Mathematics teaching development as a human practice: identifying and drawing the threads

The didactic triangle links mathematics, teachers and students in a consideration of teaching?Clearning interactions in mathematics classrooms. This paper focuses on teachers and teaching in the development of fruitful learning experiences for students with mathematics. It recognises primarily that teachers are humans with personal characteristics, subject to a range of influences through the communities of which they are a part, and considers aspects of teachers?? personhood, identity and agency in designing teaching for the benefit of their students. Teaching is seen as a developmental process in which inquiry plays a central role, both in doing mathematics in the classroom and in exploring teaching practice. The teacher-as-inquirer in collaboration with outsider researchers leads to growth of knowledge in teaching through development of identity and agency for both groups. The inclusion of the outsider researcher brings an additional node into the didactic triangle.     

Random walks on almost connected locally compact groups: Boundary and convergence

We prove that, given an arbitrary spread out probability measure μ on an almost connected locally compact second countable groupG, there exists a homogeneous spaceG/H, called the μ-boundary, such that the space of bounded μ-harmonic functions can be identified withL ^∞ (G/H). The μ-boundary is an amenable contractive homogeneous space. We also establish that the canonical projection onto the μ-boundary of the right random walk of law μ always converges in probability and, whenG is amenable, it converges almost surely. The μ-boundary can be characterised as the largest homogeneous space among those homogeneous spaces in which the canonical projection of the random walk converges in probability.     

Entropy of electrochemical formation ofp-semiquinones in dimethylformamide

Entropy of formation of anion radicals at the mercury electrode in dimethyl-formamide for 1,4-benzoquinone, 1,4-naphthoquinone, 9,10-anthraquinone, and 5,12-tetracenequinone has been determined from the temperature dependences of the reversible half-wave potentials using a nonisothermal cell. The linear relationship between the entropy and the calculated spin density at the oxygen atom of semiquinones was established. Similarly the linear correlation of the literature reaction entropy for electrooxidation of a few heteroaromatic compounds versus the experimental spin density at a heteroatom (N, S or Se) from ESR measurements was also observed.

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Die Entropie der elektrochemischen Bildung vonp-Semichinonen in Dimethylformamid

Zusammenfassung Die Bildungsentropie von Anionenradikalen an der Quecksilberelektrode wurde in Dimethylformamid für 1,4-Benzochinon, 1,4-Naphthochinon, 9,10-Anthrachinon und 5,12-Tetracenchinon aus der Temperaturabhängigkeit der reversiblen Halbwellenpotentiale unter Benutzung einer nichtisothermen Zelle bestimmt. Es wurde eine lineare Abhängigkeit zwischen der Entropie und der berechneten Spindichte am Sauerstoffatom der Semichinone festgestellt. Eine ähnliche lineare Korrelation konnte für die Literaturdaten von Reaktionsentropien der Elektrooxidation einiger heteroaromatischer Verbindungen gegenüber den experimentellen Spindichten am Heteroatom (N, S, Se; ESR-Messungen) beobachtet werden.

     

Activation enthalpy for ligand-bridged electroreduction of cobalt(III) complexes in mixtures of water with acetone

The activation enthalpy for the reduction of two pentaamminecobalt(III) complexes attached to a mercury electrode by thiophenecarboxylate ligands was evaluated from the temperature dependence of the unimolecular rate constants measured in a nonisothermal cell arrangement. The data obtained for the reactant with a bridge featuring uninterrupted conjugation support our previous opinion that the decrease of the rate constant on addition of acetone to the mixtures results from a decrease in the nuclear frequency factor owing to the overdamped solvent friction. The unimolecular frequency factors and the adiabaticity of the processes are discussed.     

Quasi-periodic solutions of the sine-Gordon equation

Two-phase quasi-periodic solutions of the sine-Gordon equation are discussed. It is shown that the solutions expressed by the multidimensional θ-function form a broader class than the well-known factorised solutions 4 arctan [f(u)g(v)]. Expressions describing the spatial and temporal period are also given.     

Temperature dependences of dual fluorescence as criterion for determining character of photoreactions

We performed comparative studies of the temperature quenching of dual fluorescence of acetonitrile solutions of several molecular probes with proton transfer reaction in an excited singlet state of 4′-(dieth-ylamino)-3-hydroxyflavone (FET), 1-methyl-2-(4-methoxy)phenyl-3-hydroxy-4(1H)-quinolone (QMOM), and 3-hydroxyflavone (3HF) parent molecule at different energies of excitation quanta. In accordance with expressions obtained from balance equations for photoreactions of the kinetic and thermodynamic character, the intensity ratios of fluorescence bands as functions of the degree of quenching behave differently. Namely, the quenching increases the relative intensity of bands normal form/tautomer for reactions of the kinetic type, retaining this ratio unchanged for reactions of the thermodynamic character. Our experimental studies showed that, for fluorescent probes with the kinetic reaction (3HF and QMOM), the intensity ratio fluorescence bands increases almost linearly with the degree of quenching, whereas, in the thermodynamic case (FET), this ratio is independent of this parameter. Conclusions about the character of reactions that we obtained in this work agree well with data of independent investigations of these molecules by laser spectroscopy with high time resolution, and the obtained relations allow us also to judge the mechanism of temperature quenching in the case of the reaction of the kinetic type. The method can be used for comparatively simple express selection of molecular probes, candidate for new applications.     

High performance Na-doped PbTe-PbS thermoelectric materials: electronic density of states modification and shape-controlled nanostructures

Thermoelectric heat-to-power generation is an attractive option for robust and environmentally friendly renewable energy production. Historically, the performance of thermoelectric materials has been limited by low efficiencies, related to the thermoelectric figure-of-merit ZT. Nanostructuring thermoelectric materials have shown to enhance ZT primarily via increasing phonon scattering, beneficially reducing lattice thermal conductivity. Conversely, density-of-states (DOS) engineering has also enhanced electronic transport properties. However, successfully joining the two approaches has proved elusive. Herein, we report a thermoelectric materials system whereby we can control both nanostructure formations to effectively reduce thermal conductivity, while concurrently modifying the electronic structure to significantly enhance thermoelectric power factor. We report that the thermoelectric system PbTe-PbS 12% doped with 2% Na produces shape-controlled cubic PbS nanostructures, which help reduce lattice thermal conductivity, while altering the solubility of PbS within the PbTe matrix beneficially modifies the DOS that allow for enhancements in thermoelectric power factor. These concomitant and synergistic effects result in a maximum ZT for 2% Na-doped PbTe-PbS 12% of 1.8 at 800 K.     

Multiband fluorescence spectral properties of QMOM

The spectral characteristics of the 1-methyl-2-(4-methoxyphenyl)-3-hydroxy-4(1H)-quinolone (QMOM) dye with dual fluorescence in acetonitrile were studied under selective excitation in a wide temperature range. This dye is a structural analog of 3-hydroxyflavone and exhibits excited-state proton transfer, which forms a fluorescent tautomeric form, while the solution is characterized by dual fluorescence. The thermal behavior of the relative band intensities revealed the kinetic character of the proton transfer. The third form showed itself as a maximum between the bands of the normal and tautomeric forms upon excitation in several regions of the absorption spectrum and became dominant in solution at 60–80°C. The characteristics of the third form were studied. Additional experiments showed that this was possibly the anionic form of the dye.