Invariants of knots coming from twisted signatures have played a central role in the study of knot concordance. Unfortunately, except in the simplest of cases, these signature invariants have proven exceedingly difficult to compute. As a consequence, many knots which presumably can be detected by these invariants are not a well understood as they should be. We study a family of signature invariants of knots and show that they provide concordance information. Significantly, we provide a tractable means for computing these signatures. Once armed with these tools we use them first to study the knot concordance group generated by the twist knots which are of order 2 in the algebraic concordance group. With our computational tools we can show that with only finitely many exceptions, they form a linearly independent set in the concordance group. We go on to study a procedure given by Cochran-Harvey-Leidy which produces infinite rank subgroups of the knot concordance group which, in some sense are extremely subtle and difficult to detect. The construction they give has an inherent ambiguity due to the difficulty of computing some signature invariants. This ambiguity prevents their construction from yielding an actual linearly independent set. Using the tools we develop we make progress to removing this ambiguity from their procedure.

Due to increasing numbers of non-coding RNA (ncRNA) being discovered recently, there is interest in identifying homologs of a given structured RNA sequence. Exhaustive homology searching for structured RNA molecules using covariance models is infeasible on genome-length sequences. Hence, heuristic methods are employed, but they largely ignore structural information in the query. We present a novel method, which uses secondary structure information, to perform homology searches for a structured RNA molecule. We define the concept of a <em>pair seed</em> and theoretically model alignments of random and related paired regions to compute expected sensitivity and specificity. We show that our method gives theoretical gains in sensitivity and specificity compared to a BLAST-based heuristic approach. We provide experimental verification of this gain. <br /><br /> We also show that pair seeds can be effectively combined with the spaced seeds approach to nucleotide homology search. The hybrid search method has theoretical specificity superior to that of the BLAST seed. We provide experimental evaluation of our hypotheses. Finally, we note that our method is easily modified to process pseudo-knotted regions in the query, something outside the scope of covariance model based methods.

Due to increasing numbers of non-coding RNA (ncRNA) being discovered recently, there is interest in identifying homologs of a given structured RNA sequence. Exhaustive homology searching for structured RNA molecules using covariance models is infeasible on genome-length sequences. Hence, heuristic methods are employed, but they largely ignore structural information in the query. We present a novel method, which uses secondary structure information, to perform homology searches for a structured RNA molecule. We define the concept of a <em>pair seed</em> and theoretically model alignments of random and related paired regions to compute expected sensitivity and specificity. We show that our method gives theoretical gains in sensitivity and specificity compared to a BLAST-based heuristic approach. We provide experimental verification of this gain. <br /><br /> We also show that pair seeds can be effectively combined with the spaced seeds approach to nucleotide homology search. The hybrid search method has theoretical specificity superior to that of the BLAST seed. We provide experimental evaluation of our hypotheses. Finally, we note that our method is easily modified to process pseudo-knotted regions in the query, something outside the scope of covariance model based methods.

In the field of proteomic mass spectrometry, proteins can be sequenced by two independent yet complementary algorithms: de novo sequencing which uses no prior knowledge and database search which relies upon existing protein databases. In the case where an organism’s protein database is not available, the software Spider was developed in order to search sequence tags produced by de novo sequencing against a database from a related organism while accounting for both errors in the sequence tags and mutations. This thesis further develops Spider by using the concept of reconstruction in order to predict the real sequence by considering both the sequence tags and their matched homologous peptides. The significant value of these reconstructed sequences is demonstrated. Additionally, the runtime is greatly reduced and separated into independent caching and matching steps. This new approach allows for the development of an efficient algorithm for search. In addition, the algorithm’s output can be used for new applications. This is illustrated by a contribution to a complete protein sequencing application.

The problem of functional annotation of novel sequences has been a sigfinicant issue for many laboratories that decided to apply next generation sequencing techniques to less studied species. In particular experiments such as transcriptome analysis heavily suer from this problem due to the impossibility of ascribing their results in a relevant biological context. Several tools have been proposed to solve this problem through homology annotation transfer. The principle behind this strategy is that homologous genes share common functions in dierent organisms, and therefore annotations are transferable between these genes. Commonly, BLAST reports are used to identify a suitable homologousgene in a well annotated species and the annotation is then transferred fromthe homologue to the novel sequence. Not all homologues, however, possess valid functional annotations. The aim of this project was to devise an algorithm to process BLAST reports and provide a criterion to discriminate between homologues with a biologically informative and uninformative annotation, respectively. In addition, all data obtained from the BLAST report isto be stored in a relational database for ease of consultation and visualization. In order to test the solidity of the system, we utilized 750 novel sequences obtained through application of next generation sequencing techniques to Avena sativa samples. This species particularly suits our needs as it represents the typical target for homology annotation transfer: lack of a reference genome and diculty in attributing functional annotation. The system was able to perform all the required tasks. Comparisons between best hits asdetermined by BLAST and best hits as determined by the algorithm showed a significant increase in the biological significance of the results when thealgorithm sorting system was applied.

