Discussion about math, puzzles, games and fun. Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ • π ƒ ¹ ² ³ °
 

You are not logged in. #2 20080228 05:12:59
#4 20080228 10:09:09#5 20110729 04:27:22
Re: a and r"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  Buddha? "Data! Data! Data!" he cried impatiently. "I can't make bricks without clay." #6 20110729 05:57:36
Re: a and rHi tony123, wonderful Jane, and gAr the great who finally got me to learn how to differentiate under the integral, wunderbar! We could deflate out the two roots they found and solve a 4th degree equation. If this were 1670 that would be cool. Powerful computational techniques make life easier. We use a Cauchy bound to bound the roots. They are found to be. To find out how many roots are in the interval { 3 , 3 } we form a Sturm chain. Go down the first column and count the number of sign changes and call it A. Go down the second column and count the number of sign changes and call it B. The theorem says the number of real roots in { 3, 3 } is A  B = 2. Multple roots count as 1 root. So Jane and gAr have found the only 2 real roots of r, the rest are complex. In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #7 20110729 13:06:36
Re: a and rHi bobbym, "Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  Buddha? "Data! Data! Data!" he cried impatiently. "I can't make bricks without clay." #8 20110729 16:16:01
Re: a and rHi gAr; Using the Cauchy bound we get: We see that all the real roots are located in the closed interval [12398,12398]. Not as tight a bounds as the other but adequate. We form the Sturm chain. A = 4 and B = 2, so A  B = 2. There are two real roots in that interval. So Jane's and gAr's are the only real roots of both r and a. We are done! In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. 