Steve Butler
Songting Luo
Derrick Stolee

# Fall 2013

December 3 - Y.T. Poon (ISU) Spectral inequalities and quantum marginal problems

# Abstracts

Y.T. Poon (ISU) Spectral inequalities and quantum marginal problems

The characterization of the spectra of Hermitian matrices $A$, $B$ and $C$ satisfying $A+B=C$ began with Weyl's inequalities in 1912.  The problem was solved 15 years ago with conditions given by a set of linear inequalities.  Recently, similar inequalities arise in the solution of some quantum marginal problems.  However, in both cases, the number of inequalities grows exponentially with the size of the matrices.
In this talk, we will discuss the connection among these and other related problems.  In particular, we are interested in specific problems for which the solution can be given by a small set of inequalities.

Misun Min(Argonne National Lab) joint with CAM Scalable high-order algorithms and simulations for electromagnetics and fluids

We demonstrate scalable high-order algorithms based on spectral element, spectral element discontinuous Galerkin, and spectral element discontinuous Galerkin lattice Boltzmann method approaches for transport simulations in electromagnetics and fluids. For electromagnetic systems, we consider the time-domain and frequency-domain formulations for wave scattering and absorption problems. For fluid systems, we consider high-order lattice Boltzmann approaches for turbulence and heat transfer simulations. Algorithmic efforts include efficient parallelization and performance analysis, and realistic simulations with validation. Our computational methodologies will be further extended for efficient and accurate predictive modeling for finding the right materials and the optimal structure of solar cells possessing high energy-conversion efficiency and for understanding heat transfer mechanisms for multiphase flow systems.

JIm Cannon (BYU)

There are ideas about set size (measure) that I could have easily understood as a graduate student but somehow missed.

I will explain my favorites -- my favorite episodes in the search for the ideal measure --  with highlights including, for example, Archimedes's discovery of the volume of a ball; Wallis's product formula for $\pi$, the role it played in Fourier's work on heat, which in turn prompted Riemann's formal definition of integration; the infamous Hausdorff-Banach-Tarski paradox in response to perceived difficulties with Lebesgue integration, with the reluctant conclusion that some sets are simply too complicated to be assigned a size; the fat and space-filling curves of Peano, Hilbert, and Polya; and current research on the constrained isoperimetric inequality.

Joe Mileti (Grinell College) (Non)computable algebra

There is an obvious, but inefficient, procedure to determine whether a given natural number is prime:  simply check whether any of the finitely many smaller numbers evenly divide into it.  Despite the fact that such a naive approach does not work in integral domains like Z[x] or the Gaussian integers Z[i], with a bit of theory one can still develop computational procedures that work to determine the prime elements in these cases.  In some settings, such as when working with the ring of integers in algebraic number fields, one often prefers to work with ideals over elements, but then it becomes desirable to determine computationally whether a given element is in an ideal.  We investigate the question of whether certain ideals and/or the set of prime elements are necessarily computable in certain classes of computable rings.

Wei Wang, (Florida International Univ) Multiscale discontinuous Galerkin methods for second order elliptic equations

In this talk we will introduce the multiscale discontinuous Galerkin (DG) method for solving a class of second order elliptic problems. When the equations contain small scales or oscillating coefficients, traditional numerical methods require extremely refined meshes to resolve the small scale structure of solutions,  which brings numerical difficulties. The main ingredient of our method is to incorporate the small scales into finite element basis functions so that the method can capture the multiscale solution on coarse meshes.

James Rossmanith (ISU)Numerical methods for hyperbolic conservation laws with application to plasma physics

Hyperbolic conservation laws are systems of fi rst-order partial diff erential equations that model the dynamics of linear and nonlinear wave phenomena.

Numerical methods for approximately solving such equations must satisfy certain basic conditions in order to achieve physically meaningful results, including numerical conservation and various consistency requirements.

In specific application areas, many additional properties, including conservation of ancillary quantities and the satisfaction of constraints, must be satisfi ed in order to achieve accurate numerical results.

In this work we are concerned with the development and application of numerical methods for various models in plasma physics. Plasma is a gas to which suffi cient energy has been supplied to dissociate electrons from their nuclei, thus forming a collection of positively and negatively charged ions. The presence of charge carriers makes the plasma electrically conducting so that it responds strongly to electromagnetic fields.

Specifi cally, we describe in this talk recent work on three classes of plasma models.
1. Magnetohydrodynamics: we develop a high-order constrained transport approach, motivated by Whitney forms from discrete exterior calculus, to guarantee a globally divergence-free magnetic field.
2. Two-fl uid models: we develop high-order schemes to solve a class of quadrature-based moment-closure models and apply this to the problem of magnetic reconnection. 3. Kinetic Vlasov-Poisson: we develop high-order schemes to solve the Vlasov-Poisson system with the property that mass, total energy, and positivity of the distribution function are maintained.

