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Discrete and Continuous Analysis in Appalachia

About the REU site

Discrete Continuous and Analysis in Appalachia (DCAA) is an REU focused on introducing students to different aspects of mathematical analysis through differential equations, difference equations, and the general theory of time scales calculus. Participants will receive a $4,800 stipend and paid housing for 8 weeks during Summer 2022.

Dates and deadlines

Dates of REU: 6 June 2022 - 29 July 2022
Application deadline: All participants for Summer 2022 have been selected. Please check back at this page next year (around August) for the new page for the Summer 2023 REU program!!

Location

Participants will be housed in a dormitory at Fairmont State University. Housing is 100% paid for by the grant (i.e. it does not come out of the stipend!).

Fairmont sits at the epicenter of West Virginia's "High Tech Corridor" and Fairmont itself is home to the I-79 Technology park, which houses NASA's Independent Verification and Validation facility, a NOAA supercomputer lab, and numerous other high-tech government contractors and businesses. West Virginia University, an R1 research institution, is 25 minutes to the north in Morgantown, which is also home to a number of high-tech businesses. The FBI's Criminal Justice Information Systems building is 20 minutes to the south in Clarksburg. Local infrastructure exists to facilitate travel for participants without cars. There is a bus service in Fairmont that runs six days a week; rates for this service are affordable and includes routes to local recreation, shopping, and dining locations in Fairmont as well as the nearby cities of Clarksburg, Bridgeport, and Morgantown. Those respective cities also have their own transit authorities with comparable services. Fairmont also hosts a Greyhound bus station for travel to more distant locations.

How to apply

As per NSF rules, participants must be US Citizens or Permanent Residents of the United States and have not yet received their bachelors degree during the program (if you graduate in Spring 2022, then you are unfortunately not eligible to participate). Students who are women, underrepresented minorities, first-generation college students, and those whose home institution have limited research opportunities in mathematics are encouraged to apply. Participants are expected to work on research and related activities full time (40 hours a week)! The ideal applicant will have had at least a course in one of differential equations, linear algebra, or calculus-based probability; but we will entertain applicants with any typical "sophomore-level or higher" mathematical background.

Please fill out our application form here

Projects

All participants will learn the time scales calculus as well as being instructed in the Python programming language and typesetting with $\LaTeX$. From this common background, deeper projects will be investigated by the participants, detailed below.

Special functions (Dr. Cuchta)

Participants will be introduced to the generalized hypergeometric series ${}_p\mathcal{F}_q$ and the related generalized discrete hypergeometric series. The ${}_pF_q$ functions are a large historically important class of special functions whose study can be traced back to works of Euler and Gauss. One research track is to consider new special cases of the discrete hypergeometric function. Previous similar studies include discrete analogues of Chebyshev polynomials, Legendre polynomials, as well as discrete Bessel functions which were used to establish qualitative properties of solutions to certain partial difference equations and semidiscrete partial differential equations. Participants will do a search across special functions literature for other special functions that have representations as classical hypergeometric series, pick one, and investigate its discrete analogue. Another avenue for this track is to consider the discrete matrix analogue of any of these functions. Particularly interesting would be projects that attempt to find partial difference or partial differential equations that their discrete special function solves. Projects of these types will require writing software to express their new special functions in order to create plots and check derived properties.

A second promising track is the development of hypergeometric functions on different time scales. The classical hypergeometric series can be thought of as the generalized hypergeometric function "on the time scale $\mathbb{T}=\mathbb{R}$" while the discrete hypergeometric series is "on the time scale $\mathbb{T}=\mathbb{Z}$". It is not known in-general what "hypergeometric series on a general time scale" would look like, and there are some analytical problems to overcome to define them. Participants would begin with a particular time scale that is not $\mathbb{R}$ or $\mathbb{Z}$; popular time scales might include the quantum time scales $q^{\mathbb{N}_0}=\{1,q,q^2,\ldots\}$ for $q>1$ or exotic time scales such as the Cantor set. Once a time scale $\mathbb{T}$ is chosen, its "polynomial shift operator" $\mathscr{U}_{\mathbb{T}}$ must be determined. In the likely scenario that the REU participant cannot find a functional relationship that represents $\mathscr{U}_{\mathbb{T}}$, numerical simulations using the timescalecalculus Python package will be used for computational investigation of special generalized hypergeometric functions on time scales. Numerous old questions in time scales become accessible through this lens: for instance, how does the behavior of a "time scales Bessel function" change as it travels along an arc in the hyperspace of time scales? Such questions can be related to numerical approximations of functions, but are nonetheless interesting by themselves.

Finally, students may opt to look at generalizations of hypergeometric series. For example, a discrete analogue of the Meijer-$G$ functin is currently under investigation, defined by a contour integral. Such integrals can be approached with the residue theorem and series manipulations from a typical Calculus 2 course. The discrete hypergeometric series are a special case of discrete Meijer-$G$. Students interested in this could initiate new studies on the $G$ function itself, investigate previously studied special functions from the contour integral definition, attempt to find analogues of $G$ on new time scales, consider matrix analogues. Particularly ambitious REU participants could even try to discover further generalizations of $G$ to perhaps a "discrete Fox $H$" function.

Dynamic optimal control (Dr. Wintz)

Students will become acquainted with the linear quadratic regulator (LQR) in the continuous and discrete cases. Since its inception, the LQR has become a powerful tool in electrical and mechanical engineering, economics, and mathematical biology, among others. The goal of such problems is to find an optimal control $u$ that minimizes a quadratic cost functional associated with some dynamic system. Students will be introduced to the time scales LQR problem, where the generalized state equation $x^{\Delta}(t)=Ax(t)+Bu(t)$ is associated with the cost functional $J(u)=\dfrac{1}{2}x^{T}(t_f)S(t_f)x(t_f)+\dfrac{1}{2}\displaystyle\int_{t_0}^{t_f}\Big(x^{T}Qx+u^{T}Ru\Big)\Delta\tau$. Similarly, students will become familiar with the corresponding tracking problem.

