AWM at JMM 2015 Abstracts

Special Sessions Abstracts

Monday January 12, 2015, 6:00 p.m.-7:15 p.m. Bridge Hall, Convention Center

AWM Workshop Poster Presentations and Reception

  • Organizers:
  • Gizem Karaali, Pomona College
  • Lerna Pehlivan, York University
  • Brooke Shipley, University of Illinois at Chicago

On Monotonicity for Strain-Limiting Theories of Elasticity

Tina Mai*, Texas A&M University
Jay R. Walton, Texas A&M University (1106-74-150)

This presentation addresses certain notions of convexity for strain-limiting theories of elasticity in which the Green- St.Venant strain tensor is written as a nonlinear response function of the second Piola-Kirchhoff stress tensor. Previous results on strong ellipticity for special strain-limiting theories of elasticity required invertibility of the Fr ́echet derivative of the response function as a fourth-order tensor. The present contribution generalizes the theory to cases in which the Fr ́echet derivative of the response function is not invertible, by studying a weaker rank-1 convexity notion, monotonicity, applied to a general class of nonlinear strain-limiting models. It is shown that the generalized monotonicity holds for Green-St.Venant strains with sufficiently small norms, and fails (through demonstration by counterexample) when the small strain constraint is relaxed.

Semidualizing complexes over tensor products

Hannah Lee Altmann*, North Dakota State University (1106-13-155)>

Let R be a commutative, noetherian ring with identity. A finitely generated R-module C is semidualizing if the homothety map ΧCR : R → HomR(C,C) is an isomorphism and ExtiR(C,C) = 0 for all i > 0. For example, R is semidualizing over R, as is a dualizing module, if R has one. In some sense the number of semidualizing modules gives a measure of the “complexity” of R. We are interested in that number. We will discuss constructing semidualizing modules over tensor products of rings over a field. In particular, this gives us a lower bound on the number of semidualizing modules over the tensor product.

Bratteli diagrams for weak solenoids

Jessica C. Dyer*, University of Illinois at Chicago (UIC) (1106-37-200)

A weak solenoid, in the sense of McCord and Schori, induces a minimal equicontinuous action of a finitely generated group G on a Cantor space X. We use the coding method for such actions, as developed in the paper ”Homogeneous matchbox manifolds” in Transactions AMS, 2013 by Clark and Hurder, to construct an ”almost finite presentation” representing (X, G). We then use this presentation to construct a Bratteli diagram with group actions that captures the partially homogeneous dynamics of weak solenoids.
This work is joint with Steven Hurder and Olga Lukina. (Received August 08, 2014)

Extremal Questions for Matchings

Lauren Keough*, University of Nebraska – Lincoln (1106-05-201)

In recent years there has been increased interest in extremal problems for “counting” parameters of graphs. For example, the Kahn-Zhao theorem gives an upper bound on the number of independent sets in a d-regular graph. In the same spirit, the Upper Matching Conjecture claims an upper bound on the number of k-matchings in a d-regular graph. We are interested in finding which graphs on n vertices with e edges have the minimum number of matchings.
We first solve this question for bipartite graphs. We show that the lex bipartite graph has the fewest matchings of all sizes among bipartite graphs with fixed part sizes and a given number of edges. To prove this result we use both previously known and previously unknown facts about rook placements in Young diagrams. Then we consider both matchings and matchings of fixed sizes in graphs with a given number vertices and edges. We prove that the graph with the fewest matchings is either the lex or the colex graph. Similarly, for fixed k, the graph with the fewest k-matchings is either the lex or the colex graph.

From graphs to Lie algebras to nilmanifolds

Allie Ray*, University of Texas-Arlington (1106-53-218)

I will present if and only if conditions for extending a certain two-step nilpotent Lie algebra associated with a colored, directed graph to a three-step nilpotent Lie algebra. The two-step construction is a generalization of a method used by S. Dani and M. Mainkar. Three step nilpotent Lie algebras are more delicate to construct since the Jacobi equation becomes a consideration. In addition, starting with pairs of Schreier graphs of a Gassmann-Sunada triple, I will consider issues of isospectrality and isometry of the associated nilmanifolds. Methods used include graph theory, combinatorics, Lie theory, group actions, and differential geometry.

