## 2014 ICIAM Scientific Workshop

### Organizers

Jose Cuminato
Applied Mathematics and Statistics, University of Sao Paulo
Maria J. Esteban
CEREMADE, CNRS & University Paris-Dauphine
Alistair Fitt
Senior Management Team, Oxford Brookes University
Barbara Keyfitz
Department of Mathematics, The Ohio State University
Taketomo Mitsui
Mathematical Science, Doshisha University
Mario Primicerio
Mathematics, Universit`a di Firenze

The occasion of the annual Board meeting of the International Council for Industrial and Applied Mathematics (ICIAM) provides a confluence of distinguished applied mathematicians from around the world.  This workshop provides a forum to exchange ideas, to review recent developments in applied mathematics, and to allow the local community of mathematical scientists to share this international perspective.

The theme of the meeting will be broad, reflecting the range of expertise of these scientists.

The workshop is hosted by the Mathematical Biosciences Institute at OSU, with additional funding provided by the Mathematics Research Institute of OSU and by the Institute for Mathematics and its Applications (University of Minnesota).

The grant from the IMA allows us to support speakers and participants from neighboring institutions in Ohio and throughout the Midwest. In particular, we would like to invite graduate students to attend.

Partial support is available for students and junior participants.  We solicit contributions for a poster session.

### Accepted Speakers

Grégoire Allaire
Applied Mathematics, Ecole Polytechnique
Weizhu Bao
Mathematics, National University of Singapore
Peter Benner
Computational Methods in Systems and Control, Max Planck Institute for Dynamics of Complex Technical Systems
Jean-Paul Berrut
Departement de Mathematiques, Universite de Fribourg
Sean Bohun
Science, University of Ontario Institute of Technology
José A. Cuminato
Applied Mathematics and Statistics, University of Sao Paulo
Iain S. Duff
SCD, Rutherford Appleton Laboratory
Alistair Fitt
Senior Management Team, Oxford Brookes University
Irene Fonseca
Department of Mathematical Sciences, Carnegie Mellon University
Ian Frigaard
Mechanical Engineering and Mathematics, UBC
Marty Golubitsky
Mathematical Biosciences Institute, The Ohio State University
Michael Günther
Applied Mathematics / Numerical Analysis, University of Wuppertal
Robert Kass
Department of Statistics, Carnegie-Mellon University
Hiroshi Kokubu
Department of Mathematics, Kyoto University
Chang-Ock Lee
Mathematical Sciences, KAIST
Yuan Lou
Department of Mathematics, The Ohio State University
Pierangelo Marcati
Information Engineering,Computer Science and Mathematics, University of LAquila
Taketomo Mitsui
Mathematical Science, Doshisha University
Helena J. Nussenzveig Lopes
Mathematics, Universidade Federal do Rio de Janeiro
Shin'ichi Oishi
Applied Mathematics, Waseda University
Tomás C. Rebollo
Differential Equations and Numerical Analysis, University of Sevilla
Lê Hùng Sơn
Applied Mathematics and Informatics, Hanoi University of Science and Technology
Daniel Thompson
Mathematics, The Ohio State University
Pingwen Zhang
School of Mathematical Sciences, Peking University
Thursday, May 15, 2014
Time Session
07:45 AM

Shuttle to MBI

08:00 AM
09:00 AM

Breakfast

09:00 AM
09:30 AM

Remarks (Peter, Marty, MBI staff)

09:30 AM
10:00 AM
Marty Golubitsky - Binocular Rivalry and Symmetry-Breaking

In binocular rivalry a subject is presented with two different images --- one to each eye. Usually, the subject perceives alternation between these two images. However, in a number of binocular rivalry experiments, subjects report perceiving surprising combinations of the two presented images. Wilson has proposed a class of neuronal networks that admit multiple competing patterns. We show that symmetry-breaking in appropriately constructed Wilson-type networks predicts the surprising perceived images in the rivalry experiments. This is joint work with Casey Diekman and Yunjiao Wang.

