Examples of vector spaces - Revision history

en>Niceguyedc: WPCleaner v1.33 - Repaired 1 link to disambiguation page - (You can help) - Subfield

2014-08-17T19:23:53Z

WPCleaner v1.33 - Repaired 1 link to disambiguation page - (You can help) - Subfield

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	In [~~[mathematics~~]~~], '''Birkhoff interpolation''' is an extension of [[polynomial interpolation]]. It refers to the problem finding~~ a ~~polynomial ''p''~~ of ~~degree ''d'' such that certain [~~[~~derivative]]s have specified values at specified points~~:
	~~:<math> p^{(n_i)}(x_i)~~ = ~~y_i \qquad\mbox{for } i~~=~~1,\ldots,d~~, ~~</math>~~
	~~where~~ the ~~data points~~ <~~math~~>~~(x_i,y_i)</math> and~~ the ~~nonnegative integers <math>n_i</math> are given~~. ~~It differs from [[Hermite interpolation]] in that it is possible~~ to ~~specify derivatives of ''p'' at some points without specifying the lower derivatives~~ or ~~the polynomial itself~~. ~~The name refers to~~ [~~[George David Birkhoff]~~]~~, who first studied the problem in {{harvtxt\|Birkhoff\|1906}}~~.

	~~In contrast to Lagrange interpolation and Hermite interpolation,~~ a ~~Birkhoff interpolation problem does~~ not ~~always~~ have ~~a unique solution. For instance, there is no quadratic polynomial ''p'' such that ''p''(−1) = ''p''(1) = 0 and ''p''′(0) = 1. On the other hand,~~ the ~~Birkhoff interpolation problem where the values~~ of ~~''p''′(−1), ''p''(0)~~ and ~~''p''′(1)~~ are ~~given always has a unique solution {{harv\|Passow\|1983}}~~.

	~~An important problem in the theory of Birkhoff interpolation is to classify those problems that have a unique solution~~. ~~{{harvtxt\|Schoenberg\|1966}} formulates the problem as follows. Let ''d'' denote the number of conditions (as above)~~ and ~~let ''k'' be the number of interpolation points. Given a ''d''-by-''k'' matrix ''E'', all of whose entries are either 0~~ or ~~1, such that exactly ''d'' entries are 1, then the corresponding problem~~ is to ~~determine ''p'' such~~ that
	:~~<math> p^{(j)}(x_i) = y_{i,j} \qquad\text{for all } (i,j) \text{ with } e_{ij} = 1. <~~/~~math>~~
	~~The matrix ''E'' is called the incidence matrix~~. ~~For example, the incidence matrices for the interpolation problems mentioned in the previous paragraph are:~~
	~~:<math> \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{bmatrix} \quad\text{and}\quad \begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{bmatrix}~~. ~~</math>~~
	~~Now the question is: does~~ a ~~Birkhoff interpolation problem~~ with ~~a given incidence matrix~~ have ~~a unique solution for any choice of the interpolation points?~~

	~~The case with ''k'' = 2 interpolation points was tackled by {{harvtxt\|Pólya\|1931}}~~. ~~Let ''S~~<~~sub~~>m<~~/sub~~>~~'' denote the sum of the entries in the first ''m'' columns of the incidence matrix:~~
	:<~~math~~> ~~S_m = \sum_{i=1}^k \sum_{j=1}^m e_{ij}.~~ <~~/math~~>
	Then the Birkhoff interpolation problem with ''k'' = 2 has a unique solution if and only if ''S<sub>m</sub>'' ≥ ''m'' for all ''m''. {{harvtxt\|Schoenberg\|1966}} showed that this is a necessary condition for all values of ''k''.

	~~==References==~~
	* {{Citation \| last1=Birkhoff \| first1=George David \| author1-~~link=George David Birkhoff \| title=General mean value and remainder theorems with applications to mechanical differentiation and quadrature \| jstor=1986339 \| year=1906 \| journal=[~~[~~Transactions of the American Mathematical Society]] \| issn=0002-9947 \| volume=7 \| pages=107–136 \| issue=1 \| publisher=American Mathematical Society}}.~~
	* {{Citation \| last1=Passow \| first1=Eli \| title=Book Review: ~~Birkhoff interpolation by G. G. Lorentz, K. Jetter and S. D. Riemenschneider \| doi=10.1090~~/~~S0273-0979-1983-15204-7 \| year=1983 \| journal=American Mathematical Society. Bulletin. New Series \| issn=0002-9904 \| volume=9 \| issue=3 \| pages=348–351}}.~~
	* {{Citation \| doi=10.1002/zamm.19310110620 \| last1=Pólya \| first1=George \| author1-link=George Pólya \| title=Bemerkung zur Interpolation und zur Naherungstheorie der Balkenbiegung \| year=1931 \| journal=[[Journal of Applied Mathematics and Mechanics]] \| issn=0044-2267 \| volume=11 \| pages=445–449}}.
	* {{Citation \| last1=Schoenberg \| first1=Isaac Jacob \| author1-link=Isaac Jacob Schoenberg \| title=On Hermite-Birkhoff interpolation \| doi=10.~~1016/0022-247X(66)90160-0 \| year=1966 \| journal=Journal of Mathematical Analysis and Applications \| issn=0022-247X \| volume=16 \| pages=538–543}}.~~

	~~[[Category:Interpolation]~~]

en>AsphyxiateDrake: /* Matrices */

2013-12-19T23:34:00Z

Matrices

← Older revision		Revision as of 01:34, 20 December 2013
Line 1:		Line 1:
	~~When you have ever had a panic assault~~, ~~you don~~'~~t sit up for one other one~~. ~~The "combat-or flight" response in your body is involuntarily turned on and your nervous system alerts your adrenal glands~~ to ~~begin pumping out stress hormones to get you able to both run away or flip and combat.~~<br><br>~~To begin with~~ the ~~process of naturally beating nervousness attacks~~, ~~take into consideration one particular situation~~ that is ~~leading~~ to ~~your blood strain degree~~ to ~~skyrocket sending you into a real panic attack~~. ~~This can be done as prevention~~ to a ~~full blown assault~~, ~~since you~~'~~ll be able to cease it earlier than it actually takes place~~. ~~Think about all of~~ the ~~stuff you can lose in case you permit~~ the ~~worries take over~~, ~~if you ever really feel like it~~'~~s getting out of control~~. ~~One~~ of ~~the vital necessary keys~~ to ~~self-control is realizing~~ that ~~you are~~ the ~~one who has control over your physique~~.~~<br><br>If you~~'~~re attempting to resolve whether or not or not to treat your anxiety with treatment, it~~'~~s necessary to weigh~~ the ~~professionals~~ and ~~cons together with your doctor. It~~'~~s also important to learn concerning~~ the ~~frequent unwanted effects~~ of ~~the anxiety medication you are contemplating~~. ~~Unwanted effects~~ of ~~tension remedy vary from gentle nuisances similar to dry mouth to more severe issues similar to acute nausea~~ or ~~pronounced weight achieve. For any anxiety remedy~~, ~~you will have to stability~~ the ~~unwanted effects towards the benefits. Questions~~ to ~~ask yourself and a psychological well being professional~~<br><br>~~During your scorching flash, it~~'~~s possible you~~'~~ll breathe too quickly and lightly and feel as if~~ the ~~wind is being pressed out of your chest~~. ~~Of course anxiousness is~~ the ~~root reason~~ for the ~~recent flash, but being~~ in ~~a room with a high temperature can worsen~~ the ~~state of affairs. Spicy foods set off ache receptors~~ and ~~set off chemical chain-reactions which are very very similar to sizzling flashes~~. ~~The fundamental act of eating a spicy meal can be comparable enough to~~ the ~~experience of~~ a ~~sizzling flush that your physique is stimulated into producing an precise anxiety scorching flush. The recognizable hot flash feeling may additionally make you anxious, which then results in~~ a ~~genuine hot flash from nervousness. Those are just~~ a ~~few examples~~ of ~~medicines and~~ the ~~harmful effects they have on your body~~. ~~Concern of meeting new people~~<br><br>~~These hormonal elevations can result~~ in ~~an increase in blood stress , a quicker heart charge, and a constriction~~ of the ~~blood vessels~~. ~~When~~ the ~~annoying condition is over, the system gradually returns to baseline,~~ and ~~blood stress returns to normal~~. ~~As said above, frequent short-term spikes in blood pressure can have adverse consequences~~ that ~~are just like these~~ of ~~persistent high blood pressure Also, individuals who undergo~~ with ~~despair or a Normal Anxiousness Disorder (GAD) may turn~~ to ~~sure crutches to assist them deal with a disturbing state~~ of ~~dash diet affairs~~. ~~Including how to lose weight fast consuming excessively, or selecting the unsuitable foods, smoking, or consuming too much alcohol, which might~~ of ~~course enhance the risk~~ of ~~higher blood strain~~ and ~~heart problems.<br><br>Should you beloved this post as well as you would want to receive details about simple diet burn fat generously go to the webpage~~.		{{Multiple issues\|lead missing=December 2010\|confusing=December 2010}}
			In [[mathematics]], '''Birkhoff interpolation''' is an extension of [[polynomial interpolation]]. It refers to the problem finding a polynomial ''p'' of degree ''d'' such that certain [[derivative]]s have specified values at specified points:
			:<math> p^{(n_i)}(x_i) = y_i \qquad\mbox{for } i=1,\ldots,d, </math>
			where the data points <math>(x_i,y_i)</math> and the nonnegative integers <math>n_i</math> are given. It differs from [[Hermite interpolation]] in that it is possible to specify derivatives of ''p'' at some points without specifying the lower derivatives or the polynomial itself. The name refers to [[George David Birkhoff]], who first studied the problem in {{harvtxt\|Birkhoff\|1906}}.

