Safet Penjić, mr sc Filozofski fakultet u Zenici
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1 Safe Penjć, mr sc Flozofs faule u Zenc Uloga speralnh projeora u rješavanju ssema homogenh dferencjalnh jednačna sa onsannm oefcjenma jedna zanmljva prmjena u bologj Sažea Za podprosore, prosora ažemo da su omplemenarn adgod = + Y = {0} u om slučaju za ažemo da je drena suma od, ovo označavamo sa =. Ovo je evvalenno sa vrdnjom da za sva v posoje jednsven veor x y av da v= x+ y. Za prozvoljan aav v možemo defnsa operaor P sa Pv = x ovo je jednsven lnearn operaor sa osobnom Pv = x ( v= x+ y, x y ) oj je pozna pod menom projeor na paralelno sa. U ovom radu posmara ćemo speralne projeore sors njhove osobne za rješavanje ssema dferencjalnh jednačna obla y'= Ay čje rješenje reba zadovoljava ncjaln uslov y(0) = c, razmara ćemo slučaj samo ada je marca ssema A djagonablna. Da b smo došl do odgovarajućeg rješenja prvo smo uvel defncju funcje na djagonablnm marcama. Poslje oga je prezenrana jedna njhova zanmljva prmjena - prmjena od dfuzje (šrenja) ćelja u medcn bologj. Na raju rada je prezenran MaLab od funcje oja daje rješenje ssema dferencjalnh jednačna ada nam je daa marcu ssema A, dmenzja 3 3 marca olona daog uslova c. Ključne rječ: speraln projeor, djagonablna marca, ssem homogenh lnearnh dferencjalnh jednačna Role of specral projecors n solvng sysems of homogeneous lnear dfferenal equaons wh consan coeffcens and one neresng applcaon n bology Absrac Subspaces, of a space are sad o be complemenary whenever = + and Y ={ 0}, n whch case s sad o be he drec sum of and, and hs s denoed by wrng =. Ths s equvalen o sayng ha for each v here s unque vecors x and y such ha v= x+ y. For arbrary such v we can defne operaor P by Pv = x, and hs operaor s unque lnear operaor wh propery Pv = x ( v= x+ y, x and y ) and s called he projecor ono along. In hs paper we wll consder specral projecors and we wll use her properes o solve a sysems of frs-order lnear dfferenal equaons, whch can be wre n marx form as y'= Ay wh gven nal value y(0) = c, and we wll examne only case when marx A of gven sysem s dagonalzable. For hese purpose we had frs defned funcons of dagonalzable marces. Afer ha we have presened one neresng applcaon - an applcaon o dffuson of cells n medcne and bology.
2 In he end of paper we gve MaLab code of funcon whch solves sysems of frs-order lnear dfferenal equaons when we have marx of sysem A of form 3 3 and marxcolumn of nal value c. Keywords: specral projecors, dagonalzable marx, sysems of homogeneous lnear dfferenal equaons 0 Osnovne defncje rezula Defnšmo oznae oje ćemo ors u nasavu esa prsjemo se neh osnovnh defncja eorema z Lnearne algebre. Za n n marcu A, salare veore x = n 0 oj zadovoljavaju jednaos Ax= x, zovemo redom, svojsvene vrjednos svojsven veor marce A, blo oj aav par (, x ) nazvamo svojsven par marce A. Sup svh razlčh svojsvenh vrjednos ćemo označava sa σ ( A ). Za dvje n n marce A B ažemo da su slčne adgod posoj nesngularna marca P ava da je P AP = B. Prozvod P AP se nazva ransformacja slčnos na A. Za vadranu marcu A ažemo da je djagonablna adgod je A slčna djagonalnoj marc. Popun sup svojsvenh veora marce A, oja je obla n n, je blo oj sup od n lnearno nezavsnh svojsvenh veora marce A. Algebarsa všesruos svojsvene vrjednos je broj pua u ojoj se ona ponov ao orjen araersčnog polnoma. Algebarsu všesruos od ćemo označava sa algmul A ( ). Geomerjsa všesruos svojsvene vrjednos je dm er( A I ). Geomerjsu všesruos od ćemo označava sa geomul A ( ). Tvrdnj (a.)-(a.5) su poznae od ranje (z Lnearne algebre) a zaneresranog čaoca možemo upu na [] (l [5] l [3]). (a.) σ( A ) A I je sngularna marca de( A I ) = 0 (a.) { x= 0 x er( A I )} je sup svh svojsvenh veora prdruženh. Prosor E = er( A I ) se nazva svojsven prosor marce A. (a.3) Marca A, obla n n, je djagonablna ao samo ao A posjeduje popun sup svojsvenh veora. (a.4) Marca A obla n n sa razlčh svojsvenh vrjednos σ( A ) = {,,..., } je djagonablna ao samo ao posoje marce { G, G,... G } ave da A= G+ G G () gdje marce G maju sljedeće osobne () G je projeor na er( A I) paralelno sa m( A I). () GG 0 gdje je = j. () G+ G G = I. Razvoj () je pozna pod menom speralna deompozcja marce A, a marce G se nazvaju speraln projeor prdružen marc A. (a.5) Marca A, obla n n, je djagonablna ao samo ao je algmul ( )= geomul ( ) A A.
