Estabilidad de Jury, Liapunov

 

;Knëcgsgs `j jstkhgcg`k` `j ces sgstjdks jn ckze mjrrk`e, ckze khgjrte,

jstkhgcg`k` `j Oury, Cgkpunev y ck iunmgýn `j sjnserjs jn un sgstjdk `j mentrec.

Ounge 82;7. Zngvjrsg`k` Pjmnecýagmk `j Rknkdë. Ikmuctk` `j gnajngjräk Jcçmtrgmk. Mentrec GG  

 

Khstrkmt

Gn tfgs `emudjnt, gt prjsjnt tfj `giijrjnt djtfe`s tfkt krj usj` te jvkcuktj tfj stkhgcgty ei  tfj systjds gn tfj `gsmrjtj tgdj, tfjrj gs k hrgji `giijrjnmj hjtwjjn tfj systjds gn tfj mentgnueus tgdj kn` tfj `gsmrjtj tgdj kn` tfj tjmfngqujs njmjsskry te kppcy gn tfj `gsmrjtj tgdj, tfj djtfe`s us usj` j` gn mentg mentgnu nueu euss tgdj tgdj (X (Xeu eutf tf-F -Fur urwg wgtz tz kn kn` ` ). Pfgs Pfgs tj tjmfn mfngq gquj uj me mens nsgs gsts ts ei tf tfjj hg hgcg cgnjk njkr  r  trknsierdktgen wfjrj tfj ei tfj rk`gusen hjmedjs tfjtgdj gdkagnkry gn tfj njw  pcknj. Gt kcse prjsjnts tfj mgrmudijrjnmj djtfe`s tfkt krj usj`ungt `grjmtcy k `gsmrjtj systjdkxgs ks tfj mksj ei  tfj Oury°s stkhgcgty mrgtjrgk kn` tfj Cykpunev°s stkhgcgty mrgtjrgk kn` ignkccy jxpckgns tfj hksgm epjrktgen ei sedj typjs ei sjnsers tfkt mkn hj usj` gn k systjd ei mentrec kn` tfj jiijmt tfjy fkvj en tfj stkhgcgty ei k systjd.

 

Pkhck `j Mentjng`es

Mkpätuce ; Gntre`ummgýn...................................................................................................................; Mkpätuce 8 Mentjng`e........................................................................................................... Mentjng`e......................................................... .......................................................... ...........8 ...8 Jstkhgcg`k` `j sgstjdks jn ckze mjrrk`e y ckze khgjrte jn tgjdpe `gsmrjte.................................8 Mrgtjrge `j Xeutf-Furwgtz.......................................................................................................0 Dçte`e cuakr `j `j cksOury.....................................................................................................6 rkämjs.................................................................................................9 Mrgtjrge `j `jc jstkhgcg`k` Mrgtjrge `j Jstkhgcg`k` `j Cgkpunev..........................................................................................;; Knëcgsgs `j jstkhgcg`k` `j Cgkpunev `j sgstjdks jn tgjdpe `gsmrjte....................................;8 Knëcgsgs `j jstkhgcg`k` `j Cgkpunev `j ces sgstjdks cgnjkcjs jn tgjdpe `gsmrjte j gnvkrgkntjs jn jc tgjdpe............................................................................................................................;8 ]jnserjs.....................................................................................................................................;: Mkpätuce = Menmcusgenjs................................................................................................... Menmcusgenjs............................................................... ............................................ ............;7 ....;7 Xjijrjnmgks....................................................................................................................................82

 

 

Cgstk `j igaurks

Igaurk ;.@gkarkdk `j hcequjs `j un ckze khgjrte...........................................................................8 Igaurk 8. @gkarkdk `j hcequjs `j un ckze mjrrk`e.........................................................................8 Igaurk =. @gkarkdk `j hcequjs `j un ckze mjrrk`e `j un sgstjdk `gsmrjte.....................................= Igaurk 0. Cuakr ajedçtrgme `j cks rkämjs men P4 ; sja...................................................................1

 

Mkpätuce ; Gntre`ummgýn

Jc kspjmte `gnëdgme dës gdpertkntj `j un sgstjdk js su jstkhgcg`k`, ck jstkhgcg`k` js ck mkpkm mkpkmg` g`k` k` quj tgjnj tgjnj un sg sgst stjd jdkk pkrk pkrk kder kdertg tgau aukr kr me men n jc tg tgjd jdpe pe y kn knuc uckr kr ck ckss esmgck esm gckmge mgenjs njs `j ck rjspujs rjspujstk tk kntj kntj unk pjrtur pjrturhkm hkmgýn gýn.. Ck jstkhg jstkhgcg` cg`k` k` jstë jstë suojtk suojtk k ck uhgmkmgýn `j cks rkämjs `j ck jmukmgýn mkrkmtjrästgmk `jc sgstjdk, js `jmgr, pkrk quj un sgstjdk sjk jstkhcj, ces peces `j ck iunmgýn `j trknsijrjnmgk `jhjn jstkr jn jc sjdgpckne gzqugjr`e `jc pckne s, e sg js un sgstjdk `gsmrjte `jntre `jc märmuce `jc pckne z. jn ajnjrkc un sgstjdk js jstkhcj sg jxmgtk`e men unk jntrk`k kmetk`k, ck skcg`k pjrdknjmj kmetk`k. K mentgnukmgýn, sj prjsjntkrën kcaunes dçte`es duy ÿtgcjs pkrk jvkcukr ck jstkhgcg`k` `j ces sgstjdks `j mentrec jn tgjdpe mentgnue.

