formulasearchengine - User contributions [en]

YIQ

2015-01-08T03:31:00Z

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{{Unreferenced|date=November 2009}}
In mathematics, '''generalized means''' are a family of functions for aggregating sets of numbers, that include as special cases the [[arithmetic mean|arithmetic]], [[geometric mean|geometric]], and [[harmonic mean|harmonic]] [[mean]]s. The generalized mean is also known as '''power mean''' or '''Hölder mean''' (named after [[Otto Hölder]]).

==Definition==
If ''p'' is a non-zero [[real number]], we can define the '''generalized mean''' or '''power mean''' with exponent ''p'' of the positive real numbers <math>x_1,\dots,x_n</math> as:

:<math>M_p(x_1,\dots,x_n) = \left( \frac{1}{n} \sum_{i=1}^n x_i^p \right)^{\frac{1}{p}}</math>

While for ''p'' = 0 we assume that it's equal to the geometric mean (which is, in fact, the limit of means with exponents approaching zero, as proved below for the general case):

:<math>M_0(x_1, \dots, x_n) = \sqrt[n]{\prod_{i=1}^n x_i}</math>

Furthermore, for a sequence of positive weights ''wi'' with sum <math>\sum w_i = 1</math> we define the '''weighted power mean''' as:

:<math>\begin{align}
M_p(x_1,\dots,x_n) &= \left(\sum_{i=1}^n w_i x_i^p \right)^{\frac{1}{p}} \\
M_0(x_1,\dots,x_n) &= \prod_{i=1}^n x_i^{w_i}
\end{align}</math>

The unweighted means correspond to setting all ''wi'' = 1/''n''. For exponents equal to positive or negative infinity the means are maximum and minimum, respectively, regardless of weights (and they are actually the limit points for exponents approaching the respective extremes, as proved below):

:<math>\begin{align}
M_{ \infty}(x_1, \dots, x_n) &= \max(x_1, \dots, x_n) \\
M_{-\infty}(x_1, \dots, x_n) &= \min(x_1, \dots, x_n)
\end{align}</math>

:{| class="toccolours collapsible collapsed" width="90%" style="text-align:left"
!Proof of <math>\textstyle \lim_{p \to 0} M_p = M_0</math> (geometric mean)
|-
|
We can rewrite the definition of ''Mp'' using the [[exponential function]]

:<math>M_p(x_1,\dots,x_n) = \exp{\left( \ln{\left[\left(\sum_{i=1}^n w_ix_{i}^p \right)^{1/p}\right]} \right) } = \exp{\left( \frac{\ln{\left(\sum_{i=1}^n w_ix_{i}^p \right)}}{p} \right) }</math>

In the limit ''p'' → 0, we can apply [[L'Hôpital's rule]] to the exponential component,

:<math>\lim_{p \to 0} \frac{\ln{\left(\sum_{i=1}^n w_ix_{i}^p \right)}}{p} = \lim_{p \to 0} \frac{\sum_{i=1}^n w_i x_i^p \ln{x_i}}{\sum_{i=1}^n w_i x_i^p} = \sum_{i=1}^n w_i \ln{x_i} = \ln{\left(\prod_{i=1}^n x_i^{w_i} \right)}</math>

By the continuity of the exponential function, we can substitute back into the above relation to obtain

:<math>\lim_{p \to 0} M_p(x_1,\dots,x_n) = \exp{\left( \ln{\left(\prod_{i=1}^n x_i^{w_i} \right)} \right)} = \prod_{i=1}^n x_i^{w_i} = M_0(x_1,\dots,x_n)</math>

as desired.
|}

:{| class="toccolours collapsible collapsed" width="90%" style="text-align:left"
!Proof of <math>\textstyle \lim_{p \to \infty} M_p = M_\infty</math> and <math>\textstyle \lim_{p \to -\infty} M_p = M_{-\infty}</math>
|-
|
Assume (possibly after relabeling and combining terms together) that <math>x_1 \geq \dots \geq x_n</math>. Then

:<math>\lim_{p \to \infty} M_p(x_1,\dots,x_n) = \lim_{p \to \infty} \left( \sum_{i=1}^n w_i x_i^p \right)^{1/p} = x_1 \lim_{p \to \infty} \left( \sum_{i=1}^n w_i \left( \frac{x_i}{x_1} \right)^p \right)^{1/p} = x_1 = M_\infty (x_1,\dots,x_n).</math>

