In an earlier paper [6] we considered the application of Lagrangian methods to the enumeration of plane rooted trees with given colour partition. We obtained an expression which generalised Tutte’s result [9], and a correspondence, which, when specialised, gives the de Bruijn-van Aardenne Ehrenfest-Smith-Tutte Theorem [1]. A corollary of these results is a one-to-one correspondence [4], between trees and generalised derangements, for which no combinatorial description has yet been found.

In this paper we extend these methods to the enumeration of rooted labelled trees to demonstrate how another pair of well-known and apparently unrelated theorems may be obtained as the result of a single enumerative approach. In particular, we show that a generalisation of Good's result [3], also considered by Knuth [7], and the matrix tree theorem [8] have a common origin in a single system of functional equations, and that they correspond to different coefficients in the power series solution.

Let G be a symmetric connected graph without loops. Denote by b(G) the maximum number of edges in a bipartite subgraph of G. Determination of b(G) is polynomial for planar graphs ([6], [8]); in general it is an NP-complete problem ([5]). Edwards in [1], [2] found some estimates of b(G) which give, in particular,

for a connected graph G of n vertices and m edges, where

and ﹛x﹜ denotes the smallest integer ≧ x.

We give an 0(V3) algorithm which for a given graph constructs a bipartite subgraph B with at least f(m, n) edges, yielding a short proof of Edwards’ result.

Further, we consider similar methods for obtaining some estimates for a particular case of the satisfiability problem. Let Φ be a Boolean formula of variables x1, …, xn.

The purpose of this paper is to study three concepts that deal with the topologies on ideals of commutative integral domains. We call a domain R prime-injective if for each torsion free R-module M, and all non-zero prime ideals

commutes implies that M is injective. From [6, Theorem 1 and the technique of Example 6] this is equivalent to all non-zero ideals of R being open in the topology defined by finite products of non-zero prime ideals as a base of neighborhoods around zero.

A domain is strongly prime-injective if for each (torsion theory) topology and for ϕ the set of primes in , ϕ-injective implies -injective for torsion free modules (see [6, 8] for notation). As in the prime-injective case, this is equivalent to being the topology generated by ϕ for all topologies .

Throughout this paper L will be an orthomodular lattice and the set of all maximal Boolean subalgebras, also called blocks [4], of L. For every x ∈ L, C(x) will be the set of all elements of L which commute with x. Let n ≧ 1 be a natural number. In this paper we consider the following conditions for L:

An: L has at most n blocks,

Bn: there exists a covering of L by at most n blocks,

Cn: the set ﹛C(x)| x ∈ L﹜ has at most n elements,

Dn: out of any n + 1 elements of L at least two commute.

We also consider quantified versions of these statements, namely the statements A, B, C, D defined by: A ⇔ ∃ nAn, B ⇔ ∃ nBn, C ⇔ ∃ nCn and D ⇔ ∃ nDn.

In this paper we classify finite-dimensional flexible division algebras over the real numbers. We show that every such algebra is either (i) commutative and of dimension one or two, (ii) a slight variant of a noncommutative Jordan algebra of degree two, or (iii) an algebra defined by putting a certain product on the 3 × 3 complex skew-Hermitian matrices of trace zero. A precise statement of this result is given at the end of this section after we have developed the necessary background and terminology. In Section 3 we show that, if one also assumes that the algebra is Lie-admissible, then the structure follows rapidly from results in [2] and [3].

All algebras in this paper will be assumed to be finite-dimensional. A nonassociative algebra A is called flexible if (xy)x = x(yx) for all x, y ∈ A.

The Betti numbers βn(k) of the residue class field k = R/m of a commutative local ring (R, m) have been studied for about 20 years, primarily as the coefficients of the Poincaré series of E . Several authors have obtained results about the growth of the sequence {βn(k)}.For example, Gulliksen [3] showed that when R is non-regular, the sequence is non-decreasing. More recently, Avramov [1] studied asymptotic properties of {βn(k)} and found that under certain conditions the growth is exponential, i.e., there is a natural number p such that for all n, βpn(k) ≧ 2n.

