5(=21$162:(:=0$&1,$&=(02&<:,(/.,(-
&=6727/,:2&,

:352:$'=(1,(

5H]RQDQVRZH Z]PDFQLDF]H PRF\ ZF] VWDQRZL QDMF]FLHM

VWRSQLH NRFRZH XU]G]H JHQHUDF\MQ\FK OXE QDGDZF]\FK D LFK

]DGDQLHP MHVW GRVWDUF]HQLH GR RGELRUQLND HQHUJLL RGSRZLHGQLR GX*HM

PRF\ X*\WHF]QHM V\JQDáX SU]\ PDá\FK ]QLHNV]WDáFHQLDFK L GX*HM

VSUDZQRFLRHQHUJHW\F]QHMXNáDGX0RJRQHSUDFRZDüZNODVLH$$%

% D WDN*H Z NODVLH & SU]\ NFLH SU]HSá\ZX 2

Θ

VSHáQLDMF\P

QLHUyZQRü 0 2

180

≤

<

Θ

$

0LPRVLOQLHQLHOLQLRZ\FKZDUXQNyZSUDF\

HOHPHQWXDNW\ZQHJRSU]HELHJQDSLFLDZ\MFLRZHJRPR*HE\ü]EOL*RQ\

GR VLQXVRLGDOQHJR G]LNL ILOWUDF\MQ\P ZáDFLZRFLRP VHOHNW\ZQHJR

REZRGX UH]RQDQVRZHJR :H Z]PDFQLDF]DFK PRF\ ZF] SUDFXMF\FK

ZNODVLH&DZLFEGF\FKXNáDGDPLVLOQLHQLHOLQLRZ\PLVWRVRZDQLH

DQDOL]\ PDáRV\JQDáRZHM MHVW Z ]DVDG]LH QLHGRSXV]F]DOQH 1DMF]FLHM

NRU]\VWD VL ] PHWRG NRPSXWHURZ\FK OXE JUDILF]QRDQDOLW\F]Q\FK

VWRVXMF OLQHDU\]DFM RGFLQNRZ QLHOLQLRZ\FK FKDUDNWHU\VW\N :H

Z]PDFQLDF]DFK PRF\ ZF] GX*\FK L UHGQLFK PRF\ SRZ\*HM N:

]UHJXá\VVWRVRZDQHODPS\HOHNWURQRZHWHWURG\VWUXPLHQLRZHSU]\

PRFDFK SRQL*HM G]LHVLWNyZ N: RUD] WULRG\ SU]\ ZLNV]\FK PRFDFK

%LSRODUQH L XQLSRODUQH WUDQ]\VWRU\ PRF\ Z F] ]QDMGXM ]DVWRVRZDQLD

ZH Z]PDFQLDF]DFK PQLHMV]\FK PRF\ Z ]DNUHVLH G]LHVLWNyZ : GR

N:

1DV]HUR]ZD*DQLDRJUDQLF]\P\GRNUyWNLHJRVFKDUDNWHU\]RZDQLD

Z]PDFQLDF]\ PRF\ ZF] SUDFXMF\FK Z NODVLH & UHDOL]RZDQ\FK QD

HSLSODQDUQ\FK WUDQ]\VWRUDFK PRF\ ZF] RUD] GR RSLVX ]DVDG G]LDáDQLD

Z]PDFQLDF]\ZF]SUDFXMF\FKZNODVLH'L($QDOL]DLSURMHNWRZDQLH

W\FK Z]PDFQLDF]\ V RSDUWH ] NRQLHF]QRFL QD ]QDF]Q\FK

XSURV]F]HQLDFK L SU]\EOL*HQLDFK SR]ZDODMF\FK UDF]HM QD MDNRFLRZH

Z\MDQLHQLH]MDZLVNIL]\F]Q\FKDZPQLHMV]\PVWRSQLXQDLFKLORFLRZH

XMFLH 3U]\F]\Q WHJR MHVW ZSá\Z HIHNWyZ F]VWRWOLZRFLRZ\FK

SDUDPHWU\F]Q\FK ]DOH*QRü SRMHPQRFL ]áF]RZ\FK L G\IX]\MQ\FK RG

QDSLü L SUGyZ ZSá\Z\ WHPSHUDWXURZH SU]\ GX*\FK PRFDFK

WUDFRQ\FK Z WUDQ]\VWRU]H RJUDQLF]HQLD PRF\ SUGyZ L QDSLü PDá\

SR]LRP LPSHGDQFML ZHMFLRZHM Z SU]\SDGNX WUDQ]\VWRUD ELSRODUQHJR

RUD] VLOQLH QLHOLQLRZH ]DOH*QRFL SRPLG]\ SUGDPL L QDSLFLDPL Z

XNáDG]LH

=$6$'$35$&<,2*Ï/1(:à$612&,5(=21$162:<&+

:=0$&1,$&=<02&<./$6<&

2JyOQ\ VFKHPDW EORNRZ\ Z]PDFQLDF]D PRF\ NODV\ &

SU]HGVWDZLRQRQDU\V

+

-

X

L

X

6

( )

X

W

L

Ω

8

%%

<

0

X

2

+

-

&

/

HOHPHQW

Z]PDFQLDM F\

5

/

Ω

X

L

2

L

6

8

&&

>

0

5\V6FKHPDWEORNRZ\Z]PDFQLDF]DNODV\&

$QDOL]DWHJRXNáDGX]RVWDQLHSU]HSURZDG]RQDSU]\QDVWSXMF\FK

]DáR*HQLDFKXSUDV]F]DMF\FK

−

&KDUDNWHU\VW\ND

( )

i u

O

S

F]\OL

( )

i u

C

BE

Z SU]\SDGNX WUDQ]\VWRUD

ELSRODUQHJR MHVW OLQL SRMHG\QF] L PR*H E\ü DSURNV\PRZDQD

RGFLQNDPL SURVWHM 2]QDF]D WR *H REZyG UH]RQDQVRZ\ VWDQRZLF\

REFL*HQLH Z]PDFQLDF]D MHVW GRVWURMRQ\ GR SXOVDFML V\JQDáX

VWHUXMFHJR NWyUD MHVW QD W\OH PDáD *H PR*QD SRPLQü HIHNW\

EH]ZáDGQRFLRZH WUDQ]\VWRUD SRMHPQRFL ZHMFLRZH L Z\MFLRZH

PRJE\üGRáF]RQHGRRGSRZLHGQLFKREZRGyZELHUQ\FKDZSá\Z

UHDNWDQFML ZHZQWU]QHJR VSU]*HQLD ]ZURWQHJR NROHNWRU ED]D MHVW

SRPLMDOQLHPDá\2]QDF]DWRWDN*H*HQDFK\OHQLHFKDUDNWHU\VW\NLMHVW

VWDáH]DWHPWUDQVNRQGXNWDQFMD

J

FRQVW

P

=

−

,PSHGDQFMD ZHMFLRZD QLHOLQLRZD Z]PDFQLDF]D MHVW GX*D Z

SRUyZQDQLX ] LPSHGDQFM JHQHUDWRUD VWHUXMFHJR F]\OL VWHURZDQLH

PR*HP\X]QDü]DQDSLFLRZH

−

,PSHGDQFMD Z\MFLRZD QLHOLQLRZD MHVW GX*D Z VWRVXQNX GR

LPSHGDQFMLREFL*HQLD

−

1DSLFLHZ\MFLRZHMHVWVLQXVRLGDOQHREZyGUH]RQDQVRZ\RG]QDF]D

VLWDNGX*GREURFL Q *HGODSXOVDFMLGRVWURMHQLD

Ω

UHSUH]HQWXMH

UH]\VWDQFM G\QDPLF]Q

R

L

RUD] ]ZDUFLH GOD LQQ\FK SXOVDFML

KDUPRQLF]Q\FK k

Ω

RUD]SRGKDUPRQLF]Q\FK

Ω

/ .

k

1D U\V SU]HGVWDZLRQR FKDUDNWHU\VW\N URERF]

( )

i u

O

S

RUD]

SU]HELHJL QDSLFLD ZHMFLRZHJR

( )

u

t

i

Ω

VWHUXMFHJR

( )

u

t

S

Ω

QDSLFLD

QDREZRG]LHUH]RQDQVRZ\P

( )

u

t

Ω

LSUGXZ\MFLRZHJR

( )

i

t

O

Ω

.

