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This paper is a contribution to the study of countable Borel equivalence relations on standard Borel spaces. We concentrate here on the study of the nature of translation equivalence. We study these known hyperfinite spaces in order to gain insight into the approach necessary to classify certain variables as either being hyperfinite or not. In Chapter 1, we will give the basic definitions and examples of spaces used in this work. The general construction of marker sets is developed in this work. These marker sets are used to develop several invariant tilings of the equivalence classes of specific variables . Some properties that are equivalent to hyperfiniteness in the certain space are also developed. Lastly, we will give the new result that there is a continuous injective embedding from certain defined variables.

The problem with which this paper is concerned is that of investigating a class of topological properties commonly called separation properties. A topological space which satisfies only the definition may be very limited in open sets. By use of the separation properties, specific families of open sets can be guaranteed.

Using postulations formulated by Harlow, very few investigators have experimented with discriminative learning in relation to various levels of human mental abilities to the pattern of forming a set. The present study was designed to investigate the effects of different levels of mental abilities on the formation of these sets, using mental retardates, and analyzing the formation of these sets and the abilities of these retardates to shift dimension of cues by reversing the response conditions.

Given a C*-algebra $\\A$, an irredundant set in $\\A$ is a subset $\\mathcal$ of $\\A$ such that no $a\\in \\mathcal$ belongs to the C*-subalgebra generated by $\\mathcal\\setminus\\{a\\}$. Every separable C*-algebra has only countable irredundant sets and we ask if every nonseparable C*-algebra has an uncountable irredundant set. For commutative C*-algebras, if $K$ is the Kunen line then $C(K)$ is a consistent example of a nonseparable commutative C*-algebra without uncountable irredundant sets. On the other hand, a result due to S. Todorcevic establishes that it is consistent with ZFC that every nonseparable C*-algebra of the form $C(K)$, for a compact 0-dimensional space $K$, has an uncountable irredundant set. By the method of forcing, we construct a nonseparable and noncommutative scattered C*-algebra $\\A$ without uncountable irredundant sets and with no nonseparable abelian subalgebras. On the other hand, we prove that it is consistent that every C*-subalgebra of $\\B(\\ell_2)$ of density continuum has an irredundant set of size continuum. / Given a C*-algebra $\\A$, an irredundant set in $\\A$ is a subset $\\mathcal$ of $\\A$ such that no $a\\in \\mathcal$ belongs to the C*-subalgebra generated by $\\mathcal\\setminus\\{a\\}$. Every separable C*-algebra has only countable irredundant sets and we ask if every nonseparable C*-algebra has an uncountable irredundant set. For commutative C*-algebras, if $K$ is the Kunen line then $C(K)$ is a consistent example of a nonseparable commutative C*-algebra without uncountable irredundant sets. On the other hand, a result due to S. Todorcevic establishes that it is consistent with ZFC that every nonseparable C*-algebra of the form $C(K)$, for a compact 0-dimensional space $K$, has an uncountable irredundant set. By the method of forcing, we construct a nonseparable and noncommutative scattered C*-algebra $\\A$ without uncountable irredundant sets and with no nonseparable abelian subalgebras. On the other hand, we prove that it is consistent that every C*-subalgebra of $\\B(\\ell_2)$ of density continuum has an irredundant set of size continuum.

The context for the development of this work is constructive mathematics without the axiom of countable choice. By constructive mathematics, we mean mathematics done without the law of excluded middle. Our original goal was to give a list of axioms for the real numbers R by only considering the order on R. We instead develop a theory of ordered sets and their completions and a theory of ordered abelian groups. / Includes bibliography. / Dissertation (Ph.D.)--Florida Atlantic University, 2018. / FAU Electronic Theses and Dissertations Collection

