On Extremal k-Graphs Without Repeated Copies of 2-Intersecting Edges

The problem of determining extremal hypergraphs containing at most r isomorphic copies of some element of a given hypergraph family was first studied by Boros et al. in 2001. There are not many hypergraph families for which exact results are known concerning the size of the corresponding extremal hypergraphs, except for those equivalent to the classical Turan numbers. In this paper, we determine the size of extremal k-uniform hypergraphs containing at most one pair of 2-intersecting edges for k in {3,4}. We give a complete solution when k=3 and an almost complete solution (with eleven exceptions) when k=4.Comment: 17 pages, 5 figure

Edge coloring BIBDS and constructing MOELRs

Chapter 1 is used to introduce the basic tools and mechanics used within this thesis. Some historical uses and background are touched upon as well. The majority of the definitions are contained within this chapter as well. In Chapter 2 we consider the question whether one can decompose λ copies of monochromatic Kv into copies of Kk such that each copy of the Kk contains at most one edge from each Kv. This is called a proper edge coloring (Hurd, Sarvate, [29]). The majority of the content in this section is a wide variety of examples to explain the constructions used in Chapters 3 and 4. In Chapters 3 and 4 we investigate how to properly color BIBD(v, k, λ) for k = 4, and 5. Not only will there be direct constructions of relatively small BIBDs, we also prove some generalized constructions used within. In Chapter 5 we talk about an alternate solution to Chapters 3 and 4. A purely graph theoretical solution using matchings, augmenting paths, and theorems about the edgechromatic number is used to develop a theorem that than covers all possible cases. We also discuss how this method performed compared to the methods in Chapters 3 and 4. In Chapter 6, we switch topics to Latin rectangles that have the same number of symbols and an equivalent sized matrix to Latin squares. Suppose ab = n2. We define an equitable Latin rectangle as an a × b matrix on a set of n symbols where each symbol appears either [b/n] or [b/n] times in each row of the matrix and either [a/n] or [a/n] times in each column of the matrix. Two equitable Latin rectangles are orthogonal in the usual way. Denote a set of ka × b mutually orthogonal equitable Latin rectangles as a k–MOELR(a, b; n). We show that there exists a k–MOELR(a, b; n) for all a, b, n where k is at least 3 with some exceptions

Annual Report Of the State Geologist, 1925-1926, Vol. 32

What we do: Since 1892, the Iowa Geological and Water Survey (IGWS) has provided earth, water, and mapping science to all Iowans. We collect and interpret information on subsurface geologic conditions, groundwater and surface water quantity and quality, and the natural and built features of our landscape. This information is critical for: Predicting the future availability of economic water supplies and mineral resources. Assuring proper function of waste disposal facilities. Delineation of geologic hazards that may jeopardize property and public safety. Assessing trends and providing protection of water quality and soil resources. Applied technical assistance for economic development and environmental stewardship. Our goal: Providing the tools for good decision making to assure the long-term vitality of Iowa’s communities, businesses, and quality of life. Information and technical assistance are provided through web-based databases, comprehensive Geographic Information System (GIS) tools, predictive groundwater models, and watershed assessments and improvement grants. The key service we provide is direct assistance from our technical staff, working with Iowans to overcome real-world challenges. This report describes the basic functions of IGWS program areas and highlights major activities and accomplishments during calendar year 2011. More information on IGWS is available at http://www.igsb.uiowa.edu/