精彩无限，尽在智力快车

1．请问哪里的佛像最少？A．南半球B．北半球C．赤道.2．3个人3天用3桶水,9个人9天用几桶水？A．3桶B．9桶C．27桶.3．什么动物的牙齿最好？A．狼B．老虎C．老鼠.4．读完北京大学需要多久？A．一秒钟B．4年C．一辈子.5．关羽为什么比张飞死得早？A．身体虚弱B．奋战沙场C．红颜薄命。     

排球的发球问题

实际问题：在排球比赛中,地运动员站在后场发球．已知：排球的直径为0．21米（规定在20．69cm至21．327cm之间）,排球场总长为18米,网高2．5米,女子网高为2．24米;为计算方便不妨取2．5米）．运动员采用跳发球的方式以水平方向发球,为了使发出的球落在界内．     

同分布的NA序列加权和的强大数律

讨论了同分布NA随机变量序列加权和的强大数律,所得结果推广了Z．D．Bai和P．E．Cheng及S．H．Sung的结果．     

Dickson多项式与差集

1998年,Maschietti用超卵形线构造了几个循环差集．R．Evans,H．D．L．Holloman, C．Krattnthaler与Qing Xiang等给出了其对应的二元序列具有良好自相关函数的简单代数证明．在本文中,证明了超卵形线与二对一映射有着紧密的联系,并且推广了Maschietti的结果．     

学校体育的困局与破局——在天津市学校体育工作会议上的报告

一、缘起我校2009届高三有20个班级,笔者连续三年施教（20）班的数学．2009年高考,我班文科数学均分127．8,理科数学均分112．6（2009届学校实行高三“加一”学科走班制,数学在同一个班级文理并存）,均为全区第一,分别超校平均18．8和11．6分,超市平均33．1和19．9分;超市实验性示范性高中平均分12．7和7．8分．个中原因,值得回味．     

贝卡的历险日记

担心着．担心着．北风来了,冰雪的脚步近了。一切都像没睡醒的样子．昏昏然紧闭着喉。山冷冻起来了．水凉起来了太阳在地平线下躲起来了．海豹偷偷地从水里钻出来．傻傻的．胖胖的．海洋里．浮冰上．瞧去．一大片一大片满是的．坐着．漂着．打两个滚．踢几块．水．赛几趟泳．捉几回迷藏。     

到复Grassmann流形的多重调和映照的构造

本文给出了一些到复Ｇｒａｓｓｍａｎｎ流形的多重调和映照的构造定理,从而推广了莫小欢,Ｂｕｒｓｔａｌｌ　Ｆ． Ｅ．,Ｗｏｏｄ Ｊ． Ｃ．和 Ｕｄａｇａｗａ Ｓ．的结果．     

HERM ITE-FEJR TYPE INTERPOLATION OF HIGHER ORDER

§１ＬｅｔＸｖ＝｛ｘｎｋ,ｙｎμ｝ｂｅａｍａｔｒｉｘｏｆｎｏｄｅｓｓａｔｉｓｆｙｉｎｇ－１≤ｘｎ１＜ｘｎ２＜．．．＜ｘｎｎ≤１,　－１≤ｙｎ１＜ｙｎ２＜．．．＜ｙｎｌ≤１,ｘｎｋ≠ｙｎμ,ｋ＝１,．．．,ｎ,μ＝１,．．．,ｌ,ｎ≥１,ｌｖ＝ｌ（ｎ）≥０（１）　　Ｆｏｒｓｉｍｐｌｉｃｉｔｙｗｅｓｈａｌｌｏｆｔｅｎｏｍｉｔｔｈｅｓｕｐｅｒｆｌｕｏｕｓｎｏｔａｔｉｏｎｓ．ＤｅｆｉｎｅｂｙＰｎｔｈｅｓｅｔｏｆｐｏｌｙｎｏｍｉａｌｓｏｆｄｅｇｒｅｅ≤ｎ．Ｎｏｗｆｏｒａｎａｒｂｉｔｒａｒｙｆｉｘｅｄｉｎｔ…     

