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. 相似文献
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... 相似文献
A novel SBA-15-based fluorescent sensor, SBA-PI: mesoporous SBA-15 structure modified with iminostilbene groups, was designed, synthesized, and characterized by Fourier transform-infrared spectroscopy (FT-IR), ultraviolet–visible spectroscopy, field emission scanning electron microscopy (FESEM), transmission electron microscopy (TEM), energy-dispersive X-ray spectroscopy (EDS), thermogravimetric analysis (TGA), low-angle X-ray diffraction techniques (low-angle XRD), and N2 adsorption–desorption techniques. The SBA-PI as a sensor with a selective behavior for detection of Cu2+ comprises iminostilbene carbonyl as the fluorophore group. The SBA-PI sensor displays an excellent fluorescence response in aqueous solutions and the fluorescence intensity quenches remarkably upon addition of Cu2+. Other common interfering ions even at high concentration ratio showed either no or very small changes in the fluorescence intensity of SBA-PI in the absence of Cu2+. A limit of detection of 8.7 × 10−9 M for Cu2+ indicated that this fluorescence sensor has a high sensitivity and selectivity toward the target copper (II) ion. The fabricated Cu2+ sensor was successfully applied for the determination of the Cu2+ in human blood samples without any significant interference. With the selective analysis of Cu2+ ions down to 0.9 nM in blood, the sensor is a promising and a novel detection candidate for Cu2+ and can be applied in the clinical laboratory. A reversibility and accuracy in the fluorescence behavior of the sensor was found in the presence of I¯ that was described as a masking agent for Cu2+.