Solving Leontief Input-Output Model with Trapezoidal Fuzzy Numbers and Gauss-Seidel Algorithm

In this work a fuzzy linear system is used to solve Leontief input-output model with fuzzy entries. For solving this model, we assume that the consumption matrix from di erent sectors of the economy and demand are known. These assumptions heavily depend on the information obtained from the industries. Hence uncertainties are involved in this information. The aim of this work is to model these uncertainties and to address them by fuzzy entries such as fuzzy numbers and LR-type fuzzy numbers (triangular and trapezoidal). Fuzzy linear system has been developed using fuzzy data and it is solved using Gauss-Seidel algorithm. Numerical examples show the e ciency of this algorithm. The famous example from Prof. Leontief, where he solved the production levels for U.S. economy in 1958, is also further analyzed.

A quantitative view on fuzzy numbers

Since its introduction, fuzzy set theory has become a useful tool in the mathematical modelling of problems in Operations Research and many other fields. The number of applications is growing continuously. In this thesis we investigate a special type of fuzzy set, namely fuzzy numbers. Fuzzy numbers (which will be considered in the thesis as possibility distributions) have been widely used in quantitative analysis in recent decades. In this work two measures of interactivity are defined for fuzzy numbers, the possibilistic correlation and correlation ratio. We focus on both the theoretical and practical applications of these new indices. The approach is based on the level-sets of the fuzzy numbers and on the concept of the joint distribution of marginal possibility distributions. The measures possess similar properties to the corresponding probabilistic correlation and correlation ratio. The connections to real life decision making problems are emphasized focusing on the financial applications. We extend the definitions of possibilistic mean value, variance, covariance and correlation to quasi fuzzy numbers and prove necessary and sufficient conditions for the finiteness of possibilistic mean value and variance. The connection between the concepts of probabilistic and possibilistic correlation is investigated using an exponential distribution. The use of fuzzy numbers in practical applications is demonstrated by the Fuzzy Pay-Off method. This model for real option valuation is based on findings from earlier real option valuation models. We illustrate the use of number of different types of fuzzy numbers and mean value concepts with the method and provide a real life application.

Numbers and Languages

The thesis presents results obtained during the authors PhD-studies. First systems of language equations of a simple form consisting of just two equations are proved to be computationally universal. These are systems over unary alphabet, that are seen as systems of equations over natural numbers. The systems contain only an equation X+A=B and an equation X+X+C=X+X+D, where A, B, C and D are eventually periodic constants. It is proved that for every recursive set S there exists natural numbers p and d, and eventually periodic sets A, B, C and D such that a number n is in S if and only if np+d is in the unique solution of the abovementioned system of two equations, so all recursive sets can be represented in an encoded form. It is also proved that all recursive sets cannot be represented as they are, so the encoding is really needed. Furthermore, it is proved that the family of languages generated by Boolean grammars is closed under injective gsm-mappings and inverse gsm-mappings. The arguments apply also for the families of unambiguous Boolean languages, conjunctive languages and unambiguous languages. Finally, characterizations for morphisims preserving subfamilies of context-free languages are presented. It is shown that the families of deterministic and LL context-free languages are closed under codes if and only if they are of bounded deciphering delay. These families are also closed under non-codes, if they map every letter into a submonoid generated by a single word. The family of unambiguous context-free languages is closed under all codes and under the same non-codes as the families of deterministic and LL context-free languages.

Composite artistry in the Book of Numbers : a study in biblical narrative conventions

The present thesis discusses the coherence or lack of coherence in the book of Numbers, with special regard to its narrative features. The fragmented nature of Numbers is a well-known problem in research on the book, affecting how we approach and interpret it, but to date there has not been any thorough investigation of the narrative features of the work and how they might contribute to the coherence or the lack of coherence in the book. The discussion is pursued in light of narrative theory, and especially in connection to three parameters that are typically understood to be invoked in the interpretation of narratives: 1) a narrative paradigm, or ‘story,’ meaning events related to each other temporally, causally, and thematically, in a plot with a beginning, middle, and end; 2) discourse, being the expression plane of a narrative, or the devices that an author has at hand in constructing a narrative; 3) the situation or languagegame of the narrative, prototypical examples being factual reports, which seeks to depict a state of affairs, and storytelling narratives, driven by a demand for tellability. In view of these parameters the present thesis argues that it is reasonable to form four groups to describe the narrative material of Numbers: genuine narratives (e.g. Num 12), independent narrative sequences (e.g. Num 5:1-4), instrumental scenes and situations (e.g. Num 27:1-5), and narrative fragments (e.g. Num 18:1). These groups are mixed throughout with non-narrative materials. Seen together, however, the narrative features of these groups can be understood to create an attenuated narrative sequence from beginning to end in Numbers, where one thing happens after another. This sequence, termed the ‘larger story’ of Numbers, concerns the wandering of Israel from Sinai to Moab. Furthermore, the larger story has a fragmented plot. The end-point is fixed on the promised land, Israel prepares for the wandering towards it (Num 1-10), rebels against wandering and the promise and is sent back into the wilderness (Num 13-14), returns again after forty years (Num 21ff.), and prepares for conquering the land (Num 22-36). Finally, themes of the promised land, generational succession, and obedience-disobedience, operate in this larger story. Purity is also a significant theme in the book, albeit not connected to plot in the larger story. All in all, sequence, plot, and theme in the larger story of Numbers can be understood to bring some coherence to the book. However, neither aspect entirely subsumes the whole book, and the four groups of narrative materials can also be understood to underscore the incoherence of the work in differentiating its variegated narrative contents. Numbers should therefore be described as an anthology of different materials that are loosely connected through its narrative features in the larger story, with the aim of informing Israelite identity by depicting a certain period in the early history of the people.