abstract book - ntnutms2018.math.ntnu.edu.tw/html/2018tms...

D e c . 8 & 9 國立臺灣師範大學數學系 Department of Mathematics, National Taiwan Normal University

TMS Annual Meeting

數學年會2 0 1 8

數學年會

大會手冊A b s t r a c t B o o k

2018年中華民國數學年會 2018 Taiwan Mathematical Society Annual Meeting

會議時間：2018年 12 月8日 (星期六) 至9日 (星期日)

會議地點：國立臺灣師範大學公館校區

主辦單位：中華民國數學會

承辦單位：國立臺灣師範大學數學系

協辦單位：台灣數學教育學會、科技部數學研究推動中心、

國立臺灣師範大學、李梅樹紀念館

贊助廠商：昊青公司

學術委員會Scientific Committee

淡江大學數學系郭忠勝召集人

國立臺灣師範大學數學系林俊吉

國立臺灣大學數學系王偉仲

國立中央大學數學系洪盟凱

國立中興大學應用數學系施因澤

國立政治大學應用數學系陳隆奇

國立清華大學數學系蔡志強

承辦單位籌備人員(組織委員會Organizing Committee)

國立臺灣師範大學數學系林俊吉

國立臺灣師範大學數學系王婷瑩

國立臺灣師範大學數學系呂翠珊

國立臺灣師範大學數學系林延輯

國立臺灣師範大學數學系張毓麟

國立臺灣師範大學數學系郭君逸

國立臺灣師範大學數學系郭庭榕

國立臺灣師範大學數學系陳賢修

國立臺灣師範大學數學系黃聰明

國立臺灣師範大學數學系楊青育

國立臺灣師範大學數學系楊凱琳

國立臺灣師範大學數學系劉容真

國立臺灣師範大學數學系樂美亨

國立臺灣師範大學數學系謝世峰

國立臺灣師範大學數學系謝豐瑞

國立臺灣師範大學數學系朱啓台

國立臺灣師範大學數學系李小慧

國立臺灣師範大學數學系李君柔

國立臺灣師範大學數學系張珈華

國立臺灣師範大學數學系莊雲閔

國立臺灣師範大學數學系陳秉君

國立臺灣師範大學數學系游珮詩

國立臺灣師範大學數學系黃鴻霖

國立臺灣師範大學數學系劉欣怡

Sessions

數論與代數

Number

Theory and

Algebra

幾何

Geometry

偏微分方程

Partial

Differential

Equations

離散數學

Discrete

Mathematics

計算數學Computational

Mathematics

機率

Probability

最佳化

Optimization

動態系統與

生物數學Dynamical

Systems and

Biomathematics

分析

Analysis

統計

Statistics

數學科普

Popular

Mathematics

數學教育

Mathematics

Education

Room G003圖書館

B102理學院

M210數學館

G001圖書館

M212數學館

B103理學院

M310數學館

M417數學館

G002圖書館

M211數學館

H301綜合館

B101理學院

8:30-9:30

9:30-10:00

10:00-10:50

10:50-11:0511:05-11:20

11:20-12:05

主持人:夏良忠

演講者:王姿月

主持人: 江孟蓉

演講者: 崔茂培

主持人:吳恭儉

演講者:劉太平

主持人: 董立大

演講者: 李渭天

主持人:鄧君豪

演講者:吳金典

主持人:黃啟瑞

演講者:許順吉

主持人:陳界山

演講者:Yongdo Lim

主持人:陳國璋

演講者:吳昌鴻

主持人:沈俊嚴

演講者:林欽誠

主持人:蔡碧紋

演講者:姚怡慶

主持人:賴以威

演講者:秋山仁

主持人:劉柏宏

演講者:單維彰*** *** ***

12:05-13:30

13:30-13:55

主持人:劉家新

演講者:劉承楷

主持人: 何南國

演講者: 王業凱

主持人:李志豪

演講者:江金城

主持人: 張薰文

演講者: 林晉宏

主持人:李勇達

演講者:嚴健彰

主持人:陳冠宇

演講者:須上苑

主持人: 張毓麟

演講者:黃淑琴***

14:00-14:25

主持人:劉家新

演講者:彭勇寧


演講者: 蔡忠潤

主持人:李志豪

演講者:林英杰

主持人: 張薰文

演講者: 鄭硯仁

主持人:李勇達

演講者:游承書

主持人:陳冠宇

演講者:洪芷漪

主持人: 張毓麟

演講者: 胡承方***

14:30-14:55

主持人:劉家新

演講者:姚為成


演講者: 邱聖夫*** 主持人: 張薰文

演講者: 商珍綾

主持人:吳金典

演講者:黃杰森*** 主持人: 張毓麟

演講者: 杜威仕***

14:55-15:20

15:20-15:45

主持人:劉承楷

演講者:陳光武*** *** 主持人: 郭君逸

演講者: 黃鵬瑞*** *** 主持人: 杜威仕

演講者: 蔡豐聲

主持人: 張志鴻

演講者: 張覺心

15:10-15:50主持人:沈俊嚴

演講者:蔡明誠

主持人: 蔡碧紋

演講者: 李百靈*** *** *** *** ***

16:00-16:50

16:50-18:1018:30-21:00

2018年中華民國數學年會2018 Taiwan Mathematical Society Annual Meeting

2018年12月8日(星期六)

中正堂

H301, 綜合館

H301, 綜合館

Registration

Opening CeremonyChair：President Jong-Shenq Guo

Plenary Lecture by Professor Fang-Hua LinChair ：Professor Jenn-Nan Wang

報到註冊

年會開幕式

主持人郭忠勝理事長

大會演講林芳華教授

主持人王振男教授

遊戲,思考,學習

數學普及特展

Math Museum

中正堂

2018年中華民國數學年會晚宴

H301, 綜合館

中正堂

H301, 綜合館

13:30-14:15主持人:沈俊嚴

演講者:黃毅青

14:15-14:55主持人:沈俊嚴

演講者:王雅書

13:30-14:15主持人:陳賢修

演講者:Eliot Fried

14:30-14:55主持人: 鄭昌源

演講者: 王埄彬

13:30-15:00Fun學奠數

1. 國中組

2. 國小組

Group Photo

Coffee Break

茶會 Coffee Break

14:00-16:00數學會是您

的好夥伴!

Plenary Lecture by Professor Motoko KotaniChair：Professor Jung-Kai Chen

大會演講 Motoko Kotani 教授

主持人陳榮凱教授

團體照 H301, 綜合館

中華民國數學會會員大會暨頒獎典禮

13:30-14:15主持人:姚怡慶

演講者:王維菁

14:20-14:45主持人: 呂翠珊

演講者: 陳春樹

14:45-15:10主持人: 呂翠珊

演講者: 黃世豪

茶會

午餐 Lunch與韓國數學會雙邊會談 Bilateral Talk between KMS & TMS

13:30-15:00論壇：

因應AI 時代學生該

具備哪些數學能力

主持人兼與談人：

李華倫

其他與談人：

樂美亨

曾俊雄

魏澤人

13:30-14:15秘數淘學

真實秘數遊戲

14:15-15:00秘數淘學

真實秘數遊戲

更新日期: 2018年12月6日

Sessions

數論與代數

Number

Theory and

Algebra

幾何

Geometry

偏微分方程

Partial

Differential

Equations

離散數學

Discrete

Mathematics

計算數學Computational

Mathematics

機率

Probability

最佳化

Optimization

動態系統與

生物數學Dynamical

Systems and

Biomathematics

分析

Analysis

統計

Statistics

數學科普

Popular

Mathematics

數學教育

Mathematics

Education

Room G003圖書館

B102理學院

M210數學館

G001圖書館

M212數學館

B103理學院

M310數學館

M417數學館

G002圖書館

M211數學館

H301綜合館

上午: B101下午:B101(主場),B102, B103(分場)

理學院

8:30-9:00

9:00-9:50

9:50-10:20

10:20-11:05

主持人:王姿月

演講者:粘珠鳳

主持人: 林俊吉

演講者:Eliot Fried

主持人: 游森棚

演講者: 周文賢

主持人:鄧君豪

演講者:吳宗信

主持人:許順吉

演講者:陳冠宇

主持人: 許瑞麟

演講者: 孫德鋒

主持人: 陳賢修

演講者: 班榮超*** *** *** *** ***

11:10-11:35

主持人:王姿月

演講者:魏福村

主持人: 崔茂培

演講者: 陳志偉

主持人: 林延輯

演講者: 俞韋亘

主持人:鄧君豪

演講者:李勇達

主持人:陳隆奇

演講者:劉聚仁

主持人:孫德鋒

演講者: 許瑞麟

主持人: 陳賢修

演講者: 林得勝*** *** ***

11:40-12:05

主持人:王姿月

演講者:康明軒*** 主持人: 林延輯

演講者: 徐祥峻

主持人:鄧君豪

演講者:許佳璵

主持人:陳隆奇

演講者:黃建豪

主持人:孫德鋒

演講者: 陳鵬文

主持人:陳賢修

演講者:王琪仁*** *** ***

12:05-13:30

13:30-13:55 *** *** *** *** *** *** 主持人: 黃建豪

演講者: 許為明

主持人:曾睿彬

演講者:李俊憲*** ***

14:00-14:25 *** *** *** *** *** ***

主持人:黃建豪

演講者:Chieu Thanh

Nguyen

主持人:曾睿彬

演講者:梁育豪*** ***

14:30-14:55 *** *** *** *** *** ***

主持人: 黃建豪

演講者:Jan Harold

Mercado Alcantara

*** *** ***

11:10-11:55主持人:左台益

演講者:郭伯臣

賦歸

中正堂

14:00-16:00數學會是您

的好夥伴!

13:30-15:00工作坊：

數學素養評量

主持人:楊凱琳

帶領人：

洪有情

謝豐瑞

林素微

13:30-14:10主持人:沈俊嚴

演講者:莊智升

14:15-14:55主持人:沈俊嚴

演講者:吳希淳

10:20-10:45主持人: 蔡碧紋

演講者: 呂翠珊

10:45-11:10主持人: 蔡碧紋

演講者: 趙維雄

11:10-11:35主持人: 蔡碧紋

演講者: 邱詠惠

11:35-12:00主持人: 蔡碧紋

演講者:楊洪鼎

10:20-11:05主持人:沈俊嚴

演講者:王昆湶

11:10-11:55主持人:沈俊嚴

演講者:簡茂丁

13:30-14:15秘數淘學

1. 真實秘數遊

戲

2. 賽局理論教

你思考

14:15-15:00秘數淘學

1. 真實秘數遊

戲

2. 賽局理論教

你思考

Plenary Lecture by Professor Wen-Ching Winnie LiChair：Professor Shun-Jen Cheng

大會演講李文卿教授

主持人程舜仁教授

13:30-15:00Fun學奠數

1. 國中組

2. 國小組

午餐 Lunch

茶會 Coffee Break

H301, 綜合館

11:10-11:55主持人:陳宏賓

演講者:施宣光

10:20-10:45主持人:郭鴻文

演講者:陳逸昆

10:55-11:20主持人:郭鴻文

演講者:張覺心

遊戲,思考,學習

數學普及特展

Math Museum

Registration 報到註冊中正堂

2018 Taiwan Mathematical Society Annual Meeting2018年12月9日(星期日)

2018年中華民國數學年會

更新日期: 2018年12月6日

SessionsNumber

Theory andAlgebra

GeometryPartial

DifferentialEquations

DiscreteMathematics

ComputationalMathematics Probability Optimization

DynamicalSystems and

BiomathematicsAnalysis Statistics Popular

MathematicsMathematics

Education

Room G003Library

B102College of Science

M210Math Building

G001Library

M212Math Building


M310Math Building

M417Math Building

G002Library

M211Math Building

H301General Hall


8:30-9:30

9:30-10:00

10:00-10:50

10:50-11:0511:05-11:20

11:20-12:05

Chair:Liang-Chung Hsia

Speaker:Tzu-Yueh Wang

Chair:River Chiang

Speaker:Mao-Pei Tsui

Chair:Kung-Chien Wu

Speaker:Tai-Ping Liu

Chair:Li-Da Tong

Speaker: Wei-Tien Li

Chair:Chun-Hao Teng

Speaker:Chin-Tien Wu

Chair:Chii-Ruey Hwang

Speaker:Shun-Chi Hsu

Chair:Jein-Shan Chen

Speaker:Yongdo Lim

Chair:Kuo-Chang Chen

Speaker:Chang-Hong Wu

Chair:Chun-Yen Shen

Speaker:Chin-Cheng Lin

Chair:Pi-Wen Tsai

Speaker:Yi-Ching Yao

Chair:I-Wei LaiSpeaker:

Jin Akiyama

Chair:Po-hung Liu

Speaker:Wei-Chang Shann

*** *** ***

12:05-13:30

13:30-13:55

Chair:Chia-Hsin Liu

Speaker:Cheng-Kai Liu

Chair:Nan-Kuo Ho

Speaker:Ye-Kai Wang

Chair:Jyh-Hao Lee

Speaker:Chin-Cheng Chiang

Chair:Hsun-Wen Chang

Speaker:Jephian C.-H. Lin

Chair:Yung-Ta Li

Speaker:Chien-Chang Yen

Chair:Guan-Yu Chen

Speaker:Shang-Yuan Hsu

Chair:Yu-Lin Chang

Speaker:Shu-Chin Huang

***

14:00-14:25

Chair:Chia-Hsin Liu

Speaker:Yung-Ning Peng

Chair:Nan-Kuo Ho

Speaker:Chung-Jun Tsai

Chair:Jyh-Hao Lee

Speaker:Ying-Chieh Lin


Speaker:Yen-Jen Cheng

Chair:Yung-Ta Li

Speaker:Cheng-Shu You

Chair:Guan-Yu Chen

Speaker:Jyy-I Hong

Chair:Yu-Lin Chang

Speaker:Cheng-Feng Hu

***

14:30-14:55

Chair:Chia-Hsin Liu

Speaker:Wei-Chen Yao

Chair:Nan-Kuo Ho

Speaker:Sheng-Fu Chiu

***


Speaker:Jen-Ling Shang

Chair:Chin-Tien Wu

Speaker:Chieh-Sen Huang

***

Chair:Yu-Lin Chang

Speaker:Wei-Shih Du

***

14:55-15:20

15:20-15:45

Chair:Cheng-Kai Liu

Speaker:Kwang-Wu Chen

*** ***

Chair:Junyi GuoSpeaker:

Peng-Ruei Huang

*** ***

Chair:Wei-Shih Du

Speaker:Feng-Sheng Tsai

Chair:Chih-Hung Chang

Speaker:Chueh-Hsin Chang

15:10-15:50Chair:

Chun-Yen ShenSpeaker:

Ming-Cheng Tsai

Chair: Pi-Wen TsaiSpeaker:

Pai-Ling Li*** *** *** *** ***

16:00-16:50

16:50-18:10

18:30-21:00

Group Photo

Plenary Lecture by Professor Motoko KotaniChair：Professor Jung-Kai Chen

LunchBilateral Talk between KMS & TMS

13:30-14:15Chair:Yi-Ching Yao

Speaker:Wei-Jing Wang

14:20-14:45Chair:Tsui-Shan Lu

Speaker:Chun-Shu Chen

14:45-15:10Chair:Tsui-Shan Lu

Speaker:Shih-Hao Huang

13:30-14:15Chair:


Ngai-Ching Wong

14:15-14:55Chair:


Ya-Shu Wang

13:30-14:15Chair:

Shyan-Shiou ChenSpeaker:

Eliot Fried

14:30-14:55Chair:

Chang-yuan ChengSpeaker:

Feng-Bin Wang

Coffee Break

H301, General Hall

14:00-16:00Join Your

MathWorld!!

13:30-15:00Forum:

What mathematicalcompetences shouldwe prepare for AI

generations?

Chair and Panelist:Hua-Lun Li

other Panelists:Mei-Heng Yueh

Chun-Hsiung TsengTzer-jen Wei

13:30-14:15Mathemagic

EscapeAuthentic

MathemagicGames


Escape(2nd round)

13:30-15:00Just Fun

Math(1)

Junior highlevel(2)

Elementarylevel

Coffee Break

2018 Taiwan Mathematical Society Annual Meeting8 Dec. 2018(Saturday)

Jhong-Jheng Hall

H301, General Hall

H301, General Hall

Math Museum

Jhong-Jheng Hall

Registration

Opening CeremonyChair：President Jong-Shenq Guo

Plenary Lecture by Professor Fang-Hua LinChair ：Professor Jenn-Nan Wang

H301, General Hall

Jhong-Jheng Hall

H301, General Hall

TMS Meeting & Award Ceremony

Banquet

6 Dec. 2018 Updated

SessionsNumber

Theory andAlgebra

GeometryPartial

DifferentialEquations

DiscreteMathematics

ComputationalMathematics Probability Optimization

DynamicalSystems and

BiomathematicsAnalysis Statistics Popular

MathematicsMathematics

Education

Room G003Library


M210Math Building

G001Library

M212Math Building


M310Math Building

M417Math Building

G002Library

M211Math Building

H301General Hall

Morning: B101Afternoon:

B101(Main)B102, B103(parallel)College of Science

8:30-9:00

9:00-9:50

9:50-10:20

10:20-11:05

Chair:Tzu-Yueh Wang

Speaker:Chu-Feng Nien

Chair:Chun-Chi Lin

Speaker:Eliot Fried

Chair:Sen-Peng Eu

Speaker:Wun-Seng Chou

Chair:Chun-Hao Teng

Speaker:Jong-Shinn Wu

Chair:Shun-Chi Hsu

Speaker:Guan-Yu Chen

Chair:Ruey-Lin Sheu

Speaker:Defeng Sun

Chair:Shyan-Shiou Chen

Speaker:Jung-Chao Ban

*** *** *** *** ***

11:10-11:35

Chair:Tzu-Yueh Wang

Speaker:Fu-Tsun Wei

Chair:Mao-Pei Tsui

Speaker:Chih-Wei Chen

Chair:Yen-chi Roger Lin

Speaker:Wei-Hsuan Yu

Chair:Chun-Hao Teng

Speaker:Yung-Ta Li

Chair:Lung-Chi Chen

Speaker:Gi-Ren Liu

Chair:Defeng Sun

Speaker:Ruey-Lin Sheu


Speaker:Jung-Chao Ban

*** *** ***

11:40-12:05

Chair:Tzu-Yueh Wang

Speaker:Ming-Hsuan Kang

***

Chair:Yen-chi Roger Lin

Speaker:Hsiang-Chun Hsu

Chair:Chun-Hao Teng

Speaker:Chia-Yu Hsu

Chair:Lung-Chi Chen

Speaker:Chien-Hao Huang

Chair:Defeng Sun

Speaker:Peng-Wen Chen


Speaker:Chi-Jen Wang

*** *** ***

12:05-13:30

13:30-13:55 *** *** *** *** *** ***

Chair:Chien-Hao Huang

Speaker:Wei-Ming Hsu

Chair:Jui-Pin Tseng

Speaker:Chun-Hsieh Li

*** ***

14:00-14:25 *** *** *** *** *** ***


Speaker:Chieu Thanh

Nguyen

Chair:Jui-Pin Tseng

Speaker:Yu-Hao Liang

*** ***

14:30-14:55 *** *** *** *** *** ***


Speaker:Jan Harold

Mercado Alcantara

*** *** ***

Plenary Lecture by Professor Wen-Ching Winnie LiChair：Professor Shun-Jen Cheng

Registration

H301, General Hall

Coffee Break

Closing

Math Musium

14:00-16:00Join Your

MathWorld!!

