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Fm(m)p Fn(+1)1084 4180 y Fv(.)51 b(V)-8 b(ektorsk)j(a)36 b(je)g(funk)m(cija)f(analiti)m(\024)-46 b(ck)-5 b(a)32 b(ak)m(o)k(je)f(takv)-5 b(a)35 b(sv)-5 b(ak)g(a)36 b(njezina)-165 4298 y(k)m(omp)s(onen)m(ta.)-165 4461 y Fw(T)-9 b(eorem)39 b(1.)106 b(\(Cauc)m(h)m(y\))35 b Fi(Ak)m(o)g(je)g(funk)m(cija)g Fz(f)41 b Fi(analiti)m(\024)-46 b(ck)-5 b(a)32 b(na)i(nek)m(o)5 b(j)37 b(ok)m(olini)32 b(to)m(\024)-46 b(ck)m(e)36 b Fv(\()p Fq(t)3167 4476 y Fn(0)3210 4461 y Fq(;)17 b Fw(x)3313 4476 y Fn(0)3355 4461 y Fv(\))p Fi(,)35 b(onda)-165 4580 y(p)s(osto)5 b(ji)27 b(jedinstv)m(eno)j(rje)-5 b(\024)-44 b(senje)30 b Fw(x)f Fi(gornje)f(Cauc)m(h)m(yjev)m(e)k(zada)m(\023)-46 b(ce,)30 b(k)m(o)5 b(je)30 b(je)f(analiti)m(\024)-46 b(ck)m(o)26 b(na)j(nek)m(o)5 b(j)30 b(ok)m(olini)-165 4699 y(to)m(\024)-46 b(ck)m(e)33 b Fq(t)124 4714 y Fn(0)167 4699 y Fi(.)p 3632 4741 30 43 v 57 4861 a Fv(Na)h(temelju)g(Cauc)m(h)m (yjev)m(og)j(teorema)d(imamo)e(sljede)m(\023)-46 b(ci)34 b(p)s(ostupak)h(za)g(rje)-5 b(\024)-44 b(sa)m(v)-5 b(anje)35 b(Cauc)m(h)m(yjev)m(e)-165 4980 y(zada)m(\023)-46 b(ce.)42 b(Pretp)s(osta)m(vimo)26 b(da)g(se)h(nep)s(oznata)g(funk)m(cija)f(mo)m (\024)-46 b(ze)26 b(razviti)f(u)h(k)m(on)m(v)m(ergen)m(tan)j(red)e(p)s (otencija)-165 5099 y(i)i(ta)5 b(j)31 b(red)g(uvrstimo)e(u)i(jednad)m (\024)-46 b(zbu,)33 b(u)d(k)m(o)5 b(jo)g(j)32 b(smo)e(sv)-5 b(aku)31 b(funk)m(ciju)g(k)m(o)5 b(ja)31 b(se)h(p)s(o)5 b(ja)m(vljuje)30 b(razvili)e(u)j(red)-165 5217 y(p)s(otencija.)89 b(Izjedna)m(\024)-46 b(ca)m(v)-5 b(anjem)49 b(dobiv)-5 b(amo)46 b(algebarsk)m(e)j(jednad)m(\024)-46 b(zb)s(e)49 b(iz)e(k)m(o)5 b(jih)48 b(u)h(na)m(\024)-46 b(celu)48 b(mo)m(\024)-46 b(zemo)-165 5336 y(o)s(drediti)30 b(k)m(o)s(e\014cien)m (te)k(u)f(razv)m(o)5 b(ju)34 b(rje)-5 b(\024)-44 b(senja.)57 5455 y(Cauc)m(h)m(yjev)30 b(teorem)e(zah)m(tijev)-5 b(a)28 b(prev)m(eliku)g(regularnost.)41 b(Matemati)m(\024)-46 b(cari)26 b(se)j(na)5 b(j)m(\024)-46 b(ce)-5 b(\024)-44 b(s)m(\023)e(ce)30 b(p)s(oziv)-5 b(a)5 b(ju)-165 5574 y(na)22 b(Picardo)m(v)g(teorem,)i(u)e(k)m(o)5 b(jem)22 b(se)h(pretp)s(osta)m(vlja)e(da)h(je)g(funk)m(cija)g Fz(f)29 b Fv(neprekin)m(uta)23 b(p)s(o)e Fq(t)i Fv(i)e(Lipsc)m(hitzo)m (v)-5 b(a)-165 5692 y(p)s(o)36 b Fw(x)p Fv(.)58 b(T)-8 b(a)5 b(j)37 b(se)h(teorem)f(dok)-5 b(azuje)38 b(s)f(p)s(omo)m(\023)-46 b(cu)37 b(Banac)m(ho)m(v)m(og)h(teorema)e(o)f(\024)-46 b(cvrsto)5 b(j)38 b(to)m(\024)-46 b(cki,)38 b(k)m(o)5 b(ji)37 b(da)5 b(je)37 b(i)-165 5811 y(prakti)m(\024)-46 b(cn)m(u)33 b(meto)s(du)f(za)g(k)m(onstruk)m(ciju)i(\(Picardo)m(vih\))e (aproksimacija)f(rje)-5 b(\024)-44 b(senja.)-165 6010 y Ft(2)p eop %%Page: 3 8 3 7 bop 1928 -116 a Fr(Cauc)n(h)n(yjev)-5 b(a)26 b(zada)n(\023)-39 b(ca)26 b(za)h(obi)n(\024)-39 b(cn)n(u)26 b(diferencijaln)n(u)h(jednad) n(\024)-39 b(zbu)71 83 y Fw(T)-9 b(eorem)35 b(2.)99 b(\(Picard\))29 b Fi(Ak)m(o)j(je)f(funk)m(cija)g Fz(f)38 b Fi(neprekin)m(uta)32 b(p)s(o)e(v)-5 b(arijabli)28 b Fq(t)k Fi(i)e(Lipsc)m(hitzo)m(v)-5 b(a)31 b(p)s(o)f Fw(x)i Fi(na)71 202 y(nek)m(o)5 b(j)40 b(ok)m(olini)c(to)m(\024)-46 b(ck)m(e)40 b Fv(\()p Fq(t)994 217 y Fn(0)1036 202 y Fq(;)17 b Fw(x)1139 217 y Fn(0)1182 202 y Fv(\))p Fi(,)40 b(onda)e(p)s(osto)5 b(ji)38 b(jedinstv)m(eno)h (rje)-5 b(\024)-44 b(senje)40 b(Cauc)m(h)m(yjev)m(e)i(zada)m(\023)-46 b(ce,)41 b(k)m(o)5 b(je)39 b(je)71 320 y(diferencijabilna)30 b(funk)m(cija)i(na)h(nek)m(o)5 b(j)34 b(ok)m(olini)c(to)m(\024)-46 b(ck)m(e)34 b Fq(t)2125 335 y Fn(0)2167 320 y Fi(.)p 3868 363 30 43 v 293 489 a Fv(Na)29 b(kra)5 b(ju)29 b(sp)s(omenimo)e (da)i(se)h(pretp)s(osta)m(vk)m(e)h(na)e(funk)m(ciju)g Fz(f)36 b Fv(mogu)28 b(jo)-5 b(\024)-44 b(s)29 b(oslabiti,)e(ali)g(se)i (pritom)71 607 y(gubi)j(jedinstv)m(enost.)71 775 y Fw(T)-9 b(eorem)46 b(3.)124 b(\(P)m(eano\))40 b Fi(Ak)m(o)i(je)f Fz(f)47 b Fi(neprekidna)42 b(funk)m(cija,)h(onda)d(lok)-5 b(alno)39 b(p)s(osto)5 b(ji)40 b(barem)g(jedno)71 894 y(rje)-5 b(\024)-44 b(senje)34 b(Cauc)m(h)m(yjev)m(e)i(zada)m(\023)-46 b(ce.)p 3868 937 V 623 1260 a Fw(Generalizirana)37 b(rje)-6 b(\024)-50 b(senja)39 b(obi)m(\024)-53 b(cnih)36 b(diferencijalnih)g (jednad)m(\024)-53 b(zbi)293 1428 y Fv(Rezultat)27 b(P)m(eano)m(v)m(og) i(teorema)d(mo)m(\024)-46 b(zemo)27 b(dobiti)e(za)e(\024)-44 b(siru)27 b(klasu)g(funk)m(cija.)42 b(Nek)-5 b(a)28 b(je)f Fz(f)35 b Fv(:)27 b Fq(S)34 b Fo(!)27 b Fw(R)3854 1392 y Fm(d)71 1546 y Fv(izmjeriv)-5 b(a)30 b(funk)m(cija,)j(gdje)f(je)1152 1763 y Fq(S)i Fv(=)27 b Fo(h)p Fq(t)1423 1778 y Fn(0)1488 1763 y Fo(\000)c Fq(a;)17 b(t)1718 1778 y Fn(0)1782 1763 y Fv(+)22 b Fq(a)p Fo(i)g(\002)h Fv(K\()p Fw(x)2265 1778 y Fn(0)2308 1763 y Fq(;)17 b(b)p Fv(\))27 b Fo(\022)i Fw(R)2648 1722 y Fn(1+)p Fm(d)2789 1763 y Fq(:)76 1954 y Fv(\024)71 1979 y(Zelimo)h(rije)-5 b(\024)-44 b(siti)30 b(Cauc)m(h)m(yjevu)35 b(zada)m(\023)-46 b(cu)71 2253 y(\(1\))1572 2112 y Fl(\032)1722 2178 y Fv(_)1706 2179 y Fw(x)q Fv(\()p Fq(t)p Fv(\))27 b(=)h Fz(f)7 b Fv(\()p Fq(t;)17 b Fw(x)p Fv(\()p Fq(t)p Fv(\)\))1664 2323 y Fw(x)p Fv(\()p Fq(t)1796 2338 y Fn(0)1839 2323 y Fv(\))27 b(=)h Fw(x)2067 2338 y Fn(0)71 2532 y Fv(na)33 b(nek)m(o)5 b(j)35 b(ok)m(olini)c Fq(t)821 2547 y Fn(0)863 2532 y Fv(,)j(te)f(pretp)s(osta)m(vljamo)f(ograni)m(\024)-46 b(cenje)33 b Fo(j)p Fz(f)7 b Fv(\()p Fq(t;)17 b Fw(x)p Fv(\))p Fo(j)28 b Fh(6)h Fq(M)10 b Fv(\()p Fq(t)p Fv(\))30 b(\(ss)j Fq(t)p Fv(\))p Fq(;)17 b Fw(x)29 b Fo(2)g Fv(K\()p Fw(x)3579 2547 y Fn(0)3622 2532 y Fq(;)17 b(b)p Fv(\),)34 b(za)71 2651 y(neku)g(funk)m(ciju)g Fq(M)40 b Fo(2)29 b Fv(L)972 2615 y Fn(1)1014 2651 y Fv(\()p Fw(R)p Fv(\).)45 b(Pro)-5 b(\024)-44 b(sirimo)31 b(li)h Fz(f)39 b Fv(n)m(ulom)32 b(v)-5 b(an)34 b Fq(S)6 b Fv(,)33 b(pretp)s(osta)m(vk)m(e)g(\023)-46 b(ce)33 b(biti)f(ispunjenje)i(na)68 2770 y(\024)-46 b(cita)m(v)m(om)32 b Fw(R)521 2734 y Fn(1+)p Fm(d)662 2770 y Fv(,)h(te)g(stoga)f(b)s(ez)h (smanjenja)f(op)m(\023)-46 b(cenitosti)32 b(mo)m(\024)-46 b(zemo)31 b(promatrati)g Fq(S)i Fv(=)28 b Fw(R)21 b Fo(\002)i Fv(K\()p Fw(x)3670 2785 y Fn(0)3712 2770 y Fq(;)17 b(b)p Fv(\))q(.)293 2889 y(F)-8 b(unk)m(ciju)48 b Fq(\032)53 b Fo(2)g Fv(C)1000 2852 y Fp(1)1000 2913 y Fm(c)1082 2889 y Fv(\()p Fw(R)1204 2852 y Fm(d)1247 2889 y Fv(;)17 b Fw(R)1375 2847 y Fn(+)1375 2918 y(0)1438 2889 y Fv(\))48 b(takvu)g(da)g(je)f(supp)18 b Fq(\032)53 b Fo(\022)h Fv(K\()p Fw(0)p Fq(;)17 b Fv(1\))o(\))47 b(i)2998 2808 y Fl(R)3045 2923 y Fg(R)3111 2902 y Fj(d)3171 2889 y Fq(\032)53 b Fv(=)g(1)47 b(naziv)-5 b(amo)71 3007 y(izglad)p 288 2955 30 2 v -1 w(iv)g(a)m(\024)-46 b(cem.)42 b(T)-8 b(om)31 b(nazivu)h(je)g(p)s(orijeklo)f(u)h(tome)26 b(\024)-44 b(sto)32 b(pri)g(k)m(on)m(v)m(oluciji)f(nek)m(e)j(proizv)m(oljne)d (funk)m(cije)71 3126 y Fq( )h Fo(2)c Fv(L)321 3090 y Fp(1)321 3158 y Fn(lo)r(c)421 3126 y Fv(\()p Fw(R)543 3090 y Fm(d)586 3126 y Fv(\))33 b(s)g Fq(\032)p Fv(:)1059 3302 y Fq( )1122 3317 y Fm(\032)1166 3302 y Fv(\()p Fw(x)p Fv(\))28 b(=)f(\()p Fq( )f Fo(\003)c Fq(\032)p Fv(\)\()p Fw(x)p Fv(\))28 b(=)1986 3166 y Fl(Z)2041 3391 y Fg(R)2107 3370 y Fj(d)2167 3302 y Fq( )t Fv(\()p Fw(x)22 b Fo(\000)h Fw(y)r Fv(\))p Fq(\032)p Fv(\()p Fw(y)r Fv(\))17 b Fq(d)p Fw(y)g Fq(;)71 3545 y Fv(dobiv)-5 b(amo)33 b Fq( )578 3560 y Fm(\032)654 3545 y Fo(2)f Fv(C)822 3509 y Fp(1)904 3545 y Fv(\()p Fw(R)1026 3509 y Fm(d)1069 3545 y Fv(\),)k(jer)f(deriv) -5 b(acije)34 b(parcijalnom)e(in)m(tegracijom)h(prelaze)i(na)f Fq(\032)i Fv(u)f(in)m(tegralu.)71 3664 y(Izglad)p 295 3612 V(iv)-5 b(a)m(\024)-46 b(ci)28 b(\023)-46 b(ce)33 b(biti)28 b(\024)-46 b(cesto)33 b(k)m(ori)-5 b(\024)-44 b(steni)32 b(u)h(sljede)m(\023)-46 b(cem)33 b(p)s(ogla)m(vlju.)293 3782 y(Promatramo,)d(za)j(neki)g Fq(")27 b(>)h Fv(0,)k(regularizaciju)e (\(izglad)p 2368 3731 V -1 w(enje\))j(funk)m(cije)g Fz(f)1268 4049 y(f)1298 4064 y Fm(")1337 4049 y Fv(\()p Fq(t;)17 b Fw(x)p Fv(\))28 b(=)1682 3913 y Fl(Z)1738 4139 y Fg(R)1804 4117 y Fj(d)1864 4049 y Fz(f)7 b Fv(\()p Fq(t;)17 b Fw(x)k Fo(\000)i Fq(")p Fw(y)r Fv(\))p Fq(\032)p Fv(\()p Fw(y)r Fv(\))17 b Fq(d)p Fw(y)g Fq(;)71 4333 y Fv(za)45 b(k)m(o)5 b(ju,)43 b(\024)-44 b(sto)45 b(se)h(lak)m(o)e(vidi,)i(vrijedi)e(ista)g (ograda)g Fo(j)p Fz(f)2121 4348 y Fm(")2160 4333 y Fv(\()p Fq(t;)17 b Fw(x)p Fv(\))p Fo(j)48 b Fh(6)h Fq(M)10 b Fv(\()p Fq(t)p Fv(\))49 b(\(ss)33 b Fq(t)p Fv(\))p Fq(;)17 b Fw(x)48 b Fo(2)h Fw(R)3410 4297 y Fm(d)3454 4333 y Fv(.)79 b(Za)44 b(tak)m(o)71 4451 y(dobiv)m(en)m(u)28 b Fz(f)510 4466 y Fm(")577 4451 y Fv(imamo)d(da)j(je)f(funk)m(cija)h Fq(t)g Fo(7!)f Fz(f)1707 4466 y Fm(")1747 4451 y Fv(\()p Fq(t;)17 b Fw(x)p Fv(\()p Fq(t)p Fv(\)\))27 b(in)m(tegrabilna)e(za)j (sv)-5 b(aku)28 b(neprekin)m(utu)h(funk)m(ciju)71 4570 y Fw(x)p Fv(,)k(te)f(pritom)f(zado)m(v)m(olja)m(v)-5 b(a)32 b(o)s(cjen)m(u:)1503 4838 y Fo(jr)1614 4853 y Fg(x)1665 4838 y Fz(f)1695 4853 y Fm(")1734 4838 y Fv(\()p Fq(t;)17 b Fw(x)p Fv(\))p Fo(j)28 b Fh(6)2119 4770 y Fq(C)7 b(M)j Fv(\()p Fq(t)p Fv(\))p 2119 4815 293 4 v 2242 4906 a Fq(")2438 4838 y(;)71 5078 y Fv(pri)29 b(\024)-46 b(cem)m(u)33 b(je)1616 5241 y Fq(C)h Fv(=)1824 5106 y Fl(Z)1940 5241 y Fo(jr)2051 5256 y Fg(y)2103 5241 y Fq(\032)p Fo(j)17 b Fq(d)p Fw(y)h Fq(:)71 5462 y Fv(Uzmimo)30 b(fiksan)j Fq(")27 b(>)h Fv(0.)43 b(Pretho)s(dne)33 b(o)s(cjene)g(nam)f(omogu)m (\023)-46 b(cuje)32 b(da,)h(za)f Fq(T)47 b Fv(tak)-5 b(a)m(v)33 b(da)f(je)1581 5747 y Fq(C)1674 5612 y Fl(Z)1774 5638 y Fm(T)1730 5837 y Fp(\000)p Fm(T)1865 5747 y Fq(M)10 b Fv(\()p Fq(t)p Fv(\))p Fq(dt)29 b(<)f(")17 b(;)3817 6010 y Ft(3)p eop %%Page: 4 9 4 8 bop -165 -116 a Fr(Nelinearni)27 b(zak)n(oni)f(sa)n(\024)-39 b(cuv)-5 b(anja)-165 83 y Fv(rije)g(\024)-44 b(simo)30 b(in)m(tegraln)m(u)i(jednad)m(\024)-46 b(zbu)33 b(na)g([)p Fo(\000)p Fq(T)8 b(;)17 b(T)d Fv(]:)-165 374 y(\(2\))1134 b Fw(x)1153 389 y Fm(")1193 374 y Fv(\()p Fq(t)p Fv(\))28 b(=)g Fw(x)1495 389 y Fn(0)1559 374 y Fv(+)1657 238 y Fl(Z)1757 264 y Fm(t)1713 464 y Fn(0)1806 374 y Fz(f)1836 389 y Fm(")1875 374 y Fv(\()p Fq(s;)17 b Fw(x)2062 389 y Fm(")2102 374 y Fv(\()p Fq(s)p Fv(\)\))p Fq(ds)g(;)-165 649 y Fv(k)m(o)5 b(ja)27 b(je,)h(zasad)g(formalno,)d(ekviv)-5 b(alen)m(tna)27 b(Cauc)m(h)m(yjev)m(o)5 b(j)30 b(zada)m(\023)-46 b(ci)26 b(\(2\))h(uk)m(olik)m(o)f(funk)m(ciju)h Fz(f)33 b Fv(zamijenimo)-165 768 y(s)g(izglad)p 123 716 30 2 v -2 w(jenjem)g Fz(f)500 783 y Fm(")539 768 y Fv(.)44 b(Rje)-5 b(\024)-44 b(senje)33 b(tra)m(\024)-46 b(zimo)31 b(p)s(omo)m(\023)-46 b(cu)32 b(uzastopnih)h(aproksimacija)1047 1053 y Fw(x)1106 1012 y Fm(k)1152 1053 y Fv(\()p Fq(t)p Fv(\))28 b(=)g Fw(x)1454 1068 y Fn(0)1519 1053 y Fv(+)1617 917 y Fl(Z)1716 944 y Fm(t)1672 1143 y Fn(0)1765 1053 y Fz(f)1795 1068 y Fm(")1834 1053 y Fv(\()p Fq(s;)17 b Fw(x)2021 1012 y Fm(k)r Fp(\000)p Fn(1)2166 1053 y Fv(\()p Fq(s)p Fv(\)\))p Fq(ds;)1051 1265 y Fw(x)1110 1224 y Fn(0)1152 1265 y Fv(\()p Fq(t)p Fv(\))28 b(=)g Fw(x)1454 1280 y Fn(0)1497 1265 y Fq(:)-165 1474 y Fv(Za)k(ta)5 b(j)32 b(niz)g(jednosta)m(vno)i(se)f(dobije)f(o)s(cijena)829 1815 y Fo(j)p Fw(x)916 1774 y Fm(k)962 1815 y Fv(\()p Fq(t)p Fv(\))22 b Fo(\000)h Fw(x)1254 1774 y Fm(k)r 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b(zado)m(v)m(olja)m(v)-5 b(a)35 b(in)m(tegraln)m(u)g(jednad)m(\024)-46 b(zbu)-165 2743 y(\(2\).)42 b(Stoga)32 b(je)f Fw(x)463 2758 y Fm(")535 2743 y Fv(apsolutno)g(neprekin)m(uta)i(za)e Fo(j)p Fq(t)p Fo(j)d Fh(6)g Fq(T)45 b Fv(i)31 b(vrijedi)2341 2742 y(_)2325 2743 y Fw(x)2384 2758 y Fm(")2424 2743 y Fv(\()p Fq(t)p Fv(\))d(=)g Fz(f)2697 2758 y Fm(")2736 2743 y Fv(\()p Fq(t;)17 b Fw(x)2912 2758 y Fm(")2952 2743 y Fv(\()p Fq(t)p Fv(\)\))32 b(sk)m(oro)g(svuda.)