ZFC的模型的上限序数的一个性质

S1 引言 Forcing方法假设存在ZFC的一个可数可传的模型M。记满足αM的最小序数α为,显然M中一切序数所成的集合即。由于M是ZFC的模型,故应具有某些性质。本文证明了它满足关系,故为ε数或1级关键数,进而证明了是H级关键数(H为任意自然数)。文中的记号等引用。     

关于球的表面积的教学的一些尝试

(一)导言球的表面积公式是旋转体一章的重点内容,从教材的科学性看是没有问题的——在给出预备定理的基础上,再建立球面积的计算公式。但跳出传授知识之外,在教学中我们总有点感到不满足。因为按照现行教材的体系。我们无法回答这样几个问题: 已经有一整套圆柱、圆锥、圆台的侧面积公式,为什么还需要这个统一公式? 这个预备定理起什么作用?人们又是怎样想到引入这个定理的? 讲授预备定理是不是仅仅为了学习球面积公式提供敲门砖,还是也要让学生获得别的什么? 针对这些问题,我们的试验小组对此作了初步的探讨。现将我们讨论的意见和处理方案扼要介绍如下。不当之处请批评指正。     

图的邻域复形的同调群的不变性

本文研究了图的邻域复形同调群的不变性质。设G是一个简单连通图,x是G的一个顶点,以G/x表示G中剔去点v及其关联边而得到的图,给出了G和G/x的邻域复形的同阶同调群同构的充要条件。     

扰动的Couette－Poiseuille流的特征值的计算

本文考虑的问题是二维粘性渠流。对0到2000之间的雷诺数,计算了平稳扰动的Couette-Poiseuille流的下游特征值,其特征方程类似于Orr-Sommerfeld方程。所用的方法是谱方法和初值方法(复合矩阵方法).就几种有趣的流量,给出了相应的特征值的计算结果。这些特征值确定了扰动的衰减率。     

中学函数的极限的教学

十年制中学教材中关于函数的极限这一单元的内容写得很简单,要求也不很高,对函数的极限这个概念,只作描述性的定义。函数的连续性也只简单地提一下。但是要使学生对这部分的内容有个比较明确的了解,却是教学上的一个难题。讲浅了,不明不白;讲深了,又超出教材要求。下面仅就函数的极限这个概念的教学,谈谈我的一点意见。     

图的树宽的分解定理

图的树宽问题是名的ＮＰ－困难问题。其分解原则在确定树宽的一般算法和特殊算法中有重要应用。本给出这方面的若干定理。     

具有数据丢失的网络化控制系统的基于观测器的镇定

研究具有数据丢失的网络化控制系统的基于观测器的镇定问题。受控对象是由T-S模糊模型描述的非线性时滞系统。把数据丢失过程建模成独立同分布过程。利用模糊Lyapunov函数方法得到了闭环系统随机稳定的充分条件,其基于观测器的反馈控制器可通过求解一组线性矩阵不等式来设计。一个仿真例子说明方法的有效性。     

三角函数的定义域的教学

現行高中平面三角課本在§7同角的三角函数間的关系的注中写着“上面的关系式都是对于使它的两边具有意义的那些角而說的,以后遇到的关系式也是这样”。这是一段非常重要的話。学生如果对这段話沒有充分的注意和深刻的理解,在以后对待三角函数的恆等变換时就会不注意自变量的允許值的扩大与縮小;运用三角公式时往往不注意公式的适用范围;解三角方程时不能根据函数定义域的扩大和縮小鉴别增根和收回遺根。应該肯定:三角函数的定义域的教学不仅是理解三角方程增、減根的基础,更是加强函数观念所不可缺少的課題。为了加强三角函数定义域的教学,笔者对現行高中平面三角課本作了某些修改,进行了試驗,現将試教情况介紹如下,請指正。     

