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]).

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 …

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.

The product on the stable homotopy ring of spheres π*scan be defined by composing, smashing or joining maps. Each of these three points of view is used in Section 2 to define Massey products on π*s. In fact we define composition and smash Massey products (x1, … , xt)where X1, … ,xt-1∈ π*s, xt∈ π*(E) and E is a spectrum. In Theorem 3.2, we prove that these three types of Massey products are equal. Consequently, a theorem which is easy to prove for one of these Massey products is also valid for the other two. For example, [3, Theorem 8.1] which relates algebraic Massey products in the Adams spectral sequence to Massey smash products in π*s is now also valid for Massey composition products in π*s

Our first result is a decomposition theorem for free groups relative to a set of elements. This enables us to formulate several algebraic conditions, some necessary and some sufficient, for various surfaces in the boundary of a 3-dimensional handlebody to be incompressible. Moreover, we show that there exists an algorithm to determine whether or not these algebraic conditions are met.

Many of our algebraic ideas are similar to those of Shenitzer [3]. Conversations with Professor Roger Lyndon were helpful in the initial development of these results, and he reviewed an earlier version of this paper, suggesting Theorem 1 (iii) and its proof. Our notation and techniques are standard (cf. [1], [2]). A set X of elements in a finitely generated free group F is a basis if it is a minimal generating set, and X±l denotes the set of all elements in X, together wTith their inverses.

We consider the following problem: If KG is the group ring of a torsion free group over a field K,show that KG has no divisors of zero. At characteristic zero, major progress was made by Brown [2], who solved the problem for G abelian-by-finite, and then by Farkas and Snider [4], who considered Gpolycyclic-by-finite. Here we present a solution at nonzero characteristic for polycyclic-by-finite groups. We also show that if Khas characteristic p > 0 and G is polycyclic-by-finite with only p-torsion, then KG has no idempotents other than 0 or 1. Finally we show that if R is a commutative ring of nonzero characteristic without nontrivial idempotents and G is polycyclic-by-finite such that no element different from 1 in G has order invertible in R, then RG has no nontrivial idempotents. This is proved at characteristic zero in [3].

A major goal of this paper is to give a proof of the following isotropy criterion: Let X = (X,G) be a space of orderings in the terminology of [9] or [10], and let f be a form defined over G.Then f is anisotropic over X if and only if f is anisotropic over some finite subspace of X.This is the content of Theorem 1.4, and generalizes [1, Corollary 3.4]. Moreover, in view of the known structure of finite spaces (see [9]), this has, essentially, the strength of [2, Satz 3.9] or [12, Theorem 8.12]. The technique used to prove this criterion is roughly patterned on that of [6], and yields some interesting by-products: An interesting invariant of a space of orderings called the chain length is introduced (Definition 1.1) and spaces of orderings with finite chain length are classified (Theorem 1.6).

In what follows, a theorem on blocking sets is generalized to higher dimensions. The result is then used to study maximal partial spreads of odd-dimensional projective spaces.

Notation. The number of elements in a set X is denoted by |X|. Those elements in a set A which are not in the set Bare denoted by A — B. In a projective space Σ = PG(n, q) of dimension n over the field GF(q) of order q, ┌d(Ωd, Λd, etc.) will mean a subspace of dimension d. A hyperplane of Σ is a subspace of dimension n — 1, that is, of co-dimension one.

A blocking set in a projective plane π is a subset S of the points of π such that each line of π contains at least one point in S and at least one point not in S. The following result is shown in [1], [2].

In this paper we study the nonlinear initial boundary value problem

(1.1) ωtt— αΔ ωt— Δω= f(ω), t> 0

ω(x, 0) = ϕ(x), x∈ Ω

ωt(x, 0) = ψ (x), x∈ Ω

ω(x, t ) = 0, x ∈ ∂Ω, t ≥ 0.

In (1.1) Ω is a smooth bounded domain in Rn, n = 1, 2, 3, α > 0, and f ∈ C1(R;R) with f‘(x) ≦ co for all x ∈ R (where c0 is a nonnegative constant), lim sup|x|→+∞f(x)/x ≦0, and f(0) = 0. Our objective will be to establish the existence of unique strong global solutions to (1.1) and investigate their behavior as t→ +∞.

Our approach takes advantage of the semilinear character of (1.1) and reformulates the problem as an abstract ordinary differential equation in a Banach space.

Homeomorphic embeddings of Knin the m-cube were investigated in [6]. In particular, it was proved that any homeomorph of Kn+1embedded in the m-cube has at least n2edges. Furthermore, homeomorphic embeddings of Kn+1having exactly n2edges are unique up to isomorphism. In this paper a similar problem for the complete bipartite graph is considered.

We adopt the notation and terminology of [5].

All graphs considered are without loops and multiple edges.

