The proof of Kontsevich's periodicity conjecture on noncommutative birational transformations

For an arbitrary associative unital ring RR, let J1J1 and J2J2 be the following noncommutative, birational, partly defined involutions on the set M3(R)M3(R) of 3×33×3 matrices over RR: J1(M)=M−1J1(M)=M−1 (the usual matrix inverse) and J2(M)jk=(Mkj)−1J2(M)jk=(Mkj)−1 (the transpose of the Hadamard inverse).

We prove the surprising conjecture by Kontsevich that (J2∘J1)3(J2∘J1)3 is the identity map modulo the DiagL×DiagRDiagL×DiagR action (D1,D2)(M)=D−11MD2(D1,D2)(M)=D1−1MD2 of pairs of invertible diagonal matrices. That is, we show that, for each MM in the domain where (J2∘J1)3(J2∘J1)3 is defined, there are invertible diagonal 3×33×3 matrices D1=D1(M)D1=D1(M) and D2=D2(M)D2=D2(M) such that (J2∘J1)3(M)=D−11MD2(J2∘J1)3(M)=D1−1MD2.

Examination of the flow rheological and textural properties of polymer gels composed of poly(methylvinylether-co-maleic anhydride) and poly(vinylpyrrolidone): rheological and mathematical interpretation of textural parameters.

The purpose of this study was to mathematically characterize the effects of defined experimental parameters (probe speed and the ratio of the probe diameter to the diameter of sample container) on the textural/mechanical properties of model gel systems. In addition, this study examined the applicability of dimensional analysis for the rheological interpretation of textural data in terms of shear stress and rate of shear. Aqueous gels (pH 7) were prepared containing 15% w/w poly(methylvinylether-co-maleic anhydride) and poly(vinylpyrrolidone) (PVP) (0, 3, 6, or 9% w/w). Texture profile analysis (TPA) was performed using a Stable Micro Systems texture analyzer (model TA-XT 2; Surrey, UK) in which an analytical probe was twice compressed into each formulation to a defined depth (15 mm) and at defined rates (1, 3, 5, 8, and 10 mm s-1), allowing a delay period (15 s) between the end of the first and beginning of the second compressions. Flow rheograms were performed using a Carri-Med CSL2-100 rheometer (TA Instruments, Surrey, UK) with parallel plate geometry under controlled shearing stresses at 20.0°?±?0.1°C. All formulations exhibited pseudoplastic flow with no thixotropy. Increasing concentrations of PVP significantly increased formulation hardness, compressibility, adhesiveness, and consistency. Increased hardness, compressibility, and consistency were ascribed to enhanced polymeric entanglements, thereby increasing the resistance to deformation. Increasing probe speed increased formulation hardness in a linear manner, because of the effects of probe speed on probe displacement and surface area. The relationship between formulation hardness and probe displacement was linear and was dependent on probe speed. Furthermore, the proportionality constant (gel strength) increased as a function of PVP concentration. The relationship between formulation hardness and diameter ratio was biphasic and was statistically defined by two linear relationships relating to diameter ratios from 0 to 0.4 and from 0.4 to 0.563. The dramatically increased hardness, associated with diameter ratios in excess of 0.4, was accredited to boundary effects, that is, the effect of the container wall on product flow. Using dimensional analysis, the hardness and probe displacement in TPA were mathematically transformed into corresponding rheological parameters, namely shearing stress and rate of shear, thereby allowing the application of the power law (??=?k?n) to textural data. Importantly, the consistencies (k) of the formulations, calculated using transformed textural data, were statistically similar to those obtained using flow rheometry. In conclusion, this study has, firstly, characterized the relationships between textural data and two key instrumental parameters in TPA and, secondly, described a method by which rheological information may be derived using this technique. This will enable a greater application of TPA for the rheological characterization of pharmaceutical gels and, in addition, will enable efficient interpretation of textural data under different experimental parameters.

Experimental and Modelling Studies of the Thermodynamics and Kinetics of Phase and Structural Transformations in a Gamma TiAl-based Alloy

This work combines microscopy, synchrotron radiation X-ray diffraction, differential scanning calorimetry and thermodynamic calculations in the characterisation of phase transformation behaviour of a Ti–46Al–1.9Cr–3Nb alloy upon continuous heating at constant rates. It has been found that the Ti–46Al–1.9Cr–3Nb alloy after being forged at 1200 °C without further treatment has a duplex microstructure consisting of fine equiaxed and lamellar ? grains with a small amount of a2 plates and particles and about 1 wt.% B2 phase. Differential scanning calorimetry revealed reproducibly several thermal effects upon heating of the as-forged alloy. These thermal effects are related to the equilibration and homogenisation of the sample, change of phase ratios between a2, ? and B2 phases in particular the increase of B2 in respect to a2 and ?, and the following five phase transformations: a2 + ? + B2 a + ? + B2, a + ? + B2 a + ?, ? + a a, a a + ß, a + ß a + ß + L. The observation of these transformations by differential scanning calorimetry is largely in agreement with literature phase diagrams and thermodynamic calculations, though care is needed to consider the different alloy compositions. Kinetics of the ? + a a phase transformation in the Ti–46Al–1.9Cr–3Nb alloy has been quantitatively derived from the calorimetry data, giving phase compositions at any point during the transformation upon continuous heating.

