Lapkova [‘On the average number of divisors of reducible quadratic polynomials’, J. Number Theory 180 (2017), 710–729] uses a Tauberian theorem to derive an asymptotic formula for the divisor sum $\sum _{n\leq x}d(n(n+v))$ where $v$ is a fixed integer and $d(n)$ denotes the number of divisors of $n$. We reprove this result with additional terms in the asymptotic formula, by investigating the relationship between this divisor sum and the well-known sum $\sum _{n\leq x}d(n)d(n+v)$.

Let $G$ be a finite abelian group, $A$ a nonempty subset of $G$ and $h\geq 2$ an integer. For $g\in G$, let $R_{A,h}(g)$ denote the number of solutions of the equation $x_{1}+\cdots +x_{h}=g$ with $x_{i}\in A$ for $1\leq i\leq h$. Kiss et al. [‘Groups, partitions and representation functions’, Publ. Math. Debrecen85(3) (2014), 425–433] proved that (a) if $R_{A,h}(g)=R_{G\setminus A,h}(g)$ for all $g\in G$, then $|G|=2|A|$, and (b) if $h$ is even and $|G|=2|A|$, then $R_{A,h}(g)=R_{G\setminus A,h}(g)$ for all $g\in G$. We prove that $R_{G\setminus A,h}(g)-(-1)^{h}R_{A,h}(g)$ does not depend on $g$. In particular, if $h$ is even and $R_{A,h}(g)=R_{G\setminus A,h}(g)$ for some $g\in G$, then $|G|=2|A|$. If $h>1$ is odd and $R_{A,h}(g)=R_{G\setminus A,h}(g)$ for all $g\in G$, then $R_{A,h}(g)=\frac{1}{2}|A|^{h-1}$ for all $g\in G$. If $h>1$ is odd and $|G|$ is even, then there exists a subset $A$ of $G$ with $|A|=\frac{1}{2}|G|$ such that $R_{A,h}(g)\not =R_{G\setminus A,h}(g)$ for all $g\in G$.

For a given set $S\subset \mathbb{N}$, $R_{S}(n)$ is the number of solutions of the equation $n=s+s^{\prime },s<s^{\prime },s,s^{\prime }\in S$. Suppose that $m$ and $r$ are integers with $m>r\geq 0$ and that $A$ and $B$ are sets with $A\cup B=\mathbb{N}$ and $A\cap B=\{r+mk:k\in \mathbb{N}\}$. We prove that if $R_{A}(n)=R_{B}(n)$ for all positive integers $n$, then there exists an integer $l\geq 1$ such that $r=2^{2l}-1$ and $m=2^{2l+1}-1$. This solves a problem of Chen and Lev [‘Integer sets with identical representation functions’, Integers16 (2016), A36] under the condition $m>r$.

Let $F$ be an integral linear recurrence, $G$ an integer-valued polynomial splitting over the rationals and $h$ a positive integer. Also, let ${\mathcal{A}}_{F,G,h}$ be the set of all natural numbers $n$ such that $\gcd (F(n),G(n))=h$. We prove that ${\mathcal{A}}_{F,G,h}$ has a natural density. Moreover, assuming that $F$ is nondegenerate and $G$ has no fixed divisors, we show that the density of ${\mathcal{A}}_{F,G,1}$ is 0 if and only if ${\mathcal{A}}_{F,G,1}$ is finite.

In this paper, we consider the Diophantine equations

$$\begin{eqnarray}\displaystyle F_{n}^{q}\pm F_{m}^{q}=y^{p} & & \displaystyle \nonumber\end{eqnarray}$$

$q,p\geq 2$

$\gcd (F_{n},F_{m})=1$

$F_{k}$

$q=2$

$q$

$q\equiv 3\;(\text{mod}\;4),3<q<1087$

$q=3$

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.

We consider the function $f(n)$ that enumerates partitions of weight $n$ wherein each part appears an odd number of times. Chern [‘Unlimited parity alternating partitions’, Quaest. Math. (to appear)] noted that such partitions can be placed in one-to-one correspondence with the partitions of $n$ which he calls unlimited parity alternating partitions with smallest part odd. Our goal is to study the parity of $f(n)$ in detail. In particular, we prove a characterisation of $f(2n)$ modulo 2 which implies that there are infinitely many Ramanujan-like congruences modulo 2 satisfied by the function $f.$ The proof techniques are elementary and involve classical generating function dissection tools.

Let $\mathfrak{D}$ be a residually finite Dedekind domain and let $\mathfrak{n}$ be a nonzero ideal of $\mathfrak{D}$. We consider counting problems for the ideal chains in $\mathfrak{D}/\mathfrak{n}$. By using the Cauchy–Frobenius–Burnside lemma, we also obtain some further extensions of Menon’s identity.

We compute Betti numbers for a Cohen–Macaulay tangent cone of a monomial curve in the affine $4$-space corresponding to a pseudo-symmetric numerical semigroup. As a byproduct, we also show that for these semigroups, being of homogeneous type and homogeneous are equivalent properties.

We consider the relationship between structural information of a finite group $G$ and $\text{cd}_{\unicode[STIX]{x1D6FC}}(G)$, the set of all irreducible projective character degrees of $G$ with factor set $\unicode[STIX]{x1D6FC}$. We show that for nontrivial $\unicode[STIX]{x1D6FC}$, if all numbers in $\text{cd}_{\unicode[STIX]{x1D6FC}}(G)$ are prime powers, then $G$ is solvable. Our result is proved by classical character theory using the bijection between irreducible projective representations and irreducible constituents of induced representations in its representation group.

