In [2] Brungs shows that every ring T between a principal (right and left) ideal domain R and its quotient field is a quotient ring of R. In this note we obtain similar results without assuming the ascending chain conditions. For a (right and left) Bezout domain R we show that T is a quotient ring of R which is again a Bezout domain; furthermore Tis a valuation domain if and only if T is a local ring.

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.

Edelblute [1] has given a method of finding an inverse of a nonsingular matrix by rank annihilation. The purpose of this paper is to show that the method can be extended in the case of a singular matrix. This method will produce a singular inverse satisfying condition (3) of Penrose [3].

The study of systems of conies and other algebraic curves was initiated in the middle of the nineteenth century by Cayley, Hesse, Cremona, and others. Most of the investigations from that time to the present have been concerned with extensions to algebraic varieties and systems of higher orders or dimensions, or with associated algebraic curves such as Jacobians and Hessians.

This paper attempts to generalize a property of regular rings, namely,I2=I for every right (left) ideal. Rings with this property are called right (left) weakly regular. A ring which is both left and right weakly regular is called weakly regular. Kovacs in [6] proved that, for commutative rings, weak regularity and regularity are equivalent conditions and left open the question whether for arbitrary rings the two conditions are equivalent. We show in §1 that, in general weak regularity does not imply regularity. In fact, the class of weakly regular rings strictly contains the class of regular rings as well as the class of biregular rings.

Several authors have recently considered the problem of obtaining sufficient conditions for the oscillation of the quasilinear matrix differential equation

(1)

and the associated inequality VTLV ≤ 0 (as a form). Here A, B, and V are m x m matrix functions, A(x) is symmetric, positive semidefinite and continuous in an interval [a, ∞) and B(x,V, V') is symmetric and continuous in a≤ x < ∞ for all V and V'.

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).

Recently Luedeman studied certain idempotent topologizing families of left ideals in semi-group rings AS which arise from such families of left ideals of A. Let ∑ be an idempotent topologizing family of left ideals in A and G a group, let ∑G denote the family of left ideals of AG which contain left ideals of the form LG, L ∈ ∑.

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.

Let denote the automorphism group of the tournament T. Let g(n) be the maximum of taken over all tournaments of order n. It was noted in [3] that g(n) is also the order of the subgroups of Sn of maximum odd order where Sn denotes the symmetric group of degree n.

In this paper the author characterizes images of class- spaces (as defined by Ishii, Tsuda and Kunugi, Proc. Japan Acad. 44, (1968), 897-903) under almost-open maps, bi-quotient maps, pseudoopen maps and quotient maps.

In the past three decades the problem of generating (balanced incomplete block) designs by difference sets has received much attention. Bose [2] gave the two "fundamental theorems of the method of differences". Bose, Sprott [9], Lehmer [7], Chowla [4], Takeuchi [10] and others have given specific classes of difference sets.

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.

In Meir and Keeler [3], the authors proved a fixed point theorem in a complete metric space (X, d) for a mapping f that satisfies the following condition of weakly uniformly strict contraction:

Given ∊ > 0, there exists δ > 0 such that

(A)

It is proved, for any uncountable cardinal λ, that a λ-complete Boolean ring is λ-self-injective. An example shows that the converse need not hold.

The problem of finding the largest set of nodes in a d-cube of side 3 such that no three nodes are collinear was proposed by Moser. Small values of d (viz., d ≤,3) resulted in elegant symmetric solutions. It is shown that this does not remain the case in 4 dimensions where at most 43 nodes can be chosen, and these must not include the center node.

The basic solvability theorems of Farkas [2] and Levinson [4] were recently extended in different directions by Ben-Israel [1] and Kaul [3].

The theorem stated in this note generalizes both results of Ben-Israel and Kaul and is applicable to nonlinear programming over complex cones.

Let(X1, Y1), (X2, Y2),…, (Xn, Yn) be n mutually independent pairs of random variables with absolutely continuous (hereafter, a.c.) pdf given by

(1)

where f(ρ) denotes the conditional pdf of X given Y, g(y) the marginal pdf of Y, e(ρ)→ 1 and b(ρ)→0 as ρ→0 and,

(2)

We wish to test the hypothesis

(3)

against the alternative

(4)

For the two-sided alternative we take — ∞< b < ∞. A feature of the model (1) is that it covers both-sided alternatives which have not been considered in the literature so far.

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.