Isserlis' theorem - Revision history

en>Rjwilmsi: /* References */Added 2 dois to journal cites using AWB (10094)

2014-05-05T08:33:16Z

References: Added 2 dois to journal cites using AWB (10094)

← Older revision		Revision as of 10:33, 5 May 2014
Line 1:		Line 1:
	'''Positive-real functions''', often abbreviated to '''PR function''', are a kind of mathematical function that first arose in [[Network analysis (electrical circuits)\|electrical network analysis]]. They are [[Complex analysis#Complex functions\|complex functions]], ''Z''(''s''), of a complex variable, ''s''. A [[rational function]] is defined to have the PR property if it has a positive real part and is analytic in the right halfplane of the complex plane and takes on real values on the real axis.

	~~In symbols the definition is,~~

	~~:<math> \begin{align}~~		Race is a sport that a lot of frustrated racers would want to participate to. Alternatively, not all are fortunate enough to join the grand prix. Hence, there is a way to enjoy the excitement that racing should bring to drivers. Online racing games makes it possible. One of the [http://www.Google.com/search?q=racing+video&btnI=lucky racing video] gaming that is becoming favorite among racing fanatics can be CSR Racing. As a matter regarding fact, this game has taken ostentatious 100 free iPhone and iPad games category. This only shows the way furiously addictive this sport is. An array of upgradable cars is available, in addition to this short and exciting drag races.<br><br>This ingenious game has given lots of people the opportunity to dwell out their imagination. A variety of automobile components and equipment can be purchased, which allows players to customize their car. These upgrades in your car will help increase your car’s efficiency. It will also put you before game. Winning a race will allow you to earn some gold and coins that you'll require to fuel up and buying certain parts. Though succeeding races appears simple, one must face the that you cannot win on a regular basis. This is apparent particularly when your opponent’s car is actually more improved than your own house. Well, there is a method to boost your efficiency and also win in each get race; that is through CSR Auto racing cheats.<br><br>The use of CSR Race hacks is viewed by some individuals as wrong. Making use of the actual CSR Racing cheats in your great advantage is an imaginative move, and there is you don't need to feel guilty about the idea. This is created to aid gamers boost their gambling experience. With it, the games became a lot more thrilling. Don’t hesitate to make use of the Cheat Codes for CSR Rushing. After all, an incredible number of gamers are selecting it. Enabling other gamers to defeat you is obviously not an option especially if there are things which can be done to become the safe bet.<br><br>Look into the credibility on the CSR Racing [http://Faceria.com/home/index.php?do=/profile-217953/info/ Blitz Brigade hack download] instrument v1. 6, before you download it. Well, there is nothing you must be concerned with since there are numerous up-to-date CSR Racing cheats available. These can help you've got the fuel and money that will put your virtual race car further forward in the ethnic background.<br><br>If you are going to have that powerhouse benefits right in the beginning of the game, then these CSR Racing cheat are suitable for you. This game is absolutely thrilling, but you can still take it one stage further. Don’t worry, lightning won’t hit you in case you bend the rules slightly.
	~~& \Re[Z(s)]>0 \quad\text{if}\quad \Re(s) > 0 \\~~
	~~& \Im[Z(s)]=0 \quad\text{if}\quad \Im(s)=0~~
	~~\end{align} </math>~~

	~~In electrical network analysis, ''Z''(''s'') represents an [[electrical impedance\|impedance]] expression and ''s''~~ is ~~the [[complex frequency]] variable~~, ~~often expressed as its real and imaginary parts;~~

	~~:<math>s=\sigma+i\omega \,\!</math>~~

	~~in which terms the PR condition can be stated;~~

	~~:<math> \begin{align}~~
	~~& \Re[Z(s)]>0 \quad\text{if}\quad \sigma > 0 \\~~
	~~& \Im[Z(s)]=0 \quad\text{if}\quad \omega=0~~
	~~\end{align} </math>~~

