20.500.12556/DKUM-73192 Ekstremalni problemi psa in žoge Dog and ball extremal problems V magistrskem delu so predstavljene optimalne poti, ki jih mora pes preteči oziroma preplavati, da pride do žoge, ki se nahaja v morju, pri čemer se spreminja položaj psa in oblika obale. Obravnavo ekstremalnih problemov pričnemo z najosnovnejšim problemom, ko želi pes, ki se nahaja na ravni obali, najhitreje priti do žoge, ki se nahaja v vodi. Problem rešimo računsko, z iskanjem globalnega minimuma obravnavane funkcije, in geometrijsko, s konstrukcijo optimalnih poti. Razvijemo tudi lemo, ki govori o razmerju med časom, ki ga pes potrebuje za plavanje in tek na določeni razdalji, ter konstruiramo kot, pod katerim mora plavati od obale proti žogi pri optimalni poti. Nadaljujemo s prvo izpeljanko osnovnega ekstremalnega problema, in sicer psa z ravne obale prestavimo v morje. Tudi ta problem rešimo računsko in geometrijsko. Poiščemo mejno množico točk, pri kateri je direktno plavanje enako hitro kot optimalna pot preko kopnega. Pri tem prvič za točko bifurkacije pojmujemo dolžino obale, drugič pa položaj žoge. Pri drugem izpeljanem ekstremalnem problemu psa z ravne obale prestavimo na kopno. Pri reševanju naletimo na analogijo z lomnim zakonom pri prehodu svetlobe iz ene snovi v drugo. Nalogo zaključimo z obravnavo tretjega izpeljanega ekstremalnega problema, kjer se pes nahaja na neravni obali. Problem rešimo geometrijsko, s konstrukcijo optimalnih poti. The examination of the dog and ball extremal problems begins with the most basic problem - the dog situated on a flat coast wants to get to the ball in the water as fast as possible. The problem is solved computationally, by searching for the global minimum of the examined function, and geometrically, by constructing optimal paths. We also establish a lemma that conveys the ratio between the time the dog needs for swimming and running a certain distance. We also construct the angle at which he has to swim from the coast to the ball on the optimal path. We continue with the first variation of the basic problem, which moves the dog form the flat coast into the sea. We solve this problem computationally and geometrically as well. We also find the boundary point set in which the direct swimming is as quick as the optimal path through land. The first time the coast length is regarded as the point of bifurcation, and the second time the ball position is regarded as the point of bifurcation. With the second variation of the problem, we move the dog from the flat coast to the land. Through the solving, we encounter the analogy of the law of refraction at the transition of light from one substance to another. We conclude with the examination of the third variation of the problem in which the dog is situated on an uneven coast. We solve the problem geometrically, by constructing optimal paths. ekstremalni problem globalni minimum optimalna pot točka bifurkacije bifurkacijska krivulja lomni zakon extremal problem global minimum optimal path point of bifurcation bifurcation locus Snell's law true true false [A. Rozman] Slovenski jezik Angleški jezik Magistrsko delo/naloga 2019-03-04 09:51:15 2019-04-10 12:41:30 2022-08-02 00:04:03 2019-07-11 19:48:00 2019 0 Maribor 0 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 517.54(043.2) 24473864 URN:SI:UM:DK:VWXBLEMA MAG_Rozman_Ana_2019.pdf MAG_Rozman_Ana_2019.pdf 1 0C8F623AAAF41B4492276AD0842161B9 80bdf6a2b33f1e010d06cfb3b1284d63433d54877de183c13098c6fb421fce9f 9f12ec37-7c10-11eb-bb7a-00155d0001ca 20.500.12556/dkum/c72f6203-1f2d-4a04-9a1e-d63c751cb8d6 https://dk.um.si/Dokument.php?lang=slv&id=132851 Fakulteta za naravoslovje in matematiko 0 0 0