formulasearchengine - User contributions [en]

Hilbert space

2014-02-04T01:42:45Z

72.172.11.140: /* Second example: sequence spaces */

'''Membrane fusion''' is a key biophysical process that is essential for the functioning of life itself. It is defined as the event where two [[lipid bilayers]] approach each other and then merge to form a single continuous structure.<ref>Yang et al.,Science,2002,297,1877</ref> In living beings, cells are made of an outer coat made of lipid bilayers; which then cause fusion to take place in events such as [[fertilization]], [[embryogenesis]] and even infections by various types of [[bacteria]] and [[viruses]].<ref>Jahn et al.,Current Opinion in Cell Biology 2002,14,488</ref> It is therefore an extremely important event to study. From an evolutionary angle, fusion is an extremely controlled phenomenon. Random fusion can result in severe problems to the normal functioning of the human body. Fusion of [[biological membranes]] is mediated by [[membrane fusion protein|protein]]s. Regardless of the complexity of the system, fusion essentially occurs due to the interplay of various interfacial forces, namely hydration repulsion, hydrophobic attraction and [[van der Waals force]]s.<ref>Israelachvili et al.,Biochemistry,1992,31,1794</ref>
__TOC__
[[Image:Membrane fusion via stalk formation.jpg|900px|center]]

==Inter-bilayer forces==
[[Lipid bilayers]] are structures of [[lipid]] molecules consisting of a [[hydrophobic]] tail and a [[hydrophilic]] head group. Therefore, these structures experience all the characteristic Interbilayer forces involved in that regime.

===Hydration repulsion===
Two hydrated bilayers experience strong repulsion as they approach each other. These forces have been measured using the [[Surface forces apparatus]] (S.F.A), an instrument used for measuring forces between surfaces. This repulsion was first proposed by [[Irving Langmuir|Langmuir]] and was thought to arise due to water molecules that [[hydrate]] the bilayers. Hydration repulsion can thus be defined as the work required in removing the water molecules around [[hydrophilic]] molecules (like [[lipid]] head groups) in the bilayer system.<ref>R.P Rand,Annual Reviews of Biophysics and Bioengineering,1981,10,277</ref> As water molecules have an affinity towards [[hydrophilic]] head groups, they try to arrange themselves around the head groups of the [[lipid]] molecules and it becomes very hard to separate this favorable combination.

Experiments performed through SFA have confirmed that the nature of this force is an exponential decline.<ref>McIntosh et al.,Biochemistry,1987,26,7325</ref> The [[potential]] ''VR'' is given by<ref>Ruckenstein et al.,2001,17,2455</ref>

:<math>V_R = C_R \cdot \exp\!\left[{-z\over\lambda_R}\right]</math>

where ''CR'' (>0) is a measure of the hydration interaction energy for [[hydrophilic]] molecules of the given system, ''λR'' is a characteristic length scale of hydration repulsion and ''z'' is the distance of separation. In other words, it is on distances up to this length that molecules/surfaces fully experience this repulsion.

===Hydrophobic attraction===
[[Hydrophobic]] forces are the attractive forces between any two hydrophobic groups in aqueous media, e.g. the forces between two long hydrocarbon chains in aqueous solutions. The magnitude of these forces depends on the [[hydrophobicity]] of the interacting groups as well as the distance separating them (they are found to decrease roughly exponentially with the distance). The physical origin of these forces is a debated issue but they have been found to be long-ranged and are the strongest among all the physical interaction forces operating between biological surfaces and molecules.<ref name="Israel">Israelachvili et al.,Quarterly Reviews of Biophysics,2001,34,2,105</ref> Due to their long range nature, they are responsible for rapid [[coagulation]] of [[hydrophobic]] particles in water and play important roles in various biological phenomena including folding and stabilization of macromolecules such as [[proteins]] and fusion of cell membranes.

The potential ''VA'' is given by<ref name="Israel"/>

:<math>V_A = C_A \cdot \exp\!\left[{-z\over\lambda_A}\right]</math>

where ''CA'' (<0) is a measure of the [[hydrophobic]] interaction energy for the given system, ''λA'' is a characteristic length scale of [[hydrophobic]] attraction and ''z'' is the distance of separation.

===van der Waals forces in bilayers===
[[Image:Lipid bilayer fluid.JPG|right|300px]]

These forces arise due to [[dipole-dipole interaction]]s (induced/permanent) between molecules of bilayers. As molecules come closer, this attractive force arises due to the ordering of these dipoles; like in the case of magnets that align and attract each other as they approach.<ref name="Israel"/> This also implies that any surface would experience a van der waals attraction. In bilayers, the form taken by van der Waals interaction potential ''VVDW'' is given by<ref>Petrarche et al.,Physical Review E,1998,57,6,7014</ref>

:<math>V_{VDW} = -{H\over 12\pi} * \left( {1\over z^2} - {2\over(z+D)^2} + {1\over(z+2D)^2} \right)</math>

where ''H'' is the [[Hamaker constant]] and ''D'' and ''z'' are the bilayers thickness and the distance of separation respectively.

==Background==
For fusion to take place, it has to overcome huge repulsive forces due to the strong hydration repulsion between [[hydrophilic]] [[lipid]] head groups.<ref name="Israel"/> However, it has been hard to exactly determine the connection between [[adhesion]], fusion and interbilayer forces. The forces that promote [[cell adhesion]] are not the same as the ones that promote membrane fusion. Studies show that by creating a [[stress (mechanics)|stress]] on the interacting bilayers, fusion can be achieved without disrupting the interbilayer interactions. It has also been suggested that membrane fusion takes place through a sequence of structural rearrangements that help to overcome the barrier that prevents fusion.<ref name="Israel"/> Thus, interbilayer fusion takes place through

* local approach of membrane
* structural rearrangements causing [[Hydrate|hydration]] repulsion forces to be overcome
* complete merging to form a single entity

==Interbilayer interactions during membrane fusion==
When two [[lipid bilayers]] approach each other, they experience weak van der Waals attractive forces and much stronger repulsive forces due to hydration repulsion.<ref>Leikin et al., Journal of Theoretical Biology,1987,129,411</ref> These forces are normally dominant over the [[hydrophobic]] attractive forces between the membranes. Studies done on membrane bilayers using [[Surface forces apparatus]] (SFA) indicate that membrane fusion can instantaneously occur when two bilayers are still at a finite distance from each other without them having to overcome the short-range repulsive force barrier.<ref name="Israel"/> This is attributed to the molecular rearrangements that occur resulting in the bypassing of these forces by the membranes. During fusion, the [[hydrophobic]] tails of a small patch of [[lipids]] on the [[cell membrane]] are exposed to the aqueous phase surrounding them. This results in very strong [[hydrophobic]] attractions (which dominate the repulsive force) between the exposed groups leading to membrane fusion.<ref>Israelachvili et al.,Science,1989,246,4932</ref> The attractive van der Waals forces play a negligible role in membrane fusion. Thus, fusion is a result of the hydrophobic attractions between internal hydrocarbon chain groups that are exposed to the normally inaccessible aqueous environment. Fusion is observed to start at points on the membranes where the membrane stresses are either the weakest or the strongest.<ref name="Israel"/>

==Applications==
Interbilayer forces play a key role in mediating membrane fusion, which has extremely important biomedical applications.<ref>Chen at al,Science,2005,308,369</ref>

* The most important application of membrane fusion is in the production of [[hybridomas]] which are cells that arise as a result of the fusion of [[antibody]]-secreting and immortal [[B-cells]]. [[Hybridomas]] are used in the industry for the production of [[monoclonal antibodies]].

* Membrane fusion also has a major role in [[cancer immunotherapy]]. Currently, one of the approaches in cancer immunotherapy involves [[vaccination]] of [[dendritic cells]] which express a specific [[tumor]] [[antigen]] on their membranes. Instead, the hybrid cells obtained from the fusion of dendritic cells with tumor cells can be used. These hybrids would help in the expression of a range of tumor-associated antigens on their membranes.

* Understanding membrane fusion better can also lead to improvements in [[gene therapy]].

==See also==
* [[Lipid bilayers]]
* [[Hydrophobic effect]]
* [[Surface forces apparatus]]
* [[Cell membrane]]
* [[Hydrate]]

==References==
{{reflist}}

[[Category:Intermolecular forces]]
[[Category:Membrane biology]]
[[Category:Surface chemistry]]
[[Category:Biophysics]]

Norm (mathematics)

2014-02-03T22:15:08Z

72.172.11.140: /* Examples */

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A '''divisibility rule''' is a shorthand way of determining whether a given number is divisible by a fixed divisor without performing the division, usually by examining its digits. Although there are divisibility tests for numbers in any [[radix]], and they are all different, this article presents rules and examples only for [[decimal]] numbers.

==Divisibility rules for numbers 1–20==
The rules given below transform a given number into a generally smaller number, while preserving divisibility by the divisor of interest. Therefore, unless otherwise noted, the resulting number should be evaluated for divisibility by the same divisor. In some cases the process can be iterated until the divisibility is obvious; for others (such as examining the last ''n'' digits) the result must be examined by other means.

For divisors with multiple rules, the rules are generally ordered first for those appropriate for numbers with many digits, then those useful for numbers with fewer digits.

Note: To test divisibility by any number that can be expressed as 2''n'' or 5''n'', in which ''n'' is a positive integer, just examine the last ''n'' digits.
{| class="wikitable"
! Divisor
! Divisibility condition
! Examples
|-
| '''[[1 (number)|1]]'''
| No special condition. Any integer is divisible by 1.
| 2 is divisible by 1.
|-
| '''[[2 (number)|2]]'''
| The last digit is even (0, 2, 4, 6, or 8).<ref name="Pascal's-criterion">This follows from Pascal's criterion. See Kisačanin (1998), {{Google books quote|id=BFtOuh5xGOwC|page=101|text=A number is divisible by|p. 100–101}}</ref><ref name="last-m-digits">A number is divisible by 2''m'', 5''m'' or 10''m'' if and only if the number formed by the last ''m'' digits is divisible by that number. See Richmond & Richmond (2009), {{Google books quote|id=HucyKYx0_WwC|page=105|text=formed by the last|p. 105}}</ref>
| 1,294: 4 is even.
|-
|rowspan=2| '''[[3 (number)|3]]'''
| Sum the digits. If the result is divisible by 3, then the original number is divisible by 3.<ref name="Pascal's-criterion"/><ref name="apostol-1976">Apostol (1976), {{Google books quote|id=Il64dZELHEIC|page=108|text=sum of its digits|p. 108}}</ref><ref name="Richmond-Richmond-2009">Richmond & Richmond (2009), {{Google books quote|id=HucyKYx0_WwC|page=102|text=divisible by|Section 3.4 (Divisibility Tests), p. 102–108}}</ref>
| 405 → 4 + 0 + 5 = 9 and 636 → 6 + 3 + 6 = 15 which both are clearly divisible by 3. 16,499,205,854,376 → 1+6+4+9+9+2+0+5+8+5+4+3+7+6 sums to 69 → 6 + 9 = 15 → 1 + 5 = 6, which is clearly divisible by 3.
|-
| Subtract the quantity of the digits 2, 5 and 8 in the number from the quantity of the digits 1, 4 and 7 in the number.
| Using the example above: 16,499,205,854,376 has '''four''' of the digits 1, 4 and 7; '''four''' of the digits 2, 5 and 8; ∴ Since 4 − 4 = 0 is a multiple of 3, the number 16,499,205,854,376 is divisible by 3.
|-
|rowspan=3| '''[[4 (number)|4]]'''
| Examine the last two digits.<ref name="Pascal's-criterion"/><ref name="last-m-digits"/>
| 40832: 32 is divisible by 4.
|-
| If the tens digit is even, the ones digit must be 0, 4, or 8. If the tens digit is odd, the ones digit must be 2 or 6.
| 40832: 3 is odd, and the last digit is 2.
|-
| Twice the tens digit, plus the ones digit.
| 40832: 2 × 3 + 2 = 8, which is divisible by 4.
|-
| '''[[5 (number)|5]]'''
| The last digit is 0 or 5.<ref name="Pascal's-criterion"/><ref name="last-m-digits"/>
| 495: the last digit is 5.
|-
| '''[[6 (number)|6]]'''
| It is divisible by 2 and by 3.<ref name="product-of-coprimes">Richmond & Richmond (2009), {{Google books quote|id=HucyKYx0_WwC|page=102|text=divisible by the product|Section 3.4 (Divisibility Tests), Theorem 3.4.3, p. 107}}</ref>
| 1,458: 1 + 4 + 5 + 8 = 18, so it is divisible by 3 and the last digit is even, hence the number is divisible by 6.
|-
|rowspan=5| '''[[7 (number)|7]]'''
| Form the [[alternating sum]] of blocks of three from right to left.<ref name="Richmond-Richmond-2009"/><ref name="alternating-sum-of-blocks-of-three">Kisačanin (1998), {{Google books quote|id=BFtOuh5xGOwC|page=101|text=third criterion for 11|p. 101}}</ref>
| 1,369,851: 851 − 369 + 1 = 483 = 7 × 69
|-
| Subtract 2 times the last digit from the rest. (Works because 21 is divisible by 7.)
| 483: 48 − (3 × 2) = 42 = 7 × 6.
|-
| Or, add 5 times the last digit to the rest. (Works because 49 is divisible by 7.)
| 483: 48 + (3 × 5) = 63 = 7 × 9.
|-
| Or, add 3 times the first digit to the next. (This works because 10''a'' + ''b'' − 7''a'' = 3''a'' + ''b'' − last number has the same remainder)
| 483: 4×3 + 8 = 20 remainder 6, 6×3 + 3 = 21.
|-
| Multiply each digit (from right to left) by the digit in the corresponding position in this pattern (from left to right): 1, 3, 2, -1, -3, -2 (repeating for digits beyond the hundred-thousands place). Then sum the results.
| 483595: (4 × (-2)) + (8 × (-3)) + (3 × (-1)) + (5 × 2) + (9 × 3) + (5 × 1) = 7.
|-
|rowspan=5| '''[[8 (number)|8]]'''
|style="border-bottom: hidden;"| If the hundreds digit is even, examine the number formed by the last two digits.
|style="border-bottom: hidden;"| 624: 24.
|-
| If the hundreds digit is odd, examine the number obtained by the last two digits plus 4.
| 352: 52 + 4 = 56.
|-
| Add the last digit to twice the rest.
| 56: (5 × 2) + 6 = 16.
|-
| Examine the last three digits.<ref name="Pascal's-criterion"/><ref name="last-m-digits"/>
| 34152: Examine divisibility of just 152: 19 × 8
|-
| Add four times the hundreds digit to twice the tens digit to the ones digit.
| 34152: 4 × 1 + 5 × 2 + 2 = 16
|-
| '''[[9 (number)|9]]'''
| Sum the digits. If the result is divisible by 9, then the original number is divisible by 9.<ref name="Pascal's-criterion"/><ref name="apostol-1976"/><ref name="Richmond-Richmond-2009"/>
| 2,880: 2 + 8 + 8 + 0 = 18: 1 + 8 = 9.
|-
| '''[[10 (number)|10]]'''
| The last digit is 0.<ref name="last-m-digits"/>
| 130: the last digit is 0.
|-
|rowspan=5| '''[[11 (number)|11]]'''
| Form the alternating sum of the digits.<ref name="Pascal's-criterion"/><ref name="Richmond-Richmond-2009"/>
| 918,082: 9 − 1 + 8 − 0 + 8 − 2 = 22.
|-
| Add the digits in blocks of two from right to left.<ref name="Pascal's-criterion"/>
| 627: 6 + 27 = 33.
|-
| Subtract the last digit from the rest.
| 627: 62 − 7 = 55.
|-
| If the number of digits is even, add the first and subtract the last digit from the rest.
| 918,082: the number of digits is even (6) → 1808 + 9 − 2 = 1815: 81 + 1 − 5 = 77 = 7 × 11
|-
| If the number of digits is odd, subtract the first and last digit from the rest.
| 14,179: the number of digits is odd (5) → 417 − 1 − 9 = 407 = 37 × 11
|-
|rowspan=2| '''[[12 (number)|12]]'''
| It is divisible by 3 and by 4.<ref name="product-of-coprimes"/>
| 324: it is divisible by 3 and by 4.
|-
| Subtract the last digit from twice the rest.
| 324: 32 × 2 − 4 = 60.
|-
|rowspan=3| '''[[13 (number)|13]]'''
| Form the [[alternating sum]] of blocks of three from right to left.<ref name="alternating-sum-of-blocks-of-three"/>
| 2,911,272: −2 + 911 − 272 = 637
|-
| Add 4 times the last digit to the rest.
| 637: 63 + 7 × 4 = 91, 9 + 1 × 4 = 13.
|-
| Multiply each digit (from right to left) by the digit in the corresponding position in this pattern (from left to right): -3, -4, -1, 3, 4, 1 (repeating for digits beyond the hundred-thousands place). Then sum the results.<ref>http://www.tavas.net/index.php?op=NEArticle&sid=3358 New divisibility by 13 rule was found by Ethem Deynek, Turkish teacher</ref>
| 30,747,912: (2 × (-3)) + (1 × (-4)) + (9 × (-1)) + (7 × 3) + (4 × 4) + (7 × 1) + (0 × (-3)) + (3 × (-4)) = 13.
|-
|rowspan=2| '''[[14 (number)|14]]'''
| It is divisible by 2 and by 7.<ref name="product-of-coprimes"/>
| 224: it is divisible by 2 and by 7.
|-
| Add the last two digits to twice the rest. The answer must be divisible by 14.
| 364: 3 × 2 + 64 = 70. 1764: 17 × 2 + 64 = 98.
|-
| '''[[15 (number)|15]]'''
| It is divisible by 3 and by 5.<ref name="product-of-coprimes"/>
| 390: it is divisible by 3 and by 5.
|-
|rowspan=4| '''[[16 (number)|16]]'''
| If the thousands digit is even, examine the number formed by the last three digits.
| 254,176: 176.
|-
| If the thousands digit is odd, examine the number formed by the last three digits plus 8.
| 3,408: 408 + 8 = 416.
|-
| Add the last two digits to four times the rest.
| 176: 1 × 4 + 76 = 80.

1168: 11 × 4 + 68 = 112.
|-
| Examine the last four digits.<ref name="Pascal's-criterion"/><ref name="last-m-digits"/>
| 157,648: 7,648 = 478 × 16.
|-
| '''[[17 (number)|17]]'''
| Subtract 5 times the last digit from the rest.
| 221: 22 − 1 × 5 = 17.
|-
| '''[[18 (number)|18]]'''
| It is divisible by 2 and by 9.<ref name="product-of-coprimes"/>
| 342: it is divisible by 2 and by 9.
|-
| '''[[19 (number)|19]]'''
| Add twice the last digit to the rest.
| 437: 43 + 7 × 2 = 57.
|-
|rowspan=2| '''[[20 (number)|20]]'''
| It is divisible by 10, and the tens digit is even.
| 360: is divisible by 10, and 6 is even.
|-
| If the number formed by the last two digits is divisible by 20.
| 480: 80 is divisible by 20.
|}

==Step-by-step examples==

===Divisibility by 2===
First, take any even number (for this example it will be 376) and note the last digit in the number, discarding the other digits. Then take that digit (6) while ignoring the rest of the number and determine if it is divisible by 2. If it is divisible by 2, then the original number is divisible by 2.

'''Example'''
# 376 (The original number)
# <s>37</s> 6 (Take the last digit)
# 6 ÷ 2 = 3 (Check to see if the last digit is divisible by 2)
# 376 ÷ 2 = 188 (If the last digit is divisible by 2, then the whole number is divisible by 2)

===Divisibility by 3===
First, take any number (for this example it will be 492) and add together each digit in the number (4 + 9 + 2 = 15). Then take that sum (15) and determine if it is divisible by 3. The original number is divisible by 3 if and only if the final number is divisible by 3.

If a number is a multiplication of 3 consecutive numbers then that number is always divisible by 3. This is useful for when the number takes the form of (''n'' × (''n'' − 1) × (''n'' + 1))

'''Ex.'''
# 492 (The original number)
# 4 + 9 + 2 = 15 (Add each individual digit together)
# 15 is divisible by 3 at which point we can stop. Alternatively we can continue using the same method if the number is still too large:
# 1 + 5 = 6 (Add each individual digit together)
# 6 ÷ 3 = 2 (Check to see if the number received is divisible by 3)
# 492 ÷ 3 = 164 (If the number obtained by using the rule is divisible by 3, then the whole number is divisible by 3)

'''Ex.'''
# 336 (The original number)
# 6 × 7 × 8 = 336
# 336 ÷ 3 = 112

===Divisibility by 4===
The basic rule for divisibility by 4 is that if the number formed by the last two digits in a number is divisible by 4, the original number is divisible by 4;<ref name="Pascal's-criterion"/><ref name="last-m-digits"/> this is because 100 is divisible by 4 and so adding hundreds, thousands, etc. is simply adding another number that is divisible by 4. If any number ends in a two digit number that you know is divisible by 4 (e.g. 24, 04, 08, etc.), then the whole number will be divisible by 4 regardless of what is before the last two digits.

Alternatively, one can simply divide the number by 2, and then check the result to find if it is divisible by 2. If it is, the original number is divisible by 4. In addition, the result of this test is the same as the original number divided by 4.

