In a cellular automaton, a finite pattern is called a sawtooth if its population grows without bound but does not tend to infinity. In other words, a sawtooth is a pattern with population that reaches new heights infinitely often, but also infinitely often drops below some fixed value. Their name comes from the fact that their plot of population versus generation number looks roughly like an ever-increasing sawtooth wave.
== In rules with small replicators ==
For instance, in Rule 90, a one-dimensional elementary cellular automaton, the population size starting from a single live cell follows Gould...
In mathematical analysis, a space-filling curve is a curve whose range contains the entire 2-dimensional unit square (or more generally an n-dimensional unit hypercube). Because Giuseppe Peano (1858–1932) was the first to discover one, space-filling curves in the 2-dimensional plane are sometimes called Peano curves, but that phrase also refers to the Peano curve, the specific example of a space-filling curve found by Peano.
== Definition ==
Intuitively, a continuous curve in 2 or 3 (or higher) dimensions can be thought of as the path of a continuously moving point. To eliminate the inhe...
1.27 Theorem: Suppose $Y$ is a subspace of a topological vector space $X$, and $Y$ is an $F$-space (in the topology inherited from $X$). Then $Y$ is a closed subspace of $X$.
Here is Rudin's proof:
I don't see why $y_0$ being in $\overline{x + W}$ for every open nbhd $W$ of $0$ implie...