Let x = uv be an edge of a graph G; x will be called subdivided if it is replaced by a vertex ω and by edges uω and ωv. A graph G’ is called a subdivision of G if it is obtained from G by a subdivision of an edge of G. A refinement Ĝ of G, is a graph isomorphic to a graph obtained from G by a finite sequence of subdivisions.

In [1], Solovay constructed a model of ZFC in which every set of reals in OD (R) is Lebesgue measurable. Here we construct a model in which every equivalence class of sets of reals modulo null sets that is in OD(R) consists of Lebesgue measurable sets. This result immediately implies Solovay's, since the equivalence class of any set of reals in OD (R) is itself OD (R). As a consequence, one cannot provably explicitly define a nonmeasurable set modulo null sets within ZFC. We do not know whether this holds in the model Solovay uses (where an inaccessible cardinal is collapsed to ω 1). Instead, our model is a generic extension of his model.

In the model we construct, a somewhat stronger statement holds: every set in OD(R) of sets of reals which has < 2C inequivalent elements modulo null sets, consists entirely of Lebesgue measurable sets of reals.

This paper studies sequential order convergence and the associated completion in vector lattices of continuous functions. Such a completion for lattices C(X) is related to certain topological properties of the space X and to ring properties of C(X). The appropriate topological condition on the space X equivalent to this type of completeness for the lattice C(X) was first identified, for compact spaces X, in [6]. This condition is that every dense cozero set S in X should be C*-embedded in X (that is, all bounded continuous functions on S extend to X). We call Tychonoff spaces X with this property quasi-F spaces (since they generalize the F-spaces of [12]).

In Section 1, the notion of a completion with respect to sequential order convergence is first described in the setting of a commutative lattice group G.

Heine [7, p. 93] gave the following representation for the Legendre Polynomial {Pn(x)}∞n=o

where fo,n = 1 and

Szegö [7, p. 96] generalized this result to the Ultraspherical Polynomial set ﹛Cnλ(x)﹜∞n=o and obtained

where

In this paper a new type of inclusion theorem concerning Abel, Borel and Lambert summability is established. To state our results we need some definitions and notations. With a formal series ∑k=0∞ak, ak∈ C, and its partial sums snwe associate the series

Then ∑k=0∞ ak is said to be summable to the value s

(a) by Abel's method, if (1.1) is convergent for |v| > 1 and limv→1+A(v)= s,

(b) by Lambert's method, if (1.2) is convergent for |v| > 1 and limv→1+L(v)= s,

(c) by Borel's method, if (1.3) is convergent for all x ∈ R and limx→+∞B(x)= s,

We begin with some preliminary definitions. Let S be a subset of Rd. For points x and y in S, we say x sees y via S if and only if the corresponding segment [x, y] lies in S. The set Sis said to be starshaped if and only if there is some point p in S such that, for every x in S, p sees x via S. The collection of all such points p is called the kernel of S, denoted ker S. Furthermore, if we define the star of x in S by Sx = {y: [x, y] ⊆ S}, it is clear that ker S = ⋂{Sx: x in S}.

Several interesting results indicate a relationship between ker S and the set E of (d – 2)-extreme points of S. Recall that for d ≧ 2, a point x in S is a (d – 2)-extreme point of S if and only if x is not relatively interior to a (d – 1)-dimensional simplex which lies in S. Kenelly, Hare et al. [4] have proved that if S is a compact starshaped set in Rd, d ≧ 2, then ker S = ⋂{Se: eE}. This was strengthened in papers by Stavrakas [6] and Goodey [2], and their results show that the conclusion follows whenever S is a compact set whose complement ~S is connected.

1. Introduction.Let p be a prime number. A finite group G = (G, B, N, R, U) is called a split(B, N)-pair of characteristic p and rank n if

(i) G has a (B, N)-pair (see [3, Definition 2.1, p. B-8]) where H= B ⋂ N and the Weyl group W= N/H is generated by the set R= ﹛ω 1,… , ω n) of “special generators.”

(ii) H= ⋂n∈N n-1Bn

(iii) There exists a p-subgroup U of G such that B = UH is a semidirect product, and H is abelian with order prime to p.

A (B, N)-pair satisfying (ii) is called a saturated (B, N)-pair. We call a finite group G which satisfies (i) and (iii) an unsaturated split (B, N)- pair. (Unsaturated means “not necessarily saturated”.)

Given a finite undirected graph G and A ⊆ E(G), G(A) denotes the subgraph of G having edge-set A and having no isolated vertices. For a partition {E1, E2} of E(G), W(G; E1) denotes the set V(G(E1)) ⋂ V(G(E2)). We say that Gis non-separable if it is connected and for every proper, non-empty subset A of E(G), we have |W(G; A)| ≧ 2. A split of a non-separable graph Gis a partition {E1, E2} of E(G) such that

|E1| ≧ 2 ≧ |E2| and |W(G; E1)| = 2.

Where {E1, E2} is a split of G, W(G; E2) = {u, v}, and e is an element not in E(G), we form graphs Gii= 1 and 2, by adding e to G(Ei) as an edge joining u to v.