Let $(Z_{t})_{t \geq 0}$ be a planar Brownian motion running in some domain W and denote by $\tau _{W}$ the exit time of $Z_{t}$ from W. To establish the finiteness of $\mathbf {E}(\sup _{0\leq t\leq \tau _{W}}|Z_{t}|^{p})$ from the finiteness of $\mathbf {E}(|Z_{\tau _{W}}|^{p})$ for some $p>0$, Burkholder [‘Exit times of Brownian motion, harmonic majorization, and Hardy spaces’, Adv. Math. 26(2) (1977), 182–205] imposed an additional condition on the exit time $\tau _{W}$, namely the finiteness of $\mathbf {E}(\log (\tau _{W}))$. Such a condition is typically difficult to verify, since the law of the exit time is often delicate. In this paper, we revisit Burkholder’s condition and propose an alternative viewpoint. Our approach is purely analytic, weaker and formulated in terms of proper analytic maps rather than exit times themselves. This provides a more flexible framework for further applications.

We prove that the zero-divisor graph $\Gamma (P)$ of a Boolean poset P is both well-covered and Cohen–Macaulay. Furthermore, for a poset $\mathbf {P} = \prod _{i=1}^{n} P_i \ (n \ge 3)$, where each $P_i$ is a finite bounded poset satisfying $Z(P_i) = \{0\}$ for all i and $2 \le |P_1| \le |P_2| \le \cdots \le |P_n|,$ we show that the zero-divisor graph $\Gamma (\mathbf {P})$ is Cohen–Macaulay if and only if $\mathbf {P}$ is a Boolean lattice.

We develop a representation-theoretic refinement of the Iwasawa theory of finite Cayley graphs. Building on analogies between graph zeta functions and number-theoretic L-functions, we study $\mathbb {Z}_\ell $-towers of Cayley graphs and the asymptotic growth of their Jacobians. The main result establishes that the Iwasawa polynomial associated to such a tower admits a canonical factorisation indexed by the irreducible representations of the underlying group. This leads to the definition of representation-theoretic Iwasawa polynomials, whose properties are studied.

We investigate solutions to the matrix Diophantine equation $ X^n + Y^n = n!Z^n $ within specific classes of $2 \times 2$ integer matrices. Firstly, we introduce a class of $2\times 2$ matrices $G_2(d)$ and show that $X^n + Y^n = n!Z^n$ admits a solution in $G_2(d)$ if and only if the classical Diophantine equation $x^n + y^n = n!z^n$ has a solution in $\mathbb {Q}(\sqrt {d})$. We further present a constructive theorem for generating matrix solutions over a general number field K. Finally, we introduce a determinant-fixed class of matrices $G_2(d, l)$, and prove that the Fermat-type equation $X^n+Y^n=2Z^n$ has only the trivial solution in $G_2(d, l)$ whenever $n>2$.

We study a pair of finite multisets of nonnegative integers which have coinciding representation functions. We show that there is a close connection to factorisation of polynomials into cyclotomic polynomials. As an application, we give a complete characterisation of pairs of such sets whose union is the interval $[0,m]$ with two elements removed.

In a recent work of Moreno-Fernandez, Wierstra and the present author [‘A recognition principle for iterated suspensions as coalgebras over the little cubes operad’, Preprint, 2022, arXiv:2210.00839], a coendomorphism operad in the category of pointed topological spaces endowed with the wedge sum was introduced. In this paper, we construct an analogue completely internal to the category of simplicial sets with the goal of defining simplicial coalgebras. As an application, we show that simplicial n-fold suspensions are coalgebras up to coherent homotopy over the Barratt–Eccles $E_n$-operad provided they have finitely many nondegenerate simplices.

We study the congruence classes attained by positive integers D with a prescribed period for the continued fraction of $\sqrt D$. As an application, we refine the results on large ranks of universal quadratic forms over real quadratic fields by imposing congruence conditions on their discriminants.

We solve three problems raised in recent articles by Longstaff [Bull. Aust. Math. Soc. 102 (2020), 226–236; 103 (2021), 260–270; 104 (2021), 78–93]. We show that: (1) the length of an irreducible family of $n\times n$ rank-one matrices can take any value $1,2,\ldots ,n$; (2) the slot length of an irreducible pair of $n\times n$ matrices can exceed $2n-4$; and (3) we give bounds on the $1$-minimum spanning lengths of irreducible pairs.

We study the long-time behaviour of solutions to a free boundary model in river networks with small water flow speeds. First, we establish the well-posedness of our model. Under suitable assumptions, we then establish a spreading–vanishing dichotomy result for solutions to our model. By introducing a parameter $\sigma $ in the initial data, we derive sharp threshold behaviours.

Let A be a subset of an additive abelian semigroup S and let $hA$ be the h-fold sumset of A. The following question is considered. Let $(A_q)_{q=1}^{\infty }$ be a strictly decreasing sequence of sets in S and let $A = \bigcap _{q=1}^{\infty } A_q$. When does one have

$$ \begin{align*} hA = \bigcap_{q=1}^{\infty} hA_q \end{align*} $$

for some or all $h \geq 2$?

