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.

Let $\mathbb {T}$ be the unit circle and ${\Gamma \backslash G}$ the $3$ -dimensional Heisenberg nilmanifold. We consider the skew products on $\mathbb {T} \times {\Gamma \backslash G}$ and prove that the Möbius function is linearly disjoint from these skew products which improves the recent result of Huang, Liu and Wang [‘Möbius disjointness for skew products on a circle and a nilmanifold’, Discrete Contin. Dyn. Syst. 41(8) (2021), 3531–3553].

We define a graph encoding the structure of contact surgery on contact $3$ -manifolds and analyse its basic properties and some of its interesting subgraphs.

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.

For a continuous and positive function $w(\lambda )$ , $\lambda>0$ and $\mu $ a positive measure on $(0,\infty )$ , we consider the integral transform

$$ \begin{align*} \mathcal{D}( w,\mu ) ( T) :=\int_{0}^{\infty }w(\lambda) ( \lambda +T) ^{-1}\,d\mu ( \lambda ) , \end{align*} $$

where the integral is assumed to exist for T a positive operator on a complex Hilbert space H. We show among other things that if B, $A>0,$ then $\mathcal {D}( w,\mu ) $ is operator subadditive on $(0,\infty ) $ , that is,

$$ \begin{align*} \mathcal{D}( w,\mu ) ( A) +\mathcal{D}( w,\mu) ( B) \geq \mathcal{D}( w,\mu )(A+B). \end{align*} $$

From this, we derive that if $f:[0,\infty )\rightarrow \mathbb {R}$ is an operator monotone function on $[0,\infty )$ , then the function $[ f( t) -f( 0) ] t^{-1}$ is operator subadditive on $( 0,\infty ) .$ Also, if $f:[0,\infty )\rightarrow \mathbb {R}$ is an operator convex function on $[0,\infty )$ , then the function $[ f( t) -f( 0) -f_{+}^{\prime }( 0) t ] t^{-2}$ is operator subadditive on $( 0,\infty ) .$

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 $\mathbb {Z}_p^k$ .

We present novel approaches for constructing linear codes over $\mathbb {Z}_{4}$ from the known ones. We obtain new linear codes, many of which are optimal. In particular, we find all optimal codes of type $4^{k_{1}}2^{k_{2}}$ for $k_{1}=2,~k_{2}=0$ and many optimal codes for $k_{1}=3,~k_{2}=0.$

We prove that a Banach algebra B that is a completion of the universal enveloping algebra of a finite-dimensional complex Lie algebra $\mathfrak {g}$ satisfies a polynomial identity if and only if the nilpotent radical $\mathfrak {n}$ of $\mathfrak {g}$ is associatively nilpotent in B. Furthermore, this holds if and only if a certain polynomial growth condition is satisfied on $\mathfrak {n}$ .

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.

Given $E \subseteq \mathbb {F}_q^d \times \mathbb {F}_q^d$ , with the finite field $\mathbb {F}_q$ of order q and the integer $d\,\ge \, 2$ , we define the two-parameter distance set $\Delta _{d, d}(E)=\{(\|x-y\|, \|z-t\|) : (x, z), (y, t) \in E \}$ . Birklbauer and Iosevich [‘A two-parameter finite field Erdős–Falconer distance problem’, Bull. Hellenic Math. Soc. 61 (2017), 21–30] proved that if $|E| \gg q^{{(3d+1)}/{2}}$ , then $ |\Delta _{d, d}(E)| = q^2$ . For $d=2$ , they showed that if $|E| \gg q^{{10}/{3}}$ , then $ |\Delta _{2, 2}(E)| \gg q^2$ . In this paper, we give extensions and improvements of these results. Given the diagonal polynomial $P(x)=\sum _{i=1}^da_ix_i^s\in \mathbb F_q[x_1,\ldots , x_d]$ , the distance induced by P over $\mathbb {F}_q^d$ is $\|x-y\|_s:=P(x-y)$ , with the corresponding distance set $\Delta ^s_{d, d}(E)=\{(\|x-y\|_s, \|z-t\|_s) : (x, z), (y, t) \in E \}$ . We show that if $|E| \gg q^{{(3d+1)}/{2}}$ , then $ |\Delta _{d, d}^s(E)| \gg q^2$ . For $d=2$ and the Euclidean distance, we improve the former result over prime fields by showing that $ |\Delta _{2,2}(E)| \gg p^2$ for $|E| \gg p^{{13}/{4}}$ .

We prove that for two connected sets $E,F\subset \mathbb {R}^2$ with cardinalities greater than $1$ , if one of E and F is compact and not a line segment, then the arithmetic sum $E+F$ has nonempty interior. This improves a recent result of Banakh et al. [‘The continuity of additive and convex functions which are upper bounded on non-flat continua in $\mathbb {R}^n$ ’, Topol. Methods Nonlinear Anal. 54(1)(2019), 247–256] in dimension two by relaxing their assumption that E and F are both compact.

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}$ .