soxumis saxelmwifo universitetis S r o m e b i VII

soxumis saxelmwifo universitetisoxumis saxelmwifo universitetisS r o m e b iVIImaTematikisa da kompiuterulmecnierebaTa seriasoxumis saxelmwifo universitetis gamomcemlobaTbilisi – 2009

Sromebis VII tomi (maTematikisa da kompiuterul mecnierebaTaseria) eZRvneba profesor revaz absavas xsovnas (gardacvalebidan 5 wlis-Tavs) da masSi, ZiriTadad, warmodgenilia albaTobiis Teoriisa da maTematikuristatistikis, aseve momijnave dargebis aqtualuri sakiTxebi.krebuli gankuTvnilia Sesabamisi dargebis specialistebisaTvis dastudentebisaTvis.SromebismTavari saredaqcio sabWo: istoriis mecnierebaTa doqtori,profesori joni afaqiZe (Tavmjdomare), filologiis mecniereba-Ta doqtori, profesori diana alania, ekonomikis mecnierebaTa doqtori,profesori revaz xarebava, fizika-maTematikis mecnierebaTadoqtori, profesori vladimer kircxalia, biologiis mecnierebaTadoqtori, profesori zaur lomTaTiZe, filologiis mecnierebaTadoqtori, profesori mariam miresaSvili, filologiis doqtori,profesori merab naWyebia, istoriis mecnierebaTa doqtori, profesorizurab papasqiri (Tavmjdomaris moadgile), fizika-maTematikismecnierebaTa doqtori, profesori Temur CilaCava, filosofiismecnierebaTa doqtori, profesori leonide jaxaia, pedagogikis mecnierebaTadoqtori, profesori jemal jinjixaZe, iuridiul mecnierebaTadoqtori, profesori zaur jinjolavamTavari saredaqcio sabWos pasuxismgebeli mdivani: asocirebuliprofesori roin beriamTavari saredaqcio sabWos mdivani: doqtori, eka esebuamaTematikisa da kompiuterul mecnierebaTa seriismaTematikisa da kompiuterul mecnierebaTa seriissaredaqcio kolegia: fizika-maTematikis mecnierebaTa doqtori,prof. Temur CilaCava (mTavari redaqtori); fizika-maTematikismecnierebaTa doqtori, prof. malxaz aSordia; fizika-maTematikismecnierebaTa doqtori, prof. uSangi goginava (Tsu), fizika-ma-Tematikis mecnierebaTa doqtori, prof. daviT gordeziani (Tsu);doqtori, prof. nana gulua; teqnikur mecnierebaTa doqtori, prof.revaz kakubava; fizika-maTematikis mecnierebaTa doqtori, prof.franCesko kriado (malagis universiteti, espaneTi); fizikamaTematikismecnierebaTa doqtori, prof. hamlet melaZe (mTavariredaqtoris moadgile); fizika-maTematikis mecnierebaTa doqtori;prof grigori sisoevi (birmingemis universiteti, didi britaneTi);doqtori, prof. aleqsandre Sangua; fizika-maTematikis mecnierebaTadoqtori, prof. oTar Wkadua.seriis pasuxismgebeli mdivani: asocirebuli profesoriinga gabisonia© soxumis saxelmwifo universiteti, 2009

SOKHUMI STATE UNIVERSITYSOKHUMI STATE UNIVERSITYPROCEEDINGSVIIMATHEMATICS AND COMPUTERSCIENCES SERIESSokhumi State University Publishing HouseTbilisi – 2009

The VII volume of “PROCEEDINGS” (Mathematics and Computer SciencesSeries) represents the researches Sokhumi State Universities emploee‘son the topical issues about actual problems of mathematics and computer sciences( prof. Revaz Absava mention).CHIEF EDITORIAL COUNCIL OF “PROCEEDINGS”Doctor of Historical Sciences, Professor John Apakidze (Head ofthe Council), Doctor of Philological Sciences, Professor Diana Alania,Doctor of Physics and Mathematics Sciences, Professor Temur Chilachava,Doctor of Philosophical Sciences, Professor Leonid Djakhaia,Doctor of Pedagogical Sciences, Professor Jemal Jinjikhadze, Doctor ofJuridical Sciences, Professor Zaur Jinjolava, Doctor of Economical Sciences,Professor Revaz Kharebava, Doctor of Physics and MathematicsSciences, Professor Vladimer Kirtskhalia, Doctor of Biological Sciences,Professor Zaur Lomtatidze, Doctor of Philological Sciences, ProfessorMariam Miresashvili, Doctor of Philology Professor MerabNachkebia, Doctor of Historical Sciences, Professor Zurab Papaskiri(Deputy head of Council)Executive Secretary of the Chief Editorial Council: Doctor, AssociateProfessor Roin BeriaSecretary of the Chief Editorial Council: Doctor Eka EsebuaMATHEMATICS AND COMPUTER SCIENCES SERIESEditorial Board: Doctor of Physics and Mathematics Sciences,Professor Temur Chilachava (Editor-in-Chief), Doctor of Physics andMathematics Sciences, Professor Malkhaz Ashordia; Doctor of Physicsand Mathematics Sciences, Professor Otar Chkadua; Professor FranciscoCriado (University of Malaga, Spain,); Professor Ushangi Goginava(Iv. Javakhishvili Tbilisi State University); Professor David Gordeziani(Iv. Javakhishvili Tbilisi State University), Associate ProfessorNana Gulua; Professor Revaz Kakubava; Professor Hamlet Meladze(Vice Editor-in-Chief), Professor Alexsander Shangua; Professor GrigoriSisoev (Birmingham University, United Kingdom).Executive Secretary: Doctor, Assistant Professor Inga Gabisonia,© Sokhumi State University – 2009

s a r C e v ielizbar nadaraia, petre babilua, grigol soxaZe.ganawilebis funqciis Sefaseba arapirdapiri SerCeviT ....................... 12elizbar nadaraia, mzia facacia, grigol soxaZe.ganawilebis parametrebis fsevdo-maqsimaluridasajerobis meTodiT Sefasebis Sesaxeb cenzurirebulidajgufebuli monacemebisaTvis ...................................................................................... 33gogi fanculaia. usasrulo-ganzomilebian paralelepipedzegansazRvruli funqciebis rimanis azriT integrebadobisSesaxeb ...................................................................................................................................................... 44uSangi goginava. organzomilebiani SeuRlebuli furieuolSismwkrivebis zomiT krebadobis Sesaxeb ........................................... 53nGML 2 – mebus-listingis ganzogadebulizedapirebis zogierTi geometriuli Tvisebis Sesaxeb damaTi kavSiri `lentisebri~ xlarTebis simravlesTan ............................. 65ilia TavxeliZe.Temur CilaCava, nugzar kereseliZe. sainformacio omisaraprevenciuli uwyveti wrfivi maTematikuri modeli ......................... 91Temur CilaCava, nugzar kereseliZe. prevenciulisainformacio omis uwyveti wrfivi maTematikuri modeli ............... 113hamlet melaZe. orSriani faqtorizebuli sxvaobiani sqemebiparaboluri tipis kerZowarmoebuliani gantolebaTamravalganzomilebiani sistemisaTvis ..................................................................... 142Tina gorozia, TinaTin daviTaSvili. sasazRvro amocanagrafebze gansazRvruli Cveulebriv diferencialurgantolebaTa erTi klasisTvis ..................................................................................... 155Temur CilaCava, ciala ZiZiguri, leila sulava,maia Cakaberia. administraciuli zewolis arawrfivimaTematikuri modeli .............................................................................................................. 1695

C O N T E N T SElizbar Nadaraya, Petre Babilua, Grigol Sokhadze.The estimation of a distribution function by an indirect sample ........... 12Elizbar Nadaraya, Mzia Patsatsia, Grigol Sokhadze. On theMaximum Pseudo-Likelihood Estimation of a DistributionParameters by Grouped Observations with Censoring ........................ 33Gogi Pantsulaia On a Riemann integrability of functions definedon infinite-diementional ctangles .......................................................... 44Ushangi Goginava Convergence in measure of two-dimentionalconjugate Walsh-Fejer means .............................................................. 53Ilia Tavkhelidze. About Some Geometric Characteristic of thenGeneralized Möbius Listing's surfaces GML 2 and its Connnectionswith set of Ribbon Links ....................................................................... 65Temur Chilachava, Nugzar Kereselidze. Non-preventivecontinuous linear mathematical model of information warfare ......... 91Temur Chilachava, Nugzar Kereselidze. Continuous linearmathematical model of preventive information warfare ................. 113Hamlet Meladze. On Two-layer Factorized Difference Schemesfor System of Differential Equations with Partial Derivativesof Parabolic Type ............................................................................... 142Tina Gorozia, Tinatin Davitashvili. Boundary value problemfor one class of ordinary differential equations, given on graphs ..... 155Temur Chilachava., Tsiala Dzidziguri, Leila Sulava,Maia Chkaberia. Nonlinear mathematical model of administrativepressure .............................................................................................. 1696

soxumis saxelmwifo universitetis Sromebit. VII, 2009maTematikisa da kompiuterul mecnierebaTa seriaprofesori revaz absavarevaz mixeilis Ze absava daibada 1947 wlis 9 Tebervalszugdidis raionis sofel grigoliSSi, mosamsaxurisojaxSi.7

1965 wels daamTavra korcxelis saSualo skola oqrosmedalze, 1969w. ki m. gorkis saxelobis soxumis saxelmwifopedagogiuri institutis fizika-maTematikis fakulteti maTematikisspecialobiT. 1969-1971 wlebSi igi muSaobda zugdidisraionis sofel jixaSkaris I saSualo skolaSi maTematikismaswavleblad, 1971 wlidan ki soxumis pedagogiuri institutis,1979 wlidan afxazeTis saxelmwifo universitetis fizika-maTematikisfakultetis algebra-geometriis kaTedrazemaswavleblis, ufrosi maswavleblis, docentis Tanamdebobebze.1989 wlidan, afxazeTis saxelmwifo universitetidanqarTuli nawilis gamoyofisa da mis bazaze axali sauniversitetocentris – ivane javaxiSvilis sax. Tbilisis saxelmwifouniversitetis soxumis filialis dafuZnebis Semdeg,is iyo am umaRlesi saswavleblis maTematikisa da kompiuterulmecnierebaTa fakultetis dekani.mSobliur institutSi samuSaod dabrunebis dRidanveganisazRvra misi pedagogiuri da samecniero muSaobis mimar-Tuleba, albaTobis Teoria da maTematikuri statistika.revaz absava, rogorc mecnieri, ivane javaxiSvilis saxelobisTbilisis saxelmwifo universitetis albaTobis Teoriisada maTematikuri statistikis kaTedris aRzrdilia.is mecnierebis did da eklian gzaze daakvaliana kaTedrisgamgem profesorma gvanji maniam, gamoCenilma mecnierma damecnierebis brwyinvale organizatorma. amave kaTedris profesoris,elizbar nadaraias xelmZRvanelobiT Sesrulda sakandidatodisertacia Temaze: `mravalganzomilebiani ganawilebissimkvrivisa da regresiis funqciis araparametruligulovani tipis SefasebaTa Sesaxeb~ (spec.: 01.01.05 – albaTobisTeoria da maTematikuri statistika), romelic brwyinvaleddaicva moskovis eleqtronuli manqanaTmSeneblobis institutisspecializirebuli sabWos (К.063.68.05) sxdomaze 1986wlis 10 oqtombers da mas mieniWa fizika-maTematikis mecnierebaTakandidatis samecniero xarisxi. xolo, 1989w. martSiumaRlesma saatestacio komisiam mianiWa docentis wodeba.Semdgom iyo politikuri krizisi afxazeTSi, omi, romelmacSeaferxa ara marto TiToeuli afxazeTelis winsvla...1993 wlidan TbilisSi, miuxedavad didi problemebisa,pedagogiuri da administraciuli muSaobis paralelurad r.absavam ganagrZo samecniero muSaoba (konsultanti: Tsu alba-Tobis Teoriisa da maTematikuri statistikis kaTedris gamge,saqarTvelos mecnierebaTa akademiis wevr-korespondenti, profesorielizbar nadaraia). kvlevis obieqti iyo: ganawilebiskanonis funqcionaluri maxasiaTeblebis zogadi saxis arap-8

arametrul SefasebaTa kvadratuli gadaxris zRvariTi ganawilebismiRebis meTodis SemuSaveba da misi realizeba; ganawilebissimkvrivis da regresiis mrudis Sesaxeb martivi darTuli hipoTezebis Sesamowmebeli kriteriumebis ageba damaTi simZlavris dadgena; klasifikaciis da identifikaciisproblemebTan dakavSirebuli zogierTi sakiTxis gadawyveta.revaz absava 30-ze meti samecniero publikaciis avtoria,maT Soris aris ori meToduri miTiTeba, saxelmZRvaneloalbaTobis TeoriaSi da monografia. revaz absavas mecnierulmakvlevebma dasrulebuli saxe miiRo sadoqtoro disertaciissaxiT: `ganawilebis kanonis funqcionaluri maxasiaTeblebisaraparametruli Sefasebebi da gamoyeneba klasifikaciisda identifikaciis amocanebSi~ (spec.: 01.01.08 – ma-Tematikuri kibernetika). fizika–maTematikis mecnierebaTa doqtorissamecniero xarisxis mosapoveblad disertacia waredginaivane javaxiSvilis sax. Tbilisis saxelmwifo universitetisPh.M.01.08 sadisertacio sabWos. dainiSna dacvis dRe...dacvamde ramdenime dRiT adre, 57 wlis asakSi wavidaCvengan batoni revazi da dagvitova didi gulistkivili.imdenad didi iyo am SemTxvevis rezonansi saqarTvelos samecnierosazogadoebaSi, rom moxda uprecedento SemTxveva maTematikosebsSoris saqarTvelos istoriaSi. daniSnul drosSedga zemoT aRniSnuli sabWos sxdoma, sadac disertantisrolSi iyo konsultanti, batoni elizbar nadaraia.naSromis mecnieruli siaxle mdgomareobs SemdegSi: ganawilebis funqcionaluri maxasiaTeblebis araparametrulSefasebaTa zogadi klasisaTvis Seswavliliaintegraluri kvadratuli gadaxris zRvariTiganawileba. arsebiTad, es klasi moicavsbevr cnobil Sefasebas, rogoricaa, magaliTad, ganawilebissimkvrivis gulovani da proeqciuli tipisSefasebebi, histogramebi, regresiis mrudis gulovaniSefaseba, regresogramebi, riskis funqciisgulovani Sefaseba da a.S. moZebnilia kvadratuli gadaxris zRvariTi gadawilebaTiTqmis yvela zemoT CamoTvlili funqcionalurimaxasiaTeblebis SefasebisaTvis friad zogadpirobebSi, amasTanave pirvelad iqna Seswavlilikvadratuli gadaxris zRvariTi ganawileba regresogramis,`SeSfoTebuli~ regresiis (bernSteinisamocana) da riskis funqciisaTvis.9

10ganawilebis simkvrivis da regresiis mrudisaTvisagebulia integraluri tipis kriteriumebi statistikurhipoTezaTa Sesamowmeblad.helingeris manZilis cnebaze dayrdnobiT agebuliaTanxmobis kriteriumi, romelic ar aris damokidebuliganawilebis simkvrivis saxeze.Seswavlilia ganawilebis simkvrivis da regresiismrudis araparametruli gulovani tipis Sefaseba-Ta gamoyeneba klasifikaciis da identifikaciis amocanebSi.kerZod, agebulia empiriuli gadamwyvetiwesebis klasi, romelTaTvisac baiesis riskis Sesabamisimimdevrobebi TiTqmis yvelgan ikribeba optimalurisaken.Seswavlilia regresiis funqciisaTvis gaser–miuleristipis araparametruli gulovani Sefasebis gamoyenebaidentifikaciis amocanebisaTvis. miRebuliaam Sefasebis kvadratuli gadaxris zRvariTiganawileba da moyvanilia maTi gamoyeneba regresiisfunqciisaTvis martivi da rTuli hipoTezis SemowmebasTandakavSirebul amocanebTan. amoxsniliai. linikis ganzogadebuli amocana. SemowmebuliahipoTeza mocemuli rigis paraboluri tipis regresiisarsebobis Sesaxeb.SemuSavebulia regresiis ori funqciis tolobishipoTezis Semowmebis kriteriumis agebis meTodika,romelic mWidro kavSirSia identifikaciis amocanebTanda Seswavlilia agebuli kriteriumis asimptoturiTvisebebi.ramdenime ganawilebis SemTxvevaSi agebulia erTgvarovnebishipoTezis Semowmebis kriteriumi, amasTanaveganixileba `maxlobeli~ alternativebis mimdevroba,romelTaTvisac SemoTavazebuli kriteriumebiufro mZlavria, vidre kolmogorov–smirnovistipis kriteriumebi.SemuSavebulia optimaluri, adapturi gulovani Sefasebisagebis meTodika mravalganzomilebiani ganawilebissimkvrivisaTvis. moZebnilia guli, romlissaSualebiT SegviZlia minimumamde daviyvanoT ganawilebissimkvrivis Sefasebis integraluri kvadratuligadaxris maTematikuri lodini. ganzogadebuliaveis-volfovicis Sedegi wertilSi ganawilebissimkvrivis Sefasebis Sesaxeb.

sadisertacio sabWom erTxmad, faruli kenWisyriT, revazabsavas mianiWa fizika-maTematikis mecnierebaTa doqtorissamecniero xarisxi sikvdilis Semdeg.aRsaniSnavia, rom gardacvalebis Semdeg, batoni elizbarnadaraias ZalisxmeviT 2005 wels iyo gamocemuli zemoTaRniSnuli monografia `dakvirvebaTa ganawilebis kanonisfunqcionaluri maxasiaTeblebis araparametrul SefasebaTaTeoriis zogierTi amocana~. Semdgom, 2008 wels, misive TanaavtorobiTTsu gamomcemlobam gamosca monografiis ganaxlebulivarianti.2005 da 2009 wlebSi Tbilisis universitetma orjer gamoscarevaz absavas TanaavtorobiT `albaTobis Teoriis~saxelmZRvanelos ganaxlebuli da Sevsebuli variantebi.revaz absavam pirovnuli TvisebebiT, saqmianobiT, mecnierulimoRvaweobiT samaradJamod ukvdavyo Tavisi saxeli.man Tavisi cxovrebiT bevrs uCvena magaliTi adamianuri Tanadgomisada uSvretelad siTbos gacemisa. man Camoayalibabrwyinvale, Zlieri, akademiuri koleqtivi, misi mSobliurimaTematikisa da kompiuterul mecnierebaTa fakulteti. isiyo fakultetis yvela wevris megobari, soxumis saxelmwifouniversitetis erT-erTi aRiarebuli lideri. yvela, vinc rezoabsavas kargad icnobda, ecodineba, rom man Tavisi xanmoklesicocxlis ganmavlobaSi SeZlo kacurad ecxovra, gamoevlinasiyvarulisa da Sromis udidesi niWi, gamxdariyomisabaZi Tavisi ganumeorebeli da unikaluri xasiaTiT...soxumis saxelmwifo universitetisSromebis MmaTematikisa dakompiuterul mecnierebaTa seriissaredaqcio kolegiaMMM11

soxumis saxelmwifo universitetis Sromebit. VII, 2009maTematikisa da kompiuterul mecnierebaTa seriaELIZBAR NADARAYA, PETRE BABILUA,GRIGOL SOKHADZETHE ESTIMATION OF A DISTRIBUTION FUNCTIONBY AN INDIRECT SAMPLEAbstract. The problem of estimation of a distribution function isconsideredwhen the observer has an access only to some indicator random values.Some basic asymptotic properties of the constructed estimates are studied. In thispaper, the limit theorems are proved for continuous functionals related to the estimateof F ˆ nxin the space C a, 1a .2010 Mathematical Subject Classification: 60F05, 62G05, 62G10,62G20.Key words and phrases: distribution function estimate, unbiased, consistency,asymptotic normality, estimate of time moments, Wiener process, randomprocess.LetX ,1. Introduction, X , 1 2 Xnbe a s ample of independent observations of arandom non-negative value X with a distribution function F x . In problemsof the theory of censored observations, sample values are pairsYi Xiti and Zi IYi Xi, i 1,n , where tiare given numbers( ti tjfor i j ) or sample values independent of Xi, i 1,n . Throughoutthe paper, I Adenotes the indicator of the set A.Our present study deals with a somewhat different case: an observ- I X t ,er has an access only to the values of random variables with ti2i1 cF, i 1,n2n , inf x 0 : Fx1 c F.i ii12

F xThe problem consists in estimating the distribution functionby means of a sample , , 1 2 , n. Such a problem arises for examplefrom a region of corrosion investigations, see [1] where an experimentrelated to corrosion is described.As an estimate for F x we consider an expression of the form 0,x 0,ˆ 1Fnx F1nxF2nx,0 x cF,(1)1,x cF,n1 x tj F1nx Kj,nh j1 h n1 x tj F2nx K,nh j1 h K is some weight function (kernel), hnh is a se-where xquence of positive numbers converging to zero.2. Conditions of asymptotic unbiasedness and consistencyIn this subsection we give the conditions of asymptotic unbiasednessand consistency and the theorems on a limiting distribution ˆ x .1nhLemma 1. Assume that1 0 . xand Kx Kx, R ,F n K is some distribution density of bounded variationx . If nh , thenx t1, (2)nc Fm 1 j m 1 m x u m 1 2 11 12 1 K Ftj K Fudu Oj1 h cFh h nh0uniformly with respect to 0x ,c F; m1, m2are natural numbers.P xbe a uniform distribution function on Proof. Let0 ,c F, andP nxbe an empirical distribution function of ―the sample‖ t 1, t 2, , tn,n1i.e., Pnx n Itj xj1. It is obvious that13

supPx Pxsup1 nnn0xcF 0xcFF2F2We haven1 m x ti m 111 2 K Finhi1 h cFh0c Fxc1 xcc F1 2 x u h 1. (3)nm11m 1tK F udu1 m x u 1 1 m21 K FudPnu Pu. (4)h 0 h Applying the integration by parts formula to the integral in theright-hand part of (4) and taking (3) into account, we obtain (2).Below it is assumed without loss of generality that the interval0,c 0,1 . F Theorem 1. Let F x be continuous and the conditions of thelemma be fulfilled. Then the estimate (1) is asymptotically unbiased andconsistent at all points x 0,1. Moreover, F ˆ nxhas an asymptoticallynormal distribution, i.e., ˆ ˆ 1dnh Fnx EFnxxN0,1 ,22 x Fx1 FxK u du,where d denotes convergence in distribution, and 0,1N a randomvalue having a normal distribution with mean 0 and variance 1.Proof. By Lemma 1 we haveand forEFF1n2nn xhx KtFx htx1h1h1 1 dt O, nh 1 nh x K du O ,0 x u h (5)14

11,x 0,1 ,1 x u KduF2x 1h h0 , x 0, x 1,2xhx1hKtFx thdt FxFxHence it follows that Fˆ xFx, 0,1E nx as n .Analogously, it is not difficult to show that1 x uVar Fˆ1 2 1 2nx K Fu1Fudu O F2nx22 nh h0nh . Hence we readily derive2., and thereforenh Var Fˆ n22x ~ x Fx1 Fx K ufor x 0,1.Thus ˆ xis a consistent estimate for F x , x 0,1F ndu(6)ˆ has an asymptotically normal dis-P Fˆx ˆ1 Fnx21as n ,x1 x2,x1,x20,1Let us now establish that F nxtribution. Since, by virtue of (5), F nx F 2xrify the condition of the Liapunov central limit theorem for xLLet us denoteand show thatn. ii2 , it remains for us to ve- x t h1ix nh KiF 1 n.n21 2n EjEjVar F1nx 0, 0. (7)j1We havenj 1 n21 2 x tj E j Ej 2Mnh KFtj,M max Kx.hxRj 1 15

Hence, taking (2) into account, we findL nnj12 1nh Ej Ej c1. (8)Using the relation (6) and the inequality (8), we establish that O nh 2 , i.e., (7) holds. In this subsection we define the conditions, under which the esti-xF x .mate163. Uniform consistencyFˆ n uniformly converges in probability (a.s.) to true Let us introduce the Fourier transform of the function K xand assume thatteitxKx2 0 . t is absolutely integrable. Following E. Parzen [2], xcan be represented asThusF1nDenoteF1nx12exiuhu1nhdxnj 1t jiuhjedu . xn1 iu 1hxEF1nx e u j Ftjdn sup FˆxTheorem 2. Letnn2nhj1et jiuh ˆ x EF x, h,1 h ,0 1nK xsatisfy conditions 1 0 and 2 0 .(a) Let xnF 1 ndu .F be continuous and n1h 2 n , thensup ˆPD F x F x ;n(b) If n1xnn 0p2 pn h , 2p , then D 0 a.s.n.

