钻井布局的数学模型

本文对钻井布局问题的研究 ,是从全局搜索入手 ,逐步深入讨论了各种算法的有效性、适用性和复杂性 ,得到不同条件下求最多可利用旧井数的较好算法 .对问题 1 ,我们给出了全局搜索模型、局部精化模型与图论模型 ,讨论了各种算法的可行性和复杂度 .得到的答案为 :最多可使用 4口旧井 ,井号为 2 ,4 ,5,1 0 .对问题 2 ,我们给出了全局搜索、局部精化和旋转矢量等模型 ,并对局部精化模型给出了理论证明 ,答案为 :最多可使用 6口旧井 ,井号为 1 ,6,7,8,9,1 1 ,此时的网格逆时针旋转 4 4.37度 ,网格原点坐标为 (0 .4 7,0 .62 ) .对问题 3,给出判断 n口井是否均可利用的几个充分条件、必要条件和充要条件及其有效算法     

钻井布局的最优设计

本文讨论了钻井布局的最优策略问题，建立了一个最优化模型，针对题目的三个问题，提出了重要定理1，必要条件1、2，在此基础上计算并证明了问题1中可利用的旧井最多为（P2，P4，P5，P10）四只。又借助最大团理论，计算机搜索得总问题2中可利用的旧井最多为（P1，P6，P7，P8，P9，P11）6口，并对此进行了证明，对于问题3；运用定理1给出并证明了利用n口旧井的充分必要条件及实现它的算法。模型求解采用MATLAB软件，计算结果表明本文采用的算法简便可行，并具有一定的推广价值。     

钻井布局的设计

本文首先给出钻井布局的数学模型 ,进一步采用全面搜索法、局部搜索法、图论法、目测法、图上作业法等不同的优化方法 ,进行了模型求解 .对于给定的数值例子 ,得到问题 (1 )的解为 4 ,可利用的旧井为P2 ,P4 ,P5和 P10 ;问题 (2 )的解为 6,可利用的旧井为 P1,P6,P7,P8,P9和 P11.最后对于问题 (3) ,本文给出了 n个旧井均可利用的充分必要条件     

钻井布局

本文将旧井的利用问题归结为 0 -1规划问题 ,由此建立了目标函数 .提出映射原理 ,将旧井的位置映射到一个单位网格中 ,从而大大地简化了模型的求解 .应用映射原理和穷举方法 ,求解出有方向约束条件下的可利用点为 4个 ,经过转化 ,推广到无方向约束条件下的可利用问题 ,解得 6个点可利用 .研究了目标成立的充分条件 ,给出了三种特殊情形下的判定方法 .提出了中垂线上的二分逼近法     

注重变式研究,深化所学内容

《代数》(必修)(上)(人民教育出版社)有如下例题:例题　把一段半径为Ｒ的圆木,锯成横截面为矩形的木料,怎样锯法才能使横截面面积最大?解　因为锯得的矩形的横截面是圆内接矩形,所以它的对角线是圆的直径,其长度为2Ｒ,设对角线与一条边的夹角为θ(如图1),则ＡＢ=2Ｒｃｏｓθ,ＢＣ=2Ｒｓｉｎθ,∴Ｓ=2Ｒｃｏｓθ·2Ｒｓｉｎθ=2Ｒ2ｓｉｎ2θ.∵ｓｉｎ2θ≤1,∴Ｓ≤2Ｒ2.当Ｓ=2Ｒ2时,θ=45°,圆内接矩形为正方形.答:以圆木的直径为对角线,锯成横截面为正方形的木料时,横截面的面积最大.本例使学生认识到建立目标函数求最值的一种新方法———…     

对一类求面积习题解法的研究

下面是两道流行的习题及解答: 题1 一平行四边形的两邻边长分别为2和4,两对角线的夹角为60°,试求其面积。设这个平行四边形的两对角线长分别为2x、2y,面积为S。则有S=4·1/2xysin60°=3~(1/3)xy,又据余弦定理得解之,得xy=6。所以,S=6(3)~(1/2)。例2 已知平行四边形的两邻边分别为2和4,其对角线的夹角为45°,求该平行四边形的面积。设法同题1.则S=4×1/2xysin45°=     

