Fixed block configuration GDDs with block size 6 and (3, r)-regular graphs

Chapter 1 is used to introduce the basic tools and mechanics used within this thesis. Most of the definitions used in the thesis will be defined, and we provide a basic survey of topics in graph theory and design theory pertinent to the topics studied in this thesis. In Chapter 2, we are concerned with the study of fixed block configuration group divisible designs, GDD(n; m; k; λ1; λ2). We study those GDDs in which each block has configuration (s; t), that is, GDDs in which each block has exactly s points from one of the two groups and t points from the other. Chapter 2 begins with an overview of previous results and constructions for small group size and block sizes 3, 4 and 5. Chapter 2 is largely devoted to presenting constructions and results about GDDs with two groups and block size 6. We show the necessary conditions are sufficient for the existence of GDD(n, 2, 6; λ1, λ2) with fixed block configuration (3; 3). For configuration (1; 5), we give minimal or nearminimal index constructions for all group sizes n ≥ 5 except n = 10, 15, 160, or 190. For configuration (2, 4), we provide constructions for several families ofGDD(n, 2, 6; λ1, λ2)s. Chapter 3 addresses characterizing (3, r)-regular graphs. We begin with providing previous results on the well studied class of (2, r)-regular graphs and some results on the structure of large (t; r)-regular graphs. In Chapter 3, we completely characterize all (3, 1)-regular and (3, 2)-regular graphs, as well has sharpen existing bounds on the order of large (3, r)- regular graphs of a certain form for r ≥ 3. Finally, the appendix gives computational data resulting from Sage and C programs used to generate (3, 3)-regular graphs on less than 10 vertices.

Edge coloring BIBDS and constructing MOELRs

Chapter 1 is used to introduce the basic tools and mechanics used within this thesis. Some historical uses and background are touched upon as well. The majority of the definitions are contained within this chapter as well. In Chapter 2 we consider the question whether one can decompose λ copies of monochromatic Kv into copies of Kk such that each copy of the Kk contains at most one edge from each Kv. This is called a proper edge coloring (Hurd, Sarvate, [29]). The majority of the content in this section is a wide variety of examples to explain the constructions used in Chapters 3 and 4. In Chapters 3 and 4 we investigate how to properly color BIBD(v, k, λ) for k = 4, and 5. Not only will there be direct constructions of relatively small BIBDs, we also prove some generalized constructions used within. In Chapter 5 we talk about an alternate solution to Chapters 3 and 4. A purely graph theoretical solution using matchings, augmenting paths, and theorems about the edgechromatic number is used to develop a theorem that than covers all possible cases. We also discuss how this method performed compared to the methods in Chapters 3 and 4. In Chapter 6, we switch topics to Latin rectangles that have the same number of symbols and an equivalent sized matrix to Latin squares. Suppose ab = n2. We define an equitable Latin rectangle as an a × b matrix on a set of n symbols where each symbol appears either [b/n] or [b/n] times in each row of the matrix and either [a/n] or [a/n] times in each column of the matrix. Two equitable Latin rectangles are orthogonal in the usual way. Denote a set of ka × b mutually orthogonal equitable Latin rectangles as a k–MOELR(a, b; n). We show that there exists a k–MOELR(a, b; n) for all a, b, n where k is at least 3 with some exceptions.

IDENTIFICATION OF GENES CONTROLLING BIOLOGICAL PROCESSES AND PATHWAYS THROUGH STATISTICAL ANALYSIS AND NETWORK RECONSTRUCTION

