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Title: Generalized Polygonal Designs
Authors: Tahir, Muhammad Hussain
Keywords: Natural Sciences
Probabilties & applied mathematics
Issue Date: 2010
Publisher: The Islamia University of Bahawalpur Bahawalpur 63100, PAKISTAN
Abstract: Polygonal designs, a class of partially balanced incomplete block designs (PBIBDs) with regular polygons, are useful in survey sampling in terms of balanced sam- pling plans excluding contiguous units (BSECs) and balanced sampling plans to avoid the selection of adjacent units (BSAs), when neighboring (contiguous or ad- jacent) units in a population provide similar information. The reason for using such designs is that the units that are physically close might be more similar than the distant units. By the use of such designs or plans we can select the units over the entire experimental region by avoiding the selection of units that provide es- sentially redundant information. In other words, these neighboring units are de- liberately excluded from being sampled under the idea that they provide little new information to the sampling effort. Searches for polygonal designs may be divided into two broad categories: those which attempts to prove the existence of polygonal designs with a given set of pa- rameters (v, k, λ, α), and those which attempts to construct (or enumerate) polygo- nal designs with a given set of parameters (v, k, λ, α). In this thesis, the construction of cyclic polygonal designs is generalized for the parameters: the distance α (or m), the concurrence (or index) parameter λ and the treatments v. The major reasons for introducing generalized cyclic polygonal designs in this thesis are that: (i) the existing literature considers the existence and the construction of cyclic polygonal designs only for the limited distance α, the concurrence param- eter λ and the treatments v; iii(ii) the existence and the construction of unequal block sized cyclic polygonal de- signs for distance α ≥ 1 has not been attempted in literature. In Chapter 1, an introduction to polygonal designs is given. A brief review on the existing work on polygonal designs is presented, and some limitations in the existing work are pointed out. In Chapter 2, the method of cyclic shifts is briefly described, and explained that how this method helps in the development of concurrence matrix (or concurrence vector) which is the main tool for the detection of the properties of cyclic polygonal designs. The distinguishing feature of this method is that the properties of a design can easily be obtained from the sets of shifts instead of constructing the actual blocks of the design. The pattern of off-diagonal zero elements (in bold form) from the main diagonal in a concurrence matrix (or in a concurrence vector) is useful in the identification of the distance in a cyclic polygonal design. In Chapter 3, minimal cyclic polygonal designs with block size k = 3 and λ = 1 are constructed for distance α = 1, 2, 3, . . . , 16 and for v < 100 treatments. In Chapter 4, the existence and construction of cyclic polygonal designs with block size k = 3, for λ = 1, 2, 3, 4, 6, 12 and for α = 1, 2, 3, 4, 5, 6 is considered, and complete solutions for v ≤ 100 treatments are presented. In Chapter 5, the existence and construction of minimal cyclic polygonal designs with unequal-sized blocks and λ = 1 is first ever introduced for distance α ≥ 1. In Chapter 6, the thesis is summarized and future directions for the extension of cyclic polygonal designs are proposed.
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