球面域上高阶拉普拉斯的特征值估计

本文研究了球面域上高阶拉普拉斯的特征值问题. 利用Rayleigh-Ritz不等式, 获得了球面域上高阶拉普拉斯的第(k+1)个特征值的上界估计, 这个估计式由前k个特征值给出.     

关于广义Volterra拉普拉斯与傅立叶变换的关系

本是献[1]工作的继续，研究了广义Volterra拉普拉斯的性质及其与傅立叶变换的关系并进行了研讨．     

神奇的常数——e

法国著名数学家拉普拉斯说：“阅读欧拉,读懂欧拉,他是我们所有人的大师．”其实,拉普拉斯的名言可以改为：“学习e,研究e,它是数中之大师．”正如高斯在他同时代人中赢得了“数学王子”的称号,e也可冠以“常数王子”的称号．当然,欧拉是会同意的,要不然他怎么会用自己名字（Euler）的第一个字母表示呢？其实,e最初被表达为下式的极限     

一维p-拉普拉斯四点边值问题拟对称正解的多重性

应用凸锥上的一个不动点定理,讨论了一类一维p-拉普拉斯四点边值问题在非线性项f依赖于未知函数的一阶导数的情况下拟对称正解的多重性,得到了这类边值问题存在多个拟对称正解的充分条件.     

对"新课标"中概率与统计内容及要求的认识

在自然界与人类的社会活动中会出现各种各样的现象，既有确定性现象，又有随机现象．随机现象在日常生活中随处可见，概率是研究随机现象规律的学科，它为人们认识客观世界提供了重要的思维模式和解决问题的方法．拉普拉斯说：“生活中最重要的问题，其中绝大多数在实质上只是概率的问题．”由此可见概率的重要性。     

初中概率教学“鸡肋说”的根源分析——兼谈一个教学案例的启示

19世纪法国著名数学家拉普拉斯说:"生活中最重要的问题,其中绝大多数在实质上只是概率的问题".然而现实却是初中概率教学一直备受冷落,有的教师甚至说这部分内容如同"鸡肋",弃之可惜,食之无味.这种状况是怎样造成的?作为教师的我们又该如何改变概率教学无奈的现状?本文就此与大家一起探讨.     

图的拟拉普拉斯谱(英文)

设Ｇ是一简单无向图,Ｃ（Ｇ）表示 Ｇ的无向关联矩阵,Ｑ（Ｇ）＝Ｃ（Ｇ）Ｃ（Ｇ）～Ｔ． Ｑ（Ｇ）的特征值称为图Ｇ的拟拉普拉斯谱．在这篇文章,我们研究图的拟拉普拉斯谱,表明Ｇ＋ｅ,Ｌ（Ｇ）和Ｇ_１ＶＧ_２拟拉普拉斯的谱．     

开心农场趣事多

趁着暑假,爸爸带我和妈妈去奶奶家体验了一下真实的开心农场.奶奶见我们到了,二话不说,在门前的自制大棚里拉了两根细绳,把一块菜地分成了四小块(如下图). 奶奶乐呵呵地对我们说:“最大的这块就我来吧,你们自由选择其他的!”我迫不及待地选了面积最小的2号地,爸爸选了1号地,妈妈选择了4号.     

剩余格上的n-维模糊滤子

在剩余格上提出n-维模糊滤子,n-维模糊EIMTL-滤子和n-维模糊t-滤子概念并研究它们的一些性质。给出n-维模糊滤子,n-维模糊EIMTL-滤子和n-维模糊t-滤子的一些等价刻画。证明了n-维模糊集f是n-维模糊滤子(n-维模糊EIMTL-滤子,n-维模糊t-滤子)当且仅当对任意的λ∈[0,1],fλ是1-维模糊滤子(1-维模糊EIMTL-滤子,1-维模糊t-滤子)。此外,得到了剩余格上n-维模糊Divisible滤子和n-维模糊Strong滤子的等价刻画。     

填充维数的一个等价定义

本文研究了填充维数与上盒维数的关系.利用Cantor-Bendixson定理的方法,得到了由上盒维数给出的填充维数的等价定义.并证明了齐次Moran集对上盒维数和填充维数的连续性.     

Dynamically equilibrium shapes and directions of motion of ocean current rings

The dynamically equilibrium shapes of a uniform-density rotating mass of liquid (a ring) in the surface layer of a quiescent stratified ocean are determined. The examination is carried out in a plane tangential to the Earth, taking into account the vertical and horizontal projections of the angular velocity of its rotation. Exact solutions of the equations of motion of an ideal incompressibe fluid are obtained, making it possible, for a linearly stratified ocean, to determine the dynamic all equilibrium shape of the interfaces of water masses and the free boundaries of cyclonic and antocyclonic rings. These shapes comprise second-order surfaces inclined to the water level in the meridian plane, the type of surfaces depending on the governing parameters of the problem. Expressions are obtained for the angles of inclination of the principal axes. For small deviations from equilibrium, due to a difference in the gravitational forces and Archimedes forces, motion of the ring occurs, governed by the inclination of the principal axes and the nature of change (increase or reduction) in the average density of the ring, determined by the ratio of the rates of diffusion of heat and salt. The displacement along the parallel comprises geostrophic motion, for the velocity of which an analytical expression is obtained. The displacement along the meridian comprises motion over an inclined plane. An analytical expression is given that relates the change in the depth of the centre of mass of the ring to the velocity of motion along the meridian through the angle of inclination of the principal axes of the ring. This explains the motion of both types of Gulf Stream ring to the south-west and of the Oyasio ring to the north-east.     