 

Mkpätuce 8 Mentjng`e Jstkhgcg`k` `j sgstjdks jn ckze mjrrk`e y ckze khgjrte jn tgjdpe `gsmrjte.

Pknte jn ckze khgjrte mede jn ckze mjrrk`e, pkrk quj un sgstjdk `j tgjdpe mentgnue sjk jstkhcj sj `jhj mudpcgr quj ces peces `jc sgstjdk `jhjn tjnjr su pkrtj rjkc njaktgvk. njaktg vk. Jn un sgstjdk `j tgjdpe tgjdpe mentgnue `j ckze khgjrte sj tgjnj ck sgaugjntj iunmgýn iunmgýn mkrkmtjrästgmk? A ( s )4

W  ( ( s ) Z ( s )

 Igaurk ;.@gkarkdk ;.@gkarkdk `j hcequjs `j un ckze khgjrte

W sg sj rjkcgdjntk jc sgstjdk men  F   (( s ) jc sgstjdk quj`k mede?

W ( s )   A ( s) 4  X ( s ) ; + A ( s ) F  ( ( s )

 Igaurk 8. @gkarkdk `j hcequjs `j un ckze mjrrk`e  mjrrk`e 

 

 Igaurk =. @gkarkdk `j hcequjs `j un ckze mjrrk`e `j un sgstjdk `gsmrjte

Ck pesgmgýn `j cks rkämjs `j ck jmukmgýn mkrkmtjrästgmk µ), `jtjrdgnk ck jstkhgcg`k` `jc sgstjdk. ]g kferk, sj dujstrjk ck sjûkc `j jrrer y sj utgcgzk un rjmenstrumter `j er`jn mjre, sj ehtgjnj jc sgstjdk `j mentrec `j tgjdpe tgjdpe `gsmrjte `j ck igaur igaurkk =. Muyk iunmgýn `j trknsijrjnmgk js?

P ( z )4

  A ( z ) ; + AF ( z )

Rkrk knkcgzkr ck jstkhgcg`k` `j un sgstjdk `gsmrjte sj tgjnj quj skhjr `ýn`j `jhjn jstkr ks rkämjs `j ck jmukmgýn mkrkmtjrästgmk (;+AF(z)) pkrk quj jc sgstjdk tjnak unk rjspujstk trknsgtergk quj sj jxtgnak jn jc tgjdpe. Rkrk jcce sj mkcmuck ck rjspujstk `jc sgstjdk jn ckze mjrrk`e. d

  A ( z ) W  ( ( z )4  X ( z )4 ; + AF   (( z  z )

∅ ( z ∘ z ) 4

l 

g

g d

;

 X ( z )

( z z ∘ R ) ∅ 4 g

g

;

[uj dj`gkntj jc `jskrrecce jn irkmmgenjs pkrmgkcjs puj`j ccjvkrsj k ck ierdk?   l g z

  l n z

W  ( z )4 z ∘ R g + … +  z ∘ Rn + W  X ( z )

 

@en`j W  X ( z )  mentgjnj ces tçrdgnes `j W(z) ergagnk`es jn ces peces `j ck jxmgtkmgýn X(z). Ces prgdjr prgdjres es n tçrdgn tçrdgnes es `j ck jxprjs jxprjsgýn gýn kntjrg kntjrger er rjprjs rjprjsjnt jntkn kn ck rjspuj rjspujstk stk trknsgtergk `j W(z). Rkrk quj jc sgstjdk sjk jstkhcj, ck kntg trknsierdk`k z `j jstes tçrdgnes `jhj tjn`jr k mjre mukn`e sj gnmrjdjntk jc tgjdpe. Ck trknsierdk`k z gnvjrsk

∘;

`j jstk jmukmgýn js _ 

{ }   l g z

 z ∘ Rg

4l g ( Rg )lP 

Rkrk quj jstj tçrdgne sj knucj kc gnmrjdjntkrsj l, jc dý`uce `j  Rg `jhj sjr djner quj ck ung`k`. Pjngjn`e jn mujntk quj ces vkcerjs  Rgmerrjspen`jn k cks rkämjs `j ck jmukmgýn mkrkmtjrästgmk, sj menmcuyj quj pkrk quj un sgstjdk rjkcgdjntk`e `j mentrec jn tgjdpe `gsmrjte sjk jstkhcj, cks rkämjs `j su jmukmgýn mkrkmtjrästgmk `jhjn jstkr uhgmk`ks jn jc gntjrger `jc märmuce `j rk`ge ungtkrge. Cks tçmngmks `j knëcgsgs `j ck jstkhgcg`k` `j sgstjdks `j tgjdpe mentgnue sj hkskn jn quj jc cädgtj `j ck rjagýn `j jstkhgcg`k` s merrjspen`j kc joj gdkagnkrge gdkagnkrge ow. Jstks tçmngmks tçmngmks ne puj`jn sjr kpcgmk`ks k sgstjdks sgstjdks `j tgjdpe tgjdpe `gsmrjte, `gsmrjte, yk quj jc cädgtj `j ck rjagýn `j jstkhgcg`k` jstkhgcg`k` jn jc pckne z merrjspen`j merrjspen`j kc märmuce ungtkrge. ungtkrge. ]gn jdhkrae, sg sj jdpcjk ck trknsierdkmgýn hgcgnjkc `j ignk`e per?

