• Aucun résultat trouvé

{N ord, Sud, Est, Ouest}

N/A
N/A
Protected

Academic year: 2022

Partager "{N ord, Sud, Est, Ouest}"

Copied!
32
0
0

Texte intégral

(1)

! "## "$%# &'(#!

)* +

# ,--.

/0123 435 60789:35

; <=>?=@ AB CD>=ECFFGHIJCKG ;

LML NOPQRSRTQ UVW XYZY[VW Y\ST]YWWV^_`YQSW M M M M M M M M M M M M M M L

LM a bcV^X`VW UdYWWV^_`Y[VW M M M M M M M M M M M M M M M M M M M M M M M e

LMf ghYZY\c XhOiOUVQSW M M M M M M M M M M M M M M M M M M M M M M M M M M j

LMfML kY`RUYSRTQ VcXOhR^VQSY`V U\ ^TUl`V M M M M M M M M M M M M j

LMfMa mOQOhYSRTQ UVnTh^VW\QRo\VW M M M M M M M M M M M M M M M M j

p q=H=>rs>?GF t

aML uVWvT^TSvOSRVW VS `Y QTSRTQ UV UwQY^Ro\V M M M M M M M M M M M M x

aM a yTQURSRTQ zy VS iTQWSh\iSRTQ Ud\QVvT^TSvOSRV M M M M M M M M M M La

aMf {Q iT^XR`YSV\h UdvT^TSvOSRVW YXXhTivOVW M M M M M M M M M M M M M Lj

aMfML kVhW `dY\ST]WR^R`YhRSO M M M M M M M M M M M M M M M M M M M M M aL

| }GF JC@KCKGF ~=HGF FC>DsGF €=HHG H=AJG AG €CJ€DJ p|

‚ ƒGF„G€>?…GF pt

eML †YZY[VW U\ X`YQ M M M M M M M M M M M M M M M M M M M M M M M M M M M ax

eM a gY\c UV XYhY``O`RW^V M M M M M M M M M M M M M M M M M M M M M M M M M ax

eMf b‡VSW [OT^OShRo\VW M M M M M M M M M M M M M M M M M M M M M M M M M M aˆ

(2)
(3)

478

uVW XhTiVWW\W UdY\ST]YWWV^_`Y[V WTQS T^QRXhOWVQSW UYQW `Y QYS\hV o\V iV

WTRSXT\h`Y nTh^YSRTQ UVihRWSY\cT\ UdTh[YQRW^VWZRZYQSW XYhSRhUdNMuVW

ShYZY\c UV RQnhVV jUdUV`^YQL VS UVzTSvV^\QU eTQUTQQO\Q

iYUhVSvOThRo\V iVSSVQTSRTQiV`\RUVWXYZY[VWY\ST]YWWV^_`YQSWM`WTQSOS\URO

`VW`RVQW VQShViVSSVQTSRTQ SvOThRo\VVS UVWhOW\`SYSWVcXOhR^VQSY\cMudT_VSUV

iV WSY[V OSYRS UdOS\URVh \Q XV\ X`\W VQ XhTnTQUV\h `Y SvOThRV UV iVW XYZY[VW

Y\ST]YWWV^_`YQSWM

uVW ShYZY\c XhOiOUVQSW WdYSSYivYRVQS `dYWWV^_`Y[V UV nTh^VW \QRo\VW VS

ivVhivYRVQSUTQihVQUhV`VXhTiVWW\WUOSVh^RQRWSVMT\WQT\WWT^^VWYSSYivOW

`Y [OQOhYSRTQ Ud\Q `YQ[Y[VUV nTh^VXhTiVWW\W QTQ UOSVh^RQRWSV iV o\R QT\W

Y XVh^RWUVUO[Y[Vh`V h`VXYhSRi\`RVhT\OXYh `VWvT^TSvOSRVWo\R XVh^VSSVQS

UV UOPQRh \QV QTSRTQ UV iT^X`VcRSOM T\W QT\W WT^^VW Y\WWR XYhY``l`V^VQS

RQSOhVWWOW `Y QTSRTQ UV o\R XVh^VS UV iYXS\hVh`Y QYS\hVXYhY``l`V

UV`dY\ST]YWWV^_`Y[VM

NYQW iV hYXXThS QT\W ZVhhTQW ST\S UdY_ThU `Y QTSRTQ UV UwQY^Ro\V VS UV

ShYQWnTh^YSRTQ UV UwQY^Ro\V o\R QT\W XVh^VS UV \[Vh WR UV\c YWWV^_`Y[VW

XYh UVW V\c UV S\R`VW UR‡OhVQSW WV nTQS UV ^YQRlhV iT^XYhY_`VM udOS\UV UV

iVSSV UwQY^Ro\V VS QTQ X`\W WR^X`V^VQS UVW nTh^VW VQ[VQUhOVW XYh `dY\ST]

YWWV^_`Y[V XVh^VS UV iYhYiSOhRWVh `VW ShYQWnTh^YSRTQW o\R XhOWVhZVQS `Y nYTQ

UTQS `VW XYZY[VW WdYWWV^_`VQSM†Yh^R `VW ShYQWnTh^YSRTQW UV UwQY^Ro\VWQT\W

OS\URVhTQW X`\W XYhSRi\`RlhV^VQS `VW vT^TSvOSRVW o\R WTQS `VW X`\W WR^X`VW UVW

ShYQWnTh^YSRTQW QTQ ShRZRY`VW iV o\R QT\W XVh^VSShY UV ZTRh o\V `Y QTSRTQ UV

`TiY`RSO QdVWS XYW WYQWXTWVh UV XhT_`l^VWM

T\W ZVhhTQW VQW\RSV iT^^VQS `VW XYZY[VW Y\ST]YWWV^_`YQSW XV\ZVQS ShV

\SR`RWOW iT^^V ^TUl`V UV iY`i\`M†T\h iV`YTQ ivVhivVhY iYhYiSOhRWVh `VW VQ]

WV^_`VW UV nTh^VW XT\h `VWo\V`W R` VcRWSV \Q XhTiVWW\W UdY\ST]YWWV^_`Y[V VS

QT\WZVhhTQWo\V`VWivYQ[V^VQSWUdOivV``VidVWS URhV`VWvT^TSvOSRVW XVh]

^VSSVQS UV WV hYXXhTivVh UVW ^TUl`VW UV iY`i\` i`YWWRo\VW g\hRQ[]X\RWWYQSWM

udYXXhTivV XYh `YQ[Y[V X`\SS o\V XYh [OQOhYSRTQ UV nTh^VW \QRo\VW XVh^VS

UdT_SVQRh UVWhOW\`SYSW X`\W WR^X`VWo\V `VW hOW\`SYSW UO iTQQ\WM

bQPQ QT\W UTQQVhTQW \Q YXVh\ UVW W\RSVW XTWWR_`VW UV iV ShYZYR` QT]

SY^^VQS UYQW `dOS\UV UVW XYZY[VW UV ST\S `V X`YQ UTQS TQ XVQWV o\V `dY\ST]

YWWV^_`Y[VXVh^VS UV UO[Y[VhiVhSYRQVWXhTXhROSOWMbQ XYhSRi\`RVhTQ XV\SZTRh

`dY\ST]YWWV^_`Y[ViT^^V \Q iY`i\` XYhY``l`VShlW YWwQivhTQVVS YRQWRSVQSVh UV

UO[Y[Vh UVWXhTXhROSOWUV !" #$ UVW XYZY[VWM

% &

78 4'07(05536120)3

*+* ,-./0102/ 345 6787945 7:12;7554<=>7/15

uVW XYZY[VW Y\ST]YWWV^_`YQSW WTQS \Q ^OiYQRW^V UV iY`i\` UOPQRW XYhSRh

Ud\Q V\ UV S\R`VM yVV\ UV S\R`VW VWS \QV\ UVS\R`VW UV YQ[ o\R WdYWWV^_`V

VQ \QXYZY[VMuYXYhSRV Y\ST]YWWV^_`Y[VYXXYhY?SUYQW`Y ^VW\hVT@`VWS\R`VW

(4)

Y`[ThRSv^VXT\h`VW YWWV^_`VhM

kTwTQW UdY_ThU iT^^VQS WdYWWV^_`V\Q XYZY[VUV YQ[M

s@?>?=@ ;

GD AG >D?JG AG C@K !#

C

"!

