formulasearchengine - User contributions [en]

Spin magnetic moment

2015-01-12T11:57:35Z

193.231.19.53: /* Selected books */ isbn

Nice to meet you, my name is Refugia. Body developing is one of the issues I adore most. Minnesota is where he's been residing for years. For many years he's been operating as a receptionist. Also visit my web page - home std test ([http://mcb-law.net/candidiasis-tips-you-have-to-remember/ mouse click the following web page])

Electron magnetic dipole moment

2015-01-12T11:05:40Z

193.231.19.53: /* Bibliography */ added ref

[http://authoritynutrition.com/how-to-lose-weight-as-fast-as-possible/ authoritynutrition.com]Do you need a low-cost technique for losing weight that really works? Searching for one of the best programs around where you may drop over a pound or two every week? Fat Loss 4 Idiots is gaining floor in communities throughout the world. People are knowing the pros attained while using this system. So, is there a plan all about? This program is located following several distinct guidelines. The primary internet site works on a foods generation resource to decide on what food you are able to eat. Men and women eat each day, four to five periods. Enjoying comes about right up until completely satisfied, and never entire or overstuffed. The diet system happens for 11 days, accompanied by a 3 morning break. On the split, men and women can actually eat something they demand. Other principle includes walking some each day. This program will teach the individual the best way to consume properly letting our bodies to lose weight appropriately. Fat Loss 4 Idiots also trains the owner maintaining the losses occurring. Using the master plan frequently, the entire body continues to drop the load weekly. What types of foods are ingested on the 11 times? - Several different vegatables and fruits - Meats in trim shape - Excess fat that comes in whole grain food items - Very little starchy carbohydrates The blueprint lets anyone to enjoy a minimum of 4 times on a daily basis. Enough time span between foods is most optimum at two and fifty percent several hours somewhere between. The diet plan permits our bodies to feel like it is actually staying fed regularly, but not famished. Exactly what are the software advantages? - No keeping track of carbs or excess calories - Take in until finally pleased - Portion sizes are certainly not very little - A few day time cheat enabled between cycles - Body system is not going to are aware of it is inside significant burn option - Choose from a listing of desired foods - Intensive facts permitting the user to completely see the process - Vegan ideas accessible on top of that - Functions moving calories process The Fat Loss 4 Idiots method is easy to follow along with. Light exercise is included in the blueprint to help the body consistently burn up the body weight out of. This system is absolutely not strict for your user, making use of common sense in meals alternatives for the entire day. Every day food can be purchased out of your regional grocery store. The plan is rather easy for a person to comply with. Why does the master plan do the job so well for users? The plan does not require the specific to number unhealthy calories, carbs, or excess fat gr. The program readies the eating plan in your case based on the chosen possibilities designed. Customers usually do not worry about obtaining unique eating habits foods. Consumers have tried it and located remarkable outcomes, because the strategy is extremely inexpensive. Persons feel happy when slimming down. Their health are not robbed of simple carbohydrate food, or intake of body fat. The software program fat loss 4 idiots - [http://fatloss4idiots.customerreviewshq.com/fat-loss-4-idiots-review/ fatloss4idiots.customerreviewshq.com] - educates the patient the best time to actually eat what forms of meals. The plan actually works with human hormones in the body that will help to lose excess weight. Who are able to makes use of the Fat Loss 4 Idiots strategy? - Over weight most women - Fat guys - Overweight young people - People with a medical history of ingesting problems - These having to lose smaller sized amounts of body weight Nearly now you can make use of the program to help the body to effectively lose weight. Individuals who have put into practice the program feel better about their selves, get words of flattery rather then snide remarks, and look good. As with any important weight reducing system, it is important to achieve a health care real prior to starting dieting. This is applicable to any kind of losing weight regimen. If you are dealing with any special sickness, this is especially true. What are the issues to the method? - You might need to buy new clothing much more frequently - A number of people might not realize you just after giving up greater numbers of pounds - If foods are banned in the work environment, you might need to change the planned plan - Many people can find the 11 day stretch challenging to maintain, based upon their will ability to shed pounds How come weight reduction crucial? Weight loss is extremely important for the obese man or woman. Any additional excess weight taken on your body causes it to become more difficult for your system to function. Oftentimes excess weight relates to cardiovascular disease and also diabetes. Other health issues could happen resulting from excessive weight. Additionally, magnify the situation. A nutritious body weight is much better for those individual. He will feel much better, be able to work more effective in physical form and psychologically. The electricity gained with shedding pounds will enable the patient so that you can do a lot more points in everyday life. Clothes will probably be easier to find and most importantly anyone are going to be more healthy after some time. Ratings for fat loss 4 idiots - [http://fatloss4idiots.customerreviewshq.com/fat-loss-4-idiots-review/ fatloss4idiots.customerreviewshq.com] - Quite a few ratings are accessible indicating end user final results. Unique remarks show that almost all folks who take advantage of the system can reduce the load effortlessly. This software permits unique nutritional desires such as vegetarian, and so forth. Plenty of men and women may use the cheat days or weeks, however not overdo it in consuming throughout these times. System readers go to a significant decline in belly fat when using the eating plan. The math is taken out of the diet program world, by lacking to number energy eaten in one day. Pen and pieces of paper will not be found it necessary to collection the amount of carbs used, neither excess fat grams. This software does the majority of the meet your needs exactly and will help within the setting up method. Compact substitutions are made by a lot of individuals that has not affected the number of weightloss. For instance, a single particular may have a boiled egg cell incorporated into food for lunchtime time. She truly does unlike boiled ovum, and determines to eat a scrambled ovum instead with no interfering inside the weight reduction never-ending cycle. Fat Loss 4 Idiots fees in at 9.5 outside of 10 details. This is certainly purely for the belief that an 11 day time stretch might be hard for a few end users. Should they could use exactly the same ideas to create a 5 moment extend, followed by a one cheat moment, it is always good. This might be much better for individuals who have a problem with the hold up quickly. This is usually a good system and is also truly worth your smaller expense. Internet message boards of recent individuals are revealing to the storyplot of being successful. Provide the solution a go currently and commence your trip to fat loss!

