SlideShare a Scribd company logo
2018/4/14
*d¥*±h¥±kte & #X## EPEE sttt
'
EEE' .
slit -
4¥ HE ¥A a K¥+5
@ TakeTo 1024
Contents
0 .
t
'
7th # ksuc .
1 .
4¥ ' HtTHE a hat # at #¥x .
2 . Euler ¥ta a A¥# Aiz 'l¥s4t¥a .
3 .
PEEMEIE'EF# rmz .
4 .
4¥ HE #a a
' '
IHI
' '
.
* =a7¥Eirt Ih¥±± F)
"
¥eF±t5
"
2 h#¥El£F .
5*h{ KIBE,
tkukrt # Fta 1- t.EE#TEeIFFET In .
O .
t
'
7th # knuz
i 7th # sit
¥F⇒sb'ts3#¥h#
a # ¥n±kt9kh# Eat A "¥n2n3ta .
531 M =
s2 I a f¥tt9HlFh E = TM .
(Tangent bundle )
TM
~
# '
* Beta .
I>
y
=
¥µTpM :# '
LIMIT
p 1122 : 7 } hli
'
-
4pm
YY
: EIIERIF
( pkaF¥isHvtEp¥l )
←
n7HH¥aaF#±⇒
'
(
ninth '¥ X e * an a × :
t.Y.IT#ptas
# HE .
)
"
Etants
"
n' 9th #
E¥4¥a '
EMIT E = Mx
Rr
it €53hMknt9Hv#n .
= th 2 As #± A t
'
9th # z Eta # At 9th
'
# said .
^3Ni'
.
4¥ , , A¥# Egg ;qµ*z .
' trivial Vector bundle )
By M =
s
'
In ELKE # .
FZTHTEAKH F¥e*U×R' at 's .
FT IT # RA '
# rtn9HlFkath¥kE£h3 .
RR
-
R
d
s
' -
•
•
•
•
•
€1- - -
4) E = S
'
x IR 2) E :
open
Mobius band
C trivial ) ( non . Trivial )
2) to Etna Itau tea IEIAA .
i ) El E = s
'
x IR tiztnktt
' '
.
3.
'
E. Ii
'
n' 7Hubt -
ha ( to ) { 53
'
'Ei7Hv±¥s to 2123 ttf
lb 't FEE # a z #
2324M¥
the a
'k#¥ b 's HF
"
0 { he ,2lF 's .
qp¥.
" rahman
• n' 9th # A 3F÷kh# e i7HvBE¥n
"
tslikk TELEFE
"
.
95k¥ HE M a
"
#'s
"
a .
Zn F¥ Fk K Ls ERk2a3 .
4¥ K TM A
'
LFAA Azi 's b'
IF ,
M 2 'E#i3 It E¥#a9 .
931 .
s
'
a F¥ # a # a A .
*
•
•
¥ "
Fit 'U
•
~
•
=
•
•
•
•
- - -
TS
'
5×121
'
931 .
5 a t¥ # T5 is Etna itsu .
→ →
tl #aAts5 ,
5 Ii non -
zero then 7khE¥ b"
h¥k3 rtd .
→ Is
→
I→I ( b 'l
'
# 1£ zfnbt i Etsu ! ( taei 'EEAAF3 )
→
€ a 4- It fi -
N KH AT
'
7ft (£EHBHA=) b "
ZEZLF 5 .
a # in 'a#⇒
-15¥ 5×1122
Kd .
K - 52 T
'
a ¥¥ # It Etaa .
T2 = s
'
× s
'
,
TT2= Ts
'
×Ts
'
= -12×(112×112) F) ,
¥,
¥
% "
€a4=L⇐k .
574£ EIEHZFFHL 5123 .
T -12=5×1122
( £4 e :X na £¥X )
-
n' 9th Eh it 3F¥eh# ± a n' 7th ±¥s * HYEXBFKEl
'
{ FR 's than ¥¥¥ .
- ni 9th # on # AA i tahr he ¥Eti¥ K FR 23 .
-
A- a A 1 ¥ F Eta A I n 9th # 2 EB 't It " .
A #¥ His Hv In a H hit
"
Al
"
he
"
{
'
¥ a 3 ft hi Makin .
it 'E±¥ )
¥ t Et a A Tt a 7th # it suzrt . EnKEEFE FE ( Bikaata ) t ÷s¥t# tkn .
1 .
4¥44 * A ah #¥ as tE±¥
"
HE # A f
"
Tsai
'
Fti tEF¥5 It
43$33 s ( II ( 2 k£ £ .
HE # East .
3. THE # M tansth # t : E ¥ M KKHZ .
Mn ITHEIII -
¥¥ a "¥tEi CCE ) e HIM ; @ ) { t.tk#2FhF5az
"
.
Az#± to A7Hv # lift hit Ali h#Lz It 'Fi+t3ta .
input output
E
| µ
B
"
) W
°
DE ) e HIM ; %)
n'
'
7th # .
IKEA Ii Fs¥a¥t¥a .
( "FEFE ) KERI - .
ITHEDI -
rmz
FYFE # M a Th Eat
'
-
¥¥ elf ,
Ma IKE K
£H3A#
Eta th 'm { 5-23 A' ¥2 at # FE .
BI .
Hals
'
) =
|
.
( k=° ) HKCT ) =
|
•
( k=o )
0 ( k = 1 ) g
( k=l )
.
( k = 2) ;
( k= 2)
,
-
- -
-
,
•
it.
52 -12
( "FEFE ) Thea I - .
IF # a I -
rmz
FYFE # a ] THE DI
'
-
¥¥ e it ,
M a TMNK Hk hT# z
t.tk#3FE7*tixti5Ai3ht*ExAS2FFx
.
HE ' '
HE a
iktt Hit a Ifk k£3 bags a :c :
/
IK , <
sd .
this) =
|
. →
2 h=o )
( ±*YYY*Yn⇒¥IFIt¥¥I*F¥*n#⇒
0 ( k = 1 )
.
→ 2 ) ( k = 2)
] # EDI -
#¥ k If
Cup
45¥ K JR I'¥a tt3 Eh ,
t
At2 K
Then I -
¥4 zy t±# bits F¥a4⇐ z to .
4¥ # ¥57 sit ( # HE )
Fy THE # M Ian "9th # E httlz .
Mn ITHEDI -
¥¥ a "¥tEi CCE ) e HEM ; @ ) { t.tk#2FhF5az
"
.
Az#± to A7Hv # lift hit At ht# z It 'Fi±t3€a .
input output
f
Ka'
tkn
E
( µ
"3
"
) W
°
DE ) e HIM ; @ )
t.IE#iazY
( Helen )
i 7th # .
ITHERI
'
-
¥¥a¥t¥a .
1) Euler #a
FYFE # Mta A±HH5hEi7Hv # t : E ¥ M lift .
E a Euler ¥57 It ,
ECE ) e
HTM
; 7L )
zi .
4nF {
IEAEFEN
.
D e ( ft E) =
fte (E) I AYE 't )
2 e ( EDF ) =
ec E) .
e (F )
3 E : A- a A ⇒ EC E) =
0
4 E : E ATEGTE ⇒ ec E) = -
ECE ) .
2)
Stiefel -
Whitney #A
FYFE # Mta i7Hv # t : E ¥ M KFH ,
E a Stiefel -
Whitney # A { Witt
)3F=
.
H .
WICE ) E H
:(
M ; 22 ) ( i. 0,1 ,
...
,r
)
zi .
4nF { is
'¥kF€n.
00 Wo( E) =
1 .
1
Wic# E) =ftw:( E) l⇐7*H± )
2 W ( EDF )
=W(E) -
WCF) ( w=1 + wit . - .
+ Was )
3
W
,(8 ,
"
) = L ( Le HYRPIZ. ) =2z
,
4=AiE )
3)
Chern #I
FYFE # Mta
kFE¥ai7Hv
# t : E #M kttl .
E a Chern ¥52 {
Cite
)
}F=°
H .
CICE
) E H
"
( M ; 2 ) ( i. 0,1 ,
...
,r
)
zi .
4nF {
IEAEFEN.
@ Co (E) = 1 .
D
Cicft E) =ftc:(E) IEMIIHE)
2
c(EDF ) =
CCE) .
CCF) ( c=nta+ ... +
cn )
3
c.( 8 ,
'
) =
d ( de H4 # 2) a 2 ,
EIAIE )
l 7¥¥uEgttkR¥ .
)
75k¥ # MKIH . Mta nsth # a
R#±¥k¥¥¥
{
tfk±t2B¥I
:
Vectt ) : Mtd →
Sets
a a
M i→ Vea (M ) =
{ E → M }/±
4¥ '
Kt ¥2 et, =aF¥t 't $5 IAEA 's
"
.
P¥f4 na FFYEI '¥## a :{ :
µf+E
E
a
f*
a
Vectcm ) Vect CN )
f : M → N tr c  c
3- A¥t#E¥n¥h÷k
f*
HTCMIA ) HTCN ;A )
6
c # E) =
ftctt )
"
CCE )
( ETYIIHI )
2 .
Euler #E a t¥¥Ai z
'
pasta .
* .
) ] I It Bbl 55kt ¥EE¥ Hani Fta
.
Euler #aa#I¥Ai .
Rr
FFEHHSKK A 9th # TL :
E M K # L ,
Eo :=
E -
Mx { o }
SEE
,
Mta # Eva
"
a 7tN±'¥
"
s : M → E Ezy ,
 -
Annett -47*4 ( section ) 2n3 .
EaThom*a :
TC
E)
TIE
)
A i* 9
HTE ,Eo ) > HRCE )
= Thom #¥
' I s*
