半圆区域的包含测度

本文研究了半圆区域内定长线段的包含测度问题.利用广义支撑函数和限弦函数,得到了这类区域的包含测度.关于非多边形凸域的包含测度的现有结果不多,本文的结果丰富了凸体包含测度的具体结果.     

截面体与凸体的包含测度

本文建立了E^n中的凸体与其位似体的包含测度的相等关系，同时给出了凸体Minkowski和的包含测度的估计。最后证明了E^3中的关于原点对称的旋转椭球与其截面体相似的充要条件是K为以原点为心的球。     

凸域的外平行集的包含测度

本文研究了凸域内定长线段的包含测度问题,利用广义支持函数和限弦函数,得到了一类特殊凸域的包含测度.     

欧氏空间R3中的几个Bonnesen型不等式

本文研究欧氏空间R3中关于一个凸体的Bonnesen型不等式.利用包含测度的方法,获得了几个Bonnesen型不等式.     

多凸域型网格的Buffon问题

本文研究了以多个凸域的并为基本区域的网格的Buffon问题．利用包含测度的有关结论，获得了Buffon问题的普遍公式，并作为举例具体计算了一种三个凸域的情况的表达式和特殊化的数值．     

凸域内矩形的运动测度

本文研究了凸域内矩形的运动测度,通过对凸域内定长线段运动测度的推广,建立了包含在凸域内且长、宽都确定的矩形运动测度的一般公式,利用此公式得到了圆域和矩形域内此类矩形的运动测度,并以此为基础得到了推广后的Buffon投针问题的一些结果.     

ε-迷向测度的刻画及应用

球面上的迷向测度刻画了凸体与其John椭球的关系,并在凸体的极值问题研究中起着核心作用.本文在非典范的欧式内积的结构下对迷向测度进行了新的刻画并给出一个测度是ε-迷向测度的充分必要条件,并用ε-迷向测度来刻画John定理.     

R4中的Willmore泛函与包含问题

设∑为Euclid空间R4中的凸超曲面,其中曲率为H,我们得到了Willmore泛函∫∑H2dσ的一个几何下界估计．这个下界是一个涉及∑的面积、∑所界的凸体K的体积、以及K的．Minkowski均值积分的不变量．还得到了Euclid空间R4中一凸体包含另一凸体的充分条件．     

凸体的锥体积测度的一些注记

锥体积测度在凸几何分析中扮演着重要的角色.给出了"正交补变换"的定义,并且证明了关于凸体的锥体积测度的两个命题等价.最后,给出了相关的性质及应用.     

基于凸函数的风险测度

利用凸函数构造了一种新的风险测度,发现它是包含了损失概率、损失期望值、绝对离差、绝对半离差,下偏矩、(α,t)模型、ES等常见方法的更为广泛的风险测度.对其性质的研究发现该风险测度满足凸性和协调性,考虑到凸性以及协调性在投资组合以及风险管理中的重要意义,因此对该风险测度的研究就具有一定的实践和学术价值.     

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.     

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.     

播挪运算及其性质

在现有的播挪定义基础上,引入了对偶播挪的概念和任意论域子集族的内并、内交两种新运算,提出了十个定理.它们是：平凡播挪空间定理,最小播挪定理,播挪内运算封闭性定理,播挪任意交定理,播挪上确界定理,播挪运算分配性定理,播挪基判定定理,播挪基闭包定理,播挪基约简定理,播挪基内运算封闭性定理.这些定理对于播挪空间的研究具有一定的理论价值.     

中国科大数学系是怎样办学的

中国科学技术大学原副校长、数学系教授龚昇先生2011年1月10日不幸病逝.本刊编委会、编辑部于2011年1月第1期发表《沉痛悼念著名数学家龚？教授》一文,简短介绍了他的生平和学术成就.本期特转载他为《中国科学技术大学数学五十年》一书所作的序,以纪念这位杰出的数学家和数学教育家.本文标题为编者所加.1958年,经中共中央...     

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.     