'''Ex.''' 
'''General rule'''
# 2092 (The original number)
# <s>20</s> 92 (Take the last two digits of the number, discarding any other digits)
# 92 ÷ 4 = 23 (Check to see if the number is divisible by 4)
# 2092 ÷ 4 = 523 (If the number that is obtained is divisible by 4, then the original number is divisible by 4)

'''Alternative example'''
# 1720 (The original number)
# 1720 ÷ 2 = 860 (Divide the original number by 2)
# 860 ÷ 2 = 430 (Check to see if the result is divisible by 2)
# 1720 ÷ 4 = 430 (If the result is divisible by 2, then the original number is divisible by 4)

===Divisibility by 5===
Divisibility by 5 is easily determined by checking the last digit in the number (47'''5'''), and seeing if it is either 0 or 5. If the last number is either 0 or 5, the entire number is divisible by 5.<ref name="Pascal's-criterion"/><ref name="last-m-digits"/>

If the last digit in the number is 0, then the result will be the remaining digits multiplied by 2. For example, the number 40 ends in a zero (0), so take the remaining digits (4) and multiply that by two (4 × 2 = 8). The result is the same as the result of 40 divided by 5(40/5 = 8).

If the last digit in the number is 5, then the result will be the remaining digits multiplied by two (2), plus one (1). For example, the number 125 ends in a 5, so take the remaining digits (12), multiply them by two (12 × 2 = 24), then add one (24 + 1 = 25). The result is the same as the result of 125 divided by 5 (125/5=25).

'''Ex.''' 
''' If the last digit is 0'''
# 110 (The original number)
# <s>11</s> 0 (Take the last digit of the number, and check if it is 0 or 5)
# 11 <s>0</s> (If it is 0, take the remaining digits, discarding the last)
# 11 × 2 = 22 (Multiply the result by 2)
# 110 ÷ 5 = 22 (The result is the same as the original number divided by 5)

'''If the last digit is 5'''
# 85 (The original number)
# <s>8</s> 5 (Take the last digit of the number, and check if it is 0 or 5)
# 8 <s>5</s> (If it is 5, take the remaining digits, discarding the last)
# 8 × 2 = 16 (Multiply the result by 2)
# 16 + 1 = 17 (Add 1 to the result)
# 85 ÷ 5 = 17 (The result is the same as the original number divided by 5)

===Divisibility by 6===
Divisibility by 6 is determined by checking the original number to see if it is both an even number ([[#Divisibility by 2|divisible by 2]]) and [[#Divisibility by 3|divisible by 3]].<ref name="product-of-coprimes"/> This is the best test to use.

Alternatively, one can check for divisibility by six by taking the number (246), dropping the last digit in the number (24 <s>6</s>, adding together the remaining number (24 becomes 2 + 4 = 6), multiplying that by four (6 × 4 = 24), and adding the last digit of the original number to that (24 + 6 = 30). If this number is divisible by six, the original number is divisible by 6.

If the number is divisible by six, take the original number (246) and divide it by two (246 ÷ 2 = 123). Then, take that result and divide it by three (123 ÷ 3 = 41). This result is the same as the original number divided by six (246 ÷ 6 = 41).

'''Ex.''' 
'''General rule'''
# 324 (The original number)
# 324 ÷ 3 = 108 (Check to see if the original number is divisible by 3)
# 324 ÷ 2 = 162 '''OR''' 108 ÷ 2 = 54 (Check to see if either the original number or the result of the previous equation is divisible by 2)
# 324 ÷ 6 = 54 (If either of the tests in the last step are true, then the original number is divisible by 6. Also, the result of the second test returns the same result as the original number divided by 6)

 '''Finding a remainder of a number when divided by 6''' 
6 − (1, −2, −2, −2, −2, and −2 goes on for the rest) No period.
 Minimum magnitude sequence 
(1, 4, 4, 4, 4, and 4 goes on for the rest)
 Positive sequence 
Multiply the right most digit by the left most digit in the sequence and multiply the second right most digit by the second left most digit in the sequence and so on. Next, compute the sum of all the values and take the remainder on division by 6.
 Example: What is the remainder when 1036125837 is divided by 6? 
Multiplication of the rightmost digit = 1 × 7 = 7
 Multiplication of the second rightmost digit = 3 × −2 = −6 
Third rightmost digit = −16
 Fourth rightmost digit = −10 
Fifth rightmost digit = −4
 Sixth rightmost digit = −2 
Seventh rightmost digit = −12
 Eighth rightmost digit = −6 
Ninth rightmost digit = 0
 Tenth rightmost digit = −2 
Sum = −51
 −51 modulo 6 = 3 
Remainder = 3

===Divisibility by 7===
{{Cleanup|section|date=August 2010}}
Divisibility by 7 can be tested by a recursive method. A number of the form 10''x'' + ''y'' is divisible by 7 if and only if ''x'' − 2''y'' is divisible by 7. In other words, subtract twice the last digit from the number formed by the remaining digits. Continue to do this until a number known to be divisible by 7 is obtained. The original number is divisible by 7 if and only if the number obtained using this procedure is divisible by 7. For example, the number 371: 37 − (2×1) = 37 − 2 = 35; 3 − (2 × 5) = 3 − 10 = −7; thus, since −7 is divisible by 7, 371 is divisible by 7.

Another method is multiplication by 3. A number of the form 10''x'' + ''y'' has the same remainder when divided by 7 as 3''x'' + ''y''. One must multiply the leftmost digit of the original number by 3, add the next digit, take the remainder when divided by 7, and continue from the beginning: multiply by 3, add the next digit, etc. For example, the number 371: 3×3 + 7 = 16 remainder 2, and 2×3 + 1 = 7. This method can be used to find the remainder of division by 7.

A more complicated algorithm for testing divisibility by 7 uses the fact that 100 ≡ 1, 101 ≡ 3, 102 ≡ 2, 103 ≡ 6, 104 ≡ 4, 105 ≡ 5, 106 ≡ 1, ... (mod 7). Take each digit of the number (371) in reverse order (173), multiplying them successively by the digits '''1''', '''3''', '''2''', '''6''', '''4''', '''5''', repeating with this sequence of multipliers as long as necessary (1, 3, 2, 6, 4, 5, 1, 3, 2, 6, 4, 5, ...), and adding the products (1×'''1''' + 7×'''3''' + 3×'''2''' = 1 + 21 + 6 = 28). The original number is divisible by 7 if and only if the number obtained using this procedure is divisible by 7 (hence 371 is divisible by 7 since 28 is).<ref name="7Divis1">{{cite web
| last = Su
| first = Francis E.
| title = "Divisibility by Seven" ''Mudd Math Fun Facts''
| url = http://www.math.hmc.edu/funfacts/ffiles/10005.5.shtml
| accessdate = 2006-12-12 }}</ref>

This method can be simplified by removing the need to multiply. All it would take with this simplification is to memorize the sequence above (132645...), and to add and subtract, but always working with one-digit numbers.

The simplification goes as follows:

*Take for instance the number '''371'''
*Change all occurrences of '''7''', '''8''' or '''9''' into '''0''', '''1''' and '''2''', respectively. In this example, we get: '''301'''. This second step may be skipped, except for the left most digit, but following it may facilitate calculations later on.
*Now convert the first digit (3) into the following digit in the sequence '''13264513...''' In our example, 3 becomes '''2'''.
*Add the result in the previous step (2) to the second digit of the number, and substitute the result for both digits, leaving all remaining digits unmodified: 2 + 0 = 2. So ''30''1 becomes '''''2''1'''.
*Repeat the procedure until you have a recognizable multiple of 7, or to make sure, a number between 0 and 6. So, starting from 21 (which is a recognizable multiple of 7), take the first digit (2) and convert it into the following in the sequence above: 2 becomes 6. Then add this to the second digit: 6 + 1 = '''7'''.
*If at any point the first digit is 8 or 9, these become 1 or 2, respectively. But if it is a 7 it should become 0, only if no other digits follow. Otherwise, it should simply be dropped. This is because that 7 would have become 0, and numbers with at least two digits before the decimal dot do not begin with 0, which is useless. According to this, our 7 becomes '''0'''.

If through this procedure you obtain a '''0''' or any recognizable multiple of 7, then the original number is a multiple of 7. If you obtain any number from '''1''' to '''6''', that will indicate how much you should subtract from the original number to get a multiple of 7. In other words, you will find the [[remainder]] of dividing the number by 7. For example take the number '''186''':

*First, change the 8 into a 1: '''116'''.
*Now, change 1 into the following digit in the sequence (3), add it to the second digit, and write the result instead of both: 3 + 1 = ''4''. So ''11''6 becomes now '''''4''6'''.
*Repeat the procedure, since the number is greater than 7. Now, 4 becomes 5, which must be added to 6. That is '''11'''.
*Repeat the procedure one more time: 1 becomes 3, which is added to the second digit (1): 3 + 1 = '''4'''.

Now we have a number lower than 7, and this number (4) is the remainder of dividing 186/7. So 186 minus 4, which is 182, must be a multiple of 7.

Note: The reason why this works is that if we have: '''a+b=c''' and '''b''' is a multiple of any given number '''n''', then '''a''' and '''c''' will necessarily produce the same remainder when divided by '''n'''. In other words, in 2 + 7 = 9, 7 is divisible by 7. So 2 and 9 must have the same reminder when divided by 7. The remainder is 2.

Therefore, if a number ''n'' is a multiple of 7 (i.e.: the remainder of ''n''/7 is 0), then adding (or subtracting) multiples of 7 cannot possibly change that property.

What this procedure does, as explained above for most divisibility rules, is simply subtract little by little multiples of 7 from the original number until reaching a number that is small enough for us to remember whether it is a multiple of 7. If 1 becomes a 3 in the following decimal position, that is just the same as converting 10×10''n'' into a 3×10''n''. And that is actually the same as subtracting 7×10''n'' (clearly a multiple of 7) from 10×10''n''.

Similarly, when you turn a 3 into a 2 in the following decimal position, you are turning 30×10''n'' into 2×10''n'', which is the same as subtracting 30×10''n''−28×10n, and this is again subtracting a multiple of 7. The same reason applies for all the remaining conversions:

* 20×10''n'' − 6×10''n''='''14'''×10''n''
* 60×10''n'' − 4×10''n''='''56'''×10''n''
* 40×10''n'' − 5×10''n''='''35'''×10''n''
* 50×10''n'' − 1×10''n''='''49'''×10''n''

'''First method example''' 
1050 → 105 − 0=105 → 10 − 10 = 0. ANSWER: 1050 is divisible by 7.

'''Second method example''' 
1050 → 0501 (reverse) → 0×'''1''' + 5×'''3''' + 0×'''2''' + 1×'''6''' = 0 + 15 + 0 + 6 = 21 (multiply and add). ANSWER: 1050 is divisible by 7.

'''Vedic method of divisibility by osculation''' 
Divisibility by seven can be tested by multiplication by the ''Ekhādika''. Convert the divisor seven to the nines family by multiplying by seven. 7×7=49. Add one, drop the units digit and, take the 5, the ''Ekhādika'', as the multiplier. Start on the right. Multiply by 5, add the product to the next digit to the left. Set down that result on a line below that digit. Repeat that method of multiplying the units digit by five and adding that product to the number of tens. Add the result to the next digit to the left. Write down that result below the digit. Continue to the end. If the end result is zero or a multiple of seven, then yes, the number is divisible by seven. Otherwise, it is not. This follows the Vedic ideal, one-line notation.<ref>Page 274, '''Vedic Mathematics: Sixteen Simple Mathematical Formulae''', by Swami Sankaracarya, published by Motilal Banarsidass, Varanasi, India, 1965, Delhi, 1978. 367 pages.</ref>

'''Vedic method example:'''
Is 438,722,025 divisible by seven? Multiplier = 5.
4 3 8 7 2 2 0 2 5
42 37 46 37 6 40 37 27
YES

'''Pohlman–Mass method of divisibility by 7''' 
The Pohlman–Mass method provides a quick solution that can determine if most integers are divisible by seven in three steps or less. This method could be useful in a mathematics competition such as MATHCOUNTS, where time is a factor to determine the solution without a calculator in the Sprint Round.

Step A:
If the integer is 1,000 or less, subtract twice the last digit from the number formed by the remaining digits. If the result is a multiple of seven, then so is the original number (and vice versa). For example:

112 -> 11 − (2×2) = 11 − 4 = 7 YES
98 -> 9 − (8×2) = 9 − 16 = −7 YES
634 -> 63 − (4×2) = 63 − 8 = 55 NO

Because 1,001 is divisible by seven, an interesting pattern develops for repeating sets of 1, 2, or 3 digits that form 6-digit numbers (leading zeros are allowed) in that all such numbers are divisible by seven. For example:

001 001 = 1,001 / 7 = 143
010 010 = 10,010 / 7 = 1,430
011 011 = 11,011 / 7 = 1,573
100 100 = 100,100 / 7 = 14,300
101 101 = 101,101 / 7 = 14,443
110 110 = 110,110 / 7 = 15,730

01 01 01 = 10,101 / 7 = 1,443
10 10 10 = 101,010 / 7 = 14,430

111,111 / 7 = 15,873
222,222 / 7 = 31,746
999,999 / 7 = 142,857

576,576 / 7 = 82,368

For all of the above examples, subtracting the first three digits from the last three results in a multiple of seven. Notice that leading zeros are permitted to form a 6-digit pattern.

This phenomenon forms the basis for Steps B and C.

Step B:
If the integer is between 1,001 and one million, find a repeating pattern of 1, 2, or 3 digits that forms a 6-digit number that is close to the integer (leading zeros are allowed and can help you visualize the pattern). If the positive difference is less than 1,000, apply Step A. This can be done by subtracting the first three digits from the last three digits. For example:

341,355 − 341,341 = 14 -> 1 − (4×2) = 1 − 8 = −7 YES
67,326 − 067,067 = 259 -> 25 − (9×2) = 25 − 18 = 7 YES

The fact that 999,999 is a multiple of 7 can be used for determining divisibility of integers larger than one million by reducing the integer to a 6-digit number that can be determined using Step B. This can be done easily by adding the digits left of the first six to the last six and follow with Step A.

Step C:
If the integer is larger than one million, subtract the nearest multiple of 999,999 and then apply Step B. For even larger numbers, use larger sets such as 12-digits (999,999,999,999) and so on. Then, break the integer into a smaller number that can be solved using Step B. For example:

22,862,420 − (999,999 × 22) = 22,862,420 − 21,999,978 -> 862,420 + 22 = 862,442
862,442 -> 862 − 442 (Step B) = 420 -> 42 − (0×2) (Step A) = 42 YES

This allows adding and subtracting alternating sets of three digits to determine divisibility by seven. Understanding these patterns allows you to quickly calculate divisibility of seven as seen in the following examples:

'''Pohlman–Mass method of divisibility by 7, examples:'''

Is 98 divisible by seven?
98 -> 9 − (8×2) = 9 − 16 = −7 YES (Step A)

Is 634 divisible by seven?
634 -> 63 − (4×2) = 63 − 8 = 55 NO (Step A)

Is 355,341 divisible by seven?
355,341 − 341,341 = 14,000 (Step B) -> 014 − 000 (Step B) -> 14 = 1 − (4×2) (Step A) = 1 − 8 = −7 YES

Is 42,341,530 divisible by seven?
42,341,530 -> 341,530 + 42 = 341,572 (Step C)
341,572 − 341,341 = 231 (Step B)
231 -> 23 − (1×2) = 23 − 2 = 21 YES (Step A)

Using quick alternating additions and subtractions:
42,341,530 -> 530 − 341 = 189 + 42 = 231 -> 23 − (1×2) = 21 YES

'''Multiplication by 3 method of divisibility by 7, examples:'''

Is 98 divisible by seven?
98 -> 9 remainder 2 -> 2×3 + 8 = 14 YES

Is 634 divisible by seven?
634 -> 6×3 + 3 = 21 -> remainder 0 -> 0×3 + 4 = 4 NO

Is 355,341 divisible by seven?
3 * 3 + 5 = 14 -> remainder 0 -> 0×3 + 5 = 5 -> 5×3 + 3 = 18 -> remainder 4 -> 4×3 + 4 = 16 -> remainder 2 -> 2×3 + 1 = 7 YES

Find remainder of 1036125837 divided by 7
1×3 + 0 = 3
3×3 + 3 = 12 remainder 5
5×3 + 6 = 21 remainder 0
0×3 + 1 = 1
1×3 + 2 = 5
5×3 + 5 = 20 remainder 6
6×3 + 8 = 26 remainder 5
5×3 + 3 = 18 remainder 4
4×3 + 7 = 19 remainder 5
Answer is 5

'''Finding remainder of a number when divided by 7'''

7 − (1, 3, 2, −1, −3, −2, cycle repeats for the next six digits) Period: 6 digits.
Recurring numbers: 1, 3, 2, −1, −3, −2
 Minimum magnitude sequence 
(1, 3, 2, 6, 4, 5, cycle repeats for the next six digits) Period: 6 digits.
Recurring numbers: 1, 3, 2, 6, 4, 5
 Positive sequence

Multiply the right most digit by the left most digit in the sequence and multiply the second right most digit by the second left most digit in the sequence and so on and so for. Next, compute the sum of all the values and take the modulus of 7.
 Example: What is the remainder when 1036125837 is divided by 7? 
 Multiplication of the rightmost digit = 1 × 7 = 7 
 Multiplication of the second rightmost digit = 3 × 3 = 9 
 Third rightmost digit = 8 × 2 = 16 
 Fourth rightmost digit = 5 × −1 = −5 
 Fifth rightmost digit = 2 × −3 = −6 
 Sixth rightmost digit = 1 × −2 = −2 
 Seventh rightmost digit = 6 × 1 = 6 
 Eighth rightmost digit = 3 × 3 = 9 
 Ninth rightmost digit = 0 
 Tenth rightmost digit = 1 × −1 = −1 
 Sum = 33 
 33 modulus 7 = 5 
 Remainder = 5

'''Digit pair method of divisibility by 7'''

This method uses '''1''', '''−3''', '''2''' pattern on the ''digit pairs''. That is, the divisibility of any number by seven can be tested by first separating the number into digit pairs, and then applying the algorithm on three digit pairs (six digits). When the number is smaller than six digits, then fill zero’s to the right side until there are six digits. When the number is larger than six digits, then repeat the cycle on the next six digit group and then add the results. Repeat the algorithm until the result is a small number. The original number is divisible by seven if and only if the number obtained using this algorithm is divisible by seven. This method is especially suitable for large numbers.

''Example 1:'' 
The number to be tested is 157514.
First we separate the number into three digit pairs: 15, 75 and 14. 
Then we apply the algorithm: '''1''' × 15 '''− 3''' × 75 + '''2''' × 14 = 182 
Because the resulting 182 is less than six digits, we add zero’s to the right side until it is six digits. 
Then we apply our algorithm again: '''1''' × 18 '''− 3''' × 20 + '''2''' × 0 = −42 
The result −42 is divisible by seven, thus the original number 157514 is divisible by seven.

''Example 2:'' 
The number to be tested is 15751537186. 
('''1''' × 15 '''− 3''' × 75 + '''2''' × 15) + ('''1''' × 37 '''− 3''' × 18 + '''2''' × 60) = −180 + 103 = −77 
The result −77 is divisible by seven, thus the original number 15751537186 is divisible by seven.

===Divisibility by 13===
Remainder Test
13 (1, −3, −4, −1, 3, 4, cycle goes on.)
If you are not comfortable with negative numbers, then use this sequence. (1, 10, 9, 12, 3, 4)

Multiply the right most digit of the number with the left most number in the sequence shown above and the second right most digit to the second left most digit of the number in the sequence. The cycle goes on.

Example: What is the remainder when 321 is divided by 13? 
Using the first sequence, 
Ans: '''1''' × 1 + '''2''' × −3 + '''3''' × −4 = 9 
Remainder = −17 mod 13 = 9

Example: What is the remainder when 1234567 is divided by 13? 
Using the second sequence, 
Answer: '''7''' × 1 + '''6''' × 10 + '''5''' × 9 + '''4''' × 12 + '''3''' × 3 + '''2''' × 4 + '''1''' × 1 = 178 mod 13 = 9 
Remainder = 9

==Beyond 20==
Divisibility properties can be determined in two ways, depending on the type of the divisor.

=== Composite divisors ===
A number is divisible by a given divisor if it is divisible by the highest power of each of its [[prime number|prime]] factors. For example, to determine divisibility by 24, check divisibility by 8 and by 3.<ref name="product-of-coprimes"/> Note that checking 4 and 6, or 2 and 12, would not be sufficient. A [[table of prime factors]] may be useful.

A [[Composite number|composite]] divisor may also have a rule formed using the same procedure as for a prime divisor, given below, with the caveat that the manipulations involved may not introduce any factor which is present in the divisor. For instance, one can not make a rule for 14 that involves multiplying the equation by 7. This is not an issue for prime divisors because they have no smaller factors.

=== Prime divisors ===
The goal is to find an inverse to 10 modulo the prime (not 2 or 5) and use that as a multiplier to make the divisibility of the original number by that prime depend on the divisibility of the new (usually smaller) number by the same prime.
Using 17 as an example, since 10 × (−5) = −50 = 1 mod 17, we get the rule for using ''y'' − 5''x'' in the table above. In fact, this rule for prime divisors besides 2 and 5 is ''really'' a rule
for divisibility by any integer relatively prime to 10 (including 21 and 27; see tables below). This is why the last divisibility condition in the tables above and below for any number relatively prime to 10 has the same kind of form (add or subtract some multiple of the last digit from the rest of the number).

