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en>Mild Bill Hiccup: Cleaned up using AutoEd; atleast → at least

2014-10-24T00:40:08Z

Cleaned up using AutoEd; atleast → at least

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	~~The '''ramp function''' is a~~ [~~[unary function\|unary]] [[real function]], easily computable as the [[arithmetic mean\|mean]] of the [[independent variable]] and its [[absolute value]].~~		Nice to [http://Www.allstate.com/tools-and-resources/car-insurance/whats-in-a-new-car-warranty.aspx satisfy] you, my name is Ling and I completely dig that name. Interviewing is what [http://www.chillister.com/blogs/post/27388 extended auto warranty] she does auto warranty in her day occupation but soon her husband and her will start their personal business. Kansas is our beginning place and my parents live close by. One of the issues [http://appbutton.co.kr/vp/xe/vp/248180 extended car warranty] I love most is greeting [https://Www.Geico.com/getaquote/auto/mechanical-breakdown-insurance/ extended] car warranty card collecting but [http://www.dmv.org/buy-sell/auto-warranty/extended-warranty.php I don't] have the time recently.<br><br>My web blog ... auto warranty - [http://Www.Darkasylum.org/wiki/KermitbpEspinozatd Going to Www.Darkasylum.org],

	~~This function is applied in engineering (e.g., in the theory of [[Digital signal processing\|DSP]]). The name ''ramp function'' can be derived by the look of its graph.~~

	~~== Definitions ==~~
	~~[[Image~~:~~Ramp_function~~.~~svg\|[[Graph of a function\|Graph]] of the ramp function\|thumb\|260px\|right]]~~

	~~The ramp function (<math> R(x): \mathbb{R} \rightarrow \mathbb{R}</math>) may be defined analytically in several ways~~. ~~Possible definitions are:~~
	~~:<math>R(x) := \begin{cases} x, & x \ge 0; \\ 0, & x<0 \end{cases} <~~/~~math>~~
	* The mean of a straight line with unity gradient and ~~its modulus:~~
	~~:<math>R(x) := \frac{x+\|x\|}{2} <~~/~~math>~~
	~~this can be derived by noting the following definition of <math> \operatorname{max}(a,b) <~~/~~math>,~~
	~~: <math> \operatorname{max}(~~a,~~b) = \frac{a+b+\|a-b\|}{2} </math>~~
	~~for which <math>a = x</math>~~ and ~~<math>b = 0</math>~~
	* The [~~[Heaviside step function]] multiplied by a straight line with unity gradient:~~
	~~: <math>R\left( x \right)~~ :~~= xH\left( x \right)<~~/~~math>~~
	* The [[convolution]] of the Heaviside step function with itself:
	~~: <math>R\left( x \right) := H\left( x \right) * H\left( x \right)<~~/~~math>~~
	* The [[integral]] of the Heaviside step function:
	~~: <math>R(x) := \int_{-\infty}^{x} H(\xi)\,\mathrm{d}\xi</math>~~
	* [[Macaulay brackets]]:
	~~: <math>R(x) := \langle x\rangle</math>~~

	~~== Analytic properties ==~~

	~~=== Non-negativity ===~~

	~~In the whole [[domain of a function\|domain]] the function is non-negative, so its [[absolute value]] is itself, i~~.e.

	~~<math>\forall x \in \mathbb{R}: R(x) \geqslant 0 <~~/~~math>~~

	~~and~~

	~~<math>\left\| R \left( x \right) \right\| = R\left( x \right)<~~/~~math>~~
	* Proof: by the mean of definition [2] ~~it is non-negative~~ in ~~the I. quarter,~~ and ~~zero in the II~~.~~; so everywhere it~~ is ~~non-negative~~.