13:30-15:00Workshop:Assessing

MathematicsLiteracy

Chair:Kai-Lin Yang

Facilitators:Yu-Ching HungFeng-Jui Hsieh

Su-Wei Lin

13:30-14:10Chair:


Chih-Sheng Chuang

14:15-14:55Chair:


Hsi-Chun Wu

10:20-10:45Chair: Pi-Wen Tsai

Speaker:Tsui-Shan Lu


Speaker:Wei-Hsiung Chao


Speaker:Yung-Huei Chiou


Speaker:Hong-Ding Yang

10:20-11:05Chair:


Kun-Chuan Wang

11:10-11:55Chair:


Mao-Ting Chien


Escape(1) AuthenticMathemagic

Games(2) Game

Theory InspiredThinking


Escape(2nd round)

Lunch

Jhong-Jheng Hall

13:30-15:00Just Fun

Math(1)

Junior highlevel(2)

Elementarylevel

11:10-11:55Chair:

Hong-Bin ChenSpeaker:

Hsuan-Kuang Shih

10:20-10:45Chair:

Hung-Wen KuoSpeaker:

I-Kun Chen

10:55-11:20Chair:

Hung-Wen KuoSpeaker:

Chueh-Hsin Chang

11:10-11:55Chair:

Tai-Yih TsoSpeaker:

Bor-Chen Kuo

Jhong-Jheng Hall

2018 Taiwan Mathematical Society Annual Meeting9 Dec. 2018(Sunday) 6 Dec. 2018 Updated

目錄 Table of Contents

【邀請演講 Plenary Speeches】 .......................................................................... 1

【研究群論文發表 Sessions and Abstracts】 ................................................... 4

01. 數論與代數 Number Theory and Algebra ...................................................... 4

02. 分析 Analysis ................................................................................................ 14

03. 幾何 Geometry .............................................................................................. 25

04. 動態系統與生物數學 Dynamical Systems and Biomathematics ................. 32

05. 偏微分方程 Partial Differential Equations ................................................... 42

06. 離散數學 Discrete Mathematics ................................................................... 49

07. 計算數學 Computational Mathematics ......................................................... 61

08. 機率 Probability ............................................................................................ 69

09. 最佳化 Optimization ..................................................................................... 76

10. 統計 Statistics ............................................................................................... 95

11. 數學科普 Popular Mathematics .................................................................. 105

12. 數學教育 Mathematics Education .............................................................. 109

【附錄】

1. 論文發表注意事項 Guidelines for the speakers ............................................. 112

2. 大會會場示意圖 Site Map .............................................................................. 113

Plenary Speech Fang-Hua Lin

Silver Professor, Courant Institute of Mathematical Sciences New York University

E-mail: [email protected]

Extremum Problems for Laplacian Eigenvalues

Eigenvalue Problems for Laplacians are among most studied ones in classical analysis, partial differential equations, calculus of variations and mathematical physics. In this lecture I shall discuss some recent progress on a couple extremum problems involving Dirichlet eigenvalues of the Laplacian. These problems have origins in shape optimization, pattern formation,..., and even the data science. We will show how they are related to harmonic maps into singular spaces and some recent works on free boundary value problems involving vector valued functions.

Prof. Lin is currently a Silver Professor of Mathematics at Courant Institute of Mathematical Sciences. He is a world-renowned mathematician in the field of nonlinear analysis and partial differential equations, making important contributions in geometric measure theory, mathematical theory of liquid crystal, homogenization theory, unique continuation property, etc.

Prof Lin graduated from Zhejiang University, China in 1981 and obtained his PhD from University of Minnesota in 1985. His research output includes more than 160 research papers and three books of lecture notes.

Prof Lin received numerous honors and awards, notably the Alfred P. Sloan Research Fellowship, the Presidential Young Investigator Award, AMS Bôcher Prize, a member of American Academy of Arts and Sciences, and S.S. Chern Prize at ICCM.

1

mailto:[email protected]

Plenary Speech Motoko Kotani

Professor, Mathematical Institute, Tohoku University E-mail: [email protected]

Mathematical challenge to understand Materials structure

Materials is a complex system with hierarchical structure; namely materials consists of atoms, atoms form atomic clusters, atomic clusters from their networks, entangled networks form nano-structures. Therefore it is a challenge to understand relations between structures and functions of materials. I would like to discuss how mathematics, discrete geometric analysis, contributes in the challenges. Non-equilibrium materials, topological materials, carbon materials are our targets.

Prof. Kotani got her degree from Tokyo Metropolitan University. After several years as a researcher and lecturer, she joined the Tohoku University in 1999. Kotani's research is in geometry, mainly on discrete geometry and symmetry of figures. Her research on discrete geometric analysis, aiming to understand micro-scale structures and macro-scale properties of materials, finds interesting connection of geometry with crystal lattices.

In 2005, she was awarded Saruhashi Prize, awarded to a Japanese female researcher in natural science. In 2012, she was appointed as director of Advanced Institute for Materials Research (AIMR) of Tohoku University. She was President of Math. Soc. of Japan during 2015-2016.

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Plenary Speech Wen-Ching Winnie Li

Professor, Department of Mathematics, Pennsylvania State University E-mail: [email protected]

The Riemann Hypothesis, Ramanujan conjecture and applications

The Riemann Hypothesis and Ramanujan conjecture are two celebrated open problems in number theory which have important and far reaching consequences. They also appear in different settings. In this talk we discuss their relations and show some applications derived from the occasions where the conjectures hold. This is a survey talk highlighting the interplay between number theory and combinatorics.

Prof. Winnie Li is currently a Distinguished Professor of Mathematics at the Pennsylvania State University. She is highly recognized for her works in number theory, where, in particular, she has made outstanding contributions to automorphic forms. She is also known for her influential works in applying number theory to areas such as coding theory and spectral graph theory.

Li graduated from the National Taiwan University in 1970. She received her Ph.D. in mathematics from the University of California, Berkeley, in 1974, and joined the PSU faculty in 1979. She was appointed director of the National Center of Theoretical Sciences in Taiwan from 2009 to 2014.

Li was awarded the Chern Prize at the International Congress of Chinese Mathematicians in 2010. She became a fellow of the American Mathematical Society in 2010 and was chosen to give the 2015 Noether Lecture.

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數論與代數Number Theory and Algebra地點：G 0 0 3 圖書館

TMS Annual Meeting數學年會2

018

數學年會D e c . 8 / 0 9 : 3 0 - 2 1 : 0 0D e c . 9 / 0 9 : 3 0 - 1 5 : 5 0

演講摘要Speech Abstracts

D e c . 8 / 0 9 : 3 0 - 2 1 : 0 0

1 1 : 2 0 - 1 2 : 0 5

1 3 : 3 0 - 1 3 : 5 5

1 4 : 0 0 - 1 4 : 2 5

1 4 : 3 0 - 1 4 : 5 5 Fq t

1 5 : 2 0 - 1 5 : 4 5

D e c . 9 / 0 9 : 3 0 - 1 5 : 5 0

1 0 : 2 0 - 1 1 : 0 5

1 1 : 1 0 - 1 1 : 3 5

1 1 : 4 0 - 1 2 : 0 5

1 3 : 3 0 - 1 3 : 5 5

1 4 : 0 0 - 1 4 : 2 5

1 4 : 3 0 - 1 4 : 5 5

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Diophantine Approximation for Subvarieties andGCD problems

Tzu-Yueh WangInstitute of Mathematics

Academia SinicaE-mail: [email protected]

A fundamental problem in Diophantine approximation is to study how wellan algebraic number can be approximated by rational numbers. The celebratedRoth’s theorem states that for a fixed algebraic number α, ϵ > 0, and C > 0,there are only finitely many p/q ∈ Q, where p and q are relatively prime integers,such that |α− p

q | ≤C

|b|2+ϵ .There are two kinds of generalization of Roth’s theorem. One is to approx-

imate an algebraic point by rational points in an arbitrary projective varietieswhich is done in a recent work of McKinnon and Roth. Another is in the di-rection of Schmidt’s subspace theorem to study Diophantine approximation ofrational points to a set of hyperplanes in projective spaces, or more generally aset of divisors in an arbitrary projective variety.

I will discuss a Diophantine inequality in terms of subschemes which is ajoint work with Min Ru. In the case of points, it recovers a result of McKinnonand Roth. In the case of divisors, it connects Schmidt’s subspace theorem andthe recent Diophantine approximation results obtained by Autissier, Corvarja,Evertse, Faltings, Ferretti, Levin, Ru, W ustholz, Vojta, Zannier, and etc. I willthen discuss possible application of the above result to the study of gcd problemwhich is a joint work with Ji Gou.

5

The structure of triple homomorphisms ontoprime algebras

Cheng-Kai LiuDepartment of Mathematics

National Changhua University of EducationE-mail: [email protected]

A well-known result of Kaup states that a linear bijection between two JB∗-triples is an isometry if and only if it is a triple isomorphism. Fundamental examples of JB∗-triples are C∗-algebras and JB∗-algebras. From the viewpoint of associative algebras, we characterize the structure of triple homomorphisms from an arbitrary ⋆-algebra onto a prime ∗-algebra. The analogous results for prime C∗-algebras factor von Neumann algebras and standard operator ∗-algebras on Hilbert spaces are also described. As an application, we show that every triple homomorphism from a Banach ⋆-algebra onto a prime semisimple idempotent Banach ∗-algebra or a prime C∗-algebra is automatically continuous.

Keywords: Triple homomorphism, prime algebra, Banach algebra, C∗-algebra, standard operator algebra.

6

Schur-Weyl duality and its variationsYung-Ning Peng

Department of MathematicsNational Central University


Let V = Cn be the standard representation of the general linear Lie algebraG = gl(C). Then for any positive integer d, the tensor space V = V ⊗d isnaturally a G-module and hence a U(G) is the universal enveloping algebra ofG. On the other hand, the symmetric group Sd also acts naturally on V bypermuting the tensor factors. As a result, V, can be viewed as a C[Sd]-moduleas well, where C[Sd] means the group algebra of Sd. The celebrated Schur-Weylduality implies that the images of these two actions in the endomorphism spaceEnd V actually form a full centralizer of each other.

In this short talk, we will try to introduce a few variations of the Schur-Weylduality by replacing the role of G by other structure (in particular, the periplec-tic Lie superalgebra pn) and suitably modifying V. In particular, an interestingalgebra Ad and it’s affine version, called the affine pereplectic Brauer algebraP−d , will show up. If time permitted, we will give a diagrammatic realization

of P−d together with a PBW basis which is very similar to that of the Brauer

algebras and that of the Nazarov-Wenzl algebras. This talk is based on a jointwork with Prof. Chih-Whi Chen (Xiamen University).

Keywords: Schur-Weyl duality, Brauer algebra, pereplectice Lie superal-gebra.

References[1] Richard Brauer, On algebras which are connected with the semisimple

continuous groups, Annals of Mathematics, 38 (1937) 857-872.

[2] Chih-Whi Chen, Yung-Ning Peng, Ane periplectic Brauer algebras, Journalof Algebra, 501 (2018), 345-372.

[3] Dongho Moon, Tensor product representations of the Lie superalgebra p(n)and their centralizers, Communications in Algebra, 31 (2003), 2095-2140.

[4] Maxim Nazarov, Youngs orthogonal form for Brauers centralizer algebra,Journal of Algebra, 182 (1996), 664-693.

7

Representations Functions of Definite BinaryQuadratic Forms over Fq[t]

Wei-Chen YaoDepartment of Mathematics

University of TaipeiE-mail: [email protected]

Let Fq be a finite field of odd characteristic. For A,B,C ∈ Fq[t], a binaryquadratic form f(x, y) = Ax2 +Bxy+Cy2 is called definite if the discriminantmathcalD = B2−4AC has either odd degree or has even degree and non-squareleading coefficient in Fq. For m ∈ Fq[t] and f is a definite binary quadraticform, we define N(f,m) by the number of representations of m in f . Let{f1, · · · , fh, · · · , fH} be a representative set of properly equivalent classes ofdefinite binary quadratic forms for given discriminant D and f1, · · · , fh areprimitive. We define

R(D,m) =1

2

H∑i=1

N(fi,m) and r(D,m) =1

2

h∑i=1

N(fi,m).

In this talk, we will discuss properties of these functions and present formulasfor these functions.

8

Generalized Harmonic Number Sums andSymmetric Functions

Kwang-Wu ChenDepartment of Mathematics

University of TaipeiE-mail: [email protected]

We express some general type of infinite series such as

∞∑n=1

F (H(1)n (z),H

(2)n (z), . . . , H

(ℓ)n (z))

(n+ z)s1(n+ 1 + z)s2 · · · (n+ k − 1 + z)sk,

where F (x1, . . . , xℓ) ∈ Q[x1, . . . , xℓ], H(m)n (z) =

∑nj=1 1/(j+z)m, and s1, . . . , sk

are nonnegative integers with s1+· · ·+sk ≥ 2, as a linear combination of multipleHurwitz zeta functions and harmonic functions.

Keywords: Symmetric functions, multiple Hurwitz zeta functions

References[1] S. Akiyama, H. Ishikawa, On analytic continuation of multiple L-functions

and related zeta-functions, Analytic Number Theory, C. Jia and K. Mat-sumoto (eds.), Developments in Math. Vol. 6, Kluwer, 2002, pp. 1–16. DOI:10.1007/978-1-4757-3621-2 1.

[2] K.-W. Chen, Generalized Harmonic Numbers and Euler Sums, Int. J. Num-ber Theory, 13 (2) (2017), 513–528. DOI:10.1142/S1793042116500883.

[3] K.-W. Chen, C.-L. Chung, M. Eie, Sum formulas of multiple zeta valueswith arguments multiples of a common positive integer, J. Number Theory,177 (2016), 479–496.

[4] M.-A. Coppo, B. Candelpergher, The Arakawa-Kaneko zeta function, Ra-manujan J., 22.2 (2010), 153–162.

[5] R. L. Graham, D. E. Knuth, O. Patashnik, Concrete Mathematics, secondedition, Addison-Wesley, 1994.

[6] M. E. Hoffman, On multiple zeta values of even arguments, arXiv:1205.7051v4 (2016).

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[7] M. E. Hoffman, Harmonic-number summation identities, symmetric func-tions, and multiple zeta values, Ramanujan J., 42 (2) (2017), 501–526.

[8] J. P. Kelliher, R. Masri, Analytic continuation of multiple Hurwitz zetafunctions, Math. Proc. Camb. Phil. Soc., 145 (2008), 605–617.

[9] I. G. MacDonald, Symmetric Functions and Hall Polynomials, 2nd edition,Claredon Press, 1995.

[10] J. Mehta, G. K. Viswanadham, Analytic continuation of multiple Hurwitzzeta functions, J. Math. Soc. Janpan, 69 (4) (2017), 1431–1442.

[11] A. Sofo, Harmonic number sums in higher powers, J. Math. Anal., 2 (2)(2011), 15–22.

[12] A. Sofo, M. Hassani, Quadratic harmonic number sums, Appl. Math. E-Notes, 12 (2012), 110–117.

[13] R. P. Stanley, Enumerative Combinatorics, vol. 2, Cambridge UniversityPress, 1999.

[14] J. Zhao, Sum formula of multiple Hurwitz-zeta values, Forum Mathe-maticum, 27 (2) (2015), 929-936.

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Converse theorem on gamma factorsChu-Feng Nien

College of Mathematics and StatisticsHunan Normal University


The talk is about converse theorem of gamma factors. After Jacquet’s con-jecture (n × [n2 ] converse theorem) is confirmed, we wonder what informationabout representations is encoded in n × 1 gamma factors in finite field caseand its counterpart of level zero cuspidal representations in p-adic case. In ajoint work with Lei Zhang, we use number-theoretic result of Gauss sums toverify n × 1 Local Converse Theorem of cuspidal representations of GLn(Fp),for prime p and n ≤ 5. After the communication with Zhiwei Yun, he appliedgeometric method and established n × 1 Local Converse Theorem for genericrepresentations of GLn(Fq), when n < q−1

2√q + 1 and q is a prime power.

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Green’s functions on Mumford curvesFu-Tsun Wei

Department of MathematicsNational Tsing Hua University


In this talk, we shall introduce an analogue of Green’s function on Mumfordcurves. The special value of Green’s function at s = 0 interprets the volumeof the corresponding curve. Using harmonic analysis on Bruhat-Tits trees, weconnect the derivative of Green’s functions at s = 0 with the Manin-Drinfeldtheta functions, which enables us to show that the special derivative here isequal to twice of the Neron’s local height with sign changed.

12

Geometric Decomposition of Affine Weyl GroupsMing-Hsuan Kang

Department of Applied MathematicsNational Chiao-Tung University


We will introduce a geometric decomposition of affine Weyl groups aris-ing from their actions on Coxeter complexes. This decomposition is length-preserving and gives some new results on invariant theory of affine Weyl groups.

