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  2. Cartographic generalization - Wikipedia

    Generalization is often defined simply as removing detail, but it is based on the notion, originally adopted from Information theory, of the volume of information or detail found on the map, and how that volume is controlled by map scale, map purpose, and intended audience.

  3. Homology (mathematics) - Wikipedia

    Background Origins. Homology theory can be said to start with the Euler polyhedron formula, or Euler characteristic. This was followed by Riemann's definition of genus and n-fold connectedness numerical invariants in 1857 and Betti's proof in 1871 of the independence of "homology numbers" from the choice of basis.

  4. Kakutani fixed-point theorem - Wikipedia

    A function with infinitely many fixed points. The function: () = [/, /], shown on the figure at the right, satisfies all Kakutani's conditions, and indeed it has many fixed points: any point on the 45° line (dotted line in red) which intersects the graph of the function (shaded in grey) is a fixed point, so in fact there is an infinity of fixed points in this particular case.

  5. Grounded theory - Wikipedia

    Grounded theory has been criticized based on the scientific idea of what a theory is. Thomas and James, for example, distinguish the ideas of generalization, over-generalization, and theory, noting that some scientific theories explain a broad range of phenomena succinctly, which grounded theory does not. Thomas and James observed that "The ...

  6. Unique factorization domain - Wikipedia

    A has a divisor theory in which every divisor is principal. A is a Krull domain in which every divisorial ideal is principal (in fact, this is the definition of UFD in Bourbaki.) A is a Krull domain and every prime ideal of height 1 is principal. In practice, (2) and (3) are the most useful conditions to check.

  7. Ensemble learning - Wikipedia

    Gating is a generalization of Cross-Validation Selection. It involves training another learning model to decide which of the models in the bucket is best-suited to solve the problem. Often, a perceptron is used for the gating model. It can be used to pick the "best" model, or it can be used to give a linear weight to the predictions from each ...

  8. Young's inequality for products - Wikipedia's_inequality_for...

    Generalization using Fenchel–Legendre transforms. If f is a convex function and its Legendre transformation (convex conjugate) is denoted by g, then + (). This follows immediately from the definition of the Legendre transform.

  9. Euler's theorem - Wikipedia's_theorem

    Subsequently, Euler presented other proofs of the theorem, culminating with his paper of 1763, in which he proved a generalization to the case where n is not prime. [2] The converse of Euler's theorem is also true: if the above congruence is true, then a {\displaystyle a} and n {\displaystyle n} must be coprime.