=== Notable examples ===
The following table provides rules for a few more notable divisors:

{| class="wikitable"
!Divisor
!Divisibility condition
!Examples
|-
|'''[[21 (number)|21]]'''
|Subtract twice the last digit from the rest.
|168: 16 − (8×2) = 0, 168 is divisible. 1050: 105 − (0×2) = 105, 10 − (5×2) = 0, 1050 is divisible.
|-
|'''[[23 (number)|23]]'''
|Add 7 times the last digit to the rest.
|3128: 312 + (8×7) = 368, 368 ÷ 23 = 16.
|-
|'''[[25 (number)|25]]'''
|The number formed by the last two digits is divisible by 25.<ref name="last-m-digits"/>
|134,250: 50 is divisible by 25.
|-
|rowspan=2|'''[[27 (number)|27]]'''
|Sum the digits in blocks of three from right to left. If the result is divisible by 27, then the number is divisible by 27.
|2,644,272: 2 + 644 + 272 = 918 = 27×34.
|-
|Subtract 8 times the last digit from the rest.
|621: 62 − (1×8) = 54.
|-
|'''[[29 (number)|29]]'''
|Add three times the last digit to the rest.
|261: 1×3 = 3; 3 + 26 = 29
|-
|'''[[31 (number)|31]]'''
|Subtract three times the last digit from the rest.
|837: 83 − 3×7 = 62
|-
|rowspan=4|''' [[32 (number)|32]] '''
| style="border-bottom: hidden;" |The number formed by the last five digits is divisible by 32.<ref name="Pascal's-criterion"/><ref name="last-m-digits"/>
| style="border-bottom: hidden;" |25,135,520: 35,520=1110×32
|-
| style="border-bottom: hidden;" |If the ten thousands digit is even, examine the number formed by the last four digits.
| style="border-bottom: hidden;" |41,312: 1312.
|-
|If the ten thousands digit is odd, examine the number formed by the last four digits plus 16.
|254,176: 4176+16 = 4192.
|-
|Add the last two digits to 4 times the rest.
|1,312: (13×4) + 12 = 64.
|-
|rowspan=2|''' [[33 (number)|33]] '''
|Add 10 times the last digit to the rest; it has to be divisible by 3 and 11.
|627: 62 + 7 × 10 = 132, 13 + 2 × 10 = 33.
|-
|Add the digits in blocks of two from right to left.
|2,145: 21 + 45 = 66.
|-
|''' [[35 (number)|35]] '''
|Number must be divisible by 7 ending in 0 or 5.
|700 is divisible by 7 ending in a 0.
|-
|rowspan=2|''' [[37 (number)|37]] '''
|Take the digits in blocks of three from right to left and add each block, just as for 27.
|2,651,272: 2 + 651 + 272 = 925. 925 = 37×25.
|-
|Subtract 11 times the last digit from the rest.
|925: 92 − (5×11) = 37.
|-
|'''[[39 (number)|39]]'''
|Add 4 times the last digit to the rest.
|351: 1 × 4 = 4; 4 + 35 = 39
|-
|'''[[41 (number)|41]]'''
|Subtract 4 times the last digit from the rest.
|738: 73 − 8 × 4 = 41.
|-
|rowspan=2|'''[[43 (number)|43]]'''
|Add 13 times the last digit to the rest.
|36,249: 3624 + 9 × 13 = 3741, 374 + 1 × 13 = 387, 38 + 7 × 13 = 129, 12 + 9 × 13 = 129 = 43 × 3.
|-
|Subtract 30 times the last digit from the rest.
|36,249: 3624 - 9 × 30 = 3354, 335 - 4 × 30 = 215 = 43 × 5.
|-
|'''[[45 (number)|45]]'''
|The number must be divisible by 9 ending in 0 or 5.<ref name="product-of-coprimes"/>
|495: 4 + 9 + 5 = 18, 1 + 8 = 9; (495 is divisible by both 5 and 9.)
|-
|'''[[47 (number)|47]]'''
|Subtract 14 times the last digit from the rest.
|1,642,979: 164297 − 9 × 14 = 164171, 16417 − 14 = 16403, 1640 − 3 × 14 = 1598, 159 − 8 × 14 = 47.
|-
|'''[[49 (number)|49]]'''
|Add 5 times the last digit to the rest.
|1,127: 112+(7×5)=147. 147: 14 + (7×5) = 49
|-
|'''[[50 (number)|50]]'''
|The last two digits are 00 or 50.
|134,250: 50.
|-
|'''[[51 (number)|51]]'''
|Subtract 5 times the last digit to the rest.
|204: 20-(4×5)=0
|-
|'''[[55 (number)|55]]'''
|Number must be divisible by 11 ending in 0 or 5.<ref name="product-of-coprimes"/>
|935: 93 − 5 = 88 or 9 + 35 = 44.
|-
|'''[[59 (number)|59]]'''
|Add 6 times the last digit to the rest.
|295: 5×6 = 30; 30 + 29 = 59
|-
|'''[[61 (number)|61]]'''
|Subtract 6 times the last digit from the rest.
|732: 73-(2×6)=61
|-
|'''[[64 (number)|64]]'''
|The number formed by the last six digits must be divisible by 64.<ref name="Pascal's-criterion"/><ref name="last-m-digits"/>
|2,640,000 is divisible by 64.
|-
|'''[[65 (number)|65]]'''
|Number must be divisible by 13 ending in 0 or 5.<ref name="product-of-coprimes"/>
|130 is divisible by 13 ending in 0.
|-
|'''[[66 (number)|66]]'''
|Number must be divisible by 6 and 11.<ref name="product-of-coprimes"/>
|132 is divisible by 6 and 11.
|-
|'''[[69 (number)|69]]'''
|Add 7 times the last digit to the rest.
|345: 5×7 = 35; 35 + 34 = 69
|-
|'''[[71 (number)|71]]'''
|Subtract 7 times the last digit from the rest.
|852: 85-(2×7)=71
|-
|'''[[75 (number)|75]]'''
|Number must be divisible by 3 ending in 00, 25, 50 or 75.<ref name="product-of-coprimes"/>
|825: ends in 25 and is divisible by 3.
|-
|'''[[77 (number)|77]]'''
|Form the alternating sum of blocks of three from right to left.
|76,923: 923 - 76 = 847.
|-
|'''[[79 (number)|79]]'''
|Add 8 times the last digit to the rest.
|711: 1×8 = 8; 8 + 71 = 79
|-
|'''[[81 (number)|81]]'''
|Subtract 8 times the last digit from the rest.
|162: 16-(2×8)=0
|-
|'''[[89 (number)|89]]'''
|Add 9 times the last digit to the rest.
|801: 1×9 = 9; 80 + 9 = 89
|-
|rowspan=2|'''[[91 (number)|91]]'''
|Subtract 9 times the last digit from the rest.
|182: 18-(2×9)=0
|-
|Form the alternating sum of blocks of three from right to left.
|5,274,997: 5 - 274 + 997 = 728
|-
|'''[[99 (number)|99]]'''
|Add the digits in blocks of two from right to left.
|144,837: 14 + 48 + 37 = 99.
|-
|'''[[100 (number)|100]]'''
|Ends with at least two zeros.
|900 ends with 2 zeros
|-
|'''[[101 (number)|101]]'''
|Form the alternating sum of blocks of two from right to left.
|40,299: 4 - 2 + 99 = 101.
|-
|'''[[111 (number)|111]]'''
|Add the digits in blocks of three from right to left.
|1,370,184: 1 + 370 + 184 = 555
|-
|'''[[125 (number)|125]]'''
|The number formed by the last three digits must be divisible by 125.<ref name="last-m-digits"/>
|2125 is divisible by 125.
|-
|'''[[128 (number)|128]]'''
|The number formed by the last seven digits must be divisible by 128.<ref name="Pascal's-criterion"/><ref name="last-m-digits"/>
|11,280,000 is divisible by 128.
|-
|'''[[143 (number)|143]]'''
|Form the alternating sum of blocks of three from right to left.
|1,774,487: 1 - 774 + 487 = -286
|-
|'''[[256 (number)|256]]'''
|The number formed by the last eight digits must be divisible by 256.<ref name="Pascal's-criterion"/><ref name="last-m-digits"/>
|225,600,000 is divisible by 256.
|-
|'''[[333 (number)|333]]'''
|Add the digits in blocks of three from right to left.
|410,922: 410 + 922 = 1,332
|-
|'''[[512 (number)|512]]'''
|The number formed by the last nine digits must be divisible by 512.<ref name="Pascal's-criterion"/><ref name="last-m-digits"/>
|1,512,000,000 is divisible by 512.
|-
|'''989'''
|Add the last three digits to eleven times the rest.
|21758: 21 × 11 = 231; 758 + 231 = 989
|-
|'''[[999 (number)|999]]'''
|Add the digits in blocks of three from right to left.
|999,999: 999 + 999 = 1,998
|-
|'''[[1000 (number)|1000]]'''
|Ends with at least three zeros.
|2000 ends with 3 zeros
|-
|}

==Generalized divisibility rule==

To test for divisibility by ''D'', where ''D'' ends in 1, 3, 7, or 9, the following method can be used.<ref>Dunkels, Andrejs, "Comments on note 82.53—a generalized test for divisibility", ''[[Mathematical Gazette]]'' 84, March 2000, 79-81.</ref> Find any multiple of ''D'' ending in 9. (If ''D'' ends respectively in 1, 3, 7, or 9, then multiply by 9, 3, 7, or 1.) Then add 1 and divide by 10, denoting the result as ''m''. Then a number ''N'' = 10''t'' + ''q'' is divisible by ''D'' if and only if ''mq + t'' is divisible by ''D''.

For example, to determine if 913 = 10×91 + 3 is divisible by 11, find that ''m'' = (11×9+1)÷10 = 10. Then ''mq+t'' = 10×3+91 = 121; this is divisible by 11 (with quotient 11), so 913 is also divisible by 11. As another example, to determine if 689 = 10×68 + 9 is divisible by 53, find that ''m'' = (53×3+1)÷10 = 16. Then ''mq+t'' = 16×9 + 68 = 212, which is divisible by 53 (with quotient 4); so 689 is also divisible by 53.

== Proofs ==

=== Proof using basic algebra ===

Many of the simpler rules can be produced using only algebraic manipulation, creating [[binomial]]s and rearranging them. By writing a number as the [[positional notation|sum of each digit times a power of 10]] each digit's power can be manipulated individually.

'''Case where all digits are summed'''

This method works for divisors that are factors of 10 − 1 = 9.

Using 3 as an example, 3 divides 9 = 10 − 1. That means <math>10 \equiv 1 \pmod{3}</math> (see [[modular arithmetic]]). The same for all the higher powers of 10: <math>10^n \equiv 1^n \equiv 1 \pmod{3}</math> They are all [[congruence relation|congruent]] to 1 modulo 3. Since two things that are congruent modulo 3 are either both divisible by 3 or both not, we can interchange values that are congruent modulo 3. So, in a number such as the following, we can replace all the powers of 10 by 1:

:<math>100\cdot a + 10\cdot b + 1\cdot c \equiv (1)a + (1)b + (1)c \pmod{3}</math>

which is exactly the sum of the digits.

'''Case where the alternating sum of digits is used'''

This method works for divisors that are factors of 10 + 1 = 11.

Using 11 as an example, 11 divides 11 = 10 + 1. That means <math>10 \equiv -1 \pmod{11}</math>. For the higher powers of 10, they are congruent to 1 for even powers and congruent to −1 for odd powers:

:<math>10^n \equiv (-1)^n \equiv \begin{cases} 1, & \mbox{if }n\mbox{ is even} \\ -1, & \mbox{if }n\mbox{ is odd} \end{cases} \pmod{11}.</math>

Like the previous case, we can substitute powers of 10 with congruent values:

:<math>1000\cdot a + 100\cdot b + 10\cdot c + 1\cdot d \equiv (-1)a + (1)b + (-1)c + (1)d \pmod{11}</math>

which is also the difference between the sum of digits at odd positions and the sum of digits at even positions.

'''Case where only the last digit(s) matter'''

This applies to divisors that are a factor of a power of 10. This is because sufficiently high powers of the base are multiples of the divisor, and can be eliminated.

For example, in base 10, the factors of 101 include 2, 5, and 10. Therefore, divisibility by 2, 5, and 10 only depend on whether the last 1 digit is divisible by those divisors. The factors of 102 include 4 and 25, and divisibility by those only depend on the last 2 digits.

'''Case where only the last digit(s) are removed'''

Most numbers do not divide 9 or 10 evenly, but do divide a higher power of 10''n'' or 10''n'' − 1. In this case the number is still written in powers of 10, but not fully expanded.

For example, 7 does not divide 9 or 10, but does divide 98, which is close to 100. Thus, proceed from

:<math>100 \cdot a + b</math>

where in this case a is any integer, and b can range from 0 to 99. Next,

:<math>(98+2) \cdot a + b</math>

and again expanding

:<math>98 \cdot a + 2 \cdot a + b,</math>

and after eliminating the known multiple of 7, the result is

:<math>2 \cdot a + b,</math>

which is the rule "double the number formed by all but the last two digits, then add the last two digits".

'''Case where the last digit(s) is multiplied by a factor'''

The representation of the number may also be multiplied by any number relatively prime to the divisor without changing its divisibility. After observing that 7 divides 21, we can perform the following:

:<math>10 \cdot a + b,</math>

after multiplying by 2, this becomes

:<math>20 \cdot a + 2 \cdot b,</math>

and then

:<math>(21 - 1) \cdot a + 2 \cdot b.</math>

Eliminating the 21 gives

:<math> -1 \cdot a + 2 \cdot b,</math>

and multiplying by −1 gives

:<math> a - 2 \cdot b.</math>

Either of the last two rules may be used, depending on which is easier to perform. They correspond to the rule "subtract twice the last digit from the rest".

=== Proof using modular arithmetic ===

This section will illustrate the basic method; all the rules can be derived following the same procedure. The following requires a basic grounding in [[modular arithmetic]]; for divisibility other than by 2's and 5's the proofs rest on the basic fact that 10 mod ''m'' is invertible if 10 and ''m'' are relatively prime.

'''For 2''n'' or 5''n'':'''

Only the last ''n'' digits need to be checked.

:<math>10^n = 2^n \cdot 5^n \equiv 0 \pmod{2^n \mathrm{\ or\ } 5^n}</math>

Representing ''x'' as <math>10^n \cdot y + z,</math>

:<math>x = 10^n \cdot y + z \equiv z \pmod{2^n \mathrm{\ or\ } 5^n}</math>

and the divisibility of ''x'' is the same as that of ''z''.

'''For 7:'''

Since 10 × 5  ≡  10 × (−2)  ≡ 1 (mod 7) we can do the following:

Representing ''x'' as <math>10 \cdot y + z,</math>

:<math>-2x \equiv y -2z \pmod{7},</math>

so ''x'' is divisible by 7 if and only if ''y'' − 2''z'' is divisible by 7.

== Notes ==
<references />

== References ==
* {{cite book |last1=Apostol |first1=Tom M. |authorlink1= |last2= |first2= |authorlink2= |title=Introduction to analytic number theory |url= |edition= |series=Undergraduate texts in mathematics |volume=1 |year=1976 |publisher=Springer-Verlag |location= |isbn=978-0-387-90163-3 |id= }}
* {{cite book |last1=Kisačanin |first1=Branislav |authorlink1= |last2= |first2= |authorlink2= |title=Mathematical problems and proofs: combinatorics, number theory, and geometry |url= |edition= |series= |volume= |year=1998 |publisher=Plenum Press |location= |isbn=978-0-306-45967-2 |id= }}
* {{cite book |last1=Richmond |first1=Bettina |authorlink1= |last2=Richmond |first2=Thomas |authorlink2= |title=A Discrete Transition to Advanced Mathematics |url= |edition= |series=Pure and Applied Undergraduate Texts |volume=3 |year=2009 |publisher=American Mathematical Soc. |location= |isbn=978-0-8218-4789-3 |id= }}

== External links ==
* [http://www.mathwarehouse.com/arithmetic/numbers/divisibility-rules-and-tests.php Interactive Divisibility Lesson on these rules]
* [http://www.cut-the-knot.org/blue/divisibility.shtml Divisibility Criteria] at [[cut-the-knot]]
* [http://www.cut-the-knot.org/Generalization/div11.shtml Divisibility by 9 and 11] at [[cut-the-knot]]
* [http://www.cut-the-knot.org/Generalization/div11.shtml#div7 Divisibility by 7] at [[cut-the-knot]]
* [http://www.cut-the-knot.org/Generalization/81.shtml Divisibility by 81] at [[cut-the-knot]]
* [http://www.apronus.com/math/threediv.htm Divisibility by Three Explained]
* [http://webspace.ship.edu/msrenault/divisibility/index.htm Stupid Divisibility Tricks] Divisibility rules for 2-100.

{{DEFAULTSORT:Divisibility Rule}}
[[Category:Elementary number theory]]
[[Category:Division]]
[[Category:Articles containing proofs]]

Vacuum state

2014-02-03T01:01:57Z

72.172.11.140: /* Notations */

{{Refimprove|date=July 2009}}
'''Creation and annihilation operators''' are [[Operator (mathematics)|mathematical operators]] that have widespread applications in [[quantum mechanics]], notably in the study of [[quantum harmonic oscillator]]s and many-particle systems.<ref name='Feynman1998p151'>{{harv|Feynman|1998|p=151}}</ref> An annihilation operator lowers the number of particles in a given state by one. A creation operator increases the number of particles in a given state by one, and it is the [[Hermitian adjoint|adjoint]] of the annihilation operator. In many subfields of [[physics]] and [[chemistry]], the use of these operators instead of [[wavefunction]]s is known as [[second quantization]].

Creation and annihilation operators can act on states of various types of particles. For example, in [[quantum chemistry]] and [[many-body theory]] the creation and annihilation operators often act on [[electron]] states.
They can also refer specifically to the [[ladder operators]] for the [[quantum harmonic oscillator]]. In the latter case, the raising operator is interpreted as a creation operator, adding a quantum of energy to the oscillator system (similarly for the lowering operator). They can be used to represent [[phonons]].

The [[mathematics]] for the creation and annihilation operators for [[bosons]] is the same as for the [[ladder operators]] of the [[quantum harmonic oscillator]].<ref name='Feynman1998p167'>{{harv|Feynman|1998|p=167}}</ref> For example, the [[commutator]] of the creation and annihilation operators that are associated with the same boson state equals one, while all other commutators vanish. However, for [[fermions]] the mathematics is different, involving [[Commutator#Anticommutator|anticommutators]] instead of commutators.<ref name='Feynman1998p174-5'>{{harv|Feynman|1998|pp=174–5}}</ref>

==Ladder operators for quantum harmonic oscillator==
In the context of the quantum harmonic oscillator, we reinterpret the ladder operators as creation and annihilation operators, adding or subtracting fixed [[quantum|quanta]] of energy to the oscillator system. Creation/annihilation operators are different for [[boson]]s (integer spin) and [[fermion]]s (half-integer spin). This is because their [[wavefunction]]s have different [[identical particles|symmetry properties]].

First consider the simpler bosonic case of the phonons of the quantum harmonic oscillator.

Start with the [[Schrödinger equation]] for the one dimensional time independent [[quantum harmonic oscillator]]
:<math>\left(-\frac{\hbar^2}{2m} \frac{d^2}{d x^2} + \frac{1}{2}m \omega^2 x^2\right) \psi(x) = E \psi(x)</math>

Make a coordinate substitution to [[nondimensionalization|nondimensionalize]] the differential equation
:<math>x \ = \ \sqrt{ \frac{\hbar}{m \omega}} q</math>.

and the Schrödinger equation for the oscillator becomes
:<math> \frac{\hbar \omega}{2} \left(-\frac{d^2}{d q^2} + q^2 \right) \psi(q) = E \psi(q)</math>.

Note that the quantity <math> \hbar \omega = h \nu </math> is the same energy as that found for light [[quantum|quanta]] and that the parenthesis in the [[Hamiltonian (quantum mechanics)|Hamiltonian]] can be written as
:<math> -\frac{d^2}{dq^2} + q^2 = \left(-\frac{d}{dq}+q \right) \left(\frac{d}{dq}+ q \right) + \frac {d}{dq}q - q \frac {d}{dq} </math>

The last two terms can be simplified by considering their effect on an arbitrary differentiable function f(q),
:<math>\left(\frac{d}{dq} q- q \frac{d}{dq} \right)f(q) = \frac{d}{dq}(q f(q)) - q \frac{df(q)}{dq} = f(q) </math>
which implies,
:<math>\frac{d}{dq} q- q \frac{d}{dq} = 1 </math>

Therefore
:<math> -\frac{d^2}{dq^2} + q^2 = \left(-\frac{d}{dq}+q \right) \left(\frac{d}{dq}+ q \right) + 1 </math>
and the Schrödinger equation for the oscillator becomes, with substitution of the above and rearrangement of the factor of 1/2,
:<math> \hbar \omega \left[\frac{1}{\sqrt{2}} \left(-\frac{d}{dq}+q \right)\frac{1}{\sqrt{2}} \left(\frac{d}{dq}+ q \right) + \frac{1}{2} \right] \psi(q) = E \psi(q)</math>.
If we define
:<math>a^\dagger \ = \ \frac{1}{\sqrt{2}} \left(-\frac{d}{dq} + q\right)</math> as the "creation operator" or the "raising operator" and
:<math> a \ \ = \ \frac{1}{\sqrt{2}} \left(\ \ \ \frac{d}{dq} + q\right)</math> as the "annihilation operator" or the "lowering operator"
then the Schrödinger equation for the oscillator becomes
:<math> \hbar \omega \left( a^\dagger a + \frac{1}{2} \right) \psi(q) = E \psi(q)</math>
This is ''significantly'' simpler than the original form. Further simplifications of this equation enables one to derive all the properties listed above thus far.

