If $X$ is a subgroup of a group $G$, the cardinal number $\min \{ \vert X: X_{G}\vert , \vert {X}^{G} : X\vert \} $ is called the normal oscillation of $X$ in $G$. It is proved that if all subgroups of a locally finite group $G$ have finite normal oscillation, then $G$ contains a nilpotent subgroup of finite index.

We relate a one-parametric generating function for the squares of Legendre polynomials to an arithmetic hypergeometric series whose parametrisation by a level 7 modular function was recently given by Cooper. By using this modular parametrisation we resolve a subfamily of identities involving $1/ \pi $ which was experimentally observed by Sun.

We use bounds of mixed character sum to study the distribution of solutions to certain polynomial systems of congruences modulo a prime $p$. In particular, we obtain nontrivial results about the number of solutions in boxes with the side length below ${p}^{1/ 2} $, which seems to be the limit of more general methods based on the bounds of exponential sums along varieties.

Let $R$ be a ring and $Z(R)$ be the set of all zero-divisors of $R$. The total graph of $R$, denoted by $T(\Gamma (R))$ is a graph with all elements of $R$ as vertices, and two distinct vertices $x, y\in R$ are adjacent if and only if $x+ y\in Z(R)$. Let the regular graph of $R$, $\mathrm{Reg} (\Gamma (R))$, be the induced subgraph of $T(\Gamma (R))$ on the regular elements of $R$. In 2008, Anderson and Badawi proved that the girth of the total graph and the regular graph of a commutative ring are contained in the set $\{ 3, 4, \infty \} $. In this paper, we extend this result to an arbitrary ring (not necessarily commutative). We also prove that if $R$ is a reduced left Noetherian ring and $2\not\in Z(R)$, then the chromatic number and the clique number of $\mathrm{Reg} (\Gamma (R))$ are the same and they are ${2}^{r} $, where $r$ is the number of minimal prime ideals of $R$. Among other results, we show that if $R$ is a semiprime left Noetherian ring and $\mathrm{Reg} (R)$ is finite, then $R$ is finite.

Let $\alpha : 1, 1, \sqrt{x} , \mathop{( \sqrt{u} , \sqrt{v} , \sqrt{w} )}\nolimits ^{\wedge } $ be a backward 3-step extension of a recursively generated weighted sequence of positive real numbers with $1\leq x\leq u\leq v\leq w$ and let ${W}_{\alpha } $ be the associated weighted shift with weight sequence $\alpha $. The set of positive real numbers $x$ such that ${W}_{\alpha } $ is quadratically hyponormal for some $u, v$ and $w$ is described, solving an open problem due to Curto and Jung [‘Quadratically hyponormal weighted shifts with two equal weights’, Integr. Equ. Oper. Theory 37 (2000), 208–231].

In this paper we provide a new proof of strong convergence of resolvent operators associated with boundary value problems on thin domains.

Answering a problem posed by John Michael Rassias, we study the functional inequality

$$\begin{eqnarray*}f(x+ y+ xy)\leq f(x)+ f(y)+ f(xy),\end{eqnarray*}$$

$f$

Consider a map of class ${C}^{3} $ with nonflat critical points and with all periodic points hyperbolic repelling. We show that the ‘backward contracting condition’ implies the summability condition. This result is the converse of Theorem 3 of Bruin et al. [‘Large derivatives, backward contraction and invariant densities for interval maps’, Invent. Math. 172 (2008), 509–533].

Let $p$ be a real number greater than one and let $\Gamma $ be a graph of bounded degree. We investigate links between the $p$-harmonic boundary of $\Gamma $ and the ${D}_{p} $-massive subsets of $\Gamma $. In particular, if there are $n$ pairwise disjoint ${D}_{p} $-massive subsets of $\Gamma $, then the $p$-harmonic boundary of $\Gamma $ consists of at least $n$ elements. We show that the converse of this statement is also true.

Let $(a, b, c)$ be a primitive Pythagorean triple satisfying ${a}^{2} + {b}^{2} = {c}^{2} . $ In 1956, Jeśmanowicz conjectured that for any given positive integer $n$ the only solution of $\mathop{(an)}\nolimits ^{x} + \mathop{(bn)}\nolimits ^{y} = \mathop{(cn)}\nolimits ^{z} $ in positive integers is $x= y= z= 2. $ In this paper, for the primitive Pythagorean triple $(a, b, c)= (4{k}^{2} - 1, 4k, 4{k}^{2} + 1)$ with $k= {2}^{s} $ for some positive integer $s\geq 0$, we prove the conjecture when $n\gt 1$ and certain divisibility conditions are satisfied.

Suppose that $k_0\geq 3.5\times 10^6$ and $\mathcal{H}=\{h_1,\ldots,h_{k_0}\}$ is admissible. Then, for any $m\geq 1$, the set $\{m(h_j-h_i):\, h_i<h_j\}$ contains at least one Polignac number.

If $T$ is any finite higher commutator in an associative ring $R$, for example, $T= [[R, R] , [R, R] ] $, and if $T$ has minimal cardinality so that the ideal generated by $T$ is infinite, then $T$ is in the centre of $R$ and ${T}^{2} = 0$. Also, if $T$ is any finite, higher commutator containing no nonzero nilpotent element then $T$ generates a finite ideal.

In this short paper, we characterise graphs of order $pq$ with $p, q$ prime which are self-complementary and vertex-transitive.

The primitive finite permutation groups containing a cycle are classified. Of these, only the alternating and symmetric groups contain a cycle fixing at least three points. This removes a primality condition from a classical theorem of Jordan. Some applications to monodromy groups are given, and the contributions of Jordan and Marggraff to this topic are briefly discussed.

We show that the permutation module over $ \mathbb{C} $ afforded by the action of ${\mathrm{Sp} }_{2m} ({2}^{f} )$ on its natural module is isomorphic to the permutation module over $ \mathbb{C} $ afforded by the action of ${\mathrm{Sp} }_{2m} ({2}^{f} )$ on the union of the right cosets of ${ \mathrm{O} }_{2m}^{+ } ({2}^{f} )$ and ${ \mathrm{O} }_{2m}^{- } ({2}^{f} )$.

We mainly prove that, assuming $\mathfrak{b}= \mathfrak{c}$, every regular star-compact space with a strong rank 1-diagonal is metrisable.

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.

We prove that the hypotheses in the Pigola–Rigoli–Setti version of the Omori–Yau maximum principle are logically equivalent to the assumption that the manifold carries a ${C}^{2} $ proper function whose gradient and Hessian (Laplacian) are bounded. In particular, this result extends the scope of the original Omori–Yau principle, formulated in terms of lower bounds for curvature.