The use of geometric invariants has recently played an important role in the solution of classification problems in non-commutative ring theory. We construct geometric invariants of non-commutative projectivizataions, a significant class of examples in non-commutative algebraic geometry. More precisely, if $S$ is an affine, noetherian scheme, $X$ is a separated, noetherian $S$-scheme, $\mathcal{E}$ is a coherent ${\mathcal{O}}_{X}$-bimodule and $\mathcal{I} \subset T(\mathcal{E})$ is a graded ideal then we develop a compatibility theory on adjoint squares in order to construct the functor $\Gamma_{n}$ of flat families of truncated $T(\mathcal{E})/\mathcal{I}$-point modules of length $n+1$. For $n \geq 1$ we represent $\Gamma_{n}$ as a closed subscheme of ${\mathbb{P}}_{X^{2}}({\mathcal{E}}^{\otimes n})$. The representing scheme is defined in terms of both ${\mathcal{I}}_{n}$ and the bimodule Segre embedding, which we construct. Truncating a truncated family of point modules of length $i+1$ by taking its first $i$ components defines a morphism $\Gamma_{i} \rightarrow \Gamma_{i-1}$ which makes the set $\{\Gamma_{n}\}$ an inverse system. In order for the point modules of $T(\mathcal{E})/\mathcal{I}$ to be parameterizable by a scheme, this system must be eventually constant. In [20], we give sufficient conditions for this system to be constant and show that these conditions are satisfied when ${\mathsf{Proj}} T(\mathcal{E})/\mathcal{I}$ is a quantum ruled surface. In this case, we show the point modules over $T(\mathcal{E})/\mathcal{I}$ are parameterized by the closed points of ${\mathbb{P}}_{X^{2}}(\mathcal{E})$.

Contributes to the theory of Borel equivalence relations, considered up to Borel reducibility, and measures preserving group actions considered up to orbit equivalence. This title catalogs the actions of products of the free group and obtains additional rigidity theorems and relative ergodicity results in this context.

A paper that studies two types of integral transformation associated with fractional Brownian motion. They are applied to construct approximation schemes for fractional Brownian motion by polygonal approximation of standard Brownian motion. This approximation is the best in the sense that it minimizes the mean square error.

Introduction Dessins d'enfants Discrete Dessins via circle packing Uniformizing Dessins A menagerie of Dessins d'enfants Computational issues Additional constructions Non-equilateral triangulations The discrete option Appendix: Implementation Bibliography.

Generating function techniques are used to study the probability that an element of a classical group defined over a finite field is separable, cyclic, semisimple or regular. The limits of these probabilities as the dimension tends to infinity are calculated in all cases, and exponential convergence to the limit is proved. These results complement and extend earlier results of the authors, G. E. Wall, and Guralnick & Lubeck.