formulasearchengine - User contributions [en]

Economic order quantity

2015-01-06T14:00:47Z

129.217.183.66: /* The Total Cost function */

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2014-11-26T19:37:06Z

129.217.106.58:

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Preparata code

2014-07-01T11:24:20Z

129.217.151.245: /* Construction */

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Incidence structure

2014-05-09T15:41:19Z

129.217.132.38: /* Example: Fano plane */

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Oriented matroid

2014-03-26T11:17:23Z

129.217.151.246: /* Chirotope axioms */

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Order of accuracy

2013-10-31T13:30:59Z

129.217.84.126: nicer

In [[number theory]], the '''crank of a partition of an integer''' is a certain [[integer]] associated with the [[Partition (number theory)|partition]]. The term was first introduced without a definition by [[Freeman Dyson]] in a paper published in [[Eureka (University of Cambridge magazine)|Eureka]], a journal published by the Mathematics Society of [[Cambridge University]].<ref>{{cite journal|last=Freeman J. Dyson|title=Some Guesses in The Theory of Partitions|journal=Eureka (Cambridge)|year=1944|volume=8|pages=10–15|url=http://books.google.com/books?id=nnyNUidX1OMC&pg=PA51&lpg=PA51}}</ref> Dyson then gave a list of properties this yet to be defined quantity should have. [[George E. Andrews]] and F.G. Garvan in 1988 discovered a definition for crank satisfying the properties hypothesized for it by Dyson.<ref name=def/>

==Dyson's crank==

Let ''n'' be a non-negative integer and let ''p''(''n'') denote the number of partitions of ''n'' (''p''(0) is defined to be 1). [[Srinivasa Ramanujan]] in a paper<ref>{{cite journal|last=Srinivasa|first=Ramanujan|title=Some properties of ''p''(''n''), number of partitions of ''n''|journal=Proceedings of the Cambridge Philosophical Society|year=1919|volume=XIX|pages=207–210.}}</ref> published in 1918 stated and proved the following congruences for the [[partition function (number theory)|partition function]] ''p''(''n''), since known as [[Ramanujan congruences]].

* ''p''(5''n'' + 4) &equiv; 0 (mod 5)
* ''p''(7''n'' + 5) &equiv; 0 (mod 7)
* ''p''(11''n'' + 6) &equiv; 0 (mod 11)

These congruences imply that partitions of numbers of the form 5''n'' + 4 (respectively, of the forms 7''n'' + 5 and 11''n'' + 6 ) can be divided into 5 (respectively, 7 and 11) subclasses of equal size. The then known proofs of these congruences were based on the ideas of generating functions and they did not specify a method for the division of the partitions into subclasses of equal size.

In his Eureka paper Dyson proposed the idea of the [[rank of a partition]]. The rank of a partition is the integer obtained by subtracting the number of parts in the partition from the largest part in the partition. For example, the rank of the partition λ = { 4, 2, 1, 1, 1 } of 9 is 4 − 5 = − 1. Denoting by ''N''(''m'' , ''q'', ''n''), the number of partitions of ''n'' whose ranks are congruent to ''m'' modulo ''q'', Dyson considered ''N''(''m'' , 5 , 5 ''n'' + 4) and ''N''(''m'' , 7 , 7 ''n'' + 5) for various values of ''n'' and ''m''. Based on empirical evidences Dyson formulated the following conjectures known as ''rank conjectures''.

For all non-negative integers ''n'' we have:
* ''N''(0 , 5 , 5''n'' + 4) = ''N''(1 , 5 , 5''n'' + 4) = ''N''(2 , 5 , 5''n'' + 4) = ''N''(3 , 5 , 5''n'' + 4) = ''N''(4 , 5 , 5''n'' + 4).
* ''N''(0 , 7 , 7''n'' + 5) = ''N''(1 , 7 , 7''n'' + 5) = ''N''(2 , 7 , 7''n'' + 5) = ''N''(3 , 7 , 7''n'' + 5) = ''N''(4 , 7 , 7''n'' + 5) = ''N''(5 , 7 , 7''n'' + 5) = ''N''(6 , 7 , 7 ''n'' + 5)

Assuming that these conjectures are true, they provided a way of splitting up all partitions of numbers of the form 5''n'' + 4 into five classes of equal size: Put in one class all those partitions whose ranks are congruent to each other modulo 5. The same idea can be applied to divide the partitions of integers of the from 7''n'' + 6 into seven equally numerous classes. But the idea fails to divide partitions of integers of the form 11''n'' + 6 into 11 classes of the same size, as the following table shows.

