Samuel CoskeyDepartment of Mathematics, Boise State University, 1910 University Dr, Boise, ID 83725-1555.
https://scoskey.github.io/
Sat, 12 Aug 2017 21:14:56 +0000Sat, 12 Aug 2017 21:14:56 +0000Jekyll v3.5.1Real and linear analysis<p>Math 515, Fall 2017 (<a href="http://github.com/scoskey/m515">site</a>)<!--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>
Sat, 12 Aug 2017 00:00:00 +0000
https://scoskey.github.io/1718f-515
https://scoskey.github.io/1718f-515real-analysiscourseHonors calculus I<p>Math 170H, Fall 2017<!--more--></p>
<p><em>Catalog description</em>: Definitions of limit, derivative, and integral. Computation of the derivative, including logarithmic, exponential and trigonometric functions. Applications of the derivative, approximations, optimization, mean value theorem. Fundamental theorem of calculus, brief introduction to the applications of the integral and to computations of antiderivatives. Intended for students in engineering, mathematics and the sciences.</p>
Sat, 12 Aug 2017 00:00:00 +0000
https://scoskey.github.io/1718s-170
https://scoskey.github.io/1718s-170calculuscourseThe classification of countable models of set theory<p>With John Clemens and Samuel Dworetzky. (<a href="https://arxiv.org/abs/1707.04660">arχiv</a>)<!--more--></p>
<p><em>Abstract</em>: We study the complexity of the classification problem for countable models of set theory (ZFC). We prove that the classification of arbitrary countable models of ZFC is Borel complete, meaning that it is as complex as it can conceivably be. We then give partial results concerning the classification of countable well-founded models of ZFC.</p>
Mon, 17 Jul 2017 00:00:00 +0000
https://scoskey.github.io/zfcmod
https://scoskey.github.io/zfcmodclassification,set-theorypublicationOn the classification of vertex-transitive structures<p>With John Clemens and Stephanie Potter. (<a href="https://arxiv.org/abs/1707.02383">arχiv</a>)<!--more--></p>
<p><em>Abstract</em>: We consider the classification problem for several classes of countable structures which are “vertex-transitive”, meaning that the automorphism group acts transitively on the elements. (This is sometimes called homogeneous.) We show that the classification of countable vertex-transitive digraphs and partial orders are Borel complete. We identify the complexity of the classification of countable vertex-transitive linear orders. Finally we show that the classification of vertex-transitive countable tournaments is properly above $E_0$ in complexity.</p>
Mon, 10 Jul 2017 00:00:00 +0000
https://scoskey.github.io/vertex-transitive
https://scoskey.github.io/vertex-transitiveclassificationpublicationThe classification problem for models of set theory<p>A master’s thesis by Samuel Dworetzky, Spring 2017 (<a href="http://scholarworks.boisestate.edu/td/1253/">link</a>)<!--more--></p>
<p><em>Abstract</em>: Models of set theory are ubiquitous in modern day set theoretic research. There are many different constructions that produce countable models of ZFC via techniques such as forcing, ultraproducts, and compactness. The models that these techniques produce have many different characteristics; thus it is natural to ask whether or not models of ZFC are classifiable. We will answer this question by showing that models of ZFC are unclassifiable and have maximal complexity. The notions of complexity used in this thesis will be phrased in the language of Borel complexity theory.</p>
<p>In particular, we will show that the class of countable models of ZFC is Borel complete. Most of the models in the construction as it turns out are ill-founded. Thus, we also investigate the sub problem of identifying the complexity for well-founded models. We give partial results for the well-founded case by identifying lower bounds on the complexity for these models in the Borel complexity hierarchy.</p>
Mon, 15 May 2017 00:00:00 +0000
https://scoskey.github.io/the-classification-problem-for-models-of-zfc
https://scoskey.github.io/the-classification-problem-for-models-of-zfcclassification,zfcmasters-thesisOn the classification of vertex-transitive structures<p>A master’s thesis by Stephanie Potter, Spring 2017 (<a href="http://scholarworks.boisestate.edu/td/1279/">link</a>)<!--more--></p>
<p><em>Abstract</em>: When one thinks of objects with a significant level of symmetry it is natural to expect there to be a simple classification. However, this leads to an interesting problem in that research has revealed the existence of highly symmetric objects which are very complex when considered within the framework of Borel complexity. The tension between these two seemingly contradictory notions leads to a wealth of natural questions which have yet to be answered.</p>
