Mathematics Commons^™

Connectivity Differences Between Gulf War Illness (Gwi) Phenotypes During A Test Of Attention, Tomas Clarke, Jessie Jamieson, Patrick Malone, Rakib U. Rayhan, Stuart Washington, John W. Vanmeter, James N. Baraniuk

Department of Mathematics: Faculty Publications

One quarter of veterans returning from the 1990–1991 Persian Gulf War have developed Gulf War Illness (GWI) with chronic pain, fatigue, cognitive and gastrointestinal dysfunction. Exertion leads to characteristic, delayed onset exacerbations that are not relieved by sleep. We have modeled exertional exhaustion by comparing magnetic resonance images from before and after submaximal exercise. One third of the 27 GWI participants had brain stem atrophy and developed postural tachycardia after exercise (START: Stress Test Activated Reversible Tachycardia). The remainder activated basal ganglia and anterior insulae during a cognitive task (STOPP: Stress Test Originated Phantom Perception). Here, the role of attention …

Intermediate C∗-Algebras Of Cartan Embeddings, Jonathan H. Brown, Ruy Exel, Adam H. Fuller, David R. Pitts, Sarah A. Reznikoff

Department of Mathematics: Faculty Publications

Let A be a C*-algebra and let D be a Cartan subalgebra of A. We study the following question: if B is a C*-algebra such that D B A, is D a Cartan subalgebra of B? We give a positive answer in two cases: the case when there is a faithful conditional expectation from A onto B, and the case when A is nuclear and D is a C*-diagonal of A. In both cases there is a one-to-one correspondence between the intermediate C*-algebras B, and a class of open subgroupoids of the groupoid G, where ! G is the twist …

A “Rule-Of-Five” Framework For Models And Modeling To Unify Mathematicians And Biologists And Improve Student Learning, C. Diaz Eaton, H. C. Highlander, K. D. Dahlquist, G. Ledder, M.D. Lamar, R.C. Schugart

Department of Mathematics: Faculty Publications

Despite widespread calls for the incorporation of mathematical modeling into the undergraduate biology curriculum, there is lack of a common understanding around the definition of modeling, which inhibits progress. In this paper, we extend the “Rule-of-Four,” initially used in calculus reform efforts, to a “Rule-of-Five” framework for models and modeling that is inclusive of varying disciplinary definitions of each. This unifying framework allows us to both build on strengths that each discipline and its students bring, but also identify gaps in modeling activities practiced by each discipline. We also discuss benefits to student learning and interdisciplinary collaboration.

Predicting Impacts Of Chemicals From Organisms To Ecosystem Service Delivery: A Case Study Of Insecticide Impacts On A Freshwater Lake, Nika Galic, Chris J. Salice, Bjorn Birnir, Randall J.F. Bruins, Virginie Ducrot, Henriette I. Jager, Andrew Kanarek, Roberto Pastorok, Richard Rebarber, Pernille Thorbek, Valery E. Forbes

Department of Mathematics: Faculty Publications

Assessing and managing risks of anthropogenic activities to ecological systems is necessary to ensure sustained delivery of ecosystem services for future generations. Ecological models provide a means of quantitatively linking measured risk assessment end points with protection goals, by integrating potential chemical effects with species life history, ecological interactions, environmental drivers and other potential stressors. Here we demonstrate how an ecosystem modeling approach can be used to quantify insecticide-induced impacts on ecosystem services provided by a lake from toxicity data for organism-level endpoints. We used a publicly available aquatic ecosystem model AQUATOX that integrates environmental fate of chemicals and their …

Determinants Of Incidence And Hessian Matrices Arising From The Vector Space Lattice, Saeed Nasseh, Alexandra Seceleanu, Junzo Watanabe

Department of Mathematics: Faculty Publications

Let V = ⁿi= _o V^I bethe lattice of subspaces of the n-dimensional vector space over the finite field Fq, and let A be the graded Gorenstein algebra defined over Q which has V as a Q basis. Let F be the Macaulay dual generator for A. We explicitly compute the Hessian determinant j 2F= Xi Xj j, evaluated at the point X1 = X2 = ... = XN = 1, and relate it to the determinant of the incidence matrix between V1 and Vn-1. Our exploration is motivated by the fact that both of these matrices naturally …