Letting <math>p = - i \frac{d}{dq}</math>, where "p" is the nondimensionalized [[momentum operator]]
then we have
:<math> [q, p] = i \,</math>
and
:<math>a = \frac{1}{\sqrt{2}}(q + i p) = \frac{1}{\sqrt{2}}\left( q + \frac{d}{dq}\right) </math>
:<math>a^\dagger = \frac{1}{\sqrt{2}}(q - i p) = \frac{1}{\sqrt{2}}\left( q - \frac{d}{dq}\right)</math>.

Note that these imply that

:<math> [a, a^\dagger ] = \frac{1}{2} [ q + ip , q-i p] = \frac{1}{2} ([q,-ip] + [ip, q]) = \frac{-i}{2} ([q, p] + [q, p]) = 1 </math>

in contrast to the so-called "[[normal operators]]" of mathematics, which have a similar representation (e.g. <math>A= W_1 + i\, W_2)\,,</math> with self-adjoint <math>W_i\,.</math> But in the case of normal operators one would be dealing with [[commuting]] <math> W_i\,,</math> i.e. with <math>W_1W_2=W_2W_1\,,</math> so that the 1 at the extreme r.h.s. of the previous equation would be replaced by 0, which would have the consequence of one-and-the-same set of eigenfunctions (and/or eigendistributions) for both <math> W_1</math> and <math> W_2</math>, whereas here common eigenfunctions or eigendistributions of the operators p and q don't exist.

Thus, although in the present case one is explicitly dealing with non-normal operators, by the commutation relation given above, the Hamiltonian operator can be expressed as

:<math>\hat H = \hbar \omega \left( a \, a^\dagger - \frac{1}{2}\right).</math>
:<math>\hat H = \hbar \omega \left( a^\dagger \, a + \frac{1}{2}\right).</math>

And <math>a</math> and <math>a^\dagger\,,</math> operators give the following commutation relations with the Hamiltonian<ref>{{cite web|last=Branson|first=Jim|title=Quantum Physics at UCSD|url=http://quantummechanics.ucsd.edu/ph130a/130_notes/node170.html#section:HOraise|accessdate=16 May 2012}}</ref>

:<math>[\hat H, a ] = -\hbar \omega \, a.</math>
:<math>[\hat H, a^\dagger ] = \hbar \omega \, a^\dagger .</math>

These relations can be used to find the energy eigenstates of the quantum harmonic oscillator. Assuming that <math>\psi_n</math> is an eigenstate of the Hamiltonian <math>\hat H \psi_n = E_n\, \psi_n</math>. Using these commutation relations it can be shown that<ref>{{cite web|last=Branson|first=Jim|title=Quantum Physics at UCSD|url=http://quantummechanics.ucsd.edu/ph130a/130_notes/node170.html#section:HOraise|accessdate=16 May 2012}}</ref>

:<math>\hat H\, a\psi_n = (E_n - \hbar \omega)\, a\psi_n .</math>
:<math>\hat H\, a^\dagger\psi_n = (E_n + \hbar \omega)\, a^\dagger\psi_n .</math>

This shows that <math>a\psi_n</math> and <math>a^\dagger\psi_n</math> are also eigenstates of the Hamiltonian with eigenvalues <math>E_n - \hbar \omega</math> and <math>E_n + \hbar \omega</math>. This identifies the operators <math>a</math> and <math>a^\dagger</math> as lowering and rising operators between the eigenstates. Energy difference between two eigenstates is <math>\Delta E = \hbar \omega</math>.

The ground state can be found by assuming that the lowering operator will collapse it, <math>a\, \psi_0 = 0</math>. And then using the Hamiltonian in terms of rising and lowering operators,

:<math>a^\dagger a \psi_0 = 0 = \left(\frac{\hat H}{\hbar \omega} - \frac{1}{2} \right) \,\psi_0 = \left(\frac{E_0}{\hbar \omega} - \frac{1}{2} \right) \,\psi_0.</math>

the wave-function on the right is non-zero, thus term in brackets must be. This gives the ground state energy <math>E_0 = \hbar \omega /2</math>. This allows to identify the energy eigenvalue of any eigenstate <math>\psi_n</math> as<ref>{{cite web|last=Branson|first=Jim|title=Quantum Physics at UCSD|url=http://quantummechanics.ucsd.edu/ph130a/130_notes/node170.html#section:HOraise|accessdate=16 May 2012}}</ref>

:<math>E_n = \left(n + \frac{1}{2}\right)\hbar \omega</math>

Furthermore it can be shown that the first-mentioned operator, the '''number operator''' <math>N=a^\dagger a\,,</math> plays a most-important role in applications, while the second one, <math>a \,a^\dagger\,,</math> can simply be replaced by <math>N +1\,.</math> So one simply gets

: <math>\hbar\omega \,\left(N+\frac{1}{2}\right)\,\psi (q) =E\,\psi (q)</math>.

=== Applications ===
The ground state <math>\ \psi_0(q)</math> of the [[quantum harmonic oscillator]] can be found by imposing the condition that

:<math> a \ \psi_0(q) = 0</math>.

Written out as a differential equation, the wavefunction satisfies

:<math>q \psi_0 + \frac{d\psi_0}{dq} = 0</math>

which has the solution

:<math>\psi_0(q) = C \exp\left(-{q^2 \over 2}\right).</math>

The normalization constant ''C'' can be found to be  <math>1\over \sqrt[4]{\pi}</math>  from <math>\int_{-\infty}^\infty \psi_0^* \psi_0 \,dq = 1</math>,  using the [[Gaussian integral]].

=== Matrix representation ===
The matrix counterparts of the creation and annihilation operators obtained from the quantum harmonic oscillator model are

:<math>{a}^{\dagger}=\begin{pmatrix}
0 & 0 & 0 & \dots & 0 &\dots \\
\sqrt{1} & 0 & 0 & \dots & 0 & \dots\\
0 & \sqrt{2} & 0 & \dots & 0 & \dots\\
0 & 0 & \sqrt{3} & \dots & 0 & \dots\\
\vdots & \vdots & \vdots & \ddots & \vdots & \dots\\
0 & 0 & 0 & 0 & \sqrt{n} &\dots & \\
\vdots & \vdots & \vdots & \vdots & \vdots &\ddots \end{pmatrix}
</math>

:<math>{a}=\begin{pmatrix}
0 & \sqrt{1} & 0 & 0 & \dots & 0 & \dots \\
0 & 0 & \sqrt{2} & 0 & \dots & 0 & \dots \\
0 & 0 & 0 & \sqrt{3} & \dots & 0 & \dots \\
0 & 0 & 0 & 0 & \ddots & \vdots & \dots \\
\vdots & \vdots & \vdots & \vdots & \ddots & \sqrt{n} & \dots \\
0 & 0 & 0 & 0 & \dots & 0 & \ddots \\
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots \end{pmatrix}</math>

Substituting backwards, the laddering operators are recovered. They can be obtained via the relationships
<math>a^\dagger_{ij} = \langle\psi_i | {a}^\dagger | \psi_j\rangle</math> and
<math>a_{ij} = \langle\psi_i | {a} | \psi_j\rangle</math>. The wavefunctions are those of the quantum harmonic oscillator, and are sometimes called the "number basis".

=== Mathematical details ===
The operators derived above are actually a specific instance of a more generalized class of creation and annihilation operators. The more abstract form of the operators satisfy the properties below.

Let ''H'' be the one-particle [[Hilbert space]]. To get the [[boson]]ic [[CCR algebra]], look at the algebra generated by ''a''(''f'') for any ''f'' in ''H''. The operator ''a''(''f'') is called an annihilation operator and the map ''a''(.) is [[antilinear]]. Its [[adjoint]]{{dn|date=December 2013}} is ''a''†(''f'') which is [[linear]] in ''H''.

For a boson,
:<math>[a(f),a(g)]=[a^\dagger(f),a^\dagger(g)]=0</math>
:<math>[a(f),a^\dagger(g)]=\langle f|g \rangle</math>,

where we are using [[bra-ket notation]].

For a fermion, the [[anticommutator]]s are
:<math>\{a(f),a(g)\}=\{a^\dagger(f),a^\dagger(g)\}=0 </math>
:<math>\{a(f),a^\dagger(g)\}=\langle f|g \rangle</math>.

A [[CAR algebra]].

Physically speaking, ''a''(''f'') removes (i.e. annihilates) a particle in the state {{Dket|''f''}} whereas ''a''†(''f'') creates a particle in the state {{Dket|''f''}}.

The [[free field]] [[vacuum state]] is the state with no particles. In other words,

:<math>a(f)|0\rangle=0</math>

where {{Dket|0}} is the vacuum state.

If {{Dket|''f''}} is normalized so that {{Dbraket|''f''|''f''}} = 1, then ''a''†(''f'') ''a''(''f'') gives the number of particles in the state {{Dket|''f''}}.

=== Creation and annihilation operators for reaction-diffusion equations ===
The annihilation and creation operator description has also been useful to analyze classical reaction diffusion equations, such as the situation when a gas of molecules ''A'' diffuse and interact on contact, forming an inert product: {{nowrap|''A'' + ''A'' → ∅ .}} To see how this kind of reaction can be described by the annihilation and creation operator formalism, consider <math>n_{i}</math> particles at a site <math>i</math> on a 1-d lattice. Each particle diffuses independently, so that the probability that one of them leaves the site for short times <math>dt</math> is proportional to <math>n_{i}dt</math>, say <math>\alpha n_{i}dt</math> to hop left and <math>\alpha n_{i}dt</math> to hop right. All <math>n</math> particles will stay put with a probability <math>1-2\alpha n_{i}dt</math>.

We can now describe the occupation of particles on the lattice as a `ket' of the form {{Dket|''n1, n2, ...''}}. A slight modification of the annihilation and creation operators is needed so that
:<math>a|n\rangle= \sqrt{n} \ |n-1\rangle</math>

and

:<math>a^{\dagger}|n\rangle= \sqrt{n+1} \ |n+1\rangle</math>.

This modification preserves the commutation relation

:<math>[a,a^{\dagger}]=1</math>,

but allows us to write the pure diffusive behaviour of the particles as

:<math>\partial_{t}|\psi\rangle=-\alpha\sum(2a_{i}^{\dagger}a_{i}-a_{i-1}^{\dagger}a_{i}-a_{i+1}^{\dagger}a_{i})|\psi\rangle=-\alpha\sum(a_{i}^{\dagger}-a_{i-1}^{\dagger})(a_{i}-a_{i-1})|\psi\rangle </math>

The reaction term can be deduced by noting that <math>n</math> particles can interact in <math>n(n-1)</math> different ways, so that the probability that a pair annihilates is <math>\lambda n(n-1)dt</math> and the probability that no pair annihilates is <math>1-\lambda n(n-1)dt</math> leaving us with a term

:<math>\lambda\sum(a_{i}a_{i}-a_{i}^{\dagger}a_{i}^{\dagger}a_{i}a_{i})</math>

yielding

:<math>\partial_{t}|\psi\rangle=-\alpha\sum(a_{i}^{\dagger}-a_{i-1}^{\dagger})(a_{i}-a_{i-1})|\psi\rangle+\lambda\sum(a_{i}^{2}-a_{i}^{\dagger 2}a_{i}^{2})|\psi\rangle </math>

Other kinds of interactions can be included in a similar manner.

This kind of notation allows the use of quantum field theoretic techniques to be used in the analysis of reaction diffusion systems.

==Creation and annihilation operators in quantum field theories==

{{Main|Second quantization|Quantum_field_theory#Second_quantization}}

In [[Quantum field theory|quantum field theories]] and [[many-body problem]]s one works with creation and annihilation operators of quantum states, <math>a^\dagger_i</math> and <math>a^{\,}_i</math>. These operators change the eigenvalues of the [[number operator]],
: <math>N = \sum_i n_i = \sum_i a^\dagger_i a^{\,}_i</math>,
by one, in analogy to the harmonic oscillator. The indices (such as <math>i</math>) represent [[quantum numbers]] that label the single-particle states of the system; hence, they are not necessarily single numbers. For example, a [[tuple]] of quantum numbers <math>(n, l, m, s)</math> is used to label states in the [[hydrogen atom]].

The commutation relations of creation and annihilation operators in a multiple-[[boson]] system are,
: <math>[a^{\,}_i, a^\dagger_j] \equiv a^{\,}_i a^\dagger_j - a^\dagger_ja^{\,}_i = \delta_{i j},</math>
: <math>[a^\dagger_i, a^\dagger_j] = [a^{\,}_i, a^{\,}_j] = 0,</math>
where <math>[\ \ , \ \ ]</math> is the [[commutator]] and <math>\delta_{i j}</math> is the [[Kronecker delta]].

For [[fermion]]s, the commutator is replaced by the [[anticommutator]] <math>\{\ \ , \ \ \}</math>,
: <math>\{a^{\,}_i, a^\dagger_j\} \equiv a^{\,}_i a^\dagger_j +a^\dagger_j a^{\,}_i = \delta_{i j},</math>
: <math>\{a^\dagger_i, a^\dagger_j\} = \{a^{\,}_i, a^{\,}_j\} = 0.</math>
Therefore, exchanging disjoint (i.e. <math>i \ne j</math>) operators in a product of creation of annihilation operators will reverse the sign in fermion systems, but not in boson systems.

==See also==
* [[Bogoliubov transformation]]s - arises in the theory of quantum optics.
* [[Optical Phase Space]]
* [[Fock space]]
* [[Canonical commutation relation]]s

==References==
*{{cite book | last = Feynman | first = Richard P. | authorlink = Richard Feynman | coauthors = | title = Statistical Mechanics: A Set of Lectures | publisher = Addison-Wesley | year = 1998 | origyear = 1972 | edition=2nd | location = Reading, Massachusetts | page = | url = http://books.google.com/books?id=Ou4ltPYiXPgC&pg=Front | doi = | id = | isbn = 978-0-201-36076-9| ref=harv }}

==Footnotes==
<references/>

{{Physics operator}}

[[Category:Quantum mechanics]]
[[Category:Quantum field theory]]

Vacuum expectation value

2014-02-03T00:50:18Z

72.172.11.140: see talk page

[[File:Montreal-tower-top.thumb2.jpg|thumb|right|22{{math|}}0px|A three-element Yagi-Uda antenna used for [[amateur radio]]. The longer ''reflector'' element (left), the [[driven element]] (center), and the shorter ''director'' (right) each have a so-called ''trap'' (parallel [[LC circuit]]) inserted along their conductors on each side, allowing the antenna to be used at two different frequency bands.]]
[[File:FuG 220 and FuG 202 radar of Me 110 1945.jpg|thumb|This late-WWII Me 110 night fighter features the prominent Yagi arrays of its FuG 220 radar.]]

A '''Yagi-Uda array''', commonly known simply as a '''Yagi antenna''', is a [[directional antenna]] consisting of a [[driven element]] (typically a [[dipole antenna|dipole]] or [[folded dipole]]) and additional [[parasitic element]]s (usually a so-called ''reflector'' and one or more ''directors''). The reflector element is slightly longer (typically 5% longer) than the driven dipole, whereas the so-called directors are a little shorter. This design achieves a very substantial increase in the antenna's [[directional antenna|directionality]] and [[antenna gain|gain]] compared to a simple dipole.<ref>[http://what-is-what.com/what_is/Yagi_Uda_antenna.html What is a Yagi-Uda antenna? – An explanation of the familiar Yagi-Uda antenna from a non-technical point of view. Includes information on wi-fi applications of Yagi Antennas]</ref>

Highly directional antennas such as the Yagi-Uda are commonly referred to as "beam antennas" due to their high gain. However, the Yagi-Uda design only achieves this high gain over a rather narrow bandwidth, making it useful for specific communications bands. [[Amateur radio]] operators ("hams") frequently employ these on HF, [[VHF]], and [[UHF]] bands, often constructing antennas themselves ("[[Amateur radio homebrew|homebrewing]]"), leading to a quantity of technical papers and design software. Yagi's are not very useful for signals spread across a wide band, like [[Terrestrial television|television signals]], where the similar looking [[log-periodic dipole array]] is commonly used, which works on different principles.

The name stems from its inventors, [[Shintaro Uda]] of [[Tohoku University|Tohoku Imperial University]], [[Japan]], with a lesser role played by his colleague [[Hidetsugu Yagi]]. However the "Yagi" name has become more familiar with the name of Uda often omitted. Yagi antennas were first widely used during [[World War II]] for [[radar]] systems, and were widely used by the British, US and Germans. Large Yagi arrays were particularly evident on German [[night fighter]]s. Inter-service rivalries and the military's distrust of all things civilian resulted in no use in Japan until late in the war, when the device was re-introduced via foreign technical articles captured in Singapore.

==Description==

[[File:A8-8.jpg|thumb|400px|Yagi-Uda antenna. Viewed left to right: [[passive radiator|reflector]], driven element, [[passive radiator|director]]. Exact spacings and element lengths vary somewhat according to specific designs.]]
Yagi-Uda antennas are directional along the axis perpendicular to the dipole in the plane of the elements, from the reflector toward the driven element and the director(s). Typical spacings between elements vary from about 1/10 to 1/4 of a wavelength, depending on the specific design. The lengths of the directors are smaller than that of the driven element, which is smaller than that of the reflector(s) according to an elaborate design procedure. These elements are usually parallel in one plane, supported on a single crossbar known as a ''boom''.

The [[Bandwidth (signal processing)|bandwidth]] of a Yagi-Uda antenna refers to the frequency range over which its directional gain and impedance match are preserved to within a stated criterion. The Yagi-Uda array in its basic form is very narrowband, with its performance already compromised at frequencies just a few percent above or below its design frequency. However, using larger diameter conductors, among other techniques, the bandwidth can be substantially extended.

Yagi-Uda antennas used for [[amateur radio]] are sometimes designed to operate on multiple bands. These elaborate designs create electrical breaks along each element (both sides) at which point a parallel [[LC circuit|LC]] ([[inductor]] and [[capacitor]]) circuit is inserted. This so-called ''trap'' has the effect of truncating the element at the higher frequency band, making it approximately a half wavelength in length. At the lower frequency, the entire element (including the remaining inductance due to the trap) is close to half-wave resonance, implementing a ''different'' Yagi-Uda antenna. Using a second set of traps a "triband" antenna can be resonant at three different bands. Given the associated costs of erecting an antenna and rotor system above a tower, the combination of antennas for three amateur bands in one unit is a very practical solution. The use of traps is not without disadvantages, however, as they reduce the bandwidth of the antenna on the individual bands and reduce the antenna's electrical efficiency and subject the antenna to additional mechanical considerations (wind loading, water and insect ingress).

==Theory of operation==
[[File:Two meter yagi.jpg|thumb|right|300px|A Yagi-Uda antenna for use at 144MHz (VHF).]]
Consider a Yagi-Uda consisting of a reflector, driven element and a single director as shown here. The driven element is typically a [[Dipole antenna#Half-wave antenna|λ/2 dipole]] or [[folded dipole]] and is the only member of the structure that is directly excited (electrically connected to the [[feedline]]). All the other elements are considered ''parasitic''. That is, they reradiate power which they receive from the driven element (they also interact with each other).

One way of thinking about the operation of such an antenna is to consider a dipole element to be a normal parasitic element with a gap at its center, the feedpoint. Now instead of attaching the antenna to a load (such as a receiver) we connect it to a short circuit. As is well known in [[transmission line]] theory, a short circuit reflects all of the incident power 180 degrees out of phase. So one could as well model the operation of the parasitic element as the superposition of a dipole element receiving power and sending it down a transmission line to a matched load, and a transmitter sending the same amount of power down the transmission line back toward the antenna element. If the wave from the transmitter were 180 degrees out of phase with the received wave at that point, it would be equivalent to just shorting out that dipole at the feedpoint (making it a solid element, as it is).

The fact that the parasitic element involved is not exactly resonant but is somewhat shorter (or longer) than λ/2 modifies the phase of the element's current with respect to its excitation from the driven element. The so-called ''reflector'' element, being longer than λ/2, has an inductive reactance which means the phase of its current lags the phase of the open-circuit voltage that would be induced by the received field. The ''director'' element, on the other hand, being shorter than λ/2 has a capacitive reactance with the voltage phase lagging that of the current.<ref>[[#Poz01|Pozar (2001)]]</ref> If the parasitic elements were broken in the center and driven with the same voltage applied to the center element, then such a phase difference in the currents would implement an end-fire [[phased array]], enhancing the radiation in one direction and decreasing it in the opposite direction. Thus, one can appreciate the mechanism by which parasitic elements of unequal length can lead to a unidirectional radiation pattern.

==Analysis==

While the above qualitative explanation is useful for understanding how parasitic elements can enhance the driven elements radiation in one direction at the expense of the other, the assumptions used are quite inaccurate. Since the so-called reflector, the longer parasitic element, has a current whose phase lags that of the driven element, one would expect the directivity to be in the direction of the reflector, opposite of the actual directional pattern of the Yagi-Uda antenna. In fact, that would be the case were we to construct a phased array with rather closely spaced elements all driven by voltages in phase, as we posited.

However these elements are not driven as such but receive their energy from the field created by the driven element, so we will find almost the opposite to be true. For now, consider that the parasitic element is also of length λ/2. Again looking at the parasitic element as a dipole which has been shorted at the feedpoint, we can see that if the parasitic element were to respond to the driven element with an open-circuit feedpoint voltage in phase with that applied to the driven element (which we'll assume for now) then the ''reflected'' wave from the short circuit would induce a current 180° out of phase with the current in the driven element. This would tend to cancel the radiation of the driven element. However due to the reactance caused by the length difference, the phase lag of the current in the reflector, added to this 180° lag, results in a phase ''advance'', and vice versa for the director. Thus the directivity of the array indeed is in the direction towards the director.

[[File:Yagi en.svg|right|thumb|300px|Illustration of forward gain of a two element Yagi-Uda array using only a driven element (left) and a director (right). The wave (green) from the driven element excites a current in the passive director which reradiates a wave (black) having a particular phase shift (see explanation in text). The addition of these waves (bottom) is increased in the forward direction, but leads to cancellation in the reverse direction.]]
[[File:Zij-en.png|right|thumb|300px|Mutual impedance between parallel <math>\scriptstyle{{\lambda \over 2}}</math> dipoles not staggered as a function of spacing. Curves '''Re''' and '''Im''' are the resistive and reactive parts of the mutual impedance. Note that at zero spacing we obtain the self-impedance of a half-wave dipole, 73+j43 ohms.]]
One must take into account an additional phase delay due to the finite distance between the elements which further delays the phase of the currents in both the directors and reflector(s). The case of a Yagi-Uda array using just a driven element and a director is illustrated in the accompanying diagram taking all of these effects into account. The wave generated by the driven element (green) propagates in both the forward and reverse directions (as well as other directions, not shown). The director receives that wave slightly delayed in time (amounting to a phase delay of about 35° which will be important for the reverse direction calculations later), and generating a current that would be out of phase with the driven element (thus an additional 180° phase shift), but which is further ''advanced'' in phase (by about 70°) due to the director's shorter length. In the forward direction the net effect is a wave emitted by the director (blue) which is about 110° (180° - 70°) retarded with respect to that from the driven element (green), in this particular design. These waves combine to produce the net forward wave (bottom, right) with an amplitude slightly larger than the individual waves.

In the reverse direction, on the other hand, the additional delay of the wave from the director (blue) due to the spacing between the two elements (about 35° of phase delay traversed twice) causes it to be about 180° (110° + 2*35°) out of phase with the wave from the driven element (green). The net effect of these two waves, when added (bottom, left), is almost complete cancellation. The combination of the director's position and shorter length has thus obtained a unidirectional rather than the bidirectional response of the driven (half-wave dipole) element alone.

A full analysis of such a system requires computing the ''mutual impedances'' between the dipole elements<ref>Principles of Antenna Theory, Kai Fong Lee, 1984, John Wiley and Sons Ltd., ISBN 0-471-90167-9</ref> which implicitly takes into account the propagation delay due to the finite spacing between elements. We model element number ''j'' as having a feedpoint at the center with a voltage ''V''j and a current ''I''j flowing into it. Just considering two such elements we can write the voltage at each feedpoint in terms of the currents using the mutual impedances ''Z''ij:

: <math> V_1 = Z_{11} I_1 + Z_{12} I_2 </math>

: <math> V_2 = Z_{21} I_1 + Z_{22} I_2 </math>

''Z''11 and ''Z''22 are simply the ordinary driving point impedances of a dipole, thus 73+j43 ohms for a half-wave element (or purely resistive for one slightly shorter, as is usually desired for the driven element). Due to the differences in the elements' lengths ''Z''11 and ''Z''22 have a substantially different reactive component. Due to reciprocity we know that ''Z''21 = ''Z''12. Now the difficult computation is in determining that mutual impedance ''Z''21 which requires a numerical solution. This has been computed for two exact half-wave dipole elements at various spacings in the accompanying graph.

The solution of the system then is as follows. Let the driven element be designated 1 so that ''V''1 and ''I''1 are the voltage and current supplied by the transmitter. The parasitic element is designated 2, and since it is shorted at its "feedpoint" we can write that ''V''2 =0. Using the above relationships, then, we can solve for ''I''2 in terms of ''I''1:

: <math>0 = V_2 = Z_{21} I_1 + Z_{22} I_2 </math>
and so
: <math>I_2 = - {Z_{21} \over Z_{22}} \, I_1 </math>.
This is the current induced in the parasitic element due to the current ''I''1 in the driven element. We can also solve for the voltage ''V''1 at the feedpoint of the driven element using the earlier equation:
:<math> V_1 = Z_{11} I_1 + Z_{12} I_2 =
Z_{11} I_1 - Z_{12}{Z_{21} \over Z_{22}} \, I_1 </math>
:<math> \qquad\qquad = \left( Z_{11} - {Z_{21}^2 \over Z_{22}} \right) \, I_1 </math>
where we have substituted ''Z''12 = ''Z''21. The ratio of voltage to current at this point is the ''driving point impedance'' ''Zdp'' of the 2-element Yagi:

: <math> Z_{dp}= V_1 / I_1 = Z_{11} - {Z_{21}^2 \over Z_{22}} </math>
With only the driven element present the driving point impedance would have simply been ''Z''11, but has now been modified by the presence of the parasitic element. And now knowing the phase (and amplitude) of ''I''2 in relation to ''I''1 as computed above allows us to determine the radiation pattern (gain as a function of direction) due to the currents flowing in these two elements. Solution of such an antenna with more than two elements proceeds along the same lines, setting each ''V''j=0 for all but the driven element, and solving for the currents in each element (and the voltage ''V''1 at the feedpoint).<ref>{{cite book |url=http://www.sm.rim.or.jp/~ymushiak/sub.yubook.htm |title=Yagi-Uda Antenna |author1=S. Uda |author2=Y. Mushiake |publisher=The Research Institute of Electrical Communication, Tohoku University |location=Sendai, Japan |date=1954}}</ref>

==Design==

[[File:Yagi uda antenna.jpg|thumb|Two Yagi-Uda antennas on a single mast. The top one includes a corner reflector and 3 stacked Yagis fed in phase in order to increase gain in the horizontal direction (by cancelling power radiated toward the ground or sky). The lower antenna is oriented for vertical polarization, with a much lower resonant frequency.]]
There are no simple formulas for designing Yagi-Uda antennas due to the complex relationships between physical parameters such as element length, spacing, and diameter, and performance characteristics such as gain and input impedance. But using the above sort of analysis one can calculate the performance given a set of parameters and adjust them to optimize the gain (perhaps subject to some constraints). Since with an N element Yagi-Uda antenna, there are 2N-1 parameters to adjust (the element lengths and relative spacings), this is not a straightforward problem at all. The mutual impedances plotted above only apply to λ/2 length elements, so these might need to be recomputed to get good accuracy. What's more, the current distribution along a real antenna element is only approximately given by the usual assumption of a classical standing wave, requiring a solution of Hallen's integral equation taking into account the other conductors. Such a complete exact analysis considering all of the interactions mentioned is rather overwhelming, and approximations are inevitably invoked, as we have done in the above example.

Consequently, these antennas are often empirical designs using an element of [[trial and error]], often starting with an existing design modified according to one's hunch. The result might be checked by direct measurement or by computer simulation. A well-known reference employed in the latter approach is a report published by the National Bureau of Standards (NBS) (now the [[National Institute of Standards and Technology]] (NIST)) that provides six basic designs derived from measurements conducted at 400 MHz and procedures for adapting these designs to other frequencies.<ref>[http://tf.nist.gov/timefreq/general/pdf/451.pdf ''Yagi Antenna Design'', Peter P. Viezbicke, National Bureau of Standard Technical Note 688, December 1976]</ref> These designs, and those derived from them, are sometimes referred to as "NBS yagis."

By adjusting the distance between the adjacent directors it is possible to reduce the back lobe of the radiation pattern

==History==

The Yagi-Uda antenna was invented in 1926 by [[Shintaro Uda]] of [[Tohoku University|Tohoku Imperial University]], [[Sendai, Miyagi|Sendai]], [[Japan]], with the collaboration of [[Hidetsugu Yagi]], also of Tohoku Imperial University. Hidetsugu Yagi attempted [[wireless energy transfer]] in February 1926 with this antenna. Yagi and Uda published their first report on the wave projector directional antenna. Yagi demonstrated a [[proof of concept]], but the engineering problems proved to be more onerous than conventional systems.<ref name=Brown138>Brown, 1999, p. 138</ref>

Yagi published the first English-language reference on the antenna in a 1928 survey article on short wave research in Japan and it came to be associated with his name. However, Yagi always acknowledged Uda's principal contribution to the design, and the proper name for the antenna is, as above, the Yagi-Uda antenna (or array).

The Yagi was first widely used during [[World War II]] for airborne [[radar]] sets, because of its simplicity and directionality.<ref name=Brown138/><ref>Graf, Rudolf F. (June 1959). [http://books.google.com.my/books?id=0dsDAAAAMBAJ&pg=PA214 "Make Your Own UHF Yagi Antenna".] ''Popular Mechanics'', pp. 144–145, 214.</ref> Despite its being invented in Japan, many Japanese radar engineers were unaware of the design until very late in the war, partly due to rivalry between the Army and Navy. The Japanese military authorities first became aware of this technology after the [[Battle of Singapore]] when they captured the notes of a British radar technician that mentioned "yagi antenna". Japanese intelligence officers did not even recognise that Yagi was a Japanese name in this context. When questioned, the technician said it was an antenna named after a Japanese professor.<ref>[http://books.google.com.my/books?id=MgpWAAAAMAAJ&q=yagi+singapore&dq=yagi+singapore&cd=8 2001 IEEE Antennas and Propagation Society International Symposium By IEEE Antennas and Propagation Society. International Symposium.]</ref> (This story is analogous to the story of American intelligence officers interrogating German rocket scientists and finding out that [[Robert Goddard]] was the real pioneer of rocket technology even though he was not well known in the US at that time.)

A [[Horizontal polarization|horizontally polarized]] array can be seen under the left leading edge of Grumman [[F4F]], [[F6F]], [[TBF Avenger]] carrier-based [[US Navy]] aircraft. Vertically polarized arrays can be seen on the cheeks of the [[P-61]] and on the [[nose cone]]s of many WWII aircraft, notably the [[Lichtenstein radar]]-equipped examples of the German [[Junkers Ju 88]]R-1 [[fighter-bomber]], and the British [[Bristol Beaufighter]] night-fighter and [[Short Sunderland]] flying-boat. Indeed, the latter had so many antenna elements arranged on its back - in addition to its formidable turreted defensive armament in the nose and tail, and atop the hull - it was nicknamed the ''fliegendes Stachelschwein'', or "Flying Porcupine" by German airmen.<ref>[http://books.google.com.my/books?id=9LVOSdGUGPkC&pg=PA5&dq=%22Flying+Porcupine%22&hl=en&ei=Dk0GTKy4B4i8rAeWvM3hDA&sa=X&oi=book_result&ct=result&resnum=3&ved=0CC8Q6AEwAg#v=onepage&q=%22Flying%20Porcupine%22&f=false The Sunderland flying-boat queen, Volume 1 By John Evans, Page 5]</ref> The experimental ''Morgenstern'' German AI VHF-band radar antenna of 1943-44 used a "double-Yagi" structure from its 90° angled pairs of Yagi antennas, making it possible to fair the array within a conical, rubber-covered plywood radome on an aircraft's nose, with the extreme tips of the ''Morgenstern's'' antenna elements protruding from the radome's surface, with an [[NJG 4]] [[Ju 88]]G-6 of the wing's staff flight using it late in the war for its Lichtenstein SN-2 AI radar.<ref>{{cite web |url=http://www.hyperscale.com/images/aims_48D001%201%20-%203.jpg |title=HyperScale 48D001 Ju 88 G-6 and Mistel S-3C Collection decals |author= |date= |work= |publisher=Hyperscale.com |accessdate=April 15, 2012}}</ref>

Yagi-Uda antennas are routinely made with rather high gains (over 10dB) making them a common choice for directional antennas especially in VHF and UHF communications systems where a narrowband antenna is acceptable. Only at higher UHF and microwave frequencies are parabolic reflectors and other so-called ''aperture antennas'' of a practical size; these can easily achieve yet higher gains.

The Yagi-Uda antenna was named an [[List of IEEE milestones|IEEE Milestone]] in 1995.<ref>{{cite web |url=http://www.ieeeghn.org/wiki/index.php/Milestones:Directive_Short_Wave_Antenna,_1924 |title=Milestones:Directive Short Wave Antenna, 1924 |author= |date= |work=IEEE Global History Network |publisher=IEEE |accessdate=29 July 2011}}</ref>

== See also ==
* [[Antenna (radio)]]
* [[Larmor formula]]
* [[Numerical Electromagnetics Code]]
* [[Radio direction finder]]
* [[Radio direction finding]]

==Notes==
{{Reflist}}

==References==
* Brown, Louis (1999). [http://books.google.com.my/books?id=wpFMWeLmp4cC ''A radar history of World War II: technical and military imperatives'']. CRC Press. ISBN 0-7503-0659-9
* S. Uda, "High angle radiation of short electric waves". ''[[Proceedings of the IRE]]'', vol. 15, pp. 377–385, May 1927.
* S. Uda, "Radiotelegraphy and radiotelephony on half-meter waves". ''Proceedings of the IRE'', vol. 18, pp. 1047–1063, June 1930.
* H .Yagi, {{doi-inline|10.1109/JPROC.1997.649674|Beam transmission of ultra-shortwaves}}, Proceedings of the IRE, vol. 16, pp. 715–740, June 1928. The URL is to a 1997 IEEE reprint of the classic article. See also {{doi-inline|10.1109/JPROC.1997.649661|Beam Transmission Of Ultra Short Waves: An Introduction To The Classic Paper By H. Yagi}} by D.M. Pozar, in [[Proceedings of the IEEE]], Volume 85, Issue 11, Nov. 1997 Page(s):1857 - 1863.
* "[http://ieee.cincinnati.fuse.net/reiman/05_2004.htm Scanning the Past: A History of Electrical Engineering from the Past]". Proceedings of the IEEE Vol. 81, No. 6, 1993.
* Shozo Usami and Gentei Sato, "[http://ieeexplore.ieee.org/iel5/7598/20722/00958785.pdf?/history_center/milestones_photos/yagi.html Directive Short Wave Antenna, 1924]". IEEE Milestones, IEEE History Center, IEEE, 2005.
* {{cite book
|last=Pozar
|first=David M.
|title=Microwave and RF Design of Wireless Systems
|year=2001
|publisher=John Wiley & Sons Inc.
|isbn=978-0-471-32282-5
|page=134
|ref=Poz01
}}

==External links==
{{Commons category|Yagi-Uda antennas}}

* D. Jefferies, "[http://www.ee.surrey.ac.uk/Personal/D.Jefferies/yagiuda.html Yagi-Uda antennas]". 2004.
* [http://yagi-uda.com/ Yagi-Uda Antenna]. Simple information on basic design, project and measure of Yagi-Uda antenna. 2008
* [http://www.antenna-theory.com/antennas/travelling/yagi.php Yagi-Uda Antennas] www.antenna-theory.com

{{Japanese Electronics Industry}}

{{Antenna Types}}

{{DEFAULTSORT:Yagi Antenna}}
[[Category:Radio frequency antenna types]]
[[Category:Antennas (radio)| ]]
[[Category:Radio electronics]]
[[Category:Japanese inventions]]

QCD vacuum

2014-02-03T00:32:48Z

72.172.11.140: removed superfluous editorial comment "(perturbative?)"

{{dablink|Not to be confused with [[spherical aberration]], a loss of image sharpness that can result from spherical lens surfaces.}}
{{Use dmy dates|date=July 2013}}
{{Optical aberration}}

[[Image:Uniformity.jpg|thumb|Wine glasses create non-uniform distortion of their background]]

In [[geometric optics]] and [[cathode ray tube]] (CRT) displays, '''distortion''' is a deviation from [[rectilinear projection]], a projection in which straight lines in a scene remain straight in an image. It is a form of [[aberration in optical systems|optical aberration]].

==Radial distortion==

Although distortion can be irregular or follow many patterns, the most commonly encountered distortions are radially symmetric, or approximately so, arising from the symmetry of a [[photographic lens]]. These ''radial distortions'' can usually be classified as either ''barrel'' distortions or ''pincushion'' distortions. See van Walree.<ref name="van Walree">{{cite web| url= http://toothwalker.org/optics/distortion.html |title= Distortion |author= Paul van Walree |work= Photographic optics |accessdate=2 February 2009}}</ref>
{{clear}}
{|
|valign="top"|[[Image:Barrel distortion.svg|100px]]
|valign="top"|'''Barrel distortion''' 
In barrel distortion, image [[magnification]] decreases with distance from the [[optical axis]]. The apparent effect is that of an image which has been mapped around a [[sphere]] (or [[barrel]]). [[Fisheye lens]]es, which take hemispherical views, utilize this type of distortion as a way to map an infinitely wide object plane into a finite image area. In a [[zoom lens]] barrel distortion appears in the middle of the lens's focal length range and is worst at the wide-angle end of the range.<ref>{{cite web |url=http://www.cnet.com.au/tamron-18-270mm-f3-5-6-3-di-ii-vc-pzd-339323047.htm |title=Tamron 18-270mm f/3.5-6.3 Di II VC PZD |accessdate=20 March 2013}}</ref>
|-
|valign="top"|[[Image:Pincushion distortion.svg|100px]]
|valign="top"|'''Pincushion distortion''' 
In pincushion distortion, image magnification increases with the distance from the [[optical axis]]. The visible effect is that lines that do not go through the centre of the image are bowed inwards, towards the centre of the image, like a [[pincushion]].
|}
{|
|[[Image:Mustache distortion.svg|100px]]
|'''Mustache distortion''' 
A mixture of both types, sometimes referred to as ''mustache distortion'' (''moustache distortion'') or ''complex distortion'', is less common but not rare. It starts out as barrel distortion close to the image center and gradually turns into pincushion distortion towards the image periphery, making horizontal lines in the top half of the frame look like a [[handlebar mustache]].
|}

Mathematically, barrel and pincushion distortion are [[quadratic function|quadratic]], meaning they increase as the ''square'' of distance from the center. In mustache distortion the [[quartic function|quartic]] (degree 4) term is significant: in the center, the degree 2 barrel distortion is dominant, while at the edge the degree 4 distortion in the pincushion direction dominates. Other distortions are in principle possible – pincushion in center and barrel at the edge, or higher order distortions (degree 6, degree 8) – but do not generally occur in practical lenses, and higher order distortions are small relative to the main barrel and pincushion effects.

===Occurrence===
[[File:globe effect.gif|thumb|Simulated animation of globe effect (right) compared with a simple pan (left)]]
In photography, distortion is particularly associated with [[zoom lens]]es, particularly large-range zooms, but may also be found in prime lenses, and depends on focal distance – for example, the [[Canon EF 50mm lens|Canon EF 50mm]] {{f/}}1.4 exhibits barrel distortion at extremely short focal distances. Barrel distortion may be found in wide-angle lenses, and is often seen at the wide-angle end of zoom lenses, while telephoto distortion is often seen in older or low-end [[telephoto lens]]es. Mustache distortion is observed particularly on the wide end of some zooms, with certain [[Angenieux retrofocus|retrofocus]] lenses, and more recently on large-range zooms such as the [[Nikon]] 18–200 mm.

A certain amount of pincushion distortion is often found with visual optical instruments, e.g., [[binoculars]], where it serves to eliminate the [[globe effect]].

[[File:Archery Target 80cm.svg|thumb|Radial distortions can be understood by their effect on concentric circles, as in an archery target.]]
In order to understand these distortions, it should be remembered that these are ''radial'' defects; the optical systems in question have [[rotational symmetry]] (omitting non-radial defects), so the [[didactic]]ally correct test image would be a set of [[concentric]] circles having even separation—like a shooter's target. It will then be observed that these common distortions actually imply a nonlinear radius mapping from the object to the image: What is seemingly pincushion distortion, is actually simply an exaggerated radius mapping for large radii in comparison with small radii. A graph showing radius transformations (from object to image) will be steeper in the upper (rightmost) end. Conversely, barrel distortion is actually a diminished radius mapping for large radii in comparison with small radii. A graph showing radius transformations (from object to image) will be less steep in the upper (rightmost) end.

===Chromatic aberration===
{{Further|Chromatic aberration}}
Radial distortion that depends on wavelength is called "[[lateral chromatic aberration]]" – "lateral" because radial, "chromatic" because dependent on color (wavelength). This can cause colored fringes in high-contrast areas in the outer parts of the image. This should not be confused with ''axial'' (longitudinal) chromatic aberration, which causes aberrations throughout the field, particularly [[purple fringing]].