	~~The importance~~ to ~~network analysis of the PR condition lies in~~ the ~~realisability condition~~. ~~''Z''(''s'')~~ is ~~realisable as~~ a ~~one-port rational impedance if and only if it meets~~ the ~~PR condition. Realisable in this sense means~~ that the impedance can be constructed from a finite (hence rational) number of discrete ideal [[Passivity (engineering)\|passive]] linear elements ([[resistor]]s, [[inductor]]s and [[capacitor]]s in electrical terminology).~~<ref name=CauerMathisPauli>E. Cauer, W. Mathis, and R~~. ~~Pauli, "Life and Work of Wilhelm Cauer (1900 – 1945)", ''Proceedings~~ of the ~~Fourteenth International Symposium of Mathematical Theory of Networks and Systems (MTNS2000)'', Perpignan, June, 2000.~~ [http://www.~~cs.princeton~~.~~edu~~/~~courses/archive/fall03/cs323/links/cauer.pdf Retrieved online] 19 September 2008.</ref>~~

	~~==Definition~~==
	~~The term ''positive-real function'' was originally defined by<ref name=CauerMathisPauli /> [[Otto Brune~~]~~] to describe any function ''Z''(''s'') which<ref name=Brune1931a>Brune, O, "Synthesis of a finite two-terminal network whose driving-point impedance~~ is a ~~prescribed function of frequency"~~, ~~Doctoral thesis, MIT, 1931~~. ~~[http://dspace~~.~~mit.edu/bitstream/handle/1721.1/10661/36311006.pdf?sequence=1 Retrieved online] 3rd June 2010.</ref>~~
	*is ~~[[rational function\|rational]] (the quotient of two [[polynomials]])~~,
	*is real when ''s'' is real
	*has positive real part when ''s'' has a positive real part
	~~Many authors strictly adhere~~ to this ~~definition by explicitly requiring rationality,~~<~~ref~~>~~{{cite book \|title=Network Theory \|last=Bakshi \|first=Uday \|last2=Bakshi \|first2=Ajay \|year=2008 \|publisher=Technical Publications \|location=Pune \|isbn=978-81-8431-402-1}}~~<~~/ref~~> ~~<!--U~~.A~~.Bakshi~~, J.S.~~Chitode --> or by restricting attention to rational functions, at least in the first instance<!-- Kendall Ling-chiao Su + -->~~.~~<ref name=Wing>{{cite book \|title=Classical Circuit Theory \|last=Wing \|first=Omar \|year=2008 \|publisher=Springer \|isbn=978-0-387-09739-8}}</ref> However,~~ a ~~similar more general condition, not restricted~~ to ~~rational functions had earlier been considered by Cauer,<ref name=CauerMathisPauli />~~ and ~~some authors ascribe the term ''positive-real'~~' to ~~this type of condition<!-- Wai-Kai Chen -->~~, ~~while other consider it to be a generalization of~~ the ~~basic definition.<ref name=Wing />~~

	~~==History==~~
	The condition was first proposed by [[Wilhelm Cauer]] (1926)<ref>Cauer, W, "Die Verwirklichung der Wechselstromwiderst ände vorgeschriebener Frequenzabh ängigkeit", ''Archiv für Elektrotechnik'', '''vol 17''', pp355–388, 1926.</ref> who determined that ~~it was~~ a ~~necessary condition~~. ~~[[Otto Brune]] (1931)<ref name=Brune1931a /><ref name=Brune1931b>Brune~~, ~~O, "Synthesis of a finite two-terminal network whose driving-point impedance~~ is a ~~prescribed function of frequency", ''J. Math.~~ and ~~Phys.'', '''vol 10''', pp191–236, 1931~~.<~~/ref~~> ~~coined the term positive-real for the condition and proved that it was both necessary and sufficient for realisability.~~

