A Non-linear Singular Perturbation Problem Arising in the Study of Chemical Flow Reactors

This paper presents a method of obtaining the complete asymptoticsolution of boundary value problems of the form for x [0,1] where b(x) is strictly positive andfor small and positive. Physically, the problem arises in determiningthe steady-state concentration of a substance in a chemicalflow reactor. A "two-variable" expansion procedure is used.     

The preparation and reactions of C(α),N-dilithiocarboalkoxyhydrazones

C(α),N-Carboalkoxyhydrazones were metalated with an excess of lithium diisopropylamide (LDA), and the resulting dianions were condensed with methyl iodide, an unsubstituted ketone, a benzoate ester, salicylates, a benzoylacetate, or ethyl chloroformate.     

DISTRIBUTIONS OF FREE RADICALS AMONG AMINO ACIDS FROM LYOPHILIZED ULTRAVIOLET-IRRADIATED PROTEINS

Abstract— The effects of u.v.-irradiation at 254 nm upon lyophilized ribonuclease, lysozyme, insulin, and chymotrypsinogen have been investigated by electron spin resonance (ESR). enzymatic assay, and labeling of free radical sites with tritiated hydrogen sulfide (HST). The ESR signal of the irradiated protein diminishes on exposure to HST, and tritium becomes covalently bound to carbon. The distribution of tritium among the amino acids of each protein. studied as an indicator of the carbon free radical distribution, differs markedly from those observed previously to result from exposure to gamma radiation, electrical discharge. or hydrogen atoms. However, the earlier observation that the tritium distribution is influenced by protein conformation holds true as well for u.v.-irradiation. Moreover, the distributions of tritium among the amino acids of u.v.-irradiated proteins indicate a broad scattering of free radicals. Tyrosine and phenylalanine, residues that absorb light energy in the region of the wavelength employed, are not particularly important as radical carriers. Thus, for ribonuclease, these residues incorporated 3.8 and 1.5 per cent of the total tritium, but they absorb 51 and 12 per cent of the light, respectively. These results, together with the observed low recoveries of methionine, an amino acid that does not absorb at 254 nm, add weight to the concept that a migration of energy ensues after the initial absorption of light energy and that photolytic damage may thus be due to destruction of amino acids other than those initially absorbing the u.v.-radiation.     

Synthesis, characterization, and X-ray crystal structures of W(VI) alkyl complexes with chelating diamide and imido co-ligands

The complex W(NPh)Cl₂[o-(NSiMe₃)₂C₆H₄] 3 was synthesized from PhN = WCl₄ · OEt₂ and N,N′-(Li₂[o-(NSiMe₃)₂C₆H₄] and reacts with Lewis bases to form the adducts W(NPh)Cl₂[o-(NSiMe₃)₂C₆H₄](L) (L = PMe₃, THF, 3-picoline, ^tBuNC, MeCN) 4a–e. Crystals of 4a are triclinic, space group P1¯, with a = 9.562(1), b = 10.277(1), c = 14.920(2) Å, = 82.15(1), β = 80.18(1), γ = 80.41(1)°, and Z = 2. The structure was solved by the heavy atom method and refined to R = 0.0408 for 4224 observed (I > 2σ(I)) reflections. The dialkyl complexes W(NPh)R₂[o-(NSiMe₃)₂C₆H₄] (R = Me, Et, CH₂Ph, CH₂CMe₃, CH₂CMe₂Ph) 5–9 are formed through subsequent reactions of 3 with the corresponding Grignard reagent. Crystals of complex 5 are monoclinic, space group P2(1)/n, with a = 10.3545(2), b = 17.9669(1), c = 13.3168(1) Å, β = 103.826(1)°, and Z = 4. The structure of complex 5 was solved by direct methods in SHELXTL5 and refined to R = 0.0247 for 4572 observed reflections. Compound 5 has a square pyramidal geometry in which the imido ligand occupies the apical position and reacts with PMe₃ to form the adduct W(NPh)Me₂[o-(NSiMe₃)₂C₆H₄](PMe₃) 5a. Crystals of complex 5a are monoclinic, space group C2/m, with a = 13.5336(1), b = 14.4291(1), c = 15.3785(1) Å, β = 110.365(1)°, and Z = 4. The structure of compound 5a was solved by direct methods in shelxtl5 and refined to R = 0.0272 for 3057 observed reflections. Crystals of the bis-neopentyl complex 8 are monoclinic, space group P2(1)/n, with a = 10.6992(4), b = 18.3144(7), c = 16.0726(6) Å, β = 92.042(1)°, and Z = 4. The structure of 8 was solved by direct methods in shelxtl5 and refined to R = 0.0261 for 5881 observed reflections. Complex 8 has a trigonal bipyramidal geometry with both neopentyl groups and one amido nitrogen in the equatorial plane.     

Soap films and Kelvin's curved, truncated octahedron

Here, the Weierstrass representation theorem for minimal surfaces is used to derive parametrizations of two soap films spanning a rectangular prism with square base. The parametrizations are explicit, although the exact values of the height of the prism and the edge length of the square base can only be determined numerically from the formulas. The more famous of these two soap films can be extended to a partition of ℝ³ by curved, truncated octahedra. Furthermore, if the height of the prism is taken to be , then the partition is that discovered by Lord Kelvin in the late 19th century.     

A Type Calculus for Executable Modelling Languages

There is considerable current interest in the design and constructionof directly executable modelling languages for mathematicalprogramming. The present research extends contemporary modellinglanguages by specifying a type calculus for an extended dimensionalsystem that determines if the model is well formed in the sensethat the objective function and constraints consist of homogeneouscomponents. Each variable, coefficient, constant, function,constraint, input, and output of the model is assigned a typethat consists of its concepts, quantities, and units of measurementwith optional scale factors. In checking the composition offunctions and constraints, the system can do automatic unitconversions and apply user-supplied conversions. The systemallows a hierarchy of concepts that provides inheritance ofquantities and automatic concept conversion. In addition, eachset in a model is typed so the system can check the validityof operations on indices.