Swisher [‘On the supercongruence conjectures of van Hamme’, Res. Math. Sci. 2 (2015), Article no. 18] and He [‘Supercongruences on truncated hypergeometric series’, Results Math. 72 (2017), 303–317] independently proved that Van Hamme’s (G.2) supercongruence holds modulo $p^4$ for any prime $p\equiv 1\pmod {4}$ . Swisher also obtained an extension of Van Hamme’s (G.2) supercongruence for $p\equiv 3 \pmod 4$ and $p>3$ . In this note, we give new one-parameter generalisations of Van Hamme’s (G.2) supercongruence modulo $p^3$ for any odd prime p. Our proof uses the method of ‘creative microscoping’ introduced by Guo and Zudilin [‘A q-microscope for supercongruences’, Adv. Math. 346 (2019), 329–358].

We use a linear algebra interpretation of the action of Hecke operators on Drinfeld cusp forms to prove that when the dimension of the $\mathbb {C}_\infty $ -vector space $S_{k,m}(\mathrm {{GL}}_2(\mathbb {F}_q[t]))$ is one, the Hecke operator $\mathbf {T}_t$ is injective on $S_{k,m}(\mathrm {{GL}}_2(\mathbb {F}_q[t]))$ and $S_{k,m}(\Gamma _0(t))$ is a direct sum of oldforms and newforms.

We consider the set of elements in a translation of the middle-third Cantor set which can be well approximated by algebraic numbers of bounded degree. A doubling dimensional result is given, which enables one to conclude an upper bound on the dimension of the set in question for a generic translation.

Let $0\leq \alpha \leq \infty $ , $0\leq a\leq b\leq \infty $ and $\psi $ be a positive function defined on $(0,\infty )$ . This paper is concerned with the growth of $L_{n}(x)$ , the largest digit of the first n terms in the Lüroth expansion of $x\in (0,1]$ . Under some suitable assumptions on the function $\psi $ , we completely determine the Hausdorff dimensions of the sets

$$\begin{align*}E_\psi(\alpha)=\bigg\{x\in(0,1]: \lim\limits_{n\rightarrow\infty}\frac{\log L_n(x)}{\log\psi(n)}=\alpha\bigg\} \end{align*}$$

and

$$\begin{align*}E_\psi(a,b)=\bigg\{x\in(0,1]: \liminf\limits_{n\rightarrow\infty}\frac{\log L_n(x)}{\log\psi(n)}=a, \limsup\limits_{n\rightarrow\infty}\frac{\log L_n(x)}{\log\psi(n)}=b\bigg\}. \end{align*}$$

We study Ohno–Zagier type relations for multiple t-values and multiple t-star values. We represent the generating function of sums of multiple t-(star) values with fixed weight, depth and height in terms of the generalised hypergeometric function $\,_3F_2$ . As applications, we get a formula for the generating function of sums of multiple t-(star) values of maximal height and a weighted sum formula for sums of multiple t-(star) values with fixed weight and depth.

By analogy with the trace of an algebraic integer $\alpha $ with conjugates $\alpha _1=\alpha , \ldots , \alpha _d$ , we define the G-measure $ {\mathrm {G}} (\alpha )= \sum _{i=1}^d ( |\alpha _i| + 1/ | \alpha _i | )$ and the absolute ${\mathrm G}$ -measure ${\mathrm {g}}(\alpha )={\mathrm {G}}(\alpha )/d$ . We establish an analogue of the Schur–Siegel–Smyth trace problem for totally positive algebraic integers. Then we consider the case where $\alpha $ has all its conjugates in a sector $| \arg z | \leq \theta $ , $0 < \theta < 90^{\circ }$ . We compute the greatest lower bound $c(\theta )$ of the absolute G-measure of $\alpha $ , for $\alpha $ belonging to $11$ consecutive subintervals of $]0, 90 [$ . This phenomenon appears here for the first time, conforming to a conjecture of Rhin and Smyth on the nature of the function $c(\theta )$ . All computations are done by the method of explicit auxiliary functions.