	~~==Properties==~~
	*The sum of two PR functions is PR.<~~!-- too obvious to mention? --~~>
	*The ~~[[function composition\|composition]]~~ of ~~two PR functions~~ is PR. ~~In particular, if ''Z''(''s'')~~ is PR, ~~then so are 1/''Z''(''s'')~~ and '~~'Z''(1/''s'').~~
	*All the ~~[[pole (complex analysis)\|poles]] and [[zero (complex analysis)\|zeros]] of a PR function are in the left half plane or on its boundary the imaginary axis~~.
	*Any poles and zeroes on the imaginary axis are [[zero (complex analysis)#Multiplicity_of_a_zero\|simple]] (have a [[multiplicity (mathematics)\|multiplicity]] of one).
	*Any poles on the imaginary axis have real strictly positive [[residue (complex analysis)\|residues]], and similarly at any zeroes on the imaginary axis, the ~~function has~~ a ~~real strictly positive derivative~~.
	*Over the right half plane, the minimum value of the real part of a PR function occurs on the imaginary axis (because the real part of an analytic function constitutes a [[harmonic function]] over the plane, and therefore satisfies the [[maximum principle]]).
	*For a [[rational function\|rational]] PR function, ~~the~~ number of ~~poles and number of zeroes differ by at most one~~.

	~~==Generalizations==~~
	~~A couple of generalizations~~ are ~~sometimes made, with intention of characterizing~~ the [~~[immittance]] functions of a wider class of passive linear electrical networks~~.

	=~~==Irrational functions===~~
	~~The impedance ''Z''(''s'') of a network consisting of an infinite number of components (such as a semi~~-~~infinite [[Ladder_network#Ladder_topologies\|ladder~~]]), ~~need not~~ be ~~a rational function of~~ '~~'s'',~~ and in ~~particular may have [[branch points]] on~~ the ~~negative real ''s''-axis~~. ~~To accommodate such functions~~ in the ~~definition~~ of PR, ~~it is therefore necessary to relax the condition that the function be real~~ for ~~all real ''s'', and only require this when ''s'' is positive~~. ~~Thus, a possibly irrational function ''Z''(''s'') is PR if and only if~~
	*''Z''(''s'') is ~~analytic in the open right half ''s''-plane (Re[''s''] > 0)~~
	*''Z''(''s'') is real when ''s'' is positive and real
	*Re[''Z''(''s'')] ≥ 0 when Re[''s''] ≥ 0
	~~Some authors start from this more general definition~~, ~~and then particularize~~ it ~~to the rational case.~~

	~~===Matrix-valued functions===~~
	~~Linear electrical networks with more than~~ one ~~port may be described by [[impedance parameters\|impedance or]] [[admittance parameters\|admittance matrices]]~~. ~~So by extending the definition of PR to matrix-valued functions~~, ~~linear multi-port networks which are passive may be distinguished from those that are not. A possibly irrational matrix-valued function ''Z''(''s'') is PR if and only if~~
	*Each element of ''Z''(''s'') is analytic in the ~~open right half ''s''-plane (Re[''s''] > 0)~~
	*Each element of ''Z''(''s'') is real when ''s'' is positive and real
	*The [[Hermitian matrix\|Hermitian]] part (''Z''(''s'') + ''Z''<sup>†</sup>(''s''))/2 of ''Z''(''s'') is [[Positive-definite matrix\|positive semi-definite]] when Re[''s''] ≥ 0

	~~==References==~~
	~~{{reflist}}~~

	~~[[Category:Complex analysis]]~~
	~~[[Category:Electronic engineering]]~~
	~~[[Category:Types of functions]]~~

en>Addbot: Bot: Migrating 1 interwiki links, now provided by Wikidata on d:q4491919

