We prove that the invariably generating graph of a finite group can have an arbitrarily large number of connected components with at least two vertices.

The $p$-length of a finite $p$-soluble group is an important invariant parameter. The well-known Hall–Higman $p$-length theorem states that the $p$-length of a $p$-soluble group is bounded above by the nilpotent class of its Sylow $p$-subgroups. In this paper, we improve this result by giving a better estimation on the $p$-length of a $p$-soluble group in terms of other invariant parameters of its Sylow $p$-subgroups.

If a positive definite Hermitian lattice represents all positive integers, we call it universal. Several mathematicians, including the author, have between them found 25 universal binary Hermitian lattices. But their ad hoc proofs are complicated. We give simple and unified proofs of universality.

An eulerian chain in a graph is a continuous route which traces every edge exactly once. It may be finite or infinite, and may have 0,1 or 2 end vertices. For each kind of eulerian chain, there is a characterization of the graphs which admit such a tracing. This paper derives a uniform characterization of graphs with an eulerian chain, regardless of the kind of chain. Relationships between the edge complements of various kinds of finite subgraphs are also investigated, and hence a sharpened version of the eulerian chain characterization is derived.

We give a description of the structure of the semigroups for which each principal ideal is a retract. The globally idempotent case is solved quickly using a—suitably modified—construction which has been developed by Tully for the study of semigroups in which each ideal is a retract. The general case can be treated by a naturally obtained semigroup of subsets of the semigroup constructed for the globally idempotent case.

The differentiability, of a specified strength, of a convex function at a point, is shown to be characterised by the convergence of subdifferentials in the appropriate topology on the dual space. This is used to prove that if each gauge is densely differentiable then so is each convex function. The generic version of this is equivalent to a conjecture which, for Gateaux differentiability and Banach spaces, is the long standing open question of whether X × ℝ is Weak Asplund whenever X is. Some progress is made towards a resolution.

Khovanov homology, an invariant of links in ${ \mathbb{R} }^{3} $, is a graded homology theory that categorifies the Jones polynomial in the sense that the graded Euler characteristic of the homology is the Jones polynomial. Asaeda et al. [‘Categorification of the Kauffman bracket skein module of $I$-bundles over surfaces’, Algebr. Geom. Topol. 4 (2004), 1177–1210] generalised this construction by defining a double graded homology theory that categorifies the Kauffman bracket skein module of links in $I$-bundles over surfaces, except for the surface $ \mathbb{R} {\mathrm{P} }^{2} $, where the construction fails due to strange behaviour of links when projected to the nonorientable surface $ \mathbb{R} {\mathrm{P} }^{2} $. This paper categorifies the missing case of the twisted $I$-bundle over $ \mathbb{R} {\mathrm{P} }^{2} $, $ \mathbb{R} {\mathrm{P} }^{2} \widetilde {\times } I\approx \mathbb{R} {\mathrm{P} }^{3} \setminus \{ \ast \} $, by redefining the differential in the Khovanov chain complex in a suitable manner.

In this paper we study rings and bimodules with no known one-sided chain conditions, but whose (two-sided) ideals and subbimodules are ‘nicely’ generated. We define bi-noetherian polycentral (BPC) and bi-noetherian polynormal (BPN) rings and bimodules, large classes of (almost always) non-noetherian objects, and put on record the basic facts about them. Any BPC ring is a BPN ring. In the case of rings we reduce their properties to properties of the prime ideals, and study the d.c.c. on (two-sided) ideals. We define both the artinian and bi-artinian radicals of a BPN ring, and use them to show that for BPN rings the intersections of the powers of both the Brown-McCoy and the Jacobson radicals are zero.

Let $(R, \mathfrak{m})$ be a Cohen–Macaulay complete local ring. We will apply an inductive argument to show that for every nonprojective locally projective maximal Cohen–Macaulay object $ \mathcal{X} $ of the morphism category of $R$ with local endomorphism ring, there exists an almost split sequence ending in $ \mathcal{X} $. Regular sequences are exploited to reduce the Krull dimension of $R$ on which the inductive argument is established. Moreover, the Auslander–Reiten translate of certain objects is described.

