空中加油问题模型建立及其解法

空中加油问题是一个关于在飞机飞行过程中,辅机在空中给主机加油来提高主机直航能力的问题.该题的要求是在辅机架数n一定的情况下,确定最优作战方案及主机的最大作战半径.对于问题1和问题2,首先给出了一般情况下的飞机调度的数学模型,然后用穷举法求出了n≤4情况下的最优作战方案及主机的最大作战半径rn,然后用归纳法推导出了n为一般情况下rn的上下界,最后给出了判断最优作战方案的必要条件.问题3中,给出了与问题1、问题2类似问题的求解结果.问题4中,首先求出了n≤4时空军基地的选址和最优作战方案,然后给出了n为一般情况下,最优作战方案和基地选址的通用数学模型.问题5中,在主机最快到达目的地并返回的条件下,给出了主机的飞行路线和最优作战方案;在满足辅机架数最少的条件下,给出了作战方案,并用MATLAB求出了满足该条件时的最少辅机架数的上界为248架.另外,给出了一些新的定义方法和定理并全部给予证明.     

飞机空中加油问题的研究

讨论了飞机的空中加油问题,通过比较不同方案下主机的最大作战半径,找到了辅机给主机空中加油的最佳方案,分析了存在多个基地时各基地的位置及辅机在各基地的数量,研究了二维坐标下主机的最佳飞行路径,给出了解决问题的办法.整个思路由浅入深,通过考虑辅机数目n分别为1,2,3,4的特殊情况进而探讨一般的规律,包括辅机在基地无加油时间和有加油时间以及主机在空中盘旋等不同假设条件下的复杂情况,并将统计出的数据列表对比,获得了更直观的认识.在最后进行了展望,希望结合实际中的飞机空中加油问题综合考虑风力、风向等随机因素的影响,拓展到3维空间的更复杂模型中,使飞机的空间加油问题变得更有实际意义.     

空中加油问题当前最好的几个方案

通过比较增加一架辅机后主机飞行距离的净增量,得到当前最好的从一架到数万架辅机的可行方案;同时通过两个反例,揭示了空中加油问题新的研究方向.     

空中加油最优分配建模与实现

确定最佳空中加油方案,既有一定难度,又有重要的实用价值.本文针对加油机可以多次上天情况,对这一问题展开研究,确定了使主机飞行距离最大的空中加油方案.我们将飞机飞行过程分为出航和返航两个阶段,首先利用数学归纳法得出出航阶段的最佳加油方案;然后对返航过程建立约束方程组,求出返航阶段的最佳加油方案.利用这一模型结果可以对任意多架加油机的情况确定全程最佳加油方案.最后结合出航、返航编队及飞行姿态调整对模型作了一定范围的定量扩展,使本文的研究更具有实用价值.     

圆锥曲线焦半径长度-比值公式妙解焦半径问题

<正>圆锥曲线给定焦半径比值时,若想直接求出焦半径长度,往往需要一定计算量．本文通过焦半径比值关系得出圆锥曲线焦半径长度比值公式,该公式十分有趣且使用方便,若已知焦半径的比值,则通过圆锥曲线方程既可迅速计算对应焦半径长度,同时还能对与焦半径有关的范围问题进行研究,欢迎同学们阅读学习．     

一类带多项式约束的不确定凸优化问题的鲁棒可行性半径刻画

该文旨在刻画一类约束函数是带有不确定信息的凸多项式的不确定凸优化问题的鲁棒可行性半径的下界.首先借助鲁棒优化方法,引入了该不确定凸优化问题的鲁棒对等问题(Robust counterpart),并给出了其鲁棒可行性半径的定义.随后通过引入一类上图集和借助由不确定集所生成的Minkowski泛函,刻画了该不确定凸优化问题的鲁棒可行性半径的下界.进一步的,在不确定集是仿射不确定集以及约束函数是平方和凸多项式时,得到了该不确定优化问题的鲁棒可行性半径的一个精确公式,推广和改进了文献[10]的相应结果.     

