Affine-inscribed and affine-circumscribed polygons and polytopes

Five theorems on polygons and polytopes inscribed in (or circumscribed about) a convex compact set in the plane or space are proved by topological methods. In particular, it is proved that for every interior point O of a convex compact set in ℝ³, there exists a two-dimensional section through O circumscribed about an affine image of a regular octagon. It is also proved that every compact convex set in ℝ³ (except the cases listed below) is circumscribed about an affine image of a cube-octahedron (the convex hull of the midpoints of the edges of a cube). Possible exceptions are provided by the bodies containing a parallelogram P and contained in a cylinder with directrix P. Bibliography: 29 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 231, 1995, pp. 286–298. Translated by B. M. Bekker.     

Linear birth and death models and associated Laguerre and Meixner polynomials

We study birth and death processes with linear rates λ_n = n + α + c + 1, μ_{n + 1} = n + c, n 0 and μ₀ is either zero or c. The spectral measures of both processes are found using generating functions and the integral transforms of Laplace and Stieltjes. The corresponding orthogonal polynomials generalize Laguerre polynomials and the choice μ₀ = c generates the associated Laguerre polynomials of Askey and Wimp. We investigate the orthogonal polynomials in both cases and give alternate proofs of some of the results of Askey and Wimp on the associated Laguerre polynomials. We also identify the spectra of the associated Charlier and Meixner polynomials as zeros of certain transcendental equations.     

Partitions and pairwise sums and products

A seven cell partition of N is constructed with the property that no infinite set has all of its pairwise sums and products in any one cell. A related Ramsey Theory question is shown to have different answers for two and three cell partitions.     

Beliefs and beyond: affect and the teaching and learning of mathematics

The importance of beliefs for the teaching and learning of mathematics is widely recognized among mathematics educators. In this special issue, we explicitly address what we call “beliefs and beyond” to indicate the larger field surrounding beliefs in mathematics education. This is done to broaden the discussion to related concepts (which may not originate in mathematics education) and to consider the interconnectedness of concepts. In particular, we present some new developments at the conceptual level, address different approaches to investigate beliefs, highlight the role of student beliefs in problem-solving activities, and discuss teacher beliefs and their significance for professional development. One specific intention is to consider expertise from colleagues in the fields of educational research and psychology, side by side with perspectives provided by researchers from mathematics education.     

Competition and Cooperation between Manufacturer and Retailer

We assume that each manufacturer decides his price to the tradeand that retailers set their prices in the light of the manufacturer'sprices. We consider two hypotheses about the way in which eachmakes his decision. In the absence of cooperation, the retailerchooses a price (as a function of the trade price) which willmaximize the retail profit. The manufacturer takes this intoaccount in determining his price to maximize his profit. Ifthey cooperate (which may be illegal), they agree to a retailprice which maximizes their joint profit. The trade price isset via a discussion on how the profit is to be split betweenthem. It is clear that the cooperative profit cannot be lessthan the non-cooperative profit. In at least one set of circumstancesthe non-cooperative solution yields a smaller profit for theretailer than the manufacturer, which means that the latteris in a weak position with respect to bargaining over the additionalprofits from cooperation. More surprising is our main resultthat, for some response functions, cooperative prices are lowerand sales volumes are higher under cooperation. Examples tobe examined in detail are: (a) single manufacturer and singleretailer; (b) several manufacturers and a single retailer (ora group of identical retailers) with a linear symmetric price-volumerelationship.     

Comonotone and coconvex rational interpolation and approximation

Comonotonicity and coconvexity are well-understood in uniform polynomial approximation and in piecewise interpolation. The covariance of a global (Hermite) rational interpolant under certain transformations, such as taking the reciprocal, is well-known, but its comonotonicity and its coconvexity are much less studied. In this paper we show how the barycentric weights in global rational (interval) interpolation can be chosen so as to guarantee the absence of unwanted poles and at the same time deliver comonotone and/or coconvex interpolants. In addition the rational (interval) interpolant is well-suited to reflect asymptotic behaviour or the like.     

Navier and Stokes meet Poincare and Dulac

This paper surveys various precise (long-time) asymptotic results for the solutions of the Navier-Stokes equations with potential forces in bounded domains. It turns out that the asymptotic expansion leads surprisingly to a kind of Poincare-Dulac normal form of the Navier-Stokes equations. We will also discuss some related results and a few open issues.     

Edges and switches,tunnels and bridges

Edge casing is a well-known method to improve the readability of drawings of non-planar graphs. A cased drawing orders the edges of each edge crossing and interrupts the lower edge in an appropriate neighborhood of the crossing. Certain orders will lead to a more readable drawing than others. We formulate several optimization criteria that try to capture the concept of a “good” cased drawing. Further, we address the algorithmic question of how to turn a given drawing into an optimal cased drawing. For many of the resulting optimization problems, we either find polynomial time algorithms or NP-hardness results.     

