We consider a domain $D$ in $\mathbb{C}^{n}$ such that there is a Stein manifold $E$ which is a $\mathbb{C}$-fibration over $D$. Simple examples show that $D$ does not need to be Stein. However it cannot be arbitrarily and, in fact, we prove that $D$ is pseudoconvex of order $n-2$.

A definition of an invariant averaging for a linear representation of a group in a locally convex space is given. Main results: A group $H$ is finite if and only if every linear representation of $H$ in a locally convex space has an invariant averaging. A group $H$ is amenable if and only if every almost periodic representation of $H$ in a quasi-complete locally convex space has an invariant averaging. A locally compact group $H$ is compact if and only if every strongly continuous linear representation of $H$ in a quasi-complete locally convex space has an invariant averaging.

We give the asymptotics of Green functions $G_{\lambda \pm i0}(x, y)$ as $|x-y| \to \infty$ for an elliptic operator with periodic coefficients on $\mathbf{R}^{d}$ in the case where $d \geq 2$ and the spectral parameter $\lambda$ is close to and greater than the bottom of the spectrum of the operator. The main tools are the Bloch representation of the resolvent and the stationary phase method. As a by-product, we also show directly the limiting absorption principle. In the one dimensional case, we show that Green functions are written as products of exponential functions and periodic functions for any $\lambda$ in the interior of the spectrum or the resolvent set.

In this work some direct and inverse theorems of approximation theory in the weighted Smirnov-Orlicz classes, defined in the domains with a Dini-smooth boundary, are proved. In particular, a constructive characterization of the generalized Lipschitz classes $Lip^{*}\alpha (M,\omega )$, $\alpha >0$, is obtained.

A result due to Serrin assures that a graph with constant mean curvature $H \neq 0$ in Euclidean space $\mathbb{R}^{3}$ cannot keep away a distance $1/|H|$ from its boundary. When the distance is exactly $1/|H|$, then the surface is a hemisphere. Following ideas due to Meeks, in this note we treat the aspect of the equality in the Serrin’s estimate as well as generalizations in other situations and ambient spaces.

We study convex subsets of buildings, discuss some structural features and derive several characterizations of buildings.

We consider a class of higher order hyperbolic equations with finite degeneracy.

We give sufficient conditions in order the Cauchy problem to be well-posed in $\mathcal{C}^{\infty}$ and in Gevrey classes.

We consider a sufficient condition that a knot self-concordance surgery introduced in [3] gives rise to the diffeomorphic manifolds. The main theorem in the present paper is a certain generalization of Akbulut’s result in [3] and author’s result in [6].

In this paper, the sufficient conditions for the existence of twin positive solutions to boundary value problems for functional differential equations are presented. The results are obtained by using a fixed point theorem (See [1, Theorem 12.3]).

We consider a first-passage percolation problem on the square lattice, where the distribution function of time coordinates of horizontal edges may be different from that of vertical edges. Some basic limit theorems for first-passage times and minimal lengths of optimal paths are obtained. Especially, we show that as long as the system is in the supercritical phase, the expectation of first-passage time from the origin to a point with distance $n$ converges to a finite constant, which is independent of the directions, as $n \to \infty$.