At IWU we have a dynamic group of students, as well as a faculty with diverse research expertise. This combination leads to many different opportunities for students to engage in independent research projects. Not only do we encourage doing research, but also communicating this research to others. Projects have led to our students presenting their findings at undergraduate research conferences, writing honors theses, presenting at national professional mathematics meetings, and strengthening their applications for jobs and/or internships.

There are opportunities for students to receive funding to live on campus over the summer in order to conduct research guided by faculty members. The Eckley Summer Scholars and Artists program is a university-wide competitive program that awards funding to 5 students per year. As part of our Beling Summer Student program, the math department provides funding to approximately 3 students per year to conduct on-campus research during the summer. This funding is made possible by the Earl and Marian Beling Professorship of Natural Sciences held by Prof. Tian-Xiao He.

Students also have the opportunity to earn the Research Honors distinction on their diploma. See below for descriptions of many of the research honors projects completed in mathematics.

## Student Research Projects

**Wuyang (Tony) Liu **'18 (Beling summer student 2016)

He is currently working with **Prof. Shallue** on a research project in algorithmic number theory.

**Hang (Jade) Phung** '17

Her research is in the field of graph multidecompositions. In particular, she obtained necessary and sufficient conditions for the existence of a multidecomposition of the complete graph into the graph pair formed by a 7-cycle and its complement. During the summer of 2016 Jade is the Mark Israel '91 research fellow, which is awarded to one student each year by the Department of Economics at IWU.

**Jinze Zheng **'17 (Beling summer student 2015)

She has been prolific with respect to the number of research areas that she has investigated. She has done research in group-antimagic labelings of graphs. This was completed as two projects throughout the 2014-15 academic year. Over the summer of 2015 she worked on another project in the area of enumerative combinatorics. She has given numerous presentations and posters at the JWP undergraduate research conference. Jinze is currently working on a problem in the field of Riordan arrays.

**Jacob Nowakowski** '17 (Beling summer student 2015)

He conducted research in maximum packings of graphs during the fall of 2014. During the summer of 2015, he will begin working in algorithmic number theory.

**Brian Bollivar** '16 (Beling summer student 2015)

Over the summer of 2015 he conducted research in algorithmic number theory.

**Yizhe Gao **'16

His research is in the field of graph decompositions. In particular, he has prepared a paper entitled Multidecompositions of Complete Graphs into a Graph Pair of Order 6.

He also did a project in spring 2013 which culminated in a poster presented at JWP entitled A Look at Multi-Decompositions of Complete Graphs into Graph Pairs of Order 4.

**Kimberly Wenger** '15 (honors)(Beling summer student 2014)

She completed an honors research project during spring 2015. Her research area is graph labelings and their implications for graph decompositions. The title of her paper is Two Rosa-type Labelings of Uniform k-distant Trees and a New Class of Trees.

Over summer 2013 she completed a project in graph labeling. During the fall 2013 semester she prepared a paper for publication entitled Uniform k-distant Trees are Graceful.

Kimberly has presented her research at various stages at the JWP undergraduate research conference, the ISMAA annual meeting, and as a colloquium at IWU.

**Anh Phan** '15

He designed an algorithm to assist with determining whether two graphs are isomorphic or not. Also, during summer 2013 he classified all graph pairs of order 6. Anh presented his research at the JWP undergraduate research conference.

**Anqi Zhang** '15

She has worked on group-antimagic labelings of a few classes of graphs including caterpillars and banana trees. Her project included characterizing the Integer Antimagic Spectrum of various classes of graphs.

**Tom Simmons **'15 (honors)(Eckley scholar 2014)

He completed an honors research project during spring 2015. His research area is hashing onto elliptic curves. The title of his paper is A Computational Study of Icart's Function. Tom presented his research both at the JWP undergraduate research conference and as a colloquium at IWU.

