Recent content by SweatingBear

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...

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...

Would you look at that, it was already treated unbeknownst to me (which just nullifies this thread, I'll have it depreciated). Thanks!

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?

I was actually a bit uncertain about that. How else would one go about?

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 $(p+1)^3 \geqslant 125$...

Fair enough, thanks!

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...

Thanks!

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...

Much appreciated you could share your thoughts, Ackbach.

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...