Philip Warren Anderson

2015-01-12T10:37:54Z

193.231.19.53: /* Biography */ [[]]

A lagging computer is absolutely annoying and is quite a headache. Almost every individual that utilizes a computer faces this problem several time or the different. If the computer equally suffers from the same issue, there are it difficult to continue working as normal. In such a situation, the thought, "what should I do to create my PC run quicker?" is recurring plus infuriating. There's a solution, though! If you registry gets cluttered up with a lot of junk we don't use, the PC may run slower. Therefore it's prudent which you regularly get a registry cleaned. One of the most overlooked factors a computer usually slow down is considering the registry has become corrupt. The registry is essentially your computer's operating system. Anytime you're running a computer, entries are being made and deleted from the registry. The impact this has is it leaves false entries inside a registry. So, your computer's resources should function about these false entries. Always see to it which you have installed antivirus, anti-spyware plus anti-adware programs plus have them updated on a regular basis. This can help stop windows XP running slow. So to fix this, we only have to be capable to create all of the registry files non-corrupted again. This might dramatically speed up the loading time of the computer and will allow we to a big number of things on it again. And fixing these files couldn't be easier - you only require to employ a tool called a [http://bestregistrycleanerfix.com/tune-up-utilities tuneup utilities]. Let's begin with all the bad sides initially. The initial cost of the product is truly inexpensive. But, it only comes with 1 year of updates. After which you need to subscribe to monthly updates. The advantage of which is the fact that ideal optimizer has enough money plus resources to analysis errors. This way, you may be ensured of safe fixes. The disk requires area inside order to run smoothly. By freeing up some space from the disk, you'll be capable to speed up the PC a bit. Delete all file inside the temporary internet files folder, recycle bin, well-defined shortcuts plus icons from your desktop that we never utilize and remove programs we never use. Another important system you'll like to receive is a registry cleaner. The registry is a big list of everything installed on a computer, and Windows references it whenever it opens a program or uses a device attached to your computer. Whenever you delete a program, its registry entry should moreover be deleted, however, often it's not. A registry cleaner can receive rid of these older entries thus Windows may search the registry faster. It also deletes or corrects any entries that viruses have corrupted.

Anomalous magnetic dipole moment

2014-11-27T10:37:49Z

193.231.19.53: /* Notes */ bibref

Nice to satisfy you, I am Marvella Shryock. Doing ceramics is what her family and her enjoy. Years ago we moved to North Dakota. Bookkeeping is her working day occupation now. My weblog :: std testing at home ([http://www.gaysphere.net/blog/297540 simply click the following web site])

Ion-association

2014-10-14T09:42:31Z

193.231.19.53: /* External links */ book

My name is Gregorio and I am studying Theatre and English Literature at Offenbach Am Main / Germany. Feel free to surf to my webpage; [http://www.boardgamemeetup.com/profile_info.php?ID=25647 почему бы не узнать больше]

Orbit equation

2014-06-24T08:41:36Z

193.231.19.53: /* See also */ vel

Greetings. Allow me begin by telling you the author's name - Phebe. The std testing at home preferred hobby for my kids and me is to play baseball and I'm attempting to make it a occupation. I used to be unemployed but now I am a [http://www.idph.state.il.us/public/hb/hbcrabs.htm librarian] and the wage has std home test been really satisfying. California is exactly where her house is but she needs to transfer std testing at home because of her family members. Also visit my weblog - [http://www.rachat-points.com/node/5780286 std home test]

Nuclear magnetic moment

2014-01-14T12:03:16Z

193.231.19.53: /* See also */ proton spin

{{redirect|Convergent Series|the short story collection|Convergent Series (short story collection)}}
In [[mathematics]], a '''series''' is the [[summation|sum]] of the terms of a [[sequence]] of numbers.

Given a sequence <math>\left \{ a_1,\ a_2,\ a_3,\dots \right \}</math>, the ''n''th [[partial sum]] <math>S_n</math> is the sum of the first ''n'' terms of the sequence, that is,

:<math>S_n = \sum_{k=1}^n a_k.</math>

A [[series (mathematics)|series]] is [[limit of a sequence|convergent]] if the sequence of its partial sums <math>\left \{ S_1,\ S_2,\ S_3,\dots \right \}</math> converges; in other words, it approaches a given number. In more formal language, a series converges if there exists a [[Limit of a sequence|limit]] <math>\ell</math> such that for any arbitrarily small positive number <math>\varepsilon > 0</math>, there is a large [[integer]] <math>N</math> such that for all <math>n \ge \ N</math>,

:<math>\left | S_n - \ell \right \vert \le \ \varepsilon.</math>

A series that is not convergent is said to be [[Divergent series|divergent]].

== Examples of convergent and divergent series ==

* The reciprocals of the [[natural number|positive integers]] produce a [[divergent series]] ([[harmonic series (mathematics)|harmonic series]]):
*: <math>{1 \over 1}+{1 \over 2}+{1 \over 3}+{1 \over 4}+{1 \over 5}+{1 \over 6}+\cdots \rightarrow \infty. </math>
* Alternating the signs of the reciprocals of positive integers produces a convergent series:
*: <math>{1\over 1} -{1\over 2} + {1\over 3} - {1\over 4} + {1\over 5} \cdots = \ln(2)</math>
* Alternating the signs of the reciprocals of positive [[parity (mathematics)|odd integers]] produces a convergent series (the [[Leibniz formula for pi]]):
*: <math>{1 \over 1}-{1 \over 3}+{1 \over 5}-{1 \over 7}+{1 \over 9}-{1 \over 11}+\cdots = {\pi \over 4}.</math>
* The reciprocals of [[prime number]]s produce a divergent series (so the set of primes is "[[Small set (combinatorics)|large]]"):
*: <math>{1 \over 2}+{1 \over 3}+{1 \over 5}+{1 \over 7}+{1 \over 11}+{1 \over 13}+\cdots \rightarrow \infty.</math>
* The reciprocals of [[triangular number]]s produce a convergent series:
*: <math>{1 \over 1}+{1 \over 3}+{1 \over 6}+{1 \over 10}+{1 \over 15}+{1 \over 21}+\cdots = 2.</math>
* The reciprocals of [[factorial]]s produce a convergent series (see [[e (mathematical constant)|e]]):
*: <math>\frac{1}{1} + \frac{1}{1} + \frac{1}{2} + \frac{1}{6} + \frac{1}{24} + \frac{1}{120} + \cdots = e.</math>
* The reciprocals of [[square number]]s produce a convergent series (the [[Basel problem]]):
*: <math>{1 \over 1}+{1 \over 4}+{1 \over 9}+{1 \over 16}+{1 \over 25}+{1 \over 36}+\cdots = {\pi^2 \over 6}.</math>
* The reciprocals of [[power of two|powers of 2]] produce a convergent series (so the set of powers of 2 is "[[Small set (combinatorics)|small]]"):
*: <math>{1 \over 1}+{1 \over 2}+{1 \over 4}+{1 \over 8}+{1 \over 16}+{1 \over 32}+\cdots = 2.</math>
* Alternating the signs of reciprocals of [[power of two|powers of 2]] also produces a convergent series:
*: <math>{1 \over 1}-{1 \over 2}+{1 \over 4}-{1 \over 8}+{1 \over 16}-{1 \over 32}+\cdots = {2\over3}.</math>
* The reciprocals of [[Fibonacci number]]s produce a convergent series (see [[reciprocal Fibonacci constant|ψ]]):
*: <math>\frac{1}{1} + \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{5} + \frac{1}{8} + \cdots = \psi.</math>

