formulasearchengine - User contributions [en]

Constant of motion

2015-01-05T22:37:56Z

128.227.38.253: /* Derivation */

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Theorem of corresponding states

2014-12-05T15:00:48Z

128.227.98.159: /* See also */

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Isothermal process

2014-12-03T19:20:03Z

128.227.142.158: /* Details for an ideal gas */

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Functional response

2014-11-24T21:32:42Z

128.227.135.69: /* Type II */

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Diversity gain

2014-11-18T21:27:56Z

128.227.2.255:

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Spheroid

2014-02-27T20:15:38Z

128.227.20.42:

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Semi-supervised learning

2014-02-13T15:22:22Z

128.227.194.85:

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Nanoindentation

2013-11-20T17:17:44Z

128.227.37.84: /* Software */

The '''persistence length''' is a basic mechanical property quantifying the [[stiffness]] of a [[polymer]].

Informally, for pieces of the polymer that are shorter than the persistence length, the molecule behaves rather like a flexible elastic rod, while for pieces of the polymer that are much longer than the persistence length, the properties can only be described statistically, like a three-dimensional [[random walk]].

Formally, the persistence length, ''P'', is defined as the length over which correlations in the direction of the tangent are lost. In a more chemical based manner it can also be defined as the average sum of the projections of all bonds j ≥ i on bond i in an indefinitely long chain.<ref>{{cite book |title=Statistical Mechanics of Chain Molecules |first=Paul J. |last=Flory |publisher=Interscience Publishers |location=New York |year=1969 |isbn=0-470-26495-0 |url=http://openlibrary.org/b/OL5613440M/Statistical_mechanics_of_chain_molecules}}</ref>

Let us define the angle ''θ'' between a vector that is tangent to the polymer at position 0 (zero) and a tangent vector at a distance ''L'' away from position 0, along the contour of the chain. It can be shown that the expectation value of the cosine of the angle falls off exponentially with distance,<ref>{{cite book |title=The Theory of Polymer Dynamics |first1=M. |last1=Doi |first2=S.F. |last2=Edwards |publisher=Clarendon, Oxford |year=1986 |page=317}}</ref>

:<math> \langle\cos{\theta}\rangle = e^{-(L/P)} \, </math>

where ''P'' is the persistence length and the angled brackets denote the average over all starting positions.

In polymer science jargon, the persistence length is considered to be one half of the [[Kuhn length]], the length of hypothetical segments that the chain can be considered as freely joined. The persistence length equals the average projection of the [[end-to-end vector]] on the tangent to the chain contour at a chain end in the limit of infinite chain length.<ref>{{cite web |title=Compendium of Chemical Terminology |publisher=IUPAC |url=http://www.iupac.org/goldbook/P04515.pdf#search=%22persistence%20length%22}}</ref>

The persistence length can be also expressed using the bending stiffness <math> B_s </math>, the [[Young's modulus]] ''E'' and knowing the section of the polymer
chain.<ref>{{cite book |title=Cytoskeletal mechanics: models and measurements
|first=M.R.K. |last=Mofrad |publisher=Cambridge Univ Pr |url=http://books.google.ch/books?hl=en&lr=&id=HI8xTBUhnuYC&oi=fnd&pg=PA18&dq=Cytoskeletal+mechanics:+models+and+measurements&ots=EQKON4nMwi&sig=lvdjSlj8vTNcPp-70AQJLVscHhU}}</ref>
:<math> P=\frac{B_s}{k_BT}\, </math>

:<math> B_s=EI\, </math>

In the case of a rigid and uniform rod ''I'' can be expressed as:
:<math> I=\frac{\pi a^4}{4}\, </math>
where ''a'' is the radius.

For example a piece of uncooked [[spaghetti]] has a persistence length on the order of <math> 10^{18} </math> m (taking in consideration a Young modulus of
5 GPa and a radius of 1 mm).<ref>{{cite web |title=Brittle failure of dry spaghetti
|first=G. V. |last=Guinea |publisher=Engineering Failure Analysis |url=http://linkinghub.elsevier.com/retrieve/pii/S1350630704000123}}</ref> Double-helical [[DNA]] has a persistence length of about 500 [[Angstrom]]s.

