No abstract

The concepts of the Mandelbrot set and the definition of the stability regions of cycles for rational maps require careful investigation. The standard definition of the Mandelbrot set for the map f : z → z2+ c (the set of c values for which the iteration of the critical point at 0 remains bounded) is inappropriate for meromorphic maps such as the inverse square map. The notion of cycle sets, introduced by Brooks and Matelski [1978] for the quadratic map and applied to meromorphic maps by Yin [1994], facilitates a precise definition of the Mandelbrot parameter space for these maps. Close scrutiny of the cycle sets of these maps reveals generic fractal structures, echoing many of the features of the Mandelbrot set. Computer representations confirm these features and allow the dynamical comparison with the Mandelbrot set. In the parameter space, a purely algebraic result locates the stability regions of the cycles as the zeros of characteristic polynomials. These maps are generalized to quaternions. The powerful theoretical support that exists for complex maps is not generally available for quaternions. However, it is possible to construct and analyze cycle sets for a class of quaternionic rational maps (QRM). Three-dimensional sections of the cycle sets of QRM are nontrivial extensions of the cycle sets of complex maps, while sharing many of their features.

No abstract

The aim of this paper is to look at some important educational aspects of complexity decision making in a multidisciplinary manner from the perspective of General Systems Theory (GST). First, the major issues involved in complexity management and decision making are summarized as they are viewed in literature, and a review of GST and Systems Thinking is given. The discussion in the paper is developed within the context of GST in general, but concentrated on decision making in the three trends of GST: Operations Research, Cybernetics, and Managerial Cybernetics. Here, the role of Cybernetics in complexity decision making is particularly emphasized. The discussion is then extended to the latest developments in complexity decision making in Science of Complexity and Soft Systems Thinking. The study also includes a framework which is expected to guide instructors who are planning to offer contemporary courses on decision making. The framework provides some clues for assessing the level of complexity for a given situation and selecting the appropriate methodology for solution development.

Decision making in most universities is taught within the conventional OR/MS (Operations Research/Management Science) paradigm.  This paradigm is known to be “hard” since it is consisted of mathematical tools, and normally suitable for solving structured problems. In complex situations the conventional OR/MS paradigm proves to be inadequate; decisions must be based on systems thinking which provides the decision maker(s) the opportunity to address the problematic situation in its full system context. The aim here is to develop a framework and a new course to teach “decision making through systems thinking” to industrial engineering students. The new course is to be taken after the existing course which covers the traditional OR/MS paradigm. It is believed that such an “extended” framework will help students to improve their skills in handling complex decision situations.