B-splines and optimal stability

It is proved that, among all nonnegative bases of its space, the B-spline basis is optimally stable for evaluating spline functions.

     

面板半参数空间自回归模型的变量选择——基于STIRPAT模型的碳排放影响因素分析

变量选择有助于简化模型,提高估计和预测的精度,但目前鲜有涉及面板半参数空间自回归模型变量选择的研究。本文在ALASSO的基础上提出了SSAR-ALASSO法,该法的核心在于惩罚函数的选择和目标函数的构建。SSAR-ALASSO在变量和参数的对应关系、惩罚函数的选择、特殊参数的取值区间以及适用模型等方面与ALASSO存在差异。模拟结果显示,SSAR-ALASSO法在变量选择的准确性和参数估计的精度两方面均表现良好,随着样本容量的增加表现效果更佳。本文在碳排放量影响因素实证中采用SSAR-ALASSO法对STIRPAT模型进行变量选择。研究结果表明人均财富、技术水平、产业结构、所有制结构和产业集聚显著影响碳排放量,城市化、对外开放、能源价格和环境政策对碳排放量无显著影响。     

A Generalized Smoother for Linear Ordinary Differential Equations

Ordinary differential equations (ODEs) are equalities involving a function and its derivatives that define the evolution of the function over a prespecified domain. The applications of ODEs range from simulation and prediction to control and diagnosis in diverse fields such as engineering, physics, medicine, and finance. Parameter estimation is often required to calibrate these theoretical models to data. While there are many methods for estimating ODE parameters from partially observed data, they are invariably subject to several problems including high computational cost, complex estimation procedures, biased estimates, and large sampling variance. We propose a method that overcomes these issues and produces estimates of the ODE parameters that have less bias, a smaller sampling variance, and a 10-fold improvement in computational efficiency. The package GenPen containing the Matlab code to perform the methods described in this article is available online.     

Dielectric relaxation of α-cyclodextrin—polyiodide complexes (α-cyclodextrin)2 · LiI3 · I2 · 8H2O and (α-cyclodextrin)2 · Cd0.5 · I5 · 26H2O

The frequency and temperature dependence of real (?′) and imaginary (?″) parts of the dielectric constant of polycrystalline complexes (α-CD)₂ · LiI₃ · I₂ · 8H₂O and (α-CD)₂ · Cd_0.5 · I₅ · 26H₂O (α-CD = α-cyclodextrin) has been investigated over the frequency and temperature ranges of 0–100 kHz and 12–300 K. The dielectric behaviour is described well by Debye type relaxation (α-dispersion). Both systems exhibit an additional Ω dispersion at low frequencies which is attributed to ionic conductance and is much greater in the case of Li due to the greater mobility of cations Li⁺. The temperature dependence of ?′ reveals the existence of two kinds of water molecule in the case of the (α-CD)₂ · Cd_0.5 · I₅ · 26H₂O complex; these can be classified as tiqhtly bound and easily movable water molecules that cause two steps in ?′ versus T plots. In the case of the (α-CD)₂ · LiI₃ · I₂ · 8H₂O complex the water molecules are tightly bound and as a result only one step is observed in these graphs. These finding are also confirmed from the ?″_max versus T plots, which exhibit the same number of steps with ?′, and from calorimetric measurements. The order-disorder transition or the transformation of normal hydrogen bonds to flip-flop type has been observed as a peak in ?″ versus T plots that is more intense and narrow in the case of Li and less high but more broad in Cd. The relaxation time vanes in a α-like curve (from 120 K to 240 K) and rises rapidly for temperatures greater than 240 K, indicating the existence of a new process involving the breaking of hydrogen bonds (normal or flip-flop type). The calculated values of activation energy (0.35–0.62 k_BT_trans) reveal the greater stability of the Li compared with the Cd complex. The starting value of 8.2–8.4 μs for τ is the same as observed in β-CD complexes with guest 4-t-butylbenzyl alcohol (TERB). However, the activation energies of these are greater (1.1–1.7k_BT_trans), indicating greater stability for β-CD complexes.     

