formulasearchengine - User contributions [en]

Jacobi elliptic functions

2015-01-05T20:24:03Z

129.13.72.196: /* Definition as trigonometryhttp://www.und.edu/instruct/schwalm/MAA_Presentation_10-02/handout.pdf */

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Dehn function

2014-11-20T16:46:46Z

129.13.72.197: /* Growth types of functions */ typo

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Uniform boundedness

2014-10-10T12:13:41Z

129.13.72.197: /* Real line and complex plane */

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Power-law fluid

2014-10-08T14:44:05Z

129.13.72.195: /* Dilatant fluids */ fluids don´t care about absolute deformation

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Weissenberg number

2014-10-08T06:55:32Z

129.13.72.195: explain symbols. but isn´t relaxation time usually \tau, not \lambda ?

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Hu Washizu principle

2014-04-29T09:16:28Z

129.13.72.198:

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Extensional viscosity

2014-02-27T11:39:43Z

129.13.72.198:

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DEVS

2013-10-07T12:02:27Z

129.13.72.198: /* Other Formalisms */

{{more footnotes|date=October 2013}}
{{Infobox person
| name = Albert Francis Birch
| image = Albert Francis Birch00.jpg
| image_size = 250px
| caption =
| birth_date = {{Birth date|1903|8|22}}
| birth_place = [[Washington D.C.]], [[United States]]
| death_date = {{Death date and age|1992|1|30|1903|8|22|df=yes}}
| death_place = United States
| occupation = Geophysicist
| spouse =
}}
'''Albert Francis Birch''' (August 22, 1903-January 30, 1992) was an American [[geophysicist]].<ref>{{cite journal|author=Shankland, Thomas|author2=O'Connell, Richard|title=Obituary: Francis Birch|journal=Physics Today|date=November 1992|volume=45|issue=11|pages=105–106|url=http://www.physicstoday.org/resource/1/phtoad/v45/i11/p105_s2?bypassSSO=1|doi=10.1063/1.2809890}}</ref>

==Biography==
He was born in Washington DC, United States to a [[Virginia]] born father involved in business and real estate and a church [[choir]] singer and soloist mother at [[Cathedral of St. Matthew the Apostle (Washington, D.C.)|St. Matthew's Cathedral]] in Washington DC. He graduated from [[Harvard University]] and became a full time professor at Harvard in 1946.

Albert Francis Birch is known for his experimental work on the properties of Earth-forming [[minerals]] at high pressure and temperature, in 1952 he published a well-known paper in the [[Journal of Geophysical Research]],<ref name="JGR52">Birch, F. (1952),
Elasticity and constitution of the Earth's interior, ''J. Geophys. Res., 57,'' 227-286.</ref>
where he demonstrated that the [[Mantle (geology)|mantle]] is chiefly composed of [[silicate minerals]], the upper and lower mantle are separated by a thin transition zone associated with silicate [[phase transitions]], and the inner and outer [[Structure of the Earth#Core|core]] are alloys of crystalline and molten [[iron]]. The most famous portion of the paper, however, is a humorous footnote he included in the introduction:

<blockquote>
Unwary readers should take warning that ordinary language undergoes modification to a high-pressure form when applied to the interior of the Earth. A few examples of equivalents follow:
<center>
{| class="wikitable"
|-
! High Pressure Form
! Ordinary Meaning
|-
| Certain
| Dubious
|-
| Undoubtedly
| Perhaps
|-
| Positive proof
| Vague suggestion
|-
| Unanswerable argument
| Trivial objection
|-
| Pure iron
| Uncertain mixture of all the elements
|}
</center>
</blockquote>

==Nuclear bombing==
During [[World War II]] he took a leave of absence from Harvard and was assigned to the [[Manhattan Project]], first in New York City, then in New Mexico. Because of his knowledge of metals and metallurgy he was in charge of designing and engineering the triggering mechanism for the Hiroshima atomic bomb code-named [[Little Boy]]; and went to Tinian to supervise loading it onto ''[[Enola Gay]]'' (the [[Boeing B-29 Superfortress]] tasked with dropping the bomb) and assembling the bomb before takeoff. He devised the 'double plug' system that allowed for actually arming the bomb after ''Enola Gay'' took off (so that if it crashed, there would not be a nuclear explosion on Tinian).

In 1947, he adapted the isothermal Murnaghan equation of state, which had been developed for infinitesimal strain, for [[Eulerian]] finite strain, developing what is now known as the [[Birch-Murnaghan equation of state]].<ref name="PhysRev47">Birch, F. (1947), Finite elastic strain of cubic crystals, ''Phys. Rev., 71,'' 809-824.</ref>

In 1961, Birch published two papers <ref name="JGR61">Birch, F. (1961), The velocity of compressional waves in rocks to 10 kilobars. Part 2. ''J. Geophys. Res., 66,''2199-2224.</ref>
<ref name="GJRAS61">Birch, F. (1961), Composition of the Earth's mantle, ''Geophys. J. R. astron. Soc., 4,'' 295-311.</ref> on [[compressional wave]] velocities establishing a linear relation (now called [[Birch's law]]) of the compressional wave velocity Vp of rocks and minerals of a constant average [[atomic weight]] <math>\bar{ M}</math> with density <math>\rho</math> as:

<math> V_p = a (\bar{ M}) + b \rho </math>.