Ω

W

8

RP

( )

X

W

2

Ω

L X

8

7

8

%%

X

6

L

2

J

L

X

P

2

6

=

∂

∂

FRQVW

( )

X

W

L

Ω

8

VP

2

Θ

0

Θ

π

π

2

3

2

π

Ω

W

Ω

W

,

RP

−Θ

0

Θ π

2

π

3

2

π

2

π

( )

X

W

6

Ω

8

7

8

%%

( ) ( )

X

W

X

W 8

6

L

%%

Ω

Ω

=

+

Θ

0

π

π

2

3

2

π

( )

L

W

2

Ω

2

π

8

&&

( )

X W

Ω

0

5\V &KDUDNWHU\VW\ND URERF]D

( )

L X

2

6

L SU]HELHJL F]DVRZH Z LGHDOQ\P

Z]PDFQLDF]XUH]RQDQVRZ\PNODV\&

( )

X

W

L

Ω

QDSLFLHZHMFLRZH

( )

X

W

6

Ω

QDSLFLH VWHUXMFH

( )

L

W

2

Ω

SUG Z\MFLRZ\

( )

X

W

Ω

QDSLFLH QD REZRG]LH

UH]RQDQVRZ\P

( )

X

W

2

Ω

QDSLFLHQDHOHPHQFLHDNW\ZQ\P

3UG Z\MFLRZ\ PD NV]WDáW LPSXOVyZ NRVLQXVRLGDOQ\FK R

DPSOLWXG]LH

I

om

]DOH*QHM RG DPSOLWXG\ QDSLFLD VWHUXMFHJR

U

sm

,

QDSLFLDSRODU\]DFML

U

BB

LQDSLFLDSURJRZHJR

U

T

FKDUDNWHU\VW\NL

(

)

,

8

8

8 J

8

8

8

8

RP

VP

%%

7

P

VP

%%

7

VP

%%

7

=

+

−

≥ −

+

≤ −

+







GOD8

GOD8

0

1DSLFLHVWHUXMFH

( )

u

t

S

Ω

GOD

Ω

Θ

W

=

RVLJDZDUWRü

8

8

8

VP

%%

7

FRV

Θ

= −

+

=]DOH*QRFLRUD]Z\QLND

(

)

,

8

J

RP

VP

P

=

−

1

FRV

Θ

3U]HSá\Z SUGX

( )

i

t

O

Ω

SU]H] REZyG UH]RQDQVRZ\ R SXOVDFML

UH]RQDQVRZHMGRVWURMRQHMGRSXOVDFMLV\JQDáXVWHUXM FHJR

Ω

SRZRGXMH

SRZVWDQLH QD QLP QDSLFLD VLQXVRLGDOQHJR

( )

u

t

Ω

.

3U]HELHJL QDSLFLD

Z\MFLRZHJR

( )

u

t

O

Ω

L QDSLFLD QD REZRG]LH UH]RQDQVRZ\P

( )

u

t

Ω

SU]HGVWDZLRQRQDU\V

=JRGQLH]SU]HGVWDZLRQ\PLZ\NUHVDPLQDSL FLHVWHUXMFH

u

S

RUD]

QDSLFLHZ\MFLRZH

u

O

RSLVXM]DOH*QRFL

(

)

u

U

U

t

U

u

U

U

t

S

BB

sm

BB

O

CC

om

=

+

<

=

−

cos

cos

Ω

Ω

0

1D SRGVWDZLH ]DOH*QRFL PR*HP\ RNUHOLü QDSLFLH

Z\MFLRZH

u

O

ZIXQNFMLQDSLFLDVWHUXMFHJR

u

S

(

)

u

U

U

u

U

U

O

om

sm

S

BB

CC

= −

−

+

:XNáDG]LHZVSyáU]GQ\FK

( )

u

u

O

S

i

i

const

S

0

,

=

W]QZSROXFKDUDNWHU\VW\N

L]RSUGRZ\FK FKDUDNWHU\VW\ND URERF]D VWDQRZL SURVW SU]HFKRG]F

SU]H]SXQNW\

(

)

u

U

u

U

S

BB

O

CC

=

=

,

RUD]

(

u

U

U

S

BB

sm

=

+

)

u

U

U

O

CC

om

=

−

=]DáR*HQLDGZXRGFLQNRZHMDSURNV\PDFMLFKDUDNWHU\VW\NL

( )

i u

O

S

Z\QLND*HLPSXOVSUGX

i

O

PR*QDRSLVDü]DOH*QRFL

L

$ ,

W

2

RP

=

FRV

Ω

%

SU]\F]\PVWDáH$L%Z\]QDF]\P\]ZDUXQNyZ

(

)

(

)

L

W

,

L

W

2

RP

2

Ω
Ω

Θ

=

=







0

VWG

$ ,

% ,

$ ,

%

RP

RP

RP

− =

− =







FRV

Θ

0

=]DOH*QRFLRUD]RWU]\PXMHP\

(

)

L

,

W

W

W

2

RP

=

−

−

− ≤

≤

≤

≤









1
0

FRV

FRV

FRV

Θ

Ω

Θ

Θ Ω

Θ

Θ Ω

GOD
GOD

π

JG]LH

Θ

MHVW NWHP SU]HSá\ZX SUGX Z\MFLRZHJR NWyU\ PR*HP\

REOLF]\ü]]DOH*QRFL

Θ =

−

+











DUFFRV

8

8

8

%%

7

VP

.W\SU]HSá\ZXSUGXZRGSRZLHGQLFKNODVDFKSUDF\Z\QRV]

NODVD$

Θ = π

NODVD$%

π

π

2

< <

Θ

NODVD%

Θ = π

2

NODVD&

0

2

≤ <

Θ π

5H]RQDQVRZ\ REZyG Z\MFLRZ\ GRNRQXMH VHOHNFML ZLGPD

LPSXOVX SUGX Z\MFLRZHJR

i

O

ÄZ\ELHUDMF´ ] MHJR VNáDGRZ\FK

VNáDGRZ SRGVWDZRZ ,QWHUHVXMFH QDV VNáDGRZH SUGX

i

O

PR*HP\

Z\]QDF]\ü QD SRGVWDZLH UR]NáDGX )RXULHUD SU]HELHJX

i

O

6NáDGRZD

VWDáD

,

2

RUD]DPSOLWXG\NW\FKKDUPRQLF]Q\FK

,

RPN

VUyZQH

( )

( )

( )

( )

(

)

(

)

( )

,

,

L

,

,

,

,

,

,

,

N

N

N

N N

,

GOD N

2

&&

R

RP

RP

RP

RP

RP

RPN

RP

RP N

=

= =

=

−

−

=

=

−

−

=

=

−

−

−

=

=

≥

π

α

π

α

π

α

VLQ

FRV

FRV

VLQ FRV

FRV

VLQ

FRV

FRV

VLQ

FRV

Θ Θ

Θ

Θ

Θ

Θ

Θ

Θ

Θ

Θ

Θ

Θ

Θ

Θ

Θ

Θ

1

1

2

1 1

2

0

1

1

2



























:VSyáF]\QQLNL

( )

α

N

Θ

QRV] QD]Z ZVSyáF]\QQLNyZ UR]NáDGX LPSXOVX

NRVLQXVRLGDOQHJR:VSyáF]\QQLN

( )

( )

( )

J

1

1

0

Θ

Θ

Θ

=

α

α

QRVL QD]Z ZVSyáF]\QQLND NV]WDáWX =DOH*QRFL ZVSyáF]\QQLNyZ

( )

( )

α

k

g

Θ

Θ

,

1

SU]HGVWDZLRQRQDU\V

-DN ZLGDü ] U\V SUDFD Z NODVLH & ]ZáDV]F]D SU]\ PDá\FK

NWDFK SU]HSá\ZX JáERND NODVD & RG]QDF]D VL ERJDW\P ZLGPHP

KDUPRQLF]Q\FK LPSXOVX SUGX Z\MFLRZHJR

i

O

0LPR WR ]JRGQLH ]

SRF]\QLRQ\PZF]HQLHM]DáR*HQLHPZZ\QLNXVHOHNW\ZQHJRG]LDáDQLD

Z\MFLRZHJR REZRGX UH]RQDQVRZHJR R GX*HM GREURFL QDSLFLH

Z\MFLRZH MHVW ]EOL*RQH GR SU]HELHJX VLQXVRLGDOQHJR R PDNV\PDOQHM

DPSOLWXG]LH

8

RP

Z\QRV]FHM

8

,

5

(

RP

RP

/

&

=

=

1

ξ

JG]LH

ξ =

=

+

U

U

U

U

U

om

CC

om

om

O min

MHVWZVSyáF]\QQLNLHPZ\NRU]\VWDQLDQDSLFLDZ\MFLRZHJR

ξ

]DOH*\P

LQRGREFL*HQLD

R

L

30

60

90

120

( )

α Θ

Θ

150

180

0

α

1

α

2

α

0

α

3

α

4

α

5

α

6

J

1

1

0

=

α

α

0

01

0 5

0 4

0 3

0 2

0

0 4

0 8

−

01

12

2

16

( )

J

1

Θ

&

$%

%

$

5\V:VSyáF]\QQLNLUR]NáDGXLPSXOVXNRVLQXVRLGDOQHJR

=DOH*QRFLHQHUJHW\F]QHZXNáDG]LH

=QDMRPRü NV]WDáWX L SDUDPHWUyZ LPSXOVX SUGX Z\MFLRZHJR

( )

,

RPN

N

α Θ

SDUDPHWUyZ Z\MFLRZHJR REZRGX UH]RQDQVRZHJR L

REFL*HQLD

R

L

RUD] ZDUXQNyZ ]DVLODQLD

U

CC

XPR*OLZLD Z\]QDF]HQLH

PRF\ZXNáDG]LHZ\MFLRZ\P0RFZ\MFLRZRNUHOD]DOH*QRü

( )

3

, 8

,

8

/

RP

RP

RP

&&

=

=

1

2

1

2

1

1

α

ξ

Θ

0RFZ\MFLRZD]DOH*\Z\áF]QLHRGDPSOLWXG\VNáDGRZHMSRGVWDZRZHM

JG\ REZyG UH]RQDQVRZ\ GOD VNáDGRZ\FK KDUPRQLF]Q\FK L

SRGKDUPRQLF]Q\FKVWDQRZL]ZDUFLH

0RF GRVWDUF]RQD ]H (UyGáD ]DVLODQLD

8

&&

GR REZRGX Z\MFLRZHJR

Z\QRVL

( )

3

, 8

,

8

'

&&

&&

RP

&&

=

=

α

0

Θ

JG]LH

I

I

CC

=

0

MHVW UHGQL ZDUWRFL SUGX SREUDQHJR ]H (UyGáD

]DVLODQLD

U

CC

:\NRU]\VWXMF ]DOH*QRFL RUD] PR*HP\

Z\]QDF]\üVSUDZQRüZ]PDFQLDF]D

( )

