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{{DISPLAYTITLE:G<sub>2</sub> manifold}}
{{Infobox scientist
In [[differential geometry]], a '''''G''<sub>2</sub> manifold''' is a seven-dimensional [[Riemannian manifold]] with [[holonomy group]] [[G2 (mathematics)|''G''<sub>2</sub>]]. The [[group (mathematics)|group]] <math>G_2</math> is one of the five exceptional [[simple Lie group]]s. It can be described as the [[automorphism group]] of the [[octonions]], or equivalently, as a proper subgroup of SO(7) that preserves a [[spinor]] in the eight-dimensional spinor representation or lastly as the subgroup of GL(7) which preserves a ''positive, nondegenerate'' 3-form, <math>\phi_0</math>. The later definition was used by R. Bryant. Non-degenerate may be taken to be one whose orbit has maximal dimension in <math>\Lambda^3(\Bbb R^7)</math>. The stabilizer of such a non-degenerate form necessarily preserves an inner product which is either positive definite or of signature <math>(3,4)</math>. Thus, <math>G_2</math> is a subgroup of <math>SO(7)</math>. By covariant transport, a manifold with holonomy <math>G_2</math> has a Riemannian metric and a parallel (covariant constant) 3-form, <math>\phi</math>, the associative form. The Hodge dual, <math>\psi=*\phi</math> is then a parallel 4-form, the coassociative form. These forms are [[calibrated geometry|calibrations in the sense of Harvey-Lawson]], and thus define special classes of 3 and 4 dimensional submanifolds, respectively. The deformation theory of such submanifolds was studied by McLean.
|name              = Yuan-Cheng Fung<br />馮元楨
|image            =
|birth_date        = {{Birth date and age|1919|9|15|mf=yes}}
|birth_place      = [[Changzhou]], [[Jiangsu]], China
|death_date        =
|death_place      =
|residence        = [[USA]]
|citizenship      = [[United States|American]]
|nationality      = [[USA|American]]
|ethnicity        = [[Han Chinese|Chinese]]
|field            = [[Bioengineering]]<br />[[Biomechanics]]
|work_institutions = [[Caltech]]<br />[[University of California, San Diego|UCSD]]
|alma_mater        = [[Nanjing University]]<br />[[Caltech]]
|doctoral_advisor  =  [[Ernest Edwin Sechler|Ernest Sechler]]
|doctoral_students =
|known_for        = [[bioengineering]]<br />[[biomechanics]]
|author_abbrev_bot =
|author_abbrev_zoo =
|influences        =
|influenced        =
|prizes            = {{no wrap|[[Theodore von Karman Medal|von Karman Medal]] (1976) <br /> [[Otto Laporte Award]] (1977) <br />[[Timoshenko Medal]] (1991) <br>[[National Medal of Science]] (2000)<br />[[Jordan Allen Medal]] (1991)<br />[[Russ Prize]] (2007)}}
|religion          =
|footnotes        =
|signature        =
}}


== Properties ==
'''Yuan-Cheng "Bert" Fung''' (born 1919) is an American bio-engineer. He is regarded as a founding figure of [[bioengineering]], [[tissue engineering]], and the "Founder of Modern [[Biomechanics]]".<ref>[http://www.techscience.com/mcb_pdf/v1n1/pdf/184288277842.pdf YC “Bert” Fung: The Father of Modern Biomechanics (pdf)]</ref>
If ''M'' is a <math>G_2</math>-manifold, then ''M'' is:
* [[Ricci-flat]]
* [[orientable]]
* a [[spin manifold]]


== History ==
==Biography==
Manifold with holonomy <math>G_2</math> was firstly introduced by Edmond Bonan in 1966, who constructed the parallel 3-form, the parallel 4-form and showed that this manifold was Ricci-flat. The first complete, but noncompact 7-manifolds with holonomy <math>G_2</math> were constructed by [[Robert Bryant (mathematician)|Robert Bryant]] and Salamon in 1989.  The first compact 7-manifolds with holonomy <math>G_2</math> were constructed by [[Dominic Joyce]] in 1994, and compact <math>G_2</math> manifolds are sometimes known as "Joyce manifolds", especially in the physics literature.
Fung was born in [[Jiangsu]] Province, [[China]] in 1919. He earned a Bachelor's degree in 1941 and a Master's degree in 1943 from the [[National Central University]] (later renamed [[Nanjing University]] in [[mainland China]] and reinstated in [[Taiwan]]), and earned a Ph.D. from the [[California Institute of Technology]] in the [[United States]] in 1948.  


