## MBI Videos

### 2014 ICIAM Scientific Workshop

• Pingwen Zhang

Defects in liquid crystals (LCs) are of great practical and theoretical importance. Recently there is a growing interest in LCs materials under topological constrain and/or external force, but the defects pattern and dynamics are still poorly understood. We investigate three-dimensional spherical droplet within the Landau-de Gennes model under different boundary conditions. When the Q-tensor is uniaxial, the model degenerates to vector model (Oseen-Frank), but Q-tensor model is superior to vector model as the former allows biaxial in the order parameter. Using numerical simulation, a rich variety of defects pattern are found, and the results suggest that, line disclinations always involve biaxial, or equivalently, uniaxial only admits point defects. Then we believe that Q-tensor model is essential to include the disclinations line which is a common phenomena in LCs. The mathematical implication of this observation will be discussed in this talk.

• Akitoshi Takayasu

For an initial-boundary value problem of semilinear parabolic equations, a computational method is proposed to rigorously prove that the exact solution is enclosed in a ball. Central to our method is to use the semigroup theory with fully discretized approximations organized by Galerkin method and the backward Euler method, which is the most elementary scheme for parabolic equations. Using the scheme as it is, a step-by-step approach has been investigated. Our method is capable of verifying the weak solutions by using the Galerkin method.

It implies that the domain is allowed to be a polygonal or polyhedral domain with arbitrary shape. This work is joint with Mr. Makoto Mizuguchi, Dr. Takayuki Kubo, and Prof. Shin'ichi Oishi

• Pierangelo Marcati

The QHD system is used to describe models in superfluidity, in superconductivity, in Bose Einstein condensates and also to model carrier transport for semiconductor devices at nanoscale. Several mathematical questions need to be addressed to fully understand the various hydrodynamic quantities and their relationships with the Schroedinger and the Wigner pictures.

The mathematics need to agree with the requirement from the physics on various issues. In particular the meaning of the velocity in presence of vacuum, the presence of quantum vortices, the presence of the collision terms in semiconductors modeling.

Our approach is devised from an the analysis of the dispersive behavior of the solutions, on the factorization of the wave functions (polar decomposition) and stability of null forms.

There will be sketched recent advances on Magneto Quantum Hydridynamics and on other dispersive models like Euler Korteweg systems. Weak Strong uniqueness via relative entropy methods will also be mentioned. Finally we discuss perspectives related to the study of quantum turbulence.

• Sean Bohun

Modelling and simulation of real-life problems involve the very real dilemma of having to incorporate crucial mathematical structure that is locked within other disciples.

An example of a particularly pervasive process is the carbonate chemical system, being responsible for ocean acidification due to green house gases, limestone cave formation, degradation of monuments and even arterial blood gas makeup.

A problem from the oil and gas industry concerning the reproducibility of the characterization of carbonate rock formations forms the backdrop for an application of this chemical system.

The analysis of the resulting reaction-diffusion-advection system of equations that describe the experimental process benefits in two distinct ways: 1) it provides the form of the reaction terms and 2) it infers a natural rescaling of the chemical species. This second point results in an asymptotic analysis that is uniformly applicable across the complete chemical reactivity of the process. An explanation is offered for the variability in the observed experiments and other future problems are posed that capitalize on these new insights.

• Robert Kass

Statistical Assessment of Network Connectivity: The Case of Neural Synchrony

• Iain S. Duff
We study the preconditioning of the augmented system formulation of the least squares problem $min_x || b - A x ||^2_2$, viz. $$left[egin{array}{cc}I_m&AA^T&0end{array} ight];left[egin{array}{c}rxend{array} ight]=left[egin{array}{c}bend{array} ight],$$ 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 $P A = left[ egin{array}{c} BN end{array} ight],$ where $P$ is a permutation matrix, and we use the preconditioner $$M = left[ egin{array}{cc} I & 0 0 & B^{-T} end{array} ight]$$ to symmetrically precondition the system to obtain, after a simple block Gaussian elimination, the reduced symmetric quasi-definite (SQD) system $$egin{eqnarray*} left[ egin{array}{cc} I_{m-n} & N B^{-1} B^{-T}N^T & -I_n end{array} ight] ; left[ egin{array}{c} r_N B x end{array} ight] = left[ egin{array}{c} b_N-b_B end{array} ight] . end{eqnarray*}$$ 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.
• Gregoire Allaire

In the context of structural optimization via a level-set method we propose a framework to handle geometric constraints related to a notion of local thickness. The local thickness is calculated using the signed distance function to the shape. We formulate global constraints using integral functionals and compute their shape derivatives. We discuss different strategies and possible approximations to handle the geometric constraints. We implement our approach in two and three space dimensions for a model of linearized elasticity.

This is a joint work with F. Jouve and G. Michailidis.

• Hiroshi Kokubu

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 Mirskyâ€™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.

• Irene Fonseca

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.

• Yuan Lou

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 â€œoptimallyâ€? in heterogeneous environments? I will discuss some recent development on the evolution of dispersal, focusing on finding evolutionarily stable strategies (ESS) for dispersal.