===Origin of terms===
The names for these distortions come from familiar objects which are visually similar.
<gallery widths="150" heights="180">
File:Barrel (PSF).png|In barrel distortion, straight lines bulge ''outwards'' at the center, as in a [[barrel]].
File:Cushion.jpg|In pincushion distortion, corners of squares form elongated points, as in a [[cushion]].
File:Villianc transparent background.svg|In mustache distortion, horizontal lines bulge up in the center, then bend the other way as they approach the edge of the frame (if in the top of the frame), as in curly [[handlebar mustache]]s
</gallery>

==Software correction==
Radial distortion, whilst primarily dominated by low order radial components,<ref name="devilliers">{{cite conference |last1 = de Villiers | first1 = J. P. | last2 = Leuschner | first2 = F.W. | last3 = Geldenhuys| first3 = R. | title = Centi-pixel accurate real-time inverse distortion correction| booktitle = 2008 International Symposium on Optomechatronic Technologies | publisher = SPIE | date = 17–19 November 2008 | url = http://researchspace.csir.co.za/dspace/bitstream/10204/3168/1/De%20Villiers_2008.pdf |doi=10.1117/12.804771}}</ref> can be corrected using Brown's distortion model.<ref name="brown">{{Cite journal| last=Brown |first=Duane C. |title=Decentering distortion of lenses |journal=Photogrammetric Engineering. |volume=32 |issue=3 |pages= 444–462 |date=May 1966 |url=https://eserv.asprs.org/PERS/1966journal/may/1966_may_444-462.pdf}}</ref> Brown's model corrects both for radial distortion and for tangential distortion caused by physical elements in a lens not being perfectly aligned. The latter is also known as ''decentering distortion''. See Zhang <ref name="zhang">{{cite techreport |first=Zhengyou |last=Zhang |title=A Flexible New Technique for Camera Calibration |number=MSR-TR-98-71 |institution=Microsoft Research |year=1998 |url=http://research.microsoft.com/en-us/um/people/zhang/Papers/TR98-71.pdf }}</ref> for radial distortion discussion.

:<math> x_\mathrm{d} = x_\mathrm{u} + x_\mathrm{u}(1 + K_1r^2 + K_2r^4 + \cdots) +
(P_1(r^2 + 2x_\mathrm{u}^2) + 2P_2 x_\mathrm{u}y_\mathrm{u})(1 + P_3r^2 + P_4r^4 \cdots)
</math>

:<math>
y_\mathrm{d} = x_\mathrm{u} + y_\mathrm{u}(1 + K_1r^2 + K_2r^4 + \cdots) +
(P_2(r^2 + 2y_\mathrm{u}^2) + 2P_1 x_\mathrm{u}y_\mathrm{u})(1 + P_3r^2 + P_4r^4 \cdots)
</math>

where:
:<math>(x_\mathrm{d},\ y_\mathrm{d})</math> = distorted image point as projected on image plane using specified lens,
:<math>(x_\mathrm{u},\ y_\mathrm{u})</math> = undistorted image point as projected by an ideal pin-hole camera,
:<math>(x_\mathrm{c},\ y_\mathrm{c})</math> = distortion center (assumed to be the [[principal point]]),
:<math>K_n</math> = <math>n^{\mathrm{th}}</math> radial distortion coefficient,
:<math>P_n</math> = <math>n^{\mathrm{th}}</math> tangential distortion coefficient,
:<math>r</math> = <math>\sqrt{x_\mathrm{u}^2 + y_\mathrm{u}^2}</math>, and
:<math>...</math> = an infinite series.

Barrel distortion typically will have a positive term for <math>K_1</math> whereas pincushion distortion will have a negative value. Moustache distortion will have a non-[[monotonic]] radial [[geometric series]] where for some <math>r</math> the sequence will change sign.

Software can correct those distortions by [[image warping|warping]] the image with a reverse distortion. This involves determining which distorted pixel corresponds to each undistorted pixel, which is non-trivial due to the [[nonlinear|non-linearity]] of the distortion equation.<ref name="devilliers"/> Lateral chromatic aberration (purple/green fringing) can be significantly reduced by applying such warping for red, green and blue separately.

An alternative method iteratively computes the undistorted pixel position.<ref>{{cite web|title=A Four-step Camera Calibration Procedure with Implicit Image Correction |url=http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=609468 |accessdate=19 January 2011}}</ref>

===Calibrated===
Calibrated systems work from a table of lens/camera transfer functions:
* Adobe Photoshop Lightroom and Photoshop CS5 can correct complex distortion.
* PTlens is a Photoshop plugin or standalone application which corrects complex distortion. It not only corrects for linear distortion, but also second degree and higher nonlinear components.<ref>{{cite web |url=http://epaperpress.com/ptlens/ |title=PTlens |accessdate=2 Jan 2012}}</ref>
* [http://lensfun.berlios.de/ Lensfun] is a free to use database and library for correcting lens distortion.<ref>{{cite web |url=http://svn.berlios.de/wsvn/lensfun/trunk/README?rev=246 |title=lensfun - Rev 246 - /trunk/README |accessdate=13 Oct 2013}}</ref>
* [[DxO Labs]]' Optics Pro can correct complex distortion, and takes into account the focus distance.
* The [[Micro Four Thirds system]] cameras and lenses perform automatic distortion correction using correction parameters that are stored in each lens's firmware, and are applied automatically by the camera and RAW converter software. The optics of most of these lenses feature substantially more distortion than their counterparts in systems that don't offer such automatic corrections, but the software-corrected final images show noticeably less distortion than competing designs.<ref>{{cite web|last=Wiley |first=Carlisle |url=http://www.dpreview.com/articles/distortion/ |title=Articles: Digital Photography Review |publisher=Dpreview.com |date= |accessdate=2013-07-03}}</ref>

===Manual===
Manual systems allow manual adjustment of distortion parameters:
* [[Photoshop]] CS2 and [[Photoshop Elements]] (from version 5) include a manual Lens Correction filter for simple (pincushion/barrel) distortion
* [[Corel Paint Shop Pro Photo]] include a manual Lens Distortion effect for simple (barrel, fisheye, fisheye spherical and pincushion) distortion.
* The [[GIMP]] includes manual lens distortion correction (from version 2.4).
* [[PhotoPerfect]] has interactive functions for general pincushion adjustment, and for fringe (adjusting the size of the red, green and blue image parts).
* [[Hugin (software)|Hugin]] can be used to correct distortion, though that is not its primary application.<ref>{{cite web|title=Hugin tutorial – Simulating an architectural projection |url=http://hugin.sourceforge.net/tutorials/architectural/en.shtml |accessdate=9 September 2009}}</ref>

==Related phenomena==
Radial distortion is a failure of a lens to be [[rectilinear lens|rectilinear]]: a failure to image lines into lines. If a photograph is not taken straight-on then, even with a perfect rectilinear lens, rectangles will appear as [[trapezoid]]s: lines are imaged as lines, but the angles between them are not preserved (tilt is not a [[conformal map]]). This effect can be controlled by using a [[perspective control lens]], or [[Perspective control|corrected]] in post-processing.

Due to [[Perspective (visual)|perspective]], cameras image a cube as a square [[frustum]] (a truncated pyramid, with trapezoidal sides)—the far end is smaller than the near end. This creates perspective, and the rate at which this scaling happens (how quickly more distant objects shrink) creates a sense of a scene being deep or shallow. This cannot be changed or corrected by a simple transform of the resulting image, because it requires 3D information, namely the depth of objects in the scene. This effect is known as [[Perspective distortion (photography)|perspective distortion]]; the image itself is not distorted, but is perceived as distorted when viewed from a normal viewing distance.

Note that if the center of the image is closer than the edges (for example, a straight-on shot of a face), then barrel distortion and wide-angle distortion (taking the shot from close) both increase the size of the center, while pincushion distortion and telephoto distortion (taking the shot from far) both decrease the size of the center. However, radial distortion bends straight lines (out or in), while perspective distortion does not bend lines, and these are distinct phenomena. [[Fisheye lens]]es are wide-angle lenses with heavy barrel distortion and thus exhibit ''both'' these phenomena, so objects in the center of the image (if shot from a short distance) are particularly enlarged: even if the barrel distortion is corrected, the resulting image is still from a wide-angle lens, and will still have a wide-angle perspective.

==See also==
* [[Anamorphosis]]
* [[Angle of view]]
* [[Cylindrical perspective]]
* [[Distortion]]
* [[Texture gradient]]
* [[Underwater vision]]
* [[Vignetting]]

==References==
{{Commons category|Distortion}}
<references/>

==External links==
*[http://www.ipol.im/pub/algo/ags_algebraic_lens_distortion_estimation/ Lens distortion estimation and correction] with source code and online demonstration
*[http://gearoracle.com/articles/lens-distortion-correction-on-post-processing/ Lens distortion correction on post-processing]

{{DEFAULTSORT:Distortion (Optics)}}
[[Category:Optics]]

Uncertainty principle

2014-02-03T00:23:00Z

72.172.11.140: /* Robertson–Schrödinger uncertainty relations */ corrected link

{{Use mdy dates|date=October 2012}}
{{Infobox planet
| name = Umbriel
| alt_names = Uranus II
| pronounced = {{IPAc-en|ˈ|ʌ|m|b|r|i|ə|l}} {{respell|UM|bree-əl}}<ref name="dict-def" />
| adjectives = Umbrielian
| image = [[File:PIA00040 Umbrielx2.47.jpg|250px|alt=A round spherical body with its left half illuminated. The surface is dark and has a low contrast. There are only a few bright patches. The terminator is slightly to the right from the center and runs from the top to bottom. A large crater with a bright ring on its floor can be seen at the top of the image near the terminator. A pair of large craters with bright central peaks can be seen along the terminator in the upper part of the body. The illuminated surface is covered by a large number of craters.]]
| caption = Umbriel as seen by ''Voyager 2'' in 1986. At the top is the large crater [[Wunda (crater)|Wunda]], whose walls enclose a ring of bright material.
| discovery = yes
| discoverer = [[William Lassell]]
| discovered = October 24, 1851
| orbit_ref =<ref name="orbit" />
| semimajor = {{val|266000|u=km}}
| eccentricity = {{val|0.0039}}
| period = {{val|4.144|u=[[Day|d]]}}
| inclination = {{val|0.128|s=°}} (to Uranus's equator)
| satellite_of = [[Uranus]]
| physical_characteristics = yes
| mean_radius = {{val|584.7|2.8|u=km}} (0.092 Earths)<ref name="Thomas 1988" />
| surface_area = {{val|4296000|u=km2}} (0.008 Earths){{efn|name=surface area|Surface area derived from the radius ''r'' : <math>4\pi r^2</math>.}}
| volume = {{val|837300000|u=km3}} (0.0008 Earths){{efn|name=volume|Volume ''v'' derived from the radius ''r'' : <math>4\pi r^3/3</math>.}}
| mass = {{val|1.172|0.135|e=21|u=kg}} (2{{Esp|−4}} Earths)<ref name="Jacobson Campbell et al. 1992" />
| density = {{val|1.39|0.16|u=g/cm3}}<ref name="Jacobson Campbell et al. 1992" />
| surface_grav = {{Gr|1|dateform=mdy.172|584.7|2}} [[Acceleration|m/s2]] (~ 0.023 [[g-force|g]]){{efn|name=surface gravity|Surface gravity derived from the mass ''m'', the [[gravitational constant]] ''G'' and the radius ''r'' : <math>Gm/r^2</math>.}}
| escape_velocity = {{V2|1.172|584.7|2}} km/s{{efn|name=escape velocity|Escape velocity derived from the mass ''m'', the [[gravitational constant]] ''G'' and the radius ''r'' : <math>\sqrt{\frac{2Gm}{r}}</math>.}}
| rotation = presumed [[Synchronous rotation|synchronous]]<ref name="Smith Soderblom et al. 1986" />
| axial_tilt = 0<ref name="Smith Soderblom et al. 1986" />
| albedo =
{{plainlist |
* 0.26 (geometrical)
* 0.10 (Bond)<ref name="Karkoschka 2001, Hubble" />
}}
| magnitude = 14.5 (V-band, opposition)<ref name="NASAspp" />
| temp_name1 = solstice<ref name="Grundy Young et al. 2006" />
| mean_temp_1 = ≈ 75 [[Kelvin|K]]
| max_temp_1 = 85 K
| min_temp_1 = ?
| atmosphere = no
| surface_pressure = zero
| note = no
}}

'''Umbriel''' is a [[moons of Uranus|moon of Uranus]] discovered on October 24, 1851, by [[William Lassell]]. It was discovered at the same time as [[Ariel (moon)|Ariel]] and named after a character in [[Alexander Pope]]'s poem ''[[The Rape of the Lock]]''. Umbriel consists mainly of [[ice]] with a substantial fraction of [[rock (geology)|rock]], and may be differentiated into a rocky [[core (geology)|core]] and an icy [[mantle (geology)|mantle]]. The surface is the darkest among Uranian moons, and appears to have been shaped primarily by impacts. However, the presence of canyons suggests early [[Endogeny|endogenic]] processes, and the moon may have undergone an early endogenically driven resurfacing event that obliterated its older surface.

Covered by numerous [[impact crater]]s reaching {{convert|210|km|mi|abbr=on}} in diameter, Umbriel is the second most heavily cratered satellite of Uranus after [[Oberon (moon)|Oberon]]. The most prominent surface feature is a ring of bright material on the floor of [[Wunda (crater)|Wunda]] crater. This moon, like all moons of Uranus, probably formed from an [[accretion disk]] that surrounded the planet just after its formation. The Uranian system has been studied up close only once, by the spacecraft ''[[Voyager 2]]'' in January 1986. It took several images of Umbriel, which allowed mapping of about 40% of the moon’s surface.

== Discovery and name ==
Umbriel, along with another Uranian satellite, [[Ariel (moon)|Ariel]], was discovered by [[William Lassell]] on October 24, 1851.<ref name="Lassell 1851" /><ref name="Lassell, letter 1851" /> Although [[William Herschel]], the discoverer of [[Titania (moon)|Titania]] and [[Oberon (moon)|Oberon]], claimed at the end of the 18th century that he had observed four additional moons of Uranus,<ref name="Herschel 1798" /> his observations were not confirmed and those four objects are now thought to be spurious.<ref name="Struve 1848" />

All of Uranus's moons are named after characters created by [[William Shakespeare]] or [[Alexander Pope]]. The names of all four satellites of Uranus then known were suggested by [[John Herschel]] in 1852 at the request of Lassell.<ref name="Lassell 1852" /> Umbriel is the 'dusky melancholy sprite' in Alexander Pope's ''[[The Rape of the Lock]]'',<ref name="Kuiper 1949" /> and the name suggests the [[Latin]] ''[[umbra]]'', meaning ''shadow''.  The moon is also designated '''Uranus II'''.<ref name="Lassell, letter 1851" />

== Orbit ==
Umbriel orbits Uranus at the distance of about {{convert|266000|km|mi|abbr=on}}, being the third farthest from the planet among its [[moons of Uranus|five major moons]].{{efn|The five major moons are [[Miranda (moon)|Miranda]], [[Ariel (moon)|Ariel]], Umbriel, Titania and Oberon.}} Umbriel's orbit has a small [[orbital eccentricity|eccentricity]] and is [[orbital inclination|inclined]] very little relative to the [[equator]] of Uranus.<ref name="orbit" /> Its [[orbital period]] is around 4.1 Earth days, coincident with its [[rotational period]]. In other words, Umbriel is a [[synchronous orbit|synchronous]] or [[tidally locked]] satellite, with one face always pointing toward its parent planet.<ref name="Smith Soderblom et al. 1986" /> Umbriel's orbit lies completely inside the [[Magnetosphere of Uranus|Uranian magnetosphere]].<ref name="Grundy Young et al. 2006" /> This is important, because the trailing [[Sphere#Hemisphere|hemispheres]] of airless satellites orbiting inside a magnetosphere (like Umbriel) are struck by magnetospheric [[Plasma (physics)|plasma]], which co-rotates with the planet.<ref name="Ness Acuña et al. 1986" /> This bombardment may lead to the darkening of the trailing hemispheres, which is actually observed for all Uranian moons except Oberon (see below).<ref name="Grundy Young et al. 2006" /> Umbriel also serves as a sink of the magnetospheric charged particles, which creates a pronounced dip in energetic particle count near the moon's orbit as observed by ''Voyager 2'' in 1986.<ref name="Krimigis Armstrong et al. 1986" />

Because Uranus orbits the [[Sun]] almost on its side, and its moons orbit in the planet's equatorial plane, they (including Umbriel) are subject to an extreme seasonal cycle. Both northern and southern [[Poles of astronomical bodies#Geographic poles|poles]] spend 42 years in complete darkness, and another 42 years in continuous sunlight, with the sun rising close to the [[zenith]] over one of the poles at each [[solstice]].<ref name="Grundy Young et al. 2006" /> The ''Voyager 2'' flyby coincided with the southern hemisphere's 1986 summer solstice, when nearly the entire northern hemisphere was unilluminated. Once every 42 years, when Uranus has an [[equinox]] and its equatorial plane intersects the Earth, mutual [[occultation]]s of Uranus's moons become possible. In 2007–2008 a number of such events were observed including two occultations of Titania by Umbriel on August 15 and December 8, 2007 as well as of Ariel by Umbriel on August 19, 2007.<ref name="Miller Chanover 2009" /><ref name="Arlot Dumas et al. 2008" />

Currently Umbriel is not involved in any [[orbital resonance]] with other Uranian satellites. Early in its history, however, it may have been in a 1:3 resonance with [[Miranda (moon)|Miranda]]. This would have increased Miranda's orbital eccentricity, contributing to the internal heating and geological activity of that moon, while Umbriel's orbit would have been less affected.<ref name="Tittemore Wisdom 1990" /> Due to Uranus's lower [[oblate spheroid|oblate]]ness and smaller size relative to its satellites, its moons can escape more easily from a mean motion resonance than those of [[Jupiter]] or [[Saturn]]. After Miranda escaped from this resonance (through a mechanism that probably resulted in its anomalously high orbital inclination), its eccentricity would have been damped, turning off the heat source.<ref name="Tittemore Wisdom 1989" /><ref name="Malhotra Dermott 1990" />

== Composition and internal structure ==
Umbriel is the third largest and fourth most massive of Uranian moons.{{efn|Due to the current [[observational error]], it is not yet known for certain whether [[Ariel (moon)|Ariel]] is more massive than Umbriel.<ref name="JPLSSD"/>}} The moon's density is 1.39 g/cm3,<ref name="Jacobson Campbell et al. 1992"/> which indicates that it mainly consists of [[Ice|water ice]], with a dense non-ice component constituting around 40% of its mass.<ref name="Hussmann Sohl et al. 2006"/> The latter could be made of [[rock (geology)|rock]] and [[carbon]]aceous material including heavy [[organic compound]]s known as [[tholin]]s.<ref name="Smith Soderblom et al. 1986" /> The presence of water ice is supported by [[infrared]] [[spectroscopic]] observations, which have revealed [[crystalline]] water ice on the surface of the moon.<ref name="Grundy Young et al. 2006" /> Water ice [[absorption band]]s are stronger on Umbriel's leading hemisphere than on the trailing hemisphere.<ref name="Grundy Young et al. 2006" /> The cause of this asymmetry is not known, but it may be related to the bombardment by charged particles from the [[magnetosphere of Uranus]], which is stronger on the trailing hemisphere (due to the plasma's co-rotation).<ref name="Grundy Young et al. 2006" /> The energetic particles tend to [[sputtering|sputter]] water ice, decompose [[methane]] trapped in ice as [[clathrate hydrate]] and darken other organics, leaving a dark, carbon-rich [[residue (chemistry)|residue]] behind.<ref name="Grundy Young et al. 2006" />

Except for water, the only other compound identified on the surface of Umbriel by the infrared spectroscopy is [[carbon dioxide]], which is concentrated mainly on the trailing hemisphere.<ref name="Grundy Young et al. 2006" /> The origin of the carbon dioxide is not completely clear. It might be produced locally from [[carbonate]]s or organic materials under the influence of the energetic charged particles coming from the magnetosphere of Uranus or the solar [[ultraviolet]] radiation. This hypothesis would explain the asymmetry in its distribution, as the trailing hemisphere is subject to a more intense magnetospheric influence than the leading hemisphere. Another possible source is the [[outgassing]] of the [[Primordial element|primordial]] CO2 trapped by water ice in Umbriel's interior. The escape of CO2 from the interior may be a result of past geological activity on this moon.<ref name="Grundy Young et al. 2006" />

Umbriel may be differentiated into a rocky [[core (geology)|core]] surrounded by an icy [[mantle (geology)|mantle]].<ref name="Hussmann Sohl et al. 2006" /> If this is the case, the radius of the core (317 km) is about 54% of the radius of the moon, and its mass is around 40% of the moon’s mass—the parameters are dictated by the moon's composition. The pressure in the center of Umbriel is about 0.24 [[GPa]] (2.4 [[kbar]]).<ref name="Hussmann Sohl et al. 2006" /> The current state of the icy mantle is unclear, although the existence of a subsurface ocean is considered unlikely.<ref name="Hussmann Sohl et al. 2006" />

== Surface features ==

[[File:Umbriel usgsx2.jpg|thumb|Map of Umbriel showing polygons|alt=A spherical blueish body with its surface covered by craters and polygons. The lower right part is smooth.]]

