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Mathematics, Applied Mathematics and Science . . . . . . . . . . . . . . . . . . . . Weinan E 6
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Full-ranked Decomposition for 2-D Polynomial Matrices . . . . . . Ü�< Ü � 14
Connections and Covariant Derivative in Vector Bundles . . . . . . . . . . . . . ?�¤ 18
A conjecture about Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � d 23
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gà§Ï�·��õ�)êÆ" — B.Russell
Mathematics, Applied Mathematics and Science
Weinan E
Department of Mathematics and PACM, Princeton University
What is the relation between mathematics and science? For mathematicians, it is
tempting to argue that mathematics is the foundation of science. After all, it provides
the language in terms of which scientific laws are stated. It provides the tools and
techniques with which scientific calculations are carried out. But besides these, it has
its own set of questions as well as an intrinsic structure, the pursuit of which is driving
much of mathematics today. There is no argument that a very impressive amount
of mathematics was developed as a result of the quest for internal completeness, of
studying the fundamentals such as numbers, equations and shapes.
The issue is therefore not whether mathematics will survive, but how to make
it grow. In principle, mathematics should be in an advantageous position compared
with other scientific disciplines for attracting talents, funding, and public support. The
intellectual ability of a child is often first revealed from his/her ability in mathematics.
Most parents have a deep appreciation for the importance of their child.s success
in mathematics. Mathematics is a necessary background course for most science and
engineering majors. It is heavily used in scientific research, industrial design, and a
host of other applications. However, over the years a gap has been created between
mathematics education, application, and mathematics research. While mathematics
education/application is understood to be important by almost everyone, mathematics
research remains largely a mystery to even the most educated public including our
colleagues in other departments. There is a lot to be done before the mathematics
community will be able to fully capitalize on the advantages mentioned above.
Computers have impacted our lives in a very fundamental way. They have changed
the way that scientific and engineering research is carried out. Computation has become
a major scientific tool in conducting research, playing a comparable role to experiment
and theory. When a new problem comes along, one of the first things to try is to
find its mathematical formulation so that it can be modeled on the computer. Even
though such a process of mathematicalization was also an essential part of scientific
6
1 60 Ï Mathematics, Applied Mathematics and Science 7
research in the old days, what happens now and what will happen in the future differs
essentially from our past experience in at least two fundamental ways. The first is the
time scale. It no longer takes years or decades to translate our understanding of nature
into laws formulated in mathematical terms and have them checked quantitatively. The
demand is that this process should happen in days or weeks. As a result, modeling
and computation have become a much more interactive process. The faster time scale
also means that if mathematicians do not act quickly enough, they become irrelevant
to such a process. The second is the form, variety and increased complexity of the
problems. Mathematical models are no longer polished when they are presented to us.
They are not necessarily clean. They certainly do not necessarily fall into the standard
categories that we have set up for mathematical problems. This means that if we want
to make an impact, we should be prepared to get our hands dirty.
Naturally the task of bridging mathematics with science and engineering falls
in the hands of applied mathematicians, as it has been traditionally. Indeed applied
mathematics has contributed greatly, in developing and analyzing the basic computa-
tional methods, in applications to fluid mechanics, structural mechanics, and a host of
other areas. Yet as the basic computational techniques become mature, more and more
scientific disciplines are developing their own computational tools. Consequently com-
putation as a whole is moving closer and closer to modeling. Can applied mathematics
meet the new challenges and find and foster its new position in scientific research? Or
will it adopt the current style of traditional pure mathematics and look into itself for fu-
ture development? What are the new challenges in applied mathematics today? These
are important questions that face all of us in mathematics, pure or applied. These
questions can no longer be swept under the rug. As has happened in the past for pure
mathematics, applied mathematics also requires some/soul-searching0.
Research
Among the many interesting new directions in applied mathematics, we will discuss
a few topics that we think will enjoy fast growth: first principle-based modeling, discrete
models, stochastic effects and the combination of data analysis and modeling.
First principle-based approach to modeling. Much of the physical modeling
relies on empirical laws based on physical intuition, or experimental results. It is
astonishing that basic conservation laws plus the simplest linear constitutive relations
describe so well the behavior of fluids in such a wide variety of situations, from creeping
to turbulent flow, from water waves in a river to blood flow in a blood vessel. There
is little need to refer to the underlying behavior of the molecules that make up the
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Logic merely sanctions the conquests of the intuition. —Jacques Hadamard
8 A�v �´
fluid. The same can be said for much of chemistry. The basic properties of chemical
elements were found and the periodic table was discovered before its foundation was
understood using quantum mechanics. The success of such empirical methods provided
a strong push to extend them to more complex systems, with however mixed results. For
example, constructing empirical constitutive relations for polymeric liquids and plastic
deformations proved to be a very difficult task. In many areas, scientists have now
realized the limitations of the empirical approaches that bypass the microscopic details
of the processes, and increasingly favor approaches that directly take into account
the microscopics. Such a first-principle based approach is likely going to play bigger
roles in the future for several reasons. One is that the improvement of computational
power and computational methods will make it more feasible. The second reason is
that the demand for more accuracy in our models, particularly for systems that fall in
between scales described by well-established theories, such as nano-systems, will make
it a necessity. The third is simply the quest for understanding problems in a more
fundamental way. The need for solving practical problems often makes it necessary to
simplify the first-principle based models, by/sweeping things under the rug0. But
this does not mean that there is no value in understanding the details that were swept
under the rug. To the contrary, the quest for deeper and deeper understanding is the
heart of scientific research.
Discrete models. In applied mathematics, we are very used to modeling physical
process using differential equations, i.e., the continuum models. While differential
equations will continue to play a very pivotal role in applied mathematics, discrete
models will certainly claim their role in the coming years. This is simply because
many physical processes are naturally described by discrete models, such as discrete
stochastic processes, molecular dynamics, and kinetic Monte Carlo models. Examples
are abundant in biology, ecology, materials sciences, and chemistry.
Discrete models bring out a host of new questions that should be addressed from
points of view that are quite foreign to us. Take the example of speeding up molecular
dynamics. The traditional approach from the viewpoint of applied mathematics is to
design ODE methods that allow large time steps or to process the models so that
certain degrees of freedom that require small time steps can be eliminated. There is
an alternative viewpoint, which is based on the observation that for many examples
modeled by molecular dynamics the system spends most of its time vibrating around
local equilibrium states, with occasional sudden hops to different local equilibriums.
The dynamics of the systems is characterized by these hopping events. It is therefore
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Every mathematical discipline goes through three periods of development: the naive, the formal, and the
critical. —David Hilbert
1 60 Ï Mathematics, Applied Mathematics and Science 9
tempting to approximate the original molecular dynamics by a Markov chain that
captures correctly the hopping events.
Our interest in analyzing these discrete models will bring us closer to another
important area of mathematics that has so far remained tangential to core applied
mathematics, and that is mathematical physics. It is likely that mathematical physics
will become a main ally for applied mathematics among the areas of pure mathematics,
together with differential equations.
Stochastic effects. For historical reasons, stochastic analysis and stochastic
methods have not become a standard tool for a large part of the applied mathematics
community interested in scientific/engineering problems. Indeed when our main con-
cern was fluid dynamics at intermediate scales or structural mechanics, there was little
need to think about stochastic effects. However, things are different when we turn to
material sciences, chemistry, biology, and ecology. In these areas, stochastic effects are
an intrinsic part of the problem. In some cases, they seem to have become the main
obstacle for mathematicians to make further contributions.
Stochastic ideas also bring new tools into applied mathematics. A classical ex-
ample is the Monte Carlo method for numerical integration. Other examples include
global optimization techniques and kinetic Monte Carlo methods. The performance of
these methods is much less understood from the point of view of numerical analysis,
and there is certainly a lot of room for important contributions.
We should note that discrete models and stochastic methods themselves are not
foreign to applied mathematics. They play important roles in areas such as network
models, finance, statistics, and control theory. One important direction of research will
be to integrate the knowledge we have learned from these areas with applications to
sciences.
Modeling and data analysis. Another interesting new direction of research
is combining data analysis with modeling. In many applications, the underlying laws
of nature are not known or not known at the scales of interest. It can also be that
some of the important parameters are not known. In such cases, one might want
to extract the governing laws or parameters from available data. One particularly
attractive approach is to start the simulation with a complex, microscopic model and
to then extract a simplified macroscopic model as the computation goes on using the
computed data. In other words, the numerical algorithm learns in the process of the
computations. Such/learning algorithms0should combine scientific modeling with
data processing techniques. These ideas already exist in various forms, but they should
##########################################################################
Mathematics is a dangerous profession; an appreciable proportion of us goes mad. —J E Littlewood
10 A�v �´
be explored in much larger scale in computational sciences.
Education
At a time when applied mathematics should be aggressively moving into new
areas of science, we also have to think carefully about how to train our students to best
prepare them for the many different new challenges that they will face.
Students in applied mathematics should be cultivated both in mathematics and
other sciences. They should receive a solid training in both the fundamentals of pure
mathematics and the fundamentals of sciences. This should be our basic principle in
education. This is undoubtedly a very difficult task. But perhaps it is not more difficult
than the task Landau faced when he formulated the basic curriculum for theoretical
physics, of which mathematics is an essential part.
Our current graduate curriculum is still very much in tune with/traditional0applied
mathematics with a strong emphasis on differential equations, continuum mechanics,
and numerical methods. Some applied mathematics graduate curricula contain only
these topics. While they will no doubt continue to play key roles in future graduate
curricula, appropriate weights will also have to be given to new emerging topics such
as stochastic and statistical methods, and the basic principles of science. Some well-
established courses, such as numerical methods, have to be modified in order to give
more emphasis to areas such as molecular dynamics and Monte Carlo methods. For
applied mathematics students interested in science and engineering, we propose a set of
four courses as the basic core graduate curriculum. These are: computational methods,
applied differential equations, applied stochastic methods, and introduction to scientific
modeling.
Computational Methods. This is perhaps the most well-established course
among the four proposed courses. However, current teaching of this course needs to
be modified in at least two aspects. 1. It needs to be streamlined, to be taught more
efficiently, in order to make room for other new courses. 2. More emphasis has to be
put on discrete simulations such as Monte Carlo methods and molecular dynamics, as
well as computations based on quantum mechanics.
Applied Differential Equations. This course should cover rigorous analysis of
prototypical equations, qualitative techniques such as bifurcation analysis and invariant
manifolds, analytical techniques such as transform methods and asymptotic methods.
It should also cover prototypical equations from applications such as fluid mechanics,
nonlinear diffusion, material sciences, etc.
Applied Stochastic Methods. As we discussed earlier, stochastic analysis and
##########################################################################
Mathematics is the art of giving the same name to different things. — Henri Poincare
1 60 Ï Mathematics, Applied Mathematics and Science 11
stochastic methods will become a major tool in applied mathematics, along with nu-
merical methods and differential equations. It is important to develop a course that
is tailored to the needs of applied mathematics students interested in science. Such a
course may contain the following list of topics: A quick introduction to random vari-
ables and limit theorems, Markov chains and Markov processes, stochastic differential
equations, Fokker-Planck equations, path integrals, Monte Carlo methods, and rare
events. A course that covers these topics has been developed at Princeton University.
Introduction to Scientific Modeling. It is difficult to decide on a best title
for this course. We intentionally avoided calling it/Mathematical Modeling0since
this course is intended to be a systematic introduction to the basic theoretical tools
for modeling scientific problems. But we also have in mind to select those topics that
are more mathematical, with a clear distinction between first-principle based methods
and empirical methods. Much of these will be physics, since it provides most of the
theoretical tools that are now used in every scientific discipline. But the teaching of it
can be tailored to the needs of mathematicians.
Among the four courses discussed, this is perhaps the most difficult course to
develop and mature. The purpose of this course is to teach students basic principles,
techniques, and languages in the science. Such a course is needed for several reasons.
Our students may work in a variety of scientific disciplines and they may change their
interest later on in their career, therefore a course limited to say, fluid mechanics, is
not sufficient for the preparation of their scientific background. Currently students are
encouraged to take such courses from individual departments outside of mathematics.
While this will continue to be an important way that our students learn science, two
factors have to be considered when we send our students to other departments. The
first is that this is often time-consuming. Many topics covered in these courses are of
little interest and/or importance to our students. If a student is interested in phase
transformation in solids, he/she may not be able to afford the time to take one course in
continuum mechanics, one course in statistical mechanics, and one course in quantum
mechanics or solid state physics. The other factor is that our students are often un-
comfortable with the way courses are taught in other departments. They are unhappy
about the lack of precision, the readiness to resort to empirical solutions rather than
the analysis of the detailed process. While the complexity of real systems often do
not leave us a second choice besides sweeping things under the rug, a more complete
picture about the successful techniques should be presented to the students before they
are asked to accept the ad hoc approaches. Moreover, science courses in other depart-
##########################################################################
Science is built up with facts, as a house is with stones. But a collection of facts is no more a science than a
heap of stones is a house. — Henri Poincare
12 A�v �´
ments are often taught with an eye on minimizing mathematical complexity. This is
one short-cut that our students do not always need.
