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1500 questions
77
votes
9 answers
Best Algebraic Topology book/Alternative to Allen Hatcher free book?
Allen Hatcher seems impossible and this is set as the course text? So was wondering is there a better book than this? It's pretty cheap book compared to other books on amazon and is free online. Any good intro to Algebraic topology books? I can find…
simplicity
- 3,854
77
votes
3 answers
Combinatorial proof that $\sum \limits_{k=0}^n \binom{2k}{k} \binom{2n-2k}{n-k} (-1)^k = 2^n \binom{n}{n/2}$ when $n$ is even
In my answer here I prove, using generating functions, a statement equivalent to $$\sum_{k=0}^n \binom{2k}{k} \binom{2n-2k}{n-k} (-1)^k = 2^n \binom{n}{n/2}$$ when $n$ is even. (Clearly the sum is $0$ when $n$ is odd.) The nice expression on the…
Mike Spivey
- 57,148
77
votes
1 answer
Difference between Analytic and Holomorphic function
A function $f : \mathbb{C} \rightarrow \mathbb{C}$ is said to be holomorphic in an open set $A \subset \mathbb{C}$ if it is differentiable at each point of the set $A$. The function $f : \mathbb{C} \rightarrow \mathbb{C}$ is said to be analytic if…
Supriyo
- 6,339
77
votes
3 answers
What do prime ideals in $k[x,y]$ look like?
Suppose that $k$ is an algebraically closed field. Then what do the prime ideals in the polynomial ring $k[x,y]$ look like? As far as I know, the maximal ideals of $k[x,y]$ are of the form $(x-a,y-b)$ where $a,b\in k$. What can we say about the…
user14242
- 3,030
77
votes
4 answers
Why does the google calculator give $\tan 90^{\circ} = 1.6331779e^{+16}$?
I typed in $\tan 90^{\circ}$ in Google and it gave $1.6331779\mathrm{E}16$ . How did it come to this answer? Limits? Some magic?
Gizmo
- 939
77
votes
4 answers
Difference between basis and subbasis in a topology?
I was reading Topology from Munkres and got confused by the definition of a subbasis. What is/are the difference between basis and subbasis in a topology?
Grobber
- 3,376
77
votes
1 answer
Is $\sqrt1+\sqrt2+\dots+\sqrt n$ ever an integer?
Related: Can a sum of square roots be an integer? Except for the obvious cases $n=0,1$, are there any values of $n$ such that $\sum_{k=1}^n\sqrt k$ is an integer? How does one even approach such a problem? (This is not homework - just a problem I…
Mario Carneiro
- 28,431
- 4
- 74
- 162
77
votes
10 answers
Is learning (theoretical) physics useful/important for a mathematician?
I'm starting to read The Princeton Companion to Mathematics , at the beginning it says: A proper appreciation of pure mathematics requires some knowledge of applied mathematics and theoretical physics. Some of my professors have told me that modern…
Vicfred
- 2,957
77
votes
2 answers
Numerical phenomenon. Who can explain?
I was doing some software engineering and wanted to have a thread do something in the background to basically just waste CPU time for a certain test. While I could have done something really boring like for(i < 10000000) { j = 2 * i } , I ended up…
Jake Mirra
- 3,323
77
votes
1 answer
Why is TREE(3) so big? (Explanation for beginners)
I am not a mathematician but I am interested in big numbers. I find them to be really interesting, almost god-like. I am watching a series of videos from David Metzler on YouTube. I have a basic understanding of some fast growing functions. David…
Josh Kerr
- 977
77
votes
6 answers
Proof a graph is bipartite if and only if it contains no odd cycles
How can we prove that a graph is bipartite if and only if all of its cycles have even order? Also, does this theorem have a common name? I found it in a maths Olympiad toolbox.
Asinomás
- 107,927
77
votes
0 answers
Determinant of a matrix that contains the first $n^2$ primes.
Let $n$ be an integer and $p_1,\ldots,p_{n^2}$ be the first prime numbers. Writing them down in a matrix $$ \left(\begin{matrix} p_1 & p_2 & \cdots & p_n \\ p_{n+1} & p_{n+2} & \cdots & p_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ \cdots & \cdots…
Rofl Ukulus
- 932
77
votes
5 answers
Under what condition we can interchange order of a limit and a summation?
Suppose f(m,n) is a double sequence in $\mathbb R$. Under what condition do we have $\lim\limits_{n\to\infty}\sum\limits_{m=1}^\infty f(m,n)=\sum\limits_{m=1}^\infty \lim\limits_{n\to\infty} f(m,n)$? Thanks!
zzzhhh
- 909
77
votes
11 answers
Why is $\Gamma\left(\frac{1}{2}\right)=\sqrt{\pi}$?
It seems as if no one has asked this here before, unless I don't know how to search. The Gamma function is $$ \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
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