易拉罐形状和尺寸的最优设计

从用料最省的角度研究了易拉罐的形状和尺寸的优化设计问题,首先通过多次测量取平均值的方法得到了题目所需的数据．然后就问题二和问题三分别建立了优化模型,并借助数学软件进行了求解,得到了最优设计的尺寸．最后设计出了椭球形状的易拉罐作为自己的最优设计．     

易拉罐形状和尺寸的最优设计模型

用游标卡尺测量了当前流行的几种易拉罐饮料尺寸参数,然后建立微分方程模型和规划模型,借助MATLAB 6.5,LINGO8.0编程求解出了易拉罐为正圆柱体、圆台和圆柱体的组合体时的最优设计.最后综合经济、美观、实用等因素,运用非线性规划和层次分析法得出设想中易拉罐的最佳设计,对2006"高教社杯"全国数学建模竞赛C题中的各问题作出了完整的解答.     

凸体的对偶迷向常数与LYZ椭球

研究了凸体处于对偶迷向位置时的解析特征,并建立了凸体对偶迷向常数的新的下界;其次,证明了关于原点中心对称凸体的LYZ椭球与John椭球相等的充要条件;最后,举例具体计算了几个凸多边形的LYZ椭球和John椭球,以进一步认清两者的差别.     

黏性分层流中椭球体俯仰振荡研究

以黏性密度分层流下椭球体自由俯仰振荡衰减过程为研究内容,建立了密度连续分层流数值计算模型.通过对经典小球黏性绕流场的数值模拟和增阻系数的计算验证了数值模型的正确性.以初始45°攻角下的椭球体俯仰振荡过程为研究对象,采用基于Aitken亚松弛适应算法的双向流固耦合方法,数值模拟了不同内Froude(弗汝德)数Fri下椭球体俯仰衰减振荡的动态绕流场.数值研究结果表明,俯仰振荡将上下搅动周围流体,在椭球体上下两侧对称形成四个密度涡环,密度的垂向分层限制了涡环的垂向传播,也加速了涡环的消失,这种限制助长了水平运动的发展,远场尾涡流场将以水平波动的形式传播.在较高的内Froude数Fri和Reynolds(雷诺)数Re下,双向耦合抑制了数值震荡.研究还发现,随着来流速度的增加,阻力系数不增反降,这说明,对于自由俯仰振荡的椭球体,负阻尼现象仍然会出现.     

一种考虑孔洞形状变化的损伤细观模型及其在孔洞闭合过程中的应用

本文在一种特殊的坐标系下,建立了非线性的基体材料,有限大的椭球体中含椭球形孔洞的损伤细观模型,考虑了孔洞形状的影响.得出的粘性约束方程(或称屈服面方程)除应力∑_ij,孔隙度f,幂硬化指数m外,还与孔洞的形状有关.通过曲线拟合的方法,对Gurson方程进行了修正,使之适合于非线性的基体材料、变形状孔洞的情形.最后将此模型用于分析非线性材料内部孔洞的闭合过程.     

易拉罐的设计方案

在日常生活中,我们总会买些易拉罐装的饮料或食品.殊不知,这易拉罐的设计便包含了一定的数学道理. 对易拉罐的设计,经营者总是考虑让成本最低.如:设计一个体积固定为V的圆柱形易拉罐,什么样的设计方案最优? 要比较易拉罐优劣的标准,有下面两种不同的标准进行考虑. 第一种标准:根据制造过程中消耗铁皮的多少来判别优劣,即最优易拉罐应该具有的最小表面S. 分析设易拉罐的高为h,底面半径为r.由圆柱的体积公式V=πr2h,得h=V/πr2.又易拉罐的表面积S=2πr2 2πrh ①     

清晰理解数学模型 合理探求线段最值

<正>笔者整理了近几年中考中关于"线段最值"的试题,归纳出三种数学模型.从"形"的角度构造"三角形两边之和大于第三边"和"垂线段最短"这两种几何模型,以及从"数"的角度建立函数模型.现举例加以分析.模型一、运用"三角形两边之和大于第三边"模型     

椭球体的带任意Abel系数群的辛同调群

本文中，我们建立了辛空间Ｒ~２ｎ中椭球体的带任意Ａｂｅｌ系数的辛同调群理论．从而给出了椭球体的一个辛不变量．     

设备空间磁场预测的传感器布阵优化

本文研究了铁磁性设备周围空间传感器布阵的问题。我们建立了关于传感器位置和数量优化的数学模型,并通过遗传算法对模型进行求解。首先,本文选用对传感器数量和距离要求较少的旋转椭球体作为磁场远场换算的模型。在旋转椭球体模型中,传感器分布位置不当会导致磁场计算系数矩阵的条件数过大,模型将出现病态,因而计算得到的远场磁场结果不可靠。所以,本文以旋转椭球体模型中的系数矩阵条件数为优化目标,建立数学模型优化单个设备上方传感器的数量与位置分布,并利用遗传算法对模型求解。其次,通过实验验证了本模型对于单个设备的传感器位置和数量优化是有效的,且所用传感器数量少,计算结果可靠。最后,将单个设备传感器位置和数量的优化模型推广到多个设备,以两个设备为代表用同时优化和分别优化两种方法计算传感器位置,根据实验计算这两种方法都具有较高的远场磁场计算精度,但分开优化的方法在实际计算更加简便、容易操作。     

受轴向集中压力的椭球体的应力分析

用积分方程法和光弹性方法分析受轴向集中压力的椭球体.在弹性全空间z=-c轴上的[a,∞)和[-a,-∞)区间上,与z=0平面对称地分布集度为X₁(c)=X₁(-c)的集中力、集度为X₂(c)=X₂(-c)的挤压中心,以及迭加一对平行z轴、等值反向、分别作用于z=α及z=-α上的集中力,就能使受轴向集中压力的椭球体问题归结为两个联立的Fredholm第一种积分方程.然后,便能方便地进行数值计算.三维光弹性“冻结”切片法用于详细分析两个椭球体的模型,给出几个切面的应力分布,所得结果σ_z与积分方程法相近,并将结果应用于分析不规则岩石力学试件实测资料的整理.     

