Reducing Exhausters

The notions of upper and lower exhausters were introduced by Demyanov (Optimization 45:13–29, 1999). Upper and lower exhausters can be employed to study a very wide range of positively homogeneous functions, for example, various directional derivatives of nonsmooth functions. Exhausters are not uniquely defined; hence, the problem of minimality arises naturally. This paper describes some techniques for reducing exhausters, both in size and amount of sets. We define also a modified convertor which provides much more flexibility in converting upper exhausters to lower ones and vice versa, and allows us to obtain much smaller sets.     

Quasidifferentials and maximal normal operators

Quasidifferentials are studied with the theory of maximal normal operators. The quasidifferential of a normally quasidifferentiable function is a pair of upper and lower semicontinuous operators, which are special maximal normal operators. The function sum of the upper and lower semicontinuous operators is the Clarke subdifferential of this function. Basic calculus and minimal forms of quasidifferentials are also discussed.     

Exhausters, optimality conditions and related problems

The notions of exhausters were introduced in (Demyanov, Exhauster of a positively homogeneous function, Optimization 45, 13–29 (1999)). These dual tools (upper and lower exhausters) can be employed to describe optimality conditions and to find directions of steepest ascent and descent for a very wide range of nonsmooth functions. What is also important, exhausters enjoy a very good calculus (in the form of equalities). In the present paper we review the constrained and unconstrained optimality conditions in terms of exhausters, introduce necessary and sufficient conditions for the Lipschitzivity and Quasidifferentiability, and also present some new results on relationships between exhausters and other nonsmooth tools (such as the Clarke, Michel-Penot and Fréchet subdifferentials).     

Exhausters af a positively homogeneous function *

Notions of upper exhauster and lower exhauster of a positively homogeneous (of the first degree) function h: ? ⁿ →? are introduced. They are linked to exhaustive families of upper convex and lower concave approximations of the function h. The pair of an upper exhauster and a lower exhauster is called a biexhauster of h. A calculus for biexhausters is described (in particular, a composition theorem is formulated). The problem of minimality of exhausters is stated. Necessary and sufficient conditions for a constrained minimum and a constrained maximum of a directionally differentiable function f: ? ⁿ →? are formulated in terms of exhausters of the directional derivative of f. In general, they are described by means of exhausters of the Hadamard upper and lower directional derivatives of the function f. To formulate conditions for a minimum, an upper exhauster is employed while conditions for a maximum are formulated via a lower exhauster of the respective directional derivative (the Hadamard lower derivative for a minimum and the Hadamard upper derivative for a maximum).

If a point x _o is not stationary then directions of steepest ascent and descent can also be calculated by means of exhausters.     

Representative of Quasidifferentials and Its Formula for a Quasidifferentiable Function

The quasidifferential of a quasidifferentiable function in the sense of Demyanov and Rubinov is not uniquely defined. Xia proposed the notion of the kernelled quasidifferential, which is expected to be a representative for the equivalent class of quasidifferentials. In the 2-dimensional case, the existence of the kernelled quasidifferential was shown. In this paper, the existence of the kernelled quasidifferential in the n-dimensional space (n>2) is proved under the assumption that the Minkowski difference and the Demyanov difference of subdifferential and minus superdifferential coincide. In particular, given a quasidifferential, the kernelled quasidifferential can be formulated. Applications to two classes of generalized separable quasidifferentiable functions are developed. Mathematics Subject Classifications (2000) 49J52, 54C60, 90C26. This work was supported by Shanghai Education Committee (04EA01).     

A Comparison Between Chebyshev and Ultraspherical Expansions

This paper examines the norms of the projections from C[–1,1] to P_M[–1, 1] which are obtained by truncating Chebyshevand ultraspherical expansions after n terms. The norms are evaluatednumerically for n = 1, 2, 3 , ... , 10 and –0.1 5 wherethe ultraspherical weight function is (1 – x²)^x–.In particular the data show that the norm of the Chebyshev projectionis not the smallest in this range for 1 n 10.     

