This paper is about basic set theory and cardinalities for infinite sets. One of the results are that the line R and the plane R2 contains exactly the same number of points. Because of that the set theory is described with a formal language this the paper has an appendix about formal languages. / Denna uppsats behandlar grundläggande mängdlära och inriktar sig sedan på kardinaliteter för oändliga mängder. Bland de resultat som redovisas finns bland annat resultatet som säger att linjen R och planet R2 innehåller precis lika många punkter. Då mängdläran beskrivs av ett formellt språk så innehåller uppsatsen en bilaga om formella språk.

Due to increase global competitive pressure, shortened product life cycles and ease of imitation, firms must continue to innovate to maintain their competitiveness. Technological innovation has become the primary basis of productivity improvements, sales volume growth, and competitiveness of firms, especially for the high-tech companies. Thus, identification and evaluation of technologies from a variety of perspectives now play important roles in the effective technological sources management. Traditionally, technological innovation studies stressed single model or variable having effects on firm productivity and performance. However, the challenge for business environment is continually changing; single model or variable is not good enough to explain the overall impact of technological innovation. The most difficult aspect of technological innovation performance measurement is the identification of appropriate metrics and approaches that provide information concerning these facets. In this study, the researcher tried to develop a technological innovation performance measurement model and determine tangible and intangible factors from the systematical perspective. That is, technological innovation in its nature is multi-dimensional and multi-criteria. Furthermore, technology innovation performance measurement can be conceptualized as multi-criteria a complex problem which involves the simultaneous consideration of multiple quantitative and qualitative requirements. In this empirical study, the researcher firstly utilizes the Delphi technique to build a hierarchical network structure model for evaluating the technological innovation performance measurement of high tech firms. Secondly, analytic network process (ANP) was applied to determine the importance weights of each dimension and criterion while exists interdependencies among criteria within the same dimension. Thirdly, Non-additive fuzzy integral method was then applied for information fusion and calculates the synthetic performance on a hierarchical network model structure for which criteria are interdependent and interactive. This study applied fuzzy measure and non-additive fuzzy integral method to derive the synthetic performance values of each dimension and firm. Through the technological innovation performance evaluation model can provide firms with an overview of their strengths and weaknesses with regards to technological innovation management. Furthermore, R&D managers and senior managers can apply this model to evaluate and determine the technological innovation capabilities of a firm to improve its technological innovation performance. Finally, this model may provide the useful information for managers and to reduce the overall technological innovation uncertainty.

<p>This paper is about basic set theory and cardinalities for infinite sets. One of the results are that the line R and the plane R2 contains exactly the same number of points. Because of that the set theory is described with a formal language this the paper has an appendix about formal languages.</p> / <p>Denna uppsats behandlar grundläggande mängdlära och inriktar sig sedan på kardinaliteter för oändliga mängder. Bland de resultat som redovisas finns bland annat resultatet som säger att linjen R och planet R2 innehåller precis lika många punkter. Då mängdläran beskrivs av ett formellt språk så innehåller uppsatsen en bilaga om formella språk.</p>

In this dissertation, we present a collection of manuscripts describing the development of prognostic models designed to assist clinical decision making. This work is motivated by limitations of commonly used techniques to produce accessible prognostic models with easily interpretable and clinically credible results. Such limitations hinder prognostic model widespread utilization in medical practice. Our methodology is based on Rough Set Theory (RST) as a mathematical tool for clinical data anal- ysis. We focus on developing rule-based prognostic models for end-of life care decision making in an effort to improve the hospice referral process. The development of the prognostic models is demonstrated using a retrospective data set of 9,103 terminally ill patients containing physiological characteristics, diagnostic information and neurological function values. We develop four RST-based prognostic models and compare them with commonly used classification techniques including logistic regression, support vector machines, random forest and decision trees in terms of characteristics related to clinical credibility such as accessibility and accuracy. RST based models show comparable accuracy with other methodologies while providing accessible models with a structure that facilitates clinical interpretation. They offer both more insight into the model process and more opportunity for the model to incorporate personal information of those making and being affected by the decision.

