Fifth graders’ understanding of fractions on the number line

The goal of this research was to examine fifth graders’ understanding of fractions on the number line. This case‐study design focused on the various ways that students represented fractions on number lines. Students responded to task‐based interview questions by identifying fractions as a number on the number line as well as equivalency and problem solving. The tasks were administered individually to 26 fifth‐grade students over a 15‐minute time frame in their respective schools. The two groups of 10‐year‐old students answered most questions in written form with pencil and paper and were often asked to explain how they arrived at an answer. Student performance was highest when instructed to plot ½ on a number line of 0 to 1 as well as naming a fraction less than ½. The students performed lowest when they attempted to plot ½, ¼, and 1 on a number line with a predetermined unit 0 to 1/3. Other low performing concepts consisted of plotting ¼ on a number line from 0‐3, identifying ¼ on a non‐routine number line, and plotting a unit fraction with an equivalent fraction as well as an improper fraction on a common number line.     

The catwalk task: Reflections and synthesis: Part 1

In this article I recount my experiences with a series of encounters with the catwalk task and reflect on the professional growth that these opportunities afforded. First, I reflect on my own mathematical work on the catwalk task, including my efforts to fit various algebraic models to the data. Second, I reflect on my experiences working with a group of high school students on the catwalk task and my interpretations of their mathematical thinking. Finally, I reflect on the entire experience with the catwalk problem, as a mathematics learner, as a teacher, and as a professional.     

Explicitly given pairs of dual frames with compactly supported generators and applications to irregular B-splines

We consider systems of functions appearing by letting a class of modulations act on a countable collection of functions. These systems correspond to shift-invariant systems, considered on the Fourier side. We provide sufficient conditions for the system to be a frame, as well as an explicit construction of a class of frames and associated duals. We use the result to construct frames based on B-splines with knot sequences satisfying a natural condition, as well as explicitly given duals.     

The motion of a ball on a grassy lawn

The motion of a ball of constant spherical shape on a grassy lawn represented by a set of deformable rods with one end rigidly attached to a plane, while the other end is free and can contact the ball, is investigated. The contact area on the ball surface is represented by a spherical sector, whose dimensions and location on the ball surface depend on the position of the centre of the ball and its orientation. The projection of the contact area onto the plane is a circle, whose centre and radius vary as the centre of the ball moves. When the rods come into contact with the ball, they experience longitudinal and flexural strains, as well as impact actions on the boundary of the contact area. The translational–rotational motion of a ball on a lawn of rods without friction, as well as motion with viscous friction at the points of contact of the ends of the rods with the ball, is considered.     

Decay of the energy for the Benjamin–Bona–Mahony equation posed on bounded intervals and on a half‐line

This paper concerns nonlinear dissipative initial boundary value problems for the Benjamin–Bona–Mahony equation posed on a half‐line and on bounded intervals. We prove the existence and uniqueness of global solutions and decay of the energy as time tends to infinity as well as convergence of solutions on bounded intervals to a solution on a half‐line. Copyright © 2012 John Wiley & Sons, Ltd.     

Exploring Images of Parental Participation in Mathematics Education: Challenges and Possibilities

In this article, we draw on research within a large project on parental involvement in mathematics education in working-class Latino communities. Our research is situated within a sociocultural framework and, in particular, the concept of funds of knowledge. We also draw on research on parental involvement in education, particularly that which critically examines issues of power and perceptions of parents. We build on the concept of dialogic learning and on the characterization of parents as intellectual resources and present a model for parental involvement in mathematics in which parents engage as (a) parents, (b) learners, (c) facilitators, and (d) leaders. In particular, in this article, we focus on the third component—parents as facilitators of mathematics workshops for the community at large—centering on some of the challenges as parents and teachers engage in this type of collaboration. We also look at the possibilities afforded by a model for parental involvement that views parents as intellectual resources. By looking at examples of interactions among parents and teachers, and among parents and children, in mathematics workshops, we challenge conventional notions about parental involvement—in particular, as they apply to working-class, language/ethnic "minority" parents.     

