Tag Archives: Mathematical Sciences

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.

Higher Dimensional Smooth Data Interpolation Techniques from Computational Geometry

Ariel Herbert-Voss, University of Utah

Mathematical Sciences

A typical problem in numerical analysis is finding a smooth interpolation of a given data set such that information at extended positions can be evaluated. When extended to higher dimensions, there are few such algorithms available for practical use. Drawing from techniques used in geometric modeling we developed a practical algorithm with improved complexity by implementing the techniques in a query model as part of a MATLAB software package. From initial input data the algorithm builds a d-dimensional cell complex using Delaunay triangulation. Each cell has an associated interpolation function that satisfies Lipschitz continuity for each new point. During query time the user specifies a query point and the algorithm returns the interpolated function value. To reduce complexity related to point location within the cell complex, we implemented a binary tree search based on hyperplane decision criteria. Efficiency analysis completed using benchmark data sets indicated that the decision tree algorithm improved the efficiency from O(N) to O(N log N). This algorithm is the first of its kind that can be used on actual data sets and is the first implemented as a MATLAB package.

Residues and Independence Numbers of Graphs

Grant Molnar, Brigham Young University

Mathematical Sciences

One important attribute of a network of points, or graph, is its independence number: the maximum size of a set of points (vertices) in which no two vertices are joined by an edge. The degree sequence of a graph is a list recording the number of edges that meet at each vertex. The residue of a graph is a number computed by an algorithm that reduces the degree sequence to a string of zeroes. Calculating the independence number of large graphs in general is believed to be computationally hard. However, the residue can be calculated much more easily and is a lower bound for the independence number. In many cases res(G)=ind(G), and it is an open question when equality holds.

Our research focuses on when res(G)≠ind(G). We study this by analyzing the minimal graphs with residues differing from their independence numbers, and by considering how a degree sequence can be modified without changing its residue. We present results regarding these graphs and degree sequences that lead to many conditions under which the residue of a graph equals its independence number. We generate all minimal graphs of order less than 9 for which res(G)≠ind(G) and discuss the form of larger minimal graphs. We also discuss certain cases in which res(G)=ind(G) in spite of having one of these minimal graphs as a subgraph. We showcase a variety of transformations that can be applied to a degree sequence without changing its residue, and examine the implications of these transformations for the construction of larger minimal graphs. Our findings offer insight into the open question of when the residue of a graph is equal to its independence number, and this improves our understanding of when the independence number can be efficiently computed.

Examining the rainbow effect of metamaterial droplets

Nirdosh Chapagain, Brigham Young University

Mathematical Sciences

Rainbow is an optical phenomenon created by reflection and refraction of light at the boundaries of water droplets. Descartes was the first to provide a geometric explanation for the optics of the rainbow. We use Descartes’ method to examine if rainbow effect is possible with metamaterial droplets. Metamaterials are artificial materials whose permittivity and permeability can be simultaneously negative hence, giving them negative index of refraction for certain frequencies. The recent extraordinary level of output in the field of metamaterials has resulted in examinations of applications of these substances to a variety of fields, including the arrow of time and cloaking. Many applications of metamaterials can create physical effects that were previously assumed impossible. In this study we have considered that our hypothetical droplet has negative refractive index for visible band of the electromagnetic spectrum. We also examine the effect of using composite metamaterial droplets.

Models for Dementia Diagnoses with Distributed Learning

Samantha Smiley, Brigham Young University

Mathematical Sciences

Dementia is a clinical syndrome characterized by an overall loss of cognitive ability. There are multiple forms of dementia with various causes and various impacts on the suffering individuals. Accurate diagnosis is essential to effective intervention and treatment. Currently, clinicians lack a biological marker that definitively distinguishes the different forms of dementia. Hence, they rely on physical exams, neuropsychological tests, and patient report to provide a diagnosis. Recent advances in brain imaging make it possible to obtain detailed maps of brain activity, which in turn may offer insight into many conditions such as dementia. Developing a predictive model from patient data, including brain scans, would greatly enhance the ability of clinicians to provide accurate diagnosis, and hence appropriate treatment, to their patients. Doing so, however, is not trivial as patient data is heterogeneously and non-uniformly distributed across sites, where some sites have far more data than others and calibration varies among scanners used. We report on the development of novel predictive models based on distributed learning for the effective diagnosis of dementia.

Space Filling Curves and Their Applications With Metamaterials

Steffan Larsen, Brigham Young University

Mathematical Sciences

The popularity of metamaterials has exploded with in the last decade. Metamaterials are materials that exhibit interesting properties not found in nature; one of the most widely known features being a negative refractive index. Metamaterials are composites different types of materials that give them their interesting properties. In addition to being composed of several element types, metamaterials also contain certain inclusions that influence their electromagnetic properties. Among these are space filling curves. Space filling curves are curves that are entirely contained within a specific area and yet can become infinitely long. In my research I investigated the properties of space filling curves and their application/benefit to the research surrounding metamaterials, specifically metamaterial antennas.