Mathematical Sciences

Mathematical Sciences Professor Elected to Lead Role in Faculty Senate

Neil Calkin was elected secretary of the Faculty Senate for its 2016-17 term during the March 7 Faculty Senate meeting. The mathematical sciences professor joined Clemson in 1997. In his three years on the senate, Calkin said that he enjoyed being part of the conversation and bringing issues to the forefront. Calkin plans to work closely with Faculty Senate leadership to address issues such as the status of special faculty, Clemson’s college reorganization and the university’s planned child care center.

 

See more about the new Faculty Senate leadership here:

https://blogs.clemson.edu/inside-clemson/inside-news/lawton-rauh-calkin-elected-to-lead-roles-in-faculty-senate/.

calkin

Mathematical Sciences 2015-2016 Graduates

In Fall 2015 and Spring 2016, Clemson’s Graduate Program in Mathematical Sciences awarded six Ph.D. degrees and eleven M.S. degrees. Some graduates of the program will apply their knowledge and skills to positions in academia and industry, while others will continue their studies at a higher level. These degrees are the culmination of years of hard work, unyielding determination, intense mathematical ability, and excellent faculty advising.

 

The graduates’ names, advisors’ names, and disquisition titles are listed below. Photographs and current/future positions are also included where available.

 

Please join us in congratulating our newest graduates for their outstanding accomplishments!

 

PhD Fall 2015

 

Anuradha

Hewa Arachchige Anuradha Priyadarshani

Advisors: Dr. R. Lund and Dr. Y. Li

Dissertation title: Bayesian Minimum Description Length Techniques for Multiple Changepoint Detection

Current position: Senior lecturer grade II, University of Kelaniya, Dalugama, Sri Lanka

 

shiyi

Shiyi Tu

Advisor: Dr. X. Sun

Dissertation title: Objective Bayesian Analysis on the Quantile Regression Advisor

Current position: Financial Analyst at Bank of America

 

PhD Spring 2016

 

dowlingMichael Dowling

Advisor: Dr. S. Gao

Dissertation title: Exponder Graphs and Good Codes

 

GrotheerRachel Grotheer

Advisor: Dr. T. Khan

Dissertation title: Hyperspectial Diffuse Optical Tomography using the Reduced Basis Method and Sparsity Constraints

Fall position: Assistant Professor of Mathematics at Goucher College in Towson, MD.

 

 

 

 

Jason Hedetniemi

Advisor: Dr. K. James

Dissertation title: Problems in Domination and Graph Products

 

xuHonghai Xu

Advisor: Dr. W. Goddard

Dissertation title: Generalized Colorings of Graphs

 

 

MS Fall 2015

 

Kevin Wayne Dettman

Advisor: Dr. M. Wiecek

Thesis title: Bi-Objective Quadratic Optimization with Application to Portfolio Selection

 

liHaodong Li

Advisor: Dr. M. Mitkovski

Thesis title: Multiplier Operators on Framed Hilbert Space

Current position: Clemson Mathematical Sciences PhD program

 

HernandezMonica Deni Moralex-Hernandez

Advisor: Dr. L. Rebholz

Thesis title: Some new Results for Leray -x discretization

Current position: Clemson Mathematical Sciences PhD program

 

 

 

 

 

 

MS Spring 2016

 

Baumbaugh_2016Travis Baumbaugh

Advisor: Dr. F. Manganiello

Thesis title: Results on Common Left/Right Divisors of Skew Polynomials

Current position: Clemson Mathematical Sciences PhD program

 

 

 

 

 

 

Knoll

 

Rebecca Knoll

Advisor: Dr. L. Rebholz

Thesis title: On the Optimal Parameter Choice of Steady NSE with Grad-Div

Current/future position: Mathematics instructor, Rockbridge Academy in Millersville, MD

 

 

 

 

lamoreuxMichael Lamoreux

Advisor: Dr. A. Brown

Thesis title: Empirical Null Estimation via Central Matching with Application to Functional Magnetic Resonance Imaging

 

 

menardMatthew Menard

Advisor: Dr. C. Williams

Thesis title: Study of the Fisher-Tippett Theorem with Applications to Problems in Actuarial Science

Current position: Clemson Mathematical Sciences PhD program

 

Stefani Mokalled

 

Stefani Mokalled

Advisor: Dr. C. McMahan

Thesis title: Estimating Biomarker Distributions via Pooled Assessments

Current/future position: Research Assistant in biostatistics, Faculty of Public Health at the American University of Beirut, Lebanon.