3 Funcje defnsane na djagonablnm marcama Ša b za vadranu marcu A značlo da napšemo sna, e A, lna slčno. Navan prsup b mogao b da jednosavno prmjenmo danu funcju na sva elemen marce A ao da a? a sn a sn a sn =. a a sn a sn a Al radeć ovao n jedna osobna od marčnh funcja se neće polop sa njhovm salarnm olegama. Na prmjer, ao je sn x+ cos x= za sve salare x, voljel b da naša defncja od sna cosa ao rezula dadne analogn marčn dene sn A+ cos A= I za sve vadrane marce A. Jasno je da prsup po vrjednosma () neće ma ove osobne. Drug načn da defnšemo marčnu funcju oja će ma se osobne ao njhove salarne olege je da orsmo razvoje funcja u besonačne redove. Na prmjer, posmarajmo esponencjalnu funcju 3 z z z z e = = + z !! 3! =0 Formalno zamjenjvanje salarnog argumena z sa vadranom marcom A ( z 0 = je zamjenjeno sa A 0 = I) ao rezula daje besonačan red marca 3 A A A e = I+ A (3)! 3! oj se nazva marčn esponencjal. Iao ovaj prsup ma rezula da marčne funcje maju se osobne ao njhove salarne olege, vela mana ovavog prsupa je da sva pu moramo razmšlja o onvergencj, sva pu se suobljavamo sa problemom opsa elemenaa pomoću lmesa. Međum, ao je A djagonablna marca, ada je A= PDP = P dag(,..., n ) P, A = PD P = P dag(,..., n ) P gdje je dag(,..., n ) =, pa je n e = A = PD P A = P D P = Pdag( e,..., e ) P. =0! =0! =0! Drugm rječma, ne moramo ors besonačan red (3) da defnšemo e A. Umjeso oga, D defnšemo e = dag( e, e,..., e n ) savmo A D = = (,,..., ). e e dag e e e n P P P P Ova deja se može poopš na blo oju funcju f( z ) oja je defnsana na svojsvenm vrjednosma djagonablne marce A= PDP defnšuć f ( D ) da bude f ( D ) = dag( f ( ), f ( ),..., f ( )) savljajuć n f ( A) = Pf ( DP ) = Pdag( f ( ), f ( ),..., f ( n )) P. Puno vše o funcjama defnranm na djagonablnm marcama možee proča u [] na srancama Ovdje samo još sumrajmo vrdnje oje ćemo ors u nasavu esa: 3 ()
4 Nea je A= PDP djagonablna marca gdje su svojsvene vrjednos u D= dag( I, I,..., I ) gruprane po ponavljanju. Za funcju f( z ) oja ma onačnu vrjednos za sva σ( A ) defnšmo gdje su f ( ) I f ( )... 0 f( )= f( ) = I A P DP P P f ( ) I = f( ) G + f( ) G f( ) G, (4) G -ev -e speralne projecje oje se mogu zračuna na ne od sljedećh načna: () G je projeor na er( A I ) paralelno sa m( A I ); Τ Τ () G = XY gdje olone marce X formraju bazu za er( A I ), do su Y Τ Y Τ Y dobjen z marce P = gdje je P= ( X Τ Y X... X ) ; () PP = I= G + G G ; (v) G= ( A ji ), gdje je π = ( j). π Prmjena lnearne algebre u