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1500 questions
280
votes
6 answers
What is the practical difference between a differential and a derivative?
I ask because, as a first-year calculus student, I am running into the fact that I didn't quite get this down when understanding the derivative: So, a derivative is the rate of change of a function with respect to changes in its variable, this much…
Faqa
- 2,801
279
votes
3 answers
Meaning of Rays in Polar Plot of Prime Numbers
I recently began experimenting with gnuplot and I quickly made an interesting discovery. I plotted all of the prime numbers beneath 1 million in polar coordinates such that for every prime $p$, $(r,\theta) = (p,p)$. I was not expecting anything in…
dwymark
- 2,801
278
votes
10 answers
How to read a book in mathematics?
How is it that you read a mathematics book? Do you keep a notebook of definitions? What about theorems? Do you do all the exercises? Focus on or ignore the proofs? I have been reading Munkres, Artin, Halmos, etc. but I get a bit lost usually around…
Pax
- 6,000
277
votes
13 answers
Given an infinite number of monkeys and an infinite amount of time, would one of them write Hamlet?
Of course, we've all heard the colloquialism "If a bunch of monkeys pound on a typewriter, eventually one of them will write Hamlet." I have a (not very mathematically intelligent) friend who presented it as if it were a mathematical fact, which got…
Jason
- 1,415
277
votes
9 answers
Evaluating $\lim\limits_{n\to\infty} e^{-n} \sum\limits_{k=0}^{n} \frac{n^k}{k!}$
I'm supposed to calculate: $$\lim_{n\to\infty} e^{-n} \sum_{k=0}^{n} \frac{n^k}{k!}$$ By using WolframAlpha, I might guess that the limit is $\frac{1}{2}$ , which is a pretty interesting and nice result. I wonder in which ways we may approach it.
user 1591719
- 45,325
276
votes
11 answers
Derivative of sigmoid function $\sigma (x) = \frac{1}{1+e^{-x}}$
In my AI textbook there is this paragraph, without any explanation. The sigmoid function is defined as follows $$\sigma (x) = \frac{1}{1+e^{-x}}.$$ This function is easy to differentiate because $$\frac{d\sigma (x)}{d(x)} = \sigma (x)\cdot…
Bryan Glazer
- 3,144
275
votes
40 answers
Fun but serious mathematics books to gift advanced undergraduates.
I am looking for fun, interesting mathematics textbooks which would make good studious holiday gifts for advanced mathematics undergraduates or beginning graduate students. They should be serious but also readable . In particular, I am looking for…
Samuel Handwich
- 2,831
274
votes
32 answers
Your favourite application of the Baire Category Theorem
I think I remember reading somewhere that the Baire Category Theorem is supposedly quite powerful. Whether that is true or not, it's my favourite theorem (so far) and I'd love to see some applications that confirm its neatness and/or power. Here's…
Rudy the Reindeer
- 48,632
272
votes
6 answers
Eigenvectors of real symmetric matrices are orthogonal
Can someone point me to a paper, or show here, why symmetric matrices have orthogonal eigenvectors? In particular, I'd like to see proof that for a symmetric matrix $A$ there exists decomposition $A = Q\Lambda Q^{-1} = Q\Lambda Q^{T}$ where…
Phonon
- 4,152
271
votes
5 answers
What is the difference between linear and affine function?
I am a bit confused. What is the difference between a linear and affine function? Any suggestions will be appreciated.
user34790
- 4,462
270
votes
7 answers
Why do we care about dual spaces?
When I first took linear algebra, we never learned about dual spaces. Today in lecture we discussed them and I understand what they are, but I don't really understand why we want to study them within linear algebra. I was wondering if anyone knew a…
WWright
- 5,730
270
votes
4 answers
What is the maximum volume that can be contained by a sheet of paper?
I was writing some exercises about the AM-GM inequality and I got carried away by the following (pretty nontrivial, I believe) question: Q: By properly folding a common $210mm\times 297mm$ sheet of paper, what is the maximum amount of water such a…
Jack D'Aurizio
- 372,937
- 42
- 421
- 888
270
votes
28 answers
Too old to start math
I'm sorry if this question goes against the meta for posting questions - I attached all the "beware, this is a soft-question" tags I could. This is a question I've been asking myself now for some time. In most areas, there's a "cut off age" to be…
Thomas Nesbitt
- 624
269
votes
1 answer
Why are rings called rings?
I've done some search in Internet and other sources about this question. Why the name ring to this particular object? Just curiosity. Thanks.
leo
- 10,929
269
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
- 41,890