Pure mathematics. The emphasis in applications does not mean that there is
less need for mathematics itself. To the contrary, the core values of mathematics are
very crucial to our success in other areas. Mathematicians have developed a distinct
style for approaching a problem, symbolized by its precision and its ability to extract
the essence of the matter - the ability to abstract. This style of thinking is needed
ultimately in all other areas of science.
Academic Standards
Evaluating the work in applied mathematics can be a difficult, frustrating task.
It is not surprising that opinions about particular pieces of contributions can be highly
non-uniform. The main reason is quite simple. In applied mathematics, we are forced
to use/double standards0: The mathematical standards and the scientific standards.
Certainly rigorous proofs of existence of solutions to nonlinear systems of conservation
laws should be considered an important contribution. But so is the fast Fourier trans-
form, which does not use more than high school trigonometry. While mathematics has
over the years developed a rather complete set of standards on the grounds of pure
mathematics, most mathematicians, including applied mathematicians, are rather un-
comfortable or unfamiliar with scientific standards. This is compounded by the fact
that applied mathematics is continuously moving into new territories, leaving us little
past experience that can be used to evaluate new work.
How do we resolve this situation? First and foremost, we should realize that
while mathematics does and should have its own standards, ultimately our work will
be put into the context of human knowledge and be judged in that broader context.
Secondly, to be able to exercise scientific judgment, we as a community should become
mature scientifically. We should educate ourselves. This is not just the task of applied
mathematicians, but the task of the whole mathematics community. Realizing that
there is more out there than the theorems we can prove and being able to adjust
ourselves to that fact will ultimately lead the mathematics community to a new level
of maturity.
At the same time, it is equally important for the applied mathematics community
to become more cultivated in the basic values of core mathematics. After all, applied
mathematics is still part of mathematics. It is different from engineering. Mathematical
beauty, structure, and techniques should be among its most important goals, and should
also be used as a basic standard in evaluating its work.
##########################################################################
Facts do not speak. — Henri Poincare
1 60 Ï Mathematics, Applied Mathematics and Science 13
Concluding Remarks
The coming of the information age provides a great opportunity but also a great
challenge to mathematics. Whether mathematics will grow or shrink depends on
whether the mathematics community will be able to adapt to the needs of the ap-
plications. To deal with their own problems, the applied areas will use more and
more heavily mathematics. Without help from the mathematics community, they will
develop the necessary mathematics on their own. This will lead to a separation of
mathematics with the rest of the sciences.
Applied mathematics should naturally take up the task of bridging mathematics
with other scientific disciplines. To be able to meet such a challenge, we must attract
the best talent to work in applied mathematics, to develop a solid curriculum, and to
develop a balanced view of mathematics and the sciences. Most importantly it should
always be in touch with the frontiers of science. Nothing is more damaging to applied
mathematics than isolating itself from the applications.
Acknowledgement: This is based on an article entitled /Mathematics and
Science0, written by the author for/Beijing Intelligencer0, published during ICM
2002.
####################
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In mathematics you don’t understand things. You just get used to them. — John von Neumann
Full-ranked Decomposition for 2-D Polynomial
Matrices
Xiangxiong Zhang and Zhi Zhang
Supervised by Prof. Jiansong Deng
In this paper, we apply the theory of syzygy modules to affirm the full-ranked
decomposition for bivariate polynomial matrices. An efficient algorithm is presented
and illustrated with an example.
Keywords:Polynomial matrix; Bivariate polynomial matrix; Syzygy; Full-ranked
decomposition
Introduction
There is a classical conclusion in Linear Algebra as follows:
Let A be a m × n matrix with real entries. The rank is r. Then there exists a
m× r matrix B and a r × n matrix C satisfying that A = BC.
We call the property above full-ranked decomposition. What we are interested in
is:
1. Does the property remain when we discuss polynomial matrices? That is to say:
Let A be a m × n polynomial matrix. The rank is r. Does there exist a m × r
polynomial matrix B and a r × n polynomial matrix C satisfying that A = BC?
2. If it is right, how can we calculate B and C?
The main difficulty of dealing with polynomial matrices lies in that the entries
are restricted within a ring rather than a field. So elementary row operations can not
be used. Fortunately, all 1-D polynomials form an Euclidean Domain. It is easy to
answer the two questions for 1-D polynomial matrices if we use the division algorithm
in Euclidean domains when following the proof of the conclusion in Linear Algebra. But
it is not so easy for 2-D and n-D(n > 3) cases any more. For n-D case, counterexamples
to the questions have been presented by others. We will give the results of 2-D case in
the next section, which is not trivial.
14
1 60 Ï Full-ranked Decomposition for 2-D Polynomial Matrices 15
Main Result
Let K be a field, and let K[s, t] donate the polynomial ring in two variables over
K. Let Km×n[s, t] donate the union of all m×n matrices with entries in K[s, t]. Other
related concepts such as syzygy module and greatest common right divisor can be found
in references.
Before we answer the questions raised in the first section, the following lemma is
required. It can be deduced by the results in [1].
Lemma 1 Let A ∈ Km×n[s, t] and the rank is r. Then there exists a generating matrix
H ∈ Kn×(n−r)[s, t] of Syz(A). Moreover, H is of rank (n− r).
Now we can prove the following important result:
Theorem 1 (Full-ranked decomposition for 2-D case) Let A ∈ Km×n[s, t] and
the rank is r. Then there exist a m × r polynomial matrix B and a r × n polynomial
matrix C satisfying that A = BC.
Proof Without loss of generality, we may assume that r < m 6 n.
By lemma 1, there exists a generating matrix H ∈ Kn×(n−r)[s, t] of Syz(A) and
its rank is (n − r). Lemma 1 applies to HT , which is the transpose of H. Then there
exists a generating matrix F ∈ Kn×r[s, t] of Syz(HT ) and its rank is r.
Let AT = (a1, . . . ,am), F = (f1, . . . , fr), and a1, . . . ,am, f1, . . . , fr ∈ Kn[s, t] are
column vectors. Since H is the generating matrix of Syz(A), we have AH = 0, which
implies that HTAT = 0. Hence a1, . . . ,am ∈ are Syzygies of HT . So there exist
tij ∈ K[s, t],i = 1, · · · ,m,j = 1, · · · , r, satisfying thatµ
ai = ti1f1 + · · · + tirfr,
Let
T =
t11 · · · · · · t1r
t21 · · · · · · t2r
. . . . . . . . . . . . . . . . . .
tm1 · · · · · · tmr
,
then T T ∈ Kr×m[s, t] and (a1, . . . ,am) = FT T , namely AT = FT T .
LetB = T,C = F TP T2 P
−11 , thenA = BC. It is obvious that B and C are polyno-
mial matrices. 2
By theorem 1, we have answered the first question for 2-D case. According to the
proof above, we have to calculate two generating matrices of syzygy module to obtain
B and C. This method consumes too much time because the algorithm to calculate
⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆
If only I had the theorems! Then I should find the proofs easily enough. —Bernhard Riemann
16 ïÄ?Ø �´
generating matrices is very complicated. By further discussion, we find a much more
efficient way to calculate B and C. It is presented as the following algorithm:
Algorithm 1 (Full-ranked decomposition for 2-D case)
Input A: A ∈ Km×n[s, t], the rank is r, and r < m 6 n"
Output B,C: B ∈ Km×r[s, t] and C ∈ Kr×n[s, t], satisfying that A = BC.
Step
1. SupposeD0 is a r×r full-ranked submatrix of A.The row indices ofD0 are i1, · · · , ir,and the column indices are j1, · · · , jr. Let A donate the submatrix of A consisting
of the i1th, · · · , irth rows of A. Let F donate the submatrix consisting of the
j1th, · · · , jrth columns of A.
2. Get rid of the j1th, · · · , jrth columns of A, and let N0 donate the matrix consisting
of the remaining columns. Calculate the Greatest Common Right Divisor(GCRD)
of DT0 and NT
0 , donated by M .
3. Calculate B = FD−10 MT , C = (MT )−1A.
The algorithm finishes.
Proof Notice that A and H satisfying that AH = 0. By using this equation, it is easy
to obtain the conclusion in the algorithm above when calculating generating matrices
of syzygy module. The detail of the proof is omitted.
Calculating two generating matrices of syzygy module is replaced by calculating a
GCRD of two matrices. That is why this algorithm provides much more convenience.
2
Example Let A=
0 st s− t
−st 0 s
t− s −s 0
. We can obtain the following result by using
the algorithm above:
0 st s− t
−st 0 s
t− s −s 0
=
s −ts 0
−1 1
(−t 0 1
−s −s 1
).
It is easy to see that the rank of A is 2, so it is exactly the full-ranked decompo-
sition. 2
Reference
[1] GZhiping Lin, On syzygy modules for polynomial matrices, Linear Algebra and its Appli-
cations, Vol.298, 1999, 73–86.
⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆
The early study of Euclid made me a hater of geometry. — James Joseph Sylvester
1 60 Ï Full-ranked Decomposition for 2-D Polynomial Matrices 17
[2] John P. Guiver and N. K. Bose, Polynomial matrix primitive factorization over arbitrary
coefficient field and related results, IEEE Transactions on Circuits and Systems, Vol. Cas-
29, No.10, 1982, 649–657.
[3] Zhiping Lin, On matrix fraction description of multivariable linear n-D systems, IEEE
Transactions on Circuits and Systems, Vol.35, No.10, 1988, 1317–1322.
[4] Martin Morf, Bernard C. Levy, and Sun-Yuan Kung, New results in 2-D systems theory,
Part I: 2-D polynomial matrices, factorization, and coprimeness, Proceedings of the IEEE,
Vol.65, No.6, 1979, 861–872.
[5] Michael Sebek, One more counterexample in n-D systems — Unimodular versus elementary
operations, IEEE Transactions on Automatic Control, Vol.33, No.5, 1988, 502–503.
[6] B. L. van der Waerden, Modern Algebra, Vol.II, New York, Ungar, 1950.
[7]Dante C. Youla and G. Gnavi, Notes on n-dimensional system theory, IEEE Transactions
on Circuits and Systems, Vol. Cas-26, No.2, 1979, 105–111.
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All stable processes we shall predict. All unstable processes we shall control. — John von Neumann
Connections and Covariant Derivative in Vector
Bundles
0201 Dacheng Xiu
In this article, we briefly introduce a direct method of defining a connection in a vector
bundle, and then make an effort to prove the equivalence between covariant derivative and
connections in vector bundles.
There are various definitions of connections in vector bundles. For example, we can per-
ceive a vector bundle as the associate bundle of a principle one. Then a connection in principle
bundles induces a connection in vector bundles with the help of parallel displacement. However,
we will introduce another direct definition, which is not very common in most textbooks.
We begin our discussion with the concept of covariant derivative, which is in accordance
with general definition.
Definition 1 Let M be a differentiable manifold, E a vector bundle over M. A covariant deriv-
ative is a map D : Γ(E)⊗
Γ(TM) → Γ(E) with the following properties:
DX(V ) := DVX, for V ∈ Γ(TM), X ∈ Γ(E)
1. D is tensorial in V.
DU+V X = DUX +DVX, for U, V ∈ Γ(TM)
DfV X = fDVX, for V ∈ Γ(TM), f ∈ C∞(M,R)
2. D is a derivation in Γ(E).
DV (X + Y ) = DVX +DV Y, for V ∈ Γ(TM), X, Y ∈ Γ(E)
DV (fX) = V (f) ·X + fDVX, for f ∈ C∞(M,R)
Definition 2 Let π : E → M be a vector bundle. A connection H on E is a distribution on
TE, the tangent bundle of E, i.e. a map which assigns each point of E a subspace Hu of TuE,
1. π∗u : Hu → Tπ(u)M is an isomorphism for all u ∈ E.
18
1 60 Ï Connections and Covariant Derivative in Vector Bundles 19
2. µa∗Hu = Hau, where µa(u) = a · u is multiplication by a ∈ R.
Remark 1 Let Vu be the kernel of π∗u : TuE → Tπ(u)M . Then the statement (1) of Definition
2 is equivalent to TuE = Hu ⊕ Vu, for each u ∈ E. Thus a vector X in TE decomposes as
X = Xh +Xv, where Xh ∈ H and Xv ∈ V . Indeed, since π∗u : TuE → Tπ(u)M is a surjective
map, then TuE/kerπ∗u → Tπ(u)M is an isomorphism. Thus, π∗u : Hu → Tπ(u)M is an iso-
morphism ⇔ TuE/kerπ∗u ⋍ Hu ⇔ TuE = Hu ⊕ kerπ∗u.
