Tag Archives: Mathematical Sciences

A Digital Database of Integrable Systems and Their Properties

Thomas Hill, Utah State University

Nonlinear evolution equations are a class of partial differential equations used to model many important physical phenomena such as water waves, transmission of signals in optical fibers, and magnetic fields. In the late 1960‰Ûªs Zabusky, Kruskal, and others discovered that some special evolution equations, such as the Korteweg-de Vries (KdV), AKNS, nonlinear Schrodinger, and Harry Dym equations, admit localized traveling wave solutions which are stable under interactions. These traveling wave solutions are called solitons. This discovery created a revolution in mathematics that resulted in a new field of study called integrable systems and soliton theory. The goals of my research are the following: provide a database of the most well known integrable systems and their properties; create computational tools to determine if an integrable system has one of the specific properties listed above; and create software that will allow the calculation of one or more of these properties from the others. Building this database and studying these properties in general requires enormous amounts of computation that have done using computer algebra software packages developed at USU by Dr. Ian Anderson.

Using principle components to estimate representative curves

W. Zachary Horton, Brigham Young University

Recently it was discovered that differences in movement patterns exist among subjects that suffer from chronic ankle instability (CAI). To learn about differences, a study was conducted in that subjects were asked to perform a jump-cut maneuver. During the maneuver angles associated with the knee, ankle, and hip were measured from two planar perspectives over time. This resulted in six curves measured on each individual. Having one representative movement curve for each plane would facilitate interpretation and as a result would be very appealing to practitioners. We show how this can be carried out using functional principle components and curve registration.

Hydrazine Dendrimers

Aleksei Ananin, Southern Utah University

Dendrimers constitute a family of branching polymers. Every consecutive addition of a monomer creates a new generation of dendrimers. Unlike linear polymers, synthesis of dendrimers faces obstacles like steric hindrance when we move towards larger-sized generations. Idea is to create a molecule with unique structure and the greatest possible size. Project utilizes cyanuric chloride as a core molecule, hydrazine as linking units and piperidine as surface groups. Such chemical polymer is unique and never has seen before.

MASS AND ENERGY EXPENDITURE DURING INDOOR TREADWALL ROCK CLIMBING

Taylor Clement, Southern Utah University

PURPOSE: The purpose of this study is to determine to what extent a climber‰Ûªs body mass or added mass impacts the energy expenditure during sport climbing on an indoor treadwall climbing trainer. METHODS: For this study we recruited 8 participants (n=5 men, 3 women) with at least one year of rock climbing experience. Each climber participated in four separate sessions. In session 1, we measured and recorded their baseline weight, height, and body composition. The final three sessions consisted of steady state climbing for four minutes on a treadwall climbing trainer, set at a 6-degree angle under three different loading conditions. Each climber had different loading conditions, set in a random order, consisting of: body weight (BW) ,BW+5-pound weighted vest, and BW+10-pound weighted vest. We measured energy expenditure using a small, portable indirect calorimetry gas analyzer system that was placed in a small backpack (CosMed K4b2, about 1 kg). Subjects also wore a Polar HR monitor during each climbing session to determine relative intensity. Before and after each climbing session we measured their blood lactate levels and handgrip strength using a blood lactate analyzer and a Jamar handgrip dynamometer. We ensured that each climber had at least 24-48 hours of rest in between sessions. The three climbing conditions were compared using ANOVA with a significance set at p=0.05. RESULTS: Among the 8 subjects that participated, the average caloric expenditure and HR were 40.34å±7.3 Kcals and 154.4å±21.4 bpm. When calories per meter of climbing were evaluated, the differences among loading conditions was not significant (p=0.74: BW 1.53å±.27, BW+5: 1.65å±.42, BW+10: 1.1.59å±.28 Kcals‰Û¢m-1). There were no significant differences among loading conditions for hand-grip strength or lactate pre to post. CONCLUSIONS: Climbers are innately aware of exertion and energy expenditure and tend to compensate for added loads by modifying the speed of climbing. This innate modification of effort would suggest that climbers would have similar caloric expenditure for a given climb duration regardless of mass.

