Real cubic algebraic surfaces may be described by either implicit or parametric equations. One particularly useful representation is the rational parametrization, where the three spatial coordinates are given by rational functions of two parameters. These parametrizations take on different forms for different classes of cubic surfaces. Classification of real cubic algebraic surfaces into five families for the nonsingular case is based on the configuration of 27 lines on them. We provide a method of extracting all these lines by constructing and solving a polynomial of degree 27. Simple roots of this polynomial correspond to real lines on the surface, and real skew lines are used to form rational parametrizations for three of these families. Complex conjugate skew lines are used to parametrize surfaces from the fourth family. The parametrizations for these four families involve quotients of polynomials of degree no higher than four. Each of these parametrizations covers the whole surface except for a few points, lines, or conic sections. The parametrization for the fifth family, as noted previously in the literature, requires a square root. We also analyze the image of the derived rational parametrization for both real and complex parameter values, together with "base" points where the parametrizations are ill-defined.
Abstract. Human supragingival dental plaque was collected from patients with various degrees of caries and periodontal disease. Plaque extracts, prepared in five different solutions (four varied from pH 1.8 to 12.7; one contained urea), were analyzed by polyacrylamide gel electrophoresis, and tested for amylase and lysozyme enzyme activity. Because no qualitative or quantitative advantages of using the extremes of pH or urea were observed, all subsequent extracts were prepared in phosphate buffered saline at pH 7.3. Concentrated extracts were fractionated by gel filtration and characterized by polyacrylamide gel electrophoresis, peptide mapping, molecular weight estimation, determination of enzymatic activities and amino acid and carbohydrate analyses. Regions of similarity among the gels were revealed by comparing the electrophoretic patterns of pooled plaque extract, normal serum and whole saliva. The elution pattern of pooled plaque extract from a standardized Sephadex G‐200 column indicated the presence of both high and low molecular weight proteins that might have correlated with the components of normal serum and saliva. A predominant and dialyzable third fraction had no correlate in either serum or saliva. The small peptides in this fraction were subjected to amino acid, carbohydrate and peptide map analyses. The most abundant amino acids were alanine, glutamic acid, glycine, valine, leucine, lysine and serine. These small components contained no neutral or amino sugars. Pooled plaque extract and the small peptides exhibited similar peptide maps.
Abstract. From collected supragingival plaque, extracts were prepared for immunochemical analyses. Extracted sediments were examined by fluorescein‐labeled antibodies for the presence of immunoglobulins. Precipitation with monospecific and polyvalent antisera revealed IgA, IgG, secretory component, C3, α2macroglobulin, lactoferrin, and albumin in the extracts. Gel filtration of pooled plaque extract yielded two fractions that contained the aforementioned proteins and a prominent, dialyzable third fraction that was immunochemically nonreactive. IgA, IgG, secretory component, and light chains were shown, by immunofluorescence, to be present in washed, pooled plaque sediment. Release of these immunoglobulins by urea treatment indicated their probable participation in immune complexes.
The determination of iron(III) employing 8-hydroxyquinoline combines liquid / liquid extraction with the use of a pH meter as well as introducing the student to the formation and properties of complexes.
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