Physical Sciences and Mathematics Commons^™

Research Needs And Challenges In The Few System: Coupling Economic Models With Agronomic, Hydrologic, And Bioenergy Models For Sustainable Food, Energy, And Water Systems, Catherine L. Kling, Raymond W. Arritt, Gray Calhoun, David A. Keiser, John M. Antle, Jeffery G. Arnold, Miguel Carriquiry, Indrajeet Chaubey, Peter Christensen, Baskar Ganapathysubramanian, Philip Gassman, William Gutowski, Thomas W. Hertel, Gerritt Hoogenboom, Elena Irwin, Madhu Khanna, Pierre Mérel, Daniel J. Phaneuf, Andrew Plantinga, Stephen Polasky, Paul Preckel, Sergey Rabotyagov, Ivan Rudik, Silvia Secchi, Aaron Smith, Andrew Vanloocke, Calvin Wolter, Jinhua Zhao, Wendong Zhang

Andy VanLoocke

On October 12–13, a workshop funded by the National Science Foundation was held at Iowa State University in Ames, Iowa with a goal of identifying research needs related to coupled economic and biophysical models within the FEW system. Approximately 80 people attended the workshop with about half representing the social sciences (primarily economics) and the rest from the physical and natural sciences. The focus and attendees were chosen so that findings would be particularly relevant to SBE research needs while taking into account the critical connectivity needed between social sciences and other disciplines. We have identified several major gaps in …

Organohalogen Pollutants And Human Health, Prasada Rao S. Kodavanti, Bommanna Loganathan

Bommanna Loganathan

Enhanced Lasso Recovery On Graph, Xavier Bresson, Thomas Laurent, James Von Brecht

Thomas Laurent

This work aims at recovering signals that are sparse on graphs. Compressed sensing offers techniques for signal recovery from a few linear measurements and graph Fourier analysis provides a signal representation on graph. In this paper, we leverage these two frameworks to introduce a new Lasso recovery algorithm on graphs. More precisely, we present a non-convex, non-smooth algorithm that outperforms the standard convex Lasso technique. We carry out numerical experiments on three benchmark graph datasets.

An Incremental Reseeding Strategy For Clustering, Xavier Bresson, Huiyi Hu, Thomas Laurent, Arthur Szlam, James Von Brecht

Thomas Laurent

In this work we propose a simple and easily parallelizable algorithm for multiway graph partitioning. The algorithm alternates between three basic components: diffusing seed vertices over the graph, thresholding the diffused seeds, and then randomly reseeding the thresholded clusters. We demonstrate experimentally that the proper combination of these ingredients leads to an algorithm that achieves state-of-the-art performance in terms of cluster purity on standard benchmarks datasets. Moreover, the algorithm runs an order of magnitude faster than the other algorithms that achieve comparable results in terms of accuracy. We also describe a coarsen, cluster and refine approach similar to GRACLUS and …

Characterization Of Radially Symmetric Finite Time Blowup In Multidimensional Aggregation Equations, Andrea L. Bertozzi, John B. Garnett, Thomas Laurent

Thomas Laurent

This paper studies the transport of a mass $\mu$ in $\mathbb{R}^d, d \geq 2,$ by a flow field $v= -\nabla K*\mu$. We focus on kernels $K=|x|^\alpha/ \alpha$ for $2-d\leq \alpha<2$ for which the smooth densities are known to develop singularities in finite time. For this range we prove the existence for all time of radially symmetric measure solutions that are monotone decreasing as a function of the radius, thus allowing for continuation of the solution past the blowup time. The monotone constraint on the data is consistent with the typical blowup profiles observed in recent numerical studies of these singularities. We prove monotonicity is preserved for all time, even after blowup, in contrast to the case $\alpha >2$ where radially symmetric solutions are known to lose monotonicity. In the case of the Newtonian potential ($\alpha=2-d$), under the assumption of radial symmetry the equation can be transformed into the inviscid Burgers equation on a half line. This enables us to prove preservation of monotonicity using the classical theory of conservation laws. In the case $2 -d < \alpha < 2$ and at the critical exponent p we exhibit initial data …

A Method Based On Total Variation For Network Modularity Optimization Using The Mbo Scheme, Huiyi Hu, Thomas Laurent, Mason A. Porter, Andrea L. Bertozzi

Thomas Laurent

The study of network structure is pervasive in sociology, biology, computer science, and many other disciplines. One of the most important areas of network science is the algorithmic detection of cohesive groups of nodes called “communities.” One popular approach to finding communities is to maximize a quality function known as modularity to achieve some sort of optimal clustering of nodes. In this paper, we interpret the modularity function from a novel perspective: we reformulate modularity optimization as a minimization problem of an energy functional that consists of a total variation term and an $\ell_2$ balance term. By employing numerical techniques …

An Adaptive Total Variation Algorithm For Computing The Balanced Cut Of A Graph, Xavier Bresson, Thomas Laurent, David Uminsky, James H. Von Brecht

