Web: http://www.ima.umn.edu | Email: ima-staff@ima.umn.edu | Telephone: (612) 624-6066 | Fax: (612) 626-7370
Additional newsletters available at http://www.ima.umn.edu/newsletters

IMA Newsletter #402

April 2010

2009-2010 Program

See http://www.ima.umn.edu/2009-2010/ for a full description of the 2009-2010 program on Complex Fluids and Complex Flows.

News and Notes

IMA Events

IMA Workshop

Physical Knotting and Linking and its Applications

April 9, 2010

Organizers: Dorothy E. Buck (Imperial College London), Isabel K. Darcy (University of Iowa), Kenneth C. Millett (University of California), Eric J. Rawdon (University of St. Thomas)

IMA Tutorial

Transport and Mixing in Complex and Turbulent Flows

April 11, 2010

Organizers: Charles R. Doering (University of Michigan), Peter R. Kramer (Rensselaer Polytechnic Institute), William Roy Young (Scripps Research Institute)

IMA Annual Program Year Workshop

Transport and Mixing in Complex and Turbulent Flows

April 12-16, 2010

Organizers: Peter Constantin (University of Chicago), Charles R. Doering (University of Michigan), Peter R. Kramer (Rensselaer Polytechnic Institute), David H. Sharp (Los Alamos National Laboratory), William Roy Young (Scripps Research Institute)

Public Lecture

Nancy Reid: Can Chocolate Save Your Life?

April 22, 2010

Speakers: Nancy Reid (University of Toronto)
Schedule

Thursday, April 1

10:45am-11:15amCoffee breakLind Hall 400
3:30pm-4:30pmTag Team Tutorials: Transport & Mixing in Incompressible Fluid FlowsCharles R. Doering (University of Michigan)
Jean-Luc Thiffeault (University of Wisconsin)
Lind Hall 305
5:00pm-6:30pmMathematics awareness month lecture - Mathematics that swings: the math behind golfDouglas N. Arnold (University of Minnesota)Campus Center, Macalester College

Friday, April 2

10:45am-11:15amCoffee breakLind Hall 400

Monday, April 5

10:45am-11:15amCoffee breakLind Hall 400

Tuesday, April 6

10:45am-11:15amCoffee breakLind Hall 400
11:15am-12:15pmPostdoc seminar: Coupled boundary layers for the primitive equations of atmosphereDongjuan Niu (Capital Normal University)Lind Hall 305 PS

Wednesday, April 7

10:45am-11:15amCoffee breakLind Hall 400

Thursday, April 8

10:45am-11:15amCoffee breakLind Hall 400
3:30pm-4:30pmTag Team Tutorials: Transport & Mixing in Incompressible Fluid FlowsCharles R. Doering (University of Michigan)
Jean-Luc Thiffeault (University of Wisconsin)
Lind Hall 305

Friday, April 9

10:45am-11:15amCoffee breakLind Hall 400
1:15pm-1:45pmRegistration and coffeeLind Hall 400 SW4.9.10
1:25pm-2:25pm Applying numerical grid generation to the visualization of complex function dataBonita V. Saunders (National Institute of Standards and Technology)Vincent Hall 570 IPS
1:45pm-2:00pmWelcome and introductionFadil Santosa (University of Minnesota)Lind Hall 305 SW4.9.10
2:00pm-3:00pmUsing topology to understand chromosome organization across organismsF. Javier Arsuaga (San Francisco State University)Lind Hall 305 SW4.9.10
3:00pm-4:00pmBiological applications that utilize DNA Topology
Lynn Zechiedrich (Baylor College of Medicine)Lind Hall 305 SW4.9.10
4:00pm-4:10pmGroup Photo SW4.9.10
4:10pm-6:10pmPoster Session and Refreshments
Poster submissions welcome from all participants
Instructions
Lind Hall 400 SW4.9.10
Solving a system of four tangle equationsLauren M. Beaumont (University of Iowa)
Dianne Smith (University of Iowa)
Engineering multiple site-specific modifications in supercoiled DNAsAnusha Bharadwaj (University of Texas at Dallas)
Three dimensional reconstructions of knotted particlesJohn Collins (San Francisco State University)
Table of rational links and their invariantsIsabel K. Darcy (University of Iowa)
Guanyu Wang (University of Iowa)
Modeling local knots in proteins caused by random crossing changesJohn R. Froehlig (University of Iowa)
Direct entropy calculations for discrete wormlike chainsStefan M. Giovan (University of Texas at Dallas)
Braid indices in a class of closed braidsKenneth E. Hinson (University of North Carolina - Charlotte)
High school level introduction to knotsJeffrey Hunt (University of Iowa)
Symmetries of knots and links Matt Mastin (University of Georgia)
Energetics of DNA tangling in complex nucleoprotein assembliesMary Therese Padberg (University of Iowa)
Gregory Witt (University of Iowa)
Invariance of the sign of the average space writhe of free and confined knotted polygonsJuliet Portillo (San Francisco State University)
Minimal step number of cubic lattice knots in thin slabsRobert Glenn Scharein (San Francisco State University)
Symmetry-breaking in cumulative measures of shapes of polymer modelsVy T. Tran (University of St. Thomas)
Polygonal cable links Rolland Trapp (California State University)
Tangle tabulationDanielle Washburn (University of Iowa)
The local and global shape of regular embedded polygons: Theoretical and experimentalLaura K. Zirbel (University of California, Santa Barbara)

Sunday, April 11

9:30am-10:15amRegistration and coffeeLind Hall 400 T4.11.10
10:15am-11:45amIntroduction to dynamics and tracer dispersion in geophysical fluids

(1) Geophysical fluids: phenomenology and dynamics of rotating, stratified flow.

(2) Potential vorticity: Dynamic significance and kinematic interpretation.

(3) Importance of mixing for maintenance of ocean pycnocline and global overturning circulation: ocean energetics, a little on thermocline theories and/or Sandstrom's theory.

(4) Tracer Observations in the ocean (and atmosphere): Turbulent mixing from dye releases, maybe something on surface stirring/mixing diagnostics from floats.

(5) Tracer cascades: theory and observational support.

Thomas W. N. Haine (Johns Hopkins University)Lind Hall 305 T4.11.10
11:45am-1:15pmLunch T4.11.10
1:15pm-2:45pmMathematical models and methods for characterizing turbulent diffusionPeter R. Kramer (Rensselaer Polytechnic Institute)Lind Hall 305 T4.11.10
2:45pm-3:00pmBreakLind Hall 400 T4.11.10
3:00pm-4:30pmShear dispersion

(1) interaction of molecular diffusion with simple unidirectional shear flows (bounded and unbounded domains).

(2) Limitation of the effective diffusion approximation to long times and small domains, and low moments of the tracer distribution.

(3) Perhaps a geophysical example: shear diffusion in the internal gravity wave field. The vertical tracer cascade of Haynes & Anglade.

(4) Some examples of "pre-asymptotic" anomalous shear diffusion e.g.,〈x2〉 ∼ t ln t due to the no-slip condition, and 〈x2〉 ∼ t 3/2 in the example of de Marsily & Matheron.

William Roy Young (Scripps Research Institute)Lind Hall 305 T4.11.10

Monday, April 12

8:00am-8:45amRegistration and coffeeEE/CS 3-176 W4.12-16.10
8:45am-9:00amWelcome to the IMAFadil Santosa (University of Minnesota)EE/CS 3-180 W4.12-16.10
9:00am-9:45amTransit-time distributions: A tool to diagnose rates and pathways of tracer transport in advective/diffusive flowThomas W. N. Haine (Johns Hopkins University)EE/CS 3-180 W4.12-16.10
9:45am-10:00amDiscussionEE/CS 3-180 W4.12-16.10
10:00am-10:45amThe inverse problem of inferring transit-time distributions from tracer observations in the oceanFrançois W. Primeau (University of California, Irvine)EE/CS 3-180 W4.12-16.10
10:45am-11:00amDiscussionEE/CS 3-180 W4.12-16.10
11:00am-11:45amExit time problem in an incompressible flowAlexei Novikov (Pennsylvania State University)EE/CS 3-180 W4.12-16.10
11:45am-12:00pmDiscussionEE/CS 3-180 W4.12-16.10
12:00pm-2:00pmLunch W4.12-16.10
2:00pm-2:45pmDynamics of shallow water layer models: Stability, wave breaking and mixingPaul Milewski (University of Wisconsin)EE/CS 3-180 W4.12-16.10
2:45pm-3:00pmDiscussionEE/CS 3-180 W4.12-16.10
3:00pm-3:45pmModelling streaming by surface acoustic wavesJacques Vanneste (University of Edinburgh)EE/CS 3-180 W4.12-16.10
3:45pm-4:00pmDiscussionEE/CS 3-180 W4.12-16.10
4:00pm-4:15pmGroup Photo W4.12-16.10
4:15pm-6:00pmReception and Poster Session
Poster submissions welcome from all participants
Instructions
Lind Hall 400 W4.12-16.10
A numerical study of the effect of diffusion on a fast chemical reaction in a two-dimensional turbulent flowFarid Ait Chaalal (McGill University)
The spectrally-hyperviscous Navier-Stokes equations Joel D. Avrin (University of North Carolina - Charlotte)
Rayleigh-Taylor turbulence: a simple model for heat transfer in thermal convection Guido Boffetta (Università di Torino)
A fast explicit operator splitting method for passive scalar advectionAlina Chertock (North Carolina State University)
Charles R. Doering (University of Michigan)
Alexander Kurganov (Tulane University)
Low-dimensional models from upper bound and energy stability theoryGregory P. Chini (University of New Hampshire)
Chaotic granular mixing in quasi-two-dimensional tumblers: streamline jumping, piecewise isometries and strange eigenmodesIvan Christov (Northwestern University)
Alternate powers in Serrin's swirling vortex solutions 2Doug Dokken (University of St. Thomas)
A class of Hamilton-Jacobi PDE in space of measures and its associated compressible Euler equationsJin Feng (University of Kansas)
Active and hibernating turbulence in channel flow of Newtonian and viscoelastic fluidsMichael D. Graham (University of Wisconsin)
Bridging the Boussinesq and primitive equations through spatio-temporal filteringTraian Iliescu (Virginia Polytechnic Institute and State University)
Homogenization and mixing measures for a replenishing passive scalar field Shane Keating (New York University)
Miscible and immiscible Rayleigh-Taylor turbulenceAndrea Mazzino (Università di Genova)
On the effect of initial velocity field and phase shifting of an initial binary perturbation for Rayleigh-Taylor instabilityBertrand Rollin (Los Alamos National Laboratory)
Numerical studies in shallow moist convectionJoerg Schumacher (Ilmenau University of Technology)
Lagrangian dynamics in stochastic inertia-gravity wavesWenbo Tang (Arizona State University)
Super-convergence for the 3D Navier-StokesGiordano Tierra Chica (University of Sevilla)
Fast chemical reactions in chaotic flows: Reaction rate and mixdown timeYue-Kin Tsang (University of California, San Diego)
Estimating generalised Lyapunov exponents for random flowsJacques Vanneste (University of Edinburgh)
Shear cell rupture of nematic droplets in viscous fluids Xiaofeng Yang (University of South Carolina)