数学基础课是通向数学主流的门户

2007年．作者翻译T．Needham所写的《可视的复分析》（Visual Complex Analysis．Oxford Univ．Press．2006）一书．以为这是一本很好的教本．值得许多同志花一点时间读一下．译后有一些感想．因此写了一篇较长的译后记．主要思想即认为数学基础课是通向数学主流的门户．译本将由人民邮电出版社出版．现在征得出版社的同意,将这篇译后记发表于此．在发表时又作了一点补充．补写的文字放在方括号内．     

3-齐次群与旗传递5-（v，k，3）设计

旗传递t-设计的分类是代数组合学的一个重要课题．本文主要讨论了旗传递5-（v,k,3）设计．由P．J．Cameron和C．E．Praeger的结论可知,此时设计的自同构群是3-齐次群．本文利用3-齐次群的分类,证明了设计的自同构群不能是仿射型群．     

The generalized hierarchical product of graphs

A generalization of both the hierarchical product and the Cartesian product of graphs is introduced and some of its properties are studied. We call it the generalized hierarchical product. In fact, the obtained graphs turn out to be subgraphs of the Cartesian product of the corresponding factors. Thus, some well-known properties of this product, such as a good connectivity, reduced mean distance, radius and diameter, simple routing algorithms and some optimal communication protocols, are inherited by the generalized hierarchical product. Besides some of these properties, in this paper we study the spectrum, the existence of Hamiltonian cycles, the chromatic number and index, and the connectivity of the generalized hierarchical product.     

Rolling-ball method for estimating the boundary of the support of a point-process intensity

We suggest a generalisation of the convex-hull method, or ‘DEA’ approach, for estimating the boundary or frontier of the support of a point cloud. Figuratively, our method involves rolling a ball around the cloud, and using the equilibrium positions of the ball to define an estimator of the envelope of the point cloud. Constructively, we use these ideas to remove lines from a triangulation of the points, and thereby compute a generalised form of a convex hull. The radius of the ball acts as a smoothing parameter, with the convex-hull estimator being obtained by taking the radius to be infinite. Unlike the convex-hull approach, however, our method applies to quite general frontiers, which may be neither convex nor concave. It brings to these contexts the attractive features of the convex hull: simplicity of concept, rotation-invariance, and ready extension to higher dimensions. It admits bias corrections, which we describe and illustrate through implementation.     

Analysis of the Stress State of the Surface Layer of a Polymeric Mass upon Filling of Bulky Compression Molds

The stress state of the surface layer of a polymeric mass during filling of bulky compression molds is analyzed. It is shown that, at particular rheological characteristics of the mass, temperature, and filling rates, cracking of the surface layer occurs, which leads to defects in the finished products. A physical analysis of this process makes it possible to conclude that the cracks arise due to the normal stresses operating in the front region of the moving polymeric mass. It is found that, under certain flow conditions, areas with a pressure lower than the atmospheric one appear on the surface of the polymer. If the tensile stresses arising in these local regions are higher than the tensile strength of the mass, the continuity of the composition is broken in the direction determined by the greatest rate of the normal deformation. To confirm the reliability of the crack-formation mechanism proposed, the distribution of the pressure and normal stresses over the free surface is calculated based on a numerical method. These calculations show that, by comparing the stress level achieved in the front region with the tensile-strength characteristics of the polymeric composition, it is possible to predict, with a sufficient accuracy, the possibility of crack formation in the surface layer of such a mass under given flow conditions and thus to solve the question on flawless manufacturing of products.     

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.     

Development of the Tangent Modulus from the Euler-Problem to Nonsmooth Materials

History and development of the tangent modulus from the origins to the recent nonsmooth damaging versions are presented. Load history and stability analyses of structures of nonlinear reversible or irreversible materials are based on the concept of tangent modulus. Generally, instantaneously changing tangent modulus is needed and the solution yields iteration process. In the case of inelastic problems, the switch from loading to unloading of the material behaviour results in nonsmooth material functions. Nonsmooth, generally saw-tooth like behaviour happens in composite, laminated or rock type materials, or in the interaction of concrete and the reinforcement, too. Recently, damage and localization are in the focus of structural analyses, extending the tangent modulus to the negative cases, as well. Consequently, an overview of the history and development of the tangent modulus containing the recent modifications seems to be necessary. On the other hand, the more than a century long history of the tangent modulus is a marvellous study of the parallel development of mechanics and mathematics, by following the mutual inspiring effect of them through the activity of such pioneers like P.D. Panagiotopoulos in creating Nonsmooth Mechanics.     

