We derive a simple continuum model of multiwalled carbon nanotubes that takes into account both strong covalent bonds within graphene layers and weak bonds between graphene layers. The model predicts polygonization of cross sections of large multiwalled nanotubes.

Deformation theory studies the behavior of mathematical objects, such as representations or modules, under small perturbations. This theory is useful in both pure and applied mathematics and has been used in the proof of many long-standing problems. In particular, in number theory Wiles and Taylor used universal deformation rings of Galois representations in the proof of Fermat's Last Theorem. The main motivation for determining universal deformation rings of modules for finite dimensional algebras is that deep results from representation theory can be used to arrive at a better understanding of deformation rings. In this thesis, I study the universal deformation rings of certain modules for algebras of dihedral type of polynomial growth which have been completely classified by Erdmann and Skowroński using quivers and relations.More precisely, let k be an algebraically closed field and let Λ be a k-algebra of dihedral type which is of polynomial growth. In this thesis, I first classify all Λ-modules whose stable endomorphism ring is isomorphic to k and which are given combinatorially by strings, and then I determine the universal deformation ring of each of these modules. Abstract Approved:Thesis Supervisor

In this paper, we provide a natural bijection between a special family of block circulant graphs and the graphs of critical pairs of the posets known as generalized crowns. In particular, every graph in this family of block circulant graphs we investigate has a generating block row that follows a symmetric growth pattern of the all ones matrix. The natural bijection provides an upper bound on the chromatic number for this infinite family of graphs.