Manin matrix - Revision history

en>Rjwilmsi: Added 2 dois to journal cites using AWB (10159)

2014-05-14T12:21:45Z

Added 2 dois to journal cites using AWB (10159)

← Older revision		Revision as of 14:21, 14 May 2014
Line 1:		Line 1:
	Today technologies has achieved new height. Everything is obtaining quicker and smoother. Even these days if you have cellular phone in your pocket, then anytime from anyplace you can avail cash in your credit score crunch. This is possible simply because of the mobile textual content loans that provide the ~~cash approval within a few hours~~.~~<br><br><br><br>A secured personal mortgage is a secured mortgage offered to satisfy personal needs of the UK citizens. To avail this loan a borrower needs to~~ place collateral towards the loan. Your vehicle, home or even a conserving bank account can work as collateral. Secured mortgage helps debtors in creating the best use of the equity stored in his or her property that assists him in borrowing a larger quantity of loan and that as well for a lengthier loan term.<br><br>Mobile textual content loans UK are extremely simple in acceptance if you are a UK citizen and 18 many years old. Nonetheless you must be working permanently and have a bank account. To get the approval you also must have the mobile telephone and an e-mail deal with.<br><br><br><br>There are some essential factors that you should remember prior to you use for the mortgage. ~~Your vehicle ought to be at your name or the individual who~~ is making use of for this facility. Your vehicle should not be more than eight many years previous. You ought to be having a great paying occupation and that should be fixed. You should have paid all the money owed associated to your car and it ought to be tax totally free.<br><br>The great component about availing these loans is that you will be able to buy a car of any make or design. These loans will offer you with 90-one hundred%25 of the complete funds needed. Furthermore you can also use the amount to purchase a utilized car, supplied it ought to be 5-six many years previous, as following this the cost of the marketplace worth of the car starts falls and needs a ~~great deal of upkeep~~.~~<br><br>These text loans uk have become extremely well-liked as these loans are produced according to the needs of young people. You don~~'t need to know much to get text loans. You just have to know about the loan providers. These short applications when verified by the mortgage supplier are granted loans within the shortest possible time body. But the primary thing about these loans is cash will provided in the applicant account in very secure and safe.<br><br>Same working day mortgage UK is available to these who need instant cash for their financially disaster. It is accessible for all type of budgets. You can get assured money on the exact same working day of your submission of application inside 24 hrs of time.<br><br>In UK there are numerous financial companies who give individual loans on reduced interest rate in but ~~we have~~ to ~~lookup. So the next benefit of personal loan is the reduced curiosity rate. Because most of the~~ time ~~the borrower invested their hard attained cash in paying the interest rate. So this is the next benefit of personal loans UK i.e. low interest~~.~~<br><br>~~If you ~~liked this article and you would like~~ to ~~get~~ much more ~~data with regards to~~ [http://~~topnocreditcheckloans~~.com/~~reviews/100~~-~~day-uk-loans-review wmflabs~~.~~org~~] ~~kindly take a look at the webpage.~~		They contact the author Nicole Earp. Arizona has usually been her living place and she has everything that she needs there. Meter studying is how I make a residing. It's not a common factor but what I like doing is baking but I struggle to find time for it. If you want to discover out much more verify out my website: http://[http://Www.youtube.com/watch?v=jOGgdu4f-H8 Www.youtube.com]/watch?v=jOGgdu4f-H8

