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1500 questions
132
votes
8 answers
Are half of all numbers odd?
Plato puts the following words in Socrates' mouth in the Phaedo dialogue: I mean, for instance, the number three, and there are many other examples. Take the case of three; do you not think it may always be called by its own name and also be called…
Jon Ericson
- 1,449
131
votes
1 answer
Can $x^{x^{x^x}}$ be a rational number?
If $x$ is a positive rational number, but not an integer, then can $x^{x^{x^x}}$ be a rational number ? We can prove that if $x$ is a positive rational number but not an integer, then $x^x$ can not be rational: Denote…
lsr314
- 16,098
131
votes
5 answers
What concept does an open set axiomatise?
In the context of metric (and in general first-countable) topologies, it's reasonably clear what a closed set is: a set $F$ is closed if and only if every convergent sequence of points in $F$ converges to a point also in $F$. This naturally…
Zhen Lin
- 98,824
131
votes
4 answers
Could someone explain conditional independence?
My understanding right now is that an example of conditional independence would be: If two people live in the same city, the probability that person A gets home in time for dinner, and the probability that person B gets home in time for dinner are…
Ryan
- 1,733
131
votes
1 answer
Simplicial Complex vs Delta Complex vs CW Complex
I am a little confused about what exactly are the difference(s) between simplicial complex, $\Delta$-complex, and CW Complex. What I roughly understand is that $\Delta$-complexes are generalisation of simplicial complexes (without the requirement…
yoyostein
- 20,598
131
votes
7 answers
What is the difference between a class and a set?
I know what a set is. I have no idea what a class is. As best as I can make out, every set is also a class, but a class can be "larger" than any set (a so-called "proper class"). This obviously makes no sense whatsoever, since sets are of unlimited…
MathematicalOrchid
- 6,495
131
votes
1 answer
Application of Hilbert's basis theorem in representation theory
In Smalø: Degenerations of Representations of Associative Algebras, Milan J. Math., 2008 there is an application of Hilbert's basis theorem that I don't understand: Two orders are defined on the set of $d$ -dimensional modules over an algebra…
Julian Kuelshammer
- 9,800
130
votes
7 answers
Are all algebraic integers with absolute value 1 roots of unity?
If we have an algebraic number $\alpha$ with (complex) absolute value $1$ , it does not follow that $\alpha$ is a root of unity (i.e., that $\alpha^n = 1$ for some $n$ ). For example, $(3/5 + 4/5 i)$ is not a root of unity. But if we assume that…
Jonas Kibelbek
- 7,438
130
votes
19 answers
Nice proofs of $\zeta(4) = \frac{\pi^4}{90}$?
I know some nice ways to prove that $\zeta(2) = \sum_{n=1}^{\infty} \frac{1}{n^2} = \pi^2/6$. For example, see Robin Chapman's list or the answers to the question " Different methods to compute $\sum_{n=1}^{\infty} \frac{1}{n^2}$ ?" Are there any…
Mike Spivey
- 57,148
130
votes
16 answers
Olympiad Inequality $\sum\limits_{cyc} \frac{x^4}{8x^3+5y^3} \geqslant \frac{x+y+z}{13}$
$x,y,z >0$ , prove $$\frac{x^4}{8x^3+5y^3}+\frac{y^4}{8y^3+5z^3}+\frac{z^4}{8z^3+5x^3} \geqslant \frac{x+y+z}{13}$$ Note: Often Stack Exchange asked to show some work before answering the question. This inequality was used as a proposal problem for…
HN_NH
- 4,521
130
votes
3 answers
When is the closure of an open ball equal to the closed ball?
It is not necessarily true that the closure of an open ball $B_{r}(x)$ is equal to the closed ball of the same radius $r$ centered at the same point $x$. For a quick example, take $X$ to be any set and define a metric $$ d(x,y)= \begin{cases}…
Alex Lapanowski
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129
votes
1 answer
Motivation of irrationality measure
I have a question about the irrationality of $e$: In proving the irrationality of $e$, one can prove the irrationality of $e^{-1}$ by using the series $$e^x = 1+x+\frac{x^2}{2!} + \cdots + \frac{x^n}{n!}+ \cdots$$ So the series for $e^{-1}$ is $$s_n…
Damien
- 1
129
votes
38 answers
"Almost true": non-trivial claims that have exactly one counterexample
Question: What are some non-trivial claims that have exactly one counterexample? Example 1: "There is no Pythagorean triple whose numbers are consecutive terms in a row of Pascal's triangle." There is exactly one counterexample:…
Dan
- 41,295
129
votes
7 answers
Is there a function that grows faster than exponentially but slower than a factorial?
In big-O notation the complexity class $O(2^n)$ is named "exponential". The complexity class $O(n!)$ is named "factorial". I believe that $f(n) = O(2^n)$ and $g(n) = O(n!)$ means that $\dfrac{f(n)}{g(n)}$ goes to zero in the limit as $n$ goes to…
Robert L. Read
- 1,217
129
votes
6 answers
Why are There No "Triernions" (3-dimensional analogue of complex numbers / quaternions)?
Since there are complex numbers (2 dimensions) and quaternions (4 dimensions), it follows intuitively that there ought to be something in between for 3 dimensions ("triernions"). Yet no one uses these. Why is this?
thecat
- 1,888