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1500 questions
189
votes
9 answers
There are apparently $3072$ ways to draw this flower. But why?
This picture was in my friend's math book: Below the picture it says: There are $3072$ ways to draw this flower, starting from the center of the petals, without lifting the pen. I know it's based on combinatorics, but I don't know how to show that…
user265554
- 2,943
189
votes
21 answers
How do people perform mental arithmetic for complicated expressions? (In particular, $\frac{10^2+11^2+12^2+13^2+14^2}{365}$)
This is the famous picture " Mental Arithmetic. In the Public School of S. Rachinsky. " by the Russian artist Nikolay Bogdanov-Belsky . The problem on the blackboard is: $$ \dfrac{10^{2} + 11^{2} + 12^{2} + 13^{2} + 14^{2}}{365} $$ The answer is…
Vlad
- 6,956
189
votes
12 answers
The median minimizes the sum of absolute deviations (the $ {\ell}_{1} $ norm)
Suppose we have a set $S$ of real numbers. Show that $$\sum_{s\in S}|s-x| $$ is minimal if $x$ is equal to the median. This is a sample exam question of one of the exams that I need to take and I don't know how to proceed.
hattenn
- 2,297
188
votes
7 answers
Is there a quick proof as to why the vector space of $\mathbb{R}$ over $\mathbb{Q}$ is infinite-dimensional?
It would seem that one way of proving this would be to show the existence of non-algebraic numbers. Is there a simpler way to show this?
Elchanan Solomon
- 32,154
188
votes
0 answers
Does every ring of integers sit inside a ring of integers that has a power basis?
Given a finite extension of the rationals, $K$ , we know that $K=\mathbb{Q}[\alpha]$ by the primitive element theorem, so every $x \in K$ has the form $$x = a_0 + a_1 \alpha + \cdots + a_n \alpha^n,$$ with $a_i \in \mathbb{Q}$ . However, the ring of…
Eins Null
- 2,307
187
votes
6 answers
What are the differences between rings, groups, and fields?
Rings, groups, and fields all feel similar. What are the differences between them, both in definition and in how they are used?
cobbal
- 2,165
187
votes
2 answers
Can we ascertain that there exists an epimorphism $G\rightarrow H$?
Let $G,H$ be finite groups. Suppose we have an epimorphism $$G\times G\rightarrow H\times H$$ Can we find an epimorphism $G\rightarrow H$?
Kerry
- 1,879
187
votes
8 answers
Why is Euler's Gamma function the "best" extension of the factorial function to the reals?
There are lots (an infinitude) of smooth functions that coincide with $f(n)=n!$ on the integers. Is there a simple reason why Euler's Gamma function $\Gamma (z) = \int_0^\infty t^{z-1} e^{-t} dt$ is the "best"? In particular, I'm looking for reasons…
pbrooks
- 1,973
187
votes
6 answers
An Introduction to Tensors
As a physics student, I've come across mathematical objects called tensors in several different contexts. Perhaps confusingly, I've also been given both the mathematician's and physicist's definition, which I believe are slightly different. I…
Noldorin
- 6,818
186
votes
6 answers
Symmetry of function defined by integral
Define a function $f(\alpha, \beta)$, $\alpha \in (-1,1)$, $\beta \in (-1,1)$ as $$ f(\alpha, \beta) = \int_0^{\infty} dx \: \frac{x^{\alpha}}{1+2 x \cos{(\pi \beta)} + x^2}$$ One can use, for example, the Residue Theorem to show that $$ f(\alpha,…
Ron Gordon
- 142,110
186
votes
15 answers
Why is gradient the direction of steepest ascent?
$$f(x_1,x_2,\dots, x_n):\mathbb{R}^n \to \mathbb{R}$$ The definition of the gradient is $$ \frac{\partial f}{\partial x_1}\hat{e}_1 +\ \cdots +\frac{\partial f}{\partial x_n}\hat{e}_n$$ which is a vector. Reading this definition makes me consider…
Jing
- 2,567
185
votes
18 answers
How to prove Euler's formula: $e^{it}=\cos t +i\sin t$?
Could you provide a proof of Euler's formula: $e^{it}=\cos t +i\sin t$ ?
Jichao
- 8,270
185
votes
6 answers
What were some major mathematical breakthroughs in 2016?
As the year is slowly coming to an end, I was wondering which great advances have there been in mathematics in the past 12 months. As researchers usually work in only a limited number of fields in mathematics, one often does not hear a lot of news…
YukiJ
- 2,579
185
votes
53 answers
Is there any integral for the Golden Ratio?
I was wondering about important/famous mathematical constants, like $e$ , $\pi$ , $\gamma$ , and obviously the golden ratio $\phi$ . The first three ones are really well known, and there are lots of integrals and series whose results are simply…
user266764
184
votes
4 answers
Evaluate $\int_0^1 \frac{\log \left( 1+x^{2+\sqrt{3}}\right)}{1+x}\mathrm dx$
I am trying to find a closed form for $$\int_0^1 \frac{\log \left( 1+x^{2+\sqrt{3}}\right)}{1+x}\mathrm dx = 0.094561677526995723016 \cdots$$ It seems that the answer is $$\frac{\pi^2}{12}\left( 1-\sqrt{3}\right)+\log(2) \log \left(1+\sqrt{3}…
Shobhit Bhatnagar
- 11,906