( )

( )

η

ξ α

α

ξ

=

=

=

3

3

J

/

'

1

2

1

2

1

0

1

Θ

Θ

Θ

= ]DOH*QRFL L REOLF]\P\ PRF VWUDW Z NROHNWRU]H

WUDQ]\VWRUD

(

)

P

P

P

P

P

P

P

C

D

L

D

D

D

L

=

−

=

−

= −

= −

η

η

η

η

1

1

=DOH*QRFLJUDILF]QHPRF\Z\MFLRZHM

P

L

,

PRF\GRVWDUF]RQHM

P

D

L PRF\ WUDFRQHM

P

C

RUD] VSUDZQRFL

η

RG NWD SU]HSá\ZX

Θ

SU]HGVWDZLRQR QD U\V :\NUHV\ WH SU]HGVWDZLRQR Z SRVWDFL

]QRUPDOL]RZDQHMSU]\F]\P

−

PRFZ\MFLRZ

P

L

UHSUH]HQWXMHZ\UD*HQLH

( )

α

1

Θ

( )

2

1

3

, 8

/

RP

&&

ξ

α

=

Θ

−

PRFGRVWDUF]RQUHSUH]HQWXMHZ\UD*HQLH

( )

2

0

α Θ

( )

2

2

0

3

, 8

'

RP

&&

=

α Θ

−

VSUDZQRüZ]PDFQLDF]DUHSUH]HQWXMHZ\UD*HQLH

( )

0 5

1

, g

Θ

( )

η

ξ

=

0 5

1

J

Θ

−

PRF WUDFRQ Z NROHNWRU]H WUDQ]\VWRUD GOD

ξ

UHSUH]HQWXMH

Z\UD*HQLH

(

)

( )

( )

2

2

2

2

2

1

0

1

3

, 8

3

3

, 8

3

, 8

3

, 8

&

RP

&&

'

/

RP

&&

'

RP

&&

/

RP

&&

=

−

+

−

=

α

α

ξ

Θ

Θ

GOD

-H*HOL ]QDQD MHVW FKDUDNWHU\VW\ND SUGX VWHUXMFHJR

( )

i u

S

S

RUD]

NV]WDáW LPSXOVX WHJR SUGX ZUD] ]H ZVSyáF]\QQLNDPL UR]NáDGX WR

DQDORJLF]QH UR]ZD*DQLD SURZDG] GR Z\]QDF]HQLD VNáDGRZ\FK

KDUPRQLF]Q\FKSUGX

i

S

DGDOHMGRPRF\ZHMFLRZHM

P

in

]ZDQHMPRF

Z]EXG]HQLD

P

I

U

in

sm

sm

=

1

2

1

JG]LH

I

sm1

DPSOLWXGD VNáDGRZHM SRGVWDZRZHM LPSXOVX SUGX

ZHMFLRZHJR

U

sm

DPSOLWXGDQDSLFLDVWHUXMFHJR

u

s

30

60

90

120

150

180

0

0

122

π

4

( )

( )

2

2

0

1

3

, 8

&

RP &&

=

−

α

α

Θ

Θ

GOD

ξ

( )

2

2

0

3

, 8

'

RP &&

=

α Θ

( )

2

1

3

, 8

/

RP

&&

ξ

α

=

Θ

( )

η

ξ

=

J

Θ

Θ

5\V=DOH*QRüPRF\LVSUDZQRFLRGNWDSU]HSá\ZX

1DSLFLH

X

6

MHVWQDSLFLHPNRVLQXVRLGDOQ\P

(

)

u

u

U

S

i

BE

= +

RSXOVDFML

Ω

.

=DWHPKDUPRQLF]QHVUyZQH]HUX

U

U

sm

sm

1

=

U

smk

=

0

'ODWHJR

,

3

, 8

,

8

LQ

VP

VP

VPN

VPN

N

=

+

=

∞

∑

1

2

1

2

0

1

1

2

LZNRQVHNZHQFMLRWU]\PXMHVL]DOH*QRü

0RF GRVWDUF]RQD ]D (UyGáD QDSLFLD

U

BB

Z REZRG]LH ZHMFLRZ\P

Z\QRVL

P

I

U

B

B

B B

=

0

JG]LH

I

B 0

MHVW SUGHP SRELHUDQ\P ]H (UyGáD SRODU\]DFML

U

BB

,

F]\OL

UHGQLZDUWRFLSUGXVWHUXMFHJR

i

S

0RF WUDFRQD QD ZHMFLX

P

Din

MHVW VXP PRF\ Z]EXG]HQLD

P

in

L

PRF\

P

B

GRVWDUF]RQHM]H(UyGáD

U

BB

.

P

P

P

Din

in

B

=

+

1DOH*\SRGNUHOLü*HMH*HOL

U

BB

<

0,

WRPRFGRVWDUF]RQD

P

B

MHVWXMHPQD

SRQLHZD* (UyGáR ]DVLODQLD

8

%%

SRELHUD D QLH GRVWDUF]D PRF\ :WHG\

EDWHULD

U

BB

PR*HE\ü]DVWSLRQDUH]\VWRUHP

R

U

I

B

BB

B

=

0

]ERF]QLNRZDQ\P SRMHPQRFL

C

B

R WDNLHM ZDUWRFL DE\ UHDNWDQFMD

NRQGHQVDWRUD

C

B

GOD VNáDGRZHM SRGVWDZRZHM E\áD GRVWDWHF]QLH PDáD

8NáDG WDNL MHVW QD]\ZDQ\ XNáDGHP SRODU\]DFML G\QDPLF]QHM REZRGX

ZHMFLRZHJR

:QLRVNLZ\QLNDMFH]DQDOL]\SUDF\Z]PDFQLDF]DNODV\&

3U]HSURZDG]RQDDQDOL]DSUDF\Z]PDFQLDF]DNODV\&RSLHUDáDVL

QD V]HUHJX ]DáR*H XSUDV]F]DMF\FK -HGQDN*H Z RSDUFLX R QL PR*QD

SU]HGVWDZLü NLOND ZQLRVNyZ VáXV]Q\FK WDN*H GOD U]HF]\ZLVWHJR

Z]PDFQLDF]DNODV\&
3U]\ VWDáHM ZDUWRFL V]F]\WRZHM LPSXOVX NRVLQXVRLGDOQHJR SUGX

Z\MFLRZHJR

(

)

,

FRQVW

RP

=

RUD] SU]\ VWDá\P ZVSyáF]\QQLNX

ξ

Z\NRU]\VWDQLDQDSLFLD]DVLODMFHJR

U

CC

PRFZ\MFLRZD

P

L

URQLH

JG\ NW SU]HSá\ZX SUGX Z\MFLRZHJR

Θ

URQLH =DOH*QRü

( )

3

/

Θ

MHVW DQDORJLF]QD GR ]DOH*QRFL

( )

α

1

Θ

'OD SU]\MWHM RGFLQNRZR

OLQLRZHM DSURNV\PDFML FKDUDNWHU\VW\NL

( )

i u

O

S

PDNVLPXP PRF\

Z\MFLRZHM SU]\SDGD GOD

Θ

RSW

R

=

122

NODVD $% =PQLHMV]DMF NW

SU]HSá\ZXSUGX

Θ

WDNDE\PRFZ\MFLRZDE\áDVWDáD

(

)

P

const

L

=

,

QDOH*\]ZLNV]\üZDUWRüV]F]\WRZLPSXOVXSUGXZ\MFLRZHJR

I

om

.

'OD X]\VNDQLD GX*HM PRF\ Z\MFLRZHM

P

L

QDOH*\ VWRVRZDü QLH]E\W

PDáHZDUWRFLNWDSU]HSá\ZX

Θ

F]\OLQLH]E\WJáERNNODV&

3U]\VWDáHMZDUWRFLV]F]\WRZHMLPSXOVX

I

om

SUGXZ\MFLRZHJRPRF

GRVWDUF]RQD

P

D

URQLH JG\ NW SU]HSá\ZX

Θ

URQLH =DOH*QRü

( )

3

'

Θ

MHVW DQDORJLF]QD GR ]DOH*QRFL

( )

α

0

Θ

:]URVW PRF\

GRVWDUF]RQHM

P

D

]H Z]URVWHP NWD SU]HSá\ZX

Θ

MHVW V]\EV]\ QL*

Z]URVWPRF\Z\MFLRZHM

P

L

.

3U]\VWDáHMZDUWRFLZVSyáF]\QQLNDZ\NRU]\VWDQLDQDSLFLD]DVLODQLD

(

)

ξ =

const

VSUDZQRüZ]PDFQLDF]D

η

URQLHJG\NWSU]HSá\ZX

Θ

PDOHMH =DOH*QRü

( )

η Θ

MHVW DQDORJLF]QD GR ]DOH*QRFL

ZVSyáF]\QQLND NV]WDáWX

( )

J

1

Θ

: V]F]HJyOQRFL JG\

Θ

→

L

ξ

→

WR

η

η

=

→

=

1 2

1

1

PD[

J

*UDQLF]QD WHRUHW\F]QD VSUDZQRü

Z]PDFQLDF]DZNODVLH&Z\QRVLGODNODV\%

η

max

π

RUD]

GODNODV\$

η

max

MHGQDNPRFZ\MFLRZDPDOHMHZWHG\GR]HUD

= ZQLRVNyZ L Z\QLND *H Z\PDJDQLD RGQRQLH GX*HM PRF\

Z\MFLRZHM

P

L

L GX*HM VSUDZQRFL

η

V VSU]HF]QH L Z ZDUXQNDFK

U]HF]\ZLVW\FKZ\PDJDQ\MHVWNRPSURPLV

:]PDFQLDF] NODV\ & FKDUDNWHU\]XMH VL QLHOLQLRZ ]DOH*QRFL

( )

I

f U

om

sm

1

=

.

