特殊化处理——解决数学问题的点睛之笔

解决数学问题的关键是把握问题的特征 ,然后对症下药 ,突破难点 ;而所求问题的特殊情形 ,往往集中了我们所要研究问题的关键信息 ,由此入手 ,简洁有效 ,事半功倍 .笔者列举了几类常见的特殊情形 ,以飨读者 .(一 )利用特殊值代入 ,或直接得出结果 ,或直接否定命题结论　　在问题成立的有效区域内 ,能否恰当地选择特殊值 ,是考查学生观察能力的良好途径 ,因而倍受高考出题者青睐 .例 1　下列是一组能用特殊值解决的高考试题1 )若函数 f(x)的图象可由 y=lg(x +1 )的图象绕坐标原点O逆时针旋转 π2 得到 ,则f(x) =.A) 1 0 -x- 1　　　　　B) 1…     

Robin型离散Schwarz波形松弛算法的收敛性分析

Schwarz波形松弛(Schwarz waveform relaxation,SWR)是一种新型区域分解算法,是当今并行计算研究领域的焦点之一,但针对该算法的收敛性分析基本上都停留在时空连续层面.从实际计算角度看,分析离散SWR算法的收敛性更重要.本文考虑SWR研究领域中非常流行的Robin型人工边界条件,分析时空离散参数t和x、模型参数等因素对算法收敛速度的影响.Robin型人工边界条件中含有一个自由参数p,可以用来优化算法的收敛速度,但最优参数的选取却需要求解一个非常复杂的极小-极大问题.本文对该极小-极大问题进行深入分析,给出最优参数的计算方法.本文给出的数值实验结果表明所获最优参数具有以下优点:(1)相比连续情形下所获最优参数,利用离散情形下获得的参数可以进一步提高Robin型SWR算法在实际计算中的收敛速度,当固定t或x而令另一个趋于零时,利用离散情形下所获参数可以使算法的收敛速度具有鲁棒性(即收敛速度不随离散参数的减小而持续变慢).(2)相比连续情形下所获收敛速度估计,离散情形下获得的收敛速度估计可以更加准确地预测算法的实际收敛速度.     

回收量不确定下考虑技术授权的闭环供应链产品定价研究

越来越多的企业开始采取再制造的运作模式以缓解资源浪费和环境危机。原始设备制造商(OEM)为专注于新产品制造,可通过技术授权的方式委托第三方再制造商(TPR)进行再制造活动。本文研究了回收量不确定且OEM技术授权TPR进行再制造情形下闭环供应链的产品定价问题。研究发现:(1)当废旧品回收量在较小范围波动时,OEM和TPR若想获得不低于确定情形下的利润需考虑企业风险规避程度的影响;(2)当废旧品回收量在较大范围波动时,OEM和零售商在分别制定产品批发价和零售价时需按废旧品回收量波动方向相反的方向进行;(3)在废旧品回收量波动情形下,OEM应优先选择与风险中性的TPR合作,再适当降低技术授权费用使得TPR有利可图,这样既有利于促进废旧品的回收,也益于双方合作关系的持续发展。     

带有延迟时间下界的k-(n1,1,…,1)-排序问题的拟多项式时间算法

讨论Wikum的关于带有延迟时间下界的k-(n1,1,…,1)-链形结构排序问题的拟多项式时间算法,其中当n1=2的情况已由Yin等人(1999)解决,这里主要以n1=3的情形为例作更加细致的分析,然后给出较Yin等人(1999)的算法更加有效的拟多项式时间算法.为了保持文章的连续性,也将列出Yin等人(1999)的n1=2的算法加以比较.     

