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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 …

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- …

In computability theory, the function $r\mapsto j_r$ is called the principal function of the set $J=\{j_1<\dots<j_m\}$ and denoted $p_J$. The relation with your function being that $p_r=j_r-r$, i.e., …

This is in Downey and Hirschfeldt: Algorithmic randomness and complexity, Theorem 6.1.3, which cites Chaitin, G. Information-theoretical characterizations of recursive infinite strings, Theoretical C …

It appears in the Hitting Set Problem, and so it would make some sense to call $\tau(\omega)$ the hitting set cardinality.

There are some references and some more information in the entry on the number of asymmetric graphs on n nodes in the OEIS (Online Encyclopedia of Integer Sequences): F. Harary and E. M. Palmer, Grap …

It depends on the universal machine. Consider length 0, the empty string could have complexity 455, say.

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\ …

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 …

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

I suppose the larger surface area plane will have a greater hitting probability. But what would be a rigorous way of proving that? Depends on whether for each plane $A$, the center of $A$ is the …

This is interesting. I think a number could be of low complexity in terms of approximating it to within smaller and smaller $\epsilon $, while of high complexity in terms of finding its binary represe …

Note that $y''(x)=\sqrt{1+y'(x)^2}$ so $u=y'$ satisfies $u'(x)=\sqrt{1+u^2}$. This is separable, so we get $$ \frac{du}{\sqrt{1+u^2}}=dx,\quad \sinh^{-1}(u)=x + C. $$

Shreve, Stochastic Calculus for Finance, volumes 1 and 2.

I guess somehow we should rule out spurious ways to depend on parameter, such as "a graph is $k$-cliqueish if either $k=1$ and the graph is connected, or $k\ge 2$ and the graph has a clique of s …