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.     

Small-Angle Neutron Scattering at the Pulsed Reactor IBR-2: Current Status and Prospects

Crystallography Reports - Neutron diffraction studies on the small-angle neutron spectrometer YuMO (Joint Institute for Nuclear Research, Dubna), based on the IBR-2 pulsed reactor, have been...     

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....     

Application of Synchrotron Radiation to Analyze the Friction-Induced Structural and Phase Transformations of Chrome-Nickel Austenitic Steel

Russian Physics Journal -     

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     

Versatile Near-Infrared Super-Resolution Imaging of Amyloid Fibrils with the Fluorogenic Probe CRANAD-2

CRANAD-2 is a fluorogenic curcumin derivative used for near-infrared detection and imaging in vivo of amyloid aggregates, which are involved in neurodegenerative diseases. We explore the performance of CRANAD-2 in two super-resolution imaging techniques, namely stimulated emission depletion (STED) and single-molecule localization microscopy (SMLM), with markedly different fluorophore requirements. By conveniently adapting the concentration of CRANAD-2, which transiently binds to amyloid fibrils, we show that it performs well in both techniques, achieving a resolution in the range of 45–55 nm. Correlation of SMLM with atomic force microscopy (AFM) validates the resolution of fine features in the reconstructed super-resolved image. The good performance and versatility of CRANAD-2 provides a powerful tool for near-infrared nanoscopic imaging of amyloids in vitro and in vivo.     

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...     

Effect of External Factors on the Ferromagnetic Resonance Line Width in Exchange-Biased Structures

Physics of the Solid State - The external factors which influence the ferromagnetic resonance (FMR) line width in bilayer (ferromagnet/antiferromagnet) exchange-biased systems are studied. The...