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Math Forum / Mathematics / General Topics / July 2009max min discrete functions
My problem is: assume a discrete function, find the value of the variable for which the function becomes extreme. My question arises from an exam i've just had, in which i had to find the value of n (a discrete variable) for which cos((2*n-1)*Pi/(2*a)*x) or sin(n*Pi/a*x) corresponds the best to A*x*(a^2-x^2) with x between -a and a. In this example its quite simple to see that the function will correspond to a sine function with a period of 2*a (if you just plot the function). But i started thinking. Normally to find a solution to this problem you would had to take a dot product between these two functions (as an example the integral of their product between -a and a) and find for which n this dot product becomes extreme. But to do so, you need a method to find an extreme value of a discrete function. But i've never heard of such a thing, and i can't find anything that would help me on the internet. Is there a method to find such an extreme, or do i really have to check all the values and find the n for which the function becomes extreme? I was for example thinking of sin(20^(1/2)*n), the period is non-rational, so you can't just see the solution to this problem. > But i started thinking. Normally to find a solution to this problem you would had to take a dot product between these two functions (as an example the integral of their product between -a and a) and find for which n this dot product becomes extreme. But to do so, you need a method to find an extreme value of a discrete function. But i've never heard of such a thing, and i can't find anything that would help me on the internet. Is there a method to find such an extreme, or do i really have to check all the values and find the n for which the function becomes extreme? > > I was for example thinking of sin(20^(1/2)*n), the period is non-rational, so you can't just see the solution to this problem. Method depends on problem. For the particular one above, sin (n sqrt 20), one can prove that the numbers n sqrt 20 are dense in the reals modulo 2 pi, which implies that n sqrt 20 gets arbitrarily close to (but never equals) numbers x such that sin x = 1. So the supremum is 1 but it is never achieved. In other words, sin (n sqrt 20) has no maximum; it gets as close to 1 as you like but never equals 1. -- GM Thanks, I had already thought that the limes superior of that row would be one, but i couldn't prove it so i wasn't sure. > Thanks, > > I had already thought that the limes superior of that row would be one, but i couldn't prove it so i wasn't sure. One way to prove it is via Dirichlet's Theorem on Diophantine Approximation, which you can find in elementary Number Theory textbooks (and at various websites, no doubt). -- GM |
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