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1500 questions
76
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
- 3,112
- 4
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76
votes
18 answers
How to convince a layperson that the $\pi = 4$ proof is wrong?
The infamous "$\pi = 4$" proof was already discussed here:
Is value of $\pi = 4$?
And I have read all the answers, yet I think that they will not be of much help to me if I try to explain this thing to a non mathematician. The main missing point, in…
Gadi A
- 19,265
76
votes
2 answers
Fourier Transform of Derivative
Consider a function $f(t)$ with Fourier Transform $F(s)$. So $$F(s) = \int_{-\infty}^{\infty} e^{-2 \pi i s t} f(t) \ dt$$
What is the Fourier Transform of $f'(t)$? Call it $G(s)$.So $$G(s) = \int_{-\infty}^{\infty} e^{-2 \pi i s t} f'(t) \…
NebulousReveal
- 13,935
76
votes
7 answers
Bag of tricks in Advanced Calculus/ Real Analysis/Complex Analysis
I am studying for an exam and I have been studying my butt off
during the winter break for it. During the course of my study I have written
down quite a number of tricks, which in my opinion were 'outrageous' :-). Meaning there
was no way I would…
Cousin
- 3,525
76
votes
9 answers
Probability of 3 people in a room of 30 having the same birthday
I have been looking at the birthday problem (http://en.wikipedia.org/wiki/Birthday_problem) and I am trying to figure out what the probability of 3 people sharing a birthday in a room of 30 people is. (Instead of 2).
I thought I understood the…
irl_irl
- 863
76
votes
5 answers
Why should I care about adjoint functors
I am comfortable with the definition of adjoint functors. I have done a few exercises proving that certain pairs of functors are adjoint (tensor and hom, sheafification and forgetful, direct image and inverse image of sheaves, spec and global…
DBr
- 4,780
76
votes
4 answers
"Closed" form for $\sum \frac{1}{n^n}$
Earlier today, I was talking with my friend about some "cool" infinite series and the value they converge to like the Basel problem, Madhava-Leibniz formula for $\pi/4, \log 2$ and similar alternating series etc.
One series that popped into our…
user17762
76
votes
5 answers
What exactly is Laplace transform?
I've been working on Laplace transform for a while. I can carry it out on calculation and it's amazingly helpful. But I don't understand what exactly is it and how it works. I google and found out that it gives "less familiar" frequency view.
My…
hasExams
- 2,285
76
votes
5 answers
Do Diagonal Matrices Always Commute?
Let $A$ be an $n \times n$ matrix and let $\Lambda$ be an $n \times n$ diagonal matrix. Is it always the case that $A\Lambda = \Lambda A$? If not, when is it the case that $A \Lambda = \Lambda A$?
If we restrict the diagonal entries of $\Lambda$ to…
Jeff Scott
- 769
76
votes
20 answers
Proof of the hockey stick/Zhu Shijie identity $\sum\limits_{t=0}^n \binom tk = \binom{n+1}{k+1}$
After reading this question, the most popular answer use the identity
$$\sum_{t=0}^n \binom{t}{k} = \binom{n+1}{k+1},$$
or, what is equivalent,
$$\sum_{t=k}^n \binom{t}{k} = \binom{n+1}{k+1}.$$
What's the name of this identity? Is it the identity of…
hlapointe
- 1,610
76
votes
7 answers
Functions that are their own inverse.
What are the functions that are their own inverse?
(thus functions where $ f(f(x)) = x $ for a large domain)
I always thought there were only 4:
$f(x) = x , f(x) = -x , f(x) = \frac {1}{x} $ and $ f(x) = \frac {-1}{x} $
Later I heard about a fifth…
Willemien
- 6,582
76
votes
0 answers
Is there a "ping-pong lemma proof" that $\langle x \mapsto x+1,x \mapsto x^3 \rangle$ is a free group of rank 2?
Let $f,g\colon \mathbb R \to \mathbb R$ be the permutations defined by $f\colon x \mapsto x+1$ and $g\colon x \mapsto x^3$, or maybe even have $g\colon x \mapsto x^p$, $p$ an odd prime.
In the book, by Pierre de la Harpe, Topics in Geometric Group…
user29123
75
votes
7 answers
What is the main difference between a vector space and a field?
In my opinion both are almost same. However there should be some differenes like any two elements can be multiplied in a field but it is not allowed in vector space as only scalar multiplication is allowed where scalars are from the field.
Could…
Vineel Kumar Veludandi
- 1,035
- 1
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- 16
75
votes
3 answers
An Explanation of the Kalman Filter
In the past 3 months, I've been trying to understand the Kalman Filter. I have tried to implement it, I have watched YouTube tutorials, and I have read some papers about it and its operation (update, predicate, etc.). However, I still am unable to…
xsari3x
- 187
75
votes
4 answers
Limit $\frac{x^2y}{x^4+y^2}$ is found using polar coordinates but it is not supposed to exist.
Consider the following 2-variable function:
$$f(x,y) = \frac{x^2y}{x^4+y^2}$$
I would like to find the limit of this function as $(x,y) \rightarrow (0,0)$.
I used polar coordinates instead of solving explicitly in $\mathbb R^2 $, and it went as…
mesllo
- 1,440