Orthogonal sequences constructed from quasi-orthogonal ultraspherical polynomials

Numerical Algorithms - Let $\displaystyle \{x_{k,n-1}\}_{k=1}^{n-1}$ and $\displaystyle \{x_{k,n}\}_{k=1}^{n},$ $n \in \mathbb {N}$ , be two sets of real, distinct points satisfying the interlacing...     

Grain orientation fragmentation in hot-deformed aluminium: Experiment and simulation

Grain orientation fragmentation is studied in a set of 176 individual grains of an aluminium polycrystal deformed in plane strain compression at 400 °C to a strain of $\epsilon =1.2$. Experimental observations were made by EBSD at successive strains of 0, 0.42, 0.77 and 1.2 on the internal surface of a split sample. Statistics of the in-grain orientation spreads were computed based on approximately 3000 orientation measurements per grain. A high-resolution finite element simulation (about 1000 elements per grain) was carried out on a polycrystal whose grains were assigned the initial experimental crystal orientations. The experimental and simulation results were compared in terms of the fractions of grains that exhibit fragmentation and the lattice orientations of the fragmenting grains. The numbers of fragmented grains increase with strain, reaching values of 10% in the experiment (2-D characterization) and 20% in the simulation (3-D characterization) at $\epsilon =1.2$. For both experiment and simulation, fragmentation is more likely in grains whose lattice is symmetrically oriented with respect to the loading axes. Under plane strain compression, the orientations of the fragmented grains coincide with regions of orientation space in which the reorientation velocity field in the plane perpendicular to the reorientation velocity direction is unstable.     

Direct transacylation of 2,2,2-trihaloethyl esters with amines and alcohols using phosphorus (III) reagents for reductive fragmentation and in situ activation

Amides and esters have been synthesized from 2,2,2-trihaloethyl esters in one pot using phosphorus-(III) reagents as reductants, with resultant carboxylate activation as an acyloxyphosphonium intermediate, and in situ trapping by amine or alcohol nucleophiles. Secondary and tertiary amides were synthesized, including a dipeptide, in good yields using hexamethylphosphorous triamide (Me2N)3P, as reducing agent. Optimal yields of esters derived from primary and secondary alcohols were obtained using tributylphosphine and DMAP. Tribromoethyl esters provided yields superior to those obtained with trichloroethyl esters.     

Diastereoselective silacyclopropanations of functionalized chiral alkenes

Lithium reduction of di-tert-butyldichlorosilane and thermal silylene transfer (105-125 degrees C) are complementary methods for the highly diastereoselective silacyclopropanations of a range of functionalized chiral olefins to afford complex silacycles. We have shown that functionalized cyclohexenes, cyclopentenes, norbornenes, and 1,1-disubstituted alkenes undergo silacyclopropanation with excellent diastereoselectivity (92:8 to >99:1). Our results demonstrate that steric interactions, rather than oxygen-directing effects, control the approach of the silylene or silylenoid intermediate to the olefin. We believe that the sterically demanding nature of the di-tert-butylsilylene species prevents coordination to the oxygen functionality. Thermal silylene transfer conditions exhibit broad functional group tolerance; the elevated temperatures for silylene transfer, however, cannot be employed for the silacyclopropanation of substituted cyclohexenes and 1,1-disubstituted alkenes. Elaboration of the resulting functionalized silacyclopropanes provides an efficient route to polyoxygenated products.