published_or_final_version / Mathematics / Doctoral / Doctor of Philosophy

Thesis. 1975. Ph.D.--Massachusetts Institute of Technology. Dept. of Mathematics. / Vita. / Includes bibliographical references. / Ph.D.

A “biomedical event” is a broad term used to describe the roles and interactions between entities (such as proteins, genes and cells) in a biological system. The task of biomedical event extraction aims at identifying and extracting these events from unstructured texts. An important component in the early stage of the task is biomedical trigger classification which involves identifying and classifying words/phrases that indicate an event. In this thesis, we present our work on biomedical trigger classification developed using the multi-level event extraction dataset. We restrict the scope of our classification to 19 biomedical event types grouped under four broad categories - Anatomical, Molecular, General and Planned. While most of the existing approaches are based on traditional machine learning algorithms which require extensive feature engineering, our model relies on neural networks to implicitly learn important features directly from the text. We use natural language processing techniques to transform the text into vectorized inputs that can be used in a neural network architecture. As per our knowledge, this is the first time neural attention strategies are being explored in the area of biomedical trigger classification. Our best results were obtained from an ensemble of 50 models which produced a micro F-score of 79.82%, an improvement of 1.3% over the previous best score.

Graduation date: 1993

We discuss the problem of embedding planar graphs onto the plane with pre-specified vertex locations. In particular, we introduce a method for constructing such an embedding for both the case where the mapping from the vertices onto the vertex locations is fixed and the case where this mapping can be chosen. Moreover, the technique we present is sufficiently abstract to generalize to a method for constructing simultaneous planar embeddings with fixed vertex locations. In all cases, we are concerned with minimizing the number of bends per edge in the embeddings we produce. In the case where the mapping is fixed, our technique guarantees embeddings with at most 8n - 7 bends per edge in the worst case and, on average, at most 16/3n - 1 bends per edge. This result improves previously known techniques by a significant constant factor. When the mapping is not pre-specified, our technique guarantees embeddings with at most O(n^(1 - 2^(1-k))) bends per edge in the worst case and, on average, at most O(n^(1 - 1/k)) bends per edge, where k is the number of graphs in the simultaneous embedding. This improves upon the previously known O(n) bound on the number of bends per edge for k at least 2. Moreover, we give an average-case lower bound on the number of bends that has similar asymptotic behaviour to our upper bound.

We discuss the problem of embedding planar graphs onto the plane with pre-specified vertex locations. In particular, we introduce a method for constructing such an embedding for both the case where the mapping from the vertices onto the vertex locations is fixed and the case where this mapping can be chosen. Moreover, the technique we present is sufficiently abstract to generalize to a method for constructing simultaneous planar embeddings with fixed vertex locations. In all cases, we are concerned with minimizing the number of bends per edge in the embeddings we produce. In the case where the mapping is fixed, our technique guarantees embeddings with at most 8n - 7 bends per edge in the worst case and, on average, at most 16/3n - 1 bends per edge. This result improves previously known techniques by a significant constant factor. When the mapping is not pre-specified, our technique guarantees embeddings with at most O(n^(1 - 2^(1-k))) bends per edge in the worst case and, on average, at most O(n^(1 - 1/k)) bends per edge, where k is the number of graphs in the simultaneous embedding. This improves upon the previously known O(n) bound on the number of bends per edge for k at least 2. Moreover, we give an average-case lower bound on the number of bends that has similar asymptotic behaviour to our upper bound.

Embedding theories are concerned with the embedding of a lower dimensional manifold (dim = n, say) into a higher dimensional one (usually dim = n+1, but not necessarily so). We are concerned with the particular case of embedding 4D spherically symmetric equations into 5D Einstein spaces. This scenario is of particular relevance to contemporary cosmology and astrophysics. Essentially, they are 5D vacuum field equations with initial data given on a 4D spacetime hypersurface. The equations that arise in this framework are highly nonlinear systems of ordinary differential equations and they have been particularly resistant to solution techniques over the past few years. As a matter of fact, to date, despite theoretical results for the existence of solutions for embedding classes of 4D space times, no general solutions to the local embedding equations are known. The Lie theory of extended groups applied to differential equations has proved to be very successful since its inception in the nineteenth century. More recently, it has been successfully utilized in relativity and has provided solutions where none were previously found, as well as explaining the existence of ad hoc methods. In our work, we utilize this method in an attempt to find solutions to the embedding equations. It is hoped that we can place the analysis of these equations onto a firm theoretical basis and thus provide valuable insight into embedding theories. / Thesis (M.Sc.)-University of KwaZulu-Natal, Westville, 2010.

The first result of this thesis is a partial result in the direction of Steinberg's Conjecture. Steinberg's Conjecture states that any planar graph without cycles of length four or five is three colorable. Borodin, Glebov, Montassier, and Raspaud showed that planar graphs without cycles of length four, five, or seven are three colorable and Borodin and Glebov showed that planar graphs without five cycles or triangles at distance at most two apart are three colorable. We prove a statement that implies the first of these theorems and is incomparable with the second: that any planar graph with no cycles of length four through six or cycles of length seven with incident triangles distance exactly two apart are three colorable. The third and fourth chapters of this thesis are concerned with the study of Pfaffian orientations. A theorem proved by William McCuaig and, independently, Neil Robertson, Paul Seymour, and Robin Thomas provides a good characterization for whether or not a bipartite graph has a Pfaffian orientation as well as a polynomial time algorithm for that problem. We reprove this characterization and provide a new algorithm for this problem. In Chapter 3, we generalize a preliminary result needed to reprove this theorem. Specifically, we show that any internally 4-connected, non-planar bipartite graph contains a subdivision of K3,3 in which each path has odd length. In Chapter 4, we make use of this result to provide a much shorter proof using elementary methods of this characterization. In the fourth and fifth chapters we investigate flat embeddings. A piecewise-linear embedding of a graph in 3-space is flat if every cycle of the graph bounds a disk disjoint from the rest of the graph. We provide a structural theorem for flat embeddings that indicates how to build them from small pieces in Chapter 5. In Chapter 6, we present a class of flat graphs that are highly non-planar in the sense that, for any fixed k, there are an infinite number of members of the class such that deleting k vertices leaves the graph non-planar.

Thesis (Ph. D.)--University of Hong Kong, 2009. / Includes bibliographical references (leaves 53-54) Also available in print.

Thesis (Ph. D.)--University of Washington, 1998. / Vita. Includes bibliographical references (p. [59]-61).