"数老哥"与"形老弟"

在数学王国里,住蔷一对亲兄弟"数老哥"与"形老弟",他们本来形影不离,亲密无间,可就在前些时候,教学王国的一些居民风言风语地说他们不是亲兄弟,说他们之间根本就设有什么关系,数就是数,形就是形.听了这些流言蜚语,"数老哥"与"形老弟"受不了,相互之闻竟然猜疑起来,好长时间都不来往.     

骄傲的"x"

"x"这个符号很神秘,长得也很特别,与众不同.在数学王国里你需要几它就会给你几.因为它的这项特异功能,很多数字朋友都很崇拜它,"x"自己也觉得自己的本领最大,于是,它就越来越骄傲了. 有一天,"1"、"2"、"3"、"4"等几个数字在一起比大小."x"突然出现在它们面前,斜着眼睛说:"你们不要再比啦!你们谁也不是我的对手!""1"听了,生气地说:"你怎么能和我们比呢?你连自己是谁都不知道,还敢来这里和我们比赛.     

"上课"教师不要表现得太"聪明"

经常参加听课,往往有种感觉:授课教师为了表现自己高超的教学艺术,常对教学过程及每一问题都一一作了精心设计安排,让学生都掉进了教师精心设计的"圈套".无疑,在这样的课堂中,学生成了学习的旁观者,课堂民主成了假民主,主体参与成了虚假的被动配合,独立思考成了"牵引着赶路"在2007年4月宁波数学高级研修班上,有幸也参加了此会议,东北师范大学史宁中校长的报告中有一句话:老师,上课时不要表现得太聪明,才能让学生显得更聪明.让我感慨不已:在新课程积极创导学生主体参与、培养创新能力的课堂中,老师是否可以抛弃一些虚假的"聪明",放下一点架子,在上课中表现得"愚钝"一些,参与到课堂学生的学习讨论之中呢?     

"精确到"与"精确度"

新教材引入了二分法求函数的近似值,这就涉及到近似计算中的两个概念:"精确到"与"精确度".教材上用的是精确度,一般如精确度为0.1,是指函数f(x)在区间[a,b]上,有f(a)·f(b)<0,则在(a,b)上存在零点,如∣a-b∣<0.1,在区间上任取一点,包括两个端点,其精确度即为0.1.而"精确到",以精确到0.1为例,即我们平常所说的精确到小数点后一位,是指与精确值的误差不超过±0.05.     

寻求"必要条件"解题

数学解题大多追求题设成立的"充要条件",但如果能抓住"必要条件"解题,往往比利用"充要条件"解题更迅速、更方便,思路更清晰.而使用"必要条件"解题,终归不如利用充要条件的思路来得顺畅,因此很多参考书忽略了利用必要条件及利用必要条件时必要的解题步骤.     

正确运用"或"与"且"

"或"与"且"是数学中常用的名词和逻辑联结词,这里的"或"与日常语言中的"或者"、"可能…也可能…"和"也许…,也许…"意思相当,而且是可兼容的(即两者中至少一个成     

关于高中数学"阅读材料"的思考

根据<普通高中数学课程标准(实验)>编写的教材中的"阅读材料"融知识性、科学性、趣味性与教育性于一体,丰富了教材也为高中数学的教学注入了新的活力.但<课标>未将"阅读材料"规定为教学内容,因此,教材将"阅读材料"设置为选学内容,作为正文的补充和延伸.     

折叠纸片"搭台" 解析几何"唱戏"

近几年的高考数学试题,设置了一些数学学科内的综合题,它们的新颖性、综合性值得我们重视.在知识网络交汇处设计试题是高考命题改革的一个方向.以"折叠纸片"为依托的解析几何题正是在这种背景下"闪亮登场".这类题目将探究解析几何有关问题融于操作实验中,形态鲜活,旨在倡导探索、创造、发现,体现了新课程改革的新理念--研究性学习,具有素质教育的良好导向功能.……     

容易混淆的"恒成立"问题与"存在成立"问题

不等式是中学数学的基础知识和重要部分,一直是各类考试、考查的热点与重点.不等式"恒成立"问题与"存在成立"问题,又是不等式中常见的题型.在各地的自主招生、高考、模拟考试中屡见不鲜.此二类问题对学生掌握基本数学思想与方法提出了较高的要求.学生对此二类问题往往感到容易混淆.     

