The rate of convergence for the cyclic projections algorithm III: Regularity of convex sets

The cyclic projections algorithm is an important method for determining a point in the intersection of a finite number of closed convex sets in a Hilbert space. That is, for determining a solution to the “convex feasibility” problem. This is the third paper in a series on a study of the rate of convergence for the cyclic projections algorithm. In the first of these papers, we showed that the rate could be described in terms of the “angles” between the convex sets involved. In the second, we showed that these angles often had a more tractable formulation in terms of the “norm” of the product of the (nonlinear) metric projections onto related convex sets.In this paper, we show that the rate of convergence of the cyclic projections algorithm is also intimately related to the “linear regularity property” of Bauschke and Borwein, the “normal property” of Jameson (as well as Bakan, Deutsch, and Li’s generalization of Jameson’s normal property), the “strong conical hull intersection property” of Deutsch, Li, and Ward, and the rate of convergence of iterated parallel projections. Such properties have already been shown to be important in various other contexts as well.     

Global smoothness and uniform convergence of smooth Gauss–Weierstrass singular operators

In this article we continue with the study of smooth Gauss–Weierstrass singular integral operators over the real line regarding their simultaneous global smoothness preservation property with respect to the L_p norm, 1≤p≤∞, by involving higher order moduli of smoothness. Also we study their simultaneous approximation to the unit operator with rates involving the modulus of continuity with respect to the uniform norm. The produced Jackson type inequalities are almost sharp containing elegant constants, and they reflect the high order of differentiability of the engaged function.     

A note on wavelet density deconvolution for weakly dependent data

In this paper we investigate the performance of a linear wavelet-type deconvolution estimator for weakly dependent data. We show that the rates of convergence which are optimal in the case of i.i.d. data are also (almost) attained for strongly mixing observations, provided the mixing coefficients decay fast enough. The results are applied to a discretely observed continuous-time stochastic volatility model.     

基于最新势能面的C(1D)+H2反应的环聚合分子动力学研究

本文利用环聚合分子动力学方法对C(¹D)+H₂反应开展了详细的理论研究. 计算中使用了最近构建的Zhang-Ma-Bian(ZMB)从头算势能面,该势能面对锥形交叉附近区域以及范德华区域均有精确的描述. 环聚合分子动力学计算得到的热反应速率常数与最新实验值吻合很好. 与前人计算结果比较,发现在?¹A′电子基态的ZMB-a势能面上获得的反应速率常数远大于前人构建的RKHS势能面上的结果,这是由于ZMB势能面上的范德华鞍具有与之前势能面上的范德华阱完全不同的动态学作用,表明环聚合分子动力学方法能够处理范德华作用引起的势能面拓扑结构所导致的动态学效应. 本文还揭示了b¹A′′电子激发态ZMB-b势能面以及量子效应对反应的重要性.     

Rate enhancement by water in a TiCl4-mediated stereoselective vinylogous Mukaiyama aldol reaction

A remarkable enhancement in the rate of a TiCl₄-mediated stereoselective vinylogous Mukaiyama aldol reaction (VMAR) using vinylketene silyl N,O-acetal 1 was observed in the presence of water.     

激发态Cs2和H2的电子-振转能级的碰撞转移

利用相干反斯托克斯拉曼光谱(CARS)探测技术, 研究了激发态Cs2与H2间的电子-振转能级的碰撞转移。用波长为532 nm和中心波长为716 nm的两束激光同时聚焦到样品池中, 扫描CARS谱确认了H2分子的S支(△v =1, △J=2)仅在v=1, J=4,5及v=2, J=3,4能级上有布居, 用n1、n2、n3、n4分别表示(2,4)、(2,3)、(1,4)及(1,5)上的粒子数密度。从CARS线的峰值得到n2/n1、n3/n1、n4/n1分别为6.34±1.27、3.66±0.73和1.45±0.29。转移能配置到振动、转动和平动的比例分别为0.44、0.06和0.50, 能量主要分配在振动和平动上。在T=523 K和PH2=2.5×103 Pa条件下, 通过求解速率方程组和对时间分辨CARS线轮廓的分析, 得到碰撞转移速率系数k1=(6.0±1.2)×10-14 cm-3s-1和k2=(4.0±0.8)×10-13cm-3 s-1。     

双分子水和氨气催化CF3OH分子裂解的理论研究

利用G3和CBS-QB3的理论方法研究CF₃OH分子裂解成FCFO和HF,并考虑大气中双分子和氨气对CF₃OH分子裂解的催化作用. 理论计算表明：由于在G3的理论水平下,计算的能垒为188.52 kJ/mol,所以CF₃OH分子在大气条件下不可能发生单分子裂解;当氨气和双分子水被加入时,能垒都被降到25.08 kJ/mol,起了强的催化作用. 除此之外,应用过渡态理论对速率常数进行了计算,计算结果表明：氨气催化CF₃OH分子的速率常数是单分子和双分子催化CF₃OH分子裂解速率常数的10⁹和10⁵倍. 考虑到大气中这些物质的浓度,计算结果预测了氨气催化CF₃OH分子裂解在大气中起到重要的作用.     

A fast dual proximal gradient algorithm for convex minimization and applications

We consider the convex composite problem of minimizing the sum of a strongly convex function and a general extended valued convex function. We present a dual-based proximal gradient scheme for solving this problem. We show that although the rate of convergence of the dual objective function sequence converges to the optimal value with the rate O(1/k²)$O(1/k^2)$, the rate of convergence of the primal sequence is of the order O(1/k)$O(1/k)$.     

Extending the applicability of Secant methods and nondiscrete induction

We use nondiscrete mathematical induction to extend the applicability of the Secant methods for solving equations in a Banach setting. Our approach has the following advantages over earlier works under the same information: weaker sufficient convergence conditions; tighter error bounds on the distances involved and more information on the location of the solution. Numerical examples where our results apply but earlier ones fail to solve nonlinear equation as well as tighter error bounds are also provided in this study.     

Approximation by Chlodowsky-Taylor polynomials

The subject of this paper is to study the problem of the approximation of function of two variables by means of Chlodowsky-Taylor polynomials in a rectangular domain.