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1.
Let be a plane curve given by an equation , and let be the affine plane curve given by . Let denote a cyclic covering of determined by . The number is called the Albanese dimension of . In this article, we shall give examples of with the Albanese dimension 2.

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2.
A hollow axis-aligned box is the boundary of the cartesian product of compact intervals in . We show that for , if any of a collection of hollow axis-aligned boxes have non-empty intersection, then the whole collection has non-empty intersection; and if any of a collection of hollow axis-aligned rectangles in have non-empty intersection, then the whole collection has non-empty intersection. The values for and for are the best possible in general. We also characterize the collections of hollow boxes which would be counterexamples if were lowered to , and to , respectively.

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3.
Using smooth one-fixed-point actions on spheres and a result due to Bob Oliver on the tangent representations at fixed points for smooth group actions on disks, we obtain a similar result for perfect group actions on spheres. For a finite group , we compute a certain subgroup of the representation ring . This allows us to prove that a finite perfect group has a smooth -proper action on a sphere with isolated fixed points at which the tangent representations of are mutually nonisomorphic if and only if contains two or more real conjugacy classes of elements not of prime power order. Moreover, by reducing group theoretical computations to number theory, for an integer and primes , we prove similar results for the group , , or . In particular, has Smith equivalent representations that are not isomorphic if and only if , , .

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4.
For let be complex numbers such that is bounded. For define , where . Then the excesses in the sense of Paley and Wiener satisfy .

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5.
Erdös and Szemerédi proved that if is a set of positive integers, then there must be at least integers that can be written as the sum or product of two elements of , where is a constant and . Nathanson proved that the result holds for . In this paper it is proved that the result holds for and .

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6.
We prove that if is an integer greater than one, and are any positive rationals such that for all integers , then

is irrational and is not a Liouville number.

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7.
As an application of the analogue of C-S. Chen's kinematic formula in the 3-dimensional space of constant curvature , that is, Euclidean space , -sphere , hyperbolic space (, respectively), we obtain sufficient conditions for one domain to contain another domain in either an Euclidean space , or a -sphere or a hyperbolic space .

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8.
It is an observation due to J. J. Kohn that for a smooth bounded pseudoconvex domain in there exists such that the -Neumann operator on maps (the space of -forms with coefficient functions in -Sobolev space of order ) into itself continuously. We show that this conclusion does not hold without the smoothness assumption by constructing a bounded pseudoconvex domain in , smooth except at one point, whose -Neumann operator is not bounded on for any .

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9.
Fix integers such that and , and let be the set of all integral, projective and nondegenerate curves of degree in the projective space , such that, for all , does not lie on any integral, projective and nondegenerate variety of dimension and degree . We say that a curve satisfies the flag condition if belongs to . Define where denotes the arithmetic genus of . In the present paper, under the hypothesis , we prove that a curve satisfying the flag condition and of maximal arithmetic genus must lie on a unique flag such as , where, for any , denotes an integral projective subvariety of of degree and dimension , such that its general linear curve section satisfies the flag condition and has maximal arithmetic genus . This proves the existence of a sort of a hierarchical structure of the family of curves with maximal genus verifying flag conditions.

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10.
Let be a complex polynomial of degree having zeros in a disk . We deal with the problem of finding the smallest concentric disk containing zeros of . We obtain some estimates on the radius of this disk in general as well as in the special case, where zeros in are isolated from the other zeros of . We indicate an application to the root-finding algorithms.

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