We give a quite general geometric criterion for a function analytic in the unit disc to be a polynomial of a univalent function, and hence a criterion for multivalence. We believe that this is the essence why multivalent close-to-convex functions enjoy the latter decomposition property. As another application, we study, as suggested by T. Sheil-Small ‘9’, the geometry of classes of analytic functions which arise from his recent investigation of multivalent harmonic mappings.

In general, the class of a nilpotent linear algebra of dimension n is at most n + 1, and the index, or derived length, of a solvable linear algebra of dimension n is at most n. In this note it is shown that, for a nilpotent linear algebra of dimension n satisfying x2 = 0 for all x, the class is at most n; and bounds are obtained for the indices of solvable Lie algebras.

This paper presents a straightforward proof of the Gel'fand-Naimark theorem for non-commutative C*-algebras with identity, established by Dauns and Hofmann(2) in the context of fields of C*-algebras, by considering instead C*-algebras in categories of sheaves. The proof differs from that of (2,3,4) in obtaining an isometric *-isomorphism

from the C*-algebra A to the C*-algebra of sections of a C*-algebra Ax in the category of sheaves on the maximal ideal space X of the centre of A, without invoking any arguments which involve completeness (3, Theorem 7·9). Instead, the results of (7) yield immediately the existence of an algebraic isomorphism, the compactness of the maximal ideal space X then being used to prove that Ax is indeed a C*-algebra in the category of sheaves on X and that the isomorphism is isometric. One recovers the representation of (2) by noting (8) that any C*-algebra in the category of sheaves on X is isomorphic to the sheaf of sections of a canonical field of C*-algebras on X.

In (3), Cor. 2 to Th. 1, the first author proved that in the complex jet space Jk (n, 1) the orbits of simple singularities form canonical strata. By definition the canonical stratification is contact invariant so the proof consisted essentially of two steps: firstly, any two functions in the same canonical stratum are C0-equivalent by right-left changes of coordinates, and secondly, the right codimension (and hence the Milnor number) of an isolated complex singularity is a topological invariant.

We discuss a general method for classifying certain subcategories of the category of finite-dimensional modules over a finite-dimensional co-commutative Hopf algebra B. Our method is based on that of Benson–Carlson–Rickard [BCR1], who classify such subcategories when B = kG, the group ring of a finite group G over an algebraically closed field k. We get a similar classification when B is a finite sub-Hopf algebra of the mod 2 Steenrod algebra, with scalars extended to the algebraic closure of F2. Along the way, we prove a Quillen stratification theorem for cohomological varieties of modules over any B, in terms of quasi-elementary sub-Hopf algebras of B.

In chapter 7 of (2) conditions were given for a ring to be embeddable in a skew field; in particular, it was shown that any semifir has a universal field of fractions, over which all full matrices can be inverted. This was generalized in two different directions, by Bergman (in a letter to one of the authors in 1971) and by Dicks and Sontag(7). Dicks and Sontag characterized those rings having a field of fractions in which all full matrices are inverted; they showed that this is equivalent to Sylvester's law of nullity, and further showed that this forces the ring to have weak global dimension not exceeding 2 and all finitely generated projective modules to be free. Bergman on the other hand investigated weakly semihereditary rings having a rank function on projective modules which takes values in the natural numbers. He showed that there was a homomorphism from any such ring to a field of fractions in which every full map between finitely generated projective modules is inverted. Weakly semihereditary rings with a rank function to the natural numbers are the analogue of semifirs and so it is natural to look for a characterization of rings with a rank function on projective modules such that all full maps between projective modules become invertible in a suitable field of fractions. We shall find that, as before, this is the case if and only if Sylvester's law of nullity holds with respect to the rank function, for maps between projective modules. Further, the ring must have weak global dimension at most two. This is the content of Sections 2 and 3.

The effect of heat transfer on boundary-layer growth is considered for the problem of a main stream of small Mach number which is instantaneously released past a heated cylinder, or alternatively the cylinder is accelerated impulsively into a stream at rest. In particular the effect of different values of the given wall temperature on the interval to separation and the thickness of the boundary layer is investigated. It is found that the former is decreased and the latter increased by the presence of heat transfer.