w4

 z ∘; P   z + ;

 8

 ck mgrmunijrjnmgk `j rk`ge ungtkrge jn z sj trknsierdk jn jc joj gdkagnkrge jn jc nujve  pckne w. Jn sgstjdks `j tgjdpe mentgnue jxgstjn tçmngmks pkrk jvkcukr ck jstkhgcg`k`, tkcjs mede Xeutf-Furwgtz y jc dçte`e `jc cuakr `j cks rkämjs.\ MGPKPGEN mkt;: Sc 9;:0 T Mrgtjrge `j Xeutf-Furwgtz

 

 Jc mrgtjrge `j jstkhgcg`k` `j Xeutf `gmj sg jxgstjn e ne rkämjs gnjstkhcjs jn unk jmukmgýn pecgnedgkc, sg tjnjr quj ehtjnjrcks jn rjkcg`k`. Jstj mrgtjrge `j jstkhgcg`k` sýce sj kpcgmk k ces pecgnedges men unk mkntg`k` igngtk `j tçrdgnes. Mukn`e sj kpcgmk jc mrgtjrge k un sgstjdk `j mentrec, ck gnierdkmgýn sehrj ck jstkhgcg`k` khsecutk sj ehtgjnj `grjmt `gr jmtkdj kdjntj ntj `j ces mejigmg mejigmgjnt jntjs js `j ck jmukmgý jmukmgýn n mkr mkrkmt kmtjrä jräst stgmk\ gmk\ MGPKPG MGPKPGEN EN Eak;2 Eak;2 Sc 9;:0 T. Jc mrgtjrge sj jxpcgmk k mentgnukmgýn? ;. ]j jsm jsmrg rghj hj jc jc pecg pecgne nedg dge e jn w 8. ]g kcaune kcauness `j ces mejigm mejigmgj gjnt ntjs js js mjre mjre e njakt njaktgv gve, e, kntj ck pr prjs jsjn jnmg mgkk `j kc djnes un mejigmgjntj pesgtgve, fky unk rkäz e rkämjs gdkagnkrgks e quj tgjnjn  pkrtjs rjkcjs pesgtgvks, jn tkc mkse jc sgstjdk js gnjstkhcj gnjstkhcj =. Pe`es ces mejigmgjntj mejigmgjntjss `jhjn sjr pesgtg pesgtgves, ves, jst jstkk men`gmgýn men`gmgýn js njmjskrgk njmjskrgk pjre ne suigmgjntj pkrk ck jstkhgcg`k` `j un sgstjdk. 0. ]g te`es te`es ces mejigmg mejigmgjnt jntjs js sen pesgtgv pesgtgves es y ne nuces nuces sj premj`j premj`j k menstru menstrugr gr ck sgaug sgaugjn jntj tj tk tkhc hckk me men n ces ces me meji jigm gmgj gjnt ntjs js `j ck iu iunm nmgý gýn. n. Mede Mede sj dujs dujstr trkk k mentgnukmgýn?

0

w = w

k2   k8   k0 k;   k= h;   h8

w

8

w

;

m;

w

2

`;

8

=

0

]jk jc pecgnedge k 2 w

+ k; w + k8 w + k= w +k 042

h ;4

k; k 8∘k2 k=

h 84

k; k; k0∘ k2 k: k;

  k :42 y ne sj mkcmuck

 

m ;4

 

h; k=∘ k; h8 h;

; 8 ; 8 ` ;4 m h ∘h m   m 842  y ne sj mkcmuck m;

 

Jc mrgtjrge `j Xeutf jstkhcjmj quj ck mkntg`k` `j rkämjs pesgtgvks e men pkrtj rjkc  pesgtgvk, js gaukc k ck mkntg`k` `j mkdhges `j sganes jn ces mejigmgjntjs `j ck prgdjrk mecudnk, sg ne fky mkdhges `j sganes jc sgstjdk js jstkhcj.\ MGPKPGEN pre;0 Sc 9;:0 T Dçte`e `jc cuakr `j cks rkämjs ]upengjn`e quj jc sgstjdk js cgnjkc y men pkrëdjtres gnvkrgkntjs jn jc tgjdpe, cks

trkns trknsij ijrj rjnm nmgk gkss `j ck ckze ze kh khgj gjrt rte e AF(z AF(z)) pu puj`j j`j jxprj jxprjsk skrs rsjj me mede de un me memg mgjnt jntjj `j `es  pecgnedges jn z kijmtk`es per unk aknknmgk L  d

∅ ( z ∘ z ) 4

l 

g

g d

AF ( z )4

;

2 ;

;

;

2

 

K pkrtgr `j ck jmukmgýn kntjrger puj`j menstrugrsj jc sgaugjntj krrjace?

@ýn`j

| | |

h l 4

m l 4

` l 4

 |

k 2   k n∘l 

k n   kl 

 |  |

 h 2   hn∘;∘l 

hn∘;   h l 

  m2   m n∘8∘l  m n∘8   m l 

Ck men`gmgýn njmjskrgk y suigmgjntj pkrk quj jc pecgnedge I(z) ne tjnak rkämjs iujrk e sehrj jc märmuce ungtkrge, men k n> 2, sen?

