Algebraic Geometry and Statistical Learning Theory

Sure to be influential, Watanabe's book lays the foundations for the use of algebraic geometry in statistical learning theory. Many models/machines are singular: mixture models, neural networks, HMMs, Bayesian networks, stochastic context-free grammars are major examples. The theory achieved here underpins accurate estimation techniques in the presence of singularities.

This remarkable reference book tells the story of science from earliest times to the present day, taking in everything from ancient Greek geometry to quantum physics, and the wedge to the worldwide web. Exploring science in a thematic, highly approachable manner, each spread takes as its theme a specific event, discovery, invention, experiment, theory, or individual and explains why this subject was so significant in the development of scientific thought and what its impact on history has been.

There are incredibly rich connections between classical analysis and number theory. For instance, analytic number theory contains many examples of asymptotic expressions derived from estimates for analytic functions, such as in the proof of the Prime Number Theorem. In combinatorial number theory, exact formulas for number-theoretic quantities are derived from relations between analytic functions. Elliptic functions, especially theta functions, are an important class of such functions in this context, which had been made clear already in Jacobi's Fundamenta nova. 

This book reviews current theories of the sound-structure of words and syllables. Dr. Coleman presents technical arguments showing that the contemporary theories are too complex and that a simpler theory, Declarative Phonology, is adequate. This theory is exemplified with detailed accounts of the sound-structure of words and syllables in English and Japanese.

In this provocative work, Luigi Burzio argues that many common assumptions within stress theory, and phonological theory more generally, are in fact rather arbitrary. He proposes radical departures from recent tradition. In Part I he analyzes stress in the underived English lexicon, arguing that the basic accentual groups or "feet" are not monosyllabic or bisyllabic, as often assumed, but rather bisyllabic or trisyllabic.