Been a while since I stopped by here...
There's one thing about optimization on non-compact sets that's been bugging me for quite a while and I'd love to hear how you perceive things.
Say we are optimizing a partially differentiable (and thus continuous) function $f:\mathbb{R^2} \to...
Let [x] be the floor function i.e. it produces the integral part of x. So for example if x = 1.5 then [x] = 1. I recently saw the claim
[x] \geq x - 1
The strict part of the inequality makes perfect sense, but when does equality occur? Does it even occur at all? I have not been able to find an...
The given statement can be written $ |z-w| < |1 - \overline{z}w| $, which equivalently is $ |z-w|^2 < |1 - \overline{z}w|^2 $.
Let $ z = a +bi $ and $ w = c + di $. Thus $a^2 + b^2 + c^2 + d^2 < (a^2+b^2)(c^2 + d^2) + 1$, or equivalently, $|z|^2 + |w|^2 < 1 + |z|^2 |w|^2 $. That statement is...
Here's a fun problem proof I came across. Show that
\left| \frac { z- w }{1 - \overline{z}w} \right| < 1
given |z|<1, |w|<1. I attempted writing z and w in rectangular coordinates (a+bi) but to no avail. Any suggestions, forum?
We wish to show that $3^n > n^3 \, , \ \forall n \geqslant 4 $. Base case $n = 4$ yields
$3^4 = 81 > 4^3 = 64 $
Assume the inequality holds for $n = p $ i.e. $3^p > p^3$ for $p \geqslant 4$. Then
$3^{p+1} > 3p^3$
$p \geqslant 4$ implies $3p^3 \geq 192$, as well as $(p+1)^3 \geqslant 125$...
Why is it that the norm of a vector is written as a "double" absolute value sign instead of a single one? I.e. why is the norm written as $ || \vec{v} || $ and not $ | \vec{v} | $? I think $ | \vec{v} | $ is appropriate enough, why such emphasis on $ || \vec{v} || $? I think it's rather natural...
Yes that's great but I am afraid it does not answer the main question: In the case where we have several variables, how can one tell which variable one should optimize with respect to?
We have the following trapezoid:
The question is to find the length of the fourth side when the area of trapezoid is maximized.
I realize we will not be able to find a numerical value for the fourth side due to the given information (rather, lack thereof). So we are essentially going have...
Oh right of course, the trigonometric identities! But here is a follow-up question: How would the issue be resolve if it was the case that we could not take advantage of a trigonometric identity?
How did you end up with $9v = n \cdot 2 \pi$? $9v$ must equal the argument of the number in the right-hand side i.e. $\pi + n \cdot 2\pi$, I see no other way.
Excellent! Thanks for that perspective, however I will not rest until I have figured out the nitty and gritty details of my approach...