The formula for <math>M_{-\infty}</math> follows from <math>M_{-\infty} (x_1,\dots,x_n) = \frac{1}{M_\infty (1/x_1,\dots,1/x_n)}.</math>
|}

==Properties==

* Like most [[Mean#Properties|mean]]s, the generalized mean is a [[homogeneous function]] of its arguments ''x''1, ..., ''xn''. That is, if ''b'' is a positive real number, then the generalized mean with exponent ''p'' of the numbers <math>b\cdot x_1,\dots, b\cdot x_n</math> is equal to ''b'' times the generalized mean of the numbers ''x''1, …, ''xn''.
* Like the [[quasi-arithmetic mean]]s, the computation of the mean can be split into computations of equal sized sub-blocks.

:: <math>M_p(x_1, \dots, x_{n \cdot k}) = M_p(M_p(x_1, \dots, x_{k}), M_p(x_{k + 1}, \dots, x_{2 \cdot k}), \dots, M_p(x_{(n - 1) \cdot k + 1}, \dots, x_{n \cdot k}))</math>

=== Generalized mean inequality ===
In general,
: if ''p'' < ''q'', then <math>M_p(x_1,\dots,x_n) \le M_q(x_1,\dots,x_n)</math>

and the two means are equal if and only if ''x''1 = ''x''2 = ... = ''xn''.

The inequality is true for real values if ''p'' and ''q'', as well as positive and negative infinity values.

It follows from the fact that, for all real ''p'',

: <math>\frac{\partial}{\partial p}M_p(x_1, \dots, x_n) \geq 0</math>

which can be proved using [[Jensen's inequality]].

In particular, for ''p'' in {−1, 0, 1}, the generalized mean inequality implies the [[Pythagorean means]] inequality as well as the [[inequality of arithmetic and geometric means]].

== Special cases ==
[[Image:MathematicalMeans.svg|thumb|right|A visual depiction of some of the specified cases for ''n=2''.]]
{| style="border:none; padding:20px"
|<math>M_{-\infty}(x_1,\dots,x_n) = \lim_{p\to-\infty} M_p(x_1,\dots,x_n) = \min \{x_1,\dots,x_n\}</math>
|style="padding:0px 0px 0px 20px"| [[minimum]]
|-
|<math>M_{-1}(x_1,\dots,x_n) = \frac{n}{\frac{1}{x_1}+\dots+\frac{1}{x_n}}</math>
|style="padding:0px 0px 0px 20px"| [[harmonic mean]]
|-
|<math>M_0(x_1,\dots,x_n) = \lim_{p\to0} M_p(x_1,\dots,x_n) = \sqrt[n]{x_1\cdot\dots\cdot x_n}</math>
|style="padding:0px 0px 0px 20px"| [[geometric mean]]
|-
|<math>M_1(x_1,\dots,x_n) = \frac{x_1 + \dots + x_n}{n}</math>
|style="padding:0px 0px 0px 20px"| [[arithmetic mean]]
|-
|<math>M_2(x_1,\dots,x_n) = \sqrt{\frac{x_1^2 + \dots + x_n^2}{n}}</math>
|style="padding:0px 0px 0px 20px"| quadratic mean, a.k.a. [[root mean square]]
|-
|<math>M_{+\infty}(x_1,\dots,x_n) = \lim_{p\to\infty} M_p(x_1,\dots,x_n) = \max \{x_1,\dots,x_n\}</math>
|style="padding:0px 0px 0px 20px"| [[maximum]]
|}

==Proof of power means inequality==
We will prove weighted power means inequality, for the purpose of the proof we will assume the following without loss of generality:

:<math>\begin{align}
w_i \in [0; 1] \\
\sum_{i=1}^nw_i = 1
\end{align}</math>

Proof for unweighted power means is easily obtained by substituting ''wi'' = 1/''n''.

===Equivalence of inequalities between means of opposite signs===
Suppose an average between power means with exponents ''p'' and ''q'' holds:

:<math>\sqrt[p]{\sum_{i=1}^nw_ix_i^p}\geq \sqrt[q]{\sum_{i=1}^nw_ix_i^q}</math>

applying this, then:

:<math>\sqrt[p]{\sum_{i=1}^n\frac{w_i}{x_i^p}}\leq \sqrt[q]{\sum_{i=1}^n\frac{w_i}{x_i^q}}</math>

We raise both sides to the power of −1 (strictly decreasing function in positive reals):

:<math>\sqrt[-p]{\sum_{i=1}^nw_ix_i^{-p}}=\sqrt[p]{\frac{1}{\sum_{i=1}^nw_i\frac{1}{x_i^p}}}\geq \sqrt[q]{\frac{1}{\sum_{i=1}^nw_i\frac{1}{x_i^q}}}=\sqrt[-q]{\sum_{i=1}^nw_ix_i^{-q}}</math>

We get the inequality for means with exponents −''p'' and −''q'', and we can use the same reasoning backwards, thus proving the inequalities to be equivalent, which will be used in some of the later proofs.

