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Description
This paper focuses on the Szemerédi regularity lemma, a result in the field of extremal graph theory. The lemma says that every graph can be partitioned into bounded equal parts such that most edges of the graph span these partitions, and these edges are distributed in a fairly uniform way. Definitions and notation will be established, leading to explorations of three proofs of the regularity lemma. These are a version of the original proof, a Pythagoras proof utilizing elemental geometry, and a proof utilizing concepts of spectral graph theory. This paper is intended to supplement the proofs with background information about the concepts utilized. Furthermore, it is the hope that this paper will serve as another resource for students and others to begin study of the regularity lemma.
ContributorsByrne, Michael John (Author) / Czygrinow, Andrzej (Thesis director) / Kierstead, Hal (Committee member) / Barrett, The Honors College (Contributor) / School of Mathematical and Statistical Sciences (Contributor) / Department of Chemistry and Biochemistry (Contributor)
Created2015-05
Description
Sickle cell disease is a genetic disorder that can cause substantial helath problems. It is the result of a mutation in the DNA coding for hemoglobin. As a result of changes in two important amino acids, a person suffering from sickle cell disease will have erythrocytes that do not maintain the typical biconcave shape and instead for a crescent shape. Individuals with sickle cell disease may have many health problems tied to their irregular hemoglobin. The unusual shape of the erythrocytes leads to a much shorter cell life, which means that even though bone marrow remains active long past childhood to try to keep up with the loss of erythrocytes, the body is still unable to accommodate the rapid death of erythrocytes. The malformed erythrocytes can also cause vascular occlusion, blocking blood vessels and slowing blood flow. While sickle cell disease has the potential to spread worldwide, it is particularly common in Africa. This may be because people with the sickle cell trait have a high resistance to malaria, making them more likely to survive that ubiquitous disease and pass on their traits to their offspring. However, the mortality rate in young children with sickle cell disease is very high, in part because the spleen, already stressed by filtering out dead erythrocytes, has difficulties filtering out bacteria. One of the keys to stopping the spread of the disease is neonatal screening, but this requires specialized equipment that is fairly uncommon in rural areas, as can be seen in Kenya. Therefore, it would be highly beneficial to develop a more cost-effective and widely available method for testing for sickle cell disease.
ContributorsWold, John (Author) / Caplan, Michael (Thesis director) / LaBelle, Jeffrey (Committee member) / Snyder, Jan (Committee member) / Barrett, The Honors College (Contributor)
Created2012-05
Description
Sickle Cell Disease (SCD) is a prevalent genetic disease in Africa, and specifically in Kenya. The lack of available relevant disease education and screening mean that most don't understand the importance of getting testing and many children die before they can get prophylactic care. This project was designed to address the lack of knowledge with supplemental educational materials to be partnered with an engineering capstone project that provides a low cost diagnostic test.
ContributorsShawver, Jamie Christine (Author) / Caplan, Michael (Thesis director) / Snyder, Jan (Committee member) / Barrett, The Honors College (Contributor) / Department of Chemistry and Biochemistry (Contributor) / Harrington Bioengineering Program (Contributor)
Created2014-05
Description
Mathematics education, defined briefly by both students' understanding and teacher instruction, is a cause for concern in the United States. A 1998 comprehensive study conducted by The Third International Mathematics and Science Study (TIMSS) shows that preadolescent mathematics education is comparatively less effective in this country than it is in other countries. The purposes of the present investigation were to understand why mathematics education has its short-comings in the United States, to analyze the most effective ways to help middle grade students learn mathematics, and to examine instructional methods for improving student understanding. The focus is on effective instructional methods because this is an aspect that teachers can directly control and influence. A thorough review of neurological findings and learning theories strongly gave insight into how the preadolescent brain learns best and the investigation further examined the effectiveness of research-based findings by executing a lesson in a 6th grade mathematics classroom and analyzing student results.
ContributorsPatel, Jay Narendra (Author) / Brass, Amber (Thesis director) / White, Darcy (Committee member) / Klem-Deleon, Olga (Committee member) / Barrett, The Honors College (Contributor) / Department of Chemistry and Biochemistry (Contributor) / School of Historical, Philosophical and Religious Studies (Contributor)
Created2013-05
Description
Dividing the plane in half leaves every border point of one region a border point of both regions. Can we divide up the plane into three or more regions such that any point on the boundary of at least one region is on the border of all the regions? In fact, it is possible to design a dynamical system for which the basins of attractions have this Wada property. In certain circumstances, both the Hénon map, a simple system, and the forced damped pendulum, a physical model, produce Wada basins.
ContributorsWhitehurst, Ryan David (Author) / Kostelich, Eric (Thesis director) / Jones, Donald (Committee member) / Armbruster, Dieter (Committee member) / Barrett, The Honors College (Contributor) / School of Mathematical and Statistical Sciences (Contributor) / Department of Chemistry and Biochemistry (Contributor)
Created2013-05