It has been known since the last century that a single quadratic form in at least five variables has a nontrivial zero in any p-adic field, but the analogous question for systems of quadratic forms remains unanswered. It is plausible that the number of variables required for solubility of a system of quadratic forms simply is proportional to the number of forms; however, the best result to date, from an elementary argument of Leep [6], is that the number of variables needed is at most a quadratic function of the number of forms. The purpose of this paper is to show how these elementary arguments can be used, in a certain class of fields including the p-adic fields, to refine the upper bound for the number of variables needed to guarantee solubility of systems of quadratic forms. This result partially addresses Problem 6 of Lewis' survey article [7] on Diophantine problems.

Let G be a finite group, and let IG be the augmentation ideal of ℤG. We denote by d(G) the minimum number of generators for the group G, and by d(IG) the minimum number of elements of IG needed to generate IG as a G-module. The connection between d(G) and d(IG) is given by the following result due to Roggenkamp ]14&]:

formula here

where pr(G) is a non-negative integer, called the presentation rank of G, whose definition comes from the study of relation modules (see [4&] for more details).

Let Γ⊂SL2(ℤ) be a subgroup of finite index, and let [Hscr ] denote the Hecke algebra of Γ. The aim of this note is to give some information about the action of [Hscr ] on spaces of modular forms for certain noncongruence subgroups Γ, which can be deduced from the geometric results of [9].

Cappell conjectured the following in ]2]; compare ]11, Problem 2].

Conjecture 0.1. Let π be the fundamental group of a knot complement in [ ]3. Then Lhn(π)≃ Lhn(ℤ), for all n, where Lhnis the Wall–Novikov surgery group for homotopy equivalence, and the isomorphism is induced from the abelianization homomorphism π→ℤ.

The descent algebra of the symmetric group, over a field of non-zero characteristic p, is studied. A homomorphism into the algebra of generalised p-modular characters of the symmetric group is defined. This is then used to determine the radical, and its nilpotency index. It also allows the irreducible representations of the descent algebra to be described.

A set A of real numbers is called universal (in measure) if every measurable set of positive measure necessarily contains an affine copy of A. All finite sets are universal, but no infinite universal sets are known. Here we prove some results related to a conjecture of Erdős that there is no infinite universal set. For every infinite set A, there is a set E of positive measure such that (x+tA)⊆E fails for almost all (Lebesgue) pairs (x, t). Also, the exceptional set of pairs (x, t) (for which (x+tA)⊆E) can be taken to project to a null set on the t-axis. Finally, if the set A contains large subsets whose minimum gap is large (in a scale-invariant way), then there is E⊆R of positive measure which contains no affine copy of A.

Yang proved in [10] that if f and f(k) have no fix-points for every f∈F, where F is a family of meromorphic functions in a domain G and k a fixed integer, then F is normal in G. In this paper we prove normality for families F for which every f∈F omits ψ1 and f(k) omits ψ2, where ψ1 and ψ2 are analytic functions with ψ(k)1[nequiv ]ψ2.

We introduce several successively stronger conditions which are all necessary for a sequence in the unit ball to be interpolating for the space of bounded holomorphic functions, and which all imply Varopoulos' classical necessary condition.

Belyi˘'s Theorem implies that a Riemann surface X represents a curve defined over a number field if and only if it can be expressed as U/Γ, where U is simply-connected and Γ is a subgroup of finite index in a triangle group. We consider the case when X has genus 1, and ask for which curves and number fields Γ can be chosen to be a lattice. As an application, we give examples of Galois actions on Grothendieck dessins.