Proof. We have 11h 1sup x u1 Kdu Kudu Kudu 0. (9)xn1 h0 h hThis and (5) imply thatsup 2x 1 , (10)F nxn 0i.e., due to the uniform convergence for any 0 , 0 1,and0sufficiently large n0to x . Thereforedpnndnn , we have2x 1011 sup F xEF x0120 F nuniformly with respect1nh1iuh1 u e du, Ft.0xn1n1nnj1Hence, by Hölder‘s inequality, we obtain 1EdpnThuspnt j p 1iuh0 2 u e dupjj 1p q. j kE cos u j,k h jtjjp1 1 q u du , 1,p 221t t c , p, u du nh pjk, (11)whereDenotec p 1,0udup .2p, 1 tj k cosu j,k h A uThen by (11) we writep112 t .pp , p, u EAu 2 du u E AuEAu 2 du pEdn 2 c 0pnh. (12)Using Whittle‘s inequality [3] for moments of quadratic form, weobtainjkp q pij17

18E A u3 p 2 p p 4212 j k 2 2 EA u 2 2 c c p 2 cosup p where1i,jt th j k, 2pp2 p 1kp Ek 1, cs . 2 Hence it follows thatpp 2E A u EA u 2 On(13) u , . It is also clear thatuniformly with respect to pp 2EA u 2 On(14) uniformly with respect to u ,.Having combined the relations (12), (13) and (14), we obtain 1 pEd n O ,p n h p 2 .ThereforecP FˆnxEFˆ3supnx xp n h pn . (15)Furthermore, we have1 sup EFˆnxFx sup EF1nxFx sup 1F2nx .xn10 xnxn (16)By virtue of (10), the second summand in the right-hand part of(16) tends to 0, whereas the first summand is estimated as follows: 1 sup EF1 nxFx S1n S2n O , (17)xn nh SS1n2n1 sup0x1h10F 1 sup 1xnhy10 Fx x y Kdy, h p x y Kdy, h p

and, by virtue of (9),S2 n 0 as n . (18)Let us now consider . Note thatsup 0x1S1nS 1 n sup sup10x10x0x1x1FFy Fx1 x y Kdyh h 1h u h x uFx K du 1 u Fx uFxKdu. (19)h h Assume that 0 and divide the integration domain in (19) intotwo domains u and u . ThenxRu S1n sup0x1u supFx uFx0x1u Fx uFx 2 Ku1 u K du h h 1h u h x uFx K du sup sup Fdu . (20)u hBy a choice of 0 the first summand in the right-hand part of(20) can be made arbitrarily small. Choosing 0 and letting n ,we find that the second summand tends to zero. Thereforelim S 1 0 . (21)nnFinally, from the relations (15)-(18) and (21) the proof of the theoremfollows.Remark 1.K , x 1and 1(1) If x0then S2n 0.(2) In the conditions of Theorem 2 , i.e., h1hn, ,19

ˆsup F nxa,bxFx 0in probability (a.s.) for any fixed interval a, b0,1n such that a, bn, n n0.since there may exist0. The conditions of Theorem 2 are ful-Assume thatfilled:andh n , 0 n1121n h n if 0 ,2p22 pp n hn if 0 , p 2 .2 p4. Estimation of momentsIn considering the problem, there naturally arises a question of estimationof the integral functionals of F x, for example, moments m,m 1:1t m tm 1 1F dt .As estimates for we consider the statisticsmm0 t tn 1hm 1 m1j 1ˆ nm 1j t KF2ntdt.n j1h hhTheorem 3. Let xK x0 outside the interval 1,1 K satisfy condition 1 0 and, in addition to this, . If nh as n , then ˆnkisan asymptotically unbiased, consistent estimate for mand moreoverthatn ˆ E ˆnmProof. SincenmdNK xhas 1,1 12 2 2m20,1 , m t Ft1 Ft0dt . as a support, we establish from (5)20

1 F2nn 1O nh uniformly with respect to xh, 1h.Hence, by Lemma 1 we haven1hm 1 m1 t tj E ˆ nm 1 Ftj t KFn j 1h hh 1m 1m1hh1h1h h1110K t u KF h udut12nm1tdt 1 dt O nh 1 nh m1vFt vhdvtdt O 1 1 mtm 111 KvFt vhdvdt Oh O . 22) nh0 1By the Lebesgue theorem on majorized convergence, from (22) weestablish thatVar ˆ 1t11 Ftdt , m 1m1m1E ˆ 1m F t dt m t. (23)nm00Therefore ˆnmis an asymptotically unbiased estimate for m.Further, analogously to (22), it can be shown thatnmmnwhere2 10Ft1Ftt2m2 1t K1 K 1 h ,vvu K du .By the same Lebesgue theorem we see that1ttth2m h 1 dt O O2 n nh 2 2 2m2n Var ˆ ~ m t F 1F dt . (24)nm0PTherefore (23) and (24) imply that ˆ nmm.To complete the proof of the theorem it remains to show that then ˆ E ˆ have an asymptotically normal distributionstatistics nmnm21

with mean 0 and dispersion 1. For this it suffices to show that the Liapunovfraction L 0. Indeed,Ln n c6n c n7n1h2 22m1j 1m E ˆjF tjt K F2ndt Varnm22 E ˆjF tjVarnm1nj 1nj 11 ˆ2Var 2 On.nmThe theorem is proved.1hh 1 2 t th2 1 2 ˆ .5. Limit theorems of functionals related to the estimate xIn this subsection the kernel K x0 is chosen so that it would bea function of finite variation and satisfy the conditionsF n u Ku, Kudu1,Ku 0 for u 1K .Theorem 4. Let g x0 , xa1aand bounded function.Tnwheren(a) If a 01aag, ,10 a , be a measurable22F and nh as n , thend2 dx N0 x FˆxEFˆx1 nn,g1 ;t 1tx gx Fx,t1, (25)2(b) If F a 0 , nh , nh 4 0 as n andF xhas bounded derivatives up to second order, then asn 22

Tn2 1aang1a2ag1d2 dxN0, ,x FˆxFxu du.nRemark 2. We have introduced a 0 in (25) in order to avoid theboundary effect of the estimate F ˆ nxsince near the interval boundarythe estimate F ˆ nxbeing a kernel type estimate behaves worse in thesense of order of bias tendency to zero than on any inner interval1a , 1 a0,1,0 a .2Proof of Theorem 4. We haven1a11 u tj Tn j Ftj Kg2nudu,n j1ha h where1g2nu g1uF2nu.Hencen1a2 12 1 u tj 2 n VarTn F tj Kgnu du. (26)n j 1 ha h Since K uhas 1,1 as a support and 0 a u 1a, it can beeasily verified that 1 1 F2nu 1O and g2nu g1uO nh nh u a, 1a . Therefore from (26) we haveuniformly on n1a2 1 21 u tj 1 n F tj Kg1u du O .n j 1 h h nhaBy virtue of Lemma 1, we can easily show that2223

1n24nj121a1 j Ftj Kg1u hThereforea10u t2h 1 hdu 1a2 u t h 1 nhFtdt K g u du O .1a1a2 21 u t 1 2 1 n F t dt Kg1u du n n O2 , (27) h h nhaa1n2na011a22 1 ha1a u t h 1Ftdt K g u du ,2 1 ha1aSince by F u1 Fu , u c8 u t h 1Ftdt K g u du .14ag andFu 1 , a u aF1 ,it follows that1u c9a1 F1 ag , we have1at a h 1n c10dt Kudu, (28)0 at h where a t 0 and 1at 0. The first inequality is obvious,whereas the second one follows from the inequalities 0 t a and10 a .2Therefore1ath 0,0 t a,lim Kudu 1n at , t a.h2By the Lebesgue theorem on bounded convergence, from the latterexpression and (28) we obtain1 0 as n . (29)n212222

Analogously,1n 0 as n . (30)Now let us establish that1aa2We have1aa2 1 h1a2 2Ftdt K g udu g uduas n a u t 1 h .21a22Ftdt g uKdu Ftgt cSince1a12a 1 h 1dt h1aa1aag1 c1 u t h 1a11a2 u t h 21aa 1h1aFtdtg uKdu g tuKdu g t K du 1a1aa11h1a1a1 u t c13 Kdu1dt A1n A2nh ha a 1aa1 u t h u t h dt1 . (31)1 u t K du 1ha h for all t a, 1a, we haveA2 n 0 as n . (32)Further, we continue the function g 1uso that that outsidea, 1 ait has zero values and denote the continued function by g 1u.Then1A 1 n c140 1 c 1511 gg1x yg y11 1 x dyKdx h h y uhg y dyKuduuhKudu 0 as n 1 c15 , (33)125

26wherey 11 g y xg x dx .The (33) holds by virtue of the Lebesgue theorem on majorized uh g and uh 0 asconvergence and the fact that 1 ,n 2L 1. Thereby, taking (27)-(33) into account, we have proved that1au2 2 2 g du . (34)naNow let us verify the fulfillment of the conditions of the centrallimit theorems for the sumsn1Tn ajnj Ftj,nWe havesince c162Var T n .ajn 1 n1aaj11 x tKh hjg2ntx dx.2 2 2n ajnE j Fjj L n,a jn , Ft 1j12 On 2VarT nj2E for all 1 j n andFinally, the statement (b) of the theorem follows from (a) if we takeinto account thatn1aag11a1 dx n g x Ku Fx uhFxx EFˆx Fxn 1 OThe theorem is proved.Lemma 2.s2 h n h Oa n1 dudx . (35)(1) In the conditions of the item (a) of Theorem 4,s1a 217g udu, s 2ETn c . (36) a

(2) In the conditions of the item (b) of Theorem 4,ss1a 218g udu, s 2ETn c . (37) a Proof. Tnis the linear form of j j F tj, Ej 0 , 1 j n .Hence to prove (36) we use Whittle‘s inequality [2].It is obvious that E 1, j 1,n . Therefore by Whittle‘s inequalityETnE Tss n 1as 1 s u tE Tn cs2 K2 h j1ha 1where g2nu g1uF2nu.This, by virtue of Lemma 1, yieldsjj g2nu2 du 122 1a1as 1 u t 1 cs2 K g2nudu dt O g2 2 nu du n hnh0 aa . (38)Further, sinceu 1 1 guO c guFaF a1 nh 19, 11 a u afrom (38) it follows thatg2 n,ns c20 1sup0t1 h1a 1 O nh1a c 21 g u du a Next we obtaina2 u t Kg h s2s21aag2n2nuudu 1a1 u t Kgh h 1 o1 c gu du , s 2.22du s2a10dt1aas22ns2,s2udus227

E Tsn 2 c E T1as1 sn n g ˆ1 na231aag c 24The lemma is proved.s2u EF uFu1a2udu O n h gudu1aags2u du.Let us introduce the following random processes:assdu TTnnt Fut n Fˆu EFˆuat Fut n Fˆu Fuannn du.du,Theorem 5.1 0 . Let the conditions of the item (a) of Theorem 4 be fulfilled.Then for all continuous functionals C a, 1a ,f on the distribution f Tntconverges to the distributionf W t awhere Wt a, a t 1 a , is a Wiener processwith a correlation function rs, t min t a,s a,W t a 0 , t a .2 0 . Let the conditions of the item (b) of Theorem 4 be fulfilled.Then for all continuous functionals f on Ca, 1a,the distribution f Tn t converges to the distributionf W t a . Proof. First we will show that the finite-dimensional distributionsof processes T nt converge to the finite-dimensional distribution of a process taW , t a . Let us consider one moment of time t 1. We haveto show that28

Tndt W t a11. (39)To prove (39), it suffices to take gx I xvirtue of Theorem 4,Tn in (25). Then, bya,t 11a1 0 a,t1,1adtN, I xdx N0t aLet us now consider two moments of time t 1, t2, t1 t2. We haveto show that.dTt T t Wt a W t an1 n 21,2, .(40)To prove (40), it suffices to take in (25)gx I xI x ,1 2 a,t1 2 t1, t2where 1and 2are arbitrary finite numbers. Then, by virtue ofTheorem 4,d22t T tN 0 t a tt1Tn1 2 n 2,1 2 12 2 1.On the other hand,ta Wt a W t aW W t aWt a1W12 21 2 10 22122is distributed as N 0, t a t t 1 2 12 2 1. Therefore (40)holds. The case of three and more number of moments is considered analogously.Therefore the finite-dimensional distributions of processesT nt converge to the finite-dimensional distributions of a Wiener process tW a , a t 1 a with a correlation functionrtt min t a,t a,Wt a0,t a1,21 2.29

Now we will show that the sequence T nt is dense, i.e., the sequenceof the corresponding distributions is dense. For this it suffices toshow that for any t1 , t2a,1aand all nE Tss 2nt1 Tnt2 c25t1 t2, s 2 .Indeed, this inequality is obtained from (36) for gx I x .Further, taking (35), (37) and the statement (b) of Theorem 4 intoaccount, we easily ascertain that the finite-dimensional distributions ofprocesses T nt converge to the finite-dimensional distributions of a Wiener process taW , and also thatE Tss 2nt1 Tnt2 c26t1t2, s 2 .Hence, from Theorem 2 of the monograph [3, p. 583] the proof ofthe theorem follows.t1 , t26. ApplicationBy virtue of Theorem 5 and the Corollary of Theorem 1 from [3,p. 371] we can write thatPTn max Tnt Gat1a22exp 1 2a 21 2a dxx2(a is a prescribed number, 0 a 1 2) as n .This result makes it possible to construct tests of a level ,0 1, for testing the hypothesis H0by whichH lim EFˆx F x , a x 1 a0:n 0,nin the presence of the alternative hypothesis301an 0dx0Fx lim EFˆxF xH1 : 0 .an

Let be the critical value, G . If as a result of the experimentit turns out that Tn , then the hypothesis H0must be rejected.Remark 3. Let tibe the partitioning points of an interval 0 ,c F,2 j 1c F inf x 0: Fx1 , chosen from the relation H tj ,2nj 1,n, whereHxx 0hudu ,h uis some known density of a distribution on ,c Fh x 0 for all 00 andx ,c F. In that case, by a reasoning analogous tothat used above we can obtain a generalization of the results of thepresent study.Remark 4. Some ideas of the proof of Theorem 4 are borrowedfrom the interesting paper by A. V. Ivanov [5].REFERENCES1. K. V. Mandzhgaladze. On an estimator of a distribution functionand its moments. (Russian) – Soobshch. Akad. Nauk Gruzin. SSR,124 (1986), No. 2, 261-263.2. E. Parzen. On estimation of a probability density function andmode. – Ann. Math. Statist. 33 (1962), 1065-1076.3. P. Whittle. Bounds for the moments of linear and quadratic forms inindependent variables. – Teor. Verojatnost. i Primenen. 5 (1960),331-335.4. I. I. Gikhman and A. V. Skorokhod. Introduction to the theory ofrandom processes (in Russian). Gosudarstv. Izdat. Fiz.-Mat. Lit.Moskva, 1965.31

5. A. V. Ivanov. Properties of a nonparametric estimate of the regressionfunction (in Russian). – Dokl. Akad. Nauk Ukrain. SSR, Ser. ANo. 7 (1979), 499-502, 589.elizbar nadaraia, petre babilua,grigol soxaZeganawilebis funqciis Sefasebaarapirdapiri SerCeviTˆ SefasebasTan dakavSirebuli naSromSi agebulia ganawilebis funqciis Sefaseba, rodesacdamkvirveblisTvis misawvdomia zogierTi indikatoruliSemTxveviTi sididis mniSvnelobebi. Seswavlilia agebuliSefasebis zogierTi ZiriTadi Tviseba. naSromSi aseve damtkicebuliaF nxC a, 1a sivrce-Si uwyveti funqcionalebisaTvis zRvariTi Teoremebi.32

soxumis saxelmwifo universitetis Sromebit. VII, 2009maTematikisa da kompiuterul mecnierebaTa seriaELIZBAR NADARAYA, MZIA PATSATSIA,GRIGOL SOKHADZEON THE MAXIMUM PSEUDO-LIKELIHOODESTIMATION OF A DISTRIBUTION PARAMETERSBY GROUPED OBSERVATIONSWITH CENSORINGAbstract. The paper considers the problem of estimation of probabilitydistribution parameters by using grouped observations. In somegroups, observation involves not individual values of a random variable,but only their total quantity, whereas some groups are censored for theobserver so that even the quantity of elements contained in them is unknown.For such a sampling, the maximum pseudo-likelihood method isused, the question of asymptotic consistency and asymptotic effectivenessof such estimators is investigated. The consideration is concretizedfor the estimator of a mean normal distribution.2010 Mathematical Subject Classification: 60F05, 62G05,62G10, 62G20.Key words and phrases: Pseudo-Likelihood Estimation,Distribution Parameters, incomplete observation.1. IntroductionIn mathematical statistics, in particular in the point estimationtheory, a special place is held by the maximum likelihood method (in thesequel abbreviated to ―mlm‖). Various modifications of this method andits generalizations are successfully applied in many situations. In thecourse of many years the mlm has been used for grouped samplings, aswell as in censoring problems and estimator truncation problems (see [1]and [5] and the references therein). The results obtained are satisfactoryfrom both the mathematical and the computational standpoint.33

In the present paper, we consider the problem of estimation ofprobability distribution parameters by using a specific variant of observations.It is assumed that observations are grouped and there might be casesof partial or full non-observation of individual realization variables.We propose to use a somewhat more refined (in a certain sense) versionof the mlm and show that it leads to asymptotically consistent and effectiveestimators.342. Statement of the problem and the method its solution. Let X be a random variable with a distribution functionF x F x, , where is an unknown vector parameter in a finite-qdimensional space R . Assume that is a compactum. We need toconstruct the consistent estimator using observation data on the randomvariable X . The experiment is set up so that we do not know theactual number of realizations, but know only a part of them.Let the fixed points t1 t2 tn be given on thestraight line R (we do not exclude the case where the first or the lastpoint takes an infinite value). These points form intervals which may beof three categories:0) an interval интервал t i, t i 1 belongs to the zero-th category if inthis interval neither individual values of the sample nor the total quantityof sample values of the random value X are known:1) an interval t i, t i 1 belongs to the first category if in this categoryindividual values of the sampling are unknown, but the number ofsample values of the random value X is known. As usual, this number isdenoted by ni.t belongs to the second category if in this in-i, t i 12) an interval terval individual values of the samplingx ,i1, xi2, xinare known.iIn the sequel, summation or integration over the intervals of the zero-th,first or second categories will be denoted by the symbols (0), (1)или (2), respectively.We call a sampling of this type a partially grouped sampling withcensoring. Censored samplings of both types, as well as truncated samplingsare obviously a particular case of the problem stated. The absenceof information in the intervals of the zero-th category creates a difficulty

which we will try to overcome by assuming that we know the type of theF x, and the quantity of sample values not occurring indistribution an interval of the first category:iLet tt ii, i1n .1,2n iA be an interval of the zero-th category. Denote bym the quantity of sampling terms occurring inthe total quantity of observations. Note thatis a relative frequencyof the occurrence of X indistribution function, thenmi Fˆrt iFˆ1 rn m0i tinAi. Thenmi0mA . If Fˆx Fˆx,iir n is0m irris an empiricaland, by the strengthened Bernoulli law of large numbers, it reducesto pi Fti 1, Fti, with probability 1.By summing equalities (1) over all intervals of the zero-th categorywe findtFˆtFˆr i1r ii 0mi n.01Fˆrti1Frtii 0Hence we obtainFˆ ˆr( ti1)Fr( ti)mi n. (2)1Fˆ (1) ˆrtiFr( ti) i(0)Let us apply the maximum pseudo-likelihood method. Assume thatthe random value X has a distribution density f x f x, with respectto the Lebesgue measure. Then a likelihood function has the formLnwhereiminix FtFt FtFt f xi(0)i1ii(1)i1in; . (3)mi,j1il(1)i 0, are defined by (2). The finding of points of amaximum of the function x;L is complicated by the difficulty of35

studying the smoothness properties of empirical functions. For this reasonwe consider a slightly modified likelihood functionLnx FtFti(0)n1i1 i1ti1FtiF tFtFini ii FtFt f x; . (4)Lemma. Let the following conditions be fulfilled:(a) the distribution function F x, is continuous with respectto both variables and has the continuous derivativeF x, f x, ;x(b) the function L nx, has the absolute maximum n.Then nis an asymptotically consistent and asymptotically effectiveestimator of the true value of the parameter 0.i(1)The proof follows from the respective theorems of [6], [7].i1inj1il 3. Estimation of a mean for a normal distributionwith incomplete observation.t1 222Let X be a normally distributed random value with densityp t e , where is the unknown mean and is known..2Let the interval [ a , b]be inaccessible ( a and b may be infinite, too) forthe observer, and also the quantity of individual observations in this intervalbe unknown. However we have observations outside this interval:X1, X2,...,X n. It is required to estimate by these observations. For thiswe use maximum pseudo-likelihood estimators.To construct the likelihood function, note that if we denote by kthe number of terms of the total sampling which have occurred in [ a , b],kthen will be the frequency of occurrences in the interval [ a , b].k nTherefore, by the Bernoulli-Kholmogorov theorem,k pk na.s.,36

wheretaket u2t b a 1p , () e23du . Thus, as k , we b a n kˆ . b a 1 Since the probability that exactly k elements from the sampling b a will occur in the «black hole » is the pseudo-likelihood function in the formL ni11 Xi b a 2t2where as usual we denote t e1 b an b a1 k, we can write, (5). Note that the power in2(5) may turn out to be a non-integral number. However it is always positive.Note also that the multiplication sign is related to the expression infront of the large bracket.From (5) we obtainln L nlnn1 2X 22i1i2 b a n b a ln b a 1 .Hence we calculated 1ln L 2dni1Xin2 37

38 ababababnln1112 ababababnln112 ababababn11 ababnnXnii11212 21lnabababn bababnnXnii 1112212 abaln . (6)

Let us study the expression (6) for . Denote, for the sakeb a of brevity, t , s and note that for we obtain tt , s and 1. Therefores b a b a 0 , 0 and b a ln . By the Cauchy mean value theorem,ttss where s t . Note that for there exists the limit 1.tThe denominator in (6) tends to 1 and the first two summands inthe braces also tend to 1. So, we have to calculate the limit of the expression0 limlim ts lnts as .By l‘Hopital‘s rule,tsln t s3tst sttsslimtln limTherefore tlnL dlimd .tttt 1ss2ss 1 lim1 lim ttttttAnalogously, for we have t , s andlim t s ln t s 0 . ssss2sts 0.39

Therefored lim ln L .d The continuous functiond lndLchanges the sign and thereforedthere exists a point ˆ such that ln L 0 . Let us verify that thedˆsecond derivative at this point is negative.We write the second derivative in the formn 2n 22dln L 2dttss1t sln ts 1t s31t snn 2 2 ts t s t s t s221 t s2ts 1 t sln t s31 t sWe have to show that these four fractions, when summed, have anegative value. The first fraction is negative. For the third fraction2 1 ts1 n t s2 1t s 2we have a negative value since it is obvious that ts 1. Itremains to show that2 01 t sln t s ttss1 t s2ts .Since for 0 x 1we have 1tlnt 0 , we need to prove thatttss1 t s 2ts 2 0 . (7).40

may be negative. This mayhappen in two cases: when s 1and t 0 or s 0 and t 1. Becauseof symmetry we consider the second case, in which it is assumed ˆ athat . We use Kulldorf‘s inequality (see [1, p. 134]), by which ˆ b 2tt ss t s t s t s t s .In this expression only t t ss Since ts ts1 , we have2ttss1 t s 2ts ttsst s2 2ts t ststs 0.The second case with s 1and t 0 is considered in analogousmanner.2dThus ln L 0 and therefore ˆ is a unique maximum point for2dthe likelihood function (5). Taking Lemmas 1 and 2 into account, we canstate that the following theorem is valid.Theorem. Assume that we have the sampling of a normal randomvalue X1, X2,...,Xnwith the unknown mathematical expectation and2the known dispersion . Observation is carried out outside the interval[ a , b]where neither sampling terms nor their quantity are registered.Then the estimator of a maximum pseudo-likelihood exists for and isthe unique root of the equation b a n 1 n n b Xi 1 222 i1 b a 1 a b a ln 0. (8) Moreover, this estimator is asymptotically consistent and effective..41

Remark 1. While proving the theorem, we see that a or b may beinfinite. The case a and b correspond to the case of leftcensoring, whereas for a and b we have the right censoring.Example 1. If a and b 0 , then we obtain the following equationfor defining ˆ : 2n 1 ln 1 X i. n n i1Example 2. As has been shown in proving the theorem, forn1the equation (8) implies the classical case ˆ .X in i1b aREFERENCES1. G. Kulldorff. To the theory of estimation from grouped and partiallygrouped samples. Nauka. Moscow, 1966.2. L. Weiss, J. Wolfowitz. Maximum likelihood estimation of a translationparameter of a truncated distribution. – Ann. Statist. 1 (1973),944-947.3. N. Balakrishnan, Sh. Gupta, S. Panchapakesan. Estimation of thelocation and scale parameters of the extreme value distribution basedon multiply type-II censored samples. – Comm. Statist. TheoryMethods. 24 (1995), No. 8, 2105-2125.4. J. M. Wooldridge. Econometric analysis of cross section and paneldata. – Cambridge, MA: MIT Press, 2002.5. Zh. Zhang, H. E. Rockette. Semiparametric maximum likelihoodfor missing covariates in parametric regression. – Ann. Inst. Statist.Math. 58 (2006), No. 4, 687-706.6. T. S. Ferguson. A course in large sample theory. – Texts in StatisticalScience Series. Chapman & Hall. London, 1996.7. J. V. Dillon, G. Lebanon. Statistical and computational tradeoffs instochastic compose likelihood. ArXiv: 1003.0691v1., 2010, 29 p.42

elizbar nadaraia, mzia facacia,grigol soxaZeganawilebis parametrebis fsevdo-maqsimaluridasajerobis meTodiT Sefasebis Sesaxebcenzurirebuli dajgufebulimonacemebisaTvisnaSromSi ganxilulia albaTuri ganawilebis parametrebisSefaseba dajgufebuli monacemebisaTvis. amasTan, zogierTjgufebSi dakvirvebadia SemTxveviTi sididis ara individualurimniSvnelobebi, aramed maTi saerTo raodenoba, xolozogierT jgufSi damkvirveblisaTvis miuwvdomelia maTiraodenobac ki. aseTi tipis SerCevisaTvis gamoyenebulia fsevdo-maqsimaluridasajerobis meTodi, Seswavlilia Sefasebebisasimptoturad Zaldebuloba da asimptoturad efeqturoba.magaliTis saxiT ganxilulia normaluri ganawileba.43

soxumis saxelmwifo universitetis Sromebit. VII, 2009maTematikisa da kompiuterul mecnierebaTa seriaGOGI PANTSULAIAON A RIEMANN INTEGRABILITY OF FUNCTIONS DEFINEDON INFINITE-DIEMENSIONAL RECTANGLESAbstract. We announce a result asserted that for a Riemann integrablefunction f defined on the infinite-dimensional rectanglei1[ a , b]ОВ , an infinite-dimensional version of the Weyl theorem isiivalid. This fact allows us to give an effective algorithm for a calculationof the Riemann integralтҐХ[ ai, bi]i=1( R) f ( x) dl( x), where l denotes an infinite-dimensional‗Lebesgue measure‘ constructed by R.Baker in 1991.2010 Mathematical Subject Classification: Primary 28Axx,28Cxx, 28Dxx; Secondary 28C20, 28D10, 28D99.Key words and phrases: Riemann integrability, Lebesguemeasure, infinite-dimensional rectangles1. IntroductionVarious questions combinatorial or discrete type frequently arise indifferent domains of modern mathematics (especially, in Mathematicalanalysis, Measure theory, Differential equations, Game theory, Set Theory,Graph Theory etc.) and are important from the theoretical viewpointand from the view-point of their numerous applications. In particular,these questions play a key role in applications of algorithms andcomputer science. For example, the notion of a equidistributed, or uniformlydistributed sequences ( an)nО Nin an interval [ ab , ] describes acertain discrete mathematical structure which has various interesting44

applications from the view-point of its applications in the theory of algorithmsand computer science. In particular, by Weyl well known theorem, for every Rieman integrable function f on [ ab, , ] the followingequalityn1( R) f ( x)dxalimn®Ґе f( ak)=n b-ak=1тbholds if and only if the sequence ( )annО Nis equidistributed, or uniformlydistributed in an interval [ ab , ] , where ( R) т f ( x)dx denotesaRiemann integral of the f over [ ab. , ] Moreover, in common cases, it ispossible to estimate the velocity of such a convergence.To various applications of a equidistributed, or uniformly distributedsequences is devoted the well known monograph of L. Kuipersand H. Niederreiter [1]. Firstly, such sequences have been used byHardy and Littlewood [2] in Diophantine approximation theory.It is natural to considerProblem 1.1. Whether one can elaborate a theory of uniformlydistributed sequences for infinite-dimensional rectangles ?.Since the theory of uniformly distributed sequences for finitedimensionalrectangles essentially implies the technique of the Lebesguemeasure, here arises the followingProblem 1.2. What measures in infinite-dimensional topologicalvector space R Ґ can be assumed as partial analogs of the n -dimensional classical Lebesgue measure (defined on the Euclidean vectorspace R n ) ?In this direction, we must say that a partial analog of the Lebesguemeasure on R Ґ is not defined uniquely. Problem 1.2 was solved byAfrican mathematician R.Baker [3]. He introduced a notion of "Lebesguemeasure" on R as follows: a measure being a completion of ashift-invariant Borel measure on R is called a "Lebesgue measure" onR if for any infinite-dimensional rectangle [ ai, bi]( ai bi ) with 0 ( biai) i1the equalityi1b45

