Mathematics Faculty Publications and PresentationsCopyright (c) 2021 University of Portland All rights reserved.
https://pilotscholars.up.edu/mth_facpubs
Recent documents in Mathematics Faculty Publications and Presentationsen-usFri, 08 Jan 2021 10:51:59 PST3600The Case for Biocalculus: Design, Retention, and Student Performance
https://pilotscholars.up.edu/mth_facpubs/18
https://pilotscholars.up.edu/mth_facpubs/18Tue, 05 Mar 2019 12:16:04 PST
Calculus is one of the primary avenues for initial quantitative training of students in all science, technology, engineering, and mathematics fields, but life science students have been found to underperform in the traditional calculus setting. As a result, and because of perceived lack of its contribution to the understanding of biology, calculus is being actively cut from biology program requirements at many institutions. Here, we present an alternative: a model for learning mathematics that sees the partner disciplines as crucial to student success. We equip faculty with information to engage in dialogue within and between disciplinary departments involved in quantitative education. This includes presenting a process for interdisciplinary development and implementation of biology-oriented Calculus I courses at two institutions with different constituents, goals, and curricular constraints. When life science students enrolled in these redesigned calculus courses are compared with life science students enrolled in traditional calculus courses, students in the redesigned calculus courses learn calculus concepts and skills as well as their traditional course peers; however, the students in the redesigned courses experience more authentic life science applications and are more likely to stay and succeed in the course than their peers who are enrolled in traditional courses. Therefore, these redesigned calculus courses hold promise in helping life science undergraduate students attain Vision and Change recommended competencies.
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Carrie Diaz Eaton et al.Keys to successful mentoring of undergraduate research teams with an emphasis in applied mathematics research
https://pilotscholars.up.edu/mth_facpubs/17
https://pilotscholars.up.edu/mth_facpubs/17Tue, 05 Mar 2019 12:15:57 PST
Independent of institution size and faculty research expectations, a growing number of colleges and universities encourage their undergraduates to engage in some form of research experience. To meet the demand of students seeking such experiences and to ensure these experiences are of high quality, it is imperative to have qualified mentors. While senior faculty rely on years of experience in mentoring research projects, professors stepping into these undergraduate mentoring roles at the graduate student or junior faculty level may not be as equipped to handle the potential hurdles unique to working with teams of undergraduates. This article is aimed at such an audience. Although much of the article is relevant to mentoring projects in any area of mathematics, some comments and suggestions are directed more to working with students in applied mathematics. This article includes advice gleaned from the National Science Foundation-sponsored Center for Undergraduate Research in Mathematics (CURM) faculty workshop in conjunction with personal experiences from the author, a CURM mini-grant recipient. The primary goals of the paper are to answer questions one might have when starting a project with undergraduates and to provide the reader with concrete steps to follow in planning and successfully completing such a project.
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Hannah Callender HighlanderModeling the Spread of the Zika Virus at the 2016 Olympics
https://pilotscholars.up.edu/mth_facpubs/16
https://pilotscholars.up.edu/mth_facpubs/16Wed, 30 May 2018 13:45:57 PDT
The Zika Virus is an arbovirus that is spread by mosquitoes of the Aedes genus and causes mild fever-like symptoms. It is strongly associated with microcephaly, a condition that affects development of fetal brains. With the recent emergence of Zika in Brazil, we develop an agent-based model to track mosquitoes and humans throughout the 2016 Olympics in Rio de Janeiro to investigate how the Olympics might affect the spread of the virus. There are many unknowns regarding the spread and prevalence of Zika, with approximately 80% of infected individuals unaware of their infectious status. We therefore discuss results of experiments where several unknown parameters were varied, including the rate at which mosquitoes successfully bite humans, the percentage of initially infected mosquitoes, and the sizes of the human and mosquito populations. From these experiments, we make initial predictions regarding effective control measures for the spread of Zika.