We show in this note the following statement which is an improvement over a result of R. M. Dudley and which is also of independent interest. Let X be a set of a Hilbert space with the property that there are constants ρ, σ>0, and for each n∈ℕ, the set X can be covered by at most n balls of radius ρn−σ. Then, for each n∈ℕ, the convex hull of X can be covered by 2n balls of radius cn−½−σ. The estimate is best possible for all n∈ℕ, apart from the value c=c(ρ, σ, X). In other words, let N(ε, X), ε>0, be the minimal number of balls of radius ε covering the set X. Then the above result is equivalent to saying that if N(ε, X)=O(ε−1/σ) as ε↓0, then for the convex hull conv (X) of X, N(ε, conv (X)) =O(exp(ε−2/(1+2σ))). Moreover, we give an interplay between several covering parameters based on coverings by balls (entropy numbers) and coverings by cylindrical sets (Kolmogorov numbers).

Requirements for boundary conditions are found which imply sharp by order estimates in L∞-metrics of eigenfunctions of the ordinary differential operator

formula here

It is shown that the inequalities

formula here

for the eigenfunctions {yλ} yield certain restrictions on the spectrum.

For each d[ges ]2 we construct a connected open set Ω⊂ℝd such that Ω=int (clos(Ω)), and for each k[ges ]1 and each p∈[1, +∞), the subset Wk, ∞(Ω) fails to be dense in the Sobolev space Wk, p(Ω), in the norm of Wk, p(Ω).

In the 1970s, a question of Kaplansky about discontinuous homomorphisms from certain commutative Banach algebras was resolved. Let A be the commutative C*-algebra C(Ω), where Ω is an infinite compact space. Then, if the continuum hypothesis (CH) be assumed, there is a discontinuous homomorphism from C(Ω) into a Banach algebra [2, 7]. In fact, let A be a commutative Banach algebra. Then (with (CH)) there is a discontinuous homomorphism from A into a Banach algebra whenever the character space ΦA of A is infinite [3, Theorem 3] and also whenever there is a non-maximal, prime ideal P in A such that [mid ]A/P[mid ]=2ℵ0 [4, 8]. (It is an open question whether or not every infinite-dimensional, commutative Banach algebra A satisfies this latter condition.)

A C*-algebra A is said to be monotone (respectively monotone σ-) complete if every increasing net (respectively increasing sequence) of elements in the ordered space Ah of all hermitian elements of A has a supremum in Ah. It is straightforward to verify that every monotone complete C*-algebra is an AW*-algebra. For type I AW*-algebras, the converse is known to be true. However, for general AW*-algebras, this question is still open, although an impressive attack on the problem was made by E. Christensen and G. K. Pedersen, who showed that properly infinite AW*-algebras are monotone σ-complete [4].

Le théorème fondamental de la théorie locale des surfaces affirme que toute immersion C3 d'une surface dans R3 est déterminée, è congruence près, par sa première et deuxième forme fondamentale. Nous montrerons dans cette note qu'une surface complète C5 de R3 avec [mid ]dH[mid ]≠0 est en fait essentiellement détermineé, à congruence près, par la première forme fondamentale et la trace de la seconde, i.e. par la métrique et la fonction courbure moyenne H.

Let X be a space that has the homotopy type of a finite simply connected CW complex. We denote by [Escr ](X) the group of homotopy classes of self-homotopy equivalences of X. This group has been extensively studied (see [1] for a survey). In this paper we consider the subgroup [Escr ]n#(X) consisting of homotopy classes of self-homotopy equivalences of X that induce the identity on the homotopy groups πi(X) for i[les ]n, and the subgroup [Escr ]#(X) consisting of homotopy classes of self-homotopy equivalences of X that induce the identity on all the homotopy groups. Our first result is as follows.

Frank Adams was born in Woolwich on 5 November 1930. His home was in New Eltham, about ten miles east of the centre of London. Both his parents were graduates of King's College, London, which was where they had met. They had one other child — Frank's younger brother Michael — who rose to the rank of Air Vice-Marshal in the Royal Air Force.

In his creative gifts and practical sense, Frank took after his father, a civil engineer, who worked for the government on road building in peace-time and airfield construction in war-time. In his exceptional capacity for hard work, Frank took after his mother, who was a biologist active in the educational field.