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1500 questions
78
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 is:…
ShungChing
- 891
78
votes
3 answers
An explanation of the Kalman filter
In the past 3 months, I've been trying to understand the Kalman filter. I have tried to implement it, I have watched YouTube tutorials, and I have read some papers about it and its operation (update, predicate, etc.). However, I still am unable to…
KnowledgeDude
- 217
78
votes
5 answers
How to prove that a compact set in a Hausdorff topological space is closed?
How to prove that a compact set $K$ in a Hausdorff topological space $\mathbb{X}$ is closed? I seek a proof that is as self contained as possible. Thank you.
Elias Costa
- 15,400
78
votes
6 answers
Why can't you pick socks using coin flips?
I'm teaching myself axiomatic set theory and I'm having some trouble getting my head around the axiom of choice . I (think I) understand what the axiom says, but I don't get why it is so 'contentious', which probably means I haven't yet digested it…
MGA
- 9,869
78
votes
32 answers
Interesting and unexpected applications of $\pi$
$\text{What are some interesting cases of $\pi$ appearing in situations that do not seem geometric?}$ Ever since I saw the identity $$\displaystyle \sum_{n = 1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}$$ and the generalization of $\zeta (2k)$, my…
MT_
- 20,051
- 10
- 45
- 88
78
votes
4 answers
Integral $\int_1^\infty\operatorname{arccot}\left(1+\frac{2\pi}{\operatorname{arcoth}x-\operatorname{arccsc}x}\right)\frac{\mathrm dx}{\sqrt{x^2-1}}$
Consider the following integral: $$\mathcal{I}=\int_1^\infty\operatorname{arccot}\left(1+\frac{2\,\pi}{\operatorname{arcoth}x\,-\,\operatorname{arccsc}x}\right)\frac{\mathrm dx}{\sqrt{x^2-1}}\,,$$ where $\operatorname{arccsc}$ is the inverse…
Vladimir Reshetnikov
- 32,880
78
votes
2 answers
Integration of forms and integration on a measure space
In Terence Tao's PCM article: DIFFERENTIAL FORMS AND INTEGRATION , it is pointed out that there are three concepts of integration which appear in the subject (single-variable calculus): the indefinite integral $\int f$ (also known as the…
user9464
78
votes
10 answers
Why is the Traveling Salesperson Problem "Difficult"?
The Traveling Salesperson Problem is originally a mathematics/computer science optimization problem in which the goal is to determine a path to take between a group of cities such that you return to the starting city after visiting each city exactly…
stats_noob
- 4,375
78
votes
4 answers
Laplace, Legendre, Fourier, Hankel, Mellin, Hilbert, Borel, Z...: unified treatment of transforms?
I understand "transform methods" as recipes, but beyond this they are a big mystery to me. There are two aspects of them I find bewildering. One is the sheer number of them. Is there a unified framework that includes all these transforms as special…
kjo
- 14,954
78
votes
2 answers
Wild automorphisms of the complex numbers
I read about so called "wild" automorphisms of the field of complex numbers (i.e. not the identity nor the complex conjugation). I suppose they must be rather weird and I wonder whether someone could explain in the simplest possible way (please) how…
Gerard
- 1,651
78
votes
16 answers
Dot Product Intuition
I'm searching to develop the intuition (rather than memorization) in relating the two forms of a dot product (by an angle theta between the vectors and by the components of the vector ). For example, suppose I have vector $\mathbf{a} = (a_1,a_2)$…
nerdy
- 3,504
78
votes
14 answers
What are some mathematical topics that involve adding and multiplying pictures?
Let me give you an example of what I mean. Flag algebras are a tool used in extremal graph theory which involve writing inequalities that look like: (It's not too important to my question what this inequality means, but let me give you some context.…
Misha Lavrov
- 162,248
78
votes
5 answers
How are mathematicians taught to write with such an expository style?
I wasn't sure if this question was appropriate for MSE. One of the major complaints we see in industry is a person's ability to communicate which includes writing. We see the same thing on questions that are posted on MSE by younger students.…
Amzoti
- 56,689
78
votes
3 answers
Prove: If a sequence converges, then every subsequence converges to the same limit.
I need some help understanding this proof: Prove: If a sequence converges, then every subsequence converges to the same limit. Proof: Let $s_{n_k}$ denote a subsequence of $s_n$. Note that $n_k \geq k$ for all $k$. This easy to prove by induction:…
CodeKingPlusPlus
- 7,598
78
votes
1 answer
Overview of basic results on cardinal arithmetic
Are there some good overviews of basic formulas about addition, multiplication and exponentiation of cardinals (preferably available online)?
Martin Sleziak
- 56,568