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.     

Mesomorphism of Dichotomous Compounds; X-Ray Diffraction Analysis of 4-Chlorophenyl Hexadecyl Ether

Crystallography Reports - The crystal and molecular structures of Cl–C6H4–O–C16H33 (I) have been studied, and its differential scanning calorimetry (DSC) study is performed....     

Baer sums for a natural class of monoid extensions

Semigroup Forum - It is well known that the set of isomorphism classes of extensions of groups with abelian kernel is characterized by the second cohomology group. In this paper we generalise this...     

Correction to: Impact of leg bending in the patient-specific computational fluid dynamics of popliteal stenting

Acta Mechanica Sinica - A Correction to this paper has been published: https://doi.org/10.1007/s10409-021-01066-2     

Effect of Plasmon Resonance of Silver Nanoparticles on the Fluorescent Properties of Doxycycline

Journal of Analytical Chemistry - Silver nanoparticles modified with Eu3+ ions increase the intensity of the fluorescence signal by a factor of 125 in an analytical system containing doxycycline as...     

Comparative Study of LiInSe2 Single Crystals for Thermal-Neutron Detection

Crystallography Reports - Two LiInSe2 single crystals, grown under different conditions, have been studied. Characteristics of these crystals for neutron detection have been compared using...     

A study for the selection of NPE-free cocktails for LSC routine measurements

Journal of Radioanalytical and Nuclear Chemistry - In this paper a study for the selection of an NPE (nonylphenol ethoxylates)—free cocktail is discussed in order to be used for our routine...     

Tandem silylation—desilylation reaction in the synthesis of N-methyl carboxamides

Russian Chemical Bulletin - The N-methylation of primary carboxamides is based on two consecutive reactions. The silylation of the corresponding carboxamides with chloro(chloromethyl)dimethylsilane...     

Operator Interpolation and Systems of Linear Equations and Inequalities in Euclidean Spaces

Ukrainian Mathematical Journal - We establish new criteria of compatibility for a linear system of equations (equivalent to the Kronecker–Capelli theorem) or inequalities (equivalent to the...