en>Quebec99: fixed references

2014-03-04T23:38:28Z

fixed references

← Older revision		Revision as of 01:38, 5 March 2014
Line 1:		Line 1:
	~~In the field of [[ordinary differential equation]]s, the '''Mingarelli identity''' (coined by [[Philip Hartman]]<ref name="Hartman">{{harvnb\|Clark D~~.N., G. ~~Pecelli, and R~~. ~~Sacksteder\|1981}}~~<~~/ref~~>) is a ~~theorem that provides criteria for the [[oscillation theory\|oscillation]] and [[oscillation theory\|non-oscillation]]~~ of ~~solutions of some [[linear differential equation]]s in~~ the ~~real domain~~. ~~It extends the [[Picone identity]] from two~~ to ~~three or more differential equations of~~ the ~~second order~~. ~~Its most basic form appears here~~.		Today technologies has achieved new height. Everything is obtaining quicker and smoother. Even these days if you have cellular phone in your pocket, then anytime from anyplace you can avail cash in your credit score crunch. This is possible simply because of the mobile textual content loans that provide the cash approval within a few hours.<br><br><br><br>A secured personal mortgage is a secured mortgage offered to satisfy personal needs of the UK citizens. To avail this loan a borrower needs to place collateral towards the loan. Your vehicle, home or even a conserving bank account can work as collateral. Secured mortgage helps debtors in creating the best use of the equity stored in his or her property that assists him in borrowing a larger quantity of loan and that as well for a lengthier loan term.<br><br>Mobile textual content loans UK are extremely simple in acceptance if you are a UK citizen and 18 many years old. Nonetheless you must be working permanently and have a bank account. To get the approval you also must have the mobile telephone and an e-mail deal with.<br><br><br><br>There are some essential factors that you should remember prior to you use for the mortgage. Your vehicle ought to be at your name or the individual who is making use of for this facility. Your vehicle should not be more than eight many years previous. You ought to be having a great paying occupation and that should be fixed. You should have paid all the money owed associated to your car and it ought to be tax totally free.<br><br>The great component about availing these loans is that you will be able to buy a car of any make or design. These loans will offer you with 90-one hundred%25 of the complete funds needed. Furthermore you can also use the amount to purchase a utilized car, supplied it ought to be 5-six many years previous, as following this the cost of the marketplace worth of the car starts falls and needs a great deal of upkeep.<br><br>These text loans uk have become extremely well-liked as these loans are produced according to the needs of young people. You don't need to know much to get text loans. You just have to know about the loan providers. These short applications when verified by the mortgage supplier are granted loans within the shortest possible time body. But the primary thing about these loans is cash will provided in the applicant account in very secure and safe.<br><br>Same working day mortgage UK is available to these who need instant cash for their financially disaster. It is accessible for all type of budgets. You can get assured money on the exact same working day of your submission of application inside 24 hrs of time.<br><br>In UK there are numerous financial companies who give individual loans on reduced interest rate in but we have to lookup. So the next benefit of personal loan is the reduced curiosity rate. Because most of the time the borrower invested their hard attained cash in paying the interest rate. So this is the next benefit of personal loans UK i.e. low interest.<br><br>If you liked this article and you would like to get much more data with regards to [http://topnocreditcheckloans.com/reviews/100-day-uk-loans-review wmflabs.org] kindly take a look at the webpage.

	~~== The identity ==~~
	~~Consider~~ the ~~<math>n</math> solutions~~ of the ~~following (uncoupled) system~~ of ~~second order linear differential equations over the ''t''-interval [''~~a~~'', ''b'']~~.
	<~~math~~>~~(p_i(t) x_i^\prime)^\prime + q_i(t) x_i = 0, \,\,\,\,\,\,\,\,\,\, x_i(~~a~~)=1,\,\, x_i^\prime(~~a~~)=R_i\,</math>~~
	~~where <math>i=1,2, \ldots, n</math>~~. ~~Let <math>\Delta</math> denote~~ the ~~forward difference operator, i.~~e., <~~math~~>~~\Delta x_i = x_{i+1}-x_i.~~<~~/math~~> ~~The second order difference operator is found by iterating the first order operator as in~~ <~~math~~>~~\Delta^2 (x_i) = \Delta(\Delta x_i) = x_{i+2}-2x_{i+1}+x_{i}~~<~~/math~~>~~, with a similar definition~~ for the ~~higher iterates~~.

	~~Leaving out~~ the ~~independent variable ''t''~~ for ~~convenience,~~ and ~~assuming~~ the <~~math~~>~~x_i(t) \ne 0~~<~~/math~~> ~~on (''~~a~~'', ''b''], there holds~~ the ~~identity~~,~~<ref name="Mingarelli">{{harvnb\|Mingarelli\|1979\|p=223}}</ref>~~
	~~: <math>~~
	~~\begin{align}~~
	~~x_{n-1}^2\Delta^{n-1}(p_1r_1)]_a^b & = \int_a^b (x^\prime_{n-1})^2 \Delta^{n-1}(p_1)~~ - ~~\int_a^b x_{n-1}^2 \Delta^{n-1}(q_1)~~
	~~- \sum_{k=0}^{n-1} C(n-1~~,~~k)(-1)^{n-k-1}\int_a^b p_{k+1} W^2(x_{k+1},x_{n-1})/x_{k+1}^2,~~
	~~\end{align}~~
	~~</math>~~

	~~where <math>r_i = x^\prime_i/x_i</math> is~~ a ~~[[logarithmic derivative]],~~ <~~math~~>~~W(x_i, x_j) = x^\prime_ix_j - x_ix^\prime_j~~<~~/math~~>~~, is a [[Wronskian]] and the <math>C(n~~-~~1,k)</math>~~ are ~~[[binomial coefficients]]~~. ~~When <math>n=2</math> this reduces~~ to the ~~[[Picone identity]]~~.

	~~The above identity leads quickly to~~ the ~~following comparison theorem for three linear differential equations,<ref name="Mingarelli">{{harv\|Mingarelli\|1979\|Theorem 2}}</ref> extending~~ the ~~[[Sturm–Picone comparison theorem]]~~.

	~~Let <math>p_i,\, q_i,\,</math> ''i'' = 1, 2, 3 be real-valued continuous functions on~~ the ~~interval [''a'', ''b'']~~ and ~~let~~
	#<~~math~~>~~(p_1(t) x_1^\prime)^\prime + q_1(t) x_1 = 0, \,\,\,\,\,\,\,\,\,\, x_1(a)=1,\,\, x_1^\prime(a)=R_1\,~~<~~/math>~~
	~~#<math>(p_2(t) x_2^\prime)^\prime + q_2(t) x_2 = 0, \,\,\,\,\,\,\,\,\,\, x_2(a)=1,\,\, x_2^\prime(a)=R_2\,</math>~~
	~~#<math>(p_3(t) x_3^\prime)^\prime + q_3(t) x_3 = 0, \,\,\,\,\,\,\,\,\,\, x_3(a)=1,\,\, x_3^\prime(a)=R_3\,</math>~~
	~~be three homogeneous linear second order differential equations in [[self-adjoint form]] with~~
	~~:<math>p_i(t)>0\,</math~~> for ~~each i and~~ for all ~~''t'' in [''a'', ''b''], and where~~ the <~~math~~>~~R_i~~<~~/math~~> are ~~arbitrary real numbers.~~

	~~Assume that for all ''t''~~ in ~~[''a'', ''b'']~~ we have,
	:<~~math~~>~~\Delta^2(q_1) \ge 0~~ <~~/math~~>,
	:~~<math>\Delta^2(p_1) \le 0 <~~/~~math>,~~
	~~:<math>\Delta^2(p_1(a)R_1) \le 0 <~~/~~math>~~.
	~~If <math>x_1(t) > 0<~~/~~math> on [''a'', ''b''], and <math>x_2(b)=0<~~/~~math>, then any solution <math>x_3(t)</math> has at least one zero in [''a'', ''b''].~~

	~~== References ==~~

	~~{{Reflist}}~~

	* {{cite book
	~~\|last=Clark D.N., G. Pecelli, and R. Sacksteder~~
	~~\|first=~~
	~~\|year=1981~~
	~~\|author~~-~~link=~~
	~~\|author2~~-~~link= G. Pecelli~~
	~~\|author3~~-~~link=R~~. ~~Sacksteder~~
	~~\|language=~~
	~~\|title=Contributions to Analysis and Geometry~~
	~~\|journal=~~
	~~\|series=~~
	~~\|volume=~~
	~~\|pages=~~
	~~\|location= Baltimore, USA~~
	~~\|publisher= Johns Hopkins University Press~~
	~~\|isbn=~~
	~~\|url=~~
	~~\|ref=harv}}~~

	* {{cite journal
	~~\|last=Mingarelli~~
	~~\|first= Angelo B.~~
	~~\|year=1979~~
	~~\|author-link=~~
	~~\|author2-link=~~
	~~\|language=~~
	~~\|title= Some extensions of~~ the ~~Sturm–Picone theorem~~
	~~\|journal= Comptes Rendus Math~~. ~~Rep. Acad. Sci. Canada~~
	~~\|series=~~
	~~\|volume=1 (4)~~
	~~\|pages=223–226~~
	~~\|location= Toronto, Ontario, Canada~~
	~~\|publisher= The Royal Society of Canada~~
	~~\|isbn=~~
	~~\|url=~~
	~~\|ref=harv}}~~