HOCM ) -
HTM )e a
1 ECE ) fruits .
K -2,2 E a Euler #a el E) e HIM ;2 ) { 'E¥xF3 .
the# 1 E bl
"
non -
zero section Etosit ,
ECE ) = 0 .
4¥ 's E A
"
Et a A Fsstt
'
ECE ) =
0 .
i
'
) s : M → Eo E non -
zero section
{(2
.
e (E) a #¥aAikikZAn32 . ( E. Eo ) a
cohomology ex ,
seq
.
LT ,
des HE ' °
)A A A A
HTEEO ) HTE ) HMEO )
A
s* s*
HIM )
a
ECE ) =
0
B'
# I # 2 77 FE # Ma Euler #x { XCM) 5h32 .
←
Ma¥¥¥a
(
ECTM
) ,
[ M ] > =
XCM ) (
fuetlu
) =
XCM ) )
^Mn¥*
# a Euler #A
MnEler¥I7 EEEAA RB )
97 F¥h# M a Euler #x XCM ) H ,
XCM ) =
Io c- isi rank CHICM ) )
i 'a¥x±k3¥¥#k .
4¥ K M A
"
= # FIB '¥'t HE F¥ # At a ±n¥h± .
(M ) = V -
E + F .
X
aa¥nh##* ) Katana ) M#ah#¥D
tz .
KTFIHF ( a # At ) if 8 a I # FK Bas't s 5-23 .
•
XC s
'
) =
4 -
6+4 =
2 .
•
•
•
He I # 2 gy ,
eCTs2 ) =
0 s if 1274¥ An .
tkbtr theft 1 sy Ts2 If FIAA itsu
!
-
to ,
XCT ) =
0 g) ,
t*hL ECTT
'
) =
0 { ta2n3 .
( It ) iaiz $15 TTZ of EFAA iTa3
'
- st kFBt2u .
)
ni 7th # a Et AAHI a TXFTEH't HE a- a A # at ,A¥h4 ) 2£23
AE'¥E A H¥skt= !
( Fun { Hina Eatx )
-
7¥ 'HE ¥k ,
E. a ! EID
# In it to { ÷zGH=n izt '
teeth h3 . . . ( Ftxk Ethic ,zH2 ! )
-
cxl
A¥¥ Ai I ÷*d ... ( SW #a .
Chern#aaA¥# Air Er Hit )
he
"
k¥57 a chtttn { R > It 5k£ ?
HEE )
3. PEE '¥FE¥÷t¥ i= suz
per
¥ Et night # IT : E → M A
"
# AA b 'i5b '
# 2312 ;
non -
zero section s : M → Eo I zkzdlz # HE .
#
E
Rr i
:
-
Mx{o } : 11h ftp.lntznjk -23 .
it
?
If :
S
µ
EH
#aAt£5€53h
s It FTFSFZ ( A=aAita3t=aaXT¥FHt )
# EE As,
r=2 I M HE# FIFA'¥'t
Ikzeisza
.
sz ¥¥th#lz2£ 's .
( o )
Fai Ma Jake. As a # 95k M → Eo zhF3 .
;•,⇐#
EPi
, 1 1
'
-
Mx{ 0 }
"Ytiltii'
'
l '
.
' ' ' '
,
1
.
1
i
, ,
i t
i
, I I
1
itufAsI
ML,
'
,
1 1 1
1
1
1
•
' 1
*.
:-/,
=kE #a5¥u Its 's a # both M
"
'→Eo
{ 94-3 .
t.sk the #a3⇐acz A±A5n. - .
# baa,
#a5Eu it YBK Faa Eb ' ?
t#•
⇐
•
s
( )
¥¥#
" " " }
•
OK
of )
" "
a
FX
.
Fy.
• •
IFFFH, a TIEKF * Mx { o } e # Litt 's e .
¥42 it IHNI't KFFAFE it An !
# EE { it
'
Ma # IFAFI's little ,
EE'Re bi Mx{o } a As DE
FI # F3 A { EAF P¥f#£ c : cz ( µ ) →
2 2¥23 .
i.
⇐ . as
E
• • 2
T- • 1
¥14×{0 } . °
• - 1
sf
• :
•
PH. a•
c = o 135 it
"
Ehn FEEEHFEAE !
at 0 not ,
Eitik s z FEBEAFZ PEETE ( obstruction ) 4
'
'ta3 .
# EE in F '¥#x c : GCM ) → 2 IF 2- cochair 2ta3 .
±5k
Sc(D3
) =
cc
D8
) =
0 ztay ,
2-
cocyde 2ta3 .
i.
€ . as
E
• • 2
T- • 1
;t.to::
•
< •
YT#ML•
Letsie [ c ] EHYM ;2 ) an 's cohomology '¥kbtek3 .
c it s NEK '
) 'FkhFc3bt , [ c ] rshfustan ! ( EaAbI¥± )
-
#k¥F# the Hit 5kt a 9th # a : E # M re HL ,
E n non -
zero section s : M → Eo It
M a v. 1
t.hn#a=tiaEeFhh4i=FFtathtiiI3
.
stir
#aFE
£iF¥aAi
it 3M£ a¥s 1 PEE '¥¥k
[ c ] e HRCM ; 2)
k£2 K£3 .
> FT ,
Is : Mcr ,
→ Eo [ c ] =
0
T c=o tss { a££FEBEaiE3 .
A
'
ht
'
'
) 'I > ! [ c ] =o 175 ,
-
¥4Ey£iE3izz
'
( r¥#a£ifEiEiE3 ! cost # au )
4 .
4¥ 't'¥¥B a
"
I #
"
Euler #A a Ih#
HE ## A -±hfH5hk n' 7th # t : E
*
M KJHL .
E a Euler ¥72
ECE ) e HRCM ; 2)
IF ,
E a non .
zero section E hF3Eh a
* np¥a'¥¥A e -
Exa3 !
CEIIAAIIE )
= KK £ , 2 Euler ¥52 { hE¥x at En .
f3{ . ( E :#a A ⇒ ECE ) .
.
0 ] if ha¥xb 's AA
5101473.
EL F¥k
r=dimM
at¥h¥ .
s : M → Eo e (E) =0
'
Ee !
.
:) Ma ✓ Manta it M { atafib 's . B
[ TI En # kktitvkt Xtistutsti 'E3 .
: ) rank ( TM ) = dim M F) ,
In # HA
'
As '
I 'Is .
XCS
'
) = 2 F) easy to Ita 's ,
5 I KIF non -
zero vector field ttzktsu . DK
Stiefel .
Whitney #ZaIh#
a-
EL
FEEE ninth # E * M ⇐ HL ,
f , T
•
. - -
→
Ea k -
FFKYE VKCE ) E it E In k ] a
!
12k$ '&'IHA9kka¥E $ 's # 373mi
'
.
# .
I
i M
p
E a
kikStiefel -
Whitney ¥57,
WRCE ) e HKCM ; 22 )
If ,
Vr.
kt , (E) a MK 't a section ehF3Eh a
* 1p¥IH÷**k ( mod 2) s -
ZQFZ .
% WICE ) = 0 c⇒ E it A IHTHFIFAE
.
:) W , (E) =
0 # M
' ' '
± zi Vr .
it , (E) = VRCE ) a
section lot 2123 .
c ⇒ M
" '
s F¥aAih3E±a sink -
7012€ ,
A⇐4eEs⇒. i. ¥hEI3 Ea ¥± # A ek3 .
←
AEHAFEKXZHE's .
( ⇒ Mta KEE. ah .
s°±i .
ftp.g#$ygy,y*⇐&*
⇒ " ¥453 Ea¥± # signs . #
Chern #2aIh*
4ft # KEEFE ninth # E
Is
M EHL ,
E a kBa¥,
k -
HER VKCE ) at ,
E In k ] a
1 -
12km , 'ItaA9Hua4E a 's Ai 3 73mi
'
-
# .
E a kihChan ¥57,
CKCE) e
HYM ; 2)
It ,
Vr
.
kt , (E) a MMI a section ktthf
* 1 EE '¥*k e -
2ixF3 .
( £4 c ¥iF¥5 )
-
4¥ HE # a it n' 7th # a Btk¥E .
> Fy
' '
EFAAHEFZEEVH
'
'
{ tttttu ⇒ ,
Ali 's { FEI ,
-43 FED #x state
!
-
ZEZE 3 Th Eat
'
-
EE
Ka
kER¥7 At A 31 , by "
) E #E4IF3
F¥f**k # took hEhia3 Fkttai ,
{ it 4¥ 'tE#I ol
'
ta5hn3 at 22€ AEK .
-
n' 7th 5ha Bnikb
"
It #sFaA a ] # Eaii -
EF a 4h Task R3 aii
FIFA tabspart at kit ¥-445 A 9th # c b '
Ft # Liliana €45613 .
-
# in 114+714 Blitz " 3 AIFAF a Iitt .
"
h543 faith
"
to "¥nERit3 .
→
6^(18 ) :
F¥PQzkE Grossmann 957.3kt# ,
A 7th Tan B¥A # Part
¥,¥¥e
Characteristic Classes
Milnor, Stasheff
微分位相幾何学
田村一郎
Lecture Notes in Algebraic Topology
Davis, Kirk
available online!
Thanked
@ Take To 9024 .