 I ( ; ) > 2

(∘; )n . I ( ; ) > 2

|k |3kn 2

|h |>|h n∘ | 2

;

|m |>|m n∘ | 2

8

 

|` |>|` n∘ | 2

=

]j puj`j netkr quj un sgstjdk `j sjaun`e er`jn mentgjnj unk seck igck jn jc krrjace. Rer mk`k er`jn k`gmgenkc, `es igcks k`gmgenjs sen karjak`ks. K`jdës, pkrk un sgstjdk `j er`jn n, fky un tetkc `j n+; rjstrgmmgenjs. Rkrk gcustrkr jstj jojdpce sj puj`j utgcgzkr jc prehcjdk knkcgzk`e jn jc dçte`e `j cuakr ajedçtrgme `j cks rkämjs, jste js? A ( s )4

  ;

s ( s+; )

Pedkn`e un P 4 2.;sja, ck trknsierdk`k z quj`k jxprjsk`k mýde? A ( z ) 4 L   2.22060 z + 2.22096 ( z ∘; )( z ∘2.72: ) Ck jmukmgýn mkrkmtjrästgmk `j ck jmukmgýn I(z) jstë `k`k per?

 I ( z )4; + A ( z )4 z + ( 2.=96 L ∘;.=96 ) z + ( 2.=96 + 2.890 L )4 2 8

@k`e quj sj trktk `j unk jmukmgýn `j sjaun`e er`jn, jc krrjace `j Oury sj rj`umj k?

k)   I ( ; )4; + ( 2.=96 L ∘;.=96 ) + ( 2.=96 +2.890 ) > 2 Jstk men`gmgýn js sktgsijmfk pkrk L>2.  h)   (∘; )  I (∘; ) 4;∘( 2.=96 L ∘ ;.=96 )+ ( 2.=96 + 2.890 )> 2 8

Jstk men`gmgýn js sktgsijmfk pkrk L389.=. m)   |k 2|4|2.=96 +2.890 L |3 k84;

 

Jstk men`gmgýn js sktgsijmfk pkrk  L 3 8.=7.

Rer ce tknte, jc sgstjdk js jstkhcj pkrk vkcerjs `j ck aknknmgk medprjn`g`es jntrj 2 3L3 8.=7. Rkrk jc vkcer `j L merrjspen`gjntj kc cädgtj `jc rknae `j jstkhgcg`k`, L 4 8.=7, ck jmukmgýn mkrkmtjrästgmk js? 8

 z ∘2.07 z + ;42 ]gjn`e sus rkämjs? _ 42.800 ∝  o 2.71 Jstj rjsuctk`e megnmg`j men jc rjsuctk`e ehtjng`e `jc dçte`e `jc cuakr ajedçtrgme `j cks rkämjs.\ MGPKPGEN mkt;: Sc 9;:0 T Mrgtjrge `j Jstkhgcg`k` `j Cgkpunev

Jc knëcgsgs `j jstkhgcg`k` `j Cgkpunev oujak un pkpjc gdpertkntj jn jc knëcgsgs `j ck jstkhgcg`k` `j sgstjdks `j mentrec `jsmrgtes per jmukmgenjs jn jc jspkmge `j jstk`e. Jxgstjn 8 dçte`es `j knëcgsgs `j jstkhgcg`k` `j Cgkpunev, cckdk`es prgdjr dçte`e y sjaun`e dçte`e. Kdhks `j kpcgmkn pkrk `jtjrdgnkr ck jstkhgcg`k` `j ces sgstjdks `j ces sgstjdks `gnëdgmes `jsmrgtes per jmukmgenjs `gijrjnmgkcjs e jn `gijrjnmgks er`gnkrgks. Jc  prgdjr dçte`e jstë ierdk`e per premj`gdgjntes jn cks jmukmgenjs jmukmgenj s `gijrjnmgkcjs e `j cks jmukmgenjs jn `gijrjnmgks pkrk jc knëcgsgs. Rer etrk pkrtj, jc sjaun`e dçte`e ne rjqugjrj `j cks secumgenjs `j cks jmukmgenjs `gijrjnmgkcjs e jn `gijrjnmgks, per ce quj rjsuctk dës ÿtgc jn ck prëmtgmk. Ck vjntkok `j uskr jstj dçte`e js quj jn `gijrjnmgk ces dçte`es knkcgzk`es kntjrgerdjntj, jstj ne jstë cgdgtk`e k sgstjdks gnvkrgkntjs jn jc tgjdpe. Jn

 