Diversity and Popularity in Organizations and Communities

This paper examines social groupings whose structure depends only on the distribution of ability to attract attention. When people‘s attention is a scarce resource, those individuals who are rated highest by a large number of other individuals will have to ration their attention, resulting in constraints on the social structure of the group. The incidence of popular individuals by that definition reflects the extent to which individuals agree on who their highest-rated colleague is. We propose three basic distributions or ways to generate the matrix of perceived ability so as to yield popularity profiles that can be parametrically adjusted to match observations. We demonstrate that each of these assumption sets leads to a slightly different correlation between the value of the assumption‘s parameter and the set of observable popularity patterns. Since popularity, in real life, often determines such things as power, centrality, over-utilization and perhaps reduced accessibility, having more realistic ways of representing it is important for modeling and understanding virtual organizations and communities.     

Reduction and realization in toda and volterra

We construct a new symplectic, bi-Hamiltonian realization of the KM-system by reducing the corresponding one for the Toda lattice. The bi-Hamiltonian pair is constructed using a reduction theorem of Fernandes and Vanhaecke. In this paper we also review the important work of Moser on the Toda and KM-systems.      

Singular quasilinear and Hessian equations and inequalities

We solve the existence problem in the renormalized, or viscosity sense, and obtain global pointwise estimates of solutions for quasilinear and Hessian equations with measure coefficients and data, including the following model problems:

     

On exact and inexact differentials and applications

Considering the important role played by mathematical derivatives in the study of physical–chemical processes, this paper discusses the different possibilities and formulations of this concept and its application. In particular, in Chemical Thermodynamics, we study exact differentials associated with the so-called state functions and inexact differentials associated with path functions. As an application, we discuss the equation of state associated with a gas.     

Uncertainty and Imprecision: Modelling and Analysis

Any statistical analysis or decision analysis contains numerical inputs of which we are unsure. Some of our uncertainty arises from physical randomness which we can model in various ways, ideally through probability. Some uncertainty relates to judgemental estimates of quantities about which we may be unsure in many different respects. There are other uncertainties involved, however: some relate to ambiguity and imprecision of meaning; some relate to lack of clarity in the objectives which the analysis seeks to meet; some relate to the numerical accuracy of calculations. How should the uncertainty arising from ambiguity be modelled? Other uncertainties can also impact on an analysis. Why is the analysis being conducted? Are the objectives clear?     

C_(pk)写控制图的研究与应用

控制图与过程能力指数Cp(Cpk)都是用来进行质量分析与处理的重要工具。文 [1]谈及控制图的两类错误 (其概率分别为α、β) ,文 [2 ]讨论了Cpk与α的关系 ,但涉及Cpk与控制图检出力 (1- β)关系的文章却不多。本文以 X控制图为例 ,讨论了α、β、1- β、与Cpk的关系 ,得出了联合应用它们会对工序状态有较好的了解 ,对其适当的调整有助於提高产品质量     

National Standards and School Science and Mathematics

We explore the relationship between national standards as policy recommendations and the transformation of those recommendations into programs for school science and mathematics. We provide a brief introduction to national standards, propose a strategic framework for standards-based reform, and offer a discussion of specific challenges associated with the improvement of contemporary science and mathematics education. A set of features common to the NCTM and the NRC standards is described. The strategic-framework focuses on dissemination, interpretation, implementation, evaluation, and revision. Challenges include implementing school programs for all students, understanding inquiry and problem solving; thinking differently about curriculum; increasing instructional effectiveness; and aligning assessment strategies with standards.     

Dual and triple equations and q-orthogonal polynomials

In this paper, we solve dual and triple sequences involving q-orthogonal polynomials. We also introduce and solve a system of dual series equations when the kernel is the q-Laguerre polynomials. Examples are included.     

Fuzzy programming and profit and cost allocation for a production and transportation problem

In this paper, we deal with a real problem on production and transportation in a housing material manufacturer, and consider a production and transportation planning under the assumption that the manufacturer makes multiple products at factories in multiple regions and the products are in demand in each of the regions. First, we formulate mixed zero–one programming problems such that the cost of production and transportation is minimized subject to capacities of factories and demands of regions. Second, to realize stable production and satisfactory supply of the products in fuzzy environments, fuzzy programming for the production and transportation problem is incorporated. Finally, under the optimal planning of production and transportation, we show a profit and cost allocation by applying a solution concept from game theory. Using actual data, we show usefulness of the fuzzy programming and a rational allocation scheme of the profit and cost.     

Proper and Unit Trapezoid Orders and Graphs

We show that the class of trapezoid orders in which no trapezoid strictly contains any other trapezoid strictly contains the class of trapezoid orders in which every trapezoid can be drawn with unit area. This is different from the case of interval orders, where the class of proper interval orders is exactly the same as the class of unit interval orders.     

Bloch and normal functions and their relation

Following a brief introduction to Bloch and normal functions, several conditions, including a convergence theorem, are shown for determining them. In addition, since an exponential of any constant multiple of a Bloch function is always normal, we investigate whether or not the converse holds, and construct an example of a non-Bloch function such that the exponential of any constant multiple of it is normal.