DOCSLIB.ORG

WAVE MAP with POTENTIAL and HYPERSURFACE FLOW Jian Zhai

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
WAVE MAP with POTENTIAL and HYPERSURFACE FLOW Jian Zhai DISCRETE AND CONTINUOUS Website: http://AIMsciences.org DYNAMICAL SYSTEMS Supplement Volume 2005 pp. 940{946 WAVE MAP WITH POTENTIAL AND HYPERSURFACE FLOW Jian Zhai, Jianping Fang and Lanjun Li Department of Mathematics, Zhejiang University, Hangzhou 310027, PRC Abstract. The simpli¯ed equation of the dynamics of weak ferromagnets magneti- zation is related to wave maps with potential. The global existence and behavior of solutions as parameter ² ! 0 are obtained. 1. Introduction. We consider the equation 8 > @2u 1 > ¡ ¢u ¡ B(u)(@ u; @®u) + (W (u) < @t2 ® ²2 u ¡ (W (u) ¢ n(u))n(u)) = 0 in Rn £ R+ (1:1) > u :> u : Rn+1 ! N: Here B is the second fundamental form of N and W : N ! R is a smooth function in a neighborhood of N. When N = S2, (1.1) is a simpli¯ed equation of the dynamics of weak ferromagnets magnetization. The ferromagnetic theory states that below a critical temperature, a su±ciently large ferromagnetic body breaks up into small uniformly magnetized regions separated by thin transition layers which are often called Bloch walls or domain walls ([1], [8]). The equation (1.1) can be regarded as wave map equation with potential. The existence of wave maps and development of singularities are researched in [2], [3], [4], [5] etc. We interest in simple wave solutions of (1.1). In this paper, the Bloch wall motions are governed by hyperbolic equation. It is di±cult to prove the convergence of solutions to (2.2) (see section 2), since the third term may change its sign. In Theorem 4.1 (see section 4), we shall prove that the limit of the solutions of the equation (2.2), in some sense, satis¯es following Hamilton-Jacobi equation 2 2 (rt) ¡ jrrj + 1 = 0: (1:2) It means that the Bloch wall motions are governed by the Hamilton-Jacobi equation (1.2) as the parameter ² ! 0. 2. Simple wave. The solutions of equation (2.1) are called the simple waves of the equation (1.1) ( Rss + B(R)(Rs;Rs) ¡ Wu(R) + (Wu(R) ¢ n(R))n(R) = 0 (2:1) R(¡1) = R¡;R(1) = R+: 2000 Mathematics Subject Classi¯cation. Primary: 35L70,35Q60,35F20,78A25. Key words and phrases. wave maps with potential; Hamilton-Jacobi equation; weak ferromagnets. 