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The 10th Japan-Taiwan Joint Workshop for Young Scholars in Applied Mathematics Program & Abstract

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The 10th Japan-Taiwan Joint Workshop for Young Scholars

in Applied Mathematics

Program & Abstract

February 27 – March 1, 2019 at Ryukoku University

Lecture Room 101, Building 8, Seta Campus

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February 27 (Wednesday)

8:50–9:00 Opening

9:00-10:00 Session 1 Chair: Shih-Hsin Chen (National Taiwan University) 9:00–9:15 Yusaku Shimoji (Meiji University, M2)

9:15–9:30 Po-Yu Lin (National Taiwan University, M2)

9:30–9:45 Ching-Lun Lu (National Cheng Kung University, M2) 9:45–10:00 Kana Mizuno (Shimane University, M2)

10:10-11:10 Session 2 Chair: Shun-Chieh Wang (National Taiwan University) 10:10–10:25 Yuka Fukase (Meiji University, M2)

10:25–10:40 Takanori Kawase (Ryukoku University, M2) 10:40–10:55 Kazuki Ikeda (Meiji University, M2)

10:55–11:10 Yi-Ting Chen (National Chiao Tung University, M2)

11:20-12:20 Session 3 Chair: Wei-Chen Chang (National Taiwan University) 11:20–11:35 Shota Yamakawa (Ryukoku University, M2)

11:35–11:50 Yuta Shimazaki (Meiji University, M2)

11:50–12:05 Po-Hsin Cheng (National Cheng Kung University, M2) 12:05–12:20 Hiroto Kawata (Meiji University, M1)

12:20–12:35 Yuma Watabe (Hiroshima University, M1) 12:35–12:45 Group photo

12:45–13:40 Lunch

13:40-14:55 Session 4 Chair: Takahiro Hiraga (Hiroshima University) 13:40–13:55 Chengen Lee (National Taiwan University, M2)

13:55–14:10 Liu Yang (Northeast Normal University / Shimane University, M2) 14:10–14:25 Yi-Juan Du (National Chiao Tung University, M1)

14:25–14:40 Hiroaki Ishiyama (Ryukoku University, M2)

14:40–14:55 Le Nguyen Thuy Van (National Central University, M1) 15:15-16:30 Session 5 Chair: Lorenzo Contento (Meiji University)

15:15–15:30 Wataru Makita (Ryukoku University, M1)

15:30–15:45 Kuan-Jhan Lin (National Taiwan University, M1)

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15:45–16:00 Daichi Inoue (Meiji University, M1)

16:00–16:15 Fei-Hung Huang (National University of Tainan, M1) 16:15–16:30 Naoto Ichikawa (Shimane University, M1)

16:50-18:00 Session 6 Chair: Chueh-Hsin Chang (Tunghai University) 16:50–17:00 Hiromu Gion (Shimane University, B4)

17:00–17:10 Yu-Kai Lin (National Central University, B3) 17:10–17:20 Miu Nodagashira (Ryukoku University, B4)

17:20–17:30 Pei-Shan Fang (National Cheng Kung University, B4) 17:30–17:40 Kaito Fujihara (Shimane University, B4)

17:40–17:50 Ming-Hsiu Lu (National Central University, B4) 17:50–18:00 Ryu Fujiwara (Meiji University, B4)

18:10–18:40 Parallel discussion

February 28 (Thursday)

8:50–9:00 Announcements

9:00-10:00 Session 7 Chair: Yung-Hsiang Huang (National Taiwan University) 9:00–9:15 Riku Kanai (Meiji University, M2)

9:15–9:30 Hiroki Komoto (Ryukoku University, M2)

9:30–9:45 Bing-Ze Lu (National Cheng Kung University, M2) 9:45–10:00 Yuki Yagasaki (Meiji University, M2)

10:10-11:10 Session 8 Chair: Shunsuke Kobayashi (Meiji University) 10:10–10:25 Wen-Hao Yang (National Taiwan University, M2)

10:25–10:40 Yu-Yu Weng (National Tsing Hua University, M1) 10:40–10:55 Akira Machishaku (Ryukoku University, M2) 10:55–11:10 Mana Futamura (Meiji University, M1)

11:20-12:20 Session 9 Chair: Huai-Hua Lu (National Taiwan University) 11:20–11:35 Eigo Fukada (Shimane University, M1)

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11:50–12:05 Tomohiro Nakahara (Hiroshima University, M1) 12:05–12:20 Yuyuan Yuan (National Taiwan University, M1) 12:30–13:30 Lunch

13:30-14:30 Session 10 Chair: Ryo Ito (Meiji University) 13:30–13:50 Wei-Chen Chang (National Taiwan University, D4) 13:50–14:10 Shunsuke Kobayashi (Meiji University, D2)

14:10–14:30 Huai-Hua Lu (National Taiwan University, D4)

14:50-16:10 Session 11 Chair: Yan-Yu Chen (Tamkang University) 14:50–15:10 Yung-Hsiang Huang (National Taiwan University, D4) 15:10–15:30 Takahiro Hiraga (Hiroshima University, D1)

15:30–15:50 Shih-Hsin Chen (National Taiwan University, D4) 15:50–16:10 Shun-Chieh Wang (National Taiwan University, D1)

16:30-17:10 Session 12 Chair: Yasufumi Yamada (Hiroshima University) 16:30–16:50 Ryo Ito (Meiji University, PD)

16:50–17:10 Lorenzo Contento (Meiji University, PD) 18:00–20:00 Awards Ceremony and Party

March 1st (Friday)

9:30–18:00 Free Discussion

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This workshop is sponsored by a project of Ryukoku Joint Research Center for Sci- ence and Technology, “Mathematical studies on emergence of localized patterns, propagations, cross-diffusion and non-local effect” (S. Yotsutani), and partially sup- ported by Meiji University, Ryukoku Center for Mathematical Sciences, Mathemati- cal Society of the Republic of China, Taiwan (TMS), National Center for Theoretical Science, Taiwan (NCTS), Ministry of Science and Technology, Taiwan (MOST), JST Presto,”Multiphase shape optimization in phononic crystal design ” (E. Ginder), JSPS KAKENHI Grant-in-Aid for Scientific Research (B) Grant Number 16KT0022 (H. Ninomiya), 18H01139 (Y. Morita), Grant-in-Aid for Scientific Research (C) Grant Number 15K04972 (S. Yotsutani), Grant-in-Aid for Young Scientists (B) Grant Number 16K17629 (T. Kawakami), Grant-in-Aid for challenging Exploratory Research Grant Number 16K13778 (H. Ninomiya).

