On the spectral norm of algebraic numbers

In this paper we continue to study the spectral norms and their completions ([4]) in the case of the algebraic closure $ \overline {\mathbb Q} $ of ? in ?. Let $ \widetilde{\overline{\mathbb{Q}}} $ be the completion of $ \overline {\mathbb Q} $ relative to the spectral norm. We prove that $ \widetilde{\overline{\mathbb{Q}}} $ can be identified with the R‐subalgebra of all symmetric functions of C(G), where C(G) denotes the ?‐Banach algebra of all continuous functions defined on the absolute Galois group G = Gal$ {\overline {\mathbb Q}} / {\mathbb Q} $. We prove that any compact, closed to conjugation subset of ? is the pseudo‐orbit of a suitable element of $ \widetilde{\overline{\mathbb{Q}}} $. We also prove that the topological closure of any algebraic number field in $ \widetilde{\overline{\mathbb{Q}}} $ is of the form $\widetilde{\mathbb{Q}[x]}$ with x in $ \widetilde{\overline{\mathbb{Q}}} $.     

Enlightened sonochemistry

Sonochemistry and photochemistry are initiated by high-energy transient species, which may be prone to mutual interaction. Electronic excitation of solutes by energy transfer from high energy species generated in collapsing bubbles is already supported by experimental evidence. The rates of photochemical reactions can be affected by ultrasound-induced mixing of liquids caused by microstreaming near pulsating cavitation bubbles and shockwaves due to bubble collapse. This may not only improve light absorption but also modify the pathway of reaction by increasing the contact between reagents. Finally, one may speculate about a potentially new chemistry of photoexcited solutes under the extreme conditions inside cavitation microreactors. This work reviews research on the excitation of solutes by sonoluminescence, the combined effects of ultrasound and light on liquid systems and the effect of ultrasound on photocatalytic reactions.     

A common generalization for MV-algebras and Łukasiewicz–Moisil algebras

We introduce the notion of n-nuanced MV-algebra by performing a Łukasiewicz–Moisil nuancing construction on top of MV-algebras. These structures extend both MV-algebras and Łukasiewicz–Moisil algebras, thus unifying two important types of structures in the algebra of logic. On a logical level, n-nuanced MV-algebras amalgamate two distinct approaches to many valuedness: that of the infinitely valued Łukasiewicz logic, more related in spirit to the fuzzy approach, and that of Moisil n-nuanced logic, which is more concerned with nuances of truth rather than truth degree. We study n-nuanced MV-algebras mainly from the algebraic and categorical points of view, and also consider some basic model-theoretic aspects. The relationship with a suitable notion of n-nuanced ordered group via an extension of the Γ construction is also analyzed.     

Immediate extensions of filtered rings

Summary A separated filtered ring has a maximally complete immediate extension. This is an analogue of Krull Theorem from valuation ring theory in the frame of filtered rings.

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Riassunto Un anello separato filtrato ha una immediata estensione massimale completa. Questo risultato è un analogo del teorema di Krull della teoria degli anelli di valutazione nell’ambito degli anelli filtrati.

     

4-Methylpyrimidinium ylides. Part 7: 3+2 Dipolar cycloadditions to non-symmetrical substituted alkenes and alkynes

The ability of 4-methylpyrimidinium ylides (as 1,3-dipoles) to react with activated non-symmetrical substituted dipolarophiles (alkenes and alkynes) is presented. 4-Methylpyrimidinium ylides did not react with alkenes. With alkynes the reactions are regiospecific, a single regioisomer being obtained. A possible mechanism for the reaction pathway is proposed. For the first time in the pyrimidinium ylides series both isomers resulting from bonding to the 2-and 6-positions of the heterocycle ring were obtained. The appropriate conditions in order to increase the selectivity of one of the isomers were determined.     

Organotin and tin(IV) derivatives of dimethyldithioarsinic acid

Organotin derivatives of dimethyldithioarsinic (dithocacodylic) acid have been obtained from the appropriate organotin chloride and the sodium salt of the latter. Tin(IV) chloride and NaS₂AsMe₂ · 2 H₂O yielded only two products, namely Cl₂Sn(S₂AsMe₂)₂ and Sn (S₂AsMe₂)₄, regardless of the reagent ratio. Spectroscopic characterization of the compounds (infrared and¹H NMR) provides structural information suggesting that the dimethyldithioarsinato group behaves as monodentate (or anisobidentate) ligand in Me₂Sn(S₂AsMe₂)₂, Bu₂Sn-(S₂AsMe₂)₂ and Cy₃Sn(S₂AsMe₂), as bidentate in Ph₂Sn(S₂AsMe₂)₂, Ph₃Sn(S₂AsMe₂) and Cl₂As(S₂AsMe₂)₂, whereas Sn(S₂AsMe₂)₄ contains both mono- and bidentate ligands, presumably in a six-coordinate structure.     

PMR-Spektren einiger Alkoxy-9-chlor-6-nitroacridine

The spectra of some 9-chloro-6-nitroacridines with methoxy substituents in position 1,4; 2,4; 3,4; 2,3; and ethoxy substituents in positions 1, 2, 3, were recorded. The influence of electron releasing alkoxy groups and electron withdrawing nitro groups on chemical shifts of protons in the 9-chloroacridinic cycle was discussed and PMR parameters were determined, comparatively with those of acridine and 9-chloroacridine. Good agreement between calculated and experimental values is observed. The spectrum of 9-chloro-1,2,3-triethoxyacridine shows for the four protons of the ringC. (H-5, H-6, H-7, H-8) anAMNX coupling pattern whose parameters were determined.