13

分析Analysis

地點：G 0 0 2 圖書館


018

數學年會D e c . 8 / 0 9 : 3 0 - 2 1 : 0 0D e c . 9 / 0 9 : 3 0 - 1 5 : 5 0


D e c . 8 / 0 9 : 3 0 - 2 1 : 0 0

1 1 : 2 0 - 1 2 : 0 5

1 3 : 3 0 - 1 4 : 1 5

1 4 : 1 5 - 1 4 : 5 5

1 5 : 1 0 - 1 5 : 5 0

D e c . 9 / 0 9 : 3 0 - 1 5 : 5 0

1 0 : 2 0 - 1 1 : 0 5

1 1 : 1 0 - 1 1 : 5 5

:

1 3 : 3 0 - 1 4 : 1 0

1 4 : 1 5 - 1 4 : 5 5

:

14

p

p

Hardy spaces associated with Monge-Ampereequation

Chin-Cheng LinDepartment of MathematicsNational Central University


We study the boundedness of singular integrals related to the Monge-Ampere equation established by Caffarelli and G utierrez. They obtained the L 2 bound-edness. Since then the Lp, 1 < p < ∞, weak (1, 1) and the boundedness for these operators on atomic Hardy space were obtained by several authors. It was well known that the geometric conditions on measures play a crucial role in the theory of the Hardy space. In this talk, we establish the Hardy space H via the Littlewood-Paley theory with the Monge-Ampere measure satisfying the

F

doubling property together with the noncollapsing condition, and show the H boundedness of Monge-Ampere singular integrals. The approach is based on

F

the L2 theory and the main tool is the discrete Calderon reproducing formula associated with the doubling property only.

15

The No Trade Principle and the Characterizationof Compact Beliefs

Ngai-Ching WongDepartment of Applied Mathematics

National Chung Hsing UniversityE-mail: [email protected]

We establish the no trade principle, i.e., the no trade theorem and its con-verse, for any dual pair of bet and extended belief spaces, defined on a givenmeasurable space. A key condition is that, except perhaps one of the agents,everyone else has (weak∗) compact sets of beliefs. We find out that in most ofthe models of uncertainty adopted in the economic literature, roughly speaking,the epistemic statement that an agent has a compact set of beliefs is equiv-alent to the economic statement that he has an open cone of positive bets.This improves our understanding of what compactness actually means withinan economic context.

This is a joint work with Man-Chung Ng.

16

2-local isometries on vector-valued Lipschitzfunction spaces

Ya-Shu WangDepartment of Applied Mathematics

National Chung Hsing UniversityE-mail: [email protected]

Let E and F be Banach spaces. Let S be a subset of the space L(E,F )of all continuous linear maps from E into F . A (non-necessarily linear norcontinuous) mapping ∆ : E → F is a 2-local S-map if for any x, y ∈ E, thereexists Tx,y ∈ S, depending on x and y, such that

∆(x) = Tx,y(x) and ∆(y) = Tx,y(y).

In this talk, I will present a description of the 2-local isometries on the Ba-nach space Lip(X,E) of vector-valued Lipschitz functions from compact metricspace X to Banach space E in terms of a generalized composition operator.Also, I will show when every 2-local (standard) isometry on Lip(X,E) is bothlinear and surjective.

Co-author(s): Jimenez-Vargas, L. Li, A. M. Peralta and L. Wang.

17

Maps between different matrix spaces preservingdisjoint matrix pairs

Ming-Cheng TsaiGeneral Education Center

National Taipei University of TechnologyE-mail: [email protected]

In this talk, the structure of the linear maps between two different matrixspaces preserving disjointness is characterized. Moreover, some related resultsfor zero product preservers between matrix algebras are obtained. This is a joinwork with Chi-Kwong Li, Ya-Shu Wang, Ngai-Ching Wong.

Keywords: disjointness preserver, zero product, matrix spaces, matrix al-gebras

18

The Convergence of Calderón ReproducingFormulae of Two Parameters on Some Classical

Function SpacesKun-chuan Wang

Department of Applied MathematicsNational Dong Hwa University


The Calderón reproducing formula is the most important in the study ofharmonic analysis, which has the same property as the one of approximateidentity in many special function spaces. In this talk, we use the idea of sep-aration variables and molecular decomposition to extend single parameter intotwo-parameters and discuss the convergence of Calderón reproducing formula oftwo-parameters in some generalized function spaces of two parameters. Mainly,we focus on Besov spaces in two-parameter and show that these spaces are well-defined by Plancherel-Pôlya inequalities. Consequently, we obtain the normequivalence between Besov spaces and corresponding sequence space in two-parameter. Also we show the convergence of Calderón reproducing formula inBesov space.

Keywords: atomic decomposition, Calderón reproducing formula, Littlewood-Paley, Plancherel-Pôlya inequality

References[1] Bui, H.-Q., Paluszyński M. and Taibleson M.H., A maximal function char-

acterization of weighted Besov-Lipschitz and Triebel-Lizorkin spaces, Stu-dia Math. 119 (1996), 219-246.

[2] Deng, D.G., Han, Y.-S. and Yang, D.C., Besov spaces with non-doublingmeasures. Trans. Amer. Math. Soc., 358 (2006), 2965-3001.

[3] Frazier, M., Jawerth, B., Decomposition of Besov spaces, Indiana Math.J. 34 (1985), 777-799.

[4] Frazier, M. and Jawerth, B., A discrete transform and decompositions ofdistribution spaces, J. Funct. Anal, 93 (1990), 34-170.

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[5] Han, Y.-S., Lee, M.-Y., Lin, C.-C. and Lin, Y.-C., Calderón-Zygmundoperators on product Hardy spaces, J. Funct. Anal. 258 (2010), 2834-2861.

[6] Han, Y.-S., Lu, S.Z., Yang, D.C., Inhomogeneous Besov and Triebel-Lizorkin spaces on spaces of homogeneous type, Approx. Theory Appl(N.S.), 15 (1999), no. 3, 37-65.

[7] Han, Y., Yang, D.C., New characterizations and applications of inhomoge-neous Besov and Triebel-Lizorkin spaces on homogeneous type spaces andfractals, Dissertationes Math. (Rozprawy Mat.) 403 (2002), 102 pp.

[8] Johnson, R., Temperatures, Riesz potentials, and the Lipschitz spaces ofHerz, Proc. Lond. Math. Soc. 27 (1973), 290-316.

[9] Peetre, J., New thoughts on Besov spaces, Duke University MathematicsSeries, No. 1. Mathematics Department, Duke University, Durham, NC,1976.

[10] Taibleson, M. H., On the Theory of Lipschitz Spaces of Distributions onEuclidean n-Space: I. Principal Properties, J. Math. Mech., 13 (1964),407-479.

[11] Triebel, H., Theory of Function Spaces, Monographs in Mathematics, 78Birkhauser Verlag, Basel, 1983.

[12] Weisz, F., On duality problems of two-parameter martingale Hardy spaces,Bull. Sci. Math. 114 (1990), no. 4, 395-410.

[13] Weisz, F., Interpolation between two-parameter martingale Hardy spaces,the real method, Bull. Sci. Math. 115 (1991), no. 3, 253-264.

[14] Weisz, F. The boundedness of the two-parameter Sunouchi operators onHardy spaces, Acta Math. Hungar. 72 (1996), no. 1-2, 121-152.

[15] Yuan, W., Sawano, Y. and Yang, D.C., Decompositions of Besov-Hausdorffand Triebel-Lizorkin-Hausdorff spaces and their applications, J. Math.Anal. Appl. 369 (2010), no. 2, 736-757.

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Linear matrix representations of ternary formsMao-Ting Chien

Department of MathematicsSoochow [email protected]

Peter Lax (1958) conjectured that every hyperbolic ternary form F (t, x, y)of degree n admits a determinantal linear matrix representation, i.e., there existn× n real symmetric matrices H and K satisfying F (t, x, y) = det(tIn + xH +yK). Helton and Vinnikov confirmed in 2007 the Lax conjecture is true. In thistalk, we study the linear matrix representations of the hyperbolic ternary formsassociated to some matrices.

Keywords: Hyperbolic ternary form, determinantal representation, Laxconjecture.

References[1] Mao-Ting Chien, Hiroshi Nakazato, Unitary similarity of the determinan-

tal representation of unitary bordering matrices, Linear Algebra Appl.,541(2018), 13-35.

[2] Mao-Ting Chien, Hiroshi Nakazato, Symmetric representation of ternaryforms associated to some Toeplitz matrices, Symmetry, 10(2018), 55;doi:10.3390/sym10030055

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Douglas-Rachford Algorithm for The GeneralizedDC Programming in Hilbert Spaces

Chih-Sheng ChuangDepartment of Applied Mathematics

National Chiayi UniversityE-mail: [email protected], [email protected]

In this paper, we first study the generalized DC programming in real Hilbert space:

(GDCP) argminx∈H{f(x) = φ(x) + g(x) − h(x)},

where H is a real Hilbert space, φ, g : H → (−∞, ∞] are proper, strongly con-vex, and lower semicontinuous functions, and h : H → (−∞, ∞] is a Fréchet dif-ferentiable with the gradient ∇f(x). Next, we give the Douglas-Rachford algo-rithm and Peaceman-Rachford algorithm to study the generalized DC program-ming. Next, we also study the split generalized DC programming by proposing a split Douglas-Rachford algorithm.

Keywords: Douglas-Rachford algorithm, Peaceman-Rachford algorithm, DC programming, strongly convexity, strongly monotonicity.

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Poincaré Lemma on Some SubRiemannianManifoldsHsi-Chun Wu



Let X = {X1, X2, . . . , Xm} be m linearly independent vector fields definedon an n-dimensional manifold Mn with m ≤ n, and assume that X satisfies thebracket generating property: the vector fields X and finitely many steps of theirLie brackets span TMn. Therefore, Mn can be recognized as a subRiemannianmanifold by Chow’s theorem [6] and the Carnot-Carathéodory distance. LetV = (a1, a2, . . . , am) be a vector-valued function defined on Mn where aj ,j = 1, . . . ,m are smooth functions. The function V is said to be conservative ifthere exists a function f , called the potential function, that satisfies the followingsystem

X1f = a1, X2f = a2, · · · Xmf = am.

It is known that in [2], in virtue of the curl operator [7], a characterization ofconservative vector fields, called the integrability condition, on the Heisenberggroup H1 is provided as{

X21b = (X1X2 + [X1, X2])a,

X22a = (X2X1 + [X2, X1])b,

where X1 = ∂x − 2y∂z, X2 = ∂y + 2x∂z, a, b are smooth functions, and [·, ·] isthe Lie bracket. In [3], a potential function on H1 is given by

f(x, y, z) =

∫ 1

0

⟨U(γ(t)), γ(t)⟩dt,

where ⟨·, ·⟩ is a subRiemannian metric, U = aX1 + bX2, and γ is a geodesic connecting (x, y, z) and the origin.

In this talk, I will discuss integrability conditions and potential functions on two important examples in subRiemannian manifolds: the Heisenberg groups Hn and the quaternion Heisenberg group qH1 [4, 5]. The integrability conditions can be found by using the curl tensor [1]

(curl U)(X, Y ) = Y g(U, X) − Xg(U, Y ) + g(U, [X, Y ]),

23

where U,X, Y are vector fields and g is a Riemannian metric. The potentialfunctions related to conservative vector fields are able to be solved explicitly inintegral forms.

Keywords: Bracket generating property, Heisenberg group, Curl, Integra-bility condition, Poincaré lemma

References[1] O. Calin and D. C. Chang, Geometric mechanics on Riemannian manifolds:

applications to partial differential equations, Birkhäuser Boston, 2005. doi:https://doi.org/10.1007/b138771

[2] O. Calin, D. C. Chang, and M. Eastwood, Integrability conditions forHeisenberg and Grushin-type distributions, Anal. Math. Phys., 4 (2014),99-114. doi: 10.1007/s13324-014-0073-1

[3] O. Calin, D. C. Chang, and J. Hu, Poincaré’s lemma onthe Heisenberg group, Adv. in Appl. Math., 60 (2014), 90-102.http://dx.doi.org/10.1016/j.aam.2014.08.003

[4] D. C. Chang, Y. S. Lin, H. C. Wu, and N. Yang, Poincaré lemma onHeisenberg groups, Applied Analysis and Optimization, 1 (2017), 283-300.

[5] D. C. Chang, N. Yang, and H. C. Wu, Poincaré lemma on quaternion-like Heisenberg groups, Canad. Math. Bull., 61 (2018), 495-508.http://dx.doi.org/10.4153/CMB-2017-027-4

[6] W. L. Chow, Uber Systeme van Linearen partiellen Differentialgleichungenerster Ordnung, Math. Ann., 117 (1939), 98-105.

[7] B. Franchi, N. Tchou, and M. C. Tesi, Div-curl type theorem, H-convergenceand Stokes formula in the Heisenberg group, Comm. Contemp. Math., 8(2006), 67-99.

24

幾何Geometry

地點：B 1 0 2 理學院


018

數學年會D e c . 8 / 0 9 : 3 0 - 2 1 : 0 0D e c . 9 / 0 9 : 3 0 - 1 5 : 5 0


D e c . 8 / 0 9 : 3 0 - 2 1 : 0 0

1 1 : 2 0 - 1 2 : 0 5

1 3 : 3 0 - 1 3 : 5 5

1 4 : 0 0 - 1 4 : 2 5

1 4 : 3 0 - 1 4 : 5 5

1 5 : 2 0 - 1 5 : 4 5

D e c . 9 / 0 9 : 3 0 - 1 5 : 5 0

1 0 : 2 0 - 1 1 : 0 5

1 1 : 1 0 - 1 1 : 3 5

1 1 : 4 0 - 1 2 : 0 5

1 3 : 3 0 - 1 3 : 5 5

1 4 : 0 0 - 1 4 : 2 5

1 4 : 3 0 - 1 4 : 5 5

25

Deformation of area decreasing maps by meancurvature flow

Mao-Pei TsuiDepartment of MathematicsNational Taiwan University


In this talk, I will present some recent results about the the evolution of anarea decreasing map induced by its mean curvature. I will explain the key esti-mates in proving the longtime existence and convergence results under suitablecurvature conditions.

26

Revisit the small sphere limits of Wang-Yauquasi-local mass

Ye-Kai WangDepartment of Mathematics

National Cheng Kung UniversityE-mail: [email protected]

A fundamental problem in general relativity is to measure the energy ofgravitational field in an extended but finite region. The Wang-Yau quasi-localmass is proposed in 2008 to tackle the problem. Besides positivity, theWang-Yauquasi-local mass verifies various classical limits: It recovers ADM mass, Bondi-Sachs energy-momentum for large spheres at spatial infinity and null infinity,and Bel-Robinson tensor for small spheres in vacuum spacetimes. In this talk,we present another approach to the small sphere limits using the idea fromthe proof of positive mass theorem. This is a joint work with Po-Ning Chen,Mu-Tao Wang, and Shing-Tung Yau.

27

Global uniqueness of the minimal sphere in theAtiyah–Hitchin manifold

Chung-Jun TsaiDepartment of MathematicsNational Taiwan UniversityE-mail: [email protected]

In a hyper-Kahler 4-manifold, holomorphic curves are stable minimal sur-faces. One may wonder whether those are all the stable minimal surfaces.

Micallef gave an affirmative answer in many cases. However, this cannotbe true in general. Micallef and Wolfson found that the minimal sphere inthe Atiyah–Hitchin manifold is strictly stable, but cannot be holomorphic withrespect to any compatible complex structure. The minimal sphere in the Atiyah–Hitchin manifold is conjectured to be quite rigid.

In this talk, we will first review the construction of the Atiyah–Hitchin man-ifold, and then explain the uniqueness of that minimal sphere. This is based ona joint work with Mu-Tao Wang.

28

Introduction to Hochschild (co)homology

Sheng-Fu ChiuInstitute of Mathematics


In this talk I will introduce the notion of Hochschild (co)homology of cate-gories satisfying certain microlocal conditions. If time permits, I will discuss itsapplication to symplectic geometric quantities such as displacement energy.

Keywords: Hochschild cohomology, displacement energy, microlocal con-dition.

29

Möbius Kaleidocycles:A New Class of Everting Ring Lnkages

Eliot FriedMathematics, Mechanics, and Materials Unit

Okinawa Institute of Science and Technology Graduate UniversityE-mail: [email protected]

Many mechanical devices from scissors to robotic arms are characterizableas ‘linkages’ or sets of rigid links connected by moving joints. Known linkagesdisplaying a single degree-of-freedom, which facilitates control, have hithertoconsisted of six or fewer links. In this talk, we will introduce a new class of one-degree-of-freedom ring linkages: ‘Möbius kaleidocycles’. These objects consistof seven or more identical hinge-joined links and might thus serve as build-ing blocks in positioning, extrusion, and pultrusion systems, to instance a fewamong many promising applications in engineering, architecture, robotics, andchemistry. They also pose a myriad of intriguing fundamental geometrical andtopological questions, some of which will be touched upon in this talk. This isjoint work with postdoctoral scholar Johannes Schönke.

Keywords: spatial mechanisms, nonorientability, single degree-of-freedom

30

Geometry in Big DataChih-Wei Chen

Department of Applied MathematicsNational Sun Yet-sen University


I will talk about the interactions between geometry and data analysis, espe-cially nonlinear dimensionality reductions.

Keywords: dimensionality reduction, manifold learning

31

動態系統與生物數學Dynamical Systems and Biomathematics

地點：M 4 1 7 數學館


018

數學年會D e c . 8 / 0 9 : 3 0 - 2 1 : 0 0D e c . 9 / 0 9 : 3 0 - 1 5 : 5 0


D e c . 8 / 0 9 : 3 0 - 2 1 : 0 0

1 1 : 2 0 - 1 2 : 0 5

1 3 : 3 0 - 1 4 : 1 5

1 4 : 3 0 - 1 4 : 5 5

1 5 : 2 0 - 1 5 : 4 5

D e c . 9 / 0 9 : 3 0 - 1 5 : 5 0

1 0 : 2 0 - 1 1 : 0 5

1 1 : 1 0 - 1 1 : 3 5

1 1 : 4 0 - 1 2 : 0 5

1 3 : 3 0 - 1 3 : 5 5

1 4 : 0 0 - 1 4 : 2 5

1 4 : 3 0 - 1 4 : 5 5

32

Travelling curved waves in two dimensionalexcitable media

Chang-Hong WuDepartment of Applied Mathematics

National University of TainanE-mail: [email protected]

Wave propagation can occur in various area such as physics, biology, chemi-cal kinetics, and so on. In particular, excitable media, which are often modeledby nonlinear PDEs, can support abundant spatiotemporal dynamics. In thistalk, we focus on a free boundary problem in two-dimensional excitable mediaarising from a singular limiting problem of a FitzHugh-Nagumo-type reaction-diffusion system. The existence, uniqueness and stability of traveling curvedwaves will be discussed. This is a joint work with Hirokazu Ninomiya (MeijiUniversity).