Tuesday, April 13

8:30am-9:00amCoffeeEE/CS 3-176 W4.12-16.10
9:00am-9:45amProgress & problems in the analysis of turbulent transport & mixing Charles R. Doering (University of Michigan)EE/CS 3-180 W4.12-16.10
9:45am-10:00amDiscussionEE/CS 3-180 W4.12-16.10
10:00am-10:45amSetting limits on turbulence - balancing rigor with practicality Rich R. Kerswell (University of Bristol)EE/CS 3-180 W4.12-16.10
10:45am-11:00amDiscussionEE/CS 3-180 W4.12-16.10
11:00am-11:45amApproximating the rate of heat transportXiaoming Wang (Florida State University)EE/CS 3-180 W4.12-16.10
11:45am-12:00pmDiscussionEE/CS 3-180 W4.12-16.10
12:00pm-2:30pmLunch W4.12-16.10
2:30pm-3:15pmRadial basis functions for geofluid modeling Natasha Flyer (National Center for Atmospheric Research)EE/CS 3-180 W4.12-16.10
3:15pm-3:30pmDiscussionEE/CS 3-180 W4.12-16.10
3:30pm-4:15pmA Stokesian viscoelastic flow: Transition to mixing and oscillationsBecca Thomases (University of California, Davis)EE/CS 3-180 W4.12-16.10
4:15pm-4:30pmDiscussionEE/CS 3-180 W4.12-16.10

Wednesday, April 14

8:30am-9:00amCoffeeEE/CS 3-176 W4.12-16.10
9:00am-9:45amMixing by eddies in the atmosphere and ocean Emily F. Shuckburgh (British Antarctic Survey)EE/CS 3-180 W4.12-16.10
9:45am-10:00amDiscussionEE/CS 3-180 W4.12-16.10
10:00am-10:45amThe three-dimensional structure of turbulent geostrophic stirring K. Shafer Smith (New York University)EE/CS 3-180 W4.12-16.10
10:45am-11:00amDiscussionEE/CS 3-180 W4.12-16.10
11:00am-11:45amDo fish stir the ocean?Jean-Luc Thiffeault (University of Wisconsin)EE/CS 3-180 W4.12-16.10
11:45am-12:00pmDiscussionEE/CS 3-180 W4.12-16.10

Thursday, April 15

8:30am-9:30amCoffeeEE/CS 3-176 W4.12-16.10
9:30am-10:15amSpatial structures of chaotically advected reactive tracers: The role of a delay time Alexandra Tzella (École Normale Supérieure)EE/CS 3-180 W4.12-16.10
10:15am-10:30amDiscussionEE/CS 3-180 W4.12-16.10
10:30am-11:15amBroadcast spawning: A new class of reaction-mixing problemsJeffrey B. Weiss (University of Colorado)EE/CS 3-180 W4.12-16.10
11:15am-11:30amDiscussionEE/CS 3-180 W4.12-16.10
11:30am-12:15pmMixing and enhanced relaxation in fluid flows Alexander Kiselev (University of Wisconsin)EE/CS 3-180 W4.12-16.10
12:15pm-12:30pmDiscussionEE/CS 3-180 W4.12-16.10
12:30pm-3:00pmLunch W4.12-16.10
3:00pm-3:45pmPassive scalar advection in parallel shear flows: WKBJ mode sorti on intermediate times and the evolution of skewnessRichard M. McLaughlin (University of North Carolina)EE/CS 3-180 W4.12-16.10
3:45pm-4:00pmDiscussionEE/CS 3-180 W4.12-16.10
4:00pm-4:45pmMixing: Visualization, norms, and controlIgor Mezic (University of California, Santa Barbara)EE/CS 3-180 W4.12-16.10
4:45pm-5:00pmDiscussionEE/CS 3-180 W4.12-16.10
6:00pm-8:00pmWorkshop dinner at Pagoda RestaurantPagoda Restaurant
1417 4th St. SE
Minneapolis, MN
612-378-4710
W4.12-16.10

Friday, April 16

8:30am-9:00amCoffeeEE/CS 3-176 W4.12-16.10
9:00am-9:45amBounds on mixing in stratified shear flowsColm P. Caulfield (University of Cambridge)EE/CS 3-180 W4.12-16.10
9:45am-10:00amDiscussionEE/CS 3-180 W4.12-16.10
10:00am-10:45amJet dynamics in stratified mediaJuan Mario Restrepo (University of Arizona)EE/CS 3-180 W4.12-16.10
10:45am-11:00amDiscussionEE/CS 3-180 W4.12-16.10
11:00am-11:45amIs turbulence stable?William Roy Young (Scripps Research Institute)EE/CS 3-180 W4.12-16.10
11:45am-12:00pmDiscussionEE/CS 3-180 W4.12-16.10
12:00pm-12:15pmClosing remarksEE/CS 3-180 W4.12-16.10

Monday, April 19

10:45am-11:15amCoffee breakLind Hall 400

Tuesday, April 20

11:15am-12:15pmSpecial seminar: Wave and curvesGregory R. Baker (Ohio State University)Lind Hall 305 PS

Wednesday, April 21

10:45am-11:15amCoffee breakLind Hall 400

Thursday, April 22

10:45am-11:15amCoffee breakLind Hall 400
7:00pm-8:00pmCan chocolate save your life? Nancy Reid (University of Toronto)Willey Hall 175 PUB4.22.10

Friday, April 23

10:45am-11:15amCoffee breakLind Hall 400

Monday, April 26

10:45am-11:15amCoffee breakLind Hall 400

Tuesday, April 27

10:45am-11:15amCoffee breakLind Hall 400
11:15am-12:15pmTBAChiun-Chang Lee (National Taiwan University)Lind Hall 305 PS

Wednesday, April 28

10:45am-11:15amCoffee breakLind Hall 400

Thursday, April 29

10:45am-11:15amCoffee breakLind Hall 400

Friday, April 30

10:45am-11:15amCoffee breakLind Hall 400
1:25pm-2:25pmOptimal chemical spectroscopyAnthony José Kearsley (National Institute of Standards and Technology)Vincent Hall 570 IPS
Abstracts
Farid Ait Chaalal (McGill University) A numerical study of the effect of diffusion on a fast chemical reaction in a two-dimensional turbulent flow
Abstract: Stratospheric Climate-Chemistry Models neglect the effects of sub-grid flow structures on chemistry. Several previous studies have pointed out that such unresolved small scales could significantly affect the chemistry . However this problem has not been thoroughly studied from a theoretical point of view. To fulfill this gap, we investigate the interactions between advection, diffusion and chemistry for a simple bimolecular reaction between two initially unmixed reactants, within the framework of two-dimensional isotropic and homogeneous turbulence. This is a highly simplified representation of quasi-isentropic mixing in the stratosphere. Our goal here is to describe and understand how the production rate is affected by the size of the smallest scales of the tracer field, as determined by the tracer diffusion. We focus on the case of an infinitely fast chemical reaction. Our results show a strong dependence of the total production on the diffusion coefficient. This production scales like the diffusion to the power of p(t), where p(t) is a positive decreasing function of time. This dependence is particularly important during an initial transient regime and is affected by the separation between the reactants at the initial time. This first regime is characterized by an exponential lengthening of the boundary between the reactants. The evolution of the tracer gradients along this interface explains the dependence of the chemistry on the diffusion. For larger times, our simulations suggest the appearance of an asymptotic strange eigenmode that controls the decay of the reactants.
Douglas N. Arnold (University of Minnesota) Mathematics awareness month lecture - Mathematics that swings: the math behind golf
Abstract: Mathematics is everywhere and the golf course is no exception. Many aspects of the game of golf can be illuminated or improved through mathematical modeling and analysis. We will discuss a few examples, employing mathematics ranging from simple high school algebra to computational techniques at the frontiers of contemporary research.
F. Javier Arsuaga (San Francisco State University) Using topology to understand chromosome organization across organisms
Abstract: Topological methods have been mostly used to study the action of enzymes and properties of naked DNA molecules. However topology can also be used to study chromosome organization. In this talk I will present three problems in which topology can be used to study complex organization of DNA. First I will present the problem of DNA knotting in bacteriophages. Understanding the organization of the genome in bacteriophages is important because bacteriophages are good models for DNA organization in some animal viruses (such as herpex viruses) and in DNA lipo-complexes used in gene therapy. Our approach is based on work by Liu, Calendar, Wang and colleagues that showed that DNA extracted from bacteriophage P4 is knotted. We have investigated these knots and shown that they are informative of the organization of the genome inside the capsid. I will present models that have been derived from these knots as well as the mathematical problems that this biological problem has generated. Second I will discuss the problem of chromosome intermingling of chromosome territories in the eukaryotic cell. During the G0/G1 phase of the cell cycle the eukaryotic genome is organized into chromosome territories. The positions of these territories as well as the structure along their surface are believed to play a major role in the formation of recurrent aberrations found in some genetic diseases and in some cancers. I will present some of our results on the topological implications of the Interchromosomal Network Model proposed by Branco and Pombo. In particular I will introduce our estimation of the linking probability of two neighboring chromosome territories assuming that chromatin fibers follow random trajectories. Third I will present some new and unpublished results on the linking of mitochondrial DNA in trypanosomes. Trypanosomatid parasites, trypanosoma and lishmania, are the cause of disease and death in many third world countries. One of the most unusual features of these organisms is the 3 dimensional organization of their mitochondrial DNA into maxi and minicircles. Minicircles are confined into a small volume and are interlocked forming a huge network. Some initial models for the organization of this network were proposed by Cozzarelli and Englund. Here we discuss some of the possible pathways for the formation and maintenance of this network as well as the mathematical results that derive from this problem. This work is in collaboration with: Y. Diao (UNC Charlotte), R. Scharein (SFSU), R. Kaplan (SFSU) and M. Vazquez (SFSU).
Joel D. Avrin (University of North Carolina - Charlotte) The spectrally-hyperviscous Navier-Stokes equations
Abstract: We regularize the 3-D Navier-Stokes equations with hyperviscosity of degree alpha, but applied only to the high wavenumbers past a cutoff m; such a technique is also designed to approximate the subgrid-scale modeling effects of spectral eddy viscosity. Attractor estimates stay within the Landau-Lifschitz degrees-of-freedom estimates even for very large m. An inertial manifold exists for m large enough whenever alpha is at or above 3/2. Galerkin-convergence and inviscid-limit results are optimized for the high wavenumbers; the latter case is defined to mean that nu goes to zero while the spectral hyperviscous term stays fixed. Computational studies over many runs produce parameter choices that facilitate close-to-parallel agreement (over a good-sized portion of the inertial range) with the Kolmogorov energy-spectrum power law for high (up to 107) Reynolds numbers.
Gregory R. Baker (Ohio State University) Special seminar: Wave and curves
Abstract: Water waves are perhaps the most notable feature of the planet, and they have occupied the attention of scientists since the birth of civilization. Yet they remain incompletely understood. Despite recent theoretical advances, the generic mathematical behavior of water waves eludes description. I will present a different view of water waves traveling in two-dimensions, a view based on the relationship between the curvature and the arclength. The curvature has simple poles in the complex arclength plane that travel about while retaining their form. They can approach closely to the real axis during wave breaking and are associated with the tip of the plunging breaker. A different view of wave breaking is the presence of a square-root singularity in the surface height as a function of the horizontal coordinate that reaches the real axis in finite time when the slope becomes vertical. Even in the absence of wave breaking, these singularities are present and strongly affect the wave spectra.
Lauren M. Beaumont (University of Iowa), Dianne Smith (University of Iowa) Solving a system of four tangle equations
Abstract: A tangle consists of strings properly embedded within a 3-dimensional ball. Solutions of tangle equations have proven quite useful when applied to recombinases. Recombinases are enzymes that cut DNA strands and interchange the ends, changing the topology of the DNA. The recombinase action will be mathematically modeled by replacing the zero tangle with the tangle t/w, resulting in a new DNA product. If we model experiments involving two topologically different substrates and/or two topologically different products, we have a corresponding system of four tangle equations. Given a1, a2, b1, b2, z1, z2, v1, and v2, we are solving the following system of four tangle equations for t/w: N(j1/p1 + 0/1) = N(a1/b1)
N(j1/p1 + t/w) = N(z1/v1)
N(j2/p2 + 0/1) = N(a2/b2)
N(j2/p2 + t/w) = N(z2/v2).
Anusha Bharadwaj (University of Texas at Dallas) Engineering multiple site-specific modifications in supercoiled DNAs
Abstract: Joint work with Matthew R. Kesinger*, Massa J. Shoura*, Alexandre Vetcher*, and Stephen D. Levene*†. Biological processes such as DNA recombination, replication, and gene expression involve specific interactions between one or more DNA-binding proteins and multiple protein-binding sites along a single DNA molecule. Such interactions lead to the formation of a topologically closed DNA loop between protein-recognition sites, whose energetics depends on the structure and the flexibility of the intervening DNA, the degree of supercoiling, and the binding of additional proteins such as HU and Fis in bacterial systems or histones and HMG proteins in the case of eukaryotic cells. We present here a novel technique for incorporating multiple modifications such as covalently attached fluorescent probes to multiple defined sites within covalently closed DNA molecules. Applications of this technology include the use of two- and three-color FRET to investigate effects of DNA supercoiling on lac-repressor DNA interactions both in vitro and in vivo. Departments of Molecular and Cell Biology* and Physics
University of Texas at Dallas
Richardson, TX 75080
Guido Boffetta (Università di Torino) Rayleigh-Taylor turbulence: a simple model for heat transfer in thermal convection
Abstract: I will discuss turbulent mixing within the framework of Rayleigh-Taylor geometry. Large scale properties of mixing are described by a simple non-linear diffusion model, derived within the general framework of Prandtl mixing theory, which fits very well the evolution of turbulent profiles obtained from numerical simulations. The effect of polymer additives is then discussed and on the basis of numerical simulations of complete viscoelastic models we obtain clear evidence that the heat transport is enhanced up to 50% with respect to the Newtonian case. This phenomenon is accompanied by a speed up of the mixing layer growth.
Colm P. Caulfield (University of Cambridge) Bounds on mixing in stratified shear flows
Abstract: Keywords: Turbulent mixing, Rigorous bounds, stratified shear flows Abstract: Parameterizing the mixing of a stratified fluid subject to shear is a fundamental challenge for models of environmental and industrial flows. In particular, it is of great value to parameterize the efficiency of turbulent mixing, in the sense of the proportion of the kinetic energy converted into potential energy (through irreversible mixing of fluid of different density) compared to the total amount converted to both potential energy and internal energy (through viscous dissipation). Various competing models have been presented to relate the mixing efficiency to bulk properties of the flow, especially through different Richardson numbers, which quantify the relative importance of buoyancy and shear within the flow. One promising approach is to construct rigorous bounds on the long-time average of the buoyancy flux (i.e. the mixing rate) within simple model stratified shear flows, imposing physically reasonable constraints on the model flow fields. In this talk, we apply this technique to stably stratified Couette flow. By identifying the stratification which leads to maximal buoyancy flux, we make a prediction of what bulk stratification (as a function of the shear) is optimal for turbulent mixing. A previous attempt to do this failed due to an unexpected degeneracy in the variational problem. Here, we overcome this issue by parameterizing the variational problem implicitly with the overall mixing efficiency which is then optimized across to return a rigorous upper bound on the buoyancy flux. We discuss the implications of our results for various classical stratified shear turbulence models. Joint work with W. Tang (Arizona State University) & R. R. Kerswell (University of Bristol).
Alina Chertock (North Carolina State University), Charles R. Doering (University of Michigan), Alexander Kurganov (Tulane University) A fast explicit operator splitting method for passive scalar advection
Abstract: Joint work with Alina Chertock, Charles R. Doering and Eugene Kashdan. The dispersal and mixing of scalar quantities such as concentrations or thermal energy are often modeled by advection-diffusion equations. Such problems arise in a wide variety of engineering, ecological and geophysical applications. In these situations a quantity such as chemical or pollutant concentration or temperature variation diffuses while being transported by the governing flow. In the passive scalar case, this flow prescribed and unaffected by the scalar. Both steady laminar and complex (chaotic, turbulent or random) time-dependent flows are of interest and such systems naturally lead to questions about the effectiveness of the stirring to disperse and mix the scalar. The development of reliable numerical methods for advection-diffusion equations is crucial for understanding their properties, both physical and mathematical. In this work, we extend a fast explicit operator splitting method, recently proposed in [A. Chertock, A. Kurganov, and G. Petrova, International Journal for Numerical Methods in Fluids, 59 (2009), pp. 309-332] for solving deterministic convection-diffusion equations, to the problems with random velocity fields and singular source terms. A superb performance of the method is demonstrated on several two-dimensional examples.
Gregory P. Chini (University of New Hampshire) Low-dimensional models from upper bound and energy stability theory
Abstract: Joint work with N. Dianati, Z. Zhang, and C. Doering.

