We characterise regular bipartite locally primitive graphs of order 2pe, where p is prime. We show that either p=2 (this case is known by previous work), or the graph is a binormal Cayley graph or a normal cover of one of the basic locally primitive graphs; these are described in detail.

A cohomology theory for locally trivial, locally compact topological groupoids with coefficients in vector bundles is constructed, generalizing constructions of Hochschild and Mostow (1962) for topological groups and Higgins (1971) for discrete groupoids. It is calculated to be naturally isomorphic to the cohomology of the vertex groups, and is thus independent of the twistedness of the groupoid. The second cohomology space is accordingly realized as those “rigid” extensions which essentially arise from extensions of the vertex group; the cohomological machinery now yields the unexpected result that in fact all extensions, satisfying some natural weak conditions, are rigid.

The automorphisms of the canonical core UHF subalgebra ℱn of the Cuntz algebra 𝒪n do not necessarily extend to automorphisms of 𝒪n. Simple examples are discussed within the family of infinite tensor products of (inner) automorphisms of the matrix algebras Mn. In that case, necessary and sufficient conditions for the extension property are presented. Also addressed is the problem of extending to 𝒪n the automorphisms of the diagonal 𝒟n, which is a regular maximal abelian subalgebra with Cantor spectrum. In particular, it is shown that there exist product-type automorphisms of 𝒟n that do not extend to (possibly proper) endomorphisms of 𝒪n.

We give a survey of some of the realisations that have been given of monogenic inverse semigroups and discuss their relation to one another. We then analyse the representations by bijections, combined under composition, of monogenic inverse semigroups, and classify these into isomorphism types. This provides a particularly easy way of classifying monogenic inverse semigroups into isomorphism types. Of interest is that we find two quite distinct representations by bijections of free monogenic inverse semigroups and show that all such representations must contain one of these two representations. We call a bijection of the form ai ↦ ai+1, i = 1,2,…, r − 1, a finite link of length r, and one of the form ai ↦ ai+1, i = 1,2…, a forward link. The inverse of a forward link we call a backward link. Two bijections u: A → B and r: C → D are said to be strongly disjoint if A ∩ C, A ∩ D, B ∩ C and B ∩ D are each empty. The two distinct representations of a free monogenic inverse semigroup, that we have just referred to, are first, such that its generator is the union of a counbtable set os finite links that are pairwise storongly disjoint part of any representation of a free monogenic inverse semigroup, the remaining part not affecting the isomorphism type. Each representation of a monogenic inverse semigroup that is not free contains a strongly disjoint part, determining it to within isomorphism, that is generated by either the strongly disjoint union of a finite link and a permutation or the strongly disjoint union of a finite and a forward link.

Let aπ denote the spectral idempotent of a generalized Drazin invertible element a of a ring. We characterize elements b such that 1 − (bπ − aπ)2 is invertible. We also apply this result in rings with involution to obtain a characterization of the perturbation of EP elements. In Banach algebras we obtain a characterization in terms of matrix representations and derive error bounds for the perturbation of the Drazin Inverse. This work extends recent results for matrices given by the same authors to the setting of rings and Banach algebras. Finally, we characterize generalized Drazin invertible operators A, B ∈ (X) such that pr(Bπ) = pr(Aπ + S), where pr is the natural homomorphism of (X) onto the Calkin algebra and S ∈(X) is given.

2000 Mathematics subject classification: primary 16A32, 16A28, 15A09.

We consider a regression relation of the from wherein y(t) and x(t) are real (column) vectors of q and p components and e(t) is real and is generated by a stationary generalised vector process of q components with zero mean and covariance function (a q rowed matrix) Γ(t–s) = E{x(s)x(t)′}. (See Hannan (1970; pages 23–26, 91–94) and references therein for definitions of terms used.) We assume e(t) to be independent of x(s) for all s, t. Thus we may regard x(t) as a fixed time function and not stochastic and we shall henceforth do that. We take Γ(t) to be continuous and to correspond to an absolutely continuous spectral function with spectral density which is uniformly bounded and continuous. Then we have We do not exclude the possibility that for theyth diagonal element, fjj, of fwe have

We develop techniques to compute the homology of Quillen's complex of elementary abelian p-subgroups of a finite group in the case where the group has a normal subgroup of order divisible by p. The main result is a long exact sequence relating the homologies of these complexes for the whole group, the normal subgroup, and certain centralizer subgroups. The proof takes place at the level of partially-ordered sets. Notions of suspension and wedge product are considered in this context, which are analogous to the corresponding notions for topological spaces. We conclude with a formula for the generalized Steinberg module of a group with a normal subgroup, and give some examples.

Let $p:\mathbb{C}\rightarrow \mathbb{C}$ be a polynomial. The Gauss–Lucas theorem states that its critical points, $p^{\prime }(z)=0$, are contained in the convex hull of its roots. We prove a stability version whose simplest form is as follows: suppose that $p$ has $n+m$ roots, where $n$ are inside the unit disk,

$$\begin{eqnarray}\max _{1\leq i\leq n}|a_{i}|\leq 1~\text{and}~m~\text{are outside}~\min _{n+1\leq i\leq n+m}|a_{i}|\geq d>1+\frac{2m}{n};\end{eqnarray}$$

$p^{\prime }$

$n-1$

$m$

$(dn-m)/(n+m)>1$

$n$

$m$

${\sim}n^{-1}$

The direct product $\mathbb{N}\times \mathbb{N}$ of two free monogenic semigroups contains uncountably many pairwise nonisomorphic subdirect products. Furthermore, the following hold for $\mathbb{N}\times S$, where $S$ is a finite semigroup. It contains only countably many pairwise nonisomorphic subsemigroups if and only if $S$ is a union of groups. And it contains only countably many pairwise nonisomorphic subdirect products if and only if every element of $S$ has a relative left or right identity element.