rješavanju ssema dferencjalnh jednačna Sad razmaramo ao rješ ssem lnearnh dferencjalnh jednačna prvog reda sa onsannm oefcjenma uz da ncjaln uslov oršenjem svojsvenh vrjednos svojsvenh veora. Za onsane a j, clj nam je da rješmo sljedeć ssem po nepoznam funcjama y (). y = ay + a y a nyn; y (0) = c, y = ay + a y anyn; y(0) = c, (5) y = a y + a y a y ; y (0) = c. n n n nn n n n α Kao je salarn esponen y ()= e cjednsveno rješenje jedne dferencjalne jednačne obla y ()= α y () sa ncjalnm uslovom y(0) = c, sasvm je prrodno da poušamo sors marčn esponen na popuno s načn da rješmo ssem dferencjalnh jednačna. Započe ćemo ao šo ćemo napsa ssem (5) u marčnom oblu y'= Ay, y(0) = c gdje je 4
5 y ( ) a a... a n c y ( ) a a... a c y A c y ( ) a a... a c n =, =, =. n n n nn n Ao je marca A djagonablna sa σ( A ) = {,,..., } ada (4) garanuje da je A e = e G + e G +... e G. Sljedeće jednaos se zvode z osobna marca A () = ( = )( = ) 5 G oje smo navel u (a.4). de / d = e G = G e G = A e A () Ae = e A A (zbog slčnh argumenaa). A A A A 0 () e e = e e == I e (zbog slčnh argumenaa). A A Osobna ( ) znad de / d = A e nam osgurava da je funcja y= A e A c jedno od rješenja ssema y'= Ay sa ncjalnm uslovom y(0) = c. Da b vdjel da je y= e A c jedno rješenje, preposavmo da je v () neo drugo rješenje ao da je v'= Av sa v(0) = c. Dferencrajuć e A v dob ćemo A de [ v] = A A e Av+ e v '= 0, d pa je e A A 0 v onsana za sve. Za =0 mamo e v =0= e v(0) = Ic= c, me e A v= c za sve. Množeć obe srane ove jednaos sa e A A A 0 orseć osobnu ( ) znad e e == I e možemo zaljuč da je v= e A c. Tme smo dobl da je y= e A c jednsveno rješenje ssema dferencjalnh jednačna y'= Ay sa ncjalnm uslovom y(0) = c. Na raju prmjemo da je v= Gc er ( A I ) svojsven veor prdružen svojsvenoj vrjednos, ao da je rješenje za y'= Ay, y(0) = c u svar y= e v + e v e v ovo rješenje je popuno određeno svojsvenm parovma (, v ). Može se doaza da se y aođer može zraz u oblu blo ojeg popunog supa nezavsnh svojsvenh veora. U sljedećoj eorem sumrajmo prehodnu prču. (.0) Teorema (rješenje ssema dferencjalnh jednačna) Ao je A marca obla n n oja je djagonablna sa σ( A ) = {,,..., } ada jednsveno rješenje ssema dferencjalnh jednačna y'= Ay oje zadovoljava uslov y(0) = c je dano sa gdje je A y= e c= e v + e v e v v svojsven veor v = Gc, a G je - speraln projeor. (.0) Problem Rješ ssem dferencjalnh jednačna x =3x y+ z y = x+ 5y z z = x y+ 3z ao da rješenja zadovoljavaju ncjalne uslov x (0) =, y (0) = z (0) = 3.
