Consistency in Differential Equations

  1. Commutative diagrams.
    1. If we interchange the order of two procedures, the result will be the same.
      Examples. Sum + Dual = Dual + Sum [War, p.59, l.!12-l.!11]. Wedge + Transpose = Transpose + Wedge [War, p.62, '2.13] (Warner fails to express this point).
    2. The identification in stages agrees with the identification achieved in one step [Mas, p.248, (a)].
    3. Homomorphisms of covering spaces [Mas, p.158, l.!8].

    4. What can we say about a noncommutative diagram?
          Which structure is stronger if the diagram is not commutative? Under what condition will the diagram become commutative?
      Examples. Quotient + product [Mas, p.249, sec.3]; subspace +quotient [Mas, pp.250-251, sec.4].

  2. Agreement.
    1. From abstract to concrete: [War, p.12, Definition 1.14]
      [War, p.14, Definition 1.19]. For agreement, see [War, p.15, l.!15].
    2. From general to standard: [War, p.14, Definition 1.19] [War, p.18, l.!14]. For agreement, see [War, p.18, l.12].

  3. From local to global.
        We may use local structures to build a global structure. Sometimes there are many local structures corresponding to the same global structure: Ex.1 (Topology, local base of nbds) [Po3, p.57, F] & Ex.2 (Differential structure, Atlas) [Arn1, p.289, l.1]. The emphasis should have been the agreement of specific local structures (e.g. cubic nbds versus spherical nbds), but mathematicians like to make a fuss about the side issue C correspondence. They use the following methods to change the above many-to-one correspondence to 1-1 correspondence:
    Equivalence class: Ex.1 [Po3, p.57, F]; Ex.2 [Arn1, p.290, Definition 4].
    Maximal representative: Ex.1 [Po3, p.55, l.25]; Ex.2 [War, p.6, l.4].

  4. One way to make two contradictory statements compatible is to make them meet at infinity.
        The concept of free boundary and that of fixed boundary are mutually exclusive. One might wonder how to make these two contradictory concepts compatible, i.e., to include a fixed boundary as a special case of free boundaries. In fact, the problem with a fixed boundary is the limiting case of the problems with free boundaries [Cou, vol. 1, p.211, l.6-l.7].

  5. We can show that f*f @p ex1 either using an argument similar to that in [Mun, p.325, l.1-l.7] or replacing f in the formula in [Mun, p.324, l.-7] by f [Mun, p.325, l.9-l.11].

  6. [Mun00, p.444, Theorem 73.4] shows that under proper assumptions the following statement is true: On a topological space the operation of pasting and the operation of constructing a fundamental group commute [Mun00, p.439, Fig. 72.1] if the cardinal number of points being pasted in the topological space equals the the cardinal number of the elements in the corresponding quotient group.

  7. To what basis we refer when we check consistency?
        To make a theory consistent is a never ending battle because we must maintain constant vigilance to ensure that every possible contradictory statement cannot occur in the present or in the future. If a theory contains only a few physical quantities, it is usually not a problem to check consistency. However, if the theory contains many quantities, we must know what basis we refer to when we check consistency.
    Example 1. The basis of the notation Tan (z, l) is [Gon1, (5.7-2)]. The basis of the notation sn (z, k) is [Gon1, (5.17-10)]. The definition k1 = x2 [Gon1, p.438, l.1] is consistent with the basis of the notation sn (z, k) because [Gon1, p.437, l.-5] and the last term of [Gon1, (5.17-10)] have the same form. Although k1 and k [Gon1, p.437, l.7] have similar appearances, they are unrelated with respect to consistency and we should not try to build a consistency link between them.
    Example 2. The notation K'(k') is consistent with the notation K(k) [Gon1, p.438, l.3-l.4] because of [Gon1, (5.9-3)] and the similarity between (5.11-10) and (5.11-5)].

  8. If we want to examine whether or not two systems are consistent, it suffices to check their consistency at the beginning stage of their constructions.
        In order to incorporate one system into another system, it suffices to check if the two systems have similar constructions [Lan8, p.131, Theorem 299] at the beginning stage. For example, in order to incorporate the real numbers into the system of complex numbers, all we need to do is prove [Lan8, p.131, Theorem 299]. This is because we start with natural numbers when we construct real numbers [Lan8, p.40, Theorem 113].

  9. Devices for consistency
    Example. The statement given in [Har, p.88, l.6-l.7] and that of [Har, p.89, l.1-l.2] are used as devices for consistency.

  10. Advantages of consistency [1]

  11. Links {1, 2}.