Umbriel's surface is the darkest of the Uranian moons, and reflects less than half as much light as Ariel, a sister satellite of similar size.<ref name="JPLSSD" /> Umbriel has a very low [[Bond albedo]] of only about 10% as compared to 23% for Ariel.<ref name="Karkoschka 2001, Hubble" /> The reflectivity of the moon's surface decreases from 26% at a phase angle of 0° ([[geometric albedo]]) to 19% at an angle of about 1°. This phenomenon is called [[opposition surge]]. The surface of Umbriel is slightly blue in color,<ref name="Bell McCord 1991" /> while fresh bright impact deposits (in [[Wunda (crater)|Wunda]] crater, for instance)<ref name="Plescia 1987" /> are even bluer. There may be an asymmetry between the leading and trailing hemispheres; the former appears to be redder than the latter.<ref name="Buratti Mosher 1991" /> The reddening of the surfaces probably results from [[space weathering]] from bombardment by charged particles and [[Micrometeoroid|micrometeorites]] over the age of the [[Solar System]].<ref name="Bell McCord 1991" /> However, the color asymmetry of Umbriel is likely caused by accretion of a reddish material coming from outer parts of the Uranian system, possibly, from [[irregular satellite]]s, which would occur predominately on the leading hemisphere.<ref name="Buratti Mosher 1991" /> The surface of Umbriel is relatively homogeneous—it does not demonstrate strong variation in either albedo or color.<ref name="Bell McCord 1991" />

Scientists have so far recognized only one class of geological feature on Umbriel—[[Impact crater|craters]].<ref name="usgs" /> The surface of Umbriel has far more and larger craters than do Ariel and [[Titania (moon)|Titania]] and shows the least geological activity.<ref name="Plescia 1987" /> In fact, among the Uranian moons only Oberon has more impact craters than Umbriel. The observed crater diameters range from a few kilometers at the low end to 210 kilometers for the largest known crater, Wokolo.<ref name="Plescia 1987" /><ref name="usgs" /> All recognized craters on Umbriel have central peaks,<ref name="Plescia 1987" /> but no crater has [[Ray system|rays]].<ref name="Smith Soderblom et al. 1986" />

{| class="wikitable plainrowheaders sortable" style="float: left;"
|+ Named craters on Umbriel<ref name="usgs" />{{efn|name=spirits|Surface features on Umbriel are named for evil or dark spirits taken from various mythologies.<ref name="Strobell & Masursky 1987" />}}
! scope="col" | Crater
! scope="col" | Named after
! scope="col" class="unsortable" | Coordinates
! scope="col" | Diameter (km)
|-
! scope="row" | Alberich
| [[Alberich]] ([[Norse mythology|Norse]])
| {{coord|33.6|S|42.2|E|dim:52.0km_globe:umbriel_type:landmark}}
| 52.0
|-
! scope="row" | Fin
| [[Fin (troll)|Fin]] ([[Danish folklore|Danish]])
| {{coord|37.4|S|44.3|E|dim:43.0km_globe:umbriel_type:landmark}}
| 43.0
|-
! scope="row" | Gob
| Gob ([[Paganism|Pagan]])
| {{coord|12.7|S|27.8|E|dim:88.0km_globe:umbriel_type:landmark}}
| 88.0
|-
! scope="row" | Kanaloa
| [[Kanaloa]] ([[Polynesian mythology|Polynesian]])
| {{coord|10.8|S|345.7|E|dim:86.0km_globe:umbriel_type:landmark}}
| 86.0
|-
! scope="row" | Malingee
| Malingee ([[Australian Aboriginal mythology]])
| {{coord|22.9|S|13.9|E|dim:164.0km_globe:umbriel_type:landmark}}
| 164.0
|-
! scope="row" | Minepa
| [[Minepa]] ([[Makua (people)|Makua]] people of [[Mozambique]])
| {{coord|42.7|S|8.2|E|dim:58.0km_globe:umbriel_type:landmark}}
| 58.0
|-
! scope="row" | Peri
| [[Peri]] ([[Islamic mythology|Persian]])
| {{coord|9.2|S|4.3|E|dim:61.0km_globe:umbriel_type:landmark}}
| 61.0
|-
! scope="row" | Setibos
| [[Setibos]] ([[Patagonia]]n)
| {{coord|30.8|S|346.3|E|dim:50.0km_globe:umbriel_type:landmark}}
| 50.0
|-
! scope="row" | [[Skynd (crater)|Skynd]]
| [[Skynd]] ([[Danish folklore|Danish]])
| {{coord|1.8|S|331.7|E|dim:72.0km_globe:umbriel_type:landmark}}
| 72.0
|-
! scope="row" | [[Vuver (crater)|Vuver]]
| [[Vuver]] ([[Finnish mythology|Finnish]])
| {{coord|4.7|S|311.6|E|dim:98.0km_globe:umbriel_type:landmark}}
| 98.0
|-
! scope="row" | [[Wokolo (crater)|Wokolo]]
| Wokolo ([[Bambara people]] of West Africa)
| {{coord|30|S|1.8|E|dim:208.0km_globe:umbriel_type:landmark}}
| 208.0
|-
! scope="row" | [[Wunda (crater)|Wunda]]
| Wunda (Australian Aboriginal mythology)
| {{coord|7.9|S|273.6|E|dim:131.0km_globe:umbriel_type:landmark}}
| 131.0
|-
! scope="row" | Zlyden
| [[Zlyden]] ([[Slavic mythology|Slavic]])
| {{coord|23.3|S|326.2|E|dim:44.0km_globe:umbriel_type:landmark}}
| 44.0
|}

Near Umbriel's equator lies the most prominent surface feature: Wunda crater, which has a diameter of about 131 km.<ref name="usgsWunda" /><ref name="hunt" /> Wunda has a large ring of bright material on its floor, which appears to be an impact deposit.<ref name="Plescia 1987" /> Nearby, seen along the [[terminator (solar)|terminator]], are the craters [[Vuver (crater)|Vuver]] and [[Skynd (crater)|Skynd]], which lack bright rims but possess bright central peaks.<ref name="Smith Soderblom et al. 1986" /><ref name="hunt" /> Study of limb profiles of Umbriel revealed a possible very large impact feature having the diameter of about 400 km and depth of approximately 5 km.<ref name="Moore Schenk et al. 2004" />

Much like other moons of Uranus, the surface of Umbriel is cut by a system of canyons trending northeast–southwest.<ref name="Croft1989" /> They are not, however, officially recognized due to the poor imaging resolution and generally bland appearance of this moon, which hinders [[Geologic map|geological mapping]].<ref name="Plescia 1987" />

Umbriel's heavily cratered surface has probably been stable since the [[Late Heavy Bombardment]].<ref name="Plescia 1987" /> The only signs of the ancient internal activity are canyons and dark polygons—dark patches with complex shapes measuring from tens to hundreds of kilometers across.<ref name="Helfenstein Thomas et al. 1989" /> The polygons were identified from precise photometry of ''Voyager 2'''s images and are distributed more or less uniformly on the surface of Umbriel, trending northeast–southwest. Some polygons correspond to depressions of a few kilometers deep and may have been created during an early episode of tectonic activity.<ref name="Helfenstein Thomas et al. 1989" /> Currently there is no explanation for why Umbriel is so dark and uniform in appearance. Its surface may be covered by a relatively thin layer of dark material (so called ''[[umbra]]l material'') excavated by an impact or expelled in an explosive volcanic eruption.{{efn|While a co-orbiting population of dust particles is another possible source of the dark material, this is considered less likely because other satellites were not affected.<ref name="Smith Soderblom et al. 1986" />}}<ref name="Buratti Mosher 1991" /> Alternatively, Umbriel's crust may be entirely composed of the dark material, which prevented formation of bright features like crater rays. However, the presence of the bright feature within Wunda seems to contradict this hypothesis.<ref name="Smith Soderblom et al. 1986" />

== Origin and evolution ==
Umbriel is thought to have formed from an [[accretion disc]] or subnebula; a disc of gas and dust that either existed around Uranus for some time after its formation or was created by the giant impact that most likely gave Uranus its large [[Axial tilt|obliquity]].<ref name="Mousis 2004" /> The precise composition of the subnebula is not known; however, the higher density of Uranian moons compared to the [[moons of Saturn]] indicates that it may have been relatively water-poor.{{efn|For instance, [[Tethys (moon)|Tethys]], a Saturnian moon, has a density of 0.97 g/cm3, which suggests that over 90% of its composition is water.<ref name="Grundy Young et al. 2006" />}}<ref name="Smith Soderblom et al. 1986" /> Significant amounts of [[nitrogen]] and [[carbon]] may have been present in the form of [[carbon monoxide]] (CO) and [[nitrogen|molecular nitrogen]] (N2) instead of [[ammonia]] and methane.<ref name="Mousis 2004" /> The moons that formed in such a subnebula would contain less water ice (with CO and N2 trapped as clathrate) and more rock, explaining the higher density.<ref name="Smith Soderblom et al. 1986" />

Umbriel's accretion probably lasted for several thousand years.<ref name="Mousis 2004" /> The impacts that accompanied accretion caused heating of the moon's outer layer.<ref name="Squyres Reynolds et al. 1988" /> The maximum temperature of around 180 K was reached at the depth of about 3 km.<ref name="Squyres Reynolds et al. 1988" /> After the end of formation, the subsurface layer cooled, while the interior of Umbriel heated due to decay of [[radioactivity|radioactive elements]] present in its rocks.<ref name="Smith Soderblom et al. 1986" /> The cooling near-surface layer contracted, while the interior expanded. This caused strong [[Stress (mechanics)|extensional stresses]] in the moon's crust, which may have led to cracking.<ref name="Hillier & Squyres 1991" /> This process probably lasted for about 200 million years, implying that any endogenous activity ceased billions of years ago.<ref name="Smith Soderblom et al. 1986" />

The initial [[accretion (astrophysics)|accretional heating]] together with continued decay of radioactive elements may have led to melting of the ice<ref name="Squyres Reynolds et al. 1988" /> if an [[antifreeze]] like ammonia (in the form of [[hydrate|ammonia hydrate]]) or some salt was present.<ref name="Hussmann Sohl et al. 2006" /> The melting may have led to the separation of ice from rocks and formation of a rocky core surrounded by an icy mantle.<ref name="Plescia 1987" /> A layer of liquid water (ocean) rich in dissolved ammonia may have formed at the core–mantle boundary. The [[eutectic temperature]] of this mixture is 176 K. The ocean, however, is likely to have frozen long ago.<ref name="Hussmann Sohl et al. 2006" /> Among Uranian moons Umbriel was least subjected to endogenic resurfacing processes,<ref name="Plescia 1987" /> although it may like other Uranian moons have experienced a very early resurfacing event.<ref name="Helfenstein Thomas et al. 1989" />

== Exploration ==

{{further|Exploration of Uranus}}

So far the only close-up images of Umbriel have been from the ''[[Voyager 2]]'' probe, which photographed the moon during its flyby of Uranus in January 1986. Since the closest distance between ''Voyager 2'' and Umbriel was {{convert|325000|km|mi|abbr=on}},<ref name="Stone 1987" /> the best images of this moon have a spatial resolution of about 5.2 km.<ref name="Plescia 1987" /> The images cover about 40% of the surface, but only 20% was photographed with the quality required for [[geological mapping]].<ref name="Plescia 1987" /> At the time of the flyby the southern hemisphere of Umbriel (like those of the other moons) was pointed towards the Sun, so the northern (dark) hemisphere could not be studied.<ref name="Smith Soderblom et al. 1986" /> No other spacecraft has ever visited Uranus (and Umbriel), and no mission to Uranus and its moons are planned.

== Notes ==

{{notes
| colwidth = 30em
| notes =

}}

== References ==

{{reflist
| colwidth = 30em
| refs =

<ref name="dict-def">
{{cite web
| title = Umbriel
| publisher = Dictionary.com
| url = http://dictionary.reference.com/browse/Umbriel
| accessdate = 2010-01-14
}}
</ref>

<ref name="Smith Soderblom et al. 1986">
{{cite doi | 10.1126/science.233.4759.43 }}
</ref>

<ref name="Karkoschka 2001, Hubble">
{{cite doi | 10.1006/icar.2001.6596 }}
</ref>

<ref name="Hussmann Sohl et al. 2006">
{{cite doi | 10.1016/j.icarus.2006.06.005 }}
</ref>

<ref name="Jacobson Campbell et al. 1992">
{{cite doi | 10.1086/116211 }}
</ref>

<ref name="Grundy Young et al. 2006">
{{cite doi | 10.1016/j.icarus.2006.04.016 }}
</ref>

<ref name="Squyres Reynolds et al. 1988">
{{cite doi | 10.1029/JB093iB08p08779 }}
</ref>

<ref name="Moore Schenk et al. 2004">
{{cite doi | 10.1016/j.icarus.2004.05.009 }}
</ref>

<ref name="Mousis 2004">
{{cite doi | 10.1051/0004-6361:20031515 }}
</ref>

<ref name="Croft1989">
{{cite conference
| last = Croft
| first = S. K.
| title = New geological maps of Uranian satellites Titania, Oberon, Umbriel and Miranda
| year = 1989
| publisher = Lunar and Planetary Sciences Institute, Houston
| work = Proceeding of Lunar and Planetary Sciences
| volume = 20
| page = 205C
| url = http://adsabs.harvard.edu/abs/1989LPI....20..205C
}}
</ref>

<ref name="Herschel 1798">
{{cite doi | 10.1098/rstl.1798.0005 }}
</ref>

<ref name="Lassell 1851">
{{cite journal
| last = Lassell
| first = W.
| title = On the interior satellites of Uranus
| journal = Monthly Notices of the Royal Astronomical Society
| volume = 12
| year = 1851
| pages = 15–17
| bibcode = 1851MNRAS..12...15L
}}
</ref>

<ref name="Plescia 1987">
{{cite doi | 10.1029/JA092iA13p14918 }}
</ref>

<ref name="Kuiper 1949">
{{cite doi | 10.1086/126146 }}
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<ref name="Ness Acuña et al. 1986">
{{cite doi | 10.1126/science.233.4759.85 }}
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<ref name="orbit">
{{cite web
| title = Planetary Satellite Mean Orbital Parameters
| publisher = Jet Propulsion Laboratory, California Institute of Technology
| url = http://ssd.jpl.nasa.gov/?sat_elem
}}
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<ref name="Lassell, letter 1851">
{{cite doi | 10.1086/100198 }}
</ref>

<ref name="Stone 1987">
{{cite doi | 10.1029/JA092iA13p14873 }}
</ref>

<ref name="Bell McCord 1991">
{{cite conference
| last1 = Bell
| first1 = J. F., III
| last2 = McCord
| first2 = T. B.
| year = 1991
| title = A search for spectral units on the Uranian satellites using color ratio images
| conference = Lunar and Planetary Science Conference, 21st, Mar. 12–16, 1990
| publisher = Lunar and Planetary Sciences Institute
| location = Houston, TX, United States
| format = Conference Proceedings
| pages = 473–489
| bibcode = 1991LPSC...21..473B
}}
</ref>

<ref name="Lassell 1852">
{{cite journal
| last = Lassell
| first = W.
| year = 1852
| language = German
| title = Beobachtungen der Uranus-Satelliten
| journal = Astronomische Nachrichten
| volume = 34
| page = 325
| bibcode = 1852AN.....34..325.
}}
</ref>

<ref name="Hillier & Squyres 1991">
{{cite doi | 10.1029/91JE01401 }}
</ref>

<ref name="Miller Chanover 2009">
{{cite doi | 10.1016/j.icarus.2008.12.010 }}
</ref>

<ref name="Arlot Dumas et al. 2008">
{{cite doi | 10.1051/0004-6361:200810134 }}
</ref>

<ref name="JPLSSD">
{{cite web
| title = Planetary Satellite Physical Parameters
| publisher = Jet Propulsion Laboratory (Solar System Dynamics)
| url = http://ssd.jpl.nasa.gov/?sat_phys_par
| accessdate = 2009-05-28
}}
</ref>

<ref name="Thomas 1988">
{{cite doi | 10.1016/0019-1035(88)90054-1 }}
</ref>

<ref name="Tittemore Wisdom 1990">
{{cite doi | 10.1016/0019-1035(90)90125-S }}
</ref>

<ref name="Tittemore Wisdom 1989">
{{cite doi | 10.1016/0019-1035(89)90070-5 }}
</ref>

<ref name="Malhotra Dermott 1990">
{{cite doi | 10.1016/0019-1035(90)90126-T }}
</ref>

<ref name="Struve 1848">
{{cite journal
| last = Struve
| first = O.
| year = 1848
| title = Note on the Satellites of Uranus
| journal = Monthly Notices of the Royal Astronomical Society
| volume = 8
| issue = 3
| pages = 44–47
| bibcode = 1848MNRAS...8...43.
}}
</ref>

<ref name="Buratti Mosher 1991">
{{cite doi | 10.1016/0019-1035(91)90064-Z }}
</ref>

<ref name="hunt">
{{cite book
| last = Hunt
| first = Garry E.
| coauthors = Patrick Moore
| year = 1989
| title = Atlas of Uranus
| publisher = Cambridge University Press.
| isbn = 978-0-521-34323-7
| url = http://books.google.com/?id=DTc9AAAAIAAJ&pg=PA82&lpg=PA82&dq=Umbriel+crater+Skynd&q=Umbriel%20crater%20Skynd
}}
</ref>

<ref name="Helfenstein Thomas et al. 1989">
{{cite doi | 10.1038/338324a0 }}
</ref>

<ref name="usgsWunda">
{{cite web
| title = Umbriel:Wunda
| publisher = United States Geological Survey, Astrogeology
| work = Gazetteer of Planetary Nomenclature
| url = http://planetarynames.wr.usgs.gov/jsp/FeatureNameDetail.jsp?feature=66756
| accessdate = 2009-08-08
}}
</ref>

<ref name="usgs">
{{cite web
| title = Umbriel Nomenclature Table Of Contents
| publisher = United States Geological Survey, Astrogeology
| work = Gazetteer of Planetary Nomenclature
| url = http://planetarynames.wr.usgs.gov/jsp/FeatureTypes2.jsp?system=Uranus&body=Umbriel&systemID=7&bodyID=36&sort=AName&show=Fname&show=Lat&show=Long&show=Diam&show=Stat&show=Orig
| accessdate = 2009-09-26
}}
</ref>

<ref name="Krimigis Armstrong et al. 1986">
{{cite doi | 10.1126/science.233.4759.97 }}
</ref>

<ref name="NASAspp">
{{cite web
| title = Planetary Satellite Physical Parameters
| publisher = NASA/JPL
| url = http://ssd.jpl.nasa.gov/?sat_phys_par
| accessdate = June 6, 2010
}}
</ref>

<ref name="Strobell & Masursky 1987">
{{cite journal
| last = Strobell
| first = M. E.
| last2 = Masursky
| first2 = H.
|date=March 1987
| title = New Features Named on the Moon and Uranian Satellites
| journal = Abstracts of the Lunar and Planetary Science Conference
| volume = 18
| pages = 964–965
| bibcode = 1987LPI....18..964S
}}
</ref>

}}

== External links ==

{{Commons category}}

* {{cite web | url = http://solarsystem.nasa.gov/planets/profile.cfm?Object=Ura_Umbriel | title = Umbriel Profile | publisher = NASA's Solar System Exploration | accessdate = 2009-10-10 }}
* {{cite journal | bibcode = 1851AN.....33..259L | title = Entdeckung von 2 neuen Uranus Trabanten | volume = 33 | issue = 17 | year = 1852 | pages = 259–262 | first = Herrn W. | last = Lassell | journal = Astronomische Nachrichten | language = German | doi = 10.1002/asna.18520331707 }}
* {{cite web | url = http://www.eso.org/public/news/eso0737/ | title = Edge-on! | publisher = [[Very Large Telescope]] | date = August 23, 2007 | accessdate = 2010-01-14 }}
* [http://www.solarviews.com/eng/umbriel.htm Umbriel page] (including a [http://www.solarviews.com/raw/uranus/umbmap1.jpg labelled map of Umbriel]) at ''Views of the Solar System''
* [http://planetarynames.wr.usgs.gov/Page/UMBRIEL/target Umbriel Nomenclature] from the [http://planetarynames.wr.usgs.gov/ USGS Planetary Nomenclature web site]

{{Uranus}}
{{Moons of Uranus}}
{{Solar System moons (compact)}}

{{Featured article}}

{{DEFAULTSORT:Umbriel (Moon)}}
[[Category:Moons of Uranus]]
[[Category:Astronomical objects discovered in 1851]]
[[Category:Planemos]]
[[Category:Umbriel (moon)| ]]

{{Link FA|fr}}
{{Link GA|ru}}

QED vacuum

2014-02-02T23:50:47Z

72.172.11.140: /* Virtual particles */

'''Photoelectron photoion coincidence spectroscopy''', '''PEPICO''' for short, is a combination of photoionization [[mass spectrometry]] and [[Photoemission spectroscopy|photoelectron spectroscopy]].<ref>{{cite book |last1=Baer | first1=Tomas |last2=Booze | first2=Jon |last3=Weitzel | first3=Karl-Michael | editor-last=Ng |editor-first=Cheuk-Yiu |title=Vacuum Ultraviolet Photoionization and Photodissociation of Molecules and Clusters |publisher=World Scientific Pub Co Inc |date=1991-02 |pages=259–296 |chapter=Photoelectron Photoion Coincidence Studies of Ion Dissociation Dynamics |isbn=981-02-0430-2}}</ref> Gas phase sample, i.e. isolated molecules are ionized by incident vacuum [[ultraviolet]] (VUV) radiation. In the ensuing [[Photoelectrochemical_processes#Photoionization|photoionization]], a [[Ion|cation]] and a photo[[electron]] are formed for each sample molecule. The mass of the photoion is determined by [[time-of-flight mass spectrometry]], whereas, in current setups, photoelectrons are typically [[Photofragment-ion_imaging#Velocity_Map_Imaging|velocity map imaged]] onto a [[Microchannel_plate_detector#Delay_line_detector|position sensitive detector]]. Electron times-of-flight are three orders of magnitude smaller than ion ones, which means that the electron detection can be used as a time stamp for the ionization event, starting the clock for the ion time-of-flight analysis. In contrast with pulsed experiments, such as [[REMPI]], in which the light pulse must act as the time stamp, this allows for the use of continuous light sources, e.g. a [[Gas-discharge lamp|discharge lamp]] or a [[synchrotron]] light source. No more than a few ion/electron pairs are present simultaneously in the instrument, and the electron/ion pairs belonging to a single photoionization event can be identified and detected in delayed coincidence.