The limiting state of a rigidly fixed nonlinearly elastic multilayer rod

The loss of the load-carrying capacity of a nonlinearly elastic multilayer rod is investigated. The rod, whose layers have various thickness and are made of different materials, is rigidly fixed at both its ends. Rigid contact conditions between the layers are assumed. The problem posed is solved by using the variational method of mixed type in combination with the Rayleigh-Ritz method. The initial analysis is reduced to the solution of the Cauchy problem for a nonlinear ordinary differential equation solved for the first derivative. As the initial condition, the maximum initial eccentricity of the rod is assumed. In the case of zero eccentricity, the Shanley critical force for an axially compressed rod is determined. For a three-layer rod whose outer layers have equal thickness and are made of the same material, numerically, for various degrees of nonlinearity, the effect of physicomechanical and geometric parameters on the critical load of buckling instability is determined. It is found that, by matching the heterogeneity of the rod, it is possible to raise its load-carrying capacity. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 42, No. 3, pp. 347–360, May–June, 2006.     

Dynamic properties of composite laminates and panels with highly contrasting directions of anisotropy

An asymptotic analysis of the dynamic stress-strain state of a thin laminar packet of anisotropic layers is presented. The statement is nonclassical, since in layer materials the ratio of elastic moduli in the longitudinal and transverse directions can generate small parameters comparable to the relative half-thickness of the packet, as, for example, in high-strength unidirectional composites. Alternation of strong load-carrying layers and a relatively soft filler with a similar difference in the elastic moduli between the layers is also allowed. The averaged two-dimensional equations and the total stress tensor in the layers are determined. The results are classified with respect to the types of anisotropy and the indices of differences in the elastic moduli. It is shown that first-approximation models lead to kinematic relations similar to those of the theories of high-order shear strains. Institute of Problems of Mechanics, Russian Academy of Sciences, Moscow, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 35, No. 5, pp. 605–614, September–October, 1999.     

Some curious properties and loci problems associated with cubics and other polynomials

The article begins with a well-known property regarding tangent lines to a cubic polynomial that has distinct, real zeros. We were then able to generalize this property to any polynomial with distinct, real zeros. We also considered a certain family of cubics with two fixed zeros and one variable zero, and explored the loci of centroids of triangles associated with the family. Some fascinating connections were observed between the original family of the cubics and the loci of the centroids of these triangles. For example, we were able to prove that the locus of the centroid of certain triangles associated with the family of cubics is another cubic whose zeros are in arithmetic progression. Motivated by this, in the last section of the article, we considered families of cubic polynomials whose zeros are in arithmetic progression, along with the loci of the special points of certain triangles arising from such families. Special points include the centroid, circumcentre, orthocentre, and nine-point centre of the triangles. Throughout the article, we used the computer algebra system, Mathematica®, to form conjectures and facilitate calculations. Mathematica® was also used to create various animations to explore and illustrate many of the results.     

Dispersed failure of an anisotropic flywheel

Based on a hereditary damage model of solids, the strength of a cylindrically anisotropic flywheel is calculated. By using a failure criterion, the location and time of initial failure is determined in relation to an anisotropy parameter, for which the ratio of rigidities in the tangential and radial directions is taken. The process of dispersed failure depends on the expansion intensity of the damaged zone. The boundary of the zone is the failure front, whose equations of motion are obtained in the cases of absence and presence of a residual strength for the material behind the failure front. In the second case, the damaged material is modeled by an isotropic elastic medium with considerably reduced values of strength and rigidity characteristics, and variations in the pressure on the failure front are also determined. Graphs of the radial coordinate of failure front as a function of time are constructed and analyzed for different values of the anisotropy parameter, the degree of residual strength and density behind the failure front, and proportions of geometrical sizes of the flywheel. The critical failure times are found. A system of restrictions on the values of mechanical and geometrical parameters is revealed which makes possible the realization of the process of dispersed failure investigated.Translated from Mekhanika Kompozitnykh Materialov, Vol. 41, No. 1, pp. 93–108, January–February, 2005.