Three-dimensional flops and noncommutative rings

Two- and Three-dimensional Rings in Drugs

Using small, flat aromatic rings as components of fragments or molecules is a common practice in fragment-based drug discovery and lead optimization. With an increasing focus on the exploration of novel biological and chemical space, and their improved synthetic accessibility, 3D fragments are attracting increasing interest. This study presents a detailed analysis of 3D and 2D ring fragments in...

Three Examples of Gröbner Reduction over Noncommutative Rings

Orderings for Noncommutative Rings

Introduction and Notation. The goal of this paper is to present the beginnings of a theory of real algebraic geometry for noncommutative rings. For a basic introduction to the commutative theory, see Lam [L]. The word field will be used in this paper to mean a (generally noncommutative) skewfield; we shall specify a commutative field when we need to. R will denote a noncommutative ring with 1. ...

Three-Dimensional Dynamics of Narrow Planetary Rings

Narrow planetary rings are eccentric and inclined. Particles within a given ring must therefore share the same pericenter and node. We solve for the threedimensional geometries and mass distributions that enable the Uranian α and β rings, and the Saturnian Maxwell and Colombo (Titan) rings, to maintain simultaneous apsidal and nodal lock. Ring self-gravity, interparticle collisions, and the qua...

Noncommutative Extensions of Hilbert Rings