2013-03-20T11:31:56Z

Bot: Migrating 1 interwiki links, now provided by Wikidata on d:q4491919

← Older revision		Revision as of 13:31, 20 March 2013
Line 1:		Line 1:
	~~The key golf energy~~, ~~unknown~~ to ~~many lovers~~ of the ~~game could be~~ the ~~health~~ and ~~strengthening of~~ the ~~muscles in your system~~, ~~which you use to play golf. The weaker~~ and ~~unprepared this core golf strength~~ is, the ~~weaker your game of golf will~~ be. <br><br>~~But let us explain why~~ this ~~should~~ be ~~the core tennis energy~~ of ~~any player~~, ~~whether amateur or professional~~. ~~The sport~~ of ~~golf requires exceptionally unpleasant movements for~~ the ~~body~~. ~~This novel~~ [http://www.~~truecoresupplements~~.~~com~~/ http://~~www~~.~~truecoresupplements~~.~~com~~/] ~~site~~ has ~~collected striking cautions for~~ when to ~~allow for~~ this ~~idea.~~ <br><br>~~This is one of many reasons why the golf swing is indeed difficult~~ to ~~master. It"s mainly because even as you try you far better get your swing right~~, the ~~behaviors of your body and the muscles is against your efforts~~. <br><br>~~I-t thus makes plenty of sense~~ to ~~situation~~ and ~~exercise those muscles because as you improve them~~ to ~~cope with the golf swing for example~~, the ~~motion can come more naturally to you~~. The ~~inevitable result is~~ that ~~your round of golf will improve significantly~~. <br><br>~~And it will all be due to your respect for the core golf strength~~. ~~Several people think that the core golf strength is another thing~~. ~~Probably swing method~~, ~~or even experience~~. <br><br>The ~~consequence~~ is ~~that their golf game gets more annoying by~~ the ~~morning because they find it difficult to work~~ on the ~~wrong key golf energy that can become having very little influence if~~ any, on the ~~caliber~~ of ~~their game~~. ~~<br><br>One other reason~~ the ~~actual key tennis strength is shunned or ignored~~ by ~~many is~~ the ~~fact many inexperienced people hate the mere thought~~ of ~~exercise~~. The ~~idea~~ of ~~dripping rigorous exercise~~ and ~~sweat is scary to numerous who~~ have ~~taken to this so~~-~~called gentlemans game~~. ~~<br><br>So terrifying that they rarely bother to learn the facts~~ in ~~regards to~~ the ~~nature~~ of the ~~exercises or why older persons with back dilemmas control them so well~~ that ~~they sometimes also end-up reducing some irritating physical pains like continual back pains~~. ~~<br><br>These are~~ a ~~few of~~ the ~~reasons why~~ this ~~core tennis strength remains undiscovered~~ by ~~many~~.~~. Learn supplementary info on this related URL~~ by ~~browsing~~ to [~~http://www.truecoresupplements.com/ http://truecoresupplements.com~~].<br><br>~~If you loved this posting and you would like to receive far more details about~~ [~~http~~:~~//www.fizzlive.com/magnificentwoma27 purchase health insurance~~] ~~kindly check out the website.~~		'''Positive-real functions''', often abbreviated to '''PR function''', are a kind of mathematical function that first arose in [[Network analysis (electrical circuits)\|electrical network analysis]]. They are [[Complex analysis#Complex functions\|complex functions]], ''Z''(''s''), of a complex variable, ''s''. A [[rational function]] is defined to have the PR property if it has a positive real part and is analytic in the right halfplane of the complex plane and takes on real values on the real axis.

			In symbols the definition is,

			:<math> \begin{align}
			& \Re[Z(s)]>0 \quad\text{if}\quad \Re(s) > 0 \\
			& \Im[Z(s)]=0 \quad\text{if}\quad \Im(s)=0
			\end{align} </math>

			In electrical network analysis, ''Z''(''s'') represents an [[electrical impedance\|impedance]] expression and ''s'' is the [[complex frequency]] variable, often expressed as its real and imaginary parts;

			:<math>s=\sigma+i\omega \,\!</math>

			in which terms the PR condition can be stated;

			:<math> \begin{align}
			& \Re[Z(s)]>0 \quad\text{if}\quad \sigma > 0 \\
			& \Im[Z(s)]=0 \quad\text{if}\quad \omega=0
			\end{align} </math>