940 WAVE MAP WITH POTENTIAL AND HYPERSURFACE FLOW 941 We want to seek the simple wave solutions of (1.1) by the simple waves. Let u(x; t) = R(r(x; t)=²) in (1.1). We get the equation for r(x; t) (lnjR (r(x; t)=²)j2) r ¡ ¢r + s s ((r )2 ¡ jrrj2 + 1) = 0: (2:2) tt 2² t 2 2 2 tr Hereafter, we take N = S , W (u) = 2(u2+u3) and u3 ´ 0, and R§ = (§1; 0; 0) . Then R(s) = (q1(s); q2(s); 0) e4s ¡ 1 §2e2s q (s) = ; q (s) = : (2:3) 1 e4s + 1 2 1 + e4s and 2q (r=²) r ¡ ¢r ¡ 1 ((r )2 ¡ jrrj2 + 1) = 0: (2:4) tt ² t 3. Global existence. By a modi¯ed method for the wave maps, we have following existence result. For s 2 N0, we let Hs(Rn;N) = fv 2 Hs;2(Rn; Rm); v 2 Ng and s+1 2 n _ n m 1 n s+1 n (¹v; v^) 2 Lloc(R ;TN) :v ¹ 2 H (R ; R );W (¹v) 2 L (R ) Hc (R ;TN) = f g: v^ 2 Hs(Rn; Rm); spt(rv;¹ v^) ½½ Rn m¡1 1 n Theorem 3.1. Let n ¸ 1 and N = S . For ('; Ã) 2 Hc (R ;TN), there exists 1 a global weak solution u of (1.1) of class H_ . Theorem 3.1 can be proved by a penalty method developed in [3]. We ¯rst n m consider the relaxation of the equation (1.1) for u¸ : R £ R ! R 8 2 > @ u¸ > ¡ ¢u + ¸(ju j2 ¡ 1)u <> @t2 ¸ ¸ ¸ 1 (3:1) + (W (u ) ¡ (W (u ) ¢ u )u ) = 0 in Rn £ R+ > 2 u ¸ u ¸ ¸ ¸ > ² : n u¸(x; 0) = '(x);@tu¸(x; 0) = Ã(x); 8x 2 R : Let Z 1 2 1 2 E¸(u¸)(t) := j@tu¸(x; t)j + jru¸(x; t)j Rn 2 2 1 ¸ + W (u ) + (ju j2 ¡ 1)2dx: (3:2) ²2 ¸ 4 ¸ d We have E¸(u¸)(t) = 0 and dt Z 1 2 1 2 1 E¸(u¸)(t) = E¸(u¸)(0) = jÃj + jr'j + 2 W (')dx (3:3) Rn 2 2 ² uniformly for ¸ and t. Then we have Claim 1. There is a subsequence (denoted also by u¸) such that 1 2 n @tu¸ *@tu weak -* in L (R; L (R )) 1 2 n @xi u¸ *@xi u weak -* in L (R; L (R )) (3:4) 2 n u¸ ¡! u strongly in Lloc(R ), uniformly in t: 942 JIAN ZHAI, JIANPING FANG AND LANJUN LI By Fatou's lemma, for any t, Z Z 2 2 2 2 1 (ju(x; t)j ¡ 1) dx · lim inf (ju¸(x; t)j ¡ 1) dx · lim sup E¸(u¸)(t) = 0; Rn ¸!1 Rn ¸!1 ¸ so we get Claim 2. u(x; t) 2 Sm¡1. Note that the ¯rst equation of (3.1) is equivalent to @2u 1 ( ¸ ¡ ¢u + (W (u ) ¡ (W (u ) ¢ u )u )) £ u = 0: (3:5) @t2 ¸ ²2 u ¸ u ¸ ¸ ¸ ¸ That is 1 @®(@ u £ u ) + W (u ) £ u = 0: (3:6) ® ¸ ¸ ²2 u ¸ ¸ Since ® ® @ (@®u¸ £ u¸) ! @ (@®u £ u) in the sense of distributions and Wu(u¸) £ u¸ ! Wu(u) £ u 1 in Lloc, we obtain that u(x; t) weakly solves (1.1). 1 n We show that the initial conditions are also satis¯ed. For any Á 2 C0 (R £ R), Z 1 Z ® dt (@®u¸ £ u¸ ¡ @®u £ u)@ Ádx n 0 Z RZ 1 1 1 + dt f 2 Wu(u¸) £ u¸ ¡ 2 Wu(u) £ ugÁdx n ² ² Z0 R = (Ã(x) £ '(x) ¡ @tu(x; 0) £ '(x))Á(x; 0)dx ! 