DOCSLIB.ORG

INVERSE PROBLEMS for NONLINEAR HYPERBOLIC EQUATIONS Gunther Uhlmann Jian Zhai 1. Introduction. Many Physical Phenomena Are Model

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read 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. To solve inverse problems for linear hyperbolic equations, a powerful method, called the boundary control (BC) method, was developed by Belishev starting with the acoustic wave type equation ([2]). The BC method relies on a sharp unique continuation result for the wave equation that was proven later by Tataru [33]. However, this result is only valid if the coefficients of the equation depend ana- lytically on time. One way to avoid using Tataru's result is to reduce the inverse problems to some geometrical inverse problems. We refer to [29, 27, 31, 30] for 2020 Mathematics Subject Classification. Primary: 35R30; Secondary: 35A27, 35L70. Key words and phrases. Inverse problems, nonlinear hyperbolic equations, Lorentzian manifolds. The first author was partially supported by NSF, a Walker Professorship at UW and a Si-Yuan Professorship at IAS, HKUST. ∗ Corresponding author: [email protected]. 455 456 GUNTHER UHLMANN AND JIAN ZHAI this direction of work. Microlocal analysis is then used as a powerful tool to ana- lyze the propagation of singularities of solutions. Nonlinear waves have even richer geometric behaviors, which can be utilized to solve inverse problems. For an inverse problem for a nonlinear equation, usually the nonlinear equation is linearized, and the results for the linearized equation are applied when available. For example, a semilinear parabolic equation is treated in [15]; the nonlinear Calder´on problem in the isotropic case is reduced to the linear one [32]; the inverse problem for the nonlinear Navier-Stokes equation is reduced to the problem for the linear Stokes equation [24]. Since the paper [20], the nonlinearity has been used to solve some inverse prob- lems, whereas the results for the associated linearized equations are not available. What is used is the interaction of several waves to produce new waves that give information not available for the linearized equation. The technique of higher order linearization is a necessary part in the treatment. We shall briefly explain this. The higher order linearization can be explained as follows. Assume Λ(f) is the measured response of a medium with an input f. For example, Λ can be the Dirichlet-to- Neumann map associated to some partial differential equation. One can take an PN input of the form f = i=1 ifi, where i, i = 1; ··· ;N are small parameters. If the response of the medium to the input is nonlinear, then Λ is a nonlinear map, and PN consequently Λ( i=1 ifi) contains extra information other than what is contained in fΛ(fi)gi=1;··· ;N . Therefore useful information can possibly be extracted from the higher order linearization of the map Λ: N @N X Λ( ifi): @ ··· @ =···= =0 1 N 1 N i=1 For hyperbolic equations modeling waves, one usually chooses input sources of the PN form f = i=1 ifi such that fi, i = 1; 2; ··· N produce distorted plane waves or Gaussian beams. The outline of this paper is as follows. In Section 2 we consider the inverse problem of determining a globally hyperbolic Lorentzian metric by measuring the source-to-solution map in a neighborhood of a time-like geodesic for the wave equa- tion plus a quadratic term. The result of [17] is reviewed. The concept of earliest light observation set is defined and the result that the earliest light observation sets determine the conformal class of a globally hyperbolic Lorentzian metric in 3+1 dimensions is reviewed. In Section 3 more general semilinear equations are con- sidered. In Section 4, several results for other equations like Einstein equations, a simpler analog of Yang-Mills equations and nonlinear elasticity are formulated. 2. Quadratic non-linearities. We first introduce some notations and definitions. Let (M; g) be a C1-smooth (1 + 3)-dimensional globally hyperbolic Lorentzian manifold, where the metric g is of signature (−; +; +; +). By [3], (M; g) is isometric to a smooth manifold (R × N; h), where N is a 3-dimensional manifold and the metric h has the form h = −β(t; y)dt2 + κ(t; y): (1) Here β : R × N ! (0; +1) is a smooth function and κ(t; ·) is a Riemannian metric on N depending smoothly on t 2 R. A smooth curve µ :(a; b) ! M is called causal if g(_µ(s); µ_ (s)) ≤ 0 andµ _ (s) 6= 0 for all s 2 (a; b). Given p; q 2 M, we denote p ≤ q if p = q or p can be joined to q by a future-pointing causal curve. We say p < q if p ≤ q + and p 6= q. We denote the causal future of p 2 M to be Jg (p) = fq 2 M : p ≤ qg NONLINEAR HYPERBOLIC EQUATIONS 457 − and the causal past of q 2 M to be Jg (q) = fp 2 M : p ≤ qg. The curve µ is called time-like if g(_µ(s); µ_ (s)) < 0 for all s 2 (a; b). We denote p q if p 6= q and there is a future-pointing time-like path from p to q. Then the chronological + future of p 2 M is the set Ig (p) = fq 2 M; p qg and the chronological past of − + − q 2 M is Ig (q) = fp 2 M; p qg. We also denote Jg(p; q) := Jg (p) \ Jg (q) and + − Ig(p; q) := Ig (p) \ Ig (q). Take t0 > 0 to be a fixed number and denote M(t0) = (−∞; t0) × N: Consider the semilinear wave equations with source terms 2 gu(x) + a(x)u(x) = f(x); on M(t0); + (2) u = 0 in M(t0) n Jg (supp(f)); where g is the Laplacian-Beltrami operator on (M; g) and a(x) is nowhere vanish- ing. In local coordinates, 1 p jk g := @j( j det gjg @k): pj det gj For p 2 M, denote the set of light-like vectors at p by LpM = fζ 2 TpM n f0g : g(ζ; ζ) = 0g: ∗ Also, the set of light-like covectors at p is denoted by LpM. The future and past + − ∗;+ ∗;− light-like vectors (covectors) are denoted by Lp M and Lp M (Lp M and Lp M). Assume V ⊂ M(t0) is an open, relatively compact, connected neighborhood of a future-pointing time-like pathµ ^ =µ ^([−1; 1]) ⊂ M(t0).Take −1 < s− < s+ < 1, ± and denote p =µ ^(s±). 6 Definition 2.1. Let W = ff 2 C0 (V ); kfkC6(V ) < "g, where " > 0 is so small 2 that the equation (2) has a unique solution u 2 C (M(t0)) for all f 2 W. The source-to-solution map LV : W! C(V ) is the nonlinear operator defined as LV : f 7! ujV ; f 2 W; (3) where u satisfies the wave equation (2) on (M(t0); g). Kurylev, Lassas and Uhlmann [20] first studied the inverse problem of determin- ing the Lorentzian metric g from the source-to-solution operator LV . The main result is: Theorem 2.2. Let (M (j); g(j)), j = 1; 2 be two smooth, globally hyperbolic Lorentzian manifolds of dimension (1 + 3) that are represented in the form M (j) = R × N (j) withe a metric of the form (1). (j) + − Let µ^j :[−1; 1] ! M (t0) be smooth time-like paths, pj =µ ^j(s+), pj =µ ^j(s−), (j) where −1 < s− < s+ < 1, and Vj ⊂ M be neighborhoods of µ^j([−1; 1]). Let LVj , j = 1; 2 be the source-to-solution maps for wave equations (2) on man- (j) (j) (j) ifolds (M ; g ) with a = aj, where aj : M ! R n f0g are nowhere vanishing C1-smooth functions, j = 1; 2. − − Assume that there is a diffeomorphism Φ: V1 ! V2 such that Φ(p1 ) = p2 , + + Φ(p1 ) = p2 and the source-to-solution maps satisfy −1 ∗ ∗ ((Φ ) ◦ LV1 ◦ Φ )f = LV2 f 6 for f 2 W, where W is a neighborhood of the zero function in C0 (V2). 458 GUNTHER UHLMANN AND JIAN ZHAI − + − + Then there is a diffeomorphism Ψ: Ig(1) (p1 ; p1 ) ! Ig(2) (p2 ; p2 ) and the metric ∗ (2) (1) − + (1) (1) Ψ g is conformal to g in Ig(1) (p1 ; p1 ) ⊂ M , that is, there is b : M ! R+ (1) ∗ (2) − + such that g (x) = b(x)(Ψ g )(x) in Ig(1) (p1 ; p1 ).