=PLDQDDPSOLWXG\V\JQDáXVWHUXMFHJR

U

sm

SRZRGXMH

]PLDQ\ NWD SU]HSá\ZX

Θ

RUD] ]PLDQ\ V]F]\WRZHM ZDUWRFL SUGX

Z\MFLRZHJR

,

RP

:DUXQHN OLQLRZRFL MHVW VSHáQLRQ\ GRSLHUR SU]\

SUDF\ Z NODVLH % OXE $ ZyZF]DV ERZLHP NW SU]HSá\ZX

Θ

QLH

]DOH*\RG

U

sm

:WHG\]PLDQD

U

sm

SRZRGXMHSURSRUFMRQDOQGRQLHM

]PLDQ

I

om

D ]DWHP UyZQLH*

I

om1

'OD

U

BB

8

7

U

BB

>

0

NW

SU]HSá\ZX

Θ = π

2 Z]PDFQLDF] SUDFXMH Z NODVLH % SU]\ F]\P

DPSOLWXG\SUGyZVUyZQH

I

U

g

I

U g

om

sm

m

om

sm

m

1

2

=

=









'OD

8

8

8

%%

7

VP

=

+

NW SU]HSá\ZX

Θ = π

:WHG\ Z]PDFQLDF]

SUDFXMHZNODVLH$]DPSOLWXGDPLSUGyZ

I

U

g

I

U

g

m

sm

m

om

sm

m

0 1

2

=

=







:]PDFQLDF]H SUDFXMFH SU]\ ]PLHQQHM DPSOLWXG]LH V\JQDáX

X*\WHF]QHJR QS Z]PDFQLDF]H SU]HELHJyZ PRGXORZDQ\FK

DPSOLWXGRZR PXV] ]DWHP SUDFRZDü Z NODVLH % OXE Z NODVLH $

DE\ ]DSHZQLü OLQLRZ ]DOH*QRü QDSLFLD Z\MFLRZHJR RG QDSLFLD

VWHUXMFHJR:W\PSU]\SDGNXQLHMHVWPR*OLZHZ\NRU]\VWDQLHSUDF\

ZNODVLH&LX]\VNDQLHGX*HMVSUDZQRFLHQHUJHW\F]QHM]ZL]DQHM]

NODV&

1D SRGVWDZLH SRZ\*V]\FK XZDJ PR*HP\ RNUHOLü ]DOH*QRFL

HQHUJHW\F]QHZ]PDFQLDF]DZNODVLH%WMGOD

Θ = π

2=DOH*QRFLWH

]RVWDá\ RNUHORQH ZF]HQLHM GOD GRZROQHJR NWD SU]HSá\ZX

Θ

:\NRU]\VWXMF]DOH*QRFLLRWU]\PXMHP\

α

π

π

α

π

ξ

α

π

0

1

1

1

2

1

2

0 5

2

Θ

Θ

Θ

=







 =

=







 =

=

=

=

=

























8

8

,

5

, 5

RP

&&

RP

/

RP

/

6WG

,

8

5

3

,

8

8

5

3

,

8

8
5

J

RP

&&

/

/

RP

&&

&&

/

'

RP

&&

&&

/

=

=









=

=









=

=

=









=

=

=







 =



























ξ

α

π

α

π

ξ ξ

α

π

π

ξ

η

π ξ π ξ

1

1

2

2

0

2

1

2

1

2

2

2

2

2

1

2

2

4

Θ

Θ

Θ

Θ

: SRGREQ\ VSRVyE PR*HP\ RNUHOLü ]DOH*QRFL HQHUJHW\F]QH GOD

NODV\$WMGOD

Θ = π

:WHG\

(

)

(

)

(

)

α

π

α

π

π

0

1

1

0 5

1

Θ

Θ

Θ

=

=

=

=

=

=







J

:]PDFQLDF] Z NODVLH & PR*H E\ü X*\W\ MDNR NNURWQ\ SRZLHODF]

F]VWRWOLZRFL F]\OL Z]PDFQLDF] NWHM KDUPRQLF]QHM :WHG\

Z\MFLRZ\REZyGUH]RQDQVRZ\MHVWGRVWURMRQ\GR NWHMKDUPRQLF]QHM

V\JQDáX VWHUXMFHJR .W SU]HSá\ZX SU]\ NWyU\P DPSOLWXGD NWHM

KDUPRQLF]QHM SUGX Z\MFLRZHJR

I

omk

RVLJD PDNVLPXP ZWHG\

WDN*H PRF Z\MFLRZD RVLJD PDNVLPXP SU]\ ]DáR*HQLX VWDáHM

ZDUWRFLV]F]\WRZHMSUGXZ\MFLRZHJR

I

om

LVWDá\PZVSyáF]\QQLNX

ξ

Z\NRU]\VWDQLDQDSLFLD]DVLODQLDZ\QRVL

Θ

RSW

N

=

2

3

π

3U]\

Θ

RSW

Z\VWSXM PDNVLPD ZVSyáF]\QQLNyZ UR]NáDGX

NRVLQXVRLGDOQHJR

( )

α

N

Θ

0DNV\PDOQD PRF Z\MFLRZD PDOHMH ]H

Z]URVWHPVWRSQLDSRZLHODQLDN:SUDNW\FHVWRVXMHVL SRZLHODF]HR

VWRSQLXSRZLHODQLDQLHZLNV]\PQL*

2EZyG UH]RQDQVRZ\ R LPSHGDQFML = ]DSHZQLD Z\RGU EQLHQLH

VNáDGRZHM SRGVWDZRZHM L ILOWUDFM SR]RVWDá\FK KDUPRQLF]Q\FK

=DZDUWRüNWHMKDUPRQLF]QHMQDSLFLDZ\MFLRZHJRZ\QRVL

(

)

(

)

( )

( )

8

8

=

N

,

=

,

4 N

N

RPN

RP

R

RPN

R

RP

N

1

1

1

1

1

=

=

=

≈

−









Ω

Ω

Ω Ω

Θ

Θ

α

α

=DWHP ILOWUDFM GDQHM KDUPRQLF]QHM PR*QD SRSUDZLü VWRVXMF

RGSRZLHGQLR GREUDQ ZDUWRü NWD SU]HSá\ZX 1S

Θ = π

2

]DSHZQLDHOLPLQDFMQLHSDU]\VW\FKKDUPRQLF]Q\FK

:3à<: &=6727/,:2&, 6<*1$à8 67(58-&(*2 1$

35$&:=0$&1,$&=$:./$6,(&

$E\ SU]HSURZDG]Lü SHáQ DQDOL] SUDF\ Z]PDFQLDF]D NODV\ &

XZ]JOGQLDMF ZV]\VWNLH LVWRWQH L ]áR*RQH F]\QQLNL ]ZL]DQH ]

QLHOLQLRZRFL

FKDUDNWHU\VW\N

WUDQ]\VWRUD

Uy*QHJR

URG]DMX

RJUDQLF]HQLDPL SUGX QDSLFLD L PRF\ QLHOLQLRZ\PL HIHNWDPL

SDUDPHWU\F]Q\PL Z ]DNUHVLH ZF] LWS QDOH*\ Z\NRU]\VWDü V\PXODFM

NRPSXWHURZ XNáDGX QS Z SURJUDPLH 63,&( $E\ DQDOL]D E\áD

ZLDU\JRGQD PRGHOH HOHPHQWyZ DNW\ZQ\FK Z\NRU]\VW\ZDQ\FK GR

EXGRZ\ Z]PDFQLDF]D PXV] XZ]JOGQLDü ZV]\VWNLH Z\*HM

ZVSRPQLDQHHIHNW\QLHOLQLRZH'DOV]HUR]ZD *DQLDRJUDQLF]\P\MHG\QLH

GR MDNRFLRZHJR Z\MDQLHQLD ]MDZLVN IL]\F]Q\FK EH] LFK LORFLRZHJR

XMFLD

:Sá\Z F]VWRWOLZRFL QD SUDF Z]PDFQLDF]D UR]SDWU]\P\ Z

WU]HFKSU]HG]LDáDFKGOD

Ω

Ω

Ω

<

< <<

<

ω

ω

ω

ω

β

β

7

7

D=DNUHVVWRVXQNRZRPDá\FKF]VWRWOLZRFL

Ω <

ω

β

:W\P]DNUHVLHPR*HP\NRU]\VWDü]FKDUDNWHU\VW\NVWDW\F]Q\FKL

URERF]\FKWUDQ]\VWRUDRUD]]PHWRGJUDILF]Q\FK

=DNáDGDMF *H GREURü REZRGX UH]RQDQVRZHJR MHVW GX*D ]D

UH]\VWDQFMD (UyGáDVWHUXMFHJRMHVWPDáD

(

)