Thomas Laurent

We propose an adaptive version of the total variation algorithm proposed in [3] for computing the balanced cut of a graph. The algorithm from [3] used a sequence of inner total variation minimizations to guarantee descent of the balanced cut energy as well as convergence of the algorithm. In practice the total variation minimization step is never solved exactly. Instead, an accuracy parameter is specified and the total variation minimization terminates once this level of accuracy is reached. The choice of this parameter can vastly impact both the computational time of the overall algorithm as well as the accuracy of …

Convergence Of A Steepest Descent Algorithm For Ratio Cut Clustering, Xavier Bresson, Thomas Laurent, David Uminsky, James H. Von Brecht

Thomas Laurent

Unsupervised clustering of scattered, noisy and high-dimensional data points is an important and difficult problem. Tight continuous relaxations of balanced cut problems have recently been shown to provide excellent clustering results. In this paper, we present an explicit-implicit gradient flow scheme for the relaxed ratio cut problem, and prove that the algorithm converges to a critical point of the energy. We also show the efficiency of the proposed algorithm on the two moons dataset.

The Regularity Of The Boundary Of A Multidimensional Aggregation Patch, Andrea L. Bertozzi, John B. Garnett, Thomas Laurent, Joan Verdera

Thomas Laurent

We consider solutions to the aggregation equation with Newtonian potential where the initial data are the characteristic function of a domain with boundary of class $C^{1+\gamma}$ ,$0<\gamma<1$. Such initial data are known to yield a solution that, going forward in time, retains a patch-like structure with a constant time-dependent density inside an evolving region, which collapses on itself in a finite time, and which, going backward in time, converges in an $L^1$ sense to a self-similar expanding ball solution. In this work, we prove $C^{1+\gamma}$ regularity of the domain's boundary on the time interval on which the solution …

Deep Learning Methods For Protein Torsion Angle Prediction, Haiou Li, Jie Hou, Badri Adhikari, Qiang Lyu, Jianlin Cheng

Badri Adhikari

No abstract provided.

Cyclipostins And Cyclophostin Analogs As Promising Compounds In The Fight Against Tuberculosis, Phuong Chi Nguyen, Vincent Delorme, Anaïs Bénarouche, Benjamin P. Martin, Rishi Paudel, Giri R. Gnawali, Abdeldjalil Madani, Rémy Puppo, Valérie Landry, Laurent Kremer, Priscille Brodin, Christopher D. Spilling, Jean-François Cavalier, Stéphane Canaan

Christopher Spilling

Intrinsic Linking And Knotting Of Graphs In Arbitrary 3–Manifolds, Erica Flapan, Hugh Howards, Don Lawrence, Blake Mellor

Blake Mellor

We prove that a graph is intrinsically linked in an arbitrary 3–manifold MM if and only if it is intrinsically linked in S3. Also, assuming the Poincaré Conjecture, we prove that a graph is intrinsically knotted in M if and only if it is intrinsically knotted in S3.

Tree Diagrams For String Links, Blake Mellor

Blake Mellor

In previous work, the author defined the intersection graph of a chord diagram associated with string links (as in the theory of finite type invariants). In this paper, we classify the trees which can be obtained as intersection graphs of string link diagrams.

The Forbidden Number Of A Knot, Alissa S. Crans, Blake Mellor, Sandy Ganzell

Blake Mellor

Every classical or virtual knot is equivalent to the unknot via a sequence of extended Reidemeister moves and the so-called forbidden moves. The minimum number of forbidden moves necessary to unknot a given knot is an invariant we call the forbid- den number. We relate the forbidden number to several known invariants, and calculate bounds for some classes of virtual knots.

On The Existence Of Finite Type Link Homotopy Invariants, Blake Mellor, Dylan Thurston

Blake Mellor

We show that for links with at most 5 components, the only finite type homotopy invariants are products of the linking numbers. In contrast, we show that for links with at least 9 components, there must exist finite type homotopy invariants which are not products of the linking numbers. This corrects previous errors of the first author.

Tree Diagrams For String Links Ii: Determining Chord Diagrams, Blake Mellor

Blake Mellor

In previous work, we defined the intersection graph of a chord diagram associated with a string link (as in the theory of finite type invariants). In this paper, we look at the case when this graph is a tree, and we show that in many cases these trees determine the chord diagram (modulo the usual 1-term and 4-term relations).

Spatial Graphs With Local Knots, Erica Flapan, Blake Mellor, Ramin Naimi

Blake Mellor

It is shown that for any locally knotted edge of a 3-connected graph in S3, there is a ball that contains all of the local knots of that edge and is unique up to an isotopy setwise fixing the graph. This result is applied to the study of topological symmetry groups of graphs embedded in S3.

The Intersection Graph Conjecture For Loop Diagrams, Blake Mellor

Blake Mellor

Vassiliev invariants can be studied by studying the spaces of chord diagrams associated with singular knots. To these chord diagrams are associated the intersection graphs of the chords. We extend results of Chmutov, Duzhin and Lando to show that these graphs determine the chord diagram if the graph has at most one loop. We also compute the size of the subalgebra generated by these "loop diagrams."