A novel model reduction strategy for forced-dissipative infinite-dimensional nonlinear dynamical systems is described. Unlike popular but empirical methods (e.g. based on the Proper Orthogonal Decomposition), this new approach does not require extensive data sets from experiments or full PDE simulations. Instead, truly predictive reduced-order models are constructed via Galerkin projection of the governing PDEs onto a-priori basis functions. This basis set is obtained by solving a constrained eigenvalue problem drawn from energy stability and upper bound theory. Within the context of porous medium convection, we show that these eigenfunctions contain information about boundary layers and other complex dynamic features and, thus, are well suited for the low-order description of highly nonlinear phenomena. Crucially, our analysis reveals a gap in the eigenvalue spectrum that persists even for strongly supercritical forcing conditions, thereby enabling the identification of a rational truncation scheme. We demonstrate the efficacy of our approach via comparisons with Fourier--Galerkin approximations of various orders.

Ivan Christov (Northwestern University) Chaotic granular mixing in quasi-two-dimensional tumblers: streamline jumping, piecewise isometries and strange eigenmodes
Abstract: The singular limit of a vanishing flowing layer in a tumbled granular flow is studied numerically, analytically and experimentally. We formulate the no-shear-layer dynamical system as a piecewise isometry (PWI) and show that the mechanism of streamline jumping leads to mixing. In the special case of a half-full tumbler, chaotic behavior is shown to disappear completely in the singular limit. Experimental results are compared to the zeroth-order PWI model and a realistic continuum model, showing that the no-shear-layer dynamics form the skeleton of the granular flow. Even though, in this limit, stretching in the sense of shear strain is replaced by spreading due to streamline jumping, finite-time Lyapunov exponents and Lagrangian coherent structures still reveal the manifold structure of the flow. Finally, using the simplified mapping method, the asymptotic mixing pattern in a tumbled granular flows is decomposed into eigenmodes, showing a significant number of strange eigenmodes. These appear align with the unstable manifolds, which have been shown to outline the shape of segregation patterns in bidisperse granular mixing experiments in polygonal tumblers. Co-authors: Julio M. Ottino (Northwestern Univeristy, http://mixing.chem-eng.northwestern.edu) and Richard M. Lueptow (Northwestern University, http://www.mech.northwestern.edu/lueptow).
John Collins (San Francisco State University) Three dimensional reconstructions of knotted particles
Abstract: Single Particle Reconstructions are commonly used to elucidate structures of different types of small particles with precision approaching that of x-ray techniques. Such reconstructions assume a homogeneity of data or a small heterogeneous collection of homogeneous subgroups. Current models of the packing of small capsid viruses like bacteriophages suggest a spooling type model with some non-uniformity resulting in small numbers of knots. Three dimensional reconstructions have been used to justify different models but such assumptions assuming uniform homogeneity of a data set disregard slight variations which could be present. We present a look at the single particle reconstruction process as a whole as well as a view of two different sets of data. We ask, "what happens if single particle reconstruction is used to reconstruct a single model from a data set in which each particle is very similar but no two are exactly alike?".
Isabel K. Darcy (University of Iowa), Guanyu Wang (University of Iowa) Table of rational links and their invariants
Abstract: Joint work with Thomas LeHew and Joe Eichholz. We are creating a webpage which will allow users to create tables of links, knots and their invariants. Our plan is to provide a platform which visualizes the information about knots and links on a table that will satisfy most types of users. The best platform for this plan is a webpage. The webpage would be separated into three main components. First, the actually html document that the user will see and interact with to properly generate the specific table of knots/links that the user would like to see. Second, a series of scripts will be set up to take in the input provided to retrieve the information needed from the database and then output the information in an easy to read table for the user to see. Last, a component specific to our project is the desire to allow others to contribute to the table. Livingston and Cha’s KnotInfo is an outstanding webpage for creating knot tables. However, for our project, we put more emphasis on links and their relevant invariants. It will also handle composition of links, orientation, and mirror images.
Charles R. Doering (University of Michigan) Progress & problems in the analysis of turbulent transport & mixing
Abstract: Keywords: turbulent energy dissipation, eddy viscosity, effective diffusion, mixing efficiency Abstract: Several open problems in the analysis of turbulent transport and mixing are described. Among these are the determination of physically relevant bounds on turbulent dissipation and eddy viscosities in a variety of flow configurations, the accurate estimation of active and passive scalar transport and effective diffusivites, and questions of effective mixing.
Doug Dokken (University of St. Thomas) Alternate powers in Serrin's swirling vortex solutions 2
Abstract: Joint work with Kurt Scholz, and Misha Shvartsman, University of St. Thomas, St. Paul, MN, USA. Motivated by results of Cai(2005, Monthly Weather Review), we consider alternate power dependencies in Serrin's Swirling Vortex model. We also give a heuristic argument to justify Cai's power law for tornados.
Jin Feng (University of Kansas) A class of Hamilton-Jacobi PDE in space of measures and its associated compressible Euler equations
Abstract: We introduce a class of action integrals defined over probability measure-valued path space. We show that minimal action exists and satisfies a compressible Euler equation in weak sense. Moreover, we prove that both Cauchy and resolvent formulations of the associated Hamilton-Jacobi equation, in the space of probability measures, are well posed. There are two key arguments which involves relaxation and regularization in formulation of the problem. They are probabilistically motivated. This is a joint work with Truyen Nguyen.
Natasha Flyer (National Center for Atmospheric Research) Radial basis functions for geofluid modeling
Abstract: Keywords: meshless methods, radial basis functions, geosciences Abstract: The most classical approach for solving PDEs numerically is finite difference methods (FD). Although they are easy to implement, their accuracy is often low. In contrast, pseudospectral methods (PS) can give spectral (exponential) convergence but suffer from severe geometric restrictions. On the other hand, radial basis function methods (RBF), based on expansions of translates of a single radially symmetric function, combine algorithmic simplicity with spectral accuracy while generalizing to arbitrary node layouts. Since they do not depend on any grid, RBFs allow for high geometric flexibility, permitting local node refinements in critical areas. Only within the last few years have they been applied to non-trivial PDE systems, illustrating their potential in the geosciences. Here, the application of the RBF method to three common geoscience benchmark cases of increasing complexity will be discussed. The first case is idealized cyclogenesis, which models the wrap-up of vortices as they traverse the sphere. For this case, we will also show the simplicity of implementing local node refinement. The second case is unsteady nonlinear flows described by the shallow water equations. The third case is thermal convection in a 3-D spherical shell, a situation of interest in modeling the earth's mantle. Current research topics focus on developing fast and efficient RBF algorithms that are parallelizable.
John R. Froehlig (University of Iowa) Modeling local knots in proteins caused by random crossing changes
Abstract: Proteins are linear chains of amino acids. Proteins are composed of secondary structure units called alpha helices and beta sheets, which are energetically stable, and random coils, which are not. Many diseases are caused by protein folding disorders. Local knots in proteins are much rarer than is expected for a long polymer. As of 2006, only thirty-nine proteins out of 9,553 proteins with determined structures contain local knots. Eighteen of those thirty-nine proteins contain shallow knots which can deform to the unknot with the removal of five to ten residues from the N- and C-termini, which are the ends of the protein. The most complicated knot in proteins with known structures is the 5_2 knot found in ubiquitin hydrolase (pdb code 1xd3). The purpose of this project is to engineer knots into proteins with known structures that currently do not contain knots. The archive for all current structures of proteins and nucleic acids is the Protein Data Bank (PDB). For the purpose of this project, the central carbon atom will signify the amino acid. We will use a program called KnotPlot to graph the coordinates of each alpha carbon and join the two termini. From here, we will perform crossing changes in random coils on the chain and determine whether this creates a local knot that is not the unknot. Since most proteins are linear strands and not closed loops, it is not generally possible to talk about mathematical knots in proteins. We will start with choosing a method to close the gap between the N- and C-termini. As a positive control, we will start by using proteins with local knots to see if the algorithm works. We will then perform crossing changes caused by changing random coils. We will create a protocol to do these functions using KnotPlot, and then write a program to do this automatically so that we can discover all places that can be knotted.
Stefan M. Giovan (University of Texas at Dallas) Direct entropy calculations for discrete wormlike chains
Abstract: Joint work with Stephen D. Levene*†. The thermodynamic properties of a semi-flexible linear polymer, DNA for example, are examined using discrete wormlike chains (dWLCs) as a model. Monte Carlo ensembles of dWLCs were generated to investigate the effect of excluded volume on the configurational entropy of the chain, SC, which is calculated based on the Schlitter approximation. We examined the dependence of absolute and relative entropies on the cylinder diameter and also practical aspects of this approach such as fluctuations in computed SC values as a function of ensemble size. Future applications include estimating the free energy of DNA looping in complex nucleoprotein assemblies. Departments of Molecular and Cell Biology* and Physics
University of Texas at Dallas
Richardson, TX 75080
Michael D. Graham (University of Wisconsin) Active and hibernating turbulence in channel flow of Newtonian and viscoelastic fluids
Abstract: Turbulent channel flow of drag-reducing polymer solutions is simulated in minimal flow geometries. Even in the Newtonian limit, we find intervals of ``hibernating'' turbulence that display many features of the universal maximum drag reduction (MDR) asymptote observed in polymer solutions: weak streamwise vortices, nearly nonexistent streamwise variations and a mean velocity gradient that quantitatively matches experiments. As viscoelasticity increases, the frequency of these intervals also increases, while the intervals themselves are unchanged, leading to flows that increasingly resemble MDR.
Thomas W. N. Haine (Johns Hopkins University) Transit-time distributions: A tool to diagnose rates and pathways of tracer transport in advective/diffusive flow
Abstract: Keywords: Transit-time distribution (TTD); tracer transport rates and pathways Abstract: I will explain the concept of transit-time distributions (TTDs) to diagnose passive tracer transport in geophysical fluids, like the Earth's ocean and atmosphere. The TTD is directly related to the Green's function to the advection/diffusion equation for the concentration of a dynamically-passive trace substance. Diagnosing and interpreting the TTD, rather than the passive tracer field itself, focuses attention on the advective/diffusive transport properties of the underlying flow, and removes the influence of the tracer sources and sinks. The TTD therefore quantifies fundamental transport rate and pathway information about the flow. Various basic mathematical constraints on the TTD function exist, which may illuminate traditional diagnostics of tracer stirring and mixing. Applications of the TTD approach are presented in ocean circulation models, and using ocean tracer measurements.
Thomas W. N. Haine (Johns Hopkins University) Introduction to dynamics and tracer dispersion in geophysical fluids