<center>
'''Partitions of the integer 6 ( 11''n'' + 6 with ''n'' = 0 ) divided into classes based on ranks'''

{| class="wikitable"
|-
! rank &equiv; 0 (mod 11) !! rank &equiv; 1 (mod 11) !! rank &equiv; 2 (mod 11) !! rank &equiv; 3 (mod 11)!! rank &equiv; 4 (mod 11) !! rank &equiv; 5 (mod 11) !! rank &equiv; 6 (mod 11) !! rank &equiv; 7 (mod 11) !! rank &equiv; 8 (mod 11) !! rank &equiv; 9 (mod 11) !! rank &equiv; 10 (mod 11)
|-
| {3,2,1} || {4,1,1} || {4,2} || {5,2} || || {6} || {1,1,1,1,1,1} || || {2,1,1,1,1} || {2,2,1,1} || {2,2,2}
|-
| || {3,3} || || || || || || || || || {3,1,1,1}
|}
</center>

Thus the rank cannot be used to prove the theorem combinatorially. However, Dyson wrote,

'' I hold in fact :''

* '' that there exists an arithmetical coefficient similar to, but more recondite than, the rank of a partition; I shall call this hypothetical coefficient the "crank" of the partition and denote by M(m , q , n) the number of partitions of n whose crank is congruent to m modulo q;''

* ''that M(m , q , n) = M(q − m , q , n);''

* ''that M(0 , 11 , 11n + 6) = M(1 , 11 , 11n + 6) = M(2 , 11 , 11n + 6) = M(3 , 11 , 11n + 6) = M(4 , 11 , 11n + 6);''

* ''that . . . ''

''Whether these guesses are warranted by evidence, I leave it to the reader to decide. Whatever the final verdict of posterity may be, I believe the "crank" is unique among arithmetical functions in having been named before it was discovered. May it be preserved from the ignominious fate of the planet [[Vulcan (hypothetical planet)|Vulcan]]. ''

==Definition of crank==
In a paper<ref name=def>{{cite journal|last=George E. Andrews|coauthors=F.G. Garvan|title=Dyson's crank of a partition|journal=Bulletin (New Series) of The American Mathematical Society|year=1988|month=April|volume=18|issue=2|url=http://www.ams.org/journals/bull/1988-18-02/S0273-0979-1988-15637-6/S0273-0979-1988-15637-6.pdf|accessdate=26 November 2012}}</ref> published in 1988 George E. Andrews and F. G. Garvan defined the crank of a partition as follows:

: For a partition λ, let ''l''(λ) denote the largest part of λ, ω(λ) denote the number of 1's in λ, and μ(λ) denote the number of parts of λ larger than ω (λ). The crank ''c''(λ) is given by
:<math>c(\lambda)=\begin{cases} l(\lambda) & \text{ if } \omega(\lambda)=0 \\ \mu(\lambda)-\omega(\lambda) &\text{ if } \omega(\lambda)>0.\end{cases}</math>

The cranks of the partitions of the integers 4, 5, 6 are computed in the following tables.

<center>
'''Cranks of the partitions of 4'''
{| class="wikitable"
|-
!Partition λ !! Largest part ''l''(''λ) !! Number of 1's ω(λ)!! Number of parts larger than ω(λ) μ(λ) ||Crank ''c''(λ)
|-
| {4} || 4 || 0 || 1 || 4
|-
| {3,1} || 3 || 1 || 1 || 0
|-
| {2,2} || 2 || 0 || 2 || 2
|-
| {2,1,1} || 2 || 2 || 0 || −2
|-
| {1,1,1,1} || 1 || 4 || 0 || −4
|}