== Connections to physics ==
Fung currently is Professor Emeritus and Research Engineer at the [[University of California, San Diego|University of California, San Diego (UCSD)]].
These manifolds are important in [[string theory]]. They break the original [[supersymmetry]] to 1/8 of the original amount. For example, [[M-theory]] compactified on a <math>G_2</math> manifold leads to a realistic four-dimensional (11-7=4) theory with N=1 supersymmetry. The resulting low energy effective [[supergravity]] contains a single supergravity [[supermultiplet]], a number of [[chiral supermultiplet]]s equal to the third [[Betti number]] of the <math>G_2</math> manifold and a number of U(1) [[vector supermultiplet]]s equal to the second Betti number.
 
==Research==
He is the author of numerous books including Foundations of Solid Mechanics, Continuum Mechanics, and a series of books on Biomechanics. He is also one of the principal founders of the ''Journal of Biomechanics'' and was a past chair of the [[American Society of Mechanical Engineers|ASME]] International [[Applied Mechanics Division]]. In 1972, Fung established the Biomechanics Symposium under the American Society of Mechanical Engineers. This biannual summer meeting, first held at the Georgia Institute of Technology, became the annual Summer Bioengineering Conference. Fung and colleagues were also the first to recognize the importance of residual stress on arterial mechanical behavior.<ref name="residual">{{cite journal
|author=Chuong,C.J. and Y.C. Fung
|title=On Residual Stress in Arteries
|journal=Journal of Biomechanics
|year=1986
|volume=108
|pages=189–192
|doi=10.1115/1.3138600
|issue=2}}</ref>
 
===Fung's Law===
Fung's famous exponential strain [[constitutive equation]] for preconditioned [[soft tissues]] is
:<math>w = \frac{1}{2}\left[q + c\left( e^Q -1 \right) \right]</math>
with
:<math>q=a_{ijkl}E_{ij}E_{kl} \qquad Q=b_{ijkl}E_{ij}E_{kl}</math>
quadratic forms of [[strain (mechanics)|Green-Lagrange strains]] <math>E_{ij}</math> and <math>a_{ijkl}</math>, <math>b_{ijkl}</math> and <math>c</math> material constants.<ref name="Fung">{{cite book
|author=Fung, Y.-C.
|title=Biomechanics: Mechanical Properties of Living Tissues
|publisher=Springer-Verlag
|location=New York
|year=1993
|pages= 568
|isbn=0-387-97947-6}}</ref> <math>w</math> is a [[strain energy]] function per volume unit, which is the mechanical strain energy for a given temperature. Materials that follow this law are known as '''Fung-elastic'''.<ref name="Humphrey">{{cite journal
|author=Humphrey, Jay D.  
|title=Continuum biomechanics of soft biological tissues
|journal=Proceedings of the Royal Society of London A
|editor=The Royal Society
|year=2003
|volume=459
|pages=3–43
 
|doi=10.1098/rspa.2002.1060
|url=http://rspa.royalsocietypublishing.org/content/459/2029/3.full.pdf
|bibcode=2003RSPSA.459....3H
|issue=2029}}</ref>
 