	~~=== Derivative ===~~
	~~Its derivative is~~ the [~~[Heaviside function]]~~:

	~~<math>R'(x) = H(x)\ \mathrm{if}\ x \ne 0<~~/~~math>~~

	~~<!--Ugyanis~~
	* ha x<0, akkor R(x)=0 konstans, tehát ezen a tartományon (ℝ<sup>-</~~sup>) R'(x)=0 (konstans deriváltja 0); ami megegyezik a Heaviside-függvénnyel~~.
	* ha x>0, akkor R(x)=x, tehát ezen a tartományon (ℝ<sup>+</~~sup>) R'(x)=1 (a valós számokon értelmezett [[identitás~~]~~] deriváltja 1); ami megegyezik a Heaviside-függvénnyel.~~
	* 0-ban a függvénynek [~~[töréspont]]ja van, tehát nem deriválható (jobbról deriválva 0-t, balról deriválva 1-et kapunk, holott a deriválhatóság feltétele, hogy a jobb és bal oldali derivált megegyezzen)~~.~~-->~~

	~~From this property definition [5]~~. ~~goes.~~

	~~=== [[Fourier transform]] ===~~
	~~<center> <math> \mathcal{F}\left\{ R(x) \right\}(f) <~~/~~math> <math> = <~~/~~math> <math> \int_{~~-~~\infty}^{\infty}R(x) e^{~~-~~2\pi ifx}dx <~~/~~math> <math> = <~~/~~math> <math> \frac{i\delta '(f)}{4\pi}-\frac{1}{4\pi^{2}f^{2}} <~~/~~math> <~~/~~center>~~

	~~Where <code>δ(x)<~~/~~code> is the [[Dirac delta]] (in this formula, its [[derivative]] appears).~~

	~~=== [[Laplace transform]] ===~~

	~~The single~~-~~sided [[Laplace transform]] of <math>R(x)<~~/~~math> is given as follows,~~

	~~<center> <math> \mathcal{L}\left\{ R\left( x \right)\right\} (s) = \int_{0}^{\infty} e^{~~-~~sx}R(x)dx = \frac{1}{s^2}~~. ~~</math> </center>~~

	~~== Algebraic properties ==~~
	~~=== Iteration invariance ===~~
	~~Every [[iterated function~~]~~] of~~ the ~~ramp mapping is itself, as<br>~~
	~~<center><math> R \left( R \left( x \right) \right) = R \left( x \right) </math>~~. ~~</center>~~<br>
	* Proof: <math> R(R(x)):= \frac{R(x)+\|R(x)\|}{2} = \frac{R(x)+R(x)}{2} </math> <math>=</math> <br> ~~<math>=</math> <math> \frac{2R(x)}{2} = R(x) </math>~~.

	~~We applied the [[#Non~~-~~negativity\|non-negative property]].~~

	~~== References ==~~
	* [http://~~mathworld~~.~~wolfram~~.~~com~~/~~RampFunction~~.~~html Mathworld]~~

	~~[[Category:Real analysis]]~~
	~~[[Category:Special functions]~~]

en>ChrisGualtieri: /* References */Remove stub template(s). Page is start class or higher. Also check for and do General Fixes + Checkwiki fixes using AWB

2013-12-16T12:40:31Z

References: Remove stub template(s). Page is start class or higher. Also check for and do General Fixes + Checkwiki fixes using AWB

← Older revision		Revision as of 14:40, 16 December 2013
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	~~Hi there~~, ~~I am Sophia~~. ~~One~~ of the ~~extremely very best issues~~ in the ~~world~~ for ~~him is doing ballet~~ and ~~he'll be starting something else along~~ with it. ~~Some time~~ in the ~~past she selected to live~~ in ~~Alaska~~ and ~~her mothers and fathers live nearby~~. ~~Distributing production~~ is ~~how he makes~~ a ~~residing~~.<br><br>~~Look into my page accurate psychic predictions~~ ([~~http:~~//~~165~~.~~132~~.39.93/xe/~~visitors~~/~~372912~~ http://~~165~~.~~132~~.39.~~93/xe/visitors/372912~~])		The '''ramp function''' is a [[unary function\|unary]] [[real function]], easily computable as the [[arithmetic mean\|mean]] of the [[independent variable]] and its [[absolute value]].

			This function is applied in engineering (e.g., in the theory of [[Digital signal processing\|DSP]]). The name ''ramp function'' can be derived by the look of its graph.

			== Definitions ==
			[[Image:Ramp_function.svg\|[[Graph of a function\|Graph]] of the ramp function\|thumb\|260px\|right]]

			The ramp function (<math> R(x): \mathbb{R} \rightarrow \mathbb{R}</math>) may be defined analytically in several ways. Possible definitions are:
			:<math>R(x) := \begin{cases} x, & x \ge 0; \\ 0, & x<0 \end{cases} </math>
			* The mean of a straight line with unity gradient and its modulus:
			:<math>R(x) := \frac{x+\|x\|}{2} </math>
			this can be derived by noting the following definition of <math> \operatorname{max}(a,b) </math>,
			: <math> \operatorname{max}(a,b) = \frac{a+b+\|a-b\|}{2} </math>
			for which <math>a = x</math> and <math>b = 0</math>
			* The [[Heaviside step function]] multiplied by a straight line with unity gradient:
			: <math>R\left( x \right) := xH\left( x \right)</math>
			* The [[convolution]] of the Heaviside step function with itself:
			: <math>R\left( x \right) := H\left( x \right) * H\left( x \right)</math>
			* The [[integral]] of the Heaviside step function:
			: <math>R(x) := \int_{-\infty}^{x} H(\xi)\,\mathrm{d}\xi</math>
			* [[Macaulay brackets]]:
			: <math>R(x) := \langle x\rangle</math>

			== Analytic properties ==

			=== Non-negativity ===

			In the whole [[domain of a function\|domain]] the function is non-negative, so its [[absolute value]] is itself, i.e.

			<math>\forall x \in \mathbb{R}: R(x) \geqslant 0 </math>

			and

			<math>\left\| R \left( x \right) \right\| = R\left( x \right)</math>
			* Proof: by the mean of definition [2] it is non-negative in the I. quarter, and zero in the II.; so everywhere it is non-negative.

			=== Derivative ===
			Its derivative is the [[Heaviside function]]:

			<math>R'(x) = H(x)\ \mathrm{if}\ x \ne 0</math>

			<!--Ugyanis
			* ha x<0, akkor R(x)=0 konstans, tehát ezen a tartományon (ℝ<sup>-</sup>) R'(x)=0 (konstans deriváltja 0); ami megegyezik a Heaviside-függvénnyel.
			* ha x>0, akkor R(x)=x, tehát ezen a tartományon (ℝ<sup>+</sup>) R'(x)=1 (a valós számokon értelmezett [[identitás]] deriváltja 1); ami megegyezik a Heaviside-függvénnyel.
			* 0-ban a függvénynek [[töréspont]]ja van, tehát nem deriválható (jobbról deriválva 0-t, balról deriválva 1-et kapunk, holott a deriválhatóság feltétele, hogy a jobb és bal oldali derivált megegyezzen).-->

			From this property definition [5]. goes.

			=== [[Fourier transform]] ===
			<center> <math> \mathcal{F}\left\{ R(x) \right\}(f) </math> <math> = </math> <math> \int_{-\infty}^{\infty}R(x) e^{-2\pi ifx}dx </math> <math> = </math> <math> \frac{i\delta '(f)}{4\pi}-\frac{1}{4\pi^{2}f^{2}} </math> </center>

			Where <code>δ(x)</code> is the [[Dirac delta]] (in this formula, its [[derivative]] appears).

			=== [[Laplace transform]] ===

			The single-sided [[Laplace transform]] of <math>R(x)</math> is given as follows,

			<center> <math> \mathcal{L}\left\{ R\left( x \right)\right\} (s) = \int_{0}^{\infty} e^{-sx}R(x)dx = \frac{1}{s^2}. </math> </center>

			== Algebraic properties ==
			=== Iteration invariance ===
			Every [[iterated function]] of the ramp mapping is itself, as<br>
			<center><math> R \left( R \left( x \right) \right) = R \left( x \right) </math>. </center><br>
			* Proof: <math> R(R(x)):= \frac{R(x)+\|R(x)\|}{2} = \frac{R(x)+R(x)}{2} </math> <math>=</math> <br> <math>=</math> <math> \frac{2R(x)}{2} = R(x) </math>.

			We applied the [[#Non-negativity\|non-negative property]].

			== References ==
			* [http://mathworld.wolfram.com/RampFunction.html Mathworld]

			[[Category:Real analysis]]
			[[Category:Special functions]]

en>Gracefoo: Typo fixing, typos fixed: , → , using AWB

2012-06-25T21:39:31Z

Typo fixing, typos fixed: , → , using AWB

New page

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