Mathematical prerequisites for learning statistics in Psychology: Assessing core skills of numeracy and mathematical reasoning among undergraduates.

This study sought to extend earlier work by Mulhern and Wylie (2004) to investigate a UK-wide sample of psychology undergraduates. A total of 890 participants from eight universities across the UK were tested on six broadly defined components of mathematical thinking relevant to the teaching of statistics in psychology - calculation, algebraic reasoning, graphical interpretation, proportionality and ratio, probability and sampling, and estimation. Results were consistent with Mulhern and Wylie's (2004) previously reported findings. Overall, participants across institutions exhibited marked deficiencies in many aspects of mathematical thinking. Results also revealed significant gender differences on calculation, proportionality and ratio, and estimation. Level of qualification in mathematics was found to predict overall performance. Analysis of the nature and content of errors revealed consistent patterns of misconceptions in core mathematical knowledge , likely to hamper the learning of statistics.

Mineralogical characteristics and transformations during long-term operation of a zero-valent iron reactive barrier

Received for publication October 31, 2002. Design and operation of Fe0 permeable reactive barriers (PRBs) can be improved by understanding the long-term mineralogical transformations that occur within PRBs. Changes in mineral precipitates, cementation, and corrosion of Fe0 filings within an in situ pilot-scale PRB were examined after the first 30 months of operation and compared with results of a previous study of the PRB conducted 15 months earlier using X-ray diffraction and scanning electron microscopy employing energy dispersive X-ray and backscatter electron analyses. Iron (oxy)hydroxides, aragonite, and maghemite and/or magnetite occurred throughout the cores collected 30 mo after installation. Goethite, lepidocrocite, mackinawite, aragonite, calcite, and siderite were associated with oxidized and cemented areas, while green rusts were detected in more reduced zones. Basic differences from our last detailed investigation include (i) mackinawite crystallized from amorphous FeS, (ii) aragonite transformed into calcite, (iii) akaganeite transformed to goethite and lepidocrocite, (iv) iron (oxy)hydroxides and calcium and iron carbonate minerals increased, (v) cementation was greater in the more recent study, and (vi) oxidation, corrosion, and disintegration of Fe0 filings were greater, especially in cemented areas, in the more recent study. If the degree of corrosion and cementation that was observed from 15 to 30 mo after installation continues, certain portions of the PRB (i.e., up-gradient entrance of the ground water to the Fe0 section of the PRB) may last less than five more years, thus reducing the effectiveness of the PRB to mitigate contaminants. Abbreviations: EDX, energy dispersive X-ray • Fe0, zerovalent iron • PRB, permeable reactive barrier • SEM, scanning electron microscopy • XRD, X-ray diffraction

Synchrotron X-ray diffraction study of the phase transformations in titanium alloys

High-resolution synchrotron X-ray diffraction was used to study the phase transformations in titanium alloys. Three titanium alloys were investigated: Ti-6Al-4V, Ti-6Al-2Sn-4Zr-2Mo-0.08Si and beta21s. Both room and high temperature measurements were performed. The room temperature experiments were performed to study the structure of the alloys after different heat treatments, namely as received (AR), furnace cooling (FC), water quenching (WQ) and water quenching followed by ageing. The alpha, alpha', alpha'' and beta phases were observed in different combinations depending on the heat treatment conditions and the alloy studied. A multicomponent hexagonal close packed (hcp) alpha phase, with different c and the same a lattice parameters, was detected in Ti-6Al-4V after FC. High temperature synchrotron X-ray diffraction was used for 'in situ' study of the transformations on the sample surface at elevated temperatures. The results were used to trace the kinetics of surface oxidation and the concurrent phase transformations taking place under different conditions. The influence of the temperature and oxygen content on the lattice parameters of the alpha phase was derived and new data obtained on the coefficients of thermal expansion in the different directions of the hcp alpha phase, for Ti-6Al-4V and Ti-6Al-2Sn-4Zr-2Mo-0.08Si.