== History ==

[[File:IPEPICO by Jonelle Harvey.jpg|thumb|A painting featuring the PEPICO endstation at the [[Swiss Light Source]].]]

Brehm and von Puttkammer published the first PEPICO study on methane in 1967.<ref>{{cite journal |last1=Brehm |first1=B. |last2= von Puttkammer |first2= E. |title=Koinzidensmessung von Photoionen und Photoelektronen bei Methan |journal=Zeitschrift Für Naturforschung. Teil A, Astrophysik, Physik, Physikalische Chemie |volume=22 |year=1967 |issue=1 |page=8}}</ref> In the early works, a fixed energy light source was used, and the electron detection was carried out using retarding grids or [[media:ARPESgeneral.png|hemispherical analyzers]]: the mass spectra were recorded as a function of electron energy. Tunable vacuum ultraviolet light sources have been used in later setups,<ref>{{cite journal |last1=Stockbauer |first1=R. |title=Threshold electron-photoion coincidence mass spectrometric study of CH4, CD4, C2H6, and C2D6 |journal=Journal of Chemical Physics |volume=58 |year=1973 |issue=9 |pages=3800–3815|bibcode = 1973JChPh..58.3800S |doi = 10.1063/1.1679733 }}</ref><ref>{{cite journal |last1=Werner |first1=AS. |last2=Baer | first2=T. |title=Absolute unimolecular decay rates of energy selected C4H6+ metastable ions |journal=Journal of Chemical Physics |volume=62 |year=1975 |issue=7 |pages=2900–2910|bibcode = 1975JChPh..62.2900W |doi = 10.1063/1.430828 }}</ref> in which fixed, mostly zero kinetic energy electrons are detected, and the mass spectra are recorded as a function of photon energy. Detecting zero kinetic energy or threshold electrons in threshold photoelectron photoion coincidence spectroscopy, TPEPICO, has two major advantages. Firstly, no kinetic energy electrons are produced in energy ranges with poor [[Franck–Condon principle|Franck–Condon]] factors in the photoelectron spectrum, but threshold electrons can still be emitted via other ionization mechanisms.<ref>{{cite doi|10.1063/1.445141}}</ref> Second, threshold electrons are stationary and can be detected with higher collection efficiencies, thereby increasing signal levels.

Threshold electron detection was first based line-of-sight, i.e. a small positive field is applied towards the electron detector, and kinetic energy electrons with perpendicular velocities are stopped by small apertures.<ref>{{cite doi|10.1016/0009-2614(69)80174-0}}</ref> The inherent compromise between resolution and collection efficiency was resolved by applying velocity map imaging<ref>{{cite doi|10.1063/1.1148310}}</ref> conditions.<ref>{{cite doi|10.1063/1.1593788}}</ref> Most recent setups offer meV or better (0.1 kJ mol–1) resolution both in terms of photon energy and electron kinetic energy.<ref>{{cite doi|10.1063/1.3079331}}</ref><ref>{{cite doi|10.1063/1.3082016}}</ref>

The 5–20 eV (500–2000 kJ mol–1, [[Wavelength|''λ'']] = 250–60 nm) energy range is of prime interest in [[Valence_electron|valence]] photoionization. Widely tunable light sources are few and far between in this energy range. The only laboratory based one is the H2 discharge lamp, which delivers quasi-continuous radiation up to 14 eV.<ref>{{cite doi|10.1364/AO.10.001904}}</ref> The few high resolution [[laser]] setups for this energy range are not easily tunable over several eV. Currently, VUV [[beamline]]s at third generation [[synchrotron]] light sources are the brightest and most tunable photon sources for valence ionization. The first high energy resolution PEPICO experiment at a synchrotron was the pulsed-field ionization setup at the Chemical Dynamics Beamline of the [[Advanced Light Source]].<ref>{{cite doi|10.1063/1.1150009}}</ref>

== Overview ==

[[File:Scheme - Photoelectron photoion coincidence apparatus.png|thumb|left|Velocity map imaging photoelectron photoion coincidence apparatus. Electrons with different kinetic energies are shown as well as ions with a room temperature kinetic energy distribution.]]

The primary application of TPEPICO is the production of internal [[Photoelectron photoion coincidence spectroscopy#Energy selection|energy selected ions]] to study their [[Unimolecular ion decomposition|unimolecular dissociation dynamics]] as a function of internal energy. The electrons are extracted by a continuous electric field and are e.g. velocity map imaged depending on their initial kinetic energy. Ions are accelerated in the opposite direction and their mass is determined by time-of-flight mass spectrometry. The data analysis yields dissociation thresholds, which can be used to derive new [[Photoelectron photoion coincidence spectroscopy#Thermochemical applications|thermochemistry]] for the sample.<ref>{{cite doi|10.1039/b502051d}}</ref>

The electron imager side can also be used to record photoionization cross sections, photoelectron energy and angular distributions. With the help of circularly polarized light, photoelectron [[circular dichroism]] (PECD) can be studied.<ref>{{cite doi|10.1039/b714095a}}</ref> A thorough understanding of PECD effects could help explain the homochirality of life.<ref>{{cite doi|10.1063/1.2336432}}</ref> Flash pyrolysis can also be used to produce free radicals or intermediates, which are then characterized to complement e.g. combustion studies.<ref>{{cite doi|10.1016/j.ijms.2006.09.023}}</ref><ref>{{cite doi|10.1002/cphc.201000892}}</ref> In such cases, the photoion mass analysis is used to confirm the identity of the radical produced.

Photoelectron photoion coincidence spectroscopy can be used to shed light on reaction mechanisms,<ref>{{cite doi|10.1255/ejms.943}}</ref> and can also be generalized to study double ionization in (photoelectron) photoion photoion coincidence ((PE)PIPICO),<ref>{{cite doi|10.1080/00268978700101421}}</ref> fluorescence using photoelectron photon coincidence (PEFCO),<ref>{{cite doi|10.1016/0301-0104(80)80106-6}}</ref> or photoelectron photoelectron coincidence (PEPECO).<ref>{{cite doi|10.1016/j.chemphys.2003.08.001}}</ref> Ion–electron velocity vector correlation functions can be obtained in double imaging setups, in which the ion detector also delivers position information.<ref>{{cite doi|10.1063/1.1458063}}</ref>

== Energy selection ==

[[File:PEC Scheme.tif|thumb|Potential energy diagram for dissociative photoionization. When only zero kinetic energy electrons are detected, the photon energy above the adiabatic ionization energy is converted into the internal energy of the photoion AB+]]

The relatively low intensity of the ionizing VUV radiation guarantees one-photon processes, in other words only one, fixed energy photon will be responsible for photoionization. The energy balance of photoionization comprises the internal energy and the [[Ionization energy|adiabatic ionization energy]] of the neutral as well as the photon energy, the kinetic energy of the photoelectron and of the photoion. Because only threshold electrons are considered and the [[Momentum#Conservation_of_linear_momentum|conservation of momentum]] holds, the last two terms vanish, and the internal energy of the photoion is known:

<math>E_{\text{int}}^{\text{ion}} = E_{\text{int}}^{\text{neutral}} + h \nu - IE_{\text{ad}}</math>

Scanning the photon energy corresponds to shifting the internal energy distribution of the parent ion. The parent ion sits in a potential energy well, in which the lowest energy exit channel often corresponds to the breaking of the weakest [[chemical bond]], resulting in the formation of a fragment or daughter ion. A mass spectrum is recorded at every photon energy, and the fractional ion abundances are plotted to obtain the breakdown diagram. At low energies no parent ion is energetic enough to dissociate, and the parent ion corresponds to 100% of the ion signal. As the photon energy is increased, a certain fraction of the parent ions (in fact according to the [[cumulative distribution function]] of the neutral internal energy distribution) still has too little energy to dissociate, but some do. The parent ion fractional abundances decrease, and the daughter ion signal increases. At the dissociative photoionization threshold, ''E''0, all parent ions, even the ones with initially 0 internal energy, can dissociate, and the daughter ion abundance reaches 100% in the breakdown diagram.

If the potential energy well of the parent ion is shallow and the complete initial thermal energy distribution is broader than the depth of the well, the breakdown diagram can also be used to determine adiabatic ionization energies.<ref>{{cite doi|10.1021/jp208018r}}</ref>

== Data analysis ==

The data analysis becomes more demanding if there are competing parallel dissociation channels or if the dissociation at threshold is too slow to be observed on the time scale (several μs) of the experiment. In the first case, the slower dissociation channel will appear only at higher energies, an effect called competitive shift, whereas in the second, the resulting kinetic shift means that the fragmentation will only be observed at some excess energy, i.e. only when it is fast enough to take place on the experimental time scale. When several dissociation steps follow sequentially, the second step typically occurs at high excess energies: the system has much more internal energy than needed for breaking the weakest bond in the parent ion. Some of this excess energy is retained as internal energy of the fragment ion, some may be converted into the internal energy of the leaving neutral fragment (invisible to mass spectrometry) and the rest is released as kinetic energy, in that the fragments fly apart at some non-zero velocity.

More often than not, dissociative photoionization processes can be described within a [[Statistical mechanics|statistical]] framework, similarly to the approach used in [[collision-induced dissociation]] experiments. If the [[ergodic hypothesis]] holds, the system will explore each region of the [[phase space]] with a probability according to its volume. A [[transition state]] (TS) can then be defined in the phase space, which connects the dissociating ion with the dissociation products, and the dissociation rates for the slow or competing dissociations can be expressed in terms of the TS phase space volume vs. the total phase space volume. The total phase space volume is calculated in a [[microcanonical ensemble]] using the known energy and the density of states of the dissociating ion. There are several approaches how to define the transition state, the most widely used being [[RRKM theory]]. The unimolecular dissociation [[Reaction rate|rate]] curve as a function of energy, ''k''(''E''), vanishes below the dissociative photoionization energy, ''E''0.<ref>{{cite book |last1= Baer |first1= Tomas |last2= Hase |first2= William L. |title= Unimolecular Reaction Dynamics: Theory and Experiments |publisher= Oxford University Press |year= 1996 |isbn= 0-19-507494-7}}</ref>

Statistical theory can also be used in the microcanonical formalism to describe the excess energy partitioning in sequential dissociation steps, as proposed by Klots<ref>{{cite doi|10.1063/1.1679153}}</ref> for a canonical ensemble. Such a statistical approach was used for more than a hundred systems to determine accurate dissociative photoionization onsets, and derive thermochemical information from them.<ref>{{cite doi|10.1002/jms.1813}}</ref>

== [[Thermochemistry|Thermochemical]] applications ==

Dissociative photoionization processes can be generalized as:

AB + ''hν'' <math>\rightarrow</math> A+ + B + e–

If the [[Standard_enthalpy_of_formation|enthalpies of formation]] of two of the three species are known, the third can be calculated with the help of the dissociative photoionization energy, ''E''0, using [[Hess's law]]. This approach was used, for instance, to determine the enthalpy of formation of the [[Methyl_group#Methyl_cation|methyl ion]], CH3+,<ref>{{cite doi|10.1063/1.480169}}</ref> which in turn was used to obtain the enthalpy of formation of [[iodomethane]], CH3I as 15.23 kJ mol–1, with an uncertainty of only 0.3 kJ mol–1.<ref>{{cite doi|10.1039/b915400k}}</ref>

If different sample molecules produce shared fragment ions, a complete thermochemical chain can be constructed, as was shown for some methyl trihalides,<ref>{{cite doi|10.1021/jp8056459}}</ref> where the uncertainty in e.g. the CHCl2Br, ([[Halomethane|Halon-1021]]) heat of formation was reduced from 20 to 2 kJ mol–1. Furthermore, dissociative photoionization energies can be combined with [[Computational chemistry|calculated]] [[isodesmic reaction]] energies to build thermochemical networks. Such an approach was used to revise primary alkylamine enthalpies of formation.<ref>{{cite doi|10.1021/jp064739s}}</ref>

== References ==
{{Reflist}}

== External links ==
* [http://www.psi.ch/sls/vuv/endstations PEPICO endstation at the Swiss Light Source]
* [http://www.synchrotron-soleil.fr/Soleil/ToutesActualites/2009/DELICIOUS2 DELICIOUS2: a PEPICO experiment at SOLEIL, France]
* [http://www1.pacific.edu/~bsztaray/research/pepico/ PEPICO page at the University of the Pacific]
* [http://physchem.ox.ac.uk/~jhde/techniq.html PEPICO page at Oxford University]


[[Category:Physical_chemistry]]
[[Category:Chemical_kinetics]]
[[Category:Thermochemistry]]

Expectation value (quantum mechanics)

2014-02-02T23:41:12Z

72.172.11.140: Added mention that not only is expectation value not "most probable" but it may have 0 probability of ever occuring.

The '''Levy–Mises equations''' (also called '''flow rules''') describe the relationship between [[Shear stress|stress]] and [[Strain (materials science)|strain]] for an ideal [[Plasticity (physics)|plastic]] [[solid]] where the [[Elasticity (physics)|elastic]] strains are negligible.

The generalized Levy–Mises equation can be written as:

: <math>\frac{\mathbf{d}\varepsilon_1}{\sigma'_1}=\frac{\mathbf{d}\varepsilon_2}
{\sigma'_2}=\frac{\mathbf{d}\varepsilon_3}{\sigma'_3}=\mathbf{d}\lambda</math>

{{DEFAULTSORT:Levy-Mises equations}}
[[Category:Materials science]]
[[Category:Continuum mechanics]]
[[Category:Solid mechanics]]

{{mathapplied-stub}}

Spin–statistics theorem

2014-02-02T23:14:15Z

72.172.11.140: /* General discussion */ added mention of "system"

In the [[theory of computation]], a '''Moore machine''' is a [[finite-state machine]] whose output values are determined solely by its current [[state (computer science)|state]]. This is in contrast to a [[Mealy machine]], whose output values are determined both by its current state and by the values of its inputs. The Moore machine is named after [[Edward F. Moore]], who presented the concept in a 1956 paper, “Gedanken-experiments on Sequential Machines.”<ref name="gedanken">{{cite journal| last=Moore| first=Edward F| title=Gedanken-experiments on Sequential Machines| pages=129–153| journal=Automata Studies,Annals of Mathematical Studies| issue=34| publisher=Princeton University Press| location=Princeton, N.J.| year=1956}}</ref>

== Formal definition ==

A Moore machine can be defined as a [[N-tuple|6-tuple]] ( S, ''S''0, Σ, Λ, ''T'', ''G'' ) consisting of the following:
* a finite set of [[State (computer science)|states]] ( ''S'' )
* a start state (also called initial state) ''S''0 which is an element of (''S'')
* a finite set called the input [[Alphabet (computer science)|alphabet]] ( Σ )
* a finite set called the output [[Alphabet (computer science)|alphabet]] ( Λ )
* a transition [[function (mathematics)|function]] (''T'' : ''S'' × Σ → ''S'') mapping a state and the input alphabet to the next state
* an output function (''G'' : ''S'' → Λ) mapping each state to the output alphabet

A Moore machine can be regarded as a restricted type of [[finite state transducer]].

==Visual representation==

===Table===
[[State transition table]] is a table showing relation between an input and a state.

===Diagram===
The [[state diagram]] for a Moore machine or Moore diagram is a diagram that associates an output value with each state.

==Relationship with Mealy machines==

The difference between Moore machines and [[Mealy machine]]s is that in the latter, the output of a transition is determined by the combination of current [[state (computer science)|state]] and current input. In other words: in a '''[[state diagram|diagram]]'''
* for a Moore machine, each node (state) is labeled with an output value;
* for a Mealy machine, each arc (transition) is labeled with an output value.

Every Moore machine M is equivalent to the Mealy machine with the same states and transitions
and the output function that takes each state-input pair (q,x) to GM(q), where GM is M's output function.

Conversely, every Mealy machine M is equivalent to the Moore machine with as its states all pairs of S × Λ of M, and as its transitions ((q,o),y) → (TM(q,y),GM(q,y)), and as output function (q,o) → o, where TM and GM are M's transition resp. output functions.

==Examples==
Types according to number of inputs/outputs.

===Simple===

Simple Moore machine have one input and one output:
* [[Edge detection|edge detector]] using [[XOR]]
* [[wikibooks:Fractals/Mathematics/group/Binary adding machine|binary adding machine]]
* [[Clocked_sequential_system#Clocked_sequential_system|clocked sequential systems]] (a restricted form of Moore machine where the state changes only when the global clock signal changes)

Most digital electronic systems are designed as [[Clocked_sequential_system#Clocked_sequential_system|clocked sequential systems]]. Clocked sequential systems are a restricted form of Moore machine where the state changes only when the global clock signal changes. Typically the current state is stored in [[Flip-flop (electronics)|flip-flops]], and a global clock signal is connected to the "clock" input of the flip-flops. Clocked sequential systems are one way to solve [[Metastability in electronics|metastability]] problems. A typical electronic Moore machine includes a [[combinational logic]] chain to decode the current state into the outputs (lambda). The instant the current state changes, those changes ripple through that chain, and almost instantaneously the output gets updated. There are design techniques to ensure that no [[glitch]]es occur on the outputs during that brief period while those changes are rippling through the chain, but most systems are designed so that glitches during that brief transition time are ignored or are irrelevant. The outputs then stay the same indefinitely ([[LED]]s stay bright, power stays connected to the motors, [[solenoid]]s stay energized, etc.), until the Moore machine changes state again.

<gallery>
File:Moore-Automat-en.svg|Moore machine in combinational logic

</gallery>

===Complex===
More complex Moore machines can have multiple inputs as well as multiple outputs.

== Gedanken-experiments ==

In Moore's paper "[[Thought experiment|Gedanken-experiments]] on Sequential Machines",<ref name="gedanken"/> the '''(n;m;p)''' automata (or machines) '''S''' are defined as having '''n''' states, '''m''' input symbols and '''p''' output symbols. Nine theorems are proved about the structure of '''S''', and experiments with '''S'''. Later, '''''S''' machines'' became known as ''Moore machines''.

At the end of the paper, in Section '''Further problems''', the following task is stated: ''Another directly following problem is the improvement of the bounds given at the theorems 8 and 9''.

Moore's '''Theorem 8''' is formulated as:
''Given an arbitrary '''(n;m;p)''' machine '''S''', such that every two of its states are distinguishable from one another, then there exists an experiment of length '''<math>\frac{n(n-1)}{2}</math>'''
''which determines the state of '''S''' at the end of the experiment.''

In 1957, [[Anatolii Alexeevitch Karatsuba|A. A. Karatsuba]] proved the following two theorems, which completely solved Moore's problem on the improvement of the bounds of the experiment length of his '''Theorem 8'''.

'''Theorem ''A''.'''
''If '''S''' is an '''(n;m;p)''' machine, such that every two of its states are distinguishable from one another, then there exists a branched experiment of length at most '''<math>\frac{(n-1)(n-2)}{2} + 1</math>''' through which one may determine the state of '''S''' at the end of the experiment.''

'''Theorem ''B''.'''
''There exists an '''(n;m;p)''' machine, every two states of which are distinguishable from one another, such that the length of the shortest experiments establishing the state of the machine at the end of the experiment is equal to '''<math>\frac{(n-1)(n-2)}{2} + 1</math> '''.''

Theorems '''A''' and '''B''' were used for the basis of the course work of a student of the fourth year, [[Anatolii Alexeevitch Karatsuba|A. A. Karatsuba]], '''"On a problem from the automata theory"''' which was distinguished by testimonial
reference at the competition of student works of the faculty of mechanics and mathematics of Moscow Lomonosow State University in 1958. The paper by [[Anatolii Alexeevitch Karatsuba|A. A. Karatsuba]] was given to the journal Uspekhi Mat. Nauk on 17 December 1958 and was published there in June 1960.<ref>{{cite journal| last=Karatsuba| first=A. A.| title=Solution of one problem from the theory of finite automata | pages=157–159| journal=Usp. Mat. Nauk| issue=15:3| year=1960}}</ref>

Until the present day (2011), Karatsuba's result on the length of experiments is the only exact nonlinear result, both in automata theory, and in similar problems of computational complexity theory.

== See also ==
* [[Synchronous circuit]]
* [[Mealy machine]]
* [[Algorithmic State Machine]]

== References ==
{{reflist}}

== Further reading ==
* Conway, John Horton (1971): Regular Algebra and Finite Machines, Chapman and Hall.
* Moore E. F. Gedanken-experiments on Sequential Machines. Automata Studies, Annals of Mathematical Studies, 34, 129–153. Princeton University Press, Princeton, N.J.(1956).
* Karatsuba A. A. Solution of one problem from the theory of finite automata. Usp. Mat. Nauk, 15:3, 157–159 (1960).
* Karacuba A. A. Experimente mit Automaten (German) Elektron. Informationsverarb. Kybernetik, 11, 611–612 (1975).
* Karatsuba A. A. ''[http://www.mi.ras.ru/~karatsuba/list_e.html List of research works]''

[[Category:Models of computation]]

Talk:Invariant mass

2013-05-31T01:11:11Z

72.172.11.228: /* Edit needed - rest mass is NOT invariant mass */ new section

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