The developmental processes and functions of an organism are controlled by the genes and the proteins that are derived from these genes. The identification of key genes and the reconstruction of gene networks can provide a model to help us understand the regulatory mechanisms for the initiation and progression of biological processes or functional abnormalities (e.g. diseases) in living organisms. In this dissertation, I have developed statistical methods to identify the genes and transcription factors (TFs) involved in biological processes, constructed their regulatory networks, and also evaluated some existing association methods to find robust methods for coexpression analyses. Two kinds of data sets were used for this work: genotype data and gene expression microarray data. On the basis of these data sets, this dissertation has two major parts, together forming six chapters. The first part deals with developing association methods for rare variants using genotype data (chapter 4 and 5). The second part deals with developing and/or evaluating statistical methods to identify genes and TFs involved in biological processes, and construction of their regulatory networks using gene expression data (chapter 2, 3, and 6). For the first part, I have developed two methods to find the groupwise association of rare variants with given diseases or traits. The first method is based on kernel machine learning and can be applied to both quantitative as well as qualitative traits. Simulation results showed that the proposed method has improved power over the existing weighted sum method (WS) in most settings. The second method uses multiple phenotypes to select a few top significant genes. It then finds the association of each gene with each phenotype while controlling the population stratification by adjusting the data for ancestry using principal components. This method was applied to GAW 17 data and was able to find several disease risk genes. For the second part, I have worked on three problems. First problem involved evaluation of eight gene association methods. A very comprehensive comparison of these methods with further analysis clearly demonstrates the distinct and common performance of these eight gene association methods. For the second problem, an algorithm named the bottom-up graphical Gaussian model was developed to identify the TFs that regulate pathway genes and reconstruct their hierarchical regulatory networks. This algorithm has produced very significant results and it is the first report to produce such hierarchical networks for these pathways. The third problem dealt with developing another algorithm called the top-down graphical Gaussian model that identifies the network governed by a specific TF. The network produced by the algorithm is proven to be of very high accuracy.

Syntheses and structures of molybdenum and tungsten complexes capable of epoxidaton and copper coordination polymers and dendrimers

We are interested in the syntheses of new complexes and in their characterization by single crystal X-ray diffraction techniques. Once we understand the structures, studies aimed at understanding uses of these complexes in the field of catalytic epoxidation using complexes soluble in water and syntheses of thin films (not assessed) were conducted. The syntheses, characterization and catalytic properties of a series of mononuclear, dinuclear and tetranuclear molybdenum and tungsten oxo complexes are described. The syntheses and structural characterization of two copper coordination polymers with 3,5-dihydroxylbenzoate ligand, and five paddlewheel shaped copper dendrimers coordinated with Fréchet-type dendrons are also detailed. The background of this dissertation is outlined in Chapter 1. Chapter 2 describes the syntheses, and characterization of two new mononuclear molybdenum(VI) and tungsten(VI) oxo complexes, MoO2Cl2(OPPh2CH2OH)2, and WO2Cl2(OPPh2CH2OH)2, bearing hydrophilic phosphine oxide ligand. The catalytic properties of these complexes for the epoxidation of cis-cyclooctene were also studied. Two new dinuclear molybdenum(VI) and tungsten(VI) oxo complexes Mo2O4Cl2[(HOCH2)PhPOO]2, and (CH3O)2(O)W(μ-O)(μ-O2PPh2)2W(O)(CH3O)2, bearing organophosphinate ligand are described in Chapter 3 and 4. Chapter 4 and 5 describes the syntheses and characterization of tetranuclear molybdenum(V) oxo complexes bearing various organophosphinate ligands. The catalytic abilities of these complexes for the epoxidation of cis-cyclooctene in the presence of hydrogen peroxide as oxidant were explored as well. Various spectroscopic methods, such as IR, UV-vis, and NMR are used to characterize the nature of these complexes. Crystal structures of compounds MoO2Cl2(OPPh2CH2OH)2, WO2Cl2(OPPh2CH2OH)2, Mo2O4Cl2[(HOCH2)PhPOO]2, (CH3O)2(O)W(μ-O)(μ-O2PPh2)2W(O)(CH3O)2, and Mo4(µ3-O)4(µ-O2PR2)4O4 (R=Ph, Me, ClCH2, o-C6H4(CH2)2) are also presented. The syntheses, and structural characterization of three copper(II) coordination polymers bearing 3,5-dihydroxybenzoate ligand are described in Chapter 6. Two copper(II) coordination polymers, [Cu2(3,5-dhb)2(pyridine)4]n, and [Cu2(3,5-dhb)4]n were afforded based on different amount of pyridine used in the reaction. The structures of these complexes are further built into 2D or 3D networks via inter or intra hydrogen bonds. The syntheses and structural characterization of the zinc(II) monomer, Zn(3,5-dhb)2(pyridine)2 is also described in this Chapter. Chapter 7 describes the syntheses, and characterization of five dendronized dicopper complexes bearing different generations of Fréchet-type dendrons. The structures of 3,5- bis(benzoyloxl)benzoic acid, 3,5-(PhCOO)2PhCOOH (G1), Cu2(3,5-dhb)4(THF)2, Cu2(G1)4(pyridine)2, and Cu2(G1)4(CH3OH)2 were characterized unambiguously by single X-ray diffraction. In addition, all compounds were characterized by FT-IR, UV-vis spectroscopy and elemental analyses.