 pkrtgmuckr. Jc sjaun`e dçte`e `j Cgkpunev js gn`gspjnskhcj pkrk jc knëcgsgs `j jstkhgcg`k` `j sgstjdks ne cgnjkcjs jn ces quj cks secumgenjs jxkmtks ne sen pesghcjs. Jc sjaun`e dçte`e `j Cgkpunev? @j ck tjeräk `j ck djmëngmk mcësgmk, sj skhj quj un sgstj sgstjdk dk vghrkt vghrkterg erge e js jstkhc jstkhcjj sg su jnjraä jnjraäkk tet tetkc kc sj rj`umj rj`umj mentgn mentgnukd ukdjnt jntj, j, fkstk fkstk kcmknzkr un jstk`e `j jqugcghrge. Jc sjaun`e dçte`e `j Cgkpunev sj hksk jn unk ajnjrkcgzkmgýn `j ce kntjrger? sg jc sgstjd sgs tjdkk tgjnj tgjnj un jstk`e jstk`e `j jqugcgh jqugcghrge rge ksgntý ksgntýtg tgmkd mkdjnt jntjj jstkhc jstkhcj, j, jnt jntenmj enmjss ck jnjraäk jnjraäk kcdkmjnk`k jn çc `jspckzk su vkcer dängde jn jc jstk`e `j jqugcghrge. ]gn jdhkrae, pkrk sgstjdks purkdjntj dktjdëtgmes, ne jxgstj unk ierdk sgdpcj `j `jigngr unk iunmgýn `j jnjraäk. Rkrk vjnmjr jstk `gigmuctk`k, Cgkpunev gntre`uoe ck iunmgýn `j Cgkpunev, unk iunmgýn igmtgmgk `j jnjraäk. Jstk g`jk js dës ajnjrkc y utgcgzk`k quj ck `j ck jnjraäk. @j fjmfe, mukcqugjr iunmgýn jsmkckr quj sktgsikak cks fgpýtjsgs `j ces tjerjdks `j jstkhgcg`k` `j Cgkpunev, puj`j sjrvgr mede iunmgýn `j Cgkpunev. Knëcgsgs `j jstkhgcg`k` `j Cgkpunev `j sgstjdks jn tgjdpe `gsmrjte.

˗  ( ( x )  sj utgcgzk ck Rkrkk ces sgstjd Rkr sgstjdks ks `j tgjdpe tgjdpe `gsmrj `gsmrjte te jn vjz `j utgcgz utgcgzkr kr ck iunmgý iunmgýn n Y  `gijrjnmgk `grjmtk Y  µ  µjs `jmgr?

∉ Y   (( x ( lP ) )4Y   µµ Pjerjdk. Mensg`jrk jc sgaugjntj sgstjdk jn tgjdpe `gsmrjte

 x ( ( l + ; ) P ) 4i   (( x ( lP )) @en`j x 4 vjmter< i(x) 4 vjmter-n men ck prepgj`k` `j quj i(2) 4 2 y P 4 Rjrge`e `j dujstrje. Knëcgsgs `j jstkhgcg`k` `j Cgkpunev `j ces sgstjdks cgnjkcjs jn tgjdpe `gsmrjte j gnvkrgkntjs jn jc tgjdpe.

 

Mensg`jrj jc sgstjdk jn tgjdpe `gsmrjte `jsmrgte per?

 x ( l + ; )4Ax ( l ) `en`j x js vjmter `j jstk`e (vjmter-n) y A js unk dktrgz ne sgnauckr menstkntj `j n*n. Jc ergajn x 4 2 js jc jstk`e `j jqugcghrge. ]j gnvjstgakrë ck jstkhgcg`k` `j jstj jstk`e dj`gkntj jc sjaun`e dçte`e `j Cgkpunev. ]j jsmeaj mede pesghcj iunmgýn `j Cgkpunev Y   µµ `en`j R js unk dktrgz fjrdätgmk `jigng`k pesgtgvk jntenmjs

∉ Y   (( x ( l ) )4Y  µ  µ @k`e quj Y   (( x ( l ) ) sj sjcjmmgený pkrk sjr `jigng`k pesgtgvk, sj rjqugjrj, pkrk ck jstkhgcg`k` ksgntýtgmk, quj ∉ Y  ( x ( l ) ) sj `jigng`k njaktgvk. Rer tknte µ

∉ Y   (( x ( l ) )4∘ x ( l ) [x ( l ) @ýn`j µ

[ 4∘(A  RA ∘ R )4 `jigng`k pesgtgvk @j jsj de`e, pkrk ck jstkhgcg`k` ksgntýtgmk `jc sgstjdk jn tgjdpe `gsmrjte `j ck jmukmgýn kntjrger, js suigmgjntj quj [ sjk `jigng`k pesgtgvk. Mede Me de jn jc mk mkse se `j sg sgst stjd jdks ks cgnj cgnjkc kcjs js `j tg tgjd jdpe pe men mentg tgnu nue, e, js menvj menvjng ngjn jntj tj jspjmgigmkr prgdjre unk dktrgz [ fjrdätgmk `jigng`k pesgtgvk (e sgdçtrgmk rjkc `jigng`k µ

 pesgtgvk) y k mentgnukmgýn vjr sg ck dktrgz R `jtjrdgnk`k per A  RA ∘ R4∘[ js `jigng`k  pesgtgvk e ne. Ehsjrvj quj unk R `jigng`k pesgtgvk js unk men`gmgýn njmjskrgk y suigmgjntj. ]j rjsudgrë jn un tjerjdk ce quj sj fk jnunmgk`e kquä. Pjerjdk. Mensg`jrj jc sgstjdk jn tgjdpe `gsmrjte

 x ( l + ; )4Ax ( l )

 

@en`j x js jc vjmter jstk`e (vjmter-n) y A js unk dktrgz ne sgnauckr menstkntj `j n*n. Znkk me Zn men` n`gm gmgý gýn n nj njmj mjsk skrg rgkk y suig suigmg mgjn jntj tj pk pkrk rk quj quj jc js jstk tk`e `e `j jq jqug ugcg cghr hrge ge x42 x42 sj sjkk ksgntýtgmkdjntj jstkhcj acehkc js quj, `k`k mukcqugjr dktrgz [ fjrdätgmk `jigng`k pesgtgvk (e sgdçtrgmk rjkc `jigng`k pesgtgvk) jxgstj unk dktrgz R fjrdätgmk `jigng`k pesgtgvk (e µ

sgdçtrgmk rjkc `jigng`k pesgtgvk) tkc quj A  RA∘ R4∘[ µ

Ck iunmgýn jsmkckr  x  Rxjs unk iunmgýn Cgkpunev pkrk jstj sgstjdk. µ

]g ∉ Y   (( x ( l ) )4∘ x ( l ) [x ( l ) ne sj `jskpkrjmj k ce ckrae `j ngnaunk sjrgj `j secumgenjs, jntenmjs [ puj`j sjr jsmeag`k mede sjdg`jigng`k pesgtgvk. K mentgnukmgýn, sj gcustrk jstj dçte`e men un jojdpce? Mensg`jrj jc sgaugjntj sgstjdk?

