Incorrect but instructive

This note is a possible contribution to the teaching of mathematics. We present a variety of examples of wrong proofs, misinterpreted definitions and the mistaken use of theory. The examples are based on our experience of teaching mathematics to engineering students. They include elementary examples and more advanced ones, and are taken from different subjects. In analysing the mistakes, we try to use them to improve the students' understanding.     

Learning to prove: from examples to general statements

In this paper we describe a method for teaching students to prove some mathematical statements independently, by using specially designed auxiliary assignments. The assignments are designed as homework problems and can be adapted for online learning. We illustrate our method using examples from calculus and differential equations.     

Spectral properties and solubility of the lanthanide complexes with substituted and annelated porphyrazines

An analysis of the electronic optical properties and solubility in chloroform and DMF of the erbium and ytterbium complexes with porphyrazine and its octaphenyl- and octa(p-alkylsulfamoylphenyl)-substituted derivatives and tetrapyrazinoporphyrazine and its octaethyl-, tetra(tert-butyl)-, and octaphenyl-substituted derivatives is presented. The dependence of the electronic optical properties and solubility on the composition of the macrocycle of the studied complexes, namely, on the nature of the extraligand and peripheral environment of the porphyrazine macrocycle, is shown.     

Effects of mixed conductivity of nanocomposite membranes MF-4SC/PANI

Changes in the conducting and hydrophilic properties of composites MF-4SC/polyaniline (PAni) under conditions of prolonged synthesis have been studied. A maximum of PAni content of about 0.20 by weight, which can be incorporated into the matrix of MF-4SC under these conditions of synthesis, is determined. Percolation behavior of electrical conductivity of the composites after drying was observed. The conductivity of PAni salt inside MF-4SC was estimated within the frames of the percolation model. Using the fibrous cluster model of the membrane and the conductivity data on individual PAni, theoretical assessment of the electrical conductivity of nanocomposite MF-4SC/PAni has been performed. Reasons for a significant reduction in the conductivity of PAni during its integration into the structure of the initial matrix were discussed. A scale of membrane conductivity, reflecting changes in the electrical conductivity of composites at various stages of synthesis, was drawn.     

Asymmetry of voltammetric characteristics of perfluorinated MF-4SK membranes with polyaniline-modified surface

The effect of asymmetry of the voltammetric characteristics is studied on anisotropic composites obtained using the technique of polyaniline matrix synthesis in the surface layer of the MF-4SK membrane. On the basis of studies of the ohmic voltammetric curve section in a series of isomolar HCl-NaCl solutions, independence of the membrane conductivity on the fraction of protons in the solution is established in the case when the modified membrane side is turned towards the counterion flow. In the case of the reverse orientation, the membrane conductivity is changed symbatically at an increase in the proton content in the mixed solution. An explanation of this effect is suggested with account for the regularities of the concentration profile formed in the bilayer membrane under polarization, water decomposition on the inner bipolar boundary, and also a change in the energy state of water in the composite membrane.     

Effect of polyaniline on the current passing through structural fragments of ion-exchange sulfonic-cationite resins and membranes

Concentration dependences of the specific electrical conductivity of ion-exchange resin KU-2 and ion-exchange heterogeneous membrane MK-40 made thereof are obtained and compared prior to and after the membrane modification with polyaniline. It is shown that the introducing of the polyaniline to polymer matrices of ionites results in the change of the specific electrical conductivity of both the resin KU-2 and the membrane MK-40, which correlates with the earlier data obtained for the MF-4SK/polyaniline composites. The quantitative evaluation of the polyaniline effect on the mechanism of current passing through structural fragments of initial and modified ionites is carried out. To this purpose, model parameters were used, which reflect the current paths and the conducting phases’ volume fractions and spatial orientation.     

Electrochemical behavior of <Emphasis Type="Italic">meso</Emphasis>-substituted iron porphyrins in alkaline aqueous media

The effect meso-substitution in iron porphyrin complexes has on their redox behavior in alkaline aqueous solutions is studied via cyclic voltammetry. The voltammetric features of the reduction of iron pyridylporphyrins suggest that the sites of electron transfer lie at the ligand, the metal ion, and the pyridyl moieties. The electron transfer reactions between the different forms of these compounds, including the oxygen reduction reaction they mediate, are outlined to show the sequence and potential ranges in which they occur in alkaline aqueous media. Under our experimental conditions, the iron porphyrins exist as μ-oxo dimmers whose activity for the electrocatalytic reduction of oxygen displays a considerable dependence on the nature of the substitutents and nitrogen isomerization (for pyridylporphyrins) and grows in the order (Fe(ms-Ph)₄P)₂О, (Fe[ms-(Py-3)Ph₃]P)₂O, (Fe[ms-(Py-4)₄]P)₂O, and (Fe[ms-(Py-3)₄]P)₂O.     

Electrochemical behavior of MF-4SK/polyaniline composites as investigated by membrane voltammetry

Membrane voltammetry was used to investigate composites consisting of perfluorinated MF-4SK membranes and polyaniline (PANI) and synthesized under various conditions. A theoretical analysis of the influence of transport-structure parameters of the ion-exchange membranes on their selectivity and limiting-current value is carried out. For MF-4SK/PANI composites, the increase in the exchange potential at the overlimiting state is found to be more than 2V compared to the original membrane. An analysis of how various factors influence the parameters of a current-voltage curve shows that the presence of anilinium and Fe³⁺ ions in the electromembrane system has no effect on the value of the potential at the onset of the overlimiting state. A phenomenological model is proposed to account for the increased plateau length at the limiting current through a change in the energetic state of water in the perfluorinated-membrane matrix resulting from the template synthesis of polyaniline.     

Synthesis and study of macroheterocyclic compounds derived from 1<Emphasis Type="Italic">H</Emphasis>-phenalene-1,3(2<Emphasis Type="Italic">H</Emphasis>)-diimine

Symmetrical macroheterocyclic compounds are synthesized by the reaction of 1H-phenalene-1,3(2H)-diimine with m-phenylenediamine, 2,6-diaminopyridine, and 2,5-diamino-1,3,4-thiadiazole. IR and UV spectral data are given.Translated from Zhurnal Obshchei Khimii, Vol. 74, No. 8, 2004, pp. 1355–1359.Original Russian Text Copyright © 2004 by Berezina, Vorobev, Vorobeva, Andreikina.This revised version was published online in April 2005 with a corrected cover date.