Synchronization of random walks with reflecting boundaries

Reflecting boundary conditions cause two one-dimensional random walks to synchronize if a common direction is chosen in each step. The mean synchronization time and its standard deviation are calculated analytically. Both quantities are found to increase proportional to the square of the system size. Additionally, the probability of synchronization in a given step is analyzed, which converges to a geometric distribution for long synchronization times. From this asymptotic behavior the number of steps required to synchronize an ensemble of independent random walk pairs is deduced. Here the synchronization time increases with the logarithm of the ensemble size. The results of this model are compared to those observed in neural synchronization.Comment: 10 pages, 7 figures; introduction changed, typos correcte

Phase Transitions of Neural Networks

The cooperative behaviour of interacting neurons and synapses is studied using models and methods from statistical physics. The competition between training error and entropy may lead to discontinuous properties of the neural network. This is demonstrated for a few examples: Perceptron, associative memory, learning from examples, generalization, multilayer networks, structure recognition, Bayesian estimate, on-line training, noise estimation and time series generation.Comment: Plenary talk for MINERVA workshop on mesoscopics, fractals and neural networks, Eilat, March 1997 Postscript Fil

Precise calculation of the threshold of various directed percolation models on a square lattice

Using Monte Carlo simulations on different system sizes we determine with high precision the critical thresholds of two families of directed percolation models on a square lattice. The thresholds decrease exponentially with the degree of connectivity. We conjecture that $p_{c}$ decays exactly as the inverse of the coodination number.Comment: 2 pages, 2 figures and 1 tabl

Space Representation of Stochastic Processes with Delay

We show that a time series $x_t$ evolving by a non-local update rule $x_t = f (x_{t-n},x_{t-k})$ with two different delays $k<n$ can be mapped onto a local process in two dimensions with special time-delayed boundary conditions provided that $n$ and $k$ are coprime. For certain stochastic update rules exhibiting a non-equilibrium phase transition this mapping implies that the critical behavior does not depend on the short delay $k$. In these cases, the autocorrelation function of the time series is related to the critical properties of directed percolation.Comment: 6 pages, 8 figure

Mutual learning in a tree parity machine and its application to cryptography

Mutual learning of a pair of tree parity machines with continuous and discrete weight vectors is studied analytically. The analysis is based on a mapping procedure that maps the mutual learning in tree parity machines onto mutual learning in noisy perceptrons. The stationary solution of the mutual learning in the case of continuous tree parity machines depends on the learning rate where a phase transition from partial to full synchronization is observed. In the discrete case the learning process is based on a finite increment and a full synchronized state is achieved in a finite number of steps. The synchronization of discrete parity machines is introduced in order to construct an ephemeral key-exchange protocol. The dynamic learning of a third tree parity machine (an attacker) that tries to imitate one of the two machines while the two still update their weight vectors is also analyzed. In particular, the synchronization times of the naive attacker and the flipping attacker recently introduced in [1] are analyzed. All analytical results are found to be in good agreement with simulation results

Critical behavior for mixed site-bond directed percolation

We study mixed site-bond directed percolation on 2D and 3D lattices by using time-dependent simulations. Our results are compared with rigorous bounds recently obtained by Liggett and by Katori and Tsukahara. The critical fractions $p_{site}^c$ and $p_{bond}^c$ of sites and bonds are extremely well approximated by a relationship reported earlier for isotropic percolation, $(\log p_{site}^c/\log p_{site}^{c^*}+\log p_{bond}^c/\log p_{bond}^{c^*} = 1)$, where $p_{site}^{c^*}$ and $p_{bond}^{c^*}$ are the critical fractions in pure site and bond directed percolation.Comment: 10 pages, figures available on request from [email protected]

An examination of thermal modeling affects to the numerical prediction of large-scale cavitating fluid flow

The importance of modeling thermal effects in cavitatingfluid is examined in the context of computational fluid dynamics. Simulations of cavitation in water are used to study the effects of thermal versus and pressure variations in the fluid properties, and their impact on predictions. These studies are extended to evaluate energyconserving approaches compared to isothermal ones, to assess the underlying thermal models influence on the predicted cavities occurring in water. Results indicate that the thermal effects remain important, but only for specific applications that need high-frequency phenomena from the numerical simulation. Low-frequency measures, needed for loading analysis, appear to be relatively insensitive to thermal effects. Lastly, various thermally driven cavitation problems requiring energy-equation conservation are presented to display applications requiring such a formulation.http://deepblue.lib.umich.edu/bitstream/2027.42/84311/1/CAV2009-final137.pd

Cryptography based on neural networks - analytical results

Mutual learning process between two parity feed-forward networks with discrete and continuous weights is studied analytically, and we find that the number of steps required to achieve full synchronization between the two networks in the case of discrete weights is finite. The synchronization process is shown to be non-self-averaging and the analytical solution is based on random auxiliary variables. The learning time of an attacker that is trying to imitate one of the networks is examined analytically and is found to be much longer than the synchronization time. Analytical results are found to be in agreement with simulations

Biologically inspired learning in a layered neural net

A feed-forward neural net with adaptable synaptic weights and fixed, zero or non-zero threshold potentials is studied, in the presence of a global feedback signal that can only have two values, depending on whether the output of the network in reaction to its input is right or wrong. It is found, on the basis of four biologically motivated assumptions, that only two forms of learning are possible, Hebbian and Anti-Hebbian learning. Hebbian learning should take place when the output is right, while there should be Anti-Hebbian learning when the output is wrong. For the Anti-Hebbian part of the learning rule a particular choice is made, which guarantees an adequate average neuronal activity without the need of introducing, by hand, control mechanisms like extremal dynamics. A network with realistic, i.e., non-zero threshold potentials is shown to perform its task of realizing the desired input-output relations best if it is sufficiently diluted, i.e. if only a relatively low fraction of all possible synaptic connections is realized

Air entrainment mechanisms from artificial supercavities: Insight based on numerical simulations

Using multiphase computational simulations based on the Navier-Stokes equations, we examine the internal gaseous flows of artificially ventilated supercavities. These simulations indicate that air shear layers that develop on the cavity-wall (the air-liquid interface surrounding the cavity) are an important mechanism of air entrainment. This corroborates previous theory developed for toroidal cavities, and indicates that similar mechanisms occur in twin-vortex cavities and cavities closing on bodies. The importance of these shear layers on the cavity behavior potentially impacts computational simulations, experiments, and design-level models. Lastly, a more inclusive, semi-empirical air entrainment model is presented that attempts to accommodate the observed processes.http://deepblue.lib.umich.edu/bitstream/2027.42/84310/1/CAV2009-final136.pd