Smart Workshop
Exploring Collaborative Mathematics

 Date:March 13-14, 2012
 Venue:
Large Lecture Hall (2F), Graduate School of Information Sciences,Tohoku University
 Organizing Committee:
Nobuaki Obata, Motoko Kotani, Keiji Miura, Michiaki Onodera,
Masaya Maeda
 Sponsored byTohoku University's Focused Research Project“Interdisciplinary Mathematics Toward Smart Innovations”

Program
March 13 (Tuesday)
10:00-10:20

Takehisa Hasegawa (Graduate School of Information Sciences, Tohoku Univ.)
Novel percolation transition in complex networks and nonamenable graphs

10:20-10:40

Keiji Miura (Graduate School of Information Sciences, Tohoku Univ.)
An unbiased estimator of noise correlations under signal drift

11:00-11:20

Hirotoshi Kuroda (Graduate School of Science, Tohoku Univ.)
The density of state of the Tomonaga-Luttinger liquid with geometricconditions
11:20-11:40

Kaname Matsue (Graduate School of Science, Tohoku Univ.)
Rigorous numerical verification of local dynamics around equilibria ofdynamics in infinite dimensions<Lunch>13:10

Michitaka Kameyama (Graduate School of Information Sciences, Dean)
Welcome Address

13:20-13:40 Masaya Maeda (Graduate School of Science, Tohoku Univ.)
Concentration of the least energy solution in a thin domain
13:40-14:00 Michiaki Onodera (Graduate School of Science, Tohoku Univ.)
A new geometric flow of surfaces and its applications
14:00-14:20

Kanako Suzuki (Graduate School of Information Sciences, Tohoku Univ.)
Patterns in activator-inhibitor systems based on the diffusion driven

instability<Short break>
14:45-15:35 Uzy Similansky (Weizmann Institute of Physical Sciences, Israel)
Oscillations and Vibrations --- The Sturm and Courant Theorems Revisited<Short break>
16:00-16:50 Yoshitaka Kimura (International Advanced Research and Education Organization,Tohoku Univ.)
Mathmatical formulars of gene-protein netowrk

March 14 (Wednesday)
Etsuo Segawa (Graduate School of Information Science and Technology, Tokyo Univ.)
Brief invitation to quantum walks

10:00-10:30<Short break>
11:00-11:20

Ryokichi Tanaka (WPI-AIMR Tohoku Univ.)
Mendel-Naor's ultrametric subsets and geometric data analysis

11:20-11:40

Natsuto Yoshinaga (WPI-AIMR Tohoku Univ.)
Self-propulsion of a drop driven by Marangoni flow

<Lunch>13:30-14:20

Hiroya Nakao (Graduate School of Information Science and Engineering,Tokyo Institute of Technology)
Spatio-temporal pattern formation in reaction-diffusion systems on complex networks

The 26th AMF Workshop
15:00-15:50

Arvind Kumar (University of Freiburg, Germany)
The critical role of striatal inhibition in shaping the oscillatory activity in thebasal ganglia

16:10-17:00 Uzy Smilansky (Weizmann Institute of Physical Sciences, Israel)
Nonlinear Schroedinger Equation on Networks

Abstracts
Novel percolation transition in complex networks and nonamenable graphs
Takehisa Hasegawa (Graduate School of Information Sciences, Tohoku Univ.)
We study phase transition of bond percolation on a nonamenable graph (NAG) and several
network models. The phase transitions of these networked systems are quite new: thesesystems exhibit the critical phase, which is never observed on the Euclidean lattices. First,
we investigate bond percolation on the enhanced binary tree (EBT), which is given by adding
intra-generation links to the binary tree, as an example of NAG. Here NAG is defined to be agraph with positive Cheeger constant. For bond percolation on a NAG, three phases appear
according to open bond probability p: (i) nonpercolating phase where there is no infinite
cluster for 0 < p < pc1, (ii) critical phase where there are infinitely many infinite clusters for
pc1 < p < pc2, and (iii) percolating phase where there is a unique infinite cluster for pc1 < p <1. We perform Monte-Carlo simulations to give a physical picture. We also consider bondpercolation on complex networks. Critical phenomena of bond percolation on complex
networks crucially depend on their network topology. In particular, some network models
under growth mechanisms show the critical phase. We investigate the phase diagram ofbond percolation on several networks via Monte-Carlo simulation and renormalization grouptechnique.