Symmetries Of Embedded Complete Bipartite Graphs, Erica Flapan, Nicole Lehle, Blake Mellor, Matt Pittluck, Xan Vongsathorn

Blake Mellor

We characterize which automorphisms of an arbitrary complete bipartite graph Kn,m can be induced by a homeomorphism of some embedding of the graph in S3.

A Few Weight Systems Arising From Intersection Graphs, Blake Mellor

Blake Mellor

No abstract provided.

Complete Graphs Whose Topological Symmetry Groups Are Polyhedral, Eric Flapan, Blake Mellor, Ramin Naimi

Blake Mellor

We determine for which m the complete graph Km has an embedding in S3 whose topological symmetry group is isomorphic to one of the polyhedral groups A4, A5 or S4.

A Geometric Interpretation Of Milnor's Triple Invariants, Blake Mellor, Paul Melvin

Blake Mellor

Milnor's triple linking numbers of a link in the 3-sphere are interpreted geometrically in terms of the pattern of intersections of the Seifert surfaces of the components of the link. This generalizes the well known formula as an algebraic count of triple points when the pairwise linking numbers vanish.

Finite Type Link Concordance Invariants, Blake Mellor

Blake Mellor

This paper is a generalization of the author's previous work on link homotopy to link concordance. We show that the only real-valued finite type link concordance invariants are the linking numbers of the components.

Finite Type Link Homotopy Invariants, Blake Mellor

Blake Mellor

Bar-Natan used Chinese characters to show that finite type invariants classify string links up to homotopy. In this paper, I construct the correct spaces of chord diagrams and Chinese characters for links up to homotopy. I use these spaces to show that the only rational finite type invariants of link homotopy are the pairwise linking numbers of the components.

Colorings, Determinants And Alexander Polynomials For Spatial Graphs, Terry Kong, Alec Lewald, Blake Mellor, Vadim Pigrish

Blake Mellor

A {\em balanced} spatial graph has an integer weight on each edge, so that the directed sum of the weights at each vertex is zero. We describe the Alexander module and polynomial for balanced spatial graphs (originally due to Kinoshita \cite{ki}), and examine their behavior under some common operations on the graph. We use the Alexander module to define the determinant and p-colorings of a balanced spatial graph, and provide examples. We show that the determinant of a spatial graph determines for which p the graph is p-colorable, and that a p-coloring of a graph corresponds to a representation of …

Counting Links In Complete Graphs, Thomas Fleming, Blake Mellor

Blake Mellor

We find the minimal number of non-trivial links in an embedding of any complete kk-partite graph on 7 vertices (including K₇, which has at least 21 non-trivial links). We give either exact values or upper and lower bounds for the minimal number of non-trivial links for all complete kk-partite graphs on 8 vertices. We also look at larger complete bipartite graphs, and state a conjecture relating minimal linking embeddings with minimal book embeddings.

Chord Diagrams And Gauss Codes For Graphs, Thomas Fleming, Blake Mellor

Blake Mellor

Chord diagrams on circles and their intersection graphs (also known as circle graphs) have been intensively studied, and have many applications to the study of knots and knot invariants, among others. However, chord diagrams on more general graphs have not been studied, and are potentially equally valuable in the study of spatial graphs. We will define chord diagrams for planar embeddings of planar graphs and their intersection graphs, and prove some basic results. Then, as an application, we will introduce Gauss codes for immersions of graphs in the plane and give algorithms to determine whether a particular crossing sequence is …

Drawing A Triangle On The Thurston Model Of Hyperbolic Space, Curtis D. Bennett, Blake Mellor, Patrick D. Shanahan

Blake Mellor

In looking at a common physical model of the hyperbolic plane, the authors encountered surprising difficulties in drawing a large triangle. Understanding these difficulties leads to an intriguing exploration of the geometry of the Thurston model of the hyperbolic plane. In this exploration we encounter topics ranging from combinatorics and Pick’s Theorem to differential geometry and the Gauss-Bonnet Theorem.

Alexander And Writhe Polynomials For Virtual Knots, Blake Mellor

Blake Mellor

We give a new interpretation of the Alexander polynomial Δ0 for virtual knots due to Sawollek and Silver and Williams, and use it to show that, for any virtual knot, Δ0 determines the writhe polynomial of Cheng and Gao (equivalently, Kauffman's affine index polynomial). We also use it to define a second-order writhe polynomial, and give some applications.

Finite Type Link Homotopy Invariants Ii: Milnor's Invariants, Blake Mellor

Blake Mellor

We define a notion of finite type invariants for links with a fixed linking matrix. We show that Milnor's triple link homotopy invariant is a finite type invariant, of type 1, in this sense. We also generalize the approach to Milnor's higher order homotopy invariants and show that they are also, in a sense, of finite type. Finally, we compare our approach to another approach for defining finite type invariants within linking classes.