Formal proofs of operator identities by a single formal computation

A formal computation proving a new operator identity from known ones is, in principle, restricted by domains and codomains of linear operators involved, since not any two operators can be added or composed. Algebraically, identities can be modelled by noncommutative polynomials and such a formal computation proves that the polynomial corresponding to the new identity lies in the ideal generated by the polynomials corresponding to the known identities. In order to prove an operator identity, however, just proving membership of the polynomial in the ideal is not enough, since the ring of noncommutative polynomials ignores domains and codomains. We show that it suffices to additionally verify compatibility of this polynomial and of the generators of the ideal with the labelled quiver that encodes which polynomials can be realized as linear operators. Then, for every consistent representation of such a quiver in a linear category, there exists a computation in the category that proves the corresponding instance of the identity. Moreover, by assigning the same label to several edges of the quiver, the algebraic framework developed allows to model different versions of an operator by the same indeterminate in the noncommutative polynomials.     

On finite {1,2,3}-semiaffine planes

In this paper, we investigate {1,2,3}-semiaffine planes. All such planes of order n >51 shall be classified. It turns out that they are embeddable into projective planes of the same order n in the most natural way.Work supported by National Research Project on Strutture Geometriche Combinatoria, loro applicazioni of Italian M.P.I. and G.N.S.A.G.A. of C.N.R.     

Synthesis,Structure and Reactivity of PH‐Phosphoraneiminato‐ and Iminophosphoraneiminato Group 4 Transition‐Metal Complexes

The versatile coordination chemistry of the well‐investigated phosphoraneiminato‐ligand R₃PN^— ( I ) was extended by the successive introduction of protons to the phosphorus atom. The position of the resulting equilibrium between the NH‐phosphanylamido‐ [R₂P‐NH^—] and the PH‐phosphoraneiminato‐form [R₂HP=N^—] is affected by the Lewis acidity of the coordinated metal fragment. Experimental studies on complexes with various substitution patterns at the group 4 metal center R₂HP=N[M] ( II ) were unambiguously confirmed by DFT‐calculations. The isolation of group 4 PH‐dihydrido‐phosphoraneiminato‐complexes RH₂P‐N[M] ( III ) is prevented by the low thermodynamic stability of the target molecules, also supported by the results of ab initio calculations. However, an access to the by then unknown transition‐metal substituted iminophosphanes RP=N[M] ( IV ) was verified for the first time. Within extensive studies on the coordination chemistry of bis(imino)phosphoranes RP(=NR′)(=NR″), several species of group 4 complexes R(R′N=)P=N[M] ( V ) were isolated and structurally characterized. In this case, investigations on the NH/PH‐tautomerism were performed exclusively on theoretical level, because the required educts are experimentally non‐accessible due to their kinetic instability.     

Ruthenium complexes of analogues of the antitumor antibiotic streptonigrin

The complexes Ru(L1-CH3)(CO)3Cl, RuL2(CO)2Cl2, and RuL3(CO)2Cl2 (L1= 6-methoxy-5,8-quinolinedione, L2 = 7-amino-6-methoxy-5,8-quinolinedione, L3 = 6,6'-dimethoxycarbonyl-2,2'-bipyridine) were prepared by reaction of L1-L3 with the tricarbonyldichlororuthenium(II) dimer. L1-L3 act as bidentates through the ortho oxygen atoms, the pyridyl nitrogen and the adjacent quinone oxygen, and the bipyridyl nitrogens, respectively. RuL3(CO)2Cl2 is characterized by X-ray crystallography. 15N NMR correlation spectra give upfield shifts of around 60 ppm for the pyridyl nitrogens that are coordinated to the metal, while 13C NMR correlation spectra give a downfield shift of 10 ppm for the quinone carbonyl group that is coordinated to the metal. The electrochemistry of RuL2(CO)2Cl2 is examined, and the implications for the formation of metal complexes of the antitumor antibiotic streptonigrin, which cleaves DNA in the presence of metal ions, are discussed.