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...     

Wet carbon-based solid acid/NaNO_3 as a mild and efficient reagent for nitration of aromatic compound under solvent free conditions

Nitro aromatic compound can be obtained in high yields via nitration of aromatic compound with wet carbon-based solid acid and NaNO_3 under solvent free oxidation at room temperature.     

Solvent extraction of mercury using Alamine 304

Summary A study has been undertaken to develop a solvent extraction system for mercury using Alamine 304 in aqueous chloride medium. The parameters investigated included reagent concentration, acid concentration, pH, aqueous to organic phase ratio, and rate of extraction and stripping. The study shows that mercury can be extracted rapidly and efficiently from aqueous chloride solutions. A number of aqueous stripping reagents removed more than 90% of the mercury from the organic phase with only one equilibrium. These include HNO₃, NaOH and EDTA.

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Extraktion von Quecksilber mit Alamin 304

Zusammenfassung Die Extraktion von Quecksilber aus der wäßrigen Lösung des Chlorids mit Alamin 304 wurde untersucht. Die Konzentration des Reagens, der Säure, das pH, das Verhältnis der wäßrigen zur organischen Phase sowie das Extraktionsverhältnis wurden geprüft. Die Extraktion verläuft rasch und wirkungsvoll. Eine Reihe wäßriger Reagenzien entzieht mehr als 90% des Quecksilbers aus der organischen Phase. Dazu gehören Salpetersäure, Natronlauge und EDTA.

     

Quaternized amino functionalized cross-linked polyacrylamide as a new solid-liquid phase transfer catalyst in reduction of carbonyl compounds with NaBH4

Poly[N-(2-aminoethyl)acrylamido]trimethyl ammonium chloride resin was developed as a new polymeric phase transfer catalyst. This quaternized polyacrylamide catalyzed the chemoselective reduction of aldehydes and ketones by NaBH₄ to give corresponding alcohols in high yields under mild conditions.