We demonstrate that many properties of topological spaces connected with the notion of resolvability are preserved by the relation of similarity between topologies. Moreover, many of them can be characterised by the properties of the algebra of sets with nowhere dense boundary and the ideal of nowhere dense sets. We use these results to investigate whether a given pair of an algebra and an ideal is topological.

Let $G$ be a compact group. The aim of this note is to show that the only continuous *-homomorphism from $L^{1}(G)$ to $\ell ^{\infty }\text{-}\bigoplus _{[{\it\pi}]\in {\hat{G}}}{\mathcal{B}}_{2}({\mathcal{H}}_{{\it\pi}})$ that transforms a convolution product into a pointwise product is, essentially, a Fourier transform. A similar result is also deduced for maps from $L^{2}(G)$ to $\ell ^{2}\text{-}\bigoplus _{[{\it\pi}]\in {\hat{G}}}{\mathcal{B}}_{2}({\mathcal{H}}_{{\it\pi}})$.

In this paper, we present characterisations of the sets of infima and efficient solutions and we give also a multiplier rule for these kinds of points. The results are established for a vector optimisation problem with C-convexlike criterion, C being a polyhedral cone.

Two fixed point theorems for a subset C of a normed vector space X are established by using the concept of centre. These results differ from previous fixed point theorems in that X is assumed to have a topology T as well as a norm. The norm is required to be lower semi-continuous with respect to T and C is required to be convex, bounded with respect to the norm and compact with respect to T.

A closed curve f: S1 → ℝ2 in general position gives rise to a word whose letters are the self-intersection points, each of them appearing exactly twice. Such a word is called a Gauss code. The problem of determining whether a given Gauss code is realisable or not was first proposed by Gauss and it has been settled a long time ago. The analogous question for immersions f: S2 → ℝ3 in general position is settled only in a special case when the immersion has no triple points. We give a necessary condition for a system of curves to be realisable by a general immersion f: S2 → ℝ3.

In this note we consider commutative rings with identity over which every unitary module is a zero-divisor module. We call such rings Universally Zero-divisor (UZD) rings. We show (1) a Noetherian ring R is a UZD if and only if R is semilocal and the Krull dimension of R is at most one, (2) a Prüfer domain R is a UZD if and only if R has only a finite number of maximal ideals, and (3) if a ring R has Noetherian spectrum and descending chain condition on prime ideals then R is a UZD if and only if Spec (R) is a finite set. The question of ascent and descent of the property of a ring being a UZD with respect to integral extension of rings has also been answered.

It is shown by enumeration that for any two non-isomorphic limbs with the same number of points the number of trees with p points that contain one limb equals the number of trees with p points that contain the other.

We study the following Dirichlet problem for the degenerate elliptic Monge-Ampère equation: Given , f ≥ 0 and , find a solution , t ≥ 2, satisfying in Ω and u = g on ∂Ω. Since f is nonnegative, we cannot apply any standard elliptic methods. In this paper, we use an iteration scheme of Nash-Moser type and a priori estimates for degenerate elliptic operators, and solve the Dirichlet problem for a certain class of f and g.

A monic polynomial $f(x)\in {\mathbb Z}[x]$ of degree N is called monogenic if $f(x)$ is irreducible over ${\mathbb Q}$ and $\{1,\theta ,\theta ^2,\ldots ,\theta ^{N-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(\theta )$, where $f(\theta )=0$. We use the classification of the Galois groups of quartic polynomials, due to Kappe and Warren [‘An elementary test for the Galois group of a quartic polynomial’, Amer. Math. Monthly 96(2) (1989), 133–137], to investigate the existence of infinite collections of monogenic quartic polynomials having a prescribed Galois group, such that each member of the collection generates a distinct quartic field. With the exception of the cyclic case, we provide such an infinite single-parameter collection for each possible Galois group. We believe these examples are new and we provide evidence to support this belief by showing that they are distinct from other infinite collections in the literature. Finally, we devote a separate section to the cyclic case.

A relationship is pointed out between the results in a recent paper of Shin's and those in a previously published paper by M.J.C. Gover.