Mathematics Commons^™

Why Ellipsoids In Mechanical Analysis Of Wood Structures, F. Niklas Schietzold, Julio Urenda, Vladik Kreinovich, Wolfgang Graf, Michael Kaliske

Departmental Technical Reports (CS)

Wood is a very mechanically anisotropic material. At each point on the wooden beam, both average values and fluctuations of the local mechanical properties corresponding to a certain direction depend, e.g., on whether this direction is longitudinal, radial or tangential with respect to the grain orientation of the original tree. This anisotropy can be described in geometric terms, if we select a point x and form iso-correlation surfaces -- i.e., surfaces formed by points y with the same level of correlation ρ(x,y) between local changes in the vicinities of the points x and y. Empirical analysis shows that for each …

Can We Preserve Physically Meaningful "Macro" Analyticity Without Requiring Physically Meaningless "Micro" Analyticity?, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Physicists working on quantum field theory actively used "macro" analyticity -- e.g., that an integral of an analytical function over a large closed loop is 0 -- but they agree that "micro" analyticity -- the possibility to expand into Taylor series -- is not physically meaningful on the micro level. Many physicists prefer physical theories with physically meaningful mathematical foundations. So, a natural question is: can we preserve physically meaningful "macro" analyticity without requiring physically meaningless "micro" analyticity? In the 1970s, an attempt to do it was made by using constructive mathematics, in which only objects generated by algorithms are …

Several Years Of Practice May Not Be As Good As Comprehensive Training: Zipf's Law Explains Why, Francisco Zapata, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Many professions practice certifications as a way to establish that a person practicing this profession has reached a certain skills level. At first glance, it may sound like several years of practice should help a person pass the corresponding certification test, but in reality, even after several years of practice, most people are not able to pass the test, while after a few weeks of intensive training, most people pass it successfully. This sounds counterintuitive, since the overall number of problems that a person solves during several years of practice is much larger than the number of problems solved during …

Two- And Three-Loop Data For Groomed Jet Mass, Adam Kardos, Andrew J. Larkoski, Zoltán Trócsányi

Portland Institute for Computational Science Publications

We discuss the status of resummation of large logarithmic contributions to groomed event shapes of hadronic final states in electron-positron annihilation. We identify the missing ingredients needed for next-to-next-to-next-to-leading logarithmic (NNNLL) resummation of the mMDT groomed jet mass in e⁺e⁻ collisions: the low-scale collinear-soft constants at two-loop accuracy, c⁽²⁾_Sc, and the three-loop non-cusp anomalous dimension of the global soft function, γ⁽²⁾_S. We present a method for extracting those constants using fixed-order codes: the EVENT2 program to obtain the color coefficients of c⁽²⁾_Sc, and MCCSM for extracting γ^{(2) …}

Quantifying Cds Sortability Of Permutations By Strategic Pile Size, Marisa Gaetz, Bethany Flanagan, Marion Scheepers, Meghan Shanks

Mathematics Faculty Publications and Presentations

The special purpose sorting operation, context directed swap (CDS), is an example of the block interchange sorting operation studied in prior work on permutation sorting. CDS has been postulated to model certain molecular sorting events that occur in the genome maintenance program of some species of ciliates. We investigate the mathematical structure of permutations not sortable by the CDS sorting operation. In particular, we present substantial progress towards quantifying permutations with a given strategic pile size, which can be understood as a measure of CDS non-sortability. Our main results include formulas for the number of permutations in S_n with …

Enhancing Produce Safety: State Estimation-Based Robust Adaptive Control Of A Produce Wash System, Vahid Azimi, Daniel Munther, Mojtaba Sharifi, Patricio A. Vela

Mathematics and Statistics Faculty Publications

The rapid introduction of fresh-cut produce into a produce wash system can dramatically decrease the free chlorine (FC) concentration level in the wash water, resulting in potential widespread cross-contamination throughout the entire wash system. To minimize such contamination, a sufficient level of FC must be maintained in the wash water. This paper presents a state estimation-based robust adaptive sliding mode (RASM) control strategy for the wash system to stabilize the FC concentration level during fresh-cut iceberg lettuce washing. This feedback control law for FC dosing is suggested to provide a sufficient FC injection rate (FCIR) to the wash system in …

Boolean Network Topologies And The Determinative Power Of Nodes, Bronson W. Wacker, Mihaela T. Velcsov, Jim A. Rogers

