When Ramification points meet

Summary This paper contains three applications of the technique of limit series (our [1986]) to the theory of ramification of linear series on smooth curves, and curves of compact type, overC.Let {L _t t|<}, be a family of linear series on a smooth family of smooth curves {C _t }, and letp ₁(t),p ₂(t)C _t be sections of the family which coincide (only) att=0. Setp=p ₁())=p ₂(0)C ₀.We first give a condition related to the Schubert calculus which must be satisfied by the ramification series and the . We then take up the converse problem: In what ways can a given ramification point arise as a limit? We show that if the ramification point isdimensionally proper in the sense of our [1986], then families of every kind allowed by the Schubert calculus condition can actually be constructed. Finally, we prove that dimensional propriety is in a strong sense an open condition, so that ramification points constructed as above are again dimensionally proper.In the body of the paper we work not with pairs of points, as above, but with arbitrary finite collections of points approaching (possibly) several points of the limit curve. Further, by their nature, the results are valid for families of curves of compact type.     

On Bateman's method for second kind integral equations

Summary Under suitable conditions, we prove the convergence of the Bateman method for integral equations defined over bounded domains inR ^d ,d1. The proof makes use of Hilbert space methods, and requires the integral operator to be non-negative definite. For one-dimensional integral equations over finite intervals, estimated rates of convergence are obtained which depend on the smoothness of the kernel, but are independent of the inhomogeneous term. In particular, for aC kernel andn reasonably spaced Bateman points, the convergence is shown to be faster than any power of 1/n. Numerical calculations support this result.     

Ternary rare-earth alumo-silicides—single-crystal growth from Al flux, structural and physical properties

A number of rare-earth alumo-silicides (R-Al-Si) have been synthesized from the corresponding elements by high-temperature reactions, carried out in excess of aluminum to serve as a flux. Under these experimental conditions, large single crystals of all R-Al-Si ternary phases were readily produced. The crystal structures these ternaries adopt were studied by means of powder and single-crystal X-ray diffraction and were classified as follows: (1) the early rare-earths (R=La, Ce, Pr, Nd, Sm, Gd) yield RAl_xSi_2−x, x∼1, non-stoichiometric ternary derivatives of the body-centered α-ThSi₂-type; (2) the late rare-earths (R=Tb, Dy, Ho, Er, Tm) form stoichiometric R₂Al₃Si₂ compounds that crystallize in the C-centered monoclinic Y₂Al₃Si₂-type; (3) the divalent Eu and Yb produce EuAl₂Si₂ and YbAl₂Si₂ with the trigonal CaAl₂Si₂-type, whereas the last lanthanide element, Lu, forms LuAlSi with C-centered orthorhombic YAlGe-type. These structural trends are reviewed, and the evolution of the basic physical properties such as magnetism, heat capacity and electrical resistivity when moving across the series is described in detail.     

Statistical-overlap theory for elliptical zones of high aspect ratio in comprehensive two-dimensional separations

The low prediction by statistical-overlap theory of the numbers of singlets and peaks in two-dimensional separations containing zones represented by either circles of small number or eccentric ellipses of any number is shown to result from use of probability expressions for unbound spaces of infinite extent. An exact theory is derived for the probability of singlet formation in a reduced two-dimensional space of unit length, width, and area. The probability is a weighted sum of the probabilities of singlet formation in the interior, edge, and corner regions of the space, which depend only on saturation. The weighting factors are the fractions of area associated with each region and depend on the number of zones, the aspect ratio, the saturation, and the ellipse's spatial orientation. The average numbers of doublets, triplets, and peaks in the space are approximated by combining these results with Roach's equations describing the clustering of circles in an unbound two-dimensional space. Simulations show that theory predicts the number of singlets, doublets, triplets, and peaks, when the number of zones is 25 or more, the aspect ratio is 100 or less, and the saturation is 2 or less. The relationship is derived between the aspect ratios of ellipses in the reduced space and actual separation space. Calculations are presented for comprehensive two-dimensional gas chromatography.     

Capillary electrophoresis: An integrated system with a unique split-flow sample introduction mechanism

The performance of an integrated capillary electrophoresis system with a novel split-flow sample injection mechanism and special high sensitivity UV absorbance detector is described. Sample introduction into the capillary is accomplished with a standard HPLC-type microliter syringe. The injected sample is divided proportionally between the separation capillary and an adjustable splitvent. The volume of sample introduced into the capillary can be manipulated by varying the length or the i.d. of the splitvent tubing; or the volume of sample injected. Data are presented showing reproducibility of retention time, peak height, and peak area; minimum detectability; and operation at short UV wavelengths.     

