Mathematics Commons^™

The Averaging Lemma, Ronald A. Devore, Guergana Petrova

Faculty Publications

No abstract provided.

Biological Implications Of A Discrete Mathematical Model For Collagen Deposition And Alignment In Dermal Wound Repair, J. C. Dallon, J. A. Sherratt, P. K. Maini, M. W. Ferguson

Faculty Publications

We develop a novel mathematical model for collagen deposition and alignment during dermal wound healing. We focus on the interactions between fibroblasts, modelled as discrete entities, and a continuous extracellular matrix composed of collagen and a fibrin based blood clot. There are four basic interactions assumed in the model: fibroblasts orient the collagen matrix, fibroblasts produce and degrade collagen and fibrin and the matrix directs the fibroblasts and determines the speed of the cells. Several factors which influence the alignment of collagen are examined and related to current anti-scarring therapies using transforming growth factor " The most influential of these …

An Approximation To Miscible Fluid Flows In Porous Media With Point Sources And Sinks By An Eulerian-Lagrangian Localized Adjoint Method And Mixed Finite Element Methods, Hong Wang, Liang Dong, Richard E. Ewing, Stephen L. Lyons, Guan Qin

Faculty Publications

We develop an Eulerian–Lagrangian localized adjoint method (ELLAM)-mixed finite element method (MFEM) solution technique for accurate numerical simulation of coupled systems of partial differential equations (PDEs), which describe complex fluid flow processes in porous media. An ELLAM, which was shown previously to outperform many widely used methods in the context of linear convection-diffusion PDEs, is presented to solve the transport equation for concentration. Since accurate fluid velocities are crucial in numerical simulations, an MFEM is used to solve the pressure equation for the pressure and Darcy velocity. This minimizes the numerical difficulties occurring in standard methods for approximating velocities caused …

An Optimal-Order Error Estimate For An Ellam Scheme For Two-Dimensional Linear Advection-Diffusion Equations, Hong Wang

Faculty Publications

An Eulerian-Lagrangian localized adjoint method (ELLAM) is presented and an- alyzed for two-dimensional linear advection-diffusion partial differential equations (PDEs). An optimal-order error estimate in the L^2 norm and a superconvergence estimate in a discrete H^1 norm are derived. Numerical experiments are performed to verify the theoretical estimates.

Numerical Aspects Of Discrete And Continuum Hybrid Models In Cell Biology, J. C. Dallon

Faculty Publications

In this paper we introduce a method of modeling which mixes continuum and discrete variables, and explain two models in cell biology that use this method. The first application deals with wound healing, more specifically the collagen alignment in scar tissue formation and the second models early aggregation in the cellular slime mold Dictyostelium discoideum. We solve these models using numerical techniques similar to the particle-in-cell method which requires that the discrete and continuum variables are interpolated one to the other. The implementational and numerical details are discussed in an informal and practical manner with particular attention given to the …

An Extension Of Zermelo's Model For Ranking By Paired Comparisons, Christopher P. Grant, Gregory R. Conner

Faculty Publications

In 1929, Zermelo proposed a probabilistic model for ranking by paired comparisons and showed that this model produces a unique ranking of the objects under consideration when the outcome matrix is irreducible. When the matrix is reducible, the model may yield only a partial ordering of the objects. In this paper, we analyse a natural extension of Zermelo's model resulting from a singular perturbation. We show that this extension produces a ranking for arbitrary (nonnegative) outcome matrices and retains several of the desirable properties of the original model. In addition, we discuss computational techniques and provide examples of their use.

A Mathematical Model For Spatially Varying Extracellular Matrix, J. C. Dallon, J. A. Sherratt

Faculty Publications

Orientation of extracellular matrix fibers in the skin is a key ingredient of tissue appearance and function, and differences in fiber alignment are one of the main distinctions between scar tissue and normal skin. In this paper, the authors develop a mathematical model for alignment of collagen fibers and the fibroblast cells that remodel them; the model extends previous work in which spatial variation was excluded. Numerical simulations of the model are presented, which show spatial variations in alignment over long transients, but with spatially uniform behavior in the long term. This is investigated further via asymptotic analysis, using the …

Intersecting Chains In Finite Vector Spaces, Eva Czabarka

Faculty Publications

We prove an Erdős–Ko–Rado-type theorem for intersecting k-chains of subspaces of a finite vector space. This is the q-generalization of earlier results of Erdős, Seress and Székely for intersecting k-chains of subsets of an underlying set. The proof hinges on the author's proper generalization of the shift technique from extremal set theory to finite vector spaces, which uses a linear map to define the generalized shift operation. The theorem is the following.