== Convergence tests ==
{{main|Convergence tests}}

There are a number of methods of determining whether a series converges or [[divergent series|diverges]].
[[Image:Comparison test series.svg|thumb|250px|right|If the blue series, <math>\Sigma b_n</math>, can be proven to converge, then the smaller series, <math>\Sigma a_n</math> must converge. By contraposition, if the red series, <math>\Sigma a_n</math> is proven to diverge, then <math>\Sigma b_n</math> must also diverge.]]
'''[[Comparison test]]'''. The terms of the sequence <math>\left \{ a_n \right \}</math> are compared to those of another sequence <math>\left \{ b_n \right \}</math>. If,

for all ''n'', <math>0 \le \ a_n \le \ b_n</math>, and <math>\sum_{n=1}^\infty b_n</math> converges, then so does <math>\sum_{n=1}^\infty a_n.</math>

However, if,

for all ''n'', <math>0 \le \ b_n \le \ a_n</math>, and <math>\sum_{n=1}^\infty b_n</math> diverges, then so does <math>\sum_{n=1}^\infty a_n.</math>

'''[[Ratio test]]'''. Assume that for all ''n'', <math>a_n > 0</math>. Suppose that there exists <math>r</math> such that

:<math>\lim_{n \to \infty} |{\frac{a_{n+1}}{a_n}}| = r.</math>

If ''r'' < 1, then the series converges. If {{nowrap|''r'' > 1,}} then the series diverges. If {{nowrap|1=''r'' = 1,}} the ratio test is inconclusive, and the series may converge or diverge.

'''[[Root test]]''' or ''' ''n''th root test'''. Suppose that the terms of the sequence in question are [[non-negative]]. Define ''r'' as follows:

:<math>r = \limsup_{n\rightarrow\infty}\sqrt[n]{|a_n|},</math>

:where "lim sup" denotes the [[limit superior]] (possibly ∞; if the limit exists it is the same value).

If ''r'' < 1, then the series converges. If {{nowrap|''r'' > 1,}} then the series diverges. If {{nowrap|1=''r'' = 1,}} the root test is inconclusive, and the series may converge or diverge.

The ratio test and the root test are both based on comparison with a geometric series, and as such they work in similar situations. In fact, if the ratio test works (meaning that the limit exists and is not equal to 1) then so does the root test; the converse, however, is not true. The root test is therefore more generally applicable, but as a practical matter the limit is often difficult to compute for commonly seen types of series.

'''[[Integral test for convergence|Integral test]]'''. The series can be compared to an integral to establish convergence or divergence. Let <math>f(n) = a_n</math> be a positive and [[monotonic function|monotone decreasing function]]. If

:<math>\int_{1}^{\infty} f(x)\, dx = \lim_{t \to \infty} \int_{1}^{t} f(x)\, dx < \infty,</math>

then the series converges. But if the integral diverges, then the series does so as well.

'''[[Limit comparison test]]'''. If <math>\left \{ a_n \right \}, \left \{ b_n \right \} > 0</math>, and the limit <math>\lim_{n \to \infty} \frac{a_n}{b_n}</math> exists and is not zero, then <math>\sum_{n=1}^\infty a_n</math> converges [[if and only if]] <math>\sum_{n=1}^\infty b_n</math> converges.

'''[[Alternating series test]]'''. Also known as the ''Leibniz criterion'', the [[alternating series test]] states that for an [[alternating series]] of the form <math>\sum_{n=1}^\infty a_n (-1)^n</math>, if <math>\left \{ a_n \right \}</math> is monotone [[decreasing]], and has a limit of 0 at infinity, then the series converges.

'''[[Cauchy condensation test]]'''. If <math>\left \{ a_n \right \}</math> is a positive monotone decreasing sequence, then
<math> \sum_{n=1}^\infty a_n </math> converges if and only if <math> \sum_{k=1}^\infty 2^k a_{2^{k}} </math> converges.

'''[[Dirichlet's test]]'''

'''[[Abel's test]]'''

'''[[Raabe's test]]'''

== Conditional and absolute convergence ==

[[Image:ExpConvergence.gif|thumb|right|Illustration of the absolute convergence of the power series of Exp[''z''] around 0 evaluated at {{nowrap|1=''z'' = Exp[{{frac|i|3}}]}}. The length of the line is finite.]]

[[Image:LogConvergenceAnim.gif|thumb|right|Illustration of the conditional convergence of the power series of log(''z''+1) around 0 evaluated at {{nowrap|1=''z'' = exp((π−{{frac|1|3}})''i'').}} The length of the line is infinite.]]

For any sequence <math>\left \{ a_1,\ a_2,\ a_3,\dots \right \}</math>, <math>a_n \le \ \left | a_n \right \vert</math> for all n. Therefore,

:<math>\sum_{n=1}^\infty a_n \le \ \sum_{n=1}^\infty \left | a_n \right \vert.</math>

This means that if <math>\sum_{n=1}^\infty \left | a_n \right \vert</math> converges, then <math>\sum_{n=1}^\infty a_n</math> also converges (but not vice-versa).

If the series <math>\sum_{n=1}^\infty \left | a_n \right \vert</math> converges, then the series <math>\sum_{n=1}^\infty a_n</math> is [[absolutely convergent]]. An absolutely convergent sequence is one in which the length of the line created by joining together all of the increments to the partial sum is finitely long. The power series of the [[exponential function]] is absolutely convergent everywhere.

If the series <math>\sum_{n=1}^\infty a_n</math> converges but the series <math>\sum_{n=1}^\infty \left | a_n \right \vert</math> diverges, then the series <math>\sum_{n=1}^\infty a_n</math> is [[conditionally convergent]]. The path formed by connecting the partial sums of a conditionally convergent series is infinitely long. The power series of the [[logarithm]] is conditionally convergent.

