A unified view on bipartite species-reaction and interaction graphs for chemical reaction networks

The Jacobian matrix of a dynamic system and its principal minors play a prominent role in the study of qualitative dynamics and bifurcation analysis. When interpreting the Jacobian as an adjacency matrix of an interaction graph, its principal minors correspond to sets of disjoint cycles in this graph and conditions for various dynamic behaviors can be inferred from its cycle structure. For chemical reaction systems, more fine-grained analyses are possible by studying a bipartite species-reaction graph. Several results on injectivity, multistationarity, and bifurcations of a chemical reaction system have been derived by using various definitions of such bipartite graph. Here, we present a new definition of the species-reaction graph that more directly connects the cycle structure with determinant expansion terms, principal minors, and the coefficients of the characteristic polynomial and encompasses previous graph constructions as special cases. This graph has a direct relation to the interaction graph, and properties of cycles and sub-graphs can be translated in both directions. A simple equivalence relation enables to decompose determinant expansions more directly and allows simpler and more direct proofs of previous results.Comment: 27 pages. submitted to J. Math. Bio

Partial norms and the convergence of general products of matrices

Motivated by the theory of inhomogeneous Markov chains, we determine a sufficient condition for the convergence to 0 of a general product formed from a sequence of real or complex matrices. When the matrices have a common invariant subspace $H$, we give a sufficient condition for the convergence to 0 on $H$ of a general product. Our result is applied to obtain a condition for the weak ergodicity of an inhomogeneous Markov chain. We compare various types of contractions which may be defined for a single matrix, such as paracontraction, $l$--contraction, and $H$--contraction, where $H$ is an invariant subspace of the matrix

Control of Towing Kites for Seagoing Vessels

In this paper we present the basic features of the flight control of the SkySails towing kite system. After introduction of coordinate definitions and basic system dynamics we introduce a novel model used for controller design and justify its main dynamics with results from system identification based on numerous sea trials. We then present the controller design which we successfully use for operational flights for several years. Finally we explain the generation of dynamical flight patterns.Comment: 12 pages, 18 figures; submitted to IEEE Trans. on Control Systems Technology; revision: Fig. 15 corrected, minor text change

Stability of the decagonal quasicrystal in the Lennard-Jones-Gauss system

Although quasicrystals have been studied for 25 years, there are many open questions concerning their stability: What is the role of phason fluctuations? Do quasicrystals transform into periodic crystals at low temperature? If yes, by what mechanisms? We address these questions here for a simple two-dimensional model system, a monatomic decagonal quasicrystal, which is stabilized by the Lennard-Jones-Gauss potential in thermodynamic equilibrium. It is known to transform to the approximant Xi, when cooled below a critical temperature. We show that the decagonal phase is an entropically stabilized random tiling. By determining the average particle energy for a series of approximants, it is found that the approximant Xi is the one with lowest potential energy.Comment: 7 pages, 2 figures, Proceedings of Quasicrystals - The Silver Jubile