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Generalization in Differential Equations

  1. Generalization.

    1. Ambiguity is the drawback of generalization because we do not know which case the author refers to. However, flexibility is the merit of generalization because we are not restricted to one specific case.

    2. There are two typical cases for stabilities: [Pon, p.117, Fig.28(a)] and [Arn1, p.56, Fig.45]. [Pon, p.208, Theorem 19] is a story with the picture of [Pon, p.117, Fig.28(a)] in mind. [Pon, p.284, Theorem 25] is a story with the picture of [Arn1, p.56, Fig.45] in mind. After formalization, the argument of [Pon, p.208, Theorem 19] can be seen as a special case of that of [Pon, p.284, Theorem 25]. More precisely, [Pon, p.208, Theorem 19]=[Bir, p.129, Poincar -Liapunov theorem] is a special case of [Pon, p.265, (C)]. However, the above two pictures should not be mixed up and the former typical case can not be omitted.

    3. The general solution of Cauchy's problem [Joh, p.11, chap.1, sec.5] provides a geometric picture [Joh, p.12, l.7]. For a specific case [Joh, p.15, Example (1)], the general solution should agree with the solution from simple consideration [Joh, p.15, l.14].

    4.  
      1. Goal.
            Generalization should be used as a means for us to determine whether a method in a system is more promising than others [Wan2, p.92, l.19].
      2. Method.
            The duty should be uniformly distributed to all variables which will constitute a cycle. For example, Jacobi's method [Sne, p.80] can be directly generalized to the case for n variables, while Charpit's method cannot [Sne, p.69].

    5. General theory versus specific cases.
          In order to interpret various physical phenomena, we search for the specific closed form solution of a DE. To obtain a solution in closed form relies on the features of a DE. A general DE may extend its scope of application, but lose its features. When a DE is generalized to the extreme, the only choice left to solve it is through numerical approach: Find an approximate solution and minimize its error [Joh, p.4, chap.1, sec.3]. Sometimes a general solution may recover a specific case [Joh, p.15, l.14]. However, for most cases, it won't.

    6. Ideal generalization.
      1. (Starting point) We start from a primitive model.
      2. (Development) There are a few impediments which require more sophisticated concepts to overcome.
      3. (Step-by-step agreement). When the sophisticated concepts are reduced to the primitive case, they should agree with the corresponding concepts in the original model.
        Example.
        1. Characteristic direction → Monge cone (For agreement, see [Joh, p.20, l.23]).
        2. (Characteristic equation) Characteristic curves [Joh, p.9, (4.3)] → Characteristic strips [Joh, p.22, (7.10 a,b), (7.14 a,b)] (For agreement, see [Joh, p.22, l.2]).
      Remark. For first-order PDE's, Sneddon discusses the general case first, and then the quasi-linear case [Sne, chap.2, sec.3 & 4]. His order of discussion deprives us of the opportunity to track the development of generalization.

    7. The 1-dim wave equation.
      d'Alembert's solution [Sne, p.215, (5)] → The Riemann-Volterration solution [Sne, p.223, (12)] (For agreement, see [Sne, p.224, l.2]).

    8. Requirement and purpose.
          A generalization requires at least two cases which have the same feature. We may use the same method [Dug, p.128, Ex.2] to construct the topology on a cone [Dug, p.126, Definition 5.1] and the topology on a suspension [Dug, p.127, Definition 5.3].

    9. The generalization from a sequence to a net serves many purposes. Of course, the more purposes a generalization serves, the more valuable it is. More importantly, a generalization is so specific to each individual purpose that it fits exactly what the situation demands.
      1. Inadequacy of sequences for topology [Dug, p.215, l.6-l.8]. Nets to a general topological space are like sequences to a 1E countable space [Dug, p.215, Theorem 4.1 & p.218, Theorem 6.2]. Accumulation points: Compare [Dug, p.214, Corollary 3.3(2)] with [Dug, p.214, l.−7]. Closure: Compare [Dug, p.215, Theorem 4.1] with [Dug, p.215, Ex. 1].
      2. Inadequacy of sequences for analysis. Riemann integrals [Dug, p.210, l.13-l.20 & p.212, Ex.7].

    10. A generalization can be too abstract to visualize the big picture.
          The exponential map can be defined on a real line, n x n matrices [Che, p.5, Definition 1] or a Lie algebra. Warner begins the definition with an abstract setting [War, p.102, (5)]. It would be much easier to invoke the properties of the exponential map [War, p.103, Theorem 3.31] if he were to start the definition with a concrete setting.

    11. Integration of a differential form. Each generalization of integration in [War, chap.4] serves a purpose. Unlike a telescope which expands in a single direction, different purposes may lead to different directions of expansion. We may take a subclass of one generalization [War, p.150, (7)] and expand it in another direction [War, p.150, (8)]. However, whenever two generalizations have a common domain, the resulting integrals must agree there [War, p.150, l.−15]. Most importantly, we must identify each generalization with its purpose: Manifold → From local to global [War, p.145, §4.8]. Chains → Triangulation of a domain [War, p.141, §4.6]. Lie Group → Left invariance of integral [War, p.151, §4.11]. Remark. The usefulness of a generalization depends on its direction of expansion. A blind generalization may be useless.