Definition 3 The subspace Hu is called the horizontal subspace at u, and vectors in Hu called
horizontal. The subspace Vu is called the vertical subspace, and vectors called vertical.
As it is known to all, covariant derivative is equivalent to connections in vector bundles.
After some preparation, we shall explain how covariant derivative coincides with connections
(as defined above) in vector bundles, which in turn illustrates the rationality of Definition 2.
Proposition 1 Let π : E → M be a vector bundle, where E, M are respectively n, m dimen-
sional manifolds. π(u) = p, and i : π−1(p) → E denotes the inclusion. Then Vu = i∗π−1(p)u :=
{i∗Juv|v ∈ π−1(p)}, where Ju : π−1(p) → TuE denotes the isomorphism identifying v ∈ π−1(p)
with xi(v) ∂∂xi (u), where x : π−1(p) → Rn is any isomorphism between the two n-dimensional
vector spaces.
Proof Since π−1(p) ⋍ {p} × Rn−m, dim i∗π−1(p)u = dimπ−1(p) = n − m = dimTuE −
dimTπ(u)M = dim kerπ∗u, then it suffices to prove that i∗π−1(p)u ⊂ kerπ∗u. Actually, for any
φ ∈ C∞(M,R) and V ∈ TuE, π∗u(i∗V )(φ) = (π ◦ i)∗pV (φ) = V (φ ◦ π ◦ i) = V (φ(p)) = 0 . 2
Definition 4 The connection map κ : TE → E is given by:
κ(X) = (i∗Ju)−1Xv, for X ∈ TuE.
We now define an operator ∇ in terms of κ.
Definition 5 Let H denote a connection on vector bundle π : E → M with a connection map
κ. Given a section X of E and V ∈ Γ(TM), we define an operator ∇ by:
∇V X(p) := ∇V (p)X := κX∗V (p) ∈ Γ(E).
In the sequel, we shall find that ∇ is nothing but D. As a preparation, the proof requires
a few lemmas.
Lemma 1 Let (x, U) be a chart around p in a manifold M . Then any tangent vector V ∈ TpM
can be uniquely written as a linear combination,V = V (xi)( ∂∂xi )p.
Proof The proof is so common that it is omitted here. 2
Lemma 2 κ ◦ µa∗ = µa ◦ κ.××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××If there is a problem you can’t solve, then there is an easier problem you can solve: find it. —George Polya
20 ïÄ?Ø �´
Proof Since both sides vanish when applied to horizontal vectors, if suffices to consider vertical
ones. Let v ∈ π−1(p),
κ ◦ µa∗(i∗Juv) = κ ◦ (µa ◦ i)∗(Juv) = κ ◦ i∗µa∗Juv = κ ◦ i∗(Jauav) = av,
µa ◦ κ(i∗Juv) = µav = av.
Thus, κ ◦ µa∗ = µa ◦ κ. 2
Let (π, φ) : π−1(U) → U × Rn−m be a local trivialization of vector bundle π : E → M .
(x, U) is a chart of M. Put y = x ◦ π, so that (y, φ) : π−1(U) → x(U) × Rn−m is a coordinate
map of E. Since φ : π−1(U) → U × Rn−m is a diffeomorphism, the basis {ej} of Rn−m yields a
basis µj(y) := φ−1(y, ej) of π−1(U) at any point y ∈ U .
Lemma 3 Suppose u, v ∈ π−1(p), y(p) = 0, then
κ((∂
∂yi)u+v) = κ((
∂
∂yi)u) + κ((
∂
∂yi)v), 1 6 i 6 m;
κ((∂
∂φj)u) = µj ◦ π(u), 1 6 j 6 n−m.
Proof Let f : π−1(p) → π−1(p), with f(u) = κ(( ∂∂yi )u), since µa∗(
∂∂yi )u = ( ∂
∂yi )au. By
Lemma 2, we have f(tu) = κ(( ∂∂yi )tu) = κ(µa∗(
∂∂yi )u) = µt ◦ κ(( ∂
∂yi )u) = tf(u). Then by
applying ddt |t=0 to both sides, we have f(u) = uf ′(0), hence f is linear in u. Suppose {ej} is a
basis of Rn−m, {Dj} is a basis of the tangent space of Rn−m. Then,
(∂
∂φj)u = i∗φ
−1∗Dj(φ(u)) = i∗φ
−1∗
Jφ(u)ej = i∗Juφ−1ej = i∗Juµj(p),
where φ = φ|π−1(p). The statement is established by Definition 4. 2
Theorem 1 Let H be a connection on E with an operator ∇. For any section X,Y of E, for
any vector U, V ∈ TpM , we have,
1. ∇V (X + Y ) = ∇V X + ∇V Y .
2. ∇fV X = f∇V X, for f ∈ R.
3. ∇U+V X = ∇UX + ∇V X.
4. ∇V fX = V (f)X(p) + f(p)∇V X, for f ∈ C∞(M,R).
Proof Since (i∗Ju)−1 and Xv are linear operators, then (2) and (3) follow clearly from that
∇V X is linear in V. To prove (1), we continue to use the local trivialization as defined before.
According to Lemma 1, we have
X∗V = X∗V (yi)(∂
∂yi)X(p) +X∗V (φj)(
∂
∂φj)X(p)
= V (yi ◦X)(∂
∂yi)X(p) + V (φj ◦X)(
∂
∂φj)X(p)
= V (xi)(∂
∂yi)X(p) + V (Xj)(
∂
∂φj)X(p)
××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××John von Neumann was the only student I was ever afraid of. —George Polya
1 60 Ï Connections and Covariant Derivative in Vector Bundles 21
Similarly,
(X + Y )∗V = V (xi)(∂
∂yi)X(p)+Y (p) + V (Xj + Y j)(
∂
∂φj)X(p)+Y (p)
Applying Lemma 2 and Lemma 3,
κ(X + Y )∗V = κV (xi)(∂
∂yi)X(p)+Y (p) + κV (Xj + Y j)(
∂
∂φj)X(p)+Y (p)
= V (xi)κ(∂
∂yi)X(p)+Y (p) + V (Xj + Y j)µj(p)
= V (xi)κ(∂
∂yi)X(p) + V (xi)κ(
∂
∂yi)Y (p) + (V (Xj) + V (Y j))µj(p)
= κX∗V + κY∗V
The last statement can be verified as follows,
(fX)∗V = V (xi)(∂
∂yi)f(p)X(p) + (V (f)Xj(p) + f(p)V (Xj))(
∂
∂φj)f(p)X(p)
= V (xi)µf(p)∗(∂
∂yi)X(p) + f(p)V (Xj)(
∂
∂φj)f(p)X(p) + V (f)Xj(p)(
∂
∂φj)f(p)X(p)
Thus,
κ(fX)∗V = V (xi)f(p)κ(∂
∂yi)X(p) + f(p)V (Xj)µj(p) + V (f)Xj(p)µj(p)
= f(p)κ(X∗V ) + V (f)X(p)
2
Finally, we make some efforts to reverse the above process and illuminate the relationship
between covariant derivative and connections in vector bundles.
Theorem 2 Let π : E → M be a vector bundle with a covariant derivative operator D as
defined above. Put Hu = {X∗V : V ∈ TpM,X ∈ Γ(E), X(p) = u,DVX = 0}. Then H is a
connection and the operator ∇ induced by H is D.
The proof of Theorem 2 requires a lemma:
Lemma 4 Given p ∈ M , for any V ∈ TM , there exists a section X ∈ Γ(E), satisfying
X(p) = u and ∇V X(p) = 0.
Proof of Lemma 4. Choose a local trivialization ψ over the neighborhood U of p. Let
{ ∂∂xi } be a coordinate vector field of M. Let X ∈ Γ(E), locally, we write X(y) = ak(y)µk(y),
where {µk} is a basis of Γ(E). Let c : I → M be a smooth curve, with c(0) = p, and
V (t) = c(t) := c∗tD(t) = ci′
(t) ∂∂xi (c(t)), µ(t) := µ(c(t)). Then,
DV (t)X(t) = (ak ◦ c)′(t)µk(c(t)) + Γjik(c(t))ci
′
(t)ak ◦ c(t)µj(c(t))
Thus, DV (t)X(t) = 0 determines a group of first-order ordinary differential equations for the
coefficients ak ◦ c(t) of X(t), which can be uniquely solved for a given initial vector Vp. 2
Proof of Theorem 2. Given vectors Au, Bu ∈ Hu, assume π∗Au = Vp and π∗Bu = Wp.
We can choose X with X(p) = u and ∇VpX = ∇Wp
X = 0 by Lemma 4, then we also have
××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××For Bourbaki, Poincare was the devil incarnate. For students of chaos and fractals, Poincare is of course God
on Earth. —Marshall Stone
22 ïÄ?Ø �´
∇λVp+WpX = 0, and λAu + Bu = X∗(λVp) +X∗(Wp) = X∗(λVp +Wp), for any λ ∈ R. Thus,
Hu is a subspace of TuE.
Consider π∗u|Hu: Hu → TpM . For any X∗V ∈ Hu, π∗u(X∗V ) = 0 ⇐ :V = 0, which
implies kerπ∗u|Hu= {0}. On the other hand, given Vp ∈ TpM , according to Lemma 4, there
exists a section X satisfying ∇VpX = 0, hence X∗pV ∈ Hu and π∗uX∗pV = V . Thus, the map
π∗u|Huinduces an isomorphism, which is in accordance with the first requirement of Definition
2.
As to the second requirement, since dimµa∗Hu = dimHu, it suffices to prove µa∗Hu ⊆ Hau.
Indeed, for any X∗pV ∈ Hu, µa∗X∗pV = (µX)∗pV , µX(p) = au, and ∇V µX = µ∇V X = 0, so
(µX)∗pV ∈ Hau, which establishes the claim.
At last, suppose ∇ is induced by H. Then by definition,
∇VX = κX∗V = (i∗Ju)−1(X∗V )v,
then we safely arrive at the conclusion by noticing that
∇VX = 0 ⇔ X∗V ∈ Hu ⇔ DVX = 0
2
Reference
[1] Genard Walschap, Metric Structures in Differential Geometry. Springer,2004 (Graduate
Texts in Mathematics 224).
[2] Jurgen Jost. Riemannian Geometry and Geometric Analysis. Springer, 1995.
[3] Kobayashi&Nomizu, Foundations of Differential Geometry (Volume 1 and Volume 2), John
Wiley&Sons, Inc, 1996.
[4] Jurgen Jost. Nonlinear Methods in Riemannian and Kahlerian Geomery. DMV Seminar
Band 10, Birkhauser, 1986.
[5] Spivak. A Comprehensive Introduction to Differential Geometry.
××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××Technical skill is mastery of complexity while creativity is mastery of simplicity. — Chris Zeeman
A Conjecture About Dynamical Systems
0201 Sun Jun
At the ICM 1998 Berlin,the famous mathematician Michael Herman proposed a conjecture:
Let f : z ∈ R2n −→ Az +O(z2) ∈ R2n be a germ of symplectic diffeomorphisms such that
A ∈ Sp(2n,R) is conjugated in Sp(2n,R) to rα1 × rα2 × · · · × rαn, α = (α1, · · · , αn) ∈ DC.
Conjecture:
If f is real analytic,then f leaves invariant,in any small neighborhood of O,a set of positive
Lebesgue measure of Lagrangian tori.
Notion:
Sp(2n,R) the set of symplectic transformation
rα1 × rα2 × · · · × rαn(rα1 × rα2 × · · · × rαn
)(z1, · · · , zn) = (eiα1z1, · · · , eiαnzn)
DC diophantine condition
Definition 1 A diffeomorphism f is said to be a symplectic diffeomorphism if it satisfies
the following equation: (Jf)′
J(Jf) = J
where J =
(O −InIn O
).
Remark 1 Let n=1, J =
(a b
c d
),then
(a b
c d
)′(0 −1
−1 0
)(a b
c d
)=
(0 bc− ad
ad− bc 0
)
Obviously, f : R2 −→ R2 is a symplectic diffeomorphism if and only if the determinant of
Jf is equal to 1 for any z ∈ R2.
Remark 2 In the case of multidimensional conditions, we also have the result that the deter-
minant of Jf is equal to 1 if f is a symplectic diffeomorphism.
Definition 2 A diffeomorphism f is said to be a germ of symplectic diffeomorphism at
the point of z0 if f is a symplectic diffeomorphism in a neighborhood of z0.
Definition 3 A ∈ Sp(2n,R) is said to be conjugated in Sp(2n,R) to B if there exists a
homeomorphism H ∈ Sp(2n,R), which converts A into B=H−1 ◦A◦ H.
