The binary tensor product, for modules over a commutative ring, has two different aspects: its connection with universal bilinear maps and its adjointness to the internal hom-functor. Furthermore, in the special situation of finite-dimensional vector spaces, the tensor product can also be described in terms of dual spaces and the internal hom-functor. The aim of this paper is to investigate these relationships in the setting of arbitrary concrete categories.

In this paper we investigate the structure of certain spaces of operator valued measures and the Lp spaces they generate. The work is motivated by our earlier paper [1] in which we studied the Lp spaces generated by matrix valued measures. The present results can thus be regarded as a generalization of this “finite dimensional” situation.

On the zeros of . If not stated otherwise, we assume throughout that κ > 0, c > 1, and that k < κ ≤ k + 1 where k = 0, 1, 2, … We reserve the symbol x to denote real numbers, and define C* = C-{x:x <-1}, C being the complex plane. Let

Leeming [3] has defined a sequence of polynomials {Q4n(x)} and a sequence of integers {Q4n} by means of

1.1

and

1.2

Thus

1.3

and equivalently

1.4

A friendship graph is a graph in which every two distinct vertices have exactly one common neighbour. Finite friendship graphs were characterized by Erdös, Rényi, and Sós [1] as those for which the vertices can be enumerated as u, υ1,…υk, w1,…wk in such a way that the only edges are uυiuwi and υiwi (i = 1,…,k). Thus finite friendship graphs are rather rare. In contrast, we shall show that there are as many nonisomorphic friendship graphs of given infinite cardinal as there are nonisomorphic graphs of that cardinal altogether. In fact, we do a little more.

Let G be a real Lie group, A a closed subgroup of G and B an analytic subgroup of G. Assume that B normalizes A and that AB is closed in G. Then our main result (Theorem 1) asserts that .

This result generalizes Lemma 2 in the paper [4], G. Hochschild has pointed out to me that the proof of that lemma given in [4] is not complete but that it can be easily completed.

Let f be a function denned in the unit disk D(|z| < 1). For each point eiθ on the unit circle C(|z| = 1) and each subset S of D, we denote by Cs(f, eiθ) the cluster set of f at eiθ relative to s, i.e.

where N(eiθ, j) = {z∈D:|z-eiθ| <1/j}.

C. Feffermann and E. M. Stein [2] have shown that the continuity property of the Hardy-Littlewood maximal functions between Lp-spaces, 1 < p < ∞, extends to ℓr-valued functions on ℝn. Specifically, if f = (f1, f2,…) is a sequence of functions defined on Rn, let for l<∞, |f(x)|r be given by

The use of decreasing rearrangement formulas, and particularly that of the weak N inequality, is illustrated by deriving from Eτ |f-f(τ -)|≤Eτu (where ft is some stochastic process and τ arbitrary stopping time) the estimate ||f||≤Const||u|| in the class of structureless norms with finite dual Hardy bound.

The basic estimate is

In this paper, the cyclotomic polynomial is generalized and several of its properties based on the Môbius inversion are derived. It is deduced that a polynomial whose roots are the roots of a cyclotomic polynomial multiplied by those of another cyclotomic polynomial is the product of cyclotomic polynomials. Character sums and finite Fourier series are employed for the latter result.

It is well known that every point of the closed unit interval I can be expressed as the midpoint of two points of the Cantor ternary set D. See [2, p. 549] and [3, p. 105]. Regarding J as a one dimensional compact convex set, it seems natural to try to generalize the above result to higher dimensional convex sets. We prove in section 3 that every convex polytope K in Euclidean space Rd contains a topological copy C of D such that each point of K is expressible as a midpoint of two points of C. Also, we give necessary and sufficient conditions on a planar compact convex set for it to contain a copy of D with the midpoint property above. In the final section we prove a result on minimal midpoint sets.

Let θ = θ(k) be the positive root of θ2 + (k-2)θ-k = 0. Let f(n) = [(n + l)θ]-[nθ] for positive integers n, where [x] denotes the greatest integer in x. Then the elements of the infinite sequence (f(l), f(2), f(3),…) can be rapidly generated from the finite sequence (f(l), f(2),…,f(k)) by means of certain shift operators. For k = 1 we can generate (the characteristic function of) the sequence [nθ] itself in this manner.

A map f: X → Y is said to be universal if for every map g:X → Y there exists an x ∈ X such that f(x) = g(x). In [2] W. Holsztynski observed that if B is a Boltyanskiĭ continuum (see [1]), then there exists a universal map f:B→ I2 such that the product map fxf:BxB→I2×I2 is not universal. Using this he showed that B can be replaced by a two-dimensional polyhedron. He did not, however, give a concrete example. We exhibit explicitly a two-dimensional polyhedron K and a universal map f:K→ I2 such that f×f:K×K→ I2×I2 is not universal.

Following Kay and Womble [2] an abstract convexity structure on a set X is a collection ξ of subsets of X which includes the empty set, X and is closed under arbitrary intersections. One of the natural problems that arises in convexity structures is to give necessary and sufficient conditions for the existance of a linear structure on X such that the collection of all convex sets in the resulting linear space is precisely ξ. An associated problem is to consider a set with a convexity structure and a topology and find necessary and sufficient conditions for the existance of a linear structure on X such that X becomes a linear topological space with again ξ the collection of convex sets.

If X and Y are infinite-dimensional Banach spaces, then Y is the inductive limit of Banach spaces each isomorphic to X.

In this note the Erdös-Renyi law of large numbers is extended to stationary Gaussian sequences.

Locally quasi-uniform spaces are studied, and it is shown that a topological space (X, t) admits exactly one compatible locally quasi-uniform structure if and only if t is finite.

Let Γn = {z = (z1, z2,…, zn): and Re(zi)>0, ∀i}. For a multisequence Mj, = (j1, j2,…,jn) and 0 < M(j)≤∞, let , q(r) = sup|j||{r|j|Mj} and .