Visualisation and proof: a brief survey of philosophical perspectives

The contribution of visualisation to mathematics and to mathematics education raises a number of questions of an epistemological nature. This paper is a brief survey of the ways in which visualisation is discussed in the literature on the philosophy of mathematics. The survey is not exhaustive, but pays special attention to the ways in which visualisation is thought to be useful to some aspects of mathematical proof, in particular the ones connected with explanation and justification.     

Trace Equivalence in SU(2, 1)

In this paper it is shown that one can choose an arbitrarily large number of inconjugate elements of the group Z/2Z*Z/2Z*Z/2Z which have the property that, under all representations of the group in SU(2,1) as a discrete complex hyperbolic ideal triangle group, the elements are hyperbolic and correspond to closed geodesics of equal length on the associated complex hyperbolic surface. This is an analogue of the geometric fact that the multiplicity of the length spectrum of a Riemann surface is never bounded or the equivalent algebraic phenomenon that an arbitrarily large number of conjugacy classes in a free group can have the same trace under all representations in SL(2,R ).     

Molecular and crystal structure of the intracomplex 2-[(dibutylboryloxy)(Butyl) borylamino]-4-methylpyridine with boron atoms of differing hybridization in the chelate ring

X-ray diffraction has been used to establish the crystalline and molecular structure of the intracomplex 2-[(dibutylboryloxy)(butyl)borylamino]4-methylpyridine. In the crystal, the molecule forms weak intermolecular N(H)...O hydrogen bonds. The bicyclic portion of the molecule is nearly planar. From the bond lengths in the ring, which incorporates boron atoms of differing hybridization (sp³ and sp²), it follows that delocalization of electron density extends only over the amidine fragment. Quantum chemical calculations (MO LCAO, version CNINDO/2) show that the electron density at the endocyclic nitrogen of the aminopyridine system coordinated with boron is lower than that at the endocyclic nitrogen of the aminopyridine.Translated from Izvestiya Akademii Nauk SSSR, Seriya Khimicheskaya, No. 1, pp. 173–177, January, 1991.     

The oxidative decarboxylation of polyaminocarboxylic acids

Summary The rates of oxidation of four chelating agents, NTA, EDTA, CDTA, and DTPA with Ce(IV), in sulfuric acid media, were determined spectrophotometrically by a stopped-flow technique. The reductive ability is in the order CDTA > EDTA > DTPA > NTA. The influence of varying the acidity of the medium was studied, and in each case a maximum in the rate constant vs. [H⁺] plot was observed. A possible interpretation of the reactivities and the influence of acidity is advanced.

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Oxydative Decarboxylierung von PolyaminocarbonsäurenII. Vergleichende kinetische Untersuchung der Oxydation von NTA, ÄDTA, CDTA und DTPA mit Ce(IV) in saurer Lösung

Zusammenfassung Die Oxydationsgeschwindigkeiten von 4 Chelaten (NTA, ÄDTA, CDTA und DTPA) mit Ce(IV) in saurer Lösung wurden spektrophotometrisch mit Hilfe der stopped-flow-Technik bestimmt. Die Reduzierfähigkeit nimmt in der Reihenfolge CDTA > ÄDTA > DTPA > NTA ab. Der Einfluß verschiedener Säuregehalte in der Lösung wurde untersucht, und in jedem Fall wurde ein Maximum in der graphischen Darstellung der Geschwindigkeitskonstante gegen [H⁺] beobachtet. Eine mögliche Erklärung des Reaktionsvermögens und des Säureeinflusses wird gegeben.

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Part I: Z. Anal. Chem. 246, 231 (1969).     

DSC,FT-IR and NIR with Chemometric Assessment Using PCA and HCA for Estimation of the Chemical Stability of Oral Antidiabetic Drug Linagliptin in the Presence of Pharmaceutical Excipients

Pharmaceutical excipients should not interact with active substances, however, in practice, they sometimes do it, affecting the efficacy, stability and safety of drugs. Thus, interactions between active substances and excipients are not desirable. For this reason, two component mixtures of oral antidiabetic drug linagliptin (LINA) with four excipients of different reactivity, i.e., lactose (LAC), mannitol (MAN), magnesium stearate (MGS) and polyvinylpyrrolidone (PVP), were prepared in a solid state. A high temperature and a high humidity of 60 °C and 70% RH, respectively, were applied as stressors in order to accelerate the potential interactions between LINA and excipients. Differential scanning calorimetry (DSC) as well as Fourier transform infrared (FT-IR) and near infrared (NIR) spectroscopy were used to estimate the changes due to potential interactions. In addition, chemometric computation of the data with principal component analysis (PCA) and hierarchical cluster analysis (HCA) was applied to adequately interpret the findings. Of the excipients used in the present experiment, all of them were not inert in relation to LINA. Some of the interactions were shown without any stressing, whereas others were observed under high-temperature/high-humidity conditions. Thus, it could be concluded that selection of appropriate excipients for LINA is very important question to minimize its degradation, especially when new types of formulations with LINA are being developed and manufactured.     

Charge transport in anodic TiO2 nanotubes studied by terahertz spectroscopy

We report on the photoelectrochemical and terahertz measurements, of the charge transport properties of 1 μm thick self‐organized TiO₂ nanotube layers, prepared by the anodization of titanium. We provide evidence regarding the complexity of electron transport, and dynamics in the nanotubes. Shortly after photoexcitation, charge mobilites in amorphous and crystalline nanotubes are similar, but still lower compared to the bulk anatase. The mobility subsequently decreases due to trapping‐detrapping processes. The recombination rate in anatase nanotubes is much slower than in the amorphous ones, enabling the material to reach an internal photon to electron conversion efficiency exceeding 60%.