Another example:<ref>http://www.physicsforums.com/showthread.php?t=510330</ref> 
Imagine a long cord that is slightly flexible. At short distance scales, the cord will basically be rigid. If you look at the direction the cord is pointing at two points that are very close together, the cord will likely be pointing in the same direction at those two points (i.e. the angles of the tangent vectors are highly correlated). If you choose two points on this flexible cord (imagine a piece of cooked spaghetti that you've just tossed on your plate) that are very far apart, however, the tangent to the cords at those locations will likely be pointing in different directions (i.e. the angles will be uncorrelated). If you plot out how correlated the tangent angles at two different points are as a function of the distance between the two points, you'll get a plot that starts out at 1 (perfect correlation) at a distance of zero and drops exponentially as distance increases. The persistence length is the characteristic length scale of that exponential decay.

==See also==
* [[Polymer]]
* [[Worm-like chain]]
* [[Freely Jointed Chain]]
* [[Kuhn length]]
* [[Paul Flory]]

==References==
{{Reflist}}

[[Category:Physical quantities]]
[[Category:Polymer physics]]

Dual matroid

2013-10-22T21:11:49Z

128.227.35.87: /* Basic properties */ added link for flats

The '''Tetrahedral hypothesis''' is an [[Superseded scientific theories|obsolete scientific theory]] attempting to explain the arrangement of the [[Earth]]'s [[continent]]s and [[ocean]]s by referring to the geometry of a [[tetrahedron]]. Although it was a historically interesting theory in the late 19th and early 20th century, it was superseded by the concepts of [[continental drift]] and modern [[plate tectonics]].

==Theory==

This idea, described as ‘"ingenious" by geologist [[Arthur Holmes]],<ref>{{cite book |author=[[Arthur Holmes]] |title=Principles of physical geology |url=http://books.google.com/books?id=XUJRAAAAMAAJ |year=1965 |publisher=Nelson |page=32 |isbn=0-17-448020-2}}</ref> is now of historical interest only, being finally refuted by that same Holmes.<ref>''Nature'' magazine, June 6th. 1925, p.873.</ref> It attempted to explain apparent anomalies in the distribution of land and water on the Earth's surface:<ref>{{cite book |author= A. Z. Bukhari |title=Encyclopedia of nature of geography |url=http://books.google.com/books?id=vef_5GExuUgC&pg=PA109 |year=2005 |publisher=Anmol Publications PVT. Limited |isbn=978-81-261-2443-5 |chapter= Continental Drift |pages=109–113 }}</ref>
#More than 75% of the Earth's land area is in the [[northern hemisphere]].
#Continents are roughly triangular.
#Oceans are roughly triangular.
#The north pole is surrounded by water, the south pole by land.
#Exactly opposite the Earth from land is almost always water.
#The [[Pacific Ocean]] occupies about one third of the Earth's surface.

To understand its appeal, consider the "regular solids": the [[sphere]] and the 5-member set of [[Platonic Solids]]. The solid with the lowest number of sides is the [[tetrahedron]] (four equilateral triangles); progressing through the [[cube]], and the [[icosahedron]] (20 sides), the sphere can be considered to have an infinite number of sides. All six regular solids share many symmetries.

Now, for each regular solid, we may relate its surface area and volume by the equation:
: <math>V = k\times A \times\sqrt{A}</math>
where'' k'' is a characteristic of each solid, ''V'' its volume, and ''A'' its area. As we traverse the set in order of increasing number of faces, we find that k increases for each member; it is 0.0227 for a tetrahedron and 0.0940 for a sphere. Thus the tetrahedron is the regular solid with the largest surface area for a given volume, and makes a reasonable endpoint for a shrinking spherical Earth.<ref>{{cite book |title=Principles of Physical Geography |publisher=APH Publishing |year=2004 |isbn= 978-81-7648-731-3 |url=http://books.google.com/?id=TXkdMyGF_wgC&pg=PA84&lpg=PA84 |pages= 84–85 |author9=S. A. Qazi }}</ref>