23
24 ïÄ?Ø �´
Definition 4 α = (α1, · · · , αn) ∈ T n is said to be satisfying a diophantine condition (we write
it α ∈ DC) if there exist γ > 0, β > 0 ,such that
|e2πi<k,α> − 1| ≥ γ
(n∑
j=1
|kj |)β
, ∀k ∈ Zn\0
Definition 5 We say f leaves invariant in a neighborhood U of O if f (z) ∈ U for any z ∈ U .
Our fundamental idea is to use the similar method applied in the proof of Siegel’s theorem
to reduce f to its normal form which we have not yet known. ( cf: V.I.Arnold: Geometrical
Methods in the Theory of Ordinary Differential Equations.)
First,let’s consider some simple conditions, for example, let n=1 and A = rα.
Note that A is equal to rα, that is, Az = eiαz = (cosα + i sinα)(x + iy) = (x cosα −y sinα) + i(x sinα+ y cosα), then we have
A
(x
y
)=
(cosα − sinα
sinα cosα
)(x
y
)
So far,I have three directions to consider this conjecture. Unfortunately,none of them has
derived essential progress so far.
(I) Let
u : R2 −→ C1
(x, y) 7−→ z = x+ iy, z = x− iy
we have x = z + z2 , y = z − z
2i , then,
u−1 : C1 −→ R2
z 7−→ (z + z
2,z − z
2i)
Assume f (z)=Az+a(z), where a(z) =
(a1
a2
)= O(|z|2). Let
f = u ◦ f ◦ u−1 : C1 −→ C1
f (z) = (u ◦ f)(z + z
2,z − z
2i)
= u(z + z
2cosα− z − z
2isinα+ a1,
z + z
2sinα− z − z
2icosα+ a2)
= eiαz + a1 + a2i.
⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣Technical skill is mastery of complexity while creativity is mastery of simplicity. — Chris Zeeman
1 60 Ï A Conjecture About Dynamical Systems 25
f
(x
y
)= f
(z + z
2z − z
2i
)=
(z + z
2 cosα− z − z2i sinα+ a1
z + z2 sinα− z − z
2i cosα+ a2
)
Jf =
(cosα
2 − sinα2i + a1z
cosα2 + sinα
2i + a1z
sinα2 + cosα
2i + a2zsinα
2 − cosα2i + a2z
)
Note that f is a sympletic diffeomorphism, we have
|Jf | = (cosα
2− sinα
2i+ a1z)(
sinα
2− cosα
2i+ a2z)
−(sinα
2+
cosα
2i+ a2z)(
cosα
2+
sinα
2i+ a1z) = 1
that is,
(a1z − a2zi)e−iα − (a1z − a2z)e
iα + 2i(a1za2z − a1za2z) = 2i.
Difficulties about (I): Although f is real analytic, f is not necessarily complex analytic.
This is the essential difference compared with Siegel’s theorem.
(II) Assume f (z)=Az+a(z),where a(z) = O(|z|2).
Let x = r cos θ, y = r sin θ, a =
(a1
a2
),
f
(r cos θ
r sin θ
)=
(cosα − sinα
sinα cosα
)(r cos θ
r sin θ
)+
(a1
a2
)=
(r cos(θ + α) + a1
r sin(θ + α) + a2
)
Jf =
(cos(θ + α) + a1r −r sin(θ + α) + a1θ
sin(θ + α) + a2r r cos(θ + α) + a2θ
)(1)
Our goal is to find a ur(θ) : T 1 −→ R2 for some r,such that f (ur(θ)) = ur(θ + α). Let
ur(θ) =
(u1
u2
), then
f (ur(θ)) =
(u1(θ) cosα− u2(θ) sinα+ a1
u1(θ) sinα+ u2(θ) cosα+ a2
)=
(u1(θ + α)
u2(θ + α)
)(2)
From (1),(2) and f is a symplectic diffeomorphism, we can get that
r +
∣∣∣∣∣ru1r(θ + α) + u2θ(θ + α) cosα
ru1r(θ) − u2θ(θ) cos(θ + α)
∣∣∣∣∣
+
∣∣∣∣∣ru2r(θ + α) − u1θ(θ + α) sinα
ru2r(θ) − u1θ(θ) sin(θ + α)
∣∣∣∣∣
+
∣∣∣∣∣u1r(θ + α) u1θ(θ + α)
u2r(θ + α) u2θ(θ + α)
∣∣∣∣∣+∣∣∣∣∣
u1r(θ) u1θ(θ)
u2r(θ) u2θ(θ)
∣∣∣∣∣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣Do not imagine that mathematics is hard and crabbed, and repulsive to common sense. It is merely the
etherialization of common sense. —William Thomson
26 ïÄ?Ø �´
+(
∣∣∣∣∣u1r(θ) u1θ(θ)
u1r(θ + α) u1θ(θ + α)
∣∣∣∣∣+∣∣∣∣∣
u2r(θ) u2θ(θ)
u2r(θ + α) u2θ(θ + α)
∣∣∣∣∣) sinα
+(
∣∣∣∣∣u2r(θ) u2θ(θ)
u1r(θ + α) u1θ(θ + α)
∣∣∣∣∣+∣∣∣∣∣
u2r(θ + α) u2θ(θ + α)
u1r(θ) u1θ(θ)
∣∣∣∣∣) cosα
= 1 (3)
Difficulties about (II): The most difficult aspect about (II) is the complexity of the equation
(3).
(III) Let f (z)=Az+a(z), where a(z) = O(|z|2),a =
(a1
a2
). Let T : x =
√2I cos θ, y ==
√2I sin θ, 0 < I ≪ 1. Then,we have
JT =
cos θ√2I
−√
2I sin θ
sin θ√2I
√2I cos θ
We can deduce the determinant of JT is equal to 1 from that T is a symplectic diffeomor-
phism.
Let f = T−1 ◦ f ◦ T =
(I
θ
), and
(x
y
)= (f ◦ T )(I, θ) = f
( √2I cos θ√2I sin θ
)
=
( √2I cos(θ + α) + a1(
√2I cos θ,
√2I sin θ)√
2I sin(θ + α) + a2(√
2I cos θ,√
2I sin θ)
)
From x =√
2I cos θ, y =√
2I sin θ, we can get that
I =x2 + y2
2, cos θ =
x√2I, sin θ =
y√2I
So,
I =x2 + y2
2= I + a1(
√I, θ), cos θ =
x√2I
= cos(θ + α) + a2(√I, θ)
where
a1(√I, θ) =
√2I(a1 cos(θ + α) + a2 sin(θ + α) +
1
2(a2
1 + a22) = O(I
32 )
a2(√I, θ) =
1√2Ia1(
√2I cos θ,
√2I sin θ) − 1
2Ia1(
√I, θ) cos(θ + α) +O(I) = O(
√I)
Then θ = θ + α+ a2(√I, θ), where a2 = O(
√I).
Now, we have get that
⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣⊣I used to measure the Heavens, now I measure the shadows of Earth. The mind belonged to Heaven, the body’s
shadow lies here. —Johannes Kepler
1 60 Ï A Conjecture About Dynamical Systems 27
I = I + a1(√I, θ)
θ = θ + α+ a2(√I, θ)
where
a1(√I, θ) = O(I
32 ), a2(
√I, θ) = O(I
12 )
Difficulties about (III): By far, the difficulty is how to expand a1 and a2 in the Fourier
series or Taylor series with concrete coefficients.
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u´, ∀ε > 0, ∃N , ¦�
|f(λ) − fN(λ)| ≤ ‖f − fN‖L1(R) < ε/2,
≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎
ÑWU-u½8¤�~§±x¦<ü%�8§�yUÄ<%u§óƦ<¼��¦§�Æ�UõÔ�)¹§
�êÆU��±þ���" — F. Klein
30 �(�¡ �´
é?¿� λ ¤á"q ∃A > 0§� |λ| > A �§|fN (λ)| < ε/2§u´§
|f(λ)| ≤ |f(λ) − fN(λ)| + |fN (λ)| < ε,
=
|f(λ)| → 0, |λ| → ∞.
2
ùp2Jø�«�é�{B�y², ù��{ée¡�J��,�«C/��k
�:
(�{ 2) -
Sλ =
∫ +∞
−∞
f(x) cosλxdx
=
∫ +∞
−∞
f(x+ π/λ) cosλ(x + π/λ) dx
= −∫ +∞
−∞
f(x+ π/λ) cosλxdx,
|2Sλ| = |∫ +∞
−∞
(f(x) − f(x+ π/λ)) cosλxdx|
≤∫ +∞
−∞
|f(x) − f(x+ π/λ)| → 0, λ→ ∞.
���Úd LebesgueÈ©�²þëY5¤�y" 2
íØ 2 £éÈ©«��,�«í2¤e f ∈ L1(−∞,+∞)§(aλ, bλ) ´��ê λ k'�«
m§K
limλ→+∞
∫ bλ
aλ
f(x) cosλxdx = 0.
y² ^þ¡��{ 2§-
Sλ =
∫ bλ
aλ
f(x) cosλxdx
=
∫ bλ−π/λ
aλ−π/λ
f(x+ π/λ) cosλ(x+ π/λ) dx
= −∫ bλ−π/λ
aλ−π/λ
f(x+ π/λ) cosλxdx,
≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎
·�¹e��·�kéõÀÜvk��, —Abel
1 60 Ï Riemann− Lebesgue Ún9ÙA�í2/ª 31
K
2Sλ =
∫ bλ−π/λ
aλ
(f(x) − f(x+ π/λ)) cosλxdx
−∫ aλ
aλ−π/λ
f(x+ π/λ) cosλxdx
+
∫ bλ
bλ−π/λ
f(x) cosλxdx,
� λ→ ∞ �§
|2Sλ| ≤∫
R
|f(x) − f(x+ π/λ)| dx +
∫ aλ+π/λ
aλ
|f(x)| dx +
∫ bλ
bλ−π/λ
|f(x)| dx→ 0.
1��ªu 0 d Lebesgue È©�²þëY5¤�y§�¡ü�ªu 0 d Lebesgue È
©�ýéëY5¤�y" 2
íØ 3 £é sin Ú cos ¼ê�í2¤e {gn(x)}n=+∞
n=1 ´ [a,b] þ��ÿ¼ê��÷vµ
(i)|gn(x)| ≤M (x ∈ [a, b]) (n = 1, 2, · · ·);
(ii) é ∀c ∈ [a, b] k
limn→+∞
∫
[a,c]
gn(x) dx = 0,
Ké ∀f ∈ L1[a, b]§k
limn→+∞
∫
[a,b]
f(x)gn(x) dx = 0.
y² �5¢C¼êØ6§±¬r§P197§~ 4" 2
�z
[1] ±¬r"5¢C¼êØ6§�®�ÆÑ��"
[2] ~�ó§¤L~"5êÆ©Û�§61nþ§ô���Ñ��"
[3] L.C. Evans, Partial Differencial Equations, American Mathematical Society.
≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎≎
«2pk�´�m�>E" —Erdo s
üëÏ�þ>.éA�n�n)9í2
0201 ð
3���EC¼ê�§¥, ·�ÆL�~�� Riemann K�½n"·���éuE
²¡ C ¥�üëÏ� D§eÙ>.õu�:§K�±�/N��ü �� ∆ þ"@o§ù
�N�´Ä�½Uòÿ�>.þº·���� D � Jordan 4��¤�«���±��"
�©Ò´�ÏL�à�*:§)û��� D � ∆ ��/N�3>.��¹"
��>.:� Koebe ½n
·�ÄkÚ\��>.:�Vgµ éu z0∈ ∂D, e�3 Jordan � l(t),
t ∈ [0, 1], l(1) = z0, ¿� l\{z0} ⊂ D, K¡ (z0, l) � D ���lë:"·�`ü�lë
: (z1, l1),(z2, l2) �d§XJ z1 = z2 � ∀z1 ��� U§∃ l γ ⊂ (U⋂D) ë� l1,l2"w,
ù´�d'X"@o·�rlë:��da¡� D ���>.:§(z0, l) ¤3�daP�
[z0, l]. � z0 Ñ�k����>.:§½öز(�Ñ l qØu)· �§�^ z0 L«"w
,§éu��lë:§l �±� D ¥?�:�Ù��à:�ë� z0 ��"
e¡·�ÚÑ Koebe �(ص
½n 1 D �k.üëÏ«�§△ �ü �§f(z) � D � △ ��/N�§Kµ
(1)D �z���>.: [z0, l] ÑéAuü �±���: w0, ¦�� D ¥ l þ�:ª�
u z0 �§Ù3 f e��ªu w0
(2)D �ü�ØÓ���>.:éA ∂△ þü�ØÓ:
(3) P D ��N��>.:éA�8Ü� F§K F 3 ∂△ þ�È�"
�y²ù�½n§·�kÚ\ü�Ún:
Ún 1 Jordan lS� {Υk}∞k=1 uü � ∆ S§�3�: 0 ��� G §Υk �à:
z1k,z2k ©OÂñu ∂△ þü�ØÓ: w1,w2"ek.�)Û¼ê f 3 {Υk}∞k=1 þ��ªu
0§K f(z) ≡ 0"
32
1 60 Ï üëÏ�þ>.éA�n�n)9í2 33
y² 1◦·�^�y{,b� ∆�3:¦� f(z) 6= 0. �I�Ä f(0) 6= 0��¹"Ï�e 0�
f � m ":,�±�Ä g(z) = f(z)/zm, g(0) 6= 0,∞"K g 73 0 ���� B(0, δ) ⊂ Gk
.§ 1/zm3 ∆�K B(0, δ)¥k.,l g3 ∆¥k.",§{Υk}∞k=1 ⊂ ∆\B(0, δ)§f 3
��þ��ªu 0§�� g3 {Υk}∞k=1þ��ªu 0. ·���±b� w1 = eπi4 , w2 = e
5πi4 "
Ï�o�±^ ∆ �gÓ���§ EÜgÓ�é(ØvkK�"
2◦ du f 3 {Υk}∞k=1 þ��ªu 0, K ∀ǫ > 0, ∃N , ¦� f(z) < ǫ éu ∀z ∈ ΥN . du
ΥN ⊂ (∆\G), �ü�à:©O3 eπi4 , e
5πi4 NC, K��§�¢¶J¶©O�u P ,Q ü:"
2� P ,Q 'u�:�é¡:©O� P ′, Q′. P PQ � ΥN þë� PQ �Ü©l§ §'u
¢¶J¶±9�:�é¡�©OP� PQ′, P ′Q, P ′Q′. §��¤ ∆ ¥�¹ G �«� K"
3◦ ·�3�¼ê F (z) = f(z)f(−z)f(z)f(z)§w, F(z) 3 ∆ �X§l 3 K þ�X"�
|f | 3 ∆ þkþ. M"K3 PQ þ§|f(z)| < ǫ, |f(−z)| < M, |f(z)| < M , |f(z)| < M , l
|F (z)| < ǫM3" 3 PQ′, P ′Q, P ′Q′ þ§©Ok |f(z)| < ǫ, |f(−z)| < ǫ, |f(z)| < ǫ, l
|F (z)| < ǫM3 3 ∂K þ¤á"d����n§|F (z)| < ǫM3 3 K þ¤á, �,3 0 ?¤
á"|F (0)| = |f(0)|4 < ǫM3. - ǫ→ 0, f(0) = 0, gñ"l §3 ∆ ¥§f(z) ≡ 0" 2
·��I�XeÚnµ
Ún 2 � D �k.üëÏ«�§)Û¼ê f(z) 3 D ¥k.§¿��3 D �k¡>.:
a 9Ù���� U , ¦�� z l D �Sܪu ∂D⋃U �§f(z) ªu~ê c, K3 D Sk
f(z) ≡ c.
y² 1◦ Äkb� a � D �:�4�:"�Ø�� c = 0, ÄK�Ä f(z) − c =�"�
B(a, ρ) ⊂ U . K ∀zo ∈ B(a, ρ2 )⋂D, ∃ r > 0 ¦� B(z0, r) ⊂ B(a, ρ), ¿�¦� ∂B(z0, r) þk
D �:"(ù�:d |z0 − a| < ρ2 9 a ∈ ∂D w,) Ï §|z − z0| = r ¥7�¹�ãl γ 3
D �Ü"·�À� γ§�±¦Ù�Ý� 2πrm , Ù¥ m �,����ê"
2◦ d^�·��� ∀ǫ > 0, � z l D �Sܪu B(z0, r)⋂∂D �§|f(z)| < ǫ"¿��
3 D S |f(z)| < M ,M ��~ê"e¡·��y |f(z)| ≤M( ǫM )
1m .
ò«�D7: z0_��^=�Ý2πm , 4π
m , 6πm . . . 2(m−1)π
m ,K��«�D1, D2, D3, . . . Dm−1"
2P V = D⋂D1
⋂D2
⋂. . .⋂Dm−1"duzg^=¦��Ý�
2πrm �l3 G §��
z0 ∈ G ⊂ B(z0, r) ⊂ B(a, ρ) ⊂ D"¿�·���� G �>.d m �Ü©|¤§z�Ü©�
,� Di �>.§=��2πrm ��l"
�e5§�ìÚn 2 �y²§�E¼ê F (z) = f(z)f(e2πim )f(e
4πim ) · · · f(e
(m−1)πi
m ), F (z)
3 G S)Û§¿�´� |F (z)| < ǫMm−1, � z → ∂G. d����n§|F (z)| ≤ ǫMm−1,
∀z ∈ G"AO�§3 z0 ?§|F (z0)|m ≤ ǫMm−1"= |f(z0)| ≤M( ǫM )
1m"
⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳
�X©Æp�<���a�)��§êÆKéu<�����ín" — Chancellor,W.E.
34 �(�¡ �´
3◦ dþ¡·��� ∀zo ∈ B(a, ρ2 )⋂D,∃m(��ê), 2 ∀ǫ > 0,k |f(z0)| ≤M( ǫ
M )1m"-
ǫ→ 0,�� f(z0) = 0. d z0 �?¿5§��3 B(a, ρ2 )⋂D ¥§f(z) = 0,l 3 D ¥�´"
4◦ � a Ø´ D �:�4�:§du D Ø�¹ a 9 ∞, K�±�√z − a 3 D þ�
ü�)Û©| h(x), ¦ h(D) �k.üëÏ�"w,§0 ∈ ∂h(D), � 0 � h(D) �:�4
�:"£Ï� ∀ǫ > 0,ǫ,−ǫ, iǫ,−iǫ¥7k�:� h(D) :¤"�Ä f(a + ξ2) 3 h(D) þ§d
h(z)� D � h(D) ��/N�§��Ún^�E,÷v§l f(a+ ξ2) ≡ c 3 h(D) þ§l
f(z) ≡ c 3 D S" 2
e¡·�B5y² Koebe ½n 1µ
y² 1◦ �yé�½���>.: [z0, l], � z l D �SÜ÷ l ªu>. z0 �§f(l(t)) ù
^ ∆ S��ªu>.þ�:§��¤���L� l Ã'"
ÄkyÙªu>.§XJ f ◦ l ªu ∆ S: ø§Kd f−1 ��/N�§f−1(ø) ∈ D, �
z0 ∈ ∂D gñ"
Ùg§·�yÙªu��:"^�y{§� ∃w1, w2 ∈ ∂∆§¿� w1 6= w2, ¦� l þ
�:� {z1k}∞k=1, {z2k}∞k=1 Âñ� z0§ 3 f e��©OÂñ� w1, w2"P lk ⊂ l �ë�
z1k, z2k ��ã§K {lk}∞k=1 7,��Âñ� z0. 2P {lk}∞k=1 3 f e��� ∆ ¥��
{γk}∞k=1, §�ü�à:�¤�:�©OÂñ� w1, w2 ∈ ∂∆"� f(z1) = 0, K�3 z1 ��
� G§¦� k ¿©��§lk 3 G �"l Ø�� G⋂{lk}∞k=1 = ∅§K 0 ∈ f(G) ⊂ ∆ �
f(G)⋂{γk}∞k=1 = ∅"2� g(w) = f−1(w) − z0, du D �k.üëÏ«�§f � D � ∆ �
�/§K g(w) � ∆ þ�k.)Û¼ê§�3 {γk}∞k=1 þ��ª�u 0"@o§é g(w) ^Ú
n 2§�� g(w) ≡ 0, K f−1(w) ≡ z0, � f ��/gñ"
¯¢þ§dþ¡·�y²é D ¥?¿�^� z0 �ë��§§3 f e��þªu
∂∆ þÓ��:"Ï�ù��ü^�¥�g�±é�:�ªu z0§ qd��>.:�½
§o�±é�ë�ùü�:���Âñ� z0 ���"2^þ¡�{=�"
2◦ e¡�y D �ü�ØÓ���>.: [z1, l1], [z2, l2], éA ∂∆ þü�ØÓ�:
w1, w2"
·�Ø�b� l1⋂l2 = {a}, a ∈ D, =ùü^�ë(¤�^üà:3>.� D ¥�
Jordan�§r D ©�ü�üëÏ�"2� λ1 = f(l1), λ2 = f(l2)"K�, λ1, la2 �ë�¤
r ∆ ©�¤ü�üëÏÜ©� Jordan l, üà� w1, w2"
·�æ^�y{§b� w1 = w2"@o λ = λ1
⋃λ2 ��^ Jordan4�§� λ
⋂∂∆ =
{w1}. � λ �¤�üëÏ«�� ∆1 ⊂ ∆§¿�Ù3 f e����üëÏ«� D1, �, D1
´ l1⋃l2 r D ©¤ü°¥��°"¿�w, ∂D1 ⊂ (∂D
⋃l1⋃l2)"
⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳
¯K´êÆ�%9" —P.R.Halmos
1 60 Ï üëÏ�þ>.éA�n�n)9í2 35
e ∂D1 = (l1⋃l2)§K`² z1 = z2 �� l1, l2 þ��é�:¿©�C z1 �§o´�±
3 D1 ¥é��ë(Tü:§l � [z1, l1], [z2, l2] �ü�ØÓ���>.:gñ"
@o ∂D\(l1⋃l2)��§K�3 ∂D þ�ã Jordanl l ¦ ∂D\(l1
⋃l2) = l"du l1, l2, l
þ� Jordan�§3î¼Ýþ¿Âe§∃b ∈ l ±9Ù�� V , ¦� V⋂
(l1⋃l2) = ∅"� z l
V⋂D1 ⊂ D ¥ª�u V
⋂∂D1 = V
⋂∂D �§Ù3 f e��òª�u ∂∆
⋂∂∆1 = {w1}"
l ·��±^Ún 3§��3 D Sk f(z) ≡ c"gñ"l w1 6= w2"
3◦ P D ��N��>.:éA�8Ü� F ⊂ ∂∆§e¡·��I�y F 3 ∂∆þ�È
�"
e¡�´^�y{§b� F 3 ∂∆¥ØÈ�§K�3: w0 ∈ ∂∆±9ml Γ ⊂ ∂∆§w0 ∈
Γ§¦� Γ⋂F = ∅"�:� {wn}∞n=1 ⊂ ∆§¿� wn → w0, n→ ∞"P zn = f−1(wn)§K:�
{zn}∞n=1 3 D ¥kÂñf� {znk}∞nk=1"d 1o �§znk
→ z0 ∈ ∂∆, nk → ∞"�l znk� z0
���§§Ø�½�¹3 D ¥§��½¬� ∂D ��§�±Pl znk�1���:���Ü
©� lnk"@o§lnk
\{z0} ⊂ D§��, lnk(½����>.:"du znk
→ z0, nk → ∞§
�±�� lnk��Âñ� z0.