 

 

 

 

phillipsLee Phillips

Advisor: Dr. W. Adams

Thesis title: Solvable Cases of the QAP Explained by the Level – 1 RLT

 

 

 

Sijun Shen

Advisor: Dr. X. Sun

Thesis title: Evaluating an Individuals use of the Internet and Online Chat rooms to learn about Health Information

 

Zerfas

 

Camille Zerfas

Advisor: Dr. L. Rebholz

Thesis title: Sensitivity of the Filtering Radius of the rNS Turbulence Model

Current position: Clemson Mathematical Sciences PhD program

 

Mathematical Sciences Sponsors 13th Calculus Challenge

The Department of Mathematical Sciences sponsored the 13th annual Clemson Calculus Challenge on Friday, April 15, 2016. The Clemson Calculus Challenge invites high school calculus students from South Carolina, Northeast Georgia, Western North Carolina, and Eastern Alabama to compete in a one-day event. There were 233 students from 34 regional high schools that took the 2016 Challenge. 20160415 CalcChallenge (26)

Students take an individual test in the morning and then participate in a team competition in the afternoon. After the team competition, the schools attended a short research presentation entitled “Don’t be fooled by computer models: they don’t always tell the truth!,” delivered by Dr. Sez Atamturktur, Distinguished Professor of Intelligent Infrastructure, Department of Civil Engineering.

 

Several prizes were presented at the awards ceremony. Each school was placed into one of three divisions based on their school’s enrollment. Trophies were handed out in three categories:  Individual, Team, and School.  Five hundred dollar scholarships to Clemson University were given to the first and second place finishers on the morning exam in each school division.  Mu Alpha Theta (the national high school and two-year college honor society) presented $100 to the top scorer and $50 to the second place scorer on the morning exam.  The honor society also conferred three 1-year licenses to Mathematica upon the top morning exam scorers in each school division.  Lastly, one team in each school division was recognized as having the most creative team name.  These teams were given gift cards to Barnes & Nobles.  The awardees for the 2016 Clemson Calculus Challenge are listed below:

 

School Awards

 

Place Division I Division II Division III
First Christ Church Episcopal School D.W. Daniel High School Spring Valley High School
Second The Altamont School North Oconee High School Mountain View High School
Third Southside Christian School Oconee County High School South Forsyth High School

 

Team Awards

 

Place Division I Division II Division III
First CCES squared (Christ Church Episcopal School) ~ i (North Oconee High School) Laplace’s Opimus Primes (Mountain View High School)
Second SDS A (Spartanburg Day School) We are # sin (π/2) (D.W. Daniel High School) Math Destruction (Spring Valley High School)
Third Feel the Curve (Southside Christian School) Differentiated from the crowd (D.W. Daniel High School) Calcaholics Anonymous (Mauldin High School)

 

Individual Awards

 

Place Division I Division II Division III
First Andy Xu***, Top Score Morning Exam (Christ Church Episcopal School) Sarah Baum (D.W. Daniel High School) Albert Huang*, Second-place Score Morning Exam (Spring Valley High School)
Second William Tang (The Altamont School) Ian Ruohoniemi** (D.W. Daniel High School) Jenning Chen* (Spring Valley High School)
Third John Staubes** (Academic Magnet High School) Hank Morris* (Oconee County High School) Harish Kamath* (South Forsyth High School)

*Indicates junior status in high school   **Indicates sophomore status in high school

***Indicates freshman status in high school

Most Creative Team Name

 

