101.98.214.38: /* Odd number of points */

2014-11-14T01:39:57Z

Odd number of points

en>PBenard at 12:59, 9 October 2013

2013-10-09T12:59:33Z

← Older revision		Revision as of 14:59, 9 October 2013
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	It is ~~time to address~~ the ~~slow computer issues even if you do not learn how~~. ~~Just because a computer~~ is ~~functioning thus slow or keeps freezing up; does not signify~~ to ~~not address~~ the ~~issue plus fix it~~. ~~You~~ may or ~~can not be aware which any computer owner should learn that there are certain details~~ which a ~~computer requires~~ to ~~keep~~ the ~~greatest performance. The sad fact~~ is ~~that~~ a ~~lot~~ of ~~folks that own~~ a ~~program have no idea which it requires routine repair really like their vehicles.<br><br>However registry is easily corrupted plus damaged whenever you~~'~~re utilizing~~ a ~~computer. Overtime~~, ~~without right repair, it~~ is ~~loaded with mistakes plus incorrect or even lost information that can make the system unable to work correctly or apply~~ a ~~certain task. And when your program cannot find~~ the ~~correct information, it will likely not recognize what to do~~. ~~Then it freezes up! That~~ is ~~the real cause~~ of the ~~trouble~~.<br><br>~~The error~~ is ~~basically a outcome of problem with Windows Installer package~~. ~~The Windows Installer is~~ a ~~tool utilized to install~~, ~~uninstall plus repair~~ the ~~many programs on a computer. Let you discuss a limited items which helped a lot~~ of ~~folks who facing~~ the ~~synonymous problem.<br><br>The issue with almost all~~ of the ~~persons is the fact that they never wish to spend money~~. ~~In the cracked variation one refuses to have to pay anything plus will download it from internet easily. It is easy to install too. But~~, the ~~problem comes whenever it really~~ is ~~not able to identify all possible viruses~~, ~~spyware~~ and ~~malware inside~~ the ~~program. This~~ is ~~because it is very obsolete inside nature plus refuses to get any standard changes within the site downloaded~~. ~~Thus,~~ a ~~system~~ is ~~accessible to difficulties like hacking.~~<br><br>~~Whenever it comes to software~~, ~~this really is the vital piece because~~ it ~~is the one running your system in addition to additional programs needed inside~~ a ~~works~~. ~~Always keep~~ the ~~cleanliness of your system from obsolete data by getting a advantageous~~ [~~http~~://~~bestregistrycleanerfix.com/tune~~-up-~~utilities tuneup utilities]. Protect it from~~ a ~~virus on the net by providing~~ a ~~workable virus protection~~ system~~. You should furthermore have~~ a ~~monthly clean up by running your defragmenter system. This technique it will enhance the performance~~ of the ~~computer plus for we to avoid any mistakes~~. ~~If you think anything is incorrect with~~ the ~~software~~, and ~~we don'~~t ~~recognize how to fix it then refer to a technician~~.<br><br>~~The most probable cause of your trouble is the program problem - Registry Errors! That is the reason why folks who already have more than~~ 2 ~~G RAM on their computers are nevertheless continually bothered by the problem.~~<br><br>~~Across the best of the scan results display page you see the tabs... Registry~~, ~~Junk Files~~, ~~Privacy~~, ~~Bad Active X~~, Performance, etc. Each of these tabs may show you the results of that area. The Junk Files are mainly temporary files like web information, pictures, web pages... And they are merely taking up storage space.<br><br>~~So in summary~~, ~~when comparing registry cleaning~~, ~~make certain that that the one you choose offers you~~ the ~~following~~.~~A backup plus restore center, quickly procedure, automatic deletion center, start-up administration, an easy technique~~ of ~~contact along with a cash back guarantee.~~		{{Other uses\|Caustic (disambiguation)}}
			[[Image:Circle_caustic.png\|right\|frame\|Reflective Caustic generated from a [[circle]] and parallel rays]]

			In [[differential geometry]] and [[geometric optics]], a '''caustic''' is the [[envelope (mathematics)\|envelope]] of [[Line (mathematics)#Ray\|ray]]s either [[Reflection (mathematics)\|reflected]] or [[refracted]] by a [[manifold]]. It is related to the concept of [[Caustic (optics)\|caustics in optics]]. The ray's source may be a point (called the radiant) or parallel rays from a point at infinity, in which case a direction vector of the rays must be specified.