qP λnk= f(lnk
)§K lnk���à:� wnk
§d 2o ��§,��à: wnk3 ∂∆\Γ
þ"(5¿ Γ¥vk:éA��>.:")d ∂∆�;5§�3 {wnk}∞nk=1 �f�Âñ� w0§
dueI���§Ø�� wnk→ w0, nk → ∞"d w0 ∈ (∂∆\Γ) 9 w0 ∈ Γ§��§w0 6= w0"
·��±b� {wnk}∞nk=1 3 0 :�,����§Ï�§������Âñ���>
.:" é λnkÑ� á�¦Ù� ∆ ¥�4�§�ù���4��ü�à:©OÂñ
� ∂∆ þü: w0, w0£w0 6= w0¤"é g(w) = f−1(z)− z0 ^Ún 2§�±�� f−1(z) ≡ z0 3
∆ ¥"l gñ"l F 3 ∂∆ þÈ�" 2
d Koebe½n§éulk.üëÏ«� D �ü � ∆��/N�§·��±r§½Â
� D ���>.:þ"@oéu@Ø���/�Qº·���?nK D >.þ@ØÐ
�/�§��¦ù���/N�½Â��õ�>.Ü©§ÒI�r«�?1;z"�d§·
��±Ú\�à�Vg"ù�Vg5 u Caratheodory"
�à
Äk½ÂÄ�ó"éu,�üëÏ� D§XJ γ : [0, 1] → D ��^ Jordan�§ü�
à:3 ∂∆§Ù{þ3 D ¥§@o§7,r D ©¤ü�üëÏ«�"òÙ¥��«�P�
N(γ)§,��P� M(γ). ·�ò��ù�� D ¥� {γn}∞n=1 ¡�Ä�ó§XJµ
(1) N(γn+1) $ N(γn)§n = 1, 2, 3 . . . (2) diam(γn)→ 0, n→ ∞
⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳
êÆ[ÎØ�9(²½ß�§¦�==�â½ÂÚún§¿^ØyÚín5üÌz��¯"—Reid,Thomas
36 �(�¡ �´
·K 1 D �üëÏ«�§∆ �ü �§f �l D � ∆ ��/N�"K3 f �¿Âe§D
¥Ä�ó {γn}∞n=1 éAu ∆ ���>.:"
y² P {λn}∞n=1 � {γn}∞n=1 3 f e��§Kd Koebe ½n�y²L§±9 f ��/5ë
Y5��§λn �à:3 ∂∆ þ§eP N(λn) = f(N(γn))§@ok N(λn+1) $ N(λn)§n =
1, 2, 3 . . .§� diam(λn) → 0, n→ ∞"d ∆ �AÏ/G§∂N(λn)\λn ⊂ ∂∆§¿�|^� n ¿
©��§λn�ü�à:� ∂N(λn)þ:�ålþØ�u diam(λn)§�±�� diam(N(λn)) →
0, n→ ∞"é {N(λn)}∞n=1 ^48@½n§��∞⋂
n=1
N(λn) ��: w0§¿� w0 ∈ ∂∆"£Ï�
§�Óu ∂∆ þ��¥üN�¹'X��»ª�u 0 �4���¤w0 =� {γn}∞n=1 éA
�:" 2
éuü�Ä�ó§{γn}∞n=1 � {γn}∞n=1, ¡§���d�§XJéz� n5`§N(γn)Ñ
�¹,� N(γm)§¿� N(γn) Ñ�¹,� N(γk)"N´wÑù´�d'X"¿�·�kµ
·K 2 f,D Óþ§D �ü�Ä�ó�d ⇔ §�éAu ∂∆ þÓ��:"
y² 1◦ e {γn}∞n=1 � {γn′}∞n=1 �d§§�3 f eéA��� {λn}∞n=1 � {λn′}∞n=1§Kd
f ��/N�§éz� n 5`§N(λn) Ñ�¹,� N(λm′)§N(λn′) Ñ�¹,� N(λk)"d
·K 1 ±948@½n§w,�7�5¤á"
2◦ e {γn}∞n=1 � {γn′}∞n=1 éAu ∂∆ þÓ�:§KéA� ∆ ¥k§∞⋂
n=1N(λn) =
∞⋂n=1
N(λn′)"lù�ªf�±��§é ∀n§Ñk N(λn) ⊃∞⋂
n=1N(λn′)§qdý�¹'X§�
� ∃m, N(λn) ⊃ N(λm′)"l N(γn) ⊃ N(γm′)§K N(λn) ⊃ N(λm′)"Ón§�� ∃k ¦�
N(λn′) ⊃ N(λk)"l �� {γn}∞n=1 � {γn′}∞n=1 �d" 2
dd§·�½Â D ¥Ä�ó��da��à"e¡�ÑÌ�½nµ
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��éA"
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�ü�:� {wn}∞n=1, {wn′}∞n=1§£::ØÓ¤¦§�ÑÂñ� w0§¿�©OéA ∂D ���
>.:� {[zn, ln]}∞n=1, {[zn′, ln′]}∞n=1"� ln, ln′ ©OéA ∆ ¥� λn, λn′"�,§wn � λn
�à:§wn′ � λn′ �à:"⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳
vk?Û¯K�±�á@����>Ä�a§é�kO�*gU�á@�-yn��)Lk¤J�g
�§�vk?ÛÙ¦VgU�á@�I�\±�²" — D. Hilbert
1 60 Ï üëÏ�þ>.éA�n�n)9í2 37
éuz� n§� λn �± wn �à:��Ü©�ã γn§¦� γn þ�:��� wnwn′
�ål§�u��� wn−1wn−1′ �ål§��u��� wn+1wn+1′ �ål"é λn′ Ó�?
n��§�± wn′ �à:���ã γn′"
� ∆ Së� γn � γn′ ü�à:���§¦�§� γn � γn′ |¤l wn � wn′ �!©
� ∆ �ü�üëÏ«�� Jordan � hn"�d hn ©�Ñ� ∆ �ü�üëÏ«�¥§±
w0 �>.:�@�� N(hn)"
dc¡��E�{§·�N´�� {hn}∞n=1 ¥��üüØ��§¿��Âñ� w0§
� N(hn+1) $ N(hn), n = 1, 2 . . ."K {hn}∞n=1 3 D ¥����¤��Ä��"l ��
d w0 Ñu§·������§éA��à" 2
ù�·���k.üëÏ«� D ��à� ∆ �>.:���éA§�Ò´`§·�
r D � ∆��/N�½Â� D ��N�àþ"@o§�à� D �>.q´�o'XQº
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γn �üà:þ�� z0, �¦ γn � l o´��§@o§ù���>.:ÒéA����
�à"��§�±@���>.:��Ò´���à"Ó�§d�à�N� ∂∆ ��éA§
±9z���>.:Ñ��éAu ∂∆ þ��:§�±��z��à�o�±w¤���
�>.:§�o���>.:Ã'§Ø�Uk���àéAü���>.:"ù��5§Ò
��u��>.:´�à8Ü�È�f8"
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>.:§�oéAü���>.:"dd§·��±@�§�à8Ü�|¤k 3 «¤©µ
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D §Kr z ���à¶
(2) éu��>.: [z, l]§eØ÷v (1)§Kr z ��ü�:�\�à¶
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|ÜÏ ∂D É"
ù�·����à� D �>.�'X"
±þ·�ob� D �k.üëÏ�§e D Ã.§K·��±kïá§� ∆ ��/N
� f§,�r D �,�k.üëÏ<Ñ5§XJ§�¹ D �Ü©>.§d f ���éA�
ëY5§o�±rþ¡�½nA^�ù>�Ü©>."ù�o�±r D �>.ÛÜ�?n"
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�éA�"
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êÆ¥��{w½näkù��A5: §�4´l¯¢¥8BÑ5,�y²%Ûõ�4�. —Gauss
38 �(�¡ �´
�{ü�íØ
c¡·�)û?¿üëÏ«� D �ü � ∆��/N�e§>.éA�¯K" ù
«*:§·��A#"À~��>.éA¯K"
íØ 1 D,∆, f Óþ§� D ��Ü©>.� Jordan ml Γ§§þ: D :�4�: �
Γ �?¿�S:Ø´Ù{� ∂D ¥:�4�:�§f o´�±½Â3 Γ þ§ � f 3 D⋃
Γ
þëY§Ó�u ∆⋃f(Γ)"
y² ^�à�*:w§Γþ�:�éA����>.:§�ÒþéA���à"K�
I�y²ëY5"Ï�dëY59��éA§B���Ó��(Ø"
f 3 D S�,ëY§l D � Γ déA'X����ëY§@o�I�y²µ∀z0 ∈
Γ, limξ→z0,ξ∈Γ
f(ξ) = f(z0).
d®�^�±9 f l D � Γ �ëY��§∀ǫ > 0, ∃δ1 > 0§¦� B(z0, δ1)⋂∂D ⊂ Γ§
¿�� z ∈ B(z0, δ1)⋂D �§|f(z0) − f(z)| < ǫ.
2� ξ ∈ Γ⋂B(z0, δ1)§Ó z0 ���n§·��� ∃δ2§¦� B(ξ, δ2)
⋂∂D ⊂ Γ§��
z ∈ B(ξ, δ2)⋂D �§|f(ξ) − f(z)| < ǫ.
u´§� z ∈ B(ξ, δ2)⋂B(z0, δ1)
⋂D§Bk |f(z0)−f(ξ)| < |f(z0)−f(z)|+ |f(z)−f(ξ)| <
2ǫ. - ǫ→ 0§B��� ξ → z0 �§f(ξ) → f(z0)"l �� f 3 D⋃
Γ ëY" 2
²w�§XJ D �>.��^ Jordan4�§@o ∂D ¥z�:þ����à"Óí
Ø 7 ��§·�Bkµ
íØ 2 D �d�^ Jordan 4��¤�«�§K D � ∆ ��/N� f �±òÿ� D �
∆ �Ó�N�"
3þãü�íØ¥§XJ>.�)Û� Jordan l§K(Øg,�±�Ч=�±N´
�� f �X½ÂuT>.þ"
dd§·�^�à�*:§�±��ß�@£üëÏ��ü �m�/N�3>.þ½
Â��¹"
�z
[1] Ahlfors L.V.5E©Û6§1n�§þ°�ÆEâÑ��§1984"
[2] ÷,"5{²E©Û6§�®�ÆÑ��§1996"
[3] o§"5E©Û�Ú6§�®�ÆÑ��§2004"
⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳
�â´<a�£��P§�N´���P���©|¶, §����c����Ù�²��ýn´�
��ë�" — H.J.S. ¤�d
1 60 Ï üëÏ�þ>.éA�n�n)9í2 39
[4] ¤L~§4�^"5EC¼ê6§¥I�ÆEâ�ÆÑ��§1998"
[5] ªIT"5�/N��>�¯K6§p���Ñ��§1985"
[6] ÜH�§�~¨"5EC¼êØÀù6§�®�ÆÑ��§1995"
⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳
wD���x§ �AÛ©�iZ"ÎÙíá§Y^XY±/�~063§%�ª6£p?§XdÌ�
E¶�mm©�,§þe!�m!SÏÏ6�§%¿÷{"
⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳
Mathematics, rightly viewed, posses not only truth, but supreme beauty; a beauty cold and austere, like that
of sculpture — Bertrand Russell
�« Riemann ¡þ�ÄåXÚ
0201 �;¿
·�3EC¼ê¥Q²ÆL*¿E²¡ C þ�üëÏ«�)ÛÓ�u D(ü ��)§ C
(E²¡)§C (*¿E²¡) nö��"3 Riemann ¡þkaq�(ا= (Poincare − Klein − Koebe)
½n§§´`üëÏ Riemann ¡)ÛÓ�ue¡n«;. Riemann ¡��µ(1)*¿E²¡ C = C∪∞; (2) E²¡ C; (3) ü �� D={z ∈ C : |z| < 1}"
dd·�|^ Riemann ¡��'�£�±í�§é?¿ Riemann ¡ S§(S, π) ´ S ��kCX§P
G �ÙCXC�+§K: (1)S = C �§G = id; (2)S = C �§G = C ½ C\{0} ½�¡ T = C/Λ, Ù¥
Λ = {z → z + m1 +n2 : n, m ∈ Z}, 1, 2 ∈ C� Im(1/2) 6= 0¶(3)S = D �§G´��ÃL� Fuchs
+. 3�©¥§·�=é T = C/Λ þ�ÄåXÚ�?Ø"
Äk0�A�Ä�Vgµ
½Â 1 U ´ C ¥�m8§fn : U → C ´�X¼ê (�©Ø«©�X¼êÚæX¼ê)§¡ F = {fα : U →C, α ∈ A} ´�5x§XJ§é F ¥�?¿f�Ñ�3S4��Âñ�f�"½Â 2 � S ´ Riemann ¡§f : S → S ´�~��XN�"- fn : S → S ´ f � n gS�"éu�½�
z0 ∈ S, e�3 z0 ��� U ¦� {fn|U} ´�5x§K¡ z0 áu f � Fatou 8 (P� F (f))¶eù���
�Ø�3§K¡ z0 áu f � Julia 8 (P� J(f))"
½Â 3 z ¡� f �±Ï:´��3 n ∈ N ¦� fn(z) = z§��� n ¡� z �±Ï"λ = (fn)′
(z), e
|λ| > 1,K¡±Ï:�½5�"½5±Ï:Ñ3 J(f)¥§�3Ù¥È�"d½Â´�§Fatou8´m8§Julia
8´48§§�p�{8"
±e·�?\Ì�¯K"
½n 1 z��XN� f : T → T Ñ´��N�§= f(z) ≡ αz + c(modΛ); éA� J(f) � |α| ≤ 1 �´�
8§� |α| > 1 �´��� T"
y² Äky² f(z) äk αz + c �/ª"
Ø�� Λ ��� 1 Ú τ ¤)¤�§Ù¥ τ 6∈ R"Ï� T ± (C, π) �Ù�kCX§J,�3"=�3�
XN� F : C → C ¦� π ◦ F = f ◦ π"
�é?¿ z ∈ C,
π ◦ F (z + 1) = f ◦ π(z + 1) = f ◦ π(z) = π ◦ F (z),
l F (z + 1) ≡ F (z)(modΛ)§= H(z) = F (z + 1) − F (z) = λ ∈ Λ"
d H(z) � C þ�ëY¼ê§C´ëÏ�§Λ´lÑ8§� H(z) ´~ê"= ∃ λ1 ∈ Λ, ¦� F (z + 1) −F (z) = λ1 ∈ Λ"
Ón§∃ λ2 ∈ Λ, ¦� F (z + τ) − F (z) = λ2 ∈ Λ"
- g(z) = F (z) − λ1z, K
g(z + 1) = F (z + 1) − λ1(z + 1) = g(z),
g(z + τ) = F (z + τ) − λ1(z + τ) = g(z) + (λ2 − λ1τ),
l
g(z + nτ + m) = g(z + nτ) = g(z) + n(λ2 − λ1τ),
w,§g 3 C ��kn���Ø�"d Picard �½n� g �~ê§�� c1, � F (z) = λ1z + c1,
f ◦ π(z) = π ◦ F (z) = λ1π(z) + π(c1),
40
1 60 Ï �« Riemann ¡þ�ÄåXÚ 41
l §é?¿ z ∈ Λ
f(z) ≡ λ1z + c (modΛ).
