13
votes
Accepted
The maximum area of a pentagon inside a circle
Consider any two consecutive sides $AB$, $BC$ of a convex polygon inscribed in a given circle. Suppose to keep all vertices fixed apart $B$: the area of the polygon is maximum when $B$ is the midpoint ...
- 52.3k
5
votes
The maximum area of a pentagon inside a circle
Let's look at general (convex) $n$-sided polygons inscribed in a unit circle. Join the centre of the circle to each vertex of the polygon to form $n$ isosceles triangles. Say their apex angles are $\...
- 2,638
3
votes
Largest Area Triangle in the Vesica Piscis
Here is a solution in the "17th century spirit" where extremal solutions were found based on the computation of infinitesimal quantities.
I assume that we look for an optimal solution under ...
- 83.9k
2
votes
Find sum of factorials divisible by the largest possible prime squared
I used such approach:
for given $n$ (currently, $n=32$), loop through prime numbers $p$ starting from certain value $p_0$ to, theoretically, $\sqrt{\sum_{k=1}^n k!}$;
and for these $(n,p)$ construct 2 ...
- 17.4k
1
vote
Accepted
Constrained optimization with restriction
Both methods will yield the same result, it's just a matter of when you perform the algebraic manipulation.
In your first case, you first calculate the derivative of $f$ and $r$ and then combine them,...
- 124k
1
vote
How to rigorously show that the maximum variance of an hermitian matrix is $\left( \frac{h_{max}-h_{min}}{2} \right) ^2$?
Without loss of generality we may assume that $H$ is real diagonal. In that case, you have--denoting $\lambda_j$ for the (positive, real) eigenvalues,
$$
F(v) = \sum_{j} \lambda_j^2 v_j^2 - \Big(\...
- 3,774
1
vote
Minimizing a function using matrix calculus
The basic idea is to find $dJ(K)$ in terms of $dK$. I'll demonstrate for the first term $$J(K) = x^T C^T K^T G K Cx.$$
We have
\begin{align}
dJ(K) &= x^T C^T dK^T G K C x + x^T C^T K^T G dK C x \\
...
- 2,254
1
vote
Optimal permutation of transition probabilities in random walk to minimize expected stopping time
I have a way of reformulating the problem that may be helpful:
Let $t_r$ be the expected stopping time starting from position $r$. Then we have, for $2 \le r \le n$:
$$t_r=p_r t_{r+1}+(1-p_r)t_{r-1}+1$...
- 5,633
Only top scored, non community-wiki answers of a minimum length are eligible