5

U

J

EE

<

PR*HP\SU]\Mü*H

QDSLFLH ZHMFLRZH

X

EH

L Z\MFLRZH

X

FH

V VLQXVRLGDOQH 3U]\SDGHN

QDSLFLRZHJRVWHURZDQLDWUDQ]\VWRUDMHVWU]DGNRVSHáQLRQ\ZSUDNW\FH

SR]ZDOD MHGQDN QD MDNRFLRZH Z\MDQLHQLH ]MDZLVN ]DFKRG]F\FK

ZWUDQ]\VWRU]H

&KDUDNWHU\VW\NURERF]XNáDGX2(]JRGQLH]UyZQDQLHP

RNUHOD]DOH*QRü

(

)

u

U

U

U

u

U

CE

CC

cem

bem

BE

BB

−

= −

+

=DOH*QRüWSU]HGVWDZLRQRJUDILF]QLHZSROXFKDUDNWHU\VW\NWUDQ]\VWRUD

( )

i

f u

C

CE

=

SU]\

u

const

BE

=

QDU\V

X

&(

L

&

Ω

W

Θ

8

&&

Θ

Θ

L L

&

%

L

&

L

%

L

%

L

%

L

&

L

&

6

6

6

6

6

3

*

8

&( PLQ

4

6

X

%(

8

8

8

%(

EHP

%%

PD[

=

−

Ω

W

5\V&KDUDNWHU\VW\NDURERF]DZSROXFKDUDNWHU\VW\NNROHNWRURZ\FKWUDQ]\VWRUD

*UDQLF]QD OLQLD * VWDQX QDV\FHQLD QD U\V UR]G]LHOD VWDQ\

SUDF\ QLHGRZ]EXG]RQHM L SU]HZ]EXG]RQHM WUDQ]\VWRUD 3XQNWRZL

V]F]\WRZHPX

S

QDFKDUDNWHU\VW\FHURERF]HMRGSRZLDGDVWDQNU\W\F]Q\

6WDQRZL SU]HZ]EXG]RQHPX RGSRZLDGD SUDFD Z REV]DU]H QDV\FHQLD

SXQNW

6

]D VWDQRZL QLHGRZ]EXG]RQHPX RGSRZLDGD SUDFD Z

REV]DU]HDNW\ZQ\PSXQNW

S'

: VWDQLH SU]HZ]EXG]RQ\P Z RWRF]HQLX SXQNWX V]F]\WRZHJR

Z\VWSXMH FKDUDNWHU\VW\F]QH RGNV]WDáFHQLH LPSXOVX SUGX NROHNWRUD

6SRZRGRZDQH MHVW WR ]QDF]Q\P Z]URVWHP SUGX ED]\ ]áF]D ED]D

HPLWHU L ED]D NROHNWRU ]RVWDM VLOQLH VSRODU\]RZDQH Z NLHUXQNX

SU]HZRG]HQLD]D (UyGáDQDSLFLRZHJRNRV]WHPSUGXNROHNWRUD3UG

HPLWHUD

i

i

i

E

C

B

= +

MHVWZ\PXV]RQ\SU]H]REZyGUH]RQDQVRZ\]DWHP

JG\

i

B

JZDáWRZQLH URQLH WR

i

C

PDOHMH 3UDFD Z VWDQLH VLOQLH

SU]HZ]EXG]RQ\P QLH MHVW NRU]\VWQD SRQLHZD * SRZRGXMH ]PQLHMV]HQLH

PRF\Z\MFLRZHMLVSUDZQRFLRUD]Z]URVWPRF\ZHMFLRZHM

5yZQLH* SUDFD Z VWDQLH QLHGRZ]EXG]RQ\P FKDUDNWHU\]XMH VL

]PQLHMV]HQLHP PRF\ Z\MFLRZHM L VSUDZQRFL XNáDGX SRQLHZD*

JZDáWRZQHPX ]PQLHMV]HQLX XOHJD ZVSyáF]\QQLN Z\NRU]\VWDQLD

QDSLFLRZHJR

ξ

E=DNUHVUHGQLFKF]VWRWOLZRFL

ω

ω

β

< <<

Ω

7

: UR]SDWU\ZDQ\P ]DNUHVLH F]VWRWOLZRFL RJUDQLF]HQLH

PLQLPDOQHJR QDSLFLD

U

CE min

VWDQRZL W]Z SDUDPHWU\F]QD OLQLD

JUDQLF]QD3U\V:REV]DU]HÄSDUDPHWU\F]Q\P´F]\OLQDOHZRRG

FKDUDNWHU\VW\NL JUDQLF]QHM * F]VWRWOLZRü JUDQLF]QD WUDQ]\VWRUD

ω

T

PDOHMH C

jc

URQLHSRQDGWRSDUDPHWU\WHVZVLOQ\PVWRSQLXQLHOLQLRZH

L X]DOH*QLRQH ]DUyZQR RG SUGX MDN L RG QDSLFLD NROHNWRUD RSUyF]

WHJRVSRGF]DVSUDF\IXQNFMDPLF]DVX

: REV]DU]H DNW\ZQHM SUDF\ WUDQ]\VWRUD ]DF]\QD RGJU\ZDü URO

VNRF]RQ\ F]DV SU]HORWX QRQLNyZ SUGX SU]H] ED] (IHNW\ZQH

QDSLFLH HIHNW\ZQLH VWDUXMFH SUDF WUDQ]\VWRUD MHVW RSy(QLRQH

Z]JOGHP ]HZQWU]QHJR QDSLFLD VWHUXMFHJR RUD] MHVW ]QLHNV]WDáFRQH

ZVNXWHN FDáNXMFHJR G]LDáDQLD XUHGQLRQHM SRMHPQRFL G\IX]\MQHM

(IHNWHPW\FK]MDZLVNMHVW

−

RSy(QLHQLH SUGX NROHNWRUD

i

C

L QDSLFLD

u

B E

'

QD ]áF]X ED]D

ZHZQWU]QDHPLWHUWUDQ]\VWRUDZVWRVXQNXGRQDSLFLDVWHUXMFHJR

u

BE

−

]ZLNV]HQLHNWDSU]HSá\ZX

Θ

SUGXNROHNWRUDZVWRVXQNXGRMHJR

ZDUWRFL

Θ

SU]\PF]

−

]PQLHMV]HQLH ZDUWRFL V]F]\WRZHM

I

cm

LPSXOVX SUGX NROHNWRUD Z

SRUyZQDQLXGRMHJRZDUWRFLSU]\PF]

−

Z]URVWDPSOLWXG\LF]DVXWUZDQLDLPSXOVXSUGXED]\

3XOVDFMDJUDQLF]QD

ω

g

SRZ\*HMNWyUHMF]DVSU]HORWXQRQLNyZRGJU\ZD

LVWRWQ URO MHVW Z SU]\EOL*HQLX RNUHORQD VWDá F]DVRZ REZRGX

ZHMFLRZHJRWUDQ]\VWRUDZVWDQLHDNW\ZQ\PW]Q

ω

ω

g

bb

e

T

eb

bb

r C

r

r

≈

=

1

'

'

'

Ω

W

Θ

Θ Θ

+

RS

Θ

RS

Θ

L

&

X

L

%(

&

J

Ω

ω

=

L

&

J

Ω

ω

=

L

&

J

Ω

ω

=

L

&

J

Ω

ω

=

Ω

W

Θ

Θ

L

%

X

%(

L

%

J

Ω

ω

≥

L

%

J

Ω

ω

=

L

%

J

Ω

ω

=

L

%

J

Ω

ω

≈

L

%

J

Ω

ω

=

5\V3U]HELHJLF]DVRZHQDSLüLSUGyZWUDQ]\VWRUDSUDFXMFHJRZNODVLH&SU]\

Uy*Q\FKF]VWRWOLZRFLDFKURERF]\FK

1DU\VSU]HGVWDZLRQRSRJOGRZRSU]HELHJL

( )

X

W

%(

Ω

( )

L

W

&

Ω

RUD]

( )

L

%

W

Ω

GODNLONXZDUWRFLSDUDPHWUX

Ω

ω

J

F=DNUHVZLHONLFKF]VWRWOLZRFL

Ω

<

ω

7

: W\P ]DNUHVLH RERN Z\*HM Z\PLHQLRQ\FK HIHNWyZ ]DF]\QDM

RGJU\ZDü URO LQQH QS LQGXNF\MQRü HPLWHUD 3RZRGXMH WR GDOV]H

]PQLHMV]HQLHZDUWRFLV]F]\WRZHM

I

om

SUGXNROHNWRUDRUD]]ZLNV]HQLH

NWD SU]HSá\ZX Z NRQVHNZHQFML ]PQLHMV]HQLH PRF\ Z\M FLRZHM

VSUDZQRFLRUD]Z]URVWPRF\ZHMFLRZHM

52=:,=$1,$8.à$'2:(:=0$&1,$&=<./$6<&

1DU\VXQNXSU]HGVWDZLRQRSU]\NáDG\UR]ZL]DXNáDGRZ\FK

UH]RQDQVRZ\FKZ]PDFQLDF]\PRF\NODV\&

5

%

+

8

&&

a)

&

%

5

/

/

&

&

E

'

Z F]

á

5

%

+

8

&&

&

%

5

/

/

&

&

E

'

Z F]

á

b)

5

/

&

%

5

%

&

%

5

%

&

/

'

Z F]

á

&

E

+

8

&&

c)

5\V 5H]RQDQVRZH Z]PDFQLDF]H NODV\ & D Z]PDFQLDF] Z NRQILJXUDFML 2(

EZ]PDFQLDF]ZNRQILJXUDFML2%FZ]PDFQLDF]SU]HFLZVREQ\

: SU]HGVWDZLRQ\FK XNáDGDFK ]DVWRVRZDQR SRODU\]DFM

G\QDPLF]Q UHDOL]RZDQ SU]H] HOHPHQW\

R

C

B

B

,

:]PDFQLDF] Z

NRQILJXUDFML 2% SRVLDGD PQLHMV]H ]DVWRVRZDQLH ]H Z]JOGX QD MHJR

PQLHMV]H Z]PRFQLHQLH PRF\ -HGQDN*H XNáDG 2% SRVLDGD OHSV]H

ZáDFLZRFLF]VWRWOLZRFLRZHLSR]ZDODQDX]\VNDQLHQLHFRZLNV]HM

PRF\Z\MFLRZHMZ]DNUHVLHUHGQLFKF]VWRWOLZRFL

5(=21$162:( :=0$&1,$&=( 02&< :&= ./$6< '