Paired Electrolysis of Aldoses in a Divided Flow Cell

Simultaneous anodic oxidation and cathodic reduction of aldoses in a divided flow cell were studied. The stream of the anolyte was an aqueous solution containing D-glucose, sodium bromide, and sodium bicarbonate. The stream of the catholyte was also an aqueous solution containing xylose and sodium sulfite. The factors which affected both the anodic and cathodic reactions were studied. The results indicate that the flow rates and temperatures of the anolyte and the catholyte, concentrations of the aldoses, pH values and the material of electrodes significantly affect both anodic and cathodic yields. The selectivities of gluconic acid in the anode and xylitol in the cathode were very high. The power consumption of paired electrolysis in the flow system was less than paired electrolysis in a batch system.     

Effect of oil substitution in chiral microemulsion electrokinetic chromatography

In a previous publication (Pascoe, R., Foley, J. P., Analyst 2002, 127, 710-714), a novel chiral microemulsion based on 1.0% w/v dodecoxycarbonylvaline (DDCV), 0.50% v/v ethyl acetate and 1.2% v/v 1-butanol, was shown to provide rapid enantiomeric separations of various pharmaceutical compounds. The two deficiencies noted with this method were that the peak shapes obtained were asymmetric and the efficiencies were lower than those previously obtained using DDCV micelles (Peterson, A. G., Ahuja, E. S., Foley, J. P., J. Chromatogr. B 1996, 683, 15-28). This study examines the use of three alternative low-interfacial-tension oils (methyl acetate, methyl propionate, and methyl formate), in combination with DDCV, to characterize their effect on the elution range, efficiency, resolution, and enantioselectivity of various pharmaceutical enantiomers. The oils were evaluated in both the same volume percentage and the same molar concentration as ethyl acetate in the original DDCV microemulsion system. Including ethyl acetate, a total of seven microemulsion systems were examined. For the compounds that were separated, average enantioselectivities ranged from 1.09 to 1.28, with corresponding efficiencies of 14,000-20,000. While some interesting differences were observed, ethyl acetate still proved to be the most advantageous in terms of enantioselectivity, resolution, and elution range.     

Electroreductive polymerization of trans-[RuCl 2(vpy) 4] on Nd-Fe-B magnets: electrochemical impedance spectroscopy interpretation, Raman spectroscopy, X-ray photoelectron spectroscopy and scanning electron microscopy analysis

The electropolymerization of trans-[RuCl₂(vpy)₄] (vpy=4-vinylpyridine) monomer on Nd-Fe-B magnets was studied by electrochemical impedance spectroscopy (EIS). Impedance diagrams obtained during the polymerization process were used to monitor film formation. The EIS results gave insight into the electrochemical phenomena occurring at the magnet surface as the polymerization process progressed. The film structure and morphology were also studied by X-ray photoelectron spectroscopy, scanning electron microscopy and Raman spectroscopy. The Raman spectroscopy results showed that the polymerization takes place at the vinyl groups of the monomer and also that the redox polymer structure is very similar to that of the monomer. The ratio of the intensity of the XPS peaks for fluorine (from the electrolyte PF₆ ^–) and ruthenium present in the film showed that the polymer on Nd-Fe-B contained an equal proportion of Ru²⁺ and Ru³⁺, indicating that part of the film is positively charged, i.e. {[RuCl₂(vpy)₄]⁺} _n .     

Relationship between tetrahedron shape measures

Tetrahedron shape measures are used for evaluating the quality of tetrahedra in finite element meshes. Three shape measures, theminimum solid angle _min theradius ratio , and themean ratio , are discussed in this paper. A new formula for the computation of a solid angle of tetrahedron is derived. For different shape measures andv (with values 1), we establish a relationship between andv of the form wherec ₀,c ₁,e ₀, ande ₁ are positive constants. This means that if one measure approaches zero for a poorly-shaped tetrahedron, so does the other. Combined with the property that each measure attains a maximum value only for the regular tetrahedron, this means that the shape measures are equivalent.This work was partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada.