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A Proof of Finite Sum Theorem Using the Concept of Ultrafilter . . . . . X 7 9
A general approach rate to the strong law of large numbers . . . . . . . . . . . ² 12
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A Proof of Finite Sum Theorem Using the
Concept of Ultrafilter
0201 Lei Tao
In this discussion we aim towards a proof of a celebrated result in combinatorial
number theory, the Finite Sum Theorem. Originally proven by Professor N.Hindman
with number theoretic techniques, we here show a proof that involves a unique synthesis
of topology and algebra. We state the theorem:
Theorem 1 (Finite Sum Theorem(Hindman)) Let Aiki=1 be a finite partition
of the set of natural numbers N . There exists an Ai that contains an infinite sequence
whose finite, nonrepeating sums remain in Ai.
We will need the very useful concept of ultrafilter from set-theroetic topology.
Definition 1 Let X be a set. A nonempty family F ⊂ P(X) is a filter provided:
(1) F ∈ F implies F 6= ∅(2) F,G ∈ F implies F ∩ G ∈ F(3) F ∈ F , G ∈ P(X) and G ⊃ F imply G ∈ F (Superset Property)
Definition 2 A filter F on a set X is an ultrafilter if it is a maximal element in the
set of all filters on X, partially ordered by inclusion. I.e, F is not contained in any
strictly larger filter on X.
If U is an ultrafilter then it contains so many subsets of X that for all A ⊂ X
either A ∈ U or X\A ∈ U .
Let βN = U : U is an ultrafilter on N .For a given n ∈ N , U = A ⊂ N : n ∈ A is an ultrafilter on N. These are principle
ultrafilters.
A Topological Structure on βN :
For A ⊂ N , set A∗ = U ∈ βN : A ∈ U. Then the set A∗ : A ⊂ N forms a basis
9
10 ïÄ?Ø ´
for a compact Hausdorff topology on βN . Next, we would like to identify the elements
of N with certain elements of βN , so that we can think of N as being contained in
βN . We do this using the principle ultrafilters. We note that for n ∈ N , n∗ consists
of one ultrafilter. [Suppose there exist U and V in n∗. If A ∈ U , then n ∈ A. As
n ∈ V, we have A ∈ V by the superset property of V. This shows U ⊂ V. Similarly,
V ⊂ U . So, U = V.] Let n∗ be the unique element of n∗. We then identify n ∈ N
with n∗ ∈ βN . With this identification, it turns out that N is dense in βN and βN is
the Stone-Cech Compactification of N.
An Algebraic Structure on βN :
We now define an algebraic operation (+) on βN that extends the ordinary sum
on N, i.e. n∗ + m∗ = (n + m)∗. For A ⊂ N , define A − n = k ∈ N : k + n ∈ A. This
is simply the elements of A shifted to the left n units. For U ,V ∈ βN , define:
U + V = A ⊂ N : n ∈ N : A − n ∈ U ∈ V.
Theorem 2 n∗ + m∗ = (n + m)∗, for all n,m ∈ N
Proof
n∗ + m∗ = A ⊂ N : k ∈ N : A − k ∈ n∗ ∈ m∗= A ⊂ N : k ∈ N : n ∈ A − k ∈ m∗= A ⊂ N : k ∈ N : n + k ∈ A ∈ m∗= A ⊂ N : A − n ∈ m∗= A ⊂ N : m ∈ A − n= A ⊂ N : n + m ∈ A= (n + m)∗
2
It can be shown that (+) defined above is closed and associative, making (βN,+) a
semigroup. Furthermore, the topological and algebraic structures on βN interact well:
For a fixed U ∈ βN , the left translation map LU : βN → βN defined by LU (V) = U +Vis continuous. Then, we finally have the necessary structure on βN for the proof of the
Finite Sum Theorem: (βN,+) is a compact, left-topological semigroup.
A very important theorem is known about such structures:
Theorem 3 (Auslander-Ellis) Every compact left-topological semigroup has an idem-
potent, i.e. an element a such that a + a = a.
⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳⊲⊳
In great mathematics there is a very high degree of unexpectedness, combined with inevitability and economy.
— G H Hardy
1 59 Ï A Proof of Finite Sum Theorem Using the Concept of Ultrafilter 11
As (βN,+) is such a structure, the following is an immediate corollary:
Theorem 4 (Glazer) There exists U ∈ βN such that U + U = U .
Lemma 1 (Galvin) Let U be an idempotent element of βN . Then for all A ∈ U there
exists an infinite sequence B ⊂ A such that all nonempty finite sums of elements in B
remains in A.
Proof of Finite Sum Theorem . Consider the partition N = ∪ki=1Ak, and let U
be an idempotent ultrafilter on N. There exists some i ∈ 1, · · · , k such that Ai ∈ U .
By Galvin’s Lemma, there exists an infinite sequence B ⊂ Ai such that all nonempty
finite sums of elements of B remain in Ai. 2
As a final remark, we note that techniques similar to the above can be used to prove
other results in combinatorial number theory, notably Van der Waerden’s Theorem and
the Hales-Jewett Theorem.
::::::::::::::::::::
[øøÙ
¡ ¡
[ø´±\<êÆ[
A general approach rate to the strong law of
large numbers
0217 Hu Ming
We obtain the strong growth rate for the sum Sn of random variables by using a
Hajek-Renyi type maximal inequality. Under the same conditions as that in Fazekas
and Klesov(Theory Probab. Appl. 45 (2000) 436), we get sharper results for some
dependent sums.
Introduction
The strong law of large numbers (SLLN) asserts that a sequence of cumulative
sums of random variables becomes ”nonrandom” by normalizing it by an appropri-
ate sequence of nonrandom numbers and approaching the limit. Many results of this
type were obtained for both independent and dependent summands forming cumulative
sums.
There are two basic approaches to prove the strong law of large numbers. The first
is to prove the desired result for a sub sebsequence and then reduce the problem for
the whole sequence to that for the subsequence. In so doing, a maximal inequality for
cumulative sums is usually needed for the second step. Note that maximal inequalities
make up a well-developed branch of probability theory and many inequalities are known
for different classes of random variables.
The second approach is to use directly a maximal inequality for normed sums.
Inequalities of this kind are said to be of Hajek-Renyi type, referring to the paper by
Hajek and Renyi devoted to independent summands. However, after a Hajek-Renyi
inequality is obtained, the proof of the strong law of large numbers becomes an obvious
problem.
In this paper, our goals are to show that a Hajek-Renyi type inequality is , in fact,
a consequence of an appropriate maximal inequality for cumulative sums and to show
that the latter automatically implies the strong law of large numbers. Most important,
we made no restriction on the dependence structure of random variables.
12
1 59 Ï A general approach rate to the strong law of large numbers 13
Recently, Fazekas and Klesov obtained a Hajek-Renyi type maximal inequality.
Then they proved the strong law of large numbers for general summands, and give
some applications for dependent summands. In this paper, we study the strong growth
rate for sums of random variable under the same conditions as that in Fazekas and
Klesov. We get sharper results for some dependent random variable sums.
Hajek-Renyi type maximal inequality and strong law of large numbers
Lemma 1 (Hajek-Renyi type maximal inequality) Let β1, · · · , βn be a nondecreasing
sequence of positive numbers. Let α1, · · · , αn be nonnegative numbers. Let r be a fixed
positive number. Assume that for each m with 1 ≤ m ≤ n
E
[
max1≤l≤m
|Sl|]r
≤m∑
l=1
αl (1)
Then
E
[
max1≤l≤n
|Sl
βl|]r
≤ 4
n∑
l=1
αl
βrl
(2)
Proof First, we multiply βr1 on both side of (2). Since β1 > 0, (2) holds true if and
only if
E
[
max1≤l≤n
| Sl
βl/β1|]r
≤ 4
n∑
l=1
αl
(βl/β1)r
So we can suppose that β1 = 1.
Let c = 21/r. Consider the sets
Ai = k : ci ≤ βk < ci+1, i = 0, 1, 2, · · ·
Because β1 = c0 = 1, we notice that 1 ∈ A0, A0 6= ∅. Using Ai, we can seperate
1, · · · , n into some pieces.
Denote by i(n) the index of the last nonempty Ai. When i is large enough such
that βn < ci, Ai = ∅. So i(n) < ∞. If Ai is nonempty, let k(i) = maxk : k ∈ Ai,i = 0, 1, 2, · · · , while k(i) = k(i − 1) if Ai is empty. Let k(−1) = 0.
If Al is nonempty, let
δl =
k(l)∑
j=k(l−1)+1
αj , l = 0, 1, 2, · · ·
where δl is considered to be zero if Al is empty.
∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼Abel has left mathematicians enough to keep them busy for 500 years. —Charles Hermite
14 ïÄ?Ø ´
For a fixed ω0, there exists some l0, Ai0 such that l0 ∈ Ai0 , and:
max1≤l≤n
|Sl
βl|r(ω0) = |Sl0
βl0
|r(ω0) ≤ maxl∈Ai0
|Sl
βl|r(ω0) ≤
i(n)∑
i=0
maxl∈Ai
|Sl
βl|r(ω0)
The inequality above holds true for almost every ω0 ∈ Ω. After take expectation on
both sides, the inequaltiy still holds true. We have:
E
[
max1≤l≤n
|Sl
βl|r]
≤ E
i(n)∑
i=0
maxl∈Ai
|Sl
βl|r
=
i(n)∑
i=0
E
[
maxl∈Ai
|Sl
βl|r]
Since i(n) < ∞, we can change the sum and expectation. According to the definition
of Ai, when l ∈ Ai, βl ≥ ci, so β−rl ≤ c−ir. We have:
i(n)∑
i=0
E
[
maxl∈Ai
|Sl
βl|r]
≤i(n)∑
i=0
c−irE
[
maxl∈Ai
|Sl|r]
Since k(i) = maxk : k ∈ Ai, i = 0, 1, 2, · · · , we notice that Ai ⊂ 1, · · · k(i) Using
the condition (1), we have:
E
[
maxl∈Ai
|Sl|r]
≤ E
[
max1≤l≤k(i)
|Sl|r]
≤k(i)∑
l=1
αl
i(n)∑
i=0
c−irE
[
maxl∈Ai
|Sl|r]
≤i(n)∑
i=0
c−ir
k(i)∑
k=1
αk
According to the defintion of δl and change the order of sum, we have:
i(n)∑
i=0
c−ir
k(i)∑
k=1
αk =
i(n)∑
i=0
c−iri∑
l=0
δl =
i(n)∑
l=0
δl
i(n)∑
i=l
c−ir
Now we get the inequality:
E
[
max1≤l≤n
|Sl
βl|r]
≤i(n)∑
l=0
δl
i(n)∑
i=l
c−ir ≤i(n)∑
l=0
δl
∞∑
i=l
c−ir =1
1 − c−r
i(n)∑
l=0
δlc−lr
According to the definition of δl, we have:
1
1 − c−r
i(n)∑
l=0
δlc−lr =
1
1 − c−r
i(n)∑
l=0
c−lr
k(l)∑
j=k(l−1)+1
αj
∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼On earth there is nothing great but man; in man there is nothing great but mind.
—William Rowan Hamilton
1 59 Ï A general approach rate to the strong law of large numbers 15
Using the defintion of Ai, when j ∈ Al, βj < cl+1, we have:
E
[
max1≤l≤n
|Sl
βl|r]
≤ 1
1 − c−r
i(n)∑
l=0
c−lr
k(l)∑
j=k(l−1)+1
αj
≤ 1
1 − c−r
i(n)∑
l=0
c−lr
k(l)∑
j=k(l−1)+1
c(l+1)r αj
βrj
=cr
1 − c−r
i(n)∑
l=0
k(l)∑
j=k(l−1)+1
αj
βrj
= 4n∑
k=1
αk
βrk
This completes the proof.
Lemma 2 (Fazekas-Klesov strong law of large numbers) Let b1, b2, · · · be a nonder-
creasing unbounded sequence of positive numbers. Let α1, α2, · · · be nonnegative num-
bers. Let r be a fixed positive number. Assume that for each n ≥ 1
E
[
max1≤l≤n
|Sl|]r
≤n∑
l=1
αl (3)
if∞∑
l=1
(al/brl ) < ∞, then
limn→∞
Sn
bn= 0 a.s. (4)
Proof Case 1: We assume that there exists an integer m, such tha αn = 0, n ≥ m.
Using condition (3), for all n > 1, we have:
E
[
max1≤l≤n
|Sl|]r
≤m∑
l=1
αl
Let n → ∞. By monotone convergence theorem, we get change the order of limitation
and expectatioin:
limn→∞
E
[
max1≤l≤n
|Sl|]r
= E
[
limn→∞
max1≤l≤n
|Sl|]r
= E
[
supn≥1
|Sn|]r
≤m∑
l=1
αl < ∞
So we get:
supn≥1
|Sn|r < ∞ a.s.
For almost every ω0 ∈ Ω,
|Sn(ω0)|bn
≤supn≥1
|Sn(ω0)|
bn→ 0 (n → ∞)
∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼What we know is not much. What we do not know is immense. — Pierre-Simon Laplace
16 ïÄ?Ø ´
So we have the conclusion:
limn→∞
Sn
bn= 0, a.s.
Case 2: We assume that αn > 0 for an infinite number of indices n. Set:
vn =
∞∑
k=n
αk
brk
, βn = max1≤k≤n
bkv1/2rk
It is easy to see that 0 < vn < ∞, for all n ≥ 1, and vn is a decreasing sequence. Using
the mean value theorem: ∀ 0 < α < 1, there exists ξ ∈ (vn+1, vn), such that:
v1−αn − v1−α
n+1 = (1 − α)(vn − vn+1)ξ−α ≥ (1 − α)
αn
brnvα
n
Sum from 1 to ∞, we have:
(1 − α)
∞∑
n=1
αn
brnvα
n
< v1−α1 − lim
n→∞v1−αn+1 = v1−α
1 < ∞
At the same time, we notice that vk → 0, since∞∑
k=1
vk < ∞. ∀ε > 0, there exists
N ∈ N, such that when i > N , v1/2ri < ε When k is large enough, we have:
βk
bk≤
max1≤i≤N
biv1/2ri
bk+
maxN+1≤i≤k
biv1/2ri
bk≤
max1≤i≤N
biv1/2ri
bk+ ε
Let k → ∞, we have βk/bk < ε. Let ε → 0, we prove that limk→∞
βk/bk = 0
So the sequence βn, n ≥ 1 is such that
(a) βk < βk+1, k = 1, 2, · · · ;(b)
∞∑
k=1
αk/βrk ≤
∞∑
k=1
αk/brkv
1/2k < ∞
(c) limk→∞
βk/bk = 0
Then Lemma 1 implies that (2) is satisfiled. Therefore, E[supl≥1
|Sl/βl|]r ≤ 4∞∑
l=1
αl/βrl <
∞. This implies supl≥1
|Sl/βl| < ∞, a.s.
Finally,
0 ≤∣∣∣∣
Sl
bl
∣∣∣∣=
∣∣∣∣
Sl
βl
∣∣∣∣
βl
bl≤
supl≥1
∣∣∣∣
Sl
βl
∣∣∣∣
βl
bl→ 0, a.s. as l → ∞
The lemma is proved. 2
∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼Number theorists are like lotus-eaters - having tasted this food they can never give it up.
—Leopold Kronecker
1 59 Ï A general approach rate to the strong law of large numbers 17
Strong growth rate of random sequence under moment conditions
Lemma 3 (Dini Theorem) Let c1, c2, · · · be nonnegative numbers, vn =∞∑
k=n
ck, if 0 <
cn < ∞ for n ≥ 1,then∞∑
n=1
cn
vδn
< ∞, ∀0 < δ < 1 (5)
Proof Let f(x) = x1−δ, x > 0, 0 < δ < 1, then f ′(x) = (1− δ)/xδ . By the mean value
theroem, there exists a ξ ∈ (b, a), such that f(b) − f(a) = f ′(ξ)(b − a). Take b = vn,
a = vn+1, then we have:
vδn − vδ
n+1 = (1 − δ)(vn − vn+1)/ξδ ≥ (1 − δ)cn/vδ
n, ξ ∈ (vn, vn+1)
Thus:
(1 − δ)
∞∑
n=1
cn
vδn
≤ vδ1 − lim
n→∞vδn+1 = vδ
1 < ∞
Then the lemma is proved. 2
Theorem 1 Let b1, b2, · · · be a nondecreasing unbounded sequence of positive numbers
and α1, α2, · · · be nonnegative numbers. Let r and C be fixed positive numbers. Assume
that for each n ≥ 1
E
(
max1≤l≤n
|Sl|)r
< C
n∑
l=1
αl (6)
∞∑
l=1
αl
brl
< ∞ (7)
then
limn→∞
Sn
bn= 0 a.s. (8)
and with the growth rateSn
bn= O
(βn
bn
)
a.s. (9)
where
βn = max1≤k≤n
bkvδ/rk ∀0 < δ < 1, vn =
∞∑
k=n
αk
brk
, limn→∞
βn
bn= 0 (10)
Proof 1) First we assume that there exists an integer m such that αn = 0 for n ≥ m,
thus βn = max1≤k≤m
bkvδ/rk for n ≥ m. By monotone convergence theorem, we have
E
(
supn≥1
|Sn|)r
= limn→∞
E
(
max1≤l≤n
|Sl|)r
≤ Cm∑
l=1
αl < ∞,
thus supn≥1
|Sn| < ∞, a.s., and this easily gives (8)-(10).
∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼God made the integers, all else is the work of man. —Leopold Kronecker
18 ïÄ?Ø ´
2) We assume that αn > 0 for infinitely many n. By (7) and Lemma 3, we
know that∞∑
n=1αn/br
nvδn < ∞, it is easy to see that 0 < βk ≤ βk+1 for k ≥ 1, and
∞∑
n=1αn/βr
n ≤∞∑
n=1αn/br
nvδn < ∞,
βk
bk≤
max1≤l<k1
blvδ/rl
bk+
maxk1≤l≤k
blvδ/rl
bk≤
max1≤l<k1
blvδ/rl
bk+ v
δ/rk1
,
by (7) and limn→∞
bn = ∞, we get limn→∞
βn/bn = 0 . (6) and Lemma 2 imply that
E
(
max1≤l≤n
∣∣∣∣
Sl
βl
∣∣∣∣
)r
≤ 4Cn∑
l=1
αl
βrl
≤ 4C∞∑
l=1
αl
βrl
< ∞,
hence by monotone convergence theorem, we have
E
(
supn≥1
∣∣∣∣
Sn
βn
∣∣∣∣
)r
= limn→∞
E
(
max1≤l≤n
∣∣∣∣
Sl
βl
∣∣∣∣
)r
≤ 4C
∞∑
n=1
αn
βrn
< ∞,
so that supn≥1
|Sn/βn| < ∞, a.s., and
0 ≤∣∣∣∣
Sn
bn
∣∣∣∣≤ βn
bnsupn≥1
∣∣∣∣
Sn
βn
∣∣∣∣= O
(βn
bn
)
, a.s.,
this completes the proof.
2
Reference
[1] Hajek, J., Renyi, A., 1955. Generalization of an inequality of Kolmogorov, Acta Math.
Acad. Sci. Hungar., 6, 281-283.
[2] Fazekas, I., Klesov, O., 2000. A general approach to the strong law of large numbers.
Theory Probab. Appl. 45, 436-449.
∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼∼Try a hard problem. You may not solve it, but you will prove something else. — J E Littlewood
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Ò±üëêÿ/x"du U ÷ γ ©þ¿ØK.ݧ
´UC÷rÝ"ùué γ ?1#ëêz" U⊥ â´é
ÝCzkzþ"
⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄¤¦<h²§y¦<ŧêƦ<°[§óƦ<Û§¦<î§Ü6?`¦<õF"
—
1 59 Ï lÛÜáNá 21
½Â 1 p ∈ M,X ∈ Mp, expp : Mp → M . ¡ d expp 3 X ?òz§XJ3 γ(t) =
expp(tX)þ3Øð"~ Jcaobi| U§÷v U(p) = 0 = U(expp(X)).
ù¡ X êN3 p ?Ý:§¡ expp X p ÷X γ = expp tX Ý
:"
e¡ÑC©úª"Ø t γ lëê§u´ |γ| = 1, L(u) =∫ ba |γu(t)|dt"
du T = γu(t), U = dγ( ∂∂u ), [T,U ] = 0 µ
L′(u) =
∫ b
a
d
du
√
< γu(t), γu(t) >dt =
∫ b
a
1
T< T,DUT > dt =
∫ b
a
1
T< T,DT U > dt
AO§u=0l1C©úªµ
L′(0) =
∫ b
a< γ,DγU > dt =< γ(t), U(t) > |ba −
∫ b
a< γ(t), U(t) > dt
XJ.à:½§= U(a) = U(b) = 0§K L′(0) = −∫ ba < γ(t), U(t) > dt"
L′(0) = 0, ·ÿ/ γ Ý3C÷ U 6ÄÝC
z¥´."ä γ ´Ä´á§IÄlC© L′′(0):
L′′(u) =
∫ b
a
d
du(1
T< T,DT U >)dt
L′′(0) =
∫ b
a(|U (t)|2− < γ,RUγU > −[< γ(t), U(t) >′]2+ < DUU, γ >′)dt.
±e·ob U ´à:½~ Jacobi |§OÑþªµ
L′′(0) =
∫ b
a[|U (t)|2− < RγU γ, U >]dt.
y3í2ù½Âþ|"é¤k÷ γ ÷v X(t) ⊥ γ(t) ∀t ∈(a, b) ©ã C∞ þ| X NP C∞
p [a, b]"±Ú\I/ªµ
I(X,X) =
∫ b
a[|X(t)|2− < RγX γ,X >]dt.
£é C∞p ¼êÈ©À31w¡þÈ©Ú§ X = DγX.¤
O I é R 5µI(aX, aX) = a2I(X,X)"½ÂAg
.µI(X,Y ) = 12 [I(X + Y,X + Y ) − I(X,X) − I(Y, Y )],∀ X,Y ∈ C∞
p "Ðmµ
I(X,Y ) =
∫ b
a[< X(t), Y (t) > − < Rγ(t),X(t)γ(t), Y (t) >]dt.
ù´é¡g.§¡§I/ª"ù´·l γ C©pÑ'u
îþ|g."ùrC©©)÷îþ|.§
3îþ|.U5zïÄ"ù´éUõ§·3©AÛ¥
~ùo"
⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄»§âf§»§zó|§/¥C§)Ô§F^§Ã?Ø^êÆ" —uÛ
22 ïÄ?Ø ´
I/ªÿ/á5
M þ^l p q ^ÿ/ γ : [a, b] → M"γ[a, t] 3 t ¿©C a
· γ 3¤k M þë( γ(a) γ(t) ´á" t r'
ÿ§¹ÒkUu)Cz"~~fÒ´¥¡"ª:Lå©
:é»:§@o·r´ÒØ´á"e¡·¬å©:é»:
Ò´§Ý:"
Ún 2 5ÿ/ γ : [a, b] → M . XJ γ(a) Ý γ(b),U ÷ γ ¦ U(a) =
U(b) = 0 ~ Jacobi |§K I(U,U) = 0"
y² I(U,U) =∫ ba (|U |2+ < U,U >)dt =< U,U > |ba = 0" 2
e¡Qã'ªµ X,Y ∈ C∞[a, b]§ X,Y ⊥ γ
I(X,Y ) =< X(t), Y (t) > |ba −∫ b
a< X(t) + Rγ(t),X(t)γ(t), Y (t) > dt (1)
UìÎk) [1] ¥PÒÚ\µ
V= ¤k÷ γ ÷v X(t) ⊥ γ(t) X ∈ C∞p þ| X N"
V0 = X ∈ V : X(a) = X(b) = 0
·K 2 5ÿ/ γ : [a, b] → M,U ∈ V. K I(U, V ) = 0,∀V ∈ V ⇐ :U Jacobi
|"
·K 3 XJ γ[a, b] þع γ(a) Ý:§KI/ª3 V0 þ½"
·K 4 γ(b) ´÷ γ 'u γ(a) 1Ý:§KI/ª3 V0 þ´
½ ½"
·K 5 γ(c) ´÷ γ 'u γ(a) 1Ý:§a < c < b§KI/ª3 V0
þ´Ø½"
ù5y²3 [1]pé"'X1§dúª (1)§XJ U
Jacobi |§K I(U, V ) = 0,∀V ∈ V0. § [a,b] þ©ã1wK¼
ê f§§T3Ø1w: 0"Ä V = f ∗ (U + Rγ,U γ) ∈ V0 K
0 = I(U, V ) = −∑
i
∫ ai+1
ai
f(t)|Ui + Rγ,U γ|2dt
⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄Ü6´ØÔ§ÏéÜ6¦^Ü6" —ÙA°
1 59 Ï lÛÜáNá 23
l U ÷ γ Jacobi |"ù·±aLÙ¦y²§5¼þ¡*
)º";.~f´nm¥ü ¥¡M = S2§éN´: p ∈ S2
é»: q AT´§Ý:§ù´o¿gQº
Ä¥¡3 p :m TpS2 9ÙþêN exp : TpS
2 → S2"
±b ∃X ∈ TpS2, |X| = 1, s.t.q = exp(πX). m©Ø¡§H4Ú4"
ùwm TpS2 ¥^AÏ Ct = Y ∈ TpS
2, |Y | = 1 3êNeµexp(πY ) : Y ∈ Ct"dé¡5± exp(tY )´lH4Ñu÷ Y
²§¿3 t = π Ó4"ÄêN3 X ∈ TpS2 ?N
dγq = d expq"du3 X ∈ TpS2 ?m TpS
2 Ó§ TpS2 ´d X,X1 Ü
