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.

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.

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.

Throughout let be faithful.

1.1. A U-morphism with domain X is a pair (e, A), where e ∈ Hom (X, UA). A [U-morphism (e, A) is called U-epi ( = generating) provided that r, s ∈ Hom (A, A’) and (Ur)e = (Us)e imply that r = s.

1.2. A U-source is a pair (X, (fi,Ai)I), (written more simply (X, (fi,Ai)I),, where (fi,Ai)I 7 is a family of U-morphisms each with domain X.

1.3. A factorization of a U-source (X,fi,Ai)I is a triple (e,A,gi)I such that (e, A) is a U-morphisms with domain X and for each i ∈ I gi ∈ Hom (A,Ai) and (Ugi)e = fi.

In this paper we construct modular forms from combinatorial designs, and codes over finite fields. We construct codes from designs, and lattices from codes. Then we use the combinatorial properties of the designs and the weight (or shape) structures of the codes to study the theta functions of the associated lattices. These theta functions are shown to be modular forms for the modular group or for various congruence subgroups. The levels of the forms we examine are determined by the dimensions of the codes and the characteristics of the fields. Using the Lee polynomial of the codes we can write the theta functions as homogeneous polynomials in modified Jacobi theta functions. By extending the underlying combinatorial structure, a modular form of half-integral weight is associated with a modular form of integral weight.

A Banach space B is called an isometric predual, or simply a predual, of a Banach space X if the dual B* of B is isometrically isomorphic to X. A Banach space X is said to have a unique (isometric) predual if X has a predual and all preduals are mutually isometrically isomorphic. In general a Banach space does not have a unique predual even if it has a predual. A simple example of this is the space l1, because c0 and c are isometric preduals of l1 but not isometrically isomorphic. A. Grothendieck [3] first noticed that L∞-spaces have unique preduals, and then S. Sakai generalized this to von Neumann algebras (see p. 30 of [9]).

In this note we discuss how the saturation I X J, where I, J are k-complete ideals on a regular uncountable cardinal K, depends on the saturation of I and J. We show that if 2k = k+ then the saturation of I X J is completely determined by the saturation of I and J . A consequence of a negative saturation result is that NSk X NSk is not k+-saturated, where NSk is the nonstationary ideal on k (even though it is still open whether NSk can be K+- saturated). We also discuss the preservation of precipitousness under certain products, obtaining a simple example of an ideal on k that is precipitous but not k+-saturated.

In this paper we study the question “which generalized metric spaces can be written as the union of k (closed) metrizable subspaces, where k is a cardinal number with k ≤ c?“ Questions of this type first arose in [16] where J. Nagata asked for examples of certain generalized metric spaces which could not be written as the union of countably many closed metrizable subspaces. Using Baire Category arguments, Fitzpatrick provided the required examples in [12]. We begin this paper by sharpening Fitzpatrick's examples, showing in Section 2 that there is a Moore space which is not the union of countably many metrizable subspaces of any kind. Then in Section 3 we present a positive result, proving that any cr-space, and a fortiori any Moore space, can be written as the union of c = 2ω0 closed metrizable subspaces.

Let be a probability space, J a directed set filtering to the right. (X t) t∈J is a family of random variables adapted to an increasing family of σ-algebras . Vitali conditions V and V' on the σ-algebras, abstracting classical assumptions in Lebesgue's derivation theory, were made to insure essential convergence of martingales and submartingales (under proper boundedness assumptions). In reality these conditions, guaranteeing the existence of certain disjoint and properly measurable sets Bi, are better suited for study of amarts, since the sets Bi are a natural habitat and breeding ground for stopping times, thriving, as well known, precisely on disjoint and properly measurable sets. Thus K. Astbury [1] showed that the condition V, proved by K. Krickeberg to be sufficient for convergence of martingales (see [20] or Neveu [26], p. 98) is both necessary and sufficient for convergence of amarts. (We follow Neveu denoting by V the condition Krickeberg denotes by V+∞.) The Vitali condition V’, proved by Krickeberg [21] to be sufficient for convergence of submartingales, is shown here to be both necessary and sufficient for convergence of ordered amarts, defined similarly to amarts, except that the stopping times are ordered. We also introduce the controlled Vitali condition Ve, properly weaker than V’, and show that Ve is sufficient for convergence of controlled amarts, including submartingales.

A classical result in the theory of convex polyhedra is that every bounded polyhedral convex set can be expressed either as the intersection of half-spaces or as a convex combination of extreme points. It is becoming increasingly apparent that a full understanding of a class of convex polyhedra requires the knowledge of both of these characterizations. Perhaps the earliest and neatest example of this is the class of doubly stochastic matrices. This polyhedron can be defined by the system of equations

Birkhoff [2] and Von Neuman have shown that the extreme points of this bounded polyhedron are just the n × n permutation matrices. The importance of this result for mathematical programming is that it tells us that the maximum of any linear form over P will occur for a permutation matrix X.

A lattice L is called transferable if and only if, whenever L can be embedded in the ideal lattice I(K) of a lattice K, L can be embedded in K. L is called sharply transferable if and only if, for every lattice embedding ψ(x) , there exists an embedding such that for x, y ϵ L, if and only if x ≤ y. Finite sharply transferable lattices were characterized in [3]. In this paper we extend the characterization to the infinite case. We begin by revising some of the terminology of [3].

