The distinct Hausdorff compactifications δX of a completely regular (Hausdorff) space X are in one-one correspondence with the admissible proximity relations δ on X, or alternatively, with the admissible totally bounded uniform structures for X. (See [1], [2].) Thus, δX is the Smirnov compactification of (X, δ). Generalized uniform structures for X will be described by means of pseudometrics on X (cf. [5], [7], [13]). Let where is in the proximity class π(δ) associated with (X, δ). Then a subset S of X is σ-discrete of gauge if , for all x, y ∈ S where x ≠ y.

Let L(s, χ) be a Dirichlet L-function and let N(σ, T, χ) denote the number of zeros ρ of L(s, χ), counted according to multiplicity, in the rectangle σ ≦ Re(ρ) ≦ 1, |Im(ρ)| ≦ T, (T ≦ 1). In this paper we shall prove several new estimates for the sum

where Σ* denotes summation over primitive characters only. These estimates will all be of the type

(1)

where denotes any fixed positive quantity.

F. B. Jones (1937) conjectured that every normal Moore space is metrizable. He also denned a particular kind of topological space (now known as Jones' spaces), proved that they were all non-metrizable Moore spaces, but was unable to decide whether or not Jones’ spaces are normal. J. H. Silver (1967) proved that a positive solution to Jones’ conjecture was not possible, and W. Fleissner (1973) obtained an alternative proof by showing that it is not possible to prove the non-normality of Jones’ spaces. These results left open the possibility of resolving the questions from the GCH. In this paper we show that if CH be assumed, then Jones’ spaces are not normal (Devlin, Shelah, independently) and that the GCH does not lead to a positive solution to the Jones conjecture (Shelah). A brief survey of the progress on the problem to date is also included.

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.

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.

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:

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}.

α-Recursion theory was invented simultaneously by Kripke [15] and Platek [22] and served to generalize the theories of Takeuti [34], Machover [20], Kreisel and Sacks [14] and others. Kripke (in [16]) derived machinery to construct an analogue to Kleene's T-predicate enabling him to assert that all of unrelativized ordinary recursion theory (as found in Kleene [13]) lifted to α-recursion theory. As a result, we were able to set down in [8] α-analogues to Blum's [1] well-studied axioms, thus, introducing the study of α-computational complexity theory.

Let K be a field. The Schur subgroup, S(K), of the Brauer group, B(K), consists of all classes [△] in B(K) some representative of which is a simple component of one of the semi-simple group algebras, KG, where G is a finite group such that char K ∤ G. Yamada ([11], p. 46) has characterized S(K) for all finite extensions of the p-adic number field, Qp. If p is odd, [△] ∈ S(K) if and only if

where c is the tame ramification index of k/Qp, k the maximal cyclotomic subfield of K, and s = ((p – 1)/c, [K : k]). invp △ is the Hasse invariant. Yamada showed this by proving first that S(K) is the group of classes containing cyclotomic algebras and then determining the invariants of such algebras.

R. N. Ball (unpublished) and G. E. Davis and C. D. Fox [1] established that if Ω is a doubly homogeneous totally ordered set, the l-group A (Ω) of all orderpreserving permutations of Ω endures a compatible tight Riesz order. Specifically T = {g ∈ A(Ω)+ : supp (g) is dense in Ω} is a compatible tight Riesz order for A(Ω). Using this fact, I inserted Theorem 3.7 into [2; MR 53 (1977), #13070] at the galley proof stage. (It was also included in MR 54 (1977), #7350 and [3; p. 472].) Theorem 3.7 stated: Let Ω be homogeneous. Then A(Ω) endures a compatible tight Reisz order if and only if Ω is dense. I stated that it was obvious that if Ω were homogeneous and discrete, A(Ω) could not endure a compatible tight Riesz order. This “obvious” is neither obvious nor true.

1. Introduction. Throughout this paper, G is a connected linear algebraic group over an algebraically closed field whose characteristic is denoted p. For any closed subgroup H of G, denotes the Lie algebra of H and H0 denotes the connected component of the identity of H.

A Borel subalgebra of is the Lie algebra of some Borel subgroup B of G. A maximal torus of is the Lie algebra of some maximal torus T of G. In [4], it is shown that the maximal tori of are the maximal commutative subalgebras of consisting of semisimple elements, and the question was raised in § 14.3 as to whether the set of Borel subalgebras of is the set of maximal triangulable subalgebras of .

1. In their paper “Permutation Problems and Special Functions,” Askey and Ismail [1] give the following striking identity. Consider three boxes containing j, k, m distinguishable balls, and consider all possible rearrangements of these balls such that each box still has the same number of balls; i.e., j end up in the first, k in the second, m in the third. One disregards the order of the balls within a box so there are (j + k + m)!/(j!k!m!) possible rearrangements. Let RE be the number of rearrangements where an even number of balls change boxes and R0 the number of rearrangements where an odd number change boxes. The identity is

(1.1)

where

(1.2)

is the jth Laguerre polynomial. These polynomials are orthonormal with respect to the weight function e–x; i.e.