	~~[[Category:Ordinary differential equations]]~~
	~~[[Category:Mathematical identities]]~~

en>Myasuda: sp

2013-12-21T05:14:01Z

← Older revision		Revision as of 07:14, 21 December 2013
Line 1:		Line 1:
	~~Unfortunately there was not other available room~~, the ~~hour was last so we had to stay~~.<br>~~One color~~ that ~~never goes out~~ of ~~style and always seems~~ to ~~make a splash is white and we're already seeing this all over~~ the ~~red carpet~~. ~~Today's look~~ of the ~~day features~~ a ~~white hot look on fantastic actress Ginnifer Goodwin~~.<br>~~http:~~//~~www~~. ~~If you cherished this report and you would like to obtain a lot more data pertaining to [http:~~/~~/www~~.~~restaurantcalcuta~~.~~com~~/~~outlet~~/~~ugg.asp Ugg Outlet Store] kindly take~~ a ~~look at our website~~. ~~shlgrouptr.com~~/~~warm/?~~p=15 <br /> ~~http~~://~~www.shlgrouptr.com~~/~~warm~~/?p=~~215~~ <br /> ~~http:~~//~~www~~.~~shlgrouptr~~.~~com~~/~~warm~~/?p=~~282~~ <br /> ~~http~~://~~www~~.~~shlgrouptr.com~~/~~warm/?p=348~~ <br /> ~~http~~:/~~/www~~.~~shlgrouptr.com~~/~~warm~~/~~?p=88~~ <br />		In the field of [[ordinary differential equation]]s, the '''Mingarelli identity''' (coined by [[Philip Hartman]]<ref name="Hartman">{{harvnb\|Clark D.N., G. Pecelli, and R. Sacksteder\|1981}}</ref>) is a theorem that provides criteria for the [[oscillation theory\|oscillation]] and [[oscillation theory\|non-oscillation]] of solutions of some [[linear differential equation]]s in the real domain. It extends the [[Picone identity]] from two to three or more differential equations of the second order. Its most basic form appears here.

			== The identity ==
			Consider the <math>n</math> solutions of the following (uncoupled) system of second order linear differential equations over the ''t''-interval [''a'', ''b''].
			<math>(p_i(t) x_i^\prime)^\prime + q_i(t) x_i = 0, \,\,\,\,\,\,\,\,\,\, x_i(a)=1,\,\, x_i^\prime(a)=R_i\,</math>
			where <math>i=1,2, \ldots, n</math>. Let <math>\Delta</math> denote the forward difference operator, i.e., <math>\Delta x_i = x_{i+1}-x_i.</math> The second order difference operator is found by iterating the first order operator as in <math>\Delta^2 (x_i) = \Delta(\Delta x_i) = x_{i+2}-2x_{i+1}+x_{i}</math>, with a similar definition for the higher iterates.

			Leaving out the independent variable ''t'' for convenience, and assuming the <math>x_i(t) \ne 0</math> on (''a'', ''b''], there holds the identity,<ref name="Mingarelli">{{harvnb\|Mingarelli\|1979\|p=223}}</ref>
			: <math>
			\begin{align}
			x_{n-1}^2\Delta^{n-1}(p_1r_1)]_a^b & = \int_a^b (x^\prime_{n-1})^2 \Delta^{n-1}(p_1) - \int_a^b x_{n-1}^2 \Delta^{n-1}(q_1)
			- \sum_{k=0}^{n-1} C(n-1,k)(-1)^{n-k-1}\int_a^b p_{k+1} W^2(x_{k+1},x_{n-1})/x_{k+1}^2,
			\end{align}
			</math>

			where <math>r_i = x^\prime_i/x_i</math> is a [[logarithmic derivative]], <math>W(x_i, x_j) = x^\prime_ix_j - x_ix^\prime_j</math>, is a [[Wronskian]] and the <math>C(n-1,k)</math> are [[binomial coefficients]]. When <math>n=2</math> this reduces to the [[Picone identity]].

			The above identity leads quickly to the following comparison theorem for three linear differential equations,<ref name="Mingarelli">{{harv\|Mingarelli\|1979\|Theorem 2}}</ref> extending the [[Sturm–Picone comparison theorem]].