两次"插空"要当心

在解决排列组合应用题中,经常会遇到一类"不相邻"的问题,学生们往往会采用"插空法"来解决,一般会迎刃而解,但笔者在教学中作了如下探究性的挖掘:将一类"一次插空"题演变为"二次插空"题,然后让他们再去做,90%以上的同学均做错了.     

Creep of polymer concrete in the nonlinear region

Two polyester-based polymer concretes with various volume content of diabase as an extender and aggregate are tested in creep under compression at different stress levels. The phenomenological and structural approaches are both used to analyze the experimental data. Common features of changes in the instantaneous and creep compliances are clarified, and a phenomenological creep model which accounts for the changes in the instantaneous compliance and in the retardation spectrum depending on the stress level is developed. It is shown that the model can be used to describe the experimental results of stress relaxation and creep under repeated loading. Modeling of the composite structure and subsequent solution of the optimization problem confirm the possibility of the existence of an interphase layer more compliant than the binder. A direct correlation between the interphase volume content and the instantaneous compliance of the composite is revealed. It is found that the distinction in nonlinearity of the viscoelastic behavior of the two polymer concretes under investigation can be due to the difference in their porosity. Submitted to the 11th International Conference on Mechanics of Composite Materials (Riga, June 11–15, 2000.) Translated from Mekhanika Kompozitnykh Materialov, Vol. 36, No. 2, pp. 147–164, 2000.     

Peridynamics via finite element analysis

Peridynamics is a recently developed theory of solid mechanics that replaces the partial differential equations of the classical continuum theory with integral equations. Since the integral equations remain valid in the presence of discontinuities such as cracks, the method has the potential to model fracture and damage with great generality and without the complications of mathematical singularities that plague conventional continuum approaches. Although a discretized form of the peridynamic integral equations has been implemented in a meshless code called EMU, the objective of the present paper is to describe how the peridynamic model can also be implemented in a conventional finite element analysis (FEA) code using truss elements. Since FEA is arguably the most widely used tool for structural analysis, this implementation may hasten the verification of peridynamics and significantly broaden the range of problems that the practicing analyst might attempt. Also, the present work demonstrates that different subregions of a model can be solved with either the classical partial differential equations or the peridynamic equations in the same calculation thus combining the efficiency of FEA with the generality of peridynamics. Several example problems show the equivalency of the FEA and the meshless peridynamic approach as well as demonstrate the utility and robustness of the method for problems involving fracture, damage and penetration.     

Variation of the structure and permeability of tensioned fiber layers during impregnation with a thermoplastic polymer melt

The influence of displacements of tensioned fibers on the impregnation of fibrous layers with a polymer melt and on the final composite structure is studied. Using computer simulation, it is shown that, during impregnation, the structure of tensioned fibrous layers changes considerably depending on the initial arrangement and tensioning of fibers. The consolidated regions formed under the melt front move inside the impregnated layer with the advancing melt front. Displacement of the tensioned fibers as well as the formation of “washouts” favors the impregnation of internal layers, but cause significant inhomogeneity of the polymer structure. The surface (on the side of the melt flow) regions are more saturated with the polymer than the internal ones. A difference in the melt percolation mechanisms at various impregnation regimes is revealed. The effective permeability coefficients of a tensioned fiber layer are not constant but depend on the conditions and regimes of impregnation. Submitted to the 11th Conference on the Mechanics of Composite Materials (Riga, June 11–15, 2000). Translated from Mekhanika Kompozitnykh Materialov, Vol. 36, No. 2, pp. 259–270, March–April, 2000.     

The Cauchy problem for ut=Δu+|∇u|

With q a positive real number, the nonlinear partial differential equation in the title of the paper arises in the study of the growth of surfaces. In that context it is known as the generalized deterministic KPZ equation. The paper is concerned with the initial-value problem for the equation under the assumption that the initial-data function is bounded and continuous. Results on the existence, uniqueness, and regularity of solutions are obtained.     