-165 2862 y(Budu)m(\023)-46 b(ci)36 b(da)g(su)h(realne)g(vrijednosti)e Fw(x)1278 2877 y Fm(")1318 2862 y Fv(\()p Fo(\006)p Fq(T)14 b Fv(\))37 b(dobro)f(definirane,)g(to)g(mo)m(\024)-46 b(zemo)36 b(rje)-5 b(\024)-44 b(sa)m(v)-5 b(ati)35 b(in)m(tegralne)-165 2980 y(jednad)m(\024)-46 b(zb)s(e)31 b(s)f(njima)e(k)-5 b(ao)29 b(p)s(o)m(\024)-46 b(cetnim)29 b(uvijetima)e(dok)k(ne)f (dobijemo)e Fw(x)2446 2995 y Fm(")2486 2980 y Fv(\()p Fq(t)p Fv(\))i(za)g Fo(j)p Fq(t)p Fo(j)d Fq(<)g(a)p Fv(,)k(za)f(k)m(o)5 b(ji)29 b(vrijedi)1165 3257 y Fo(j)p Fw(x)1252 3272 y Fm(")1292 3257 y Fv(\()p Fq(t)p Fv(\))22 b Fo(\000)h Fw(x)1584 3272 y Fn(0)1627 3257 y Fo(j)k Fh(6)1787 3122 y Fl(Z)1887 3148 y Fm(a)1843 3347 y Fp(\000)p Fm(a)1964 3257 y Fq(M)10 b Fv(\()p Fq(s)p Fv(\))p Fq(ds)17 b(;)1389 3462 y Fo(j)1433 3461 y Fv(_)1417 3462 y Fw(x)1476 3477 y Fm(")1516 3462 y Fv(\()p Fq(t)p Fv(\))p Fo(j)27 b Fh(6)h Fq(M)10 b Fv(\()p Fq(t)p Fv(\))-165 3676 y(za)34 b(sk)m(oro)h(sv)-5 b(aki)35 b Fq(t)c Fo(2)g(h\000)p Fq(a;)17 b(a)p Fo(i)p Fv(.)50 b(Kad)34 b(bi)g(dobili)e(da)i(p)s(osto)5 b(ji)34 b Fq(a)2124 3691 y Fn(0)2197 3676 y Fq(<)d(a)k Fv(tak)-5 b(a)m(v)35 b(da)f(je)h Fw(x)2959 3691 y Fm(")3033 3676 y Fv(definiran)e(samo)-165 3795 y(za)39 b Fq(t)h(<)g(a)208 3810 y Fn(0)251 3795 y Fv(,)h(imali)36 b(bismo)j(omed)p 1054 3743 30 2 v(en)m(u)i(apsolutno)e(neprekin)m(utu)i(funk)m(ciju)f Fw(x)2718 3810 y Fm(")2797 3795 y Fv(k)m(o)5 b(ju)41 b(bi)e(bilo)f(mogu)m(\023)-46 b(ce)-165 3914 y(do)s(definirati)24 b(i)j(na)g(rubu)h(za)g Fq(t)g Fv(=)f Fq(a)1105 3929 y Fn(0)1148 3914 y Fv(.)42 b(P)m(ona)m(vljanjem)27 b(pretho)s(dnog)g (argumen)m(ta)g(dolazimo)e(do)j(tra)m(\024)-46 b(zene)-165 4033 y(funk)m(cije)33 b Fw(x)257 4048 y Fm(")329 4033 y Fv(definirane)f(na)g Fo(h\000)p Fq(a;)17 b(a)p Fo(i)p Fv(.)57 4152 y(Uo)m(\024)-46 b(cimo)26 b(da)h(su)h(o)s(cjene)g(dobiv)m (ene)h(za)e Fw(x)1530 4167 y Fm(")1598 4152 y Fv(neo)m(visne)h(o)f Fq(")p Fv(,)i(pa)24 b(\023)-46 b(cemo)27 b(dobiti)f(iste)h(o)s(cjene)h (jednolik)m(o)-165 4271 y(na)k(\()p Fw(x)67 4286 y Fm(")107 4271 y Fv(\))145 4288 y Fm(")p Fp(2)p Fm(K)339 4271 y Fv(k)-5 b(ad)32 b(je)h Fq(K)i Fo(\022)28 b Fw(R)933 4235 y Fn(+)997 4271 y Fv(.)57 4391 y(Dakle,)45 b(\()p Fw(x)472 4406 y Fm(")512 4391 y Fv(\))550 4414 y Fm(")p Fp(2)p Fg(R)703 4392 y Fk(+)806 4391 y Fv(je)e(uniformno)f(ograni)m(\024)-46 b(cen)42 b(i)h(ekvineprekidan,)k(iz)40 b(\024)-46 b(cega)44 b(nam)e(p)s(o)h(Arzel\023)-49 b(a-)-165 4509 y(Ascolijev)m(om)44 b(teorem)m(u)i(slijedi)d(pretk)m(ompaktnost.)84 b(Za)45 b(nek)m(o)h(gomili)-5 b(\024)-44 b(ste)43 b Fw(x)i Fv(dobiv)m(eno)h(k) -5 b(ao)45 b(limes)-165 4628 y(funk)m(cija)37 b Fw(x)267 4643 y Fm(")303 4658 y Fj(k)384 4628 y Fv(gdje)g(je)g(\()p Fq(")791 4646 y Fm(k)837 4628 y Fv(\))g(niz)g(p)s(ozitivnih)e(bro)5 b(jev)-5 b(a)37 b(k)m(o)5 b(ji)37 b(te)m(\024)-46 b(zi)36 b(u)h(n)m(ulu,)i(imamo)34 b(da)j(je)g(neprekidna)-165 4747 y(funk)m(cija.)67 b(Stoga)40 b(je)h(dobro)g(je)f(definirana)f (deriv)-5 b(acija)1985 4746 y(_)1969 4747 y Fw(x)41 b Fv(u)g(smislu)e(teorije)h(distribucija.)66 b(Za)40 b(test)-165 4865 y(funk)m(ciju)33 b Ff( )d Fo(2)f Fv(C)480 4829 y Fp(1)480 4890 y Fm(c)561 4865 y Fv(\()p Fw(R)p Fv(;)17 b Fw(R)811 4829 y Fm(d)854 4865 y Fv(\))32 b(dobiv)-5 b(amo)1241 5137 y Fo(jh)1324 5136 y Fv(_)1308 5137 y Fw(x)1367 5152 y Fm(")1403 5167 y Fj(k)1447 5137 y Fq(;)17 b Ff( )t Fo(i)o(j)27 b Fh(6)1770 5001 y Fl(Z)1886 5137 y Fq(M)10 b Fo(j)p Ff( )t Fo(j)p Fq(dt)17 b(;)-171 5402 y Fv(\024)-44 b(sto)33 b(na)g(limesu)e(p)s(o)m(vla)m(\024)-46 b(ci)1282 5568 y Fo(j)o(h)1364 5567 y Fv(_)1348 5568 y Fw(x)p Fq(;)17 b Ff( )t Fo(i)o(j)27 b Fh(6)1729 5432 y Fl(Z)1846 5568 y Fq(M)10 b Fo(j)p Ff( )t Fo(j)p Fq(dt)17 b(;)-165 5811 y Fv(iz)29 b(\024)-46 b(cega)32 b(zaklju)m(\024)-46 b(cujemo)32 b(da)h(je)1000 5810 y(_)984 5811 y Fw(x)27 b Fo(2)h Fv(L)1225 5775 y Fn(1)1268 5811 y Fv(\()p Fw(R)p Fv(;)17 b Fw(R)1518 5775 y Fm(d)1561 5811 y Fv(\),)32 b(i)g(da)g(m)m(u)h(je)f(norma)g(manja)f(o)s(d)h Fq(M)44 b Fv(sk)m(oro)33 b(svuda.)-165 6010 y Ft(4)p eop %%Page: 5 10 5 9 bop 1928 -116 a Fr(Cauc)n(h)n(yjev)-5 b(a)26 b(zada)n(\023)-39 b(ca)26 b(za)h(obi)n(\024)-39 b(cn)n(u)26 b(diferencijaln)n(u)h(jednad) n(\024)-39 b(zbu)293 83 y Fv(P)m(osta)m(vlja)31 b(se)h(pitanje)f(na)g (k)m(o)5 b(ji)31 b(na)m(\024)-46 b(cin)30 b(tak)-5 b(a)m(v)32 b Fw(x)f Fv(zado)m(v)m(olja)m(v)-5 b(a)31 b(diferencijaln)m(u)f(jednad) m(\024)-46 b(zbu.)44 b(Kad)71 202 y(bi)32 b Fz(f)39 b Fv(bila)30 b(Carath)m(\023)-46 b(eo)s(doryjev)-5 b(a)33 b(\(za)g(sk)m(oro)g(sv)-5 b(aki)33 b Fq(t)f Fv(neprekidna)i(na)e Fw(R)2680 166 y Fm(d)2724 202 y Fv(\),)g(tada)g(bismo)g(imali)1607 401 y Fz(f)1637 416 y Fm(")1677 401 y Fv(\()p Fq(t;)17 b Fw(x)p Fv(\))60 b Fo(!)g Fz(f)7 b Fv(\()p Fq(t;)17 b Fw(x)p Fv(\))71 599 y(p)s(o)32 b(to)m(\024)-46 b(ck)-5 b(ama,)32 b(za)g Fq(")c Fo(&)f Fv(0)32 b(na)h Fq(S)6 b Fv(,)27 b(\024)-44 b(sto)33 b(op)s(et)f(da)5 b(je)33 b(jedno)g(pro)-5 b(\024)-44 b(sirenje)33 b(P)m(eano)m(v)m(og)h (teorema.)293 718 y(Prije)d(nego)h(li)e(nasta)m(vimo)h(s)h(daljnjim)d (razmatranjem,)i(na)m(\024)-46 b(cinit)27 b(\023)-46 b(cu)32 b(malu)f(digresiju.)42 b(Mo)m(\024)-46 b(zemo)71 837 y(primijetiti)42 b(da)47 b(je)g(za)f(sv)-5 b(aku)48 b(izmjerivu)d(funk)m(ciju)i Fq(f)63 b Fv(:)52 b Fw(R)2346 801 y Fn(2)2439 837 y Fo(!)g Fw(R)46 b Fv(i)g(ess)17 b(sup)3079 860 y Fm(y)3140 837 y Fq(f)11 b Fv(\()p Fo(\001)p Fq(;)17 b(y)t Fv(\))50 b(:)h Fw(R)h Fo(!)f Fw(R)71 955 y Fv(izmjeriv)-5 b(a)30 b(funk)m(cija.)44 b(T)-8 b(o)32 b(slijedi)f(iz)h(F)-8 b(oubinijev)m(og)31 b(teorema)h(tak)m(o)h(da)f (primijetimo)d(da)j(je)1425 1184 y Fq(e)1470 1142 y Fn(ess)14 b(sup)1694 1161 y Fj(y)1746 1142 y Fm(f)8 b Fn(\()p Fp(\001)p Fm(;y)s Fn(\))1967 1184 y Fv(=)2070 1069 y Fl(\015)2070 1129 y(\015)2070 1189 y(\015)2125 1184 y Fq(e)2170 1142 y Fm(f)g Fn(\()p Fp(\001)p Fm(;y)s Fn(\))2363 1069 y Fl(\015)2363 1129 y(\015)2363 1189 y(\015)2418 1253 y Fp(1)2516 1184 y Fq(;)71 1410 y Fv(o)s(dnosno)32 b(imamo)e(da)j(je)1346 1552 y(ess)17 b(sup)i Fq(f)11 b Fv(\()p Fo(\001)p Fq(;)17 b(y)t Fv(\))25 b(=)j(ln)2133 1438 y Fl(\015)2133 1498 y(\015)2133 1557 y(\015)2188 1552 y Fq(e)2233 1511 y Fm(f)8 b Fn(\()p Fp(\001)p Fm(;y)s Fn(\))2426 1438 y Fl(\015)2426 1498 y(\015)2426 1557 y(\015)2481 1621 y Fp(1)2595 1552 y Fq(;)71 1754 y Fv(a)32 b Fo(k)22 b(\001)g(k)324 1769 y Fp(1)437 1754 y Fv(je)33 b(izmjeriv)-5 b(a)30 b(funk)m(cija.)293 1872 y(Definira)5 b(jmo)30 b(funk)m(ciju)1244 2071 y Fq(H)8 b Fv(\()p Fq(t;)17 b Fw(x)p Fq(;)g Ff(\030)s Fv(\))27 b(=)35 b(lim)1775 2140 y Fm(\016)s Fp(&)p Fn(0)1981 2071 y Fv(ess)17 b(sup)1976 2163 y Fp(j)p Fg(x)p Fp(\000)p Fg(y)q Fp(j)p Fm(<\016)2303 2071 y Fz(f)7 b Fv(\()p Fq(t;)17 b Fw(y)r Fv(\))k Fo(\001)h Ff(\030)d Fq(;)71 2355 y Fv(za)35 b(k)m(o)5 b(ju)36 b(se)g(lak)m(o)e(vidi)g(da)h(je)g(k)m(on)m(v)m(eksna) j(i)c(p)s(ozitivno)g(homogena)g(p)s(o)h Ff(\030)s Fv(,)g(pa)g(p)s(o)g ([H\177)-49 b(o2,)35 b(Theorem)h(5])71 2474 y(zaklju)m(\024)-46 b(cujemo)34 b(da)h(je)g(p)s(o)s(dupiru)m(\023)-46 b(ca)35 b(za)g(neki)g(zatv)m(oren)h(k)m(on)m(v)m(eksan)i(p)s(o)s(dskup)e Fw(R)3110 2437 y Fm(d)3188 2474 y Fv(k)m(o)5 b(ji)35 b(je)g(pridru)m(\024)-46 b(zen)71 2592 y(\()p Fq(t;)17 b Fw(x)p Fv(\),)32 b(i)g(k)m(o)5 b(jeg)33 b(ozna)m(\024)-46 b(cimo)32 b(s)h Fq(F)14 b Fv(\()p Fq(t;)j Fw(x)p Fv(\);)32 b(o)s(dnosno)g(vrijedi)1438 2791 y Fq(H)8 b Fv(\()p Fq(t;)17 b Fw(x)p Fq(;)g Ff(\030)s Fv(\))27 b(=)111 b(sup)1969 2883 y Fg(y)q Fp(2)p Fm(F)11 b Fn(\()p Fm(t;)p Fg(x)p Fn(\))2300 2791 y Fw(y)24 b Fo(\001)e Ff(\030)d Fq(:)71 3075 y Fv(Budu)m(\023)-46 b(ci)40 b(da)g(je)g Fq(H)8 b Fv(\()p Fq(t;)17 b Fw(x)p Fq(;)g Ff(\030)s Fv(\))39 b(o)s(dozgo)g(p)s(oluneprekin)m(uta)h(funk)m(cija)g(u)g Fw(x)p Fv(,)i(to)d(je)h Fq(F)14 b Fv(\()p Fq(t;)j Fw(x)p Fv(\))40 b(o)s(dozgo)f(p)s(olu-)71 3194 y(neprekin)m(uta)f(skup)s(o)m (vna)h(funk)m(cija)e(p)s(o)g Fw(x)p Fv(.)1721 3168 y(\024)1710 3194 y(Cak)h(i)f(vi)-5 b(\024)-44 b(se,)39 b Fq(F)14 b Fv(\()p Fq(t;)j Fw(x)p Fv(\))36 b(je)i(na)5 b(jmanji)35 b(zatv)m(oren)k(k)m(on)m(v)m(eksan)71 3312 y(skup)32 b(tak)-5 b(a)m(v)31 b(da)f(sv)-5 b(ak)g(a)31 b(njego)m(v)-5 b(a)31 b(ok)m(olina)e(sadr)m(\024)-46 b(zi)31 b(sk)m(oro)g(sv)-5 b(aki)31 b Fz(f)7 b Fv(\()p Fq(t;)17 b Fw(y)r Fv(\))29 b(za)i Fw(y)h Fv(iz)e(nek)m(e)j(ok)m(oline)d(to)m(\024)-46 b(ck)m(e)31 b Fw(x)p Fv(.)293 3431 y(P)m(ogleda)5 b(jmo)31 b(funk)m(ciju)i Fw(x)1258 3446 y Fm(")1331 3431 y Fv(u)g(nekim)e(to)m (\024)-46 b(ck)-5 b(ama)32 b Fq(t)2130 3446 y Fn(1)2173 3431 y Fq(;)17 b(t)2252 3446 y Fn(2)2322 3431 y Fo(2)28 b(h\000)p Fq(a;)17 b(a)p Fo(i)p Fv(,)33 b(pri)c(\024)-46 b(cem)m(u)32 b(je)h Fq(t)3321 3446 y Fn(1)3392 3431 y Fh(6)28 b Fq(t)3532 3446 y Fn(2)1059 3700 y Fv(\()p Fw(x)1156 3715 y Fm(")1195 3700 y Fv(\()p Fq(t)1268 3715 y Fn(2)1311 3700 y Fv(\))22 b Fo(\000)h Fw(x)1530 3715 y Fm(")1570 3700 y Fv(\()p Fq(t)1643 3715 y Fn(1)1686 3700 y Fv(\)\))f Fo(\001)f Ff(\030)31 b Fv(=)2018 3564 y Fl(Z)2118 3590 y Fm(t)2146 3602 y Fk(2)2074 3790 y Fm(t)2102 3802 y Fk(1)2204 3700 y Fz(f)2234 3715 y Fm(")2273 3700 y Fv(\()p Fq(t;)17 b Fw(x)2449 3715 y Fm(")2489 3700 y Fv(\()p Fq(t)p Fv(\)\))22 b Fo(\001)g Ff(\030)d Fq(dt)e(:)71 3964 y Fv(Za)43 b(neki)g Fq(\016)50 b(>)d Fv(0)c(o)s(dab)s(erimo)e Fq(")i Fv(tak)-5 b(a)m(v)44 b(da)f(vrijedi)g Fq(")29 b Fv(+)g Fo(j)p Fw(x)2334 3979 y Fm(")2374 3964 y Fv(\()p Fq(t)p Fv(\))h Fo(\000)g Fw(x)p Fv(\()p Fq(t)p Fv(\))p Fo(j)46 b Fq(<)g(\016)t Fv(,)41 b(\024)-44 b(sto)44 b(je)g(mogu)m(\023) -46 b(ce,)46 b(jer)71 4083 y Fw(x)130 4098 y Fm(")166 4113 y Fj(k)238 4083 y Fo(!)28 b Fw(x)p Fv(.)43 b(T)-8 b(ada)33 b(lak)m(o)f(dolazimo)e(do)i(o)s(cjene)942 4356 y(\()p Fw(x)1039 4371 y Fm(")1079 4356 y Fv(\()p Fq(t)1152 4371 y Fn(2)1195 4356 y Fv(\))22 b Fo(\000)g Fw(x)1413 4371 y Fm(")1453 4356 y Fv(\()p Fq(t)1526 4371 y Fn(1)1569 4356 y Fv(\)\))g Fo(\001)g Ff(\030)30 b Fh(6)1903 4221 y Fl(Z)2003 4247 y Fm(t)2031 4259 y Fk(2)1959 4446 y Fm(t)1987 4458 y Fk(1)2134 4356 y Fv(ess)17 b(sup)2089 4448 y Fp(j)p Fm(y)s Fp(\000)p Fg(x)p Fn(\()p Fm(t)p Fn(\))p Fp(j)p Fm(<\016)2479 4356 y Fz(f)2509 4371 y Fm(")2549 4356 y Fv(\()p Fq(t;)g(y)t Fv(\))k Fo(\001)g Ff(\030)f Fq(dt)d(;)66 4635 y Fv(\024)-44 b(sto)33 b(s)g(obzirom)e(da)h (je)h(ess)17 b(sup)34 b(izmjeriv)-5 b(a)30 b(funk)m(cija,)j(u)g(limesu) e Fq(")c Fo(&)g Fv(0,)33 b(a)f(p)s(otom)f(i)h Fq(\016)f Fo(&)d Fv(0,)k(da)5 b(je)1186 4898 y(\()p Fw(x)p Fv(\()p Fq(t)1356 4913 y Fn(2)1399 4898 y Fv(\))22 b Fo(\000)h Fw(x)p Fv(\()p Fq(t)1691 4913 y Fn(1)1733 4898 y Fv(\)\))f Fo(\001)g Ff(\030)31 b Fh(6)2068 4762 y Fl(Z)2167 4788 y Fm(t)2195 4800 y Fk(2)2123 4987 y Fm(t)2151 4999 y Fk(1)2253 4898 y Fq(H)8 b Fv(\()p Fq(t;)17 b Fw(x)p Fq(;)g Ff(\030)s Fv(\))p Fq(dt)g(;)71 5162 y Fv(o)s(dnosno)46 b(u)g(to)m(\024)-46 b(ck)-5 b(ama)45 b(gdje)h Fw(x)g Fv(ima)e(deriv)-5 b(aciju)1972 5161 y(_)1956 5162 y Fw(x)p Fv(\()p Fq(t)p Fv(\))32 b Fo(\001)e Ff(\030)54 b Fh(6)d Fq(H)8 b Fv(\()p Fq(t;)17 b Fw(x)p Fq(;)g Ff(\030)s Fv(\),)43 b(\024)-44 b(sto)46 b(op)s(et)g(p)s(o)m(vla)m(\024)-46 b(ci)45 b(da)g(je)87 5280 y(_)71 5281 y Fw(x)p Fv(\()p Fq(t)p Fv(\))28 b Fo(2)g Fq(F)14 b Fv(\()p Fq(t;)j Fw(x)p Fv(\).)293 5400 y(Iz)33 b(izlaganja)d(zaklju)m(\024)-46 b(cujemo)32 b(da)h(vrijedi)e(sljede)m(\023)-46 b(ci)32 b(teorem.)71 5564 y Fw(T)-9 b(eorem)42 b(4.)115 b Fi(Uz)38 b(pretp)s(osta)m(vk)m(e)i(s)e(p)s(o)m(\024)-46 b(cetk)-5 b(a)38 b(p)s(ogla)m(vlja)e(Cauc)m(h)m(yjev)-5 b(a)39 b(zada)m(\023)-46 b(ca)38 b Fv(\(1\))g Fi(ima)d(rje)-5 b(\024)-44 b(senje)39 b(u)71 5683 y(smislu)h(da)h(je)653 5682 y Fv(_)637 5683 y Fw(x)p Fv(\()p Fq(t)p Fv(\))i Fo(2)g Fq(F)14 b Fv(\()p Fq(t;)j Fw(x)p Fv(\))p Fi(,)43 b(gdje)e(je)h Fq(F)14 b Fv(\()p Fq(t;)j Fw(x)p Fv(\))40 b Fi(na)5 b(jmanji)40 b(zatv)m(oren)i(k)m(on)m(v)m(eksan)j(skup)d(tak) -5 b(a)m(v)42 b(da)71 5801 y(sv)-5 b(ak)g(a)33 b(njego)m(v)-5 b(a)32 b(ok)m(olina)f(sadr)m(\024)-46 b(zi)32 b(sk)m(oro)i(sv)-5 b(aki)32 b Fz(f)7 b Fv(\()p Fq(t;)17 b Fw(y)r Fv(\))32 b Fi(za)g Fw(y)j Fi(iz)d(nek)m(e)i(ok)m(oline)d(o)s(d)i Fw(x)p Fi(.)p 3868 5844 30 43 v 3817 6010 a Ft(5)p eop %%Page: 6 11 6 10 bop -165 -116 a Fr(Nelinearni)27 b(zak)n(oni)f(sa)n(\024)-39 b(cuv)-5 b(anja)59 63 y Fv(\024)57 88 y(Sto)32 b(nam)g(k)m(onkretno)i (da)5 b(je)33 b(o)m(v)-5 b(a)5 b(j)33 b(teorem?)57 207 y(Pretp)s(osta)m(vimo)43 b(da)g(je)h Fz(f)53 b Fv(:)46 b Fw(R)1226 170 y Fn(1+)p Fm(d)1414 207 y Fo(!)g Fw(R)1644 170 y Fm(d)1731 207 y Fv(funk)m(cija)e(neprekin)m(uto)g (diferencijabilna)d(na)j(zat-)-165 325 y(v)-5 b(ara)m(\024)-46 b(cu)48 b(sv)-5 b(ak)m(e)49 b(strane)g(nek)m(e)g(glatk)m(e)f(C)1354 289 y Fn(1)1445 325 y Fv(plohe)g Fq(P)61 b Fv(na)48 b(k)m(o)5 b(jo)g(j)49 b Fz(f)54 b Fv(ima)46 b(prekid)j(prv)m(e)g(vrste.)91 b(S)48 b(\012)3597 340 y Fn(+)-165 444 y Fv(o)s(dnosno)34 b(\012)289 459 y Fp(\000)387 444 y Fv(ozna)m(\024)-46 b(cimo)32 b(te)j(otv)m(orene)g(skup)s(o)m(v)m(e,)h(a)e(s)g Fz(f)1914 459 y Fp(\000)2012 444 y Fv(i)f Fz(f)2103 459 y Fn(+)2201 444 y Fv(limese)f(funk)m(cije)j Fz(f)40 b Fv(na)34 b(plohi)e Fq(P)48 b Fv(iz)33 b(\012)3597 459 y Fn(+)-165 563 y Fv(i)e(\012)-36 578 y Fp(\000)28 563 y Fv(.)43 b(Pretp)s(osta)m(vimo)31 b(dalje)g(da)h(su)h(za)f(rubne)g (vrijednosti)f(\(v)m(ektore\))j Fz(f)2578 578 y Fp(\000)2673 563 y Fv(i)d Fz(f)2762 578 y Fn(+)2858 563 y Fv(s)h(ob)5 b(je)32 b(strane)h(plohe)-165 681 y(v)m(ektorsk)-5 b(a)45 b(p)s(olja)d(\(1)p Fq(;)17 b Fz(f)696 696 y Fp(\006)759 681 y Fv(\))44 b(transv)m(erzalna)h(\(normalna)c(k)m(omp)s(onen)m(ta)j (je)g(razli)m(\024)-46 b(cita)42 b(o)s(d)h(n)m(ule\))h(na)g Fq(P)14 b Fv(.)