More Related Content

What's hot

Nippondanji氏に怒られても仕方ない、配列型とJSON型の使い方
Nippondanji氏に怒られても仕方ない、配列型とJSON型の使い方Nippondanji氏に怒られても仕方ない、配列型とJSON型の使い方
Nippondanji氏に怒られても仕方ない、配列型とJSON型の使い方
kwatch
 
集約署名
集約署名集約署名
集約署名
MITSUNARI Shigeo
 
純粋関数型アルゴリズム入門
純粋関数型アルゴリズム入門純粋関数型アルゴリズム入門
純粋関数型アルゴリズム入門
Kimikazu Kato
 
Координати в просторі
Координати в просторіКоординати в просторі
Координати в просторі
Людмила Кирилюк
 
SQLアンチパターン - ナイーブツリー
SQLアンチパターン - ナイーブツリーSQLアンチパターン - ナイーブツリー
SQLアンチパターン - ナイーブツリー
ke-m kamekoopa
 
C++ Data Structure PPT.ppt
C++ Data Structure PPT.pptC++ Data Structure PPT.ppt
C++ Data Structure PPT.ppt
Mukesh Thakur
 
たのしい高階関数
たのしい高階関数たのしい高階関数
たのしい高階関数
Shinichi Kozake
 
PHP AST 徹底解説
PHP AST 徹底解説PHP AST 徹底解説
PHP AST 徹底解説
do_aki
 
PRML勉強会@長岡 第4章線形識別モデル
PRML勉強会@長岡 第4章線形識別モデルPRML勉強会@長岡 第4章線形識別モデル
PRML勉強会@長岡 第4章線形識別モデル
Shohei Okada
 
Effective Modern C++ 勉強会#3 Item16
Effective Modern C++ 勉強会#3 Item16Effective Modern C++ 勉強会#3 Item16
Effective Modern C++ 勉強会#3 Item16
Mitsuru Kariya
 
関数型プログラミング入門 with OCaml
関数型プログラミング入門 with OCaml関数型プログラミング入門 with OCaml
関数型プログラミング入門 with OCaml
Haruka Oikawa
 
複数時点の単語出現頻度を 扱う時系列データモデリング
複数時点の単語出現頻度を 扱う時系列データモデリング複数時点の単語出現頻度を 扱う時系列データモデリング
複数時点の単語出現頻度を 扱う時系列データモデリング
奈良先端大 情報科学研究科
 
Презентація:Квадратний корінь з числа. Арифметичний квадратний корінь.
Презентація:Квадратний корінь з числа. Арифметичний квадратний корінь.Презентація:Квадратний корінь з числа. Арифметичний квадратний корінь.
Презентація:Квадратний корінь з числа. Арифметичний квадратний корінь.
sveta7940
 
Advanced Sorting Algorithms
Advanced Sorting AlgorithmsAdvanced Sorting Algorithms
Advanced Sorting Algorithms
Damian T. Gordon
 
Doubly circular linked list
Doubly circular linked listDoubly circular linked list
Doubly circular linked list
Roshan Chaudhary
 
公開鍵暗号8: 有限体上の楕円曲線の位数計算
公開鍵暗号8: 有限体上の楕円曲線の位数計算公開鍵暗号8: 有限体上の楕円曲線の位数計算
公開鍵暗号8: 有限体上の楕円曲線の位数計算
Joe Suzuki
 
StructuresPointers.pptx
StructuresPointers.pptxStructuresPointers.pptx
StructuresPointers.pptx
Bharati Vidyapeeth COE, Navi Mumbai
 
Les03 Single Row Function
Les03 Single Row FunctionLes03 Single Row Function
Поняття функції
Поняття функціїПоняття функції
Поняття функції
Oksana Bryk
 
ちいさなオブジェクトでドメインモデルを組み立てる
ちいさなオブジェクトでドメインモデルを組み立てるちいさなオブジェクトでドメインモデルを組み立てる
ちいさなオブジェクトでドメインモデルを組み立てる
増田 亨
 

What's hot (20)

Nippondanji氏に怒られても仕方ない、配列型とJSON型の使い方
Nippondanji氏に怒られても仕方ない、配列型とJSON型の使い方Nippondanji氏に怒られても仕方ない、配列型とJSON型の使い方
Nippondanji氏に怒られても仕方ない、配列型とJSON型の使い方
 
集約署名
集約署名集約署名
集約署名
 
純粋関数型アルゴリズム入門
純粋関数型アルゴリズム入門純粋関数型アルゴリズム入門
純粋関数型アルゴリズム入門
 
Координати в просторі
Координати в просторіКоординати в просторі
Координати в просторі
 
SQLアンチパターン - ナイーブツリー
SQLアンチパターン - ナイーブツリーSQLアンチパターン - ナイーブツリー
SQLアンチパターン - ナイーブツリー
 
C++ Data Structure PPT.ppt
C++ Data Structure PPT.pptC++ Data Structure PPT.ppt
C++ Data Structure PPT.ppt
 
たのしい高階関数
たのしい高階関数たのしい高階関数
たのしい高階関数
 
PHP AST 徹底解説
PHP AST 徹底解説PHP AST 徹底解説
PHP AST 徹底解説
 
PRML勉強会@長岡 第4章線形識別モデル
PRML勉強会@長岡 第4章線形識別モデルPRML勉強会@長岡 第4章線形識別モデル
PRML勉強会@長岡 第4章線形識別モデル
 
Effective Modern C++ 勉強会#3 Item16
Effective Modern C++ 勉強会#3 Item16Effective Modern C++ 勉強会#3 Item16
Effective Modern C++ 勉強会#3 Item16
 
関数型プログラミング入門 with OCaml
関数型プログラミング入門 with OCaml関数型プログラミング入門 with OCaml
関数型プログラミング入門 with OCaml
 
複数時点の単語出現頻度を 扱う時系列データモデリング
複数時点の単語出現頻度を 扱う時系列データモデリング複数時点の単語出現頻度を 扱う時系列データモデリング
複数時点の単語出現頻度を 扱う時系列データモデリング
 
Презентація:Квадратний корінь з числа. Арифметичний квадратний корінь.
Презентація:Квадратний корінь з числа. Арифметичний квадратний корінь.Презентація:Квадратний корінь з числа. Арифметичний квадратний корінь.
Презентація:Квадратний корінь з числа. Арифметичний квадратний корінь.
 
Advanced Sorting Algorithms
Advanced Sorting AlgorithmsAdvanced Sorting Algorithms
Advanced Sorting Algorithms
 
Doubly circular linked list
Doubly circular linked listDoubly circular linked list
Doubly circular linked list
 
公開鍵暗号8: 有限体上の楕円曲線の位数計算
公開鍵暗号8: 有限体上の楕円曲線の位数計算公開鍵暗号8: 有限体上の楕円曲線の位数計算
公開鍵暗号8: 有限体上の楕円曲線の位数計算
 
StructuresPointers.pptx
StructuresPointers.pptxStructuresPointers.pptx
StructuresPointers.pptx
 
Les03 Single Row Function
Les03 Single Row FunctionLes03 Single Row Function
Les03 Single Row Function
 
Поняття функції
Поняття функціїПоняття функції
Поняття функції
 
ちいさなオブジェクトでドメインモデルを組み立てる
ちいさなオブジェクトでドメインモデルを組み立てるちいさなオブジェクトでドメインモデルを組み立てる
ちいさなオブジェクトでドメインモデルを組み立てる
 

Similar to 特性類の気持ち

Swift で数学研究のススメ
Swift で数学研究のススメSwift で数学研究のススメ
Swift で数学研究のススメ
Taketo Sano
 
DM_s.2023_008.pdf
DM_s.2023_008.pdfDM_s.2023_008.pdf
DM_s.2023_008.pdf
JerielMaalihan1
 
RPMS IPCRF Memo 2023 DM_s2023_008.pdf
RPMS IPCRF Memo 2023 DM_s2023_008.pdfRPMS IPCRF Memo 2023 DM_s2023_008.pdf
RPMS IPCRF Memo 2023 DM_s2023_008.pdf
Glenn Rivera
 
DM_s2023_008-NEW-GUIDELINES-FOR-CLASSROOM-OBSERVATION.pdf
DM_s2023_008-NEW-GUIDELINES-FOR-CLASSROOM-OBSERVATION.pdfDM_s2023_008-NEW-GUIDELINES-FOR-CLASSROOM-OBSERVATION.pdf
DM_s2023_008-NEW-GUIDELINES-FOR-CLASSROOM-OBSERVATION.pdf
famesians
 
DM_s2023_008.pdf
DM_s2023_008.pdfDM_s2023_008.pdf
DM_s2023_008.pdf
MarilouSacristiaRaza
 
