Patterns in Differential Equations
 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 Gspaces [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.17l.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].

[Per, p.91, Theorem 5.5.5] and
[Per, p.133, Theorem 8.2.2] have similar proof patterns.
 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].
 Patterns of converse
It would be much easier to state the converse of a theorem by using the
language of set theory.
 (The basic pattern) The converse of A ÌB is B
ÌA.
 Variants.
 The converse of AÞB is BÞA.
 The converse of f (A) ÌB is B
Ìf (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].