Eames [2], and Jeffery [5], consider separation of sets in a measure space and show that, if A is separated from B, then

where m*denotes outer measure.

In this paper we consider the class, , of nonnegative bounded real-valued functions of a real variable.

Let (Ω,ℱ, P) be a probability space. Let R denote the set of real numbers and the set of all random variables defined on Ω. Throughout this work, random variables which differ only on a set of probability zero will be considered identical. EX represents, as usual, the expectation of .

Let L/k be a Galois extension of fields with group G and let A be the category of k-algebras isomorphic to finite products of finite field subextensions of L/k. It is known that, with appropriately defined covers, A is dual to the underlying category of a Grothendieck topology T [5, Ch. I, Theorem 4.2] and that (strict) cohomological dimension of G may be characterized via TCech cohomology with coefficients in either additive (product-preserving) functors or sheaves [5, Ch. I, Theorems 4.3 and 5.9].

Let f be a real valued function on an open subset of R2. It is assumed that f satisfies Carathéodory's conditions: f (t,x) is continuous in x for each t, Lebesgue measurable in t for each x and there is a locally integrable function m(t) such that |f(t, x)| ≤ m(t) uniformly in x. A proof will be given of the following theorem.

If (Xl, X2,…, Xn), is a set of n independent random variables, such that EXi=0, Var and if t is a real positive number and , then Bernstein [2] has given an upper bound for Pr when the X's are bounded. The best English language discussion of Bernstein's work is probably by Bennett [1].

In this note we generalize the following classical theorem: If μ and ν are finite real-valued measures such that ν(A) = 0 implies μ(A) = 0, then for every ε > 0, there exists δ > 0 such that μ(A)<ε whenever ν(A)< δ.

The purpose of this note is to present a characterization of m-paracompact normal spaces in terms of ω-mappings into m-separable metric spaces; this result is almost contained in Morita's original paper [2] and is implicitly contained in Shapiro's thesis paper [5].

Given a pair of maps

in a category, we would like to know whether they form part of a pullback diagram as follows:

I am indebted to Basil Rattray for mentioning the solution of this problem for the category of sets. Here we shall solve it for any regular category in the sense of Barr [1].

It will be useful to make the following definition.

An m-bounded extension of a topological space is an m-bounded space which contains the original as a dense subspace. m-bounded spaces have been studied by Gulden, Fleischman, and Weston [4], Saks and Stephenson [6], and Woods [8].

In the series of simple groups 2F4(q),q =2 2m+1, discovered by Ree, Tits [4] showed that the group 2F4(2) was not simple but contained a simple subgroup of index 2. In this note we extend the characterization of obtained by the author in [3].

Satyanarayana establishes in [6] that a semiprime right selfinjective ring with ACC on annihilator right ideals is semisimple Artinian, thereby extending a similar result of Koh [5] for prime rings. A theorem of Faith [3, Theorem 5.2], shows that the annihilator chain condition on either side implies that a right selfinjective semiprime ring is semisimple Artinian. Noting that any selfinjective ring has torsionless injective envelope we consider the possibility of replacing selfinjectivity by torsionless together with an annihilator condition.

We shall consider scalar transport phenomena in which the mass distribution of a large number of interacting mass particles varies as a result of multiple coalescence mechanisms. Four models will be considered; all the particles are of the same kind in the first three. In the first model an integer n (n≥2) is given and the coalescence mechanism is such that under suitable conditions n particles of masses xl…, xn combine to form one particle of mass we refer to this as the n/1 transport.

Given a pair of functors F, G:A→B, Lambek defines [1] the subequalizing category, E of (F, G) as the category with objects, ordered pairs (A, b) with A ∈ |A| and b:FA→GA a morphism of B. The morphisms of E from (A, b) to (A′, b′) are ordered triples (b, a, b′) where a:A→A′ is a morphism of A and G(a)b = b′F(a).

It is well known that a large proportion of the locally convex spaces encountered in distribution theory are nuclear (Grothendieck [4], Treves [10], Schaeffer [8].) In [1] Beurling introduced spaces of test functions more general than those previously used. In this paper we shall show that many of these spaces, and resulting spaces of distributions, are also nuclear spaces.

This note, motivated by [2], [3], and [4], is devoted to an investigation of properties related to equicontinuity in function spaces of topological spaces. In §2, we study the property (G) defined in [3], and the regularity defined in [4]. A sufficient condition for the simultaneous continuity of a function of two variables, which is analogous to a well known result in equicontinuity, is given at the end of the section. In §3, we relate the regularity with the semi-equicontinuity defined in [2], by localizing the semi-equicontinuity in an obvious way which leads us to weaken some of the hypotheses used in [2]. By the way of constructing an example, we also obtained a sufficient condition for a regular semitopological group to be a topological group.

Let f be an L integrable real valued function of period 2π and let

(1)

be its Fourier series. It is known that if f is of bounded variation then all nan and nbn(n=1,2,3,…) lie in the interval [-V(F)/π, V(F)/π;] where V(f) is the total variation of f. M. Izumi and S. Izumi [3] have recently asserted the following theorem A about the density of the positive and negative Fourier sine coefficients of a function of bounded variation.

Strong oscillation criteria will be obtained for the linear elliptic partial differential equation

(1)

in unbounded domains R of general type in n-dimensional Euclidean space En. It will be assumed throughout that B and each Aij are real-valued continuous functions in R, and that the matrix (Aij(x)) is symmetric and positive definite in R.

It is well known (see [3](1)) that no continuum (i.e. compact, connected, Hausdorff space) can be written as a countable disjoint union of its (nonvoid) closed subsets. This result can be generalized in two ways into the setting of locally compact, connected, Hausdorff spaces. Using the one point compactification of a locally compact, connected, Hausdorff space X one can easily show that X cannot be written as a countable disjoint union of compact subsets. If one makes the further assumption that X is locally connected, then one can show that X cannot be written as a countable disjoint union of closed subsets.(2)

In [1], Willard proves the following

If a regular paracompact space X has a dense Lindelöfsubspace, then X is Lindelöf.

Willard notes that the above is a generalization of the standard theorem: A separable paracompact space is Lindelöf. Actually, it is a standard fact ([2, p. 24]) that a separable metacompact space is Lindelöf. Moreover, one discovers that if a separable space X is such that each open cover of X has a point-countable open refinement, then X is Lindelöf.

We are concerned with the nth-order linear differential equation

1

where the coefficients are assumed to be continuous. Hartman [1] proved that (see Definition 2) the first conjugate point η1(t) of t satisfies

2

Hartman actually proved a more general result which has very important applications in nonlinear differential equations.