For c = 0, 1, consider k-chains of subspaces of an n-dimensional vector space over GF(q), such that the smallest subspace in any chain has dimension at least …

Mathematical Modelling Of Extracellular Matrix Dynamics Using Discrete Cells: Fiber Orientation And Tissue Regeneration, J. C. Dallon, J. A. Sherratt, P. K. Maini

Faculty Publications

Matrix orientation plays a crucial role in determining the severity of scar tissue after dermal wounding. We present a model framework which allows us to examine the interaction of many of the factors involved in orientation and alignment. Within this framework, cells are considered as discrete objects, while the matrix is modeled as a continuum. Using numerical simulations, we investigate the effect on alignment of changing cell properties and of varying cell interactions with collagen and fibrin.

An Ellam Scheme For Advection-Diffusion Equations In Two Dimensions, Hong Wang, Helge K. Dahle, Richard E. Ewing, Magne S. Espedal, Robert Sharpley, Shushuang Man

Faculty Publications

We develop an Eulerian--Lagrangian localized adjoint method (ELLAM) to solve two-dimensional advection-diffusion equations with all combinations of inflow and outflow Dirichlet, Neumann, and flux boundary conditions. The ELLAM formalism provides a systematic framework for implementation of general boundary conditions, leading to mass-conservative numerical schemes. The computational advantages of the ELLAM approximation have been demonstrated for a number of one-dimensional transport systems; practical implementations of ELLAM schemes in multiple spatial dimensions that require careful algorithm development are discussed in detail in this paper. Extensive numerical results are presented to compare the ELLAM scheme with many widely used numerical methods and to …

Importance Of Convection And Damping On Rates Of Convergence For The Lax-Wendroff Method, Joseph Kolibal

Faculty Publications

It is well known that in solving steady state problems using hyperbolic time-stepping methods the intent is to drive the transients to zero as quickly as possible. In this paper the convergence to steady state of the Lax-Wendroff method applied to solving the equations of gas dynamics is analyzed for the Laval nozzle problem by comparing the relative rates of damping and convection using a linearized eigenmode analysis. This analysis is developed for the simpler isenthalpic system and then extended to the full Euler equations. Finally, this allows a comparison between these systems. For both models, useful analytical information can …

Interior Blowup In A Convection-Diffusion Equation, Christopher P. Grant

Faculty Publications

This paper addresses the qualitative behavior of a nonlinear convection-diffusion equation on a smooth bounded domain in Rn, in which the strength of the convection grows superlinearly as the density increases. While the initial-boundary value problem is guaranteed to have a local-in-time solution for smooth initial data, it is possible for this solution to be extinguished in nite time. We demonstrate that the way this may occur is through nite-time "blow up," i.e., the unboundedness of the solution in arbitrarily small neighborhoods of one or more points in the closure of the spatial domain. In special circumstances, such as the …

A Continuum Analysis Of The Chemotactic Signal Seen By Dictyostelium Discoideum, J. C. Dallon, H. G. Othmer

Faculty Publications

We develop a mathematical model of cell-to-cell-signalling in Dictyostelium discoideum that predicts the cAMP signal seen by individual cells in early aggregation. The model employs two cells on a plane and is designed to predict the space-time characteristics of both the extracellular cAMP signal seen by one cell when a nearby cell relays, and the intracellular cAMP response produced by the stimulus in the receiving cell. The effect of membrane bound phosphodiesterase is studied and it is shown that cells can orient effectively even in its absence. Our results give a detailed picture of how the spatio-temporal characteristics of the …

The Sharp Sobolev Inequality And The Banchoff-Pohl Inequality On Surfaces, Ralph Howard

Faculty Publications

No abstract provided.

Perturbation Results For Projecting A Point Onto A Linear Manifold, Jiu Ding

Faculty Publications

Some new results will be presented on the perturbation analysis for the orthogonal projection of a point onto a linear manifold. The obtained perturbation upper bound is with respect to the distance from the perturbed solution to the unperturbed manifold.

Recognizing Constant Curvature Discrete Groups In Dimension 3, J. W. Cannon, E. L. Swenson

Faculty Publications

We characterize those discrete groups Gwhich can act properly discontinuously, isometrically, and cocompactly on hyperbolic 3-space H^3 in terms of the combinatorics of the action of G on its space at infinity. The major ingredients in the proof are the properties of groups that are negatively curved (in the large) (that is, Gromov hyperbolic), the combinatorial Riemann mapping theorem, and the Sullivan-Tukia theorem on groups which act uniformly quasiconformally on the 2-sphere.

Some Harmonic N-Slit Mappings, Michael Dorff

Faculty Publications

The class SH consists of univalent, harmonic, and sense-preserving functions f in the unit disk, ∆, such that f = h+g where h(z) = (see PDF), g(z) = (see PDF) . SOH will denote the subclass with b1 = 0. We present a collection of n-slit mappings (n ≥ 2) and prove that the 2-slit mappings are in SH while for n ≥ 3 the mappings are in SOH. Finally we show that these mappings establish the sharpness of a previous theorem by Clunie and Sheil-Small while disproving a conjecture about the inner mapping radius.

