About a hundred years ago, the author of Through the Looking-Glass wrote another book called Euclid and his Modern Rivals. Were these rivals Lobachevsky and Bolyai, Riemann and Schlafli? No, they were merely the authors of dull school textbooks that would soon be forgotten. The sad truth is that, in 1873, hardly anyone in England knew of the breakthrough that had occurred on the Continent some fifty years before: only Cayley in Cambridge, Clifford in London, and a few students. Even if Cayley or Clifford had visited Oxford and given a lecture there, it is doubtful that he would have succeeded in convincing the conservative Dodgson that Euclid's postulates could be modified to yield two new worlds, surpassing in strangeness the worlds of the two Alice books and yet just as logically consistent as Euclid.

Let R denote an associative ring with 1, let n be a positive integer, and let k = 1, 2, or 3. The ring R will be called an (n, k)-ring if it satisfies the identities

for all integers m with n ≤ m ≤ n + k - 1. It was shown years ago by Herstein (See [2], [9], and [10]) that for n >1, any (n, l)-ring must have nil commutator ideal C(R). Later Luh [12] proved that primary (rc, 3)-rings must in fact be commutative, and Ligh and Richoux [11] recently showed that all (n, 3)-rings are commutative.

We give a necessary and sufficient condition on weight functions u and v such that for l≤p≤q≤∞ there exists a constant C for which

A corresponding dual result is also given. This extends a result of B. Muckenhoupt which appeared in Studia Math., 34 (1972).

Sums of the form where f(n) is a multiplicative arithmetical function and denotes summation over those values of n for which f(n)>0 and f(n) ≠1, were studied by De Koninck [2], De Koninck and Galambos [3], Brinitzer [1] and Ivič [5]. The aim of this note is to give an asymptotic formula for a certain class of multiplicative, positive, primeindependent functions (an arithmetical function is prime-independent if f(pv) = g(v) for all primes p and v = 1, 2, …). This class of functions includes, among others, the functions a(n) and τ(e)(n), which represent the number of nonisomorphic abelian groups of order n and the number of exponential divisors of n respectively, and none of the estimates of the above-mentioned papers may be applied to this class of functions. We prove the following.

An intrinsic characterization is given of those von Neumann algebras which are injective objects in the category of C*-algebras with completely positive maps. For countably generated von Neumann algebras several such characterizations have been given, so it is in fact enough to observe that an injective von Neumann algebra is generated by an upward directed collection of injective countably generated sub von Neumann algebras. The present work also shows that three of the intrinsic characterizations known in the countably generated case hold in general.

IfT is a 1-1 bimeasurable measure-preserving aperiodic transformation on a probability space X which is a Lebesgue space, then {A:A⊂X and for almost every pair of finite sets F and G there is an n∈N satisfying F⊂TnA and G ∩ TnA=ϕ} is dense in the σ-algebra of measurable sets.

Fix an integer r ≥ 2 and positive numbers b1, …, br. Write σ = b1+ …+br Let . In this note we evaluate the constant A (when it exists) for which

1

where the sum is over all vectors

2

We also obtain upper and lower bounds for the sum in (1).

The theory of analytic differential systems in Banach algebras has been investigated by E. Hille and others, see for instance Chapter 6 in [4].

In this paper we show how a projection method used by W. A. Harris, Jr., Y. Sibuya, and L. Weinberg [3] can be applied to study a class of functional differential equations in this setting. The method, based on functional analysis, had been used extensively by L. Cesari [1] in similar forms for boundary value problems, and by J. K. Hale, S. Bancroft, and D. Sweet [2]. We also obtain as corollaries several results for ordinary differential equations in Banach algebras which were proved in a different way by Hille.

For any real number θ let where [x] denotes the greatest integer not exceeding x. A method is given for computing fθ from its first few terms. A similar method is given for computing the characteristic function gθ(n) of [nθ]. The given methods converge rapidly, and generalize previous results of Bernoulli, Markorf, and Stolarsky. Note that either of the sequences fθ and gθ determines the sequence [nθ] (n = 1, 2,…).

Let Xl, …, Xn be independent random variables with the same distribution function (df) F(x) and let Xln≤X2n≤…≤nr be the corresponding order statistics. The (df) of Xkn will be denoted always by Fkn(x). Many authors have investigated the asymptotic behaviour of the maximal term Xnn as n → ∞. Gnedenko [3] proved the following

The middle graph of a graph G=( V, E) is the graph M(G) = (V∪E, E′), in which two vertices u, v are adjacent if either M is a vertex in V and v is an edge in E containing u, or u and v are edges in E having a vertex in common. Middle graphs have been characterized in terms of line graphs by Hamada and Yoshimura [7], who also investigated their traversability and connectivity properties. In this paper another characterization of middle graphs is presented, in which they are viewed as a class of intersection (representative) graphs of hereditary hypergraphs. Graph theoretic parameters associated with the concepts of vertex independence, dominance, and irredundance for middle graphs are discussed, and equalities relating the chromatic number of a graph to these parameters are obtained.

Sufficient conditions for a semilattice to be a 0- distributive are obtained. Some equivalent formulations of 0- distributivity in a semilattice are given. Further, disjunctive 0- distributive semilattices are also characterized.

A strong version of Levine′s decomposition of continuity leads to the result that a closed graph weakly continuous function into a rim-compact space is continuous. This result implies a closed graph theorem: every almost continuous closed graph function into a strongly locally compact space is continuous. An open problem of Shwu-Yeng T. Lin and Y.-F. Lin asks if every almost continuous closed graph function from a Baire space to a second countable space is necessarily continuous. This question is answered in the negative by an example.

Let V be a finite dimensional vector space over k, a field of characteristic 0, L be an algebraic Lie-subalgebra of Endk(V), with the latter a Lie algebra in the canonical way, and let V be an L-module in the canonical way. For X ∈ V, let LX = {AX | A ∈ L{. Call V a principal L-module if ∃ X ∈ V such that LX= V; X will be called a principal generator of the L-module V.

Let G be a periodic group and ZG its integral group ring. The elements ±g(g∈G) are called the trivial units of ZG. In [1], S. D. Berman has shown that if G is finite, then every unit of finite order is trivial if and only if G is abelian or the direct product of a quaternion group of order 8 and an elementary abelin 2-group. By comparison, Losey in [7] has shown that if ZG contains one non-trivial unit of finite order, then it contains infinitely many.

Given a measurable set A of real numbers with measure mA > 0, and a sequence {dn} of real numbers converging to zero, is there always an x such that x + dn ∈ A for all n sufficiently large?

The answer to this question, which was posed to the authors by P. Erdös, is NO.

In [5], Mal'cev generalized the group theoretical results of H. Neumann (see [6] Chapter 2) to produce the notion of the product, of two subclasses of a given variety of algebras, Following the group theorectic example, members of were called extensions of algebras in by algebras in

Consider the identity

where aj, …, am are positive real numbers. Then for r = 1, 2, 3, … Tr =Tr(a1, …, am) is called the rth complete symmetric function in a1, …, am (T0=l).

The papers [2] and [3] study the function g(k, n), defined for integers k > 1 and n > 1 as the smallest r with the property that every integer is a sum of r kth powers mod n. This note identifies g′(k), defined as the maximum over all n of g(k, n), with the function Γ(k) studied by Hardy and Littlewood [1] fifty years ago in connection with Waring's problem.