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1500 questions
263
votes
11 answers
"Integral milking": working backward to construct nontrivial integrals
I begin this post with a plea: please don't be too harsh with this post for being off topic or vague. It's a question about something I find myself doing as a mathematician, and wonder whether others do it as well. It is a soft question about…
Franklin Pezzuti Dyer
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262
votes
6 answers
What books must every math undergraduate read?
I'm still a student, but the same books keep getting named by my tutors (Rudin, Royden).
I've read Baby Rudin and begun Royden though I'm unsure if there are other books that I "should" be working on if I want to study beyond Masters. I'm not there…
Adam
- 2,058
259
votes
16 answers
Is there an elementary proof that $\sum \limits_{k=1}^n \frac1k$ is never an integer?
If $n>1$ is an integer, then $\sum \limits_{k=1}^n \frac1k$ is not an integer.
If you know Bertrand's Postulate, then you know there must be a prime $p$ between $n/2$ and $n$, so $\frac 1p$ appears in the sum, but $\frac{1}{2p}$ does not. Aside from…
Anton Geraschenko
- 4,870
258
votes
3 answers
Why study Algebraic Geometry?
I'm going to start self-stydying algebraic geometry very soon. So, my question is why do mathematicians study algebraic geometry? What are the types of problems in which algebraic geometers are interested in? And what are some of the most beautiful…
Mohan
- 15,494
256
votes
27 answers
Best Fake Proofs? (A M.SE April Fools Day collection)
In honor of April Fools Day $2013$, I'd like this question to collect the best, most convincing fake proofs of impossibilities you have seen.
I've posted one as an answer below. I'm also thinking of a geometric one where the "trick" is that it's…
Potato
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256
votes
9 answers
What is the importance of the Collatz conjecture?
I have been fascinated by the Collatz problem since I first heard about it in high school.
Take any natural number $n$. If $n$ is even, divide it by $2$ to get $n / 2$, if $n$ is odd multiply it by $3$ and add $1$ to obtain $3n + 1$. Repeat the…
Dan Brumleve
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256
votes
4 answers
The Integral that Stumped Feynman?
In "Surely You're Joking, Mr. Feynman!," Nobel-prize winning Physicist Richard Feynman said that he challenged his colleagues to give him an integral that they could evaluate with only complex methods that he could not do with real…
Argon
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256
votes
26 answers
Does the square or the circle have the greater perimeter? A surprisingly hard problem for high schoolers
An exam for high school students had the following problem:
Let the point $E$ be the midpoint of the line segment $AD$ on the square $ABCD$. Then let a circle be determined by the points $E$, $B$ and $C$ as shown on the diagram. Which of the…
Sid
- 4,422
254
votes
7 answers
Can you answer my son's fourth-grade homework question: Which numbers are prime, have digits adding to ten and have a three in the tens place?
My son Horatio (nine years old, fourth grade) came home with some fun math homework exercises today. One of his problems was the following little question:
I am thinking of a number...
It is prime.
The digits add up to $10.$
It has a $3$ in the…
JDH
- 45,373
253
votes
8 answers
Proof that the trace of a matrix is the sum of its eigenvalues
I have looked extensively for a proof on the internet but all of them were too obscure. I would appreciate if someone could lay out a simple proof for this important result. Thank you.
JohnK
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253
votes
10 answers
Exterior Derivative vs. Covariant Derivative vs. Lie Derivative
In differential geometry, there are several notions of differentiation, namely:
Exterior Derivative, $d$
Covariant Derivative/Connection, $\nabla$
Lie Derivative, $\mathcal{L}$.
I have listed them in order of appearance in my education/in…
Michael Albanese
- 104,029
251
votes
26 answers
What are some examples of when Mathematics 'accidentally' discovered something about the world?
I do not remember precisely what the equations or who the relevant mathematicians and physicists were, but I recall being told the following story. I apologise in advance if I have misunderstood anything, or just have it plain wrong. The story is as…
Trogdor
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248
votes
8 answers
What are the Differences Between a Matrix and a Tensor?
What is the difference between a matrix and a tensor? Or, what makes a tensor, a tensor? I know that a matrix is a table of values, right? But, a tensor?
Aurelius
- 2,881
247
votes
28 answers
Good books and lecture notes about category theory.
What are the best books and lecture notes on category theory?
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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