Numerical Techniques for the Solution of the Molecular Weight Distribution in Polymerization Mechanisms,State of the Art

The molecular weight distribution (MWD) is possibly the most important characteristic of a polymer. Polymers derive many of their physical properties from their MWD. Therefore, since the origins of polymer science, the theory provides a link between the kinetic mechanism and the mathematical expression of the MWD, and there are analytical solutions for ideal cases. However, the MWD formed in real‐life polymerization processes is usually more complex; the solution of the mathematical models that describe them can be quite challenging and has been the focus of enormous research efforts. These models may consist of systems of very large dimension: thousands of differential equations, often stiff, which demand special numerical techniques for their solution. In this paper the numerical techniques that can be used to solve this challenging problem are reviewed and contrasted, including weighted residual methods, direct integration, numerical inversion of transformed equations, and lumping methods. Stochastic techniques are also surveyed.     

Convergence of minimum norm elements of projections and intersections of nested affine spaces in Hilbert space

We consider a Hilbert space, an orthogonal projection onto a closed subspace and a sequence of downwardly directed affine spaces. We give sufficient conditions for the projection of the intersection of the affine spaces into the closed subspace to be equal to the intersection of their projections. Under a closure assumption, one such (necessary and) sufficient condition is that summation and intersection commute between the orthogonal complement of the closed subspace, and the subspaces corresponding to the affine spaces. Another sufficient condition is that the cosines of the angles between the orthogonal complement of the closed subspace, and the subspaces corresponding to the affine spaces, be bounded away from one. Our results are then applied to a general infinite horizon, positive semi-definite, linear quadratic mathematical programming problem. Specifically, under suitable conditions, we show that optimal solutions exist and, modulo those feasible solutions with zero objective value, they are limits of optimal solutions to finite-dimensional truncations of the original problem.     

条件数学期望与随机变量独立性的一个充要条件

随机试验的独立性、随机事件的独立性、随机变量的独立性均是概率统计中的重要概念,不少学者都在这些方面有所讨论.本文作者就二维离散形随机向量(ξ,η)中两个分量ξ与η的相互独立性展开讨论.先是证明了三个引理,其中引理1在一般概率论教科书中均有介绍,但为使读者方便,作者也作了证明.利用三个引理,作者找到随机变量独立性的一个充要条件.     

B矩阵对策

提出了一类局中人都设定有得失控制值的二人零和有限对策,即B矩阵对策.引入了稳妥策略、弱稳妥策略等概念.给出了B矩阵对策的数学模型、有关理论和求解方法.最后,还给了一个计算例子.     

计算主梁绝对最大挠度的数学模型与0.618法

纵横梁桥面系统中主梁的跨度较长,纵梁上直接承受的车队荷载数量多且数值大,主梁设计中绝对最大挠度的确定是关键内容.研究了一组平行车队荷载直接沿着纵梁移动时,主梁承受结点活载下绝对最大挠度数学模型的建立;并给出了相应的计算方法,以计算机为工具,适用于任意有限多个平行移动荷载作用工况,对于主梁的设计计算与安全评估,有一定的实用价值.     

Nonlinear One-Dimensional Seismodynamic Model of a Solid with Shock Absorbing Supports

A generalized method for constructing a nonlinear one-dimensional mathematical model of a solid on massive shock-absorbing supports of five different types is justified. The periods of small oscillations of the solid about the equilibrium position are determined. The equations of motion of the solid-support system with broken frictional bond are set up__________Translated from Prikladnaya Mekhanika, Vol. 41, No. 3, pp. 135–144, March 2005.     

Maximum-entropy technique with logarithmic constraints: Estimation of atomic radial densities

Maximum-entropy (ME) approximations to density functions involving logarithmic constraints are studied. It is proved the existence and uniqueness of the ME approximation constrained by the normalization, the geometric mean and (i) a moment of arbitrary order, or (ii) the logarithmic uncertainty. A numerical analysis of the accuracy of these ME approximations is carried out for the radial electron densities of neutral atoms in both position and momentum spaces. Received 23 November 1998 and Received in final form 20 May 1999     

合理利率的初探

本文应用 Cobb-Doulas生产函数 ,建立了一类描述国民经济收入和银行存款利率的数学模型 ,并利用该模型 ,通过适当简化 ,求出了维持国民经济健康稳定快速发展的最优利率 .     

Numerical simulation of the consequences of a mechanical action on a human brain under a skull injury

A two-dimensional mathematical model of the mechanical response of a human head to a shock action is proposed. It describes the spatial distribution of the mechanical loads on the brain. Some numerical results obtained using the grid characteristic methods on unstructured triangular grids are presented.