33

Dynamics on membranes mediatedby bulk diffusion

Eliot FriedMathematics, Mechanics, and Materials Unit

Okinawa Institute of Science and Technology Graduate [email protected]

Coupling between an active membrane and bulk diffusion arises in many bio-logical and chemical processes, cell polarization, protein activation waves on cellmembranes, cell-to-cell signaling, membrane bound Turing patterns, Belousov-Zhabotinsky reactions on beads, and so on. Although experimental observa-tions demonstrate the presence of collective synchronous and asynchronous be-haviors along with other more complicated spatiotemporal pattern dynamics,very little theoretical research has been directed at exploring the processes thatgovern these phenomena. Assuming Van der Pol oscillator dynamics on themembrane, we identify two essential parameters governing the system. In thistwo-dimensional parameter space we observe various stable asymptotic dynam-ics, including synchronous, asynchronous, and decaying behavior. In additionto parameter regions where only one type of asymptotic dynamics occurs, thereare also bistable domains. The parameter space contains a degenerate criticalpoint which is considered to be the organizing center of the various dynamicalregimes. This is joint work with Johannes Schönke and Toshiyuki Ogawa.

Keywords: pattern formation, collective behavior, bistability

34

Threshold dynamics of a periodic parabolicsystem modeling the influence of salinity and

nutrient recycling on the growth of algaeFeng-Bin Wang

Department of Natural Science Center for General EducationChang Gung University

E-mail: [email protected], [email protected]

In this talk, we shall study a periodic advection-dispersion-reaction system incorporating the effects o f s alinity, n utrient r ecycling, t emperature, a nd spa-tial variations for the growth of harmful algae in riverine ecosystems. We can introduce the basic reproduction number R0 for algae and show that R0 serves as a threshold value for persistence and extinction of the algae. More precisely, we prove that the washout state is globally attractive if R0 < 1, while there exists a positive periodic state and the algae is uniformly persistent if R0 > 1. This talk is based on ongoing projects joint with Drs. James P. Grover and Xiao-Qiang Zhao.

Keywords: harmful algae, salinity, nutrient recycling, the basic reproduc-tion number

35

Existence and Instability of Traveling Pulses ofGeneralized Keller-Segel Equations

Chueh-Hsin ChangDepartment of Applied Mathematics

Tunghai UniversityE-mail: [email protected]

Keller-Segel Equations exhibit the phenomenon of chemotaxis. It is difficultto find the traveling pulse solutions for the minimal model. In this talk, wetalk about the existence and instability of some Keller-Segel Equations withnonlinear chemical gradients and small diffusions by the geometric singular per-turbation theory and spectral analysis. This is a joint work with Y. S. Chen,John. M. Hong, and B. C. Huang.

Keywords: Keller-Segel model, traveling wave solution

36

On The Dynamics of Shifts of Finite Type onGroups

Jung-Chao BanDepartment of Applied Mathematics

National Dong Hwa UniversityE-mail: [email protected]

In this talk, we consider the dynamics of shifts of finite type on groups (G-SFTs) and focus on case that G is monoid or free group. By using the nonlinearrecursive formula, the complete characterization for the entropy of GSFTs isestablished. Besides, we discuss various kinds of mixing GSFTs, and providethe connection between positive-entropy and mixing GSFTs.

37

Continuation methods and numerical bifurcationanalysis

Te-Sheng LinDepartment of Applied Mathematics

National Chiao Tung UniversityE-mail: [email protected]

A numerical continuation method is developed to follow time-periodic travelling-wave solutions of both local and non-local evolution partial differential equations(PDEs). It is found that the equation for the speed of the moving coordinatecan be derived naturally from the governing equations together with a condi-tion that breaks the translational symmetry. The derived system of equationsallows one to follow the branch of travelling-wave solutions as well as solutionsthat are time-periodic in a frame of reference travelling at a constant speed.We then show examples of the bifurcation and stability analysis of long-wavemodels of electrified falling films as well as films on a rotating cylinder. Finally,we present our recent work on spontaneous autophoretic motion of colloidalparticles in two-dimensional space.

Keywords: numerical continuation method, time-periodic travelling-wavesolution, numerical bifurcation analysis

38

3D Structure Prediction of Alpha-1,4FucosyltransferaseChi-Jen Wang and Ching-Ching Yu

Department of Mathematics and Department of Chemistry and BiochemistryNational Chung Cheng University

[email protected]

Fucosyltransferase (FucT) is an enzyme that transfers fucose sugar to a sugaror protein. We are interested in one FucT from Helicobacter pylori which livesin gastric and duodenal. The function and structure of α-1,3 FucT NCTC11639has been reported a decade ago, the dual functionality of α-1,3/4 FucT UA948has been reported in 2017, however the structure of α-1,4 FucT has not. Theprior experiment shows that α-1,4 FucT DSM6709 also have α-1,3 linkage from.Further, its gene sequence also indicates this enzyme might have the same func-tion as UA948. Our goal is to predict the structures of DSM6709, and comparethese structures to other FucTs to verify the possibility of the dual functionali-ties of α-1,4 FucT DSM6709.

Keywords: α-1,4 Fucosyltransferase, diverse linkages, structure prediction.

39

Backward bifurcation of a network-based SISepidemic model with saturated treatment

functionChun-Hsien Li

Department of MathematicsNational Kaohsiung Normal University


In this talk, we present a study on a network-based SIS epidemic model with a saturated treatment function to characterize the saturation phenomenon of limited medical resources. In this model, we first o btain a t hreshold value R0, which is the threshold condition for the stability of the disease-free equi-librium. We show that a backward bifurcation occurs under certain conditions. More precisely, the saturated treatment function leads to a such bifurcation. In this case, R0 < 1 is not sufficient to eradicate the disease from the population. Numerical simulations are conducted to validate the theoretical results. This is a joint work with Yi-Jie Huang.

Keywords: complex networks, epidemic model, saturated treatment func-tion, backward bifurcation.

40

The influence of awareness on the spreading ofinfectious diseases

Yu-Hao LiangDepartment of Applied Mathematics

National University of KaohsiungE-mail: [email protected]

In this talk, I will discuss the influence of awareness on the epidemic spread-ing. We will propose a multiplex network where the spreading of the diseaseand information occurs, respectively, in two different layers of networks, namelythe physical network and the virtual network. In addition, these two diffusiveprocesses are assumed to interact and affect each other. Such concept wouldcreate a multiplex SIS-UAU epidemic model. Some recent results on these twomodels are to be introduced. This is a joint work with Jonq Juang.

41

偏微分方程Partial Differential Equations地點：M 2 1 0 數學館


018

數學年會D e c . 8 / 0 9 : 3 0 - 2 1 : 0 0D e c . 9 / 0 9 : 3 0 - 1 5 : 5 0


D e c . 8 / 0 9 : 3 0 - 2 1 : 0 0

1 1 : 2 0 - 1 2 : 0 5

1 3 : 3 0 - 1 3 : 5 5

1 4 : 0 0 - 1 4 : 2 5

1 4 : 3 0 - 1 4 : 5 5

1 5 : 2 0 - 1 5 : 4 5

D e c . 9 / 0 9 : 3 0 - 1 5 : 5 0

1 0 : 2 0 - 1 0 : 4 5

1 0 : 5 5 - 1 1 : 2 0

1 1 : 2 0 - 1 2 : 0 5

1 3 : 3 0 - 1 3 : 5 5

1 4 : 0 0 - 1 4 : 2 5

1 4 : 3 0 - 1 4 : 5 5

*掃描QRCODE，點擊講者/標題查看

42

On Well-posedness of Weak SolutionsTai-Ping Liu

Institute of MathematicsAcademia Sinica


There have been very substantial progresses on non-uniqueness of weak so-lutions for incompressible Navier-Stokes and Euler equations, and compressibleEuler equations. The well-posedness problem is a fundamental problem in thetheory of partial differential equations. This talk aims at proposing a differentwell-posedness theory. We illustrate this new theory with a recent study of theauthor with Shih-Hsien Yu on compressible Navier-Stokes equations, and also re-call the celebrated well-posedness theory for hyperbolic conservation laws. Thistalk gives concrete meaning to the author’s talk in annual differential equationsmeeting at Chung-San University few years ago on the same topic.

43

Sharp regularizing estimates for the gain term ofthe Boltzmann collision operator

Jin-Cheng JiangDepartment of Mathematics

National Tsing Hua UniversityE-mail: [email protected]

We prove the sharp regularizing estimates for the gain term of the Boltzmanncollision operator including hard sphere, hard potential and Maxwell moleculemodels. Our new estimates characterize both regularization and convolutionproperties of the gain term which were studied by Lions [4], Wennberg [6],Bouchut & Desvillettes [2], Mouhot & Villani [5] etc. and Duduchava & Kirsch& Rjasanow [3], Alonso & Carneiro & Gamba [1] etc. respectively. The newestimates have the following features. The regularizing exponent is sharp bothin the L2 based inhomogeneous and homogeneous Sobolev spaces which is exactthe exponent of the kinetic part of collision kernel. The functions in these esti-mates belong to a wider scope of (weighted) Lebesgue spaces than the previousregularizing estimates. Furthermore, for the estimates in homogeneous Sobolevspaces, we only need functions lying in Lebesgue spaces instead of weightedLebesgue spaces, i.e., no loss of weight occurs in this case.

Keywords: Boltzmann collision operator, Gain term, Regularizing, hardsphere, hard potential, Maxwell molecule, Fourier integral operator

References[1] R. Alonso, E. Carneiro and I.M. Gamba,Convolution inequalities

for the Boltzmann collision operator, Comm. Math. Physics, 298 (2010),pp. 293-322.

[2] F. Bouchut and L. Desvillettes, A proof of the smoothing propertiesof the positive part of Boltzmann’s kernel, Rev. Mat. Iberoamericana, 14(1998), pp. 47-61.

[3] R. Duduchava, R. Kirsch and S. Rjasanow, On estimates of theBoltzmann collision operator with cutoff, J. Math. Fluid Mech., 8 (2006),pp. 242-266.

44

[4] P.-L. Lions, Compactness in Boltzmann’s equation via Fourier integraloperators and applications. I, II, J. Math. Kyoto Univ., 34 (1994), pp. 391-427, pp. 429-461.

[5] C. Mouhot and C.Villani, Regularity Theory for the Spatially Homoge-neous Boltzmann Equation with Cut-Off, Arch. Rational Mech. Anal., 173(2004), pp. 169-212.

[6] B. Wennberg, Regularity in the Boltzmann equation and the radon trans-form, Comm. Partial Differential Equations, 19 (1994), pp. 2057-2074.

45

Concentration of source terms in generalizedGlimm scheme for initial-boundary problem of

nonlinear hyperbolic balance lawsYing-Chieh Lin

Department of Applied MathematicsNational University of Kaohsiung


In this talk, we consider the initial-boundary value problem for a nonlinear hyperbolic system of balance laws with sources axg and ath. We assume that the boundary data satisfy a inear or smooth nonlinear relation. Generalized Riemann and boundary Riemann problems are provided with the variation of a concentrated on a thin T -shaped region of each grid. We generalize Goodman’s boundary interaction estimates and introduce a new version of Glimm scheme to construct the approximation solutions and its stability is proved by consid-ering two types of conditions on a. The global existence of entropy solutions is established. Under some sampling condition, we find t he e ntropy solutions converge to its boundary values in L1

loc as x → 0+ and the boundary values satisfy the boundary condition almost everywhere in t.

Keywords: nonlinear balance laws, initial-boundary value problem, Rie-mann problem, generalized Glimm scheme, concentration of source, wave inter-action estimates, entropy solutions.

46

Smoothing effect due to mixing in kinetic theoremand its application to the optical tomography

I-Kun ChenInstitute of Applied Mathematical Science

National Taiwan UniversityE-mail: [email protected]

In kinetic theorem, it is known that the combination of collision or averagingand transport can result gaining of regularity, e.g., the celebrated Velocity Av-eraging Lemma by Golse, Perthame, and Sentis 1985 and the Mixture Lemmaby Liu and Yu 2004. For stationary solution in a bounded convex domain, wefind this effect can be realized by interplaying between velocity and space. Wecan decompose the solution to functions of different level of regularity due todifferent times of mixing and use it for optical tomography.

47

Weak Interaction between Traveling Waves in theThree-species Competition-diffusion Systems

Chueh-Hsin ChangDepartment of Applied Mathematics

Tunghai UniversityE-mail: [email protected]

In this talk we consider the weakly interaction between two traveling wave so-lutions of the threes-species Lotka-Volterra competition-diffusion systems. Eachof the two traveling wave solutions has one trivial component (called trivialwaves). By the asymptotic behavior of the trivial waves and the existence re-sults of the two-species traveling waves, we can observe the dynamics of thedistance between the two trivial waves. We proved that there exists an unsta-ble three-species traveling wave solution which is close to the two trivial waves.This is a joint work with Prof. Chiun-Chuan Chen and Prof. Shin-Ichiro Ei.

Keywords: Lotka-Volterra system, traveling wave solution

48

離散數學Discrete Mathematics

地點：G 0 0 1 圖書館


018

數學年會D e c . 8 / 0 9 : 3 0 - 2 1 : 0 0D e c . 9 / 0 9 : 3 0 - 1 5 : 5 0


D e c . 8 / 0 9 : 3 0 - 2 1 : 0 0

1 1 : 2 0 - 1 2 : 0 5

1 3 : 3 0 - 1 3 : 5 5

1 4 : 0 0 - 1 4 : 2 5

1 4 : 3 0 - 1 4 : 5 5

1 5 : 2 0 - 1 5 : 4 5

D e c . 9 / 0 9 : 3 0 - 1 5 : 5 0

1 0 : 2 0 - 1 1 : 0 5

1 1 : 1 0 - 1 1 : 3 5

1 1 : 4 0 - 1 2 : 0 5

1 3 : 3 0 - 1 3 : 5 5

1 4 : 0 0 - 1 4 : 2 5

1 4 : 3 0 - 1 4 : 5 5

49

Rainbow Ramsey Number for PosetsWei-Tian Li

Department of Applied MathematicsNational Chung Hsing University

[email protected]

We address the following rainbow Ramsey problem: For posets P , Q whatis the smallest number n such that any coloring of the elements of the Booleanlattice Bn either admits a monochromatic copy of P or a rainbow copy of Q.We consider both weak and strong (non-induced and induced) versions of thisproblem.

Keywords: Ramsey number, rainbow coloring, Boolean lattices

References[1] F.-H. Chang, D. Gerbner, W.-T. Li, A. Methuku, D. Nagy, B.

Patkós, M. Vizer, Rainbow Ramsey problems for the Boolean lattice,arXiv:1809.08629v1, submitted.

[2] H.-B. Chen, Y.-J. Cheng, W.-T. Li, C.-A. Liu, The Rainbow Ramsey Num-ber for Boolean Lattices, manuscript.

50

Comparability and Cocomparability BigraphsJephian C.-H. Lin

Department of Applied MathematicsNational Sun Yat-sen UniversityEmail: [email protected]

Let F be a family 0, 1-matrix. A 0, 1-matrix M is symmetrically F-free ifthere is a permutation matrix P such that P⊤MP does not contain any S ∈ Fas a submatrix. For a given graph G, the neighborhood matrix of G is definedas A(G) + I, where A(G) is the adjacency matrix and I is the identity matrix.Several important graph classes are known to have a characterization from thematrix point of view. For example, let

Γ =

[1 11 0

]and slash =

[0 11 0

].

Thus, the strongly chordal graphs are the graphs whose neighborhood matrixis symmetrically {Γ}-free; the cocomparability graphs are the graphs whoseneighborhood matrix can be permuted to avoid slash on the main diagonal; andthe interval graphs are the graphs whose neighborhood matrix is symmetrically{Γ, slash}-free. Note that the set of interval graphs is the intersection of theset of strongly chordal graphs and the set of cocomparability graphs. Thereare bipartite analogues for the strongly chordal graphs and the interval graphs,namely, the bipartite chordal graphs and the interval containment graphs. Inthis talk, we introduce the cocomparability bigraphs from the matrix perspectiveas a bipartite analogue to the cocomparability graphs.

This is a joint work with Pavol Hell, Jing Huang, and Ross M. McConnell.Keywords: submatrix avoiding, comparability graph, bigraph

51

On the determinant of distance matrices ofgraphs

Yen-Jen ChengDepartment of Mathematics

National Taiwan Normal UniversityE-mail: [email protected]

For a connected graph G = (V ;E), the distance matrix D(G) = (dij) is asquare matrix with index set V and dij the distance between i and j. In 1971,Graham and Pollak proved that if T is a tree, then det(D(T )) only dependson the order of T . In this talk, I will give new classes of graphs such thatdet(D(G)) is a constant among each class. In addition, I will introduce theaddressing problem and find the addressing number for these new graphs. Thisis a joint work with Jephian Chin-Hung Lin.

Keywords: CP graph, distance matrix, determinant, inertia

References[1] R. B. Bapat. Graphs and Matrices. Second edition. Springer 2014. (Chapter

9)

[2] R. B. Bapat and S. Sivasubramanian. The second immanant of some com-binatorial matrices. Trans. Comb., 4 (2015) 23-35.

[3] Y.-J. Cheng and J. C.-H. Lin. On the distance matrices of the CP graphs.ArXiv:1805.10269, submitted.

[4] J. H. van Lint and R. M. Wilson. A Course in Combinatorics. Secondedition. Cambridge 2001. (Chapter 9)

52

Conjectures on status sequences andbranch-weight sequences of trees

Jen-Ling ShangDepartment of Banking and Finance

Kainan UniversityE-mail: [email protected]

The status [1, 2, 3, 4, 5] of a vertex in a graph is the sum of the distances between the vertex and each vertex in the graph. The status sequence [1, 2, 3, 4, 5] of a graph is the list of the statuses of all vertices arranged in nondecreasingorder. A graph is called status injective [1, 2, 3, 4, 5] if the status sequenceconsists of distinct numbers. A tree is called weakly status injective [5] if anytwo vertices of the tree having the same status are endvertices. A tree T issaid to be status unique [3, 5] in the family of all trees if whenever T ′ is a treewith the same status sequence as T , then T ′ and T are isomorphic. Recently[5] shows that a weakly status injective tree is status unique in the family of alltrees. We have the following conjecture.

Conjugate 1: A tree in which all non-endvertices having distinct statuses is status unique in the family of all trees.

