追赶航班

正弗莱德和家人打算出去度假,可他们要乘坐的飞机已经开始登机了。你能帮助他们迅速穿越停车场,到达A出口吗?然后帮助他们穿过机场,找到相对应的航班吗?     

乐天警署

贝卡他们看到街上的情形。立刻冲了出去。有人打电话报了警,伤者被送进了医院,贝卡他们和街边一些行人作为目击证人被带到了警署去录口供。     

"愉快教育"对数学教育影响的分析和思考

2002年以来,作者参与了中小学教师的培训工作,培训对象是省级小学骨干教师和中学教师,他们是数学教育第一线的优秀者.其间,在与他们接触的过程中可以感受到他们的精神,听到、看到他们的品格和业绩.同时也发现了数学教育中的一些新情况,如“愉快教育”.本文就此谈一点自己的看法     

被欺负的鲨鱼

贝卡和乐天蚁告别海藻们,在辽阔的大海里继续探索。这是个生机无限的奇妙世界,每一种生物都让他们惊奇。他们按照海藻的指示。去寻找鱼类竞赛的地点·忽然,前方出现了一只巨大的……那是——鲨鱼!贝卡和乐天蚁大惊失色。慌忙准备逃走,却见鲨鱼对他们的到来无动于衷,而且,他们分明听到……鲨鱼在烦闷地叹息？!     

被挖走的果苗

在Lucky他们的启发下,亚美家和大伯裘终于“雨过天晴”,化解矛盾了。瞧他们其乐融融的样子,这“一人两蚁”的功劳还真不小!     

思考和应用相结合 巧开几何学习之门

怎样才能帮助学生学习好初二的几何呢?笔者认为,可以从帮助他们积累一定的方法,来提高他们解决问题的能力,帮助他们形成清晰的解题思路等方面入手.笔者结合自己的教学实践,做了一些尝试和探索.一、指导学生进行与数学学科知识的相关积累     

伪造的遗书

小试身手 杰森和助手里克匆匆赶到一家医院,给他们打电话的青年约翰迎接了他们。 “是这样的：我父亲昨天出了车祸住进了医院,我听到消息后立即从外地赶回来,但还是晚了一步,父亲在我到达前三小时去世了。     

由一道试题的解法谈一类函数题的命制

一道高质量有创新的数学试题,一定承载着命题者很多的思考和构想,这也许是他们对某些问题长久的思考和探究而得来,也许是他们思维的灵光一现而得来,也许是他们以课本题、模拟题、高考题为蓝本重新考量而得来.笔者最近碰到一题,对该题的解法作了一番探究,以此来揣摩命     

对破坏者的审判

通过“初级探险者测试”后,贝卡和乐天蚁在热带雨林中度过了一段快乐的时光。尽管凶险无数。雨林的美还是让两只蚂蚁无比惊叹。他们学会了射箭、扎鱼。也了解了土著的很多令人讶异的习俗。这天,他们听到了一个坏消息——有些唯利是圈的人正在疯狂地破坏热带雨林。雨林里的居民们发誓要严惩作恶者,因为失去森林他们就会失去一切——家、食物和生活用品。贝卡和乐天蚁看着眼前荒凉的树桩,心里也充满愤怒。     

节约飞船费的方法

正1.桃太郎有三个儿子,分别叫做桃一郎、桃二郎、桃三郎。2.当年桃太郎勇敢地救下公主后,得到了许多酬金。三兄弟长大了,桃太郎把这些钱分给了他们,现在他们想用这些钱到嘎啦星球上去旅行。3.他们的目的地分别是:离嘎啦飞船站最近的嘎啦火山、第二远的嘎啦摩天轮和最远的嘎啦玫瑰园。     

Multi-index Stochastic Collocation Convergence Rates for Random PDEs with Parametric Regularity

We analyze the recent Multi-index Stochastic Collocation (MISC) method for computing statistics of the solution of a partial differential equation (PDE) with random data, where the random coefficient is parametrized by means of a countable sequence of terms in a suitable expansion. MISC is a combination technique based on mixed differences of spatial approximations and quadratures over the space of random data, and naturally, the error analysis uses the joint regularity of the solution with respect to both the variables in the physical domain and parametric variables. In MISC, the number of problem solutions performed at each discretization level is not determined by balancing the spatial and stochastic components of the error, but rather by suitably extending the knapsack-problem approach employed in the construction of the quasi-optimal sparse-grids and Multi-index Monte Carlo methods, i.e., we use a greedy optimization procedure to select the most effective mixed differences to include in the MISC estimator. We apply our theoretical estimates to a linear elliptic PDE in which the log-diffusion coefficient is modeled as a random field, with a covariance similar to a Matérn model, whose realizations have spatial regularity determined by a scalar parameter. We conduct a complexity analysis based on a summability argument showing algebraic rates of convergence with respect to the overall computational work. The rate of convergence depends on the smoothness parameter, the physical dimensionality and the efficiency of the linear solver. Numerical experiments show the effectiveness of MISC in this infinite dimensional setting compared with the Multi-index Monte Carlo method and compare the convergence rate against the rates predicted in our theoretical analysis.     

Boundary Value Problems for Third-Order Nonlinear Ordinary Differential Equations

In this paper, we describe how to analyze boundary value problems for third-order nonlinear ordinary differential equations over an infinite interval. Several physical problems of interest are governed by such systems. The seminumerical schemes described here offer some advantages over solutions obtained by using traditional methods such as finite differences, shooting method, etc. These techniques also reveal the analytic structure of the solution function. For illustrative purposes, several physical problems, mainly drawn from fluid mechanics, are considered; they clearly demonstrate the efficiency of the techniques presented here.     

