I will be going over writing representations of both permutation and monomial progenitors, which include 2^{*4} : D_4, 2^(*7) :L_2 (7) as permutation progenitors, and monomial progenitors 7^(*2) :_m S_3 \times 2, 11^{*2} :_m (5:2)^{*}5, 11^{*3} :_m (25:3), 11^{*4} :_m (4 : 5)^{*}5. Also, the images of these different progenitors at both lower and higher fields and orders. \\ We will also do the double coset enumeration of S5 over D6, S6 over 5 : 4, A_5 x A_5 over (5:2)^{*}5, and go on to also do the double coset enumeration over maximal subgroups for larger constructions. We will also do the construction of sporadic group M22 over maximal subgroup A7, and also J1 with the monomial representation 7^(*2) :_m S_3 \times 2 over maximal subgroup PSL(2,11). We will also look at different extension problems of composition factors of different groups, and determine the isomorphism types of each extension.

This dissertation develops and analyzes differential equation-based mathematical models and efficient numerical methods and algorithms for genetic regulatory network identification. The primary objectives of the dissertation are to design, analyze, and test a general variational framework and numerical methods for seeking its approximate solutions for reverse engineering genetic regulatory networks from microarray datasets using the approach based on differential equation modeling. In the proposed variational framework, no structure assumption on the genetic network is presumed, instead, the network is solely determined by the microarray profile of the network components and is identified through a well chosen variational principle which minimizes a biological energy functional. The variational principle serves not only as a selection criterion to pick up the right biological solution of the underlying differential equation model but also provide an effective mathematical characterization of the small-world property of genetic regulatory networks which has been observed in lab experiments. Five specific models within the variational framework and efficient numerical methods and algorithms for computing their solutions are proposed and analyzed in the dissertation. Model validations using both synthetic network datasets and real world subnetwork datasets of Saccharomyces cerevisiae (yeast) and E. Coli are done on all five proposed variational models and a performance comparison vs some existing genetic regulatory network identification methods is also provided. As microarray data is typically noisy, in order to take into account the noise effect in the mathematical models, we propose a new approach based on stochastic differential equation modeling and generalize the deterministic variational framework to a stochastic variational framework which relies on stochastic optimization. Numerical algorithms are also proposed for computing solutions of the stochastic variational models. To address the important issue of post-processing computed networks to reflect the small-world property of underlying genetic regulatory networks, a novel threshholding technique based on the Random Matrix Theory is proposed and tested on various synthetic network datasets.

We performed ab initio studies of the electronic excitation spectra of the ferro- magnetic, Mott-insulator YTiO3 using density functional theory (DFT) and time- dependent density functional theory (TDDFT). In the ground state description, we included a Hubbard U to account for the strong correlations present within the d states on the cation. The excitation spectra was calculated using TDDFT linear response formalism in both the optical limit and the limit of large wavevector transfer. In order to identify the local d-d transitions in the response, we also computed the density response of YTiO3 using a novel technique where the basis included Wannier functions generated for the Ti and Y sites. Also, we describe the first implementation of the all-electron Kohn-Sham density functional equations in a periodic system using multi-wavelets and fast integral equations using MADNESS (multiresolution adaptive numerical environment for scientific simulation; http://code.google.com/p/m-a-d-n- e-s-s). This implementation is highlighted by the real space lattice sums involved in the application of the Coulomb and bound state Helmlholtz integral operators.

This paper provides an overview of the finite difference method and its application to approximating financial partial differential equations (PDEs) in incomplete markets. In particular, we study German’s [6] stochastic volatility PDE derived from indifference pricing. In [6], it is shown that the first order- correction to derivatives valued by indifference pricing can be computed as a function involving the stochastic volatility PDE itself. In this paper, we present three explicit finite difference models to approximate the stochastic volatility PDE and compare the resulting valuations to those generated by an Euler- Maruyama Monte Carlo pricing algorithm. We also discuss the significance of boundary condition choice for explicit finite difference models.

This thesis consists of six chapters. In the first chapter, we review some basic definitions and concepts of fractional calculus. Then we introduce fractional difference equations involving the Riemann-Liouville operator of real number order between zero and one. In the second chapter, we apply the Brouwer fixed point and Contraction Mapping Theorems to prove that there exists a solution for up to the first order nabla fractional difference equation with an initial condition. In chapter three, we define a lower and an upper solution for up to the first order nabla fractional difference equation with an initial condition. Under certain assumptions we prove that a lower solution stays less than an upper solution. Some examples are given to illustrate our findings in this chapter. Then we give constructive proofs of existence of a solution by defining monotone sequences. In the fourth chapter, we derive a continuous form of the Mittag-Leffler function. Then we use successive approximations method to calculate a discrete form of the Mittag-Leffler function. In the fifth chapter, we focus on finding the model which fits best for the data of tumor growth for twenty-eight mice. The models contain either three parameters (Gompertz, Logistic) or four parameters (Weibull, Richards). For each model, we consider continuous, discrete, continuous fractional and discrete fractional forms. Nihan Acar who is a former graduate student in mathematics department has already worked on Gompertz and Logistic models [1]. Here we continue and work on Richards curve. The difference between Acar’s work and ours is the number of parameters in each model. Gompertz and Logistic models contain three parameters and an alpha parameter. The Richards model has four parameters and an alpha parameter. In addition, we use statistical computation techniques such as residual sum of squares and cross-validation to compare fitting and predictive performance of these models. In conclusion, we put three models together to conclude which model is fitting best for the data of tumor growth for twenty-eight mice. In the last chapter, we conclude this thesis and state our future work.

