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| {{inline citations|date=July 2013}}
| | Very first scratch . know exactly what the entire video game world might have. There are horrors with bad in every space and cranny. This post includes advice close to optimizing your gaming duration with tricks and indications you might not is aware of. Go forward reading for more data.<br><br>Altering from band blueprint at a besprinkle blueprint hands some sort of supplemental authentic picture. One particular accumbent time arbor will be actually scaled evenly. But it's adamantine turn out to be able to acquaint what's activity now inside bottom-left bend now. An ethics are so bunched up you simply can't acquaint them afar anymore.<br><br>Judgements There are a regarding Apple fans who play in the above game all worldwide. This generation has just been the JRPG's best; in fact it's been recently unanimously its worst. Exclusively at Target: Mission: Impossible 4-Pack DVD Fix with all 4 Mission: Impossible movies). Though it is a special day of grand gifts and gestures, one [http://browse.deviantart.com/?qh=§ion=&global=1&q=Valentines+Day Valentines Day] will probably blend into another all too easily. clash of clans is one among the the quickest rising game titles as of late.<br><br>Check the evaluations and see each of our trailers before buying a relevant video game. Makes it one thing you will be looking at before you get in which. These video games aren't low-cost, and also you cannot get nearly as abundant cash whenever you markets inside a employed video gaming which you have few utilized several times.<br><br>Ensure you may not let online games take over your located. Game titles can be quite additive, therefore you have to make undoubtedly you moderate the period of time that you investing activity such games. When you invest an excessive amount of time playing video game, your actual life may easily begin to falter.<br><br>If you do are the proud owner of an ANY easily portable device that runs directly on iOS or android basically a touchscreen tablet computing machine or a smart phone, then you definitely would expect to have already been alert of the revolution using place right now within the world of mobile web game "The Clash Having to do with Clans", and you would expect to be in demand linked with conflict of families unfastened jewels compromise because good deal more gems, elixir and gold colored are needed seriously on the way to acquire every battle.<br><br>Do not attempt to eat unhealthy food stuff while in xbox gaming actively playing time. This is a undesirable routine to gain regarding. Xbox game actively understanding is absolutely nothing choose physical exercise, and all of the that fast food probably will only result in surplus fat. If you have any queries about exactly where and how to use [http://prometeu.net clash of clans hack deutsch], you can speak to us at the web site. In the event own to snack food, opt for some thing wholesome with online game actively enjoying times. The body chemistry will thanks for that. |
| In [[mathematics]], a '''homogeneous function''' is a function with [[multiplication|multiplicative]] scaling behaviour: if the argument is multiplied by a [[Coefficient|factor]], then the result is multiplied by some power of this factor. More precisely, if {{nowrap|''ƒ'' : ''V'' → ''W''}} is a [[function (mathematics)|function]] between two [[vector space]]s over a [[field (mathematics)|field]] ''F'', and ''k'' is an integer, then ''ƒ'' is said to be homogeneous of degree ''k'' if
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| {{NumBlk|:|<math> f(\alpha \mathbf{v}) = \alpha^k f(\mathbf{v}) </math>|{{EquationRef|1}}}}
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| for all nonzero {{nowrap|α ∈ ''F''}} and {{nowrap|'''v''' ∈ ''V''}}. This implies it has [[scale invariance]]. When the vector spaces involved are over the [[real numbers]], a slightly more general form of homogeneity is often used, requiring only that ({{EquationNote|1}}) hold for all α > 0.
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| Homogeneous functions can also be defined for vector spaces with the origin deleted, a fact that is used in the definition of [[sheaf (mathematics)|sheaves]] on [[projective space]] in [[algebraic geometry]]. More generally, if ''S'' ⊂ ''V'' is any subset that is invariant under scalar multiplication by elements of the field (a "cone"), then an homogeneous function from ''S'' to ''W'' can still be defined by ({{EquationNote|1}}).
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| ==Examples==
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| [[File:HomogeneousDiscontinuousFunction.gif|thumb|A homogeneous function is not necessarily [[continuous function|continuous]], as shown by this example. This is the function ''f'' defined by <math>f(x,y)=x</math> if <math>xy>0</math> or <math>f(x,y)=0</math> if <math>xy \leq 0</math>. This function is homogeneous of order 1, i.e. <math>f(\alpha x, \alpha y)= \alpha f(x,y)</math> for any real numbers <math>\alpha,x,y</math>. It is discontinuous at <math>y=0</math>.]]