群上同余

本引进了群上同余的概念，得到了群上的等价关系为群上同余的一个充分必要条件，并讨论了群上同余与群上正规子群之间的关系以及群上同余与群上同态之间的关系．     

The catwalk task: Reflections and synthesis: Part 2

In this article we recount our experiences with a series of encounters with the catwalk task and reflect on the professional growth that these opportunities afforded. First, we individually reflect on our own mathematical work on the catwalk task. Second, we reflect on our experiences working with a group of community college students on the catwalk task and our interpretations of their mathematical thinking. In so doing we also detail a number of innovative and novel student-generated representations of the catwalk photos. Finally, we each individually reflect on the entire experience with the catwalk problem, as mathematics learners, as teachers, and as professionals.     

Hamiltonian systems with a stochastic force:nonlinear versus linear,and a girsanov formula

We discuss stochastic perturbations of classical Hamiltonian systems by a white noise force. We prove existence and uniqueness results for the solutions of the equation of motion under general conditions on the classical system, as well as their continuous dependence on the initial conditions. We also prove that the process in phase space is a diffusion with transition probability densities, and Lebesgue measure as c-finite invariant measure. We prove a Girsanov formula relating the solution for a nonlinear force with the one for a linear force, and give asymptotic estimates on functions of the phase space process     

Spanning trees with variable degree bounds

In this paper, we introduce and study a generalization of the degree constrained minimum spanning tree problem where we may install one of several available transmission systems (each with a different cost value) in each edge. The degree of the endnodes of each edge depends on the system installed on the edge. We also discuss a particular case that arises in the design of wireless mesh networks (in this variant the degree of the endnodes of each edge depend on the transmission system installed on it as well as on the length of the edge). We propose three classes of models using different sets of variables and compare from a theoretical perspective as well as from a computational point of view, the models and the corresponding linear programming relaxations. The computational results show that some of the proposed models are able to solve to optimality instances with 100 nodes and different scenarios.     

Spaces to discuss mathematics: communities of practice on an online discussion board

In theories of learning that adopt a situated stance to knowledge the notion of identity is vital; how learners position themselves in relation to, and are mutually positioned by, the situation within which they are learning will have a strong bearing on the learning outcomes. One of the challenges for learning mathematics in school is that learners position themselves, and are positioned, as pupils rather than as mathematicians. This paper focuses on discussion boards designed for secondary school mathematics students, and we use Wenger's (1998) model of communities of practice, building on earlier work by the authors (Back and Pratt 2007; Pratt and Kelly 2007) in which ‘idealised communities’ are constructed and used, to consider a case study of one participant who engages in developing his identity as a mathematician doing mathematics, as well his identity as a learner and a teacher of mathematics.     

Regular reduction of controlled Hamiltonian system with symplectic structure and symmetry

In this paper, our goal is to study the regular reduction theory of regular controlled Hamiltonian (RCH) systems with symplectic structure and symmetry, and this reduction is an extension of regular symplectic reduction theory of Hamiltonian systems under regular controlled Hamiltonian equivalence conditions. Thus, in order to describe uniformly RCH systems defined on a cotangent bundle and on the regular reduced spaces, we first define a kind of RCH systems on a symplectic fiber bundle. Then we introduce regular point and regular orbit reducible RCH systems with symmetry by using momentum map and the associated reduced symplectic forms. Moreover, we give regular point and regular orbit reduction theorems for RCH systems to explain the relationships between RpCH-equivalence, RoCH-equivalence for reducible RCH systems with symmetry and RCH-equivalence for associated reduced RCH systems. Finally, as an application we regard rigid body and heavy top as well as them with internal rotors as the regular point reducible RCH systems on the rotation group SO(3) and on the Euclidean group SE(3), as well as on their generalizations, respectively, and discuss their RCH-equivalence. We also describe the RCH system and RCH-equivalence from the viewpoint of port Hamiltonian system with a symplectic structure.     