Let $\alpha $ be a totally positive algebraic integer of degree d, with conjugates $\alpha _1=\alpha , \alpha _2, \ldots , \alpha _d$ . The absolute $S_k$ -measure of $\alpha $ is defined by $s_k(\alpha )= d^{-1} \sum _{i=1}^{d}\alpha _i^k$ . We compute the lower bounds $\upsilon _k$ of $s_k(\alpha )$ for each integer in the range $2\leq k \leq 15$ and give a conjecture on the results for integers $k>15$ . Then we derive the lower bounds of $s_k(\alpha )$ for all real numbers $k>2$ . Our computation is based on an improvement in the application of the LLL algorithm and analysis of the polynomials in the explicit auxiliary functions.

Let Q be a quiver of type $\tilde {A}_n$ . Let $\alpha =\alpha _1+\alpha _2+\cdots +\alpha _s$ be the canonical decomposition. For the polynomials $M_Q(\alpha ,q)$ that count the number of isoclasses of representations of Q over ${\mathbb F}_q$ with dimension vector $\alpha $ , we obtain a precise relation between the degree of $M_Q(\alpha ,q)$ and that of $\prod _{i=1}^{s} M_Q(\alpha _i,q)$ for an arbitrary dimension vector $\alpha $ .

We describe finite soluble nonnilpotent groups in which every minimal nonnilpotent subgroup is abnormal. We also show that if G is a nonsoluble finite group in which every minimal nonnilpotent subgroup is abnormal, then G is quasisimple and $Z(G)$ is cyclic of order $|Z(G)|\in \{1, 2, 3, 4\}$ .

In this note, we investigate some products of subgroups and vanishing conjugacy class sizes of finite groups. We prove some supersolubility criteria for groups with restrictions on the vanishing conjugacy class sizes of their subgroups.

A group is called quasihamiltonian if all its subgroups are permutable, and we say that a subgroup Q of a group G is permutably embedded in G if $\langle Q,g\rangle $ is quasihamiltonian for each element g of G. It is proved here that if a group G contains a permutably embedded normal subgroup Q such that $G/Q$ is Černikov, then G has a quasihamiltonian subgroup of finite index; moreover, if G is periodic, then it contains a Černikov normal subgroup N such that $G/N$ is quasihamiltonian. This result should be compared with theorems of Černikov and Schlette stating that if a group G is Černikov over its centre, then G is abelian-by-finite and its commutator subgroup is Černikov.

In 1844, Joseph Liouville proved the existence of transcendental numbers. He introduced the set $\mathcal L$ of numbers, now known as Liouville numbers, and showed that they are all transcendental. It is known that $\mathcal L$ has cardinality $\mathfrak {c}$, the cardinality of the continuum, and is a dense $G_{\delta }$ subset of the set $\mathbb {R}$ of all real numbers. In 1962, Erdős proved that every real number is the sum of two Liouville numbers. In this paper, a set W of complex numbers is said to have the Erdős property if every real number is the sum of two numbers in W. The set W is said to be an Erdős–Liouville set if it is a dense subset of $\mathcal {L}$ and has the Erdős property. Each subset of $\mathbb {R}$ is assigned its subspace topology, where $\mathbb {R}$ has the euclidean topology. It is proved here that: (i) there exist $2^{\mathfrak {c}}$ Erdős–Liouville sets no two of which are homeomorphic; (ii) there exist $\mathfrak {c}$ Erdős–Liouville sets each of which is homeomorphic to $\mathcal {L}$ with its subspace topology and homeomorphic to the space of all irrational numbers; (iii) each Erdős–Liouville set L homeomorphic to $\mathcal {L}$ contains another Erdős–Liouville set $L'$ homeomorphic to $\mathcal {L}$. Therefore, there is no minimal Erdős–Liouville set homeomorphic to $\mathcal {L}$.