**Tung Nguyen** '15 (honors)(Beling summer student 2014)

He completed an honors research project during spring 2015. His research area is wavelets and framelets. The title of his paper is Construction of Spline Type Orthogonal Scaling Functions and Wavelets.

Tung presented his work at various stages at the JWP undergraduate research conference, ISMAA annual meeting, Rose-Hulman undergraduate research conference, and at colloquia at IWU.

He also conducted research in economics.

**Linh Nguyen** '15 (Beling summer student 2014)

Her research field is Cryptography and Elliptic Curves. She completed a project during summer 2014 and presented her work at the JWP undergraduate research conference.

**Yulin Deng** '12

In summer 2012 he created code for proving primality of integers p, given factorization of p+1.

**Steve Hayman** '12 (honors)

In summer 2010 he studied irreducibility algorithms for polynomials over GF(2).

In summer 2011 he created code for constructing large Carmichael numbers.

During the 2011-2012 school year he finished his project on irreducibility algorithms. Presented at JWP, accepted research honors thesis (see below).

## Student Honors Papers

The Student Honors Papers collection represents exemplary work in mathematics at Illinois Wesleyan University. The Ames Library is proud to archive these and other honors projects in Digital Commons @ IWU, the University's online archive of student, faculty and staff scholarship and creative activity.

Testing Irreducibility of Trinomials over GF(2)

*by Steven Hayman '12*

The focus of this paper is testing the irreducibility of polynomials over finite fields. In particular there is an emphasis on testing trinomials over the finite field GF(2). We also prove a the probability of a trinomial satisfying Swan's theorem is asymptotically 5/8 as n goes to infinity.

LaSalle's Invariance Principle on Measure Chains

*by Anders Floor '00*

It was in 1892 that Lyapunov published his paper giving his "second method". The basic guiding principle was that we might be able to know something about the stability of the system from the form of the equations describing it. Specifically, the idea was that it would not be necessary to know the solutions of the equations involved. This is of course very useful since in most cases solutions are extremely difficult or even impossible to find. Lyapunov's insight was that if a function could be found with, among other properties, a negative rate of change along the solution of the system except in the equilibrium case, then disturbances from the equilibrium solution would return to that solution. (In the equilibrium case, the solution is constant.) The kind of function involved is called a Lyapunov function, and it is defined in such a way that it mimics the energy function. In fact, it was the energy function which originally inspired these ideas. There is an intuitive physical appeal about the assertion that systems that lose energy "fall" to an equilibrium state. And in many cases, the expression for energy ends up being our choice for Lyapunov function. The historical data above can be found in [5].

Steiner Trees Over Generalized Checkerboards

*by Meta M. Voelker '97*

To minimize the length of a planar network, we can build a Steiner minimal tree that is, a tree consisting of the original network points, as well as additional, strategically-placed (Steiner) points. Chung, Gardner and Graham [2] investigated building Steiner trees over grids of unit squares. We generalize their ideas to grids of rhombuses, and show that two near-optimal Steiner trees exist for each grid, one built from Steiner trees over rhombuses and one built from Steiner trees over isosceles triangles. Further, we conjecture that for grids with an odd number of layers, only the small angle of the rhombus drives which tree is shorter; for grids with an even number of layers, the small angle is the most important factor in determining which scheme to use.

Macroelements and Orthogonal Multiresolutional Analysis

*by Jonathan M. Corbett '96*

Orthogonal multiresolutional wavelet analysis in a two dimension setting furnishes a basis for wavelet analysis. Bernstein-Bezier polynomials over simplexes provide elegant expressions of the necessary and sufficient conditions for a shift invariant space generating an orthogonal multiresolution analysis.