关于6 Sigma与3 Sigma的比较

在企业推广6σ的过程中,经常有学员不明白为什么6σ比3σ好,甚至有人会说6σ的规格限比3σ的规格限宽,当然6σ的合格率比3σ的合格率高。本文认为文献[1]、[2]关于6σ与3σ的比较图是引起这种误会的根源。为此本文给出另外的比较和解释。     

Navier-Stokes方程与Euler方程的稳定性比较

比较了Navier-Stokes 方程和Euler方程的稳定性;并以它们的典型初值问题为例,分析了Navier-Stokes方程和Euler方程稳定性不同的原因．     

修正贝叶斯控制图与累积和控制图的比较分析

面向多品种、小批量制造过程的小波动检验是目前统计过程控制图发展和研究的方向.在对修正贝叶斯控制图与累积和控制图原理进行比较后,通过一组过程仿真数据对两种控制图进行了分析,指出了两种控制图的特性和应用前景.     

Comparison Between Baumann and Admissible Simplex Forms in Interval Analysis

Two ways for bounding n-variables functions over a box, based on interval evaluations of first order derivatives, are compared. The optimal Baumann form gives the best lower bound using a center within the box. The admissible simplex form, proposed by the two last authors, uses point evaluations at n + 1 vertices of the box. We show that the Baumann center is within any admissible simplex and can be represented as a linear convex combination of its vertices with coefficients equal to the dual variables of the linear program used to compute the corresponding admissible simplex lower bound. This result is applied in a branch-and-bound global optimization and computational results are reported.     

Comparison Results Between Preconditioned Jacobi and the AOR Iterative Method

The large scale linear systems with M-matrices often appear in a wide variety of areas of physical,fluid dynamics and economic sciences.It is reported in[1]that the convergence rate of the IMGS method,with the preconditioner I S_α,is superior to that of the basic SOR iterative method for the M-matrix.This paper considers the preconditioned Jacobi(PJ)method with the preconditioner P=I S_α S_β,and proves theoretically that the convergence rate of the PJ method is better than that of the basic AOR method.Numerical examples are provided to illustrate the main results obtained.     

深埋隧洞围岩应力的精确解与近似解的对比分析

对不同断面形状的深埋隧洞进行了分析,比较了隧洞围岩应力解析解与通过当量半径方法得到的近似解之间的差别.首先,应用复变函数的基本理论,给出圆形、椭圆、矩形、直墙拱形等几种常见深埋隧洞围岩应力的解析表达式.其次,应用当量半径的折算形式,将其任意形状的边界转化为标准圆形断面,利用Lamé解答得到了各围岩应力分量.最后,考虑隧洞断面形状参数的变化,通过数值算例对精确解和近似解进行了比较,分析了当量半径折算形式的精确度.在此基础上,应用有限元方法验证了复变函数解析解的精确性,以椭圆、矩形和直墙拱形的复变函数解验证当量半径精确度.结果表明,当量半径的折算形式解答与精确解答之间相似程度与隧洞的断面形状和几何参数之间有着密切的关系.     

Comparison Between the Mean-Variance Optimal and the Mean-Quadratic-Variation Optimal Trading Strategies

ABSTRACT

We compare optimal liquidation policies in continuous time in the presence of trading impact using numerical solutions of Hamilton–Jacobi–Bellman (HJB) partial differential equations (PDEs). In particular, we compare the time-consistent mean-quadratic-variation strategy with the time-inconsistent (pre-commitment) mean-variance strategy. We show that the two different risk measures lead to very different strategies and liquidation profiles. In terms of the optimal trading velocities, the mean-quadratic-variation strategy is much less sensitive to changes in asset price and varies more smoothly. In terms of the liquidation profiles, the mean-variance strategy is much more variable, although the mean liquidation profiles for the two strategies are surprisingly similar. On a numerical note, we show that using an interpolation scheme along a parametric curve in conjunction with the semi-Lagrangian method results in significantly better accuracy than standard axis-aligned linear interpolation. We also demonstrate how a scaled computational grid can improve solution accuracy.     