0 as ¸ ! 1: Rn Then n (@tu(x; 0) ¡ Ã(x)) £ '(x) = 0; 8a:e:x 2 R : (3:7) But (@tu(x; 0) ¡ Ã(x)) ¢ '(x) = 0; (3:8) so (3.7)-(3.8) imply n @tu(x; 0) = Ã(x) 8a:e:x 2 R : (3:9) For any f 2 L1(R) and g 2 L2(Rn), Z 1 Z f(t)dt (@tu¸(x; t) ¡ @tu(x; t))g(x)dx ! 0; as ¸ ! 1; 0 Rn then Z 2 n lim (@tu¸(x; t) ¡ @tu(x; t))g(x)dx = 0; 8g 2 L (R ) t#0 Rn and as t # 0 2 n @tu¸(x; t) ! Ã(x) strongly in L (R ): Thus as t # 0 2 n @tu(x; t) *Ã(x) weakly in L (R ): (3:10) WAVE MAP WITH POTENTIAL AND HYPERSURFACE FLOW 943 Note that Z 1 2 1 2 1 lim sup j@tu(x; t)j + jr'(x)j + 2 W (')dx n 2 2 ² t!0 R Z 1 2 1 2 1 · lim sup j@tu(x; t)j + jru(x; t)j + 2 W (u(x; t))dx t!0 Rn 2 2 ² · lim sup lim inf E¸(u¸)(t) ¸!1 Z t!0 1 2 1 2 1 · jÃj + jr'j + 2 W (')dx: Rn 2 2 ² So Z Z 1 2 1 2 lim sup j@tu(x; t)j dx · jÃ(x)j dx: (3:11) t!0 2 Rn 2 Rn From (3.10)-(3.11), we get as t # 0 2 n @tu(x; t) ! Ã(x); strongly in L (R ): 4. Convergence. De¯ne Z ² @u(x; t) 2 ² 2 1 E²(u)(t) = j j + jru(x; t)j + W (u)dx: Rn 2 @t 2 ² Lemma 4.1. Assume that u is a classical solution of (1.1). Then d E (u)(t) = 0: (4:1) dt ² Assume that R(r(x; t)=²) is a solution of (1.1) with the initial data R(r0(x)=²) satisfying r² E (R(r(x; t)=²)) ´ E (R( ))(0) · C : (4:2) ² ² ² 2 Then, we have Z 2 2 2 ²C2 ¸ 2q2(r=²)((rt) + jrrj + 1)dx ZRn 2 2 2 = 2(1 ¡ q1(r=²))((rt) + jrrj + 1)dx (4:3) ZRn 2 ¸ 2(1 ¡ q1(r=²))dx: Rn For s 2 (0; 1), we de¯ne n 2 ² s !s(²; t) := fx 2 R j 1 ¡ q1(r (x; t)=²) ¸ ² g (0 < s < 1); (4:4) T Let ²(x; t) be the characteristic function of the set [t=0(!s(²; t) £ ftg) and p ² 1 + 1 ¡ ²s r+(²) = ln p : (4:5) 4 1 ¡ 1 ¡ ²s Thus we get Lemma 4.2. For any t > 0, n n 2 s 1¡s L [!s(²; t)] = jfx 2 R j 1 ¡ q1(r(x; t)=²) ¸ ² gj · ² C2=2 (4:6) with 0 < s < 1. Moreover, ² x 2 !s(²; t) () ¡r+(²) · r (x; t) · r+(²): (4:7) 944 JIAN ZHAI, JIANPING FANG AND LANJUN LI P roof: (4.6) is obtained by (4.3). To prove (4.7), let r§ satisfy r p q ( § ) = § 1 ¡ ²s: 1 ² Noting (2.3) we have p ² 1 + 1 ¡ ²s r+ = ln p ; 4 1 ¡ 1 ¡ ²s and r¡ = ¡r+. We shall use the notation ­ denoting a smooth bounded domain of Rn and [t0; t1] the subset of [0;T ]. Recall that Â(x; t) is the characteristic function of the T set [ (!s(²; t) £ ftg). That is, t=0 ( T 1; 8(x; t) 2 [t=0(!s(²; t) £ ftg) Â(x; t) = T 0; 8(x; t) 62 [t=0(!s(²; t) £ ftg): Theorem 4.1.
