Most Popular
1500 questions
69
votes
6 answers
Chance of meeting in a bar
Two people have to spend exactly 15 consecutive minutes in a bar on a given day, between 12:00 and 13:00. Assuming uniform arrival times, what is the probability they will meet?
I am mainly interested to see how people would model this formally. I…
Beltrame
- 3,186
68
votes
4 answers
Why we consider log likelihood instead of Likelihood in Gaussian Distribution
I am reading Gaussian Distribution from a machine learning book. It states that -
We shall determine values for the unknown parameters $\mu$ and
$\sigma^2$ in the Gaussian by maximizing the likelihood function. In practice, it is more convenient…
Kaidul Islam
- 783
- 1
- 6
- 7
68
votes
8 answers
Why do people lose in chess?
Zermelo's Theorem, when applied to chess, states:
"either white can force a win, or black can force a win, or both sides can force at least a draw [1]"
I do not get this. How can it be proven?
And why do people lose in chess then, if they can…
Brika
- 1,127
- 1
- 8
- 12
68
votes
9 answers
When to give up on a hard math problem?
I practice olympiad problems from books like Putnam and Beyond. Often I come across a problem that I simply can't solve. After $\sim30$ minutes of deep thinking it feels like I'm ramming my head into a brick wall, since I've exhausted all avenues of…
user1299784
- 2,079
68
votes
3 answers
Definitions of Hessian in Riemannian Geometry
I am wondering if there is any quick way to see the following two definitions of Hessian coincide with each other without using local coordinates.
$\operatorname{Hess}(f)(X,Y)= \langle \nabla_X \operatorname{grad}f,Y \ \rangle$;…
user17150
- 1,055
68
votes
6 answers
Can someone please explain the Riemann Hypothesis to me... in English?
I've read so much about it but none of it makes a lot of sense. Also, what's so unsolvable about it?
Letseatlunch
- 839
68
votes
13 answers
Sum of the alternating harmonic series $\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k} = \frac{1}{1} - \frac{1}{2} + \cdots $
I know that the harmonic series $$\sum_{k=1}^{\infty}\frac{1}{k} = \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \cdots + \frac{1}{n} + \cdots \tag{I}$$ diverges, but what about the alternating harmonic series…
Isaac
- 37,057
68
votes
3 answers
Why is $\text{Hom}(V,W)$ the same thing as $V^* \otimes W$?
I have a couple of questions about tensor products:
Why is $\text{Hom}(V,W)$ the same thing as $V^* \otimes W$?
Why is an element of $V^{*\otimes m}\otimes V^{\otimes n}$ the same thing as a multilinear map $V^m \to V^{\otimes n}$?
What is the…
Eric Auld
- 28,997
68
votes
9 answers
Do all square matrices have eigenvectors?
I came across a video lecture in which the professor stated that there may or may not be any eigenvectors for a given linear transformation.
But I had previously thought every square matrix has eigenvectors.
sash
- 823
68
votes
2 answers
Category-theoretic limit related to topological limit?
Is there any connection between category-theoretic term 'limit' (=universal cone) over diagram, and topological term 'limit point' of a sequence, function, net...?
To be more precise, is there a category-theoretic setting of some non-trivial…
Rafael Mrden
- 2,279
68
votes
7 answers
$\sqrt{7\sqrt{7\sqrt{7\sqrt{7\sqrt{7\cdots}}}}}$ approximation
Is there any trick to evaluate this or this is an approximation, I mean I am not allowed to use calculator.
$$\sqrt{7\sqrt{7\sqrt{7\sqrt{7\sqrt{7\cdots}}}}}$$
user2378
- 1,071
- 1
- 8
- 9
68
votes
4 answers
Integral $\int_{-1}^{1} \frac{1}{x}\sqrt{\frac{1+x}{1-x}} \log \left( \frac{(r-1)x^{2} + sx + 1}{(r-1)x^{2} - sx + 1} \right) \, \mathrm dx$
Regarding this problem, I conjectured that
$$ I(r, s) = \int_{-1}^{1} \frac{1}{x}\sqrt{\frac{1+x}{1-x}} \log \left( \frac{(r-1)x^{2} + sx + 1}{(r-1)x^{2} - sx + 1} \right) \, \mathrm dx = 4 \pi \operatorname{arccot} \sqrt{ \frac{2r + 2\sqrt{r^{2} -…
Sangchul Lee
- 181,930
68
votes
9 answers
Refuting the Anti-Cantor Cranks
I occasionally have the opportunity to argue with anti-Cantor cranks, people who for some reason or the other attack the validity of Cantor's diagonalization proof of the uncountability of the real numbers, arguably one of the most beautiful ideas…
Keshav Srinivasan
- 10,804
68
votes
1 answer
Formal definition of conditional probability
It would be extremely helpful if anyone gives me the formal definition of conditional probability and expectation in the following setting, given probability space
$ (\Omega, \mathscr{A}, \mu ) $ with $\mu(\Omega) = 1 $, and a random variable $ X :…
smiley06
- 4,327
68
votes
3 answers
What is Haar Measure?
Is there any simple explanation for Haar Measure and its geometry?
How do we understand the analogy between Lebesgue measure and Haar Measure?
How to show integration with respect to Haar Measure?
What do we mean by integrating with respect to…
Milan Amrut Joshi
- 1,205