In the global construction market, most construction companies are willing to undertake international projects in order to maximise their profitability by taking advantage of attractive emerging markets and minimise dependence on unfavorable domestic market conditions. In order to be awarded a contract in highly competitive global construction market, companies should excel in choosing the most attractive markets and prepare winning bids for the selected construction projects in those markets. While preparing bids, the major concern of companies is to offer an optimum price that will enable them to earn enough profits and win the contract at the same time, where profit making ability is strongly correlated with proper estimation of a risk premium that is added onto the estimated cost of the project. Due to the nature of construction works, there are lots of uncertainties associated with the project, market and country conditions. Therefore, how the profitability of the project changes with occurrence of various risk events, in other words, the sensitivity of project costs to risk events, should be estimated by bidders realistically. In this study, fuzzy set theory is used to estimate cost overrun risk in international projects at the bidding stage. The objective is to propose a methodology which can be used by bidders to quantify cost overrun risk so that a realistic risk premium may be determined. A fuzzy risk rating approach is proposed to quantify cost overrun risk rating, which takes into account of risks characterised in international construction projects. For this purpose, risk sources have been identified and a risk model is put forward by using influence diagramming method. Based on this risk model, a fuzzy risk rating algorithm has been defined and software has been developed to conduct fuzzy risk rating calculations easily. After a decision-maker inserts the necessary inputs related with project and country risk factors, the output of the software is a rating that takes into account of all factors that may affect cost overrun risk in international construction projects. The reliability of the algorithm and developed software have been tested by an application on a real construction project. The proposed methodology and decision support tool have been proved to be reliable for the estimation of cost overrun risk while giving bidding decisions in international markets.

The thesis consist of three independent parts. In the first part, we investigate the height sequence of an element of a partially ordered set. Let $x$ be an element of the partially ordered set $P$. Then $h_i(x)$ is the number of linear extensions of $P$ in which $x$ is in the $i$th lowest position. The sequence ${h_i(x)}$ is called the height sequence of $x$ in $P$. Stanley proved in 1981 that the height sequence is log-concave, but no combinatorial proof has been found, and Stanley's proof does not reveal anything about the deeper structure of the height sequence. In this part of the thesis, we provide a combinatorial proof of a special case of Stanley's theorem. The proof of the inequality uses the Ahlswede--Daykin Four Functions Theorem. In the second part, we study two classes of segment orders introduced by Shahrokhi. Both classes are natural generalizations of interval containment orders and interval orders. We prove several properties of the classes, and inspired by the observation, that the classes seem to be very similar, we attempt to find out if they actually contain the same partially ordered sets. We prove that the question is equivalent to a stretchability question involving certain sets of pseudoline arrangements. We also prove several facts about continuous universal functions that would transfer segment orders of the first kind into segments orders of the second kind. In the third part, we consider the lattice whose elements are the subsets of ${1,2,ldots,n}$. Trotter and Felsner asked whether this subset lattice always contains a monotone Hamiltonian path. We make progress toward answering this question by constructing a path for all $n$ that satisfies the monotone properties and covers every set of size at most $3$. This portion of thesis represents joint work with David M.~Howard.

A critical set in a Latin square of order n is a set of entries in an n x n array which can be embedded in precisely one Latin square of order n, with the property that if any entry of the critical set is deleted, the remaining set can be embedded in more than one Latin square of order n. The number of critical sets grows super-exponentially as the order of the Latin square increases. It is difficult to find patterns in Latin squares of small order (order 5 or less) which can be generalised in the process of creating new theorems. Thus, I have written many algorithms to find critical sets with various properties in Latin squares of order greater than 5, and to deal with other related structures. Some algorithms used in the body of the thesis are presented in Chapter 3; results which arise from the computational studies and observations of the patterns and subsequent results are presented in Chapters 4, 5, 6, 7 and 8. The cardinality of the largest critical set in any Latin square of order n is denoted by lcs(n). In 1978 Curran and van Rees proved that lcs(n)<=n2-n. In Chapter 4, it is shown that lcs(n)<=n2-3n+3. Chapter 5 provides new bounds on the maximum number of intercalates in Latin squares of orders mX2^alpha (m odd, alpha>=2) and mX2^alpha+1 (m odd, alpha>=2 and alpha not equal to 3), and a new lower bound on lcs(4m). It also discusses critical sets in intercalate-rich Latin squares of orders 11 and 14. In Chapter 6 a construction is given which verifies the existence of a critical set of size n2 divided by 4 + 1 when n is even and n>=6. The construction is based on the discovery of a critical set of size 17 for a Latin square of order 8. In Chapter 7 the representation of Steiner trades of volume less than or equal to nine is examined. Computational results are used to identify those trades for which the associated partial Latin square can be decomposed into six disjoint Latin interchanges. Chapter 8 focusses on critical sets in Latin squares of order at most six and extensive computational routines are used to identify all the critical sets of different sizes in these Latin squares.

