The study of group types was completed by Meldrum [1]. The concept of ring type described here is based on analogous definitions.

The series R = R0 ⊃ R1 ⊃ … ⊃ Rα = Rα+1is the lower central series for the ring R if Rγ+1=RRγ + RγR for ordinal number γ and Rγ∩δ<γ Rδ if γ is a limit ordinal. The upper central series for R is the series 0=J0 ⊂ J1 ⊂ … ⊂jβ = Jβ+1 where Jγ+1={x ∊ R:xR + Rx ⊆ Jγ} for every ordinal number γ and Jγ = ∪ ∩δ<γ Jδ if γ is a limit ordinal. The length of the upper central series is the smallest ordinal number γ for which Jβ = Jβ+1.The length of the lower central series is defined similarily. We shall say the ring has type (β,α) if the length of the upper central series is β and the length of the lower central series is α.

The spectra of the Toeplitz operators on the weighted Hardy space ${{H}^{2}}\,(W\,d\theta \,/\,2\pi )$ and the Hardy space ${{H}^{p}}\,(d\theta \,/\,2\pi )$, and the singular integral operators on the Lebesgue space ${{L}^{2}}\,(d\theta \,/\,2\pi )$ are studied. For example, the theorems of Brown-Halmos type and Hartman-Wintner type are studied.

Simple structure theorems or representation theorems make their appearance at a very elementary level in subjects such as algebra, but one infrequently encounters such theorems in elementary topology. In this note we offer one such theorem, for finite topological spaces, which involves only the most elementary concepts: discrete space, indiscrete space, product space and homeomorphism. Our theorem describes the structure of those finite spaces which are homogeneous.

W-groups are treated in the more general context of near-ring modules. It is shown that they share properties of d.g. modules with respect to central series. In particular, results of Magomaev are obtained and used to show that the group-theoretic and module-theoretic central series coincide for W-groups.

We prove that for a generic hypersurface in ${{\mathbb{P}}^{2n+1}}$ of degree at least $2\,+\,2/n$, the $n$-th Picard number is one. The proof is algebraic in nature and follows from certain coherent cohomology vanishing.

On behalf of the Canadian Mathematical Congress, I wish to thank the University of Toronto for its hospitality, the members of the Local Arrangements Committee (especially Chandler Davis) for the many comforts and pleasures they have provided, and our nine distinguished visitors for the courses of lectures they gave at our Seminar.

We present another example of a 3-variable polynomial defining a $K3$-hypersurface and having a logarithmic Mahler measure expressed in terms of a Dirichlet $L$-series.

Suppose f is a continuous endomorphism of an interval which has a periodic orbit, p0 < P1 < … < pn, that defines a permutation a by f(pi) = pσ(i). If σ is irreducible the topological entropy of f is bounded below by the logarithm of the spectral radius of an n x n matrix which is induced by σ.

The generalized binomial function can be obtained as the solution of the equation y = 1 +zyα which satisfies y(0) = 1 where α ≠ 1 is assumed to be real and positive. The technique of Lagrange inversion can be used to express as a series which converges for |z| < α-α|a — l|α-1. We obtain a representation of the function as a contour integral and show that if α > 1 it is an analytic function in the complex z plane cut along the nonnegative real axis. For 0 < α < 1 the region of analyticity is the sector |arg(—z)| < απ. In either case defined by the series can be continued beyond the circle of convergenece of the series through a functional equation which can be derived from the integral representation.

Let f(x) be a real-valued function defined on an interval Ia: [ 0, a ]. For each point x in Ia we form the series , where u0 and un+1 = f(un) for n ≥ 0. If the series converges, x will be called a point of convergence; if this series diverges, x will be called a point of divergence. La this note several properties of sets of convergence2 will be obtained.

In this paper we introduce several properties closely related to the fixed point property of a partially ordered set P: the comparability property, the fixed point property for cones, and the fixed point extension property. We apply these properties to the sets of common bounds of the minimal (maximal) elements of the partially ordered set P in order to derive fixed point theorems for P.

Suppose $G$ is a connected complex Lie group and $H$ is a closed complex subgroup. Then there exists a closed complex subgroup $J$ of $G$ containing $H$ such that the fibration $\pi :G/H\to $$G/J$ is the holomorphic reduction of $G/H$i.e., $G/J$ is holomorphically separable and $\mathcal{O}(G/H)\cong $${{\pi }^{*}}\mathcal{O}(G/J)$. In this paper we prove that if $G/H$ is pseudoconvex, i.e., if $G/H$ admits a continuous plurisubharmonic exhaustion function, then $G/J$ is Stein and $J/H$ has no non-constant holomorphic functions.

Self-complementary digraphs, and oriented type of these were counted by Read [4] and Sridharan [5] respectively. In [3] Palmer obtained an asymptotic formula for the number of self-complementary digraphs following a method of Oberschelp [2]. An asymptotic formula for the number of self-complementary oriented graphs is given here. We refer to [1] for definitions and details not mentioned here.

A 2-cell embedding of an Eulerian digraph $D$ into a closed surface is said to be directed if the boundary of each face is a directed closed walk in $D$. In this paper, a method is developed with the purpose of enumerating unlabelled embeddings for an Eulerian digraph. As an application, we obtain explicit formulas for the number of unlabelled embeddings of directed bouquets of cycles ${{B}_{n}}$, directed dipoles $O{{D}_{2n}}$ and for a class of regular tournaments ${{T}_{2n+1}}$.

A classical question for a Toeplitz matrix with given symbol is how to compute asymptotics for the determinants of its reductions to finite rank. One can also consider how those asymptotics are affected when shifting an initial set of rows and columns (or, equivalently, asymptotics of their minors). Bump and Diaconis obtained a formula for such shifts involving Laguerre polynomials and sums over symmetric groups. They also showed how the Heine identity extends for such minors, which makes this question relevant to Random Matrix Theory. Independently, Tracy and Widom used the Wiener–Hopf factorization to express those shifts in terms of products of infinite matrices. We show directly why those two expressions are equal and uncover some structure in both formulas that was unknown to their authors. We introduce a mysterious differential operator on symmetric functions that is very similar to vertex operators. We show that the Bump–Diaconis–Tracy–Widom identity is a differentiated version of the classical Jacobi–Trudi identity.

Free binary systems are shown not to admit idempotent means. This refutes a conjecture of the author. It is also shown that the extension of Hindman’s theorem to nonassociative binary systems formulated and conjectured by the author is false.

In this work, we study the existence of solutions of nonlinear fractional coupled system of $\varphi $-Hilfer type in the frame of Banach spaces. We improve a property of a measure of noncompactness in a suitably selected Banach space. Darbo’s fixed point theorem is applied to obtain a new existence result. Finally, the validity of our result is illustrated through an example.

We provide a novel proof of the existence of regulator indecomposables in the cycle group $C{{H}^{2}}\left( X,\,1 \right)$, where $X$ is a sufficiently general product of two elliptic curves. In particular, the nature of our proof provides an illustration of Beilinson rigidity.

We define a refined motivic dimension for an algebraic variety by modifying the definition of motivic dimension by Arapura. We apply this to check and recheck the generalized Hodge conjecture for certain varieties, such as uniruled, rationally connected varieties and a rational surface fibration.