			The importance to network analysis of the PR condition lies in the realisability condition. ''Z''(''s'') is realisable as a one-port rational impedance if and only if it meets the PR condition. Realisable in this sense means that the impedance can be constructed from a finite (hence rational) number of discrete ideal [[Passivity (engineering)\|passive]] linear elements ([[resistor]]s, [[inductor]]s and [[capacitor]]s in electrical terminology).<ref name=CauerMathisPauli>E. Cauer, W. Mathis, and R. Pauli, "Life and Work of Wilhelm Cauer (1900 – 1945)", ''Proceedings of the Fourteenth International Symposium of Mathematical Theory of Networks and Systems (MTNS2000)'', Perpignan, June, 2000. [http://www.cs.princeton.edu/courses/archive/fall03/cs323/links/cauer.pdf Retrieved online] 19 September 2008.</ref>

			==Definition==
			The term ''positive-real function'' was originally defined by<ref name=CauerMathisPauli /> [[Otto Brune]] to describe any function ''Z''(''s'') which<ref name=Brune1931a>Brune, O, "Synthesis of a finite two-terminal network whose driving-point impedance is a prescribed function of frequency", Doctoral thesis, MIT, 1931. [http://dspace.mit.edu/bitstream/handle/1721.1/10661/36311006.pdf?sequence=1 Retrieved online] 3rd June 2010.</ref>
			*is [[rational function\|rational]] (the quotient of two [[polynomials]]),
			*is real when ''s'' is real
			*has positive real part when ''s'' has a positive real part
			Many authors strictly adhere to this definition by explicitly requiring rationality,<ref>{{cite book \|title=Network Theory \|last=Bakshi \|first=Uday \|last2=Bakshi \|first2=Ajay \|year=2008 \|publisher=Technical Publications \|location=Pune \|isbn=978-81-8431-402-1}}</ref> <!--U.A.Bakshi, J.S.Chitode --> or by restricting attention to rational functions, at least in the first instance<!-- Kendall Ling-chiao Su + -->.<ref name=Wing>{{cite book \|title=Classical Circuit Theory \|last=Wing \|first=Omar \|year=2008 \|publisher=Springer \|isbn=978-0-387-09739-8}}</ref> However, a similar more general condition, not restricted to rational functions had earlier been considered by Cauer,<ref name=CauerMathisPauli /> and some authors ascribe the term ''positive-real'' to this type of condition<!-- Wai-Kai Chen -->, while other consider it to be a generalization of the basic definition.<ref name=Wing />

			==History==
			The condition was first proposed by [[Wilhelm Cauer]] (1926)<ref>Cauer, W, "Die Verwirklichung der Wechselstromwiderst ände vorgeschriebener Frequenzabh ängigkeit", ''Archiv für Elektrotechnik'', '''vol 17''', pp355–388, 1926.</ref> who determined that it was a necessary condition. [[Otto Brune]] (1931)<ref name=Brune1931a /><ref name=Brune1931b>Brune, O, "Synthesis of a finite two-terminal network whose driving-point impedance is a prescribed function of frequency", ''J. Math. and Phys.'', '''vol 10''', pp191–236, 1931.</ref> coined the term positive-real for the condition and proved that it was both necessary and sufficient for realisability.

			==Properties==
			*The sum of two PR functions is PR.<!-- too obvious to mention? -->
			*The [[function composition\|composition]] of two PR functions is PR. In particular, if ''Z''(''s'') is PR, then so are 1/''Z''(''s'') and ''Z''(1/''s'').
			*All the [[pole (complex analysis)\|poles]] and [[zero (complex analysis)\|zeros]] of a PR function are in the left half plane or on its boundary the imaginary axis.
			*Any poles and zeroes on the imaginary axis are [[zero (complex analysis)#Multiplicity_of_a_zero\|simple]] (have a [[multiplicity (mathematics)\|multiplicity]] of one).
			*Any poles on the imaginary axis have real strictly positive [[residue (complex analysis)\|residues]], and similarly at any zeroes on the imaginary axis, the function has a real strictly positive derivative.
			*Over the right half plane, the minimum value of the real part of a PR function occurs on the imaginary axis (because the real part of an analytic function constitutes a [[harmonic function]] over the plane, and therefore satisfies the [[maximum principle]]).
			*For a [[rational function\|rational]] PR function, the number of poles and number of zeroes differ by at most one.

