In this paper, we characterise all maximal elements in the semigroup S(X,Y)=\{f\in T(X) : f(Y)\subseteq Y\}$S(X,Y)=\{f\in T(X) : f(Y)\subseteq Y\}$ with respect to the natural partial order. Our results correct an error in the work of Sun and Wang [‘Natural partial order in semigroups of transformations with invariant set’, Bull. Aust. Math. Soc. 87(1) (2013), 94–107].

Kam Cheong Au [‘Wilf–Zeilberger seeds and non-trivial hypergeometric series’, Journal of Symbolic Computation 130 (2025), Article no. 102241] discovered a powerful methodology for finding new Wilf–Zeilberger (WZ) pairs. He calls it WZ seeds and gives numerous examples of applications to proving longstanding conjectural identities for reciprocal powers of \pi $\pi$ and their duals for Dirichlet L-values. In this note, we explain how a modification of Au’s WZ pairs together with a classical analytic argument yields simpler proofs of these results. We illustrate our method with examples elaborated with assistance of Maple code that we have developed.

For a group G, a subgroup U \leqslant G$U \leqslant G$ and a group A such that \mathrm {Inn}(G) \leqslant A \leqslant \mathrm {Aut}(G)$\mathrm {Inn}(G) \leqslant A \leqslant \mathrm {Aut}(G)$, we say that U is an A-covering group of G if G = \bigcup _{a\in A}U^a$G = \bigcup _{a\in A}U^a$. A theorem of Jordan (1872), implies that if G is a finite group, A = \mathrm {Inn}(G)$A = \mathrm {Inn}(G)$ and U is an A-covering group of G, then U = G$U = G$. Motivated by a question concerning Kronecker classes of field extensions, Neumann and Praeger (1990) conjectured that, more generally, there is an integer function f such that if G is a finite group and U is an A-covering subgroup of G, then |G:U| \leqslant f(|A:\mathrm {Inn}(G)|)$|G:U| \leqslant f(|A:\mathrm {Inn}(G)|)$. A key piece of evidence for this conjecture is a theorem of Praeger [‘Kronecker classes of fields and covering subgroups of finite groups’, J. Aust. Math. Soc. 57 (1994), 17–34], which asserts that there is a two-variable integer function g such that if G is a finite group and U is an A-covering subgroup of G, then |G:U|\leqslant g(|A:\mathrm {Inn}(G)|,c)$|G:U|\leqslant g(|A:\mathrm {Inn}(G)|,c)$, where c is the number of A-chief factors of G. Unfortunately, the proof of this theorem contains an error. In this paper, using a different argument, we give a correct proof of the theorem.

In this work, we investigate the arithmetic properties of b_{5^k}(n)$b_{5^k}(n)$, which counts the partitions of n where no part is divisible by 5^k$5^k$. By constructing generating functions for b_{5^k}(n)$b_{5^k}(n)$ across specific arithmetic progressions, we establish a set of Ramanujan-type congruences.

A fundamental extremality result due to Sidorenko [‘A partially ordered set of functionals corresponding to graphs’, Discrete Math. 131(1–3) (1994), 263–277] states that among all connected graphs G on k vertices, the k-vertex star maximises the number of graph homomorphisms of G into any graph H. We provide a new short proof of this result using only a simple recursive counting argument for trees and Hölder’s inequality.

Andrews and El Bachraoui [‘On two-colour partitions with odd smallest part’, Preprint, arXiv:2410.14190] explored many integer partitions in two colours, some of which are generated by the mock theta functions of third order of Ramanujan and Watson. They also posed questions regarding combinatorial proofs for these results. In this paper, we establish bijections to provide a combinatorial proof of one of these results and a companion result. We give analytic proofs of further companion results.

We examine bicoset digraphs and their natural properties from the point of view of symmetry. We then consider connected bicoset digraphs that are X-joins with collections of empty graphs, and show that their automorphism groups can be obtained from their natural irreducible quotients. We further show that such digraphs can be recognised from their connection sets.

We establish two new truncated identities for theta series. Their partition-theoretic interpretations are also given.

Let s be a fixed positive integer constant and let \varepsilon $\varepsilon$ be a fixed small positive number. Then, provided that a prime p is large enough, we prove that, for any set {\mathcal M}\subseteq \mathbb {F}_p^*${\mathcal M}\subseteq \mathbb {F}_p^*$ of size |{\mathcal M}|= \lfloor { p^{14/29}}\rfloor $|{\mathcal M}|= \lfloor { p^{14/29}}\rfloor$ and integer H=\lfloor {p^{14/29+\varepsilon }}\rfloor $H=\lfloor {p^{14/29+\varepsilon }}\rfloor$, any integer \lambda $\lambda$ can be represented in the form

\begin{align*} \frac{m_1}{x_1^s}+\frac{m_2}{x_2^s}+\frac{m_3}{x_3^s}\equiv \lambda \bmod p \quad\text{with } m_i\in {\mathcal M} \text{ and } 1\leqslant x_i\leqslant H, \, i=1,2,3. \end{align*} $$\begin{align*} \frac{m_1}{x_1^s}+\frac{m_2}{x_2^s}+\frac{m_3}{x_3^s}\equiv \lambda \bmod p \quad\text{with } m_i\in {\mathcal M} \text{ and } 1\leqslant x_i\leqslant H, \, i=1,2,3. \end{align*}$$