1.1. Definition, (a) Let 〈P; ≧〉 be a poset and X, Y ⊆ P. Then X dominates Y (written X Dom Y) if and only if, for every y ∈ F, there exists x ∈ X such that y ∈ x. Dually, X supports Y (written X Spp Y) if and only if, for every y ∈ Y, there exists x ∈ X such that x ≦ y.

The aim of this paper is the characterization of PSL(2, 31) in terms of its action on a certain polygonal graph. A polygonal graph is a pair (, ) consisting of a graph which is regular, connected and has girth m for some m ≧ 3, and a set of m-gons (circuits of length m) of such that every 2-claw (i.e. path of length 2) of is contained in a unique element of , (See Section 2 for the definitions of the terms used here.) If is the set of all m-gons of H, so that there is in a unique m-gon on every one of its 2-claws, then we write for (, ) and call a strict polygonal graph. If we wish to emphasize the integer m, then we call (, ) an m-gon-graph (respectively, a strict m-gon-graph). For convenience, a strict 5-gon-graph will be called a pentagraph.

Soit R un anneau et soit Gop un sous groupe de Aut (R). Formons l'anneau de groupe à gauche (“skew group ring”) RSG, en imposant, sur le module libre à gauche avec une base G, la multiplication rg = grg. (On écrit Gop au lieu de G car, si G < Inn (R) = le groupe d'automorphismes intérieurs, la condition g–rg = rg; implique que le homomorphisme naturel est vraiment un anti-homomorphisme).

Nous considérons le cas où R = MnF, F un corps, MnF l'anneau de matrices, d'ordre n, et Gop un sous-groupe (généralement) fini, de PGL(n, F) = Inn (R). Ce travail est consacré à l'étude de la simplicité de RSG.

An n-knot (Sn+2, Sn) is amphicheiral if there is an orientation reversing autohomeomorphism of Sn+2 leaving Sn invariant as a set. It is invertible if there is an orientation preserving autohomeomorphism of Sn+2 whose restriction to Sn is an orientation reversing autohomeomorphism of Sn onto itself. In 1961 Fox [8, Problem 35] asked if there exist non-amphicheiral locally flat 2-knots. We will prove the following THEOREM 1. For any integer n there are smooth n-knots which are neither amphicheiral nor invertible.

A knot (Sn+2, Sn) is + amphicheiral (resp. —amphicheiral) if there is an orientation reversing autohomeomorphism f of Sn+2 leaving Sn invariant such that f| Snpreserves (resp. reverses) orientation.

A remote point of a metric space X is a point in βX |Xnot in the βX-closure of any discrete subset of X. Remote points have been studied by Fine and Gillman [2], Plank [4], Robinson [6], Woods [9, 10], Van Douwen [7] and others. The main results concerning the existence of remote points are listed in Section 1. In this paper we determine some structural properties of the set of remote points of a metric space which has no isolated points. The notation of Gillman and Jerison [3] and Walker [8] will be used. All spaces are assumed to be Tychonoff.

Let X be a space. The general remote points of X, denoted TX, are those points in βX |X which are not in the βX-closure of any nowhere dense sets of X.

Throughout this paper all rings considered will be commutative, noetherian with identity. If R is such a ring and M is a finitely generated R-module, we shall use v(M) to denote that non-negative integer with the property that M can be generated by v(M) elements but not by fewer.

Since every ideal in a noetherian ring is finitely generated, it is a natural question to ask what v(I) is for a given ideal I. Hilbert's Nullstellensatz may be viewed as the first general theorem dealing with this question, answering it when I is a maximal ideal in a polynomial ring over an algebraically closed field.

More recently, it has been noticed that the properties of an R-ideal I are intertwined with those of the R-module I/I2.

Spreads of curves were introduced by Grunbaum in [1]. A spread of curves is a continuous family of simple arcs in the real plane, every two of which intersect in exactly one point. A spread is the continuous analogue of a finite arrangement of pseudolines in the plane. Sylvester's problem for finite arrangements of pseudolines asks if every non-trivial arrangement has a simple vertex, that is a point contained in exactly two pseudolines of the arrangement. This question was answered in the affirmative by Kelly and Rottenberg [5]. One interesting feature of this result is that it does not depend on the pseudolines being straight lines.

Here we settle Sylvester's problem for spreads. We show that every nontrivial spread of line segments has uncountably many simple vertices. But we also give examples of non-trivial spreads with no simple vertices.

Let R be a principal ideal domain, i.e., a commutative ring without zero divisors in which every ideal is principal. The invariant factors of a matrix A with entries in R are the diagonal elements when A is converted to a diagonal form D = UAV, where U, V have entries in R and are unimodular (invertible over R), and the diagonal entries d1 …, dn of D form a divisibility chain: d1|d2| … |dn. Very little has been proved about how invariant factors may change when matrices are added. This is in contrast to the corresponding question for matrix multiplication, where much information is now available [6].

If H is a closed subgroup of a locally compact group G, with G/H having finite G-invariant measure, then, as observed by Atle Selberg [8], for any neighborhood U of the identity in G and any element g in G, there is an integer n > 0 such that gn is in U·H·U. A subgroup satisfying this latter condition is said to be an S-sub group, or satisfies property (S). If G is a solvable Lie group, then the converse of Selberg's result has been proved by S. P. Wang [10]: If H is a closed S-subgroup of G, then G/H is compact. Property (S) has been used by A. Borel in the important “density theorem” (see Section 2 or [1]).