On Vertices, focal curvatures and differential geometry of space curves

The focal curve of an immersed smooth curve γ : θ ↦ γ (θ), in Euclidean space ℝ^m+1, consists of the centres of its osculating hyperspheres. This curve may be parametrised in terms of the Frenet frame of γ (t, n₁, . . . , n_m), as C_γ (θ) = (γ +c₁n₁+ c₂n₂ + • • • + c_mn_m)(θ), where the coefficients c₁, . . . , c_m-1 are smooth functions that we call the focal curvatures of γ . We discovered a remarkable formula relating the Euclidean curvatures κ_i , i = 1, . . . ,m, of γ with its focal curvatures. We show that the focal curvatures satisfy a system of Frenet equations (not vectorial, but scalar!). We use the properties of the focal curvatures in order to give, for ℓ = 1, . . . ,m, necessary and sufficient conditions for the radius of the osculating ℓ-dimensional sphere to be critical. We also give necessary and sufficient conditions for a point of γ to be a vertex. Finally, we show explicitly the relations of the Frenet frame and the Euclidean curvatures of γ with the Frenet frame and the Euclidean curvatures of its focal curve C_γ.     

逻辑斯蒂曲线的几个推广模型与应用

本给出逻辑斯蒂曲线的三个推广模型，即广义逻辑斯蒂曲线模型、含季节性变化的逻辑斯蒂曲线模型、合成逻辑斯蒂曲线模型及其模型参数估计的方法，井对我国汽车生产量作出了很好的预测。     

邻苯二钾酸氢铊晶体积分衍射效率的标定

 以北京同步辐射实验室4B7中能束线为光源，在2.1~6.0 keV的范围内对邻苯二钾酸氢铊(TlAP)平面晶体一、二 、三级衍射的积分衍射效率进行了精确的实验标定。标定结果表明：TlAP晶体有较高的峰值衍射率； 其一级衍射的积分衍射效率向低能端有增加的趋势，而在2.6~5.4 keV的范围内大约为1.53×10^-4 rad；随着衍射级次的提高，衍射效率逐渐减小，二级衍射的积分衍射效率约为一级衍射效率的1/4，三级衍射结果比一级衍射下降一个量级。实验所标定的TlAP晶体可用于平晶谱仪对激光等离子体X射线光谱的定量分析。     

圆锥体上截取二次曲线的误差分析

建立了采用轨迹成形法加工二次非球面光学零件的数学模型,给出了相关变量的误差公式,并以某炸弹的导引头上的非球面透镜作为实例,分析了相关变量对轨迹形状误差的影响。     

A theoretical study of the modal dispersion characteristics of an annular circular waveguide with helical winding as the outer cladding

In this present communication, we introduce a new type of annular optical waveguide whose outer cladding is made of sheath helix. The sheath helix is made up of the dielectric material of lower refractive index than the core materials. Using vector approach, the general characteristic equation for the proposed waveguide has been derived. The modal dispersion characteristics for the lowest-order modes for different pitch angles are determined and analyzed. We observe that there is no effect of pitch angles on dispersion curve nearly up to 89° but drastic change is observed at 90° and above pitch angle. On comparison of helically inner cladded annular circular waveguide, we have found that there is not the existence of the negative dispersion curve and photonic band gap in the helically outer cladded annular circular waveguide. It means only the inner cladding is responsible for the negative dispersion as well as photonic band gap. There is no effect of the outer cladding.     

单摆的角振幅与其周期的关系

当角振幅的变化范围为0到π时,讨论了单摆的角振幅与其周期的关系,从而对单摆的运动有了一个完整的认识;同时得到了一些重要的结论。     

Observation of a diverse deviation from macropore-formation theory in silicon electrochemistry

Via anodizing patterned and unpatterned samples with a high HF concentration （[HF]）, the degree of deviation from pore-formation theory was found to be markedly different. Based on the analysis of scanning electron microscope （SEM） micrographs and current-voltage （I - V） curves, the variation of physical and chemical parameters of patterned and unpatterned substrates was found to be crucial to the understanding of the observations. Our results indicate that the initial surface morphology of samples can have a considerable influence upon pore formation. The electric-field effect as well as current-burst-model was employed to interpret the underlying mechanism.     

Mathematical analysis of affinity membrane chromatography

A mathematical model including convection, diffusion and Freundlich adsorption is developed. To examine the validity of the model, the affinity membranes were prepared by coating chitosan on the nylon membranes, a ligand of poly-L-lysine was bound to the chitoan-coating membranes, and the adsorption behavior of bilirubin through the stacked affinity membranes was investigated. The agreements between the theoretical and experimental results are exceptional. Using our new model, we show that: (1) As Pe increases, the breakthrough curves become sharper. For Pe greater than 30, the effect of axial diffusion is insignificant; (2) As m increases, the time of total saturation is delayed and the loading capacity at the point of breakthrough is increased; (3) As n decreases, the time of total saturation is delayed and the loading capacity at the point of breakthrough is increased; (4) As r increases, both the time of total saturation and the loading capacity at the point of breakthrough are increased; (5) adsorption rate influences the time of total saturation strongly but contributes little to the loading capacity.     

Adsorption isotherm for a solid electrode: The link between the differential surface tension and capacitance curves

Investigation of thermodynamically equilibrium single-component adsorption from a liquid solution on a solid electrode with allowance made for elastic deformation of its surface is continued. A full electrocapillarity equation is derived from thermodynamics equations for an interphase layer in the absence of irreversible processes. Thermodynamic aspects of the Shuttleworth equation are discussed and the equation is compared with two-dimensional Murnaghan formulas for elastic isotropic media. An adsorption isotherm equation and compatibility equations that had been derived previously are examined in a special case where the derivative of a surface concentration with respect to depends solely on (=()) and a rigorous solution of these is obtained for a deformed electrode ( 0). The effect of and dimensionless electrode potential on the extreme (at an infinitely high adsorbate concentration) value of is studied. The model of two parallel capacitors is considered in detail for a general case. Owing to the use of capacitance curves for an elastically stretched electrode, a formula that expresses the differential surface tension of a nondeformed electrode through such curves is derived for the first time ever.To the memory of my motherTranslated from Elektrokhimiya, Vol. 41, No. 1, 2005, pp. 20–34.Original Russian Text Copyright © 2005 by Podgaetskii.     

Luminescence of Higher Plants upon Excitation by Laser Radiation with Different Energy and Time Parameters

The luminescence response of the leaves of higher plants exposed to laser radiation with different energy and time parameters has been investigated at a wavelength of 685 nm. The plants were irradiated by laser radiation at 1060, 632, and 530 nm. Radiation at = 632 nm was used as the control. It is shown that radiation at = 1060 nm suppresses photosynthesis, whereas that at = 530 nm stimulates it.