A Residual Based Aposteriori Error Estimation in a Fully Automatic hp–FEM for The 2 and 3-D Stokes Model Problem

Arezou Ghesmati*, Texas A&M University;
Wolfgang Bangerth, Texas A&M University (1106-35-236)

Aposteriori error estimator as a computable quantity in terms of known quantities such as approximated solution and the data, gives a useful tool to assess the approximation quality in order to improve the solution adaptively. In this research we present a fully automatic hp-adaptive refinement strategy for Finite Element Method, using a residual based aposteriori error estimation which is based on the solution and the data of local boundary value problems. The reliability and also the efficiency for this introduced estimator has been proved. Moreover the contraction convergence is shown and verified in theoretical part. Our results out of implementation for Stokes problem indicates the exponential rate of convergence in our hp-adaptive algorithm.

Topology of Configurations on Graphs

Safia Chettih*, University of Oregon

The homology and cohomology groups of ordered configurations are known on a number of simple graphs. I will give explicit presentations for the (co)homology of ordered and unordered configurations of two points, along with intersection pairings, on k-pronged graphs and trivalent trees, and discuss the geometrical/combinatorial structures which relate the presentations.

Braid groups and euclidean simplices

Elizabeth Leyton Chisholm*, University of California, Santa Barbara;
Jon McCammond, University of California, Santa Barbara (1106-20-289)

When Krammer and Bigelow independently proved that braid groups are linear, they used the Lawrence-Krammer- Bigelow representation for generic values of its variables q and t. The t variable is closely connected to the traditional Garside structure of the braid groups and it plays a major role in Krammer’s proof. The q variable, associated with the dual Garside structure of the braid groups, has received less attention.
In the special case t = 1 and q real, we show that there is an elegant geometric interpretation of the LKB representation that highlights the role of the q variable, at least when it is viewed in Krammer’s original basis. Concretely, braid group elements can be viewed as acting on and systematically reshaping euclidean simplices. In fact, for each simple element in the dual Garside structure, the reshaping is an elementary operation that we call edge rescaling.

Rational maps with Qp critical points

Bianca A Thompson*, University of Hawaii at Manoa

In prior work by Eremenko and Gabrielov it is shown that if all critical points of a rational function ϕ are real, then ϕ is equivalent to a real rational function. We can reframe the question in the local field Qp. We prove a rational map phi of degree d ≥ 2 with exactly 2 distinct critical points in Qp is equivalent to a Qp−rational function. Similarly, if ϕ is a degree 3 map with 4 critical points in Qp it is equivalent to a Qp −rational function.

Tame filling invariants, examples, and closure properties

Anisah Nu’Man*, University of Nebraska-Lincoln (1106-20-334)

Filling invariants are quasi-isometry invariants for groups with finite presentations defined using properties of van Kampen diagrams. Intrinsic and extrinsic tame filling functions are a recent pair of asymptotic invariants that are a strengthening of the intrinsic diameter (i.e., isodiametric) function and the extrinsic diameter function. Mihalik and Tschantz defined the related concept of tame comb able groups, and Brittenham and Hermiller showed that the existence of a (finite-valued) tame filling function implies that the group is tame combable. Here we give examples of tame filling functions and how they behave under group constructions.

Coupled groundwater-surface water flow: effect of small parameters and numerical methods

Marina Moraiti*, University of Pittsburgh (1106-65-368)

We study the effect of small parameters in the fully evolutionary Stokes-Darcy problem that models the interaction between groundwater and surface water flows. In particular, we look at the effect of the specific storage parameter as it approaches zero as well as at its effect, along with the hydraulic conductivity parameter, on stability and convergence properties of numerical schemes. Further, we present a new numerical method for the coupled problem that is strongly stable – uniformly in all parameters – and second order convergent in space and time.

A Spatio-Temporal Point Process Model for Ambulance Demand

Zhengyi Zhou*, Center for Applied Math, Cornell University;
David S. Matteson, Department of Statistical Science, Cornell University;
Dawn B. Woodard, School of Operations Research and Information Engineering, Cornell University;
Shane G. Henderson, School of Operations Research and Information Engineering, Cornell University;
Athanasios C. Micheas, Department of Statistics, University of Missouri-Columbia, Columbia (1106-62-878)

Ambulance demand estimation at fine time and location scales is critical for fleet management and dynamic deployment. We estimate the spatial distribution of ambulance demand in Toronto, Canada, as it changes over discrete 2-hour intervals. This large-scale dataset is sparse at the desired temporal resolutions and exhibits location-specific temporal dependencies. We address these challenges by introducing a novel characterization of time-varying Gaussian mixture models. We fix the mixture component distributions across all time periods to overcome data sparsity and accurately describe Toronto’s spatial structure, while representing the complex spatio-temporal dynamics through time-varying mixture weights. We constrain the mixture weights to capture weekly seasonality, and apply autoregressive priors on the mixture weights to represent location-specific serial dependence and daily seasonality. While we can use a fixed number of mixture components, we also extend to estimate the number of components using birth-and-death Markov chain Monte Carlo. The proposed model gives higher statistical predictive accuracy and reduces the error in predicting the industry’s operational performance by as much as two-thirds compared to a typical industry practice.