DM_s2023_008 (1).pdf
DM_s2023_008 (1).pdfDM_s2023_008 (1).pdf
DM_s2023_008 (1).pdf
LanceAng3
 
DM_s2023_008 rpms.pdf
DM_s2023_008 rpms.pdfDM_s2023_008 rpms.pdf
DM_s2023_008 rpms.pdf
JeanisilCereno2
 
Dasar dasar network planning
Dasar dasar network planningDasar dasar network planning
Dasar dasar network planning
BagusSunantoA5
 
COSTEP Student Loan Brochure
COSTEP Student Loan BrochureCOSTEP Student Loan Brochure
COSTEP Student Loan Brochure
The University of Texas (UTRGV)
 
Mtk khoirul bab 1
Mtk khoirul bab 1Mtk khoirul bab 1
Mtk khoirul bab 1
Zinoa
 
ROP BKB champs_MVN
ROP BKB champs_MVNROP BKB champs_MVN
ROP BKB champs_MVN
Brett Fisher
 
ARIMA
ARIMAARIMA
ARIMA
Amine Khalil
 
SSC Cert
SSC CertSSC Cert
SSC Cert
Nikhil Khankar
 
Vampiro -a_mascara_-_compendi
Vampiro  -a_mascara_-_compendiVampiro  -a_mascara_-_compendi
Vampiro -a_mascara_-_compendi
Thiago Albuquerque
 
;e€.E9-Ee E i--u-=3--,=El= E fnEo!2-= =i --=--=--= a.docx
;e€.E9-Ee E i--u-=3--,=El= E fnEo!2-= =i --=--=--= a.docx;e€.E9-Ee E i--u-=3--,=El= E fnEo!2-= =i --=--=--= a.docx
;e€.E9-Ee E i--u-=3--,=El= E fnEo!2-= =i --=--=--= a.docx
evonnehoggarth79783
 
Limit &amp; maxima
Limit &amp; maximaLimit &amp; maxima
Limit &amp; maxima
Zahidul Islam
 
Limit &amp; maxima
Limit &amp; maximaLimit &amp; maxima
Limit &amp; maxima
Zahidul Islam
 
Identifying A Unifying Mechanism for the Implementation of Concurrency Abstra...
Identifying A Unifying Mechanism for the Implementation of Concurrency Abstra...Identifying A Unifying Mechanism for the Implementation of Concurrency Abstra...
Identifying A Unifying Mechanism for the Implementation of Concurrency Abstra...
Stefan Marr
 
Petunjuk teknis lhkpn a
Petunjuk teknis lhkpn aPetunjuk teknis lhkpn a
Petunjuk teknis lhkpn a
vision5999
 
Petunjuk pengisian lhkpn model a
Petunjuk pengisian lhkpn model aPetunjuk pengisian lhkpn model a
Petunjuk pengisian lhkpn model a
Golden Saragih
 

Similar to 特性類の気持ち (20)

Swift で数学研究のススメ
Swift で数学研究のススメSwift で数学研究のススメ
Swift で数学研究のススメ
 
DM_s.2023_008.pdf
DM_s.2023_008.pdfDM_s.2023_008.pdf
DM_s.2023_008.pdf
 
RPMS IPCRF Memo 2023 DM_s2023_008.pdf
RPMS IPCRF Memo 2023 DM_s2023_008.pdfRPMS IPCRF Memo 2023 DM_s2023_008.pdf
RPMS IPCRF Memo 2023 DM_s2023_008.pdf
 
DM_s2023_008-NEW-GUIDELINES-FOR-CLASSROOM-OBSERVATION.pdf
DM_s2023_008-NEW-GUIDELINES-FOR-CLASSROOM-OBSERVATION.pdfDM_s2023_008-NEW-GUIDELINES-FOR-CLASSROOM-OBSERVATION.pdf
DM_s2023_008-NEW-GUIDELINES-FOR-CLASSROOM-OBSERVATION.pdf
 
DM_s2023_008.pdf
DM_s2023_008.pdfDM_s2023_008.pdf
DM_s2023_008.pdf
 
DM_s2023_008 (1).pdf
DM_s2023_008 (1).pdfDM_s2023_008 (1).pdf
DM_s2023_008 (1).pdf
 
DM_s2023_008 rpms.pdf
DM_s2023_008 rpms.pdfDM_s2023_008 rpms.pdf
DM_s2023_008 rpms.pdf
 
Dasar dasar network planning
Dasar dasar network planningDasar dasar network planning
Dasar dasar network planning
 
COSTEP Student Loan Brochure
COSTEP Student Loan BrochureCOSTEP Student Loan Brochure
COSTEP Student Loan Brochure
 
Mtk khoirul bab 1
Mtk khoirul bab 1Mtk khoirul bab 1
Mtk khoirul bab 1
 
ROP BKB champs_MVN
ROP BKB champs_MVNROP BKB champs_MVN
ROP BKB champs_MVN
 
ARIMA
ARIMAARIMA
ARIMA
 
SSC Cert
SSC CertSSC Cert
SSC Cert
 
Vampiro -a_mascara_-_compendi
Vampiro  -a_mascara_-_compendiVampiro  -a_mascara_-_compendi
Vampiro -a_mascara_-_compendi
 
;e€.E9-Ee E i--u-=3--,=El= E fnEo!2-= =i --=--=--= a.docx
;e€.E9-Ee E i--u-=3--,=El= E fnEo!2-= =i --=--=--= a.docx;e€.E9-Ee E i--u-=3--,=El= E fnEo!2-= =i --=--=--= a.docx
;e€.E9-Ee E i--u-=3--,=El= E fnEo!2-= =i --=--=--= a.docx
 
Limit &amp; maxima
Limit &amp; maximaLimit &amp; maxima
Limit &amp; maxima
 
Limit &amp; maxima
Limit &amp; maximaLimit &amp; maxima
Limit &amp; maxima
 
Identifying A Unifying Mechanism for the Implementation of Concurrency Abstra...
Identifying A Unifying Mechanism for the Implementation of Concurrency Abstra...Identifying A Unifying Mechanism for the Implementation of Concurrency Abstra...
Identifying A Unifying Mechanism for the Implementation of Concurrency Abstra...
 
Petunjuk teknis lhkpn a
Petunjuk teknis lhkpn aPetunjuk teknis lhkpn a
Petunjuk teknis lhkpn a
 
Petunjuk pengisian lhkpn model a
Petunjuk pengisian lhkpn model aPetunjuk pengisian lhkpn model a
Petunjuk pengisian lhkpn model a
 

More from Taketo Sano

Divisibility of Lee’s class and its relation with Rasmussen’s invariant / 201...
Divisibility of Lee’s class and its relation with Rasmussen’s invariant / 201...Divisibility of Lee’s class and its relation with Rasmussen’s invariant / 201...
Divisibility of Lee’s class and its relation with Rasmussen’s invariant / 201...
Taketo Sano
 
トポロジーと圏論の夜明け
トポロジーと圏論の夜明けトポロジーと圏論の夜明け
トポロジーと圏論の夜明け
Taketo Sano
 
(意欲的な中高生のための)トポロジー・圏論・コンピュータ
(意欲的な中高生のための)トポロジー・圏論・コンピュータ(意欲的な中高生のための)トポロジー・圏論・コンピュータ
(意欲的な中高生のための)トポロジー・圏論・コンピュータ
Taketo Sano
 
Swift で数学のススメ 〜 プログラミングと数学は同時に学べ
Swift で数学のススメ 〜 プログラミングと数学は同時に学べSwift で数学のススメ 〜 プログラミングと数学は同時に学べ
Swift で数学のススメ 〜 プログラミングと数学は同時に学べ
Taketo Sano
 
山手線は丸いのか?プログラマのためのトポロジー入門
山手線は丸いのか?プログラマのためのトポロジー入門山手線は丸いのか?プログラマのためのトポロジー入門
山手線は丸いのか?プログラマのためのトポロジー入門
Taketo Sano
 
何もないところから数を作る
何もないところから数を作る何もないところから数を作る
何もないところから数を作る
Taketo Sano
 
「数える」とは何か? 〜 「とは何か?」を問う、AI時代の数学
「数える」とは何か? 〜 「とは何か?」を問う、AI時代の数学「数える」とは何か? 〜 「とは何か?」を問う、AI時代の数学
「数える」とは何か? 〜 「とは何か?」を問う、AI時代の数学
Taketo Sano
 
情報幾何学 #2.4
情報幾何学 #2.4情報幾何学 #2.4
情報幾何学 #2.4
Taketo Sano
 
情報幾何学 #2 #infogeo16
情報幾何学 #2 #infogeo16情報幾何学 #2 #infogeo16
情報幾何学 #2 #infogeo16
Taketo Sano
 
objc2swift 〜 Objective-C から Swift への「コード&パラダイム」シフト
objc2swift 〜 Objective-C から Swift への「コード&パラダイム」シフトobjc2swift 〜 Objective-C から Swift への「コード&パラダイム」シフト
objc2swift 〜 Objective-C から Swift への「コード&パラダイム」シフト
Taketo Sano
 