# # !"#!

{N ord, Sud, Est, Ouest}

C

# # #

†T\h ivYo\V S\R`V

t

TQ QTSVhY

t.N

XT\h

N ord(t)

VS UV ^^V

t.S, t.E, t.W

XT\h hVWXViSRZV^VQS

Sud(t)

Est(t)

Ouest(t)

s@?>?=@ p =HG

! # !"#!

f : Z 2 → {0, 1}

#

f −1 (1)

!# # !

f −1 (1)

! #

! # # #!

! # !"

!#

Z 2

s@?>?=@ | =>?~ !# # !"#! #

Z 2

#

J

# !# $ # #!

!# !# !"#! # " "

! !# # " !#

Q

YXXV``VhY `R[QVW `VW VQWV^_`VW UV XTRQSW UV

Z 2

UOPQRW XYh \QV Oo\YSRTQ

U\ SwXV

y = a

VS iT`TQQVW iV\c UOPQRWXYh \QVOo\YSRTQ

x = a

M

s@?>?=@ ‚ ƒC…CKG „C AGF >D?JGF AG C@K !#

J

#

#

m

!# !

m

# "!#!

J

! #!#

(x, y)

!#

m

!

m(x, y).N = m(x, y + 1).S

!

m(x, y + 1)

# $

m(x, y).S = m(x, y − 1).N

!

m(x, y − 1)

# $

m(x, y).E = m(x + 1, y).W

!

m(x + 1, y)

# $

m(x, y).W = m(x − 1, y).E

!

m(x − 1, y)

# $

Q

R^XTWV UTQi o\V `VW iT\`V\hW XThSOVW XYh UVW iSOW YUYiVQSW UVW S\R`VW

iTQiRUVQSM NV SV``VW iTQShYRQSVW UV XYZY[VW XVh^VSSVQS UdVcXhR^Vh UV ^YQRlhV

`TiY`VUVWiTQShYRQSVW[`T_Y`VWW\h`VW^TSRnWT_SVQ\WMyVXVQUYQSTQ QVURWXTWV

XYW UYQW `VV\ UV S\R`VW Ud\Q Y`[ThRSv^V URWYQS iT^^VQS `dYWWV^_`VhM yVS Y`]

[ThRSv^V OZVQS\V` VWS VcSOhRV\h Y\ XYZY[VM udRUOV UV `dY\ST]YWWV^_`Y[V VWS UV

iTQWSh\RhV\QV UVWihRXSRTQ UV`dY`[ThRSv^VVQ ^^V SV^XW o\V `dTQ YWWV^_`V`V

XYZY[VM†T\h iV`YTQ YT\SV Y\c iT\`V\hW \QV QTSRTQ UViT``VW

s@?>?=@ GD AG >D?JGF CD>=ECFFGHIJC@>F # #!!

# # # !"#! "!

"" "! !" # iT``V "! #"! #$

# nThiV "! #

bQ X`\W UV iVW iT``VW `dY\ST]YWWV^_`Y[V VWS hO[R XYh \Q XYhY^lShV [`T_Y`

`Y #$#M` WdY[RS Ud\Q VQSRVhMudY\ST]YWWV^_`Y[V Ud\QV nTh^V WV nYRS XYh

`dRQSVh^OURYRhV UVShYQWRSRTQWM

(5)

WVQSVh`VWV\cUVS\R`VWY\ST]YWWV^_`YQSWMP[\hVLyvYo\VS\R`VVWShVXhOWVQSOV

XYh\Q iYhhOM uVQT^_hV UV [T\SSVW W\h ivYo\V iSOiThhVWXTQU `Y nThiVUV

`Y iT``VVS `V\h iT\`V\h `Y iT\`V\h UV`Y iT``V

Q YT\SVhY XYhnTRW\Q Ww^_T`V

T\ \QV `VSShV W\h `VW [T\SSVW XT\h XhOiRWVh `Y iT\`V\h UV `Y iT``VMuY iT\`V\h

UVWS\R`VWV``VW]^^V QdY XYW UV WR[QRPiYSRTQ XYhSRi\`RlhVM

L hVXhOWVQSYSRTQ UVWS\R`VW Ud\Q XYZY[V Y\ST]YWWV^_`YQS

s@?>?=@ C@F?>?=@ !#

c

#

c 0

"!#!

J

# #!! # #!

c

c 0

#

(x, y)

#

c

#

c 0

"!"#

(x, y)

#

c 0 (x, y)

!#

! !" "! !# "#$ "#

(x, y)

c 0

!# $ #$# #! "! "#$ "

!## "! !"

0

$

yT^^V

c

VS

c 0

UTRZVQS ShV UVW iTQP[\hYSRTQW TQ QdRQShTU\RS XYW UdVhhV\hW

W\h `VWiT\`V\hWM

RQWR`dY`[ThRSv^VUdYWWV^_`Y[VUOPQRXYh `dY\ST]YWWV^_`Y[VVWS`VW\RZYQS

UYQW\QViTQP[\hYSRTQhVXOhVh `VW WRSVWYiSRnW idVWS URhViV\c XT\h `VWo\V`W

`Y WT^^V UVW iT``VW W\h `VW iSOW YU YiVQSW UOXYWWV `Y SV^XOhYS\hV VQ ivTRWRh

\QivTRWRh \QVS\R`V iYQURUYSVXT\h iVWRSV idVWS URhVQdRQShTU\RWYQSXYWUV

URWiThUYQiVUYQW`VWiT\`V\hWVS`dYT\SVhMydVWS\QY`[ThRSv^VQTQUOSVh^RQRWSVM

NYQW ivYo\VV\ UVS\R`VWTQ URWSRQ[\V\QVS\R`V XYhSRi\`RlhV`Y WT\hiVM Q

URhY o\d\QV iTQP[\hYSRTQ

c

VWS YSSVR[QY_`V WdR` VcRWSV \QV W\RSV UV ShYQWRSRTQW

XYhSYQSUV `Y iTQP[\hYSRTQ

[(0, 0) 7→

`Y WT\hiV

]

VS YhhRZYQS

c

M

yVSSV OZT`\SRTQ Ud\Q XYZY[V Y\ST]YWWV^_`YQS XYhSRh UV `Y iTQP[\hYSRTQ

hOU\RSV `Y WT\hiV VWS YXXV`OV `Y UwQY^Ro\V U\ V\ UV S\R`VWM bQ ZTRiR \QV

UOPQRSRTQ nTh^V``V o\R QT\W XVh^VSShY XYh `Y W\RSV UV iYhYiSOhRWVh `V nYRS o\V

UV\cV\c UV S\R`VW WV iT^XThSVQS UV^YQRlhV YQY`T[\VM

s@?>?=@ @CH?DG !#

J

# !#

E

! ##

J

E

" !## #

# ! !#

$ # !# $ UwQY^Ro\V

J

" !#

yVSSVZRWRTQWT\WnTh^VUVUwQY^Ro\VWXVh^VSUdT_SVQRh\QiVhSYRQWQT^_hVW

UV `V^^VW o\R QT\W hVQWVR[QVQS W\h `Y nYTQ UTQS WV UOhT\`V `dYWWV^_`Y[VUd\Q

(6)

\SR`VM

}GHHG;

!#

J

# #

D

!#

b

#

b 0

""

D

## "!#!

c

# # "!!

#!

c 0

! # #! "!! #

b

#

b 0

!# !

$! ##! R UV\c_hYQivVWiTQ\VQSV``VWTQSQOiVWWYRhV^VQS `VW ^^VW

ShYQWRSRTQW `dThUhV XhlW X\RWo\V TQ QV XV\S Y^YRW VQ`VZVh UV S\R`VW VS o\V

WR V``VW YhhRZVQS Y\ ^^V ^TSRn V``VW TQS XTWO `VW ^^VW S\R`VW Y\c ^^VW

VQUhTRSWM

udYT\S UV iT``VW XVh^VS UV iTQSh`Vh `dYWWV^_`Y[V VS UV hVWShVRQUhV `dVQ]

WV^_`VUVWnTh^VWT_SVQ\VWXYh \QV\ UVS\R`VW UTQQOM` XVh^VSUTQiUdT_SVQRh

\Q Y`[ThRSv^V UV UVWWRQ UV nTh^VW XYhSRhUd\QV\ UV S\R`VW UV YQ[

*+ 4<6>45 37554<=>7945

GH

„JG JGF €CsF

kTwTQW \Q VcV^X`V Ud\QV iTQWSh\iSRTQ Ud\Q `YQ[Y[V

UVnTh^VWM` WdY[RS UV`Y iTQWSh\iSRTQ U\ `YQ[Y[VUVW iYhhOWUV zO^R`Y LLMQ

ivVhivV \Q V\ UV S\R`VW

J

o\R UTQS `VW iTQP[\SYSRTQW PQY`VW SV^XOhYS\hV

2

WTRVQS UVW iYhhOWM Q ZV\S Y\WWR o\V XT\h ST\S VQSRVh

s ≥ 2

`V iYhhO UV SYR``V

s

WTRS `Y nTh^VUd\QV iTQP[\hYSRTQ PQY`VUViVSSV UwQY^Ro\VM

uVV\ UV S\R`VW o\R `V XVh^VS VWS `V W\RZYQS in P[\hV aR` WV iT^XTWV UV

ShTRW SwXVW UV S\R`VW UVW S\R`VW UV hV^X`RWWY[V h UVW S\R`VW o\R ZTQS WVhZRh

ShYiVh `Y URY[TQY`V U\ iYhhO UVS UVW S\R`VW _T\ivTQW _M

a V\ UV S\R`VW o\R YWWV^_`V \Q iYhhOM uY S\R`V YZVi \QV OSTR`V VWS `Y

WT\hiVM UhTRSV`dYWWV^_`Y[VT_SVQ\M

_

WVhZTQW`dYWWV^_`Y[VUd\QVP[\hV TQ XYhSUV`Y S\R`VWT\hiV iV``Vo\RVWS

^Yho\OV Ud\QV OSTR`V UYQW iVSSV iTQP[\hYSRTQ`VW WV\`VW WRSVW XTWWR_`VW XT\h

`dYT\S Ud\QVS\R`V WTQS

(1, 0), (−1, 0), (0, 1), (0, −1)

M†Yh^R iV\c]iR`VWV\` YT\S

o\R XV\S WV XhTU\RhV VWS iV`\R Ud\QV S\R`V

d1

VQ

(0, 1)

MbQ V‡VS ST\W `VW Y\ShVW

iSOWUV`Y WT\hiVXThSVQSUVW iT``VWUVnThiV

1

o\R QV W\hTQS XYW nYRhViT``Vh

(7)

\QV S\R`V X\RWo\V `Y SV^XOhYS\hV VWS UV

2

M \h iV iSO idVWS _RVQ \QV S\R`V

d1

o\R iT``ViYh idVWS `Y WV\`V YZTRh `Y _TQQV iT\`V\h W\h WTQ iSO \UM uVWS\R`VW

d

VS

b

WTQS `VW WV\`VW YZTRh \QV iT``V UV nThiV

2

UTQi `VW WV\`VW XT\ZTRh ShV

X`YiOVW UYQW \QV `R[QV T\ \QV iT`TQQV T@ R` Qdw Y XYW VQiThV UV S\R`VWM Q

Y

UTQiUV\c YWWV^_`Y[VWVQXYhY``l`V Ud\QVXYhS`VWS\R`VW

d

UVWWRQVQS\Q R[ Y[

`V `TQ[ UV `Y UhTRSV

x = y

M` VWS XTWWR_`V ivYo\V OSYXV T@ `dTQ UVZhYRS X`YiVh

\QV S\R`V

d2

UV `Y hV^X`YiVh XYh \QV S\R`V

b

Y\o\V` iYW `Y iTQWSh\iSRTQ UV `Y URY[TQY`V WdYhhSVM NdY\ShV XYhS o\YQU `Y URY[TQY`V VWS ShYiOV\Wo\dVQ

(a, a)

UYQW `Y TQV

[0 ≤ x ≤ a; 0 ≤ y ≤ a]

iVhSYRQW V^X`YiV^VQSW `R_hVW WdR` w VQ Y

TQS UV\c ZTRWRQW Tii\XOW XYh UVW S\R`VW

d

T\

r

M iVW V^X`YiV^VQSW TQ XV\S

X`YiVh \QV S\R`V

r

M

uY nTh^V Ud\QV iTQP[\hYSRTQ PQY`V VWS UTQi \Q iYhhO UTQS `V iSO VWS UO]

SVh^RQO XYh `Y `TQ[\V\h UV `Y URY[TQY`V idVWS URhV `V QT^_hV UV nTRW T@ TQ Y

X`YiO \QV S\R`V

d2

X`\SS o\d\QV S\R`V

b

M YQW `VW iT``VW idVWS URhV WR `dTQ QV

iTQWRUlhVo\V`VV\ UVS\R`VW UV YQ[`VW YWWV^_`Y[VWXTWWR_`VWWTQS_RVQ X`\W

QT^_hV\cX\RWo\V hRVQ QdV^XivV UVX`YiVhUVW S\R`VW

r

UVW VQUhTRSWT@ V``VW

QdTQS o\d\Q ZTRWRQiV o\R XVh^VSUVWThSRhU\ iYhhO`R^RSO XYh `Y URY[TQY`VM

GH

„JG ?G„?@

F

udYT\S UVW iT``VW XVh^VS Y\WWROSYQS UTQQO \Q V\

UVS\R`VWUV YQ[ o\RXYZV`VX`YQUVWdYWW\hVho\V`dTQRQShTU\RhY XYWUdRQiTvO]

hVQiVWVQ`dYWWV^_`YQSMbQ iVWVQW`dY\ST]YWWV^_`Y[VUTQQV\Q Y`[ThRSv^VXT\h

XYZVh `V X`YQ YZVi \Q V\ UV S\R`VW UV YQ[ UTQQOM kTRiR P[\hV f \Q V\ UV

S\R`VWY\ST]YWWV^_`YQSVWXVh^VSSYQSUVXYZVh`Vo\YhSThU]bWSU\ X`YQYZVi\Q

^TSRn UVShRYQ[`VUV RVhXRQWwMyVV\ UV S\R`VW Y OSO RQShTU\RS XYhzTSvV^\QU

UYQW aM

TQShTQWST\SUdY_ThUo\VST\SXYZY[ViThhViSU\o\YhSUVX`YQ

x ≥ 0, y ≥ 0

XYh iVV\ UV S\R`VW "! # UVWWRQV `V ShRYQ[`V UV

RVhXRQWw WR `dTQ iT`ThRV`VW S\R`VW UV`Y ^YQRlhV W\RZYQSV `VW S\R`VW

u 10

VS

u 01

VQ QTRh VS `VWS\R`VW

z 00

VS

z 11

VQ _`YQiM

_WVhZTQW o\V `V ShRYQ[`V UV RVhXRQWw WdT_SRVQS XYhSRh U\ ShRYQ[`V UV

†YWiY`VQiT`ThRYQS`VWQT^_hVWVQnTQiSRTQUV`V\h XYhRSO R^XYRhWVQQTRhXYRhW

VQ _`YQiM †T\h iTQWSh\RhV `V ShRYQ[`V UV RVhXRQWw TQ ZY UTQi iTQWSh\RhV \Q

ShRYQ[`V UV†YWiY` VQ nYRWYQS `VW YUURSRTQW^TU\`T

2

M

†T\h iV`YTQ iTQWRUlhV \Q XYZY[V U\ o\YhS UV X`YQ XYh iVV\ UV S\R`VW

XV\ R^XThSV iT^^VQS R` Y OSO YWWV^_`OX\RWo\V `dTQ iTQWRUlhV `VV\ UV S\R`VW

iT^^V \Q V\ UV YQ[M uY `R[QV

y = 0

VS `Y iT`TQQV

x = 0

WTQS hViT\ZVhSVW

UVS\R`VW

v

VS

h

M uV hVWSV U\ o\YhSUVX`YQ

{(x, y)|x ≥ 0, y ≥ 0}

VWS hViT\ZVhSUV

S\R`VW

u

VS

z

M

Q XV\S YWWTiRVh ivYi\QVUVW S\R`VW

u

VS

z

\QV YUURSRTQ ^TU\`T

2

z 00 0 + 0 = 0

u 01 0 + 1 = 1 u 10 1 + 0 = 1 z 11 1 + 1 = 1

yvYi\QV UV WVW S\R`VW XhVQU `VW Yh[\^VQSW UV `dYUURSRTQ o\dV``V hVXhOWVQSV

W\h WVW iSOW W\U VS T\VWS VS XThSV \QV iT\`V\h iThhVWXTQUYQS WTQ hOW\`SYS

(8)