===Geometric mean===
For any ''q'' the inequality between mean with exponent ''q'' and geometric mean can be transformed in the following way:

:<math>\begin{align}
\prod_{i=1}^nx_i^{w_i} &\leq \sqrt[q]{\sum_{i=1}^nw_ix_i^q} \\
\sqrt[q]{\sum_{i=1}^nw_ix_i^q} &\leq \prod_{i=1}^nx_i^{w_i}
\end{align}</math>

(the first inequality is to be proven for positive ''q'', and the latter otherwise)

We raise both sides to the power of ''q'':

:<math>\prod_{i=1}^nx_i^{w_i\cdot q} \leq \sum_{i=1}^nw_ix_i^q</math>

in both cases we get the inequality between weighted arithmetic and geometric means for the sequence <math>x_i^q</math>, which can be proved by [[Jensen's inequality]], making use of the fact the [[logarithm]]ic function is concave:

:<math>\begin{align}
\sum_{i=1}^nw_i\log(x_i) &\leq \log\left( \sum_{i=1}^nw_ix_i \right) \\
\log \left(\prod_{i=1}^nx_i^{w_i} \right) &\leq \log\left( \sum_{i=1}^nw_ix_i \right)
\end{align}</math>

By applying the [[exponential function]] to both sides (and observing that as a strictly increasing function it preserves the sign of the inequality) we get

:<math>\prod_{i=1}^nx_i^{w_i} \leq \sum_{i=1}^nw_ix_i</math>

Thus for any positive ''q'' it is true that:

:<math>\sqrt[-q]{\sum_{i=1}^nw_ix_i^{-q}}\leq \prod_{i=1}^nx_i^{w_i} \leq \sqrt[q]{\sum_{i=1}^nw_ix_i^q}</math>

thus we have proved the inequality between geometric mean and any power mean.

===Inequality between any two power means===
We are to prove that for any ''p'' < ''q'' the following inequality holds:

:<math>\sqrt[p]{\sum_{i=1}^nw_ix_i^p}\leq \sqrt[q]{\sum_{i=1}^nw_ix_i^q}</math>

if ''p'' is negative, and ''q'' is positive, the inequality is equivalent to the one proved above:

:<math>\sqrt[p]{\sum_{i=1}^nw_ix_i^p}\leq \prod_{i=1}^nx_i^{w_i} \leq\sqrt[q]{\sum_{i=1}^nw_ix_i^q}</math>

The proof for positive ''p'' and ''q'' is as follows: Define the following function: ''f'' : '''R'''+ → '''R'''+ <math>f(x)=x^{\frac{q}{p}}</math>. ''f'' is a power function, so it does have a second derivative:

: <math>f''(x) = \left(\frac{q}{p} \right) \left( \frac{q}{p}-1 \right)x^{\frac{q}{p}-2}</math>

which is strictly positive within the domain of ''f'', since ''q'' > ''p'', so we know ''f'' is convex.

Using this, and the Jensen's inequality we get:

:<math>\begin{align}
f \left( \sum_{i=1}^nw_ix_i^p \right) &\leq \sum_{i=1}^nw_if(x_i^p) \\
\sqrt[\frac{p}{q}]{\sum_{i=1}^nw_ix_i^p} &\leq \sum_{i=1}^nw_ix_i^q
\end{align}</math>

after raising both side to the power of 1/''q'' (an increasing function, since 1/''q'' is positive) we get the inequality which was to be proven:

:<math>\sqrt[p]{\sum_{i=1}^nw_ix_i^p}\leq\sqrt[q]{\sum_{i=1}^nw_ix_i^q}</math>

Using the previously shown equivalence we can prove the inequality for negative ''p'' and ''q'' by substituting them with, respectively, −''q'' and −''p'', QED.

