We compute the Galois groups of the reductions modulo a prime number p of the generating series of Apéry numbers, Domb numbers and Almkvist–Zudilin numbers. We observe in particular that their behaviour is governed by congruence conditions on p.

Jabłonowski [‘On biquandle-based invariant of immersed surface-links, Yoshikawa oriented fifth move, and ribbon 2-knots’, Preprint, 2025, arXiv:2505.14724] proved that the knot quandles of Suciu’s n-knots, which share isomorphic knot groups, are mutually nonisomorphic, and Yasuda [‘Knot quandles distinguish Suciu’s ribbon knots’, Preprint, 2025, arXiv:2508.15129] later gave a different proof. We present yet another proof of this result by analysing the conjugacy classes of certain automorphisms of the free group of rank two.

The notion of $\theta $-congruent numbers generalises the classical congruent number problem. A positive integer n is $\theta $-congruent if it is the area of a rational triangle with an angle $\theta $ whose cosine is rational. Das and Saikia [‘On $\theta $-congruent numbers over real number fields’, Bull. Aust. Math. Soc. 103(2) (2021), 218–229] established criteria for numbers to be $\theta $-congruent over certain real number fields and concluded their work by posing four open questions regarding the relationship between $\theta $-congruent and properly $\theta $-congruent numbers. In this work, we provide complete answers to those questions. Indeed, we remove a technical assumption from their result on fields with degrees coprime to six, provide a definitive answer for real cubic fields without congruence restrictions, extend the analysis to fields of degree six and examine the exceptional cases $n=1, 2, 3$ and $6$.

The set of sums of two squares plays a significant role in number theory. We establish the existence of several rich monochromatic configurations in the natural numbers by exploiting algebraic structures induced by the set of sums of two squares. The proofs rely on algebraic properties arising from the induced structures on the Stone–Čech compactification of the natural numbers.

The planar Skorokhod embedding problem was first proposed and solved by Gross [‘A conformal Skorokhod embedding’, Electron. Commun. Probab. 24 (2019), 11 pages; doi:10.1214/19-ECP272]. Gross worked with probability distributions having finite second moment. Boudabra and Markowsky [‘Remarks on Gross’ technique for obtaining a conformal Skorokhod embedding of planar Brownian motion’, Electron. Commun. Probab. 25 (2020), 13 pages; doi:10.1214/20-ECP300] extended the solution to all distributions with a finite pth moment for $p>1$. The case $p=1$ has remained uncovered since then. In this note, we show that the planar Skorokhod embedding problem is solvable for $p=1$ when the Hilbert transform of its quantile function is integrable, effectively closing this line of investigation.

Let $\lambda (G)$ be the maximum number of subgroups in an irredundant cover of the finite group G. We establish bounds on the order, exponent and derived length of the group in terms of this invariant.

Let $\mathbb {N}$ be the set of all nonnegative integers. For a set $A\subseteq \mathbb {N}$, let $R_2(A,n)$ and $R_3(A,n)$ be the number of solutions of the equation $n=a_1+a_2$ with $a_1<a_2, a_1,a_2\in A$ and with $a_1\le a_2, a_1,a_2\in A$, respectively. If $-N\le g\le N$, Yan [‘On the structure of partition which the difference of their representation function is a constant’, Period. Math. Hungar. 82 (2021), 149–152] showed that there is a set $A\subseteq \mathbb {N}$ such that $R_i(A,n)-R_i(\mathbb {N}\setminus A,n)=g$ for all integers $n\ge 2N-1$, where N is a positive integer. In this paper, we prove that if $g_1,g_2$ are nonnegative integers with $g_1\neq g_2$, then there does not exist $A\subseteq \mathbb {N}$ such that $R_i(A,2n)-R_i(\mathbb {N}\setminus A,2n)=g_1$ and $R_i(A,2n+1)-R_i(\mathbb {N}\setminus A,2n+1)=g_2$ for all sufficiently large integers n.

A family of $n\times n$ matrices over a field $\mathbb {F}$ is irreducible if it has no common nontrivial invariant subspace, and minimally irreducible if it is irreducible but has no proper irreducible subfamily. If $\mathbb {F}$ is algebraically closed and $n\ge 2$, a minimally irreducible family has at most $2n-1$ elements. We show that for complex $n\times n$ matrices, $n\ge 3$, a family of minimally irreducible (i) matrix units, (ii) rank one projections, (iii) unicellular matrices and (iv) orthoatomic matrices has k elements where respectively (i) $n\le k\le 2n-2$, (ii) $k=n$, (iii) $2\le k\le n-1$ and (iv) $2\le k\le n-1$. All of the values of k in these ranges are attained. If $n=2$, each such minimally irreducible family has $2$ elements.

For a group G and $m\ge 1$, $G^m$ denotes the subgroup generated by the elements $g^m$, where g runs through G. The subgroups not of the form $G^m$ are called nonpower subgroups. We extend the classification of groups with few nonpower subgroups from groups with at most nine nonpower subgroups to groups with at most 13 nonpower subgroups.

Several authors have investigated the structure of groups in which each subgroup satisfies a property $\mathcal {X}$ or a property which is antagonistic to $\mathcal {X}$. This point of view will be adopted here, considering groups in which each subgroup is either nearly normal or contranormal.