A memetic algorithm for the virtual network mapping problem

The Internet has ossified. It has lost its capability to adapt as requirements change. A promising technique to solve this problem is the introduction of network virtualization. Instead of directly using a single physical network, working just well enough for a limited range of applications, multiple virtual networks are embedded on demand into the physical network, each of them perfectly adapted to a specific application class. The challenge lies in mapping the different virtual networks with all the resources they require into the available physical network, which is the core of the virtual network mapping problem. In this work, we introduce a memetic algorithm that significantly outperforms the previously best algorithms for this problem. We also offer an analysis of the influence of different problem representations and in particular the implementation of a uniform crossover for the grouping genetic algorithm that may also be interesting outside of the virtual network mapping domain. Furthermore, we study the influence of different hybridization techniques and the behaviour of the developed algorithm in an online setting.     

Symbolic Differential Elimination for Symmetry Analysis

Differential problems are ubiquitous in mathematical modeling of physical and scientific problems. Algebraic analysis of differential systems can help in determining qualitative and quantitative properties of solutions of such systems. In this tutorial paper we describe several algebraic methods for investigating differential systems.     

Numerical approximation for a variable-order nonlinear reaction–subdiffusion equation

Fractional reaction–subdiffusion equations are widely used in recent years to simulate physical phenomena. In this paper, we consider a variable-order nonlinear reaction–subdiffusion equation. A numerical approximation method is proposed to solve the equation. Its convergence and stability are analyzed by Fourier analysis. By means of the technique for improving temporal accuracy, we also propose an improved numerical approximation. Finally, the effectiveness of the theoretical results is demonstrated by numerical examples.     

On steady state Euler-Poisson models for semiconductors

This paper is concerned with an analysis of the Euler-Poisson model for unipolar semiconductor devices in the steady state isentropic case. In the two-dimensional case we prove the existence of smooth solutions under a smallness assumption on the prescribed outflow velocity (small boundary current) and, additionally, under a smallness assumption on the gradient of the velocity relaxation time. The latter assumption allows a control of the vorticity of the flow and the former guarantees subsonic flow. The main ingredient of the proof is a regularization of the equation for the vorticity.Also, in the irrotational two- and three-dimensional cases we show that the smallness assumption on the outflow velocity can be replaced by a smallness assumption on the (physical) parameter multiplying the drift-term in the velocity equation. Moreover, we show that solutions of the Euler-Poisson system converge to a solution of the drift-diffusion model as this parameter tends to zero.     

New Exact Solutions of the Navier–Stokes Equations for Axisymmetric Automodel Flows of Fluids

We investigate self-similar solutions of the Navier–Stokes equations for the axisymmetric flow of a viscous incompressible fluid. The original equations are transformed by the Slezkin method. On the basis of analysis of physical properties of the flow and the Slezkin general equation, we show that, in parallel with the known solutions of this equation, there exist several other solutions with physical meaning. We consider the simplest case of irrotational flows for which current lines may be circles, ellipses, parabolas, and hyperbolas. Unlike the Landau and Squire solutions, these flows are interpreted as nonjet flows of fluid flowing into and out of a homogeneous porous axially symmetric body.     

A fast subspace method for image deblurring

Image deblurring problems appear frequently in astronomical image analysis. For image deblurring problems, it is reasonable to add a non-negativity constraint because of the physical meaning of the image. Previous research works are mainly full-space methods, i.e., solving a regularized optimization problem in a full space. To solve the problem more efficiently, we propose a subspace method. We first formulate the problem from full space to subspace and then use an interior-point trust-region method to solve it. The numerical experiments show that this method is suitable for ill-posed image deblurring problems.     

A stochastic collocation method for the second order wave equation with a discontinuous random speed

In this paper we propose and analyze a stochastic collocation method for solving the second order wave equation with a random wave speed and subjected to deterministic boundary and initial conditions. The speed is piecewise smooth in the physical space and depends on a finite number of random variables. The numerical scheme consists of a finite difference or finite element method in the physical space and a collocation in the zeros of suitable tensor product orthogonal polynomials (Gauss points) in the probability space. This approach leads to the solution of uncoupled deterministic problems as in the Monte Carlo method. We consider both full and sparse tensor product spaces of orthogonal polynomials. We provide a rigorous convergence analysis and demonstrate different types of convergence of the probability error with respect to the number of collocation points for full and sparse tensor product spaces and under some regularity assumptions on the data. In particular, we show that, unlike in elliptic and parabolic problems, the solution to hyperbolic problems is not in general analytic with respect to the random variables. Therefore, the rate of convergence may only be algebraic. An exponential/fast rate of convergence is still possible for some quantities of interest and for the wave solution with particular types of data. We present numerical examples, which confirm the analysis and show that the collocation method is a valid alternative to the more traditional Monte Carlo method for this class of problems.     

Asymptotic analysis for linear difference equations

We are concerned with asymptotic analysis for linear difference equations in a locally convex space. First we introduce the profile operator, which plays a central role in analyzing the asymptotic behaviors of the solutions. Then factorial asymptotic expansions for the solutions are given quite explicitly. Finally we obtain Gevrey estimates for the solutions. In a forthcoming paper we will develop the theory of cohomology groups for recurrence relations. The main results in this paper lay analytic foundations of such an algebraic theory, while they are of intrinsic interest in the theory of finite differences.