Inspired by Xiao’s work on Hankel measures for Hardy and Bergman spaces [‘Pseudo-Carleson measures for weighted Bergman spaces’. Michigan Math. J. 47 (2000), 447–452], we introduce Hankel measures for Fock space $F^p_\alpha $ . Given $p\ge 1$ , we obtain several equivalent descriptions for such measures on $F^p_\alpha $ .

Recently, Lin and Liu [‘Congruences for the truncated Appell series $F_3$ and $F_4$ ’, Integral Transforms Spec. Funct. 31(1) (2020), 10–17] confirmed a supercongruence on the truncated Appell series $F_3$ . Motivated by their work, we give a generalisation of this supercongruence by establishing a q-supercongruence modulo the fourth power of a cyclotomic polynomial.

We establish local-in-time Strichartz estimates for solutions of the model case Dirichlet wave equation inside cylindrical convex domains $\Omega \subset \mathbb {R}^ 3$ with smooth boundary $\partial \Omega \neq \emptyset $ . The key ingredients to prove Strichartz estimates are dispersive estimates, energy estimates, interpolation and $TT^*$ arguments. Strichartz estimates for waves inside an arbitrary domain $\Omega $ have been proved by Blair, Smith and Sogge [‘Strichartz estimates for the wave equation on manifolds with boundary’, Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), 1817–1829]. We provide a detailed proof of the usual Strichartz estimates from dispersive estimates inside cylindrical convex domains for a certain range of the wave admissibility.

Let W be a real vector space and let V be an orthogonal representation of a group G such that $V^{G} = \{0\}$ (for the set of fixed points of G). Let $S(V)$ be the sphere of V and suppose that $f: S(V) \to W$ is a continuous map. We estimate the size of the $(H, G)$ -coincidences set if G is a cyclic group of prime power order $\mathbb {Z}_{p^k}$ or a p-torus $\mathbb {Z}_p^k$ .

Let $(M,g)$ be a closed Riemannian $4$ -manifold and let E be a vector bundle over M with structure group G, where G is a compact Lie group. We consider a new higher order Yang–Mills–Higgs functional, in which the Higgs field is a section of $\Omega ^0(\text {ad}E)$ . We show that, under suitable conditions, solutions to the gradient flow do not hit any finite time singularities. In the case that E is a line bundle, we are able to use a different blow-up procedure and obtain an improvement of the long-time result of Zhang [‘Gradient flows of higher order Yang–Mills–Higgs functionals’, J. Aust. Math. Soc. 113 (2022), 257–287]. The proof relies on properties of the Green function, which is very different from the previous techniques.

Wu and Shi [‘A note on k-Galois LCD codes over the ring $\mathbb {F}_q + u\mathbb {F}_q$ ’, Bull. Aust. Math. Soc. 104(1) (2021), 154–161] studied $ k $ -Galois LCD codes over the finite chain ring $\mathcal {R}=\mathbb {F}_q+u\mathbb {F}_q$ , where $u^2=0$ and $ q=p^e$ for some prime p and positive integer e. We extend the results to the finite nonchain ring $ \mathcal {R} =\mathbb {F}_q+u\mathbb {F}_q+v\mathbb {F}_q+uv\mathbb {F}_q$ , where $u^2=u,v^2=v $ and $ uv=vu $ . We define a correspondence between the $ l $ -Galois dual of linear codes over $ \mathcal {R} $ and the $ l $ -Galois dual of their component codes over $ \mathbb {F}_q $ . Further, we construct Euclidean LCD and $ l $ -Galois LCD codes from linear codes over $ \mathcal {R} $ . We prove that any linear code over $ \mathcal {R} $ is equivalent to a Euclidean code over $\mathbb {F}_q$ with $ q>3 $ and an $ l $ -Galois LCD code over $ \mathcal {R}$ with $0<l<e$ and $p^{e-l}+1\mid p^e-1$ . Finally, we investigate MDS codes over $ \mathcal {R}$ .