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.     

PANI-Fe3O4@ZnO nanocomposite as magnetically recoverable organometallic nanocatalyst promoted synthesis of new Azo chromene dyes and evaluation of their antioxidant and antimicrobial activities

Molecular Diversity - A new series of azo chromene dyes were synthesized via a facile cyclocondensation reaction of (E)-1,2-diphenyl-1-diazene and 4-aminocoumarin with 1:2 molar ratio catalyzed by...     

Incorporation of silica nanoparticles and polyurethane into hybrid composites for increase of char residue

Journal of Thermal Analysis and Calorimetry - Effect of different chain extenders, silica nanoparticle loading, and using sol–gel method on the thermal properties of polyurethane (PU) hybrid...     

Turbulent combined forced and natural convection of nanofluid in a 3D rectangular channel using two-phase model approach

Journal of Thermal Analysis and Calorimetry - The nanofluid is a mixture of base fluid and solid nanoparticles in nanosize. The heat transfer generated by the nanofluid is more than the base fluid...     

B-, N-doped and BN codoped C60 heterofullerenes for environmental monitoring of NO and NO2: a DFT study

ABSTRACT

Using density functional theory calculations, we investigate the gas sensing performance of B-, N-doped and BN-codoped C₆₀ fullerenes towards NO and NO₂ molecules. The calculated adsorption energies and net charge-transfer values indicate that NO and NO₂ molecules have a stronger interaction with the BN-codoped fullerenes compared to the B- or N-doped ones. It is also found that the electronic properties of the BN-codoped C₆₀ exhibit a larger sensitivity towards NO and NO₂ molecules. An increase in the concentration of doped/co-doped B and N atoms tends to weaken the gas sensing ability of these systems.     

Design,preparation, and characterization of Fe3O4 nanoparticles encapsulating β-cyclodextrin-bearing guanidine as a highly efficient and reusable heterogeneous base catalyst for synthesis of 3,4-dihydropyrano[3,2-c]chromenes

Chromene substructure is an important structural motif present in a variety of medicines, natural products, and materials showing biological activities. Here, a simple and convenient procedure for the synthesis of 3,4-dihydropyrano[3,2-c]chromene derivatives is described. For this purpose, Fe₃O₄ nanoparticles supported on β-cyclodextrin-guanidine were successfully prepared and used as catalyst. The structure of this catalyst was assigned by Fourier transform infrared spectroscopy, X-ray diffraction, scanning electron microscopy, thermal gravimetric analysis, and vibrating sample magnetometer techniques. The prepared nanocomposites were used as a highly active, heterogeneous, and reusable nanocatalyst for the one-pot, three-component reaction of 4-hydroxycoumarin, aromatic aldehydes, and ethyl cyanoacetate. This method has advantages such as mild conditions, high yields, easy workup and simple purification of products, little catalyst loading, cost efficiency, and reusability of the catalyst.