The [[Riemann series theorem]] states that if a series converges conditionally, it is possible to rearrange the terms of the series in such a way that the series converges to any value, or even diverges.

== Uniform convergence ==

{{Main|uniform convergence}}

Let <math>\left \{ f_1,\ f_2,\ f_3,\dots \right \}</math> be a sequence of functions.
The series <math>\sum_{n=1}^\infty f_n</math> is said to converge uniformly to ''f''
if the sequence <math>\{s_n\}</math> of partial sums defined by

: <math> s_n(x) = \sum_{k=1}^n f_k (x)</math>

converges uniformly to ''f''.

There is an analogue of the comparison test for infinite series of functions called the [[Weierstrass M-test]].

== Cauchy convergence criterion ==

The '''[[Cauchy's convergence test|Cauchy convergence criterion]]''' states that a series
:<math>\sum_{n=1}^\infty a_n</math>
converges [[if and only if]] the sequence of [[partial sum]]s is a [[Cauchy sequence]].
This means that for every <math> \varepsilon > 0, </math> there is a positive integer <math>N</math> such that for all <math>n \geq m \geq N</math> we have
:<math> \left| \sum_{k=m}^n a_k \right| < \varepsilon, </math>
which is equivalent to
:<math>\lim_{n \to \infty \atop m\to \infty} \sum_{k=n}^{n+m} a_k = 0.</math>

== See also==
* [[Convergent sequence]]
* [[Normal convergence]]
* [[List of mathematical series]]
* [[Divergent series]]

==External links==
* {{springer|title=Series|id=p/s084670}}
* Weisstein, Eric (2005). [http://mathworld.wolfram.com/RiemannSeriesTheorem.html Riemann Series Theorem]. Retrieved May 16, 2005.

{{Series (mathematics)}}

[[Category:Mathematical series]]
[[Category:Convergence (mathematics)]]

[[de:Konvergenzkriterium]]
[[sv:Konvergens (matematik)]]

Electric field gradient

2014-01-14T11:29:42Z

193.231.19.53: [[]]

{{Other uses}}
In [[computer science]] and [[computer programming]], a '''continuation''' is an [[Abstraction (computer science)|abstract representation]] of the [[Control flow|control state]] of a [[computer program]]. A continuation [[Reification (computer science)|reifies]] the program control state, i.e. the continuation is a data structure that represents the computational process at a given point in the process' execution; the created data structure can be accessed by the programming language, instead of being hidden in the [[Run-time system|runtime environment]]. Continuations are useful for encoding other control mechanisms in programming languages such as [[Exception handling|exception]]s, [[Generator (computer science)|generators]], [[coroutine]]s, and so on.

The "'''current continuation'''" or "continuation of the computation step" is the continuation that, from the perspective of running code, would be derived from the current point in a program's execution. The term ''continuations'' can also be used to refer to '''first-class continuations''', which are constructs that give a [[programming language]] the ability to save the execution state at any point and return to that point at a later point in the program.

==History==
The earliest description of continuations was made by [[Adriaan van Wijngaarden]] in September 1964. Wijngaarden spoke at the IFIP Working Conference on Formal Language Description Languages held in Baden bei Wien, Austria. As part of a formulation for an [[Algol 60]] preprocessor, he called for a transformation of proper procedures into continuation-passing style.<ref name="history_of_continuations">{{harvnb|Reynolds|1993}}</ref>

[[Christopher Strachey]], [[Christopher P. Wadsworth]] and [[John C. Reynolds]] brought the term ''continuation'' into prominence in their work in the field of [[denotational semantics]] that makes extensive use of continuations to allow sequential programs to be analysed in terms of [[functional programming]] semantics.<ref name="history_of_continuations"/>

[[Steve Russell]] invented the continuation in his second [[Lisp (programming language)|Lisp]] implementation for the [[IBM 704]], though he did not name it.<ref>{{cite web | url = http://www.computernostalgia.net/articles/steveRussell.htm | title = Steve "Slug" Russell | work = Computer History}}</ref> However there is no cross references about that and the given reference is a mere claim.

A complete history of the discovery of continuations is given by {{harv|Reynolds|1993}}.

==First-class continuations==


First-class continuations are a language's ability to completely control the execution order of instructions. They can be used to jump to a function that produced the call to the current function, or to a function that has previously exited. One can think of a first-class continuation as saving the state of the program. However, it is important to note that true first-class continuations do not save program data, only the execution context. This is illustrated by the "continuation sandwich" description:

<blockquote>
''Say you're in the kitchen in front of the refrigerator, thinking about a sandwich. You take a continuation right there and stick it in your pocket. Then you get some turkey and bread out of the refrigerator and make yourself a sandwich, which is now sitting on the counter. You invoke the continuation in your pocket, and you find yourself standing in front of the refrigerator again, thinking about a sandwich. But fortunately, there's a sandwich on the counter, and all the materials used to make it are gone. So you eat it. :-)''<ref name="cont-sandwich">{{cite web|url=http://groups.google.com/group/perl.perl6.language/msg/b0cfa757f0ce1cfd|title=undo()? ("continuation sandwich" example)|last=Palmer|first=Luke|date=June 29, 2004|work=perl.perl6.language (newsgroup)|accessdate=2009-10-04}}</ref>
</blockquote>

[[Scheme (programming language)|Scheme]] was the first full production system, providing first "catch"<ref name="history_of_continuations"/> and then [[call-with-current-continuation|call/cc]]. Bruce Duba introduced call/cc into [[Standard ML|SML]].

Continuations are also used in models of computation including [[denotational semantics]], the [[Actor model]], [[process calculi]], and [[lambda calculus]]. These models rely on programmers or semantics engineers to write mathematical functions in the so-called [[continuation-passing style]]. This means that each function consumes a function that represents the rest of the computation relative to this function call. To return a value, the function calls this "continuation function" with a return value; to abort the computation it returns a value.

Functional programmers who write their programs in [[continuation-passing style]] gain the expressive power to manipulate the flow of control in arbitrary ways. The cost is that they must maintain the invariants of control and continuations by hand, which is a highly complex undertaking.

===Uses===
Continuations are useful because they simplify and clarify the implementation of several other common [[programming design pattern]]s, including [[coroutine]]s (aka [[green thread]]s), and [[exception handling]]. Continuations provide the basic, low-level primitive unifying these otherwise seemingly unconnected patterns. Continuations can provide elegant solutions to some difficult high-level problems, for example the problem of programming a web server that supports multiple pages, accessed by the user's use of the forward and back buttons and by following links. The [[Seaside (software)|Seaside]] web framework in [[Smalltalk]] uses continuations to great effect, allowing one to program the web server in procedural style, by switching continuations when switching pages.

