We consider finite dimensional Lie algebras over an algebraically closed field F of arbitrary characteristic. Such an algebra L will be called a centralizer nilpotent Lie algebra (abbreviated c.n.) provided that the centralizer C(x) is a nilpotent subalgebra of L for all nonzero x ∈ L.

For each algebraically closed F, there is a unique simple Lie algebra of dimension 3 over F which we shall denote S(F). This algebra has a basis e−1, e0, e1 such that [e−1e0] = e−1, [e−1e1] = e0 and [e0e1] = e1. (If char(F) ≠ 2, then S(F) ≅ sl2(F).) It is trivial to check that S(F) is a c.n. algebra for all F.

There are two other types of simple Lie algebras we consider. If char (F) = 3, construct the octonion (Cayley) algebra over F.

Let D be a bounded plane domain and let Lp(D) stand for the usual Lebesgue spaces of functions with domain D, relative to the area Lebesque measure dσ(z) = dxdy. The class of all holomorphic functions in D will be denoted by H(D) and we write Bp(D) = Lp(D) ∩ H(D). Bp(D) is called the Bergman p-space and its norm is given by

Let be the Bergman kernel of D and consider the Bergman projection

(1.1)

It is well known that P is not bounded on Lp(D), p = 1, ∞, and moreover, it can be shown that there are no bounded projections of L∞(Δ) onto B∞(Δ).

Let K be a field of characteristic zero. Let V be an n-dimensional vector space over K. A linear automorphism of V is said to be of type i if it leaves fixed a subspace of dimension i. A reflection is a linear automorphism of type n − 1 which has finite order. A finite reflection group is a finite group of linear automorphisms which is generated by reflections. These groups are especially interesting because the full group of symmetries of a regular poly tope is always a finite reflection group. There is also a strong connection between these groups and Lie groups.

Let (X, ) be a topological space equipped with a partial order ≦ and let C (≦) denote the continuous increasing functions mapping X into R (a function f : X → R is increasing provided f(x) ≦ f(y) whenever x ≦ y) Then (X,, ≦) is an N-space (in the terminology of [16], a completely regular order space) provided is the weak topology of C (≦) and if x ≦ y is false, then there is an f ∈ C (≦) such that f(y) < f(x). L. Nachbin's introduction of N-spaces was perspicacious, for these spaces now find application in a wide spectrum of mathematics.

We extend a theorem of L. E. Dubins on “purely finitely additive disintegrations” of measures (cf. [4]) and apply this result to the disintegrations of extremal Gibbs states with respect to the asymptotic algebra enlarging another result of L. E. Dubins on the symmetric coin tossing game.

We recall the following definition of L. E. Dubins (cf. [3], [4]): Let (X , , μ) be a measure space, a sub σ-algebra of . A real function σx (A), is called a measurable-disintegration of μ if:

• (i) ∀x ∊ X , σx(.) is a finitely additive measure .

• (ii) ∀A ∊ , σ. (A) is constant on each -atom.

• (iii) For each A ∊ , σ. (A) is measurable with respect to the completion of by μ and

• (iv)σx(B) = 1 if x ∊ B ∊ .

1. Introduction. One of the basic objects of study in harmonic analysis is the Fourier transform (or Fourier-Stieltjes transform) μ of a bounded (complex) measure μ on the real line R:

(1.1)

More generally, if μ is a bounded measure on a locally compact abelian group G, then its Fourier transform is the function

(1.2)

where Ĝ is the dual group of G and One answer to the question “Which functions can be represented as Fourier transforms of bounded measures?” was given by the following criterion due to Schoenberg [11] for the real line and Eberlein [5] in general: f is a Fourier transform of a bounded measure if and only if there is a constant M such that

(1.3)

for all ϕ ∈ L1(G) where

The Szász and Baskakov approximation operators are given by

1.1

1.2

respectively. For continuous functions on [0, ∞) with exponential growth (i.e. ‖ƒ‖A ≡ supx\ƒ(x)e–Ax\ < M) the modulus of continuity is defined by

1.3

where ƒ ∈ Lip* (∝, A) for some 0 < ∝ ≦ 2 if w2(ƒ, δ, A) ≦ Mδ∝ for all δ < 1. We shall find a necessary and sufficient condition on the rate of convergence of An(ƒ, x) (representing Sn(ƒ, x) or Vn(ƒ, x)) to ƒ(x) for ƒ(x) ∈ Lip* (∝, A). In a recent paper of M. Becker [1] such conditions were found for functions of polynomial growth (where (1 + \x\N)−1 replaced e–Ax in the above). M. Becker explained the difficulties in treating functions of exponential growth.