Organizers:

Tatsuki Kawakami (Ryukoku University) Yoshihisa Morita (Ryukoku University) Shoji Yotsutani (Ryukoku University) Kota Ikeda (Meiji University)

Hirokazu Ninomiya (Meiji University) Yuichi Togashi (Hiroshima University)

Chiun-Chuan Chen (National Taiwan University, NTU) Jann-Long Chern (National Central University, NCU) Yung-Fu Fang (National Cheng Kung University, NCKU) Yu-Chen Shu (National Cheng Kung University, NCKU) Yu-Yu Liu (National Cheng Kung University, NCKU) Dong-Ho Tsai (National Tsing Hua University, NTHU) Ming-Chih Lai (National Chiao Tung University, NCTU) Chang-Hong Wu (National University of Tainan, NUTN) Chueh-Hsin Chang (Tunghai University, THU)

Yan-Yu Chen (Tamkang University, TKU)

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Numerical computation of magnetic fluid instabilities in a Hele-Shaw cell with time-dependent gap by the method of

fundamental solutions

Yusaku Shimoji Meiji University

E-mail-shimocchi.0739@gmail.com

The purpose of my talk is to track the boundary of magnetic fluid in the Hele- Shaw cell numerically. The model equations which describe the motion of magnetic fluid under the condition of external energy supply due to the upper plate of the Hele-Shaw cell in the presence of a normal magnetic field is formulated as follows:



















△p = 12ηht(t)

h(t)3 in Ω(t), p = σk− 3η ˙h(t)

h(t)3∥x∥2 on ∂Ω(t), V =−h(t)2

12η (

∇p − 2M h(t)∇φm

)

· n on ∂Ω(t).

Here p is an unknown pressure function, h(t) is a time-dependent gap between two parallel plates in the z-direction and φm is the value of magnetostatic field potential created by the magnetic fluid on the boundary of the Hele-Shaw cell. We apply the method of fundamental solutions (MFS) to simulate the magnetic fluid instabilities, since the MFS is known as the fast and mesh-free numerical solver for potential problems such as Laplace equations, Poisson equations, etc.

References

[1] K. Sakakibara & S. Yazaki, A charge simulation method for the computation of Hele-Shaw problems, RIMS Kˆokyˆuroku 1957, (2015) 116-133

[2] S. Yazaki, A numerical scheme for the Hele-Shaw flow with a time-dependent gap by a curvature adjusted method, Advanced Studies in Pure Mathematics 64, (2015) 253-261

[3] A. Tatulchenkov & A. Cebers, Magnetic fluid labyrinthine instability in Hele- Shaw cell with time dependent gap, Physics of fluids 20, (2008)

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On Numerical Experiments for Extended Kalman Filtering

Po-Hsin Cheng

National Cheng Kung University E-mail: wilhelmzimmermann1942@gmail.com

In this talk, numerical simulations of Kalman filtering for two given examples illustrate that signals cannot be tracking provided the target system is undetectable.

This is the main drawback for applying the extended Kalman filtering to a nonlinear problem. To overcome the drawback, a regularization strategy based on dimension lifting is developed. Numerical experiments conclude the efficiency of the regular- ization strategy.

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Isogeometric analysis and splines

Ching-Lun Lu

National Cheng Kung University E-mail: l16074075@ncku.edu.tw

This is my study on the CAD(Computer aide design) and FEM(Finite element method). I will introduce the relation between computer graphic and CAD. The important part is NURBS(Non-Uniform Rational Bezier Splines), which is used widely in CAD and computer graphics.

Spline is a function defined by piecewise polynomials. I study some linear prob- lems using NURBS, and it is the concept of isogeometric analysis.

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A Model for Dynamic Pattern Formation in Cuttlefish

Kana Mizuno (1), Eigo Fukada (2), Mayuko Iwamoto (3)

1: Interdisciplinary Graduate School of Science and Engineering, Shimane University;

2: Graduate School of Natural Science and Technology, Shimane University;

3: Interdisciplinary Faculty of Science and Engineering, Shimane University;

s179413@matsu.shimane-u.ac.jp

Body pattern of cephalopods changes instantaneously [1], and the time scale is fast compared with Turing pattern formation. A purpose of this study is to understand the mechanism of pattern formation in cuttlefish. By capturing the characteristics of cuttlefish’s skin structure [2], we build a mathematical model which is described by self-driven springs and dampers with FitzHugh-Nagumo model. In this presentation, we will show the simulation results of the non-uniform patterns with our model.

References

[1] R. Nakajima, Y. Ikeda, A catalog of the chromatic, postural, and locomotor be- haviors of the pharaoh cuttlefish (Sepia pharaonis) from Okinawa Island, Japan, Marine Biodiversity Volume 47(2017), 735-753.

[2] L. F. Deravi, et al., The structure-function relationships of a natural nanoscale photonic device in cuttlefish chromatophores, J. R. Soc. Interface Volume 11(2014), 20130942.

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Travelling Waves in A Rabies Propagation Model

Yuka Fukase Meiji University cs171010@meiji.ac.jp

Rabies is an infectious viral disease transmitted from animals to humans. If a bite from a rabid animal goes untreated and rabies develops, it is almost always fatal.

Today, this disease occurs in more than 150 countries and territories. Therefore, it is important to study how the distribution of the disease propagates. Macdonald [1] studied the spread of rabies in the northeastern part of France in the 1970’s, according to that regions of an epidemic phase and a silent phase appear alternately in the process of spreading the disease. As one of mathematical models expressing such propagation of rabies, we introduce a two-component reaction-diffusion system motivated by Murray et al. [2]. This model constructed by a density of susceptible S(x, t) and the one of infectious I(x, t) to explain the propagation of rabies, where S and I are functions of space x and time t. Then, our model is as follows:







St = ε (

1 S K

)

S− βSI,

It = dIxx+ βSI − γI.

Here, ε is the birth rate, β is the contact rate, γ is the removal rate, K is a car- rying capacity and d is a diffusion coefficient, all being positive constants. In this presentation, we will discus the travelling wave solution of this model.

References

[1] Macdonald, D. W. Rabies and Wildlife A biologist’s perspective., Oxford Univer- sity Press (1980).

[2] Murray, J. D., Stanley, E. A. and Brown, D. L. (1986). On the Spatial Spread of Rabies among Foxes. The Royal Society, B 229, 111-150

[3] Ai, S., Du, Y. and Peng, R. (2017). Traveling waves for a generalized Holling- Tanner predator-prey model. J. Differential Equations, 263 (11), 7782-7814

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A new method in ecology:

Estimating a relationship by Time series data

Takanori Kawase (1), Kazutaka Kawatsu (2), Daisuke Kyogoku (2), Michio Kondoh (2) 1: Ryukoku University; 2: Tohoku University

t17m079@mail.ryukoku.ac.jp Introduction.

In order to understand the structure of ecosystem, interspecific interactions should be in- vestigated through the methods in population ecology. However, interspecific interaction changes its direction and strength along with time. In addition, interspecific interactions defined based on individual-level behavioral observation may be not always a driver of population dynamics. Thus, it is difficult to quantify the interaction for population ecol- ogy. Recently, a statistical method, Empirical Dynamic Modeling (EDM), which is based on theory of non-linear dynamical systems, has been developed for estimating temporally- changing interactions from time-series data. We used one of the EDM methods, Con- vergent Cross Mapping (CCM), which can estimate causal-effect relationships between potentially-interacting multiple time series, by distinguishing causality from correlation.

In addition, it can estimate the timing when the causality happened. We used this method for evaluating whether behaviorally-defined interactions (i.e., resource competition and re- productive interference) really the cause of population dynamics.