¤§Ù¥ X1 ⊥ X"du tlu π O\u π §exp(tX)BLÙ§:
BL q :"· |dγq(X)| = |∂γ(t)∂t |t=π = 1 =3 X Øòz§
3 X1 %´òz§Ï exp(πY ) = q,∀Y 3m¥ X R"dÝ
:½Â γ C¥Ø´á"
3AÛþwù´w,§ÏlÙ§²ÑurLål´"dA
Tk÷ γ" Jacobi| U,U(a) = U(b) = 0¦ I(U,U) = 0"e¡Oe
Jacobi§" e1(t) = γ(t), e2(t) ⊥ e1(t)±½"KU(t) = sin(t)e2(t), t ∈ [0, π]
=´3à:§÷v Jacobi§µf(t) + 1 ∗ f(t) = 0(ÏdÇ 1)
îþ|"Ù¢ùîþ|3 e´"=¤k3à
:îþ|µU(t) = a sin(t)e2(t)£´d§)Ñ5¿÷vЩ^¤"
ÿ/rLÝ:§=¦3C¥§Ø´á"~X·
r5 3¥¡þ§rà:½3H4§,à:4£Ä§
rL4§5Ò¬¥,ý "ù`²3ÊÏÝþe§3§NCk
áëüॡ´"y3·éÝ:k:)§3-1Ý:
cÿ/´3C¥á§rL1Ý:ÒØ´á§=´3
C¥k'§á"?Ú·¬¯µoÿÿ/´Ná
Qº
Ná9:¿Â
e¡Äÿ/oÿ´Ná"·é5â5
äNá5"Äk¬¯: XJv-Ý:§@o^ÿ/´Ná
íº2w~~fµÎ¡"Bå§Ñ§3 R3 ¥IL
⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄´Sê§Ñ %§k/u§kêí" —yÀ
24 ïÄ?Ø ´
«µx2 + y2 = 1,∀z. P p = (−1, 0, 0), q = (1, 0, 0). X ´ p ?þ X =
(0, 1, 0), γ(0) = p, γ(0) = X, K γ ´l p Ñu q ÿ/§ exp(πX) = q"l
mΡêN??òz§l Ý:8´8"ùqkÿ
/Ø´á§'X γ rL q :5 q′§,3C¥§´á
§´r exp(−tX)w,¬'r exp(tX)@ q′"y3·2ѽµ
½Â 2 M Riemann 6/§γ : R+ → M ^5ÿ/"½öé¤k
t§γ[0, t]Ѵ᧽ö3, t0, γ[0, t0]´á§é?Û s > t0,
Ñk' γ[0, s] áë( γ(0) γ(s) ´"31«¹§¡ γ(t0) γ
'u γ(0) :"
ù½Âdu`µ3-:c§ÿ/Ñ´Ná"XJÝ:
Ú:ÑÑy§ ¬k-:"k§Ü§'X3¥¡þ§§Ñ
´é»:"kÝ:8´8"'Xþ¡Î¡§ù p ¤k:T´
x = 1, y = 0,∀z ù^"3v-§c§l p Ñuÿ/Ñ´Ná
"
é p ∈ M§P C(p) l p Ñu-¤k:8ܧ¡,"P V(p)
,SÜ (ù´¹ p ëÏm8)"l p Ñuÿ/3,SÜØ
¬-:§3 TpM ¥¹:m8 V (p)§¦ exp : V (p) → V(p)
©Ó§qk V (p) ©Óuü m¥£Ó´éw,¤"P V (p) >
. C(p)"
½n 1 Riemann 6/ M ´;= ∀x ∈ M,C(x) ÑÓuü
¥¡ Sx ⊂ TxM
y² e M ;§P d(M) = δ. u´5ÿ/ γ : R+ → M,γ(0) = x 3rL δ
ÒØ´á§3 ǫ > 0 ¦ γ[0, ǫ] o´á§
=:C(x) ∈ Bx(0, δ + ǫ)\Bx(0, ǫ)
l C(x)Óu Sx ∈ TxM . §C(x)Óu Sx ∈ TxM ,K V (x)∪C(x);"
- d = maxx∈M
maxX∈C(x)
|X|, K d(M) = d < ∞, l M ;" 2
½Â 3 ¡ U ⊂ M :à ('u x ∈ U)§XJ U ¥?¿:T±^^
áÿ/ x ë("
d½Â V(p) 'u p :à8"©(·OA
:à8ü~f"
⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄·d)vko¢Ã§k¿Øù§§ØL´·¼ê" —.KF
1 59 Ï lÛÜáNá 25
~ 1
1). î¼m Rn",´§V(p) = Rn§'X R3"
2). ¥¡ Sn"=k p é»: q ´:£½´Ý:¤§V(p) = Sn\q§'X S2"
3). R3 ¥Î¡ Mµx2 + y2 = 1,∀z"V(−1, 0, 0) = M\x = 1, y = 0"
4). R3 ¥¡ Mµ(√
x2 + y2 − 2)2 + z2 = 1"V(−3, 0, 0) = M\x2 + y2 = 1, z =
0\(x − 2)2 + z2 = 1, y = 0"
·w¡¹k:E,§Ù¢ x2 +y2 = 1, z = 0∪(x−2)2 +z2 = 1, y = 0Tд (−3, 0, 0) :8"ù´~g,§·e÷,r¡m§
ùL§Tд·3ÿÀþ¡_L§"u´ù;6/Àl
4ü ¥ÏLé>.,/ÊÜ0£3î¿Âþı
Ýþ¤"é;6/§ù´é"ù§V (p) ∪ C(p) ©Óu4ü
¥§expp : V (p) ∪ C(p) → M ´÷"ùêNÒué>. C(p) ?1
Êܧl C(p)"·e¥¡Ú¡§ù´ü²~f"
5y² [1]"
ëÖ [1] ´©ÌëÔ§¤kÑy²þ3Ù¥é"ù´
É~Ðá"XJé6/ÿÀm'Xk¡@£§¿
sm§[2]´ýÐëÖ"
ë©z
[1] ÎõÚ§!Xn§xó"5iùAÛÐÚ6§®ÆÑ"
[2] ¿"56/ÿÀÆ6§ÉÇÆÑ"
⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄þ2MEê§Ù¦Ñ´<E" —ÛS
Hodge (f9Ù5
0201 d
Hodge (f´©AÛ¥©f§k Hodge (f½
§·Ò±½Â©f§? ½Â6/þ Laplace-Beltramif§ª
Ͷ Hodge ©)½n"©Ì8´0 Hodge (f½Â9
ÙÄ5"
£M,g¤´ m ½iù6/§ÙiùÝþ g"Ä M þ?Iã
£U,ϕ, xi¤§·k
g = gijdxi ⊗ dxj (11)
gij = g(∂
∂xi,
∂
∂xj)
±eeÃAO`²§Io´ 1, · · · ,m"
´ M NÈ η ±L«¤
η =√
Gdx1 ∧ · · · ∧ dxm, G = det(gij) (12)
BuO§·Ú\ Kronecker ÎÒ
δi1···irj1···jr
= det
δi1j1
δi1j2
· · · δi1jr
δi2j1
δi2j2
· · · δi2jr
· · · · · · · · ·δirj1
δirj2
· · · δirjr
(13)
r=1§ùÒ´Ï~Kronecker delta δij§d½Âª£3¤wѧKronecker
ÎÒäk±e5µ
1. δi1···irj1···jr
'uI (i1, · · · , ir)½ (j1, · · · , jr) ´¡;
2. e i1 < · · · < ir,j1 < · · · < jr,K δi1···irj1···jr
= δi1j1· · · δir
jr;
3. δi1···irj1···jr
=
0, e (j1, · · · , jr)Ø´ (i1, · · · , ir);
1, e (j1, · · · , jr)´ (i1, · · · , ir) ó;
−1, e (j1, · · · , jr)´ (i1, · · · , ir) Û"
26
1 59 Ï Hodge (f9Ù5 27
NÈ η M m g/ª§¤
η =∑
i<
ηi1···imdxi1 ∧ · · · ∧ dxim =1
m!ηi1···imdxi1 ∧ · · · ∧ dxim (14)
Ù¥∑
i<
≡∑
i1<···<im
L«IU^Sü¦Ú§
ηi1···im =√
Gδ1···mi1···im (15)
w, dη=0§du ηi1···im ´ (0,m) .Üþ©þ§
ηi1···im,k =∂ηi1···im
∂xk−
m∑
t=1
Γlitkηi1···it−ilit+1···im
(0,m + 1) .Üþ©þ"q dimM = m§k
ηi1···im,k = 0 (16)
^ Ar(M) L« M þ r /ª¤þm§·½Â Hodge (f ∗ :
Ar(M) −→ Am−r(M) Xeµ
½Â 1 α ∈ Ar(M),0 ≤ r ≤ m, ÛÜL«
α =∑
i<
αi1···irdxi1 ∧ · · · ∧ dxir =1
r!αi1···irdxi1 ∧ · · · ∧ dxir (17)
½Â ∗α ∈ Am−r(M) ´
∗α =∑
j<
∗αjr+1···jmdxjr+1 ∧ · · · ∧ dxjm (18)
Ù¥
∗αjr+1···jm=∑
i<
ηi1···irjr+1···jmαi1···ir (19)
αi1···ir = gi1k1 · · · girkrαk1···kr, (gij) = (gij)
−1
∗α ¡ α /ª"
≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖
êÆ3u§gd" —x÷
28 (¡ ´
β ∈ Ar(M) §
β =∑
i<
βi1···irdxi1 ∧ · · · ∧ dxir (20)
