Hi again! Only one (1) of my four (4) proofs produces a result in two (2) variables. I know that:
$f(x) = \Theta(f(x))$
In my one proof, I get:
$S_k = \frac{\log (n + 1)}{\log (n + k) - \log k}$
Can I say:
$S_k = \Theta[\frac{\log (n + 1)}{\log (n + k) - \log k}]$
?
Thanks!
-James
Thanks! (How do I "thank" you for your help here in this thread?)
I went back to the original mathematics, and proved the four (4) $\Theta$-notations in four (4) long proofs. I don't want to type them in here. It'll take too long.
Three (3) of the proofs produced the expected result. One (1)...
It's true: my notation was sloppy. I apologize. I'm still out of town, but am working on the nitty gritty mathematics. Nothing I want to share yet. But I have a stupid question :
If $a(x) = \Theta(f(x))$, and $b(y) = \Theta(g(y))$, is $a(x)/b(y) = \Theta(f(x)/g(y))$?
Thanks!
-James
I have to leave town, and will reply in greater detail when I return, but...
I disagree that $\Theta(x)$ is a set of functions. Looking at the definition of $\Theta$-notation:
$\Theta(x) = x$
because:
$c_1 x \leq x \leq c_2 x$
Thanks!
-James
Re: Growth of Functions - Big Theta
Ok, I think I've got it worked out. For my research, there's three $f(x)$'s and $g(y)$'s I am specifically interested in. In each case, I want to show $\Theta(f(x))/\Theta(g(y)) = \Theta(f(x)/g(y))$. I think I have done this. Here's my reasoning in each...
Re: Growth of Functions - Big Theta
Yes, my question is: Does $\Theta(x/y) = \Theta(x)/\Theta(y)$?
Or, more generally: Does $\Theta(f(x)/g(y)) = \Theta(f(x))/\Theta(g(y))$?
Thanks!
I'm brand new to this website, and couldn't figure out how to start a new thread, but I also have a question about Big Theta notation.
Is big-theta(x/y) = big-theta(x)/big-theta(y)?
I know big-o(x/y) = big-o(x)/big-o(y), but I don't know if big-omega(x/y) = big-omega(x)/big-omega(y).
I can...