Bruck nets,codes, and characters of loops

Numerous computational examples suggest that if _{k-1 k} are (k- 1)- and k-nets of order n, then rank _{p k} - rank _{p k-1} n - k + 1 for any prime p dividing n at most once. We conjecture that this inequality always holds. Using characters of loops, we verify the conjecture in case k = 3, proving in fact that if p ^e n, then rank _{p 3} 3n - 2 - e, where equality holds if and only if the loop G coordinatizing ₃ has a normal subloop K such that G/K is an elementary abelian group of order p ^e . Furthermore if n is squarefree, then rank _p = 3n - 3 for every prime p ¦ n, if and only if ₃ is cyclic (i.e., ₃ is coordinated by a cyclic group of order n).The validity of our conjectured lower bound would imply that any projective plane of squarefree order, or of order n 2 mod 4, is in fact desarguesian of prime order.     

Completeness for intersection classes

An intersection representation of a graph is a function gf mapping vertices to sets such that for any u ≠ v, u is adjacent to v iff gf(u) ∩ gf(v) ≠ . The intersection class defined by a set of sets ∑ is the set of all graphs having an intersection representation using sets from ∑. Interval graphs and chordal graphs are well-studied examples of intersection classes.

This paper introduces the notion of completeness for intersection classes. That is, determining precisely what restrictions can be made on the allowable sets without losing the power to represent any graph in the intersection class. The solution to this problem is presented for the chordal graphs.     

New insights into the intercalation chemistry of Al(OH)3

This paper reports a number of recent developments in the intercalation chemistry of Al(OH)(3). From Rietveld refinement and solid-state NMR, it has been possible to develop a structural model for the recently reported [M(II)Al(4)(OH)(12)](NO(3))(2)·yH(2)O family of layered double hydroxides (LDHs). The M(2+) cations occupy half of the octahedral holes in the Al(OH)(3) layers, and it is thought that there is complete ordering of the metal ions while the interlayer nitrate anions are highly disordered. Filling the remainder of the octahedral holes in the layers proved impossible. While the intercalation of Li salts into Al(OH)(3) is facile, it was found that the intercalation of M(II) salts is much more capricious. Only with Co, Ni, Cu, and Zn nitrates and Zn sulfate were phase-pure LDHs produced. In other cases, there is either no reaction or a phase believed to be an LDH forms concomitantly with impurity phases. Reacting Al(OH)(3) with mixtures of M(II) salts can lead to the production of three-metal M(II)-M(II)'-Al LDHs, but it is necessary to control precisely the starting ratios of the two M(II) salts in the reaction gel because Al(OH)(3) displays selective intercalation of M nitrate (Li > Ni > Co ≈ Zn). The three-metal M(II)-M(II)'-Al LDHs exhibit facile ion exchange intercalation, which has been investigated in the first energy dispersive X-ray diffraction study of a chemical reaction system performed on Beamline I12 of the Diamond Light Source.     

A Family of Antipodal Distance-Regular Graphs Related to the Classical Preparata Codes

A new family of distance-regular graphs is constructed. They are antipodal 2^2t–1-fold covers of the complete graph on 2^2t vertices. The automorphism groups are determined, and the extended Preparata codes are reconstructed using walks on these graphs.There are connections to other interesting structures: the graphs are equivalent to certain generalized Hadamard matrices; and their underlying 3-class association scheme is formally dual to the scheme of a system of linked symmetric designs obtained from Kerdock sets of skew matrices in characteristic two.     

Supergravity symmetries in superspace

We show the extension of the Forgacs-Manton Killing symmetry for gauge theories, to solutions, extended to superspace, of the d = 11 supergravity theory.     

KN interactions in the resonance region (III). Resonance spectra

The parameters of the resonances from an energy-dependent multichannel analysis of two-body $\text{K}$N data are extracted in several models and compared with results from other recent analyses. A critical review is given of the status of the hyperon (S = ?1) spectrum below 2 GeV, and some comments made on classifications in simple quark models.     

RESTRICTED LIE ALGEBRAS WITH MAXIMAL 0-PIM

In this paper it is shown that the projective cover of the trivial irreducible module of a finite-dimensional solvable restricted Lie algebra is induced from the one dimensional trivial module of a maximal torus. As a consequence, the number of the isomorphism classes of irreducible modules with a fixed p-character for a finite-dimensional solvable restricted Lie algebra L is bounded above by p ^MT(L), where MT(L) denotes the maximal dimension of a torus in L. Finally, it is proved that in characteristic p > 3 the projective cover of the trivial irreducible L-module is induced from the one-dimensional trivial module of a torus of maximal dimension, only if L is solvable.     

The growing applications of click chemistry

Click chemistry, the subject of this tutorial review, is a modular synthetic approach towards the assembly of new molecular entities. This powerful strategy relies mainly upon the construction of carbon-heteroatom bonds using spring-loaded reactants. Its growing number of applications are found in nearly all areas of modern chemistry from drug discovery to materials science. The copper(I)-catalysed 1,2,3-triazole forming reaction between azides and terminal alkynes has become the gold standard of click chemistry due to its reliability, specificity and biocompatibility.     

PSL(2,q) as a collineation group of projective planes of small order

Let Π be a projective plane of order n admitting a collineation group G≅PSL(2, q) for some prime power q. It is well known for n=q that Π must be Desarguesian. We show that if n<q then only finitely many cases may occur for П, all of which are Desarguesian. We obtain some information in case n=q ² with q odd, notably that G acts irreducibly on П for q≠3, 5, 9. The material herein was presented to the University of Toronto in partial fulfillment of the requirements for the degree of Doctor of Philosophy. The author is grateful to Professor Chat Y. Ho, presently at the University of Florida, for guidance in this research.