Virtual knot theory is an extension of classical knot theory based on a combinatorial presentation of crossing information. The appropriate extensions of braid groups and string link monoids have also been studied. While some previously known knot invariants can be evaluated for virtual objects, entirely new techniques can also be used, for example, the concept of index of a crossing, and its resulting (Gaussian) parity theory. In general, a parity is a rule which assigns 0 or 1 to each crossing in a knot or link diagram. Recently, they have also been defined for virtual braids. Here, novel parities for knots, braids, and string links are defined, some of their applications are explored, most notably, defining a new subgroup of the virtual braid groups. / Thesis / Master of Science (MSc)

The Kitaev model has become a source of much excitement in the field of condensed matter. It is a two dimensional model of spins ½ on a honeycomb lattice with bond-dependent interactions. Its interesting properties include a quantum spin liquid ground state and anyonic excitations. These properties could lead to exciting applications in quantum computing if materials were found to behave similarly to the Kitaev model. Such materials have been found, however the Kitaev model is too simple to describe these materials and additional interactions must be considered. The Heisenberg interaction is one such additional interaction. As such, we can define the Kitaev-Heisenberg model by combining the Kitaev and Heisenberg interactions. We can now ask ourselves if the quantum spin liquid ground state and anyonic excitations still exist in the Kitaev-Heisenberg model. To answer this question, a non-local string order parameter has been defined which is non-zero inside the quantum spin liquid phase and zero outside of it. This string order parameter was shown to exist and survive the Heisenberg interaction on the 2-leg ladder. In this thesis, we look to expand this result to multileg ladders such as the 3-leg, 4-leg, and 5-leg ladders to see if the string order parameter survives in the Kitaev-Heisenberg model in 2 dimensions. Our results show that the string order parameter does exist in multileg ladders, however the phase space window in which it survives the Heisenberg interaction is narrower than in the 2-leg ladder. / Thesis / Master of Science (MSc)

In today’s world, ontologies are being widely used for data integration tasks and solving information heterogeneity problems on the web because of their capability in providing explicit meaning to the information. The growing need to resolve the heterogeneities between different information systems within a domain of interest has led to the rapid development of individual ontologies by different organizations. These ontologies designed for a particular task could be a unique representation of their project needs. Thus, integrating distributed and heterogeneous ontologies by finding semantic correspondences between their concepts has become the key point to achieve interoperability among different representations. In this thesis, an advanced instance-based ontology matching algorithm has been proposed to enable data integration tasks in ocean sensor networks, whose data are highly heterogeneous in syntax, structure, and semantics. This provides a solution to the ontology mapping problem in such systems based on machine-learning methods and string-based methods.

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