A critical set in a Latin square of order n is a set of entries in an n x n array which can be embedded in precisely one Latin square of order n, with the property that if any entry of the critical set is deleted, the remaining set can be embedded in more than one Latin square of order n. The number of critical sets grows super-exponentially as the order of the Latin square increases. It is difficult to find patterns in Latin squares of small order (order 5 or less) which can be generalised in the process of creating new theorems. Thus, I have written many algorithms to find critical sets with various properties in Latin squares of order greater than 5, and to deal with other related structures. Some algorithms used in the body of the thesis are presented in Chapter 3; results which arise from the computational studies and observations of the patterns and subsequent results are presented in Chapters 4, 5, 6, 7 and 8. The cardinality of the largest critical set in any Latin square of order n is denoted by lcs(n). In 1978 Curran and van Rees proved that lcs(n)<=n2-n. In Chapter 4, it is shown that lcs(n)<=n2-3n+3. Chapter 5 provides new bounds on the maximum number of intercalates in Latin squares of orders mX2^alpha (m odd, alpha>=2) and mX2^alpha+1 (m odd, alpha>=2 and alpha not equal to 3), and a new lower bound on lcs(4m). It also discusses critical sets in intercalate-rich Latin squares of orders 11 and 14. In Chapter 6 a construction is given which verifies the existence of a critical set of size n2 divided by 4 + 1 when n is even and n>=6. The construction is based on the discovery of a critical set of size 17 for a Latin square of order 8. In Chapter 7 the representation of Steiner trades of volume less than or equal to nine is examined. Computational results are used to identify those trades for which the associated partial Latin square can be decomposed into six disjoint Latin interchanges. Chapter 8 focusses on critical sets in Latin squares of order at most six and extensive computational routines are used to identify all the critical sets of different sizes in these Latin squares.

Thesis (Ph. D.)--State University of New York at Binghamton, Thomas J. Watson School of Engineering and Applied Science, Department of Systems Science and Industrial Engineering, 2009. / Includes bibliographical references.

Computing with Words based Question Answering (CWQA) system provides a foundation to develop futuristic search engines where more of reasoning and less of pattern matching and statistical methods are used for information retrieval. In order to perform successful reasoning, these systems should analyze the semantic of the query and the related information in the Knowledge Base. The concept of Computing with Words (CW) which is a kind of perception based reasoning where manipulation of perceptions using fuzzy set theory and fuzzy logic play a key role in recognition, decision and execution processes can be utilized for this purpose. Two concepts that were introduced by Computing with Words are the Generalized Constraint Language (GCL) and the Generalized Theory of Uncertainty (GTU) . In GCL propositions, i.e. perceptions in natural language, are denoted using generalized constraints. The Generalized Theory of Uncertainty (GTU) uses GCL to express proposition drawn from natural language as a generalized constraint. The GCL plays a fundamental role in GTU by serving as a precisiation language for propositions, commands and questions in natural language. In GTU, deduction rules are used to propagate generalized constraints to accomplish reasoning under uncertainty. In the previous work a CW-based QA-system methodology was introduced which uses a knowledge tree data structure, called as a Constraint Propagation Tree (CPT) that utilizes the concepts briefed above. The realization of Constraint Propagation Tree, the first phase, and partial implementation of constraint propagation and node combination, the second phase, is the main goal of this work.