Student-Invented Numeration Systems: Pattern-Analysis and Mathematical Understanding

Thirteen fifth graders were given an assignment to invent their own numeration systems, following a unit on bases and a look at early events in the history of numbers. The task presented options that required the students to make decisions (such as whether to use a base, which base to use, design of symbols, etc.), and build a rationale for the elements of their system. Analyses of patterns embedded in their invented systems provided an assessment of student understanding of numeration. The progression of more and less complex thinking related to the student's choice of a base other than 10, consistency of logic throughout the system in words and symbols, rationale for change, and perception of real life examples that would change if the system was adopted. The invention task is presented as another way to make connections.     

Increasing the accuracy of eigenvalue computation in the spectral problem

A simple method is proposed for increasing the accuracy of computation of the eigenvalues of elliptic operators, which does not require high-order accurate schemes.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 63, pp. 63–68, 1987.     

Porphyrins 34. Mass spectrometric study of ethane- and ethylene-bis-porphyrins

The mass spectral behavior of 1, 2-di (mesoetioporphyr inyl-1)- and 1, 2-di (mesooctaethylporphyrlnyl) ethanes and ethylenes, their Cu and Ni complexes, as well as Ni Schiff bases of 1, 2-di (mesooctaethylporphyrinyl)ethane complexes, under ionization by electron impact and bombardment by accelerated atoms, was studied. Peaks of molecular ions, for which the basic decomposition pathways were established on the basis of an analysis of the parent and daughter ions, were observed in the electron impact mass spectra of all these compounds.For Communication 29, see [1].Translated from Khimiya Geterotsiklicheskikh Soedinenii, No. 1, pp. 24–29, January, 1994.     

Interaction of 4-aminobenzofurazan with aryldiazonium salts

The reaction of 4-aminobenzofurazan with aryldiazonium salts leads to the formation of 4-amino-5-aryl-azobenzofurazans and 5-amino-2-aryl-4-nitroso-2H-benzotriazoles, products of the rearrangement of the initially formed 4-amino-7-(arylazo)benzofurazans. Oxidation of the benzofurazan as well as of the triazole derivatives gives 7-aryl-1,2,3-triazol[4,5-e]benzofurazans. The chemical properties of some of the compounds obtained have been investigated.Novosibirsk Institute of Organic Chemistry, Siberian Branch, Russian Academy of Sciences, Novosibirsk 630090. Translated from Khimiya Geterotsiklicheskikh Soedinenii, No. 10, pp. 1432–1438, October, 1994. Original article submitted July 25, 1994.     

Use of ultrasound in heterocyclic chemistry (review)

We review literature data on synthesis and conversions of heterocyclic compounds using ultrasonic treatment.We analyze the effect of ultrasound on organic reactions involving participation of heterocyclic compounds.A. N. Kosygin State Technical Academy, Moscow 117918. Translated from Khimiya Geterotsiklicheskikh Soedinenii, No. 6, pp. 723–755, June 1994. Original article submitted December 22, 1993.     

Synthesis,structure, and chemical properties of some N-(3-chloro-2-quinoxalyl)arylsulfonamides

We have developed a method for synthesis of N-(3-clzloro-2-quinoxalyl)sulfonamides by reaction of 2,3-dichloroquinoxaline with substituted arylsufonamides. Based on the IR spectra, we have established that in the solid state, the synthesized compounds exist in the form of amide tautomers. Alkylation of these compounds leads to N-metliyl-N-(3-chloro-2-quinoxalyl)arylsulfonamides. We demonstrate the possibility of nucleophilic substitution of the halogen upon treatment with O- and N-nucleophiles. The use of bifunctional nucleophiles leads to condensed quinoxalines.Translated from Khimiya Geterotsiklicheskikh Soedinenii, Vol. 30, No. 3, pp. 387–392, March, 1994.     

Synthesis and ring-chain isomerism of acid chlorides and amides of 2-(2-pyridyl and 2-quinolylcarbonyl) benzoic acids

The reaction of 2-(2 pyridytcarbonyl)benzoic acid with thionyl chloride affords an unexpected product of the intramolecular acylation of the pyridine nitrogen atom, namely, 6,11-dioxo-6,11-dihydrobenzo[blquinotizinium chloride. At the same time, 2-(2-quinotylcarbonyl)benzoic acid forms the expected cyclic acid chloride, namely, 3-(2-gitinotyl)-3-chlorophthalide in this reaction. Both compounds acylate ammonia and primary amines, including those with bulky alkyl groups (tert-butyl, 1-adamantyl, and 1,1,3,3-tetramethylbutyl) with the formation of 2-R-3-hydroxy-3-(2pyridyl- or 2-quinolyl)isoindolines. The protonation of the pyridine nitrogen atom of N-(1,1,3,3-tetramethylbutyl)-2-(2pyridylcarbonyl)benzamide, obtained in the open amide form, is accompanied by the closing of the isoindotinone ring; the deprotonation is accompanied by ring opening.Riga Technical University, Riga LV-1048. Translated from Khimiya Geterotsiklicheskikh Soedinenii, No. 4, pp. 499–504, April, 1994. Original article submitted March 17, 1994.     

O-Alkylation of furan and thiophene ketoximes in conditions of interphase catalysis in a liquid/solid system

Alkylation of 2 furanyl- and 2-thienylalkyl ketoximes with alkyl, allyl, and propargyl halides in the two phase catalytic system sol. K₂CO₃/C₆H₆/18-crown-6 at room temperature causes the formation of mixtures of E- and Z-isomers of the corresponding O-ethers with yields of 32–74%. Partial E/Z-isomerization takes place during the reaction.Translated from Khimiya Geterotsiklicheskikh Soedinenii, No. 1, pp. 18–23, January, 1994.