½Â 2 r g/ª α Ú β ÛÜSÈ < α, β > ½Â
< α, β >=∑
i<
αi1···irβi1···ir =1
r!αi1···irβi1···ir (21)
±þ·þ®b αi1···ir (½ βi1···ir) 'u i1, · · · , ir ´é¡"
zO, ·kÛÜ5Ie| ei 9ÙéóIe| wi,ù
gij =< ei, ej >= δij , gij = δi
j ,√
G = 1
XJ
α =∑
i<
αi1···irωi1 ∧ · · · ∧ ωir
Kd½Â 1, k
∗α =∑
j<
∗αjr+1···jmωjr+1 ∧ · · · ∧ ωjm
Ù¥
∗αjr+1···jm=
∑
i<
δ1···mi1···irjr+1···jm
αi1···ir
=1
r!δ1···mi1···irjr+1···jm
αi1···ir
u´,35Ie|e,NȤ
η = ω1 ∧ · · · ∧ ωm = ∗1 (22)
|^eã·K´:
∗η = ∗(ω1 ∧ · · · ∧ ωm) = 1
·K 1 α, β ∈ Ar(M), f ∈ C∞(M), Kk
(i) ∗ (α + β) = ∗α + ∗β, ∗fα = f(∗α)
(ii) ∗ ∗α = ∗(∗α) = (−1)r(m−r)α
(iii)α ∧ ∗β = β ∧ ∗α =< α, β > η
≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖
êÆ´£ó䧽´Ù§£óä "¤kïÄ^SÚÝþÆþÚêÆk'" —(k
1 59 Ï Hodge (f9Ù5 29
y² Bå§ÛÜ5Ie|§
α =∑
i<
αi1···irωi1 ∧ · · · ∧ ωir
β =∑
i<
βi1···irωi1 ∧ · · · ∧ ωir
K
gij = δij,√
G = 1, ηi1···im = δ1···mi1···im, αi1···ir = αi1···ir
(i) d½Âw,"
(ii) d½Â 1 µ
∗α =∑
j<
∗αjr+1···jmωjr+1 ∧ · · · ∧ ωjm
=∑
j,i<
δ1···mi1···irjr+1···jm
αi1···irωjr+1 ∧ · · · ∧ ωjm
k
∗(∗α) =∑
j,i<
δ1···mi1···irjr+1···jm
αi1···ir ∗ (ωjr+1 ∧ · · · ∧ ωjm)
P γ = ωjr+1 ∧ · · · ∧ ωjm,K
γlr+1···lm=
1, lr+1 = jr+1, · · · lm = jm ;
0, Ù§.
u´§
∗γ =∑
k<
∗γk1···krωk1 ∧ · · · ∧ ωkr
Ù¥§
∗γ =∑
l<
δ1···mlr+1···lmk1···kr
γlr+1···lm
=∑
l<
(−1)r(m−r)δ1···mk1···krlr+1···lm γlr+1···lm
= (−1)r(m−r)δ1···mk1···krjr+1···jm
≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖
êÆ[Òô< · · · · · · êÆ[n§¦òl¥Ñ\7L@(اlù(ئ
q¬Ñ,(Ø" —´S|
30 (¡ ´
k
∗(∗α) =∑
j,i<
δ1···mi1···irjr+1···jm
αi1···ir
∑
k<
(−1)r(m−r)δ1···mk1···krjr+1···jm
ωk1 ∧ · · · ∧ ωkr
=∑
j,i,k<
(−1)r(m−r)αi1···irδ1···mi1···irjr+1···jm
δ1···mk1···krjr+1···jm
ωk1 ∧ · · · ∧ ωkr
du i1 < · · · < ir, k1 < · · · < kr,Úª¥"÷v i1, · · · , ir = k1, · · · , kr =
1, · · · ,m − jr+1, · · · , jm, k i1 = k1, · · · , ir = kr. (i1, · · · , ir) ½
§jr+1, · · · , jm (½"
∗(∗α) =∑
i,j<
(−1)r(m−r)αi1···irωi1 ∧ · · · ∧ ωir
= (−1)r(m−r)∑
i<
αi1···irωi1 ∧ · · · ∧ ωir = (−1)r(m−r)α
(iii)
α ∧ ∗β = (∑
i<
αi1···irωi1 ∧ · · · ∧ ωir) ∧ (
∑
j<
∗βjr+1···jmωjr+1 ∧ · · · ∧ ωjm)
=∑
i,j,k<
αi1···ir βk1···krδ1···mk1···krjr+1···jm
ωi1 ∧ · · · ∧ ωir ∧ ωjr+1 ∧ · · · ∧ ωjm
d δ1···mk1···krjr+1···jm
6= 09 ωi1 ∧ · · · ∧ ωir ∧ ωjr+1 ∧ · · · ∧ ωjm 6= 0Ó (ii) ¥?ا
Úª¥"÷v i1 = k1, · · · , ir = kr. (i1, · · · , ir)½§jr+1, · · · , jm
(½"u´k
α ∧ ∗β =∑
i,j<
αi1···ir βi1···irδ1···mi1···irjr+1···jm
ωi1 ∧ · · · ∧ ωir ∧ ωjr+1 ∧ · · · ∧ ωjm
= (∑
i<
αi1···ir βi1···ir)ω1 ∧ ω2 ∧ · · · ∧ ωm
d½Â 2 9 βi1···ir = βi1···ir , ω1 ∧ ω2 ∧ · · · ∧ ωm = η
α ∧ ∗β =< α, β > η
Óny β ∧ ∗α =< β,α > η =< α, β > η"·Ky." 2
≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖
½N\±Ø&þ2§´\ØØ&êƶÃØ^oØy§\Ñvy\Øuo§§
ûØUuÊ. —x%
1 59 Ï Hodge (f9Ù5 31
íØ 1 α ∧ ∗α = 0 = α = 0"
Ù/n) Hodge (f§·5wA~f"
~ 1 α = α1dx1,m = 3, ¦ ∗α"
)µ d½Â 1§
∗α =∑
j1<j2
∗αj1j2dxj1 ∧ dxj2
∗αj1j2 = ηij1j2αi = ηij1j2g
ikαk
= α1ηij1j2gi1 = α1
√Gδ123
ij1j2
∗α =∑
j1<j2
α1
√Gδ123
ij1j2dxj1 ∧ dxj2
= α1
√G(g31dx1 ∧ dx2 + g11dx2 ∧ dx3 − g21dx1 ∧ dx3)
~ 2 α = α12dx1 ∧ dx2,m = 3, ¦ ∗α"
)µ Äkò α ¤é¡/ªµ
α =1
2(α12dx1 ∧ dx2 − α12dx2 ∧ dx1)
K ∗α =∑
j ∗αjdxj "Ù¥µ
∗αj =∑
i1<i2
ηi1i2jαi1i2 =
∑
i1<i2
√Gδ123
i1i2jgi1k1gi2k2αk1k2
5¿µ
αij=
α12, i = 1, j = 2 ;
−α12, i = 2, j = 1 ;
0, Ù§"
∗α3 =√
Gg1k1g2k2αk1k2= α12
√G(g11g22 − g12g21)
∗α1 =√
Gg2k1g3k2αk1k2= α12
√G(g21g32 − g22g31)
≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖
·Qfk<`·´êÆéö§´êÆ'<§vk<'·êƧϧ¤·ØQ
Ù¤Ò1" —x
32 (¡ ´
∗α2 = −√
Gg1k1g3k2αk1k2= −α12
√G(g11g32 − g12g31)
l µ
∗α = α12
√G[(g21g32 − g22g31)dx1 − (g11g32 − g12g31)dx2 + (g11g22 − g12g21)dx3]
= α12
√Gdet
g11 g12 dx1
g21 g22 dx2
g31 g32 dx3
~ 3 α = dx1 ∧ dx2 ∧ dx3,m = 3, ¦ ∗α")µ Äkò α ¤é¡/ªµ
α = dx1 ∧ dx2 ∧ dx3
=1
3![dx1 ∧ dx2 ∧ dx3 − dx1 ∧ dx3 ∧ dx2 + dx3 ∧ dx1 ∧ dx2
− dx3 ∧ dx2 ∧ dx1 + dx2 ∧ dx3 ∧ dx1 − dx2 ∧ dx1 ∧ dx3]
u´ βijk = δ123ijk . d½Â 1
∗(dx1 ∧ dx2 ∧ dx3) =∑
i1<i2<i3
ηi1i2i3βi1i2i3 =
√Gβ123
=√
Gg1i1g2i2g3i3βi1i2i3
=√
G[g11g22g33 − g11g23g32 + g12g23g31
− g12g21g33 + g13g21g32 − g13g22g31]
=√
Gdet
g11 g12 g13
g21 g22 g23
g31 g32 g33
=√
G · 1
G=
1√G
AO/, 3ÛÜIee,dxi = ωi, i = 1, 2, 3, G = det(gij) = 1, k
∗(ω1 ∧ ω2 ∧ ω3) = 1
.
