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1500 questions
77
votes
13 answers
What's a good place to learn Lie groups?
Ok so I read the following article the other day: http://www.aimath.org/E8/ and I wanted to learn more about lie groups. Using my exceptional deduction skills I thought "oh it must have something to do with groups" So I picked up a copy of Dummit…
Asinomás
- 105,651
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
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
- 3,170
77
votes
9 answers
Do "Parabolic Trigonometric Functions" exist?
The parametric equation
$$\begin{align*}
x(t) &= \cos t\\
y(t) &= \sin t
\end{align*}$$
traces the unit circle centered at the origin ($x^2+y^2=1$). Similarly,
$$\begin{align*}
x(t) &= \cosh t\\
y(t) &= \sinh t
\end{align*}$$
draws the right part…
Argon
- 25,303
76
votes
5 answers
Difference between axioms, theorems, postulates, corollaries, and hypotheses
I've heard all these terms thrown about in proofs and in geometry, but what are the differences and relationships between them? Examples would be awesome! :)
Gordon Gustafson
- 3,365
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
- 9,636
76
votes
2 answers
Cardinality of Borel sigma algebra
It seems it's well known that if a sigma algebra is generated by countably many sets, then the cardinality of it is either finite or $c$ (the cardinality of continuum). But it seems hard to prove it, and actually hard to find a proof of it. Can…
Syang Chen
- 3,416
76
votes
32 answers
Interesting and unexpected applications of $\pi$
$\text{What are some interesting cases of $\pi$ appearing in situations that do not seem geometric?}$
Ever since I saw the identity $$\displaystyle \sum_{n = 1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}$$
and the generalization of $\zeta (2k)$, my…
MT_
- 19,603
- 9
- 40
- 81
76
votes
1 answer
Is it possible for a function to be smooth everywhere, analytic nowhere, yet Taylor series at any point converges in a nonzero radius?
It is well-known that the function
$$f(x) = \begin{cases} e^{-1/x^2}, \mbox{if } x \ne 0 \\ 0, \mbox{if } x = 0\end{cases}$$
is smooth everywhere, yet not analytic at $x = 0$. In particular, its Taylor series exists there, but it equals $0 + 0x +…
The_Sympathizer
- 19,651
76
votes
3 answers
Eigenvalues of the rank one matrix $uv^T$
Suppose $A=uv^T$ where $u$ and $v$ are non-zero column vectors in ${\mathbb R}^n$, $n\geq 3$. $\lambda=0$ is an eigenvalue of $A$ since $A$ is not of full rank. $\lambda=v^Tu$ is also an eigenvalue of $A$ since
$$Au = (uv^T)u=u(v^Tu)=(v^Tu)u.$$…
user9464
76
votes
2 answers
Dantzig's unsolved homework problems
From Wikipedia:
An event in George Dantzig's life became the origin of a famous
story in 1939 while he was a graduate student at UC Berkeley. Near the
beginning of a class for which Dantzig was late, professor Jerzy
Neyman wrote two examples…
Max
- 987
76
votes
2 answers
Integration of forms and integration on a measure space
In Terence Tao's PCM article: DIFFERENTIAL FORMS AND INTEGRATION, it is pointed out that there are three concepts of integration which appear in the subject (single-variable calculus):
the indefinite integral $\int f$ (also known as the…
user9464
76
votes
5 answers
What's the intuition with partitions of unity?
I've been studying Spivak's Calculus on Manifolds and I'm really not getting what's behind partitions of unity. Spivak introduces the topic with the following theorem:
Let $A\subset \Bbb R^n$ and let $\mathcal{O}$ be an open cover of $A$. Then…
Gold
- 26,547