Division I Division II Division III
Mary Had a Little Lambda (Rockdale Magnet School) Calcoholics (Crescent High School) Southern L’Hopitality (West Forsyth High School)

 

20160415 CalcChallenge (31) 20160415 CalcChallenge (84)

 

 

 

 

 

 

The 2016 Clemson Calculus Challenge received funding from the Department of Mathematical Sciences and the College of Engineering and Science. Terri Johnson and Shari Prevost coordinated the 2016 competition.  They were aided greatly by many volunteers from the faculty, staff and students of the Department of Mathematical Sciences.

 

 

Mathematical Sciences Departmental Awards

svetlana

Assistant Professor Svetlana Poznanovikj has been awarded the Department of Mathematical Sciences Teaching Award for 2015-2016. As one of her former students writes, “Dr. Poznanovikj is the kind of instructor and mentor that pushes you to be a better student.” Professor Poznanovikj is well-know with her students for providing “high expectations for her students, but provid[ing] them with all of the possible resources for them to succeed” and for creating “an environment that foster[s] questions.” One student writes:

This class was the first time I transitioned from learning math for the sake of a good grade to exploring math due to genuine curiosity. This new outlook was in large part due to Dr. Poznanovijk’s teaching…. As I sat and listened to her explanations of the theorems, my mind was blown by the ideas and my interest in mathematics was increased by the minute. We could all tell that Dr. Poznanovijk was truly passionate about the topics she covered and our understanding of them.

Cawood IMG_6256

 

Senior Lecturer Mark Cawood received a National Scholars Program Award of Distinction for his “commitment to the intellectual, professional, and personal development of Clemson National Scholars.” Each senior in Clemson University’s National Scholars Program chooses one faculty or staff member for this honor, and Mark was nominated by Kaitlin Carter. She presented him with the award at the National Scholars Program Awards of Distinction Dinner on Tuesday, April 19, 2016, at the Madren Center. He writes, “It was a great honor from a truly outstanding student and dedicated campus leader.”

 

 

 

Other awards in the department include:

Departmental Graduate Student Awards

Grad Student Awardees IMG_6263-2

  • Nathan Adelgren: Outstanding Graduate Research
  • Brandon Goodell: Outstanding TA award
  • Chase Joyner: Outstanding MS Student Award
  • Fiona Knoll: Excellence in Graduate Teaching
  • Drew Lipman: Outstanding Citizenship Award
  • Stefani Mokalled: Outstanding MS Student Award
  • Stella Watson: Outstanding MS Student Award
  • Anastasia Williams: Outstanding Graduate Research
  • Honghai Xu: Excellence in Graduate Research
  • Yibo Xu: Outstanding MS Student Award

 

Departmental Undergraduate Student Awards

20160402 UG Freshman Awardees (48)-2 20160402 UG Jr Awardees (55)-2 20160402 UG Soph Awardees (49)-2 SR UG Awardees IMG_6245-2

  • Jarryd Boyle: Sue King Dunkle Award
  • Luna Bozeman: Mathematical Sciences Faculty Sophomore Award
  • Patrick Dynes: John Charles Harden Junior Award
  • Hayden Girard: Mathematical Sciences Faculty Freshman Award
  • Sarah Hicklin: Samuel Maner Martin Award
  • Sarah Malick: Mathematical Sciences Faculty Senior Award
  • Jake Marshall: John Charles Harden Junior Award
  • Hannah Mccall: Mathematical Sciences Faculty Freshman Award
  • Mary Mell: Mathematical Sciences Faculty Freshman Award
  • Sloan Nietert: Mathematical Sciences Faculty Freshman Award
  • Polly Payne: Alice Louise Gray Fulmer Award
  • Hannah Rollins: John Charles Harden Junior Award
  • Andrew Shore: Mathematical Sciences Faculty Sophomore Award
  • Alexander Stoll: John Charles Harden Junior Award
  • Emily Anne Thompson: Mathematical Sciences Faculty Senior Award
  • Katy Wrenn: Mathematical Sciences Faculty Senior Award

 

Congratulations to all of this year’s awardees!

