Browsing Doctoral Dissertations by Title
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Stojković, Vojislav (Belgrade , 1981)[more][less]
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Krgović, Dragica (Belgrade , 1982)[more][less]
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Bogdanović, Stojan (Novi Sad)[more][less]
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Janjić, Slobodanka (Belgrade , 1985)[more][less]
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Shkodra, Sadri (Priština)[more][less]
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Jovanović, Aleksandar (Beograd)[more][less]
Abstract: The dissertation consists of five chapters. The Chapters 1 and 2 contain the known results from the model theory and the set theory which are used in the other chapters. A classification of the properties of filters is given in Chapter 3. Some connections between combinatoric properties are made and the theorems about existence are also given. Ultraproducts are studied in Chapter 4. The structure of ultraproducts is connected with the structure of ultrafilters and cardinality of ultraproducts. Moreover, some other problems are studied, as the 2- cardinality problem. In the case of a measurable cardinal the connection with continuum problem is presented and several theorems of the cardinality of ultraproducts are proved. The problems about the real measure are studied in Chapter 5. The forcing is presented and by using results from Chapter 3 and Chapter 4 several properties are proved. The notion of norm of measure is introduced and some possible relations between additivity and norm of a measure are studied. Real large measurable cardinals are introduced analogously as the other large cardinals. The inspiration for this introduction were Solovay’s results of equconsistency of the theory ZDF + ”there is a measurable cardinal” and the theory ZDF+”there is a real measurable cardinal”. The relative consistency of the real large measurable cardinals with respect to ZDF+”the corresponding large cardinal” is proved by a generalization of Solovay’s forcing. URI: http://hdl.handle.net/123456789/683 Files in this item: 1
work001AleksandarJovanovic.pdf ( 16.02Mb ) -
Jovanov, Đurica (Beograd , 1991)[more][less]
URI: http://hdl.handle.net/123456789/4128 Files in this item: 1
Varijacione_nejednacine.PDF ( 910.5Kb ) -
Angelov, Trajče (Belgrade , 1980)[more][less]
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Janković, Vladimir (Belgrade , 1983)[more][less]
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Simić, Slavko (Belgrade , 1997)[more][less]
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Lukić, Mirko (Belgrade)[more][less]
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Kulenović, Mustafa (Sarajevo)[more][less]
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Radojčić, Nina (Beograd , 2018)[more][less]
Abstract: In this dissertation, three NP-hard optimization problems are studied and va- rious computational intelligence methods are considered for solving them, with a special emphasis on the possibilities of applying fuzzy logic in order to improve the performances of proposed methods. In addition, it is shown how fuzzy logic can be incorporated into a model to make it more adequate to real world applications. The first problem considered is the Risk-Constrained Cash-in-Transit Vehicle Routing Problem (RCTVRP) that represents a special case of the vehicle routing problem (VRP). Similar to the classical VRP, the aim is to determine the collection routes from one depot to a number of customers in order to minimize the overall travel distance (or cost). Additionally, the safety aspect of the routed risk constraints are introduced in the case of RCTVRP. The RCTVRP concerns the issue of secu- rity during the transportation of cash or valuable goods (e.g. in the cash-in-transit industry). The other two problems studied in this dissertation belong to the class of loca- tion problems: the Load Balancing Problem (LOBA) and the Max-Min Diversity Problem (MMDP). The goal of the LOBA problem is to