Jan Oderfeld & Hugo Steinhaus

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Jan Oderfeld & Hugo Steinhaus Dr. Jan Oderfeld (1908 – 2010) was a renowned Polish probabilist and statistician; I was fortunate to have him as my PhD advisor in applied mathematics. https://pl.wikipedia.org/wiki/Jan_Oderfeld Dr. Oderfeld’s PhD advisor was Hugo Steinhaus, a world-famous mathematician, one of the founders of probability theory and game theory. https://en.wikipedia.org/wiki/Hugo_Steinhaus

A Bit More About Steinhaus Hugo Steinhaus (1887 – 1972) was an early contributor to, and co-founder of, probability theory, which at the time was in its infancy and not even considered an actual part of mathematics. He provided the first axiomatic measure-theoretic description of coin-tossing, which was to influence the full axiomatization of probability by the Russian mathematician Andrey Kolmogorov a decade later. Steinhaus was also the first to offer precise definitions of what it means for two events to be "independent", as well as for what it means for a random variable to be "uniformly distributed".

Krzysztof Burdzy Krzysztof Burdzy is an eminent probabilist from the University of Washington: http://www.math.washington.edu/~burdzy/ Recently he has been interested in the philosophy of probability. He published 6 books and about 160 papers (in two of these I am his co-author): Brownian motion with inert drift, but without flux: a model Physica A 384 (2007) Fractal trace of earthworms Phys. Rev. E 87 no. 5, May 2013

Fractal Trace of Earthworms Abstract We investigate a process of random walks of a point particle on a two-dimensional square lattice of size n × n with periodic boundary conditions. A fraction p 20% of the lattice is occupied by holes (p represents macroporosity). A site not occupied by a hole is occupied by an obstacle. Upon a random step of the walker, a number of obstacles, M, can be pushed aside. The system approaches equilibrium in (n log n)^2 steps. We determine the distribution of M pushed in a single move at equilibrium. The distribution F(M) is given by Mγ where γ = −1.18 for p = 0.1, decreasing to γ = −1.28 for p = 0.01. Irrespective of the initial distribution of holes on the lattice, the final equilibrium distribution of holes forms a fractal with fractal dimension changing from a = 1.56 for p = 0.20 to a = 1.42 for p = 0.001 (for n = 4,000). The trace of a random walker forms a distribution with expected fractal dimension 2.