It is a well known fact, that for certain polynomials $f$ the relaxed Newton's method $N_{f,h}(z) = z - h\frac{f(z)}{f'(z)}$ associated with $f$ has some extraneous attracting cycles. In the case of cubic polynomials the set of these bad conditioned polynomials has been intensively studied and described by means of quasi--holomorphic surgery and holomorphic motions, cf.~\cite{haeseler:1988}. In the present paper we will generalize this description to polynomials of higher degree.

In this paper we construct all the surfaces defined by $n$-valued entire algebroid functions having at least $n$ + 1 exceptional values. And we investigate the number of exceptional values of entire functions on the surfaces. Furthermore we determine the Picard constants of the surfaces under certain conditions.

Let $\hatX$ be a smooth connected subvariety of complex projective space $\pn n$. The question was raised in \cite{CHS} of how to characterize $\hatX$ if it admits a reducible hyperplane section $\hatL$. In the case in which $\hatL$ is the union of $r \geq 2$ smooth normal crossing divisors, each of sectional genus zero, classification theorems were given for $\dim \hatX \geq 5$ or $\dim X=4$ and $r=2$. This paper restricts attention to the case of two divisors on a threefold, whose sum is ample, and which meet transversely in a smooth curve of genus at least $2$. A finiteness theorem and some general results are proven, when the two divisors are in a restricted class including $\pn 1$-bundles over curves of genus less than two and surfaces with nef and big anticanonical bundle. Next, we give results on the case of a projective threefold $\hatX$ with hyperplane section $\hatL$ that is the union of two transverse divisors, each of which is either $\pn 2$, a Hirzebruch surface $\eff_r$, or $\widetilde{\eff_2}$.

Certain types of singular foliations on a manifold have Leibniz algebra structures on the space of multivector fields. Each of them has a structure of a central extension of a Lie algebra in the sense of Leibniz algebra. To a specific Leibniz cohomology class, there corresponds an isomorphism class of central extension of a Leibniz algebra similarly as in the case of Lie algebra.