-165 800 y(Ak)m(o)31 b(su)h(oba)e(p)s(olja)f(usmjerena)j (na)e(istu)h(stran)m(u)h(plohe,)f(tada)c(\023)-46 b(ce)32 b(in)m(tegralne)d(krivulje)i(prolaziti)d(kroz)-165 919 y(nju.)51 b(Nadalje,)36 b(ak)m(o)f(su)h(usmjerena)g(na)f(stran)m(u)h (razli)m(\024)-46 b(citu)33 b(o)s(d)i(one)g(na)g(k)m(o)5 b(jo)g(j)36 b(su)g(de\014nirana,)g(v)m(ektori)-165 1038 y(\(1)p Fq(;)17 b Fz(f)-4 1053 y Fp(\006)59 1038 y Fv(\))38 b(su)h(usmjereni)f(prema)g(skup)s(o)m(vima)g(\012)1582 1053 y Fp(\007)1646 1038 y Fv(,)i(tada)e(in)m(tegralne)f(krivulje)h (osta)5 b(ju)38 b(u)h(plohi)d Fq(P)52 b Fv(k)-5 b(ao)-165 1156 y(in)m(tegralne)31 b(krivulje)h(jednad)m(\024)-46 b(zb)s(e)1313 1274 y(_)1297 1275 y Fw(x)28 b Fv(=)f Fq(\025)p Fz(f)1574 1290 y Fn(+)1660 1275 y Fv(+)22 b(\(1)g Fo(\000)h Fq(\025)p Fv(\))p Fz(f)2092 1290 y Fp(\000)2172 1275 y Fq(;)-165 1449 y Fv(gdje)28 b(je)g Fq(\025)g Fo(2)g(h)p Fv(0)p Fq(;)17 b Fv(1)p Fo(i)27 b Fv(o)s(dabran)g(tak)m(o)h(da)g(je)h (\(1)p Fq(;)17 b(\025)p Fz(f)1606 1464 y Fn(+)1682 1449 y Fv(+)c(\(1)g Fo(\000)g Fq(\025)p Fv(\))p Fz(f)2086 1464 y Fp(\000)2149 1449 y Fv(\))28 b(tangencijalan)f(na)h Fq(P)14 b Fv(,)28 b(i)f(na)h(kra)5 b(ju)29 b(ak)m(o)-165 1568 y(su)35 b(usmjereni)f(o)s(d)g Fq(P)14 b Fv(,)35 b(in)m(tegralne)e(krivulje)h(mogu)g(te)m(\023)-46 b(ci)34 b(kroz)h Fq(P)48 b Fv(do)35 b(nek)m(e)h(to)m(\024)-46 b(ck)m(e)35 b(u)g(k)m(o)5 b(jo)g(j)35 b(napu)-5 b(\024)-44 b(sta)5 b(ju)-165 1687 y(ploh)m(u,)32 b(te)h(idu)f(na)g(neku)i(stran)m (u)f(k)-5 b(ao)32 b(standarna)h(in)m(tegralna)e(krivulja.)57 1805 y(Dok)-5 b(a)m(\024)-46 b(zimo)25 b(sada)j(teorem)f(jedinstv)m (enosti)h(k)m(o)5 b(ji)25 b(\023)-46 b(ce)28 b(u)f(sk)-5 b(alarnom)26 b(slu)m(\024)-46 b(ca)5 b(ju)28 b(dopu)-5 b(\024)-44 b(stati)27 b(negativne)-165 1924 y(sk)m(ok)m(o)m(v)m(e)35 b(k)m(o)s(d)e(funk)m(cije)g Fz(f)7 b Fv(.)-165 2091 y Fw(T)-9 b(eorem)35 b(5.)99 b Fi(Ak)m(o)32 b(uz)g(pretp)s(osta)m(vk)m(e) h(pretho)s(dnog)e(teorema,)g(za)h Fw(y)r Fq(;)17 b Fw(z)27 b Fo(2)h Fw(R)2730 2055 y Fm(d)2804 2091 y Fi(te)k Fq(t)c Fo(2)g Fv([)p Fq(t)3136 2106 y Fn(0)3198 2091 y Fo(\000)20 b Fq(a;)d(t)3425 2106 y Fn(0)3488 2091 y Fv(+)i Fq(a)p Fv(])-165 2210 y Fi(vrijedi)31 b(i)-165 2423 y Fv(\(3\))961 b(\()p Fw(y)23 b Fo(\000)g Fw(z)p Fv(\))f Fo(\001)g Fv(\()p Fz(f)7 b Fv(\()p Fq(t;)17 b Fw(y)r Fv(\))k Fo(\000)i Fz(f)7 b Fv(\()p Fq(t;)17 b Fw(z)p Fv(\)\))26 b Fh(6)j Fq(C)7 b Fo(j)p Fw(y)23 b Fo(\000)g Fw(z)p Fo(j)2490 2381 y Fn(2)2548 2423 y Fq(;)-165 2635 y Fi(tada)g(obi)m(\024)-46 b(cna)24 b(diferencijalna)e(jednad)m(\024)-46 b(zba)25 b Fv(\(1)1484 2650 y Fn(1)1526 2635 y Fv(\))p Fi(,)h(s)e(p)s(o)m(\024) -46 b(cetnim)23 b(uvjetom)h Fw(x)p Fv(\(0)p Fq(;)17 b Fw(y)r Fv(\))27 b(=)h Fw(y)r Fi(,)d(ima)d(jedinstv)m(eno)-165 2754 y(rje)-5 b(\024)-44 b(senje)33 b Fw(x)p Fv(\()p Fq(t;)17 b Fw(y)r Fv(\))p Fi(,)33 b(te)g(za)f Fq(t)c Fo(2)g Fv([)p Fq(t)979 2769 y Fn(0)1022 2754 y Fq(;)17 b(t)1101 2769 y Fn(0)1166 2754 y Fv(+)22 b Fq(a)p Fv(])32 b Fi(vrijedi)-165 2978 y Fv(\(4\))1037 b Fo(j)p Fw(x)p Fv(\()p Fq(t;)17 b Fw(y)r Fv(\))22 b Fo(\000)g Fw(x)p Fv(\()p Fq(t;)17 b Fw(z)p Fv(\))p Fo(j)28 b Fh(6)g Fq(e)1891 2937 y Fm(C)5 b Fn(\()p Fm(t)p Fp(\000)p Fm(t)2097 2949 y Fk(0)2133 2937 y Fn(\))2167 2978 y Fo(j)p Fw(y)24 b Fo(\000)f Fw(z)p Fo(j)17 b Fq(:)-165 3215 y Fz(Dem.)93 b Fv(Na)5 b(jprije)32 b(primijetimo)c(da)k(za)h(sv)-5 b(aki)33 b Fq(")27 b(>)g Fv(0)33 b(vrijedi)1048 3427 y(\()p Fq(a)23 b Fv(+)f Fq(b)p Fv(\))1337 3386 y Fn(2)1407 3427 y Fh(6)28 b Fv(\(1)22 b(+)g Fq(")p Fv(\))p Fq(a)1854 3386 y Fn(2)1919 3427 y Fv(+)g(\(1)g(+)2234 3388 y Fn(1)p 2234 3404 39 4 v 2235 3461 a Fm(")2282 3427 y Fv(\))p Fq(b)2361 3386 y Fn(2)2421 3427 y Fq(:)-165 3640 y Fv(Nek)-5 b(a)32 b(su)i Fq(";)17 b(\016)31 b(>)c Fv(0)p Fq(;)49 b Fo(j)p Fw(y)685 3603 y Fp(0)732 3640 y Fo(\000)22 b Fw(y)r Fo(j)27 b Fq(<)h(\016)36 b Fv(i)c Fo(j)p Fw(z)1268 3603 y Fp(0)1315 3640 y Fo(\000)22 b Fw(z)p Fo(j)27 b Fq(<)h(\016)t Fv(.)43 b(T)-8 b(ada)33 b(p)s(o)f(\(3\))g(vrijedi)407 3856 y(\()p Fw(y)506 3815 y Fp(0)553 3856 y Fo(\000)22 b Fw(z)702 3815 y Fp(0)727 3856 y Fv(\))g Fo(\001)g Fv(\()p Fz(f)7 b Fv(\()p Fq(t;)17 b Fw(y)1090 3815 y Fp(0)1114 3856 y Fv(\))22 b Fo(\000)h Fz(f)7 b Fv(\()p Fq(t;)17 b Fw(z)1478 3815 y Fp(0)1502 3856 y Fv(\)\))28 b Fh(6)g Fq(C)7 b Fo(j)p Fw(y)1877 3815 y Fp(0)1923 3856 y Fo(\000)23 b Fw(z)2073 3815 y Fp(0)2098 3856 y Fo(j)2126 3815 y Fn(2)1606 3999 y Fh(6)28 b Fq(C)7 b Fv(\(\(1)21 b(+)h Fq(")p Fv(\))p Fo(j)p Fw(y)i Fo(\000)e Fw(z)p Fo(j)2404 3958 y Fn(2)2468 3999 y Fv(+)g(\(1)g(+)2783 3960 y Fn(1)p 2783 3976 V 2784 4034 a Fm(")2832 3999 y Fv(\)4)p Fq(\016)2966 3958 y Fn(2)3008 3999 y Fv(\))17 b Fq(;)-165 4205 y Fv(o)s(dnosno)224 4417 y(\()p Fw(y)24 b Fo(\000)f Fw(z)p Fv(\))e Fo(\001)h Fv(\()p Fz(f)7 b Fv(\()p Fq(t;)17 b Fw(y)857 4376 y Fp(0)881 4417 y Fv(\))22 b Fo(\000)h Fz(f)7 b Fv(\()p Fq(t;)17 b Fw(z)1245 4376 y Fp(0)1269 4417 y Fv(\)\))28 b Fh(6)g Fq(C)7 b Fv(\(\(1)22 b(+)g Fq(")p Fv(\))p Fo(j)p Fw(y)h Fo(\000)g Fw(z)p Fo(j)2172 4376 y Fn(2)2236 4417 y Fv(+)f(\(1)g(+)2551 4378 y Fn(1)p 2551 4395 V 2552 4452 a Fm(")2599 4417 y Fv(\)4)p Fq(\016)2733 4376 y Fn(2)2775 4417 y Fv(\))g(+)g(4)p Fq(\016)t(M)10 b Fv(\()p Fq(t)p Fv(\))p Fq(:)-165 4630 y Fv(U)32 b(limesu)f(k)-5 b(ad)33 b Fq(\016)e Fo(&)d Fv(0)k(zaklju)m(\024)-46 b(cujemo)32 b(da)h(za)f Fq(Y)49 b Fo(2)28 b Fq(F)14 b Fv(\()p Fq(t;)j Fw(y)r Fv(\))32 b(i)g Fq(Z)i Fo(2)28 b Fq(F)14 b Fv(\()p Fq(t;)j Fw(z)p Fv(\))32 b(vrijedi)946 4842 y(\()p Fw(y)24 b Fo(\000)e Fw(z)p Fv(\))g Fo(\001)g Fv(\()p Fq(Y)43 b Fo(\000)23 b Fq(Z)7 b Fv(\))28 b Fh(6)g Fq(C)7 b Fv(\(1)22 b(+)g Fq(")p Fv(\))p Fo(j)p Fw(y)h Fo(\000)f Fw(z)p Fo(j)2464 4801 y Fn(2)2523 4842 y Fq(;)-165 5055 y Fv(iz)29 b(\024)-46 b(cega)32 b(u)h(limesu)e(k)-5 b(ad)33 b Fq(")27 b Fo(&)g Fv(0)33 b(slijedi)1127 5267 y(\()p Fw(y)24 b Fo(\000)f Fw(z)p Fv(\)\()p Fq(Y)43 b Fo(\000)22 b Fq(Z)7 b Fv(\))28 b Fh(6)g Fq(C)7 b Fo(j)p Fw(y)24 b Fo(\000)e Fw(z)p Fo(j)2283 5226 y Fn(2)2342 5267 y Fq(:)-165 5480 y Fv(Nek)-5 b(a)39 b Fw(x)p Fv(\()p Fq(t;)17 b Fw(y)r Fv(\))39 b(ozna)m(\024)-46 b(ca)m(v)-5 b(a)40 b(rje)-5 b(\024)-44 b(senje)41 b(za)f(p)s(o)m(\024) -46 b(cetni)39 b Fw(y)r Fv(.)63 b(Ak)m(o)41 b(p)s(ok)-5 b(a)m(\024)-46 b(zemo)38 b(\(4\))h(jedinstv)m(enost)i(rje)-5 b(\024)-44 b(sena)38 b(\023)-46 b(ce)-165 5599 y(slijediti)29 b(iz)j Fw(y)e Fv(=)d Fw(z)p Fv(.)43 b(Definira)5 b(jmo)1210 5811 y Fq(R)q Fv(\()p Fq(t)p Fv(\))28 b(=)f Fo(j)p Fw(x)p Fv(\()p Fq(t;)17 b Fw(y)r Fv(\))22 b Fo(\000)g Fw(x)p Fv(\()p Fq(t;)17 b Fw(z)p Fv(\))p Fo(j)g Fq(:)-165 6010 y Ft(6)p eop %%Page: 7 12 7 11 bop 1928 -116 a Fr(Cauc)n(h)n(yjev)-5 b(a)26 b(zada)n(\023)-39 b(ca)26 b(za)h(obi)n(\024)-39 b(cn)n(u)26 b(diferencijaln)n(u)h(jednad) n(\024)-39 b(zbu)71 83 y Fv(Za)32 b(sk)m(oro)h(sv)-5 b(aki)32 b Fq(t)h Fv(imamo:)897 328 y Fq(R)q Fv(\()p Fq(t)p Fv(\))1110 260 y Fq(d)p 1093 305 86 4 v 1093 396 a(dt)1189 328 y(R)q Fv(\()p Fq(t)p Fv(\))27 b(=)1516 260 y(1)p 1516 305 49 4 v 1516 396 a(2)1602 260 y Fq(d)p 1585 305 86 4 v 1585 396 a(dt)1681 328 y(R)1756 286 y Fn(2)1798 328 y Fv(\()p Fq(t)p Fv(\))1402 504 y(=)h(\()p Fw(x)p Fv(\()p Fq(t;)17 b Fw(y)r Fv(\))22 b Fo(\000)g Fw(x)p Fv(\()p Fq(t;)17 b Fw(y)r Fv(\)\))22 b Fo(\001)g Fv(\()2379 503 y(_)2363 504 y Fw(x)p Fv(\()p Fq(t;)17 b Fw(y)r Fv(\))22 b Fo(\000)2775 503 y Fv(_)2759 504 y Fw(x)p Fv(\()p Fq(t;)17 b Fw(y)r Fv(\)\))1402 647 y(=)28 b(\()p Fw(x)p Fv(\()p Fq(t;)17 b Fw(y)r Fv(\))22 b Fo(\000)g Fw(x)p Fv(\()p Fq(t;)17 b Fw(y)r Fv(\)\))22 b Fo(\001)g Fv(\()p Fq(Y)43 b Fo(\000)23 b Fq(Z)7 b Fv(\))17 b Fq(;)71 859 y Fv(gdje)33 b(su)g Fq(Y)49 b Fo(2)28 b Fq(F)14 b Fv(\()p Fq(t;)j Fw(y)r Fv(\))31 b(i)h Fq(Z)j Fo(2)28 b Fq(F)14 b Fv(\()p Fq(t;)j Fw(z)p Fv(\),)27 b(\024)-44 b(sto)32 b(upra)m(v)m(o)i(p)s(o)m(vla)m(\024)-46 b(ci)1504 1129 y Fq(R)q Fv(\()p Fq(t)p Fv(\))1718 1061 y Fq(d)p 1700 1106 V 1700 1197 a(dt)1796 1129 y(R)q Fv(\()p Fq(t)p Fv(\))28 b Fh(6)g Fq(C)7 b(R)2267 1087 y Fn(2)2310 1129 y Fv(\()p Fq(t)p Fv(\))17 b Fq(;)71 1380 y Fv(o)s(dnosno)32 b(sk)m(oro)i(svuda)f(tamo)f(gdje)g Fq(R)q Fv(\()p Fq(t)p Fv(\))c Fo(6)p Fv(=)g(0)k(imamo)1646 1582 y Fq(d)p 1628 1626 V 1628 1717 a(dt)1724 1649 y(R)q Fv(\()p Fq(t)p Fv(\))c Fh(6)g Fq(C)7 b(R)q Fv(\()p Fq(t)p Fv(\))17 b Fq(;)66 1894 y Fv(\024)-44 b(sto)33 b(k)m(ona)m(\024)-46 b(cno)33 b(da)5 b(je)1550 2013 y Fq(R)q Fv(\()p Fq(t)p Fv(\))27 b Fh(6)i Fq(e)1914 1972 y Fm(C)5 b Fn(\()p Fm(t)p Fp(\000)p Fm(t)2120 1984 y Fk(0)2156 1972 y Fn(\))2190 2013 y Fq(R)q Fv(\()p Fq(t)2338 2028 y Fn(0)2381 2013 y Fv(\))71 2190 y(za)33 b Fq(t)27 b Fo(2)i Fv([)p Fq(t)415 2205 y Fn(0)457 2190 y Fq(;)17 b(t)536 2205 y Fn(0)601 2190 y Fv(+)22 b Fq(a)p Fv(])3561 2309 y Fw(Q.E.D.)3817 6010 y Ft(7)p eop %%Page: 8 13 8 12 bop eop %%Page: 9 14 9 13 bop 2125 5211 a Fu(I)t(I.)54 b(Burgerso)l(v)-9 b(a)52 b(jednad)l(\024)-76 b(zba)p eop %%Page: 10 15 10 14 bop -165 -116 a Fr(Nelinearni)27 b(zak)n(oni)f(sa)n(\024)-39 b(cuv)-5 b(anja)57 83 y Fv(U)36 b(o)m(v)m(om)h(p)s(ogla)m(vlju)32 b(\023)-46 b(cemo)36 b(prou)m(\024)-46 b(ca)m(v)-5 b(ati)35 b(sv)m(o)5 b(jstv)-5 b(a)38 b(zak)m(ona)g(sa)m(\024)-46 b(cuv)-5 b(anja.)55 b(Zak)m(oni)36 b(za)m(\024)-46 b(cuv)-5 b(anja)37 b(su)-165 202 y(kv)-5 b(azilinearne)31 b(hip)s(erb)s(oli)m (\024)-46 b(ck)m(e)31 b(jednad)m(\024)-46 b(zb)s(e)34 b(oblik)-5 b(a)1337 387 y Fq(@)1388 402 y Fm(t)1420 387 y Fq(u)22 b Fv(+)g Fq(@)1647 402 y Fm(x)1695 387 y Fv(\()p Fq(f)33 b Fo(\016)22 b Fq(u)p Fv(\))27 b(=)g(0)-165 572 y(na)34 b Fw(R)56 535 y Fn(+)142 572 y Fo(\002)24 b Fw(R)p Fv(,)34 b(pri)c(\024)-46 b(cem)m(u)35 b(je)f Fq(f)41 b Fv(:)30 b Fw(R)g Fo(!)g Fw(R)j Fv(glatk)-5 b(a)33 b(funk)m(cija.)48 b(F)-8 b(unk)m(ciju)34 b Fq(f)45 b Fv(obi)m(\024)-46 b(cno)33 b(naziv)-5 b(amo)32 b Fz(funk)m(cija)-165 690 y(\015uksa)p Fv(.)48 b(T)-8 b(o)m(\024)-46 b(cnije,)35 b(prou)m(\024)-46 b(ca)m(v)-5 b(at)32 b(\023)-46 b(cemo)33 b(Cauc)m(h)m(yjevu)k(zada)m(\023)-46 b(cu)35 b(za)f(zak)m(on)h(sa)m (\024)-46 b(cuv)-5 b(anja)35 b(dobiv)m(en)m(u)g(tak)m(o)f(da)-165 809 y(se)f(p)s(osta)m(vi)f(p)s(o)m(\024)-46 b(cetni)32 b(uvjet)i Fq(u)p Fv(\(0)p Fq(;)17 b Fo(\001)p Fv(\))26 b(=)h Fq(u)1320 824 y Fn(0)1363 809 y Fv(,)32 b(i)g(to)g(na)g(p)s (osebnom)h(primjeru)e(Burgerso)m(v)m(o)5 b(j)34 b(jednad)m(\024)-46 b(zbi.)57 928 y(Prije)40 b(nego)g(li)f(prijed)p 845 876 30 2 v(emo)h(na)g(prou)m(\024)-46 b(ca)m(v)-5 b(anje)42 b(nelinarne)d(jedna)m(\024)-46 b(zb)s(e)42 b(ukratk)m(o)c(\023)-46 b(cemo)40 b(raspra)m(viti)-165 1046 y(sv)m(o)5 b(jstv)-5 b(a)45 b(linearne)f(jednad)m(\024)-46 b(zb)s(e.)80 b(Unato)m(\024)-46 b(c)44 b(sv)m(o)5 b(jo)g(j)46 b(jednosta)m(vnosti)f(ona)f(nas)g(priro)s (dno)f(na)m(v)m(o)s(di)h(na)-165 1165 y(meto)s(du)32 b(k)-5 b(arakteristik)g(a)31 b(k)m(o)5 b(ju)31 b(\023)-46 b(cemo)32 b(k)m(oristiti)e(i)i(k)-5 b(asnije)32 b(u)h(kv)-5 b(azilinernom)30 b(slu)m(\024)-46 b(ca)5 b(ju.)1268 1416 y Fw(Linearna)39 b(jednad)m(\024)-53 b(zba)57 1612 y Fv(Promatramo)26 b(Cauc)m(h)m(yjev)32 b(problem)27 b(za)i(sk)-5 b(alarn)m(u)29 b(linearn)m(u)f(jednad)m(\024)-46 b(zbu)30 b(prv)m(og)f(reda)g(na)g Fw(R)3506 1576 y Fn(+)3584 1612 y Fo(\002)-165 1731 y Fw(R)p Fv(:)-165 1954 y(\(1\))938 1814 y Fl(\032)1029 1880 y Fq(@)1080 1895 y Fm(t)1112 1880 y Fq(u)22 b Fv(+)g Fq(a)p Fv(\()p Fq(t;)17 b(x)p Fv(\))p Fq(@)1600 1895 y Fm(x)1649 1880 y Fq(u)27 b Fv(=)g Fq(b)p Fv(\()p Fq(t;)17 b(x)p Fv(\))p Fq(u)23 b Fv(+)f Fq(f)11 b Fv(\()p Fq(t;)17 b(x)p Fv(\))1029 2024 y Fq(u)p Fv(\(0)p Fq(;)g Fo(\001)p Fv(\))26 b(=)i Fq(u)1468 2039 y Fn(0)1526 2024 y Fq(:)-165 2213 y Fv(Pritom)i(pretp)s(osta)m(vljamo)i (da)g(su)h Fq(a;)17 b(@)1294 2228 y Fm(x)1342 2213 y Fq(a;)g(b)33 b Fv(i)f Fq(f)43 b Fv(neprekidne)34 b(realne)e(funk)m (cije)h(na)f Fw(R)3027 2172 y Fn(+)3027 2243 y(0)3113 2213 y Fo(\002)22 b Fw(R)p Fv(.)43 b(Karak-)-165 2332 y(teristik)m(e)33 b(parcijalne)g(diferencijalne)f(jednad)m(\024)-46 b(zb)s(e)35 b(\(1)1805 2347 y Fn(1)1848 2332 y Fv(\))e(de\014niramo)g (k)-5 b(ao)33 b(in)m(tegralne)g(krivulje)g(obi)m(\024)-46 b(cne)-165 2451 y(diferencijalne)31 b(jednad)m(\024)-46 b(zb)s(e)1416 2569 y(_)i Fq(x)q Fv(\()p Fq(t)p Fv(\))28 b(=)f Fq(a)p Fv(\()p Fq(t;)17 b(x)p Fv(\()p Fq(t)p Fv(\)\))p Fq(:)-165 2731 y Fv(Nek)-5 b(a)30 b(je)h Fq(x)g Fv(funk)m(cija)g(takv) -5 b(a)30 b(da)h(je)f Fq(x)p Fv(\()p Fo(\001)p Fq(;)17 b(y)t Fv(\))30 b(k)-5 b(arakteristik)g(a)29 b(k)m(o)5 b(ja)31 b(prolazi)e(kroz)i(to)m(\024)-46 b(cku)31 b(\(0)p Fq(;)17 b(y)t Fv(\),)29 b(o)s(dnosno)-165 2850 y(rje)-5 b(\024)-44 b(senje)33 b(Cauc)m(h)m(yjev)m(oga)i(problema)1250 2952 y Fl(\032)1372 3019 y Fv(_)-44 b Fq(x)p Fv(\()p Fq(t;)17 b(y)t Fv(\))27 b(=)32 b Fq(a)p Fv(\()p Fq(t;)17 b(x)p Fv(\()p Fq(t;)g(y)t Fv(\)\))1341 3162 y Fq(x)p Fv(\(0)p Fq(;)g(y)t Fv(\))27 b(=)32 b Fq(y)20 b(:)-165 3335 y Fv(T)-8 b(akv)j(a)33 b(funk)m(cija)h Fq(x)g Fv(p)s(osto)5 b(ji)32 b(na)i(nek)m(o)5 b(j)34 b(ok)m(olini)e Fq(x)p Fv(-osi,)h(gdje)g(je)h(i)e(neprekidno)i(diferencijabilna.)44 b(Ak)m(o)-165 3454 y(pretp)s(osta)m(vimo)34 b(i)h(da)h(je)f(funk)m (cija)h Fq(a)g Fv(omed)p 1443 3402 V(ena,)h(onda)e(rje)-5 b(\024)-44 b(senje)37 b Fq(x)f Fv(p)s(osto)5 b(ji)35 b(na)g Fw(R)2926 3412 y Fn(+)2926 3483 y(0)3014 3454 y Fo(\002)25 b Fw(R)p Fv(.)52 b(Iz)36 b(teorije)-165 3572 y(obi)m(\024)-46 b(cnih)40 b(diferencijalnih)e(jednad)m(\024)-46 b(zbi)42 b(slijedi)d(da)i(tada)g(kroz)h(sv)-5 b(aku)42 b(to)m(\024)-46 b(cku)42 b(\()p Fq(t;)17 b(x)p Fv(\))42 b Fo(2)h Fw(R)3166 3531 y Fn(+)3166 3602 y(0)3258 3572 y Fo(\002)28 b Fw(R)41 b Fv(pro-)-165 3691 y(lazi)33 b(to)m(\024)-46 b(cno)35 b(jedna)h(k)-5 b(arakteristik)g(a,)34 b(pa)i Fq(x)p Fv(\()p Fq(t;)17 b(y)t Fv(\))34 b(o)s(dred)p 1832 3639 V 1 w(uje)h(v)-5 b(aljan)m(u)35 b(zamjen)m(u)g(v)-5 b(arijabli.)48 b(Preina)m(\024)-46 b(cimo)-165 3810 y(na)-5 b(\024)-44 b(s)29 