More complex constructs exist as well, for which ''"continuations provide an elegant description"''.<ref name="history_of_continuations"/> For example, in [[C (programming language)|C]], [[setjmp]] can be used to jump from the middle of one [[Function (computer science)|function]] to another function, provided the second function lies deeper in the stack (if it is waiting for the first function to return, possibly among others). Other more complex examples include [[coroutine]]s in [[Simula|Simula 67]], [[Lua (programming language)|Lua]], and [[Perl]]; tasklets in [[Stackless Python]]; [[Generator (computer science)|generators]] in [[Icon (programming language)#Generators|Icon]] and [[Python (programming language)|Python]]; continuations in [[Scala (programming language)|Scala]] (starting in 2.8); [[Fiber (computer science)|fibers]] in [[Ruby (programming language)|Ruby]] (starting in 1.9.1); the [[backtracking]] mechanism in [[Prolog]]; [[Monad (functional programming)|monads]] in [[functional programming]]; and [[Thread (computer science)|threads]].

===Examples===

The [[Scheme (programming language)|Scheme]] programming language includes the control operator [[call-with-current-continuation]] (abbreviated as: call/cc) with which a Scheme program can manipulate the flow of control:

<source lang="scheme">
(define the-continuation #f)

(define (test)
(let ((i 0))
; call/cc calls its first function argument, passing
; a continuation variable representing this point in
; the program as the argument to that function.
;
; In this case, the function argument assigns that
; continuation to the variable the-continuation.
;
(call/cc (lambda (k) (set! the-continuation k)))
;
; The next time the-continuation is called, we start here.
(set! i (+ i 1))
i))
</source>

Defines a function <code>test</code> that sets <code>the-continuation</code> to the future execution state of itself:

<source lang="scheme">
> (test)
1
> (the-continuation)
2
> (the-continuation)
3
> ; stores the current continuation (which will print 4 next) away
> (define another-continuation the-continuation)
> (test) ; resets the-continuation
1
> (the-continuation)
2
> (another-continuation) ; uses the previously stored continuation
4
</source>

For a gentler introduction to this mechanism, see [[call-with-current-continuation]].

====Coroutines====

This example shows a possible usage of continuations to implement [[coroutines]] as separate threads.<ref>Haynes, C. T., Friedman, D. P., and Wand, M. 1984. Continuations and coroutines. In Proceedings of the 1984 ACM Symposium on LISP and Functional Programming (Austin, Texas, United States, August 06–08, 1984). LFP '84. ACM, New York, NY, 293-298.</ref>

<source lang="scheme">
;;; A naive queue for thread scheduling.
;;; It holds a list of continuations "waiting to run".

(define *queue* '())

(define (empty-queue?)
(null? *queue*))

(define (enqueue x)
(set! *queue* (append *queue* (list x))))

(define (dequeue)
(let ((x (car *queue*)))
(set! *queue* (cdr *queue*))
x))

;;; This starts a new thread running (proc).

(define (fork proc)
(call/cc
(lambda (k)
(enqueue k)
(proc))))

;;; This yields the processor to another thread, if there is one.

(define (yield)
(call/cc
(lambda (k)
(enqueue k)
((dequeue)))))

;;; This terminates the current thread, or the entire program
;;; if there are no other threads left.

(define (thread-exit)
(if (empty-queue?)
(exit)
((dequeue))))
</source>

The functions defined above allow for defining and executing threads through [[Computer_multitasking#Cooperative_multitasking|cooperative multitasking]], i.e. threads that yield control to the next one in a queue:

<source lang="scheme">
;;; The body of some typical Scheme thread that does stuff:

(define (do-stuff-n-print str)
(lambda ()
(let loop ((n 0))
(format #t "~A ~A\n" str n)
(yield)
(loop (1+ n)))))

;;; Create two threads, and start them running.
(fork (do-stuff-n-print "This is AAA"))
(fork (do-stuff-n-print "Hello from BBB"))
(thread-exit)
</source>

The previous code will produce this output:
This is AAA 0
Hello from BBB 0
This is AAA 1
Hello from BBB 1
This is AAA 2
Hello from BBB 2
...

===Implementation===

A program must allocate space in memory for the variables its functions use. Most programming languages use a [[call stack]] for storing the variables needed because it allows for fast and simple allocating and automatic deallocation of memory. Other programming languages use a [[Dynamic memory allocation|heap]] for this, which allows for flexibility at a higher cost for allocating and deallocating memory. Both of these implementations have benefits and drawbacks in the context of continuations.<ref>{{cite web | url = http://community.schemewiki.org/?call-with-current-continuation-for-C-programmers | title = Call with current continuation for C programmers | work = Community-Scheme-Wiki | date = 12 October 2008}}</ref>

===Programming language support===
Many programming languages exhibit first-class continuations under various names; specifically:

*[[Common Lisp]]: [http://common-lisp.net/project/cl-cont/ cl-cont]. One can also use custom macros
*[[C Sharp (programming language)|C#]] / [[VB.NET]]: <code>async</code> and <code>await</code>: “sign up the rest of method as the continuation, and then return to your caller immediately; the task will invoke the continuation when it completes.” [http://msdn.microsoft.com/en-us/vstudio/gg316360 Asynchronous Programming for C#]
*[[Java (programming language)|Java]]: [http://lightwolf.sourceforge.net/index.html Lightwolf]
*[[Rhino (JavaScript engine)|JavaScript Rhino]] : <code>Continuation</code>
*[[Factor (programming language)|Factor]]: <code>callcc0</code> and <code>callcc1</code>
*[[Haskell (programming language)|Haskell]]: The Continuation Monad in <code>[http://hackage.haskell.org/packages/archive/mtl/2.0.1.0/doc/html/Control-Monad-Cont.html Control.Monad.Cont]</code>
*[[Haxe]]: [https://github.com/Atry/haxe-continuation haxe-continuation]
*[[Icon (programming language)|Icon]], [[Unicon (programming language)|Unicon]] : <code>create, suspend, @</code> operator: coexpressions
*[[Parrot virtual machine|Parrot]]: <code>Continuation</code> PMC; uses [[continuation-passing style]] for all control flow
*[[Perl]]: [https://metacpan.org/module/Coro Coro] and [https://metacpan.org/module/Continuity Continuity]
*[[Pico (programming language)|Pico]]: <code>call(exp())</code> and <code>continue(aContinuation, anyValue)</code>
*[[Ruby (programming language)|Ruby]]: <code>callcc</code>
*[[Scala (programming language)|Scala]]: <code>scala.util.continuations</code> provides <code>shift</code>/<code>reset</code>
*[[Scheme (programming language)|Scheme]]: <code>[[call-with-current-continuation]]</code> (commonly shortened to <code>call/cc</code>)
*[[Smalltalk]]: <code>Continuation currentDo:</code>, in most modern Smalltalk environments continuations can be implemented without additional VM support.
*[[Standard ML of New Jersey]]: <code>SMLofNJ.Cont.callcc</code>
*[[Unlambda]]: <code>c</code>, the flow control operation for call with current continuation


