A Riemannian manifold (M, g) is said to be conformally flat if every point has a neighborhood conformai to an open set in Euclidean space. Over the past thirty years, many papers have appeared attacking, with varying degrees of success, the problem of classifying the conformally flat spaces which occur as hypersurfaces in Euclidean space. Most of these start from the following pointwise result of Schouten.

The main object of this paper is to study when infective noetherian modules are artinian. This question was first raised by J. Fisher and an example of an injective noetherian module which is not artinian is given in [9]. However, it is shown in [4] that over commutative rings, and over hereditary noetherian P.I rings, injective noetherian does imply artinian. By combining results of [6] and [4] it can be shown that the above implication is true over any noetherian P.I ring. It is shown in this paper that injective noetherian modules are artinian over rings finitely generated as modules over their centers, and over semiprime rings of Krull dimension 1. It is also shown that every injective noetherian module over a P.I ring contains a simple submodule. Since any noetherian injective module is a finite direct sum of indecomposable injectives it suffices to study when such injectives are artinian. If Q is an injective indecomposable noetherian module, then Q contains a non-zero submodule Q0 such that the endomorphism rings of Q0 and all its submodules are skewfields. Over a commutative ring, such a Q0 is simple. In the non-commutative case very little can be concluded, and many of the difficulties seem to arise here.

Let S be a semigroup. An element a ∈ S is said to be a left quasi-zero if <a>x ⋂ <a> ≠ ∅ for all x ∈ S, where <a> denotes the cyclic sub-semigroup of S generated by a. In a recent study [6] of semigroups with a maximum right congruence, such elements proved to be useful in providing characterizations of these semigroups. Left quasizeroes have appeared in the literature under different names in a variety of situations. In the context of semigroup radicals, left quasi-zeroes are called right quasi-regular elements, where an element is defined to be right quasi-regular if it is not a left identity for any right congruence other than the universal congruence (see [4], [5], [2], [7], and [8]).

The problem of determining necessary and sufficient conditions bearing upon the numbers a2 and a3 in order that the polynomial z + a2z2 + a3z3 be univalent in the unit disk |z| < 1 was solved by Brannan ([3], [4]) and by Cowling and Royster [6], at about the same time. For his investigation Brannan used the following result due to Dieudonné [7] and the well-known Cohn rule [9].

THEOREM A (Dieudonné criterion). The polynomial

1

is univalent in |z| < 1 if and only if for every Θ in [0, π/2] the associated polynomial

2

does not vanish in |z| < 1. For Θ = 0, (2) is to be interpreted as the derivative of (1).

The procedure of Cowling and Royster was based on the observation that is univalent in |z| < 1 if and only if for all α such that 0 ≧ |α| ≧ 1, α ≠ 1 the function

is regular in the unit disk.

Denote by C0 the collection of complex-valued functions which are continuous and compactly supported on (0, ∞). The transformations of the title are defined on C0 by

1.1

and

1.2

respectively, where Yv(x) is the Bessel function of the second kind, and Hv(x) is the Struve function; see [1; 7.5.4(55)]. The two transformations are studied briefly in [6; §8.4]; tables of transform pairs are given in [2; Chapters IX and XI], where it is also stated that, for – , each of the transformations is the inverse of the other.

Studies of optimization problems and certain kinds of differential equations have led in recent years to the development of a generalized theory of differentiation quite distinct in spirit and range of application from the one based on L. Schwartz's “distributions.” This theory associates with an extended-real-valued function ƒ on a linear topological space E and a point x ∈ E certain elements of the dual space E* called subgradients or generalized gradients of ƒ at x. These form a set ∂ƒ(x) that is always convex and weak*-closed (possibly empty). The multifunction ∂ƒ: x →∂ƒ(x) is the sub differential of ƒ.

Rules that relate ∂ƒ to generalized directional derivatives of ƒ, or allow ∂ƒ to be expressed or estimated in terms of the subdifferentials of other functions (whenƒ = ƒ1 + ƒ2,ƒ = g o A, etc.), comprise the sub differential calculus.

Let 5 be a subset of Euclidean space. The set 5 is said to be m-convex, m ≥ 2, if and only if for every m distinct points of S, at least one of the line segments determined by these points lies in 5. Clearly any union of m mdash — 1 convex sets will be m-convex, yet the converse is false. However, several decomposition theorems have been proved which allow us to write any closed planar m-convex set as a finite union of convex sets, and actual bounds for the decomposition in terms of m have been obtained ([6], [4], [3]). Moreover, with the restriction that (int cl S) ∼ S contain no isolated points, an arbitrary planar m-convex set S may be decomposed into a finite union of convex sets ([1].

