Perturbed unstable variational inequalities with unbounded operators on approximately given sets

The stability and convergence of the solutions of perturbed and regularized variational inequality to the solutions of the primary (unstable a priori) variational inequality with proper monotone operator are investigated. All the objects of inequality: the operatorA, the right-hand partf and the set of constrains are to be perturbed. At the same time no assumptions of boundedness and smoothness of the operatorA are used. The connection between the parameters of perturbations, which guarantees strong convergence of approximate solutions, is established. It is proved that the existence of the solution to the unperturbed variational inequality is necessary and sufficient condition for convergence of the regularized perturbed inequality solutions.This research was supported in part by the Ministry of Science Grant 3481-1-91 and by the Ministry of Absorption Center for Absorption in Science.     

Banach algebras of singular integral operators with piecewise continuous coefficients. General contour and weight

The Banach algebra generated by one-dimensional linear singular integral operators with matrix valued piecewise continuous coefficients in the spaceL _p (,) with an arbitrary weight is studied. The contour consists of a finite number of closed curves and open arcs with satisfy the Carleson condition. The contour may have a finite number of points of selfintersection. The symbol calculus in this algebra is the main result of the paper.     

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.     

The erdös-dushnik-miller theorem for topological graphs and orders

A topological graph is a graph G=(V, E) on a topological space V such that the edge set E is a closed subset of the product space V x V. If the graph contains no infinite independent set then, by a well-known theorem of Erdös, Dushnik and Miller, for any infinite set LV, there is a subset LL of the same oardinality |L| = |L| such that the restriction G L is a complete graph. We investigate the question of whether the same conclusion holds if we weaken the hypothesis and assume only that some dense subset AV does not contain an infinite independent set. If the cofinality cf (|L|)>|A|, then there is an L as before, but if cf (|L|)<-|A|, then some additional hypothesis seems to be required. We prove that, if the graph GA is a comparability graph and A is a dense subset, then for any set LV such that cf (|L|)>, there is a subset LL of size |L|=|L| such that GL is complete. The condition cf (|L|)> is needed.Research supported by NSERC grant #A5198.     

5－（4－乙酰氧基苯基），10，15，20－三苯基卟啉金属配合物修饰的铂支撑双层脂膜的IV性质

合成并表征了５－（４－乙酰氧基苯基），１０，１５，２０－三苯基卟啉及其金属配合物，将它们作为修饰物，制得铂支撑的双层脂膜，用循环伏安法研究了抗坏血酸的氧化还原反应，并对其机理作了探讨。     

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.     

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.     

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.