Congruence criteria for finite subsets of complex projective and complex hyperbolic spaces

We investigate congruence classes and direct congruence classes of m-tuples in the complex projective space ℂP ⁿ . For direct congruence one allows only isometries which are induced by linear (instead of semilinear) mappings. We establish a canonical bijection between the set of direct congruence classes of m-tuples of points in ℂP ⁿ and the set of equivalence classes of positive semidefinite Hermitean m×m-matrices of rank at most n+1 with 1's on the diagonal. As a corollary we get that the direct congruence class of an m-tuple is uniquely determined by the direct congruence classes of all of its triangles, provided that no pair of points of the m-tuple has distance π/2. Examples show that the situation changes drastically if one replaces direct congruence classes by congruence classes or if distances π/2 are allowed. Finally we do the same kind of investigation also for the complex hyperbolic space ℂH ⁿ . Most of the results are completely analogous, however, there are also some interesting differences. Received: 15 January 1996     

The shape invariant of triangles and trigonometry in two-point homogeneous spaces

We define a fourth basic invariant, which, besides the lengths of the three sides of a triangle, determines a triangle in the complex and quaternion projective spaces P ⁿ and P ⁿ (n2) uniquely up to isometry. We give inequalities describing the exact range of the four basic invariants. We express the angular invariants of a triangle with our basic invariants, giving a new completely elementary proof of the laws of trigonometry. As a corollary we derive a large number of congruence theorems. Finally we get, in exactly the same way, the corresponding results for triangles in the complex and quaternion hyperbolic spaces H ⁿ and H ⁿ (n2).     

Immobilizing a Fluorescent Dye Offers Potential to Investigate the Glass/Resin Interface

Silane coupling agents are commonly applied to glass fibers to promote fiber/resin adhesion and enhance durability in composite parts. In this study, a coupling agent multilayer on glass was doped with trace levels of the dimethylaminonitrostilbene (DMANS) fluorophore. The fluorophore was immobilized on the glass surface by tethering the molecule to a triethoxy silane coupling agent, creating the DMANS/silane coupling agent molecule (DMSCA). DMSCA was then diluted with commonly used coupling agents and grafted to a glass microscope coverslip to create a model composite interface. A 53-nm blue shift in fluorescence from the immobilized DMSCA can be followed during cure of an epoxy resin overlayer, giving this technique potential to monitor the properties of the fiber/resin interface during composite processing. Contact angle measurements on these coupling agent layers were similar in the presence or absence of the DMSCA molecule, suggesting that trace levels of the fluorescent probe did not affect the structure of the layer. The immobilized DMSCA molecule behaved similarly to the DMANS precursor in solution. Both showed longer wavelength fluorescence in more polar environments. Copyright 2000 Academic Press.     

Binding N2, N2H2, N2H4, and NH3 to transition-metal sulfur sites: modeling potential intermediates of biological N2 fixation

In the quest for low-molecular-weight metal sulfur complexes that bind nitrogenase-relevant small molecules and can serve as model complexes for nitrogenase, compounds with the [Ru(PiPr(3))('N(2)Me(2)S(2)')] fragment were found ('N(2)Me(2)S(2)'(2-)=1,2-ethanediamine-N,N'-dimethyl-N,N'-bis(2-benzenethiolate)(2-)). This fragment enabled the synthesis of a first series of chiral metal sulfur complexes, [Ru(L)(PiPr(3))('N(2)Me(2)S(2)')] with L=N(2), N(2)H(2), N(2)H(4), and NH(3), that meet the biological constraint of forming under mild conditions. The reaction of [Ru(NCCH(3))(PiPr(3))('N(2)Me(2)S(2)')] (1) with NH(3) gave the ammonia complex [Ru(NH(3))(PiPr(3))('N(2)Me(2)S(2)')] (4), which readily exchanged NH(3) for N(2) to yield the mononuclear dinitrogen complex [Ru(N(2))(PiPr(3))('N(2)Me(2)S(2)')] (2) in almost quantitative yield. Complex 2, obtained by this new efficient synthesis, was the starting material for the synthesis of dinuclear (R,R)- and (S,S)-[micro-N(2)[Ru(PiPr(3))('N(2)Me(2)S(2)')](2)] ((R,R)-/(S,S)-3). (Both 2 and 3 have been reported previously.) The as-yet inexplicable behavior of complex 3 to form also the R,S isomer in solution has been revealed by DFT calculations and (2)D NMR spectroscopy studies. The reaction of 1 or 2 with anhydrous hydrazine yielded the hydrazine complex [Ru(N(2)H(4))(PiPr(3))('N(2)Me(2)S(2)')] (6), which is a highly reactive intermediate. Disproportionation of 6 resulted in the formation of mononuclear diazene complexes, the ammonia complex 4, and finally the dinuclear diazene complex [micro-N(2)H(2)[Ru(PiPr(3))('N(2)Me(2)S(2)')](2)] (5). Dinuclear complex 5 could also be obtained directly in an independent synthesis from 1 and N(2)H(2), which was generated in situ by acidolysis of K(2)N(2)(CO(2))(2). Treatment of 6 with CH(2)Cl(2), however, formed a chloromethylated diazene species [[Ru(PiPr(3))('N(2)Me(2)S(2)')]-micro-N(2)H(2)[Ru(Cl)('N(2)Me(2)S(2)CH(2)Cl')]] (9) ('N(2)Me(2)S(2)CH(2)Cl'(2-) =1,2-ethanediamine-N,N'-dimethyl-N-(2-benzenethiolate)(1-)-N'-(2-benzenechloromethylthioether)(1-)]. The molecular structures of 4, 5, and 9 were determined by X-ray crystal structure analysis, and the labile N(2)H(4) complex 6 was characterized by NMR spectroscopy.     

(Z)-3-p-tolylsulfinylacrylonitrile as a chiral dienophile: diels-alder reactions with furan and acyclic dienes

The behavior of (Z)-3-p-tolylsulfinylacrylonitrile (1) as a chiral dienophile has been evaluated from its reactions with furan and acyclic dienes. Electrostatic interactions of the cyano group with the sulfinyl one restrict the conformational mobility around the C-S bond, thus controlling the pi-facial selectivity, which is almost complete in all cases, the approach of the diene from the less-hindered face of the dienophile (that bearing the lone electron pair) in the predominant rotamer being the favored one. The regioselectivity is also completely controlled by the cyano group. Additionally, the reactivity of compound 1 as well as its endo-selectivity are both higher than those observed for the corresponding (Z)-3-sulfinylacrylates, thus proving the potential of sulfinylnitriles as chiral dienophiles.