The de Bruijn—van Aardenne Ehrenfest— Smith—Tutte theorem [1] is a theorem which connects the number of Eulerian dicircuits in a directed graph with the number of rooted spanning arborescences. In this paper we obtain a proof of this theorem by considering sequences over a finite alphabet, and we show that the theorem emerges from the generating function for a certain type of sequence. The generating function for the set of sequences is obtained as the solution of a linear system of equations in Section 2. The power series expansion for the solution of this system is obtained by means of the multivariate form of the Lagrange theorem for implicit functions, and is given in Section 3, together with a restatement of the theorem as a matrix identity.

An equidistant permutation array (EPA) A(r λ v) is a v × r array defined on a set V of r symbols such that every row is a permutation of V and any two distinct rows have precisely λ common column entries. Define R(r, λ) to be the largest value of v for which there exists an A (r, λ; v). Deza [2] has shown that

where n = r – λ. Bolton [1] has shown that

(*)

In this paper, we show that equality holds in (*) for λ > ┌n/3┐(n2 + n). In order to do this we require several more definitions.

Let Λ be an artin algebra, and denote by mod Λ the category of finitely generated Λ-modules. All modules we consider are finitely generated.

We recall from [6] that a nonsplit exact sequence in mod A is said to be almost split if A and C are indecomposable, and given a map h: X → C which is not an isomorphism and with X indecomposable, there is some t: X → B such that gt = h.

Almost split sequences have turned out to be useful in the study of representation theory of artin algebras. Given a nonprojective indecomposable Λ-module C (or an indecomposable noninjective Λ-module A), we know that

there exists a unique almost split sequence [6, Proposition 4.3], [5, Section 3].

1.1. Introduction. Soit G un sous-groupe de Lie, de dimension q, du groupe GLP(R). Soient Gi(n) = Ci0,e(Rn, G) le groupe des germes en 0 des applications g de classe Ci de Rn dans G telles que g(0) = e (où e est l'élément neutre de G); Diffi(n) le groupe des germes en 0 des difféomorphismes τ d'un voisinage de 0 dans Rn sur un voisinage de 0 dans Rn tels que τ(0) = 0; C0i(Rn, Rp) l'ensemble des germes en 0 des applications de classe Ci de Rn dans Rp; l'anneau des germes en 0 des fonctions numériques de classe C∞ sur Rn et m son idéal maximal. Pour ƒ ∈ C0∞(Rn, Rp), nous désignerons par jr(ƒ) le jet d'ordre r de ƒ en 0. On munit l'ensemble de la structure de groupe définie par la multiplication:

Generalized axisymmetric potentials Fα (GASP) are regular solutions to the generalized axisymmetric potential equation

(1.1)

in some neighborhood Ω of the origin where they are subject to the initial data

(1.2)

along the singular line y = 0. In Ω, these potentials may be uniquely expanded in terms of the complete set of normalized ultraspherical polynomials

(1.3)

defined from the symmetric Jacobi polynomials Pn(α, α)(ξ) of degree n with parameter α as Fourier series

(1.4)

The question of which finite CW-complexes are if-spaces has been studied for many years. Since a finite CW-complex is an H-space if and only if its localization at each prime p is an H-space [21], an examination of finite local cell complexes as H-spaces yields results concerning CW-complexes. On the other hand, if it is known that a particular CW-complex is not an H-space, one would like to know for which primes p its localization at p fails to be an H-space. The main result of this paper gives a condition equivalent to a three cell local CW-complex's being an H-space for a prime p > 3.