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]
  • 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]
  • 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]
  • SDA: Software-Defined Accelerator for General-Purpose Big Data Analysis System
    SDA: Software-Defined Accelerator for general-purpose big data analysis system Jian Ouyang([email protected]), Wei Qi, Yong Wang, Yichen Tu, Jing Wang, Bowen Jia Baidu is beyond a search engine • Search • O2O/autonomous car/cloud/finance… 1/22 Outline • The overview of big data and its system • Motivations and challenge of accelerating big data processing • Our solution : SDA for big data – Design goals – Design and implementation – Performance evaluation • Conclusion 2/22 The overview of big data and its system Total data: ~1EB Processing data : ~100PB/day Total web pages: ~1000 Billion Web pages updated: ~10Billion/day Data is big Requests: ~10Billion/day Total logs : ~100PB Logs updated: ~1PB/day 3/22 The overview of big data and its system Knowledge Application Multimedia NLP CTR Recommend Graph • Applications are diverse Management Data warehouse Machine learning • Systems are complex Software infrastructure (Data center OS, Real time storage and computation) system architecture Basic facilities (Data center, Network, Server) 4/22 Motivation of accelerating big data processing 16 15 14 • CPU typical configuration in data 12 center • 2009, dual-core, ~3.0Ghz 10 • 2015, 8-core, ~2.0Ghz 8 • Scale of business (Baidu revenue) 6 • 2009, ~4.4B RMB • 2015,~66B RMB 4 4 2 1 0 2009 2015 CPU capability scale of business • Computing = data scale * algorithm complexity • Trends – Data scale increase rapidly – Algorithm complexity increase rapidly – Performance of CPU increase slowly • Need to bridge the gap 5/22 Challenges of accelerating general-purpose
    [Show full text]
  • The Muslim Emperor of China: Everyday Politics in Colonial Xinjiang, 1877-1933
    The Muslim Emperor of China: Everyday Politics in Colonial Xinjiang, 1877-1933 The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters Citation Schluessel, Eric T. 2016. The Muslim Emperor of China: Everyday Politics in Colonial Xinjiang, 1877-1933. Doctoral dissertation, Harvard University, Graduate School of Arts & Sciences. Citable link http://nrs.harvard.edu/urn-3:HUL.InstRepos:33493602 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 The Muslim Emperor of China: Everyday Politics in Colonial Xinjiang, 1877-1933 A dissertation presented by Eric Tanner Schluessel to The Committee on History and East Asian Languages in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the subject of History and East Asian Languages Harvard University Cambridge, Massachusetts April, 2016 © 2016 – Eric Schluessel All rights reserved. Dissertation Advisor: Mark C. Elliott Eric Tanner Schluessel The Muslim Emperor of China: Everyday Politics in Colonial Xinjiang, 1877-1933 Abstract This dissertation concerns the ways in which a Chinese civilizing project intervened powerfully in cultural and social change in the Muslim-majority region of Xinjiang from the 1870s through the 1930s. I demonstrate that the efforts of officials following an ideology of domination and transformation rooted in the Chinese Classics changed the ways that people associated with each other and defined themselves and how Muslims understood their place in history and in global space.
