A food web is defined as an acyclic graph where vertices represent different species and there is a directed edge from species x to species y if species x preys on species y. Food webs are important in Biology because they model the flow of energy in an ecosystem. A competition graph has the same vertex set as a food web, but now two vertices are adjacent if they prey on a common species in the food web. Most of these competition graphs are interval graphs. Interval graphs are graphs where vertices can be represented as intervals of the real line such that vertices are adjacent if and only if their intervals overlap. In this talk, we will explain these structures and their relationships with real examples from Biology.
The logarithmic spiral, also known as the growth spiral, is an interesting form in mathematics that happens to be very applicable to the natural world. We explore the structure of this curve, and how this spiral can be used to model the flight pattern of a moth. Specifically, we investigate assumptions used to construct a model for insect flight. Analysis of these underlying assumptions gives insight into possible improvements to the model and limitations of specific models.
The eigenvectors of an electrical network are voltages that, when placed at the boundary vertices, produce boundary currents that are a scalar multiple of the boundary voltage. The objective of this research is to gain information about a given electrical network using eigensystems. In particular, eigensystems are useful when an edges or multiple edges in a network have zero resistance or infinite resistance. Using the eigensystems of an electrical network which are based on the resistance and connectivity of edges in the network, one can determine the location of a dysfunctional resistor in a network or determine the expected number of dysfunctional resistors.
We all enjoy that time of day when we get to sit down and enjoy our favorite meal. We typically eat three or more times per day. However, being a hard-working college student and living on a college budget often severely restricts the amount of money we can spend. Essentials like lunch, dinner, and second dinner could be much cheaper than what we currently spend. Everyone knows that coupons save us money on our everyday needs, but how many people know how to use mathematical logic to solve these expensive cravings? It is possible to reduce the size of your food budget by using equations to calculate the savings from coupons. Let's apply math so we can buy more late-night grub and keep up with the massive amounts of math homework!
My talk will focus on how mathematics can be used in the real world relating to theatre arts, and I will support my discussion by giving real-world examples. Actors and other theatre individuals should be able to take math seriously when the math is involved with a theatrical performance. I will also share some personal stories about theatre regarding how I have found math useful in theatre for budgets, costume designs, ticket sales, and stage design.
First I will give an introduction to fractals, a naturally occurring mathematical set that repeats a detailed pattern visible at any scale. Then I will connect fractals to real-world applications in Geology and Earth Science. I will discuss how fractals are used by geologists with the use of mathematical equations to find the dimensions of detailed forms ranging from bigger surfaces such as coastlines and islands to smaller forms such as leaves. Come find out how scientists relate these findings to tectonic plate slipping and the formation of mountain ranges!
Time: Saturday, April 2, 2016 - 11:55
Room Number: STAG 162
Talk Subject: Numerical Analysis and Scientific Computing
One of the fundamental computations in numerical linear algebra is the QR factorization. A QR factorization decomposes a matrix into the product of an orthogonal matrix and an upper triangular matrix. The algorithms that compute these decompositions can often be costly, and at times, do not perform well for particular matrices. We investigate different methods of computing a QR factorization on a tall and skinny matrix, that is a matrix with more rows than columns. We discuss algorithmic variants and the move to a new family of algorithms based on tiles.
The Lagrange-Relative Equilibrium and the Figure-8 Equilibrium are the only known periodic solutions of the Three-Body problem in the case of equal masses. These equilibria were analyzed by using a finite-difference method to approximate their perturbation-response for various quantities. The resultant behavior is discussed, explained, and classified.
In recent years sparse coding has been employed to efficiently process images. Since recovering sharp images from images corrupted with noise is a well-known ill-posed problem, small perturbations in the image lead to large deviations in the reconstructed image. We look to combine research in superresolution with that of sparse coding for elucidation.
This paper explores the work of Frank P. Ramsey who founded Ramsey’s Theorem and is centered on the fact that complete disorder is impossible. The goal is to dig into Ramsey’s Theory by examining various Ramsey Numbers and bounds. Through this examination of Ramsey Numbers we will begin to see how we can reach structures arbitrarily large and be able to guarantee substructure through the proof of Ramsey’s Theorem.
Encoding Sudoku puzzles as partially colored graphs, we state and prove Akman’s theorem regarding the associated partial chromatic polynomial; we count the 4x4 sudoku boards, in total and fundamentally distinct; we count the diagonally distinct 4x4 sudoku boards; and we classify and enumerate the different structure types of 4x4 boards.
A well-known conjecture of Erd\H os and S\'{o}s states that the Tur\'{a}n number for paths is enough for any tree i.e. a graph $G$ on $n$ vertices and more than $\frac{k-2}{2}n$ edges contains any tree on $k$ vertices.