locate a fixed number of facilities, such that the difference between the maximum and minimum number of customers served by each facility is balanced. The LOBA model is useful in cases where customers naturally choose the closest facility. The MMDP consists of se- lecting a subset of a fixed number of elements from a given set in such a way that the diversity among the selected elements is maximized. This problem also arises in real world situations encompassing a variety of fields, particularly the social and biological sciences. In order to solve the RCTVRP, a fuzzy GRASP (Greedy Randomized Adaptive Search Procedure) is hybridized with Path Reliking (PR) methodology. Carefully adjusted fuzzy modification incorporated into the proposed GRASP for the RC- TVRP improved its performance. Moreover, in this dissertation a new PR structure is implemented and can be used for other vehicle routing problems. To improve the algorithm’s time complexity, a new data structure for the RCTVRP is incor- porated. The proposed fuzzy GRASP with PR hybrid shows better computational performance compared to its non-fuzzy version. Furthermore, computational results on publicly available data sets indicate that the proposed algorithm outperforms all existing methods from the literature for solving the RCTVRP. For solving the LOBA problem two efficient hybrid metaheuristic methods are proposed: a combination of reduced and standard variable neighborhood search met- hods (RVNS-VNS) and hybridization of evolutionary algorithm and VNS approach (EA-VNS). The proposed hybrid methods are first benchmarked and compared to all the other methods on existing test instances for the LOBA problem with up to 100 customers and potential suppliers. In order to test the effectiveness of the pro- posed methods, we modify several large-scale instances from the literature with up to 402 customers and potential suppliers. Exhaustive computational experiments show that the proposed hybrid methods quickly reach all known optimal solutions while providing solutions on large-scale problem instances in short CPU times. Re- garding solution quality and running times, we conclude that the proposed EA-VNS approach outperforms other considered methods for solving the LOBA problem. EA approach is also proposed for solving the MMDP. Computational experi- ments on a smaller benchmark data set showed that the classic EA quickly reached all optimal solutions obtained previously by an exact solver. However, some of the larger instances of MMDP were challenging for classic EA. Although researchers have established the most commonly used parameter setting for EA that has good performance for most of the problems, it is still challenging to choose the adequate values for the parameters of the algorithm. One approach to overcome this is chan- ging parameter values during the algorithm run. As part of this dissertation this problem was addressed by extending the evolutionary algorithm by adding a fuzzy rule formulated from EA experts’ knowledge and experience. The implemented fuzzy rule changes the mutation parameter during the algorithm run. The results on tested instances indicate that the proposed fuzzy approach is more suitable for solving the MMDP than classic EA. For all three problems addressed whereas the smaller instances that CPLEX was able to solve, obtained optimal solutions were used for comparison with proposed methods and all of the proposed methods obtained these optimal solutions. Moreover, in this dissertation it has been shown that fuzzy logic is a successful tool in modeling the RCTVRP. In this problem the risk constraints are set by using a risk threshold T on each route and thus, the routes with risk larger than T are forbidden. However, in this dissertation the aim is to take into account the probability of being robbed along each route instead of just allowing solutions with routes that satisfy the risk constraints. A new fuzzy model for the RCTVRP is developed which takes into account the value of the risk index of each route and the solutions with lower values of risk indexes