矩阵谱半径上界的估值及非负矩阵谱半径的一个公式

<正> 本文建立了循环矩阵和非负矩阵谱半径的公式,并提出几个不等式.用这些不等式估计矩阵谱半径的上界,可得到比一般方法更精确的估计,把这些不等式作为矩阵敛散的判据,则可得到比[2]、[3]更精确、应用范围更广的结果.由于估出了谱半径的上界,故能了解矩阵特征值分布的区域.对于估计循环矩阵谱半径的上界,我们提出了一个比较精确的公式,它有时能定出循环矩阵谱半径的上确界.     

Poisson方程差分格式迭代解法中一些最优参数的有理拟合方法(英文)

本文针对二维Poisson方程五点和九点差分格式,导出了求解这些格式的SOR方法中最优松弛因子与区域剖分数的有理拟合公式,给出了Jacobi结合Chebyshev加速方法中Jacobi迭代矩阵谱半径的有理拟合公式.实际计算表明这些公式计算效果良好.     

与二次曲线相切于顶点的“最大圆”的不等式求法

在平面解析几何中,有一类问题是在二次曲线的某一侧求出一个与该已知二次曲线和切于顶点的最大圆。这类问題的常规解法是通过联立该二次曲线及与该二次曲线相切于顶点半径为R的圆的方程,来确定这个“最大圆”的半径R_0。应当指出,用这种方法所确定的半径R_0,往往还需要证明它是“最大圆”的半径,也就是说,还必须证明如下两点:     

关于圆锥曲线焦半径的两类公式

圆锥曲线上一点与其焦点的连线叫做焦半径．与它有关的问题是各类考试的热点之一,故在圆锥曲线学习中,值得我们总结与研究,为此,本文介绍两类焦半径公式．     

The construction of a nose shape of minimum drag for specified external dimensions and volume using Euler equations

The problem of constructing the axisymmetric nose shape which gives minimum wave drag for a specified volume and external dimensions is solved by a direct method using Euler equations. As in the Newton's formula approximation, the optimum contours together with the front faces – a segment of the boundary extremum along the longitudinal coordinate and the gently sloping segment of the bilateral extremum – may contain a cylindrical end part with a horizontal segment of the boundary extremum with respect to the maximum admissible radial coordinate. In the direct method, the required parameters (“controls”), which define the shape of the optimum contour, are the radii corresponding to the points of the segment of a bilateral extremum, including the radius of the face for fixed abscissas. For each aspect ratio (the ratio of the length to the radius of the base), when a certain value of the volume coefficient (the ratio of the volume to the volume of a cylinder of maximum external dimensions) is exceeded, the optimum nose shape is completed by a rear cylindrical part. The optimum nose shape, which begins from a certain initial contour, that satisfies the limitations of the problem, is constructed after a finite number of cycles. In each cycle, all the controls are corrected, and together with the directions of the change, their increments are found, while the information necessary for this for any number of controls is obtained after three direct calculations. One other advantage of the method is the rapid, close to quadratic, convergence. The nose shapes constructed are compared with the nose shapes that are optimum in the Newton's formula approximation.     

Calculation for thin-walled closed contours of composite materials in torsion

We consider the torsion of closed torsion-box structures made of composite materials (such as airplane wings) under mechanical loads inducing only torsion of the construction without bending. The problem is solved by an energy method and using a conforming displacement method. The derived distribution law for the tangential stresses contains two terms and gives the Bredt formula for the special case of pure torsion. Since in solving the problem by a conforming displacement method the axial warping distribution law for torsion of a simple closed contour is obtained automatically, this method may be considered to be ecact in the sense of satisfying the continuity condition for the structural material.Deceased.Moscow State Aviation Institute (Technical University), Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 33, No. 2, pp. 235–241, March–April, 1997.     