10:00 AM
10:30 AM
Shin'ichi Oishi
10:30 AM
11:00 AM

Poster and Coffee Break

11:00 AM
11:30 AM
Tomás C. Rebollo - Some remarks on the numerical approximation of turbulence models with wall laws

This talk deals with the numerical approximation of Large Eddy Simulation (LES) and Projection-based Variational Multi-Scale (VMS) turbulence models by the finite element method. We consider mixed boundary conditions that combine Dirichlet and non-linear wall laws. We prove convergence to the continuous targeted models. We prove density results by finite element $C^0$ spaces, for polyhedric domains, that replace the usual ones by smooth functions. We study the uniform well-posedness with respect to the discretization parameters and the asymptotic energy balance. We finally present some numerical results for 3D benchmark flows: Cavity and Turbulent Channel flow.

11:30 AM
12:00 PM
Daniel Thompson - Coding Sequence Density Estimation via Topological Pressure

I will describe an approach to coding sequence (CDS) density estimation in genomic analysis introduced recently by myself and David Koslicki. Our approach is based on the topological pressure, which is a measure of ‘weighted information content’ adapted from ergodic theory. We use the topological pressure (with suitable training data) to give ab initio predictions of CDS density on the genomes of Mus Musculus, Rhesus Macaque and Drososphilia Melanogaster. While our method is not sufficiently precise to predict, for example, the exact locations of genes, we demonstrate that our method gives reasonable estimates for the ‘coarse scale’ problem of predicting CDS density. This is joint work with David Koslicki (Oregon State).

12:00 PM
01:30 PM

Lunch Break

01:30 PM
02:00 PM
Helena J. Nussenzveig Lopes
02:00 PM
02:30 PM
Pierangelo Marcati
02:30 PM
03:00 PM
Ian Frigaard
03:00 PM
03:30 PM

Poster and Coffee Break

03:30 PM
04:00 PM
Sean Bohun
04:00 PM
04:30 PM
Jean-Paul Berrut
04:30 PM
05:00 PM
Grégoire Allaire
05:15 PM

Shuttle pick-up from MBI

06:30 PM
08:00 PM

Reception/dinner at Marty and Barbara's home. Directions will be provided.

Friday, May 16, 2014
Time Session
07:45 AM

Shuttle to MBI

08:00 AM
09:00 AM

Breakfast

09:00 AM
09:30 AM
Robert Kass
09:30 AM
10:00 AM
Iain S. Duff - Preconditioning of Least-Squares Problems by Identifying Basic Variables
We study the preconditioning of the augmented system formulation of the least squares problem $\min_x || b - A x ||^2_2$, viz. $$\left[ \begin{array}{cc} I_m & A\\ A^T & 0 \end{array} \right] \; \left[ \begin{array}{c} r\\x \end{array} \right] = \left[ \begin{array}{c} b\\0 \end{array} \right],$$ where A is a sparse matrix of order $m imes n$ with full column rank and $r$ is the residual vector equal to $b - Ax$. We split the matrix $A$ into basic and non-basic parts so that $P A = \left[ \begin{array}{c} B\\N \end{array}\right],$ where $P$ is a permutation matrix, and we use the preconditioner $$M = \left[ \begin{array}{cc} I & 0\\ 0 & B^{-T} \end{array}\right]$$ to symmetrically precondition the system to obtain, after a simple block Gaussian elimination, the reduced symmetric quasi-definite (SQD) system $$\begin{eqnarray*} \left[ \begin{array}{cc} I_{m-n} & N B^{-1}\\ B^{-T}N^T & -I_n \end{array} \right] \; \left[ \begin{array}{c} r_N\\ B x \end{array} \right] = \left[ \begin{array}{c} b_N\\-b_B \end{array} \right] . \end{eqnarray*}$$ We discuss the conditioning of the SQD system with some minor extensions to standard eigenanalysis, show the difficulties associated with choosing the basis matrix $B$, and discuss how sparse direct techniques can be used to choose it. We also comment on the common case where A is an incidence matrix and the basis can be chosen graphically.
10:00 AM
10:30 AM
Pingwen Zhang
10:30 AM
11:00 AM

