Knowledge Spaces and Learning Spaces

How to design automated procedures which (i) accurately assess the knowledge of a student, and (ii) efficiently provide advices for further study? To produce well-founded answers, Knowledge Space Theory relies on a combinatorial viewpoint on the assessment of knowledge, and thus departs from common, numerical evaluation. Its assessment procedures fundamentally differ from other current ones (such as those of S.A.T. and A.C.T.). They are adaptative (taking into account the possible correctness of previous answers from the student) and they produce an outcome which is far more informative than a crude numerical mark. This chapter recapitulates the main concepts underlying Knowledge Space Theory and its special case, Learning Space Theory. We begin by describing the combinatorial core of the theory, in the form of two basic axioms and the main ensuing results (most of which we give without proofs). In practical applications, learning spaces are huge combinatorial structures which may be difficult to manage. We outline methods providing efficient and comprehensive summaries of such large structures. We then describe the probabilistic part of the theory, especially the Markovian type processes which are instrumental in uncovering the knowledge states of individuals. In the guise of the ALEKS system, which includes a teaching component, these methods have been used by millions of students in schools and colleges, and by home schooled students. We summarize some of the results of these applications

On Verifying and Engineering the Well-gradedness of a Union-closed Family

Current techniques for generating a knowledge space, such as QUERY, guarantees that the resulting structure is closed under union, but not that it satisfies wellgradedness, which is one of the defining conditions for a learning space. We give necessary and sufficient conditions on the base of a union-closed set family that ensures that the family is well-graded. We consider two cases, depending on whether or not the family contains the empty set. We also provide algorithms for efficiently testing these conditions, and for augmenting a set family in a minimal way to one that satisfies these conditions.Comment: 15 page

Primary Facets Of Order Polytopes

Mixture models on order relations play a central role in recent investigations of transitivity in binary choice data. In such a model, the vectors of choice probabilities are the convex combinations of the characteristic vectors of all order relations of a chosen type. The five prominent types of order relations are linear orders, weak orders, semiorders, interval orders and partial orders. For each of them, the problem of finding a complete, workable characterization of the vectors of probabilities is crucial---but it is reputably inaccessible. Under a geometric reformulation, the problem asks for a linear description of a convex polytope whose vertices are known. As for any convex polytope, a shortest linear description comprises one linear inequality per facet. Getting all of the facet-defining inequalities of any of the five order polytopes seems presently out of reach. Here we search for the facet-defining inequalities which we call primary because their coefficients take only the values -1, 0 or 1. We provide a classification of all primary, facet-defining inequalities of three of the five order polytopes. Moreover, we elaborate on the intricacy of the primary facet-defining inequalities of the linear order and the weak order polytopes

Note: Axiomatic Derivation of the Doppler Factor and Related Relativistic Laws

The formula for the relativistic Doppler effect is investigated in the context of two compelling invariance axioms. The axioms are expressed in terms of an abstract operation generalizing the relativistic addition of velocities. We prove the following results. (1) If the standard representation for the operation is not assumed a priori, then each of the two axioms is consistent with both the relativistic Doppler effect formula and the Lorentz-Fitzgerald Contraction. (2) If the standard representation for the operation is assumed, then the two axioms are equivalent to each other and to the relativistic Doppler effect formula. Thus, the axioms are inconsistent with the Lorentz-FitzGerald Contraction in this case. (3) If the Lorentz-FitzGerald Contraction is assumed, then the two axioms are equivalent to each other and to a different mathematical representation for the operation which applies in the case of perpendicular motions. The relativistic Doppler effect is derived up to one positive exponent parameter (replacing the square root). We prove these facts under regularity and other reasonable background conditions.Comment: 12 page

Territorial grids: space versus population

International audienc

Simuler la baisse de fécondité en Inde

In the second half of the twentieth century, the limitation of fertility became a real concern for developing countries. In fact, the fertility rate of the majority of the countries of the world decreased during that period. India will see a multitude of different policies applied on a local and regional scale. The decline of fertility observed since 1951 can be likened to a diffusion of innovation, the origin of which is the South of the country. The decision to have fewer children remains individual, although it can be influenced by various cultural and social criteria. Individual decisions at a micro scale are thus at the origin of a phenomenon of diffusion at a macro scale. To understand the phenomenon of diffusion from a local scale, it is advisable to situate oneself at an individual‐centered scale. The difficulty of observing behaviors based on hypotheses placed at an individual‐centered scale can be surmounted by the recourses to multi‐agent systems.Dans la seconde moitié du XXe siècle, la limitation de la fécondité devient une réelle préoccupation pour les pays en voie de développement. De fait, le taux de fécondité de la majorité des pays du monde diminue pendant cette période. L'Inde verra une multitude de politiques différentes appliquées à l'échelle locale et régionale. La baisse de fécondité observée depuis 1951 peut être assimilée à une diffusion de l'innovation, dont l'origine est au sud du pays. La décision de faire moins d'enfants reste individuelle, bien qu'elle puisse être influencée par différents critères culturels et sociaux. Des décisions individuelles sont donc à l'origine d'un phénomène de diffusion à une échelle supérieure. Afin de saisir le phénomène de diffusion à partir d'une échelle locale, il convient de se placer à une échelle individucentrée. La difficulté que représente l'observation de comportements basés sur des hypothèses posées à une échelle individu‐centrée peut être surmontée par le recours aux Systèmes Multi‐Agents