Let S denote a compact, connected, orientable surface with genus g and h boundary components. We refer to S as a surface of genus g with h holes. Let [Mscr ]S denote the mapping class group of S, the group of isotopy classes of orientation-preserving homeomorphisms S → S.

Let G be a group. G is hopfian if every homomorphism from G onto itself is an automorphism. G is residually finite if for every g ∈ G with g ≠ 1 there exists a normal subgroup of finite index in G which does not contain g. Every finitely generated residually finite group is hopfian ([11, 12]). A group G is hyperhopfian ([2, 3]) if every homomorphism ψ G → G with ψ(G) normal in G and G/ψ(G) cyclic is an automorphism. As observed in [14], examples of hopfian groups which are not hyperhopfian are afforded by the fundamental groups of torus knots.

By a result of Grossman [5], [Mscr ]S is residually finite. Since [Mscr ]S is also finitely generated, it is hopfian. It is a natural question to ask whether [Mscr ]S is hyperhopfian. In this paper, we shall answer a more general question. We say that a group G is ultrahopfian if every homomorphism ψ: G → G with ψ(G) normal in G and G/ψ(G) abelian is an automorphism. Note that an ultrahopfian group is hyperhopfian. We shall prove the following result.

Let p be an odd prime number. Let k be a [pfr ]-number field and [ofr ] the ring of all integers of k with a prime element π. Let K/k be a cyclic extension with Galois group G, and [Afr ] the associated order of the ring [Ofr ] of all integers in K:

[Afr ]={f∈kG[mid ]f[Ofr ]⊆[Ofr ]}.

F. Bertrandias and M-J. Ferton [1] obtained necessary and sufficient conditions that [Ofr ] is [Afr ]-free in the case K/k is of degree p. The purpose of this paper is to study such conditions in case K/k is a cyclic totally ramified Kummer extension of degree n=pm.

The present work has its motivation in the papers [2] and [6] on distinguished Fréchet function spaces. Recall that a Fréchet space E is distinguished if it is the projective limit of a sequence of Banach spaces En such that the strong dual E′β is the inductive limit of the sequence of the duals E′n. Clearly, the property of being distinguished is inherited by complemented subspaces and in [6] Taskinen proved that the Fréchet function space C(R) ∩ L1(R) (intersection topology) is not distinguished, by showing that it contains a complemented subspace of Moscatelli type (see Section 1) that is not distinguished. Because of the criterion in [1], it is easy to decide when a Frechet space of Moscatelli type is distinguished. Using this, in [2], Bonet and Taskinen obtained that the spaces open in RN) are distinguished, by proving that they are isomorphic to complemented subspaces of distinguished Fréchet spaces of Moscatelli type.

Let ${\cal A}$ be the mod 2 Steenrod algebra and $D_k$ the Dickson algebra of $k$ variables. We study the Lannes–Zarati homomorphisms \[ \varphi_k:{\rm Ext}_{{\cal A}}^{k, k+i}({\bb F}_2,{\bb F}_2)\rightarrow ({\bb F}_2\otimes_{{\cal A}}D_k)^{*}_{i}, \] which correspond to an associated graded of the Hurewicz map $H:\pi^s_*(S^0)\cong\pi_*(Q_0S^0)\rightarrow H_*(Q_0S^0)$. An algebraic version of the long-standing conjecture on spherical classes predicts that $\varphi_k = 0$ in positive stems, for $k > 2$. That the conjecture is no longer valid for $k = 1$ and $2$ is respectively an exposition of the existence of Hopf invariant one classes and Kervaire invariant one classes.

This conjecture has been proved for $k = 3$ by Hu'ng ‘9’. It has been shown that $\varphi_k$ vanishes on decomposable elements for $k > 2$ ‘14’ and on the image of Singer's algebraic transfer for $k > 2$ ‘9, 12’. In this paper, we establish the conjecture for $k = 4$. To this end, our main tools include (1) an explicit chain-level representation of $\varphi_k$ and (2) a squaring operation ${\rm Sq}^0$ on $({\bb F}_2\otimes_{{\cal A}}D_k)^*$, which commutes with the classical ${\rm Sq}^0$ on ${\rm Ext}_{{\cal A}}^k({\bb F}_2,{\bb F}_2)$ through the Lannes–Zarati homomorphism.

