The purpose of this thesis is to investigate a theory of integration of real-valued functions defined on fields of sets.

This thesis investigates some inequalities and some relationships between function properties and integral properties.

The purpose of this paper is to examine properties of convex sets in linear topological spaces with special emphasis on compact convex sets.

Gives the basic definitions and theorems of similar partially ordered sets; studies finite partially ordered sets, including the problem of combinatorial analysis; and includes the ideas of complete, dense, and continuous partially ordered sets, including proofs.

Wong Man-Lung. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2002. / Includes bibliographical references (leaves 99-103). / Abstracts in English and Chinese. / Abstract --- p.i / Acknowledgments --- p.iii / List of Figures --- p.viii / List of Tables --- p.ix / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Background --- p.1 / Chapter 1.2 --- Objectives --- p.5 / Chapter 2 --- Cartesian Representation of Membership Function --- p.7 / Chapter 2.1 --- The Cartesian Representation --- p.8 / Chapter 2.2 --- Region of Well-defined Membership Functions --- p.10 / Chapter 2.3 --- Similarity Triangle Interpolation Method --- p.12 / Chapter 2.4 --- The Interpolation Example --- p.18 / Chapter 2.5 --- Further Issues --- p.23 / Chapter 2.6 --- Conclusions --- p.24 / Chapter 3 --- Membership Function as Elements in Function Space --- p.26 / Chapter 3.1 --- L2[0，2] Representation --- p.27 / Chapter 3.2 --- "The Inner Product Space of L2[0,2]" --- p.31 / Chapter 3.3 --- The Similarity Triangle Interpolation Method --- p.32 / Chapter 3.4 --- The Interpolation Example --- p.36 / Chapter 3.5 --- Conclusions --- p.48 / Chapter 4 --- Radius of Influence of Membership Functions --- p.50 / Chapter 4.1 --- Previous Works on Mountain Method --- p.51 / Chapter 4.2 --- Combining Mountain Method and Cartesian Representation --- p.56 / Chapter 4.3 --- Extensibility Function and Weighted-Sum-Averaging Equation --- p.61 / Chapter 4.4 --- Radius of Influence --- p.62 / Chapter 4.5 --- Combining Radius of Influence and Fuzzy Interpolation Technique --- p.64 / Chapter 4.6 --- Model Identification Example --- p.66 / Chapter 4.7 --- Eliminative Extraction --- p.67 / Chapter 4.8 --- Eliminative Extraction Example --- p.70 / Chapter 4.9 --- Conclusions --- p.71 / Chapter 5 --- Fuzzy Inferencing --- p.73 / Chapter 5.1 --- Fuzzy Inferencing and Interpolation in Cartesian Representation --- p.74 / Chapter 5.2 --- Sparse Rule Extraction via Radius of Influence and Elimination --- p.77 / Chapter 5.3 --- Single Input and Single Output Case --- p.78 / Chapter 5.4 --- Multiple Input and Single Output Case --- p.81 / Chapter 5.5 --- Application --- p.89 / Chapter 5.6 --- Conclusions --- p.94 / Chapter 6 --- Conclusions --- p.96 / Appendix --- p.99 / Bibliography --- p.99

Xiao, Chang. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 37-[38]). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts also in Chinese.

Li, Chung Fai. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (p. 43). / Abstracts in English and Chinese. / Acknowledgement --- p.ii / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Construction of the State Spaces --- p.5 / Chapter 2.1 --- The Sierpinski Gasket --- p.5 / Chapter 2.1.1 --- Neighbourhood in the Sierpinski Gasket --- p.7 / Chapter 2.2 --- The Sierpinski Carpet --- p.9 / Chapter 2.2.1 --- Neighbourhood in the Sierpinski Carpet --- p.10 / Chapter 3 --- Preliminary Random Processes on Each Level --- p.12 / Chapter 3.1 --- The Sierpinski Gasket --- p.12 / Chapter 3.1.1 --- Definitions --- p.12 / Chapter 3.1.2 --- Properties of the Random Walk --- p.13 / Chapter 3.1.3 --- Preparations for convergence and continuity --- p.16 / Chapter 3.2 --- The Sierpinski Carpet --- p.19 / Chapter 3.2.1 --- The Brownian Motion Bn on Cn --- p.19 / Chapter 3.2.2 --- Properties of Bm(t) --- p.20 / Chapter 3.2.3 --- Exit time for Bn --- p.27 / Chapter 4 --- The limiting process --- p.29 / Chapter 4.1 --- The Sierpinski Gasket --- p.29 / Chapter 4.1.1 --- Convergence and continuity --- p.29 / Chapter 4.1.2 --- Extension from to G --- p.31 / Chapter 4.1.3 --- Markov property --- p.33 / Chapter 4.2 --- The Sierpinski Carpet --- p.34 / Chapter 4.2.1 --- Continuity --- p.34 / Chapter 4.2.2 --- Existence of Markov process on C --- p.37 / Chapter 4.2.3 --- Piecing Together --- p.38 / Bibliography --- p.43

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.

Thesis (M. Phil.)--Chinese University of Hong Kong, 1998. / Description based on contents viewed Mar. 13, 2007; title from title screen. Includes bibliographical references (p. 99-103). Also available in print.

Thesis (Ph. D.) in Physics--University of Maine, 2003. / Includes vita. Includes bibliographical references (leaves 129-143).