formulasearchengine - User contributions [en]

Speedcubing

2014-02-24T17:02:30Z

81.155.215.177: /* World records */

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Random close pack

2014-02-01T23:23:00Z

81.155.214.158: typo fix

In [[knot theory]], a '''Lissajous knot''' is a [[knot (mathematics)|knot]] defined by [[parametric equations]] of the form

:<math>x = \cos(n_x t + \phi_x),\qquad y = \cos(n_y t + \phi_y), \qquad z = \cos(n_z t + \phi_z),</math>

[[Image:Lissajous 8 21 Knot.png|thumb|A Lissajous 821 knot]]
where <math>n_x</math>, <math>n_y</math>, and <math>n_z</math> are [[integer]]s and the [[phase shift]]s <math>\phi_x</math>, <math>\phi_y</math>, and <math>\phi_z</math> may be any [[real number]]s.<ref name="Bogle">M.G.V. Bogle, J.E. Hearst, V.F.R. Jones, L. Stoilov, "Lissajous knots", Journal of Knot Theory and Its Ramifications, 3(2), 1994, 121–140.</ref>

The projection of a Lissajous knot onto any of the three coordinate planes is a [[Lissajous curve]], and many of the properties of these knots are closely related to properties of Lissajous curves.

Replacing the cosine function in the parametrization by a [[triangle wave]] transforms every Lissajous
knot isotopically into a billiard curve inside a cube, the simplest case of so-called ''billiard knots''.
Billiard knots can also be studied in other domains, for instance in a cylinder.<ref name="Cylinder">C. Lamm, D. Obermeyer. "Billiard knots in a cylinder", Journal of Knot Theory and Its Ramifications, 8(3), 1999, 353–366.</ref>

== Form ==
Because a knot cannot be self-intersecting, the three integers <math>n_x, n_y, n_z</math> must be pairwise [[relatively prime]], and none of the quantities

:<math>n_x \phi_y - n_y \phi_x,\quad n_y \phi_z - n_z \phi_y,\quad n_z \phi_x - n_x \phi_z</math>

may be an integer multiple of [[pi]]. Moreover, by making a substitution of the form <math>t' = t+c</math>, one may assume that any of the three phase shifts <math>\phi_x</math>, <math>\phi_y</math>, <math>\phi_z</math> is equal to zero.

== Examples ==
Here are some examples of Lissajous knots,<ref>{{cite book |author=Cromwell, Peter R. |title=Knots and links |publisher=Cambridge University Press |location=Cambridge, UK |year=2004 |pages=13 |isbn=0-521-54831-4}}</ref> all of which have <math>\phi_z=0</math>:
<gallery>
Image:Lissajous 5_2 Knot.png|[[Three-twist knot]] <math>(n_x,n_y,n_z)=(3,2,7)</math> <math>(\phi_x,\phi_y)=(0.7,0.2)</math>
Image:Lissajous Stevedore Knot.png|[[Stevedore knot (mathematics)|Stevedore knot]] <math>(n_x,n_y,n_z)=(3,2,5)</math> <math>(\phi_x,\phi_y)=(1.5,0.2)</math>
Image:Lissajous Square Knot.png|[[Square knot (mathematics)|Square knot]] <math>(n_x,n_y,n_z)=(3,5,7)</math> <math>(\phi_x,\phi_y)=(0.7,1.0)</math>
Image:Lissajous 8_21 Knot.png|821 knot <math>(n_x,n_y,n_z)=(3,4,7)</math> <math>(\phi_x,\phi_y)=(0.1,0.7)</math>
</gallery>
There are infinitely many different Lissajous knots,<ref>C. Lamm. "There are infinitely many Lissajous knots." ''Manuscripta Math.'', 93:29–37, 1997, [http://www.springerlink.com/content/67427263811l501q Springerlink]</ref> and other examples with 10 or fewer [[crossing number (knot theory)|crossings]] include the 74 knot, the 815 knot, the 101 knot, the 1035 knot, the 1058 knot, and the composite knot 52* # 52,<ref name="Bogle"/> as well as the 916 knot, 1076 knot, the 1099 knot, the 10122 knot, the 10144 knot, the [[granny knot (mathematics)|granny knot]], and the composite knot 52 # 52.<ref>{{cite arXiv |author=A. Boocher, J. Daigle, J. Hoste, W. Zheng |eprint=0707.4210 |title=Sampling Lissajous and Fourier knots |year= 2007 }}</ref> In addition, it is known that every [[twist knot]] with [[Arf invariant]] zero is a Lissajous knot.<ref>{{cite arXiv |last1=Hoste | first1=Jim | last2=Zirbel | first2=Laura |eprint=math.GT/0605632|title=Lissajous knots and knots with Lissajous projections |year=2006 }}</ref>

== Symmetry ==
Lissajous knots are highly symmetric, though the type of symmetry depends on whether or not the numbers <math>n_x</math>, <math>n_y</math>, and <math>n_z</math> are all odd.

=== Odd case ===
If <math>n_x</math>, <math>n_y</math>, and <math>n_z</math> are all odd, then the [[point reflection]] across the origin <math>(x,y,z)\mapsto (-x,-y,-z)</math> is a symmetry of the Lissajous knot which preserves the knot orientation.

In general, a knot that has an orientation-preserving point reflection symmetry is known as '''strongly plus [[amphicheiral knot|amphicheiral]]'''.<ref>{{cite arXiv |last=Przytycki |first=Jozef H. |eprint=math/0405151 |title=Symmetric knots and billiard knots |year=2004 }}</ref> This is a fairly rare property: only three [[prime knot]]s with twelve or fewer crossings are strongly plus amphicheiral [[prime knot]], the first of which has [[crossing number (knot theory)|crossing number]] ten.<ref>Jim Hoste, Morwen Thistlethwaite, and Jeff Weeks. "The first 1,701,936 knots." ''Math. Intelligencer'',
20(4):33–48, 1998.</ref> Since this is so rare, most Lissajous knots lie in the even case.

=== Even case ===
If one of the frequencies (say <math>n_x</math>) is even, then the 180° rotation around the ''x''-axis <math>(x,y,z)\mapsto (x,-y,-z)</math> is a symmetry of the Lissajous knot. In general, a knot that has a symmetry of this type is called '''2-periodic''', so every even Lissajous knot must be 2-periodic.

=== Consequences ===
The symmetry of a Lissajous knot puts severe constraints on the [[Alexander polynomial]]. In the odd case, the Alexander
polynomial of the Lissajous knot must be a perfect [[square (algebra)|square]].<ref>R. Hartley and A Kawauchi. "Polynomials of amphicheiral knots." ''Math. Ann.'', 243:63–70, 1979.</ref> In the even case, the Alexander polynomial must be a perfect square [[modular arithmetic|modulo]] 2.<ref>K. Murasugi. "On periodic knots." ''Comment. Math.Helv.'', 46:162–174, 1971.</ref> In addition, the [[Arf invariant]] of a Lissajous knot must be zero. It follows that:
* The [[trefoil knot]] and [[figure-eight knot (mathematics)|figure-eight knot]] are not Lissajous.
* No [[torus knot]] can be Lissajous.
* No [[fibered knot|fibered]] [[2-bridge knot]] can be Lissajous.

== References ==
{{reflist}}

{{DEFAULTSORT:Lissajous Knot}}
[[Category:Knots (knot theory)]]

Twistor space

2014-01-29T00:38:03Z

81.155.214.158: Linkified name

In [[statistics]], the '''Shapiro–Wilk test''' tests the [[null hypothesis]] that a [[statistical sample|sample]] ''x''1, ..., ''x''''n'' came from a [[normal distribution|normally distributed]] population. It was published in 1965 by Samuel Shapiro and [[Martin Wilk]].<ref>{{cite journal
|last=Shapiro |first=S. S.
|last2=Wilk |first2=M. B. |authorlink2=Martin Wilk
|year=1965
|title=An analysis of variance test for normality (complete samples)
|journal=[[Biometrika]]
|volume=52 |issue=3-4 |pages=591–611
|doi=10.1093/biomet/52.3-4.591 |jstor=2333709 | mr = 205384
}}</ref>

The [[test statistic]] is:

:<math>W = {\left(\sum_{i=1}^n a_i x_{(i)}\right)^2 \over \sum_{i=1}^n (x_i-\overline{x})^2}</math>

where

* <math>x_{(i)}</math> (with parentheses enclosing the subscript index ''i'') is the ''i''th [[order statistic]], i.e., the ''i''th-smallest number in the sample;
* <math>\overline{x} = \left( x_1 + \cdots + x_n \right) / n</math> is the sample mean;
* the constants <math>a_i</math> are given by<ref>{{cite journal
|last=Shapiro |first=S. S.
|last2=Wilk |first2=M. B. |authorlink2=Martin Wilk
|year=1965
|title=An analysis of variance test for normality (complete samples)
|journal=[[Biometrika]]
|volume=52 |issue=3-4 |pages=591–611
|doi=10.1093/biomet/52.3-4.591 |jstor=2333709 | mr = 205384
}} p. 593</ref>

::<math>(a_1,\dots,a_n) = {m^\top V^{-1} \over (m^\top V^{-1}V^{-1}m)^{1/2}}</math>

:where

::<math>m = (m_1,\dots,m_n)^\top\,</math>

:and <math>m_1,\ldots,m_n</math> are the [[expected value]]s of the [[order statistic]]s of [[independent and identically distributed random variables]] sampled from the standard normal distribution, and <math>V</math> is the [[covariance matrix]] of those order statistics.

The user may reject the null hypothesis if <math>W</math> is too small.<ref>{{cite journal
|last=Shapiro |first=S. S.
|last2=Wilk |first2=M. B. |authorlink2=Martin Wilk
|year=1965
|title=An analysis of variance test for normality (complete samples)
|journal=[[Biometrika]]
|volume=52 |issue=3-4 |pages=591–611
|doi=10.1093/biomet/52.3-4.591 |jstor=2333709 | mr = 205384
}} p. 605</ref>

It can be interpreted via a [[Q-Q plot]].