f V\ UV S\R`V o\R YWWV^_`V \Q ShRYQ[`V UV RVhXRQWwM udYWWV^_`Y[VVWS

hVXhOWVQSOYZVi\QVhTSYSRTQ UdYQ[`V

+π/4

MuVW S\R`VW WTQSUYQW `dThUhV`Y S\R`V

h

`Y WT\hiV`Y S\R`V

v

`VW S\R`VW

u 01

z 00

u 10

z 11

W\h WVW iSOW ThU VS bWSM yT^^V `VW iSOW ThU UVW S\R`VW

h

VS bWS UVW S\R`VW

v

XThSVQS `Y iT``V iThhVWXTQUYQS

1

\Q XYZY[V XYh WVW S\R`VW iTQSRVQS _RVQ `Y

iTQWSh\iSRTQ U\ ShRYQ[`V UV†YWiY` UYQW

Z/2Z

M

uY WXOiRPiYSRTQ UVW iT\`V\hWUVW iT``VWW\SUTQi R^XTWVh o\V`VWXYZY[VW

T_SVQ\W YZVi iVV\ UV S\R`VW WTRVQS `V ShRYQ[`V UV RVhXRQWwM bQ YT\SYQS UVW

iT``VW TQ T_SRVQS \Q Y`[ThRSv^V XT\h YWWV^_`Vh iV ^TSRnM udY`[ThRSv^V UTQQV

ivYo\VOSYXV \Q VQWV^_`V UdV^X`YiV^VQSW

(x, y)

SV` o\V WR `dTQ w YT\SV \QV

S\R`V XYh^R iV``VW UTQS `VW iT\`V\hW iThhVWXTQUVQS iV``VW UV iVW ZTRWRQVW UO

XTWOVW TQ QdRQShTU\RS XYW UdVhhV\h UYQW `dYWWV^_`Y[V idVWS URhV o\V WR `V

^TSRn VWS iT^X`OSY_`V XT\h UTQQVh `V ShRYQ[`V UV RVhXRQWw R` `V hVWSV YXhlW

YT\S UV `Y S\R`VM

udY`[ThRSv^VVWS`V W\RZYQS

h

y = 0

v

x = 0

u

z

yVS Y`[ThRSv^VVWS VQiTUO UV `Y ^YQRlhV W\RZYQSV UYQW `VW S\R`VW `VW S\R`VW

h

TQS UVW iT``VW UVnThiV

2

W\h `V\hW iSOW bWS VS \VWS iYW L`VW S\R`VW

v

TQS

UVW iT``VWUVnThiV

2

W\h`V\hW iSOWThU VS\U iYWa`VWS\R`VW

u

VS

z

TQSUVW

iT``VWUVnThiV

1

W\h ST\W `V\hW iSOW iYW fM

iR TQ Qd\SR`RWV XYW `VW iT``VW XT\h ivYQ[Vh `dVQWV^_`V UVW ^TSRnW o\dR` VWS

XTWWR_`V UdYWWV^_`Vh ^YRW XT\h WdYWW\hVh UV `dYWWV^_`Y[ViThhViS UVW ^TSRnW VQ

o\VWSRTQM

(9)

s@?>?=@ AGF „C…CKGF CD>=ECFFGHIJC@>F uVW XYZY[VW Y\ST]YWWV^_`YQSW

TQS OSO RQShTU\RSW XYh RQnhVV UYQWWY SvlWVj iT^^VVcV^X`V UV^OiYQRW^V

XVh^VSSYQSUVhOY`RWVhUVWiY`i\`W\QRZVhWV`WYZViUV`dNM` iT^XYhV`V^OiY]

QRW^V UV`dY\ST]YWWV^_`Y[V iV`\R UV `Y nTh^YSRTQ UVW ihRWSY\c VS RQShTU\RS `Y

o\VWSRTQUVWnTh^VWT\^TSRnWY\ST]YWWV^_`Y_`VWM`OS\URVVQW\RSV`YnYRWY_R`RSO

U\ ^TUl`VVS WTQ hYXXThSYZVi`dY\ST]YWWV^_`Y[VUV ^T`Oi\`VW UdNMudT_VS

UV WTQ ShYZYR` VWS UV ^TQShVh o\V `dTXOhYSRTQ UV #! UVW ^T`Oi\`VW UdN

VWSiV``Vo\RXVh^VS UVhOY`RWVhUVWiY`i\`W \QRZVhWV`WX`\SS o\V`VWTXOhYSRTQW

UV WOXYhYSRTQUV n\WRTQ VS UdY^X`RPiYSRTQ o\R OSYRVQS \SR`RWOVW\Wo\dY`ThWM

NVX`\WUYQWWYSvlWV RQnhVVOS\URV`Vh`VUV`YUR^VQWRTQUV`dVWXYiVUYQW

`Y X\RWWYQiVUV`dY\ST]YWWV^_`Y[VM`^TQShVo\V`dY\ST]YWWV^_`Y[VVWS \QRZVhWV`

VQ UR^VQWRTQ

2

VS X`\W^YRW o\dVQ UR^VQWRTQ

1

R` Y `Y ^^V X\RWWYQiV o\V`VW

Y\ST^YSVW PQRW `VW`YQ[Y[VWY\ST]YWWV^_`OWVQ UR^VQWRTQ

1

WTQS hO[\`RVhWM

NYQW WY SvlWVe zTSvV^\QU Y Y\WWR OS\URO `V h`V UV `Y SV^XOhYS\hV W\h

`VW YWWV^_`Y[VWM` Y ^TQShO o\V `Y SV^XOhYS\hV

1

XVh^VSSYRS UdT_SVQRhY\SYQS

o\V `Y SV^XOhYS\hV

2

UVW WR^\`YSRTQW UdY\ST^YSVW iV``\`YRhVW ^YRW XT\h \QV

QTSRTQ UV WR^\`YSRTQ UR‡OhVQSV `Y WR^\`YSRTQ Ud\Q iY`i\` iThhVWXTQU QTQ XYW

`dYWWV^_`Y[V Ud\Q ^TSRn ^YRW \Q `YQ[Y[V UV ^TSRnWM ` ^TQShV Y\WWR o\V

XT\h `Y iTQWSh\iSRTQ UV nTh^VW `Y SV^XOhYS\hV

2

VWS _RVQ X`\W ViYiV o\V `Y

SV^XOhYS\hV

1

MNV X`\WidVWS SV^XOhYS\hV

2

o\dYXXYhYRWWVQS`VW XhT_`l^VWUV

WwQivhTQRWYSRTQVSUVXhRThRSOo\VQT\WYZTQWOS\UROY\ShYZVhWUVWivYQ[V^VQSW

UdOivV``VM

; |; CJ?AC>?=@ G „

s?HG@>CJG AD H=AJG

uV ^TUl`V UVW XYZY[VW Y\ST]YWWV^_`YQSW Y OSO RQShTU\RS XYh RQnhVV o\R Y

^TQShO UYQW j o\dR` iThhVWXTQUYRS `dYWWV^_`Y[V UV ^T`Oi\`VW UdNM RQ]

nhVV VS Y`M UYQW ˆ ^TQShVQS XhOiRWO^VQS iT^^VQS hOY`RWVh UVW XYZY[VW Y\ST]

YWWV^_`YQSW XYhSRh UV ^T`Oi\`VW UdNVS o\V `V XYhY^lShV UV SV^XOhYS\hV

iThhVWXTQU `Y nTRW `Y SV^XOhYS\hV UV `Y WT`\SRTQ VS Y\c iTQiVQShYSRTQW R`

WdY[RS VQ nYRS Ud\QV iTQWSYQSV UV hOYiSRTQM` ^TQShV o\dVQ XhYSRo\V WV\`VW `VW

SV^XOhYS\hVW

1

VS

2

WTQS hOY`RWY_`VWVS o\V `Y SV^XOhYS\hV

1

hV\QV iVhSYRQV

`YSRS\UV o\YQS Y\c iTQURSRTQW VcXOhR^VQSY`VW Y`ThW o\V `Y SV^XOhYS\hV

2

VWS

X`\WUO`RiYSV T_SVQRhM

NV X`\WXT\h OS\URVh `VW hYXXThSW YZVi `dVcXOhRVQiV UYQW L VS x`V ^T]

