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        riddles >> hard >> Some super hard math problems
        
(Message started by: Weazel on Nov 18th, 2004, 7:01pm)
Title: Some super hard math problems
Post by Weazel on Nov 18th, 2004, 7:01pm
My finite teacher asked us to see if we could solve some math problems. I really have no idea how to do it as I haven't learned it yet. So I decided to pose the questions to you guys.

Find the area under the curve to ten decimal places:
                1
 lim [smiley=smallint.gif][smiley=in.gif] x -1cos(x-1log10x)dx
[smiley=in.gif] [smiley=to.gif] 0



Find the global minimum to ten decimal places:

f(x,y) = esin(50x) + sin[60ey] + sin[70sin(x)] + sin[sin(80y)] - sin{10(x + y)] + 1/4(x2 + y2)
Title: Re: Some super hard math problems
Post by THUDandBLUNDER on Nov 19th, 2004, 2:05am
Weazel, methinks thou art aptly-named.
If this is what your 'finite teacher' gives you, what does your 'infinite teacher' give you?

Answers to your 'homework' can be found here (http://mathworld.wolfram.com/Hundred-DollarHundred-DigitChallengeProblems.html).

While your 2nd question is identical to #4 on the given page, you don't seem to have copied #1 correctly.  ::)  

Title: Re: Some super hard math problems
Post by Weazel on Nov 20th, 2004, 5:25pm
I can assure you I did not copy it from that page - It is the first I have seen of it. My teacher claims he was emailed by a professor (I think from Cambridge, I don't remember for sure, but it is from whatever college those questions came from).

If you did not know, finite mathematics does not deal with math in the finite realm; it is actually dealing with finances and economics: Minimization/maximization, interest, income, and the like for instance.

The reason the two questions were brought up were because my teacher showed us a program that can be used to solve minimization/maximization problems with 50 constraints, and wanted to see if we can devise a program to solve the given problems.
Title: Re: Some super hard math problems
Post by Icarus on Nov 20th, 2004, 9:24pm
Actually it strikes me as strange that the first limit should even exist. As x [to] 0, the x-1 climbs to infinity at a non-integrable rate, whereas the cos term starts ocillating rapidly back and forth between 1 and -1. Without a closer examination, I can't guarantee it, but this has all the earmarks of a non-convergent limit.
Title: Re: Some super hard math problems
Post by Obob on Nov 20th, 2004, 9:34pm
The limit does in fact exist.  The fact that the cosine is oscillating so rapidly means that there is a lot of cancellation going on.

I'm not sure whether or not the limit of the integral of the absolute value of the function would exist or not.  It probably wouldn't be too hard to figure out using Mathematica, though.

If you have Mathematica, there is an interesting file you can download where the authors use Mathematica to solve this problem and 9 other problems which are very difficult numerical type problems.  Here's the link, its worth a look: http://library.wolfram.com/infocenter/Conferences/5353/
Title: Re: Some super hard math problems
Post by Barukh on Nov 21st, 2004, 6:42am

on 11/20/04 at 21:34:36, Obob wrote:
The limit does in fact exist.  The fact that the cosine is oscillating so rapidly means that there is a lot of cancellation going on.

This does not prove that the limit exists. The following page (http://www-m8.ma.tum.de/m3/bornemann/challengebook/The%20Story/) presents more than a dozen teams that won the first prize, and only one of them tried proving that the integral converges (Experimental Math!)

I wonder if the following book (http://www-m8.ma.tum.de/m3/bornemann/challengebook/) gives the proof.


Quote:
If you have Mathematica, there is an interesting file you can download where the authors use Mathematica to solve this problem and 9 other problems which are very difficult numerical type problems.  Here's the link, its worth a look: http://library.wolfram.com/infocenter/Conferences/5353/

When I try to go by the link, I get “File not found”…  :(
Title: Re: Some super hard math problems
Post by BNC on Nov 21st, 2004, 9:22am

on 11/21/04 at 06:42:06, Barukh wrote:
When I try to go by the link, I get “File not found”…  :(


Do you mean the "siam100" zip file? I downloaded it without a problem. Let me know if you want it, and will "yousendit" to you.
Title: Re: Some super hard math problems
Post by Barukh on Nov 21st, 2004, 11:02am

on 11/21/04 at 09:22:01, BNC wrote:
Do you mean the "siam100" zip file? I downloaded it without a problem. Let me know if you want it, and will "yousendit" to you.

Yes, please.  :)
Title: Re: Some super hard math problems
Post by Icarus on Nov 21st, 2004, 3:35pm
Concerning the convergence of the limit in the first problem. As Barukh indicates the oscillation itself is not sufficient to show the limit converges. What is also required is that the magnitude of the oscillations converge to zero (sort of like the alternating series test). The area under each "hump" of the cosine portion is inversely proportional to the rate of growth of the argument, in this case to d/dx(x-1ln x) = (1-ln x)/x2 (throwing out the log conversion constant). Inverting and multiplying by the other multiplicand gives that each oscillation adds or subtracts an amount roughly proportional to x/(1-ln x), which clearly converges to 0. So Obob is right about the convergence (as I expected, given the original source of the problem), but it by no means clear simply from looking at it.
Title: Re: Some super hard math problems
Post by BNC on Nov 22nd, 2004, 1:18am

on 11/21/04 at 11:02:50, Barukh wrote:
Yes, please.  :)


Here you go :)
clicky (http://s4.yousendit.com/d.aspx?id=A0CEDBBBA7C56BE68C94B7456FEDBD2B)


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