==Interpretation==
Recalling that the null hypothesis is that the population is normally distributed, if the [[p-value]] is less than the chosen [[alpha level]], then the null hypothesis is rejected (i.e. one concludes the data are not from a normally distributed population). If the p-value is greater than the chosen alpha level, then one does not reject the null hypothesis that the data came from a normally distributed population. E.g. for an alpha level of 0.05, a data set with a p-value of 0.32 does not result in rejection of the hypothesis that the data are from a normally distributed population.<ref>{{cite web |url= http://www.jmp.com/support/faq/jmp2085.shtml |title=How do I interpret the Shapiro-Wilk test for normality?
|first= |last= |work=JMP |year=2004 |accessdate=March 24, 2012}}</ref>

==See also==
* [[Anderson–Darling test]]
* [[Cramér–von Mises criterion]]
* [[Kolmogorov–Smirnov test]]
* [[Normal probability plot]]
* [[Ryan–Joiner test]]
* [[Watson test]]

{{primary sources|date=May 2012}}

==References==
{{Ibid|date=May 2012}}
<references/>

==External links==
* [http://www.answers.com/topic/samuel-sanford-shapiro Samuel Sanford Shapiro]
* [http://lib.stat.cmu.edu/apstat/R94 Algorithm AS R94 (Shapiro Wilk) FORTRAN code]
* [http://cran.us.r-project.org/doc/manuals/R-intro.html#Examining-the-distribution-of-a-set-of-data Shapiro–Wilk Normality Test in R]

{{Statistics}}

{{DEFAULTSORT:Shapiro-Wilk Test}}
[[Category:Normality tests]]

Blum's speedup theorem

2014-01-18T04:59:38Z

81.155.198.68:

== Summary ==
Reupload of [[:Image:Cube root of positive X.gif]] because we prefer pngs in Wikipedia and I forgot.

Created in [[Mathematica]]. Plots:

<math>y = \sqrt[3]{x}</math>

I'm not sure about the copyright status since this seems awfully mundane...
== Licensing ==
{{PD-self|date=October 2006}}

{{Copy to Wikimedia Commons|bot=Fbot|date=March 2012}}

Free variables and bound variables

2014-01-14T01:01:31Z

81.155.198.68: /* Formal explanation */

{{Unreferenced|date=December 2009}}
In [[mathematics]], the '''linearity of differentiation''' is a most fundamental property of the [[derivative]], in [[differential calculus]]. It follows from the [[sum rule in differentiation]] and the [[constant factor rule in differentiation]]. Thus it can be said that the act of differentiation is [[Linear map|linear]], or the [[differential operator]] is a [[Linear map|linear operator]].

Let ''f'' and ''g'' be functions, with <math>\alpha</math> and <math>\beta</math> fixed. Now consider:

:<math>\frac{\mbox{d}}{\mbox{d} x} ( \alpha \cdot f(x) + \beta \cdot g(x) ) </math>

By the [[sum rule in differentiation]], this is:

:<math>\frac{\mbox{d}}{\mbox{d} x} ( \alpha \cdot f(x) ) + \frac{\mbox{d}}{\mbox{d} x} (\beta \cdot g(x))</math>

By the [[constant factor rule in differentiation]], this reduces to:

:<math>\alpha \cdot f'(x) + \beta \cdot g'(x)</math>

This in turn leads to:

:<math>\frac{\mbox{d}}{\mbox{d} x}(\alpha \cdot f(x) + \beta \cdot g(x)) = \alpha \cdot f'(x) + \beta \cdot g'(x)</math>

Omitting the [[bracket]]s, this is often written as:

:<math>(\alpha \cdot f + \beta \cdot g)' = \alpha \cdot f'+ \beta \cdot g'</math>

{{DEFAULTSORT:Linearity Of Differentiation}}
[[Category:Differential calculus]]

Relation algebra

2014-01-13T00:49:57Z

81.155.198.68: /* Historical remarks */

{{userbox
| float = {{{float|left}}}
| border-c = {{{border-color|{{{1|black}}}}}}
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| id-c = {{{logo-background|{{{1|white}}}}}}
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| info = This user is an '''[[electronic engineer]]'''.
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}}<noinclude>
[[Category:Profession user templates|Electronic engineer]]
</noinclude>

Choice function

2014-01-11T22:24:11Z

81.155.198.68: /* Refinement of the notion of choice function */

{{Multiple issues|
{{More footnotes|date=March 2011}}
{{Refimprove|date=March 2011}}
}}
In [[statistics]] and [[signal processing]], an '''autoregressive''' ('''AR''') '''model''' is a representation of a type of [[random process]]; as such, it describes certain time-varying processes in [[natural science|nature]], [[economics]], etc. The autoregressive model specifies that the output variable depends [[linear prediction|linearly]] on its own previous values. It is a special case of the more general [[Autoregressive–moving-average model|ARMA]] model of [[time series]].

==Definition==

The notation AR(''p'') indicates an autoregressive model of order ''p''. The AR(''p'') model is defined as

:<math> X_t = c + \sum_{i=1}^p \varphi_i X_{t-i}+ \varepsilon_t \,</math>

where <math>\varphi_1, \ldots, \varphi_p</math> are the '''''parameters''''' of the model, <math>c</math> is a constant, and <math>\varepsilon_t</math> is [[white noise]]. This can be equivalently written using the [[backshift operator]] ''B'' as

:<math> X_t = c + \sum_{i=1}^p \varphi_i B^i X_t + \varepsilon_t </math>

so that, moving the summation term to the left side and using polynomial notation, we have

:<math>\phi (B)X_t= c + \varepsilon_t \, .</math>

An autoregressive model can thus be viewed as the output of an all-[[pole (complex analysis)|pole]] [[infinite impulse response]] filter whose input is white noise.

Some constraints are necessary on the values of the parameters of this model in order that the model remains [[wide-sense stationary]]. For example, processes in the AR(1) model with |''φ''1| ≥ 1 are not stationary. More generally, for an AR(''p'') model to be wide-sense stationary, the roots of the polynomial <math>\textstyle z^p - \sum_{i=1}^p \varphi_i z^{p-i}</math> must lie within the [[unit circle]], i.e., each root <math>z_i</math> must satisfy <math>|z_i|<1</math>.

==Intertemporal effect of shocks==

In an AR process, a one-time shock affects values of the evolving variable infinitely far into the future. For example, consider the AR(1) model <math> X_t = c + \varphi_1 X_{t-1} + \varepsilon_t</math>. A non-zero value for <math>\varepsilon_t</math> at say time ''t''=1 affects <math>X_1</math> by the amount <math>\varepsilon_1</math>. Then by the AR equation for <math>X_2</math> in terms of <math>X_1</math>, this affects <math>X_2</math> by the amount <math>\varphi_1 \varepsilon_1</math>. Then by the AR equation for <math>X_3</math> in terms of <math>X_2</math>, this affects <math>X_3</math> by the amount <math>\varphi_1^2 \varepsilon_1</math>. Continuing this process shows that the effect of <math>\varepsilon_1</math> never ends, although if the process is [[stationary process|stationary]] then the effect diminishes toward zero in the limit.

Because each shock affects ''X'' values infinitely far into the future from when they occur, any given value ''X''''t'' is affected by shocks occurring infinitely far into the past. This can also be seen by rewriting the autoregression

:<math>\phi (B)X_t= \varepsilon_t \,</math>

(where the constant term has been suppressed by assuming that the variable has been measured as deviations from its mean) as

:<math>X_t= \frac{1}{\phi (B)}\varepsilon_t \, .</math>

When the [[polynomial long division|polynomial division]] on the right side is carried out, the polynomial in the backshift operator applied to <math>\varepsilon_t</math> has an infinite order—that is, an infinite number of lagged values of <math>\varepsilon_t</math> appear on the right side of the equation.

==Characteristic polynomial==
The [[autocorrelation function]] of an AR(''p'') process can be expressed as {{Citation needed|date=October 2011|reason=a_k not defined and seems wrong}}
:<math>\rho(\tau) = \sum_{k=1}^p a_k y_k^{-|\tau|} ,</math>

where <math>y_k</math> are the roots of the polynomial
: <math>\phi(B) = 1- \sum_{k=1}^p \varphi_k B^k </math>

where ''B'' is the [[backshift operator]], where <math>\phi(.)</math> is the function defining the autoregression, and where <math>\varphi_k</math> are the coefficients in the autoregression.

The autocorrelation function of an AR(''p'') process is a sum of decaying exponentials.
* Each real root contributes a component to the autocorrelation function that decays exponentially.
* Similarly, each pair of complex conjugate roots contributes an exponentially damped oscillation.

==Graphs of AR(''p'') processes==

[[File:ArTimeSeries.svg|thumb|right|alt="Figure has 5 plots of AR proceses. AR(0) and AR(0.3) are white noise or look like white noise. AR(0.9) has some large scale oscillating structure."|AR(0); AR(1) with AR parameter 0.3; AR(1) with AR parameter 0.9; AR(2) with AR parameters 0.3 and 0.3; and AR(2) with AR parameters 0.9 and −0.8]]
The simplest AR process is AR(0), which has no dependence between the terms. Only the error/innovation/noise term contributes to the output of the process, so in the figure, AR(0) corresponds to white noise.

For an AR(1) process with a positive <math>\varphi</math>, only the previous term in the process and the noise term contribute to the output. If <math>\varphi</math> is close to 0, then the process still looks like white noise, but as <math>\varphi</math> approaches 1, the output gets a larger contribution from the previous term relative to the noise. This results in a "smoothing" or integration of the output, similar to a low pass filter.

For an AR(2) process, the previous two terms and the noise term contribute to the output. If both <math>\varphi_1</math> and <math>\varphi_2</math> are positive, the output will resemble a low pass filter, with the high frequency part of the noise decreased. If <math>\varphi_1</math> is positive while <math>\varphi_2</math> is negative, then the process favors changes in sign between terms of the process. The output oscillates.

==Example: An AR(1) process==

An AR(1) process is given by:

:<math>X_t = c + \varphi X_{t-1}+\varepsilon_t\,</math>

where <math>\varepsilon_t</math> is a white noise process with zero mean and constant variance <math>\sigma_\varepsilon^2</math>.
(Note: The subscript on <math>\varphi_1</math> has been dropped.) The process is [[wide-sense stationary]] if <math>|\varphi|<1</math> since it is obtained as the output of a stable filter whose input is white noise. (If <math>\varphi=1</math> then <math>X_t</math> has infinite variance, and is therefore not wide sense stationary.) Consequently, assuming <math>|\varphi|<1</math>, the mean <math>\operatorname{E} (X_t)</math> is identical for all values of ''t''. If the mean is denoted by <math>\mu</math>, it follows from

:<math>\operatorname{E} (X_t)=\operatorname{E} (c)+\varphi\operatorname{E} (X_{t-1})+\operatorname{E}(\varepsilon_t),
</math>
that
:<math> \mu=c+\varphi\mu+0,</math>

and hence

:<math>\mu=\frac{c}{1-\varphi}.</math>

In particular, if <math>c = 0</math>, then the mean is 0.

