# Master of Science in Mathematical Sciences and Master of Science in Applied and Computational Mathematics Plan B Project Papers

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This collection contains some of the Plan B project papers produced by master's degree students in the Master of Science in Mathematical Sciences graduate program or in its predecessor, the Master of Science in Applied and Computational Mathematics graduate program. Please note that students in these programs complete either a Plan A (thesis-based) program or a Plan B (project-based) program. Only Plan B project papers are included here; Plan As (theses) can be found in the University of Minnesota Twin Cities Dissertations and Theses collection.

To see Plan B project papers for specific graduate degrees, click the links below:

Search within Master of Science in Mathematical Sciences and Master of Science in Applied and Computational Mathematics Plan B Project Papers

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Item (1+1) Evolutionary Algorithm on Random Planted Vertex Cover Problems(2024-03) Kearney, JackEvolutionary Algorithms are powerful optimization tools that use the power of randomness and inspiration from biology to achieve results. A common combinatorial optimization problem is the recovery of a minimum vertex cover on some graph 𝐺 = (𝑉, 𝐸). In this work, an evolutionary algorithm will be employed on specific instances of the minimum vertex cover problem containing a random planted solution. This situation is common in data networks and translates to a core set of nodes and larger fringe set that are connected to the core. This study introduces a parameterized analysis of a standard (1+1) Evolutionary Algorithm applied to the random planted distribution of vertex cover problems. When the planted cover is at most logarithmic, restarting the (1+1) EA every 𝑂(𝑛 log 𝑛) steps will, within polynomial time, yield a cover at least as small as the planted cover for sufficiently dense random graphs (𝑝 > 0.71). For superlogarithmic planted covers, the (1+1) EA is proven to find a solution within fixed-parameter tractable time in expectation. To complement these theoretical investigations, a series of computational experiments were conducted, highlighting the intricate interplay between planted cover size, graph density, and runtime. A critical range of edge probability was also investigated.Item Constructing Confidence Intervals for L-statistics Using Jackknife Empirical Likelihood(2020-06-16) Wang, FuliThe linear function of order statistics which is quite known as L-statistics has been widely used in non-parametric statistic such as location estimation and construction of tolerance level. The L-statistics include a family of statistics. The trimmed mean, Gini’s mean difference, and discard-deviation are all important L-statistics which have been well-investigated in relevant research. In order to make inference on L-statistics, we apply jackknife method to L-statistics and generate jackknife pseudo samples. There are two significant advantages of jackknifing the data. First, observations from the jackknife samples behave as if they were independent and identically distributed (iid) random variables. Second, the central limit theorem holds for jackknife samples under mild conditions, see, e.g Cheng [1], so the normal approximation method can be applied to the new sample to estimate the true values of L-statistics. In addition to normal approximation, we also apply jackknife empirical likelihood method to construct the confidence intervals for L-statistics. Our simulation and real-data application results both indicate that the jackknife empirical likelihood-based confidence intervals performs better than the normal approximation-based confidence intervals in terms of coverage probability and the length of confidence intervals.Item The Amazing Composobot: Music Information Retreval and Algorithmic Composition(2018-05) Walker, MarcusMusic has powerful and inscrutable effects on the human mind, and we are far from fully understanding how that magic works. But music is not random: there are patterns in the sounds and rhythms of a piece that can be analyzed, things that can be learned! In this work I will review relevant research on the subject of Music Information Retrieval and then introduce Composobot, an original program that incorporates and extends the lessons of that research. Together we will examine how Composobot prepares musical pieces for processing, analyzes them to extract systems of patterns and dependencies, and then composes novel musical pieces based on what it has learned. Finally, we will discuss how much of the magic that is in the music we love can be captured by learning patterns the way Composobot does, and how those methods might be tweaked to capture an even greater share of it.Item Pebbling of Oriented Graphs(2017-09) DeVries, Jerad SIn traditional graph pebbling a move across an edge is made by removing two pebbles from one vertex and adding one pebble to an adjacent vertex. We extend this concept to oriented graphs by subtracting three pebbles when moving against an edge orientation and two pebbles when moving with an edge orientation. The cover pebbling number of an oriented graph is the minimum number of pebbles such that given any initial placement of these pebbles we can simultaneously place a pebble on every vertex. In this paper we will look at pebblings of oriented paths.Item Multivariate Zero-Inflated Poisson Regression(2017-06) Wang, YangIn this report, we develop a procedure to analyze the relationship between the ob- served multi-dimensional counts and a set of explanatory variables. The counts follow a multivariate Poisson distribution or a multivariate zero-inflated Poisson distribution. Maximum