An interpretation of viète's “calculus of triangles” as a precursor of the algebra of complex numbers

Viète introduced operations over right triangles which are directly related to the multiplication and division of complex numbers. He determined the relationships between the angles and sides of the triangles concerned. His first operation is characterised by what corresponds to the fact that arguments of complex numbers are added when the latter are multiplied, and in a similar theorem he shows what we now can see as the anticipation of the analogous characteristic for division of complex numbers. Viète finds powers of an arbitrary right triangle which correspond to powers of complex numbers. One of his theorems is structurally similar to the modern formulation of De Moivre's theorem. The investigation of this subject promises further results.     

Multiple positive solutions of some quasilinear neumann problems

Using the fibering method we prove the existence of at least two positive solutions for a class of non-monotone quasilinear elliptic equations with nonlinear Neuman boundary conditions     

A search for dichlorocarbene ether solvent interactions

The absolute rate constant of reaction of dichlorocarbene with pyridine and with tetramethylethylene was determined in Freon-113 (CF₂ClCFCl₂), dioxane, and tetrahydrofuran. No evidence of specific solvation was obtained.     

Synthesis of 2′(3′)-0-aminoacyl triribonucleoside diphosphates with the sequence of the acceptor terminus of AA-tRNA

The specific synthesis of the 2′(3′)-0-aminoacyl triribonucleoside diphosphates, analogs of the 3′-terminus of AA-tRNA is described $\text{via}$ phosphotriester intermediates.     

Structure of 1,4,10,19,25,41-C70(CF3)6, isomer with unique arrangement of addends

A novel isomer of C₇₀(CF₃)₆ has been isolated by HPLC from a mixture prepared by trifluoromethylation of C₇₀ with CF₃COOAg. The X-ray structure revealed an unprecedented arrangement of CF₃ groups forming a p³mp ribbon. This result provides additional evidence of the preferable formation of trifluoromethylated fullerene molecules comprising a single continuous ribbon of edge-sharing para- and meta-C₆(CF₃)₂ hexagons.     

Nature of interaction energy anisotropy in the Li(<Superscript>2</Superscript>S)–HF (˜X<Superscript>1</Superscript>Σ<Superscript>+</Superscript>) van der Waals complex. A theoretical study

The potential-energy surface for the Li(²S)–HF (? X¹Σ⁺ interaction, where HF is kept rigid, is calculated using the supermolecular unrestricted fourth-order Møller–Plesset perturbation theory. The basis set superposition error corrected potential indicates two minima. The global minimum occurs for the bent Li...FH structure at R=1.95 Å and θ=70° with a relatively deep well of D_e=1,706 cm^?1 and the secondary minimum is found for the linear Li...HF configuration at R=4.11 Å with a well depth ofD_e=288 cm^?1. A barrier of 177 cm^?1 (with respect to the secondary linear minimum) separates these two minima. In this study 27 bound states of the bent Li...FH minimum and eight bound states of the linear Li...HF minimum up to the Li+HF dissociation threshold are calculated. The energy partitioning using the intermolecular perturbation theory scheme shows that the origins of the stability of the structures studied are entirely different. The global minimum is stabilised using the attractive Coulombic interaction and unrestricted Hartree–Fock deformation energy. The latter term originates from the mutual electric polarisation effects. The secondary linear minimum is mostly determined by the anisotropy of the repulsive Heitler–London exchange-penetration and attractive dispersion energies.     

Capillary zone electrophoresis in wide bore capillary tubes with fiber-coupled diode array detection

This feasibility study deals with the use of a wide bore (320 μm I.D.) capillary tube for the detection and identification of capillary zone electrophoresis (CZE) analytes by optical fiber-coupled diode array detection. A 250-μm mean effective pathlength of the detection cell with an inherently enhanced photon flux through the cell were significant contributors in reaching 0.2–1 μmol/l concentration detectabilities of the CZE analytes by this combination. Experiments with model analytes (p-sulfanilic, sorbic and naphthalene-2-sulfonic acids, tryptophan and asulam) revealed that spectral confirmations of their identities were still possible when their concentrations in the loaded samples (200 nl) were 1–5 μmol/l. Here, chemometry procedures (target transformation factor analysis, fixed size moving window-target transformation factor analysis, fixed size moving window-evolving factor analysis and orthogonal projection approach) employed in the data processing effectively contributed to reliable confirmation of the identities of the analytes also in critical situations (e.g. peak overlaps). The CZE separations were carried out in tandem-coupled columns of identical I.D. This made it possible to use, in the first column of the tandem, carrier electrolyte solutions that provide the desired separative effects, while in the second (detection) column the compositions of the carrier electrolyte solutions employed could reflect favorable conditions for obtaining spectral data. Mixtures containing model constituents at significantly differing concentrations and Maillard’s reaction products spiked with tryptophan enantiomers were employed in experiments aimed at assessing practical applicabilities and limits of the present approach to the analysis of samples characterized by complex ionic matrices.     

Replacement of 2,2'-bipyridine by 1,4-diazabutadiene acceptor ligands: why the bathochromic shift for [(N empty set N)IrCl(C5Me5)]+ complexes but the hypsochromic shift for (N empty set N)Ir(C5Me5)?

Replacement of 2,2'-bipyridine (bpy) by substituted 1,4-diazabutadiene (R-DAB) alpha-diimine ligands N empty set N leads to a substantial hypsochromic shift of about 0.8 eV for the long-wavelength absorption band in compounds (N empty set N)Ir(C(5)Me(5)) but to a bathochromic absorption shift of about 0.4 eV for the complex ions [(N empty set N)IrCl(C(5)Me(5))](+). DFT calculations on model complexes based on experimental (R-DAB compounds) and geometry-optimized structures (bpy systems) reveal that the low-energy transitions of the cationic chloro complexes are largely of ligand-to-ligand charge-transfer character L'LCT (L = alpha-diimine, L' = Cl) whereas the neutral compounds exhibit pi --> pi transitions between the considerably mixed metal d(pi) and alpha-diimine pi orbitals. The much more pronounced metal-ligand orbital interaction for the R-DAB complexes causes the qualitatively different shifts on replacing the stronger basic bpy by the better pi-acceptors R-DAB. Only the LUMO of the neutral compounds is destabilized on replacement of bpy by R-DAB whereas the LUMO of [(N empty set N)IrCl(C(5)R'(5))](+) and both HOMOs are stabilized through this change.     

Application of one-particle Green functions technique for calculation of vertical ionization potentials of closed-shell molecules described by CNDO/2 semiempirical Hamiltonian

A simple method of infinite summations of some dominant diagrams in the framework of the one-particle Green functions technique is suggested. This method for the calculation of the lowlying vertical ionization potentials of some simple closed-shell molecules described by CNDO/2 semiempirical Hamiltonian is applied. The obtained results are in quite-satisfactory agreement with the experimental values of the vertical ionization potentials measured by the photo-electron spectroscopy technique.