The [[variance]] is

:<math>\textrm{var}(X_t)=\operatorname{E}(X_t^2)-\mu^2=\frac{\sigma_\varepsilon^2}{1-\varphi^2},</math>
where <math>\sigma_\varepsilon</math> is the standard deviation of <math>\varepsilon_t</math>. This can be shown by noting that
:<math>\textrm{var}(X_t) = \varphi^2\textrm{var}(X_{t-1}) + \sigma_\varepsilon^2,</math>
and then by noticing that the quantity above is a stable fixed point of this relation.

The [[autocovariance]] is given by

:<math>B_n=\operatorname{E}(X_{t+n}X_t)-\mu^2=\frac{\sigma_\varepsilon^2}{1-\varphi^2}\,\,\varphi^{|n|}.</math>

It can be seen that the autocovariance function decays with a decay time (also called [[time constant]]) of <math>\tau=-1/\ln(\varphi)</math> [to see this, write <math>B_n=K\phi^{|n|}</math> where <math>K</math> is independent of <math>n</math>. Then note that <math>\phi^{|n|}=e^{|n|\ln\phi}</math> and match this to the exponential decay law <math>e^{-n/\tau}</math>].

The [[spectral density]] function is the [[Fourier transform]] of the autocovariance function. In discrete terms this will be the discrete-time Fourier transform:

:<math>\Phi(\omega)=
\frac{1}{\sqrt{2\pi}}\,\sum_{n=-\infty}^\infty B_n e^{-i\omega n}
=\frac{1}{\sqrt{2\pi}}\,\left(\frac{\sigma_\varepsilon^2}{1+\varphi^2-2\varphi\cos(\omega)}\right).
</math>

This expression is periodic due to the discrete nature of the <math>X_j</math>, which is manifested as the cosine term in the denominator. If we assume that the sampling time (<math>\Delta t=1</math>) is much smaller than the decay time (<math>\tau</math>), then we can use a continuum approximation to <math>B_n</math>:

:<math>B(t)\approx \frac{\sigma_\varepsilon^2}{1-\varphi^2}\,\,\varphi^{|t|}</math>

which yields a [[Cauchy distribution|Lorentzian profile]] for the spectral density:

:<math>\Phi(\omega)=
\frac{1}{\sqrt{2\pi}}\,\frac{\sigma_\varepsilon^2}{1-\varphi^2}\,\frac{\gamma}{\pi(\gamma^2+\omega^2)}</math>

where <math>\gamma=1/\tau</math> is the angular frequency associated with the decay time <math>\tau</math>.

An alternative expression for <math>X_t</math> can be derived by first substituting <math>c+\varphi X_{t-2}+\varepsilon_{t-1}</math> for <math>X_{t-1}</math> in the defining equation. Continuing this process ''N'' times yields

:<math>X_t=c\sum_{k=0}^{N-1}\varphi^k+\varphi^NX_{t-N}+\sum_{k=0}^{N-1}\varphi^k\varepsilon_{t-k}.</math>

For ''N'' approaching infinity, <math>\varphi^N</math> will approach zero and:

:<math>X_t=\frac{c}{1-\varphi}+\sum_{k=0}^\infty\varphi^k\varepsilon_{t-k}.</math>

It is seen that <math>X_t</math> is white noise convolved with the <math>\varphi^k</math> kernel plus the constant mean. If the white noise <math>\varepsilon_t</math> is a [[Gaussian process]] then <math>X_t</math> is also a Gaussian process. In other cases, the [[central limit theorem]] indicates that <math>X_t</math> will be approximately normally distributed when <math>\varphi</math> is close to one.

==Choosing the maximum lag==
{{Main|Partial autocorrelation function}}

==Calculation of the AR parameters==

There are many ways to estimate the coefficients, such as the [[ordinary least squares]] procedure, [[Method of moments (statistics)|method of moments]] (through Yule Walker equations), or [[Markov chain Monte Carlo]] methods.{{citation needed|date=July 2012}}

The AR(''p'') model is given by the equation

:<math> X_t = \sum_{i=1}^p \varphi_i X_{t-i}+ \varepsilon_t.\,</math>

It is based on parameters <math>\varphi_i</math> where ''i'' = 1, ..., ''p''. There is a direct correspondence between these parameters and the covariance function of the process, and this correspondence can be inverted to determine the parameters from the autocorrelation function (which is itself obtained from the covariances). This is done using the Yule-Walker equations.

===Yule-Walker equations===


The Yule-Walker equations are the following set of equations .{{citation needed|date=July 2012}}

:<math>
\gamma_m = \sum_{k=1}^p \varphi_k \gamma_{m-k} + \sigma_\varepsilon^2\delta_{m,0},

</math>

where {{nowrap|''m'' = 0, ..., ''p''}}, yielding {{nowrap|''p'' + 1}} equations. Here <math>\gamma_m</math> is the autocovariance function of Xt, <math>\sigma_\varepsilon</math> is the standard deviation of the input noise process, and <math>\delta_{m,0}</math> is the [[Kronecker delta function]].

Because the last part of an individual equation is non-zero only if {{nowrap|''m'' = 0}}, the set of equations can be solved by representing the equations for {{nowrap|''m'' > 0}} in matrix form, thus getting the equation

:<math>\begin{bmatrix}
\gamma_1 \\
\gamma_2 \\
\gamma_3 \\
\vdots \\
\gamma_p \\
\end{bmatrix}

=

\begin{bmatrix}
\gamma_0 & \gamma_{-1} & \gamma_{-2} & \dots \\
\gamma_1 & \gamma_0 & \gamma_{-1} & \dots \\
\gamma_2 & \gamma_{1} & \gamma_{0} & \dots \\
\vdots & \vdots & \vdots & \ddots \\
\gamma_{p-1} & \gamma_{p-2} & \gamma_{p-3} & \dots \\
\end{bmatrix}

\begin{bmatrix}
\varphi_{1} \\
\varphi_{2} \\
\varphi_{3} \\
\vdots \\
\varphi_{p} \\
\end{bmatrix}

</math>

which can be solved for all <math>\{\varphi_m; m=1,2, \cdots ,p\}.</math> The remaining equation for ''m'' = 0 is

:<math>
\gamma_0 = \sum_{k=1}^p \varphi_k \gamma_{-k} + \sigma_\varepsilon^2 ,
</math>

which, once <math>\{\varphi_m ; m=1,2, \cdots ,p \}</math> are known, can be solved for <math>\sigma_\varepsilon^2 .</math>

An alternative formulation is in terms of the [[autocorrelation function]]. The AR parameters are determined by the first p+1 elements <math>\rho(\tau)</math> of the autocorrelation function. The full autocorrelation function can then be derived by recursively calculating
<ref name=Storch>{{Cite book
| publisher = Cambridge Univ Pr
| isbn = 0-521-01230-9
| last = Von Storch
| first = H.
| coauthors = F. W Zwiers
| title = Statistical analysis in climate research
| year = 2001
}}{{Page needed|date=March 2011}}</ref>
: <math>\rho(\tau) = \sum_{k=1}^p \varphi_k \rho(k-\tau)</math>
Examples for some Low-order AR(''p'') processes
* p=1
** <math>\gamma_1 = \varphi_1 \gamma_0</math>
** Hence <math>\rho_1 = \gamma_1 / \gamma_0 = \varphi_1</math>
* p=2
** The Yule-Walker equations for an AR(2) process are
**: <math>\gamma_1 = \varphi_1 \gamma_0 + \varphi_2 \gamma_{-1}</math>
**: <math>\gamma_2 = \varphi_1 \gamma_1 + \varphi_2 \gamma_0</math>
*** Remember that <math>\gamma_{-k} = \gamma_k</math>
*** Using the first equation yields <math>\rho_1 = \gamma_1 / \gamma_0 = \frac{\varphi_1}{1-\varphi_2}</math>
*** Using the recursion formula yields <math>\rho_2 = \gamma_2 / \gamma_0 = \frac{\varphi_1^2 - \varphi_2^2 + \varphi_2}{1-\varphi_2}</math>

===Estimation of AR parameters===

The above equations (the Yule-Walker equations) provide several routes to estimating the parameters of an AR(''p'') model, by replacing the theoretical covariances with estimated values.{{citation needed|date=July 2012}} Some of these variants can be described as follows:

*Estimation of autocovariances or autocorrelations. Here each of these terms is estimated separately, using conventional estimates. There are different ways of doing this and the choice between these affects the properties of the estimation scheme. For example, negative estimates of the variance can be produced by some choices.

*Formulation as a [[least squares regression]] problem in which an ordinary least squares prediction problem is constructed, basing prediction of values of ''X''''t'' on the ''p'' previous values of the same series. This can be thought of as a forward-prediction scheme. The [[normal equations]] for this problem can be seen to correspond to an approximation of the matrix form of the Yule-Walker equations in which each appearance of an autocovariance of the same lag is replaced by a slightly different estimate.

*Formulation as an extended form of ordinary least squares prediction problem. Here two sets of prediction equations are combined into a single estimation scheme and a single set of normal equations. One set is the set of forward-prediction equations and the other is a corresponding set of backward prediction equations, relating to the backward representation of the AR model:
::<math> X_t = c + \sum_{i=1}^p \varphi_i X_{t-i}+ \varepsilon^*_t \,.</math>
:Here predicted of values of ''X''''t'' would be based on the ''p'' future values of the same series. This way of estimating the AR parameters is due to Burg,<ref name=Burg/> and call the Burg method:<ref name=Brockwell/> Burg and later authors called these particular estimates "maximum entropy estimates",<ref name=Burg1/> but the reasoning behind this applies to the use of any set of estimated AR parameters. Compared to the estimation scheme using only the forward prediction equations, different estimates of the autocovariances are produced, and the estimates have different stability properties. Burg estimates are particularly associated with [[maximum entropy spectral estimation]].<ref name=Bos/>

Other possible approaches to estimation include [[maximum likelihood estimation]]. Two distinct variants of maximum likelihood are available: in one (broadly equivalent to the forward prediction least squares scheme) the likelihood function considered is that corresponding to the conditional distribution of later values in the series given the initial ''p'' values in the series; in the second, the likelihood function considered is that corresponding to the unconditional joint distribution of all the values in the observed series. Substantial differences in the results of these approaches can occur if the observed series is short, or if the process is close to non-stationarity.

