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     

Theoretical Study on the Structures and Electronic Spectra of the Derivatives of C_(60)-P-2,4,6-Triphenyl Borazinc

1 INTRODUCTION The electron-transfer reaction of C60 derivatives, especially light-induced electron-transfer reaction, has been an active research field for a long time. Many researches are focused on the long-live charge- separated state caused by electron-transfer[1]. And several electron-transfers between electron donor and C60 or intramolecular electron-transfer have been confirmed. As an example, TTF-C60 is proved to be a short-live charge-separated compound[2].Due to the excelle…     

Bifunctional pyridyl alcohols with the bicyclo[3.3.0]octane scaffold in the asymmetric addition of diethylzinc to aldehydes

Some new pyridyl alcohols with the cis-bicyclo[3.3.0]octane scaffold were synthesized and used as chiral ligands for the enantioselective addition of diethylzinc to aldehydes. Ligands 4 were found to be far superior to the C₂-symmetric ligands 2 in terms of enantioselectivities. Quantitative yields and enantiomeric excesses of up to 92% were obtained when the ligand 4 was used. The carbonyl function in 4 proved to be beneficial for the high enantioselectivities in the addition of diethylzinc to aldehydes. Conversion of the carbonyl group into oxime or oxime ether group led to a sort of more active ligands, which catalyzed the same reaction with rate acceleration.     

Biological generation of tritiated vitamin D3 metabolites and their purification by high-performance liquid chromatography

Summary A simple, reproducible method for the biological synthesis of tritiated 24,25-dihydroxycholecalciferol, 25,26-dihydroxycholecalciferol and 1,25-dihydroxycholecalciferol is described. Kidney homogenates from both vitamin D deficient and replete chicks were usedin vitro to generate these dihydroxylated metabolites using 25 (23,24-³H) hydroxycholecalciferol as the substrate. Tritiated products were purified by Sephadex LH 20 chromatography followed by high-performance liquid chromatography; the identity of each metabolite was established by chromatography with authentic crystalline preparations.Presented at the Symposium organised by the Chromatography Discussion Group, held at Hatfield Lodge on 29 November 1979.     

A Laplace-like method for solving linear difference equations

This paper is concerned with the problem of solving linear difference equations of ordern with constant coefficients and with given initial conditions in which the variable runs not only through the integers but over . The main idea is the introduction of a suitable commutative ring of functions with discrete convolution as multiplication rule which works, although it is not a field. The existence of inverses is studied and, after the introduction of suitable functions, the problem is reduced by means of a Laplace-like relation to an algebraic equation. Examples of application are given. Finally some remarks make the connection with the Operational Calculus of Mikusinski and with the Operational Calculus of Fenyö. The advantages of this method lie in the fact that it is applicable to functions others than the step functions, in its simplicity from the theoretical point of view, in its usefulness even when computation is required and in its formal similarity to the classical treatment of linear differential equations with constant coefficients.     

Quartic X-splines

Summary The quartic periodic and nonperiodic X-spline are separated from the class of all piecewise-quartic interpolatory polynomials and their orders of convergence, smoothness and complexity of construction are examined. In particular, error estimates of interpolation of smooth functions at uniformly spaced knots by eight quartic X-splines of special interest are presented. The results are illustrated by a numerical example.     

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.     

Approximation under an arc length constraint

Let f C[a, b]. LetP be a subset ofC[a, b], L b – a be a given real number. We say thatp P is a best approximation tof fromP, with arc length constraintL, ifA[p] _b ^a [1 + (p(x)) ²]dx L andp – f q – f for allq P withA[q] L. represents an arbitrary norm onC[a, b]. The constraintA[p] L might be interpreted physically as a materials constraint.In this paper we consider the questions of existence, uniqueness and characterization of constrained best approximations. In addition a bound, independent of degree, is found for the arc length of a best unconstrained Chebyshev polynomial approximation.The work of L. L. Keener is supported by the National Research Council of Canada Grant A8755.     

On the singular graph and the Weyr characteristic of anM-matrix

LetA be anM-matrix in standard lower block triangular form, with diagonal blocksA _ii irreducible. LetS be the set of indices such that the diagonal blockA is singular. We define the singular graph ofA to be the setS with partial order defined by > if there exists a chain of non-zero blocksA _i, A_ij, , A_l.Let ₁ be the set of maximal elements ofS, and define thep-th level _p ,p = 2, 3, , inductively as the set of maximal elements ofS \( _{1 p-1}). Denote by _p the number of elements in _p . The Weyr characteristic (associated with 0) ofA is defined to be (A) = ( ₁, ₂,, _h ), where ₁ + + _p = dim KerA ^p ,p = 1, 2, , and _h > 0, _h+1 = 0.Using a special type of basis, called anS-basis, for the generalized eigenspaceE(A) of 0 ofA, we associate a matrixD withA. We show that(A) = ( ₁, , _h) if and only if certain submatricesD _p,p+1 ,p = 1, , h – 1, ofD have full column rank. This condition is also necessary and sufficient forE(A) to have a basis consisting of non-negative vectors, which is a Jordan basis for –A. We also consider a given finite partially ordered setS, and we find a necessary and sufficient condition that allM-matricesA with singular graphS have(A) = ( ₁, , _h). This condition is satisfied ifS is a rooted forest.The work of the second-named author was partly supported by the National Science Foundation, under grant MPS-08618 A02.