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1d

comment 
notable inductive proofs relating to fractals
That paper showed what the spectrum is. Incidentally the proof for the spectrum actually can be used to also prove the limit Laplacian also exists. But that is an older result than this paper. 
1d

comment 
notable inductive proofs relating to fractals
Fixed the links and added a longer description of what I was trying to say with the induction vs. iteration divide. Hope that helps. 
1d

revised 
notable inductive proofs relating to fractals
added 363 characters in body 
2d

answered  notable inductive proofs relating to fractals 
Feb 23 
awarded  Necromancer 
Feb 14 
answered  Can Hausdorff dimension make sets into a Tropical Semiring? 
Dec 17 
comment 
Modification of stochastic processes vs. generalized stochastic processes
It seems to me that the direct analogy of a modification to a generalized process would require a measure on $\mathcal{D}$ so you could talk about $\langle X,\varphi \rangle = \langle Y, \varphi \rangle$ for almost all $\varphi \in \mathcal{D}$. It is not clear to me that this interpretation would give anything particularly useful however. Or even be meaningful. 
Dec 17 
comment 
Is a semigroup always an exponential?
The answer to the first part of your question is to point you at the HilleYosida Theorem and its various cousins. They should be in any standard functional analysis textbook. More can't be said about the spectrum unless you have more detail about the operators $S(t)$. 
Dec 5 
awarded  Yearling 
Nov 6 
awarded  Nice Answer 
Nov 6 
answered  Can this informal argument (for the fact that almost all reals in the unit interval are irrational) be saved? 
Oct 14 
awarded  Caucus 
Jul 3 
awarded  Civic Duty 
Jun 14 
answered  How to define a differential form on a fractal? 
May 21 
comment 
Is there any result concerning on the metric dimension of inverse limit?
@Bingbing Liang, it is not clear to me right now whether the nonzero lower box counting dimension always holds for any compatible sequence of metrics. There are lots of metrics out there and I am not certain enough to rule out a counter example. 
May 19 
comment 
Is there any result concerning on the metric dimension of inverse limit?
@Misha thank you for making me think a bit longer about this. @Bingbing Liang yes, the idea is to do that and then take the limit. Z_2 is just the easiest example of this to write down. 
May 19 
revised 
Is there any result concerning on the metric dimension of inverse limit?
added 22 characters in body 
May 19 
answered  Is there any result concerning on the metric dimension of inverse limit? 
Apr 28 
comment 
Mathematical properties of financial prices
@Stefan, the same objection could be used against any physical science. Yet, as a community, we do not give up hope of modeling in the face of other very complicated systems like particle physics or the global climate. 
Apr 21 
comment 
Sierpinski Triangle and the Chaos Game
If you choose some initial point $x_0$ and then applying some sequence of the three maps is equivalent to picking one of the points in the image of the IFS which by the Hausdorff convergence must approaching the Sierpinski gasket. So my original answer was not quite complete. 