A natural extension of the problem is the determination of the Tur\'{a}n number of forests. Erd\H os and Gallai considered the graph $H$ consisting of $k$ independent edges. Brandt generalized Erd\H os and Gallai's result by proving that the Tur\'{a}n number for $k$ independent edges is enough for any forest on $k$ edges without isolated vertices. Bushaw and Kettle found the Tur\'{a}n number and extremal graph for the forest with components, which are paths, of the same order $l>2$. \\
Recently, Lidick\'{y}, Liu and Palmer generalized these results by finding the Tur\'{a}n number and extremal graph for the forests with components of paths with arbitrary length. Also, they investigated Tur\'{a}n number for a star forest $F=\cup_{i=1}^{k}{S_{i}}$ where $d_{i}$ is the maximum degree of $S_{i}$ and $d_{1}\geq \dots \geq d_{k}$, and the Tur\'{a}n number for forests with all components of order 4.\\
In this paper, we determine the Tur\'{a}n number of the forest $F=a\cdot P_{\ell}\cup b\cdot S_{t}$ i.e. $a$ disjoint copies of path $P_{\ell}$ and $b$ disjoint copies of star $S_{t}$. In addition, we obtain the Tur\'{a}n number for forest with all components of order 5.
The City of Burns, Oregon has a serious air quality issue. The city frequently experiences very high levels of PM2.5. PM2.5 consists of a variety of particulates whose size is less than 2.5 micrometers. Such particulates can be inhaled and generally accumulate in the lungs of humans, displaying a strong positive correlation with the instance of lung cancers. Field burning and wood burning for heat combined with the unique atmospheric conditions in Burns appear to be contributing to these high levels. In this talk we discuss the methods used in establishing a regression model to predict PM2.5 for Burns. In particular, we worked in collaboration with The National Oceanic and Atmospheric Administration (NOAA) as well as The Oregon Department of Environmental Quality (ODEQ), who had established a preliminary regression equation. Improvements we made to this model were adding interaction terms and including data gathered from the previous day. The resulting equation improved the predictions of PM2.5 by a 10% higher R-squared adjusted and displayed a narrower range of errors. Our model is now implemented as part of a pilot air quality alert system for the City of Burns sent out by ODEQ in collaboration with NOAA.
It is standard practice in analytical chemistry to use linear regression, particularly to calibrate analytical instruments. If a regression line were used to estimate the output of the instrument for a known concentration of analyte, all would be well. However, chemists use this line in reverse, estimating the concentration of an unknown analyte from the output of the instrument. In this talk we explore how this misuse of the regression line influences the estimation of concentration.
Oregon is the U.S.'s top lumber producer and the industry makes up a large portion of the jobs in the Oregon workforce. I will be discussing the use of absorbing Markov chains to model the growth of trees in a stand, then analyzing the model to develop a reliable and steady harvesting schedule.
Come hear about the intricacies of the Rubik's Cube! I will discuss the history of God's number, which is the least amount of moves it takes to solve a scrambled Rubik’s Cube. There are many techniques to solve a scrambled Rubik's Cube. One solution technique which can be modeled by group theory will also be discussed in a way that is understandable to college students who have not yet taken abstract algebra. As an added bonus, I will demonstrate solving a Rubik’s Cube in 10 seconds!
When Newton and Leibniz first developed calculus, they did so by using infinitesimals (really really small numbers). Infinitesimals were used until calculus was made more rigorous by Weierstass. The calculus that we are taught today is based on Weierstass’s ϵ-δ definition of the limit. However, people have been arguing that we go back to an infinitesimal-based calculus, not only for its historical roots, but because many proofs and concepts seem to be much cleaner when using infinitesimals. Using Keisler’s “Elementary Calculus: An Infinitesimal Approach,” our group set out to relearn calculus using infinitesimals. First we will define the hyperreal number line (an extension of the real line that contains the infinitesimals). Then we will walk through the familiar ideas and concepts of single variable calculus, such as limits, derivatives, and integrals, reformulated in terms of hyperreals.
Typically we are free to use associative and commutative properties without problems with real numbers. However, under certain conditions, these can break in infinite series. This talk will look into what happens when these conditions are met.
In the study of Vassiliev Knot Invariants, the algebra of chord diagrams plays a key role. A chord diagram of order n is a circle with 2n vertices around it with chords through the circle connecting the vertices pairwise. The algebra of such diagrams is isomorphic to closed Jacobi diagrams which are trivalent graphs with 2n vertices and a circle around the edge but here the vertices are inside the circle. I used the relationship between these two algebras to find a basis for each in order 5. I write up my results as art pieces so this talk will include many pieces of my art.
