Geometric Algebra in Linear Algebra and Geometry

This article explores the use of geometric algebra in linear and multilinear algebra, and in affine, projective and conformal geometries. Our principal objective is to show how the rich algebraic tools of geometric algebra are fully compatible with and augment the more traditional tools of matrix algebra. The novel concept of an h-twistor makes possible a simple new proof of the striking relationship between conformal transformations in a pseudo-Euclidean space to isometries in a pseudo-Euclidean space of two higher dimensions. The utility of the h-twistor concept, which is a generalization of the idea of a Penrose twistor to a pseudo-Euclidean space of arbitrary signature, is amply demonstrated in a new treatment of the Schwarzian derivative.     

Unitary Geometric Algebra

The geometric significance of the imaginary unit in a complex geometric algebra has troubled the author for 40?years. In the unitary geometric algebra presented here, the imaginary i is a unit (pseudo) vector with square minus one which anti commutes with all of the real vectors. The resulting natural hermitian inner product and hermitian outer product induce a grading of the algebra into complex k-vectors. Basic orthogonality relationships are studied.     

Solvent effects on the intramolecular charge transfer character of N,N-diaryl dihydrophenazine catalysts for organocatalyzed atom transfer radical polymerization

The nature of intramolecular charge transfer (CT) of N,N-diaryl dihydrophenazine photocatalysts (PCs) in different solvents is explored in context of their performance in organocatalyzed atom transfer radical polymerization (O-ATRP). PCs having a computationally predicted lowest energy excited state exhibiting CT character can operate a highly controlled O-ATRP in a wide range of solvent polarities, from non-polar hexanes to highly polar N,N-dimethylacetamide. For PCs having a computationally predicted lowest energy excited state not possessing CT character, their ability to operate a controlled O-ATRP is decreased. This study confirms the importance of CT character in the excited state for N,N-diaryl dihydrophenazine PCs, and a deeper understanding of the activity of CT PCs has enabled the synthesis of polymers of low dispersity (<1.10) in a controlled fashion. © 2017 Wiley Periodicals, Inc. J. Polym. Sci., Part A: Polym. Chem. 2017 , 55, 3017–3027     

Conformational analysis of N,N-diisopropylamides by combined use of n.m.r. lanthanide-induced shifts and conformational energy calculations

A new method of conformational analysis has been developed, in which energy minimization calculations are combined with lanthanide-induced shift data. First, exhaustive energy calculations are carried out on the free molecules in order to determine the conformations of lowest energy. Then, the coordinates of all low energy conformations or pairs of conformations are used in the pseudocontact shift equation for lanthanide-induced shifts in order to find which of the theoretically obtained conformation(s) gives the best agreement with experiment. The molecules complexed to the lanthanide shift reagent were N,N-diisopropylformamide (DIPF) and N,N-diisopropylacetamide (DIPA). Two different lanthanide shift reagents were used, Eu(fod)₃

1 Fod is the anion of 1,1,1,2,2,3,3-heptafluoro-7,7-dimethyl-4,6-octanedione-d₂₇.

and Pr(fod)₃, in order to check the validity of the method. Proton magnetic resonance spectra were taken at 6 °C in carbon tetrachloride solution. The principal conformation found was different for each amide. DIPF was found to exist as a mixture of I (39 mol%) and II (61 mol%) with Eu(fod)₃, and a mixture of I (37%) and II (63%) with Pr(fod)₃. DIPA was found to exist as a mixture of I (79%) and IV (21%) with Eu(fod)₃ and a mixture of I (87%) and IV (13%) with Pr(fod)₃. For both molecules, the two conformations of lowest computed energy were also the pair which gave the best fit to the lanthanide shift reagent data. The location of the principal magnetic axis of the complex was found to lie between 0° and 14° from the lanthanide atom–oxygen atom bond axis. The technique of combining lanthanide shift reagent data with energy calculations shows great promise in conformational analysis.     

Finding the convex hull facet by facet

Finding the convex hull of a finite set of points in Euclidean space was one of the first problems explored in the field of computational geometry. In two dimensions a variety of algorithms have been developed and analyzed. For higher dimensions the problems are less well understood. Several “convex hull” problems are defined, solved, and analyzed here in terms of the size of the input and output. In all cases these solutions are the best of the known algorthms. The problems include enumerating the facets of the convex hull, computing the facial lattice of the convex hull and producing a new compact structure representing the combinatorial type of the convex hull. In addition, deciding the combinatorial equivalence of two polytopes is shown to be coNP-hard.     

Total synthesis of 8,12-iso-iPF3alpha-VI, an EPA-derived isoprostane: stereoselective introduction of the fifth asymmetric center

[reaction: see text] A new and stereoselective approach for the synthesis of all-syn isoprostanes is reported. This method, which is based on acid-catalyzed Diels-Alder reaction, allows the introduction of the side chain with a predetermined stereochemistry of the hydroxy group. The first total synthesis of an eicosapentaenoic acid (EPA)-derived iP, 8,12-iso-iPF3alpha-VI 10, was performed using this approach.