(1) Geophysical fluids: phenomenology and dynamics of rotating, stratified flow.

(2) Potential vorticity: Dynamic significance and kinematic interpretation.

(3) Importance of mixing for maintenance of ocean pycnocline and global overturning circulation: ocean energetics, a little on thermocline theories and/or Sandstrom's theory.

(4) Tracer Observations in the ocean (and atmosphere): Turbulent mixing from dye releases, maybe something on surface stirring/mixing diagnostics from floats.

(5) Tracer cascades: theory and observational support.

Abstract: No Abstract
Kenneth E. Hinson (University of North Carolina - Charlotte) Braid indices in a class of closed braids
Abstract: A long-standing problem in knot theory concerns the additivity of crossing numbers of links under the connected sum operation. It is conjectured that if A and B are links, then Cr(A#B)=Cr(A)+Cr(B), but so far this has been proved only for certain classes of links. One such class is the zero-deficiency links, which includes some alternating links and some non-alternating, such as torus knots. In this paper the known realm of zero-deficiency links is expanded to include some cases of links represented by alternating closed braids. It is shown that for a link L represented by a reduced, alternating, k-string closed braid diagram D having at most three sequences of consecutive crossings between each pair of adjacent strings, the braid index of L is k. This result makes use of a well-known property of the HOMFLY polynomial, which provides a lower bound for the braid index of a link. It is then seen that the deficiency of L is zero. It seems likely that this result can be extended to more complex alternating closed braids.
Jeffrey Hunt (University of Iowa) High school level introduction to knots
Abstract: Joint work with Bruce MacTaggart. Our educational lesson plans focus on elementary properties of knots and are meant to be a mini-unit in basic knot theory for high school students. There are five total lessons in our introductory unit with a summative assessment on the fifth day. The lessons address knot notation, basic definitions, knot equivalence, and knot arithmetic. There are also various activities with hands-on manipulatives for modeling knots and activities involving the program KnotPlot. We believe that since knot theory is a relatively new field of both mathematics and biology it is important to generate interest with younger mathematics students.
Traian Iliescu (Virginia Polytechnic Institute and State University) Bridging the Boussinesq and primitive equations through spatio-temporal filtering
Abstract: For many realistic geophysical flows, the numerical discretization of the Boussinesq equations yields a prohibitively high computational cost. Thus, a significant research effort has been directed at generating mathematical models that are more computationally efficient than the Boussinesq equations, yet are physically accurate. The tool of choice in generating these simplified models has been scaling. In this note, we put forth spatio-temporal filtering as an alternative methodology for generating simplified mathematical models for the ocean and atmosphere. In particular, we show that spatio-temporal filtering represents a natural approach for bridging the Boussinesq equations and the primitive equations.
Anthony José Kearsley (National Institute of Standards and Technology) Optimal chemical spectroscopy
Abstract: Chemical spectroscopy is an invaluable tool in commerce, public safety, health care, national security, and scientific research. In most cases a measurement expert with considerable experience using an ill-defined catalog of heuristic rules is required to optimize the instrument and to interpret the data. Using an expert is expensive, not always reproducible, and introduces uncontrolled bias to the measurement process. Numerical algorithms for nonlinear optimization can supplement, and perhaps even replace, the knowledge of the expert operator. I will discuss my work with one of the most complex and fastest growing of the new chemical spectroscopies: matrix assisted laser desorption/ionization time of flight mass spectrometry (MALDI TOF MS). This technique finds broad application in both the biological and materials sciences, from drug discovery to the development of high performance plastics. I will demonstrate the application of an optimization method applied to the analysis of a synthetic polymer, one of the most difficult species to analyze by MALDI TOF MS.
Shane Keating (New York University) Homogenization and mixing measures for a replenishing passive scalar field
Abstract: The efficiency with which an incompressible flow mixes a passive scalar field that is continuously replenished by a steady source-sink distribution has been quantified using the suppression of the mean scalar variance below the value it would attain in the absence of the stirring. We examine the relationship this mixing measure has to the effective diffusivity obtained from homogenization theory, particularly establishing precise connections in the case of a stirring velocity field that is periodic in space and time and varies on scales much smaller than that of the source. We explore theoretically and numerically via the Childress-Soward family of flows how the mixing measures lose their linkage to the homogenized diffusivity when the velocity and source field do not enjoy scale separation. Some implications for homogenization-based parameterizations of mixing by flows with finite scale separation are discussed.
Rich R. Kerswell (University of Bristol) Setting limits on turbulence - balancing rigor with practicality
Abstract: Keywords: Turbulent energy dissipation, eddy viscosity, stability Abstract: Variational techniques have been successful in establishing limits on transport in turbulent flow systems (e.g heat in convection, mass in pressure-driven shear flow or momentum in boundary-driven flows). However, there is typically a significant discrepancy between the limit derived and observations as well as a disconnect between the theoretical `optimal flow' solution and what is actually seen. After giving examples of this, I will discuss some past work directed at closing this gap and motivate the use of plausible stability criteria.
Alexander Kiselev (University of Wisconsin) Mixing and enhanced relaxation in fluid flows
Abstract: Keywords: passive scalar, enhanced diffusion, mixing properties Abstract: We consider passive scalar equation on a compact domain or manifold. The fluid flow can aid diffusion and increase the speed of convergence of the initial distribution to its average. We consider either stationary or time periodic flows, and derive a sharp characterization of flows that are particularly effective in enhancing the relaxation speed to mean value. The characterization links enhanced relaxation with spectral properties of the dynamical system generated by flow. The results also provide an indication that time dependence of the flow may improve relaxation enhancing properties. Methods used involve a mix of PDE techniques and functional analysis. A key role is played by estimates similar to ones used in quantum dynamics to measure the rate of wavepacket propagation. The talk is based on works joint with P. Constantin, L. Ryzhik, R. Shterenberg and A. Zlatos.
Peter R. Kramer (Rensselaer Polytechnic Institute) Mathematical models and methods for characterizing turbulent diffusion
Abstract: No Abstract
Matt Mastin (University of Georgia) Symmetries of knots and links
Abstract: Two links are equivalent if, roughly speaking, one can be physically deformed into the other. However, we have a choice as to what information we are keeping track of. For example, if we label the components of a link we could ask whether or not the components can be permuted. A labeling of components could arise naturally in application, for example if the components are different polymers. The poster describes a method of recording all of the symmetry information of links as a certain group. We also preview an upcoming paper in which the symmetry groups for prime links through 8 crossings are computed and discuss future directions including the tabulation of composite links.
Andrea Mazzino (Università di Genova) Miscible and immiscible Rayleigh-Taylor turbulence
Abstract: The Rayleigh–Taylor (RT) instability is a fluid-mixing mechanism occurring at the interface between two fluids of different density when subjected to an external acceleration. The relevance of this mixing mechanism embraces several phenomena occurring in different contexts: astrophysical supernovae and solar-flare development are some examples. Although this instability has been known since 1883, much remains unknown especially on the turbulent regime. A deeper understanding of the mechanism of flows driven by RT instability would thus shed light on the many processes that underpin fully developed turbulence. Along this direction, we performed 2D and 3D direct numerical simulations in order to investigate the statistical properties of turbulent mixing in both miscible and immiscible situations. An introduction to this instability will be provided and some of our numerical results discussed.
Richard M. McLaughlin (University of North Carolina) Passive scalar advection in parallel shear flows: WKBJ mode sorti on intermediate times and the evolution of skewness
Abstract: Keywords: mixing, shear dispersion, Taylor dispersion Abstract: The evolution of a passive scalar diffusing in simple parallel shear flows is a problem with a long history. In 1953, GI Taylor showed theoretically and experimentally that on long times, the passive scalar experiences an enhanced diffusion in the longitudinal direction. On shorter times the scalar evolution is anomalous, characterized by second moments growing faster than linear in time as we show by analysis of the stochastic differential equations underlying the passive scalar equation. The spatial structures associated with this intermediate time evolution are predicted using WKBJ analysis of an associated non-self adjoint eigenvalue problem. This analysis predicts a sorting of wall modes and interior modes with specific predictions of the decay and propagation rates as a function of the Peclet number. Monte Carlo simulations demonstrate non-trivial skewness evolution, and skewness is studied in the new WKBJ modes. Time permitting, new behavior distinguishing channel from pipe flow will be presented along with comparisons between some of these predictions and experiments in the pipe geometry. This is joint work with Roberto Camassa, Zhi Lin, Keith Mertens, Nick Moore, and Claudio Viotti.
Igor Mezic (University of California, Santa Barbara) Mixing: Visualization, norms, and control
Abstract: Keywords: mixing, mix-norm, optimal stirring, ergodicity Abstract: I will discuss several issues in analyzing kinematics of a purely advective mixing process: 1) Determine whether the scalar quantity, such as dye, introduced into the flow field is - asymptotically - in time thoroughly mixed. 2) Determine how good is the mixture at any finite time. 3) Provide methods for open-loop optimization or feedback control of the mixture. The concept of ergodic partition allows us to discuss 1) precisely, and I will discuss some new results that allow us to compute it effectively. Concerning 2), the problem of an effective norm for mixing has attracted a lot of work over the last decade. I will discuss one family of norms - the so called mix-norm that connects to negative Sobolev space norms - that allows us to pursue study of opimization and control, thus covering 3). I will also discuss the question of ergodicity of a system and how to measure it. This is a departure from the standard off-on definition of ergodicity providing a measure of how close to ergodicity a system is. In all of the above, ergodic theory plays a prominent role.
Paul Milewski (University of Wisconsin) Dynamics of shallow water layer models: Stability, wave breaking and mixing
Abstract: Keywords: Shallow water, Layer model, hydraulic jumps, shocks, mixing, stability Abstract: Shallow-fluid models are often the first step in modeling many geophysical flows. These models apply when the horizontal scales of motion are much larger than the vertical scales. For a single fluid layer with a free-surface, the shallow water approximation results in hyperbolic equations which are well understood and broadly applied. Generically, waves steepen and break creating hydraulic jumps which satisfy the PDEs in a weak sense. Physically one must ensure that certain conserved quantities - usually mass and momentum - are preserved across shocks. For the single layer case this results in a prediction of the small scale energy dissipation at the shock. In layered shallow water models the situation is complicated by at least two issues: that the flow may be shear unstable (the Kelvin-Helmholtz instability), and that breaking waves may mix the fluids. We shall discuss some physically motivated mathematical results on these issues.
Dongjuan Niu (Capital Normal University) Postdoc seminar: Coupled boundary layers for the primitive equations of atmosphere
Abstract: In this talk I will discuss the boundary layers problems of the primitive equations of atmosphere. It is proved that, for well-prepared initial data, the smooth solutions of the primitive equations converge to smooth solutions of quasi-geostrophic equations as the Rossby number, the vertical viscosity and the vertical heat conductivity tend to zero. The new ingredient is that the velocity boundary layer and thermal layer are considered simultaneously.
Alexei Novikov (Pennsylvania State University) Exit time problem in an incompressible flow
Abstract: Consider a Brownian particle in a deterministic time-independent incompressible flow in a bounded domain. We are interested how flow affects the expected exit time, the time the particle needs to reach the boundary of the domain. In particular, whether the presence of the flow decreases the maximum of this expected exit time. One would expect that any stirring improves mixing, thus decreasing the expected exit time. We will show that generally it is not true in two dimensions. This is a joint work with G.Iyer, L.Ryzhik, and A.Zlatos.