A branch [6, 7, 8, 9] at a vertex b in a tree is a maximal subtree containing b as an endvertex. Let B1, B2, . . . , Bm (M ≥ 1) be several distinct branches at b in the tree. Assume that U is the subtree induced by V (B1)∪V (B2)∪·∪V (Bm). Then U is called a union branch [5] at b. The branch-weight [6, 7, 8, 9] of b is the maximum number of edges in any branch at b. The branch-weight sequence [6, 7, 8] of a tree is the list of the branch-weights of all vertices arranged in non-increasing order. Let T be a tree and x, y be a pair of distinct non-endvertices in T with the same status. Suppose that A is a union branch at x and B is a union branch at y, where V (A) ∩ V (B) = ϕ and |V (A)| = |V (B)|. Let C be the subgraph of T induced by the vertex set (V (T ) − (V (A) ∪ V (B))) ∪ {x, y}. Let T ′ be the tree constructed from A, B, and C by identifying the vertex x in A with the vertex y in C, and identifying the vertex y in B with the vertex x in C. It is shown that T ′ and T have the same status sequence [4]. We call such a tree T ′ a status-retained transfer of T. We now propose the following conjectures.

Conjugate 2: Assume that T0 is a tree and there is at least a pair of distinct non-endvertices of T0 with the same status. Let T be a tree. Then, T has the

53

same status sequence as T0 if and only if there exist trees T1, T2, . . . , Tk (k ∈ N)such that Ti is a status-retained transfer of Ti−1 for i = 1, 2, . . . , k and Tk ≃ T .

Conjugate 3: Two trees have the same branch-weight sequence if they havethe same status sequence.

Remark: Supported by the Ministry of Science and Technology of R.O.C. un-der grants MOST 105-2115-M-424-001, MOST 106-2115-M-424-001.

Keywords: status, status sequence, branch-weight, branch-weight sequence,tree.

References[1] F. Buckley, F. Harary, Unsolved problems on distance in graphs, Electron.

Notes Discrete Math. 11 (2002) 89-97.

[2] L. Pachter, Constructing status injective graphs, Discrete Appl. Math. 80(1997) 107-113.

[3] J.-L. Shang, C. Lin, Spiders are status unique in trees, Discrete Math. 311(2011) 785-791.

[4] J.-L. Shang, On constructing graphs with the same status sequence, ArsComb. 113 (2014) 429-433.

[5] J.-L. Shang, T.-W. Shyu, C. Lin, Weakly status injective trees are statusunique in trees, Ars Comb. 139 (2018) 133-143.

[6] J.-L. Shang, C. Lin, An uniqueness result for the branch-weight sequencesof spiders, Whampoa-An Interdisciplinary Journal 54 (2008) 31-38.

[7] R. Skurnick, Extending the concept of branch-weight centroid number tothe vertices of all connected graphs via the Slater number, Graph TheoryNotes of New York 33 (1997) 28-32.

[8] R. Skurnick, A characterization of the centroid sequence of a caterpillar,Graph Theory Notes of New York 41 (2001) 7-13.

[9] P.J. Slater, Accretion centers: A generalization of branch weight centroids,Discrete Appl. Math. 3 (1981) 187-192.

54

Nonnegative Roots of MatricesPeng-Ruei Huang

Graduate School of Science and TechnologyHirosaki University


The root of matrices is a classical problem in matrix theory which can betraced back to the work of Arthur Cayley in 1858. However, not much is knownabout the question of existence of entrywise nonnegative square roots for anonnegative matrix. We will consider the nonnegative roots of rank-one matricesand circulant matrices, etc. The necessary and sufficient conditions for theexistence of the nonnegative pth root of a circulant matrix with the order 3 and4 will be given. Moreover, it is proved that the roots of a circulant matrix arecirculant if and only if its eigenvalues are all distinct.

Keywords: Circulant matrix, nonnegative matrix, matrix roots

References[1] A. Cayley, A memoir on the theory of matrices, Phil. Trans. Roy. Soc.

London, 148 (1858), 17-37.

[2] A. Cayley, On the extraction of the square root of a matrix of the thirdorder, Proc. Roy. Soc. Edinburgh, 7 (1872), 675-682.

[3] N.J. Higham and L.-J. Lin, On pth roots of stochastic matrices, LinearAlgebra Appl., 435 (2011), 448-463.

[4] P.-R. Huang, Nonnegative roots for circulant matrices of order less thanfour, submitted.

[5] R. Loewy and D. London, A note on an inverse problems for nonnegativematrices, Linear Multilinear Algebra, 6 (1978), 83-90.

[6] B.-S. Tam and P.-R. Huang, Nonnegative square roots of matrices, LinearAlgebra Appl., 498 (2016), 404-440.

55

On the Roots of Certain Dickson PolynomialsAart Blokhuis, Xiwang Cao, Wun-Seng Chou, and Xiang-Dong Hou

Institute of MathematicsAcademia Sinica


Let n be a positive integer, q = 2n, and let Fq be the finite field with qelements. For each positive integer m, let Dm(X) be the Dickson polynomial ofthe first kind of degree m with parameter 1. Assume that m > 1 is a divisor ofq + 1. We study the existence of α ∈ F∗

q such that Dm(α) = Dm(α−1) = 0. Wealso explore the connections of this question to an open question by Wiedemannand a game called “Button Madness”.

Keywords: absolutely irreducible, button madness, Dickson polynomials,Fermat number, finite field, reciprocal polynomial

References[1] A. Blokhuis and A. E. Brouwer, Button madness, available at

http://www.win.tue.nl/∼aeb/preprints/madaart2c.pdf.

[2] W.-S. Chou, The factorization of Dickson polynomials over finite fields,Finite Fields Appl. 3 (1997), 84-96.

[3] W.-S. Chou, J. Gomez-Calderon and G. L. Mullen, Value sets of Dicksonpolynomials over finite fields, J. Number Theory 30 (1988), 334–344.

[4] M. Freedman, Priviate communication.

[5] G. H. Hardy and E. M. Wright, The Theory of Number, Oxford UniversityPress, Oxford, UK, 1971.

[6] X. Hou, G. L. Mullen, J. A. Sellers, J. L. Yucas, Reversed Dickson polyno-mials over finite fields, Finite Fields Appl. 15 (2009), 748-773.

[7] R. Lidl, G.L. Mullen and G. Turnwald, Dickson Polynomials, PitmanMonographs and Surveys in Pure and Applied Mathematics, 65, LongmanGroup UK Limited 1993.

[8] R. Lidl, H. Niederreiter, Finite Fields, Encyclopedia Math. Appl. Vol. 20,Addison-Wesley, Reading, 1983.

56

[9] H. Meyn, On the construction of irreducible self-reciprocal polynomialsover finite fields, Applicable Algebra in Engineering, Communication andComputing, 1 (1990), 43-53.

[10] The Online Encyclopedia of Integer Sequences, A001122, A093179,http://oeis.org/

[11] D. Wiedemann, An iterated quadratic extension of GF(2), Fibonacci Quart.26 (1988), 290-295.

[12] http://www.fermatsearch.org/factors/composite.php

57

Recent progress on equiangular linesWei-Hsuan Yu



In this talk, I will talk about the history and background knowledge ofequiangular line problems. Then, I will talk our contribution in this area. Thistalk is based on the joint work with Dr. Yen-Chi Lin.

58

A folding phenomenon on partitionsHsiang-Chun Hsu

Department of MathematicsTamkang University


In this talk we will introduce the signed q-counting over partitions whoseFerrers diagrams fit inside a given partition, where the sign is the parity of thesize and the enumerator statistic is the length. We will introduce several q-identities, exhibiting a certain pattern which we called the folding phenomenon.

Keywords: partition, Ferrers diagram, q-analogue, folding phenomenon

References[1] R.M. Adin, Y. Roichman, Equidistribution and sign-balance on 321-

avoiding permutations, Sémin. Loth. Combin. 51 (2004) B51d.

[2] W.Y.C. Chen, L.W. Shapiro, L.L.M. Yang, Parity reversing involution onplane trees and 2-Motzkin paths, European J. Combin. 27 (2006) 283–289.

[3] J. Désarménien, D. Foata, The signed Eulerian numbers, Discrete Math.99 (1992) 49–58.

[4] S.-P. Eu, S.-C. Liu, Y.-N. Yeh, Odd or even on plane trees, Discrete Math.281 (2004) 189–196.

[5] S.-P. Eu, T.-S. Fu, Y.-J. Pan, C.-T. Ting, Sign-balance identities of Adin-Roichman type on 321-avoiding alternating permutations, Discrete Math.312 (2012) 2228–2237.

[6] S.-P. Eu, T.-S. Fu, Y.-J. Pan, C.-T. Ting, Baxter Permutations, Maj-balances, and Positive Braids, Electronic J. Combin. 19 Issue 3 (2012)P26.

[7] S.-P. Eu, T.-S. Fu, Y.-J. Pan, C.-T. Ting, Two refined major-balance iden-tities on 321-avoiding involutions European J. Combin. 49 (2015) 250–264.

[8] S.-P. Eu, T.-S. Fu, Y.-J. Pan, P.-L. Yan, More on double Simsun permuta-tions, non-published manuscript.

[9] T. Mansour, Equidistribution and sign-balance on 132-avoiding permuta-tions, Sémin. Loth. Combin. 51 (2004) B51e.

59

[10] A. Reifegerste, Refined sign-balance on 321-avoiding permutations, Euro-pean J. Combin. 26 (2005) 1009–1018.

[11] A. Robertson, D. Saracino, D. Zeilberger, Refined restricted permutations,Ann. Comb. 6 (2002), 427–444.

[12] M. Shattuck, Parity theorems for statistics on permutations and Catalanwords, Integers: Electronic J. Combin. Number Theory 5 (2005) #A07.

[13] R. Simion, F.W. Schmidt, Restricted permutations, European J. Combin.6 (1985) 383–406.

[14] R. Stanley. Some remarks on sign-balanced and maj-balanced posets. Adv.Appl. Math. 34(4) (2005) 880–902.

[15] C.-T. Ting, Folding phenomena of some classes of permutations, Thesis,2017.

[16] M. Wachs, An involution for signed Eulerian numbers, Discrete Math. 99(1992) 59–62.

60

計算數學Computational Mathematics地點：M 2 1 2 數學館


018

數學年會D e c . 8 / 0 9 : 3 0 - 2 1 : 0 0D e c . 9 / 0 9 : 3 0 - 1 5 : 5 0


主辦單位/ 協辦單位/

D e c . 8 / 0 9 : 3 0 - 2 1 : 0 0

1 1 : 2 0 - 1 2 : 0 5 吳金典Chin-Tien Wu

Mathematics in 3D imaging and its applications

1 3 : 3 0 - 1 3 : 5 5

黃杰森Chieh-Sen Huang

Von Neumann stable, implicit finite volume WENO schemes for hyperbolic conservation laws

1 4 : 0 0 - 1 4 : 2 5

嚴健彰Chien-Chang Yen

Self-Gravitational Force Calculation Using A Direct Method for Adaptive Mesh Refinement

1 4 : 3 0 - 1 4 : 5 5

游承書Cheng-Shu You

A finite difference scheme for strongly coupled systems of singularly perturbed equations

1 5 : 2 0 - 1 5 : 4 5

D e c . 9 / 0 9 : 3 0 - 1 5 : 5 0

1 0 : 2 0 - 1 1 : 0 5

1 1 : 1 0 - 1 1 : 3 5

1 1 : 4 0 - 1 2 : 0 5

1 3 : 3 0 - 1 3 : 5 5

1 4 : 0 0 - 1 4 : 2 5

1 4 : 3 0 - 1 4 : 5 5

吳宗信Jong-Shinn Wu

RAPIT (Rigorous Advanced Plasma Integration Testbed): A Parallel Scientific Computational Platform

李勇達Yung-Ta Li

A pseudospectral method for the solution of the Helmholtz equation

許佳璵Chia-Yu Hsu

The Study of Schooling Pattern of Lampreys

61

Mathematics in 3D imaging and its applicationsChin-Tien Wu

Department of Applied MathematicsNational Chiao-Tung UniversityE-mail: [email protected]

In this talk, I shall introduce some basics mathematics in 3D imaging andits applications. Principles in 3D scanning, mesh generation from point cloud,mesh processing, texture mapping and color tuning, etc. will be presented. Iwould like to share my recent works in this area with audience, especially thosechallenges and difficulties that I am still struggling with.

62

Von Neumann stable, implicit finite volumeWENO schemes for hyperbolic conservation laws

Chieh-Sen HuangDepartment of Applied Mathematics

National Sun Yat-sen UniversityE-mail: [email protected]

We present a new approach to defining implicit WENO ( iWENO) schemes for systems of hyperbolic conservation laws. The approach leads to schemes that are simple to implement, high order accurate, maintain local mass conservation, apply to general computational meshes, and appear to be fairly robust. We present third and fifth order finite volume schemes in one and two space dimen-sions. We show that these iWENO schemes are unconditionally stable in the sense of von Neumann stability analysis, assuming the solution is smooth. The solution is approximated efficiently by two or three degrees of freedom per com-putational mesh element, independent of the spatial dimension. In space, the degrees of freedom are reconstructed implicitly to give high order approximation, while avoiding shocks and steep fronts due to the WENO framework. In time, high order quadrature is employed to produce a one step scheme. The approach is quite general, and we apply it to advection-diffusion-reaction equations with simple diffusion a nd r eaction t erms. N umerical r esults o n n onuniform meshes in one and two space dimensions are presented. These explore the properties of the new schemes for solving hyperbolic conservation laws, advection-diffusion equations, advection-reaction equations, and the Euler system.

63

Self-Gravitational Force Calculation Using ADirect Method for Adaptive Mesh Refinement

Chien-Chang YenDepartment of MathematicsFu-Jen Catholic University


A direct approach for self-gravitational force calculation of second order accuracy based on uniform grid discretization has been proposed by Yen et al. The method improves the N-body calculation on accuracy using the exact integration of kernel functions and employs fast Fourier transform (FFT) to reduce the computational complexity to nearly linear. This direct approach is also free artificial boundary conditions. However, the uniform discretization is a limitation. Due to computational facility or power has been improved during the past decade, it promotes us to investigate the direct method for non-uniform grid discretization preserving second order accuracy and simulations in reality with the help of graphic process units (GPU) to speed up computational time. The proposed method is more flexible on grid discretization and has the potential to be applied to studies the gaseous morphology of disk galaxies and the planetary migration. This is a join work with Yao-Huan Tseng and Hsien Shang.

64

A finite difference scheme for strongly coupledsystems of singularly perturbed equations

Cheng-Shu YouDepartment of Applied Mathematics

Feng Chia UniversityE-mail: [email protected]

In this talk, we will consider the strongly coupled systems of singularly per-turbed convection-diffusion equations, where strong coupling means that the so-lution components in the system are coupled together through their first deriva-tives. By decomposing the coefficient matrix of convection term into the Jordan canonical form, we fist construct a so-called Il’in-Allen-Southwell (IAS) scheme for 1D systems and then extend the scheme to 2D systems by employing an al-ternating direction technique. From the numerical results, we can observe that when the perturbation parameter ε is small enough, the developed IAS scheme is fist order convergent in the discrete maximum norm uniformly in ε on uniform meshes. This is a joint work with Po-Wen Hsieh and Suh-Yuh Yang.

Keywords: boundary and interior layers, Il’in-Allen-Southwell scheme, sin-gularly perturbed convection-diffusion e quation, s trongly c oupled s ystem, uni-form convergence.

65

RAPIT (Rigorous Advanced Plasma Integration Testbed):A Parallel Scientific Computational Platform

Jong-Shinn WuDepartment of Mechanical Engineering

National Chiao Tung UniversityE-mail: [email protected]

Many important and challenging science and engineering problems require modeling of com-plex plasma and flow physics applying hybridization of different continuum- and/or particle-basedsolvers. Examples may include plume analysis of reaction control thrusters on upper-stage rocketand satellite in orbit, rocket plume analysis at high altitude, aerodynamic analysis of atmospheric-pressure dielectric barrier discharge (DBD) actuator, radical distribution of atmospheric-pressureplasma jet, ion thruster plume analysis, and plasma distribution in etching and thin-film depo-sition chambers at low pressure, to name a few. These studies often utilize independent solversdeveloped previously and integrate them in a non-self-consistent approach, which makes theirapplications and future extension highly inflexible. Thus, a highly flexible simulation platform,which allows straightforward addition and integration of different solvers with a self-consistentapproach while maintaining efficient computations, is strongly needed to tackle some challengingproblems with complex plasma/flow physics.

In this talk, I will report our recent development of a new C++ object-oriented multi-physicssimulation platform named Rigorous Advanced Plasma Integration Testbed (RAPIT) usingunstructured-grid finite-volume method with parallel computing through MPI (message passinginterface) on distributed-memory PC clusters. The proposed RAPIT with both embedded PDEand particle solver related objects can easily accommodate continuum- and/or particle-basedsolvers with some proper hybridization algorithm in a self-consistent way. For the former, it mayinclude, but not limited to, the Navier-Stokes (NS) equation solver for general gas flow modelingand the plasma fluid modeling code for general low-temperature plasma modeling. For the latter,it may include the particle-in-cell Monte Carlo collision (PIC-MCC) solver for very low-pressuregas discharge simulation and the direct simulation Monte Carlo (DSMC) solver for rarefied neu-tral gas flow modeling. Many distinct features of RAPIT include single or multiple mesh(es) fordifferent solvers or species with automatic interpolation relation, essentially the same source codefor 2D and 3D problems due to nearly operator-like programming style, and embedded parallelimplementation, among others. Some preliminary results of DSMC, PIC-MCC and NS equa-tion and fluid modeling solvers in many practical engineering problems will be presented in thistalk. In addition, a byproduct of RAPIT, ultraMPP (ultra-fast Massively Parallel Processing),which is a parallel computing platform for PDE related solvers, will also be briefly introduced. Itis designed to greatly reduce the development time of parallel 2D/3D codes from years to weekswhile maintaining a highly manageable and consistent source coding framework for researchers.Some major findings along with outlook are summarized at the end of this presentation.

Remark: Co-authors

Y. M.∼Lee and M.∼H. HuPlasma Taiwan Innovative Corp.

Juh-bei City, Hsunchu County, Taiwan

66

A pseudospectral method for the solution of theHelmholtz equation

Yung-Ta LiDepartment of MathematicsFu-Jen Catholic University


In this talk, we present a pseudospectral method for the Helmholtz equa-tion. The key of the numerical algorithm is to choose a suitable basis associated with the Legendre polynomials that has the following two features: (1) bound-ary conditions are met and (2) the linear system arising from discretizing the Helmholtz equation under the basis is easily solved. To interpretate the proce-dure of constructing such a basis, we first introduce two matrix decompositions which are the discrete analogues of the recursion formula and the orthogonal property of the Legendre polynomials, respectively. Subsequently the basis can be constructed through performing row/column operations on the matrix de-compositions. Numerical experiments are presented to validate the proposed method.

