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1500 questions
62
votes
3 answers
Unexpected approximations which have led to important mathematical discoveries
On a regular basis, one sees at MSE approximate numerology questions like
Prove $\log_{{1}/{4}} \frac{8}{7}> \log_{{1}/{5}} \frac{5}{4}$,
Prove $\left(\dfrac{2}{5}\right)^{{2}/{5}}<\ln{2}$,
Comparing $2013!$ and $1007^{2013}$
or yet the…
Start wearing purple
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62
votes
1 answer
Is OEIS A248049 an integer sequence?
The OEIS sequence A248049 is defined by
$$ a_n \!=\! \frac{(a_{n-1}\!+\!a_{n-2})(a_{n-2}\!+\!a_{n-3})}{a_{n-4}} \;\text{with }\; a_0\!=\!2, a_1\!=\!a_2\!=\!a_3\!=\!1.$$
is apparently an integer sequence but I have no proofs. I have numerical…
Somos
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62
votes
5 answers
When is the closure of an intersection equal to the intersection of closures?
We know $\overline{\bigcap A_{\alpha}}\subseteq\bigcap\overline{A}_{\alpha} $, but when is the reverse inclusion true? Can you give some properties of the underlying space that would guarantee this?
Forever Mozart
- 9,294
62
votes
5 answers
Number of onto functions
What are the number of onto functions from a set $\Bbb A $ containing m elements to a set $\Bbb B$ containing n elements.
I found that if $m = 4$ and $n = 2$ the number of onto functions is $14$.
But is there a way to generalise this using a…
IcyFlame
- 875
62
votes
4 answers
A circle rolls along a parabola
I'm thinking about a circle rolling along a parabola. Would this be a parametric representation?
$(t + A\sin (Bt) , Ct^2 + A\cos (Bt) )$
A gives us the radius of the circle, B changes the frequency of the rotations, C, of course, varies the…
futurebird
- 6,248
62
votes
3 answers
Why is the localization at a prime ideal a local ring?
I would like to know, why $ \mathfrak{p} A_{\mathfrak{p}} $ is the maximal ideal of the local ring $ A_{\mathfrak{p}} $, where $ \mathfrak{p} $ is a prime ideal of $ A $ and $ A_{\mathfrak{p}} $ is the localization of the ring $ A $ with respect to…
Bryan
- 827
62
votes
10 answers
A linear operator commuting with all such operators is a scalar multiple of the identity.
The question is from Axler's "Linear Algebra Done Right", which I'm using for self-study.
We are given a linear operator $T$ over a finite dimensional vector space $V$. We have to show that $T$ is a scalar multiple of the identity iff $\forall S \in…
abeln
- 555
62
votes
5 answers
Can squares of infinite area always cover a unit square?
This is a claim one of my students made without justification on his exam. It definitely wasn't the right way to approach the problem, but now I've been nerdsniped into trying to figure out if it is true.
Let $a_i$ be a sequence of positive reals…
David E Speyer
- 61,724
62
votes
5 answers
Understanding the intuition behind math
I'm currently a Calculus III student. I enjoy math a lot, but only when I understand its beauty and meaning. However, so many times I have no idea what it is I am learning about, althought I am still able to solve problems pertaining to those…
Snowman
- 2,664
62
votes
4 answers
Why can we use induction when studying metamathematics?
In fact I don't understand the meaning of the word "metamathematics". I just want to know, for example, why can we use mathematical induction in the proof of logical theorems, like The Deduction Theorem, or even some more fundamental proposition…
183orbco3
- 1,441
62
votes
0 answers
Determinant of a matrix that contains the first $n^2$ primes.
Let $n$ be an integer and $p_1,\ldots,p_{n^2}$ be the first prime numbers. Writing them down in a matrix
$$
\left(\begin{matrix}
p_1 & p_2 & \cdots & p_n \\
p_{n+1} & p_{n+2} & \cdots & p_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
\cdots & \cdots…
Rofl Ukulus
- 782
62
votes
6 answers
Find three non-constant, pairwise unequal functions $f,g,h:\mathbb R\to \mathbb R$...
I've been stumped by this problem:
Find three non-constant, pairwise unequal functions $f,g,h:\mathbb R\to \mathbb R$ such that
$$f\circ g=h$$
$$g\circ h=f$$
$$h\circ f=g$$
or prove that no three such functions exist.
I highly suspect, by…
Franklin Pezzuti Dyer
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62
votes
6 answers
Create unique number from 2 numbers
is there some way to create unique number from 2 positive integer numbers? Result must be unique even for these pairs: 2 and 30, 1 and 15, 4 and 60. In general, if I take 2 random numbers result must be unique(or with very high probability…
drizzt
- 689
62
votes
4 answers
Intersection of finite number of compact sets is compact?
Is the the intersection of a finite number of compact sets is compact? If not please give a counter example to demonstrate this is not true.
I said that this is true because the intersection of finite number of compact sets are closed. Which…
John Buchta
- 631
62
votes
3 answers
Is there any conjecture that we know is provable/disprovable but we haven't found a proof of yet?
I know that there are a lot of unsolved conjectures, but it could possible for them to be independent of ZFC (see Could it be that Goldbach conjecture is undecidable? for example).
I was wondering if there is some conjecture for which we have proved…
Abc
- 541