Boundary crossing through in-service online mathematics teacher education: the case of scenarios and half-baked microworlds

We address online mathematics teacher education as a means of immersing teachers into new kinds of collectives where professional development may be enhanced by crossing the boundaries of their habitual communities and the norms within these. We analyse the ways in which newly trained teacher educators interacted around their designs and productions for in-service courses they were giving to colleagues. We focus on their uses of scenarios and half-baked microworlds as two kinds of artefacts we had designed to play the role of improvable boundary objects. We consider these interactions as a forum for challenge and for professional development through frequent and relevant boundary crossing.     

Abduction-induction (generalization) processes of elementary majors on figural patterns in algebra

The article deals with issues concerning the abductive-inductive reasoning of 42 preservice elementary majors on patterns that consist of figural and numerical cues. We discuss: ways in which the participants develop generalizations about classes of abstract objects; abductive processes they exhibit which support their induction leading to a generalization; ways they justify their generalizations in the abductive stage, and; the effects of figural and numerical cues in the manner they construct a plausible abductive generalization. Two types of abductions are explored, model-based and manipulative. A proposed abductive-inductive reasoning process for pattern sequences is presented and discussed in the concluding section.     

Mathematics professional development as design for boundary encounters

This theoretical paper examines a process for researchers and teachers to exchange knowledge. We use the concepts of communities of practice, boundary encounters, and boundary objects to conceptualize this process within mathematics professional development (MPD). We also use the ideas from design research to discuss how mathematics professional development researchers can make professional development the focus of their research. In particular, we examine the question: How can MPD be conceptualized and designed around research-based knowledge in ways that promote knowledge exchange about students’ mathematics and mathematics learning among researchers and teachers to improve the practices of both the research and the teaching communities? We propose that MPD is a premier space for researchers and teachers to exchange knowledge from their communities, impacting both researchers’ and teachers’ practices without reducing the importance of either.     

Inflow Treatment for the Simulation of Subsonic Jets with Turbulent Nozzle Boundary Layers

The state of the boundary layer at the nozzle exit of a circular nozzle-jet configuration has an important influence on the development of the shear layer and the emitted sound. Of special interest is the acoustic near-field obtained when the nozzle exit boundary layer is fully turbulent. The turbulent inflow generation and the inflow boundary treatment are important issues to be addressed. We use the Synthetic Eddy Method (SEM) to generate a turbulent inflow which reproduces mean flow and Reynolds stress profiles of specified reference data. The spatially and temporally varying synthetic fluctuations are imposed in the simulation by a forcing term added to the governing equations which is active in a small region downstream of the inflow boundary. This forcing in combination with characteristic boundary conditions allows for passing of upstream-propagating acoustic waves and avoids an uncontrolled drift of mean-flow quantities. We employ this inflow boundary treatment for a subsonic nozzle-jet flow simulation at a Reynolds number of ∼ 9500 and Mach number of 0.9. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Improved boundary treatment for the smoothed particle hydrodynamics method using implicit surfaces

The boundary treatment for modeling complex geometries in smoothed particle hydrodynamics (SPH) methods is always challenging, especially for objects with sharp corners. In this paper, we propose an improved boundary treatment technique to handle boundary conditions in the SPH method where objects are represented as implicit surfaces. The improved boundary treatment technique provides a full treatment for complicated geometries and can handle sharp edges without any special treatment. The boundary particles are uniformly distributed in compliance with the curvature of the objects, therefore taking into consideration the effect of the boundary. Several simulations are presented to validate and demonstrate the applicability and versatility of the proposed technique.     

Online paths in mathematics teacher training: new resources and new skills for teacher educators

This paper considers the work carried out by online teacher educators and their professional development. We use the theoretical perspective of the documentational approach that focuses, in this case, on the interaction between teacher educators and the resources they use for their online training work. We thus study the following issues: (1) What kinds of resources do online teacher educators need, and how are such resources modified according to the educators?? specific skills and needs? (2) What specific skills are needed for setting up online training for mathematics teachers and how do these skills evolve as teacher education resources are used? We consider both questions simultaneously, while presenting results from a study within a specific teacher training programme in France that proposes ??training paths?? on a national platform. These ??paths?? are resources designed for teacher educators. We follow the appropriation of two training paths by two educator teams. The ways in which these educator teams were able to appropriate the paths give insights into the teacher educators?? skills and, as well, into the resources they need. By looking at their use of resources (as online mathematics teacher educators), we observe and analyse professional geneses, leading to the development of new skills.     