			More generally, especially as applied to [[symplectic geometry]] and [[singularity theory]], a caustic is the [[critical value\|critical value set]] of a [[Lagrangian mapping]] {{nowrap\|1=(''π'' ○ ''i'') : ''L'' ↪ ''M'' ↠ ''B'';}} where {{nowrap\|1=''i'' : ''L'' ↪ ''M''}} is a [[Symplectic_manifold#Lagrangian_mapping\|Lagrangian immersion]] of a [[Symplectic_manifold#Lagrangian_and_other_submanifolds\|Lagrangian submanifold]] ''L'' into a [[symplectic manifold]] ''M'', and {{nowrap\|1=''π'' : ''M'' ↠ ''B''}} is a [[Symplectic_manifold#Lagrangian_fibration\|Lagrangian fibration]] of the symplectic manifold ''M''. The caustic is a [[subset]] of the Lagrangian [[fibration]]'s [[base space]] ''B''.<ref name="Arnold">{{Cite book\|first=V. I.\|last=Arnold\|first2=A. N.\|last2=Varchenko\|first3=S. M.\|last3=Gusein-Zade\|title=The Classification of Critical Points, Caustics and Wave Fronts: Singularities of Differentiable Maps, Vol 1\|publisher=Birkhäuser\|year=1985\|isbn=0-8176-3187-9}}</ref>

			==Catacaustic==
			A '''catacaustic''' is the reflective case.

			With a radiant, it is the [[evolute]] of the [[orthotomic]] of the radiant.

			The planar, parallel-source-rays case: suppose the direction vector is <math>(a,b)</math> and the mirror curve is parametrised as <math>(u(t),v(t))</math>. The normal vector at a point is <math>(-v'(t),u'(t))</math>; the reflection of the direction vector is (normal needs special normalization)
			:<math>2\mbox{proj}_nd-d=\frac{2n}{\sqrt{n\cdot n}}\frac{n\cdot d}{\sqrt{n\cdot n}}-d=2n\frac{n\cdot d}{n\cdot n}-d=\frac{
			(av'^2-2bu'v'-au'^2,bu'^2-2au'v'-bv'^2)
			}{v'^2+u'^2}</math>
			Having components of found reflected vector treat it as a tangent
			:<math>(x-u)(bu'^2-2au'v'-bv'^2)=(y-v)(av'^2-2bu'v'-au'^2).</math>
			Using the simplest [[envelope (mathematics)\|envelope]] form
			:<math>F(x,y,t)=(x-u)(bu'^2-2au'v'-bv'^2)-(y-v)(av'^2-2bu'v'-au'^2)</math> <math>=x(bu'^2-2au'v'-bv'^2)
			-y(av'^2-2bu'v'-au'^2)
			+b(uv'^2-uu'^2-2vu'v')
			+a(-vu'^2+vv'^2+2uu'v')</math>
			:<math>F_t(x,y,t)=2x(bu'u''-a(u'v''+u''v')-bv'v'')
			-2y(av'v''-b(u''v'+u'v'')-au'u'')
			+b( u'v'^2 +2uv'v'' -u'^3 -2uu'u'' -2u'v'^2 -2u''vv' -2u'vv'')
			+a(-v'u'^2 -2vu'u'' +v'^3 +2vv'v'' +2v'u'^2 +2v''uu' +2v'uu'')</math>
			which may be unaesthetic, but <math>F=F_t=0</math> gives a [[linear system]] in <math>(x,y)</math> and so it is elementary to obtain a parametrisation of the catacaustic. [[Cramer's rule]] would serve.