			==Generalizations==
			A couple of generalizations are sometimes made, with intention of characterizing the [[immittance]] functions of a wider class of passive linear electrical networks.

			===Irrational functions===
			The impedance ''Z''(''s'') of a network consisting of an infinite number of components (such as a semi-infinite [[Ladder_network#Ladder_topologies\|ladder]]), need not be a rational function of ''s'', and in particular may have [[branch points]] on the negative real ''s''-axis. To accommodate such functions in the definition of PR, it is therefore necessary to relax the condition that the function be real for all real ''s'', and only require this when ''s'' is positive. Thus, a possibly irrational function ''Z''(''s'') is PR if and only if
			*''Z''(''s'') is analytic in the open right half ''s''-plane (Re[''s''] > 0)
			*''Z''(''s'') is real when ''s'' is positive and real
			*Re[''Z''(''s'')] ≥ 0 when Re[''s''] ≥ 0
			Some authors start from this more general definition, and then particularize it to the rational case.

			===Matrix-valued functions===
			Linear electrical networks with more than one port may be described by [[impedance parameters\|impedance or]] [[admittance parameters\|admittance matrices]]. So by extending the definition of PR to matrix-valued functions, linear multi-port networks which are passive may be distinguished from those that are not. A possibly irrational matrix-valued function ''Z''(''s'') is PR if and only if
			*Each element of ''Z''(''s'') is analytic in the open right half ''s''-plane (Re[''s''] > 0)
			*Each element of ''Z''(''s'') is real when ''s'' is positive and real
			*The [[Hermitian matrix\|Hermitian]] part (''Z''(''s'') + ''Z''<sup>†</sup>(''s''))/2 of ''Z''(''s'') is [[Positive-definite matrix\|positive semi-definite]] when Re[''s''] ≥ 0

			==References==
			{{reflist}}

			[[Category:Complex analysis]]
			[[Category:Electronic engineering]]
			[[Category:Types of functions]]

en>Melcombe: /* Theorem statement */link to Isserlis

2012-05-28T21:37:55Z

Theorem statement: link to Isserlis

New page

The key golf energy, unknown to many lovers of the game could be the health and strengthening of the muscles in your system, which you use to play golf. The weaker and unprepared this core golf strength is, the weaker your game of golf will be. But let us explain why this should be the core tennis energy of any player, whether amateur or professional. The sport of golf requires exceptionally unpleasant movements for the body. This novel [http://www.truecoresupplements.com/ http://www.truecoresupplements.com/] site has collected striking cautions for when to allow for this idea. This is one of many reasons why the golf swing is indeed difficult to master. It"s mainly because even as you try you far better get your swing right, the behaviors of your body and the muscles is against your efforts. I-t thus makes plenty of sense to situation and exercise those muscles because as you improve them to cope with the golf swing for example, the motion can come more naturally to you. The inevitable result is that your round of golf will improve significantly. And it will all be due to your respect for the core golf strength. Several people think that the core golf strength is another thing. Probably swing method, or even experience. The consequence is that their golf game gets more annoying by the morning because they find it difficult to work on the wrong key golf energy that can become having very little influence if any, on the caliber of their game. One other reason the actual key tennis strength is shunned or ignored by many is the fact many inexperienced people hate the mere thought of exercise. The idea of dripping rigorous exercise and sweat is scary to numerous who have taken to this so-called gentlemans game. So terrifying that they rarely bother to learn the facts in regards to the nature of the exercises or why older persons with back dilemmas control them so well that they sometimes also end-up reducing some irritating physical pains like continual back pains. These are a few of the reasons why this core tennis strength remains undiscovered by many.. Learn supplementary info on this related URL by browsing to [http://www.truecoresupplements.com/ http://truecoresupplements.com]. If you loved this posting and you would like to receive far more details about [http://www.fizzlive.com/magnificentwoma27 purchase health insurance] kindly check out the website.