Poster and Coffee Break

11:00 AM
11:30 AM
Chang-Ock Lee
11:30 AM
12:00 PM
Lê Hùng Sơn - Applications of the Initial value Problems in weather and nature catastrophe forecasts
Many problems of weather and nature catastrophe forecasts are reduced to the Initial Value Problem (IVP) of the type:$$$$\partial _t u = L\left( {t,x,u,\partial _{x_j } u} \right)$$$$ $$$$u(0, x) = {u_0}(x)$$$$ where $x = (x_{1}, \ldots., x_{n}) \in \Omega \subset \mathbb{R}^n$, $t \geq 0$ is time variable, $u = u(t, x) \in C^1$ is the unknown vector function and $L$ in (\ref{eq:1}) is a differential operator of the first order. \\ The abstracts Cauchy-Kovalevskaya theorem states that the IVP (\ref{eq:1}) and (\ref{eq:2}) is uniquely solved if the initial data ${u_0}(x)$ satisfies the supplement condition $\ell u = 0$, where $\ell$ is an elliptic differential operator and associated to the operator $L$\\ In this paper $\ell$ is defined by $$$$\ell u:=\sum\limits_{j=1}^{3}A_j\frac{\partial u}{\partial x_j},$$$$ where $$A_1= \begin{pmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \\ 0 & 0 & 0 \end{pmatrix}, A_2=\begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & -1 \end{pmatrix}, A_3=\begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix}, \dfrac{\partial u}{\partial x_j}= \begin{pmatrix} \dfrac{\partial u_1}{\partial x_j} \\ \dfrac{\partial u_2}{\partial x_j}\\ \dfrac{\partial u_3}{\partial x_j} \end{pmatrix},$$ $u = (u_1, u_2, u_3)$ is the unknown vector function.\\ \\ $L$ are the operators of following type $$$$Lu:=\sum_{k=1}^{3}B_k\frac{\partial u}{\partial x_k}+Cu+D$$$$ where $B_k=[b_{ij}^k]_{3 \times 3}$, $C = [c_{ij}]_{3 \times 3}$, $D=[d_1, d_2, d_3]^T$. The matrix elements are the continuously differentiable functions up to second order of the space-variables $x_1, x_2, x_3$ and continuously differentiable up to first order of the time variable $t$.\\ $\ell$ and $L$ are called an associated operator if $\ell u = 0$\ \Rightarrow \ $\ell(Lu) = 0$.\\ The general theorem for problem (\ref{eq:1}) and (\ref{eq:2}) states that if $L$ is associated with $\ell$ then the problem (\ref{eq:1}) and (\ref{eq:2}) is uniquely solvable with ${u_0}$ belongs to root space of $\ell u = 0$.\\ In his Ph.D. dissertation (2013) Le Cuong given the necessary and sufficient conditions so that the $L$ is associated to the operator $\ell$ Based on the results of Le Cuong we will use the scientific computing software Mathematica to build a program to find all $L$ operators of type (\ref{eq:4}) associated with $\ell$ of type (\ref{eq:3})\\ Therefore we can describe all differential operators $L$ so that the IVP (\ref{eq:1}) and (\ref{eq:2}) is uniquely solved.\\ Keywords: Initial Value Problem; Associated space; Interior estimate; Mathematica; 2000 MR Subject Classifications; 35B45; 35F10; 47H10
12:00 PM
01:00 PM

Lunch Break

01:00 PM
01:30 PM
NOORE ZAHRA
01:30 PM
02:00 PM
Hiroshi Kokubu - Detecting Morse Decompositions of the Global Attractor of Regulatory Networks by Time Series Data

Complex network structure frequently appear in biological systems such as gene regulatory networks, circadian rhythm models, signal transduction circuits, etc. As a mathematical formulation of such biological complex network systems, Fiedler, Mochizuki and their collaborators (JDDE 2013) recently defined a class of ODEs associated with a finite directed graph called a regulatory network, and proved that its dynamics on the global attractor can in principle be faithfully monitored by information from a (potentially much) fewer number of vertices of the graph called the feedback vertex set.

In this talk, I will use their theory to give a method for detecting a more detailed information on the dynamics of regulatory networks, namely the Morse decomposition of its global attractor. The main idea is to take time series data from the feedback vertex set of a regulatory network, and construct a combinatorial multi-valued map, to which we apply the so-called Conley-Morse Database method. As a test example, we study Mirsky’s mathematical model for mammalian circadian rhythm which can be represented as a regulatory network with 21 vertices. This is a joint work with B. Fielder, A. Mochizuki, G. Kurosawa, and H. Oka.