b(p)s(o)m(\024)-46 b(cetni)29 b(Cauc)m(h)m(yjev)j (problem)c(\(1\))h(tak)m(o)24 b(\024)-44 b(sto)27 b(\023)-46 b(cemo)28 b(umjesto)h(\()p Fq(t;)17 b(x)p Fv(\))30 b(uv)m(esti)g(no)m (v)m(e)g(v)-5 b(arijable)27 b(\()p Fq(t;)17 b(y)t Fv(\))-165 3929 y(\(namjesto)32 b(\()p Fq(t;)17 b(x)p Fv(\()p Fq(t;)g(y)t Fv(\)\))31 b(promatramo)g(\()p Fq(t;)17 b(y)t Fv(\)\);)31 b(na)5 b(jprije)32 b(de\014nira)5 b(jmo)1283 4103 y Fq(U)10 b Fv(\()p Fq(t;)17 b(y)t Fv(\))27 b(=)h Fq(u)p Fv(\()p Fq(t;)17 b(x)p Fv(\()p Fq(t;)g(y)t Fv(\)\))g Fq(;)1281 4246 y(B)5 b Fv(\()p Fq(t;)17 b(y)t Fv(\))26 b(=)i Fq(b)p Fv(\()p Fq(t;)17 b(x)p Fv(\()p Fq(t;)g(y)t Fv(\)\))g Fq(:)1283 4390 y(F)d Fv(\()p Fq(t;)j(y)t Fv(\))26 b(=)i Fq(f)11 b Fv(\()p Fq(t;)17 b(x)p Fv(\()p Fq(t;)g(y)t Fv(\)\))g Fq(:)-165 4568 y Fv(Koriste)m(\023)-46 b(ci)31 b(\(1\))h(dobiv)-5 b(amo)759 4719 y Fm(d)p 745 4736 68 4 v 745 4793 a(dt)823 4759 y Fq(U)10 b Fv(\()p Fq(t;)17 b(y)t Fv(\))27 b(=)1260 4719 y Fm(d)p 1246 4736 V 1246 4793 a(dt)1324 4759 y Fq(u)p Fv(\()p Fq(t;)17 b(x)p Fv(\()p Fq(t;)g(y)t Fv(\)\))1133 4902 y(=)27 b Fq(@)1287 4917 y Fm(t)1320 4902 y Fq(u)p Fv(\()p Fq(t;)17 b(x)p Fv(\()p Fq(t;)g(y)t Fv(\)\))k(+)h Fq(a)p Fv(\()p Fq(t;)17 b(x)p Fv(\))p Fq(@)2224 4917 y Fm(x)2272 4902 y Fq(u)p Fv(\()p Fq(t;)g(x)p Fv(\()p Fq(t;)g(y)t Fv(\)\))1133 5046 y(=)27 b Fq(b)p Fv(\()p Fq(t;)17 b(x)p Fv(\()p Fq(t;)g(y)t Fv(\)\))p Fq(u)p Fv(\()p Fq(t;)g(x)p Fv(\()p Fq(t;)g(y)t Fv(\)\))k(+)h Fq(f)11 b Fv(\()p Fq(t;)17 b(x)p Fv(\()p Fq(t;)g(y)t Fv(\)\))1133 5190 y(=)27 b Fq(B)5 b Fv(\()p Fq(t;)17 b(y)t Fv(\))p Fq(U)10 b Fv(\()p Fq(t;)17 b(y)t Fv(\))k(+)h Fq(F)14 b Fv(\()p Fq(t;)j(y)t Fv(\))g Fq(:)-165 5368 y Fv(Rje)-5 b(\024)-44 b(sa)m(v)-5 b(anje)27 b(Cauc)m(h)m(yjev)m(og)j (problema)c(\(1\))g(sv)m(o)s(di)i(se)f(na)h(rje)-5 b(\024)-44 b(sa)m(v)-5 b(anje)27 b(familije)d(obi)m(\024)-46 b(cnih)26 b(diferencijalnih)-165 5487 y(jednad)m(\024)-46 b(zbi;)32 b(to)m(\024)-46 b(cnije,)33 b(za)f(sv)-5 b(aki)33 b Fq(y)d Fo(2)f Fw(R)j Fv(rje)-5 b(\024)-44 b(sa)m(v)-5 b(amo)32 b(Cauc)m(h)m(yjevu)j(zada)m(\023)-46 b(cu:)953 5597 y Fl(\032)1068 5632 y Fm(d)p 1055 5649 V 1055 5706 a(dt)1132 5672 y Fq(U)10 b Fv(\()p Fq(t;)17 b(y)t Fv(\))27 b(=)h Fq(B)5 b Fv(\()p Fq(t;)17 b(y)t Fv(\))p Fq(U)10 b Fv(\()p Fq(t;)17 b(y)t Fv(\))k(+)h Fq(F)14 b Fv(\()p Fq(t;)j(y)t Fv(\))1045 5815 y Fq(U)10 b Fv(\(0)p Fq(;)17 b(y)t Fv(\))26 b(=)i Fq(u)1528 5830 y Fn(0)1570 5815 y Fv(\()p Fq(y)t Fv(\))17 b Fq(;)-165 6010 y Ft(10)p eop %%Page: 11 16 11 15 bop 3102 -116 a Fr(Burgerso)n(v)-5 b(a)25 b(jednad)n(\024)-39 b(zba)71 83 y Fv(k)m(o)5 b(ja)33 b(ima)e(rje)-5 b(\024)-44 b(senje)33 b(dano)g(form)m(ulom)855 356 y Fq(U)10 b Fv(\()p Fq(t;)17 b(y)t Fv(\))27 b(=)h Fq(u)1325 371 y Fn(0)1367 356 y Fv(\()p Fq(y)t Fv(\))p Fq(e)1540 254 y Fe(R)1590 275 y Fj(t)1575 342 y Fk(0)1632 315 y Fm(B)t Fn(\()p Fm(s;y)s Fn(\))p Fm(ds)1953 356 y Fv(+)2051 220 y Fl(Z)2151 246 y Fm(t)2107 446 y Fn(0)2199 356 y Fq(F)14 b Fv(\()p Fq(l)r(;)j(y)t Fv(\))p Fq(e)2524 315 y Fp(\000)2596 254 y Fe(R)2647 275 y Fj(t)2632 342 y(l)2689 315 y Fm(B)t Fn(\()p Fm(s;y)s Fn(\))p Fm(ds)2988 356 y Fq(dl)h(:)71 619 y Fv(V)-8 b(ratimo)21 b(li)h(se)i(na)g(p)s(olazne)f(v)-5 b(arijable)21 b(\()p Fq(t;)c(x)p Fv(\),)26 b(dobiv)-5 b(amo)22 b(jedinstv)m(eno)i(neprekidno)h(rje)-5 b(\024)-44 b(senje)25 b(na)e(skupu)71 737 y(k)m(o)5 b(jeg)33 b(p)s(okriv)-5 b(a)5 b(ju)32 b(k)-5 b(arakteristik)m(e.)293 856 y(P)m(ok)g(a)m(\024) -46 b(zimo)26 b(teorem)i(jedinstv)m(enosti)g(k)m(o)5 b(ji)25 b(\023)-46 b(ce)28 b(biti)e(v)-5 b(a)m(\024)-46 b(zan)28 b(za)g(nelinearan)f(slu)m(\024)-46 b(ca)5 b(j.)42 b(Za)27 b(neki)h(fiksan)71 975 y Fq(T)41 b Fo(2)28 b Fw(R)347 939 y Fn(+)443 975 y Fv(s)33 b Fq(S)574 991 y Fm(T)666 975 y Fv(ozna)m(\024)-46 b(cimo)32 b(prugu)g Fo(h)p Fv(0)p Fq(;)17 b(T)d Fo(i)21 b(\002)i Fw(R)p Fv(.)71 1141 y Fw(T)-9 b(eorem)36 b(1.)100 b Fi(Nek)-5 b(a)33 b(su)g Fq(a;)17 b(b)28 b Fo(2)g Fv(L)1349 1105 y Fp(1)1430 1141 y Fv(\()p Fq(S)1528 1158 y Fm(T)1588 1141 y Fv(\))o Fi(,)33 b(i)f(nek)-5 b(a)33 b(vrijedi)e(jednostrani)h(Lipshitzo)m(v)h (uvjet)71 1350 y Fv(\(2\))1156 b Fq(a)p Fv(\()p Fq(t;)17 b(x)p Fv(\))22 b Fo(\000)h Fq(a)p Fv(\()p Fq(t;)17 b(y)t Fv(\))27 b Fo(\024)h Fq(C)7 b Fv(\()p Fq(t)p Fv(\)\()p Fq(x)22 b Fo(\000)h Fq(y)t Fv(\))71 1558 y Fi(za)36 b Fq(t)d Fo(2)h(h)p Fv(0)p Fq(;)17 b(T)d Fo(i)35 b Fi(i)g Fq(x)e(>)h(y)t Fi(,)i(gdje)g(je)g Fq(C)7 b Fv(\()p Fq(t)p Fv(\))36 b Fi(nerastu)m(\023)-46 b(ca)36 b(funk)m(cija)g(definirana)f (na)g Fw(R)3063 1522 y Fn(+)3127 1558 y Fi(.)53 b(Nadalje,)36 b(nek)-5 b(a)36 b(je)71 1677 y Fq(u)27 b Fo(2)h Fv(L)309 1641 y Fp(1)390 1677 y Fv(\()p Fq(S)488 1694 y Fm(T)548 1677 y Fv(\))k Fi(funk)m(cija)h(takv)-5 b(a)32 b(da)h(rje)-5 b(\024)-44 b(sa)m(v)-5 b(a)33 b(Cauc)m(h)m(yjev)i(problem)1541 1808 y Fl(\032)1632 1875 y Fq(@)1683 1890 y Fm(t)1715 1875 y Fq(u)22 b Fv(+)g Fq(@)1942 1890 y Fm(x)1990 1875 y Fv(\()p Fq(au)p Fv(\))28 b(=)f Fq(bu)1632 2018 y(u)p Fv(\(0)p Fq(;)17 b Fo(\001)p Fv(\))26 b(=)i(0)71 2230 y Fi(u)33 b(slab)s(om)d(smislu,)i(o)s(dnosno)g(da)h(za)f(sv)-5 b(aku)34 b Fq(')27 b Fo(2)i Fv(C)1988 2194 y Fp(1)1988 2254 y Fm(c)2069 2230 y Fv(\()p Fw(R)2191 2194 y Fn(2)2233 2230 y Fv(\))k Fi(k)m(o)5 b(ja)33 b(i)-5 b(\024)-44 b(s)m(\024)e(ceza)m (v)-5 b(a)33 b(za)f Fq(t)c Fo(\025)g Fq(T)47 b Fi(vrijedi)71 2488 y Fv(\(3\))1261 2353 y Fl(Z)1316 2578 y Fm(S)1363 2592 y Fj(T)1436 2488 y Fq(u)p Fv(\()p Fq(@)1581 2503 y Fm(t)1613 2488 y Fq(')22 b Fv(+)g Fq(a@)1899 2503 y Fm(x)1948 2488 y Fq(')g Fv(+)g Fq(b')p Fv(\))17 b Fq(dxdt)27 b Fv(=)h(0)17 b Fq(:)293 2760 y Fi(T)-8 b(ada)33 b(je)f Fq(u)27 b Fv(=)h(0)k Fi(sk)m(oro)i(svuda.)71 2902 y Fz(Dem.)104 b Fv(Nek)-5 b(a)37 b(su)g(\()p Fq(a)858 2917 y Fm(n)909 2902 y Fv(\))f(i)g(\()p Fq(b)1126 2917 y Fm(n)1176 2902 y Fv(\))h(nizo)m(vi)f(C)1607 2866 y Fp(1)1724 2902 y Fv(funk)m(cija)h(k)m(o)5 b(ji)36 b(k)m(on)m(v)m(ergira)5 b(ju)37 b(prema)f Fq(a)h Fv(i)f Fq(b)g Fv(sk)m(oro)i(svuda)71 3021 y(k)-5 b(ad)34 b Fq(n)d Fo(!)g(1)p Fv(,)j(a)g(zado)m(v)m(olja)m(v) -5 b(a)5 b(ju)35 b(i)e(iste)h(ograde,)h(uklju)m(\024)-46 b(cuju)m(\023)g(ci)34 b(i)f(\(2\).)49 b(T)-8 b(akv)m(e)36 b(nizo)m(v)m(e)f(mo)m(\024)-46 b(zemo)33 b(dobiti)71 3140 y(tak)m(o)g(da)f Fq(a)h Fv(i)e Fq(b)i Fv(k)m(on)m(v)m(oluiramo)f (sa)g(standardnim)g(izglad)p 2151 3088 30 2 v -1 w(iv)-5 b(a)m(\024)-46 b(cem)1276 3431 y Fq(\032)1326 3446 y Fm(n)1377 3431 y Fv(\()p Fq(x)p Fv(\))28 b(=)1639 3261 y Fl(\()1746 3365 y Fm(n)1792 3333 y Fk(2)p 1746 3381 84 4 v 1758 3439 a Fm(C)1840 3404 y Fq(e)1885 3339 y Fp(\000)2058 3308 y Fk(1)p 1954 3322 241 4 v 1954 3379 a(1)p Fd(\000j)p Fj(nx)p Fd(j)2158 3362 y Fk(2)2209 3404 y Fq(;)98 b Fo(j)p Fq(nx)p Fo(j)27 b Fq(<)h Fv(1)1769 3523 y(0)390 b Fq(;)99 b Fv(ina)m(\024)-46 b(ce,)71 3788 y(gdje)33 b(je)f(k)m(onstan)m(ta)i Fq(C)g Fv(=)1041 3707 y Fl(R)1088 3822 y Fp(j)p Fm(x)p Fp(j)p Fm(<)p Fn(1)1293 3788 y Fq(e)1338 3722 y Fp(\000)1491 3692 y Fk(1)p 1408 3706 200 4 v 1408 3762 a(1)p Fd(\000j)p Fj(x)p Fd(j)1571 3746 y Fk(2)1621 3788 y Fv(,)f(o)s(dnosno)g(o)s(dabrana)f(tak)m(o)g(da) h(vrijedi)1542 3939 y Fl(Z)1597 4164 y Fp(j)p Fm(nx)p Fp(j)p Fm(<)p Fn(1)1848 4075 y Fq(\032)1898 4090 y Fm(n)1949 4075 y Fv(\()p Fq(x)p Fv(\))17 b Fq(dx)28 b Fv(=)f(1)17 b Fq(:)71 4369 y Fv(T)-8 b(ak)m(od)p 299 4317 30 2 v 1 w(er,)41 b(zah)m(tijev)-5 b(amo)38 b(da)i(je)f Fq(b)1325 4384 y Fm(n)1376 4369 y Fv(\()p Fq(t;)17 b(x)p Fv(\))39 b(=)g(0)g(k)-5 b(ad)40 b(je)f Fq(t)g(<)2332 4329 y Fn(1)p 2328 4346 47 4 v 2328 4403 a Fm(n)2385 4369 y Fv(,)d(\024)-44 b(sto)39 b(se)h(mo)m(\024)-46 b(ze)39 b(p)s(osti)m(\023)-46 b(ci)38 b(tak)m(o)i(da)f(se)h(ne)71 4487 y(k)m(on)m(v)m(oluira)32 b(funk)m(cija)h Fq(b)p Fv(,)g(nego)f(funk)m(cija)h Fq(b)1652 4451 y Fp(\003)1652 4512 y Fm(n)1735 4487 y Fv(dobiv)m(ena)g(rezanjem)g (funk)m(cije)g Fq(b)p Fv(:)1613 4755 y Fq(b)1654 4713 y Fp(\003)1654 4779 y Fm(n)1733 4755 y Fv(=)1837 4614 y Fl(\032)1928 4704 y Fq(b;)105 b(t)28 b(<)2282 4665 y Fn(2)p 2278 4681 V 2278 4739 a Fm(n)1928 4823 y Fv(0)p Fq(;)97 b Fv(ina)m(\024)-46 b(ce.)71 5022 y(Nek)-5 b(a)43 b(je)g Fq( )49 b Fo(2)d Fv(C)735 4986 y Fp(1)735 5047 y Fm(c)816 5022 y Fv(\()p Fq(S)914 5039 y Fm(T)974 5022 y Fv(\).)74 b(Rije)-5 b(\024)-44 b(simo)40 b(Cauc)m(h)m(yjev)46 b(problem)c(unazad)h(za)g(sljede)m(\023)-46 b(cu)43 b(prijenosn)m(u)h (jed-)71 5141 y(nad)m(\024)-46 b(zbu)33 b(na)g Fq(S)608 5158 y Fm(T)667 5141 y Fv(:)71 5412 y(\(4\))1212 5272 y Fl(\032)1327 5308 y Fm(@)p 1314 5324 73 4 v 1314 5381 a(@)t(t)1396 5347 y Fq(')1460 5362 y Fm(nm)1600 5347 y Fv(+)22 b Fq(a)1749 5362 y Fm(n)1832 5308 y(@)p 1810 5324 89 4 v 1810 5381 a(@)t(x)1908 5347 y Fq(')1972 5362 y Fm(nm)2113 5347 y Fv(+)g Fq(b)2252 5362 y Fm(m)2324 5347 y Fq(')2388 5362 y Fm(nm)2534 5347 y Fv(=)27 b Fq( )2084 5490 y(')2148 5505 y Fm(nm)2266 5490 y Fv(\()p Fq(T)8 b(;)17 b(x)p Fv(\))28 b(=)f(0)17 b Fq(:)71 5692 y Fv(Iz)38 b(raspra)m(v)m(e)h(k)m(o)5 b(ja)39 b(je)e(pretho)s(dila)f(teorem)m(u)i (slijedi)e(da)i(rje)-5 b(\024)-44 b(senje)39 b(\(4\))e(p)s(osto)5 b(ji,)38 b(a)g(zb)s(og)f(regularnosti)71 5811 y(k)m(o)s(eficijenata)31 b(jednad)m(\024)-46 b(zb)s(e)34 b(u)f(\(4\))f(imamo)d(da)k(je)g Fq(')1967 5826 y Fm(nm)2112 5811 y Fo(2)g Fv(C)2281 5775 y Fp(1)2363 5811 y Fv(.)3770 6010 y Ft(11)p eop %%Page: 12 17 12 16 bop -165 -116 a Fr(Nelinearni)27 b(zak)n(oni)f(sa)n(\024)-39 b(cuv)-5 b(anja)57 83 y Fv(Nek)g(a)29 b(je)g Fq(x)454 98 y Fm(n)505 83 y Fv(\()p Fq(t)p Fv(;)17 b Fq(s;)g(y)t Fv(\))27 b(k)-5 b(arakteristik)g(a)28 b(k)m(o)5 b(ja)29 b(prolazi)f(kroz)h(\()p Fq(s;)17 b(y)t Fv(\),)28 b(o)s(dnosno)h(rje)-5 b(\024)-44 b(senje)31 b(obi)m(\024)-46 b(cne)28 b(dife-)-165 202 y(rencijalne)j(jednad)m(\024)-46 b(zb)s(e)1148 320 y(_)i Fq(x)1186 335 y Fm(n)1237 320 y Fv(\()p Fq(t)p Fv(;)17 b Fq(s;)g(y)t Fv(\))26 b(=)i Fq(a)1715 335 y Fm(n)1765 320 y Fv(\()p Fq(t;)17 b(x)1937 335 y Fm(n)1988 320 y Fv(\()p Fq(t)p Fv(;)g Fq(s;)g(y)t Fv(\)\))g Fq(;)-165 494 y Fv(uz)34 b Fq(x)21 509 y Fm(n)72 494 y Fv(\()p Fq(s)p Fv(;)17 b Fq(s;)g(y)t Fv(\))29 b(=)h Fq(y)t Fv(.)47 b(Budu)m(\023)-46 b(ci)34 b(da)g(smo)g Fq(')1381 509 y Fm(nm)1533 494 y Fv(fiksirali)d(na)i Fo(f)p Fq(T)14 b Fo(g)23 b(\002)h Fw(R)p Fv(,)34 b(rje)-5 b(\024)-44 b(senje)35 b(se)g(o)s(dred)p 3135 442 30 2 v(uje)g(unatrag)-165 612 y(du)m(\024)-46 b(z)33 b(k)-5 b(arakteristik)g(a.)42 b(Stoga)32 b(je)h(rje)-5 b(\024)-44 b(senje)34 b(Cauc)m(h)m(yjev)m(e)h (zada)m(\023)-46 b(ce)34 b(\(4\))e(dano)g(form)m(ulom)682 897 y Fq(')746 912 y Fm(nm)864 897 y Fv(\()p Fq(t;)17 b(x)p Fv(\))28 b(=)g Fo(\000)1300 762 y Fl(Z)1399 788 y Fm(T)1355 987 y(t)1476 897 y Fq( )t Fv(\()p Fq(s)p Fv(;)17 b Fq(x)1726 912 y Fm(n)1776 897 y Fv(\()p Fq(s)p Fv(;)g Fq(t;)g(x)p Fv(\)\))p Fq(e)2159 796 y Fe(R)2210 817 y Fj(s)2195 883 y(t)2259 856 y Fm(b)2292 864 y Fj(m)2355 856 y Fn(\()p Fm(\034)9 b Fn(;)p Fm(t;x)p Fn(\))p Fm(d\034)2674 897 y Fq(ds)17 b(:)-165 1162 y Fv(Vidimo)29 b(da)k(je)f Fq(')483 1177 y Fm(nm)601 1162 y Fv(\()p Fq(t;)17 b(x)p Fv(\))28 b(=)g(0)k(k)-5 b(ad)32 b(je)g Fq(t)c(>)g(T)35 b Fo(\000)22 b Fq(")p Fv(,)32 b(za)h(neki)f Fq(")27 b(>)h Fv(0.)43 b(Budu)m(\023)-46 b(ci)32 b(da)h(za)f(sv)-5 b(aki)32 b Fq(a)3291 1177 y Fm(n)3374 1162 y Fv(imamo)-165 1281 y(istu)27 b(uniformn)m(u)h(o)s(cjen)m(u,)h(to)f(zaklju)m(\024)-46 b(cujemo)28 b(da)g(p)s(osto)5 b(ji)27 b(k)m(ompaktan)h(skup)i(tak)-5 b(a)m(v)28 b(da)g(je)h(za)f(sv)-5 b(aki)28 b Fq(n)g Fv(i)-165 1399 y Fq(m)h Fv(imamo)e(da)i(je)h(nosa)m(\024)-46 b(c)30 b(funk)m(cije)g Fq(')1188 1414 y Fm(nm)1335 1399 y Fv(sadr)m(\024)-46 b(zan)30 b(u)g(njem)m(u.)42 b(Primjena)28 b(jednak)m(osti)i(\(3\))f(na) g Fq(')3332 1414 y Fm(nm)3480 1399 y Fv(da)5 b(je)486 1524 y Fl(Z)541 1749 y Fm(S)588 1763 y Fj(T)661 1660 y Fq(u )t(dtdx)27 b Fv(=)1107 1524 y Fl(Z)1162 1749 y Fm(S)1209 1763 y Fj(T)1282 1660 y Fq(u)1355 1579 y Fl(\000)1400 1660 y Fv(\()p Fq(a)1489 1675 y Fm(n)1562 1660 y Fo(\000)22 b Fq(a)p Fv(\))1782 1620 y Fm(@)p 1760 1637 89 4 v 1760 1694 a(@)t(x)1859 1660 y Fq(')1923 1675 y Fm(nm)2063 1660 y Fv(+)g(\()p Fq(b)h Fo(\000)f Fq(b)2403 1675 y Fm(m)2476 1660 y Fv(\))p Fq(')2578 1675 y Fm(nm)2696 1579 y Fl(\001)2774 1660 y Fq(dtdx)17 b(:)57 1932 y Fv(Izv)m(edimo)32 b(uniformne)e(o)s(cjene)j(za)f Fq(')1419 1947 y Fm(nm)1537 1932 y Fv(,)g(k)m(o)5 b(je)30 b(\023)-46 b(ce)33 b(p)s(o)m(vla)m(\024) -46 b(citi)29 b(da)j(desna)h(strana)f(te)m(\024)-46 b(zi)31 b(u)i(n)m(ulu)e(k)-5 b(ad)-165 2051 y(pustimo)31 b(na)5 b(jprije)33 b(da)f Fq(n)d Fo(!)e(1)p Fv(,)33 b(pa)g(onda)f(da)h Fq(m)28 b Fo(!)g(1)p Fv(.)44 b(Time)29 b(\023)-46 b(cemo)32 b(p)s(ok)-5 b(azati)32 b(da)g(je)h(lijev)-5 b(a)31 b(strana)-165 2170 y(uvijek)f(n)m(ula,)25 b(\024)-44 b(sto)28 b(\023)-46 b(ce)31 b(dati)e(tvrdnju)i(teorema.)42 b(Prema)29 b(T)-8 b(eorem)m(u)31 b(5)f(iz)g(pretho)s(dnog)g(p)s(ogla)m(vlja,)e(iz)i (\(2\))-165 2289 y(\(za)i Fq(a)48 2304 y Fm(n)99 2289 y Fv(\))g(dobiv)-5 b(amo)31 b(sljede)m(\023)-46 b(cu)33 b(uniformn)m(u)f(ogradu)g(k)-5 b(ad)33 b(je)f Fq(s)c Fo(2)g(h)p Fq(t;)17 b(T)d