Statistical Studies in Biomolecular Sequences and Topics in Total Positivity

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This research will concentrate on three areas: (i) Theoretical studies of recombination stochastic models incorporating interference effects. These include characterizations of multimarker crossover distributions and analysis of higher order crossover interference forms. Apart from their intrinsic interest as a class of stochastic processes, these should be helpful in constructing genetic maps of multiple genetic markers. (ii) Statistical studies of molecular sequences emphasizing score based methods. These play a role in identifying distinctive sequence features in DNA and protein sequences and in identifying significant common segments in protein sequence comparisons. (iii) New directions and developments in the theory of multivariate total positivity motivated by applications related to random walks on graphs, especially on the n-dimensional hypercube. The general problem of deciding when a probability distribution on the hypercube is multivariate totally positive is fundamental in characterizing positive and negative interference for recombination stochastic structures. The research on recombination processes is of importance in developing genetic maps (for example, mapping of disease genes in relation to other easily observable markers). The studies on majorization and stochastic comparisons can provide insights in developing computer algorithms relevant in combinatorics and of interest in queuing systems, branching phenomena, and birth and death stochastic processes. Statistical studies of patterns in biomolecular sequences can be used as benchmarks for distinguishing chance occurrences from proper functional and structural attributes in DNA and protein sequences.