Loosely speaking, a function (meromorphic or harmonic) from the hyperbolic disk of the complex plane to the Riemann sphere is normal if its dilatation is bounded. We call a function strongly normal if its dilatation vanishes at the boundary. A sequential property of this class of functions is proved. Certain integral conditions, known to be sufficient for normality, are shown to be in fact sufficient for strong normality.

In contrast to the very complicated structure of the lattice of equational classes of commutative semigroups (see [5]), the lattice of equational classes of commutative monoids (semigroups with unit) is isomorphic with N × N* with a unit adjoined, where N is the lattice of natural numbers with the usual order and N* is the lattice of natural numbers ordered by division. (See [4].) However, the lattice of equational classes of commutative semigroups-with-zero is not so simple to describe.

If $C=C\left( R \right)$ denotes the center of a ring $R$ and $g\left( x \right)$ is a polynomial in $C\left[ x \right]$, Camillo and Simón called a ring $g\left( x \right)$-clean if every element is the sum of a unit and a root of $g\left( x \right)$. If $V$ is a vector space of countable dimension over a division ring $D$, they showed that $\text{en}{{\text{d}}_{\,D}}V$ is $g\left( x \right)$-clean provided that $g\left( x \right)$ has two roots in $C\left( D \right)$. If $g\left( x \right)=x-{{x}^{2}}$ this shows that $\text{en}{{\text{d}}_{\,D}}V$ is clean, a result of Nicholson and Varadarajan. In this paper we remove the countable condition, and in fact prove that $\text{en}{{\text{d}}_{\,R}}M$ is $g\left( x \right)$-clean for any semisimple module $M$ over an arbitrary ring $R$ provided that $g\left( x \right)\in \left( x-a \right)\left( x-b \right)C\left[ x \right] $ where $a,b\in C$ and both $b$ and $b-a$ are units in $R$.

Mahler's measure of a monic polynomial is equal to the product of modules of its roots which lie outside the unit circle. By classical theorem of Kronecker it is strictly greater than 1 for any polynomial that is not a product of cyclotomic factors. In this case a number of lower bounds of the measure, depending either on the degree of the polynomial or on the number of its non-zero coefficients, has been found. Here is given an improvement of the bound of the latter type previously found by the author, A. Schinzel and W. Lawton.

We completely classify three-dimensional Lorentz manifolds, curvature homogeneous up to order one, equipped with Einstein-like metrics. New examples arise with respect to both homogeneous examples and three-dimensional Lorentz manifolds admitting a degenerate parallel null line field.

Let $\unicode[STIX]{x1D707}$ be a positive finite Borel measure on the unit circle and ${\mathcal{D}}(\unicode[STIX]{x1D707})$ the associated harmonically weighted Dirichlet space. In this paper we show that for each closed subset $E$ of the unit circle with zero $c_{\unicode[STIX]{x1D707}}$-capacity, there exists a function $f\in {\mathcal{D}}(\unicode[STIX]{x1D707})$ such that $f$ is cyclic (i.e., $\{pf:p\text{ is a polynomial}\}$ is dense in ${\mathcal{D}}(\unicode[STIX]{x1D707})$), $f$ vanishes on $E$, and $f$ is uniformly continuous. Next, we provide a sufficient condition for a continuous function on the closed unit disk to be cyclic in ${\mathcal{D}}(\unicode[STIX]{x1D707})$.

Let L denote a semi-simple, finite dimensional Lie algebra over an algebraically closed field K of characteristic zero. If denotes a Cartan subalgebra of L and denotes the centralizer of in the universal enveloping algebra U of L, then it has been shown that each algebra homomorphism (called a "mass-function" on ) uniquely determines a linear irreducible representation of L. The technique involved in this construction is analogous to the Harish-Chandra construction [2] of dominated irreducible representations of L starting from a linear functional . The difference between the two results lies in the fact that all linear functionals on are readily obtained, whereas since is in general a noncommutative algebra the construction of mass-functions is decidedly nontrivial.

P. M. Cohn [7] calls a submodule P of the left A-module M pure iff 0 → E ⊗ P → E ⊗ M is exact for all right modules E. This concept has been studied in [11] and [12]. We will call a non-zero module pure simple iff its only pure submodules are 0 and itself, and the ring A left pure simple iff it is pure simple as a left A-module. We relate these concepts to the PP and PF rings of Hattori [13], and give several new characterizations of these rings. In order to establish these, we use the following known result: the Jacobson radical of any module is the sum of all its small submodules.

Parts of this paper are contained in the author's doctoral thesis [10] at McGill University.

All our matrices are square with real elements. The Schur product of two n × n matrices B = (bij) and C = (cij) (i, j, = 1, 2, …, n), is an n × n matrix A = (aij) with aij = bij cij, (i, j = 1, 2, …, n).

A result due to Schur [1] states that if B and C are symmetric positive definite matrices then so is their Schur product A. A question now a rises. Can any symmetric positive definite matrix be expressed as a Schur product of two symmetric positive definite matrices? The answer is in the affirmative as we show in the following theorem.

