Mathematics & Statisticshttp://hdl.handle.net/10211.3/1224902018-12-12T04:46:03Z2018-12-12T04:46:03ZModeling the Spread of Information on TwitterIgnatius, Darianhttp://hdl.handle.net/10211.3/2066802018-11-27T20:45:19Z2018-11-27T00:00:00ZModeling the Spread of Information on Twitter
Ignatius, Darian
Compartmental models have been used in epidemiology for many years to study the spread of infectious diseases throughout the world. In this thesis, we are recreating and extending the work done by others in [1] to apply one of these models, the SEIZ model, to the spread of news and rumors on Twitter. After deriving the model and discussing its background, we obtained data regarding 6 events, 3 real news stories and 3 rumors. We showed that the method used to minimize the error between the model and the actual data was quite accurate, and that the model was able to work very early on in a story or with limited information. We also attempted to find several combinations of parameters which could distinguish the stories between news and rumors, but no consistent results were found. Finally, we restricted the amount of data fed into the model, and took a look at its ability to estimate the number of tweets in the future.
2018-11-27T00:00:00ZMoment Closure Techniques for the Stochastic Lotka-Volterra ModelTrakoolthai, Tanawathttp://hdl.handle.net/10211.3/2065502018-11-06T18:47:16Z2018-11-06T00:00:00ZMoment Closure Techniques for the Stochastic Lotka-Volterra Model
Trakoolthai, Tanawat
The deterministic Lotka-Volterra model is a simple predator-prey model that classically portrays the interaction between two species, leading to closed curves in the predator-prey phase plane. Using the probability generating function, we develop a corresponding stochastic version of this model, which has the form of a simple birth-death process. This stochastic model involves the expected values of the populations, which are governed by a system of differential equations almost identical in form to the deterministic system. However, we find that the stochastic model is no longer a closed system. To gain a more intuitive understanding of this model, we turn to the moment closure approximation technique, which captures the main features of the stochastic model. We use the moment closure technique for two different distributions--the multivariate normal and the multivariate log-normal distribution. With our use of the moment closure technique, we are able to obtain a closed system of differential equations. To continue the exploration of the moment closure technique, we consider several other well-known deterministic models, in which the corresponding stochastic models are developed.
2018-11-06T00:00:00ZOn Deep Learning and Neural NetworksLyche, Samuelhttp://hdl.handle.net/10211.3/2058072018-08-24T22:52:49Z2018-08-24T00:00:00ZOn Deep Learning and Neural Networks
Lyche, Samuel
This thesis gives an introductory overview of neural networks and deep learning. A new wave of computational power has brought about a fundamental change in the way that software is written. Only recently have machines become powerful enough to efficiently implement these algorithms. Due also to the massive increase in data storage capabilities and data production, artificial intelligence, of which neural networks form part of the foundation, will soon be affiliated with nearly every industry in the world. We give a high-level overview of the mathematical machinery necessary to implement neural networks, as well as the Python code to actually run the algorithms. Finally, we will show an example of how neural networks can be used to find cell nuclei in medical images as we compete in the 2018 Kaggle Data Science Bowl.
Ch1: History of neural networks. Ch2: Mathematics of perceptrons, convolutional networks, activation functions, etc. Ch3: Intro to programming perceptrons and convolutional networks in Python with Keras and TensorFlow. Ch4: Competing in the 2018 Kaggle Data Science Bowl using U-net for cell nuclei segmentation.
2018-08-24T00:00:00ZMetric Dimension of Cayley Graphs Symmetric Groups and Their TranspositionsHomier, Samanthahttp://hdl.handle.net/10211.3/2055312018-08-17T20:55:28Z2018-08-17T00:00:00ZMetric Dimension of Cayley Graphs Symmetric Groups and Their Transpositions
Homier, Samantha
Pretend that you cannot remember where you parked your car in the parking lot of the grocery store, but you do remember some of the cars parked near you. One could construct a graph based on your memory of the cars and then use the idea of the metric dimension to find your car. The metric dimension was introduced by PJ Slater in 1975 and has since been applied in fields such as chemistry, optimization, navigation, and more. There is no general/standard metric dimension for every graph, however, there are known metric dimensions for families of graphs. In this paper we study the metric dimension of Cayley graphs, which are graphs based on groups that have convenient algebraic properties. Our main goal is to find the metric dimension of the Cayley graph associated with the symmetric group $S_4$ and its set of transpositions $T_4$.
2018-08-17T00:00:00Z