ë©z
[1] xI, !WÍ"5iùAÛÐÚ6, pÑÑ"
≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖≖
3êÆ+¥§JѯK²â')¯K²â" —x÷
¡g,IeIe
0201 ?¤
3©AÛ§¥§·|^K¡g,IeIeü«¹ÄIeïÄ
¡AÛ§¿(Ü$ħí Gauss−Codazzi§§ddÚÑ¡nØ¥X
\(J"ü«IeÑ^5£ã¡AÛ§/ÏuØÓÎÒ"g,Ie$^E
, Christoffel ÎÒ§§Ie$^'©/ª§òE,$ħÚ
(§±`/ªÐyÑ5"·ØB¬§Q,ü«Lãó´d§@oü«
ÎÒm'X´NQº
Äk§4·£e¡g,IeÚIeÄVg"
½Â 1 E3¥ëê¡ r = r (u1, u2)g,Ie|´N r; r1, r2, nIe¤± r (u1, u2)
:¹ÄIe|"Ù¥§r1,r2 ´¡Iþ§r3 = r1∧r2
|r1∧r2| ´ S ü þ"
½Â 2 E3 ¥ëê¡ r = r (u1, u2) :²¡þþ e1,e2, ¦ (e1, e1) = (e2, e2) =
1,(e1, e2) = 0, ¿ e1, e2 'u (u1, u2) 1w" e3 = e1 ∧ e2 ¡ü þ|§K
r; e1, e2, e3 ¤¡ S Ie"ùIeN¤¡ S Ie|"
e¡Ñü«¹ÄIee¡$ħ§ù´·íÑu:"
½n 1 ¡$ħ
g,Ieeµ∂r
∂uα= rα (M1)
∂rα
∂uβ= Γγ
αβrγ + bαβr3 (M2)
∂r
∂uα= −bβ
αrβ (M3)
Ieeµ
dr = ωα eα, α = 1, 2 (M4)
deη = ωkη ek, η, k = 1, 2, 3 (M5)
33
34 (¡ ´
Ún 1 rα = aηαeη, α, β, γ, η = 1, 2, @o¡ü«Iee$ħ¥XêkXe'
Xµ
alγΓγ
αβduβ − dalα − aη
αωlη = 0 (1.1)
bαβduβ = aηαω3
η (1.2)
y²
3 rα = aηαeη ü>é uβ ¦µ
∂rα
∂uβ= aη
α
deη
duβ+
∂aηα
∂uβeη
(Ü (M2)§(M5) ªµ
Γγαβrγ + bαβr3 =
∂aηα
∂uβeη + aη
α
ωkη
duβek
aηα aλ
η = δλα, 2d rα = aη
α eη µ
rλaλη = eη, e3 = r3, λ = 1, 2
\þª=µ
Γγαβrγ + bαβr3 =
∂aηα
∂uβrλaλ
η + aηα
ωkη
duβrλ aλ
k + aηα
ω3η
duβr3 l, k = 1, 2, 3
nµ
(Γγαβ − ∂aη
α
∂uβaγ
η − aηα aγ
k
ωkη
duβ)rγ + (bαβ − aη
α
ω3η
duβ)r3 = 0 γ = 1, 2
d r1, r2, r3 5Ã'5µ
Γγαβ − ∂aη
α
∂uβaγ
η − aηαaγ
k
ωkη
duβ= 0
bαβ − aηα
ω3η
duβ= 0
^©/ªÑ=µ
Γγαβduβ − daη
αaγη − aη
αaγkωk
η = 0
bαβduβ − aηαω3
η = 0
α, β, η, γ, k = 1, 2
31ªü>¦± alr =(Ø" 2
ÏL±þØ㧷±wÑÛ¹3©/ª ωkη E, Christoffel ÎÒ§dd
±wÑ|^©/ªL$ħ`5"
[1] ¥Ñ e1 = r1√E
, e2 = r2√G§g,IeIee Gauss− Codazzi úª
d5"e¡·|^þ¡Úny²/§ü|úª5"
::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
ÆSêÆõSK§>>g¢"kÙ,§,Ù¤±," —Ú
1 59 Ï ¡g,IeIe 35
½n 2 Gauss-Codazzi §
g,Ieeµ
∂Γξαβ
∂uγ− ∂Γξ
αγ
∂uβ+ Γη
αβΓξηγ − Γη
αγΓξηβ − bαβbξ
γ + bαγbξβ = 0 (G1)
∂bαβ
∂uγ− ∂bαγ
∂uβ+ Γξ
αβbξγ − Γξαγbξβ = 0 (C1)
Ieeµ
dωβα = ω3
α ∧ ω3β (G2)
dω3k = ωl
k ∧ ω3l (C2)
½n 3 g,Iee Gauss § (G) Ú Codazzi § (C) IeeA§´
d.