Biological Sensitivity: Using Computations to Uncover Chaotic Behavior in Population Models

In mathematics, a system is said to be chaotic if it is extremely sensitive to change in initial conditions. In a joint project, http://arxiv.org/abs/1604.06951, Mathematical Sciences faculty members Elena Dimitrova and Lea Jenkins, Computer Science faculty member Brian Dean, and graduate students Sherli Koshy (Math Sciences) and Akshay Galande (Computer Science) developed a computational framework that helps researchers acquire better understanding of the causes of chaotic behavior in dynamical systems and connections between chaos and sustainability of biological species. The platform they built allows for efficient systematic characterization and visualization of factors leading to chaotic behavior.

In the 1963, Edward Lorenz developed a simplified model for atmospheric convection which exhibits chaotic behavior: the famous butterfly-shaped strange attractor in Figure 1. The model is a system of three ordinary differential equations now known as the Lorenz equations: equations

 

Here x, y, and z make up the system state, t is time, and r , s , and b are the system parameters. The Lorenz equations also arise in simplified models for lasers, dynamos, thermosyphons, brushless DC motors, electric circuits, chemical reactions, and forward osmosis.

 

 

lorenzFigure 1: A sample solution in the Lorenz attractor when r = 28, s = 10, and b = 8/3.

 

Soon after, mathematical biologists began searching for similar behavior in biological systems and quickly found it in simple population models. The possibility of chaotic behavior in such models created quite a stir in the 1970s [6]. It suggested that the intuitive view that complicated population dynamics can arise only from complicated interactions and environmental fluctuations would had to be abandoned. However, further work on models of both laboratory and real-world insect populations led scientists to doubt that chaotic behavior was actually present in real population dynamics. While presently there is little data of sufficiently high quality and long enough duration to test the idea that simple population systems exhibit chaotic behavior, there is little doubt that more complex population models do. The first such discovery was made by Cushing et al. in 2001 [2], when they announced that a laboratory population of the flour beetle tribolium exhibits chaotic dynamics. A few years later, Becks et al. demonstrated existence of chaotic and non-chaotic states in an experimental study of a microbial system with a nutrient source [1] and were able to document transitions between chaotic and non-chaotic states after changing the strength of the nutrient source. Other studies followed, demonstrating that chaotic behavior is present in a variety of species populations [5, 9, 3, 7, 4].

Models of population dynamics give us means of better understanding chaotic dynamics in biological systems. However, it is hard to decide if a system is actually chaotic since it can exhibit a wide range of chaotic and non-chaotic behavior, related in some nontrivial way to certain combinations of system parameters and initial conditions, making it hard for scientists without enough mathematical sophistication to identify and analyze the presence of chaotic behavior in their systems of interest.

 

To address this difficulty, the authors provided a framework based on the well-known Metropolis-Hastings (MH) sampling algorithm that allows researchers to efficiently search over a design space including parameter values and initial conditions and discover possible connections between values of these constants and chaotic dynamics. They also developed an interactive visualization platform to help the user understand the structure of the parameter space which produces chaotic behavior — a challenging task, due to the potentially complex, high-dimensional nature of this set.  The platform is based on parallel coordinates, a popular means of visualizing high-dimensional data.

 

pc_example

Figure 2: Chaotic samples visualized using parallel coordinates. These results were obtained using one of the systems described in the work by Sprott [8].  Note the wide range of values for  and , while the range for is smaller.

 

Figure 2 shows an example of chaotic sample points generated from the equations of Case N in Table I of [8] and visualized using parallel coordinates.