ai bi bi aii1 i1( ( , )) ( )holds, where( b a ): lim ( b a).i i ni ii1 i1nIn 2004, simultaneously have been published articles [4] and [5]which also contain solution of Problem 1.2. In [4] has been posed aquestion asking whether measures [3] and [4] coincide. The negativeanswer to this question has been obtained in [6]. In this manuscript hasbeen introduced the notion of generators of shy sets in Polish topologicalvector spaces and has been done their various interesting applications. Inparticular, here has been demonstrated that this class contains specificmeasures which naturally generate early implicitly introduced classes ofnull sets. For example, here has been constructed : 1) Mankiewicz generatorwhich generate exactly the class of all cube null sets; 2) Preiss -Tiser generators which generate exactly the class of all Preiss -Tiser nullsets , etc. Moreover, such measures (unlike s -finite Borel measures)possess many interesting, sometimes unexpected, geometric properties.New concepts of the ―Lebesgue measure‖ on R Ґ (the so called, a -standard and a -ordinary Lebesgue measures) have been proposed andtheir some realizations in the ZFC theory have been considered in [7] .Also, it has been shown that Baker's both measures [3] and [4], Mankiewiczand Preiss - Tiser generators [6] and the measure [5] are not ana -standard Lebesgue measure on R Ґ for a = (1,1, L ) .Under such a rich class of shifp-invariant Borel measures inR Ґ it is natural to consider the following problems:Problem 1.3. Whether one can elaborate the theory of Riemannintegration in R Ґ ?Problem 1.4. In terms of partial analogs of the Lebesgue measurein R Ґ , whether one can elaborate theory of uniformly destributed sequencesin infinite-dimensional rectangles?Note that the solution of Problems 1.3-1.4 assumes an infinite generalizationof well known classical results (for example, Lebesgue theoremabout Riemann integrability, Weyl theorem, etc) formulated in termsnof Lebesgue measure (on R ) to an infinite-dimensional topologicalvector spaces of all real valued sequences R Ґ (equipped with Tikhonovtopology ) in terms of partial analogs of the Lebesgue measure (for ex-46

ample, Mankiewicz generator, Baker measures, standard Lebesgue measure,etc).The purpose of the present manuscript is to consider a certain conceptfor a partial solution of the Problems 1.3-1.4 in terms of the measurel [3] and to announce main results established in [9].The paper is organized as follows.In Section 2 we give some auxiliary definitions from the Riemannintegrability theory. In Section 3 we announce our main results establishedin [9].2. Auxiliary definitions and notionsIn order to solve Problem 1.3 for "Lebesgue measure" l [3],we need a notion of Riemann integrability on infinite-dimensional rectanglein terms of the measure l .We denote by В a class of all measurable rectangle [ ai, bi]( ai bi ) with 0 ( biai) .i1It is clear that an equalityi1 holds. ai bi bi ai i1 i1( ( , )) ( )A set U is called an elementary rectangle in theadmits the following representation [ ai, bi]if iti1mҐХk= 1 k k Х k= m+1 k kU = ][ c , d ][ ґ [ a , b ]where ak Ј ck < dk Ј bkfor 1Ј kЈ m ( mО N).It is obvious thatlmҐХk= 1 k k Х k= m+1 k k( U) = ( d - c ) ґ ( b - a )for the elementary rectangle U .Also, a number dU ( ), defined byd - c b - adU ( ) = +2 (1 ) 2 (1 )mҐk k k kk+ 1 k+1k= 1 + dk - ck k= m+1 + bk - akе е ,47

is diameter of the U with respect to Tikhonov metric in R .A family of pairwise disjoint elementary rectanglest = ( U k) 1 Ј k Ј nof the [ ai, bi]ifi1i1i1 [ ai, bi]is called Riemann partition of thei1n [ a , b ] U .i i k1kA number d ( t ), defined byd( t ) = max{ d( Uk) :1 Ј k Ј n}is called mesh or norm of the Riemann partition t .Let f be a real-valued bounded function defined onDefinition 2.1 [ ai, bi].i1We say that the f is Riemann-integrable on [ ai, bi]if there exists a real number s such that for every positive realnumber e there exists a real number d> 0 such that, for every Riemannpartition t = ( U k)1Ј k Ј nof the [ ai, bi]with d( t ) < d and forevery sample ( tk )1Ј kЈ nwith t ОU (1 Ј k Ј n), we havekki1еn| f ( t ) l ( U )- s |

( ii ) (" x)( x О Ў Ґ \ И B Ю f ( x) = 0) ;kОN kе т Bis absolutely convergent.k BkОNk( iii ) The series ( R) f | ( x) d l ( x)If for the function f the conditions ( i) - ( iii)fulfilled, then anumberе тkОN( R) f | ( x) dl( x)BkҐis called Riemann integral of the f over Ў with respect to land is denoted by( R) т f ( x) dl( x).ЎҐNote, that a first step to forward for resolution of Problem 1.3 willbe an investigation of the followingProblem 2.1 Describe in terms of the measure l a class of all real-valued Riemann integrable functions onBk [ ai, bi].Definition 2.3. An increasing sequence ( Y ) Оof finite subsets ofthe infinite-dimensional rectangledistributed in thei1[ a , b], we haveiii1i1i1nn N[ a , b]ОВ is said to be uniformlyii[ a , b], if , for every elementary rectangle U in theilimi#( YnI U) l ( U)=.#( Yn)l ( [ a, b])n ®Ґ ҐХi=1ii2. Main ResultsPresent section we announce the main results obtained in [9].49

Theorem 3.1Let[ ai, bi]ОВ . Leti1distributed in the interval [ a , b ] for kОN. We setnn ( k)n= Х( j=1 j) ґ Х{ k}k= 1k>nkk( k )(n)n NxОbe uniformlyY U x a . Then ( Y ) Оis uniformly distributed inthe [ ai, bi],i1Theorem 3.2 Let f be a continuous (w.r.t. Tikhonov metric) functiononni1n N[ a , b]. Then the f is Riemann-integrable onii [ ai, bi].We have the following infinite-dimensional version of the Lebesguetheorem (see, [10], Lebesgue Theorem , p.359).Theorem 3.3 Let f be a bounded real-valued function oni1[ a , b]. Then f is Riemann integrable oniif is l -almost continuous (w.r.t. Tikhonov metric) onTheorem 3.4 Fori1i1i1 [ ai, bi]if and only if [ ai, bi].i1[ ai, bi]ОВ , let ( Yn)n О Nbe an increasingfamily its finite subsets. Then ( Y ) Оis uniformly distributed in thei1nn N [ ai, bi]if and only if for every continuous (w.r.t. Tikhonov metric)function f on [ ai, bi]the following equalityi1holds.limеf( y)тҐХ [ ai, bi]yОYni=1n®Ґ =Ґ#( Yn)( R) f ( x) dl( x)Хl ( [ a , b])i=1ii50

REFERENCES1. L. Kuipers and H. Niederreiter. Uniform distribution of sequences.– Pure and Applied Mathematics. Wiley-Interscience [John Wiley& Sons], New York-London-Sydney, 1974.2. G. H. Hardy, J. E. Littlewood. Some problems of diophantine approximation.– Acta Math. 41 no. 1 (1916)).3. R. Baker. "Lebesgue measure" on R . – Proc. Amer. Math. Soc.,113(4) (1991), 1023-1029.4. R. Baker. "Lebesgue measure" on R . II. – Proc. Amer. Math. Soc.,132(9), (2004), 2577-25915. G. Pantsulaia. Relations between Shy sets and Sets of np-MeasureZero in Solovay's Model. – Bull. Polish Acad.Sci., vol.52, No.1,(2004). 63-696. G. Pantsulaia. On generators of shy sets on Polish topological vectorspaces. – New York J. Math., 14 ( 2008), 235-2617. G. Pantsulaia. On ordinary and Standard Lebesgue Measures onҐЎ . – Bull. Polish Acad. Sci.73(3) (2009), 209-222.8. G. Pantsulaia. Riemann Integrability and Uniform Distribution inInfinite-Dimensional Rectangles. – Georg. Inter. J. Sci. Tech., NovaScience Publishers. Volume 2, Issue 3 (2010).9. G. Pantsulaia. On uniformly distributed sequences of an increasingfamily of finite sets in infinite-dimensional rectangles (Submittedfor consideration to publish).10. S. M. Nikolski. Course of mathematical analysis (in Russian), № 1,Moscow, 1983.51

gogi fanculaiausasrulo-ganzomilebian paralelepipedzegansazRvruli funqciebis rimanis azriTintegrebadobis SesaxebnaSromSi anonsirebulia is faqti, rom usasrulo-ganzomilebiani1[ a , b]ОВiiparalelepipedze gansazRvruli rimanisazriT integrebadi f funqciisaTvis marTebulia veilisTeoremis usasrulo-ganzomilebiani analogi. aRniSnuli faqtiiZleva rimanisт( R) f ( x) dl( x)ҐХ[ ai, bi]i=1integralis gamoTvlis efeqtur algoriTms, sadac laRniSnavs r. beikeris mier 1991 wels agebul usasrulo-ganzomilebian`lebegis zomas~.52

soxumis saxelmwifo universitetis Sromebit. VII, 2009maTematikisa da kompiuterul mecnierebaTa seriaUSHANGI GOGINAVACONVERGENCE IN MEASURE OF TWO-DIMENSIONALCONJUGATE WALSH-FEJER MEANSAbstract. The main aim of this paper is to prove that for any Orlicz2space, which is not a subspace of L log L(I ) , the set of the functions thatquadratic conjugate Fejer means ( = = 2/3) of the double Walsh-Fourierseries converges in measure is of first Baire category.2010 Mathematical Subject Classification: 42C10.Key words and phrases: Conjugate Walsh-Fourier series,Orlicz space, Convergence in measure1. IntroductionThe partial sums S n( f ) of the Walsh-Fourier series of a functionf L( I),I = [0,1) converges in measure on I [9, 14]. The condition 22f LlnL(I ) provides convergence in measure on I of therectangular partial sums S n , m( f ) of double Fourier-Walsh series [19].2The first example of a function from classes wider than Lln L(I ) with2S n , n( f ) divergent in measure on I was obtained by Getsadze in [5].Moreover, Tkebuchava [15] proved that in each Orlicz space wider than2Lln L(I ) the set of functions with quadratic Walsh-Fourier sums2converge in measure on I is of first Baire category.Weisz [16, 17] proved that conjugate Fejйr means of double, Walsh-Fourier f , , [0,1)converges a. e. for eachnm , 2 2f LlnL(I ). In particular, the condition f LlnL(I ) provides the2 ,convergence in measure on I of the f nm ,, , [0,1). In this paperwe prove that the ConjugateFejйr means of the double Walsh-Fourier series does not improvethe convergence in measure. In other words, we prove that for any Orlicz53

2space, which is not a subspace of L log L(I ) , the set of the functions thatquadratic conjugate Fejer means ( = = 2/3) of the double Walsh-Fourier series converges in measure is of first Baire category.For results with respect to divergence of measure of double Walsh-Fourier seriessee [2- 4, 6- 8, 10, 12].2. Definitions and Notation0 0 2We denote by L = L ( I ) the Lebesque space of functions that are2measurable and finite almost everywhere on I = [0,1) [0,1). mes (A)is2the Lebesque measure of the set A I . The constants appearing in thearticle denoted by c .2Let L = L( I ) be the Orlicz space [11] generated by Youngfunction , i.e. is convex continuous even function such that (0) = 0 andlimuuThis space is endowed with the normu= ,limu0 uu= 0. fL (2I )= inf { k > 0:2(If ( x,y) / k)dxdy 1}.In particular in case if ( u)= u ln (1u),u > 0 , then2corresponding space will denote by Lln L(I ).It is well-known [11] thatLetL L r 0xbe a function defined bylimuuu> 0. 1, if x [0,1/2)r 01,if x [1/2,1)The Rademacher system is defined bynr x = r 02 x , n 1and x nx , r x1 = r .0=0x [0,1).54

Let ,...kw represent the Walsh functions [13], i.e. w = 1, w 0 1ksif k = 21 2 is a positive integer with k k > > 0 thenw0 x1> 2k skx = rkxr x.1ksEvery point [0,1)can be written in the following way:= k , = 0,1.k 21 kk =0In case there are two different forms we choose the one for whichlim = 0.kkThe Walsh-Dirichlet kernel is defined byDnn 1 k =0x = w x.kandRecall thatDxn2, if= 0, ifx 0,1/22nn,nx 1/2 ,1.We consider the double system wn( x) wm( y) : n,m = 0,1,2,... on2the unit square I .The rectangular partial sums of double Fourier series with respectto the Walsh system are defined bywhere the numberSM 1N1M , N m nym=0n=0fˆ f , x,y = fˆm,n w ( x)w ( ),m,n = f x,yw( x)w ( y)dxdyI2is m, nth Walsh-Fourier coefficient.nn1Let 0:= r 0, k:= rnif 2 < 2 .sums of the conjugate transforms is given byk Then the M M1N1, ˆ mnN, th partialS f , x, y = f k, l w ( x) w ( y).n,m k l k lk=0 l=0The conjugate Fejér means of a function f are defined be55

n m1, , := , , .nm , , n, mf x y Sn,m f x yi=1 j=1It is simple to show that , f , x, y = f t, u K x t K y u dtdu,where N , M M NI2M 1 K := Mx Djx,nj1j=1 := D x r D x D xj i2i12ii=0and denotes dyadic addition (see [9, 14]).3. Main ResultsThe main results of this paper are presented in the followingproposition.Theorem 1 Let L ( I2) be an Orlicz space, such thatL I LlnL I2 2( ) ( ).Then the set of the functions from the Orlicz space L ( I2) with2/3,2/3conjugate Fejйr means f convergent in measure onBaire category in ( I2 ).L nn ,2I is of firstCorollary 1 Let :[0,[ [0, [be a nondecreasing functionsatisfying for x the condition ( x)= o(x log x).56Then there exists the function f L( I2 ) such that

a)measureonb) conjugate Fejйr means2I .2( f ( x,y) ) dxdy < ;I f of the function f diverges in2/3,2/3nn ,4. Auxiliary Results 1 1 Lemma 1 Let x , , l = 1,2,..., m.l l 1 Them 2 2 Proof. Letn12/3K 2mcxx . := D x r D x D xn j2j12jj=0n1j=01 jD x D x= =n 1 j=0On the other handj121 jr xDx.jj2j2Hencen1 rxDxn11k k2kk = 0k =0 2=1n1n111= rk2k2 k =02 k =01 1 = D 1 .22nx Dnx2krkxD2kxDx k 1r xDxk2kn1x D x = D x 1 2 r x D x , n2n k k2kk =057

1 m m m n1 1 2K := D x = D x 1r x D x m n2n k k2km n=1 m n=1 m n=1 k =01=mmn=1D2n21mm1k =0m kr xDx. 1 1 Let x , , l = 1,2,..., m.l l 1 Then from (1) we have 2 2 l1 l11 1K x = D x 1 2 m k r x D x . 2m 2n k k22km n=1 m k=02It is evident that for = we have3Let l is odd number. Thenkkk2k 1, k = 2 j= 0,k = 2 j 1l1l1 /22/3 1 12m 2n 2k 22k2mn=1 m k=0 K x = D x 1 2m 2 k r x D x ,1 1 1K m l D m kl1l1 /212/3l1 2k2m 2 2 1 2 2 222n m m n=1 m k=01 l 2 2 12m3ll2 1 l c2c .xLet l is even number. Then58

1 1 1K m l D m kl1l2 /212/3l2 2k2m 2 22 1 2 2 222n m m n=1 m k=0 2l2l c2c .xl11 l 2 2 12m3Combining (2) and (3) we complete the proof of Lemma 1.We apply the reasoning of [1] formulated as the followingproposition in particular case.1 2 0 2Lemma 2 [1]Let H : L ( I ) L ( I ) be a linear continuousoperator, which commutes with family of translations , i. e.E f L1 ( I2 ) HEf = EHf . Let f = 1 and > 1. Then forL1( I2)21any 1 r under condition mes{x,y I :| Hf |> } there existr' 'E1,..., Er, E1,...,Er and i= 1, i = 1,..., r such thatr2' 1mes{x,y I :| H ( if ( Eix,Eiy)|> } .8Lemma 3 [18]Let{m}m=1i=1H be a sequence of linear continuesoperators, acting from Orlicz space L ( I2) in to the space L 0 ( I 2 ).Suppose that there exists the sequence of functionsS (0,1) of space L ( I2) , sequences of integersincreasing to infinity such that = inf mes{{k}k =1 from unit bull{k}k =1m and{ k}k =12x,y I :| H x,y|> } > 0.0 mkkkkThen B - the set of functions f from space L ( I2) , for which thesequence { H mf } converges in measure to an a. e. finite function is offirst Baire category in space L ( I2) .59

Lemma 4 [18]Let Lbe an Orlicz space and let :[0,) [0, )be measurable function with condition x = oxas x .Thenthere exists Orlicz space x = o x as x , andxx for x c 0.L , such that 5. Proof of the TheoremProof of Theorem 1. By Lemma 3 the proof of Theorem 1 will becomplete if we show that there exists sequences of integers { : k 1}and { : k 1}increasing to infinity, and a sequence of functionsk from the unit bull 0,1{k: k 1}for all k2S of Orlicz space IL m k, such that 1mes x yI x y8(4)First we prove that there exists c > 0 such that2/3,2/3 mmes x yI2 2 D 2D 2x yc2(5)Denotem 1 1 Gm= , .l l1l=12 2 Since2 2/3,2/3{ , : 2,2k; , > k} .mkmk2m{ , : m,m; , > 2 } > .2m2m m2/3,2/3 2/3 2/3( D D , x, y) = F ( x) F ( y),2 m,2m 22m 22m 2m 2mthen forestimation( x,y)G mGwe have from Lemma 1 the followingm2/3,2/3 2/3 2/3c2 m,2m ( D22m D 22m , x, y) = F2 m( x) F2m( y) .xyIt is easy to show that60

mes x y I D D x y c2 2/3,2/3m{( , ) : 2 m,2m (22m 22m, , ) > 2 }m mes{( x, y) G G : ( D D , x, y) > c2 }2/3,2/3m m 2 m,2m 22m 22mc mmes{( x, y) : > c2 }xym m1 cmc .slml=1 s=ml2 2Hence (5) is proved.From the condition of the theorem we write [11](u)liminf = 0.uu log uConsequently, there exists a sequence of integersincreasing to infinity, such that{ mk}k =1From (5) we writelim (2k2mk)22mkm1k= 0,(22mk) 22mk,k. mmes x y I D D x y c c2'Then by the virtue of Lemma 2 there exists e1,..., e ,1,...,' re er[0,1]and ,..., = 1 such that1 r2 2/3,2/3mkk{( , ) :| 2 m ,2m (2m 2m , , ) |> 2 } > .k k m2 k 2 k kmes x y I D D x e y er22/3,2/3' mk( , ) :| i 2 m ,2m (2m 2m , i, i) |> 2 > ,k k=12 k 2 ki8m k2 where r = 1.cmkDenoteand3mk12k=2mkr2161

whereMkkx2mk1, y2mkr1ir2= 2M x,y,'xy = D e xDe y.,2m2=12 kim2 kikiThus, we obtain (4).Moreover, since is convex, we have 0,14mk S kestimation 2 and M 1 impliesM kL ( I2)k 1(2L I ). Indeed, the2m1 k2 , 21 Mkx yk 21.( )22 dxdyLIIm k(2 ) Theorem 1 is proved.The validity of Corollary 1 follows immediately from Theorem 1and Lemma 4.2We note that = = implies that3and | ii 1|= i=1 | i i 1|= If series (6) and (7) are finite then it easy to show that 2converge of the metric of space Ii=12converge in measure for every f LI.We can now formulate the following, f nm ,2L for every f LI, in particularProblem 1 Whether the conditions (6) and (7) guarantees justice ofTheorem 1 and Corollary 1?62

REFERENCES1. A. Garsia. Topic in almost everywhere convergence. Chicago, 1970.2. G. Gat, U. Goginava and G. Tkebuchava. Convergence of logarithmicmeans of multiple Walsh-Fourier series. – Analysis in Theoryand Application. 21, 4(2005), 326-338.3. G. Gat, U. Goginava and G. Tkebuchava. Convergence in measureof logarithmic means double Walsh-Fourier series. – Georg. Math. J.12(2005), 607-618.4. G. Gat, U. Goginava and G. Tkebuchava. Convergence in measureof logarithmic means of quadratical partial sums of double Walsh-Fourier series. – J. Math. Anal. Appl. 323(2006) 535-549.5. R. Getsadze. On the divergence in measure of multiple Fourier seties.– Some problems of functions theory, 4 (1988), 84-117(in Russian).6. R. Getsadze. On the convergence in measure of Norlund logarithmicmeans of multiple orthogonal Fourier series. – East J. Approx. 11(2005), no. 3, 237--256.7. R. Getsadze. On the boundedness in measure of sequences ofsuperlinear operators in classes L L. – Acta Sci. Math. (Szeged)71 (2005), no. 1-2, 195--226.8. R. Getsadz. The sets of convergence in measure of multiple orthogonalFourier series. – J. London Math. Soc. (2) 72 (2005), no. 1,239-257.9. B. I. Golubov, A. V. Efimov, and V. A. Skvortsov. Series and transformationsof Walsh, Nauka, Moscow, 1987 (in Russian); Englishtransl.: Kluwer Acad. publ; 1991.10. S. A. Konjagin. On subsequences of partial Fourier-Walsh series. –Mat. Notes, 54 4(1993), 69-75.11. M. A. Krasnosel'skii and Ya. B. Rutickii. Convex functions andOrlicz space (English translation), P. Noorhoff (Groningen, 1961).12. S. F. Lukomski. On subsequences of multiple Fourier-Walsh series.– Mat. Sb. 186 7(1995), 133-146.13. A. Paley. A remarkable series of orthogonal functions. Proc. LondonMath. Soc. 34 (1932), 241-279.14. F. Schipp, W. R. Wade, P. Simon, and J. Pàl. Walsh series,Introduction to dyadic harmonic analysis. – Adam Hilger, Bristoland New York, 1990.15. G. Tkebuchava. Subsequence of partial sums of multiple Fourierand Fourier-Walsh series. – Bull. Georg. Acad. Sci, 169 2(2004),252-253.63

16. F. Weisz. Summability of multi-dimensional Fourier series andHardy space. – Kluwer Academic. Dordrecht, 2002.17. F. Weisz. The maximal C ,, operator of two-paremeter Walsh-Fourier series. – J. Fourier Anal. Appl. 6(2000), 389-401.18. L. V. Zhizhiashvili and G. Tkebuchava. On the properties of someclassical operators that arise in Fourier analysis. (English. Russianoriginal). – Sb. Math. 195, No. 10, 1395-1412 (2004); translationsfrom Mat. Sb. 195, No. 10, 3-20 (2004).19. L. V. Zhizhiashvili. Some problems of multidimensional harmonicanalysis. Tbilisi, TGU, 1996 (in Russian).uSangi goginavaorganzomilebiani SeuRlebuli furie-uolSismwkrivebis zomiT krebadobis SesaxebnaSromSi mtkicdeba, rom nebismier orliCis sivrceSi,2romelic ar aris L log L(I ) sivrcis qvesivrce, im funqciebissimravle, romelTa ormagi furie-uolSis SeuRlebuli mwkrivebisfeieris saSualoebi ( = = 2/3) aris zomiT krebadiaqvs beris pirveli kategoria.64

soxumis saxelmwifo universitetis Sromebit. VII, 2009maTematikisa da kompiuterul mecnierebaTa seriaILIA TAVKHELIDZEABOUT SOME GEOMETRIC CHARACTERISTIC OF THEGENERALIZED MOBIUS LISTING'SnSURFACES GML 2 AND ITS CONNECTIONSWITH SET OF RIBBON LINKSAbstract. We consider the cutting process of a Generalized Möbius-Listing‘ssurfaces GML 2 along a set of lines ―parallel‖ to its ―basicnline‖. We show connection of the resulting mathematical objects with theset of Ribbon knots and links.2000Mathematics Subject Classification. 53A05, 51B10, 57M25.Key words and phrases. Möbius strip, Möbius-Listing's surfaces, Firstfundamental form, Second fundamental form, Mean curvature, GaussianCurvature, Hyperbolic point, Parabolic point, Knots, Links, Ribbon links.1. IntroductionIn previous articles [1-5] a wide class of geometric figures – ―GeneralizedTwisting and Rotated‖ bodies (sometimes called ―surface ofnRevolution‖ see [9]) - shortly GTR m - was defined through their analyticrepresentation. In particular cases, this analytic representation gives backmany classical objects (torus, helicoid, helix, Möbius strip ... etc.). Aimof this article is to consider some geometric properties of a wide subclassof the already defined surfaces, by using their analytical representation.In previous articles [1-5] a set of the Generalized Möbius Listing's bodiesn n- shortly GML m , which are a particular case of the GTR m bodies, havebeen defined. In the present paper we show some geometric properties of65

Generalized Twisting and Rotated - surfaces and relationships betweenthe set and the sets of Ribbon Knots and Links.2. Notations and DefinitionsIn this article we use following notations:• X , Y , Z , or x , y,z - is the ordinary notation for coordinates;• , , - are space values (local coordinates or parameters inparallelogram);1.2.3.* [ , ], where usually are non negative constants;* [0,2];* [0,2h],where h R ( Real);(1)But sometimes, as a special case, we suppose that* ** [ , ](1*)• Pm A 1 A 2 Am- denotes an ―Plane figure with m-symmetry‖, in particular P m is a ―regular polygon‖ and m is the numberof its angles or vertices. In the general case the edges of ―regularpolygons‖ are not always straight lines ( A i A i1may be, for example:edge of epicycloid, or edge of hypocycloid, or part of lemniscate ofBernoulli, and so on) (see e.g. Fig. 1f);• PRm A1 A2AmA1A2Amdenotes an orthogonal prism,whose ends A 1 A 2 Am and A 1 A2 Am are ―Plane m-symmetricfigures‖ P m (see e.g. Fig. 1b);For example:- PR 0 - is a segment and P 0 is a point;- PR 1 - is an orthogonal cylinder, whose cross section is a P 1 -plane figure without symmetry;- PR2 A1A2A1A2is a rectangle, if P2 A1A2is a segment ofstraight line; but also PR 2 maybe a cylinder with cross section P 2(ellipse, or lemniscate of Bernoulli and so on);66

