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§1unimodaluri funqciebis minimizacia intervalTa gamoricxvis meTodiT
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| §1unimodaluri funqciebis minimizacia intervalTa gamoricxvis meTodiT.
1. ganmartebebi da intervalTa gamoricxvis wesi ganmarteba. funqcias ewodeba unimodaluri segmentze, Tu igi uwyvetia -ze da arseboben ricxvebi da , , iseTebi, rom:
SevniSnoT, rom Semdegi sami segmentidan: , , , erTi an ori SeiZleba gadagvardes wertilSi. nebismier SemTxvevaSi, unimodaluri funqciis ganmartebidan gamomdinare, misi lokaluri minimalebis simravle emTxveva globaluri minimalebis simravles da is aris an erTi wertili ( SemTxvevaSi), an mTeli segmenti. qvemoT moyvanilia unimodaluri funqciis ramdenime magaliTi. g da erTi araunimodaluric arsebobs unimodaluri funqciebis sxva ganmartebebic, romlebic gansxvavdeba aq moyvanilisgan (magaliTad [3]). kerZod, moxsnilia uwyvetobis moTxovna. -ze unimodaluri funqciebis simravle aRvniSnoT -Ti. funqciis -sadmi mikuTvnebis dasadgenad, gavrcelebulia ori kriteriumi.
unimodaluri funqciis minimizaciis ricxviTi meTodebi eyrdnobian unimodaluri funqciis Tvisebas, rom minimalis marjvniv funqcia zrdadia, xolo minimalis marcxniv _ klebadi. amitom, or gansxvavebul wertilSi aseTi funqciis mniSvnelobebis SedarebiT SegviZlia davadginoT, am wertilebiT da segmentis boloebiT Sedgenil romel intervalSi ar aris moTavsebuli minimali, gamovricxoT igi da amiT SevamciroT Ziebis intervali. rodesac unimodaluri funqcia ar aris warmoebadi, SegviZlia gamoviyenoTYmxolod pirdapiri meTodebi, romlebic saWiroeben miznis funqciis mniSvnelobebis gamoTvlas specialurad SerCeul wertilebSi da ar saWiroeben warmoebulis mniSvnelobebis gamoTvlas. praqtikidan wamosul eqstremalur amocanebSi sakmaod xSiria SemTxveva, rodesac funqciis mniSvnelobis gamoTvla wertilSi niSnavs garkveuli fizikuri eqsperimentis Catarebas (magaliTad, temperaturuli velis gazomvas). roca mosaxerxebelia funqciis mniSvnelobebis gamoTvla wertilSi erTdroulad da maTi mniSvnelobebis gamoTvla mimdevrobiT dakavSirebulia garkveul siZneleebTan, gamoiyeneba pasiuri Ziebis meTodi. im SemTxvevaSi, roca SesaZlebelia wina wertilebSi funqciis mniSvnelobebis gamoyenebiT momdevno wertilebis arCeva, maSin minimalis Zieba, cxadia, ufro efeqturi iqneba. aseTi tipis meTodebs mimdevrobiTi meTodebi ewodeba.. pirdapiri meTodebi bevria, amitom mniSvnelovania gvesmodes, Tu ras niSnavs erTi pirdapiri meTodis upiratesoba meoreze. amitom sakiTxi ismeba ase: Tu gvaqvs saSualeba mxolod -jer gamovTvaloT -ze unimodaluri f funqciis mniSvneloba, maSin ukeTesia is algoriTmi, romelic am pirobebSi mogvcems minimumis wertilis saukeTeso miaxloebas, anu minimalis Semcvel umcires intervals, anu umcires maqsimalur cdomilebas. SeiZleba sakiTxi asec davsvaT: ukeTesia is algoriTmi, romlisTvisac mocemuli cdomilebiT, minimalis miaxloebiTi mniSvnelobis sapovnelad, dagvWirdeba miznis funqciis mniSvnelobebis naklebi raodenobis gamoTvla. umartivesi pirdapiri meTodi aris pasiuri Ziebis meTodi, romlis mixedviTac cali wertili Tanabrad nawildeba -ze, gamoiTvleba miznis funqciis mniSvnelobebi maTSi da sadac funqciis mniSvneloba minimaluria, is wertili gamocxaddeba minimalis saukeTeso miaxloebad. am dros maqsimaluri cdomileba ar aRemateba sidides, raSic gvarwmunebs Semdegi Teorema, romelsac ewodeba intervalTa gamoricxvis wesi.