In this paper, we examine the sequence {βn(M)} for arbitrary finitely generated non-free modules M over any commutative local artin ring R. We establish the following bounds:

1

2

3

where l(X) is the length of X.

Soit {f1(x),f2(x), …,fm(x)} un système de Tchebycheff de dimension m, c'est-à-dire les fonctions fj(x) sont des fonctions réelles continues définies sur un intervalle ouvert (a, b) et toute fonction de la forme admet au plus m — 1 racines dans (a, b) lorsqu'au moins un des coefficients Cj diffère de 0. Notre point de départ est un théorème dû à Krein [3]. On se limite au cas où la dimension m est paire, m = 2n. Ce théorème dit que la totalité Ω des points de R2n qui se représentent ainsi sont n points distincts de (a, b) et où pi > 0, 1 ≦ i≦n, forme effectivement un ouvert convexe de R2n. Les systèmes d'équations que nous voulons traiter sont justement les équations:

I

The central result is the existence and uniqueness for an arbitrary l-group G of two hulls, Ḡ and Ḡω, which in the representable case coincide with the orthocompletion and strongly protectable hull of G. This is done by introducing two notions of extension, written ≼ and ≼ω, and proving that each G has a maximal ≼ extension Ḡ and a maximal ≼ω extension Ḡω. Two constructions of Ḡ and Ḡω are-given: an l-permutation construction leads to descriptive structural information, and a construction by “consistent maps” leads to a strong universal mapping property.

The notion of a strongly projectable hull has a lengthy history. The concept of an orthocompletion, together with the first proof of its existence and uniqueness, is due to Bernau [4]. Conrad summarized and extended all these results in an important paper [10].

It is convenient and not without precedent (see [2], [1], and also [6]) to call a loop which is isomorphic to all of its loop isotopes a G-loop. Since all groups are readily seen to be G-loops, the only interest in such loops, from a loop-theoretic standpoint, resides with those which are not associative. Examples and ad hoc constructions of such loops have appeared sporadically in the literature (see, for instance, [1], [2], [4], [6], [8], [9], and [13]).

Any finite loop of order n < 5 is a group and, hence, must also be a G-loop. R. L. Wilson [11, 12, 13] proved that a finite G-loop of prime order is necessarily a group; he also exhibited for each even integer n > 5 a G-loop of order n which is not associative and then raised questions concerning the existence of finite G-loops which are not groups for every possible composite order n > 5.

Let A be a uniform algebra on a compact Hausdorff space X. The spectrum, or the maximal ideal space, MA, of A is given by

We define the measure spectrum, SA, of A by

SA is the set of all representing measures on X for all Φ ∈ MA. (A representing measure for Φ ∈ MA is a probability measure μ on X satisfying

The concept of representing measure continues to be an effective tool in the study of uniform algebras. See for example [12, Chapters 2 and 3], [5, pp. 15-22] and [3]. Most of the known results on the subject of representing measures, however, concern measures associated with a single homomorphism.

Let R be a commutative ring, C a finite abelian group, S a Galois extension of R with group C, in the sense of [1]. Viewing S as an RC-module defines the Picard invariant map [4] from the Harrison group Gal (R, C) of isomorphism classes of Galois extensions of R with group C to CI (RC), the class group of RC. The image of the Picard invariant map is known to be contained in the subgroup h Cl (RC) of primitive elements of CI (RC) (for definition see below). Characterizing the image of the Picard invariant map has been of some interest, for the image describes the extent of failure of Galois extensions to have normal bases.

Let R be the ring of integers of an algebraic number field K.

A theorem of Eisenbud, Griffith, and Robson states that if R is hereditary and noetherian (on both the left and right) then every proper homomorphic image of R is a generalized unserial ring (see, for example, [3, p. 244]). Singh [11, p. 883] states a converse: If R is a right bounded, noetherian prime ring, all of whose proper homomorphic images are generalized uniserial rings, then (every divisible right R-module is injective, so) R is right hereditary. (Actually, Singh omitted the clearly necessary “bounded” condition.) Singh's theorem generalizes results of [9, Proposition 15], [2, Theorem 2.1], and [8], about commutative rings.