Mathematics Faculty Publications

Boolean networks have been used extensively for modeling networks whose node activity could be simplified to a binary outcome, such as on-off. Each node is influenced by the states of the other nodes via a logical Boolean function. The network is described by its topological properties which refer to the links between nodes, and its dynamical properties which refer to the way each node uses the information obtained from other nodes to update its state. This work explores the correlation between the information stored in the Boolean functions for each node in a property known as the determinative power and …

Harmonic Equiangular Tight Frames Comprised Of Regular Simplices, Matthew C. Fickus, Courtney A. Schmitt

Faculty Publications

An equiangular tight frame (ETF) is a sequence of unit-norm vectors in a Euclidean space whose coherence achieves equality in the Welch bound, and thus yields an optimal packing in a projective space. A regular simplex is a simple type of ETF in which the number of vectors is one more than the dimension of the underlying space. More sophisticated examples include harmonic ETFs which equate to difference sets in finite abelian groups. Recently, it was shown that some harmonic ETFs are comprised of regular simplices. In this paper, we continue the investigation into these special harmonic ETFs. We begin …

Quasinormality Of Powers Of Commuting Pairs Of Bounded Operators, Raul E. Curto, Sang H. Lee, Jasang Yoon

School of Mathematical & Statistical Sciences Faculty Publications

We study jointly quasinormal and spherically quasinormal pairs of commuting operators on Hilbert space, as well as their powers. We first prove that, up to a constant multiple, the only jointly quasinormal 2-variable weighted shift is the Helton-Howe shift. Second, we show that a left invertible subnormal operator T whose square T2 is quasinormal must be quasinormal. Third, we generalize a characterization of quasinormality for subnormal operators in terms of their normal extensions to the case of commuting subnormal n-tuples. Fourth, we show that if a 2-variable weighted shift W(α,β) and its powers W(α,β)(2,1) and W(α,β)(1,2) are all spherically quasinormal, …

Expected Resurgences And Symbolic Powers Of Ideals, Eloísa Grifo, Craig Huneke, Vivek Mukundan

Department of Mathematics: Faculty Publications

We give explicit criteria that imply the resurgence of a self-radical ideal in a regular ring is strictly smaller than its codimension, which in turn implies that the stable version of Harbourne's conjecture holds for such ideals. This criterion is used to give several explicit families of such ideals, including the defining ideals of space monomial curves. Other results generalize known theorems concerning when the third symbolic power is in the square of an ideal, and a strong resurgence bound for some classes of space monomial curves

Three-Dimensional Viscoelastic Instabilities In A Four-Roll Mill Geometry At The Stokes Limit, Paloma Gutierrez-Castillo, Adam Kagel, Becca Thomases

Mathematics Sciences: Faculty Publications

Three-dimensional numerical simulations of viscoelastic fluids in the Stokes limit with a four-roll mill background force (extended to the third dimension) were performed. Both the Oldroyd-B model and FENE-P model of viscoelastic fluids were used. Different temporal behaviors were observed depending on the Weissenberg number (non-dimensional relaxation time), model, and initial conditions. Temporal dynamics evolve on long time scales, and simulations were accelerated by using a Graphics Processing Unit (GPU). Previously, parameter explorations and long-time simulations in 3D were prohibitively expensive. For a small Weissenberg number, all the solutions are constant in the third dimension, displaying strictly two-dimensional temporal evolutions. …

Fibrations, Unique Path Lifting, And Continuous Monodromy, Hanspeter Fischer, Jacob D. Garcia

Mathematics Sciences: Faculty Publications

Given a path-connected space X and H≤π1(X,x0), there is essentially only one construction of a map pH:(X˜H,x˜0)→(X,x0) with connected and locally path-connected domain that can possibly have the following two properties: (pH)#π1(X˜H,x˜0)=H and pH has the unique lifting property. X˜H consists of equivalence classes of paths starting at x0, appropriately topologized, and pH is the endpoint projection. For pH to have these two properties, T1 fibers are necessary and unique path lifting is sufficient. However, pH always admits the standard lifts of paths. We show that pH has unique path lifting if it has continuous (standard) monodromies toward a T1 …

Analysis In Metric Spaces, Mario Bonk, Luca Capogna, Piotr Hajłasz, Nageswari Shanmugalingam, Jeremy Tyson

Mathematics Sciences: Faculty Publications

No abstract provided.