Keywords: Helmholtz equation; pseudospectral method; Legendre polyno-mials; tridiagonal matrix.

67

The Study of Schooling Pattern of LampreysChia-Yu Hsu

Department of Applied MathematicsFeng Chia University


The schooling pattern [1] in marine ecology is a common migration patternfor fishes of different swimming styles, such as carangiform of makrells, sub-carangiform of salmonids or anguiliform of eels [2]. This pattern is not only tomove for food or survival, but also to avoid the predators and save the bodyenergy loss, such as diadromous fishes of eels or salmons. In this talk, a model ofmultiple annugiliform swimmers, such as lamprey [3], is created to simulate theschooling pattern. The adaptive mesh refinement immersed boundary methodis used for the numerical solution for the simulations. Moreover, the factors ofbody [4], such as body stiffness, spacing or body waveform, for swimming inschooling pattern will be investigated.

Keywords: lamprey, schooling pattern, adaptive mesh refinement immersedboundary method

References[1] A.D. Becker, H. Masoud, J. W. Newbolt1, M. Shelley, L. Ristroph1 Hydro-

dynamic schooling of apping swimmers Natural Communication (2015),1-8

[2] Eric D. Tytell The hydrodynamics of eel swimming, II. Effect of swimmingspeed J. of Exp. Biol., 207 (2004), 3265-3279.

[3] F. W. H. Beamish Swimming Performance of Adult Sea Lamprey, Petromyzon marinus, in Relation, to Veight and Temperature Trans. Amer. Fish.Soc., (1974), NO.2, 355-358

[4] E. D. Tytell, M. C. Leftwich, C-Y Hsu, B. E. Griffth, A. H. Cohen, A.J.Smits and C. Hamlet and L. J. Fauci Role of body stiffness in undulatoryswimming: Insights from robotic and computational models Phys. Rev.Flu., 1, (2016), 073202

68

機率Probability

地點：B 1 0 3 理學院


018

數學年會D e c . 8 / 0 9 : 3 0 - 2 1 : 0 0D e c . 9 / 0 9 : 3 0 - 1 5 : 5 0


D e c . 8 / 0 9 : 3 0 - 2 1 : 0 0

1 1 : 2 0 - 1 2 : 0 5

1 3 : 3 0 - 1 3 : 5 5

1 4 : 0 0 - 1 4 : 2 5

1 4 : 3 0 - 1 4 : 5 5

1 5 : 2 0 - 1 5 : 4 5

D e c . 9 / 0 9 : 3 0 - 1 5 : 5 0

1 0 : 2 0 - 1 1 : 0 5

1 1 : 1 0 - 1 1 : 3 5

1 1 : 4 0 - 1 2 : 0 5

1 3 : 3 0 - 1 3 : 5 5

1 4 : 0 0 - 1 4 : 2 5

1 4 : 3 0 - 1 4 : 5 5

l

69

Some Stochastic Control Problems in the Studyof Finance

Shuenn-Jyi SheuDepartment of Mathematics and Department of Applied Mathematics

National Central University and National Chengchi UniversityE-mail: [email protected]

Merton(1969) studied a continuous time portfolio optimization problem us-ing dynamic programming approach to solve the problem. In Merton (1971)a general Markovian model was discussed and the HJB equation was derivedwhich is a nonlinear PDE with complicated nonlinearity. Solving the HJB equa-tion left open for many years. The development of martingale method in Pliska(1986) and Karatzas-Lehoczky-Sethi (1986) provides a powerful alternative ap-proach to find a solution when the market is complete. For the case of incom-plete market, market completion, as a consequence of martingale method, hasbeen studied in Pages(1987), He-Pearson(1991) and Karatzas-Lehoczky-Shreve-Xu(1991), and also some recent works of Haugh-Kogan-Wang(2006), Rogers(2003), Klein-Rogers and Rogers-Zacvkowski(2013).

In this talk, a review is given to the recent developments of the studies ofMerton portfolio optimization problems following the dynamic programmingapproach. They include risk-sensitive portfolio optimization problem, upsidechance and downside risk probabilities optimization and optimal consumptionproblem. The line of developments follow the ideas of Fleming(1995), which sug-gests to reformulate the risk-sensitive optimization problem as a risk-sensitivestochastic control problem. We will also discuss recent ideas to find a solutionfor the finite time consumption problem by rewriting the HJB equation as aninf-sup type Isaacs equation, suggested by the idea of market completion. TheMerton problem for a model with risk income will be also discussed. When themarket is complete, a solution of the HJB equation can be found. When themarket is incomplete, the solution of the HJB equation remains a challenge.

We also mention some investment problems with risky income from insur-ance, and for model with delay. The talk is based on joint works with H. Nagai,H. Hata, L.H Sun and Z. Zhang.

Keywords: Merton problem; dynamic programming; Hamilton-Jacobi-Bellmanequationsstochastic control.

70

On the Stochastic Heat EquationsShang-Yuan Shiu

Department of MathmaticsNational Central University


We consider the following stochastic heat equation:

∂tut(x) = ∂xxut(x) + σ(ut(x))W (t, x),

x ∈ (−∞,∞) or x ∈ [−1, 1] with certain boundary conditions subject to initialdata u0(x). We will discuss how initial data and the noise term effect thebehaviors of the solution. This is based on several different papers.

Keywords: Stochastic heat equations, intermittency, dissipation.

71

Some Limit Distributions of DiscountedBranching Random Walks

Jyy-I HongDepartment of Applied Mathematics

National Chengchi UniversityE-mail: [email protected]

We consider a Galton-Watson discounted branching random walk {Zn, ζn}n≥0,where zn is the population of the nth generation and ζn is a collection of thepositions on R of the the Zn individuals in the nth generation, and let Yn bethe position of a randomly chosen individual from the nth generation and Zn(x)be the number of points in ζn that are less than or equal to x, for x ∈ R. Inthis talk, we present the limit theorems for the distributions of Yn and Zn(x)

Znin

both supercritical and explosive cases.Keywords: branching random walks, branching processes, coalescence, su-

percritical, explosive

References[1] K. B. Athreya. Discounted branching random walks. Advanced Applied

Probability, 17, 1985, 53-66.

[2] K. B. Athreya and J.-I. Hong. An application of the coalescence theory tobranching random walks. Journal of Applied Probability, 50, 2013, 893-899.

72

The max-ℓ2 mixing of reversible Markov chainsGuan-Yu Chen

Department of Applied MathematicsNational Chiao Tung University


The ℓ2-distance is one frequently used measurement to analyze the conver-gence of Markov chains to their stationarity. For reversible Markov chains, theirℓ2-distances can be formulated by the spectral information of their transitionmatrices. The corresponding mixing time and cutoff phenomenon for reversibleMarkov chains were first systemically studied by C. and Saloff-Coste in [3]. Laterin [1], C., Hsu and Sheu revealed more intrinsic mechanisms of ℓ2-cutoffs andpolished the cutoff criterion of C. and Saloff-Coste. Such a refinement makessome further theoretical analyses feasible including the comparison of ℓ2-cutoffsbetween discrete time chains and continuous time chains.

The max-ℓ2 distance was first considered by C. and Saloff-Coste in [2]. Anequivalent condition for the max-ℓ2 cutoff was then built on the product of themax-ℓ2 mixing time and the spectral gap of the transition matrix. Based onthe theoretical work in [1], we derive another proof for the max-ℓ2 cutoff crite-rion in [2] and provide a formula of its cutoff time using the spectral information.

Keywords: Markov chains, reversibility, ℓ2-distance

References[1] Guan-Yu Chen, Jui-Ming Hsu, and Yuan-Chung Sheu. The L2-cutoffs for

reversible Markov chains. Ann. Appl. Probab., 27(4):2305-2341, 2017.

[2] Guan-Yu Chen and Laurent Saloff-Coste. The cutoff phenomenon for er-godic Markov processes. Electron. J. Probab., 13:no. 3, 26-78, 2008.

[3] Guan-Yu Chen and Laurent Saloff-Coste. The L2-cutoff for reversibleMarkov processes. J. Funct. Anal., 258(7):2246-2315, 2010.

73

State-Dependent M/G/1 Queue with MultipleVacationsGi-Ren Liu

Department of MathematicsNational Cheng Kung University


In this talk, we consider an M/G/1 queue, whose state depends on theworkload. When the workload is equal to zero, the state will switch to the sleepstate from the awake state. In order to avoid the ping-pong effect, the durationof the sleep state is determined by a sleep timer and the accumulated workloadduring the sleeping period. Hence, the waiting time for each service requestdepends on the system state. The random Sleep-Awake schedule has been usedextensively in the design of modern communication systems for reducing theenergy cost. For example, during the power saving mode, the message deliverywill be delayed at some level. In view of that there exists a trade-off betweenthe energy saving ratio and the service-communication delay, the waiting timeanalysis can provide a guideline for the operators to set up the length of sleeptime for the Sleep-Awake scheme to maintain reasonable service delay and reducethe impact of buffer overflow. A recent work on the waiting time analysis fora restricted-length M/G/1 queue with multiple vacations will be presented inthis talk. The details of its application on the design of green routers can bereferred to [1] and [2].

Keywords: Waiting Time Analysis, Sleep-Awake scheme, M/G/1 queue,multiple vacations.

References[1] G.-R. Liu, P. Lin and M. K. Awad. Modeling Energy Saving Mechanism

for Green Routers. IEEE Transactions on Green Communications and Net-working (2018). Vol. 2, 817-829.

[2] M. K. Awad, P. Lin and G.-R. Liu. Distributed and Load Adaptive EnergyManagement Algorithm for Ethernet Green Routers. Journal of InternetTechnology (2018). Vol. 19, 781-794.

74

The localized phase transition of a polymerChien-Hao Huang

Department of MathematicsNational Taiwan University


A polymer is penalized by long excursions. We discuss the free energy andthe order of phase transitions. The behavior of infinite-volume system is alsodiscussed.

Keywords: Polymers, phase transitions, localization

75

最佳化Optimization



018

數學年會D e c . 8 / 0 9 : 3 0 - 2 1 : 0 0D e c . 9 / 0 9 : 3 0 - 1 5 : 5 0


D e c . 8 / 0 9 : 3 0 - 2 1 : 0 0

1 1 : 2 0 - 1 2 : 0 5

1 3 : 3 0 - 1 3 : 5 5

1 4 : 0 0 - 1 4 : 2 5

1 4 : 3 0 - 1 4 : 5 5

1 5 : 2 0 - 1 5 : 4 5

D e c . 9 / 0 9 : 3 0 - 1 5 : 5 0

1 0 : 2 0 - 1 1 : 0 5

1 1 : 1 0 - 1 1 : 3 5

1 1 : 4 0 - 1 2 : 0 5

1 3 : 3 0 - 1 3 : 5 5

1 4 : 0 0 - 1 4 : 2 5

1 4 : 3 0 - 1 4 : 5 5

76

Strong Convexity of Sandwiched Entropies andRelated Optimization Problems

Yongdo LimDepartment of MathematicsSungkyunkwan University


We present several theorems on strict and strong convexity for sandwichedquasi-relative entropy (a parametrised version of the classical fidelity). Theseare crucial for establishing global linear convergence of the gradient projectionalgorithm for optimization problems for these functions. The case of the classi-cal fidelity is of special interest for the multimarginal optimal transport problem(the n-coupling problem) for Gaussian measures. This is joint work with Ra-jendra Bhatia and Tanvi Jain.

77

KKM theorems in Hadamard manifoldsShue-Chin Huang

Department of Applied MathematicsNational Dong Hwa University


The purpose of this talk is to present a fixed point theorem for generalizedKKM mappings in the Hadamard manifold settings. We derive the finite inter-section property of this class of mappings. As an application of this property,we also discuss the existence conditions on the generalized equilibrium prob-lem. This research is supported by a grant MOST 106-2115-M-259-005 fromthe Ministry of Science and Technology of Taiwan.

78

Network data envelopment analysis with commonweights

Cheng-Feng HuDepartment of Applied Mathematics

National Chiayi UniversityE-mail: [email protected]

Common weight models can combat the computational burden of data en-velopment analysis (DEA) in the big data environment. This work considers studying a common-weights general network DEA model which is applicable to most network systems, except those with feedbacks and cycles. It shows that the general network DEA model with common weights can be reduced into an auxiliary fuzzy bi-objective mathematical programming problem by applying the basic principle of compromise of TOPSIS. The case of Taiwanese non-life insurance companies is utilized for illustration and comparison purposes. Our results show that the proposed common-weights network DEA model not only compares DMUs on a common base, but also produces reliable results in mea-suring efficiencies.

79

New generalizations of Ekeland’s variationalprinciple and well-known fixed point theoremswith applications to nonconvex optimization

problemsWei-Shih Du

Department of MathematicsNational Kaohsiung Normal University

Email: [email protected]

In this talk, we establish new generalizations of Ekeland’s variational princi-ple, Caristi’s fixed point theorem, Takahashi’s nonconvex minimization theorem and nonconvex maximal element theorem for uniformly below sequentially lower semicontinuous from above functions and essential distances. New simultane-ous generalizations of fixed point theorems of Mizoguchi-Takahashi type, Nadler type, Banach type, Kannan type, Chatterjea type and others are also presented. As applications, we concentrate on studying nonconvex optimization and mini-max theorems in metric spaces.

Keywords: Nonconvex optimization, minimax theorem, Ekeland’s varia-tional principle, Caristi’s (common) fixed point theorem, Takahashi’s noncon-vex minimization theorem, nonconvex maximal element theorem, MT -function (or R-function), MT (λ)-function, uniformly below sequentially lower semicon-tinuous from above, essential distance, Mizoguchi-Takahashi’s fixed point theo-rem, Nadler’s fixed point theorem, Banach contraction principle, Kannan’s fixed point theorem, Chatterjea’s fixed point theorem.

80

Deep Learning for Region of Interest BasedClustering of White Matter Fibers

Feng-Sheng TsaiDepartment of Biomedical Imaging and Radiological Science

China Medical UniversityE-mail: [email protected]

To cluster white matter fibers in whole-brain tractography, anatomical re-gions of interest (ROIs) are selected manually in brain diffusion MRI. ThoseROIs are used to isolate tracts and cluster fiber bundles accordingly. Deeplearning approaches may be applied to voxel-based ROI segmentation imme-diately; however, the number of voxels in ROIs is extremely smaller than thenumber of voxels in whole brain images, so they always suffer from the class im-balance problem when extracting related voxels of ROIs for training. Here wepropose a hierarchical sampling technique to resolve the class imbalance problemof deep learning. ROI segmentation with deep learning is divided into hierar-chical sub-tasks, from 2-dimensional objective-plane explorations to restricted,bounded hot-zone locations, and then to voxel-based discrimination. Samplingdatasets in all sub-tasks are more balanced for training. Specifically, two ROIs forclustering arcuate fasciculus in whole-brain tractography are presented.

81

A block symmetric Gauss-Seidel decompositiontheorem and its applications in big data

nonsmooth optimizationDe-Feng Sun

Department of Applied MathematicsThe Hong Kong Polytechnic University


The Gauss-Seidel method is a classical iterative method of solving the linear system Ax = b. It has long been known to be convergent when A is symmet-ric positive definite. I n t his t alk, w e s hall f ocus o n i ntroducing a symmetric version of the Gauss-Seidel method and its elegant extensions in solving big data nonsmooth optimization problems. For a symmetric positive semidefinite linear system Ax = b with x = (x1, . . . , xs) being partitioned into s blocks, we show that each cycle of the block symmetric Gauss-Seidel (block sGS) method exactly solves the associated quadratic programming (QP) problem but added with an extra proximal term. By leveraging on such a connection to optimiza-tion, one can extend the classical convergent result, named as the block sGS decomposition theorem, to solve a convex composite QP (CCQP) with an addi-tional nonsmooth term in x1. Consequently, one is able to use the sGS method to solve a CCQP. In addition, the extended block sGS method has the flexi-bility of allowing for inexact computation in each step of the block sGS cycle. At the same time, one can also accelerate the inexact block sGS method to achieve an iteration complexity of O(1/k2) after performing k block sGS cycles. As a fundamental building block, the block sGS decomposition theorem has played a key role in various recently developed algorithms such as the proxi-mal ALM/ADMM for linearly constrained multi-block convex composite conic programming (CCCP) and the accelerated block coordinate descent method for multi-block CCCP.

82

Strong duality in minimizing a quadratic formsubject to two homogeneous quadratic

inequalities over the unit sphereRuey-Lin Sheu

Department of MathematicsNational ChengKung UniversityE-mail: [email protected]

This problem, called (P), is a contrast with a simpler version (P′) which also minimizes a quadratic form but has just one homogeneous quadratic constraint over the unit sphere. The inclusion of an additional homogeneous quadratic constraint can cause (P) to have a positive duality gap, although the simpler version (P′) has been proved to adopt strong duality under Slater’s condition. On the surface the underlined problem (P) appears to be different f rom the CDT (Celis-Dennis-Tapia) problem. Their SDP relaxations, however, share a very similar format. The minute observation turns out to be valuable in deriving a necessary and sufficient condition for (P) to admit strong du ality. We will see that, in the sense of strong duality results, problem (P) is a generalization of the CDT problem. Many nontrivial examples are constructed in the paper to help understand the mechanism. Finally, as the strong duality in quadratic optimization is closely related to the S-lemma, we derive a new extension of the S-Lemma with three homogeneous quadratic inequalities over the unit sphere,with and without the Slater condition.

Keywords: Quadratically constrained quadratic programming, CDT prob-lem, S-lemma, Slater condition, Joint numerical range

83

Phase retrieval algorithms with random masksPeng-Wen Chen

Department of Applied MathematicsNational Chung Hsing UniversityE-mail: [email protected]

Phase retrieval aims to recover one unknown vector from its magnitudemeasurements, e.g., coherent diffractive imaging, where phase information ismissing. The recovery of phase information can be formulated as one minimiza-tion problem subject to a non convex high-dimensional torus set. In theory,uniqueness of solutions can be obtained under random masks. The introductionof random masks actually breaks the symmetry of Fourier matrices and cre-ates spectral gap for the local convergence of many phase retrieval algorithms,including alternative projection methods and Fourier Douglas-Rachford algo-rithms. The spectral gap is related to the local convergence rate.

