Mock theta functions and asymptotics for partition-theoretic functions

This thesis contains research articles on various topics in the theory of mod- ular forms and integer partitions. First we revisit Ramanujan’s original defini- tion of a mock theta function. We solve the general problem of understanding Ramanujan’s definition explicitly for the universal mock theta function g3, an- swering a question of Rhoades. After that we study a new spt function and its crank function. We investigate asymptotic aspects of this crank function and confirm a positivity conjecture of the crank. We further analyze a sign pattern of the crank and obtain linear congruences of the spt function via its mock modularity. Finally we provide the asymptotic formula for so-called odd-even partitions whose generating function appears in Ramanujan’s identities. We also study their overpartition analogue odd-even overpartitions

Algorithms Seminar, 2002-2004

These seminar notes constitute the proceedings of a seminar devoted to the analysis of algorithms and related topics. The subjects covered include combinatorics, symbolic computation, and the asymptotic analysis of algorithms, data structures, and network protocols

Neuroengineering of Clustering Algorithms

Cluster analysis can be broadly divided into multivariate data visualization, clustering algorithms, and cluster validation. This dissertation contributes neural network-based techniques to perform all three unsupervised learning tasks. Particularly, the first paper provides a comprehensive review on adaptive resonance theory (ART) models for engineering applications and provides context for the four subsequent papers. These papers are devoted to enhancements of ART-based clustering algorithms from (a) a practical perspective by exploiting the visual assessment of cluster tendency (VAT) sorting algorithm as a preprocessor for ART offline training, thus mitigating ordering effects; and (b) an engineering perspective by designing a family of multi-criteria ART models: dual vigilance fuzzy ART and distributed dual vigilance fuzzy ART (both of which are capable of detecting complex cluster structures), merge ART (aggregates partitions and lessens ordering effects in online learning), and cluster validity index vigilance in fuzzy ART (features a robust vigilance parameter selection and alleviates ordering effects in offline learning). The sixth paper consists of enhancements to data visualization using self-organizing maps (SOMs) by depicting in the reduced dimension and topology-preserving SOM grid information-theoretic similarity measures between neighboring neurons. This visualization\u27s parameters are estimated using samples selected via a single-linkage procedure, thereby generating heatmaps that portray more homogeneous within-cluster similarities and crisper between-cluster boundaries. The seventh paper presents incremental cluster validity indices (iCVIs) realized by (a) incorporating existing formulations of online computations for clusters\u27 descriptors, or (b) modifying an existing ART-based model and incrementally updating local density counts between prototypes. Moreover, this last paper provides the first comprehensive comparison of iCVIs in the computational intelligence literature --Abstract, page iv