- PR - is an orthogonal cylinder, whose cross section is a P-circle.x = p(, ), z = q(, )(2)orx = p( , )cosz = p( , )sin(2*)are the analytic representations of a “Plane figure with m-symmetry” m p (0,0) = = 0 and the point (0,0) is thecenter of symmetry of this polygon (see [1-5]).For example, when the function p ( , ) in formula (2 * ) has theformm 1p(, ) ( i ),i=0(2‘)where the arguments and are defined in (1); i [0,2 ) aresome constants for each i = 1, m 1, with i j if i j , and0if = i ( i) 1if i,then the corresponding plane figure P (some time in this article*P m ) is:1. a “simple star” with m ―wings‖ or ―vertices‖ when * 0 (seee.g. Fig. 1.a. i = 0,6);2. a set of m segments of straight lines lying on the radiuses of acircle centered at the origin when > 0 (see e.g. Fig. 1.e.)).*2i*Conclusion 1. In the case when i , i = 0, m 1, then P m isma “Regular simple star” (see Figs. 1 b., d., c. ); if m = 2 and 0, then (2‘) is a representation of P (see e.g.* Fig. 1.c.)), which is a segment of straight line [ , ]. In thisparticular case we setm* **267

p ( , ) (2**)where the ―argument‖ satisfies (1 * ) ;68• D ( p,q)or D (p)- diameter of plane figure P m ;• OO - axis of symmetry of the prism PR m ;• L - Family of lines situated on the plane, whose parametricrepresentations areX= f1(,)*L = [0, ), Y = f2(,) [0,2h], hZ (3)orX( ,)= 1()cosL = Y(,)= 2()sin(3*)andWe assume the following hypotheses:i) For any parameters 1 , 2 [0, ], 1 2, the linesL have not intersection.2ii) If - are 2 p -periodic functions fi( , 2) = fi(,),(i = 1,2) g ( ) - be an arbitrary sufficiently smooth function*L 1L is a closed curve, then for every fixed [0, ] f i .*

g( ) :[0,2h] [0,2h](4)and if h = 1, then for every [0,2 ] there exists [0,2 ],such that = g();• mod m (n)- natural number < m ; for every two numbersm N (natural) and n Z (integer) there exists a unique representationn = km j km mod m(n), where k Z andj mod m ( n) N {0};n/m,when mN and n Z (5) n when m = and n Z ( or n R (Real))n• Generalized Twisting and Rotated bodies - shortly GTR m(sometimes called “Surfaces of revolution” see[9]) are defined by theparametric representations:X ( , , )= fY( , , )= f21Z( , , )= Q() p( , )sin( g()) q( , )cos( g( )),orX ( , , ) = ( ) p( )cos( g( )) cos( )1,,R p( , )cos( g()) q( , )sin( g())R p( , )cos( g()) q( , )sin( g())Y( , , ) = ( ) p( )cos( g( )) sin( )2Z( , , ) = Q( ) p( )sin( g( )),( *(6)(6*)where, respectively:- the arguments ( , , )are defined in (1);- the functions f 1 and f 2 or 1()and 2( ) in (3) or3 ) define the L R “Shape of plane basic line”, more precisely ―Shapeof orthogonal projection on the plane XOY of the basic line‖ ofcorresponding body (see e.g.: circle in – Figs. 2b, 2c, 2g; ellips in – Fig.2e; spiral in – Figs. 2d, 2f, 2i and square in – Fig. 2h. );- R is a some fixed real number, which defines the “Radius” ofthe ―plane basic line‖ L ;R69

- Functions p ( , ) and q ( , ) or p ( ) in (2) or ( 2 )define the “Shape of the radial cross section” of corresponding figure.In general case this functions may be depends from arguments (see forexample (2') ) and , i.e. ―Shape of the radial cross section‖ dependsfrom the ―place‖ of this cross section (see e.g. Fig 2i.);- The function g ( ) from (4) defines the “Rule of twistingaround basic line”;- The number in (5) defines the “Characteristic of twisting”;- Q ( ) is a smooth function which defines the “Law of verticalstretching of figure”.nTherefore, this parametric representation defines a GTR m body(some examples are shown in Fig. 2) with the following restrictions:1) The OO -axis of symmetry (middle line) of the prism PR m istransformed into a “Basic line” (sometimes called “Profile curve”) -( L R , Q( )) ;2) Rotation at the end of the prism (2) or ( 2 ) is semi-regularalong the middle line OO , or the twisting of the shape of radial crosssection around the basic line is semi-regular (depending from g ( ) ) .**70

•nGeneralized Möbius Listing's body - shortly GML m - isobtained by identifying the opposite ends of the prism PR m in such away that:A) For any integer n Z and i = 1, , m each vertex A icoincides with Ai n Amod( in), and each edge A i A i1with the edgecorrespondingly;mA inA AAin1 modm( in)modm( in1)B) The integer n Z denotes the number of rotations of the endof the prism with respect to the axis OO before the identification. Ifn > 0, the rotations are counter-clockwise, and if n < 0 then rotationsnare clockwise. Some particular examples of GML m and its graphicalrealizations can be found in [2-5] (see e.g. Fig 2e.).Conclusion 2. We can assert that:a.) ThenmGML body is a particular case of thenGTR m body.nb.) The basic line L R of a GML m body, is always a closed lineand the number is such that the boundary of this body is a closedsurface (see [2]).71

72c.) The functions f 1 , f 2 , 1(), 2( ), Q ( ) in parametricrepresentation (6) and ( 6 * ) of a GML body are always 2-periodicfunctions (see e.g. Fig. 3a) or Q ( ) 0, (some examples are shown inFigs. 3).n Some additional information about the classification of GRT mbodies are reported in [2,4].1. Some Geometric properties of a “Regular”nmnGML 2 surfaces.In this part of our article we study some geometric characteristic ofna ``Regular" Generalized Möbius-Listing's surfaces GML 2 , with circleas basic line. This means that the parametric representations of these*surfaces (6) or ( 6 ) have the following simple form nX ( , ) = Rcos cos( )2 nY ( , ) = Rcos sin( )2 (7) nZ( , ) = sin , 2 where, respectively:- R - radius of basic circle - is constant (see e.g. Figs. 4a., 4b., 4c.);- In the general case (6) , defined in (1), is variable, but now,according to the notation (2**) (Corollary 1.), the argument always* *belongs to the interval [ , ], see (1*) where < R is some nonnegativeconstant. So that, when n = 1 formula (7) is the classical (wellknown) form of analytic representation of the Möbius strip ([2],[9]).*nActually, 2 is the width of the surface GML 2 ;- the variable is defined in (1) and in this case h 1, i.e. [0,2 ];- The ―rule of twisting around basic line‖ is “Regular”, i.e. thefunction, which is defined by eq. (4) is g ( );*

n- n - the “Number or twisting” of GML 2 - is an arbitraryinteger number, i.e. the number defined by eq. (5) is n/2(see e.g.n=1- “Möbius strip” Fig. 4a; n=2 Fig. 4b; n=14 Fig. 4c; n=6 Fig. 4d;n=0 - Figs. 4e, 4f, 4g, 4h, 4i, 4j. );- in the present case, is a constant defined in (1) (but whenn = 0, the number in eq. (7) defines even the type of thecorresponding surface, for example: if = 0 , then the “Regular”0Generalized Möbius-Listing's surfaces GML 2 , with basic line a circle, is**a Ring ( R > ) (see. e.g. Fig. 4e) or Disk ( R = )(see. e.g. Fig. 4g),and if = 0 , then GML 2 is a cylinder (see. e.g. Fig. 4h), in other cases2these surfaces are cones or truncated cones (see. e.g. Fig. 4i) (see [3-6])).73

Remark 1 Note that:a.) For every integer number n eq. (7) defines a one to onecorrespondence between the strip [ , ] [0,2) and the surfacenGML 2 .b.) If n is an even number, then each function ( X , Y,Z)in therepresentation (7) is a 2-periodic function of the argument .c.) If n is a odd number, then each function ( X , Y,Z)in therepresentation (7) is a 4-periodic function satisfying the followingproperties (Möbius property, see [8])X ( , 2); Y ( , 2); Z( , 2) = X ( , ); Y ( , ); Z(, ) (M*) .According to the representation (7), the tangential vectors of then―Regular‖ Generalized Möbius-Listing's surface GML 2 , with circle asbasic line, are correspondingly n n nr = cos cos( );cos sin ( );sin (8) 2 2 2 and**74

nn n Rcos sin( ) sin cos( );2 2 2 nn n r = Rcos cos( ) sin sin(); .2 (9) 2 2 n ncos 2 2 It is easy to check, that the scalar product of these two vectors (8)and (9) is( r , r) = 0.Remark 2 For any integer number n two tangential vectors of anregular GML 2 , with circle as basic line, are always orthogonal, i.e. thelocal system of coordinates ( , )in this surface is an orthogonalsystem.Also we may check that(x,y) n n= cos cos ;( , )R 2 2 (z,x) n n n= R cos sin sin( ) cos( );( , ) 2 (10) 2 2(y,z) n n n= R cos sin( cos( ) sin( );( , ) 2 2 2and the module of the vector product of these two vectors is n( n)| r r|= Rcos .2(11) 4Therefore, the we may rewrite unit normal vector of a ―Regular‖nGeneralized Möbius-Listing's surfaces GML 2 .Remark 3 Note that:a.) If n is an even number, then the unit normal vector ( , ) is a2-periodic vector and consequently the22nGML 2 body is a two-sided75

nsurface; i.e. to each point of the GML 2 corresponds one (external orinternal) normal vector of this surface;b.) If n is a odd number, then the unit normal vector ( , ) is a4-periodic vector function, with the Möbius property ( , 2) = ( , )(12)so that thenGML 2 is a one-sided surfaces; i.e. to each point of thenGML 2 surface correspond two normal vectors to this surface and, by thegeometric point of view, it is impossible to ―distinguish‖ the externalfrom the internal normal vector to this surface.The first fundamental form of a regular generalized MöbiusnListing's surfaces GML 2 , with circle as basic line, is given byE( , ) = 1;F( , ) = 0; nG( , ) = Rcos 2 so that, it is very easy to see that2( n)42,(13)Remark 4 Each point of the corresponding surface (7) is regular,i.e, for each point ( , )the corresponding forms satisfy the conditions:E ( , )> 0, G( , )> 0 and EG F > 0 .The second fundamental form of a regular generalized MöbiusnListing's surfaces GML 2 , with circle as basic line, is given by276

L( , ) = 0;M ( , ) =2N( , ) =nR nR cos 2 nR cos 2 nR cos 2 222( n)4( n)22( n)422; nsin . 2 (14)So that we may rewrite the mean and Gaussian curvatures of anregular GML 2 , with circle as basic line, in the formH ( , ) = nRcos 2 n Rcos 2 22( n)2( n)42232sinn2 (15)andK( , ) =4 ncos R 2 2 n R222n22.(16)Remark 5 Each point of Regular generalized Möbius-Listing'snsurfaces GML 2 , with circle as basic line, is:a.) Hyperbolic (saddle) point if the number n 0, i.e. alwaysK ( ,)< 0;b.) Parabolic point if the number n 0 , i.e. always K ( ,) 0 ;77

4. Relations between the set of Generalized Möbius-Listing'sSurfaces and the sets of Knots and LinksWe use the following definitions and notations:Definition 1. A closed line (similar to the basic or border's line)nwhich is situated on a GML 2 and is “parallel” to the basic (ornborder's) line of the GML 2 - i.e. the distance between this line and basicor border's lines is constant - is called a “Slit line” or shortly an “sline”(see e.g. Fig. 5.d.). If the distance between an s-line and the basic line is zero, thenthis s-line coincides with the basic line (and sometimes is called “Bline”)(seee.g. Fig. 5.c.).nDefinition 2. A domain situated on the surface GML 2 and suchthat its border's lines are slit lines, is called a “Slit zone” or shortly an“s-zone”. The distance between the border's lines of an s-zone is the``width" of this s-zone. If an s-zone's width equals to zero, then this zone reduces toan s-line.78

Definition 3. If the “B-line” is properly contained inside a “Slitzone” - i.e. his distance to the border's lines is strictly positive - then this“Slit zone” will be called a “B-zone”.Definition 4 The “process of cutting” or shortly the “cutting” isalways realized belong some s-lines and produces the vanishing (i.e.elimination) of the corresponding s-zone (which eventually reduces to ans-line)n. If a GML 2 surface is cut along an s-line (sometimes 1 ),then the corresponding vanishing zone will be called an s-slit (see e.g.Figs. 5.a., 5.b., 5.e.). If anGML 2 surface is cut along its B-line (sometimes )B ,then the corresponding vanishing zone will be called a B-slit(see e.g. Fig. 5.d.). If the vanishing zone - after an s-slit (a B-slit) - is given by an “szone”(a “B-zone”), then the cutting process will be called an s-zone-slit (a B-zone-slit).n If GML 2 surface is cut (k+1)-times along (k+1), k=0,1,2,…,different s-lines and none of them coincides with the B-line ( forthis process we use the symbolic notation k 1), then thenresulting object is called a (k+1)-slitting GML 2 , and the corres-79

ponding vanishing zones are (k+1)-slits. In this case the cuttingprocessis called a (k+1)-zone-slits.n If GML 2 surface is cut (k+1)-times along (k+1), k=0,1,2,…,different s-lines and one of this line coincides with the B-line (for this process we use the symbolic notation B k ), then thenresulting object is called a ( B+k)-slitting GML 2 , and thecorresponding vanishing zones are (B+k)-slits. In this case thecutting processis called a (B+k)-zone-slits. For each natural number k the segment [0, *] is divided by oneof the following rules:~ * 2 j j j 0,1,..., k; k 1k (17)~ * 2 j 1 j 1 j j 0,1,..., k;k 1korj*ˆ 2 jj 0,1,..., k;k(18)j* ˆ 2 j1j 0,1,..., k 1,kwhere [0,*) is a real number. For each natural number k the domain of definition T {[-*,*][0,2]} (see (2)) of the reprezentation formula (7) is divided byone of the following rules:~ ~T T T~Tk~T orkk ~ , ~ ~, ~ ~ , ~11 2 j2 j 1 2 j12 j0,2'2k1~, ~ ~ , ~ 2 j 12 j 2 j2 j10,2'j 1kkj1(19)80

T TˆTˆB,kB,k Tˆ2k1ˆ , ˆ ˆ , ˆ2 j 12 j 2 j2 j 10,2'j 1ˆTB, k or~ * ~T T T~T*kk B,kkˆ ˆ ˆ, ˆ ~ , ~1,1 2 j2 j 1 2 j 12 j0,2' ,*kj 1k *,~ ~, ~1 2 j2 j10,2'j1nTheorem 1. If the GML 2 surface is cut (k+1)$-times along (k+1)different (i.e. k=0,1,…) s-lines, and n is an even number, then for eachinteger numbers n,k,after (B+k)-zone-slits or (k+1)-zone-slits, an object Link-(k+2)nappears, whose each component is a GML 2 surface (knot with structure{0 1 }); The topologic group of the classic link-(k+2) in this case is atpresent unknown; only when k=0, the link-2 is of type { n 2 1}, according81(20)(21)~ 2k1~, ~ ,*Tk 2 j12 j0,2' j1Without loss of generality and for simplifying the process ofproofing, in this article we consider the following restrictions:n The GML 2 is a regular generalized Möbius-Listing's Surface(see representation (7) and notation (1*)); B-slit is symmetric (see e.g. Figs. 5.c, 5.d., 6.b., 6.d.), i.e. the―origin‖ (domain of its parametric representation) (7) of thecorresponding B-zone is the domain TˆB,0(20). For any fixed number of cutting k, the width of the eliminatednslit zones on the GML 2 surfaces are always identic andequal to 2ε/k (see e.g. Figs. 5.e., 6.c., 6.d.); For any fixed number of cutting k, the width of the remainingnslit zones on the GML 2 surfaces are always identic andequal to 2(*-)/(k+1).

the standard classification (see [6,7,8]).}; i.e. if n=2 is an even number,then for each =0,1,2,…, and k:Case A.GMLCase B.2 B k22 Link ( k 2) of ( k 2)objects GML2 (22)GML2 k 122 Link ( k 2) of ( k 2)objects GML2 (23)Proof. The representation (7) is a one to one correspondencenbetween the points of the strip T and the points of the GML 2 surface,and according to the remak 1, if n is an even number (n2), then eachof the functions X(,),Y(,),Z(,) is a 2-periodic function of theargument .Let us consider the particular case A, when k=0 - This means thatthere exists only one B-slit-zone.So that, according to (20), a the B-zone-slit corresponds to theelimination of theTˆB,0in the domain of definition T. But in this case, the domain ofdefinitionTˆB,0in (20) consists of two parts and define by (7) twondifferent objects GML 2 . The one to one correspondence (7) guaranteesthat the new objects have not self-cross points.In this case, according to the above restrictions, both new objectshave identic widths.Let us consider the particular case B, when k=0 - This means thatthere exists only one s-slit-zone.So that, according to (7) and (21), a s-slit-zone corresponds to the~ ,*elimination of the T 1 in the domain of definition T. But in this case, the~ *domain of definition T in (21) consists of two parts and define by (7)1ntwo different objects GML 2 with different widths. The widths of componentsof B-zone-slits are equal, but the widths of components of s-zone-slits are different.82

Let us consider now the general case A, when k is an arbitrarynatural number. This means that there exist (B+k)-zone-slits.– Recall that in the particular case A, after a B-zone-slit, an objectnLink-2 appears, whose both components are GML 2 surfaces. So that thefollowing cutting or the (B+1)-zone-slits (k=1) is a cutting of one of thennew objects GML 2 , which appears in the particular case A.Consequently, applying the same arguments of the particular casesnA or B to each of the new GML 2 surfaces (we do not know if thenew slit-zone includes the B-line of this surface or not, but this is notimportant, since the number n is even and both results are the same), wenfind that an object Link-2 appears, whose both components are GML 2surfaces. Therefore, the domain of definition TˆB,1in (20) consists of 3nparts and define by (7) 3 different objects GML 2 .So that, in the general case, according to (20), to a (B+k)-zones-slitor to each number k corresponds the elimination of Tˆsub-domains inB,k 1the domain T. But in this case, the domain of definition TˆB,k 1in (20)nconsists of (k+2) parts and defines, by (3), k+2 different objects GML 2 ,and the structure of these new objects is always identic to the originalsurface. The one to one correspondence (7) guarantees that the newobjects have not self-crossing points.Some examples are given in Figs. 7.83

Let us consider now the general case B, when k is an arbitrarynatural number.This means that there exist (k+1)-slit-zones. In this case, by usingthe same arguments of the previous case A, but considering the domain~ *of definition Tk 1in (21), we find identical results. Some examples aregiven in Figs. 8.84

nTheorem 2. If the GML 2 surface is cut (k+1)-times along (k+1)different (i.e. k=0,1,…,) s-lines and n is an odd number, then for eachinteger numbers n, k, after:Case A. - a (B+k)-zone-slits an object Link-(k+1) appears, whoseneach component is a GML 2 surface (Ribbon knot with structure {n 1 });The topologic group of the link- (k+1) in this case is at present unknown;only when k=0, the knot is of type {n 1 }, when n> 1, and of type {0 1 },when n=1, according the standard classification (see [6-8]); i.e. for everynatural numbers =0,1,2,…, and k21Bk44GML Link ( k 1)of ( k 1objects GML (24)2)Case B. - a (k+1)-zones-slit an object Link-(k+2) appears, whosenone component is a GML 2 surface (knot with structure {0 1 }, and eachnother component is a GML 2 surface (ribbon knot with structure {n 1 },except when n=1, since in this case the topological group is {0 1 }); Thegeneral topological group, in this case, is at present unknown; i.e. forevery natural numbers =0,1,2,…, and k,21k 12144GML Link ( k 2) of one GML and ( k 1objects GML (25)2 2)Proof. If n is an odd number (n=2+1), then the functionsX(,),Y(,),Z(,) is a 4-periodic function of the argument withproperty (M*); i.e. for each [-*,*]X , 0 X,2;Y,0 Y,2;Z,0 X,2; (26)Let us consider the particular case A, when k=0 - this means thatthere exists only one B-slit-zone. In this case, the one to onecorrespondence (3) defines a single object, in spite of the fact that thedomain of definition TˆB,0in (20) is disconnected. This new object has anew basic line (in particular, according to equations(7) and restriction (17) which is given by2285

* * nX , R cos cos 2 2 2 * * nY, R cos sin 2 2 2 * * nZ, sin 2 2 2 (27)This is a representation of a really closed line, but now [0,4](see Fig. 2.a. or Remark 2, when Q in [6]), and therefore the unitnormal vector (12) makes 2n+2 rotations around the new basic line (27),nsince it belongs to a GML 2 surface.86The following cutting of thenGML 2 body or (B+1)-zones-slit (k=1)2n2is a cutting of the new GML2surface, which appears after a B-zonesslit.Since the number of rotations 2n+2 is an even number, and by usingthe same arguments of the previous theorem, we find that, after a (B+1)-zones-slit, an object Link-(k+2) appears, whose both components2n2GML2surfaces.In general, after a (B+k)-zones-slit an object Link-(k+1) appears,2n2whose each component is a GML2surface. Some examples are givenin Figs. 9.Let us consider the particular case B, when k=0 - i.e. there existsonly one s-slit-zone. Since n is an odd number, then according to (19)after an s-zone-slit the new domain of definition (7) is given by~T (28) ~ , ~ 0,2 ~ , ~ 0,2 ~ , ~ 0,21 1 12 33 2This is a disconnected domain, which consist to three parties. Thedomain defined from the central part of the right hand side of equation(28), according to the representation formulas (7), defines an objectnGML 2 . But its width is smaller with respect to the original surface;Each of the remaining parts of the right hand side of eq. (28), similarly to2n2the case A of this theorem, define a geometrical object GML . So thatafter an s-zone-slit of alink-2, where onenGML 2 , when n is an odd number, appears a2

of the components is anGML 2 and the second one is aGML .2n22The following cutting of thenGML 2 body or 2-zones-slit (k=1) is a2n2ncut of one of the new GML2or GML 2 surfaces, which appear after aB-zones-slit.2n2If the second cut is a cutting of the GML2surface, then since2n+2 is a even number, by using the same arguments of previoustheorem, we find that after a 2-zones-slit an object Link-3 appears,nwhose one component is a GML 2 surface, and each other component is aGML2n22surface.87

nBut if second cut is a cutting of the GML 2 surface, then, since n isan odd number, by using the same argument of the particular case B ofthis theorem, we find that after a2-zones-slit an object Link-3 appears, whose one component is an2n2GML 2 surface, and all other components are GML2surfaces.Let us consider the general case B, when k is an arbitrary naturalnumber. This means that there exist (k+1)-slit-zones. In this case we canuse the previous arguments (when k=1) and remark that the domain of~definition Tk 1in (19) consists of (2k+3) sub-strips and defines an objectnLink-(k+2). Therefore, we can conclude that one object is a GML 2nsurface, and each other object is a GML 2 surface. Some examples aregiven in Figs. 10.Lastly, from the above Theorems and Remarks 1,2,3 (alsoRemarks 1,2,3,4 ofthe [13]) we can deduce the following factsRemark 6. Both the previous Theorems still hold when the basicline is aclosed sufficiently smooth space line.nAfter cutting each regular Möbius-Listing surface GML 2 , whosebasic line is a circle, appear objects (object) with the followingproperties:88

the tangential vectors of the new objects (object) r and r (8),(9) are orthogonal; each point of these objects are Hyperbolic (saddle) points if n0; each point of these objects are Parabolic points if n=0.AcknowledgementsAuthors are deeply grateful to Prof. P. E. Ricci. For constructingfigures in this article we used Matlab 7.0 - # 212817 - P.E. Ricci – ―LaSapienza‖.REFERENCES1. I. Tavkhelidze. On the some properties of one class of geometricalfigures and lines. – Reports of Enlarged Sessions of the Seminar ofI. Vekua Institute of Applied Mathematics, vol. 16, N. 1, (2001),35-38.2. I. Tavkhelidze, P. E. Ricci. Classification of a wide set of Geometricfigures, surfaces and lines (Trajectories). – Rendiconti AccademiaNazionale delle Science detta dei XL, Memorie di Matematicae Applicazioni, 124 , vol. XXX, fasc. 1, (2006), 191-212.o3. I. Tavkhelidze. Classification of a Wide Set of Geometric Figures.– Lecture Notes of TICMI, Tbilisi, Volume 8, (2007), 53-61.4. C. Cassisa, I. Tavkhelidze. About Some Geometric haracteristicnof the Generalized Möbius Listing's urfaces GML 2 . – ComputerScience and Telecommunications. Georgian Electronic ScientificJournal. No. 4(21), 2009, http://gesj.internet-academy.org.ge/ge/list_artic_ge.php?b_sec=comp pp. 54-84.5. C. Cassisa, P. E. Ricci, I. Tavkhelidze. About Connection of thegeneralized Möbius Listing‘s surfaces GML 2 with Sets of Knotsand Links. – Lecture Notes of Seminario Interdicciplinare diMatematica. Vol.9 (2010 ) pp. 73-86.6. E. W. Weisstein. The CRC concise Encyclopedia of Mathematics,(Second edition), Chapman & Hall/CRC, Boca Raton, FL, (2003).7. H. Doll and J. Hoste. A Tabulation of Oriented Links. – Math.Comput., 57, (1991), 747-761.n89

8. G. Kupenberg. Quadrisecants of Knots and Links. – J. KnotTheory Ramifications, 3, (1994), 41-50.9. A. Gray, E. Albena and S. Salamon. Modern Differential Geometryof Curves and Surfaces with Mathematica, (Third Edition), J.Capman and Hall / CRC.ilia TavxeliZenGML 2 – mebus-listingis ganzogadebulizedaprebis zogierTi geometriuli TvisebisSesaxeb da maTi kavSiri “lentisebri”xlarTebis simravlesTankvanZebis (Knots) da xlarTebis (Links) klasifikacia kargadaacnobili, magram ZiriTadad maTi struqtura `erTganzomilebiania~,anu maTi ganikveTi wertilia. ukanasknel wlebSiganixileba amgvari struqturebi – `lentisebri~ (Ribbon) xlarTebi,romelTa ganikveTi an raime wiria an mravalkuTxedi.warmodgenil naSromSi ganixileba iseTi `lentisebri~ kvanZebida xlarTebi, romlebic mebius- listingis ganzogadebulizedapirebis sabaziso wiris paraleluri wirebis `gaWrisas~warmoiSvebian. naSromSi Seswavlilia Tu ramdeni geometriuliobieqti warmoiSveba k – `gaWris~ Sedegad, ra tipis iqnebian`gaWris~ Sedegad warmoqmnili obieqtebi da zogierT Sem-TxvevaSi pasuxi gaeca SekiTxvas Tu romel klass, cnobiliklasifikaciidan, ganekuTvneba miRebuli struqtura. mebiuslistingisregularuli ganzogadebuli zedapirebis SemTxvevaSi,rodesac sabaziso wiri wrea Seswavlilia `gaWris~ SedegadmiRebuli geometriuli obieqtebis pirveli da meoreZiriTadi formebi da gausisa da normaluri simrudeebi.90

soxumis saxelmwifo universitetis Sromebit. VII, 2009maTematikisa da kompiuterul mecnierebaTa seriaTEMUR CHILACHAVA, NUGZAR KERESELIDZENON-PREVENTIVE CONTINUOUS LINEARMATHEMATICAL MODEL OF INFORMATIONWARFAREAbstract. In the given work the new direction in the theory of informationwarfare (mathematical modeling of information warfare) isoffered. In particular, the antagonism is meant ‖information warfare‖mass media (an electronic and printing press, the Internet) two states ortwo associations of the states, or two powerful economic structures (consortiums)conducting under the relation to each other purposeful misinformation,propagation. As the third side in process association of theinternational organizations (the United Nations, OSCE, EU, the WTOetc.) which effort are directed on intensity removal between the antagonisticstates, the sides and the termination of information warfare acts.The general continuous linear mathematical model of informationwarfare between two antagonistic sides which considers an antagonismcase as equipotent associations ("yak-bear"), and it is strong different -("wolf-lamb") is constructed. As required functions quantities of informationat present time, made each of the sides promoting achievement ofthe purposes, to the chosen strategy are taken. In that specific case themodels, each side identical rates wages information warfare and reacts toappeals of the international organizations. In turn, the third side in regularintervals reacts to intensity of information attacks of the antagonisticsides.Exact analytical solutions of a Cauchy ' s problem for system ofthe linear differential equations of the first order with constant factors arereceived.Parities between constants of model and initial conditions are revealed,at which:1. The antagonistic sides, despite increasing appeals of the thirdside, intensify information attacks.2. One of the antagonistic sides, under the influence of the thirdside stops, eventually, information warfare (an exit of the correspondingsolution on zero) while another strengthens it.91

