Convex function - Revision history

en>John of Reading: Typo/general fixing, replaced: vice-versa → vice versa using AWB

2015-01-02T16:22:06Z

Typo/general fixing, replaced: vice-versa → vice versa using AWB

← Older revision		Revision as of 18:22, 2 January 2015
Line 1:		Line 1:
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en>David Pal: /* Definition */

2014-02-16T22:51:25Z

Definition

← Older revision		Revision as of 00:51, 17 February 2014
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	~~In [[mathematics]], the '''inverse Laplace transform''' of a function ''F''(''s'') is the function ''f''(''t'') which has the property~~
	~~<math>\mathcal{L}\left\{ f\right\}(s) = F(s)</math>,~~
	~~or alternatively <math>\mathcal{L}_t\left\{ f(t)\right\}(s) = F(s)</math>,~~
	~~where <math>\mathcal{L}</math> denotes the [[Laplace transform]].~~

	It can be proven, that if a function <math>F(s)</math> has the inverse Laplace transform <math>f(t)</math>, i.e. <math>f</math> is a piecewise-continuous and exponentially-restricted real function <math>f</math> satisfying the condition

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	~~then <math>f(t)</math>~~ is ~~uniquely determined (considering functions which differ from each other only on a point set having Lebesgue measure zero as the same).~~

	~~The [[Laplace transform]] and the inverse Laplace transform together have a number of properties that make them useful for analysing [[linear dynamic system]]s.~~

	~~==Mellin~~'s ~~inverse formula==~~
	~~An integral formula for the inverse [[Laplace transform]], called the~~ '~~''Bromwich integral''', the '''Fourier–Mellin integral''', and '''Mellin's inverse formula''', is given by the [[line integral]]:~~

	~~:<math>f(~~t~~) = \mathcal{L}^{-1} \{F\}(t) = \mathcal{L}^{-1}_s \{F(s)\}(t) = \frac{1}{2\pi i}\lim_{T\to\infty}\int_{\gamma-iT}^{\gamma+iT}e^{st}F(s)\,ds,</math>~~

	where the integration is done along the vertical line <math>Re(s)=\gamma</math> in the [[complex plane]] such that <math>\gamma</math> is greater than the real part of all [[Mathematical singularity\|singularities]] of ''F''(''s''). ~~This ensures that the contour path is in the [[region of convergence]].~~
	~~If all singularities are in the left half-plane, or ''F''(''s'') is~~ a ~~[[smooth function]] on - ∞ < ''Re''(''s'') < ∞ (i.e. no [[Mathematical singularity\|singularities]]), then <math>\gamma</math> can be set to zero~~ and ~~the above inverse integral formula above becomes identical to the [[inverse Fourier transform]].~~

	~~In practice, computing the complex integral can be done by using the [[Cauchy residue theorem]]~~.

	It is ~~named after [[Hjalmar Mellin]], [[Joseph Fourier]] and [[Thomas John I'Anson Bromwich]].~~

	~~If F(s) is the Laplace transform of the function f(t),then f(t) is called the inverse Laplace transform of F(s).~~

	~~==Software tools==~~

	* [http://~~reference~~.~~wolfram.com/mathematica/ref/InverseLaplaceTransform.html InverseLaplaceTransform] performs symbolic inverse transforms in [[Mathematica]]~~
	* [http://library.wolfram.com/infocenter/MathSource/5026/ Numerical Inversion of Laplace Transform with Multiple Precision Using the Complex Domain] in Mathematica gives numerical solutions<ref>{{cite doi\|10.1002/nme.995}}</ref>
	* [http://www.mathworks.co.uk/~~help~~/~~symbolic/ilaplace~~.html ~~ilaplace] performs symbolic inverse transforms in [[MATLAB]]~~
	* [http://www.mathworks.co.uk/matlabcentral/fileexchange/32824-numerical-inversion-of-laplace-transforms-in-matlab Numerical Inversion of Laplace Transforms in Matlab]

	~~==See also~~==
	* [[Inverse Fourier transform]]
	* [[Post's ~~inversion formula]], an alternative formula for the inverse Laplace transform~~.

	~~== References ==~~
	~~{{Reflist}}~~
	* {{Citation \| last1=Davies \| first1=B. J. \| title=Integral transforms and their applications \| publisher=[[Springer-Verlag]] \| location=Berlin, New York \| edition=3rd \| isbn=978-0-387-95314-4 \| year=2002}}
	* {{Citation \| last1=Manzhirov \| first1=A. V. \| last2=Polyanin \| first2=Andrei D. \| title=Handbook of integral equations \| publisher=[[CRC Press]] \| location=London \| isbn=978-0-8493-2876-3 \| year=1998}}
	* {{Citation \| last1=Boas \| first1=Mary \| year=1983 \| title=Mathematical Methods in the ~~physical sciences \| publisher=[[John Wiley & Sons]] \| isbn=0-471-04409-1 \| page=662 }} (p~~.~~ 662 or search Index for "Bromwich Integral", a nice explanation showing the connection~~ to ~~the fourier transform)~~

	~~==External links==~~
	* [http://~~eqworld~~.~~ipmnet~~.ru/en/~~auxiliary/aux-inttrans~~.~~htm Tables of Integral Transforms] at EqWorld: The World~~ of ~~Mathematical Equations.~~

	~~{{PlanetMath attribution\|id=5877\|title=Mellin's inverse formula}}~~

	~~[[Category:Transforms]]~~
	~~[[Category:Complex analysis]]~~
	~~[[Category:Integral transforms]~~]

en>Zfeinst: book referenced twice, with separate pages

2014-01-22T13:03:45Z

book referenced twice, with separate pages

← Older revision		Revision as of 15:03, 22 January 2014
Line 1:		Line 1:
	~~An underweight person may have low immunity~~, ~~reduced endurance~~, ~~and a deficiency of vitamins inside his body. So~~, ~~fat control is important. There are numerous ways to decide~~ the ~~ideal weight of a individual~~. ~~Among them are~~, ~~waist to height ratio~~, ~~and height to weight ratio~~.<br><br>~~To calculate the waist~~-to-~~hip ratio~~ (~~to keep an acronym it'~~s ~~called the WTHR~~), measure ~~both your waist and hips, plus divide~~ the ~~waist size by the hip size~~. ~~And yes, this indicator realizes the difference between men~~ and ~~women, unlike~~ the ~~BMI~~. ~~For ladies~~, the ~~ratio could be no more than 0.8. In different words~~, the ~~waist ought to be small than a hips. And for men~~, ~~it ought to be 1.0 or less. That means which a man~~'s ~~waist must be~~ the ~~same measuring because his hips or smaller. The beer~~-~~and~~-~~pizza abdomen has~~ to ~~go!~~<br><br>~~So a person~~ is '~~skinny~~' (~~translation~~ - ~~not weigh~~ a ~~lot~~ on ~~a scale~~) ~~nevertheless have rather little muscle mass~~ and ~~more fat than someone that is bigger but has more muscle mass plus less fat~~. ~~For this reason~~ using ~~a BMI alone to calculate your obesity~~ is ~~not a valid way~~. ~~A guy who is 610" plus weighs 270 pounds~~ is ~~considered fat by BMI specifications. However,~~ the ~~calculation does not consider how much muscle~~ the ~~guy has~~ (~~think WW wrestlers or professional football players~~)~~. The BMI~~ is ~~helpful just whenever employed inside~~ the ~~full context~~ of ~~height, weight, fat mass, plus muscle~~.<br><br>~~A lot of persons have benefited from the~~ [http://~~safedietplansforwomen~~.~~com~~/~~bmi~~-~~calculator bmi calculator men~~] ~~over~~ the ~~past limited decades however~~, ~~there are more exact alternatives available today~~. ~~If you are not sure what method is ideal for a current condition, then you need to consult a health professional for right guidance~~.~~<br><br>Burnello plus Beatice~~ D~~'Hombres~~ of ~~the Euopean Commision , Joint Research Center inside Ispra, Italy utilized info within the European Community Home Panel, that is a compative dataset,~~ in the ~~research published in The Journal Economics plus Human Biology~~.~~<br><br>Another limitation is the fact that age is not considered whenever calculating BMI. So~~ for ~~elderly people~~, ~~whether or not~~ the ~~BMI indication is normal,~~ the ~~truth remains that most muscles have been lost due to age~~.~~<br><br>Women should usually depend on a Body mass index calculator before they begin a weight reduction system~~. ~~This might equally enable females know when they need healthcare enable with regard to alarming BMI fluctuations or a quite significant~~ / ~~low BMI~~. ~~Although it really is possible to calculate the BMI manually with all the help~~ of ~~some formulae and charts, it is very furthermore greater should you depend on calculator to know what the ideal BMI is for women~~ of ~~your height~~.		In [[mathematics]], the '''inverse Laplace transform''' of a function ''F''(''s'') is the function ''f''(''t'') which has the property
			<math>\mathcal{L}\left\{ f\right\}(s) = F(s)</math>,
			or alternatively <math>\mathcal{L}_t\left\{ f(t)\right\}(s) = F(s)</math>,
			where <math>\mathcal{L}</math> denotes the [[Laplace transform]].

			It can be proven, that if a function <math>F(s)</math> has the inverse Laplace transform <math>f(t)</math>, i.e. <math>f</math> is a piecewise-continuous and exponentially-restricted real function <math>f</math> satisfying the condition

			:<math>\mathcal{L}_t\{f(t)\}(s) = F(s),\ \forall s \in \mathbb R</math>

			then <math>f(t)</math> is uniquely determined (considering functions which differ from each other only on a point set having Lebesgue measure zero as the same).

			The [[Laplace transform]] and the inverse Laplace transform together have a number of properties that make them useful for analysing [[linear dynamic system]]s.

			==Mellin's inverse formula==
			An integral formula for the inverse [[Laplace transform]], called the '''Bromwich integral''', the '''Fourier–Mellin integral''', and '''Mellin's inverse formula''', is given by the [[line integral]]:

			:<math>f(t) = \mathcal{L}^{-1} \{F\}(t) = \mathcal{L}^{-1}_s \{F(s)\}(t) = \frac{1}{2\pi i}\lim_{T\to\infty}\int_{\gamma-iT}^{\gamma+iT}e^{st}F(s)\,ds,</math>

			where the integration is done along the vertical line <math>Re(s)=\gamma</math> in the [[complex plane]] such that <math>\gamma</math> is greater than the real part of all [[Mathematical singularity\|singularities]] of ''F''(''s''). This ensures that the contour path is in the [[region of convergence]].
			If all singularities are in the left half-plane, or ''F''(''s'') is a [[smooth function]] on - ∞ < ''Re''(''s'') < ∞ (i.e. no [[Mathematical singularity\|singularities]]), then <math>\gamma</math> can be set to zero and the above inverse integral formula above becomes identical to the [[inverse Fourier transform]].

			In practice, computing the complex integral can be done by using the [[Cauchy residue theorem]].

			It is named after [[Hjalmar Mellin]], [[Joseph Fourier]] and [[Thomas John I'Anson Bromwich]].

			If F(s) is the Laplace transform of the function f(t),then f(t) is called the inverse Laplace transform of F(s).

			==Software tools==

			* [http://reference.wolfram.com/mathematica/ref/InverseLaplaceTransform.html InverseLaplaceTransform] performs symbolic inverse transforms in [[Mathematica]]
			* [http://library.wolfram.com/infocenter/MathSource/5026/ Numerical Inversion of Laplace Transform with Multiple Precision Using the Complex Domain] in Mathematica gives numerical solutions<ref>{{cite doi\|10.1002/nme.995}}</ref>
			* [http://www.mathworks.co.uk/help/symbolic/ilaplace.html ilaplace] performs symbolic inverse transforms in [[MATLAB]]
			* [http://www.mathworks.co.uk/matlabcentral/fileexchange/32824-numerical-inversion-of-laplace-transforms-in-matlab Numerical Inversion of Laplace Transforms in Matlab]

			==See also==
			* [[Inverse Fourier transform]]
			* [[Post's inversion formula]], an alternative formula for the inverse Laplace transform.

			== References ==
			{{Reflist}}
			* {{Citation \| last1=Davies \| first1=B. J. \| title=Integral transforms and their applications \| publisher=[[Springer-Verlag]] \| location=Berlin, New York \| edition=3rd \| isbn=978-0-387-95314-4 \| year=2002}}
			* {{Citation \| last1=Manzhirov \| first1=A. V. \| last2=Polyanin \| first2=Andrei D. \| title=Handbook of integral equations \| publisher=[[CRC Press]] \| location=London \| isbn=978-0-8493-2876-3 \| year=1998}}
			* {{Citation \| last1=Boas \| first1=Mary \| year=1983 \| title=Mathematical Methods in the physical sciences \| publisher=[[John Wiley & Sons]] \| isbn=0-471-04409-1 \| page=662 }} (p. 662 or search Index for "Bromwich Integral", a nice explanation showing the connection to the fourier transform)

			==External links==
			* [http://eqworld.ipmnet.ru/en/auxiliary/aux-inttrans.htm Tables of Integral Transforms] at EqWorld: The World of Mathematical Equations.

			{{PlanetMath attribution\|id=5877\|title=Mellin's inverse formula}}

			[[Category:Transforms]]
			[[Category:Complex analysis]]
			[[Category:Integral transforms]]

en>Zfeinst: use cite doi

2012-08-29T12:49:43Z

use cite doi

New page

An underweight person may have low immunity, reduced endurance, and a deficiency of vitamins inside his body. So, fat control is important. There are numerous ways to decide the ideal weight of a individual. Among them are, waist to height ratio, and height to weight ratio. To calculate the waist-to-hip ratio (to keep an acronym it's called the WTHR), measure both your waist and hips, plus divide the waist size by the hip size. And yes, this indicator realizes the difference between men and women, unlike the BMI. For ladies, the ratio could be no more than 0.8. In different words, the waist ought to be small than a hips. And for men, it ought to be 1.0 or less. That means which a man's waist must be the same measuring because his hips or smaller. The beer-and-pizza abdomen has to go! So a person is 'skinny' (translation - not weigh a lot on a scale) nevertheless have rather little muscle mass and more fat than someone that is bigger but has more muscle mass plus less fat. For this reason using a BMI alone to calculate your obesity is not a valid way. A guy who is 610" plus weighs 270 pounds is considered fat by BMI specifications. However, the calculation does not consider how much muscle the guy has (think WW wrestlers or professional football players). The BMI is helpful just whenever employed inside the full context of height, weight, fat mass, plus muscle. A lot of persons have benefited from the [http://safedietplansforwomen.com/bmi-calculator bmi calculator men] over the past limited decades however, there are more exact alternatives available today. If you are not sure what method is ideal for a current condition, then you need to consult a health professional for right guidance. Burnello plus Beatice D'Hombres of the Euopean Commision , Joint Research Center inside Ispra, Italy utilized info within the European Community Home Panel, that is a compative dataset, in the research published in The Journal Economics plus Human Biology. Another limitation is the fact that age is not considered whenever calculating BMI. So for elderly people, whether or not the BMI indication is normal, the truth remains that most muscles have been lost due to age. Women should usually depend on a Body mass index calculator before they begin a weight reduction system. This might equally enable females know when they need healthcare enable with regard to alarming BMI fluctuations or a quite significant / low BMI. Although it really is possible to calculate the BMI manually with all the help of some formulae and charts, it is very furthermore greater should you depend on calculator to know what the ideal BMI is for women of your height.