[MUG] solving inequality

[MUG] Re: solving inequality
Author: Maple User Group    Posted: Fri, 1 Nov 2002 10:17:54 -0500 (
>> From: Maple User Group "maple_gr" | >> From: M A Suzen "mehmet.suzen" | Are there any workaround to get a solution for below in equality | in terms of n? solve seems is not able to solve it. | ineqn:=n*ln(n)-(1-alpha)*n*ln((1-alpha)*n)-n*alpha*ln(n*alpha)+ln(B)>=-4(1- | alpha)*n*ln(2): | > B:=2^((3-2*alpha)*n)-2^(4*(1-alpha)*n): | > e:=evala(ineqn): | > c:=simplify(e,symbolic): | > assume(alpha,real):solve(c,n); -=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=- Date: Tue, 29 Oct 2002 21:23:39 -0800 (PST) From: Robert Israel "israel" To: "maple-list" Subject: solving inequality I suspect you want n > 0, and also to assume 1/2 < alpha < 1, not just alpha real, in order to make ln((1-alpha)*n), ln(n*alpha) and ln(B) real when n > 0. It's easy to prove that rhs(c) - lhs(c) is an increasing function of n for such alpha, with limits -infinity and +infinity as n -> 0+ and infinity respectively, so there is exactly one n (say n_0) that makes the inequality an equality, and the solution of the inequality is n = [n_0, infinity). Not surprisingly, Maple doesn't find a closed-form solution for n_0; I doubt there is such a closed-form solution. For particular values of alpha in the interval 1/2 < alpha < 1, you can use fsolve. Robert Israel "israel" Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 -=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=- Date: Wed, 30 Oct 2002 09:35:11 -0500 To: "maple-list" From: "Gerald A. Edgar" "edgar" Subject: solving inequality Why do you think there is a formula for the solution? Subtract to get an inequality (formula) >= 0, then exponentiate to get (formula)>=1. Simplifying the formula, we get something of the form U^n-V^n >=1, where U and V depend on alpha. If we solve U^n-V^n=1, then we can solve the inequality, since U^n-V^n is a continuous function of the real variable n. But I see no reason to think there is a nice formula for solutions of U^n-V^n=1. If I am not mistaken, 1 U = 128 -------------------------------------------- (1 - alpha) alpha (2 alpha) (1 - alpha) alpha 2 1 V = 256 -------------------------------------------- (1 - alpha) alpha (4 alpha) (1 - alpha) alpha 2 -- G. A. Edgar http://www.math.ohio-state.edu/~edgar/ -=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=- From: "Daniel Willard" "willardd" To: "maple-list" Subject: solving inequality Date: Wed, 30 Oct 2002 10:49:24 -0500 I think you are going to find this is in the "too hard" box for Maple. Your "c" can be recast into n*(Complex number in alpha) <= ln( sum of exponentials in n) and this seems to say that for some range(s) of alpha n will be complex. Try first plotting something.

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