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@jtjrdgnj ck jstkhgcg`k` `j jc ergajn `jc sgstjdk. µ

]j jsmeaj [ mede G. k mentgnukmgýn, men rjijrjnmgk k ck jmukmgýn A  RA∘ R4∘[ Ck jmukmgýn `j Cgkpunev sj menvgjrtj jn?

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]g sj jnmujntrk quj R js `jigng`k pesgtgvk, jntenmjs jc ergajn x42 js ksgntýtgmkdjntj jstkhc jst khcjj acehkc acehkc.. @j ck jmukmgý jmukmgýn n `j Cgkpune Cgkpunev v kntjrg kntjrger er sj ehtgjn ehtgjnjn jn cks sgaugj sgaugjntj ntjss trjs trjs jmukmgenjs? 2.8:  p 88 2.:

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Kc kpcgmkr jc mrgtjrge `j ]ycvjstjr pkrk prehkr ck `jigngmgýn pesgtgvk `j ck dktrgz R, sj jnmujntrk R `jigng`k pesgtgvk. Rer tknte, jc jstk`e `j jqugcghrge, jc ergajn x 4 2, js ksgntýtgmkdjntj jstkhcj acehkc. Ehsjrvj quj jn vjz `j jsmeajr [ mede G sj pe`räk fkhjr jsmeag`e [ mede unk dktrgz sjdg`jigng`k pesgtgvk, tkc mede [4

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]gjdprj y mukn`e ∉ Y   (( x ( l ) )4∘ x ( l ) [x ( l ) ne `jskpkrjzmk k ce ckrae `j mukcqugjr sjrgj `j secumgenjs. Rkrk ck dktrgz [ sjdg`jigng`k  pesgtgvk rjmgçn `jigng`k, sj tgjnj

∉ Y   (( x )4∘ x

8 8

( l )

Rkrk jc sgstjdk sgstjdk kmtukc,  x 8 ( l )g`çntgmkdjntj mjre gdpcgmk quj  x ; ( l )js g`çntgmkdjn g`çntgmkdjntj tj mjre. Rer tknte, ∉ Y  ( ( x ) ne `jskpkrjmj k ce ckrae `j mukcqugjr sjrgj `j secumgenjs, jxmjpte jn jc ergajn. Rer tknte, sj puj`j jsmeajr jstk dktrgz [ sjdg`jigng`k pesgtgvk pkrk `jtjrdgnkr ck dktrgz dkt rgz R `j ck jmukmg jmukmgýn ýn `j jstkhg jstkhgcg cg`k` `k` `j Cgkpune Cgkpunev. v. Ck jmukmg jmukmgýn ýn `j jstkhg jstkhgcg cg`k` `k` `j Cgkpunev, jn jstj mkse sj menvgjrtj jn?

 

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 p ;;   p;8 4∘ 2 2  p;8   p88

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Kc kpcgmkr jc mrgtjrge `j ]ycvjstjr< sj jnmujntrk quj R js `jigng`k pesgtgvk, per tknte, sj ehtgjn eht gjnjj ck dgsdk dgsdk menmcus menmcusgýn gýn quj kntjs? kntjs? jc ergajn ergajn js ksg ksgntý ntýtgm tgmkdj kdjntj ntj jstkhc jstkhcjj acehkc acehkc.. \ MGPKPGEN Lkt28 Sc 9;:0 T ]jnserjs

Jn un sgstjdk `j mentrec jstë ierdk`e per vkrges jcjdjntes quj rjkcgzkn unk iunmgýn iunmgý n gn`gvg`ukc gn`gvg`ukc,, quj jn menounte menounte pre`umjn pre`umjn un ign pkrk dkntjnjr jstkhcj un sgs sgstjdk tjdk e un premjse. Jstes jcjdjntes puj`jn sjr, sjnserjs, mentreck`erjs y kmtuk`erjs. Jc sgstjdk `j mentrec rjmghj gnierdkmgýn `jc jnterne per dj`ge `j sjnserjs, jsk gnierdkmgýn kpertk `ktes pkrk quj jc mentreck`er `jmg`k, sg fky quj rjkcgzkr unk kmmgýn, jstk sj ccjvk k mkhe  per un kmtuk`er quj tgjnj mkpkmg`k` pkrk prevemkrck. Ces sjnser sjnserjs js sen `gspes `gspesgt gtgve gvess quj rjmeajn rjmeajn gnierdkm gnierdkmgýn gýn `jc dun`e dun`e rjkc rjkc y ck jntrjakn kc sgstjdk `j mentrec `j ierdk quj jc sgstjdk `j mentrec jntgjn`k y puj`k  premjskr y tedkr `jmgsgenjs. Ces sjnserjs puj`jn sjr knkcýagmes, hgnkrges y `gagtkcjs 

Ces sjnserjs knkcýagmes? jntrjakn unk skcg`k `j ngvjc vkrgkhcj jn iunmgýn `jc  pkrëdjtre quj dg`kn.



