A Look at Examinations

This paper discusses various aspects of mathematics examinations and gives the results of some experiments which highlight the discrepancies that arise between the marks obtained in two equivalent mathematics examination papers.     

A Look at Project AIMS

Project AIMS (Activities that Integrate Mathematics and Science) began in 1981 as a National Science Foundation grant. Twelve years later it is a flourishing program which relies on materials developed and written by teachers, and one which uses inexpensive, easily acquired equipment and materials. It is particularly noteworthy that the focus of AIMS-integration through hands-on mathematics and science-supports such recent initiatives as the NCTM Standards and Project 2061 of the AAAS, even though it predates both of them.     

A Constructive Look at Generalised Cauchy Reals

We investigate how nonstandard reals can be established constructively as arbitrary infinite sequences of rationals, following the classical approach due to Schmieden and Laugwitz. In particular, a total standard part map into Richman's generalised Dedekind reals is constructed without countable choice.     

A New Look at the Burnside-Schur Theorem

The famous Burnside–Schur theorem states that every primitivefinite permutation group containing a regular cyclic subgroupis either 2-transitive or isomorphic to a subgroup of a 1-dimensionalaffine group of prime degree. It is known that this theoremcan be expressed as a statement on Schur rings over a finitecyclic group. Generalizing the latter, Schur rings are introducedover a finite commutative ring, and an analogue of this statementis proved for them. Also, the finite local commutative ringsare characterized in permutation group terms. 2000 MathematicsSubject Classification 20B10, 20B15, 05E99.     

A Constructive Look at Functions of Bounded Variation

Functions with bounded variation and with a (total) variationare examined within Bishop's constructive mathematics. It isshown that the property of having a variation is hereditarydownward on compact intervals, and hence that a real-valuedfunction f with a variation on a compact interval can be expressedas a difference of two increasing functions. Moreover, if fis sequentially continuous, then the corresponding variationfunction, and hence f itself, is uniformly continuous. 1991Mathematics Subject Classification 26A45.     

A Look at the Stationarity of 14 Quarterly U.S. Economic Time Series

Application of single series time series methods has become very popular, due, in particular, to the short term forecasting abilities of such models. This is particularly true of the Box-Jenkins linear models. In this paper, the relationship between the forecasting ability and the number of data points used in the estimation of Box-Jenkins models, for 14 economic time series, is considered, with the result that in a large number of cases, there is very little gain in analysing long historical records. In fact, it is possible that forecasting ability may be impaired if too many observations are considered.     

A New Look at Mourre’s Commutator Theory

Mourre’s commutator theory is a powerful tool to study the continuous spectrum of self-adjoint operators and to develop scattering theory. We propose a new approach of its main result, namely the derivation of the limiting absorption principle (LAP) from a so called Mourre estimate. We provide a new interpretation of this result. Received: September 20, 2006. Revised: February 1, 2007.     

再论中国股票市场的弱有效性

本文利用检验鞅差序列的非参数统计量来检验中国股票市场的弱有效性问题,本文给出的方法不同于文献中已有的方法,实证分析表明本文使用的检验股票市场弱有效的方法比文献中的大多数方法更有效。实证结果显示,中国的股市在发展过程中,表现山渐进有效的态势;中国A股市场比B股市场更有效率:沪市比深市更趋于有效。文中的有些结论是以前的实证研究所没有的。另外,本文比较了中国股市和境外成熟股市之间有效性程度的差异,并得出香港市场对深市影响显著的结论。     

A Fresh Variational-Analysis Look at the Positive Semidefinite Matrices World

Engineering sciences and applications of mathematics show unambiguously that positive semidefiniteness of matrices is the most important generalization of non-negative real numbers. This notion of non-negativity for matrices has been well-studied in the literature; it has been the subject of review papers and entire chapters of books.     

A Geometrical Look at Iterative Methods for Operators with Fixed Points

ABSTRACT

We distinguish classes of operators T with fixed points on a real Hilbert space by comparing the distances of a point x and its image Tx to the (set of) fixed points of T; this leads to a ranking of those classes, based on a nonnegative parameter. That same parameter also lets us conclude about the sign of and an upper bound for a characteristic inner product result that arises in iterative processes to obtain a common fixed point of a set of operators. We use that parameter as the starting point for a geometrically-inclined study of specific iterative algorithms intended to find a common fixed point of operators belonging to such class.     

A Second Look at the Toric h-Polynomial of a Cubical Complex

We provide an explicit formula for the toric h-contribution of each cubical shelling component, and a new combinatorial model to prove Chan??s result on the non-negativity of these contributions. Our model allows for a variant of the Gessel-Shapiro result on the g-polynomial of the cubical lattice, this variant may be shown by simple inclusion-exclusion. We establish an isomorphism between our model and Chan??s model and provide a reinterpretation in terms of noncrossing partitions. By discovering another variant of the Gessel-Shapiro result in the work of Denise and Simion, we find evidence that the toric h-polynomials of cubes are related to the Morgan-Voyce polynomials via Viennot??s combinatorial theory of orthogonal polynomials.     

Liénard方程至多存在n个极限环的充分条件

在文[7,8]中对Lienard方程提出状态函数的概念.本文我们将极限环存在性的研究转化为微积分中连续函数(状态函数)零点的研究,给出了Lienard方程至多存在n个极限环的充分条件.最后证明了A.Lins,W.de Melo,C.C.Pugh的猜想.