Poincaré series are of fundamental importance in the theory of modular forms. For a fixed even integer $k \ge 4$, the space of modular forms of weight k can be spanned by certain Poincaré series. We provide a criterion for a modular form to have a specified number of zeros on the unit circle in the standard fundamental domain for the action of $\mathrm {SL}(2,{\mathbb {Z}})$ on the complex upper half-plane. The criterion is given in terms of the coefficients that are obtained by writing the modular form as a linear combination of the generating Poincaré series.

We use a recent result of Orevkov and Pakovich [‘On intersection of lemniscates of rational functions’, Arnold J. Math. 11 (2025), 7–26] to obtain a bound on the frequency with which elements in an orbit of polynomial iterations can fall on the unit circle.

Let $\triangle $ denote the integers represented by the quadratic form $x^2+3y^2$ and denote the numbers represented as a sum of two squares. For a nonzero integer a, let be the set of integers n such that $n \in \triangle $ and . We conduct a census of in short intervals by showing that there exists a constant $H_{a}> 0$ with

for large x. To derive this result and its generalisation, we use a theorem of Tolev [‘On the remainder term in the circle problem in an arithmetic progression’, Proc. Steklov Inst. Math. 276 (2012), 261–274].

We establish a two-part framework for analysing homotopy equivalence in periodic adelic functions. First, we introduce the concept of pre-periodic functions and define their homotopy invariant through the construction of a generalised winding number. Second, we establish a fundamental correspondence between periodic adelic functions and pre-periodic functions. By extending the generalised winding number to periodic adelic functions, we demonstrate that this invariant completely characterises homotopy equivalence classes within the space of periodic adelic functions. Building on this classification, we further obtain an explicit description of the $K_{1}$-group of the rational group $C^\ast $-algebra, $K_{1}(C^{*}(\mathbb {Q}))$. Finally, we employ a similar strategy to derive the K-theory of $C^{\ast }(\mathbb {A})$.

We consider a nonautonomous abstract evolution equation on Banach space. If the evolutionary operator and the evolutionary kernel are real analytic at the initial time there is an explicit formula for the coefficients of the operator in terms of the coefficients of the kernel. In this note we show that the converse is also true by finding an explicit formula for the coefficients of the kernel in terms of the coefficients of the operator.

Let $f(x) = (x^{2}+1)^{n} - ax^{n} \in \mathbb {Z}[x]$ and assume $f(x)$ is irreducible. Let $\theta $ be a root of $f(x)$, set $K= \mathbb {Q}(\theta )$ and denote by $\mathbb {Z}_{K}$ the ring of integers of K. The index of f, denoted $\operatorname {ind}(f)$, is the index of $\mathbb {Z}[\theta ]$ in $\mathbb {Z}_{K}$. A polynomial $f(x)$ is said to be monogenic if $\operatorname {ind}(f) = 1$. We compute the discriminant of the polynomial $f(x)$, and then derive necessary and sufficient conditions on the parameters a and n for $f(x)$ to be monogenic. Furthermore, we provide a complete description of the primes that divide $\operatorname {ind}(f)$.

We study projection algorithms for finding fixed points of nonexpansive mappings in a Hilbert space. In previous work, the well-definedness of the iterates depended on the nonemptiness of the fixed point set. We first show that the existence of the iterates generated by a general projection algorithm is independent of the existence of fixed points. Next, we introduce a multi-step inertial projection algorithm and establish results on existence, strong convergence and iteration complexity. Our results generalise and improve projection algorithms discussed in the literature.

Recently, Harrington et al. [‘Every arithmetic progression contains infinitely many b-Niven numbers’, Bull. Aust. Math. Soc. 109(3) (2024), 409–413] proved that every arithmetic progression contains infinitely many base-b Niven numbers for any fixed $b\ge 2$. We use a sparse repunit construction to treat a structured two-base version of the same problem, showing that every arithmetic progression with common difference relatively prime to b contains infinitely many integers that are simultaneously b-Niven and $b^k$-Niven (indeed, we can obtain simultaneous $b^\ell $-Niven-ness for $\ell =1,\ldots , k$).

We define the concept of slenderness for arbitrary semigroups containing at least one idempotent element, and give necessary and sufficient conditions for semigroups to be slender. We describe the semigroup of all homomorphisms of the direct product of a nonempty family of groups into a slender rectangular abelian group.

Let $\mathcal {A}$ denote the class of normalised analytic functions f in the open unit disk $\mathbb {D}:=\{z\in \mathbb {C}:|z|<1\}$ with $f(0)=0$ and $f'(0)=1$. A function $f\in \mathcal {A}$ is said to be convex if $f(\mathbb {D})$ is convex. We establish a sharp upper bound for the third Hankel determinant corresponding to the inverse coefficients of convex univalent (that is, one-to-one) functions in the unit disk $\mathbb {D}$.