The expected area of the filled planar Brownian loop is Pi/5

Let B_t be a planar Brownian loop of time duration 1 (a Brownian motion conditioned so that B_0 = B_1). We consider the compact hull obtained by filling in all the holes, i.e. the complement of the unique unbounded component of R^2\B[0,1]. We show that the expected area of this hull is Pi/5. The proof uses, perhaps not surprisingly, the Schramm Loewner Evolution (SLE). Also, using the result of Yor about the law of the index of a Brownian loop, we show that the expected areas of the regions of non-zero index n equal 1/(2 Pi n^2). As a consequence, we find that the expected area of the region of index zero inside the loop is Pi/30; this value could not be obtained directly using Yor's index description.Comment: 15 pages, 3 figure

The Radial Distribution of the Kuiper Belt

We examine the radial distribution of the Kuiper Belt objects (KBOs) using a method that is insensitive to observational bias effects. This technique allows the use of the discovery distances of all KBOs, independent of orbital classification or discovery circumstance. We verify the presence of an outer edge to the Kuiper Belt, as reported in other works, and we measure this edge to be at R = 47 ± 1 AU given any physically plausible model of the size distribution. We confirm that this outer edge is due to the classical KBOs, the most numerically dominant observationally. In addition, we find that current surveys do not preclude the presence of a second, unobserved Kuiper Belt beyond R = 76 AU

A recipe for an unpredictable random number generator

In this work we present a model for computation of random processes in digital computers which solves the problem of periodic sequences and hidden errors produced by correlations. We show that systems with non-invertible non-linearities can produce unpredictable sequences of independent random numbers. We illustrate our result with some numerical calculations related with random walks simulations.Comment: 8 pages, 5 figures, Proceedings Mochima spring school in theoretial physic