			Let <math>p_i,\, q_i,\,</math> ''i'' = 1, 2, 3 be real-valued continuous functions on the interval [''a'', ''b''] and let
			#<math>(p_1(t) x_1^\prime)^\prime + q_1(t) x_1 = 0, \,\,\,\,\,\,\,\,\,\, x_1(a)=1,\,\, x_1^\prime(a)=R_1\,</math>
			#<math>(p_2(t) x_2^\prime)^\prime + q_2(t) x_2 = 0, \,\,\,\,\,\,\,\,\,\, x_2(a)=1,\,\, x_2^\prime(a)=R_2\,</math>
			#<math>(p_3(t) x_3^\prime)^\prime + q_3(t) x_3 = 0, \,\,\,\,\,\,\,\,\,\, x_3(a)=1,\,\, x_3^\prime(a)=R_3\,</math>
			be three homogeneous linear second order differential equations in [[self-adjoint form]] with
			:<math>p_i(t)>0\,</math> for each i and for all ''t'' in [''a'', ''b''], and where the <math>R_i</math> are arbitrary real numbers.

			Assume that for all ''t'' in [''a'', ''b''] we have,
			:<math>\Delta^2(q_1) \ge 0 </math>,
			:<math>\Delta^2(p_1) \le 0 </math>,
			:<math>\Delta^2(p_1(a)R_1) \le 0 </math>.
			If <math>x_1(t) > 0</math> on [''a'', ''b''], and <math>x_2(b)=0</math>, then any solution <math>x_3(t)</math> has at least one zero in [''a'', ''b''].

			== References ==

			{{Reflist}}

			* {{cite book
			\|last=Clark D.N., G. Pecelli, and R. Sacksteder
			\|first=
			\|year=1981
			\|author-link=
			\|author2-link= G. Pecelli
			\|author3-link=R. Sacksteder
			\|language=
			\|title=Contributions to Analysis and Geometry
			\|journal=
			\|series=
			\|volume=
			\|pages=
			\|location= Baltimore, USA
			\|publisher= Johns Hopkins University Press
			\|isbn=
			\|url=
			\|ref=harv}}

			* {{cite journal
			\|last=Mingarelli
			\|first= Angelo B.
			\|year=1979
			\|author-link=
			\|author2-link=
			\|language=
			\|title= Some extensions of the Sturm–Picone theorem
			\|journal= Comptes Rendus Math. Rep. Acad. Sci. Canada
			\|series=
			\|volume=1 (4)
			\|pages=223–226
			\|location= Toronto, Ontario, Canada
			\|publisher= The Royal Society of Canada
			\|isbn=
			\|url=
			\|ref=harv}}

			[[Category:Ordinary differential equations]]
			[[Category:Mathematical identities]]

en>SchreiberBike: fix centuries to match WP:CENTURY

2012-08-29T16:13:07Z

fix centuries to match WP:CENTURY

New page

Unfortunately there was not other available room, the hour was last so we had to stay. One color that never goes out of style and always seems to make a splash is white and we're already seeing this all over the red carpet. Today's look of the day features a white hot look on fantastic actress Ginnifer Goodwin. http://www. If you cherished this report and you would like to obtain a lot more data pertaining to [http://www.restaurantcalcuta.com/outlet/ugg.asp Ugg Outlet Store] kindly take a look at our website. shlgrouptr.com/warm/?p=15 http://www.shlgrouptr.com/warm/?p=215 http://www.shlgrouptr.com/warm/?p=282 http://www.shlgrouptr.com/warm/?p=348 http://www.shlgrouptr.com/warm/?p=88