Here we strengthen the m-convexity condition to define an analogous combinatorial property for segments.

A positive bounded selfadjoint operator is in the trace class of von Neumann and Schatten ([4]) if the sum of its diagonal matrix elements with respect to some orthonormal basis is finite, and the trace is then defined to be this sum, which is independent of the basis. A bounded selfadjoint but not necessarily positive operator x is in the trace class if in the decomposition x = x+ – x−, with x+ and x− positive and x+x− = 0, both x+ and x−are in the trace class; the trace of x is then defined to be the difference of the finite traces of x+ and x−. The trace defined in this way is a linear functional on the trace class, and is unitarily invariant; if u is a unitary operator, the trace of uxu−1 is the same as the trace of x.

Throughout this paper we shall use I to denote a given interval, not necessarily bounded, of real numbers and Cn to denote the real valued n times continuously differentiable functions on I and C0 will be abbreviated to C. By a differential operator of order n we shall mean a linear function L:Cn → C of the form

1.1

where pn(x) ≠ 0 for x ∊ I and pi ∊ Cj 0 ≦ j ≦ n. The function pn is called the leading coefficient of L.

It is well known (see, for example, [2, pp. 73-74]) thai a differential operator L of order n uniquely determines both a differential operator L* of order n (the adjoint of L) and a bilinear form [·,·]L (the Lagrange bracket) so that if D denotes differentiation, we have for u, v ∊ Cn,

1.2

Il y a en mathématiques plusieurs théorèmes d'existence: ceux liés à la notion de point fixe (théorème des fonctions implicites, solution locale des équations différentielles y’= f(x, y)) ou ceux liés à la notion de cardinalité (théorèmes de Chevalley et Warning, de Sylow, existence de nombres réels transcendants) par exemple. Nous nous intéressons ici exclusivement à deux sortes de théorèmes d'existence: ceux liés au théorème de Baire (méthodes quasi-sûres) et ceux liés à la théorie des probabilités (méthodes presque sûres). Ces deux méthodes ont déjà donné lieu à de nombreux théorèmes: existence de fonctions continues partout sans dérivée, existence de fonctions C∞partout non analytiques, existence d'ensembles de Kronecker parfaits (Kaufman) par le quasi-sûr, existence de réels t tels que (tn) soit équirépartie modulo 1 (Koksma), existence de nombres normaux au sens de Borel, existence d'ensembles de nonsynthèse par le presque sûr, etc …

Every finite simplicial complex has a tangent bundle in the category of simplicial bundles (see [9]). The goal of this paper is to classify simplicial bundles, and, as an application of this result, to construct Euler classes for a large class of combinatorial manifolds. This construction is closely related to [3] and [4].

A finite transitive permutation group G is called a Frobenius group if every element of G other than 1 leaves at most one letter fixed, and some element of G other than 1 leaves some letter fixed. It is proved in [20] (and sketched below) that if k is a number field such that SL(2, 5) and one other nonsolvable group Ŝ5 of order 240 are realizable as Galois groups over k, then every Frobenius group is realizable over k. It was also proved in [20] that there exists a quadratic (imaginary) field over which these two groups are realizable. In this paper we prove that they are realizable over the rationals Q, hence we Obtain

THEOREM 1. Every Frobenius group is realizable as the Galois group of an extension of the rational numbersQ.

The purpose of this paper is to show how Fred hoi m criteria for Toeplitz operators, whose symbols lie in an algebra,A, may often be generalized to cover a larger symbol algebra generated by A and SO, the slowly oscillating functions. Mere A and SO are algebras of continuous functions on the real line, so that we are concerned principally with the effect of a single discontinuity in the symbol function.

We shall treat the cases when A is the almost periodic functions, the semi-almost periodic functions and the multiplicatively periodic functions. Sufficient criteria for Fredholmness are obtained in Section 5. The more difficult task of establishing necessary and sufficient criteria is only achieved here for the slowly oscillating almost periodic functions and this is done in Section 6.

We show that any PG(2n, q2) is a disjoint union of (q2n+1 − 1)/ (q − 1) caps, each cap consisting of (q2n+1 + 1)/(q + 1) points. Furthermore, these caps constitute the “large points” of a PG(2n, q), with the incidence relation defined in a natural way.

