Search Results

The Cantor space $\mathcal C=\{0,1\}^\mathbb N$ with the metric $$d(x,y)=2^{-\min\{n:x(n)\ne y(n)\}}$$ is totally disconnected. Yet any three distinct elements form an isosceles triangle. Namely take …

Let us consider the case $n=2$. Assume $p_k$ is at the origin and $p_0$ is at the point $(1,0)$ on the $x$-axis, and that $r<1$. Distribution of $h_1$ By subtracting a constant it suffices to find t …

Let's say that a function $d:S\times S\to [0,\infty)$ for a countable set $S$ is a metric in the limit if $$d(x,y)\le \liminf_{n\to\infty} d(x,z_n)+d(z_n,y),$$ $$\lim_{n\to\infty} d(z_n,z_n)=0, \quad\ …

Algorithm create the two-nearest-neighbors graph $N_2$ using the first data set. That is, let each station be connected to the two closest stations. for a path in the second data set, assume that th …

If $$x_1^2+\dots+x_{n+1}^2=1$$ then $$ x_1^2+\dots+x_n^2 = 1-x_{n+1}^2 \in [0,1]. $$ Now $S = \sum_{i=1}^n x_i^2$ for $-1\le x_i\le 1$ has expectation $c\cdot n$ for a certain $c>0$, and will be appro …

A naive Monte Carlo will always work, probabilistically, by the Law of Large Numbers. The problem only arises if you want guaranteed correctness, 100% chance as opposed to say 99.999%.

A quick proof sketch that the ratio goes to 0: Let $a$ and $b$ be points in the unit ball of distance 2. (The existence of such does not hold in an arbitrary metric space!) As we add more and more poi …

I don't know that your characteristic has been explicitly studied before, nor would I be surprised if it has, but it fits into a more general setting as follows. The directed graph distance $d(a,b)$ …

I'm wondering where the relative probabilistic distance or Jaccard distance was first studied: $$d(A,B) =\mathbb P(\overline A\cup\overline B\mid A\cup B)$$ where $\overline A$ is the complement of $A …