Hierarchical-matrix method for a class of diffusion-dominated partial integro-differential equations

A hierarchical matrix approach for solving diffusion-dominated partial integro-differential problems is presented. The corresponding diffusion-dominated differential operator is discretized by a second-order accurate finite-volume scheme, while the Fredholm integral term is approximated by the trapezoidal rule. The hierarchical matrix approach is used to approximate the resulting algebraic problem and includes the implementation of an efficient preconditioned generalized minimum residue (GMRes) solver. This approach extends previous work on integral forms of boundary element methods by taking into account inherent characteristics of the diffusion-dominated differential operator in the resultant algebraic problem. Numerical analysis estimates of the accuracy and stability of the finite-volume and the trapezoidal rule approximation are presented and combined with estimates of the hierarchical-matrix approximation and with the accuracy of the GMRes iterates. Results of numerical experiments are reported that successfully validate the theoretical accuracy and convergence estimates, and demonstrate the almost optimal computational complexity of the proposed solution procedure.     

Smartphone‐Based Android app for Determining UVA Aerosol Optical Depth and Direct Solar Irradiances

This research describes the development and evaluation of the accuracy and precision of an Android app specifically designed, written and installed on a smartphone for detecting and quantifying incident solar UVA radiation and subsequently, aerosol optical depth at 340 and 380 nm. Earlier studies demonstrated that a smartphone image sensor can detect UVA radiation and the responsivity can be calibrated to measured direct solar irradiance. This current research provides the data collection, calibration, processing, calculations and display all on a smartphone. A very strong coefficient of determination of 0.98 was achieved when the digital response was recalibrated and compared to the Microtops sun photometer direct UVA irradiance observations. The mean percentage discrepancy for derived direct solar irradiance was only 4% and 6% for observations at 380 and 340 nm, respectively, lessening with decreasing solar zenith angle. An 8% mean percent difference discrepancy was observed when comparing aerosol optical depth, also decreasing as solar zenith angle decreases. The results indicate that a specifically designed Android app linking and using a smartphone image sensor, calendar and clock, with additional external narrow bandpass and neutral density filters can be used as a field sensor to evaluate both direct solar UVA irradiance and low aerosol optical depths for areas with low aerosol loads.     

Evaluating UVA Aerosol Optical Depth using a Smartphone Camera

This research evaluates a smartphone complementary metal oxide semiconductor (CMOS) image sensor's ability to detect and quantify incident solar UVA radiation and subsequently, aerosol optical depth at 340 and 380 nm. Earlier studies revealed that the consumer grade CMOS sensor has inherent UVA sensitivities, despite attenuating effects of the lens. Narrow bandpass and neutral density filters were used to protect the image sensor and to not allow saturation of the solar images produced. Observations were made on clear days, free from clouds. The results of this research demonstrate that there is a definable response to changing solar irradiance and aerosol optical depth can be measured within 5% and 10% error margins at 380 and 340 nm respectively. The greater relative error occurs at lower wavelengths (340 nm) due to increased atmospheric scattering effects, particularly at higher air masses and due to lower signal to noise ratio in the image sensor. The relative error for solar irradiance was under 1% for observations made at 380 nm. The results indicate that the smartphone image sensor, with additional external narrow bandpass and neutral density filters can be used as a field sensor to evaluate solar UVA irradiance and aerosol optical depth.     

Combined finite element and spectral approximation of the Navier-Stokes equations

Summary We present a method for the numerical approximation of Navier-Stokes equations with one direction of periodicity. In this direction a Fourier pseudospectral method is used, in the two others a standard F.E.M. is applied. We prove optimal rate of convergence where the two parameters of discretization intervene independently.

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Approximation des équations de Navier-Stokes par une méthode éléments finis-spectrale Fourier

Resumé On présente une méthode d'approximation numérique des équations de Navier-Stokes possédant une direction de périodicité. Dans cette direction une méthode pseudospectrale basée sur des développements en série de Fourier est utilisée, dans les deux autres on applique une méthode d'éléments finis standard. On montre que la convergence est optimale et que les deux paramètres de discrétisation peuvent être choisis de façon indépendante.

     

Representable Lexicographic Products

A linear ordering is said to be representable if it can be order-embedded into the reals. Representable linear orderings have been characterized as those which are separable in the order topology and have at most countably many jumps. We use this characterization to study the representability of a lexicographic product of linear orderings. First we count the jumps in a lexicographic product in terms of the number of jumps in its factors. Then we relate the separability of a lexicographic product to properties of its factors, and derive a classification of representable lexicographic products. This revised version was published online in September 2006 with corrections to the Cover Date.     

Stability and accuracy of a pseudospectral scheme for the Wigner function equation

A pseudospectral scheme with centered time‐differencing for solving the Wigner function (WF) equation is investigated. Stability, second‐order accuracy in time, and spectral accuracy in space are proved for the WF equation with a potential in a periodic setting. In addition, normalization and energy conservation properties, and Ehrenfest's theorem are discussed. Numerical experiments are presented to validate the theoretical results. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 62–87, 2017     

Solar ultraviolet protection provided by human head hair

The solar erythemal UV irradiances through human hair and the protection from UV provided by human head hair have been investigated for a solar zenith angle (SZA) range of 17–51° for the conditions of a head upright in full sun, a head upright in shade and a head in full sun tilted toward the sun. The two hair lengths investigated were 49.1 ± 7.1 mm for the short type and 109.5 ± 5.5 mm for the long type. For the head upright in full sun, the irradiances through the hair ranged from 0.75 to 1.4 SED h⁻¹ for SZA <25° and <0.6 SED h⁻¹ in shade. The ultraviolet protection factor (UPF) ranged from approximately 5 to 17 in full sun, with the UPF increasing with higher SZA. The longer hair provided a lower UPF than the shorter hair and for the head oriented toward the sun, there was a marginally lower UPF than for the upright head. This research shows that the UV exposure limits to the scalp through hair can be exceeded within short timeframes and provides important information to assist employers to comply with Workplace Health and Safety legislation.