y² é (1.1) Ú (1.2) ª?1·C/µ
ωlk = aα
k alγΓγ
αβduβ − aαkdal
α (2.1)
ω3k = aα
k bαηduη (2.2)
/(C2) −→ (C1)0
(2.2) ªü>©µ
dω3k = bαηdaα
k ∧ duη + aαk dbαη ∧ duη = (bαη
∂aαk
∂uβ+ aα
k
∂bαη
∂uβ)duβ ∧ duη
ωlk ∧ ω3
l = (aαkal
γΓγαβduβ − aα
kdalα) ∧ (am
l bmηduη)
= (aαkΓm
αβbmη − aαk am
l bmη
∂alα
∂uβ)duβ ∧ duη
d (C2) dω3k = ωl
k ∧ ω3l ,
2(Ü duβ ∧ duη = −duη ∧ duβ 9 duβ ∧ duηp β < η Ã'5,
aαk (Γm
αβbmη − Γmαηbmβ +
∂bαβ
∂uη− ∂bαη
∂uβ) = bαη
∂aαk
∂uβ− bαβ
∂aαk
∂uη+ aα
k aml bmη
∂alα
∂uβ− aα
k aml bmβ
∂alα
∂uη
ü>¦± akξ§¿·N¦ÚI§=k
Γmξβbmη − Γm
ξηbmβ +∂bξβ
∂uη− ∂bξη
∂uβ= bαηak
ξ
∂aαk
∂uβ− bαβak
ξ
∂aαk
∂uη+ aα
l bαη
∂alξ
∂uβ− aα
l bαβ
∂alξ
∂uη
qdu alξa
αl = δα
ξ§k
::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
ê"/*§/"êJ\" —uÛ
36 (¡ ´
∂alξ
∂uβ+ ak
ξalµ
∂aµk
∂uβ= 0
\þª§=þªm> 0"
l
Γmξβbmη − Γm
ξηbmβ +∂bξβ
∂uη− ∂bξη
∂uβ= 0
d= Codazzi §"
/(C1) −→ (C2)0
þãy²=^§¦ÚI§l ±_í£§y"
/(G2) −→ (G1)0
Oµ
dωlk = (
∂aαk
∂uξal
γΓγαβ +
∂Γγαβ
∂uξaα
kalγ + aα
k
∂alγ
∂uξΓγ
αβ − ∂aαk
∂uξ
∂alα
∂uβ)duξ ∧ duβ
ω3k ∧ ω3
l = amk an
l bmξbnβduξ ∧ duβ
d (G2) µdωlk = −ω3
k ∧ ω3l§òÙ\þªnµ
(∂Γγ
αβ
uξ−
∂Γγαξ
uβ)aα
k alγ−am
k anl (bmβbnξ−bmξbnβ)+al
γ(Γγαβ
∂aαk
∂uξ−Γγ
αξ
∂aαk
∂uβ)+
∂aαk
∂uβ
∂alα
∂uξ−∂aα
k
∂uξ
∂alα
∂uβ= 0
ü>¦± aks at
l µ
(∂Γt
sβ
uξ− ∂Γt
sξuβ) − at
l anl (bsβbnξ − bsξbnβ) + ak
s(Γtαβ
∂aαk
∂uξ− Γt
αξ
∂aαk
∂uβ)
+aks at
l(∂aα
k
∂uβ
∂alα
∂uξ− ∂aα
k
∂uξ
∂alα
∂uβ) = 0 (A1)
5¿þª1§¿|^ gβα = aαl aβ
l
µ
anl at
l(bsβbnξ − bsξbnβ) = gnt(bsβbnξ − bsξbnβ) = bsβbtξ − bsξbt
β (A2)
(Ü®(Jµ∂aα
k
∂uξ +∂aµ
η
∂uξ aαµ aη
k = 0
\ (1) üµ
aks(Γt
αβ
∂aαk
∂uξ− Γt
αξ
∂aαk
∂uβ) + ak
s atl(
∂aαk
∂uβ
alα
∂uξ− ∂aα
k
∂uξ
alα
∂uβ)
::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
êÆ[ѧªÚúª§XÓwH!wºµ§f`N ¿©¯W"
—Êw
1 59 Ï ¡g,IeIe 37
= −Γtαβ
∂als
∂uξaα
l + Γtαξ
∂als
∂uβaα
l − atµ
∂aµα
∂uξ
∂als
∂uβaα
l + atµ
∂aµα
∂uβ
∂als
∂uξaα
l
±yµ
þª = ΓηsβΓt
ηξ − ΓηsξΓ
tηβ (A3)
nÜ (A1), (A2), (A3) ª§ Gauss §:
∂Γtsβ
∂uξ−
∂Γtsξ
∂uβ+ Γη
sβΓtηξ − Γt
sξΓtηβ − bsβbt
ξ + bsξbtβ = 0
/(G1) −→ (G2)0
aq§òþãy²£í=" 2
ë©z
[1] $[B§"5©AÛ6§pÑ"
::::::::::::::::::::
Koch Snow Curves
::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
êÚ£X»" —.x.d
[-2,2] ¥a:8È5
0201 Á[
½Â:8:
F1 = ±√
2
F2 = ±√
2 ±√
2
F3 = ±√
2 ±√
2 ±√
2...
Fn = ±√
2 ±√
2 ± . . . ±√
2︸ ︷︷ ︸
n
-
S =
∞⋃
n=1
Fn
Ù¥ Fn ¥ ± L«?KÒþ?¿|Ü, ~X
F2 = ±√
2 ±√
2 = +√
2 +√
2, +
√
2 −√
2,−√
2 +√
2,−√
2 −√
2
ØJ Fn ⊂ [−2, 2],l S =⋃∞
n=1Fn ⊂ [−2, 2].
¯K:S 3 [−2, 2]¥´ÄȺ=´Ä [−2, 2]¥?¿:þ^ S ¥:%Cº
e¡òy² [−2, 2]¥äkù«/ª:8È5. ¦´,qXêõª fii∈N
3 [−2, 2]¥":8. e¡Ñõªq fii∈N ½Â: -
f1(z) = z2 − 2
ò fn(z) ½Âd f1(z) n gSõª
fn(z) = fn1 (z) = f1 · · · f1(z)
︸ ︷︷ ︸
n
l fn(z)½Â,§´ 2ngXêõªy8Ü Fn¥:þ´ fn(z)":. 2n
gXêõªõk 2n":, |Fn| = 2n§ Fn fn(z)":8"l §·µ
38
1 59 Ï [-2,2] ¥a:8È5 39
(Ø 1õªq fii∈N ":8 S.
y²õªq fii∈N ":8 S 3 [−2, 2] ¥È5§e¡Ñù":,«
L«µC
z = ω +1
ω
K
fn(ω) = fn(ω +1
ω) = ω2
n
+1
ω2n
e¡¦ fn(ω) = 0 ":µ
ω2n
+1
ω2n = 0
m
ω2n+1
= −1 = eiπ
m
ω = cos(2k + 1)π
2n+1+ i sin
(2k + 1)π
2n+1, 0 ≤ k ≤ 2n+1 − 1
m
z = ω +1
ω= 2 cos θ, θ =
(2k + 1)π
2n+1, 0 ≤ k ≤ 2n+1 − 1
-
T = θ|θ =(2k + 1)π
2n+1, 0 ≤ k ≤ 2n+1 − 1, ∀n ∈ N
këY÷N ϕ : [0, 2π] −→ [−2, 2], θ 7−→ 2cosθ § ϕ(T ) = S. ÏØJwÑ T 3 [0, 2π]
¥È, l § S 3 [−2, 2] ¥È.
(Ø 2 S 3 [−2, 2]¥È.
aq S ½Â§∀x ∈ [−2, 2] -
H1(x) = ±x
H2(x) = ±√
2 ± x
H3(x) = ±√
2 ±√
2 ± x...
Hn(x) = ±√
2 ±√
2 ± . . .√
2 ± x︸ ︷︷ ︸
n
-
Hx =
∞⋃
n=1
Hn(x)
⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛
·m!m!9é꧷±ME»" —³|Ñ
40 (¡ ´
(Ø 2 íØ·kµ
íØ 3 ∀ x ∈ [−2, 2], Hx 3 [−2, 2]¥È.
y² Ø x ∈ [0, 2], Ïe x ∈ [−2, 0] K√
2 − x ∈ [0, 2]. : ε1 =√
x, ε2 =√
2 − x, . . . , εk =
√
2 −√
2 − . . .√
2 − x, , . . ., KØJy²
limk−→∞
εn = 2
∀ y ∈ [−2, 2] Ú§?Û U ⊂ [−2, 2], d S È57, ∃n ∈ N, z ∈ Fn st. z ∈ U
l εk ¿©C 2 , K3 z V ⊂ U ¦ V ¥äkù/ª:
±
√
2 ±√
2 ± . . .√
2 ±√εk
︸ ︷︷ ︸
n
∈ Hx,
l y U ¥3ù«/ª:. Hx 3 [−2, 2]¥È. y..
[_%K]
¡È
àÅó§!ÔnÆ[ÚêÆ[5§^jkÑ¡È"ó§
^jkѧ\¡ù´`O"ÔnÆ[òjk.m¤^§bjk
kç@å/¥o"êÆ[ÐЦ"¦^éjkrg
Cå5§,`µ/·y3´3¡"0
(£
ÔnÆ[Úó§¦X9í¥§3b¥"¦p(͵/·3
=º0L 15 ©¨§¦f£A3쥣µ/\39í¥p0ÔnÆ[
µ/@[º½´êÆ["0ó§Ø)µ/oº0ÔnÆ[µ/Ϧ^é
m§Ñ(Y§Y:^vk"0
⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛⊛
1´êƧ1´êƧ1n´êÆ" —Ô
\UÙeíº
04001 +¶ ÛÜ
N·Ñf`Lùw2µXJ\Ã/ÙƧ\´@¬["
ùé.kõnºe¡·^ÅiÄ.5&ľ£'uÅiÄ9Ù'
Vg§ëwWP?Í5VÇØ6P74-75¤"
·îIL]ê§z?ا·b]ê§0 L»§ØU2Ù"
zgÑVÇÑ p§]~ 1§3I¶Ny£Ä¶IVÇ q = 1− p§
]O\ 1§I¶þNym£Ä"
Щ] k§e]O\ m£m > k¤§KÂÃØÙ"dWP?5VÇØ6P74
~K 2.4.4 (J±§Ñ1VÇ´µ
pk =
1 − k
m, p = 1 = 1
2;
(p
q
)k
−(
p
q
)m
1 −(
p
q
)m , p 6= q.