The general system in [8] is given as

 

The varied parameters are , , , , and ; the remainder of the parameters were chosen to be zero. Each of the selected varied parameters is mapped to an individual coordinate axis, arranged in parallel from left to right, with each sample point drawn as a “poly line” that intersects each coordinate axis at the appropriate location.  All five parameters were constrained to lie in the interval [–5, 5] during initial sampling, although by dynamically dragging the upper and lower endpoint markers on each axis, the user can further restrict the display so it only shows samples generated within a smaller sub-rectangle.  Here, the upper bound on the parameter  was decreased to –2.4, filtering out some of the 500 initial sample points initially present.

 

fig_drag

Figure 3: The result (from left to right) of dragging down the upper limit on the  axis, observing a corresponding increase in the mean of the  axis due to anticorrelation between the two.

 

Parallel coordinate plots allow us to understand a number of useful properties by visual inspection and interactive manipulation. For example, in Figure 2 we see that in order to achieve chaos with  restricted to such a low range, the parameters  and  need to be anticorrelated: that is, there needs to be a negative relationship in which  increases as decreases. This is visually apparent from the “X” pattern between the  and axes, and we can also see it by dynamically dragging the upper bound on  downward, watching the marker for the mean value on the axis move upward in lock step, as shown in Figure 3. Interestingly, if one raises the lower bound on , forcing this parameter to take large values, then the pattern between  and becomes one of mostly straight horizontal lines, indicating correlation rather than anticorrelation.

 

By restricting several coordinates at a time, the user can filter an initially large number of sample points down to only a few. For example, if we restrict the range to half of each of five axes, this will on average show only  of all sample points.  In order to populate the filtered space with sufficiently many samples to understand its geometric structure, it may be necessary to re-launch a new round of MH sampling within this restricted space, requiring tight coupling between the user interface and the “back end” parallel MH sampler.

 

The ability to search the parameter space of a system and visualize the results can reveal chaotic behavior in systems not previously known to have chaotic regime and reveal the existence of parameters and initial conditions not previously known to yield chaotic behavior in studied systems. Furthermore, it allows for studying system behavior that is virtually impossible to observe in laboratory environment, thus making it useful to experimentalists.

We are currently studying the connection between chaotic dynamics and long-term survival of populations in ecological context. We focus on perhaps the most widely accepted criterion for long-term survival, permanence. Roughly speaking, a system is permanent if no orbit comes too close to the boundary. Our preliminary results demonstrate that the high responsiveness of chaotic systems allows for easier transition towards the desirable state of permanence, identifying chaos as a beneficial property of a system.

 

References

[1]  L. Becks, F.M. Hilker, H. Malchow, K. Jürgens, and H. Arndt. Experimental demonstration of chaos in a microbial food web. Nature, 435:1226– 1229, June 2005.

[2]  J.M. Cushing, S.M. Henson, R.A. Desharnais, B. Dennis, and R.F. Costantino A. King. A chaotic attractor in ecology: Theory and experimental data. Chaos Solitons Fractals, 12(2):219–234, 2001.

[3]  B.W. Kooi and M.P. Boer. Chaotic behavior of a predator-prey system in the chemostat. Dynam. Cont. Dis. Ser. B, 10(2):259–272, 2003.

[4]  M. Kot, G.S. Sayler, and T.W. Schultz. Complex dynamics in a model microbial system. B. Math. Biol., 54(4):619–648, 1992.

[5]  L.V. Kravchenko, N.S. Strigul, and I.A. Shvytov. Mathematical simul tion of the dynamics of interacting populations of rhizosphere microorganisms. Microbiology, 73(2):189–195, 2004.

[6]  R.M. May. Simple mathematical models with very complicated dynamics. Nature, 261:459–567, 1978.

[7]  F. Molz and B. Faybishenko. Increasing evidence for chaotic dynamics in the soil-plant-atmosphere system: A motivation for future research. Procedia Environmental Sciences, 19:681–690, 2013.

[8]  J.C. Sprott. Some simple chaotic flows. Phys. Rev. E, 50(2):647–650, 1994.

[9]  N.S. Strigul and L.V. Kravchenko. Mathematical modeling of PGPR inoculation into the rhizosphere. Environ. Modell. Softw., 21:1158–1171, 2006.