A modification of the invariant imbedding method for a singular boundary value problem

We consider a dynamically-consistent analytical model of a 3D topographic vortex. The model is governed by equations derived from the classical problem of the axisymmetric Taylor–Couette flow. Using linear expansions, these equations can be reduced to a differential sixth-order equation with variable coefficients. For this differential equation, we formulate a boundary value problem, which has a number of issues for numerical solving. To avoid these issues and find the eigenvalues and eigenfunctions of the boundary value problem, we suggest a modification of the invariant imbedding method (the Riccati equation method). In this paper, we show that such a modification is necessary since the boundary conditions possess singular matrices, which sufficiently complicate the derivation of the Riccati equation. We suggest algebraic manipulations, which permit the initial problem to be reduced to a problem with regular boundary conditions. Also, we propose a method for obtaining a numerical solution of the matrix Riccati equation by means of recurrence relations, which allow us to obtain a matrizer converging to the required eigenfunction. The suggested method is tested by calculating the corresponding eigenvalues and eigenfunctions, and then, by constructing fluid particle trajectories on the basis of the eigenfunctions.     

Tabu search with strategic oscillation for the maximally diverse grouping problem

We propose new heuristic procedures for the maximally diverse grouping problem (MDGP). This NP-hard problem consists of forming maximally diverse groups—of equal or different size—from a given set of elements. The most general formulation, which we address, allows for the size of each group to fall within specified limits. The MDGP has applications in academics, such as creating diverse teams of students, or in training settings where it may be desired to create groups that are as diverse as possible. Search mechanisms, based on the tabu search methodology, are developed for the MDGP, including a strategic oscillation that enables search paths to cross a feasibility boundary. We evaluate construction and improvement mechanisms to configure a solution procedure that is then compared to state-of-the-art solvers for the MDGP. Extensive computational experiments with medium and large instances show the advantages of a solution method that includes strategic oscillation.     

On the geometry of model almost complex manifolds with boundary

We study some special almost complex structures on strictly pseudoconvex domains in ℝ^{2 n} . They appear naturally as limits under a nonisotropic scaling procedure and play a role of model objects in the geometry of almost complex manifolds with boundary. We determine explicitely some geometric invariants of these model structures and derive necessary and sufficient conditions for their integrability. As applications we prove a boundary extension and a compactness principle for some elliptic diffeomorphisms between relatively compact domains.     

Rankings and values for team games

In this article we attack several problems that arise when a group of individuals is organized in several teams with equal number of players in each one (e.g., for company work, in sports leagues, etc). We define a team game as a cooperative game v that can have non-zero values only on coalitions of a given cardinality; it is further shown that, for such games, there is essentially a unique ranking among the players. We also study the way the ranking changes after one or more players retire. Also, we characterize axiomatically different ways of ranking the players that intervene in a cooperative game.     

Towards an Account of Epistemic Luck for Necessary Truths

Modal epistemologists parse modal conditions on knowledge in terms of metaphysical possibilities or ways the world might have been. This is problematic. Understanding modal conditions on knowledge this way has made modal epistemology, as currently worked out, unable to account for epistemic luck in the case of necessary truths, and unable to characterise widely discussed issues such as the problem of religious diversity and the perceived epistemological problem with knowledge of abstract objects. Moreover, there is reason to think that this is a congenital defect of orthodox modal epistemology. This way of characterising modal epistemology is however optional. It is shown that one can non-circularly characterise modal conditions on knowledge in terms of epistemic possibilities, or ways the world might be for the target agent. Characterising the anti-luck condition in terms of epistemic possibilities removes the impediment to understanding epistemic luck in the case of necessary truths and opens the door to using these conditions to shed new light on some longstanding epistemological problems.     

Generalization of Backward Differentiation Formulas for Parallel Computers

We analyze an extension of backward differentiation formulas, used as boundary value methods, that generates a class of methods with nice stability and convergence properties. These methods are obtained starting from the boundary value GBDFs class, and are in the class of EBDF-type methods. We discuss different ways of using these linear multistep formulas in order to have efficient parallel implementations. Numerical experiments show their effectiveness.     

Riemann Boundary Value Problems for Iterated Dirac Operator on the Ball in Clifford Analysis

In this paper we consider the Riemann boundary value problem for null solutions to the iterated Dirac operator over the ball in Clifford analysis with boundary data given in $\mathbb L _{p}\left(1<p<+\infty \right)$ -space. We will use two different ways to derive its solution, one which is based on the Almansi-type decomposition theorem for null solutions to the iterated Dirac operator and a second one based on the poly-Cauchy type integral operator.     

