Using video as a tool for promoting inquiry among preschool teachers and didacticians of mathematics

This paper explores the use of video as a tool for promoting inquiry among preschool teachers and didacticians. In this case, the didacticians are teacher educators who are also mathematics education researchers. Preschool teachers recorded themselves with video implementing number and geometry tasks with children and shared these recordings with other teachers and didacticians. The session where the teachers and didacticians viewed and discussed these recordings was recorded and viewed later by a group of didacticians. The multiple uses of video led to inquiry on several levels. Teachers inquired into the practice of implementing tasks with children, evaluating children’s knowledge, and the practice of using video as a tool. Didacticians inquired into their practice of research with children, their practice as teacher educators, the use of video as a tool in professional development, and the use of video in their inquiry process. Teachers’ and didacticians’ inquiries led to increased appreciation for the practice of inquiry, belonging to a community of practice, and its role in promoting both teachers’ and didacticians’ knowledge for teaching.     

Analysis of a kinetic cellular model for tumor-immune system interaction

Starting at a kinetic level from the equations for the evolution of dominance in populations of interacting organisms, and taking proliferative and destructive encounters into account, a simple model describing the competition between tumor cells and immune system is studied in some detail. Under reasonable assumptions, a closed set of macroscopic balance equations for macroscopic observables is derived by a moment procedure, and analyzed in the frame of the theory of dynamical systems. It is shown that a transcritical bifurcation of equilibria generates a region in the phase space in which, according to the model, the immune system defeats the tumor and leads to its depletion. Numerical results are presented and briefly commented on.     

Using dynamic graphs to reveal student reasoning

Using dynamic graphs, future secondary mathematics teachers were able to represent and communicate their understanding of a brief mathematical investigation in a way that a symbolic proof of the problem could not. Four different student work samples are discussed.     

Using a Volterra system model to analyze nonlinear response in video-packet transmission over IP networks

This paper presents a Volterra system-based nonlinear analysis of video-packet transmission over IP networks. With the Volterra system, which is applicable to the modeling of nonlinear dynamic systems from sets of input and output data, we applied a time-series analysis of measured data for network response evaluation. In a test-bed connected to the Internet, we measured two parameters: the time intervals between consecutive packets from a video server at the originating side, and the transmission time of packets between originating and terminating sides. We used these as input and output data for the Volterra system and confirmed that the relative error of this model changed with conditions of network systems, which suggested that the packet transmission process affected the degree of nonlinearity of the system. The proposed method can reproduce the time-series responses observed in video-packet transmission over the Internet, reflecting nonlinear dynamic behaviors such that the obtained results provided us with an effective depiction of network conditions at different times.     

Using FSBT technique with Rough Set Theory for personal investment portfolio analysis

This study proposes a novel Forward Search and Backward Trace (FSBT) technique based on Rough Set Theory to improve data analysis and extend the scope of observations made from sample data to solve personal investment portfolio problems. Rough Set Theory mathematically classifies data into class sets. The class set with the most objects may generate one decision rule. The rules generated from RST are rough and fragmented, that are very difficult to interpret the information. An empirical case is used to generate more than 85 rules by the RST method in comparison with FSBT method which only generated 14 rules. This result can show our proposed method is better than traditional RST method based on class sets that contain the most objects. Much of human knowledge is described in natural language. It is a very important thing to convert information from computer databases into normal human language. Sample data taken from features with the same backgrounds are used to compile different portfolios that investment companies and investment advisors can employ to satisfy the investor’ needs. The method not only can provide decision-making rules, but also can offer alternative strategies for better data analysis. We believe that the FSBT technique can be fully applied in research on investment marketing.     

Study of a multi-component system with failure interaction

In this paper we formulate and study a special type of failure interaction, between components, in a multi-component system. We derive expressions for the expected costs of operating the system for both finite and infinite time operation under two different operating policies.     

An algorithm for reducing the moment of inertia of a system interaction matrix

An algorithm is presented which reduces the moment of inertia of a system interaction matrix by applying a sequence of permutation transformations. The algorithm is shown to have certain finite convergence properties. The application of the technique to several system matrices is discussed with a view to revealing some of the latent structure within the model.     

