A proof of Wright's conjecture

Wright's conjecture states that the origin is the global attractor for the delay differential equation $y'(t) = - \alpha y(t-1) [ 1 + y(t) ]$ for all $\alpha \in (0,\tfrac{\pi}{2}]$. This has been proven to be true for a subset of parameter values $\alpha$. We extend the result to the full parameter range $\alpha \in (0,\tfrac{\pi}{2}]$, and thus prove Wright's conjecture to be true. Our approach relies on a careful investigation of the neighborhood of the Hopf bifurcation occurring at $\alpha =\tfrac{\pi}{2}$. This analysis fills the gap left by complementary work on Wright's conjecture, which covers parameter values further away from the bifurcation point. Furthermore, we show that the branch of (slowly oscillating) periodic orbits originating from this Hopf bifurcation does not have any subsequent bifurcations (and in particular no folds) for $\alpha\in(\tfrac{\pi}{2} , \tfrac{\pi}{2} + 6.830 \times 10^{-3}]$. When combined with other results, this proves that the branch of slowly oscillating solutions that originates from the Hopf bifurcation at $\alpha=\tfrac{\pi}{2}$ is globally parametrized by $\alpha > \tfrac{\pi}{2}$.Comment: 45 page

A comparison of two techniques for bibliometric mapping: Multidimensional scaling and VOS

VOS is a new mapping technique that can serve as an alternative to the well-known technique of multidimensional scaling. We present an extensive comparison between the use of multidimensional scaling and the use of VOS for constructing bibliometric maps. In our theoretical analysis, we show the mathematical relation between the two techniques. In our experimental analysis, we use the techniques for constructing maps of authors, journals, and keywords. Two commonly used approaches to bibliometric mapping, both based on multidimensional scaling, turn out to produce maps that suffer from artifacts. Maps constructed using VOS turn out not to have this problem. We conclude that in general maps constructed using VOS provide a more satisfactory representation of a data set than maps constructed using well-known multidimensional scaling approaches

The parameterization method for center manifolds

In this paper, we present a generalization of the parameterization method, introduced by Cabr\'{e}, Fontich and De la Llave, to center manifolds associated to non-hyperbolic fixed points of discrete dynamical systems. As a byproduct, we find a new proof for the existence and regularity of center manifolds. However, in contrast to the classical center manifold theorem, our parameterization method will simultaneously obtain the center manifold and its conjugate center dynamical system. Furthermore, we will provide bounds on the error between approximations of the center manifold and the actual center manifold, as well as bounds for the error in the conjugate dynamical system

Generalized Hopfield networks for constrained optimization

A twofold generalization of the classical continuous Hopfield neural network for modelling constrained optimization problems is proposed. On the one hand, non-quadratic cost functions are admitted corresponding to non-linear output summation functions in the neurons. On the other hand it is shown under which conditions various (new) types of constraints can be incorporated directly. The stability properties of several relaxation schemes are shown. If a direct incorporation of the constraints appears to be impossible, the Hopfield-Lagrange model can be applied, the stability properties of which are analyzed as well. Another good way to deal with constraints is by means of dynamic penalty terms, using mean field annealing in order to end up in a feasible solution. A famous example in this context is the elastic net, although it seems impossible - contrary to what is suggested in the literature - to derive the architecture of this network from a constrained Hopfield model. Furthermore, a non-equidistant elastic net is proposed and its stability properties are compared to those of the classical elastic network. In addition to certain simulation results as known from the literature, most theoretical statements of this paper are validated with simulations of toy problems while in some cases, more sophisticated combinatorial optimization problems have been tried as well. In the final section, we discuss the possibilities of applying the various models in the area of constrained optimization. It is also demonstrated how the new ideas as inspired by the analysis of generalized continuous Hopfield models, can be transferred to discrete stochastic Hopfield models. By doing so, simulating annealing can be exploited in other to improve the quality of solutions. The transfer also opens new avenues for continued theoretical research

Spintronics and thermoelectrics in exfoliated and epitaxial graphene

This thesis is about two subjects: graphene spintronics and graphene thermoelectrics. Spintronics is about the creation and manipulation of spin currents. These are electrical currents in which we can control the spin orientation (up or down) of the conduction electrons. The second subject, thermoelectrics, is about the interaction between heat and charge currents. A classic thermoelectric phenomena is the Peltier effect, the cooling or heating of an interface between two materials when an electric current is flowing through. Graphene is an interesting material for studying both spintronic and thermoelectric effects. Spins in graphene can travel unperturbed over a distance of several micrometers at room temperature, further than in any other material. And the strength of the Peltier effect in graphene can be controlled with an applied voltage, even allowing for the switching between cooling and heating the interface. In this thesis I investigate the suitability of epitaxial graphene in spintronic applications. Epitaxial graphene is a large area graphene type that can be grown on a silicon carbide (SiC) chip. I show that it is possible to produce wafer scale graphene spin devices and that spin transport behaves differently on this particular substrate, because spins are temporarily immobilized in charge traps that are present in the graphene-SiC interface. I also investigate a nanodevice that (1) can switch between heating and cooling of a graphene-gold interface and (2) can measure the induced temperature change using a nano-scaled thermocouple