likelihood estimates (MLE) for the model parameters are obtained by the Newton-Raphson (NR) iteration and the expectation-maximization (EM) algorithm, respectively. In Newton-Raphson method, the first and second derivatives of the log- likelihood function are derived to carry out the numerical evaluation. Formulas using EM algorithm are also introduced. A comparison of the estimation performance is made from simulation studies.Item Applied Time Series and Duluth Temperature Prediction(2017-06) Wan, XiangpengAutoregressive integrated moving average (ARIMA) has been one of the popular linear models in time series forecasting during the past three decades.The Triple Expo- nential Model also can be used to fit the time series data. This project takes Duluth temperature predictions as a case study, finding the best statistical model to predict the temperature. I collected 30 years of Duluth monthly maximum temperature data, from 1986 to 2016, and I fi t 29 years of them into di erent models including Triple Exponential Smoothing model, ARIMA model, and SARIMA model. Then I predicted the last year's temperature in those models, and I compared them to the true value of last year's temperature, which gave me the SSE value for each model so that I could find the best model.Item Vertex Magic Group Edge Labelings(2017-05) McKeown, Michael AA vertex-magic group edge labeling of a graph G(V;E) with |E| = n is an injection from E to an abelian group ᴦ of order n such that the sum of labels of all incident edges of every vertex x ϵ V is equal to the same element µ ϵ ᴦ. We completely characterize all Cartesian products Cn□Cm that admit a vertex-magic group edge labeling by Z2nm, as well as provide labelings by a few other finite abelian groups.Item A Connection between Analytic and Nonanalytic Singular Perturbations of the Quadratic Map(2017-05) Liu, YujiongTo explore the connection between the analytic and the nonanalytic complex dy- namics, this paper studied a special case of the singularly perturbed quadratic map: β β ƒβ‚t (z) = z2 + t — + (1 – t) — z2 – z2 which can transit from nonanalytic to analytic by varying t from 0 to 1. Other variables involved in this map are the dynamic variable z ϵ C and the main parameter β ϵ R. Our investigation shows that this transition map has much richer behaviors than the end point cases. The parameter space can be no longer subdivided into only four or three regions as shown in the studies by Devaney and Bozyk respectively. Correspondingly, in the dynamic plane, there also appear several new intermediate cases among which we identified two transitions: a connected case where the filled Julia set is connected and a disconnected case where the filled Julia set consists of infinitely many components. This paper also pointed out that ƒβ‚t (z) is semiconjugate to the four symbols shift map for the disconnected case. This semiconjugacy provides a way to largely understand the dynamical behaviors for the non escape points. Further study shows that the critical set plays an important role in the construction of the filled Julia set. Therefore, the transition of the critical set and its image set are also discussed in this paper. At the end, we presented two sets of conjectures for the bounded critical orbits and the escape critical orbits for future study.Item Statistical Analysis of Moose Habitat Behaviors Using Bayesian Hierarchical Model with Spatially Varying Coefficients(2017-06) Kroc, MatejIn the past few years interest in statistical modeling has rapidly increased for scientists in many different fields. With new technologies and the ability to collect larger amounts of data they sought a tool which would help them to get a better understanding, and eventually, prediction of behavior of subjects in their range of study. For biologists and ecologists habitat data is necessary to develop effective conservation and management strategies, and help determine what is behind the change in the population of different species. Our research is focused on the moose habitat behavior statistics. Moose, Alces alces, are the largest of all deer species. Male moose are recognizable by their huge antlers, which can spread up to 6 feet wide. Because of their tall body, they prefer to browse higher shrubs and their typical habitat is a dense mixed boreal forest in North America, including the northern United States, Canada, Alaska, and in Scandinavia and Russia. Despite their large bodies, moose are good swimmers and are often seen in lakes and rivers feeding on aquatic plants both at and below the surface. One of the reasons why moose habitat behavior is the subject of study by many biologists is recent changes in population in North America. Since the 1990's, the moose population in northern Minnesota has decline significantly. Based on a moose population survey from 2017, the population in northeastern Minnesota has dropped from about 8; 000 moose to a stable population of just under 4; 000 moose over the last 4 years. Meanwhile, the northwestern Minnesotan population practically disappeared after declining from 4; 000 to fewer than 100. The reason behind this steep drop is unknown. Many scientists believe that it could be caused by climate change. Shorter winters and longer falls give more time for parasites, especially winter ticks, to find a host. For purposes of research, moose wore GPS collars, which allow biologists to track their location and collect essential data for future work. In some cases, moose received a tiny transmitter which monitored their heart rate and temperature and notified biologists when the moose died. This work intends to utilize the Bayesian hierarchic model with spatially varying coefficients to obtain better insights into moose habitat behavior in Northern Minnesota.