It is a well known fact, that for certain polynomials f the relaxed Newton's method N_{f, h}(z)=z−h(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. [12]. In the present paper we will generalize this description to polynomials of higher degree.

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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.

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Let \hat{X} be a smooth connected subvariety of complex projective space Pⁿ. The question was raised in [2] of how to characterize \hat{X} if it admits a reducible hyperplane section \hat{L}. In the case in which \hat{L} is the union of r≥2 smooth normal crossing divisors, each of sectional genus zero, classification theorems were given for dim \hat{X}≥5 or dim \hat{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 P¹-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 \hat{X} with hyperplane section \hat{L} that is the union of two transverse divisors, each of which is either P², a Hirzebruch surface F_r, or \widetilde{F₂}.

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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.

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