Let A: a1 < a2 < … be a sequence of positive integers and let f(A:n) = f(n) denote the number of representations of n as the sum of one or more consecutive a's. Although f(n) will usually oscillate rapidly, an asymptotic estimate for its average value

can be obtained for many sequences A.

Let R be a prime ring with σ ∊ Aut (R). We determine the extended centroid of the skew polynomial ring R[x, σ] when (i) 〈σ〉 is X-outer of finite order, (ii) 〈σ〉 is X-outer and infinite, (iii) σm is X-inner and no smaller power of σ fixes the extended centroid of R.

Using ideas of S. Wassermann on non-exact ${{C}^{*}}$-algebras and property $\text{T}$ groups, we show that one of his examples of non-invertible ${{C}^{*}}$-extensions is not semi-invertible. To prove this, we show that a certain element vanishes in the asymptotic tensor product. We also show that a modification of the example gives a ${{C}^{*}}$-extension which is not even invertible up to homotopy.

Let $G$ be a finitely generated, infinite group, let $p\,>\,1$, and let ${{L}^{p}}\left( G \right)$ denote the Banach space $\left\{ \sum{_{x\in G}{{a}_{x}}x}|\sum{_{x\in G}|{{a}_{x}}{{|}^{p}}<\infty } \right\}$. In this paper we will study the first cohomology group of $G$ with coefficients in ${{L}^{p}}\left( G \right)$, and the first reduced ${{L}^{p}}$-cohomology space of $G$. Most of our results will be for a class of groups that contains all finitely generated, infinite nilpotent groups.

Let $X,Y$ be two Hilbert spaces, let E be a subset of $X,$ and let $G\colon E \to Y$ be a Lipschitz mapping. A famous theorem of Kirszbraun’s states that there exists $\tilde {G} : X \to Y$ with $\tilde {G}=G$ on E and $ \operatorname {\mathrm {Lip}}(\tilde {G})= \operatorname {\mathrm {Lip}}(G).$ In this note we show that in fact the function $\tilde {G}:=\nabla _Y( \operatorname {\mathrm {conv}} (g))( \cdot , 0)$, where

$$\begin{align*}g(x,y) = \inf_{z \in E} \Big\lbrace \langle G(z), y \rangle + \frac{\operatorname{\mathrm{Lip}}(G)}{2} \|(x-z,y)\|^2 \Big\rbrace + \frac{\operatorname{\mathrm{Lip}}(G)}{2}\|(x,y)\|^2, \end{align*}$$

$C^{1,1}$

We give a unified treatment of several fixed-point type theorems by using the concept of dismantlability, extended from ordered sets to arbitrary graphs. For a graph G and a vertex x of G we let NG(X) denote the set of neighbours of x in G. We say that x is a subdominant vertex of G if there is a vertex y of G, distinct from x, such that NG(x)∪{x} ⊆ NG(y)∪{y}. If G has n vertices we say that G is dismantlable if the vertices of G can be listed as x1, x2, ..., xi,..., xn such that, for all i = 1,2,..., n— 1, xi is a subdominant vertex of the graph Gi = G — {xj : j < i}

Writing ϕl(n) = ϕ(ϕl−1(n)), the integers n which are solutions of n = kϕl(n) are considered. The set of all such n is completely characterized.

The purpose of this paper is to locate five points Pi ( l ≤ i ≤ 5 ) in a closed unit cube C such that is as large as possible, where d(Pi, Pj) denotes the distance i j between Pi and Pj. We prove that this minimum distance cannot exceed (=m, say), and if it is equal to m, then the corresponding configuration is congruent to the set of points shown in fig. 1, namely P1 = A1 (0,0,0), P2 = A8 (1, 1, 1), P3 = B1 (0,1/2,1), P4 = B3 (1/2,1,0) and P5 = B5 (1, 0, 1/2).

Questions concerning the convergence of Padé and best rational approximations are considered from a categorical point of view in the complete metric space of entire functions. The set of functions for which a subsequence of the mth row of the Padé table converges uniformly on compact subsets of the complex plane is shown to be residual.

The speed of convergence of best uniform rational approximations and Padé approximations on the unit disc is compared. It is shown that, in a categorical sense, it is expected that subsequences of these approximants will converge at the same rate.

Likewise, it is expected that the poles of certain sequences of best uniform rational approximations wil be dense in the entire plane.

