A condition equivalent to sparseness of a set on the plane is formulated and used as a motivation for a new concept of density point on the plane. This is investigated and compared with known previous versions.

Michael A.B. Deakin (Bull. Austral. Math. Soc. 17 (1977), 161–175) states (on p. 173) a conjecture, which, if true, would lead to simpler proofs of a number of known theorems. This conjecture is, however, shown to be false by means of a counter-example.

For two given graphs $G_{1}$ and $G_{2}$, the planar Ramsey number $PR(G_{1},G_{2})$ is the smallest integer $N$ such that every planar graph $G$ on $N$ vertices either contains $G_{1}$, or its complement contains $G_{2}$. Let $C_{4}$ be a quadrilateral, $T_{n}$ a tree of order $n\geq 3$ with maximum degree $k$, and $K_{1,k}$ a star of order $k+1$. We show that $PR(C_{4},T_{n})=\max \{n+1,PR(C_{4},K_{1,k})\}$. Combining this with a result of Chen et al. [‘All quadrilateral-wheel planar Ramsey numbers’, Graphs Combin.33 (2017), 335–346] yields exact values of all the quadrilateral-tree planar Ramsey numbers.

We give an explicit upper bound for the number of zeros of Hecke–Landau zeta-functions in a rectangle.

In this paper, we find the dual relationship between solidness and the angle property of cones, which is characteristic of reflexivity for locally convex spaces.

Let G be a finite group. A subgroup A of G is said to be S-permutable in G if A permutes with every Sylow subgroup P of G, that is, $AP=PA$. Let $A_{sG}$ be the subgroup of A generated by all S-permutable subgroups of G contained in A and $A^{sG}$ be the intersection of all S-permutable subgroups of G containing A. We prove that if G is a soluble group, then S-permutability is a transitive relation in G if and only if the nilpotent residual $G^{\mathfrak {N}}$ of G avoids the pair $(A^{s G}, A_{sG})$, that is, $G^{\mathfrak {N}}\cap A^{sG}= G^{\mathfrak {N}}\cap A_{sG}$ for every subnormal subgroup A of G.

The authors determine the sharp radius of convexity for functions analytic and starlike in the unit disc having power series representation of the form where an+1 is fixed. The estimate obtained is an improvement over the corresponding fixed second coefficient result. It is expected that this approach will lead to sharpening and improvement of a number of earlier known results.

Let $G$ be a finite group. A subgroup $H$ of $G$ is said to be $E$-supplemented in $G$ if there is a subgroup $T$ of $G$ such that $G= HT$ and $H\cap T\leq {H}_{eG} $, where ${H}_{eG} $ denotes the subgroup of $H$ generated by all those subgroups of $H$ which are $S$-quasinormally embedded in $G$. In this paper, some new characterisations of $p$-supersolubility of finite groups are given under the assumption that some primary subgroups are $E$-supplemented.

For a normed infinite-dimensional space, we prove that the family of all locally convex topologies which are compatible with the original norm topology has cardinality greater or equal to $\mathfrak{c}$.

Here we derive a recursive formula for even-power moments of Kloosterman sums or equivalently for power moments of two-dimensional Kloosterman sums. This is done by using the Pless power-moment identity and an explicit expression of the Gauss sum for Sp(4,q).

Let $b_{3,5}(n)$ denote the number of partitions of $n$ into parts that are not multiples of 3 or 5. We establish several infinite families of congruences modulo 2 for $b_{3,5}(n)$. In the process, we also prove numerous parity results for broken 7-diamond partitions.

One of the important problems of the local theory of Banach Spaces can be stated in the following way. We consider a condition on finite sets in normed spaces that makes sense for a finite set any cardinality. Suppose that the condition is such that to each set satisfying it there corresponds a constant describing “how well” the set satisfies the condition.

The problem is: Suppose that a normed space X has a set of large cardinality satisfying the condition with “poor” constant. Does there exist in X a set of smaller cardinality satisfying the condition with a better constant?

In the paper this problem is studied for conditions associated with one of R.C. James's characterisations of superreflexivity.

Using the equivariant Ljusternik-Schnirelmann theory and the estimate of the upper bound of the critical value and lower bound for the collision solutions, we obtain some new results in the large concerning multiple geometrically distinct periodic solutions of fixed energy for a class of planar N-body type problems.

We show that an essentially amenable Banach algebra need not have an approximate identity. This answers a question posed by Ghahramani and Loy [‘Generalized notions of amenability’, J. Funct. Anal.  208 (2004), 229–260]. Essentially Connes-amenable dual Banach algebras are introduced and studied.

Let α1, …, αm be distinct complex numbers and τ(1), …, τ(m) be non-negative integers. We obtain conditions under which the functions

form a perfect system, that is, for every set ρ(1), …, ρ(m) of non-negative integers, there are polynomials a1 (z), …, am (z) with respective degrees exactly ρ(1)−1, …, ρ(m)−1, such that the function

has a zero of order at least ρ(1) + … + ρ(m)−1 at the origin. Moreover, subject to the evaluation of certain determinants, we give explicit formulae for the approximating polynomials a1 (z), …, am (z).

We generalise a result of Hilbert which asserts that the Riemann zeta-function ${\it\zeta}(s)$ is hypertranscendental over $\mathbb{C}(s)$. Let ${\it\pi}$ be any irreducible cuspidal automorphic representation of $\text{GL}_{m}(\mathbb{A}_{\mathbb{Q}})$ with unitary central character. We establish a certain type of functional difference–differential independence for the associated $L$-function $L(s,{\it\pi})$. This result implies algebraic difference–differential independence of $L(s,{\it\pi})$ over $\mathbb{C}(s)$ (and more strongly, over a certain field ${\mathcal{F}}_{s}$ which contains $\mathbb{C}(s)$). In particular, $L(s,{\it\pi})$ is hypertranscendental over $\mathbb{C}(s)$. We also extend a result of Ostrowski on the hypertranscendence of ordinary Dirichlet series.