1. Let V be a non-singular rational surface defined over an algebraic number field k. There is a standard conjecture that the only obstructions to the Hasse principle and to weak approximation on V are the Brauer–Manin obstructions. A prerequisite for calculating these is a knowledge of the Brauer group of V; indeed there is one such obstruction, which may however be trivial, corresponding to each element of Br V/Br k. Because k is an algebraic number field, the natural injection

is an isomorphism; so the first step in calculating the Brauer–Manin obstruction is to calculate the finite group H1 (k), Pic .

In this paper we shall consider simultaneous approximations to algebraic numbers of the form for rational t1, …, tm, v. If t1, …, tm, are sufficiently close to 1 with respect to a common denominator we shall derive an effective inequality of the form max {|θ1 – p1 / q|, …, |θm – pm / q|} > cq–1–λ for all integers p1, …, pm, q with q > 0. The technique is used to derive the inequality

We use this to show that the only solution to the simultaneous equations

are the trivial ones.

We introduce a rational-valued invariant which is capable of distinguishing between the commensurability classes of certain discrete groups, namely, the fundamental groups of smooth closed orientable aspherical manifolds of dimensional 4k(k ≥ 1) whose Euler characteristic χ(Λ) is non-zero. The invariant in question is the quotient

where Sign (Λ) is the absolute value of the signature of the intersection form

and [Λ] is a generator of H4k(Λ; ℝ).

It is well-known that a compact connected abelian group G has weight w(G) less than or equal to the cardinality c of the continuum if and only if it is monothetic; that is, if and only if it can be topologically generated by one element. Hofmann and Morris [2] extended this by showing that a compact connected (not necessarily abelian) group can be topologically generated by two elements if and only if w(G) ≤ c.

Let V be an infinite-dimensional real Hilbert space with associated C* Clifford algebra C[V]. To any state σ of the C* algebra C[V] there corresponds a skew-adjoint operator C of norm at most unity on V such that

we refer to C as the covariance of the state σ.

A matroid M will be called sign-representable if, for every basis B of M, there is a (0, 1, −l)-matrix [Ir|y] representing M over ℚ in which the first r columns correspond to the members of B. The class of sign-representable matroids, which is closely related to the important class of regular matroids, is easily seen to be closed under both duality and the taking of minors. This paper proves several characterizations of the class, including a constructive one, and shows that the excluded minors for the class are U2, 5, U3, 5, the Fano matroid and its dual, and the rank-3 whirl.

Let G be a locally compact group and let UG denote the spectrum of the C*-algebra LUC(G) of bounded left uniformly continuous complex-valued functions on G, with the Gelfand topology. Then there is a multiplication on UG extending the multiplication on G (when naturally embedded in UG) such that UG is a semigroup and for each x ∈ UG, the map y ↦ yx from UG into UG is continuous, i.e. UG is a compact right topological semigroup. Consequently UG has a unique minimal ideal K which is the union of minimal (closed) left ideals UG. Furthermore K is the union of the set of maximal subgroups of K (see [3], theorem 3·ll).

A class of regular semigroups closed under taking direct products, regular subsemigroups and homomorphic images is an e(xistence) -variety of regular semigroups. The class of all combinatorial strict regular semigroups is the e-variety generated by the five element non-orthodox completely 0-simple semigroup and consists of all regular subdirect products of combinatorial completely 0-simple semigroups and/or rectangular bands. The bifree object on the set X in is the natural concept of a ‘free object’ in the class . is generated by the set X and the set of formal inverses X′ under the two binary operations of multiplication · and forming the sandwich element ∧A. Hence is a homomorphic image of the absolutely free algebra of type 〈2, 2〉 generated by X ∪X′. In this paper we shall describe the associated congruence on F〈2, 2〉(X∪X′) and construct a model of in terms of sets and binary relations. As an application, a model of the free strict pseudosemilattice on a set X is obtained.

In this paper we show that if a group generated by two elements acts freely and without phantom inversions on a Λ-tree, then it is either free or free abelian.

Let f be a rational function of degree at least two. We shall prove that the Julia set J(f) of f is uniformly perfect. This means that there is a constant c∈(0, 1) depending on f only such that whenever z∈J(f) and 0 < r < diam J(f) then J(f) intersects the annulus .

We characterize those domains Ω in ℝN for which the positive harmonic functions on Ω separate the points of Ω.

Let U(1, n; ℂ) be the automorphism group of the Hermitian form

for . We can regard an element of U(1, n; ℂ) as a transformation acting on , where is the closure of the complex unit ball

The non-trivial elements of U(1, n; ℂ) fall into three conjugacy types, depending on the number and the location of their fixed points. Let g be a non-trivial element of U(1, n; ℂ). We call g elliptic if it has a fixed point in Bn and g parabolic if it has exactly one fixed point and this lies on the boundary ∂Bn. An element g will be called loxodromic if it has exactly two fixed points and they lie on ∂Bn.

Recent efforts have been focused on using numerical methods to estimate the Titchmarsh–Weyl m-coefficient. In this paper we look at interval analytic methods to provide provable bounds for these values.

In recent years many papers have dealt with the zero-dimensional multiple point sets of smooth immersions of closed manifolds in Euclidean spaces (see, for example, [2, 6, 7, 8, 9, 10, 13, 14, 15, 21]). In the present paper we deal with the case when the double points form a two-dimensional surface and consider the following question: Which surfaces can occur as double point surfaces of self-transverse immersions of closed n-manifolds in ℝ2n−2? We prove the following

We show that trimming a fixed number of terms from sums of i.i.d. random variables (so-called light trimming) can have only a modest effect on limiting behaviour. More specifically, the trimmed sums, after centralization and normalization, have a limit distribution, if and only if the untrimmed sums have a limit distribution (with the same centralization and normalization constants).