Let COP = 0 ∩ H∞, where 0 is the little Bloch space on the open unit disk , and A() be the disk algebra on . For non-zero functions u1,u2,. . ., uN ∈ A() and distinct analytic self-maps ϕ1,ϕ2,. . .,ϕN satisfying ϕj ∈ A() and ∥ϕj∥∞=1 for every j, it is given characterisations of which the sum of weighted composition operators ∑Nj=1ujCϕj maps COP into A().

Let M be a smooth compact manifold with boundary. Under some geometric conditions on M, a homotopical model for the pair (M, ∂M) can be recovered from the configuration category of M \ ∂M. The grouplike monoid of derived homotopy automorphisms of the configuration category of M \ ∂M then acts on the homotopical model of (M, ∂M). That action is compatible with a better known homotopical action of the homeomorphism group of M \ ∂M on (M, ∂M).

We consider a 10-dimensional family of Lehn–Lehn–Sorger–van Straten hyperkähler eightfolds, which have a non-symplectic automorphism of order 3. Using the theory of finite-dimensional motives, we show that the action of this automorphism on the Chow group of 0-cycles is as predicted by the Bloch–Beilinson conjectures. We prove a similar statement for the anti-symplectic involution on varieties in this family. This has interesting consequences for the intersection product of the Chow ring of these varieties.

Born in London on 14 July 1947, Robert Winston Keith Odoni was the eldest of the five children of Walter Anthony and Lois Marie Theresa Odoni. The family name is Italian: grandfather Alfred Odoni had come to England from Switzerland in the 1920s and set up a manufacturing business in Birmingham, later transferring to London. Robert's father continued the business (then making cycle stands) until the 1970s. His mother was from Boulder in Colorado: they had met in America whilst he was on wartime naval service.

The ‘canonical embedding approach’ was introduced by the second author and, subsequently, it has been applied several times to prove the embeddability of certain regular extensions by groups into semidirect products by groups. In the present paper this technique is generalized so that it is suitable to handle regular extensions by inverse semigroups. As an application, B. Billhardt's embedding theorem on regular extensions of semilattices by inverse semigroups is reproved.

Let S be an (ideal) extension of a completely 0-simple semigroup S0 by a completely 0-simple semigroup S1. Congruences on S can be uniquely represented in terms of congruences on S0 and S1. In this representation, for a congruence ρ on S, we express ρK,ρT,ρK and ρT, where these denote the least (greatest) congruences with the same kernel (trace) as ρ. Let κ be the least completely 0-simple congruence on S. We provide necessary and sufficient conditions, in terms of the kernel of κ, in order that the relation K be a congruence, and also that [Cscr ](S)/K be a modular lattice, where [Cscr ](S) denotes the congruence lattice of S.

In this note we derive some integrals involving confluent hypergeometric functions and analogous to Lommel's integrals for Bessel functions. Although the method of derivation is straightforward, the integrals do not seem to be mentioned in the literature.

The paper deals with the sets of numbers from [0,1] such that their binary representation is almost convergent. The aim of the study is to compute the Hausdorff dimensions of such sets. Previously, the results of this type were proved for a single summation method (e.g. Cesàro, Abel, Toeplitz). This study extends the results to a wide range of matrix summation methods.

In this paper, we first summarise the progress for the famous Chern conjecture, and then we consider n-dimensional closed hypersurfaces with constant mean curvature H in the unit sphere n+1 with n ≤ 8 and generalise the result of Cheng et al. (Q. M. Cheng, Y. J. He and H. Z. Li, Scalar curvature of hypersurfaces with constant mean curvature in a sphere, Glasg. Math. J. 51(2) (2009), 413–423). In order to be precise, we prove that if |H| ≤ ϵ(n), then there exists a constant δ(n, H) > 0, which depends only on n and H, such that if S0 ≤ S ≤ S0 + δ(n, H), then S = S0 and M is isometric to the Clifford hypersurface, where ϵ(n) is a sufficiently small constant depending on n.

This paper deals with the following problem: Can an arbitrary continuous function on [0, 1], which vanishes at the origin, be represented in some sense as a series of constant multiples of indefinite integrals of a complete orthonormal set of functions on [0,1]? Four contexts in which this problem arises naturally will be given in the introduction and the remainder of the paper will be devoted to giving a partial answer to the specific problem formulated in one of these contexts.

In this note, we show that if we write ⌊en!⌋ = s(n)u(n)2, where s(n) is square-free then has at least C log log N distinct prime factors for some absolute constant C > 0 and sufficiently large N. A similar result is obtained for the total number of distinct primes dividing the mth power-free part of s(n) as n ranges from 1 to N, where m ≥ 3 is a positive integer. As an application of such results, we give an upper bound on the number of n ≤ N such that ⌊en!⌋ is a square.

The Laplace transform

has been generalised by Varma [4] by the relation

which reduces to (1.1) when k = -m + ½ by virtue of the identity

We shall define πk, m, λ (p) by the relation

The object of this paper is to obtain some recurrence formulae and series for πk, m, λ (p) and to use them to obtain recurrence formulae and series for MacRobert's E-function.

We define the integral operators and as

and

where

and Wk, u and Mk, u are the Whittaker's confluent hyper-geometric functions. These operators, in their slightly less general form, have been dealt with in [2] and [4]. There the authors have used the fact that these integral operators can be expressed as compositions of the Kober's fractional integral operators and the modified Laplace operator. Then these operators are inverted accordingly. Generally, this type of technique has been very useful for inverting many kinds of integral equations; and a good account of the procedures involved is given [5].

