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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

##########################################################################

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

##########################################################################

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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—Joseph Fourier

Riemann − Lebesgue Ún9ÙA�í2/ª

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1 60 Ï Riemann− Lebesgue Ún9ÙA�í2/ª 29

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f(λ) → 0, |λ| → ∞.

y² d5êÆ©Û�§61nþ P268 ½n 20.13 �§

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∫ +∞

−∞

f(x) cosλxdx

b(λ) =

∫ +∞

−∞

f(x) sinλxdx

3 R þ��ëY§u´ f(λ) �3 R þ��ëY"

i) e f ∈ L2(−∞,+∞)§K ‖f(λ)‖L2(R) = ‖f(x)‖L2(R) < ∞ ([3], P183, Plancheral

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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

½n 2 3k.üëÏ«� D �ü � ∆ ��/N� f e§D �¤k�à� ∂∆ þ¤k:

��éA"

y² d·K 4 ·K 5§·�®²�� D �z��àéA�� ∂∆ ¥:§ ØÓ��àé

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·�æ^�E5�y²"d Koebe½n§P D ��N��>.:3 f e�éA:8Ü

F§k F 3 ∂∆ þ"¤±éu ∀w0 ∈ ∂∆, d ∆ �/G§��3 ∂∆ þ�±l w0 �ü>é

�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

���>.:q´�o'Xº

·�£��à�½Â§?�����>.: [z0, l]§�±r��Ä�ó {γn}∞n=1 �z�

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"

·����§����>.:�éA ∂D þ��:§ ∂D þ��:�oéA����

>.:§�oéAü���>.:"dd§·��±@�§�à8Ü�|¤k 3 «¤©µ

(1) éu��>.: [z, l]§e z �3�� V ¦� V \∂D �ü�ëÏ�§¿�k��3

D §Kr z ���à¶

(2) éu��>.: [z, l]§eØ÷v (1)§Kr z ��ü�:�\�à¶

(3) éu ∂D þØáu?Û��>.:�:?1|ܧ,�r|Ü���à§äNXÛ

|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"

�éu�� ∂D §duØ´ëÏ4­�§3Ø\\ ∞ ?1?n�§o´ØU� ∂∆ ïá�

�éA�"

⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳

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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