In any language which supports [[closure (computer science)|closures]] and [[tail recursion|proper tail calls]], it is possible to write programs in [[continuation-passing style]] and manually implement call/cc. (In [[continuation-passing style]], call/cc becomes a simple function that can be written with [[lambda calculus|lambda]].) This is a particularly common strategy in [[Haskell (programming language)|Haskell]], where it is easy to construct a "continuation-passing [[Monad (functional programming)|monad]]" (for example, the <code>Cont</code> monad and <code>ContT</code> monad transformer in the <code>mtl</code> library). The support for [[tail recursion|proper tail calls]] is needed because in [[continuation-passing style]] no function ever returns; ''all'' calls are tail calls.

==In Web development==
One area that has seen practical use of continuations is in [[Web programming]].<ref>[http://readscheme.org/xml-web/]</ref><ref>[http://www.double.co.nz/pdf/continuations.pdf]</ref> The use of continuations shields the programmer from the [[stateless server|stateless]] nature of the [[HTTP]] protocol. In the traditional model of web programming, the lack of state is reflected in the program's structure, leading to code constructed around a model that lends itself very poorly to expressing computational problems. Thus continuations enable code that has the useful properties associated with [[inversion of control]], while avoiding its problems. <cite>[http://pagesperso-systeme.lip6.fr/Christian.Queinnec/PDF/www.pdf Inverting back the inversion of control or, Continuations versus page-centric programming]</cite> is a paper that provides a good introduction to continuations applied to web programming.

Some of the more popular continuation-aware [[Web server]]s are the [[Racket (programming language)|Racket]] [http://docs.racket-lang.org/web-server/ Web Server], the [http://common-lisp.net/project/ucw UnCommon Web Framework] and [http://common-lisp.net/project/cl-weblocks/ Weblocks Web framework] for [[Common Lisp]], the [[Seaside (software)|Seaside framework]] for [[Smalltalk]], [[Ocsigen|Ocsigen/Eliom]] for [[OCaml]], [https://metacpan.org/module/Continuity Continuity] for [[Perl]], [http://github.com/mneumann/wee Wee] for [[Ruby (programming language)|Ruby]], [http://www.talesframework.org/ Tales Framework] for [[Fantom_(programming_language)|Fantom]] and the [http://www.nagare.org/ Nagare framework] for [[Python (programming language)|Python]], [http://webtoolkit.eu/wt Wt] for [[C++]], [https://github.com/agocorona/MFlow MFlow] for [[Haskell]]. The [[Apache Cocoon]] [[Web application framework]] also provides continuations (see the [http://cocoon.apache.org/2.1/userdocs/flow/continuations.html Cocoon manual]).

==Kinds==
Support for continuations varies widely. A programming language supports ''re-invocable'' continuations if a continuation may be invoked repeatedly (even after it has already returned). Re-invocable continuations were introduced by [[Peter J. Landin]] using his '''J''' (for Jump) operator that could transfer the flow of control back into the middle of a procedure invocation. Re-invocable continuations have also been called "re-entrant" in the [[Racket (programming language)|Racket]] language. However this use of the term "re-entrant" can be easily confused with its use in discussions of [[Thread (computer science)|multithreading]].

A more limited kind is the ''escape continuation'' that may be used to escape the current context to a surrounding one. Many languages which do not explicitly support continuations support [[exception handling]], which is equivalent to escape continuations and can be used for the same purposes. C's <code>[[setjmp/longjmp]]</code> are also equivalent: they can only be used to unwind the stack. Escape continuations can also be used to implement [[tail call elimination]].

One generalization of continuations are [[delimited continuation]]s. Continuation operators like <code>call/cc</code> capture the ''entire'' remaining computation at a given point in the program and provide no way of delimiting this capture. Delimited continuation operators address this by providing two separate control mechanisms: a ''prompt'' that delimits a continuation operation and a ''reification'' operator such as <code>shift</code> or <code>control</code>. Continuations captured using delimited operators thus only represent a slice of the program context.

==Disadvantages==
Continuations are the functional expression of the [[GOTO]] statement, and the same caveats apply.<ref>{{cite web | url = http://www.jquigley.com/files/talks/continuations.pdf | title =Computational Continuations | first = John | last = Quigley |date=September 2007 | format = PDF | page = 38}}</ref> While they are a sensible option in some special cases such as web programming, use of continuations can result in code that is difficult to follow. In fact, the [[esoteric programming language]] [[Unlambda]] includes [[call-with-current-continuation]] as one of its features solely because of its resistance to understanding{{Citation needed|date=July 2010}}. The external links below illustrate the concept in more detail.

==Linguistics==

In "Continuations and the nature of quantification", Chris Barker introduced the "continuation hypothesis", that
<blockquote>
some linguistic expressions (in particular, QNPs [quantificational noun phrases]) have denotations that manipulate their own continuations.<ref>Chris Barker, [http://www.semanticsarchive.net/Archive/902ad5f7/barker.continuations.pdf Continuations and the nature of quantification], 2002 Natural Language Semantics 10:211-242.</ref>
</blockquote>
Barker argued that this hypothesis could be used to explain phenomena such as ''duality of NP meaning'' (e.g., the fact that the QNP "everyone" behaves very differently from the non-quantificational noun phrase "Bob" in contributing towards the meaning of a sentence like "Alice sees [Bob/everyone]"), ''scope displacement'' (e.g., that "a raindrop fell on every car" is interpreted typically as <math>\forall c \exists r, \mbox{fell}(r,c)</math> rather than as <math>\exists r \forall c, \mbox{fell}(r,c)</math>), and ''scope ambiguity'' (that a sentence like "someone saw everyone" may be ambiguous between <math>\exists x \forall y, \mbox{saw}(x,y)</math> and <math>\forall y \exists x, \mbox{saw}(x,y)</math>). He also observed that this idea is in a way just a natural extension of [[Montague grammar|Richard Montague's approach]] in "The Proper Treatment of Quantification in Ordinary English" (PTQ), writing that "with the beneﬁt of hindsight, a limited form of continuation-passing is clearly discernible at the core of Montague’s (1973) PTQ treatment of NPs as generalized quantiﬁers".

