We prove that the class number of the imaginary quadratic field is divisible by n for any positive integers k and n with 22k < 3n, by using Y. Bugeaud and T. N. Shorey's result on Diophantine equations.

A polynomial P ∈ (kE, F) is left ℓ1-factorable if there are a polynomial Q ∈ (kE, ℓ1) and an operator L ∈ (ℓ1, F) such that P = L ○ Q. We characterise the Radon–Nikodým property by the left ℓ1-factorisation of polynomials on L1(μ). We study the left ℓ1-factorisation of nuclear, compact and Pietsch integral polynomials. For Pietsch integral polynomials, we introduce the left integral ℓ1-factorisation property, obtaining a second polynomial characterisation of the Radon–Nikodým property and showing that it plays a role somehow comparable, in this setting, to nuclearity of operators. A characterisation of 1-spaces is also given in terms of the left compact ℓ1-factorisation of polynomials.

In this paper we describe a special class of self-adjoint operators associated with the singular self-adjoint second-order differential expression ℓ. This class is defined by the requirement that the sesquilinear form q(u, v) obtained from ℓ by integration by parts once agrees with the inner product 〈ℓu, v〉. We call this class Type I operators. The Friedrichs Extension is a special case of these operators. A complete characterization of these operators is given, for the various values of the deficiency index, in terms of their domains and the boundary conditions they satisfy (separated or coupled).

The centre of the maximal p-subgroup of (pkD2p) is described as an elementary abelian p-group, where p is a prime.

Henry Richard Dowson – known to everyone as Harry – was born in Newcastle upon Tyne in England on 2 March 1939. His father Matthew Ridley Dowson came from near Penrith, Cumbria, England and was a park gardener who rose to be head gardener of Newcastle's hospitals. His mother Frances Walker Dowson (whose maiden name was McQuet, pronounced mac-yew-it) came from Bo'ness, near Linlithgow in Scotland and was trained as a lady's maid. They met when they were in the service of the same landowner.

Let B be a submodule of an R-module M. The intersection of all prime (resp. weakly prime) submodules of M containing B is denoted by rad(B) (resp. wrad(B)). A generalisation of 〈E(B)〉 denoted by UE(B) of M will be introduced. The inclusions 〈E(B)〉 ⊆ UE(B) ⊆ wrad(B) ⊆ rad(B) are motivations for studying the equalities UE(B) = wrad(B) and UE(B) = rad(B) in this paper. It is proved that if R is an arithmetical ring, then UE(B) = wrad(B). In Theorem 2.5, a generalisation of the main result of [11] is given.

Let M be an n-dimensional closed hypersurface with constant mean curvature H satisfying |H| ≤ ϵ(n) in a unit sphere Sn+1, n ≤ 7, and S the square of the length of the second fundamental form of M. 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 a Clifford hypersurface, where ϵ(n) is a sufficiently small constant depending on n and .

Let n be a positive integer. In this paper, we consider the diophantine equation We prove that this equation has only the positive integer solutions (n, x, y, z) = (1, t, 1, 1), (1, t, 3, 2), (3, 2, 2, 2). Therefore we extend the work done by Leszczyński (Wiadom. Mat., vol. 3, 1959, pp. 37–39) and Makowski (Wiadom. Mat., vol. 9, 1967, pp. 221–224).

Based on the ideas of Bessa, Jorge and Montenegro (Comm. Anal. Geom., vol. 15, no. 4, 2007, pp. 725–732) we show that a complete submanifold M with tamed second fundamental form in a complete Riemannian manifold N with sectional curvature KN ≤ κ ≤ 0 is proper (compact if N is compact). In addition, if N is Hadamard, then M has finite topology. We also show that the fundamental tone is an obstruction for a Riemannian manifold to be realised as submanifold with tamed second fundamental form of a Hadamard manifold with sectional curvature bounded below.

This paper studies the Ratliff–Rush closure of ideals in integral domains. By definition, the Ratliff–Rush closure of an ideal I of a domain R is the ideal given by Ĩ := ∪(In+1 :R In), and an ideal I is said to be a Ratliff–Rush ideal if Ĩ = I. We completely characterise integrally closed domains in which every ideal is a Ratliff–Rush ideal, and we give a complete description of the Ratliff–Rush closure of an ideal in a valuation domain.

We introduce in this paper the concepts of rings characterized by minimal one-sided ideals and concern ourselves with rings containing an injective maximal left ideal. Some known results for idempotent reflexive rings and left HI rings can be extended to left MC2 rings. As applications, we are able to give some new characterizations of regular left self-injective rings with non-zero socle and extend some known results for strongly regular rings.

We prove that almost all natural numbers satisfying certain necessary congruence conditions can be written as the sum of two cubes of primes and two cubes of P2-numbers, where, as usual, we call a natural number a P2-number when it is a prime or the product of two primes. From this result we also deduce that every sufficiently large integer can be written as the sum of eight cubes of P2-numbers.