Means,Milne’s inequality,and quadrilateral area

Aequationes mathematicae - We discuss Milne’s inequality and apply it to the sides of a convex quadrilateral to derive an approximation to the area of the quadrilateral via arithmetic and...     

On the law S ( S ( x , y ), T ( x , y )) = S ( x , y ) of fuzzy logic

Motivated by some functional models arising in fuzzy logic, when classical boolean relations between sets are generalized, we study the functional equation S(S(x, y), T(x, y)) = S(x, y), where S is a continuous t-conorm and T is a continuous t-norm. Some interesting methods for solving this type of equations are introduced.     

On a family of connectives for fuzzy sets

In this paper we find the general solution of the functional equation $\text{S}{}^1\text{(T(x, y), T(x, N(y))) = x}$, where $\text{S}{}^1$ is a t-conorm, T is a t-norm and N is a strong negation on the unit interval. In particular the result yields a family of connectives for fuzzy sets.     

Acid-catalysed,nucleophile-catalysed and thermal decomposition of 2-amino-3-(2′,2′-dimethylaziridino)-1,4-naphthoquinone

The course of the thermal, acid-catalysed and iodide-catalysed decomposition of 2-amino-3-(2′,2′-dimethylaziridino)-1,4-naphthoquinone (III) was investigated. Thermal and iodide-catalysed decompositions gave mainly 2,3-diamino-1,4-naphthoquinone (VI) and 2-amino-3-(2′-methylallylamino)-1,4-naphthoquinone (V) together with low amounts of 2,2-dimethyl-1,2,3,4,5,10-hexahydrobenzo[g]quinoxaline (IV) and 2-isopropyl-1H-naphthoimid-azole-4,9-dione (VII). The acid catalysed isomerization of the aziridinonaphthoquinone III with halohydric acids or with acetic acid readily gave the opening of the aziridine ring; the corresponding salts of 2-amino-3-(2′-haloisobutylamino)-1,4-naphthoquinones (VIIIa-c) and 2-amino-3-(2′-acetoxyisobutylamino)-1,4-naphthoqunone (X) were formed by cleavage of the carbon-nitrogen bond at the substituted carbon atom. Hypotheses on the mechanism of these reactions are given.     

Investigation on the isomerization of 2-amino-3-aziridino-1,4-naphthoquinones to benzo[g]quinoxalines

The course of the hydriodic acid-catalysed and iodide-catalysed isomerization of various 2-amino-3-substituted-aziridino-1,4-naphthoquinones (I) to 1,2,3,4,5,10-hexahydrobenzo[g]-quinoxaline-5,10-diones (III) is investigated, and steric aspects of the reaction are also considered. Only in the case of the phenylaziridino derivative (Ie) does hydriodic acid afford direct cyclization to the corresponding benzoquinoxalinedione (IIIe); in all other cases the hydriodides (V) of the cleavage products (II) are obtained, and liberation of the free bases (II) results in cyclization to the corresponding benzoquinoxalinediones (III) when the aziridine ring is monosubstituted or trans disubstituted, with retention of configuration in the latter case. In contrast, the free bases (II) obtained from cis disubstituted (I) are relatively stable and cyclize with excess iodide yielding trans disubstituted (III). Correspondingly, monosubstituted and trans disubstituted I undergo iodide-catalysed isomerization to III whereas cis disubstituted I do not react.     

Thermal decomposition of 2-amino-3-aziridino-1,4-naphthoquinones

The course of the thermal decomposition of various 2-amino-3-substituted aziridino-1,4-naphthoquinones (Ia-g) was investigated. In all the cases, the thermal decomposition gave variable amounts of 2,3-diamino-1,4-naphthoquinone (II) and of substituted 1,2,3,4,5,10-hexahydrobenzo[g]quinoxaline-5,10-diones (IIIa-g) with complete stereospecificity. The decomposition of the aziridines Ib,f also gave significative amounts of 2-amino-3-allylamino-1,4-naphthoquinones (IVb,f). In the case of 2-amino-3-(2′-phenyl-3′-ethylaziridino)-1,4-naphthoquinone (Ig), the formation of trans-1-phenyl-1-butene (V), 2-(1-phenylpropyl)-1H-naphtho-imidazole-4,9-dione (VI), 2-phenyl-3-ethyl-3,4,5,10-tetrahydrobenzo[g]quinoxaline-5,10-dione (VII), 2-phenyl-3-ethyl-5,10-dihydrobenzo[g]quinoxaline-5,10-dione (VIII), and a mixture of cis- and trans-4H-2,3,5,6-tetra-hydro-2-phenyl-3-ethyl-5-iminonaphtho[1,2-b]oxazin-6-one (IX) also occurred. Hypotheses concerning the mechanism and the steric course of this reaction are given. The reaction is a general method for the stereospecific synthesis of 2,3-disubstituted 1,2,3,4,5,10-hexahydrobenzo[g]quinoxalines.     

Flow boiling visualization in a vertical circular minichannel at high vapor quality

This paper reports on an experimental study of saturated flow boiling of R134a inside a circular vertical quartz tube coated with a transparent heater. The inner diameter of the tube was 1.33 mm and the heated length 235.5 mm. The flow pattern at high vapor qualities and the dryout of the liquid film were studied using a high speed CCD camera at the mass fluxes 47.4 and 124.4 kg/m² s in up flow at 6.425 bar. The heat fluxes ranged from 5 to 13.6 kW/m² for the lower mass flux and from 20 to 32.4 kW/m² for the higher mass flux.

The behavior of the flow close to dryout was found to be different at low and high mass flux. At low mass flux the location of the liquid front fluctuated with waves passing high up in the tube. In between the waves, a thin film was formed, slowly evaporating without breaking up.

At high mass flux the location of the liquid front was more stable. In this case the liquid film was seen to break up into liquid streams and dry zones on the tube wall.     

On a functional equation based upon a result of Gaspard Monge

We present and solve completely a functional equation motivated by a classical result of Gaspard Monge.     

Mathematics and cross-curricular activities: Bridges exist for crossing them

The paper presents some thoughts on the relations of mathematics teaching and learning and the cross-curricular activities. Some Spantsh examples are reviewed and some insights for future actions are given.