6 Rješenje: Da ssem možemo napsa u oblu x'= Ax, x(0) = b gdje je x 3 x= y, A= 5, b =. z 3 3 Lagano računanje nam daje svojsvene vrjednos marce A, σ( A ) = { = 6, = 3, 3 = }. Koršenjem formule G= ( A ji ), π za računanje speralnh projeora, gdje je π = ( ), mamo π = ( )( ) = 3 4 = ; 3 π =( )( )=( 3)= 3; 3 π = ( )( ) = ( 4) ( ) = 4 ; j 0 /6 /3 /6 G= ( A 3 I)( A I )= 3 = /3 /3 /3, 0 /6 /3 /6 3 /3 /3 /3 G = ( A 6 I)( A I ) = 3 = /3 /3 /3, /3 /3 /3 3 0 / 0 / G3= ( A 6 I)( A 3 I )= = / 0 / Lagana provjera nam poazuje da je G+ G =, = = + G3 I G G j 0 za j ao A= G + G + G 3 3 pa su dobvene marce G, G G 3 zasa speraln projeor prdružen marc A. Sad zračunajmo veore v= Gb, v= G b v3= G3b : Možemo zaljuč da je 0 v= 0, v=, v 3= 0. 0 x e e 3 ()= y 3 ()=e 3 z ()=e + e rješenje daog ssema dferencjalnh jednačna oje zadovoljava dae ncjalne uslove. 6
7 (.03) Problem Rješ ssem dferencjalnh jednačna x =x y+ z y = x+ y z z = x z+ z ao da rješenja zadovoljavaju uslov x (0) =, y (0) = 3 z (0) = 4. Rješenje: Da ssem možemo napsa u oblu x'= Ax, x(0) = b gdje je x x= y, A=, b = 3. z 4 Svojsvene vrjednos marce A su σ( A ) = { = 3, =, 3 = }. Uz pomoć formule G= ( A ji ), π za računanje speralnh projeora, gdje je π = ( ), dobjamo π = ( )( ) = ; 3 π = ( )( ) = ; 3 π = ( )( ) = ; j 0 G= ( A I)( A I )= 0 0 0, 0 G = ( A 3 I)( A I ) =, G3= ( A 3 I)( A I ) = Lagana provjera nam poazuje da je G+ G =, = = + G3 I G G j 0 za j ao A= G + G + G 3 3 pa su dobvene marce G, G G 3 zasa speraln projeor prdružen marc A. Sad zračunajmo veore v= Gb, v= G b v3= G3b : Možemo zaljuč da je 0 v= 0, v=, v 3=. 7
8 x e + e 3 ()= y e + e ()= 3 z ()= e + e + e rješenje daog ssema dferencjalnh jednačna oje zadovoljava dae ncjalne uslove. 3 Prmjena od dfuzje Važna ema u medcn bologj uljučuje panje ao droga l hemjs sasav uču na pomjeranje jedne ćelje na drugu u smslu šrenja roz zdove ćelje. Posmarajmo dvje ćelje, ao šo je prazano na Fgur, gdje su obe zguble do neog svog sasava. Jednčna olčna sasava je ubačena u prvu ćelju u vremenu =0, ao vrjeme prolaz, njezn sasav se šr prema sljedećm preposavama. FIGURA Šrenje zdova jedne ćelja u odnosu na drugu. U svaom renuu vremena omjer (olčna po seund) šrenja jedne ćelje prema drugoj je proporconalna oncenracj (olčna po jednčnoj zapremn) smjese ćelj oja preda do svog sasava - recmo omjer šrenja ćelje prema ćelj je α pua oncenracja u ćelj, a omjer šrenja ćelje prema ćelj je β pua oncenracja u ćelj. Preposavmo da je αβ, >0. (3.0) Problem Odred oncenracju smjese svae od ćelja u blo ojem renuu vremena,, posmarajuć duž vremens perod, odred oncenracju sablnog sanja. Rješenje: Ao u = u() označava oncenracju sasava u ćelj u renuu, ada vrdnja u gornjoj preposavc se može preves na sljedeć načn: du = omjer oj ulaz omjer oj zlaz = βu αu, gdje u(0) =, d du = omjer oj ulaz omjer oj zlaz = αu βu, gdje u(0) = 0. d U marčnm oznaama ovaj ssem je u'= Au, u(0) = c, gdje α β u A=, u=, c =. α β u 0 Karaersčna jednačna za A je + ( α+ β ) =0, ao da su svojsvene vrjednos 8