The stability of the equilibrium of two-phase elastic solids

The distinctive features of the loss of stability of elastic solids which undergo phase transitions are investigated for the case of small deformations. The non-uniqueness of the solution of the boundary-value problem for the describing of the thermodynamic equilibrium of a two-phase body is caused by the non-linearity associated with the unknown interface. The solution can be chosen by comparing the potential energies of the body in the two-phase and single phase states and by analysing of the local stability of the two-phase states. A linearized boundary-value problem is formulated which describes infinitesimal small perturbations of an initial two-phase state which is in thermodynamic equilibrium. Analysis of the stability of the two-phase state reduces to an investigation of the bifurcation points and the behaviour of the small solutions of the system of integrodifferential equations in terms of functions describing the perturbations of the interface. The problem of the non-uniqueness and loss of stability of centrisymmetric equilibrium two-phase deformations is investigated as an example. A theorem concerning the number of centrisymmetric solutions is proved. The energy changes accompanying the formation and development of two-phase states and the stability of the solutions obtained are investigated. The concept of topological instability as a bifurcation is introduced, as a result of which the type of geometry of a solution of the boundary-value problem changes and surfaces of separation of the phases actually appear and disappear. Macrodiagrams of the deformational are constructed which demonstrate the effect of deformation softening in the path of a phase transition.     

Inspection and maintenance optimization methods

Three methods of the optimal planning of the inspection and maintenance of offshore structures are described. The models are based on respectively: the maximization of the effect of inspections, measured by the total importance value of the errors detected, subject to a given total economical budget; the minimization of the total costs of obtaining respectively: a given importance value of errors detected or given numbers of inspections of various types. Special selections of the importance values of structural elements give problems of the maximization of the reliability of the structural system, or the minimization of the economical consequences of failures, or the minimization of the sum of the costs of inspections and failure-consequences, subject to a given total failure probability of the system.Different failure types of elements and time schedules of inspections can be included in the model.An extension of the incremental method of Fox is applied, and an evaluation measure is given for the calculation of bounds of the optimal objective value, or given numbers of inspections are planned by application of continuous linear programming with integral solutions.     

Localization of microfailure in fibrous composites

Conclusion When a fibrous composite is loaded, the process of microfailures becomes localized in consequence of the nonuiformity of internal stresses. The degree of localization can be quantitatively characterized by the magnitude of the parameter of localization whose determination was provided in the present work. The dependence of the parameter of localization on the stress applied to the specimen can be measured experimentally from the data on the location of the coordinates of the sources of AE, and it can be calculated theoretically on the basis of the model of failure of the composite. A comparison of the theoretical model with the experimental data makes it possible to determine the magnitude of the overstresses in the fibers of the composite material and the form of the distribution function of these overstresses.Translated from Mekhanika Kompozitnykh Materialov, No. 3, pp. 437–443, May–June, 1989.     

A study of the causes of the reduced strength of glass-reinforced phenolformaldehyde plastics

A study of the curing kinetics of phenolformaldehyde resin in the presence of glass and quartz has shown that one of the chief causes of the reduced strength of glass-reinforced plastics based on phenol-formal-dehyde resin is the difference in the rate and degree of cure in layers close to the fibers and in the bulk of the resin. This is caused by the presence on the surface of the fibers of a hydrate sheath with increased concentration of hydroxyl ions and by the presence of hydrogen bonds between the oxyphenyl groups of the resin and the silanol groups on the surface of the fibers. Chemical treatment of the glass fibers has the effect of diminishing those factors responsible for the reduced rate and degree of cure, and in spite of the lower surface energy of the fibers, the strength of the glass-reinforced plastic increases.Mekhanika Polimerov, Vol. 1, No. 3. pp. 8–14, 1965     

Problems of geometric non-linearity and stability in the mechanics of thin shells and rectilinear columns