Method.

We experimentally co-grew two species bean weevils (Callosobruchus maculatus [C.m here- after] and Callosobruhus chinensis [C.c hereafter]) and evaluated whether two types of in- terspecific interactions were the causes of their population dynamics from the time series data. There are two interspecific interactions occurring at the different life stage of the weevils: (1) resource competition in the larval stage and (2) reproductive interference in the adult stage. In addition, the interference as the driver of population is reported from previous observation reports that C.c affects C.m only. The estimated causality can be discuss based on its lifecycle. We counted the number of living adults of each species every week to generate time series data. In order to detect the causality for different life stages, we used CCM to the time series with no time delay to three weeks delay.

Result and Discussion.

Competitive exclusion occurred in the experiment; C.m excluded C.c CCM analysis de- tected bio-directional causality between C.m population and C.c population, implying that population dynamics of these species influences each other. Likewise, these causalities were estimated in every timing of the life stage. Based on the timing of life stages with specific time delay (no delay to three weeks delay), our results suggest that (1) the reproductive interference as the driver of population can occur from C.m to C.c. and that (2) resource competition is contest type in both species. Although the behavioral observation cannot argue how interspecific interactions affect population dynamics, the application of EDM enabled us to detect the causality of interspecific interactions.

References

[1] K. Kawatsu and S. Kishi, Identifying critical interactions in complex competition dynamics between bean beetles, Oikos, 127 (4), 553–560 (2018).

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The propagation of reaction-diffusion equation in growing region

Kazuki Ikeda

Graduate of Advanced Mathematical Sciences, Meiji University cs171007@meiji.ac.jp

Reaction-diffusion equations are usually considered in a fixed domain. However, the domain often varies or grows in biological phenomena, such as animal coat patterns and phyllotactic patterns in shoot apex. In this talk we study distributions of chemical substances that are diffusing and reacting with one another in a growing domain and are investigate the effect of the growth of domain. Especially, we treat the heat equation and the Fisher-KPP equation in a growing domain.

References

[1] D. G. Aronson and Weunberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. Math. 30(1978), 949-1032.

[2] A. Kolmogorov, I. Petrovskii, and N. Piscounov. A study of the diffusion equation with increase in the amount of substance, and its application to a biological problem. In V. M. Tikhomirov, editor, Selected Works of A. N. Kolmogorov I, pages 248-270. Kluwer 1991

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Spectral methods for modified Poisson-Boltzmann equation on different geometries

Yi-Ting Chen

National Chiao Tung University E-mail: s956816@gmail.com

To investigate the structure of the electrical double layer (EDL) in electrolyte solutions, we visit modified Poisson-Boltzmann (MPB) equation over different ge- ometries, and verify a theoretical prediction numerically by using Chebyshev and Fourier spectral method. The advantage of our approach is that the grid points are clustered close to the domain boundary so that we can capture the behavior of the boundary layer accurately.

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Solution structure of stationary solutions of a limiting SKT cross-diffusion equation

Shota Yamakawa (1)

1: Graduate School of Science and Technology Ryukoku University T17M008@mail.ryukoku.ac.jp

We consider the following stationary limiting equation

(S1 )













1 0

τ v

(

a1− b1

τ v − c1v

)

dx = 0 in (0, 1), d2vxx+ v

(

a2− b2

τ v − c2v

)

= 0 in (0, 1), vx(0) = 0, vx(1) = 0,

v(x) > 0, vx(x) > 0 in (0, 1)

for a stationary cross-diffusion equation. Here, v(x) is an unknown function, and τ is an unknown constant. The constants d2, ai, bi, ci, (i = 1, 2) are all positive. We remark that the important quantities involving the constants ai, bi, ci, (i = 1, 2),are A := a1/a2, B := b1/b2, C := c1/c2. It seems natural to consider the following two cases separately: ”weak competition” case C ≤ B and ”strong competition” case B < C.

Shigesada-Kawasaki-Teramoto [1] proposed the original cross-diffusion equation.

Lou-Ni [2, 3] derived the stationary limiting equation (S1 ) to investigate the ef- fect of cross-diffusion. Lou-Ni-Yotsutani [4] showed theorems about existence, non- existence of non-constant steady state solutions, the shape of the solution, and clarified the structure of solutions of (S1 ) for all cases C ≤ B and B < C.

We have precisely investigated the solution structure and the stability in the case B < C . In this talk, we report our numerical results on the solution structure and the stability of (S1 ) for the case C ≤ B.

This is a joint work with Y.Lou, T.Mori, W.-M.Ni and S.Yotsutani.

References

[1] N.Shigesada, K.Kawasaki, and E.Teramoto, Spatial segregation of interacting species, J.Theoret Biology 79 (1979), 83-99.

[2] Y.Lou and W.-M.Ni, Diffusion, self-diffusion and cross-diffusion, J.Differential Equations 131 (1996), 79-131.

[3] Y.Lou and W.-M.Ni, Diffusion vs cross-diffusion: An elliptic approach, J.Differential Equations 154 (1999), 159-190.

[4] Y.Lou, W.-M.Ni and S.Yotsutani, On a limiting system in the Lotka-Volterra competision with cross-diffusion, Discrete Contin. Dyn. Syst. A 10 (2004), 435- 458.

[5] T.Mori, T.Suzuki and S.Yotsutani, Numerical Approach to Existence and Sta- bility of Stationary Solutions to a SKT Cross-diffusion Equation, Mathematical Models and Methods in Applied Sciences,28(2018), 2191-2210.

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The Analysis of Economic Structure in September 2008

Yuta Shimazaki, Kazuyuki Nakamura Meiji University

cs171016@meiji.ac.jp

Several financial crises have broken out in these 20 years. Though they have large impact on our society, it is impossible to prevent damage from them because there are many factors to cause the financial crises. Therefore, financial crises are major problem and are treated by various approaches to solve the problem.

In this research, we analyze the exchange rates to clarify the structure of the financial crises. Used exchange rates are USD/JPY, EUR/USD, USD/CHF, and GBP/USD from September 1 to September 30 in 2008, where bankruptcy of Lehman Brothers happened on September 15. We divide September into 4 period and min- imum time interval is one hour. We use two analyzing methods, regime switching model [1] and relative noise contribution [2].

As the results, the stability of all exchange rates was decided by own character- istics in all periods. For example, Table 1 shows the indices how much an exchange rate is affected by other exchange rates in September 29. The indices are the aver- age of parameters from regime switching model by other exchange rates. The result shows regime change is not determined by other exchange rates, that is, interactions among exchange rate do not relate to stability. However, there is a relationship be- tween affairs and stability. Therefore, the factors affecting stability of the exchange rates seems to be external factors like government’s economic policy. These investi- gations imply that traders consider external factor to be more important than other rates when financial crises happens.

Rates

Explanatory variables

Intersept USD/JPY EUR/USD GBP/USD USD/CHF

USD/JPY 109.992 18.807 -15.907 3.926

EUR/USD 2.695 -0.073 -0.0085 -0.243

GBP/USD 2.033 -0.053 0.869 -0.551

USD/CHF 451.056 -10.861 -68.948 3.882

Table 1: The indices of other rates effects in September 29

References

[1] F. X. Diebold, J.-H. Lee, and G. C. Weinbach, “Regime Switching with Time- Varying Transition Probabilities”, in Nonstationary Time Series Analysis and Cointegration, Oxford University Press, pp. 283-302 1994.