<p>Borel complexity theory is an area of logic where the relative complexities of classification problems are studied. Within this theory, we regard a classification problem as an equivalence relation on a Polish space. An example of such is the isomorphism relation on the class of countable groups. The notion of a Borel reduction allows one to compare complexities of various classification problems.</p>
<p>The central aim of this research is determine the Borel complexities of various classes of vertex-transitive structures, or structures for which every pair or elements are equivalent under some element of its automorphism group. John Clemens has shown that the class of vertex-transitive graphs has maximum possible complexity, namely Borel completeness. On the other hand, we show that the class of vertex- transitive linear orderings does not.</p>
<p>We explore this phenomenon further by considering other natural classes of vertex- transitive structures such as tournaments and partial orderings. In doing so, we discover that several other complexities arise for classes of vertex-transitive structures.</p>
Mon, 15 May 2017 00:00:00 +0000
https://scoskey.github.io/on-the-classification-of-vertex-transitive-structures
https://scoskey.github.io/on-the-classification-of-vertex-transitive-structuresclassification,vertex-transitivemasters-thesisIntroduction to linear algebra<p>Math 301, Spring 2017<!--more--></p>
<p><em>Catalog description</em>: Linear algebra from a matrix perspective with applications from the applied sciences. Topics include the algebra of matrices, methods for solving linear systems of equations, eigenvalues and eigenvectors, matrix decompositions, vector spaces, linear transformations, least squares, and numerical techniques.</p>
Tue, 27 Dec 2016 00:00:00 +0000
https://scoskey.github.io/1617s-301
https://scoskey.github.io/1617s-301linear-algebracourseCalculus I<p>Math 170, Spring 2017<!--more--></p>
<p><em>Catalog description</em>: Definitions of limit, derivative, and integral. Computation of the derivative, including logarithmic, exponential and trigonometric functions. Applications of the derivative, approximations, optimization, mean value theorem. Fundamental theorem of calculus, brief introduction to the applications of the integral and to computations of antiderivatives. Intended for students in engineering, mathematics and the sciences.</p>
Tue, 27 Dec 2016 00:00:00 +0000
https://scoskey.github.io/1617s-170
https://scoskey.github.io/1617s-170calculuscourseSolving polynomial equations by radicals<p>A senior thesis by Mack Fox, Fall 2016<!--more--></p>
<p><em>Abstract</em>: This article will show the derivation of closed form radical expressions for polynomials of degree $n\leq4$. For degree one and two polynomials, it is simplistic to show solvability by radicals. For degree three and four polynomials however, these derivations can be quite complex. Due to this, much greater detail is shown throughout those sections. We will also introduce the reader to aspects of Group and Field theory which will serve as a stepping stone to Galois theory. We will use Galois theory to show that for polynomials of degree $n\geq5$, no closed form radical expression for the roots exists.</p>
Sat, 24 Dec 2016 00:00:00 +0000
https://scoskey.github.io/solving-polynomial-equations-by-radicals
https://scoskey.github.io/solving-polynomial-equations-by-radicalsset-theorysenior-thesisAn examination of the euclidean algorithm<p>A senior thesis by Ethan Stieha, Fall 2016<!--more--></p>
<p><em>Introduction</em>: The Euclidean Algorithm was first published in 300 B.C. yet still remains widely useful in solving the greatest common divisor of two computationally large natural numbers. The algorithm provides a step by step process to reduce natural numbers into remainders derived from the division theorem with the same common divisors. While the algorithm itself is rather simple, it has several unique behaviors that make it fascinating to study. As years pass, mathematicians consistently rely on the Euclidean algorithm to be well-conditioned, and provide accurate computational results.</p>
<p><em>Summary</em>: The thesis defines and illustrates the algorithm. It uses experimental methods to investigate the likelihood of each outcome of the algorithm. It then uses both experimental and rigorous methods to examine the case when the outcome is 1, that is, the two inputs are relatively prime.</p>
Sat, 24 Dec 2016 00:00:00 +0000
https://scoskey.github.io/an-examination-of-the-euclidean-algorithm
https://scoskey.github.io/an-examination-of-the-euclidean-algorithmset-theorysenior-thesis