Let G be a graph. Assume that to each vertex of a set of vertices $S\subseteq V(G)$ a robot is assigned. At each stage one robot can move to a neighbouring vertex. Then S is a mobile general position set of G if there exists a sequence of moves of the robots such that all the vertices of G are visited while maintaining the general position property at all times. The mobile general position number of G is the cardinality of a largest mobile general position set of G. We give bounds on the mobile general position number and determine exact values for certain common classes of graphs, including block graphs, rooted products, unicyclic graphs, Kneser graphs $K(n,2)$ and line graphs of complete graphs.

We discuss near-perfect numbers of various forms. In particular, we study the existence of near-perfect numbers in the Fibonacci and Lucas sequences, near-perfect values taken by integer polynomials and repdigit near-perfect numbers.

Let a and b be positive integers and let $\{U_n\}_{n\ge 0}$ be the Lucas sequence of the first kind defined by

$$ \begin{align*}U_0=0,\quad U_1=1\quad \mbox{and} \quad U_n=aU_{n-1}+bU_{n-2} \quad \mbox{for }n\ge 2.\end{align*} $$

We define an $(a,b)$-Wall–Sun–Sun prime to be a prime p such that $\gcd (p,b)=1$ and $\pi (p^2)=\pi (p),$ where $\pi (p):=\pi _{(a,b)}(p)$ is the length of the period of $\{U_n\}_{n\ge 0}$ modulo p. When $(a,b)=(1,1)$, such primes are known in the literature simply as Wall–Sun–Sun primes. In this note, we provide necessary and sufficient conditions such that a prime p dividing $a^2+4b$ is an $(a,b)$-Wall–Sun–Sun prime.

Let $\ell $ and p be (not necessarily distinct) prime numbers and F be a global function field of characteristic $\ell $ with field of constants $\kappa $. Assume that there exists a prime $P_\infty $ of F which has degree $1$ and let $\mathcal {O}_F$ be the subring of F consisting of functions with no poles away from $P_\infty $. Let $f(X)$ be a polynomial in X with coefficients in $\kappa $. We study solutions to Diophantine equations of the form $Y^{n}=f(X)$ which lie in $\mathcal {O}_F$ and, in particular, show that if m and $f(X)$ satisfy additional conditions, then there are no nonconstant solutions. The results apply to the study of solutions to $Y^{n}=f(X)$ in certain rings of integers in $\mathbb {Z}_{p}$-extensions of F known as constant $\mathbb {Z}_p$-extensions. We prove similar results for solutions in the polynomial ring $K[T_1, \ldots , T_r]$, where K is any field of characteristic $\ell $, showing that the only solutions must lie in K. We apply our methods to study solutions of Diophantine equations of the form $Y^n=\sum _{i=1}^d (X+ir)^m$, where $m,n, d\geq 2$ are integers.

Let $\psi $ be a decreasing function. We prove zero-infinity Hausdorff measure criteria for the set of dual $\psi $-approximable points and for the set of inhomogeneous multiplicative $\psi $-approximable points on nondegenerate planar curves. Our results extend theorems of Huang [‘Hausdorff theory of dual approximation on planar curves’, J. reine angew. Math. 740 (2018), 63–76] and Beresnevich and Velani [‘A note on three problems in metric Diophantine approximation’, in: Recent Trends in Ergodic Theory and Dynamical Systems, Contemporary Mathematics, 631 (American Mathematical Society, Providence, RI, 2015), 211–229] from s-Hausdorff measure, where $s\in \mathbb R$, to the more general g-Hausdorff measure, where g is a suitable class of dimension functions.

Recently, when studying intricate connections between Ramanujan’s theta functions and a class of partition functions, Banerjee and Dastidar [‘Ramanujan’s theta functions and parity of parts and cranks of partitions’, Ann. Comb., to appear] studied some arithmetic properties for $c_o(n)$, the number of partitions of n with odd crank. They conjectured a congruence modulo $4$ satisfied by $c_o(n)$. We confirm the conjecture and evaluate $c_o(4n)$ modulo $8$ by dissecting some q-series into even powers. Moreover, we give a conjecture on the density of divisibility of odd cranks modulo 4, 8 and 16.

Let k be a field of characteristic zero and $k^{[n]}$ the polynomial algebra in n variables over k. The LND conjecture concerning the images of locally nilpotent derivations arose from the Jacobian conjecture. We give a positive answer to the LND conjecture in several cases. More precisely, we prove that the images of rank-one locally nilpotent derivations of $k^{[n]}$ acting on principal ideals are MZ-subspaces for any $n\geq 2$, and that the images of a large class of locally nilpotent derivations of $k^{[3]}$ (including all rank-two and homogeneous rank-three locally nilpotent derivations) acting on principal ideals are MZ-subspaces.