==History==
[[Image:Tetrahedral hypothesis.jpg|thumb|alt=view of earth above Antarctica|Illustration of Tetrahedral hypothesis by [[William Lowthian Green]] from 1875]]
The theory was first proposed by [[William Lowthian Green]] in 1875.<ref>{{cite book |title=Vestiges of the molten globe, as exhibited in the figure of the earth, volcanic action and physiography |author=[[William Lowthian Green]] |publisher= E. Stanford |place=London |year=1875 |oclc=3571917 |url= http://books.google.com/books?id=9DkDAAAAQAAJ }}</ref>
It was still popular in 1917 when summarized as:

<blockquote>"The law of least action … demands that the somewhat rigid crustal portion of the earth keep in contact with the lessening interior with the least possible readjustment of its surface. … a shrinking sphere tends by the law of least action to collapse into a tetrahedron, or a tetra-hedroid, a sphere marked by four equal and equidistant triangular projections; and the earth with its three about equal and equidistant double continental masses triangular southward with three intervening depressed oceans triangular northward, its northern ocean and southern continent, with land everywhere antipodal to water, realizes the tetrahedroid status remarkably.“<ref>{{cite news |title= Tetrahedral Deformations and Intercontinental Torsions |author = [[Benjamin Kendall Emerson]] |work= Proceedings, American Philosophical Society |volume= 56 |number= 6 |year= 1917 |url=http://books.google.com/books?id=4lULAAAAIAAJ&pg=PA445 |publisher=American Philosophical Society |isbn=978-1-4223-7253-1 |pages=445–472 }}</ref></blockquote>
This is suggesting that a cooling spherical Earth might have shrunk to form a tetrahedron, with its vertices and edges forming the continents, and four oceans ([[Pacific Ocean]], [[Atlantic Ocean]], [[Indian Ocean]] and [[Arctic Ocean]]) on its faces.

By 1915 German [[Alfred Wegener]] (1880–1930) had proposed in his [[continental drift]] theory that land masses moved great distances over the Earth's history. Wegener was also at first met with hostile reactions.<ref>{{cite web |title= Alfred Wegener (1880-1930) |url= http://www.ucmp.berkeley.edu/history/wegener.html |work= [[University of California Museum of Paleontology]] web site |publisher= University of California |year= 2006 |accessdate= August 4, 2010 }}</ref>
By the mid-1920s the same Holmes developed theories on what could cause the drift.<ref>{{cite journal |title= The Origin of the Continents |author= [[Arthur Holmes]] |journal= Nature |date= June 6, 1925 |doi= 10.1038/115873a0 |pages= 873–874 |volume= 115 |issue=2901}}</ref><ref>{{cite journal |title= Theory of Continental Drift: a Symposium on the Origin and Movement of Land Masses, both Inter-Continental and Intra-Continental, as proposed by Alfred Wegener |author= [[Arthur Holmes]] |journal= Nature |date= September 22, 1928 |doi= 10.1038/122431a0 |pages= 431–433 |volume= 122 |issue=3073}}</ref>
The [[plate tectonics]] theory is now generally accepted to explain the dynamic nature of the Earth's surface; the tetrahedral shape plays no special role in modern theories.<ref>{{cite web |title= Plate Tectonics: The Rocky History of an Idea |url= http://www.ucmp.berkeley.edu/geology/techist.html |work= [[University of California Museum of Paleontology]] web site |publisher= University of California |year= 2006 |accessdate= August 4, 2010 }}</ref>
Explanations of details such as water to land ratios, the precise shape of continents and their sizes continue to be developed.

