Complex dynamics is the study of iteration of functions which map the complex plane onto itself. In general, their dynamics are quite complicated and hard to explain but for some simple classes of functions many interesting results can be proved. For example, one often studies the class of rational functions (i.e. quotients of polynomials) or, even more specifically, polynomials. Each such function f partitions the extended complex plane C into two regions, one where iteration of the function is chaotic and one where it is not. The nonchaotic region, called the Fatou Set, is the set of all points z such that, under iteration by f, the point z and all its neighbors do approximately the same thing. The remainder of the complex plane is called the Julia set and consists of those points which do not behave like all closely neighboring points. The Julia set of a polynomial typically has a complicated, self similar structure. Many questions can be asked about this structure. The one that we seek to investigate is the notion of the dimension of the Julia set. While the dimension of a line segment, disc, or cube is familiar, there are sets for which no integer dimension seems reasonable. The notion of Hausdorff dimension gives a reasonable way of assigning appropriate non-integer dimensions to such sets. Our goal is to investigate the behavior of the Hausdorff dimension of the Julia sets of a certain simple class of polynomials, namely fd,c(z) = zd + c. In particular, we seek to determine for what values of c and d the Hausdorff dimension of the Julia set varies continuously with c. Roughly speaking, given a fixed integer d > 1 and some complex c, do nearby values of c have Julia sets with Hausdorff dimension relatively close to each other? We find that for most values of c, the Hausdorff dimension of the Julia set does indeed vary continuously with c. However, we shall also construct an infinite set of discontinuities for each d. Our results are summarized in Theorem 10, Chapter 2. In Chapter 1 we state and briefly explain the terminology and definitions we use for the remainder of the paper. In Chapter 2 we will state the main theorems we prove later and deduce from them the desired continuity properties. In Chapters 3 we prove the major results of this paper.

In this research, the authors aim to identify important dimensions for newly established ventures and the way these dimensions are dealt with. However, this study is limited to small on-line new ventures. In order to achieve this purpose, the authors, after reviewing the literature in the field of e-commerce, have identified five dimensions and have proposed their importance to newly established small internet ventures. Furthermore, the authors tried to check for similarities in the strategies followed by the studied ventures. The dimensions are system qualities, promotion, security, competition and cost. The data was collected from five entrepreneurs who have founded and currently running small online new ventures. The authors found that system qualities, promotion and security were important for all ventures. Whereas, the importance of competition and cost was not shared by all the ventures.

This thesis proposes a new hybrid particle swarm optimizer, which employs landscape estimation and the cooperative behavior of different particles to significantly improve the performance of the original algorithm. The landscape estimation is to explore the landscape of the function in order to predict whether the function is unimodal or multimodal. Then we can decide how to optimize the function accordingly. The cooperative behavior is achieved by using two swarms, in which one swarm explores only a single dimension at a time, and the other explores all dimensions simultaneously. Furthermore, we also propose a movement tracking-based strategy to adjust the maximal velocity of the particles. This strategy can control the exploration and exploitation abilities of the swarm efficiency. Finally, we testify the performance of the proposed approach on a suite of unimodal/multimodal benchmark functions and provide comparisons with other recent variants of the PSO. The results show that our approach outperforms other methods in most of the benchmark problems.

Univ., Diss.--Erlangen-Nürnberg, 2005. / Parallel als CD-ROM-Ausg. erschienen.

Freiburg i. Br., Univ., Diss., 2005.

Thesis (Ph. D.)--Oregon State University, 1990. / Typescript (photocopy). Includes bibliography (leaves 65-67). Also available on the World Wide Web.

Stuttgart, Univ., Diss., 2001.

Universiẗat, Diss., 2002--Konstanz.

Universiẗat, Diss., 2004--Bremen.

Universiẗat, Diss., 1996--Frankfurt (Main).