,./$6<(

2JyOQHZáDFLZRFLZ]PDFQLDF]\ZF]NODV\'LNODV\(

:]PDFQLDF]H PRF\ NODV\ ' L NODV\ ( Z F] Z\NRU]\VWXM

LPSXOVRZ SUDF HOHPHQWyZ DNW\ZQ\FK SU]HáF]DQ\FK MDNR NOXF]H ]

GX*F]VWRWOLZRFLRGSHáQHJR]DáF]HQLDGRZ\áF]HQLD7DNLURG]DM

SUDF\ HOHPHQWyZ DNW\ZQ\FK SR]ZDOD QD X]\VNDQLH GX*HM VSUDZQRFL

HQHUJHW\F]QHMZ]PDFQLDF]\NODV\'L( NWyUD Z SU]\SDGNX LGHDOQ\FK

NOXF]\ Z\QRVL 0DNV\PDOQD PRF REFL*HQLD MHVW OLPLWRZDQD

SU]H] JUDQLF]QH ZDUWRFL SUGyZ L QDSLü ]DVWRVRZDQ\FK HOHPHQWyZ

DNW\ZQ\FKDQLHSU]H]LFKGRSXV]F]DOQPRFVWUDW6WUDW\PRF\ZWHJR

W\SXZ]PDFQLDF]DFKVNáDGDMVLJáyZQLH]GZyFKVNáDGQLNyZ

−

VWUDW\ PRF\ ]ZL]DQH ] QDSLFLHP QD ]DáF]RQ\P WUDQ]\VWRU]H

WUDQ]\VWRU QLH MHVW LGHDOQ\P NOXF]HP L Z VWDQLH ]DáF]HQLD

Z\VWSXMHQDQLPQLHZLHONLHQDSLFLH

−

VWUDW\PRF\]ZL]DQH]SURFHVHPSU]HáF]DQLDWUDQ]\VWRUDJG\MHJR

SXQNW SUDF\ SU]HPLHV]F]D VL SU]H] REV]DU DNW\ZQ\ L MHGQRF]HQLH

PRJZ\VWSLüGX*HFKZLORZHZDUWRFLQDSLFLDLSUGXWUDQ]\VWRUD

2 HIHNW\ZQRFL W\FK XNáDGyZ Z GX*\P VWRSQLX GHF\GXMH

ZáDFLZ\ GREyU HOHPHQWyZ ]H Z]JOGX QD V]\ENRü LFK G]LDáDQLD MDN

UyZQLH* ]DSURMHNWRZDQLH RGSRZLHGQLFK XNáDGyZ VWHURZDQLD

WUDQ]\VWRUDPL Z FHOX ]PLQLPDOL]RZDQLD VWUDW PRF\ ]ZL]DQ\FK ]

SURFHVDPL SU]HáF]DQLD : SU]HFLZQ\P UD]LH VNáDGQLN WHQ E\áE\

GRPLQXMF\ZRJyOQ\PELODQVLHVWUDWPRF\

5H]RQDQVRZHZ]PDFQLDF]HNODV\'LNODV\(RPRF\Z\MFLRZHM

GRNLONXVHW>:@LF]VWRWOLZRFLNLONXQDVWX>0+]@VUHDOL]RZDQHSU]\

X*\FLX ELSRODUQ\FK WUDQ]\VWRUyZ PRF\ ZF] D Z V]F]HJyOQRFL SU]\

X*\FLX SRORZ\FK WUDQ]\VWRUyZ PRF\ W\SX 9026 *áyZQLH G]LNL

ZáDFLZRFLRPWUDQ]\VWRUyZ9026SDWU]UR]G]LDáZ]PDFQLDF]H

NODV\'L(]QDOD]á\EDUG]RV]HURNLH]DVWRVRZDQLHZSUDNW\FHLVWDá\VL

JáyZQ\P VSRVREHP UHDOL]DFML UH]RQDQVRZ\FK Z]PDFQLDF]\ PRF\ ZH

ZVSRPQLDQ\FKZ\*HM]DNUHVDFKPRF\LF]VWRWOLZRFL

:SRUyZQDQLXGRZ]PDFQLDF]\NODV\&Z]PDFQLDF]HNODV\'L

NODV\(FKDUDNWHU\]XMVLQLHW\ONRZLNV]VSUDZQRFLHQHUJHW\F]QL

OHSV]\P Z\NRU]\VWDQLHP QDSLFLRZ\P L SUGRZ\P HOHPHQWyZ

DNW\ZQ\FK SR]ZDOD WR QD X]\VNDQLH GX*R ZLNV]HM PRF\ Z\MFLRZHM

SU]\W\PVDP\PHOHPHQFLHDNW\ZQ\PDOHWDN*HPQLHMV]]DZDUWRFL

Z\*V]\FKKDUPRQLF]Q\FKZV\JQDOHZ\MFLRZ\P

,VWQLHMHZLHOHRGPLDQUR]ZL]DXNáDGRZ\FKZ]PDFQLDF]\NODV\

'L(ZF]1DV]F]HJyOQHZ\Uy*QLHQLH]DVáXJXMUR]ZL]DQLDW\SX=96

DQJ ]HUR YROWDJH VZLWFKLQJ : XNáDGDFK W\FK SU]HáF]DQLH

HOHPHQWyZDNW\ZQ\FKRGE\ZDVLSU]\]HURZ\PQDSLFLX3R]ZDODWR

QD PLQLPDOL]DFM VWUDW PRF\ Z SURFHVDFK SU]HáF]DQLD 1DV]H

UR]ZD*DQLD RJUDQLF]\P\ MHG\QLH GR NUyWNLHJR SU]HGVWDZLHQLD RJyOQHM

]DVDG\ G]LDáDQLD Z]PDFQLDF]\ NODV\ ' ] NRPXWDFM QDSLFLD OXE ]

NRPXWDFM SUGX >@ RUD] QLHV\PDWU\F]QHJR Z]PDFQLDF]D NODV\ (

>@

5H]RQDQVRZHZ]PDFQLDF]HPRF\ZF]NODV\'

,VWQLHM GZD SRGVWDZRZH XNáDG\ Z]PDFQLDF]\ PRF\ NODV\ '

ZF]6WRXNáDG\]NRPXWDFMQDSLFLDOXE]NRPXWDFMSUGX

5

/V

X

LQ

Ω

W

7 2

L

'1

X

'61

0

0

0

7

Ω

W

8

'6 PLQ

Ω

W

L

'2

X

'6 2

0

0

Ω

W

Ω

W

,

P

8

''

0

Ω

W

8

'6 PLQ

L

/

&

/

X

'61

X

'6 2

+

8

''

L

'1

L

'2

X

LQ

X

/V

8

'6 PLQ

,

P

,

P

8

'6 PLQ

8

''

X

LQ

M

1

M

2

L

/

b)

a)

5\V :]PDFQLDF] NODV\ ' ] NRPXWDFM QDSLFLD D VWRSLH NRFRZ\

ESU]HELHJLF]DVRZHZXNáDG]LH

8SURV]F]RQ\VFKHPDWLGHRZ\VWRSQLDNRFRZHJRZ]PDFQLDF]DNODV\'

]NRPXWDFMQDSLFLDSU]HGVWDZLRQRQDU\VDQDWRPLDVWLGHDOL]RZDQH

SU]HELHJL F]DVRZH QDSLü L SUGyZ Z W\P XNáDG]LH SU]\ SRPLQLFLX

HIHNWyZ ]ZL]DQ\FK ] SURFHVDPL SU]HáF]DQLD WUDQ]\VWRUyZ

SU]HGVWDZLRQRQDU\VE

: F]DVLH WUZDQLD XMHPQHJR QDSLFLD

u

in

] XNáDGX VWHUXMFHJR

WUDQ]\VWRU 0

]RVWDMH ]DáF]RQ\ D WUDQ]\VWRU 0

Z\áF]RQ\ 1DSLFLH

U

DD

SRPQLHMV]RQH R QLHZLHONL VSDGHN QDSL FLD

U

DS min

QD

SU]HZRG]F\P WUDQ]\VWRU]H 0

]RVWDMH SRGDQH QD V]HUHJRZ\ REZyG

UH]RQDQVRZ\

R C L

Ls

.

