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.

MEASURING HEAD IMPACT CONTACT PRESSURE IN COLLEGIATE FOOTBALL GAMES TO CORRELATE HEAD KINEMATICS TO BRAIN KINETICS ELUCIDATING BRAIN INJURY DYNAMICS

Does a brain store thoughts and memories the way a computer saves its files? How can a single hit or a fall erase all those memories? Brain Mapping and traumatic brain injuries (TBIs) have become widely researched fields today. Many researchers have been studying TBIs caused to adult American football players however youth athletes have been rarely considered for these studies, contradicting to the fact that American football enrolls highest number of collegiate and high-school children than adults. This research is an attempt to contribute to the field of youth TBIs. Earlier studies have related head kinematics (linear and angular accelerations) to TBIs. However, fewer studies have dealt with brain kinetics (impact pressures and stresses) occurring during head-on collisions. The National Operating Committee on Standards for Athletic Equipment (NOCSAE) drop tests were conducted for linear impact accelerations and the Head Impact Contact Pressures (HICP) calculated from them were applied to a validated FE model. The results showed lateral region of the head as the most vulnerable region to damage from any drop height or impact distance followed by posterior region. The TBI tolerance levels in terms of Von-Mises and Maximum Principal Stresses deduced for lateral impact were 30 MPa and 18 MPa respectively. These levels were corresponding to 2.625 feet drop height. The drop heights beyond this value will result in TBI causing stress concentrations in human head without any detectable structural damage to the brain tissue. This data can be utilized for designing helmets that provide cushioning to brain along with providing a resistance to shear.