Boltzmann and Fokker-Planck equations modelling the Elo rating system with learning effects

In this paper we propose and study a new kinetic rating model for a large number of players, which is motivated by the well-known Elo rating system. Each player is characterised by an intrinsic strength and a rating, which are both updated after each game. We state and analyse the respective Boltzmann type equation and derive the corresponding nonlinear, nonlocal Fokker-Planck equation. We investigate the existence of solutions to the Fokker-Planck equation and discuss their behaviour in the long time limit. Furthermore, we illustrate the dynamics of the Boltzmann and Fokker-Planck equation with various numerical experiments

Filamentous supramolecular structures with polyelectrolyte and cadmium sulfide

In this study, a new type of filamentous structures consisting of a generation 9 poly(amido amine) dendrimer (G9) and CdS is reported. The linearity of the interconnected dendrimers is a result of the electrostatic repulsion between the multiply charged dendrimer macroions. Structures have been investigated by dynamic light scattering (DLS) and transmission electron microscopy (TEM). The internal structure of the CdS-fibers reveals information on the mechanism of the fiber formation. In contrast to previous systems with smaller generation poly(propylene imine)-dendrimers, Cd2+ is here found to be responsible for the interconnection of G9. Furthermore, more complex supramolecular structures were built by associating the CdS–dendrimer hybrid fibers with different ionic dyes, displaying the versatility of this system for future nanotechnology applications such as optoelectronics or energy conversion

Kinetic models for optimal control of wealth inequalities

We introduce and discuss optimal control strategies for kinetic models for wealth distribution in a simple market economy, acting to minimize the variance of the wealth density among the population. Our analysis is based on a finite time horizon approximation, or model predictive control, of the corresponding control problem for the microscopic agents' dynamic and results in an alternative theoretical approach to the taxation and redistribution policy at a global level. It is shown that in general the control is able to modify the Pareto index of the stationary solution of the corresponding Boltzmann kinetic equation, and that this modification can be exactly quantified. Connections between previous Fokker-Planck based models and taxation-redistribution policies and the present approach are also discussed

Phase Diagram and Storage Capacity of Sequence Processing Neural Networks

We solve the dynamics of Hopfield-type neural networks which store sequences of patterns, close to saturation. The asymmetry of the interaction matrix in such models leads to violation of detailed balance, ruling out an equilibrium statistical mechanical analysis. Using generating functional methods we derive exact closed equations for dynamical order parameters, viz. the sequence overlap and correlation- and response functions, in the thermodynamic limit. We calculate the time translation invariant solutions of these equations, describing stationary limit-cycles, which leads to a phase diagram. The effective retarded self-interaction usually appearing in symmetric models is here found to vanish, which causes a significantly enlarged storage capacity of $\alpha_c\sim 0.269$, compared to \alpha_\c\sim 0.139 for Hopfield networks storing static patterns. Our results are tested against extensive computer simulations and excellent agreement is found.Comment: 17 pages Latex2e, 2 postscript figure

The path-integral analysis of an associative memory model storing an infinite number of finite limit cycles

It is shown that an exact solution of the transient dynamics of an associative memory model storing an infinite number of limit cycles with l finite steps by means of the path-integral analysis. Assuming the Maxwell construction ansatz, we have succeeded in deriving the stationary state equations of the order parameters from the macroscopic recursive equations with respect to the finite-step sequence processing model which has retarded self-interactions. We have also derived the stationary state equations by means of the signal-to-noise analysis (SCSNA). The signal-to-noise analysis must assume that crosstalk noise of an input to spins obeys a Gaussian distribution. On the other hand, the path-integral method does not require such a Gaussian approximation of crosstalk noise. We have found that both the signal-to-noise analysis and the path-integral analysis give the completely same result with respect to the stationary state in the case where the dynamics is deterministic, when we assume the Maxwell construction ansatz. We have shown the dependence of storage capacity (alpha_c) on the number of patterns per one limit cycle (l). Storage capacity monotonously increases with the number of steps, and converges to alpha_c=0.269 at l ~= 10. The original properties of the finite-step sequence processing model appear as long as the number of steps of the limit cycle has order l=O(1).Comment: 24 pages, 3 figure

Transient dynamics for sequence processing neural networks

An exact solution of the transient dynamics for a sequential associative memory model is discussed through both the path-integral method and the statistical neurodynamics. Although the path-integral method has the ability to give an exact solution of the transient dynamics, only stationary properties have been discussed for the sequential associative memory. We have succeeded in deriving an exact macroscopic description of the transient dynamics by analyzing the correlation of crosstalk noise. Surprisingly, the order parameter equations of this exact solution are completely equivalent to those of the statistical neurodynamics, which is an approximation theory that assumes crosstalk noise to obey the Gaussian distribution. In order to examine our theoretical findings, we numerically obtain cumulants of the crosstalk noise. We verify that the third- and fourth-order cumulants are equal to zero, and that the crosstalk noise is normally distributed even in the non-retrieval case. We show that the results obtained by our theory agree with those obtained by computer simulations. We have also found that the macroscopic unstable state completely coincides with the separatrix.Comment: 21 pages, 4 figure

Subhydrische Böden als Schnittstelle im System Festphase-Flüssigphase-Organismus

Subhydrische Böden fungieren in Form von Gewässersedimenten als Schnittstelle von Lithosphäre, Hydrosphäre und Biosphäre. In diesem von Wechselwirkungen geprägten Kompartiment kommt es darüber hinaus zu Interaktionen mit der Anthroposphäre, beispielweise durch Eintrag, Sorption und Freisetzung von Schadstoffen in anthropogen beeinflussten Systemen.Wechselwirkungen von Schadstoffen mit Oberflächen bestimmen Abbau, Sequestration, Remobilisierung und Bioverfügbarkeit im System Festphase-Flüssigphase-Organismus. Dabei stellen die Charakterisierung und Quantifizierung der biologisch wirksamen Konzentrationen von Schadstoffen, ihrer biologischen Zugänglichkeit und ihres Remobilisierungspotentials große Herausforderungen dar. Der Beitrag gibt einen Überblick der Möglichkeiten innovativer Extraktionsmethoden zur Beantwortung von Fragen der Bindung von Schadstoffen, ihrer Remobilisierbarkeit und Bioverfügbarkeit

Symmetric sequence processing in a recurrent neural network model with a synchronous dynamics

The synchronous dynamics and the stationary states of a recurrent attractor neural network model with competing synapses between symmetric sequence processing and Hebbian pattern reconstruction is studied in this work allowing for the presence of a self-interaction for each unit. Phase diagrams of stationary states are obtained exhibiting phases of retrieval, symmetric and period-two cyclic states as well as correlated and frozen-in states, in the absence of noise. The frozen-in states are destabilised by synaptic noise and well separated regions of correlated and cyclic states are obtained. Excitatory or inhibitory self-interactions yield enlarged phases of fixed-point or cyclic behaviour.Comment: Accepted for publication in Journal of Physics A: Mathematical and Theoretica

ADI splitting schemes for a fourth-order nonlinear partial differential equation from image processing

We present directional operator splitting schemes for the numerical solution of a fourth-order, nonlinear partial differential evolution equation which arises in image processing. This equation constitutes the H−1-gradient flow of the total variation and represents a prototype of higher-order equations of similar type which are popular in imaging for denoising, deblurring and inpainting problems. The efficient numerical solution of this equation is very challenging due to the stiffness of most numerical schemes. We show that the combination of directional splitting schemes with implicit time-stepping provides a stable and computationally cheap numerical realisation of the equation