We formulate and prove a version of Paley-Wiener theorem for the inverse Fourier transforms on noncompact Riemannian symmetric spaces and Heisenberg groups. The main ingredient in the proof is the Gutzmer’s formula.

In this paper we shall show asymptotic behavior of all positive solutions of the second order nonlinear di¤erential equation written in the title. It will complete this task to obtain an analytical expression or an asymptotic form of every solution valid in a neighborhood of an end of its domain.

Using the reducible algebraic polynomial \(x^{5} - x^{4}-1 = \left( x^{2}-x + 1 \right) \left( x^{3} - x - 1 \right),\) we study two types of tiling substitutions $\tau^*$ and $\sigma^*$: $\tau^*$ generates a tiling of a plane based on five prototiles of polygons, and $\sigma^*$ generates a tiling of a Riemann surface, which consists of two copies of the plane, based on ten prototiles of parallelograms. Finally we claim that $\tau^*$-tiling of $\mathcal{P}$ equals a re-tiling of $\sigma^*$-tiling of $\mathcal{R}$ through the canonical projection of the Riemann surface to the plane.

We give an upper bound on the denominators of numerical boundary slopes and an upper bound on the di¤erences between two numerical boundary slopes for Montesinos knots.

In this paper, we give a classification of normal del Pezzo surfaces $X$ with at most three quasi-lines and determine the geometric structure of the complement of quasi-lines on $X$. Moreover, we give the complete list of compactifications $X$ of ${\Bbb C}^2$ with quasi-lines as boundaries.

Potential theory on a Cartier tree $T$ is developed on the lines of the classical and the axiomatic theories on harmonic spaces. The harmonic classifications of such trees are considered; the notion of a subordinate structure on $T$ is introduced to consider more generally the potential theory on $T$ associated with the Schrödinger equation $\Delta u\left( x\right) =Q\left(x\right) u\left( x\right) ,Q\left( x\right) \geq 0$ on $T$; polysuperharmonic functions and polypotentials on $T$ are defined and a Riesz-Martin representation for positive polysuperharmonic functions is obtained.

Let $R$ be a complete discrete valuation ring of equal characteristic $p>0$. In this paper we investigate finite and flat morphisms $f:Y\to X$ between formal $R$-schemes which have the structure of an étale $\Bbb Z/p^n\Bbb Z$-torsor above the generic fiber of $X$, for $n=1,2$, with some extra geometric conditions on $X$ and $Y$. In the case $n=1$, we prove that $f$ has the structure of a torsor under a finite and flat $R$-group scheme of rank $p$ and we describe the group schemes that arise as the group of the torsor. In the case $n=2$, we describe explicitly how the Artin-Schreier-Witt equations describing $f$ on the generic fiber, locally, degenerate. Moreover, in some cases where $f$ has the structure of a torsor under a finite and flat $R$-group scheme of rank $p^2$, we describe the group schemes of rank $p^2$ which arise in this way.