鄧國章. / Thesis (M.A.)--香港中文大學硏究院地理學部. / Includes bibliographical references (leaves 215-228). / Deng Guozhang. / Thesis (M.A.)--Xianggang Zhong wen da xue yan jiu yuan di li xue bu. / Chapter 第一章 --- 導論 --- p.1 / Chapter 1.1 --- 研究目的 --- p.1 / Chapter 1.2 --- 研究意義 --- p.7 / Chapter 1.2.1 --- 方法學上的意義 --- p.7 / Chapter 1.2.2 --- 地理學上的意義 --- p.8 / Chapter 1.2.3 --- 應用上的意義 --- p.9 / Chapter 1.3 --- 研究範圍 --- p.10 / Chapter 1.4 --- 論文結構概述 --- p.12 / Chapter 第二章 --- 文獻簡讀 --- p.15 / Chapter 2.1 --- 分類與區劃的意義 --- p.15 / Chapter 2.1.1 --- 分類與區劃 --- p.15 / Chapter 2.1.2 --- 區域的類型 --- p.18 / Chapter 2.1.3 --- 區劃的方法　 --- p.23 / Chapter 2.2 --- 氣候分類 --- p.27 / Chapter 2.2.1 --- 氣候分類的目的及意義　　 --- p.27 / Chapter 2.2.2 --- 氣候分類的類型及方法 --- p.29 / Chapter 2.2.3 --- 柯本氏氣候分類法 --- p.31 / Chapter 2.3 --- 中國氣候區劃 --- p.41 / Chapter 2.3.1 --- 中國氣候區劃歷史概況 --- p.41 / Chapter 2.3.2 --- 應用柯本氏氣候分類法於中國氣候區劃的經驗 --- p.44 / Chapter 2.4 --- 糢糊集合論於分類及區畫間題上的應用 --- p.47 / Chapter 2.4.1 --- 糢糊集合論於分類及區劃上的應用 --- p.47 / Chapter 2.4.2 --- 糢糊集合論於氣候區劃的應用 --- p.50 / Chapter 第三章 --- 研究方法 --- p.53 / Chapter 3.1 --- 應用模糊集合論於氣候分類之理論基璴 　 --- p.53 / Chapter 3.1.1 --- 集合與區域 --- p.53 / Chapter 3.1.2 --- 模糊集合的基本概念及運算 --- p.54 / Chapter 3.2 --- 氣候區的釐訂及區域界綫的劃定方法 --- p.68 / Chapter 3.2.1 --- 氣候區的描述 --- p.68 / Chapter 3.2.2 --- 氣候區的重疊與分割 --- p.75 / Chapter 3.3 --- 數據資料處理方法 --- p.80 / Chapter 3.3.1 --- 數據搜集方法 --- p.80 / Chapter 3.3.2 --- 站點分佈情況 --- p.80 / Chapter 3.3.3 --- 資料整理方法 --- p.82 / Chapter 3.4 --- 注釋 --- p.85 / Chapter 第四章 --- 柯本氏氣候分類系統的普通集合表示 --- p.86 / Chapter 4.1 --- 柯本氏氣候類型可視一組集合 --- p.86 / Chapter 4.2 --- 結果及分析 --- p.96 / Chapter 第五章 --- 柯本氏氣侯分類之模楜集合分析－隸屬度分析 --- p.99 / Chapter 5.1 --- 訂定隸屬涵數方法及區劃步驟 --- p.99 / Chapter 5.2 --- 結果及隸屬度分析 　　 --- p.108 / Chapter 5.3 --- 中國氣候區的劃定 --- p.125 / Chapter 5.4 --- 小結 --- p.140 / Chapter 第六章 --- 柯本氏氣類之模糊集合分析－分割度分析 --- p.143 / Chapter 6.1 --- 分割與區界 --- p.143 / Chapter 6.2 --- 結果及分析 --- p.146 / Chapter 6.3 --- 中國氣候區重疊地帶的劃訂 --- p.155 / Chapter 6.4 --- 小結 --- p.166 / Chapter 6.5 --- 注釋 --- p.169 / Chapter 第七章 --- 柯本氏氣候分類法的改進 --- p.170 / Chapter 7.1 --- 柯本氏氣候分類法的改善 --- p.170 / Chapter 7.2 --- 改良後的柯本氏範式之中國氣候區劃 --- p.186 / Chapter 7.3 --- 與中國植被區的劃配合情況 　 --- p.186 / Chapter 7.4 --- 與中國土壤區劃的配合情況　　 --- p.194 / Chapter 第八章 --- 後語 --- p.208 / Chapter 8.1 --- 研究結果要點重申 --- p.208 / Chapter 8.2 --- 研究限制 --- p.210 / Chapter 8.3 --- 研究展望 --- p.213 / 參考文獻 --- p.215 / 附錄 / Chapter I --- 中國各省、市、自治區志面　 --- p.229 / Chapter II --- 柯氏氣候類型的隸屬函數 --- p.235 / Chapter III --- 電腦計算程式 --- p.254 / Chapter IV --- 隸屬度計算結果 --- p.259 / Chapter （一） --- 原本的柯本氏氣候類型的隸屬度計算結果 --- p.259 / Chapter （二） --- 改良後的柯本氏氣候類型的隸屬度計算結果 --- p.265