Mary Therese Padberg (University of Iowa), Gregory Witt (University of Iowa) Energetics of DNA tangling in complex nucleoprotein assemblies
Abstract: Tangle analysis, a branch of mathematical knot theory, in conjunction with difference topology experiments has become a powerful emerging approach for the analysis of complex nucleoprotein assemblies containing DNA loops. A tangle consists of strings properly embedded in a 3-dimensional ball. The protein complex can be thought of as a 3D ball while the DNA segments bound by the protein complex can be thought of as strings embedded within the ball. At present, tangle analysis can only provide information about 2-dimensional diagrams representing the topology of DNA bound within a protein complex. Many DNA geometries can be consistent with a particular topological solution, however, limiting the value of tangle analysis in deducing biological mechanism and function. In addition, many problems of interest do not yield unique tangle solutions. Information about the relative energies of geometric solutions is badly needed to evaluate the plausibility of a particular mathematical solution both physically and biologically. We will demonstrate preliminary software for determining likely DNA geometries consistent with protein-bound DNA topologies.
Juliet Portillo (San Francisco State University) Invariance of the sign of the average space writhe of free and confined knotted polygons
Abstract: Our group studies topological properties of DNA molecules in solution. We consider highly compacted models of knotted DNA, such as DNA extracted from P4 phages. Circular DNA molecules are modeled as self-avoiding polygons (SAPs) in three-dimensional space. Using different Monte Carlo algorithms, we sample the space of knotted SAPs and study knotting probabilities. To better understand how DNA knotting is affected in confined environments, we generate knotted configurations confined inside small spheres. Writhe is a geometric invariant that measures the entanglement complexity of a given configuration. A comparison of the writhe of confined versus free knots suggests that the sign of the average writhe is invariant for each chiral knot type under varying polygonal lengths on the simple cubic lattice and in R3. We propose that the sign of the average space writhe is a robust measure of knot chirality.
François W. Primeau (University of California, Irvine) The inverse problem of inferring transit-time distributions from tracer observations in the ocean
Abstract: Keywords: Transit-time distribution (TTD); tracer transport; inverse problem; maximum-entropy deconvolution Abstract: I will discuss the inverse problem of inferring the Green's function for advective-diffusive transport (also known as the transit-time distribution) from tracer observations. Tracers with different boundary conditions and/or different radio-active decay rates probe different transport pathways and timescales. Using multiple tracers in combination can therefore help constrain the full transit-time distribution (TTD). I will review two inversion methodologies applicable to ocean tracer measurements, one based on a parametric model for the TTD and one based on a more flexible maximum-entropy deconvolution approach. Because the oceanographic inverse problem is grossly under-determined an important focus of this talk will be on quantifying the uncertainty associated with the inversion results.
Nancy Reid (University of Toronto) Can chocolate save your life?
Abstract: This question appeared in a recent newspaper headline, but was based on a study involving only 14 people. How can we interpret the statistics behind headlines? What does statistically significant really mean? How do statistics get manipulated to further an agenda? The field of statistics is essential to understanding most current issues. It informs economics, health care, and environmental protection. The speaker calls statistics mathematical social work; it helps science progress, so it is important to understand its power.
Juan Mario Restrepo (University of Arizona) Jet dynamics in stratified media
Abstract: Keywords: jet, buoyancy, stratified flow, stability Abstract: I discuss the flow structure and stability of a planar saline jet descending into a stable, density-stratified fluid. The jet retains its slender shape, largely due to the low salt diffusion. As the jet descends it entrains fresher water due to the relatively high mechanical viscous effects, when these are compared to inertial effects. This fresher water forms a recirculation cell. The jet exhibits a rapid acceleration on release, then deceleration, as it encounters the more dense surrounding fluid, and stops at a location much higher than the neutral buoyancy point. I will recount preliminary work aimed at explaining the fluid dynamics of the jet: Stratification, mechanical diffusion and nonlinear inertial effects, as well as salt diffusion are all found to be crucial to the dynamics. I will also summarize our work on characterizing the basic instability modes of the jet by numerical means. We successfully captured the inception of the most salient symmetric and anti-symmetric instabilities and their dependence on the Reynolds number and the non-dimensional stratification gradient number. This jet, though deceptively simple, is far from well understood. I will enumerate key dynamic aspects that are beyond our present understanding and worthy of further study due to their relevance to other important physical phenomena. This is joint work with Sam Schofield, Los Alamos National Laboratory, with contributions from Adriana Pesci and Raymond Goldstein, Cambridge University.
Bertrand Rollin (Los Alamos National Laboratory) On the effect of initial velocity field and phase shifting of an initial binary perturbation for Rayleigh-Taylor instability
Abstract: Starting (initial) conditions (ICs) can influence the development of hydrodynamic turbulence and material mixing in buoyancy driven flows. The overall goal of our research is to determine the extent to which starting conditions can be used to predict and design turbulent transport/material mixing. In particular, this work studies the effect of the initial velocity field and phase shifting on a binary initial perturbation. Results of an experimental investigation in which precisely defined initial conditions have been prescribed are presented. These experimental results serve as references that we try to match as closely as possible with numerical simulations. Our simulations show that the initial velocity field drives the growth of the initial perturbation in this experiment. Also, a “leaning” of the growing flow structures observed in the experiment is captured by the simulations, and linked to the phase shift.
Bonita V. Saunders (National Institute of Standards and Technology) Applying numerical grid generation to the visualization of complex function data
Abstract: Numerical grid generation, that is, structured grid generation, is the development of a generalized curvilinear coordinate system. Originally designed for solving computational fluid dynamics problems over oddly shaped domains, structured techniques have competed with various unstructured methods such as Voronoi or Delaunay triangulations and quadtree designs. However, the effectiveness of a given grid often depends on how it is used. For complex function visualization problems, the grid generation technique may be less important than how closely the grid lines follow the contours of the function. This talk looks at the use of a tensor product B-spline mapping to generate a boundary/contour fitted mesh that captures significant attributes such as zeros, poles, branch cuts and other singularities when the mesh is used to plot a complex function surface. This work has been used to create over 200 interactive 3D visualizations of complex function surfaces for the NIST Digital Library of Mathematical Functions (DLMF). The NIST DLMF and its hardcopy version, the NIST Handbook of Mathematical Functions, will replace the well-known NBS Handbook of Mathematical Functions edited by Abramowitz and Stegun and first published in 1964.
Robert Glenn Scharein (San Francisco State University) Minimal step number of cubic lattice knots in thin slabs
Abstract: We present provisional data on the minimal step number of cubic lattice knots confined to a thin slab. In particular, we investigate thin slabs of thickness 1, 2 and 3. For most knot types, several ergodicity classes are found, often with dramatically different minimal step numbers. We discuss the number of distinct minimal step embeddings found within each class. We show that in the case of the 1-slab, arbitrarily high step number representatives for each knot type may be found that are irreducible within the 1-slab. Finally, we examine recurring patterns across the entire database of minimal step knots, both in thin slabs and for the unconstrained case.
Joerg Schumacher (Ilmenau University of Technology) Numerical studies in shallow moist convection
Abstract: Convective turbulence with phase changes and latent release is an important dynamical process in the atmosphere of the Earth which causes, e.g., the formation of clouds. Here we study moist convection in simplified setting - shallow and nonprecipitating moist Rayleigh-Benard convection with a piecewise linear thermodynamics on both sides of the phase boundary. The presented model is a first nontrivial extension of the classical dry Rayleigh-Benard convection. The equations of motion and the fully developed turbulent dynamics in very flat Cartesian cells are discussed.
Emily F. Shuckburgh (British Antarctic Survey) Mixing by eddies in the atmosphere and ocean
Abstract: Keywords: eddy diffusivity, conserved tracer, reactive tracer Abstract: Eddies have an important role in transport and dynamical processes in the atmosphere and ocean. They influence the distribution of chemical species and are responsible for driving mean flows. I will discuss the quantification of eddy effects in the atmosphere and ocean. I will focus in particular on two problems. The first is how to quantify the geographic (latitude-longitude) variation of eddy diffusivity of a conserved tracer in such flows. I will describe two different techniques and discuss the implications of the results for an atmospheric and an oceanic case. The second is how to quantify the effects of eddies on the distribution of a reactive tracer. I will take the example of different tracers at the sea surface (temperature, salinity, chlorophyll, etc). Eddy stirring directly influences the distribution of such tracers, but small-scale eddies in the ocean can also influence air-sea interactions and I will describe how this latter effect may be quantified.
K. Shafer Smith (New York University) The three-dimensional structure of turbulent geostrophic stirring
Abstract: Keywords: geostrophic turbulence, baroclinic instability, stirring and mixing Abstract: Turbulent flows generated by baroclinic instability develop strong nearly-barotropic vortices and lateral strain fields, with energy accumulating in large horizontal and vertical scales. Nevertheless, away from boundaries these flows simultaneously generate strong vertical shear and vertical strain variance on small scales, resulting from the three-dimensional forward cascade of potential enstrophy. It is shown that the combined action of strain and shear generates tracer filaments that, on average, maintain a scale-independent aspect ratio proportional to N/f. The result is a submesoscale coupling between vertical mixing and horizontal stirring that allows vertical diffusion to effectively absorb the laterally driven cascade of tracer variance. Nearer the ocean's surface, geostrophic turbulence changes its character: the turbulent dynamics becomes dominated by a forward cascade of buoyancy, causing the energy spectrum to flatten, and the tracer variance spectrum to steepen. In the upper-ocean where the stratification is weak, this results in a ubiquitous generation of lateral density fronts. These dynamically active fronts are primed by mesoscale stirring, but provide a pathway to interaction with smaller-scale turbulent processes. I'll present results from a series of simulations demonstrating the dynamics of both regimes described above.
Wenbo Tang (Arizona State University) Lagrangian dynamics in stochastic inertia-gravity waves
Abstract: We perturb the analytic deterministic solution of inertia-gravity waves with stationary random noise and solve for the Fokker-Planck equation to study the evolution in time of the probability density function of passive tracers in such a flow. We find that at initial times the probability density closely follows the nonlinear background flow and non-trivial Stokes drift ensues as a result. Over finite time, we measure chaotic mixing based on the stochastic mean flow and identify non-trivial mixing structures of passive tracers, as compared to their absence in the deterministic flow. At later times, when the probability density field spreads out to sample larger regions, the mean Stokes drift approaches an asymptotic value, indicating suppression of Lagrangian mixing at long time-scales. However, the skewness of the probability density remains non-Gaussian even at large times.
Jean-Luc Thiffeault (University of Wisconsin) Do fish stir the ocean?
Abstract: Keywords: stirring, mixing, biomixing, Brownian motion. Abstract: As fish or other bodies move through a fluid, they stir their surroundings. This can be beneficial to some fish, since the plankton they eat depends on a well-stirred medium to feed on nutrients. Bacterial colonies also stir their environment, and this is even more crucial for them since at small scales there is no turbulence to help mixing. It has even been suggested that the total biomass in the ocean makes a significant contribution to large-scale vertical transport, but this is still a contentious issue. We propose a simple model of the stirring action of moving bodies through both inviscid and viscous fluids. In the dilute limit, this model can be solved using Einstein and Taylor's formula for diffusion (Brownian motion). We compare to direct numerical simulations of objects moving through a fluid. This is joint work with Steve Childress and George Lin.