We study a discrete process on planar convex bodies in which, at each step, a body is replaced by a weighted Minkowski average of itself and its rotation by a fixed angle. Up to translation and uniform scaling, this produces a rigid averaging dynamical system. We give a complete classification of the limit shapes. If the angle is an irrational multiple of $2\pi $, the iterates converge to a disk. If the angle is rational, they converge to the average of finitely many rotated copies of the initial body. We also obtain sharp convergence rates. In the rational case, the decay is uniform and exponential with an explicit constant depending only on the weight and the denominator of the angle. For irrational angles, we prove quantitative rates under a mild number-theoretic condition that holds for almost every angle: low regularity inputs have polynomial decay up to a logarithmic factor, while real analytic inputs have stretched exponential decay. For angles with bounded continued fraction coefficients, we give matching lower bounds along subsequences. These results describe the global attractors of the dynamics and indicate the absence of chaotic behaviour.

An element x of a lattice L is modular if L has no five-element sublattice isomorphic to the pentagon in which x would correspond to the lonely midpoint. We classify all modular elements of the lattice of all monoid varieties.

This paper studies the $4$-ranks of narrow class groups in certain families of quadratic fields. We prove that for any positive integer n, there exists an integer $s_\lambda (n)$ depending on n and the sign $\lambda $ of the fundamental discriminant D, such that for any choice of $s_\lambda (n)$ integers $t_1, \ldots , t_{s_\lambda (n)}$, there are infinitely many D for which the narrow class group of $\mathbb {Q}(\sqrt {D + t_i})$ has $4$-rank bounded by n for all i. This result extends previous work on $3$-ranks to the case of $4$-ranks.

If A is in the p-Schatten class on ${\mathbb {R}}^n$, $1\leq p \leq {4n}/{(2n-1)}$, then the quantum translates of A are linearly independent. Moreover, there exists a nonzero operator in the p-Schatten class on ${\mathbb {R}}^n$, $p>{4n}/{(2n-1)}$, whose quantum translates are linearly dependent.

We consider quasilinear Schrödinger equations in $\mathbb {R}^N$ of the form $-\Delta u+V(x)u-u\Delta (u^2)=g(u)$, where the potential V is allowed to be sign-changing and the nonlinearity g is sublinear at zero. Except for being subcritical, no additional condition is imposed on $g(u)$ for $|u|$ large. We obtain a sequence of solutions with negative energy and converging to zero via Clark’s theorem. We also obtain a similar result for fourth-order quasilinear Schrödinger equations in $\mathbb {R}^N$ of the form $\Delta ^2u-\Delta u+ V(x)u-u\Delta (u^2)=g(u)$.

For any integer k and any positive integer n, let $\sigma _k(n)=\sum _{d\mid n}d^k$. For any prime p and any positive integer m, let $\nu _p(m)$ be the largest integer $\alpha $ such that $p^\alpha \mid m$ and let $\lceil x\rceil $ denote the least integer not less than x. In 2021, Amdeberhan et al. [‘Arithmetic properties of the sum of divisors’, J. Number Theory 223 (2021), 325–349] proved that $\nu _2(\sigma _1(n))\le \lceil \log _2n\rceil $ for any positive integer n and that $\nu _p(\sigma _1(n))\le \lceil \log _pn\rceil $ for any odd prime p if n satisfies some conditions. Recently, Zhao and Chen [‘p-adic valuation of the sum of divisors’, Front. Math. 20(4) (2025), 795–827] proved this unconditionally. We generalise these results to all k: for any prime p, any n and any $k\ge 2$, $\nu _p(\sigma _k(n))\le \lceil k\log _p n\rceil .$ Let $p^\star $ be an odd prime. We also prove that there are an integer $k\ge 2$ and a prime q satisfying $\nu _{p^\star }(\sigma _k(q))=\lceil k\log _{p^\star }q\rceil $ if and only if $p^\star $ is a Fermat prime.

Andrews and El Bachraoui [‘On two-color partitions with odd smallest part’, Preprint (2024), arXiv:2410. 14190] recently investigated identities involving two-colour partitions, with particular emphasis on their connection to overpartitions, and posed questions regarding possible companion results. Subsequently, Chen and Zou [‘Combinatorial proofs for two-colour partitions’, Bull. Aust. Math. Soc. 113(1) (2025), to appear] obtained some companion results by employing q-series identities and generating functions. In addition, they presented a combinatorial proof for one of their own results and one of the results of Andrews and El Bachraoui. They posed questions regarding combinatorial proofs of the remaining companion results. In this paper, we provide such proofs.

Given a group G and an automorphism $\varphi $ of G, two elements $x,y\in G$ are said to be $\varphi $-conjugate if $x=gy\varphi (g)^{-1}$ for some $g\in G$. The number $R(\varphi )$ of equivalence classes with respect to this relation is called the Reidemeister number of $\varphi $ and the set $\{R(\varphi ) \mid \varphi \in \text {Aut}(G)\}$ is called the Reidemeister spectrum of G. We determine the Reidemeister spectrum of ZM-groups, extending some results of Senden [‘The Reidemeister spectrum of split metacyclic groups’, Preprint, 2022, arXiv:2109.12892].