All local solutions of the two dimensional Einstein-Weyl equations are found, and related to the compact examples obtained in [1].

A set D in a Banach space E is called limited if pointwise convergent sequences of linear functionals converge uniformly on D and E is called a GP-space (after Gelfand and Phillips) if every limited set in E is relatively compact. Banach spaces with weak * sequentially compact dual balls (W*SCDB for short) are GP-spaces and l1 is a GP-space without W*SCDB. Disproving a conjecture of Rosenthal and inspired by James tree space, Hagler and Odell constructed a class of Banach spaces ([HO]-spaces) without both W*SCDB and subspaces isomorphic to l1. Schlumprecht has shown that there is a subclass of the [HO]-spaces which are also GP-spaces. It is not clear however if any [HO]-construction yields a GP-space—in fact it is not even clear that W*SCDB[lrarr ]GP-space is false in general for the class of Banach spaces containing no subspace isomorphic to l1. In this note the example of Hagler and Odell is modified to yield a GP-space without W*SCDB and without an isomorphic copy of l1.

It is shown that an immersion of n dimensional compact oriented manifold without boundary into the n+1 dimensional Euclidean space, hyperbolic space or open half sphere is a totally umbilic immersion if one of the mean curvature function H_l does not vanish and the ratio H_k/H_l is constant, 1\leq k, l \leq n, k\ne l.

Let X be a proper geodesic metric space which is \delta-hyperbolic in the sense of Gromov. We study a class of functions on X, called horofunctions, which generalize Busemann functions. To each horofunction is associated a point in the boundary at infinity of X. Horofunctions are used to give a description of the boundary. In the case where X is the Cayley graph of a hyperbolic group \Gamma, we show, following ideas of Gromov sketched in his paper Hyperbolic groups, that the space of cocycles associated to horofunctions which take integral values on the vertices is a one-sided subshift of finite type.

We provide sufficient conditions on a positive finite rotation invariant Borel measure on {𝔻}^n which guarantee that the analogue of the Carleson measure theorem remains valid for Bergman spaces of holomorphic and n-harmonic functions on {𝔻}^n generated by the measure.

Bai [1] gave an intrinsic integral inequality for compact minimal submanifolds of constant curvature Riemannian manifolds. In this paper, we extend Bai's result to the case of pseudo-umbilical submanifolds.

This article is to discuss the automatic continuity properties and the representation of disjointness preserving linear mappings on certain normal Fréchet algebras of complex-valued functions. This class of operators is defined by the condition that any pair of functions with disjoint cozero sets is mapped to functions with disjoint cozero sets, and subsumes the class of local operators. It turns out that such operators are always continuous outside some finite singularity set of the underlying topological space. Our main emphasis is on disjointness preserving operators from Fréchet algebras of differentiable functions. Such operators are shown to admit a canonical representation that involves weighted composition for the derivatives. This result extends the classical characterization of local operators as linear partial differential operators.

The projective Schur group of a commutative ring was introduced by Lorenz and Opolka. It was revived by Nelis and Van Oystaeyen, and later by Aljadeff and Sonn. In this paper we study the intriguing question that there seems to be no adequate version of the crossed product theorem for the projective Schur group. We present a radical group R(k) (k a field) situated between the Schur group and the projective Schur group, and we prove the crossed product theorem for R(k).

An operator T on a Banach space is called ‘semi B-Fredholm’ if for some n \in {\tf="times-b"N} the range R(T\;\!^n) of T\;\!^n is closed and the induced operator T_n on R(T\;\!^n) semi-Fredholm. Semi B-Fredholm operators are stable under finite rank perturbation, and subject to the spectral mapping theorem; on Hilbert spaces they decompose as sums of nilpotent and semi-Fredholm operators. In addition some recent generalizations of the punctured neighborhood theorem turn out to be consequences of Grabiner's theory of ‘topological uniform descent’.

Every compact well-bounded operator has a representation as a linear combination of disjoint projections reminiscent of the representation of compact self-adjoint operators. In this note we show that the converse of this result holds, thus characterizing compact well-bounded operators. We also apply this result to study compact well-bounded operators on some special classes of Banach spaces such as hereditarily indecomposable spaces and certain spaces constructed by G. Pisier.

We investigate cyclic bi-embeddings in an orientable surface of Steiner triple systems on 31 points. Up to isomorphism, we show that there are precisely 2408 such embeddings. The relationship of these to solutions of Heffter's first difference problem is discussed and a procedure described which, under certain conditions, transforms one bi-embedding to another.

Suppose that G is a finite p-solvable group and let \chi \in {\rm Irr}(G) be of p^\prime -degree. In this note, we investigate when \chi remains irreducible when restricted to {\bf {N)}_{G}(P).

We compute the precise bounds, for every positive integer n, for the nilpotency class of nilpotent subgroups of {\text GL}(n,{\tf="P7DC6"Q}) and {\text GL}(n,{\tf="P7DC6"Z}).

In this paper we prove that if M_C=\pmatrix {A&#TAB;C\cr0&#TAB;B} is a 2\times 2 upper triangular operator matrix on the Hilbert space H\bigoplus K and if \sigma (A)\cap \sigma (B)=\emptyset , then \sigma is continuous at A and B if and only if \sigma is continuous at M_C, for every C\in B(K,H{\hskip1}).