Classification in Differential Equations

  1. How to tell the crucial difference between two objects.
        To prove that two given classical Lie algebras are non-isomorphic, we should use a test as simple as possible. For example, we should try to use dimension [Po3, p.502, l.-19-l.-11] or center [Po3, p.504, F)] first. However, for Bn and Cn [Po3, p.504, F)], they have the same dimension and the same center [Po3, p.504, l.23], so we have to use a much more delicate feature (root system) to tell their difference [Po3, p.502, l.-10-l.-4].

  2. The definition of asymptotic directions given in [Wea1, p.83, l.1-l.2] leads to the differential equation [Wea1, p.83, (21)]. This differential equation can be used for comparison or classification (see [Wea1, p.66, (2) & p.80, (17); Kre, p.178, l.-6-l.-1]). Thus, Weatherburn's definition emphasizes its external relationship to other concepts, i.e., the role it plays in a group of concepts. In contrast, the definition given in [Kre, p.82, l.-7] is closely related to [Kre, p.96, Fig. 28.1(C); p.97, l.-18-l.-16]. In other words, Kreyszig's definition help us visualize its individual and internal meaning. Both approaches are able to characterize the concept uniquely and are indispensable to the understanding of asymptotic directions.

  3. Only through classifications may we solve more refined problems.
        The theorem given in [Guo, p.56] originates from a simple question: if the solution of a second order differential equation does not have an essential singularity, what do the coefficients of the equation look like? [Bir, p.240, Theorem 6] only discusses one case, so Birkhoff's discussion is incomplete. Guo is somewhat lazy in aiming low and seeking a shortcut, so his results obtained from [Guo, p.56, l.-6-p.57, l.8] are not as strong as those in [Jef, p.479, l.11-l.16].
        Using the concept of analytic continuation and the properties of the natural logarithm, we may divide the solutions of a differential equation with a singularity into two categories: [Jef, p.478, (1) & (2)]. Based on the properties of the coefficients of the differential equation we may classify the solutions further into three categories. However, if we substitute [Jef, p.481, (6)] using the formal series [Jef, p.481, (5)] into the differential equation, we may have another kind of classification: [Jef, p.478, (1); p.482, (11) & (12)]. By finding the correspondence between the two classifications we may answer deeper and more sophisticated questions given the information on the coefficients of the DE: Whether one of the solutions contains the log term? If a solution contains the log term, whether c in [Jef, p.478, (2)] is a positive integer or a negative integer? If we use Guo's weak results, we would have no way to answer these more refined questions.
    Remark. Current academic systems not only encourage, but also reward haste and ignorance: Mathematicians should devote themselves only to mathematics; physicists should devote themselves only to physics; professors assign homework only to keep students busy and fail to consider whether the workload is too much for a student to fully digest the important material within the time allotted. Thus, the systems move toward a direction that would destroy rather than develop a student's talent. In my opinion, even though N. Levinson had a bachelor's degree in electrical engineering, his background in physics is not strong enough for him to write quality mathematics textbooks that can compete with Jeffreys' [Jef] or Hilbert's [Cou].

  4. Classify second degree polynomial equations of two variables according to their shape, and then reduce them to the standard form [Fin, 160].
    1. Assume D0. Does the curve have a center? (i.e. Does ab-h2 0?) If the answer is no, it is a parabola. We may reduce the equation to the standard form by the following steps: first find its vertex and axis; then choose its vertex as the new origin and choose its axis as the new positive x-axis by finding the line that passes through the vertex and is perpendicular to the parabola's axis [Fin, 158]. If the answer is yes, go to the next step.
    2. If ab-h2 > 0, it is an ellipse; if ab-h2 < 0, it is an hyperbola. We may reduce the equation to the standard form by the following steps: first find its center and axes; then choose its center as the new origin by translation and choose its axes as the new coordinate axes by rotation  [Fin, 155 & 156].

  5. Classify second degree polynomial equations of three variables according to its shape, and then reduce them to the standard form [Fin, 364].
    Remark 1. [Fin, 371-373] basically repeat [Fin, 364] three times only with more jargon. In my opinion, [Fin, 364] gives a clearer picture about how we reduce a given equation to the standard form. The methods in [Fin, 371-373] may be somewhat simpler than that in [Fin, 364] for determining the shape of the surface, but I do not think they are more effective than [Fin, 364] for reducing a given surface to the standard form.
    Remark 2. Summary of the process of classification
    1. Build a list of various types of conicoids in standard form [Fin, 327].
    2. Find the center or vertex of a conicoid [Fin 354-358].
    3. Determine the principal planes of a conicoid [Fin, 363].
    4. Find the proper coordinate axes to put the conicoid in standard form using the number of zero roots of the discriminating cubic as the key to classifying conicoids [Fin, 364].
    5. Check if the list given in [Fin, 327] is complete.

