Search Results

A somewhat different quote has been attributed to Einstein: http://izquotes.com/quote/226612 (link broken now) https://quotefancy.com/quote/764082/Albert-Einstein-The-physicist-s-greatest-tool-is- …

It sounds like you are describing a situation where $a$ is more true than $b$, $b$ is more true than $c$, but nevertheless $c$ is more true than $a$. I am not sure about the best starting point in loo …

Erdős and Selfridge (1971) state that the following is "implied by an unpublished result of Rosser" which they claim appears in a forthcoming book on sieve methods by Halberstam and Richert. I guess I …

Yes, for starters there is the arithmetical hierarchy, where enumerable = $\Sigma^0_1$ and it continues $\Pi^0_1$, $\Delta^0_2$, $\Sigma^0_2$ etc. See also the Computability Menagerie.

Here is a counterexample of size 23. Let $m=6$ and let $$P=\{0,a_1,\dots,a_m,1\}\cup\{b_{ij}: 1\le i<j\le m\}$$ where $0<a_i<b_{jk}<1$ whenever $i$ is distinct from $j$ and $k$. The cardinality of $P$ …

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\ …

It is perhaps equally natural to consider sequences wrapping around cyclically, like a de Bruijn sequence. In that case there is a simple reason why it won't work for $n=3$. Clearly 000111222, or cycl …

Yes, this should follow from the elementary bound. The point is that having a Kalmar elementary time bound is "closed under" searches through exponentially large collections. Suppose $N=W(r,k)$ is lea …

Brownian motion, i.e. Wiener measure, is a good source of ideas and examples here. For instance if $W_t$ is 1-dimensional standard Brownian motion at time $t$ and $$P(\forall x\,F(x)=x^2)=1$$ and $Y=W …

Imagine that you lay out the N (0) and E (1) moves as follows ($n=4$ shown): $$0000$$ $$1111$$ As you go along the path, color $\color{red}{red}$ the ones you have used, so that after reading either 0 …

This is studied in Reverse Mathematics as the Chain Antichain Principle (CAC) and it is observed that it follows from Ramsey's theorem.

$\mathscr F$-indistinguishability. In analogy with Topological indistinguishability.

Question 1: Inspired by the ones you found we can see that there are infinitely many solutions as follows: $$(x,y,z;n) = (k-1,\quad k(k-1),\quad k-1;\quad 2^k)$$ for any $k\ge 0$. Edit re: Question 2 …

Chebyshev's bias says that there are slightly more non-Pythagorean primes than Pythagorean primes (although the limiting frequency is the same).

Maybe the simplest counterexample? Let $\newcommand{\1}{\mathbf 1}\1=1_{[0,1]}$. Then any $f\cdot\1$ is zero outside of $[0,1]$, but $$\1*\1(x)=\int \1(t)\1(x-t)\,dt = \begin{cases}x& 0\le x\le 1\\ 2 …