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• Does a network have any parts that match a given subnetwork (see page 1038 )?
… (See page 1090 .)
• Does a matrix have a permanent of given value?
Several paradoxes associated with infinite sets were quickly noted—a 1901 example due to Bertrand Russell being to ask whether a set containing all sets that do not contain themselves in fact contains itself. … (The von Neumann–Bernays-Gödel formulation does avoid these, but at the cost of introducing additional objects more general than sets.)
If one has a sufficiently long sequence, therefore, just knowing that a run of equally spaced identical elements exists in it does not narrow down at all what the sequence actually is, and can so cannot ultimately be considered a useful regularity.
Scale invariance [in vision]
In a first approximation our recognition of objects does not seem to be much affected by overall size or overall light level.
In all cases the rules have been at least slightly more complicated than the ones I consider here, and behavior starting from simple initial conditions does not appear to have been studied before.
In all examples found so far the densest packings can always be repetitive; most can also be highly symmetrical—though in high dimensions random lattices often do not yield much worse results.
But except in rather simple cases there is practically no evidence that results obtained after Wick rotation have anything to do with physical reality—and certainly the transformation removes some very basic phenomena such as particle propagation.
And since one certainly does not even know at any given time exactly where all these molecules are, the details of their effect on the motion of the grain will inevitably seem quite random.
But what does this actually mean for the motion of a point in the material?
So in what kinds of systems do the largest such separations occur?