			In contrast to Lagrange interpolation and Hermite interpolation, a Birkhoff interpolation problem does not always have a unique solution. For instance, there is no quadratic polynomial ''p'' such that ''p''(−1) = ''p''(1) = 0 and ''p''′(0) = 1. On the other hand, the Birkhoff interpolation problem where the values of ''p''′(−1), ''p''(0) and ''p''′(1) are given always has a unique solution {{harv\|Passow\|1983}}.

			An important problem in the theory of Birkhoff interpolation is to classify those problems that have a unique solution. {{harvtxt\|Schoenberg\|1966}} formulates the problem as follows. Let ''d'' denote the number of conditions (as above) and let ''k'' be the number of interpolation points. Given a ''d''-by-''k'' matrix ''E'', all of whose entries are either 0 or 1, such that exactly ''d'' entries are 1, then the corresponding problem is to determine ''p'' such that
			:<math> p^{(j)}(x_i) = y_{i,j} \qquad\text{for all } (i,j) \text{ with } e_{ij} = 1. </math>
			The matrix ''E'' is called the incidence matrix. For example, the incidence matrices for the interpolation problems mentioned in the previous paragraph are:
			:<math> \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{bmatrix} \quad\text{and}\quad \begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{bmatrix}. </math>
			Now the question is: does a Birkhoff interpolation problem with a given incidence matrix have a unique solution for any choice of the interpolation points?

			The case with ''k'' = 2 interpolation points was tackled by {{harvtxt\|Pólya\|1931}}. Let ''S<sub>m</sub>'' denote the sum of the entries in the first ''m'' columns of the incidence matrix:
			:<math> S_m = \sum_{i=1}^k \sum_{j=1}^m e_{ij}. </math>
			Then the Birkhoff interpolation problem with ''k'' = 2 has a unique solution if and only if ''S<sub>m</sub>'' ≥ ''m'' for all ''m''. {{harvtxt\|Schoenberg\|1966}} showed that this is a necessary condition for all values of ''k''.