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Triona Matheson et al.Agent-Based Models of the Green Dot Bystander Violence Prevention Program on College Campuses
https://pilotscholars.up.edu/mth_facpubs/15
https://pilotscholars.up.edu/mth_facpubs/15Wed, 30 May 2018 13:41:33 PDT
Despite the hard work of a diverse collection of organizations committed to violence prevention, the prevalence of rape, abuse, and other forms of interpersonal violence remains startling, especially on college campuses. Here we present an agent-based model (ABM) of interpersonal violence rooted in the philosophy of the Green Dot Bystander Training Program, in the hopes of providing insight into ways in which training of students can be improved so that intervention attempts are more effective. Two models, with and without adaptive behaviors, are studied under two population sizes. Through sensitivity testing, various outcomes are analyzed to measure the effectiveness of each intervention strategy. The scenarios that result in the smallest relative number of violent acts are those with a denser population, while the adaptive models produce unexpected results that prompt questions about human behavior and our tendency toward bystander intervention.
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Parkes Kendrick et al.Proportional Insulin Infusion in Closed-Loop Control of Blood Glucose
https://pilotscholars.up.edu/mth_facpubs/14
https://pilotscholars.up.edu/mth_facpubs/14Wed, 30 May 2018 13:41:26 PDT
A differential equation model is formulated that describes the dynamics of glucose concentration in blood circulation. The model accounts for the intake of food, expenditure of calories and the control of glucose levels by insulin and glucagon. These and other hormones affect the blood glucose level in various ways. In this study only main effects are taken into consideration. Moreover, by making a quasi-steady state approximation the model is reduced to a single nonlinear differential equation of which parameters are fit to data from healthy subjects. Feedback provided by insulin plays a key role in the control of the blood glucose level. Reduced β-cell function and insulin resistance may hamper this process. With the present model it is shown how by closed-loop control these defects, in an organic way, can be compensated with continuous infusion of exogenous insulin.
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Johan Grasman et al.A Feasibility Study of Personalized Prescription Schemes for Glioblastoma Patients Using a Proliferation and Invasion Glioma Model
https://pilotscholars.up.edu/mth_facpubs/13
https://pilotscholars.up.edu/mth_facpubs/13Wed, 30 May 2018 13:41:19 PDTPurpose: This study investigates the feasibility of personalizing radiotherapy prescription schemes (treatment margins and fractional doses) for glioblastoma (GBM) patients and their potential benefits using a proliferation and invasion (PI) glioma model on phantoms. Methods and Materials: We propose a strategy to personalize radiotherapy prescription schemes by simulating the proliferation and invasion of the tumor in 2D according to the PI glioma model. We demonstrate the strategy and its potential benefits by presenting virtual cases, where the standard and personalized prescriptions were applied to the tumor. Standard prescription was assumed to deliver 46 Gy in 23 fractions to the initial, gross tumor volume (GTV_{1}) plus a 2 cm margin and an additional 14 Gy in 7 fractions to the boost GTV_{2} plus a 2 cm margin. The virtual cases include the tumors with a moving velocity of 0.029 (slow-move), 0.079 (average-move), and 0.13 (fast-move) mm/day for the gross tumor volume (GTV) with a radius of 1 (small) and 2 (large) cm. For each tumor size and velocity, the margin around GTV_{1} and GTV_{2} was varied between 0–6 cm and 1–3 cm, respectively. Equivalent uniform dose (EUD) to normal brain was constrained to the EUD value obtained by using the standard prescription. Various linear dose policies, where the fractional dose is linearly decreasing, constant, or increasing, were investigated to estimate the temporal effect of the radiation dose on tumor cell-kills. The goal was to find the combination of margins for GTV_{1} and GTV_{2} and a linear dose policy, which minimize the tumor cell-surviving fraction (SF) under a normal tissue constraint. The efficacy of a personalized prescription was evaluated by tumor EUD and the estimated survival time. Results: The personalized prescription for the slow-move tumors was to use 3.0–3.5 cm margins for GTV_{1}, and a 1.5 cm margin for GTV_{2}. For the average- and fast-move tumors, it was optimal to use a 6.0 cm margin for GTV_{1} and then 1.5–3.0 cm margins for GTV_{2}, suggesting a course of whole brain therapy followed by a boost to a smaller volume. It was more effective to deliver the boost sequentially using a linearly decreasing fractional dose for all tumors. Personalized prescriptions led to surviving fractions of 0.001–0.465% compared to the standard prescription, and increased the tumor EUDs by 25.3–49.3% and estimated survival times by 7.6–22.2 months. Conclusions: Personalizing treatment margins based on the measured proliferative capacity of GBM tumor cells can potentially lead to significant improvements in tumor cell kill and related clinical outcomes.
