Pit your wits against young minds!

This column is a place for those bits of contagious mathematics that travel from person to person in the community, because they are so elegant, suprising, or appealing that one has an urge to pass them on. Contributions are most welcome.     

Altered protein synthesis in rat kidney cells exposed to semiconductor materials

It is thought that the extensive industrial use of arsenic, gallium and indium, which have applications as the materials for III–V semiconductors, will increase human exposure to these compounds in the near future. We have undertaken the development of new biological indicators for assessing exposure to these elements. Element-specific alterations in protein synthesis patterns were expected to occur following exposure to arsenic compounds. We examined alterations in protein synthesis in primary cultures of rat kidney proximal tubule epithelial cells by sodium arsenite, gallium chloride and indium chloride, utilizing two-dimensional gel electrophoresis. After incubation with the chemicals for 20 h, newly synthesized proteins were labeled with [³⁵S]methionine. A protein with a molecular weight (M_r) of 30 000 was markedly induced on exposure to 10 μM arsenite or 300 μM gallium chloride, and synthesis of proteins with M_r values of 85 000, 71 000, 65 000, 51 000, 38 000 and 28 000 were also increased by exposure to arsenite and gallium chloride. No significant changes were observed upon exposure to indium. Some of these increased proteins could be heat-shock proteins.     

Atomistic phenomena in dense fluid shock waves

The shock structure problem is one of the classical problems of fluid mechanics and at least for non-reacting dilute gases it has been considered essentially solved. Here we present a few recent findings, to show that this is not the case. There are still new physical effects to be discovered provided that the numerical technique is general enough to not rule them out a priori. While the results have been obtained for dense fluids, some of the effects might also be observable for shocks in dilute gases.     

Approximation of transient temperatures in complex geometries using fractional derivatives

It is shown that time-dependent temperatures in a transient, conductive system can be approximately modeled by a fractional-order differential equation, the order of which depends on the Biot number. This approximation is particularly suitable for complex shapes for which a first-principles approach is too difficult or computationally time-consuming. Analytical solutions of these equations can be written in terms of the Mittag-Leffler function. The approximation is especially useful if a suitable fractional-order controller is to be designed for the system.     

Damage and fracture evaluation of granular composite materials by digital image correlation method

This paper presents the applications of digital image correlation technique to the mesoscopic damage and fracture study of some granular based composite materials including steelfiber reinforced concrete, sandstone and crystal-polymer composite. The deformation fields of the composite materials resulted from stress localization were obtained by the correlation computation of the surface images with loading steps and thus the related damage prediction and fracture parameters were evaluated. The correlation searching could be performed either directly based on the gray levels of the digital images or from the wavelet transform (WT) coefficients of the transform spectrum. The latter was developed by the authors and showed higher resolution and sensitivity to the singularity detection. Because the displacement components came from the rough surfaces of the composite materials without any coats of gratings or fringes of optical interferometry, both surface profiles and the deformation fields of the composites were visualized which was helpful to compare each other to analyze the damage of those heterogeneous materials. The project supported by the National Natural Science Foundation of China (10125211 and 10072002), the Scientific Committee of Yunnan Province for the Program of Steel Fiber Reinforced Concrete, and the Institute of Chemical Materials, CAEP at Mianyang