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1500 questions
70
votes
13 answers
What is exactly the difference between a definition and an axiom?
I am wondering what the difference between a definition and an axiom.
Isn't an axiom something what we define to be true?
For example, one of the axioms of Peano Arithmetic states that $\forall n:0\neq S(n)$, or in English, that zero isn't the…
wythagoras
- 25,726
70
votes
20 answers
What are some applications of elementary linear algebra outside of math?
I'm TAing linear algebra next quarter, and it strikes me that I only know one example of an application I can present to my students. I'm looking for applications of elementary linear algebra outside of mathematics that I might talk about in…
user98602
70
votes
8 answers
How can a set contain itself?
In Russell's famous paradox ("Does the set of all sets which do not contain themselves contain itself?") he obviously makes the assumption that a set can contain itself. I do not understand how this should be possible and therefore my answer to…
jimmyorpheus
- 835
70
votes
6 answers
Foundation for analysis without axiom of choice?
Let's say I consider the Banach–Tarski paradox unacceptable, meaning that I would rather do all my mathematics without using the axiom of choice. As my foundation, I would presumably have to use ZF, ZF plus other axioms, or an approach in which sets…
user13618
69
votes
4 answers
Does the string of prime numbers contain all natural numbers?
Does the string of prime numbers $$2357111317\ldots$$ contain every natural number as its sub-string?
Buddha
- 1,256
- 11
- 16
69
votes
2 answers
Evaluating $\int_{0}^{1}\cdots\int_{0}^{1}\bigl\{\frac{1}{x_{1}\cdots x_{n}}\bigr\}^{2}\:\mathrm{d}x_{1}\cdots\mathrm{d}x_{n}$
Here is my source of inspiration for this question.
I suggest to evaluate the following new one.
$$
I_{n}:= \int_0^1 \! \cdots \! \int_0^1
\left\{\frac{1}{x_1x_2 \cdots x_n}\right\}^{2} \:\mathrm{d}x_1\,\mathrm{d}\,x_2 \cdots \mathrm{d}x_n…
Olivier Oloa
- 122,789
69
votes
4 answers
Difference between NFA and DFA
In very simple terms please, all resources I'm finding are talking about tuples and stuff and I just need a simple explanation that I can remember easily because I keep getting them mixed up.
Ogen
- 2,295
- 8
- 34
- 44
69
votes
3 answers
The $\sigma$-algebra of subsets of $X$ generated by a set $\mathcal{A}$ is the smallest sigma algebra including $\mathcal{A}$
I am struggling to understand why it should be that the $\sigma$-algebra of subsets of $X$ generated by $\mathcal{A}$ should be the smallest $\sigma$-algebra of subsets of $X$ including $\mathcal{A}$.
Let me try to elucidate my understanding of the…
Harry Williams
- 1,829
69
votes
5 answers
Why are two permutations conjugate iff they have the same cycle structure?
I have heard that two permutations are conjugate if they have the same cyclic structure. Is there an intuitive way to understand why this is?
Han Solo
- 691
69
votes
27 answers
What are some conjectures of your own?
Background: Although this site is most-often used for specific one-off questions, many of the highest scored questions (also on MathOverflow), which gather a lot of attention to the site are about informal lists. So, in the theme of, but in contrast…
Graviton
- 4,678
69
votes
1 answer
How to prove this recurrence relation for generalized "rounding up to $\pi$"?
The webpage Rounding Up To $\pi$ defines a certain "rounding up" function by an extremely simple procedure:
Beginning with any positive integer $n$, round up to the nearest multiple of $n-1$, then up to the nearest multiple of $n-2$, and so on, up…
r.e.s.
- 15,537
69
votes
7 answers
Geometric understanding of differential forms.
I would like to understand differential forms more intuitively. I have yet to find a book which explains how the use of the exterior product in differential forms ties into the geometrical significance of it. Most books briefly introduce the…
Markus
- 693
69
votes
2 answers
What does "communicated by" mean in math papers?
[This question involves mostly math papers, and may be relevant to graduate students learning to write and cite papers, although this is my only justification for this being a math question.]
Usually papers start out with the title and then the…
user2055
69
votes
6 answers
How do you show monotonicity of the $\ell^p$ norms?
I can't seem to work out the inequality $(\sum |x_n|^q)^{1/q} \leq (\sum |x_n|^p)^{1/p}$ for $p \leq q$ (which I'm assuming is the way to go about it).
user1736
- 8,993
69
votes
15 answers
Prove if $n^2$ is even, then $n$ is even.
I am just learning maths, and would like someone to verify my proof.
Suppose $n$ is an integer, and that $n^2$ is even. If we add $n$ to $n^2$, we have $n^2 + n = n(n+1)$, and it follows that $n(n+1)$ is even. Since $n^2$ is even, $n$ is even.
Is…
user79612
- 575