Formula 1 – A Mathematical Microworld with CAS: Analysis of Learning Opportunities and Experiences with Students

In this article we describe a mathematical microworld for investigating car motion on a racing course and its use with a group of grade 12 students. The microworld is concerned with the mathematical construction of courses and functions which describe car motion. It is implemented in the computer algebra system, Maple^®, which provides the means to represent courses and functions symbolically and graphically. We describe the learning opportunities offered by the microworld in relation to the research literature on functions. Various facets and layers of the function concept are addressed in the microworld, and we suggest how work in the microworld might help in overcoming some well-known misconceptions.This revised version was published online in September 2005 with corrections to the Cover Date.     

混合效应模型中的D-最优设计

本研究的是混合效应-Ⅱ模型，首先，给出二阶模型存在D-最优回归设计点的必要条件。其次，组合D-最优回归设计和D-最优区组设计BIBD构造D-最优设计并给出寻找新的D-最优设计点的方法。以两种有代表性的组合误差分布为例，阐明新的D-最优设计点的获得过程，并给出了数值结果。     

Determining radii of meromorphy via orthogonal polynomials on the unit circle

Using a convergence theorem for Fourier–Padé approximants constructed from orthogonal polynomials on the unit circle, we prove an analogue of Hadamard's theorem for determining the radius of m-meromorphy of a function analytic on the unit disk and apply this to the location of poles of the reciprocal of Szeg functions.     

Internal standardization for the determination of cadmium,cobalt, chromium and manganese in saline produced water from petroleum industry by inductively coupled plasma optical emission spectrometry after cloud point extraction

In the present paper a procedure is proposed for the determination of traces of Cd, Co, Mn and Cr in petroleum industry produced water by inductively coupled plasma optical emission spectrometry. The procedure is based on cloud point extraction of these metals, as their dithizonate complexes, into the surfactant-rich phase of octylphenoxypolyethoxyethanol surfactant (Triton X-114). Extractions were carried out in solutions with salinities between 10‰ and 70‰. Since residual salinity in the surfactant-rich phase caused differences in its transport to the plasma, yttrium was used as an internal standard to correct for this effect. The simultaneous metal extraction procedure was optimized by response surface methodology using a Doehlert design and desirability function. Enhancement factors of 21, 21, 9 and 19, along with limits of quantification of 0.093, 0.20, 0.73 and 1.2 μg L^− 1, and precision expressed as relative standard deviation (n = 8, 20.0 μg L^− 1) of 5.8, 1.2, 1.7 and 5.7% were obtained for Cd, Co, Mn and Cr, respectively. The accuracy was evaluated by spike recovery tests on the high salinity water samples with salinity of 40 and 63‰.     

干电池供电式电子标准电池的设计

标准电池是度量电源电动势的标准仪器．利用RC4191和MC1403精密的低压稳压特性，设计一款电子稳压式的标准电池，既可克服传统汞镉标准电池的缺点，又能保证相同精度的实验性能要求．     

Pseudoaffinity, de Boor algorithm, and blossoms

In order to ensure existence of a de Boor algorithm (hence of a B-spline basis) in a given spline space with (n+1)-dimensional sections, it is important to be able to generate each spline by restriction to the diagonal of a symmetric function of n variables supposed to be pseudoaffine w.r. to each variable. We proved that a way to obtain these three properties (symmetry, n-pseudoaffinity, diagonal property) is to suppose the existence of blossoms on the set of admissible n-tuples, given that blossoms are defined in a geometric way by means of intersections of osculating flats. In the present paper, we examine the converse: do symmetry, n-pseudoaffinity, and diagonal property imply existence of blossoms?     

Improving forecasting performance by employing the Taguchi method

To satisfy the volatile nature of today’s markets, businesses require a significant reduction in product development lead times. Consequently, the ability to develop precise product sales forecasts is of fundamental importance to decision-makers. Over the years, many forecasting techniques of varying capabilities have been introduced. The precise extent of their influences, and the interactions between them, has never been fully clarified, although various forecasting factors have been explored in previous studies. Accordingly, this study adopts the Taguchi method to calibrate the controllable factors of a forecasting model. An L₉(3⁴) inner orthogonal array is constructed for the controllable factors of data period, horizon length, and number of observations required. An experimental design is then performed to establish the appropriate levels for each factor. At the same time, an L₄(2³) outer orthogonal array is used to consider the inherited parameters of forecasting method as the noise factors of Taguchi method simultaneously. An illustrated example, employing data from a power company, serves to demonstrate the thesis. The results show that the proposed model permits the construction of a highly efficient forecasting model through the suggested data collection method.     

The solution of orthogonal Procrustes problems for a family of orthogonally invariant norms

The orthogonal Procrustes problem involves finding an orthogonal matrix which transforms one given matrix into another in the least-squares sense, and thus it requires the minimization of the Frobenius matrix norm. We consider, the solution of this problem for a family of orthogonally invariant norms which includes the Frobenius norm as a special case.     

The dynamics of water in small volumes probed by incoherent quasi-elastic neutron scattering

Summary The dynamics of water molecules occupying very small volumes can be probed by incoherent quasi-elastic neutron scattering. We summarise some results obtained with wet samples of porous silica and hydrated proteins. The movements of the molecules in the vicinity of the substrate are extremely reduced as compared to those of bulk water at the same temperature. Almost only rotational movements subsist and the local diffusion is of the order of that observed in deeply supercooled water. Paper presented at the I International Conference on Scaling Concepts and Complex Fluids, Copanello, Italy, July 4–8, 1994.     

Difference Families in Z2d+1⊕Z2d+1 and Infinite Translation Designs in Z⊕Z

We analyse 3-subset difference families of Z_2d+1⊕Z_2d+1 arising as reductions (mod 2d+1) of particular families of 3-subsets of Z⊕Z. The latter structures, namely perfect d-families, can be viewed as 2-dimensional analogues of difference triangle sets having the least scope. Indeed, every perfect d-family is a set of base blocks which, under the natural action of the translation group Z⊕Z, cover all edges {(x,y),(x′,y′)} such that |x−x′|, |y−y′|≤d. In particular, such a family realises a translation invariant (G,K₃)-design, where V(G)=Z⊕Z and the edges satisfy the above constraint. For that reason, we regard perfect families as part of the hereby defined translation designs, which comprise and slightly generalise many structures already existing in the literature. The geometric context allows some suggestive additional definitions. The main result of the paper is the construction of two infinite classes of d-families. Furthermore, we provide two sporadic examples and show that a d-family may exist only if d≡0,3,8,11 (mod 12).