Ul`V VWS OSVQU\ XT\h SVQRh iT^XSV UVW hOY`RSOW XvwWRo\VW `VWS\R`VW XV\ZVQS WV

iT``Vh ^^V WR `Y o\YQSRSO UV iT``V QdVWS XYW W\WYQSV ^YRW YZVi \QV nYR_`V

XhT_Y_R`RSOV``VW XV\ZVQS WVUOiT``Vh \QV nTRW PcOVWW\h `V ^TSRnMuY hT_\WSVWWV

UV iVW YWWV^_`Y[VW ZRW ZRW UV iVW VhhV\hW UOXVQU Ud\QV ZYhRY_`V `V #

$ o\R iThhVWXTQU \QV iVhSYRQV ViYiRSOVQ SV^XWUV`dYWWV^_`Y[VM

; |p s@sC>?=@ AG ~=HGF D@?DGF

uV XhRQiRXY` XhT_`l^VY\o\V` RQnhVV VS zTSvV^\QU WV WTQS RQSOhVWWOWWTQS

`VWXhT_`l^VWUV[OQOhYSRTQUVnTh^VWPQRVWMNYQW L UV`^YQ ^TQShVo\dTXSR]

(10)

^RWVh iVSSV[OQOhYSRTQVWS\QXhT_`l^V

N P

]iT^X`VSMuY hVivVhivVWdVWSnTiY`RWOV

W\h `Y [OQOhYSRTQ UV nTh^VW PQRVW VQ XYhSRi\`RVh `VW iYhhOWM T\W YZTQW ShYQW]

XTWO`VWhOW\`SYSWT_SVQ\W Y\iYWUV`Y[OQOhYSRTQ UV`YQ[Y[VVQXYhSRi\`RVh iV\c

iTQiVhQYQS `VW ivYQ[V^VQSW UdOivV``VM bQ XYhSRi\`RVhXT\h `VW `YQ[Y[VWTQ T_]

SRVQS \Q hOW\`SYS UV

N P

]iT^X`OS\UV XT\h`V XhT_`l^V UV `dYXXYhSVQYQiVUd\QV

nTh^VY\ `YQ[Y[VVQ[VQUhOXYh\QV\ UVS\R`VW\QXhT_`l^VX`\WWR^X`V o\V`V

XhT_`l^VUdTXSR^RWYSRTQ UV L M

uVWVcV^X`VWUVnTh^VWY\ST]YWWV^_`Y_`VWiTQQ\VWOSYRVQS`VWiT^XSV\hW`VW

iYhhOWUVSYR``VPcOVefYRQWRo\ViVhSYRQVWnY^R``VWUdYh_hVWo\RXVh^VSSVQS

UYQW L UV ^TQShVh `Y

N P

]iT^X`OS\UVU\ XhT_`l^V UVWnTh^VW VQ[VQUhOVWM

NYQW iV iYUhV `VW QTSRTQW UdViYiRSO OS\UROVW OSYRVQS `V QT^_hV UV S\R`VW

QOiVWWYRhVWXT\hT_SVQRh\QVnTh^VUTQQOVVS`VSY\cUVXYhY``O`RW^VUV`dYWWV^]

_`Y[VM uY o\VWSRTQ U\ QT^_hV UV S\R`VW QOiVWWYRhVW XT\h VQ[VQUhVh \QV nTh^V

UTQQOV Y OSO `ROV `Y iT^X`VcRSO UV T`^T[ThTZ UV`Y nTh^V XYh RQnhVV UYQW

hOW\`SYS o\V QT\W YZTQW OSVQU\ Y\c `YQ[Y[VWM NV X`\W iV hOW\`SYS ShYRSYRS

UVW nTh^VW^TU\`T vT^TSvOSRVM T\W YZTQW VcvR_O\Q `RVQ VQShV iT^X`VcRSO VQ

SV^XW Ud\Q `YQ[Y[V VS hYXXThSUVW vT^TSvOSRVWM

677835

udY\ST]YWWV^_`Y[VVWS \QVY_WShYiSRTQ UVW^OiYQRW^VWXvwWRo\VW ^RW VQV\

QTSY^^VQS UYQW `Y nTh^YSRTQ UV ihRWSY\cM yVW ^OiYQRW^V WdYiiT^XY[QVQS QY]

S\hV``V^VQS Ud\Q ivYQ[V^VQSUdOivV``VM` VWS UTQiQYS\hV` UV WV UV^YQUVh WR `Y

QTSRTQUVivYQ[V^VQSUdOivV``VQVT\VXYW\Qh`VXYhSRi\`RVhMNVX`\W`VWvT^T]

SvOSRVWiTQWSRS\VQS \Q VcV^X`V WR^X`V UVShYQWnTh^YSRTQ UV

Z 2

ZRW]]ZRWU\o\V`

R` VWS RQSOhVWWYQS UdOS\URVh `V iT^XThSV^VQS UVWXYZY[VWY\ST]YWWV^_`YQSWM

+* 45 2<21-1045 41 >7 /2102/ 34 3/7<0:4

s@?>?=@ t

"!!#"$# "# !#

1/s

"#!

h s : Z 2 7→ Z 2

$

h s (x, y) = (x/s, y/s)

!

/

#$ # !##

! "

Q

\SR`RWVUVW vT^TSvOSRVW UVhYXXThS

1/s

X`\SS o\V

s

UVnYTQ YZTRh \QV

nTQiSRTQ W\hViSRZVM

QXV\SYRQWR XYh`VhUVST\SVW`VWS\R`VW UVW nTh^VW [hYQUV

OivV``VVS XYW\QRo\V^VQS UV iV``VW UTQS `VW iTThUTQQOVWWTQS ^\`SRX`VW UV

s

M

yVW vT^TSvOSRVW Y[RWWVQS W\hUVW S\R`VWVS `V\h YiSRTQ UOXVQU UV `Y XTWRSRTQ

UV`dThR[RQVM Q

YUYXSV`Y UOPQRSRTQ Y\c nTh^VW

s@?>?=@

! $!$# ! $!$# # "!!!

#"$# "# !#

M

#

m

!

m

# !

M

$""

s

# $ ##

M ¯

M

#

m ¯

m

#" # ""! "#

"!!#"$#$ # ! #!#

(x, y) ∈ Z 2

(x, y) ∈ M ¯ ⇐⇒ h s (x, y) ∈ m ¯

s@?>?=@ ; !#

J

# #

s

# !#

≡ s

#!

$ " !#

J

!# ! # "$ #

s

" #

(11)

!"#!

{1, . . . , s} 2 7→ J

#" #

s

" $ " ! !

≡ s

Q URhY YRQWR o\V UV\c iYhhOW UV SYR``V

s

TQS `Y ^^V RQSVhnYiV WR `VW ^TSRnW

UTQSR`WWTQShViT\ZVhSWWTQSOo\RZY`VQSWM Q

XYh`VhYUVWRQSVhnYiVWUd\QV# T\

Ud\Q ^TSRn RQi`\WUYQW

{1, . . . , s} 2

XT\h UOWR[QVh `VWRQSVhnYiVW UVW ^TSRnW iYhhOW

iTQSVQYQS iVSSV S\R`V T\ iV^TSRnM

s@?>?=@ ;; ! !#

s

# !#

J

#

j

# #

I

##!

J

#" #

s

!#

M

!#

J

#

m

!#

j

! #

m

# !

M

" #"

I

# "#!

f

j

I

#

p

!# !

G

! "!!#"$#

h

! #!# "#

(x, y) ∈ Z 2

#

(x, y)

m

#

t

"$

#

s h −1 (x, y)

M

! #"

f (t)

!

c ∈ j Z 2

"!#!

j

! !#

h(c) ⊂ I Z 2

!#

J

!# !

c

†T\hOSVQUhV`Y QTSRTQUV[hT\XY[VY\cXYZY[VW # # $ VS

UTQi `Y UwQY^Ro\VR` VWS QOiVWWYRhVUdT_WVhZVhST\W `VW^TSRnWRQSVh^OURYRhVW

UTQS iVhSYRQW QV WVhTQSXYW UVW R^Y[VW XYhvT^TSvOSRVW UVhYXXThS

s

M

s@?>?=@ ;p # !#! !#

J

#

K

#

K ¯

#"

K

#

f : J Z 2 → K ¯ Z 2

#

K

# "

J

f

"!#!