We will call a semi-prime ring R essentially right bounded if each essential right ideal contains a two-sided ideal which is essential as a right ideal. In case R is prime, “essentially right bounded” coincides with “right bounded”.

In our paper [3] we considered four nniteness conditions for an orthomodular lattice (abbreviated: OML) L and conjectured their equivalence. The only question left open in that paper was whether an OML L which can be covered by finitely many blocks (maximal Boolean subalgebras) has only finitely many blocks. In this paper we give an affirmative answer to this question, in fact, we prove the slightly stronger result:

There are two principal kinds of input data for infinite loop space theory, namely E∞ spaces à la Boardman-Vogt [3] and May [7] and Γ-spaces à la Segal [14]. May and Thomason [13] introduced a common generalization and used it to prove the equivalence of the output obtained from these two kinds of input.

This suggests that any invariants of one kind of input should have analogs for the other. Homology operations are among the most basic invariants of E∞ spaces, and we here establish the analogous invariants for Γ-spaces. The definition is transparently obvious from the point of view of the common generalization but is at first sight rather surprising and unnatural from the point of view of Γ-spaces alone. Probably for this reason, there is no hint of the possibility of a direct definition of homology operations for Γ-spaces in the literature.

A sequence of polynomials {Pn(x)} is orthogonal if Pn(x) is of precise degree n and there is a finite positive measure dμ such that

1.1

A necessary and sufficient condition for orthogonality [9] is that {Pn(x)} satisfies a three term recurrence

1.2

with

1.3

Given a sequence of orthogonal polynomials {Pn(x)} satisfying (1.2), the associated polynomials {Pn(γ)(x)}, γ > 0, are defined by

1.4

with P(γ)-1(x) = 0, P0(γ)(x) = 1, when An+γ and Bn+γ are well-defined.

If P(z) is a polynomial of degree n, then the inequality

1

is trivial. It was asked by Callahan [1], what improvement results from supposing that P(z) has a zero on |z| = 1 and he answered the question by showing that if P( l ) = 0, then

2

Donaldson and Rahman [3] have shown that if P(z) is a polynomial of degree n such that P(β) = 0 where β is an arbitrary non-negative number, then

3

whereas if the polynomial P(z) is such that P(l) = 0, then [4]

4

Early in the Seventies I sought the number of rooted λ-coloured triangulations of the sphere with 2p faces. In these triangulations double joins, but not loops, were permitted. The investigation soon took the form of a discussion of a certain formal power series l(y, z, λ) in two independent variables y and z.

The basic theory of l is set out in [1]. There l is defined as the coefficient of x2 in a more complicated power series g(x, y, z, λ). But the definition is equivalent to the following formula.

1

Here T denotes a general rooted triangulation. n(T) is the valency of its root-vertex, and 2p(T) is the number of its faces. P(T, λ) is the chromatic polynomial of the graph of T.

Let ω0, ω1, … denote the Walsh-Paley functions and let G denote the dyadic group introduced by Fine [3]. Recall that a subset E of G is said to be a set of uniqueness if the zero series is the only Walsh series ∑ akωk which satisfies

A subset E of G which is not a set of uniqueness is called a set of multiplicity.

It is known that any subset of G of positive Haar measure is a set of multiplicity [5] and that any countable subset of G is a set of uniqueness [2]. As far as uncountable subsets of Haar measure zero are concerned, both possibilities present themselves. Indeed, among perfect subsets of G of Haar measure zero there are sets of multiplicity [1] and there are sets of uniqueness [5].

Let E be an m × (n + 1) regular interpolation matrix with elements ei, k = (E)i, k which are zero or one, with n + 1 ones. Then for each f ∈ Cn[a, b] and each set of knots X: a ≦ x1 < … < xm ≦ b, there is a unique interpolation polynomial P(f, E, X; t) of degree ≦ n which satisfies

1

A recent paper [1] discussed the continuity of P, as a function of x1, …,xm(with coalescences allowed). We would like to study in this note the analytic character of P as a function of real or complex knots X: x1, …, xm. This is easy for the Lagrange or the Hermite interpolation. In this case P is a polynomial in x1, …, xm if f is a polynomial, and an entire function in x1, …, xm if f is entire.