Load more

Recommended publications
  • Auditory Hair Cell-Specific Deletion of P27 in Postnatal Mice Promotes
    The Journal of Neuroscience, November 19, 2014 • 34(47):15751–15763 • 15751 Development/Plasticity/Repair Auditory Hair Cell-Specific Deletion of p27Kip1 in Postnatal Mice Promotes Cell-Autonomous Generation of New Hair Cells and Normal Hearing Bradley J. Walters,* Zhiyong Liu,* Mark Crabtree,* Emily Coak, Brandon C. Cox, and Jian Zuo Department of Developmental Neurobiology, St. Jude Children’s Research Hospital, Memphis, Tennessee 38105 Hearing in mammals relies upon the transduction of sound by hair cells (HCs) in the organ of Corti within the cochlea of the inner ear. Sensorineural hearing loss is a widespread and permanent disability due largely to a lack of HC regeneration in mammals. Recent studies suggest that targeting the retinoblastoma (Rb)/E2F pathway can elicit proliferation of auditory HCs. However, previous attempts to induce HC proliferation in this manner have resulted in abnormal cochlear morphology, HC death, and hearing loss. Here we show that cochlear HCs readily proliferate and survive following neonatal, HC-specific, conditional knock-out of p27 Kip1 (p27CKO), a tumor suppressor upstream of Rb. Indeed, HC-specific p27CKO results in proliferation of these cells without the upregulation of the supporting cell or progenitor cell proteins, Prox1 or Sox2, suggesting that they remain HCs. Furthermore, p27CKO leads to a significant addition of postnatallyderivedHCsthatexpresscharacteristicsynapticandstereociliarymarkersandsurvivetoadulthood,althoughaportionofthe newly derived inner HCs exhibit cytocauds and lack VGlut3 expression. Despite this, p27CKO mice exhibit normal hearing as measured by evoked auditory brainstem responses, which suggests that the newly generated HCs may contribute to, or at least do not greatly detract from, function. These results show that p27 Kip1 actively maintains HC quiescence in postnatal mice, and suggest that inhibition of p27 Kip1 in residual HCs represents a potential strategy for cell-autonomous auditory HC regeneration.
    [Show full text]
  • Is Shuma the Chinese Analog of Soma/Haoma? a Study of Early Contacts Between Indo-Iranians and Chinese
    SINO-PLATONIC PAPERS Number 216 October, 2011 Is Shuma the Chinese Analog of Soma/Haoma? A Study of Early Contacts between Indo-Iranians and Chinese by ZHANG He Victor H. Mair, Editor Sino-Platonic Papers Department of East Asian Languages and Civilizations University of Pennsylvania Philadelphia, PA 19104-6305 USA [email protected] www.sino-platonic.org SINO-PLATONIC PAPERS FOUNDED 1986 Editor-in-Chief VICTOR H. MAIR Associate Editors PAULA ROBERTS MARK SWOFFORD ISSN 2157-9679 (print) 2157-9687 (online) SINO-PLATONIC PAPERS is an occasional series dedicated to making available to specialists and the interested public the results of research that, because of its unconventional or controversial nature, might otherwise go unpublished. The editor-in-chief actively encourages younger, not yet well established, scholars and independent authors to submit manuscripts for consideration. Contributions in any of the major scholarly languages of the world, including romanized modern standard Mandarin (MSM) and Japanese, are acceptable. In special circumstances, papers written in one of the Sinitic topolects (fangyan) may be considered for publication. Although the chief focus of Sino-Platonic Papers is on the intercultural relations of China with other peoples, challenging and creative studies on a wide variety of philological subjects will be entertained. This series is not the place for safe, sober, and stodgy presentations. Sino- Platonic Papers prefers lively work that, while taking reasonable risks to advance the field, capitalizes on brilliant new insights into the development of civilization. Submissions are regularly sent out to be refereed, and extensive editorial suggestions for revision may be offered. Sino-Platonic Papers emphasizes substance over form.
