New PDF release: Learning Spaces: Interdisciplinary Applied Mathematics

By Jean-Claude Falmagne

ISBN-10: 3642010385

ISBN-13: 9783642010385

ISBN-10: 3642010393

ISBN-13: 9783642010392

Learning areas provide a rigorous mathematical origin for varied sensible platforms of information evaluate. An instance is on the market by way of the ALEKS procedure (Assessment and studying in wisdom Spaces), a software program for the evaluate of mathematical wisdom. From a mathematical viewpoint, studying areas in addition to wisdom areas (which made the identify of the 1st version) generalize in part ordered units. they're investigated either from a combinatorial and a stochastic standpoint. the consequences are utilized to actual and simulated info. The ebook supplies a scientific presentation of analysis and extends the implications to new occasions. it's of curiosity to mathematically orientated readers in schooling, machine technological know-how and combinatorics at examine and graduate degrees. The textual content comprises a variety of examples and workouts, and an in depth bibliography.

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Example text

Thus, Q is necessarily finite. Axiom [MA] results immediately from the fact that, for any state K, there is an L1-chain from ∅ to K. Turning to the ∪-closure, we take any two states K and L in K and suppose that neither of them is empty or a subset of the other (otherwise ∪-closure holds trivially). Since ∅ ⊂ L, Axiom [L1] implies the existence of an L1–chain ∅ ⊂ {q1 } ⊂ · · · ⊂ {q1 , . . , qn } = L. Let j ∈ {1, . . , n} be the first index with qj ∈ / K. If j > 1; we have {q1 , . . , qj−1 } ⊂ K, and {q1 , .

A knowledge structure (Q, K) is called a quasi learning space if it satisfies the two following conditions. [L1∗ ] Quasi learning smoothness. 5) with p = e(K, L) and Ki = Ki−1 + {qi∗ } for some qi ∈ Q, 1 ≤ i ≤ p. 2) as a quasi L1–chain from K to L. [L2∗ ] Quasi learning consistency. If K, L are two states satisfying K ⊂ L and q is an item such that K + {q ∗ } ∈ K, then L ∪ {q ∗ } ∈ K. Our next definition introduces a nondiscriminative variant of the wellgradedness condition. 3 Definition. A family of sets F is quasi well-graded or a qwg-family if, for any two distinct states K, L ∈ F, there exists a finite sequence of states K = K0 , K1 , .

In some cases, the base can be considerably smaller than the knowledge space, which results in a substantial economy of storage in a computer memory. The extreme case is the power set of a set of n elements, where the 2n knowledge states can be subsumed by the family of the n singleton sets. 6. Other features of knowledge spaces are also important, however, and are dealt with in this chapter. In the next section, we mention in passing the problem of building a knowledge space in practice. The key idea is to code the information concerning the structure in the form of a relation R on the power set of the domain, with the following interpretation: ARB holds if failing all the items in A implies failing all those in B.

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Learning Spaces: Interdisciplinary Applied Mathematics by Jean-Claude Falmagne

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