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1500 questions
70
votes
3 answers
What does "isomorphic" mean in linear algebra?
My professor keeps mentioning the word "isomorphic" in class, but has yet to define it... I've asked him and his response is that something that is isomorphic to something else means that they have the same vector structure. I'm not sure what that…
Sujaan Kunalan
- 11,217
70
votes
4 answers
How much does symbolic integration mean to mathematics?
(Before reading, I apologize for my poor English ability.) I have enjoyed calculating some symbolic integrals as a hobby, and this has been one of the main source of my interest towards the vast world of mathematics. For instance, the integral below…
Sangchul Lee
- 186,861
70
votes
5 answers
What does it mean to integrate with respect to the distribution function?
If $f(x)$ is a density function and $F(x)$ is a distribution function of a random variable $X$ then I understand that the expectation of x is often written as: $$E(X) = \int x f(x) dx$$ where the bounds of integration are implicitly $-\infty$ and…
Jeromy Anglim
- 937
70
votes
6 answers
Is there a reason it is so rare we can solve differential equations?
Speaking about ALL differential equations, it is extremely rare to find analytical solutions. Further, simple differential equations made of basic functions usually tend to have ludicrously complicated solutions or be unsolvable. Is there some…
novawarrior77
- 860
70
votes
9 answers
How to effectively study math?
Maybe this is too general for here, but I am having a lot of difficulty studying math. Just got out of the military and I guess I am not use to this yet but when I run into a problem I have trouble with and I just can't get it I get extremely…
Adam
- 1,385
70
votes
1 answer
Is $a_0=2$, $a_1=a_2=a_3=1$, $a_n=\frac{(a_{n-1}+a_{n-2})(a_{n-2}+a_{n-3})}{a_{n-4}}$ (OEIS A248049) an integer sequence?
The OEIS sequence A248049 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 evidence…
Somos
- 37,748
70
votes
6 answers
Japanese Temple Problem From 1844
I recently learnt a Japanese geometry temple problem. The problem is the following: Five squares are arranged as the image shows. Prove that the area of triangle T and the area of square S are equal. This is problem 6 in this article. I am thinking…
Larry
- 5,180
70
votes
8 answers
Entire one-to-one functions are linear
Can we prove that every entire one-to-one function is linear?
Petey
- 701
70
votes
4 answers
What is a differential form?
can someone please informally (but intuitively) explain what "differential form" mean? I know that there is (of course) some formalism behind it - definition and possible operations with differential forms, but what is the motivation of introducing…
Check drummer
- 823
70
votes
2 answers
Explain this mathematical meme (Geometers bird interrupting Topologists bird)
My knowledge of geometry is just a little bit above high school level and I know absolutely nothing about topology. So, what is the point of this meme? (Original unedited webcomic: “Juncrow” by False Knees )
Hanlon
- 1,899
70
votes
4 answers
Why does multiplying a number on a clock face by 10 and then halving, give the minutes? ${}{}$
My daughter in grade 3 is learning about telling time at her school. She eagerly showed me this method she has discovered on her own to tell the minutes part of the time on an analogue clock. I wasn't sure at first because I have never heard about…
SMa
- 657
70
votes
1 answer
Simplicial homology of real projective space by Mayer-Vietoris
Consider the $n$-sphere $S^n$ and the real projective space $\mathbb{RP}^n$. There is a universal covering map $p: S^n \to \mathbb{RP}^n$, and it's clear that it's the coequaliser of $\mathrm{id}: S^n \to S^n$ and the antipodal map $a: S^n \to S^n$.…
Zhen Lin
- 98,824
70
votes
29 answers
Understandable questions which are hard for non-mathematicians but easy for mathematicians
A friend of mine has set me the challenge of finding an example of the following: Is there a question, that everyone (both mathematicians and non-mathematicians) can understand, that most mathematicians would answer correctly, instantly, but that…
Zestylemonzi
- 4,213
70
votes
3 answers
Is $n \sin n$ dense on the real line?
Is $\{n \sin n | n \in \mathbb{N}\}$ dense on the real line? If so, is $\{n^p \sin n | n \in \mathbb{N}\}$ dense for all $p>0$? This seems much harder than showing that $\sin n$ is dense on [-1,1], which is easy to show. EDIT: This seems a bit…
Per Alexandersson
- 3,696
70
votes
1 answer
Are there infinitely many "super-palindromes"?
Let me first explain what I call a "super-palindrome": Consider the number $99999999$. That number is obviously a palindrome. ${}{}{}{}$ The largest prime factor of $99999999$ is $137$. If you divide $99999999$ by $137$, you get $729927$. This…
celtschk
- 44,737