In this survey, we give a neat summary of the applications of the multi-resolution analysis to the studies of Besov-Q type spaces {\dot{B}}_{p,q}^{{\gamma }_1,{\gamma }_2}({\text{ℝ}}^n) and Triebel-Lizorkin-Q type spaces {\dot{B}}_{p,q}^{{\gamma }_1,{\gamma }_2}({\text{ℝ}}^n). We will state briefly the recent progress on the wavelet characterizations, the boundedness of Calderón-Zygmund operators, the boundary value problem of {\dot{B}}_{p,q}^{{\gamma }_1,{\gamma }_2}({\text{ℝ}}^n) and {\dot{F}}_{p,q}^{{\gamma }_1,{\gamma }_2}({\text{ℝ}}^n). We also present the recent developments on the well-posedness of fluid equations with small data in {\dot{B}}_{p,q}^{{\gamma }_1,{\gamma }_2}({\text{ℝ}}^n) and {\dot{F}}_{p,q}^{{\gamma }_1,{\gamma }_2}({\text{ℝ}}^n).

In this paper we introduce the history and present situation of the computation of the cohomology rings of Kac-Moody groups, their flag manifolds and classifying spaces, and give some problems and conjectures that deserve further study.

Let (X, d, μ) be a metric measure space with non-negative Ricci curvature. This paper is concerned with the boundary behavior of harmonic function on the (open) upper half-space X\times {\text{ℝ}}_{+}. We derive that a function f of bounded mean oscillation (BMO) is the trace of harmonic function u(x,t) on X\times {\text{ℝ}}_{+},u(x,0)=f\left(x\right), whenever u satisfies the following Carleson measure condition

where \nabla =({\nabla }_x,{\partial }_t) denotes the total gradient and B(x_B,r_B) denotes the (open) ball centered at x_B with radius r_B. Conversely, the above condition characterizes all the harmonic functions whose traces are in BMO space.

A coloring of a graph G is injective if its restriction to the neighbour of any vertex is injective. The injective chromatic number X_i\left(G\right) of a graph G is the leastk such that there is an injective k-coloring. In this paper, we prove that for each planar graph with g\ge 5 and \Delta \left(G\right)\ge 20, {\chi }_i\left(G\right)\le \Delta \left(G\right)+3.

In this paper, we study the category of corepresentations of a monoidal comonad. We show that it is a semisimple category if and only if the monoidal comonad is a cosemisipmle (coseparable) comonad, and it is a braided category if and only if the monoidal comonad admit a cobraided structure. At last, as an application, the braided structure and the semisimplicity of the Hom-comodule category of a monoidal Hom-bialgebra are discussed.