Tu , maSin .
Tu , maSin . damtkiceba: ganvixiloT 1. vTqvaT, da , magram ekuTvnis –s. maSin, unimodaluri funqciis ganmartebidan gamomdinare, nebismieri minimalis marjvniv miznis funqcia zrdadia (mkacrad zrdadobas ar vgulisxmobT). amitom unda gvqondes , rac ewinaaRmdegeba pirobas. e. i. nebismieri minimali moTavsebulia -is marjvniv. 1-s meore nawili analogiurad mtkicdeba. ganvixiloT 2. niSnavs, rom (ix. unimodaluri funqciis ganmarteba) da vTqvaT iseTia rom , maSin, radgan minimalis marjvniv miznis funqcia zrdadia, amitom . SemTxveva analogiuria. bolos, 3 warmoadgens 1-s da 2-s Sedegs. intervalTa gamoricxvis wesi safuZvlad udevs minimizaciis mimdevrobiT meTodebs, romlebSic funqciis mniSvnelobebis gamoTvla xdeba ukve gamoTvlili mniSvnelobebis safuZvelze. aRvweroT ori mimdevrobiTi meTodi, romlebic gamoirCevian efeqturobiT. 2. intervalis sigrZis ganaxevrebis meTodi es meTodi xasiaTdeba imiT, rom yovel iteraciaze (garda pirvelisa) gvWirdeba funqciis mniSvnelobis gamoTvla or wertilSi da es ori mniSvneloba saSualebas gvaZlevs orjer SevamciroT Ziebis intervalis sigrZe, romelSic moTavsebulia erTi globaluri minimali mainc. funqciis mniSvnelobebis -jer gamoTvlis Semdeg Cven gvrCeba sigrZis monakveTi, romelSic moTavsebulia erTi mainc minimali da gamoTvlilia agreTve funqciis mniSvneloba bolo intervalis Sua wertilSi, romelsac viRebT –is saukeTeso miaxloebad. am meTodisaTvis maqsimaluri cdomileba ar aRemateba sidides, rac bevrad ukeTesi Sefasebaa vidre gvqonda pasiuri meTodisTvis. vTqvaT, unimodaluria. CamovayaliboT e.w. intervalis sigrZis ganaxevrebis algoriTmi romlis saSualebiTac miaxloebiT ganvsazRvravT , minimizaciis amocanis globalur minimals. mxedvelobaSi unda viqonioT, rom arsebobs agreTve monakveTis Suaze gayofis algoriTmi, romelic aq ganxilulTan SedarebiT naklebefeqturia.
biji 4). SevadaroT da .
biji 5). gamovTvaloT . Tu mcirea, anu davamTavroT Zieba da aviRoT: , , winaaRmdeg SemTxvevaSi, gadavideT me_2 bijze. magaliTi: monakveTis ganaxevrebis meTodiT moZebneT funqciis minimali SualedSi. gansazRvreT sizustiT. amoxsna: vimoqmedoT zemoT aRwerili algoriTmiT da Sedegebi (iteraciebis mixedviT) warmovadginoT cxrilis saxiT. 196102697.599100.56.2510.25299102399.75100.5101.250.0630.251.563399100.51.599.37599.75100.1250.3910.0630.016499.75100.50.7599.938100.1250.016radgan mesame iteraciis Semdeg miRebuli monakveTis sigrZe naklebia -ze, amitom , . _ sizustiT minimalis miaxloebiTi mniSvnelobis gamosaTvlelad dagvWirda funqciis mniSvnelobis gamoTvla 7-jer. 7-jer funqciis mniSvnelobis gamoTvliT miviReT sigrZis intervali. 3. oqros kveTis meTodi mimdevrobiTi meTodebidan optimaluria fibonaCis meTodi. masTan SedarebiT gacilebiT martivia oqros kveTis meTodi, romelic didi –ebisTvis igive Sedegs iZleva, rasac fibonaCis meTodi, anu aris asimptoturiad optimaluri. oqros kveTis meTodi, miznis funqciis mniSvnelobebis –jer gamoTvlis Semdeg gvitovebs sigrZis monakveTs, romelic Seicavs minimals. aq aris gantolebis dadebiTi amonaxsni . da ricxvebi, ganmartebis mixedviT, axdenen segmentis oqros kveTas. monakveTis or aratol nawilad iseT dayofas, roca mTeli monakveTis sigrZis Sefardeba misi didi nawilis sigrZesTan tolia didi nawilis sigrZis mcire nawilis sigrZis SefardebasTan, ewodeba am monakveTis oqros kveTa. rac didia , miT ufro vlindeba oqros kveTis meTodis upiratesoba zemoT ganxilul meTodebTan SedarebiT. kvlav ganvixiloT minimizaciis amocana
Tu , maSin sruldeba Semdegi biji. biji 3). SevadaroT da .