何もないところから数を作る
何もないところから数を作る何もないところから数を作る
何もないところから数を作る
Taketo Sano
 
objc2swift (続・自動変換の野望)
objc2swift (続・自動変換の野望) objc2swift (続・自動変換の野望)
objc2swift (続・自動変換の野望)
Taketo Sano
 
さらに上を目指すための iOS アプリ設計
さらに上を目指すための iOS アプリ設計さらに上を目指すための iOS アプリ設計
さらに上を目指すための iOS アプリ設計
Taketo Sano
 
基底変換、固有値・固有ベクトル、そしてその先
基底変換、固有値・固有ベクトル、そしてその先基底変換、固有値・固有ベクトル、そしてその先
基底変換、固有値・固有ベクトル、そしてその先
Taketo Sano
 
objc2swift (自動変換の野望)
objc2swift (自動変換の野望)objc2swift (自動変換の野望)
objc2swift (自動変換の野望)
Taketo Sano
 
プログラマのための線形代数再入門2 〜 要件定義から学ぶ行列式と逆行列
プログラマのための線形代数再入門2 〜 要件定義から学ぶ行列式と逆行列プログラマのための線形代数再入門2 〜 要件定義から学ぶ行列式と逆行列
プログラマのための線形代数再入門2 〜 要件定義から学ぶ行列式と逆行列
Taketo Sano
 
2015 02-18 xxx-literalconvertible
2015 02-18 xxx-literalconvertible2015 02-18 xxx-literalconvertible
2015 02-18 xxx-literalconvertible
Taketo Sano
 
let UIWebView as WKWebView
let UIWebView as WKWebViewlet UIWebView as WKWebView
let UIWebView as WKWebView
Taketo Sano
 
プログラマのための線形代数再入門
プログラマのための線形代数再入門プログラマのための線形代数再入門
プログラマのための線形代数再入門
Taketo Sano
 
コードを書けば複素数がわかる
コードを書けば複素数がわかるコードを書けば複素数がわかる
コードを書けば複素数がわかる
Taketo Sano
 

More from Taketo Sano (20)

Divisibility of Lee’s class and its relation with Rasmussen’s invariant / 201...
Divisibility of Lee’s class and its relation with Rasmussen’s invariant / 201...Divisibility of Lee’s class and its relation with Rasmussen’s invariant / 201...
Divisibility of Lee’s class and its relation with Rasmussen’s invariant / 201...
 
トポロジーと圏論の夜明け
トポロジーと圏論の夜明けトポロジーと圏論の夜明け
トポロジーと圏論の夜明け
 
(意欲的な中高生のための)トポロジー・圏論・コンピュータ
(意欲的な中高生のための)トポロジー・圏論・コンピュータ(意欲的な中高生のための)トポロジー・圏論・コンピュータ
(意欲的な中高生のための)トポロジー・圏論・コンピュータ
 
Swift で数学のススメ 〜 プログラミングと数学は同時に学べ
Swift で数学のススメ 〜 プログラミングと数学は同時に学べSwift で数学のススメ 〜 プログラミングと数学は同時に学べ
Swift で数学のススメ 〜 プログラミングと数学は同時に学べ
 
山手線は丸いのか?プログラマのためのトポロジー入門
山手線は丸いのか?プログラマのためのトポロジー入門山手線は丸いのか?プログラマのためのトポロジー入門
山手線は丸いのか?プログラマのためのトポロジー入門
 
何もないところから数を作る
何もないところから数を作る何もないところから数を作る
何もないところから数を作る
 
「数える」とは何か? 〜 「とは何か?」を問う、AI時代の数学
「数える」とは何か? 〜 「とは何か?」を問う、AI時代の数学「数える」とは何か? 〜 「とは何か?」を問う、AI時代の数学
「数える」とは何か? 〜 「とは何か?」を問う、AI時代の数学
 
情報幾何学 #2.4
情報幾何学 #2.4情報幾何学 #2.4
情報幾何学 #2.4
 
情報幾何学 #2 #infogeo16
情報幾何学 #2 #infogeo16情報幾何学 #2 #infogeo16
情報幾何学 #2 #infogeo16
 
objc2swift 〜 Objective-C から Swift への「コード&パラダイム」シフト
objc2swift 〜 Objective-C から Swift への「コード&パラダイム」シフトobjc2swift 〜 Objective-C から Swift への「コード&パラダイム」シフト
objc2swift 〜 Objective-C から Swift への「コード&パラダイム」シフト
 
何もないところから数を作る
何もないところから数を作る何もないところから数を作る
何もないところから数を作る
 
objc2swift (続・自動変換の野望)
objc2swift (続・自動変換の野望) objc2swift (続・自動変換の野望)
objc2swift (続・自動変換の野望)
 
さらに上を目指すための iOS アプリ設計
さらに上を目指すための iOS アプリ設計さらに上を目指すための iOS アプリ設計
さらに上を目指すための iOS アプリ設計
 
基底変換、固有値・固有ベクトル、そしてその先
基底変換、固有値・固有ベクトル、そしてその先基底変換、固有値・固有ベクトル、そしてその先
基底変換、固有値・固有ベクトル、そしてその先
 
objc2swift (自動変換の野望)
objc2swift (自動変換の野望)objc2swift (自動変換の野望)
objc2swift (自動変換の野望)
 
プログラマのための線形代数再入門2 〜 要件定義から学ぶ行列式と逆行列
プログラマのための線形代数再入門2 〜 要件定義から学ぶ行列式と逆行列プログラマのための線形代数再入門2 〜 要件定義から学ぶ行列式と逆行列
プログラマのための線形代数再入門2 〜 要件定義から学ぶ行列式と逆行列
 
2015 02-18 xxx-literalconvertible
2015 02-18 xxx-literalconvertible2015 02-18 xxx-literalconvertible
2015 02-18 xxx-literalconvertible
 
let UIWebView as WKWebView
let UIWebView as WKWebViewlet UIWebView as WKWebView
let UIWebView as WKWebView
 
プログラマのための線形代数再入門
プログラマのための線形代数再入門プログラマのための線形代数再入門
プログラマのための線形代数再入門
 
コードを書けば複素数がわかる
コードを書けば複素数がわかるコードを書けば複素数がわかる
コードを書けば複素数がわかる
 

Recently uploaded

H. A. Roberts: VITAL FORCE - Dr. Niranjan Bapat
H. A. Roberts: VITAL FORCE - Dr. Niranjan BapatH. A. Roberts: VITAL FORCE - Dr. Niranjan Bapat
H. A. Roberts: VITAL FORCE - Dr. Niranjan Bapat
Niranjan Bapat
 
Principles of Roods Approach!!!!!!!.pptx
Principles of Roods Approach!!!!!!!.pptxPrinciples of Roods Approach!!!!!!!.pptx
Principles of Roods Approach!!!!!!!.pptx
ibtesaam huma
 
Howe Writing Center - Orientation Summer 2024
Howe Writing Center - Orientation Summer 2024Howe Writing Center - Orientation Summer 2024
Howe Writing Center - Orientation Summer 2024
Elizabeth Walsh
 
How to Manage Access Rights & User Types in Odoo 17
How to Manage Access Rights & User Types in Odoo 17How to Manage Access Rights & User Types in Odoo 17
How to Manage Access Rights & User Types in Odoo 17
Celine George
 
The basics of sentences session 10pptx.pptx
The basics of sentences session 10pptx.pptxThe basics of sentences session 10pptx.pptx
The basics of sentences session 10pptx.pptx
heathfieldcps1
 
ENGLISH-7-CURRICULUM MAP- MATATAG CURRICULUM
ENGLISH-7-CURRICULUM MAP- MATATAG CURRICULUMENGLISH-7-CURRICULUM MAP- MATATAG CURRICULUM
ENGLISH-7-CURRICULUM MAP- MATATAG CURRICULUM
HappieMontevirgenCas
 
How to Manage Large Scrollbar in Odoo 17 POS
How to Manage Large Scrollbar in Odoo 17 POSHow to Manage Large Scrollbar in Odoo 17 POS
How to Manage Large Scrollbar in Odoo 17 POS
Celine George
 
2024 KWL Back 2 School Summer Conference
2024 KWL Back 2 School Summer Conference2024 KWL Back 2 School Summer Conference
2024 KWL Back 2 School Summer Conference
KlettWorldLanguages
 
How to Manage Early Receipt Printing in Odoo 17 POS
How to Manage Early Receipt Printing in Odoo 17 POSHow to Manage Early Receipt Printing in Odoo 17 POS
How to Manage Early Receipt Printing in Odoo 17 POS
Celine George
 
Year-to-Date Filter in Odoo 17 Dashboard
Year-to-Date Filter in Odoo 17 DashboardYear-to-Date Filter in Odoo 17 Dashboard
Year-to-Date Filter in Odoo 17 Dashboard
Celine George
 
What is Rescue Session in Odoo 17 POS - Odoo 17 Slides
What is Rescue Session in Odoo 17 POS - Odoo 17 SlidesWhat is Rescue Session in Odoo 17 POS - Odoo 17 Slides
What is Rescue Session in Odoo 17 POS - Odoo 17 Slides
Celine George
 
How to Manage Line Discount in Odoo 17 POS
How to Manage Line Discount in Odoo 17 POSHow to Manage Line Discount in Odoo 17 POS
How to Manage Line Discount in Odoo 17 POS
Celine George
 
