A Lyndon word is a primitive word which is minimum in its conjugation class, for the lexicographical ordering. These words have been introduced by Lyndon in order to find bases of the quotients of the lower central series of a free group or, equivalently, bases of the free Lie algebra [2], [7]. They have also many combinatorial properties, with applications to semigroups, pi-rings and pattern-matching, see [1], [10].

We study here the Poincaré-Birkhoff-Witt basis constructed on the Lyndon basis (PBWL basis). We give an algorithm to write each word in this basis: it reads the word from right to left, and the first encountered inversion is either bracketted, or straightened, and this process is iterated: the point is to show that each bracketting is a standard one: this we show by introducing a loop invariant (property (S)) of the algorithm. This algorithm has some analogy with the collecting process of P. Hall [5], but was never described for the Lyndon basis, as far we know.

If VK is a finite dimensional vector space over a field K and L is a lattice of subspaces of V, then, following Halmos [11], alg L is defined to be (the K-algebra of) all K-endomorphisms of V which leave every subspace in L invariant. If R ⊆ end(VK) is any subalgebra we define lat R to be (the sublattice of) all subspaces of VK which are invariant under every transformation in R. Then R ⊆ alg [lat R] and R is called a reflexive algebra when this is equality. Every finite dimensional algebra is isomorphic to a reflexive one ([4]) and these reflexive algebras have been studied by Azoff [1], Barker and Conklin [3] and Habibi and Gustafson [9] among others.

Classes of functions, meromorphic and univalent in

Δ = {z:|z|< 1}

with simple pole at z = p, 0 < p < 1, have been discussed in several places in the literature ([3], [6], [8], [10], [11], and [12]). The purpose of this paper is to discuss a class of Close-to-Convex functions with pole at p analogous to the class of Close-to-Convex functions with pole at zero studied by Libera and Robertson [9].

Shalika [6] proved the existence of germs (associated with a connected semi-simple algebraic group and a maximal torus over a non-archimedean local field), established many of their properties, and conjectured that the germ associated to the trivial unipotent class in GL(n) should be a constant. R. Howe and Harish-Chandra proved that it is a constant and J. Rogawski [5] proved that it had the value predicted previously by J. Shalika.

We denote by E the open unit disc in C and by H(E) the class of all analytic functions f on E with f(0) = 0. Let (see [3] for more complete definitions)

S = {ƒ ∈ H(E)|ƒ is univalent on E}

S0 = {ƒ ∈ H(E)|ƒ is starlike univalent on E}

TR = {ƒ ∈ H(E)|ƒ is typically real on E}.

The uniform norm on (— 1, 1) of a function ƒ ∈ H(E) is defined by

In this paper, Köthe spaces of vector-valued functions are considered. These spaces, which are generalizations of both the Lebesgue-Bochner and Orlicz-Bochner spaces, have been studied by several people (e.g., see [1], [8]). Perhaps the earliest paper concerning the rotundity of such Köthe space is due to I. Halperin [8]. In his paper, Halperin proved that the function spaces E(X) is uniformly rotund exactly when both the Köthe space E and the Banach space X are uniformly rotund; this generalized the analogous result, due to M. M. Day [4], concerning Lebesgue-Bochner spaces. In [20], M. Smith and B. Turett showed that many properties akin to uniform rotundity lift from X to the Lebesgue-Bochner space LP(X) when 1 < p < ∞. A survey of rotundity notions in Lebesgue-Bochner function and sequence spaces can be found in [19].

This paper arises out of joint work with F. R. Cohen and F. P. Peterson [5, 2, 3] on the joint structure of infinite loop spaces QX. The homology of such a space is operated on by both the Dyer-Lashof algebra, R, and the opposite of the Steenrod algebra A∗. We describe a convenient summary of these actions; let M be the algebra which is R ⊗ A∗ as a vector space and where multiplication Q1 ⊗ PJ. Q1’ ⊗ PJ’∗ is given by applying the Nishida relations in the middle and then the appropriate Adem relations on the ends. Then M is a Hopf algebra which acts on the homology of infinite loop spaces.

This paper makes a unified development of what the authors know about the existence of nearest points to closed subsets of (real) Banach spaces. Our work is made simpler by the methodical use of subderivatives. The results of Section 3 and Section 7 in particular are, to the best of our knowledge, new. In Section 5 and Section 6 we provide refined proofs of the Lau-Konjagin nearest point characterizations of reflexive Kadec spaces (Theorem 5.11, Theorem 6.6) and give a substantial extension (Theorem 5.12). The main open question is: are nearest points dense in the boundary of every closed subset of every reflexive space? Indeed can a proper closed set in a reflexive space fail to have any nearest points? In Section 7 we show that there are some non-Kadec reflexive spaces in which nearest points are dense in the boundary of every closed set.

If K is the set of all compact bounded operators and L(H) is the set of all bounded operators on a separable Hilbert space H, then L(H) is the multiplier algebra of K. In general we denote the multiplier algebra of a C*-algebra A by M(A). For more information about M(A), readers are referred to the articles [1], [3],[7], [9], [14], [18],[20], [23], [26], [27], among others. It is well known that in the Calkin algebra L(H)/K every nonzero projection is infinite. If we assume that A is a-unital (nonunital) and regard the corona algebra M(A)/A as a generalized case of the Calkin algebra, is every nonzero projection in M(A)/A still infinite? Another basic question can be raised: How does the (closed) ideal structure of A relate to the (closed) ideal structure of M(A)/A?

In the first part of this note (Sections 1 and 2) we shall give an affirmative answer for the first question if A is a simple a-unital (nonunital) C*-algebra with FS.

Let be a pair of a formal series (fps) in z1 and z2 of the form where is an fps with for i = 1,2. Then there exists a unique pair of fps which is also of the form (1.1), with This pair is called the inverse of f(z1,z2 ).