on their routes are considered superior. In order to achieve that fuzzy numbers are used in the improved model. Moreover, two mixed integer program formulations of new fuzzy model are developed and presented in this dissertation. The introduced fuzzy model is compared with the model from the literature using an adequate example and the advantages of the newly proposed fuzzy RCTVRP is demonstrated. Computational experiments are performed and the comparison of the two models given in the paper show that the newly presented approach leads to safer routes. URI: http://hdl.handle.net/123456789/4737 Files in this item: 1
tezaNinaRadojicic.pdf ( 1.665Mb ) -
Popstanojević, Zoran (Belgrade , 1963)[more][less]
URI: http://hdl.handle.net/123456789/224 Files in this item: 1
phdZoranPopstojanovic.pdf ( 1.549Mb ) -
Muzika Dizdarević, Manuela (Beograd , 2017)[more][less]
Abstract: Subject of this doctoral thesis is the application of algebraic techniques on one of the central topics of combinatorics and discrete geometry - polyomino tiling. Polyomino tilings are interesting not only to mathematicians, but also to physicists and biologists, and they can also be applied in computer science. In this thesis we put some emphasis on possibility to solve special class of tiling problems, that are invariant under the action of nite group, by using theory of Gr obner basis for polynomial rings with integer coe cients. Method used here is re ecting deep connection between algebra, geometry and combinatorics. Original scienti c contribution of this doctoral thesis is, at the rst place, in developing a techniques which enable us to consider not only ordinary Z?tiling problems in a lattice but the problems of tilings which are invariant under some subgroups of the symmetry group of the given lattice. Besides, it provides additional generalizations, originally provided by famous mathematicians J. Conway and J. Lagarias, about tiling of the triangular region in hexagonal lattice. Here is a summary of the content of the theses. In the rst chapter we give an exposition of the Gr obner basis theory. Especially, we emphasize Gr obner basis for polynomial rings with integer coe cients. This is because, in this thesis, we use algorithms for determining Gr obner basis for polynomials with integer coe cients. Second chapter provides basic facts about regular lattices in the plane. Also, this chapter provides some fundamental terms of polyomino tiling in the square and hexagonal lattice. Third chapter of this thesis is about studying Ztilings in the square lattice, which are invariant under the subgroup G of the group of all isometric transformations of the lattice which is generated by the central symmetry. One of the steps to resolve this problem was to determine a ring of invariants PG and its generators and relations among them. We use Gr obner basis theory to achieve this. Forth chapter covers the analysis of Ztilings in the hexagonal lattice which are symmetric with respect to the rotation of the plane for the angle of 120 . Main result of the fourth chapter is the theorem which gives conditions for symmetric tiling of the triangular region in plane TN, where N is the number of hexagons on each side of triangle. This theorem is one of the possible generalizations of the well known result, provided by Conway and Langarias. Fifth chapter provides another generalization of Conway and Lagarias result, but this time it is about determining conditions of tiling of triangular region TN in the hexagonal lattice not only with tribones, but with nbones. nbone is basic shape of of n connected cells in the hexagonal lattice, where n is arbitrary integer. URI: http://hdl.handle.net/123456789/4503 Files in this item: 1
muzikadizarevic.manuela.pdf ( 33.23Mb ) -
Lazarević, Ilija (Belgrade)[more][less]
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Lazović, Bojana (Beograd , 2018)[more][less]
URI: http://hdl.handle.net/123456789/4748 Files in this item: 1
B_Lazovic_Doktorska_disertacija.pdf ( 2.269Mb ) -
Šašić, Mane (Belgrade , 1966)[more][less]
