SUMMER SCHOLAR PRESENTATIONS Location: Room V206, Mathematics Building (Callaghan Campus) The University of Newcastle Dates: 12:00 pm, Tue, 3rd Feb 2015 - 4:00 pm, Tue, 3rd Feb 2015 Schedule: 12:00 Elliot Catt 12:30 Joshua Hartigan 1:00 Ghislain McKay 1:30 Lachlan O'Neil 2:00 Tom Robinson 2:30 Chris Wright 3:00 Jordan Velich 3:30 Corey Sinnamon (please note slightly altered order) Speaker: Elliot Catt Title: On normal numbers and experimental mathematics      A look into an extension on the proof of a class of normal numbers by Davenport and Erdos, as well as a leap into the world of experimental mathematics relating to the property of strong normality, in particular the strong normality of some very famous numbers. Download: Presentation by Elliot Catt (24MB) Speaker: Joshua Hartigan Title: An Investigation Into Gram Matrices of Rectangular {+1,-1} Matrices      Inspired by the Hadamard Maximal Determinant Problem, we investigate the possible Gram matrices from rectangular {+1, -1} matrices. We can fully classify and count the Gram matrices from rectangular {+1, -1} matrices with just two rows and have conjectured a counting formula for the Gram matrices when there are more than two rows in the original matrix. Download: Presentation by Joshua Hartigan (1.8MB) Speaker: Ghislain McKay Title: Short walks in higher dimensions      We build upon the ideas of short random walks in 2 dimensions in an attempt to understand the behaviours of these objects in higher dimensions. We explore the density and moment functions to find combinatorial and analytical results that generalise nicely. Download: Presentation by Ghislain McKay (480KB) Speaker: Corey Sinnamon Title: I prefer Pi      A history of Pi in the American Mathematical Monthly and the variety of approaches to understanding this stubborn constant. I will focus on the common threads of discussion over the last century, especially the changing methods for computing pi to high precision, to illustrate how we have progressed to our current state. Download: Presentation by Corey Sinnamon (644KB) Speaker: Tom Robinson Title: 1324 avoiding permutations      In this talk I will be exploring certain aspects of permutations of length n that avoid the pattern 1324. This is an interesting pattern in that it is simple yet defies simple analysis. It can be shown that there is a growth rate, yet it cannot be shown what that growth rate is; nor has a explicit formula been found to give the number of permutations of length n which avoid the pattern (whereas this has been found for every other non Wilf-equivalent length 4 pattern). Specifically, this talk will look at how an encoding technique (developed by Bona) of the 1324 avoiding permutations was cleverly used to obtain an upper bound for the growth rate of this class. Speaker: Chris Wright Title: The fairness of voting systems      The fairness of voting systems has been a topic of interest to mathematicians since 1770 when Marquis de Condorcet proposed the Condorcet criterion, and particularly so after 1951 when Kenneth Arrow proposed the Arrow impossibility theorem, which proved that no rank-order voting system can satisfy all properties one would desire. The system I have been studying is known as runoff voting. It is a method of voting used around the world, often for presidential elections such as in France. Each voter selects their favourite candidate, and if any candidate receives above 50% of the vote, then they are elected. If no one reaches this, then another election will be held, but this time with only the top 2 candidates from the previous election. Whoever receives more votes in this second round will be elected. The runoff voting system satisfies a number of desired properties, though the running of the second round can have significant drawbacks. It can be very costly, it can result in periods of time without government, and in it has been known to cause unrest in some politically unstable countries. In my research I have introduced the parameter alpha, which varies the original threshold of 50% for a candidate winning the election in the first round. I am using both analytical methods and simulation to observe how the properties change with alpha. Speaker: Jordan Velich Title: The theorem of Copeland and Erdos on normal numbers      As an extension of Copeland and Erdos' original paper of the same title, we present a clearer and more complete version of the proof that the number of integers up to $N$ ($N$ sufficiently large) which are not $\left(\eps,k\right)$ normal is less than $N^{\gd}$ where $\gd<1$. We also conjecture that the numbers formed from the concatenation of the increasing sequence $a_{1},a_{2},a_{3},\dots$ (provided the sequence is dense enough) are not strongly normal. Download: Presentation by Jordan Velich (288KB) Speaker: Lachlan O'Neil Title: Wave Scattering by a String with Masses and Springs      We consider the problem of scattering of waves by a string with attached masses, focussing on the problem in the time-domain. We propose this as a simple model for more complicated wave scattering problems which arise in the study of elastic metamaterials. We present the governing system of equations and show how we have solved them. Some numerical simulations are also presented. [Permanent link]