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We prove a sharp Bernstein inequality for general-state-space and not necessarily reversible Markov chains. It is sharp in the sense that the variance proxy term is optimal. Our result covers the classical Bernstein's inequality for independent random variables as a special case.

We establish the global existence and uniqueness of classical solutions to the Cauchy problem for the isentropic compressible Navier-Stokes equations in three spatial dimensions with smooth initial data that are of small energy but possibly large oscillations with constant state as far field, which could be either vacuum or nonvacuum. The initial density is allowed to vanish, and the spatial measure of the set of vacuum can be arbitrarily large; in particular, the initial density can even have compact support. These results generalize previous results on classical solutions for initial densities being strictly away from vacuum and are the first for global classical solutions that may have large oscillations and can contain vacuum states. © 2012 Wiley Periodicals, Inc

This monograph introduces the basic concepts of the theory of causal fermion systems, a recent approach to the description of fundamental physics. The theory yields quantum mechanics, general relativity and quantum field theory as limiting cases and is therefore a candidate for a unified physical theory. From the mathematical perspective, causal fermion systems provide a general framework for describing and analyzing non-smooth geometries and ``quantum geometries.'' The dynamics is described by a novel variational principle, called the causal action principle. In addition to the basics, the book provides all the necessary mathematical background and explains how the causal action principle gives rise to the interactions of the standard model plus gravity on the level of second-quantized fermionic fields coupled to classical bosonic fields. The focus is on getting a mathematically sound connection between causal fermion systems and physical systems in Minkowski space. The book is intended for graduate students entering the field, and is furthermore a valuable reference work for researchers in quantum field theory and quantum gravity.