F. A. Valentine in [4] proved the following two theorems.

THEOREM 1. Let S be a closed connected subset of Rd which has at most n points of local nonconvexity. Then S is an Ln+i set.

THEOREM 2. Let S be a closed connected subset of Rd whose points of local nonconvexity are decomposable into n closed convex sets. Then S is an L2n+i set.

These results have been extended by a number of authors, but always with stronger hypothesis. (See [1] and [2].) Using a minimal arc technique, new pr∞fs of Theorems 1 and 2 were given in [3].

Valentine remarks in [4] that Theorem 2 might be improved in the case that 5 is the closure of an open connected set. The goal of this paper is to give such an improvement for sets satisfying a particular local connectivity property.

Throughout this paper we use the Bourbaki [1] conventions for rings and modules: all rings are associative but not necessarily commutative and have a 1; all modules are unital.

Our purpose is to extend and simplify some recent results of Maddox [7], Megibben [8], Enochs [3], and the author [5] on absolutely pure modules by introducing several new dimensions, and using the absolutely pure dimension introduced by the author in [6], This completes some work on character modules and dimension in [5] and [6].

An A -module will be called an FFR-module if and only if it has a resolution by finitely generated free A -modules.

The main theorems concern the relation between the - compact spaces and the -regular spaces, and their analogues in uniform spaces. In either of the categories of Tychonoff spaces or uniform spaces, let be a class of spaces, let be the epi-reflective hull of se (closed subspaces of products of members of ), let be the “onto-reflective” hull of (all subspaces of products of members of ), and let r and o be the associated functors. Let be the class of spaces which admit a perfect map into a member of . Then, is epi-reflective (and in Tych, = but in Unif, the equality fails); call the functor p.

Let F be a variety (equational class) of algebras. For n ≧ 0, Vn is the variety generated by the F-free algebra on n free generators while Vn is the variety of all algebras satisfying each identity of V which has no more than n variables. (Equivalently, Vn is the class of all algebras, , such that every n-generated subalgebra of is in V.) Note that unless nullary operation symbols are specified by the similarity type of V, V0 is the variety of all one element algebras while V° is the variety of all algebras.

This paper introduces two relations both weaker than isotopism which may hold between planar ternary rings. We will concentrate on the geometric consequences rather than the algebraic properties of these relations. It is well-known that every projective plane can be coordinatized by a planar ternary ring and every planar ternary ring coordinatizes a projective plane. If two planar ternary rings are isomorphic then their associated projective planes are isomorphic; however, the converse is not true. In fact, an algebraic bond which necessarily holds between the coordinatizing planar ternary rings of isomorphic projective planes has not been found. Such a bond must, of course, be weaker than isomorphism; furthermore, it must be weaker than isotopism. Here we show that it is even weaker than the two new relations introduced.This is significant because the weaker of our relations is, in a sense, the weakest possible algebraic relation which can hold between planar ternary rings which coordinatize isomorphic projective planes.

Cardan motion in Euclidean geometry may be defined as the motion of a plane Г1 with respect to a coinciding plane Г such that two points A1, A2 of Г1 move along two orthogonal lines ai, a2 of Г1 The properties of this classical motion are well-known: the path of a point of Г1 is in general an ellipse with its center at the intersection o of a1 and a2; there are ∞ l points of Г1 (their locus being the circle C1 with A1, A2 = 2d as diameter) the paths of which are line segments. The moving polhode is the circle Ci, the fixed polhode is the circle (o; 2d). We investigate here Cardan motion-defined in the same way-in the elliptic plane.

It is known that minimally thin and semithin sets in R2 are preserved by conformai mappings (see [3]) but it is not known whether or not analogous results hold true for ordinary thin and semithin sets respectively. In an unpublished work, Brelot has shown that ordinary thin sets at the origin are preserved by mappings of the form f(z) = z∝ (∝ > 0) where one always considers the principal branch of the mapping. We shall prove this result along with an analogous one for ordinary semithin sets and will see that the implications established by Jackson (see [4, Theorem 4]), for ordinary and minimally thin sets and by Brelot (see [2, p. 152]) for semithin sets in a half plane hold true for any wedge shaped region with vertex at the origin.

The notion of a monad (triple) has become increasingly important as an extension of the classical universal algebraic approach to “algebraic” categories. Indeed the categories of algebras arising from a monad seem to be the most natural generalization of Birkhoffs equational classes. Moreover in [2], Barr's concept of a relational model of a monad also coincides nicely with both the concepts of partial algebras (when suitably restricted) and (Moore) closure systems.

In this paper, we wish to examine two particular monads determined by filters. The first is the filter monad F = (F, η, 𝛍) over Sets where FX is the set of all (not necessarily proper) filters on X.

Let J1 and J2 be two Jordan algebras with unit elements. We define various tensor products of J1 and J2. The first, which we call the Kronecker product, is the most obvious and is based on the tensor product of the vector spaces. We find conditions sufficient for its existence and for its non-existence. Motivated by the universal mapping property for the tensor product of associative algebras we define, in Section 2, tensor products of J1 and J2 by means of a universal mapping property. The tensor products always exist for special Jordan algebras and need not coincide with the Kronecker product when the latter exists. In Section 3 we construct a more concrete tensor product for special Jordan algebras. Here the tensor product of a special Jordan algebra and an associative Jordan algebra coincides with the Kronecker product of these algebras. We show that this "special" tensor product is the natural tensor product for some Jordan matrix algebras.

