Let E be the (nonelementary) plane Euclidean geometry without the Pasch axiom. (The Pasch axiom says that a line cutting one side of a triangle must also cut another side. A full list of axoms for E is given in [5].) E satisfies in particular the full second-order continuity axiom.

Szczerba [5] has recently shown using a Hamel basis for the reals over the rationals that there exists a model of E not satisfying the Pasch axiom. It is natural to ask whether the axiom of choice plays an essential role in the proof. It will turn out that it does.

In the following (Ω, α, μ) is a totally σ-finite measure space except where noted. For a sub-σ-algebra β ⊂ α, the conditional expectation E{f|β} off given β is a function measurable relative to β, such that

In [5] R.G.Douglas proved, among other things the following, in the finite case:

Suppose μ(Ω)=l. Then a linear operator T on L1(Ω, α,μ) is a conditional expect ion if and only if

1.1

1.2

1.3

The point of this note is to characterize conditional expectation in the σ-finite case (Theorems 2, 3).

Le théorème de Weierstrass affirme que toute fonction réelle continue sur l'intervalle [0, 1] est limite uniforme d'une suite de polynômes (voir [5]). Dans [4], Ch. Müntz généralise ce résultat de la façon suivante.

Soit α1, α2, α3,… une suite strictement croissante et non bornée de nombres réels.

A construction of an epi-reflector by injective hull techniques is given which applies to the class of models of any universal theory with the Amalgamation Property and there yields a weak but functorial type of algebraic closure. Various completions such as the boolean envelope and quotient field constructions are identified as such injective hulls over epimorphic injections. Forms of the Amalgamation Property are also shown to eliminate various pathologies of epimorphisms and equalizers.

Let Xl,X2,X3, … be a sequence of independent and identically distributed (i.i.d.) random variables which belong to the domain of attraction of a stable law of index α≠1. That is,

1

where

and

where L(n) is a function of slow variation; also take S0=0, B0=l.

In §2, we are concerned with the weak convergence of the partial sum process to a stable process and the question of centering for stable laws and drift for stable processes.

The relationship between bases and isomorphisms (i.e. linear homeomorphisms) between complete metrizable linear spaces has been studied with great interest by Arsove and Edwards (see [1] and [2]). We prove (Theorem 1) that in the case of B-complete barrelled spaces, similar generalized bases imply existence of an isomorphism. This result was also proved by Dyer and Johnson [4], so we do not give a proof. We show (Theorem 6) that if one assumes that the bases are Schauder and similar, then Theorem 1 holds for countably barrelled spaces. We use Theorem 1 to advantage (Theorems 2-5) to show that one can improve some results due to Davis [3].

A mapping f from a metric space (X,d) into itself is said to be contractive if x≠y implies d(f(x),f(y))<d(x,y). Theorems of Edelstein [2] state that a contractive selfmapfofa metric space X has a fixed point if, for some x0, the sequence {fn(x0)} of iterates at x0 has a convergent subsequence; moreover, the sequence {fn(x0)} converges to the unique fixed point of f. Nadler [3] observes that, from the point of view of applications, it is usually as difficult to verify the condition (for some x0 …) as it is to find the fixed point directly.

Maeda ([1, p. 29, Problème 1]) pose la question suivante: Est-ce qu'un treillis ⏊-symétrique et relativement semi-orthocomplémenté est nécessairement M-symétrique? Nous donnons une réponse affirmative à cette question.

Soit L un treillis, et a,b,y ∊ L. Si (a,b)M et y≤b, alors (y V a,b)M.

In 1892, Sylvester [7] proved that in the set of integers n, n+l,…, n+k—1, n> k > 1, there is a number containing a prime divisor greater than k. This theorem was rediscovered, in 1929, by Schur [6]. More recent results include an elementary proof by Erdös [1] and a proof of the following theorem by Faulkner [2]: Let pk be the least prime ≥2k; if n≥pk then has a prime divisor ≥pk with the exceptions and In that paper the author uses some deep results of Rosser and Schoenfeld [5] on the distribution of primes.

The following theorem is the principal result of this paper.

Let (M, d) be a metric space and T a self-mapping of M satisfying the condition for x,y ∊ M

1

where a, b, c, e,f are nonnegative and we set α=a+b+c+e+f.

A sequential procedure for the exploration of response surfaces is proposed. The procedure, which is for experiments with two factors, uses the kiss-precise configuration, i.e., the design points are on circles in mutual contact at each stage. Only three points need be added at each stage and the design points form a first-order rotatable design. A second-degree surface may be fitted when a near stationary region is reached.

We have an alternative proof of the following result of Kervaire [2]:

Let V→M be a real vector bundle with fibre dimension n≥4k+l over a compact 4k-manifold. Suppose V restricted to M — {x} is trivial. Choose a Riemann structure for V and an orthonormal frame for V restricted to M —{x}. Thus the obstruction to extending the frame smoothly over M is an element λ in π4k+1SO(n))≅Z. Then up to sign the evaluation of the kth Pontrayagin class Pk on M is ak(2k— 1)!. λ, where ak is 1 or 2 depending upon whether k is even or odd.

We assume throughout that (X, T, π) is a transformation group [2], where X is a topological space which is always assumed to be regular and Hausdorff. We call a point x ∊ X regular under T if for any open set U in X and any subset G of T such that , there exists an open set V containing x, such that VG ⊆ U [7]. Let R(X) denote the interior of the set of all the regular points of X under T, and I(X) the set of irregular points of X under T, that is the set of points which are not regular under T.

It is known that the Fitting length h(G) of a finite soluble group G is bounded in terms of the number v(G) of the conjugacy classes of its maximal nilpotent subgroups. For |G| odd, a bound on h(G) in terms of v(G) was discussed in Lausch and Makan [6]. In the case when the prime 2 divides |G|, a logarithmic bound on h(G) in terms of v(G) is obtained in [7]. The main purpose of this paper is to show that the Fitting length of a finite soluble group is also bounded in terms of the number of conjugacy classes of its maximal metanilpotent subgroups. In fact, our result is rather more general.

Let H be a Hilbert space; ( , ) and | | represent the scalar product and the norm respectively in H. Let A be a closed linear operator with domain DA dense in H and A* be its adjoint with domain DA*. DA and DA*are also Hilbert spaces under their respective graph scalar product. R(λ; A*) denotes the resolvent of A*; complex plane. We write L = D — A, L* = D — A*; D = (l/i)(d/dt).

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.

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.

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.

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.

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.