:F]DVLHWUZDQLDGRGDWQLHJRSyáRNUHVXQDSLFLD

VWHUXMFHJR

u

in

QDVWSXMH Z\áF]HQLH WUDQ]\VWRUD 0

RUD] ]DáF]HQLH

WUDQ]\VWRUD0

:Z\QLNXWDNLHMSUDF\WUDQ]\VWRUyZV]HUHJRZ\REZyG

UH]RQDQVRZ\]RVWDMHSREXG]RQ\QDSLFLHPZNV]WDáFLHV\PHWU\F]QHMIDOL

SURVWRNWQHM R ZDUWRFL PLG]\V]F]\WRZHM

U

U

DD

DS

−

2

min

6]HUHJRZ\

REZyGUH]RQDQVRZ\GRVWURMRQ\GRF]VWRWOLZRFLV\JQDáXVWHUXMFHJR

Z\G]LHOD SLHUZV] KDUPRQLF]Q ] SU]HELHJX SURVWRNWQHJR 3U]H]

WUDQ]\VWRU\0

L0

SU]HSá\ZDSUGRNV]WDáFLHSRáyZHNVLQXVRLG\RUD]

R ZDUWRFL PDNV\PDOQHM RJUDQLF]RQHM UH]\VWDQFM REFL*HQLD

R

Ls

1DSLFLD SUG\ L PRFH Z XNáDG]LH SU]\ SRPLQLFLX HIHNWyZ

SU]HáF]DQLD RNUHORQH V Z SU]\EOL*HQLX SURVW\PL ]DOH*QRFLDPL

Z\QLNDMF\PL]UR]NáDGXSU]HELHJXSURVWRNWQHJRZV]HUHJ)RXULHUD

−

1DSLFLHLSUGZREFL*HQLX

(

)

X

8

8

W

/

''

'6

=

−

2

2

PLQ

π

VLQ

Ω

(

)

L

8

8

5

W

/

''

'6

/V

=

−

2

2

PLQ

π

VLQ

Ω

−

0RFX*\WHF]QD

(

)

P

U

U

R

L

DD

DS

Ls

=

−

2

2

2

2

min

π

−

UHGQLDZDUWRüSUGXGRVWDUF]RQHJR]]DVLODF]D

(

)

I

U

U

R

DD

DD

DS

Ls

=

−

2

2

2

min

π

−

0RFGRVWDUF]RQD]]DVLODF]D

(

)

P

U

U

U

R

D

DD

DS

DD

LS

=

−

2

2

2

min

π

−

6SUDZQRüZ]PDFQLDF]D

η =

= −

P

P

U

U

L

D

DS

DD

1

2

min

=]DOH*QRFLZ\QLND*HVSUDZQRüZ]PDFQLDF]DNODV\']

NRPXWDFM QDSLFLD ]DOH*\ W\ONR RG VWRVXQNX QDSLFLD

U

DS min

GR

QDSLFLD]DVLODQLD

U

DD

1DSLFLH

U

DS min

QDSU]HZRG]F\PWUDQ]\VWRU]H

Z REV]DU]H OLQLRZ\P MHVW GOD ZVSyáF]HVQ\FK WUDQ]\VWRUyZ 9026

EDUG]R PDáH GODWHJR VSUDZQRü HQHUJHW\F]QD Z]PDFQLDF]D PR*H E\ü

GX*D -HGQDN SU]\ GX*\FK F]VWRWOLZRFLDFK SU]HáF]DQLD QLH PR*QD

SRPLQü HIHNWyZ SU]HáF]DQLD WUDQ]\VWRUyZ NWyUH WR HIHNW\ SRZRGXM

GRGDWNRZHVWUDW\HQHUJLLREQL*DMFHVSUDZQRüZ]PDFQLDF]D

8SURV]F]RQ\ VFKHPDW LGHRZ\ VWRSQLD NRFRZHJR Z]PDFQLDF]D

NODV\ ' Z F] ] NRPXWDFM SUGX SU]HGVWDZLRQR QD U\VD ]D

LGHDOL]RZDQH SU]HELHJL F]DVRZH QDSLü L SUGyZ Z XNáDG]LH SU]\

SRPLQLFLXHIHNWyZSU]HáF]DQLDQDU\VE

5

/U

X

LQ

Ω

W

7 2

L

'1

X

'61

0

0

0

7

Ω

W

8

'6 PLQ

(

)

π

8

8

''

'6

−

PLQ

Ω

W

L

'2

X

'62

0

0

Ω

W

(

)

π

8

8

''

'6

−

PLQ

Ω

W

,

''

,

''

0

Ω

W

8

'6 PLQ

(

)

π

8

8

''

'6

−

PLQ

X

/

&

/

/

1

X

'61

X

'6 2

+

8

''

,

''

L

'1

L

'2

X

LQ

X

LQ

X

/

M

1

M

2

D

E

5\V :]PDFQLDF] NODV\ ' ZF] ] NRPXWDFM SUGX D VWRSLH NRFRZ\

ESU]HELHJLF]DVRZHZXNáDG]LH

:F]DVLHWUZDQLDGRGDWQLHJRSyáRNUHVXQDSLFLDVWHUXMFHJR

u

in

]RVWDMH ]DáF]RQ\ WUDQ]\VWRU 0

D Z\áF]RQ\ 0

3U]H] ]DáF]RQ\

WUDQ]\VWRU 0

Sá\QLH SUG VWDá\ Z\PXV]RQ\ SU]H] GáDZLN /

ZLHONLHM

F]VWRWOLZRFL R GX*HM LQGXNF\MQRFL : F]DVLH WUZDQLD XMHPQHJR

SyáRNUHVX QDSLFLD VWHUXMFHJR WUDQ]\VWRU 0

]RVWDMH Z\áF]RQ\ D

WUDQ]\VWRU 0

]RVWDMH ]DáF]RQ\ SU]HMPXMF SU]HZRG]HQLH SUGX ]

]DVLODF]D Sá\QFHJR SU]H] LQGXNF\MQRü

L

1

: Z\QLNX WDNLHJR

VWHURZDQLDUyZQROHJá\REZyGUH]RQDQVRZ\]RVWDMHSREXG]RQ\SU GHP

R NV]WDáFLH V\PHWU\F]QHM IDOL SURVWRNWQHM L QD UH]\VWDQFML REFL*HQLD

RWU]\PXMH VL SU]HELHJ VLQXVRLGDOQ\ 1DSLFLH GUHQ (UyGáR

u

DS

WUDQ]\VWRUDRGFLWHJRQLHMHVWVWDáHOHF]PDNV]WDáWSRáyZNLVLQXVRLG\R

DPSOLWXG]LH

(

)

π

U

U

DD

DS

−

min

UHGQLD ZDUWRü QDSLFLD QD

LQGXNF\MQRFL

L

1

]D MHGHQ RNUHV Z\QRVL ]HUR ]D FKZLORZD ZDUWRü

QDSLFLD QD LQGXNF\MQRFL

L

1

MHVW Uy*QLF PRGXáX QDSLFLD

VLQXVRLGDOQHJRLVWDáHJRQDSLFLD]DVLODMFHJR

U

DD

1DSLFLD SUG\ L PRFH Z XNáDG]LH SU]\ SRPLQLFLX HIHNWyZ

SU]HáF]DQLDRNUHODMQDVWSXMFH]DOH*QRFL

−

1DSLFLHLSUGZUH]\VWDQFMLREFL*HQLD

(

)

X

8

8

W

/

''

'6

=

−

π

PLQ

VLQ

Ω

(

)

L

8

8

5

W

/

''

'6

/U

=

−

π

PLQ

VLQ

Ω

−

0RFX*\WHF]QD

(

)

P

U

U

R

L

DD

DS

Lr

=

−

π

2

2

2

min

−

UHGQLSUGGRVWDUF]RQ\]]DVLODF]DSUGGáDZLNDZF]

/

1

(

)

I

U

U

R

DD

DD

DS

Lr

=

−

π

2

2

min

−

0RFGRVWDUF]RQD]]DVLODF]D

(

)

P

U

U

U

R

D

DD

DD

DS

Lr

=

−

π

2

2

min

−

1DSLFLHQDGáDZLNXZF]

L

1

(

)

u

U

U

t

DD

DS

1

2

1

=

−

−









min

sin

π

Ω

−

6SUDZQRüZ]PDFQLDF]D

η =

= −

P

P

U

U

L

D

DS

DD

1

min

=SRUyZQDQLD]DOH*QRFLLZ\QLND*HSU]\SRPLQLFLX

SURFHVyZSU]HáF]DQLDVSUDZQRüZ]PDFQLDF]D]NRPXWDFMSUGXMHVW

QLHFR ZLNV]D QL* Z]PDFQLDF]D ] NRPXWDFM QDSLFLD : XNáDG]LH ]

NRPXWDFMSUGXQDRGFLW\PWUDQ]\VWRU]HZ\VWSXMHGRüGX*DZDUWRü

V]F]\WRZDQDSLFLDUyZQD

(

)

π

U

U

DD

DS

−

min

1LHV\PHWU\F]Q\Z]PDFQLDF]NODV\(ZF]

: SUDNW\FH VSUDZQRü HQHUJHW\F]QD RSLVDQ\FK ZF]HQLHM

Z]PDFQLDF]\ NODV\ ' VLJD QLHFR SRZ\*HM SU]\ PDNV\PDOQHM

F]VWRWOLZRFL SUDF\ QLH SU]HNUDF]DMFHM QD RJyá NLONXQDVWX 0+]

'DOV]H]ZLNV]HQLHVSUDZQRFLHQHUJHW\F]QHMRUD]UR]V]HU]HQLH]DNUHVX

F]VWRWOLZRFLRZHJRSUDF\X]\VNDQRZHZ]PDFQLDF]DFKNODV\(ZF]

G]LNL Z\HOLPLQRZDQLX VWUDW PRF\ ]ZL]DQ\FK ] SURFHVDPL ]DáF]DQLD

WUDQ]\VWRUyZ Z]PDFQLDF]H NODV\ ( ] UyZQROHJá\P L V]HUHJRZ\P

NRQGHQVDWRUHP OXE Z\áF]DQLD WUDQ]\VWRUD Z]PDFQLDF]H NODV\ ( ]

V]HUHJRZLQGXNF\MQRFL:DGQLHV\PHWU\F]Q\FKZ]PDFQLDF]\NODV\

(MHVWGRüGX*D]DZDUWRüKDUPRQLF]Q\FKZV\JQDOHZ\MFLRZ\PJG\

GREURü UH]RQDQVRZHJR REZRGX REFL*DMFHJR MHVW PDáD FR MHVW

SRZRGRZDQH QLHV\PHWU\F]Q\P SU]HáDGRZ\ZDQLHP WHJR REZRGX :