← Older revision		Revision as of 01:38, 5 March 2014
Line 1:		Line 1:
	~~In the field of [[ordinary differential equation]]s, the '''Mingarelli identity''' (coined by [[Philip Hartman]]<ref name="Hartman">{{harvnb\|Clark D~~.N., G. ~~Pecelli, and R~~. ~~Sacksteder\|1981}}~~<~~/ref~~>) is a ~~theorem that provides criteria for the [[oscillation theory\|oscillation]] and [[oscillation theory\|non-oscillation]]~~ of ~~solutions of some [[linear differential equation]]s in~~ the ~~real domain~~. ~~It extends the [[Picone identity]] from two~~ to ~~three or more differential equations of~~ the ~~second order~~. ~~Its most basic form appears here~~.		Today technologies has achieved new height. Everything is obtaining quicker and smoother. Even these days if you have cellular phone in your pocket, then anytime from anyplace you can avail cash in your credit score crunch. This is possible simply because of the mobile textual content loans that provide the cash approval within a few hours.<br><br><br><br>A secured personal mortgage is a secured mortgage offered to satisfy personal needs of the UK citizens. To avail this loan a borrower needs to place collateral towards the loan. Your vehicle, home or even a conserving bank account can work as collateral. Secured mortgage helps debtors in creating the best use of the equity stored in his or her property that assists him in borrowing a larger quantity of loan and that as well for a lengthier loan term.<br><br>Mobile textual content loans UK are extremely simple in acceptance if you are a UK citizen and 18 many years old. Nonetheless you must be working permanently and have a bank account. To get the approval you also must have the mobile telephone and an e-mail deal with.<br><br><br><br>There are some essential factors that you should remember prior to you use for the mortgage. Your vehicle ought to be at your name or the individual who is making use of for this facility. Your vehicle should not be more than eight many years previous. You ought to be having a great paying occupation and that should be fixed. You should have paid all the money owed associated to your car and it ought to be tax totally free.<br><br>The great component about availing these loans is that you will be able to buy a car of any make or design. These loans will offer you with 90-one hundred%25 of the complete funds needed. Furthermore you can also use the amount to purchase a utilized car, supplied it ought to be 5-six many years previous, as following this the cost of the marketplace worth of the car starts falls and needs a great deal of upkeep.<br><br>These text loans uk have become extremely well-liked as these loans are produced according to the needs of young people. You don't need to know much to get text loans. You just have to know about the loan providers. These short applications when verified by the mortgage supplier are granted loans within the shortest possible time body. But the primary thing about these loans is cash will provided in the applicant account in very secure and safe.<br><br>Same working day mortgage UK is available to these who need instant cash for their financially disaster. It is accessible for all type of budgets. You can get assured money on the exact same working day of your submission of application inside 24 hrs of time.<br><br>In UK there are numerous financial companies who give individual loans on reduced interest rate in but we have to lookup. So the next benefit of personal loan is the reduced curiosity rate. Because most of the time the borrower invested their hard attained cash in paying the interest rate. So this is the next benefit of personal loans UK i.e. low interest.<br><br>If you liked this article and you would like to get much more data with regards to [http://topnocreditcheckloans.com/reviews/100-day-uk-loans-review wmflabs.org] kindly take a look at the webpage.

	~~== The identity ==~~
	~~Consider~~ the ~~<math>n</math> solutions~~ of the ~~following (uncoupled) system~~ of ~~second order linear differential equations over the ''t''-interval [''~~a~~'', ''b'']~~.
	<~~math~~>~~(p_i(t) x_i^\prime)^\prime + q_i(t) x_i = 0, \,\,\,\,\,\,\,\,\,\, x_i(~~a~~)=1,\,\, x_i^\prime(~~a~~)=R_i\,</math>~~
	~~where <math>i=1,2, \ldots, n</math>~~. ~~Let <math>\Delta</math> denote~~ the ~~forward difference operator, i.~~e., <~~math~~>~~\Delta x_i = x_{i+1}-x_i.~~<~~/math~~> ~~The second order difference operator is found by iterating the first order operator as in~~ <~~math~~>~~\Delta^2 (x_i) = \Delta(\Delta x_i) = x_{i+2}-2x_{i+1}+x_{i}~~<~~/math~~>~~, with a similar definition~~ for the ~~higher iterates~~.

	~~Leaving out~~ the ~~independent variable ''t''~~ for ~~convenience,~~ and ~~assuming~~ the <~~math~~>~~x_i(t) \ne 0~~<~~/math~~> ~~on (''~~a~~'', ''b''], there holds~~ the ~~identity~~,~~<ref name="Mingarelli">{{harvnb\|Mingarelli\|1979\|p=223}}</ref>~~
	~~: <math>~~
	~~\begin{align}~~
	~~x_{n-1}^2\Delta^{n-1}(p_1r_1)]_a^b & = \int_a^b (x^\prime_{n-1})^2 \Delta^{n-1}(p_1)~~ - ~~\int_a^b x_{n-1}^2 \Delta^{n-1}(q_1)~~
	~~- \sum_{k=0}^{n-1} C(n-1~~,~~k)(-1)^{n-k-1}\int_a^b p_{k+1} W^2(x_{k+1},x_{n-1})/x_{k+1}^2,~~
	~~\end{align}~~
	~~</math>~~

	~~where <math>r_i = x^\prime_i/x_i</math> is~~ a ~~[[logarithmic derivative]],~~ <~~math~~>~~W(x_i, x_j) = x^\prime_ix_j - x_ix^\prime_j~~<~~/math~~>~~, is a [[Wronskian]] and the <math>C(n~~-~~1,k)</math>~~ are ~~[[binomial coefficients]]~~. ~~When <math>n=2</math> this reduces~~ to the ~~[[Picone identity]]~~.

