Arithmetic-geometric mean sequences were already studied over the real and complex numbers, and recently, Griffin et al. [‘AGM and jellyfish swarms of elliptic curves’, Amer. Math. Monthly 130(4) (2023), 355–369] considered them over finite fields $\mathbb {F}_q$ for $q \equiv 3 \pmod 4$. We extend the definition of arithmetic-geometric mean sequences over $\mathbb {F}_q$ to $q \equiv 5 \pmod 8$. We explain the connection of these sequences with graphs and explore the properties of the graphs in the case where $q \equiv 5 \pmod 8$.

We examine the maximum dimension of a linear system of plane cubic curves whose $\mathbb {F}_q$-members are all geometrically irreducible. Computational evidence suggests that such a system has a maximum (projective) dimension of $3$. As a step towards the conjecture, we prove that there exists a three-dimensional linear system $\mathcal {L}$ with at most one geometrically reducible $\mathbb {F}_q$-member.

Thomassen’s chord conjecture from 1976 states that every longest cycle in a $3$-connected graph has a chord. The circumference $c(G)$ and induced circumference $c'(G)$ of a graph G are the length of its longest cycles and the length of its longest chordless cycles, respectively. Harvey [‘A cycle of maximum order in a graph of high minimum degree has a chord’, Electron. J. Combin. 24(4) (2017), Article no. 4.33, 8 pages] proposed the stronger conjecture: every $2$-connected graph G with minimum degree at least $3$ has $c(G)\geq c'(G)+2$. This conjecture implies Thomassen’s chord conjecture. We observe that wheels are the unique Hamiltonian graphs for which the circumference and the induced circumference differ by exactly one. Thus, we need only consider non-Hamiltonian graphs for Harvey’s conjecture. We propose a conjecture involving wheels that is equivalent to Harvey’s conjecture on non-Hamiltonian graphs. A graph is $\ell $-holed if all its holes have length exactly $\ell $. We prove that Harvey’s conjecture and hence also Thomassen’s conjecture hold for $\ell $-holed graphs and graphs with a small induced circumference.

Bergelson and Richter [‘Dynamical generalizations of the prime number theorem and disjointness of additive and multiplicative semigroup actions’, Duke Math. J. 171(15) (2022), 3133–3200] established a new dynamical generalisation of the prime number theorem (PNT) and the PNT for arithmetic progressions. Let $h\ge 1, k\ge 2$. Mirsky [‘Note on an asymptotic formula connected with r-free integers’, Quart. J. Math. Oxford Ser. 18 (1947), 178–182] showed that the numbers n such that $n+l_1,\ldots , n+l_h$ are k-free have a natural density for any given nonnegative integers $l_1,\ldots , l_h$. In this note, we show that the Bergelson–Richter theorem holds for the numbers studied by Mirsky.

The minimal faithful permutation degree $\mu (G)$ of a finite group G is the least integer n such that G is isomorphic to a subgroup of the symmetric group $S_n$. If G has a normal subgroup N such that $\mu (G/N)> \mu (G)$, then G is exceptional. We prove that the proportion of exceptional groups of order $p^6$ for primes $p \geq 5$ is asymptotically zero. We identify $(11p+107)/2$ such groups and conjecture that there are no others.

In our paper [‘Linking the boundary and exponential spectra via the restricted topology’, J. Math. Anal. Appl. 454 (2017), 730–745], we defined and used the restricted topology to establish certain connections among the boundary spectrum, the exponential spectrum, the topological boundary of the spectrum and the connected hull of the spectrum, and in [‘The restricted connected hull: filling the hole’, Bull. Aust. Math. Soc. 109 (2024), 388–392], we presented further properties of the restricted connected hull. We now continue our investigation of the restricted boundary. In particular, we prove a number of mapping and regularity-type properties of the restricted boundary. In addition, we use this concept to provide a new characterisation of the Jacobson radical of a Banach algebra and a variation of Harte’s theorem. Finally, we establish spectral continuity results, in particular, in ordered Banach algebras.

We say that two nonempty subsets A and B with cardinality r of a group G are noncommuting subsets if $xy\neq yx$ for every $x\in A$ and $y\in B$. We say a nonempty set $\mathcal {X}$ of subsets with cardinality r of G is an r-noncommuting set if every two elements of $\mathcal {X}$ are noncommuting subsets. If $|\mathcal {X}| \geq |\mathcal {Y}|$ for any other r-noncommuting set $\mathcal {Y}$ of G, then the cardinality of $\mathcal {X}$ (if it exists) is denoted by $w_G(r)$ and is called the r-clique number of G. In this paper, we try to find the influence of the function $w_G: \mathbb {N} \longrightarrow \mathbb {N}$ on the structure of groups.

Determining the insdel distance of linear codes is a very challenging problem. Recently, Ji et al. [‘Strict half-Singleton bound, strict direct upper bound for linear insertion-deletion codes and optimal codes’, IEEE Trans. Inform. Theory 69(5) (2023), 2900–2910] proposed their strict half-Singleton bound for linear codes that do not include the all-one vector. A natural question asks for linear codes with both large Hamming distance and insdel distance. Almost maximum distance separable (MDS) codes are a special kind of linear code with good Hamming error-correcting capability. We present a sufficient condition for the insdel distance of almost MDS codes to be bounded by the strict half-Singleton bound. In addition, we construct a class of two-dimensional near MDS codes that achieve the strict half-Singleton bound using twisted Reed–Solomon codes. Finally, we present a construction of optimal almost MDS insdel codes from Reed–Solomon codes for large dimensions.

