We present a systematic approach to the problem whether a topologically infinite-dimensional space can be made homogeneous in the Coifman–Weiss sense. The answer to the question is negative, as expected. Our leading representative of spaces with this property is $\mathbb {T}^\omega = \mathbb {T} \times \mathbb {T} \times \cdots $ with the natural product topology.

We investigate the large values of the derivatives of the Riemann zeta function $\zeta (s)$ on the 1-line. We give a larger lower bound for $\max _{t\in [T,2T]}|\zeta ^{(\ell )}(1+{i} t)|$, which improves the previous result established by Yang [‘Extreme values of derivatives of the Riemann zeta function’, Mathematika 68 (2022), 486–510].

We give effective finiteness results for the power values of polynomials with coefficients composed of a fixed finite set of primes; in particular, of Littlewood polynomials.

Girstmair [‘On an irreducibility criterion of M. Ram Murty’, Amer. Math. Monthly 112(3) (2005), 269–270] gave a generalisation of Ram Murty’s irreducibility criterion. We further generalise these criteria.

We study a question on minimal asymptotic bases asked by Nathanson [‘Minimal bases and powers of 2’, Acta Arith. 49 (1988), 525–532].

We study super weakly compact operators through a quantitative method. We introduce a semi-norm $\sigma (T)$ of an operator $T:X\to Y$ , where X, Y are Banach spaces, the so-called measure of super weak noncompactness, which measures how far T is from the family of super weakly compact operators. We study the equivalence of the measure $\sigma (T)$ and the super weak essential norm of T. We prove that Y has the super weakly compact approximation property if and and only if these two semi-norms are equivalent. As an application, we construct an example to show that the measures of T and its dual $T^*$ are not always equivalent. In addition we give some sequence spaces as examples of Banach spaces having the super weakly compact approximation property.

We correct two minor errors in Theorem 2.1 of Lukkarinen and Pakkanen [‘On the positivity of Riemann–Stieltjes integrals’, Bull. Aust. Math. Soc. 87 (2013), 400–405].

We show that two distinct singular moduli $j(\unicode[STIX]{x1D70F}),j(\unicode[STIX]{x1D70F}^{\prime })$, such that for some positive integers $m$ and $n$ the numbers $1,j(\unicode[STIX]{x1D70F})^{m}$ and $j(\unicode[STIX]{x1D70F}^{\prime })^{n}$ are linearly dependent over $\mathbb{Q}$, generate the same number field of degree at most two. This completes a result of Riffaut [‘Equations with powers of singular moduli’, Int. J. Number Theory, to appear], who proved the above theorem except for two explicit pairs of exceptions consisting of numbers of degree three. The purpose of this article is to treat these two remaining cases.

Necessary and sufficient conditions for the equality of the determinant and permanent for all powers of a given matrix are provided. The characterisation is based on a condition on merely one power.

Z.-W. Sun [‘Refining Lagrange’s four-square theorem’, J. Number Theory175 (2017), 169–190] conjectured that every positive integer n can be written as $ x^2+y^2+z^2+w^2\ (x,y,z,w\in \mathbb {N}=\{0,1,\ldots \})$ with $x+3y$ a square and also as $n=x^2+y^2+z^2+w^2\ (x,y,z,w \in \mathbb {Z})$ with $x+3y\in \{4^k:k\in \mathbb {N}\}$ . In this paper, we confirm these conjectures via the arithmetic theory of ternary quadratic forms.

Christine W. Ayoub has defined diagrams for abelian groups and has shown that the structure of an abelian group A is determined by certain subgroups Ti of A which arise naturally from a given diagram for A. In this paper it is shown that a given diagram for A is formed from certain elementary diagrams for the Ti.

We use a generalised Nevanlinna counting function to compute the Hilbert–Schmidt norm of a composition operator on the Bergman space $L_{a}^{2}(\mathbb{D})$ and weighted Bergman spaces $L_{a}^{1}(\text{d}A_{\unicode[STIX]{x1D6FC}})$ when $\unicode[STIX]{x1D6FC}$ is a nonnegative integer.

For a d-dimensional convex lattice polytope P, a formula for the boundary volume vol(∂P) is derived in terms of the number of boundary lattice points on the first ⌊d/2⌋ dilations of P. As an application we give a necessary and sufficient condition for a polytope to be reflexive, and derive formulas for the f-vector of a smooth polytope in dimensions three, four, and five. We also give applications to reflexive order polytopes, and to the Birkhoff polytope.

We show that a compact convex body in En cannot be partitioned by n + 1 hyperplanes into 2n+l – 1 subsets of equal measure, thus generalising a result in the plane due to R.C. and E.F. Buck.

It is shovn that simple algebras of characteristic not equal to 2, which contain non-trivial elements that satisfy xn = x for some fixed, minimal integer n > 1, are generated by these elements.

In this short note, considering functions, we show that taking an asymptotic viewpoint allows one to prove strong transcendence statements in many general situations. In particular, as a consequence of a more general result, we show that if $F(z)\in \mathbb{C}[[z]]$ is a power series with coefficients from a finite set, then $F(z)$ is either rational or it is transcendental over the field of meromorphic functions.

Using potential theory, we prove the existence of filling disks of an algebroid function of finite order defined in the unit disk.

We investigate the structure of the lattice of subalgebras of an infinite Boolean algebra; in particular, we make a contribution to the question as to when such a lattice is simple.