Patterns in Differential Equations

  1. An action pattern
        Sometimes our goal is remote and complicated. To make the goal accessible, an expert has to lay out a framework for us. If this framework belongs to a familiar pattern, all we need to do is fill in the details.
    Examples.
    For the setting (Group, set), we establish an isomorphism of left G-spaces [Mas, p.256, l.-11].
    For the setting (Lie group, manifold), we establish a diffeomorphism [War, p.123, Theorem 3.62].
    For the setting (Effective [Po3, p.437, l.-4] local Lie group, open set in an Euclidean space), we establish  [Po3, p.437, Definition 50 & p.443, l.-17-l.-9].
    Remark 1. An action determines the quotient structure [Manifold: War, p.124, l.10; Lie group: War, p.124, Theorem 3.64].
    Remark 2. A pattern will make it easier to see the interaction of effectiveness among its components. Assume a pattern has two components. If one component becomes more specific, so does the other. In the action pattern, if the topological space is a manifold, then the compact transformation group will automatically become a Lie group [Po3, p.343, Theorem 75].

  2. [Per, p.91, Theorem 5.5.5] and [Per, p.133, Theorem 8.2.2] have similar proof patterns.

  3. Suppose we have a pattern. If we find an example which has a similar hypothesis, we may predict its conclusion.
    Example. [Dug, p.289, Theorem 6.5] [Ru3, p.270, Corollary].

  4. Patterns of converse
        It would be much easier to state the converse of a theorem by using the language of set theory.
    1. (The basic pattern) The converse of A B is B A.
    2. Variants.
      1. The converse of AB is BA.

      2. The converse of f (A) B is B f (A).
        Example. Let A = {X is a compact group containing an element a the multiples of which are everywhere dense in X}, f (X) = The weight of X, and B = the set of all cardinal numbers not exceeding the continuum [Po3, p.256, Example 64].