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1500 questions
69
votes
4 answers
Integrals of the form ${\large\int}_0^\infty\operatorname{arccot}(x)\cdot\operatorname{arccot}(a\,x)\cdot\operatorname{arccot}(b\,x)\ dx$
I'm interested in integrals of the form
$$I(a,b)=\int_0^\infty\operatorname{arccot}(x)\cdot\operatorname{arccot}(a\,x)\cdot\operatorname{arccot}(b\,x)\ dx,\color{#808080}{\text{ for }a>0,\,b>0}\tag1$$
It's…
Vladimir Reshetnikov
- 47,122
69
votes
20 answers
What are some applications of elementary linear algebra outside of math?
I'm TAing linear algebra next quarter, and it strikes me that I only know one example of an application I can present to my students. I'm looking for applications of elementary linear algebra outside of mathematics that I might talk about in…
user98602
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
- 3,086
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
68
votes
2 answers
Why absolute values of Jacobians in change of variables for multiple integrals but not single integrals?
If $g:[a,b]\to\mathbf R$ is a change of 1D coordinates, then the formula is:
$$ \int_{g(a)}^{g(b)}\,f(x)\,dx = \int_a^b\,f(g(t))\frac{dx}{dt}\,dt.
\qquad\text{(1)}$$
If $T=\{x=f(u,v); y=g(u,v)\}$ is a change of 2D coordinates, then the formula…
ShungChing
- 791
68
votes
8 answers
Subgroup of index $2$ is Normal
Please excuse the selfishness of the following question:
Let $G$ be a group and $H \le G$ such that $|G:H|=2$. Show that $H$ is normal.
Proof:
Because $|G:H|=2$, $G = H \cup aH$ for some $a \in G \setminus H$.
Let $x\in G$. Then $x \in H$ or $x…
William T.
- 737
68
votes
2 answers
Does non-symmetric positive definite matrix have positive eigenvalues?
I found out that there exist positive definite matrices that are non-symmetric, and I know that symmetric positive definite matrices have positive eigenvalues.
Does this hold for non-symmetric matrices as well?
user19653
68
votes
5 answers
Understanding the Laplace operator conceptually
The Laplace operator: those of you who now understand it, how would you explain what it "does" conceptually? How do you wish you had been taught it?
Any good essays (combining both history and conceptual understanding) on the Laplace operator, and…
bzm3r
- 2,632
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
16 answers
Proving $1^3+ 2^3 + \cdots + n^3 = \left(\frac{n(n+1)}{2}\right)^2$ using induction
How can I prove that
$$1^3+ 2^3 + \cdots + n^3 = \left(\frac{n(n+1)}{2}\right)^2$$
for all $n \in \mathbb{N}$? I am looking for a proof using mathematical induction.
Thanks
Steve
- 419
68
votes
6 answers
Similar matrices and field extensions
Given a field $F$ and a subfield $K$ of $F$. Let $A$, $B$ be $n\times n$ matrices such that all the entries of $A$ and $B$ are in $K$. Is it true that if $A$ is similar to $B$ in $F^{n\times n}$ then they are similar in $K^{n\times n}$?
Any help…
Melesia
- 681
68
votes
3 answers
Evaluating the log gamma integral $\int_{0}^{z} \log \Gamma (x) \, \mathrm dx$ in terms of the Hurwitz zeta function
One way to evaluate $ \displaystyle\int_{0}^{z} \log \Gamma(x) \, \mathrm dx $ is in terms of the Barnes G-function.
$$ \int_{0}^{z} \log \Gamma(x) \, \mathrm dx = \frac{z}{2} \log (2 \pi) + \frac{z(1-z)}{2} + z \log \Gamma(z) - \log…
Random Variable
- 42,026
68
votes
17 answers
How can you prove that the square root of two is irrational?
I have read a few proofs that $\sqrt{2}$ is irrational.
I have never, however, been able to really grasp what they were talking about.
Is there a simplified proof that $\sqrt{2}$ is irrational?
John Gietzen
- 3,501
68
votes
7 answers
Geometric understanding of differential forms.
I would like to understand differential forms more intuitively. I have yet to find a book which explains how the use of the exterior product in differential forms ties into the geometrical significance of it. Most books briefly introduce the…
Markus
- 683
68
votes
2 answers
Is this similarity to the Fourier transform of the von Mangoldt function real?
Mathematica knows that the logarithm of $n$ is:
$$\log(n) = \lim\limits_{s \rightarrow 1} \zeta(s)\left(1 - \frac{1}{n^{(s - 1)}}\right)$$
The von Mangoldt function should then be:
$$\Lambda(n)=\lim\limits_{s \rightarrow 1} \zeta(s)\sum\limits_{d|n}…
Mats Granvik
- 7,396