    12. Convergence of a sequence of functions.
      1. (fn→f on compact subsets) [Ru3, p.32, l.6-l.8] → (fn→f in the c-topology) [Dug, p.268, Theorem 7.2 ]. For agreement on metric spaces, see [Dug, p.268, Theorem 7.2 ].
      2. A more generalized concept of convergence may add more restrictions to the setting and then loosen the restrictions somewhat: C(Ω) [Ru3, p.31, §1.44] →H(Ω) [Ru3, p.32, §1.45] →C(Ω) [Ru3, p.32, §1.46].

    13. Generalizing a theorem fails to answer the following three questions of effectiveness: If the hypothesis is strengthened, how strong will the conclusion be? Can we find a more direct relationship between the stronger hypothesis and the stronger conclusion? What is the motive for the theorem? In view of the proof in [Arn1, p.250, Corollary], [Arn1, p.250, Corollary] is just a special case of [Arn1, p.249, Corollary]. The proof emphasizes the main mechanism [Arn1, p.249, the first part of (5)] because a(t) contains too many variables. It fails to show the direct relationship between W(t) and a1(t) in [Arn1, p.250, Corollary]. However, the short cut in the proof of [Bur, p.16, Theorem 8] elegantly and pointedly explains Liouville's formula's simple relation and its effectiveness for application.

    14. When a definition is generalized, on the one hand, it should preserve the essential properties of the old definition; on the other hand, it should include something new. Furthermore, we would like to provide effective methods to create some examples that belong to the new, but not old category. Schwabik generalizes the Lebesgue integral [Lee, p.22, l.11] to Henstock-Kurzweil integral [Schw, p.2, l.8]], but he fails to show that the generalization preserves the old properties [Lee, p.36, Theorem 6.22]. Therefore, Schwabik's interpretation of the Henstock integral is incomplete. The characterization of the Henstock integral in [Lee, p.36, Theorem 6.22] also measures the number of old properties that have been lost.

    15. The purposes of generalization.
      1. Untangling complexities and determining a relationship's main components.
        Example 1. [Ru2, p.p.63, Theorem 3.2] → [Baz, p.81, Theorem 3.1.3].
        Example 2. (Haar measure) [Hew, vol.1, p.199, (15.17) (d)] → [Hew, vol. 1, p.194, l.11].
      2. Extending intuition [Baz, p.89, l.−5-p.90, l.3].

    16. At first glance, it seems that the problem in [Col, p.109, II.§6] does not fit directly into the type of problems in [Col, p.92, II. §4]. However, a wise transformation [Col, p.110, (II.60)] may allow us to use the established theory [Col, p.111, l.77] and avoid repeating a similar argument.

    17. More generalizations can be created by adding parameters.
          For a string, the natural boundary conditions for a free boundary are [Cou, vol.1, p.246, (95)]. From the viewpoint of [Cou, vol.1, p.208, l.-10], it seems that a free boundary is the most general case for natural boundary conditions. However, we can still add parameters to create more generalizations. For example, after we add two parameters k1 & k2 [Fomi, p.156, l.-11] as in the mechanical design of [Fomi, p.155, Fig. 8], the natural boundary conditions [Fomi, p.158, l.-1] are generalized to [Fomi, p.158, (23) & (24)].

    18. A limiting case [Cou, vol.1, p.211, l.3-l.7; p.248, l.-17-l.-15] is also a special case.
          Suppose a general theorem contains a parameter a. If we substitute a with any finite value, we will obtain a special case. Similarly, if we let a = +, we will have the limiting case (strictly speaking, only a + rather than a = + is allowed in logic).

    19. A general theorem should guide the common approach to a certain type of problem. In other words, a general theorem should be formulated as a unified means for answering each of the following two questions. (1). What kind of conclusions shall we expect? (2). What assumptions are required?
      Example. General theorem: the Fubini theorem [Ru2, p.150, Theorem 7.8(c)].
      Special cases.
      1. Interchange summations [Ru1, p.161, Theorem 8.3]. 
      2. Interchange summation and integration [Ru2, p.23, Theorem 1.27; p.30, Theorem 1.38].

    20. Using a stepping stone to smooth a generalization.
      Example. (Generalization of the formulae of Frenet)
          [Kre, pp.305-306, 105] generalizes the formulae of Frenet directly from R3 to an n-dimensional Riemannian manifold. The transformation is somewhat abrupt. If we insert [Kli, pp.11-15, 1.3] (the Rn case) in between, it will smooth the generalization.