The problem of finding an efficient algorithm to create a two-dimensional embedding of a knot diagram is not an easy one. Typically, knots with a large number of crossings will not nicely generate two-dimensional drawings. This thesis presents an efficient algorithm to generate a knot and to create a nice two-dimensional embedding of the knot. For the purpose of this thesis a drawing is “nice” if the number of tangles in the diagram consisting of half-twists is minimal. More specifically, the algorithm generates prime, alternating Conway algebraic knots in O(n) time where n is the number of crossings in the knot, and it derives a precise representation of the knot’s nice drawing in O(n) time (The rendering of the drawing is not O(n).). Central to the algorithm is a special type of rooted binary tree which represents a distinct prime, alternating Conway algebraic knot. Each leaf in the tree represents a crossing in the knot. The algorithm first generates the tree and then modifies such a tree repeatedly to reduce the number of its leaves while ensuring that the knot type associated with the tree is not modified. The result of the algorithm is a tree (for the knot) with a minimum number of leaves. This minimum tree is the basis of deriving a 4-regular plane map which represents the knot embedding and to finally draw the knot’s diagram.

This thesis investigated a method of barcode reconstruction to address the recovery of a blurred and convoluted one-dimensional barcode. There are a lot of types of barcodes used today, such as Code 39, Code 93, Code 128, etc. Our algorithm applies to the universal barcode, EAN 13. We extend the methodologies proposed by Iwen et al. (2013) in the journal article "A Symbol-Based Algorithm for Decoding barcodes." The algorithm proposed in the paper requires a signal measured by a laser scanner as an input. The observed signal is modeled as a true signal corrupted by a Gaussian convolution, additional noises, and an unknown multiplier. The known barcode dictionaries were incorporated into the forward map between the true barcode and the observed barcode. Unlike the one proposed by Iwen et al., we take dictionaries of different patterns into account, specifically for decoding book barcodes from images which are captured with smartphones. We also presented numerical experiments that examined the performance of the proposed algorithm and illustrated that the unique determination of barcode digits is possible even in the presence of noise.

Fourier pseudo-spectral method on equispaced grid is one of the approaches in turbulence simulation, to compute derivative of discrete data, using fast Fourier Transform (FFT) and gives low dispersion and dissipation errors. In many turbulent flows the dynamically important scales of motion are concentrated in certain regions which requires a coarser grid for higher accuracy. A coarser grid in other regions minimizes the memory requirement. This requires the use of Non-equispaced Fast Fourier Transform (NFFT) to compute the Fourier transform, by solving a system of linear equations. To achieve similar accuracy, the NFFT needs to return more Fourier coefficients than the number of non-equispaced gridpoints, making it an under-determined system. The minimum L2 norm solution of the system is refined using an iterative reconstruction algorithm, FOCUSS. The NFFT and FOCUSS algorithms yield accurate results with smaller test case of a Direct Numerical Simulation on a grid of 64 gridpoints in each dimension, using Taylor green initial condition. The computational speed for this case was found to be unacceptably slow and few methods to improve the performance have been discussed. The approach of NFFT and FOCUSS was tested on a line extracted from 3-dimensional turbulent flow field. Fourier transform of the extracted line, sampled on 1024 non-equispaced gridpoints, computed for 2048 coefficients and the corresponding numerical derivative are found to be inaccurate. It can be observed that the NFFT and FOCUSS approach works for sparse Fourier transform, but not for turbulent fields having a wideband Fourier transform.

Many practical problems in the field of engineering become so complex that they may be effectively solved only with the aid of a computer. An effective solution depends on the use of an efficient algorithm. Plotting root locus diagrams is such a problem. This thesis presents such an algorithm. Root locus design of feedback control systems is a very powerful tool. Stability of systems under the influence of variables can be easily determined from the root locus diagram. For even moderately complex systems of the type found in practical applications, determination of the locus is extremely difficult if accuracy is required. The difficulty lies in the classical method of graphically determining the location of points on the locus by trial and error. Such a method cannot be efficiently applied to a computer program. The text presents an original algorithm for plotting the root locus of a general system. The algorithm is derived using the combined methods of complex variable algebra and numerical analysis. For each abscissa desired a polynomial is generated. The real roots of this polynomial are the ordinate values for points on the root locus. Root finding methods from numerical analysis enable the solution of the problem to be one of convergent iteration rather than trial and error. Among the material presented is a computer program for solution of the general problem, an example of a completely analytic solution, and a table of solutions for more simple systems. The program inputs are the coefficients of the open loop transfer function and the range and increments of the real axis which are to be swept. The output lists the real and imaginary components of all solution points at each increment of the sweep. Also listed are the magnitude and angle components of the solution point and the value of system gain for which this is a solution. For less complex problems, the method can be applied analytically. This may result in an explicit relation between the real and imaginary components of all solution points or even in a single expression which can be analyzed using the methods of analytic geometry. As with any advance in the theory of problem solving, the ideas presented in the thesis are best applied in conjunction with previous solution methods. Specifically, an idea of the approximate location of the root locus can be obtained using sketching rules which are well known. The method presented here becomes much more efficient when even a rough approximation is known. Furthermore, the specific locations of system poles and zeros are not required, but can be helpful in planning areas in which to search for solutions.

In this thesis, we study how sovereign credit default swaps are able to measure systemic risk as well as how they can be used to construct optimal portfolios to minimize risk. We define the clustering coefficient as a proxy for systemic risk and design an optimization problem with the goal of minimizing the mean absolute deviation of the clustering coefficient on a group of nine European countries. Additionally, we define a metric we call the diversity score that measures the diversification of any given portfolio. We solve this problem for a baseline set of parameters, then spend the remainder of the thesis modifying these parameters to investigate how the optimal solution and diversity score are impacted.