In [1] and [2] T. Husain and J. Liang show the following results: RI. Every character on a Fréchet algebra with a Schauder basis (xi)i≧1 such that: (1) XiXj = XjXi = Xj if i ≦ j, (2) Pi(Xi) ≠ 0 and Pi(xi+1) = 0 (where (pi)i≧1 is a denumerable family of semi-norms defining the topology of the algebra) is continuous. R2. Every character on a Fréchet algebra with orthogonal and unconditional Schauder basis is continuous. The proofs of these last results are very long and introduce complex calculation without aid of spectral theory of locally ra-convex algebras. We give here short proofs of these results with aid of a characterization of elements of the spectrum in locally m-convex algebras with values of characters.

The relationship between H-sets and open filter adhérences is considered. The open filter adhérences of an H-closed space are shown to be H-sets; and, a necessary and sufficient condition is given for an H-set S, of a Hausdorff space X, to be an open filter adherence. A necessary condition is determined for the existence of a minimal adherent set which contains S; and, in the case that X is H-closed, sufficient conditions are determined. As a related result, an H-closed space X is shown to be seminormal if every H-set of X possesses a neighborhood base consisting of regular open sets.

A new proof of a product formula for Laguerre polynomials, due originally to Watson, is given. Considering the commutative Banach algebra of radial functions on the Heisenberg groups Hn, n ≧ 2, we observe that Watson's formula holds for z = 1,2, 3, …. Then, applying a complex function theory argument, we establish the validity of this formula for other complex values of z, i.e. for Re z > - 1/2.

We prove that Choi's map in M3 cannot be written as the sum of a 2-positive map and a 2-copositive map. We also provide other examples of positive maps in Mn which cannot be written as the sum of an n-positive map and a 2-copositive map.

Let G be a group with elements 1,…, n such that the group operation agrees with ordinary multiplication whenever the ordinary product of two elements lies in G. We show that if n is odd, then G is abelian.

Given any homogeneous complex manifold X = G/H, there exists a natural coset map π :G/H → G/K satisfying π (X1) = π (x2) if and only if dx(x1 x2) = 0, where dx denotes the Kobayashi pseudometric on X. Its typical fiber Z : = K/H is a connected complex submanifold of X. Also G/K has a (7-invariant complex structure, provided K satisfies a certain technical assumption (see Theorem 3). If Z is compact as well, then G/K is biholomorphic to a homogeneous bounded domain.

In this paper we use the Lašnev Theorem in order to give some properties of a class of metrizable spaces having compact metric shape.

compact Riemannian 4-symmetric space M can be regarded as a fibre bundle over a Riemannian 2-symmetric space with totally geodesic fibres isometric to a 2-symmetric space. Here the result of R. Crittenden for conjugate and cut points in a 2-symmetric space is extended to the focal points of the fibres of M. Also the restriction of the exponential map of M up to the first focal locus in the normal bundle of a fibre is proved to yield a covering map onto its image. It is shown that for the noncompact dual M*, the fibres have no focal points and hence the exponential map of M* restricted to the normal bundle of a fibre is a covering map. The classification of the compact simply connected 4-symmetric spaces G/L with G classical simple provides a large class of examples of these fibrations.

A two dimensional system of differential equations with a finite number of discrete delays is considered. Conditions are derived for there to be no stability switching for arbitrary such delays.

Let A, B denote sequence-to-sequence matrix methods of summability and A · B the “dot” or iteration product defined by (A · B)x = A(Bx) for all sequences x for which this exists. Some inclusion relations are given involving the methods A, B, A · B, B · A and the method defined by the matrix product AB. We take A, B to be of certain types whose products have not been studied extensively before, e.g. H* · Ck or Ck · H* where H* is quasi-Hausdorff (and hence upper triangular) and Ck is a Cesàro matrix (which is lower triangular). The investigations show also a link between the “Product Property” A ⊂ A · B and the translativity properties of A and B.

We present a quasi-normed space which is locally H∞-convex but is not locally PL-convex in the sense of Davis, Garling and Tomczak-Jaegermann.

In this paper p-local versions of the Rational Fibre and Cofibre Decomposition Theorems are given. In particular, if there exists an element in the nth Gottlieb group of a space F such that its image under the Hurewicz map has infinite order, then Sn for almost all primes p. A dual result is proved for cofibrations.

Σ q!/(h1! . . . hn!) with h1 + . . . + hn = q, the first a hj’s odd and the rest even, is expressed in terms of values of Krawtchouk polynomials.

We use tk moves on links to prove that the first homology group with Z2-coefficients of an odd sheeted cyclic cover of S3 branched over a link is even dimensional.

The self-adjoint extensions of the singular differential operator Ly = [(py’)’ + qy]/w, where p < 0, w > 0, q ≧ mw, are characterized under limit-circle conditions. It is shown that as long as the coefficients of certain boundary conditions define points which lie between two lines, the extension they help define has the same lower bound.

The C*-algebras associated with irrational rotations of the circle were classified up to strong Morita equivalence by M. A. Rieffel. As a corollary, he gave a complete classification of the C*-algebras arising from irrational or Kronecker flows on the 2-torus up to *-isomorphism. Here, we extend the result to the socalled Denjoy homeomorphisms. Specifically, we give a necessary and sufficient condition for the strong Morita equivalence of two C*-algebras arising from homeomorphisms of the circle without periodic points. As a corollary, we show that two C*-algebras arising from flows on the 2-torus obtained from such homeomorphisms by the “flow under constant function” construction are *-isomorphic if and only if the flows themselves are topologically conjugate.

The aim of this note is to clarify some statements in the book: Discrete Subgroups of Lie groups: Springer, 1972.

Given a domain D in R” and two specified points P0 and P1 in D we consider the problem of minimizing u(p1) over all functions harmonic in D with values between 0 and 1 normalised by the requirement u(P0) = 1/2. We show that when D is suitably regular the problem has a unique solution u* which necessarily takes on boundary values 0 or 1 almost everywhere on the boundary. In the process we prove that it is possible to separate P0 and P1by a harmonic function whose boundary value is supported in an arbitrary set of positive measure. These results depend on the fact that (under suitable regularity conditions) a harmonic function which vanishes on an open subset of the boundary has a normal derivative which is almost everywhere non-vanishing in that set.

It is known that among all plane convex domains of given constant width Reuleaux triangles have minimal and circular discs have maximal area. Some estimates are given concerning the following associated stability problem: If K is a convex domain of constant width w and if the area of K differs at most ∊ from the area of a Reuleaux triangle or a circular disc of width w, how close (in terms of the Hausdorff distance) is K to a Reuleaux triangle or a circular disc? Another result concerns the deviation of a convex domain M of diameter d from a convex domain of constant width if the perimeter of M is close to πd.