Hey everyone, just something somewhat interesting that I noticed. A basis for a topology generates one and only one topology, but a topology can have multiple bases, So let's say we have a set $X$, and a collection of bases for topologies $\mathcal{B_{\alpha}}$ and a collection of topologies on $X$, denoted by $\mathcal{T_{\beta}}$. There seems to be a surjective mapping $f : \{\mathcal{B_{\alpha}}\} \to \{\mathcal{T_{\beta}}\}$.
"So the isometries on R^3 form the free group which is crazy, like it's completely crazy, and then you sorta rotate your stuff in weird ways and then everything just becomes a mess."
All you do in Banach-Tarski is apply isometries - which are volume preserving. Hence the volume of one unit ball and two unit balls must be equal: $1 = 2$. :D
@TedShifrin Yeah I laughed when he said that, it was great. I think Marianna does the full proof when she teaches measure but until then I'll just roll with VSauce + "The free group is crazy" for the proof
@SteamyRoot So we know that 1 = 1!, so if 1=2, then 1! = 2, but commuting the exclamation mark and the space, we get 1 != 2. Contradiction
That reminds me of the one time someone did a proof by contradiction and finished with "and hence we end up with a contradiction: $1 = 0$!". The prof teaching the course wrote down that he didn't see a contradiction at all.
I want to learn more about classical varieties. How do I proceed?
I was looking for more information on Classical Varieties and what exactly it means. I worked through the wikipedia article - http://en.wikipedia.org/wiki/Algebraic_variety, but I do not understand what they are used for.
I have ...
yeah he is a bit confused... " have little experience in geometry (Is this geometry?) "
from the comments: sellable product, other than educating others about the math. Ok I'll check that out and page through it. If you were to explain Algebraic Varieties and its implications to an executive manager in one or a couple paragraphs, what would that contain? Here is my version, but I made up the exec details as I don't know them -- Algebraic Varieties is the study of the emergent patterns from simple shapes and rules.
- Algebraic Varieties is the study of the emergent patterns from simple shapes and rules. If a faster converging solution can be found to the GTi projection, then this would mean tighter manufacturing
$\Bbb Z$ satisfies the second one, but not the first.
Functions from $\Bbb R$ to $\Bbb Z$ are a little more interesting for this: such a function is continuous iff the fiber $f^{-1}(\{m\})$ of any element of $\Bbb Z$ is an open set.
@Rüdiger I hadn't even looked at it closely enough to see that, lol
I think cryptography is an example of a 'sellable product', so I'd count elliptic curve cryptography in that respect. But algebraic geometry is not a branch of operations research!
@Pissedofflayman because if you literally told someone to go do that in a serious manner it wouldn't actually be insulting. You'd just be telling someone to literally masturbate.
@JourneymanGeek a couple people posed some uncomfortable questions about whether "F*** you" was actually insulting by default. I don't personally see an issue. I think it was a slight overreaction. It's been over and done with for a good minute or so.
A firemen got "f you" from someone to his bidding to "open the door". His response to the "f you" and opening the door was "thank you very much". Which is imo proffessionalism in person.
@arctictern I think @Null was merely asking whether the statement was actually an insult by literal meaning. After all, many of our insults nowadays are literal things that are actually demeaning in some way.
It entertains the idea that maybe humans themselves aren't truly conscious, but only approximating consciousness. And, if that's the case, maybe it's possible to build something more conscious than us.
@AkivaWeinberger eh. didn't really say humans aren't conscious. hinted at the idea of the hosts becoming more conscious eventually. did imply that humans have their own "loops" and inner workings like hosts.
campaigning on the platform of revealing my identity would have in all likelihood won me the job at the time. not that there's anything interesting about it.
Given any set $S$, and a measure $\mu : S\rightarrow [0,1]$ $$\{\textrm{SF}\}\subseteq \{\textrm{young adult literature}\}, \mu((\{\textrm{SF}\}\cap \{\textrm{young adult literature}\})^C)=0$$
Well mathematically speaking, given (1) the set of all science fiction books and (2) the set of all young adult novels, it is said that (1) is almost a subset of (2) and their intersection is almost total (meaning that the intersection (1) and (2)) approaches that of (2), how will this be wrote in terms of set notations?
Lacking a measure, is it possible to tell how close is $A \cap B = B$ given $A \subseteq B$?
Hey guys, how do i realize that every metric space consiting of only a singleton $\{x\}$ then the induced Borel sigma algebra is always given by $\{\emptyset, \{x\} \}$ ?
I want to learn more about classical varieties. How do I proceed?
I was looking for more information on Classical Varieties and what exactly it means. I worked through the wikipedia article - http://en.wikipedia.org/wiki/Algebraic_variety, but I do not understand what they are used for.
I have ...
$\int_0^4 xe^xdx$
$u=x$ then $du=1$ $dv=e^x$ and $v=e^x$
This gives:
$\int_0^4 xe^x-\int_0^4e^x$ which then gives:
$4e^4-1-e^4-1$
