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1500 questions
64
votes
1 answer
Does Fermat's Last Theorem hold for cyclotomic integers in $\mathbb{Q(\zeta_{37})}$?
The first irregular prime is 37. Does FLT(37) $$x^{37} + y^{37} = z^{37}$$ have any solutions in the ring of integers of $\mathbb Q(\zeta_{37})$, where $\zeta_{37}$ is a primitive 37th root of unity? Maybe it's not true, but how could I go about…
quanta
- 12,783
64
votes
7 answers
What's the point of studying topological (as opposed to smooth, PL, or PDiff) manifolds?
Part of the reason I think algebraic topology has acquired something of a fearsome reputation is that the terrible properties of the topological category (e.g. the existence of space-filling curves) force us to work very hard to prove the main…
Qiaochu Yuan
- 478,279
64
votes
5 answers
What is the difference between the weak and strong law of large numbers?
I don't really understand exactly what the difference between the weak and strong law of large numbers is. The weak law says \begin{align*} \lim_{n \rightarrow \infty} \mathbb{P}[\mid \bar{X}_n - \mu \mid \leq \epsilon ] = 1, \end{align*} while the…
Stefan Falk
- 1,317
64
votes
10 answers
How to solve an $n$-th degree polynomial equation
The typical approach of solving $$ f_2(x):=ax^2+bx+c=0 $$ is to solve for the roots $$x_{1/2}=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}.$$ Here, the degree of $f$ is given to be $2$ . However, I was wondering on how to generalize this problem. For example,…
Ayush Khemka
- 1,005
64
votes
10 answers
Big List of Erdős' elementary proofs
Paul Erdős was one of the greatest mathematicians of all time and he was famous for his elegant proofs from The Book. I posted a question about one of his theorem and got a reference, and I have other questions I want to know the answer to too. But,…
Saikat
- 2,571
64
votes
3 answers
Motivation and methods for self-study
First, a little background: Beginning with calculus in my first semester of college, I fell in love with mathematics. That was the point at which the concepts became interesting to me, and I started reading up, through Wikipedia and various other…
Alex Petzke
- 9,031
64
votes
6 answers
Are commutative C*-algebras really dual to locally compact Hausdorff spaces?
Several online sources (e.g. Wikipedia , the nLab ) assert that the Gelfand representation defines a contravariant equivalence from the category of (non-unital) commutative $C^{\ast}$-algebras to the category of locally compact Hausdorff (LCH)…
Qiaochu Yuan
- 478,279
64
votes
5 answers
Why is it that $\mathbb{Q}$ cannot be homeomorphic to _any_ complete metric space?
Why is it that $\mathbb{Q}$ cannot be homeomorphic to any complete metric space? Certainly $\mathbb{Q}$ is not a complete metric space. But completeness is not a topological invariant, so why is the above statement true?
user
- 649
64
votes
1 answer
Is there a "greater than about" symbol?
To indicate approximate equality, one can use ≃, ≅, ~, ♎, or ≒. I need to indicate an approximate inequality. Specifically, I know A is greater than a quantity of approximately B. Is there a way to succinctly express this mathematically?
foobarbecue
- 753
64
votes
2 answers
Does a Fourier transformation on a (pseudo-)Riemannian manifold make sense?
the Fourier transformation of a scalar function with respect to one variable might be defined as $\mathcal{F}\left[w\right](\omega )\equiv \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}w(t)e^{-\mathrm{i}\omega t}dt$ In physics, this transformation…
Robert Filter
- 1,600
64
votes
12 answers
$\lim\limits_{n \to{+}\infty}{\sqrt[n]{n!}}$ is infinite
How do I prove that $ \displaystyle\lim_{n \to{+}\infty}{\sqrt[n]{n!}}$ is infinite?
Breton
- 1,698
64
votes
4 answers
Closed Form for $~\int_0^1\frac{\text{arctanh }x}{\tan\left(\frac\pi2~x\right)}~dx$
Does $$~\displaystyle{\int}_0^1\frac{\text{arctanh }x}{\tan\left(\dfrac\pi2~x\right)}~dx~\simeq~0.4883854771179872995286585433480\ldots~$$ possess a closed form expression ? This recent post, in conjunction with my age-old interest in Gudermannian…
Lucian
- 49,592
64
votes
13 answers
I have learned that 1/0 is infinity, why isn't it minus infinity?
My brother was teaching me the basics of mathematics and we had some confusion about the positive and negative behavior of Zero. After reading a few post on this we came to know that it depends on the context of its use. Why do we take 1/0 as…
Rishav Mehta
- 761
64
votes
7 answers
Union of the conjugates of a proper subgroup
Let $G$ be a finite group and $H$ be a proper subgroup. Prove that the union of the conjugates of $H$ is not the whole of $G$ . Thanks for any help
hmmmm
- 5,843
64
votes
3 answers
How do I exactly project a vector onto a subspace?
I am trying to understand how - exactly - I go about projecting a vector onto a subspace. Now, I know enough about linear algebra to know about projections, dot products, spans, etc etc, so I am not sure if I am reading too much into this, or if…
Spacey
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