初等函数的值域的求法

中学数学研究的函数一般只限于初等函数。下面,我们从中学数学教学实际出发,探讨求初等函数的值域的一些方法。一、最大最小值法及其推广本文所说的函数都是指初等函数。为简便计,以后函数у=f(x)的定义域用X表示,值域用Y`示,实数集用R表示。这里月是全集。     

路的补图的色唯一性

设Pn表示n阶的路。[2]中刘猜测：如果n是偶数且n≠4,则/Pn色唯一的。本得到/Pn色唯一的充要条件,从而肯定的回作了刘提出的猜测。     

Physicomechanical characteristics of a disperse-filled polytetrafluoroethylene

Composite polymer materials based on PTFE containing ultradisperse -sialon (1–10 wt.%) as the cross-linking agent are studied. It is shown that the injection of small amounts of fillers (1–2 wt.%) increases the degree of composite crystallinity. A correlation between the structure, element distribution on the surface layers of samples, and tribotechnical characteristics of composites is found. The change in the physicomechanical characteristics is associated with the effect of the interstructural plasticization.Ammosov Yakutsk State University, Yakutsk, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 35, No. 6, pp. 797–806, November–December, 1999.     

A REPRESENTATION FORMULA RELATED TO SCHR(O)DINGER OPERATORS

Schr(o)dinger operator is a central subject in the mathematical study of quantum mechanics.Consider the Schrodinger operator H = -△ V on R, where △ = d2/dx2 and the potential function V is real valued. In Fourier analysis, it is well-known that a square integrable function admits an expansion with exponentials as eigenfunctions of -△. A natural conjecture is that an L2 function admits a similar expansion in terms of "eigenfunctions" of H, a perturbation of the Laplacian (see [7], Ch. Ⅺ and the notes), under certain condition on V.     

Staircase transportation problems with superadditive rewards and cumulative capacities

A cumulative-capacitated transportation problem is studied. The supply nodes and demand nodes are each chains. Shipments from a supply node to a demand node are possible only if the pair lies in a sublattice, or equivalently, in a staircase disjoint union of rectangles, of the product of the two chains. There are (lattice) superadditive upper bounds on the cumulative flows in all leading subrectangles of each rectangle. It is shown that there is a greatest cumulative flow formed by the natural generalization of the South-West Corner Rule that respects cumulative-flow capacities; it has maximum reward when the rewards are (lattice) superadditive; it is integer if the supplies, demands and capacities are integer; and it can be calculated myopically in linear time. The result is specialized to earlier work of Hoeffding (1940), Fréchet (1951), Lorentz (1953), Hoffman (1963) and Barnes and Hoffman (1985). Applications are given to extreme constrained bivariate distributions, optimal distribution with limited one-way product substitution and, generalizing results of Derman and Klein (1958), optimal sales with age-dependent rewards and capacities.To our friend, Philip Wolfe, with admiration and affection, on the occasion of his 65th birthday.Research was supported respectively by the IBM T.J. Watson and IBM Almaden Research Centers and is a minor revision of the IBM Research Report [6].     

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.     

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.     

Coordinate and subspace optimization methods for linear least squares with non-quadratic regularization

This work addresses the problem of regularized linear least squares (RLS) with non-quadratic separable regularization. Despite being frequently deployed in many applications, the RLS problem is often hard to solve using standard iterative methods. In a recent work [M. Elad, Why simple shrinkage is still relevant for redundant representations? IEEE Trans. Inform. Theory 52 (12) (2006) 5559–5569], a new iterative method called parallel coordinate descent (PCD) was devised. We provide herein a convergence analysis of the PCD algorithm, and also introduce a form of the regularization function, which permits analytical solution to the coordinate optimization. Several other recent works [I. Daubechies, M. Defrise, C. De-Mol, An iterative thresholding algorithm for linear inverse problems with a sparsity constraint, Comm. Pure Appl. Math. LVII (2004) 1413–1457; M.A. Figueiredo, R.D. Nowak, An EM algorithm for wavelet-based image restoration, IEEE Trans. Image Process. 12 (8) (2003) 906–916; M.A. Figueiredo, R.D. Nowak, A bound optimization approach to wavelet-based image deconvolution, in: IEEE International Conference on Image Processing, 2005], which considered the deblurring problem in a Bayesian methodology, also obtained element-wise optimization algorithms. We show that the last three methods are essentially equivalent, and the unified method is termed separable surrogate functionals (SSF). We also provide a convergence analysis for SSF. To further accelerate PCD and SSF, we merge them into a recently developed sequential subspace optimization technique (SESOP), with almost no additional complexity. A thorough numerical comparison of the denoising application is presented, using the basis pursuit denoising (BPDN) objective function, which leads all of the above algorithms to an iterated shrinkage format. Both with synthetic data and with real images, the advantage of the combined PCD-SESOP method is demonstrated.     