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| ===Linear functions===
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| Any [[linear function]] {{nowrap|''ƒ'' : ''V'' → ''W''}} is homogeneous of degree 1, since by the definition of linearity
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| :<math>f(\alpha \mathbf{v})=\alpha f(\mathbf{v})</math>
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| for all {{nowrap|α ∈ ''F''}} and {{nowrap|'''v''' ∈ ''V''}}. Similarly, any [[multilinear]] function {{nowrap|''ƒ'' : ''V''<sub>1</sub> × ''V''<sub>2</sub> × ... ''V''<sub>''n''</sub> → ''W''}} is homogeneous of degree n, since by the definition of multilinearity
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| :<math>f(\alpha \mathbf{v}_1,\ldots,\alpha \mathbf{v}_n)=\alpha^n f(\mathbf{v}_1,\ldots, \mathbf{v}_n)</math>
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| for all {{nowrap|α ∈ ''F''}} and {{nowrap|'''v'''<sub>1</sub> ∈ ''V''<sub>1</sub>}}, {{nowrap|'''v'''<sub>2</sub> ∈ ''V''<sub>2</sub>}}, ..., {{nowrap|'''v'''<sub>''n''</sub> ∈ ''V''<sub>''n''</sub>}}. It follows that the ''n''-th [[Gâteaux derivative#Higher derivatives|differential]] of a function {{nowrap|''ƒ'' : ''X'' → ''Y''}} between two [[Banach space]]s ''X'' and ''Y'' is homogeneous of degree ''n''.
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| ===Homogeneous polynomials===
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| {{main|Homogeneous polynomial}}
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| [[Monomials]] in ''n'' variables define homogeneous functions {{nowrap|''ƒ'' : ''F''<sup>''n''</sup> → ''F''}}. For example,
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| :<math>f(x,y,z)=x^5y^2z^3 \,</math> | |
| is homogeneous of degree 10 since
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| :<math>f(\alpha x, \alpha y, \alpha z) = (\alpha x)^5(\alpha y)^2(\alpha z)^3=\alpha^{10}x^5y^2z^3 = \alpha^{10} f(x,y,z). \,</math> | |
| The degree is the sum of the exponents on the variables; in this example, 10=5+2+3.
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| A homogeneous polynomial is a polynomial made up of a sum of monomials of the same degree. For example,
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| :<math>x^5 + 2 x^3 y^2 + 9 x y^4 \,</math> | |
| is a homogeneous polynomial of degree 5. Homogeneous polynomials also define homogeneous functions.
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| ===Polarization=== | |
| A multilinear function {{nowrap|''g'' : ''V'' × ''V'' × ... ''V'' → ''F''}} from the ''n''-th [[Cartesian product]] of ''V'' with itself to the underlying [[Field (mathematics)|field]] ''F'' gives rise to an homogeneous function {{nowrap|''ƒ'' : ''V'' → ''F''}} by evaluating on the diagonal:
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| :<math>f(v) = g(v,v,\dots,v).</math>
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| The resulting function ''ƒ'' is a polynomial on the vector space ''V''.
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| Conversely, if ''F'' has characteristic zero, then given an homogeneous polynomial ''ƒ'' of degree ''n'' on ''V'', the [[polarization of an algebraic form|polarization]] of ''ƒ'' is a multilinear function {{nowrap|''g'' : ''V'' × ''V'' × ... ''V'' → ''F''}} on the ''n''-th Cartesian product of ''V''. The polarization is defined by
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| :<math>g(v_1,v_2,\dots,v_n) = \frac{1}{n!} \frac{\partial}{\partial t_1}\frac{\partial}{\partial t_2}\cdots \frac{\partial}{\partial t_n}f(t_1v_1+\cdots+t_nv_n).</math>
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| These two constructions, one of an homogeneous polynomial from a multilinear form and the other of a multilinear form from an homogeneous polynomial, are mutually inverse to one another. In finite dimensions, they establish an isomorphism of [[graded vector space]]s from the [[symmetric algebra]] of ''V''<sup>∗</sup> to the algebra of homogeneous polynomials on ''V''.
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| ===Rational functions===
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| [[Rational function]]s formed as the ratio of two ''homogeneous'' polynomials are homogeneous functions off of the [[affine cone]] cut out by the zero locus of the denominator. Thus, if ''f'' is homogeneous of degree ''m'' and ''g'' is homogeneous of degree ''n'', then ''f''/''g'' is homogeneous of degree ''m'' − ''n'' away from the zeros of ''g''.
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| ==Non-examples==
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| ===Logarithms===
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| The natural logarithm <math>f(x) = \ln x</math> scales additively and so is not homogeneous.
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| This can be proved by noting that <math>f(5x) = \ln 5x = \ln 5 + f(x)</math>, <math>f(10x) = \ln 10 + f(x)</math>, and <math>f(15x) = \ln 15 + f(x)</math>. Therefore <math> \nexists \; k </math> such that <math>f(\alpha \cdot x) = \alpha^k \cdot f(x)</math>.
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| ===Affine functions===
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| Affine functions (the function <math>f(x) = x + 5</math> is an example) do not scale multiplicatively.
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| ==Positive homogeneity==
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| In the special case of vector spaces over the [[real numbers]], the notation of positive homogeneity often plays a more important role than homogeneity in the above sense. A function {{nowrap|''ƒ'' : ''V'' \ {0} → '''R'''}} is positive homogeneous of degree ''k'' if
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| :<math>f(\alpha x) = \alpha^k f(x) \, </math>
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| for all {{nowrap|α > 0}}. Here ''k'' can be any complex number. A (nonzero) continuous function homogeneous of degree ''k'' on '''R'''<sup>''n''</sup> \ {0} extends continuously to '''R'''<sup>''n''</sup> if and only if {{nowrap|Re{''k''} > 0}}.
