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Query expansion is a functionality of search engines that suggests a set of related queries for a user-issued keyword query. Typical corpus-driven keyword query expansion approaches return popular words in the results as expanded queries. Using these approaches, the expanded queries may correspond to a subset of possible query semantics, and thus miss relevant results. To handle ambiguous queries and exploratory queries, whose result relevance is difficult to judge, we propose a new framework for keyword query expansion: we start with clustering the results according to user specified granularity, and then generate expanded queries, such that one expanded query is generated for each cluster whose result set should ideally be the corresponding cluster. We formalize this problem and show its APX-hardness. Then we propose two efficient algorithms named iterative single-keyword refinement and partial elimination based convergence, respectively, which effectively generate a set of expanded queries from clustered results that provides a classification of the original query results. We believe our study of generating an optimal query based on the ground truth of the query results not only has applications in query expansion, but has significance for studying keyword search quality in general.

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Representations of compact metric groups in Hilbert spaces over the quaternions are studied. A generalization of the Peter-Weyl theorem is formulated and proved. The problem of finding all the irreducible quaternionic representations of an arbitrary compact metric group is solved, and a rule is given for computing the ``Q-characters'' of all the irreducible quaternionic representations once the characters of all the irreducible complex representations are known. For the Abelian case, it is shown that every irreducible quaternionic representation is equivalent to a complex representation and hence one dimensional. An example is given of a non-Abelian group whose irreducible quaternionic representations are all one dimensional.

The proper mixing of nanoscale fillers in conventional dielectric materials leads to an enhancement in the breakdown strength and voltage endurance. In this study experimental investigations are carried out to compare the breakdown characteristics of epoxy nano-composites with that of a base epoxy resin under the influence of divergent electric fields so as to obtain inferences on its breakdown performances. This would in turn enable providing solutions to acquire more effective electrical insulation systems and explore the prospect of tapping the merits of utilizing the rapid strides made in field of fabrication of nano-dielectrics. The main objective is such studies are to enhance the electrical properties of the epoxy dielectric by employing nano-fillers such as SiO 2 , TiO 2 etc. This research envisages the use of epoxy resin mixed with nano-fillers for ascertaining the ability of the nano-composite to be utilized as a dielectric/ insulator in power apparatus. The epoxy resin is mixed with appropriate proportion of SiO 2 and TiO 2 and experimentation is carried out under the influence of divergent electric fields. Classical breakdown voltage withstand tests such as AC power frequency, DC voltage, lightning impulse and switching impulse test is carried out on epoxy dielectrics (with and without nano-fillers) and the results are compared. In addition, a non-classical breakdown voltage test (high frequency high voltage) is also devised to analyze and ascertain the breakdown characteristics due to varying frequencies so as to investigate the possibility of utilizing such nano-composites in applications related to high speed switching devices.