02:00 PM
02:30 PM
Irene Fonseca - Variational Methods for Crystal Surface Instability

Using the calculus of variations it is shown that important qualitative features of the equilibrium shape of a material void in a linearly elastic solid may be deduced from smoothness and convexity properties of the interfacial energy.

In addition, short time existence, uniqueness, and regularity for an anisotropic surface diffusion evolution equation with curvature regularization are proved in the context of epitaxially strained two-dimensional films. This is achieved by using the $H^{-1}$-gradient flow structure of the evolution law, via De Giorgi's minimizing movements. This seems to be the first short time existence result for a surface diffusion type geometric evolution equation in the presence of elasticity.

02:30 PM
03:00 PM
Michael Günther
03:00 PM
03:30 PM

Poster and Coffee Break

03:30 PM
04:00 PM
Weizhu Bao
04:00 PM
04:30 PM
Peter Benner - Parametric Model Order Reduction using Bilinear Systems

Model order reduction (MOR) nowadays is an important tool in simulation and control for dynamical systems arising in various engineering disciplines. Often, models of physical processes contain parameters, either describing material properties and geometry variations or arising from changing boundary conditions. For purposes of design, optimization and uncertainty quantification, it is often desirable to preserve these parameters as symbolic quantities in the reduced-order model (ROM). This allows the re-use of the ROM after changing the parameter so that the repeated computation of reduced-order models can be avoided. Significant savings in simulation times for full parameter sweeps, Monte Carlo simulations, or within optimization algorithms can be achieved this way.

In this talk, we study a particular approach for computing ROMs for linear parametric systems based on interpreting the reduced-order model as a bilinear system. This open the door to employ methods designed for MOR of bilinear systems in the context of parametric MOR. We will discuss the merits and pitfalls of using this approach as well as the MOR methods that become available via this re-formulation of the MOR problem. Numerical results illustrate the performance of all the methods under consideration.

04:30 PM
05:00 PM
Yuan Lou - ESS in Spatial Models for Evolution of Dispersal

From habitat degradation and climate change to spatial spread of invasive species, dispersal plays a central role in determining how organisms cope with a changing environment. How should organisms disperse “optimally” in heterogeneous environments? I will discuss some recent development on the evolution of dispersal, focusing on finding evolutionarily stable strategies (ESS) for dispersal.

