The work of H. M. Srivastava (9) on generalizations of an interesting identity of Carlson is expanded upon in order to obtain series identities analogous to all four of the Appell functions and their generalizations. Reduction formulas for some Kampé de Fériet functions are derived as consequences of the various double-series identities considered in this paper.

In (5) it is shown that if m ≽ 3 then there is no non-constant harmonic map φ: ℝm → Sn with finite energy. The method of proof makes use of the fact that the energy functional is not invariant under conformal transformations. This fact has also allowed Xin(9), to show that any non-constant-harmonic map φ:Sm → (N, h), m ≽ 3, is not stable in the sense of having non-negative second variation.

A unital C*-algebra is said to satisfy the Dixmier property if for each element x in the closed convex hull of all elements of the form u*xu, u being a unitary in , intersects the centre of ((2), 2·7). The von Neumann algebras and also some other classes of C*-algebras are known to satisfy the Dixmier property (cf. (2), (3), (4), (6)). If is a simple C*-algebra which satisfies the Dixmier property then has at most one tracial state. In (3) Archbold raised the question whether there exists a unital simple C*-algebra which has at most one tracial state without satisfying the Dixmier property. In the present note we characterize the unital simple C*-algebras with at most one tracial state in terms of a condition which is similar to the Dixmier property, but is in fact formally weaker in the framework of simple C*-algebras. This characterization relies on the method used by Pedersen in (5) in order to show that for a unital simple C*-algebra which has at most one tracial state and at least one non-trivial projection the linear span of all projections in is dense in As an application we characterize those unital simple C*-algebras with a unique tracial state which satisfy the Dixmier property.

1. Introduction. Suppose that K is a continuous function on the square Q = [ – 1, 1] x [– 1,1] satisfying , for – 1 ≤ s, t ≤ 1; then the Fredholm operator T on L2(-1,1)

is compact and symmetric. Suppose also that T is a positive operator, i.e.

then there is an eigenfunction expansion

where (λn) is a sequence of non-negative real numbers which decreases to 0 and (φn) is an orthonormal sequence in L2( – 1,1). In this paper we shall find asymptotic estimates for λn when K takes certain specific analytic forms. In all cases K will be real-valued on Q and analytic in a neighbourhood of Q in complex 2-space; for example

In this paper we extend to certain domains in m-dimensional Euclidean space Rm, m ≥ 3, some results about the boundary behaviour of harmonic functions which, in R2, are known to follow from distortion theorems for conformal mappings.

Let A and E be C*-algebras, let A ⊗ B denote the minimal C*-tensor product, and let ε A *. The right slice map R: A ⊗ B → B is the unique bounded linear mapping with the property that R (a ⊗ b) = (a)b (a ε A, b ε B)(10). A triple (A, B, D), where D is a C*-subalgebra of B, is said to have the slice map property if whenever x ε A ⊗ B and R(x) D for all ε A* then x ε A ⊗ D). It is known that (A, B, D) has the slice map property whenever A is nuclear (11,13), but it appears to be still unknown whether the nuclearity of B will suffice (unless some extra condition is placed on D (l)).

The series is said to be summable A to s if the power series converges to φ(x) for 0 ⋞ x < 1 and limx→1 φ(x) = s. If the function ⋞(x) is of bounded variation over the same interval, the series is said to be summable │A│. It is easy to prove │A│ ⊆ A, i.e. a series summable │A│ is necessarily summable A to the same sum.

A function f:X → ℝ is countably decomposable (into continuous functions) if the topological space X can be partitioned into countably many sets An with each restriction f│ An continuous. According to L. V. Keldysh(2), the question whether every Baire function is countably decomposable was first raised by N. N. Luzin, and answered by P. S. Novikov. The answer is negative even for Baire-1 functions, as is shown in (2) (see also (1). In this paper we develop a characterization of the countably decomposable functions on a separable metric space X (see Corollary 1). We deduce that when X is complete they include all functions possessing the property P defined by D. E. Peek in (3): each non-empty σ-perfect set H contains a point at which f│ H is continuous. The example given by Peek shows that not every countably decomposable Baire-1 function has property P.

Let Ll …, Lr be independent linear subspaces of Ed, with di = dim Li ≥ 1 (i = 1, …, r), and Ed = L1 + … + Lr. For each i, let K¯;i be a convex body in Li with 0 ∈ K¯i, Ki = K¯i + ti(ti ∈ Ed) any translate of K¯i, and define K = con v (K1 ∪ … ∪ Kr), and similarl K¯. Then vol K ≥ vol K¯. Necessary and sufficient conditions for equality are also obtained.

