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1500 questions
70
votes
3 answers
Set of continuity points of a real function
I have a question about subsets $$
A \subseteq \mathbb R
$$
for which there exists a function $$f : \mathbb R \to \mathbb R$$ such that the set of continuity points of $f$ is $A$. Can I characterize this kind of sets? In a topological,measurable…
Daniel
- 3,053
70
votes
5 answers
Matrices commute if and only if they share a common basis of eigenvectors?
I've come across a paper that mentions the fact that matrices commute if and only if they share a common basis of eigenvectors. Where can I find a proof of this statement?
Yaroslav Bulatov
- 5,327
70
votes
5 answers
Integers $n$ such that $i(i+1)(i+2) \cdots (i+n)$ is real or pure imaginary
A couple of days ago I happened to come across [1], where the curious fact that $i(i-1)(i-2)(i-3)=-10$ appears ($i$ is the imaginary unit). This led me to the following question:
Problem 1: Is $3$ the only positive integer value of $n$ such that…
Dave L. Renfro
- 36,843
70
votes
0 answers
Dedekind Sum Congruences
For $a,b,c \in \mathbb{N}$, let $a^{\prime} = \gcd(b,c)$, $b^{\prime} = \gcd(a,c)$, $c^{\prime} = \gcd(a,b)$ and $d = a^{\prime} b^{\prime} c^{\prime}$. Define $\mathfrak{S}(a,b,c) = a^{\prime} \mathfrak{s}( \tfrac{bc}{d}, \tfrac{a}{b^{\prime}…
user02138
- 17,064
70
votes
4 answers
How do people apply the Lebesgue integration theory?
This question has puzzled me for a long time. It may be too vague to ask here. I hope I can narrow down the question well so that one can offer some ideas.
In a lot of calculus textbooks, there is usually a chapter about "applications" after the…
user9464
70
votes
8 answers
A bestiary about adjunctions
What is your favourite adjoint? Following Mac Lane philosophy adjoints are everywhere, so I would like to draw a (possibly but unprobably) exhaustive list of adjunctions one faces in studying Mathematics. For the sake of clarity I would like you to…
fosco
- 11,814
70
votes
12 answers
How to prove every closed interval in R is compact?
Let $[a,b]\subseteq \mathbb R$. As we know, it is compact. This is a very important result. However, the proof for the result may be not familiar to us. Here I want to collect the ways to prove $[a,b]$ is compact.
Thanks for your help and any link.
Paul
- 20,553
70
votes
9 answers
How to effectively study math?
Maybe this is too general for here, but I am having a lot of difficulty studying math. Just got out of the military and I guess I am not use to this yet but when I run into a problem I have trouble with and I just can't get it I get extremely…
Adam
- 1,405
70
votes
6 answers
Japanese Temple Problem From 1844
I recently learnt a Japanese geometry temple problem.
The problem is the following:
Five squares are arranged as the image shows. Prove that the area of triangle T and the area of square S are equal.
This is problem 6 in this article.
I am…
Larry
- 5,090
70
votes
5 answers
Show that $\int_{0}^{\pi/2}\frac {\log^2\sin x\log^2\cos x}{\cos x\sin x}\mathrm{d}x=\frac14\left( 2\zeta (5)-\zeta(2)\zeta (3)\right)$
Show that :
$$
\int_{0}^{\Large\frac\pi2}
{\ln^{2}\left(\vphantom{\large A}\cos\left(x\right)\right)
\ln^{2}\left(\vphantom{\large A}\sin\left(x\right)\right)
\over
\cos\left(x\right)\sin\left(x\right)}\,{\rm d}x
={1 \over…
Ryan
- 3,945
70
votes
28 answers
Non-associative operations
There are lots of operations that are not commutative.
I'm looking for striking counter-examples of operations that are not associative.
Or may associativity be genuinely built-in the concept of an operation? May non-associative operations be of…
Hans-Peter Stricker
- 18,159
70
votes
10 answers
There are 4 cups of liquid. Three are water and one is poison. If you were to drink 3 of the 4 cups, what is the probability of being poisoned?
In Season 5 Episode 16 of Agents of Shield, one of the characters decides to prove she can't die by pouring three glasses of water and one of poison; she then randomly drinks three of the four cups. I was wondering how to compute the probability of…
student
- 942
70
votes
4 answers
Why does multiplying a number on a clock face by 10 and then halving, give the minutes? ${}{}$
My daughter in grade 3 is learning about telling time at her school. She eagerly showed me this method she has discovered on her own to tell the minutes part of the time on an analogue clock. I wasn't sure at first because I have never heard about…
Sabeen Malik
- 657
70
votes
9 answers
Given real numbers: define integers?
I have only a basic understanding of mathematics, and I was wondering and could not find a satisfying answer to the following:
Integer numbers are just special cases (a subset) of real numbers. Imagine a world where you know only real numbers. How…
Daniel A.A. Pelsmaeker
- 1,099
- 11
- 17
70
votes
11 answers
What is the largest eigenvalue of the following matrix?
Find the largest eigenvalue of the following matrix
$$\begin{bmatrix}
1 & 4 & 16\\
4 & 16 & 1\\
16 & 1 & 4
\end{bmatrix}$$
This matrix is symmetric and, thus, the eigenvalues are real. I solved for the possible eigenvalues and,…
Cloud JR K
- 2,466