A Doubly Nonlocal Laplace Operator And Its Connection To The Classical Laplacian, Petronela Radu, Kelseys Wells

Department of Mathematics: Faculty Publications

In this paper, motivated by the state-based peridynamic frame- work, we introduce a new nonlocal Laplacian that exhibits double nonlocality through the use of iterated integral operators. The operator introduces addi- tional degrees of exibility that can allow for better representation of physical phenomena at different scales and in materials with different properties. We study mathematical properties of this state-based Laplacian, including connec- tions with other nonlocal and local counterparts. Finally, we obtain explicit rates of convergence for this doubly nonlocal operator to the classical Laplacian as the radii for the horizons of interaction kernels shrink to zero.

Intermediate Cell States In Epithelial-To-Mesenchymal Transition, Yutong Sha, Daniel Haensel, Guadalupe Gutierrez, Huijing Du, Xing Dai, Qing Nie

Department of Mathematics: Faculty Publications

The transition of epithelial cells into a mesenchymal state (epithelial-to-mesenchymal transition or EMT) is a highly dynamic process implicated in various biological processes. During EMT, cells do not necessarily exist in ‘pure’ epithelial or mesenchymal states. There are cells with mixed (or hybrid) features of the two, which are termed as the intermediate cell states (ICSs). While the exact functions of ICS remain elusive, together with EMT it appears to play important roles in embryogenesis, tissue development, and pathological processes such as cancer metastasis. Recent single cell experiments and advanced mathematical modeling have improved our capability in identifying ICS and …

Associated Primes And Syzygies Of Linked Modules, Olgur Celikbas, Mohammad T. Dibaei, Mohsen Gheibi, Arash Sadeghi, Ryo Takahaski

Department of Mathematics: Faculty Publications

Motivated by the notion of geometrically linked ideals, we show that over a Gorenstein local ring R, if a Cohen-Macaulay R-module M of grade g is linked to an R-module N by a Gorenstein ideal c, such that AssR(M)\AssR(N) = ;, then M R N is isomorphic to direct sum of copies of R=a, where a is a Gorenstein ideal of R of grade g + 1. We give a criterion for the depth of a local ring (R;m; k) in terms of the homological dimensions of the modules linked to the syzygies of the residue eld k. As a …

Persistence Metrics For A River Population In A Two-Dimensional Benthic-Drift Model, Yu Jin, Qihua Huang, Julia Blackburn, Mark A. Lewis

Department of Mathematics: Faculty Publications

The study of population persistence in river ecosystems is key for understanding population dynamics, invasions, and instream flow needs. In this paper, we extend theories of persistence measures for population models in one-dimensional rivers to a benthic-drift model in two-dimensional depth- averaged rivers. We define the fundamental niche and the source and sink metric, and establish the net reproductive rate R₀ to determine global persistence of a population in a spatially heterogeneous two-dimensional river. We then couple the benthic-drift model into the two-dimensional computational river model, River2D, to study the growth and persistence of a population and its source …

Lozenge Tilings Of A Halved Hexagon With An Array Of Triangles Removed From The Boundary, Part Ii, Tri Lai

Department of Mathematics: Faculty Publications

Proctor's work on staircase plane partitions yields an exact enumeration of lozenge tilings of a halved hexagon on the triangular lattice. Rohatgi later ex- tended this tiling enumeration to a halved hexagon with a triangle cut o from the boundary. In his previous paper, the author proved a common generalization of Proctor's and Rohatgi's results by enumerating lozenge tilings of a halved hexagon in the case an array of an arbitrary number of triangles has been removed from a non-staircase side. In this paper we consider the other case when the array of tri- angles has been removed from the …

Translation Theorems For The Fourier-Feynman Transform On The Product Function Space C2 A,B, [0,T], Seung Jun Chang, Jae Gil Choi, David Skoug

Department of Mathematics: Faculty Publications

In this article, we establish the Cameron{Martin translation theo- rems for the analytic Fourier{Feynman transform of functionals on the product function space C2 a;b[0; T]. The function space Ca;b[0; T] is induced by the gener- alized Brownian motion process associated with continuous functions a(t) and b(t) on the time interval [0; T]. The process used here is nonstationary in time and is subject to a drift a(t). To study our translation theorem, we introduce a Fresnel-type class Fa;b A1;A2 of functionals on C2 a;b[0; T], which is a generaliza- tion of the Kallianpur and Bromley{Fresnel class FA1;A2 . We then …

