Some equationally defined classes of loops are identified and characterized among a class of loops which are isomorphic to all of their loop isotopes.

In 1892, Sylvester [7] proved that in the set of integers n, n+l,…, n+k—1, n> k > 1, there is a number containing a prime divisor greater than k. This theorem was rediscovered, in 1929, by Schur [6]. More recent results include an elementary proof by Erdös [1] and a proof of the following theorem by Faulkner [2]: Let pk be the least prime ≥2k; if n≥pk then has a prime divisor ≥pk with the exceptions and In that paper the author uses some deep results of Rosser and Schoenfeld [5] on the distribution of primes.

A definition of an isometric shift operator on a Banach space is given which extends the usual definition of a shift operator on a separable Hilbert space. It is shown that there is no such shift on many of the common Banach spaces of continuous functions. The associated ideas of a semi-shift and a backward shift are also introduced and studied in the case of continuous function spaces.

The local class group of a Krull domain A is the quotient group G(A) = CI(A)/Pic(A). A Krull domain A is locally factorial if and only if G(A) = 0. In this paper, we characterize the Krull domains for which G(A) is a torsion group. We evaluate the local class group of several examples and finally, we explain why every abelian group is the local class group of a Krull domain.

In this article, we establish a new atomic decomposition for $f\,\in \,L_{w}^{2}\,\bigcap \,H_{w}^{p}$, where the decomposition converges in $L_{w}^{2}$-norm rather than in the distribution sense. As applications of this decomposition, assuming that $T$ is a linear operator bounded on $L_{w}^{2}$ and $0\,<\,p\,\le \,1$, we obtain (i) if $T$ is uniformly bounded in $L_{w}^{p}$-norm for all $w-p$-atoms, then $T$ can be extended to be bounded from $H_{w}^{p}$ to $L_{w}^{2}$; (ii) if $T$ is uniformly bounded in $H_{w}^{p}$-norm for all $w-p$-atoms, then $T$ can be extended to be bounded on $H_{w}^{p}$; (iii) if $T$ is bounded on $H_{w}^{p}$, then $T$ can be extended to be bounded from $H_{w}^{p}$ to $L_{w}^{2}$.

Let p*(n) denote the number of product partitions, that is, the number of ways of expressing a natural number n > 1 as the product of positive integers ≥ 2, the order of the factors in the product being irrelevant, with p*(1) = 1. For any integer if d is an ith power, and = 1, otherwise, and let . Using a suitable generating function for p*(n) we prove that

A ring $R$ is called quasi-Baer if the right annihilator of every right ideal of $R$ is generated by an idempotent as a right ideal. We investigate the quasi-Baer property of skew group rings and fixed rings under a finite group action on a semiprime ring and their applications to ${{C}^{*}}$-algebras. Various examples to illustrate and delimit our results are provided.

Lower bounds are given for the independence ratio in graphs satisfying certain girth and maximum degree requirements. In particular, the independence ratio of a graph with maximum degree Δ and girth at least six is at least (2Δ − 1)/(Δ2 + 2Δ − 1). Sharper bounds are given for cubic graphs.

Langlands has conjectured the existence of a universal group, an extension of the absolute Galois group, which would play a fundamental role in the classification of automorphic representations. We shall describe a possible candidate for this group. We shall also describe a possible candidate for the complexification of Grothendieck's motivic Galois group.

We prove the following theorem. Suppose that $F\,=\,\left( {{f}_{1}},\,{{f}_{2}} \right)$ is a 2-dimensional, vector-valued modular form on $\text{S}{{\text{L}}_{2}}\left( \mathbb{Z} \right)$ whose component functions ${{f}_{1}}$, ${{f}_{2}}$ have rational Fourier coefficients with bounded denominators. Then ${{f}_{1}}$ and ${{f}_{2}}$ are classical modular forms on a congruence subgroup of the modular group.

This paper characterizes when a Delone set $X$ in ${{\mathbb{R}}^{n}}$ is an ideal crystal in terms of restrictions on the number of its local patches of a given size or on the heterogeneity of their distribution. For a Delone set $X$, let ${{N}_{X}}\left( T \right)$ count the number of translation-inequivalent patches of radius $T$ in $X$ and let ${{M}_{X}}\left( T \right)$ be the minimum radius such that every closed ball of radius ${{M}_{X}}\left( T \right)$ contains the center of a patch of every one of these kinds. We show that for each of these functions there is a “gap in the spectrum” of possible growth rates between being bounded and having linear growth, and that having sufficiently slow linear growth is equivalent to $X$ being an ideal crystal.