On the other hand, these alternative algorithms still could fail to generate theglobal solution effectively. To alleviate the stagnation of possible local solutions,we propose one null vector method as an initialization method for phase retrievalalgorithms. The method is motivated by the following observation: Gaussianrandom vectors in high dimensional space are always nearly orthogonal to eachother. According to magnitude data, we can construct one sub-matrix assem-bled from the sensing vectors nearly orthogonal to the unknown vector. Onecandidate for the initialization vector is given by the singular vector of the sub-matrix corresponding to the least singular value. Thanks to isometric Fouriermatrices, this vector coincides with the dominant singular vector of the com-plement sub-matrix. Empirical studies (non-ptychography and ptychography)indicate that its incredible closeness to the unknown vector, compared withother existing methods. In this talk, we present one nonasymptotic error boundin the case of random complex Gaussian matrices, which sheds some light onits superior performance in the Fourier coherent diffractive case with randommasks.

Keywords: random masks, phase retrieval, null vector method.

84

The solvabilities of SOCEiCP and SOCQEiCPWei-Ming Hsu

Department of MathematicsNational Taiwan Normal University

In this paper, we study the solvabilities of two optimization problems asso-ciated with second-order cone, including eigenvalue complementarity problemassociated with second order cone (SOCEiCP), and quadratic eigenvalue com-plementarity problem associated with second order cone (SOCQEiCP). First ofall, we try to rewrite the SOCEiCP as instances of the SOCCP. Secondly, wealso try to rewrite SOCQEiCP as instances of SOCCP. Furthermore, we studysome algorithms for solving SOCEiCP and SOCQEiCP.

Keywords: Solvability, eigenvalue, second-order cone.

References[1] A. Seeger, Eigenvalue analysis of equilibrium processes defined by linear

complementarity conditions, Linear Algebra and its Applications, vol. 292,pp. 1-14, 1999.

[2] A. Seeger, Quadratic eigenvalue problems under conic constraints, SIAMJournal on Matrix Analysis and Applications, vol. 32, no.3, pp. 700-721,2011.

[3] M. Queiroz, J. Júdice,C. Humes, The symmetric eigenvalue comple-mentarity problem, Mathmatics of Computation, vol. 73, no.248 ,pp. 1849-1863, 2003.

[4] S. Adly, H. Rammal, A new method for solving second-order cone eigen-value complementarity problems, Journal of Optimization Theory and Ap-plications, vol. 165, issue 1, pp. 563-585, 2015.

[5] C. Brás, M. Fukushima, A. Iusem, J. Júdice, On the quadratic eigen-value complementarity problem over a general convex cone, Applied Math-ematics and Computation, vol. 271, pp. 391-403, 2015.

[6] C. Brás, A. Iusem, J. Júdice, On the quadratic eigenvalue complemen-tarity problem, Journal of Global Optimization, vol. 66, issue 2, pp. 153-171,2016.

85

[7] L. Fernandes, M. Fukushima, J. Júdice, H. Sherali, The second-order cone eigenvalue complementarity problem, Optimization Methods andSoftware, vol. 31, issue 1, pp. 24-52, 2016.

[8] J. Tao, M. Gowda, Some P-Properties for Nonlinear Transformations onEuclidean Jordan Algebras, Mathematical Methods of Operations Research,vol. 30, no. 4, pp. 985-1004, 2005.

[9] S.-H. Pan, S. Kum, Y. Lim, J.-S. Chen, On the generalized Fischer-Burmeister merit function for the second-order cone complementarity prob-lem, Mathematics of Computation, vol. 83, no. 287, 1143-1171, 2014.

[10] J. Wu, J.-S. Chen, A proximal point algorithm for the monotone second-order cone complementarity problem, Computational Optimization and Ap-plications, vol. 51, no. 3, pp. 1037-1063, 2012.

[11] J.-S. Chen, S.-H. Pan, A survey on SOC complementarity functions andsolution methods for SOCPs and SOCCPs, Pacific Journal of Optimization,vol. 8, no. 1, pp. 33-74, 2012.

[12] S.-H. Pan, J.-S. Chen, A least-square semismooth Newton method for thesecond-order cone complementarity problem, Optimization Methods andSoftware, vol. 26, no. 1, pp. 1-22, 2011.

[13] S.-H. Pan, J.-S. Chen, A semismooth Newton method for SOCCPs basedon a one-parametric class of complementarity functions, ComputationalOptimization and Applications, vol. 45, no. 1, pp. 59-88, 2010.

[14] S.-H. Pan, J.-S. Chen, A linearly convergent derivative-free descentmethod for the second-order cone complementarity problem, Optimization,vol. 59, no. 8, pp. 1173-1197, 2010.

[15] J.-S. Chen, S.-H. Pan, A one-parametric class of merit functions for thesecond-order cone complementarity problem, Computational Optimizationand Applications, vol. 45, no. 3, pp. 581-606, 2010.

[16] S.-H. Pan, J.-S. Chen, A damped Gauss-Newton method for the second-order cone complementarity problem, Applied Mathematics and Optimiza-tion, vol. 59, no. 3, pp. 293-318, 2009.

[17] S.-H. Pan, J.-S. Chen, A regularization method for the second-order conecomplementarity problems with the Cartesian P0-property, Nonlinear Anal-ysis: Theory, Methods and Applications, vol. 70, no. 4, pp. 1475-1491,2009.

[18] J.-S. Chen, S.-H. Pan, A descent method for solving reformulation of thesecond-order cone complementarity problem, Journal of Computational andApplied Mathematics, vol. 213, no. 2, pp. 547-558, 2008.

86

[19] J.-S. Chen, Conditions for error bounds and bounded level sets of somemerit functions for SOCCP, Journal of Optimization Theory and Applica-tions, vol. 135, no. 3, pp. 459-473, 2007.

[20] J.-S. Chen, Two classes of merit functions for the second-order cone com-plementarity problem, Mathematical Methods of Operations Research, vol.64, no. 3, pp. 495-519, 2006.

[21] J.-S. Chen, A new merit function and its related properties for the second-order cone complementarity problem, Pacific Journal of Optimization, vol.2, no. 1, pp. 167-179, 2006.

[22] J.-S. Chen, P. Tseng, An unconstrained smooth minimization reformu-lation of second-order cone complementarity problem, Mathematical Pro-gramming, vol. 104, no. 2-3, pp. 293-327, 2005.

87

Penalty and barrier methods for second-ordercone programming

Nguyen Thanh ChieuDepartment of Mathematics

National Taiwan Normal UniversityE-mail: [email protected]

In this talk we will present penalty and barrier methods for solving convexsecond-order cone programming:

min f(x)s.t Ax− b ≼Kn 0

where A is an n×m matrix with n ≥ m, rank A = m. f : ℜm → (−∞,+∞] isa closed proper convex function. Kn is a second-order cone (SOC for short) inℜn given by

Kn :={(x1, x2) ∈ ℜ × ℜn−1 | ∥x2∥ ≤ x1

},

where ∥ · ∥ denotes the Euclidean norm.

This class of methods is an extension of penalty and barrier methods forconvex optimization which was presented by A. Auslender et al. in 1997. Withthis method, we provide under implementable stopping rule that the sequencegenerated by the proposed algorithm is bounded and that every accumulationpoint is a solution to the considered problem. Furthermore, we examine effec-tiveness of the algorithm by means of numerical experiments.

Keywords: Second-order cone, penalty and barrier methods, asymptoticfunctions, recession functions, convex analysis, smoothing functions.

88

References[1] A. Auslender, R. Cominetti and M. Haddou, Asymptotic analysis of penalty

and barrier methods in convex and linear programming, Mathematics ofOperations Research, 22, pp. 43-62, 1997.

[2] A. Auslender, Penalty and barrier methods: A unified framework, SIAMJ. Optimization, 10, pp. 211-230, 1999.

[3] A. Auslender, Variational inequalities over the cone of semidefinite positivematrices and over the Lorentz cone, Optimization methods and software,pp. 359-376, 2003.

[4] A. Auslender and M. Teboulle, Asymptotic cones and functions in opti-mization and variational inequalities, Springer monographs in mathemat-ics, Springer, Berlin Heidelberg New York, 2003.

[5] A. Auslender and H. Ramírez C, Penalty and barrier methods for con-vex semidefinite programming, Mathematics of Operations Research, 63,pp.195-219, 2006.

[6] M. S. Bazaraa, H. D. Sherali and C. M. Shetty , Nonlinear Programming:Theory and Algorithms, 3rd Edition, Wiley - Interscience, 2006.

[7] A. Ben-Tal and M. Teboulle, A smoothing technique for nondifferentiableoptimization problems, in Optimization, Fifth French German Conference,Lecture Notes in Math. 1405, Springer-Verlag, New York, pp. 1-11, 1989.

[8] A. Ben-Tal and M. Zibulevsky, Penalty-barrier multiplier methods for con-vex programming problems, SIAM J. Optim. Vol 7, No. 2, pp. 347-366,1997.

[9] J. S. Chen, T. K. Liao and S. Pan, Using Schur complement theorem toprove convexity of some soc-functions, Journal of Nonlinear and ConvexAnalysis, Vol 13, No. 3, pp. 421-431, 2012.

[10] C. Chen and O. L. Mangasarian, A class of smoothing functions for non-linear and mixed complementarity problems, Comput. Optim. Appl, 5, pp.97-138, 1996.

[11] U. Faraut and A. Koranyi, Anlysis on Symmetric Cones, Oxford Mathe-matical Monographs, Oxford University Press, New York, 1994.

[12] E.D. Dolan and J.J. More, Benchmarking optimization software with per-formance profiles, Mathematical Programming, vol. 91, pp. 201-213, 2002.

[13] M. Fukushima, Z.-Q. Luo, and P. Tseng, Smoothing functions for second-order-cone complimentarity problems, SIAM Journal on Optimization, vol.12, pp. 436-460, 2002.

89

[14] L. Mosheyev and M. Zibulevsky, Penalty-barrier multiplier algorithm forsemidefinite programming, Optimization Meth. Soft., Vol. 13, pp. 235-261,2000.

[15] N. Parikh and S. Boyd, Proximal Algorithms, Foundations and Trends inOptimization, Vol. 1, No. 3, pp. 123-231, 2013.

[16] S. H. Pan and J. S. Chen, A proximal-like algorithnm using quasi D-function for convex second-order cone programming, J. Optim. TheoryAppl., 138 , pp. 95-113, 2008.

[17] S. H Pan and J. S. Chen, A class of interior proximal-like algorithms forconvex second-order cone programming, SIAM J. Optim. Vol. 19, No. 2,pp. 883-910, 2008.

[18] R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton,NJ, 1970.

[19] L. Zhang, J. Gu and X. Xiao, A class of nonlinear Lagrangians for non-convex second order cone programming, Computational Optimization andApplications. Vol. 49, pp. 61-99, 2011.

90

Neural networks based on three classes ofNCP-functions for solving nonlinear

complementarity problemsJan Harold M. Alcantara



We consider a family of neural networks for solving nonlinear complementar-ity problems (NCP). The neural networks are based from the merit functionsinduced by three classes of NCP-functions: the generalized natural residualfunction and its two symmetrizations. We first provide a characterization of thestationary points of the induced merit functions. To describe the level sets of themerit functions, we prove some important properties related to the growth be-havior of the complementarity functions. Furthermore, we analyze the stabilityof the steepest descent-based neural network model for NCP. To illustrate thetheoretical results, we provide numerical simulations using our neural networkand compare it with other similar neural networks in the literature which arebased on other well-known NCP-functions. The numerical results suggest thatthe neural network has a better performance when their common parameter pis smaller. We also found that one among the three families of neural networkswe considered is capable of outperforming other existing neural networks.

This is a joint work with Jein-Shan Chen.

Keywords: NCP-function, Neural network, natural residual function, sta-bility.

References[1] Y.-L. Chang, J.-S. Chen, C.-Y. Yang, Symmetrization of generalized

natural residual function for NCP, Operations Research Letters, 43(2015),354-358.

[2] J.-S. Chen, C.-H. Ko, and S.-H. Pan, A neural network based on gen-eralized Fischer-Burmeister function for nonlinear complementarity prob-lems, Information Sciences, 180(2010), 697-711.

91

[3] J.-S. Chen, C.-H. Ko, and X.-R. Wu, What is the generalization of nat-ural residual function for NCP, Pacific Journal of Optimization, 12(2016),19-27.

[4] J.-S. Chen and S.-H. Pan (2008), A family of NCP functions and adescent method for the nonlinear complementarity problem, ComputationalOptimization and Applications, vol. 40, 389-404.

[5] R.W. Cottle, J.-S. Pang and R.-E. Stone,The Linear Complemen-tarity Problem, Academic Press, New York 1992.

[6] C. Dang, Y. Leung, X. Gao, and K. Chen (2004), Neural networksfor nonlinear and mixed complementarity problems and their applications,Neural Networks, vol. 17, 271-283.

[7] M. C. Ferris, O. L. Mangasarian, and J.-S. Pang, editors, Com-plementarity: Applications, Algorithms and Extensions, Kluwer AcademicPublishers, Dordrecht, 2001.

[8] F. Facchinei and J.-S. Pang, Finite-Dimensional Variational Inequal-ities and Complementarity Problems, Volumes I and II, Springer-Verlag,New York, 2003.

[9] F. Facchinei and J. Soares (1997), A new merit function for nonlin-ear complementarity problems and a related algorithm, SIAM Journal onOptimization, vol. 7, 225-247.

[10] C. Geiger, and C. Kanzow (1996), On the resolution of monotonecomplementarity problems, Computational Optimization and Applications,vol. 5, 155-173.

[11] J. J. Hopfield and D. W. Tank (1985), Neural computation of decisionin optimization problems, Biological Cybernetics, vol. 52, 141-152.

[12] X. Hu and J. Wang (2006), Solving pseudomonotone variational inequal-ities and pseudoconvex optimization problems using the projection neuralnetwork, IEEE Transactions on Neural Networks, vol. 17, 1487-1499.

[13] X. Hu and J. Wang (2007), A recurrent neural network for solving aclass of general variational inequalities, IEEE Transactions on Systems,Man, and Cybernetics-B, vol. 37, 528–539.

[14] C.-H. Huang, K.-J. Weng, J.-S. Chen, H.-W. Chu and M.-Y. Li(2017), On four discrete-type families of NCP Functions, to appear in Jour-nal of Nonlinear and Convex Analysis, 2017.

[15] C. Kanzow and H. Kleinmichel (1995), A class of Newton-type methodsfor equality and inequality constrained optimization, Optimization Methodsand Software, vol. 5, pp. 173-198.

92

[16] C. Kanzow (1996), Nonlinear complementarity as unconstrained optimiza-tion, Journal of Optimization Theory and Applications, vol. 88, 139-155.

[17] M. P. Kennedy and L. O. Chua (1988), Neural network for nonlinearprogramming, IEEE Tansaction on Circuits and Systems, vol. 35, 554-562.

[18] M. Kojima and S. Shindo (1986), Extensions of Newton and quasi-Newton methods to systems of PC1 equations, Journal of Operations Re-search Society of Japan, vol. 29, 352-374.

[19] J. P. LaSalle (1968) Stability Theory for Ordinary Differential Equations,Journal of Differential Equations, vol. 4, 57-65.

[20] L.-Z. Liao, H. Qi, and L. Qi (2001), Solving nonlinear complementarityproblems with neural networks: a reformulation method approach, Journalof Computational and Applied Mathematics, vol. 131, 342-359.

[21] R. K. Miller and A. N. Michel (1982), Ordinary Differential Equations,Academic Press.

[22] S-K. Oh, W. Pedrycz, and S-B. Roh (2006), Genetically optimizedfuzzy polynomial neural networks with fuzzy set-based polynomial neurons,Information Sciences, vol. 176, 3490-3519.

[23] L. Qi and J. Sun (1993) A nonsmooth version of Newton’s method Math.Programm. 58, 353-368.

[24] A. Shortt, J. Keating, L. Monlinier, and C. Pannell (2005), Op-tical implementation of the Kak neural network, Information Sciences, vol.171, 273-287.

[25] D. W. Tank and J. J. Hopfield (1986), Simple neural optimization net-works: an A/D converter, signal decision circuit, and a linear programmingcircuit, IEEE Transactions on Circuits and Systems, vol. 33, 533-541.

[26] S. Wiggins (2003), Introduction to Applied and Nonlinear Dynamical Sys-tems and Chaos, Springer-Verlag, New York, Inc.

[27] Y. Xia, H. Leung, and J. Wang (2002), A projection neural network andits application to constrained optimization problems, IEEE Transactions onCircuits and Systems-I, vol. 49, 447-458.

[28] Y. Xia, H. Leung, and J. Wang (2004), A genarl projection neural net-work for solving monotone variational inequalities and related optimizationproblems, IEEE Transactions on Neural Networks, vol. 15, 318-328.

[29] Y. Xia, H. Leung, and J. Wang (2005), A recurrent neural networkfor solving nonlinear convex programs subject to linear constraints, IEEETransactions on Neural Networks, vol. 16, 379-386.

93

[30] M. Yashtini and A. Malek (2007), Solving complementarity and vari-ational inequalities problems using neural networks, Applied Mathematicsand Computation, vol. 190, 216-230.

[31] S. H. Zak, V. Upatising, and S. Hui (1995), Solving linear programmingproblems with neural networks: a comparative study, IEEE Transactions onNeural Networks, vol. 6, 94-104.

[32] G. Zhang (2007), A neural network ensemble method with jittered trainingdata for time series forecasting, Information Sciences, vol. 177, 5329-5340.

94

統計Statistics



018

數學年會D e c . 8 / 0 9 : 3 0 - 2 1 : 0 0D e c . 9 / 0 9 : 3 0 - 1 5 : 5 0


D e c . 8 / 0 9 : 3 0 - 2 1 : 0 0

1 1 : 2 0 - 1 2 : 0 5

1 3 : 3 0 - 1 4 : 1 5

1 4 : 2 0 - 1 4 : 4 5

1 4 : 4 5 - 1 5 : 1 0

1 5 : 2 0 - 1 5 : 4 5

D e c . 9 / 0 9 : 3 0 - 1 5 : 5 0

1 0 : 2 0 - 1 0 : 4 5

1 0 : 4 5 - 1 1 : 1 0

1 1 : 1 0 - 1 1 : 3 5

1 1 : 3 5 - 1 2 : 0 0

1 3 : 3 0 - 1 3 : 5 5

1 4 : 0 0 - 1 4 : 2 5

1 4 : 3 0 - 1 4 : 5 5

95

A model bias problem arising from image analysisin cryogenic electron microscopy

Yi-Ching YaoInstitute of Statistical Science


Cryogenic electron microscopy (cryo-EM) is an imaging technique to con-struct the 3D structures of biological samples such as membrane proteins. Insome cases, the extremely low signal-to-noise ratio of cryo-EM images results inthe processing dictated by the reference of a model, which is known as modelbias. A well-known example showed that a blurred Einstein face emerged from1000 aligned images of pure noise (often referred as “Einstein from noise”). Toinvestigate this model bias phenomenon quantitatively, we consider a simplifiedmodel consisting of n iid p-dimensional images of pure Gaussian noise and aspecified reference image (of Einstein). The n images of pure noise are sorted interms of their cross correlation values with the reference image, and the top mimages (of pure noise) are selected and averaged. We derive asymptotic distri-butions for the cross correlation between the averaged image and the referenceimage as n, p,m → ∞ at suitable rates. (This is joint work with Shao-HsuanWang, Wei-Hau Chang and I-Ping Tu.)