# !# $$

c

# "!#! ##

J

# #

"!#! ##

K

#"

f (c)

"!#!

K

!# ! #"

f (c)

!

c

#

"!#!

J

!#

c

#

c 0

"!#!

J

#

c ≤ c 0

! ! #!

γ, γ 0

#"

f (c)

##

K

f (c 0 )

γ ≤ γ 0

!#

γ

#

γ 0

"!#!

K

#"

f (c)

#

f (c 0 )

#

f (c) ≤ f (c 0 )

!

c

#

c 0

!# "!#! ##

J

#

c ≤ c 0

uY XhV^RlhViTQURSRTQ XVh^VS UVURhVo\V

K

iTQWSh\RSVQShVY\ShV`VWR^[VW

UVW ^TSRnW iTQWSh\RSW XYh

J

M uY UV\cRl^V XVh^VS UV URhV o\V ST\W `VW ^TSRnW

PQY\c UV

K

WTQS _RVQ UVW R^Y[VW UV ^TSRnW PQY\c UV

J

M uVW UV\c UVhQRlhVW

iTQURSRTQW XVh^VSSVQS UV URhV o\V `VW WSh\iS\hVW UVW UV\c UwQY^Ro\VW WV hVW]

WV^_`VQSM

NYQW `Y W\RSVXYhY_\W UVQTSYSRTQTQQTSVhY

f (c)

XT\h \QV iTQP[\hYSRTQ

YSSVR[QY_`V UV

K

UdRQSVhnYiV

f (c)

uV `V^^V W\RZYQS ^TQShV o\V `dTQ XV\S YWWTiRVh ST\SV _hYQivV UV `Y Uw]

QY^Ro\V UV

K

\QV _hYQivVUV `Y UwQY^Ro\V UV

J

UTQS V``VVWS `dR^Y[VM

}GHHG p !#

J

#

K

#

K ¯

#"

K

#

f

#

K

!#

f

!#

B

""

(12)

K

$##

f (c)

! # ""

b

J

#

c

# ""

B 0

K

"""

f (x)

!

x ∈ b

B

#

B 0

!# "!! $

# #! ! $#

$! ##! ` W\S UV ^TQShVh o\dR` VcRWSV

B 0

o\R XYhS U\ ^^V VQUhTRS

o\V

B

VS YhhRZVVQ

f (c)

VQ XYWWYQS XYhST\W `VW

f (x)

XT\h

x ∈ b

MuV `V^^V L

yT^^V`Y UwQY^Ro\VUV

K

VWSR^Y[VXYh

f

UViV``VUV

J

R` wY \QV_hYQivV

b

UYQW `Y UwQY^Ro\V UV

J

o\R WVSVh^RQVVQ

c

MbQ hOYXX`Ro\YQS `Y UOPQRSRTQ R`

VcRWSV \QV _hYQivV

B 0

UYQW `Y UwQY^Ro\V UV

K

o\R YhhRZV VQ

f (c)

VQ XYWWYQS

XYh ST\W`VW

f (x)

XT\h

x ∈ b

M

rs=HG ; ! #$ "!!#"$# #

#

J∗

# !

s ≥ 2

#

J ∗ s

#

J ∗ s

!#

s

!"!!#"$# "

J ∗

yVV\ UVS\R`V VWS `V W\RZYQS ZTRhP[\hV e

e uVV\ UVS\R`VW iTQShV]VcV^X`V

}GHHG |

"!!## " # # # ! "

!"

(0, 0)

# #

h −1

#

h 1

# !

$

J ∗

#

{−1} × {0, . . . , h −1 }

{1} × {0, . . . , h 1 }

# !

{0} × {0, . . . , h 0 } h 0 = max{h 1 , h − 1 }

#

# "! !

x = 0

!# "# #

# #! " # "#

{0} × {0, . . . , h 0 }

ZVi`VV\ UVS\R`VW UOihRS W\h `Y P[\hV e`VW YWWV^_`Y[VWXTWWR_`VW WTQS `VW

W\RZYQSW P[\hV

\h `Y `R[QV

y = 0

XYh iTQZVQSRTQTQ X`YiV`Y WT\hiVVQ

(0, 0)

TQ QVXV\S

YZTRho\V ShTRW S\R`VW `Y WT\hiVVQ

(0, 0)

VS `VWS\R`VWY\QVW UV XYhS VSUdY\ShVM

NYQW ivYi\QV UVW UV^R]iT`TQQVW

x = 1, y ≥ 1

VS

x = −1, y ≥ 1

TQ XV\S YZTRh

\Q QT^_hVo\V`oiTQo\VUVS\R`VW_`V\VW _YZVi\QVS\R`VhT\[V hY\]UVWW\WM

bQShV `VW UV\c iT`TQQVW R` XV\S w YZTRhUVW S\R`VW ZVhSVW Z^YRWY^YRW UV\c

(13)

iSV iSVM ` XV\S Y\WWR w YZTRh UVW S\R`VW ZVhSVWUYQW `Y iT`TQQV

x = 0

WR V``VW

TQS \Q WV\` ZTRWRQ VQ

x = 1

T\

x = −1

M

NV X`\WUYQW`VW^TSRnWPQY\cR` QVXV\Sw YZTRhX`\W UVUV\cVWXYiVWZRUVW

YU YiVQSW VQShV UV\c S\R`VW ZVhSVW UYQW `Y iT`TQQV

x = 0

M bQ VS WdR` w Y ShTRW

VWXYiVWZRUVW iTQWOi\SRnWTQ XV\S X`YiVh \QV S\R`V ZVhSVUYQW iV`\R U\ ^R`RV\M

\XXTWTQW o\V `dTQ YRS \Q V\ UV S\R`VW

J∗ s

UTQS `Y UwQY^Ro\V WTRS `dR^Y[V

XYh \QV vT^TSvOSRV UV hYXXThS

s

UV iV``V UV

J ∗

M bcY^RQTQW `Y UwQY^Ro\V UV

J ∗

XYhSRh UV `Y WRS\YSRTQ W\RZYQSV

†Yh iTQZVQSRTQ `dThR[RQV VWS ivTRWRV UV nYTQ iV o\V `dR^Y[V U\ iYhhO

{0, . . . , s − 1}

WTRS `Y WT\hiV UV

J ∗

M

bcY^RQTQW iVo\R WV XYWWV UYQW`Y iT`TQQV

x = 0

M TSTQWST\S UdY_ThU o\dR`

VWS XTWWR_`V UYQW `Y UwQY^Ro\V UV

J ∗

UV X`YiVh UVW S\R`VW UYQW iVSSV iT`TQQV

^^V WR \QV WV\`V UVW UV\c iT`TQQVW ZTRWRQVW VWS hV^X`RVM uY iTQWSh\iSRTQ UVW

iYhhOW UV SYR``V

s

UYQW `Y iT`TQQV U\ ^R`RV\ XV\S UTQi iT^^VQiVh WTRS XYh `Y

[Y\ivVWTRSXYh`Y UhTRSVMbQV‡VSWR`ViSOXYh`Vo\V`iT^^VQiV`Y iTQWSh\iSRTQ

VWS UOSVh^RQOY`ThW `Y XYhSRVUV `Y UwQY^Ro\V T@ R` w Y \QV WV\`V UhTRSV o\R VWS

XhOWVQSV XTWV XhT_`l^V Y\i\QV S\R`V QV XT\hhY ZVQRh WV iT``Vh VQ

0 ≤ x ≤ s

iV o\R QViThhVWXTQU XYW `Y UwQY^Ro\VUV

J ∗

M

_WVhZTQW^YRQSVQYQSiVo\R WVXYWWVVQShV`VWUV\c iT`TQQVWM`VWSXTWWR_`V

XT\hivYo\VV^X`YiV^VQSiThhVWXTQUYQS\QVS\R`VUViT^^VQiVh`dYWWV^_`Y[V

WTRS [Y\ivVWTRS UhTRSVMNTQiXT\hivYo\ViYhhOUVSYR``V

s

UV`Y iT`TQQVU\

^R`RV\ R` VcRWSV UV\c ShYQWRSRTQW UYQW `Y UwQY^Ro\V UV

J ∗ s

`d\QV o\R iTQWRWSV

YT\SVh \QV S\R`V `V `TQ[ UV `Y iT`TQQV

x = 0

`dY\ShV `V `TQ[ UV `Y iT`TQQV

x = s − 1

M o\R iThhVWXTQUVQS Y\ iSOW UV `dR^Y[VUV `Y iT`TQQV

x = 0

M

TRVQS

c 1

VS

c 2

UV\c iYhhOWUVSYR``V

s

UV `Y iT`TQQV

0 ≤ x ≤ s − 1

YU ViVQSWM

R`dTQ YT\SV

c 1

\QVS\R`V`V`TQ[ UV`Y iT`TQQVUVUhTRSViV`Y QVivYQ[VXYW`Y

XTWWR_R`RSO UdYT\SVh \QVS\R`V [Y\ivVUYQW

c 2

X\RWo\V`VW UV\cV^X`YiV^VQSW

T@ TQS`RV\ iVW YT\SWQVWTQSXYW YU YiVQSWM

(14)

yTQWRUOhTQW `Y WRS\YSRTQ T@ iVW UV\c ShYQWRSRTQWTQS V\ `RV\M†\RWo\V `Y Uw]

