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1500 questions
77
votes
5 answers
Under what condition we can interchange order of a limit and a summation?
Suppose f(m,n) is a double sequence in $\mathbb R$. Under what condition do we have $\lim\limits_{n\to\infty}\sum\limits_{m=1}^\infty f(m,n)=\sum\limits_{m=1}^\infty \lim\limits_{n\to\infty} f(m,n)$? Thanks!
zzzhhh
- 909
77
votes
11 answers
Why is $\Gamma\left(\frac{1}{2}\right)=\sqrt{\pi}$?
It seems as if no one has asked this here before, unless I don't know how to search.
The Gamma function is
$$
\Gamma(\alpha)=\int_0^\infty x^{\alpha-1} e^{-x}\,dx.
$$
Why is
$$
\Gamma\left(\frac{1}{2}\right)=\sqrt{\pi}\text{ ?}
$$
(I'll post my own…
77
votes
24 answers
Prove $0! = 1$ from first principles
How can I prove from first principles that $0!$ is equal to $1$?
Ssegawa Victor
- 1,055
77
votes
1 answer
Divisor -- line bundle correspondence in algebraic geometry
I know a little bit of the theory of compact Riemann surfaces, wherein there is a very nice divisor -- line bundle correspondence.
But when I take up the book of Hartshorne, the notion of Cartier divisor there is very confusing. It is certainly not…
user977
77
votes
4 answers
Graph theory: adjacency vs incident
Okay, so I think if 2 vertices are adjacent to each other, they are incident to each other....or do I have it wrong? Is this just different terminology. I thought I was totally clear on this for my class, but now I am doubting myself reading the…
pqsk
- 875
77
votes
6 answers
How to generate random points on a sphere?
How do I generate $1000$ points $\left(x, y, z\right)$ and make sure they land on a sphere whose center is
$\left(0, 0, 0\right)$ and its diameter is $20$ ?.
Simply, how do I manipulate a point's coordinates so that the point lies on the sphere's…
Filip
- 779
- 1
- 6
- 5
77
votes
8 answers
What is the chance to get a parking ticket in half an hour if the chance to get a ticket is 80% in 1 hour?
This sounds more like a brain teaser, but I had some kink to think it through :( Suppose you're parking at a non-parking zone, the probability to get a parking ticket is 80% in 1 hour, what is the probability to get a ticket in half an hour? Please…
Rock
- 816
77
votes
4 answers
Teenager solves Newton dynamics problem - where is the paper?
From Ottawa Citizen (and all over, really):
An Indian-born teenager has won a research award for solving a
mathematical problem first posed by Sir Isaac Newton more than 300
years ago that has baffled mathematicians ever since.
The solution…
jnm2
- 3,260
77
votes
3 answers
how to read a mathematical paper?
I hope that this question is on-topic, though it is not quite technical.
I am curious to hear from people how they approach reading a mathematical paper.
I am not asking specific questions on purpose, though at first I had a few. But I want to keep…
normvector
- 713
77
votes
1 answer
Overview of basic results on cardinal arithmetic
Are there some good overviews of basic formulas about addition, multiplication and exponentiation of cardinals (preferably available online)?
Martin Sleziak
- 56,060
76
votes
2 answers
Determinant of a rank $1$ update of a scalar matrix, or characteristic polynomial of a rank $1$ matrix
This question aims to create an "abstract duplicate" of numerous questions that ask about determinants of specific matrices (I may have missed a few):
Characteristic polynomial of a matrix of $1$'s
Eigenvalues of the rank one matrix…
Marc van Leeuwen
- 119,547
76
votes
5 answers
How to prove that a compact set in a Hausdorff topological space is closed?
How to prove that a compact set $K$ in a Hausdorff topological space $\mathbb{X}$ is closed? I seek a proof that is as self contained as possible.
Thank you.
Elias Costa
- 15,282
76
votes
14 answers
Dividing 100% by 3 without any left
In mathematics, as far as I know, you can't divide 100% by 3 without having 0,1...% left.
Imagine an apple which was cloned two times, so the other 2 are completely equal in 'quality'. The totality of the 3 apples is 100%. Now, you can divide those…
RAO
- 1,027
76
votes
6 answers
Why can't you pick socks using coin flips?
I'm teaching myself axiomatic set theory and I'm having some trouble getting my head around the axiom of choice. I (think I) understand what the axiom says, but I don't get why it is so 'contentious', which probably means I haven't yet digested it…
MGA
- 9,788
76
votes
6 answers
Functions which are Continuous, but not Bicontinuous
What are some examples of functions which are continuous, but whose inverse is not continuous?
nb: I changed the question after a few comments, so some of the below no longer make sense. Sorry.
isomorphismes
- 4,758