The work presented here is joint with several people, including C. Helzel
(Bochum), D. Seal (Michigan State), and Y. Cheng (Wisconsin).

Mike Ferrara (UC Denver)Realization problems for degree sequences

The {\it degree} of a vertex $v$ in a graph $G$ is the number of edges incident to $v$, and the {\it degree sequence} of $G$ is the list of degrees of the vertices of $G$.  A nonnegative integer sequence $\pi$ is then said to be {\it graphic} if it is the degree sequence of some graph $G$, and in this case we call $G$ a {\it realization} of $\pi$. While there are a number of efficient ways to determine if a given sequence is graphic, it is also the case that a particular graphic sequence may have a large number of number of nonisomorphic realizations.  Thus, it is of interested to study the properties that arise over the family of realizations of a sequence.
In this talk, we will primarily focus on {\it potential} problems for graphic sequences, wherein we wish to determine when at least one realization of a given sequence has a particular property.   Specifically, we will examine potential degree sequence analogues to the Tur\'{a}n function and graph packing, both of which are widely studied in classical graph theory.  We will also discuss some interesting applications to discrete imaging and the study of complex networks.

Ryan Martin (ISU) The edit distance in graphs

In this talk, we will discuss the edit distance function, a function of a hereditary property [i]H[/i] and of [i]p[/i], which measures the maximum proportion of edges in a density-[i]p[/i] graph that need to be inserted/deleted in order to transform it into a member of [i]H[/i]. We will describe a method of computing this function and give some results that have been attained using it. The edit distance problem has applications in property testing and evolutionary biology and is closely related to well-studied Tur&aacute;n-type problems.  The results we address will include cycle-free graphs and shows a close relationship between the problem of Zarankiewicz as well as strongly regular graphs. This is joint work with many collaborators, including former graduate students Tracy McKay, Dickinson College and Chelsea Peck.

Marilyn Carlson (ASU) A research-based approach fro improving precalculus teaching and learning

The function concept is a central idea of precalculus and beginning calculus and is used for modeling in the sciences, yet many students complete courses in precalculus with weak understandings of this concept. Students who are unable to construct meaningful function formulas to relate two varying quantities have little chance of understanding ideas of derivative, accumulation and the Fundamental Theorem of Calculus. They are also unable to compose two functions and/or use the chain rule to model relationships between changing quantities in applied problems in calculus, physics, and engineering.

The Pathways Precalculus student materials and teacher resources provide one response to this problem. In this presentation I will describe the Pathways Precalculus student materials and teacher tools that are showing positive gains in student learning of the function concept and other foundational ideas for learning calculus. Our approach to developing and refining these materials should provide a generalizable model for others interested in shifting their curriculum and instruction to support student construction of mathematical practices and knowledge for continued mathematics, science, and engineering course taking and learning.

Tim McNicholl (ISU) Asymptotic density and the Ershov Hierarchy

We discuss recent work on applications of asymptotic density in computability theory. In particular, we classify the asymptotic densities of the $\Delta^0_2$ sets according to their level in the Ershov hierarchy. In particular, it is shown that for $n \geq 2$, a real $r \in [0,1]$ is the density of an $n$-c.e.\ set if and only if it is a diff erence of left-$\Pi_2^0$ reals. Further, we show that the densities of the $\omega$-c.e.\ sets coincide with the densities of the $\Delta^0_2$ sets, and there are $\omega$-c.e.\ sets whose density is not the density of an $n$-c.e. set for any $n \in \omega$. These results are joint work with Rod Downey (Wellington), Carl Jockusch (Illinois), and Paul Schupp (Illinois).

# Archived

September 17 - Tim McNicholl (ISU) Asymptotic density and the Ershov Hierarchy
Poster

September 20 (FRI - Room 150) - Marilyn Carlson (ASU) A research-based approach for improving precalculus teaching and learning
Poster

September 24 - Chelsey Lass and Allison Clark from Transamerica on the actuarial profession

October 15 - Ryan Martin (ISU) The edit distance in graphs

October 22 - Mike Ferrara (UC Denver) Realization problems for degree sequences

October 29 - James Rossmanith (ISU) Numerical methods for hyperbolic conservation laws with application to plasma physics

November 4 - Wei Wang, (Florida International Univ) Multiscale discontinuous Galerkin methods for second order elliptic equations (joint with CAM)

November 5 - Joe Mileti (Grinell College) (Non)computable algebra

November 12 - JIm Cannon (BYU) Does every set have a size?

November 19 - Misun Min (Argonne National Lab) joint with CAM Scalable high-order algorithms and simulations for electromagnetics and fluids

December 3 - Y.T. Poon (ISU) Spectral inequalities and quantum marginal problems