    [Show full text]
  • Names of Chinese People in Singapore
    101 Lodz Papers in Pragmatics 7.1 (2011): 101-133 DOI: 10.2478/v10016-011-0005-6 Lee Cher Leng Department of Chinese Studies, National University of Singapore ETHNOGRAPHY OF SINGAPORE CHINESE NAMES: RACE, RELIGION, AND REPRESENTATION Abstract Singapore Chinese is part of the Chinese Diaspora.This research shows how Singapore Chinese names reflect the Chinese naming tradition of surnames and generation names, as well as Straits Chinese influence. The names also reflect the beliefs and religion of Singapore Chinese. More significantly, a change of identity and representation is reflected in the names of earlier settlers and Singapore Chinese today. This paper aims to show the general naming traditions of Chinese in Singapore as well as a change in ideology and trends due to globalization. Keywords Singapore, Chinese, names, identity, beliefs, globalization. 1. Introduction When parents choose a name for a child, the name necessarily reflects their thoughts and aspirations with regards to the child. These thoughts and aspirations are shaped by the historical, social, cultural or spiritual setting of the time and place they are living in whether or not they are aware of them. Thus, the study of names is an important window through which one could view how these parents prefer their children to be perceived by society at large, according to the identities, roles, values, hierarchies or expectations constructed within a social space. Goodenough explains this culturally driven context of names and naming practices: Department of Chinese Studies, National University of Singapore The Shaw Foundation Building, Block AS7, Level 5 5 Arts Link, Singapore 117570 e-mail: [email protected] 102 Lee Cher Leng Ethnography of Singapore Chinese Names: Race, Religion, and Representation Different naming and address customs necessarily select different things about the self for communication and consequent emphasis.
    [Show full text]
  • “The Hereditary House of King Goujian of Yue”
    "Yuewang Goujian Shijia": An Annotated Translation Item Type text; Electronic Thesis Authors Daniels, Benjamin Publisher The University of Arizona. Rights Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author. Download date 26/09/2021 20:21:08 Link to Item http://hdl.handle.net/10150/293623 “YUEWANG GOUJIAN SHIJIA”: AN ANNOTATED TRANSLATION by Benjamin Daniels ____________________________ Copyright © Benjamin Daniels 2013 A Thesis Submitted to the Faculty of the DEPARTMENT OF EAST ASIAN STUDIES In Partial Fulfillment of the Requirements For the Degree of MASTER OF ARTS In the Graduate College THE UNIVERSITY OF ARIZONA 2013 2 STATEMENT BY AUTHOR This thesis has been submitted in partial fulfillment of requirements for an advanced degree at the University of Arizona and is deposited in the University Library to be made available to borrowers under rules of the Library. Brief quotations from this thesis are allowable without special permission, provided that an accurate acknowledgement of the source is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the copyright holder. SIGNED: Benjamin Daniels APPROVAL BY THESIS DIRECTOR This thesis has been approved on the date shown below: Dr. Brigitta Lee May 8, 2013 3 ACKNOWLEDGEMENTS First, I need to express my deepest gratitude to Dr. Enno Giele, who was my first mentor in anything related to ancient China.
    [Show full text]
  • Rethinking Chinese Kinship in the Han and the Six Dynasties: a Preliminary Observation
    part 1 volume xxiii • academia sinica • taiwan • 2010 INSTITUTE OF HISTORY AND PHILOLOGY third series asia major • third series • volume xxiii • part 1 • 2010 rethinking chinese kinship hou xudong 侯旭東 translated and edited by howard l. goodman Rethinking Chinese Kinship in the Han and the Six Dynasties: A Preliminary Observation n the eyes of most sinologists and Chinese scholars generally, even I most everyday Chinese, the dominant social organization during imperial China was patrilineal descent groups (often called PDG; and in Chinese usually “zongzu 宗族”),1 whatever the regional differences between south and north China. Particularly after the systematization of Maurice Freedman in the 1950s and 1960s, this view, as a stereo- type concerning China, has greatly affected the West’s understanding of the Chinese past. Meanwhile, most Chinese also wear the same PDG- focused glasses, even if the background from which they arrive at this view differs from the West’s. Recently like Patricia B. Ebrey, P. Steven Sangren, and James L. Watson have tried to challenge the prevailing idea from diverse perspectives.2 Some have proven that PDG proper did not appear until the Song era (in other words, about the eleventh century). Although they have confirmed that PDG was a somewhat later institution, the actual underlying view remains the same as before. Ebrey and Watson, for example, indicate: “Many basic kinship prin- ciples and practices continued with only minor changes from the Han through the Ch’ing dynasties.”3 In other words, they assume a certain continuity of paternally linked descent before and after the Song, and insist that the Chinese possessed such a tradition at least from the Han 1 This article will use both “PDG” and “zongzu” rather than try to formalize one term or one English translation.
