We give the crossing number of the join product $W_{4}+D_{n}$, where $W_{4}$ is the wheel on five vertices and $D_{n}$ consists of $n$ isolated vertices. The proof is based on calculating the minimum number of crossings between two different subgraphs from the set of subgraphs which do not cross the edges of the graph $W_{4}$ and from the set of subgraphs which cross the edges of $W_{4}$ exactly once.

We show that for all $m,k,r\in \mathbb{N}$, there is an $n\in \mathbb{N}$ such that whenever $L$ is a Latin square of order $m$ and the Cartesian product $L^{n}$ of $n$ copies of $L$ is $r$-coloured, there is a monochrome Latin subsquare of $L^{n}$, isotopic to $L^{k}$. In particular, for every prime $p$ and for all $k,r\in \mathbb{N}$, there is an $n\in \mathbb{N}$ such that whenever the multiplication table $L({\mathbb{Z}_{p}}^{n})$ of the group ${\mathbb{Z}_{p}}^{n}$ is $r$-coloured, there is a monochrome Latin subsquare of order $p^{k}$. On the other hand, we show that for every group $G$ of order $\leq 15$, there is a 2-colouring of $L(G)$ without a nontrivial monochrome Latin subsquare.

Let $p>3$ be a prime and let $a$ be a rational $p$-adic integer with $a\not \equiv 0\;(\text{mod}\;p)$. We evaluate

$$\begin{eqnarray}\mathop{\sum }_{k=1}^{(p-1)/2}\frac{1}{k}\binom{a}{k}\binom{-1-a}{k}\quad \text{and}\quad \mathop{\sum }_{k=0}^{(p-1)/2}\frac{1}{2k-1}\binom{a}{k}\binom{-1-a}{k}\end{eqnarray}$$

$p^{2}$

Under sufficiently strong assumptions about the first prime in an arithmetic progression, we prove that the number of Carmichael numbers up to $X$ is $\gg X^{1-R}$, where $R=(2+o(1))\log \log \log \log X/\text{log}\log \log X$. This is close to Pomerance’s conjectured density of $X^{1-R}$ with $R=(1+o(1))\log \log \log X/\text{log}\log X$.

For every integer $k\geq 2$ and every $A\subseteq \mathbb{N}$, we define the $k$-directions sets of $A$ as $D^{k}(A):=\{\boldsymbol{a}/\Vert \boldsymbol{a}\Vert :\boldsymbol{a}\in A^{k}\}$ and $D^{\text{}\underline{k}}(A):=\{\boldsymbol{a}/\Vert \boldsymbol{a}\Vert :\boldsymbol{a}\in A^{\text{}\underline{k}}\}$, where $\Vert \cdot \Vert$ is the Euclidean norm and $A^{\text{}\underline{k}}:=\{\boldsymbol{a}\in A^{k}:a_{i}\neq a_{j}\text{ for all }i\neq j\}$. Via an appropriate homeomorphism, $D^{k}(A)$ is a generalisation of the ratio set$R(A):=\{a/b:a,b\in A\}$. We study $D^{k}(A)$ and $D^{\text{}\underline{k}}(A)$ as subspaces of $S^{k-1}:=\{\boldsymbol{x}\in [0,1]^{k}:\Vert \boldsymbol{x}\Vert =1\}$. In particular, generalising a result of Bukor and Tóth, we provide a characterisation of the sets $X\subseteq S^{k-1}$ such that there exists $A\subseteq \mathbb{N}$ satisfying $D^{\text{}\underline{k}}(A)^{\prime }=X$, where $Y^{\prime }$ denotes the set of accumulation points of $Y$. Moreover, we provide a simple sufficient condition for $D^{k}(A)$ to be dense in $S^{k-1}$. We conclude with questions for further research.

Suppose $a^{2}(a^{2}+1)$ divides $b^{2}(b^{2}+1)$ with $b>a$. We improve a previous result and prove a gap principle, without any additional assumptions, namely $b\gg a(\log a)^{1/8}/(\log \log a)^{12}$. We also obtain $b\gg _{\unicode[STIX]{x1D716}}a^{15/14-\unicode[STIX]{x1D716}}$ under the abc conjecture.

Let $d_{3}(n)$ be the divisor function of order three. Let $g$ be a Hecke–Maass form for $\unicode[STIX]{x1D6E4}$ with $\unicode[STIX]{x1D6E5}g=(1/4+t^{2})g$. Suppose that $\unicode[STIX]{x1D706}_{g}(n)$ is the $n$th Hecke eigenvalue of $g$. Using the Voronoi summation formula for $\unicode[STIX]{x1D706}_{g}(n)$ and the Kuznetsov trace formula, we estimate a shifted convolution sum of $d_{3}(n)$ and $\unicode[STIX]{x1D706}_{g}(n)$ and show that

$$\begin{eqnarray}\mathop{\sum }_{n\leq x}d_{3}(n)\unicode[STIX]{x1D706}_{g}(n-1)\ll _{t,\unicode[STIX]{x1D700}}x^{8/9+\unicode[STIX]{x1D700}}.\end{eqnarray}$$

$d_{3}$

We give a partial answer to a question attributed to Chris Miller on algebraic values of certain transcendental functions of order less than one. We obtain $C(\log H)^{\unicode[STIX]{x1D702}}$ bounds for the number of algebraic points of height at most $H$ on certain subsets of the graphs of such functions. The constant $C$ and exponent $\unicode[STIX]{x1D702}$ depend on data associated with the functions and can be effectively computed from them.