A square matrix H = (hij) over the finite field GF(q2), q a prime power, is said to be Hermitian if hijq = hij for all i, j [1, p. 1161]. In particular, hii ∈ GF(q). If if is Hermitian, so is p(H), where p(x) is any polynomial with coefficients in GF(q).

Given a Desarguesian Projective Geometry PG(2n, q2), n > 0, we denote its points by column vectors:

All Hermitian matrices in this paper will be 2n + 1 by 2n + 1, n > 0.

Recall the classical criterion for the complete decomposability of exterior vectors: the completely decomposable vectors in ∧pRn, R a field, are precisely the “Plücker vectors,” i.e. those whose coordinates (relative to the standard bases for ∧pRn) satisfy the Plücker equations. For R an arbitrary commutative ring, completely decomposable exterior vectors are still Plücker vectors, but the converse is not generally true. Rings for which the converse is true (for all 1 ≤ p ≤ n) are called Towber rings. Noetherian Towber rings are regular and, in fact, are characterized by the property that every Plücker vector in ∧2R4 is completely decomposable. (See [10] for these two results as well as for the above mentioned facts.) The present note develops a new characterization of Towber rings, combining it with results of Kleiner [9] and Estes-Matijevic [5] in (1) below.

In the last fifteen years, topology has shown up with an increasing interest in the use of closed subbases. Starting from Frink's internal characterization of complete regularity (Frink [6]), DeGroot and Aarts used closed subbases to obtain Hausdorff compactifications of completely regular spaces, thus giving a characterization of the latter in terms of their subbases [1]. The main tool of that paper is the notion of a linked system, which naturally leads to the notions of supercompactness and superextensions [7]. After 1970, these two topics developed to indepedennt theories, with several deep results available at this moment. Most results up to 1976 are summarized in [12].

Let G be a closed subgroup of one of the classical compact groups 0(n), U(n), Sp(n). By a reflection we mean a matrix in one of these groups which is conjugate to the diagonal matrix diag (–1, 1, …, 1). We say that G is a topological reflection group (t.r.g.) if the subgroup of G generated by all reflections in G is dense in G.

It was shown recently by Eaton and Perlman [5] that, in case of 0(n), the whole group 0(n) is the unique infinite irreducible t.r.g. In this paper we solve the analogous problem for U(n) and Spin). Our method of proof is quite different from the one used in [5]. We treat simultaneously all the three cases.

We show that if Y is a metric minimal flow and θ: Y→Z in an open homomorphism that has a section (i.e., a RIM), and if S(θ)= R(θ),then °YΩ contains a dense set of transitive points, where Ω is the first uncountable ordinal

YΩ = П{Y:1 ≦ α < Ω and α not a limit ordinal}, and

°YΩ = {y ∈ YΩ:θ(yα)= θ(yβ)for 1 ≦ α,β < Ω and α, β not limit ordinals},

S(θ) is the relativized equicontinuous structure relation, and

R(θ)= {(y1,y2) ∈ Y X Y:θ(y1) = θ(y2)}.

We use this to generalize a result of Glasner that a metric minimal flow whose enveloping semigroup contains finitely many minimal ideals is PI, [5].

I would like to thank Professor T. S. Wu for making helpful suggestions, and thank the referee for his time and effort.

Nous présentons ici une étude relative aux applications entre monoïdes, que nous nommerons transductions, qui peuvent être réalisées par un transducteur (machine séquentielle généralisée). L'exposé s'oriente sur l'examen des conditions caractérisant de telles applications. Le point de départ est le théorème de Ginsburg-Rose [2], qui caractérise les transductions entre monoïdes libres de type fini, en termes de quatre conditions dont une, la condition de “bounded output”, implique la longueur des mots et, par conséquent, ne peut être formulée que dans un monoïde libre. A l'aide d'un langage propre aux structures algébriques en général, celle des congruences, nous étudions des conditions remplaçant ‘'bounded output”, pour généraliser le théorème connu aux monoïdes quelconques et aux monoïdes libres quelconques.

What invariants of a finite simplicial complex K can be computed solely from the values v0(K), V1(K), …, vi(K), … where Vi(K) is the number of i-simplexes of K? The Euler chracteristic χ(K) = Σ i (– 1)ivi(K) is a subdivision invariant and a homotopy invariant while the dimension of K is a subdivision invariant and homeomorphism invariant. In [3], Wall has shown that the Euler chracteristic is the only linear function to the integers that is a subdivision invariant. In this paper we show that the only subdivision invariants (linear or not) of K are the Euler characteristic and the dimension. More precisely we prove the following theorem.