We study orthogonal polynomials for which the weight function is a linear combination of the Jacobi weight function and two delta functions at 1 and — 1. These polynomials can be expressed as 4F3 hypergeometric functions and they satisfy second order differential equations. They include Krall’s Jacobi type polynomials as special cases. The fourth order differential equation for the latter polynomials is derived in a more simple way.

Let Vn, k denote the Stiefel manifold of k-frames in ℝn. There is a free action on Vn, k by the group . We show that if acts freely on and

then l≤k

It is shown that if L is a Dedekind complete Riesz space equipped with a locally solid topology T defined by strongly (A, 0) Riesz pseudonorms, then order intervals of L are T-complete. This is an extension of a well known theorem of Nakano. The second part of the paper gives a necessary and sufficient condition for topological completeness of order intervals in a Dedekind σ-complete Riesz space which has a weak order unit and which is equipped with a locally solid σ-Fatou topology.

A short proof of the following result of Bernstein and Ganea is given:

“Let X be a topological space which is homotopy dominated by a closed connected n-dimensional manifold M. If Hn (X; Z2 ) ≠ 0 then X has the homotopy type of M”.

It is also shown that the manifold in this theorem can be replaced by a Poincaré complex.

In [A], Arhangel'skii showed that for any T2 space X, |X|≤2L(x)χ(x), where L(X) is the Lindelöf degree of X and χ(X) is the character of X.

In [B], Bell, Ginsburg and Woods improved this result, assuming normality, by showing that for T4 spaces X, |X|≤2wL(x)χ(x), where wL(X) is the weak Lindelöf degree of X.

We introduce below a new cardinal function aL(X), the almost Lindelöf degree of X, which agrees with L(X) on T3 spaces, but which is often smaller than L(X) on T2 spaces, and show that for T2 spaces X,

G(A, X) consists of all homotopy classes of cyclic maps from a space A to another space X. If A is an H-cogroup, then G(A, X) is a group. G(A, X) preserves products in the second variable and is a contravariant functor of A from the full subcategory of H-cogroups and maps into the category of abelian groups and homomorphisms. If X is an H-cogroup, then G(X, X) is a ring.

We investigate connections between the associated Lie ring and the adjoint group of a radical ring, studying their upper central chains. Part of a conjecture of S. A. Jennings is proved, and one of our results improves a theorem of his.

Given a Prüfer domain R and a prime ideal P in R, we study some conditions which force P to be a divisorial ideal of R. This paper extends some recent work of Huckaba and Papick.

Necessary and sufficient conditions are found for the existence of two positive solutions of the semilinear elliptic equation Δu + q(|x|)u = f(x, u) in an exterior domain Ω⊂ℝn, n ≥ 1, where q, f are real-valued and locally Hölder continuous, and f(x, u) is nonincreasing in u for each fixed x∈Ω. An example is the singular stationary Klein-Gordon equation Δu — k2u = p(x)u-λ where k and λ are positive constants. In this case NASC are given for the existence of two positive solutions ui(x) in some exterior subdomain of Ω such that both |x|m exp[(-l)i-1k|x|]ui(x) are bounded and bounded away from zero in this subdomain, m = (n —1)/2, i = 1, 2.

Let R be a ring with identity. R satisfies one of the following properties for all x, y ∈ R:

1. (I) xynxmy = xm+1yn+1 and mnm! n! x≠0 except x = 0;

2. (II) xynxm = xm + 1yn + 1 and mm! n! x≠0 except x = 0;

3. (III) xmyn = ynxm and m! n! x≠0 except x = 0;

4. (IV) (xpyQ)n = xpnyqn for n = k, k + 1 and N(p, q, k) x≠0 except x = 0, where N(p, q, k) is a definite positive integer. Then R is commutative.

Let B be a σ-finite von Neumann factor of type II1 or III and let σ be an involutory *-antiautomorphism of B. We consider U(B) the unitary group of B and its subgroup G = {g ∈U(B) | σ(g) = g*}, which are unitary classical groups. In this paper, we prove that G has a unique non trivial normal subgroup, which is its centre {±1}.

In his Ph.D. thesis Allaway found all polynomials that can be represented as

and a0bn≠0. We solve the essentially equivalent problem of finding all symmetric polynomials with when are orthogonal with respect to dα(x). The polynomials are the continuous q-ultraspherical polynomials and some of their limiting cases.

In the euclidean plane let K be a compact convex set and Sl, S2,… strips of respective widths wl, w2,… Some conditions on Σ wi are given that imply that K can be covered by translates of the strips Si. These conditions involve the perimeter, the diameter, or the minimal width of K and yield improvements of previously known results.

For a linear map we introduce two notions of being open according to a subset. With these two definitions, we characterize the linear, nearly open mappings which are open. Thus we generalize a famous theorem of V. Pták.

During the last decade or so there has been a revival of interest in the analysis of error-bounds f(s)-S(s) for different classes of functions and their interpolatory splines of odd degree on a finite interval with variations on end conditions. Our object is to present a unified treatment of the asymptotic error expansion both for even and for odd degree interpolatory splines.

A short proof is given of an internal characterization of completely regular spaces due to J. Kerstan.

A new oscillation criterion is given for general superlinear ordinary differential equations of second order of the form x″(t)+ a(t)f[x(t)]=0, where a ∈ C([t0∞,)), f∈C(R) with yf(y)>0 for y≠0 and and f is continously differentiable on R-{0} with f'(y)≥0 for all y≠O. In the special case of the differential equation (γ > 1) this criterion leads to an oscillation result due to Wong [9].

In this note the Holland representation theorem for l-groups is extended to l-monoids by the following theorem: an l-monoid is distributive if and only if it may be embedded into the l-monoid of order-preserving functions on some totally ordered set. A corollary of this representation theorem is that a monoid is right orderable if and only if it is a subsemigroup of a distributive l-monoid; this result is analogous to the group theory case.

First we construct spreads consisting of analytic curves (circular arcs and segments), without points of finite multiplicity. Then we see that, in the sense of Baire categories, most such spreads have no points of finite multiplicity.

Bernstein's inequality says that if f is an entire function of exponential type τ which is bounded on the real axis then

Genchev has proved that if, in addition, hf (π/2) ≤0, where hf is the indicator function of f, then

Using a method of approximation due to Lewitan, in a form given by Hörmander, we obtain, to begin, a generalization and a refinement of Genchev's result. Also, we extend to entire functions of exponential type two results first proved for polynomials by Rahman. Finally, we generalize a theorem of Boas concerning trigonometric polynomials vanishing at the origin.