Fo(i)p Fv(:)1323 2499 y Fo(j)p Fq(@)1402 2514 y Fm(x)1450 2499 y Fq(x)1505 2514 y Fm(n)1556 2499 y Fv(\()p Fq(s)p Fv(;)j Fq(t;)g(x)p Fv(\))p Fo(j)27 b(\024)i Fq(C)2087 2514 y Fn(1)2146 2499 y Fq(;)-165 2709 y Fv(gdje)38 b Fq(C)119 2724 y Fn(1)200 2709 y Fv(o)m(visi)f(samo) h(o)g Fq(t)p Fv(.)61 b(Budu)m(\023)-46 b(ci)38 b(da)h(je)f Fq( )j Fo(2)d Fv(C)1765 2673 y Fp(1)1765 2733 y Fm(c)1847 2709 y Fv(\()p Fq(S)1945 2726 y Fm(T)2004 2709 y Fv(\),)i(to)e(imamo)e (da)i(je)g Fq( )t Fv(\()p Fq(t;)17 b(x)p Fv(\))38 b(=)f(0)h(k)-5 b(ad)39 b(je)f Fq(t)-165 2828 y Fv(do)m(v)m(oljno)d(mali,)f(o)s(dnosno) i(p)s(osto)5 b(ji)35 b Fq(\016)j(>)33 b Fv(0)j(tak)-5 b(a)m(v)36 b(da)g(je)g(\([0)p Fq(;)17 b(\016)t Fo(i)23 b(\002)i Fw(R)p Fv(\))f Fo(\\)h Fv(supp)18 b Fq(')33 b Fv(=)g Fo(;)p Fv(.)54 b(Lak)m(o)36 b(se)h(vidi)-165 2946 y(da,)29 b(p)s(o)m(\024)-46 b(cev)-5 b(\024)-44 b(si)29 b(o)s(d)f(nek)m(og)h(do)m(v)m(oljno)f(v)m(elik)m(og)g Fq(m)1530 2961 y Fn(0)1573 2946 y Fv(,)h(za)g Fq(m)f Fo(\025)g Fq(m)2053 2961 y Fn(0)2124 2946 y Fv(imamo)e Fq( )t Fv(\()p Fq(t;)17 b(x)p Fv(\))27 b(=)h(0)g(i)g Fq(b)3022 2961 y Fm(m)3094 2946 y Fv(\()p Fq(t;)17 b(x)p Fv(\))28 b(=)f(0)h(k)-5 b(ad)-165 3065 y(je)34 b Fq(t)d(<)138 3026 y Fn(1)p 124 3042 68 4 v 124 3099 a Fm(m)201 3065 y Fv(.)49 b(Iz)34 b(toga)g(zaklju)m(\024)-46 b(cujemo)33 b(da)h(se)h(in)m(tegrali)d(u)i(form)m(uli)e(za)i Fq(')2450 3080 y Fm(nm)2602 3065 y Fv(ne)h(mijenja)5 b(ju)33 b(o)s(d)g(prolask)-5 b(a)-165 3195 y(k)g(arakteristik)g(a)26 b(kroz)j(pra)m(v)-5 b(ac)28 b Fq(t)g Fv(=)1145 3156 y Fn(1)p 1131 3172 V 1131 3229 a Fm(m)1208 3195 y Fv(,)h(to)f(jest)h(za)f Fq(t)g(<)1868 3156 y Fn(1)p 1853 3172 V 1853 3229 a Fm(m)1959 3195 y Fv(imamo)d Fq(')2338 3210 y Fm(nm)2456 3195 y Fv(\()p Fq(t;)17 b(x)p Fv(\))28 b(=)g Fq(')2862 3210 y Fm(nm)2980 3195 y Fv(\()3042 3156 y Fn(1)p 3028 3172 V 3028 3229 a Fm(m)3105 3195 y Fq(;)17 b(x)3204 3210 y Fm(n)3255 3195 y Fv(\()3317 3156 y Fn(1)p 3303 3172 V 3303 3229 a Fm(m)3380 3195 y Fv(;)g Fq(t;)g(x)p Fv(\)\).)-165 3314 y(Dalje,)30 b(iz)h(form)m(ule)f(za)h(rje)-5 b(\024)-44 b(senje)33 b Fq(')1109 3329 y Fm(nm)1258 3314 y Fv(zaklju)m(\024)-46 b(cujemo)31 b(da)h(za)f(sv)-5 b(aki)32 b Fq(\034)39 b Fo(2)28 b(h)p Fv(0)p Fq(;)17 b(T)d Fv(],)31 b(uk)m(olik)m(o)f(je)i Fq(t)c Fo(2)g(h)p Fq(\034)6 b(;)17 b(T)d Fv(],)-165 3433 y(imamo)29 b(o)s(cjen)m(u)1358 3551 y Fq(@)1409 3566 y Fm(x)1457 3551 y Fq(')1521 3566 y Fm(nm)1639 3551 y Fv(\()p Fq(t;)17 b(x)p Fv(\))28 b Fo(\024)g Fq(C)2052 3566 y Fn(2)2111 3551 y Fq(;)-165 3724 y Fv(gdje)36 b(k)m(onstan)m(ta)h Fq(C)568 3739 y Fn(2)646 3724 y Fv(o)m(visi)e(samo)g(o)h Fq(m)g Fv(i)f Fq(\034)11 b Fv(,)37 b(pa)f(je)g(neo)m(visna)g(o)g Fq(n)p Fv(.)53 b(Stoga,)37 b(dalje)e(iz)d(\024)-46 b(cinjenice)35 b(da)h(za)-165 3843 y Fq(t)28 b(<)26 3804 y Fn(1)p 11 3820 V 11 3878 a Fm(m)121 3843 y Fv(imamo)1068 3973 y Fq(')1132 3988 y Fm(nm)1250 3973 y Fv(\()p Fq(t;)17 b(x)p Fv(\))28 b(=)f Fq(')1655 3988 y Fm(nm)1774 3973 y Fv(\()1836 3934 y Fn(1)p 1822 3950 V 1822 4008 a Fm(m)1899 3973 y Fq(;)17 b(x)1998 3988 y Fm(n)2049 3973 y Fv(\()2111 3934 y Fn(1)p 2097 3950 V 2097 4008 a Fm(m)2174 3973 y Fv(;)g Fq(t;)g(x)p Fv(\)\))-165 4158 y(zaklju)m(\024)-46 b(cujemo)32 b(da)g(k)-5 b(ad)33 b(je)f Fq(t)c(<)1036 4118 y Fn(1)p 1022 4135 V 1022 4192 a Fm(m)1132 4158 y Fv(vrijedi)700 4292 y Fl(Z)755 4517 y Fg(R)842 4427 y Fo(j)p Fq(@)921 4442 y Fm(x)969 4427 y Fq(')1033 4442 y Fm(nm)1151 4427 y Fv(\()p Fq(t;)17 b(x)p Fv(\))p Fo(j)g Fq(dx)27 b Fv(=)1643 4292 y Fl(Z)1698 4517 y Fg(R)1785 4427 y Fo(j)p Fq(@)1864 4442 y Fm(x)1912 4427 y Fq(')1976 4442 y Fm(nm)2094 4427 y Fv(\()2157 4388 y Fn(1)p 2142 4404 V 2142 4462 a Fm(m)2220 4427 y Fq(;)17 b(x)p Fv(\))p Fo(j)g Fq(dx)27 b Fo(\024)h Fq(C)2710 4442 y Fn(3)2769 4427 y Fq(;)-165 4686 y Fv(gdje)k Fq(C)113 4701 y Fn(3)188 4686 y Fv(o)m(visi)g(samo)g(o)g Fq(m)p Fv(.)44 b(Uz)33 b(o)m(v)m(e)h(unifomne)d(o)s(cjene,)i(k)-5 b(ad)33 b(pustimo)e Fq(n)d Fo(!)f(1)p Fv(,)33 b(dobiv)-5 b(amo)717 4805 y Fl(\014)717 4865 y(\014)717 4925 y(\014)717 4985 y(\014)750 4814 y(Z)805 5040 y Fm(S)852 5054 y Fj(T)925 4950 y Fq(u )20 b(dtdx)1256 4805 y Fl(\014)1256 4865 y(\014)1256 4925 y(\014)1256 4985 y(\014)1317 4950 y Fo(\024)28 b Fq(M)1544 4814 y Fl(Z)1599 5040 y Fp(j)p Fm(x)p Fp(j)p Fm()27 b Fv(0.)44 b(Uzmemo)31 b(li)g(prv)m(o)i(limes)e(k)-5 b(ad)33 b Fq(\034)39 b Fo(!)27 b Fv(0,)33 b(a)f(p)s(otom)f(i)h Fq(m)c Fo(!)f(1)p Fv(,)32 b(dobit)d(\023)-46 b(cemo)1383 5347 y Fl(Z)1438 5572 y Fm(S)1485 5586 y Fj(T)1558 5482 y Fq(u )20 b(dtdx)28 b Fv(=)g(0)17 b Fq(:)3324 5811 y Fw(Q.E.D.)-165 6010 y Ft(12)p eop %%Page: 13 18 13 17 bop 3102 -116 a Fr(Burgerso)n(v)-5 b(a)25 b(jednad)n(\024)-39 b(zba)1013 83 y Fw(Klasi)m(\024)-53 b(cna)36 b(rje)-6 b(\024)-50 b(senja)39 b(Burgerso)m(v)m(e)f(jednad)m(\024)-53 b(zb)s(e)293 291 y Fv(Prelazimo)31 b(na)h(prou)m(\024)-46 b(ca)m(v)-5 b(anje)33 b(Cauc)m(h)m(yjev)m(og)i(problema)c(za)i (Burgerso)m(vu)h(jednad)m(\024)-46 b(zbu)71 568 y(\(5\))1510 427 y Fl(\032)1602 492 y Fq(@)1653 507 y Fm(t)1685 492 y Fq(u)22 b Fv(+)g Fq(u@)1968 507 y Fm(x)2015 492 y Fq(u)28 b Fv(=)f(0)1792 636 y Fq(u)p Fv(\(0)p Fq(;)17 b(x)p Fv(\))27 b(=)g Fq(u)2258 651 y Fn(0)2300 636 y Fv(\()p Fq(x)p Fv(\))71 845 y(za)35 b Fq(t)e(>)g Fv(0.)52 b(Promotrimo)33 b(pripadne)i(k)-5 b(arakteristik)m(e,)37 b(dobiv)m(ene)f(k)-5 b(ao)35 b(in)m(tegralne)g(krivulje)g(diferenci-)71 964 y(jalne)d(jedand)m(\024)-46 b(zb)s(e)71 1186 y(\(6\))1605 1147 y Fm(d)p 1591 1163 68 4 v 1591 1220 a(dt)1669 1186 y Fq(x)p Fv(\()p Fq(t)p Fv(\))28 b(=)g Fq(u)p Fv(\()p Fq(t;)17 b(x)p Fv(\()p Fq(t)p Fv(\)\))g Fq(;)71 1417 y Fv(k)m(o)5 b(je)49 b(prolaze)e(kroz)h(neku)h(zadan)m(u)f(to)m(\024) -46 b(cku)49 b(iz)e Fw(R)1960 1376 y Fn(+)1960 1447 y(0)2056 1417 y Fo(\002)33 b Fw(R)p Fv(.)88 b(P)m(o)48 b(lan)m(\024)-46 b(canom)46 b(pra)m(vilu,)51 b(uv)-5 b(a)m(\024)-46 b(za)m(v)-5 b(anjem)71 1536 y(jednad)m(\024)-46 b(zb)s(e)34 b(\(5)619 1551 y Fn(1)661 1536 y Fv(\))e(slijedi)919 1758 y(_)-44 b Fq(u)o Fv(\()p Fq(t;)17 b(x)p Fv(\()p Fq(t)p Fv(\)\))28 b(=)g Fq(@)1461 1773 y Fm(t)1493 1758 y Fq(u)p Fv(\()p Fq(t;)17 b(x)p Fv(\()p Fq(t)p Fv(\)\))22 b(+)g Fq(@)2041 1773 y Fm(x)2089 1758 y Fq(u)p Fv(\()p Fq(t;)17 b(x)p Fv(\()p Fq(t)p Fv(\)\))p Fq(u)p Fv(\()p Fq(t;)g(x)p Fv(\()p Fq(t)p Fv(\)\))28 b(=)f(0)17 b Fq(:)71 1980 y Fv(Dakle,)47 b(ak)m(o)e(je)f Fq(u)h Fv(neprekidno)g(diferencijabilna,)g(tada)f(se)i (na)e(krivuljama)f(o)s(dred)p 3256 1928 30 2 v(enim)h(sa)h(\(6\))f Fq(u)71 2098 y Fv(ne)d(mijenja.)66 b(Nadalje,)42 b(u)f(\(6\))g(je)g(s)g (desne)h(strane)f(k)m(onstan)m(ta,)j(pa)d(slijedi)e(da)h(su)i(u)f(o)m (v)m(om)f(slu)m(\024)-46 b(ca)5 b(ju)71 2217 y(k)-5 b(arakteristik)m(e) 32 b(pra)m(v)m(ci.)298 2312 y(\024)293 2337 y(Zelimo)h(li)h(izra)m (\024)-46 b(cunati)34 b(vrijednost)i(funk)m(cije)g Fq(u)f Fv(u)h(nek)m(o)5 b(j)37 b(to)m(\024)-46 b(cki)35 b(\()p Fq(t;)17 b(x)p Fv(\),)37 b(trebamo)e(na)m(\023)-46 b(ci)35 b(k)-5 b(arakte-)71 2456 y(ristiku)38 b(k)m(o)5 b(ja)40 b(prolazi)d(kroz)i(tu)h(to)m(\024)-46 b(cku)39 b(i)f(o)s(drediti)g (gdje)h(ona)g(sije)m(\024)-46 b(ce)39 b(pra)m(v)-5 b(ac)39 b Fq(t)g Fv(=)g(0,)h(o)s(dnosno)f(treba)71 2574 y(na)m(\023)-46 b(ci)32 b Fq(y)j Fv(tak)-5 b(a)m(v)33 b(da)g(je)71 2796 y(\(7\))1479 b Fq(x)28 b Fv(=)g Fq(y)d Fv(+)d Fq(tu)2124 2811 y Fn(0)2166 2796 y Fv(\()p Fq(y)t Fv(\))71 3018 y(i)31 b(uzeti)71 3240 y(\(8\))1464 b Fq(u)p Fv(\()p Fq(t;)17 b(x)p Fv(\))27 b(=)h Fq(u)2113 3255 y Fn(0)2155 3240 y Fv(\()p Fq(y)t Fv(\))p Fq(:)293 3463 y Fv(Med)p 445 3411 V 1 w(utim,)53 b(tu)c(se)i(ja)m(vlja)d(pitanje)h(egzistencije) h(i)e(jedinstv)m(enosti)j(tak)m(o)m(v)m(og)f Fq(y)j Fv(\(egzistencije) 71 3582 y(i)c(jedinstv)m(enosti)i(k)-5 b(arakteristik)m(e\).)96 b(Egzistenciju)48 b(\023)-46 b(cemo)50 b(dobiti)e(k)-5 b(ao)50 b(p)s(osljedicu)f(dok)-5 b(aza)50 b(jedin-)71 3701 y(stv)m(enosti.)106 b(Sada)50 b(\023)-46 b(cemo,)57 b(dok)-5 b(azuju)m(\023)-46 b(ci)53 b(jedinstv)m(enost,)60 b(dobiti)51 b(i)h(n)m(u)m(\024)-46 b(zna)54 b(ograni)m(\024)-46 b(cenja)52 b(na)h(doseg)71 3819 y(rje)-5 b(\024)-44 b(senja.)74 b(U)42 b(\(7\))g(promotrimo)e Fq(x)j Fv(k)-5 b(ao)42 b(funk)m(ciju)h(v)-5 b(arijabli)39 b Fq(t)j Fv(i)g Fq(y)t Fv(,)i(za)f(k)m(o)5 b(ju,)46 b(uz)40 b(\024)-46 b(cvrsti)42 b Fq(t)p Fv(,)j(tra)m(\024)-46 b(zimo)71 3938 y(in)m(v)m(erz)33 b(\(jedinstv)m(eni)f Fq(y)j Fv(za)c(sv)-5 b(aki)32 b Fq(x)p Fv(\).)43 b(T)-8 b(eorem)32 b(o)g(implicitno)27 b(zadano)5 b(j)32 b(funk)m(ciji,)g(k)m(o)5 b(ji)31 b(da)5 b(je)33 b(egzisten-)71 4057 y(ciju)k(i)g(jedinstv)m(enost)j(lok)-5 b(alnog)35 b(C)1380 4021 y Fn(1)1461 4057 y Fv(in)m(v)m(erza,)41 b(tra)m(\024)-46 b(zi)37 b(da)h(je)g Fq(@)2365 4072 y Fm(y)2410 4057 y Fq(x)f Fv(=)g(1)26 b(+)g Fq(tu)2883 4021 y Fp(0)2883 4085 y Fn(0)2925 4057 y Fv(\()p Fq(y)t Fv(\))36 b Fo(6)p Fv(=)h(0.)59 b(Pitamo)37 b(se)i(za)71 4175 y(k)m(o)5 b(je)34 b Fq(t)g Fv(to)f(vrijedi.)44 b(Nara)m(vno,)34 b(zb)s(og)f(p)s(o)m(\024)-46 b(cetnog)33 b(uvjeta,)h(zanima)e(nas)i(in) m(terv)-5 b(al)32 b(k)m(o)5 b(ji)33 b(uklju)m(\024)-46 b(cuje)34 b(n)m(ulu.)71 4294 y(Za)e Fq(t)c Fv(=)f(0)33 b(je)f Fq(@)616 4309 y Fm(y)661 4294 y Fq(x)c Fv(=)g(1)f Fq(>)h Fv(0,)k(pa)h(zah)m(tijev)-5 b(amo)31 b(da)i(je)1666 4516 y(0)28 b Fq(<)f Fv(1)22 b(+)g Fq(tu)2106 4475 y Fp(0)2106 4541 y Fn(0)2148 4516 y Fv(\()p Fq(y)t Fv(\))p Fq(:)71 4738 y Fv(Kad)37 b(je)h Fq(u)454 4702 y Fp(0)454 4766 y Fn(0)496 4738 y Fv(\()p Fq(y)t Fv(\))e Fh(>)h Fv(0,)i(to)e(je)h(ispunjeno)g(za)g(sv)-5 b(aki)38 b Fq(t)f(>)f Fv(0.)59 b(Slo)m(\024)-46 b(zeniji)36 b(je)h(slu)m(\024)-46 b(ca)5 b(j)38 b(k)-5 b(ad)38 b(u)g(nek)m(o)5 b(j)39 b(to)m(\024)-46 b(cki)37 b Fq(y)71 4857 y Fv(imamo)30 b Fq(u)447 4821 y Fp(0)447 4885 y Fn(0)489 4857 y Fv(\()p Fq(y)t Fv(\))c Fq(<)i Fv(0,)k(i)g(tada)g(ta)5 b(j)33 b(zah)m(tjev)g(zapisujemo)f(u)h (obliku)1726 5128 y Fq(t)28 b(<)2030 5060 y Fv(1)p 1903 5105 303 4 v 1903 5196 a Fo(\000)p Fq(u)2036 5162 y Fp(0)2036 5226 y Fn(0)2078 5196 y Fv(\()p Fq(y)t Fv(\))2215 5128 y Fq(;)71 5406 y Fv(a)k(za)h(ograni)m(\024)-46 b(cenje)32 b(uzimamo)e(da)j(je)f Fq(t)c(<)g(T)14 b Fv(,)32 b(pri)d(\024)-46 b(cem)m(u)33 b(je)1333 5669 y Fq(T)41 b Fv(=)28 b(inf)1573 5729 y Fm(y)1806 5602 y Fv(1)p 1679 5646 V 1679 5737 a Fo(\000)p Fq(u)1812 5703 y Fp(0)1812 5767 y Fn(0)1855 5737 y Fv(\()p Fq(y)t Fv(\))2019 5669 y(=)2341 5602 y(1)p 2133 5646 467 4 v 2133 5741 a(sup)2186 5820 y Fm(y)2296 5741 y Fo(\000)p Fq(u)2429 5699 y Fp(0)2429 5765 y Fn(0)2472 5741 y Fv(\()p Fq(y)t Fv(\))2609 5669 y Fq(:)3770 6010 y Ft(13)p eop %%Page: 14 19 14 18 bop -165 -116 a Fr(Nelinearni)27 b(zak)n(oni)f(sa)n(\024)-39 b(cuv)-5 b(anja)-165 83 y Fv(Za)42 b Fq(t)47 b Fo(2)g Fv([0)p Fq(;)17 b(T)d Fo(i)42 b Fv(imamo)f(da)i(vrijedi)g Fq(@)1296 98 y Fm(y)1341 83 y Fq(x)j(>)h Fv(0,)f(i)c(zaklju)m(\024)-46 b(cujemo)43 b(da,)j(ak)m(o)e(su)g Fq(u)2914 98 y Fn(0)3000 83 y Fv(i)e Fq(u)3126 47 y Fp(0)3126 111 y Fn(0)3212 83 y Fv(ograni)m(\024)-46 b(cene)-165 202 y(na)39 b([0)p Fq(;)17 b(T)d Fo(i)26 b(\002)h Fw(R)p Fv(,)40 b(p)s(osto)5 b(ji)39 b(jedinstv)m(eno)h(C)1410 166 y Fn(1)1492 202 y Fv(rje)-5 b(\024)-44 b(senje)41 b(Cauc)m(h)m(yjev)m(og)h(problema.)62 b(Iz)40 b(\(7\))e(i)h(\(8\))g(lakim)-165 320 y(ra)m(\024)-46 b(cunom)32 b(slijedi)1171 493 y Fq(@)1222 508 y Fm(x)1270 493 y Fq(u)27 b Fv(=)1466 426 y Fq(u)1522 390 y Fp(0)1522 454 y Fn(0)1564 426 y Fv(\()p Fq(y)t Fv(\))p 1466 470 226 4 v 1503 562 a Fq(@)1554 577 y Fm(y)1599 562 y Fq(x)1729 493 y Fv(=)1945 426 y Fq(u)2001 390 y Fp(0)2001 454 y Fn(0)2043 426 y Fv(\()p Fq(y)t Fv(\))p 1843 470 430 4 v 1843 562 a(1)22 b(+)g Fq(tu)2103 527 y Fp(0)2103 591 y Fn(0)2145 562 y Fv(\()p Fq(y)t Fv(\))2298 493 y Fq(:)-165 720 y Fv(Oda)m(vdje)46 b(vidimo)e(da)h(ak)m(o)h(u)g(nek)m(om)g Fq(y)j Fv(funk)m(cija)c Fo(\000)p Fq(u)1944 684 y Fp(0)1944 748 y Fn(0)2033 720 y Fv(p)s(osti)m(\024)-46 b(ze)45 b(p)s(ozitiv)-5 b(an)44 b(maksimim)m(um,)h(onda)-165 839 y Fq(@)-114 854 y Fm(x)-66 839 y Fq(u)32 b Fv(div)m(ergira)h(k)-5 b(ad)33 b Fq(t)g Fv(te)m(\024)-46 b(zi)33 b Fq(T)14 b Fv(.)45 b(Odnosno,)34 b(neprekidno)g(diferencijabilno)c(rje)-5 b(\024)-44 b(senje)35 b(p)s(osto)5 b(ji)32 b(samo)g(na)-165 958 y([0)p Fq(;)17 b(T)d Fo(i)21 b(\002)h Fw(R)p Fv(.)44 b(Iz)33 b(ra)m(\024)-46 b(cuna)32 b(se)i(mo)m(\024)-46 b(ze)32 b(tak)m(od)p 1338 906 30 2 v 1 w(er)h(primijetiti)28 b(da)k(za)h(sv)-5 b(aki)33 b Fq(t)27 b Fo(2)i Fv([0)p Fq(;)17 b(T)d Fo(i)31 b Fv(vrijedi)1216 1143 y(sup)1268 1222 y Fm(x)1380 1143 y Fo(j)p Fq(u)p Fv(\()p Fq(t;)17 b(x)p Fv(\))p Fo(j)26 b Fv(=)i(sup)1883 1222 y Fm(x)1995 1143 y Fo(j)p Fq(u)2079 1158 y Fn(0)2121 1143 y Fv(\()p Fq(x)p Fv(\))p Fo(j)-165 1372 y Fv(i)1047 1381 y Fl(Z)1103 1606 y Fg(R)1190 1516 y Fo(j)o Fq(@)1268 1531 y Fm(x)1316 1516 y Fq(u)p Fv(\()p Fq(t;)17 b(x)p Fv(\))p Fo(j)33 b Fq(dx)28 b Fv(=)1880 1381 y Fl(Z)1936 1606 y Fg(R)2023 1516 y Fo(j)p Fq(u)2107 1475 y Fp(0)2107 1541 y Fn(0)2149 1516 y Fv(\()p Fq(y)t Fv(\))p Fo(j)p Fq(dy)18 b(:)-165 1727 y Fv(Vidimo)k(da)j(se)h(maksimalna)d(vrijednost)i(i)f(totalna)g(v) -5 