(1)
§XJk!/Ù§ÑIkU"XJÃ!/Ù§»§@q´N
/Qºò.?UµK m:áN9§ 0:áN9"·Ñ»
VÇ P"
¿§·Ð©] 1£d¡n§·¬uyЩ] k /N´l
ù«AÏ/í2¤"
Noºg,´|^ (1) ª"- k = 1§m → +∞§µ
P =
1, p > 1
2;
p
1 − p, p < 1
2.
ùoÜníº©ÛXeµ
^ AL«»§Ai L«3 i»§i = 1, 2, 3, · · ·"K A =∞⋃
i=1
Ai§Ù¥ Ai üüØ"
¤±§
P (A) =
∞∑
i=1
P (Ai) = limn→∞
n−1∑
i=1
P (Ai).
5¿n−1∑
i=1
P (Ai) ´ (1) ª¥ m = n ¦Ñ pi§4´Ün"
41
42 #)/ ´
±þ·¡"ùÏLÖþ§e¡50AÙ§
"
µ
ù´È¢"A ¿ÂØC§ Bl L«kI l g§,Ñ l + 1 g¯
§l = 0, 1, 2, · · ·"K A =∞⋃
l=0
Bl§ Bl üüØ"¤±k§
P (A) =
∞∑
l=0
P (Bl) =
∞∑
l=0
al · pl+1ql.
Ù¥ al ´ Bl rê"|^|ÜêÆ£§±Ñ
al =1
1 + l
(2l
l
)
.
dª½n
(1 − 4x)−12 =
+∞∑
i=1
(2i
i
)
xi,
ü>È©µ
−1
2
√1 − 4x + c =
+∞∑
i=0
1
1 + i
(2i
i
)
xi+1.
- x = 0§ c = 1
2"¤±kµ
P (A) =1
q
∞∑
l=0
al · (pq)l+1 =1
q
∞∑
l=0
1
l + 1
(2l
l
)
· (pq)l+1
=1
2q(1 −
√
1 − 4pq) =1
2q(1 − |1 − 2p|).
P =
1, p >1
2;
p
1 − p, p < 1
2.
XJ5¿¯¢§lI 2 ÑurI 1 Vǧul 1 ÑuUr 0 V
ǧ@oU±enµ
l k ÑuUr 0 VÇP pk§l k ÑuUr i VÇP pki"Kkµ
p2 = p21p10 = p21 (2)
,§w,k'Xªµ
p1 = p + (1 − p)p21 (3)
éá (2) Ú (3)§k
p1 = p + (1 − p)p21,
)µ
p1 = 1 ½p
1 − p.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
<ok§´§¦1[ —Ü
1 59 Ï \UÙeíº 43
1. 1
26 p §
p
1 − p> 1, p1 ∈ [0, 1],¤±p1 = 1.d=¤¦P.
2. p < 1
2§
d§p1 =∞∑
l=0
al · pl+1ql§al ´ p Ã'|Üê"
pl+1ql = pl+1(1 − p)l = p(p(1 − p))l 6 p · (1
4)l < (1
2)l+1 · (1
2)l
p = 1
2§p1 = 1"¤± p < 1
2§p1 < 1"
¤± p1 = p1−p
,d=P.
¤±k
P =
1, p > 1
2;
p
1 − p, p < 1
2.
Ónu (2)(3) ª§·k pk = pk1 , pk = p · pk−1 + (1 − p)pk+1, k > 2"dd=¦ pk§
´§ p < 1
2§k §P §=ØU»"
±þ§·lnØÓݧ^n«ØÓ§Ó(J"l?Ø(J
±§XJÑVÇØu§Ùe(J´A7,¬»"
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gêƶ<
• Music
Art Garfunkel: ùÒ´Ã<Ø"Simon & GarfunkelkͶy Scarborough
Fair. Columbia 1967 MA, UY3 Columbia g PhD, ØL¥åòÆ, ;%ÑW¯
• Literature
Lewis Carroll:/Alice in Wonderland0Ú/Through the Looking Glass0ö. ý
¶ Charles Lutwidge Dodgson, ùÿ¦Ò´Ü6Æ[
• Finance
John Maynard Keynes: ²LÆ[, Cambridge
J. P. Morgan: Õ1, gc, c´ãÞ. Ø^[`, [ www.jpmorgan.com. D`
Gottigen êÆ[Q²¿ã!`¦êÆ[
• Philosophers
Edmund Husserl: yÆI. Vienna 1883 PhD
Ludwig Wittgenstein: 20 VóÆã<. l Bertrand Russel ÆênÜ6
• Athletes
Michael Jordan: ;¥ã(. ØL¦3 junior ÿ=X
David Robinson: , NBA ã(. BS in math from Annapolis
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
ï, Øv§J%´lØv 5"J^J§=UÆTÝ(§ ØU¦ý?Ú"
—uÛ
The 65th William Lowell Putnam Mathematical
Competition
Saturday, December 4, 2004
A1 Basketball star Shanille O’Keal’s team statistician keeps track of the number, S(N), of
successful free throws she has made in her first N attempts of the season. Early in the
season, S(N) was less than 80% of N , but by the end of the season, S(N) was more than
80% of N . Was there necessarily a moment in between when S(N) was exactly 80% of N?
A2 For i = 1, 2 let Ti be a triangle with side lengths ai, bi, ci, and area Ai. Suppose that
a1 ≤ a2, b1 ≤ b2, c1 ≤ c2, and that T2 is an acute triangle. Does it follow that A1 ≤ A2?
A3 Define a sequence un∞n=0 by u0 = u1 = u2 = 1, and thereafter by the condition that
det
(
un un+1
un+2 un+3
)
= n!
for all n ≥ 0. Show that un is an integer for all n. (By convention, 0! = 1.)
A4 Show that for any positive integer n, there is an integer N such that the product x1x2 · · ·xn
can be expressed identically in the form
x1x2 · · ·xn =
N∑
i=1
ci(ai1x1 + ai2x2 + · · · + ainxn)n
where the ci are rational numbers and each aij is one of the numbers −1, 0, 1.
A5 An m×n checkerboard is colored randomly: each square is independently assigned red or
black with probability 1/2. We say that two squares, p and q, are in the same connected
monochromatic component if there is a sequence of squares, all of the same color, starting
at p and ending at q, in which successive squares in the sequence share a common side.
Show that the expected number of connected monochromatic regions is greater than mn/8.
A6 Suppose that f(x, y) is a continuous real-valued function on the unit square 0 ≤ x ≤ 1, 0 ≤y ≤ 1. Show that
∫ 1
0
(∫ 1
0
f(x, y)dx
)2
dy +
∫ 1
0
(∫ 1
0
f(x, y)dy
)2
dx
≤(∫ 1
0
∫ 1
0
f(x, y)dx dy
)2
+
∫ 1
0
∫ 1
0
(f(x, y))2 dx dy.
44
1 59 Ï The 65th William Lowell Putnam Mathematical Competition 45
B1 Let P (x) = cnxn + cn−1xn−1 + · · ·+ c0 be a polynomial with integer coefficients. Suppose
that r is a rational number such that P (r) = 0. Show that the n numbers
cnr, cnr2 + cn−1r, cnr3 + cn−1r2 + cn−2r,
. . . , cnrn + cn−1rn−1 + · · · + c1r
are integers.
B2 Let m and n be positive integers. Show that
(m + n)!
(m + n)m+n<
m!
mm
n!
nn.
B3 Determine all real numbers a > 0 for which there exists a nonnegative continuous function
f(x) defined on [0, a] with the property that the region
R = (x, y); 0 ≤ x ≤ a, 0 ≤ y ≤ f(x)
has perimeter k units and area k square units for some real number k.
B4 Let n be a positive integer, n ≥ 2, and put θ = 2π/n. Define points Pk = (k, 0) in the
xy-plane, for k = 1, 2, . . . , n. Let Rk be the map that rotates the plane counterclockwise
by the angle θ about the point Pk. Let R denote the map obtained by applying, in order,
R1, then R2, . . . , then Rn. For an arbitrary point (x, y), find, and simplify, the coordinates
of R(x, y).
B5 Evaluate
limx→1−
∞∏
n=0
(1 + xn+1
1 + xn
)xn
.
B6 Let A be a non-empty set of positive integers, and let N(x) denote the number of elements
of A not exceeding x. Let B denote the set of positive integers b that can be written in
the form b = a− a′ with a ∈ A and a′ ∈ A. Let b1 < b2 < · · · be the members of B, listed
in increasing order. Show that if the sequence bi+1 − bi is unbounded, then
limx→∞
N(x)/x = 0.
Ì?µ ¥IÆEâÆ 2002?êÆX
?µ ?¤ X 7 Ü
"vµ 7U( Ü+ d ;¿
Oµ Ü< ÈC 4 !
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