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.     

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...     

An investigation on the influence of the shape of the vortex generator on fluid flow and turbulent heat transfer of hybrid nanofluid in a channel

Journal of Thermal Analysis and Calorimetry - The purpose of this study is to numerically investigate flow field and turbulent heat transfer of hybrid nanofluid, water–DWCNT–TiO2 in a...     

Controlling a bunch of multiple filaments by means of a beam diameter

We demonstrate a three orders of magnitude increase and stability in the backscattered fluorescence signal from nitrogen molecules by terawatt femtosecond laser pulse induced air filaments using a new method. The method is based on squeezing the initial beam diameter using a telescope. The effect of laser shot-to-shot fluctuations was included in numerical simulations by a random distribution of the initial intensity in both squeezed and non-squeezed beams. Statistical processing of the simulation results shows that the average diameter of plasma channels as well as the total amount of free electrons generated in a bunch of multiple filaments in air is larger in the squeezed beam. Shot-to-shot stability of the simulated plasma density increases in the squeezed beam. The change of this plasma density with propagation distance is in good qualitative agreement with the change of the range-corrected nitrogen fluorescence signal with distance. PACS 42.65.Jx; 42.60.Jf; 42.68.Ay; 42.68.Wt     

Fabrication and Determination of Refractive Index Profile of the Planar Waveguides by Wedge Technique

Several planar waveguides have been fabricated. The waveguides have been polished for determination of their refractiveindex profiles (RIP) by wedge method. The RIP determined by inserting the sample in a Mach-Zehnder interferometer andapplying fringe analysis methods.     

An Efficient and Facile Synthesis of Quinoxaline Derivatives Catalyzed by KHSO4 at Room Temperature

Summary. A facile synthesis of quinoxaline derivatives catalyzed by KHSO₄ in very high yields at room temperature is reported.