'''Cranks of the partitions of 5'''
{| class="wikitable"
|-
!Partition λ !! Largest part ''l''(''λ) !! Number of 1's ω(λ)!! Number of parts larger than ω(λ) μ(λ) ||Crank ''c''(λ)
|-
| {5} || 5 || 0 || 1 || 5
|-
| {4,1} || 4 || 1 || 1 || 0
|-
| {3,2} || 3 || 0 || 2 || 3
|-
| {3,1,1} || 3 || 2 || 1 || −1
|-
| {2,2,1} || 2 || 1 || 2 || 1
|-
| {2,1,1,1} || 2 || 3 || 0 || −3
|-
| {1,1,1,1,1} || 1 || 5 || 0 || −5
|}

'''Cranks of the partitions of 6'''
{| class="wikitable"
|-
!Partition λ !! Largest part ''l''(''λ) !! Number of 1's ω(λ)!! Number of parts larger than ω(λ) μ(λ) ||Crank ''c''(λ)
|-
| {6} || 6 || 0 || 1 || 6
|-
| {5,1} || 5 || 1 || 1 || 0
|-
| {4,2} || 4|| 0 || 2 || 4
|-
| {4,1,1} || 4 || 2 || 1 || −1
|-
| {3,3} || 3 || 0 || 2 || 3
|-
| {3,2,1} || 3 || 1 || 2 || 1
|-
| {3,1,1,1} || 3 || 3 || 0 || −3
|-
| {2,2,2} || 2 || 0 || 3 || 2
|-
| {2,2,1,1} || 2 || 2 || 0 || −2
|-
| {2,1,1,1,1} || 2 || 4 || 0 || −4
|-
| {1,1,1,1,1,1} || 1 || 6 || 0 || −6
|}

</center>

==Notations==

For all integers ''n'' ≥ 0 and all integers ''m'', the number of partitions of ''n'' with crank equal to ''m'' is denoted by ''M''(''m'',''n'') except for ''n''=1 where ''M''(-1,1) = -''M''(0,1) = ''M''(1,1) = 1 as given by the following generating function. The number of partitions of ''n'' with crank equal to ''m'' modulo ''q'' is denoted by ''M''(''m'',''q'',''n'').

The generating function for ''M''(''m'',''n'') is given below:

: <math>\sum_{n=0}^{\infty}\sum_{m=-\infty}^\infty M(m,n) z^mq^n = \prod_{n=1}^\infty \frac{(1-q^n)}{(1-zq^n)(1-z^{-1}q^n)}</math>

== Basic result ==

Andrew and Garvan proved the following result<ref name=def/> which shows that the crank as defined above does meet the conditions given by Dyson.

* ''M''(0, 5 ,5''n'' + 4) = ''M''(1, 5, 5''n'' + 4) = ''M''(2, 5, 5''n'' + 4) = ''M''(3, 5, 5''n'' + 4) = ''M''(4, 5, 5''n'' + 4) = ''p''(5''n'' + 4) / 5
* ''M''(0, 7, 7''n'' + 5) = ''M''(1, 7, 7''n'' + 5) = ''M''(2, 7, 7''n'' + 5) = ''M''(3, 7, 7''n'' + 5) = ''M''(4, 7, 7''n'' + 5) = ''M''(5, 7, 7''n'' + 5) = ''M''(6, 7, 7''n'' + 5) = ''p''(7''n'' + 5) / 7
* ''M''(0, 11, 11''n'' + 6) = ''M''(1, 11, 11''n'' + 6) = ''M''(2, 11, 11''n'' + 6) = ''M''(3, 11, 11''n'' + 6) = . . . = ''M''(9, 11, 11''n'' + 6) = ''M''(10, 11, 11''n'' + 6) = ''p''(11''n'' + 6) / 11

The concepts of rank and crank can both be used to classify partitions of certain integers into subclasses of equal size. However the two concepts produce different subclasses of partitions. This is illustrated in the following two tables.

