Subordinate solutions and spectral measures of canonical systems

The theory of 2×2 trace-normed canonical systems of differential equations on ?⁺ can be put in the framework of abstract extension theory, cf. [9]. This includes the theory of strings as developed by I.S. Kac and M.G. Kre?n. In the present paper the spectral properties of such canonical systems are characterized by means of subordinate solutions. The theory of subordinacy for Schrödinger operators on the halfline ?⁺, was originally developed D.J. Gilbert and D.B. Pearson. Its extension to the framework of canonical systems makes it possible to describe the spectral measure of any Nevanlinna function in terms of subordinate solutions of the corresponding trace-normed canonical system, which is uniquely determined by a result of L. de Branges.     

Trace Element Determination in Diesel Particulates by Total-Reflection X-Ray Fluorescence Analysis

 Particulate matter of Diesel emissions collected from diluted exhaust during standard test runs with two Diesel fuels of different sulfur content was analysed by total-reflection X-ray fluorescence (TXRF). The trace elements determined were S, Ca, Cr, Mn, Fe, Ni, Cu, Zn and Pb. Additionally, some filter materials were tested with respect to their applicability for sample collection. Quartz fibre filters were found to be most suitable, both from a technical and a chemical point of view. A clear reduction of total particulate emissions during the whole test was observed when the fuel with very low sulfur content was used. In addition, it was found that the wear metal content of particulate matter emitted by a cold engine was higher than that observed during normal engine working temperature. Received September 10, 1998. Revision March 2, 1999     

Towards accurate statistical estimation of rounding errors in floating-point computations

A new method of estimatinga posteriori the statistical characteristics of the rounding errors of an arbitrary algorithm is presented. This method is based on a discrete model of the distribution of rounding errors which makes more accurate estimates possible. The analysis is given for both rounding and truncating arithmetic. Finally, some experimental results are reported.     

High-performance liquid chromatography--mass spectrometry and electron-capture dissociation tandem mass spectrometry of osteocalcin. Determination of gamma-carboxyglutamic acid residues

Two mass spectrometry methods, high-performance liquid chromatography combined on-line with electrospray ionization mass spectrometry (HPLC-ESI-MS) and electron-capture (EC) dissociation tandem mass spectrometry (MS-MS), were applied for structural analysis of bovine and human osteocalcins. Osteocalcin contains gamma-carboxyglutamic acid (Gla) residues, which bind metal ions, among its amino acids. Ethylenediaminetetraacetic acid (EDTA) was added to all samples in order to chelate bound metal ions. After elimination of interfering metal ions MS spectra became uncomplicated to interpret. EDTA is incompatible with ESI and it was removed from samples using either on-line HPLC or micropurification method. The number of Gla residues varies in osteocalcin. These subforms, which contain different amounts of Gla residues, were separated using the HPLC-ESI-MS method. In order to determine locations of Gla residues in human osteocalcin, which contained two Gla residues, dissociation MS-MS method was successfully applied.     

Study of M4,5N4,5N4,5 auger energy shifts of Cs and I in free CsI molecules

High-resolution electron beam excited M_4,5N_4,5N_4,5 Auger electron spectra of Cs and I have been measured from CsI vapour. The Auger energies of both Cs and I observed from gaseous CsI are higher than the corresponding free-atom energies due to extra-atomic relaxation. The molecular Auger results have been compared with corresponding photoelectron measurements and free-atom data. Estimates for extra-atomic relaxation energies have been extracted from the changes of the Auger parameter between molecular and atomic species and from the difference between experimental energies and energies calculated with a relativistic Dirac-Fock program, applying the point-charge model for the CsI molecule.     

Computations on the growth of the first factor for prime cyclotomic fields II

It is shown by computing thath(p) — the first factor of the class number of the prime cyclotomic fieldk(exp(2i/p)) — is strictly increasing for 19p1097.     

Normal Auger spectra of Br in alkali bromide molecules

Molecular Auger electron spectra following the bromine 3d ionization in gas-phase alkali bromides and in HBr were studied both experimentally and theoretically. The AES for HBr and CsBr were measured using photoexcitation, and for LiBr, NaBr, and KBr by using electron impact. These results are compared with the theoretical spectra from nonrelativistic ab initio calculations and one-center approximation and with the spectra of Br(-), computed with the multiconfiguration Dirac-Fock method.     

A Phragmén — Lindelöf principle for subsolutions of quasi — linear equations

Quasi — linear elliptic equations of homogeneous type are studied in this paper. The equation satisfies uniform ellipticity and growth conditions. A Phragmén — Lindelöf principle is proved by a combined technique of Ladyenskaya-Ural'ceva and Moser. This technique makes it possible to estimate the growth of the maximum modulus of a subsolution in general unbounded domains.     

Chaos in the one-dimensional gravitational three-body problem

We have investigated the appearance of chaos in the one-dimensional Newtonian gravitational three-body system (three masses on a line with -1/r pairwise potential). In the center of mass coordinates this system has two degrees of freedom and can be conveniently studied using Poincare sections. We have concentrated in particular on how the behavior changes when the relative masses of the three bodies change. We consider only the physically more interesting case of negative total energy. For two mass choices we have calculated 18 000 full orbits (with initial states on a 100x180 lattice on the Poincare section) and obtained dwell time distributions. For 105 mass choices we have calculated Poincare maps for 10x18 starting points. Our results show that the Poincare section (and hence the phase space) divides into three well defined regions with orbits of different characteristics: (1) There is a region of fast scattering, with a minimum of pairwise collisions. This region consists of 'scallops' bordering the E=0 line, within a scallop the orbits vary smoothly. The number of the scallops increases as the mass of the central particle decreases. (2) In the chaotic scattering region the interaction times are longer, and both the interaction time and the final state depend sensitively on the starting point on the Poincare section. For both (1) and (2) the initial and final states consist of a binary + single particle. (3) The third region consists of quasiperiodic orbits where the three masses are bound together forever. At the center of the quasiperiodic region there is a periodic orbit discovered (numerically) by Schubart in 1956. The stability of the Schubart orbit turns out to correlate strongly with the global behavior.