ùÒy²þ�Ü©"±e·�?Ø Julia 8 J(f)"
Ún 1 �3�XN� f : T → T, f(z) ≡ αz + c ��=� αΛ ⊂ Λ .
y² /=�0�Ä f(z) − c á��±wÑ"
/�0e αΛ ⊂ Λ§- f(z) = αz, ´�yÙ÷v^�" 2
Ún 2 e |α| = 1, � α 6= 1, K f ´k��gÓ�" ¢þ§��U´ 2§3§4§6"
y² dÚn 1§αΛ ⊂ Λ, AO�§α ∈ Λ, ατ ∈ Λ = ∃ m1, m2, n1, n2 ∈ Z ¦�
α = m1 + τn1, ατ = m2 + τn2
�n�
α2 − (m1 + n2)α − n1m2 + m1n2 = 0
�k (α1 + α2 = m1 + n2
α1α2 = −n1m2 + m1n2
d�§��§α1, α2 p��ÝEê§� |α|2 ´�ê"d |α| = 1 �§−1 ≤ Reα < 1, � −2 ≤ m1 + n2 < 2, l
Reα = −1,−1
2, 0,
1
2�
α = −1,−1
2±
√3
2i,±i,
1
2±
√3
2i,
�A��©O´ 2§3§4§6" 2
Ún 3 e |α| 6= 0, é ∀z0 ∈ T , �§ f(z) = z0 3 T ¥Tk |α|2 �)"y² dÚn 2 �y²� |α|2 ��ê§|α − 1|2 = (α − 1)(α − 1) = |α|2 + 1 − 2Reα �´�ê"�Iy²
c = z0 ∈ Λ ��/=�§=�Ä z =m + nτ
α3 T ¥)��ê"
Ø�� T ´d 1 Ú τ ü���)¤§dc¡�?Ø� ∃m0, n0 ¦� α = m0 + n0τ"©l z �¢ÜÚ
Jܧ¿�d 0 ≤ Imz < |Imτ | 9 Imτ =τ − τ
2i��:
|mn0 − nm0| ≤ |α|2= 0 ≤ m0n − n0m
(m0, n0)≤ |α|2
(m0, n0).
� m0n − n0m ��|α|2
(m0, n0)�ØÓ��§�ØÓ� m0n − n0m éA�)ØÓ"
e m0n−n0m = m0n′ −n0m′ �m′ + n′τ
m0 + n0τ½´��)§K
m − m′
m0=
n − n′
n0= t"@o t +
m′ + n′τ
m0 + n0τ
½´)"d n − n′ ´m0
(m0, n0)��ê§�Ñ t ´
1
(n0, m0)��ê�"
- z0 =m′ + n′τ
m0 + n0τ, K z0 +
1
(n0, m0), · · · , z0 +
(n0, m0) − 1
(n0, m0)´ T ¥�ØÓ�)" 2
½n���Ü©y²µ
dÚn 2§|α|2 ��ê§� |α| ≤ 1 ��k |α| = 0 9 |α| = 1 ü«�/"
1. e |α| = 0, = α = 0, f(z) ≡ c (modΛ)"w, ∀z ∈ T, z ∈ F (f). � J(f) = ∅"
2. e |α| = 1, dÚn 2§f(z) = az + c (modΛ)"
XJ α = 1§K f(z) = z + c (modΛ)§l {fn(z)|T } ��Âñu ∞"� z ∈ F (f)"
XJ α 6= 1§K α ��ê�U� 2,3,4,6"= fk(α) = z + ck (modΛ), k = 2, 3, 4, 6"
� {fkn|T } ��Âñ� ∞,z ∈ F (f)"J(f) = ∅"
∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅Mathematics are well and good but nature keeps dragging us around by the nose. — Albert Einstein
42 �(�¡ �´
nþ¤ã§|α| ≤ 1 � J(f) = ∅"e |α| > 1§dÚn 3§� α 6= 1 �§f(z) = z = αz + c = z, = (α − 1)z + c ≡ 0 (modΛ) Tk |αn − 1|2
�)" ù±Ï:þ�½5±Ï:§�þ3 Julia 8¥"
é z0 ∈ T , éu z0 �?¿�� B(z0, ǫ),
- fn(z) = z§K
αnz + c1 − αn
1 − α= z + k + mτ,
)��§
z =c
1 − α+
k + mτ
αn − 1,
@o
z − z0 =c
1 − α− z0 +
k + mτ
αn − 1.
- t =c
1 − α− z0 �¢Ü� Ret, JÜ� Imt"@oéu
1
αn − 1∗ k +
τ
αn − 1∗ m§� n ¿©���
ÿ§|1 + τ |αn − 1
�|1 − τ |αn − 1
þ�u ǫ"l �3 k0, m0 ¦�k0
αn − 1+
m0τ
αn − 1∈ B(t, ǫ)"�½5±Ï:3 T ¥?
?È�§l k Julia 8�½Â� J(f) = T" 2
ØØØØØØØØØØØØØØØØØØØØ
�ÚíÛ¦)
kr�º�r�ºÚÛº
∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅∅A mathematician is a device for turning coffee into theorems. — Paul Erdos
k���+�f+�ê
04001 öUÐ
·���éu?����k�)¤��+ G§o�±ò G ©)�
G ∼= Zrank(G) ⊕ At
Ù¥ At � G �Ûf+"� G ��ê�á�§Ùf+��êg,´Ã¡§·�e¡�Ä�´k���+
�f+�ê"·��
G = Zpα1 ⊕ Zpα2 ⊕ · · · ⊕ Zpαn
Ù¥ n, αi(1 6 i 6 n) ∈ N∗§�´��4Ok�S�"·��½
(∗) py1 × py2 × · · · × pyn : y
(∗) ª¥� yi ∈ [0, αi], 1 6 i 6 n§kÒ�>L«�´ G �¥��«a.� pmaxi{yi} ��§Ùd©O3 G
¥����Ú� Zpαi (1 6 i 6 n) ¥� pyj (1 6 j 6 n) ���Ú ��£pyj ��Ø�½3 Zpαi ¥�§=^
SØ�½¤"kÒm>�ê y L«Ta.� pmaxi{yi} ��3 G ¥��ê"~X Zp ⊕ Zp ¥� p1 × p0 .�
p ��k ϕ(p) + ϕ(p) = 2ϕ(p) �" yØÓa.� pmaxi{yi} ��ج��"
½n 1 G = Zpα1 ⊕ Zpα2 (α1 6 α2) � G ���f+9�A�êXeµ
pβ(0 6 β 6 α1) �f+�ê�µ
(1)
βXi=0
pi
pγ(α1 < γ 6 α2) �f+��ê�µ
(2)
α1Xi=0
pi
pσ(α2 < σ 6 α1 + α2) �f+�ê�µ
(3)
α1+α2−σXi=0
pi
G ������괵
pβ(0 6 β 6 α1) ����ê´µ
(1′) p2β − p2(β−1)
pγ(α1 < γ 6 α2) ����ê´µ
(2′) pα1(pγ − pγ−1)
43
44 #)/ �´
y² ·�k5¦�����ê"éu pβ ��§
pβ × pβ : ϕ(pβ)ϕ(pβ)
pβ × pβ−1 :�
21
�ϕ(pβ)ϕ(pβ−1)
pβ × pβ−2 :�
21
�ϕ(pβ)ϕ(pβ−2)
......
pβ × p0 :�
21
�ϕ(pβ)
�ÙÚT�µ
ϕ(pβ)2 + 2ϕ(pβ)pβ−1 = ϕ(pβ)(pβ + pβ−1) = p2β − p2(β−1)
éu pγ ��µ
pγ × pα1 : ϕ(pγ)ϕ(pα1 )
pγ × pα1−1 : ϕ(pγ)ϕ(pα1−1)
......
pγ × p0 : ϕ(pγ)
ÙÚ�µ
ϕ(pγ)pα1 = pα1(pγ − pγ−1)
l §duØÓ� pβ£½ pγ¤�Ì�+´Ø¬k�Ó� pβ ��£½ pγ ��¤�§�´� G ¥� Zpβ
.f+��ê´p2β − p2(β−1)
ϕ(pβ)= pβ + pβ−1
G ¥� Zpγ .�f+��ê´µpα1ϕ(pγ)
ϕ(pγ)= pα1
·�25ww G ¥���f+��ê"
éu pβ ��f+§§�±k±e�(�µ
Zpβ
Zpβ−1 ⊕ Zp
Zpβ−2 ⊕ Zp2 Zpβ−2 ⊕ Z2p
Zpβ−3 ⊕ Zp3 Zpβ−3 ⊕ Zp2 ⊕ Zp Zpβ−3 ⊕ Z3p
.
.....
.
..
�´5¿� G ¥��Ú�=kü�§�ê�������êØÉ�ê�������ê�K�§�þã¥
�¹kn��Ú�� pβ ��f+��êÑ´ 0"
Ïd§·�=I�Äþã¥�1��£ü��Ú�¤��/"éu?¿� x ÷v 0 ≤ x ≤ β − x �§�
Ä Zpβ−x ⊕ Zpx ¥� pβ−x ����§$^úª (2′)§Ù�ê� ϕ(pβ−x)px"3 G ¥��êØ�u px ��
�w,Ѭ3 Zpβ−x ⊕ Zpx .�f+¥"ù`²ù«a.�f+��8ج¹k pβ−x ��§½= G ¥�
pβ−x ���²©�ù«a.�f+¥"l ��ù«a.�f+3 G ¥��ê´µ
(4)p2(β−x) − p2(β−x−1)
ϕ(pβ−x)px= pβ−2x + pβ−2x−1
éu x � β kü«�¹¬u)µ
⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊The union of the mathematician with the poet, fervor with measure, passion with correctness, this surely is the
ideal. —William James
1 60 Ï k���+�f+�ê 45
(i). � x = β − x �§Z2px �f+¥w,�¹ G ¥�¤k�êØ�u px ���§�ù«a.�f+
Tk��"l §d� G ¥� pβ �f+��ê´µ
(pβ + pβ−1) + (pβ−2 + pβ−3) + · · · + (p2 + p) + 1 =
βXi=0
pi
(ii). � β �Ûê�
(pβ + pβ−1) + (pβ−2 + pβ−3) + · · · + (p + 1) =
βXi=0
pi
éu pγ �f+§Ù(��±k£E´�õ�kü��Ú���/¤µ
Zpγ , Zpγ−1 ⊕ Zp, Zpγ−2 ⊕ Zp2 . . .
?¿g,ê x§� x 6 α1 < γ − x �§Zpγ−x ⊕Zpx .�f+¥� pγ−x ����ê´ ϕ(pγ−x)px§l §ù
«a.�f+��ê´ϕ(pγ−x)pα1
ϕ(pγ−x)px= pα1−x
éu x � γ kn«�¹¬u)µ
(i). � x = α1 = γ − x �§Zpx .�f+�ê´ 1§d�§G ¥� pγ �f+��ê´µ
pα1 + pα1−1 + · · · + pα1−(α1−1) + 1 =
α1Xi=0
pi
(ii). � x = α1, γ − x > α1 �§Zpx ⊕ Zpγ−x �f+��ê� 1§x e2O\§®²vkéAa.� pγ
�f+"d�§(2) ª¤á"
(iii). � γ−x = α1, x < α1 �§Zpx ⊕Zpγ−x �f+��ê� pα1−x +pα1−x−1"Zpx+1 ⊕Zpγ−x−1 �f+
��ê´ pα1−x−2 +pα1−x−3(γ−x = α1, x < α1−2)½ 1(γ−x = α1, x = α1−2)"£� x = α1 −1, γ−x = α1
� Zpx+1 ⊕ Zpγ−x−1 .�f+®²Ú Zpx ⊕ Zpγ−x .�f+�Ó¤"x 2O���f+�ê¯Kz�� (4)
ªaq��/"� (2) ªE¤á"
é� pσ �f+��/§§®²vkÌ�f+§§�(��±kµ
Zpα2 ⊕ Zpσ−α2 , Zpα2−1 ⊕ Zpσ−α2+1 , . . .
