The Distribution of the Irreducibles in an Algebraic Number Field

The objective of this thesis is to study the distribution of the number of principal ideals generated by an irreducible element in an algebraic number field, namely in the non-unique factorization ring of integers of such a field. In particular we are investigating the size of M(x), defined as M ( x ) =∑ (α) α irred.|N (α)|≤≠ 1, where x is any positive real number and N (α) is the norm of α. We finally obtain asymptotic results for hl(x).

L'Hopital's Rule in Chemistry: Irreversible Morphing into Reversible Isothermal Expansions

L'Hopital's Rule is discussed in the cvase of an irreversible isothermal expansion.

Pathway Semantics: An Algebraic Data Driven Algorithm to Generate Hypotheses about Molecular Patterns Underlying Disease Progression

The overarching goal of the Pathway Semantics Algorithm (PSA) is to improve the in silico identification of clinically useful hypotheses about molecular patterns in disease progression. By framing biomedical questions within a variety of matrix representations, PSA has the flexibility to analyze combined quantitative and qualitative data over a wide range of stratifications. The resulting hypothetical answers can then move to in vitro and in vivo verification, research assay optimization, clinical validation, and commercialization. Herein PSA is shown to generate novel hypotheses about the significant biological pathways in two disease domains: shock / trauma and hemophilia A, and validated experimentally in the latter. The PSA matrix algebra approach identified differential molecular patterns in biological networks over time and outcome that would not be easily found through direct assays, literature or database searches. In this dissertation, Chapter 1 provides a broad overview of the background and motivation for the study, followed by Chapter 2 with a literature review of relevant computational methods. Chapters 3 and 4 describe PSA for node and edge analysis respectively, and apply the method to disease progression in shock / trauma. Chapter 5 demonstrates the application of PSA to hemophilia A and the validation with experimental results. The work is summarized in Chapter 6, followed by extensive references and an Appendix with additional material.

The algebraic density property for affine toric varieties

In this paper we generalize the algebraic density property to not necessarily smooth affine varieties relative to some closed subvariety containing the singular locus. This property implies the remarkable approximation results for holomorphic automorphisms of the Andersén–Lempert theory. We show that an affine toric variety X satisfies this algebraic density property relative to a closed T-invariant subvariety Y if and only if X∖Y≠TX∖Y≠T. For toric surfaces we are able to classify those which possess a strong version of the algebraic density property (relative to the singular locus). The main ingredient in this classification is our proof of an equivariant version of Brunella's famous classification of complete algebraic vector fields in the affine plane.

On the approximate parametrization problem of algebraic curves.

The problem of parameterizing approximately algebraic curves and surfaces is an active research field, with many implications in practical applications. The problem can be treated locally or globally. We formally state the problem, in its global version for the case of algebraic curves (planar or spatial), and we report on some algorithms approaching it, as well as on the associated error distance analysis.

Orientation-dependent and sequence-specific expansions of CTG/CAG trinucleotide repeats in Saccharomyces cerevisiae

A quantitative and selective genetic assay was developed to monitor expansions of trinucleotide repeats (TNRs) in yeast. A promoter containing 25 repeats allows expression of a URA3 reporter gene and yields sensitivity to the drug 5-fluoroorotic acid. Expansion of the TNR to 30 or more repeats turns off URA3 and provides drug resistance. When integrated at either of two chromosomal loci, expansion rates were 1 × 10−5 to 4 × 10−5 per generation if CTG repeats were replicated on the lagging daughter strand. PCR analysis indicated that 5–28 additional repeats were present in 95% of the expanded alleles. No significant changes in CTG expansion rates occurred in strains deficient in the mismatch repair gene MSH2 or the recombination gene RAD52. The frequent nature of CTG expansions suggests that the threshold number for this repeat is below 25 in this system. In contrast, expansions of the complementary repeat CAG occurred at 500- to 1,000-fold lower rates, similar to a randomized (C,A,G) control sequence. When the reporter plasmid was inverted within the chromosome, switching the leading and lagging strands of replication, frequent expansions were observed only when CTG repeats resided on the lagging daughter strand. Among the rare CAG expansions, the largest gain in tract size was 38 repeats. The control repeats CTA and TAG showed no detectable rate of expansions. The orientation-dependence and sequence-specificity data support the model that expansions of CTG and CAG tracts result from aberrant DNA replication via hairpin-containing Okazaki fragments.

