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Chebfun is a collection of algorithms and an open-source software system in object-oriented Matlab that extends familiar powerful methods of numerical computation involving numbers to continuous or piecewise-continuous functions. The success of this strategy is based on the mathematical fact that smooth functions can be represented very efficiently by polynomial interpolation at Chebyshev points or by trigonometric interpolation at equispaced points for periodic functions. More recently, the system has been extended to handle bivariate functions and vector fields. These two new classes of objects are called Chebfun2 and Chebfun2v, respectively. We will show that Chebfun2 and Chebfun2v, and can be used to accurately and efficiently perform various computations on parametric surfaces in two or three dimensions, including path trajectories and mean and Gaussian curvatures. More advanced surface computations such as mean curvature flows are also explored. This is also the first work to use the newly implemented trigonometric representation, namely Trigfun, for computations on surfaces.
ContributorsPage-Bottorff, Courtney Michelle (Author) / Platte, Rodrigo (Thesis director) / Kostelich, Eric (Committee member) / School of Mathematical and Statistical Sciences (Contributor) / Barrett, The Honors College (Contributor)
Created2016-05
Description
Mathematics is an increasingly critical subject and the achievement of students in mathematics has been the focus of many recent reports and studies. However, few studies exist that both observe and discuss the specific teaching and assessment techniques employed in the classrooms across multiple countries. The focus of this study is to look at classrooms and educators across six high achieving countries to identify and compare teaching strategies being used. In Finland, Hong Kong, Japan, New Zealand, Singapore, and Switzerland, twenty educators were interviewed and fourteen educators were observed teaching. Themes were first identified by comparing individual teacher responses within each country. These themes were then grouped together across countries and eight emerging patterns were identified. These strategies include students active involvement in the classroom, students given written feedback on assessments, students involvement in thoughtful discussion about mathematical concepts, students solving and explaining mathematics problems at the board, students exploring mathematical concepts either before or after being taught the material, students engagement in practical applications, students making connections between concepts, and students having confidence in their ability to understand mathematics. The strategies identified across these six high achieving countries can inform educators in their efforts of increasing student understanding of mathematical concepts and lead to an improvement in mathematics performance.
ContributorsAnglin, Julia Mae (Author) / Middleton, James (Thesis director) / Vicich, James (Committee member) / Barrett, The Honors College (Contributor) / School of Mathematical and Statistical Sciences (Contributor)
Created2014-12
Description
Covering subsequences with sets of permutations arises in many applications, including event-sequence testing. Given a set of subsequences to cover, one is often interested in knowing the fewest number of permutations required to cover each subsequence, and in finding an explicit construction of such a set of permutations that has size close to or equal to the minimum possible. The construction of such permutation coverings has proven to be computationally difficult. While many examples for permutations of small length have been found, and strong asymptotic behavior is known, there are few explicit constructions for permutations of intermediate lengths. Most of these are generated from scratch using greedy algorithms. We explore a different approach here. Starting with a set of permutations with the desired coverage properties, we compute local changes to individual permutations that retain the total coverage of the set. By choosing these local changes so as to make one permutation less "essential" in maintaining the coverage of the set, our method attempts to make a permutation completely non-essential, so it can be removed without sacrificing total coverage. We develop a post-optimization method to do this and present results on sequence covering arrays and other types of permutation covering problems demonstrating that it is surprisingly effective.
ContributorsMurray, Patrick Charles (Author) / Colbourn, Charles (Thesis director) / Czygrinow, Andrzej (Committee member) / Barrett, The Honors College (Contributor) / School of Mathematical and Statistical Sciences (Contributor) / Department of Physics (Contributor)
Created2014-12
Description
In many systems, it is difficult or impossible to measure the phase of a signal. Direct recovery from magnitude is an ill-posed problem. Nevertheless, with a sufficiently large set of magnitude measurements, it is often possible to reconstruct the original signal using algorithms that implicitly impose regularization conditions on this ill-posed problem. Two such algorithms were examined: alternating projections, utilizing iterative Fourier transforms with manipulations performed in each domain on every iteration, and phase lifting, converting the problem to that of trace minimization, allowing for the use of convex optimization algorithms to perform the signal recovery. These recovery algorithms were compared on a basis of robustness as a function of signal-to-noise ratio. A second problem examined was that of unimodular polyphase radar waveform design. Under a finite signal energy constraint, the maximal energy return of a scene operator is obtained by transmitting the eigenvector of the scene Gramian associated with the largest eigenvalue. It is shown that if instead the problem is considered under a power constraint, a unimodular signal can be constructed starting from such an eigenvector that will have a greater return.