aucilebelia yuradReba gavamaxviloT Semdeg momentebze: !) , amitom (cxadia, ganvixilavT SemTxvevas). !!) Tu , maSin , kerZod ki . amitom sizustiT viRebT . !!!) Tu biji 3)-Si Sesrulda , maSin viTvliT mxolod , Tumca axal -Si (anu Zvel -Si) axali unda agveRo Cveulebrivi wesiT: , Semdeg ki dagveTvala . magram es mogvcemda: (aq garkveuli analogiaa wina meTodis wertilTan mimarTebaSi), rac niSnavs rom axali SegviZlia pirdapir gavutoloT Zvel –s. magaliTi. oqros kveTis meTodis gamoyenebiT, sizustiT ganvsazRvroT globaluri minimalis wertili , , minimizaciis amocanaSi. amoxsna. visargebloT aRwerili algoriTmiT da Sedegebi warmovad-ginoT Semdeg cxrilSi, iteraciis nomris miTiTebiT: 1961023.70898.29299.7082.9170.085298.2921022.29299.708100.5840.0850.341398.292100.5841.41699.16899.7080.6920.085499.168100.5840.87599.708100.0430.0850.0018599.708100.5840.541100.043100.2490.00180.062699.708100.2490.334 amitom , xolo . cxrilSi isrebiT miTiTebulia wertilebis transformaciis da maTi mniSvnelobis SenarCunebis faqtebi. _ sizustiT minimalis miaxloebiTi mniSvnelobis gamosaTvlelad dagvWirda funqciis mniSvnelobis gamoTvla 6-jer. 6-jer funqciis mniSvnelobis gamoTvliT miviReT sigrZis intervali. savarjiSoebi amocanebSi #47 _ #49 intervalis sigrZis ganaxevrebis meTodiT, mocemuli sizustiT gansazRvreT f funqciis globaluri minimumis wertili. #47 , , . #48 , , . #49 , , . #50 ramdenjer unda gamoviTvaloT f funqciis mniSvneloba monakveTis sigrZis ganaxevrebis meTodiT, rom globaluri minimumis wertili vipovoT winaswar mocemuli sizustiT? #51 ipoveT _s maqsimaluri mniSvneloba, romlisTvisac funqcia iqneba unimodaluri segmentze. #52 ipoveT a_s maqsimaluri mniSvneloba, romlisTvisac funqcia iqneba unimodaluri segmentze. #53 ipoveT c_s maqsimaluri mniSvneloba, romlisTvisac funqcia iqneba unimodaluri segmentze. #54 vTqvaT, da aris monakveTis oqros kveTis wertilebi. aCveneT, rom aris monakveTis oqros kveTis wertilebidan marjvena (udidesi), xolo aris monakveTis oqros kveTis wertilebidan marcxena (umciresi). ipoveT da monakveTebis sigrZeebi. #55 risi tolia maqsimaluri SesaZlo cdomileba oqros kveTis meTodisTvis, Tu miznis funqciis mniSvnelobebi gamoTvlil iqna -jer? #56 ramdenjer iqneba miznis funqciis mniSvnelobebis gamoTvla saWiro, Tu viyenebiT oqros kveTis meTods da gvinda globaluri minimumis wertilis gansazRvra sizustiT? Semdeg amocanebSi oqros kveTis meTodiT gansazRvreT f funqciis globaluri minimumis wertili monakveTze sizustiT. #57 , , . #58 , , . #59 , , .
