### Browsing 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, JackShow more Evolutionary 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.Show more Item The Amazing Composobot: Music Information Retreval and Algorithmic Composition(2018-05) Walker, MarcusShow more Music 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.Show more Item Applied Time Series and Duluth Temperature Prediction(2017-06) Wan, XiangpengShow more Autoregressive 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.Show more Item Can ethology help make optimal foraging theory more realistic and useful?(1993) Green, Richard FShow more Item Can optimal foraging stabilize a predator-prey system?(1990) Green, Richard FShow more Item Combinatorial Problems Motivated by Databases (2014-02-06)(2014) Leck, UweShow more Some combinatorial problems and results will be discussed that arise in the context of restoration of lost information in distributed databases. Consider a set T of lattice points in a k x k grid and call it a configuration. You can think of the points in T as faulty nodes that need to be repaired or decoded. Performing a step of decoding means transforming T into a new configuration T' by removing all points belonging to some horizontal or vertical line L, under the constraint that only t points of T are on L (where t is some given number). T is decodable if it can be transformed into the empty set by an appropriate sequence of decoding steps. Examples of interesting questions in this context are: What is the largest size of a decodable configuration? Among all decodable configurations with the same given number of points, which are the hardest to decode (i.e., which require the most decoding steps)? What are the smallest decodable configurations that require some given number of decoding steps? How does all this generalize to higher dimensional grids?Show more Item A Connection between Analytic and Nonanalytic Singular Perturbations of the Quadratic Map(2017-05) Liu, YujiongShow more To 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.Show more Item Constant-Sum Partitions of Even Cardinality(2020) Montgomery, Grant AShow more A number n is said to have the constant-sum property if for every possible partition of n, there exists disjoint subsets of the first n integers, such that the sum of the elements in each subset have the same remainder when divided by n (This means they are congruent with some constant integer c modulo n). For the case when n is odd, Kaplan, Lev, and Roditty [1] proved in 2009 that there exists a constant-sum partition for n if and only if only there exists no more than one singleton set. For the case when n is even and split into an odd number of subsets, Freyberg [2] proved in 2019 that an n/2-sum partition exists for n if and only if there exists no more than one singleton subset. This leaves open the case when n is even and each subset has an even number of elements. My UROP project focused on the case when n is even and broken into groups that each have even cardinality, that is, they each have an even number of elements.Show more Item Constructing Confidence Intervals for L-statistics Using Jackknife Empirical Likelihood(2020-06-16) Wang, FuliShow more The 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.Show more Item Developing an Assessment Instrument for Outreach Events to Measure Impact on STEM Identity of Middle and High School Students(2020-12) Parranto, Helena; Bibelnieks, TracyShow more There is a large body of research literature that uses quantitative and qualitative methods to identify factors affecting attitudes, interest and motivation of k-12 students to engage in and pursue STEM related activities, academic programs and careers [Potvin, 2014]. Furthermore, impact of stand-alone events directed at middle and high school students [Badri, 2016] has been studied for impact on development (or reconsideration) of a students’ STEM identity in the context of their community, school and further STEM career goals. The STEM outreach events such as UMD’s Swenson College of Science and Engineering STEM Discovery Day have the potential to positively influence the STEM identity of middle and high school participants that could potentially lead to those students selecting a STEM degree in college. Through this research, baseline data from STEM Discovery Day can be used to research and create an evaluation instrument that will measure STEM Identity in future SCSE Outreach events.Show more Item Dynamics of Quadratic Families(2015-07-08) Shang, Haitao; Peckham, BruceShow more This paper is based on the readings in the author's independent study on "advanced dynamical systems", and the author's mathematics honors project. It is a combination of the survey of some classical papers and the results from the research project. In the review part, none of the results are new and even less of them are due to the author; in the research part, we mainly focus the dynamics of the quadratic family along the real line. More specifically, in this paper we review and summarize the dynamics of one- and two- dimensional real quadratic maps from both topological and statistical viewpoints, and provide global pictures for their dynamics. Meanwhile, we briefly review the main results of the dynamics of one-dimensional complex quadratic maps under holomorphic singular perturbations, and provide recent research results about its dynamics under a nonholomorphic singular perturbation.Show more Item Fashions in science(2002) Green, Richard FShow more Item Feeding, optimal foraging and Bayesian foraging(2004) Green, Richard FShow more Item Generation of Pseudoprimes(2012) Stewart, DanielleShow more Item The Gini index and other measures of inequality(2002) Green, Richard FShow more Item How much does it cost a parasitoid to be unmated?(2008) Green, Richard FShow more Item Hyperbole (1987-09-21)(University of Minnesota, Duluth, 1987-09-21) University of Minnesota, Duluth. Math ClubShow more Item Hyperbole (1987-09-28)(University of Minnesota, Duluth, 1987-09-28) University of Minnesota, Duluth. Math ClubShow more Item Hyperbole (1987-10-05)(University of Minnesota, Duluth, 1987-10-05) University of Minnesota, Duluth. Math ClubShow more