A convexity-preservingC 2 parametric rational cubic interpolation

Summary AC ² parametric rational cubic interpolantr(t)=x(t) i+y(t) j,t[t ₁,t _n] to data S={(x_j, y_j)|j=1,...,n} is defined in terms of non-negative tension parameters _j ,j=1,...,n–1. LetP be the polygonal line defined by the directed line segments joining the points (x _j ,y _j ),t=1,...,n. Sufficient conditions are derived which ensure thatr(t) is a strictly convex function on strictly left/right winding polygonal line segmentsP. It is then proved that there always exist _j ,j=1,...,n–1 for whichr(t) preserves the local left/righ winding properties of any polygonal lineP. An example application is discussed.This research was supported in part by the natural Sciences and Engineering Research Council of Canada.     

Cycloaddition de derives organiques insatures du silicium,du germanium et de l'etain

The effect of atom M and of its substituents Σ upon dienophilic activity of Σ₃MCHCH₂ and Σ₃MCCH (MC, Si, Ge, Sn) is interpreted in terms of a perturbation scheme involving the lowest unoccupied orbital of the dienophile and the highest occupied orbital of the diene. This scheme also explains the stereochemistry of Diels-Alder reactions and the regioselectivity of the ene reaction with organometallic compounds.     

Préparation de thiocétones acycliques α-insaturées. dimérisation régiosélective en 4h-dithiine-1,3

α-Unsaturated acyclic thioketones 6 a–d were prepared by a retro-Diels-Alder reaction using flash thermolysis technique. They were detected at low temperature (blue coloured pyrolysate) but dimerize in solution as soon as the temperature rose to ?60° (case 6 a–c) and ?20° (case 6 d). The structure of 6 d was established by UV and NMR spectra taken at ?60°. Dimeric products were proved to be 4H - 1,3 dithiin by spectral data and by univocal synthesis of 7 c and 7 d from the corresponding monomeric ketone. 1–3 Dithiin 7 d was isomerized at 190° into the thermodynamic isomer (8). This observed orientation of dimerisation agrees with theorical results obtained from molecular frontier orbital calculations.     

On maximal antichains containing no set and its complement

Let 1?k₁?k₂?…?k_n be integers and let S denote the set of all vectors x = (x₁, …, x_n with integral coordinates satisfying 0?x_i?k_i, i = 1,2, …, n; equivalently, S is the set of all subsets of a multiset consisting of k_i elements of type i, i = 1,2, …, n. A subset X of S is an antichain if and only if for any two vectors x and y in X the inequalities x_i?y_i, i = 1,2, …, n, do not all hold. For an arbitrary subset H of S, (i)H denotes the subset of H consisting of vectors with component sum i, i = 0, 1, 2, …, K, where K = k₁ + k₂ + …k_n. |H| denotes the number of vectors in H, and the complement of a vector x?S is (k₁-x₁, k₂-x₂, …, k_n -x_n). What is the maximal cardinality of an antichain containing no vector and its complement? The answer is obtained as a corollary of the following theorem: if X is an antichain, K is even and$\text{|(}\text{1}\text{2}\text{K)X|}$ does not exceed the number of vectors in $\text{(}\text{1}\text{2}\text{K)S}$ with first coordinate different from k₁, then

$\text{∑}\text{i=0}\text{K}\text{i≠}\text{1}\text{2}\text{K}\text{|(i)X|}\text{|(i)S|}\text{+}\text{|(}\text{1}\text{2}\text{K)X|}\text{|(}\text{1}\text{2}\text{K-1)S|}\text{?1}$

.     

A simple problem in comparative diffusion

The problem of determining the amount of a given substance that has diffused into one life form given that a known amount has diffused into a second life form under the same conditions is studied under the assumption that: free molecular diffusion is the only process involved, the mass diffusion coefficients are known constants and the external concentration of the diffusing substance remains constant. The solution of this comparative diffusion problem is obtained in terms of the external concentration value and the physical properties of each life from for the special case of a cylinder-sphere configuration of the pair. The properties of this solution are them determined for all reasonably small and sufficiently large values of the external concentration value. The solution for arbitary configurations is also obtained and on the basis of the results established here, several conjectures are proposed regarding its general properties     

Arene-cation interactions of positive quadrupole moment aromatics and arene-anion interactions of negative quadrupole moment aromatics

Intermolecular interactions involving aromatic pi-electron density are widely believed to be governed by the aromatic molecular quadrupole moment, Theta(zz). Arene-cation binding is believed to occur primarily with negative Theta(zz) aromatics, and arene-anion binding is believed to occur largely with positive Theta(zz) aromatics. We have performed quantum mechanical computations that show the cation binding of positive Theta(zz) aromatics and the anion binding of negative Theta(zz) aromatics is quite common in the gas phase. The pi-electron density of hexafluorobenzene, the prototypical positive Theta(zz ) aromatic (experimental Theta(zz) = 9.5 +/- 0.5 DA), has a Li+ binding enthalpy of -4.37 kcal/mol at the MP2(full)/6-311G**level of theory. The RHF/6-311G** calculated Theta(zz) value of 1,4-dicyanobenzene is +11.81 DA, yet it has an MP2(full)/6-311G** Li+ binding enthalpy of -12.65 kcal/mol and a Na+ binding enthalpy of -3.72 kcal/mol. The pi-electron density of benzene, the prototypical negative Theta(zz) aromatic (experimental Theta(zz) = -8.7 +/- 0.5 DA), has a F- binding enthalpy of -5.51 kcal/mol. The RHF/6-311G** calculated Theta(zz) of C6H2I4 is -10.45 DA, yet it has an MP2(full)/6-311++G** calculated F- binding enthalpy of -20.13 kcal/mol. Our results show that as the aromatic Theta(zz) value increases the cation binding enthalpy decreases; a plot of cation binding enthalpies versus aromatic Theta(zz) gives a line of best of fit with R2 = 0.778. No such correlation exists between the aromatic Theta(zz) value and the anion binding enthalpy; the line of best fit has R2 = 0.297. These results are discussed in terms of electrostatic and polarizability contributions to the overall binding enthalpies.     

Periodic DFT study of the structural and electronic properties of bulk CoAl2O4 spinel

In this study, structural and electronic properties of CoAl2O4 spinel are investigated for the first time by means of quantum chemical computational tools. Coupling supercell periodic calculations under the density functional theory formalism with a nonempirical quasi-harmonic Debye model, we examine the influence of temperature on the relative stability of several cation distributions of Co2+ and Al3+ over tetrahedral and octahedral interstices of the oxygen sublattice. Our simulations are able to reproduce the experimentally observed trend: (i) the normal spinel is calculated to be the stable structure at static and low-temperature conditions, and (ii) as the temperature increases, the preference of structures with Al3+ at tetrahedral sites (and Co2+ at octahedral sites) is found to progress following an asymptotic conduct. The effects of the cation distributions on geometrical variations of electronic and magnetic properties of CoAl2O4 can be interpreted as dominated by the local behavior of Co2+ at octahedral sites.