Explicit Ambient Metrics And Holonomy, Ian M. Anderson, Thomas Leistner, Pawel Nurowski

Mathematics and Statistics Faculty Publications

We present three large classes of examples of conformal structures whose Fefferman-Graham ambient metrics can be found explicitly. Our method for constructing these examples rests upon a set of sufficiency conditions under which the Fefferman-Graham equations are assured to reduce to a system of inhomogeneous linear partial differential equations. Our examples include conformal pp-waves and, more importantly, conformal structures that are defined by generic co-rank 3 distributions in dimensions 5 and 6.Our examples illustrate various aspects of the ambient metric construction.

The holonomy algebras of our ambient metrics are studied in detail. In particular, we exhibit a large class of …

Mathematical Modelling Of The Tuberculosis Epidemiology, Ally Yeketi Ayinla

Student Works (2020-2029)

This project analyses the tuberculosis (TB) epidemic mathematically using compartmental modelling approach. Three models are presented to discuss drug susceptible and multi-drug resistant TB. The first model presented has 4 compartments; susceptible, exposed, infectious and recovered. The relevance of the exposed class in managing TB is analysed and found to be useful in delaying the eventual onset of the infection. Compared to previous researches, our results significantly show that when efforts are made such that no infected individual bypasses the exposed class and progresses directly to the infectious, the TB epidemic is successfully combatted. Also, the model is used to …

Classification Of Radial Solutions To Equations Related To Caffarelli–Kohn–Nirenberg Inequalities, John Villavert

School of Mathematical & Statistical Sciences Faculty Publications

This article studies the qualitative and quantitative properties of radial solutions to an elliptic equation related to the Euler–Lagrange equations for certain sharp Caffarelli–Kohn–Nirenberg inequalities. Namely, we examine the equation

−div(|x|aDu)=|x|bup, u>0, in RN, where p>1, N≥2, N−2+a≥0 and b>−N. The main results establish the properties of radially symmetric solutions including existence, uniqueness, and classification results as well as results on the asymptotic and intersecting behaviour of such solutions.

Delaunay Surfaces Expressed In Terms Of A Cartan Moving Frame, Paul Bracken

School of Mathematical & Statistical Sciences Faculty Publications

Delaunay surfaces are investigated by using a moving frame approach. These surfaces correspond to surfaces of revolution in the Euclidean three-space. A set of basic one-forms is defined. Moving frame equations can be formulated and studied. Related differential equation which depend on variables relevant to the surface are obtained. For the case of minimal and constant mean curvature surfaces, the coordinate functions can be calculated in closed form. In the case in which the mean curvature is constant, these functions can be expressed in terms of Jacobi elliptic functions.

The Role Of The Asymmetric Ekman Dissipation Term On The Energetics Of The Two-Layer Quasi-Geostrophic Model, Eleftherios Gkioulekas

School of Mathematical & Statistical Sciences Faculty Publications

In the two-layer quasi-geostrophic model, the friction between the flow at the lower layer and the surface boundary layer, placed beneath the lower layer, is modeled by the Ekman term, which is a linear dissipation term with respect to the horizontal velocity at the lower layer. The Ekman term appears in the governing equations asymmetrically; it is placed at the lower layer, but does not appear at the upper layer. A variation, proposed by Phillips and Salmon, uses extrapolation to place the Ekman term between the lower layer and the surface boundary layer, or at the surface boundary layer. We …

Sufficient Dimension Folding In Regression Via Distance Covariance For Matrix‐Valued Predictors, Wenhui Sheng, Qingcong Yuan

Mathematical and Statistical Science Faculty Research and Publications

In modern data, when predictors are matrix/array‐valued, building a reasonable model is much more difficult due to the complicate structure. However, dimension folding that reduces the predictor dimensions while keeps its structure is critical in helping to build a useful model. In this paper, we develop a new sufficient dimension folding method using distance covariance for regression in such a case. The method works efficiently without strict assumptions on the predictors. It is model‐free and nonparametric, but neither smoothing techniques nor selection of tuning parameters is needed. Moreover, it works for both univariate and multivariate response cases. In addition, we …

Black And Brown Students’ Mathematics Anxiety In Elementary School: The Use Of Restorative Justice Circles And Critical Concepts Of Care, Hope, And Love, Mariana E. Winnik

Dissertations, Theses, and Capstone Projects

Children navigate their world and are constantly making meaning of their experiences. Through this meaning making, children are also constructing their identities. Black and Brown children have an added layer of identity construction compared to their White peers. Black and Brown students develop their racial identity in conjunction with multiple other identities. This paper focuses specifically on how Black and Brown students construct a "mathematics identity" that is meaningful to their racial identity in order to help lessen their mathematics anxiety. I argue that the use of Restorative Justice Circles (RJC) in classrooms will allow for students to bring their …