==Post-war==
Birch was the Sturgis Hooper Professor of Geology at [[Harvard University]] from 1949 to 1974. He served as the president of the [[Geological Society of America]] in 1963-1964 and he received numerous honors in his career, including the GSA's [[Arthur L. Day Medal]] (1950) and [[Penrose Medal]] (1969), the [[American Geophysical Union|American Geophysical Union's]] [[William Bowie Medal]] (1960), the [[National Medal of Science]] (1967), the [[Vetlesen Prize]] (1968) (shared with Sir [[Edward Bullard]]), the [[Gold Medal of the Royal Astronomical Society]] (1973), and the International Association for the Advancement of High Pressure Research's [[Percy Bridgman|Bridgman]] Medal (1983). Since 1992, the AGU's [[Tectonophysics]] section has sponsored a Francis Birch Lecture, given at its annual meeting by a noted researcher in this field.

==See also==
* [[List of geophysicists]]

==References==
{{reflist|1}}

==External links==
* [http://books.nap.edu/openbook.php?record_id=6201&page=2 Birch's biography at the National Academy of Science]
* [http://www.agu.org/inside/awards/birch.html Birch's biography at AGU]
*[http://www.nasonline.org/publications/biographical-memoirs/memoir-pdfs/birch-francis.pdf National Academy of Sciences Biographical Memoir]

{{Geophysics navbox}}

{{Authority control|VIAF=35897886}}

{{Persondata 
| NAME =Birch, Francis
| ALTERNATIVE NAMES =
| SHORT DESCRIPTION = Geophysicist
| DATE OF BIRTH = 1903
| PLACE OF BIRTH = [[Washington D.C.]], [[United States]]
| DATE OF DEATH = 1992
| PLACE OF DEATH = United States
}}
{{DEFAULTSORT:Birch, Francis}}
[[Category:1903 births]]
[[Category:1992 deaths]]
[[Category:People from Washington D.C.]]
[[Category:Harvard University faculty]]
[[Category:American geophysicists]]
[[Category:Penrose Medal winners]]
[[Category:National Medal of Science laureates]]
[[Category:Recipients of the Gold Medal of the Royal Astronomical Society]]
[[Category:Manhattan Project people]]
[[Category:People associated with the atomic bombings of Hiroshima and Nagasaki]]

Stress relaxation

2013-09-04T09:28:09Z

129.13.72.198: Maxwell and Voigt are the simplest models for liquids and solids respectively. the slsm is a rather simple derivative of a voigt model, and surely not "most accurate".

{{Nofootnotes|date=February 2009}}
{{Refimprove|date=February 2009}}
In [[mathematics]], in the area of classical [[potential theory]], '''polar sets''' are the "negligible sets", similar to the way in which sets of measure zero are the [[negligible set]]s in [[measure theory]].

== Definition ==

A set <math>Z</math> in <math>\R^n</math> (where <math>n\ge 2</math>) is a polar set if there is a non-constant [[subharmonic function]]

:<math>u</math> on <math>\R^n</math>

such that

:<math>Z \subseteq \{x: u(x) = -\infty\}.</math>

Note that there are other (equivalent) ways in which polar sets may be defined, such as by replacing "subharmonic" by "superharmonic", and <math>-\infty</math> by <math>\infty</math> in the definition above.

== Properties ==

The most important properties of polar sets are:

*A singleton set in <math>\R^n</math> is polar.
*A countable set in <math>\R^n</math> is polar.
*The union of a countable collection of polar sets is polar.
*A polar set has Lebesgue measure zero in <math>\R^n.</math>

==Nearly everywhere==
A property holds '''nearly everywhere''' in a set ''S'' if it holds on ''S''−''E'' where ''E'' is a Borel polar set. If ''P'' holds nearly everywhere then it holds [[almost everywhere]].<ref name=Ran56>Ransford (1995) p.56</ref>

==See also==
* [[Pluripolar set]]

== References ==
{{reflist}}
* {{cite book | first=Joseph L. | last=Doob | authorlink=Joseph L. Doob | title=Classical Potential Theory and Its Probabilistic Counterpart | publisher=[[Springer-Verlag]] | location=Berlin Heidelberg New York | isbn=3-540-41206-9 | zbl=0549.31001 | series=Grundlehren der Mathematischen Wissenschaften | volume=262 | year=1984 }}
* {{cite book | first=L. L. | last=Helms | year=1975 | title=Introduction to potential theory | publisher=R. E. Krieger | isbn=0-88275-224-3 }}
* {{cite book | last=Ransford | first=Thomas | title=Potential theory in the complex plane | series=London Mathematical Society Student Texts | volume=28 | location=Cambridge | publisher=[[Cambridge University Press]] | year=1995 | isbn=0-521-46654-7 | zbl=0828.31001 }}

==External links==
* {{planetmath reference|id=6020|title=Polar set}}

[[Category:Subharmonic functions]]

{{mathanalysis-stub}}

Pyrgeometer

2012-04-25T14:08:22Z

129.13.72.198: /* Measurement of long wave downward radiation */

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Clutter (radar)

2012-03-14T08:37:08Z

129.13.72.198:

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