			===Example===
			Let the direction vector be (0,1) and the mirror be <math>(t,t^2).</math>
			Then
			:<math>u'=1</math>   <math>u''=0</math>   <math>v'=2t</math>   <math>v''=2</math>   <math>a=0</math>   <math>b=1</math>
			:<math>F(x,y,t)=(x-t)(1-4t^2)+4t(y-t^2)=x(1-4t^2)+4ty-t</math>
			:<math>F_t(x,y,t)=-8tx+4y-1</math>
			and <math>F=F_t=0</math> has solution <math>(0,1/4)</math>; ''i.e.'', light entering a [[parabola\|parabolic]] mirror parallel to its axis is reflected through the focus.

			== References ==

			{{reflist}}

			==See also==
			* [[Cut locus (Riemannian manifold)]]

			==External links==
			*{{MathWorld\|title=Caustic\|urlname=Caustic}}

			{{Differential transforms of plane curves}}

			[[Category:Differential geometry]]

92.76.238.116: added link to history of fft

2012-02-26T14:01:24Z

added link to history of fft

New page

It is time to address the slow computer issues even if you do not learn how. Just because a computer is functioning thus slow or keeps freezing up; does not signify to not address the issue plus fix it. You may or can not be aware which any computer owner should learn that there are certain details which a computer requires to keep the greatest performance. The sad fact is that a lot of folks that own a program have no idea which it requires routine repair really like their vehicles. However registry is easily corrupted plus damaged whenever you're utilizing a computer. Overtime, without right repair, it is loaded with mistakes plus incorrect or even lost information that can make the system unable to work correctly or apply a certain task. And when your program cannot find the correct information, it will likely not recognize what to do. Then it freezes up! That is the real cause of the trouble. The error is basically a outcome of problem with Windows Installer package. The Windows Installer is a tool utilized to install, uninstall plus repair the many programs on a computer. Let you discuss a limited items which helped a lot of folks who facing the synonymous problem. The issue with almost all of the persons is the fact that they never wish to spend money. In the cracked variation one refuses to have to pay anything plus will download it from internet easily. It is easy to install too. But, the problem comes whenever it really is not able to identify all possible viruses, spyware and malware inside the program. This is because it is very obsolete inside nature plus refuses to get any standard changes within the site downloaded. Thus, a system is accessible to difficulties like hacking. Whenever it comes to software, this really is the vital piece because it is the one running your system in addition to additional programs needed inside a works. Always keep the cleanliness of your system from obsolete data by getting a advantageous [http://bestregistrycleanerfix.com/tune-up-utilities tuneup utilities]. Protect it from a virus on the net by providing a workable virus protection system. You should furthermore have a monthly clean up by running your defragmenter system. This technique it will enhance the performance of the computer plus for we to avoid any mistakes. If you think anything is incorrect with the software, and we don't recognize how to fix it then refer to a technician. The most probable cause of your trouble is the program problem - Registry Errors! That is the reason why folks who already have more than 2 G RAM on their computers are nevertheless continually bothered by the problem. Across the best of the scan results display page you see the tabs... Registry, Junk Files, Privacy, Bad Active X, Performance, etc. Each of these tabs may show you the results of that area. The Junk Files are mainly temporary files like web information, pictures, web pages... And they are merely taking up storage space. So in summary, when comparing registry cleaning, make certain that that the one you choose offers you the following.A backup plus restore center, quickly procedure, automatic deletion center, start-up administration, an easy technique of contact along with a cash back guarantee.

Trigonometric interpolation - Revision history

101.98.214.38: /* Odd number of points */

en>PBenard at 12:59, 9 October 2013

92.76.238.116: added link to history of fft