An example of Wehrfritz is pointed out to show that GL(4, Q) is not coherent. This answers a question of Serre. It is shown that finitely generated soluble coherent subgroups of GL(2, Q) need not be polycyclic, in sharp contrast to the fact that all soluble subgroups of GL(n, Z) are polycyclic and so automatically coherent.

In this paper, we investigate $\unicode[STIX]{x1D70B}(m,n)$, the number of partitions of the bipartite number$(m,n)$ into steadily decreasing parts, introduced by Carlitz [‘A problem in partitions’, Duke Math. J.30 (1963), 203–213]. We give a relation between $\unicode[STIX]{x1D70B}(m,n)$ and the crank statistic $M(m,n)$ for integer partitions. Using this relation, we establish some uniform asymptotic formulas for $\unicode[STIX]{x1D70B}(m,n)$.

We establish a link between the two characterizations, independently obtained in 1991 by Puchs and Robson, of complete matrix rings in terms of the existence of nilpotent elements.

Let X be a reflexive Banach space and be a family of weakly continuous operators which map X to X. Conditions are provided which guarantee the existence and the uniqueness to the Cauchy initial value problem

The goal of this paper is to study the multiplicity of positive solutions of a class of quasilinear elliptic equations. Based on the mountain pass theorems and sub-and supersolutions argument for p-Laplacian operators, under suitable conditions on nonlinearity f(x, s), we show the follwing problem: , where Ω is a bounded open subset of RN, N ≥ 2, with smooth boundary, λ is a positive parameter and ∆p is the p-Laplacian operator with p > 1, possesses at least two positive solutions for large λ.

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.

Our main theorem describes FC-soluble and linear repetitive groups. As a corollary, we characterise algebraic linear repetitive semigroups.

Let G be a permutation group on a set Ω with no fixed points in Ω and let m be a positive integer. Then we define the movement of G as, m := move(G) := supΓ{|Γg \ Γ| │ g ∈ G}. Let p be a prime, p ≥ 5. If G is not a 2-group and p is the least odd prime dividing |G|, then we show that n := |Ω| ≤ 4m – p + 3.

Moreover, if we suppose that the permutation group induced by G on each orbit is not a 2-group then we improve the last bound of n and for an infinite family of groups the bound is attained.

A Ducci sequence is a sequence of integer $n$-tuples obtained by iterating the map

$$\begin{eqnarray}D:(a_{1},a_{2},\ldots ,a_{n})\mapsto (|a_{1}-a_{2}|,|a_{2}-a_{3}|,\ldots ,|a_{n}-a_{1}|).\end{eqnarray}$$

$P(n)$

$n$

$P(n)$

$n$

We obtain several norm and eigenvalue inequalities for positive matrices partitioned into four blocks. The results involve the numerical range $W(X)$ of the off-diagonal block $X$, especially the distance $d$ from $0$ to $W(X)$. A special consequence is an estimate,

$$\begin{eqnarray}\text{diam}\,W\left(\left[\begin{array}{@{}cc@{}}A & X\\ X^{\ast } & B\end{array}\right]\right)-\text{diam}\,W\biggl(\frac{A+B}{2}\biggr)\geq 2d,\end{eqnarray}$$

We apply a mean-value inequality for positive subsolutions of the $f$-heat operator, obtained from a Sobolev embedding, to prove a nonexistence result concerning complete noncompact $f$-maximal spacelike hypersurfaces in a class of weighted Lorentzian manifolds. Furthermore, we establish a new Calabi–Bernstein result for complete noncompact maximal spacelike hypersurfaces in a Lorentzian product space.

We adjoin a point at infinity to a field with a ring topology and extend the topology by taking complements of bounded sets as neighbourhoods of infinity. We note some elementary relations between the topology on the field and the topology on the field with infinity adjoined.

This note contains a (short) proof of the following generalisation of the Friedman–Mineyev theorem (earlier known as the Hanna Neumann conjecture): if $A$ and $B$ are nontrivial free subgroups of a virtually free group containing a free subgroup of index $n$, then $\text{rank}(A\cap B)-1\leq n\cdot (\text{rank}(A)-1)\cdot (\text{rank}(B)-1)$. In addition, we obtain a virtually-free-product analogue of this result.

No group has exactly one or two nonpower subgroups. We classify groups containing exactly three nonpower subgroups and show that there is a unique finite group with exactly four nonpower subgroups. Finally, we show that given any integer k greater than $4$ , there are infinitely many groups with exactly k nonpower subgroups.

In this note, by introducing a new variant of the resonator function, we give an explicit version of the lower bound for $\log |L(\sigma ,\chi )|$ in the strip $1/2<\sigma <1$ , which improves the result of Aistleitner et al. [‘On large values of $L(\sigma ,\chi )$ ’, Q. J. Math. 70 (2019), 831–848].

We answer some questions in a paper by Kaneko and Koike [‘On modular forms arising from a differential equation of hypergeometric type’, Ramanujan J. 7(1–3) (2003), 145–164] about the modularity of the solutions of a certain differential equation. In particular, we provide a number-theoretic explanation of why the modularity of the solutions occurs in some cases and does not occur in others. This also proves their conjecture on the completeness of the list of modular solutions after adding some missing cases.

We study some properties which imply weak normal structure and thus the fixed point property. We investigate whether the latter two properties are inherited by spaces obtained by direct sum with a finite dimensional space. We exhibit a space X which satisfies Opial's condition, X ⊕ ℝ does not have weak normal structure but X ⊕ ℝ has the fixed point property.

This paper provides examples of indecomposable 2-groups with non-abelian automorphism groups in which every automorphism is central.