Direct simulation of the motion of a settling ellipsoid in Newtonian fluid

In this paper, we discuss the generalization of a Lagrange multiplier-based fictitious domain method to the simulation of the motion of general shape particles in Newtonian fluid. Preliminary numerical results of a settling ellipsoid in a narrow channel filled with a Newtonian fluid are presented. As expected, the ellipsoid turns its broadside to the stream in the simulations.     

受约束回归模型参数的统一几乎无偏估计

考虑实际回归问题中存在更多受约束条件的情况,提出了带约束的统一几乎无偏估计类,统一了常见的具有线性约束的回归模型的几乎无偏估计,进一步的研究给出了在均方误差和均方误差矩阵意义下,带约束的统一几乎无偏估计优于一般带约束的最小二乘估计的充分条件和椭球范围.     

Multivariate Gaussian Simulation Outside Arbitrary Ellipsoids

Methods for simulation from multivariate Gaussian distributions restricted to be from outside an arbitrary ellipsoidal region are often needed in applications. A standard rejection algorithm that draws a sample from a multivariate Gaussian distribution and accepts it if it is outside the ellipsoid is often employed; however, this is computationally inefficient if the probability of that ellipsoid under the multivariate normal distribution is substantial. We provide a two-stage rejection sampling scheme for drawing samples from such a truncated distribution. Experiments show that the added complexity of the two-stage approach results in the standard algorithm being more efficient for small ellipsoids (i.e., with small rejection probability). However, as the size of the ellipsoid increases, the efficiency of the two-stage approach relative to the standard algorithm increases indefinitely. The relative efficiency also increases as the number of dimensions increases, as the centers of the ellipsoid and the multivariate Gaussian distribution come closer, and as the shape of the ellipsoid becomes more spherical. We provide results of simulation experiments conducted to quantify the relative efficiency over a range of parameter settings.     

Some Results for Low-frequency Dirichlet Scattering by Arbitrary Obstacles and Their Application to the Particular Case of the Ellipsoid

It is shown that, for the case of an incident plane wave, thefirst two terms in the low frequency expansion of the scatteringcoefficient of an arbitrary soft obstacle can be obtained explicitlyin terms of the solution to one electrostatic problem. The particularcase of the ellipsoid is discussed in detail.     

从“求圆截线平面问题”的研究谈学生能力的培养

对“求圆截线平面问题”作了深入研究,给出了四种解法,得到了与椭球面交线是圆的平面的充分必要条件,交线圆的中心和半径,并结合该问题谈了基础课教学中对学生能力的培养．     

On lattice points in multidimensional ellipsoids: Problem of centers

The paper determines the exact order of the lattice remainder term for an integral ellipsoid in terms of the arithmetic character of its center.     

Reliable computation of eigenvalues of the magnetostatic integral operator

Some variational problems in magnetostatics can be reformulated as eigenvalue problems for vector surface integral operators in appropriate function spaces, e.g., the magnetostatic integral operator is of considerable interest in the theory of permanent magnetization of compact bodies. In the case that the underlying surface is either a sphere, a spheroid, or a triaxial ellipsoid, explicit expressions for eigenvalues and eigenfunctions are well known. For the ellipsoid, these quantities are given in terms of Lamé functions and surface ellipsoidal harmonics. Since there is an apparent lack in literature we provide an new effective scheme for the reliable computation of these functions and of the corresponding eigenvalues of the magnetostatic operator.     

Projectively equivalent metrics, exact transverse line fields and the geodesic flow on the ellipsoid

We give a new proof of the complete integrability of the geodesic flow on the ellipsoid (in Euclidean, spherical or hyperbolic space). The proof is based on the construction of a metric on the ellipsoid whose non-parameterized geodesics coincide with those of the standard metric. This new metric is induced by the hyperbolic metric inside the ellipsoid (Klein's model).     

The ellipsoid algorithm using parallel cuts

We present an ellipsoid algorithm using parallel cuts which is robust and conceptually simple. If the ratio fo the distance between the parallel cuts under consideration and the corresponding radius of the current ellipsoid is less than or equal to some constant, it is called the canonical case. Applying our algorithm to this case the volume of the next ellipsoid decreases by a factor which is, at worst, exp For the noncanonical case, we first add an extra constraint to make it a canonical case in a higher-dimensional space, then apply our algorithm to this canonical case, and finally reduce it back to the original space. Some interesting variants are also presented to show the flexibility of our basic algorithm.     

Electroencephalography in ellipsoidal geometry

The human brain is shaped in the form of an ellipsoid with average semiaxes equal to 6, 6.5 and 9 cm. This is a genuine 3-D shape that reflects the anisotropic characteristics of the brain as a conductive body. The direct electroencephalography problem in such anisotropic geometry is studied in the present work. The results, which are obtained through successively solving an interior and an exterior boundary value problem, are expressed in terms of elliptic integrals and ellipsoidal harmonics, both in Jacobian as well as in Cartesian form. Reduction of our results to spheroidal as well as to spherical geometry is included. In contrast to the spherical case where the boundary does not appear in the solution, the boundary of the realistic conductive brain enters explicitly in the relative expressions for the electric field. Moreover, the results in all three geometrical models reveal that to some extend the strength of the electric source is more important than its location.