Unification in Differential Equations

    Confucius said, "My theories are one." In terms of mathematics, "one" can mean two things: The first is unification, and the second is math network. [Cou, vol.1, Preface, p.v,, l.6] warns that our resources for solving a problem may dry out if we fail to reach the goal of interdisciplinary thought. Here are several examples: (1). The formulation of boundary and initial conditions [Sne, p.294, (2) & (3)] and the idea of using Green's functions [Sne, p.295, l.12-l.16] to solve PDE's both come from physics. (2). In order to intuitively predict the existence of a solution [Pet, p.4, l.22-25], it is necessary to understand a PDE's physical origin [Pet, p.2, l.7-p.4, l.13].

  1. Single piece.
        The solutions of a DE seem to be unrelated because we just see parts of a single piece. (1). du/dx=x2 has two real solutions u = -1/x (x<0 & x>0). The complex solution w = -1/z includes both of the above real solutions [Bir, p.222, l.8-l.20]. (2). Periodic linear systems [Bir, p.325, (12)]: A continuation with the initial-value vector multiplied by constants [Bir, p.325, l.-13-l.-10]. (3). Analytic continuation: [Bir, p.223, Example 1; p.224, Example 2].
        Although C1 (W)C2 (W)C (W) Cω (W) [Joh, p.64, l.-4], we often find that a math statement is valid for Ci (W), where i=1,2,...,,ω, and that their proofs repeat the same pattern. In the domain C, differentiable functions and analytic functions are the same, therefore, the above phenomenon is fully expected.
        Cusp locus, envelope, and tac locus seem to be elusive and loosely related phenomena for a particular DE. From Ince's geometric point of view [Inc, 24-26], we see that in the general case they are just branches of the p-discriminant locus.
        [Inc, pp.96-97] gives an example of how to unify different shapes of curves.

  2. Universal features. A feature may be degenerated when the domain is restricted: Any quadratic real polynomial can always be factored into complex linear factors, but not into real linear factors.

  3. Establish a relation among solutions. (1). The linear equation: [Inc, p.37, Note.5]. (2). The Riccati equation: [Inc, p.41, Note.2].

  4. The purpose of decomplexification and complexification [Arn, pp.119-124, 18].
        In mathematics, the most important thing we need to learn is a proper attitude. Our focus is not a system with a specific setting or the language we use to describe its theory. We do not want to repeat every case [Arn, p.128, Remark 2]. It is essential to identify the central idea for these cases.

  5. Hitting two targets with one shot.
        We like to use one argument to serve two purposes.
    1. Birkhoff uses [Bir, p.25, (23)] to prove both the uniqueness and the continuous dependence on initial values at the same time.

    2. Hartman uses [Har, p.9, l.-1] to prove the uniqueness and estimate the error term at the same time.
    3. [Ru2, p.130, l.18].

  6. Like a double in baseball, we use the same ideas twice to find the primitive [Sne, p.30, Natani's method: Example 11]. At the first base (dy=0), we frame the primitive's shape [Sne, p.31, (9)]. At the second base (dz=0), we determine its exact expression. Another example is given by [Sne, p.11, Example 2].

  7. One viewpoint explains it all.
        After adapting a point of view to discuss a concept, we may use different names and twist the procedure a little. Sometimes our language is so complicated that the original viewpoint becomes unrecognizable. Actually all the derivatives come from the same origin (Energy: Not finished)

  8. Parallel theoriesOne can not grow without the other.
        Since conformal mappings and Green's functions are related by simple formulas [Sne, p.196, l.5-l.16], the theory of the Riemann mapping theorem [Ru2, p.302, Theorem 14.8] and the theory of Dirichlet's problem are parallel. That is, the effectiveness of constructing a conformal mapping depends on the effectiveness of constructing a Green's function, and vice versa.

  9. (Making the sameness as strong as possible) Unification is an inductive method which applies to all the cases. [Che, p.8, Lemma 1] uses the exponential map to calculate dimensions in various cases (see [Che, p.8, Proposition 6]). Warner provides a strong version of this unifying method [War, p.104, Theorem 3.34] and uses it to determine the corresponding Lie algebra of a Lie subgroup (see [War, p.108]).