Four classes of multiple series, which can be considered as multifold counterparts of four classical summation formulae related to the Riemann zeta series, are evaluated in closed form.

In this paper, we study some connections between the polynomial ring $R[y]$ and the differential polynomial ring $R[x;D]$. In particular, we answer a question posed by Smoktunowicz, which asks whether $R[y]$ is nil when $R[x;D]$ is nil, provided that $R$ is an algebra over a field of positive characteristic and $D$ is a locally nilpotent derivation.

An example of a cocomplete abelian category that is not complete is constructed.

Given a finite group $G$, define the minimal degree $\unicode[STIX]{x1D707}(G)$ of $G$ to be the least $n$ such that $G$ embeds into $S_{n}$. We call $G$ exceptional if there is some $N\unlhd G$ with $\unicode[STIX]{x1D707}(G/N)>\unicode[STIX]{x1D707}(G)$, in which case we call $N$ distinguished. We prove here that a subgroup with no abelian composition factors is not distinguished.

According to a conjecture by Yang, if $f(z)f^{(k)}(z)$ is a periodic function, where $f(z)$ is a transcendental entire function and $k$ is a positive integer, then $f(z)$ is also a periodic function. We propose related questions, which can be viewed as difference or differential-difference versions of Yang’s conjecture. We consider the periodicity of a transcendental entire function $f(z)$ when differential, difference or differential-difference polynomials in $f(z)$ are periodic. For instance, we show that if $f(z)^{n}f(z+\unicode[STIX]{x1D702})$ is a periodic function with period $c$, then $f(z)$ is also a periodic function with period $(n+1)c$, where $f(z)$ is a transcendental entire function of hyper-order $\unicode[STIX]{x1D70C}_{2}(f)<1$ and $n\geq 2$ is an integer.

In this paper we study a class ${\mathcal{Z}}_{H}$ of harmonic mappings on the open unit disk $\mathbb{D}$ in the complex plane that is an extension of the classical (analytic) Zygmund space. We extend to the elements of this class a characterisation that is valid in the analytic case. We also provide a similar result for a closed separable subspace of ${\mathcal{Z}}_{H}$ which we call the little harmonic Zygmund space.

Suppose that $G=(V,E)$ is a finite graph with the vertex set $V$ and the edge set $E$. Let $\unicode[STIX]{x1D6E5}$ be the usual graph Laplacian. Consider the nonlinear Schrödinger equation of the form

$$\begin{eqnarray}-\unicode[STIX]{x1D6E5}u-\unicode[STIX]{x1D6FC}u=f(x,u),\quad u\in W^{1,2}(V),\end{eqnarray}$$

$G$

$f(x,u):V\times \mathbb{R}\rightarrow \mathbb{R}$

$\unicode[STIX]{x1D6FC}$

$G$

$G$

$CD(m,\unicode[STIX]{x1D709})$

$G$

$\unicode[STIX]{x1D6FC}<2\unicode[STIX]{x1D706}_{1}^{2}/m(\unicode[STIX]{x1D706}_{1}-\unicode[STIX]{x1D709})$

$\unicode[STIX]{x1D706}_{1}$

Let $S$ be a discrete inverse semigroup, $l^{1}(S)$ the Banach semigroup algebra on $S$ and $\mathbb{X}$ a Banach $l^{1}(S)$-bimodule which is an $L$-embedded Banach space. We show that under some mild conditions ${\mathcal{H}}^{1}(l^{1}(S),\mathbb{X})=0$. We also provide an application of the main result.

We introduce the notion of a distributional $k$-Hessian ($k=2,\ldots ,n$) associated with fractional Sobolev functions on $\unicode[STIX]{x1D6FA}$, a smooth bounded open subset in $\mathbb{R}^{n}$. We show that the distributional $k$-Hessian is weakly continuous on the fractional Sobolev space $W^{2-2/k,k}(\unicode[STIX]{x1D6FA})$ and that the weak continuity result is optimal, that is, the distributional $k$-Hessian is well defined in $W^{s,p}(\unicode[STIX]{x1D6FA})$ if and only if $W^{s,p}(\unicode[STIX]{x1D6FA})\subseteq W^{2-2/k,k}(\unicode[STIX]{x1D6FA})$.