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.     

THE ENERGY ASPECT OF THE RECIPROCAL INTERACTIONS OF PAIRS OF TWO DIFFERENT VIBRATION MODES OF A CLAMPED ANNULAR PLATE

The standardized mutual active and reactive sound power of a clamped plate, representing the energy aspect of the reciprocal interactions of two different in vacuo modes, has been computed. It was assumed that the vibrations are axisymmetric, elastic and time harmonic, the plate's transverse deflection is small as compared with the plate's size, and that the vibration velocity is small as compared with the acoustic wavenumber generated. The Kirchhoff-Love theory of a perfectly elastic plate was used. The integral formulae for the mutual sound power were transformed into their Hankel representations which made possible their subsequent computation. A closed path integral was used to express the integral in its Hankel representation to compute the mutual active sound power. The asymptotic stationary phase method was used to compute the two magnitudes, i.e., the mutual active and reactive sound power. The results obtained are the asymptotic formulae valid for the acoustically fast waves. The oscillating as well as the non-oscillating terms have been identified in the formulae to make possible their further separate analysis. The availability of the asymptotic formulae makes possible some fast numerical computations of the mutual sound power. Moreover, the formulae presented herein, together with those for the individual modes known from the literature, make a complete basis for further computations of the total sound power of the plate's damped and forced vibrations in fluid.     

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.     

Chelate von Chinolin-8-01-Derivaten. V [I]. Komplexbildung und Extraktion von Nickel mit alkylsubstituierten Chinolin-8-olen

Chelates of 8-Quinolinol Derivatives. V. Coordination and Extraction of Nickel with Alkylsubstituted 8-Quinolinols . The nickel chelates of 2-, 5-, and 7-methyl-8-quinolinols and of a series of 7-alkyl-8-quinolinols were synthesized in the water-free state. Spectroscopic and magnetochemical differences between the compounds with 2-, 5-, and 7-methyl-8-quinolinols were explained by structural peculiarities. For the nickel chelates with 7-alkyl-8-quinolinols a somewhat distorted square-planar structure is supposed. To characterize the influence of substituents the adduct formation with pyridine was studied. For the extraction of nickel with 8-quinolinols the kind of the alkyl or alkenyl substituent in 7-position has no distinct influence on the state of the extraction equilibrium but it impairs the quickness of the extraction with increasing hydrophobicity.     

1,8-bis(dimethylethyleneguanidino)naphthalene: tailoring the basicity of bisguanidine "proton sponges" by experiment and theory

1,8-bis(dimethylethyleneguanidino)naphthalene (DMEGN), the second example of a peralkyl guanidine "proton sponge" based on the 1,8-naphthalene backbone, was prepared and fully characterized. The crystal structure analysis of monoprotonated DMEGN reveals an unsymmetrical intramolecular hydrogen bridge. A decrease in the basicity with respect to the noncyclic parent 1,8-bis(tetramethylguanidino)naphthalene was found. Nevertheless, a new proton sponge provides a new crossbar in the ladder of highly basic neutral organic compounds. A detailed theoretical study of DMEGN and related cyclic guanidines explains this surprising experimental result. Homodesmotic reactions reveal that the intramolecular hydrogen bond contributes effectively 10 kcal/mol to proton affinity of DMEGN.