A Mathematical Model For Fibroblast And Collagen Orientation, J. C. Dallon, J. A. Sherratt

Faculty Publications

Due to the increasing importance of the extracellular matrix in many biological problems, in this paper we develop a model for fibroblast and collagen orientation with the ultimate objective of understanding how fibroblasts form and remodel the extracellular matrix, in particular its collagen component. The model uses integro-differential equations to describe the interaction between the cells and fibers at a point in space with various orientations. The equations are studied both analytically and numerically to discover different types of solutions and their behavior. In particular we examine solutions where all the fibroblasts and collagen have discrete orientations, a localized continuum …

On The Dynamic Behaviour Of A Thermoviscoelastic Body In Frictional Contact With A Rigid Obstacle, Kenneth Kuttler, K. T. Andrews, M. Shillor

Faculty Publications

We consider the dynamic behaviour of a thermoviscoelastic body which may come into frictional contact with a rigid obstacle. The frictional contact is modelled by general contact and friction laws which include as special cases the power law normal compliance condition and the corresponding generalization of Coulomb's law of dry friction. The stress-strain constitutive relation is assumed to be of Kelvin-Voigt type and the frictional heat generation on the contact surface is taken into account. In this setting we establish the existence of a solution to a weak version of the energy-elasticity system which consists of a parabolic equation coupled …

A Note On Generators Of Least Degree In Gorenstein Ideals, Matthew Miller, Rafael H. Villarreal

Faculty Publications

No abstract provided.

Paracompact Subspaces In The Box Product Topology, Peter Nyikos, Leszek Piatkiwicz

Faculty Publications

No abstract provided.

On The Equivalence Of Certain Consequences Of The Proper Forcing Axiom, Peter Nyikos, Leszek Piatkiwicz

Faculty Publications

No abstract provided.

A Reverse Isoperimetric Inequality, Stability And Extremal Theorems For Plane-Curves With Bounded Curvature, Ralph Howard, Andrejs Treibergs

Faculty Publications

No abstract provided.

Slow Motion In One-Dimensional Cahn-Morral Systems, Christopher P. Grant

Faculty Publications

In this paper we study one-dimensional Cahn-Morral systems, which are the multicomponent analogues of the Cahn-Hilliard model for phase separation and coarsening in binary mixtures. In particular, we examine solutions that start with initial data close to the preferred phases except at finitely many transition points where the data has sharp transition layers, and we show that such solutions may evolve exponentially slowly; i.e., if ε is the interaction length then there exists a constant C such that in exp(C/ε) units of time the change in such a solution is o(1). This corresponds to extremely slow coarsening of a multicomponent …

Classification Of The Tor-Algebras Of Codimension Four Almost Complete Intersections, Andrew R. Kustin

Faculty Publications

Let (R, m, k) be a local ring in which 2 is a unit. Assume that every element of k has a square root in k . We classify the algebras Tor'(R/J, k) as J varies over all grade four almost complete intersection ideals in R. The analogous classification has already been found when J varies over all grade four Gorenstein ideals [21], and when J varies over all ideals of grade at most three [5, 30]. The present paper makes use of the classification, in [21], of the Tor-algebraso f codimension four Gorenstein rings, as well as the (usually …

On The Distribution Of Sums Of Residues, Jerrold R. Griggs

Faculty Publications

No abstract provided.

Besov-Spaces On Domains In Rd, Ronald A. Devore, Robert C. Sharpley

Faculty Publications

No abstract provided.

Precision Spectroscopy Using The Lamb Dip In A Pure Ion Plasma, P. N. Barnes, Grant W. Hart

Faculty Publications

The use of the Lamb dip as a technique for precision spectroscopy in a non-neutral plasma is explored through computer modeling. Using singly ionized magnesium as the ion and under typical pure ion plasma conditions, the measurement appears to be feasible. Under the conditions calculated here, the Lamb dip is only 4% wider than the natural linewidth of the transition.

Projection Solutions Of Frobenius-Perron Operator Equations, Jiu Ding, Tien Tien Li

Faculty Publications

We construct in this paper the first order and second order piecewise polynomial finite approximation schemes for the computation of invariant measures of a class of nonsingular measurable transformations on the unit interval of the real axis. These schemes are based on the Galerkin’s projection method for L¹-spaces and are proved to be convergent for the class of Frobenius-Perron operators.

Labeling Graphs With A Condition At Distance 2, Jerrold R. Griggs, Roger K. Yeh

Faculty Publications

No abstract provided.