[2] G. Kitagawa: Introduction to Time Series Modeling, CRC Press, 2010.

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On Numerical Experiments for Extended Kalman Filtering

Po-Hsin Cheng

National Cheng Kung University E-mail: wilhelmzimmermann1942@gmail.com

In this talk, numerical simulations of Kalman filtering for two given examples illustrate that signals cannot be tracking provided the target system is undetectable.

This is the main drawback for applying the extended Kalman filtering to a nonlinear problem. To overcome the drawback, a regularization strategy based on dimension lifting is developed. Numerical experiments conclude the efficiency of the regular- ization strategy.

(17)

Automatic drum transcription by RNN considering phase information

Hiroto Kawata (1), Takeshi Hori (1), Kazuyuki Nakamura (1) 1: Graduate School of Advanced Mathematical Science, Meiji University

{cs181004, thori, knaka}@meiji.ac.jp

Automatic drum transcription (ADT) is the task of retrieving symbolic repre- sentation of drum instruments from monaural drum solo recordings. In recent years, recurrent neural network (RNN) based approach have achieved the highest evalua- tion accuracies and various architectures have been discussed by many researchers.

In the conventional RNN implementations, power spectrum in the frequency do- main have been used as the input feature. However, for the purpose of improving the evaluation accuracy, we propose a new RNN architecture with phase informa- tion added to the input layer. As a result of the evaluation experiment, by adding the phase information to the input, the accuracy of estimating activations of the drum instruments considerably increased and the effectiveness of phase information is demonstrated.

Figure 1: Our RNN architecture using power and phase spectrum of hi-hat (HH), snare drum (SD) and kick drum (KD) as input and estimating their activations.

Previous Proposed HH 0.304 0.205 SD 0.130 0.0676 KD 0.107 0.0360

Figure 2: Comparison of RMSE.

References

[1] Richard Vogl, Matthias Dorfer, and Peter Knees, “Recurrent neural networks for drum transcription”, Proc 17th ISMIR. (2016).

[2] Juan P.Bello, Chris Duxbury, Mike Davies, and Mark Sandler, “On the Use of Phase and Energy for Musical Onset Detection in the Complex Domain”, IEEE Signal Procrssing Letters, Vol.11, No. 6. (2004).

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Practical investigation of acoustic navigation employed by bats

Yuma Watabe, Yasufumi Yamada, Ryo Kobayashi

Department of Mathematical and Life Science, Hiroshima University m184493@hiroshima-u.ac.jp

Bats can avoid obstacles and other individuals using a ultrasonic sensing (echoloca- tion). Their sensing system is a very simple design which consists of 1 transmitter, 2 receivers and small brain. Although, it is not well understood how do bats re- alize advanced navigation with such a simple sensing system. In a previous study, navigation strategies employed by bats have been investigated by scientist of the animal behavior. In order to understand the behavioral principle quantitatively, it is require to evaluate the usefulness to the real environment.

Especially, during the echolocating flight of Japanese horseshoe bats, it is confirmed that bats obtain the echoes including frequency beat which was occured by interfer- ece of the emmitted pulse and doppler shiftted echoes.

We hypothesized that bats are using such a frequency beat and proposed useful object localization method for agent moving condition. By using 1 transmitter and 2 receivers with the proposed method, we evaluated localization accuracy in the real environment.

Finally, we will introduce the drone equipped with 1 trasmitter and 2 receivers which was constructed as a tool for verifying bats behavior optimized for 3D navigation.

(19)

Deep Neural Network Based Multi-modality and Multi-organ Automatic Segmentation for Brain

Radiosurgery

Cheng-En Lee

National Taiwan University E-mail: tb830930@gmail.com

Tumor and critical organ delineation is the most critical step in automatic stereo- tactic radiosurgery (SRS) treatment planning workflow for brain metastases. Recent progress in Convolutional Neural Networks(CNN) has made it feasible to produce voxel-wise predictions of volumetric images and provide a powerful tool for automatic segmentation. The present study aims to develop a CNN algorithm for tumor and multi-organ segmentation on multi-modality imaging including contrast-enhanced computed tomography (CTc) and contrast-enhanced T1-weighted magnetic reso- nance imaging (MRI-T1c). Our dataset are acquired from one brain metastases cohort (n=95) treated with SRS using CyberKnife system. Each data included volume mask for brain tumors, brain stem, optic chiasm, bilateral eyes and optic nerves with associated CTc and MRI-T1c. We develop a workflow to organize the raw data, perform image preprocessing including data augmentation and registra- tion, construct CNN models, train and test the models then visualize the results.

The results of DICE scores for tumors and multi-organs around 0.5 and 0.8 in the testing set for CTc and MRI-T1c images, respectively.

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Stability of an epidemic model with boosting of immunity

Liu Yang (1,2), Yukihiko Nakata (2)

1: Northeast Normal University, School of Mathematics and Statistics; 2: Shimane University, Interdisciplinary Faculty of Science and Engineering

yangl994@nenu.edu.cn

We formulate an epidemic model with waning and boosting of immunity by delay differential equations, following the idea by Aron [1, 2]. In the model, recovered individuals may be reinfected because the immune system does not confer long- lasting immunity against a pathogen, thus the immunity wanes. On the other hand, it is also reported that the natural immunity is enhanced by the continued exposure to infection which is called boosting effect. Our main purpose is to explore the effect of the boosting of immunity on the disease transmission dynamics using an epidemic model. We also compare the transmission dynamics with and without boosting effect by numerical simulations.

References

[1] J. L. Aron, “Dynamics of acquired immunity boosted by exposure to infection”, Math. Biosci. 64, 249–259 (1983).

[2] J. L. Aron, “Acquired immunity dependent upon exposure in an SIRS epidemic model”, Math. Biosci. 88, 37–47 (1988).

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Analysis of continuous data assimilation algorithm for Lorenz 63 model with nonlinear nudging

Yi-Juan Du

National Chiao Tung University E-mail: fish40519@yahoo.com.tw

Data assimilation is an important issue because it can process data from different sources through a series of processing and adjustment, and finally can be compre- hensively applied. Especially for dynamic systems with sensitivity to initial value and chaos, such as Lorenz 63 model, we will use the nonlinear nudging techniques to predict on this model. In this talk, we will present the analysis of continuous data assimilation algorithm for this model based on nonlinear nudging and numerical experiments will be demonstrated, too.

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Nakayama Algebra and Algebraic Stability of Persisitent Homology

Hiroaki Ishiyama (1), Michio Yoshiwaki (2, 3, 4), Emerson G. Escolar (3), Hiroe Oka (1) 1: Ryukoku University; 2: RIKEN AIP;

3: Kyoto University; 4: Osaka City University t17M001@mail.ryukoku.jp.ac

Topological Data Analysis (TDA) is method of analyzing data from the geometric characteristics of data. It is applied to machine learning, material science, data analysis and so on. Persistent Homology is the most important mathematical idea in TDA [1]. By Persistent Homology, we can visualble the geometric characteristics of data using Persistence Diagram. Here we introduce an important theorem on Persistent Homology.