Becca Thomases (University of California, Davis) A Stokesian viscoelastic flow: Transition to mixing and oscillations
Abstract: Keywords: Oldroyd-B, viscoelastic, instabilities, mixing Abstract: To understand observations of low Reynolds number mixing and flow transitions in viscoelastic fluids, we study numerically the dynamics of the Oldroyd-B viscoelastic fluid model. The fluid is driven by a simple time-independent forcing that creates a cellular flow with extensional stagnation points. We find that at O(1) Weissenberg number these flows lose their slaving to the forcing geometry of the background force, become oscillatory with multiple frequencies, and show continual formation and destruction of small-scale vortices. This drives flow mixing. These new flow states are dominated by a single large vortex, which may be stationary or move persistently from cell to cell. Increasing the number of degrees of freedom by increasing the number of driving cells broadens the temporal frequency spectrum and yields richer dynamics with no persistent vortices and improved fluid mixing.
Giordano Tierra Chica (University of Sevilla) Super-convergence for the 3D Navier-Stokes
Abstract: This work is devoted to study the stability and error estimates of a fully discrete scheme for the incompressible time-dependent Navier-Stokes Equations in three-dimensional domains. Space is discretized by using the Finite Element Method, whereas time is discretized using the Finite Difference Method. We introduce an extension to mixed elliptic problems of the negative-norm estimates for uniformly elliptic problems. Using this extension, we prove some super-convergence results in space for velocity which have been observed in several computational experiments. Furthermore, we obtain some error estimates results for the pressure without restrictions relating time and space discrete parameters.
Vy T. Tran (University of St. Thomas) Symmetry-breaking in cumulative measures of shapes of polymer models
Abstract: In a thermally agitated environment, randomly generated polygons are used to model the conformations of fluctuating polymer chains. To characterize the shapes of these polygons, we created 3D density plots of the vertex distributions of families of random 6 edge polygons. The distributions give a measure of the shapes of the polygons, and our symmetry-breaking alignment procedure is not only able to reveal their average bulk shape, but also distinguish between different knot topologies and chirality. We looked at the family of 6 edge polygons, separating them by knot type, and we also looked at 6 edge open chains.
Rolland Trapp (California State University) Polygonal cable links
Abstract: Given a polygonal knot we present an efficient construction of polygonal cables of the knot. The construction is applied to polygonal unknots to obtain results about stick numbers of torus knots. In particular, we show that (2,q) torus links can be constructed with about two-thirds q sticks. This is used to show that for q greater than 14, minimal stick representatives of (2,q) torus links are "supercoiled". Finally we show that for 2p < q < 3p the stick number of (p,q) torus links is 4p.
Yue-Kin Tsang (University of California, San Diego) Fast chemical reactions in chaotic flows: Reaction rate and mixdown time
Abstract: We study the effect of chaotic flows on the progress of fast bimolecular reactions. Simulations show that the reactant concentration decays exponentially and then crosses over to the algebraic law of chemical kinetics in the final stage of the reaction. By transforming the reactive mixing problem to an equivalent passive scalar problem, we make prediction to the crossover time and the overall reaction rate. Depending on the relative length scale between the velocity and the concentration fields, the overall reaction rate is either related to the distribution of the finite-time Lyapunov exponent or given in terms of an effective diffusivity. Preliminary results on a variation of this problem in which the reactants are initially isolated from one another is also presented. Here, we focus on the mixdown time, i.e. the time taken for the flow to bring the reactants into contact, and its dependence on the various length scales in the system.
Alexandra Tzella (École Normale Supérieure) Spatial structures of chaotically advected reactive tracers: The role of a delay time
Abstract: Keywords: reactive flows, chaos, structure functions, Holder exponents, delay differential equations, Lyapunov exponents Abstract: Motivated by the spatial heterogeneity observed in plankton distributions in the meso-scale ocean, we examine the stationary-state spatial structure of reacting tracer fields, for the case for which the reaction equations contain delay terms. The fields are advected by a flow that gives rise to chaotic parcel trajectories and the structures are maintained by a large-scale source. Previous theoretical investigations have shown that, in the absence of delay terms and in a regime where diffusion can be neglected (large Peclet number), the structures are filamental and characterized by a single scaling regime with a Holder exponent that depends on the rate of convergence of the reactive processes and the strength of the stirring measured by the average stretching rate (Lyapunov exponent). In the presence of delay terms, we show that for sufficiently small scales all interacting fields should share the same spatial structure, as found in the absence of delay terms. However, depending on the strength of the stirring and the magnitude of the delay time, two further scaling regimes that are unique to the delay system may appear at intermediate length-scales. An expression for the transition length-scale dividing small-scale and intermediate-scale regimes is obtained and the scaling behavior of the tracer field is explained. Finally, we discuss the dependence of the field's scaling exponents on the distribution of the stretching statistics. Joint work with P. H. Haynes.
Jacques Vanneste (University of Edinburgh) Modelling streaming by surface acoustic waves
Abstract: Keywords: microfluidics, acoustic mixing, surface waves, acoustic streaming Abstract: Acoustic streaming, the generation of flow by dissipating acoustic waves, provides a promising method for flow pumping in microfluidic devices. In recent years, several groups have been experimenting with a acoustic streaming induced by leaky surface waves: (Rayleigh) surface waves excited in a piezoelectric solid interact with a small volume of fluid where they generate acoustic waves and, as result of the viscous dissipation of these waves, a mean flow. We discuss the basic mathematical model that has been employed in simulations of this type of acoustic streaming and reformulate it to account for the dynamical constraints imposed by vorticity conservation. The formulation proposed makes it clear that dissipative processes in the bulk of the fluid are essential to the streaming, and separates the Eulerian and Stokes contributions to the mean flow. Particular attention is paid to the thin boundary layer that forms at the solid/liquid interface, where both the acoustic waves and their streaming effect are best computed by asymptotic means. A simple two-dimensional model of mean-flow generation by surface acoustic waves is discussed as an illustration. Joint work with Oliver Buhler (Courant).
Jacques Vanneste (University of Edinburgh) Estimating generalised Lyapunov exponents for random flows
Abstract: The generalised Lyapunov exponents (GLEs) quantify the growth of the separation between particles advected in fluid flows. They provide valuable information about mixing, in particular because, in some cases, the decay rate of passive scalars released a flow can be directly related to specific GLEs of this flow. Here we discuss some numerical and asymptotic methods for the estimation of the GLEs of random renewing flows (such as the alternating-sine flow) in which the particle separation is described by a product of random matrices. Specifically, we propose an importance-sampling Monte Carlo algorithm as a general purpose numerical method which is both efficient and easy to implement. We also discuss asymptotic approximations for the GLEs characterising extremes of stretching.
Xiaoming Wang (Florida State University) Approximating the rate of heat transport
Abstract: Keywords: convection, heat transport, Nusselt number, Constantin-Doering-Hopf technique, long time statistical properties, numerical approximation, convergence of long time statistical properties Abstract: We survey a few recent results on estimating the long time averaged rate of heat transport in the vertical direction (the Nusselt number) in Rayleigh-Bénard convection. In the first half of the talk, we recall rigorous upper bounds on the Nusselt number that are of the form of Ra1/3 modulo logarithmic correction for both the infinite Prandtl number model and the classical Boussinesq model for convection with large but finite Prandtl number. The main technique is the Constantin-Doering-Hopf approach. We also discuss the infinite Prandtl number limit in the Boussinesq model for convection, and the formal infinite Rayleigh number limit within the infinite Prandtl number model for convection. In the second half of the talk, we discuss numerical schemes (time discretization) that are able to capture the long time statistical properties of the convection problems. We first recall that the maximum long time averaged rate of heat transport in the vertical direction (true maximum Nusselt number) is a long time statistical property of the convection system. We then show that appropriate time discretizations of the systems will be able to capture the true maximum Nusselt number asymptotically. Several specific schemes that satisfy the desired properties will be presented. This numerical approach complements the Constantin-Doering-Hopf approach in the sense that it provides a computational asymptotic lower bound. Noise effects will be mentioned.
Danielle Washburn (University of Iowa) Tangle tabulation
Abstract: Like knots, tabulating tangles is done by crossing number. Tangles are similar to knots, but contain strings whose endpoints are "nailed down" on the boundary of a 3-dimensional ball. The crossing number is the minimal number of crossings needed to draw the diagram of a knot (tangle). We will discuss some basic concepts common between knots and tangles, how to code this and issues that have arisen. Finally, we will introduce why we are interested in tabulating tangles: math biology.
Jeffrey B. Weiss (University of Colorado) Broadcast spawning: A new class of reaction-mixing problems
Abstract: Keywords: mixing reaction broadcast spawning coral scaling Abstract: Coral and other marine organisms reproduce through the mechanism of broadcast spawning, where egg and sperm are released at separate locations and brought together by fluid mixing and transport. Coral fertilization is particularly important because corals are threatened by anthropogenic climate change. We idealize broadcast spawning as point sources of two reacting tracers separated by a neutral fluid. This represents a new class of problems, different from well-studied problems such as flame fronts, where two tracers fill the fluid and are separated by an interface. For the case of broadcast spawning within a vortex, we show that the vortex stirring leads to a self-similar solution with enhanced fertilization rates scaling as the Peclet number(1/3) and reduced fertilization times scaling as the Peclet number(-2/3).
Xiaofeng Yang (University of South Carolina) Shear cell rupture of nematic droplets in viscous fluids
Abstract: We model the hydrodynamics of a two-phase system of a nematic liquid crystal drop in a viscous fluid using an energetic variational approach with phase-field methods cite{YFLS04}. The model includes the coupled system for the flow field for each phase, a phase-field function for the diffuse interface and the orientational director field of the liquid crystal phase. An efficient numerical scheme following is implemented for the two-dimensional evolution of the shear cell experiment for this initial data. We simulate the deformation and rupture of nematic droplets, identifying the formation of surface topological defects, and exploring the shear and normal stress distributions that accompany the evolution. A bipolar global defect structure, with two half-integer surface point defects called boojums, emerges in every daughter droplet when tangential anchoring conditions are imposed together with Oseen-Frank distortional bulk elasticity. The fate of the original mother drop is compared for the limiting case of an immiscible viscous drop versus strength of the liquid crystal interfacial and bulk potentials.
William Roy Young (Scripps Research Institute) Is turbulence stable?
Abstract: Keywords: Jet, layer, potential vorticity, beta-plane, stratified turbulence, Cahn-Hilliard equation, negative viscosity Abstract: I'll discuss two examples of uniformly forced turbulent flows in which quasi-steady structures spontaneously form. This results in the intensity of turbulence becoming spatially inhomogeneous on length scales larger than those of the eddies i.e., as argued by Owen Phillips in 1972, a spatially homogeneous turbulent flow may be subject to a large-scale instability. The first example is stirring a fluid with strong gravitational stability due to, for example, dissolved salt. The resulting stratified turbulence produces well-mixed layers with uniform density, separated by strongly stable steps in density. The second example is the formation of zonal jets in forced-dissipative beta-plane turbulence. I'll discuss the prospects of understanding these systems using models related to the Cahn-Hilliard equation.
William Roy Young (Scripps Research Institute) Shear dispersion

