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1500 questions
209
votes
4 answers
Overview of basic results about images and preimages
Are there some good overviews of basic facts about images and inverse images of sets under functions?
Martin Sleziak
- 53,687
207
votes
9 answers
Importance of Representation Theory
Representation theory is a subject I want to like (it can be fun finding the representations of a group), but it's hard for me to see it as a subject that arises naturally or why it is important. I can think of two mathematical reasons for studying…
Eric O. Korman
- 19,137
207
votes
32 answers
Books on Number Theory for Layman
Books on Number Theory for anyone who loves Mathematics?
(Beginner to Advanced & just for someone who has a basic grasp of math)
Prasoon Saurav
- 600
207
votes
7 answers
How could we define the factorial of a matrix?
Suppose I have a square matrix $\mathsf{A}$ with $\det \mathsf{A}\neq 0$.
How could we define the following operation? $$\mathsf{A}!$$
Maybe we could make some simple example, admitted it makes any sense, with
$$\mathsf{A} =
\left(\begin{matrix}
1…
Enrico M.
- 26,114
205
votes
31 answers
Counterintuitive examples in probability
I want to teach a short course in probability and I am looking for some counter-intuitive examples in probability. I am mainly interested in the problems whose results seem to be obviously false while they are not.
I already found some things. For…
MR_BD
- 5,942
205
votes
34 answers
List of interesting math videos / documentaries
This is an offshoot of the question on Fun math outreach/social activities. I have listed a few videos/documentaries I have seen. I would appreciate if people could add on to this list.
$1.$ Story of maths Part1 Part2 Part3 Part4
$2.$ Dangerous…
user17762
204
votes
1 answer
Are $14$ and $21$ the only "interesting" numbers?
The numbers $14$ and $21$ are quite interesting.
The prime factorisation of $14$ is $2\cdot 7$ and the prime factorisation of $14+1$ is $3\cdot 5$. Note that $3$ is the prime after $2$ and $5$ is the prime before $7$.
Similarly, the prime…
Simon Parker
- 4,303
203
votes
8 answers
Intuitively, what is the difference between Eigendecomposition and Singular Value Decomposition?
I'm trying to intuitively understand the difference between SVD and eigendecomposition.
From my understanding, eigendecomposition seeks to describe a linear transformation as a sequence of three basic operations ($P^{-1}DP$) on a vector:
Rotation…
user541686
- 13,772
203
votes
2 answers
How to show $e^{e^{e^{79}}}$ is not an integer
In this question, I needed to assume in my answer that $e^{e^{e^{79}}}$ is not an integer. Is there some standard result in number theory that applies to situations like this?
After several years, it appears this is an open problem. As a non-number…
Carl Mummert
- 81,604
202
votes
14 answers
How many sides does a circle have?
My son is in 2nd grade. His math teacher gave the class a quiz, and one question was this:
If a triangle has 3 sides, and a rectangle has 4 sides,
how many sides does a circle have?
My first reaction was "0" or "undefined". But my son wrote…
Fixee
- 11,565
202
votes
28 answers
What is a good complex analysis textbook, barring Ahlfors's?
I'm out of college, and trying to self-learn complex analysis. I'm finding Ahlfors' text difficult. Any recommendations? I'm probably at an intermediate sophistication level for an undergrad. (Bonus points if the text has a section on the Riemann…
MBP
- 1,205
201
votes
91 answers
Which one result in mathematics has surprised you the most?
A large part of my fascination in mathematics is because of some very surprising results that I have seen there.
I remember one I found very hard to swallow when I first encountered it, was what is known as the Banach Tarski Paradox. It states that…
KalEl
- 3,297
201
votes
6 answers
Why can't differentiability be generalized as nicely as continuity?
The question: Can we define differentiable functions between (some class of) sets, "without $\Bbb R$"* so that it
Reduces to the traditional definition when desired?
Has the same use in at least some of the higher contexts where we would use the…
GPerez
- 6,766
199
votes
4 answers
Limit of $L^p$ norm
Could someone help me prove that given a finite measure space $(X, \mathcal{M}, \sigma)$ and a measurable function $f:X\to\mathbb{R}$ in $L^\infty$ and some $L^q$, $\displaystyle\lim_{p\to\infty}\|f\|_p=\|f\|_\infty$?
I don't know where to start.
Parakee
- 3,304
198
votes
4 answers
Some users are mind bogglingly skilled at integration. How did they get there?
Looking through old problems, it is not difficult to see that some users are beyond incredible at computing integrals. It only took a couple seconds to dig up an example like this.
Especially in a world where most scientists compute their integrals…
JessicaK
- 7,655