Symmetric spaces and Lie triple systems in numerical analysis of differential equations

A remarkable number of different numerical algorithms can be understood and analyzed using the concepts of symmetric spaces and Lie triple systems, which are well known in differential geometry from the study of spaces of constant curvature and their tangents. This theory can be used to unify a range of different topics, such as polar-type matrix decompositions, splitting methods for computation of the matrix exponential, composition of selfadjoint numerical integrators and dynamical systems with symmetries and reversing symmetries. The thread of this paper is the following: involutive automorphisms on groups induce a factorization at a group level, and a splitting at the algebra level. In this paper we will give an introduction to the mathematical theory behind these constructions, and review recent results. Furthermore, we present a new Yoshida-like technique, for self-adjoint numerical schemes, that allows to increase the order of preservation of symmetries by two units. Since all the time-steps are positive, the technique is particularly suited to stiff problems, where a negative time-step can cause instabilities

The Local Structure of Tilings and their Integer Group of Coinvariants

The local structure of a tiling is described in terms of a multiplicative structure on its pattern classes. The groupoid associated to the tiling is derived from this structure and its integer group of coinvariants is defined. This group furnishes part of the $K_0$-group of the groupoid $C^*$-algebra for tilings which reduce to decorations of $\Z^d$. The group itself as well as the image of its state is computed for substitution tilings in case the substitution is locally invertible and primitive. This yields in particular the set of possible gap labels predicted by $K$-theory for Schr\"odinger operators describing the particle motion in such a tiling.Comment: 45 pages including 9 figures, LaTe

Computational Processes and Incompleteness

We introduce a formal definition of Wolfram's notion of computational process based on cellular automata, a physics-like model of computation. There is a natural classification of these processes into decidable, intermediate and complete. It is shown that in the context of standard finite injury priority arguments one cannot establish the existence of an intermediate computational process

Universal quadratic forms, small norms and traces in families of number fields

We obtain good estimates on the ranks of universal quadratic forms over Shanks' family of the simplest cubic fields and several other families of totally real number fields. As the main tool we characterize all the indecomposable integers in these fields and the elements of the codifferent of small trace. We also determine the asymptotics of the number of principal ideals of norm less than the square root of the discriminant.Comment: 20 page

Aperiodic Ising Quantum Chains

Some years ago, Luck proposed a relevance criterion for the effect of aperiodic disorder on the critical behaviour of ferromagnetic Ising systems. In this article, we show how Luck's criterion can be derived within an exact renormalisation scheme for Ising quantum chains with coupling constants modulated according to substitution rules. Luck's conjectures for this case are confirmed and refined. Among other outcomes, we give an exact formula for the correlation length critical exponent for arbitrary two-letter substitution sequences with marginal fluctuations of the coupling constants.Comment: 27 pages, LaTeX, 1 Postscript figure included, using epsf.sty and amssymb.sty (one error corrected, some minor changes

Computation and visualization of photonic quasicrystal spectra via Blochs theorem

Previous methods for determining photonic quasicrystal (PQC) spectra have relied on the use of large supercells to compute the eigenfrequencies and/or local density of states (LDOS). In this manuscript, we present a method by which the energy spectrum and the eigenstates of a PQC can be obtained by solving Maxwells equations in higher dimensions for any PQC defined by the standard cut-and-project construction, to which a generalization of Blochs theorem applies. In addition, we demonstrate how one can compute band structures with defect states in the higher-dimensional superspace with no additional computational cost. As a proof of concept, these general ideas are demonstrated for the simple case of one-dimensional quasicrystals, which can also be solved by simple transfer-matrix techniques.Comment: Published in Physical Review B, 77 104201, 200

Negative curvature in graphical small cancellation groups

We use the interplay between combinatorial and coarse geometric versions of negative curvature to investigate the geometry of infinitely presented graphical $Gr'(1/6)$ small cancellation groups. In particular, we characterize their 'contracting geodesics', which should be thought of as the geodesics that behave hyperbolically. We show that every degree of contraction can be achieved by a geodesic in a finitely generated group. We construct the first example of a finitely generated group $G$ containing an element $g$ that is strongly contracting with respect to one finite generating set of $G$ and not strongly contracting with respect to another. In the case of classical $C'(1/6)$ small cancellation groups we give complete characterizations of geodesics that are Morse and that are strongly contracting. We show that many graphical $Gr'(1/6)$ small cancellation groups contain strongly contracting elements and, in particular, are growth tight. We construct uncountably many quasi-isometry classes of finitely generated, torsion-free groups in which every maximal cyclic subgroup is hyperbolically embedded. These are the first examples of this kind that are not subgroups of hyperbolic groups. In the course of our analysis we show that if the defining graph of a graphical $Gr'(1/6)$ small cancellation group has finite components, then the elements of the group have translation lengths that are rational and bounded away from zero.Comment: 40 pages, 14 figures, v2: improved introduction, updated statement of Theorem 4.4, v3: new title (previously: "Contracting geodesics in infinitely presented graphical small cancellation groups"), minor changes, to appear in Groups, Geometry, and Dynamic