Student research is conducted with faculty members. Our faculty have various areas
of expertise. Below are some recent faculty publications.
Assessing the censored linear regression model using martingale approximations, by
Seung-Hwan, Lee Communications in Statistics, 49, 408-418 (2020)
A weighted log-rank test for comparing two survival curves, by Seung-Hwan Lee and
Eun-Joo Lee (Millikin U.), Monte Carlo Methods and Application, 26, 253-262 (2020)
Stochastic risk assessment of urban soils contaminated by heavy metals in Kazakhstan,
by Elmira Ramazanova (Nazarbayev U., Kazakhstan), Seung-Hwan Lee, Woojin Lee (Nazarbayev
U., Kazakhstan), Science of Total Environment (accepted)
Tian-Xiao He (selected work with students)
Kurt VanNess ’13 (with Jack Maier and Tian-Xiao He), Title: Polynomials that have
golden ratio zeros, Journal: J. Adv. Math. Stud. 7 (2014), no. 1, 151–175.
Tung Nguyen ’15 (with Tian-Xiao He), Construction of spline type orthogonal scaling
functions and wavelets, J. Appl. Funct. Anal. 10 (2015), no. 3-4, 189–203.
Tung Nguyen ’15 (with Tian-Xiao He), Wavelet Analysis and Applications in Economics
and Finance, J. Statistics Math. Sciences, 1 (2015), no. 1, 22–37
Tung Nguyen ’15 (with Tian-Xiao He), A unified approach to construct a class of Daubechies
orthogonal scaling functions, J. Math. Res. Appl. 37 (2017), no. 1, 29–39.
Jinze Zheng ’17 (with Tian-Xiao He), Duals of the Bernoulli numbers and polynomials
and the Euler numbers and polynomials, Integers 17 (2017), 1–26.
Ximing Dong’2018 (with Tian-Xiao He), Permanents of Tridiagonal and Hessenberg Matrices
Representing Recursive Number and Polynomial Sequences, Journal: J. Combin. Math. Combin. Comput. 106 (2018), 209–222.
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.
Yizhe Gao '16
Graph pair decompositions.
Presenters and advisers.
Panxi Chen '20 Panxi is working on characterizing the leaves of maximum packings of complete graphs
with stars. As part of this work she is implementing the algorithmic construction
of maximum packings from a research paper in Java.
Yossif Elmadny '22 Yossif is conducting a preliminary investigation into designing orientations which
will lead to decompositions of split graphs into stars.
Patrick Ward '20 (Beling summer student 2018) Patrick works on decompositions of graphs, directed graphs, and hypergraphs. He has
participated in the ISU Research Experience for Undergraduates twice as well as worked
throughout the semesters at IWU. His investigations have included finding the spectra
for various directed graphs and hypergraphs. Has also characterized all complete
digraphs (and complete hypergraphs) that admit decompositions into various directed
graph pairs (and hypergraph pairs) for several small orders.
Zihan Nie '21 (State Farm summer research fellow 2018) Zihan has worked on projects spanning multiple areas of mathematics. She has worked
on using recursive sequences to construct the famous Girard-Waring identities, as
well as developing algorithms for constructing all possible maximum packings of complete
graphs with 6- and 7-stars.
Yutong Li '21 Yutong has worked at the intersection of graph theory and algebra. In particular,
she is interested in how graphs can be labeled with elements of Abelian groups such
that the labeling has certain desired properties.
Wuyang (Tony) Liu '18 (Beling summer student 2016) He worked with Prof. Shallue on a research project in algorithmic number theory.
Jonathan Vlk '17 Under Dr. Lee’s supervision, he published the following book: Forecasting Stock Returns using a Copula-GARCH model ISBN: 978-3-659-23357-9 Lambert Academic Publishing
This book introduces an idea of modeling dynamic dependence of stock returns and stock
forecasting methods by utilizing a copula-based time series model. For illustration,
the book uses stock market data. We started a book project when he was senior, and
it was released in winter 2017. Jonathan participated in the entire process of authoring
a book as the second author.
Jonathan Vlk '17 (second project) Under Dr. Lee’s supervision, he investigated dependence structures of stocks that
should be an important consideration in managing a stock portfolio, among other investment
strategies, which is part of a research project. The manuscript based on the research
results appeared inInternational Journal of Statistics and Probability(Vol. 6, pp 32-41, 2017).
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
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
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
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.
Philip Yulin Deng '13 Under Dr. Lee’s supervision, he performed extensive simulations to assess the adequacy
of the model developed to estimate the anticipated life span of a patient with multiple
myeloma, which is part of a research project. The manuscript based on the research
results appeared inInternational Journal of Statistics and Probability(Vol. 2, pp 44-52, 2013).
David Elliott '11 and Katlyn Mathy '12 Under Dr. Lee’s supervision, they developed some program algorithms and carried out
numerical simulations based on it to examine the effectiveness of the extreme value
distributions, which is part of a research project. The manuscript based on the research
results appeared inISRN Journal of Applied Mathematics(Vol 2011, Article ID 68743, 12 pages, 2011).
Yulin Deng '12 In summer 2012 he created code for proving primality of integers p, given factorization
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).
Leann Stuber '09 Under Dr. Lee’s supervision, she improved some existing graphical method to check
dependence structure of several variables, which is part of a research project.The manuscript based on the research results appeared inJournal of Statistical Research(Vol. 43, 3-15, 2009).
Nathan Schieve '08 and Quentin Brummet '08 Under Dr. Lee’s supervision, they performed analysis on the parameter estimation methods
for Extreme Value distribution. The Extreme Value distribution is a transformed distribution
of Weibull distribution which is often used for modeling lifetime data.
Meagan Sheehan '08 Under Dr. Lee’s supervision, she compared various estimators for Normal distribution,
theoretically and numerically, which is part of a research project. The manuscript
based on the research results appeared in journal ofMathematics and Computer Education(Vol. 42, pp 145-152, 2008).
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
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.
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 .
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  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.
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.
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.
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 addition, 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.
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.
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.
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.
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
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.