Links between power law probability distributions and marginal distributions of uniform laws on p-spheres in R show that a mathematical derivation of the Boltzmann-Gibbs distribution necessarily passes through power law ones. Results are also given that link parameters p and n to the value of the nonextensivity parameter q that characterizes these power laws in the context of non-extensive stat...

Complex systems concepts are important to learn for both scientific and pedagogical reasons. However, research has shown that these concepts are difficult for students to learn. In this paper we present a series of microworlds called “BeeSmart” for students to engage in a constructionist learning experience through manipulating NetLogo models, discussing with peers, and co-constructing meanings...

A pilot study has demonstrated that dynamic feedback systems concepts can be taught to children as young as ten and eleven years old, when nurtured by a teaching strategy that follows Bruner’s emphasis upon the teaching of basic structures. Focusing on the understanding and representation of underlying structures in the area of social studies, the author developed, implemented and evaluated a c...

The mathematical concept of time–changing continuous–time stochastic processes can be regarded as one of the standard tools for building financial models. This article reviews briefly the theory on time–changed stochastic processes and relates them to stochastic volatility models in finance. Popular models, including time–changed Lévy processes, where the time–change process is given by a subor...

Progress in the foundations of mathematics has made it possible to formulate all thinkable mathematical concepts, algorithms and proofs in one language and in an impeccable way. This is not in spite of, but partially based on the famous results of Gödel and Turing. In this way statements are about mathematical objects and algorithms, proofs show the correctness of statements and computations, a...

Proof is a construct of mathematical communities over many generations and is introduced to new generations as they develop cognitively in a social context. Here I present a practical framework for this development in simple terms that nevertheless has deep origins. The framework builds on an analysis of the growth of mathematical ideas based on genetic facilities set-before birth. It unfolds a...

Mathematics and painting are interrelated in many ways. At a technical level mathematics can be used to enhance our appreciation of paintings. This was recently demonstrated beautifully by Taylor, Micolich, and Jones in their analysis of Jackson Pollock’s drip paintings, which led them to the conclusion that Pollock’s paintings are in a way Ž gurative pictures. Relations also exist at a more ba...

This thesis introduces a new integrated formalization of the conceptual spaces framework proposed by Gärdenfors. We extend the notion of conceptual space generated by a set of prototypes by adding a weight vector to each prototype. This will allow us to represent non-uniformly scaled concepts and, as a special case, partial concepts. We also present a novel Weighted Prototypes-Based Categorizat...

Most theories of analogical transfer focus on similarities between the learning and transfer domains, where transfer is more likely between domains that share common surface features, similar elements, or common interpretations of structure. We suggest that characteristics of the learning instantiation alone can give rise to different levels of transfer. We propose that concreteness of the lear...

Abstract. We propose a mathematical derivation of Brinkman’s force for a cloud of particles immersed in an incompressible viscous fluid. Specifically, we consider the Stokes or steady Navier-Stokes equations in a bounded domain Ω ⊂ R for the velocity field u of an incompressible fluid with kinematic viscosity ν and density 1. Brinkman’s force consists of a source term 6πνj where j is the curren...