The extent to which continuations can be used to explain other general phenomena in natural language is a topic of current research.<ref>See for example Chris Barker, [http://www.cs.bham.ac.uk/~hxt/cw04/barker.pdf Continuations in Natural Language] (Continuations Workshop 2004), or Chung-chieh Shan, [http://www.cs.rutgers.edu/~ccshan/brown/paper.pdf Linguistic Side Effects]
(in "Direct compositionality,'' ed. Chris Barker and Pauline Jacobson, pp. 132-163, Oxford University Press, 2007).</ref>

==See also==
*[[Call-with-current-continuation]]
*[[Closure (computer science)|Closure]]
*[[COMEFROM]]
*[[Continuation-passing style]]
*[[Control flow]]
*[[Coroutine]]
*[[Delimited continuation]]
*[[Denotational semantics]]
*[[GOTO]]
*[[Spaghetti stack]]

==References==
{{Refimprove|date=July 2010}}
{{Reflist}}

==Further reading==
*[[Peter Landin]]. ''A Generalization of Jumps and Labels'' Report. UNIVAC Systems Programming Research. August 1965. Reprinted in Higher Order and Symbolic Computation, 11(2):125-143, 1998, with a foreword by Hayo Thielecke.
*[[Drew McDermott]] and [[Gerry Sussman]]. ''The Conniver Reference Manual'' MIT AI Memo 259. May 1972.
*[[Daniel Bobrow]]: ''A Model for Control Structures for Artificial Intelligence Programming Languages'' IJCAI 1973.
*[[Carl Hewitt]], [[Peter Bishop]] and [[Richard Steiger]]. ''A Universal Modular Actor Formalism for Artificial Intelligence'' IJCAI 1973.
*[[Christopher Strachey]] and [[Christopher P. Wadsworth]]. ''Continuations: a Mathematical semantics for handling full jumps'' Technical Monograph PRG-11. Oxford University Computing Laboratory. January 1974. Reprinted in Higher Order and Symbolic Computation, 13(1/2):135—152, 2000, with a foreword by Christopher P. Wadsworth.
*[[John C. Reynolds]]. ''Definitional Interpreters for Higher-Order Programming Languages'' Proceedings of 25th ACM National Conference, pp. 717–740, 1972. Reprinted in Higher-Order and Symbolic Computation 11(4):363-397, 1998, with a foreword.
*John C. Reynolds. ''On the Relation between Direct and Continuation Semantics'' Proceedings of Second Colloquium on Automata, Languages, and Programming. LNCS Vol. 14, pp. 141–156, 1974.
* {{cite journal|first=John C.|last=Reynolds|year=1993|url=ftp://ftp.cs.cmu.edu/user/jcr/histcont.pdf|title=The discoveries of continuations|journal=[[Lisp and Symbolic Computation]]|volume=6|issue=3/4|pages=233–248|ref=harv}}
*[[Gerald Sussman]] and [[Guy Steele]]. ''SCHEME: An Interpreter for Extended Lambda Calculus'' AI Memo 349, MIT Artificial Intelligence Laboratory, Cambridge, Massachusetts, December 1975. Reprinted in Higher-Order and Symbolic Computation 11(4):405-439, 1998, with a foreword.
*[[Robert Hieb]], [[R. Kent Dybvig]], [[Carl Bruggeman]]. ''Representing Control in the Presence of First-Class Continuations'' Proceedings of the ACM SIGPLAN '90 Conference on Programming Language Design and Implementation, pp. 66–77.
*[[Will Clinger]], [[Anne Hartheimer]], [[Eric Ost]]. ''Implementation Strategies for Continuations'' Proceedings of the 1988 ACM conference on LISP and Functional Programming, pp. 124–131, 1988. Journal version: Higher-Order and Symbolic Computation, 12(1):7-45, 1999.

==External links==
* [[Association for Computing Machinery|ACM]] [[SIGPLAN]] [http://logic.cs.tsukuba.ac.jp/cw2011/ Workshop on Continuations 2011] at the [[ICFP]].
*[http://www.intertwingly.net/blog/2005/04/13/Continuations-for-Curmudgeons Continuations for Curmudgeons] by Sam Ruby
*[http://www.ccs.neu.edu/home/dorai/t-y-scheme/t-y-scheme-Z-H-15.html#node_chap_13 Teach Yourself Scheme in Fixnum Days] by Dorai Sitaram features a nice chapter on continuations.
*[http://www.stackless.com/spcpaper.htm Continuations and Stackless Python] by Christian Tismer
*[http://www.cs.bham.ac.uk/~hxt/cw04/cw04-program.html On-line proceedings of the Fourth ACM SIGPLAN Workshop on Continuations]
*[http://www.brics.dk/~cw97/ On-line proceedings of the Second ACM SIGPLAN Workshop on Continuations]
*[http://www.cs.bham.ac.uk/~hxt/research/Logiccolumn8.pdf Continuation, functions and jumps]
*[http://okmij.org/ftp/continuations/ http://okmij.org/ftp/continuations/] by Oleg Kiselyov
*[http://wiki.apache.org/cocoon/RhinoWithContinuations Rhino With Continuations]
*[http://rifers.org/wiki/display/RIFE/Web+continuations Continuations in pure Java] from the [[RIFE]] web application framework
*[http://rifers.org/theater/debugging_continuations Debugging continuations in pure Java] from the [[RIFE]] web application framework
* [http://mail.python.org/pipermail/python-dev/1999-July/000467.html Comparison of generators, coroutines, and continuations, source of the above example]

[[Category:Continuations| ]]
[[Category:Control flow]]
[[Category:Articles with example Scheme code]]

Virial coefficient

2014-01-08T11:43:33Z

193.231.19.53: /* Further reading */ added link 3rd virial

The '''Mayer f-function''' is an auxiliary function that often appears in the series expansion of [[thermodynamic]] quantities related to classical [[many-particle system]]s.<ref name - "mcquarrie">Donald Allan McQuarrie, ''Statistical Mechanics'' ([[HarperCollins]], 1976), page 228
</ref>

==Definition==
Consider a system of classical particles interacting through a pair-wise potential
:<math>V(\mathbf{i},\mathbf{j})</math>
where the bold labels <math>\mathbf{i}</math> and <math>\mathbf{j}</math> denote the continuous degrees of freedom associated with the particles, e.g.,
:<math>\mathbf{i}=\mathbf{r}_i</math>
for spherically symmetric particles and
:<math>\mathbf{i}=(\mathbf{r}_i,\Omega_i)</math>
for rigid non-spherical particles where <math>\mathbf{r}</math> denotes position and <math>\Omega</math> the orientation parametrized e.g. by [[Euler angles]]. The Mayer f-function is then defined as
:<math>f(\mathbf{i},\mathbf{j})=e^{-\beta V(\mathbf{i},\mathbf{j})}-1</math>
where <math>\beta=(k_{B}T)^{-1}</math> the inverse absolute [[temperature]] in units of (Temperature times the [[Boltzmann constant]] <math>k_{B}</math>)-1 .