==Spectrum==
[[File:AutocorrTimeAr.svg|thumb|right]]
[[File:AutoCorrAR.svg|thumb|right]]

The [[Spectral_density#Power_spectral_density|power spectral density]] of an AR(''p'') process with noise variance <math>Var(Z_t) = \sigma_Z^2</math> is<ref name=Storch/>
: <math>S(f) = \frac{\sigma_Z^2}{| 1-\sum_{k=1}^p \varphi_k e^{-2 \pi i k f} |^2}.</math>

===AR(0)===
For white noise (AR(0))
: <math>S(f) = \sigma_Z^2.</math>

===AR(1)===
For AR(1)
: <math>S(f) = \frac{\sigma_Z^2}{| 1- \varphi_1 e^{-2 \pi i f} |^2}
= \frac{\sigma_Z^2}{ 1 + \varphi_1^2 - 2 \varphi_1 cos{2 \pi f} }</math>
*If <math>\varphi_1 > 0</math> there is a single spectral peak at f=0, often referred to as [[red noise]]. As <math>\varphi_1 </math> becomes nearer 1, there is stronger power at low frequencies, i.e. larger time lags. This is then a low-pass filter, when applied to full spectrum light, everything except for the red light will be filtered.
*If <math>\varphi_1 < 0</math> there is a minimum at f=0, often referred to as [[blue noise]]. This similarly acts as a high-pass filter, everything except for blue light will be filtered.

===AR(2)===
AR(2) processes can be split into three groups depending on the characteristics of their roots:
:<math>z_1,z_2 = \frac{1}{2}\left(\varphi_1 \pm \sqrt{\varphi_1^2 + 4\varphi_2}\right)</math>
* When <math>\varphi_1^2 + 4\varphi_2 < 0</math>, the process has a pair of complex-conjugate roots, creating a mid-frequency peak at:
:<math>f^* = \frac{1}{2\pi}\cos^{-1}\left(\frac{\varphi_1(\varphi_2-1)}{4\varphi_2}\right)</math>
Otherwise the process has real roots, and:
* When <math>\varphi_1 > 0</math> it acts as a low-pass filter on the white noise with a spectral peak at <math>f=0</math>
* When <math>\varphi_1 < 0</math> it acts as a high-pass filter on the white noise with a spectral peak at <math>f=1/2</math>.
The process is stable when the roots are within the unit circle, or equivalently when the coefficients are in the triangle <math>-1 \le \varphi_2 \le 1 - |\varphi_1|</math>.

The full PSD function can be expressed in real form as:
:<math>S(f) = \frac{\sigma_Z^2}{1 + \varphi_1^2 + \varphi_2^2 - 2\varphi_1(1-\varphi_2)\cos(2\pi f) - 2\varphi_2\cos(4\pi f)}</math>

==Implementations in statistics packages==
* [[R (programming language)|R]], the ''stats'' package includes an ''ar'' function.<ref>[http://finzi.psych.upenn.edu/R/library/stats/html/ar.html "Fit Autoregressive Models to Time Series"] (in R)</ref>
* [[Matlab (programming language)|Matlab]] and [[Octave (programming language)|Octave]]: the ''TSA toolbox'' contains several estimation functions for uni-variate, [[multivariate statistics|multivariate]] and adaptive autoregressive models.<ref>[http://pub.ist.ac.at/~schloegl/matlab/tsa/ "Time Series Analysis toolbox for Matlab and Octave"]</ref>

==''n''-step-ahead forecasting==

Once the parameters of the autoregression

:<math> X_t = c + \sum_{i=1}^p \varphi_i X_{t-i}+ \varepsilon_t \,</math>

have been estimated, the autoregression can be used to forecast an arbitrary number of periods into the future. First use ''t'' to refer to the first period for which data is not yet available; substitute the known prior values ''X''''t-i'' for ''i=''1, ..., ''p'' into the autoregressive equation while setting the error term <math>\varepsilon_t</math> equal to zero (because we forecast ''X''''t'' to equal its expected value, and the expected value of the unobserved error term is zero). The output of the autoregressive equation is the forecast for the first unobserved period. Next, use ''t'' to refer to the ''next'' period for which data is not yet available; again the autoregressive equation is used to make the forecast, with one difference: the value of ''X'' one period prior to the one now being forecast is not known, so its expected value—the predicted value arising from the previous forecasting step—is used instead. Then for future periods the same procedure is used, each time using one more forecast value on the right side of the predictive equation until, after ''p'' predictions, all ''p'' right-side values are predicted values from prior steps.

There are four sources of uncertainty regarding predictions obtained in this manner: (1) uncertainty as to whether the autoregressive model is the correct model; (2) uncertainty about the accuracy of the forecasted values that are used as lagged values in the right side of the autoregressive equation; (3) uncertainty about the true values of the autoregressive coefficients; and (4) uncertainty about the value of the error term <math>\varepsilon_t \,</math> for the period being predicted. Each of the last three can be quantified and combined to give a [[confidence interval]] for the ''n''-step-ahead predictions; the confidence interval will become wider as ''n'' increases because of the use of an increasing number of estimated values for the right-side variables.

==Evaluating the quality of forecasts==

The predictive performance of the autoregressive model can be assessed as soon as estimation has been done if [[cross-validation (statistics)|cross-validation]] is used. In this approach, some of the initially available data was used for parameter estimation purposes, and some (from available observations later in the data set) was held back for out-of-sample testing. Alternatively, after some time has passed after the parameter estimation was conducted, more data will have become available and predictive performance can be evaluated then using the new data.

In either case, there are two aspects of predictive performance that can be evaluated: one-step-ahead and ''n''-step-ahead performance. For one-step-ahead performance, the estimated parameters are used in the autoregressive equation along with observed values of ''X'' for all periods prior to the one being predicted, and the output of the equation is the one-step-ahead forecast; this procedure is used to obtain forecasts for each of the out-of-sample obefefeffeservations. To evaluate the quality of ''n''-step-ahead forecasts, the forecasting procedure in the previous section is employed to obtain the predictions.

Given a set of predicted values and a corresponding set of actual values for ''X'' for various time periods, a common evaluation technique is to use the [[mean squared prediction error]]; other measures are also available (see [[Forecasting#Forecasting accuracy]]).

The question of how to interpret the measured forecasting accuracy arises—for example, what is a "high" (bad) or a "low" (good) value for the mean squared prediction error? There are two possible points of comparison. First, the forecasting accuracy of an alternative model, estimated under different modeling assumptions or different estimation techniques, can be used for comparison purposes. Second, the out-of-sample accuracy measure can be compared to the same measure computed for the in-sample data points (that were used for parameter estimation) for which enough prior data values are available (that is, dropping the first ''p'' data points, for which ''p'' prior data points are not available). Since the model was estimated specifically to fit the in-sample points as well as possible, it will usually be the case that the out-of-sample predictive performance will be poorer than the in-sample predictive performance. But if the predictive quality deteriorates out-of-sample by "not very much" (which is not precisely definable), then the forecaster may be satisfied with the performance.

==See also==
* [[Moving average model]]
* [[Predictive analytics]]
* [[Linear predictive coding]]

==Notes==
{{Reflist | refs=

<ref name=Brockwell>{{cite journal| first1=Peter J.|last1=Brockwell |first2= Rainer|last2= Dahlhaus| first3=A. Alexandre|last3= Trindade
|journal=Statistica Sinica | volume= 15 |year=2005 |pages=197–213
|title=Modified Burg Algorithms for Multivariate Subset Autoregression
|url=http://www3.stat.sinica.edu.tw/statistica/oldpdf/A15n112.pdf}}</ref>

<ref name=Bos>{{cite doi|10.1109/TIM.2002.808031}}</ref>

<ref name=Burg>Burg, J. P. (1968). "A new analysis technique for time series data". In ''Modern Spectrum Analysis'' (Edited by D. G. Childers), NATO Advanced Study Institute of Signal Processing with emphasis on Underwater Acoustics. IEEE Press, New York.</ref>

<ref name=Burg1>Burg, J.P. (1967) "Maximum Entropy Spectral Analysis", ''Proceedings of the 37th Meeting of the Society of
Exploration Geophysicists'', Oklahoma City, Oklahoma.</ref>

}}

==References==
*Mills, Terence C. (1990) ''Time Series Techniques for Economists.'' Cambridge University Press
*Percival, Donald B. and Andrew T. Walden. (1993) ''Spectral Analysis for Physical Applications.'' Cambridge University Press
*Pandit, Sudhakar M. and Wu, Shien-Ming. (1983) ''Time Series and System Analysis with Applications.'' John Wiley & Sons
*[[Udny Yule|Yule, G. Udny]] (1927) [http://visualiseur.bnf.fr/Visualiseur?Destination=Gallica&O=NUMM-56031 "On a Method of Investigating Periodicities in Disturbed Series, with Special Reference to Wolfer's Sunspot Numbers"], ''[[Philosophical Transactions of the Royal Society]] of London'', Ser. A, Vol. 226, 267–298.]
*Walker, Gilbert (1931) [http://visualiseur.bnf.fr/Visualiseur?Destination=Gallica&O=NUMM-56224 "On Periodicity in Series of Related Terms"], ''[[Proceedings of the Royal Society]] of London'', Ser. A, Vol. 131, 518–532.

==External links==
*[http://paulbourke.net/miscellaneous/ar/ AutoRegression Analysis (AR) by Paul Bourke]

{{Stochastic processes}}

[[Category:Noise]]
[[Category:Time series models]]
[[Category:Signal processing]]

Transformation semigroup

2013-10-09T20:11:47Z

81.155.221.114:

The '''Customer Average Interruption Frequency Index (CAIFI)'''<Ref>{{cite web
|url =http://www.ee.iastate.edu/~jdm/ee653/DistributionReliabilityPredictive.ppt
|title = Distribution System Reliability Evaluation
|first = Sree
|last = Yeddanapudi
|publisher= Iowa State University
|accessdate = 18 June 2011
}}</ref> is a popular index used in electrical [[reliability analysis]].<ref>[http://books.google.com/books?id=9EShPwTRnoUC&pg=PA112&lpg=PA112&dq=caifi+%22four+most+popular%22&source=web&ots=SngUPzIEDK&sig=Iv7PKqW_0YusKTYJgXz3PHeTWsg&hl=en&sa=X&oi=book_result&resnum=1&ct=result Power Distribution Planning Reference Book, H. Lee Willis, published by CRC Press, 2004, page 112]</ref> It is designed to show trends in customers interrupted and helps to show the number of customers affected out of the whole customer base.

