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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
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
Description
The deficiency of American primary and secondary schools as compared to schools worldwide has long been documented. The Teaching Gap highlights exactly where our problems might lie, and lays out a plan for how to deal with it. Progress would be slow, but tangible. Our country, however, seems to prefer vast and immediate overhauls that have historically failed (see: New math, etc.). If we had implemented the changes in The Teaching Gap in the decade in which it was written, we would be seeing results by now. Instead, every change we make gets reverted. The Common Core State Standards will prove over the next few years to be either another one of these attempts or a large step in the right direction. It might finally be the latter, as its creation was informed by practices that work best in every state in the US, as well as high- performing countries around the world.
ContributorsMcKee, Emily (Author) / Sande, V. Carla (Thesis director) / Ashbrook, Mark (Committee member) / Schroeder, Darcy (Committee member) / Barrett, The Honors College (Contributor) / College of Liberal Arts and Sciences (Contributor)
Created2012-12
Description
Identifying associations between genotypes and gene expression levels using next-generation technology has enabled systematic interrogation of regulatory variation underlying complex phenotypes. Understanding the source of expression variation has important implications for disease susceptibility, phenotypic diversity, and adaptation (Main, 2009). Interest in the existence of allele-specific expression in autosomal genes evolved with the increased awareness of the important role that variation in non-coding DNA sequences can play in determining phenotypic diversity, and the essential role parent-of-origin expression has in early development (Knight, 2004). As new implications of high-throughput sequencing are conceived, it is becoming increasingly important to develop statistical methods tailored to large and formidably complex data sets in order to maximize the biological insights derived from next-generation sequencing experiments. Here, a Bayesian hierarchical probability model based on the beta-binomial distribution is proposed as a possible approach for quantifying allele-specific expression from whole genome (WGS) and whole transcriptome (RNA-seq) data. Pipeline for the analysis of WGS and RNA-seq data sets from ten samples was developed and implemented, while allele-specific expression (ASE) was quantified from both haplotypes using individuals heterozygous at the tested variants utilizing the described methodology. Both computational and statistical framework applied accurately quantified ASE, achieving high reproducibility of already described allele-specific genes in the literature. In conclusion, described methodology provides a solid starting point for quantifying allele specific expression across whole genomes.
ContributorsMalenica, Ivana (Author) / Craig, David (Thesis director) / Rosenberg, Michael (Committee member) / Szelinger, Szabolcs (Committee member) / Barrett, The Honors College (Contributor) / College of Liberal Arts and Sciences (Contributor)
Created2012-12