<center>
'''Classification of the partitions of the integer 9 based on cranks'''
{| class="wikitable"
|-
! Partitions with crank &equiv; 0 (mod 5) !! Partitions with crank &equiv; 1 (mod 5) !!Partitions with crank &equiv; 2 (mod 5) !! Partitions with crank &equiv; 3 (mod 5) !! Partitions with crank &equiv; 4 (mod 5)
|-
| bgcolor="red"|{ 8, 1 } || { 6, 3 } || { 7, 2 } || { 6, 1, 1, 1 } || { 9 }
|-
| bgcolor="red"|{ 5, 4 } || { 6, 2, 1 } || { 5, 1, 1, 1, 1 } || { 4, 2, 1, 1, 1 } || { 7, 1, 1 }
|-
| bgcolor="red"|{ 5, 2, 2 } || { 5, 3, 1 } || { 4, 2, 2, 1 } || { 3, 3, 3 } || { 5, 2, 1, 1 }
|-
| bgcolor="red"|{ 4, 3, 1, 1 } || { 4, 4, 1 } || { 3, 3, 2, 1 } || { 3, 2, 2, 2 } || { 4, 3, 2 }
|-
| bgcolor="red"|{ 4, 1, 1, 1, 1, 1 } || { 3, 2, 1, 1, 1, 1 } || { 3, 3, 1, 1, 1 } ||{ 2, 2, 2, 2, 1 } || { 3, 2, 2, 1, 1 }
|-
|bgcolor="red"| { 2, 2, 1, 1, 1, 1, 1 } || { 1, 1, 1, 1, 1, 1, 1, 1, 1 } || { 2, 2, 2, 1, 1, 1 } || { 2, 1, 1, 1, 1, 1, 1, 1}|| { 3, 1, 1, 1, 1, 1, 1 }
|}

'''Classification of the partitions of the integer 9 based on ranks'''
{| class="wikitable"
|-
! Partitions with rank &equiv; 0 (mod 5) !! Partitions with rank &equiv; 1 (mod 5) !!Partitions with rank &equiv; 2 (mod 5) !! Partitions with rank &equiv; 3 (mod 5) !! Partitions with rank &equiv; 4 (mod 5)
|-
| { 7, 2 } || bgcolor="red"|{ 8, 1 } || { 6, 1, 1, 1 } || { 9 } || { 7, 1, 1 }
|-
| { 5, 1, 1, 1, 1 } || { 5, 2, 1, 1 } || { 5, 3, 1} || { 6, 2, 1 } || { 6, 3 }
|-
| bgcolor="red"|{ 4, 3, 1, 1 } || { 4, 4, 1 } ||bgcolor="red"| { 5, 2, 2 } || bgcolor="red"|{ 5, 4 } || { 4, 2, 1, 1, 1 }
|-
| { 4, 2, 2, 1 } || { 4, 3, 2 } || { 3, 2, 1, 1, 1, 1 } || { 3, 3, 1, 1, 1 } || { 3, 3, 2, 1 }
|-
| { 3, 3, 3 } || { 3, 1, 1, 1, 1, 1, 1 } || { 2, 2, 2, 2, 1 } ||bgcolor="red"|{ 4, 1, 1, 1, 1, 1 } || { 3, 2, 2, 2 }
|-
|bgcolor="red"| { 2, 2, 1, 1, 1, 1, 1 } || { 2, 2, 2, 1, 1, 1 } || { 1, 1, 1, 1, 1, 1, 1, 1, 1 } || { 3, 2, 2, 1, 1}|| { 2, 1, 1, 1, 1, 1, 1, 1 }
|}
</center>

==References==
{{reflist}}

[[Category:Number theory]]
[[Category:Combinatorics]]
[[Category:Arithmetic functions]]
[[Category:Srinivasa Ramanujan]]

Semibatch reactor

2013-04-14T12:00:12Z

129.217.177.224: /* Reactor Choice */

{{stack|
[[Image:Triggered spark gap.jpg|thumb|Spark between anode and cathode, triggered by the third electrode inside the inner quartz tube. The inner tube serves as a guide for the spark and to cool it even faster. The outer tube muffles the explosive sound that the spark produces.]]
[[Image:Bullet coming from S&W.jpg|thumb|A photo of a revolver firing, taken with the flash above. The photo was taken in a darkened room, with camera's shutter open and the flash was triggered by the sound of the shot using a microphone.]]
[[File:.308 Winchester bullet (150 grain FMJ).JPG|thumb|Ultra-high-speed photo of a bullet travelling at about {{convert|2850|ft/s|m/s}}.]]
[[Image:Microflash-spectrum.jpg|thumb|Air-gap flash spectrum generated with a [[Diffraction grating|grating]].]]
[[Image:Microflash-quartz-phosphorescence.jpg|thumb| Upper half shows the air-gap in daylight. Lower half shows the [[phosphorescence]] of the quartz ignition tube in blue in a darkened environment after a flash has occurred.]]
}}

An '''air-gap flash''' is a photographic light source capable of producing sub-microsecond light flashes, allowing for (ultra) [[high-speed photography]]. This is achieved by a high-voltage (20 kV typically) electric discharge between two electrodes over a quartz (or glass) tube's surface. The distance between the electrodes is such that a spontaneous discharge does not occur. To start the discharge a high-voltage pulse is applied on an electrode inside the quartz tube.