An unbiased estimator of noise correlations under signal drift
Keiji Miura (Graduate School of Information Sciences, Tohoku Univ.)
It is hard to estimate noise correlations from nonstationary multivariate time series because
the conventional correlogram is contaminated by time-dependent signals (means), which are
typically unknown. Here I derive an estimator of covariance matrix of Gaussian noises which
work whatever the time course of the signal is.

The density of state of the Tomonaga-Luttinger liquid with geometric conditions
Hirotoshi Kuroda (Graduate School of Science, Tohoku Univ.)
A Tomonaga-Luttinger liquid (TLL) is a theoretical model which describes interacting
electrons in one-dimensional case. It is well known that Fermi liquid theory breaks down for
TLL model and power-law type singularities appear in the density of state, becauseinteractions have drastic effects compared to higher dimension. In this talk we consider thismodel on a thin two-dimensional tubular surface with periodic radius modulation to study
the effects of geometric conditions.

Rigorous numerical verification of local dynamics around equilibria of dynamics
in infinite dimensions
Kaname Matsue (Graduate School of Science, Tohoku Univ.)
We briefly show a criterion for analyzing local dynamics of equilibria of dissipative PDEs. Our
method can be applied to various parabolic PDEs with computer assistance, and thisapproach will be extended to analyze further dynamical information of PDEs such as
time-periodic orbits or connecting orbits.

Concentration of the least energy solution in a thin domain
Masaya Maeda (Graduate School of Science, Tohoku Univ.)
We consider a nonlinear elliptic equation with small diffusion term in a thin domain with a
Neumann boundary condition. It is shown that the least energy solution of the elliptic
equation concentrates on a line in the domain. This is contrast to the case of ordinary
domain which the least energy solution concentrates at a single point on the boundary of thedomain.

A new geometric flow of surfaces and its applications
Michiaki Onodera (Graduate School of Science, Tohoku Univ.)
Hele-Shaw flow is an incompressible viscous fluid flow in an experimental device which
consists of two closely placed parallel plates. An interesting feature of Hele-Shaw flow is thatthe evolution of a fluid domain under the flow produced by injection of fluid does not change
its geometric moments in time, while the area increases. We will introduce a new geometricflow which has an analogous property, that is, the moments of the boundary of the domain
are preserved under the flow. Applications to other problems will also be mentioned.

Patterns in activator-inhibitor systems based on the diffusion driven instability
Kanako Suzuki (Graduate School of Information Sciences, Tohoku Univ.)
We consider non-homogeneous stationary solutions of an activator-inhibitor system which
has the diffusion-driven instability. In the system, it is assumed that the inhibitor diffusesmuch faster than the activator. It is expected that stable spatial patterns emerge, but all
non-homogeneous stationary solutions can be unstable if the activator does not diffuses. We
can see the importance of diffusion to obtain stable spatial patterns.

Oscillations and Vibrations --- The Sturm and Courant Theorems Revisited
Uzy Similansky (Weizmann Institute of Physical Sciences, Israel)

Consider the Laplace (Schrödinger) operator on a domain Ω⊂Rd. Dirichlet boundaryconditions are assumed on ∂Ω. Arrange the spectrum as a non-decreasing sequence andconsider the n'th eigenfunction f(n). The number of nodal domains νn is the number ofmaximally connected subdomains where the eigenfunction has a constant sign. For d=1
Sturm's oscillation theory states that the number of sign changes of the wave function φn=n-1 and consequently n=νn. For d>1 φn is not defined, and Courant's theorem states thatn≥νn . In the present talk I shall discuss the nodal deficiency = n-ν, and show that it contains
valuable information on the geometry of ∂Ω. In particular I shall present recent results whichshow that nodal deficiency can be derived from a new variational approach, and that it isequal to the Morse index (the number of unstable directions) of a properly defined Energyfunctional at its critical points. The generalization of Sturm's oscillation theorem will bepresented for the discrete Schrödinger operator on graphs, where the variational principle
mentioned above takes a simple form.