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Minsun Kim et al.Modeling epidemics on a regular tree graph
https://pilotscholars.up.edu/mth_facpubs/12
https://pilotscholars.up.edu/mth_facpubs/12Wed, 15 Mar 2017 14:25:46 PDT
We will first provide a brief introduction to models of disease transmission on so-called contact networks, which can be represented by various structures from the mathematical field of graph theory. These models allow for exploration of stochastic effects and incorporation of more biological detail than the classical compartment-based ordinary differential equation models, which usually assume both homogeneity in the population and uniform mixing. In particular, we use an agent-based modelling platform to compare theoretical predictions from mathematical epidemiology to results obtained from simulations of disease transmission on a regular tree graph. We also demonstrate how this graph reveals connections between network structure and the spread of infectious diseases. Specifically, we discuss results for how certain properties of the tree graph, such as network diameter and density, alter the duration of an outbreak.
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Claire Seibold et al.Exceptional automorphisms of (generalized) super elliptic surfaces
https://pilotscholars.up.edu/mth_facpubs/11
https://pilotscholars.up.edu/mth_facpubs/11Tue, 10 May 2016 13:02:29 PDT
A super-elliptic surface is a compact, smooth Riemann surface S with a conformal automorphism w of prime order p such that S/ has genus zero, extending the hyper-elliptic case p=2. More generally, a cyclic n-gonal surface S has an automorphism w of order n such that S/ has genus zero. All cyclic n-gonal surfaces have tractable defining equations. Let A = Aut(S) and N be the normalizer of C = in A. The structure of N, in principal, can be easily determined from the defining equation. If the genus of S is sufficiently large in comparison to n, and C satisfies a generalized super-elliptic condition, then A = N. For small genus A - N may be non-empty and, in this case, any automorphism h ∈ A - N is called exceptional. The exceptional automorphisms of all general cyclic n-gonal surfaces seems to be hard. We focus on generalized super-elliptic surfaces in which n is composite and the projection of S onto S/C is fully ramified. Generalized super-elliptic surfaces are easily identified by their defining equations. In this paper we discuss an approach to the determination of generalized super-elliptic surfaces with exceptional automorphisms.
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S. Allen Broughton et al.Gaps in the space of skeletal signatures
https://pilotscholars.up.edu/mth_facpubs/10
https://pilotscholars.up.edu/mth_facpubs/10Tue, 16 Jun 2015 09:45:14 PDT
Skeletal signatures were introduced in Anderson and Wootton (see [1]) as a tool to describe the space of all signatures with which a group can act on a surface of genusσ ≥ 2.In the present paper, we provide an essentially complete description of the regular gaps that appear in the space of skeletal signatures, together with proofs of those parts of the conjectures posed in Anderson and Wootton (see [1]) related to these regular gaps.
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James W. Anderson et al.The Rank of Recurrence Matrices
https://pilotscholars.up.edu/mth_facpubs/9
https://pilotscholars.up.edu/mth_facpubs/9Wed, 25 Feb 2015 11:20:14 PST
A recurrence matrix is defined as a matrix whose entries (read left-to-right, row-by-row) are sequential elements generated by a linear recurrence relation. The maximal rank of this matrix is determined by the order of the corresponding recurrence. In the case of an order-two recurrence, the associated matrix fails to have full rank whenever the ratio of the two initial values of the sequence is an eigenvalue of the relation.
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Christopher Lee et al.Toric Integrable Geodesic Flows in Odd Dimensions
https://pilotscholars.up.edu/mth_facpubs/8
https://pilotscholars.up.edu/mth_facpubs/8Tue, 03 Dec 2013 16:10:38 PST
Let Q be a compact, connected n-dimensional Riemannian manifold, and assume that the geodesic flow is toric integrable. If n ≠ 3 is odd, or if π1(Q) is infinite, we show that the cosphere bundle of Q is equivariantly contactomorphic to the cosphere bundle of the torus T^{n}. As a consequence, Q is homeomorphic to T^{n}.
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Christopher R. Lee et al.Trivial, Strongly Minimal Theories Are Model Complete After Naming Constants
https://pilotscholars.up.edu/mth_facpubs/7
https://pilotscholars.up.edu/mth_facpubs/7Fri, 15 Nov 2013 16:12:30 PST
We prove that if M is any model of a trivial, strongly minimal theory, then the elementary diagram Th(M_{M}) is a model complete L_{M}-theory. We conclude that all countable models of a trivial, strongly minimal theory with at least one computable model are 0"-decidable, and that the spectrum of computable models of any trivial, strongly minimal theory is Σ^{0}_{5}.
