Mathematics

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1500 questions
248
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
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246
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
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245
votes
1 answer

Is this really a categorical approach to integration?

Here 's an article by Reinhard Börger I found recently whose title and content, prima facie , seem quite exciting to me, given my misadventures lately (like this and this ); it's called, " A Categorical Approach to Integration ". The Abstract: "We…
Shaun
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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
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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…
Tomek Tarczynski
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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
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238
votes
8 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
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238
votes
2 answers

Does the open mapping theorem imply the Baire category theorem?

A nice observation by C.E. Blair 1, 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…
t.b.
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234
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
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233
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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
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232
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
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231
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
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231
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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 are…
AlanSTACK
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231
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, then: $$ P(A)=\begin{bmatrix}…
ASKASK
  • 9,281
231
votes
10 answers

What does $2^x$ really mean when $x$ is not an integer?

We all know that $2^5$ means $2\times 2\times 2\times 2\times 2 = 32$, but what does $2^\pi$ mean? How is it possible to calculate that without using a calculator? I am really curious about this, so please let me know what you think.
David G
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