Mathmatical formulars of gene-protein netowrk
Yoshitaka Kimura (International Advanced Research and Education Organization, Tohoku
Univ.)
細胞内で繰り広げられる多因子系の化学反応には、核内でおきる DNA から mRNA への転写と核外の細胞質で起きる mRNA から蛋白質への変化、および、蛋白質同士の相互作用ネットワークがある。これらは、一部核内に移行し転写因子として核内の巨大蛋白を形成し遺伝子発現そのものを制御したり、細胞質で酵素反応をおこし、代謝産物を生産し、細胞構造を変化させたり細胞壁から分泌したりする。この時起きる種々の反応の時間変化の特徴は、作用主体である遺伝子、蛋白ネットワークと生成産物である代謝産物分泌のネットワークにおいて決定的に違う構造が考えられる。ここでは、これらの時間構造の違いからカオスの縁としての遺伝子、蛋白ネットワークの実態に迫る。

Brief invitation to quantum walks
Etsuo Segawa (Graduate School of Information Science and Technology, Tokyo Univ.)
Since high efficiency of quantum walks to quantum speed up algorithms was shown around2000, nowadays the quantum walk has been intensively studied by many researchers fromvarious viewpoints: for example, fundamental physics, spectral analysis on the unit circle,
stochastic behavior comparing with its corresponding random walk, the Anderson
localization, and photosynthesis as its mathematical model. In this talk, we briefly review on
the above works and two important stochastic properties of the quantum walk i.e., ballistic
spreading and localization, which are considered to ensure its nice effectiveness ofapplications.

Mendel-Naor's ultrametric subsets and geometric data analysis
Ryokichi Tanaka (WPI-AIMR, Tohoku Univ.)
Mendel and Naor show that, given the distortion bound, for any compact metric space there
exists a subset which can be embedded into an ultrametric space with “large” Hausdorff
dimension. I introduce their results and discuss with relation to data analysis.

Self-propulsion of a drop driven by Marangoni flow
Natsuto Yoshinaga (WPI-AIMR, Tohoku Univ.)
Spontaneous motion or self-propulsion have been attracting attention in last decades for its
potential application to biological problems such as cell motility and wound healing.
Recently several model experiments showing spontaneous motion have been proposed. Thesystems
in these works consist of relatively simple ingredients for instance oil drops in water
nevertheless the motion is as if the drops are alive. The key questions are why the particle
moves without external force and why it breaks symmetry and chooses one direction. The
first point has been discussed in hydrodynamics of the Marangoni effect in which a liquid
droplet is driven by a gradient of surface tension. The mechanism is that the gradientinduces convective flow inside and outside of a drop, which leads to swimming motion of thedrop itself. The second point was less discussed, but has been discussed in the field of
nonlinear dynamics as drift instability. Thus far there are only few attempts to discuss the
spontaneous symmetry breaking from hydrodynamics. In this presentation, we show thenonlinear amplitude equations exhibiting drift instability, which can be derived fromhydrodynamic and reaction-diffusion equations including the advection term.

Spatio-temporal pattern formation in reaction-diffusion systems on complex
networks
Hiroya Nakao (Graduate School of Information Science and Engineering, Tokyo Institute ofTechnology)
Spatio-temporal pattern formation in network-organized reaction-diffusion systems isinvestigated. The Turing instability in activator-inhibitor systems is a classical framework ofnon-equilibrium pattern formation in chemical and biological systems. Extending the pastwork by Othmer & Scriven (1971) on network Turing instability, we investigate Turing
patterns on large scale-free networks with strong degree heterogeneity. The instability leadsto entirely different Turing patterns from those in ordinary continuous media. No spatiallyperiodic patterns are observed, but rather the network nodes differentiate into distinct
dynamical states depending on their degrees. The resulting patterns can qualitatively beunderstood by adopting the mean-field approximation of the network and considering thebifurcation diagrams of the individual nodes driven by the mean field. Similarly, coupledlimit-cycle oscillators on scale-free networks can undergo Benjamin-Feir instability, whichleads to high dimensional chaos on the network.
[1] H.Nakao & A.S.Mikhailov, Turing patterns in network-organized activator-inhibitor systems.
Nature Phys. (2010), 544.
[2] H.Nakao & A.S.Mikhailov, Diffusion-induced instability and chaos in random oscillator
networks. Phys. Rev. E 79 (2009), 036214.

