The Brownian Fan

Brownian Motion: The Link Between Probability and Mathematical Analysis

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The fan theorem

Walking Through The Gröbner Fan

The Gröbner Walk is an algorithm that transforms a polynomial ideal’s Gröbner basis under one term order into a basis under another order. Although we could always directly compute such a basis using Buchberger’s algorithm, the Gröbner Walk is of special interest because, in many cases, it is much more efficient to convert a basis using this algorithm than to compute it directly. We first discu...

Fan Homeomorphisms in the Plane

In this paper x, a denote real numbers and p, q denote points of E2 T. The following propositions are true: (1) If |x| < a, then −a < x and x < a. (2) If a ­ 0 and (x− a) · (x + a) < 0, then −a < x and x < a. (3) For every real number s1 such that −1 < s1 and s1 < 1 holds 1 + s1 > 0 and 1− s1 > 0. (4) For every real number a such that a2 ¬ 1 holds −1 ¬ a and a ¬ 1. (5) For every real number a s...

Brownian Brownian Motion – I

A classical model of Brownian motion consists of a heavy molecule submerged into a gas of light atoms in a closed container. In this work we study a 2D version of this model, where the molecule is a heavy disk of mass M ≫ 1 and the gas is represented by just one point particle of mass m = 1, which interacts with the disk and the walls of the container via elastic collisions. Chaotic behavior of...