The intent of this paper is to present the reader will enough information to spark a curiosity in to the subject. By no means is the following a complete formulation of any of the topics covered. I want to give the reader a tour of the mathematical landscape. There are plenty of further details to explore in each section, I have just touched the tip the iceberg. The work is basically in four sections: Numbers, Geometry, Functions, Sets and Logic, which are the basic building blocks of Math. The first sections are a exposition into the mathematical objects and their algebras. The last section dives into the foundation of math, sets and logic, and develops the ``language'' of Math. My hope is that after this, the reader will have the necessary (maybe not sufficient) information needed to talk the language of Math.

A previous study on students approach to problems of Set Theory in Iran brought me somehypotheses. These encouraged me to test the hypotheses in this supplementary study. Mypurpose of this study was to consider some students' cognitive difficulties in Set Theory. Iinvestigated students’ conceptual understanding of two major concepts of Set Theory – theconcepts of inclusion and belonging. I also studied the ways students use Venn diagram tofigure out problems in Set Theory. I wanted to examine how students figure out the differentmeanings of words in the natural and formal language. To do so, I analyzed six experiments. Icompared three experiments with the experiments of my previous study in Iran. My researchquestions suggested using a qualitative research method. My theoretical framework built aroundtheories of semiotic activities, which were used in the analysis.The results indicated that students can make unfortunate use of Euler–Venn diagrams. A set ofsets was a difficult concept for students. Some words in natural language, even the word "set" inboth natural and formal language, caused confusion for students when introduced in Settheoretic contexts. Students failed sometimes to distinguish between sets and elements. Theexperiments showed that students’ cultural context had affect on students' cognition regardingmathematical objects.

Typically, design is approached as a sequence of decisions in which designers select what they believe to be the best alternative in each decision. While this approach can be used to arrive at a final solution quickly, it is unlikely to result in the most-preferred solution. The reason for this is that all the decisions in the design process are coupled. To determine the most preferred alternative in the current decision, the designer would need to know the outcomes of all future decisions, information that is currently unavailable or indeterminate. Since the designer cannot select a single alternative because of this indeterminate (interval-based) uncertainty, a set-based design approach is introduced. The approach is motivated by the engineering practices at Toyota and is based on the structure of the Branch and Bound Algorithm. Instead of selecting a single design alternative that is perceived as being the most preferred at the time of the decision, the proposed set-based design approach eliminates dominated design alternatives: rather than selecting the best, eliminate the worst. Starting from a large initial design space, the approach sequentially reduces the set of non-dominated design alternatives until no further reduction is possible ??e remaining set cannot be rationally differentiated based on the available information. A single alternative is then selected from the remaining set of non-dominated designs. In this thesis, the focus is on the elimination step of the set-based design method: A criterion for rational elimination under interval-based uncertainty is derived. To be efficient, the criterion takes into account shared uncertainty ??certainty shared between design alternatives. In taking this uncertainty into account, one is able to eliminate significantly more design alternatives, improving the efficiency of the set-based design approach. Additionally, the criterion uses a detailed reference design to allow more elimination of inferior design sets without evaluating each alternative in that set. The effectiveness of this elimination is demonstrated in two examples: a beam design and a gearbox design.

Real-world problems especially the ones that involve natural systems are complex and they are composed of many non-deterministic components. Uncertainties associated with these non-deterministic components may originate from randomness or from imprecision due to lack of information. Until recently, uncertainty, regardless of its nature or source has been treated by probability concepts. However, uncertainties associated with real-world systems are not limited to randomness. Imprecise, vague or incomplete information may better be represented by other mathematical tools, such as fuzzy set theory, possibility theory, belief functions, etc. New approaches which allow utilization of probability theory in combination with these new mathematical tools found applications in various engineering fields. Uncertainty modeling in human health risk assessment and groundwater resources management areas are investigated in this thesis. In the first part of this thesis two new approaches which utilize both probability theory and fuzzy set theory concepts to treat parameter uncertainties in carcinogenic risk assessment are proposed. As a result of these approaches fuzzy health risks are generated. For the fuzzy risk to be useful for practical purposes its acceptability with respect to compliance guideline has to be evaluated. A new fuzzy measure, the risk tolerance measure, is proposed for this purpose. The risk tolerance measure is a weighed average of the possibility and the necessity measures which are currently used for decision making purposes. In the second part of this thesis two decision making frameworks are proposed to determine the best groundwater resources management strategy in the Savannah region, Georgia. Groundwater resources management problems, especially ones in the coastal areas are complex and require treatment of various uncertain inputs. The first decision making framework proposed in this study is composed of a coupled simulation-optimization model followed by a fuzzy multi-objective decision making approach while the second framework includes a groundwater flow model in which the parameters of the flow equation are characterized by fuzzy numbers and a decision making approach which utilizes the risk tolerance measure proposed in the first part of this thesis.

