Recently the authors extended to positive semi-definite matrices converses of matrix inequalities corresponding to ratios and differences of means and of Hölder's inequality. Here, further generalisations are given to the case of several vectors with appropriate weights.

Let $w$ be a group-word. For a group $G$, let $G_{w}$ denote the set of all $w$-values in $G$ and let $w(G)$ denote the verbal subgroup of $G$ corresponding to $w$. The group $G$ is an $FC(w)$-group if the set of conjugates $x^{G_{w}}$ is finite for all $x\in G$. It is known that if $w$ is a concise word, then $G$ is an $FC(w)$-group if and only if $w(G)$ is $FC$-embedded in $G$, that is, the conjugacy class $x^{w(G)}$ is finite for all $x\in G$. There are examples showing that this is no longer true if $w$ is not concise. In the present paper, for an arbitrary word $w$, we show that if $G$ is an $FC(w)$-group, then the commutator subgroup $w(G)^{\prime }$ is $FC$-embedded in $G$. We also establish the analogous result for $BFC(w)$-groups, that is, groups in which the sets $x^{G_{w}}$ are boundedly finite.

This paper is concerned with projective metabelian -groups. where is the variety of all abelian groups, - of all abelian groups of exponent n. Let P be a projective -group. Since the group P/P′ is a free abelian group. Define rank P = rank P/P′. It is shown that for all numbers r, n > 1, except r = n = 2, there exists a protective nonfree -group of rank r with r + 1 generators.

In this paper the subgroup-closed saturated Fitting formations of radical locally finite groups with min-p for all p are fully characterised. Moreover the study of a class of generalised nilpotent groups introduced by Ballester-Bolinches and Pedraza is continued.

Gunningham [‘Spin Hurwitz numbers and topological quantum field theory’, Geom. Topol.20(4) (2016), 1859–1907] constructed an extended topological quantum field theory (TQFT) to obtain a closed formula for all spin Hurwitz numbers. In this note, we use a gluing theorem for spin Hurwitz numbers to re-prove Gunningham’s formula. We also describe a TQFT formalism naturally induced by the gluing theorem.

A subgroup $H$ is called a weak second maximal subgroup of $G$ if $H$ is a maximal subgroup of a maximal subgroup of $G$. Let $m(G,H)$ denote the number of maximal subgroups of $G$ containing $H$. We prove that $m(G,H)-1$ divides the index of some maximal subgroup of $G$ when $H$ is a weak second maximal subgroup of $G$. This partially answers a question of Flavell [‘Overgroups of second maximal subgroups’, Arch. Math.64(4) (1995), 277–282] and extends a result of Pálfy and Pudlák [‘Congruence lattices of finite algebras and intervals in subgroup lattices of finite groups’, Algebra Universalis11(1) (1980), 22–27].

A classical theorem of Brudno, dealing with the consistency of summability with regular matrices is shown by example not to hold over a non-archimedian field.

It has been shown by D. Stephen that the number N of open sets in a non-discrete topology on a finite set with n elements is not greater than 3 × 2n-2.We show that for admissable topologies on a finite group N ≦ 2n/r, where r is the least order of its non-trivial normal subgroups. This is clearly a sharper bound.

We show that the λ-multiplier convergence of operator series depends completely upon the AK property of the sequence space λ, and thus present a lot of new important theorems.

We present a family of continuous piecewise linear maps of the unit interval into itself that are all chaotic in the sense of Li and Yorke [‘Period three implies chaos’, Amer. Math. Monthly82 (1975), 985–992] and for which almost every point (in the sense of Lebesgue) in the unit interval is an eventually periodic point of period $p,p\geq 3$, for a member of the family.

Let V denote the symmetric monoidal closed category of limit-space abelian groups and let L denote the full subcategory of locally compact Hausdorff abelian groups. Results of Samuel Kaplan on extension of characters to products of L–groups are used to show that each closed subgroup of a product of L-groups is a limit of L–groups. From this we deduce that the limit closure of L in V is reflective in V and has every group Pontryagin reflexive with respect to the structure of continuous convergence on the character groups. The basic duality L ≃ Lop is then extended.

In this article we show that, under suitable conditions a quadratic functional attains its minimum on a closed convex cone (in a finite dimensional real Hilbert space) whenever it is bounded below on the cone. As an application, we solve Generalised Linear Complementarity Problems over closed convex cones.

In this note, we provide refined estimates of two sums involving the Euler totient function,

$$\begin{eqnarray}\mathop{\sum }_{n\leq x}\unicode[STIX]{x1D719}\biggl(\biggl[\frac{x}{n}\biggr]\biggr)\quad \text{and}\quad \mathop{\sum }_{n\leq x}\frac{\unicode[STIX]{x1D719}([x/n])}{[x/n]},\end{eqnarray}$$

$[x]$

$x$

We prove the analogue of the Baer Criterion for injectivity in the category AbShℒ of abelian groups in a topos of sheaves on a locale, that is, we show A is injective in AbShℒ if and only if it is injective relative to all S ↣ Zℒ where Zℒ is the group of integers in Shℒ. for a well-ordered locale we describe the injective hulls in AbShℒ in terms of injective hulls in Ab. Further we show that the global functor A → AE preserves injective hulls if and only if ℒ is a finite boolean locale. Finally We characterise injectives in AbShℒ for some special locales.

The set of all complex lines of the right-handed Dirac spinor bundle of a standard six-sphere is the total space of the twistor fibration. The twistor space, endowed with its natural Kähler structure, is recognised to be a six-dimensional complex quadric. The relevant group is Spin(7), which acts transitively on the six-quadric, as a group of fiber-preserving isometries. We use a result due to Berard-Bérgery and Matsuzawa to show the existence of a non-Kähler, non symmetric, Hermitian-Einstein metric on the six-quadric, which is Spin(7)-invariant.

Let R be an integral domain and A a cellular algebra over R with a cellular basis {CλS,T∣λ∈Λ and S,T∈M(λ)}. Suppose that A is equipped with a family of Jucys–Murphy elements which satisfy the separation condition in the sense of Mathas [‘Seminormal forms and Gram determinants for cellular algebras’, J. reine angew. Math.619 (2008), 141–173, with an appendix by M. Soriano]. Let K be the field of fractions of R and AK=A⨂ RK. We give a necessary and sufficient condition under which the centre of AK consists of the symmetric polynomials in Jucys–Murphy elements. We also give an application of our result to Ariki–Koike algebras.

It is well-known that every sufficiently large positive integer is the sum of seven cubes. Both proofs of this result, due to Linnik and Watson, are ineffective. Here we show that Watson's proof may be made effective.

A class of second-order asymptotically stationary random fields is shown to contain the class of almost harmonisable random fields. A continuity theorem which leads to the spectral representation for the covariance function of asymptotically stationary random fields is established. A mean ergodic theorem for the fields is also given. When stationarity is assumed, the results reduce to the well-known corresponding theorems for stationary random fields.