Bochner has introduced the idea of almost automorphy in various contexts (see for example [1] and [2]). We shall use the following definition:

A measurable real valued function f of a real variable will be called almost automorphic if from every given infinite sequence of real numbers we can extract a subsequence {αn} such that

• (i) exits for every real t but no kind of uniformity of convergence is stipulated;

• (ii) exits for every t;

• (iii) for every t.

We prove some results concerning convolutions, additive energies, and sumsets of convex sets and their generalizations. In particular, we show that if a set $A\,=\,{{\{{{a}_{1}},\,.\,.\,.\,,\,{{a}_{n}}\}}_{<}}\,\subseteq \,\mathbb{R}$ has the property that for every fixed $1\,\le \,d\,<\,n$, all differences ${{a}_{i}}\,-\,{{a}_{i-d}},\,d\,<\,i\,<n$, are distinct, then $\left| A\,+\,A \right|\,\gg \,{{\left| A \right|}^{3/2+c}}$ for a constant $c\,>\,0$.

Let Q(x) denote the number of squarefree integers ≤ x. Recently K. Rogers [ l ] has shown that Q(x) ≥ 53x / 88 for all x, with equality only at x = 176. Define R(x) to be Q(x) - 6/п2 X. (We observe that and ) Our objective will be to examine R(x). In particular, we show that for all x and observe that for x ≥ 8.

Let Hp(0<p≤∞) be the Hardy class of functions holomorphic in D={|z|<1}, and let f be a function holomorphic in D. It follows from the well known theorem [1, Theorem 3.11, p. 42] that if (I) , then (II) f can be extended continuously to the closed disk {|z|≤1}, and consequently, (III) f is normal in D in the sense of O. Lehto and K. I. Virtanen [2]. Since there is a considerable gap between (II) and (III), one would suspect that the condition (I) might be much stronger than necessary to obtain the conclusion (III). However, it is curious that the implication (I)⇒(III) is sharp as the following theorem shows.

In this paper we introduce the concept of Boolean near-rings. Using any Boolean ring with identity, we construct a class of Boolean near-rings, called special, and determine left ideals, ideals, factor near-rings which are Boolean rings, isomorphism classes, and ideals which are near-ring direct summands for these special Boolean near-rings.

Blackett [6] discusses the near-ring of affine transformations on a vector space where the near-ring has a unique maximal ideal. Gonshor [10] defines abstract affine near-rings and completely determines the lattice of ideals for these near-rings. The near-ring of differentiable transformations is seen to be simple in [7], For near-rings with geometric interpretations, see [1] or [2].

Let (X, d) be a (nonempty) metric space. bc(X) will denote the family of all nonempty bounded closed subsets of X endowed with the Hausdorff metric D induced by d [2, pp. 205]. Let/be a map of X into bc(X). f is nonexpansive at a point x in X if for all y in X.

The densest lattice packings of spheres in Euclidean spaces En of n dimensions are known for n ≤ 8 (for full n — references see [6]). However, it i s not known for any n ≥ 3 whether there can be any non-lattice sphere packing with density exceeding that of the corresponding densest lattice packing.

We are concerned with an algorithm for determining the index (when it is finite) of a subgroup H of a group K when K is specified by a finite set of generators and relations and H is specified as generated by a finite set of words in the generators of K. A systematic computational attack on the problem was discovered by Coxeter and Todd [l], and has proved to be a very useful tool in problems involving generators and relations in groups [2]. Although the method was not completely formalized it was clearly possible to convert it into a computer program, and this has been done by a number of people. Leech has given a survey of this work in [3], where further references may be found.

We explicitly identify the possible probability entrance laws for a class of measure-valued processes that are constructed by taking a particular measure-valued Markov branching process and conditioning it to stay away from the zero measure trap. The set of extreme points of the entrance space is larger than the state space of the conditioned process, and contains elements which correspond to starting the conditioned process at the zero measure.

Let P be a Markov operator on L∞(X, Σ, m) which does not disappear (i.e., P1A ≡ 0 => 1A ≡ 0 ) . We study the relationship between the σ-algebras

(the deterministicσ-algebra), and the asymptoticσ-algebra

When m is a σ-finite invariant measure, measurable iff p*npnf = f, and also iff Pnf has the same distribution as f . The case of a convolution operator on a locally compact group is considered.

Let A be a real n × n matrix and define the Hadamard ratio h(A) to be the absolute value of det A divided by the product of the Euclidean norms of the columns of A. It is shown that if A is a random variable whose distribution satisfies some simple symmetry properties then the random variable log h(A) has mean and variance . In particular, for each ε > 0, the probability that h(A) lies in the range tends to 1 as n tends to ∞.