Intereting Posts

$\sum_{k=0}^n \binom{n}{k}^2 = \binom{2n}{n}?$
Joint probabilities, conditional probabilities with the chain rule.
Help understanding the complexity of my algorithm (summation)
Isomorphism of Elliptic Curves:
Fourier cosine transforms of Schwartz functions and the Fejer-Riesz theorem
Is $ \frac{\mathrm{d}{x}}{\mathrm{d}{y}} = \frac{1}{\left( \frac{\mathrm{d}{y}}{\mathrm{d}{x}} \right)} $?
Countable set of truth assignments satisfying set of well formed formulas
Neatest proof that set of finite subsets is countable?
Example of different topologies with same convergent sequences
Difference between ZFC & NBG
How is the Fourier transform “linear”?
Existence of real roots of a quartic polynomial
Determining the coefficients of the reciprocal of a Dirichlet series
Examples of metric vector spaces but not normed ? Normed but not prehilbertian?
Show that if f is analytic in $|z|\leq 1$, there must be some positive integer n such that $f(\frac{1}{n})\neq \frac{1}{n+1}$

**Recipe.** Do the following.

- Throw $N$ random points $(x_0,y_0),(x_1,y_1),x_2,y_2),\cdots,(x_{N-1},y_{N-1})$ in the plane.

Define $(x_N,y_N)=(x_0,y_0)$ : enumeration is $\mod N$ . - These points are joined in a number of ways to form a polygon with $N$ edges. Define “a number of ways” as permutations $\pi$ of the vertex numbering. Vertex $(0)$

can be left in place without loss of generality. - Select the polygon with minimal perimeter i.e. the smallest sum of the edges lengths:

$$\sum_{k=0}^{N-1}\sqrt{(x_{\pi(k+1)}-x_{\pi(k)})^2+(y_{\pi(k+1)}-y_{\pi(k)})^2}

= \mbox{minimum}(\pi)$$

Picture on the left: Area maximized (and self intersecting).

(Where the area of the polygon is defined as in

this answer)

Picture on the right: Perimeter minimized (same vertices).

Vertex numbering: digit on the left is original, digit on the right is permutation.

- Uniform thickness border around skewed ellipse?
- Proof that the Convex Hull of a finite set S is equal to all convex combinations of S
- Polyhedra from number fields
- What is a composition of two binary relations geometrically?
- Formal proof for detection of intersections for constrained segments
- How to find the intersection of the area of multiple triangles

Can someone prove or disprove this conjecture?

- How to tile a sphere with points at an even density?
- How do we define arc length?
- Cleverest construction of a dodecahedron / icosahedron?
- The vertices of an equilateral triangle are shrinking towards each other
- Geometry question regarding existence of a quadrilateral
- $7$ points inside a circle at equal distances
- Find this angle, in terms of variables
- $\arctan (x) + \arctan(1/x) = \frac{\pi}{2}$
- Calculate the intersection points of two ellipses
- Can a rectangle be written as a finite almost disjoint union of squares?

- How can we apply the Borel-Cantelli lemma here?
- How to prove $\sum_{i=1}^k(\frac{1}{\alpha_i}\prod_{j\neq i}^k\frac{\alpha_j}{\alpha_j-\alpha_i})=\sum_{i=1}^k\frac{1}{\alpha_i}$?
- Chernoff Bounds. Solve the probability
- Find Number Of Roots of Equation $11^x + 13^x + 17^x =19^x $
- Neutral element in $\hom_C(A, B)$
- Field Norm Surjective for Finite Extensions of $\mathbb{F}_{p^k}$
- $C$ is NOT a Banach Space w.r.t $\|\cdot\|_2$
- Refining the central limit theorem on discrete random vars
- “Binomiable” numbers
- Compute $e^{e^z}$, where z is a complex number
- The n-th prime is less than $n^2$?
- Limit of a stochastic integral
- If you draw two cards, what is the probability that the second card is a queen?
- For which values of $n$ is the polynomial $p(x)=1+x+x^2+\cdots+x^n$ irreducible over $\mathbb{F}_2$?
- Bob starts with \$20. Bob flips a coin. Heads = Win +\$1 Tails = Lose -\$1. Stops if he has \$0 or \$100. Probability he ends up with \$0?