== Generalized ''f''-mean ==
{{Main|Generalized f-mean|l1=Generalized ƒ-mean}}

The power mean could be generalized further to the [[generalized f-mean|generalized ''f''-mean]]:

:<math> M_f(x_1,\dots,x_n) = f^{-1} \left({\frac{1}{n}\cdot\sum_{i=1}^n{f(x_i)}}\right) </math>

Which covers the geometric mean without using a limit with ''f''(''x'') = ''log''(''x''). The power mean is obtained for ''f''(''x'') = ''xp''.

== Applications ==
===Signal processing===
A power mean serves a non-linear [[moving average]] which is shifted towards small signal values for small ''p'' and emphasizes big signal values for big ''p''. Given an efficient implementation of a [[lowpass|moving arithmetic mean]] called <code>smooth</code> you can implement a moving power mean according to the following [[Haskell (programming language)|Haskell]] code.

<source lang="haskell">
powerSmooth :: Floating a => ([a] -> [a]) -> a -> [a] -> [a]
powerSmooth smooth p = map (** recip p) . smooth . map (**p)
</source>

* For big ''p'' it can serve an [[envelope detector]] on a [[rectifier|rectified]] signal.
* For small ''p'' it can serve an [[Baseline (spectrometry)|baseline detector]] on a [[mass spectrum]].

==See also==
* [[Arithmetic mean]]
* [[Arithmetic-geometric mean]]
* [[Average]]
* [[Geometric mean]]
* [[Harmonic mean]]
* [[Heronian mean]]
* [[Inequality of arithmetic and geometric means]]
* [[Lehmer mean]] – also a mean related to [[Power (mathematics)|powers]]
* [[Root mean square]]

==External links==
*[http://mathworld.wolfram.com/PowerMean.html Power mean at MathWorld]
*[http://people.revoledu.com/kardi/tutorial/BasicMath/Average/Generalized%20mean.html Examples of Generalized Mean]
*A [http://planetmath.org/?op=getobj&from=objects&name=ProofOfGeneralMeansInequality proof of the Generalized Mean] on [[PlanetMath]]

[[Category:Means]]
[[Category:Inequalities]]
[[Category:Articles with example Haskell code]]

Talk:Lane–Emden equation

2013-10-25T07:25:37Z

24.130.146.226: /* n = 0 case */ new section

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Transport of structure

2012-09-29T01:54:56Z

24.130.99.89:

'''[[K]]''' is the eleventh letter of the Latin alphabet.

'''K''' may also refer to:

{{TOC right}}
== General uses ==
* [[K (programming language)]], an array processing language developed by Arthur Whitney and commercialized by Kx Systems
* [[K (cider)]], a British draft cider manufactured and distributed by the Gaymer Cider Company of Bath, England
* [[K band]], radio frequency from 12 to 93 GHz
* [[K computer]], a Japanese supercomputer
* [[K-factor (disambiguation)]], several unrelated terms in physics, engineering, telecommunications and chess
* [[K (Broadway Brooklyn Local)]], earlier KK, discontinued in 1976
* [[K (Eighth Avenue Local)]], a defunct line on the New York City Subway, which was known as the AA until 1985
* [[Vitamin K]], a group of vitamins that are needed to promote blood coagulation
* [[Kappa]] (Κ) (Greek alphabet)
* [[Ka (Cyrillic)]] (К) (Cyrillic alphabet)
* [[Chrysler K platform]], 1981–1995 car design used by Chrysler
* [[Low-K]], the dielectric constant in semiconductors, electronics, and physics
* [[Operation K]], a Japanese World War II naval operation
* [[The K Project]], a series of five nuclear tests by the Soviet Union
* [[K–12 education]], a designation for the sum of primary and secondary education, used mainly in North America
* K, a prefix for [[North American call sign]]s used by broadcast stations in the US west of the Mississippi River
* [[K College]], in Tonbridge, Kent

==People==
* [[Adem K]], Australian musician
* [[Yasmin K.]], German singer
* [[K (music director)]], an Indian music director primarily based on [[Tamil cinema]]
* [[K (singer)]] (born 1983), a Korean singer who works in Japan as a J-pop singer

== In fiction and literature ==
* [[K (TV miniseries)|''K'' (TV miniseries)]], a Filipino television series

* [[K (2002 film)|''K'' (2002 film)]]

