Title: Approximate Polynomial Greatest Common Divisor Author: Ján Eliaš Department: Department of Numerical Mathematics, MFF UK Supervisor: Doc. RNDr. Jan Zítko, CSc., Department of Numerical Mathematics, MFF UK Abstract: The computation of polynomial greatest common divisor (GCD) ranks among basic algebraic problems with many applications. The Euclidean algorithm is the oldest and usual technique for computing GCD. However, the GCD computation problem is ill-posed, particularly when some unknown noise is applied to the polyno- mial coefficients. Since the Euclidean algorithm is unstable, new methods have been extensively studied in recent years. Methods based on the numerical rank estimation represent one group of current meth- ods. Their disadvantage is that the numerical rank cannot be computed reliably due to the sensitivity of singular values on noise. The aim of the work is to overcome the ill-posed sensitivity of GCD computation in the presence of noise. Keywords: AGCD, Sylvester matrix, numerical rank, TLS

In coding theory we wish to find as many codewords as possible, while simultaneously maintaining high distance between codewords to ease the detection and correction of errors. For linear codes, this translates to finding high-dimensional subspaces of a given metric space, where the induced distance between vectors stays above a specified minimum. In this work I describe the recent advances of this problem in the contexts of lexicodes and Ferrers diagram rank-metric codes. In the first chapter, we study lexicodes. For a ring R, we describe a lexicographic ordering of the left R-module Rn. With this ordering we set up a greedy algorithm which sequentially selects vectors for which all linear combinations satisfy a given property. The resulting output is called a lexicode. This process was discussed earlier in the literature for fields and chain rings. We describe a generalization of the algorithm to finite principal ideal rings. In the second chapter, we investigate Ferrers diagram rank-metric codes, which play a role in the construction of subspace codes. A well-known upper bound for dimension of these codes is conjectured to be sharp. We describe several solved cases of the conjecture, and further contribute new ones. In addition, probabilities for maximal Ferrers diagram codes and MRD codes are investigated in a new light. It is shown that for growing field size, the limiting probability depends highly on the Ferrers diagram.

The main focus of this paper is the study of elliptic curves, non-singular projective curves of genus 1. Under a geometric operation, the rational points E(Q) of an elliptic curve E form a group, which is a finitely-generated abelian group by Mordell’s theorem. Thus, this group can be expressed as the finite direct sum of copies of Z and finite cyclic groups. The number of finite copies of Z is called the rank of E(Q). From John Tate and Joseph Silverman we have a formula to compute the rank of curves of the form E: y2 = x3 + ax2 + bx. In this thesis, we generalize this formula, using a purely group theoretic approach, and utilize this generalization to find the rank of curves of the form E: y2 = x3 + c. To do this, we review a few well-known homomorphisms on the curve E: y2 = x3 + ax2 + bx as in Tate and Silverman's Elliptic Curves, and study analogous homomorphisms on E: y2 = x3 + c and relevant facts.

Microarrays have been widely used to study differential gene expression at the genomic level. They can also provide genome-wide co-expression information. Robust approaches are needed for integration and validation of independently-collected datasets which may contribute to a common hypothesis. Previously, attempts at meta-analysis have contributed to solutions to this problem. As an alternative, for microarray data from multiple highly similar biological experimental designs, a more direct combined approach is possible. In this thesis, a novel approach is described for microarray combined analysis, including gene-level unification into a virtual platform followed by normalization and a method for ranking candidate genes based on co-expression information - called Coex-Rank. We applied this approach to our Sppar (a PPARγ mutant) dataset, which illustrated an improvement in statistical power and a complementary advantage of the Coex-Rank method from a biological perspective. We also performed analysis to other PPARγ-related microarray datasets. From the perspective of gene sets, we observed that up-regulated genes from mice treated with the PPARγ ligand rosiglitazone were significantly down-regulated in mice with a global knock-in dominant-negative mutation of PPARγ. Integrated with publicly available PPRE (PPAR Response Element) datasets, we found that the genes which were most up-regulated by rosiglitazone treatment and which were also down-regulated by the global knock-in mutation of PPARγ were robustly enriched in PPREs near transcription start sites. In addition, we identified several potential PPARγ targets in the aorta and mesenteric artery for further experimental validation, such as Rhobtb1 and Rgs5.

Vectors and matrices over the Boolean (0,1) semiring have been studied extensively along with their applications to graph theory. The Boolean (0,1) semiring has been generalized to a class of semirings called chain semirings. This class includes the fuzzy interval. Vectors and matrices over chain semirings are examined. Rank-1 sets of vectors are defined and characterized. These rank-1 sets of vectors are then used to construct spaces of matrices (rank-1 spaces) with the property that all nonzero matrices in the space have semiring rank equal to 1. Finally, three classes of maximal (relative to containment) rank-1 spaces are identified.

Demand for domestic tobacco has decreased over the past two decades. In 2004, the tobacco buyout program terminated marketing quotas and price support established under the federal tobacco program in 1938. Additionally, in 2003, the World Health Organization Framework Convention on Tobacco Control (WTO FCTC) acknowledged the importance of supply control in conjunction with demand control strategies to eliminate or reduce the consumption of tobacco products. According to the Census of Agriculture, the number of tobacco farms in the US fell by 40 percent between 2002 and 2007. Tobacco farmers are looking for alternative on-farm and off-farm sources of income. This study uses a rank-ordered logit model (ROLM) to explore factors affecting farmers’ perceptions about the potential for grain crops, cotton, peanuts, hay, fruits and vegetables, cow/calf, dairy, beef cattle and, other crops/livestock to replace tobacco production. Results suggest that hay is one of the on-farm enterprises perceived as having the highest potential to replace tobacco among burley tobacco farmers. Age, education, farm size and farm cash receipts were found to affect farmers’ perceptions about the potential for different alternative enterprises to replace tobacco. Additionally, results suggest that researchers should be careful when designing ranking questions in order to maximize rate of response and quality of the data obtained from this type of questions.

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Ten years ago, J. Villadsen constructed the first examples of simple C*-algebras with stable rank other than one or infinity. Villadsen's examples all had a unique tracial state. It is natural to ask whether examples can be found of simple C*-algebras with higher stable rank and more than one tracial state; by building on Villadsen's construction, we describe such examples that admit arbitrary tracial state spaces.

Ten years ago, J. Villadsen constructed the first examples of simple C*-algebras with stable rank other than one or infinity. Villadsen's examples all had a unique tracial state. It is natural to ask whether examples can be found of simple C*-algebras with higher stable rank and more than one tracial state; by building on Villadsen's construction, we describe such examples that admit arbitrary tracial state spaces.