1. Introduction. A space of orderings will refer to what was called a “set of quasi-orderings” in [5]. That is, a space of orderings is a pair (X, G) where G is an elementary 2-group (i.e. x2 = 1 for all x ∈ G) with a distinguished element – 1 ∈ G, and X is a subset of the character group x(G) = Horn (G, {1, –1};) satisfying the following properties:

01: X is a closed subset of χ(G).

02: σ(−l) = −1 holds for all σ ∊ X.

03: X⊥ = {a ∊ G|χa = 1 for all a ∊ X} = 1.

04: If f and g are forms over G and if x ∊ Df⊗g, then there exist y ∊ Df and z ∊ Dg such that x ∊ D(y, z).

1. Introduction. An unrestricted section of a sequence x is any sequence of the form Σk∈Fxkδk, where F is some finite subset of the natural numbers. The notion of boundedness of the set of unrestricted sections of a sequence in a K-space was studied in [10], and called unconditional section boundedness (UAB). It was shown in [10] (Theorem 7) that the class of FK-spaces in which every element has UAB consists of those FK-spaces that are invariant under coordinatewise multiplication by the convergent sequences.

Introduction and notations. Let Ω be a bounded region in Rn. In this note we discuss the existence of weak solutions (see [4, Section 2]) of the Dirichlet problem

(I)

where Δ is the Laplacian operator, g : Ω × R → R and f : Ω × Rn+1 → R are functions satisfying the Caratheodory condition (see [2, Section 3]), and ∇ is the gradient operator.

We let λ1 < λ2 ≦ … ≦ λm ≦ … denote the sequence of numbers for which the problem

(II)

has nontrivial weak solutions.

This paper continues the investigations in [1] and [2] and extends the results of the latter paper to functions of several complex variables. Namely let λ : R>0 —> R>0 be monotonically increasing. (If λ is non-constant, we also require that log r = 0(λ(r)). We write / = 0(g) to mean that there is a constant C > 0 such that f(r) ≦ Cg(r) except possibly on a set of intervals of finite total length.) Let 0\ denote the set of meromorphic / on Cn for which the Nevanlinna characteristic function [5, p. 174] T(f, r) satisfies T(f, r) = 0(λ(r)).

1. Introduction and statement of results. If pn(z) is a polynomial of degree at most n, then according to a famous result known as Bernstein's inequality (for references see [4])

(1)

Here equality holds if and only if pn(z) has all its zeros at the origin and so it is natural to seek for improvements under appropriate assumptions on the zeros of pn(z). Thus, for example, it was conjectured by P. Erdôs and later proved by Lax [2] that if pn(z) does not vanish in │z│ < 1, then (1) can be replaced by

(2)

On the other hand, Turán [5] showed that if pn(z) is a polynomial of degree n having all its zeros in │z│ ≦ 1, then

(3)

The decomposition of mappings into a minimal number of simple mappings is a common sight in geometry. One well-known instance is the representation of a plane motion by three reflections (see e.g. H. S M. Coxeter [3]) or the representation of equiaffinities by a minimal number of shears or reflections ([14], [5], [7], [8]). Theorems of this nature not only give valuable insight into the nature of the mapping, but they are also often used as a base for characterization theories (see e.g. F. Bachmann [2], M. Götzky [10]). A more abstract version of the same type of results is the famous Cartan-Dieudonné theorem. Its usefulness is indisputable. P. Scherk [13] gave a refined version of this theorem.

Noting that certain restrictions are placed on homeomorphisms of the pseudo-arc, since it is hereditarily indecomposable, in 1955 [4] R. H. Bing asked if the identity is the only stable homeomorphism of the pseudo-arc. In this paper we prove the following theorem.

THEOREM. Let U be an open subset of the pseudo-arc P. Let p and q be distinct points of P such that the subcontinuum M irreducible between p and q does not intersect cl(U). Then there exists a homeomorphism h : P → P with h (p) = q andh│U = 1U.

1. Definitions. A chain C is a collection of open sets C = {C(i))}i≧n such that C(i) ∩ C(j) ≠ ∅ if and only if |i – j| ≦ 1, cl|(C(i)) ∩ cl(C(j)) ≠ ∅ if and only if |i – j| ≦ 1, C(0) – C(l) ≠ ∅, and C(n) – C(n – 1) ≠ ∅. Each C(i) is called a link of C.

Dans ce travail, nous désignons par la catégorie des groupes localement nilpotents. Si P est un ensemble de premiers, nous disons que n est un P-nombre et nous écrivons n ∈ P si tous les facteurs premiers de n appartiennent à P; on convient toujours que 1 ∈ P. Dans tout ce qui suit, il est souvent commode de ne pas faire explicitement la distinction dans la notation entre un ensemble P de premiers et l'ensemble de tous les entiers naturels ayant tous leurs facteurs premiers dans P; par exemple, n ∈ P signifiera toujours que n est un P-nombre mais pas nécessairement un premier. L'ensemble de tous les premiers n'appartenant pas à P est désigné par P′; on convient que 1 appartient à la fois à P et à P′.