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1500 questions
217
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
- 14,298
215
votes
29 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
- 1,235
214
votes
22 answers
Is it faster to count to the infinite going one by one or two by two?
A child asked me this question yesterday:
Would it be faster to count to the infinite going one by one or two by two?
And I was split with two answers:
In both case it will take an infinite time.
Skipping half of the number should be really…
Thomas Ayoub
- 1,645
213
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
- 19,927
213
votes
15 answers
Identification of a quadrilateral as a trapezoid, rectangle, or square
Yesterday I was tutoring a student, and the following question arose (number 76):
My student believed the answer to be J: square. I reasoned with her that the information given only allows us to conclude that the top and bottom sides are parallel,…
The Chaz 2.0
- 10,606
212
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
- 3,474
212
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…
user266764
211
votes
32 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
- 6,307
210
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
210
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
- 4,421
210
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
- 84,178
209
votes
24 answers
Taking Seats on a Plane
This is a neat little problem that I was discussing today with my lab group out at lunch. Not particularly difficult but interesting implications nonetheless
Imagine there are a 100 people in line to board a plane that seats 100. The first person in…
crasic
- 5,139
209
votes
7 answers
Do we have negative prime numbers?
Do we have negative prime numbers?
$..., -7, -5, -3, -2, ...$
user103028
207
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
207
votes
8 answers
How can we sum up $\sin$ and $\cos$ series when the angles are in arithmetic progression?
How can we sum up $\sin$ and $\cos$ series when the angles are in arithmetic progression? For example here is the sum of $\cos$ series:
$$\sum_{k=0}^{n-1}\cos (a+k \cdot d) =\frac{\sin(n \times \frac{d}{2})}{\sin ( \frac{d}{2} )} \times \cos \biggl(…
Quixotic
- 22,817