Parametric study of the class-shape-refinement-transformation method

The application of a novel parametrization technique to the optimization of aircraft shapes is presented. This class-shape-refinement transformation (CSRT) technique combines an analytical function (class function), a set of Bernstein polynomials (shape function) and a B-spline (refinement function) and can be used to model various aircraft components. It allows for both global and local control of a shape and forms a very efficient and intuitive way of mathematically describing aircraft parts. A parametric study was performed that shows the behaviour of the shape as a function of a number of different parameters, such as total number of shape variables and Bernstein/B-spline coefficient ratio. The CSRT method was used to approximate a typical aircraft wing cross-section and the results showed a very non-linear relationship between the number of shape variables and the error of the approximation, expressed in terms of a correlation factor. This behaviour has been thoroughly analysed. Additionally, optimization results were obtained that show that the CSRT method was successfully coupled to an aerodynamic flow solver. The objective of the optimization runs was to maximize lift-to-drag ratio, but in principle any objective function could be used as long as its input follows from the aerodynamic analysis. The optimization algorithm is capable of largely removing the shock wave on an airfoil at a typical cruise Mach number.     

A shape-preserving approximation by weighted cubic splines

This paper addresses new algorithms for constructing weighted cubic splines that are very effective in interpolation and approximation of sharply changing data. Such spline interpolations are a useful and efficient tool in computer-aided design when control of tension on intervals connecting interpolation points is needed. The error bounds for interpolating weighted splines are obtained. A method for automatic selection of the weights is presented that permits preservation of the monotonicity and convexity of the data. The weighted B-spline basis is also well suited for generation of freeform curves, in the same way as the usual B-splines. By using recurrence relations we derive weighted B-splines and give a three-point local approximation formula that is exact for first-degree polynomials. The resulting curves satisfy the convex hull property, they are piecewise cubics, and the curves can be locally controlled with interval tension in a computationally efficient manner.     

Bivariate box splines and smooth pp functions on a three direction mesh

Let S denote the space of bivariate piecewise polynomial functions of degree ? k and smoothness ρ on the regular mesh generated by the three directions (1, 0), (0, 1), (1, 1). We construct a basis for S in terms of box splines and truncated powers. This allows us to determine the polynomials which are locally contained in S and to give upper and lower bounds for the degree of approximation. For $\text{ρ = ?}\text{(2k ? 2)}\text{3}\text{?}$, k ? 2 (3), the case of minimal degree k for given smoothness ρ, we identify the elements of minimal support in S and give a basis for $\text{S}{}_{\mathrm{<mtext>loc</mtext>}}\text{= {f ∈ S:}\text{supp}\text{f ? Ω}}$, with Ω a convex subset of $\text{R}$².     

一类双k次B样条曲面的G1连续性条件

本文针对两个k×k次B样条曲面的节点向量为端点插值、内部是单节点的情形 ,给出它们之间的G1光滑拼接条件 ,同时得到它们的公共边界曲线的控制顶点所要满足的本征方程 .其中本征方程是B样条曲面片所独有的现象 .     

Product integration of logarithmic singular integrands based on cubic splines

This paper is concerned with the practical evaluation of the product integral ∫¹_{? 1}f(x)k(x)dx for the case when k(x) = In|x - λ|, λ? (?1, +1) and f is bounded in [?1, +1]. The approximation is a quadrature rule

where the weights {w_n,n,i} are chosen to be exact when f is given by a linear combination of a chosen set of functions {φ_n,j}. In this paper the functions {φ_n,j} are chosen to be cubic B-splines. An error bound for product quadrature rules based on cubic splines is provided. Examples that test the performance of the product quadrature rules for different choices of the function are given. A comparison is made with product quadrature rules based on first kind Chebyshev polynomials.     

由基样条构造有限基集计算的超精细结构常数(英文)

多体微扰论有效算符方法应用于超精细结构的计算 .由HF波函数计算零阶超精细常数 .使用基样条构造了薛定谔方程的有限基集 .使用这些有限基集计算了原子实极化和关联 ,以及 7Li,2 3 Na ,3 9K和4 3 Ca离子的s1/ 2 ,p1/ 2 和p3 / 2 态的超精细结构常数和4 3 Ca离子的d3 / 2和d5/ 2 态的超精细常数 . The effective-operator form of many-body theory is applied to the calculation of hyperfine structure. The zeroth order hyperfine constants are evaluated with Hartree-Fock wavefunction. Τhe finite basis sets of Schrdinger s equation are constructed by using B-splines. With the finite basis sets, we have calculated the core polarization, and the correlation diagrams. The hyperfine constants of the s 1/2, p 1/2and p 3/2 states of 7Li, 23Na, 39K, 43Ca + as well as the d 3/2...