Let A be a ring (not necessarily associative) in which 2x = a has a unique solution for each a ∈ A. Then it is known that if A contains an identity element 1 and an involution j : x ↦ x and if Ja is the canonical involution on An determined by

where the ai al−l, 1 ≦ i ≦ n are symmetric elements in the nucleus of A then H(An, Ja), the set of symmetric elements of An, for n ≧ 3 is a Jordan ring if and only if either A is associative or n = 3 and A is an alternative ring whose symmetric elements lie in its nucleus [2, p. 127].

Dans ce travail, nous désignons par la catégorie des groupes localement nilpotents. Si P est un ensemble de premiers, nous disons que n est un P-nombre et nous écrivons n ∈ P si tous les facteurs premiers de n appartiennent à P; on convient toujours que 1 ∈ P. Dans tout ce qui suit, il est souvent commode de ne pas faire explicitement la distinction dans la notation entre un ensemble P de premiers et l'ensemble de tous les entiers naturels ayant tous leurs facteurs premiers dans P; par exemple, n ∈ P signifiera toujours que n est un P-nombre mais pas nécessairement un premier. L'ensemble de tous les premiers n'appartenant pas à P est désigné par P′; on convient que 1 appartient à la fois à P et à P′.

1. Introduction. Our interest here lies in the following theorem:

THEOREM 1. Assume there is defined on Rn (n ≧ 3) a “square-distance” of the form

where (gij) is a given symmetric non-singular matrix over the reals and x = (x1, …, xn), y = (y1, …, yn) ∈ Rn. Assume further that f is a bijection ofRnwhich preserves a given fixed square-distance ρ, i.e. d(x, y) = ρ if and only if d(ƒ(x),ƒ(y)) = ρ. Then (unless ρ = 0 and (gij) is positive or negative definite) ƒ(x) = Lx + ƒ(0), where L is a linear bijection ofRnsatisfying d(Lx, Ly) = ±d(x, y) for all x, y ∈ Rn (the – sign is possible if and only if ρ = 0 and (gij) has signature 0).

In 1972, Lelek introduced the notion of Class (W) in his seminar at the University of Houston [see below for definitions of concepts mentioned here]. Since then there has been much interest in classifying and characterizing continua in Class (W). For example, Cook has a result [5, Theorem 4] which implies that any hereditarily indecomposible continuum is in Class (W) Read [21, Theorem 4] showed that all chainable continua are in Class (W), and Feuerbacher proved the following result:

(1.1) THEOREM [7, Theorem 7]. A non-chainable circle-like continuum is in Class (W) if and only if it is not weakly chainable

In [14, 4.2 and section 6], a covering property (denoted here and in [18] by CP) was defined and studied primarily for the purpose of proving that indecomposability is a Whitney property for the class of chainable continua [14, 4.3].

It is well known that a translation plane can be represented in a vector space over a field F where F is a subfield of the kernel of a quasifield which coordinatizes the plane [1; 2; 4, p.220; 10]. If II is a finite translation plane of order qr (q = pn, p any prime), then II may be represented in V2r(q), the vector space of dimension 2r over GF(q), as follows:

(i) The points of II are the vectors in V = V2r(q)

(ii) The lines of II are

(a) A set of qr + 1 mutually disjoint r-dimensional subspaces of V.

(b) All translates of in V.

(iii) Incidence is inclusion.

0. Introduction. A surface of order three, F, in the real projective threespace P3 is met by every line, not in F, in at most three points. F is biplanar if it contains exactly one non-differentiable point v and the set of tangents of F at v is the union of two distinct planes, say τ1 and τ2. In the present paper, we classify and describe those biplanar F which contain the line τ1 ∩ τ2.

We describe a surface by determining the tangent plane sections of the surface at the differentiable points. This approach was introduced in [1] and it is based upon A. Marchaud's definition of “surfaces of order three” in [4].

Let G denote an unbounded domain of Euclidean m-space Em with regular boundary, and let L be a self-adjoint operator generated in L2(G) by a second order elliptic expression. We denote by S(L) the spectrum of L, by µ the least point of the essential spectrum Se(L) and by N(L) the number of bound states of L; that is, the number of points in (–∞, µ) ∩ S(L). There are many results in the literature dealing with the localization, significance and properties of µ, of Se(L) and of (–∞, µ)⌒ S(L), with most of the emphasis on the cases where G = Em or G is the exterior of a closed surface in Em. We refer the reader to the books by Glazman [12], Schechter [19], Reed and Simon [18], and Paris [9], where extensive references are also found.

If (Y, µ) is an H-Space (here all our spaces are assumed to be finitely generated) with homotopy associative multiplication µ. and X is a finite CW complex then [X, Y] has the structure of a nilpotent group. Using this and the relationship between the localizations of nilpotent groups and topological spaces one can demonstrate various properties of [X,Y] (see [1], [2], [6] for example). If µ is not homotopy associative then [X, Y] has the structure of a nilpotent loop [7], [9]. However this algebraic structure is not rich enough to reflect certain significant properties of [X, Y]. Indeed, we will show that there is no theory of localization for nilpotent loops which will correspond to topological localization or will restrict to the localization of nilpotent groups.

Let K be a field and L an extension field. L. Fuchs [2, Problem 98] has suggested studying the change in multiplicative groups in going from K* to L*. We wish to indicate difficulties that arise in trying to relate the group theoretic structure of L* to that of K*, even when K* has particularly simple structure and the extension is quadratic.

First let us note a trivial fact. If [L : K] = n < ∞ and K* has a free direct factor A, then L* has a free direct factor isomorphic to A. To see this, let ϕ be the composite L* → K* → A of the norm map followed by the projection map. Then L* has a free direct factor isomorphic to ϕ(L*). But the image of the norm map contains (K*)n, hence ϕ(L*) ≅ A.

1. Introduction. Let (X, , m) be a σ-finite measure space and let T be a positive linear operator on L1 = L1(X, , m). T is called Markovian if

(1.1)

T is called sub-Markovian if

(1.2)

All sets and functions are assumed measurable; all relations and statements are assumed to hold modulo sets of measure zero.

For a sequence of L1+ functions (ƒ0, ƒ1, ƒ2, …), let

(ƒn) is called a super additive sequence or process, and (sn) a super additive sum relative to a positive linear operator T on L1 if

(1.3)

and

(1.4)