An analysis of the current state of the geometrically non-linear theory of elasticity and of thin shells is presented in the case of small deformations but large displacements and rotations, the ratios of which are known as the ratios of the non-linear theory in the quadratic approximation. It is shown that they required specific revision and correction by virtue of the fact that, when they are used in the solution of problems, spurious bifurcation points appear. In view of this, consistent geometrically non-linear equations of the theory of thin shells of the Timoshenko type are constructed in the quadratic approximation which enable one to investigate in a correct formulation both flexural as well as previously unknown non-classical forms of loss of stability (FLS) of thin plates and shells, many of which are encountered in practice, primarily in structures made of composite materials with a low shear stiffness. In the case of rectilinear elastic whereas, which are subjected to the action of conservative external forces and are made of an orthotropic material, the three-dimensional equations of the theory of elasticity are reduced to one-dimensional equations by using the Timoshenko model. Two versions of the latter equations are derived. The first of these corresponds to the use of the consistent version of the three-dimensional, geometrically non-linear relations in an incomplete quadratic approximation and the Timoshenko model without taking account of the transverse stretching deformations, and the second corresponds to the use of the three- dimensional relations in the complete quadratic approximation and the Timoshenko model taking account of the transverse stretching deformations. A series of new non-classical problems of the stability of columns is formulated and their analytical solutions are found using the equations which have been derived with the aim of analyzing their richness of content. Among these are problems concerning the torsional, flexural and shear FLS of a column in the case of a longitudinal axial, bilateral transverse and trilateral compression, a flexural-torsional FLS in the case of pure bending and axial compression together with pure bending and, also, a flexural FLS of a column in the case of torsion and a flexural-torsional FLS under conditions of pure shear. Five FLS of a cylindrical shell under torsion are investigated using the linearized neutral equilibrium equations which have been constructed: 1) a torsional FLS where the solution corresponding to it has a zero variability of the functions in the peripheral direction, 2) a purely beam bending FLS that is possible in the case of long shells and is accompanied by the formation of a single half-wave along the length of the shell while preserving the initial circular form of the cross-section, 3) a classical bending FLS, which is accompanied by the formation of a small number of half-waves in the axial direction and a large number of half-waves in a peripheral direction which is true in the case of long shells, 4) a classical bending FLS which holds in the case of short and medium length shells (the third and fourth types of FLS have already been thoroughly studied in the case of isotropic cylindrical shells), 5) a non-classical FLS characterized by the formation of a large number of shallow depressions in the axial as well as in the peripheral directions; the critical value of the torsional moment corresponding to this FLS is practically independent of the relative thickness of the shell. It is established that the well-known equations of the geometrically non-linear theory of shells, which were formulated for the case of the mean flexure of a shell, do not enable one to reveal the first, second and fifth non-classical FLS.     

The existence of smooth solutions in a problem of the optimal control of the rotation of an axisymmetric rigid body

The problem of the existence of a solution in the problem of the optimal control of the rotation of an axisymmetric rigid body for the arbitrary case of angular velocity boundary conditions is studied. A square integrable functional, which is consistent with the symmetry of the rotating body and characterizes the power consumption, is chosen as the criterion. The principal moment of the applied external forces serves as the control and the time of termination of a manoeuvre can be both specified as well as free. In the case of a specified termination time, it is shown that the solution (control) belongs to the class of infinitely-differentiable functions of time. The reasoning is based on the use of the singularities of the structure of the differential equations and the possibility of reducing the initial problem to two successive variational problems. The existence of a solution of the first of these problems in the class of square integrable functions is proved using the Cauchy–Bunyakovskii inequality. The second problem reduces to a search for the minimum of a functional which is weakly lower semi-continuous on a weakly compact set and the existence of its solution in the same class of functions follows from the Weierstrass theorem. The required conclusion concerning the smoothness of the solution of the optimal control problem is obtained from the necessary conditions of Pontryagin's maximum principle. In the case of a free termination time, one of the minimizing sequence can be constructed and it can be shown that, in the general case, there is no solution in the class of measurable controls.     

Some qualitative effects in the exact solutions of the Lamé problem for large deformations

Qualitative effects in the solution of a number of radially symmetric and plane axisymmetric problems for bodies made of non-linearly elastic incompressible materials are analysed for large deformations. In the case of problems of the axisymmetric plane deformation of cylindrical bodies, the lack of uniqueness of the solution for a given follower load in the case of a Bartenev–Khazanovich material and the existence of a limiting load in the case of a Treloar (neo-Hookian) material have been studied in detail and the dependences of the limiting load on the ratio of the external and internal radii of a hollow cylinder in the undeformed state have been presented. A similar study has been carried out for constitutive relations of a special form that well describe the properties of rubber. For this material, the lack of uniqueness of the solution is revealed for fairly high loads. The axisymmetric problem of the plane stress state of a circular ring made of a Bartenev–Khazanovich material has been solved and a lack of uniqueness of the solution for a given follower load was discovered in the case when the dimensions of the ring are given in the undeformed state. Similar studies have been carried out for Chernykh and Treloar materials in the case of the problem of the radially symmetric deformation of a spherical shell. It was established that, in the case of a Chernykh material, the lack of uniqueness of the solution depends considerably on the constant characterizing the physical non-linearity. The limit case of the deformation of a spherical cavity in an infinitely extended body has been investigated. The effect of an unbounded increase in the boundary stresses is observed for finite external loads, that appears in the case of the problem of the plane axisymmetric deformation of a cylindrical cavity in an infinitely extended body made of a Bartenev–Khazanovich material and in the case of the problem of the radially symmetric deformation of an infinitely extended body made of a Chernykh material with a spherical cavity.