9 marce A =0 = ( α β) +. Prmjemo da se svojsvene vrjednos ne ponavljaju, pa je marca A djagonablna. Korseć funcju aže da je f( z)= e z, speralna reprezenacja (4) nam + + A e = f( A)= f( ) G f( ) G = e G e G. Speraln projeor G G se mogu zračuna po formul G= ( A ji ), gdje je π π = ( ), gdje ćemo dob da je j A I β β A α β G= = G = =, α + β α α α + β α β pa je ( ) α β β β ( α β) α β A + + e = G+ e G = + e α + β α α α β Sad mamo β β ( ) α β A ()= e = e α + u c + β, α + β α α α β 0 z čega sljed β α ( α+ β) α ( α+ β) u( ) = + e u( ) = ( e ). α + β α + β α + β Posmarajuć duž perod, oncenracja u ćelj će se sablzra u smslu da β α lmu()= lmu()=. α + β α + β 4 MaLab od Kod funcje oja ao ulazne vrjednos ma marcu ssema A, olona marcu ncjalnog uslova b je: funcon [rjesenje] = rjes_ssem(a,b) % MaLab od funcje oja daje rješenje % ssema dferencjalnh jednačna ada nam % je daa marcu ssema $A$, dmenzja % $3\mes 3$ marca olona daog uslova $c$ I=[ 0 0; 0 0; 0 0 ]; syms x; a=[ ]; ara_pol=de(a-x*i); % zvadmo oefcjene z araersčnog polnma [oefcjen,]=coeffs(ara_pol); % dodajmo oefcjene u veor a for =:numel(oefcjen) a()=oefcjen(); end % odredmo nule araersčnog polnoma orjen=roos(a); lambda=(orjen()); 9
10 lambda=(orjen()); lambda3=(orjen(3)); % zaoružmo svojsvene vrjednos na 4 decmalna mjesa lambda=round(lambda*000)/000; lambda=round(lambda*000)/000; lambda3=round(lambda3*000)/000; % zračunajmo p_, p_ p_3 p_=(lambda-lambda)*(lambda-lambda3); p_=(lambda-lambda)*(lambda-lambda3); p_3=(lambda3-lambda)*(lambda3-lambda); % zračunajmo G, G G3 G=sym(/p_)*(A-lambda*I)*(A-lambda3*I); G=sym(/p_)*(A-lambda*I)*(A-lambda3*I); G3=sym(/p_3)*(A-lambda*I)*(A-lambda*I); % zračunajmo veore v, v, v3 v=g*b; v=g*b; v3=g3*b; % spšmo rješenje dferencjalne jednačne syms ; dsp('rješenje daog ssema u marčnom oblu je'); rjesenje=exp(lambda*)*v+exp(lambda*)*v+exp(lambda3*)*v3; prey(rjesenje) end Pažljv čalac će prmje da da MaLab od ne provjerava da l je unesena marca djagonablna. Naravno funcja bez og djela rad dovoljno dobro a umeanje og djela oda osavljamo ao zanmljvu vježbu. Leraura [] J. L. Massey: "Appled Dgal Informaon Theory II", blješe sa predavanja, srpa oja je oršena na predavanjma oje je držao prof. dr James L. Massey od 98 do 997 na ETF-u u Crhu, snuo sa hp:// sr. -59 [] C. D. Meyer: "Marx analyss and appled lnear algebra", SIAM, 000., sr [3] L. Mrsy: "An nroducon o lnear algebra", Oxford Unversy Press, 955., sr [4] P. J. Olver C. Shaban: "Appled lnear algebra", Pearson Prence Hall, 006., sr [5] V. Perć: "Algebra I (prsen modul, lnearna algebra)", Svjelos, 987., sr [6] V. V. Prasolov: "Problems and heorems n Lnear Algebra", Amercan Mahemacal Socey, 000., sr 97-0
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EERAIED BU-MAU YTEM ITH A FREQUECY AD A EVERITY CMET A IDIVIDUA BAI I AUTMBIE IURACE* BY RAHIM MAHMUDVAD AD HEI HAAI ABTRACT Frangos and Vronos (2001) proposed an opmal bonus-malus sysems wh a frequency
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Inersymol nererence ISI ISI s a sgnal-dependen orm o nererence ha arses ecause o devaons n he requency response o a channel rom he deal channel. Example: Bandlmed channel Tme Doman Bandlmed channel Frequency
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