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.     

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.     

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.     

Distributive and Multiplication Modules and Rings

We study rings in which every ideal is a finitely generated multiplication right ideal.     

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.     

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.      

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?     

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.     

ERP与ISO的结合与集成实施

ERP和ISO9000的一体化结合是一个值得重视的研究课题.文中首先分析了ERP和ISO9000的对应点,指出两者可以相互支撑.最后,探讨了两者的集成实施问题.     

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.     

On b-connectedness and b-disconnectedness and their applications

In this paper, by using b-open (=γ-open) sets we study the concept of b-separated sets. With this concept we study the notion of b-connected sets and strongly b-connected sets. We give some properties of such concepts with some b-separation axioms and compact spaces. Finally, we construct a new topological space on a connected graph.     

树自动机的同余与同态及其语言

从代数的角度出发研究树自动机的同余与同态,得出树自动机的同态基本定理和同构基本定理,进而研究它们之间语言的相关性质.