b(arijacija)22 b(rje)-5 b(\024)-44 b(senja)26 b(ne)g(mijenja,)19 b(\024)-44 b(sto)26 b(da)5 b(je)26 b(smisao)-165 1846 y(tra)m(\024)-46 b(zenju)37 b(ograni)m(\024)-46 b(cenog)37 b(omed)p 957 1794 V(ene)h(v)-5 b(arijacije)35 b(nak)m(on)j(nastank)-5 b(a)38 b(singularnosti.)55 b(Pitanje)37 b(je)h(k)-5 b(ak)m(o)37 b(u)-165 1964 y(tom)31 b(slu)m(\024)-46 b(ca)5 b(ju)33 b(sh)m(v)-5 b(a)m(\023)-46 b(cati)32 b(diferencijaln)m(u)g(jednad)m (\024)-46 b(zbu.)44 b(Odgo)m(v)m(or)33 b(le)m(\024)-46 b(zi)32 b(u)h(slab)s(o)5 b(j)31 b(form)m(ulaciji.)787 2168 y Fw(Slaba)38 b(form)m(ulacija)f(Burgerso)m(v)m(e)g(jednad)m(\024) -53 b(zb)s(e)57 2329 y Fv(Ka)m(\024)-46 b(zemo)25 b(da)h(je)f Fq(u)j Fo(2)g Fv(L)890 2293 y Fp(1)890 2361 y Fn(lo)r(c)990 2329 y Fv(\()p Fw(R)1112 2288 y Fn(+)1112 2359 y(0)1197 2329 y Fo(\002)23 b Fw(R)p Fv(\))i Fz(slab)s(o)h(rje)-6 b(\024)-43 b(senje)26 b Fv(Burgerso)m(v)m(e)i(jednad)m(\024)-46 b(zb)s(e)27 b(\(5\),)g(ak)m(o)f(za)f(sv)-5 b(aku)-165 2448 y(funk)m(ciju)33 b Fq(')27 b Fo(2)h Fv(C)464 2411 y Fp(1)464 2472 y Fm(c)546 2448 y Fv(\()p Fw(R)668 2411 y Fn(+)753 2448 y Fo(\002)23 b Fw(R)p Fv(\))32 b(vrijedi)1020 2682 y Fo(\000)1114 2547 y Fl(Z)1169 2772 y Fg(R)1235 2751 y Fk(2)1294 2602 y Fl(\000)1339 2682 y Fq(u@)1446 2697 y Fm(t)1478 2682 y Fq(')22 b Fv(+)1672 2643 y Fn(1)p 1672 2660 39 4 v 1672 2717 a(2)1720 2682 y Fq(u)1776 2641 y Fn(2)1819 2682 y Fq(@)1870 2697 y Fm(x)1918 2682 y Fq(')1982 2602 y Fl(\001)2060 2682 y Fq(dtdx)28 b Fv(=)g(0)17 b Fq(:)-165 2927 y Fv(Pro)-5 b(\024)-44 b(sirimo)24 b Fq(u)h Fv(n)m(ulom)g(na)i Fw(R)859 2891 y Fp(\000)931 2927 y Fo(\002)9 b Fw(R)p Fv(.)42 b(Ka)m(\024)-46 b(zemo)26 b(da)g(je)g(takv)m(o)h Fq(u)f Fv(slab)s(o)g(rje)-5 b(\024)-44 b(senje)27 b(Cauc)m(h)m(yjev)m(og)i(problema)-165 3046 y(ak)m(o)j(za)h(sv)-5 b(aku)34 b Fq(')27 b Fo(2)h Fv(C)665 3010 y Fp(1)665 3071 y Fm(c)747 3046 y Fv(\()p Fw(R)869 3010 y Fn(2)911 3046 y Fv(\))k(vrijedi)831 3281 y Fo(\000)925 3145 y Fl(Z)980 3371 y Fg(R)1046 3349 y Fk(2)1104 3200 y Fl(\000)1150 3281 y Fq(u@)1257 3296 y Fm(t)1289 3281 y Fq(')22 b Fv(+)1483 3242 y Fn(1)p 1483 3258 V 1483 3315 a(2)1531 3281 y Fq(u)1587 3240 y Fn(2)1629 3281 y Fq(@)1680 3296 y Fm(x)1729 3281 y Fq(')1793 3200 y Fl(\001)1871 3281 y Fq(dtdx)28 b Fv(=)2195 3145 y Fl(Z)2250 3371 y Fg(R)2337 3281 y Fq(u)2393 3296 y Fn(0)2435 3281 y Fq(')17 b(dx)g(:)57 3514 y Fv(Prije)25 b(nego)g(li)f(prijed)p 800 3462 30 2 v(emo)g(na)i(prou)m(\024)-46 b(ca)m(v)-5 b(anje)26 b(sv)m(o)5 b(jsta)m(v)-5 b(a)27 b(slabih)d(rje)-5 b(\024)-44 b(senja,)27 b(isk)-5 b(azat)23 b(\023)-46 b(cemo)24 b(k)m(oristan)-165 3633 y(teorem)32 b(iz)g(v)m(ektorsk)m(e)j (analize)c(\(vidi)h([Ev2,)h(App.)44 b(C,)33 b(Theorem)g(1]\).)-165 3794 y Fw(T)-9 b(eorem)34 b(2.)56 b(\(Gauss-Green\))32 b Fi(Nek)-5 b(a)31 b(je)g Fv(\012)g Fi(omed)p 1767 3742 V(en)h(otv)m(oren)f(p)s(o)s(dskup)h(o)s(d)e Fw(R)2892 3758 y Fm(d)2967 3794 y Fi(i)g(nek)-5 b(a)31 b(m)m(u)f(je)h(rub)-165 3913 y Fq(@)5 b Fv(\012)34 b Fi(klase)e Fv(C)306 3877 y Fn(1)349 3913 y Fi(.)44 b(Ak)m(o)33 b(je)g Fq(u)27 b Fo(2)h Fv(C)976 3877 y Fn(1)1019 3913 y Fv(\()p 1057 3833 71 4 v(\012\))p Fi(,)33 b(tada)f(vrijedi)1242 4012 y Fl(Z)1297 4238 y Fn(\012)1373 4148 y Fq(@)1424 4163 y Fm(i)1455 4148 y Fq(u)17 b(dx)27 b Fv(=)1765 4012 y Fl(Z)1820 4238 y Fm(@)t Fn(\012)1941 4148 y Fq(u)p Fz(n)2047 4163 y Fm(i)2094 4148 y Fq(dS)22 b(;)-165 4381 y Fi(gdje)32 b(je)h Fz(n)g Fi(jedini)m(\024)-46 b(cna)31 b(v)-5 b(anjsk)g(a)33 b(normala)d(na)j Fq(@)5 b Fv(\012)p Fi(.)p 3632 4424 30 43 v 57 4560 a Fv(U)28 b(k)-5 b(akv)m(o)5 b(j)30 b(su)f(v)m(ezi)g (klasi)m(\024)-46 b(cno)28 b(i)g(slab)s(o)f(rje)-5 b(\024)-44 b(senje?)44 b(Nek)-5 b(a)29 b(je)f Fq(u)h Fv(klase)f(C)2592 4524 y Fn(1)2635 4560 y Fv(\()p Fw(R)2757 4519 y Fn(+)2757 4590 y(0)2843 4560 y Fo(\002)22 b Fw(R)p Fv(\))29 b(slab)s(o)e(rje)-5 b(\024)-44 b(senje)-165 4679 y(Cauc)m(h)m(yjev)m(og)35 b(problema.)42 b(Za)32 b Fq(')27 b Fo(2)h Fv(C)1266 4643 y Fn(1)1309 4679 y Fv(\()p Fw(R)1431 4643 y Fn(2)1473 4679 y Fv(\))33 b(ra)m(\024)-46 b(cunamo)111 4778 y Fl(Z)167 5004 y Fg(R)254 4914 y Fq(u)310 4929 y Fn(0)352 4914 y Fq(')17 b(dx)27 b Fv(=)g Fo(\000)763 4778 y Fl(Z)819 5004 y Fg(R)885 4982 y Fk(2)943 4833 y Fl(\000)989 4914 y Fq(u@)1096 4929 y Fm(t)1128 4914 y Fq(')22 b Fv(+)1322 4875 y Fn(1)p 1322 4891 39 4 v 1322 4948 a(2)1370 4914 y Fq(u)1426 4873 y Fn(2)1468 4914 y Fq(@)1519 4929 y Fm(x)1567 4914 y Fq(')1631 4833 y Fl(\001)1710 4914 y Fq(dtdx)566 5173 y Fv(=)27 b Fo(\000)763 5037 y Fl(Z)819 5263 y Fg(R)885 5241 y Fk(2)943 5092 y Fl(\000)989 5173 y Fq(@)1040 5188 y Fm(t)1072 5173 y Fv(\()p Fq(u')p Fv(\))22 b(+)g Fq(@)1439 5188 y Fm(x)1487 5173 y Fv(\()1535 5133 y Fn(1)p 1535 5150 V 1535 5207 a(2)1583 5173 y Fq(u)1639 5132 y Fn(2)1681 5173 y Fq(')p Fv(\))1783 5092 y Fl(\001)1862 5173 y Fq(dtdx)g Fv(+)2174 5037 y Fl(Z)2229 5263 y Fg(R)2295 5241 y Fk(2)2354 5173 y Fq(')2435 5092 y Fl(\000)2480 5173 y Fq(@)2531 5188 y Fm(t)2563 5173 y Fq(u)g Fv(+)g Fq(@)2790 5188 y Fm(x)2838 5173 y Fv(\()2886 5133 y Fn(1)p 2886 5150 V 2886 5207 a(2)2934 5173 y Fq(u)2990 5132 y Fn(2)3032 5173 y Fv(\))3070 5092 y Fl(\001)3149 5173 y Fq(dtdx)17 b(:)-165 5406 y Fv(P)m(o)37 b(Gauss-Greeno)m(v)m(om)g (teorem)m(u,)h(uv)-5 b(a)m(\024)-46 b(za)m(v)-5 b(a)5 b(ju)m(\023)-46 b(ci)37 b(da)g(je)g Fq(')g Fv(funk)m(cija)g(s)h(k)m (ompaktnim)d(nosa)m(\024)-46 b(cem)37 b(i)f(da)-165 5525 y(je)c Fq(u)c Fv(=)f(0)32 b(za)h Fq(t)28 b(<)f Fv(0,)33 b(iz)f(prv)m(og)h(in)m(tegrala)e(na)h(desno)5 b(j)34 b(strani)e(dobiv)-5 b(amo)664 5754 y Fo(\000)758 5619 y Fl(Z)814 5844 y Fg(R)880 5823 y Fk(2)938 5674 y Fl(\000)984 5754 y Fq(@)1035 5769 y Fm(t)1067 5754 y Fv(\()p Fq(u')p Fv(\))21 b(+)h Fq(@)1433 5769 y Fm(x)1482 5754 y Fv(\()1530 5715 y Fn(1)p 1530 5731 V 1530 5789 a(2)1578 5754 y Fq(u)1634 5713 y Fn(2)1676 5754 y Fq(')p Fv(\))1778 5674 y Fl(\001)1857 5754 y Fq(dtdx)27 b Fv(=)2180 5619 y Fl(Z)2235 5844 y Fg(R)2322 5754 y Fq(u)p Fv(\(0)p Fq(;)17 b(x)p Fv(\))p Fq(')g(dx)g(:)-165 6010 y Ft(14)p eop %%Page: 15 20 15 19 bop 3102 -116 a Fr(Burgerso)n(v)-5 b(a)25 b(jednad)n(\024)-39 b(zba)71 83 y Fv(Pretho)s(dne)33 b(rezultate)g(mo)m(\024)-46 b(zemo)31 b(za)5 b(jedno)34 b(zapisati)d(u)i(sljede)m(\023)-46 b(cem)32 b(obliku)71 312 y(\(9\))719 177 y Fl(Z)775 402 y Fg(R)845 312 y Fv(\()p Fq(u)p Fv(\(0)p Fq(;)17 b(x)p Fv(\))22 b Fo(\000)g Fq(u)1340 327 y Fn(0)1382 312 y Fv(\()p Fq(x)p Fv(\)\))p Fq(')17 b(dx)22 b Fv(+)1858 177 y Fl(Z)1914 402 y Fg(R)1980 381 y Fk(2)2038 312 y Fq(')2119 231 y Fl(\000)2164 312 y Fq(@)2215 327 y Fm(t)2247 312 y Fq(u)g Fv(+)g Fq(@)2474 327 y Fm(x)2522 312 y Fv(\()2570 273 y Fn(1)p 2570 289 39 4 v 2570 346 a(2)2619 312 y Fq(u)2675 271 y Fn(2)2717 312 y Fv(\))2755 231 y Fl(\001)2833 312 y Fq(dtdx)28 b Fv(=)g(0)17 b Fq(:)71 545 y Fv(Budu)m(\023)-46 b(ci)29 b(da)g(u)g(tu)f(jednak)m(ost)j(mo)m(\024)-46 b(zemo)27 b(uvrstiti)h(proizv)m(oljn)m(u)g(glatku)g(funk)m(ciju)i Fq(')e Fv(takvu)i(da)f(je)g Fq(')e Fv(=)h(0)71 664 y(za)35 b Fq(t)d Fv(=)g(0,)j(zaklju)m(\024)-46 b(cujemo)35 b(da)g(jednad)m (\024)-46 b(zba)36 b(\(5\))e(vrijedi)g(u)i(jak)m(om)e(\(klasi)m(\024) -46 b(cnom\))33 b(smislu.)50 b(Za)34 b(op)m(\023)-46 b(cenite)71 783 y Fq(')p Fv(,)32 b(uzev)-5 b(\024)-44 b(si)34 b(u)f(obzir)e(jednad)m(\024)-46 b(zbu)34 b(\(5\),)e(dobiv)-5 b(amo)31 b(i)h(da)g(je)h(p)s(o)m(\024)-46 b(cetni)32 b(uvjet)h(ispunjen.)293 902 y(Sad)24 b(\023)-46 b(cemo)25 b(dok)-5 b(azati)25 b(op)m(\023)-46 b(cenitu)27 b(lem)m(u)e(k)m(o)5 b(ja)24 b(\023)-46 b(ce)27 b(nam)e(o)s(drediti)g(p)s(ona)-5 b(\024)-44 b(sanje)26 b(funk)m(cija)g(ok)m(o)h(disk)m(on-)71 1020 y(tin)m(uiteta.)41 b(Pretp)s(osta)m(vljat)28 b(\023)-46 b(cemo)30 b(da)g(disk)m(on)m(tin)m(uiteti)f(le)m(\024)-46 b(ze)31 b(na)f(parametrizabilnim)c(plohama,)j(k)m(o)5 b(je)71 1139 y(naziv)-5 b(amo)46 b Fz(frontama)41 b(\024)-43 b(sok)m(ova)p Fv(.)89 b(Kad)48 b(radimo)d(u)j(jedno)5 b(j)48 b(prostorno)5 b(j)48 b(dimenziji,)c(\024)-44 b(sok)49 b(je)f(krivulja)71 1258 y Fq(x)37 b Fv(=)g Fq(g)t Fv(\()p Fq(t)p Fv(\).)528 1233 y(\024)525 1258 y(Sto)m(vi)-5 b(\024)-44 b(se,)40 b(u)e(tom)f(slu)m(\024)-46 b(ca)5 b(ju)36 b(\023)-46 b(cemo)38 b(za)g(\()p Fq(t;)17 b(x)p Fv(\))38 b(tak)-5 b(a)m(v)39 b(da)f(je)g Fq(x)f(<)g(g)t Fv(\()p Fq(t)p Fv(\))h(re)m(\023)-46 b(ci)38 b(da)g(je)g(lijev)m(o)f(o) s(d)66 1376 y(\024)-44 b(sok)-5 b(a,)33 b(o)s(dnosno)f(za)h Fq(x)28 b(>)f(g)t Fv(\()p Fq(t)p Fv(\))32 b(da)h(je)g(desno.)71 1537 y Fw(Lema)40 b(1.)56 b Fi(Nek)-5 b(a)35 b(su)g(funk)m(cije)g Fq(u;)17 b(v)34 b Fv(:)d Fw(R)1581 1501 y Fn(2)1655 1537 y Fo(!)f Fw(R)k Fi(glatk)m(e)h(sa)f(sv)-5 b(ak)m(e)36 b(strane)g(glatk)m(e)e(krivulje)g Fq(x)d Fv(=)g Fq(g)t Fv(\()p Fq(t)p Fv(\))p Fi(,)71 1656 y(na)45 b(k)m(o)5 b(jo)g(j)46 b(p)s(osto)5 b(je)46 b(njiho)m(vi)e(limesi)e(i)j(limesi)d (njiho)m(vih)j(prvih)g(deriv)-5 b(acija,)47 b(k)m(o)5 b(je)46 b(u)f(slab)s(om)f(smislu)71 1775 y(zado)m(v)m(olja)m(v)-5 b(a)5 b(ju)71 1959 y Fv(\(10\))1423 b Fq(@)1719 1974 y Fm(t)1751 1959 y Fq(u)22 b Fv(+)g Fq(@)1978 1974 y Fm(x)2026 1959 y Fq(v)32 b Fv(=)27 b(0)17 b(;)71 2144 y Fi(o)s(dnosno,)33 b(nek)-5 b(a)32 b(za)h(sv)-5 b(aku)34 b(funk)m(ciju)f Fq(')27 b Fo(2)h Fv(C)1731 2108 y Fp(1)1731 2169 y Fm(c)1813 2144 y Fv(\()p Fw(R)1935 2108 y Fn(2)1977 2144 y Fv(\))k Fi(vrijedi)1469 2379 y Fo(\000)1563 2243 y 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Fo(2)g Fw(R)1773 3205 y Fn(2)1862 3241 y Fv(:)h Fq(x)f(<)g(g)t Fv(\()p Fq(t)p Fv(\))p Fo(g)c Fv(i)g(\012)2558 3256 y Fn(+)2670 3241 y Fv(=)k Fo(f)p Fv(\()p Fq(t;)17 b(x)p Fv(\))47 b Fo(2)g Fw(R)3297 3205 y Fn(2)3387 3241 y Fv(:)g Fq(x)g(>)g(g)t Fv(\()p Fq(t)p Fv(\))p Fo(g)71 3360 y Fv(otv)m(orene)36 b(skup)s(o)m(v)m(e)g(na)c(\024)-46 b(cijim)32 b(zatv)-5 b(ara)m(\024)-46 b(cima)33 b(se)i Fq(u)f Fv(i)g Fq(v)k Fv(mogu)c(do)s(de\014nirati)f(do)h(C)3123 3323 y Fn(1)3166 3360 y Fv(funk)m(cija.)50 b(Nek)-5 b(a)34 b(je)71 3478 y Fq(')e Fv(proizv)m(oljna)g(funk)m(cija)g(iz)g(C)1210 3442 y Fp(1)1210 3503 y Fm(c)1291 3478 y Fv(\()p Fw(R)1413 3442 y Fn(2)1456 3478 y Fv(\))g(takv)-5 b(a)33 b(da)f(je)h(supp)17 b Fq(')28 b Fo(\022)g Fv(\012)2517 3493 y Fp(\000)2581 3478 y Fv(.)44 b(Ra)m(\024)-46 b(cunamo:)721 3713 y(0)27 b(=)h Fo(\000)995 3577 y Fl(Z)1050 3803 y Fn(\012)1105 3813 y Fd(\000)1166 3713 y Fv(\()p Fq(u@)1311 3728 y Fm(t)1343 3713 y Fq(')22 b Fv(+)g Fq(v)t(@)1629 3728 y Fm(x)1677 3713 y 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Fd(\000)1768 5737 y Fq(')p Fv(\()p Fq(@)1921 5752 y Fm(t)1953 5737 y Fq(u)21 b Fv(+)h Fq(@)2179 5752 y Fm(x)2228 5737 y Fq(v)t Fv(\))17 b Fq(dtdx)g(;)3770 6010 y Ft(15)p eop %%Page: 16 21 16 20 bop -165 -116 a Fr(Nelinearni)27 b(zak)n(oni)f(sa)n(\024)-39 b(cuv)-5 b(anja)-165 83 y Fv(a)40 b(k)-5 b(ak)m(o)41 b(je)g Fq(')f Fv(bio)g(proizv)m(oljan,)h(dobiv)-5 b(amo)39 b(da)i(je)f(\(10\))g(ispunjeno)h(u)g(jak)m(om)f(smislu)f(na)i(\012)3364 98 y Fp(\000)3428 83 y Fv(.)68 b(Isti)-165 202 y(p)s(ostupak)33 b(mo)m(\024)-46 b(zemo)31 b(primijeniti)e(za)k(supp)17 b Fq(')28 b Fo(\022)g Fv(\012)1726 217 y Fn(+)1790 202 y Fv(,)g(\024)-44 b(sto)33 b(da)5 b(je)33 b(prvu)g(tvrdnju.)57 324 y(Sad)f(uzimamo)f(proizv)m(oljn)m(u)h(funk)m(ciju)h Fq(')27 b Fo(2)h Fv(C)1797 288 y Fp(1)1797 348 y Fm(c)1879 324 y Fv(\()p Fw(R)2001 288 y Fn(2)2043 324 y Fv(\))k(i)g(ra)m(\024)-46 b(cunamo:)494 602 y(0)27 b(=)22 b Fo(\000)768 467 y Fl(Z)823 692 y Fg(R)889 671 y Fk(2)931 602 y Fv(\()p Fq(u@)1076 617 y Fm(t)1108 602 y Fq(')g Fv(+)g Fq(v)t(@)1394 617 y Fm(x)1442 602 y Fq(')p Fv(\))17 b Fq(dtdx)570 861 y Fv(=)22 b Fo(\000)768 725 y Fl(Z)823 951 y Fn(\012)878 961 y Fd(\000)939 861 y Fv(\()p Fq(u@)1084 876 y Fm(t)1116 861 y Fq(')g Fv(+)g Fq(v)t(@)1402 876 y Fm(x)1450 861 y Fq(')p Fv(\))17 b Fq(dtdx)k Fo(\000)1882 725 y Fl(Z)1937 951 y Fn(\012)1992 961 y Fk(+)2052 861 y Fv(\()p Fq(u@)2197 876 y Fm(t)2229 861 y Fq(')h Fv(+)g Fq(v)t(@)2515 876 y Fm(x)2563 861 y Fq(')p Fv(\))17 b Fq(dtdx)570 1137 y Fv(=)22 b Fo(\000)768 1001 y Fl(Z)823 1227 y Fn(\012)878 1237 y Fd(\000)939 1137 y Fv(\()p Fq(@)1028 1152 y Fm(t)1060 1137 y Fv(\()p Fq(u')p Fv(\))g(+)g Fq(@)1427 1152 y Fm(x)1475 1137 y Fv(\()p Fq(v)t(')p Fv(\)\))17 b Fq(dtdx)k Fv(+)2032 1001 y Fl(Z)2087 1227 y Fn(\012)2142 1237 y Fd(\000)2220 1137 y Fq(')p Fv(\()p Fq(@)2373 1152 y Fm(t)2405 1137 y Fq(u)h Fv(+)g Fq(@)2632 1152 y Fm(x)2680 1137 y Fq(v)t Fv(\))17 b Fq(dtdx)668 1413 y Fo(\000)768 1277 y Fl(Z)823 1503 y Fn(\012)878 1513 y Fk(+)938 1413 y Fv(\()p 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Fm(@)t Fn(\012)784 2356 y Fd(\000)861 2256 y Fq(')p Fv(\()p Fq(un)1077 2271 y Fn(1)1142 2256 y Fv(+)22 b Fq(v)t(n)1349 2271 y Fn(2)1391 2256 y Fv(\))17 b Fq(dS)27 b Fv(+)1682 2120 y Fl(Z)1738 2346 y Fm(@)t Fn(\012)1838 2356 y Fk(+)1914 2256 y Fq(')p Fv(\()p Fq(un)2130 2271 y Fn(1)2194 2256 y Fv(+)22 b Fq(v)t(n)2401 2271 y Fn(2)2444 2256 y Fv(\))17 b Fq(dS)k(:)-165 2556 y Fv(Uzmemo)31 b(li)g(u)i(obzir)f(da)g(je)h Fq(')27 b Fv(=)h(0)k(svuda)i(na)e(rub)s(o)m(vima)g(\012)2095 