We develop an isomorphism invariant for limit algebras: an extension of Power's strong algebraic order on the scale of the K0-group (Power, J. Operator Theory 27 (1992), 87–106). This invariant is complete for a certain family of limit algebras: inductive limits of digraph algebras (a.k.a. finite dimensional CSL algebras) satisfying two conditions: (1) the inclusions of the digraph algebras respect the order-preserving normalisers, and (2) the digraph algebras have chordal digraphs. The first condition is also used to show that the invariant depends only on the limit algebra and not the direct system. We give an intrinsic characterisation of the limit algebra and not the direct system. We give an intrinsic characterisation of the limit algebras satisfying both (1) and (2).

Soit Θ un automorphisme analytique de , Θ ayant un point fixe attractif zo (c'está-dire il existe un voisinage V de zo tel que pour tout élément z de V, on ait Alors d'aprés un résultat bien connu, il existe une fonction F entiére de dans , injective, telle que Ceci provient du fait qu'il existe une fonction F, définie au voisinage de z0, telle que F'(zo)=I, vérifiant Fequation fonctionnelle est un automorphisme analytique de vérifiant pour tout z élément de .

This paper contains a new proof of a theorem of D. A. R. Wallace [5] in case of a p-solvable group. An alternative proof has been given by K. Motose [4].

Let G be a finite group, F a field with prime characteristic p, P a Sylow p-subgroup of G. JFG will denote the Jacobson radical of FG.

Let R be a noetherian prime PI ring and let P be a prime ideal of R. There is a set of prime ideals, the fundamental prime ideals, associated with the injective hull of R/P and denoted by Fund(P). The set Fund(P) is finite, by a result of Miiller. In this paper we give a natural description of Fund(P) in terms of the trace ring of R which strengthens Miiller's result by establishing a uniform bound for the size of Fund(P) for all primes P in the ring.

Cette note a pour objet de familiariser les élèves avec l'emploi des coordonnées tangentielles, en appliquant ces coordonnées, concurremment avec les coordonnées ponctuelles, à la résolution d'un certain nombre de questions, d'ordre très général, concernant les surfaces du second ordre.

Let the straight line be denned by the coordinates (α β γ) of a point on it and by its direction cosines l, m, n. It may be referred to as the line (α β γ l, m, n). "Write, for shortness, the equation to the quadric surface in the form F(x, y, z)=0.

For any element a of a semigroup (S,·), we may define a “sandwich” operation ∘ on the set S by x ∘ y = xay (x, y ∈ S). Under this operation the set S is again a semigroup; we denote this semigroup by (S, a) and call it a variant of S. Variants of semigroups of binary relations have been studied by Chase [6, 7]. In this paper we consider variants of arbitrary semigroups.

In a general metric space of four dimensions, with an interval given by , where – ds2 = gμνdxμdxν, where g = ║gμν║<0, we can choose locally galilean coordinates at any point. The initial directions of the axes can be fixed in an absolute fashion as the directions of the principal axes of the quadric Gμνdxμdxν = const., is the contracted Riemann-Christoffel tensor.

In a recent paper I discussed, for a given series Σan the relation between the conditions

where 0 < ρ < 1 and p is a positive integer. It was there proved (Theorem 14) that (2) implies (1) but that the converse is not necessarily true. My object in this note is to render more precise the connection between (1) and (2) by showing that (2) is equivalent to (1′), where (1′) is (1) with the additional restriction that Σn−1wn is convergent. The proof of this equivalence relation is based on a technique which was introduced and developed by Andersen and is similar to the proofs in my former paper. The note concludes with a brief discussion of the corresponding results for the case ρ = 1.

The chief object of this Note is to develope some simple and rather interesting properties of the operators α β γ referred to in my Note read at last meeting. I take for brevity the symbol μ. to denote the compound operator

A semigroup S with identity is (left) perfect if every unitary left S-system has a projective cover. This is the semigroup analogue of the definition of left perfect rings introduced in (1). The investigation of perfect semigroups was initiated by Isbell (4), who proved that a semigroup is perfect if and only if it satisfies two conditions referred to as conditions A and D.

The continuant referred to is that in which the elements of the main diagonal are all equal (to x, say), the elements of the one minor diagonal all equal (to b, say), and the elements of the other minor diagonal all equal (to c, say). It may be denoted by F(b, x, c, n) when it is of the nth order. Professor Wolstenholme has recently given two elegant theorems regarding the condensation of F(1, x, 1, n). I wish to establish the analogous theorems for F(b, x, c, n).

If A1 and B3 (Fig. 17), B1 and C3, C1 and D3, D1 and A3, A2 and C2, B2 and D2 be pairs of points on the edges

respectively of a tetrahedron ABCD, such that the two points on any edge are concyclic with the two points on any other edge—a manifestly possible condition of things—then the twelve points lie on a sphere.

1. Introduction. The problem of extending Dirac's equation of the electron to general relativity has been attacked by many authors, by methods which fall roughly into either of two classes according as the formulation does or does not require the introduction of a local Galilean system of coordinates at each point of space-time. As examples of the former class we mention the methods of Fock (1929) and of Cartan (1938), and as representing the latter class the method described by Ruse (1937). Also, Whittaker (1937) discovered a vector whose vanishing is completely equivalent to the Dirac equations, but this method, unlike the others in the second category, does not apply the Riemannian technique to spinors but only to vectors and tensors derived from these. Now Cartan has denied the possibility of fitting a spinor into Riemannian Geometry if his point of view of spinors is adhered to, and this he argues accounts for the “choquant” properties with which they have been endowed by the geometricians in order to enable them to write down an expression of the usual form for the covariant derivative of a spinor. Consequently, doubt has been cast on the compatibility of the various methods, so in this paper an attempt is made to clarify the matter by working out explicitly the case of the general metric by some of the more important of these methods.

1. The following condition may be new; it does not appear in any of the books:—

The condition that the straight line

should be a normal to the general conic,

referred to rectangular axes, is

where Σ is written for

and Δ, A, B, C, F, G, H have their usual meanings.

Let S be a compact Hausdorff space and let Φ: C(S)→E be a linear operator defined on the space of real-valued continuous functions on S and taking values in a (real) topological vector space E. Then Φ is called exhaustive (7) if given any sequence of functions fn ∈ C(S) such that fn ≧ 0 and

then Φ(fn)→0 If E is complete then it was shown in (7) that exhaustive maps are precisely those which possess regular integral extensions to the space of bounded Borel functions on S; this is equivalent to possessing a representation

where μ is a regular countably additive E-valued measure defined on the σ-algebra of Borel subsets of S.