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Vukobrat, Mirko (Belgrade , 1978)[more][less]
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Lukačević, Mirjana (Beograd , 1974)[more][less]
Abstract: Relativisti6ka hidromehanika je po6ela da se razvija 1914. godine; kada je Ajatajn [1] formulisao izraz za tenzor energije relativisti6kog idealnog fluida i izveo njegove diferencijalne dna6ine kretanja,•ograni6iv6i se pri tome na adijabatska strujanja, kada za fluid vazi karakteristi6na jedna6ina koja vezuje pritisak i gustinu. Jednaine kretanja takvog fluida ispitivao je 1924. godine Ajzenhart (Eisenhart) [2]. On je pokaZao da je, .kada se radio adj.• jabatskom strujanju idealnog fluida, mogu6e prona6i. metriku kola je konformna metrici'prostor-vremena, u odnosu na koju strujne:iinije fluida predstavljaju geodezijske linije. Pri tome je utvrdio i oblik skalarne funkcije pomo6u koje se pomenuta konformna metrika uvodi To su, medjutim, bili samo pojedina6ni, iako veoma zna6aj-. ni rezultati iz oblasti relativisti6ke mehanike fluida. Tek 1937. godine pojavljuje se•prvi rad.koji sistematski.izlaZe relativisti6ku teoriju-idealnog•fluidapo uzoru na odgovaraju, 6u klasi6nu teoriju. To je ops6irni Singov.( . L. Synge) rad nRela-r tivisti6ka hidrodinamika" [3]. U njemu actor pre svega izlaze ono to je do tada o relativisti6kom idealnom fluidu bilo poznato, Tret7 postavljajuoi kao i njegovi prethodnici da za fluid van.. karakteri- Virtual Library of Faculty of Mathematics - University of Belgrade elibrary.matf.bg.ac.rs stiena jodnacina koja vezuje sopotveni pritioak i sopstvenu'guStinu. on prihvata od Ajzenharta skalarnu funkciju pomo6u koje se . strujne linije idealnogfluida preslikavaju na geodezijske linije jednog prostora oija je metrika konformna metrici prostor-vremena, i pomoCu nje uvodi svoju funkciju; tzv. funkciju-indeksl ) , koja u relativistiCkoj hidrodinamici igra veoma vaInu ulogu. Pored toga, Sing je u• svome radu uveo i 6itav niz novih pojmova: definisao je kinemati6ku i'dinami6ku cirkulaciju, uveo pojam.kinemati6kog i dinamiekog tenzora vrtlolenja, kinemati6kog dinamiCkog vektor-vrtloga, i definisao je i ispitivao bezvrtloZno strujanje. Izvestan broj definicija iz ()yoga' rada je docnije pretrpeo izmene, pa se 6ak neki pojmovi definisani u njemu i ne pomi- . nju u docnijim radovima, kako Singovim tako i ostalih autora, jer uu pokazalo da 110 predutavijaju relativiutiaa uop6tonja odgovarajuaih klasi6nih pojmova: Tako, recimo, udanaSnjiM radovima ne molemo najoi . na Singovu kinemati6ku'cirkulaciju; kao izraz koji predstavlja relativistieko uopftenje klasi6nog pojma cirkulacije vektora brzine prihvadena je samo njegova dinami6ka cirkulacija, pod imenom cirkulacije vektora toka. Ipak, ovaj Singov rad predstavlja zna6ajan doprinos razvoju relativisti6ke teorije fluida, buduai da je posluZio kao osnovica na kojoj su docniji autori, ukljusoujuOi i samoga Singa4 gradili dalje to teoriju. 1) Funkcija-indeks koju Sing uvodi, i koju tako naziva zbog njene analogije sa klasi6nim indeksom prelamanja.date transparentne sredine, ne razlikuje se bitno od pomenute AjzenhartoVe funkci -• je: samo iz formalnih razloga, umesto da koristi . Ajzenhartovu funkciju qq= ft. 41i1.. leNtil- (gde je ft/ sopstveni pritisak:fluida, a S njegova sopstvena gustina), Sing kao funkciju-indeks defini..6e (f,)=exp(A(p4). U svojim radovima Lihnerovi6 prihvata Singovu funkciju i naziva je indeks fluida. Virtual Library of Faculty of Mathematics - University of Belgrade elibrary.matf.bg.ac.rs -3_ Posle izvesne praznine u literaturi koju je doneo drugi sve-. tski rat, krajem 6etrdesetih, a zatim.pedesetih i 6ezdesetih godina, pojavljuje se atav niz.veoma zna6ajnih radova iz ove oblasti. PomenuCemo; pre svega, Taubove. (A. H. Taub) radove [4],[5] i [6] Koristeei Ekartove (C. Eckart) [7] ideje.o relativisti6koj -termodinamici koje prenosi iz specijalne u .op6tu teoriju relativnosti, Taub se ne ogranieava na adijabatska i izotermi6ka strujanja, pri kojima za fluid vaZi karakteristi6na jednaina kCja vezuje sopstveni pritisak i sopstvenu gustinu, veo posmatra opgiti slu6aj, kada su dve