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The two most important notions of fractal dimension are Hausdorff dimension, developed by Haus-dorff (1919), and packing dimension, developed independently by Tricot (1982) and Sullivan (1984). Both dimensions have the mathematical advantage of being defined from measures, and both have yielded extensive applications in fractal geometry and dynamical(More)
The two most important notions of fractal dimension are Hausdorff dimension, developed by Hausdorff (1919), and packing dimension, developed independently by Tricot (1982) and Sullivan (1984). Both dimensions have the mathematical advantage of being defined from measures, and both have yielded extensive applications in fractal geometry and dynamical(More)
If qk is the extinction probability of a slightly supercritical branching process with offspring distribution [pkr:r = 0, 1, 2, ...], then it is shown that if supk sigma r r3pkr less than infinity, inf sigma 2k greater than 0, and mk----1, then 1-qk approximately 2(mk - 1)sigma -2k, where mk = sigma r rpkr, sigma 2k = k sigma r r2pkr - m2k. This provides a(More)
For Markov chains that can be generated by iteration of i.i.d. random maps from the state space X into itself (this holds if X is Polish) it is shown that the Doeblin minorization condition is necessary and suucient for the method by Propp and Wilson for \perfect" sampling from the stationary distribution to be successful. Using only the transition(More)
Consider a network of sites growing over time such that at step n a newcomer chooses a vertex from the existing vertices with probability proportional to a function of the degree of that vertex, i.e., the number of other vertices that this vertex is connected to. This is called a preferential attachment random graph. The objects of interest are the growth(More)