Knot Theory is the study of simple closed curves in three dimensions. Complicated structures can be analyzed using three basic moves and knots can be shown to be equivalent. We discuss ways of using physical manipulation to get middle school students interested in mathematics. We also, explore the basic mathematical postulates and theoretical foundation of Link/Knot Theory.
The Cantor Set is a famous set in point-set topology. There are a wide variety of types of Cantor Sets. However, we will only cover the ternary, or standard Cantor Set. In this talk, we will define and discuss what it means for a set to be separable and prove that the Cantor Set satisfies this definition.
We will examine the relationship between Klein links and torus links, using both diagrammatic techniques and link invariants. We begin with definitions of these links and some basic results about Klein links.
We will examine the relationship between Klein links and torus links, using both diagrammatic techniques and link invariants. We determine the types of components in a Klein link, and use this result to look at which Klein links are torus links and which are not.
Time: Saturday, April 2, 2016 - 15:05
Room Number: STAG 160
Talk Subject: Ordinary Differential Equations and Dynamical Systems
Bacteria growth is really important in our life. Some bacteria cells help us overcome different diseases, while others bring those diseases to us. We have learned to produce medicine with help of bacteria growth, like insulin, to help those who are ill. Modeling bacteria growth is an important part of understanding it. Consider a situation of bacteria doubling every half an hour starting with one cell, how much time will it take to fill all the oceans on Earth? Questions like this are answered by developing mathematical models of bacteria growth. We examine modeling bacteria growth using differential equations. Our focus is on model construction and building realistic models that match empirical data.
Prior to the invention of Chua's circuit by Leon Chua in 1983, it was generally believed that it was not possible to design an electronic oscillator which exhibited chaos. In this talk, we provide an overview and analysis of Chua’s chaotic circuit. Principally, we will show the nondimensionalization of the model and a bifurcation analysis for the case where all passive circuit elements have positive values. This analytical approach will be further illustrated by numerical solutions to the state equations for selected parameter values. Finally, we will compare our theoretical results with the output of a circuit simulator and also with measured results from a physical implementation of Chua's circuit. Previous experience with circuit analysis is not necessary to understand this talk.
Dynamical systems both linear and non-linear have the power to describe intricate behavior and provide analysis. In this paper, linear and non-linear models are employed to replicate the interaction between individuals with varying romantic styles. Using traditional analysis methods the goal was to examine the models laid out in Sprott of the dynamic phenomena of love. The graphical outputs and implications between the simple linear and two-dimensional non-linear models were compared; despite identical initial conditions, results varied. This showcases the impact that a variation of parameters has on the system. The non-linear model had an additional logistic function, which made the system more realistic by adding the possibility for emotional reactions. Predictions suggest that with the addition of more parameters, the dynamical system will diverge to chaos.
We discuss multiple methods for modelling the temperature of a bathtub taking into account conductance, evaporation, and turbulence. One model uses a system of ordinary differential equations to represent the temperature with respect to time of multiple materials in the system, while another discretizes the space into uniform cubes propagating heat at discrete time steps.
No one likes a cold bath. When the bath water starts to get cold a person might turn on a constant trickle of water to keep the water tepid. We attempted to model, and optimize, this behavior with differential equations. To do this we simplified the situation and started with a bathtub that had only water in it. We were able to model the heat flow through the bath water using a simple “lumped system” analysis that broke the tub up into 10 equal “plates” of water. The first of these plates was assumed to stay at a constant temperature because of the continual introduction of hot water from the tap. That first plate would then pass some of its heat to the next plate according to thermal conductance and Newton’s Law of Cooling. This model assumed that no heat was lost anywhere in the system except to the next plate of water. To add an element of realism we then added heat loss to the air above each plate. We then attempted to add a person to the tub. We treated the person as if they were a “Heat Sink”. Meaning that they take in heat but their temperature never changes. The rate of heat transfer between the water and the person required different physics from those that we had been using to model the heat flow through the homogeneous tub. We approximated this different rate by using the ratio of the thermal conductance's of water and a person. We also distributed the person’s mass evenly in each plate. This model worked reasonably well. Showing that the water in the tub would all reach the temperature of the incoming water. We then attempted to distribute the person’s mass throughout the tub in a more realistic way, with more mass towards the back of the tub. To do this we introduced a mass ratio term to our equations. This ratio of water mass to person mass did not have the desired results. The water at the back of the tub would continue to decline in temperature, seemingly unaffected by the incoming hot water. This discrepancy is mostly likely caused by the person absorbing more heat than they actually would in reality. The effect of the person’s movements was not included in our models. The shape of the tub was also not explored; our models assume either a perfectly rectangular tub or a perfectly hemispherical tub.
