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Juergen Jost

Publications and source records attributed to Juergen Jost.

At least 55 records · Page 3Linked to original sources
arXiv2005

Global Analysis of Synchronization in Coupled Maps

We introduce a new method for determining the global stability of synchronization in systems of coupled identical maps. The method is based on the study of invariant measures. Besides the simplest non-trivial example, namely two symmetrically coupled tent maps, we also treat the case of two asymmetrically coupled tent maps as well as a globally coupled network. Our main result is the identification of the precise value of the coupling parameter where the synchronizing and desynchronizing transitions take place.

nlin.CD↗
arXiv2005

Temporal correlation based learning in neuron models

We study a learning rule based upon the temporal correlation (weighted by a learning kernel) between incoming spikes and the internal state of the postsynaptic neuron, building upon previous studies of spike timing dependent synaptic plasticity (\cite{KGvHW,KGvH1,vH}). Our learning rule for the synaptic weight $w_{ij}$ is $$ \dot w_{ij}(t)= ε\int_{-\infty}^\infty \frac{1}{T_l} \int_{t-T_l}^t \sum_μδ(τ+s-t_{j,μ}) u(τ) dτ Γ(s)ds $$ where the $t_{j,μ}$ are the arrival times of spikes from the presynaptic neuron $j$ and the function $u(t)$ describes the state of the postsynaptic neuron $i$. Thus, the spike-triggered average contained in the inner integral is weighted by a kernel $Γ(s)$, the learning window, positive for negative, negative for positive values of the time diffence $s$ between post- and presynaptic activity. An antisymmetry assumption for the learning window enables us to derive analytical expressions for a general class of neuron models and to study the changes in input-output relationships following from synaptic weight changes. This is a genuinely non-linear effect (\cite{SMA}).

q-bio.NC↗
arXiv2005

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.

math.DG↗
arXiv2005

Fractal Stationary Density in Coupled Maps

We study the invariant measure or the stationary density of a coupled discrete dynamical system as a function of the coupling parameter ε(0 < ε< 1/4). The dynamical system considered is chaotic and unsynchronized for this range of parameter values. We find that the stationary density, restricted on the synchronization manifold, is a fractal function. We find the lower bound on the fractal dimension of the graph of this function and show that it changes continuously with the coupling parameter

nlin.CD↗
arXiv2005

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 non-synchronizing networks having a prescribed degree distribution.

nlin.AO↗
arXiv2005

Cohomologies of unipotent harmonic bundles over quasi-projective varieties I: The case of noncompact curves

Let $S$ be a Riemann surface obtained by deleting a finite number of points, called cusps, from a compact Riemann surface. Let $ρ: π_1(S)\to Sl(n, \mathbb{C})$ be a semisimple linear representation of $π_1(S)$ which is unipotent near the cusps. We investigate various cohomologies associated to $ρ$ of $\bar S$ with degenerating coefficients $L_ρ$ (considered as a local system -- a flat vector bundle, a Higgs bundle, or a $\mathcal{D}$-module, depending on the context): the Čech cohomology of $j_*L_ρ$, the $L^2$-cohomology, the $L^2$-Dolbeault cohomology, and the $L^2$-Higgs cohomology, and the relationships between them. This paper is meant to be a part of the general program of studying cohomologies with degenerating coefficients on quasiprojective varieties and their Kählerian generalizations. The general aim here is not restricted to the case of curves nor to the one of representations that are unipotent near the divisor. The purpose of this note therefore is to illuminate at this particular case where many of the (analytic and geometric) difficulties of the general case are not present what differences will appear when we consider unipotent harmonic bundles instead of Variations of Hodge Structures where the results are known.

math.AG↗
arXiv2004

Dirac-Harmonic Maps

We introduce a functional that couples the nonlinear sigma model with a spinor field: $L=\int_M[|dϕ|^2+(ψ,\Dψ)]$. In two dimensions, it is conformally invariant. The critical points of this functional are called Dirac-harmonic maps. We study some geometric and analytic aspects of such maps, in particular a removable singularity theorem.

math.DG↗
arXiv2004

Regularity Theorems and Energy Identities for Dirac-Harmonic Maps

We study a new set of coupled field equations motivated by the non-linear supersymmetric sigma model of quantum field theory. These equations couple a map into a Riemannian manifold controlled by a harmonic map like action with a spinor field along that map. We study the solutions which we call Dirac-harmonic maps from a Riemann surface to a sphere $§^n$. We show that a weakly Dirac-harmonic map is in fact smooth, and prove that the energy identity holds during the blow-up process.

math.DG↗
arXiv2004

Individual strategies in complementarity games and population dynamics

We introduce and study an evolutionary complementarity game where in each round a player of population 1 is paired with a member of population 2. The game is symmetric, and each player tries to obtain an advantageous deal, but when one of them pushes too hard, no deal at all can be concluded, and they both loose. The game has many equilibria, and which of them is reached depends on the history of the interactions as the players evolve according to a fitness function that measures their gains across the interactions. We can then break the symmetry by assigning different strategy spaces to the populations, varying in particular with respect to the information available to the agents. The agents can, for example, adapt to the behavior of their opponents met in previous rounds, or they can try to copy the strategies of their successful friends. It turns out that, in general, the more restricted strategy spaces, that is, those that utilize less information, are more advantageous for a population as a whole as their adoption drives the equilibrium in a direction advantageous to that population. One reason is that a simpler strategy can be learned faster in an evolutionary setting, another is that it is good for a population to have some individuals that are unfit in the sense that they make offers that are individually unsuccessful, but have a systematic effect on the strategies of their opponents. All these effects are demonstrated through systematic simulations.

nlin.AO↗
arXiv2003

Maps with prescribed tension fields

We consider maps into Riemannian manifolds of non-positive curvature and start developing a systematic PDE theory. We control the Sobolev $H^{2,2}$-norm of such a map in terms of its energy, the $L^2$-norm of its tension field and a topological term depending on the homotopy class. We also solve a Dirchlet problem without an underlying variational structure, as an extension of the topic of harmonic maps with potentials.

math.DG↗
arXiv2003

Kähler manifolds and fundamental groups of negatively $δ$-pinched manifolds

The fundamental group of a Riemannian manifold with $δ$-pinched negative curvature, $δ>1/4$, cannot be the fundamental group of a quasicompact Kähler manifold. The proof also implies that a non-uniform lattice in $F_{4(-20)}$ cannot be the fundamental group of a quasicompact Kähler manifold. We also construct examples in the spirit of Gromov-Thurston to show that our result is a non-trivial extension of the previously known result that a non-uniform lattice in real hyperbolic space in dimension at least 3 cannot be the fundamental group of a quasicompact Kähler manifold.

math.DG↗
arXiv2003

Conley Index Theory and Novikov-Morse Theory

We derive general Novikov-Morse type inequalities in a Conley type framework for flows carrying cocycles, therefore generalizing our results in [FJ2] derived for integral cocycle. The condition of carrying a cocycle expresses the nontriviality of integrals of that cocycle on flow lines. Gradient-like flows are distinguished from general flows carrying a cocycle by boundedness conditions on these integrals.

math.GT↗
arXiv2003

Entanglement of Formation for a Class of Quantum States

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 $16\times 16$ density matrices are calculated.

quant-ph↗
arXiv2003

Harmonic bundles on quasi-compact Kaehler manifolds

We define the C^*-action on moduli spaces of reductive representations of fundamental groups of quasi-compact Kaehler manifolds by solving Hermitian-Yang-Mills equation. As applications in algebraic geometry we show a non-abelian Hodge (p,q)-type theorem for families of quasi-projective manifolds. We also prove that any discrete sub group of a Lie group of rank >1 that is not of Hodge type can't be the fundamental group of a quasi-compact Kaehler manifold.

math.AG↗