termodinamifte Veli6ine nezavisno 'promenllive. U svojim ra- ___ . _ doVima on, izmedju ostalog, fotmulibe i jedan nov oblik tenzora energije relativisti6kog idealnog fluida. To je oblik u kome se unutra-6nja energija pojaVljuje eksplicitno, aokoji je.veoma pogodan za rad kada uz diferencijalne jedna6ine kretanja• fluida . i uz jedna_inu kohtinuiteta treba, pored .karakteristione jednaane, koristiti i druge termodinami6ke veze. Uporedo sa Taubovim radovima, u literaturi iz toga perioda nailazimo na niz radova A. LihneroviOa. (A. Lichnerowilpz) [8],[93 i njegovih u6enika. U njima autori uzimaju u obzir pojave provodjenja toplote i elektriciteta kroz idealnu fluidnu sredinu, a takodje posmatraju naelektrisani iii namagnetisani idealni fluid.. Tenzor energije koji nalazimo u.njihovim radovima dopunjen je 61anovima koji se odnose na pojave koje smo pomenuli. Veoma znaajno mesto u tim radovima zauzima i Kogijey problem_koji,se postavlja i ispituje u razli6itim slu6ajevima_materijalnih.sredina. Najzad, A. Lihnerovic je prvi koji . je u relativisti6koj teoriji uzeo u obzir unutra6nje trenje, koje postoji izmedju 6estica fluida pri njihovom relativnom pomeranju. Po6to je postavio izraz za tenzor energije relativistiekog viskoznog,fluida, on je, pola7 zea oduslova konzervativnosti toga tenzora, izveo i'diferencijalne jednaane kretanja takve . materijalne sredine i.ispitivao nje- Virtual Library of Faculty of Mathematics - University of Belgrade elibrary.matf.bg.ac.rs no bezvrtloZno strujanje. Relativistieki viskozni fluid koji je uz to i naelektrisan . u svojoj studiji [10] Pion (G. Pichon). Tenzor vis- Prou6avao je koznosti koji nalazimo u njegovom radu ne poklapa se sa izrazom koji za taj tenzor daje Lihnerovie. U odeljku 1.1 ovoga rada govorieemo detaljnije o tome zbog 6ega te izrazi za tenzor viskoznosti kod dvojice pomenutih autora razlikuju. Primena varijacionih principa u relativistiekoj mehanici fluida takodje je razmatrana od nekoliko autora o ko.jima smo ovde govorili. Koliko nam je poznato, ve6 pomenuti Ajzenhartov rad [2] je prvi rad u kome su diferencijalne jedna6ine kretanja relativistiekog idealnog fluida za koji vaIi adijabatska jednanna propene staftja izvedene primenom jednog varijacionog principa. Potpunu. formalnu analogiju toga principa sa Fermaovim principom geometrijske optike istakao je u svome radu [3] Sing. Godine 1954. je Taub [11] formulisao varijacioni princip koji dovodi do jedna6ina polja opfte relativnosti i do jednanna kretanja relativistincog idealnog fluida. Najzad, jedan interesan7 tan prilaz tome pitanju razmatrao je 1970. godine Bernard Suc (Bernard G. Schutz, Jr.) [12] v drle6i se i dalje granica idealnog fluida. Iz ovog kratkog istorijskOg prikaza razvoja relativistieke hidromehanike mote se zapaziti da.je u literaturi uglavnom prow:- eavan relativistieki.idealni fluid; dok upadljivo mali broj radova uzima u - obzir pojavu viskozhosti. Osim dva rada, koje smo ovde citirali, u 6itavoj relativisti6koj:literaturi koja nam je bila • 'dostupna nismo naigli ni na jedan drugi rad u kome se razmatra unutragnje trenje izmedju 6estica .fluida pri njihovom relativnom pomeranju. Pri tome se, kao gto smo Ve64stakli, izrazi za tenzor viskoznosti u dva pomenuta rada razlikuju, izvodjenju jedna6i- ,•a kretanja posmatrane sredine autori polaze od uslova konzervativ- Virtual Library of Faculty of Mathematics - University of Belgrade elibrary.matf.bg.ac.rs nosti tenzora energije, to zna6i da koriste put kojiAe u rela- . tivisti6koj teoriji fluida uobi6ajen.. U ovome radu eemo posmatrati relativistieki viskozni flu- . , sa ciljem da pri izvodjenju diferencijalnih id jednaeina kretanja takve materijalne sredine primenimo tzv. Pfafovu metodu. Primeni Pfafove metode u mehanici i teorijskoj fizici po-. ove6en je veliki broj radova objavljenih u Beogradu krajem•eetrdesetih i pedesetih.godina. Ovde 6emo pomenuti pre svega radove A. Bilimovi6a [133 i [143. Baveoi se pitanjem utvrdjivanja fenomenolOgke osnove za Pfafovu metodu, on je u svojoj monografiji [14] f.ormulisao jedan op'Sti fenomenolaki diferencijalni'princip: Taj princip