Surface chemical analysis of raw cotton fibres and associated materials

The surface chemical composition of raw unscoured cotton was successfully investigated by the surface analytical techniques X-ray Photoelectron Spectroscopy (XPS) and Time of Flight Secondary Ion Mass Spectrometry (ToF-SIMS). The presence of non-cellulosic material at the fibre surface was established and determined to be a complex mixture of fatty acids, alcohols, alkanes, esters and glycerides. The effect of scouring and bleaching was to reduce the surface concentration of these materials but even after aqueous processing some non-cellulosic material residue was still detected at the fibre surface.     

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.     

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.     

Characterization of the normal distribution by constant regression of quadratic statistics on linear statistics

This paper contains applications of theorems of [1] for quadratic statistics which have constant regression on linear statistics. Two theorems are proved. The first is a sufficient condition which assumes that the characteristic function of a sample is an entire function. The second gives a new characterization of the normal distribution.     

Secondary and Tertiary Structure of Polysaccharides in Solutions and Gels

Many polysaccharide chains can adopt ordered helical and ribbon-like secondary structures. It seems however that these chains are often so stiff and extended that the cooperative interactions necessary for stability in the solvent environment can only be achieved when inter-chain as well as intra-chain interactions are favorable. Hence we commonly find two-or more-stranded associations of helices, of ribbons, or of helices with ribbons. These can be regarded as tertiary and higher levels of structure. The ordered secondary structure characteristically requires a regular repeating sequence of sugar residues, and the termination of this sequence by insertion of a residue of different type may also terminate the secondary structure and hence the association in which it is involved. This is the mechanism by which native polysaccharides may link up to form three dimensional networks, or gels, in which state they perform their natural roles in maintaining the hydration and integrity of biological tissues. For several polysaccharides there is evidence that the mechanism of biological control over the fine topology and properties of the gel network is mediated by enzymes which modify sugar residues at the polymer level to change the pattern of “interrupting” sugar residues.     

Linear combination of concomitants of order statistics with application to testing and estimation

Summary Let (X ₁,Y ₁), (X ₂,Y ₂),…, (X _n,Y _n) be i.i.d. as (X, Y). TheY-variate paired with therth orderedX-variateX _rn is denoted byY _rn and terms the concomitant of therth order statistic. Statistics of the form are considered. The asymptotic normality ofT _n is established. The asymptotic results are used to test univariate and bivariate normality, to test independence and linearity ofX andY, and to estimate regression coefficient based on complete and censored samples.     

On coverings of plane convex sets by translates of strips

In the euclidean planeE ² letS ₁,S ₂, ... be a sequence of strips of widthsw ₁,w ₂, .... It is shown thatE ² can be covered by translates of the stripsS _i if w ₁ ^3/2 = . Further results concern conditions in order that a compact convex domain inE ² can be covered by translates ofS ₁,S ₂, ....This research was supported by National Science Foundation Research Grant MCS 76-06111.     

Sur la forme canonique des formes extérieures

Sans résumé

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The article is the text of a talk given at the Symposium on Differential Geometry in Debrecen, Hungary, on August 28–September 3, 1975.     

A central limit theorem for age- and density-dependent population processes

Consider a population consisting of one type of individual living in a fixed region with area A. In [8], we constructed a stochastic population model in which the death rate is affected by the age of the individual and the birth rate is affected by the population density P_A(t), i.e., the population size divided by the area A of the given region. In [8], we proposed a continuous deterministic model which in general is a nonlinear Volterra type integral equation and proved that under appropriate conditions the sequence P_A(t) would converge to the solution P(t) of our integral equation in the sense that

$\text{lim}\text{→∞}\text{P}\text{sup}\text{0?s?t}\text{|P}{}_{\mathrm{A}}\text{(s) ? P(s)|>ε}\text{=0}\text{for every}\text{ε > 0}$

.In this paper, we obtain a “central limit theorem” for the random element √A(P_A(t)?P(t)). We prove that under appropriate conditions √A(P_A(t)?P(t)) will converge to a Gaussian process. (See Theorem 3.4 for the explicit formula of this Gaussian process.)