What do multistep methods approximate?

Summary Standard analysis of multistep methods for ODE's assumes the application of an initialization routine that generates the starting points. Here ak-step method is considered directly as a mappingR ^kn R ⁿ . It is shown to approximate a mapping which is expressible directly in terms of the flow of the vector field. Some useful properties of that mapping are shown and for strictly stable methods these are applied to the question of invariant circles near a hyperbolic periodic solution.     

Convergence of Krylov methods for sums of two operators

We discuss the convergence of Krylov subspace methods for equationsx =Tx +f whereT is a sum of two operators,T =B +K, whereB is bounded andK is nuclear. Bounds are given for inf Q _k (B+K) and for inf p _k (B+K), where the infimum is over all polynomials of degreek, such thatQ _k is monic andp _k is normalized:p _k (1) = 1.     

Stability and accuracy of time discretizations for initial value problems

Summary This paper continues earlier work by the same authors concerning the shape and size of the stability regions of general linear discretization methods for initial value problems. Here the treatment is extended to cover also implicit schemes, and by placing the accuracy of the schemes into a more central position in the discussion general method-free statements are again obtained. More specialized results are additionally given for linear multistep methods and for the Taylor series method.This research has been supported by Swiss National Foundation, Grant No. 82-524.077This research has been supported by the Heinrich-Hertz-Stiftung, B 32 No. 203/79     

Shortcut to a Fermi-degenerate gas of molecules via cooperative association

We theoretically examine the creation of a Fermi-degenerate gas of molecules by considering a photoassociation or Feshbach resonance applied to a degenerate Bose-Fermi mixture of atoms. This problem raises interest because, unlike bosons, fermions in general do not behave cooperatively, so that the collective conversion of a degenerate gas atoms into a macroscopic number of diatomic molecules is not to be expected. Nevertheless, we find that the coupled Fermi system displays collective Rabi-like oscillations and a rapid adiabatic passage between atoms and molecules, thereby mimicking Bose-Einstein statistics. Cooperative association of a degenerate mixture of Bose and Fermi gases could therefore serve as a shortcut to a degenerate gas of Fermi molecules.     

Covalently Bound Antibody on Polystyrene Latex Beads: Formation, Stability, and Use in Analyses of White Blood Cell Populations

CD4 or CD8 antibodies were covalently bound to latex beads by reaction of activated CD4 or CD8 monoclonal antibodies with 2-μm-diameter, 1,3-diaminopropane (DAP) coupled, polystyrene aldehyde/sulfate latex beads. Spectrophotometric analyses of the filtrates of the antibody-bead conjugation mixtures for unreacted antibody allowed construction of binding curves of antibody for the polystyrene bead surface and evaluation of binding constants for association of antibody with bead, ranging from 1.5x10(7) to 1.6x10(7) M(-1) for CD4 and CD8 antibodies. The reaction of the antibody thiol group with the activated maleimide group on the bead at pH 7.2-7.3 was complete within 10-15 min. The kinetics of CD4 or CD8 monoclonal antibody displacement from the surface of covalently conjugated antibody-polystyrene latex beads was followed as a function of temperature (5, 22, and 37 degrees C) and the nature of the final diluent for the antibody-coated beads by measuring the concentration of antibody in the filtrates of conjugated beads by an ELISA (enzyme-linked immunosorbent assay). The displacement reaction showed a pseudo-zero-order dependence of the rate, with constants, k(1), ranging from 0.65x10(-17) to 270x10(-17) M s(-1). The functionality of antibody-coated beads suspended in various media was also monitored in a biological cell assay with whole blood. The cell assay depends on forming a layer of beads around targeted lymphocytes to distinguish them from nontargeted lymphocytes by differences in dc or rf conductivity or median angle light scatter. Covalently bound CD4 and CD8 antibody beads stored in one set of media at 5, 22, and 37 degrees C over a period of 16 weeks showed excellent results in the STKS assay with various blood donors, which correlated well (correlation coefficients of 0.99 for CD4 data and 0.93 for CD8 data) with reference results obtained with fluorescent markers by flow cytometry. Covalently bound CD4/CD8 beads stored for 2 weeks in BSA buffer at 5-37 degrees C performed equally well in providing accurate values of the percentage of CD4- or CD8-positive cells in the total white blood cell population, whereas the same beads stored in the 47-50 degrees C range showed some failures in performance. Comparison with antibody concentrations in filtrates of adsorbed antibody-bead suspensions showed 2- to 10-fold greater amounts of free antibody at comparable elapsed time, media, and temperature conditions. A threshold of 1-2 μg/mL of free antibody was necessary before adverse effects on the biological cell assay were noticeable. Copyright 2001 Academic Press.     

Sets of convergence and stability regions

This paper continues the authors' study of the convergence of dynamic iteration methods for large systems of linear initial value problems. We ask for convergence on [0, ) and show how the convergence can be reduced to a graphical test relating the splitting of the matrix to the stability properties of the discretization method.     

Quasinilpotency of the operators in picard-lindelöf iteration

The superlinear convergence of Picard-Lindelöf iteration on finite time intervals is studied by considering the resolvent operator as an operator-valued entire function. The main conclusions of this paper are related to the following fact: the convergence properties of the iteration when applied to a large and sparse system are largely determined by the graph properties of the decomposition and are partly insensitive in terms of the values of the elements of the matrices.