Free Groups from Fields

In the nineteenth century, field theory brilliantly resolveda number of questions that had taxed mathematicians for centuries;for example, ‘The circle cannot be squared’ by straightedge and compass, and solving polynomial equations by radicalsis not always possible. These successes have continued to beheld up as superb examples of the power of mathematical thought,and are demonstrated at an undergraduate level. The purposeof this article is to provide another such natural example whichleads to a concrete realisation of the free group on 2 generators.     

On the invariant measure for the quasi‐linear Lasota equation

The problem of the existence of the invariant measure is important considering its connections with chaotic behaviour. In the papers (Zesz. Nauk. Uniw. Jagiellońskiego, Pr. Mat. 1982; 23 :117–123; Ann. Pol. Math. 1983; XLI :129–137; J. Differential Equations 2004; 196 :448–465) the existence of invariant and ergodic measures according to the dynamical system generated by the Lasota equation was proved, i.e. the equation describing the dynamics and becoming different of the population of cells. In this paper, the existence of such measure for the quasi‐linear Lasota equation is proved. This measure is the carriage of the measure described by Dawidowicz (Zesz. Nauk. Uniw. Jagiellońskiego, Pr. Mat. 1982; 23 :117–123). Copyright © 2006 John Wiley & Sons, Ltd.     

Relevance of Reflection in Static and Dynamic Light Scattering Experiments

Using a monodisperse PMMA dispersion, it was shown that light reflection at the sample cuvette walls may greatly influence the results of both static (SLS) and dynamic (DLS) light scattering experiments. Considering SLS, this reflection phenomenon mostly causes an overestimation of the scattered intensity at high scattering angles, which may give rise to the emergence of an additional, artificial peak in the lower region of the particle size distribution. On the other hand, the influcence of reflection on DLS measurements was shown to be particularly important in the upper region of the particle size distribution. The experimentally observed phenomena were explained from the basic principles of both particle sizing methods. Finally, it was shown that the disturbing effect of reflection could be avoided by modifying either the hardware or the software of the static and dynamic light scattering technique.     

基于变芯径光纤的光学系统在光造型中的应用

光学系统采用变芯径光纤，可大幅度提高光的传输效率，增加光学子系统柔性。描述了光造型工作原理及其理论依据，在介绍所研制的光的成型系统光学子系统的基础上，阐述了紫外光源的选择与设计，及变芯径光纤的设计、传输、耦合与聚焦等技术。该套基于变芯径光纤的光学子系统成本低、体积小、效率高，巳成功应用于所研制的台式低成本快速成型系统中，为该系统的进一步推广奠定了技术基础。     

Characterization of heat-assisted magnetic probe recording on CoNi/Pt multilayers

A method of heat-assisted magnetic recording (HAMR) potentially suitable for probe-based storage systems is characterized. In this work, field emission current from a scanning tunneling microscope (STM) tip is used as the heating source. Pulse voltages of 2–7 V were applied to a CoNi/Pt multilayered film fabricated on either bare silicon or oxidized silicon substrates. Different types of Ir/Pt and W STM tips were used in the experiment. The results show that thermally recorded magnetic marks are formed with a nearly uniform mark size of 170 nm on the film fabricated on bare silicon substrate when the pulse voltage is above a threshold voltage. The mark size becomes 260 nm when they are written on the identical film fabricated on an oxidized silicon substrate. The threshold voltage depends on the material work function of the tip, with W having a threshold voltage about 1 V lower than Pt. A synthesized model, which contains the calculation of the emission current, the simulation of heat transfer during heating, and the study of magnetic domain formation, was introduced to explain experimental results. The simulation agrees well with the experiments.