Interfaces in Differential Equations

    An interface serves to establish an intertwining close relationship between two theories by constantly shifting back and forth from one theory to the other.
  1. Systematically interface the theory of differential equations with linear algebra.
        The interface helps us gain insight into the solutions of a differential equation.
    1. vectors functions.
    2. matrices linear operators.
    3. differential equations differential operators [Arf, p.538, (9.1)].
    4. the quadratic form for a matrix <u|L|u>, where L is a differential operator [Arf, p.538, l.15].
    5. self-adjoint differential equations the more general self-adjoint differential equations [Arf, p.540, (9.8)] self-adjoint differential operators [Arf, p.539, l.5].
      Remark. A self-adjoint operator is similar to a symmetric matrix [Arf, p.539, l.2].
    6. boundary conditions [Arf, p.544, (9.20)] Hermitian operators [Arf, p.546, (9.25)].
      Remark. The above boundary conditions are usually satisfied when we deal with a real physical system [Arf, p544, l.16].

  2. Since 1900, only three great books on differential equations have been published: [Inc1], [Cou] and [Guo]. It is necessary to let the duty of book writing on this subject pass between physicists and mathematicians every 25 years. Thereby, the subject can continue to interface with the new development in both areas and we can remain up-to-date with current thinking about differential equations.

    1. (Reducing the problem of classifying compact surfaces to a problem in group theory) [Mun00, p.477, l.1-l.3; p.478, l.1-l.12]
      Remark. [1].
    2. (Reducing a problem in group theory to a problem in the theory of covering spaces [Mun00, p.501, l.1-l.8].
          This approach provides us with concrete resources and enables us to gain the insight (geometric meaning) of the lengthy proof  when we attack an abstract problem.
      1. [Mun00, p.514, Theorem 85.1].
        Remark. (free abelian group vs. free group) [Mun00, p.424, l.1-l.6; Jaco, vol. 2, p.78, Theorem 4]. The proof of [Jaco, vol. 2, p.78, Theorem 4] uses mathematical induction alone, while the proof of [Mun00, p.514, Theorem 85.1] uses the axiom of choice in two places [Mun00, p.503, l.-6; p.509, l.-13]. However, these two usages can be avoided if we restrict our consideration to a free group with a finite number of generators and a finite linear graph. When we formulate a theory and weigh the priorities, we should adopt the following order: quality first (use dubious axioms as seldom as possible); practicability (common usage) second; generality third.
      2. [Mun00, p.515, Theorem 85.3]
        Remark 1. [Mun00, p.514, Lemma 85.2] provides a direct link between the system of free generators in group theory and the Euler number in topology.
        Remark 2. (free abelian group vs. free group) [Mun00, p.424, l.7-l.11].
        Remark 3. Let F be the fundamental group of a figure 8 and H be the fundamental group of its covering space given in [Mun00, p.490, Fig. 81.2]. Then both H [Mun00, p.515, l.19; p.348, Lemma 55.1] and F/H [Mun00, p.515, l.17; p.346, Theorem 54.6 (b)] are infinite.

  4. Links {1}.