==Honors and awards==
* [[Theodore von Karman Medal]], 1976
* [[Otto Laporte Award]], 1977
* Worcester Reed Warner Medal, 1984 <ref>[http://www.asme.org/Governance/Honors/SocietyAwards/Worcester_Reed_Warner_Medal.cfm WORCESTER REED WARNER MEDAL RECIPIENTS]</ref>
* Jean-Leonard-Marie Poiseuille Award, 1986<ref>[http://www.coe.ou.edu/isb/awardees/1986.htm The International Society of Biorheology: Yuan-Cheng Fung, 1986 Recipient of the Jean-Leonard-Marie Poiseuille Award]</ref>
* [[Timoshenko Medal]], 1991
* Lissner Award for Bioengineering, from ASME
* Borelli Medal, from ASB
* Landis Award, from Microcirculation Society
* Alza Award, from BMES
* Melville Medal, 1994 <ref>[http://www.asme.org/Governance/Honors/SocietyAwards/Melville_Medal.cfm MELVILLE MEDAL RECIPIENTS]</ref>
* United States National Academy of Engineering Founders Award (NAE Founders Award), 1998
* [[National Medal of Science]], 2000
* [[Russ Prize|Fritz J. and Dolores H. Russ Prize]], 2007 ("for the characterization and modeling of human tissue mechanics and function leading to prevention and mitigation of trauma.")<ref>[http://www.nae.edu/nae/awardscom.nsf/weblinks/JMAN-6X5NY5?OpenDocument Recipient of the Fritz J. and Dolores H. Russ Prize]</ref>
 
Fung was elected to the [[United States National Academy of Science]] (1993), the [[National Academy of Engineering]] (1979), the [[Institute of Medicine]] (1991), the [[Academia Sinica]] (1968), and is a Foreign Member of the [[Chinese Academy of Sciences]] (1994 election).


''See also'': [[Calabi-Yau manifold]], [[Spin(7) manifold]]
==References==
==References==
{{reflist}}


*{{citation | first =| last = E. Bonan, | title = Sur les variétés riemanniennes à groupe d'holonomie G2 ou Spin(7)| journal = C. R. Acad. Sci. Paris | volume =262| year = 1966  | pages = 127&ndash;129}}.
==External links==
*{{citation | last = Bryant | first = R.L. | title = Metrics with exceptional holonomy | journal = Annals of Mathematics | issue = 2 | volume = 126 | year = 1987 | pages = 525&ndash;576 | doi = 10.2307/1971360 | publisher = Annals of Mathematics | jstor = 1971360}}.
*[http://www.jacobsschool.ucsd.edu/faculty/faculty_bios/findprofile.sfe?department=BENG&fmp_recid=21 Profile at UCSD]
*{{citation | last = Bryant | first = R.L. | first2 = S.M. | last2 = Salamon | title = On the construction of some complete metrics with exceptional holonomy | journal = Duke Mathematical Journal | volume = 58 | year = 1989 | pages = 829&ndash;850}}.
*Y.C. Fung, [http://amresearch.blogspot.com/2006/07/1991-timoshenko-medal-lecture-by-yuan.html Mechanics of Man], Acceptance Speech for the Timoshenko Medal.
*{{citation | first = R. | last = Harvey | first2 = H.B. | last2 = Lawson | title = Calibrated geometries | journal = Acta Mathematica | volume = 148 | year = 1982 | pages = 47&ndash;157 | doi = 10.1007/BF02392726}}.
*[http://www.asme.org/Governance/Honors/SocietyAwards/YC_Fung_Young_Investigator.cfm YC Fung Young Investigator Award]
*{{citation | first = D.D. | last = Joyce | title = Compact Manifolds with Special Holonomy | series = Oxford Mathematical Monographs | publisher = Oxford University Press |  isbn = 0-19-850601-5 | year = 2000}}.
*[http://www.techscience.com/mcb_pdf/v1n1/pdf/171267552381.pdf YC Fung: A Scientific Giant and a Kind Man (pdf)]
*{{citation | first = R.C. | last = McLean | title = Deformations of calibrated submanifolds | journal = Communications in Analysis and Geometry | volume = 6 | year = 1998 | pages = 705&ndash;747}}.
*[http://www.techscience.com/books/fung.html Molecular & Cellular Biomechanics: In Honor of The 90th Birthday of Professor Yuan Cheng Fung ]
*{{citation | first = Spiro | last = Karigiannis | title = What Is . . . a ''G''<sub>2</sub>-Manifold? | journal = AMS Notices | volume = 58 | issue = 04 | pages = 580–581 | year = 2011
    | url = http://www.ams.org/notices/201104/rtx110400580p.pdf }}.