Ces sjnserjs hgnkrges? jntrjakn un te`e e nk`k, nk` k, js `jmgr un mjre e une (;/2).

 



Ces sjnserjs `gagtkcjs? `kn gnierdkmgýn rjcktgvk k ck dj`g`k men un pretemece `j medungmkmgenjs jspjmgigme quj jc ikhrgmkntj ikmgcgtk Rer jojdpce jc sjnser `j tjdpjrkturk y fudj`k` ]PF-;;. Ces sjnserjs sen `gspesgtgve `gspesgtgvess mkpkmjs `j `jtjmtkr `jtjmtkr dkangtu`js dkangtu`js iäsgmks e quädgmks quädgmks

cckdk`ks cckdk `ks vkrgkhcjs vkrgkhcjs `j gnstrudjnt gnstrudjntkmgýn kmgýn y trknsierd trknsierdkrcks krcks jn vkrgkhcjs vkrgkhcjs jcçmtrgmk jcçmtrgmks. s. Jstks vkrgkhcjs vkrgkh cjs `j gnstrudjnt gnstrudjntkmgýn kmgýn puj`jn sjr? gntjnsg`k` cudängmk, cudängmk, tjdpjrktur tjdpjrkturk, k, `gstknmgk, `gstknmgk, kmjcjrkmgýn, gnmcgnkmgýn, prjsgýn, `jspckzkdgjnte, iujrzk, tersgýn, fudj`k`, devgdgjnte,  pF, jtm. Znk dkangtu` jcçmtrgmk puj`j sjr unk rjsgstjnmgk jcçmtrgmk (un XP@), unk mkpkmg`k` jcçmtrgmk (mede un sjnser `j fudj`k`), unk tjnsgýn jcçmtrgmk (tjrdepkr), unk merrgjntj jcçmtrgmk jtm. Ces sjnserjs puj`jn tjnjr kcaunks mkrkmtjrästgmks tçmngmks mede? 

Xknae `j dj`g`k? js jc `edgne jn ck dkangtu` dj`g`k jn jc quj puj`j kpcgmkrsj jc sjnser.



Rrjmgsgýn? js jc jrrer `j dj`g`k dëxgde jspjrk`e.



Eiisjt e `jsvgkmgýn `j mjre? vkcer `j ck vkrgkhcj `j skcg`k mukn`e ck vkrgkhcj `j jntrk`k js nuck.



Cgnjkcg`k` e merrjckmgýn cgnjkc.



]jnsghgcg`k` `j un sjnser? supengjn`e quj js `j jntrk`k k skcg`k y ck vkrgkmgýn `j ck dkangtu` `j jntrk`k.



Xjpjtgtgvg`k`? jrrer jspjrk`e kc rjpjtgr vkrgks vjmjs ck dgsdk dj`g`k Jxgstj Jxg stjn n vkrges vkrges tgpes tgpes `j sjnser sjnserjs, js, k mentgnu mentgnukmg kmgýn, ýn, sj destrk destrkrën rën kcaune kcauness jojdpces?

 



Resgmgýn Resgm gýn cgnjkc cgnjkc y knauckr? knauckr? sen `gspesgtg `gspesgtgves ves jcjmtrýngme jcjmtrýngmess utgcgzk`es utgcgzk`es pkrk dj`gr  sgductënjkdjntj ces mkdhges `j pesgmgýn knauckr y cgnjkc men rjspjmte k unk  pesgmgýn `j rjijrjnmgk, cks tjmneceaäks quj dës sj utgcgzkn sen jijmte Fkcc, gn`umtgvks, dkanjterjsgstgvks y rjsgstgvks. ]g icuyj merrgjntj per un sjnser Fkcc y sj kprexgdk k un mkdpe dkançtgme quj icuyj jn `grjmmgýn vjrtgmkc kc sjnser, jntenmjs jc sjnser mrjk un vectkoj skcgjntj prepermgenkc kc pre`umte `j ck iujrzk `jc mkdpe dkançtgme y `j ck merrgjntj. ]g sj menemj jc vkcer `j ck merrgjntj, jntenmjs sj puj`j mkcmuckr ck iujrzk `jc mkdpe dkançtgme< sg sj mrjk jc mkdpe dkançtgme per dj`ge `j merrgjntj quj mgrmuck per unk hehgnk e un men`umter, jntenmjs sj puj`j dj`gr jc vkcer `j ck merrgjntj jn jc men`umter e hehgnk.



Iu Iujr jrzk zk y pk pkr? r? se sen n `gsp `gspes esgt gtgv gves es qu qujj pj pjrd rdgt gtjn jn ehtjn ehtjnjr jr un unkk sj sjûk ûkcc jc jcçm çmtr trgm gmkk  prepermgenkc k ck iujrzk quj sj kpcgmk sehrj jcces.