Modeling Association In Microbial Communities With Clique Loginear Models, Adrian Dobra, Camilo Valdes, Dragana Ajdic, Bertrand S. Clarke, Jennifer Clarke

Department of Mathematics: Faculty Publications

There is a growing awareness of the important roles that microbial communities play in complex biological processes. Modern investigation of these often uses next generation sequencing of metagenomic samples to determine community composition. We propose a statistical technique based on clique loglinear models and Bayes model averaging to identify microbial components in a metagenomic sample at various taxonomic levels that have significant associations. We describe the model class, a stochastic search technique for model selection, and the calculation of estimates of posterior probabilities of interest. We demonstrate our approach using data from the Human Microbiome Project and from a study …

Cartan Triples, Allan P. Donsig, Adam H. Fuller, David R. Pitts

Department of Mathematics: Faculty Publications

We introduce the class of Cartan triples as a generalization of the notion of a Car- tan MASA in a von Neumann algebra. We obtain a one-to-one correspondence between Cartan triples and certain Clifford extensions of inverse semigroups. Moreover, there is a spectral theorem describing bimodules in terms of their support sets in the fundamental inverse semigroup and, as a corollary, an extension of Aoi’s theorem to this setting. This context contains that of Fulman’s generalization of Cartan MASAs and we discuss his generalization in an appendix.

Predicting Impacts Of Chemicals From Organisms To Ecosystem Service Delivery: A Case Study Of Endocrine Disruptor Effects On Trout, Valery E. Forbes, Steve Railsback, Chiara Accolla, Bjorn Birnir, Randall J.F. Bruins, Virginie Ducrot, Nika Galic, Kristina Garber, Bret C. Harvey, Henriette I. Jager, Andrew Kanarek, Robert Pastorok, Richard Rebarber, Pernille Thorbek, Chris J. Salice

Department of Mathematics: Faculty Publications

We demonstrate how mechanistic modeling can be used to predict whether and how biological responses to chemicals at (sub)organismal levels in model species (i.e., what we typically measure) translate into impacts on ecosystem service delivery (i.e., what we care about). We consider a hypothetical case study of two species of trout, brown trout (Salmo trutta; BT) and greenback cutthroat trout (Oncorhynchus clarkii stomias; GCT). These hypothetical populations live in a high-altitude river system and are exposed to human-derived estrogen (17α‑ethinyl estradiol, EE2), which is the bioactive estrogen in many contraceptives. We use the individual based model in STREAM …

Building Long-Term Support For Faculty Through Graduate Student Instructor Professional Development, Nathan Wakefield, Karina Uhing, Mitchell Hamidi

Department of Mathematics: Faculty Publications

Improving university-level instruction is an important step to improving instruction at all levels, and in order to improve university-level instruction, instructors need to master more effective models of instruction and be able to draw on education literature as they continue to develop as instructors. Well-trained, informed instructors are well equipped to be agents of change when they take on faculty positions. However, this mastery requires training and practice.

Lozenge Tilings Of A Halved Hexagon With An Array Of Triangles Removed From The Boundary, Tri Lai

Department of Mathematics: Faculty Publications

Proctor's work on staircase plane partitions yields an enumeration of lozenge tilings of a halved hexagon on the triangular lattice. Rohatgi recently extended this tiling enumeration to a halved hexagon with a triangle removed from the boundary. In this paper, we prove a generalization of the results of Proctor and Rohatgi by enumerating lozenge tilings of a halved hexagon in which an array of an arbitrary number of adjacent triangles has been removed from the boundary.

Examples Of Finite Free Complexes Of Small Rank And Small Homology, Srikanth B. Iyengar, Mark E. Walker

Department of Mathematics: Faculty Publications

In this paper we construct counterexamples to five related conjectures concerning the rank an homology of finite free complexes over commuatitive noetherian rings, and, in particular, over group algebras of elementary abelian groups.