Superapproximation and Commutator Properties of Discrete Orthogonal Projections for Continuous Splines

This paper builds upon the L_p-stability results for discrete orthogonal projections on the spaces S_h of continuous splines of order r obtained by R. D. Grigorieff and I. H. Sloan in (1998, Bull. Austral. Math. Soc.58, 307–332). Properties of such projections were proved with a minimum of assumptions on the mesh and on the quadrature rule defining the discrete inner product. The present results, which include superapproximation and commutator properties, are similar to those derived by I. H. Sloan and W. Wendland (1999, J. Approx. Theory97, 254–281) for smoothest splines on uniform meshes. They are expected to have applications (as in I. H. Sloan and W. Wendland, Numer. Math. (1999, 83, 497–533)) to qualocation methods for non-constant-coefficient boundary integral equations, as well as to the wide range of other numerical methods in which quadrature is used to evaluate L₂-inner products. As a first application, we consider the most basic variable-coefficient boundary integral equation, in which the constant-coefficient operator is the identity. The results are also extended to the case of periodic boundary conditions, in order to allow appplication to boundary integral equations on closed curves.     

Effect of the adhesion interaction and the degree of filling on the internal stresses in finely divided crystalline fillers of epoxy composites

Two kinds of model composite materials with finely divided (1) crystal fillers—LiF or polyethylene-filled epoxy resin cured by polyethylenepolyamine — are investigated by X-ray diffractometry. It is found that tensile stresses arise in LiF crystals, which show a strong adhesion interaction with the binder, for all degrees of filling (from =2.2 to =74 vol.%) examined. Their values remain constant up to a degree of filling at which the boundary layers come into contact with one another. Then, the inner stresses decrease with increasing . In the crystalline regions of polyethylene, where the adhesion between the binder and crystals is weak, the inner stresses are compressive. It is shown that the thickness of the boundary layer of the matrix on the surface of filler particles can be evaluated by the method used.Institute of Polymer Mechanics, Latvian University, Riga, LV-1006 Latvia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 35, No. 6, pp. 807–820, November–December, 1999.     

Rigid isotopy classification of real algebraic curves of bidegree (3,3) on quadrics

A rigid isotopy of nonsingular real algebraic curves on a quadric is a path in the space of such curves of a given bidegree. We obtain the rigid isotopy classification of nonsingular real algebraic curves of bidegree (3, 3) on a hyperboloid and on an ellipsoid. We also study of the space of real algebraic curves of bidegree (3, 3) with a single node or cusp. Translated fromMatematicheskie Zametki, Vol. 66, No. 6, pp. 810–815, December, 1999.     

Embeddings of the classesH ω

In this paper we study functions belonging to the classesV _ε and ΛBV, which are encountered in the theory of Fourier trigonometric series. Necessary and sufficient conditions for the embedding of the classesH _ω in the classesV _ϕ and ABV are obtained. Translated fromMatematicheskie Zametki, Vol. 64, No. 5, pp. 713–719, November, 1998. This research was supported by the program “Leading Scientific Schools” under grant No. 96/97-15-96073.