The 26th AMF Workshop
The critical role of striatal inhibition in shaping the oscillatory activity in the basal
ganglia
Arvind Kumar (University of Freiburg, Germany)
Movement disorders in Parkinson’s disease (PD) are commonly associated with slow
oscillations and increased synchrony of neuronal activity in the basal ganglia (subthalamicnucleus - STN, and globus pallidus external- GPe).
We investigated the dynamics of the basal ganglia using a reduced mean field model, which
resembles the Lotka-Volterra equations used in modeling predator-prey relations inpopulation biology. This model allowed us to isolate several biological plausible mechanismthat could induce oscillations. Specifically, I will discuss how both firing rate and correlations
among inhibitory inputs from the striatum to the GPe (i.e. increased inhibition of inhibitory
network) control the oscillations in the basal ganglia. Consistent with experimental
observations, we found that increase in either of these can unleash the oscillations in thebasal ganglia, similar to that observed in the PD.
This observation allows us to propose a unified explanation for different phenomena:absence of oscillation in the healthy state of the basal ganglia, oscillations in
dopamine-depleted state and quenching of oscillations under deep-brain-stimulation (DBS).
Finally, studying the model behavior under transient increase of activity of the striatal
neurons projecting to the indirect pathway, we are able to account for both motor
impairment in PD patients and for reduced response inhibition in DBS implanted patients.

Nonlinear Schroedinger Equation on Networks
Uzy Smilansky (Weizmann Institute of Physical Sciences, Israel)
Transmission through a complex network of nonlinear one-dimensional leads is discussed byextending the stationary scattering theory on quantum graphs to the nonlinear regime. The
resonances which dominate linear scattering are shown to be extremely sensitive to thenonlinearity and display multi-stability and hysteresis. This work provides a framework forthe study of light propagation in complex optical networks, and for studying universal
properties of Bose-Einstein Condensate (BEC) in connected chaotic traps.

東北大学における数学連携の取り組み
1) 応用数学連携フォーラム（平成 19 年 9 月～）
代表：尾畑、副代表：小谷、事務局：東北大学大学院情報科学研究科数学連携推進室
http://www.dais.is.tohoku.ac.jp/~amf/2) CREST「離散幾何学から提案する新物質創成と物性発現の解明」（平成 20～25 年度、研究代表者：小谷）
http://www.mathmate.tohoku.ac.jp/
科学技術振興機構 CREST 研究領域「数学と諸分野の協働によるブレークスルーの探索」（領域総括：北海道大学電子科学研究所教授・西浦廉政）
http://www.math.jst.go.jp/
3) 東北大学重点戦略支援プログラム「数学をコアとするスマート・イノベーション融合研究共通基盤の構築と展開」（平成 22～26 年度、代表：尾畑）
http://www.dais.is.tohoku.ac.jp/~smart4) 東北大学原子分子材料科学高等研究機構（WPI-AIMR）数学ユニット（平成 23 年度～）
http://www.wpi-aimr.tohoku.ac.jp/jp/
5) 東北大学大学院理学研究科数学専攻数学連携推進室（平成 23 年 1 月～）
6) 東北大学大学院情報科学研究科数学連携推進室（平成 23 年 1 月～）
http://www.is.tohoku.ac.jp/introduction/cmru/

応用数学連携フォーラム（AMF）へようこそ
フォーラムメンバー募集中！
数学は諸科学に共通する言語とも道具とも言われます。
数学は実にさまざま研究分野でいろいろな形で使われています。
応用数学連携フォーラムでは、数学と諸分野との連携に興味のある研究者たちが気軽に情報交換したり、研究交流したりするための場を提供しています。ちょっとのぞいてみたいという方は、随時開催しているワークショップなどはいかがでしょうか。
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