关于Weierstrass函数图像K—维数的简单证明

Ｗｅｉｅｒｓｔｒａｓｓ函数图像的豪斯道夫维数是一个未解决的数学难题,为了解决此难题,给出了Ｗｅｉｅｒｓｔｒａｓｓ函数图像的Ｋ－维数的简洁证明,从而进一步揭示出盒维数,Ｋ－维数和豪斯夫维数三者之间的关系。     

Correlation-crystal-field ‘δ-function' analysis of Pr(4f) energy-level structure

Inclusion of two-electron correlation-crystal-field (CCF) contributions to the model (energy-level) Hamiltonian has been shown to resolve disparities between calculated and observed energy-level splittings within problematic multiplet manifolds of Nd³⁺ and Er³⁺ systems. Application of these CCF terms to other problematic multiplets, such as the ¹G₄ and ¹D₂ multiplets of Pr³⁺ has not been feasible, owing to the fact that the number of applicable CCF terms far exceed the number of observed levels in these multiplets. However, this problem may be obviated by use of a simplified ‘δ-function' CCF model that assumes the dominant contributions to the CCF interactions come from paired electrons within the same orbital. In the present study, we examine the 4f²(Pr³⁺) electronic energy-level structures reported for Pr³⁺:LaCl₃, Pr³⁺:GdCl₃, Pr³⁺:Cs₂NaYCl₆ and Cs₂NaPrCl₆ systems and show that inclusion of δ-function CCF interaction parameters resolves the disparities between observed and calculated energy-level splittings for the problematic ¹G₄ and ¹D₂ multiplets of these systems.     

A Note on the Spectral Collocation Approximation of Some Differential Equations with Singular Source Terms

The solution of differential equations with singular source terms contains the local jump discontinuity in general and its spectral approximation is oscillatory due to the Gibbs phenomenon. To minimize the Gibbs oscillations near the local jump discontinuity and improve convergence, the regularization of the approximation is needed. In this note, a simple derivative of the discrete Heaviside function H _c (x) on the collocation points is used for the approximation of singular source terms δ(x−c) or δ ⁽ⁿ⁾(x−c) without any regularization. The direct projection of H _c (x) yields highly oscillatory approximations of δ(x−c) and δ ⁽ⁿ⁾(x−c). In this note, however, it is shown that the direct projection approach can yield a non-oscillatory approximation of the solution and the error can also decay uniformly for certain types of differential equations. For some differential equations, spectral accuracy is also recovered. This method is limited to certain types of equations but can be applied when the given equation has some nice properties. Numerical examples for elliptic and hyperbolic equations are provided. The current address: Department of Mathematics, State University of New York at Buffalo, Buffalo, NY 14260-2900, USA.     

Espaces métriques rationnellement présentés et complexité, le cas de l'espace des fonctions réelles uniformément continues sur un intervalle compact

We define the notion of rational presentation of a complete metric space, in order to study metric spaces from the algorithmic complexity point of view. In this setting, we study some representations of the space C[0,1] of uniformly continuous real functions over [0,1] with the usual norm: ||f||_∞ = Sup{|f(x)|; 0x1}. This allows us to have a comparison of global kind between complexity notions attached to these presentations. In particular, we get a generalization of Hoover's results concerning the Weierstrass approximation theorem in polynomial time. We get also a generalization of previous results on analytic functions which are computable in polynomial time.     

一类Weierstrass函数图像的盒维数

函数图像的分形维数是分形几何研究中的一个非常重要并且在当前十分活跃的课题。本文研究了一类形如∑∞k=1aktk（bkx＋θk）的Weierstrass函数,证明了它的连续而不可微性,并给出了其图像的盒维数。     

Catalytic Performance of B2O3/TiO2–ZrO2 for Vapor-Phase Beckmann Rearrangement of Cyclohexanone Oxime: The Effect of Boria Loading

A series of boria catalysts supported on titania–zirconia mixed oxide (B₂O₃/TiO₂–ZrO₂) with different boria loadings (8–20 wt%) were prepared and characterized by X-ray diffraction, adsorption of nitrogen, ¹¹B magic angle spinning (MAS) NMR measurements and temperature-programmed desorption (TPD) of ammonia. The catalytic performance of B₂O₃/TiO₂–ZrO₂ for vapor-phase Beckmann rearrangement of cyclohexanone oxime to -caprolactam was studied at 300°C. It was found that the lactam selectivity increased with increasing of boria loading, whereas a maximum oxime conversion was obtained at the boria loading of 12 wt%. The acid sites of medium strength on the surface of the catalyst play an important role in the selective formation of lactam.     

压电陶瓷执行器的动态模型辨识与控制

为了提高精密定位系统中压电陶瓷的控制精度,研究了压电执行器的动态模型及逆模型。根据Weierstrass第一逼近定理,提出了以多项式函数逼近Duhem模型中的分段连续函数f(·)和g(·),并应用递推最小二乘算法辨识Duhem模型的参数α及f(·)和g(·)的多项式系数,建立了压电陶瓷执行器的非线性参数化动态模型。利用辨识结果建立压电陶瓷执行器的动态逆模型,避免对压电陶瓷执行器进行复杂的模型求逆;介绍了通过逆补偿和PID复合控制对压电陶瓷系统进行的控制。实验结果表明:仅通过逆补偿,可在0~200μm使得控制绝对误差小于0.8μm;在前馈逆补偿和PID环控制下,绝对误差可小于40nm,结果验证了算法的有效性。该算法结构简单,适应性强,便于工程实现。