<math>\mbox{CAIFI} = \frac{\mbox{total number of customer interruptions}}{\mbox{total number of customers who had at least one interruption}}</math>
==References==
{{reflist}}

==See also==
*[[SAIDI]]
*[[CAIDI]]
*[[MAIFI]]
*[[SAIFI]]

[[Category:Electric power]]

Łoś–Tarski preservation theorem

2013-10-09T16:03:41Z

81.155.221.114:

[[Magnetohydrodynamics]] (MHD) deals with what is a quasi-neutral fluid with very high conductivity. The fluid approximation implies that the we focus at macro length and time scales which are much larger than the collision length and collision time respectively. In this article we will discuss MHD turbulence which is observed when the [[Reynolds number]] of the magnetofluid is large.

== Incompressible MHD equations ==

The incompressible MHD equations are

<math>
\begin{array}{lcl}
\frac{\partial \mathbf{u}}{\partial t} + \mathbf{u} \cdot \nabla \mathbf{u} & = & -\nabla p + \mathbf{B} \cdot \nabla \mathbf{B} + \nu \nabla^2 \mathbf{u} \\

\frac{\partial \mathbf{B}}{\partial t} + \mathbf{u} \cdot \nabla \mathbf{B} & = & \mathbf{B} \cdot \nabla \mathbf{u} +
\eta \nabla^2 \mathbf{B} \\

\nabla \cdot \mathbf{u} & = & 0 \\
\nabla \cdot \mathbf{B} & = & 0.

\end{array}
</math>

where '''u''', '''B''', ''p'' represent the velocity, magnetic, and total pressure (thermal+magnetic) fields, <math>\nu </math> and <math>\eta </math> represent [[kinematic viscosity]] and [[magnetic diffusivity]]. The third equation is the [[Incompressible flow|incompressibility condition]]. In the above equation, the [[magnetic field]] is in Alfvén units (same as velocity units).

The total magnetic field can be split into two parts: <math> \mathbf{B} = \mathbf{B_0} + \mathbf{b} </math> (mean + fluctuations).

The above equations in terms of Elsässer variables (<math> \mathbf{z}^{\pm} = \mathbf{u} \pm \mathbf{b} </math>) are

<math>
\frac{\partial {\mathbf{z}^{\pm}}}{\partial t}\mp\left(\mathbf {B}_0\cdot{\mathbf \nabla}\right){\mathbf z^{\pm}} + \left({\mathbf z^{\mp}}\cdot{\mathbf \nabla}\right){\mathbf z^{\pm}} = -{\mathbf \nabla}p
+ \nu_+ \nabla^2 \mathbf{z}^{\pm} + \nu_- \nabla^2 \mathbf{z}^{\mp}
</math>

where <math> \nu_\pm = \nu \pm \eta </math>. Nonlinear interactions occur between the Alfvénic fluctuations <math>
z^{\mp} </math>.

The important nondimensional parameters for MHD are

<math>
\begin{array}{lcl}
\text{Reynolds number } Re & = & U L /\nu \\
\text{Magnetic Reynolds number } Re_M & = & U L /\eta \\
\text{Magnetic Prandtl number } P_M & = & \nu / \eta.
\end{array}
</math>

The [[magnetic Prandtl number]] is an important property of the fluid. Liquid metals have small magnetic Prandtl numbers, for example, liquid sodium's <math> P_M </math> is around <math> 10^{-5} </math>. But plasmas have large <math> P_M </math>.

The Reynolds number is the ratio of the nonlinear term <math> \mathbf{u} \cdot \nabla \mathbf{u} </math> of the Navier-Stokes equation to the viscous term. While the magnetic Reynolds number is the ratio of the nonlinear term and the diffusive term of the induction equation.

In many practical situations, the Reynolds number <math> Re </math> of the flow is quite large. For such flows typically the velocity and the magnetic fields are random. Such flows are called to exhibit MHD turbulence. Note that <math> Re_M </math> need not be large for MHD turbulence. <math> Re_M </math> plays an important role in dynamo (magnetic field generation) problem.

The mean magnetic field plays an important role in MHD turbulence, for example it can make the turbulence anisotropic; suppress the turbulence by decreasing energy cascade etc. The earlier MHD turbulence models assumed isotropy of turbulence, while the later models have studied anisotropic aspects. In the following discussions will summarize these models. More discussions on MHD turbulence can be found in Biskamp<ref>D. Biskamp (2003), Magnetohydrodynamical Turbulence, (Cambridge University Press, Cambridge.)</ref> and Verma.<ref name="mkv-physrep">M. K. Verma (2004), Statistical theory of magnetohydrodynamic turbulence, Phys. Rep., 401, 229.</ref>

== Isotropic models ==

Iroshnikov<ref>P. S. Iroshnikov (1964), Turbulence of a Conducting Fluid in a Strong Magnetic Field, Soviet Astronomy, 7, 566.</ref> and Kraichnan<ref>R. Kraichnan(1965), Inertial-Range Spectrum of Hydromagnetic Turbulence, Physics of Fluids, 8, 1385.</ref> formulated the first phenomenological theory of MHD turbulence. They argued that in the presence
of a strong mean magnetic field, <math> z^+ </math> and <math> z^- </math> wavepackets travel in opposite directions with
the phase velocity of <math>B_0</math>, and interact weakly. The relevant time scale is Alfven time <math>(B_0 k)^{-1}</math>. As a results the energy spectra is

:<math eq:Kraichnan>
E^u(k) \approx E^b(k) \approx A (\Pi V_A)^{1/2} k^{-3/2}.
</math>

where <math> \Pi </math> is the energy cascade rate.

Later Dobrowolny et al.<ref>M. Dobrowlny, A. Mangeney, P. Veltri (1980), Fully developed anisotropic hydromagnetic turbulence in interplanetary plasma, Phys. Rev. Lett., 45, 144.</ref> derived the following generalized formulas for the cascade rates of <math> z^{\pm} </math> variables:
:<math eq:Dobrowolny>
\Pi^+ \approx \Pi^{-} \approx \tau^{\pm}_k E^{+}(k) E^{-}(k) k^4 \approx E^{+}(k) E^{-}(k) k^3 / B_0
</math>

where <math> \tau^{\pm} </math> are the interaction time scales of <math> z^{\pm} </math> variables.

Iroshnikov and Kraichnan's phenomenology follows once we choose <math> \tau^{\pm} \approx 1/(k V_A) </math>.

Marsch<ref>E. Marsch (1990), Turbulence in the solar wind, in: G. Klare (Ed.), Reviews in Modern Astronomy, Springer, Berlin, p. 43.</ref> chose the nonlinear time scale <math> T_{NL}^{\pm} \approx (k z_k^{\mp})^{-1} </math> as the interaction time scale for the eddies and derived Kolmogorov-like energy spectrum for the Elsasser variables:
:<math eq:Kolm>
E^{\pm}(k) = K^{\pm} (\Pi^{\pm})^{4/3} (\Pi^{\mp})^{-2/3} k^{-5/3}
</math>
where <math> \Pi^+ </math> and <math> \Pi^- </math> are the energy cascade rates of <math> z^+ </math> and <math> z^- </math> respectively, and <math> K^{\pm} </math> are constants.

Matthaeus and Zhou<ref>W. H. Matthaeus, Y. Zhou (1989), Extended inertial range phenomenology of magnetohydrodynamic turbulence, Phys. Fluids B, 1, 1929.</ref> attempted to combine the above two time scales by postulating the interaction time to be the harmonic
mean of Alfven time and nonlinear time.

The main difference between the two competing phenomenologies (-3/2 and -5/3) is the chosen time scales for the interaction time.
The main underlying assumption in that Iroshnikov and Kraichnan's phenomenology should work for strong mean magnetic field,
whereas Marsh's phenomenology should work when the fluctuations dominate the mean magnetic field (strong turbulence).

However, as we will discuss below, the solar wind observations and numerical simulations tend to favour -5/3 energy spectrum
even when the mean magnetic field is stronger compared to the fluctuations. This issue was resolved by Verma<ref>M. K. Verma (1999), Mean magnetic field renormalization and Kolmogorov’s energy spectrum in magnetohydrodynamic turbulence, Phys. Plasmas 6, 1455.</ref> using [[renormalization]] group analysis by showing that the Alfvénic fluctuations are affected by scale-dependent "local mean magnetic field". The local mean magnetic field scales as <math> k^{-1/3} </math>, substitution of which in Dobrowolny's equation yields Kolmogorov's energy spectrum for MHD turbulence.

Renormalization group analysis have been also performed for computing the renormalized viscosity and resistivity. It was shown that these diffusive quantities scale as <math> k^{-4/3} </math> that again yields <math> k^{-5/3} </math> energy spectra consistent with Kolmogorov-like model for MHD turbulence. The above renormalization group calculation has been performed for both zero and nonzero cross helicity.

The above phenomenologies assume isotropic turbulence that is not the case in the presence of a mean magnetic field. The mean magnetic field typically suppresses the energy cascade along the direction of the mean magnetic field.<ref>J. V. Shebalin, W. H. Matthaeus, D. Montgomery (1983), Anisotropy in mhd turbulence due to a mean magnetic field, J. Plasma Phys., 29, 525.</ref>

== Anisotropic models ==

Mean magnetic field makes turbulence anisotropic. This aspect has been studied in last two decades. In the limit
<math> \delta z^{\pm} \ll B_0 </math>, Galtier et al.<ref>S. Galtier, S. V. Nazarenko, A. C. Newell, A. Pouquet (2000), A weak turbulence theory for incompressible magnetohydrodynamics, Journal of Plasma Physics, 63, 447</ref> showed using kinetic equations that

<math>

E(k) \sim (\Pi B_0)^{1/2} k_{||}^{1/2} k_{\perp}^{-2}

</math>

where <math> k_{||} </math> and <math> k_{\perp} </math> are components of the wavenumber parallel and perpendicular to mean magnetic field. The above limit is called the '''weak turbulence limit'''.

