We denote by AN₂ the set of all absolute normalized norms on $\mathbb R$². It is known that the set AN₂ and the set of all continuous convex functions ψ on [0,1] with max{1−t,t} ≤ ψ(t) ≤ 1 for t ∈ [0,1] (denoted by Ψ₂) are in a one to one correspondence under the equation ψ(t) = ||(1−t,t)||. Recently, we characterized extreme points of AN₂ by considering Ψ₂. In this paper we give another proof of this result.

The linearized stability of stationary solutions for surface diffusion is studied. We consider three hypersurfaces that lie inside a fixed domain and touch its boundary with a right angle and fulfill a non-flux condition. Additionally they meet at a triple line with prescribed angle conditions and further boundary conditions resulting from the continuity of chemical potentials and a flux balance have to hold at the triple line. We introduce a new specific parametrization with two parameters corresponding to a movement in tangential and normal direction to formulate the geometric evolution law as a system of partial differential equations. For the linearized stability analysis we identify the problem as an H⁻¹-gradient flow, which will be crucial to show self-adjointness of the linearized operator. Finally we study the linearized stability of some examples.

We define that a ring extension S/R is weakly separable or weakly quasi-separable by using R-derivations of S, and give the necessary and sufficient condition that the extension R[X]/(Xⁿ − aX − b) of a commutative ring R is weakly separable. Since the notions of weakly separability and weakly quasi-separability coincide for commutative ring extensions, we treat a quotient ring R[x; *] = R[X; *]/f(X)R[X; *] of a skew polynomial ring R[X; *], and show that if R is a commutative domain, then the extension R[x; *]/R is always weakly quasi-separable, where * is either a ring automorphism or a derivation of R. We also treat the weakly separability of R[x; *]/R and give various types of examples of these extensions.

Among the topological conjugacy classes of the continuous flows {ϕ^t} whose orbit foliations are the planar Reeb foliation, there is one special class called the standard Reeb flow. We show that {ϕ^t} is conjugate to the standard Reeb flow if and only if {ϕ^t} is conjugate to {ϕ^λt} for any λ > 0.

This paper is concerned with the final value problem for a system of semilinear wave equations. The main issue is to solve the problem when the nonlinearity is of a long-range type. By assuming that the solution is spherically symmetric, we shall show global solvability of the final value problem around a suitable final state, and hence, the generalized wave operator and long range-scattering operator can be constructed.

It is well known that Leray's Inequality holds under stringent outflow condition (SOC). But Leray's Inequality does not hold under general outflow condition (GOC). This fact has been proved by Takeshita [8]. But, Takeshita's argument is very complicated. The author succeeds in giving an alternative proof which is simpler than Takeshita's. Moreover, the result is an improvement of Takeshita's result.

Given a CR manifold D, we shall show that existence of a CR local frame of a certain CR vector bundle over D is equivalent to the local imbeddability of D. This will imply that there exists a CR vector bundle which doesn't have CR local frames. Using this bundle, we shall construct CR line bundles over 3-dimensional non-imbeddable CR manifolds which don't have CR local frames.

We obtain the Fefferman-Stein vector-valued maximal inequalities on Morrey spaces generated by weighted Lebesgue spaces. Using these inequalities, we introduce and define the weighted Hardy-Morrey spaces by using the Littlewood-Paley functions. We also establish the non-smooth atomic decompositions for the weighted Hardy-Morrey spaces and, as an application of the decompositions, we obtain the boundedness of a class of singular integral operators on the weighted Hardy-Morrey spaces.