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The article presents a study to examine the solvability of second-order equations with partially BMO coefficients. It analyses the Lp-solvability of second-order elliptic and parabolic equations in nondivergence form using divergence-form equations. It mentions the leading coefficients has been assumed to be in locally BMO spaces.

In this paper we let $G$ be a simple algebraic group and $r$ be a natural number, and consider the codimension in $G$ of the variety of elements $g\in G$ satisfying $g^r=1$. We shall obtain a lower bound for this codimension which is independent of characteristic, and show that it is attained if $G$ is of adjoint type. [ABSTRACT FROM AUTHOR]

This paper deals with the existence and nonexistence of global positive solutions to $u_t=\Delta\ln(1+u)$ in $\Omega \times (0, +\infty)$, \[\frac{\partial\ln(1+u)}{\partial n}=\sqrt{1+u}(\ln(1+u))^{\alpha} \quad\text{on} \partial \Omega \times (0, +\infty),] and $u(x, 0)=u_0(x)$ in $\Omega$. Here $\alpha\geq 0$ is a parameter, $\Omega\subset\mathbb{R}^N$ is a bounded smooth domain. After pointing out the mistakes in {\em Global behavior of positive solutions to nonlinear diffusion problems with nonlinear absorption through the boundary}, SIAM J. Math. Anal. {\bf 24} (1993), 317--326, by N. Wolanski, which claims that, for $\Omega=B_R$ the ball of $\mathbb{R}^N$, the positive solution exists globally if and only if $\alpha\leq 1$, we reconsider the same problem in general bounded domain $\Omega$ and obtain that every positive solution exists globally if and only if $\alpha\leq {1/2}$. [ABSTRACT FROM AUTHOR]

Motivated by a result of Ky Fan in 1965, we establish a characterization of a left amenable F-algebra (which includes the group algebra and the Fourier algebra of a locally compact group and quantum group algebras, or more generally the predual algebra of a Hopf von Neumann algebra) in terms of a finite-dimensional invariant subspace property. This is done by first revealing a fixed point property for the semigroup of norm one positive linear functionals in the algebra. Our result answers an open question posted in Tokyo in 1993 by the first author. We also show that the left amenability of an ideal in an F-algebra may determine the left amenability of the algebra. [ABSTRACT FROM AUTHOR]