	~~The above identity leads quickly to~~ the ~~following comparison theorem for three linear differential equations,<ref name="Mingarelli">{{harv\|Mingarelli\|1979\|Theorem 2}}</ref> extending~~ the ~~[[Sturm–Picone comparison theorem]]~~.

	~~Let <math>p_i,\, q_i,\,</math> ''i'' = 1, 2, 3 be real-valued continuous functions on~~ the ~~interval [''a'', ''b'']~~ and ~~let~~
	#<~~math~~>~~(p_1(t) x_1^\prime)^\prime + q_1(t) x_1 = 0, \,\,\,\,\,\,\,\,\,\, x_1(a)=1,\,\, x_1^\prime(a)=R_1\,~~<~~/math>~~
	~~#<math>(p_2(t) x_2^\prime)^\prime + q_2(t) x_2 = 0, \,\,\,\,\,\,\,\,\,\, x_2(a)=1,\,\, x_2^\prime(a)=R_2\,</math>~~
	~~#<math>(p_3(t) x_3^\prime)^\prime + q_3(t) x_3 = 0, \,\,\,\,\,\,\,\,\,\, x_3(a)=1,\,\, x_3^\prime(a)=R_3\,</math>~~
	~~be three homogeneous linear second order differential equations in [[self-adjoint form]] with~~
	~~:<math>p_i(t)>0\,</math~~> for ~~each i and~~ for all ~~''t'' in [''a'', ''b''], and where~~ the <~~math~~>~~R_i~~<~~/math~~> are ~~arbitrary real numbers.~~

	~~Assume that for all ''t''~~ in ~~[''a'', ''b'']~~ we have,
	:<~~math~~>~~\Delta^2(q_1) \ge 0~~ <~~/math~~>,
	:~~<math>\Delta^2(p_1) \le 0 <~~/~~math>,~~
	~~:<math>\Delta^2(p_1(a)R_1) \le 0 <~~/~~math>~~.
	~~If <math>x_1(t) > 0<~~/~~math> on [''a'', ''b''], and <math>x_2(b)=0<~~/~~math>, then any solution <math>x_3(t)</math> has at least one zero in [''a'', ''b''].~~

	~~== References ==~~

	~~{{Reflist}}~~

	* {{cite book
	~~\|last=Clark D.N., G. Pecelli, and R. Sacksteder~~
	~~\|first=~~
	~~\|year=1981~~
	~~\|author~~-~~link=~~
	~~\|author2~~-~~link= G. Pecelli~~
	~~\|author3~~-~~link=R~~. ~~Sacksteder~~
	~~\|language=~~
	~~\|title=Contributions to Analysis and Geometry~~
	~~\|journal=~~
	~~\|series=~~
	~~\|volume=~~
	~~\|pages=~~
	~~\|location= Baltimore, USA~~
	~~\|publisher= Johns Hopkins University Press~~
	~~\|isbn=~~
	~~\|url=~~
	~~\|ref=harv}}~~

	* {{cite journal
	~~\|last=Mingarelli~~
	~~\|first= Angelo B.~~
	~~\|year=1979~~
	~~\|author-link=~~
	~~\|author2-link=~~
	~~\|language=~~
	~~\|title= Some extensions of~~ the ~~Sturm–Picone theorem~~
	~~\|journal= Comptes Rendus Math~~. ~~Rep. Acad. Sci. Canada~~
	~~\|series=~~
	~~\|volume=1 (4)~~
	~~\|pages=223–226~~
	~~\|location= Toronto, Ontario, Canada~~
	~~\|publisher= The Royal Society of Canada~~
	~~\|isbn=~~
	~~\|url=~~
	~~\|ref=harv}}~~

	~~[[Category:Ordinary differential equations]]~~
	~~[[Category:Mathematical identities]]~~