Department of Mathematics and Statistics
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Historical note: Prior to the 1986-1987 academic year, the Department of Mathematics and Statistics was known as the Department of Mathematical Sciences.
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Browsing Department of Mathematics and Statistics by Subject "Department of Mathematics and Statistics"
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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 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 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 Can ethology help make optimal foraging theory more realistic and useful?(1993) Green, Richard FItem Can optimal foraging stabilize a predator-prey system?(1990) Green, Richard FItem 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 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 Fashions in science(2002) Green, Richard FItem Feeding, optimal foraging and Bayesian foraging(2004) Green, Richard FItem The Gini index and other measures of inequality(2002) Green, Richard FItem How much does it cost a parasitoid to be unmated?(2008) Green, Richard FItem Hyperbole (1987-09-21)(University of Minnesota, Duluth, 1987-09-21) University of Minnesota, Duluth. Math ClubItem Hyperbole (1987-09-28)(University of Minnesota, Duluth, 1987-09-28) University of Minnesota, Duluth. Math ClubItem Hyperbole (1987-10-05)(University of Minnesota, Duluth, 1987-10-05) University of Minnesota, Duluth. Math ClubItem Hyperbole (1987-10-12)(University of Minnesota, Duluth, 1987-10-12) University of Minnesota, Duluth. Math ClubItem Hyperbole (1987-10-26)(University of Minnesota, Duluth, 1987-10-26) University of Minnesota, Duluth. Math ClubItem Hyperbole (1987-11-02)(University of Minnesota, Duluth, 1987-11-02) University of Minnesota, Duluth. Math ClubItem Hyperbole (1987-11-09)(University of Minnesota, Duluth, 1987-11-09) University of Minnesota, Duluth. Math ClubItem Hyperbole (1987-12-14)(University of Minnesota, Duluth, 1987-12-14) University of Minnesota, Duluth. Math Club