by Lam Wai. / Thesis (M.Ph.)--Chinese University of Hong Kong, 1988. / Bibliography: leaves 216-220.

by Yeung Wai Leung. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1995. / Includes bibliographical references (leaves 92-[96]). / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Fuzzy Inference and Real Intelligent Mapping --- p.1 / Chapter 1.2 --- Organization of the thesis --- p.5 / Chapter 2 --- Fuzzy Logic inference --- p.7 / Chapter 2.1 --- Fuzzy logic --- p.7 / Chapter 2.1.1 --- Fuzzy sets --- p.7 / Chapter 2.1.2 --- Operations on fuzzy sets --- p.10 / Chapter 2.2 --- Fuzzy Inference --- p.11 / Chapter 3 --- Weaknesses of fuzzy inference --- p.17 / Chapter 3.1 --- Is the use of linguistic fuzzy if-then rules and membership func- tions a good means of representing human expert knowledge? --- p.17 / Chapter 3.2 --- Role of conventional fuzzy inference doubtful if the expert knowl- edge is in the form of sampled input-output data --- p.21 / Chapter 3.3 --- Computational requirements --- p.23 / Chapter 3.4 --- Low transparency --- p.24 / Chapter 3.5 --- Analytical difficulties --- p.25 / Chapter 4 --- Real Intelligent Mapping --- p.27 / Chapter 5 --- Design of Real Intelligent Mapping Systems Using Dirichlet Tessellation --- p.33 / Chapter 5.1 --- Dirichlet tessellation for function approximation --- p.34 / Chapter 5.2 --- Identification of a DT based RIM system by least-squares --- p.42 / Chapter 5.3 --- Examples --- p.48 / Chapter 5.3.1 --- Defining the problem --- p.48 / Chapter 5.3.2 --- Balancing an inverted pendulum --- p.49 / Chapter 5.3.3 --- Balancing an inverted pendulum with cart --- p.53 / Chapter 5.3.4 --- Truck backing-up --- p.56 / Chapter 5.3.5 --- Chaotic time series prediction --- p.60 / Chapter 5.4 --- Interactive CAD platform for RIM systems design --- p.63 / Chapter 6 --- Analysis of Dirichlet tessellation based Real Intelligent Mapping Systems --- p.67 / Chapter 6.1 --- Local Stability Analysis of DT Based RIM Systems --- p.69 / Chapter 6.1.1 --- Balancing an inverted pendulum --- p.71 / Chapter 6.1.2 --- Truck backing-up --- p.73 / Chapter 6.2 --- Global stability analysis of DT based RIM systems --- p.74 / Chapter 6.3 --- Design of a stable DT based RIM system --- p.79 / Chapter 6.4 --- A method for analyzing Second order DT based RIM systems --- p.82 / Chapter 6.5 --- Piecewise-polynomial real domain representation of a class of fuzzy controller and its stability --- p.85 / Chapter 7 --- Conclusion --- p.90 / Bibliography --- p.92

by Kwok-leung Ho. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1994. / Includes bibliographical references (leaves 60-63). / Chapter Chapter 1 --- Introduction --- p.1 / Chapter Chapter 2 --- Model --- p.7 / Chapter 2.1 --- Basic Beliefs on the vaguely classified variable --- p.7 / Chapter 2.2 --- Properties of the contingency table --- p.13 / Chapter 2.3 --- Mathematical model --- p.21 / Chapter Chapter 3 --- Simulation --- p.32 / Chapter 3.1 --- Likelihood function for the model and the simulation method --- p.32 / Chapter 3.2 --- Simulation result --- p.48 / Chapter Chapter 4 --- Discussion --- p.57 / Reference --- p.60