A subgroup $H$ is called a weak second maximal subgroup of $G$ if $H$ is a maximal subgroup of a maximal subgroup of $G$. Let $m(G,H)$ denote the number of maximal subgroups of $G$ containing $H$. We prove that $m(G,H)-1$ divides the index of some maximal subgroup of $G$ when $H$ is a weak second maximal subgroup of $G$. This partially answers a question of Flavell [‘Overgroups of second maximal subgroups’, Arch. Math.64(4) (1995), 277–282] and extends a result of Pálfy and Pudlák [‘Congruence lattices of finite algebras and intervals in subgroup lattices of finite groups’, Algebra Universalis11(1) (1980), 22–27].

Let $f$ be analytic in $\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}$ and given by $f(z)=z+\sum _{n=2}^{\infty }a_{n}z^{n}$. We give sharp bounds for the initial coefficients of the Taylor expansion of such functions in the class of strongly Ozaki close-to-convex functions, and of the initial coefficients of the inverse function, together with some growth estimates.

This paper deals with a non-self-adjoint differential operator which is associated with a diffusion process with random jumps from the boundary. Our main result is that the algebraic multiplicity of an eigenvalue is equal to its order as a zero of the characteristic function $\unicode[STIX]{x1D6E5}(\unicode[STIX]{x1D706})$. This is a new criterion for determining the multiplicities of eigenvalues for concrete operators.

Let $G$ be a separable locally compact group with type $I$ left regular representation, $\widehat{G}$ its dual, $A(G)$ its Fourier algebra and $f\in A(G)$ with compact support. If $G=\mathbb{R}$ and the Fourier transform of $f$ is compactly supported, then, by a classical Paley–Wiener theorem, $f=0$. There are extensions of this theorem for abelian and some unimodular groups. In this paper, we prove that if $G$ has no (nonempty) open compact subsets, $\hat{f}$, the regularised Fourier cotransform of $f$, is compactly supported and $\text{Im}\,\hat{f}$ is finite dimensional, then $f=0$. In connection with this result, we characterise locally compact abelian groups whose identity components are noncompact.

Our main result states that whenever we have a non-Euclidean norm $\Vert \cdot \Vert$ on a two-dimensional vector space $X$, there exists some $x\neq 0$ such that for every $\unicode[STIX]{x1D706}\neq 1$, $\unicode[STIX]{x1D706}>0$, there exist $y,z\in X$ satisfying $\Vert y\Vert =\unicode[STIX]{x1D706}\Vert x\Vert$, $z\neq 0$ and $z$ belongs to the bisectors $B(-x,x)$ and $B(-y,y)$. We also give several results about the geometry of the unit sphere of strictly convex planes.

We study the classical Rosenthal–Szasz inequality for a plane whose geometry is determined by a norm. This inequality states that the bodies of constant width have the largest perimeter among all planar convex bodies of given diameter. In the case where the unit circle of the norm is given by a Radon curve, we obtain an inequality which is completely analogous to the Euclidean case. For arbitrary norms we obtain an upper bound for the perimeter calculated in the anti-norm, yielding an analogous characterisation of all curves of constant width. To derive these results, we use methods from the differential geometry of curves in normed planes.

We consider a curvature flow $V=\unicode[STIX]{x1D705}+A$ in a two-dimensional undulating cylinder $\unicode[STIX]{x1D6FA}$ described by $\unicode[STIX]{x1D6FA}:=\{(x,y)\in \mathbb{R}^{2}\mid -g_{1}(y)<x<g_{2}(y),y\in \mathbb{R}\}$, where $V$ is the normal velocity of a moving curve contacting the boundaries of $\unicode[STIX]{x1D6FA}$ perpendicularly, $\unicode[STIX]{x1D705}$ is its curvature, $A>0$ is a constant and $g_{1}(y),g_{2}(y)$ are positive smooth functions. If $g_{1}$ and $g_{2}$ are periodic functions and there are no stationary curves, Matano et al. [‘Periodic traveling waves in a two-dimensional cylinder with saw-toothed boundary and their homogenization limit’, Netw. Heterog. Media1 (2006), 537–568] proved the existence of a periodic travelling wave. We consider the case where $g_{1},g_{2}$ are general nonperiodic positive functions and the problem has some stationary curves. For each stationary curve $\unicode[STIX]{x1D6E4}$ unstable from above/below, we construct an entire solution growing out of it, that is, a solution curve $\unicode[STIX]{x1D6E4}_{t}$ which increases/decreases monotonically, converging to $\unicode[STIX]{x1D6E4}$ as $t\rightarrow -\infty$ and converging to another stationary curve or to $+\infty /-\infty$ as $t\rightarrow \infty$.

Given a one-dimensional downwards transient diffusion process $X$, we consider a random time $\unicode[STIX]{x1D70C}$, the last exit time when $X$ exits a certain level $\ell$, and detect the optimal stopping time for it. In particular, for this random time $\unicode[STIX]{x1D70C}$, we solve the optimisation problem $\inf _{\unicode[STIX]{x1D70F}}\mathbb{E}[\unicode[STIX]{x1D706}(\unicode[STIX]{x1D70F}-\unicode[STIX]{x1D70C})_{+}+(1-\unicode[STIX]{x1D706})(\unicode[STIX]{x1D70C}-\unicode[STIX]{x1D70F})_{+}]$ over all stopping times $\unicode[STIX]{x1D70F}$. We show that the process should stop optimally when it runs below some fixed level $\unicode[STIX]{x1D705}_{\ell }$ for the first time, where $\unicode[STIX]{x1D705}_{\ell }$ is the unique solution in the interval $(0,\unicode[STIX]{x1D706}\ell )$ of an explicitly defined equation.

We establish a separation principle for a class of fractional order time-delay nonlinear differential systems. We show that a nonlinear time-delay observer is globally convergent and give sufficient conditions under which the observer-based controller stabilises the system.