[[Category:Differential geometry]]
{{Winners of the National Medal of Science|engineering}}
[[Category:Riemannian geometry]]
[[Category:Structures on manifolds]]


[[sv:G2-mångfald]]
{{Authority control|VIAF=85337796}}
{{Persondata <!-- Metadata: see [[Wikipedia:Persondata]]. -->
| NAME              = Fung, Yuan-Cheng
| ALTERNATIVE NAMES =
| SHORT DESCRIPTION = American bio engineer
| DATE OF BIRTH    = September 15, 1919
| PLACE OF BIRTH    = [[Changzhou]], [[Jiangsu]], China
| DATE OF DEATH    =
| PLACE OF DEATH    =
}}
{{DEFAULTSORT:Fung, Yuan-Cheng}}
[[Category:1919 births]]
[[Category:American engineers]]
[[Category:Biological engineering]]
[[Category:Bioengineers]]
[[Category:California Institute of Technology alumni]]
[[Category:California Institute of Technology faculty]]
[[Category:Chinese emigrants to the United States]]
[[Category:Guggenheim Fellows]]
[[Category:Living people]]
[[Category:Members of Academia Sinica]]
[[Category:Members of the United States National Academy of Sciences]]
[[Category:National Medal of Science laureates]]
[[Category:Nanjing University alumni]]
[[Category:National Central University alumni]]
[[Category:People from Changzhou]]
[[Category:University of California, San Diego faculty]]
[[Category:Members of the United States National Academy of Engineering]]

Revision as of 13:33, 12 August 2014

Template:Infobox scientist

Yuan-Cheng "Bert" Fung (born 1919) is an American bio-engineer. He is regarded as a founding figure of bioengineering, tissue engineering, and the "Founder of Modern Biomechanics".[1]

Biography

Fung was born in Jiangsu Province, China in 1919. He earned a Bachelor's degree in 1941 and a Master's degree in 1943 from the National Central University (later renamed Nanjing University in mainland China and reinstated in Taiwan), and earned a Ph.D. from the California Institute of Technology in the United States in 1948.

Fung currently is Professor Emeritus and Research Engineer at the University of California, San Diego (UCSD).

Research

He is the author of numerous books including Foundations of Solid Mechanics, Continuum Mechanics, and a series of books on Biomechanics. He is also one of the principal founders of the Journal of Biomechanics and was a past chair of the ASME International Applied Mechanics Division. In 1972, Fung established the Biomechanics Symposium under the American Society of Mechanical Engineers. This biannual summer meeting, first held at the Georgia Institute of Technology, became the annual Summer Bioengineering Conference. Fung and colleagues were also the first to recognize the importance of residual stress on arterial mechanical behavior.[2]

Fung's Law

Fung's famous exponential strain constitutive equation for preconditioned soft tissues is

w=12[q+c(eQ−1)]

with

q=aijklEijEklQ=bijklEijEkl

quadratic forms of Green-Lagrange strains Eij and aijkl, bijkl and c material constants.[3] w is a strain energy function per volume unit, which is the mechanical strain energy for a given temperature. Materials that follow this law are known as Fung-elastic.[4]

Honors and awards

Fung was elected to the United States National Academy of Science (1993), the National Academy of Engineering (1979), the Institute of Medicine (1991), the Academia Sinica (1968), and is a Foreign Member of the Chinese Academy of Sciences (1994 election).

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

External links

Template:Winners of the National Medal of Science

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  1. ↑ YC “Bert” Fung: The Father of Modern Biomechanics (pdf)
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    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  5. ↑ WORCESTER REED WARNER MEDAL RECIPIENTS
  6. ↑ The International Society of Biorheology: Yuan-Cheng Fung, 1986 Recipient of the Jean-Leonard-Marie Poiseuille Award
  7. ↑ MELVILLE MEDAL RECIPIENTS
  8. ↑ Recipient of the Fritz J. and Dolores H. Russ Prize
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