    [Show full text]
  • Gateless Gate Has Become Common in English, Some Have Criticized This Translation As Unfaithful to the Original
    Wú Mén Guān The Barrier That Has No Gate Original Collection in Chinese by Chán Master Wúmén Huìkāi (1183-1260) Questions and Additional Comments by Sŏn Master Sǔngan Compiled and Edited by Paul Dōch’ŏng Lynch, JDPSN Page ii Frontspiece “Wú Mén Guān” Facsimile of the Original Cover Page iii Page iv Wú Mén Guān The Barrier That Has No Gate Chán Master Wúmén Huìkāi (1183-1260) Questions and Additional Comments by Sŏn Master Sǔngan Compiled and Edited by Paul Dōch’ŏng Lynch, JDPSN Sixth Edition Before Thought Publications Huntington Beach, CA 2010 Page v BEFORE THOUGHT PUBLICATIONS HUNTINGTON BEACH, CA 92648 ALL RIGHTS RESERVED. COPYRIGHT © 2010 ENGLISH VERSION BY PAUL LYNCH, JDPSN NO PART OF THIS BOOK MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, GRAPHIC, ELECTRONIC, OR MECHANICAL, INCLUDING PHOTOCOPYING, RECORDING, TAPING OR BY ANY INFORMATION STORAGE OR RETRIEVAL SYSTEM, WITHOUT THE PERMISSION IN WRITING FROM THE PUBLISHER. PRINTED IN THE UNITED STATES OF AMERICA BY LULU INCORPORATION, MORRISVILLE, NC, USA COVER PRINTED ON LAMINATED 100# ULTRA GLOSS COVER STOCK, DIGITAL COLOR SILK - C2S, 90 BRIGHT BOOK CONTENT PRINTED ON 24/60# CREAM TEXT, 90 GSM PAPER, USING 12 PT. GARAMOND FONT Page vi Dedication What are we in this cosmos? This ineffable question has haunted us since Buddha sat under the Bodhi Tree. I would like to gracefully thank the author, Chán Master Wúmén, for his grace and kindness by leaving us these wonderful teachings. I would also like to thank Chán Master Dàhuì for his ineptness in destroying all copies of this book; thankfully, Master Dàhuì missed a few so that now we can explore the teachings of his teacher.
    [Show full text]
  • Houqua and His China Trade Partners in the Nineteenth Century
    Global Positioning: Houqua and His China Trade Partners in the Nineteenth Century The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters Citation Wong, John. 2012. Global Positioning: Houqua and His China Trade Partners in the Nineteenth Century. Doctoral dissertation, Harvard University. Citable link http://nrs.harvard.edu/urn-3:HUL.InstRepos:9282867 Terms of Use This article was downloaded from Harvard University’s DASH repository, and is made available under the terms and conditions applicable to Other Posted Material, as set forth at http:// nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of- use#LAA © 2012 – John D. Wong All rights reserved. Professor Michael Szonyi John D. Wong Global Positioning: Houqua and his China Trade Partners in the Nineteenth Century Abstract This study unearths the lost world of early-nineteenth-century Canton. Known today as Guangzhou, this Chinese city witnessed the economic dynamism of global commerce until the demise of the Canton System in 1842. Records of its commercial vitality and global interactions faded only because we have allowed our image of old Canton to be clouded by China’s weakness beginning in the mid-1800s. By reviving this story of economic vibrancy, I restore the historical contingency at the juncture at which global commercial equilibrium unraveled with the collapse of the Canton system, and reshape our understanding of China’s subsequent economic experience. I explore this story of the China trade that helped shape the modern world through the lens of a single prominent merchant house and its leading figure, Wu Bingjian, known to the West by his trading name of Houqua.
    [Show full text]
  • Negotiation Philosophy in Chinese Characters
    6 Ancient Wisdom for the Modern Negotiator: What Chinese Characters Have to Offer Negotiation Pedagogy Andrew Wei-Min Lee* Editors’ Note: In a project that from its inception has been devoted to second generation updates, it is instructive nonetheless to realize how much we have to learn from the past. We believe Lee’s chapter on Chinese characters and their implications for negotiation is groundbreaking. With luck, it will prove to be a harbinger of a whole variety of new ways of looking at our field that will emerge from our next round of discussion. Introduction To the non-Chinese speaker, Chinese characters can look like a cha- otic mess of dots, lines and circles. It is said that Chinese is the most difficult language in the world to learn, and since there is no alpha- bet, the struggling student has no choice but to learn every single Chinese character by sheer force of memory – and there are tens of thousands! I suggest a different perspective. While Chinese is perhaps not the easiest language to learn, there is a very definite logic and sys- tem to the formation of Chinese characters. Some of these characters date back almost eight thousand years – and embedded in their make-up is an extraordinary amount of cultural history and wisdom. * Andrew Wei-Min Lee is founder and president of the Leading Negotiation Institute, whose mission is to promote negotiation pedagogy in China. He also teaches negotiation at Peking University Law School. His email address is an- [email protected]. This article draws primarily upon the work of Feng Ying Yu, who has spent over three hundred hours poring over ancient Chinese texts to analyze and decipher the make-up of modern Chinese characters.
