Jekyll2019-08-15T10:48:17+00:00http://scoskey.org/feed.xmlSamuel CoskeyDepartment of Mathematics, Boise State University, 1910 University Dr, Boise, ID 83725-1555.
Computable reducibility of equivalence relations2019-05-15T00:00:00+00:002019-05-15T00:00:00+00:00http://scoskey.org/masters-thesis/computable-reducibility-of-equivalence-relations<p>A master’s thesis by Gianni Krakoff, Spring 2019<!--more--></p>
<p><em>Abstract</em>: Computable reducibility of equivalence relations is a tool to compare the complexity of equivalence relations on natural numbers. Its use is important to those doing Borel equivalence relation theory, computability theory, and computable structure theory. In this thesis, we compare many naturally occurring equivalence relations with respect to computable reducibility. We will then define a jump operator on equivalence relations and study proprieties of this operation and its iteration. We will then apply this new jump operation by studying its effect on the isomorphism relations of well-founded computable trees.</p>A master’s thesis by Gianni Krakoff, Spring 2019Fractals, what are they?2019-05-01T00:00:00+00:002019-05-01T00:00:00+00:00http://scoskey.org/senior-thesis/fractals-what-are-they<p>A senior thesis by Nicole Reese, Spring 2019<!--more--></p>
<p><em>Abstract</em>: This presentation explores the concept of a fractal, the Hausdorff dimension of a fractal, and fractals generated by iterated function systems.</p>A senior thesis by Nicole Reese, Spring 2019The hyperreals; do you prefer non-standard analysis over standard analysis?2019-05-01T00:00:00+00:002019-05-01T00:00:00+00:00http://scoskey.org/senior-thesis/hyperreal-numbers<p>A senior thesis by Chloe Munroe, Spring 2019<!--more--></p>
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<td><em>Abstract</em>: The hyperreal number system $\ast\mathbb R$ forms an ordered field that contains $\mathbb R$ as a subfield as well as infinitely large and small numbers. A number is defined to be infinitely large if $</td>
<td>x</td>
<td>>n$ for all $n = 1, 2, 3, \ldots$ and infinitely small if $</td>
<td>x</td>
<td><1/n$ for all $n = 1, 2, 3\ldots$ This number system is built out of the real number system analogous to Cantor’s construcion of $\mathbb R$ out of $\mathbb Q$. The new entities in $\ast\mathbb R$ and the relationship between the reals and hyperreals provides an appealing alternate approach to real (standard) analysis referred to as nonstandard analysis. This approach is based around that principle that if a property holds for all real numbers then it holds for all hypereal numbers, known as the transfer principle. By only using the fact that $\ast\mathbb R$ is an ordered field that has $\mathbb R$ as a subfield, includes unlimited numbers and satisfies the transfer principle the topics of analysis can be explored.</td>
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</table>A senior thesis by Chloe Munroe, Spring 2019The nametag problem2019-05-01T00:00:00+00:002019-05-01T00:00:00+00:00http://scoskey.org/senior-thesis/the-nametag-problem<p>A senior thesis by Christian Carley, Spring 2019<!--more--></p>
<p><em>Abstract</em>: This paper explores what has been termed, ``The Name Tag Problem’’ (NTP). The problem is framed thusly. A group of $n$ people sit around a table and to each person a name tag has been assigned. How can you assign the nametags so that just one person has the correct name tag assigned, and no matter how many times the table is rotated, still just one person has the correct name tag assigned?</p>A senior thesis by Christian Carley, Spring 2019On splitting and splittable families2019-03-07T00:00:00+00:002019-03-07T00:00:00+00:00http://scoskey.org/splitsplit<p>With Bryce Frederickson, Samuel Mathers, and Hao-Tong Yan.<!--more--></p>
<p><em>Abstract</em>: A set $A$ is said to <em>split</em> a finite set $B$ if exactly half the elements of $B$ (up to rounding) are contained in $A$. We study the dual notions: (1) a <em>splitting family</em>, a collection of sets such that any subset of ${1,\ldots,k}$ is split by a set in the family, and (2) a <em>splittable family</em>, a collection of sets such that there is a single set $A$ that splits each set in the family.</p>
<p>We study the minimum size of a splitting family on ${1,\ldots,k}$, as well as the structure of splitting families of minimum size. We use a mixture of computational and theoretical techniques. We additionally study the related notions of $\mathord{\leq}4$-splitting families and $4$-splitting families, and we provide lower bounds on the minimum size of such families.</p>