  6. Tools for classifying compact surfaces
    1. (Refinements)
      1. [Mas, p.29, Theorem 7.2; p.33, Theorem 8.2] use Euler characteristics classify compact surfaces.
      2. Simply connectedness can distinguish between R2 and R3, between S2 and T, but will not distinguish between T and T#T [Mun00, p.322, l.3-l.22]. The condition of simply connectedness is the special case when the topological space's  fundamental group is trivial. The concept of fundamental groups using homotopy can distinguish between T and T#T. However, it is still difficult to tell whether the fundamental group of Tn and that of Pm are isomorphic if we use homotopy theory alone [Mun00, p.452, Theorem 74.3;  p.453, Theorem 74.4]. If we add the new tool of the first homology group (abelianized fundamental group) [Mun00, p.455, l.18], the problem can be easily solved [Mun00, pp.456-457, Theorem 75.3 & Theorem 75.4]. Similarly, we can use the first homology group to prove that Tm and Tn have nonisomorphic fundamental groups if mn [Mun00, p.456, Theorem 75.3]. Therefore, they are not of the same homotopy type [Mun00, p.364, Theorem 58.7]. This is a stronger result than that proved in a, where it was shown that these spaces are not homeomorphic [Mas, p.132, l.-6]. Consequently, the tool of fundamental group is a more refined than the tool of Euler characteristics because the former one enables us to strengthen the differences between Tm and Tn.
    2. (Limitations) Although the concept of fundamental groups is useful for classifying the topologies of compact surfaces ([Mas, p. 10, Theorem 5.1; chap. 4, 5] or [Mun00, p.457, Theorem 75.5; p.469, Theorem 77.5; p.472, Theorem 78.1; p.475, Theorem 78.2]), [Mun00, p.364, Theorem 58.7] shows that a fundamental group has its own characteristics other than those of a homeomorphism.
      Remark. Different tools serve different purposes. The concept of a fundamental group of path-homotopy classes essentially deals with holes [Mun00, p.434, Theorem 71.1; p.437, Lemma 71.4; p.436, Theorem 71.3]. Thus, its application is restricted to the following related problems: winding numbers, covering spaces, and singular points.

  7. (Classification of covering spaces) [Mun00, p.482, Theorem 79.4 & p.495, Theorem 82.1] show that if B is semilocally simply connected, there is a bijective correspondence from equivalent classes of coverings of B to conjugacy classes of subgroups of p1 (B, b0) [Mun00, p.494, l.-15-l.-13]. In group theory, the concepts such as conjugate subgroups, normal subgroups, and normalizers appear as disorganized stipulations. In addition, no example is provided to unite all the above concepts. In contrast, for covering spaces, these concepts are tools that play a definite role in facilitating the classification of covering spaces. For example, conjugacy classes are the tools for establishing [Mun00, p.482, Theorem 79.4]; normal subgroups are tools for establishing the claim given in [Mun00, p.490, l.-16-l.-14]; normalizers are tools for establishing [Mun00, p.488, Theorem 81.2]. Ignorant algebraists might say these terms in group theory originate from beauty or art. Actually, it is the above roles that enable us to set them in special places in group theory. In order to see their origins or their relationships, we must trace back to these roles. Furthermore, the theory of covering spaces gives group theory rich resources and concrete pictures to work with.
        The discussion given in [Mun00, p.478, l.1-l.12] is similar to the discussion given in [Jaco, vol. 3, p.24, l.17-l.23].

  8. (Organize possibilities and then present them rather than present all the possibilities and then try to organize them)
        In [Wat1, pp.284-285], Watson lists 24 possible solutions of Riemann's P-equation in terms of hypergeometric functions. Then he tries to seek relations among solutions. First, he reduces the general case {a, b, c} to the standard case {0, 1, } [Wat1, p.286, l.3-l.10]. However, he fails to distinguish between the two fundamental solutions of Riemann's P-equation. Consequently, his search for organized relations fails [Wat1, p.286, l.-15-l.-14]. He should have drawn a rectangle around 1-g as in [Guo, p.140, l.2] if he wanted to specify that exponent. [Guo, p.140, l.-2-l.-1] gives a way to distinguish between two fundamental solutions. Finding the fundamental solutions for each regular singular points and then using [Guo, p.143, (11)] are the keys to acquiring completely organized relations [Guo, p.143, l.-1]. The connection among the four solutions given in [Wat1, p.286, l.18-l.24] is not as strong and meaningful as that given in [Guo, p.143, (11)].

  9. Under the hypothesis of a theorem, our classification should be based on exhaustive cases rather than results. If our classification is based on the results given in [Cod, p.254, Problem 1(a)], then the cases will not be exhaustive. One may wonder why other results do not exist. Thus, our classification for this problem should be based the following exhaustive cases:
    I. If y()= and there exists a t0>0 such that y'(t)>0 for tt0, then y'()=.
    II. If y is bounded above and there exists a t0>0 such that y'(t)>0 for tt0, then y()=y'()=0.
    III. If y()=- and there exists a t0>0 such that y'(t)<0 for tt0, then y'()=-.
    IV. If y is bounded below and there exists a t0>0 such that y'(t)<0 for tt0, then y()=y'()=0.
    Remark. Case II and Case IV have the same conclusion as well as almost identical proofs.

  10. Belonging to the same class: [Cou, vol. 1, p.59, l.5-l.11]