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A real-valued stochastic process {B(t):t>=0} is a Brownian motion which starts at x in R if the following properties are satisfied: 1. B(0)=x. 2. For all times ...

A Markov chain is collection of random variables {X_t} (where the index t runs through 0, 1, ...) having the property that, given the present, the future is conditionally ...

The term "higher dimensional group theory" was introduced by Brown (1982), and refers to a method for obtaining new homotopical information by generalizing to higher ...

A box integral for dimension n with parameters q and s is defined as the expectation of distance from a fixed point q of a point r chosen at random over the unit n-cube, ...

The two-dimensional space consisting of the set of triples {(a,b,c):a,b,c in K, not all zero}, where triples which are scalar multiples of each other are identified.

Triangle picking is the process of picking triangles at random within a given shape in the plane, in space, or in a higher dimension. The most natural definition of a random ...

Intuitively, a model of d-dimensional percolation theory is said to be a Bernoulli model if the open/closed status of an area is completely random. In particular, it makes ...

A d-dimensional discrete percolation model is said to be inhomogeneous if different graph edges (in the case of bond percolation models) or vertices (in the case of site ...

A fair coin is tossed an even 2n number of times. Let D=|H-T| be the absolute difference in the number of heads and tails obtained. Then the probability distribution is given ...

What is the probability that a chord drawn at random on a circle of radius r (i.e., circle line picking) has length >=r (or sometimes greater than or equal to the side length ...