Comments on: Interpolation and the Chinese Remainder Theorem http://www.arsmathematica.net/archives/2007/01/16/interpolation-and-the-chinese-remainder-theorem/ Dedicated to the mathematical arts. Mon, 13 Oct 2008 00:37:13 +0000 http://wordpress.org/?v=2.5.1 By: Prahlad http://www.arsmathematica.net/archives/2007/01/16/interpolation-and-the-chinese-remainder-theorem/#comment-2092 Prahlad Mon, 22 Jan 2007 11:55:25 +0000 http://www.arsmathematica.net/archives/2007/01/16/interpolation-and-the-chinese-remainder-theorem/#comment-2092 I have been reading a bit recently about orthogonal functions, and the chinese reminder theorem reminds me a lot of the the theorem of existence of D-sets for linear spaces. In fact the proof and even the solution have a form that is very similar to those I have been reading about. (see http://books.google.com/books?id=ecS5DKstwREC&dq=fourier+series+and+orthogonal+functions&psp=1) I have been reading a bit recently about orthogonal functions, and the chinese reminder theorem reminds me a lot of the the theorem of existence of D-sets for linear spaces. In fact the proof and even the solution have a form that is very similar to those I have been reading about.

(see http://books.google.com/books?id=ecS5DKstwREC&dq=fourier+series+and+orthogonal+functions&psp=1)

]]> By: PeterMcB http://www.arsmathematica.net/archives/2007/01/16/interpolation-and-the-chinese-remainder-theorem/#comment-1721 PeterMcB Thu, 18 Jan 2007 00:03:07 +0000 http://www.arsmathematica.net/archives/2007/01/16/interpolation-and-the-chinese-remainder-theorem/#comment-1721 <i>"Mathematics is like drugs, but cheaper." </i> Not sure about this -- see <a href="http://www.amazon.co.uk/Sketches-Elephant-Theory-Compendium-Oxford/dp/0198534256/sr=1-3/qid=1169078511/ref=sr_1_3/202-9859913-3926251?ie=UTF8&s=books" rel="nofollow">here</a>. “Mathematics is like drugs, but cheaper.”

Not sure about this — see here.

]]> By: Walt http://www.arsmathematica.net/archives/2007/01/16/interpolation-and-the-chinese-remainder-theorem/#comment-1717 Walt Wed, 17 Jan 2007 22:47:16 +0000 http://www.arsmathematica.net/archives/2007/01/16/interpolation-and-the-chinese-remainder-theorem/#comment-1717 The p-adics can only be reduced mod p^n, not by any other prime. The hypernaturals are close to what I had in mind, but they of course stabilize eventually, just after an infinite number of steps. :-) The p-adics can only be reduced mod p^n, not by any other prime. The hypernaturals are close to what I had in mind, but they of course stabilize eventually, just after an infinite number of steps. :-)

]]> By: Wendell http://www.arsmathematica.net/archives/2007/01/16/interpolation-and-the-chinese-remainder-theorem/#comment-1716 Wendell Wed, 17 Jan 2007 22:20:33 +0000 http://www.arsmathematica.net/archives/2007/01/16/interpolation-and-the-chinese-remainder-theorem/#comment-1716 Hmmm... Frankly, I don't think Wendell's teaching himself very much. But the X-Y reference... <i>that's</i><i>, helpful. I can see how that fits. It would also be helpful if Wendell learned some math that deals with sets. (This is a guy who still finds division exciting and mysterious.) :)</i> Hmmm… Frankly, I don’t think Wendell’s teaching himself very much. But the X-Y reference… that’s, helpful. I can see how that fits. It would also be helpful if Wendell learned some math that deals with sets. (This is a guy who still finds division exciting and mysterious.)

:)

]]> By: a little night musing http://www.arsmathematica.net/archives/2007/01/16/interpolation-and-the-chinese-remainder-theorem/#comment-1714 a little night musing Wed, 17 Jan 2007 20:45:31 +0000 http://www.arsmathematica.net/archives/2007/01/16/interpolation-and-the-chinese-remainder-theorem/#comment-1714 <a href="http://mathforum.org/library/drmath/view/65286.html" rel="nofollow">Yup.</a> Yup.

]]> By: Kenny Easwaran http://www.arsmathematica.net/archives/2007/01/16/interpolation-and-the-chinese-remainder-theorem/#comment-1662 Kenny Easwaran Wed, 17 Jan 2007 06:55:59 +0000 http://www.arsmathematica.net/archives/2007/01/16/interpolation-and-the-chinese-remainder-theorem/#comment-1662 Isn't the fact that the value eventually stabilizes what makes the integer an integer instead of some funny hypernatural, or p-adic thing? And of course, there must be some funny way you can glue these functions together to get a scheme that's locally a lot like Spec(Z) but got some funny global behavior. At least, that's what I remember from my first year algebraic geometry class. Isn’t the fact that the value eventually stabilizes what makes the integer an integer instead of some funny hypernatural, or p-adic thing? And of course, there must be some funny way you can glue these functions together to get a scheme that’s locally a lot like Spec(Z) but got some funny global behavior. At least, that’s what I remember from my first year algebraic geometry class.

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