Price and incomes policy: a ‘Sraffian’ perspective

A Sraffa-type model of price determination is proposed that will serve as a characterization of the consequences of some of the activities of a prices board that has been charged with attempting to control the rate of increase of the absolute level of commodity prices by controlling rates of increase of factor incomes (wages and profits). Specifically, the board chooses new elements for the diagonal matrix of rates of profit and for the vector of labor incomes of a set of (Sraffa-type) equations of production that have been cast into matrix form. The unknowns of these equations are the commodity prices, and with a new set of factor incomes, a new solution set of commodity prices will be generated. The problem of fulfilment of a strict positivity condition for the price solution and that of ensuring the viability of the input-output structure are examined. A possible solution to these problems derives from exclusive concentration by the board on the ‘basic’ (in Sraffa's sense) sector of the economy. A switch from attempting to control rates of increase of factor incomes to attempting to control, directly, the rate of increase in the absolute level of commodity prices is also considered.     

饱和粘土非线性渗流规律与径向固结

以多孔介质等效渗流概念得出粘土平均孔隙微尺度数量级范围在0.01 μm至0.1μm,与测试结果一致．实验结果表明饱和粘土微尺度孔隙渗流为非线性流．理论推导表明固液界面作用与渗透率平方根或孔隙半径成反比,固液界面相互作用是导致饱和粘土非线性渗流的重要原因．提出了精确描述饱和粘土微尺度孔隙非线性渗流基本规律的数学模型,其参数量纲明确,物理意义清楚．建立了饱和粘土非线性渗流新定律,统一描述了从较低到较高水力梯度全过程渗流曲线特征,达西定律是其特例．基于新定律,建立了饱和粘土非线性渗流定流量径向固结数学模型．以粘性边界层思想与稳态依序替换法及积分方程法,导出了粘土非线性渗流平均质量守恒方程及活动边界运动方程,给出了饱和粘土非线性渗流超孔隙水压力分布公式与平均固结度计算公式,获得了粘土层压力分布规律和平均固结度随时间变化规律．结果表明:饱和粘土非线性渗流使活动边界运动速度减小．研究结果为粘土地质工程与岩土工程应用提供了新的科学依据．达西渗流径向固结计算是新的非线性渗流固结计算的特例．     

On Discrete Limits of Sequences of Approximately Continuous Functions and tae-Continuous Functions

We show that the classes of all discrete limits of sequences of ap- proximately continuous functions, of all discrete limits of sequences of derivatives and of all discrete limits of sequences of Baire 1 functions are the same. We describe also the discrete limits of sequences of quasicontinuous functions, and of sequences of almost everywhere continuous functions, and we present anec- essary condition which must be satisfied by the discrete limits of sequences of T_ae -continuous functions.     

Contact interaction of cylindrical shells of different lengths

We solve a problem of determination of the contact rigidity of a structural joint of a steel thin-walled pipe under the action of internal pressure with a shroud made of a composite material using a special technology. A mathematical model of the contact interaction of the pipe and shroud modeled by cylindrical shells of different length was constructed using the classical Kirchhoff–Love theory of shells. We obtain an analytic solution of the contact problem under conditions of ideal contact of the elements of the structural joint by the conjugation method. A numerical analysis of the influence of geometric and physicomechanical characteristics of the shroud on the contact pressure and rigidity of the shrouded pipe is presented.     

Good lattice points,discrepancy, and numerical integration

Summary Methods based on Diophantine approximations lead to a simple and easy construction of rational vectors the multiple of which, reduced modulo 1, form finite sequences of points with certain properties of equipartition over the unit square. Given a function of bounded variation over this square, it is suggested that computing the average of its values at the points of such a sequence can be a practical method of numerical integration. Precise bounds for the error ore obtained. In the general case, these are of the order of the product of the reciprocal and of the logarithm of the number of points; in the case of a function satisfying stated conditions of regularity and periodicity, they are of the order of the logarithm of the number of points divided by an appropriately high power of this number. A slight sharpening of some well-known results on equipartiton is obtained incidentally. A part of this paper was read at a meeting of the Accademia Nazionale di Scienze Lettere e Arti in Modena on April 7th. 1965.