Let A and A0 be rings with a surjective homomorphism A → A0. Given a flat extension B0 of A0, a deformation of B0/A0 over A is a flat extension B of A such that B ⊗AA0 is isomorphic to B0. We show that such a deformation will exist if A0 is an Artin local ring, A is noetherian, and the homological dimension of B0 over A0 is ≤ 2. We also show that a deformation will exist if the kernel of A is nilpotent and if A0 is a finitely generted A0-algebra whose defining ideal is a local complete intersection.

In a real normed linear space X, properties of a non-empty closed set K are closely related to those of the distance function d which it generates. If X has a uniformly Gâteaux (uniformly Fréchet) differentiable norm, then d is Gâteaux (Fréchet) differentiable at x ∈ X/K if there exists an such that

and is Géteaux (Fréchet) differentiable on X / K if there exists a set P+(K) dense in X/K where such a limit is approached uniformly for all x ∈ P+(K). When X is complete this last property implies that K is convex.

We investigate the circumstances under which the distance function to a closed set in a Banach space having a one-sided directional derivative equal to 1 or −1 implies the existence of nearest points. In reflexive spaces we show that at a dense set of points outside a closed set the distance function has a directional derivative equal to 1.

For the twice-punctured unit disc Ωp = {z: |z| < 1, z ≠ ±p}, where 0 < p < 1, we obtain precise descriptions for p near 0 of various parameters associated with the uniformisation of Ωp by the upper half-plane U = {τ: Im τ > 0}. These parameters include the hyperbolic length of the geodesic surrounding ±p, the so-called “accessory parameters”, and the “proximity parameter” which determines the behaviour of the hyperbolic density near the punctures of Ωp.

For a graph G, define the bisection width bw(G) of G as min { eG(A,B): {A,B} partitions V(G) with ‖A| − |B‖ ≤ 1 } where eG(A, B) denotes the number of edges in G with one end in A and one end in B. We show almost every cubic graph G of order n has bw(G) ≥ n/11 while every such graph has bw(G) ≤ (n + 138)/3. We also show that almost every r-regular graph G of order n has bw(G) ≥ crn where cr → r/4 as r → ∞. Our last result is asymtotically correct.

Some new sufficient conditions are obtained for the oscillation of the neutral differential equation

where r(t) > 0, 0 < c < 1, p(t) ≥ 0, σ(t) > τ > 0 and α = 1 or 0 < α < 1.

In this paper we prove the existence of an optimal admissible state-control pair for a nonlinear distributed parameter system, with control constraints of feedback type and with an integral cost criterion. An example is also worked in detail.

Let S denote the class of functions of the form that are analytic and univalent in |z| < 1. Given f ∈ S and a, b, c, real numbers other than 0, −1, −2,…, let Ω(a, b, C; f) = F(a, b, C; z)* f(z) where is a hypergeometric Gauss function with (a)0 = 1 and (a)k = a(a + 1) … (a + k − 1) and * denotes the Hadamard product. For qn(z) = z + a2z2 + … + anzn (an ≠ 0, n = 5,6) in S, it is shown that , is univalent in |z| < 1. This extends the result previously known for n = 3 and n = 4. Also, we obtain a necessary and sufficient condition involving a, b, and c such that Ω(a, b, c;·) preserves the subclass of S consisting of starlike functions of order α, 0 ≤ α ≤ 1, with ak 0.

Let G be a finite group and let k be a field. We determine the smallest possible rank of a free kG-module that contains submodules of every possible dimension. As an application, we obtain various criteria for the wreath product of two finite groups to be a CLT-group.

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.

It is shown that for every positive integer n there exists a finite group of derived length n in which all Sylow subgroups are abeian and in which the defect of subnormal subgroups is at most 3.

Let Fq be the field with q elements and let G = PGLn(Fq) or PSLn(Fq) act on Fq(x1,…,xn−1), the rational function field of n − 1 variables. Then Fq(x1,…,xn−1)G is purely transcendental over Fq. In fact, a set of n − 1 generators of Fq(x1,…xn−1)G, over Fq is exhibited. The case n = 2 is treated by direct computation.

We will prove local and global Besov-type characterisations for the Bloch space and the little Bloch space. As a special case we obtain that the Bloch space consists of those analytic functions on the unit disc whose restrictions to pseudo-hyperbolic discs (of fixed pseudo-hyperbolic radius) uniformly belong to the Besov space. We also generalise the results to Bloch functions and little Bloch functions on the unit ball in . Finally we discuss the related spaces BMOA and VMOA.

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.

It is shown that the category of fixed points of a left exact idempotent functor on a topos is again a topos. As well as a direct proof, a bicategorical proof is given which shows that the result only depends on certain bicategorical exactness properties.

Let I denote a compact real interval and let f ∈ C0(I, I). In this note we show that if f chaotic in the sense of Li and Yorke, then there is an uncountable perfect δ-scrambled set S for f in the recurrent set of f. Furthermore, the ω-limit set of every x ∈ S under f contains S and contains infinitely many periodic points of f with arbitrarily large periods.

We conjecture that if the rate of area growth of a geodesic disc of radius r on a smooth simply-connected complete surface with non-positive Gaussian curvature is faster than r2(logr)1+e for some ε ≥ 0, then the surface is hyperbolic. We prove this under an additional assumption that the surface is rotationally symmetric.

Semilinear singular perturbation problems are solved numerically by using finite–difference schemes on non-equidistant meshes which are dense in the layers. The fourth order uniform accuracy of the Hermitian approximation is improved by the Richardson extrapolation.

In this paper varieties are investigated which are generated by graph algebras of undirected graphs and—in most cases—contain Murskii's groupoid (that is the graph algebra of the graph with two adjacent vertices and one loop). Though these varieties are inherently nonfinitely based, they can be finitely based as graph varieties (finitely graph based) like, for example, the varitey generated by Murskii's groupoid. Many examples of nonfinitely based graph varities containing Murskii's groupoid are given, too. Moreover, the coatoms in the subvariety lattice of the graph variety of all undirected graphs are described. There are two coatoms and they are finitely graph based.

Let M be an R-module of finite length. For a simple R-module A, let ℓA denote the nuber of times the isomorphism type of A appears in a composition chain of M, and let σ denote the maxinium of the ℓA, A ranging over all simple submodules of M. Let S be the endomorphism ring of M. We show that the Loewy length of S is bounded by σ.