Let D be a division ring and N be a subnormal subgroup of the multiplicative group $D^*$. We show that if N contains a nonabelian solvable subgroup, then N contains a nonabelian free subgroup.

Suppose that R is an associative unital ring and that $E=(E^0,E^1,r,s)$ is a directed graph. Using results from graded ring theory, we show that the associated Leavitt path algebra $L_R(E)$ is simple if and only if R is simple, $E^0$ has no nontrivial hereditary and saturated subset, and every cycle in E has an exit. We also give a complete description of the centre of a simple Leavitt path algebra.

We prove that if a solvable group A acts coprimely on a solvable group G, then A has a relatively ‘large’ orbit in its corresponding action on the set of ordinary complex irreducible characters of G. This improves an earlier result of Keller and Yang [‘Orbits of finite solvable groups on characters’, Israel J. Math. 199 (2014), 933–940].

Let G be a p-group for some prime p. Recall that the Hughes subgroup of G is the subgroup generated by all of the elements of G with order not equal to p. In this paper, we prove that if the Hughes subgroup of G is cyclic, then G has exponent p or is cyclic or is dihedral. We also prove that if the Hughes subgroup of G is generalised quaternion, then G must be generalised quaternion. With these results in hand, we classify the tidy p-groups.

A generating set S for a group G is independent if the subgroup generated by $S\setminus \{s\}$ is properly contained in G for all $s \in S.$ We describe the structure of finite groups G such that there are precisely two numbers appearing as the cardinalities of independent generating sets for G.

We prove a reversed Hardy–Littlewood–Pólya inequality with finite terms. We also give the limit of the best constant.

In 2011, Guillera [‘A new Ramanujan-like series for $1/\pi ^2$’, Ramanujan J. 26 (2011), 369–374] introduced a remarkable rational ${}_{7}F_{6}( \frac {27}{64} )$-series for ${1}/{\pi ^2}$ using the Wilf–Zeilberger (WZ) method, and Chu and Zhang later proved this evaluation using an acceleration method based on Dougall’s ${}_{5}F_{4}$-sum. Another proof of Guillera’s ${}_{7}F_{6}( \frac {27}{64} )$-series was given by Guillera in 2018, and this subsequent proof used a recursive argument involving Dougall’s sum together with the WZ method. Subsequently, Chen and Chu introduced a q-analogue of Guillera’s ${}_{7}F_{6}( \frac {27}{64} )$-series. The many past research articles concerning Guillera’s ${}_{7}F_{6}( \frac {27}{64} )$-series for ${1}/{\pi ^2}$ naturally lead to questions about similar results for other mathematical constants. We apply a WZ-based acceleration method to prove new rational ${}_{7}F_{6}( \frac {27}{64} )$- and ${}_{6}F_{5}( \frac {27}{64} )$-series for $\sqrt {2}$.

Vertically vibrating a liquid bath may allow a self-propelled wave-particle entity to move on its free surface. The horizontal dynamics of this walking droplet, under the constraint of an external drag force, can be described adequately by an integro-differential trajectory equation. For a sinusoidal wave field, this equation is equivalent to a closed three-dimensional system of nonlinear ODEs. We explicitly define a stability boundary for the system and a quantised criterion for its partial integrability in the meromorphic category.

We present a systematic approach to the problem whether a topologically infinite-dimensional space can be made homogeneous in the Coifman–Weiss sense. The answer to the question is negative, as expected. Our leading representative of spaces with this property is $\mathbb {T}^\omega = \mathbb {T} \times \mathbb {T} \times \cdots $ with the natural product topology.

We prove Reilly-type upper bounds for the first nonzero eigenvalue of the Steklov problem associated with the p-Laplace operator on submanifolds of manifolds with sectional curvature bounded from above by a nonnegative constant.

Erdős proved that every real number is the sum of two Liouville numbers. A set W of complex numbers is said to have the Erdős property if every real number is the sum of two members of W. Mahler divided the set of all transcendental numbers into three disjoint classes S, T and U such that, in particular, any two complex numbers which are algebraically dependent lie in the same class. The set of Liouville numbers is a proper subset of the set U and has Lebesgue measure zero. It is proved here, using a theorem of Weil on locally compact groups, that if $m\in [0,\infty )$, then there exist $2^{\mathfrak {c}}$ dense subsets W of S each of Lebesgue measure m such that W has the Erdős property and no two of these W are homeomorphic. It is also proved that there are $2^{\mathfrak {c}}$ dense subsets W of S each of full Lebesgue measure, which have the Erdős property. Finally, it is proved that there are $2^{\mathfrak {c}}$ dense subsets W of S such that every complex number is the sum of two members of W and such that no two of these W are homeomorphic.