arXiv subjects

Willem L. Fouche

Publications and source records attributed to Willem L. Fouche.

3 recordsLinked to original sources

arXiv2014

Diophantine properties of Brownian motion: recursive aspects

We use recent results on the Fourier analysis of the zero sets of Brownian motion to explore the diophantine properties of an algorithmically random Brownian motion (also known as a complex oscillation). We discuss the construction and definability of perfect sets which are linearly independent over the rationals directly from Martin-Löf random reals. Finally we explore the recent work of Tsirelson on countable dense sets to study the diophantine properties of local minimisers of Brownian motion.

cs.LO↗

arXiv2014

Kolmogorov complexity and the geometry of Brownian motion

In this paper, we continue the study of the geometry of Brownian motions which are encoded by Kolmogorov-Chaitin random reals (complex oscillations). We unfold Kolmogorov-Chaitin complexity in the context of Brownian motion and specifically to phenomena emerging from the random geometric patterns generated by a Brownian motion.

cs.CC↗

arXiv2012

Martin-Lof randomness, invariant measures and countable homogeneous structurs

We use ideas from topological dynamics (amenability), combinatorics (structural Ramsey theory) and model theory (Fra\" {i}ss\' e limits) to study closed amenable subgroups $G$ of the symmetric group $S_\infty$ of a countable set, where $S_\infty$ has the topology of pointwise convergence. We construct $G$-invariant measures on the universal minimal flows associated with these groups $G$ in, moreover, an algorithmic manner. This leads to an identification of the generic elements, in the sense of being Martin-L\" of random, of these flows with respect to the constructed invariant measures. Along these lines we study the random elements of $S_\infty$, which are permutations that transform recursively presented universal structures into such structures which are Martin-L\" of random.

math.LO↗