The distribution theory of runs and patterns has a long and rich history. In this dissertation we study the distribution of some run-related statistics in sequences and random permutations of arbitrary multi-sets. Using the finite Markov chain imbedding technique (FMCI), which was proposed by Fu and Koutras (1994), we proposed an alternative method to calculate the exact distribution of the total number of adjacent increasing and adjacent consecutive increasing subsequences in sequences. Fu and Hsieh (2015) obtained the exact distribution of the length of the longest increasing subsequence in random permutations. To the best of our knowledge, little or no work has been done on the exact distribution of the length of the longest increasing subsequence in random permutations of arbitrary multi-sets. Here we obtained the exact distribution of the length of the longest increasing subsequence in random permutations of arbitrary multi-sets. We also obtain the the exact distribution of the length of the longest increasing subsequence for the set of all permutations of length N generated from {1,2,...,n}. / February 2016

Poverty analysis in fisheries is dominated by assumptions of a linear relationship between fishing, income and poverty. Poverty is seen as a function of income, and income as a function of fish catch. Thus, the analytical frameworks to understand poverty in fisheries, and the policies enacted to reduce it, have focused on issues of overexploitation, regulatory mechanisms to maximize rent extraction, and technological innovation to improve fisheries’ productivity. This set of relations is underpinned by the assumption that improving fish catch per se would reduce fishers’ poverty. The study of fishing livelihoods on Lake Kossou in Côte d’Ivoire problematizes some of these assumptions. I revisit the “essentialization” of fishers with fish by utilizing the Sustainable Livelihood Approach as a lens of analysis, and by demonstrating that fishers’ livelihoods are based on a diversified portfolio of activities that span multiple sectors. Looking at livelihoods also questions the validity of the conventional “sites” of poverty analysis in fisheries (i.e. the boat, the landing site) and how these lead to misrepresentations of fishers’ livelihoods by emphasizing the upstream elements (catches) to the detriment of downstream activities in the value chain (processing and trading) that are crucial in the realization of fishers’ sustainable livelihoods. Looking at the complexity of fishers’ livelihoods sheds light on the relations between poverty (as an outcome variable) and vulnerability as a constant condition that is linked to access to multiple types of assets, the institutional contexts in which they operate, and the ways in which access to natural resources is constantly re-negotiated. To that effect, this study shows how access to Lake Kossou took a completely new meaning when the coffee-cocoa economy collapsed and young Ivorians saw it as an opportunity being stolen from them by Malian fishers. The context of post-colonial national identity formation (epitomized in the search for “Ivoirité”) served as political justification for claiming new rights to natural resources that had been relatively unimportant until then in economic terms. Finally, this study provides an innovative approach to poverty analysis by emphasizing its multiple dimensions, and by utilizing the statistical fuzzy sets methodology to construct multidimensional poverty indices.

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The main purpose of this thesis is to find the number of convex sets on a product of two totally ordered spaces. We will give formulas to find this number for specific cases and describe a process to obtain this number for all such spaces. In the first chapter we briefly discuss the motivation behind the work presented in this thesis. Also, the definitions and notation used throughout the paper are introduced here The second chapter starts with examining the product spaces of the form {1; 2; : : : ;n} × {1; 2}. That is, we begin by analyzing a two-row by n-column space for n > N. Three separate approaches are discussed, and verified, to find the total number of convex sets on the space. A general formula is presented to obtain this total for all n. In the third chapter we take the same {1; 2; : : : ;n} × {1; 2} spaces from Chapter 2 and consider all the scenarios for adding a second disjoint convex set to the space. Adding a second convex set gives a collection of two mutually disjoint sets. Again, a general formula is presented to obtain this total number of such collections for all n. The fourth chapter takes the idea from Chapter 2 and expands it to product spaces {1; 2; : : : ;n} × {1; 2; : : : ;m} consisting of more than two rows. Here the creation of convex sets having z rows from those having z − 1 rows is exploited to obtain a model that will give the total number of z-row convex sets on any n × m space, provided the set occupies z adjacent rows. Finally, the fifth chapter describes all possible scenarios for convex sets to be placed in the {1; 2; : : : ;n}×{1; 2; : : : ;m} space. This chapter then explains the process needed to acquire a count of all convex sets on any such space as well. Chapter 5 ends by walking through this process with a concrete example, breaking it down into each scenario. We conclude by briefly summarizing the results and specifying future work we would like to further investigate, in Chapter 6.

A consequence of the Hahn-Banach theorem is the classical bipolar theorem which states that the bipolar of a subset of a locally convex vector pace equals its closed convex hull. The space $\L$ of real-valued random variables on a probability space $\OF$ equipped with the topology of convergence in measure fails to be locally convex so that - a priori - the classical bipolar theorem does not apply. In this note we show an analogue of the bipolar theorem for subsets of the positive orthant $\LO$, if we place $\LO$ in duality with itself, the scalar product now taking values in $[0, \infty]$. In this setting the order structure of $\L$ plays an important role and we obtain that the bipolar of a subset of $\LO$ equals its closed, convex and solid hull. In the course of the proof we show a decomposition lemma for convex subsets of $\LO$ into a "bounded" and "hereditarily unbounded" part, which seems interesting in its own right. (author's abstract) / Series: Working Papers SFB "Adaptive Information Systems and Modelling in Economics and Management Science"

An examination of 14 indexes for determining the number of clusters is conducted on artificial binary data sets being generated according to various design factors. To provide a variety of clustering solutions the data sets are analyzed by different non hierarchical clustering methods. The purpose of the paper is to present the performance and the ability of an index to detect the proper number of clusters in a binary data set under various conditions and different difficulty levels. (author's abstract) / Series: Working Papers SFB "Adaptive Information Systems and Modelling in Economics and Management Science"

Thesis (Ph. D.)--State University of New York at Binghamton, Department of Systems Science and Industrial Engineering, Thomas J. Watson School of Engineering and Applied Science, 2007. / Includes bibliographical references.

Thesis (Ph.D.)--Australian National University, 2000.