In this project, we demonstrate our discovery of original symmetric presentations and constructions of important groups, including nonabelian simple groups, and groups that have these as factor groups. The target nonabelian simple groups include alternating, linear, and sporadic groups. We give isomorphism types for each finite homomorphic image that has been found. We present original symmetric presentations of $M_{12}$, $M_{21}:(2 \times 2)$, $L_{3}(4):2^2$, $2:^{\cdot}L_{3}(4):2$, $S(4,3)$, and $S_{7}$ as homomorphism images of the progenitors $2^{*20}$ $:$ $A_{5}$, $2^{*10}$ $:$ $PGL(2,9)$, $2^{*10}$ $:$ $Aut(A_{6})$, $2^{*10}$ $:$ $A_{6}$, $2^{*10}$ $:$ $A_{5}$, and $2^{*24}$ $:$ $S_{5}$, respectively. We also construct $M_{12}$, $M_{21}:(2 \times 2)$, $L_{3}(4):2^2$, $L_{3}(4):2^2$, $2:^{\cdot}L_{3}(4):2$, $S(4,3)$, and $S_{7}$ over $A_{5}$, $PGL(2,9)$, $Aut(A_{6})$, $A_{6}$, $A_{5}$, and $S_{5}$, respectively, using our technique of double coset enumeration. All of the symmetric presentations given are original to the best of our knowledge.

In this thesis, we consider approaches to enumeration problems in the parameterized complexity setting. We obtain competitive parameterized algorithms to enumerate all, as well as several of, the solutions for two related problems Neighbour String and Kemeny Rank Aggregation. In both problems, the goal is to find a solution that is as close as possible to a set of inputs (strings and total orders, respectively) according to some distance measure. We also introduce a notion of enumerative kernels for which there is a bijection between solutions to the original instance and solutions to the kernel, and provide such a kernel for Kemeny Rank Aggregation, improving a previous kernel for the problem. We demonstrate how several of the algorithms and notions discussed in this thesis are extensible to a group of parameterized problems, improving published results for some other problems.

We study the leading order asymptotics of a Random Matrix theory partition function related to colored triangulations. This partition function comes from a three Hermitian matrix model that has been introduced in the physics literature. We provide a detailed and precise description of the combinatorial objects that the partition function counts that has not appeared previously in the literature. We also provide a general framework for studying the leading order asymptotics of an N dimensional integral that one encounters studying the partition function of colored triangulations. The results are obtained by generalizing well know results for integrals coming from Hermitian matrix models with only one matrix that give the leading order asymptiotics in terms of a finite dimensional variational problem. We apply these results to the partition function for colored triangulations to show that the minimizing density of the variational problem is unique, and agrees with the one proposed in the physics literature. This provides the first complete mathematically rigorous description of the leading order asymptotics of this matrix model for colored triangulations.

In this thesis, we consider approaches to enumeration problems in the parameterized complexity setting. We obtain competitive parameterized algorithms to enumerate all, as well as several of, the solutions for two related problems Neighbour String and Kemeny Rank Aggregation. In both problems, the goal is to find a solution that is as close as possible to a set of inputs (strings and total orders, respectively) according to some distance measure. We also introduce a notion of enumerative kernels for which there is a bijection between solutions to the original instance and solutions to the kernel, and provide such a kernel for Kemeny Rank Aggregation, improving a previous kernel for the problem. We demonstrate how several of the algorithms and notions discussed in this thesis are extensible to a group of parameterized problems, improving published results for some other problems.

Thesis (M.S.)--West Virginia University, 2005. / Title from document title page. Document formatted into pages; contains iii, 33 p. Includes abstract. Includes bibliographical references (p. 33).

Thesis (M.S.)--Rochester Institute of Technology, 2009. / Typescript. Includes bibliographical references (leaves 42-43).

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A new algorithm for the enumeration of derivative superstructures of a crystal is presented. The algorithm will help increase the efficiency of computational material design methods such as cluster expansion by increasing the size and diversity of the types of systems that can be modeled. Modeling potential alloys requires the exploration of all possible configurations of atoms. Additionally, modeling the thermal properties of materials requires knowledge of the possible ways of displacing the atoms. One solution to finding all symmetrically unique configurations and displacements is to generate the complete list of possible configurations and remove those that are symmetrically equivalent. This approach, however, suffers from the combinatoric explosion that happens when the supercell size is large, when there are more than two atom types, or when atomic displacements are included in the system. The combinatoric explosion is a problem because the large number of possible arrangements makes finding the relatively small number of unique arrangements for these systems impractical. The algorithm presented here is an extension of an existing algorithm [Hart & Forcade (2008a), Hart & Forcade (2009a), Hart et al. (2012a) Hart, Nelson, & Forcade] to include the extra configurational degree of freedom from the inclusion of displacement directions. The algorithm makes use of another recently developed algorithm for the Pólya [Pólya & Read (1987), Pólya (1937), Rosenbrock et al.(2015) Rosenbrock, Morgan, Hart, Curtarolo, & Forcade] counting theorem to inform the user of the total number of unique arrangements before performing the enumeration and to ensure that the list of unique arrangements will fit in system memory. The algorithm also uses group theory to eliminate large classes of arrangements rather than eliminating arrangements one by one. The three major topics of this paper will be presented in this order, first the Pólya algorithm, second the new algorithm for eliminating duplicate structures, and third the algorithms extension to include displacement directions. With these tools, it is possible to avoid the combinatoric explosion and enumerate previously inaccessible systems, including those that contain displaced atoms.

In this report, we study the enumeration of spanning trees in graphs, using two methods withinlinear algebra, Kirchhoff’s Matrix Tree Theorem and an alternative method, also referred to asLemma 1, derived by S. Klee and M.T Stamps in [KS20]. Along with introducing preliminary tools from linear algebra, we also study the Laplace matrix ofa graph and use its properties in the two methods to derive combinatorical expressions on spanningtree enumeration of different graph families. We discuss properties of the Laplace matrix obtainedfrom different graph structures, and determine which method is more suitable in each case, withrespect to linear algebra. Specifically, complete graphs, Ferrers graphs and Windmill graphs areconsidered.

The current costs of drug discovery are extremely high and need to be addressed if diseases such as AIDS and malaria are to be combated. The major reasons for the high costs are the use of expensive in vitro methods and the high failure rate of drugs at clinical testing phases. In silico techniques hold tremendous potential in addressing the high costs. In silico drug design can be done at a fraction of the cost of in vitro techniques, and can be used in synergy with in vitro techniques by doing much of the screening before any experimental studies, thereby reducing the chemical space to be searched experimentally. In silico techniques can also enhance the quality of drug candidates sent to clinical phases, increasing the probability of success. In this study techniques were investigated to build analogous ligand libraries with scaffolds and molecular building blocks through a user guided process, including the development of the LIGLIB program, which is an Open Source package for lead development. The Markush molecular enumeration technique was implemented in C++ with a Python front-end extending it to the Python molecular visualization tool, Chimera. The software makes use of chemical graphs to make permutations according to user inputs, generating an output library in SMILES and Mol2 format, the later of which is generated by Corina. As part of the validation of the software it was used in a lead discovery experiment which targeted Plasmodium falciparum Glutathione S-transferase. The developed software was able to generate a series of suitable molecules, thereby validating the Markush molecular enumeration technique as well as its implementation in LIGLIB. / Dissertation (MSc (Bioinformatics))--University of Pretoria, 2007. / Biochemistry / unrestricted