éu?¿� x, α2−x > α1 > σ−α2 +x§Zpα2−x ⊕Z
pσ−α2+x .�f+�ê´ pα1−(σ−α2+x) = pα1+α2−σ−x§
éu x, σ k±e�¹µ
(i). � α2 − x = α1 = σ − α2 + x �§Z2pα1 .f+��ê´ 1"� §(ؤá"
(ii). � α2 − x > α1 = σ − α2 + x �§Zpα2−x ⊕ Zpα1 .�f+��ê´ pα1−α1 = 1"(ؽ¤á"
(iii). � α2 − x = α1 > σ − α2 + x �§Zpα1+1 ⊕ Zpσ−α1−1 .�f+��ê´ pα1−(σ−α1−1) =
p2α1−σ+1§Zpα2−x ⊕Z
pσ−α2+x = Zpα1 ⊕Zpσ−α1 .�f+��ê´ pα1−(σ−α1) +pα1−(σ−α1)−1 = p2α1−σ +
p2α1−σ−1§x 2O�§Òz�� (4) ªaq��/§ �(ؤá"
½n 1 y." 2
Ún 1 G = Zpα1 ⊕ Zpα2 ⊕ · · · ⊕ Zpαn £�ÎÒ�ìcã�½¤¥� pβ(1 6 β 6 α1) ����ê´µ
(5′) pnβ − pn(β−1)
pβ �Ì�f+��ê´µ
(5)nX
j=1
p(n−1)β+1−j
⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊Theory attracts practice as the magnet attracts iron. —Gauss
46 #)/ �´
pγ(∃i, 1 6 i 6 n, s.t.αi < γ 6 αi+1) ����ê´µ
(6′) p
iPk=1
αk
(p(n−i)γ − p(n−i)(γ−1))
pγ �Ì�f+��ê´µ
(6)
n−iXj=1
p(n−i−1)γ+
iPk=1
αk−j+1
y² æ^êÆ8B{"
Äky² pβ ���/"� n = 1 �§Ù pβ �Ì�+��ê´ 1"pβ ����ê´ ϕ(pβ)§(5′) Ú (5)
ª3 n = 1 �¤á"
� n = 2 �§½n 1 `² (5′) 9 (5) ªd�¤á"
b� (5′) 9 (5) ª3 n 6 k − 1, k ∈ N∗\{1} �¤á§K� n = k �§- x1, x2, . . . , xk−1 ´3 [0, β]m�
4~�g,êk�ê�§Kkµ
pβ × px1 × px2 × · · · × pxk−1| {z }(k−1)�
:�
k1
�p(k−1)(β−1)ϕ(pβ)
pβ × pβ × px1 × · · · × pxk−2| {z }(k−2)�
:�
k2
�p(k−2)(β−1)ϕ(pβ)2
..
. :...
pβ × · · · × pβ| {z }i�
× px1 × · · · × pxk−i| {z }(k−i)�
:�
ki
�p(k−i)(β−1)ϕ(pβ)i
.
.. :...
pβ × · · · × pβ| {z }k�
:�
kk
�ϕ(pβ)k
l §±þ�a.� pβ ����ê´
[pβ−1 + ϕ(pβ)]k −�
k
0
�pk(β−1) = pkβ − pk(β−1)
� pβ �Ì�+��ê´
pkβ − pk(β−1)
ϕ(pβ)=
kXj=1
p(k−1)β+1−j
2y pγ ���/"
� n = 1, 2 �§(ؤá"�� n 6 k − 1, k ∈ N∗\{1} �(ؤá§� n = k �§- x1, x2, . . . , xk ´©
O3 [0, αj ], 1 6 j 6 n m�4~g,êk�ê�§·�k5wwéu?¿� d 6 k − 1 − i§px1 × px2 × · · · ×pxi × pxi+1 × · · · × pxi+d × pγ × pγ · · · × pγ .� pγ ����êµ
1 +
α1Xb=1
(p(i+d)b − p(i+d)(b−1))+
+
i−2Xg=0
α1+g+1Xf=α1+g+1
24p
1+gPm=1
αm
(p(i+d−1−g)f − p(i+d−1−g)(f−1))
35+
+
γ−1Xf=αi+1
24p
iPm=1
αm
(p(i+d−i)f−p(i+d)(f−1))
35= p
iPm=1
αm+d(γ−1)
⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊The bitter and the sweet come from the outside, the hard from within, from one’s own efforts.
—Augustus De Morgan
1 60 Ï k���+�f+�ê 47
2é d ¦Úµ
k−1−iXd=0
�k − i
d
�ϕ(pγ)k−i−dp
iPm=1
αm+d(γ−1)
= p
iPm=1
αm��
ϕ(pγ) + pγ−1�k−i − p(γ−1)(k−i)
�= p
iPm=1
αm�p(k−i)γ − p(k−i)(γ−1)
�l §Ù pγ Ì�f+��ê=´ (6) ª"
Ún 1 y." 2
3dÚ?��¼ê HG(pγ)§L«Ún 1 ¥� pγ ����ê§= (6′) ª"Ó�§·�w�§� β 6 α1
�§éu?¿� αi§β ÑØ�u§"·�À HG(pγ) ¥� p
iPk=1
αi
� p0 = 1§pγ = pβ§i = 0§K�� (5′)
ª"i L��´ G �©)ª¥ αj , 1 6 j 6 n§�u γ ��ê§� γ = β �§g, i = 0§n L« G �(�¥
��Ú��ê"·�� p ��ê§r HG(pγ), γ ∈ N ¡� “ê pγ é(uk���+ G �2 Euler ¼ê”§
{¡ “pγ − G ¼ê”"
(∗∗) HG(pγ) = p
iPk=1
αk
(p(n−i)γ − p(n−i)(γ−1))
Ún 2 G = Zpα1 ⊕ · · · ⊕ Zpαn ¥?ü��8Ø� {0} Ì�f+7k�Ó� p �f+"� αi < αi+1 �§G
¥� pαi �Ì�f+k�=k (pn−i − 1)/ϕ(p) � p �f+�¹3 pαi+1 �Ì�f+¥"
y² G ¥�?ü��8Ø� {0} Ì�f+7k�Ó��ê�u p ���§l dÌ�f+�5�9+�
�8E´+B�§�7½¬k�Ó� p ��§l Òk�Ó� p �f+"
� αi < αi+1 �§G ¥� pαi+1 �Ì�f+¥� p ��=U5g G ¥�� n − i ��Ú�¥� p ��
�c¡��Ú�¥�£��¤0�Ú ¤§ÏdÙ�ê��u G ¥� Zpαi+1 ⊕ · · · ⊕ Zpαn ¥� p ����
ê§= (pn−i − 1)/ϕ(p)" ù p ��w,Ñ3§���8¥" 2
Ún 3 G = Zpα1 ⊕ · · · ⊕ Zpαn ¥²þz p(n−1)(β−1) � pβ(β > 0)£ÎÒ�c¤�Ì�f+Òk���Ó�
p �f+§²þz p
nPk=i
αk+(i−1)(γ−1)
� pγ �Ì�f+Òk���Ó� p �f+"
y² ��e G¥� p���a.§eTa.� p��?3,� pβ �Ì�f+¥§Kù� p���U´
p1 × p1 × · · · × p1| {z }j�
×p0 · · · × p0, j ∈ N∗
w,ù«a.� p ��=U3d “�«a.” � pβ ��)¤�Ì�f+¥§£d��½ β > 1§Ï� β = 1
�§(Øw,¤á¤
(T ) pβ × pβ × · · · × pβ| {z }j�
×px1 · · · × pxn−j , j ∈ N∗, xk ∈ {0, 1, . . . , β − 1}, k ∈ {1, . . . , n − j}
“ù«a.” � pβ ����ê´�n
j
�ϕ(pβ)j(p(n−j)(β−1) − p(n−j)(β−2) + · · · + 1) =
�n
j
�ϕ(pβ)j(p(n−j)(β−1))
�A� p ����ê´ �n
j
�ϕ(p)j
duz� p �f+A�äk�Ó�ê�±Ù��8� pβ �Ì�f+§� §�nj
�ϕ(pβ)j(p(n−j)(β−1))/ϕ(pβ)�
nj
�ϕ(p)j/ϕ(p)
= p(n−1)(β−1)
⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊It is not enough to have a good mind. The main thing is to use it well. —Cauchy
48 #)/ �´
=1��(ؤá"
éu pγ �Ì�f+��/��Ó��?أѤ" 2
½n 2 G = Zpα1 ⊕ · · · ⊕ Zpαn ¥äk±ea.�f+
Zpx1 ⊕ Zpx2 ⊕ · · · ⊕ Zpxt
Ù¥ x1, x2, . . . , xt 6 α1 ´î�4~��"£k�¤g,ê�§t ∈ N∗ � t 6 n"Ù�ê´
tYi=1
pn−i+1 − 1
p − 1p(n−i)(xi−1)−
tPk=i+1
xk
Ù¥¦ÚtP
k=i+1xk � i = t �ÀÙ�� 0"
y² ·��IÀJ·����Ì�f+§,�2r§���Ú=�§�7IüØE��/"duz
����Ì�f+Ѭk�Ó� p �f+§�·��±ù�5�§k?À�� Zpx1 .�f+§�±k
(pnx1 − pn(x1−1))/ϕ(px1 ) «À{§,�2À�� Zpx2 .�f+§�ØU�c¡�f+k�Ó� p �f+§
�±k
pnx2 − pn(x2−1)
ϕ(px2)− p(n−1)(x2−1) p2−1 − 1
ϕ(p)
«�À{§ �X§À��� Zpx3 .�f+§�±k
pnx3 − pn(x3−1)
ϕ(px3)− p(n−1)(x3−1) p3−1 − 1
ϕ(p)
«À{§ù����e�§·�Ò��
(7)tY
i=1
pnxi − pn(xi−1)
ϕ(pxi )− p(n−1)(xi−1) pi−1 − 1
ϕ(p)
!«�{"�ù�Ø´·����ê§Ï�Ù¥�kE��¹u)§�Ò´3
Zpx1 ⊕ Zpx2 ⊕ · · · ⊕ Zpxt
SÜ�EÀ�§EÓ±þ�©Û�{§�ÙE��ê�µ
(8)tY
i=1
pixi − pi(xi−1)
ϕ(pxi)− p(i−1)(xi−1) pi−1 − 1
ϕ(p)
!p
tPk=i+1
xk
^ (7) ªØ± (8) ª=�¤¦" 2
·�2Ú?��ÎÒ “Λ”§ ±L«ØE�¦Ú§X Λtk=i+1xk L« k l i + 1 � t§é xk ¥�ØÓ
�ê¦Ú§�� i = t§=�I k �å©��uª��§TÚ��½Â� 0"·�êþ�±��½n 2 ��
�íص
íØ 1 G = Zpα1 ⊕ · · · ⊕ Zpαn ¥äk±ea.�f+
Zpx1 ⊕ Zpx2 ⊕ · · · ⊕ Zpxn
Ù¥ x1, x2, . . . , xt 6 α1 ´4~�£k�¤g,ê�§Ù�ê´
tYi=1
pn−i+1 − 1
p − 1p(n−i)(xi−1)−Λt
k=i+1xk
⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊Proof is an idol before whom the pure mathematician tortures himself. — Arthur Eddington
1 60 Ï k���+�f+�ê 49
�u§xj , j = 1, . . . , n Ø�½Ñ' α1 ����/§·���±Uìù«�{5�§�e��´�¡�O�
®§·�ùpÒØ2[ã"£¤ka.�f+�(�¥��Ú�ج�L n¤
�d��§·��²�±`´)ûk���+�f+�ê�¯K"·���2£LÞ5ww (∗∗)ª§·�y3���§3 px, x ∈ N ?�½Â§éu����ê��/§·�êþÒ�±w�§�½Â§§
kX¤¢� “�¦5”"éu?����k���+ G§�§�ê� n =rQ
i=1pαi
i §éÙ?1 Sylowf+©)§
G = Gp1 ⊕ Gp2 ⊕ · · · ⊕ Gpr
éuz� Sylow f+§·�®²��§§�������ê§�m =rQ
i=1pβi
i , βi 6 αi§·�á=��§G¥
� m ����ê´
HG(m) =rY
i=1
HGpi(p
βii ) =
rYi=1
HG(pβii )
�z
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⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊⋊The moving power of mathematical invention is not reasoning but imagination. —Augustus De Morgan
Ì?µ ¥I�ÆEâ�Æ 2002?êÆX
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