==References==
{{Reflist}}

==Further reading==
* {{cite book |title= The Tetrahedral Earth |work= The Book of Earths: Hollow Earth, Ancient Maps, Atlantis, and Other Theories |url= http://www.sacred-texts.com/earth/boe/boe34.htm |author= Edna Kenton |publisher= Forgotten Books|isbn= 978-1-60506-415-4 |accessdate= August 4, 2010 }}

[[Category:Obsolete scientific theories]]
[[Category:Plate tectonics]]

Ramsey problem

2013-05-08T17:42:32Z

128.227.123.202:

In [[music theory]] and [[musical tuning]] the '''Holdrian comma''', also called '''Holder's comma''', and sometimes the '''Arabian comma''',<ref name="Touma">[[Habib Hassan Touma]] (1996). ''The Music of the Arabs'', p.23. trans. Laurie Schwartz. Portland, Oregon: Amadeus Press. ISBN 0-931340-88-8.</ref> is a small [[interval (music)|musical interval]] of approximately 22.6415 [[cent (music)|cents]],<ref name="Touma"/> equal to one step of [[53 equal temperament]], or <math>\sqrt[53]{2}</math> ({{Audio|Holdrian comma on C.mid|play}}). The name [[comma (music)|comma]] is misleading, since this interval is an irrational number and does not describe the compromise between intervals of any tuning system; it assumes this name because it is an approximation of the [[syntonic comma]] (21.51 cents)({{Audio|Syntonic comma on C.mid|play}}), which was widely used as a measurement of tuning in [[William Holder]]'s time.

'''Mercator's comma''' is a name often used for a closely related interval because of its association with [[Nicholas Mercator]]. One of these intervals was first described by Ching-Fang in 45 BCE.<ref name="Touma"/>

== Mercator's comma and the Holdrian comma ==
Mercator applied logarithms to determine that <math>\sqrt[55]{2}</math> (≈ 21.8182 cents) was nearly equivalent to a syntonic comma of ≈ 21.5063 cents (a feature of the prevalent sixth-comma [[meantone temperament]] tuning system of the time). He also considered that an "artificial comma" of <math>\sqrt[53]{2}</math> might be useful, because 31 octaves could be practically approximated by a cycle of 53 [[just fifth]]s. [[William Holder]], for whom the Holdrian comma is named, favored this latter unit because the intervals of 53 equal temperament are closer to [[just intonation]] than that of 55. Thus '''Mercator's comma''' and the Holdrian comma are two distinct but related intervals.

== Arabian comma ==
The name "Arabian comma" may be inaccurate; the comma has been employed mainly in Turkish music theory by [[Kemal Ilerici]], and by the Turkish composer [[Erol Sayan]]. The name of this comma is "Holder koması" in Turkish.

For instance, the [[makam]] ''rast'' (similar to the Western [[major scale]]) may be considered in terms of Holdrian commas:
c d e f g a b c'
commas: 9 8 5 9 9 8 5

while in contrast, the makam ''nihavend'' (similar to the Western [[minor scale]]):
c d e{{music|b}} f g a{{music|b}} b{{music|b}} c'
commas: 9 4 9 9 4 9 9
has [[medium second]]s between d–e, e–f, g–a, a–b, and b–c', a medium second being somewhere in between 8 and 9 commas.<ref name="Touma"/>

==References==
{{reflist}}

==Further reading==
*Holder, William, ''A Treatise on the Natural Grounds, and Principles of Harmony'', facsimile of the 1694 edition, Broude Brothers, New York, 1967. (Original pp. 103–106.)

==External links==
*Monzo, Joe (2005). [http://www.tonalsoft.com/enc/m/mercator-comma.aspx "Mercator's Comma"], ''Tonalsoft''.

{{Intervals|state=expanded}}

{{DEFAULTSORT:Holdrian Comma}}
[[Category:Commas]]

UF Training Reactor

2013-02-12T17:35:32Z

128.227.238.19:

In aeronautics, the '''load factor''' is defined as the [[ratio]] of the [[Lift (force)|lift]] of an [[aircraft]] to its [[weight]]
<ref name="Clancy">Clancy, section 5.22</ref>
<ref name="Hurt">Hurt, page 37</ref> and represents a global measure of the [[stress (mechanics)|stress]] ("load") to which the structure of the aircraft is subjected:
:<math>n = \frac {L}{W}</math>
where:
:''n'' = Load factor
:''L'' = Lift
:''W'' = Weight
Since the load factor is the ratio of two forces, it is dimensionless. However, its units are traditionally referred to as '''g''', because of the relation between load factor and apparent acceleration of gravity felt on board the aircraft. A load factor of one, or 1 g, represents conditions in straight and level flight, where the lift is equal to the weight. Load factors greater or less than one (or even negative) are the result of maneuvers or wind gusts.<ref>McCormick, p.464-468</ref>

== Load factor and g ==
The fact that the load factor is commonly expressed in ''g'' units does not mean that it is dimensionally the same as the [[Earth's gravity|acceleration of gravity]], also indicated with ''g''. The load factor is strictly non-dimensional.

