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There are a myriad of people that I know and to whom I would like to extend my thanks. First and foremost, I am immensely thankful and indebted to my advisor, Dr. Shari L. Moskow, for her continued support and constant mentoring. I would also like to thank her for her belief in me. This work would not have been possible without her guidance. I would also like to thank Vladimir Druskin for his invaluable suggestions and ideas from time to time. This work was supported by the National Science Foundation under grants SCREMS-0619080, DMS-0605021, DMS-0713833. Next, I would like to thank my parents for believing in me and being patient with me throughout these years. My knowledge is incomparable to their vast experience and it is with this experience that they have guided me all my life helping me tackle both academic and personal problems that life has thrown at me over these years. I am extremely grateful to my co-advisor, Dr. Jayadeep Gopalakrishnan, who extended his selfless support to me throughout my journey. I am thankful to Dr. William Hager, Dr. Sergei Pilyugin, and Dr. Babette Brumback for serving on my committee. I am also thankful to my beautiful wife, Alefiya, who showed great patience with me in my final stages of completing my doctoral studies