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1500 questions
71
votes
3 answers
If I know the order of every element in a group, do I know the group?
Suppose $G$ is a finite group and I know for every $k \leq |G|$ that exactly $n_k$ elements in $G$ have order $k$. Do I know what the group is? Is there a counterexample where two groups $G$ and $H$ have the same number of elements for each order,…
Stanley
- 3,244
71
votes
1 answer
Is this determinant identity known?
Let $A$ be an $n \times n$ matrix that is 'almost upper triangular' in the following sense: entries on and above the main diagonal can be whatever they want, entries on the diagonal just below the main diagonal all equal -1 and entries on even lower…
Vincent
- 11,280
71
votes
14 answers
Pseudo Proofs that are intuitively reasonable
What are nice "proofs" of true facts that are not really rigorous but give the right answer and still make sense on some level? Personally, I consider them to be guilty pleasures. Here are examples of what I have in mind:
$s=\sum_{i=0}^\infty…
Michael Greinecker
- 34,865
70
votes
1 answer
Evaluating sums and integrals using Taylor's Theorem
Taylor's theorem states that
$$f(x)-\sum_{k=0}^n\frac{f^{(k)}(a)}{k!}(x-a)^k = \int_a^x \frac{f^{(n+1)} (t)}{n!} (x - t)^n \, dt $$
We can use this to evaluate integrals. For example, consider $f(x)=\frac{b!x^{b+n+1}}{(b+n+1)!}$. This has…
Pauly B
- 5,302
70
votes
7 answers
What is the oldest open problem in geometry?
Geometry is one of the oldest branches of mathematics, and many famous problems have been proposed and solved in its long history.
What I would like to know is: What is the oldest open problem in geometry?
Also (soft questions): Why is it so hard?…
Sudoku Polo
- 693
70
votes
4 answers
Taylor series of $\ln(1+x)$?
Compute the taylor series of $\ln(1+x)$
I've first computed derivatives (up to the $4$th) of $\ln(1+x)$
$f^{'}(x)$ = $\frac{1}{1+x}$
$f^{''}(x) = \frac{-1}{(1+x)^2}$
$f^{'''}(x) = \frac{2}{(1+x)^3}$
$f^{''''}(x) = \frac{-6}{(1+x)^4}$
Therefore the…
Minu
- 1,021
70
votes
5 answers
Are functions of independent variables also independent?
It's a really simple question. However I didn't see it in books and I tried to find the answer on the web but failed.
If I have two independent random variables, $X_1$ and $X_2$, then I define two other random variables $Y_1$ and $Y_2$, where $Y_1$…
LLS
- 867
70
votes
4 answers
What lies beyond the Sedenions
In the construction of types of numbers, we have the following sequence:
$$\mathbb{R} \subset \mathbb{C} \subset \mathbb{H} \subset \mathbb{O} \subset \mathbb{S}$$
or:
$$2^0 \mathrm{-ions} \subset 2^1 \mathrm{-ions} \subset 2^2 \mathrm{-ions}…
Willem Noorduin
- 2,739
70
votes
5 answers
Does every prime divide some Fibonacci number?
I am tring to show that $\forall a \in \Bbb P\; \exists n\in\Bbb N : a|F_n$, where $F$ is the fibonacci sequence defined as $\{F_n\}:F_0 = 0, F_1 = 1, F_n = F_{n-1} + F_{n-2}$ $(n=2,3,...)$. How can I do this?
Originally, I was trying to show that…
Chanhee Jeong
- 1,048
70
votes
3 answers
Evaluating the log gamma integral $\int_{0}^{z} \log \Gamma (x) \, \mathrm dx$ in terms of the Hurwitz zeta function
One way to evaluate $ \displaystyle\int_{0}^{z} \log \Gamma(x) \, \mathrm dx $ is in terms of the Barnes G-function.
$$ \int_{0}^{z} \log \Gamma(x) \, \mathrm dx = \frac{z}{2} \log (2 \pi) + \frac{z(1-z)}{2} + z \log \Gamma(z) - \log…
Random Variable
- 44,816
70
votes
4 answers
Finite Groups with exactly $n$ conjugacy classes $(n=2,3,...)$
I am looking to classify (up to isomorphism) those finite groups $G$ with exactly 2 conjugacy classes.
If $G$ is abelian, then each element forms its own conjugacy class, so only the cyclic group of order 2 works here.
If $G$ is not abelian, I am…
RHP
- 2,613
70
votes
2 answers
Are the sums $\sum_{n=1}^{\infty} \frac{1}{(n!)^k}$ transcendental?
This question is inspired
by my answer to the question
How to compute $\prod_{n=1}^\infty\left(1+\frac{1}{n!}\right)$? .
The sums
$f(k) = \sum_{n=1}^{\infty} \frac{1}{(n!)^k}$
(for positive integer $k$)
came up,
and I noticed that
$f(1) = e-1$ was…
marty cohen
- 110,450
70
votes
3 answers
What does "isomorphic" mean in linear algebra?
My professor keeps mentioning the word "isomorphic" in class, but has yet to define it... I've asked him and his response is that something that is isomorphic to something else means that they have the same vector structure. I'm not sure what that…
Sujaan Kunalan
- 11,194
70
votes
4 answers
How much does symbolic integration mean to mathematics?
(Before reading, I apologize for my poor English ability.)
I have enjoyed calculating some symbolic integrals as a hobby, and this has been one of the main source of my interest towards the vast world of mathematics. For instance, the integral…
Sangchul Lee
- 181,930
70
votes
5 answers
What does it mean to integrate with respect to the distribution function?
If $f(x)$ is a density function and $F(x)$ is a distribution function of a random variable $X$ then I understand that the expectation of x is often written as:
$$E(X) = \int x f(x) dx$$
where the bounds of integration are implicitly $-\infty$ and…
Jeromy Anglim
- 937