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Sergey S. Goncharov et al.Computable Categoricity of Trees of Finite Height
https://pilotscholars.up.edu/mth_facpubs/6
https://pilotscholars.up.edu/mth_facpubs/6Fri, 15 Nov 2013 16:12:28 PST
We characterize the structure of computably categorical trees of finite height, and prove that our criterion is both necessary and sufficient. Intuitively, the characterization is easiest to express in terms of isomorphisms of (possibly infinite) trees, but in fact it is equivalent to a Σ^{0}_{3}-condition. We show that all trees which are not computably categorical have computable dimension ω. Finally, we prove that for every n ≥ 1 in ω, there exists a computable tree of finite height which is ∆^{0}_{n+1}-categorical but not ∆^{0}_{n}-categorical.
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Steffen Lempp et al.Weakly Smooth Continua
https://pilotscholars.up.edu/mth_facpubs/5
https://pilotscholars.up.edu/mth_facpubs/5Fri, 15 Nov 2013 16:12:27 PST
We define and investigate a class of continua called weakly smooth. Smooth dendroids, weakly smooth dendroids, generalized trees, and smooth continua are all examples of weakly smooth continua. We generalize characterizations of the above mentioned examples to weakly smooth continua. In particular, we characterize them as compact Hausdorff spaces which admit a quasi order satisfying certain properties.
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Lewis LumMonotone Retracts and Some Characterizations of Dendrites
https://pilotscholars.up.edu/mth_facpubs/4
https://pilotscholars.up.edu/mth_facpubs/4Fri, 15 Nov 2013 16:12:25 PST
Let M be a metric continuum containing a fixed point p. The following conditions are shown to be equivalent. (i) M is a dendrite. (ii) Each subcontinuum of M is a monotone retract of M. (iii) M is arcwise connected and each subcontinuum of M containing p is a monotone retract of M.
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G. R. Gordh Jr. et al.The Cyclic Connectivity of Homogeneous Arcwise Connected Continua
https://pilotscholars.up.edu/mth_facpubs/3
https://pilotscholars.up.edu/mth_facpubs/3Fri, 15 Nov 2013 16:12:22 PST
A continuum is cyclicly connected provided each pair of its points lie together on some simple closed curve. In 1927, G. T. Whyburn proved that a locally connected plane continuum is cyclicly connected if and only if it contains no separating points. This theorem was fundamental in his original treatment of cyclic element theory. Since then numerous authors have obtained extensions of Whyburn's theorem. In this paper we characterize cyclic connectedness in the class of all Hausdorff continua.
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David P. Bellamy et al.Arc-Smooth Continua
https://pilotscholars.up.edu/mth_facpubs/2
https://pilotscholars.up.edu/mth_facpubs/2Fri, 15 Nov 2013 16:12:20 PST
Continua admitting arc-structures and arc-smooth continua are introduced as higher dimensional analogues of dendroids and smooth dendroids, respectively. These continua include such spaces as: cones over compacta, convex continua in l_{2}, strongly convex metric continua, injectively metrizable continua, as well as various topological semigroups, partially ordered spaces, and hyperspaces. The arc-smooth continua are shown to coincide with the freely contractible continua and with the metric K-spaces of Stadtlander. Known characterizations of smoothness in dendroids involving closed partial orders, the set function T, radially convex metrics, continuous selections, and order preserving mappings are extended to the setting of continua with arc-structures. Various consequences of the special contractibility properties of arc-smooth continua are also obtained.
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J. B. Fugate et al.Logarithmic bounds for infinite Prandtl number rotating convection
https://pilotscholars.up.edu/mth_facpubs/1
https://pilotscholars.up.edu/mth_facpubs/1Fri, 15 Nov 2013 16:12:18 PST
Convection refers to fluid motion that is induced by buoyancy. In thermal convection buoyancy is due to temperature differences and one of the interesting questions is how much of the total heat transfer is due to convection. The natural measure of this quantity is the Nusselt number, N, and many experiments and numerical simulations have been performed to discern the relationship between N and the various parameters which describe the system. Much of this research has focused on the forcing parameter, although it has been observed that rotation plays a nontrivial role as well.
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Peter Constantin et al.