    [Show full text]
  • Toponymic Culture of China's Ethnic Minorities' Languages
    E/CONF.94/CRP.24 7 June 2002 English only Eighth United Nations Conference on the Standardization of Geographical Names Berlin, 27 August-5 September 2002 Item 9 (c) of the provisional agenda* National standardization: treatment of names in multilingual areas Toponymic culture of China’s ethnic minorities’ languages Submitted by China** * E/CONF.94/1. ** Prepared by Wang Jitong, General-Director, China Institute of Toponymy. 02-41902 (E) *0241902* E/CONF.94/CRP.24 Toponymic Culture of China’s Ethnic Minorities’ Languages Geographical names are fossil of history and culture. Many important meanings are contained in the geographical names of China’s Ethnic Minorities’ languages. I. The number and distribution of China’s Ethnic Minorities There are 55 minorities in China have been determined now. 53 of them have their own languages, which belong to 5 language families, but the Hui and the Man use Chinese (Han language). There are 29 nationalities’ languages belong to Sino-Tibetan family, including Zang, Menba, Zhuang, Bouyei, Dai, Dong, Mulam, Shui, Maonan, Li, Yi, Lisu, Naxi, Hani, Lahu, Jino, Bai, Jingpo, Derung, Qiang, Primi, Lhoba, Nu, Aching, Miao, Yao, She, Tujia and Gelao. These nationalities distribute mainly in west and center of Southern China. There are 17 minority nationalities’ languages belong to Altaic family, including Uygul, Kazak, Uzbek, Salar, Tatar, Yugur, Kirgiz, Mongol, Tu, Dongxiang, Baoan, Daur, Xibe, Hezhen, Oroqin, Ewenki and Chaoxian. These nationalities distribute mainly in west and east of Northern China. There are 3 minority nationalities’ languages belong to South- Asian family, including Va, Benglong and Blang. These nationalities distribute mainly in Southwest China’s Yunnan Province.
    [Show full text]
  • Social Mobility in China, 1645-2012: a Surname Study Yu (Max) Hao and Gregory Clark, University of California, Davis [email protected], [email protected] 11/6/2012
    Social Mobility in China, 1645-2012: A Surname Study Yu (Max) Hao and Gregory Clark, University of California, Davis [email protected], [email protected] 11/6/2012 The dragon begets dragon, the phoenix begets phoenix, and the son of the rat digs holes in the ground (traditional saying). This paper estimates the rate of intergenerational social mobility in Late Imperial, Republican and Communist China by examining the changing social status of originally elite surnames over time. It finds much lower rates of mobility in all eras than previous studies have suggested, though there is some increase in mobility in the Republican and Communist eras. But even in the Communist era social mobility rates are much lower than are conventionally estimated for China, Scandinavia, the UK or USA. These findings are consistent with the hypotheses of Campbell and Lee (2011) of the importance of kin networks in the intergenerational transmission of status. But we argue more likely it reflects mainly a systematic tendency of standard mobility studies to overestimate rates of social mobility. This paper estimates intergenerational social mobility rates in China across three eras: the Late Imperial Era, 1644-1911, the Republican Era, 1912-49 and the Communist Era, 1949-2012. Was the economic stagnation of the late Qing era associated with low intergenerational mobility rates? Did the short lived Republic achieve greater social mobility after the demise of the centuries long Imperial exam system, and the creation of modern Westernized education? The exam system was abolished in 1905, just before the advent of the Republic. Exam titles brought high status, but taking the traditional exams required huge investment in a form of “human capital” that was unsuitable to modern growth (Yuchtman 2010).
