Abstract: In this work we introduce the reader to the program of reverse mathematics. This is done by discussing second order arithmetic and constructing the big five subsystems of second order arithmetic used in reverse mathematics. These five subsystems may be used to classify mathematical theorems in terms of their logical strength. A theorem independent of this classication is considered as well. The work concludes with an original article by Hirst and Hughes in which marriage theorems are analyzed via the language of reverse mathematics.
Abstract: Using Brian J. McCartin's "Prelude to Musical Geometry" as a guide, we will look at the geometric link between mathematics and tonal music. When a collection of pitches is heard, either successively (melodically) or simultaneously (harmonically), forming scales or chords, we nd varying intervallic relationships. One of our main goals in investigating this musical link is to understand the mathematical implications of the interval relationships among the members of any set of pitches and systems. Arranging the pitches of an n-tone system in a circle of pitches similar to a clock, we are able to plot the pitches of a chord on our musical clock to see the geometric distances, or semitones, between each note. These pitches are joined together by line segments, forming a polygon that we then try to rotate (transpose) and/or reflect (invert). From what is known about the 12-tone system, we hope to see the same geometric and mathematical representation in a 20-tone and then other general n-tone systems. We ultimately hope to nd analogies between dierent n-tone systems the applications of the same geometric and mathematical representations in order to see which, if any, microtonal systems preserve the same chordal structure and properties present in the 12-tone system that makes up the musical world we know.
Abstract: Voting is a fundamentally mathematical concept; the mathematical field of voting theory was pioneered by Donald Saari. In their paper "Voting, the Symmetric Group, and Representation Theory," Daugherty, Eustis, Minton, and Orrison build off Saari's approach by applying representation theory to the system that Saari has developed. In this paper, we reexamine the results of Daugherty, Eustis, Minton, and Orrison by using a more linear algebra based approach. Specifically, we consider ways to "weight" a voting profile, or, in other words, to score a particular ballot in a vote. This question is important from a practical standpoint in considering the effectiveness or objectivity of voting procedures. We will prove an important result using linear algebra, namely that there are mathematically infinitely ways people can vote that, when coupled with a specific weighting system, could lead to a specific numerical result. However, this particular conclusion is not always applicable to practical voting situations, and therefore we will show what these theoretically infinite ways people could vote actually means in a real-world scenario. We also look at the converse of our original question– whether a particular way that people vote and a particular result can be connected via a weighting system. We also look at the real-world reasonability of this question, and some implications that come from it. While this paper will only consider the mathematics behind this voting theory, the results will certainly be of interest to anyone interested in ensuring that voting accurately represents the interests of the parties involved, as these results will emphasize that the selection of a weight system for a vote is often more important than the actual vote itself.
Abstract: Representation theory of symmetric spaces is an increasingly important field of mathematics that gives useful insight into many areas of science and technology. To better understand generalized symmetric spaces for the special Coxeter groups known as Weyl groups, we investigate the symmetric group, the Weyl groups of the special linear Lie algebras. Given a Weyl group W and an involution theta, we dene the extended symmetric space of W as the set of all elements in W such that theta of w is equal to the inverse of w. In studying this extended symmetric space, we have identied a specic process by which to construct visual representations of the space for any theta-twisted involution of S4. We were able to show an isomorphic relationship between each diagram of S4 generated in this way, which we then extended to give an isomorphic relationship between each diagram for theta-twisted involutions in Sn.
Abstract: Large point sources account for as much as 60% of the carbon dioxide emissions for some countries. Further, in the US one third of all CO2 emissions come from only 311 point sources (power plants, industrial sites, etc.). Because CO2 emissions are seldom measured directly but are generally estimated from related, proxy, and re-purposed data; we also need to understand the uncertainty of these estimates. Simply stated, given a geographic and temporal space on the Earth, what are the CO2 emissions from that space and what is the uncertainty in this estimate? While the US data on large point sources is largely assumed to have no spatial uncertainty, the actual locations of these sources differ by 0.84km on average from their reported locations. Analysis also reveals quantifiable trends in the uncertainty based on simple characteristics such as proximity to water sources, reported location within political boundaries, local and population density. This paper presents a metric to quantify spatial uncertainty in point sources based on the results of this analysis, and explains why point source data cannot be described with traditional methods. To incorporate resolution and placement within a grid cell, a Monte Carlo simulation is used to calculate expected values for emissions for each point source. The spatial uncertainty is then derived from the simulation output to give a picture of the potential spatial spread of the emissions. This is output as gridded data at the desired resolution and can then be incorporated into other data products reporting estimated emissions from point sources.