QY^Ro\V UV

J∗ s

VWS \QV R^Y[V UV iV``V UV

J ∗

XYhSRh UV iVSSV WRS\YSRTQ TQ

YhhRZV \QV iTQP[\hYSRTQ PQY`VM yVSSV iTQP[\hYSRTQ PQY`V VWS \Q

f (c)

XT\h

\QV iTQP[\hYSRTQ

c

M †\RWo\V iVW UV\c iYhhOW iTQSRVQQVQS Y\ ^TRQW \QV S\R`V ivYi\Q UYQW iVSSV iTQP[\hYSRTQ

f (c)

R`W WTQS hV^X`RWM NTQi

c

iTQSRVQS UV\c

S\R`VW YU YiVQSVW UYQW`Y iT`TQQVU\ ^R`RV\ iV o\R VWS R^XTWWR_`VM

NTQi

J ∗ s

QdVcRWSV XYWM

+ 2/30102/ 41 2/51:102/ 3 :/4 2<21-104

}C €=@A?>?=@ D@G@=>?=@ AG J=€CJ?>s„JDF „s€?FG uVWivYQ[V^VQSW

UdOivV``VWdYiiT^XY[QVQSUd\QivYQ[V^VQSUV`YQTSRTQUV`TiY`RSOVSUTQiXT\h

`VWXYZY[VWY\ST]YWWV^_`YQSWUVWR^\`SYQORSOMyVhSYRQVWiTQURSRTQWQVWTQSX`\W

ZOhRPY_`VWiYhV``VWhVXTWYRVQSW\h\QVQTSRTQ UV`TiY`RSOo\RQdVWSXYWRQZYhRYQSV

XYh OivV``VXYhVcV^X`VVQShVUV\c iSOWTXXTWOWUd\QV^^VS\R`VMNVX`\WWR

XT\h \Q ^^V V^X`YiV^VQSR` w Y X`\WRV\hW URhViSRTQW UdYWWV^_`Y[V XTWWR_`V

TQVWST_`R[OWUVnYRhVUVWvwXTSvlWVWUTQiUViT^^VQiVhUVWS\R`VWY`ThWo\dTQ

VWS XYWWhWo\V `dTQ XT\hhY SVh^RQVh `V\h YWWV^_`Y[VM

Q

ivVhivV UTQi iYhYiSOhRWVh iVhSYRQW XYZY[VW Y\ST]YWWV^_`YQSW XT\h `VW]

o\V`W`YQTSRTQUV`TiY`RSOXYWWV_RVQ `dOivV``VMQV^Xh\QSV`YiTQURSRTQ

RC

zTSvV^\QU eM b``V VWS \SR`RWOV XT\h iYhYiSOhRWVh`VW YWWV^_`Y[VWUOSVh^RQRWSVM

yVSSVQTSRTQQViTQSRVQSST\SVnTRWo\d\QVXYhSRVU\ UOSVh^RQRWSV\QYWWV^_`Y[V

RC

QdVWS XYW QOiVWWYRhV^VQS UOSVh^RQRWSV ^YRW TQ WYRS ST\T\hW XT\h ivYo\V

(15)

\Q XYWWY[V `dOivV``VTQ XV\SYWWTiRVh ivYo\VV^X`YiV^VQSXVSRSVOivV``V

\Q \QRo\V WRSV UYQW `Y UwQY^Ro\V [hYQUV OivV``V o\R iThhVWXTQU `Y ^^V

ShYQWRSRTQM

s@?>?=@ ;| }C €=@A?>?=@

RC

!#

J

# #

J

## "!#!

RC

! "" !# $

J

! "" ! # !" # "

"! !"

2

! "#$

! "" # !" # "

"! !"

2

! "#$ !

! "" "! ! ! # # !" #

" "! !"

2

! "#$ #

! "" "! ! # # !" # "

"! !"

2

! "#$ ! #

yVW iTQURSRTQW XVh^VSSVQS UdYSShR_\Vh XT\h ivYo\V S\R`V Ud\Q ^TSRn \QV UR]

hViSRTQ YZVi `Y WO^YQSRo\V W\RZYQSV WR \QV S\R`V Y \QV URhViSRTQ ThSvT[TQY`V

ThU \U bWS T\

\

VWSV``V VWS `Y WV\`VS\R`V UV WY iT`TQQV T\ UV WY `R[QV

XThSVh \QV iT``V UV nThiV

2

UYQW `Y URhViSRTQ TXXTWOVM RQWR \QV S\R`V

N

VWS

`Y WV\`V UV WY `R[QV Y XThSVh \QV iT``V UV nThiV

2

W\h WTQ iSO \UiV o\R WR[QR]

PV o\dV``V Y XVh^RW Y\ XYZY[V UV WdOSVQUhV W\h \QV QT\ZV``V `R[QV Y\ ThUM R

V``V Y \QV URhViSRTQ URY[TQY`V ThU]bWS \U]bWS \U]\VWS T\ ThU]\VWS

Y`ThW V``V Y OSVQU\ `V XYZY[V UYQW `Y URhViSRTQ iThhVWXTQUYQSV YRQWR \QV S\R`V

ThU]bWSWdVWSPcOV VQ WV iT``YQS WVW ZTRWRQW \U VS \VWSM

rs=HG p #

J

$ "!#!

RC

! #!

$## !$ !# $

J

!" "" # "#!

{N, S, E, W, N E, N W, SE, SW }

"! ! "#!

# $#

$! ##! TRS

J

\Q V\ UV S\R`VW ZOhRPYQS `Y iTQURSRTQ

RC

M TRS

f

\QV

nTh^V VQ[VQUhOV XYh

J

M Q UTQQV UdY_ThU Y\c S\R`VW YZVi\QV iT``V UV nThiV

2

\QV URhViSRTQ ThSvT[TQY`V

N, S, E, W

XT\h ivYo\V iT`TQQV hVWXViSRZV^VQS

`R[QV WRS\OV UYQW \QV URhViSRTQ

d ∈ {N, S, E, W }

UV `Y WT\hiV TQ YSShR_\V `Y

URhViSRTQ

d

`Y S\R`VUV `Y iT`TQQV hVWXViSRZV^VQS`R[QV o\RXThSV \QV iT``VUV

nThiV

2

UYQW`Y URhViSRTQ TXXTWOV

d

M

bQW\RSV TQ YSShR_\V `Y URhViSRTQ

N E

Y\c S\R`VW o\R WTQS `dVWS UV `Y S\R`V

N

WY\n WR V``VW WTQS Y\ W\U UV `Y S\R`V

E

UV`V\h iT`TQQVM Q UTQQV Y\WWR iVSSV

URhViSRTQ Y\c S\R`VW WRS\OVW Y\ QThU UV `Y S\R`V

E

UV `V\h iT`TQQV WY\n WR V``VW

WTQS `dT\VWS UV`Y S\R`V

N

UV `V\h iT`TQQVM

Q XhTilUV UV ^^V XT\h `VW Y\ShVW URhViSRTQW iV o\R UTQQV `V SY_`VY\

W\RZYQS

ThU UV

E

\U UV

E

ThU UV

W

\U UV

W

bWS UV

N N E SE

N E

^XTWWR_`V

bWS UV

S N E SE

^XTWWR_`V

SE

\VWS UV

N N W

^XTWWR_`V

N W SW

\VWS UV

S

^XTWWR_`V

SW N W SW

(16)