* [[K′]], also known as "K Dash" or "K Prime", a ''King of Fighters'' character
* ''K'', the title of a 1915 novel by [[Mary Roberts Rinehart]]
* K, the protagonist of Franz Kafka's 1926 novel ''[[The Castle (novel)|The Castle]]''
* K, a character in the manga and anime ''[[Gravitation (manga)|''Gravitation'']]''
* K, the main character in the TV series ''Robotto Keiji'' (''[[Robot Detective]]'')
* K, Todd and Riley's adoptive superspy mother on the Disney program ''[[The Replacements (TV series)|The Replacements]]''
* [[Agent K]], the character played by Tommy Lee Jones in ''Men in Black'', ''Men in Black II'', and ''Men in Black 3''
* K, the protagonist played by John Abraham in ''[[No Smoking (2007 film)|No Smoking]]''
* K, a constructed language invented by [[Robert Dessaix]]
* K, a character in Natsume Soseki's novel ''[[Kokoro]]''
* [[K (anime)|''K'' (anime)]], a Japanese anime
* Joseph K., the protagonist of Kafka's 1925 novel, ''[[The Trial]]''

== In mathematics ==
* <math>K_n</math> and <math>K_{m,n}</math>, the [[complete graph|complete]] and [[complete bipartite graph]]
* [[K-theory]], a branch of topology and algebra
* Blackboard bold <math>\mathbb K</math> is a commonly used symbol for a [[field (mathematics)|field]], as an abbreviation of the German ''Körper''.

== In music ==
* [[K (album)|''K'' (album)]], 1996 album by Kula Shaker
* [[K Records]], independent record company based in Olympia, Washington
* [[K (song)| "K" (song)]], 2006 song by The Tutts
* K, [[Köchel catalogue]], chronological catalogue of compositions by Wolfgang Amadeus Mozart
* K, a song by the Japanese rock band [[Bump of Chicken]]

== In politics ==
* K, the ''Keskustapuolue'', the [[Centre Party (Finland)]]
* K, the politics and followers of President of Argentina [[Néstor Kirchner]] (born 1950)

== As a symbol or abbreviation ==
* k, [[1000 (number)]]
* K, [[ketamine]], a street abbreviation for a recreational drug
* K, [[Kyocera]]' s mobile phones in Japan
* K, a [[strikeout]] in baseball scorekeeping
* K, a contract in [[legal shorthand]]
* k, a [[voiceless velar plosive]] in the International Phonetic Alphabet
* K, [[kosher]]
* K, key color (i.e., black) in the [[CMYK color model]], especially for printing
* k, abbreviation of [[okay]], commonly used in instant messaging, or in S.M.S. messages
* K, the [[construction point]] of a ski jumping hill
* k, a shorthand for a [[knighthood]] in the British honours system
* K, the ticker symbol on the New York Stock Exchange for the [[Kellogg Company]]

===Unit===
* k, symbol of the [[kilo-]] prefix in the SI and other systems of unit that denotes multiples of 1000
* K, [[International System of Units|SI]] symbol for [[kelvin]], the SI unit of thermodynamic temperature
* K (upper case), informal abbreviation for [[kibibyte]] or 1024 bytes
* k (lower case), informal abbreviation for [[kilobyte]] or 1000 bytes
* K, [[kilobit]] (more correctly kbit)
* K, informal abbreviation for [[kilometre]] ([[International System of Units|SI]] symbol: km)

===In the natural sciences===
* K, one-letter designation for the amino acid [[lysine]]
* K, for [[carrying capacity]] of an environment
* K [[carat (purity)]], measurement applied to gold
* K, the [[Cretaceous]] geological period

====In the physical sciences====
* K, [[potassium]] (from its Latin name ''kalium'')
* ''k'', [[Wavenumber#In wave equations|angular wavenumber]] of a wave
* K, the symbol that represents a [[kaon]] in particle physics
* ''k'', the [[spring constant]] in Hooke's law relating deformation (strain) and force applied (stress) to a material body
* ''k'' or ''k''{{sub|B}}, the [[Boltzmann constant]], the physical constant relating energy and temperature at the particle level
* <math>\kappa</math> or ''K'', [[relative static permittivity]] (dielectric constant)
* ''ke'', [[Coulomb's constant]]
* ''K'', [[equilibrium constant]]
* k, [[thermal conductivity]]
* k, often used for some [[proportionality constant]]

==See also==
* [[K class (disambiguation)]]
* {{lookfrom|K}}
* {{intitle|K}}

{{disambiguation}}

[[nn:K]]

36 (number)

2012-08-23T02:54:19Z

24.130.146.227: reverting vandalism

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2012-08-17T05:49:25Z

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2012-08-12T07:40:29Z

24.130.33.118: /* Implementation example in Python */ Remove extraneous mod N

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2011-11-26T18:02:05Z

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