Microalgae and Cyanobacteria Harvesting using Electrostatic Potential

Anastasiia Matkovska, Austin Bettridge, Jeff Keller, Utah Valley University

Utah Lake has long been plagued by toxic cyanobacteria that strangle out the other living organisms in the lake. The goal for this project, headed by Dr. Kevin Shurtleff of the UVU Chemistry Department, is to produce a low energy collection device that can gather microalgae and cyanobacteria from Utah Lake. The most effective solution is to remove the microalgae as they grow. This both lowers the population and decreases the concentration of the compounds that they thrive on, such as fertilizers and other pollutants. The biomass collected can also be used in productive ways, particularly when converted to a carbon neutral biofuel. Our hypothesis is that if we put a large voltage across two insulated metal plates, the negatively charged microalgae will be drawn to the positive plate, allowing its collection via a porous conveyor belt running along the top of the positive plate. This process has been shown to work in many applications, such as mining and water desalination. This microalgae harvesting method would be ideal, since we could hypothetically concentrate the microalgae in one continuous process, as opposed to batches using a filter or centrifuge, and at a much lower cost. Our end goal is to be able to mount the entire apparatus onto a boat that can collect microalgae directly from the lake.

Counting Integer Points in Scaled Polytopes

Christopher Vander Wilt and Daniel Gulbrandsen

Utah Valley University

Let nP denote the polytope obtained by expanding the convex integral polytope P⊂R^d by a factor of n in each dimension. Ehrhart [1] proved that the number of lattice (integer) points contained in nP is a rational polynomial of degree d in n. What happens if the polytope is expanded by not necessarily the same factor in each dimension? In this talk a partial answer to this question will be provided, using powers of n as different factors to expand the polytope. It will be shown that the number of lattice points contained in the polytope formed by expanding P by multiplying each vertex coordinate by such a factor is a quasi-polynomial in n. Quasi-polynomials are a generalization of polynomials, where the coefficients of the quasi-polynomials are periodic functions with integral period. Furthermore, particular cases where the number of such lattice points is a polynomial will be presented. In addition, the period of these quasi-polynomials as well as the Law of Reciprocity will be addressed. At the end, future work will be discussed. [1] E. Ehrhart, “Sur les Poly`edres Rationnels Homoth ´etiques `a n Dimensions,” C. R. Acad. Sci. Paris 254 (1962)

Study the Stability of Steady Solutions for a Model of Mutualism

Amy Gifford and Brennon Bauer, Southern Utah University

Mathematical Sciences

Mutualism is the way two organisms of different species exist in a relationship in which each individual benefits from the activity of the other. We study a mathematical model of mutualism. The stability of the steady state solutions of this system will be analyzed. Also, we give some numerical experiments that verify the theoretical results for those steady solutions.

The Divisibility of p^(n)-1 for p>5, p a Prime Number

Jason Adams, Nathan Jewkes and Tyrell Vance, Southern Utah University

Mathematical Sciences

We will study the divisibility of p^(n)-1 where p is a prime number larger than 5 and n is a positive integer. We will generalize the result by considering the case where n is odd and two cases where n is even. We show that when n=2^(k), k an integer greater than 1, 2^(k+2)∙3∙5 is a factor of p^(n)-1. We also show that when n=2^(m)∙l for m a positive integer greater than 1 and l an odd positive integer greater than 1, 2^(m+2)∙3∙5 is a factor of p^(n)-1.

The Musical Phiquence: Finding Phi in Musical Progressions

Sergio Arellano, Snow College

Mathematical Sciences

Why is music so pleasant to? In Western music, we have used what is called the “tempered scale” for centuries, and even though it has gone through changes, it is still largely based around the mathematical principles that Pythagoras created two thousand years ago. Is it possible that the explanation to this phenomenon has to do with the mathematical basis of music, which is unconsciously perceived by the brain? The human brain is known for detecting the underlying mathematical patterns present in many non-music related disciplines, such as visual arts. Independent of the music world, there is a proportion called the Golden Ratio or Phi. It is found in art and geometry, because human eyes tend to find that this proportion produces beauty. This is not a coincidence; this proportion is found everywhere around us, especially in nature. With this in mind, it makes sense to search for the Golden Ratio in the tempered scale to explain the mind’s intuitive appreciation of music. This research discovered the surprising fact that there is a Phi relationship between the first and third degree of the major scale, in terms of the frequency of sounds. The relationship is this: the sum of the frequencies of the notes of the minor III chord divided by the sum of the frequencies of the major I chord in any given major key, tends to be Phi. Another very surprising result was the importance of the number 24 in the frequencies of the major scale. These two particularities help to shed light on why human ears have an untaught comprehension of music; the underlying perception of mathematical relationships by the mind are related to the natural appreciation of it.

Numerical solutions for problems in seepage flow

Ammon Washburn, Brigham Young University

Mathematical Sciences

In many problems with seepage flow, there are non-linear problems that don’t have an easy analytical solution. There is already good research on what can be done in certain situations with these problems. I will present on numerical methods that have been proven to solve certain conditions and then present other solutions for similar problems where the numerical method isn’t so readily available in past research. I will implement the algorithms and compare results.