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The seek for a new universal formulation for describing various non-equilibrium processes is a central task of modern non-equilibrium thermodynamics. In this paper, a novel steady-state thermodynamic formalism was established for general Markov processes described by the Chapman-Kolmogorov equation. Furthermore, corresponding formalisms of steady-state thermodynamics for master equation and Fokker-Planck equation could be rigorously derived in mathematics. To be concrete, we proved that: 1) in the limit of continuous time, the steady-state thermodynamic formalism for the Chapman-Kolmogorov equation fully agrees with that for the master equation; 2) a similar one-to-one correspondence could be established rigorously between the master equation and Fokker-Planck equation in the limit of large system size; 3) when a Markov process is restrained to one-step jump, the steady-state thermodynamic formalism for the Fokker-Planck equation with discrete state variables also goes to that for master equations, as the discretization step gets smaller and smaller. Our analysis indicated that, with respect to the steady state, general Markov processes admit a unified and self-consistent non-equilibrium thermodynamic formulation, regardless of underlying detailed models.

Degree theory has been developed as a tool for checking the solution existence of nonlinear equations. In his classic paper published in 1983, Browder developed a degree theory for mappings of monotone type f+ T, where f is a mapping of class (S)+ from a bounded open set in a reflexive Banach space X into its dual X, and T is a maximal monotone mapping from X into X. This breakthrough paved the way for many applications of degree theoretic techniques to several large classes of nonlinear partial differential equations. In this paper we continue to develop the results of Browder on the degree theory for mappings of monotone type f+ T. By enlarging the class of maximal monotone mappings and pseudo-monotone homotopies we obtain some new results of the degree theory for such mappings.

For a linear isometry T: C 0 (X) C 0 (Y) of finite corank, there is a cofinite subset Y 1 of Y such that Tf| Y 1= h f is a weighted composition operator and X is homeomorphic to a quotient space of Y 1 modulo a finite subset. When X= Y, such a T is called an isometric quasi-n-shift on C 0 (X). In this case, the action of T can be implemented as a shift on a tree-like structure, called a T-tree, in M (X) with exactly n joints. The T-tree is total in M (X) when T is a shift. With these tools, we can analyze the structure of T.

Error variance estimation plays an important role in statistical inference for high-dimensional regression models. This article concerns with error variance estimation in high-dimensional sparse additive model. We study the asymptotic behavior of the traditional mean squared errors, the naive estimate of error variance, and show that it may significantly underestimate the error variance due to spurious correlations that are even higher in nonparametric models than linear models. We further propose an accurate estimate for error variance in ultrahigh-dimensional sparse additive model by effectively integrating sure independence screening and refitted cross-validation techniques. The root <i>n</i> consistency and the asymptotic normality of the resulting estimate are established. We conduct Monte Carlo simulation study to examine the finite sample performance of the newly proposed estimate. A real data example is used to