Keywords: Digital image, noise, correlation, asymptotic distribution

96

評估大學不同入學管道之統計模型

陳佩珊王維菁洪慧念統計學研究所國立交通大學

電子信箱: [email protected]

對於評估大學多元入學管道之探討，多數文獻皆著重於針對特定校系之資料分析。本研究之重點在於建立機率統計模型以描述大學入學申請與指考管道的篩選過程，可以調控多種因子，包含大環境的考生數與招生比例、考試的鑑別度、考生選填志願與是否接受分發結果的行為等等。我們透過模擬分析調整模型參數值以設計各種情境，比較經由申請入學與考試分發入學錄取的學生的差異。教育當局可利用此統計模型 (或修正版本) 預測不同政策可能導致的結果。

關鍵詞: 入學管道、申請入學、指考分發、統計模型。

97

Measuring stabilization in model selectionChun-Shu Chen

Institute of Statistics and Information ScienceNational Changhua University of Education


Model selection and model averaging are essential to regression analysis, but determining which of the two approaches is the more appropriate and un-der what circumstances remains an active research topic. In this paper, we focus on geostatistical regression models for spatially referenced environmental data. For a general information criterion, we develop a new perturbation-based criterion that measures the uncertainty of spatial model selection, as well as an empirical rule for choosing between model selection and model averaging. Statistical inference based on the proposed model selection instability measure is justified b oth i n t heory a nd v ia a s imulation s tudy. T he p redictive perfor-mance of model selection and model averaging can be quite different when the uncertainty in model selection is relatively large, but the performance becomes more comparable as this uncertainty decreases. For illustration, a precipitation data set in the state of Colorado is analysed. This is a joint work with Jun Zhu and Tingjin Chu.

Keywords: Information criterion, model complexity, spatial prediction

98

Optimal designs for binary response models withmultiple nonnegative variables

Shih-Hao Huang, Mong-Na Lo Huang, and Cheng-Wei LinDepartment of Mathematics and Department of Applied Mathematics

National Central University and National Sun Yat-sen UniversityE-mail: [email protected]

In this work, we consider optimal approximate designs for binary responsemodels with nonnegative explanatory variables. With respect to the Schur or-dering, we construct an essentially complete class consisting of designs with asimple structure. In particular, we explicitly identify locally D-optimal designswithin the class for logit and probit models. When the nonnegative explanatoryvariables have more restrictions, such as factorial and mixture experiments, wealso provide an informative iteration algorithm to search an optimal design.

Keywords: ϕp-optimality, D-optimality, essentially complete class, logitmodel, probit model, schur ordering

99

Functional data classification usingcovariate-adjusted subspace projection

Pai-Ling LiDepartment of Statistics

Tamkang UniversityE-mail: [email protected]

We propose a covariate-adjusted subspace projection method for classifying functional data, where the covariate effects on the response functions influence the classification o utcome. The proposed method i s a subspace c lassifier based on functional projection, and the covariates affect the response function through the mean of a functional regression model. We assume that the response func-tions in each class are embedded in a class specific s ubspace s panned b y a covariate-adjusted mean function and a set of eigenfunctions of the covariance kernel through the covariate-adjusted Karhunen-Loève expansion. A newly ob-served response function is classified i nto t he o ptimally p redicted c lass that has the minimal distance between the observation and its projection onto the subspaces among all classes. The covariate adjustment is useful for functional classification, e specially w hen t he c ovariate e ffects on th e me an fu nctions are significantly d ifferent am ong th e cl asses. Nu merical pe rformance of th e pro-posed method is demonstrated by simulation studies, with an application to a data example. This is a joint work with Jeng-Min Chiou and Yu Shyr.

Keywords: classification, discriminant analysis, functional data analysis, functional principal component analysis

100

Statistical inference for the accelerated failuretime model under multivariate outcome

dependent sampling designTsui-Shan Lu



Researchers are always seeking for cost-effective d esigns d ue t o a limited budget, especially for large biomedical or epidemiological studies. An outcome-dependent sampling (ODS) design, a retrospective sampling scheme, has been shown to improve the study efficiency wh ile eff ectively red ucing the monetary burden. Under the ODS design, one observes the covariates with a probability depending on the outcome and selects several supplemental samples from the most informative and appealing segments. Lu, Longnecker, and Zhou (2017) extended the ODS design to incorporate multivariate data often appeared in the recent studies and proposed a further generalization of the biased sampling.

In this talk, we consider a multivariate ODS (MODS) design for time-to-different-events data under the framework of a semiparametric accelerated fail-ure time (AFT) model, allowing multiple disease outcomes with clustered failure times. We establish an estimating equation approach to estimate parameters based on induced smoothing. The asymptotic properties of the proposed esti-mators are developed. Simulation results show that the proposed design is more efficient and powerful than other existing ap proaches. The proposed method is illustrated with a real data set.

Keywords: outcome-dependent sampling, multivariate, AFT model, semi-parametric

101

Test Statistics of Pearson-Fisher’s Type withSome Remarks on the Degrees of Freedom

Wei-Hsiung ChaoDepartment of Applied Mathematics

National Dong Hwa UniversityE-mail: [email protected]

Pearson-Fisher’s tests have been widely used for assessing thet of a model for the categorical response in settings of a single multinomial

or product multinomials. The statistic used in these tests can be viewed as aquadratic form in the differences between the observed totals and fitted totalswhich uses as a weighting matrix a particular nonsingular generalized inversefor the singular variance-covariance matrix of the differences. Using propertiesof inner product spaces and the rank condition, we demonstrate an alternativeway to determine the degrees of freedom of the asymptotic null distribution ofthese Pearson-Fisher statistics.

To assess the fit of polytomous regression models with only categorical co-variates, it is also appropriate to use the Pearson-Fisher’s test for product multi-nomials since the response observations within each covariate pattern are homo-geneous so that their total can be viewed as a single non-sparse multinomial. Inthe presence of continuous covariates, direct use of this method is not appropri-ate since the response observations within each categorical covariate pattern canbe quite heterogeneous. To overcome this limitation, many ad-hoc extensions ofPearson-Fishers chi-squared statistics have been proposed for binary and ordinallogistic regression models using some sorts of grouping strategies. For example,Hosmer and Lemeshow (1980) suggested partitioning the observations into ggroups with equal size based on the fitted probabilities. Their statistic is thenformed as a sum of Pearson’s statistics over all groups. With a small numberof groups, these statistics are not close to a chi-square distribution since thewithin-group observations are more heterogeneous so that the observed totalswithin a group are actually underdispersed relative to multinomial distribution.Through extensive simulations, these authors showed that their statistic has achi-square null distribution for a certain range of numbers of groups and certaincovariate distributions being considered. We will also discuss the degrees offreedom of their statistic from the view point of the rank condition.

Keywords: goodness of fit, Pearson’s chi-square test, rank condition.

102

On Fixed Effects Estimation for SpatialRegression Under the Presence of Spatial

ConfoundingYung-Huei Chiou

Department of MathematicsNational Changhua University of Education


Spatial regression models are often used to analyze the ecological and en-vironmental data sets over a continuous spatial support. Issues of collinearity among covariates are often considered in modeling, but only rarely in discussing the relationship between covariates and unobserved spatial random processes. Past researches have shown that ignoring this relationship (or, spatial confound-ing) would have significant influences on the estimation of regression parameters. To improve this problem, an idea of restricted spatial regression is used to ensure that the unobserved spatial random process is orthogonal to covariates, but the related inferences are mainly based on Bayesian frameworks. In this thesis, an adjusted generalized least squares estimation method is proposed to estimate regression coefficients, resulting in the estimators that perform better than the conventional methods. Under the frequentist framework, statistical inferences of the proposed methodology are justified b oth i n t heories a nd v ia simulation studies. Finally, an application of a water acidity data set in the Blue Ridge region of the eastern U.S. is analyzed for illustration. This is a joint work with Hong-Ding Yang and Chun-Shu Chen.

Keywords: Bias, Generalized least squares, Maximum likelihood estimate, Random effects, Restricted spatial regression

103

Estimation and selection for spatial regressionunder the presence of spatial confounding

Hong-Ding YangInstitute of Statistics and Information ScienceNational Changhua University of Education


The spatial random effects m odel i s p opular i n a nalyzing s patially refer-enced data. The model includes spatially observed covariates and unobserved spatial random effects, w hich i f n ot d eal p roperly w ith t he c onfounding be-tween the two components, parameter estimation and spatial prediction had been demonstrated to be unreliable. In this research, we focus on discussing the estimation of regression coefficients an d th e se lection of co variates fo r spatial regression under the presence of spatial confounding. We first introduce an ad-justed estimation method of regression coefficients an d th e co nsequent spatial predictor when spatial confounding exists. From a prediction point of view, we then propose a generalized conditional Akaike information criterion to select a subset of covariates, resulting in variable selection and spatial prediction that are satisfactory. Statistical inferences of the proposed methodology are justified theoretically and numerically. This is a joint work with Yung-Huei Chiou and Chun-Shu Chen.

Keywords: conditional information criterion, mean squared prediction er-ror, restricted spatial regression, spatial prediction, variable selection.

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數學科普Popular Mathematics

地點：綜合館H 3 0 1


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數學年會D e c . 8 / 0 9 : 3 0 - 2 1 : 0 0D e c . 9 / 0 9 : 3 0 - 1 5 : 5 0


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SL

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What Does it Take to Get Ichiro and HanakoExcited About Mathematics?

Jin AkiyamaResearch Institute for Math Education

Tokyo University of ScienceE-mail: [email protected]

What does it take to appreciate a musical piece? Most people must hear themusic played. And if it is played by a full orchestra, then perhaps there willbe excitement. What does it take to appreciate a recipe? The dish must beprepared and tasted. If it is beautifully presented and taken in the ambianceof a great restaurant, then maybe there will be excitement. The senses mustbe engaged. It is the same with mathematics. A mathematical concept canbe exciting if it can be represented physically in a model that can be seen,manipulated, and, if possible, heard.

In this talk, we discuss mathematical models that can be used to teachstandard mathematics in a non-standard way. These models can be broughtinto the mathematics classroom so students can work with them, experiment,discover, and gain a deep understanding of mathematical concepts. The modelscan be used to demonstrate the following:

1. Area and Volume

2. Pytagorean Theorem

3. Sum of Integers

4. Applications of Conic Sections

5. Figures with constant width

6. Error Correcting Code

7. Math Magic

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The Art and Mathematics of Self-interlocking SL

BlocksShen-Guan Shih

Department of ArchitectureNational Taiwan University of Science and Technology


SL block is an octocube that may interlock with other SL blocks to forminfinite variations of stable structures. The property of self-interlocking makesSL block expressive to explore the beauty of symmetry, which has been re-garded as an essence of art and mathematics by many. This paper describes amathematical representation that maps polynomial expressions to compositionsof SL blocks. The use of polynomials, functions and hierarchical definitionssimplifies the creation, communication and manipulation of complex structuresby making abstractions over symmetrical parts and relationships. The discov-ery of SL block and its mathematical representation lead the way towards thedevelopment of an expressive language of forms and structures which is at thesame time, rich and compact, free and disciplined.

References[1] Y. Estrin, A.V.Dyskin, E. Pasternak, 2011. “Topological Interlock-

ing as a Material Design Concept.” Mater. Sci. Eng. C, Princi-ples and Development of Bio-Inspired Materials 31, Pages1189–1194.doi:10.1016/j.msec.2010.11.011

[2] A.J. Kanel-Belov, A.V. Dyskin, Y. Estrin, E. Pasternak, I.A. Ivanov-Pogodaev, 2008. “Interlocking of Convex Polyhedra: Towards a GeometricTheory of Fragmented Solids.” Moscow Mathematical Journal, V10, N2,April–June 2010, Pages 337–342.

[3] S. G. Shih, “On the Hierarchical Construction of ‘SL‘ Blocks – A Genera-tive System That Builds Self-interlocking Structures.” Sigrid Adriaenssens,F. Gramazio, M. Kohler, A. Menges, and M. Pauly Eds. Advances in Archi-tectural Geometry 2016, Pages 124-137, Hochschulverlag AG an der ETHZürich, DOI 10.3218/3778-4, ISBN 978-3-7281-3778-4

[4] P. Song, C. Fu, D. Cohen-Or, 2012. “Recursive Interlocking Puzzles.”ACM Trans. Graph. 31 6, Article 128 (November 2012), 10 pages. DOI

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= 10.1145/2366145.2366147http://doi.acm.org/10.1145/2366145.2366147

[5] O. Tessmann, 2013. “Topological Interlocking Assemblies.” Presented atthe 30th International Conference on Education and Research in ComputerAided Architectural Design in Europe (eCAADe),SEP 12-14, 2012, CzechTech Univ, Fac Architecture, Prague, Pages 211–219.

[6] S.-Q. Xin, C.-F. Lai, C.-W. Fu, T.-T. Wong, Y. H3, and D. Cohen-Or,2011. “Making Burr Puzzles from 3D models.” ACM Tran. on Graphics(SIGGRAPH) 30, 4. Article 97.

[7] H.T.D. Yong, 2011. “Utilisation of Topologically-interlocking OsteomorphicBlocks for Multi-purpose Civil Construction.” (PhD). University of WesternAustralia

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數學教育Mathematics Education

地點：理學院B 1 0 1


018

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1 1 : 2 0 - 1 2 : 0 5

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1 1 : 1 0 - 1 1 : 5 5

1 3 : 3 0 - 1 5 : 0 0

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微積分了沒？

微積分教材嚴格化的反思以及對高中和大一

課程的啟示

單維彰 Wei-Chang ShannDepartment of MathematicsNational Central University


Many grade-12 students in the course Elective Math A find Calculus actuallyeasier to handle with than the required materials they had in grades 10 and 11.Many math teachers agree with the students. Calculus is generally consideredhard and deep since the later part of the 20th century, therefore it is isolatedand reserved to the last part of the school math curriculum. We must agree thatthere are hard concepts in Calculus. However, those are the rigorous aspectsof the subject after Cauchy, Weierstrass, and others. The naive Calculus at itsearly stage in the 17th century was rather intuitive and it was the source ofmany intriguing mathematical ideas. It is a matter of simple fact that Newton’sPrincipia was published only 50 years after Descartes’ La Géométrie. We allknow that La Géométrie spawned Cartesian coordinates and the later fertilizedCalculus and mathematical analysis. Closer historical investigations suggestedthat the idea of Calculus occurred to Newton and Leibniz within a decadesince they were exposed to coordinate systems and early analytical geometry.Calculus in that era was much more primitive than some materials in grades 10and 11: space vectors, matrices and linear transformations, much of the contentsin probability and statistics, to name a few.

In speaker’s opinion, to cut Calculus from the rest of math curriculum wasthe most unfortunate strategy made for the math education. Because calculusis the key for the sense-making math curriculum. It plays the central role thatmotivates and links almost all topics of school math: the very idea of ratesand ratios and functions, to begin with, and it provides the reason-to-be formany topics covered in high school: the polynomial inequalities, the radian, thetrigonometric and exponential functions.

In this session, the speaker will elaborate on the foregoing remarks and pro-pose a curriculum design that incorporates Calculus with the current materialsnaturally, organically, and painlessly.

Keywords: math education, school math curriculum, calculus

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人工智慧與學習分析在數學教育上的應用

郭伯臣教育資訊與測驗統計研究所

國立臺中教育大學E-mail: [email protected]

台灣教育部於 2017 年啟動了一個適性學習平台「因材網」來協助教師進行適性教學與學生適性學習，在因材網中有國小與國中數學學習內容，包含教學影片、練習題、動態評量、對話式智慧教學系統，本演講將以因材網中的數學數位學習內容與工具為基礎，說明人工智慧與學習分析在數學學習上的應用，包含：一、數學智慧教學系統設計與成效；二、學習分析模式與在因材網數學學習資料分析；三、因材網應用在數學補救教學方式與成效。

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大會會場示意圖 Site Map

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2018中華民國數學年會論文發表注意事項 Guidelines for the Speakers at 2018 TMS Annual Meeting

1. 發表語言：中文或英文。

The conference language is Mandarin Chinese or English.

2. 簡報檔測試：

Test for the presenting files:

(1) 使用大會電腦: 請提早將簡報傳送給大會([email protected]) ，最遲

請於該場發表會開始前15-30分鐘至預定發表場次將簡報檔轉存至會場電腦

並確認簡報資料可順利投影。

Using the facilities provided by the conference: Please send your presenting filesto the e-mail box [email protected] in advance. Or, please make surethe presenting file are ready for projecting to the screen 15-30 minutes prior tothe whole session.

(2) 使用個人電腦: 請於該場發表會開始前15-30分鐘將個人電腦攜至會場，確

認可順利與大會投影系統連接。本投影系統僅接受VGA(D-sub)輸入，若您

的電腦視訊輸出並非VGA(D-sub)，請自行準備轉接線，或提早洽詢大會是

否可提供支援。

Using your own laptop: Please take your laptop to the venue, have it connectedto the projecting facilities and make sure the projection work well 15-30 minutesprior to the whole session. The projectors could only receive VGA(D-sub) signal.Please prepare the appropriate connector on your own, or consult the conferencestaff for possible help in advance.

3. 發表時間：20-45分鐘不等(請依據議程表)，最後預留5分鐘為討論時間，發表結

束前5分鐘舉牌一次(「剩下5分鐘」)，發表時間結束再舉牌一次(「時間到(Timeis Up)」)，請發表人結束發表，交由主持人進行交流時間。

There will be at least 5 minutes of discussion for each talk. So the time forpresentation will be the total time minus 5 minutes. Signs of “5 Min left” and “Time isUp” will show up when it’s time respectively.

4. 現場備有電腦、單槍投影機、麥克風、簡報筆、VGA訊號連接線。如有其他設

備需求，請提早洽詢大會是否可提供支援。

Computers, projectors, microphones, pointers with remote control and VGAconnectors are available in the conference. Should any further equipment is in need,please consult the conference staff for possible help in advance.

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