FHOX Z\HOLPLQRZDQLD WHM ZDG\ UHDOL]RZDQH V UyZQLH* SU]HFLZVREQH

Z]PDFQLDF]HNODV\(ZF]

1DV]H

UR]ZD*DQLD

RJUDQLF]\P\

GR

QLHV\PHWU\F]QHJR

Z]PDFQLDF]D NODV\ ( ] UyZQROHJá\P NRQGHQVDWRUHP NWyUHJR

XSURV]F]RQ\ VFKHPDW LGHRZ\ ZUD] ] MHJR PRGHOHP SU]HGVWDZLRQR QD

U\VD1DU\VESU]HGVWDZLRQRLGHDOL]RZDQHSU]HELHJLF]DVRZH

ZXNáDG]LHRGSRZLDGDMFHRSW\PDOQ\PZDUXQNRPSUDF\Z]PDFQLDF]D

SU]\NWyU\FK]RVWDMFDáNRZLFLHZ\HOLPLQRZDQHVWUDW\PRF\]ZL]DQH]

]DáF]DQLHPWUDQ]\VWRUD :HZ]PDFQLDF]XW\PVWUDW\PRF\]ZL]DQH]

]DáF]DQLHP WUDQ]\VWRUD ]RVWDá\ Z\HOLPLQRZDQH G]LNL RGSRZLHGQLR

GREUDQHPX REZRGRZL GUHQRZHPX NWyU\ ]DSHZQLD *H SU]HMFLH

WUDQ]\VWRUD ]H VWDQX RGFLFLD GR VWDQX ]DáF]HQLD RGE\ZD VL ]

SRPLQLFLHPVWDQXDNW\ZQHJR6SURZDG]HQLHW\FKVWUDWGR]HUDZ\PDJD

MHGQRF]HVQHJRVSHáQLHQLDZDUXQNyZDE\QDSLFLHGUHQX

u

DS

LSRFKRGQD

WHJRQDSLFLDZFKZLOLZáF]DQLDWUDQ]\VWRUDE\á\UyZQH]HUX>@

( )

(

)

( )

( )

X

W

8

Q

GX

W

G

W

'6

'6

'6

W

Q

ω

π

ω

ω

π

=

=

=










=

2

0

0

2

Q

Ω

-DN SRND]DQR QD U\VF VSHáQLHQLH SRZ\*V]\FK ZDUXQNyZ

SRZRGXMH *H Z FKZLOL ]DáF]DQLD WUDQ]\VWRUD SUG GUHQX ]DF]\QD

áDJRGQLHQDUDVWDüRG]HUDSU]\QDSLFLX

u

DS

=

0

FRZHIHNFLHSURZDG]L

GR Z\HOLPLQRZDQLD SURFHVX SU]HMFLRZHJR $E\ X]\VNDü QDMZLNV]

PRFZ\MFLRZZ]PDFQLDF]DZVSyáF]\QQLNZ\SHáQLHQLDSURVWRNWQHJR

SU]HELHJXVWHUXMFHJRSRZLQLHQZ\QRVLü

X

'6

X

LQ

,

''

L

6

L

'

&

1

&

2

/

2

5

/

/

G á

' Z F]

á

+

8

''

X

5

X

;

X

6

&

1

5

/

X

5

X

;

X

6

X

'6

8

&&

L

'

,

''

/

G á

L

6

&

2

/

2

;

/

=

ω

π

2

π

X

LQ

.OXF]

]ZDUW\

.OXF]

UR]ZDUW\

ω

W

π

2

π

ω

W

,

''

π

2

π

ω

W

π

2

π

ω

W

L

6

L

'

π

2

π

ω

W

π

2

π

ω

W

8

'6 PD[

X

'6

0

0

0

0

0

0

,

' PD[

L

&1

a)

b)

c)

L

&1

L

&1

U

'6

5\V 1LHV\PHWU\F]Q\ Z]PDFQLDF] NODV\ ( ] UyZQROHJá\P NRQGHQVDWRUHP

DVWRSLHNRFRZ\EXSURV]F]RQ\PRGHOZ]PDFQLDF]DFLGHDOL]RZDQH

SU]HELHJLF]DVRZHZXNáDG]LH

3U]HSURZDG]RQD DQDOL]D WHRUHW\F]QD XNáDGX Z SUDFDFK > @

SURZDG]L GR QDVWSXMF\FK ]ZL]NyZ PLG]\ ZDUWRFLDPL HOHPHQWyZ

REZRGXGUHQRZHJRSU]\NWyU\FKVSHáQLRQHVZDUXQNL

&

5

4

4

/

/

/

1

2

1

5 447

1

0 81

4

=

+

+











ω

C

Q

R

Q

L

L

L

2

1

1

111

1 7879

=

+

−











ω

,

,

L

Q R

L

2

=

ω

JG]LH Q

L

R

L

= ω

2

GREURüV]HUHJRZHJRREZRGXUH]RQDQVRZHJR

-DN ZLGDü ] SRZ\*V]\FK ]DOH*QRFL REZyG NROHNWRURZ\

GUHQRZ\GODF]VWRWOLZRFLSUDF\SRVLDGDFKDUDNWHULQGXNF\MQ\WM

X

L

X

C

L

C

=

>

=

ω

ω

2

2

1

: FHOX RNUHOHQLD SROD FKDUDNWHU\VW\N HOHPHQWX DNW\ZQHJR

SUDFXMFHJRMDNRNOXF]SU]\' PR*HP\Z\NRU]\VWDüQDVWSXMFH

]ZL]NL

−

PDNV\PDOQ\SUGWUDQ]\VWRUD

I

I

I

D

DD

DD

max

,

=

+ +









 ≈

π

2

4

2

1

2 8611

−

PDNV\PDOQHQDSLFLHQDWUDQ]\VWRU]H

U

U

U

DS

DD

DS

max

min

,

,

=

−

3 562

2 562

JG]LH

I

DD

SUG]DVLODQLD

U

DS min

VSDGHNQDSLFLDQD]DáF]RQ\PWUDQ]\VWRU]H

3DUDPHWU\Z\MFLRZHZ]PDFQLDF]DRNUHODM]DOH*QRFL

−

DPSOLWXGDQDSLFLDZ\MFLRZHJR

(

)

U

U

U

Lm

DD

DS

=

+

−

4

4

2

π

min

−

DPSOLWXGDSUGXZ\MFLRZHJR

I

U

R

U

U

R

Lm

Lm

L

DD

DS

L

=

=

+

−

4

4

2

π

min

−

PRFZ\MFLRZD

P

U

I

U

U

R

L

Lm

Lm

DD

DS

L

=

=

+

−

2

8

4

2

2

π

(

)

min

−

SUG]DVLODQLD

I

U

U

R U

DD

DD

DS

L

DD

=

−

+

8

4

2

2

(

)

(

)

min

π

η

−

VSUDZQRüZ]PDFQLDF]D

( )

( )

( )

η

π

π

π

=

−

−









 + −











−

1

2

3

1

2

3

1

2

2

2 3

A

U

U

A

A

A

DS

DD

min

/

JG]LH

A

Q

f t

f

= +

+







1

0 82

1 1525

,

,

D

t

f

F]DVRSDGDQLDSUGXGUHQXSU]\Z\áF]DQLXWUDQ]\VWRUD

−

PRF]DVLODQLD

P

P

U

U

R

D

L

DD

DS

L

=

=

+

−

η π

η

8

4

2

2

(

)

min

2JUDQLF]HQLDF]VWRWOLZRFLRZHZ]PDFQLDF]DNODV\(ZL*VL]

SRMHPQRFL Z\MFLRZ WUDQ]\VWRUD NWyUD áF]QLH ] SRMHPQRFL

GáDZLND ]DVLODMFHJR L SRMHPQRFL PRQWD*X WZRU]\ SHZQ PLQLPDOQ

SRMHPQRü

C

1min

SU]\ NWyUHM PR*OLZH MHVW MHV]F]H VNRQVWUXRZDQLH

REZRGX GUHQRZHJR R RSW\PDOQ\FK ZDUWRFLDFK HOHPHQWyZ

0DNV\PDOQD F]VWRWOLZRü SUDF\ Z]PDFQLDF]D NODV\ ( MHVW RNUHORQD

Z]RUHP>@

(

)

f

R C

R C

L

L

max

min

min

,

=

+

≈

4

4

0 02922

2

2

1

1

π π

: SUDNW\FH VSUDZQRü Z]PDFQLDF]D NODV\ ( Z F] PR*H RVLJDü

ZDUWRü QDZHW ']LNL VZ\P ZáDFLZRFLRP UH]RQDQVRZH

Z\VRNRVSUDZQH Z]PDFQLDF]H NODV\ ' L ( Z F] V FRUD] V]HU]HM

Z\NRU]\VW\ZDQHZUy*QRURGQ\FKXU]G]HQLDFKGRSU]HWZDU]DQLDHQHUJLL

Z\VRNRVSUDZQH UH]RQDQVRZH SU]HWZRUQLFH QDSL FLD VWDáHJR

Z]PDFQLDF]H UH]RQDQVRZH Z F] SURVWRZQLNL UH]RQDQVRZH ZF]

VWHURZDQHSURVWRZQLNLV\QFKURQLF]QHZF]DZDU\MQH]DVLODF]HQDSL FLD

SU]HPLHQQHJR LWS FR SR]ZDOD QD PLQLDWXU\]DFM L ]ZLNV]HQLH

QLH]DZRGQRFLW\FKXU]G]H