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

Mediatic graphs

Any medium can be represented as an isometric subgraph of the hypercube, with each token of the medium represented by a particular equivalence class of arcs of the subgraph. Such a representation, although useful, is not especially revealing of the structure of a particular medium. We propose an axiomatic definition of the concept of a `mediatic graph'. We prove that the graph of any medium is a mediatic graph. We also show that, for any non-necessarily finite set S, there exists a bijection from the collection M of all the media on a given set S (of states) onto the collection G of all the mediatic graphs on S.Comment: Four axioms replaced by two; two references added; Fig.6 correcte

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

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

Projections of a learning space

Any subset Q' of the domain Q of a learning space defines a projection of that learning space on Q' which is itself a learning space consistent with the original one. Moreover, such a construction defines a partition of Q having each of its classes defining a learning space also consistent with the original learning space. We give a direct proof of these facts which are instrumental in parsing large learning spaces.Comment: 13 pages, 1 figur

On a bounded version of Holder's Theorem and an application to the permutability equation

The permutability equation G(G(x,y),z) = G(G(x,z),y) is satisfied by many scientific and geometric laws. A few examples among many are: The Lorentz-FitzGerald Contraction, Beer's Law, the Pythagorean Theorem, and the formula for computing the volume of a cylinder. We prove here a representation theorem for the permutability equation, which generalizes a well-known result. The proof is based on a bounded version of Holder's Theorem.Comment: This paper will be submitted as a chapter in an edited volume honoring the 90th birthday of Patrick Suppes. The author presented the paper at a conference honoring Suppes 90th birthday at Stanford Universit

Consistency of Monomial and Difference Representations of Functions Arising from Empirical Phenomena

AbstractChoice probabilities in the behavioral sciences are often analyzed from the standpoint of a differencerepresentation such as P(x,x,y)=F[u(x,x)−g(y)]. Here, x and y are real, positive vector variables, x is a positive real variable, P(x,x,y) is the probability of choosing alternative (x,x) over alternative y, and u, g and F are real valued, continuous functions, strictly increasing in all arguments. In some situations (e.g. in psychophysics), the researchers are more interested in the functions u and g than in the function F. In such cases, they investigate the choice phenomenon by estimating empirically the value x such that P(x,x,y)=ρ, for some values of ρ, and for many values of the variables involved in x and y. In other words, they study the function ξ satisfying ξ(x,y;ρ)=x⇔P(x,x,y)=ρ. A reasonable model to consider for the function ξ is offered by the monomialrepresentationξx,y;ρ=∏n−1i=1x−ηi(ρ)i∏mj=1yζj(ρ)jCρ,in which the ηi's, the ζj's and C are functions of ρ. In this paper we investigate the consistency of these difference and monomial representations. The main result is that, under some background conditions, if both the difference and the monomial representations are assumed, then: (i) all functions ηi (1≤i≤n−1) must be constant; (ii) if one of the functions ζj is nonconstant, then all of them must be of the form ζj(ρ)=θjexp[δF−1(ρ)], for some constants θj>0 (1≤j≤m) and δ≠0, where F−1 is the inverse of the function F of the difference representation. Surprisingly, F can be chosen rather arbitrarily

Extensions of set functions

We establish a necessary and suficient condition for a function defined on a subset of an algebra of sets to be extendable to a positive additive function on the algebra. It is also shown that this condition is necessary and sufficient for a regular function defined on a regular subset of the Borel algebra of subsets of a given compact Hausdorff space to be extendable to a measure

The lattice dimension of a graph

We describe a polynomial time algorithm for, given an undirected graph G, finding the minimum dimension d such that G may be isometrically embedded into the d-dimensional integer lattice Z^d.Comment: 6 pages, 3 figure