1-NLC-MATH7-Consolidation-Lesson1 2024.pptx
1-NLC-MATH7-Consolidation-Lesson1 2024.pptx1-NLC-MATH7-Consolidation-Lesson1 2024.pptx
1-NLC-MATH7-Consolidation-Lesson1 2024.pptx
AnneMarieJacildo
 
DANH SÁCH THÍ SINH XÉT TUYỂN SỚM ĐỦ ĐIỀU KIỆN TRÚNG TUYỂN ĐẠI HỌC CHÍNH QUY N...
DANH SÁCH THÍ SINH XÉT TUYỂN SỚM ĐỦ ĐIỀU KIỆN TRÚNG TUYỂN ĐẠI HỌC CHÍNH QUY N...DANH SÁCH THÍ SINH XÉT TUYỂN SỚM ĐỦ ĐIỀU KIỆN TRÚNG TUYỂN ĐẠI HỌC CHÍNH QUY N...
DANH SÁCH THÍ SINH XÉT TUYỂN SỚM ĐỦ ĐIỀU KIỆN TRÚNG TUYỂN ĐẠI HỌC CHÍNH QUY N...
thanhluan21
 
Configuring Single Sign-On (SSO) via Identity Management | MuleSoft Mysore Me...
Configuring Single Sign-On (SSO) via Identity Management | MuleSoft Mysore Me...Configuring Single Sign-On (SSO) via Identity Management | MuleSoft Mysore Me...
Configuring Single Sign-On (SSO) via Identity Management | MuleSoft Mysore Me...
MysoreMuleSoftMeetup
 
NC Public Schools Involved in NCDPI, Zipline Partnership
NC Public Schools Involved in NCDPI, Zipline PartnershipNC Public Schools Involved in NCDPI, Zipline Partnership
NC Public Schools Involved in NCDPI, Zipline Partnership
Mebane Rash
 
"DANH SÁCH THÍ SINH XÉT TUYỂN SỚM ĐỦ ĐIỀU KIỆN TRÚNG TUYỂN ĐẠI HỌC CHÍNH QUY ...
"DANH SÁCH THÍ SINH XÉT TUYỂN SỚM ĐỦ ĐIỀU KIỆN TRÚNG TUYỂN ĐẠI HỌC CHÍNH QUY ..."DANH SÁCH THÍ SINH XÉT TUYỂN SỚM ĐỦ ĐIỀU KIỆN TRÚNG TUYỂN ĐẠI HỌC CHÍNH QUY ...
"DANH SÁCH THÍ SINH XÉT TUYỂN SỚM ĐỦ ĐIỀU KIỆN TRÚNG TUYỂN ĐẠI HỌC CHÍNH QUY ...
thanhluan21
 
Individual Performance Commitment Review Form-Developmental Plan.docx
Individual Performance Commitment Review Form-Developmental Plan.docxIndividual Performance Commitment Review Form-Developmental Plan.docx
Individual Performance Commitment Review Form-Developmental Plan.docx
monicaaringo1
 
modul ajar kelas x bahasa inggris 2024-2025
modul ajar kelas x bahasa inggris 2024-2025modul ajar kelas x bahasa inggris 2024-2025
modul ajar kelas x bahasa inggris 2024-2025
NurFitriah45
 
How to Create a New Article in Knowledge App in Odoo 17
How to Create a New Article in Knowledge App in Odoo 17How to Create a New Article in Knowledge App in Odoo 17
How to Create a New Article in Knowledge App in Odoo 17
Celine George
 

Recently uploaded (20)

H. A. Roberts: VITAL FORCE - Dr. Niranjan Bapat
H. A. Roberts: VITAL FORCE - Dr. Niranjan BapatH. A. Roberts: VITAL FORCE - Dr. Niranjan Bapat
H. A. Roberts: VITAL FORCE - Dr. Niranjan Bapat
 
Principles of Roods Approach!!!!!!!.pptx
Principles of Roods Approach!!!!!!!.pptxPrinciples of Roods Approach!!!!!!!.pptx
Principles of Roods Approach!!!!!!!.pptx
 
Howe Writing Center - Orientation Summer 2024
Howe Writing Center - Orientation Summer 2024Howe Writing Center - Orientation Summer 2024
Howe Writing Center - Orientation Summer 2024
 
How to Manage Access Rights & User Types in Odoo 17
How to Manage Access Rights & User Types in Odoo 17How to Manage Access Rights & User Types in Odoo 17
How to Manage Access Rights & User Types in Odoo 17
 
The basics of sentences session 10pptx.pptx
The basics of sentences session 10pptx.pptxThe basics of sentences session 10pptx.pptx
The basics of sentences session 10pptx.pptx
 
ENGLISH-7-CURRICULUM MAP- MATATAG CURRICULUM
ENGLISH-7-CURRICULUM MAP- MATATAG CURRICULUMENGLISH-7-CURRICULUM MAP- MATATAG CURRICULUM
ENGLISH-7-CURRICULUM MAP- MATATAG CURRICULUM
 
How to Manage Large Scrollbar in Odoo 17 POS
How to Manage Large Scrollbar in Odoo 17 POSHow to Manage Large Scrollbar in Odoo 17 POS
How to Manage Large Scrollbar in Odoo 17 POS
 
2024 KWL Back 2 School Summer Conference
2024 KWL Back 2 School Summer Conference2024 KWL Back 2 School Summer Conference
2024 KWL Back 2 School Summer Conference
 
How to Manage Early Receipt Printing in Odoo 17 POS
How to Manage Early Receipt Printing in Odoo 17 POSHow to Manage Early Receipt Printing in Odoo 17 POS
How to Manage Early Receipt Printing in Odoo 17 POS
 
Year-to-Date Filter in Odoo 17 Dashboard
Year-to-Date Filter in Odoo 17 DashboardYear-to-Date Filter in Odoo 17 Dashboard
Year-to-Date Filter in Odoo 17 Dashboard
 
What is Rescue Session in Odoo 17 POS - Odoo 17 Slides
What is Rescue Session in Odoo 17 POS - Odoo 17 SlidesWhat is Rescue Session in Odoo 17 POS - Odoo 17 Slides
What is Rescue Session in Odoo 17 POS - Odoo 17 Slides
 
How to Manage Line Discount in Odoo 17 POS
How to Manage Line Discount in Odoo 17 POSHow to Manage Line Discount in Odoo 17 POS
How to Manage Line Discount in Odoo 17 POS
 
1-NLC-MATH7-Consolidation-Lesson1 2024.pptx
1-NLC-MATH7-Consolidation-Lesson1 2024.pptx1-NLC-MATH7-Consolidation-Lesson1 2024.pptx
1-NLC-MATH7-Consolidation-Lesson1 2024.pptx
 
DANH SÁCH THÍ SINH XÉT TUYỂN SỚM ĐỦ ĐIỀU KIỆN TRÚNG TUYỂN ĐẠI HỌC CHÍNH QUY N...
DANH SÁCH THÍ SINH XÉT TUYỂN SỚM ĐỦ ĐIỀU KIỆN TRÚNG TUYỂN ĐẠI HỌC CHÍNH QUY N...DANH SÁCH THÍ SINH XÉT TUYỂN SỚM ĐỦ ĐIỀU KIỆN TRÚNG TUYỂN ĐẠI HỌC CHÍNH QUY N...
DANH SÁCH THÍ SINH XÉT TUYỂN SỚM ĐỦ ĐIỀU KIỆN TRÚNG TUYỂN ĐẠI HỌC CHÍNH QUY N...
 
Configuring Single Sign-On (SSO) via Identity Management | MuleSoft Mysore Me...
Configuring Single Sign-On (SSO) via Identity Management | MuleSoft Mysore Me...Configuring Single Sign-On (SSO) via Identity Management | MuleSoft Mysore Me...
Configuring Single Sign-On (SSO) via Identity Management | MuleSoft Mysore Me...
 
NC Public Schools Involved in NCDPI, Zipline Partnership
NC Public Schools Involved in NCDPI, Zipline PartnershipNC Public Schools Involved in NCDPI, Zipline Partnership
NC Public Schools Involved in NCDPI, Zipline Partnership
 
"DANH SÁCH THÍ SINH XÉT TUYỂN SỚM ĐỦ ĐIỀU KIỆN TRÚNG TUYỂN ĐẠI HỌC CHÍNH QUY ...
"DANH SÁCH THÍ SINH XÉT TUYỂN SỚM ĐỦ ĐIỀU KIỆN TRÚNG TUYỂN ĐẠI HỌC CHÍNH QUY ..."DANH SÁCH THÍ SINH XÉT TUYỂN SỚM ĐỦ ĐIỀU KIỆN TRÚNG TUYỂN ĐẠI HỌC CHÍNH QUY ...
"DANH SÁCH THÍ SINH XÉT TUYỂN SỚM ĐỦ ĐIỀU KIỆN TRÚNG TUYỂN ĐẠI HỌC CHÍNH QUY ...
 
