We give a process to construct what we call the Newton-Puiseux approximation for a system of germs (at the origin and at infinity) and indicate how the Newton-Puiseux approximation may be used to obtain formulas for the Łojasiewicz exponents.

Let $R$ be a Riemann surface of infinite type such that the injectivity radius at any point in $R$ is less than a positive constant $M$, and $f$ a conformal automorphism of $R$ fixing a compact subset in $R$. We show that the order of $f$ is less than a certain constant depending on $M$.

Let $X$ be a smooth projective surface defined over the complex number field and let $D$ be an effective divisor on $X$. In this paper we will propose a special class of effective divisors which has some properties similar to that of the case where $D$ is ample and we will study this divisor.

A remarkable infinite product formula was recently posed as a problem by Wilf [6]. Subsequently, Choi and Seo [2] proved Wilf's formula as well as three other analogous product formulas. The main object of this sequel to these earlier works is to present several general infinite product formulas which include, as their special cases, the aforementioned product formulas of Wilf [6] and Choi and Seo [2]. Some other related results, involving the Riemann Zeta function, are also considered briefly.

This paper provides new results on a class of totally umbilical lightlike submanifolds in semi-Riemannian manifolds of constant curvature. We prove that the induced Ricci tensor of any such submanifold is symmetric if and only if its screen distribution is integrable.

Every non-trivial closed curve $C$ on a compact Riemann surface $R$ is freely homotopic to the $r$-fold iterate ${C_0}^r$ of some primitive closed geodesic $C_0$ on $R$. We call $r$ the multiplicity of $C$, and denote it by $N_{R} (C)$. Let $f$ be a non-constant holomorphic map of a compact Riemann surface $R_1$ of genus $g_1$ onto another compact Riemann surface $R_2$ of genus $g_2$ with $g_1 \geq g_2 > 1$, and $C$ a simple closed geodesic of hypebolic length $l_{R_1} (C)$ on $R_1$. In this paper, we give an upper bound for $N_{R_2} (f(C))$ depending only on $g_1$, $g_2$ and $l_{R_1} (C)$.

B.-Y. Chen established a sharp relationship between the Ricci curvature and the squared mean curvature for a submanifold in a Riemannian space form with arbitrary codimension. The Lagrangian version of this inequality was proved by the same author.\newline In this article, we obtain a sharp estimate of the Ricci tensor of a slant submanifold $M$ in a complex space form $\widetilde M(4c)$, in terms of the main extrinsic invariant, namely the squared mean curvature. If, in particular, $M$ is a Kaehlerian slant submanifold which satisfies the equality case identically, then it is minimal.

In the paper we discuss the value distribution of the product of a meromorphic function and its derivative and we improve a recent result of K.W.Yu.

In this paper, we discuss the stability and the pointwise gap phenomenen of $p$-harmonic maps with potential. Stability theorems of $p$-$H$-harmonic maps from or into general submanifolds of the shpere and the Euclidean space are established, and Sealey's quantum theorem is extended. We also discuss the conservation law and the Liouville theorems of $p$-$H$-harmonic maps. As a consequence of our stability theorem, we not only generalize Leung's stability theorem to rather general case, but also improve it by replacing the sectional curvature bound by a Ricci curvature bound. In order to discuss the gap property of $p$-harmonic maps, we establish a Bochner-typed formula which is used by some authors in a uncorrect form.

Cosmologists are taking a renewed interest in multiconnected spherical 3-manifolds (spherical spaceforms) as possible models for the physical universe. To understand the formation of large scale structures in such a universe, cosmologists express physical quantities, such as density fluctuations in the primordial plasma, as linear combinations of the eigenmodes of the Laplacian, which can then be integrated forward in time. This need for explicit eigenmodes contrasts sharply with previous mathematical investigations, which have focused on questions of isospectrality rather than eigenmodes. The present article provides explicit orthonormal bases for the eigenmodes of lens and prism spaces.