Tor Flå

Nonlinear Landau damping of weakly dispersive circularly polarized MHD waves

The combined effect of modulational instability and nonlinear Landau damping on a circularly polarized MHD wave is studied. When the wave gets modulated, the magnetic mirror force as well as the parallel electric field associated with the modulations, transfer energy to the charged particles with velocity near the Alfven velocity, to cause a nonlinear Landau damping. Our mathematical model is a DNLS equation extended with a nonlocal term representing the resonant particles. This irreversible model allows a conservation law which in a slowly varying wave train limit corresponds to conservation of wave action. In addition to the modulational instability already known from the DNLS model, the Landau damping term introduces a modulational instability in the wave number range where the wave would otherwise be stable. We present numerical studies of the development of the instabilities and the subsequent wave damping in both ranges. One of our principal findings is that the wave frequency decreases in the same proportion as the energy density. This can be understood in terms of the conservation of action law.

Fully adaptive algorithms for multivariate integral equations using the non-standard form and multiwavelets with applications to the Poisson and bound-state Helmholtz kernels in three dimensions

We have developed and implemented a general formalism for fast numerical solution of time-independent linear partial differential equations as well as integral equations through the application of numerically separable integral operators in d ≥ 1 dimensions using the non-standard (NS) form. The proposed formalism is universal, compact and oriented towards the practical implementation into a working code using multiwavelets. The formalism is applied to the case of Poisson and bound-state Helmholtz operators in d = 3. Our algorithms are fully adaptive in the sense that the grid supporting each function is obtained on the fly while the function is being computed. In particular, when the function g = O f is obtained by applying an integral operator O, the corresponding grid is not obtained by transferring the grid from the input function f. This aspect has significant implications that will be discussed in the numerical section. The operator kernels are represented in a separated form with finite but arbitrary precision using Gaussian functions. Such a representation combined with the NS form allows us to build a sparse, banded representation of Green’s operator kernel. We have implemented a code for the application of such operators in a separated NS form to a multivariate function in a finite but, in principle, arbitrary number of dimensions. The error of the method is controlled, while the low complexity of the numerical algorithm is kept. The implemented code explicitly computes all the 22d components of the d-dimensional operator. Our algorithms are described in detail in the paper through pseudo-code examples. The final goal of our work is to be able to apply this method to build a fast and accurate Kohn–Sham solver for density functional theory.

DeltaProt: a software toolbox for comparative genomics

BackgroundStatistical bioinformatics is the study of biological data sets obtained by new micro-technologies by means of proper statistical methods. For a better understanding of environmental adaptations of proteins, orthologous sequences from different habitats may be explored and compared. The main goal of the DeltaProt Toolbox is to provide users with important functionality that is needed for comparative screening and studies of extremophile proteins and protein classes. Visualization of the data sets is also the focus of this article, since visualizations can play a key role in making the various relationships transparent. This application paper is intended to inform the reader of the existence, functionality, and applicability of the toolbox.ResultsWe present the DeltaProt Toolbox, a software toolbox that may be useful in importing, analyzing and visualizing data from multiple alignments of proteins. The toolbox has been written in MATLAB™ to provide an easy and user-friendly platform, including a graphical user interface, while ensuring good numerical performance. Problems in genome biology may be easily stated thanks to a compact input format. The toolbox also offers the possibility of utilizing structural information from the SABLE or other structure predictors. Different sequence plots can then be viewed and compared in order to find their similarities and differences. Detailed statistics are also calculated during the procedure.ConclusionsThe DeltaProt package is open source and freely available for academic, non-commercial use. The latest version of DeltaProt can be obtained from http://services.cbu.uib.no/software/deltaprot/. The website also contains documentation, and the toolbox comes with real data sets that are intended for training in applying the models to carry out bioinformatical and statistical analyses of protein sequences.Equipped with the new algorithms proposed here, DeltaProt serves as an auxiliary analysis tool capable of visualizing and comparing orthologus protein sequences. The framework of the algorithms also enables easy incorporation of extra information on structure, if such data is available.

The effect of resonant particles on Alfvén solitons

Abstract The effect of resonant particles on an Alfven soliton is studied. Soliton perturbation theory based on the inverse scattering data formalism is applied to an extended DNLS equation, including a term which represents resonant particles. The results are compared with a numerical solution of the same equation. The numerical results show that when the initial soliton state is in the normal regime, the perturbation method gives an adequate description of the evolution, while the discrepancy between numerical and perturbation results increases as the initial soliton state is moved into the anomalous regime. Both approaches indicate that the solitons generally develop into the normal regime, and that the effective damping takes place around the transition from the anomalous to the normal regime.

