The Burnside ring of the infinite cyclic group and its relations to the necklace algebra, λ-rings, and the universal ring of Witt vectors

AbstractIt is shown that well-known product decompositions of formal power series arise from combinatorially defined canonical isomorphisms between the Burnside ring of the infinite cyclic group on the one hand and Grothendieck's ring of formal power series with constant term 1 as well as the universal ring of Witt vectors on the other hand

1/fα1/f^\alpha spectra in elementary cellular automata and fractal signals

We systematically compute the power spectra of the one-dimensional elementary cellular automata introduced by Wolfram. On the one hand our analysis reveals that one automaton displays $1/f$ spectra though considered as trivial, and on the other hand that various automata classified as chaotic/complex display no $1/f$ spectra. We model the results generalizing the recently investigated Sierpinski signal to a class of fractal signals that are tailored to produce $1/f^{\alpha}$ spectra. From the widespread occurrence of (elementary) cellular automata patterns in chemistry, physics and computer sciences, there are various candidates to show spectra similar to our results.Comment: 4 pages (3 figs included

Sierpinski signal generates 1/fα1/f^\alpha spectra

We investigate the row sum of the binary pattern generated by the Sierpinski automaton: Interpreted as a time series we calculate the power spectrum of this Sierpinski signal analytically and obtain a unique rugged fine structure with underlying power law decay with an exponent of approximately 1.15. Despite the simplicity of the model, it can serve as a model for $1/f^\alpha$ spectra in a certain class of experimental and natural systems like catalytic reactions and mollusc patterns.Comment: 4 pages (4 figs included). Accepted for publication in Physical Review

Subword complexes, cluster complexes, and generalized multi-associahedra

In this paper, we use subword complexes to provide a uniform approach to finite type cluster complexes and multi-associahedra. We introduce, for any finite Coxeter group and any nonnegative integer k, a spherical subword complex called multi-cluster complex. For k=1, we show that this subword complex is isomorphic to the cluster complex of the given type. We show that multi-cluster complexes of types A and B coincide with known simplicial complexes, namely with the simplicial complexes of multi-triangulations and centrally symmetric multi-triangulations respectively. Furthermore, we show that the multi-cluster complex is universal in the sense that every spherical subword complex can be realized as a link of a face of the multi-cluster complex.Comment: 26 pages, 3 Tables, 2 Figures; final versio

A multinomial identity for Witt vectors

Dress A, Siebeneicher C. A multinomial identity for Witt vectors. Advances in Mathematics. 1990;80(2):250-260

An application of Burnside rings in elementary finite group theory

Dress A, Siebeneicher C, Yoshida T. An application of Burnside rings in elementary finite group theory. Advances in Mathematics. 1992;91(1):27-44