The growing problem of spam has generated a need for reliable anti-spam filters. There are many filtering techniques along with machine learning and data miming used to reduce the amount of spam. Such algorithms can achieve very high accuracy but with some amount of false positive tradeoff. Generally false positives are prohibitively expensive in the real world. Much work has been done to improve specific algorithms for the task of detecting spam, but less work has been report on leveraging multiple algorithms in email analysis. This study presents an alliance-based approach to classify, discovery and exchange interesting information on spam. Furthermore, the spam filter in this study is build base on the mixture of rough set theory (RST), genetic algorithm (GA) and XCS classifier system. RST has the ability to process imprecise and incomplete data such as spam. GA can speed up the rate of finding the optimal solution (i.e. the rules used to block spam). The reinforcement learning of XCS is a good mechanism to suggest the appropriate classification for the email. The results of spam filtering by alliance-based approach are evaluated by several statistical methods and the performance is great. Two main conclusions can be drawn from this study: (1) the rules exchanged from other mail servers indeed help the filter blocking more spam than before. (2) a combination of algorithms improves both accuracy and reducing false positives for the problem of spam detection.

The research focuses on the application of site selection for cage aquaculture in Taiwan by developing the site evaluation DSS (Decision Support System). The modeling aspect of the system belongs to the domain of multi-criteria decision theories, which AHP (Analytic Hierarchy Process) and Fuzzy Set theory were used. Two case studies based on real world and hypothetical data were conducted to verify the integrity of the system. According to the literature review and the interview with several domain experts, various impact factors were identified first. The corresponding weights of each factor were then decided by analyzing the questionnaires designed based on the concept of AHP. The following work was to evaluate those impact factors based on the experience of domain experts using some appropriate approaches. To represent the domain knowledge, it is appropriate to use rule based inference system. Besides, fuzzy set theory was chosen to describe the antecedent and consequence of the rule base due to the considerations of uncertainty from human experts and ocean field data. Several related mythologies derived from the fuzzy set theory were used, such as the operation of fuzzy composition, determination of suitable membership function, fuzzy relationship matrix, fuzzy inference, defuzzification, and fuzzy pattern classification. All impact factors were categorized into three different types of membership functions that were designed specifically for the site selection of cage aquaculture. The consequence in the rule base, which is the site suitability, was also represented as the unique membership function. To calculate the fuzzy relationship matrix, the current research found that the operation of ¡§algebraic product and bounder sum¡¨ would produce better results than the commonly used ¡§max-min¡¨ operation. Each impact factor would have the associated fuzzy relationship matrix derived from the rule base. The site suitability in term of a fuzzy set can then be inferred by the fuzzy composition of current situation of the factor and the relationship matrix. By multiplying the AHP weight and the fuzzy suitability, the final site suitability index, taking all the impact factors into consideration, can therefore be derived. The real data in Feng-Gang, located in the southern Taiwan, were collected and evaluated using the site selection DSS. The results show Feng-Gang is suitable for the development of cage aquaculture, which is validated by the current prosperous business locally in cage aquaculture. As for the evaluation of multiple sites, 18 hypothetical sites near shore around Taiwan were chosen to calculate the corresponding suitability indexes, which were then be partitioned into several groups using the fuzzy pattern classification. Based on the results, the sites that were classified in the same group have similar cultivation conditions, which also proves the applicability of the site evaluation DSS.

