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We give an extensive discussion of sphere-preserving maps defined on subdomains of Euclidean -space, and their relationship to Möbius maps and to the preservation of cross-ratios. In the case (the complex plane) we also relate these ideas to the solutions of certain functional equations.
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Ritt has shown that any complex polynomial p can be writtenas the composition of polynomials p1,...,pm, where each pj isprime in the sense that it cannot be written as a non-trivialcomposition of polynomials. The factors pj are not unique butthe number m of them is, as is the set of the degrees of thepj. The paper extends Ritt's theory and, in particular, a thirdinvariant of the decomposition is introduced. 相似文献
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It has been known for a long time that any real sequence y
1
, . . . ,y
n-1
is the sequence of critical values of some real polynomial. Here we show that any complex sequence w
1
, . . . ,w
n-1
is the sequence of critical values of some complex polynomial. 相似文献
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A. F. Beardon 《Proceedings of the American Mathematical Society》2000,128(5):1389-1390
A simple example is given to show that the space of germs obtained by analytic continuation of a given germ need not be a covering space in the topological sense.
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The Schwarz-Pick Lemma for derivatives 总被引:2,自引:0,他引:2
A. F. Beardon 《Proceedings of the American Mathematical Society》1997,125(11):3255-3256
The Schwarz-Pick Lemma states that any analytic function of the unit disc into itself is a contraction with respect to the hyperbolic metric. In this note a related result is proved for the derivative of an analytic function.
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Alan F. Beardon 《Geometriae Dedicata》2003,96(1):205-209
Let D be a disc in the complex plane, and let s
n
be a sequence of Möbius maps each of which maps D into itself. In 1965 Hillam and Thron proved (essentially) that if the points s
n
(), n=12,..., lie in a compact subset of D then the functions s
1^...^s
n
converge locally uniformly to a constant on D. They then made applications to continued fractions. In this paper, the corresponding results are proved in all dimensions. 相似文献
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Let f be a function that is analytic in the unit disc. We givenew estimates, and new proofs of existing estimates, of theEuclidean length of the image under f of a radial segment inthe unit disc. Our methods are based on the hyperbolic geometryof plane domains, and we address some new questions that follownaturally from this approach. 相似文献