Probabilistic Model for the Growth of Thesauri

The paper deals with a mathematical model which describes how the collection of key phrases (and key words) evolves as a field of science develops. The experimental material is based on statistical observations on the sets of key phrases which have been assigned to papers in representative major journals in the field in question. Asymptotic properties of the model are considered, as well as estimators for the parameters of the model with particular emphasis on estimators that can indicate at what stage a collection of key phrases can be assumed as complete for the field in question at a given moment in time.     

基于和积和最大积的信念传播算法的收敛性分析

在介绍马尔可夫网络和标准信念传播(Belief Propagation,BP)算法的基础上,以具有4个隐含结点的无环马尔可夫网络为例,详细分析了基于和积和最大积的BP算法收敛性,结果表明,在网络为无环马尔可夫网络时,基于和积的BP算法收敛于全局最优解,而基于最大积的BP算法在任何初值条件下,只要经过有限次迭代,BP算法必收敛到唯一固定点,若算法有唯一最优点,则此固定点为最优点.以实证分析为工具,研究BP算法收敛性,可作为BP算法基础理论研究的一次有益尝试.     

The center of an effect algebra

An effect algebra is a partial algebra modeled on the standard effect algebra of positive self-adjoint operators dominated by the identity on a Hilbert space. Every effect algebra is partially ordered in a natural way, as suggested by the partial order on the standard effect algebra. An effect algebra is said to be distributive if, as a poset, it forms a distributive lattice. We define and study the center of an effect algebra, relate it to cartesian-product factorizations, determine the center of the standard effect algebra, and characterize all finite distributive effect algebras as products of chains and diamonds.     

Static and Dynamic Equilibria in Games With Continuum of Players

This paper is a study of a general class of deterministic dynamic games with an atomless measure space of players and an arbitrary time space. The payoffs of the players depend on their own strategy, a trajectory of the system and a function with values being finite dimensional statistics of static profiles. The players' available decisions depend on trajectories of the system.The paper deals with relations between static and dynamic open-loop equilibria as well as their existence. An equivalence theorem is proven and theorems on the existence of a dynamic equilibrium are shown as consequences.     

The design of branch and bound algorithms for a class of nonlinear integer programs

This paper is concerned with computational experimentation leading to the design of effective branch and bound algorithms for an important class of nonlinear integer programming problems, namely linearly constrained problems, which are used to model several real-world situations. The main contribution here is a study of the effect of node and branching variable selection and storage reduction strategies on overall computational effort for this class of problems, as well as the generation of a set of adequate test problems. Several node and branching variable strategies are compared in the context of a pure breadth-first enumeration, as well as in a special breadth and depth enumeration combination approach presented herein. Also, the effect of using updated pseudocosts is briefly addressed. Computational experience is presented on a set of eighteen suitably-sized nonlinear test problems, as well as on some random linear integer programs. Some of the new rules proposed are demonstrated to be significantly superior to previously suggested strategies; interestingly, even for linear integer programming problems.     

On lexicographic optimal solutions in transportation problems

Lexicographic versions of the cost minimizing transportation problem (CMTP) and the time minimizing transportation problem (TMTP) are presented in this paper. In addition to minimizing the quantity sent on the costliest routes in a cost minimizing transportation problem. an attempt is made to minimize the quantity transported on the second-costliest routes. if the shipment on the costliest routes is as small as possible and the quantity shipped on the third-costliest routes, if the shipments on the costliest and the second- costliest routes are as small as possible. and so on. In a lexicographic time minimizing transportation problem one is not only interested in minimizing the transportation cost on the routes of the longest duration but also on the routes of second longest, third-longest duration and so on. For finding lexicographic optimal solutions (LOS) of lexicographic cost minimizing and time minimizing transportation problems a standard cost minimizing transportation problem is formulated whose optimal solution is shown to provide the answer. Some extensions are also discussed