Custom Search

Constructions in Differential Equations

1. (Universal covering spaces) Construction by imitating an old model
   1. Finding the necessary condition: If a universal covering space exists, then it is semilocally simply connected [Mas, p.174, l.18].

   2. The necessary condition is a sufficient condition [Mas, p.175, l.16-p.177, l.9].

   3. The necessary condition is widely applicable [Mas, p.175, l.1-l.4].

   4. (Agreement) If a universal covering space is given, then the newly constructed universal covering space is isomorphic to the old one [Mas, p.160, l.-8].
   Remark. [Mas, p.165, l.5-l.6] gives a similar example.
   
   
2. Assume we may construct a new structure by imitating an old model [Dug, p.82, theorem 9.3]. For agreement, see [Dug, p.131, Ex.2]. To give a theory for the new structure, we may directly apply the old theory [Dug, p.83, Theorem 9.4] to the new structure or, once again, build a new theory [Dug, p.132, 8.3] by imitating the old one.
   
   
3. A constructive method may allow uncertainty for its input, but the method itself should be deterministic.
   Example. Fix uÎR³, for every 2-dimensional subspace S in R³, there exists a vector vÎS such that (u,v)=0.
   Proof 1 (Reduction to absurdity by analysis of dimensions).
   If [u]^{^} ÇS={0}, then dim ([u]^{^} + S)=dim[u]^{^} +dim S = 4>3.
   Proof 2 (Geometric).
   [u]^{^} and S are two planes, so their intersection is a line.
   Proof 3 (Algebraic).
   Choose v₁, v₂ÎS. Let a₁= (u,v₁) and a₂= (u,v₂) and v = a₂v₁-a₁v₂.
   Remark. Proof 1 is not constructive, while Proof 2 and proof 3 are constructive, even though S can be arbitrary and v₁ & v₂ can be arbitrarily chosen in a fixed subspace S.
   
   

4. (Effective constructions) The goal of a construction determines what method we should use. We should not add any more structures than necessary as we proceed toward our goal. This approach allows us to easily recognize the role of each piece (e.g., the agreement of [Lan8, p.78, Definition 54] with [Lan8, p.53, Definition 35]) of the construction and helps us gain insight into the structures of irrational numbers. For example, when we try to extend the set of positive rational numbers to the set of positive real numbers, our goal is to warrantee the validity of following statement:
   Suppose S is a subset of positive real numbers. Then there exists a least upper bound of S if S is bounded above.
       Thus, the above problem is essentially a problem concerning ordering or positiveness. Therefore, the tool of Dedekind's cuts [Lan8, p.43, Definition 28] is a more appropriate choice for us to construct irrational numbers than the tool of Cauchy sequences [Wae, vol.1, p.212, l.-18]. The tool of Cauchy sequences is designed for the completion of a metric space rather than the construction of irrational numbers. In other words, it is designed for topological extensions rather than algebraic extensions. In fact, only after we detach unnecessary complications such as negative numbers, topology, metric spaces, and infinitesimals may we firmly grasp the essence of positive irrational numbers [Lan8, p.67, Definition 42]. In view of [Lan8, p.9, Definition 2], it is incorrect to say that ordering is a non-algebraic concept [Wae, vol.1, p.209, l.2].¬¬

5. How we retrace the design of a construction
       If one just reads a construction without retracing its design, one will become bewildered when one tries to design on his own. One should follow the following guidelines to retrace a construction's design:
   1. Identify and articulate the goal of construction (e.g., [Zyg, vol.1, p.82, l.6-l.10])
   2. Look for formulas and select the most important one
          Among the six formulas between [Zyg, vol.1, p.82, l.-6] and [Zyg, vol.1, p.83, l.3], we select the most important one:
      P(x) = (ò_[0,x] Rk(t) dt)/(ò_[0,1] R_k(t) dt).
   3. Analyze the key formula
          The denominator of the key formula is not less than k^-1 [Zyg, vol.1, p.82, l.-1]. In order to satisfy [Zyg, vol.1, p.82, l.9, condition (ii)], all we have to do is find aÎ(0, 1) such that max_{xÎ[0, (1/2)-h]} R_k(x) £ a^k.
   4. Fill the details.
   
   
6. (Maximal solutions) [Har, p.25, Lemma 2.1]
      The definition F given in [Cod, p.46, l.5] is problematic because there exists an initial value problem (E) that we know only some of its solutions, but not all of them. In such a case, we may find a supremum of some solutions, but we cannot find the supremum of all solutions. Someone may argue that the shortcoming is amended by proving F = lim _m®¥ j_1/m [Cod, p.47, l.2-l.3]. However, j_1/m is not accessible because j_e is defined by means of j_i (i=0,1,…,n-1) and j_i's depend on F which is not accessible. The construction given in [Har, p.25, (2.4)] enables us to avoid considering all the solutions.



7. Links {1}.