Physical Sciences and Mathematics Commons^™

Identifying Prevalent Mathematical Pathways To Engineering In South Carolina, Eliza Gallagher, Christy Brown, D. Andrew Brown, Kristin Kelly Frady, Patrick Bass, Michael A. Matthews, Thomas T. Peters, Robert J. Rabb, Ikhalfani Solan, Ronald W. Welch, Anand K. Gramopadhye

Faculty Publications

National data indicate that initial mathematics course placement in college is a strong predictor of persistence to degree in engineering, with students placed in calculus persisting at nearly twice the rate of those placed below calculus. Within the state of South Carolina, approximately 95% of engineering-intending students who initially place below calculus are from in-state. In order to make systemic change, we are first analyzing system-wide data to identify prevalent educational pathways within the state, and the mathematical milestones along those pathways taken by students in engineering and engineering-related fields. This paper reports preliminary analysis of that data to understand …

Numerical Methods For A Two-Species Competition-Diffusion Model With Free Boundaries, Shuang Liu, Xinfeng Liu

Faculty Publications

The systems of reaction-diffusion equations coupled with moving boundaries defined by Stefan condition have been widely used to describe the dynamics of spreading population and with competition of two species. To solve these systems numerically, new numerical challenges arise from the competition of two species due to the interaction of their free boundaries. On the one hand, extremely small time steps are usually needed due to the stiffness of the system. On the other hand, it is always difficult to efficiently and accurately handle the moving boundaries especially with competition of two species. To overcome these numerical difficulties, we introduce …

Hydrodynamic Theories For Flows Of Active Liquid Crystals And The Generalized Onsager Principle, Xiaogang Yang, Jun Li, M. Gregory Forest, Qi Wang

Faculty Publications

We articulate and apply the generalized Onsager principle to derive transport equations for active liquid crystals in a fixed domain as well as in a free surface domain adjacent to a passive fluid matrix. The Onsager principle ensures fundamental variational structure of the models as well as dissipative properties of the passive component in the models, irrespective of the choice of scale (kinetic to continuum) and of the physical potentials. Many popular models for passive and active liquid crystals in a fixed domain subject to consistent boundary conditions at solid walls, as well as active liquid crystals in a free …

Explicit Constructions Of Rip Matrices And Related Problems, Jean Bourgain, S J. Dilworth, Kevin Ford, Sergei Konyagin, Denka Kutzarova

Faculty Publications

We give a new explicit construction of n×N matrices satisfying the Restricted Isometry Property (RIP). Namely, for someε > 0, largeN, and any n satisfyingN1−ε ≤ n ≤ N, we construct RIP matrices of order k ≥ n1/2+ε and constant δ = n−ε. This overcomesthe natural barrier k = O(n1/2) for proofs based on small coherence, which areused in all previous explicit constructions of RIP matrices. Key ingredients in ourproof are new estimates for sumsets in product sets and for exponential sums with theproducts of sets possessing special additive structure. We also give a construction ofsets of n complex numbers whose …

On The Convergence Of Greedy Algorithms For Initial Segments Of The Haar Basis, S J. Dilworth, E Odell, Th Schlumprecht, Andras Zsak

Faculty Publications

We consider the X-Greedy Algorithm and the Dual Greedy Algorithm in a finite-dimensional Banach space with a strictly monotone basis as the dictionary. We show that when the dictionary is an initial segment of the Haar basis in Lp[0, 1] (1< p < 1) then the algorithms terminate after finitely many iterations and that the number of iterations is bounded by a function of the length of the initial segment. We also prove a more general result for a class of strictly monotone bases.

Cyclic Shifts Of The Van Der Corput Set, Dmitriy Bilyk

Faculty Publications

In 1980, K. Roth showed that the expected value of the L2 discrepancy of the cyclic shifts of the N-point van der Corput set is bounded by a constant multiple of √logN, thus guaranteeing the existence of a shift with asymptotically minimal L2 discrepancy. In the present paper, we construct a specific example of such a shift.