Let ℬ = {bi:b1 <b2<…} be an infinite sequence of positive integers that exceed 1 and are pairwise coprime, so that

Assume also that

Let A=Aℬ denote the sequence of ℬ-free numbers, that is, of positive integers divisible by no element of ℬ. This concept, generalizing square-free and k-free numbers, derives from Erdös [2] who proved in 1966 that there exists a constant c, 0<c<l, independent of ℬ, such that the interval (x, x+xc) contains elements of A provided only that x is large enough. This result of Erdös was shown by Szemeredi [7] in 1973 to hold with c=½+ε, if x≥xo(ε, ℬ), and quite recently Bantle and Grupp [1] have sharpened Szemeredi's result to c=9/20+ε.

Let M = (M, J, g) be an almost Hermitian manifold and U(M)the unit tangent bundle of M. Then the holomorphic sectional curvature H = H(x) can be regarded as a differentiable function on U(M). If the function H is constant along each fibre, then M is called a space of pointwise constant holomorphic sectional curvature. Especially, if H is constant on the whole U(M), then M is called a space of constant holomorphic sectional curvature. An almost Hermitian manifold with an integrable almost complex structure is called a Hermitian manifold. A real 4-dimensional Hermitian manifold is called a Hermitian surface. Hermitian surfaces of pointwise constant holomorphic sectional curvature have been studied by several authors (cf. [2], [3], [5], [6] and so on).

A ring R is called radical if it coincides with its Jacobson radical, which means that Rforms a group under the operation a ° b = a + b + ab for all a and b in R. This group is called the adjoint group R° of R. The relation between the adjoint group R° and the additive group R+ of a radical rin R is an interesting topic to study. It has been shown in [1] that the finiteness conditions “minimax”, “finite Prufer rank”, “finite abelian subgroup rank” and “finite torsionfree rank” carry over from the adjoint group to the additive group of a radical ring. The converse is true for the minimax condition, while it fails for all the other above finiteness conditions by an example due to Sysak [6] (see also [2, Theorem 6.1.2]). However, we will show that the converse holds if we restrict to the class of nil rings, i.e. the rings R such that for any a є R there exists an n = n(a) with an = 0.

Let Mn, n ≥ 3, be a complete hypersurface in $\mathbb{S}$n+1. When Mn is compact, we show that Mn is a homology sphere if the squared norm of its traceless second fundamental form is less than $\frac{2(n-1)}{n}$. When Mn is non-compact, we show that there are no non-trivial L2 harmonic p-forms, 1 ≤ p ≤ n − 1, on Mn under pointwise condition. We also show the non-existence of L2 harmonic 1-forms on Mn provided that Mn is minimal and $\frac{n-1}{n}$-stable. This implies that Mn has only one end. Finally, we prove that there exists an explicit positive constant C such that if the total curvature of Mn is less than C, then there are no non-trivial L2 harmonic p-forms on Mn for all 1 ≤ p ≤ n − 1.

A ring R is called left QF-3 if it has a minimal faithful left R-module. The endomorphism ring of such a module has been recently studied in [7], where conditions are given for it to be a left PF ring or a QF ring. The object of the present paper is to study, more generally, when the endomorphism ring of a Σ-quasi-projective module over any ring R is left QF-3. Necessary and sufficient conditions for this to happen are given in Theorem 2. An useful concept in this investigation is that of a QF-3 module which has been introduced in [11]. If M is a finitely generated quasi-projective module and σ[M] denotes the category of all modules isomorphic to submodules of modules generated by M, then we show that End(RM) is a left QF-3 ring if and only if the quotient module of M modulo its torsion submodule (in the torsion theory of σ[M] canonically defined by M) is a QF-3 module (Corollary 4). Finally, we apply these results to the study of the endomorphism ring of a minimal faithful R-module over a left QF-3 ring, extending some of the results of [7].

The theory of quadratic congruences modulo an integer is dominated by the Quadratic Law of Reciprocity (see § 1), which makes it possible to decide in a very short time whether a quadratic congruence

is solvable or not. The law was first proved by Gauss.* It took him over a year to obtain his first proof, which depends on a tedious lemma in elementary number theory. He subsequently obtained seven further proofs, and today more than fifty proofs are known, most of them based on the ideas of Gauss. The object of the present paper is to present a proof which is a modernised version of Gauss's seventh proof, applying the ideas of that proof to a finite set of objects, the elements of a finite or Galois field.

The purpose of this paper is to exhibit various Q-algebras (quotients of uniform algebras) which are Jacobson radical. We begin by noting easy examples of nilpotent Q-algebras and Q-algebras with dense nil radical. Then we describe two ways of constructing semiprime, Jacobson radical Q-algebras. The first is by directly constructing a uniform algebra and an ideal. This produces a nasty Q-algebra as the quotient of a nice uniform algebra (in the sense that it is a maximal ideal of R(X) for some X ⊆ ℂ). The second way is by using results of Craw and Varopoulos to show that certain weighted sequence algebras are Q-algebras. In fact we show that a weighted sequence algebra is Q if the weights satisfy (i) w(n+1)/w(n)↓0 and (ii) (w(n+l)/w(n))∊lp for some p≧ 1, but may be non-Q if either (i) or (ii) fails. This second method produces nice Q-algebras which are quotients of rather horrid uniform algebras as constructed by Craw's Lemma.