This monograph surveys the theory of quantitative homogenization for second-order linear elliptic systems in divergence form with rapidly oscillating periodic coefficients in a bounded domain. It begins with a review of the classical qualitative homogenization theory, and addresses the problem of convergence rates of solutions. The main body of the monograph investigates various interior and boundary regularity estimates that are uniform in the small parameter e>0. Additional topics include convergence rates for Dirichlet eigenvalues and asymptotic expansions of fundamental solutions, Green functions, and Neumann functions. The monograph is intended for advanced graduate students and researchers in the general areas of analysis and partial differential equations. It provides the reader with a clear and concise exposition of an important and currently active area of quantitative homogenization.

This monograph surveys the theory of quantitative homogenization for second-order linear elliptic systems in divergence form with rapidly oscillating periodic coefficients in a bounded domain. It begins with a review of the classical qualitative homogenization theory, and addresses the problem of convergence rates of solutions. The main body of the monograph investigates various interior and boundary regularity estimates that are uniform in the small parameter e>0. Additional topics include convergence rates for Dirichlet eigenvalues and asymptotic expansions of fundamental solutions, Green functions, and Neumann functions. The monograph is intended for advanced graduate students and researchers in the general areas of analysis and partial differential equations. It provides the reader with a clear and concise exposition of an important and currently active area of quantitative homogenization.

The phonological history of Chinese can be traced back to two main traditions: one starting with the Qieyun of 601, and the other starting with the Zhongyuan Yinyun of 1324. The former marks the beginning of Middle Chinese, and the latter marks the beginning of Old Mandarin. Both of these systems, as well as reconstructed Old Chinese, should be understood as ideal phonological standards and composite in nature. Until modern times, phonological standards were never based strictly on the phonology of a single dialect. This book provides the first study written in English, of the phonological history of Chinese. It provides information about the standard phonological systems for each of the language's major historical periods, drawing on a range of historical materials such as dictionaries, rhyming tables and poetry, and is the reference book for understanding the key developments in the Chinese sound system.