Existing And Regularity Of Minimizers For Nonlocal Energy Functionals, Mikil D. Foss, Petronela Radu, Cory Wright

Department of Mathematics: Faculty Publications

In this paper, we consider minimizers for nonlocal energy functionals generalizing elastic energies that are connected with the theory of peridynamics [19] or nonlocal diffusion models [1]. We derive nonlocal versions of the Euler-Lagrange equations under two sets of growth assumptions for the integrand. Existence of minimizers is shown for integrands with joint convexity (in the function and nonlocal gradient components). By using the convolution structure, we show regularity of solutions for certain Euler-Lagrange equations. No growth assumptions are needed for the existence and regularity of minimizers results, in contrast with the classical theory.

Diagnostic Effects Of An Early Mastery Activity In College Algebra And Precalculus, Nathan Wakefield, Joe Champion, Jessalyn Bolkema, Douglas Dailey

Department of Mathematics: Faculty Publications

The purpose of this study was to investigate implementation of an early intervention mastery activity during the first two weeks of college algebra and precalculus courses at a large U.S. public university. Statistical modeling of (N = 935) students’ performance in the courses, including a logistic regression model of pass/fail course achievement with students’ high school rank, ACT Mathematics scores, and performance on the intervention as explanatory variables, suggested significant independent differences in course performance across performance levels on the early mastery activity. An evaluation of diagnostic validity for the model yielded a 19% false negative rate (predicted to …

Perfect Matchings Of Trimmed Aztec Rectangles, Tri Lai

Department of Mathematics: Faculty Publications

We consider several new families of subgraphs of the square grid whose matchings are enumerated by powers of several small prime numbers: 2, 3, 5, and 11. Our graphs are obtained by trimming two opposite corners of an Aztec rectangle. The result yields a proof of a conjecture posed by Ciucu. In addition, we reveal a hidden connection between our graphs and the hexagonal dungeons introduced by Blum.

Existence And Rapid Convergence Results For Nonlinear Caputo Nabla Fractional Difference Equations, Xiang Liu, Baoguo Jia, Lynn Erbe, Allan Peterson

Department of Mathematics: Faculty Publications

This paper is concerned with finding properties of solutions to initial value problems for nonlinear Caputo nabla fractional difference equations. We obtain existence and rapid convergence results for such equations by use of Schauder’s fixed point theorem and the generalized quasi-linearization method, respectively. A numerical example is given to illustrate one of our rapid convergence results.

Detecting Finite Flat Dimension Of Modules Via Iterates Of The Frobenius Endomorphism, Douglas J. Dailey, Srikanth B. Iyengar, Thomas Marley

Department of Mathematics: Faculty Publications

It is proved that a module M over a Noetherian ring R of positive characteristic p has finite flat dimension if there exists an integer t > 0 such that Tor R (M, fe R) = 0 for t < i< t + dim R and infinitely many e. This extends results of Herzog, who proved it when M is finitely generated. It is also proved that when R is a Cohen-Macaulay local ring, it suffices that the Tor vanishing holds for one e > logp e(R) is the multiplicity of R.

A Framework For Predicting Impacts On Ecosystem Services From (Sub)Organismal Responses To Chemicals, Valery E. Forbes, Chris J. Salice, Bjorn Birnir, Randy J.F. Bruins, Peter Calow, Virginie Ducrot, Nika Galic, Kristina Garber, Bret C. Harvey, Henriette Jager, Andrew Kanarek, Robert Pastorok, Steve F. Railsback, Richard Rebarber, Pernille Thorbek

Department of Mathematics: Faculty Publications

Protection of ecosystem services is increasingly emphasized as a risk-assessment goal, but there are wide gaps between current ecological risk-assessment endpoints and potential effects on services provided by ecosystems. The authors present a framework that links common ecotoxicological endpoints to chemical impacts on populations and communities and the ecosystem services that they provide. This framework builds on considerable advances in mechanistic effects models designed to span multiple levels of biological organization and account for various types of biological interactions and feedbacks. For illustration, the authors introduce 2 case studies that employ well-developed and validated mechanistic effects models: the inSTREAM individual-based …

Gene Expression Noise Enhances Robust Organization Of The Early Mammalian Blastocyst, William R. Holmes, Nabora Soledad Reyes De Mochel, Qixuan Wang, Huijing Du, Tao Peng, Michael Chiang, Olivier Cinquin, Ken Cho, Qing Nie