One research track is to study the time scales LQR with a prescribed degree of stability by appending exponential weight $e_{\alpha}(t,t_0)$. Students will use their acquired programming skills to obtain numerical solutions to these problems. Another important application in the linear quadratic framework is non-cooperative games, where one player (the evader) seeks to minimize a given cost while another player in pursuit seeks to maximize this same cost. In this project, students would study the pursuit-evasion system $\left\{ \begin{array}{rcl} x^\Delta(t) &=&Ax(t)+Bu(t)\\ y^\Delta(t) &=&Cy(t)+Dv(t), \end{array}\right.$ associated with the cost $J(u,v)=\dfrac{1}{2}||x-y||_{M}(t_f)+\dfrac{1}{2}\displaystyle\int_{t_0}^{t_f}\left(||x-y||_{Q}+||u||_{P}-||v||_{E}\right)(\tau)\Delta \tau$. In this setting, students would develop the methods to solve such min-max problems which result in saddle-point solutions. Applications include pursuit-evasion games in defense, economics, and mathematical biology. Students may also study the effects of adding multiple pursuers. A third option for REU researchers interested in continuous nonlinear dynamics is to consider the case where the pursuer/evader is given by a bilinear state equation $x'=Ax+Dxu+bu$. Such systems are called bilinear due to the middle term representing an interaction between the state $x$ and control $u$. Bilinear systems have multiple applications in frequency modulation, chemotherapy models, and chemical reactor models.

Probability (Dr. Niichel)

One option for students to research is dynamic stochastic models on time scales. First, these students will be introduced $\mu$-discretization methods: we start with a continuous, linear, time-invariant system $x'=Ax+Bu$. Using a sequence of random variables $\{\mu_k\}$, we discretize the system to a discrete stochastic time scale $\tilde{\mathbb{T}}$ using a modified form of Euler's method. The resulting system is of the form $x^{\Delta}(t)=\mathcal{A}(\mu)x+\mathcal{B(\mu)}u$, where $\mathcal{A}$ and $\mathcal{B}$ are both given by power series. One novel project for students is to extend Poulsen's work on the LQR to tracking. Here, students will seek an optimal control that minimizes the expected value of a certain cost functional. In this project, students would find an optimal control with two components: a feedback component in terms of the current state and a feedforward component that "anticipates" the desired state. Both components evolve by separate recursive equations. Students will provide numerical simulations of their results where the sequence of time steps is given by different classical probability distributions.

A related novel project is the study of biological models using $\mu$-discretization. For instance, consider the Gompertz model for given by $y'(t)=-ry(t)\log\left(\frac{y(t)}{K}\right)$, where $y$ is the size of the population and $r$ and $K>0$ are the (perhaps stochastic) growth rate and carrying capacity. The substitution $z=\log(y)$ transforms the model into linear form $z'(t)=-rz(t)+b$. This linear model can be discretized. Participants would study the behavior of solutions numerically. Students could provide a similar treatment for the Beverton–Holt model, Lotka–Volterra models, epidemic models, and evolutionary models.

Another area of inquiry for research is the study of the Kalman filter with hybrid measurements. Some state processes cannot be accurately studied due to missing or corrupt information and thus require a state estimator to make meaningful predictions. One of the most common ways to do so is by the Kalman filter. The filter itself is a predictor-corrector algorithm that seeks to minimize the corresponding mean square error. Since its inception, the Kalman filter has become a powerful tool for largely two reasons. The first being the relative ease to implement and create new variants, which all mirror the original classical discrete filter. The second reason is the wide range of applications that the filter can be designed to handle, including navigation, climate study, ballistics, finance, and biomechanical models, to name a few.

One novel project for students is the creation and implementation of the corresponding information filter. This filter is helpful when the dimension of the measurement is large compared to that of the process noise. Unlike the Kalman filter which runs forward in time, the information filter runs backward in time. Such a filter will be of mathematical interest to a wide range of researchers, as calculations will involve the expectation of inverse matrices. Once the corresponding information filter is built, it can be compared with the existing $\mu$-discretized Kalman filter to create a smoothing algorithm. Students will run both algorithms on the same generated time scale.

A third option for projects is the investigation of probability distributions on time scales. Related to the study of special functions on time scales, probability distributions on time scales asks how the properties of known (discrete and continuous) distributions can be extended to time scale-valued random variables. Related to the study of special functions on time scales, probability distributions on time scales asks how the properties of known (discrete and continuous) distributions can be extended to time scale-valued random variables. Currently studied distributions on time scales include the uniform, exponential, gamma, and Erlang distributions. Recently, a Gompertz distribution on time scales, which was shown to unify the continuous Gompertz distribution with the "$q$-geometric distribution of the second kind". The continuous Gompertz distribution is used for modeling populations in biology while the $q$-geometric distribution of the second kind models the number of successes until the first failure, where the probability of success varies geometrically as more successes accumulate. This relationship shows there is potential for studies using time scales calculus to connect apparently unrelated distributions to each other. Participants will be encouraged to explore time scales analogues of probability distributions. This could mean characterizing when time scales distributions obey a certain property, as is being done for the time scales memoryless properties, or starting with a known continuous and discrete distribution and unifying it across all time scales as was done with the Gompertz distribution.




The opinions, findings, and conclusions or recommendations expressed are those of the involved faculty and do not necessarily reflect the views of the National Science Foundation.