The Mathematicians’ Happy Hunting Ground: Einstein’s General Theory of Relativity

There is hardly any doubt that for physics special relativity theory is of much greater consequence than the general theory. The reverse situation prevails with respect to mathematics: there special relativity theory had comparatively little, general relativity theory very considerable, influence, above all upon the development of a general scheme for differential geometry. —Hermann Weyl, “Relativity as a Stimulus to Mathematical Research,” pp. 536–537.     

Mathematical models as artefacts for research: Felix Klein and the case of Kummer surfaces

Already as a student in Bonn, Felix Klein was exposed to model-making through his teacher, Julius Plücker, who used models to visualize the properties of special surfaces that arise in line geometry. Afterward, Klein attended Kummer’s seminar in Berlin, where he began to explore the connections between general Kummer surfaces and so-called complex surfaces, first studied by Plücker. In the late 1870s, Klein and Alexander Brill supervised the work of several students at the Munich Technische Hochschule who designed a large collection of models there. There followed a new era in model production, based in Munich but marketed through the Darmstadt firm owned by Brill’s brother. By 1900 the plaster models of L. Brill could be found at leading universities around the world. Our story here centres on Kummer’s famous models, which began as research artefacts in Berlin, before passing to Munich and then proliferating to points beyond.     

Structure-based modelling in reproductive toxicology: (Q)SARs for the placental barrier†

The replacement of animal testing for endpoints such as reproductive toxicity is a long-term goal. This study describes the possibilities of using simple (quantitative) structure-activity relationships ((Q)SARs) to predict whether a molecule may cross the placental membrane. The concept is straightforward, if a molecule is not able to cross the placental barrier, then it will not be a reproductive toxicant. Such a model could be placed at the start of any integrated testing strategy. To develop these models the literature was reviewed to obtain data relating to the transfer of molecules across the placenta. A reasonable number of data were obtained and are suitable for the modelling of the ability of a molecule to cross the placenta. Clearance or transfer indices data were sought due to their ability to eliminate inter-placental variation by standardising drug clearance to the reference compound antipyrine. Modelling of the permeability data indicates that (Q)SARs with reasonable statistical fit can be developed for the ability of molecules to cross the placental barrier membrane. Analysis of the models indicates that molecular size, hydrophobicity and hydrogen-bonding ability are molecular properties that may govern the ability of a molecule to cross the placental barrier.     

The phase shift index for marking functional asynchrony in Alzheimer's disease patients using fMRI

Our previous study suggested that the functional magnetic resonance imaging MRI (fMRI) COSLOF Index (CI) could be used as a quantitative biomarker for Alzheimer's disease (AD). The fMRI CI was lowest in the AD group (0.13+/-0.10), followed by the mild cognitive impairment (MCI) group (0.20+/-0.05) and the control group (0.34+/-0.09). The current study continues an investigation into which of the following two factors has a dominant role in determining the CI: the signal-to-noise ratio (SNR) or the phase shift of spontaneous low-frequency (SLF) components. By using a theoretical model for SLF components, we demonstrated that the normalized CI does not depend on the SNR of the SLF components. Further analysis shows that by taking the ratio of the cross-correlation coefficient to the maximum-shifted cross-correlation coefficient, the SNR factor can be canceled. Therefore, the determination of the phase shift index (PSI) method is independent of the SNR, and the PSI provides an accurate measure of the phase shift between SLF components. By applying this PSI method to the control, MCI and AD groups of subjects, experimental results demonstrated that the PSI was highest in the AD group (72.6+/-11.3 degrees ), followed by the MCI group (58.6+/-5.7 degrees ) and, finally, the control group (40.6+/-8.4 degrees ). These results suggest that the larger is the PSI value, the more asynchrony exists between SLF components.     

Fast implementation of sparse iterative covariance-based estimation for source localization

Fast implementations of the sparse iterative covariance-based estimation (SPICE) algorithm are presented for source localization with a uniform linear array (ULA). SPICE is a robust, user parameter-free, high-resolution, iterative, and globally convergent estimation algorithm for array processing. SPICE offers superior resolution and lower sidelobe levels for source localization compared to the conventional delay-and-sum beamforming method; however, a traditional SPICE implementation has a higher computational complexity (which is exacerbated in higher dimensional data). It is shown that the computational complexity of the SPICE algorithm can be mitigated by exploiting the Toeplitz structure of the array output covariance matrix using Gohberg-Semencul factorization. The SPICE algorithm is also extended to the acoustic vector-sensor ULA scenario with a specific nonuniform white noise assumption, and the fast implementation is developed based on the block Toeplitz properties of the array output covariance matrix. Finally, numerical simulations illustrate the computational gains of the proposed methods.     

High-temperature emission spectroscopy of methane

A high-enthalpy source (HES) has been developed in Rennes either to heat gases up to 2000 K in local thermodynamic equilibrium (LTE) or to generate hypersonic expansions. The HES prototype has been associated with a high-resolution Bruker IFS 120 HR Fourier transform spectrometer to record emission spectra of hot gases, in LTE conditions. A series of emission spectra of methane has been obtained at 1005, 1365, 1485, 1625 and 1820 K in the pentad spectral region located around 3000 cm⁻¹, at Doppler-limited resolution (0.02 cm⁻¹). Spectra have been corrected for the transmission function that strongly affects the infrared radiation emitted by the hot gas. Line-integrated absorption cross sections have been extracted from the corrected spectra using an improved procedure for the calculation of the total partition function Q of methane at high temperature. This calculation included anharmonicity and rovibrational interaction effects, and was based on a multi-resolution fully converged direct partition sum. It has been shown that, as the temperature increases above 1000 K, the commonly used harmonic and rigid rotor double approximation to estimate Q leads to underestimated values.