The use of ''g'' units refers to the fact that an observer on board an aircraft will experience an ''apparent'' acceleration of gravity (i.e. relative to his frame of reference) equal to load factor times the acceleration of gravity. For example, an observer on board an aircraft performing a turn with a load factor of 2 (i.e. a 2 g turn) will see objects falling to the floor at twice the normal acceleration of gravity.

In general, whenever the term ''load factor'' is used, it is formally correct to express it using numbers only, as in "a maximum load factor of 4". If the term ''load factor'' is omitted then ''g'' is used instead, as in "pulling a 3 g turn".<ref>Clancy, section 14.3</ref>

== Positive and negative load factors ==
[[Image:Load factor in a turn.gif|thumb|400px|Variation of the load factor ''n'' with the bank angle ''θ'', during a coordinated turn.]]
The load factor, and in particular its sign, depends not only on the forces acting on the aircraft, but also on the orientation of its vertical axis.

During straight and level flight, the load factor is +1 if the aircraft is flown "the right way up",<ref>Clancy, page 90</ref> whereas it becomes -1 if the aircraft is flown "upside-down" (inverted). In both cases the lift vector is the same (as seen by an observer on the ground), but in the latter the vertical axis of the aircraft points downwards, making the lift vector's sign negative.

In turning flight the load factor is normally greater than +1. For example, in a turn with a 60° [[Banked turn#Aviation|angle of bank]] the load factor is +2. Again, if the same turn is performed with the aircraft inverted, the load factor becomes -2. In general, in a balanced turn in which the angle of bank is ''θ'', the load factor ''n'' is related to the [[Trigonometric function|cosine]] of ''θ'' by the formula:<ref name="Hurt"/><ref>Clancy, page 407</ref>
:<math>n = \frac {1}{\cos\,\theta}</math>
Another way to achieve load factors significantly higher than +1 is to pull on the [[Elevator (aircraft)|elevator]] control at the bottom of a dive, whereas strongly pushing the stick forward during straight and level flight is likely to produce negative load factors, by causing the lift to act in the opposite direction to normal, i.e. downwards.

== Load factor and lift ==
In the definition of load factor, the lift is not simply that one generated by the aircraft's [[wing]], instead it is the vector sum of the lift generated by the wing, by the fuselage and by the [[tailplane]],<ref>Clancy, page 395</ref> or in other words it is the component perpendicular to the airflow of the sum of all aerodynamic forces acting on the aircraft.

The lift in the load factor is also intended as having a sign, which is positive if the lift vector points in the same direction, or close to, as the aircraft's vertical axis, or negative if it points in the opposite direction, or close to opposite, to the vertical axis.<ref>{{Cite web
| last = Gardiner
| first = Dave
| title = Groundschool - Theory of Flight. Manoeuvring forces
| publisher = RA-Aus
| url = http://www.recreationalflying.net/tutorials/groundschool/umodule1b.html#g_load
| accessdate = 25 March 2010}}</ref>

== Design standards ==
Excessive load factors must be avoided because of the possibility of exceeding the structural strength of the aircraft.

