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Chinese art songs—which are derived from over four thousand years of Chinese culture and combine unique Chinese literary and musical flavors with Western compositional techniques—remain largely unknown in the Western world. Chinese art song, a treasure of a genre, deserves more attention from Western singers and scholars. With a population of 1.41 billion, China has the largest potential market for bel canto singing. Not only will studying Chinese art songs enrich a singer's worldview and repertoire choices, doing so will also be advantageous when singing internationally. In order to assist non-Chinese-speaking singers overcome the difficult language barrier that often prevents them from studying Chinese art songs, I have chosen six songs by the renowned early twentieth century Chinese composer, Huang Zi (1904–1938) to analyze. Although he remains virtually unknown in the Western scholarly world, Huang's Chinese art songs, which include both ancient and contemporary Chinese poetry as well as Chinese and Western musical characteristics, are representative of this genre. By analyzing the poetry and musical settings of six Chinese art songs by Huang Zi, providing pronunciation instructions for Mandarin Chinese, word-for-word and poetic translations, practical International Phonetic Alphabet (IPA) transcriptions from the Pinyin system, and creating an interpretive guide of these songs, this project aims to help singers worldwide to prepare and perform Chinese art songs in a more expressive and authentic way. Taking this project as the first step, singers and scholars worldwide will be able to study and research more Chinese art songs and explore a new artistic world.

We introduce a rank reduction method for computing stationary distributions of Markov chains for which low-rank iteration matrices can be formed. We first prove that, for an irreducible Markov chain, a necessary and sufficient condition for convergence in a single iteration is that the iteration matrix have rank one. Since most iteration matrices have rank larger than 1, we also consider the Wedderburn rank-1 reduction formula and develop a rank reduction procedure that takes an initial iteration matrix with rank greater than one and modifies it in successive steps, under the constraint that the exact solution be preserved at each step, until a rank-1 iteration matrix is obtained. When the iteration matrix has rank r, the proposed algorithm has time complexity O(r∧2n). Secondly we investigate the relationship among lumpability, weak lumpability, quasi-lumpability and near complete decomposability. These concepts are important in aggregating and disaggregating Markov chains. White's algorithm for identifying all possible lumpable partitions for Markov chains is improved by incorporating lumpability tests on special state orderings. Finally, instead of computing exact stationary distributions, we design stochastic-ordering-based techniques to bound them. Upper bounds can be obtained by using constructive algorithms developed recently. We observe that the more lumpable partitionings, the more accurate the upper bound for the state of interest both with matrix transformation and without matrix transformation. Lastly we combine the approaches of state permutation, matrix transformation and state partitioning to improve the quality of the upper bound for the state of interest.