Under the strong turbulence limit, <math> \delta z^{\pm} \sim B_0 </math>, Goldereich and Sridhar<ref name=GS95>Goldreich, P. & Sridhar, S. (1995), Toward a theory of interstellar turbulence. 2: Strong Alfvénic turbulence, Astrophysical Journal, 438, 763</ref> argue that <math> k_{\perp} z_{k_{\perp}} \sim k_{||} B_0 </math> ("critical balanced state") which implies that

<math>
\begin{array}{lcl}
E(k) & \propto & k_{\perp}^{-5/3}; \\
k_{||} & \propto & k_{\perp}^{2/3}
\end{array}
</math>

The above anisotropic turbulence phenomenology has been extended for large cross helicity MHD.

== Solar wind observations ==

Solar wind plasma is in turbulent state. Researchers have calculated the energy spectra of the solar wind plasma from the data
collected from the spacecraft. The kinetic and magnetic energy spectra, as well as <math> E^{\pm} </math> are closer to
<math> k^{-5/3} </math> compared to <math> k^{-3/2} </math>, thus favoring Kolmogorov-like phenomenology for MHD
turbulence.<ref>W. H. Matthaeus, M. L. Goldstein (1982), Measurement of the rugged invariants of magnetohydrodynamic turbulence in the solar wind, J. Geophys. Res., 87, 6011.</ref><ref>D. A. Roberts, M. L. Goldstein (2001), Turbulence and waves in the solar wind, Rev. Geophys., 29, 932.</ref> The interplanetary and interstellar electron density fluctuations also provide
a window for investigating MHD turbulence.

== Numerical simulations ==

The theoretical models discussed above are tested using the high resolution direct numerical simulation (DNS). Number of recent simulations report the spectral indices to be closer to 5/3.<ref>W.-C. Müller, D. Biskamp (2000) , Scaling properties of three-dimensional magnetohydrodynamic turbulence, Phys. Rev. Lett., 84, 475.</ref> There are others that report the spectral indices near 3/2. The regime of power law is typically less than a decade. Since 5/3 and 3/2 are quite close numerically, it is quite difficult to ascertain the validity of MHD turbulence models from the energy spectra.

Energy fluxes <math> \Pi^{\pm} </math> can be more reliable quantities to validate MHD turbulence models.
When <math> E^+(k) \gg E^-(k) </math>
(high cross helicity fluid or imbalanced MHD) the energy flux predictions of Kraichnan and Iroshnikov model is very different from that of Kolmogorov-like model. It has been shown using DNS that the fluxes <math> \Pi^{\pm} </math> computed from the numerical simulations are in better agreement with Kolmogorov-like model compared to Kraichnan and Iroshnikov model.<ref>M. K. Verma, D. A. Roberts, M. L. Goldstein, S. Ghosh, W. T. Stribling (1996), A numerical study of the nonlinear cascade of energy in magnetohydrodynamic turbulence, J. Geophys. Res., 101, 21619.</ref>

Anisotropic aspects of MHD turbulence have also been studied using numerical simulations. The predictions of Goldreich and Sridhar<ref name="GS95"/> (<math> k_{||} \sim k_{\perp}^{2/3} </math>) have been verified in many simulations. Some of the recent simulations report dynamic alignment of velocity and magnetic field fluctuations in the inertial range, and <math> k^{-3/2} </math> energy spectra.<ref>J. Mason, F. Cattaneo, S. Boldyrev (2008), Numerical measurements of the spectrum in magnetohydrodynamic turbulence, Phys. Rev. E, 77, 036403.</ref>

== Energy transfer ==

Energy transfer among various scales between the velocity and magnetic field is an important problem in MHD turbulence. These quantities
have been computed both theoretically and numerically.<ref name=mkv-physrep /> These calculations show a significant energy transfer from the
large scale velocity field to the large scale magnetic field. Also, the cascade of magnetic energy is typically forward. These results have critical
bearing on dynamo problem.

----

There are many open challenges in this field that hopefully will be resolved in near future with the help of numerical simulations, theoretical modelling, experiments, and observations (e.g., solar wind).

== See also ==
* [[Magnetohydrodynamics]]
* [[Turbulence]]
* [[Alfvén wave]]
* [[Solar dynamo]]
* [[Reynolds number]]
* [[Navier–Stokes equations]]
* [[Computational magnetohydrodynamics]]
* [[Computational fluid dynamics]]
* [[Solar wind]]
* [[Magnetic flow meter]]
* [[Ionic liquid]]
* [[List of plasma (physics) articles]]

==References==
{{reflist}}

[[Category:Magnetohydrodynamics]]
[[Category:Turbulence]]

Template:Highland Football League

2013-10-06T11:53:00Z

81.155.30.122:

Not only are folks shopping for second homes all world wide but they're additionally transferring to countries which are sunnier. Singapore is among the countries the place many people relocate to and it's due to this that there has been a rise within the demand for Singapore property agent. So who is a Singapore property agent? Property brokers for Singapore Land Authority masking a number of the authorities colonial properties for lease. 201, Kim Tian Road, #03-four hundred, Singapore 160201. Expat relocation, sale and rental of personal properties, condos, homes and a Singapore property picture information of the properties that are obtainable. Freehold Condo near Orchard The Edge on Cairnhill – [http://dev.lizandcode.com/318081-surge-of-undertaking-launches-anticipated-19-april-2014 Condominium For Sale] I want to relate a very sad, and I'd say almost unethical home buying experience of ours in Singapore which just happened immediately. We as consumers in Singapore and I'm positive we're not alone right here feeling this fashion; are annoyed that there isn't any real regulatory body in Singapore to vary legislature or effectively regulate the true estate industry right here to protect buyers. NEW LAUNCHES It is important to perceive that the vendor and the agent are working collectively to sell the home at the absolute best price. The partnership between the agent and the proprietor is a key factor to a clean and enjoyable sales process. Home sellers should so far as doable give correct and full information about the home. This will help the agent in analysing the energy and weak spot of your home and current it in a optimistic gentle to a potential purchaser. If there must be any slight misunderstanding, it ought to be resolved with the agent as soon as attainable. Home homeowners have to place belief into their agent. Therefore get a responsible agent and one that you may work well with. Proper planning - Pen down your objectives and the steps you will take to attain them. Train some creativity and let your finest concepts bring you results. Comply with up – So essential however usually ignored. Some of the most profitable brokers have prospects who give them referrals, so they do not have to keep prospecting new purchasers. Why? It's as a result of they stay in touch with their clients. It may be as simple as month-to-month SMS messages to remind your customer that you're still in the business. Such an motion grows your referral base. The drop out rate in our business is relatively excessive. After about 3 to 6 months, beginner agents lose their steam and their preliminary "Can do" perspective turns into a "Can I do it?" perspective. Get together Commission / Charge Payable to Agency Achievers are at all times the first to check new applied sciences & services that may help them enhance their sales. When property guru first began, many brokers didn't believe in it till they began hearing different colleagues getting implausible results. Most agents desires to see proof first, before they dare to take step one in trying. These are often the late comers or late adopters. There is a purpose why prime achievers are heading the wave or heading the way. Just because they fight new issues ahead of others. The rest just comply with after! Secondly, a Negligent Misrepresentation occurs when a statement made by the Representor truthfully believing in its fact (falling outdoors the scope of fraudulent misrepresentation) however without having reasonable grounds for thus believing. Referring to the same example above, considering now that estate agent A, when informing the potential patrons, honestly believed that the tenure of the landed property is freehold. But A had no affordable ground to believe that it's such. A might have simply cleared his doubts by simply conducting a title search on the stated property. In other words, A made an trustworthy but careless illustration to the prospective consumers about the tenure of the landed property. It's also generally free to tenants. Nevertheless tenants paying lower than $3,500 per 30 days in rental typically pay a half month commission for each year of rental. For leases above $three,500, tenants don't pay any commission. Does the purchase process differ for residential, retail, office and industrial properties? Our Marketing consultant will help you to have a head start within the SEO bandwagon by performing the right SEARCH ENGINE MARKETING implementation with our customized SEARCH ENGINE MARKETING ranking tatics. Fundamental Search Engine Optimization Skill so as to add / edit or delete Project Change Ordering of Mission Unpublish or Publish a Project Skill so as to add / edit or delete a banner of the Mission I believe what amazes us firstly is that both owner and agent have to be there with us! Month-to-month rent is $3,888

Toeplitz matrix

2013-10-05T04:02:36Z

81.155.54.84: Undid revision 573868946 by Mark viking (talk)--different Bareiss alg

{{Expand German|Rasterung von Linien|fa=yes|topic=sci|date=December 2009}}
A '''line drawing algorithm''' is a graphical [[algorithm]] for approximating a line segment on discrete graphical media. On discrete media, such as [[pixel]]-based [[computer display|display]]s and [[computer printer|printer]]s, line drawing requires such an approximation (in nontrivial cases).

On continuous media, by contrast, no algorithm is necessary to draw a line. For example, [[oscilloscope]]s use natural phenomena to draw lines and curves.

The Cartesian slope-intercept equation for a straight line is
Y= mx+b
With m representing the slope of the line and b as the y intercept. Given that the two endpoints of the line segment are specified at positions (x1,y1) and (x2,y2). we can determine values for the slope m and y intercept b with the following calculations,

m=(y2-y1)/(x2-x1)

so, b=y1-m.x1

==A naïve line-drawing algorithm==
<code>
dx = x2 - x1
dy = y2 - y1
'''for''' x '''from''' x1 '''to''' x2 {
y = y1 + dy * (x - x1) / dx
plot(x, y)
}</code>
It is assumed here that the points have already been ordered so that <math>x_2 > x_1</math>.
This algorithm works just fine when <math>dx >= dy</math> (i.e., slope is less than or equal to 1), but if <math>dx < dy</math> (i.e., slope greater than 1), the line becomes quite sparse with lots of gaps, and in the limiting case of <math>dx = 0</math>, only a single point is plotted.

The naïve line drawing algorithm is inefficient and thus, slow on a digital computer. Its inefficiency stems from the number of operations and the use of floating-point calculations. Line drawing algorithms such as [[Bresenham's line algorithm|Bresenham]]'s or [[Xiaolin Wu's line algorithm|Wu]]'s are preferred instead.