im ganmartebis mixedviT, romelsac Cveni kursi eyrdnoba, unimodaluri funqcia uwyvetia. vaierStrasis erT-erTi Teoremis Tanaxmad, segmentze uwyveti funqciis aproqsimacia nebismieri sizustiT aris SesaZlebeli polinomis saSualebiT№. Tu aproqsimacia sakmaod zustia, miznis funqciis minimali SegviZlia SevafasoT polinomis globaluri minimalis Sefasebis gziT da es Sefaseba, vaierStrasis Teoremis saafuZvelze, SegviZlia gavaumjobesoT ornairad: ufro maRali rigis polinomis gamoyenebiT an saaproqsimacio Sualedis SemcirebiT. meore gza ufro bunebrivia, radgan sawyisi Sualedi sakmaod swrafad SegviZlia SevamciroT (magaliTad, oqros kveTis meTodiT), xolo Semdeg gamoviyenoT kvadratul (an kubur) aproqsimaciaze dafuZnebuli meTodebi. 1. Sefasebis meTodi, romelic iyenebs kvadratul aproqsimacias. vTqvaT, vixilavT amocanas , , vipovoT unimodaluri f funqciis minimali segmentze mocemuli sizustiT. vTqvaT, da funqciis mniSvnelobebi am wertilebSia: ganvsazRvroT koeficientebi ise, rom kvadratuli polinomis
grafikma gaiaros wertilebze. № vaierStrasis Teorema. Tu f funqcia uwyvetia segmentze, maSin yoveli -Tvis, arsebobs algebruli polinomi iseTi, rom yoveli -Tvis sruldeba utoloba, pirvel rigSi, radgan , amitom Semdeg, radgan , amitom
. bolos, roca , gvaqvs: , saidanac
. Tu . maSin vixilavT axal amocanas , , (3.10) da vipoviT –s minimals mocemul segmentze. is iqneba, an , gantolebis amonaxsni: (3.11) an segmentis erT-erTi bolo an . roca , , wertilebi isea SerCeuli, rom
maSin da saaproqsimacio mravalwevrs, romelic gadis wertilebze aqvs globaluri minimali -s SigniT da SegviZlia gamoviyenoT pauelis meTodi. Tu (3.12) piroba ar sruldeba, maSin viqceviT Semdegnairad: vamcirebT Ziebis intervals (mag, oqros kveTis meTodiT) da vamowmebT (3.12)-s ( -s rolSi SeiZleba aviRoT oqros kveTis romelime wertili). rogorc ki Sesruldeba (3.12) piroba, viyenebT kvadratul aproqsimaciaze damyarebul meTods. Tu piroba ar Sesrulda da Ziebis intervali gaxda mocemul -ze naklebi, maSin vasrulebT minimalis Ziebas oqros kveTis meTodiT.
amocanaSi sizustiT. amoxsna. aviRoT ( -ad Sua wertils viRebT). gamoviTvaloT funqciis mniSvnelobebi: da agreTve maaproqsimirebeli polinomis koeficientebi: , ( , formalurad, ar monawileobs -is warmodgenaSi). bolos, . radgan 1<1,565<5, amitom viRebT (SedarebisaTvis, zusti mniSvneloba minimalisa aris ). 2. pauelis meTodi es meTodi mimdevrobiTia da yovel iteraciaze iyenebs kvadratul aproqsimaciaze dafuZnebul meTods. vTqvaT, wertilebi isea SerCeuli, rom da (am utolobebis Sedegad miiReba ) meTodis algoriTmi aseTia: biji 1). , da iyos is wertili, sadac miiRweva . biji 2). -is mixedviT gamoviTvaloT , kvadratul aproqsima-ciaze dafuZnebuli meTodis gamoyenebiT. biji 3). -iT aRvniSnoT , wertilebs Soris is, romelSic miznis funqcia nakleb mniSvnelobas iRebs. biji 4). SevamowmoT Ziebis damTavrebis pirobebi: , Tu orive sruldeba, algoriTmi asrulebs muSaobas da , winaaRmdeg SemTxvevaSi gadavdivarT Semdeg bijze. biji 5). wertilis orive mxares davalagoT masTan yvelaze axlos mdgomi ori wertili, gadavnomroT es wertilebi zrdadobis mixedviT da gadavideT biji 1) –ze. gavakeToT ramdenime aucilebeli SeniSvna. 1) pirvel iteraciaze . 2) zogjer iTxoven, rom SeCerebis piroba zedized Sesruldes ramdenime momdevno iteraciisaTvis (magaliTad, samjer), SemTxveviTi efeqtebis Tavidan asacileblad. 3) pauelis meTods minimalis siaxloves aqvs kvadratuli krebadoba. es aris misi upiratesoba oqros kveTis meTodTan, magram Tu sawyisi intervali sakmaod didia, kvadratuli polinomiT miaxloeba SeiZleba aRmoCndes sakmaod uxeSi, rac Seamcirebs pauelis meTodis krebadobas.