05:00 PM

Shuttle to MBI

06:30 PM
08:00 PM

Conference Banquet at Crowne Plaza

Name Affiliation
Allaire, Gregoire smai-president@emath.fr Applied Mathematics, Ecole Polytechnique
Bao, Weizhu bao@math.nus.edu.sg Mathematics, National University of Singapore
Benner, Peter benner@mpi-magdeburg.mpg.de Computational Methods in Systems and Control, Max Planck Institute for Dynamics of Complex Technical Systems
Berrut, Jean-Paul jean-paul.berrut@unifr.ch Departement de Mathematiques, Universite de Fribourg
Bohun, Sean sean.bohun@uoit.ca Science, University of Ontario Institute of Technology
Brezzi, Franco brezzi@imati.cnr.it Science and Technology, IUSS
ChacÃ³n Rebollo, TomÃ¡s chacon@us.es Differential Equations and Numerical Analysis, University of Sevilla
Chen, Zhiming zmchen@lsec.cc.ac.cn CSCM, Acadamy of Mathematics and Systems Science
Conti, Sergio sergio.conti@uni-bonn.de Institute for Applied Mathematics, University of Bonn
Crowley, James jcrowley@siam.org EDO, SIAM
Cuminato, Jose jacumina@icmc.usp.br Applied Mathematics and Statistics, University of Sao Paulo
Damlamian, Alain damla@univ-paris12.fr Dept. of Mathematics, Universite Paris Est Creteil Val de Marne
Dawes, Adriana dawes.33@osu.edu Department of Mathematics / Department of Molecular Genetics, The Ohio State University
Duff, Iain S. iain.duff@stfc.ac.uk SCD, Rutherford Appleton Laboratory
Esteban, Maria J. esteban@ceremade.dauphine.fr CEREMADE, CNRS & University Paris-Dauphine
Fitt, Alistair afitt@brookes.ac.uk Senior Management Team, Oxford Brookes University
Fonseca, Irene fonseca@andrew.cmu.edu Department of Mathematical Sciences, Carnegie Mellon University
Frigaard, Ian frigaard@math.ubc.ca Mechanical Engineering and Mathematics, UBC
Gao, Xiaoshan xgao@mmrc.iss.ac.cn Chinese Academy of Sciences, Academy of Mathematics and Systems Science, CAS
GÃ¼nther, Michael guenther@math.uni-wuppertal.de Applied Mathematics / Numerical Analysis, University of Wuppertal
Golubitsky, Marty mg@mbi.osu.edu Mathematical Biosciences Institute, The Ohio State University
Guan, Bo guan@math.osu.edu Mathematics, Ohio State University
Hsu, Ting-Hao hsu@math.ohio-state.edu Mathematics, Ohio State University
Iyiola, Olaniyi samuel@kfupm.edu.sa Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals
Jin, Yu yjin6@unl.edu Mathematics, University of Nebraska-Lincoln
Kass, Robert kass@stat.cmu.edu Department of Statistics, Carnegie-Mellon University
Keyfitz, Barbara bkeyfitz@math.ohio-state.edu Department of Mathematics, The Ohio State University
Kokubu, Hiroshi kokubu@math.kyoto-u.ac.jp Department of Mathematics, Kyoto University
Lee, Chang-Ock colee@kaist.edu Mathematical Sciences, KAIST
Lou, Yuan lou@math.ohio-state.edu Department of Mathematics, The Ohio State University
Marcati, Pierangelo pierangelo.marcati@univaq.it Information Engineering,Computer Science and Mathematics, University of LAquila
Marini, Donatella donatella.marini@unipv.it Mathematics, Pavia University
Mitsui, Taketomo tamitsui@mail.doshisha.ac.jp Mathematical Science, Doshisha University
Nussenzveig Lopes, Helena J. hlopes@ime.unicamp.br Mathematics, Universidade Federal do Rio de Janeiro
Oishi, Shin'ichi oishi@waseda.jp Applied Mathematics, Waseda University
Park, Chunjae cjpark@konkuk.ac.kr Mathematics, Konkuk University
Phillips, Cynthia caphill@sandia.gov Analytics, Sandia National Laboratories
Pipher, Jill jpipher@math.brown.edu math, Brown University
Quy, Tong Dinh son.lehung@hust.edu.vn School of Applied Mathematics and Informatics, Hanoi University of Science and Technology
Rousseau, Christiane rousseac@DMS.UMontreal.CA Mathematics and Statistics, University of Montreal
Sloan, Ian i.sloan@unsw.edu.au Applied Mathematics, The University of New South Wales
Son, Le Hung son.lehung@hust.edu.vn Applied Mathematics and Informatics, Hanoi University of Science and Technology
Thompson, Daniel thompson@math.osu.edu Mathematics, The Ohio State University
Xu, Xuejun xxj@lsec.cc.ac.cn
Xue, Chuan cxue@mbi.osu.edu Mathematics, The Ohio State University
Yan, Guiying yangy@amss.ac.cn
Ying, Hao ying.32@osu.edu Mathematics, The Ohio State University
ZAHRA, NOORE noor_zahra_india@yahoo.co.in School of Engineering and Technology, Sharda University
Zhang, Pingwen pzhang@pku.edu.cn School of Mathematical Sciences, Peking University
Parametric Model Order Reduction using Bilinear Systems

Model order reduction (MOR) nowadays is an important tool in simulation and control for dynamical systems arising in various engineering disciplines. Often, models of physical processes contain parameters, either describing material properties and geometry variations or arising from changing boundary conditions. For purposes of design, optimization and uncertainty quantification, it is often desirable to preserve these parameters as symbolic quantities in the reduced-order model (ROM). This allows the re-use of the ROM after changing the parameter so that the repeated computation of reduced-order models can be avoided. Significant savings in simulation times for full parameter sweeps, Monte Carlo simulations, or within optimization algorithms can be achieved this way.

In this talk, we study a particular approach for computing ROMs for linear parametric systems based on interpreting the reduced-order model as a bilinear system. This open the door to employ methods designed for MOR of bilinear systems in the context of parametric MOR. We will discuss the merits and pitfalls of using this approach as well as the MOR methods that become available via this re-formulation of the MOR problem. Numerical results illustrate the performance of all the methods under consideration.

