We prove a product estimate that allows to estimate the quadratic first order nonlinearity of the harmonic map flow in the Lp norm. Then the parabolic analogue of Weyl’s lemma for the Lapace operator is established. Both results are applied to prove regularity for the heat flow by parabolic bootstrapping.

We study a class of fractional semilinear elliptic equations and formulate the corresponding Calder\'on problem. determine nonlinearity from exterior partial measurements Dirichlet-to-Neumann map by using first order linearization Runge approximation property.

This paper examines the standard map with sawtooth nonlinearity when the eigenvalues of the Jacobian lie on the unit circle. This is an area-preserving map of the torus to itself that is linear except on a line on which it is discon-tinuous. We discuss the closure of the set of images of the discontinuity and present numerical evidence that its Lebesgue measure is positive. Moreover, we present...

Microarray datasets are often too large to visualise due to the high dimensionality. The self-organising map has been found useful to analyse massive complex datasets. It can be used for clustering, visualisation, and dimensionality reduction. However for visualisation purposes the SOM uses colouring schemes as a means of marking cluster boundaries on the map. The distribution of the data and t...

A dynamical systems based model of computation is studied. We demonstrate the computational capability of a class of dynamical systems called switching map systems. There exists a switching map system with two types of baker's map to emulate any Turing machines. The baker's maps are corresponding to the elementary operations of Turing machines such as left/right head-moving and read/write symbo...

It has been shown earlier that the Self-Organizing Map (SOM) can be applied to the analysis and visualization of similarities of words in their usage in short contexts formed of adjacent words. These SOMs of words, called word category maps, have many potential applications. One of the application areas is information retrieval and data mining of textual document collections where the word cate...

In recent neurophysiological experiments stimulus-related neuronal oscillations were discovered in various species. The oscillations are not persistent during the whole time of stimulation, but instead seem to be restricted to rather short periods, interrupted by stochastic periods. In this contribution we argue, that these observations can be explained by a bistability in the ensemble dynamics...

Jones matrix computation is one of the widely used methods in nonlinearitycalculation in laser interferometers. In this paper, the nonlinearity error in developedthree-longitudinal mode heterodyne interferometer (DTLMI) has been mathematicallymodeled at various speeds, by using Jones matrix calculations. This review has been donedespite the fact that simultaneously the main factors including no...

We investigate pattern formation and evolution in coupled map lattices when advection is incorporated, in addition to the usual diffusive term. All patterns may be suitably grouped into five classes: three periodic, supporting static patterns and traveling waves, and two nonperiodic. Relative frequencies are determined as a function of all model parameters: diffusion, advection, local nonlinear...

A wide range of coupled map lattices is shown to have identical pattern sequences. It is found that the wave type solutions in their respective pattern selection regimes scale linearly with the coupling range indicating the existence of a continuum limit. The findings are used to introduce a very simple novel type of spatially extended map which can serve as a representative for this class of c...