We introduce the Banach space of vector valued sequences lp, q(E), 1 ≤ p, q ≤ ∞, where E is a Banach space. Then we study the relation between lp, q(E) and the Schur multipliers of lpE, where E is taken to be some lr.

We prove first that a logical fraction functor from a topos to a topos must be a filter-power functor, then we prove that such functors can have adjoints only when the filter is principal. Finally we refine this so that we are able to prove that the filter-power of a Grothendieck topos is Grothendieck if and only if the filter is principal.

We give a brief survey of the results obtained by numerous authors in so-called almost-Hermite-Fejér-interpolation and deal mainly with new quantitative assertions.

These are based upon more general theorems for certain continuous linear operators which yield estimates involving different types of moduli of continuity.

Our paper shows that in the case of almost-Hermite-Fejér-interpolation the underlying general technique can be used to treat three essentially different cases: sequences of positive operators, which converge uniformly for every continuous function on [−1, 1], sequences of non-positive operators doing the same, and sequences of operators which converge on proper subspaces of C[−1, 1] only.

Let μ and ν be non-σ-finite pre-Radon measures on topological spaces X and Y respectively. Then there exists a unique pre-Radon measure λ on the product space X × Y which satisfies λ(A × B) = μ(A)ν(B) for all Borel sets A in X and B in Y such that μ(A) < ∞ and ν(B) < ∞.

The main purpose of this paper is to show that there exists a Souslin line if and only if there exists a countable chain condition space which is not weak-separable but has a generic π-base. If I is the closure of the isolated points in a space X, then X is said to be weak-separable if a first category set is dense in X – I. A π-base is said to be generic if, whenever a member of is included in the disjoint union of members of it is included in one of them.

This paper establishes a weak maximum principle for the difference u − v of solutions to nonlinear degenerate parabolic differential inequality

The full collineation group of the flag transitive plane of order 27 constructed by Hering is determined. It is shown that the stabilizer of the origin of this plane is of order 2184.

We apply recent results of Vaaler that simultaneously bound linear forms so as to obtain a ‘Siegel lemma’, sharper than those that have appeared in the literature, and in a shape convenient for application in transcendence theory.

A graph is called point determining if distinct points have distinct neighborhoods. In this note we provide a sufficient condition for a point determining graph to have a 1-factor. This is an extension of some results obtained by David P. Sumner.

It is shovn that simple algebras of characteristic not equal to 2, which contain non-trivial elements that satisfy xn = x for some fixed, minimal integer n > 1, are generated by these elements.

In a metric space (X, d) a ball B(x, ε) is separated if d(B(x, ε), X\B(x, ε)] > 0. If the separated balls form a sub-base for the d-topology then Ind X = 0. The metric is gap-like at x if dx(X) is not dense in any neighbourhood of 0 in [0, ∞). The usual metric on the irrational numbers, P, is the uniform limit of compatible metrics (dn), each dn being gap-like on P. In a completely metrizable space X if each dense Gδ is an Fσ then Ind X = 0.

Sufficient conditions are obtained for the existence and linear stability of time independent age distributions in two species competition with age and time lagged density dependent mortality and fertility functions.

Let G1, G2, be locally compact groups and let S1, S2, be Segal algebras on G1, G2, respectively. Under certain conditions on G1, G2, and S1, S2, we prove that if there is a bipositive or isometric isomorphism between S1, S2, or between their multiplier algebras then G1, and G2, are topologically isomorphic.

If μ and γ are ordinal numbers such that for every α ≤ γ·2, then γ < ω. This localizes a result due to Patai.

Let a function be regular in the disk |z| < 1. The radius of univalence 0.164 … of the family of f with |an| ≤ n (n ≥ 2) is, actually, the radius of star-likeness. The radius of univalence 1 - [k/(l+K)]½ of the family of f with |an| ≤ k (n ≥ 2), where K > 0 is a constant, is, actually, the radius of starlikeness. The radii of convexity of the two families are estimated from below.

Let B be a commutative, semi-simple, regular, Tauberian Banach algebra with noncompact maximal ideal space Δ(B). Suppose B has the property that there is a constant C such that for every compact subset K of Δ(B) there exists a f ∈ B with = 1 on K, ‖f‖B ≤ C and has compact support. We prove that if A is a proper abstract Segal algebra over B then for every positive integer n there exists f ∈ B such that fn ∉ A but fn+1 ∈ A. As a consequence of this result we prove that if G is a nondiscrete locally compact abelian group, μ a positive unbounded Radon measure on Γ (the dual group of G), 1 ≤ p < q < ∞ and , then .

There are two Hardy and Littlewood theorems which describe the rate of growth of functions in Hp on the unit circle T. In this paper we first establish their analogues on Euclidean space Rn and then apply them to solve multiplier and factorization problems on Hp(Rn).