(1) interaction of molecular diffusion with simple unidirectional shear flows (bounded and unbounded domains).

(2) Limitation of the effective diffusion approximation to long times and small domains, and low moments of the tracer distribution.

(3) Perhaps a geophysical example: shear diffusion in the internal gravity wave field. The vertical tracer cascade of Haynes & Anglade.

(4) Some examples of "pre-asymptotic" anomalous shear diffusion e.g.,〈x2〉 ∼ t ln t due to the no-slip condition, and 〈x2〉 ∼ t 3/2 in the example of de Marsily & Matheron.

Abstract: No Abstract
Lynn Zechiedrich (Baylor College of Medicine) Biological applications that utilize DNA Topology

Abstract: The long, rich history of topology in mathematics has proven extremely useful for the study of DNA. DNA, the genetic blueprint for life, undergoes tremendous flux as it is packaged, replicated, segregated, transcribed, recombined and repaired. Extremely long and skinny, DNA is prone to entanglement. Every time it is copied, the two resulting "daughter chromosomes" are entangled. And nearly all organisms maintain duplex DNA in a slightly underwound state. Linking number (Lk), the major descriptor for DNA apart from base pair sequence, defines the three forms of DNA topology, which are known to biologists as knots, catenanes, and supercoils. Changes in Lk have dramatic effects on biological processes. In this talk I will provide an overview of DNA topology and the biological ramifications of topology, including exciting new developments in the application to medicine. The following authors have contributed to the work: Jonathan M. Fogg1, Daniel J. Catanese, Jr.1, Donald Schrock, II1, Richard W. Deibler1,2,3, Jennifer K. Mann1,4, De Witt L. Sumners4, Brian E. Gilbert1, Youli Zu5, Nianxi Zhao5. 1Departments of Molecular Virology & Microbiology, Biochemistry and Molecular Biology, and Pharmacology, Baylor College of Medicine, Houston, TX 77025 2Interdepartmental Program in Cell and Molecular Biology, Baylor College of Medicine, Houston, TX 77030 3Department of Systems Biology, Harvard Medical School, Boston, MA 02115 4Department of Mathematics, Florida State University, Tallahassee, FL 32306 5Department of Pathology, The Methodist Hospital Research Institute, Houston, TX 77030 USA
Laura K. Zirbel (University of California, Santa Barbara) The local and global shape of regular embedded polygons: Theoretical and experimental
Abstract: We consider $mathcal{P}_n$, the space of equilateral, n-sided polygons embedded in $mathbb{R}^3$. There are several descriptions of the global shape of a polygon $P in mathcal{P}_n$, including the convex hull volume, miniball radius, asphericity and radius of gyration. We sought a description of shape that was sensitive to both local and global behavior, and to look at average trends over both the whole population of $mathcal{P}_n$, as well as finding the average over sub populations of a specific knot type. We developed two such descriptions. For a given $P in mathcal{P}_n$, we find the average of the squared distance between vertex $i$ and $i+k$ for all $1 leq i leq n$. We call this the Average Squared End to End Distance of length $k$ of $P$. Similarly, we find the squared radius of gyration for all sub-segments of $P$ of length $k$, and we call the average of these values the Average Squared Radius of Gyration of length $k$ of $P$. We determine the theoretical averages of these values, taken over all of $mathcal{P}_n$, in terms of $n$ and $k$. In addition, we examine specific examples of embedded polygons, to determine the effect of knotting of these descriptions of shape.
Visitors in Residence
Farid Ait Chaalal McGill University 4/11/2010 - 4/17/2010
Douglas N. Arnold University of Minnesota 4/1/2010 - 4/1/2010
F. Javier Arsuaga San Francisco State University 4/9/2010 - 4/9/2010
Joel D. Avrin University of North Carolina - Charlotte 4/11/2010 - 4/18/2010
Gregory R. Baker Ohio State University 4/19/2010 - 4/21/2010
Nusret Balci University of Minnesota 9/1/2009 - 8/31/2010
Claude Bardos Université de Paris VI (Pierre et Marie Curie) 2/15/2010 - 4/25/2010
Lauren M. Beaumont University of Iowa 4/9/2010 - 4/10/2010
Jennifer Beichman University of Michigan 9/1/2009 - 5/31/2010
Michael Berglund University of Georgia 4/9/2010 - 4/11/2010
Andrew Joel Bernoff Harvey Mudd College 4/6/2010 - 4/11/2010
Hakima Bessaih University of Wyoming 4/11/2010 - 4/17/2010
Anusha Bharadwaj University of Texas at Dallas 4/9/2010 - 4/11/2010
Animikh Biswas University of North Carolina - Charlotte 4/10/2010 - 4/15/2010
Katarina Bodova Komensky (Comenius) University of Bratislava 4/11/2010 - 4/17/2010
Guido Boffetta Università di Torino 4/10/2010 - 4/16/2010
Olus N. Boratav Corning Incorporated 4/10/2010 - 4/16/2010
James Joseph Brannick Pennsylvania State University 4/10/2010 - 4/15/2010
Susan Brooks University of Iowa 4/9/2010 - 4/11/2010
Dorothy E. Buck Imperial College London 4/9/2010 - 4/9/2010
Maria-Carme T. Calderer University of Minnesota 9/1/2009 - 6/30/2010
Jason Cantarella University of Georgia 4/9/2010 - 4/9/2010
Colm P. Caulfield University of Cambridge 4/10/2010 - 4/16/2010
Chi Hin Chan University of Minnesota 9/1/2009 - 8/31/2010
Xianjin Chen University of Minnesota 9/1/2008 - 8/31/2010
Alina Chertock North Carolina State University 4/11/2010 - 4/16/2010
Stephen Childress New York University 4/11/2010 - 4/16/2010
Gregory P. Chini University of New Hampshire 4/10/2010 - 4/17/2010
Ivan Christov Northwestern University 4/11/2010 - 4/16/2010
John Collins San Francisco State University 4/9/2010 - 4/11/2010
Gedeon Dagan Tel Aviv University 4/11/2010 - 4/15/2010
Domenico D'Alessandro Iowa State University 4/15/2010 - 6/30/2010
Isabel K. Darcy University of Iowa 4/9/2010 - 4/11/2010
Elizabeth Denne Smith College 4/9/2010 - 4/9/2010
Yuanan Diao University of North Carolina - Charlotte 4/9/2010 - 4/9/2010
Charles R. Doering University of Michigan 8/15/2009 - 6/15/2010
Doug Dokken University of St. Thomas 4/12/2010 - 4/16/2010
Michael Dupuis University of St. Thomas 4/9/2010 - 4/9/2010
Robert S. Eisenberg Rush University Medical Center 4/11/2010 - 4/13/2010
Claus Ernst Western Kentucky University 4/9/2010 - 4/9/2010
Randy H. Ewoldt University of Minnesota 9/1/2009 - 8/31/2010
Jin Feng University of Kansas 4/11/2010 - 4/17/2010
Aldo Fiori Università di Roma "La Sapienza" 4/10/2010 - 4/16/2010
Natasha Flyer National Center for Atmospheric Research 4/11/2010 - 4/14/2010
Eliot Fried McGill University 4/11/2010 - 4/17/2010
John R. Froehlig University of Iowa 4/9/2010 - 4/11/2010
Boris Gershgorin New York University 4/11/2010 - 4/16/2010
Stefan M. Giovan University of Texas at Dallas 4/9/2010 - 4/11/2010
Michael D. Graham University of Wisconsin 4/11/2010 - 4/16/2010
Thomas C. Hagen University of Memphis 4/9/2010 - 4/17/2010
Thomas W. N. Haine Johns Hopkins University 4/10/2010 - 4/16/2010
Peter Haynes University of Cambridge 4/10/2010 - 4/16/2010
Kenneth E. Hinson University of North Carolina - Charlotte 4/9/2010 - 4/11/2010
Tzyy-Leng Allen Horng Feng Chia University 4/11/2010 - 4/17/2010
Jeffrey Hunt University of Iowa 4/9/2010 - 4/11/2010
Vera Mikyoung Hur University of Illinois at Urbana-Champaign 2/18/2010 - 5/31/2010
Yunkyong Hyon University of Minnesota 9/1/2008 - 8/31/2010
Traian Iliescu Virginia Polytechnic Institute and State University 4/11/2010 - 4/16/2010
Mark Iwen University of Minnesota 9/1/2008 - 8/31/2010
Gautam Iyer Stanford University 4/11/2010 - 4/16/2010
Srividhya Jeyaraman University of Minnesota 9/1/2008 - 8/31/2010
Lijian Jiang University of Minnesota 9/10/2008 - 8/31/2010
Garrett Jones University of Iowa 4/9/2010 - 4/11/2010
Mihailo Jovanovic University of Minnesota 9/11/2009 - 6/10/2010
Ning Ju Oklahoma State University 1/4/2010 - 6/30/2010
Anthony José Kearsley National Institute of Standards and Technology 4/29/2010 - 5/1/2010
Shane Keating New York University 4/10/2010 - 4/16/2010
Markus Keel University of Minnesota 7/21/2008 - 6/30/2010
James Patrick Kelliher University of California, Riverside 4/11/2010 - 4/15/2010
Thomas W Kephart Vanderbilt University 4/9/2010 - 4/9/2010
