Jekyll2019-03-18T03:23:23+00:00http://scoskey.org/feed.xmlSamuel CoskeyDepartment of Mathematics, Boise State University, 1910 University Dr, Boise, ID 83725-1555.
On 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 2018On splitting families2018-10-12T00:00:00+00:002018-10-12T00:00:00+00:00http://scoskey.org/presentation/on-splitting-families<p>Boise Algebra, Geometry, and Combinatorics seminar, October 2018<!--more--></p>
<p><em>Abstract</em>: A set A splits an even-sized set B if A contains exactly half the elements of B. For a natural number k, a splitting family on k is a collection of sets that splits any even-sized subset of {1,…,k}. Variations on the concept of splitting families have appeared in applications of combinatorial search. We investigate the number of sets needed to make a splitting family on k. We give some examples and computational results, as well as theoretical partial results identifying the exact number under certain assumptions. This represents a portion of the work from the Summer 2018 REU CAD, with Bryce Frederickson, Sam Mathers, and Hao-Tong Yan.</p>Boise Algebra, Geometry, and Combinatorics seminar, October 2018Vertex-transitive graphs and linear orders2018-10-01T00:00:00+00:002018-10-01T00:00:00+00:00http://scoskey.org/presentation/vertex-transitive-graphs-and-linear-orders<p>Boise Set Theory Seminar, October 2018<!--more--></p>
<p><em>Abstract</em>: John Clemens proved that the classification of countable vertex-transitive graphs is Borel complete. Extending the terminology from graphs, let’s say that a structure A is vertex-transitive if Aut(A) acts transitively on A. In this talk we will discuss the classification of countable vertex-transitive directed graphs and linear orders. This is joint work with John Clemens and Stephanie Potter.</p>Boise Set Theory Seminar, October 2018Discrete and foundational mathematics2018-08-01T00:00:00+00:002018-08-01T00:00:00+00:00http://scoskey.org/course/1819f-187<p>Math 187, Fall 2018<!--more--></p>
<p><em>Catalog description</em>: An introduction to the language and methods of reasoning used throughout mathematics. Topics include propositional and predicate logic, elementary set theory, proof techniques including mathematical induction,functions and relations, combinatorial enumeration, permutations and symmetry.</p>Math 187, Fall 2018Real and linear analysis2018-08-01T00:00:00+00:002018-08-01T00:00:00+00:00http://scoskey.org/course/1819f-515<p>Math 515, Fall 2018<!--more--></p>
<p><em>Catalog description</em>: Lebesgue measure on the reals, construction of the Lebesgue integral and its basic properties. Advanced linear algebra and matrix analysis. Fourier analysis, introduction to functional analysis.</p>Math 515, Fall 2018