ContributorsJones, Scott Robert (Author) / Cochran, Douglas (Thesis director) / Diaz, Rodolfo (Committee member) / Barrett, The Honors College (Contributor) / Electrical Engineering Program (Contributor) / School of Mathematical and Statistical Sciences (Contributor)
Created2014-05
Description
Using object-oriented programming in MATLAB, a collection of functions, named Fourfun, has been created to allow quick and accurate approximations of periodic functions with Fourier expansions. To increase efficiency and reduce the number of computations of the Fourier transform, Fourfun automatically determines the number of nodes necessary for representations that are accurate to close to machine precision. Common MATLAB functions have been overloaded to keep the syntax of the Fourfun class as consistent as possible with the general MATLAB syntax. We show that the system can be used to efficiently solve several differential equations. Comparisons with Chebfun, a similar system based on Chebyshev polynomial approximations, are provided.
ContributorsMcleod, Kristyn Noelle (Author) / Platte, Rodrigo (Thesis director) / Gelb, Anne (Committee member) / Barrett, The Honors College (Contributor) / School of Mathematical and Statistical Sciences (Contributor) / School of International Letters and Cultures (Contributor)
Created2014-05
Description
Many forms of programmable matter have been proposed for various tasks. We use an abstract model of self-organizing particle systems for programmable matter which could be used for a variety of applications, including smart paint and coating materials for engineering or programmable cells for medical uses. Previous research using this model has focused on shape formation and other spatial configuration problems, including line formation, compression, and coating. In this work we study foundational computational tasks that exceed the capabilities of the individual constant memory particles described by the model. These tasks represent new ways to use these self-organizing systems, which, in conjunction with previous shape and configuration work, make the systems useful for a wider variety of tasks. We present an implementation of a counter using a line of particles, which makes it possible for the line of particles to count to and store values much larger than their individual capacities. We then present an algorithm that takes a matrix and a vector as input and then sets up and uses a rectangular block of particles to compute the matrix-vector multiplication. This setup also utilizes the counter implementation to store the resulting vector from the matrix-vector multiplication. Operations such as counting and matrix multiplication can leverage the distributed and dynamic nature of the self-organizing system to be more efficient and adaptable than on traditional linear computing hardware. Such computational tools also give the systems more power to make complex decisions when adapting to new situations or to analyze the data they collect, reducing reliance on a central controller for setup and output processing. Finally, we demonstrate an application of similar types of computations with self-organizing systems to image processing, with an implementation of an image edge detection algorithm.
ContributorsPorter, Alexandra Marie (Author) / Richa, Andrea (Thesis director) / Xue, Guoliang (Committee member) / School of Music (Contributor) / Computer Science and Engineering Program (Contributor) / School of Mathematical and Statistical Sciences (Contributor) / Barrett, The Honors College (Contributor)
Created2016-12
Description
Dividing the plane in half leaves every border point of one region a border point of both regions. Can we divide up the plane into three or more regions such that any point on the boundary of at least one region is on the border of all the regions? In fact, it is possible to design a dynamical system for which the basins of attractions have this Wada property. In certain circumstances, both the Hénon map, a simple system, and the forced damped pendulum, a physical model, produce Wada basins.