3. Both antagonistic sides, after achievement of a maximum ofactivity, reduce it under the influence of the third side, andthrough final time, and at all stop information attacks (an exitof solutions on zero).In the first case, it is necessary to expect transformation of informationwarfare in a hot phase, in the second – it is less probable, in the third- it is at all excluded.The offered model of information warfare, except theoretical interesthas as well the important practical meaning. She allows, on the basisof supervision and the analysis, already at an early stage of informationattacks, to establish true intentions of each of the sides and character ofdevelopment of information warfare.2000Mathematics Subject Classification: 93A30, 00A71Key words and phrases: information warfare, antagonistic sides,continuous linear mathematical model, ignoring of an opposite side, nonpreventivemodel, hot phase.1. IntroductionThe theory of the information warfare, which formalization has begunall three-four ten years ago, now has already wide applied meaning.It is considered actively by many countries by working out of informationsafety. For the first time in the USA the presidential commission forprotection of a so-called critical infrastructure has been created. Then onthe basis of the conclusion of this commission has been developed № 63instruction of the president which in 1998, became a basis of the governmentalpolicy of maintenance of information safety [1]. The leadingcountries already have begun purposeful preparation of narrow experts ofinformation war. In the USA, at national university of defence the schoolof information warfare and strategy operates. At the Californian sea schoolto group of information warfare read courses of lectures: principles of informationoperations; psychological operations; information warfare: planningand an estimation; an estimation of information warfare.Russia, the truth with delay, but too operates in this direction. TheMinistry of Defence of Russia has created the information and propagandacentre which along with other problems, will prepare khakers attackson information resources of the opponent. This decision of the Ministryof Defence of Russia, was the answer to the task of the president of Russia- prepare the offers connected with creation of the centre of prepara-92

tion of experts which can to wage information warfare by the newesttechnologies [2, 3].According to some sources problems of the information and propagandacentre will be: intimidation of the opponent, destruction of its informationcommunications and preservation of the, creation informationand misinformation parts and rendering of influence on public opinionboth to the conflict and during a confrontation. To necessity of creationof information armies the management of armed forces of Russia wasresulted by the analysis of war of 2008 with Georgia.Originally the term ‖information warfare‖ Thomas Rona has appliedin 1976 in the report‖systems weapons and information warfare‖ which intended forcompany Boeing [4]. T.Rona has noticed that by then, the informationinfrastructure became a central component of economy of the USA andsimultaneously the idle time, less protected purpose both in military andin a peace time.For the present the uniform definition of the term ‖informationwarfare‖ is not accepted, but intuitively it is considered that informationwarfare is purposeful actions on creation of the information superiority,by means of destruction of the information, information systems of anopposite side, thus simultaneously there is a process of protection of owninformation and information systems.By information warfare also mean a complex of actions for creationof information influence on public consciousness to change behaviour ofpeople, to impose them the purposes which do not enter into their interests.On the other hand, protection against the same influence is necessary.As the state information resources often become objects of an attackand protection, the state is compelled to give to an information technologya great attention. Accordingly, in the theory of information warthe great number of researches is devoted safety of the information, informationsystems and processes.On the other hand, studying of information streams as the informationstream which to fall upon mass consciousness, in most cases, allowsto manipulate people is essential [5].The description a mathematical apparatus various a component ofinformation warfare and its studying already is included into sphere ofinterests of many scientists. In this direction it is necessary to note use ofthe mathematical theory of an information transfer on communicationchannels for construction of model of information influence. Attempts ofestimations of efficiency of concrete information influences are underta-93

ken [6]. By means of the theory of counts and games models of informationnetworks and information warfare are made, and for the contradictoryparties the mixed strategy are found [7, 8]. Strategy, basically, are calculatedon deducing out of operation information infrastructures or theirprotection by means of as physical and program (viruses, Trojans, cyberattacks) influences.Our approachIn the present work as our purpose studying of quantity of informationstreams by means of new mathematical models of information warfarewas [9]. We mean an antagonism by information warfare by mass media(an electronic and printing press, the Internet) two states or two associationsof the states, or the economic structures (consortiums) conductingunder the relation to each other purposeful misinformation, propagation.In a world information field for ideological, political and economictargets the purposeful information and the misinformation, which allocationfrom the general background in most cases for the unprepared personvery difficult is actively used.The purposes of information warfare can be:– Drawing of a loss to image of the opposite country – creationfrom it an image of the enemy.– Discredit of a management of the opposite country.– Demoralization of staff of armed forces and the peace populationof other country.– Public opinion creation, both in the country, and behind itslimits, for the argument and the justification of possible poweractions in the future.– Counteraction to geopolitical ambitions of an opposite sideetc.The international organizations react to occurring processes in themodern world in this or that form and activity. Therefore in the course ofinformation warfare as the third side we consider association of the internationalorganizations (the United Nations, OSCE, EU, the WTO etc.)which effort are directed on intensity removal between the antagonisticstates, the parties and the termination of information warfare.In the given work we have constructed the general continuous linearmathematical model of information warfare between two antagonisticsides, in the presence of the third, peace-making party. The model94

considers an antagonism case as equipotent associations ("yak-bear"),and is strong different ("wolf-lamb").We consider that information warfare is conducted against eachother by the first and second sides, and by the third side mean the internationalorganizations. In that specific case the models, each side identicalrates wages information warfare and reacts to appeals of the internationalorganizations. In turn, the third side in regular intervals reacts to intensityof information attacks of the antagonistic sides. Exact analytical solutionsare found in model of ignoring of an opposite side. By means of the analysisof solutions character of actions of the sides in information warfare,depending on that what starting conditions of the sides, and parities betweenindexes of aggression, peace-making readiness and peace-makingactivity is established.2. The system of the equations and initial conditionsAll three sides involved in process of information warfare extendinformation for object in view achievement. At the moment of timet 0,quantity of the information extended by each of the sides wewill designate accordingly through N1(t),N2(t), N 3(t ) . The quantity ofinformation at the moment of time t , is defined as the sum, all provokinginformation which are extended by each of the sides all mass media.The detailed analysis of mathematical models of dynamics of populations,and also of Lanchester ' s models of military operations [10], hasled us to the following general continuous linear mathematical model ofinformation warfare:dN1(t) 1N1(t) 2N2( t)3N3( t)dtdN2( t) 1N1(t) 2N2( t) 3N3( t) dtdN3( t) 1N1(t) 2N2( t) 3N3( t)dt(2.1)95

with initial conditionsN1( 0 ) = N10, N2(0 ) = N 20, N3(0 ) = N30, (2.2)where, 1, 3, 2, 3 0 , i 0 i 1, 3 , 2, 1- constantfactors.These constant factors are model constants, thus we name 1, 2factors of growth of provoking statements according to the first and thesecond the sides in the absence of the third side (relative growth rates ofquantity of statements).These constant factors are model constants, thus we name factors ofgrowth of provoking statements according to the first and the second thesides in the absence of the third side (relative growth rates of quantity ofstatements).In the general linear model (2.1) speed of change of quantity of theinformation spread by the first and second sides linearly depends onquantity of information extended by the sides and the international peacemakingorganizations.Speed of change of quantity of pacifying information extended bythe third side linearly grows or is directly proportional to quantity of informationspread by all three sides.In initial conditions (2.2), N10, N20, N30non-negative constants,thus:If N 100 , N 0 20, then both sides are initiators of informationwarfare.If N 100 , N 0 20, then the first side is the initiator of informationwarfare.If N 100 , N 0 20, then the second side is the initiator of informationwarfare.The third side initially does not spread any information ( N 0 )30 or does preventive character peace-making statements ( N 300 ) andthen starts to react to the provocative information extended by the antagonisticsides.96

3. Model of ignoring of an opposite side.The antagonistic sides which with identical intensity conduct informationwarfare, opposite side spread information, but thus both sidescan ignore should listen equally to appeals of the third – the peacemakingside.In this case, in the general linear model (2.1) some factors can beput equal to zero. In particular, 2and 1also are equal to zero. We willput also that 30 or the third side equally reacts only to widespreadprovoking information antagonistic sides.Thus, we will put 1 2 ,3 3 , 1 2 .Then the system (2.1) will written as follows (assume the followingair):ddtddtddtNNN123t Nt Nt1t NtNtt NtNt12233(3.1)The solution of system (3.1) in 0,area will write down as follows( N 300 , international organizations "are not awake" and react onlyto already launched information warfare):a) D = 2 8> 0 N10 N20N3(t )=Dt te 1 2 e(3.2)N10 NN1(t )=20 t2 N N e 1020+ 2De 1t N N 10 20 te 2 D1(3.3)97

N20 N10tN (t )= 22 N N e 1020+ 2De 1t N N 10 20 te 2 D1(3.4)22 8 81 0 , 2 0.22b) D 2 8= 0t2N3( t) ( N10 N20)te (3.5)N ( t)1N10 N220N Nte 10 20+ ( 4t)2te 2(3.6)NN ( t)220 N210N Nte 10 20+ ( 4t)2te 2(3.7)c) D = 2 8

N20 NN2(t)=10 te +2( N N ) t 210 202e 2 8sin D 2t (3.10)4. The Analysis of the received results.In model of ignoring of an opposite side (3.1) it is possible to consider as an indicator (index) of aggression of the antagonistic sides, - an indicator of their readiness for the world, to listen to peace-makingappeals of the international organizations, - an indicator of peacemakingactivity of the international organizations. As it will be shownmore low, depending on that is more – an aggression index, or indexes ofreadiness for the world and peace-making activity, character and developmentof information war essentially varies.Let's consider a case when the international organizations have notaccepted preventive a measure and we investigate development of informationwarfare at various values D .4.I. D = 2 8> 0 .In this case, a square of an index of aggression more than eightfoldproduct of indexes of readiness for the world and peace-making activitythat unequivocally specifies in high aggression of the antagonistic sidesin information warfare.4.I.I. ( N10= N20). In that case when the international organizationshave not undertaken preventive a measure, and the antagonisticsides have begun information warfare under equal starting conditions influenceof the international organizations on the first and second side, isineffectual – they strengthen information attacks.Really, from (3.3), (3.4) and with the account N10= N 20we will receive:1N1(t )= 2N10(D1te22te1) , (4.1)99

1N2(t )= 2N10(D1te22te1) , (4.22)when t .It is enough to show justice (4.1) for N1(t ), as owing to (4.2)N2(t ) it is identically equal N1(t ).Really on a semi interval [0 , + ) function N1(t ) positive, increasingand unlimited from above.Positivity N1( t ) follows from following parities:andN1( 0 ) = N10>0 ;1 N > 0 , as , >0 ,D2101te22te1>01 1Follows from parities: 1> 2>0, 0 , 21e 1t > e 2t .Positivity of a derivative N1(t ) is an indicator of its increase. 2N ' 1t =2t2 N10 2e1 12) te2D 121> 0 ,As in this expression all factors are positive.N (t ) it is unlimited from above, since1N1(t ) 210 N 1D1te2, when t .As to the international organizations,100N (t )=32 t te 2e1 N 10D 0 ,

At t [ 0, ), N3(t ) , when t .N3( 0 )= 0 and then increases together with t , as its derivativepositive.tN / 3 = 22N2t112 ee10 t2DN3(t ) unlimited from above, as1> 02N N3(t ) 10 te 1 , at t .DThus, in not preventive model of information warfare, at,D = 2 8> 0 and equal starting a condition ( N = N10 20) the antagonisticsides, they strengthen the activity. Functions -1N (t ),N (t ),2N3(t ) monotonously increase – i.e. information war faredoes not stop,and all expands.4.I.2. (10N > N20). If at the antagonistic sides different launchingsites and a starting condition of the first side more than the second, functionN10 NN1(t )=20 t2 N10 N 1 1 2 e ee + 20 1ttD 21, (4.3)at t [ 0, )positive, increasing also it is unlimited from above.The second member of expression (4.3) we already investigated inIV.I.I and it positive, increasing and is unlimited from above. To it functionwith positive factor20N10 N>0 , >0 , which also is positive is2added exhibitor, increasing and is unlimited from above.Accordingly and their sum, i.e. N1(t ) is positive, increasing andunlimited from above.As to N2(t ) , it in a point t =0 is positive N2(0 )= N20>0 , andits derivative, proceeding from system (3.1) and entry conditions (2.2)/positive - N20= N 20>0 , therefore in a certain vicinity on the right,N (t ) is positive and increasing.2101

.N20 NN2(t )=10 t2 N10 N 1 1 2 e ee + 20 1ttD 21(4.4)But with increase t increases and N2(t ) which reaches themaximum, and then it starts to decrease monotonously and aspiresto .Really, an expression signN (t )=2e tN220N 10N2 N t 10 20 + e 10 20 te 2D N N1 D1(4.5)Defines a sign on a factor bracketedN( ) 20 NF t 10+ N N 10 20 2te 2 D2 N tN10 20 11De(4.6)Absolute value of the second and third member (4.6) becomes asmuch as small for enough big t , thus, when t the sign (4.6) is definedby a sign on the first composed10N20 N. I.e. it will be negative2N20 Nsince10

N220N 10+ N210 N20 t2De N10 N201t1De++ N10 N201tDe N tN10 20 2De= 0 (4.8)As to the international organizations, the third side, it strengthensthe activity10 1t tN (t )= e e 3 ( N N ) 20 2D , (4.9)when t .Att =0 N3(0) 0 , equals to zero, then N3(t ) increases. Really/ t = 20 t te e 1 N 3 ( N10 N )21 2D> 0 (4.10)N3(t ) it is unlimited from above, it is valid - N10 N20N3(t ) D te 1 , at t .4.I.3. ( N10< N 20). If at the antagonistic sides different launchingsites and starting conditions of the second side more the first, i.e.N10< N20, then the sides change roles and is received symmetric results,under the relation of the previous point – already second the side strengthensinformation attacksN20 NN2(t )=10 t2 N N e 1020+ 2De 1t N N 10 20 te 2 D1 , (4.11)when t . As to the first side, it makes active in the beginninginformation attacks, leaves on a maximum, further reduces, and then andat all stops information warfare (in t leaves on zero)103

N10 NN1(t )=20 t2 N N e 1020+ 2De 1t N N 10201D , (4.12)when t . t represents the solution of the transcendental equationN210N 20+ N20N10 2t2De N tN10 20 11De= 0 (4.13)The third side strengthens the activity and for it it is fair ( N N t tN3(t )= )1020 12e e , (4.14)Dat t .Thus, it is necessary to notice that the third side at not the preventiveapproach ( N30= 0 ), can have partial influence on a course of informationwarfare. In particular, influences one of the sides if the antagonisticsides have begun information warfare under unequal starting conditions( N10 N20). Thus, the third side influences that antagonistic sidewhich launching site is ―weaker". i.e. it is less. And this side, throughcertain time stops information warfare though initially actively joins in it,leaves on a maximum of actions, however then, reduces information attacks,and in the end and at all them stops.4.II. D 2 8=0 .In this case the aggression index is still high and there are analogiesto a caseD = 2 8> 0 . Really.4.II.1. ( N10 N 20) .In that case when the international organizationshave not taken preventive measures ( N 300 ), and the antagonisticsides have begun information warfare at equal starting a condition( N 10= N 20), influence international the organizations on the first andsecond sides without results – the last strengthen information attacks andstarting with(3.5) - (3.7)N1(t ) , N2(t ) , N3(t ) , при t .104

Let's really copy (3.5) - (3.7) as followsNN1(t)10 ( t 2)2te 2 NN2(t)10 ( t 2)2te 2 (4.15)(4.16)t2N3(t)2N10te (4.17)On a semi interval [ 0 ,+ ) functions N1(t ), N2(t ) are positive,increasing and are unlimited from above, since, are product defined on asemi interval [ 0 ,+ ), positive, and unlimited from above functionsN 10 ( t 2)2tand e 2 . The Same it is possible to tell and about functionN (t ) on an interval ( 0 , + ).34.II.2. ( N10> N20). If at the antagonistic sides different startingconditions, and starting conditions of the first side surpass the second, ona semi interval [ 0 ,+ ) function N1(t ), which has the followingappearanceN10 N20 tN1(t) e N N+ 20( t2)24te 2 (4.18)Positive, increasing also it is unlimited from above since representsthe sum and positive, increasing and unlimited from above functionsN10 N 20 te N Nand 10 20( t2)22te 2 defined on a semi interval [ 0 ,+ ).As to N2( t ), it in a point t =0 is positive - N2(0 )= N20>0 . Inthis point its derivative owing to system (3.1), and entry conditions (2.2),/is positive - N 20= N 20>0 . Therefore, on some right vicinity of apoint 0, N2(t ) it is positive and increasing.NN ( t)220 N210N N4te + 10 20( tt2)e 2(4.19)105

But already with increaset , N2(t ) reaches the maximum value,and then monotonously decreases and aspires to .Really, an expression signNt N20 N N Nt) e + 10 20( t2)2 4102( te 2) (4.20)defines a sign on the factor bracketedN( ) 20 NF t 10 N N+ 10 20( t2)2 4 te 2(4.21)Value of the second member (4.21) becomes as much as smallat bigt , therefore when t the sign (4.21) defines a sign the on theN20 Nfirst composed10N20 N, i.e. will be negative, for10

As to the international organizations, they strengthen the activity(see (3.5)).4.II.3.( N10 N 20) . If at the antagonistic sides different startingconditions, and starting position of the second side more the first the firstand second sides change roles and is had symmetric results - already thesecond side strengthens information attacks.N (t ) , (4.24)2att . And the first side at first makes active information attacks,leaves on a maximum, then reduces, further and at all stops informationwarfare.4.III. D 2 8< 0 .In this case, a square of an index of aggression less than eightfoldproduct of indexes of readiness for the world and peace-making activity,i.e. it is expected that peace-making activity "will pacify" (will block)aggression.4.III.1. ( N 10= N20). Really, if the international organizations havenot taken preventive measures, and the antagonistic sides have beguninformation war at equal launching sites, then influence the internationalorganizations on the first and the second the sides the productive.Really, functions, N1(t ) and N2(t ) owing to (3.9), (3.10)N1(t )=2N102 De 2 tsin 282t (4.25)2Nt 10N2(t)= 2 e 2 sin D Leave on zero, when282t (4.26)t =2 Dwhere, for fair, (4.27)107

arctg D(4.28)As to the international organizations,N3(t )=4N10 Dte 2sin 282t (4.29) 2That they leave on zero in t = , after time moment t , i.e. Dthe third side finishes distributions of peace-making appeals after, theantagonistic sides will finish information warfare.4.III.2. ( N10> N 20). If at the antagonistic sides different launchingsites and thus, starting conditions of the first side more than the secondfrom (3.9), having equated to zero function N1(t ), we will find timewhen the first side stops information warfareN10 N t 20e 2 +2( N210 N ) D20sin 282t = 0 , (4.30)sin From (4.30) we will receive2 8t N10 N20 D t= e2 2 N10 N20 2 2he solution (4.31) exists only in that case, when(4.31)1 NN1010 N N20202 D2te 2 1(4.32)Or, something, in our case0

2 N0 t ln N2010 N N1020828(4.33)The solution (5.1.3.7)t1- should satisfy to a condition0 t12 N ln N2010 N N1020828, (4.34)andt = 10 202arcsin e 12 N N D 1 D N10 N208t 2 D2 D(4.35)Performance of conditions (4.34) and (4.35) is possible at selection , product should be enough great number.As to the second side, at big t - the sign N2(t ) in (3.10) coincidesN20 Nwith a sign10, i.e. negative, therefore at N2(t)is available zero2N20 N 10 te +2( N N ) t 2 10 202e 2 8sint =0 (4.36) D 2 t2 , should satisfy to conditionsThe solution of the equation (4.35) -0 t22 N ln N2010 N N1020828(4.37)and109

t2=2 N10 N20 D t2arcsineD N10 N2082 (4.38)Thus, the international organizations stop the peace-making effortsafter end of information warfare by the antagonistic sides.N3(t )=2( N10 N20) Dte 2sin 28 2t =0 (4.39)The solution of this equation will bet =228(4.40)I.e., the third side leaves on zero in a point t , and is had a bilateralinequality.4.III.3. (10N < N20). If at the antagonistic sides different startingconditions, and starting position of the second side more the first, the firstand second sides change roles and is had symmetric results for N1(t )and N2( t ).REFERENCES1. Joint Pub 3 – 13 ―Information Operations‖, DOD US. December,1998.2. G. G. Pochentsov. Propagation and counter-propaganda. Publishhouse "Center", M., 2004.3. www. securitylab.ru4. T. Rona. ―Weapon Systems and Information War‖. Boeing AerospaceCo., Seattle, WA, 1976.110

5. M. Hananashvili. Information stress. National Academy of sciencesof Georgia. Tbilisi, 2008 (in Georgian).6. M. Bogdanov, V. V. Mohor. Mathematical model of informationinfluence. http://www.it-safety.org.ua/biblioteka/bogd_mohor_1.htm- 63k 2002. — Vip. 27. J. R. Cares. An Information Age Combat model. – AlidadeConsulting Technical Paper, March, 2001.8. J. Jormakka, J. V. E. Molsa. Modelling Information Warfare as aGame. – Journal of Information Warfare. 2005, 4 (2): 12 - 25.9. T. Chilachava, N. Kereselidze. About one mathematical model ofthe information warfare. – Fifth congress of mathematicians ofGeorgia. Abstracts of contributed talks. Batumi/Kutaisi, October 9-12, 2009, p. 85.10. T. Chilachava, T. Dzidziguri. Mathematical modelling. Tbilisi,2008, 440 p. (in Georgian).Temur CilaCava, nugzar kereseliZesainformacio omis araprevenciuli uwyvetiwrfivi maTematikuri modelinaSromSi SemoTavazebulia sainformacio omis Teoriisaxali mimarTuleba (sainformacio omis maTematikuri modelireba).`sainformacio omis~ qveS igulisxmeba ori saxelmwifosan saxelmwifoTa ori gaerTianebis, anac ori mZlavriekonomikuri struqturis (konsorciumebis) mier masobrivisainformacio saSualebebis (beWvdiTi da eleqtronulipresis, internetis) gamoyenebiT erTmaneTis winaaRmdegmizanmimarTuli dezinformaciisa Tu propagandis warmoeba.procesSi mesame mxared gvevlineba saerTaSoriso organizaciaTagaerTianeba (gaero, euTo, evrokavSiri, mso da sxva),romlis Zalisxmeva mimarTulia antagonistur saxelmwifoTa,mxareTa Soris daZabulobis neitralizaciisa da sainformacioomis Sewyvetisaken. agebulia sainformacio omis zogadi,wrfivi uwyveti maTematikuri modeli, romelic iTvaliswinebsrogorc Tanabari (`iaki – daTvi~), aseve mkveTrad gans-111

xvavebuli (`kravi – mgeli~) siZlieris mqone gaerTianebebisdapirispirebas.saZebn funqciebad aRebulia drois mocemul momentSiinformaciis is raodenoba, romelsac avrcelebs TviToeulimxare, dasaxuli miznis, arCeuli strategiis misaRwevad. mowinaaRmdegemxaris ignorirebis modelSi, orive mxare erTida igive tempiT awarmoebs sainformacio oms da reagirebssaerTaSoriso organizaciebis mowodebebze. Tavis mxriv,mesame mxare, romelic moqmedebs araprevenciulad, Tanabradreagirebs antagonistur mxareTa sainformacio Setevebis intensivobaze.pirveli rigis wrfivi mudmivkoeficientian diferencialurgantolebaTa sistemisaTvis koSis amocanis zusti amonaxsnisSeswavlam, gamoavlina modelis konstantebsa da sawyispirobebs Soris is Tanafardobebi, romlis drosac: antagonisturi mxareebi, miuxedavad mesame mxarismowodebisa, aZliereben sainformacio Setevebs; erT-erTi antagonisturi mxare, mesame mxaris zemoqmedebiT,garkveuli drois Semdeg, asrulebs sainformaciooms (Sesabamisi amonaxsnis nulze gasvla),maSin roca meore, mas aZlierebs; orive antagonisturi mxare, mas mere rac miaRwevsaqtiurobis maqsimums, mesame mxaris zemoqmedebiT amcirebssainformacio Setevebs da sasrulo droisSemdeg Sewyveten sainformacio oms (amonaxsnTa nulzegasvla).pirvel SemTxvevaSi, unda velodoT sainformacio omiscxel fazaSi transformacias, meoreSi es transformacianaklebad savaraudoa, mesameSi ki – gamoricxuli.SemoTavazebul models Teoriuli interesis garda gaaCniapraqtikuli mniSvnelobac. igi gvaZlevs saSualebas, sainformacioSetevebis ukve sawyis etapze, dakvirvebebisa daanalizis saSualebiT, davadginoT mxareTa WeSmariti ganzraxvebida sainformacio omis ganviTarebis xasiaTi.112

soxumis saxelmwifo universitetis Sromebit. VII, 2009maTematikisa da kompiuterul mecnierebaTa seriaTEMUR CHILACHAVA, NUGZAR KERESELIDZECONTINUOUS LINEAR MATHEMATICAL MODELOF PREVENTIVE INFORMATION WARFAREAbstract. Recently the working out of mathematical models of informationwarfare has become under which the struggle between thestates by using information arsenal solely, i.e. an information technologywhich is based on industrial production, propagation and information imposingis mainly meant.In the given work the new direction in the theory of informationwarfare (mathematical modeling of information warfare) is offered. Inparticular, ‖information warfare‖ means the struggle between the twostates or two associations of the states, or two powerful economic structures(consortiums) conducting purposeful disinformation, propagation toeach other by means of mass media (an electronic and printing press, theInternet). The association of the international organizations (the UnitedNations, OSCE, EU, the WTO etc.) act as the side the efforts of whichare directed towards the removal of tension between the antagonisticstates, the sides and the cessation of information warfare acts.There is constructed the general continuous linear mathematicalmodel of information warfare between two antagonistic sides which considersthe case of confrontation as equipotent associations ("yak-bear"),as well as strong differences - ("wolf - lamb").The number of information at present time, made by each of thesides is taken as an unknown quantity promoting the achievement of purposes,the chosen strategy . In that specific case the model, each sideconducts information warfare and reacts to the appeals of internationalorganizations at the same pace. In its turn, the third side equally reacts tothe intensity of information attacks of the antagonistic sides.Exact analytical solutions to a Cauchy ' s problem for the system ofthe linear differential equations of the first order with constant factors arereceived.Parities between the constants of the model and initial conditionsare revealed, at which:113