			==References==
			* {{Citation \| last1=Birkhoff \| first1=George David \| author1-link=George David Birkhoff \| title=General mean value and remainder theorems with applications to mechanical differentiation and quadrature \| jstor=1986339 \| year=1906 \| journal=[[Transactions of the American Mathematical Society]] \| issn=0002-9947 \| volume=7 \| pages=107–136 \| issue=1 \| publisher=American Mathematical Society}}.
			* {{Citation \| last1=Passow \| first1=Eli \| title=Book Review: Birkhoff interpolation by G. G. Lorentz, K. Jetter and S. D. Riemenschneider \| doi=10.1090/S0273-0979-1983-15204-7 \| year=1983 \| journal=American Mathematical Society. Bulletin. New Series \| issn=0002-9904 \| volume=9 \| issue=3 \| pages=348–351}}.
			* {{Citation \| doi=10.1002/zamm.19310110620 \| last1=Pólya \| first1=George \| author1-link=George Pólya \| title=Bemerkung zur Interpolation und zur Naherungstheorie der Balkenbiegung \| year=1931 \| journal=[[Journal of Applied Mathematics and Mechanics]] \| issn=0044-2267 \| volume=11 \| pages=445–449}}.
			* {{Citation \| last1=Schoenberg \| first1=Isaac Jacob \| author1-link=Isaac Jacob Schoenberg \| title=On Hermite-Birkhoff interpolation \| doi=10.1016/0022-247X(66)90160-0 \| year=1966 \| journal=Journal of Mathematical Analysis and Applications \| issn=0022-247X \| volume=16 \| pages=538–543}}.

			[[Category:Interpolation]]

en>Incnis Mrsi: /* Trivial or zero vector space */ {{main|Zero vector space}}

2012-02-27T16:59:17Z

Trivial or zero vector space: {{main|Zero vector space}}

New page

When you have ever had a panic assault, you don't sit up for one other one. The "combat-or flight" response in your body is involuntarily turned on and your nervous system alerts your adrenal glands to begin pumping out stress hormones to get you able to both run away or flip and combat. To begin with the process of naturally beating nervousness attacks, take into consideration one particular situation that is leading to your blood strain degree to skyrocket sending you into a real panic attack. This can be done as prevention to a full blown assault, since you'll be able to cease it earlier than it actually takes place. Think about all of the stuff you can lose in case you permit the worries take over, if you ever really feel like it's getting out of control. One of the vital necessary keys to self-control is realizing that you are the one who has control over your physique. If you're attempting to resolve whether or not or not to treat your anxiety with treatment, it's necessary to weigh the professionals and cons together with your doctor. It's also important to learn concerning the frequent unwanted effects of the anxiety medication you are contemplating. Unwanted effects of tension remedy vary from gentle nuisances similar to dry mouth to more severe issues similar to acute nausea or pronounced weight achieve. For any anxiety remedy, you will have to stability the unwanted effects towards the benefits. Questions to ask yourself and a psychological well being professional During your scorching flash, it's possible you'll breathe too quickly and lightly and feel as if the wind is being pressed out of your chest. Of course anxiousness is the root reason for the recent flash, but being in a room with a high temperature can worsen the state of affairs. Spicy foods set off ache receptors and set off chemical chain-reactions which are very very similar to sizzling flashes. The fundamental act of eating a spicy meal can be comparable enough to the experience of a sizzling flush that your physique is stimulated into producing an precise anxiety scorching flush. The recognizable hot flash feeling may additionally make you anxious, which then results in a genuine hot flash from nervousness. Those are just a few examples of medicines and the harmful effects they have on your body. Concern of meeting new people These hormonal elevations can result in an increase in blood stress , a quicker heart charge, and a constriction of the blood vessels. When the annoying condition is over, the system gradually returns to baseline, and blood stress returns to normal. As said above, frequent short-term spikes in blood pressure can have adverse consequences that are just like these of persistent high blood pressure Also, individuals who undergo with despair or a Normal Anxiousness Disorder (GAD) may turn to sure crutches to assist them deal with a disturbing state of dash diet affairs. Including how to lose weight fast consuming excessively, or selecting the unsuitable foods, smoking, or consuming too much alcohol, which might of course enhance the risk of higher blood strain and heart problems. Should you beloved this post as well as you would want to receive details about simple diet burn fat generously go to the webpage.