Theorem 1 (Algebraic Stability Theorem, [2]) M and N are p.f.d persistent ho- mology, if f : M → N(δ) is δ-interleaving morphism then B(M) and B(N) are δ matching. In particular,

dB(B(M), B(N)) ≤ dI(M, N ).

From this theorem, for example M is true data, N is observation data, if there is not much noise, we can analyze M from N . Therefore, there is no influence on the essential analysis result.

In this talk, We got the result which Algebraic Stability Theorem of Persistent Homology is established on Nakayama Algebra, so I will report it.

References

[1] H. Edelsbrunner, D Letscher and A. Zomorodian, Topological persistence and simplification, In: Discrete and Computatinal Geometry and Graph Drawing, Columbia, SC, 2001(eds, F. Shahrokhi and L. Szekely), Discrete Comput.

Geom., 28, Springer, 2002, pp.511-533.

[2] Ulrich Bauer, Michael Lesnick. Induced Matchings of Barcodes and the Al- gebraic Stability of Persistence (2013).

(23)

Dynamics of the Spruce budworm–bird interaction with the effect of insecticide

Le Nguyen Thuy Van National Central University E-mail: lntv2605@gmail.com

We study the dynamics of the spruce budworm system with the effect of bird predation and insecticide. If the growth rate is large enough, we shall show the globally asymptotical stability of equilibrium. The existence of limit cycle of dy- namical system is also shown under certain conditions. Moreover, if insecticide is applied many times or the maximum budworm mortality rate due to the action of the insecticide is very high, then the uniqueness of limit cycle is shown. In real life, in addition to birds, the number of budworm also depends on many factors such as the state of the forest and the effectiveness of insecticide in different life stages of the budworms. Thus, we would like to study the model for the dynamics of the budworm-bird-forest interaction in the future.

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Bifurcation of sticky circle packing

Wataru Makita (1), Yoshikazu Yamagishi (1) 1: Ryukoku University

t18m004@mail.ryukoku.ac.jp, yg@rins.ryukoku.ac.jp Let H = {x +

−1y : y > 0} be a upper half-plane. Let D(w, r) = {ζ :

|ζ − w| ≤ r} be a disk with the center w and the radius r > 0. A sticky circle packing Λ(r, θ) for a given pair of parameters r > 0 and θ, is the family of disks {Dj,k = D(jθ− k +√

−1yj, r) : j, k ∈ Z, j ≥ 0} such that yj = min{y ≥ 0 : Pj(y)}, where Pj(y) is the condition that the family of disks {D(sθ − k +√

−1ys, r) : k Z, s = 0, 1, . . . , j− 1} and {D(jθ − k +√

−1y, r) : k ∈ Z} do not overlap.

A fraction ma is called a (local) parastichy index of the disk Dj,k if it is tangent to the disk Dj+m,k+a. The fraction ma is called a parastichy index of the packing Λ(r, θ) if it is a parastichy index of Dj,k for sufficiently large j. The fraction ma is called a semi-parastichy index of Λ(r, θ) if it is a parastichy index of Dj,k for infinitely many j.

Let S(ma) be the set of (r, θ) such that the packing Λ(r, θ) has a parastichy index

a

m. Let S(ma;nb) be the set of (r, θ) ∈ S(ma) such that the packing Λ(r, θ) has a semi-parastichy index nb ̸= ma. Let S(ma;∅) be the set of (r, θ) ∈ S(ma) such that the packing Λ(r, θ) has no semi-parastichy index nb ̸= ma.

Any packing Λ(r, θ) has a parastichy index, so H =

a/mS(ma). For distinct regions S(ma), S(nb), we have S(ma)∩ S(nb) ̸= ∅ if and only if mb − na = ±1. The region S(ma) is connected, simply-connected, and has a boundary point ma on the real axis.

In this talk, we observe that S(a

m) = ∪

mb−na=±1

S(a m; b

n)∪ S(a m,∅)

for the case a = 1. We shall present the bifurcation diagram for the semi-parastichy indices in the (r, θ)-parameter space.

(25)

The synchronization analysis of Kuramoto model for the identical oscillators

Kuan-Jhan Lin National Taiwan University E-mail: z950069@gmail.com

In this talk, I will introduce the concept of synchronization and the Kuramoto model. Furthermore, I investigate the synchronization behavior of the solution math- ematically. My talk will consist of two parts: Part I is analysis of Kuramoto model and the stationary solution of N identical oscillators. Part II is analysis of the non-stationary solution of N identical oscillators and future works.

(26)

Three-Variable Model of the Photosensitive Belousov-Zhabotinsky Reaction

Daichi Inoue Meiji University cs181002@meiji.ac.jp

Various pattern dynamics, found in physiology, ecology and so on, have been cap- tured by the theory based on reaction diffusion systems. The Belousov-Zhabotinsky (BZ) reaction is one of the model reactions used to study nonlinear oscillation phe- nomena, which occur in vivo. The BZ reaction is brought about by mixing ap- propriate amount of oxidant, reductant, acid and metal catalyst. In the process of oxidizing the reductant, concentrations of the reaction intermediates oscillate pe- riodically. Accordingly, the oxidation-reduction potential also oscillates. Although the BZ reaction is a complex process involving more than 100 elementary reactions, the main part of reaction is summarized as the Oregonator model [1], which is the ki- netic model of the BZ reaction. The Oregonator model describes the mechanism and dynamics of the oscillation of the BZ reaction. On the other hand, the BZ reaction is known to be stimulated by light irradiation. In fact, it can be realized by using photosensitive metal catalyst. Several reaction models have been proposed with stimulations by light irradiation. However, since most of previous models contains more than four variables, these are not so easy to analyze.

Our strategy is reducing the previous four-variable model [2] by nondimension- alization and the fast-slow method to obtain a simpler model. The results of sim- ulations guarantee the validity of the simplified model. In addition, plural period- doubling bifurcations, which have not been known in the basic BZ reaction, are found by bifurcation analysis in this research.

References

[1] R. J. Field and R. M. Noyes, J. Phys. Chem. A 1974, 21, 1877.

[2] T. Amemiya, T. Ohmori and T. Yamaguchi, J. Phys. Chem. A, 2000, 104, 336.

(27)

Why marriages are so difficult: a mathematical aspect

Fei-Hung Huang

National University of Tainan E-mail: feihunghuang22@gmail.com

Most couples generally plan enduring relationships. Nevertheless, the high di- vorce rates massively reported show a resounding failure in their implementation.

The phenomenon of couple disruption is considered epidemic in the world. It is interesting to understand why so many couples end in divorce while some others do not (but may complaint why their sustained effort cannot rouse their feeling very much) since the social change induced by marriage deeply affects the social structure and the well-being of their members. Our aim is to offer a consistent explanation for the facts of marriage by modifying a feeling-effort dynamical system proposed by Rey (2010 PLoS ONE), where the optimal control modeling brings a mathematical approach to the analysis of marriage and close relationships.