1. The antagonistic sides, despite increasing appeals of the thirdside, intensify information attacks.2. One of the antagonistic sides, under the influence of the thirdside stops information warfare (an exit of the correspondingsolution on zero) while another strengthens it.3. Both antagonistic sides, after achieving maximum activity,reduce it under the influence of the third side, and through finitetime, stop information attacks at all (an exit of solutionson zero).In the first case, the transformation of information warfare into ahot phase is expected, in the second – it is less probable, in the third - it isexcluded at all.The offered model of information warfare, except theoretical interesthas as well an important practical meaning. At allows, on the basis ofobservation and the analysis, at an early stage of information attacks, toestablish true intentions of each side and the character of the developmentof information warfare.2000Mathematics Subject Classification: 93A30, 00A71Key words and phrases: information warfare, antagonistic sides,continuous linear mathematical model, ignoring of an opposite side, preventivemodel, hot phase.1. IntroductionThe theory of the information warfare, which formalization ofwhich began thirty-forty years ago, now has a wide applied meaning. It isactively considered by many countries at elaboration of information safety.The presidential commission for protection of a so-called critical infrastructurewas created in the USA at first. Then on the basis of the conclusionof this commission directive № 63 of the president was elaboratedwhich in 1998, became the basis of the governmental policy ofmaintenance of information safety [1]. The leading countries have alreadybegun purposeful preparation of experts in a narrow field of informationwarfare. In the USA, at national university of defence operatesthe school of information warfare and strategy. At Californian marineschool courses of lectures on: principles of information operations; psychologicaloperations; information warfare: planning and an estimation;an estimation of information warfare are delivered to the group of informationwarfare.114

Russia, though with delay, operates in this direction to. The Ministryof Defense of Russia has created the information and propaganda centrewhich along with tasks will prepare hacker attacks on information resourcesof the opponent. This decision of the Ministry of Defense of Russiawas the answer to the task of the president of Russia - to prepare theoffers concerning the creation of the centre of preparation of expertswhich will be able to conduct information warfare using the newest technologies[2, 3].According to some sources tasks of the information and propagandacentre will be: intimidation of the opponent, destruction of its informationcommunications and preservation of its own, creation of informationand disinformation parts and rendering of influence on public opinionbefore the conflict as well as during the confrontation. For the necessityof creation of information forces the management of armed forces ofRussia cited the analysis of war with Georgia in 2008.Originally the term ‖information warfare‖ was used Thomas Ronain his report in 1976 ―Systems of weapons and information warfare‖which meant for company Boeing [4]. T. Rona noted that by then, theinformation infrastructure was becoming the central and at same timesimple component of the economy of the USA, less protected target bothin state and in peaceful time.The common definition of the term ‖information warfare‖ has notyet been accepted, but intuitively it is considered that information warfareis a purposeful action for creation of information superiority, bymeans of destruction of information, information systems of an oppositeside, while the protection of its own information and information systemsis in process.Information warfare also implies complex of actions for creation ofinformation influence on public consciousness to change behaviour ofpeople, to impose purposes on them which are not their interests. On theother hand, protection against the same influence is necessary.As the state information resources often become objects of an attackand protection, the state is compelled to pay a great attention to informationtechnologies. Accordingly, in the theory of information warfarethe great number of researches is devoted to the safety of information,information systems and processes.On the other hand, studying of information streams is essential, asthe information stream which falls upon mass consciousness, in mostcases, allows manipulating people [ 5 ].115

The description of various components of information warfare andits study by a mathematical apparatus is already included into the sphereof interests of many scientists. In this regard the usage of the mathematicaltheory of an information transfer by communication channels for theconstruction of the model of information influence should be noted. Attemptsof estimations of efficiency of specific information influences areundertaken [ 6 ]. By means of the theory of graphs and games models ofinformation networks and information warfare are drawn up, as for thecontradictory sides mixed strategies are found [7, 8]. Strategies, basically,are calculated for extermination of information infrastructures or theirprotection by means of physical and as well as program (viruses, trojans,cyber attacks) influences.116Our approachIn the present work the studying of quantity of information streamsby means of new mathematical models of information warfare was ourpurpose [ 9 ] . By information warfare we mean an antagonism by meansof mass media (an electronic and printing press, the Internet) between thetwo states or the two associations of states, or the economic structures(consortiums) conducting purposeful misinformation, propagation againsteach other.In a world information field for ideological, political and economictargets the purposeful information and the misinformation is activelyused, from which in most cases the separation of the general backgroundfor the unprepared person is very difficult.The aims of information warfare can be:– Infliction of losses to the image of the antagonist country –creating the image of the enemy.– Discredit of the management of the antagonist country.– Demoralization of the personnel of the armed forces and thecivilians of the antagonist country.– Creation of public opinion, inside and outside of the country,for justification of argumentation of possible military operations.– Opposition to the geopolitical ambitions of the antagonistcountry etc.International organizations react to occurring processes in the modernworld in this or that form and activity. Therefore as the third side inthe course of information warfare we consider association of internation-

al organizations (the United Nations, OSCE, EU, the WTO etc.) effortsof which are directed on remonal of tension between the rival states, thesides and the cessation of information warfare.In the given work we have constructed the general continuous linearmathematical model of information warfare between two antagonisticsides, in the presence of the third, peace-making side. The model considersthe case of confrontation between the unions of equal ("yak-bear")as well as strong different ("wolf-lamb") strength.We consider that information warfare is conducted against eachother by the first and second sides, and by the third side we mean the internationalorganizations. In that specific case the model, each side conductsinformation warfare and reacts to the appeals of the internationalorganizations equally. The third side, in its turn reacts equally to the intensityof information attacks of the antagonistic sides. Exact analyticalsolutions are found in the model of ignoring of an opposite side. Bymeans of the analysis of solutions the character of actions of the sides ininformation warfare is establish, depending on starting conditions of thesides, and parities between indexes of aggression, peace-making readinessand peace-making activity.2. The system of the equations and initial conditionsAll the three sides involved in the process of information warfarespread information for the achievement of the set goal. At the moment oftime t 0,quantity of the information spread by each of the sideswe will accordingly designate by N1(t),N2(t), N 3(t ) . The quantity ofinformation at the moment of time t , is defined as the sum of all provokinginformation which is spread by each of the sides through every kindof mass media.The detailed analysis of mathematical models of dynamics of populationas well as Lanchester ' s models of military operations [10 - 12], hasled us to the following general continuous linear mathematical model ofinformation warfare:117

dN1(t) 1N1(t) 2N2( t)3N3( t)dtdN2( t) 1N1(t) 2N2( t) 3N3( t) dtdN3( t) 1N1(t) 2N2( t) 3N3( t)dt(2.1)with initial conditionsN1( 0 ) = N10, N2( 0 ) = N 20, N3(0 ) = N30, (2.2)where, 1, 3,2,3,1,2 0 , 30 , 2, 1- constant factors.These constant factors are constants of model, thus we name 1, 2factors of growth of provoking statements by the first and secondsides according in the absence of the third side (relative growth rates ofquantity of statements).In the general linear model (2.1) speed of quantity change of the informationspread by the first and second sides linearly depends on quantityof information spread by the sides and the international peace-makingorganizations.The speed of quantity change of pacifying information spread bythe third side linearly grows or is directly proportional to the quantity ofinformation spread by all three sides.In initial conditions (2.2), N10, N20, N30non-negative constants,thus:If N 100 , N 0 20, then both sides are initiators of informationwarfare.If N 100 , N 0 20, then the first side is the initiator of informationwarfare.If N 100 , N 0 20, then the second side is the initiator of informationwarfare.118

The third side initially does not spread any information ( N 0 )30 or does preventive peace-making statements ( N 300 ) and then starts toreacting to the provocative information spread by the antagonistic sides.3. The model of ignoring of an opposite sideThe antagonistic sides which wage information warfare with thesame intensity may ignore information spread by the opposite side, butmeanwhile both sides should equally listen to the appeals of the third –the peace-making side.In this case, in the general linear model (2.1) some factors can beconsidered as equal to zero. In particular, 2and 1are equal to zero.Let‘s consider that 30 or the third side equally reacts only to provokinginformation spread by antagonistic sides.Thus, let‘s consider 1 2 ,3 3 , 1 2 .Then the system (2.1) will be the following :ddtddtddtNNN123t Nt Nt1t NtNtt NtNt12233(3.1)The solution of the system (3.1) with initial conditions (2.2) in0,area will be written down as follows:1. D = 2 8> 0119

N 20N (t )=3N10 2N30D N e 1t N10 20 1N30D te 2(3.2)N10 NN1(t )=20 t2 N N N e 10 20+ D2230e 1t N N N 10 20 D1130 te 2(3.3)N20 NN2(t )=10 t2 N N N e 10 20+ D2230e 1t N N N 10 20 D1130 te 2(3.4)22 8 81 0 , 2 0.222. D = 2 8= 0t2N3( t)N30 ( N10 N20 N30)t e2 (3.5)N ( t)1N10 N220te +N N22 ( N10 2010 N20 N230) t te 2(3.6)N20 N10tN2( t) e +2 N10 N20 2 + ( N10 N20 N30)t 2 2 te 2(3.7)120

3. D = 2 8< 0N230N3(t )=2N10 N2028N302e 2 tsin 282t , (3.8) = arctg2N3028N10 N20N30.N10 NN1(t )=20 te +2( N210 N ) D20e 2 tsin 282t (3.9)N20 NN2(t)=10 te +2( N210 N ) D20e 2 tsin 282t (3.10)4. The analysis of the obtaining results.In model of ignoring of an opposite side (3.1) can be consideredas an indicator of aggression of the antagonistic sides, - an indicator oftheir readiness for peace, listening to peace-making appeals of internationalorganizations, - an indicator of peace-making activity of the internationalorganizations. As it will further be shown, depending onwhich is more – an aggression index, or indexes of readiness for peaceand peace-making activity, character and development of informationwarfare essentially varies.Influence on a course of information warfare from international organizationsis more effective, even at a big index of aggression from theantagonistic sides if their action has a preventive character ( N30>0 ).Let‘s study a course of information warfare for various D .121

IV.I. D = 2 8> 0 .Let's consider different cases of starting position of the antagonisticsides.IV.I.1. ( N10 N20).In case when international organizations havetaken preventive measures, and the antagonistic sides have begun informationwarfare under equal starting conditions, the influence of internationalorganizations on the first and second side is expressed equally as -N1( t) N2(t).From (3.2) - (3.4) we will obtain:N 3(t ) = 210 2NDN30e 1t2N10 1N30 te 2D(4.1.1)2N10 2NN1(t)= D230e 1t2N10 1N30 t e 2 D1(4.1.2)2N10 2NN2(t)= D230e 1t2N10 1N30 t e 2 D1(4.1.3)Functions N1(t ), N2(t ) in a point t =0 are equal and positiveN10> 0 , and at big t become negative, ifN30>2N210(4.1.4)Indeed, according to (4.1.2), tN1(t )= e 2 D 2N10 2N301 2e t22N10 N1130(4.1.5)122

2N10 2N30The sign N1(t)at big t defines the factorbeforee 1 2t, it is negative when 2N10 2N30< 0 i.e. it is fair (4.1.4). In2this case, the continuous function of N1( t ) changes the sign on a semiinterval[ 0 ,+ ), i.e. in some point t of this semi-interval is has zero. tis from the equation which it is obtained, having equated (4.1.5) to zeroand has the following appearance: 1t =D 2Nln 2N1010 N1 N23030 21(4.1.6)(4.1.6) it makes sense, since when it is fair (4.1.4), thus 2N 2N1010 N1 N230302>1 (4.1.7)1i.e. 1t =D 2Nln 2N1010 N1 N23030 1 ln 2(4.1.8)Analogical research we will establish that under condition of(4.1.4), the third side finishes actions ( N3(t ) comes to zero), however abit later 1t =D 2Nln 2N1010 N1 N23030(4.1.9)Thus, if international organizations match a measure preventiveN30, for the accomplishment of the condition (4.1.4) all the three123

functions N1(t ), N2(t ), N3(t ) come to zero – i.e. information warfarecomes to an end. If the condition (4.1.4) is not carried out, informationwarfare proceeds, and moreover, amplifies, as from (4.1.1) - (4.1.3) follows:N1(t ) , N2(t ) , N3(t ) , when t .IV.I.2. ( N10> N20). If the antagonistic sides have different startingposition and at the same time starting conditions of the first side are morethan of the second one, the function N3( t ) comes to zero. Really, let‘srewrite (3.2) as follows2N (t )= 12t N N N e N N 3e tD10202301020 N130(4.1.10) 1 2teIt is clear that at big t , the sign N ( ) defines factor before, in particular33 tN (t ) is negative, if N N N 0 i.e.10 20 2 30N30 N210 20 (4.1.11) NFor N30, obeying (4.1.11) N3(t ) changes the sign at a semiinterval[ 0 , + ) from positive N3(0 ) = N 300 to negative. Thus, continuousfunction of N3(t ) has zero in some pointt of this semiintervalwhich is the solution of the equation N3(t ) =0 N10 N20 2N30 1 2te N10N N (4.1.12)= 20 1 30From (4.1.12) we have124

1 N10 N20t = Dln N30N 10N202N301(4.1.13)Let's notice that the expression under the logarithm is more thanunit when, N30obeys the condition (4.2.1.11) since 1 2 N N1010 N N2020 N1 N23030 1Thus, N2(t ) equals zero at point t1 of the semi-interval [ 0 , + ).Really, continuous function N2(t ) (3.4) changes the sign at semiinterval[ 0 , + ): N2(0 ) = N 200 , and for accordingly big t it is negativeN2(t )= e t N (220N 10 N N+ 10 20 D2 N230 N te 2 N 10120D N130 te 1 ), (4.1.14)as, in this case, the sign N2(t ) defines the first composedN20 N 10of the second factor of expression (4.1.14), which is negative,2and the other members of this factor become any smaller in absolute value,for accordingly big t . Thus, t1 is the solution of the transcendentalequation NN10220N 10 N120D N+ N13010 N220D N230te 2 te 1 =0 (4.1.15)125

NF =t210N 20e 1t [( N 30 ( N10+ N 20)) Dt D 2N 30e ( ( N 10+ N 20))] (4.1.18) 1If the functionat the same point,i.e. functionF t crosses an abscissa and N1(t ) will also cross itF t has zero, and owing to (4.1.17) there will be zerosof the function N1(t ) at the same points. So, we can investigateand then use these results for N1(t ).N1( 0 )= F 0= N10> 0 .Let's enter designations:F tA N 30 ( N10N20)>0 (4.1.19) 2B N 30 ( N10+ N20)>0 (4.1.20) 1Let's notice that B > A, then with the account of (4.1.18), (4.1.19),F t will be copied in the following way:(4.1.20), F t=N210N 20e 1t [ ADte B] (4.1.21)DLet's find stationary pointsF t from the equation tF / =0F / t =N210N 201e 1tDt A e =0 (4.1.22)Let's divide (4.1.22) into eDt, we will obtain127

whenceN210N 20 1e 2t- A= 0 ,e 2t = N N 201102A(4.1.23)The equation (4.1.23) at t >0 has the solution when the right side ismore than unit, and it will occur when2 ( N 30 ( N10+ N 20))>(10 2N20N ) 1,i.e.N > N 10 N20302N+210N 201=N (4.1.24)30wayAt (4.1.24), the solution (4.1.23) will be written in the followingt0=1 2Aln2N10 N 20 1(4.1.25)will presentNow let‘s investigateF / tF / t in the neighborhood of the point t 0. Wein the following waytF / = eDtN210N 2012e t2A ( N N11020) (4.1.26)128All factors (4.1.26), are positive, except2e t2A ( N N11020. And)this last expression sign-variable in the neighborhood of t 0: to the left oft0it is negative, and to the right of t 0- positive, in t 0- it is equal to zero,i.e. t 0is a point of local minimum.

Lemma 1. There are such values of N30, for which functionF t in a point t 0of minimum is not positive:The proof. Let‘s introduce the designationK N N 20 1102.Then at big N30- from (4.1.21) we can obtainFt 0 0 .F t 0=N210N 20 1KA 2 D[ AKA 2 BD ] 0 (4.1.27)obtainReally, when30N >> N 10 N20A N30, B N30.2, from (4.1.19), (4.1.20) we willOwing to (4.1.21)F t 0=D 230NN 1],210N 20F t 0 looks like: 1 1K 2 N 230 DN30[ 2DK D30], D D1DF t 0= N 230K 22 [ K N 21DF t 0= N 230K 2 [ KN30 212 1D ] 0 , D D1D129

As in the right part of the last parity all factors, except the last arepositive, and last – negative for big N30. W.D.P.Thus (4.1.27) it is fair, whenN30 N 30, thus there is equalityF t 0, N 30 N 30 = 0 . We will also notice that N 30 N30.As F t in a point t 0is not positive owing to lemma 1 it means thatit is either equal to zero in this point , or is negative. On the other hand itmeans that N1( t ) equals to zero either at a point t0, or at some pointt1( t 1< t 0).Thus, the first side, as well as the second and the third ones, finishinformation warfare.IV.I.3. ( N10 N 20) .If at the antagonistic sides have different startingconditions, and the starting position of the second one are more, thefirst and the second sides change roles and we get symmetric results forN1(t ) and N2(t ). In this case the analogue N30 will be designatedthrough N30 .IV.II. D 2 8=0 .In this case the aggression index is still high.IV.II.1. ( N10 N 20) . In case when international organizationshave taken preventive measures, and the antagonistic sides have beguninformation warfare under equal starting conditions influence of internationalorganizations on the first and the second sides is the same and takinginto account (3.5) - (3.7) involvement of each side into informationwarfare will be described as following:N1(t )= N 2(t )=[ N10+( N10 N30)t ]2e 2 t(4.2.1)N3(t )=[ N30+(2 N 102 tN30)t ] e 2(4.2.2)All the three functions (4.2.1), (4.2.2) come to zero if the factor beforet is negative, which is reached at (4.2.1), when130

N > 302 N (4.2.3)10and at (4.2.2), whenN304> N10(4.2.4)Let's notice that the right sides (4.2.3) and (4.2.4) - are equal whenD = 0 . Functions N1(t ) and N2(t ) come to zero at some point t , where N10N10t ==, (4.2.5)N 30102 N ( N30 N10)2and N( ) comes to zero in t3 twheret =N10N30 2N210=( N230N30 N210)(4.2.6)It is obvious that in case of (4.2.3) or (4.2.4) t4> D =0 , N10= 20N и N30N10, all the three sides finish informationwarfare.Thus, if the bilateral inequality is carried out> t . Thus, whenN 10N >102N> 30, (4.2.7)then the antagonistic sides make their attacks active at first, butthen, after definite time, under the pressure of international organizations,reduce, and then and stop information warfare at all.If the inequality (4.2.7) is not carried out in the left part, i.e. there isN30> N10,131

then, the antagonistic sides under the pressure of international organizationsreduce information influence from the very beginning andcease information warfare once and for all. If the inequality (4.2.7) is notcarried out in the right part, i.e.0 N30102N ,then according to (4.2.1), (4.2.2), information warfare developsN ( ) , N ( ) , N ( ) , when t .1t2t3 tIV.II.2. ( N10> N 20).If at the antagonistic sides have different startingposition and at the same time the starting conditions of the first sideare more than of the second one, under certain conditions all the threerequired functions come to zero.For N3(t ) coming to zero is reached proceeding from (3.5) whenthe factor before t will be negative, i.e. in case ofN > N 10 N 20304, (4.2.8)thus N3(t ) comes to zero at a point tt =N230N30 N10 N20(4.2.9)As for the function N1(t ) it becomes any bigger for bigt . With theaccount of N10> N 20, it is well visible from the following record of N1(t )NN10 N N10 N20 2 t) { +2 ( N10 N20 N30)t 2 2 201( te 2} e t(4.2.10)132

Let's write down the equation N1(t ) = 0 , taking into account(4.2.10) and division intoe 2 t, then( N ) 2010N te 2 = N + 2 N N N t10N 20Let's enter a designationF(t) N )(10N20as1 tN ( ) = 21t 3010 20 (4.2.11)2 3010 20 (4.2.12)2 te 2 +( N 10+ N 20) - 2 N N N te 2 F t ,therefore zeroequalityF t will be the zero of N1(t ), and there is alsoF / t = 2F 0= 2 N 10> 0 .Let‘s find stationary points( N ) 2010Ne 2 twhence we will obtain thatF t:/ F 0= 2 N30 N 10 -2 N30 N10 N20 , (4.2.13 ) 2 302 N (4.2.14)10F t begins from zero. TheAt performance (4.2.14), the decreasedecrease interval F is calculated from the following inequalityt133

1

The solution to inequality (4.2.17) concerning N 30, we will searchas followsn N410 N20t= N30and match such n that F 0 . ThenF t= n ( N10+)n 1N10 NN20( N 10+ N 20)( lnN Nn ( n -1) n 1 N 10 N20ln[ N N1020n 1) 20] ( N 10+ N 20)[ n ( 1020n 1) ln 1= ( N 10+ N 20)n ] 0 (4.2.18)The inequality (4.2.18) for n >1 is just, ifn ( Let's designaten 1) ln 1n 0ThenG(n) n ( n 1) ln 1n .G(n)1+, при n 1 ;/G ( n)< 0 , при n >2;/G ( n)> 0 , когда 1 n 2 ;/G ( n)=0, когда n =2,i.е. n =2 is the point of a local maximum for function G (n).It‘s clear that there is such point asG ( n ) = 0 ,n , for whichas G ( 4) 0 , G ( 5) 0 , that 4 n 5 , more preciselyn 4,5911.135

t0F .Thus, when,30The lemma is proved.4N n N N 1020i.e.n n , G ( n) 0 andIt is obvious that as there is (4.2.17), the function N1(t ) has zero.As for N2(t), it is for any N30and big t aspires to , and consequentlyits zerot, is the solution of the following transcendental equation NN20 NN2( t)2 N20 2 ( N102 10 N2010te + N230) tte 2 =0 .IV.II.3. ( N10< N 20). If at the antagonistic sides have different startingconditions and the starting position of the second side is more, thefirst and the second sides change roles and there are symmetric results for1N1(t ) and N2(t ). N30 will be the analogue for N30.IV.III. D = 2 8< 0 .IV.III.1. ( N10= N 20). In case when the antagonistic sides have beguninformation warfare under equal starting conditions the influence ofinternational organizations on the first and second sides is productive. Inthis case, taking into account (3.8) - (3.10), functions N ( ) , N ( ) ,N ( ) , will become2t3 t1tN3(t )=N2304N10N82302e 2 tsin 282t 1, (4.3.1)1= arctg2N308.4NN1030136

N1(t )= N 2(t )=2N2304N10N)82302e 2 t2 8sin t 1, (4.3.2) 2 = arctg28.point tAccording to (4.3.2), functions N1(t ) and N2(t ) come to zero att =2 1 D, ( 4.3.3)and N ( ) in t3 t, which is more than t 2t = 1 D(4.3.4)and information warfare stops.IV.III.2. ( N10> N 20). If at the antagonistic sides have differentstarting positions and thus, the starting conditions of the first side aremore than of the second one the function N3(t ) comes to zero at pointt1 according to (3.8)2t = 1 D,and N2(t ) , at t , therefore the function N2(t ) haszero at pointt1, which is from the equationN 2( t 1) =0,where - N ( ) , looks like (3.10).2t137

As for N1(t ), N1(t ) , at t , but at selection N 30, thefunction N1( t ) can come to zero.Indeed, the parityN1(t) 0owing to (3.9), gives the following equationsin 2 t8 N 20N10t e22/ 2 22N N2210 20 30N30(4.3.5)28N)According to (4.3.5), the condition of existence of zero for N1(t ),leads to the following inequality0 2N230N10 N2N1020e Nt220) N28302 2 ,e 2 t2 2N230(2( NN1010 N N202028) N30)2 2 (2( N10 N20)N2 N3022 28t ln N10 N2030)2(4.3.6)138

The solution (4.3.5) concerning t , which satisfies (4.3.6), alwaysexists for big N30.Thus the function N1( t ) has a zero, hence the first side finishes informationwarfare too.IV.III.3.( N10< N20). If at the antagonistic sides have differentstarting conditions and the starting position of the second side is more thefirst and second sides change roles and there are symmetric results forN1(t ) and N2(t ).Thus, the analysis of the received results shows that functionsN ( ) , N ( ) and N ( ) come to zero at selection of corresponding1t2t3 tN30and (with the help of prevention and increase in peace-makingactivity). If there are no preventive activities from the international organizations,then coming to zero of all the three required functions ispossible only in case, when D

