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Copyright of Educacao e Pesquisa is the property of Faculdade de Educacao da Universidade de Sao Paulo and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The concept of complex numbers (CNs) is used in many disciplines. In many cases, students find it difficult to understand the logic behind CNs. Rotations, vibrations, and oscillations result in sine or cosine waves. Mathematical representation of rotation/vibration/oscillation is done in two ways—trigonometry and complex numbers. But the algebraic calculation is easier if CNs are used instead of trigonometric functions. The use of CNs as an effective representation of sinusoidal variations is discussed in this article. [ABSTRACT FROM AUTHOR]

The goal of this development is introducing a reader the solution of one class comprising trigonometric equations in the Teaching Course of Mathematics by using trigonometric form of the complex numbers. An exemplary approach to solving these equations, suitable for students from 11th to 12th grade, as well as for pupils participating in mathematical camps, olympiads, mathematical competitions, computer mathematics olympiads is considered. [ABSTRACT FROM AUTHOR]

Copyright of Hacettepe University Journal of Education is the property of Hacettepe University Journal of Education and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Abstract: A critical point is a point on which the derivatives of an error function are all zero. It has been shown in the literatures that the critical points caused by the hierarchical structure of the real-valued neural network could be local minima or saddle points, whereas most of the critical points caused by the hierarchical structure are saddle points in the case of complex- valued neural networks. Several studies have demonstrated that that kind of singularity has a negative effect on learning dynamics in neural networks. In this paper, we will demonstrate via some examples that the decomposition of high- dimensional NNs into real-valued NNs equivalent to the original NNs yields the NNs that do not have critical points based on the hierarchical structure. [Copyright &y& Elsevier]

A complex neural network is obtained from an ordinary network by extending the (real-valued) parameters, such as the weights and the thresholds, to complex values. Applications to problems involving complex numbers, such as communications systems, are expected. This paper presents the following uniqueness theorem. When a complex function is given, the three-layered neural network that approximates the function is uniquely determined by a certain finite group, if it is irreducible. The above finite group specifies the redundancy of the parameters in the complex neural network, but has a structure which is different from that of the real-valued neural network. The order of the finite group is examined, and it is shown that the redundancy of the complex-valued neural network is an exponent multiple of the redundancy of the real-valued neural network. Analysis of the redundancy is important in the theoretical investigation of the basic characteristics of complex-valued neural networks, such as the local minimum property. A sufficient condition is derived for the given three-layered complex-valued neural network to be minimal. The above results are shown, in essence, by extending the approach of Sussmann for real-valued neural networks. © 2003 Wiley Periodicals, Inc. Syst Comp Jpn, 34(14): 54–62, 2003; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/scj.10363 [ABSTRACT FROM AUTHOR]