Let $H^\infty ({\mathbb {D}}\times {\mathbb {N}})$ be the Banach algebra of bounded holomorphic functions defined on the disjoint union of countably many copies of the open unit disk ${\mathbb {D}}\subset {{\mathbb C}}$ . We show that the dense stable rank of $H^\infty ({\mathbb {D}}\times {\mathbb {N}})$ is $1$ and, using this fact, prove some nonlinear Runge-type approximation theorems for $H^\infty ({\mathbb {D}}\times {\mathbb {N}})$ maps. Then we apply these results to obtain a priori uniform estimates of norms of approximating maps in similar approximation problems for the algebra $H^\infty ({\mathbb {D}})$ .

We introduce certain $C^*$ -algebras and k-graphs associated to k finite-dimensional unitary representations $\rho _1,\ldots ,\rho _k$ of a compact group G. We define a higher rank Doplicher-Roberts algebra $\mathcal {O}_{\rho _1,\ldots ,\rho _k}$ , constructed from intertwiners of tensor powers of these representations. Under certain conditions, we show that this $C^*$ -algebra is isomorphic to a corner in the $C^*$ -algebra of a row-finite rank k graph $\Lambda $ with no sources. For G finite and $\rho _i$ faithful of dimension at least two, this graph is irreducible, it has vertices $\hat {G}$ and the edges are determined by k commuting matrices obtained from the character table of the group. We illustrate this with some examples when $\mathcal {O}_{\rho _1,\ldots ,\rho _k}$ is simple and purely infinite, and with some K-theory computations.

We obtain an effective analytic formula, with explicit constants, for the number of distinct irreducible factors of a polynomial $f \in \mathbb {Z}[x]$ . We use an explicit version of Mertens’ theorem for number fields to estimate a related sum over rational primes. For a given $f \in \mathbb {Z}[x]$ , our result yields a finite list of primes that certifies the number of distinct irreducible factors of f.

This paper is concerned with the growth rate of the product of consecutive partial quotients relative to the denominator of the convergent for the continued fraction expansion of an irrational number. More precisely, given a natural number $m,$ we determine the Hausdorff dimension of the following set:

$$ \begin{align*} E_m(\tau)=\bigg\{x\in [0,1): \limsup\limits_{n\rightarrow\infty}\frac{\log (a_n(x)a_{n+1}(x)\cdots a_{n+m}(x))}{\log q_n(x)}=\tau\bigg\}, \end{align*} $$

where $\tau $ is a nonnegative number. This extends the dimensional result of Dirichlet nonimprovable sets (when $m=1$ ) shown by Hussain, Kleinbock, Wadleigh and Wang.

A noncomplete graph is $2$ -distance-transitive if, for $i \in \{1,2\}$ and for any two vertex pairs $(u_1,v_1)$ and $(u_2,v_2)$ with the same distance i in the graph, there exists an element of the graph automorphism group that maps $(u_1,v_1)$ to $(u_2,v_2)$ . This paper determines the family of $2$ -distance-transitive Cayley graphs over dihedral groups, and it is shown that if the girth of such a graph is not $4$ , then either it is a known $2$ -arc-transitive graph or it is isomorphic to one of the following two graphs: $ {\mathrm {K}}_{x[y]}$ , where $x\geq 3,y\geq 2$ , and $G(2,p,({p-1})/{4})$ , where p is a prime and $p \equiv 1 \ (\operatorname {mod}\, 8)$ . Then, as an application of the above result, a complete classification is achieved of the family of $2$ -geodesic-transitive Cayley graphs for dihedral groups.

Taking residual finiteness as a starting point, we consider three related finiteness properties: weak subsemigroup separability, strong subsemigroup separability and complete separability. We investigate whether each of these properties is inherited by Schützenberger groups. The main result of this paper states that for a finitely generated commutative semigroup S, these three separability conditions coincide and are equivalent to every $\mathcal {H}$ -class of S being finite. We also provide examples to show that these properties in general differ for commutative semigroups and finitely generated semigroups. For a semigroup with finitely many $\mathcal {H}$ -classes, we investigate whether it has one of these properties if and only if all its Schützenberger groups have the property.