Juergen Jost

Synchronization of networks with prescribed degree distributions

We show that the degree distributions of graphs do not suffice to characterize the synchronization of systems evolving on them. We prove that, for any given degree sequence satisfying certain conditions, there exists a connected graph having that degree sequence for which the first nontrivial eigenvalue of the graph Laplacian is arbitrarily close to zero. Consequently, complex dynamical systems defined on such graphs have poor synchronization properties. The result holds under quite mild assumptions, and shows that there exists classes of random, scale-free, regular, small-world, and other common network architectures which impede synchronization. The proof is based on a construction that also serves as an algorithm for building nonsynchronizing networks having a prescribed degree distribution.

Network synchronization: Spectral versus statistical properties

We consider synchronization of weighted networks, possibly with asymmetrical connections. Focusing on causal relations rather than the observed correlations, we show that the synchronizability of networks cannot be directly inferred from their statistical properties. Small local changes in the network structure can sensitively affect the eigenvalues relevant for synchronization, while the gross statistical network properties remain essentially unchanged. Consequently, commonly used statistical properties, including the degree distribution, degree homogeneity, average degree, average distance, degree correlation and clustering coefficient, can fail to characterize the synchronizability of networks in terms of causal relations, despite the observed correlations. (c) 2006 Elsevier B.V. All rights reserved.

Some Aspects of the global Geometry of Entire Space-Like Submanifolds

We prove some Bernstein type theorems for entire space-like subma-nifolds in pseudo-Euclidean space and as a corollary, we give a new proof of the Calabi-Pogorelov theorem for Monge-Ampère equations.

Liouville theorems for Dirac-harmonic maps

We prove Liouville theorems for Dirac-harmonic maps from the Euclidean space Rn, the hyperbolic space Hn, and a Riemannian manifold Sn (n⩾3) with the Schwarzschild metric to any Riemannian manifold N.

Entanglement of Formation for a Class of Quantum States

Abstract Entanglement of formation for a class of higher-dimensional quantum mixed states is studied in terms of a generalized formula of concurrence for N -dimensional quantum systems. As applications, the entanglement of formation for a class of 16×16 density matrices are calculated.

Complexity measures from interaction structures.

We evaluate information-theoretic quantities that quantify complexity in terms of kth-order statistical dependences that cannot be reduced to interactions among k-1 random variables. Using symbolic dynamics of coupled maps and cellular automata as model systems, we demonstrate that these measures are able to identify complex dynamical regimes.

Super-Liouville Equations on Closed Riemann Surfaces

Motivated by the supersymmetric extension of Liouville theory in the recent physics literature, we couple the standard Liouville functional with a spinor field term. The resulting functional is conformally invariant. We study geometric and analytic aspects of the resulting Euler–Lagrange equations, culminating in a blow up analysis.

Synchronized chaos in networks of simple units

We study synchronization of non-diffusively coupled map networks with arbitrary network topologies, where the connections between different units are, in general, not symmetric and can carry both positive and negative weights. We show that, in contrast to diffusively coupled networks, the synchronous behavior of a non-diffusively coupled network can be dramatically different from the behavior of its constituent units. In particular, we show that chaos can emerge as synchronized behavior although the dynamics of individual units are very simple. Conversely, individually chaotic units can display simple behavior when the network synchronizes. We give a synchronization criterion that depends on the spectrum of the generalized graph Laplacian, as well as the dynamical properties of the individual units and the interaction function. This general result is then applied to coupled systems of tent and logistic maps and to two models of neuronal dynamics. Our approach yields an analytical understanding of how simple model neurons can produce complex collective behavior through the coordination of their actions.

Some explicit constructions of Dirac-harmonic maps

We construct explicit examples of Dirac-harmonic maps (φ, ψ) between Riemannian manifolds (M, g) and (N, g ′ ) which are non-trivial in the sense that φ is not harmonic. When dim M = 2, we also produce examples where φ is harmonic, but not conformal, and ψ is non-trivial.

Synchronization in discrete-time networks with general pairwise coupling

We consider complete synchronization of identical maps coupled through a general interaction function and in a general network topology where the edges may be directed and may carry both positive and negative weights. We define mixed transverse exponents and derive sufficient conditions for local complete synchronization. The general non-diffusive coupling scheme can lead to new synchronous behaviour, in networks of identical units, that cannot be produced by single units in isolation. In particular, we show that synchronous chaos can emerge in networks of simple units. Conversely, in networks of chaotic units simple synchronous dynamics can emerge; that is, chaos can be suppressed through synchrony.