含有批处理机的三机流水作业加工总长问题在某些情形下的强NP困难性

本文研究含有批处理机的三台机器流水作业加工总长问题在某些情形下的计算复杂性。在批处理机上同时加工的工件组成一个工件批，一个工件批的所有工件同时开始、同时结束。当批处理机的容量有限时，我们证明了下列情形为强NP困难的：第一台机器是批处理机、其余两台机器是单机；第二台机器是单机、其余两台机器是批处理机；第三台机器是批处理机、其余两台机器是单机。     

On a class of Riemann surfaces

It is considered the class of Riemann surfaces with dimT1 = 0, where T1 is a subclass of exact harmonic forms which is one of the factors in the orthogonal decomposition of the spaceΩH of harmonic forms of the surface, namely The surfaces in the class OHD and the class of planar surfaces satisfy dimT1 = 0. A.Pfluger posed the question whether there might exist other surfaces outside those two classes. Here it is shown that in the case of finite genus g, we should look for a surface S with dimT1 = 0 among the surfaces of the form Sg\K , where Sg is a closed surface of genus g and K a compact set of positive harmonic measure with perfect components and very irregular boundary.     

A model for predicting the shear bearing capacity of FRP-strengthened beams

The shear failure of reinforced concrete beams needs more attention than the bending failure since no or only small warning precedes the failure. For this reason, it is of utmost importance to understand the shear bearing capacity and also to be able to undertake significant rehabilitation work if necessary. In this paper, a design model for the shear strengthening of concrete beams by using fiber-reinforced polymers (FRP) is presented, and the limitations of the truss model analogy are highlighted. The fracture mechanics approach is used in analyzing the bond behavior between the FRP composites and concrete. The fracture energy of concrete and the axial rigidity of the FRP are considered to be the most important parameters. The effective strain in the FRP when the debonding occurs is determined. The limitations of the anchorage length over the cross section are analyzed. A simple iterative design method for the shear debonding is finally proposed. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 44, No. 3, pp. 357–372, May–June, 2008.     

Dynamical Spin Susceptibility in the t–J Model in the Superconducting Phase

Dynamical spin susceptibility is calculated for the t–J model in the superconducting phase using the memory function method in terms of the Hubbard operators. The self-consistent system of equations for the memory function is obtained within the mode-coupling approximation. Both itinerant hole excitations and localized spin fluctuations contribute to the memory function. Moreover, the itinerant contribution itself consists of two parts, i.e., the contribution of Bogoliubov quasiparticles and that of Cooper pairs. The spin dynamics is diffusive in the hydrodynamic limit, but the itinerant part does not contribute to the spin diffusion. In the high frequency region, spin–wave-like excitations continue to exist. We discuss our analytic results in the light of neutron scattering experiments performed on the cuprate superconductors.     

Gaussian Versus Optimal Integration of Analytic Functions

We consider error estimates for optimal and Gaussian quadrature formulas if the integrand is analytic and bounded in a certain complex region. First, a simple technique for the derivation of lower bounds for the optimal error constants is presented. This method is applied to Szeg?-type weight functions and ellipses as regions of analyticity. In this situation, the error constants for the Gaussian formulas are close to the obtained lower bounds, which proves the quality of the Gaussian formulas and also of the lower bounds. In the sequel, different regions of analyticity are investigated. It turns out that almost exclusively for ellipses, the Gaussian formulas are near-optimal. For classes of simply connected regions of analyticity, which are additionally symmetric to the real axis, the asymptotic of the worst ratio between the error constants of the Gaussian formulas and the optimal error constants is calculated. As a by-product, we prove explicit lower bounds for the Christoffel-function for the constant weight function and arguments outside the interval of integration. September 7, 1995. Date revised: October 25, 1996.     

基于VAR风险指标的投资组合模糊优化

在二目标有价证券选择基础上 ,引入目前流行的风险指标 VAR,以收益率与风险损失为目标 ,将模糊概念运用于有价证券组合选择 ,按投资者给定的期望目标及容差 ,讨论了 S型隶属函数模型 .通过 VAR的给定 ,将投资者所能承受的最大损失锁定 ,更好地反映出投资者对目标值的取值意图 .依据深圳股票市场9只股票收益率数据 ,采用进化规划进行优化计算 ,并验证模型的有效性 .