Green's theorem, for line integrals in the plane, is well known, but proofs of it are often complicated. Verblunsky [1] and Potts [2] have given elegant proofs, which depend on a lemma on the decomposition of the interior of a closed rectifiable Jordan curve into a finite collection of subregions of arbitrarily small diameter. The following proof, for the case of Riemann integration, avoids this requirement by making a construction closely analogous to Goursat's proof of Cauchy's theorem. The integrability of Qx—Py is assumed, where P(x, y) and Q(x, y) are the functions involved, but not the integrability of the individual partial derivatives Qx, and py this latter assumption being made by other authors. However, P and Q are assumed differentiable, at points interior to the curve.

It is well-known that for any set X, px, the semigroup of all partial transformations on X, can be embedded in Jx∪a for some a ∉ X (see for example Clifford and Preston (1967) and Ljapin (1963)). Recently Magill (1967) has considered a special case of what we call ‘generalised partial transformation semigroups’. We show here that any such semigroup can always be embedded in a full transformation smigroup in which the operation is not in general equal to the usual composition of mappinas. We then examine conditions under which such a semigroup, (J x, θ), is isomorphic to the semigroup, under composition, of all transformations on the same set X.

The class of $\lambda $-synchronizing subshifts generalizes the class of irreducible sofic shifts. A $\lambda $-synchronizing subshift can be presented by a certain $\lambda $-graph system, called the $\lambda $-synchronizing $\lambda $-graph system. The $\lambda $-synchronizing $\lambda $-graph system of a $\lambda $-synchronizing subshift can be regarded as an analogue of the Fischer cover of an irreducible sofic shift. We will study algebraic structure of the ${C}^{\ast } $-algebra associated with a $\lambda $-synchronizing $\lambda $-graph system and prove that the stable isomorphism class of the ${C}^{\ast } $-algebra with its Cartan subalgebra is invariant under flow equivalence of $\lambda $-synchronizing subshifts.

Pointwise bounds for characters of representations of the compact, connected, simple, exceptional Life groups are obtained. It is a classical result that if μ is a central, continuous measure on such a group, then μdimG is absolutely continuous. Our estimates on the size of characters allow us to prove that the exponent, dimension of G, can be replaced by approximately the rank of G. Similar results were obtained earlier for the classical, compact Lie groups.

A subgroup H of a group G is said to be a permutablesubgroup of G if HK = KH 〈H, K〉 for all subgroups K of G. It is known that a core-free permutable subgroup H of a finite group G is always nilpotent [5]; and even when G is not finite, H is always a subdirect product of finite nilpotent groups [11]. Thus nilpotency is a measure of the extent to which a permutable subgroup differs from being normal. Examples of non-abelian, core-free, permutable subgroups are rare and difficult to construct. The first, due to Thompson [12], had class 2. Further examples of class 2 appeared in [8]. More recently Bradway, Gross and Scott [1] have constructed corresponding to each positive integer c and each prime p ≥ c, a finite p-group possessing a core-free permutable subgroup of class c. In [3] Gross succeeded in dispensing with the requirement p ≥ c.

Several effective upper bounds are known for the solutions of Thue equations, Thue-Mahler equations and superelliptic equations. One of the basic parameters occurring in these bounds is the height of the polynomial involved in the equation. In the present paper it is shown that better (and, in certain important particular cases, best possible) upper bounds can be obtained in terms of the height, if one takes into consideration also the discriminant of the polynomial.

For a real vector space V acted on by a group K and fixed x and y in V, we consider the problem of finding the minimum (respectively, maximum) distance, relative to a K-invariant convex function on V, between x and elements of the convex hull of the K-orbit of y. We solve this problem in the case where V is a Euclidean space and K is a finite reflection group acting on V. Then we use this result to obtain an analogous result in the case where K is a maximal compact subgroup of a reductive group G with adjoint action on the vector component ρ of a Cartan decomposition of Lie G. Our results generalize results of Li and Tsing and of Cheng concerning distances to the convex hulls of matrix orbits.

The representation theory of categories is used to embed each promonoidal monad in a monoidal biclosed monad. The existence of a promonoidal structure on the ordinary Eilenberg- Moore category generated by a promonoidal monad is examined. Several results by previous authors (notably A. Kock and F. E. J. Linton) are reproved and extended.

In a paper [1] of the same title Barnes considered the problem of finding an upper bound for the infimum m+(f) of the non-negative values1 of an indefinite quadratic form f in n variables, of given determinant det(f) ≠ 0 and of signature s. In particular it was announced (and later proved — see [2]) that m+(f) ≦ (16/5)+ for ternary quadratic forms of determinant 1 and signature — 1. A simple consequence of this result is that m+(f) ≦ (256/135)+ for quaternary quadratic forms of determinant — 1 and signature — 2.

Let S be a Riemann surface of type (p,n) with 3p+n>4 and n≥1. We investigate products of some pseudo-Anosov maps θ and Dehn twists tα on S, and prove that under certain conditions the products tkα∘θ are pseudo-Anosov for all integers k. We also give examples that show that tkα∘θ are not pseudo-Anosov for some integers k.

Let M be an n-dimensional complete Riemannian manifold with Ricci curvature bounded from below. Let be an N-dimensional (N < n) complete, simply connected Riemannian manifold with nonpositive sectional curvature. We shall prove in this note that if there exists an isometric immersion φ of M into with the property that the immersed manifold is contained in a ball of radius R and that the mean curvature vector H of the immersion has bounded norm ∥H∥ > H0, (H0 > 0) then R > H−10.