    21. If we use the device of tensors to prove a special case, then the general case is automatically valid. In other words, it would be unnecessary to prove the general case because the special case would suffice to justify the general case.
      Example. A special case: [Lau, p.126, (11.14)];  the general case: [Lau, p.126, (11.15)].
      Remark. Although the proof of the general case is unnecessary, the generalization requires a careful explanation. The explanation given in [Lau, p.125, l.-13-l.-7] is short and clear, while the explanation given in [Kre, 106] is long and vague.

    22. [Fin, 79 A, 79 B & 79 C] give three proofs for the formula of a tangent line. The proof in [Fin, 79 B] can be generalized to the general plane curve of the second degree [Fin, 171]. It also specifies the equation of a secant line. Even this side result is useful. The proof in [Fin, 79 C] can be generalized to any plane curve. The language used in [Fin, 79 C] is very rough. The meaning of its argument only focuses on one point the conclusion. For a given situation, we would like to use language that is as specific as possible so that we may get more information out of the material.

    23. Careful observations in a simple case are the key to generalization.
          The algebraic meaning of [Sne, p.54, (19)] is given by Cramer's Rule [Usp, p.232, l.19-l.24]. When n=2, it reduces to [Sne, p.52, (8)], whose geometric meaning is given by [Fin, 286].
          The derivation of [Sne, p.54, (21)] is similar to that of [Sne, p.47, (10)]. [Sne, p.54, (21)] stipulates the general rules. After these complicated rules are translated for the simple case such as [Sne, p.47, (10)], they can be easily recognized as the direct consequences of a determinant's properties. This observation will greatly facilitate the generalization.

    24. It seems that if we generalize a theorem, it will become ineffective. This is not necessarily true. When we generalize a theorem, we may preserve the theorem's effectiveness by remaining flexible. The proof given in [Gon, p.690, l.-6-p.691, l.3] and that given in [Gon, p.691, l.4-l.-1] use the same method, but apply the method to different figures. The former proof applies the method to a semicircle, while the latter applies the method to a more flexible shape: a sector. Using this flexible figure we remove an unnecessary restriction [Gon, p.689, l.11] in the hypotheses of [Gon, p.692, (9.11-8)]. See [Gon, p.692, l.17, Note]. Being inflexible would restrict the power of the method.

    25. When we generalize a theorem, the proof of the generalized theorem should preserve the insight and essence of the original theorem.
      (Cauchy's theorem) [Ahl, p.109, Theorem 2] (for a rectangle) [Ahl, p.143, Theorem 16] (for a simply connected region) [Ahl, p.145, Theorem 18] (for a multiply connected region)
          The essence of Cauchy's theorem is to construct an exact differential [Ahl, p.141, l.-11-l.-6]. In preparation [Ahl, p.142, Fig. 24] for proving [Ahl, p.141, Theorem 15], we should formulate the prototype of Cauchy's theorem in terms of rectangles [Ahl, p.109, Theorem 2] rather than triangles [Ru2, p221, Theorem 10.13]. The proof of [Ahl, p.143, Theorem 16] preserves the essence of Cauchy's theorem, while the proof of [Ru2, p.235, Theorem 10.35] and that of [Sak, p.177, Theorem 2.3] do not. The construction of U(z) given in [Ahl, p.141, l.-10] is independent of the choice of s [Ahl, p.141, l.-8], while the construction of h given in [Ru2, p.236, (7)] depends on the cycle G. The use of Runge's theorem in the proof of [Sak, p.177, Theorem 2.3] obscures the essence of Cauchy's theorem. As stated in [Ahl, p.113, l.-10-l.-8], the shape of a region [Ru2, p.223, Theorem 10.14] and the type of a regular closed curve [Sak, p.189, Fig.9] are trivia which only divert our attention from the essence of Cauchy's theorem. The approximation of a cycle by polygons is useful only in proving Cauchy's theorem for a multiply connected region [Ahl, p.145, l.4].

    26. The proof of the general case may not apply to that of a special case [Perr, p.18, l.-13-p.19, l.-3].


    27. Generalization usually makes the hypothesis weaker, but in the following case it makes the hypothesis stronger by imposing a restriction: [Inc1, p.199, l.-4-p.200, l.2].


    28. Links {1, 2, 3}.

  2. Degeneralization.

        After a case is first generalized, and then degeneralized, it may no longer be the original case. The general case may descend to a new case instead.
    Example. Homogeneous wave equations (Hadamard's method of descent).
    1-dim [Joh, p.41, (4.13)] → 3-dim [Joh, p.129, (1.14)] (see [Joh, p.129, l.2]. For agreement, see [Pet, p.83, l.10])
    2-dim [Joh, p.134, (1.24a)]←3-dim.

  3. The general case may be decomposed into basic cases because we may differentiate under an integral sign.
    Example 1. The method of Fourier transforms [Joh, p.158, l.−2].
    Example 2. The method of plane waves [Joh, p.160, l.−12].
    Example 3. By decomposing the initial function φ (x) [Sne, p.283, l.−9] into [Sne, p.282, (2)].