[[National aviation authority|Aviation authorities]] specify the load factor limits within which different classes of aircraft are required to operate without damage. For example, the US [[Federal Aviation Regulations]] prescribe the following limits (for the most restrictive case):
*For commercial transport airplanes, from -1 to +2.5 (or up to +3.8 depending on design takeoff weight) <ref>{{Cite web
| title = Part 25 - Airworthiness Standards: Transport Category Airplanes
| publisher = FAA
| url = http://rgl.faa.gov/Regulatory_and_Guidance_Library%5CrgFAR.nsf/0/5D3CEA6C015DDA8685256672004EC387?OpenDocument
| accessdate = 29 March 2010}}</ref>
*For light airplanes, from -1.5 to +3.8 <ref name="Far23">{{Cite web
| title = Part 23 - Airworthiness Standards: Normal, Utility, Acrobatic, and Commuter Category Airplanes
| publisher = FAA
| url = http://rgl.faa.gov/Regulatory_and_Guidance_Library%5CrgFAR.nsf/0/D35E170180C8CB2185256687006D0C90?OpenDocument
| accessdate = 29 March 2010}}</ref>
*For aerobatic airplanes, from -3 to +6 <ref name="Far23" />
*For helicopters, from -1 to +3.5 <ref>{{Cite web
| title = Part 27 - Airworthiness Standards: Normal Category Rotorcraft
| publisher = FAA
| url = http://rgl.faa.gov/Regulatory_and_Guidance_Library%5CrgFAR.nsf/0/AEAD1A7505EF922F852565F6006C1678?OpenDocument
| accessdate = 29 March 2010}}</ref><ref>{{Cite web
| title = Part 29 - Airworthiness Standards: Transport Category Rotorcraft
| publisher = FAA
| url = http://rgl.faa.gov/Regulatory_and_Guidance_Library%5CrgFAR.nsf/0/9DEF90770C26A3EC85256613006B2A8B?OpenDocument
| accessdate = 29 March 2010}}</ref>
However, many aircraft types, in particular [[Aerobatics|aerobatic]] airplanes, are designed so that they can tolerate load factors much higher than the minimum required. For example, the [[Sukhoi Su-26]] family have load factors limits of -10 to +12.<ref>{{Cite web
| title = Su-26, 29, 31 - Historical background
| publisher = Sukhoi Company
| url = http://www.sukhoi.org/eng/planes/civil/su-26/history/
| accessdate = 25 March 2010}}</ref>

The maximum load factors, both positive and negative, applicable to an aircraft are usually specified in the pilot's operating handbook.

== Human perception of load factor ==
When the load factor is +1, all occupants of the aircraft feel that their weight is normal. When the load factor is greater than +1 all occupants feel heavier than usual. For example, in a 2 g maneuver all occupants feel that their weight is twice normal. When the load factor is zero, or very small, all occupants feel weightless.<ref>Clancy, page 398</ref> When the load factor is negative, all occupants feel they are upside down.

Human beings have limited ability to withstand a load factor significantly greater than 1, both positive and negative. [[Unmanned aerial vehicles]] can be designed for much greater load factors, both positive and negative, than conventional aircraft because these vehicles can be used in maneuvers which would be incapacitating for a human pilot.

== See also ==
* [[g-force]]
*[[G-LOC]] ''Loss of consciousness due to excessive G (also known as blackout)''
*[[Greyout]] ''Incapacitation due to excessive positive G''
*[[Redout]] ''Incapacitation due to excessive negative G''

== Notes ==
{{reflist}}

== References ==
{{refbegin}}
* Clancy, L.J. (1975). ''Aerodynamics''. Pitman Publishing Limited. London ISBN 0-273-01120-0.
* Hurt, H.H. (1960). ''Aerodynamics for Naval Aviators''. A National Flightshop Reprint. Florida.
* McCormick, Barnes W. (1979). ''Aerodynamics, Aeronautics and Flight Mechanics''. John Wiley & Sons. New York ISBN 0-471-03032-5.
{{refend}}

[[Category:Aerospace engineering]]

[[de:Lastvielfache]]

Markov decision process

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2012-08-22T19:23:45Z

128.227.181.142: /* Overview */

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UF Training Reactor

2012-08-20T16:08:51Z

128.227.87.90: update website to program

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2012-07-30T22:14:09Z

128.227.103.90: Batalin–Vilkovisky formalism has nothing to do with the quantum master equation here so I removed the linking to Batalin–Vilkovisky formalism.

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Ferromagnetic material properties

2012-07-03T14:52:15Z

128.227.34.4: /* Terms */

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