正n边形中四边形计数问题

文[1]探究了正n边形中三角形计数问题,受其启发笔者探究了正n边形中四边形计数问题.引理1圆内接四边形为平行四边形(矩形),当且仅当该四边形的两条对角线为该外接圆的两条直径.引理2圆内接四边形为菱形(正方形),当且仅当该四边形的两条对角线为该外接圆的两条互相垂直的直径.引理1,引理2由简单的平面几何知识即可得证,在此从略.问题1以正八边形的八个顶点为顶点可作多少个四边形?其中含有多少个梯形?多少平行四边形(含矩形)?多少个菱形(含正方形)?分析1)此正八边形的八个顶点中任意四点即可构成一个四边形,故四边形个数为C4=70.2)若构成梯…     

初二几何第一学期期中自测题

A组 一、填空题(每小题4分,共40分) (1)在□ABCD中,AC=6cm,BD=10cm,则AB的长度取值范围是_. (2)矩形的两条对角线的交角为60°,一条对角线和较短边的和为15,则对角线长为_,较长边的长为_.(3)已知菱形ABCD周长为20cm,BD=5cm,则菱形各角的度数为_.     

正方形两个重要性质的应用、推广及探究

我们知道:矩形平面内一点到不同对角线两端点距离的平方和相等.这个性质经常在处理矩形的相关问题时被巧妙应用(见文[1]文[2]).本文笔者将给出正方形的两个类似重要性质,并在此基础上加以直接应用和变式推广,同时对一些相关问题进行深入探究,现整理出来和读者一起分享.     

中考中的中点四边形

<正>一、中点四边形及性质顺次连接多边形各边中点所得的新多边形叫做原多边形的中点多边形.性质1中点四边形的形状取决于原四边形对角线的关系:(1)任意四边形的中点四边形是平行四边形;(2)对角线垂直的四边形的中点四边形是矩形;(3)对角线相等的四边形的中点四边形是菱形;(4)对角线垂直且相等的四边形的中点四边形是正方形.     

解带有二次约束非凸二次规划问题的一个分枝缩减方法

在这篇论文里,有机地把外逼近方法与分枝定界技术结合起来,提出了解带有二次约束非凸二次规划问题的一个分枝缩减方法;给出了原问题的一个新的线性规划松弛,以便确定它在超矩形上全局最优值的一个下界;利用超矩形的一个深度二级剖分方法,以及超矩形的缩减和删除技术,提高算法的收敛速度;证明了在知道原问题可行点的条件下,该算法在有限步里就可以获得原问题的一个全局最优化解,并且用一个例子说明了该算法是有效的.     

Modified auxiliary boundary conditions method for an ill-posed problem for the homogeneous biharmonic equation

In this paper, we are interested in the inverse problem for the biharmonic equation posed on a rectangle, which is of great importance in many areas of industry and engineering. We show that the problem under consideration is ill-posed; therefore, to solve it, we opted for a regularization method via modified auxiliary boundary conditions. The numerical implementation is based on the application of the semidiscrete finite difference method for a sequence of well-posed direct problems depending on a small parameter of regularization. Numerical results are performed for a rectangle domain showing the effectiveness of the proposed method.     

On order‐reducible sinc discretizations and block‐diagonal preconditioning methods for linear third‐order ordinary differential equations

By introducing a variable substitution, we transform the two‐point boundary value problem of a third‐order ordinary differential equation into a system of two second‐order ordinary differential equations (ODEs). We discretize this order‐reduced system of ODEs by both sinc‐collocation and sinc‐Galerkin methods, and average these two discretized linear systems to obtain the target system of linear equations. We prove that the discrete solution resulting from the linear system converges exponentially to the true solution of the order‐reduced system of ODEs. The coefficient matrix of the linear system is of block two‐by‐two structure, and each of its blocks is a combination of Toeplitz and diagonal matrices. Because of its algebraic properties and matrix structures, the linear system can be effectively solved by Krylov subspace iteration methods such as GMRES preconditioned by block‐diagonal matrices. We demonstrate that the eigenvalues of certain approximation to the preconditioned matrix are uniformly bounded within a rectangle on the complex plane independent of the size of the discretized linear system, and we use numerical examples to illustrate the feasibility and effectiveness of this new approach. Copyright © 2013 John Wiley & Sons, Ltd.     