Tu unimodaluri miznis funqcia warmoebadic aris, maSin SegviZlia gamoviyenoT wertilovan Sefasebaze damokidebuli meTodebi. cxadia, Tu da a a) , b) , g) , sadac aRniSnavs minimals amocanaSi. Sua wertilis meTodis algoriTmi aseTia: biji 1). gamoviTvaloT da ; biji 2) Tu , viRebT da vamTavrebT Ziebas. winaaRmdeg SemTxvevaSi: a) Tu , viRebT da gadavdivarT biji 1) –ze. b) Tu , viRebT da gadavdivarT biji 1) –ze.
amocanaSi sizustiT. amoxsna. , rac niSnavs, rom minimali aris [a,b] segmentis Siga wertili da SegviZlia Sua wertilis meTodis gamoyeneba. I iteracia biji 1). da . biji 2) , , amitom da gadavdivarT biji 1)–ze. II iteracia biji 1). , . biji 2) , , amitom da gadavdivarT biji 1)–ze. III iteracia biji 1). , . biji 2) , , e.i. da gadavdivarT biji 1)–ze. IV iteracia biji 1). , . biji 2) , , e.i. da gadavdivarT biji 1)–ze. V iteracia biji 1). , . biji 2) , amitom , xolo . savarjiSoebi #60 amocanebSi #47 - #49 da #57 – #59, moZebneT minimali kvadratuli aproqsimaciis gamoyenebiT da SeadareT adre miRebul Sedegebs. #61 Sua wertilis meTodis gamoyenebiT gansazRvreT f funqciis minimali [a,b] segmentze sizustiT:
#62 amocanebSi, #47 - #49 da #57 – #59, moZebneT minimali Sua wertilis meTodis gamoyenebiT da SeadareT adre miRebul Sedegebs gluvi eqstremaluri amocanebi mraval cvladze damokidebuli miznis funqciiT 1. kritikuli wertilebis gansazRvra. Teorema 1 /eqstremalurobis I rigis aucilebeli piroba mravali cvladis SemTxvevaSi/. vTqvaT, aris Ria simravle, uwyvetad warmoebadi funqciaa wertilis romeliRac midamoSi da aris lokaluri eqstremumis wertili amocanaSi. maSin, . damtkiceba. vTqvaT . Tu aris funqciis lokaluri eqstremumis wertili, maSin iqneba , erTi cvladis funqciis, lokaluri eqstremumis wertili, amasTan, radgan funqcia gansazRvrulia Ria simravleze, amitom funqciac gansazRvrulia Ria simravleze da misTvis SegviZlia gamoviyenoT eqstremalurobis aucilebeli pirobebi. analogiuria mtkiceba -Tvis ganmarteba. ganxiluli Teoremis pirobebSi, , amocanisTvis kritikuli wertilebis simravle K ganisazRvreba Semdegnairad: . magaliTi. ipoveT funqciis eqstremumis wertilebi. amoxsna. CavweroT amocana standartuli saxiT: . (4.1) (4.1) amocanisaTvis ganvsazRvroT kritikuli wertilebis simravle. am mizniT SevadginoT da amovxsnaT sistema . e.i.: saidanac vpoulobT: Semdeg punqtebSi avxsniT, Tu rogor xdeba kritikuli wertilebis gamokvleva da davasrulebT am magaliTsac. 2. kritikuli wertilebis gamokvleva vaierStrasis Teoremis Sedegebis gamoyenebiT vaierStrasis Teoremis Sedegebi saSualebas gvaZlevs gavarkvioT, Tu romeli kritikuli wertili aris globaluri eqstremumis wertili. vaierStrasis Teoremis Sedegebis gamoyenebiT Tu davadgenT, rom
amocanaSi , maSin SegviZlia SemovifargloT gacilebiT martivi , (4.3) amocanis ganxilviT, romelsac aqvs igive minimali, rac (4.2) –s, radgan . analogiurad vmoqmedebT globaluri maqsimumis Ziebis SemTxvevaSic. magaliTi. gavagrZeloT Semdegi magaliTis amoxsna: . rogorc vnaxeT, radgan radgan K erTelementiania, amitom (1,0) aris globaluri minimali amocanaSi. am meTodSi ZiriTadi sirTule mdgomareobs absoluturi eqstremumis wertilis arsebobis dadgenaSi. Cvens magaliTSi, radgan amitom