    [Show full text]
  • UNIVERSITY of CALIFORNIA Santa Barbara Scribes in Early Imperial
    UNIVERSITY OF CALIFORNIA Santa Barbara Scribes in Early Imperial China A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in History by Tsang Wing Ma Committee in charge: Professor Anthony J. Barbieri-Low, Chair Professor Luke S. Roberts Professor John W. I. Lee September 2017 The dissertation of Tsang Wing Ma is approved. ____________________________________________ Luke S. Roberts ____________________________________________ John W. I. Lee ____________________________________________ Anthony J. Barbieri-Low, Committee Chair July 2017 Scribes in Early Imperial China Copyright © 2017 by Tsang Wing Ma iii ACKNOWLEDGEMENTS I wish to thank Professor Anthony J. Barbieri-Low, my advisor at the University of California, Santa Barbara, for his patience, encouragement, and teaching over the past five years. I also thank my dissertation committees Professors Luke S. Roberts and John W. I. Lee for their comments on my dissertation and their help over the years; Professors Xiaowei Zheng and Xiaobin Ji for their encouragement. In Hong Kong, I thank my former advisor Professor Ming Chiu Lai at The Chinese University of Hong Kong for his continuing support over the past fifteen years; Professor Hung-lam Chu at The Hong Kong Polytechnic University for being a scholar model to me. I am also grateful to Dr. Kwok Fan Chu for his kindness and encouragement. In the United States, at conferences and workshops, I benefited from interacting with scholars in the field of early China. I especially thank Professors Robin D. S. Yates, Enno Giele, and Charles Sanft for their comments on my research. Although pursuing our PhD degree in different universities in the United States, my friends Kwok Leong Tang and Shiuon Chu were always able to provide useful suggestions on various matters.
    [Show full text]
  • Arxiv:2005.10447V2 [Math.AP] 26 Jan 2021 K=2 As Neumann Data, We Take F Which Are Small in Cm+1 for fixed Large M
    AN INVERSE BOUNDARY VALUE PROBLEM FOR A SEMILINEAR WAVE EQUATION ON LORENTZIAN MANIFOLDS PETER HINTZ, GUNTHER UHLMANN, AND JIAN ZHAI Abstract. We consider an inverse boundary value problem for a semilinear wave equation on a time-dependent Lorentzian manifold with time-like boundary. The time-dependent coefficients of the nonlinear terms can be recovered in the interior from the knowledge of the Neumann-to-Dirichlet map. Either distorted plane waves or Gaussian beams can be used to derive uniqueness. 1. Introduction Let (M; g) be a (1 + 3)-dimensional Lorentzian manifold with boundary @M, where the metric g is of signature (−; +; +; +). We assume that M = R × N where N is a manifold with boundary @N, and write the metric g as (1) g = −β(t; x0)dt2 + κ(t; x0); 0 0 1 2 3 where x = (t; x ) = (x ; x ; x ; x ) are local coordinates on M; here, β : R×N ! (0; 1) is a smooth function and κ(t; ·) is a Riemannian metric on N depending smoothly on t 2 R. The boundary @M = R × @N of M is then timelike. Let ν denote the unit outer normal vector field to @M. Assume that @M is null-convex, which means that II(V; V ) = g(rV ν; V ) ≥ 0 for all null vectors V 2 T (@M); see [18] for a discussion of this condition. We consider the semilinear wave equation on M gu(x) + H(x; u(x)) = 0; on M; (2) @νu(x) = f(x); on @M; u(t; x0) = 0; t < 0; −1=2 p jk where g = j det gj @j( j det gjg @k) is the wave operator (d'Alembertian) on (M; g).
    [Show full text]