    [Show full text]
  • INVERSE PROBLEMS for NONLINEAR HYPERBOLIC EQUATIONS Gunther Uhlmann Jian Zhai 1. Introduction. Many Physical Phenomena Are Model
    DISCRETE AND CONTINUOUS doi:10.3934/dcds.2020380 DYNAMICAL SYSTEMS Volume 41, Number 1, January 2021 pp. 455{469 INVERSE PROBLEMS FOR NONLINEAR HYPERBOLIC EQUATIONS Gunther Uhlmann∗ Department of Mathematics, University of Washington, Seattle, WA 98195, USA Institute for Advanced Study, The Hong Kong University of Science and Technology Kowloon, Hong Kong, China Jian Zhai Institute for Advanced Study, The Hong Kong University of Science and Technology Kowloon, Hong Kong, China Abstract. There has been considerable progress in recent years in solving inverse problems for nonlinear hyperbolic equations. One of the striking aspects of these developments is the use of nonlinearity to get new information, which is not possible for the corresponding linear equations. We illustrate this for several examples including Einstein equations and the equations of nonlinear elasticity among others. 1. Introduction. Many physical phenomena are modeled by nonlinear hyperbolic equations (see for example [40]). In recent years, starting with the paper [20] sev- eral inverse problems for nonlinear hyperbolic equations have been solved, ranging from semilinear wave equations to nonlinear elastic equations and Einstein equa- tions. The nonlinearity of the equation is used in an essential way. The interaction of waves, caused by the nonlinearity of the equation, produces new waves giving information that is not available for the corresponding linear equation. We will survey in this paper some recent progress on inverse problems for nonlinear hyper- bolic equations. The most exciting breakthrough is the possibility of recovering time-dependent coefficients for nonlinear evolution equations. The unique recovery of a time-dependent coefficient in a linear hyperbolic equation is in general still not known.
    [Show full text]
  • Chinese Zheng and Identity Politics in Taiwan A
    CHINESE ZHENG AND IDENTITY POLITICS IN TAIWAN A DISSERTATION SUBMITTED TO THE GRADUATE DIVISION OF THE UNIVERSITY OF HAWAI‘I AT MĀNOA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN MUSIC DECEMBER 2018 By Yi-Chieh Lai Dissertation Committee: Frederick Lau, Chairperson Byong Won Lee R. Anderson Sutton Chet-Yeng Loong Cathryn H. Clayton Acknowledgement The completion of this dissertation would not have been possible without the support of many individuals. First of all, I would like to express my deep gratitude to my advisor, Dr. Frederick Lau, for his professional guidelines and mentoring that helped build up my academic skills. I am also indebted to my committee, Dr. Byong Won Lee, Dr. Anderson Sutton, Dr. Chet- Yeng Loong, and Dr. Cathryn Clayton. Thank you for your patience and providing valuable advice. I am also grateful to Emeritus Professor Barbara Smith and Dr. Fred Blake for their intellectual comments and support of my doctoral studies. I would like to thank all of my interviewees from my fieldwork, in particular my zheng teachers—Prof. Wang Ruei-yu, Prof. Chang Li-chiung, Prof. Chen I-yu, Prof. Rao Ningxin, and Prof. Zhou Wang—and Prof. Sun Wenyan, Prof. Fan Wei-tsu, Prof. Li Meng, and Prof. Rao Shuhang. Thank you for your trust and sharing your insights with me. My doctoral study and fieldwork could not have been completed without financial support from several institutions. I would like to first thank the Studying Abroad Scholarship of the Ministry of Education, Taiwan and the East-West Center Graduate Degree Fellowship funded by Gary Lin.
    [Show full text]
  • Ideophones in Middle Chinese
    KU LEUVEN FACULTY OF ARTS BLIJDE INKOMSTSTRAAT 21 BOX 3301 3000 LEUVEN, BELGIË ! Ideophones in Middle Chinese: A Typological Study of a Tang Dynasty Poetic Corpus Thomas'Van'Hoey' ' Presented(in(fulfilment(of(the(requirements(for(the(degree(of(( Master(of(Arts(in(Linguistics( ( Supervisor:(prof.(dr.(Jean=Christophe(Verstraete((promotor)( ( ( Academic(year(2014=2015 149(431(characters Abstract (English) Ideophones in Middle Chinese: A Typological Study of a Tang Dynasty Poetic Corpus Thomas Van Hoey This M.A. thesis investigates ideophones in Tang dynasty (618-907 AD) Middle Chinese (Sinitic, Sino- Tibetan) from a typological perspective. Ideophones are defined as a set of words that are phonologically and morphologically marked and depict some form of sensory image (Dingemanse 2011b). Middle Chinese has a large body of ideophones, whose domains range from the depiction of sound, movement, visual and other external senses to the depiction of internal senses (cf. Dingemanse 2012a). There is some work on modern variants of Sinitic languages (cf. Mok 2001; Bodomo 2006; de Sousa 2008; de Sousa 2011; Meng 2012; Wu 2014), but so far, there is no encompassing study of ideophones of a stage in the historical development of Sinitic languages. The purpose of this study is to develop a descriptive model for ideophones in Middle Chinese, which is compatible with what we know about them cross-linguistically. The main research question of this study is “what are the phonological, morphological, semantic and syntactic features of ideophones in Middle Chinese?” This question is studied in terms of three parameters, viz. the parameters of form, of meaning and of use.