Introduction. Every orthomodular lattice (abbreviated : OML) is the union of its maximal Boolean subalgebras (blocks). The question thus arises how conversely Boolean algebras can be amalgamated in order to obtain an OML of which the given Boolean algebras are the blocks. This question we deal with in the present paper.

The problem was first investigated by Greechie [6, 7, 8, 9]. His technique of pasting [6] will also play an important role in this paper. A case solved completely by Greechie [9] is the case that any two blocks intersect either in the bounds only or have the bounds, an atom and its complement in common. This is, of course, a very special situation. The more surprising it is that Greechie's methods, if skillfully applied, yield considerable insight into the structure of OMLs and provide a seemingly unexhaustible source for counter-examples.

Cet article contient une classification détaillée des symétries des C*-algèbres uniformément hyperfinies. Soient une symétrie de ; on associe à θ un entier généralisé N(θ), un réel Tr (θ) ∈ [0, 1], et on a les résultats suivants:

Il existe une symétrie unique à conjugaison près qui possède une suite centrale (Un) d'unitaires vérifiant ‖θ(Un) + Un‖ → 0(n → ∞). Cette symétrie est la seule dont l'algèbre des points fixes est uniformément hyperfinie.

Noether lattices were introduced by R. P. Dilworth in [5] and constitute a natural abstraction of the lattice of ideals of a Noetherian ring. In his definitive work, Dilworth showed that a minimal prime of an element generated by n principal elements has rank ≦ n. Following standard ring theoretical terminology, a local Noether lattice with (unique) maximal element M is said to be regular if M has rank n and can be generated by n principal elements.

In [3], K. P. Bogart showed that a distributive regular local Noether lattice of Krull dimension n is isomorphic to RLn, the sublattice of all ideals generated by monomials of any polynomial ring (k a field). In a later paper [4], Bogart extended his results on distributive regular local Noether lattices by showing that any distributive local Noether lattice is the image of a multiplicative map θ which preserves joins, and can in fact be thought of as the related congruence lattice.

Introduction. We shall consider functions of the form

where {ri} and {si} are sets of positive integers. Such functions were studied by E. Grosswald in [2], who took {si} to be pairwise relatively prime, and asked the following two questions:

(a) When is ƒ(t) a polynomial?

(b) When does ƒ(t) have positive coefficients?

These questions arise naturally from the work of Allday and Halperin, who show in [1] that under suitable circumstance ƒ(t) will be the Poincare polynomial of the orbit space of a certain Lie group action. Grosswald gives a complete answer to (a), but (b) is a much harder question, and a complete answer is provided only for the case m = 2. His treatment involves the representation of the coefficients of ƒ(t) by partition functions, and uses a classical description by Sylvester of the semigroup generated by {si}.

Graphs which are 2-connected have been called blocks in the graph theoretic literature [3] and stars in the parlance of statistical mechanics [1]. They are also called nonseparable graphs, and in this paper include the complete graph K2 on p = 2 points. The standard terminology of graphical enumeration can be found in [3].

When its points are colored using two distinct colors in such a way that adjacent points receive different colors, a graph is called 2-colored. If the colors are considered interchangeable, such a graph is called bicolored. A graph which can be bicolored is called bicolorable. It is obvious that a connected bicolorable graph has a unique bicoloring, and so one can speak of the sizes of its color classes.

In this paper we investigate the action of finite groups G on finite polygonal graphs. The notion of a polygonal graph was introduced in [17]: 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 of such that every 2-claw of is contained in an unique element of (See Section 2 for the définitions of the terms used here.) If is the set of all m-gons of , so that there is in an 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).

An orientable map is often presented as a realization of a finite connected graph G in an orientable surface so that the complementary domains of G, the “faces” of the map are topological open discs. This is not the definition to be used in the paper. But let us contemplate it for a while.

On each edge of G we can recognize two opposite directed edges, or “darts”. Let θ be the permutation of the dart-set S that interchanges each dart with its opposite. The darts radiating from a vertex v occur in a definite cyclic order, fixed by a chosen positive sense of rotation on the surface. The cyclic orders at the various vertices are the cycles of a permutation P of S. The choice of P rather than P–l, which corresponds to the other sense of rotation, makes the map “oriented”.