2571 y Fp(\000)2191 2556 y Fv(i)g(\012)2321 2571 y Fn(+)2385 2556 y Fv(,)h(osim)e(na)i Fq(x)28 b Fv(=)f Fq(g)t Fv(\()p Fq(t)p Fv(\),)32 b(dobiv)-5 b(amo)447 2834 y(0)28 b(=)616 2699 y Fl(Z)671 2924 y Fm(x)p Fn(=)p Fm(g)s Fn(\()p Fm(t)p Fn(\))922 2834 y Fq(')p Fv(\()p Fq(g)1075 2793 y Fp(0)1099 2834 y Fv(\()p Fq(t)p Fv(\))p Fq(u)1266 2849 y Fp(\000)1352 2834 y Fo(\000)23 b Fq(v)1499 2849 y Fp(\000)1563 2834 y Fv(\))17 b Fq(dS)27 b Fv(+)1854 2699 y Fl(Z)1909 2924 y Fm(x)p Fn(=)p Fm(g)s Fn(\()p Fm(t)p Fn(\))2160 2834 y Fq(')p Fv(\()p Fo(\000)p Fq(g)2390 2793 y Fp(0)2415 2834 y Fv(\()p Fq(t)p Fv(\))p Fq(u)2582 2849 y Fn(+)2667 2834 y Fo(\000)c Fq(v)2814 2849 y Fn(+)2878 2834 y Fv(\))17 b Fq(dS)524 3112 y Fv(=)616 2976 y Fl(Z)671 3202 y Fm(x)p Fn(=)p Fm(g)s Fn(\()p Fm(t)p Fn(\))922 3112 y Fq(')p Fv(\()p Fq(g)1075 3070 y Fp(0)1099 3112 y Fv(\()p Fq(t)p Fv(\)\()p Fq(u)1304 3127 y Fn(+)1390 3112 y Fo(\000)23 b Fq(u)1546 3127 y Fp(\000)1609 3112 y Fv(\))f Fo(\000)h Fv(\()p Fq(v)1854 3127 y Fn(+)1940 3112 y Fo(\000)f Fq(v)2086 3127 y Fp(\000)2150 3112 y Fv(\)\))17 b Fq(dS)22 b(;)-171 3407 y Fv(\024)-44 b(sto)33 b(zb)s(og)g(proizv)m(oljnosti)d Fq(')j Fv(da)5 b(je)33 b(i)f(drugu)g(tvrdnju.)3324 3529 y Fw(Q.E.D.)57 3708 y Fv(Jednak)m(ost)24 b(\(11\))f(iz)f(leme)g(naziv) -5 b(a)23 b(se)h Fz(Rankine-Hugoniotov)e Fv(uvjet.)41 b(Ko)s(d)22 b(Burgerso)m(v)m(e)j(jednad)m(\024)-46 b(zb)s(e)-165 3865 y(ta)5 b(j)32 b(uvjet)h(da)5 b(je)462 3810 y Fm(u)506 3779 y Fk(2)506 3827 y(+)p 462 3842 100 4 v 493 3899 a Fn(2)594 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b(p)s(o)m(\024)-46 b(cetni)32 b(uvjet)1321 5228 y Fq(u)1377 5243 y Fn(0)1447 5228 y Fv(=)1550 5087 y Fl(\032)1642 5164 y Fq(u)1698 5179 y Fp(\000)1761 5164 y Fq(;)98 b(x)28 b(<)f Fv(0)1642 5283 y Fq(u)1698 5298 y Fn(+)1761 5283 y Fq(;)98 b(x)28 b(>)f Fv(0)17 b Fq(:)-165 5468 y Fv(Zb)s(og)31 b(Rankine-Hugonioto)m(v)m(og)g(uvjeta)i(k)-5 b(ao)33 b(rje)-5 b(\024)-44 b(senje)33 b(imamo)1138 5750 y Fq(u)p Fv(\()p Fq(t;)17 b(x)p Fv(\))27 b(=)1535 5609 y Fl(\032)1626 5686 y Fq(u)1682 5701 y Fp(\000)1746 5686 y Fq(;)97 b(x)23 b Fo(\000)f Fq(ct)28 b(<)g Fv(0)1626 5805 y Fq(u)1682 5820 y Fn(+)1746 5805 y Fq(;)97 b(x)23 b Fo(\000)f Fq(ct)28 b(>)g Fv(0)17 b Fq(;)-165 6010 y Ft(16)p eop %%Page: 17 22 17 21 bop 3102 -116 a Fr(Burgerso)n(v)-5 b(a)25 b(jednad)n(\024)-39 b(zba)71 92 y Fv(pri)41 b(\024)-46 b(cem)m(u)46 b(je)f Fq(c)k Fv(=)844 44 y Fm(u)888 54 y Fk(+)943 44 y Fn(+)p Fm(u)1047 54 y Fd(\000)p 844 69 260 4 v 954 127 a Fn(2)1113 92 y Fv(.)81 b(Mud)p 1384 40 30 2 v 2 w(utim,)46 b(k)-5 b(ad)45 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V 1742 4836 a(')1880 4768 y Fv(+)22 b(2)p Fq(\026)2086 4727 y Fn(2)2138 4701 y Fq(@)2189 4716 y Fm(xx)2281 4701 y Fq(')p 2138 4745 207 4 v 2210 4836 a(')2355 4627 y Fl(\023)2456 4768 y Fv(=)27 b(0)17 b Fq(;)71 5043 y Fv(o)s(dnosno)32 b(imamo)e(da)j(za)f (neku)i(funk)m(ciju)f Fq(\015)5 b Fv(\()p Fq(t)p Fv(\))33 b(vrijedi)1376 5310 y Fo(\000)p Fv(2)p Fq(\026)1571 5242 y(@)1622 5257 y Fm(t)1655 5242 y Fq(')p 1571 5287 148 4 v 1613 5378 a(')1751 5310 y Fv(+)22 b(2)p Fq(\026)1957 5269 y Fn(2)2009 5242 y Fq(@)2060 5257 y Fm(xx)2151 5242 y Fq(')p 2009 5287 207 4 v 2080 5378 a(')2253 5310 y Fv(=)27 b Fq(\015)2412 5269 y Fp(0)2437 5310 y Fv(\()p Fq(t)p Fv(\))17 b Fq(:)71 5595 y Fv(Kad)32 b(bismo)g(zamijenili)c Fq(')p Fv(\()p Fq(t;)17 b(x)p Fv(\))33 b(s)g Fq(')p Fv(\()p Fq(t;)17 b(x)p Fv(\))p Fq(e)1703 5559 y Fp(\000)p Fm(\015)t Fn(\()p Fm(t)p Fn(\))1898 5595 y Fv(,)33 b(dobili)d(bismo)71 5811 y(\(13\))1446 b Fq(@)1742 5826 y Fm(t)1774 5811 y Fq(')27 b Fv(=)h Fq(\026@)2079 5826 y Fm(xx)2170 5811 y Fq(')17 b(:)3770 6010 y Ft(17)p eop %%Page: 18 23 18 22 bop -165 -116 a Fr(Nelinearni)27 b(zak)n(oni)f(sa)n(\024)-39 b(cuv)-5 b(anja)-165 83 y Fv(Med)p -13 31 30 2 v 1 w(utim,)30 b(iz)g(form)m(ule)g(za)h Fq(u)f Fv(je)h(jasno)g(da)g(ta)g(funk)m(cija)g Fq(\015)36 b Fv(ne)31 b(pridonosi)f(na)-5 b(\024)-44 b(sem)31 b(rje)-5 b(\024)-44 b(senju,)32 b(pa)f(nam)f(je)-165 202 y(do)m(v)m(oljno)i(rije)-5 b(\024)-44 b(siti)30 b(upra)m(v)m(o)k (tu)e(diferencijaln)m(u)g(jednad)m(\024)-46 b(zbu.)44 b(U)33 b(daljem)e(pretp)s(osta)m(vljamo)g(da)i(je)1330 317 y Fl(Z)1430 344 y Fm(x)1386 543 y Fn(0)1494 453 y Fq(u)1550 468 y Fn(0)1592 453 y Fv(\()p Fq(\030)5 b Fv(\))p Fq(d\030)31 b Fv(=)d Fq(o)p Fv(\()p Fq(x)2085 412 y Fn(2)2128 453 y Fv(\))-165 708 y(k)-5 b(ada)36 b Fo(j)p Fq(x)p Fo(j)e(!)g(1)p Fv(,)k(a)e(to)g(je,)i(na)f(primjer,)f(zado)m(v)m(oljeno) h(k)-5 b(ada)36 b(je)h Fq(u)2305 723 y Fn(0)2381 708 y Fo(2)e Fv(L)2543 672 y Fp(1)2624 708 y Fv(\()p Fw(R)p Fv(\).)55 b(T)-8 b(ada)37 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Fq(t;)17 b(x)p Fv(\))28 b(=)1450 1567 y(1)p 1333 1611 285 4 v 1333 1631 a Fo(p)p 1416 1631 202 4 v 73 x Fv(4)p Fq(\031)t(\026t)1644 1498 y Fl(Z)1743 1525 y Fp(1)1699 1724 y(\0001)1856 1634 y Fq(')1920 1649 y Fn(0)1963 1634 y Fv(\()p Fq(y)t Fv(\))p Fq(e)2136 1587 y Fp(\000)2206 1549 y Fk(\()p Fj(x)p Fd(\000)p Fj(y)r Fk(\))2384 1524 y(2)p 2205 1569 215 4 v 2263 1612 a(4)p Fj(\026t)2433 1634 y Fq(dy)20 b(:)-165 1896 y Fv(Dano)31 b(rje)-5 b(\024)-44 b(senje)34 b(je)f(p)s(ozitivno)e(i)h(p)s(o)g([id.,)g(Theorem)h(2.3.7])f (jedinstv)m(eno.)45 b(V)-8 b(ratimo)30 b(se)j(na)g Fq(u)p Fv(.)57 2014 y(Za)41 b Fq(\026)j(>)g Fv(0)e(ozna)m(\024)-46 b(ca)m(v)-5 b(at)40 b(\023)-46 b(cemo)42 b(s)h Fq(u)1377 2029 y Fm(\026)1469 2014 y Fv(rje)-5 b(\024)-44 b(senje)43 b(Cauc)m(h)m(yjev)m(og)i(problema)c(za)h(jednad)m(\024)-46 b(zbu)44 b(\(12\).)-165 2133 y(Imamo)30 b(da)j(je)826 2438 y Fq(u)882 2453 y Fm(\026)960 2438 y Fv(=)27 b Fo(\000)p Fv(2)p Fq(\026@)1299 2453 y Fm(x)1364 2438 y Fv(ln)16 b Fq(')28 b Fv(=)1667 2167 y Fl(Z)1767 2193 y Fp(1)1722 2392 y(\0001)1890 2235 y Fq(x)22 b Fo(\000)h Fq(y)p 1890 2279 229 4 v 1986 2371 a(t)2128 2302 y(e)2173 2255 y Fp(\000)2243 2217 y Fj(F)9 b Fk(\()p Fj(t;x;y)r Fk(\))p 2243 2238 241 4 v 2327 2280 a(2)p Fj(\026)2514 2302 y Fq(dy)p 1667 2415 950 4 v 1791 2433 a Fl(Z)1891 2459 y Fp(1)1847 2659 y(\0001)2004 2569 y Fq(e)2049 2521 y Fp(\000)2119 2483 y Fj(F)g Fk(\()p Fj(t;x;y)r Fk(\))p 2119 2504 241 4 v 2202 2547 a(2)p Fj(\026)2390 2569 y Fq(dy)2643 2438 y(;)-165 2790 y Fv(pri)29 b(\024)-46 b(cem)m(u)32 b(je)978 2970 y Fq(F)14 b Fv(\()p Fq(t;)j(x;)g(y)t Fv(\))26 b(=)1501 2902 y(\()p Fq(x)c Fo(\000)h Fq(y)t Fv(\))1806 2866 y Fn(2)p 1501 2947 347 4 v 1632 3038 a Fv(2)p Fq(t)1880 2970 y Fv(+)1978 2834 y Fl(Z)2077 2860 y Fm(y)2033 3059 y(x)2138 2970 y Fq(u)2194 2985 y Fn(0)2237 2970 y Fv(\()p Fq(\030)5 b Fv(\))17 b Fq(d\030)i(:)57 3236 y Fv(U)46 b(in)m(tegralima)d Fq(e)747 3188 y Fp(\000)828 3158 y Fj(F)p 817 3172 74 4 v 817 3214 a Fk(2)p Fj(\026)951 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Fq(C)1382 5826 y Fn(3)1441 5811 y Fo(\000)17 b Fq(")28 b(>)f Fv(0.)43 b(Nek)-5 b(a)30 b(su)g Fq(y)2242 5826 y Fn(1)p Fm(;)p Fn(2)2374 5811 y Fv(n)m(ulto)m(\024)-46 b(ck)m(e)31 b(p)s(olinoma)c Fq(C)3252 5826 y Fn(1)3311 5811 y Fo(\000)17 b Fq(C)3475 5826 y Fn(2)3518 5811 y Fq(y)j Fv(+)-165 6010 y Ft(18)p eop %%Page: 19 24 19 23 bop 3102 -116 a Fr(Burgerso)n(v)-5 b(a)25 b(jednad)n(\024)-39 b(zba)71 84 y Fv(\()p Fq(C)179 99 y Fn(3)246 84 y Fo(\000)24 b Fq(")p Fv(\))p Fq(y)483 48 y Fn(2)525 84 y Fv(.)53 b(T)-8 b(ada)36 b(za)g(sv)-5 b(aki)36 b(\()p Fq(t;)17 b(x)p Fv(\))33 b Fo(2)h Fq(K)43 b Fv(i)35 b Fq(y)h(>)d Fv(max)p Fo(f)p Fq(R)q(;)17 b Fo(j)p Fq(y)2383 99 y Fn(1)2424 84 y Fo(j)p Fq(;)g Fo(j)p Fq(y)2572 99 y Fn(2)2613 84 y Fo(jg)36 b Fv(imamo)d(da)j(je)g Fq(F)14 b Fv(\()p Fq(t;)j(x;)g(y)t Fv(\))32 b Fq(>)h Fv(0.)71 203 y(Isto)g(mo)m(\024)-46 b(zemo)31 b(p)s(osti)m(\023)-46 b(ci)32 b(i)f(za)i Fq(y)e(<)d Fo(\000)p Fq(R)q Fv(,)33 b(te)g(vidimo)e(da)h(k)-5 b(ad)33 b(ograni)m(\024)-46 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Fk(\))p Fd(\000)p Fj(m)p Fk(\()p Fj(t;x)p Fk(\))p 2092 4751 486 4 v 2298 4793 a(2)p Fj(\026)2609 4815 y Fq(dy)2737 4685 y(:)71 5077 y Fv(Lak)m(o)33 b(se)g(vidi)e(da)i (za)g(sv)-5 b(aki)32 b Fq(\026)27 b(>)h Fv(0)k(i)g(\()p Fq(t;)17 b(x)p Fv(\))28 b Fo(2)g Fw(R)1869 5041 y Fn(+)1954 5077 y Fo(\002)23 b Fw(R)32 b Fv(imamo)e(da)i(je)1519 5192 y Fl(Z)1619 5219 y Fp(1)1574 5418 y(\0001)1732 5328 y Fq(')1796 5343 y Fm(\026)1846 5328 y Fv(\()p Fq(t;)17 b(x;)g(y)t Fv(\))g Fq(dy)29 b Fv(=)f(1)71 5578 y(i)j(da)i(je)1239 5747 y Fq(u)1295 5762 y Fm(\026)1345 5747 y Fv(\()p Fq(t;)17 b(x)p Fv(\))28 b(=)1687 5612 y Fl(Z)1786 5638 y Fp(1)1742 5837 y(\0001)1909 5680 y Fq(x)23 b Fo(\000)g Fq(y)p 1909 5724 229 4 v 2006 5816 a(t)2148 5747 y(')2212 5762 y Fm(\026)2262 5747 y Fv(\()p Fq(t;)17 b(x;)g(y)t Fv(\))g Fq(dy)h(:)3770 6010 y Ft(19)p eop %%Page: 20 25 20 24 bop -165 -116 a Fr(Nelinearni)27 b(zak)n(oni)f(sa)n(\024)-39 b(cuv)-5 b(anja)-165 83 y Fv(Za)44 b Fq(")49 b(>)g Fv(0)44 b(p)s(osto)5 b(ji)45 b Fq(\016)53 b(>)c Fv(0)44 b(tak)-5 b(a)m(v)46 b(da)f(za)g(sv)-5 b(aki)45 b(\()p Fq(s;)17 b(z)t Fv(\))45 b(iz)g(K\(\()p Fq(t;)17 b(x)p Fv(\))p Fq(;)g(\016)t Fv(\))45 b(imamo)d(da)j(su)h(sv)m(e)g(to)m(\024)-46 b(ck)m(e)-165 202 y(minim)m(uma)29 b(funk)m(cije)k Fq(F)14 b Fv(\()p Fq(s;)j(z)t(;)g Fo(\001)p Fv(\))32 b(manje)g(o)s(d)g Fq(y)1555 217 y Fn(+)1618 202 y Fv(\()p Fq(t;)17 b(x)p Fv(\))23 b(+)f Fq(")p Fv(.)43 b(Ra)m(\024)-46 b(cunamo:)-50 425 y Fq(u)6 440 y Fm(\026)56 425 y Fv(\()p Fq(s;)17 b(z)t Fv(\))28 b(=)403 290 y Fl(Z)502 316 y Fp(1)458 515 y(\0001)625 358 y Fq(z)f Fo(\000)c Fq(y)p 625 402 223 4 v 714 494 a(s)858 425 y(')922 440 y Fm(\026)972 425 y Fv(\()p Fq(s;)17 b(z)t(;)g(y)t Fv(\))g Fq(dy)299 721 y Fv(=)403 585 y Fl(Z)502 611 y Fm(y)540 621 y Fk(+)595 611 y Fn(\()p Fm(t;x)p Fn(\)+)p Fm(")458 811 y Fp(\0001)874 653 y Fq(z)27 b Fo(\000)22 b Fq(y)p 874 698 V 962 789 a(s)1106 721 y(')1170 736 y Fm(\026)1221 721 y Fv(\()p Fq(s;)17 b(z)t(;)g(y)t Fv(\))g Fq(dy)23 b Fv(+)1769 585 y Fl(Z)1869 611 y Fp(1)1825 811 y Fm(y)1863 821 y Fk(+)1918 811 y Fn(\()p Fm(t;x)p Fn(\)+)p Fm(")2196 653 y Fq(z)k Fo(\000)c Fq(y)p 2196 698 V 2284 789 a(s)2429 721 y(')2493 736 y Fm(\026)2543 721 y Fv(\()p Fq(s;)17 b(z)t(;)g(y)t Fv(\))g Fq(dy)299 1028 y Fh(>)414 961 y Fq(z)27 b Fo(\000)c Fv(\()p Fq(y)672 976 y Fn(+)735 961 y Fv(\()p Fq(t;)17 b(x)p Fv(\))23 b(+)f Fq(")p Fv(\))p 414 1005 735 4 v 759 1096 a Fq(s)1176 892 y Fl(Z)1275 919 y Fm(y)1313 929 y Fk(+)1368 919 y Fn(\()p Fm(t;x)p Fn(\)+)p Fm(")1231 1118 y Fp(\0001)1637 1028 y Fq(')1701 1043 y Fm(\026)1751 1028 y Fv(\()p Fq(s;)17 b(z)t(;)g(y)t Fv(\))g Fq(dy)24 b Fv(+)2300 892 y Fl(Z)2399 919 y Fp(1)2355 1118 y Fm(y)2393 1128 y Fk(+)2448 1118 y Fn(\()p Fm(t;x)p Fn(\)+)p Fm(")2727 961 y Fq(z)i Fo(\000)d Fq(y)p 2727 1005 223 4 v 2815 1096 a(s)2959 1028 y(')3023 1043 y Fm(\026)3073 1028 y Fv(\()p Fq(s;)17 b(z)t(;)g(y)t Fv(\))g Fq(dy)i(:)-165 1268 y Fv(Iz)38 b(izb)s(ora)e Fq(\016)41 b Fv(jasno)d(je)f(da)h(za)g Fq(y)h Fo(2)d Fv([)p Fq(y)1238 1283 y Fn(+)1302 1268 y Fv(\()p Fq(t;)17 b(x)p Fv(\))26 b(+)f Fq(";)17 b Fo(1i)36 b Fv(imamo)f(da)i(je)h Fq(F)14 b Fv(\()p Fq(s;)j(z)t(;)g(y)t Fv(\))24 b Fo(\000)i Fq(m)p Fv(\()p Fq(s;)17 b(z)t Fv(\))37 b Fq(>)f Fv(0,)i(pa)-165 1387 y Fq(')-101 1402 y Fm(\026)-51 1387 y Fv(\()p Fq(s;)17 b(z)t(;)g(y)t Fv(\))42 b(i)381 1342 y Fm(z)s Fp(\000)p Fm(y)p 381 1364 140 4 v 433 1422 a(s)530 1387 y Fq(')594 1402 y Fm(\026)645 1387 y Fv(\()p Fq(s;)17 b(z)t(;)g(y)t Fv(\))41 b(k)m(o)m(v)m(ergira)5 b(ju)44 b(p)s(o)e(to)m(\024)-46 b(ck)-5 b(ama)42 b(n)m(uli)f(k)-5 b(ad)43 b Fq(\026)h Fo(!)g Fv(0.)74 b(P)m(o)43 b(Lesb)s(egueso)m(v)m (om)-165 1506 y(teorem)m(u)32 b(o)h(dominirano)5 b(j)30 b(k)m(o)m(v)m(ergenciji)j(zaklju)m(\024)-46 b(cujemo)32 b(da)g(k)m(on)m(v)m(ergira)5 b(ju)34 b(i)d(in)m(tegrali:)1166 1588 y Fl(Z)1266 1615 y Fp(1)1222 1814 y Fm(y)1260 1824 y Fk(+)1315 1814 y Fn(\()p Fm(t;x)p Fn(\)+)p Fm(")1583 1724 y Fq(')1647 1739 y Fm(\026)1697 1724 y Fv(\()p Fq(s;)17 b(z)t(;)g(y)t Fv(\))g Fq(dy)30 b Fo(!)d Fv(0)-165 1956 y(i)1045 1970 y Fl(Z)1145 1996 y Fp(1)1100 2196 y Fm(y)1138 2206 y Fk(+)1193 2196 y Fn(\()p Fm(t;x)p Fn(\)+)p Fm(")1472 2038 y Fq(z)g Fo(\000)22 b Fq(y)p 1472 2083 223 4 v 1560 2174 a(s)1704 2106 y(')1768 2121 y Fm(\026)1819 2106 y Fv(\()p Fq(s;)17 b(z)t(;)g(y)t Fv(\))g Fq(dy)29 b Fo(!)e Fv(0)-165 2333 y(k)-5 b(ad)32 b Fq(\026)g Fv(te)m(\024)-46 b(zi)33 b(n)m(uli.)42 b(Zaklju)m(\024)-46 b(cujemo)31 b(da)i(za)f(proizv)m(oljn)m(u)g(to)m(\024)-46 b(cku)34 b(\()p Fq(s;)17 b(z)t Fv(\))28 b Fo(2)g Fv(K\(\()p Fq(t;)17 b(x)p Fv(\))p Fq(;)g(\016)t Fv(\))32 b(imamo)e(da)i(je)978 2557 y(lim)17 b(inf)1033 2626 y Fm(\026)p Fp(&)p Fn(0)1265 2557 y Fq(u)1321 2572 y Fm(\026)1371 2557 y Fv(\()p Fq(s;)g(z)t Fv(\))28 b Fh(>)1729 2490 y Fq(z)f Fo(\000)c Fv(\()p Fq(y)1987 2505 y Fn(+)2050 2490 y Fv(\()p Fq(t;)17 b(x)p Fv(\))22 b(+)g Fq(")p Fv(\))p 1729 2534 735 4 v 2074 2625 a Fq(s)2491 2557 y(:)-165 2782 y Fv(U)32 b(limesu)f(\()p Fq(s;)17 b(z)t Fv(\))28 b Fo(!)g Fv(\()p Fq(t;)17 b(x)p Fv(\))32 b(dobiv)-5 b(amo)976 3006 y(lim)17 b(inf)942 3066 y Fk(\()p Fj(s;z)r Fk(\))p Fd(!)p Fk(\()p Fj(t;x)p Fk(\))1041 3131 y Fj(\026)p Fd(&)p Fk(0)1306 3006 y Fq(u)1362 3021 y Fm(\026)1412 3006 y Fv(\()p Fq(s;)g(z)t Fv(\))28 b Fh(>)1770 2939 y Fq(x)23 b Fo(\000)f Fv(\()p Fq(y)2033 2954 y Fn(+)2097 2939 y Fv(\()p Fq(t;)17 b(x)p Fv(\))22 b(+)g Fq(")p Fv(\))p 1770 2983 741 4 v 2123 3075 a Fq(t)2537 3006 y(:)-165 3284 y Fv(U)38 b(limesu)f Fq(")g Fo(!)g Fv(0)h(dobiv)-5 b(amo)36 b(lijevu)h(nejednak)m(ost)j(iz)e(\(14\);)i (analogno)d(se)i(mo)m(\024)-46 b(ze)38 b(dok)-5 b(azati)37 b(i)g(desna)-165 3403 y(nejednak)m(ost.)