\QV URhViSRTQM

` VWSi`YRho\V `VWVQShOVW U\ SY_`VY\ WRS\OVWW\h `Y URY[TQY`V WTQSiThhViSVWM

uVW iYW ^Yho\OW iT^^V R^XTWWR_`VW `V WTQS hOV``V^VQS VQ V‡VSUdYXhlW `Y

iTQURSRTQ

RC

XT\h o\V `dYWWV^_`Y[V WTRS XTWWR_`VR` nY\UhYRS o\V ivYi\QV UVW

UV\c S\R`VW XThSYQS UVW URhViSRTQW ThSvT[TQY`VW YRS OSO X`YiOV YZYQS `dY\ShV iV

o\R VWS R^XTWWR_`VM

j †`YiV^VQSW RQiT^XYSR_`VW YZVi`Y iTQURSRTQ zy

†T\h ZTRh o\V `dOSRo\VSY[V VWS iThhViS UYQW `VW Y\ShVW iYW VcY^RQTQW `V iYW

^Yho\O

M uY S\R`V QV XV\S XYW YZTRh OSVQU\ `V XYZY[V ZVhW `V

N E

iYh XT\h

iV`Y R` nY\UhYRS o\V WTQ ZTRWRQ W\U YRS OSO X`YiO YZYQS V``VM YRW Y`ThW `Y S\R`V

^Yho\OV

E

o\RVWSUYQWWYiT`TQQVQdY\hYRSXYWOSO`YXhV^RlhVX`YiOVUYQWiVSSV

iT`TQQV iV o\R iTQShVURS `Y iTQURSRTQ

RC

MuY S\R`V Y UTQi OSVQU\ `V XYZY[VY\

W\U]VWSM

rs=HG | # $" ! #

J

!$ ! $#

"!#!

RC

$! ##! uY XhV\ZV WV nYRS XYh \QV WR^\`YSRTQ UdY\ST^YSV iV``\`YRhVV

QV`Y UOSYR``VhYR XYWM

yVhOW\`SYS ^TQShV o\V^Y`vV\hV\WV^VQS`Y iTQURSRTQ

RC

QVXV\S XYWShV

YWW\hOVXYh\QY`[ThRSv^VMyV`Y WR[QRPVUd\QVXYhSo\dR`VWSURiR`VUdYXXhOiRVh

XhRThR `Y nYTQ UTQS WV UOhT\`V `dY\ST]YWWV^_`Y[VVS o\dR` VWS URiR`V UVWOXYhVh

`VW V\ UV S\R`VW o\R TQS \Q iT^XThSV^VQS WR^X`V `Y iTQURSRTQ

RC

R^X`Ro\V

\QV hO[\`YhRSO UYQW `Y UwQY^Ro\V UV iV\c XT\h `VWo\V`W `dYWWV^_`Y[V VWS X`\W

iT^X`VcVM`nY\SiVXVQUYQSQTSVho\VXT\hUVQT^_hV\cV\cUVS\R`VW`YXhV\ZV

o\dR`W WTQS

RC

VWS YWWV WR^X`V nYRhVM ydVWS `V iYW XYh VcV^X`V XT\h `VW UV\c

V\c UVS\R`VW UTQQOWVQ VcV^X`V UYQW `dRQShTU\iSRTQM

rs=HG ‚ !#

J

# ! $# "!#!

RC

#

s ≥ 2

# ! #$#

2

# #

J s

#

(17)

J

# "!!#"$# !#

s

J s

!#

$

J

!# ! "

J s

$! ##! TRS

J

\QV\ UVS\R`VWVS

s

\Q VQSRVhM

Q UOPQRS`VWV\ UVS\R`VW

J s

UV`Y ^YQRlhVW\RZYQSV

J s = Seed ∪ Replicas

M

Replicas

VWS\QVQWV^_`VUVS\R`VWiTQSVQYQSUVWS\R`VWUV`YnTh^VW\RZYQSV

XT\h ivYo\V S\R`V UV

J

TQ `Y UOiT\XV VQ iYhhOW UV SYR``V

1/s

M Q UTQQV UVW

iT\`V\hW RQSVhQVW ivYi\Q UV iVW iYhhOW UV SYR``V

1/s

M †T\h iV`Y TQ Q\^OhTSV

ivYi\QUVWiSOWUVWS\R`VWo\RWTQS`dRQSOhRV\hU\iYhhO YZVi\QiTUV iT`TQQV]

`R[QV XT\h `VW iSOW ZVhSRiY\c VS `R[QV]iT`TQQV XT\h `VW iSOW vThR TQSY\cM

Q

YWWTiRVVQW\RSV ivYi\Q UV WVW iSOW\QV iT``V XhTXhVUd\QV XYhS `Y S\R`V

UdY\ShVXYhS `dV^X`YiV^VQSMx

NYQW `V iYWT@ \QVS\R`V Y \QV iT``V UV nThiV

2

W\h \Q UV WVW iSOW URWTQW

`dT\VWS x WY iT`TQQV `Y X`\WY\ QThU `dVWS XT\h `VWS\R`VW vThR TQSY`VWY UVW

iT``VWRQSVhQVW UVnThiV

2

YRQWR o\d\QViT``VUVnThiV

2

`dVQUhTRS`VX`\WY\QThU

hVWXViSRZV^VQS `dVWS UV WTQ iSO YwYQS`Y iT``V UV nThiV

2

M

NYQW`ViYWUd\QVS\R`V

t

YwYQSX`\W Ud\QViT``VUVnThiV

2

TQYT\SV`VWiT``VW

UVnThiV

2

o\V`dTQ Y\hYRS^RWWOXYhO^VQSXT\h ivYi\QVUVWiT``VW UVnThiV

2

o\R

YXXYhYRWWVQS W\h

t

M

x yTQWSh\iSRTQ UV

J s

UYQW `dThUhV`VWS\R`VW URY[TQY`VW`VW S\R`VWThSvT]

[TQY`VWVS`V iYWUV `Y WT\hiV

Seed

VWS \Q VQWV^_`V UV S\R`VW XYh^VSSYQS UV iTQWSh\RhV \Q iYhhO UV SYR``V

s

XThSYQS W\h WTQ XOhR^lShV `VW ^^VW iT\`V\hW o\V `Y S\R`V iThhVWXTQUYQSVUV

Replica

MinP[\hV x

uVW RQSVhnYiVW WTQS UOPQRVW XYh `Y iTQWSh\iSRTQ UVW S\R`VW UV

Replica

`VW

iYhhOWUVSYR``V

s

RWW\WU\UOiT\XY[VUd\QV^^VS\R`VUV

J

TQS`Y^^VRQSVhnYiVM

uVW Y\ShVW iYhhOW UV SYR``V

s

XTWWR_`VW TQS \QV ^^V RQSVhnYiV

JU N K

M XYhS

JU N K

ivYo\V RQSVhnYiV QV iTQSRVQS UTQi o\d\Q ^TSRnM NV X`\W ivYo\V S\R`V

Références

Documents relatifs

● Poursuivre la programmation des Ateliers écolos avec des ateliers visant à outiller les résident.e.s pour prolonger la durée de vie de leurs objets, en collaboration avec

couvertures(pour pas que les animaux aient froid),stylos , crayons , gommes (pour que les personnes de la S.P.A aient ce qu’il faut pour écrire),et des journaux (pour mettre

Office du Tourisme Bayonne Office du Tourisme Ile de Ré Office du Tourisme Périgueux Centre des Congrès de Jonzac Pau Pyrénées Tourisme Pierre &amp; Vacances Prestige

Mieux, à Taolambiby, près de Betioky, Alan Walker a retrouvé dans la couche supérieure des subfossiles trois tessons de poterie non décorée et à Lamboara une dent de Aye-Aye percée

• Une stratégie d’intervention basée sur une vision collective et un plan proposant des actions concrètes pour améliorer la qualité de vie des quartiers.. LOCALISATION

En effet, dans cette application numérique relativement réaliste, le nombre total de gouttes qui viendront s’écraser sur nous est fonction décroissante de notre vitesse (cette

149 stocks soumis au Système de gestion par contingents. Alors que les premiers TAC de 1986 ont été fixés en dessous des captures historiques pour favoriser une reconstitution

En 15 ans, le RESO a multiplié les innovations et les projets pour développer une culture de formation dans les entreprises du Sud-Ouest : ateliers de formation de formateurs en