Let N > 1 and let AN be the polydisc algebra, i.e. the algebra of all continuous functions on the closed polydisc δ¯N ⊂ N, analytic on the open polydisc δN, with sup norm. Call a closed set F ⊂ δ¯N a peak interpolation set for AN if given any f ε C(F), f ≠ 0, there is an extension f ε AN of f such that ¦f˜(z)¦ < ‖ f ‖ (z ε δ¯N - F); call F a norm preserving interpolation set for AN if given any f ε C(F) there is an extension f˜ ε AN of f such that ‖f˜‖ = ‖f‖. The paper gives a complete description of norm preserving interpolation sets for AN in terms of peak interpolation sets for AM, M ≤ N.

It is a matter of some interest to determine which closed manifolds have the same homology as a closed aspherical manifold (7) (9). This paper is concerned with the more restrictive question of which spheres have the same homology as some closed aspherical manifold.

Let M be a closed simply-connected 4-manifold. All manifolds will be assumed smooth and oriented. The purpose of this paper is to classify up to conjugacy the topological subgroups of Diff(M) isomorphic to the 2-dimensional torus T2 (Theorem 1), and to give an explicit formula for the number of such conjugacy classes (Theorem 2). Such a conjugacy class corresponds uniquely to a weak equivalence class of effective T2-actions on M. Thus the classification problem is trivial unless M supports an effective T2-action. Orlik and Raymond showed that this happens if and only if M is a connected sum of copies of ± ;P2 and S2 × S2 (2), and so this paper is really a study of the different T2-actions on these manifolds.

Using the decomposition of a certain vector space under the action of the structure group of Riemannian almost product manifolds, A. M. Naveira (9) has found thirty-six distinguished classes of these manifolds. In this article, we prove that this decomposition is irreducible by computing a basis of the space of invariant quadratic forms on such a space.

At the 1977 Georgia Topology Conference, Cappell and Shaneson(6) announced the following theorem.

Let M be a closed oriented smooth four-manifold.

Let n points be drawn independently and uniformly from a compact convex set K. The distribution of the shape of the resulting n–ad is determined and studied in the region corresponding to near alignment. Special attention is given to the case n ≥ 4 and a table provided to help in the assessment of practical data (e.g. megalithic ‘alignments’). The Broadbent factor, representing the effect of stretching the parent distribution, is computed explicitly.

Throughout, the word knot means a subspace of the 3-sphere S3 homeomorphic with the 1-sphere S1. Any knot can be expressed as a connected sum of a finite number of prime knots in a unique way (13), we consider the trivial knot a non-prime knot. (For higher dimensional knots, factorization and uniqueness have been studied in (1).) However given a knot it is difficult to determine if it is prime or not. We prove that totally knotted knots, see definition in §2, are prime in theorem 1, give a class of examples in theorem 2 and investigate how the last result can be applied to the conjecture that the family Y of unknotting number one knots are prime. (See problem 2 in (5).) At the end, prime tangles as defined by W. B. R. Lickerish are used to prove that in a certain family of knots, related somewhat to Y, there is just one non-prime knot: the square knot. The paper should be interpreted as being in the piecewise linear category. Standard definitions of 3-manifolds and knot theory may be found in (6) and (11) respectively.

Problems dealing with the generation of internal waves at the surface separating two fluids involves the consideration of different types of singularities in one of the two fluids. In this paper the velocity potentials describing line and point multipoles are obtained for the case when each fluid is of finite constant depth, neglecting effects of surface tension at the surface of separation.

If a second order ordinary differential equation has a simple or a double pole at a point zfl, then the standard Liouville-Green approximation could sometimes be valid near that point. In this paper we present an asymptotic series solution that is always valid near a double pole. A solution for that of a simple pole is also indicated. Asymptotic validity is proved.

For any integer N ≽ 3 we will construct a 3-manifold which can be described as + 1 surgery on N distinct knots in S3. We will also give examples of 3-manifolds which are N-fold cyclic branched covers of S3 over 2 distinct knots. Brakes (2) discovered the first examples of 3-manifolds with multiple knot surgery descriptions. Our construction is much different and follows directly from the construction used by Lickerish (6) to describe a manifold which has 2 distinct knot surgery descriptions. Giller (5) has given examples of 3-manifolds which arise as cyclic branched covers over distinct knots in S3. Qur construction is similar, but the knots are much easier to distinguish, being iterated torus knots.

A random graph on n vertices is a random subgraph of the complete graph on n vertices. By analogy with this, the present paper studies the asymptotic properties of a random submatroid ωr of the projective geometry PG(r−l, q). The main result concerns Kr, the rank of the largest projective geometry occurring as a submatroid of ωr. We show that with probability one, for sufficiently large r, Kr takes one of at most two values depending on r. This theorem is analogous to a result of Bollobás and Erdös on the clique number of a random graph. However, whereas from the matroid theorem one can essentially determine the critical exponent of ωr, the graph theorem gives only a lower bound on the chromatic number of a random graph.