A new approach for calculating the real stability radius

We present a new fast algorithm to compute the real stability radius with respect to the open left half plane which is an important problem in many engineering applications. The method is based on a well-known formula for the real stability radius and the correspondence of singular values of a transfer function to pure imaginary eigenvalues of a three-parameter Hamiltonian matrix eigenvalue problem. We then apply the implicit determinant method, used previously by the authors to compute the complex stability radius, to find the critical point corresponding to the desired singular value. This corresponds to a two-dimensional Jordan block for a pure imaginary eigenvalue in the parameter dependent Hamiltonian matrix. Numerical results showing quadratic convergence of the algorithm are given.     

The load transfer between two rigid spherical inclusions in an elastic medium

The paper is concerned with the load transfer problem between two rigid spherical inclusions in an elastic matrix. A reflection-type formula is developed which is accurate up to and including terms ofO(a/R)⁴, wherea is the maximum radius of the inclusions, andR is the centre-to-centre distance between the two inclusions. Asymptotic results are derived at near touching showing a weak logarithmic singularity in the load transfer. The results are verified by a direct numerical calculation using a boundary collocation method. The numerical method uses Kelvin's general solution as the basis functions for the approximate solution and is highly accurate and efficient, even at near touching.     

A linear-time algorithm for finding optimal vehicle refueling policies

We explore a fixed-route vehicle refueling problem as a special case of the inventory-capacitated lot-sizing problem, and present a linear-time greedy algorithm for finding optimal refueling policies.     

On the spectral radius of the Jacobi iteration matrix for a rectangular region with two different media

The spectral radius of the Jacobi iteration matrix plays an important role to estimate the optimum relaxation factor, when the successive overrelaxation (SOR) method is used for solving a linear system. The specific systems are finite difference forms of the Laplace equation satisfied on a rectanglar region with two different media. Though the potential function for the inhomogeneous closed region is continuous, the first order derivative is not continuous. So this requires internal boundary conditions or interface conditions. In this paper, the spectral radius of the Jacobi iteration matrix for the inhomogeneous rectangular region is formulated and the approximation for the explicit formula, suitable for the computation of the spectral radius, is deduced. It is also found by the proposed formula that the spectral radius and the optimum relaxation factor rigorously depend on the inhomogeneity or the internal boundary conditions in the closed region, and especially vary with the position of the internal boundary. These findings are also confirmed by the numerical results of the power method.The stationary iterative method using the proposed formula for calculating estimates of the spectral radius of the Jacobi iteration matrix is compared with Carré's method, Kulstrud's method and the stationary iterative method using Frankel's theoretical formula, all for the case of some numerical models with two different media. According to the results our stationary iterative method gives the best results ffor the estimate of the spectral radius of the Jacobi iteration matrix, for the required number of iterations to calculate solutions, and for the accuracy of the solutions.As a numerical example the microstrip transmission line is taken, the propating mode of which can be approximated by a TEM mode. The cross section includes inhomogeneous media and a strip conductor. Upper and lower bounds of the spectral radius of the Jacobi iteration matrix are estimated. Our method using these estimates is also compared with the other methods. The upper bound of the spectral radius of the Jacobi iteration matrix for more general closed regions with two different media might be given by the proposed formula.     

Inversion of the local Pompeiu transformation on Riemannian symmetric spaces of rank one

We study the problem of inversion of the local Pompeiu transform on Riemannian symmetric spaces of rank one. The explicit formula for the reconstruction of a function by its averages over balls and spheres with a single fixed radius is obtained.     

The minimum equitable radius location problem with continuous demand

We analyze the location of p facilities satisfying continuous area demand. Three objectives are considered: (i) the p-center objective (to minimize the maximum distance between all points in the area and their closest facility), (ii) equalizing the load service by the facilities, and (iii) the minimum equitable radius – minimizing the maximum radius from each point to its closest facility subject to the constraint that each facility services the same load. The paper offers three contributions: (i) a new problem – the minimum equitable radius is presented and solved by an efficient algorithm, (ii) an improved and efficient algorithm is developed for the solution of the p-center problem, and (iii) an improved algorithm for the equitable load problem is developed. Extensive computational experiments demonstrated the superiority of the new solution algorithms.