Computing monodromy via continuation methods on random Riemann surfaces

We consider a Riemann surface X defined by a polynomial f(x,y) of degree d, whose coefficients are chosen randomly. Hence, we can suppose that X is smooth, that the discriminant δ(x) of f has d(d−1) simple roots, Δ, and that δ(0)≠0, i.e. the corresponding fiber has d distinct points {y₁,…,y_d}. When we lift a loop 0∈γ⊂C−Δ by a continuation method, we get d paths in X connecting {y₁,…,y_d}, hence defining a permutation of that set. This is called monodromy.Here we present experimentations in Maple to get statistics on the distribution of transpositions corresponding to loops around each point of Δ. Multiplying families of “neighbor” transpositions, we construct permutations and the subgroups of the symmetric group they generate. This allows us to establish and study experimentally two conjectures on the distribution of these transpositions and on transitivity of the generated subgroups.Assuming that these two conjectures are true, we develop tools allowing fast probabilistic algorithms for absolute multivariate polynomial factorization, under the hypothesis that the factors behave like random polynomials whose coefficients follow uniform distributions.     

新疆开都河小山口二级水电站吊车梁设计

水电站厂房的吊车主要用于吊装水轮机与发电机。承受吊车荷载的吊车梁是厂房上部承重的重要结构之一,必须严格按照要求进行设计,以保证吊车梁的质量和使用效果。采用理正结构求解器对新疆开都河小山口二级水电站吊车粱进行了设计和结构计算分析,得出吊车梁的结构与配筋满足设计要求。此吊车梁设计的成功为今后类似工程提供了很好的参考。     

Weak Schur numbers and the search for G.W. Walker’s lost partitions

A set A of integers is weakly sum-free if it contains no three distinct elements x,y,z such that x+y=z. Given k≥1, let WS(k) denote the largest integer n for which {1,…,n} admits a partition into k weakly sum-free subsets. In 1952, G.W. Walker claimed the value WS(5)=196, without proof. Here we show WS(5)≥196, by constructing a partition of {1,…,196} of the required type. It remains as an open problem to prove the equality. With an analogous construction for k=6, we obtain WS(6)≥572. Our approach involves translating the construction problem into a Boolean satisfiability problem, which can then be handled by a SAT solver.     

Fourth-order accurate BLAGE iterative method for the solution of two-dimensional elliptic equations in polar co-ordinates

In this article, we report on the block alternating group explicit (BLAGE) iterative method for solving the block symmetric linear system derived from the fourth-order accurate nine-point discretisation of a two-dimensional elliptic equation in cylindrical polar co-ordinates. The error analysis of the BLAGE method is discussed briefly. The performance of this more accurate BLAGE method is compared with the corresponding block successive over relaxation (BSOR) method by considering two test problems, wherein the significance of the role played by two parameters ω₁ and ω₂ of the BLAGE method becomes evident in providing both convergence and accuracy of the computed solution.     

The generalized Riemann problem method for the shallow water equations with bottom topography

This paper extends the generalized Riemann problem method (GRP) to the system of shallow water equations with bottom topography. The main contribution is that the generalized Riemann problem method (J. Comput. Phys. 1984; 55 (1):1–32) is used to evaluate the midpoint values of solutions at each cell interface so that the bottom topography effect is included in numerical fluxes, and at the same step the source term is discretized with an interface method in which only mid‐point values are plugged in. This scheme is well balanced between the flux gradient and bottom topography when incorporating the surface gradient method (SGM) (J. Comput. Phys. 2001; 168 (1):1–25) into data reconstruction step, and it is also suitable for both steady and unsteady flow simulations. We illustrate the accuracy of this scheme by several 1‐D and 2‐D numerical experiments. Copyright © 2005 John Wiley & Sons, Ltd.     

关于倒上阶乘函数的一个展开式

导出了一个用第二类Stirling数表示的倒上阶乘函数fn(x)=1/x(x 1)…(x n)关于1/x的幂级数异型了fn(x)=∑v=n^∞(-1)^n vS2(v,n)1/x^v 1，其对Riemann Zeta函数的研究有重要意义。     

R-M不稳定性数值模拟方法

通常,界面追踪方法都是由界面将计算区域分割为多个部分来分别处理,同时用求解黎曼问题来追踪全部波系的发展。然而在多维情况下,波系间相互作用的黎曼问题将非常复杂且目前尚处于分析阶段,故在实际应用中只能追踪界面本身。提出了一种新的界面追踪方法,该方法只追踪界面的发展而不再追踪波系,使计算量和复杂性大大减少并保持了原界面追踪方法精确捕捉界面的特点,数值结果令人满意。     

Detecting infeasibility and generating cuts for mixed integer programming using constraint programming

We study a hybrid MIP/CP solution approach in which CP is used for detecting infeasibilities and generating cuts within a branch-and-cut algorithm for MIP. Our framework applies to MIP problems augmented by monotone constraints that can be handled by CP. We illustrate our approach on a generic multiple machine scheduling problem, and present a number of computational experiments.     

基于变量决策层的启发式变量选择策略

启发式分支策略是SAT求解器中不可或缺的一部分，直接影响求解器的效率。早期的启发式分支决策需要遍历整个子句数据库，效率比较低。随着独立变量状态衰减和（Variable State Independent Decaying Sum, VSIDS）分支策略的出现，SAT求解器的效率有所提高，但VSIDS策略以及它的延伸策略中变量的增量都只是与变量的冲突次数有关，没有考虑变量的决策层在分支策略中的影响。因此当发生冲突时，如果与冲突有关的变量的得分相同而决策层不同时，对于变量的选择就具有随机性。基于此，本文在阐述变量的决策层的重要性之后在VSIDS策略的基础上，提出一种基于变量决策层的启发式变量选择策略--HSVDL策略。然后通过实例显示HSVDL策略在变量决策阶段选择决策层低的变量的可能性比选择决策层高的变量的可能性要大，而且得分比较小，减少了内存的占用。最后通过实验表明HSVDL策略能够求解出更多的实例，求解器的效率也有所提高，说明该策略有一定的优势。     

CMRH method as iterative solver for boundary element acoustic systems

Disceretization of boundary integral equations leads to complex and fully populated linear systems. One inherent drawback of the boundary element method (BEM) is that, the dense linear system has to be constructed and solved for each frequency. For large-scale problems, BEM can be more efficient by improving the construction and solution phases of the linear system. For these problems, the application of common direct solver is inefficient. In this paper, the corresponding linear systems are solved more efficiently than common direct solvers by using the iterative technique called CMRH (Changing Minimal Residual method based on Hessenberg process). In this method, the generation of the basis vectors of the Krylov subspace is based on the Hessenberg process instead of Arnoldi's one that the most known GMRES (Generalized Minimal RESidual) solver uses. Compared to GMRES, the storage requirements are considerably reduced in CMRH.