The methods of papers I and II (1), (2) are applied to a problem in classical elastodynamics.

Let S(ℝn) be the space of Schwartz class functions and S'(ℝn) be the dual space of S(ℝn). Given a k-dimensional manifold M in ℝn with area (in case k = 1, M is a curve with length), then for 1 ≤ p ≤ ∞ we say that M has the p-approximate property of for each TεS'(ℝn) with supp T⊆M, and εLp(ℝn), we can find a sequence of measures {Tj,j = 1,2,…} on M, absolutely continuous with respect to the area measure on M, such that ∥j-∥p → 0 as j → ∞.

This paper is concerned with the small transverse displacement of an infinite thin plane isotropic plate due to the application of a transverse force applied at an arbitrary point of the plate. The plate has its outer edge free, and is clamped along and bounded internally by a closed curve that can be mapped onto the unit circle by means of a polynomial. Three polynomials are considered and in each of these cases the deflexion is obtained in finite terms. Circular and elliptic holes as well as curvilinear polygonal holes are included as special cases.

Ardakov-Wadsley defined the sheaf $\wideparen{\Ncal{D}}$X of p-adic analytic differential operators on a smooth rigid analytic variety by restricting to the case where X is affinoid and the tangent sheaf admits a smooth Lie lattice. We generalise their results by dropping the assumption of a smooth Lie lattice throughout, which allows us to describe the sections of $\wideparen{\Ncal{D}}$ for arbitrary affinoid subdomains and not just on a suitable base of the topology. The structural results concerning $\wideparen{\Ncal{D}}$ and coadmissible $\wideparen{\Ncal{D}}$-modules can then be generalised in a natural way.

The main ingredient for our proofs is a study of completed tensor products over normed K-algebras, for K a discretely valued field of mixed characteristic. Given a normed right module U over a normed K-algebra A, we provide several exactness criteria for the functor $U\widehat{\otimes}_A$ - applied to complexes of strict morphisms, including a necessary and sufficient condition in the case of short exact sequences.

The aim of this paper is to give a new description of Giraud's nonabelian cohomology set H2(L), [7], defined for a band L of a Grothendieck topos E as the set of L-equivalence classes of L-gerbes over E. Our description is similar to that for H1 by Čech cohomology, but with cocycles taken from the stack of bitorsors over E. We first continue to study the construction of gerbes by cocycle bitorsors or bouquets [13], originally given in [7] and [4]; Duskin [4, 5] showed that bouquets B and B′ of E give rise to equivalent gerbes if and only if there exist essential equivalences B″ → B and B″ → B′ for another bouquet B″.

An error in Dieudonné's paper, Sur les espaces de Köthe, published in Journal d' Analyse Mathématique, 1 (1951), 81–115, is pointed out and a correction put forward.

A random submatroid ωr of the projective geometry PG(r − 1, q) is obtained from PG(r − 1, q) by deleting elements so that each element has, independently of all other elements, probability 1 − p of being deleted and probability p of being retained. The properties of such random structures were studied in [5] and [6]. In the first of these papers, p was kept fixed, while in the second, motivated by Erdös and Rényi's work ([3), [4]) on random graphs, p was taken to be a function of r. A recent paper of Bollobás[2] strengthens and extends a number of the results of Erdös and Rényi. In this paper we prove matroid analogues of several of Bollobàs's results thereby extending some of the results of [6].

We study the scaling scenery of Gibbs measures for subshifts of finite type on self-conformal fractals and applications to Falconer's distance set problem and dimensions of projections. Our analysis includes hyperbolic Julia sets, limit sets of Schottky groups and graph-directed self-similar sets.

Let 0 < λ1 < λ2 < … be an increasing sequence of positive reals tending to infinity, and let ν1, ν2 be two (finite) signed measures on ℝ+ such that

for all k. It was originally proved by Müntz (12) that, if

then ν1 = ν2, and, conversely, if this condition fails, then there exist distinct ν1 and ν2 satisfying (1).