Theq-harmonic oscillator and the Al-Salam and Carlitz polynomials

One more model of aq-harmonic oscillator based on theq-orthogonal polynomials of Al-Salam and Carlitz is discussed. The explicit form ofq-creation andq-annihilation operators,q-coherent states and an analog of the Fourier transformation are established. A connection of the kernel of this transform with a family of self-dual biorthogonal rational functions is observed.     

Variable step size destabilizes the Störmer/leapfrog/verlet method

For fixed step-sizeh the Störmer method is stable for the standard test equationÿ= ² y,>0, if and only ifh<2. We show that for variable step sizeh _n there does not exist a (positive) limit onh which ensures stability. Nor can we guarantee stability if, in addition, we limit the step size ratioh _n/h _n–1.This work was supported in part by National Science Foundation Grant DMS 90 15533.     

Functional connectedness and Darboux property of multivalued functions

To 80^th birthday of Paul Erds     

Counterexamples in optimal quadrature

It is widely believed that order of exactness is a good measure of the quality of an algorithm for numerical quadrature. We show that this is not the case, by exhibiting a situation in which the optimal algorithm does not even integrate constants exactly. We also show that there are situations in which the penalty for using equidistant nodes is unbounded. Finally, we show that the complexity of obtaining an -approximation can be an arbitrary function of , i.e., there is no hardest quadrature problem.     

Miniature electrophoresis for speed and productivity

Attempts to miniaturize electrophoresis (EP) to save time or enhance productivity and efficiency remains a challenge for science and industry. Other advantages of miniaturization include: increased sensitivity, saving of reagents, greater yield of data, and enabling studies where only small samples are available. Since electrophoresis that takes hours may be reduced to a matter of minutes, the limitations of miniaturization in clinical, industrial, and research applications are evaluated. Clinical electrophoresis (EP) on cellulose acetate media can be performed in 3.5 min instead of 20–45 min and on SDS polyacrylamide gels in 15–30 min compared with conventional 3–8 h.     

Primary methods of measurement in chemical analysis

 Primary methods of measurement have a central function in metrology. They are an essential component in the realisation of the SI units and therefore are indispensable for establishing traceability of measurements of all kinds of physical quantities to the corresponding SI units. This is also true for chemical analysis. Gravimetry, titrimetry, coulometry, and isotope dilution mass spectrometry (IDMS) are evaluated with regard to their potential to be primary methods according to a general definition of primary methods recently given by the Comité Consultatif pour la Quantité de Matière (CCQM). Optical absorption spectrometry and methods based on colligative properties are also considered. A general scheme for establishing traceability of chemical measurements to the SI units using primary methods is discussed. Received: 17 April 1997 · Accepted: 9 August 1997     

Simultaneous determination of noradrenaline and dopamine in Portulaca oleracea L. by capillary zone electrophoresis

A simple and rapid capillary electrophoretic method was developed for the simultaneous determination of noradrenaline (NA) and dopamine (DA) in Portulaca oleracea L. The buffer solution used in this method was 40 mM tris (hydroxymethyl) aminomethane (Tris)-H3PO4 at pH 2.00 containing 15% methanol. The effects of pH value, organic modifier, and applied voltage were investigated. The linear ranges of NA and DA were 0.5-100 microg/mL (r=0.9952) and 6.25-200 microg/mL (r=0.9992), respectively. The relative standard deviations of the corrected peak area were 6.73% and 4.26%, respectively. NA and DA in Portulaca oleracea L. were simultaneous determined successfully within 5.6 min. In this way, the contents of NA and DA in different parts (stem, leaves, and seeds) of P. oleracea L. and in different extracts of leaves with different solvents (distilled water, 50% methanol, and methanol) were studied.     

Mono- and Dioses of Glycyrrhiza glabra Root

Simple sugars were isolated from the ethanol extract of Glycyrrhiza glabra L. root. The component composition of the carbohydrate fraction was established using GC-MS (gas-chromatographic separation and mass-spectrometric analysis).     

On a pair of functional equations of combinatorial interest

The equations of the title appear in the author's paper Chromatic Sums for Rooted Planar Triangulations, V: Special Equations. (Canadian Journal of Mathematics, 26 (1974), 893–907). They appear in that paper as Equations (24) and (25). They are simultaneous equations for two unknown functionsl andy ₂ of two variablesy ₁ andz. A parameter is involved. The main result is that for = 2 cos (2/n), wheren is a positive integer >1, the two equations can be reduced to a single equation (numbered (49)). Solutions of this are known forn <7. From such solutions we can expect to get information about the averaged chromatic polynomials of planar triangulations with a given number of triangles.The present work is basically an expository paper on the theory given in Chromatic Sums, V, but it includes some new results and many simplifications.