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21h
comment Equilateral triangle
Hint: Use mirror symmetry to find the $y$ coordinate, and then Pythagoras for the $x$ coordinate.
1d
comment solve for $\int_{0}^{{\alpha}{b}}(a^x-1)dx=\int_{{\alpha}{b}}^{b}(a^x-1)dx$
For $\alpha=1/2$, also $a=0$ is an obvious solution.
1d
comment Is it true that $A \in A$?
No, with my definition, the sequence you give does not converge at all. It cannot converge to $\emptyset$ because $0\notin\emptyset$ but there is no $N$ so that for all $n>N$ we have $0\notin A_n$. Note that there's an "iff", not merely an "if".
1d
comment Is it true that $A \in A$?
Note that for the union definition, both formulas fail, too (indeed, here the precondition is no longer fulfilled for $\{A_i\}$ resp. $X-A_i$). Anyway, why is (for the definition I gave) $X-\lim A_i \ne \lim(X-A_i)$?
1d
comment Is it true that $A \in A$?
@Ariel: Sets without regularity would of course not be ZFC sets.
1d
comment Is it true that $A \in A$?
I think there is an obvious natural definition of set limit: $\lim_{n\to\infty}A_n = A$ if for every $x\in\bigcup_{n\in\mathbb N}A_n$ there is a natural number $N$ so that for any $n>N$ we have $x\in A_n$ iff $x\in A$. However with that definition, for the sequence in question the limit is the empty set.
2d
comment If $b \equiv 0 \pmod a$ and $c \equiv 0 \pmod b$, then $c \equiv 0 \pmod a$
Bad form $\ne$ incorrect proof. I'm all for discouraging bad form by calling it bad form. I'm all against discouraging bad form by incorrectly claiming it to be incorrect.
2d
comment Is 4 the second or third digit of pi
Another fun fact: If you express the largest distance we can observe (46 million light years) in terms of the planck length (the smallest length that actually makes sense physically), you get that it is about $2.7\cdot 10^{58}$ planck lengths. Thus with $\pi$ to about 60 digits, you should be able to calculate the circumference of every circle in the universe to the accuracy permitted by physics (of course the objection by @MarcvanLeeuwen about GR still applies; even more so, since at that precision, every local object will change the geometry).
2d
comment If $b \equiv 0 \pmod a$ and $c \equiv 0 \pmod b$, then $c \equiv 0 \pmod a$
He writes "$c=a\cdot(r\cdot s)$ so $c=ak$. That makes it absolutely clear that $k=r\cdot s$.
2d
comment If $b \equiv 0 \pmod a$ and $c \equiv 0 \pmod b$, then $c \equiv 0 \pmod a$
He rewrites the statement "$a\mid c$" he wants to prove into "$c=ak$ for some $k$". That's nothing but another way to state the same. The statement "$c=ak$ for some $k$" is the statement he's going to proof. Note that he writes "Now I start my proof" below that line, so that line is not even part of the proof, so how can it make the proof wrong?
2d
comment If $b \equiv 0 \pmod a$ and $c \equiv 0 \pmod b$, then $c \equiv 0 \pmod a$
He did define what $k$ is. It is the integer for which $c=ak$. That comes directly from the definition of $a\mid c$.
2d
comment Intuition about an orthogonal projection operator for matrices
Thanks for the upvote. I'm sorry that my answer isn't very helpful to you. Maybe if you explain why it is not helpful to you, I may be able to make it more helpful.
2d
comment Intuition about an orthogonal projection operator for matrices
BTW, looking at your user page, I notice that you didn't accept any answer to any question you asked. Did you consider none of the answers people gave to your questions helpful?
Aug
25
comment Is f(x) = x smooth?
@JiK: While he surely meant derivatives of arbitrary order, a straightforfard definition of a derivative of infinite order would be $\lim_{n\to\infty}\frac{\mathrm d^nf}{\mathrm dx^n}$ provided that limit exists. For example, the infiniteth derivative of any polynomial would be $0$, and the infinifteth derivative of $\exp(x)$ would be $\exp(x)$. Indeed, since that infiniteth derivative could well be differentiable itself (as the two examples are), one could identify this infinity with $\omega$ and continue with $\frac{\mathrm d^{\omega+1}f}{\mathrm dx^{\omega+1}}$. and so on.
Aug
25
comment As of August 2015, is the “set” of all gold medalists in the 2016 Olympics a set?
Couldn't you simply declare real-world objects as urelements?
Aug
25
comment How many ways can 6 cars ( 3 pink, 2 orange and 1 yellow) be parked in 6 parking slots in a row?
What is $_3P_2$?
Aug
25
comment Check dedekind cut for root 2
@user247327: "B is the set of all positive rational numbers whose square is larger than 2. (As Jorge said, you need "and" not "or" in your definition of B.)" Given that it is a problem statement, it is well possible that "or" is correct, and he is supposed to arrive at the conclusion that it is not a Dedekind cut.
Aug
25
comment As of August 2015, is the “set” of all gold medalists in the 2016 Olympics a set?
If using standard set theory (ZFC), the only items that can be element of a set are other sets. Assuming the 2016 Olympics gold medallists will not be sets (which I expect to be the case), they cannot be members of a ZFC set. If not using ZFC, it depends on the set theory you use.
Aug
25
comment Is f(x) = x smooth?
@zhw.: Is there any definition of "smooth" that does not apply to $f(x)=x$?
Aug
24
comment Intuition about an orthogonal projection operator for matrices
Ah, now I understand: $\mathcal P_T$ is acting on $Z\in\mathbb R^{m\times n}$. I understood your text as $\mathcal P_T$ being parametrized by $Z$. I'll edit your post to make that more clear.