ContributorsWhitehurst, Ryan David (Author) / Kostelich, Eric (Thesis director) / Jones, Donald (Committee member) / Armbruster, Dieter (Committee member) / Barrett, The Honors College (Contributor) / School of Mathematical and Statistical Sciences (Contributor) / Department of Chemistry and Biochemistry (Contributor)
Created2013-05
Description
Many programmable matter systems have been proposed and realized recently, each often tailored toward a particular task or physical setting. In our work on self-organizing particle systems, we abstract away from specific settings and instead describe programmable matter as a collection of simple computational elements (to be referred to as particles) with limited computational power that each perform fully distributed, local, asynchronous algorithms to solve system-wide problems of movement, configuration, and coordination. In this thesis, we focus on the compression problem, in which the particle system gathers as tightly together as possible, as in a sphere or its equivalent in the presence of some underlying geometry. While there are many ways to formalize what it means for a particle system to be compressed, we address three different notions of compression: (1) local compression, in which each individual particle utilizes local rules to create an overall convex structure containing no holes, (2) hole elimination, in which the particle system seeks to detect and eliminate any holes it contains, and (3) alpha-compression, in which the particle system seeks to shrink its perimeter to be within a constant factor of the minimum possible value. We analyze the behavior of each of these algorithms, examining correctness and convergence where appropriate. In the case of the Markov Chain Algorithm for Compression, we provide improvements to the original bounds for the bias parameter lambda which influences the system to either compress or expand. Lastly, we briefly discuss contributions to the problem of leader election--in which a particle system elects a single leader--since it acts as an important prerequisite for compression algorithms that use a predetermined seed particle.
ContributorsDaymude, Joshua Jungwoo (Author) / Richa, Andrea (Thesis director) / Kierstead, Henry (Committee member) / Computer Science and Engineering Program (Contributor) / School of Mathematical and Statistical Sciences (Contributor) / Barrett, The Honors College (Contributor)
Created2016-05
Description
Optimal foraging theory provides a suite of tools that model the best way that an animal will <br/>structure its searching and processing decisions in uncertain environments. It has been <br/>successful characterizing real patterns of animal decision making, thereby providing insights<br/>into why animals behave the way they do. However, it does not speak to how animals make<br/>decisions that tend to be adaptive. Using simulation studies, prior work has shown empirically<br/>that a simple decision-making heuristic tends to produce prey-choice behaviors that, on <br/>average, match the predicted behaviors of optimal foraging theory. That heuristic chooses<br/>to spend time processing an encountered prey item if that prey item's marginal rate of<br/>caloric gain (in calories per unit of processing time) is greater than the forager's<br/>current long-term rate of accumulated caloric gain (in calories per unit of total searching<br/>and processing time). Although this heuristic may seem intuitive, a rigorous mathematical<br/>argument for why it tends to produce the theorized optimal foraging theory behavior has<br/>not been developed. In this thesis, an analytical argument is given for why this<br/>simple decision-making heuristic is expected to realize the optimal performance<br/>predicted by optimal foraging theory. This theoretical guarantee not only provides support<br/>for why such a heuristic might be favored by natural selection, but it also provides<br/>support for why such a heuristic might a reliable tool for decision-making in autonomous<br/>engineered agents moving through theatres of uncertain rewards. Ultimately, this simple<br/>decision-making heuristic may provide a recipe for reinforcement learning in small robots<br/>with little computational capabilities.
ContributorsCothren, Liliaokeawawa Kiyoko (Author) / Pavlic, Theodore (Thesis director) / Brewer, Naala (Committee member) / School of Mathematical and Statistical Sciences (Contributor, Contributor) / Barrett, The Honors College (Contributor)
Created2021-05
Description
Over the years, advances in research have continued to decrease the size of computers from the size of<br/>a room to a small device that could fit in one’s palm. However, if an application does not require extensive<br/>computation power nor accessories such as a screen, the corresponding machine could be microscopic,<br/>only a few nanometers big. Researchers at MIT have successfully created Syncells, which are micro-<br/>scale robots with limited computation power and memory that can communicate locally to achieve<br/>complex collective tasks. In order to control these Syncells for a desired outcome, they must each run a<br/>simple distributed algorithm. As they are only capable of local communication, Syncells cannot receive<br/>commands from a control center, so their algorithms cannot be centralized. In this work, we created a<br/>distributed algorithm that each Syncell can execute so that the system of Syncells is able to find and<br/>converge to a specific target within the environment. The most direct applications of this problem are in<br/>medicine. Such a system could be used as a safer alternative to invasive surgery or could be used to treat<br/>internal bleeding or tumors. We tested and analyzed our algorithm through simulation and visualization<br/>in Python. Overall, our algorithm successfully caused the system of particles to converge on a specific<br/>target present within the environment.
ContributorsMartin, Rebecca Clare (Author) / Richa, Andréa (Thesis director) / Lee, Heewook (Committee member) / Computer Science and Engineering Program (Contributor) / School of Mathematical and Statistical Sciences (Contributor, Contributor) / Barrett, The Honors College (Contributor)
Created2021-05