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 B_{n} and C_{n}
[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].
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.
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].
Classify second degree polynomial equations of two variables according to
their shape, and then reduce them to the standard form
[Fin, §160].
Assume D¹0. Does the curve have a center? (i.e.
Does ab-h^{2}
¹ 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.
If ab-h^{2} > 0, it
is an ellipse; if ab-h^{2}
< 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].
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
Build a list of various types of conicoids in standard form [Fin, §327].
Find the center or vertex of a conicoid [Fin §§354-358].
Determine the principal planes of a conicoid [Fin, §363].
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].
Check if the list given in [Fin, §327] is complete.
Simply connectedness can distinguish between R^{2}
and R^{3}, between S^{2}
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 T_{n} and that of P_{m}
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 T_{m} and T_{n} have
nonisomorphic fundamental groups if m¹n [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 T_{m} and T_{n}.
(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.
(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
p_{1} (B, b_{0})
[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].
(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)].
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 t_{0}>0 such that y'(t)>0
for t³t_{0}, then y'(¥)=¥.
II. If y is bounded above and there exists a t_{0}>0 such that y'(t)>0
for t³t_{0}, then y(¥)=y'(¥)=0.
III. If y(¥)=-¥ and there exists a t_{0}>0 such that y'(t)<0
for t³t_{0}, then y'(¥)=-¥.
IV. If y is bounded below and there exists a t_{0}>0 such that y'(t)<0
for t³t_{0}, then y(¥)=y'(¥)=0.
Remark. Case II and Case IV have the same conclusion as well as almost identical
proofs.
Belonging to the same class: [Cou, vol. 1, p.59, l.5-l.11]