(28)

A Self-organized Model for Safety Autonomous Driving Cars

Naoto Ichikawa (1), Takeshi Kano (2), Mayuko Iwamoto (3), Daishin Ueyama (4) 1: Graduate School of Natural Science and Technology, Shimane University;

2: Research Institute of Electrical Communication, Tohoku University;

3:Interdisciplinary Faculty of Science and Engineering, Shimane University;

4:Faculty of Engineering, Musashino University;

n18m003@matsu.shimane-u.ac.jp

The Japanese government aims to realize a perfect autonomous driving car which does not require a driver completely by 2020. A purpose of this study is to build a simple model with prediction [1], as a first step for understanding the behavior of perfect autonomous driving cars. We modified the previous model, Social Force Model (SF model) [2], which describes pedestrian motion by the equation of motion that calculates the physical and the psychological forces by other individuals and obstacles. In this presentation, we will discuss safety with this model compared to the previous model by numerical simulations.

References

[1] T. Kano, M. Iwamoto, D. Ueyama, ”Decentralized Control for Self-driving Cars That can Freely Move on Two-dimensional Plane,” in Proceedings of Pedestrian and Evacuation Dynamics 2018 (PED2018), 2018.

[2] D. Helbing, I. Farkas, T. Vicsek, ”Simulating dynamical features of escape panic”, Nature, vol. 407, pp. 487-490, 2000.

(29)

An epidemic model with treatment capacity

Hiromu GION (1), Yasuhisa SAITO (2) Department of Mathematics, Shimane University 1: g.hiromu1997@icloud.com; 2: ysaito@riko.shimane-u.ac.jp

Treatment is an important method to decrease the spread of diseases such as measles, tuberculosis and flu. In the present work, we propose an epidemic model with a limited resource for treatment in order to understand the effect of the treat- ment capacity, which modifies [1].

References

[1] W. Wang, Backward bifurcation of an epidemic model with treatment,Math.

Biosci. 201 (2006) 58-71.

(30)

Optimization problems in signal and audio denoising

Yu-Kai Lin

National Central University E-mail: stephen359595@gmail.com

Total variation based filtering, derived by Rudin, Osher, and Fatemi at first in 1992, has nowadays been widely applied in signal denoising and further in image denoising problem, such as signal smoothing, fingerprint enhancement, and so on.

Figure 3: Signal denoising (above: the original signal; below: the smoothing signal)

This filter (in 1-dimension, for instance) can be implemented through several recursive (or non-recursive) approaches so as to reduce the objective function, which is given by

J (x) = 1 2

N k=1

|yk− xk|2+ λ

N−1 k=1

|xk+1− xk|,

where λ is the reciprocal of Lagrange multiplier, and {yk}Nk=1 is the observed signal consisting of the original signal {xk}Nk=1, combined with a normal random variable n ∼ N(0, σ2), and is defined by yk= xk+ n for each k = 1,· · · , N.

However in audio, a special kind of signals, does not play a good result based on the above approach when solving the denoising problem. We will focus on the main idea of the total variation filtering in the beginning of this talk, as well as try out to revise the model by taking frequency into consideration, followed by accessing the particular signal denoising problem as our future work.

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Closed geodesics on the beading regular icosahedra

Miu Nodagashira (1), Yoshikazu Yamagishi (1) 1: Ryukoku University

yg@rins.ryukoku.ac.jp

A planar graph is called 3-regular if each vertex is incident to three edges. A typical example of a 3-regular graph is a honeycomb graph consisting of hexagons.

A 3-regular finite planar graph G defines a beading polyhedron, which gives rise to a knot or link of strings K. In each bead, there pass through two (segments of) strings. In this task, we assume positive crossing of strings in each bead. A connected component of string is called a beading geodesic.

If G consists only of pentagons and hexagons, then the Euler formula shows that there are exactly twelve pentagons. Such G is called corner-transitive if for any pair of vertices v1, v2 of pentagons, there exists a graph automorphism φ : G→ G that maps v1 to v2. A beading regular icosahedron is defined as a beading polyhedron of a corner-transitive 3-regular finite planar graph consisting of pentagons and hexagons.

The family of beading regular icosahedra is indexed by a pair of integers (p, q).

On the other hand, it is known that the family of closed geodesic on the regular icosahedron is also indexed by a pair of integers (p, q). There exists a natural correspondence between the beading regluar icosahedron of type (p, q) and the closed geodesic on the regular icosahedron of type (p, q).

(32)

Schauder’s estimates for general elliptic equations

Pei-Shan Fang

National Cheng Kung University E-mail: c14046222@gs.ncku.edu.tw Let Ω be an open set inRn, with smooth boundary and

Wk,p(Ω) ={u ∈ Wk(Ω); Dαu∈ Lp(Ω)f orall|α| ≤ k}.

We define the operator L by

Lu = Di(aij(x)Dju + bi(x)u) + ci(x)Diu + d(x)u

whose coefficients aij, bi, ci, d(i, j = 1, ..., n) are assumed to be measurable functions on Ω. Assume the operator L satisfies the condition (1): ∀x ∈ Ω, ξ ∈ Rn and some constant λ > 0 and Λ and v ≥ 0





aij(x)ξiξj ≥ λ|ξ|2 Σ|aij(x)|2 ≤ Λ2

λ−2Σ(|bi(x)|2+|ci(x)|2) + λ−1|d(x)| ≤ v2.

(1)

In this talk, I will introduce the Schauder interior estimates of L: Suppose that fi ∈ Lq(Ω), i = 1, ..., g ∈ Lq/2(Ω) for some q > n. If u ∈ W1,2(Ω) and satisfies Lu = g + Difi in Ω, for any ball B2R(y)∈ Ω and Ω ⊂⊂ Ω and p > 1 and R > 0,





supBR(y)u(−u) ≲ R−n/p∥u+(u)Lp(B2R(y))+ k(R)

∥u∥Cα(Ω) ≲ ∥u∥L2(Ω)+ λ−1(∥f∥q+∥g∥q/2)

|u|1,α;Ω ≲ |u|0;Ω+|g|0;Ω+|f|0,α;Ω

which play an essential role in the esistence and regularity theory of second order linear elliptic equations. What’s more, the inequalities are also useful to discuss about the Dirichlet problem for continuous boundary values.

References

[1] David Gilbarg, Neil S. Trudinger, Elliptic Partial Differential Equations of Sec- ond Order.

(33)

Machine Learning and Image Classification

Kaito Fujihara (1), Mayuko Iwamoto (1)

1: Inter disciplinary Faculty of Science and Engineering, Shimane University;

s153082@matsu.shimane-u.ac.jp

Recently, Machine Learning (ML) has become the focus of attention [1] because it would be expected to enrich our life. By finding hidden features in a lot of data, ML can make a decision without human’s consideration. Supervised Machine Learning (SML) has already got many achievements in various field. However, it is difficult to collect enough labeled data for SML, so we are required to catch a structure of data by the limited information. A purpose of this research is to establish the way to classify unlabeled data. In this presentation, we will talk about image classification using the K-means clustering.

