A natural number $n$ is called $k$-perfect if $\sigma (n)= kn$. In this paper, we show that for any integers $r\geq 2$ and $k\geq 2$, the number of odd $k$-perfect numbers $n$ with $\omega (n)\leq r$ is bounded by $\left({\lfloor {4}^{r} { \mathop{ \log } \nolimits }_{3} 2\rfloor + r\atop r} \right){ \mathop{ \sum } \nolimits }_{i= 1}^{r} \left({\lfloor kr/ 2\rfloor \atop i} \right)$, which is less than ${4}^{{r}^{2} } $ when $r$ is large enough.

The main result in the earlier paper (by the first author) is improved as follows. The number of odd multiperfect numbers with at most $r$ distinct prime factors is bounded by ${4}^{{r}^{2} } / {2}^{r+ 2} (r- 1)!$.

In this paper, employing a very recent local minimum theorem for differentiable functionals, the existence of at least one nontrivial solution for a class of systems of $n$ second-order Sturm–Liouville equations is established.

We introduce an analogue of the Mertens conjecture for elliptic curves over finite fields. Using a result of Waterhouse, we classify the isogeny classes of elliptic curves for which this conjecture holds in terms of the size of the finite field and the trace of the Frobenius endomorphism acting on the curve.

We prove a hyperstability result for the Cauchy functional equation $f(x+ y)= f(x)+ f(y)$, which complements some earlier stability outcomes of J. M. Rassias. As a consequence, we obtain the slightly surprising corollary that for every function $f$, mapping a normed space ${E}_{1} $ into a normed space ${E}_{2} $, and for all real numbers $r, s$ with $r+ s\gt 0$ one of the following two conditions must be valid:

$$\begin{eqnarray*}\displaystyle \sup _{x, y\in E_{1}}\Vert f(x+ y)- f(x)- f(y)\Vert \hspace{0.167em} \mathop{\Vert x\Vert }\nolimits ^{r} \hspace{0.167em} \mathop{\Vert y\Vert }\nolimits ^{s} = \infty , &&\displaystyle\end{eqnarray*}$$

$$\begin{eqnarray*}\displaystyle \sup _{x, y\in E_{1}}\Vert f(x+ y)- f(x)- f(y)\Vert \hspace{0.167em} \mathop{\Vert x\Vert }\nolimits ^{r} \hspace{0.167em} \mathop{\Vert y\Vert }\nolimits ^{s} = 0. &&\displaystyle\end{eqnarray*}$$

A group $G$ is said to be an $FC$-group if each element of $G$ has only finitely many conjugates, and $G$ is minimal non$FC$ if all its proper subgroups have the property $FC$ but $G$ is not an $FC$-group. It is an open question whether there exists a group of infinite rank which is minimal non$FC$. We consider here groups of infinite rank in which all proper subgroups of infinite rank are $FC$, and prove that in most cases such groups are either $FC$-groups or minimal non$FC$.

We investigate inverse limits in the category $ \mathcal{CHU} $ of compact Hausdorff spaces with upper semicontinuous functions. We introduce the notion of weak inverse limits in this category and show that the inverse limits with upper semicontinuous set-valued bonding functions (as they were defined by Ingram and Mahavier [‘Inverse limits of upper semi-continuous set valued functions’, Houston J. Math.  32 (2006), 119–130]) together with the projections are not necessarily inverse limits in $ \mathcal{CHU} $ but they are always weak inverse limits in this category. This is a realisation of our categorical approach to solving a problem stated by Ingram [An Introduction to Inverse Limits with Set-Valued Functions (Springer, New York, 2012)].

We define the class of weakly approximately divisible unital ${C}^{\ast } $-algebras and show that this class is closed under direct sums, direct limits, any tensor product with any ${C}^{\ast } $-algebra, and quotients. A nuclear ${C}^{\ast } $-algebra is weakly approximately divisible if and only if it has no finite-dimensional representations. We also show that Pisier’s similarity degree of a weakly approximately divisible ${C}^{\ast } $-algebra is at most five.

Let $A, B$ be two square complex matrices of the same dimension $n\leq 3$. We show that the following conditions are equivalent. (i) There exists a finite subset $U\subset { \mathbb{N} }_{\geq 2} $ such that for every $t\in \mathbb{N} \setminus U$, $\exp (tA+ B)= \exp (tA)\exp (B)= \exp (B)\exp (tA)$. (ii) The pair $(A, B)$ has property L of Motzkin and Taussky and $\exp (A+ B)= \exp (A)\exp (B)= \exp (B)\exp (A)$. We also characterise the pairs of real matrices $(A, B)$ of dimension three, that satisfy the previous conditions.

We show that the direct sum $\mathop{({X}_{1} \oplus \cdots \oplus {X}_{r} )}\nolimits_{\psi } $ with a strictly monotone norm has the weak fixed point property for nonexpansive mappings whenever $M({X}_{i} )\gt 1$ for each $i= 1, \ldots , r$. In particular, $\mathop{({X}_{1} \oplus \cdots \oplus {X}_{r} )}\nolimits_{\psi } $ enjoys the fixed point property if Banach spaces ${X}_{i} $ are uniformly nonsquare. This combined with the earlier results gives a definitive answer for $r= 2$: a direct sum ${X}_{1} {\mathop{\oplus }\nolimits}_{\psi } {X}_{2} $ of uniformly nonsquare spaces with any monotone norm has the fixed point property. Our results are extended to asymptotically nonexpansive mappings in the intermediate sense.

We show that a finite loop, whose inner mapping group is the direct product of a dihedral $2$-group and a nonabelian group of order $pq$ ($p$ and $q$ are distinct odd prime numbers), is solvable.

For a given integer $n$ and a set $ \mathcal{S} \subseteq \mathbb{N} $, denote by ${ R}_{h, \mathcal{S} }^{(1)} (n)$ the number of solutions of the equation $n= {s}_{{i}_{1} } + \cdots + {s}_{{i}_{h} } $, ${s}_{{i}_{j} } \in \mathcal{S} $, $j= 1, \ldots , h$. In this paper we determine all pairs $( \mathcal{A} , \mathcal{B} )$, $ \mathcal{A} , \mathcal{B} \subseteq \mathbb{N} $, for which ${ R}_{3, \mathcal{A} }^{(1)} (n)= { R}_{3, \mathcal{B} }^{(1)} (n)$ from a certain point on. We discuss some related problems.

Let ${ \mathbb{F} }_{q} $ be the finite field of characteristic $p$ containing $q= {p}^{r} $ elements and $f(x)= a{x}^{n} + {x}^{m} $, a binomial with coefficients in this field. If some conditions on the greatest common divisor of $n- m$ and $q- 1$ are satisfied then this polynomial does not permute the elements of the field. We prove in particular that if $f(x)= a{x}^{n} + {x}^{m} $ permutes ${ \mathbb{F} }_{p} $, where $n\gt m\gt 0$ and $a\in { \mathbb{F} }_{p}^{\ast } $, then $p- 1\leq (d- 1)d$, where $d= \gcd (n- m, p- 1)$, and that this bound of $p$, in terms of $d$ only, is sharp. We show as well how to obtain in certain cases a permutation binomial over a subfield of ${ \mathbb{F} }_{q} $ from a permutation binomial over ${ \mathbb{F} }_{q} $.

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.

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.

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

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.

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.