Yet it turns out that in 4D and above the fundamental group can have essentially any set of generators and relations—so that the undecidability of the word problem for arbitrary groups (see page 1141 ) implies undecidability of equivalence of manifolds.

Various physical interpretations can be given of the vanishing of the Ricci tensor implied by the ordinary vacuum Einstein equations. … The fact that T 00 is energy density implies that for static matter (where E = m c 2 ) the equation is in a sense a minimal extension of Poisson's equation of Newtonian gravity theory. Note that conservation of energy and momentum implies that T μ ν must have zero divergence—a result guaranteed in the Einstein equations by the structure of the left-hand side.

The appearance in engineering of forms somewhat like those in living systems should not be taken to imply that other forms are fundamentally difficult to produce; instead I suspect that it is more a reflection of the copying of living systems for engineering purposes.

.}  Length[k]]]] There are a total of 2 m Fibonacci[m+2] black cells in the pattern obtained up to step 2 m , implying fractal dimension Log[2, 1 + Sqrt[5]] .

The Riemann mapping theorem of complex analysis then implies that even in this case, any smooth initial shape can grow into any other such shape with a suitable growth rate function.

Averaged over all nodes, however, the number of nodes at distance up to r approximates r^Log[2, 3] , implying an effective dimension of Log[2, 3] .

The transitivity of group multiplication implies that Cayley graphs always have the property of being symmetric ( above ).

fifty or so years, it has almost never been possible to demonstrate that results obtained from such approximations even correctly reproduce what the original mathematical equations would imply.

So this implies that from a computational point of view even systems with quite different underlying structures will still usually show a certain kind of equivalence, in that rules can be found for them that achieve universality—and that therefore can always exhibit the same level of computational sophistication.

For it implies that essentially any piece of complex behavior that we see corresponds to a kind of lump of computation that is at some level equivalent.