Synpolydactyly phenotypes correlate with size of expansions in HOXD13 polyalanine tract

Synpolydactyly (SPD) is a dominantly inherited congenital limb malformation. Typical cases have 3/4 finger and 4/5 toe syndactyly, with a duplicated digit in the syndactylous web, but incomplete penetrance and variable expressivity are common. The condition has recently been shown to be caused by expansions of an imperfect trinucleotide repeat sequence encoding a 15-residue polyalanine tract in HOXD13. We have studied 16 new and 4 previously published SPD families, with between 7 and 14 extra residues in the tract, to analyze the molecular basis for the observed variation in phenotype. Although there is no evidence of change in expansion size within families, even over six generations, there is a highly significant increase in the penetrance and severity of phenotype with increasing expansion size, affecting both hands (P = 0.012) and feet (P < 0.00005). Affected individuals from a family with a 14-alanine expansion, the largest so far reported, all have a strikingly similar and unusually severe limb phenotype, involving the first digits and distal carpals. Affected males from this family also have hypospadias, not previously described in SPD, but consistent with HOXD13 expression in the developing genital tubercle. The remarkable correlation between phenotype and expansion size suggests that expansion of the tract leads to a specific gain of function in the mutant HOXD13 protein, and has interesting implications for the role of polyalanine tracts in the control of transcription.

Evolution of vertebrate steroid receptors from an ancestral estrogen receptor by ligand exploitation and serial genome expansions

The evolution of novelty in tightly integrated biological systems, such as hormones and their receptors, seems to challenge the theory of natural selection: it has not been clear how a new function for any one part (such as a ligand) can be selected for unless the other members of the system (e.g., a receptor) are already present. Here I show—based on identification and phylogenetic analysis of steroid receptors in basal vertebrates and reconstruction of the sequences and functional attributes of ancestral proteins—that the first steroid receptor was an estrogen receptor, followed by a progesterone receptor. Genome mapping and phylogenetic analyses indicate that the full complement of mammalian steroid receptors evolved from these ancient receptors by two large-scale genome expansions, one before the advent of jawed vertebrates and one after. Specific regulation of physiological processes by androgens and corticoids are relatively recent innovations that emerged after these duplications. These findings support a model of ligand exploitation in which the terminal ligand in a biosynthetic pathway is the first for which a receptor evolves; selection for this hormone also selects for the synthesis of intermediates despite the absence of receptors, and duplicated receptors then evolve affinity for these substances. In this way, novel hormone-receptor pairs are created, and an integrated system of increasing complexity elaborated. This model suggests that ligands for some “orphan” receptors may be found among intermediates in the synthesis of ligands for phylogenetically related receptors.

Algebraic derivative estimation applied to nonlinear control of magnetic levitation.

The subject of this thesis is the real-time implementation of algebraic derivative estimators as observers in nonlinear control of magnetic levitation systems. These estimators are based on operational calculus and implemented as FIR filters, resulting on a feasible real-time implementation. The algebraic method provide a fast, non-asymptotic state estimation. For the magnetic levitation systems, the algebraic estimators may replace the standard asymptotic observers assuring very good performance and robustness. To validate the estimators as observers in closed-loop control, several nonlinear controllers are proposed and implemented in a experimental magnetic levitation prototype. The results show an excellent performance of the proposed control laws together with the algebraic estimators.

Modern algebraic theories,

Mode of access: Internet.

Algebraic topology.

Mode of access: Internet.

Selected topics in algebraic geometry; report of the Committee on rational transformations:

Bibliography at end of each chapter.

Selected topics in algebraic geometry

Cover-title.