    [Show full text]
  • English Versions of Chinese Authors' Names in Biomedical Journals
    Dialogue English Versions of Chinese Authors’ Names in Biomedical Journals: Observations and Recommendations The English language is widely used inter- In English transliteration, two-syllable Forms of Chinese Authors’ Names nationally for academic purposes. Most of given names sometimes are spelled as two in Biomedical Journals the world’s leading life-science journals are words (Jian Hua), sometimes as one word We recently reviewed forms of Chinese published in English. A growing number (Jianhua), and sometimes hyphenated authors’ names accompanying English- of Chinese biomedical journals publish (Jian-Hua). language articles or abstracts in various abstracts or full papers in this language. Occasionally Chinese surnames are Chinese and Western biomedical journals. We have studied how Chinese authors’ two syllables (for example, Ou-Yang, Mu- We found considerable inconsistency even names are presented in English in bio- Rong, Si-Ma, and Si-Tu). Editors who are within the same journal or issue. The forms medical journals. There is considerable relatively unfamiliar with Chinese names were in the following categories: inconsistency. This inconsistency causes may mistake these compound surnames for • Surname in all capital letters followed by confusion, for example, in distinguishing given names. hyphenated or closed-up given name, for surnames from given names and thus cit- China has 56 ethnic groups. Names example, ing names properly in reference lists. of minority group members can differ KE Zhi-Yong (Chinese Journal of In the current article we begin by pre- considerably from those of Hans, who Contemporary Pediatrics) senting as background some features of constitute most of the Chinese population. GUO Liang-Qian (Chinese Chinese names.
    [Show full text]
  • Lei Zuo, ASME Fellow, Professor, J. R. Jones III Faculty Fellow Director
    Lei Zuo, Professor, Virginia Tech, 12/20/2019 Lei Zuo, ASME Fellow, Professor, J. R. Jones III Faculty Fellow Director, Energy Harvesting and Mechatronics Research Lab Director, NSF I/UCRC Center for Energy Harvesting Materials and Systems Virginia Polytechnic Institute and State University (Virginia Tech), Blacksburg, VA 24061 Email: [email protected], Phone: (540) 231-7270, Cell: (847) 287-6887 Quick Highlights: • Awards: 2017 ASME Leonardo Da Vinci Award/Medal (for “eminent achievement in the design or invention of a product which is universally recognized as an important advance in machine design”); 2015 ASME Thar Energy Design Award (for “pioneering research in energy engineering, especially at large energy scale”); 2015 and 2011 R&D 100 Awards (for the 100 most significant technology innovations of the year in the word); 2014 SAE Ralph R. Teetor Educational Award • Funding: $14.5M (70 projects) in total as PI and co-PI since 2008 ($12M as PI, my share $9.4M). At VT: $10.4M (44 projects) in total as PI and co-PI ($9M as PI, my share $7M). • Publications: 273 papers in journals and conferences (113 in journals), including six best paper and two best student paper awards. H-index 33, i10-index 95, Google citations over 5000. • Advising: advising 8 PhD students, 7 MS students, 2 postdocs; graduated 12 PhDs (3 became faculty) and 45 MS (21 with thesis & 24 with projects); advised 11 postdocs (4 are faculty), mentored 150+ BS, and 10 high school students (5 subsequently admitted to MIT/Harvard). • Services: Technical editor of IEEE-ASME T. on Mechatronics, Associate editor of IFAC journal Mechatronics, Associate editor of ASME J.
    [Show full text]