Individual Performance Commitment Review Form-Developmental Plan.docx
Individual Performance Commitment Review Form-Developmental Plan.docxIndividual Performance Commitment Review Form-Developmental Plan.docx
Individual Performance Commitment Review Form-Developmental Plan.docx
 
modul ajar kelas x bahasa inggris 2024-2025
modul ajar kelas x bahasa inggris 2024-2025modul ajar kelas x bahasa inggris 2024-2025
modul ajar kelas x bahasa inggris 2024-2025
 
How to Create a New Article in Knowledge App in Odoo 17
How to Create a New Article in Knowledge App in Odoo 17How to Create a New Article in Knowledge App in Odoo 17
How to Create a New Article in Knowledge App in Odoo 17
 

特性類の気持ち

  • 1. 2018/4/14 *d¥*±h¥±kte & #X## EPEE sttt ' EEE' . slit - 4¥ HE ¥A a K¥+5 @ TakeTo 1024
  • 2. Contents 0 . t ' 7th # ksuc . 1 . 4¥ ' HtTHE a hat # at #¥x . 2 . Euler ¥ta a A¥# Aiz 'l¥s4t¥a . 3 . PEEMEIE'EF# rmz . 4 . 4¥ HE #a a ' ' IHI ' ' . * =a7¥Eirt Ih¥±± F) " ¥eF±t5 " 2 h#¥El£F . 5*h{ KIBE, tkukrt # Fta 1- t.EE#TEeIFFET In .
  • 4. i 7th # sit ¥F⇒sb'ts3#¥h# a # ¥n±kt9kh# Eat A "¥n2n3ta . 531 M = s2 I a f¥tt9HlFh E = TM . (Tangent bundle ) TM ~ # ' * Beta . I> y = ¥µTpM :# ' LIMIT p 1122 : 7 } hli ' - 4pm YY : EIIERIF ( pkaF¥isHvtEp¥l ) ← n7HH¥aaF#±⇒ ' ( ninth '¥ X e * an a × : t.Y.IT#ptas # HE . )
  • 5. " Etants " n' 9th # E¥4¥a ' EMIT E = Mx Rr it €53hMknt9Hv#n . = th 2 As #± A t ' 9th # z Eta # At 9th ' # said . ^3Ni' . 4¥ , , A¥# Egg ;qµ*z . ' trivial Vector bundle ) By M = s ' In ELKE # . FZTHTEAKH F¥e*U×R' at 's . FT IT # RA ' # rtn9HlFkath¥kE£h3 . RR - R d s ' - • • • • • €1- - - 4) E = S ' x IR 2) E : open Mobius band C trivial ) ( non . Trivial )
  • 6. 2) to Etna Itau tea IEIAA . i ) El E = s ' x IR tiztnktt ' ' . 3. ' E. Ii ' n' 7Hubt - ha ( to ) { 53 ' 'Ei7Hv±¥s to 2123 ttf lb 't FEE # a z # 2324M¥ the a 'k#¥ b 's HF " 0 { he ,2lF 's . qp¥. " rahman • n' 9th # A 3F÷kh# e i7HvBE¥n " tslikk TELEFE " .
  • 7. 95k¥ HE M a " #'s " a . Zn F¥ Fk K Ls ERk2a3 . 4¥ K TM A ' LFAA Azi 's b' IF , M 2 'E#i3 It E¥#a9 . 931 . s ' a F¥ # a # a A . * • • ¥ " Fit 'U • ~ • = • • • • - - - TS ' 5×121 '
  • 8. 931 . 5 a t¥ # T5 is Etna itsu . → → tl #aAts5 , 5 Ii non - zero then 7khE¥ b" h¥k3 rtd . → Is → I→I ( b 'l ' # 1£ zfnbt i Etsu ! ( taei 'EEAAF3 ) → € a 4- It fi - N KH AT ' 7ft (£EHBHA=) b " ZEZLF 5 . a # in 'a#⇒ -15¥ 5×1122
  • 9. Kd . K - 52 T ' a ¥¥ # It Etaa . T2 = s ' × s ' , TT2= Ts ' ×Ts ' = -12×(112×112) F) , ¥, ¥ % " €a4=L⇐k . 574£ EIEHZFFHL 5123 . T -12=5×1122
  • 10. ( £4 e :X na £¥X ) - n' 9th Eh it 3F¥eh# ± a n' 7th ±¥s * HYEXBFKEl ' { FR 's than ¥¥¥ . - ni 9th # on # AA i tahr he ¥Eti¥ K FR 23 . - A- a A 1 ¥ F Eta A I n 9th # 2 EB 't It " . A #¥ His Hv In a H hit " Al " he " { ' ¥ a 3 ft hi Makin . it 'E±¥ ) ¥ t Et a A Tt a 7th # it suzrt . EnKEEFE FE ( Bikaata ) t ÷s¥t# tkn .
  • 11. 1 . 4¥44 * A ah #¥ as tE±¥ " HE # A f " Tsai ' Fti tEF¥5 It 43$33 s ( II ( 2 k£ £ .
  • 12. HE # East . 3. THE # M tansth # t : E ¥ M KKHZ . Mn ITHEIII - ¥¥ a "¥tEi CCE ) e HIM ; @ ) { t.tk#2FhF5az " . Az#± to A7Hv # lift hit Ali h#Lz It 'Fi+t3ta . input output E | µ B " ) W ° DE ) e HIM ; %) n' ' 7th # . IKEA Ii Fs¥a¥t¥a .
  • 13. ( "FEFE ) KERI - . ITHEDI - rmz FYFE # M a Th Eat ' - ¥¥ elf , Ma IKE K £H3A# Eta th 'm { 5-23 A' ¥2 at # FE . BI . Hals ' ) = | . ( k=° ) HKCT ) = | • ( k=o ) 0 ( k = 1 ) g ( k=l ) . ( k = 2) ; ( k= 2) , - - - - , • it. 52 -12
  • 14. ( "FEFE ) Thea I - . IF # a I - rmz FYFE # a ] THE DI ' - ¥¥ e it , M a TMNK Hk hT# z t.tk#3FE7*tixti5Ai3ht*ExAS2FFx . HE ' ' HE a iktt Hit a Ifk k£3 bags a :c : / IK , < sd . this) = | . → 2 h=o ) ( ±*YYY*Yn⇒¥IFIt¥¥I*F¥*n#⇒ 0 ( k = 1 ) . → 2 ) ( k = 2) ] # EDI - #¥ k If Cup 45¥ K JR I'¥a tt3 Eh , t At2 K Then I - ¥4 zy t±# bits F¥a4⇐ z to .
  • 15. 4¥ # ¥57 sit ( # HE ) Fy THE # M Ian "9th # E httlz . Mn ITHEDI - ¥¥ a "¥tEi CCE ) e HEM ; @ ) { t.tk#2FhF5az " . Az#± to A7Hv # lift hit At ht# z It 'Fi±t3€a . input output f Ka' tkn E ( µ "3 " ) W ° DE ) e HIM ; @ ) t.IE#iazY ( Helen ) i 7th # . ITHERI ' - ¥¥a¥t¥a .
  • 16. 1) Euler #a FYFE # Mta A±HH5hEi7Hv # t : E ¥ M lift . E a Euler ¥57 It , ECE ) e HTM ; 7L ) zi . 4nF { IEAEFEN . D e ( ft E) = fte (E) I AYE 't ) 2 e ( EDF ) = ec E) . e (F ) 3 E : A- a A ⇒ EC E) = 0 4 E : E ATEGTE ⇒ ec E) = - ECE ) .
  • 17. 2) Stiefel - Whitney #A FYFE # Mta i7Hv # t : E ¥ M KFH , E a Stiefel - Whitney # A { Witt )3F= . H . WICE ) E H :( M ; 22 ) ( i. 0,1 , ... ,r ) zi . 4nF { is '¥kF€n. 00 Wo( E) = 1 . 1 Wic# E) =ftw:( E) l⇐7*H± ) 2 W ( EDF ) =W(E) - WCF) ( w=1 + wit . - . + Was ) 3 W ,(8 , " ) = L ( Le HYRPIZ. ) =2z , 4=AiE )
  • 18. 3) Chern #I FYFE # Mta kFE¥ai7Hv # t : E #M kttl . E a Chern ¥52 { Cite ) }F=° H . CICE ) E H " ( M ; 2 ) ( i. 0,1 , ... ,r ) zi . 4nF { IEAEFEN. @ Co (E) = 1 . D Cicft E) =ftc:(E) IEMIIHE) 2 c(EDF ) = CCE) . CCF) ( c=nta+ ... + cn ) 3 c.( 8 , ' ) = d ( de H4 # 2) a 2 , EIAIE )
  • 19. l 7¥¥uEgttkR¥ . ) 75k¥ # MKIH . Mta nsth # a R#±¥k¥¥¥ { tfk±t2B¥I : Vectt ) : Mtd → Sets a a M i→ Vea (M ) = { E → M }/± 4¥ ' Kt ¥2 et, =aF¥t 't $5 IAEA 's " . P¥f4 na FFYEI '¥## a :{ : µf+E E a f* a Vectcm ) Vect CN ) f : M → N tr c c 3- A¥t#E¥n¥h÷k f* HTCMIA ) HTCN ;A ) 6 c # E) = ftctt ) " CCE ) ( ETYIIHI )