References

[1] K.H. Zeng, W. B. Shen, D.A. Huang, M. Sun, J.C. Niebles, “Visual Forecasting by Imitating Dynamics in Natural Sequences”, in conference paper of Interna- tional Conference on Computer Vision (ICCV 2017), 2017.

(34)

Point cloud watermarking technology

Ming-Hsiu Lu

National Central University E-mail: hugo572G@outlook.com

Recently, due to the development of 3D scanning and modeling technologies, 3D model data has been widely used in various fields. Such as architecture, medicine, and manufacturing, etc. Therefore, anti-counterfeiting, copy-proof, and integrity in- spection technologies are required. 3D model data is roughly divided into parametric surface, point cloud, and polygon mesh. In practice, point cloud and polygon grid are more common. Point cloud is the data obtained through 3D scanner, which will be a set of coordinate points in three-dimensional space. Polygon mesh is usually created by software or reconstructed by point clouds. The data format is also a set of coordinate points in three-dimensional space and contains the connection between points. The watermarking technology of polygon mesh has be developed mature, and this paper expects to study the watermarking technology on point cloud. Be- cause the process of reconstructing a polygon mesh from a point cloud often depends on the quality of the original point cloud quality, especially when the object has a complex surface, there is no way to convert the point cloud and the polygon mesh well, which will greatly increase the distortion caused by the watermarking.

(35)

Localization of graph Laplacian eigenvectors on scale free networks

Ryu Fujiwara Meiji University E-mail-ev50061@meiji.ac.jp

Eludicating characteristics of large scale free networks leads to revealing struc- tures of substantial amount of networks. Among those aspects, diffusion on a net- work is depicted by the eigenvectors of its graph Laplacian, and especially in a large scale free network, the elements of each eigenvector tend to have relatively large values only on the nodes whose degrees are close. However, it is still unsolved why the localization can be observed in scale free networks in general. Here we analyze the root of localization by introducing an Lp graphon, which is a representation of limit objects of sparse graphs, and is defined by a p-integrable function on [0, 1]2. In this sense, a continuum limit of the graph Laplacian is described by an Lp graphon.

In this research, the continuum limit of the graph Laplacian generates a closed linear operator L densely-defined on L2(0, 1). Our aim is to determine the spectra of L and obtain the resolvent estimates. Moreover, we state that the singularity of the

“eigenfunctions” of L implies the origin of the localization of the graph Laplacian eigenvectors on large scale free networks in a formal way.

References

[1] P. N. McGraw and M. Menzinger, “Laplacian spectra as a diagnostic tool for network structure and dynamics”, Phys. Rev. E, 77, 031102, doi: 10.1103/Phys- RevE.77.031102 (2008).

[2] D. Kaliuzhnyi-Verbovetskyi and G. S. Medvedev, “The semilinear heat equation on sparse random graphs”, SIAM Journal on Mathematical Analysis, 49.2, 1333- 1355 (2017).

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A variational approach to the inverse imaging of composite elastic materials

Riku Kanai

Meiji University, Graduate School of Advanced Mathematical Sciences, Japan cs171002@meiji.ac.jp

We have constructed a variational method for analyzing inverse problems whose solutions describe the interior of composite elastic materials. In particular, given an outward surface velocity along the boundary of the composite, we have designed a cost functional which measures the difference between data obtained from our model equations. Using Lagrangian functionals, we are then able to obtain the gradient of the cost functional and investigate its gradient flow. Numerical methods, based on the method of discretization of time and the finite element method, were also constructed. These methods were then used within level set equations to investigate numerical solutions of the inverse problems. Our results show that the gradient flow is able to extract the interior composition of the elastic bodies using only the boundary velocity’s information. In addition to showing numerical solutions of the inverse problems, we will show imaging results involving real surface acoustic wave data. This research is joint with E. Ginder (Meiji University), O. Wright (Hokkaido University) and P. Otsuka (Hokkaido University).

Figure 4: Simulation of elastic wave propagation though a glass slab with a steel in- clusion. Magnitude shows the difference between the simulations, with and without the inclusion.

References

[1] H. Azegami. Shape Optimization Problems. Morikita Publishing, 2016.

[2] T. J. R. Hughes. The Finite Element Method , Dover Publications, 2000.

[3] V. Laude. Phononic Crystals. De Gruyter Studies in Mathematical Physics, 2015.

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Convergent cross mapping and skew prodect mapping

Hiroki Koumoto(1) , Hiroe Oka(1)

1.Ryukoku University t17m002@mail.ryukoku.ac.jp

Convergent Cross Mapping (CCM), proposed by Sugihara et al. in 2012([1]), is a procedure which detects the causality between two kinds of data X and Y . Roughly speaking, in their terminology, X causes Y means that the information of X is used to decide that of Y . More precisely, X causes Y , if the correlation coefficient between the predicted value of Y using X and the real value of Y increases according to the amount of data X and Y . This method became one of the basic methods to determine the causality in the field of ecology.

Nevertheless, from mathematical point of view, there are several unclear points in their method; for instance:

1. The definition of the causality.

2. The increase of the correlation coefficient according to the amount of data is a necessary and/or a sufficient condition for the causal- ity? What is a threshold value of the increase constant to obtain the causality?

3. What is the mechanism to explain this kind of phenomena?

4. What kind of properties are obtained concerning the correlation coefficient in case the causality exists and/or not exists?

In this talk, we consider a simple dynamical system and use it to show how the idea of the convergent cross mapping may work in a mathematically clear way. The system we study is of skew-product type given as follows:

{ x(t + 1) = x(t) + ω1 (mod 1)

y(t + 1) = y(t) + ω2+ kx(t) (mod 1) ω1 = 5, ω2 = π

Although we cannot answer the above questions even to such a simple sys- tem, we shall discuss some results concerning them.

References

[1] Sugihara, May, Ye, Hsieh, Deyle, Fogarty and Munch: ”Detecting causal- ity in complex ecosystems”, Science 338 (2012), pp 496-500

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Algorithms of Brian Segmentation from T1/T2 MRI Image

Bing-Ze Lu

National Cheng Kung University E-mail: l16064054@ncku.edu.tw

Nowadays, according to the advanced medicine progress, clinicians highly depend on the medical imaging technique which allows them to visualize the representations of the interior patient body, such as computed tomography (CT) and magnetic resonance imaging (MRI). However, these medical images are outputted to the two- dimensional pictures which may not provide the sufficient information for doing three-dimensional computation research. Due to the previous reason, we would like to collect these images from the brain MRI via the machine learning algorithms to obtain the segmentation and reconstruction of the brain imaging. During this research work, there’s some noise after processing. Therefore, we plan to develop a suitable algorithm including the variation method, with the concept of kernel method in machine learning in order to get the better results. Finally, we would like to represent an algorithm, which may work on most image to generate high-quality imaging for the object.