The Stability of Self-Organized Rule-Following Work Teams

Self-organized rule-following systems are increasingly relevant objects of study in organization theory due to such systems&2018; capacity to maintain control while enabling decentralization of authority. This paper proposes a network model for such systems and examines the stability of the networks&2018; repetitive behavior. The networks examined are Ashby nets, a fundamental class of binary systems: connected aggregates of nodes that individually compute an interaction rule, a binary function of their three inputs. The nodes, which we interpret as workers in a work team, have two network inputs and one self-input. All workers in a given team follow the same interaction rule.We operationalize the notion of stability of the team&2018;s work routine and determine stability under small perturbations for all possible rules these teams can follow. To study the organizational concomitants of stability, we characterize the rules by their memory, fluency, homogeneity, and autonomy. We relate these measures to work routine stability, and find that stability in ten member teams is enhanced by rules that have low memory, high homogeneity, and low autonomy.     

Selecting k objects from a cycle with p pairs of separation s

Recently Konvalina determined the number of ways of selecting k objects from a cycle of n objects with no two selected objects separated by exactly one object. For arbitrary integers p, k, and s, we show that a recent result of the author can be applied to determine the number of ways of selecting k objects (from a cycle of n objects) which contain exactly p pairs of selected objects separated by exactly s objects. Konvalina's result follows as the special case p = 0 and s = 1.     

Strategic oscillation for the quadratic multiple knapsack problem

The quadratic multiple knapsack problem (QMKP) consists in assigning a set of objects, which interact through paired profit values, exclusively to different capacity-constrained knapsacks with the aim of maximising total profit. Its many applications include the assignment of workmen to different tasks when their ability to cooperate may affect the results. Strategic oscillation (SO) is a search strategy that operates in relation to a critical boundary associated with important solution features (such as feasibility). Originally proposed in the context of tabu search, it has become widely applied as an efficient memory-based methodology. We apply strategic oscillation to the quadratic multiple knapsack problem, disclosing that SO effectively exploits domain-specific knowledge, and obtains solutions of particularly high quality compared to those obtained by current state-of-the-art algorithms.     

A Novel Category of Vague Abstracta

Much attention has been given to the question of ontic vagueness, and the issues usually center around whether certain paradigmatically concrete entities - cats, clouds, mountains, etc. - are vague in the sense of having indeterminate spatial boundaries. In this paper, however, I wish to focus on a way in which some abstracta seem to be locationally vague. To begin, I will briefly cover some territory already covered regarding certain types of “traditional” abstracta and the ways they are currently alleged to be vague. I then wish to discuss two types of “nontraditional” abstracta and the sense in which I think some of these objects are locationally vague. I will next reexamine some of the traditional abstracta and discuss whether any of these objects are locationally vague in the novel way suggested for the nontraditional sorts. I’ll finish by discussing objections, and conclude with some remarks about characterizing the abstract/concrete distinction.     

On surface radiation conditions for high-frequency wave scattering

A new approximation of the logarithmic derivative of the Hankel function is derived and applied to high-frequency wave scattering. We re-derive the on surface radiation condition (OSRC) approximations that are well suited for a Dirichlet boundary in acoustics. These correspond to the Engquist–Majda absorbing boundary conditions. Inverse OSRC approximations are derived and they are used for Neumann boundary conditions. We obtain an implicit OSRC condition, where we need to solve a tridiagonal system. The OSRC approximations are well suited for moderate wave numbers. The approximation of the logarithmic derivative is also used for deriving a generalized physical optics approximation, both for Dirichlet and Neumann boundary conditions. We have obtained similar approximations in electromagnetics, for a perfect electric conductor. Numerical computations are done for different objects in 2D and 3D and for different wave numbers. The improvement over the standard physical optics is verified.     

Heuristics for the mirrored traveling tournament problem

Professional sports leagues are a major economic activity around the world. Teams and leagues do not want to waste their investments in players and structure in consequence of poor schedules of games. Game scheduling is a difficult task, involving several decision makers, different types of constraints, and multiple objectives to optimize. The traveling tournament problem abstracts certain types of sport timetabling issues, where the objective is to minimize the total distance traveled by the teams. In this work, we tackle the mirrored version of this problem. We first propose a fast and effective constructive algorithm. We also describe a new heuristic based on the combination of the GRASP and iterated local search metaheuristics. A strong neighborhood based on ejection chains is also proposed and leads to significant improvements in solution quality. Very good solutions are obtained for the mirrored problem, sometimes even better than those found by other approximate algorithms for the less constrained non-mirrored version. Computational results are shown for benchmark problems and for a large instance associated with the main division of the 2003 edition of the Brazilian soccer championship, involving 24 teams.