This presentation continues on the course begun in past NUMS presentations. Properties of the Laplacian values of the Circle and Aster, and the Laplacian's role in construction of the heat kernel on both manifolds, are covered. Also covered will be properties of the Laplace Transform values of the circle and the aster. A sheet of fundamental/basic notation and a copy of the paper on which this talk is based are/will be available before March at http://jonathandavidevenboer.weebly.com/blog
Edge scales are musical scales constructed from the edges and vertices of a regular polygon. Regular polygons are polygons that have specific structure and they can be constructed from regular stars.
We will discuss this structure using elements of rational trigonometry and discuss regular stars of order n. A star of order 6 will be used to construct a regular hexagon and motivate the construction of an edge scale. Similar constructions can be used to make other types of musical scales such as stellation scales.
Imagine each directional chute and ladder in the classic board game replaced with bidirectional portals. How does the game change? Expected game play and probabilities of each position are determined by the location and type of eigenvalues. We analyzed the transition matrix and discovered Gershgorin eigenvalue bounds for each board layout were constructed from a limited set of Gershgorin disks. Further, we discovered a minimum bounding region for the eigenvalues, independent of location and number of portals on the board.
The study of the probability that two group elements commute dates back to 1968 with the work of Paul Erdos and Paul Turan. Since then, much has been deduced about these probabilities, including its bound of $5/8$ for nonabelian groups. During this talk, we will look at the associated probabilities of finite nonabelian groups and how to calculate such probabilities using several methods. Furthermore, reports will be made on known probabilities associated with dihedral groups and how to calculate probabilities with specified denominators as well as specified numerators. Finally, we will wrap up with looking at the group of $GL(2,\mathbb{Z}_p)$ matrices and deducing the probability that two of these matrices commute.
Let $F$ be a field containing a primitive $d^{th}$ root of unity $\rho$ and char$(F)\not|d$. Let $A$ be the tensor product of $n$ cyclic $F$-algebras. An element $v\in A$ is Kummer if $v^{d}\in F$. A subspace of $A$ is Kummer if every element is Kummer. Kummer spaces have been used to bound the symbol length of a central simple algebras. In this talk we will discuss some open problems concerning Kummer subspaces
Students currently learn of $\pi$ in a formulaic context; as a value needed for the effective calculation of the circumference or area of a circle. In this talk, we present an activity that uses manipulatives to help middle grade students develop an understanding of the geometrical meaning of $\pi$.
There are multiple causes to Math Anxiety ranging from actual facts to myths about math. Mathematics is a subject that many people struggle with daily. Not just mathematics involved in engineering, but the basic math skills needed for leaving a tip. Some topics discussed will be: how to think about math differently, the fact that there are multiple methods to solve any given math problem, how making mistakes is an important way to learn mathematics, and common causes of Math Anxiety according to research. I will also present several different solutions to help reduce Math Anxiety so that you can help your friends and family today!
In the game theory problem The Traveler's Dilemma, the theoretical and experimental results differ greatly. This discrepancy is explained through evolutionary game theory.
There are many types of product operations defined in graph theory. These products use the vertices and edges of two (not necessarily distinct) graphs to create a new graph. We focus on four important types of graph products: the Cartesian product, the direct product, the strong product, and the lexicographic product. Additionally, we explore ways of labeling graphs through graph coloring and list coloring, and investigate the chromatic number of a graph. Finally, we relate both topics to one another by examining the Hedetniemi conjecture and researching what work as been done to try to prove or disprove the conjecture.
Mathematician Aviezri S. Fraenkel wrote a paper on a game called The Rat Game, which involves mathematical moves and strategies. I plan to inform and expand on some of these moves and add to the game.
A method is presented for predicting future wins and losses in tournament-style games based on previous game outcomes. This method is applied to the recent Division III women's basketball season.
Taking the conclusions found in Candy Crush Combinatorics by Dana Rowland, and expanding them by one color to see how many possible combinations can be found using two rows of candy's and 3 colors.
Microscopic phytoplankton form the basis of the food chain in the earth’s oceans. A system of differential equations relates phytoplankton population and nutrient concentration in an isolated environment. The equations were modeled with MATLAB. I conducted sensitivity analyses to determine the relationships between the system of differential equations, and their respective parameters, on phytoplankton behavior. I explored the dynamics of competition and coexistence between multiple species with multiple nutrients in the system. I was able to predict competitive exclusion, and when multiple species could reach states of coexistence when the equations reach equilibrium.