Some remarks on the numerical approximation of turbulence models with wall laws

This talk deals with the numerical approximation of Large Eddy Simulation (LES) and Projection-based Variational Multi-Scale (VMS) turbulence models by the finite element method. We consider mixed boundary conditions that combine Dirichlet and non-linear wall laws. We prove convergence to the continuous targeted models. We prove density results by finite element $C^0$ spaces, for polyhedric domains, that replace the usual ones by smooth functions. We study the uniform well-posedness with respect to the discretization parameters and the asymptotic energy balance. We finally present some numerical results for 3D benchmark flows: Cavity and Turbulent Channel flow.

Preconditioning of Least-Squares Problems by Identifying Basic Variables
We study the preconditioning of the augmented system formulation of the least squares problem $\min_x || b - A x ||^2_2$, viz. $$\left[ \begin{array}{cc} I_m & A\\ A^T & 0 \end{array} \right] \; \left[ \begin{array}{c} r\\x \end{array} \right] = \left[ \begin{array}{c} b\\0 \end{array} \right],$$ where A is a sparse matrix of order $m imes n$ with full column rank and $r$ is the residual vector equal to $b - Ax$. We split the matrix $A$ into basic and non-basic parts so that $P A = \left[ \begin{array}{c} B\\N \end{array}\right],$ where $P$ is a permutation matrix, and we use the preconditioner $$M = \left[ \begin{array}{cc} I & 0\\ 0 & B^{-T} \end{array}\right]$$ to symmetrically precondition the system to obtain, after a simple block Gaussian elimination, the reduced symmetric quasi-definite (SQD) system $$\begin{eqnarray*} \left[ \begin{array}{cc} I_{m-n} & N B^{-1}\\ B^{-T}N^T & -I_n \end{array} \right] \; \left[ \begin{array}{c} r_N\\ B x \end{array} \right] = \left[ \begin{array}{c} b_N\\-b_B \end{array} \right] . \end{eqnarray*}$$ We discuss the conditioning of the SQD system with some minor extensions to standard eigenanalysis, show the difficulties associated with choosing the basis matrix $B$, and discuss how sparse direct techniques can be used to choose it. We also comment on the common case where A is an incidence matrix and the basis can be chosen graphically.
Variational Methods for Crystal Surface Instability

Using the calculus of variations it is shown that important qualitative features of the equilibrium shape of a material void in a linearly elastic solid may be deduced from smoothness and convexity properties of the interfacial energy.

In addition, short time existence, uniqueness, and regularity for an anisotropic surface diffusion evolution equation with curvature regularization are proved in the context of epitaxially strained two-dimensional films. This is achieved by using the $H^{-1}$-gradient flow structure of the evolution law, via De Giorgi's minimizing movements. This seems to be the first short time existence result for a surface diffusion type geometric evolution equation in the presence of elasticity.

Binocular Rivalry and Symmetry-Breaking

In binocular rivalry a subject is presented with two different images --- one to each eye. Usually, the subject perceives alternation between these two images. However, in a number of binocular rivalry experiments, subjects report perceiving surprising combinations of the two presented images. Wilson has proposed a class of neuronal networks that admit multiple competing patterns. We show that symmetry-breaking in appropriately constructed Wilson-type networks predicts the surprising perceived images in the rivalry experiments. This is joint work with Casey Diekman and Yunjiao Wang.

Detecting Morse Decompositions of the Global Attractor of Regulatory Networks by Time Series Data

Complex network structure frequently appear in biological systems such as gene regulatory networks, circadian rhythm models, signal transduction circuits, etc. As a mathematical formulation of such biological complex network systems, Fiedler, Mochizuki and their collaborators (JDDE 2013) recently defined a class of ODEs associated with a finite directed graph called a regulatory network, and proved that its dynamics on the global attractor can in principle be faithfully monitored by information from a (potentially much) fewer number of vertices of the graph called the feedback vertex set.

In this talk, I will use their theory to give a method for detecting a more detailed information on the dynamics of regulatory networks, namely the Morse decomposition of its global attractor. The main idea is to take time series data from the feedback vertex set of a regulatory network, and construct a combinatorial multi-valued map, to which we apply the so-called Conley-Morse Database method. As a test example, we study Mirsky’s mathematical model for mammalian circadian rhythm which can be represented as a regulatory network with 21 vertices. This is a joint work with B. Fielder, A. Mochizuki, G. Kurosawa, and H. Oka.

