As filler in an undergraduate abstract algebra class, I tossed out a question that I had encountered in a high school math puzzler involving the final state of an infinite process. In the process, chips are added and removed from a bag and the question asks what remains at the end of all additions and removals. As the class discussed the solution, more questions arose than we could answer. The question sparked enough interest that an undergraduate and I decided to spend some time exploring the questions raised. Some of the early exploration suggested basic computational techniques, but soon we found that the more interesting questions led through an introduction to cardinality theory and the point-set topology of the real line. In the talk, I will present the original question and discuss some of our results. Hopefully, you will leave the talk with some new questions for your students to ponder.
The STEM-Prep Pathway is designed as two one-semester courses created by the New Mathways Project that prepare students beginning at the elementary algebra-level to succeed in college-level calculus. All lessons are designed to be contextual and meaningful, with guided student inquiry at the core. Each lesson is 25 minutes long and is typically preceded by a short preview assignment, which students complete before class. The preview assignment reviews skills required for the upcoming lesson, and asks readiness questions that are designed to help the student determine if they need additional support prior to class. Each lesson begins with an easily accessible opening question which includes the experiences and opinions of all students. Students complete practice assignments to cement their learning. All lessons include detailed instructor notes suggesting pedagogical approaches, facilitating questions, and the lesson's constructive persistence (CP) level. Early in the course, lessons are designated as CP 1 or 2 as students build their ability to work independently. A CP 3 level promotes productive struggle with engaging problems that are more open-ended. For this talk I will share examples from the curriculum that exemplify these design principles.
We consider a typical gambling situation such as red/black roulette bets of a fixed size, with a limited bankroll, a fixed goal (such as doubling one’s money), and a maximum time of play. Through the use of Markov chains, we analyze the expected value of this game for various bet sizes. Unlike in the classical Gambler’s Ruin problem with no time constraint, we will see there really are “good” and (very!) “bad” bet sizes, which depend on the values of the various constraints.
Two players play a game, taking turns tossing a coin; the winner is the first to reach $n$ heads for some agreed upon value of $n$. What is the probability that the player to go first wins? In this talk, we will consider this question as well as its application to your lunch-time half-court basketball game.
Time: Saturday, April 2, 2016 - 11:15
Room Number: STAG 161
David Hammond*, Oregon Institute of Technology - Wilsonville
This talk will introduce a set of models for describing the behaviour of linear flocks. One application of these models is to describe groups of autonomous automobiles on a one-lane road, where each automobile controls its acceleration based on the differences of its own position and velocity from those of its neighbors. In my talk I will discuss recent theoretical results describing the propagation of perturbations across the flock, occuring when one vehicle is initially given a velocity different than all of the others. We study systems with asymmetric coupling, in which case the transients may travel with different velocities in two directions.
Over 30 arithmetic texts and 280 cyphering books utilized in the United States during the 18th and 19th centuries were examined for subtraction algorithms. A framework for different types of subtraction algorithms utilized at this time will be presented. The investigation revealed that same algorithms were not utilized in arithmetic texts and cyphering books—providing evidence of support that printed books as solitary source of historical investigations are an insufficient representation of this time period in the United States.
Poincare said of pathological functions, "logic sometimes makes monsters." We will investigate some classical examples of the monsters which are continuous but nowhere-differentiable real functions. We will take a historical perspective, with emphasis on visualizing the various examples of these functions.
The Broken Ray transform is a transform used in single-scattering tomography, and was introduced by Lucia Florescu, Vadim A. Markel, and John C. Schotland, in 2009, and shown to be invertible in the case of fixed initial and terminal directions. An inversion formula for this case, involving a second-order derivative and an integration was presented previously in 2012, along with some numerical results. In this talk, we will explore improvements to the numerical inversion, as well as view a numerical inversion from two sets of data that requires only first-order differentiation and no integration.
Point-line configurations in the plane can have many types of symmetries. We will investigate bijections of point-line configurations that preserve a certain structure (automorphisms). Of course, to 'break' these symmetries, we can simply fix all points, but we are interested in the fewest number of points that 'break' all non-trivial automorphisms, called the fixing number. We can view these point-line configurations as matroids and get some more general results.
In the September 2013 issue of Math Horizons Gary Gordon posed the following problem:
For a finite set of points in the plane, write down the following data: For each point P, record the number of 3-point lines through P, the number of 4-point lines through P, and so on. Is there a finite set of points in the plane where each point has a unique nonempty address?
Stan Wagon posted this same problem in the Macalester Problem of the Week forum. We use geometric representations of matroids to find the minimal solution to this problem. Then through construction identify a class of matroids with the property of unique addresses.