. (4.4) axla cxadia, rom radgan (4.4) –Si kvadratuli wevrebis koeficientebi dadebiTia. 3. kritikuli wertilebis gamokvleva ganmartebis safuZvelze kvlav ganvixiloT (4.2). vTqvaT, sruldeba Teorema 1 –is pirobebi da . yoveli warmovadginoT Semdegi saxiT: da ganvixiloT sxvaoba: -is niSnis h -ze damokidebulebis garkveva arsebiTia, radgan:
magaliTi. kvlav gavagrZeloT eqstremaluri amocanis ganxilva. aq aris erTaderTi kritikuli wertili. Cvens SemTxvevaSi warmodgena aris , amitom ganvixiloT sxvaoba: , rac -is nebismierobis ZaliT niSnavs, rom aris globaluri minimali. SevniSnoT agreTve, rom bolo formulidan martivad Cans rac arsebiTia vaierStrasis Teoremis tipis SedegebiT sargeblobisas. 4. kritikuli wertilebis gamokvleva eqstremalurobis II rigis sakmarisi pirobis safuZvelze winaswar moviyvanoT ramdenime faqti, rac aucilebelia am meTodis gamoyenebisas. ganmarteba. vTqvaT, mocemulia kvadratuli matrica. am matricas ewodeba:
silvestris cnobili Teorema (ix. [1]), romelsac aq daumtkiceblad moviyvanT, gvaZlevs advilad Sesamowmebel kriteriums kvadratuli matricis gansazRvrulobis dadgenisTvis. SevniSnoT, rom silvestris Teoremis sxva, ufro informatiuli versiebic arsebobs. Teorema /silvestris/. A matricis dadebiTad gansazRvrulo-bisaTvis aucilebelia da sakmarisi, rom misi yvela mTavari minori iyos dadebiTi, anu yoveli -isaTvis unda Sesruldes , sadac . savarjiSo. CamoayalibeT kriteriumi A matricis uaryofiTad gansazRvrulobisaTvis. miTiTeba. A –s uaryofiTad gansazRvruloba igivea rac _A –s dadebiTad gansazRvruloba. SemdegSi, -iT, Cveulebriv, aRvniSnavT funqciis meore warmoebulebis (heses) matricas x wertilSi: . Teorema 2 /eqstremalurobis II rigis sakmarisi piroba/. vTqvaT aris Ria simravle, funqcias aqvs meore rigis uwyveti kerZo warmoebulebi. maSin, imisaTvis rom iyos mkacri lokaluri maqsimali (minimali) , amocanaSi, sakmarisia Semdegi pirobebis Sesruleba: uaryofiTadaa gansazRvruli ( dadebiTadaa gansazRvruli). damtkiceba. davamtkicoT Teorema 2 lokaluri minimalisaTvis. radgan dadebiTadaa gansazRvruli, amitom matricis yvela mTavari minori dadebiTia: , . determinanti aris matricis elementebis uwyveti funqcia, xolo Tavis mxriv matricis elementebi aris x _is uwyveti funqciebi indeqsebis yoveli i, j wyvilisaTvis. amgvarad, mTavari minorebi aris x _is uwyveti funqciebi da es funqciebi dadebiTia _Si. amitom _is raRac midamoSi isini SeinarCuneben niSans. Tu ise aris SerCeuli, rom _is midamoSi _is mTavari minorebi inarCuneben niSans, maSin dadebiTadaa gansazRvruli. imdenad mcire SegviZlia aviRoT, rom agreTve Sesruldes . teiloris formulis Tanaxmad, roca , gvaqvs , (4.5) sadac damokidebulia x _ze. SevniSnoT, rom teiloris formulis es saxe samarTliania mxolod ricxviTi mniSvnelobebis mqone funqci-ebisaTvis. pirobis Tanaxmad, , xolo radgan , amitom dadebiTadaa gansazRvruli, anu . amgvarad (4.5) gvaZlevs: . savarjiSoebi amoxseniT Semdegi eqstremaluri amocanebi: 4.2gluvi eqstremaluri amocana tolobis tipis SezRudvebiT Yüklə 0,57 Mb. Dostları ilə paylaş:
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