Rich R. Kerswell University of Bristol 4/10/2010 - 4/16/2010
Hyejin Kim University of Minnesota 9/1/2009 - 8/31/2010
Alexander Kiselev University of Wisconsin 4/11/2010 - 4/16/2010
Tomasz Komorowski Marie Curie-Skłodowska University 4/11/2010 - 4/16/2010
Pawel Konieczny University of Minnesota 9/1/2009 - 8/31/2010
Peter R. Kramer Rensselaer Polytechnic Institute 4/10/2010 - 4/16/2010
Alexander Kurganov Tulane University 4/11/2010 - 4/16/2010
Juan C. Latorre Freie Universität Berlin 4/11/2010 - 4/16/2010
Norman Lebovitz University of Chicago 4/11/2010 - 4/16/2010
Chiun-Chang Lee National Taiwan University 10/22/2009 - 6/30/2010
Young-Ju Lee Rutgers University 4/10/2010 - 4/17/2010
Stephen D. Levene University of Texas at Dallas 4/9/2010 - 4/10/2010
Marta Lewicka University of Minnesota 9/1/2009 - 6/30/2010
Congming Li University of Colorado 1/11/2010 - 6/15/2010
Yongfeng Li University of Minnesota 9/1/2008 - 8/31/2010
Tai-Chia Lin National Taiwan University 4/8/2010 - 4/16/2010
Zhi (George) Lin University of Minnesota 9/1/2009 - 8/31/2010
Chun Liu University of Minnesota 9/1/2008 - 8/31/2010
Stefan Llewellyn Smith University of California, San Diego 4/11/2010 - 4/16/2010
Ellen K. Longmire University of Minnesota 9/1/2009 - 6/30/2010
Krishnan Mahesh University of Minnesota 9/1/2009 - 6/30/2010
Kara Lee Maki University of Minnesota 9/1/2009 - 8/31/2010
Vasileios Maroulas University of Minnesota 9/1/2008 - 8/31/2010
Nader Masmoudi New York University 4/13/2010 - 4/16/2010
Matt Mastin University of Georgia 4/9/2010 - 4/11/2010
Sarah Matz University of Wisconsin 4/9/2010 - 4/18/2010
Andrea Mazzino Università di Genova 4/10/2010 - 4/16/2010
Anna L. Mazzucato Pennsylvania State University 1/12/2010 - 6/11/2010
Richard M. McLaughlin University of North Carolina 4/11/2010 - 4/16/2010
Igor Mezic University of California, Santa Barbara 4/14/2010 - 4/16/2010
Paul Milewski University of Wisconsin 4/9/2010 - 4/13/2010
Kenneth C. Millett University of California, Santa Barbara 4/9/2010 - 4/9/2010
Yoichiro Mori University of Minnesota 9/1/2009 - 6/30/2010
Benson Muite University of Michigan 4/11/2010 - 4/16/2010
Dongjuan Niu Capital Normal University 4/1/2010 - 6/15/2010
Alexei Novikov Pennsylvania State University 4/11/2010 - 4/16/2010
Samuel Segun Okoya Obafemi Awolowo University 2/15/2010 - 5/15/2010
Cecilia Ortiz-Duenas University of Minnesota 9/1/2009 - 8/31/2010
Hans G. Othmer University of Minnesota 9/1/2009 - 6/30/2010
Carolyn Ann Otto Rice University 4/9/2010 - 4/9/2010
Mary Therese Padberg University of Iowa 4/9/2010 - 4/11/2010
Jason Parsley Wake Forest University 4/9/2010 - 4/9/2010
Benoit Pausader Brown University 4/19/2010 - 4/25/2010
Grigorios A. Pavliotis Imperial College London 4/10/2010 - 4/18/2010
Leonid Piterbarg University of Southern California 4/12/2010 - 4/17/2010
Juliet Portillo San Francisco State University 4/9/2010 - 4/11/2010
Candice Renee Price University of Iowa 4/8/2010 - 4/12/2010
François W. Primeau University of California, Irvine 4/11/2010 - 4/16/2010
Michelle Radtke University of Minnesota 4/10/2010 - 4/10/2010
Teresita Ramirez-Rosas Grand Valley State University 4/9/2010 - 4/10/2010
Eric J. Rawdon University of St. Thomas 4/9/2010 - 4/10/2010
Nancy Reid University of Toronto 4/21/2010 - 4/23/2010
Juan Mario Restrepo University of Arizona 8/11/2009 - 6/15/2010
Bertrand Rollin Los Alamos National Laboratory 4/10/2010 - 4/16/2010
Fadil Santosa University of Minnesota 7/1/2008 - 6/30/2011
Bonita V. Saunders National Institute of Standards and Technology 4/8/2010 - 4/10/2010
Robert Glenn Scharein San Francisco State University 4/8/2010 - 4/10/2010
Arnd Scheel University of Minnesota 9/1/2009 - 6/30/2010
Joerg Schumacher Ilmenau University of Technology 4/11/2010 - 4/17/2010
Sherry Euvette Scott Marquette University 4/10/2010 - 4/17/2010
George R Sell University of Minnesota 9/1/2009 - 6/30/2010
Tsvetanka Sendova University of Minnesota 9/1/2008 - 8/31/2010
Shuanglin Shao University of Minnesota 9/1/2009 - 8/31/2010
Tiffany Shaw New York University 4/11/2010 - 4/16/2010
Amy Shen University of Washington 4/11/2010 - 4/17/2010
Jie Shen Purdue University 4/6/2010 - 4/16/2010
Emily F. Shuckburgh British Antarctic Survey 4/11/2010 - 4/16/2010
Dianne Smith University of Iowa 4/9/2010 - 4/10/2010
Christine E. Soteros University of Saskatchewan 4/9/2010 - 4/9/2010
Edward A. Spiegel Columbia University 4/10/2010 - 4/16/2010
Daniel Spirn University of Minnesota 9/8/2009 - 6/1/2010
Panagiotis Stinis University of Minnesota 9/1/2009 - 6/30/2010
De Witt L. Sumners Florida State University 4/9/2010 - 4/12/2010
Vladimir Sverak University of Minnesota 9/1/2009 - 6/30/2010
Wenbo Tang Arizona State University 4/11/2010 - 4/16/2010
Jean-Luc Thiffeault University of Wisconsin 9/1/2009 - 6/30/2010
Becca Thomases University of California, Davis 2/9/2010 - 6/15/2010
Giordano Tierra Chica University of Sevilla 4/6/2010 - 7/21/2010
Edriss Saleh Titi University of California 3/28/2010 - 6/18/2010
Chad Michael Topaz Macalester College 9/1/2009 - 6/30/2010
Nathan Totz University of Michigan 4/18/2010 - 4/25/2010
Vy T. Tran University of St. Thomas 4/9/2010 - 4/9/2010
Rolland Trapp California State University 4/9/2010 - 4/11/2010
Yue-Kin Tsang University of California, San Diego 4/10/2010 - 4/17/2010
Alexandra Tzella École Normale Supérieure 4/9/2010 - 4/16/2010
Jacques Vanneste University of Edinburgh 4/11/2010 - 4/16/2010
Mariel Vazquez San Francisco State University 4/9/2010 - 4/9/2010
Changyou Wang University of Kentucky 9/1/2009 - 6/15/2010
Guanyu Wang University of Iowa 4/9/2010 - 4/11/2010
Xiaoming Wang Florida State University 1/5/2010 - 5/14/2010
Danielle Washburn University of Iowa 4/9/2010 - 4/11/2010
Darryn Waugh Johns Hopkins University 4/11/2010 - 4/16/2010
Jeffrey B. Weiss University of Colorado 4/11/2010 - 4/16/2010
Stuart Whittington University of Toronto 4/9/2010 - 4/9/2010
Kraig Winters University of California, San Diego 4/11/2010 - 4/17/2010
Gregory Witt University of Iowa 4/9/2010 - 4/11/2010
Celestine Woodruff Florida State University 4/11/2010 - 4/17/2010
Sijue Wu University of Michigan 9/1/2009 - 6/5/2010
Wei Xiong University of Minnesota 9/1/2008 - 8/31/2010
Jin Xu Shanghai University of Traditional Chinese Medicine 12/9/2009 - 6/9/2010
Xiang Xu Pennsylvania State University 1/13/2010 - 6/13/2010
Xiaofeng Yang University of South Carolina 4/11/2010 - 4/17/2010
Tsuyoshi Yoneda University of Minnesota 9/4/2009 - 8/31/2010
William Roy Young Scripps Research Institute 4/10/2010 - 4/16/2010
Lynn Zechiedrich Baylor College of Medicine 4/9/2010 - 4/9/2010
Zhifei Zhang Beijing (Peking) University 2/15/2010 - 5/15/2010
Weigang Zhong University of Minnesota 9/8/2008 - 8/31/2010
Laura K. Zirbel University of California, Santa Barbara 4/9/2010 - 4/12/2010
Legend: Postdoc or Industrial Postdoc Long-term Visitor

IMA Affiliates:
Arizona State University, Boeing, Corning Incorporated, ExxonMobil, Ford, General Motors, Georgia Institute of Technology, Honeywell, IBM, Indiana University, Iowa State University, Kent State University, Korea Advanced Institute of Science and Technology (KAIST), Lawrence Livermore National Laboratory, Lockheed Martin, Los Alamos National Laboratory, Medtronic, Michigan State University, Michigan Technological University, Microsoft Research, Mississippi State University, Motorola, Northern Illinois University, Ohio State University, Pennsylvania State University, Portland State University, Purdue University, Rice University, Rutgers University, Sandia National Laboratories, Schlumberger Cambridge Research, Schlumberger-Doll, Seoul National University, Siemens, Telcordia, Texas A & M University, University of Central Florida, University of Chicago, University of Delaware, University of Houston, University of Illinois at Urbana-Champaign, University of Iowa, University of Kentucky, University of Maryland, University of Michigan, University of Minnesota, University of Notre Dame, University of Pennsylvania, University of Pittsburgh, University of Tennessee, University of Wisconsin, University of Wyoming, US Air Force Research Laboratory, Wayne State University, Worcester Polytechnic Institute