archives@tulane.edu / In this thesis, I define and study the foundations of the new framework of graded category theory, which I propose as just one structure that fits under the general banner of what Andree Eheresman has called “dynamic category theory”. Two approaches to defining graded categories are developed and shown to be equivalent formulations by a novel variation on the Grothendieck construction. Various notions of graded categorical constructions are studied within this framework. In particular, the structure of graded categories in general is then further elucidated by studying so-called “variable-object” models, and a version of the Yoneda lemma for graded categories. As graded category theory was originally developed in order to better understand the intuitive notions of absolute and relative cardinality – these notions are applied to the problem of vindicating the Skolemite thesis that “all sets, from an absolute perspective, are countable”. Finally, I discuss some open problems in this framework, discuss some potential applications, and discuss some of the relationships of my approach to existing approaches in the literature. / 1 / Nathan bedell

The concept of a computable function is quite a well-studied one, however, it is possible to capture certain important properties of computability categorically. A special type of category used for this purpose is called a Turing category. This thesis starts with a brief overview of Turing categories, followed by a study of additional categorical structure they may contain, based on the types of structure found in the world of computable functions, and how this is reflected in the underlying combinatorial structures.

The concept of a computable function is quite a well-studied one, however, it is possible to capture certain important properties of computability categorically. A special type of category used for this purpose is called a Turing category. This thesis starts with a brief overview of Turing categories, followed by a study of additional categorical structure they may contain, based on the types of structure found in the world of computable functions, and how this is reflected in the underlying combinatorial structures.

Observations of textual document management by individuals and organizations have suggested the popularity of using categories to organize, archive and access documents. The adequacy of an existing category understandably may diminish as it includes influxes of new documents over time or retains only a part of existing documents, bringing about significant changes to its content. Thus, the existing document categories have to be evolved over time as new documents are acquired. Following an evolution-based approach for document-category management, this dissertation extends Category Evolution (CE) technique by addressing its inherent limitations. The proposed technique (namely, CE2) automatically re-organizes document categories while taking into account those previously established. Furthermore, we propose the Ontology-based Category Evolution technique (namely, ONCE) to overcome the problems of word mismatch and ambiguity encountered by the lexicon-based category evolution approach (e.g., CE and CE2). Facilitated by a domain ontology, ONCE can evolve document categories on the conceptual rather the lexical level. Finally, this dissertation further considers the evolution of category hierarchy and proposes Category Hierarchy Evolution technique (CHE) and Ontology-based Category Hierarchy Evolution technique (OCHE) to evolve from an existing category hierarchy. We empirically evaluate the effectiveness of our proposed CE2, ONCE, CHE, and OCHE in different category evolution scenarios, respectively. Our analysis results show CE2 to be more effective than CE and the category discovery approach (specifically, HAC). The ontology-based category evolution approach, ONCE, shows its advantage over CE2 which represents the lexicon-based approach. Finally, the effectiveness attained by CHE and OCHE are satisfactory; and similarly, the ontology-based approach, OCHE, also outperforms the lexicon-based one. This dissertation has contributed to the text mining, document management, and ontology learning research and practice.

The Booklet Category Test (BCT) is a modified, highly portable version of the Halstead Category Test that has been shown to be very sensitive to brain damage. The BCT is commonly used in neuropsychological assessment in Western countries, however, no information on psychometric properties of the BCT had been report in the Chinese population thus far. This is a single-center, hospital-based, prospective, case-controlled cognitive instrument validation study. The study objective is to examine the criterion, convergent and divergent validity, test-retest reliability, internal consistency and ease of administration of the BCT in Chinese. Ten healthy controls, 12 patients with focal frontal contusions and ten patients with non-frontal contusions were recruited. The Chinese BCT did not differentiate between patients with cerebral contusions from controls, or between patients with focal frontal contusions from those with non-frontal contusions using receiver operating curve analyses. However, it showed good convergent validity with tests of spatial reasoning and had acceptable divergent validity, excellent internal consistency (Cronbach’s ss= .928) and test-retest reliability (ICC = .982, p < .982) and was generally well accepted by local participants. These results showed that the BCT is a valid and reliable clinical measure of spatial reasoning applicable to the Chinese population. / published_or_final_version / Clinical Psychology / Master / Master of Social Sciences

The concept of a computable function is quite a well-studied one, however, it is possible to capture certain important properties of computability categorically. A special type of category used for this purpose is called a Turing category. This thesis starts with a brief overview of Turing categories, followed by a study of additional categorical structure they may contain, based on the types of structure found in the world of computable functions, and how this is reflected in the underlying combinatorial structures.

No description available.

Khovanov-Lauda-Rouquier algebras, or KLR algebras, are a family of algebras known to categorify the upper half of the quantized enveloping algebra of a given Lie algebra. In finite type, these algebras come with a family of standard modules, which correspond to PBW bases under this categorification. In this thesis, we show that there are no non-zero homomorphisms between distinct standard modules and that all non-zero endomorphisms of standard modules are injective. We then apply this result to obtain applications to the modular representation theory of KLR algebras. Restricting our attention to finite type A, we are then able to compute explicit projective resolutions of all standard modules. Finally, in finite type A when alpha is a positive root, we let D be the direct sum of all distinct standard modules and compute the algebra structure on Ext(D, D). This dissertation includes unpublished co-authored material.

No description available.

The concept of a computable function is quite a well-studied one, however, it is possible to capture certain important properties of computability categorically. A special type of category used for this purpose is called a Turing category. This thesis starts with a brief overview of Turing categories, followed by a study of additional categorical structure they may contain, based on the types of structure found in the world of computable functions, and how this is reflected in the underlying combinatorial structures.