Self-Doping of Gold Chains on Silicon: A New Structural Model for Si(111)5x2-Au

A new structural model for the Si(111)5x2-Au reconstruction is proposed and analyzed using first-principles calculations. The basic model consists of a "double honeycomb chain" decorated by Si adatoms. The 5x1 periodicity of the honeycomb chains is doubled by the presence of a half-occupied row of Si atoms that partially rebonds the chains. Additional adatoms supply electrons that dope the parent band structure and stabilize the period doubling; the optimal doping corresponds to one adatom per four 5x2 cells, in agreement with experiment. All the main features observed in scanning tunneling microscopy and photoemission are well reproduced.Comment: 4 pages, 4 figures, to appear in Phys. Rev. Lett. (preprint with high quality figures available at http://cst-www.nrl.navy.mil/~erwin/papers/ausi111

Singer quadrangles

[no abstract available

Ovoids and spreads of finite classical generalized hexagons and applications

One intuitively describes a generalized hexagon as a point-line geometry full of ordinary hexagons, but containing no ordinary n-gons for n<6. A generalized hexagon has order (s,t) if every point is on t+1 lines and every line contains s+1 points. The main result of my PhD Thesis is the construction of three new examples of distance-2 ovoids (a set of non-collinear points that is uniquely intersected by any chosen line) in H(3) and H(4), where H(q) belongs to a special class of order (q,q) generalized hexagons. One of these examples has lead to the construction of a new infinite class of two-character sets. These in turn give rise to new strongly regular graphs and new two-weight codes, which is why I dedicate a whole chapter on codes arising from small generalized hexagons. By considering the (0,1)-vector space of characteristic functions within H(q), one obtains a one-to-one correspondence between such a code and some substructure of the hexagon. A regular substructure can be viewed as the eigenvector of a certain (0,1)-matrix and the fact that eigenvectors of distinct eigenvalues have to be orthogonal often yields exact values for the intersection number of the according substructures. In my thesis I reveal some unexpected results to this particular technique. Furthermore I classify all distance-2 and -3 ovoids (a maximal set of points mutually at maximal distance) within H(3). As such we obtain a geometrical interpretation of all maximal subgroups of G2(3), a geometric construction of a GAB, the first sporadic examples of ovoid-spread pairings and a transitive 1-system of Q(6,3). Research on derivations of this 1-system was followed by an investigation of common point reguli of different hexagons on the same Q(6,q), with nice applications as a result. Of these, the most important is the alternative construction of the Hölz design and a subdesign. Furthermore we theoretically prove that the Hölz design on 28 points only contains Hermitian and Ree unitals (previously shown by Tonchev by computer). As these Hölz designs are one-point extensions of generalized quadrangles, we dedicate a final chapter to the characterization of the affine extension of H(2) using a combinatorial property

Mini-Workshop: Amalgams for Graphs and Geometries

[no abstract available

Benson\u27s Theorem for Partial Geometries

In 1970 Clark Benson published a theorem in the Journal of Algebra stating a congruence for generalized quadrangles. Since then this theorem has been expanded to other specific geometries. In this thesis the theorem for partial geometries is extended to develop new divisibility conditions for the existence of a partial geometry in Chapter 2. Then in Chapter 3 the theorem is applied to higher dimensional arcs resulting in parameter restrictions on geometries derived from these structures. In Chapter 4 we look at extending previous work with partial geometries with α = 2 to uncover potential partial geometries with higher values of α. Finally the theorem is extended to strongly regular graphs in Chapter 5. In addition we obtain expressions for the multiplicities of the eigenvalues of matrices related to the adjacency matrices of these graphs. Finally, a four lesson high school level enrichment unit is included to provide students at this level with an introduction to partial geometries, strongly regular graphs, and an opportunity to develop proof skills in this new context

A geometric construction of panel-regular lattices in buildings of types ~A_2 and ~C_2

Using Singer polygons, we construct locally finite affine buildings of types ~A_2 and ~C_2 which admit uniform lattices acting regularly on panels. This construction produces very explicit descriptions of these buildings as well as very short presentations of the lattices. All but one of the ~C_2-buildings are necessarily exotic. To the knowledge of the author, these are the first presentations of lattices in buildings of type ~C_2. Integral and rational group homology for the lattices is also calculated.Comment: 42 pages, small corrections and cleanup. Results are unchanged

A single cell based model for cell divisions with spontaneous topology changes

The development of multicellular organisms is accompanied by the formation of tis- sues of precise shapes, sizes and topologies. Remarkable similarities between tissue topologies, in particular proliferating epithelial topologies, in various species suggest that the mechanisms that govern the formation of tissues are conserved among species. To understand these mechanisms various models have been developed. In this thesis, we present a novel mechanical model for cell divisions and tissue for- mation. The model accounts for cell mechanics and cell-cell adhesion. In our model, each cell is treated individually, thus the changes in cell’s shape and its local rearrange- ments occur naturally as a response to the evolving cellular environment and cell-cell interactions. We introduce the processes of cell growth and divisions and numerically simulate tissue proliferation. As tissue grows starting from few cells, we follow the dynamics of the tissue growth and cell packing topologies. The outcomes are com- pared with experimental observations in Drosophila wing growth. Our model accounts for the exponential decay of the mitotic index and reproduces commonly observed cell packing topologies in proliferating epithelia. Next, we consider two biologically relevant division schemes, namely, division through asymmetric division plane and division by Hertwig’s rule. We study the im- pact of division planes on tissue growth and show that the division plane may affect cell packing topologies. Development of the tissue is accompanied by cellular rearrange- ments. We vary the extent of cellular rearrangements and analyse their effects on tissue topology. We find that when cells are allowed to move freely, more organized packing topologies emerge