<p>Next we investigate splittable families that are just on the edge of unsplittability in several senses. First, we study splittable families that have the fewest number of splitters. We give a complete characterization in the case of two sets, and computational results in the case of three sets. Second, we define the <em>splitting game</em>, and study splittable families for which a splitter cannot be found under adversarial conditions.</p>With Bryce Frederickson, Samuel Mathers, and Hao-Tong Yan.Introduction to continuous logic and model theory for metric structures.2019-03-01T00:00:00+00:002019-03-01T00:00:00+00:00http://scoskey.org/presentation/introduction-to-continuous-logic-and-model-theory-for-metric-structures<p>Boise Set Theory Seminar, March 2019<!--more--></p>Boise Set Theory Seminar, March 2019Foundations of analysis2019-01-01T00:00:00+00:002019-01-01T00:00:00+00:00http://scoskey.org/course/1819s-314<p>Math 314, Spring 2019<!--more--></p>
<p><em>Catalog description</em>: The real number system, completeness and compactness, sequences, continuity, foundations of the calculus.</p>Math 314, Spring 2019Logic and set theory2019-01-01T00:00:00+00:002019-01-01T00:00:00+00:00http://scoskey.org/course/1819s-502<p>Math 502, Spring 2019<!--more--></p>
<p><em>Catalog description</em>: Structured as three five-week components: formal logic, set theory, and topics to be determined by the instructor. The logic component includes formalization of language and proofs, the completeness theorem, and the Lowenheim-Skolem theorem. The set theory component includes orderings, ordinals, the transfinite recursion theorem, and the Axiom of Choice and some of its equivalents.</p>Math 502, Spring 2019Euclidian geometry proposed lesson plans2018-12-17T00:00:00+00:002018-12-17T00:00:00+00:00http://scoskey.org/senior-thesis/euclidean-geometry-lesson-plans<p>A senior thesis by Joe Willert, Fall 2018<!--more--></p>
<p><em>Abstract</em>: We provide several engaging lesson plans that would aid in the teaching of geometry, specifically targeting Euclidian Geometry, towards students of high school age. The audience of this piece would be high school or college students who have not yet had an introduction to geometry, but have completed the standard mathematical courses leading up to this point (i.e. algebra, elementary math, etc.). This being the case the lessons and concepts realized in Chapter 1 target a basic understanding of what Euclidian Geometry is and the subsequent chapters aim specifically at underlying properties of a geometry. The main source of reference for these lessons and this document is the book Foundations of Geometry Second Edition by Gerard A. Venema.</p>
<p>These lessons are laid out as individual lessons that could be taught at any given point of a class that was dealing with the topic of the lesson at the time. These lessons are snapshots of what would be happening in a classroom and the idea is that lessons and teaching happen in-between each of the individual lessons and ideas presented here. Each chapter will begin with a summary of the main concepts and big ideas to be addressed in the chapter. I then offer the general structure of the lesson and how it could be taught. This includes what the teacher would say in the lesson and student misconceptions and questions. My hope is that this document would act as a teaching resource for teachers looking for individual lesson plans to be implemented in their own classroom during moments that they feel are appropriate. A lesson in this paper should take one class period to teach, which I have timed out at an hour. Being that most class periods are about 45 to 50 minutes this can be shortened or it could be spread out over several days as needed and appropriate.</p>A senior thesis by Joe Willert, Fall 2018Pythagorean theorem area proofs2018-12-17T00:00:00+00:002018-12-17T00:00:00+00:00http://scoskey.org/senior-thesis/pythagorean-theorem-area-proofs<p>A senior thesis by Rachel Morley, Fall 2018<!--more--></p>
<p><em>Abstract</em>: This composition is intended to walk the reader through four proofs of the pythagorean theorem that are based on area. It could be used in a classroom to solidify the pythagorean theorem after studying Neutral and Euclidean Geometries.</p>A senior thesis by Rachel Morley, Fall 2018