函数作图时常见错误浅析

据函数解析式来作该函数的图象时,必须确保所作图象的完备性和纯碎性,即应满足:(1)凡是适合解析式的点均在所作出的图象上;(2)凡图象上任意点的坐标均适合该解析式。当所给解析式较为复杂而求其图象时,往往先化简,后再作图。但常因化简过程不是同解变形或因作图时忽视变量范围,而引起图象范围及属性的变化,使原题图象的完备性及纯碎性被破坏。笔者在给一些学生辅导这一内容时,发现以上问题带有     

关于一类常微分方程亚纯解的个数

<正> §1.引言 关于微分方程大范围解析解的个数问题一直为许多作者所关注.如所知当n=2时,即Riccati方程情形,方程(1)可具有一个复参数的亚纯解族.伹当n≥3且{P_k(Z)}是多项式情形,新近G.Gundersen和I.Laine指出方程(1)仅具有有限多个亚纯解.本文首先考虑了方     

形同质异的五组函数题

“函数”中,时有一些形同质异、使我们容易受到其中一种情境干扰的问题,剖解这些问题,对提高我们的辨析能力很有好处!1定义域不同的形同质异题例1(1)已知函数f(x)=x2-mx 1,对一切x∈R恒有f(x)>0,求实数m的取值范围;(2)已知函数f(x)=x2-mx 1,对一切x∈(0, ∞)恒有f(x)>0,求实数m的取值范围.剖析从图象看,(1)即为f(x)的图象全在x轴上方,而(2)仅要求在y轴右边的图象在x轴上方;从不等式角度看,(1)为x2 1>mx对x∈R均成立,而(2)仅为对x>0恒成立.简解(1)从图象考虑,即Δ=m2-4<0,得-20.因为x 1x≥2,当且仅当x…     

关于函数图象对称问题的相关命题研究

函数是高中数学的重点内容之一 ,函数问题的多变体现了函数的特点 .研究函数图象的对称特点 ,对更进一步理解函数的性质是十分重要的 .1　图象关于点对称问题的相关命题定理　奇函数ｙ=ｆ(ｘ) ,ｘ∈Ｒ的图象关于原点对称 .(证明见教材 ,略 .)奇函数满足ｆ(-ｘ)= -ｆ(ｘ)可写为ｆ(0 +ｘ) +ｆ(0 -ｘ) =0 ,ｘ∈Ｒ .由以上关系式拓展得如下命题 .命题 1　若一个函数ｙ =ｆ(ｘ)对任意ｘ∈Ｒ满足ｆ(ａ -ｘ) +ｆ(ａ+ｘ) =2ｂ,当且仅当它的图象关于点 (ａ,ｂ)成对称图形 .证明　设点Ｍ(ｍ ,ｎ)为函数ｆ(ｘ)图象上任意一点 ,它关于点 (ａ ,ｂ)的对称…     

对一道试题的质疑、探究和修正

<正>在进行一轮复习时,学生对资料中出现的一个题目提出质疑,发现给出的标准答案有问题,于是笔者对此题进行了一些研究.题目已知函数f(x)=log_ax(a> 0且a≠1)和函数g(x)=sinπ/2x,若f(x)与g(x)的图象有且只有3个交点,则a的取值范围是____.资料的答案为:(1/7,1/3)∪(5,9).一、错因探究笔者首先通过GeoGebra软件作出函数的图象,函数g(x)的图象是固定的曲线,当a>1时,调整a的取值,使得函数f(x)与g(x)的图象有公共切点,     

同顺序M×N排序问题的动态规划方法

排序论(Scheduliny Theory)是组合最优化理论中一个应用十分广泛的领域;而同顺序m×n排序问题则是众多的排序模型中一个成果较多的模型.1954年,S.M.Johson给出m=2情形的解法,揭开排序问题研究的序幕.一些动态规划、组合最优化和图论文献都以此作为有趣的例子竞相引用.嗣后,许多作者企图把Johnson算法推广到m≥3的情形.但1976年Garey等人证明了m≥3情形是一个“NP完全问题”.这样,要想找到“好算法”几乎是没有希望的了.近年来,m≥3的m×n排序问题的研究,主要在如下几个方面:     

A multi-scale image reconstruction algorithm for electrical capacitance tomography