<p>We study how the preimages of a mapping f : X &rarr Y between manifolds vary under perturbations. First, we consider the preimage of a single point and track the history of its connected component as this point varies in Y. This information is compactly represented in a structure that is the generalization of the Reeb graph we call the Reeb space. We study its local and global properties and provide an algorithm for its construction. Using homology, we then consider higher dimensional connectivity of the preimage. We develop a theory quantifying the stability of each homology class under perturbations of the mapping f . This number called robustness is given to each homology class in the preimage. The robustness of a class is the magnitude of the perturbation necessary to remove it from the preimage. The generality of this theory allows for many applications. We apply this theory to quantify the stability of contours, fixed points, periodic orbits, and more.</p> / Dissertation

<p>The work presented in this dissertation includes the study of cohomology and cohomological operations within the framework of Persistence. Although Persistence was originally defined for homology, recent research has developed persistent approaches to other algebraic topology invariants. The work in this document extends the field of persistence to include cohomology classes, cohomology operations and characteristic classes. </p><p>By starting with presenting a combinatorial formula to compute the Stiefel-Whitney homology class, we set up the groundwork for Persistent Characteristic Classes. To discuss persistence for the more general cohomology classes, we construct an algorithm that allows us to find the Poincar'{e} Dual to a homology class. Then, we develop two algorithms that compute persistent cohomology, the general case and one for a specific cohomology class. We follow this with defining and composing an algorithm for extended persistent cohomology. </p><p>In addition, we construct an algorithm for determining when a cohomology class is decomposible and compose it in the context of persistence. Lastly, we provide a proof for a concise formula for the first Steenrod Square of a given cohomology class and then develop an algorithm to determine when a cohomology class is a Steenrod Square of a lower dimensional cohomology class.</p> / Dissertation

Abstract Essentially, In vitro diagnostics (IVD) testing play a key role in early disease detection; effectiveness of patient treatment monitoring throughout the progress of disease and improve decision-making for healthcare system. Several studies have demonstrated that IVD testing result in huge financial and therapeutic benefit. Taiwan IVD market has been growing rapidly in last decade due to launch of National Health Insurance policy¡Bgrowing of geriatric population¡Bimprovement of living quality¡Bconsciousness of personal health. However, this is also driving Taiwan IVD market in a highly competitive market place, it anticipated that the escalation will not only jeopardize the healthcare quality but also affect entire healthcare resources utility and increasing healthcare cost as a whole. The research analyzed the causative of Taiwan In Vitro Diagnostics (IVD) industrial is based on the framework of macroeconomic environment¡Bindustrial organization structure¡Bcompetitive marketing behavior and industrial market performance. The research found that macroeconomic environment for Taiwan IVD industrial is favorable; the industrial is presenting a mild dispersed structure and the causative for escalation was mainly caused from strategic homology among industrial enterprises. In addition to this finding, the research also developed a possibly differentiate strategy to avoid escalation throughout the learning innovation process and enterprise process reengineering as to help industrial enterprises from developing future long term strategy.

This thesis provides a partial classification of all 3-stranded pretzel knots K=P(p,q,r) with unknotting number one. Scharlemann-Thompson, and independently Kobayashi, have completely classified those knots with unknotting number one when p, q, and r are all odd. In the case where p=2m, we use the signature obstruction to greatly limit the number of 3-stranded pretzel knots which may have unknotting number one. In Chapter 3 we use Greene's strengthening of Donaldson's Diagonalization theorem to determine precisely which pretzel knots of the form P(2m,k,-k-2) have unknotting number one, where m is an integer, m>0, and k>0, k odd. In Chapter 4 we use Donaldson's Diagonalization theorem as well as an unknotting obstruction due to Ozsv\'ath and Szab\'o to partially classify which pretzel knots P(2,k,-k) have unknotting number one, where k>0, odd. The Ozsv\'ath-Szab\'o obstruction is a consequence of Heegaard Floer homology. Finally in Chapter 5 we explain why the techniques used in this paper cannot be used on the remaining cases. / text

In this dissertation we prove that if an n-stranded pretzel knot K has an essential Conway sphere, then there exists an Alexander grading s such that the rank of knot Floer homology in this grading, [mathematical equation], is at least two. As a consequence, we are able to easily classify pretzel knots admitting L-space surgeries. We conjecture that this phenomenon occurs more generally for any knot in S³ with an essential Conway sphere. We also exhibit an infinite family of knots, each of which admits a nontrivial genus two mutant which shares the same total dimension of knot Floer homology, while being distinguished by knot Floer homology as a bigraded invariant. Additionally, the genus two mutation interchanges the [mathematical symbol]-graded knot Floer homology groups in [mathematical symbol]-gradings k and -k. This infinite family of examples supports a second conjecture, namely that the total rank of knot Floer homology is invariant under genus two mutation. / text