(39)

Analysis of a mathematical model for interfacial behaviors under drying processes

Yuki Yagasaki Meiji University cs171013@meiji.ac.jp

Recently, many studies have been conducted on interface behaviors of fluids under various conditions, including several mathematical studies on drying processes of nonvolatile particles and volatile solvents. The mathematical models are proposed in previous articles and reduced from the equations of motion of fluid. In those models, viscosities and densities in fluids are supposed to be constant, although these elements are actually varied with concentrations of the solvents and particles [1]. Thus, we apply the lubrication approximation to the Navier-Stokes equation and derive the following partial differential equations under the periodic boundary condition, taking account of the change in viscosities and densities of the fluid;























∂h

∂t =−∂K

∂x ρ

ρlα(1− ϕ) − D ρl

∂x (

h∂ρ

∂x )

, 0 < x < L, t > 0, h∂ϕ

∂t =−∂ϕ

∂xK + ρ ρlD

∂x (

h∂ϕ

∂x )

+ ϕρ

ρlα(1− ϕ), 0 < x < L, t > 0, K = 1

µ ( 5

24gh4∂ρ

∂x 1 3h3

∂x (

ρgh− γ∂2h

∂x2 ))

, ρ = (ρg− ρl)ϕ + ρl, µ = (µg− µl)ϕ + µl.

(2)

The function h(x, t) is the height of the fluid and ϕ(x, t) is the volume fraction of nonvolatile particles in the fluid. All parameters in (2) are nonnegative.

Our aim in this study is to examine the behavior of the solution and the exis- tence of stationary solutions with spatial pattern. It can be shown by numerical simulations that h converges to a constant stationary solution as t → ∞ in many cases. Moreover, we analyze the linear stability of steady states. When α = 0 and D > 0, any constant stationary solutions are linearly stable. On the other hand, when α = D = 0, (2) exhibits nonconstant steady states. Then, we confirm the existence of a stationary solution in a rigorous way and obtain the following results;

If α > 0 or D > 0, (2) has only constant stationary solutions. On the other hand, if α = 0 and D = 0, there exists a nonconstant stationary solution (h, ϕ), where ϕ can be given arbitrarily.

References

[1] M. Kobayashi, M. Makino, T. Okuzono, and M. Doi, Interference Effects in the Drying of Polymer Droplets on Substrate, J. Phys. Soc. Jpn., Vol. 79, No. 4 (2010), 044802.

(40)

Predicting Time Steps in Molecular Simulations

Wen-Hao Yang National Taiwan University E-mail: r06246004@ntu.edu.tw

Time scale limitation has been and remains a challenging problem in molecular simulations. Large systems such as proteins are still narrowed in scale of nano-second with the help of super computers. The limit of time may contribute to insufficient result and improper inference while the simulation is already time-consuming. A machine learning method: time series prediction is implemented to solve this prob- lem. Since features (Total force, numbers of Hydrogen bond, etc.) of protein have strong time dependence during simulation, this relationship can be used to obtain a different data by shifting original data with time. Taking these two data, we may train the machine to catch up structure of systems and achieve predicting these features in the next time step.

(41)

Determination of Wave Propagation Direction for a Class of Biological Bistable Models

Yu-Yu Weng

National Tsing Hua University E-mail: yo850914@gmail.com

Biological invasion is an important ecological phenomenon, and thus it is im- portant to determine the propagation direction in the associated biological model.

However, there are very few analytical results on the determination of propaga- tion direction in the bistable biological system. In this study, we give a complete characterization of propagation direction in a class bistable biological system.

(42)

Mathematical Model for Quadrupedal Walking and its Dynamical Pattern

Akira Machishaku

Graduate Course of Applied Mathematics and Informatics Ryukoku University

T17M006@mail.ryukoku.ac.jp

Depending on walking speeds, various walking patterns are observed in quadruped animals. Mathematical models realizing the typical patterns of quadrupedal walk, Walk, Pace, Trot and Bound, have been proposed in [1] and [2]. In this research based on the model by Tero et al. (2013)[1], we study the following reduced dynam- ical model:



˙

ρ =−αρ − β(sin ϕ − sin ψ), ϕ = ω˙ − (g + ρ) cos ϕ, ψ = ω˙ − (g − ρ) cos ψ,

where ρ and (ϕ, ψ) are related to a physical state of the body and the motion of the legs respectively. The parameters α, β, g are assumed to be positive. Applying numerical methods to this model, we identify parameter regions where the three walking patterns take place. Moreover, we show some dynamical structure of the model equation.

This is a joint work with Prof. Y. Morita (Ryukoku University). We thank Dr. M. Akiyama (Hokkaido University) for letting us know the work ([1]) and his variable comments.

References

[1] A. Tero, M. Akiyama, D. Owaki, T. Kano, A. Ishiguro, R. Kobayashi, Interlimb neural connection is not required for gait transition in quadruped locomotion, arXiv preprint, arXiv:1310.7568v1 (2013)

[2] D. Owaki, T. Kano, K. Nagasawa, A. Tero, A. Ishiguro, Simple robot suggests physical interlimb communication is essential for quadruped walking, J.R.Soc.

Interface 10:20120669. doi: 10.1163/rsif.2012.0669 (2013)

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Synchronization of Metronomes

Mana Futamura Meiji University cs181012@meiji.ac.jp

The living beings exhibit the rhythms. Synchronization phenomena play im- portant roles in rhythms. I studied the method of homogeniation following to [1] and applied to the synchronization phenomena of two metronomes on an ex- perimental and theoretical sides. I derived the phase equations by the averaging method and checked whether a synchronization phenomenon really happens to the two metronomes experimentaly.

References

[1] Hiroshi Kori, Yoshihisa Morita, Seibutsu-rizumu to Rikigakukei, Kyoritsu Shup- pan (2011). (郡宏, 森田善久, 生物リズムと力学系, 共立出版 (2011))

[2] Yuki Kuramoto, Yoji Kawamura, Dokigensyo no Kagaku, Isokijutsu niyoru apuroch, Kyoto University gakujutsu shuppankai (2017). (蔵本由紀, 河村洋史, 同期現象の科学:位相記述によるアプローチ, 京都大学学術出版会 (2017))

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FitzHugh-Nagumo Neuron Network for understanding the mechanism of Dynamic

Pattern Formation in Cuttlefish

Eigo Fukada (1), Kana Mizuno (2), Mayuko Iwamoto (3) 1: Graduate School of Natural Science and Technology, Shimane University;

2: Interdisciplinary Graduate School of Science and Engineering, Shimane University;

3: Interdisciplinary Faculty of Science and Engineering, Shimane University;

n18m019@matsu.shimane-u.ac.jp

Cuttlefishes and squids can change own body patterns to fit the surrounding environments in the situations of camouflage and communication with others, which are controlled by contraction of muscles around pigment cells with neural system [1], but it is unclear that how visual information is reflected in their body patterns. A purpose of this research is to understand the mechanism of pattern formation in cuttlefish by focusing on the neural system connecting pigment cells and muscles.

In this presentation, we will build a neuron network model using discrete FitzHugh- Nagumo model and discuss simulation results with the model.

References

[1] T. J. Wardill, P. T. Gonzalez-bellido, R. J. Crool and R. T. Hanlon, “Neural control of tuneable skin iridesecene in squid”, Proceedings of the Royal Society B, 279(1745):4243-52 (2012).

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