As in [4], a specification is a list (r, s, t1 h0, hmc (l), …, c(ho)) such that

(i) each of r, s, t1 h0, h∞ is a non-negative integer,

(ii) for each i, c(i) is a positive integer,

(iii) if h∞ = 0 then ho = ∞,

(iv) if h∞ = 1 and t1 + h0 is finite then t1 is even,

(v) r + s + t1 + h0 + h∞ = ∞.

In this note homomorphisms of (2, 3, n) = 〈x, y: x2 = y3 = (xy)n = 1) into PSL3(q) are considered. Of particular interest is (2, 3, 7) whose finite factors are referred to as Hurwitz groups. It is known (see [3]) that for certain q, PSL2(q) is a Hurwitz group, so that one might suppose that PSL3(q) is a natural place to search for new Hurwitz groups. This intuition turns out to be ill-founded, for as we shall see all Hurwitz subgroups of PSL3(q) have already been discovered in [3].

Several papers have appeared in the past few years which have explored the topic of the Riesz decomposition for amarts. Such a decomposition for amarts enjoys several special properties. See [5, p. 208–209]. While it has been proved in [6] that not every martingale in the limit has a Riesz decomposition “in the weakest form assuring uniqueness” it is the major objective of this paper to characterize a class of martingales in the limit which is strictly larger than the class of amarts but enjoys all the properties of the decomposition for amarts.

Let M be a compact connected boundaryless surface and f: M → ℝ3 a smooth immersion transverse to a straight line L. Thus there is an even number p of points xεM such that f(x)εL. Under further transversality assumptions on f (see §3) there is a finite number q of points x of M such that the plane containing f(x) and L touches f(M) at f(x). These assumptions are mild in the sense that they hold for any f in an open dense subset of the space of smooth immersions under consideration. Suppose that the Gaussian curvature of f(M) is positive at q+ of these points and negative at q−, with q = q++ q−. Then

where e(M) denotes the Euler number of M.

Let X be an F-space (complete metric linear space) and suppose g:[0, 1] → X is a continuous map. Suppose that g has zero derivative on [0, 1], i.e.

for 0≤t≤1 (we take the left and right derivatives at the end points). Then, if X is locally convex or even if it merely possesses a separating family of continuous linear functionals, we can conclude that g is constant by using the Mean Value Theorem. If however X* = {0} then it may happen that g is not constant; for example, let X = Lp(0, 1) (0≤p≤1) and g(t) = l[0,t] (0≤t≤1) (the characteristic function of [0, t]). This example is due to Rolewicz [6], [7; p. 116].

II est connu [4] que si A = U() est l'algèbre enveloppante d'une algèbre de Lie nilpotente de dimension finie sur un corps F de caractéristique 0, tout idéal (complètement) premier P a pourlocalisé R = Ap un anneau régulier au sens de [5]; c'est-à-dire que le radical de Jacobson de R est engendrè par une suite centralisante régulière de longueur n = K-dim R, soit (z1…, zn). Dans le cas très particulier où P est l'idéal d'augmentation de U() il suffit de prendre pour (z1…, zn) l'image dans U()p d'une base de sur F adaptée à la suite centrale ascendante de .

Following the notation in Faudree and Schelp [3], we write G → (F, H) to mean that every 2-coloring of E(G), the edge set of G, contains a green (the first color) F or a red (the second color) H. Then the Ramsey number r(F, H) of two graphs F and H with no isolated vertices has been defined as the minimum p such that Kp → (F, H).

Recall that a Noetherian ring R is a Hilbert ring if the Jacobson radical of every factor ring of R is nilpotent. As one of the main results of [13], J. E. Roseblade proved that if J is a commutative Hilbert ring and G is a polycyclic-by-finite group then JG is a Hilbert ring. The main theorem of this paper is a generalisation of this result in the case where all the field images of J are absolute fields—we shall say that J is absolutely Hilbert. The result is stated in terms of the (Gabriel–Rentschler–) Krull dimension; the definition and basic properties of this may be found in [5]. Let M be a finitely generated right module over the ring R. We write AnnR(M) (or just Ann(M)) for the ideal {r ∈ R: Mr = 0}, the annihilator of M in R. If M is also a left module, its left annihilator will be denoted l-AnnR(M). If R is a group ring JG, put

Let G be a finite group and p a prime number. About five years ago I. M. Isaacs and S. D. Smith [5] gave several character-theoretic characterizations of finite p-solvable groups with p-length 1. Indeed, they proved that if P is a Sylow p-subgroup of G then the next four conditions (l)–(4) are equivalent:

(1) G is p-solvable of p-length 1.