Our work applies techniques for blending and tightening solid shapes represented by sets. We require that the output contain one set and exclude a second set, and then we optimize the boundary separating the two sets. Working within that framework, we present mason, tightening, tight hulls, tight blends, and the medial cover, with details for implementation. Mason uses opening and closing techniques from mathematical morphology to smooth small features. By contrast, tightening uses mean curvature flow to minimize the measure of the boundary separating the opening of the interior of the closed input set from the opening of its complement, guaranteeing a mean curvature bound. The tight hull offers a significant generalization of the convex hull subject to volumetric constraints, introducing developable boundary patches connecting the constraints. Tight blends then use opening to replicate some of the behaviors from tightenings by applying tight hulls. The medial cover provides a means for adjusting the topology of a tight hull or tight blend, and it provides an implementation technique for two-dimensional polygonal inputs. Collectively, we offer applications for boundary estimation, three-dimensional solid design, blending, normal field simplification, and polygonal repair. We consequently establish the value of blending and tightening as tools for solid modeling.

By means of fuzzy set theory as well as Takagi-Sugeno and Mamdani fuzzy controller, this paper presents the investigation of a Resort Management System implemented by a combination of a T-S model and a Mamdani model. It demonstrates the procedure of the specific premise parameters identification and consequence parameters identification performed by regression knowledge in the T-S model, and the process of the fuzzification, the rule base creation and the defuzzification with COG technique in the Mamdani model. Therefore, an aggregation between T-S controller and Mamdani controller applied in the field of management by a novel angle is illustrated, which, as a result, devotes an improved management system that shares great convenience in the control process when combined with mathematics. Moreover, a modification of the conventional Takagi-Sugeno and Mamdani controller is demonstrated in conjunction with fuzzy operations t-norms and OWA by adjusting the -value, which is used in the calculation of final outputs in the T-S model and the computation of rule consequences in the Mamdani model. The algebraic intersection, bounded intersection as well as the -parameter t-norm are the t-norms which are going to be introduced. Besides, we have tested that t-norms generate the same alpha values when the membership degrees meet the boundary with the value of 1 or 0 while OWA can still yield a well-balanced result different from the one computing by minimum operation. Nevertheless both t-norms and OWA are able to shift the alpha-value in a well-adjusted way when the membership degrees lie in the interval [0,1]. A tendency has been shown that alpha-value tends to decrease by means of t-norms and OWA operations and consequently, the final outputs appear to be reduced.

This dissertation provides experimental evidence for the validity of an intonational phonology. The widely used Autosegmental-Metrical theory con- tends that the phonological structure of intonation can be expressed with two tonal targets (L/H tones and derivatives) and retrieved from its phonetic im- plementations. However, it has not been specifically demonstrated so far in a systematic way. This dissertation argues that this view on intonational phonol- ogy considers the phonetic forms of intonation as instances of phonologically structured intonational units forming functionally discrete categories (tones and derivatives). The model of Pattern Recognition for Intonation (PRInt) applies the concepts of categorization (vagueness, prototype, degrees of typicality) to in- tonation in order to abstract the phonological structure of intonational cate- gories from the ranking, by degree of typicality, of their variations in phonetic implementation. First, instances belonging to an intonation category are collected. Sec- ond, a pattern recognition module, relying on the 4-layer structure protocol, extracts a feature vector from the phonetic data of each instance: a sequence of structurally organized tones (L/H tones and derivatives). Third, a fuzzy classifier, using two functions (frequency and similar- ity), organizes the data from the feature vectors of all instances by degree of typicality (grade of membership of values in multisets) and generates the phonological structure of the intonation category, the prototypical pattern, ex- tracted from all instances, and that subsumes them all. It also re-creates the phonetic implementations of the phonological structure but with their features ranked by degree of typicality. This allows the model to distinguish phono- logically distinct structures from phonetic variations of the same phonological structure. The model successfully extracted the phonological intonation structure associated to three modalities of closed questions in French: neutral, doubt- ful, and surprised. It found that neutral and doubtful closed questions are phonologically distinct while surprise is a phonetic allocontour of the neutral modality, in line with prior characterizations of these patterns. It demon- strated that a bi-tonal phonological structure of intonation can be retrieved from phonetic variations. A versatile modeling tool, PRInt will be developed to use its acquired knowledge to evaluate the categorical status of novel instances and to extract multiple phonological units from mixed corpora. / text

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