ESS in Spatial Models for Evolution of Dispersal

From habitat degradation and climate change to spatial spread of invasive species, dispersal plays a central role in determining how organisms cope with a changing environment. How should organisms disperse “optimally” in heterogeneous environments? I will discuss some recent development on the evolution of dispersal, focusing on finding evolutionarily stable strategies (ESS) for dispersal.

Applications of the Initial value Problems in weather and nature catastrophe forecasts
Many problems of weather and nature catastrophe forecasts are reduced to the Initial Value Problem (IVP) of the type:$$$$\partial _t u = L\left( {t,x,u,\partial _{x_j } u} \right)$$$$ $$$$u(0, x) = {u_0}(x)$$$$ where $x = (x_{1}, \ldots., x_{n}) \in \Omega \subset \mathbb{R}^n$, $t \geq 0$ is time variable, $u = u(t, x) \in C^1$ is the unknown vector function and $L$ in (\ref{eq:1}) is a differential operator of the first order. \\ The abstracts Cauchy-Kovalevskaya theorem states that the IVP (\ref{eq:1}) and (\ref{eq:2}) is uniquely solved if the initial data ${u_0}(x)$ satisfies the supplement condition $\ell u = 0$, where $\ell$ is an elliptic differential operator and associated to the operator $L$\\ In this paper $\ell$ is defined by $$$$\ell u:=\sum\limits_{j=1}^{3}A_j\frac{\partial u}{\partial x_j},$$$$ where $$A_1= \begin{pmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \\ 0 & 0 & 0 \end{pmatrix}, A_2=\begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & -1 \end{pmatrix}, A_3=\begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix}, \dfrac{\partial u}{\partial x_j}= \begin{pmatrix} \dfrac{\partial u_1}{\partial x_j} \\ \dfrac{\partial u_2}{\partial x_j}\\ \dfrac{\partial u_3}{\partial x_j} \end{pmatrix},$$ $u = (u_1, u_2, u_3)$ is the unknown vector function.\\ \\ $L$ are the operators of following type $$$$Lu:=\sum_{k=1}^{3}B_k\frac{\partial u}{\partial x_k}+Cu+D$$$$ where $B_k=[b_{ij}^k]_{3 \times 3}$, $C = [c_{ij}]_{3 \times 3}$, $D=[d_1, d_2, d_3]^T$. The matrix elements are the continuously differentiable functions up to second order of the space-variables $x_1, x_2, x_3$ and continuously differentiable up to first order of the time variable $t$.\\ $\ell$ and $L$ are called an associated operator if $\ell u = 0$\ \Rightarrow \ $\ell(Lu) = 0$.\\ The general theorem for problem (\ref{eq:1}) and (\ref{eq:2}) states that if $L$ is associated with $\ell$ then the problem (\ref{eq:1}) and (\ref{eq:2}) is uniquely solvable with ${u_0}$ belongs to root space of $\ell u = 0$.\\ In his Ph.D. dissertation (2013) Le Cuong given the necessary and sufficient conditions so that the $L$ is associated to the operator $\ell$ Based on the results of Le Cuong we will use the scientific computing software Mathematica to build a program to find all $L$ operators of type (\ref{eq:4}) associated with $\ell$ of type (\ref{eq:3})\\ Therefore we can describe all differential operators $L$ so that the IVP (\ref{eq:1}) and (\ref{eq:2}) is uniquely solved.\\ Keywords: Initial Value Problem; Associated space; Interior estimate; Mathematica; 2000 MR Subject Classifications; 35B45; 35F10; 47H10
Coding Sequence Density Estimation via Topological Pressure

I will describe an approach to coding sequence (CDS) density estimation in genomic analysis introduced recently by myself and David Koslicki. Our approach is based on the topological pressure, which is a measure of ‘weighted information content’ adapted from ergodic theory. We use the topological pressure (with suitable training data) to give ab initio predictions of CDS density on the genomes of Mus Musculus, Rhesus Macaque and Drososphilia Melanogaster. While our method is not sufficiently precise to predict, for example, the exact locations of genes, we demonstrate that our method gives reasonable estimates for the ‘coarse scale’ problem of predicting CDS density. This is joint work with David Koslicki (Oregon State).