In a uniform domain Ω, we present a certain reverse mean value inequality and a Harnack type inequality for positive superharmonic functions satisfying a nonlinear inequality -Δu(x) ≤ cδ_Ω(x)^-αu(x)^p for x ∈ Ω, where c > 0, α ≥ 0 and p > 1 and δ_Ω(x) is the distance from a point x to the boundary of Ω. These are established by refining a boundary growth estimate obtained in our previous paper (2008). Also, we apply them to show the existence of nontangential limits of quotients of such functions and to give an extension of a certain minimum principle studied by Dahlberg (1976).

Let f: (R², 0) → (R², 0) be a finitely determined map germ. The link of f is obtained by taking a small enough representative f: U ⊂ R² → R² and the intersection of its image with a small enough sphere S_ε¹ centered at the origin in R². We will describe the topology of f in terms of the Gauss word associated to its link.

Let R be a ring, S a strictly ordered monoid and ω: S → End(R) a monoid homomorphism. The skew generalized power series ring R[[S, ω]] is a common generalization of skew polynomial rings, skew power series rings, skew Laurent polynomial rings, skew group rings, and Mal'cev-Neumann Laurent series rings. In the case where S is positively ordered we give sufficient and necessary conditions for the skew generalized power series ring R[[S, ω]] to have weak dimension less than or equal to one. In particular, for such an S we show that the ring R[[S, ω]] is right duo of weak dimension at most one precisely when the lattice of right ideals of the ring R[[S, ω]] is distributive and ω(s) is injective for every s ∈ S.

We consider the expected volume of the Wiener sausage on the time interval [0,t] associated with a closed ball. Let L(t) be the expected volume minus the volume of the ball. We obtain that L(t) is asymptotically equal to a constant multiple of t^1/2 as t tends to 0 and that it is represented as an absolutely convergent power series of t^1/2 for any t > 0 in the odd dimensional cases. Moreover, the explicit form of L(t) can be given in five and seven dimensional cases.

By using the method of equivariant differential geometry, we construct a new family of noncompact complete hypersurfaces with constant positive scalar curvature in the Euclidean spaces. To do so we make a detailed analysis of the nonlinear ODE of the constant scalar curvature equation.

The aim of this paper is to find a general class of data in which the global well-posedness for the exterior initial-boundary value problem to the Kirchhoff equation is assured. The result obtained in the present paper will be applied to the existence of scattering states. A class of weighted Sobolev spaces will be also presented in which the global well-posedness is assured. For this purpose, the method of generalized Fourier transforms is developed for some oscillatory integral associated with this equation. The crucial point is to obtain the resolvent expansion of the minus Laplacian around the origin in C, and the differentiability of the generalized Fourier transforms.

We exhibit a pseudogroup of smooth local transformations of the real line, which is compactly generated, but not realizable as the holonomy pseudogroup of a foliation of codimension 1 on a compact manifold. The proof relies on a description of all foliations with the same dynamic as the Reeb component.

In this paper, we study the reproducing formula with fractional orders on the parabolic Bloch space. As an application of the reproducing formula, we characterize the dual and pre-dual spaces of parabolic Bergman spaces. Furthermore, we generalize the integral pairing, which gives the duality between the parabolic Bloch space and the parabolic Bergman space.

A self-transverse immersion of a smooth manifold $M^{2n}$ in $\boldsymbol{R}^{4n-5}$ for $n \gt 5$ has a double point self-intersection set which is the image of an immersion of a smooth 5-dimensional manifold, cobordant to Dold manifold $V^5$ or a boundary. We will show that the double point manifold of any such immersion is a boundary. The method of proof is to evaluate the Stiefel-Whitney numbers of the double point self-intersection manifold. By a certain method these numbers can be read off from spherical elements of $H_{4n-5}QMO(2n-5)$, corresponding to the immersions under the Pontrjagin-Thom construction.

The linear Noether's problem means the rationality problem for the fixed field of linear actions on the rational function field. This paper deals with a part of our study on the four dimensional linear Noether's problem. Apart from the main part of our study, which will be published in other papers, the results which require complicated calculations by a computer are published here as a separate paper. The problem is affirmative for all of 5 non-solvable subgroups and the largest and one of the second largest subgroups of GL(4,Q). As relevant topics, we remark that PSp(3,2) (the simple group of order 1451520) has a generic polynomial over Q.

We consider the initial-boundary value problem in hyperbolic thermoelasticity with second sound in a three-dimensional exterior domain. The low frequency expansion of solutions to the corresponding stationary resolvent problem is given and the limit to the classical thermoelastic problem is investigated.

A domain in a complex 3-dimensional projective space is said to be large, if the domain contains a line, i.e., a projective linear subspace of dimension one. We study compact complex 3-manifolds defined as non-singular quotients of large domains. Any holomorphic automorphism of a large domain becomes an element of the projective linear transformations. In the first half, we study the limit sets of properly discontinuous groups acting on large domains. In the second half, we determine all compact complex 3-manifolds with positive algebraic dimensions which are quotients of large domains.