Electrical capacitance tomography (ECT) is considered as a promising process tomography (PT) technology, and its successful applications depend mainly on the precision and speed of the image reconstruction algorithms. In this paper, based on the wavelet multi-scale analysis method, an efficient image reconstruction algorithm is presented. The original inverse problem is decomposed into a sequence of inverse problems, which are solved successively from the largest scale to the smallest scale. At different scales, the inverse problem is solved by a generalized regularized total least squares (TLS) method, which is developed using a combinational minimax estimation method and an extended stabilizing functional, until the solution of the original inverse problem is found. The homotopy algorithm is employed to solve the objective functional. The proposed algorithm is tested by the noise-free capacitance data and the noise-contaminated capacitance data, and excellent numerical performances and satisfactory results are observed. In the cases considered in this paper, the reconstruction results show remarkable improvement in the accuracy. The spatial resolution of the reconstructed images by the proposed algorithm is enhanced and the artifacts in the reconstructed images can be eliminated effectively. As a result, a promising algorithm is introduced for ECT image reconstruction.     

基于字典学习方法的CT不完全投影图像重建算法

对于不完全投影角度的重建研究是CT图像重建中一个重要的问题.将压缩感知中字典学习的方法与CT重建算法ART迭代算法相结合.字典学习方法中字典更新采用K-SVD(K-奇异值分解)算法,稀疏编码采用OMP(正交匹配追踪)算法.最后通过对标准Head头部模型进行仿真实验,验证了字典学习方法在CT图像重建中对于提高图像的重建质量和提高信噪比的可行性与有效性.另外还研究了字典学习中图像块大小和滑动距离对重建图像的影响     

The Topological Gradient Method: From Optimal Design to Image Processing

The aim of this article is to review and extend the applications of the topological gradient to major image processing problems. We briefly review the topological gradient, and then present its application to the crack localization problem, which can be solved using the Dirichlet to Neumann approach. A very natural application of this technique in image processing is the inpainting problem, which can be solved by identifying the optimal location of the missing edges. Edge detection is of extreme importance, as edges convey essential information in a picture. A second natural application is then the image reconstruction. A class of image reconstruction problems is considered that includes restoration, demosaicing, segmentation and super-resolution. These problems are studied using a unified theoretical framework which is based on the topological gradient method. This tool is able to find the localization and orientation of the edges for blurred, low sampled, partially masked, noisy images. We review existing algorithms and propose new ones. The performance of our approach is compared with conventional image reconstruction processes.     

Continuous-time image reconstruction using differential equations for computed tomography

An approach for reconstructing tomographic images based on the idea of continuous dynamical methods is presented. The method consists of a continuous-time image reconstruction (CIR) system described by differential equations for solving linear inverse problems. We theoretically demonstrate that the trajectories converge to a least squares solution to the linear inverse problem. An implementation of its equivalent electronic circuit is significantly faster than conventional discrete-time image reconstruction (DIR) systems executed in a digital computer. Moreover, the merits of our CIR are demonstrated on a tomographic inverse problem where simulated noisy projection data are generated from a known phantom. Here, we numerically demonstrate that the CIR system does not produce unphysical negative pixel values if one starts out with positive initial values. Besides, CIR also recovers the phantom with almost the same quality as DIR images.     

Positron Emission Tomography and Random Coefficients Regression

We further explore the relation between random coefficients regression (RCR) and computerized tomography. Recently, Beran et al. (1996, Ann. Statist., 24, 2569–2592) explored this connection to derive an estimation method for the non-parametric RCR problem which is closely related to image reconstruction methods in X-ray computerized tomography. In this paper we emphasize the close connection of the RCR problem with positron emission tomography (PET). Specifically, we show that the RCR problem can be viewed as an idealized (continuous) version of a PET experiment, by demonstrating that the nonparametric likelihood of the RCR problem is equivalent to that of a specific PET experiment. Consequently, methods independently developed for either of the two problems can be adapted from one problem to the other. To demonstrate the close relation between the two problems we use the estimation method of Beran, Feuerverger and Hall for image reconstruction in PET.     