Online collaborative mind mapping in a mathematics teacher education program: a study on student interaction and knowledge construction

The inclusion of alternative and multimodal methods for online interaction and knowledge construction in mathematics teacher education is still an incipien     

Strong solutions to a nonlinear fluid structure interaction system

In this paper, we prove the existence of local-in-time smooth solutions to the nonlinear fluid structure interaction model first introduced in [J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, 1969] and considered in [V. Barbu, Z. Gruji?, I. Lasiecka, A. Tuffaha, Existence of the energy-level weak solutions for a nonlinear fluid-structure interaction model, in: Fluids and Waves, in: Contemp. Math., vol. 440, Amer. Math. Soc., Providence, RI, 2007, pp. 55-82; V. Barbu, Z. Gruji?, I. Lasiecka, A. Tuffaha, Smoothness of weak solutions to a nonlinear fluid-structure interaction model, Indiana Univ. Math. J. 57 (3) (2008) 1173-1207]. In particular, the strong solutions here are obtained given initial datum for the Navier-Stokes equation in the space H¹, and initial data for the wave equation w₀ and w₁ in the spaces H²(Ωe) and H¹(Ωe) respectively.     

Asymptotic fluctuations of a particle system with singular interaction

A particle system on the real line with singular interaction consisting of electrostatic repulsion and a linear restoring force is considered. The empirical measure process is known to converge weakly in a space of continuous-measure-valued functions. In this work we show that the fluctuations around the limiting process, appropriately scaled, converge weakly to a Gaussian-distribution-valued process.     

Periodic solutions for a semi-ratio-dependent predator-prey system with Holling IV functional response

The existence of a positive periodic solution for the semi-ratio-dependent predator-prey system with Holling IV functional response is established by using the continuous theorem of coincidence degree theory. And a set of easily veribible sufficient conditions is derived by constructing a Lyapunov function. Numerical simulations are also exhibited to illustrate the validity of the results in this paper.     

Geodesically convex energies and confinement of solutions for a multi-component system of nonlocal interaction equations

We consider a system of n nonlocal interaction evolution equations on \({\mathbb{R}^d}\) with a differentiable matrix-valued interaction potential W. Under suitable conditions on convexity, symmetry and growth of W, we prove \({\lambda}\)-geodesic convexity for some \({\lambda\in\mathbb{R}}\) of the associated interaction energy with respect to a weighted compound distance of Wasserstein type. In particular, this implies existence and uniqueness of solutions to the evolution system. In one spatial dimension, we further analyse the qualitative properties of this solution in the non-uniformly convex case. We obtain, if the interaction potential is sufficiently convex far away from the origin, that the support of the solution is uniformly bounded. Under a suitable Lipschitz condition for the potential, we can exclude finite-time blow-up and give a partial characterization of the long-time behaviour.     

Boundary layers for a fluid–particle interaction system with density-dependent viscosity and cylindrical symmetry

In this paper, we study the initial boundary value problem for a cylindrical symmetry fluid–particle interaction system in three dimensions. The boundary layer phenomena is investigated when the shear viscosity $\mu =\kappa {\rho }^{\beta }$ goes to zero. Furthermore, we establish the boundary layer thickness of the order $O\left({\kappa }^{\alpha }\right)$ for more general initial data when $0<\alpha <\frac{1}{2}$ and give the optimal boundary-layer thickness for the system with more general initial data. As a byproduct, this work improves the corresponding results in Yao et al. (2011) for isentropic compressible Navier–Stokes equations where $0<\alpha <\frac{1}{4}$.     

Kirkwood–Salsburg equation for a quantum lattice system of oscillators with many-particle interaction potentials

For a Gibbs system of one-dimensional quantum oscillators on a d-dimensional hypercubic lattice interacting via superstable pair and many-particle potentials of finite range, we prove the existence of a solution of the (lattice) Kirkwood–Salsburg equation for correlation functions depending on the Wiener paths. Some many-particle potentials may be nonpositive.     

Permanence for a delayed discrete ratio-dependent predator-prey system with Holling type functional response

Sufficient conditions are established for the permanence in a delayed discrete predator-prey model with Holling type III functional response: