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1500 questions
245
votes
35 answers
What are some counter-intuitive results in mathematics that involve only finite objects?
There are many counter-intuitive results in mathematics, some of which are listed here. However, most of these theorems involve infinite objects and one can argue that the reason these results seem counter-intuitive is our intuition not working…
Burak
- 3,836
245
votes
10 answers
Teaching myself differential topology and differential geometry
I have a hazy notion of some stuff in differential geometry and a better, but still not quite rigorous understanding of basics of differential topology.
I have decided to fix this lacuna once for all. Unfortunately I cannot attend a course right…
Harddaysknight
- 2,451
244
votes
3 answers
When can a sum and integral be interchanged?
Let's say I have $\int_{0}^{\infty}\sum_{n = 0}^{\infty} f_{n}(x)\, dx$ with $f_{n}(x)$ being continuous functions. When can we interchange the integral and summation? Is $f_{n}(x) \geq 0$ for all $x$ and for all $n$ sufficient? How about when $\sum…
user192837
- 2,451
242
votes
11 answers
What is the result of $\infty - \infty$?
I would say $\infty - \infty=0$ because even though $\infty$ is an undetermined number, $\infty = \infty$. So $\infty-\infty=0$.
Pacerier
- 3,459
241
votes
13 answers
Inverse of the sum of matrices
I have two square matrices: $A$ and $B$. $A^{-1}$ is known and I want to calculate $(A+B)^{-1}$. Are there theorems that help with calculating the inverse of the sum of matrices? In general case $B^{-1}$ is not known, but if it is necessary then it…
Tomek Tarczynski
- 3,194
240
votes
6 answers
When can you switch the order of limits?
Suppose you have a double sequence $\displaystyle a_{nm}$. What are sufficient conditions for you to be able to say that $\displaystyle \lim_{n\to \infty}\,\lim_{m\to \infty}{a_{nm}} = \lim_{m\to \infty}\,\lim_{n\to \infty}{a_{nm}}$? Bonus points…
asmeurer
- 9,982
236
votes
7 answers
$L^p$ and $L^q$ space inclusion
Let $(X, \mathcal B, m)$ be a measure space. For $1 \leq p < q \leq \infty$, under what condition is it true that $L^q(X, \mathcal B, m) \subset L^p(X, \mathcal B, m)$ and what is a counterexample in the case the condition is not satisfied?
Confused
- 2,369
236
votes
2 answers
Does the open mapping theorem imply the Baire category theorem?
A nice observation by C.E. Blair1, 2, 3 shows that the Baire category theorem for complete metric spaces is equivalent to the axiom of (countable) dependent choice.
On the other hand, the three classical consequences of the Baire category theorem in…
t.b.
- 80,986
232
votes
11 answers
How do I sell out with abstract algebra?
My plan as an undergraduate was unequivocally to be a pure mathematician, working as an algebraist as a bigshot professor at a bigshot university. I'm graduating this month, and I didn't get into where I expected to get into. My letters were great…
Samuel Handwich
- 2,831
231
votes
13 answers
How can a piece of A4 paper be folded in exactly three equal parts?
This is something that always annoys me when putting an A4 letter in a oblong envelope: one has to estimate where to put the creases when folding the letter. I normally start from the bottom and on eye estimate where to fold. Then I turn the letter…
Nicky Hekster
- 52,147
231
votes
24 answers
How to check if a point is inside a rectangle?
There is a point $(x,y)$, and a rectangle $a(x_1,y_1),b(x_2,y_2),c(x_3,y_3),d(x_4,y_4)$, how can one check if the point inside the rectangle?
Freewind
- 2,535
231
votes
11 answers
In simple English, what does it mean to be transcendental in math?
From Wikipedia, we have the following definitions:
A transcendental number is a real or complex number that is not algebraic
A transcendental function is an analytic function that does not satisfy a polynomial equation
However these definitions…
AlanSTACK
- 4,185
229
votes
2 answers
Is there a 0-1 law for the theory of groups?
For each first order sentence $\phi$ in the language of groups, define :
$$p_N(\phi)=\frac{\text{number of nonisomorphic groups $G$ of order} \le N\text{ such that } \phi \text{ is valid in } G}{\text{number of nonisomorphic groups of order} \le…
Dominik
- 14,660
229
votes
4 answers
How do I convince someone that $1+1=2$ may not necessarily be true?
Me and my friend were arguing over this "fact" that we all know and hold dear. However, I do know that $1+1=2$ is an axiom. That is why I beg to differ. Neither of us have the required mathematical knowledge to convince each other.
And that is why,…
Aces12345
- 2,309
229
votes
6 answers
Why does this matrix give the derivative of a function?
I happened to stumble upon the following matrix:
$$ A = \begin{bmatrix}
a & 1 \\
0 & a
\end{bmatrix}
$$
And after trying a bunch of different examples, I noticed the following remarkable pattern. If $P$ is a polynomial,…
ASKASK
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