Let $\mathcal {H}$ be the class of all analytic self-maps of the open unit disk $\mathbb {D}$. Denote by $H^n f(z)$ the nth-order hyperbolic derivative of $f\in \mathcal H$ at $z\in \mathbb {D}$. We develop a method allowing us to calculate higher-order hyperbolic derivatives in an expeditious manner. We also generalise certain classical results for variability regions of the nth derivative of bounded analytic functions. For $z_0\in \mathbb {D}$ and $\gamma = (\gamma _0, \gamma _1 , \ldots , \gamma _{n-1}) \in {\mathbb D}^{n}$, let ${\mathcal H} (\gamma ) = \{f \in {\mathcal H} : f (z_0) = \gamma _0,H^1f (z_0) = \gamma _1,\ldots ,H^{n-1}f (z_0) = \gamma _{n-1} \}$. We determine the variability region $\{ f^{(n)}(z_0) : f \in {\mathcal H} (\gamma ) \}$ to prove a Schwarz–Pick lemma for the nth derivative. We apply this result to establish an nth-order Dieudonné lemma, which provides an explicit description of the variability region $\{h^{(n)}(z_0): h\in \mathcal {H}, h(0)=0,h(z_0) =w_0, h'(z_0)=w_1,\ldots , h^{(n-1)}(z_0)=w_{n-1}\}$ for given $z_0$, $w_0$, $w_1,\ldots ,w_{n-1}$. Moreover, we determine the form of all extremal functions.

We propose a generalised version of configuration spaces defined by disallowing combinations of simultaneous collisions among the n points determined by a family of forbidden partitions. In the case where the underlying space is a finite graph, we construct a cubical complex with the same homology as this configuration space.

We give a simplified version of the proofs that, outside of their isolated vertices, the complement of the enhanced power graph and of the power graph are connected and have diameter at most $3$.

We demonstrate the existence of K-multimagic squares of order N consisting of $N^2$ distinct integers whenever $N> 2K(K+1)$. This improves our earlier result [D. Flores, ‘A circle method approach to K-multimagic squares’, preprint (2024), arXiv:2406.08161] in which we only required $N+1$ distinct integers. Additionally, we present a direct method by which our analysis of the magic square system may be used to show the existence of $N \times N$ magic squares consisting of distinct kth powers when

$$ \begin{align*}N> \begin{cases} 2^{k+1} & \text{if}\ 2 \leqslant k \leqslant 4, \\ 2 \lceil k(\log k + 4.20032) \rceil & \text{if}\ k \geqslant 5, \end{cases}\end{align*} $$

improving on a recent result by Rome and Yamagishi [‘On the existence of magic squares of powers’, preprint (2024), arxiv:2406.09364].

Let C and W be two integer sets. If $C+W=\mathbb {Z}$, then we say that C is an additive complement to W. If no proper subset of C is an additive complement to W, then we say that C is a minimal additive complement to W. We study the existence of a minimal additive complement to $W=\{w_i\}_{i=1}^{\infty}$ when W is not eventually periodic and $w_{i+1}-w_{i}\in \{2,3\}$ for all i.

Inspired by work of Andrews and Newman [‘Partitions and the minimal excludant’, Ann. Comb. 23 (2019), 249–254] on the minimal excludant or ‘mex’ of partitions, we define four new classes of minimal excludants for overpartitions and establish relations to certain functions due to Ramanujan.

Let ${\mathcal {A}}$ be a unital ${\mathbb {F}}$-algebra and let ${\mathcal {S}}$ be a generating set of ${\mathcal {A}}$. The length of ${\mathcal {S}}$ is the smallest number k such that ${\mathcal {A}}$ equals the ${\mathbb {F}}$-linear span of all products of length at most k of elements from ${\mathcal {S}}$. The length of ${\mathcal {A}}$, denoted by $l({\mathcal {A}})$, is defined to be the maximal length of its generating sets. We show that $l({\mathcal {A}})$ does not exceed the maximum of $\dim {\mathcal {A}} / 2$ and $m({\mathcal {A}})-1$, where $m({\mathcal {A}})$ is the largest degree of the minimal polynomial among all elements of the algebra ${\mathcal {A}}$. As an application, we show that for arbitrary odd n, the length of the group algebra of the dihedral group of order $2n$ equals n.

We use deformations and mutations of scattering diagrams to show that a scattering diagram with initial functions $f_1=(1+tx)^\mu $ and $f_2=(1+ty)^\nu $ has a dense region. This answers a question asked by Gross and Pandharipande [‘Quivers, curves, and the tropical vertex’, Port. Math. 67(2) (2010), 211–259] which had been proved only for the case $\mu =\nu $.

Our work owes its origin to a recent note of Ram Murty [‘Irrationality of zeros of the digamma function’, Number Theory in Memory of Eduard Wirsing (eds. H. Maier, R. Steuding and J. Steuding) (Springer, Cham, 2023), 237–243], in which he proves that all the zeros of the digamma function are irrational with at most one possible exception. We extend this investigation to higher-order polygamma functions.

Let $g(x)=x^3+ax^2+bx+c$ and $f(x)=g(x^3)$ be irreducible polynomials with rational coefficients, and let $ {\mathrm{Gal}}(f)$ be the Galois group of $f(x)$ over $\mathbb {Q}$. We show $ {\mathrm{Gal}}(f)$ is one of 11 possible transitive subgroups of $S_9$, defined up to conjugacy; we use $ {\mathrm{Disc}}(f)$, $ {\mathrm{Disc}}(g)$ and two additional low-degree resolvent polynomials to identify $ {\mathrm{Gal}}(f)$. We further show how our method can be used for determining one-parameter families for a given group. Also included is a related algorithm that, given a field $L/\mathbb {Q}$, determines when L can be defined by an irreducible polynomial of the form $g(x^3)$ and constructs such a polynomial when it exists.