An Optimal-Order Error Estimate For A Family Of Ellam-Mfem Approximations To Porous Medium Flow, Hong Wang

Faculty Publications

Mathematical models used to describe porous medium flow lead to coupled systems of time-dependent nonlinear partial differential equations, which present serious mathematical and numerical difficulties. Standard methods tend to generate numerical solutions with nonphysical oscillations or numerical dispersion along with spurious grid-orientation effect. The ELLAM-MFEM time-stepping procedure, in which an Eulerian–Lagrangian localized adjoint method (ELLAM) is used to solve the transport equation and a mixed finite element method (MFEM) is used for the pressure equation, simulates porous medium flow accurately even if large spatial grids and time steps are used. In this paper we prove an optimal-order error estimate for …

Uniform Estimates For Eulerian-Lagrangian Methods For Singularly Perturbed Time-Dependent Problems, Hong Wang, Kaixin Wang

Faculty Publications

We prove a priori optimal-order error estimates in a weighted energy norm for several Eulerian–Lagrangian methods for singularly perturbed, time-dependent convection-diffusion equations with full regularity. The estimates depend only on certain Sobolev norms of the initial and right-hand side data, but not on ε or any norm of the true solution, and so hold uniformly with respect to ε. We use the interpolation of spaces and stability estimates to derive an ε-uniform estimate for problems with minimal or intermediate regularity, where the convergence rates are proportional to certain Besov norms of the initial and right-hand side data.

On Strongly Asymptotic L(P) Spaces And Minimality, S J. Dilworth, V Ferenczi, Denka Kutzarova, E Odell

Faculty Publications

No abstract provided.

A General Theory Of Almost Convex Functions, S J. Dilworth, Ralph Howard, James W. Roberts

Faculty Publications

No abstract provided.

Adaptive Finite Element Methods For Elliptic Pdes Based On Conforming Centroidal Voronoi-Delaunay Triangulations, Lili Ju, Max Gunzburger, Weidong Zhao

Faculty Publications

A new triangular mesh adaptivity algorithm for elliptic PDEs that combines a posteriori error estimation with centroidal Voronoi–Delaunay tessellations of domains in two dimensions is proposed and tested. The ability of the first ingredient to detect local regions of large error and the ability of the second ingredient to generate superior unstructured grids result in a mesh adaptivity algorithm that has several very desirable features, including the following. Errors are very well equidistributed over the triangles; at all levels of refinement, the triangles remain very well shaped, even if the grid size at any particular refinement level, when viewed globally, …

Outerplanar Crossing Numbers, The Circular Arrangement Problem And Isoperimetric Functions, Eva Czabarka, Ondrej Sykora, Laszlo A. Szekely, Imrich Vrt'o

Faculty Publications

We extend the lower bound in [15] for the outerplanar crossing number (in other terminologies also called convex, circular and one-page book crossing number) to a more general setting. In this setting we can show a better lower bound for the outerplanar crossing number of hypercubes than the best lower bound for the planar crossing number. We exhibit further sequences of graphs, whose outerplanar crossing number exceeds by a factor of log n the planar crossing number of the graph. We study the circular arrangement problem, as a lower bound for the linear arrangement problem, in a general fashion. We …

Every Polynomial-Time 1-Degree Collapses If And Only If P=Pspace, Stephen A. Fenner, Stuart A. Kurtz, James S. Royer

Faculty Publications

No abstract provided.

Every Polynomial-Time 1-Degree Collapses If And Only If P=Pspace, Stephen A. Fenner, Stuart A. Kurtz, James S. Royer

Faculty Publications

No abstract provided.

An Extension Of Elton's L(1)(N) Theorem To Complex Banach Spaces, S J. Dilworth, Joseph P. Patterson

Faculty Publications

No abstract provided.

Development Of Cfl-Free, Explicit Schemes For Multidimensional Advection-Reaction Equations, Hong Wang, Jiangguo Liu

Faculty Publications

We combine an Eulerian–Lagrangian approach and multiresolution analysis to develop unconditionally stable, explicit, multilevel methods for multidimensional linear hyperbolic equations. The derived schemes generate accurate numerical solutions even if large time steps are used. Furthermore, these schemes have the capability of carrying out adaptive compression without introducing mass balance error. Computational results are presented to show the strong potential of the numerical methods developed.

An Ellam Scheme For Multidimensional Advection-Reaction Equations And Its Optimal-Order Error Estimate, Hong Wang, Xiquan Shi, Richard E. Ewing

Faculty Publications

We present an Eulerian-Lagrangian localized adjoint method (ELLAM) scheme for initial-boundary value problems for advection-reaction partial differential equations in multiple space dimensions. The derived numerical scheme is not subject to the Courant-Friedrichs-Lewy condition and generates accurate numerical solutions even if large time steps are used. Moreover, the scheme naturally incorporates boundary conditions into its formulation without any artificial outflow boundary conditions needed, and it conserves mass. An optimal-order error estimate is proved for the scheme. Numerical experiments are performed to verify the theoretical estimate.

The Averaging Lemma, Ronald A. Devore, Guergana Petrova

Faculty Publications

No abstract provided.

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.

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 …

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 …

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

Faculty Publications

No abstract provided.

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.

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.