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1500 questions
64
votes
5 answers
If $e^A$ and $e^B$ commute, do $A$ and $B$ commute for finite dimensional matrices?
It is known that if two matrices $A,B \in M_n(\mathbb{C})$ commute, then $e^A$ and $e^B$ commute. Is the converse true?
If $e^A$ and $e^B$ commute, do $A$ and $B$ commute?
Edit: Additionally, what happens in $M_n(\mathbb{R})$?
Nota Bene: As a…
Seirios
- 33,157
64
votes
15 answers
Why can't you flatten a sphere?
It's a well-known fact that you can't flatten a sphere without tearing or deforming it. How can I explain why this is so to a 10 year old?
As soon as an explanation starts using terms like "Gaussian curvature", it's going too far for the audience…
Joe
- 1,253
- 2
- 10
- 21
64
votes
8 answers
How does a non-mathematician go about publishing a proof in a way that ensures it to be up to the mathematical community's standards?
I'm a computer science student who is a maths hobbyist. I'm convinced that I've proven a major conjecture. The problem lies in that I've never published anything before and am not a mathematician by profession. Knowing full well that my proof may be…
Balbanna
- 683
64
votes
1 answer
Does Fermat's Last Theorem hold for cyclotomic integers in $\mathbb{Q(\zeta_{37})}$?
The first irregular prime is 37. Does FLT(37)
$$x^{37} + y^{37} = z^{37}$$
have any solutions in the ring of integers of $\mathbb Q(\zeta_{37})$, where $\zeta_{37}$ is a primitive 37th root of unity?
Maybe it's not true, but how could I go about…
quanta
- 12,425
64
votes
1 answer
How to find a total order with constrained comparisons
There are $25$ horses with different speeds. My goal is to rank all of them, by using only runs with $5$ horses, and taking partial rankings. How many runs do I need, at minimum, to complete my task?
As a partial answer, I know that is possible to…
Jack D'Aurizio
- 353,855
64
votes
2 answers
What is the integral of 1/x?
What is the integral of $\frac{1}{x}$? Do you get $\ln(x)$ or $\ln|x|$?
In general, does integrating $f'(x)/f(x)$ give $\ln(f(x))$ or $\ln|f(x)|$?
Also, what is the derivative of $|f(x)|$? Is it $f'(x)$ or $|f'(x)|$?
hollow7
- 2,455
64
votes
7 answers
What's the point of studying topological (as opposed to smooth, PL, or PDiff) manifolds?
Part of the reason I think algebraic topology has acquired something of a fearsome reputation is that the terrible properties of the topological category (e.g. the existence of space-filling curves) force us to work very hard to prove the main…
Qiaochu Yuan
- 419,620
64
votes
9 answers
Linear Algebra Versus Functional Analysis
As it is mentioned in the answer by Sheldon Axler in this post, we usually restrict linear algebra to finite dimensional linear spaces and study the infinite dimensional ones in functional analysis.
I am wondering that what are those parts of the…
Hosein Rahnama
- 14,882
64
votes
12 answers
If a coin toss is observed to come up as heads many times, does that affect the probability of the next toss?
A two-sided coin has just been minted with two different sides (heads and tails). It has never been flipped before. Basic understanding of probability suggests that the probability of flipping heads is .5 and tails is .5. Unexpectedly, you flip the…
skurwa
- 777
64
votes
3 answers
Motivation and methods for self-study
First, a little background: Beginning with calculus in my first semester of college, I fell in love with mathematics. That was the point at which the concepts became interesting to me, and I started reading up, through Wikipedia and various other…
Alex Petzke
- 8,763
64
votes
12 answers
Easy way of memorizing values of sine, cosine, and tangent
My math professor recently told us that she wanted us to be able to answer $\sin\left(\frac{\pi }{2}\right)$ in our head on the snap. I know I can simply memorize the table for the test by this Friday, but I may likely forget them after the test. So…
James Smith
- 1,733
64
votes
2 answers
How do different definitions of "degree" coincide?
I've recently read about a number of different notions of "degree." Reading over Javier Álvarez' excellent answer for the thousandth time finally prompted me to ask this question:
How exactly do the following three notions of "degree"…
Jesse Madnick
- 31,524
64
votes
13 answers
I have learned that 1/0 is infinity, why isn't it minus infinity?
My brother was teaching me the basics of mathematics and we had some confusion about the positive and negative behavior of Zero. After reading a few post on this we came to know that it depends on the context of its use.
Why do we take 1/0 as…
Rishav Mehta
- 761
64
votes
17 answers
Is the Law of Large Numbers empirically proven?
Does this reflect the real world and what is the empirical evidence behind this?
Layman here so please avoid abstract math in your response.
The Law of Large Numbers states that the average of the results from multiple trials will tend to converge…
vantage5353
- 767
63
votes
2 answers
Computation with a memory wiped computer
Here is another result from Scott Aaronson's blog:
If every second or so your computer’s
memory were wiped completely clean,
except for the input data; the clock;
a static, unchanging program; and a
counter that could only be set to 1,
2,…
Casebash
- 9,211