  10. A universal key.
        Contracting mapping theorem [Boo, p.43, Theorem 6.5] is a universal key for inverse operations: Inverse function theorem [Boo, p.45, l.1] & Existence theorem of ODE's.

  11. Unifications in many aspects.
        Let F1 be the field of p-adic numbers and F2 be the field of power series over the field of residue classes modulo p. Pontryagin shows the similarities between F1 and F2 in many aspects [Po3, p.163, l.-9-l.-5; p.164, l.-17-l.-15; p.165, l.11] and thus strengthens their bond.

  12. (Universal Criterion) The definition of invariance given in [Po3, p.449, A)] depends on substructures. Namely, it is not obvious that the invariance in a Lie algebra implies the invariance in its subalgebras. However, the criterion in [Po3, p.449, (1)] breaks down the boundary of substructures and makes the above statement obvious.

  13. (Modify what we have rather than start all over) If the proofs of Theorem A and Theorem B are similar, we would like to use the simpler proof as a foundation for developing the other proof through slight modifications [Po3, p.501, l.-14-l.-7].

  14. The definitions of curvature, torsion, and skew curvature are all based on the same concept: the arc-rate of turning [Wea1, p.11, l.-13; p.14, l.-5; p.17, l.5]. In contrast, [Kre, p.37, l.4] assigns the rule for the third curvature without any geometric interpretation.

  15. The development of unification
        Jackson uses the same technique to prove both the orthogonality of the Legendre polynomials and that of the Bessel functions of the first kind of order n in the field of electrostatics without mentioning their similarity [Jack, (3.19) & (3.94)]
    Guo uses the same technique to prove both the orthogonality of the Legendre polynomials and that of the Bessel functions of the first kind of order n in the field of special functions without mentioning their similarity [Guo, p.220, (4); p.424, (1)] (The scope is narrowed, so the complexity is reduced)
    Choudhury uses the same technique to prove both the orthogonality of the Legendre polynomials and that of the Bessel functions of the first kind of order n and points out their similarity [Chou, p.614, l.-3-p.615, l.1]
    [Bir, p.258, Theorem 1] (The precise statement of unification is established).
    Remark. Considering the case of the Bessel functions, the appropriate boundary conditions for [Bir, p.258, Theorem 1] should be those that make the right-hand side of [Bir, p.258, (8)] equal to 0. This is because these boundary conditions would be more flexible than those given in [Bir, p.256, (2)]. See [Bir, p.264, l.-6-l.-5].

  16. (Integrations in Banach spaces compared to integration on topological groups) Linear functionals versus characters.
        Even though the characters in additive number theory, the characters of a topological group, and the distributions in functional analysis are separately developed, they all inevitably fall into the same pattern.
    1. Duality. [Ru3, p.90, l.16] [Po3, p.240, Definition 37].
    2. Separation. [Ru2, p.114, Theorem 5.19] [Po3, p.245, Example 62].

  17. From the viewpoint of unification, it is insufficient to only discuss the similarity between two specific cases. We must establish the general case and then establish the connection between a specific case and the general case.
        Although [Edw1, p.499, Fig. 8.5.1; Fig. 8.5.2] show that circular functions and hyperbolic functions have similar geometric meanings, the discussion in [Edw1, 8.5] has a drawback regarding unification because it fails to present the general case [Gon1, p.389, Definition 5.9]. The discussion in [Gon1, 5.5] also has a drawback concerning unification because it fails to identify Cos with cosh [Edw1, p.498, (1)] when l = -1.

  18. The total variation of a sequence [Zyg, vol.1, p.4, l.8] and the total variation of a function [Ru1, p.117, l.18] can be unified by their common feature:
    S | f(x) - f(y) |, where x and y are consecutive points.

  19. [Cod, p.19, l.4-l.8] provides the reason that Coddington chooses |  | as the norm. However, in Cn, the norm is not compatible with that in C [Cod, p.32, l.18-l.19]. Consequently, we should unify the two norms through the associative law for product metric spaces.

  20. Links {1}.