наукових праць Інституту problems реєстрац і ї інформац і їNAN Ukrain. – К: ІПР І, 2002. – Vip. 2.7. J. R. Cares. An Information Age Combat model. Alidade ConsultingTechnical Paper, March, 2001.8. J. Jormakka, J. V. E. Molsa. Modelling Information Warfare as aGame. – Journal of Information Warfare. 2005, 4 (2): 12-25.9. T. Chilachava, N. Kereselidze. About one mathematical model ofthe information warfare. – Fifth congress of mathematicians ofGeorgia. Abstracts of contributed talks. Batumi/Kutaisi, October 9-12, 2009, p. 85.10. T. Chilachava, T. Dzidziguri. Mathematical modelling. Tbilisi,2008, 440 p. (in Georgian).11. А. Samarskii, A. Mikhailov. Mathematical Modeling: Ideas. Methods.Examples. Moscow, 1997, 320 p.Temur CilaCava, nugzar kereseliZeprevenciuli sainformacio omis uwyveti wrfivimaTematikuri modelinaSromSi agebulia or antagonistur saxelmwifoTa Sorissainformacio omis wrfivi uwyveti maTematikuri modeli,imis gaTvaliswinebiT, rom arsebobs mesame, mSvidobismyofelimxare, romelic moqmedebs prevenciulad. modeli iTvaliswinebsrogorc Tanabari (`iaki – daTvi~), aseve mkveTradgansxvavebuli (`kravi – mgeli~) siZlieris mqone gaerTianebebisdapirispirebas.saZebn funqciebad aRebulia drois mocemul momentSi informaciisis raodenoba, romelsac avrcelebs TviToeulimxare, dasaxuli miznis, arCeuli strategiis misaRwevad. modelSi,orive mxare erTi da igive tempiT awarmoebs sainformaciooms da reagirebs saerTaSoriso organizaciebis mowodebebze.Tavis mxriv, mesame mxare Tanabrad reagirebs antagonisturmxareTa sainformacio Setevebis intensivobaze.miRebulia pirveli rigis wrfivi mudmivkoeficientiandiferencialur gantolebaTa sistemisaTvis koSis amocanis140

zusti amonaxsni. analizma gamoavlina modelis konstantebsada sawyis pirobebs Soris is Tanafardobebi, romlis drosac: antagonisturi mxareebi, miuxedavad prevenciuladmoqmedi mesame mxaris mowodebisa, aZliereben sainformacioSetevebs; erT-erTi antagonisturi mxare, mesame mxaris zemoqmedebiT,garkveuli drois Semdeg, asrulebs sainformaciooms (Sesabamisi amonaxsnis nulze gasvla),maSin roca meore, mas aZlierebs; orive antagonisturi mxare, mas mere rac miaRwevsaqtiurobis maqsimums, mesame mxaris zemoqmedebiT amcirebssainformacio Setevebs da sasrulo droisSemdeg Sewyveten sainformacio oms (amonaxsnTa nulzegasvla).pirvel SemTxvevaSi, unda velodoT sainformacio omiscxel fazaSi transformacias, meoreSi es transformacianaklebad savaraudoa, mesameSi ki – gamoricxuli.SemoTavazebul models Teoriuli interesis garda gaaCniapraqtikuli mniSvnelobac. igi gvaZlevs saSualebas, sainformacioSetevebis ukve sawyis etapze, dakvirvebebisa daanalizis saSualebiT, davadginoT mxareTa WeSmariti ganzraxvebida sainformacio omis ganviTarebis xasiaTi.141

soxumis saxelmwifo universitetis Sromebit. VII, 2009maTematikisa da kompiuterul mecnierebaTa seria142HAMLET MELADZEON TWO-LAYER FACTORIZED DIFFERENCE SCHEMESFOR SYSTEM OF DIFFERENTIAL EQUATIONSWITH PARTIAL DERIVATIVESOF PARABOLIC TYPEAbstract. The mixed problem with first sort boundary conditionsfor systems of equations of parabolic type is consideredu Lu tf ,( 1) (2) ( n)where L - strong elliptic operator with variable coefficients, containingthe mixed derivatives, u u, u , ,u ,(1)(2)( n)f f , f , ,f - n -dimensional vectors. The two-layer factorizedscheme is constructed. The received algorithms can be effectively realizedfor multiprocessing computing systems. For solution of difference0 (1)scheme the aprioristic estimation on layer in norm of mesh space W 2 isreceived, on which basis convergence of solution of difference scheme tothe solution of an initial problem is proved.2000Mathematics Subject Classification: 65M12, 65M15,65M55.Key words and phrases: difference scheme, two-layer factorizedscheme, parabolis type.1. IntroductionAs it is known, at the mathematical formulation of many scientificand technical problems there are arising the differential equations withpartial derivatives of parabolic type [1,2]. Construction and investigationof the difference schemes for linear system of equations of parabolic typewith the mixed derivatives is a subject of the present article.The solution of such problems is one of challenges in numericalmathematics and demands, as a rule, a lot of computing resources. One of

the ways of reduction the time of the solution of such problems is to usethe parallel computations on multiprocessing computing systems.Usage of parallel computing systems demands construction of algorithmsin the form accessible to parallel processing of the data. In computationalmathematics enough considerable quantity of such methods isdeveloped. One of such methods is the method of decomposition of challengingtasks on more simple tasks moving implementations on parallelprocessors.At construction of the difference schemes with given properties themethod of the regularization offered by A.A.Samarskim [3,4], possessesthe big efficiency. In the present paper this method is used for constructionof the difference schemes, which can be easily implemented on parallelcomputing systems.According to the regularization method: At first, the initial difference scheme, meeting requirements ofapproximation of the given order is constructed. the regularizator, that is operator, providing absolute stabilityof the difference scheme, is selected. By means of the factorization method of the operator on theupper layer passage to the economic stable difference schemeis made.Thus, the basic attention is given to following problems:1. Reception economic difference scheme at minimal requirementson the spatial operator. It is required only that the spatialoperator will be strong elliptic.2. Absolute stability at any ∞ ( - a step on time) and h ∞(h - a step in a direction on x , = 1,2,…,p).3. Application of one-dimensional double-sweep algorithm forsolution of received difference equations. These algorithms canbe used for parallel computing systems.4. The convergence proof of difference schemes at smallersmoothness of the solution of initial system of the differentialequations that is reached by refusal of an estimation of localapproximation.To the problem of construction of difference splitting schemes theextensive literature is devoted. We will note some monograph [3,5,6]. It ispossible to find the extensive list of works in this direction in these monographs.Let's note also some works on parallel algorithms of the solution ofparabolic equations [8-12]. It is natural, that this list is incomplete.143

1. Let, D x x, x2,,x ,2. The statement of the problemp 1 p 0 x , 1,2, ,p isopen p–dimensional parallelepiped in p–dimensional Euclid space E p . -surface of parallelepiped D p and D D -- is the D p area‘s closure.GT Dp 0,T with basis E p . G D 0T p -- cylinder in (p +1)-dimensional Euclid spaceT p ,p -- is the closure of the area G T and (x, t) –any point in TC ,k G T is the set of continuous in G T functions,which have in G T continuous derivatives to an order (inclusive) on xand to an order k (inclusive) on t.In cylinder G T we considered the following problem for system ofequations of parabolic type with mixed derivatives:Let‘s find continuous in G solution of system of the equationsG . T( i)utn p i,j K ,j 1 , 1 x ( j)du dx satisfying to boundary conditionsx,t i f x,t , i 1,2,,n ,(1.1)u( i)x , t g x, t, when x 0,T( i)and to initial conditionsu( ix0 u x( i ))0 , i 1,2,,n , (1.2), , when x D p , i 1,2,,n . (1.3)Let's assume that following conditions are satisfied:I. Coefficients K i , j x, t , are Lipschitz-continuous with respect to t.Let's assume also that matrixes from coefficients of the spatial operatori,j nK , 1,2,, pK are symmetric and that the problem, i,j1(1.1)-(1.3) has the unique solution u ux,t , continuous in G T and differentiatednecessary number of times.144

II. Spatial operator of system of equations (1.1) is strong-elliptic, sot 0,Tthe following inequality is truethat for any pn p np ni 2 1 1ij j ii K , (1.4)1 1 i1, i,j2 1 i12iwhere -- any real numbers, 1 and 2 -- positive constants.3. The two layer difference scheme for problem (1.1)-(1.3)1°. In cylinder G T let‘s introduce the difference mesh. Constructionof spatial-time mesh in G T is carried out by means of onedimensionalmeshes on intervals 0, , 1,2, ,p 0 , T : and x i i h , i 0,1, ,N , N h , 1,2,,p . x i i h , i 1,,N 1,N h , 1,2,,p .The difference mesh in parallelepipedD p is constructing in followingway:ph x i1h1 , ,i php D p , i 0,1, ,N, 1 ph 1 -- set of internal points of difference grid in D p .Denote by x Nh h h \ h the set of knots, belonging to theboundary , which are named the boundary knots. On interval 0 , Tlet‘sintroduce the difference mesh j, j 0,1, , k,k T .t j.145

Let h hbe the partial-time mesh of internal knots of cylinderG T and x, t j is any knot of h. h h be the set ofknots of difference grid on lateral surface of the cylinder G and x , is knot, belonging toh h hh. Then his the partial-time mesh in G T .2. We will deal with functions of discrete argument, defined inknots of difference mesh and named mesh functions.We will consider the set of net vector-functions1 2 n j y y, y , ,y , defined on h. For them the symbol ymeans value of a mesh vector-function in knots x, t j .We will use designations:1 x,x , x h , x , x , 1, 1 1x , p1 1y y x h 1 y x , , y x ,1n1p2 2 h . 1Let's enter the difference relations:Tt jyxyx 1 1y yy y , yx ,hh1yy y1 2 yx xx 2hp0 0 0 0 y Ay,A A. 1 0y ,, besides the speci-For the mesh vector-functions defined on fied designations, we will use still the following:146x,t ,y yx,t ,y yx,t ,,yˆ 2y y ,y ˆ y j1 jj1y yyt ,ytyˆ y ,yttyhyˆ y .22t

3. Set of the mesh functions, defined on the mesh p , we will designateby H, and its subset consisting of mesh functions, becoming zeroon p - through H .2On set H we will enter the scalar productni iyv y, v Where, ,i1i ii iyv y xvxxp, H , H h .p 1It is obvious, that this scalar product induces the normy y,y0 .Denote by p p , whereat which x l . -- set of knots of border p ,-- set of knots of border p , at whichx l , x land etc. Similarly, we will designate throughof knots of border , at which x 0 and etc.Let's enter the designation required to us:npp p 1 ,2, , pi i i i i iy,v y, v ,y, v y xvx H.i1xpLet's enter the mesh space of Sobolev 2 functions of setequalityu2 -- set. Scalar product also is(1)W p, consisting theH . The norm in this space is set by means of following21 )W2p 1202021,u . u(1 u x , u x x . 147

148 (1)Let's notice that W 2 -- is full Hilbert space concerning this norm.4. Let A -- the linear self-conjugate operator and A>0, operating inHilbert space H . We will define new scalar producty, vA Ay,v . Owing to such definition H turns in new Hilbert space, which wewill denote through H A . Energetic space H A consists of the same elements,as the space H . Norm in H A we will denote by the symbolu 2 AAu,u . :A5. For problem (1.1)-(1.4) let‘s consider two-layer differenceC 4,2 of solutionsscheme, which order of approximation in a class 2(1.1) is a value of the order O h :E Ryt Ay f xt,x,th(1)G T, , (2.1)(2)where y y, y , ,y ,f f, f , ,f n -dimensional vectors, and operator A is defined by equality1Ay 21 p , ( n)Kx,ty Kx,t(1)(2)( n) y x , (2.2) xxR – operator-regularizator, which choice provides absolute stabilityof difference scheme (2.1).The vector function y x, tsatisfies the following boundary and initialconditions:x, t gx,t,if x, yx,0 u xy h 0 . (2.3)In work [4] is proved the fairness of following relations00A y,y Ay,yAy,y1 2 , or for any vector y H 0 .x001AA 2 A , (2.4)Hence, operators A and A 0 in H are energetically equivalent withconstants 1 and 2 .On the basis of the results §2 from [7] and inequalities (2.4) it ispossible to conclude, that the difference scheme (2.1) with regularizator

0R A , at 0.52 , is absolutely stabile in space H (stabile under theinitial data and the right part).6. Let's pass to construction economic factorized two-layer differencescheme, considering the scheme (2.1) as initial. AsR A0pR 10, R A , 0.52, (2.5)then, replacing the operator E R E R with the facto-prized operator E R , we receive the two-layer factorized differenceschemep 1 1E R y Ay f x,ttp 1 , (2.6)which in a canonical form can be written down as follows~E Ryt Ay f x,t,x,th, (2.7)whereR~ R Qp, andQpR R p2 R R R R . (2.8)p 1~It‘s evident, that in H the inequality R R is valid, therefore thefactorized difference scheme (2.6) or (2.7) will be absolutely stabile underthe initial data and the right part.7. We will notice, that for difference schemes (2.6) it‘s possible toconstruct the effective computing algorithm by means of decompositionpof the operator E R 1, which is easily implemented on the parallelcomputing system.The difference scheme (2.6) we will rewrite as follows149

p 1By t F,where0 0B E R E , y yxx , F f Ay . Such record of difference schemes gives the chance to construct thefollowing computing algorithmB1v(1) F,Bv() v(1)j1jy y v(p), j 1,2,3. 2,3, ,p,v(p) yt,(2.9)For definition of functions v ( ) at x h it is necessary to use thefollowing formulas:150v( ) B 1 Bp gttx,t при x [ 0, T ] . (2.10)This algorithm is especially convenient for solution of consideredproblem in case, when the function g x, tdoes not depend from t. In thiscase we receive homogeneous boundary conditions for functions v ( ) 1,2,,p.So, the initial problem (2.6) breaks into a number of subtasks, eachof which is possible to solve, using parallel computing algorithms [12,13].4. Convergence of the difference scheme1. The initial difference scheme (2.1) has an error of approxima-2tion of the order O h equations (1.1) belongs to the class C , 2 , if the solution of system of the differential4 G T . An approximation error offactorized scheme (2.6) we will present in such kind 0 1 , where -- an error of approximation of initial difference scheme (2.1), which02order in the considered class is equal to O h , and 1 Q p ut.2From here it is visible that at p>2 requirement 1 O h imposesadditional restrictions on smoothness of the solution of the system of equations(1.1). However, the aprioristic estimation and the theorem of2

convergence in norm of space (1)2W can be received at weaker restric-2tions, than the condition of local approximation of order O h .2. To proof the convergence of factorized scheme (2.6) we willconsider the vector function of error z y u, x,th, where u -- thesolution of the initial problem (1.1)-(1.3), and y - the solution of the problem(2.6), (2.3).For the net function z we will receive the following problem~E Rzzt Az ,x,0 0, zx,t 0, xh(3.1)where 0 1 -- the error of approximation of the differencescheme (2.6).For the further statement it is useful to enter following denotations:0*T0*A T or Au TTu u x x , Tu u x , Tu u x , 1p21p21p2222y T , , , 1 2 .qTy yT2,0 qTTy y T T T3,0 qp yppp,p0Theorem 3.1. Let, conditions I-II are satisfied, regularizator R is0defined by equality R A and 0.52 . Then for the solution of theproblem (3.1) on any sequence of meshes hthe aprioristic estimationis fair*j1z M1max 0A 0t'tj12s22t' M 2 max utt' ,0p0t'tj s2q s, p where M 1 , M 2 -- the constant number which is not dependent onthe mesh.Considering the estimationj1 j1 zA 1z , where - the positiveconstant, which is not dependent on the mesh, on the basis of the151

theorem 3.1 it is easily possible to receive the aprioristic estimation for (1)the solution of the problem (3.1) in space W 2 .Theorem 3.2. Let, conditions of the theorem 3.1 are satisfied.Then, for solution of the problem (3.1) on any sequence of meshes hthe aprioristic estimation is fair:j1z M1max 01 0t'tj122 2 ' 2st M max utt' ,0p0t'tj s2q s, p where M 1 , M 2 -- positive constants, which are not depend onmesh.Theorem 3.3. Let, conditions of the theorem 3.1 are satisfied. Be-4,2u C , whensides, let solution of system of the equations (1.1) G T p,2p 4 , and u C G when p 2problem (2.6), (2.3) converge in normT. Then the solutions of difference (1)2W 2 with speed O h solution of problem (1.1)-(1.3) on any sequence of grids to the h.REFERENCES1. A. A. Samarskii, A. N. Tikhonov. Equations of Mathematical Physics.Publisher: Dover Pubns, 1990, 766 p.2. A. P. Mikhailov, A. A. Samarskii. Principles of MathematicalModeling: Ideas, Methods, Examples. Publisher: CRC Pr-ILIC,Numerical insights, 3, 2002.3. A. A. Samarskii. The Theory of Difference Schemes. Publisher:Narcel Dekker inc., 2001, 761 p.4. А. А. Самарский. О регуляризации разностных схем. Журналвычислительной математики и математической физики,1967, 7, №1. pp. 62-93.5. G. I. Marchuk. Numerical Methods and Applications. Publ.: CRCPress, 1994, 288 p.152

6. A. A. Samarskii, P. N. Vabishevich, P. P. Matus. Difference SchemesWith Operator Factors. Publ.: Kluwer Academic Pub., 2002,396 c.7. A. A. Samarskii. The Classes of stabilized schemes. J. ComputationalMathematics and Mathematical Physics , 1967, 7, №5. pp.1096-1133.8. Wenrui Hao, Shaohong Zhu. Parallel iterative methods for parabolicequations. – International Journal of Computer Mathematics,Volume 86, Issue 3 March 2009, pages 431-440.9. R. K. Mohanty. D. J. Evans. Alternating group explicit parallelalgorithms for the solution of one-space dimensional non-linearsingular parabolic equations using an O(k 2 + h 4 ) difference method.– International Journal of Computer Mathematics, Volume 82,Issue 2 February 2005 , pages 203 – 218.10. A. Q. M. Khaliq, E. H. Twizell, D. A. Voss. On parallel algorithmsfor semidiscretized parabolic partial differential equations basedon subdiagonal Padé approximations. – Numerical Methods forPartial Differential Equations, Volume 9 Issue 2 (March 1993).Pages 107 – 116 (Published Online: 20 Jun 2005).11. M. S. A. Taj; E. H. Twizell. A family of third-order parallel splittingmethods for parabolic partial differential equations. – InternationalJournal of Computer Mathematics, 1029-0265, Volume67, Issue 3, 1998, Pages 411 – 433.12. G. Horton, S. Vandewalle, and P. Worley. An Algorithm withPolylog Parallel Complexity for Solving Parabolic Partial DifferentialEquations. – SIAM Journal on Scientific Computing. Volume16, Issue 3, pp. 531-541 (May 1995).13. J. Verkaik, H. X. Lin. A class of novel parallel algorithms for thesolution of tridiagonal systems. – Parallel Computing, Volume 31,Issue 6, June 2005, Pages 563-587.14. Xian-He Sun, Hong Zhang Sun, Lionel M. Ni. Parallel algorithmsfor solution of tridiagonal systems on multicomputers. –Proceedings of the 3rd international conference on Supercomputing,Crete, Greece, 1989. Publisher: ACM, ISBN:0-89791-309-4,pp: 303 – 312.15. Paul F. Fischer, Franco P. Preparata, John E. Savage. Generalizedscans and tri-diagonal systems. – Book Series “Lecture Notesin Computer Science”, Publisher: Springer Berlin-Heidelberg.Volume 900/1995, Book ―STACS 95‖, pp. 168-180.153

hamlet melaZeorSriani faqtorizebuli sxvaobiani sqemebiparaboluri tipis kerZowarmoebulianigantolebaTa mravalganzomilebianisistemisaTvisnaSromSi ganxilulia Sereuli sasazRvro amocana pirveligvaris sasazRvro pirobebiT paraboluri tipis kerZowarmoebulianigantolebaTa mravalganzomilebiani sistemisaTvis:u Lu tf ,sadac L - cvalebadkoeficientebiani, Zlierad elifsuri( 1) (2) ( n)(1) (2) ( n) f, f , ,f operatoria, xolo u u, u , ,u , f -warmoadgens n -ganzomilebian veqtorebs.0 (1)naSromSi agebulia absoluturad mdgradi sxvaobianifaqtorizebuli sqemebi. am sxvaobiani sqemebis agebis drosgamoyenebulia regularizaciis meTodi, romelic damuSavebuliiyo a. samarskis mier. miRebuli algoriTmebi SeiZlebaefeqturad iqnes realizebuli mravalprocesorebian gamoTvliTsistemebze. energetikul utolobaTa meTodis gamoyenebiTbadur W 2 sivrceSi miRebulia aprioruli Sefasebebi,romelTa safuZvelzec damtkicebulia Teorema sxvaobiani sqemiskrebadobis Sesaxeb.154

soxumis saxelmwifo universitetis Sromebit. VII, 2009maTematikisa da kompiuterul mecnierebaTa seriaTINA GOROZIA, TINATIN DAVITASHVILIBOUNDARY VALUE PROBLEM FOR ONE CLASSOF ORDINARY DIFFERENTIAL EQUATIONS,GIVEN ON GRAPHSAbstract. In the present paper the boundary value problem for oneclass of the ordinary differential equations, given on graphs, and the differencemethod for solution of this problem is investigated. The explicitsolutions of the differential problem in the form of quadratures, and alsoexact solution of the difference scheme are constructed.Computer Rev.: j.1.5, j.1.7Key words and phrases: boundary value problem, differential equationson graphs, analytical solutions, difference schemes.1. IntroductionIn papers [1,2] the mathematical model of an electropower systemwhich represents to boundary value problem for the ordinary differentialequations set on graphs is considered. The problem in view correctness isinvestigated. The correspondent finite-difference scheme is constructedand investigated. The double-sweep method type formulas for finding thesolution of the finite-difference scheme are offered.Let's mark, that the boundary value problems on graphs are not investigatedtheoreticaly to the full in the scientific literature. See, for example[3 −7] and the literature mentioned there.In the present paper the boundary value problem for one class ofthe ordinary differential equations set on graphs is considered. The explicitsolution of this problem in the form of quadratures is constructed.Further the difference method for solution of this problem is considered.The exact solution of the difference scheme, which is the difference analogueof solution of the differential problem is constructed. We will markthat it‘s important to have exact solution of this problem for testing, andalso for an estimation of accuracy and efficiency of the difference meth-155

ods used for numerical solution of the ordinary differential equations,given on graphs.3. Analytical solutions of the differential equationsgiven on graphsFor simplicity of presentation we will consider the graph that consistsof two ribs. Tops of the graph are the points a 1 , a 2 , and the ribs ofthe graph are a a ) , a a ) .0 1 ( 10 2 ( 2a 1a 0a 2Let us state the following problem: find twice continuously differentiatedfunctions u x 1 , v x 2 , where x − the local coordinate along, α=1,2, which satisfy:1) the differential equations2d udx21 , 0 b f 1( x 1)2d vx1 ,1, f 2( x 2)2dx2 , 0 b x , (1)2,2where − is length of the rib , and f( x ), =1,2, − are given,continuous on 0 ,b functions, 1, 2 ;2) the boundary conditionsu ( b 1, ( b 21)v 2) , (2)1563) and conditions of conjunctiondu dvu( 0) v(0), 0 . (3)dx dx1 x 012 x 02

It‘s easy to construct the solution of the problem (1)-(3) in the formof quadratures. We will present functions u x 1 , v x 2 in the form ofsum of two functions:u(x ) u1v(x ) v2where(1)(1)( x ) u1( x ) v2(2)(2)( x1), (4)( x )2d2u(1)dx( x211) 0,x12(1)20,b , 0, x 0,b 1dvdx( x22)22(1)(1)u ( b1) 1,2 2v ( b ) , (5)u(1)(0) v(1)(0),dudx(1)1x 01dvdx(2)2x 02 0and2 (2)2 (2)d u ( x1)d v ( x2) f1(x1), x1 0,b1, f2( x2), x2 0,b2.dxdx2122(2)(2)u ( b 1) 0 , v ( b 2) 0, (6)(2)(2) (2) du dvu (0) v (0), 0 .dx dx1x 01(2)2x 02The solutions of the problem (5) are the linear functions:157

u(1)x1 b2b1 x1( x1) 1 2, (7)b b b b1212vb x x b . (8)(1)2 2 2 1( x2) 1 2b1 b2b1 b2For construction the solution of the problem (6) let‘s integrate thefirst equation:u(2)(2)( t) u(0) f1(s)ds .t0Integrating once again the previous relation, we will receiveu(2)1tx u x u f s ds dt (2) (2)( 1)(0) 1 (0) 1() .00 xLet‘s denote byu(2)(2)(0) v (0) . Then we will haveu(2)1tb b u f s ds dt (2)( 1) 1 (0) 1() .0 0 b(2)Considering the condition u ( b 1) 0 , we will receive, thatu(2) 1(0) b b11b1t 0 0f s ds dt 1() . (9)158

u(2)Therefore,b1x x t 1 t xf s ds dt f s ds dtb b 11( 1)1 1()1() . (10) 1 1 0 0 0 0 Let's similarly receivexv(2) 1(0) b b22b2t 0 0f s ds 2() dt , (11)v(2)b2tx tx xxf s ds dt f s ds dtb b 2 2 2(2)1 2()2() . (12) 2 2 0 0 0 0 du dvTaking into account the condition 0dx dx(2)1x 01(2)2x 0relations (9), (11), it‘s possible to define the value of a constant α:2andb1tb2t 1 b 2 f1(s)ds dt b1 f2(s)ds dtb1b. (13)2 0 0 0 0 Finally, taking into account equalities (7), (8), (10), (12), (13), weobtain the explicit solution of the problem (1)-(3) in the form of quadratures:159

x b b x b x 1 2 1 1 1 1u( x1) 1 2 bb1 b2b1 b2b1( b1 b2) xbb1 t1 1t f s dsdt 1() 0 0 1 0 0xf s ds1() dtb1t 20 0f s ds1() dt bb2t 10 0 f s ds2 ( ) dt (14)b x x b b x 2 2 2 1 2 2v( x2) 1 2 bb1 b2b1 b2b2( b1 b2) xbb22t 2 0 0x 2f s dsdt 2 ( ) t 0 0f s ds2 ( ) dtb1t 20 0f s ds1() dt bb2t 10 0 f s ds2 ( ) dt (15)3. The difference scheme for problem (1) - (3)For numerical solution of problem (1)-(3) let‘s enter on segments0,b 1and 0,b 2the uniform mesh with steps h1and h2. Then replace22d u d vand by second difference derivative [8]. Then for problem22dx1dx2(1)-(3) we will receive the following difference scheme:y( i1)( i)( i1)2 y y( i) f1(x21h1( j1)( j)z 2 z z( j)) , f2( x2) , (16)2h2( j1)i1, , N1 1,N1h1 b1, j 1,, N2 1,N2h2 b2the boundary conditions( N1) 1( N2) ,y , z 2, (17)160

and conditions of conjunction(1)(0) (0) y y z zy z , 0 , (18)h h( i)1(0)where y y ih ), z z(jh ), x ih , x jh .(1)2(0)( j)( i)( j)( 12 1 1 2 2Let's construct by analogy (14)-(15) exact solution of differencescheme (16)-(18). We will present the functionsof the sum of two functions:( i)( j)y , z in the formy( i)( i)( i)y1 y2 ,z( j)( j)( j)z1 z2 ,wherey( i1)1 2y( i)121h y( i1)1( j1)( j)z1 2z1 z1 0 , 0 ,2h2( j1)i1, , N1 1,N1h1 b1, j 1,, N2 1,N2h2 b2 ,z( N )1 1 1( N )1 2 y , z 2 , (19)(1)(0) (0) y1 y1z1 z1y1 z1, 0 .h hand1(0)(1)( i1)( i)( i1)y2 2 y2 y2( i) f1(x21h1( j1)( j)( j1)2 2 z2 z2( j) f2( x2) ,2h2i 1, , N N h b , j 1, , N2 1,N2h2 b2,1 1,1112) ,(0)161

( N )2 1 ( N )2 2 y 0 , z 0 , (20)(1)(0) (0) y2 y2z2 z2y2 z2, 0 .h h1(0)It is easy to show that solution of the problem (19) are followingmesh functions:(1)2(0)yz( i)1( j)1b2b1b2b x1 b x( i)1 b2( j)22b11b1b11b x1( i)1 b x b2( j)22 ,2 ,2i 1,,N1j 1,,N1,21.(21). For this rewrite the eq-Let's find explicit expression for yuation( i)2,z( i)2( i1) 2 ( i)( i1)( i)y2 y2 y2( iy f ( x) )2x x21( i1)( i)( i)y y hf x2x2xh11111, i 1, , N1 1,in the form:and summarizing on i from 1 to k. Then we will receiveork( k1)(1)( i)y y h f xy2x2xi111k(1)( i)y h h f x1( k1)( k)2 y2 h12 x 1 1 1 1.i1Summarizing the last equation on k from 1 to j-1, we will receive:162( j)(1)y2 y2 1j1k(1)( i) j hy h h f x12x 1k1 i1111

or( j)2(0)2j1k(1)( i)y h h f x 1y y jh. (22)12xk1 i1111( N )Taking into account the condition y2 1 0 , when j N1, fromthe last equality it is possible to write:N11k(1)( i)y h h f x0(0)2 N1h12 x 1k1 i1y .111Let‘s denotey(0)2 z(0)2 . Then we will have11k(1) ( i)y h h f x2x1b11 N 11 k1 i1b111. (23)Therefore, from equality (22) and (23), we will receive: ( j)( j)x1y2 1 b 1 x( j)1b1N 11hk1k1 i1h1f1j1k( i)( i)xh h f x,1 1k1 i1111j 2,3, , N1 1, (1) (1) N 1(1) x 1 x 1 k1( i)y21 h1h1f1x1.b 1 b 1 k1 i1(24)Similarly we will receive163

21k(1) ( i)z h h f x2andx2b21 N 22 k1 i1b222(25) ( j)( j)x2z2 1 b 2 x( j)2b2N 12hk2k1 i1h2f2j1k( i)( i)xh h f x,2 2k1 i1222j 2,3, , N2 1, (26) (1) (1) N 1(1) x 2 x 2 k2( i)z21 h2h2f2x2.b 2 b 2 k1 i1Using the condition of conjunction z 0(1) (1)2 1 2x x2y and equalities(23), (25), it is possible to define value of a constant :N21k( i)( i)xb h h f x N 11k 1b 2 h1h1f11 1 2 2 2 2 b1 b2k1 i1k1 i1. (27)Taking into account equalities (21), (24), (26) and (27), we will receivethe explicit solution of the difference problem (16)-(18):164

165 .1,2,3,, 111 1)(111111 1)(11111)(111 1)(2222111 1)(11112211)(11221)(11121)(12)(2)(1)(121 NjxfhhxfhhbxxfhhbxfhhbbbbxbbbxbbbxbyyyjkkiiNkkiijNkkiiNkkiijjjjjj .11 1)(11111(1)111 1)(2222111 1)(11112211(1)11221)(11121)(12(1)2(1)1(1)121NkkiiNkkiiNkkiijjxfhhbxxfhhbxfhhbbbbxbbbxbbbxbyyy .1,2,3,, 211 1)(222211 1)(22222)(211 1)(2222111 1)(11112212)(22221)(21121)(22)(2)(1)(221NjxfhhxfhhbxxfhhbxfhhbbbbxbbbxbbbxbzzzjkkiiNkkiijNkkiiNkkiijjjjjj

z(1) z(1)1 z(1)2b2 xb2b1 bxb(1)22b2bb(1) N21k2 h22 k1 i1N112 1k1 i1h21 x bf2h(1)22( i)x.2kb11bh1f11 xN21k( i)( i)xb h h f x1 b(1)22 21 2k1 i1222 Let's mark that results of given article can be transferred for the differentialequations, given on graphs, which consists from n ribs.166REFERENCES1. D. Gorgeziani, T. Davitashvili, M. Kuprashvili, H. Meladze. Onthe Solution of Boundary Value Problem for Differential EquationsGiven in Graphs. – Applied Mathematics and Informatics. Tbilisi,2008, v.13, №.2,2. Д. Гордезиани, Т. Давиташвили, Г. Меладзе. Об одной математической модели электроэнергетических систем. soxumis saxelmwifouniversitetis Sromebi t. VII, 2009, maTematikisada kompiuterul mecnierebaTa seria, 4. 2008.57-69. pp. 75-86.3. А. И. Вольперт. Дифференциальные уравнения на графах. – Математическийсборник, т. 88(130), №4(8), 1972.4. P. Kuchment. Graphs models for waves in thin structures. – WavesRandom Media. 2002, 12, pp. 1-24.5. E. Akkermans, A. Comtet, J. Desbois. G. Montambaux. C. Texier.Spectral determinant on quantum graphs // Ann. Phys. – 2000 –284 – pp.10-51.6. Yu. V. Pokornyi, V. L. Pryadiev. On transmission conditions in theSturm-Liouville problem on a network. (in Russian). Sovrem. Mat.