==See also==
*[[Virial coefficient]]
*[[Cluster expansion]]
*[[Excluded volume]]

==Notes==
{{reflist}}

[[Category:Special functions]]

Cluster expansion

2013-10-29T15:45:40Z

193.231.19.53: /* References */ [[]]

[[File:Wea00800,1.jpg|thumb|300px|Merchant ship labouring in heavy seas as a huge wave looms astern]]

In [[fluid dynamics]], the '''wave height''' of a [[ocean surface wave|surface wave]] is the difference between the [[elevation]]s of a [[crest (physics)|crest]] and a neighbouring [[trough (physics)|trough]].<ref name=Kinsman_38>{{harvtxt|Kinsman|1984|page=38}}</ref> Wave height is a term used by [[mariner]]s, as well as in [[coastal engineering|coastal]], [[offshore construction|ocean]] and [[naval engineering]].

At sea, the term [[significant wave height]] is used as a means to introduce a well-defined and standardized [[statistic]] to denote the characteristic height of the [[random]] waves in a [[sea state]]. It is defined in such a way that it more–or–less corresponds to what a [[mariner]] observes when estimating visually the average wave height.

== Several definitions for different situations ==
[[File:Sine wave amplitude.svg|right|thumb|Wave characteristics.]]

*For a ''[[sine wave]]'', the wave height ''H'' is twice the [[amplitude]]:<ref name=Kinsman_38/>

:: <math>H = 2a. \,</math>

*For a ''[[periodic function|periodic wave]]'' it is simply the difference between the [[maximum]] and [[minimum]] of the surface elevation ''z'' = ''η''(''x'' – ''c''p ''t''):<ref name=Kinsman_38/>

:: <math>H = \max\left\{ \eta(x\,-\,c_p\,t) \right\} - \min\left\{ \eta(x - c_p\,t) \right\}, \, </math>

:with ''c''p the [[phase speed]] (or propagation speed) of the wave. The sine wave is a specific case of a periodic wave.

*In random waves at sea, when the surface elevations are measured with a [[weather buoy|wave buoy]], the ''individual wave height'' ''H''m of each individual wave—with an [[integer]] label ''m'', running from 1 to ''N'', to denote its position in a sequence of ''N'' waves—is the difference in elevation between a wave crest and trough in that wave. For this to be possible, it is necessary to first split the measured [[time series]] of the surface elevation into individual waves. Commonly, an individual wave is denoted as the [[time interval]] between two successive downward-crossings through the [[average]] surface elevation (upward crossings might also be used). Then the individual wave height of each wave is again the difference between maximum and minimum elevation in the time interval of the wave under consideration.<ref name=Holt_24_28>{{harvtxt|Holthuijsen|2007|pages=24–28}}</ref>

*''[[Significant wave height]]'' ''H''1/3, or ''H''''s'' or ''H''''sig'', in the ''[[time domain]]'', is defined as the average height of the one-third part of the measured waves—which are ''N'' in number—having the largest wave heights:<ref name=Holt_24_28/>

:: <math>H_{1/3} = \frac{1}{\frac13\,N}\, \sum_{m=1}^{\frac13\,N}\, H_m,</math>

:with ''H''m the individual wave heights, sorted in such a way that the highest wave has ''m''=1 and the lowest wave is for ''m''=''N''. Only the highest one-third is used, since this corresponds best with visual observations of experienced mariners: eyes and brain apparently focus on the higher waves seen.<ref name=Holt_24_28/>

*''Significant wave height'' ''H''m0, defined in the ''[[frequency domain]]'', is used both for measured and [[forecasting|forecast]]ed wave [[power spectrum|variance spectra]]. Most easily, it is defined in terms of the [[variance]] ''m''0 or [[standard deviation]] ''σ''''η'' of the surface elevation:<ref>{{harvtxt|Holthuijsen|2007|page=70}}</ref>

:: <math>H_{m_0} = 4 \sqrt{m_0} = 4 \sigma_\eta, \, </math>

:where ''m''0, the zeroth-[[moment (mathematics)|moment]] of the variance spectrum, is obtained by [[integration (mathematics)|integration]] of the variance spectrum. In case of a measurement, the standard deviation ''σ''''η'' is the easiest and most accurate statistic to be used.

*Another wave-height statistic in common usage is the ''[[root-mean-square]] wave height'' (or ''RMS wave height'') ''H''rms, defined as:<ref name=Holt_24_28/>

:: <math>H_\text{rms} = \sqrt{ \frac{1}{N} \sum_{m=1}^N H_m^2}, \, </math>

:with ''H''m again denoting the individual wave heights in a certain [[time series]].

== See also ==

*[[Sea state]]
*[[Significant wave height]]
*[[Wind wave]]

== Notes ==

{{reflist}}

== References ==
{{ref begin}}
*{{Citation | title=Waves in Oceanic and Coastal Waters | first=Leo H. | last=Holthuijsen | publisher=Cambridge University Press | year=2007 | isbn=0-521-86028-8 }}, 387 pages.
*{{Citation | last=Kinsman | first=Blair | title=Wind waves: their generation and propagation on the ocean surface | year=1984 | publisher=Dover Publications | isbn=0-486-49511-6 }}, 704 pages.
*{{Citation | first=Owen M. | last=Phillips | title=The dynamics of the upper ocean | publisher=Cambridge University Press | year=1977 | edition=2nd | isbn=0-521-29801-6 }}, viii & 336 pages.
{{ref end}}

{{physical oceanography}}

[[Category:Water waves]]
[[Category:Physical oceanography]]