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1500 questions
82
votes
11 answers
Infinite sets don't exist!?
Has anyone read this article?
This accomplished mathematician gives his opinion on why he doesn't think infinite sets exist, and claims that axioms are nonsense. I don't disagree with his arguments, but with my limited knowledge of axiomatic set…
Nicolas Bourbaki
- 1,686
82
votes
6 answers
Motivation for spectral graph theory.
Why do we care about eigenvalues of graphs?
Of course, any novel question in mathematics is interesting, but there is an entire discipline of mathematics devoted to studying these eigenvalues, so they must be important.
I always assumed that…
Alexander Gruber
- 26,963
82
votes
10 answers
What's a proof that the angles of a triangle add up to 180°?
Back in grade school, I had a solution involving "folding the triangle" into a rectangle half the area, and seeing that all the angles met at a point:
However, now that I'm in university, I'm not convinced that this proof is the best one (although…
Joe Z.
- 6,719
82
votes
7 answers
Poisson Distribution of sum of two random independent variables $X$, $Y$
$X \sim \mathcal{P}( \lambda) $ and $Y \sim \mathcal{P}( \mu)$ meaning that $X$ and $Y$ are Poisson distributions. What is the probability distribution law of $X + Y$. I know it is $X+Y \sim \mathcal{P}( \lambda + \mu)$ but I don't understand how to…
user31280
82
votes
2 answers
Let $X$ be an infinite dimensional Banach space. Prove that every Hamel basis of $X$ is uncountable.
Let $X$ be an infinite dimensional Banach space. Prove that every basis of $X$ is uncountable.
Can anyone help how can I solve the above problem?
mintu
- 987
82
votes
8 answers
When not to treat dy/dx as a fraction in single-variable calculus?
While I do know that $\frac{dy}{dx}$ isn't a fraction and shouldn't be treated as such, in many situations, doing things like multiplying both sides by $dx$ and integrating, cancelling terms, doing things like $\frac{dy}{dx} =…
xasthor
- 1,356
82
votes
7 answers
Is linear algebra more “fully understood” than other maths disciplines?
In a recent question, it was discussed how LA is a foundation to other branches of mathematics, be they pure or applied. One answer argued that linear problems are fully understood, and hence a natural target to reduce pretty much anything to.
Now,…
leftaroundabout
- 6,535
82
votes
6 answers
Why Zariski topology?
Why in algebraic geometry we usually consider the Zariski topology on $\mathbb A^n_k$? Ultimately it seems a not very interesting topology, infact the open sets are very large and it doesn't satisfy the Hausdorff separation axiom. Ok the basis is…
Dubious
- 13,350
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82
votes
5 answers
How to show that the commutator subgroup is a normal subgroup
It is suggested as an exercise in Serge Lang's book "Algebra" to show that the commutator subgroup $G^c$ of a group $G$ is a normal subgroup.
I'd like to do that but I am afraid I need help,
I think the first thing I need to figure out is how a…
harlekin
- 8,740
82
votes
21 answers
How to generate a random number between 1 and 10 with a six-sided die?
Just for fun, I am trying to find a good method to generate a random number between 1 and 10 (uniformly) with an unbiased six-sided die.
I found a way, but it may requires a lot of steps before getting the number, so I was wondering if there are…
P.A.
- 829
82
votes
2 answers
What is the proper way to study (more advanced) math?
Here's what happens. I get stuck on some proof or some mathematical construction and I end up staring at the problem for hours, sometimes not making any progress. I do this because sometimes I think that I'm being lazy, I'm not thinking things…
anonymous
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- 3
82
votes
9 answers
Besides proving new theorems, how can a person contribute to mathematics?
There are at least a few things a person can do to contribute to the mathematics community without necessarily obtaining novel results, for example:
Organizing known results into a coherent narrative in the form of lecture notes or a…
David Zhang
- 8,835
82
votes
8 answers
How many connected components does $\mathrm{GL}_n(\mathbb R)$ have?
I've noticed that $\mathrm{GL}_n(\mathbb R)$ is not a connected space, because if it were $\det(\mathrm{GL}_n(\mathbb R))$ (where $\det$ is the function ascribing to each $n\times n$ matrix its determinant) would be a connected space too, since…
Bartek
- 6,265
81
votes
12 answers
I lost my love of math; I'm getting it back. How can I determine if math is actually right for me?
This question has been on my mind for a very long time, and I thought I'd finally ask it here.
When I was 6, my dad pulled me out of school. The classes were too easy; the professors, too dull. My father had been man of philosophy his entire…
Chiefy
- 665
81
votes
4 answers
Are these solutions of $2 = x^{x^{x^{\:\cdot^{\:\cdot^{\:\cdot}}}}}$ correct?
Find $x$ in
$$ \Large 2 = x^{x^{x^{\:\cdot^{\:\cdot^{\:\cdot}}}}}$$
A trick to solve this is to see that
$$\large
2 = x^{x^{x^{\:\cdot^{\:\cdot^{\:\cdot}}}}}
\quad\implies\quad
2 = x^{\Big(x^{x^{x^{\:\cdot^{\:\cdot^{\:\cdot}}}}}\Big)} =…
GarouDan
- 3,418