##### Results from the WOW.Com Content Network

**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.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.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 45Â° 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.

**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 ...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.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 ...**Generalization**using Fenchelâ€“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.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.