Linear scaling Coulomb interaction in the multiwavelet basis, a parallel implementation

We present a parallel and linear scaling implementation of the calculation of the electrostatic potential arising from an arbitrary charge distribution. Our approach is making use of the multi-resolution basis of multiwavelets. The potential is obtained as the direct solution of the Poisson equation in its Greens function integral form. In the multiwavelet basis, the formally non local integral operator decays rapidly to negligible values away from the main diagonal, yielding an effectively banded structure where the bandwidth is only dictated by the requested accuracy. This sparse operator structure has been exploited to achieve linear scaling and parallel algorithms. Parallelization has been achieved both through the shared memory (OpenMP) and the message passing interface (MPI) paradigm. Our implementation has been tested by computing the electrostatic potential of the electronic density of long-chain alkanes and diamond fragments showing (sub)linear scaling with the system size and efficent parallelization.

Classification of Kink Type Solutions to the Extended Derivative Nonlinear Schrödinger Equation

The Raman Extended Derivative Non Linear Schrodinger (R-EDNLS) equation which models single mode propagation in optical fibers, is shown to possess travelling and stationary kink envelope solutions of monotonic and oscillatory type. These structures have been called optical shocks in analogy with hydrodynamical shocks or optical double layers in analogy with electrostatic double layers in plasma physics. Hydrodynamical equations for the action density and local wave number are derived and shock wave solutions of the Rankine–Hugionot type are constructed. They are consistent with the kink structures when excluding the nongeneric case that the kink envelope approaches zero in a nonmonotonic way.

A numerical energy conserving method for the DNLS equation

An implicit, numerical energy conserving method is developed for the derivative nonlinear Schrodinger (DNLS) equation for periodic boundary conditions. We find no numerical high frequency modulational instabilities in addition to the modulational instability from a linear analysis around a nonlinear state for the DNLS equation if the modulation is small and (k0 −a22)2 t < π (k0 is the wavenumber and a the amplitude). The numerical scheme is used to follow the nonlinear behavior of the DNLS modulational instability. The numerical code is also tested by the evolution for one soliton initial data. These tests show that if the modulation is not small compared to the background wave amplitude, new nonlinear numerical instabilities are introduced.

Property-Dependent Analysis of Aligned Proteins from Two Or More Populations.

Multiple sequence alignments can provide information for comparative analyses of proteins and protein populations. We present some statistical trend-tests that can be used when an aligned data set can be divided into two or more populations based on phenotypic traits such as preference of temperature, pH, salt concentration or pressure. The approach is based on estimation and analysis of the variation between the values of physicochemical parameters at positions of the sequence alignment. Monotonic trends are detected by applying a cumulative Mann-Kendall test. The method is found to be useful to identify significant physicochemical mechanisms behind adaptation to extreme environments and uncover molecular differences between mesophile and extremophile organisms. A filtering technique is also presented to visualize the underlying structure in the data. All the comparative statistical methods are available in the toolbox DeltaProt.

Deterministic and Stochastic Dynamics of Chronic Myelogenous Leukaemia Stem Cells Subject to Hill-Function-Like Signaling

Based on a discrete Markovian birth-death model including regulated symmetric and asymmetric cell division, we formulate a continuous four-dimensional stochastic (ordinary) differential equation model for the dynamics of Chronic Myelogenous Leukaemia (CML) stem cells in a bone marrow niche involving signaling and competition between active stem cells. Invoking stochastic-deterministic correspondence we then investigate two deterministic subsystems: (a) the competition between active normal and wild-type CML stem cells or also between two developing leukaemic stem cell strains is represented by a two-dimensional equation system, and (b) a three-dimensional model involving both cycling and noncycling normal stem cells as well as cycling wild-type CML stem cells is defined. The four-dimensional equation system finally includes in addition one cycling CML stem cell clone of an anti-CML-drug-resistant mutant. By totally analytic means we discuss the existence and stability of the equilibria of the three systems in the deterministic small noise limit, and establish, by numerical means, connections between these classical results and the original stochastic setting. The robust, stable finite population equilibria can be interpreted as homeostatic equilibria of normal and leukaemic stem cell populations, in the case of the four-dimensional model for the scenario of treatment of the wild-type CML clone with a CML suppressing agent, e.g., imatinib, which leads to the emergence of a resistant CML strain. The four-dimensional model thus represents a common clinical picture.

Extracting molecular diversity between populations through sequence alignments

The use of sequence alignments for establishing protein homology relationships has an extensive tradition in the field of bioinformatics, and there is an increasing desire for more statistical methods in the data analysis. We present statistical methods and algorithms that are useful when the protein alignments can be divided into two or more populations based on known features or traits. The algorithms are considered valuable for discovering differences between populations at a molecular level. The approach is illustrated with examples from real biological data sets, and we present experimental results in applying our work on bacterial populations of Vibrio, where the populations are defined by optimal growth temperature, Topt.