(2) Every irreducible complex representation in the principal p-block of G restricts irreducibly to NG(P).

(3) Every irreducible complex representation of degree prime to p in the principal p-block of G restricts irreducibly to NG(P).

(4) Every irreducible modular representation in the principal p-block of G restricts irreducibly to NG(P).

In an important recent paper [4], G. A. Elliott has given a necessary and sufficient condition for every derivation on a separable C*-algebra with identity to be inner. Indeed, Elliott's condition has since been shown, by Akemann and Pedersen, to be equivalent to the C*-algebra being a finite direct sum of C*-algebras which are either homogeneous of finite degree or simple [8, Corollary 3.10].

In this paper we study some questions proposed by B. Schein [8] regarding the semigroup of binary relations Bx for a finite set X: what is the ideal structure of Bx, what are the congruences on Bx, what are the endomorphisms of Bx? For |X| = nit is convenient to regard Bx as the semigroup Bn of n×n (0, l)-matrices under Boolean matrix multiplication.

In Schinzel [1] the following interesting question is asked: does there exist an absolute constant K such that every trinomial in ℚ[x] has a factor irreducible in ℚ[x] which has at most K terms? The only known result appears to be that of Mrs. H. Smyczek, given in the above paper, that if K exists, then K ≥ 6. We here extend this bound to K ≥ 8 by exhibiting a trinomial in ℤ[x] which splits into the product of two irreducible factors, each having 8 terms.

In [1, Corollary 5], Figiel gives an elegant demonstration that the modulus ofconvexity δ in real Banach space X is nondecreasing, where

It is deduced from this that in fact δ(ɛ)/ɛ is nondecreasing [Proposition 3]. During the course of the proof [Lemma 4] it is stated that if v ∊ Sx is a local maximum on Sx of φ ∈Sx*, then v is a global maximum (φ(v) = 1). This is false; it could be that v is a global minimum. It is easy to construct such an example in R2 endowed with the maximum norm. What is true is that v is a global maximum of |φ|.

In [1] it was shown that for a compact normal operator on a Hilbert space the numerical range was the convex hull of the point spectrum. Here it is shown that the same holds for a semi-normal operator whose point spectrum satisfies a density condition (Theorem 1). In Theorem 2 a similar condition is shown to imply that the numerical range of a semi-normal operator is closed. Some examples are given to indicate that the condition in Theorem 1 cannot be relaxed too much.

The structure of semigroups whose subsemigroups form a chain under inclusion was determined by Tamura [9]. If we consider the analogous problem for inverse semigroups it is immediate that (since idempotents are singleton inverse subsemigroups) any inverse semigroup whose inverse subsemigroups form a chain is a group. We will therefore, continuing the approach of [5, 6], consider inverse semigroups whose full inverse subsemigroups form a chain: we call these inverse ▽-semigroups.

It is known that every complete Boolean algebra of projections on a Banach space X is strongly closed and bounded and that, although the converse of this result fails in general, it is valid if X is weakly sequentially complete [1, XVII. 3, pp. 2194–2201]. In the present note it is shown that this converse is in fact valid precisely when X contains no subspace isomorphics to the sequence space c0. More explicitly, the following two results are proved. In both, X may be a real or complex space, but c0 will consist of the null sequences in the underlying scalar field.

Let R be an integral domain with quotient field K. A fractional ideal I of R is a ∨-ideal if I is the intersection of all the principal fractional ideals of R which contain I. If I is an integral ∨-ideal, at first one is tempted to think that I is actually just the intersection of the principal integral ideals which contain I.However, this is not true. For example, if R is a Dedekind domain, then all integral ideals are ∨-ideals. Thus a maximal ideal of R is an intersection of principal integral ideals if and only if it is actually principal. Hence, if R is a Dedekind domain, each integral ∨-ideal is an intersection of principal integral ideals precisely when R is a PID.

Let X be a complex Banach space and let T be a bounded linear operator on X. Then T is decomposable if for every finite open cover of σ(T) there are invariant subspaces Yi(i= 1, 2, …, n) such that

(An invariant subspace Y is spectral maximal [for T] if it contains every invariant subspace Z for which σ(T|Z) ⊂ σ(T|Y).).

1. For functions f ∈ LLoc[0, ∞) the Riemann-Liouville operator of fractional integration I∞ is defined by

and its adjoint operator, the Weyl operator Kα, is defined by

for functions f ∈ LLoc[0, ∞) having a suitable behaviour at infinity.

This note extends classical results on certain Galois groups attached to onedimensional algebraic groups. We prove that the fields arising from the division of a fixed set of rational points on the product of an elliptic curve by the multiplicative group are as “large” as possible.