Let $P$ be a transitive permutation group of order ${{p}^{m}},\,p$ an odd prime, containing a regular cyclic subgroup. The main result of this paper is a determination of the suborbits of $P$ . The main result is used to give a simple proof of a recent result by J. Morris on Cayley digraph isomorphisms.

Unlike the (classical) Kolakoski sequence on the alphabet {1, 2}, its analogue on {1, 3} can be related to a primitive substitution rule. Using this connection, we prove that the corresponding bi-infinite fixed point is a regular generic model set and thus has a pure point diffraction spectrum. The Kolakoski-(3, 1) sequence is then obtained as a deformation, without losing the pure point diffraction property.

The congruences of a finite sectionally complemented lattice $L$ are not necessarily uniform (any two congruence classes of a congruence are of the same size). To measure how far a congruence $\Theta $ of $L$ is from being uniform, we introduce Spec $\Theta $ , the spectrum of $\Theta $ , the family of cardinalities of the congruence classes of $\Theta $ . A typical result of this paper characterizes the spectrum $S=({{m}_{j}}|j<n)$ of a nontrivial congruence $\Theta $ with the following two properties:

$$({{S}_{1}})\,\,\,\,2\le n\,\,\text{and }n\ne 3.\,\,\,\,$$

$$({{S}_{2}})\,\,\,2\le {{m}_{j}}\,\,\text{and}\,\,{{m}_{j}}\ne 3,\,\,\,\text{for}\,\text{all}\,j<n.$$

We show that Poincaré inequalities with reverse doubling weights hold in a large class of irregular domains whenever the weights satisfy certain conditions. Examples of these domains are John domains.

A discrete group $G$ is called identity excluding if the only irreducible unitary representation of $G$ which weakly contains the 1-dimensional identity representation is the 1-dimensional identity representation itself. Given a unitary representation $\pi $ of $G$ and a probability measure $\mu $ on $G$ , let ${{P}_{\mu }}$ denote the $\mu $ -average $\int{\pi (g)\mu (dg)}$ . The goal of this article is twofold: (1) to study the asymptotic behaviour of the powers $P_{\mu }^{n}$ , and (2) to provide a characterization of countable amenable identity excluding groups. We prove that for every adapted probability measure $\mu $ on an identity excluding group and every unitary representation $\pi $ there exists and orthogonal projection ${{E}_{\mu }}$ onto a $\pi $ -invariant subspace such that $s-{{\lim }_{n\to \infty }}\,(P_{\mu }^{n}\,-\,\pi {{(a)}^{n}}\,{{E}_{\mu }})\,\,=\,0$ for every $a\,\in $ supp $\mu $ . This also remains true for suitably defined identity excluding locally compact groups. We show that the class of countable amenable identity excluding groups coincides with the class of $\text{FC}$ -hypercentral groups; in the finitely generated case this is precisely the class of groups of polynomial growth. We also establish that every adapted random walk on a countable amenable identity excluding group is ergodic.

We give a survey of old and new results concerning the expressibility of the real roots of a solvable polynomial over a real number field by real radicals. A characterization of Fermat primes is obtained in terms of solvability by real radicals for certain ploynomials.

Let $p$ be a prime number. Let $k$ be a finite field of characteristic $p$ . The subset $X\,+\,{{X}^{2}}k[[X]]$ of the ring $k\left[\!\left[ X \right]\!\right]$ is a group under the substitution law $\circ $ sometimes called the Nottingham group of $k$ , it is denoted by ${{\mathcal{R}}_{k}}$ . The ramification of one series $\gamma \,\in \,{{\mathcal{R}}_{k}}$ is caracterized by its lower ramification numbers: ${{i}_{m}}(\gamma )\,=\,\text{or}{{\text{d}}_{X}}({{\gamma }^{{{p}^{m}}}}\,(X)/X-1)\,$ , as well as its upper ramification numbers:

$${{u}_{m}}(\gamma )\ =\ {{i}_{0}}(\gamma )+\frac{{{i}_{1}}(\gamma )-{{i}_{0}}(\gamma )}{p}\,+\,.\,.\,.\,+\,\frac{{{i}_{m}}(\gamma )-{{i}_{m-1}}(\gamma )}{{{p}^{m}}},\,\,\,\,\,\,(m\,\in \,\mathbb{N}).$$

By Sen's theorem, the ${{u}_{m}}(\gamma )$ are integers. In this paper, we determine the sequences of integers ( ${{u}_{m}}$ ) for which there exists $\gamma \,\in \,{{\mathcal{R}}_{k}}$ such that ${{u}_{m}}(\gamma )\,=\,{{u}_{m}}$ for all integer $m\,\ge \,0$ .

Generalizing results from $[\text{MM}1]$ referring to the intersection body $\text{IK}$ and the cross-section body $CK$ of a convex body $K\,\,\subset \,\,{{\mathbb{R}}^{d}}$ , $d\,\ge \,2$ , we prove theorems about maximal $k$ -sections of convex bodies, $k\,\in \,\{1,\,\ldots ,\,d\,-\,1\}$ , and, simultaneously, statements about common maximal $(d\,-\,1)-$ and 1-transversals of families of convex bodies.

A band is a semigroup of idempotent operators. A nonnegative band $\mathcal{S}$ in $B({{L}^{2}}(X))$ having at least one element of finite rank and with rank $(S)\,>\,1$ for all $S$ in $\mathcal{S}$ is known to have a special kind of common invariant subspace which is termed a standard subspace (defined below).

Such bands are called decomposable. Decomposability has helped to understand the structure of nonnegative bands with constant finite rank. In this paper, a geometric characterization of maximal, rank-one, indecomposable nonnegative bands is obtained which facilitates the understanding of their geometric structure.

In this paper, we prove a general result computing the number of rational points of bounded height on a projective variety $V$ which is covered by lines. The main technical result used to achieve this is an upper bound on the number of rational points of bounded height on a line. This upper bound is such that it can be easily controlled as the line varies, and hence is used to sum the counting functions of the lines which cover the original variety $V$ .

We study the interplay between canonical heights and endomorphisms of an abelian variety $A$ over a number field $k$ . In particular we show that whenever the ring of endomorphisms defined over $k$ is strictly larger than $\mathbb{Z}$ there will be $\mathbb{Q}$ -linear relations among the values of a canonical height pairing evaluated at a basis modulo torsion of $A(k)$ .

This article is a short introduction to super-Brownian motion. Some of its properties are discussed but our main objective is to describe a number of limit theorems which show super-Brownian motion is a universal limit for rescaled spatial stochastic systems at criticality above a critical dimenson. These systems include the voter model, the contact process and critical oriented percolation.

In this paper we consider collections of compact operators on a real or complex Banach space including linear operators on finite-dimensional vector spaces. We show that such a collection is simultaneously triangularizable if and only if it is arbitrarily close to a simultaneously triangularizable collection of compact operators. As an application of these results we obtain an invariant subspace theorem for certain bounded operators. We further prove that in finite dimensions near reducibility implies reducibility whenever the ground field is $\mathbb{R}$ or $\mathbb{C}$ .

We prove an analogue of mean curvature comparison theorem in the case where the Ricci curvature below a positive constant is small in ${{L}^{1}}$ -norm.