Explicitly, for ${{N}_{X}}\left( T \right)$, if $R$ is the covering radius of $X$ then either ${{N}_{X}}\left( T \right)$ is bounded or ${{N}_{X}}\left( T \right)\,\ge \,T/2R$ for all $T\,>\,0$. The constant $1/2R$ in this bound is best possible in all dimensions.

For ${{M}_{X}}\left( T \right)$, either ${{M}_{X}}\left( T \right)$ is bounded or ${{M}_{X}}\left( T \right)\ge T/3$ for all $T\,>\,0$. Examples show that the constant 1/3 in this bound cannot be replaced by any number exceeding 1/2. We also show that every aperiodic Delone set $X$ has ${{M}_{X}}\left( T \right)\,\ge \,c\left( n \right)T$ for all $T\,>\,0$, for a certain constant $c\left( n \right)$ which depends on the dimension $n$ of $X$ and is $>\,1/3$ when $n\,>\,1$.

I show that if X is the hyperstonian space of Lebesgue measure on [0,1], then there are open sets in X×X which are not measurable for the simple product outer measure. This answers a question of M. C. Godfrey and M. Sion.

By a composition of a positive integer n is meant a representation of n as a sum of one or more positive integers where the order of the summands is taken into account. Thus for example 4 has the eight compositions 4 = 3 + 1 = 1 +3 = 2 + 2 = 2 + 1 + 1 = 1 + 2 + 1 = 1 + 1 + 2 = 1 + 1 + 1 + 1. Now n can be written in the form 1 + 1 + … + 1 with n - 1 plus signs. Deletion of any subset of these plus signs breaks n into parts which form a composition of n. Conversely, any composition of n corresponds to a subset of plus signs, so that the number of compositions of n is the number of subsets of a set with n - 1 elements, namely 2n - 1. In this note we obtain a number of generalizations of this rather obvious remark by making use of the notion of a weighted composition and the method of generating series.

Suppose is a sequence of functions on ℝn with Δfk = fk+1 (where Δ is the Laplacian) that satisfies the growth condition: |fk(x)| ≤ Mk{1 + |x|)a where a ≥ 0 and the constants have sublinear growth Then Δf0 = —f0- This characterizes eigenfunctions f of Δ with polynomial growth in terms of the size of the powers Δkf, —∞ < k < ∞. It also generalizes results of Roe (where a = 0, Mk = M, and n = 1 ) and Strichartz (where a = 0, Mk = M for n). The analogue holds for formally self-adjoint constant coefficient linear partial differential operators on ℝn.

Given an essentially unitary contraction and an arbitrary unitary dilation of it, there is a naturally associated spectral flow that is shown to be equal to the index of the operator. This result is interpreted in terms of the K-theory of an associated mapping cone. It is then extended to connect Z2 indices of odd symmetric Fredholm operators to a Z2-valued spectral flow.

This expository note was prompted by some questions asked by Professor P. Hilton during his lectures "Catégories non-abétiennes" at the University of Montréal, July 1964.

The descriptions of set functions as one to one and as onto can be characterized in terms of set function composition. A set function is one to one iff it has the left cancellation property, that is, f · g = f · h implies g = h.

Two important mathematical constructions are: the construction of the rational s from the integers and the construction of the reals from the rationals. The first process can be carried out for any ring, producing its maximal ring of quotients [4, 5]. The second process can be carried out for any partially ordered set producing its Dedekind-MacNeille completion [2, p. 58]. We will show that for Boolean rings, which are both rings and partially ordered sets, the two constructions coincide.

In this paper we study left invariant Einstein–Randers metrics on compact Lie groups. First, we give a method to construct left invariant non-Riemannian Einstein–Randers metrics on a compact Lie group, using the Zermelo navigation data. Then we prove that this gives a complete classification of left invariant Einstein–Randers metrics on compact simple Lie groups with the underlying Riemannian metric naturally reductive. Further, we completely determine the identity component of the group of isometries for this type of metrics on simple groups. Finally, we study some geometric properties of such metrics. In particular, we give the formulae of geodesics and flag curvature of such metrics.

We provide vector analogues of the classical interpolation theorems for lower semicontinuous functions due to Dowker and to Hahn and Katetov-Tong.

We obtain mainly by using Jensen's inequality for convex functions an integral inequality, which contains as a special case Shun's generalization of Hardy's inequality.