solutions of the spherically symmetric Nordström–Vlasov system exist globally in time for compactly supported initial data under the additional condition that there is a lower bound on the modulus of the angular momentum of the initial particle system. We emphasize that this is not a smallness condition and that our result holds for arbitrary large initial data satisfying this hypothesis.

The results on the local existence and continuation criteria obtained by G. Rein are extended to the case with a non-zero cosmological constant. It is also shown that for the spherically symmetric case and a positive cosmological constant there is a large class of initial data with global existence and inflationary asymptotics in the future as in the case of plane or hyperbolic symmetry. Furthermore we analyze the behaviour of the energy-momentum tensor at late times.