==List of line drawing algorithms==
The following is a partial list of line drawing algorithms:
*[[Digital Differential Analyzer (graphics algorithm)]] — Similar to the naive line-drawing algorithm, with minor variations.
*[[Bresenham's line algorithm]] — optimized to use only additions (i.e. no divisions or multiplications); it also avoids floating-point computations.
*[[Xiaolin Wu's line algorithm]] — can perform [[spatial anti-aliasing]]

==References==
Fundamentals of Computer Graphics, 2nd Edition, A.K. Peters by Peter Shirley

{{DEFAULTSORT:Line Drawing Algorithm}}
[[Category:Computer graphics algorithms]]

{{Link FA|de}}

Logical constant

2013-04-24T17:33:20Z

81.155.194.151:

{{Unreferenced|date=December 2009}}
[[Bond convexity]] [[closed-form formula]] (Blake and Orszag):

<math>Conv=-\frac{D}{P}\begin{Bmatrix}\frac{(m-1+a+1)(m-1+a+2)(1/(1+i))^{(m-1+a+2)}}{i}+\\2\frac{(m-1+a+2)(1/(1+i))^{(m-1+a+2)}-(1/(1+i))}{i^2}+\\2\frac{(1/(1+i))^{(m-1+a+2)}-(1/(1+i)}{i^3}\end{Bmatrix}+\frac{B}{P}\frac{(m-1+a)(m-1+a+1)}{(1+i)^{(m-1+a+2)}}</math>

 D = coupon payment per period
 P = present value (price)
 B = face value
 i = discount rate per period (half-year)
 a = fraction of a period remaining until next coupon payment
 m = number of coupon dates until maturity

Look up [[Bond duration closed-form formula]]

{{DEFAULTSORT:Bond Convexity Closed-Form Formula}}
[[Category:Fixed income analysis]]

Complete partial order

2013-04-23T23:16:22Z

81.155.194.151: /* Definitions */

[[File:Dominant seventh chord on C.png|thumb|right|Dominant seventh chord on C: C7 {{audio|Dominant seventh chord on C.mid|Play}}.]]

In [[music theory]], a '''dominant seventh chord''', or '''major minor seventh chord''',{{efn|Also written '''major-minor seventh chord'''.}} is a [[chord (music)|chord]] composed of a [[root (chord)|root]], [[major third]], [[perfect fifth]], and [[minor seventh]]. It can be also viewed as a [[Major chord|major triad]] with an additional [[minor seventh]]. It is denoted using [[popular music symbols]] by adding a superscript "7" after the letter designating the chord root.<ref>Benward & Saker (2003). ''Music: In Theory and Practice, Vol. I'', p.77. Seventh Edition. ISBN 978-0-07-294262-0.</ref>
The dominant seventh is found almost as often as the [[dominant (music)|dominant triad]].<ref>Benward & Saker (2003), p.199.</ref> In [[Roman numeral analysis|Roman numerals]] it is represented as V7. The chord can be represented by the [[Pitch class#Integer notation|integer notation]] {0, 4, 7, 10}.

{{Infobox chord|
chord_name=dominant seventh chord|
first_interval=[[Root (chord)|root]]|
second_interval=[[major third]]|
third_interval=[[perfect fifth]]|
fourth_interval=[[minor seventh]]|
tuning=[[just intonation|20:25:30:36]]<ref name="Shirlaw">Shirlaw, Matthew (1900). ''The Theory of Harmony'', p.86. ISBN 978-1-4510-1534-8.</ref>|
forte_number=4-27|
complement=8-27
}}

Of all the [[seventh chord]]s, perhaps the most important is the dominant seventh. It was the first seventh chord to appear regularly in [[classical music]]. The name comes from the fact that it occurs naturally in the [[seventh chord]] built upon the [[Dominant (music)|dominant]] (i.e. the fifth [[Scale degree|degree]]) of a given major [[diatonic scale]].
Take for example the C major scale (C, D, E, F, G, A, B, C):

[[File:Dominant seventh in C major.png|400px]]

The note G is the dominant degree of C major - its fifth note. When we arrange the notes of the C major scale in ascending pitch and use only these notes to build a seventh chord, and we start with G (not C), then the resulting chord contains the four notes G-B-D-F and is called G dominant seventh (G7). The note F is a [[minor seventh]] from G, and it is also called the dominant seventh with respect to G. However, the 'dominant' seventh is used on notes other than the dominant, such as the subdominant. {{citation needed|date=January 2012}}

==Function==
[[File:V7-I resolution.png|thumb|Dominant seventh resolving to tonic in C major (V<math>{}^6_5</math>-I). {{audio|V7-I resolution.mid|Play}}]]
[[File:Beethoven - Piano Sonata in B-flat major, Op. 22 - dominant seventh.png|thumb|Tritone resolution in [[Ludwig van Beethoven|Beethoven]]'s ''[[Piano Sonata No. 11 (Beethoven)|Piano Sonata in B-flat major]]'', Op. 22 (1800).<ref>Forte, Allen (1979). ''Tonal Harmony in Concept & Practice'', p.145. Third edition. ISBN 0-03-020756-8.</ref> {{audio|Beethoven - Piano Sonata in B-flat major, Op. 22 - dominant seventh.mid|Play}}]]

The function of the dominant seventh chord is to drive to or resolve to the tonic note or chord.

{{quote|...the demand of the V7 for resolution is, ''to our ears'', almost inescapably compelling. The dominant seventh is, in fact, the central propulsive force in our music; it is unambiguous and unequivocal.|Goldman|(1965: 35)<ref name="Goldman">Goldman, Richard Franco (1965), ''Harmony in Western Music'', p.34-35. ISBN 0-214-66680-8.</ref>}}

This basic dominant seventh chord is useful to composers because it contains both a major triad and the interval of a [[tritone]]. The major triad confers a very "strong" sound. The tritone is created by the co-occurrence of the third degree and seventh degree (e.g., in the G7 chord, the [[Interval (music)|acoustic distance]] between B and F is a tritone). In a diatonic context, the third of the chord is the [[leading-tone]] of the scale, which has a strong tendency to pull towards the tonal center, or root note, of the key (e.g., in C, the third of G7, B, is the leading tone of the key of C). The seventh of the chord acts as an upper leading-tone to the third of the scale (in C: the seventh of G7, F, is a half-step above and leads down to E).<ref name="Goldman"/> This, in combination with the strength of [[counterpoint|root movement by fifth]], and the natural resolution of the dominant triad to the tonic triad (e.g., from GBD to CEG in the key of C major), creates a resolution with which to end a piece or a section of a piece. Because of this original usage, it also quickly became an easy way to trick the listener's ear with a [[deceptive cadence]]. The dominant seventh may work as part of a [[circle progression]], preceded by the [[supertonic]].

[[File:Charlie Parker - Au Privave - circle progression dominant seventh.png|thumb|center|350px|Dominant seventh in circle progression in [[Charlie Parker]]'s "[[Au Privave]]" (1956).<ref>Benward & Saker (2003), p.202.</ref> {{audio|Charlie Parker - Au Privave - circle progression dominant seventh.mid|Play}}]]

In [[rock music|rock]] and [[popular music]] songs following, "the blues harmonic pattern," IV and V are, "almost always," major minor seventh chords, or extensions, with the tonic most often being a major triad, for example [[Bill Haley & His Comets|Bill Haley and the Comets]]' "[[Rock Around the Clock|Rock Around The Clock]]" and [[Buster Brown (musician)|Buster Brown]]'s "[[Fannie Mae (song)|Fanny Mae]]", while in [[Chuck Berry]]'s "[[Back in the U.S.A.]]" and [[Loggins and Messina]]'s "[[Your Mama Don't Dance]]" the tonic is also a major minor seventh.<ref name="Stephenson">Stephenson, Ken (2002). ''What to Listen for in Rock: A Stylistic Analysis'', p.82. ISBN 978-0-300-09239-4.</ref> Used mostly in the first fifteen years of the rock era and now sounding somewhat, "retrospective," ([[Oasis (band)|Oasis]]' "[[Roll with It (Oasis song)|Roll With It]]") other examples of tonic dominant seventh chords include [[Little Richard]]'s "[[Lucille (Little Richard song)|Lucille]]", [[the Beatles]]' "[[I Saw Her Standing There]]", [[Harry Nilsson|Nilsson]]'s "[[Coconut (song)|Coconut]]", [[Jim Croce]]'s "[[You Don't Mess Around with Jim|You Don't Mess Around With Jim]]", and [[the Drifters]]' "[[On Broadway (song)|On Broadway]]".<ref name="Stephenson"/> Chuck Berry's "[[Rock and Roll Music|Rock And Roll Music]]" uses the dominant seventh on I, IV, and V.<ref>Stephenson (2002), p.75.</ref> See: [[Twelve-bar blues]].

===Chromatic seventh===
[[File:V of V in C four-part harmony.png|thumb|V of V in C, four-part harmony {{audio|V of V in C four-part harmony.mid|Play}}.]]
However, the most important use of the dominant seventh chord in musical composition is the way that the introduction of a non-diatonic dominant seventh chord (sometimes called a ''[[chromatic]]'' seventh), which is borrowed from another key, can allow the composer to [[modulation (music)|modulate]] to that other key. This technique is extremely common, particularly since the classical period, and has led to further innovative uses of the dominant seventh chord such as [[secondary dominant]] (V7/V), [[extended dominant]] (V/V/V), and [[substitute dominant]] ({{music|b}}V7/V) chords.