Let C(X) be the ring of all real-valued continuous functions on a completely regular topological space X, and let A﹛Y) be the ring of all functions analytic on a connected non-compact Riemann surface F. The ideal theories of these two function rings have been extensively studied since the fundamental papers of E. Hewitt on C﹛X)[12] and of M. Henriksen on the ring of entire functions [10; 11]. Despite the obvious differences between these two rings, it has turned out that there are striking similarities between their ideal theories. For instance, non-maximal prime ideals of A (F) [2; 11] behave very much like prime ideals of C﹛X)[13; 14], and primary ideals of A(Y) which are not powers of maximal ideals [19] resemble primary ideals of C(X) [15]. In this paper we show that there are very good reasons for these similarities. It turns out that much of the ideal theory of A (Y) is a special case of the ideal theory of rings of continuous functions. We develop machinery that enables one almost automatically to derive results about the ideal theory of A(Y) from corresponding known results of ideal theory for rings of continuous functions.

The /z-cobordism theorem in [8], the generalized Poincaré conjecture in higher dimensions in [20] and several other results in differential topology are proved by using the following theorems of Morse theory:

(1) the elimination of critical points;

(2) the existence of nondegenerate functions for which the descending and ascending bowls have normal intersection;

(3) the alteration of function values at critical points. (For the details see below.)

We shall give short and elementary pr∞fs of these theorems together with some stronger statements than the ones given in [8-13] or [19].

The theorems are proved for noncompact manifolds rather than for compact manifolds since, by a trivial modification of the manifold (deleting the boundary or one point) the case of compact manifolds is included.

Consider the nth order linear equation

and particularly the third order equation

A nontrivial solution of (1)n is said to be oscillatory or nonoscillatory depending on whether it has infinitely many or finitely many zeros on [a, ∞). Let denote respectively the set of all solutions, oscillatory solutions, nonoscillatory solutions of (1)n. is an n-dimensional linear space. A subspace is said to be nonoscillatory or strongly oscillatory respectively if every nontrivial solution of is nonoscillatory or oscillatory. If contains both oscillatory and nonoscillatory solutions then is said to be weakly oscillatory.

We call an extension of commutative rings, R ⊂ T, open if the spec mapping from spec (T) to spec (R), which sends the prime Q of T to Q ∩ R, is an open mapping. It is easy to show, as for example in [1], that if R ⊂ T is open then it satisfies going down. In general, the converse is false, as is shown by Z ⊂ (2z) with Z the integers. To the best of this author's knowledge, it is an open question whether for an integral extension, going down and open are equivalent.

Let R be an associative ring with identity, and let denote the category of unital left R-modules. The Walkers [6] raised the question of characterizing the maximal torsion radicals of , and showed that if R is commutative and Noetherian, then there is a one-to-one correspondence between maximal torsion radicals and minimal prime ideals of R [6, Theorem 1.29]. Popescu announced [5, Theorem 2.5] that the result remains valid for commutative rings with Gabriel dimension (in the terminology of [2]). Theorem 4.6 below shows that the result holds for rings (not necessarily commutative) with Krull dimension on either the left or right, extending the previous theorem for right Noetherian rings which appeared in [1].

Let U be a complex uniform algebra, Z and dZ its maximal ideal space and its Šilov boundary, respectively. The Dirichlet (respectively Arens-Singer) deficiency of U is the codimension in CR(∂Z) of the closure of Re U (respectively of the real linear span of log|U-1|). Algebras with finite Dirichlet deficiency have many interesting properties, especially when the Arens-Singer deficiency is zero. (See, e.g. [5].) By a real uniform algebra we mean a real commutative Banach algebra A with identity 1, and norm ‖ ‖ such that ‖f2‖ = ‖f‖2 for each fin A

(a) (ᓂ1 and ᓂ2 will always denote positive bounded measures of equal mass defined on sets X and F respectively. Lp(ᓂ1) and Lp(ᓂ2) will always be complex Lp spaces.

(b) M C L∞(ᓂ1) will always denote a subalgebra of L∞(ᓂ1) containing constants.

(c) Let be a 1 inear map of ikf into Lp(ᓂ2). We shall say that T is a linear isometry in LP norm if

We shall prove the following:

THEOREM B. If 2 < p < ∞ and is a linear isometry in the Lp norm with T(l) = 1 then T is a homomorphism on M; that is

Define the self-adjoint operator

where r(x) > 0 on (0, ∞) and q and p are real-valued. The coefficient q is assumed to be differentiate on (0, ∞) and r is assumed to be twice differentia t e on (0, ∞).

The oscillatory behavior of L4 as well as the general even order operator has been considered by Leigh ton and Nehari [5], Glazman [2], Reid [7], Hinton [3], Barrett [1], Hunt and Namb∞diri [4], Schneider [8], and Lewis [6].

Let G be a finite group. (All groups considered here are finite). There exist several results which control the structure of G in terms of cd(G), the set of degrees of the irreducible complex characters of G. Here, we are concerned with the situation where only the cardinality of cd(G) is given. If |cd(G)| ≦ 3,, then it is known [9 ; 7] that G is solvable and the derived length dl (G) ≦ cd (G) |., If |cd(G)| = 4, then G need not be solvable (e.g., G = PSL(2, 2n))\ however [5], if G is solvable then dl(G) ≦4.

Let be a Hilbert space and denote the collection of (bounded, linear) operators on by . Throughout this paper, the term ‘algebra’ will refer to a subalgebra of ; unless otherwise stated, it will not be assumed to contain I or to be closed in any topology.

An algebra is said to be transitive if it has no non-trivial invariant subspaces. The following lemma has revolutionized the study of transitive algebras. For a pr∞f and a general discussion of its implications, the reader is referred to [5].

For F a field and G a group, let FG = F(G) be the group algebra of G over F. It is a class of finite abelian groups, F induces an equivalence relation on by are equivalent if and only if FG ⋍ FH. We will call two fields F and K equivalent on if they induce the same equivalence relation on We will say F is equivalent to isomorphism on if FG ⋍ FH if and only if G ⋍ H for any two elements .