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82
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4 answers
Gaussian distribution is isotropic?
I was in a seminar today and the lecturer said that the gaussian distribution is isotropic. What does it mean for a distribution to be isotropic? It seems like he is using this property for the pseudo-independence of vectors where each entry is…
Astaboom
- 1,007
82
votes
2 answers
Why are the last two numbers of this sequence never prime?
I had the idea to make a script that generates a pattern like this: 1 2 3 4 5 6 7 8 9 10 ... and so on. After that, I replaced every non-prime by a '-' character and every prime number by a '|'. The output begins like that: - || -|- |--- |-|--…
Bruno Henrique
- 873
82
votes
7 answers
Does associativity imply commutativity?
I used to think that commutativity and associativity are two distinct properties. But recently, I started thinking of something which has troubled this idea: $$(1+1)+1 = 1+ (1+1)\implies 2+1=1+2$$ Here using associativity of addition operation,…
Aritra Das
- 3,652
82
votes
4 answers
Factorial of a matrix: what could be the use of it?
Recently on this site, the question was raised how we might define the factorial operation $\mathsf{A}!$ on a square matrix $\mathsf{A}$. The answer , perhaps unsurprisingly, involves the Gamma function . What use might it be to take the factorial…
Oliphaunt
- 953
82
votes
9 answers
Do "Parabolic Trigonometric Functions" exist?
The parametric equation $$\begin{align*} x(t) &= \cos t\\ y(t) &= \sin t \end{align*}$$ traces the unit circle centered at the origin ($x^2+y^2=1$). Similarly, $$\begin{align*} x(t) &= \cosh t\\ y(t) &= \sinh t \end{align*}$$ draws the right part of…
Argon
- 26,059
82
votes
5 answers
Use of "without loss of generality"
Why do we use "without loss of generality" when writing proofs? Is it necessary or convention? What "synonym" can be used?
Pedro
- 125,989
82
votes
2 answers
Visual proof of $\sum_{n=1}^\infty \frac{1}{n^4} = \frac{\pi^4}{90}$?
In his gorgeous paper "How to compute $\sum \frac{1}{n^2}$ by solving triangles" , Mikael Passare offers this idea for proving $\sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}$: Proof of equality of square and curved areas is based on another…
VividD
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82
votes
1 answer
Numbers $n$ such that the digit sum of $n^2$ is a square
Let $S(n)$ be the digit sum of $n\in\mathbb N$ in the decimal system. About a month ago, a friend of mine taught me the following: $$S\left(9\color{red}{^2}\right)=S(81)=8+1=3\color{red}{^2}$$…
mathlove
- 155,494
82
votes
9 answers
Besides proving new theorems, how can a person contribute to mathematics?
There are at least a few things a person can do to contribute to the mathematics community without necessarily obtaining novel results, for example: Organizing known results into a coherent narrative in the form of lecture notes or a textbook…
David Zhang
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82
votes
2 answers
Distinguishing probability measure, function and distribution
I have a bit trouble distinguishing the following concepts: probability measure probability function (with special cases probability mass function and probability density function ) probability distribution Are some of these interchangeable? Which…
Marc
- 3,451
82
votes
6 answers
Why is $\omega$ the smallest $\infty$?
I am comfortable with the different sizes of infinities and Cantor's "diagonal argument" to prove that the set of all subsets of an infinite set has cardinality strictly greater than the set itself. So if we have a set $\Omega$ and $|\Omega| =…
user17762
81
votes
7 answers
What is the main difference between a vector space and a field?
In my opinion a vector space and a field are almost the same. However, there should be some differences, such as any two elements can be multiplied in a field but it is not allowed in vector space as only scalar multiplication is allowed where…
VKV
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81
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4 answers
Are these solutions of $2 = x^{x^{x^{\:\cdot^{\:\cdot^{\:\cdot}}}}}$ correct?
Find $x$ in $$ \Large 2 = x^{x^{x^{\:\cdot^{\:\cdot^{\:\cdot}}}}}$$ A trick to solve this is to see that $$\large 2 = x^{x^{x^{\:\cdot^{\:\cdot^{\:\cdot}}}}} \quad\implies\quad 2 = x^{\Big(x^{x^{x^{\:\cdot^{\:\cdot^{\:\cdot}}}}}\Big)} = x^2…
GarouDan
- 3,488
81
votes
1 answer
The closed form of $\int_0^{\pi/4}\frac{\log(1-x) \tan^2(x)}{1-x\tan^2(x)} \ dx$
What tools or ways would you propose for getting the closed form of this integral? $$\int_0^{\pi/4}\frac{\log(1-x) \tan^2(x)}{1-x\tan^2(x)} \ dx$$ EDIT : It took a while since I made this post. I'll give a little bounty for the solver of the…
user 1591719
- 45,325
81
votes
5 answers
Difference between axioms, theorems, postulates, corollaries, and hypotheses
I've heard all these terms thrown about in proofs and in geometry, but what are the differences and relationships between them? Examples would be awesome! :)
Gordon Gustafson
- 3,545