===German sixth===
[[File:German sixth equals dominant seventh.png|thumb|right|German sixth and equivalent dominant seventh {{audio|German sixth equals dominant seventh.mid|Play}}.]]
The dominant seventh is [[enharmonic|enharmonically equivalent]] to the [[Augmented sixth chord#German sixth|German sixth]], causing the chords to be spelled enharmonically, for example the German sixth G{{music|b}}-B{{music|b}}-D{{music|b}}-E and the dominant seventh F{{music|#}}-A{{music|#}}-C{{music|#}}-E.<ref>Benward & Saker (2008). ''Music in Theory and Practice'', Vol. II, p.222. ISBN 978-0-07-310188-0.</ref>

===Harmonic seventh===
[[File:Harmonic seventh chord just on C.png|thumb|right|Just harmonic seventh chord on C {{audio|Harmonic seventh chord just on C.mid|Play just}}. 7th: 968.826 cents, a [[septimal quarter tone]] lower than B{{music|b}}.]]
The dominant seventh is frequently used to approximate a [[Harmonic seventh chord]], which is one possible [[just intonation|just tuning]], in the ratios 4:5:6:7<ref name="Benitez">Benitez, J. M. (1988). ''Contemporary Music Review: Listening 2'', p.34. ISBN 3-7186-4846-6. Cites [[Leonhard Euler|Euler]] (1764).</ref> {{audio|Harmonic seventh on C.mid|Play}}, for the dominant seventh. Others include 20:25:30:36 {{audio|Just dominant seventh chord on C.mid|Play}}, found on I, and 36:45:54:64, found on V, used in [[Five-limit tuning|5-limit]] just tunings and scales.<ref name="Wright">Wright, David (2009). ''Mathematics and Music'', p.140-41. ISBN 978-0-8218-4873-9.</ref>

==History==
[[File:Monteverdi Lamento d'Arianna dominant seventh.png|thumb|Dominant seventh (in blue) handled conservatively, "prepared and resolved as a [[suspension (music)|suspension]], clearly indicating its dissonant status," in the Baroque Period (1600–1750) and Monteverdi's "Lasciatemi Morire", ''[[L'Arianna#Arianna's lament|Lamento d'Arianna]]'' (1608).<ref name="BS201">Benward & Saker (2003), p.201.</ref> {{audio|Monteverdi Lamento d'Arianna dominant seventh.mid|Play}}]]

[[File:Beethoven - Fifth Symphony - Last movement, dominant seventh.png|thumb|center|350px|Dominant seventh in Beethoven's ''[[Symphony No. 5 (Beethoven)|Fifth Symphony]]'' (1804–08), last movement.<ref>Forte (1979), p.142.</ref> {{audio|Beethoven - Fifth Symphony - Last movement, dominant seventh.mid|Play}}]]

Renaissance composers thought in terms of intervals rather than chords, "however, certain dissonant sonorities suggest that the dominant seventh chord occurred with some frequency." Monteverdi (usually credited as the first to use the V7 chord without [[preparation (music)|preparation]]<ref>Goldman (1965), p.39.</ref>) and other early baroque composers begin to treat the V7 as a chord as part of the introduction of functional harmony. The V7 was in constant use during the classical period, with similar treatment to that of the baroque. In the romantic period freer voice-leading was gradually developed, leading to the waning of functional use in the post-romantic and impressionistic periods including more dissonant dominant chords through higher extensions and lessened use of the major-minor chord's dominant function. 20th century music either consciously used functional harmony or was entirely free of V7 chords while jazz and popular musics continued to use functional harmony including V7 chords.<ref name="BS201"/>

[[File:Chopin - Mazurka in F minor, op. 68, no. 4, m. 1-4, dominant sevenths.png|thumb|center|350px|Dominant sevenths in [[Frédéric Chopin|Chopin]]'s Mazurka in F Minor (1849), op. 68, no. 4, m. 1-4: "the seventh factor had by this time [Romantic Period (1825—1900)] achieved nearly consonant status."<ref name="BS201"/> {{audio|Chopin - Mazurka in F minor, op. 68, no. 4, m. 1-4, dominant sevenths.mid|Play}}]]

However, according to Schenker, "'The dissonance is always passing, ''never a chord member'' (''Zusammenklang''),'"<ref>Schenker, Heinrich. ''Jahrbuch II'', p. 24 cited in [[Oswald Jonas|Jonas, Oswald]] (1982). ''Introduction to the Theory of Heinrich Schenker'' (1934: ''Das Wesen des musikalischen Kunstwerks: Eine Einführung in Die Lehre Heinrich Schenkers''), p. 20. Trans. John Rothgeb. ISBN 0-582-28227-6.</ref> and often (though by no means always) the [[voice leading]] suggests either a [[passing note]]:
8 7 3
5 5 1

or resolution of a (hypothetical) [[suspension (music)|suspension]]:
(8) 7 3
(4) 5 1

Today, the dominant seventh chord enjoys particular prominence in the music of [[Barbershop music|barbershop quartets]], with the [[Barbershop Harmony Society]] specifying that a song must use the chord type (built on any scale degree, not just the dominant) for 35 to 60 percent of its duration to be considered "true barbershop" (i.e. eligible for use in competitions). As barbershop singers strive to harmonize in [[Just intonation|just intonation]] to maximize the audibility of harmonic [[Overtones|overtones]], the practical sonority of the chord tends to be that of an [[Harmonic seventh chord|harmonic seventh chord]]. This chord type has become so ingrained into the fabric of the artform that it is often referred to as the "barbershop seventh chord" by those who practice it.

==Voice leading==
[[File:Dominant seventh root doubled.png|thumb|right|Dominant seventh with root doubled and missing fifth resolving to I, in C {{audio|Dominant seventh root doubled.mid|Play}}.]]
[[File:Dominant seventh tritone resolution.png|thumb|right|Dominant seventh tritone resolution in C {{audio|Tritone resolution inward.ogg|Play}}.]]
[[File:Dominant seventh tritone resolution chords.png|thumb|right|Dominant seventh tritone resolution in C, root of tonic chord tripled {{audio|Dominant seventh tritone resolution chords.mid|Play}}.]]
[[File:Incomplete dominant seventh chord in C major.png|thumb|Dominant seventh and incomplete dominant seventh in C major: G7 and b{{music|dim}} chords {{audio|Incomplete dominant seventh chord in C major.mid|Play}}.]]
{{multiple image
| footer = Dominant seventh chord on C, played on guitar in open position {{audio|Dominant seventh chord on C guitar.mid|Play}} and as a barre chord {{audio|Dominant seventh chord on C guitar barre.mid|Play}}.
| width = 125
| image1 = Dominant seventh chord on C guitar.png
| alt1 = Dominant seventh chord on C guitar open position
| image2 = Dominant seventh chord on C guitar barre.png
| alt2 = Dominant seventh chord on C guitar barre chord
}}

For common practice [[voice leading]], or "strict [[resolution (music)|resolution]]" of the dominant seventh chord:<ref name="Techniques"/>
*In the V7-I resolution, the dominant, leading note, and supertonic resolve to the tonic, whereas the subdominant resolves to the mediant.
*In the other resolutions, the dominant remains stationary, the leading note and supertonic resolve to the tonic, and the subdominant resolves to the mediant.
*All four tones may be present, though the root may be doubled and the fifth omitted.<ref name="Techniques">Benjamin, Horvit, and Nelson (2008). ''Techniques and Materials of Music'', p.46-47. ISBN 0-495-50054-2.</ref><ref name="B&S 200s">Benward & Saker (2003), p.202-204.</ref><ref name="B&S 343">Benward & Saker (2008), p.343</ref>
*The d5 resolves inwards and the A4 resolves outwards, meaning that the seventh resolves stepwise downwards<ref name="B&S 200s"/><ref name="B&S 343"/> while the third resolves (stepwise upwards) to the tonic<ref name="Techniques"/> though in such cases the root of the tonic chord may need to be tripled.<ref name="B&S 200s"/>
*The root of the V7, when in the [[bass note|bass]], resolves to the root of the I, in the bass.<ref name="Techniques"/>
*In an incomplete V7, with a missing fifth, the doubled root remains stationary.<ref name="Techniques"/>
*The "free resolution of the seventh" features the seventh in an inner voice moving stepwise upwards to the fifth of I<ref name="Techniques"/>

==Tuning==
{| class="wikitable"
|-
! Chord !! [[Ben Johnston's notation|Notation]] !! Seventh !! Ratios
|-
| Tonic seventh chord || C E G {{nowrap|B{{music|b}}}} || [[Minor seventh]] || 20:25:30:36<ref name="Shirlaw"/><ref name="Wright"/>
|-
| Harmonic seventh chord || G B D {{nowrap|F{{music|7}}+}} || [[Harmonic seventh]] || 4:5:6:7<ref name="Benitez"/>
|-
| German sixth chord || {{nowrap|A{{music|b}}}} C {{nowrap|E{{music|b}}}} {{nowrap|G{{music|7}}}}{{music|b}} || Harmonic seventh || 4:5:6:7
|-
| Dominant seventh chord || G B D F || [[Pythagorean minor seventh]] || 36:45:54:64<ref name="Wright"/>
|}

==Dominant seventh chord table==
{| class="wikitable"
!bgcolor=#dddddd|Chord
!bgcolor=#dddddd|Root
!bgcolor=#dddddd|Major Third
!bgcolor=#dddddd|Perfect Fifth
!bgcolor=#dddddd|Minor Seventh
|-
!C7
|C
|E
|G
|B{{music|b}}
|-
!C{{music|#}}7
|C{{music|#}}
|E{{music|#}} (F)
|G{{music|#}}
|B
|-
!D{{music|b}}7
|D{{music|b}}
|F
|A{{music|b}}
|C{{music|b}} (B)
|-
!D7
|D
|F{{music|#}}
|A
|C
|-
!D{{music|#}}7
|D{{music|#}}
|F{{music|##}} (G)
|A{{music|#}}
|C{{music|#}}
|-
!E{{music|b}}7
|E{{music|b}}
|G
|B{{music|b}}
|D{{music|b}}
|-
!E7
|E
|G{{music|#}}
|B
|D
|-
!F7
|F
|A
|C
|E{{music|b}}
|-
!F{{music|#}}7
|F{{music|#}}
|A{{music|#}}
|C{{music|#}}
|E
|-
!G{{music|b}}7
|G{{music|b}}
|B{{music|b}}
|D{{music|b}}
|F{{music|b}} (E)
|-
!G7
|G
|B
|D
|F
|-
!G{{music|#}}7
|G{{music|#}}
|B{{music|#}} (C)
|D{{music|#}}
|F{{music|#}}
|-
!A{{music|b}}7
|A{{music|b}}
|C
|E{{music|b}}
|G{{music|b}}
|-
!A7
|A
|C{{music|#}}
|E
|G
|-
!A{{music|#}}7
|A{{music|#}}
|C{{music|##}} (D)
|E{{music|#}} (F)
|G{{music|#}}
|-
!B{{music|b}}7
|B{{music|b}}
|D
|F
|A{{music|b}}
|-
!B7
|B
|D{{music|#}}
|F{{music|#}}
|A
|-
|}

==See also==
*[[Dominant ninth]], etc.
*[[Irregular resolution]]
*[[Nondominant seventh chord]]

==Notes==
{{notelist}}

==Sources==
{{reflist}}

{{Degrees}}
{{Chords|state=expanded}}

{{DEFAULTSORT:Dominant Seventh Chord}}
[[Category:Dominant chords]]
[[Category:Seventh chords]]

Magnetic braking

2012-07-24T14:53:22Z

81.155.218.83:

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2012-03-31T03:32:31Z

81.155.64.110: rm spam

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