:Since no actual mathematician is stepping up to the plate, I ( a humble biophysicist ) will offer my two cents, based only on visualizing a rubber sheet : " Well, yes, how could there possibly fail to be such a homeomorphism ? " I smell an existence theorem.
42.
Weinstein observed that the hypotheses of several such existence theorems could be subsumed in the condition that the level set be of contact type . ( Weinstein's original conjecture included the condition that the first de Rham cohomology group of the level set is trivial; this hypothesis turned out to be unnecessary ).
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In mathematics, the "'Kolmogorov extension theorem "'( also known as "'Kolmogorov existence theorem "'or "'Kolmogorov consistency theorem "') is a theorem that guarantees that a suitably " consistent " collection of finite-dimensional distributions will define a stochastic process.
44.
It follows from the existence theorem for limits that a category is complete if and only if it has pullbacks and binary products ( consider the pullback of ( " f ", " g " ) along the diagonal ? ), a category is complete if and only if it has pullbacks and products.
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I'm aware the fact that the complement of the semicircle is connected ( whereas the full circle is not ) is probably somehow intrinsic to the problem-however, I don't want to just prove it can be done, i want to actually exhibit such a construction, so existence theorems aren't much use here.
46.
Another approach involves defining a collection of random variables to have specific finite-dimensional distributions, and then using Kolmogorov's existence theorem to prove a corresponding stochastic process exists . says that if any finite-dimensional distributions satisfy two conditions, known as " consistency conditions ", then there exists a stochastic process with those finite-dimensional distributions.
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In fact, the model of " any " theory containing PA obtained by the systematic construction of the arithmetical model existence theorem, is " always " non-standard with a non-equivalent provability predicate and a non-equivalent way to interpret its own construction, so that this construction is non-recursive ( as recursive definitions would be unambiguous ).
48.
The general existence theorem is more subtle : it is necessary and sufficient that H be unimodular, and that the quotient H \ backslash T be of " finite volume " in a suitable sense ( which can be expressed combinatorially in terms of the action of H ), more general than the stroger condition that the quotient be finite ( as proven by the very existence of nonuniform tree lattices ).
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