A fast Poisson solver

Summary A direct method is developed for the discrete solution of Poisson's equation on a rectangle. The algorithm proposed is of the class of marching methods. The idea is to generalize the classical Cramer's method using Chebyshev matrix polynomials formalism. This results in the solution ofN independent diagonal system of linear equations in the eigenvector coordinate system. An elementary transformation to the original coordinate system is then carried out.     

A spectral solution method for a boundary-value problem for a mixed-type equation with two singularity lines

We consider a boundary-value problem for a mixed-type equation with two perpendicular singularity lines given in a domain whose elliptic part is a rectangle, while the hyperbolic one is a vertical half-strip. This problem differs from the Dirichlet one by the fact that at the left boundary of the rectangle and the half-strip we specify the vanishing order of the desired function rather than its value. We find a solution to the problem by a spectral method with the use of the Fourier–Bessel series and prove the uniqueness of the solution. We substantiate the uniform convergence of the corresponding series under certain requirements to the problem statement.     

An alternative to the Pythagorean rule? Reevaluating Problem 1 of cuneiform tablet BM 34 568

The first problem of the Seleucid mathematical cuneiform tablet BM 34 568 calculates the diagonal of a rectangle from its sides without resorting to the Pythagorean rule. For this reason, it has been a source of discussion among specialists ever since its first publication, but so far no consensus in relation to its mathematical meaning has been attained. This paper presents two new interpretations of the scribe's procedure, based on the assumption that he was able to reduce the problem to a standard Mesopotamian question about reciprocal numbers. These new interpretations are then linked to interpretations of the Old Babylonian tablet Plimpton 322 and to the presence of Pythagorean triples in the contexts of Old Babylonian and Hellenistic mathematics.     

A Branch and Bound Algorithm for Separable Concave Programming

In this paper, we propose a new branch and bound algorithm for the solution of large scale separable concave programming problems. The largest distance bisection (LDB) technique is proposed to divide rectangle into sub-rectangles when one problem is branched into two subproblems. It is proved that the LDB method is a normal rectangle subdivision(NRS). Numerical tests on problems with dimensions from 100 to 10000 show that the proposed branch and bound algorithm is efficient for solving large scale separable concave programming problems, and convergence rate is faster than ω-subdivision method.     

An optimal filtering method for the Cauchy problem of the Helmholtz equation

In this paper, we consider a Cauchy problem for the Helmholtz equation in a rectangle. An optimal filtering method is presented for approximating the solution of this problem, and the Hölder type error estimate is obtained. Numerical illustration shows that the method works effectively.     

Approximate grid solution of a nonlocal boundary value problem for Laplace’s equation on a rectangle

A nonlocal boundary value problem for Laplace’s equation on a rectangle is considered. Dirichlet boundary conditions are set on three sides of the rectangle, while the boundary values on the fourth side are sought using the condition that they are equal to the trace of the solution on the parallel midline of the rectangle. A simple proof of the existence and uniqueness of a solution to this problem is given. Assuming that the boundary values given on three sides have a second derivative satisfying a Hölder condition, a finite difference method is proposed that produces a uniform approximation (on a square mesh) of the solution to the problem with second order accuracy in space. The method can be used to find an approximate solution of a similar nonlocal boundary value problem for Poisson’s equation.     

Improved local search algorithms for the rectangle packing problem with general spatial costs

The rectangle packing problem with general spatial costs is to pack given rectangles without overlap in the plane so that the maximum cost of the rectangles is minimized. This problem is very general, and various types of packing problems and scheduling problems can be formulated in this form. For this problem, we have previously presented local search algorithms using a pair of permutations of rectangles to represent a solution. In this paper, we propose speed-up techniques to evaluate solutions in various neighborhoods. Computational results for the rectangle packing problem and a real-world scheduling problem exhibit good prospects of the proposed techniques.