Snooker Statistics and Zipf's Law

Zipf's law is well known in linguistics: the frequency of a word is inversely proportional to its rank. This is a special case of a more general power law, a common phenomenon in many kinds of real-world statistical data. Here, it is shown that snooker statistics also follow such a mathematical pattern, but with varying (estimated) parameter values. Two types of rankings (prize money earned and centuries scored), and three time-frames (all-time, decade, and year) are considered. The results indicate that the power law parameter values depend on the type of ranking used as well as the time-frame considered. Furthermore, in some cases the resulting parameter values vary significantly over time, for which a plausible explanation is provided.Comment: 7 pages, 3 figure

Autocatalytic sets in a partitioned biochemical network

In previous work, RAF theory has been developed as a tool for making theoretical progress on the origin of life question, providing insight into the structure and occurrence of self-sustaining and collectively autocatalytic sets within catalytic polymer networks. We present here an extension in which there are two "independent" polymer sets, where catalysis occurs within and between the sets, but there are no reactions combining polymers from both sets. Such an extension reflects the interaction between nucleic acids and peptides observed in modern cells and proposed forms of early life.Comment: 28 pages, 8 figure

Dynamics of a birth-death process based on combinatorial innovation

A feature of human creativity is the ability to take a subset of existing items (e.g. objects, ideas, or techniques) and combine them in various ways to give rise to new items, which, in turn, fuel further growth. Occasionally, some of these items may also disappear (extinction). We model this process by a simple stochastic birth--death model, with non-linear combinatorial terms in the growth coefficients to capture the propensity of subsets of items to give rise to new items. In its simplest form, this model involves just two parameters $(P, \alpha)$. This process exhibits a characteristic 'hockey-stick' behaviour: a long period of relatively little growth followed by a relatively sudden 'explosive' increase. We provide exact expressions for the mean and variance of this time to explosion and compare the results with simulations. We then generalise our results to allow for more general parameter assignments, and consider possible applications to data involving human productivity and creativity.Comment: 21 pages, 4 figure

The Ideal Storage Cellular Automaton Model

We have implemented and investigated a spatial extension of the orig- inal ideal storage model by embedding it in a 2D cellular automaton with a diffusion-like coupling between neighboring cells. The resulting ideal storage cellular automaton model (ISCAM) generates many interesting spatio-temporal patterns, in particular spiral waves that grow and com- pete" with each other. We study this dynamical behavior both mathemat- ically and computationally, and compare it with similar patterns observed in actual chemical processes. Remarkably, it turned out that one can use such CA for modeling all sorts of complex processes, from phase transition in binary mixtures to using them as a metaphor for cancer onset caused by only one short pulse of \u27tissue dis-organzation\u27 (changing e.g. for only one single time step the diffusion coefficient) as hypothesized in recent papers questioning the current gene/genome centric view on cancer onset by AO Ping et al

Shapes of tree representations of spin-glass landscapes

Much of the information about the multi-valley structure of disordered spin systems can be convened in a simple tree structure - a barrier tree - the leaves and internal nodes of which represent, respectively, the local minima and the lowest energy saddles connecting those minima. Here we apply several statistics used in the study of phylogenetic trees to barrier trees that result from the energy landscapes of p-spin models. These statistics give information about the shape of these barrier trees, in particular about balance and symmetry. We then ask if they can be used to classify different types of landscapes, compare them with results obtained from random trees, and investigate the structure of subtrees of the barrier trees. We conclude that at least one of the used statistics is capable of distinguishing different types of landscapes, that the barrier trees from p-spin energy landscapes are quite different from random trees, and that subtrees of barrier trees do not reflect the overall tree structure, but their structure is correlated with their ´depth' in the tree