算术观点下的林氏微积分与传统教科书中微积分比较分析

微积分是大学里普及程度非常高的一门学科,数学系学生、理工科学生、文科学生都需要学习,传统教科书中的微积分复杂度较高,使很多学生望而生畏.而算术观点下的林氏微积分复杂度保持在乘法表的水平,大大降低了微积分的门槛,且直击微积分的核心:牛顿—莱布尼茨公式.以《数学分析》中的微积分部分为例,与林群的微积分做对比,以期为大学微积分的教学改革提供思考.     

A Comparison Between Mathematics Textbook Content and a Statewide Mathematics Proficiency Test

Seven mathematics textbook series used for Grades 1 through 8 were examined. Textbook pages were counted as arithmetic, measurement, geometry, data analysis, or algebra. Percentages of mathematics textbook content were compared with percentages of mathematics content on the Ohio Ninth Grade Proficiency Test. Areas of greatest mismatch were arithmetic, measurement, and algebra. The overall percentage of arithmetic in the textbook series was about twice that on the test; the percentage of measurement was less than half that on the test; and the percentage of algebra was a third or less than that on the test. Results are also presented by grade level, and implications for teaching are discussed.     

A Comparison Theorem on Moment Inequalities Between Negatively Associated and Independent Random Variables

Let {X _i, 1in} be a negatively associated sequence, and let {X* _i , 1in} be a sequence of independent random variables such that X* _i and X _i have the same distribution for each i=1, 2,..., n. It is shown in this paper that Ef( ⁿ _i=1 X _i)Ef( ⁿ _i=1 X* _i ) for any convex function f on R ¹ and that Ef(max_1kn ⁿ _i=k X _i)Ef(max_1kn ^k _i=1 X* _i ) for any increasing convex function. Hence, most of the well-known inequalities, such as the Rosenthal maximal inequality and the Kolmogorov exponential inequality, remain true for negatively associated random variables. In particular, the comparison theorem on moment inequalities between negatively associated and independent random variables extends the Hoeffding inequality on the probability bounds for the sum of a random sample without replacement from a finite population.     

不同风险度量方法的投资组合比较研究

为了验证投资组合理论在中国证券市场的有效性,在不允许卖空情况,针对不同风险度量方法,文章运用旋转算法或结合序列二次规划法分别求解均值-方差、均值-下半方差投资组合模型、均值-半绝对偏差、均值-平均绝对偏差和均值-VaR.文章选取三年沪市六只业绩比较好的股票,依据前两年的数据作为样本数据,分别求出五个模型在不同期望收益率下的最优投资策略,将得出的最优投资策略应用到最后一年,进行模拟投资,从而计算出各模型的总收益率.以等比例投资为标准,比较五个模型的绩效.最后,证明了两个模型对于中国证券市场是适用.     

A Cross‐Cultural Lesson Comparison on Teaching the Connection Between Multiplication and Division

This study compared one lesson across four U.S. “traditional” textbook series, two U.S. reform‐based textbook series, and one Chinese mathematics textbook series in teaching the connection between multiplication and division. The results showed the differences across U.S. and Chinese lessons in both the teaching and the practice parts of the lesson across three dimensions (i.e., problem schemata, response requirement, and algebra readiness). In particular, the Chinese lesson's penetrating analysis or explanation of the topic is reflected in its deliberately constructed examples and wide range of problems (pertaining to problem types and difficulty levels) present in the teaching and practice sections of the lesson. None of analyzed U.S. lessons are comparable with the Chinese lesson with respect to the breadth and depth in teaching the topic. A deliberate emphasis, both arithmetically and algebraically, on problem schema acquisition as found in the Chinese lesson represents a promotion of symbolic or higher order of conceptual understanding. The findings are discussed within the context of teaching big ideas through problem schemata acquisition and the importance of symbolic level of conceptual understanding.