When s=1$s=1$, we show that, for almost all primes p, if |{\mathcal M}|= \lfloor p^{1/2}\rfloor $|{\mathcal M}|= \lfloor p^{1/2}\rfloor$ and H=\lfloor p^{1/2}(\log p)^{6+\varepsilon }\rfloor $H=\lfloor p^{1/2}(\log p)^{6+\varepsilon }\rfloor$, then any integer \lambda $\lambda$ can be represented in the form

\begin{align*} \frac{m_1}{x_1}+\frac{m_2}{x_2}\equiv \lambda \bmod p \quad\text{with } m_i\in {\mathcal M} \text{ and } 1\leqslant x_i\leqslant H, \, i=1,2. \end{align*} $$\begin{align*} \frac{m_1}{x_1}+\frac{m_2}{x_2}\equiv \lambda \bmod p \quad\text{with } m_i\in {\mathcal M} \text{ and } 1\leqslant x_i\leqslant H, \, i=1,2. \end{align*}$$

The concept of stability has proved very useful in the field of Banach space geometry. In this note, we introduce and study a corresponding concept in the setting of Banach algebras, which we call multiplicative stability. As we shall prove, various interesting examples of Banach algebras are multiplicatively unstable, and hence unstable in the model-theoretic sense. The examples include Fourier algebras over noncompact amenable groups, C^*$C^*$-algebras and the measure algebra of an infinite compact group.

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$\mathbb {F}_q$ for q \equiv 3 \pmod 4$q \equiv 3 \pmod 4$. We extend the definition of arithmetic-geometric mean sequences over \mathbb {F}_q$\mathbb {F}_q$ to q \equiv 5 \pmod 8$q \equiv 5 \pmod 8$. 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$q \equiv 5 \pmod 8$.

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

Thomassen’s chord conjecture from 1976 states that every longest cycle in a 3$3$-connected graph has a chord. The circumference c(G)$c(G)$ and induced circumference c'(G)$c'(G)$ 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$2$-connected graph G with minimum degree at least 3$3$ has c(G)\geq c'(G)+2$c(G)\geq c'(G)+2$. 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 $\ell$-holed if all its holes have length exactly \ell $\ell$. We prove that Harvey’s conjecture and hence also Thomassen’s conjecture hold for \ell $\ell$-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$h\ge 1, k\ge 2$. 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$n+l_1,\ldots , n+l_h$ are k-free have a natural density for any given nonnegative integers l_1,\ldots , l_h$l_1,\ldots , l_h$. In this note, we show that the Bergelson–Richter theorem holds for the numbers studied by Mirsky.

The minimal faithful permutation degree \mu (G)$\mu (G)$ of a finite group G is the least integer n such that G is isomorphic to a subgroup of the symmetric group S_n$S_n$. If G has a normal subgroup N such that \mu (G/N)> \mu (G)$\mu (G/N)> \mu (G)$, then G is exceptional. We prove that the proportion of exceptional groups of order p^6$p^6$ for primes p \geq 5$p \geq 5$ is asymptotically zero. We identify (11p+107)/2$(11p+107)/2$ 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$xy\neq yx$ for every x\in A$x\in A$ and y\in B$y\in B$. We say a nonempty set \mathcal {X}$\mathcal {X}$ of subsets with cardinality r of G is an r-noncommuting set if every two elements of \mathcal {X}$\mathcal {X}$ are noncommuting subsets. If |\mathcal {X}| \geq |\mathcal {Y}|$|\mathcal {X}| \geq |\mathcal {Y}|$ for any other r-noncommuting set \mathcal {Y}$\mathcal {Y}$ of G, then the cardinality of \mathcal {X}$\mathcal {X}$ (if it exists) is denoted by w_G(r)$w_G(r)$ 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}$w_G: \mathbb {N} \longrightarrow \mathbb {N}$ 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}$\mathcal {H}$ be the class of all analytic self-maps of the open unit disk \mathbb {D}$\mathbb {D}$. Denote by H^n f(z)$H^n f(z)$ the nth-order hyperbolic derivative of f\in \mathcal H$f\in \mathcal H$ at z\in \mathbb {D}$z\in \mathbb {D}$. 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}$z_0\in \mathbb {D}$ and \gamma = (\gamma _0, \gamma _1 , \ldots , \gamma _{n-1}) \in {\mathbb D}^{n}$\gamma = (\gamma _0, \gamma _1 , \ldots , \gamma _{n-1}) \in {\mathbb D}^{n}$, 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} \}${\mathcal H} (\gamma ) = \{f \in {\mathcal H} : f (z_0) = \gamma _0,H^1f (z_0) = \gamma _1,\ldots ,H^{n-1}f (z_0) = \gamma _{n-1} \}$. We determine the variability region \{ f^{(n)}(z_0) : f \in {\mathcal H} (\gamma ) \}$\{ f^{(n)}(z_0) : f \in {\mathcal H} (\gamma ) \}$ 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}\}$\{h^{(n)}(z_0): h\in \mathcal {H}, h(0)=0,h(z_0) =w_0, h'(z_0)=w_1,\ldots , h^{(n-1)}(z_0)=w_{n-1}\}$ for given z_0$z_0$, w_0$w_0$, w_1,\ldots ,w_{n-1}$w_1,\ldots ,w_{n-1}$. Moreover, we determine the form of all extremal functions.