Prilozh. No. 12, Differ. Uravn. Chast. Proizvod. (2004), 107-137;translation in J. Math. Sci. (N. Y.) 130 (2005), no. 5, 5013-5045.7. Y.V. Pokornyi, O. M. Penkin, A.V. Borovskikh, V. L. Pryadiev,K. P. Lazarev, S. A. Shabrov. Differential equations on geometricgraphs (in Russian). Fizmatlit. M., 2004, p.272.8. А. А. Самарский, А. В. Гулин. Численные методы. Научныймир. М., 2000.Tina gorozia, TinaTin daviTaSvilisasazRvro amocana grafebze gansazRvruliCveulebriv diferencialur gantolebaTaerTi klasisTvisnaSromSi gamokvleulia sasazRvro amocana grafebzegansazRvruli Cveulebriv diferencialur gantolebaTaerTi klasisTvis.simartivisaTvis ganvixiloT ori wibosagan Semdgarigrafi. grafis wveroebia a 1 , a 2 wertilebi, xolo wiboebiaa 0 a 1 ( 1 ) da a 0 a 2 ( 2 ) .a1a 0a 2davsvaT Semdegi amocana: vipovoT orjer uwyvetad diferencirebadifunqciebi u x 1 , v x 2 (sadac x − lokalurikoordinataa -is gaswvriv, α=1,2), romlebic akmayofilebena) Semdeg diferencialur gantolebebs:2d udx21 , 0 b f 1( x 1)2d vx1 ,1, f 2( x 2)2dx2 , 0 b x . (1)2,2167

sadac − wibos sigrZea, xolo f( x ), =1,2 − mocemulifunqciebia;b) sasazRvro pirobebsu ( b 1, ( b 21)v 2) , (2)g) SeuRlebis pirobebsdu dvu( 0) v(0), 0 . (3)dx dx1 x 012 x 02Seswavlilia dasmuli amocanis amoxsnis sxvaobiani me-Todi. agebulia diferencialuri amocanis cxadi amonaxsnebikvadraturebSi da sxvaobiani sqemis zusti amonaxsnebi. naSromSimiRebuli Sedegebi SeiZleba miRebul iqnes diferencialurigantolebebisTvis, romlebic gansazRvrulia n wibosSemcvel grafze. SevniSnoT, rom analogiuri amocanebi warmoiqmnebaeleqtroenergetikuli sistemebis, gazsadenebis,wyalsadenebis da a.S. qselebSi sxvadasxva procesebis modelirebisas.168

soxumis saxelmwifo universitetis Sromebit. VII, 2009maTematikisa da kompiuterul mecnierebaTa seriaTemur CilaCava, ciala ZiZiguri, leila sulava,maia Cakaberiaadministraciuli zewolis arawrfivimaTematikuri modelireziume. naSromSi SemoTavazebulia axali arawrfiviuwyveti maTematikuri modeli, romelsac SeuZlia aRwerosmocemul sociumSi (qveyana, saswavlo dawesebuleba, sawarmooobieqti da sxva) mmarTveli struqturebis mxridan administraciulizewola adamianebze maTi moqmedebebis gakontrolebismizniT. maTematikuri modeli aRiwereba orucnobiani(Tavisufali (aramarTvadi) da marTvadi adamianebis raodenobadrois momentSi) diferencialuri gantolebaTasistemiT. administraciuli zewola zogadad ganisazRvrebadrois mocemuli funqciiT. mudmivi administraciulizewolis pirobebSi koSis amocana arawrfividiferencialuri gantolebaTa sistemisaTvis amoxsniliaanalizurad zustad. modelis parametrebisa (Tavisuflebisxarisxis koeficienti, administraciuli zewolis Zala) dasawyis pirobebs Soris sxvadasxva damokidebulebebismixedviT miRebulia xuTi gansxvavebuli SemTxveva: miuxedavad sastarto pirobebisa Tavisufaliadamianebis raodenoba miiswrafvis garkveulwonasworobis raodenobisken, romelic saerToraodenobis naxevarze metia (susti zewola); sociumSi mudmivi administraciuli zewolisa dasxvadasxva sastarto pirobebis miuxedavaddamyardeba marTvadi da aramarTvadi adamianebisTanabari raodenoba (arasakmarisi zewola); Tavisufali adamianebis raodenoba, romelicTavdapirvelad marTvadi raodenobis meti iyo,miiswrafvis wonasworobis mdgomareobisken,romelic saerTo raodenobis naxevarze naklebia(Zlieri, magram SezRuduli zewola Tavisufaladamianebze);169

170 Tavisufali adamianebis raodenoba, romelicTavdapirvelad marTvadi raodenobis toli annaklebi iyo, miiswrafvis nulisken (Zlierizewola, sruli kontrolis SemTxveva); Tavisufali adamianebis raodenoba, romelicTavdapirvelad marTvadi raodenobis meti iyo,miiswrafvis nulisken (uZlieresi zewolaTavisufal adamianebze, sruli damorCilebismodeli).2000Mathematics Subject Classification: 93A30, 00A71Key words and phrases: administrative pressure, nonlinear mathematicalmodel, factor of degree of freedom, a complete control case, model of fullsubmission.1. Sesavalisinergetika anu TviTorganizaciis Teoria, dReisTvissaganTa Soris kavSirebis kvlevis erT-erT yvelaze popularulida perspeqtiuli midgomaa. termini “sinergetika” ber-Znulidan TargmanSi niSnavs “erTobliv moqmedebas”.Cveni mTavari mizania vaCvenoT rogor aRwevs sinergetikamniSvnelovan Sedegebs gansxvavebul mecnierebebSi (magaliTad,maTematika da sociologia). ras unda daveyrdnoTcvlilebebis epoqaSi? albaT, gamoyenebiT maTematikas, maTematikurda kompiuterul modelebs, prognozisa da monitoringisalgoriTmebs, mecnierul msofmxedvelobas.sinergetikis ganviTarebam gviCvena, rom misi Taviseburebada originaloba damokidebulia imaze, rom is samecnio sivrceSisami sferos TanakveTaze mdebareobs – sagnobrivi codnis,filosofiuri refleqtebisa da maTematikuri modelirebis.swored es samTakavSiri saSualebas gvaZlevs erT Sem-TxvevaSi efeqturad avaSenoT mecnieruli strategia, meoreSi– movaxdinoT msxvili kvleviTi proeqtebis organizeba [1-9].am naSromSi Cvens amocanas warmoadgenda iseTi socialuriprocesis aRwera maTematikur enaze (Segveqmna maTematikurimodeli), rogoric aris administraciuli zewola, romelicSeiZleba xorcieldebodes makrodoneze (saxelmwifoan misi didi regioni, makromodeli) an mikrodoneze (saswavlodawesebuleba, sawarmoo obieqti da sxva, mikromodeli)mravalferovani ideologiuri da teqnologiuri saSualebebiT.maTematikur modelSi ganixileba sami obieqti:

1. administracia (mmarTveli struqturebi, xelisufleba),romlis mizans mikro an makrodoneze warmoadgensmis daqvemdebarebaSi myofi adamianebis mar-Tva Tavisi interesebidan gamomdinare. Ees interesebiSeiZleba iyos rogorc samarTliani (swori mar-Tva) ise usamarTlo (araswori marTva).2. marTvadi adamianebi, romelzec moqmedebs administraciulizewola.3. Tavisufali anu aramarTvadi adamianebi, romlebic,miuxedavad administraciuli zewolisa, rCebian TavisufalniTavis arCevanSi an moqmedebaSi.2. gantolebaTa sistema da sawyisi pirobebipopulaciis evoluciis, saomari moqmedebebis lanCesterismodelebisa da socialuri procesebis (maT Soris, demografiuli,istoriuli, sainformacio omi da sxva) aRmwerisxvadasxva modelebis analizs mivyavarT administraciulizewolis (marTvis) Semdeg arawrfiv maTematikur modelamde [1, 10 – 12]mocemul arawrfiv maTematikur modelSi administraciulzewolas aqvs mudmivi xasiaTi da is ar aris damokidebulidroze. modelis ganxilvis periodSi adamianTa orivejgufis bunebrivi cvlilebebi ar aris gaTvaliswinebuli (ammiaxloebaSi CaTvlilia, rom Sobadobisa da sikvdilianobiskoeficientebi tolia).modeli aris uwyveti anu saZiebel funqciebs gaaCniaTsistemaSi Semavali uwyveti warmoebulebi.drois momentSi Tavisufali (aramarTvadi) adamianebisraodenobaa,171

momentSi marTvadi adamianebis raodenobaa,droisadministraciuli zewolis Zala,Tavisuflebis koeficienti.da funqciebis raodenobrivi cvlilebebidrois momentSi aRiwereba (2.1) arawrfivi gantolebaTa sistemiT.radganac gantolebaTa sistema (2.1) aris pirveli rigis,unda daisvas damatebiTi piroba (koSis piroba)(2.2)miviReT koSis amocana (2.1), (2.2). amasTan, bunebrivia, romvipovoT cxadi saxiT da funqciebi.adganac modelis ganxilvis periodSi orive jgufis bunebrivicvlilebebi iZlevian erTobliobaSi nulovan efeqts,gvaqvs:saidanac gamovsaxoTCavsvaT (2.4) sistema (2.1)-is pirvel gantolebaSi. maSinmiviRebT pirveli rigis gancalebadcvladebiani diferencialurigantolebisaTvis koSis amocanasfunqciis raodenobrivi cvlileba gamoisaxakvadratuli samwevriT172

SemoviRoT aRniSvna:ganvixiloT sami gansxvavebuli SemTxveva:DI. D 0.maSin (2.5) miiRebs saxesradganvRebulobT martiv diferencialur gantolebasromlis erTaderTi amonaxsni, sawyisi pirobis gaTvaliswinebiTCaiwerebaxolo (2.3) da (2.4) gaTvaliswinebiT, miviRebTsabolood -s mniSvnelobis gaTvaliswinebiT, miviRebTkoSis amocanis (2.1), (2.2) erTaderT zust analizur amonaxsns.173

(2.9)am SemTxvevaSi, (2.9)-is analizi gviCvenebs, rom sociumSimudmivi administraciuli zewolisa da Tundac gansxvavebulisastarto pirobebis miuxedavadgarkveulididi drois Semdeg, damyardeba marTvadi da aramarTvadiadamianebis Tanabari raodenoba (arasakmarisi zewola).II. ganvixiloT SemTxveva, rocamaSin (2.5)-dan miviRebTMmiRebuli (2.10) gantolebis zogadi amonaxsnia:mocemuli sawyisi pirobis gaTvaliswinebiT, vRebulobT174

CavsvaT C -s mniSvneloba zogad amonaxnsnSi (2.11), mivi-RebTsaxeamrigad, am SemTxvevaSi (2.1) sistemis amonaxsns eqneba(2.12)-s analizi iZleva sam gansxvavebul SemTxvevas:- Tavisufali adamianebis raodenoba, romelic TavdapirveladmarTvadi raodenobis meti iyo ( , miiswrafviswonasworobis mdgomareobisken, romelic saerToraodenobis naxevarze naklebia (Zlieri, magram SeZRudulizewola Tavisufal adamianebze);- Tavisufali adamianebis raodenoba, romelic TavdapirveladmarTvadi raodenobis toli an naklebi iyo175

, miiswrafvis nulisken (Zlieri zewola, srulikontrolis SemTxveva);- Tavisufali adamianebis raodenoba, romelic TavdapirveladmarTvadi raodenobis meti iyo ( , miiswrafvisnulisken (uZlieresi zewola Tavisufal adamianebze, sruli damorCilebis modeli).III. ganvixiloT SemTxveva, roca .maSin (2.5) miiRebs saxesmiRebuli diferencialuri gantolebis zogadi amonaxsnia:sawyisi pirobis gaTvaliswinebiTCavsvaT C -s mniSvneloba zogad amonaxsnSi, miviRebT176

saidanacaRvniSnoTtolobis orive nawili gavyoT-zesaidanac177

da sabolood miviRebT tolobasamrigad, sabolood vRebulobT koSis amocanis (2.1), (2.2)-is zust analizur amonaxsns(2.13)(2.13) analizi gviCvenebs, rom miuxedavad sastarto pirobebisaTavisufali adamianebis raodenoba miiswrafvis garkveulwonasworobis raodenobisken, romelic saerTo raodenobisnaxevarze metia (susti zewola).amrigad, miRebuli zusti analizuri amoxsna iZlevasaSualebas mxareebma (mmarTveli struqturebi, Tavisufaliadamianebi) arCeuli strategiis mixedviT Sesabamisad Sear-Cion marTvis parametrebi (administraciuli zewolis Zala,178

Tavisuflebis koeficienti) da ecadon miaRwion maTTvis sasurvelSedegs. bunebrivia, rom moqmed mmarTvel struqturebzear aris damokidebuli mxareebis sastarto pirobebi,amitom misTvis marTvis (mxolod masze damokidebuli) parametriaideologiuri Tu teqnologiuri (beWvdiTi da eleqtronulipresa, interneti) SesaZleblobebi, romlis gamoyenebisyvelanairi ekonomikuri berketi mas gaaCnia.Lliteratura:1. T. CilaCava, c. ZiZiguri. maTematikuri modelireba.Tbilisi, 2008, 440 gv.2. А. К. Гуц, Ю. В. Фролова. Математические методы всоциологии, М., 2007, 216 gv.3. K. H. Rosen. Applied Mathematical Modeling. MultidisciplinaryApproach, 1999.4. W. Weidlich, G. Haag Concepts and Models of Quantitative Sociology.Springer. Berlin, 1983.5. W. Weidlich. Physics and Social Science. The Approach of Synergetics.– Physics Reports, 1991, v. 204, p. 1 – 163.6. В. Ю. Крылов. Методологические и теоретические проблемыматематической психологии. М., 2000.7. Л. Е. Гринин, А. В. Коротаев, С. Ю. Малков. История иматематика. Модели и теории. М. 2009.8. С. Ю. Малков, Л. Е. Гринин, А. В. Коротаев. История иматематика. Процессы и модели.,М. 2009.9. А. В. Коротаев, А. С. Малков, Д. А. Халтурина. Законыистории. Математическое моделирование развития мирсистемы.– Демография, экономика, культура.М. 2007.10. А. А.Самарский, А. П. Михайлов. Математическоемоделирование. Наука. М., 320 gv.11. С. П. Капица. Очерк теории роста человечества.Демографическая революция и информационное общество.ЛЕНАНД. М., 2008, 128 gv.12. T. Chilachava, N. Kereselidze. About one mathematical model ofthe information warfare. – Fifth congress of mathematicians of Georgia.Abstracts of contributed talks. Batumi/Kutaisi, October 9 - 12,2009, p. 85.179

180TEMUR CHILACHAVA, TSIALA DZIDZIGURI,LEILA SULAVA, MAIA CHAKABERIANONLINEAR MATHEMATICAL MODELOF ADMINISTRATIVE PRESSUREAbstractIn work the new nonlinear continuous mathematical model whichcan describe in the given society (the country, an educational institution,industrial object etc.) administrative pressure upon people from outsideadministrative structures for the purpose of the control of their actions isoffered. The mathematical model is described by nonlinear system of thedifferential equations with two unknown (quantity free (non-ruled) andruled people at the moment of t time). Administrative pressure which canhave various forms, is generally defined by the given function of time. Incase of constant administrative pressure the problem of Cauchy‘s for systemof the nonlinear differential equations of the first order is solved analyticallyexactly. Depending on various correlations between model parameters(the factor of degree of freedom, force of administrative pressure)and initial conditions are received five various cases: without dependence from starting conditions the quantity of freepeople aspires to certain equilibrium value which is more thanhalf of their total quantity (weak pressure); despite constant administrative pressure and various starting conditionsin society the equal quantity of ruled and non-ruled peoplewill be established (insufficient pressure); the quantity of free people which was initially more quantitiesruled, aspires to equilibrium value which is less than half of theirtotal quantity (strong, but the limited pressure upon free people); the quantity of free people which was initially less or equallyquantities ruled, aspires to zero (strong pressure, a complete controlcase);the quantity of free people which was initially more quantitiesruled, aspires to zero (the strongest pressure upon free people,model of full submission).The offered mathematical model except theoretical interest has alsothe important practical meaning as both sides (administration, free people)can use results of mathematical model in conformity of the purposes.

a v t o r e b ielizbar nadaraia – fizika-maTematikis mecnierebaTa doqtori,saqarTvelos mecnierebaTa erovnuli akademiis wevrkorespondenti,ivane javaxiSvilis saxelobis Tbilisissaxelmwifo universitetis sruli profesoripetre babilua – maTematikis doqtori, ivane javaxiSvilissaxelobis Tbilisis saxelmwifo universitetis asistent-profesorigrigol soxaZe – fizika-maTematikis mecnierebaTa doqtori,ivane javaxiSvilis saxelobis Tbilisis saxelmwifouniversitetis asocirebuli profesorimzia facacia – soxumis saxelmwifo universitetis asocirebuliprofesorigogi fanculaia – fizika-maTematikis mecnierebaTa doqtori,saqarTvelos teqnikuri universitetis sruli profesoriuSangi goginava – fizika-maTematikis mecnierebaTa doqtori,ivane javaxiSvilis saxelobis Tbilisis saxelmwifouniversitetis sruli profesoriilia TavxeliZe – fizika-maTematikis mecnierebaTa doqtori,ivane javaxiSvilis saxelobis Tbilisis saxelmwifouniversitetis asocirebuli profesorihamlet melaZe – fizika-maTematikis mecnierebaTa doqtori,soxumis saxelmwifo universitetis maTematikis samecniero-kvleviTiinstitutis ufrosi mecnier-TanamSromeliTemur CilaCava – fizika-maTematikis mecnierebaTa doqtori,soxumis saxelmwifo universitetis sruli profesori,maTematikis samecniero-kvleviTi institutis direqtorinugzar kereseliZe – wmida andria pirvelwodebulis saxelobisqarTuli universitetis doqtorantiTina gorozia – soxumis saxelmwifo universitetis magistriTinaTin daviTaSvili – maTematikis doqtori, ivane javaxiSvilissaxelobis Tbilisis saxelmwifo universitetisasistent-profesori181

ciala ZiZiguri – soxumis saxelmwifo universitetis asocirebuliprofesorileila sulava – soxumis saxelmwifo universitetis doqtorantimaia Cakaberia – soxumis saxelmwifo universitetis doqtoranti182

A U T H O R SElizbar Nadaraya – Doctor of Physics and Mathematics Sciences,Corresponding Member Georgian National Academy of Sciences,Professor of Ivane Javakhishvili Tbilisi State UniversityPetre Babilua – Doctor, Assistant-Professor of Ivane JavakhishviliTbilisi State UniversityGrigol Sokhadze – Doctor of Physics and Mathematics Sciences,Associate Professor of Ivane Javakhishvili Tbilisi State UniversityMzia Patsatsia – Associate Professor of Sokhumi State UniversityGogi Pantsulaia – Doctor of Physics and Mathematics Sciences,Professor of Georgian Technical UniversityUshangi Goginava – Doctor of Physics and Mathematics Sciences,Professor of Ivane Javakhishvili Tbilisi State UniversityIlia Tavkhelidze – Doctor of Physics and Mathematics Sciences,Associate Professor of Ivane Javakhishvili Tbilisi State UniversityHamlet Meladze – Doctor of Physics and Mathematics Sciences,Sokhumi State University, Institute of Mathematics, Professor of St.Andrew the First Called Georgian UniversityTemur Chilachava – Doctor of Physics and Mathematics Sciences,Professor of Sokhumi State University, Director of Institute of MathematicsNugzar Kereselidze – Doctorant of St. Andrew the First CalledGeorgian UniversityTina Gorozia – Magister of Sokhumi State UniversityTinatin Davitashvili – Doctor, Assistant-Professor of IvaneJavakhishvili Tbilisi State UniversityTsiala Dzidziguri – Associate Professor of Sokhumi State UniversityLeila Sulava – Doctorant of Sokhumi State UniversityMaia Chakaberia – Doctorant of Sokhumi State University183

Jurnalis TematikaJurnali aqveynebs masalebs, romlebic Seicavs mecnierulikvlevis axal Sedegebs Semdeg Teoriul da gamoyenebiTdargebSi: maTematikuri analizi algebra, logika da ricxvTa Teoria geometria, topologia Cveulebrivi diferencialuri gantolebebi, marTvis Teoria da dinamikuri sistemebi kerZowarmoebuliani diferencialuri gantolebebi maTematikuri fizika albaTobis Teoria da maTematikuri statistika uwyvet garemoTa meqanika maTematikuri modelireba ricxviTi meTodebi maTematikuri ganaTleba da maTematikis istoria daprogrameba monacemTa da codnis bazebis marTvis sistemebi informaciuli usafrTxoeba da kriptografiaM informaciuli teqnologiebi teqnikur da socialur-ekonomikur sistemebSi xelovnuri inteleqtiMOmasalebi miiReba qarTul da inglisur enebze. masalebiqveyndeba originalis enaze.gamosaqveyneblad miRebuli yvela masala recenzirdebada dadebiTi recenziis SemTxvevaSi gamoqveyndeba beWvdiTisaxiT JurnalSi da ganTavsdeba internetSi.184

T H E M E S:The journal publishes articles containing new scientific results inthe field of theoretical and applied problems on the following sections: Mathematical Analysis Algebra, Logic and Number Theory Geometry and Topology Ordinary Differential Equations, Control Theory, Optimization and Dynamical Systems Partial Differential Equations Mathematical Physics Probability Theory and Mathematical Statistics Mechanics of Continuity Medium Mathematical Modeling Numerical Analysis and Scientific computing Mathematics education and History of MathematicsProgramming Control in data bases and knowledge bases Informational security and cryptology Informational technologies in technical and social-economicsystems Artificial intellect theoryThe journal accepts for the publication materials in Georgian andEnglish languages. The materials are published in language of the original.All incoming materials are placed in special area, reviewed and incase of positive review, published at journal.185

soxumis saxelmwifo universitetis SromebimaTematikisa da kompiuterul mecnierebaTa seriasamaxsovro avtorebisaTvis:gamosaqveyneblad miRebuli yvela masala recenzirdebada dadebiTi recenziis SemTxvevaSi gamoqveyndeba JurnalSi.masalebi miiReba qarTul an inglisur enaze. statias Tanunda axldes anotacia (reziume) statiis enaze, agreTvegafarToebuli (araumetes origverdisa) anotacia meore enaze.avtors SeuZlia publikaciisaTvis mogvawodos erTi anramdenime statia. avtorisaTvis masalebis dabruneba SemdgomidamuSavebisaTvis ar niSnavs, rom masalebi miRebulia gamosaqveyneblad.gadamuSavebuli teqstis miRebis Semdeg masalebsxelaxla ganixilavs saredaqcio kolegia. statias mi-TiTebuli eqneba redaqciaSi miRebis TariRi.statia mogvawodeT DOC(MS-Word) failis saxiT, furcliszoma – A4, Srifti AcadNusx, 12 zoma, striqonebs SorismanZili – 1,5; inglisuri teqstis Srifti – Times New Roman,zoma 12 daSoreba furclis yvela kididan – 2,5sm. grafikulimasala jpg, gif an bmp failebi.statiis Sinaarsis struqtura:`UDC~ klasifikatori, saTauri (centrirebuli, Bold,zoma 14); monacemebi avtoris (TiToeuli Tanaavtoris) Sesaxeb:gvari da saxeli (centrirebuli, zoma 12), organizaciisdasaxeleba da misamarTi (centrirebuli, zoma 10); anotacia(reziume) statiis enaze (zoma 12). anotacia ar unda Seicavdescitatebs gamoyenebuli literaturidan da rTul formulebs.sakvanZo sityvebi statiis enaze (zoma 12); statiisSinaarsi; gamoyenebuli literaturis nusxa. literaturisnusxa SeadgineT teqstSi misi gamoyenebis Tanmimdevrobis mixedviT;dasasrul, moiyvaneT reziumeebi. ostatiis moculobaar unda aRematebodes 15 gverds (reziumeebis CaTvliT). statiawarmodgenili unda iyos: 1 egzempliari amobeWdili +eleqtronuli versia. statiis gadmogzavna SeiZleba el-fostiT:ingagabi@mail.ru; temo_chilachava@yahoo.com; temo.chila@gmail.com186

gamomcemlobis redaqtorebi: robert mesxi,lela mircxulavadaibeWda: gamomcemloba `meridiani~, Tbilisi,al. yazbegis 45. tel. 39-15-22E-mail: info@meridianpub.com