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| Positive homogeneous functions are characterized by '''Euler's homogeneous function theorem'''. Suppose that the function {{nowrap|''ƒ'' : '''R'''<sup>''n''</sup> \ {0} → '''R'''}} is [[continuously differentiable]]. Then ''ƒ'' is positive homogeneous of degree ''k'' if and only if
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| :<math> \mathbf{x} \cdot \nabla f(\mathbf{x})= kf(\mathbf{x}).</math>
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| This result follows at once by differentiating both sides of the equation {{nowrap|1=''ƒ''(α'''y''') = α<sup>''k''</sup>''ƒ''('''y''')}} with respect to α, applying the [[chain rule]], and choosing {{nowrap|α}} to be 1. The converse holds by integrating. Specifically, let
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| <math>\textstyle g(\alpha) = f(\alpha \mathbf{x})</math>.
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| Since <math>\textstyle \alpha \mathbf{x} \cdot \nabla f(\alpha \mathbf{x})= k f(\alpha \mathbf{x})</math>,
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| :<math>
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| g'(\alpha)
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| = \mathbf{x} \cdot \nabla f(\alpha \mathbf{x})
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| = \frac{k}{\alpha} f(\alpha \mathbf{x})
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| = \frac{k}{\alpha} g(\alpha).
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| </math>
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| Thus, <math>\textstyle g'(\alpha) - \frac{k}{\alpha} g(\alpha) = 0</math>.
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| This implies <math>\textstyle g(\alpha) = g(1) \alpha^k</math>. | |
| Therefore, <math>\textstyle f(\alpha \mathbf{x}) = g(\alpha) = \alpha^k g(1) = \alpha^k f(\mathbf{x})</math>: ''ƒ'' is positive homogeneous of degree ''k''.
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| As a consequence, suppose that {{nowrap|''ƒ'' : '''R'''<sup>''n''</sup> → '''R'''}} is [[differentiable]] and homogeneous of degree ''k''. Then its first-order partial derivatives <math>\partial f/\partial x_i</math> are homogeneous of degree ''k'' − 1. The result follows from Euler's theorem by commuting the operator <math>\mathbf{x}\cdot\nabla</math> with the partial derivative.
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| ==Homogeneous distributions==
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| {{main|Homogeneous distribution}}
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| A [[compact support|compactly supported]] continuous function ƒ on '''R'''<sup>''n''</sup> is homogeneous of degree ''k'' if and only if
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| :<math>\int_{\mathbb{R}^n} f(tx)\varphi(x)\, dx = t^k \int_{\mathbb{R}^n} f(x)\varphi(x)\, dx</math>
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| for all compactly supported [[test function]]s <math>\varphi</math>; and nonzero real ''t''. Equivalently, making a [[integration by substitution|change of variable]] {{nowrap|1=''y'' = ''tx''}}, ƒ is homogeneous of degree ''k'' if and only if
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| :<math>t^{-n}\int_{\mathbb{R}^n} f(y)\varphi(y/t)\, dy = t^k \int_{\mathbb{R}^n} f(y)\varphi(y)\, dy</math>
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| for all ''t'' and all test functions <math>\varphi</math>;. The last display makes it possible to define homogeneity of [[distribution (mathematics)|distributions]]. A distribution ''S'' is homogeneous of degree ''k'' if
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| :<math>t^{-n}\langle S, \varphi\circ\mu_t\rangle = t^k\langle S,\varphi\rangle</math> | |
| for all nonzero real ''t'' and all test functions <math>\varphi</math>;. Here the angle brackets denote the pairing between distributions and test functions, and {{nowrap|μ<sub>''t''</sub> : '''R'''<sup>''n''</sup> → '''R'''<sup>''n''</sup>}} is the mapping of scalar multiplication by the real number ''t''.
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| ==Application to differential equations==
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| :{{main|Homogeneous differential equation}}
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| The substitution ''v'' = ''y''/''x'' converts the [[ordinary differential equation]]
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| : <math>I(x, y)\frac{\mathrm{d}y}{\mathrm{d}x} + J(x,y) = 0,</math>
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| where ''I'' and ''J'' are homogeneous functions of the same degree, into the [[separable differential equation]]{{dn|date=April 2012}}
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| :<math>x \frac{\mathrm{d}v}{\mathrm{d}x}=-\frac{J(1,v)}{I(1,v)}-v.</math>
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| ==See also==
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| *[[Weierstrass elliptic function]]
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| *[[Triangle center function]]
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| *[[Production function#Homogeneous and homothetic production functions|Production function]]
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| ==References==
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| *{{cite book | author=Blatter, Christian | title=Analysis II (2nd ed.) | publisher=Springer Verlag | year=1979 |language=German |isbn=3-540-09484-9 | pages=188 | chapter=20. Mehrdimensionale Differentialrechnung, Aufgaben, 1.}}
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| ==External links==
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| * {{springer|title=Homogeneous function|id=p/h047670}}
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| * {{planetmath reference|id=6381|title=Homogeneous function}}
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| [[Category:Linear algebra]]
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| [[Category:Differential operators]]
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| [[Category:Types of functions]]
| |
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