  • 20. 2 . Euler #E a t¥¥Ai z ' pasta . * . ) ] I It Bbl 55kt ¥EE¥ Hani Fta .
  • 21. Euler #aa#I¥Ai . Rr FFEHHSKK A 9th # TL : E M K # L , Eo := E - Mx { o } SEE , Mta # Eva " a 7tN±'¥ " s : M → E Ezy , - Annett -47*4 ( section ) 2n3 . EaThom*a : TC E) TIE ) A i* 9 HTE ,Eo ) > HRCE ) = Thom #¥ ' I s* HOCM ) - HTM )e a 1 ECE ) fruits . K -2,2 E a Euler #a el E) e HIM ;2 ) { 'E¥xF3 .
  • 22. the# 1 E bl " non - zero section Etosit , ECE ) = 0 . 4¥ 's E A " Et a A Fsstt ' ECE ) = 0 . i ' ) s : M → Eo E non - zero section {(2 . e (E) a #¥aAikikZAn32 . ( E. Eo ) a cohomology ex , seq . LT , des HE ' ° )A A A A HTEEO ) HTE ) HMEO ) A s* s* HIM ) a ECE ) = 0 B'
  • 23. # I # 2 77 FE # Ma Euler #x { XCM) 5h32 . ← Ma¥¥¥a ( ECTM ) , [ M ] > = XCM ) ( fuetlu ) = XCM ) ) ^Mn¥* # a Euler #A MnEler¥I7 EEEAA RB ) 97 F¥h# M a Euler #x XCM ) H , XCM ) = Io c- isi rank CHICM ) ) i 'a¥x±k3¥¥#k . 4¥ K M A " = # FIB '¥'t HE F¥ # At a ±n¥h± . (M ) = V - E + F . X aa¥nh##* ) Katana ) M#ah#¥D
  • 24. tz . KTFIHF ( a # At ) if 8 a I # FK Bas't s 5-23 . • XC s ' ) = 4 - 6+4 = 2 . • • • He I # 2 gy , eCTs2 ) = 0 s if 1274¥ An . tkbtr theft 1 sy Ts2 If FIAA itsu ! - to , XCT ) = 0 g) , t*hL ECTT ' ) = 0 { ta2n3 . ( It ) iaiz $15 TTZ of EFAA iTa3 ' - st kFBt2u . ) ni 7th # a Et AAHI a TXFTEH't HE a- a A # at ,A¥h4 ) 2£23 AE'¥E A H¥skt= !
  • 25. ( Fun { Hina Eatx ) - 7¥ 'HE ¥k , E. a ! EID # In it to { ÷zGH=n izt ' teeth h3 . . . ( Ftxk Ethic ,zH2 ! ) - cxl A¥¥ Ai I ÷*d ... ( SW #a . Chern#aaA¥# Air Er Hit ) he " k¥57 a chtttn { R > It 5k£ ?
  • 26. HEE )
  • 28. per ¥ Et night # IT : E → M A " # AA b 'i5b ' # 2312 ; non - zero section s : M → Eo I zkzdlz # HE . # E Rr i : - Mx{o } : 11h ftp.lntznjk -23 . it ? If : S µ EH #aAt£5€53h s It FTFSFZ ( A=aAita3t=aaXT¥FHt )
  • 29. # EE As, r=2 I M HE# FIFA'¥'t Ikzeisza . sz ¥¥th#lz2£ 's . ( o ) Fai Ma Jake. As a # 95k M → Eo zhF3 . ;•,⇐# EPi , 1 1 ' - Mx{ 0 } "Ytiltii' ' l ' . ' ' ' ' , 1 . 1 i , , i t i , I I 1 itufAsI ML, ' , 1 1 1 1 1 1 • ' 1 *. :-/, =kE #a5¥u Its 's a # both M " '→Eo { 94-3 . t.sk the #a3⇐acz A±A5n. - .
  • 30. # baa, #a5Eu it YBK Faa Eb ' ? t#• ⇐ • s ( ) ¥¥# " " " } • OK of ) " " a FX . Fy. • • IFFFH, a TIEKF * Mx { o } e # Litt 's e . ¥42 it IHNI't KFFAFE it An !
  • 31. # EE { it ' Ma # IFAFI's little , EE'Re bi Mx{o } a As DE FI # F3 A { EAF P¥f#£ c : cz ( µ ) → 2 2¥23 . i. ⇐ . as E • • 2 T- • 1 ¥14×{0 } . ° • - 1 sf • : • PH. a• c = o 135 it " Ehn FEEEHFEAE ! at 0 not , Eitik s z FEBEAFZ PEETE ( obstruction ) 4 ' 'ta3 .
  • 32. # EE in F '¥#x c : GCM ) → 2 IF 2- cochair 2ta3 . ±5k Sc(D3 ) = cc D8 ) = 0 ztay , 2- cocyde 2ta3 . i. € . as E • • 2 T- • 1 ;t.to:: • < • YT#ML• Letsie [ c ] EHYM ;2 ) an 's cohomology '¥kbtek3 . c it s NEK ' ) 'FkhFc3bt , [ c ] rshfustan ! ( EaAbI¥± )
  • 33. - #k¥F# the Hit 5kt a 9th # a : E # M re HL , E n non - zero section s : M → Eo It M a v. 1 t.hn#a=tiaEeFhh4i=FFtathtiiI3 . stir #aFE £iF¥aAi it 3M£ a¥s 1 PEE '¥¥k [ c ] e HRCM ; 2) k£2 K£3 . > FT , Is : Mcr , → Eo [ c ] = 0 T c=o tss { a££FEBEaiE3 . A ' ht ' ' ) 'I > ! [ c ] =o 175 , - ¥4Ey£iE3izz ' ( r¥#a£ifEiEiE3 ! cost # au )
  • 34. 4 . 4¥ 't'¥¥B a " I # "
  • 35. Euler #A a Ih# HE ## A -±hfH5hk n' 7th # t : E * M KJHL . E a Euler ¥72 ECE ) e HRCM ; 2) IF , E a non . zero section E hF3Eh a * np¥a'¥¥A e - Exa3 ! CEIIAAIIE ) = KK £ , 2 Euler ¥52 { hE¥x at En . f3{ . ( E :#a A ⇒ ECE ) . . 0 ] if ha¥xb 's AA 5101473.
  • 36. EL F¥k r=dimM at¥h¥ . s : M → Eo e (E) =0 ' Ee ! . :) Ma ✓ Manta it M { atafib 's . B [ TI En # kktitvkt Xtistutsti 'E3 . : ) rank ( TM ) = dim M F) , In # HA ' As ' I 'Is . XCS ' ) = 2 F) easy to Ita 's , 5 I KIF non - zero vector field ttzktsu . DK
  • 37. Stiefel . Whitney #ZaIh# a- EL FEEE ninth # E * M ⇐ HL , f , T • . - - → Ea k - FFKYE VKCE ) E it E In k ] a ! 12k$ '&'IHA9kka¥E $ 's # 373mi ' . # . I i M p E a kikStiefel - Whitney ¥57, WRCE ) e HKCM ; 22 ) If , Vr. kt , (E) a MK 't a section ehF3Eh a * 1p¥IH÷**k ( mod 2) s - ZQFZ .
  • 38. % WICE ) = 0 c⇒ E it A IHTHFIFAE . :) W , (E) = 0 # M ' ' ' ± zi Vr . it , (E) = VRCE ) a section lot 2123 . c ⇒ M " ' s F¥aAih3E±a sink - 7012€ , A⇐4eEs⇒. i. ¥hEI3 Ea ¥± # A ek3 . ← AEHAFEKXZHE's . ( ⇒ Mta KEE. ah . s°±i . ftp.g#$ygy,y*⇐&* ⇒ " ¥453 Ea¥± # signs . #
  • 39. Chern #2aIh* 4ft # KEEFE ninth # E Is M EHL , E a kBa¥, k - HER VKCE ) at , E In k ] a 1 - 12km , 'ItaA9Hua4E a 's Ai 3 73mi ' - # . E a kihChan ¥57, CKCE) e HYM ; 2) It , Vr . kt , (E) a MMI a section ktthf * 1 EE '¥*k e - 2ixF3 .
  • 40. ( £4 c ¥iF¥5 ) - 4¥ HE # a it n' 7th # a Btk¥E . > Fy ' ' EFAAHEFZEEVH ' ' { tttttu ⇒ , Ali 's { FEI , -43 FED #x state ! - ZEZE 3 Th Eat ' - EE Ka kER¥7 At A 31 , by " ) E #E4IF3 F¥f**k # took hEhia3 Fkttai , { it 4¥ 'tE#I ol ' ta5hn3 at 22€ AEK . - n' 7th 5ha Bnikb " It #sFaA a ] # Eaii - EF a 4h Task R3 aii FIFA tabspart at kit ¥-445 A 9th # c b ' Ft # Liliana €45613 . - # in 114+714 Blitz " 3 AIFAF a Iitt . " h543 faith " to "¥nERit3 . → 6^(18 ) : F¥PQzkE Grossmann 957.3kt# , A 7th Tan B¥A # Part
  • 41. ¥,¥¥e Characteristic Classes Milnor, Stasheff 微分位相幾何学 田村一郎 Lecture Notes in Algebraic Topology Davis, Kirk available online!