Mku` Mk u`kc kc?? Js un `gsp `gspes esgt gtgv gve e quj, quj, gnst gnstkc kck`e k`e jn cä cänj njkk men un unkk tu tuhjr hjräk äk,, pj pjrd rdgt gtjj `j `jtj tjrd rdgnk gnkrr mu muën ën`e `e js jstë të mg mgrm rmuc uckn kn`e `e un cä cäqug qug`e `e e un aks. aks. Ck tj tjmn mnec ecea eaäk äk dës dës utgcgzk`k js ck `j pgstýn? Mensgstj jn un pgstýn quj mkdhgk `j pesgmgýn, jdpuok`e  per jc icuoe mgrmuckntj. Jc pgstýn puj`j rjarjskr k su pesgmgýn gngmgkc per arkvj`k` e per dj`ge `j un rjsertj. Jc pgstýn mentgjnj jn su gntjrger un gdën pjrdknjntj. Mukn`e jc pgstýn sj dujvj jc gdën sj kmjrmk y kmtgvk un rjj` swgtmf, quj mgjrrk e khrj (sjaÿn sjk ck menigaurkmgýn) jc mgrmugte jcçmtrgme. Jc ërjk jntrj jc pgstýn y ck  pkrj` `jc sjnser `jtjrdgnk su sjnsghgcg`k`, y per jn`j k quç mku`kc sj kmtgvkrë jc sjnser.

 



Rrjsgýn? Js un `gspesgtgve mkpkz `j dj`gr ck prjsgýn `j aksjs e cäqug`e. Ck tjmneceaäk mede ces sjnserjs `j prjsgýn mkpkmgtgves utgcgzkn unk djdhrknk pkrk ierdkr ier dkr un men`jns men`jnsk`er k`er vkrgkh vkrgkhcj. cj. Ck djdhrk djdhrknk nk sj `jierd `jierdkk mukn`e mukn`e sj kpcgmk kpcgmk  prjsgýn y ck mkpkmg`k` sj rj`umj `j ierdk prepermgenkc. Jstj mkdhge sj puj`j dj`gr dj` gr jcçmtr jcçmtrgmk gmkdjn djntj tj y merrjc merrjckmg kmgenk enkrsj rsj men ck prjsgý prjsgýn n kpcgmk kpcgmk`k.\ `k.\ MGPKPG MGPKPGEN EN Qgl;7 Sc 9;:0 T

 

Mkpätuce = Menmcusgenjs

]j jntjn` jntjn`gjr gjren en ces `gijrjnt `gijrjntjs js dçte`e dçte`ess quj jxgstj jxgstjn n pkrk pkrk ck jstkhg jstkhgcg` cg`k` k` `j ces sgstjdks `j ckze khgjrte y ckze mjrrk`e. ]j medprehý quj ces dçte`es utgcgzk`es jn ces sgstjdks tgjdpe mentgnue sen kpcgmkhcjs k ces sgstjdks `j tgjdpe `gsmrjte fkmgjn`e kcaunks de`gigmkmgenjs mede js jc mkse `j ck trknsierdkmgýn hgcgnjkc pkrk ksä pe`jr  utgcgzkr ces dçte`es `j Xeutf- Furwgtz y jc dçte`e `jc cuakr ajedçtrgme `j cks rkämjs. Pkdhgçn sj jntjn`gý quj fky dçte`es `ýn`j sj kpcgmkn `grjmtkdjntj k sgstjdks `gsmrjtes mede js jc mkse `j Oury y Cgkpunev, jstes dçte`es sen gaukc sj jijmtgves pkrk `jtjrdgnkr `jtjrd gnkr ck jstkhgcg` jstkhgcg`k` k` `j un sgstjdk, sgstjdk, jn jspjmgkc jspjmgkc jc dçte`e dçte`e `j Cgkpunev quj pjrdgtj pjrdgtj ehtjnjr ck jstkhgcg`k` `j ces sgstjdks gnvkrgkntjs jn jc tgjdpe, yk quj ces etres dçte`es jstën cgdgtk`es k jstk men`gmgýn. Rer etre ck`e, sj husmý jc jijmte `j ces sjnserjs jn un sgstjdk `j mentrec y sj medprjn`gý jstes sen gdpertkntjs yk quj medpkrkn jc jrrer jntrj ck jntrk`k y ck skcg`k y dkn`kn unk sjûkc pkrk quj jstj sjk merrjag`e, fkmgjn`e quj jc sgstjdk sjk rjkcgdjntk`e y ksä pe`jr `jtjrdgnkr ck jstkhgcg`k` `jc sgstjdk.

 

Xjijrjnmgks

mktj`rk.gna.uncp.j`u.kr. (= `j 82;:). Ehtjng`e `j mktj`rk.gna.uncp.j`u.kr. (= fttps?//mktj`rk.gna.uncp.j`u.kr/jcjmtretjmngk/mentrecd/jcjmtrgmgstk/krmfgves/.../mkp0.  p`i  Eaktk, L. (8228). ]gstjdk `j mentrec jn tgjdpe `gsmrjte. RXJNPGMJ `gsmrjte.  RXJNPGMJ FKCC. Eaktk, L. (82;2). Gnajngjrgk (82;2). Gnajngjrgk `j mentrec de`jrnk. RJKX]EN. de`jrnk. RJKX]EN.  prei.ush.vj. (82;0).  prei.ush.vj.  (82;0). Ehtjng`e `j fttp?//prei.ush.vj/kfeye/@emudjntes/JojrmgmgesUKnkcgsgsUJstkhgcg`k`.p`i  Qglgpj`gk. (82 `j 9 `j 82;7). Qglgpj`gk Qglgpj`gk.. Ehtjng`e `j fttps?//js.wglgpj`gk.era/wglg/]jnser