The objective of this work is twofold. One, we develop a theory of iteration for complex rational functions so that the problems of overflow and indeterminacy caused by null, or almost null, denominators can be avoide...
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The objective of this work is twofold. One, we develop a theory of iteration for complex rational functions so that the problems of overflow and indeterminacy caused by null, or almost null, denominators can be avoided when developing implementations. Second, we present easily implementable methods that allow the calculation of attracting cycles as well as the graphical representation of their basins of attraction. In order to deal with our first goal we work with homogeneous complex coordinates and we take the complex projective line as a model, which is expressed as a quotient of the 3-sphere through the Hopf fibration. An irreducible representation of a rational function can now be presented as a Hopf fibration endomorphism. As well as our second goal is concerned, we use Lyapunov functions and exponents to calculate the cycles associated with endomorphisms and to graphically represent the corresponding basins of attraction. We point out that our algorithms are based on the calculation of a finite set of non-negative real constants and their calculation does not depend on the previous determination of the fixed points or attracting cycles.
This work is devoted to the class of sets in the complex plane which nowadays are known as Carathéodory sets, more precisely speaking, as Carathéodory domains and Carathéodory compact sets. These sets n...
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ISBN:
(数字)9783985475728
ISBN:
(纸本)9783985470723
This work is devoted to the class of sets in the complex plane which nowadays are known as Carathéodory sets, more precisely speaking, as Carathéodory domains and Carathéodory compact sets. These sets naturally arose many times in various research areas in Real, complex and Functional Analysis and in the Theory of Partial Differential Equations. For instance, the concept of a Carathéodory set plays a significant role in such topical themes as approximation in the complex plane, the theory of conformal mappings, boundary value problems for elliptic partial differential equations, etc. The first appearance of Carathéodory domains in the mathematical literature (of course, without the special name at that moment) was at the beginning of the 20th century, when C. Carathéodory published his famous series of papers about boundary behavior of conformal mappings. The next breakthrough result which was obtained with the essential help of this concept is the Walsh–Lebesgue criterion for uniform approximation of functions by harmonic polynomials on plane compacta (1929). Up to now the studies of Carathéodory domains and Carathéodory compact sets remains a topical field of contemporary analysis and a number of important results were recently obtained in this direction. Among them one ought to mention the results about polyanalytic polynomial approximation, where the class of Carathéodory compact sets was one of the crucial tools, and the results about boundary behavior of conformal mappings from the unit disk onto Carathéodory domains. Our aim in the present paper is to give a survey on known results related with Carathéodory sets and to present several new results concerning the matter. Starting with the classical works of Carathéodory, Farrell, Walsh, and passing through the history of complex Analysis of the 20th century, we come to recently obtained results, and to our contribution to the theory.
In relation to the determination of optimal parameters for the alternating direction implicit (ADI) method, the authors consider the third Zolotarev problem in the complex plane, which has been the topic of several re...
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In relation to the determination of optimal parameters for the alternating direction implicit (ADI) method, the authors consider the third Zolotarev problem in the complex plane, which has been the topic of several recent contributions. In the real case, Zolotarev also posed a fourth problem that involves best rational approximants in the Chebyshev sense, and Achieser proved later that the third and fourth Zolotarev problems are actually equivalent. By generalizing Achieser’s argument, it is shown that the complex third Zolotarev problem can be stated as a best Chebyshev approximation by complex rational functions. This enables the authors to use known optimality and uniqueness conditions as well as an iterative algorithm for computing the exact solutions. Some numerical results are reported for rectangular domains of the complex plane that are of practical importance in the ADI method.
It is well known that best complexrational Chebyshev approximants are not always unique and that, in general, they cannot be characterized by the necessary local Kolmogorov condition or by the sufficient global Kolmo...
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It is well known that best complexrational Chebyshev approximants are not always unique and that, in general, they cannot be characterized by the necessary local Kolmogorov condition or by the sufficient global Kolmogorov condition. Recently, Ruttan (1985) proposed an interesting sufficient optimality criterion in terms of positive semidefiniteness of some Hermitian matrix. Moreover, he asserted that this condition is also necessary, and thus provides a characterization of best approximants, in a fundamental case. In this paper we complement Ruttan's sufficient optimality criterion by a uniqueness condition and we present a simple procedure for computing the set of best approximants in case of nonuniqueness. Then, by exhibiting an approximation problem on the unit disk, we point out that Ruttan's characterization in the fundamental case is not generally true. Finally, we produce several examples of best approximants on a real interval and on the unit circle which, among other things, give some answers to open questions raised in the literature.
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