A splitting primal-dual proximity algorithm for solving composite optimization problems

Our work considers the optimization of the sum of a non-smooth convex function and a finite family of composite convex functions, each one of which is composed of a convex function and a bounded linear operator. This type of problem is associated with many interesting challenges encountered in the image restoration and image reconstruction fields. We developed a splitting primal-dual proximity algorithm to solve this problem. Furthermore, we propose a preconditioned method, of which the iterative parameters are obtained without the need to know some particular operator norm in advance. Theoretical convergence theorems are presented. We then apply the proposed methods to solve a total variation regularization model, in which the L2 data error function is added to the L1 data error function. The main advantageous feature of this model is its capability to combine different loss functions. The numerical results obtained for computed tomography (CT) image reconstruction demonstrated the ability of the proposed algorithm to reconstruct an image with few and sparse projection views while maintaining the image quality.     

On the regularity of trust region-cg algorithm: With application to deconvolution problem

Deconvolution problem is a main topic in signal processing. Many practical applications are required to solve deconvolution problems. An important example is image reconstruction. Usually, researchers like to use regularization method to deal with this problem. But the cost of computation is high due to the fact that direct methods are used. This paper develops a trust region-cg method, a kind of iterative methods to solve this kind of problem. The regularity of the method is proved. Based on the special structure of the discrete matrix, FFT can be used for calculation. Hence combining trust region-cg method with FFT is suitable for solving large scale problems in signal processing.     

Projected Subgradient Minimization Versus Superiorization

The projected subgradient method for constrained minimization repeatedly interlaces subgradient steps for the objective function with projections onto the feasible region, which is the intersection of closed and convex constraints sets, to regain feasibility. The latter poses a computational difficulty, and, therefore, the projected subgradient method is applicable only when the feasible region is “simple to project onto.” In contrast to this, in the superiorization methodology a feasibility-seeking algorithm leads the overall process, and objective function steps are interlaced into it. This makes a difference because the feasibility-seeking algorithm employs projections onto the individual constraints sets and not onto the entire feasible region. We present the two approaches side-by-side and demonstrate their performance on a problem of computerized tomography image reconstruction, posed as a constrained minimization problem aiming at finding a constraint-compatible solution that has a reduced value of the total variation of the reconstructed image.     

基于CT图像重建的多重集合分裂可行性问题应用分析

为了较好地应用CQ算法解决稀疏角度CT 图像重建的问题,提出了一种新的实时的分块逐次混合算法．首先将稀疏角度CT 图像重建的重建问题转化成分裂可行性问题．其次,通过分析非空闭凸集C和Q的不同的定义,在N维实空间中分别针对不同的CQ算法给出了7种不同的实现方案．通过试验,分别对不同算法及其方案的重建精度和收敛速度进行了对比分析,并对多重集合分裂可行性问题算法中约束权因子的选取及其对输出的影响进行了研究,从而给出了CQ算法在稀疏角度CT图像重建问题中应用的最佳凸集定义方案．以此为基础,给出了所提出算法的最佳实现方案．试验结果表明,该算法收敛速度快,重建精度高,为多重集合分裂可行性问题及其改进算法在该重建问题上的应用提供了参考．     

Coarse grid correction by aggregation / disaggregation with application in image reconstruction

In algebraic reconstruction of images in computerized tomography we are dealing with rectangular, large, sparse and ill-conditioned linear systems of equations. In this paper we describe a two-grid algorithm for solving such kind of linear systems, which uses Kaczmarz's projection method as relaxation. The correction step is performed with a special “local” aggregation / disaggregation type procedure. In this respect, we have to solve a small sized minimization problem associated to each coarse grid pixel. The information so obtained is then “distributed” to the neighbour fine grid pixels. Some image reconstruction experiments are also presented. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)