The discharge can be triggered electronically using a microphone or an interrupted laser beam to illuminate a fast event. A sub-microsecond flash is fast enough to photographically stop even a supersonic bullet in flight without noticeable motion blur.

The person credited with popularising the flash is [[Harold Eugene Edgerton]], though earlier scientists such as [[Ernst Mach]] also used a spark gap as a fast photographic lighting system. [[Henry Fox Talbot|William Henry Fox Talbot]] is said to have created the first spark-based flash photo, using a [[Leyden jar]], the original form of the capacitor. Edgerton was one of the founders of [[EG&G]] company who sold an air-gap flash under the name Microflash 549.<ref>http://people.rit.edu/andpph/text-microflash-549-manual.pdf</ref> There are several commercial flashes available today.

== Design parameters ==

The aim of a high-speed flash is to be fast and bright enough. A flash system typically consists of a capacitor that is discharged through a gas (air in this case). The speed of a flash is mainly determined by the time it takes to discharge the capacitor through the gas. This time is proportional to

<math>t_{discharge} \propto {\sqrt{LC}}</math>,

in which L is the inductance and C the capacitance of the system. To be fast, both L and C must be kept small.
The brightness of the flash is proportional to the energy stored in the capacitor:

<math>E = {{CV^2} \over 2}</math>,

where V is the voltage over the capacitor. This calls for a large capacitor and a high voltage. However, a large C makes the flash slow, so the only solution to make a sufficiently fast and bright flash is to use a very high voltage on a relatively small capacitor, with a very low inductance.
Typical values are 0.05 µF capacity, 0.02 µH inductance, 10 J energy, 0.5 µs duration and about 20 MW power.<ref>Edgerton, Harold E. (19706). ''Electronic flash, strobe'', Chapter 7, Mc Graw Hill, New-York. ISBN 007018965x / 0-07-018965-x.</ref>

Air (mainly nitrogen) is preferred as a gas because it is fast. Xenon has a much higher efficiency in turning energy into light, but is limited in speed to about 10 microseconds, caused by its own afterglow.

The spark is guided over a quartz surface to improve the light output and benefit from the cooling capacity, making the flash faster.<ref>Topler, M, Ann Physik, vol. 4, no. 27, pp 1043-1050, 1908</ref><ref>Edgerton, H.E.K, K. Cooper and J. tredwell, Submicrosecond Flash Source, J. SMTPE, vol. 70,p. 117, March, 1961</ref> This has a negative effect in form of quartz erosion because of high energy discharge.

== Spectral properties ==

Since the spark gap discharges in air generating a [[Plasma (physics)|plasma]], the spectrum shows both a continuum and [[spectral lines]], mainly of [[nitrogen]] since air is 79% nitrogen.
The spectrum is rich in [[UV]] but covers the entire visible range down to [[infra-red]].
When a [[quartz]] tube is used as ignition tube, it shows a clear [[phosphorescence]] in blue after the flash, induced by the UV.

==References==
{{reflist}}

==External links==
{{commons category}}
*[http://www.njnoordhoek.com/?p=735 Amateur air-gap flash for ultra-high-speed photography by Niels Noordhoek]
*[http://web.mit.edu/Edgerton/ MIT Edgerton center]
*[http://web.mit.edu/Edgerton/www/pdf/Article.pdf Scientific American article on air-gap flash]

{{Photography}}

[[Category:Flash photography| ]]
[[Category:Photographic lighting]]
[[Category:Light sources]]
[[Category:Photography equipment]]

Gershgorin circle theorem

2012-08-16T10:31:54Z

129.217.131.39: /* References */

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