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1500 questions
183
votes
10 answers
Is $0$ a natural number?
Is there a consensus in the mathematical community, or some accepted authority, to determine whether zero should be classified as a natural number?
It seems as though formerly $0$ was considered in the set of natural numbers, but now it seems more…
bryn
- 9,746
182
votes
24 answers
Looking for an intuitive explanation why the row rank is equal to the column rank for a matrix
I am looking for an intuitive explanation as to why/how row rank of a matrix = column rank. I've read the proof on Wikipedia and I understand the proof, but I don't "get it". Can someone help me out with this ?
I find it hard to wrap my head around…
hari_sree
- 2,081
181
votes
14 answers
How do you revise material that you already half-know, without getting bored and demotivated?
Mathematics inevitably involves a lot of self-teaching; if you're just planning to sit there and wait for the lecturer to introduce you to important ideas, you probably need to find yourself another career. So, like a lot people here, I try to…
goblin GONE
- 67,744
181
votes
7 answers
Is the product of two Gaussian random variables also a Gaussian?
Say I have $X \sim \mathcal N(a, b)$ and $Y\sim \mathcal N(c, d)$. Is $XY$ also normally distributed?
Is the answer any different if we know that $X$ and $Y$ are independent?
jamaicanworm
- 4,494
180
votes
5 answers
A math contest problem $\int_0^1\ln\left(1+\frac{\ln^2x}{4\,\pi^2}\right)\frac{\ln(1-x)}x \ \mathrm dx$
A friend of mine sent me a math contest problem that I am not able to solve (he does not know a solution either). So, I thought I might ask you for help.
Prove:
$$\int_0^1\ln\left(1+\frac{\ln^2x}{4\,\pi^2}\right)\frac{\ln(1-x)}x…
Vladimir Reshetnikov
- 47,122
180
votes
8 answers
Intuition of the meaning of homology groups
I am studying homology groups and I am looking to try and develop, if possible, a little more intuition about what they actually mean. I've only been studying homology for a short while, so if possible I would prefer it if this could be kept…
Spyam
- 4,435
179
votes
2 answers
Open problems in General Relativity
I would like to know if there are some open mathematical problems in General Relativity, that are important from the point of view of Physics.
Is there something that still needs to be justified mathematically in order to have solid foundations?
Benjamin
- 2,816
179
votes
10 answers
How far can one get in analysis without leaving $\mathbb{Q}$?
Suppose you're trying to teach analysis to a stubborn algebraist who refuses to acknowledge the existence of any characteristic $0$ field other than $\mathbb{Q}$. How ugly are things going to get for him?
The algebraist argues that the real numbers…
Alexander Gruber
- 26,963
179
votes
6 answers
Symmetry of function defined by integral
Define a function $f(\alpha, \beta)$, $\alpha \in (-1,1)$, $\beta \in (-1,1)$ as
$$ f(\alpha, \beta) = \int_0^{\infty} dx \: \frac{x^{\alpha}}{1+2 x \cos{(\pi \beta)} + x^2}$$
One can use, for example, the Residue Theorem to show that
$$ f(\alpha,…
Ron Gordon
- 138,521
179
votes
6 answers
An Introduction to Tensors
As a physics student, I've come across mathematical objects called tensors in several different contexts. Perhaps confusingly, I've also been given both the mathematician's and physicist's definition, which I believe are slightly different.
I…
Noldorin
- 6,610
178
votes
7 answers
Is there a quick proof as to why the vector space of $\mathbb{R}$ over $\mathbb{Q}$ is infinite-dimensional?
It would seem that one way of proving this would be to show the existence of non-algebraic numbers. Is there a simpler way to show this?
Elchanan Solomon
- 30,807
178
votes
7 answers
Why is Euler's Gamma function the "best" extension of the factorial function to the reals?
There are lots (an infinitude) of smooth functions that coincide with $f(n)=n!$ on the integers. Is there a simple reason why Euler's Gamma function $\Gamma (z) = \int_0^\infty t^{z-1} e^{-t} dt$ is the "best"? In particular, I'm looking for…
pbrooks
- 1,883
177
votes
0 answers
Sorting of prime gaps
Let $g_i$ be the $i^{th}$ prime gap $p_{i+1}-p_i.$
If we rearrange the sequence $ (g_{n,i})_{i=1}^n$ so that for any finite $n$, if the gaps are arranged from smallest to largest, we have a new sequence $(\hat{g}_{n,i})_{i=1}^n.$
For example, for $n…
daniel
- 10,141
177
votes
14 answers
How to prove: if $a,b \in \mathbb N$, then $a^{1/b}$ is an integer or an irrational number?
It is well known that $\sqrt{2}$ is irrational, and by modifying the proof (replacing even with divisible by $3$), one can prove that $\sqrt{3}$ is irrational, as well. On the other hand, clearly $\sqrt{n^2} = n$ for any positive integer $n$. It…
anonymous
177
votes
7 answers
Proof of $\frac{1}{e^{\pi}+1}+\frac{3}{e^{3\pi}+1}+\frac{5}{e^{5\pi}+1}+\ldots=\frac{1}{24}$
I would like to prove that $\displaystyle\sum_{\substack{n=1\\n\text{ odd}}}^{\infty}\frac{n}{e^{n\pi}+1}=\frac1{24}$.
I found a solution by myself 10 hours after I posted it, here it is:
$$f(x)=\sum_{\substack{n=1\\n\text{…
danodare
- 1,915