We show here that the commutator subgroup of a free group of finite rank poses a basis of Bachmuth's type.

A complex hyperbolic triangle group is a group generated by three involutions fixing complex lines in complex hyperbolic space. Our purpose in this paper is to improve a previous result and to discuss discreteness of complex hyperbolic triangle groups of type $(n,\,n,\,\infty ;\,k)$.

Given a centrally symmetric convex body $B$ in ${{\mathbb{E}}^{d}}$, we denote by ${{\mathcal{M}}^{d}}\left( B \right)$ the Minkowski space (i.e., finite dimensional Banach space) with unit ball $B$. Let $K$ be an arbitrary convex body in ${{\mathcal{M}}^{d}}\left( B \right)$. The relationship between volume $V\left( K \right)$ and the Minkowskian thickness (= minimal width) ${{\Delta }_{B}}\left( K \right)$ of $K$ can naturally be given by the sharp geometric inequality $V\left( K \right)\ge \alpha \left( B \right)\cdot {{\Delta }_{B}}{{\left( K \right)}^{d}}$, where $\alpha \left( B \right)>0$. As a simple corollary of the Rogers-Shephard inequality we obtain that ${{\left( _{d}^{2d} \right)}^{-1}}\le \alpha \left( B \right)/V\left( B \right)\le {{2}^{-d}}$ with equality on the left attained if and only if $B$ is the difference body of a simplex and on the right if $B$ is a cross-polytope. The main result of this paper is that for $d=2$ the equality on the right implies that $B$ is a parallelogram. The obtained results yield the sharp upper bound for the modified Banach–Mazur distance to the regular hexagon.

The following theorem is due to Walsh [2]. For another proof see [1].

Theorem A. All the zeros of the polynomial p(z) = a0 + a1z + … an-1zn-1+zn lie on the disk.

where M = Σj = 2n|an-j|1/j

We construct two non-isomorphic nuclear, stably finite, real rank zero ${{C}^{*}}$-algebras $E$ and ${{E}^{'}}$ for which there is an isomorphism of ordered groups $\Theta :{{\oplus }_{n\ge 0}}{{K}_{\bullet }}(E;\mathbb{Z}/n)\to {{\oplus }_{n\ge 0}}{{K}_{\bullet }}({E}';\mathbb{Z}/n)$ which is compatible with all the coefficient transformations. The ${{C}^{*}}$-algebras $E$ and ${{E}^{'}}$ are not isomorphic since there is no $\Theta$ as above which is also compatible with the Bockstein operations. By tensoring with Cuntz’s algebra ${{\mathcal{O}}_{\infty }}$ one obtains a pair of non-isomorphic, real rank zero, purely infinite ${{C}^{*}}$-algebras with similar properties.

In this note we define two concepts which can be thought of as a generalization of noetherian concepts.

The main result is as follows (Corollary A): If R is a ring whose countably generated (left) ideals are (left) principal, then R is a (left) principal ideal ring.

This result if obtained, more generally, for any (left) R-module and any regular cardinal ℵα (Corollary 1); a cardinal ℵα is regular whenever W(ℵα) = {ordinals γ | card γ < ℵα} has no cofinal subset of cardinality less than ℵα.

We show that derivations of the differential structure of a subcartesian space satisfy the chain rule and have maximal integral curves.

A recent issue of the Bulletin contained a paper by Robillard [13] in which results from the theory of confounded factorial designs were used to obtain some lower bounds for Ramsey numbers. We shall derive, by more elementary methods, bounds which are much better than Robillard's in every example which he considered.

Let X and Y be two independent normal variates each distributed with zero mean and a common variance. Then the quotient X/Y has the Cauchy distribution symmetrical about the origin. Of particular interest in recent years has been the converse problem and examples of non-normal distributions with a Cauchy distribution for the quotient have been illustrated by Mauldon [9], Laha [2; 3; 4] and Steck [10].

W. Kurc [5] has proved that in the unit sphere of Orlicz space LΦ(μ) generated by an Orlicz function Φ satisfying the suitable Δ2-condition and equipped with the Luxemburg norm every extreme point is strongly extreme. In this paper it is proved in the case of a nonatomic measure μ that the unit sphere of the Orlicz space LΦ(μ) generated by an Orlicz function Φ which does not satisfy the suitable Δ2-condition and equipped with the Luxemburg norm has no strongly extreme point and no H-point.

The Kestelman–Borwein–Ditor Theorem, on embedding a null sequence by translation in (measure/category) “large” sets has two generalizations. Miller replaces the translated sequence by a “sequence homotopic to the identity”. The authors, in a previous paper, replace points by functions: a uniform functional null sequence replaces the null sequence, and translation receives a functional form. We give a unified approach to results of this kind. In particular, we show that (i) Miller's homotopy version follows fromthe functional version, and (ii) the pointwise instance of the functional version follows from Miller's homotopy version.