Conjugacy Limits of the Group of Diagonal Matrices

Arielle M Leitner*, University of California, Santa Barbara (1106-22-1091)

A geometric transition is a continuous path of geometries which abruptly changes type in the limit. In this project, we explore geometric transitions of the Cartan subgroup in SLn(R). For n = 3, it turns out the Cartan subgroup has precisely 5 limits, and for n = 4, there are 15 limits, which give rise to generalized cusps on convex projective 3-manifolds.

Applications of algebraic geometry to polar coding

Sarah E Anderson*, Clemson University (1106-14-1104)

In recent groundbreaking work, Arikan developed polar codes as an explicit construction of symmetric capacity achieving codes for binary discrete memoryless channels with low encoding and decoding complexities. A specific 2 × 2 binary kernel matrix G is considered, and G⊗n is used to create 2n new channels. As the number of channels grows, each channel becomes either a noiseless channel or a pure-noise channel, and the rate of this polarization is related to the kernel matrix used. Since Arikan’s original construction, polar codes have been generalized to q-ary discrete memoryless channels, where q is a power of a prime, and other matrices have been considered as kernels. In our work, we expand on the ideas of Mori and Tanaka and Korada and Şaşoĝlu by employing algebraic geometric codes to produce kernels of polar codes, specifically codes from maximal and optimal function fields.

Deer : Forbidden Minimal Digraphs

Stacey Renae McAdams*, Louisiana Tech University (1106-05-1105)

Kuratowski’s theorem implies that a non-directed graph G is outerplanar, or has book thickness 1, if and only if G contains neither K4-minors nor K2,3-minors. In that sense, minors of K4 or K2,3 are forbidden minimals for G to be book thickness 1. We aim to extend the theorem to directed graphs D by using a directed book embedding and, in the process, found a series of directed graphs called deer that are forbidden minimals for D to be book thickness 1.

Graph Determined Symbolic Dynamics and Hybrid Systems

Kimberly D. Ayers*, Iowa State University (1106-37-1299)

In this research we explore the concept of symbolic dynamical systems whose structure is determined by a directed graph, and then discrete-continuous hybrid systems that arise from such dynamical systems. Typically, symbolic dynamics involve the study of a left shift of a bi-infinite sequence. We examine the case when the bi-infinite system is dictated by a graph; that is, the sequence is a bi-infinite path of a directed graph. We then use the concept to study a system of dynamical systems all on the same compact space M, where “switching” between the systems occurs as given by the bi-infinite sequence in question. The concepts of limit sets, chain recurrent sets, chaos, and Morse sets for these systems are explored.

An adaptive GMsFEM for high-contrast flow problems

Eric T. Chung, Chinese University of Hong Kong;
Yalchin Efendiev, Texas A&M University;
Guanglian Li*, Texas A&M University (1106-65-1566)

In this paper, we derive an a-posteriori error indicator for the Generalized Multiscale Finite Element Method (GMs- FEM) framework. This error indicator is further used to develop an adaptive enrichment algorithm for the linear elliptic equation with multiscale high-contrast coefficients. We consider two kinds of error indicators where one is based on the L2-norm of the local residual and the other is based on the weighted H−1-norm of the local residual where the weight is related to the coefficient of the elliptic equation. We show that the use of weighted H−1-norm residual gives a more robust error indicator which works well for cases with high contrast media. The convergence analysis of the method is given.

A looping-delooping adjunction for topological spaces

Martina Rovelli*, EPF Lausanne (1106-18-1699)

Farjoun and Hess introduced twisted homotopical categories, a framework for monoidal categories that come with a looping-delooping adjunction between monoids and comonoids, in which a formal theory of bundles is available. Although much of this kind of structure was inspired by classical constructions and results holding for topological spaces, it does not not seem possible to construct a full twisted homotopical structure for spaces. However, we provide a weak twisted homotopical structure, by showing that (Milnor’s model of) the loop space functor and the classifying space functor form a sort of adjunction between pointed spaces and topological groups. The argument leads to a classification of principal bundles over a fixed space that is dual to the well-known classification of bundles with a fixed group. As a consequence, it is also possible to extend Milnor’s loop space construction to a pseudofunctorial assignment