Petya Madzharova
"Time Series Analysis Of Music"
Faculty sponsor: Robin Lock Department: Mathematics

Mathematics and music have always been very closely related to each other. In this project, we are going to discuss some of the mathematical and statistical characteristics of music using techniques from time series analysis. Applying time series analysis to music can be used to understand the composition of a musical piece, to forecast where the melody would go or make a prediction about how a song would end based on the structure of the tune. Using the sheet music of different songs, we first construct a table of the pitches and lengths of the notes of the melody. We will use the statistical program R, and its add-on musical package tuneR, and will see how time series analysis can be applied to this transcription. In particular, we will look at the time series and autocorrelation plots and compare the plots of different kinds of music – for example jazz vs. pop music. Also, we will observe the differences, if any, between music transcribed from its sheet music and a transcription obtained from the actual recording of the same song.

Hilary Hartson
"Sequential Analysis"
Faculty sponsor: Michael Schuckers Department: Mathematics, Computer Science and Statistics

In this talk we discuss Wald's theory of sequential testing as it applies to testing of proportions. Sequential testing differs from traditional hypothesis testing in that it allows for three possible conclusions to be drawn after a subset of samples have been drawn. These three are: reject the null hypothesis, accept the null hypothesis, and continue testing. We derive the appropriate tests for comparing a proportion against a one-sided alternative and we present Monte Carlo simulation results of these tests.

Raluca Dragusanu
"Get Connected: Understanding Reality Through ‘Small-World Graphs"
Faculty sponsor: Dr. Patti Frazer Lock
Department: Mathematics

Power Point Presentation: Graph Theory studies the proprieties of graphs – sets of objects (vertices) connected by links (edges). Among its numerous applications, graph theory has proved to be a powerful tool in understanding the structure of complex networks, of both nature and human design. A large number of complex networks, from a social network to the human neural network, the U.S power grid, or the Internet, have the same underlying structure, which mathematicians call a “small world” graph. The name evidences a phenomenon prevalent in our social network and famous as “six degrees of separation.” A small world graph exhibits a high degree of clustering and a short characteristic path length between vertices. Before discovering the “small world” structure, mathematicians believed that high clustering is associated with ordered networks while short global separation is common for random graphs. As an intermediary between order and randomness, which was before thought to be nonexistent, the “small world graph” becomes even more fascinating. My presentation will first examine how mathematicians were able to formalize and explain phenomena sociologists were observing in our social network. It will proceed by introducing several graph theory concepts and employing them in understanding properties of graphs. The final part of the presentation will look at the most current research in network theory which is mostly concerned with dynamic networks – assuming growth in the number of nodes and preferential attachment. The presentation will end with one of the most fascinating applications of the recent research results, which is the structure of the Internet and the World Wide Web.