Department of Mathematics: Faculty Publications

A critical event in mammalian embryo development is construction of an inner cell mass surrounded by a trophoectoderm (a shell of cells that later form extraembryonic structures). We utilize multi-scale, stochastic modeling to investigate the design principles responsible for robust establishment of these structures. This investigation makes three predictions, each supported by our quantitative imaging. First, stochasticity in the expression of critical genes promotes cell plasticity and has a critical role in accurately organizing the developing mouse blastocyst. Second, asymmetry in the levels of noise variation (expression fluctuation) of Cdx2 and Oct4 provides a means to gain the benefits of …

An Early Semester Mastery Activity And Intervention In First-Year Calculus, Allan P. Donsig, Nathan Wakefield

Department of Mathematics: Faculty Publications

Success in first-year mathematics courses is essential for students to pursue STEM careers, including teaching careers. We investigate a mastery activity given during the first two weeks of a first-year calculus course at the research site. Previous work showed a model using this activity in College Algebra, together with ACT and high school rank, was predictive of student success in precalculus. Here we do a similar analysis for such an activity in calculus, including an intervention for students who do not complete the activity. We also investigate the intervention’s effectiveness. These results show that the early mastery activity, especially when …

Structure For Regular Inclusions. I, David R. Pitts

Department of Mathematics: Faculty Publications

We give general structure theory for pairs (C,D) of unital C*- algebras where D is a regular and abelian C*-subalgebra of C.

When D is maximal abelian in C, we prove existence and uniqueness of a completely positive unital map E of C into the injective envelope I(D) of D such that EjD = idD; E is a useful replacement for a conditional expectation when no expectation exists. When E is faithful, (C,D) has numerous desirable properties: e.g. the linear span of the normalizers has a unique minimal C*- norm; D norms C; and isometric isomorphisms of norm-closed subalgebras lying …

Oscillation Of Solution To Second-Order Half-Linear Delay Dynamic Equations On Time Scales, Hongwu Wu, Lynn Erbe, Allan Peterson

Department of Mathematics: Faculty Publications

This article concerns the oscillation of solutions to second-order half-linear dynamic equations with a variable delay. By using integral averaging techniques and generalized Riccati transformations, new oscillation criteria are obtained. Our results extend Kamenev-type, Philos-type and Li-type oscillation criteria. Several examples are given to illustrate our results.

Matched Metrics And Channels, Marcelo Firer, Judy L. Walker

Department of Mathematics: Faculty Publications

The most common decision criteria for decoding are maximum likelihood decoding and nearest neighbor decoding. It is well-known that maximum likelihood decoding coincides with nearest neighbor decoding with respect to the Hamming metric on the binary symmetric channel. In this work we study channels and metrics for which those two criteria do and do not coincide for general codes.

Simple Adaptive Control For Positive Linear Systems With Applications To Pest Management, Chris Guiver, Christina Edholm, Yu Jin, Markus Mueller, Jim Powell, Richard Rebarber, Brigitte Tenhumberg, Stuart Townley

Department of Mathematics: Faculty Publications

Pest management is vitally important for modern arable farming, but models for pest species are often highly uncertain. In the context of pest management, control actions are naturally described by a nonlinear feedback that is generally unknown, which thus motivates a robust control approach. We argue that adaptive approaches are well suited for the management of pests and propose a simple high-gain adaptive tuning mechanism so that the nonlinear feedback achieves exponential stabilization. Furthermore, a switched adaptive controller is proposed, cycling through a set of given control actions, that also achieves global asymptotic stability. Such a model in practice allows …

Comparing The Growth Of The Prime Numbers To The Natural Numbers, Michael A. Brilleslyper, Nathan Wakefield, A. J. Wallerstein, Bradley Warner

Department of Mathematics: Faculty Publications

We define a new method of measuring the rate of divergence for an increasing positive sequence of integers. We introduce the growth function for such a sequence and its associated growth limit. We use these tools to study the divergence rate for the natural numbers, polynomial and exponential-type sequences, and the prime numbers. We conclude with a number of open questions concerning general properties and characterizations of growth functions and the set of possible growth limits.