The B-spline Wavelet Recurrence Relation and B-spline Wavelet Interpolation

*by Patrick J. Crowley '96*

In most signal processing applications, a given range of data is best described by a set of local characteristics as opposed to a single global characteristic. In image processing, for example, a region of an image that contains numerous edges is best described as a region whose pixel color values change abruptly, i.e., they are not continuous values of color. A region of constant color, or gradually changing color, is best described as a region whose pixel values are constant, or whose values increase linearly by some factor. It is advantageous to represent this data with signals capable of adapting to these types of local characteristics, as opposed to choosing the best global characteristic. Here, the B-spline wavelet recurrence relation is presented. The B-spline wavelet recurrence relation allows a wavelet of order n +1 to be constructed from a wavelet of order n. This recurrence relation provides a mathematical tool capable of locally varying its degree of continuity. The order of differentiability of a B-spline wavelet increases as the order of the B-spline wavelet increases, and the values range from discontinuous to an arbitrary degree of continuity. A brief discussion of interpolation for splines and B-spline wavelets is introduced as a step toward a future application.

The Recurrence Relation of B-Wavelets

*by Rumi Kumazawa '94*

Our goal is to construct smooth wavelet functions. In constructing such wavelet functions, we need a smooth scaling function to begin with. B-spline functions are suitable as our scaling function because they are piecewise polynomials with compact supports and are relatively smooth. B-wavelet functions are just dilations and translations of these B-spline functions. In addtion, we can find a recurrence relation of the B-spline functions with different order. Hence B-wavelets of any order can be constructed successively from the lower order ones.

Applications of the Wavelet Transform to Signal Analysis

*by Jie Chen '93*

Like the Fourier Transform, the Wavelet Transform decomposes signals as a superposition of simple units from which the original signals can be reconstructed. The Fourier Transform decomposes signals into sine and cosine functions of different frequencies, while the Wavelet Transform decomposes signals into wavelets. Since the Fourier Transform is a global integration transform and there is no time factor in it, it cannot effectively analyze nonstationary signals whose statistical properties change with time. In order to analyze nonstationary signals, we need to decompose signals into units that are localized in both the time and frequency domains. Using the Wavelet Transform with the B-wavelet, we wrote a program package in Mathematica to implement the decomposition and reconstruction algorithms for signal processing. A data acquisition system developed in another project is used to acquire both the synthesized signals and real voice signals. Application of the Wavelet Transform on these signals will be presented.

Multisurface Method of Pattern Separation

*by Jennifer L. Jancik '93*

The recognition and separation of patterns is becoming increasingly important in modern applications. For example, it is currently being used at the University of Wisconsin Hospitals to aid in the diagnosis of breast cancer.

On the Construction of Prime Desert n-Tuplets

*by Derek M. Marusarz '92*

If we were able to show that there is an infinite number of prime desert n-tuplets of length k, for any appropriate odd k, then we would have solved the twin prime conjecture because then we would know that there would be an infinite number of prime desert 1-tuplets of length k =1, i.e., an infinite number of twin primes.

The Theorms of the Alternative

*by Kathryn L. Balsman '91*

In layman's terms, a Theorem of the Alternative is a theorem which states that given two conditions, one of the two conditions is true. It further states that if one of those conditions fails to be true, then the other condition must be true. Mathematically, this does not give a very clear picture of this group of theorems, however. Katta Murty defines a typical Theorem of the Alternative as one which shows that corresponding to any given system of linear constraints, system I, there is another associated system of linear constraints, system II, based on the same data, satisfying the property that one of the systems among I and II is feasible if and only if the other is infeasible. These theorems are not extremely well-known although they do have direct applications in the derivation of optimality conditions. The task at hand was to explore the Theorems of the Alternative as they are found in linear programming, projection theory, and linear algebra.

Some NP-Complete Problems in Linear Algebra

*by Santhosh Sastry '90*

This research project is aimed at studying the theory of NP-Completeness and determining the complexity of certain problems in linear algebra. The first chapter introduces the reader to Complexity theory and defines NP-Completeness. It is supported by Appendices 1 and 2. Appendix 3 lists some known NP-Complete problems.

**Classes of Matrices Related to the Linear Complementarity Problem**

by Teri Prokopek '89

This project led to Teri giving two talks at national professional meetings.