General evaluation of suborbital graphs

In this paper, we aim to review suborbital graphs and also give an example to an extension of directed graphs

Limit Cycles of AdaBoost

The iterative weight update for the AdaBoost machine learning algorithm may be realized as a dynamical map on a probability simplex. When learning a low-dimensional data set this algorithm has a tendency towards cycling behavior, which is the topic of this paper. AdaBoost's cycling behavior lends itself to direct computational methods that are ineffective in the general, non-cycling case of the algorithm. From these computational properties we give a concrete correspondence between AdaBoost's cycling behavior and continued fractions dynamics. Then we explore the results of this correspondence to expound on how the algorithm comes to be in this periodic state at all. What we intend for this work is to be a novel and self-contained explanation for the cycling dynamics of this machine learning algorithm

Regular coverings and parallel products of Farey maps

We examine the structure of Farey maps, which are a class of maps (graph embeddings on surfaces) that have received significant attention recently. We describe how they are related to each other through coverings and parallel products, and use these observations to find their complete spectra, recovering some known results. We then examine a similar class of maps defined by Hecke groups

Farey Maps, Spectra and Integer Continued Fractions

We examine the structure of Farey maps, a class of graph embeddings on surfaces that have received significant attention recently. When the Farey graph is embedded in the hyperbolic plane it induces a tessellation by ideal triangles. Farey maps are the quotients of this tessellation by the principal congruence subgroups of the modular group. We describe how the Farey maps of different levels are related to each other through regular coverings and parallel products, and use this to find their complete spectra. We then generalise Farey maps to include those defined by non–principal congruence subgroups of the modular group, finding their spectra and diameter. We also examine a similar class of maps defined by Hecke groups, again obtaining results for their spectra and diameter. Most of this work is the subject of [63], which has been published in Acta Mathematica Universitatis Comenianae. Fundamental to the theory of continued fractions is the fact that every infinite continued fraction with positive integer coefficients converges; however, this is not so if the coefficients are integers which are not necessarily positive. We show that integer continued fractions can be represented as paths on the Farey graph, and use this to develop a simple test that determines whether an integer continued fraction converges or diverges. In addition, for convergent continued fractions, the test specifies whether the limit is rational or irrational. This work, carried out jointly with Ian Short, is the subject of [57], which has been published in the Proceedings of the American Mathematical Society. Finally further work is described, including practical applications of our spectral results, and a search for interesting expansions of real numbers as generalised continued fractions