analizi- ra st. anje posmatranog sistema i uzroke koji izazivaju promenu toga stanja, i na osnovu to analize sastavija jedan matemati6ki izraz u obliku Pfafove forme, iz koga se dobivaju diferencijalne je7 dna6ine kretanja sistema kao Pfafove jedna6ine. On'je zatim primenio Pfafovu metodu na niz problema teorijske mehanike, nebeske • mehanike i geometrijske optike . C14]. Pri izvodjenju diferencijalnih jedna6ina kretanja krutog Lela, zatim osnovnih diferencijalnih jednaeina hidrodinamike i mehanike elasti6nih tela, ovu metodu je koristio T..Andjelie [153, [16], [173. Time je on ukazao na op6tu primenljivost Pfafove me- . tole u dinamici kako krutih tela, tako i deformabilnih sredina. Najzad, u svome radu [18] Dj.. Niu6icki je pokazao da se Pfafova metoda mote koristiti i u teorijskoj fizici. On je izvr6io . generalizaciju Pfafovog izraza i Pfafovih jedna6ina, dokazao Aa i za takav Pfafov izraz i jedna6ine vane sli6ne osobirie kao i za .6an 'Pfafov izraz jednaCine, a.zatim tako generalisana Pfafovu todu primenio u razli6itimoblastima teorijske fizike - termodina- . mici, elektromagnetizmu i'kvantnoj mehanici. . Virtual Library of Faculty of Mathematics - University of Belgrade elibrary.matf.bg.ac.rs Ovaj rad smo podelili na dve glave. U prvoj glavi nam je bio cilj da dodjemo do oblika za tenzor energije relativistiOkog viskoznog fluida. Po!ho .smo u prvom odeljku objasnili 6ta•nas je navelo da se na tome pitanju, kojc je u literaturi ve6 re6avano, ipak zadrZimo, u drugom odeljku raspravljamo koji oblik tenzora brzine deformacije i tenzora vrtlolenja treba usvojiti u relativistiftoj teoriji. Pri tome smo kao osnovicu u na s6im razmatranjima uzeli Zahtev da rezultati do kojih dolazimo moraju biti u skiadu sa postojedom Bornovom definicijom krutosti u opkoj teoriji relativnosti, a takodje i u skiadu sa rezultatima rada [193 o vrstama kretanja Bornovog relativisti6ki krutog tela. U posiednjem, tre6em odeljku prve glave, dolazimo do relativistiOkog tenzora.viskoznosti polaze6i od zahteva da se njegove prostorne komponente u odnosu na sopstveni koordinatni sistem svo- . de na komponente odgovaraju6eg klasiOnog tenzora, dok mu ostale komponente u odnOsu na taj sistem moraju biti .jednake null. Osnovnom pitanju koje u ovom radu razmatramo pitanju primene Pfafove metode u relativistiOkoj mehanici fluida - posve6ena . je druga glava rada. Posle kratkog izlaganja o tome kako je nastala tzv. Pfafova metoda u mehanici, i 6ta predstavlja njenu sadrZinu, u drugom, centralnom odeljku toga dela rada postavlja se izraz za Pfafovu formu koja odgovara posmatranom materijalnom sistemu. Iz toga izraza zatim, pOstupkbm koji Pfafova metoda nalaZe, izvode diferencijalne jednaOine kretanja relativi.stiOkog viskoznog fluida. Dobivene jednaOine, c,je predstavljaju relativistiOko uop6tenje Navije-Stoksovih jedna6ina . klasjOne hidrodinamike, razlikuju se od. odgovaraju6ih jednaina do . kojih jedo6ao Lihnerovi6 [83; to je razumljivo ako se ima u vidu da se tenzor viskoznosti koji je formulisac ne poklapa sa izrazom koji 'sm.() u ovom radu za taj ten- Virtual Library of Faculty of Mathematics - University of Belgrade elibrary.matf.bg.ac.rs zor U tre6em, poslednjem odeljku druge glave, ogranioili smo se na slaaj relativistiftog idealnog fluida za koji van adijabatska • jedna6ina promene stanja. Pokazali smo da se u tome slu6aju Pfafov izraz za deli6 fluidne sredine mote obrazovati na veoma jednostavan naCin, bez potreba da se trai relativisti6ko uop6tenje za.klasi6ni izraz koji predstavlja rad rezultuju6e sile pritiska kojom. okolina na uoceni delic deluje, prilikom njegovog pomeranja. od po- 6etnog do krajnjeg polo2aja. Pri tome fluidnu sredinu moramo posmatrati u prottoru 6ija je.metrika konformna metrici prostor-vremena, a koja se uvodi pomo6u indeksa fluida tako. da strujne linije'. fluida predstavljaju geodezijske linije toga prostora. Najzad, zakljusaujemo da bi sa taj.rezultat pod izvesnim uslovima mogao primeniti i u slu6aju ralativisti6kog . viskoznog fluida. URI: http://hdl.handle.net/123456789/4095 Files in this item: 1
Pfafova_metoda.PDF ( 9.783Mb )