Neurons learning under an unsupervised Hebbian learning rule can perform a nonlinear generalization of principal component analysis. This relationship between nonlinear PCA and nonlinear neurons is reviewed. The stable fixed points of the neuron learning dynamics correspond to the maxima of the statist,ic optimized under nonlinear PCA. However, in order to predict. what the neuron learns, knowl...

This paper presents a generalization of the Kalman Filter for linear and nonlinear fractional order discrete state-space systems. The linear and nonlinear discrete fractional order state-space systems are also introduced. The simplified Kalman Filter for linear case is called Fractional Kalman Filter and its nonlinear extension is named Extended Fractional Kalman Filter respectively. The backgr...

This paper presents a generalization of the Kalman filter for linear and nonlinear fractional order discrete state-space systems. Linear and nonlinear discrete fractional order state-space systems are also introduced. The simplified kalman filter for the linear case is called the fractional Kalman filter and its nonlinear extension is named the extended fractional Kalman filter. The background ...

This paper introduces the emph{interval unilateral quadratic matrix equation}, $IUQe$ and attempts to find various analytical results on its AE-solution sets in which $A,B$ and $CCC$ are known real interval matrices, while $X$ is an unknown matrix. These results are derived from a generalization of some results of Shary. We also give sufficient conditions for non-emptiness of some quasi-solutio...

We derive the fractional generalization of the Ginzburg–Landau equation from the variational Euler–Lagrange equation for fractal media. To describe fractal media we use the fractional integrals considered as approximations of integrals on fractals. Some simple solutions of the Ginzburg–Landau equation for fractal media are considered and different forms of the fractional Ginzburg–Landau equatio...

A generalization of the Woodward's theorem is applied to the case of random signals jointly modulated in amplitude and frequency. This yields the signal spectrum and a rather robust estimate of the bispectrum. Furthermore, higher order statistics that quantify the amount of energy in the signal due to nonlinearities, e.g., wave–wave interaction in the case of water waves, can be inferred. Consi...

We study the periodic time dependence of shear stress that occurs in a low- and a high-density fluid in a molecular dynamics algorithm for simulation of constant shear flow. We present a generalization of the linear response theory for a case when the equilibrium relaxation function and the equilibrium shear stress depend on strain. The predictions of this generalization reveal that the time de...