Following up on a paper of Balamohan et al. [‘On the behavior of a variant of Hofstadter’s $q$-sequence’, J. Integer Seq. 10 (2007)], we analyze a variant of Hofstadter’s $Q$-sequence and show that its frequency sequence is 2-automatic. An automaton computing the sequence is explicitly given.

Characterization of generalized Schur functions in terms of their Taylor coefficients was established by Krein and Langer [‘Über einige Fortsetzungsprobleme, die eng mit der Theorie hermitescher Operatoren im Raume ${\Pi }_{\kappa } $ zusammenhängen. I. Einige Funktionenklassen und ihre Darstellungen’, Math. Nachr. 77 (1977), 187–236]. We establish a boundary analogue of this characterization.

For a finite Clifford inverse algebra $A$, with natural order meet-semilattice ${Y}_{A} $ and group of units ${G}_{A} $, we show that the inverse monoid obtained as the semidirect product ${ Y}_{A}^{1} {\mathop{\ast }\nolimits}_{\rho } {G}_{A} $ has a log-polynomial free spectrum whenever $\rho $ is a term-expressible left action of ${G}_{A} $ on ${Y}_{A} $ and all subgroups of $A$ are nilpotent. This yields a number of examples of finite inverse monoids satisfying the Seif conjecture on finite monoids whose free spectra are not doubly exponential.

Conjecturally, the only real algebraic numbers with bounded partial quotients in their regular continued fraction expansion are rationals and quadratic irrationals. We show that the corresponding statement is not true for complex algebraic numbers in a very strong sense, by constructing, for every even degree $d$, algebraic numbers of degree $d$ that have bounded complex partial quotients in their Hurwitz continued fraction expansion. The Hurwitz expansion is the complex generalization of the nearest integer continued fraction for real numbers. In the case of real numbers the boundedness of regular and nearest integer partial quotients is equivalent.

This paper is concerned with a mass concentration phenomenon for a two-dimensional nonelliptic Schrödinger equation. It is well known that this phenomenon occurs when the ${L}^{4} $-norm of the solution blows up in finite time. We extend this result to the case where a mixed norm of the solution blows up in finite time.

We give continued fraction algorithms for each conjugacy class of triangle Fuchsian group of signature $(3, n, \infty )$, with $n\geq 4$. In particular, we give an explicit form of the group that is a subgroup of the Hilbert modular group of its trace field and provide an interval map that is piecewise linear fractional, given in terms of group elements. Using natural extensions, we find an ergodic invariant measure for the interval map. We also study Diophantine properties of approximation in terms of the continued fractions and show that these continued fractions are appropriate to obtain transcendence results.

A number is squareful if the exponent of every prime in its prime factorization is at least two. In this paper, we give, for a fixed $l$, the number of pairs of squareful numbers $n$, $n+l$ such that $n$is less than a given quantity.

Rooted monounary algebras can be considered as an algebraic counterpart of directed rooted trees. We work towards a characterization of the lattice of compatible quasiorders by describing its join- and meet-irreducible elements. We introduce the limit $\cB _\infty $ of all $d$-dimensional Boolean cubes $\Two ^d$ as a monounary algebra; then the natural order on $\Two ^d$ is meet-irreducible. Our main result is that any completely meet-irreducible quasiorder of a rooted algebra is a homomorphic preimage of the natural partial order (or its inverse) of a suitable subalgebra of $\cB _\infty $. For a partial order, it is known that complete meet-irreducibility means that the corresponding partially ordered structure is subdirectly irreducible. For a rooted monounary algebra it is shown that this property implies that the unary operation has finitely many nontrivial kernel classes and its graph is a binary tree.

It is well known that the regular continued fraction expansion of a quadratic irrational is symmetric about its centre; we refer to this symmetry as horizontal. However, an additional vertical symmetry is exhibited by the continued fraction expansions arising from a family of quadratics known as Schinzel sleepers. This paper provides a method for generating every Schinzel sleeper and investigates their period lengths as well as both their horizontal and vertical symmetries.

We identify the groups ${\text{PSU} }_{6} (2)$, ${\text{PSU} }_{6} (2){: }2$, ${\text{PSU} }_{6} (2){: }3$ and $\text{Aut} ({\text{PSU} }_{6} (2))$ from the structure of the centralizer of an element of order three.

We study the differentiability of the limiting distribution function associated to the normalized Euler function defined on the shifted primes.

We show that for manifolds of dimension $m\geq 5$, the flow of a Seiberg–Witten-type functional admits a global smooth solution.

We study positive integers $n$ such that $n\phi (n)\equiv 2\hspace{0.167em} {\rm mod}\hspace{0.167em} \sigma (n)$, where $\phi (n)$ and $\sigma (n)$ are the Euler function and the sum of divisors function of the positive integer $n$, respectively. We give a general ineffective result showing that there are only finitely many such $n$ whose prime factors belong to a fixed finite set. When this finite set consists only of the two primes $2$ and $3$ we use continued fractions to find all such positive integers $n$.

Let $w$ be a multilinear commutator word. We prove that if $e$ is a positive integer and $G$ is a finite group in which any nilpotent subgroup generated by $w$-values has exponent dividing $e$, then the exponent of the corresponding verbal subgroup $w(G)$ is bounded in terms of $e$ and $w$only.

We introduce the entropy of a family of planar curves in terms of the number of intersections of the family with a random line, calculate it for key examples, and discuss the entropy of a pattern of rings produced by an impulse on the surface of still water.

Let $\phi $ and $\psi $ be analytic maps on the open unit disk $D$ such that $\phi (D) \subset D$. Such maps induce a weighted composition operator $C_{\phi ,\psi }$ acting on weighted Banach spaces of type $H^{\infty }$or on weighted Bergman spaces, respectively. We study when such operators are order bounded.

We obtain the approximate functional equation for the Rankin–Selberg zeta function in the critical strip and, in particular, on the critical line $\operatorname {Re} s= \frac {1}{2}$.

We construct a family of ideals representing ideal classes of order two in quadratic number fields and show that relations between their ideal classes are governed by certain cyclic quartic extensions of the rationals.