Standardization in Differential Equations

  1. Any matrix can be standardized to a triangular form [Halm, p.107, Theorem 2]. To prove det eX = e Tr (X) [Kna, p.8, proposition 1.1(e)], it is enough to consider the triangular case. Thus standardization can be a very effective tool.

  2. Solving DE=s effectively.
    1. (Pattern identification) There are some basic patterns to solve DE.=s.
      1. We would like to identify [Inc, p.26, l.8] with [Bir, p.16, (16)].
      2. Change of variable: [Inc, Chap. 1, '13].

  3. Recognizing the existence and uniqueness intuitively.
        The rectification theorem [Arn, p.48, Theorem 7.1] is intuitively plausible ([Olv, p.30, Proposition 1.29] actually uses a pictorial argument to convince the readers of this theorem), although the actual proof of existence of the local diffeomorphism is very complicated. Therefore, we may reduce the proof of existence and uniqueness [Arn, p.50, Corollary 1 & 2] to the trivial case [Arn, p.49, (3)].

  4. Destandardization: Finding a proper setting for application.
        To simplify a theory, we would like to standardize our model. However, if we follow the argument of the standardized theory (xn=0 [Joh, p.57, (2.9)]), we may deviate from the track for solving a practical problem (φn =0 [Joh, p.59, (2.24)]). For immediate application, we have to reverse a part of the procedure of standardization. In this process of destandardization, we should measure the scope of influence precisely [Joh, p.58, (2.13)], and design a setting (Principal part [Joh, p.57, l.!3]) which has enough room for the general case and at the same time is lean enough to be efficiently used.

  5. For Cauchy Problems, we use approximate solutions [Joh, p.137, (1.46)] to standardize the spacelike [Joh, p.138, (1.54)] initial surface t=φ (x) to t=0. Uniqueness: Formula [Joh, p.129, (1.14)] (see [Joh, p.139, l.12-l.16]).

  6. The strategy of standardizing the Cauchy problem:
    1. Reducing the higher-order system to the first-order [Pet, p.16, l.4-p.18, l.3].
    2. Reducing the initial condition to 0 [Pet, p.18, l.!17-l.!6].

  7. We would like to standardize initial conditions so that we may interpret a solution term by term. Please compare [Joh, p.139, (1.14)] with [Pet, p.80, (4,12), (5,12)].

  8. (Structures) Sometimes the representative nature of a standard example can only be recognized through isomorphisms [Mas, p.256, l.!11].

  9. Standardization serves to facilitate calculations. In solving y' =Ax, all we need is to use diagonalization to standardize A (see [Rut, p.234, l.!2]). It is not necessary to require that the basis be standardized. However, in [Arn1, p.173, l.!1-p.174, l.!3], Arnol=d seems to like to standardize everything in sight. For the standard basis, he uses coordinate x [Arn1, p.174, l.!20]; for the eigenvector basis, he uses coordinate C [Arn1, p.174, l.!8]. His practice of transforming coordinates back and forth is really much ado for nothing.

  10. Standardization eliminates insignificant generalization. In a sense, a standard version is the most generalized version. For any theorem (For example, [Col, p.42, l.!15]) involving the generalized norm defined in [Col, p.38, (I.47)], we may use the standard norm [Col, p.37, (I.46)] to replace the generalized norm without invalidating the proof. We will need trivial modifications to interpret the proof.

  11. In the calculus of variations, the linear case eh can represent the general case y(x,e) without loss of generality [Cou, vol. 1, p.186, l.11-l.16].

  12. Sometimes referring to a standard object can be a random choice. In this case, it is more appropriate to define the concept by using the method of abstraction [Cou2, vol.2, p.196, footnote 1; p.200, footnote 1].

  13. We can metrize the topological vector space (C[0,1], t) by the metric d1 given in [Ru3, p.27 (1)] or the metric d2 given in [Dug, p.301, l.-15]. The problem would become quite complicated if we were to try to directly prove the equivalence of the above two metrics. However, if we treat t as the standard and prove that d1 is compatible with t [Ru3, p.28, l.1-l.12], then we use the same scheme to prove that d2 is compatible with t. This strategy applies whether the summation index n is related to the size of a domain or the order of a derivative or both.