-165 3560 y Fw(Lema)f(2.)99 b Fi(Ak)m(o)32 b(je)f Fq(x)684 3575 y Fn(1)755 3560 y Fq(<)d(x)914 3575 y Fn(2)988 3560 y Fi(onda)j(za)g(sv)-5 b(aki)32 b Fq(t)f Fi(vrijedi)f Fq(y)2008 3575 y Fn(+)2072 3560 y Fv(\()p Fq(t;)17 b(x)2244 3575 y Fn(1)2287 3560 y Fv(\))27 b Fh(6)h Fq(y)2505 3575 y Fp(\000)2569 3560 y Fv(\()p Fq(t;)17 b(x)2741 3575 y Fn(2)2784 3560 y Fv(\))p Fi(.)43 b(P)m(osebno,)33 b(za)28 b(\024)-46 b(cvrsti)-165 3679 y Fq(t)39 b Fi(imamo)d(da)i(su)i Fq(y)556 3694 y Fp(\000)619 3679 y Fv(\()p Fq(t;)17 b Fo(\001)p Fv(\))38 b Fi(i)g Fq(y)954 3694 y Fn(+)1018 3679 y Fv(\()p Fq(t;)17 b Fo(\001)p Fv(\))38 b Fi(rastu)m(\023)-46 b(ce)40 b(funk)m(cije,)h (neprekidne)f(s)f(lijev)-5 b(a,)39 b(o)s(dnosno)f(zdesna)i(na)-165 3797 y Fw(R)p Fi(.)52 b(K)36 b(tome,)g(one)g(su)g(jednak)m(e,)i(a)e (samim)d(tim)h(i)h(neprekidne,)j(osim)c(u)i(prebro)5 b(jiv)m(o)36 b(mnogo)f(to)m(\024)-46 b(cak)-5 b(a.)-165 3916 y(Nadalje,)38 b(ak)m(o)g(je)h Fq(z)j Fi(nek)-5 b(a)39 b(to)m(\024)-46 b(ck)-5 b(a)38 b(minim)m(uma)c(funk)m(cije)39 b Fq(F)14 b Fv(\()p Fq(t;)j(x;)g Fo(\001)p Fv(\))p Fi(,)38 b(onda)g(za)g(proizv)m(oljni)e Fq(\025)h Fo(2)h(h)p Fv(0)p Fq(;)17 b Fv(1)p Fo(i)-165 4035 y Fi(vrijedi)1181 4154 y Fq(y)1229 4169 y Fp(\006)1293 4154 y Fv(\()p Fq(\025t;)g(\025x)22 b Fv(+)g(\(1)g Fo(\000)h Fq(\025)p Fv(\))p Fq(z)t Fv(\))28 b(=)f Fq(z)22 b(:)-165 4327 y Fz(Dem.)95 b Fv(Ozna)m(\024)-46 b(cimo)p 602 4272 52 4 v 31 w Fq(y)654 4351 y Fn(+)746 4327 y Fv(=)29 b Fq(y)899 4342 y Fn(+)962 4327 y Fv(\()p Fq(t;)17 b(x)1134 4342 y Fn(1)1177 4327 y Fv(\).)46 b(T)-8 b(ada)33 b(p)s(o)g(de\014niciji)e(funk)m(cije)j Fq(y)2479 4342 y Fn(+)2576 4327 y Fv(imamo)c(da)j(za)h(sv)-5 b(aki)33 b Fq(y)f Fo(2)d Fw(R)-165 4446 y Fv(vrijedi)1204 4565 y Fq(F)14 b Fv(\()p Fq(t;)j(x)1453 4580 y Fn(1)1495 4565 y Fq(;)g(y)t Fv(\))26 b Fh(>)j Fq(F)14 b Fv(\()p Fq(t;)j(x)2010 4580 y Fn(1)2052 4565 y Fq(;)p 2096 4510 V 17 w(y)2147 4588 y Fn(+)2211 4565 y Fv(\))g Fq(:)-165 4719 y Fv(Tvrdimo)31 b(da)i(je)g(za)f(sv)-5 b(aki)33 b Fq(y)d(<)p 1031 4664 V 28 w(y)1082 4742 y Fn(+)1179 4719 y Fv(ispunjeno)1204 4886 y Fq(F)14 b Fv(\()p Fq(t;)j(x)1453 4901 y Fn(2)1496 4886 y Fq(;)g(y)t Fv(\))26 b Fq(>)i(F)14 b Fv(\()p Fq(t;)j(x)2009 4901 y Fn(2)2051 4886 y Fq(;)p 2095 4832 V 17 w(y)2146 4910 y Fn(+)2210 4886 y Fv(\))g Fq(;)-165 5054 y Fv(iz)25 b(\024)-46 b(cega)25 b(\023)-46 b(ce)29 b(slijediti)d(da)i(funk)m(cija) g Fq(F)14 b Fv(\()p Fq(t;)j(x)1346 5069 y Fn(2)1389 5054 y Fq(;)g Fo(\001)p Fv(\))28 b(minim)m(um)d(dosti)m(\024)-46 b(ze)28 b(na)h([)p 2439 4999 V Fq(y)2490 5078 y Fn(+)2554 5054 y Fq(;)17 b Fo(1i)p Fv(.)41 b(Sad)28 b(uz)h(pretp)s(osta)m(vk)m(e) -165 5173 y Fq(y)h(<)p 17 5118 V 28 w(y)68 5196 y Fn(+)165 5173 y Fv(i)h Fq(x)279 5188 y Fn(1)350 5173 y Fq(<)c(x)508 5188 y Fn(2)584 5173 y Fv(ra)m(\024)-46 b(cunamo)62 5335 y Fq(F)14 b Fv(\()p Fq(t;)j(x)311 5350 y Fn(2)353 5335 y Fq(;)g(y)t Fv(\))k Fo(\000)i Fq(F)14 b Fv(\()p Fq(t;)j(x)857 5350 y Fn(2)900 5335 y Fq(;)p 944 5280 V 17 w(y)994 5359 y Fn(+)1058 5335 y Fv(\))28 b Fh(>)g Fq(F)14 b Fv(\()p Fq(t;)j(x)1478 5350 y Fn(2)1521 5335 y Fq(;)g(y)t Fv(\))j(+)i Fq(F)14 b Fv(\()p Fq(t;)j(x)2022 5350 y Fn(1)2065 5335 y Fq(;)p 2109 5280 V 17 w(y)2160 5359 y Fn(+)2224 5335 y Fv(\))22 b Fo(\000)g Fq(F)14 b Fv(\()p Fq(t;)j(x)2632 5350 y Fn(1)2675 5335 y Fq(;)g(y)t Fv(\))k Fo(\000)i Fq(F)14 b Fv(\()p Fq(t;)j(x)3179 5350 y Fn(2)3221 5335 y Fq(;)p 3265 5280 V 17 w(y)3316 5359 y Fn(+)3380 5335 y Fv(\))1124 5530 y(=)1255 5463 y(1)p 1237 5507 85 4 v 1237 5599 a(2)p Fq(t)1331 5530 y Fv([\()p Fq(x)1451 5545 y Fn(2)1516 5530 y Fo(\000)23 b Fq(y)t Fv(\))1706 5489 y Fn(2)1770 5530 y Fo(\000)f Fv(\()p Fq(x)1962 5545 y Fn(1)2028 5530 y Fo(\000)g Fq(y)t Fv(\))2217 5489 y Fn(2)2281 5530 y Fv(+)g(\()p Fq(x)2472 5545 y Fn(1)2537 5530 y Fo(\000)p 2637 5475 52 4 v 23 w Fq(y)2688 5554 y Fn(+)2752 5530 y Fv(\))2790 5489 y Fn(2)2854 5530 y Fo(\000)h Fv(\()p Fq(x)3047 5545 y Fn(2)3112 5530 y Fo(\000)p 3212 5475 V 23 w Fq(y)3263 5554 y Fn(+)3327 5530 y Fv(\))3365 5489 y Fn(2)3407 5530 y Fv(])1124 5776 y(=)1237 5707 y(\()p Fq(x)1330 5722 y Fn(2)1395 5707 y Fo(\000)g Fq(x)1550 5722 y Fn(1)1593 5707 y Fv(\)\()p 1669 5652 V Fq(y)1720 5730 y Fn(+)1806 5707 y Fo(\000)g Fq(y)t Fv(\))p 1237 5753 758 4 v 1574 5844 a(2)p Fq(t)2032 5776 y(>)28 b Fv(0)17 b Fq(;)-165 6010 y Ft(20)p eop %%Page: 21 26 21 25 bop 3102 -116 a Fr(Burgerso)n(v)-5 b(a)25 b(jednad)n(\024)-39 b(zba)68 83 y Fv(\024)-46 b(cime)30 b(smo)h(dok)-5 b(azali)29 b(prvu)j(tvrdnju.)44 b(Oda)m(vdje,)33 b(iz)d(de\014nicija)h Fq(y)2468 98 y Fp(\006)2562 83 y Fv(slijedi)f(da)h(je)g(za)h(sv)-5 b(aki)31 b Fq(t)g Fv(i)f Fq(x)3625 98 y Fn(1)3696 83 y Fq(<)e(x)3855 98 y Fn(2)71 202 y Fv(ispunjeno)1035 320 y Fq(y)1083 335 y Fp(\000)1146 320 y Fv(\()p Fq(t;)17 b(x)1318 335 y Fn(1)1361 320 y Fv(\))28 b Fh(6)g Fq(y)1580 335 y Fn(+)1643 320 y Fv(\()p Fq(t;)17 b(x)1815 335 y Fn(1)1858 320 y Fv(\))28 b Fh(6)g Fq(y)2077 335 y Fp(\000)2140 320 y Fv(\()p Fq(t;)17 b(x)2312 335 y Fn(2)2355 320 y Fv(\))28 b Fh(6)g Fq(y)2574 335 y Fn(+)2637 320 y Fv(\()p Fq(t;)17 b(x)2809 335 y Fn(2)2852 320 y Fv(\))g Fq(;)66 486 y Fv(\024)-44 b(sto)34 b(nam)e(da)5 b(je)35 b(monotonost.)1238 461 y(\024)1235 486 y(Sto)m(vi)-5 b(\024)-44 b(se,)35 b(dobiv)-5 b(amo)31 b(da)j(lim)2310 504 y Fm(x)p Fp(\045)p Fm(x)2475 516 y Fk(2)2531 486 y Fq(y)2579 501 y Fp(\000)2643 486 y Fv(\()p Fq(t;)17 b(x)p Fv(\))34 b(p)s(osto)5 b(ji)32 b(i)h(da)h(je)f(manji)f(o)s(d)71 605 y Fq(y)119 620 y Fp(\000)182 605 y Fv(\()p Fq(t;)17 b(x)354 620 y Fn(2)397 605 y Fv(\).)69 b(Kad)41 b(bi)f(limes)f(bio)h(strogo)h(manji)e(do)-5 b(\024)-44 b(slo)40 b(bi)h(do)f(k)m(on)m(tradik)m(cije,)k(s)d(tim)35 b(\024)-44 b(sto)41 b(je)g Fq(m)p Fv(\()p Fq(t;)17 b(x)p Fv(\),)71 724 y(de\014nirana)32 b(k)-5 b(ao)32 b(min)865 740 y Fm(y)s Fp(2)p Fg(R)1044 724 y Fq(F)14 b Fv(\()p Fq(t;)j(x;)g(y)t Fv(\),)31 b(neprekidna)i(funk)m(cija)g(u)f(\()p Fq(t;)17 b(x)p Fv(\).)44 b(U)33 b(stv)-5 b(ari,)32 b(imali)d(bismo)590 917 y(lim)557 985 y Fm(x)p Fp(\045)p Fm(x)722 997 y Fk(2)775 917 y Fq(m)p Fv(\()p Fq(t;)17 b(x)p Fv(\))28 b(=)61 b(lim)1202 985 y Fm(x)p Fp(\045)p Fm(x)1367 997 y Fk(2)1420 917 y Fq(F)14 b Fv(\()p Fq(t;)j(x;)g(y)1761 932 y Fp(\000)1824 917 y Fv(\()p Fq(t;)g(x)p Fv(\)\))28 b Fo(6)p Fv(=)f Fq(F)14 b Fv(\()p Fq(t;)j(x;)g(y)2544 932 y Fp(\000)2607 917 y Fv(\()p Fq(t;)g(x)2779 932 y Fn(2)2822 917 y Fv(\)\))28 b(=)f Fq(m)p Fv(\()p Fq(t;)17 b(x)3286 932 y Fn(2)3329 917 y Fv(\))g Fq(:)71 1161 y Fv(Na)32 b(isti)f(na)m(\024)-46 b(cin)33 b(mo)m(\024)-46 b(ze)32 b(se)h(dok)-5 b(azati)32 b(neprekidnost)h(zdesna)h(funk)m(cije)f Fq(y)2703 1176 y Fn(+)2767 1161 y Fv(.)293 1280 y(P)m(ok)-5 b(a)m(\024)-46 b(zimo)31 b(zadnju)i(tvrdnju)h(leme.)42 b(Za)32 b(sv)-5 b(aki)32 b Fq(y)k Fv(vrijedi)1430 1473 y Fq(F)14 b Fv(\()p Fq(t;)j(x;)g(y)t Fv(\))k Fo(\000)h Fq(F)14 b Fv(\()p Fq(t;)j(x;)g(z)t Fv(\))28 b Fh(>)g Fv(0)17 b Fq(;)71 1666 y Fv(stoga)32 b(za)h Fq(y)e Fo(6)p Fv(=)c Fq(z)37 b Fv(imamo)206 1843 y Fq(F)14 b Fv(\()p Fq(\025t;)j(\025x)p Fv(+)o(\(1)22 b Fo(\000)h Fq(\025)p Fv(\))p Fq(z)t(;)17 b(y)t Fv(\))k Fo(\000)i Fq(F)14 b Fv(\()p Fq(\025t;)j(\025x)22 b Fv(+)g(\(1)g Fo(\000)g Fq(\025)p Fv(\))p Fq(z)t(;)17 b(z)t Fv(\))29 b Fh(>)672 2057 y(>)787 1990 y Fv(\()p Fq(\025x)23 b Fv(+)f(\(1)f Fo(\000)i Fq(\025)p Fv(\))p Fq(z)k Fo(\000)22 b Fq(y)t Fv(\))1622 1953 y Fn(2)1686 1990 y Fo(\000)h Fv(\()p Fq(\025t;)17 b(\025x)22 b Fv(+)g(\(1)g Fo(\000)h Fq(\025)p Fv(\))p Fq(z)j Fo(\000)d Fq(z)t Fv(\))2754 1953 y Fn(2)p 787 2034 2010 4 v 1722 2125 a Fv(2)p Fq(\025t)2829 2057 y Fo(\000)2939 1990 y Fv(\()p Fq(x)f Fo(\000)h Fq(y)t Fv(\))3244 1953 y Fn(2)3308 1990 y Fo(\000)f Fv(\()p Fq(x)h Fo(\000)g Fq(z)t Fv(\))3710 1953 y Fn(2)p 2939 2034 814 4 v 3304 2125 a Fv(2)p Fq(t)672 2310 y Fv(=)786 2243 y(\()p Fq(z)j Fo(\000)d Fq(y)t Fv(\))1085 2207 y Fn(2)1127 2243 y Fv(\(1)f Fo(\000)g Fq(\025)p Fv(\))p 786 2287 645 4 v 1037 2379 a(2)p Fq(\025t)1468 2310 y(>)27 b Fv(0)17 b Fq(:)3561 2580 y Fw(Q.E.D.)293 2743 y Fv(Uo)m(\024)-46 b(cimo)39 b(da)h(iz)h(leme)e(slijedi)g(da)h(funk)m(cija)h Fq(F)14 b Fv(\()p Fq(\025t;)j(\025x)27 b Fv(+)h(\(1)f Fo(\000)h Fq(\025)p Fv(\))p Fq(z)t(;)17 b Fo(\001)p Fv(\))41 b(ima)e Fi(jedinstv)m(eni)i Fv(strogi)71 2862 y(minim)m(um)29 b(u)k(to)m(\024)-46 b(cki)32 b Fq(z)t Fv(.)1155 3069 y Fw(T)-9 b(eorem)36 b(egzistencije)g(slabih)h(rje)-6 b(\024)-50 b(senja)293 3232 y Fv(Dok)-5 b(a)m(\024)-46 b(zimo)45 b(sad)i(teorem)g(k)m(o)5 b(ji)45 b(\023)-46 b(ce)47 b(osigurati)e(p)s(osto)5 b(janje)48 b(slab)s(og)e(rje)-5 b(\024)-44 b(senja)48 b(Burgerso)m(v)m(e)h(jed-)71 3350 y(nad)m(\024)-46 b(zb)s(e.)44 b(T)-8 b(eorem)30 b(\023)-46 b(ce)33 b(p)s(ored)g(toga)e(dati)h(i)f(k)-5 b(arakterizaciju)32 b(za)h(prih)m(v)-5 b(atljiv)g(a)29 b(rje)-5 b(\024)-44 b(senja.)71 3513 y Fw(T)-9 b(eorem)54 b(3.)147 b Fi(Pretp)s(osta)m (vimo)48 b(de)h(je)f Fq(u)1720 3528 y Fn(0)1817 3513 y Fo(2)55 b Fv(L)1999 3477 y Fp(1)1999 3545 y Fn(lo)r(c)2099 3513 y Fv(\()p Fw(R)p Fv(\))47 b Fi(takv)-5 b(a)49 b(da)f(njezin)g(in)m (tegral)f(zado)m(v)m(olja)m(v)-5 b(a)71 3632 y(sljede)m(\023)-46 b(cu)33 b(o)s(cjen)m(u)g(rasta)1570 3666 y Fl(Z)1670 3692 y Fm(y)1626 3891 y Fn(0)1731 3801 y Fq(u)1787 3816 y Fn(0)1829 3801 y Fv(\()p Fq(\030)5 b Fv(\))p Fq(d\030)31 b Fv(=)d Fq(o)p Fv(\()p Fq(y)2319 3760 y Fn(2)2360 3801 y Fv(\))71 4021 y Fi(k)-5 b(ad)41 b Fo(j)p Fq(y)t Fo(j)f Fi(te)m(\024)-46 b(zi)40 b(u)i(b)s(esk)m(ona)m(\024)-46 b(cnost.)71 b(Nek)-5 b(a)42 b(je)f Fv(\()p Fq(\026)1819 4039 y Fm(k)1865 4021 y Fv(\))g Fi(niz)g(p)s(ozitivnih)e(bro)5 b(jev)-5 b(a)41 b(k)m(o)5 b(ji)41 b(k)m(on)m(v)m(ergira)h(k)g(n)m(uli.) 71 4140 y(T)-8 b(ada)32 b(niz)h Fv(\()p Fq(u)569 4157 y Fm(k)614 4140 y Fv(\))g Fi(rje)-5 b(\024)-44 b(senja)33 b(jednad)m(\024)-46 b(zbi)1500 4333 y Fq(@)1551 4348 y Fm(t)1583 4333 y Fq(u)22 b Fv(+)g Fq(@)1810 4348 y Fm(x)1858 4333 y Fv(\()1906 4294 y Fn(1)p 1906 4310 39 4 v 1906 4367 a(2)1954 4333 y Fq(u)2010 4292 y Fn(2)2052 4333 y Fv(\))28 b(=)f Fq(\026)2280 4350 y Fm(k)2326 4333 y Fq(@)2377 4348 y Fm(xx)71 4526 y Fi(k)m(on)m(v)m(ergira)33 b(za)g(sk)m(oro)g(sv)-5 b(aki)32 b Fv(\()p Fq(t;)17 b(x)p Fv(\))33 b Fi(k)g Fq(u)p Fv(\()p Fq(t;)17 b(x)p Fv(\))32 b Fi(k)m(o)5 b(ji)33 b(je)f(rje)-5 b(\024)-44 b(senje)34 b(Burgerso)m(v)m(e)h(jedand)m(\024)-46 b(zb)s(e.)44 b(T)-8 b(o)m(\024)-46 b(cnije,)1323 4776 y Fq(u)1379 4793 y Fm(k)1425 4776 y Fv(\()p Fq(t;)17 b(x)p Fv(\))28 b Fo(!)f Fq(u)p Fv(\()p Fq(t;)17 b(x)p Fv(\))27 b(=)2197 4708 y Fq(x)c Fo(\000)f Fq(y)t Fv(\()p Fq(t;)17 b(x)p Fv(\))p 2197 4753 439 4 v 2399 4844 a Fq(t)71 5001 y Fi(za)31 b(sv)-5 b(aki)31 b Fv(\()p Fq(t;)17 b(x)p Fv(\))31 b Fi(tak)-5 b(a)m(v)32 b(da)f Fq(F)14 b Fv(\()p Fq(t;)j(x;)g Fo(\001)p Fv(\))30 b Fi(ima)f(jedinstv)m(en)m(u)k(to)m(\024)-46 b(cku)32 b(minim)m(uma)c Fq(y)t Fv(\()p Fq(t;)17 b(x)p Fv(\))p Fi(.)42 b(T)-8 b(o)31 b(je)g(ispunjeno)71 5120 y(do)39 b(na)h(skup)g(k)m(o)5 b(ji)40 b(je)f(za)e(\024)-46 b(cvrst)41 b Fq(t)e Fi(prebro)5 b(jiv,)42 b Fq(u)d Fi(je)g(neprekidna)h (u)g(sv)-5 b(ak)m(om)40 b Fv(\()p Fq(t;)17 b(x)p Fv(\))39 b Fi(za)h(k)m(o)5 b(ji)39 b(je)h(to)m(\024)-46 b(ck)-5 b(a)71 5239 y(minim)m(uma)29 b(jedinatv)m(ena.)44 b(U)27 b(\024)-44 b(sirem)32 b(smislu)71 5617 y Fv(\(15\))1354 5353 y Fl(8)1354 5443 y(>)1354 5473 y(>)1354 5503 y(<)1354 5682 y(>)1354 5712 y(>)1354 5742 y(:)1514 5483 y Fv(lim)1502 5551 y Fm(z)s Fp(\045)p Fm(x)1678 5483 y Fq(u)p Fv(\()p Fq(t;)17 b(z)t Fv(\))28 b(=)2080 5416 y Fq(x)22 b Fo(\000)h Fq(y)2305 5431 y Fp(\000)2368 5416 y Fv(\()p Fq(t;)17 b(x)p Fv(\))p 2080 5460 499 4 v 2311 5551 a Fq(t)1471 5751 y Fv(lim)1459 5820 y Fm(z)s Fp(&)p Fm(x)1635 5751 y Fq(u)p Fv(\()p Fq(t;)g(z)t Fv(\))27 b(=)2036 5684 y Fq(x)22 b Fo(\000)h Fq(y)2261 5699 y Fn(+)2324 5684 y Fv(\()p Fq(t;)17 b(x)p Fv(\))p 2036 5728 V 2268 5820 a Fq(t)2561 5751 y(;)3770 6010 y Ft(21)p eop %%Page: 22 27 22 26 bop -165 -116 a Fr(Nelinearni)27 b(zak)n(oni)f(sa)n(\024)-39 b(cuv)-5 b(anja)-165 83 y Fi(gdje)33 b(su)h Fq(y)218 98 y Fp(\000)281 83 y Fv(\()p Fq(t;)17 b(x)p Fv(\))34 b Fi(i)e Fq(y)633 98 y Fn(+)697 83 y Fv(\()p Fq(t;)17 b(x)p Fv(\))33 b Fi(na)5 b(jmanja,)33 b(o)s(dnosno)g(na)5 b(jv)m(e)m(\023)-46 b(ca,)35 b(to)m(\024)-46 b(ck)-5 b(a)33 b(minim)m(uma)d(funk)m(cije)k Fq(F)14 b Fv(\()p Fq(t;)j(x;)g Fo(\001)p Fv(\))p Fi(.)-165 202 y(Ak)m(o)33 b(je)g Fq(x)199 217 y Fn(1)269 202 y Fh(6)28 b Fq(x)429 217 y Fn(2)505 202 y Fi(tada)-165 448 y Fv(\(16\))970 b(lim)947 516 y Fm(x)p Fp(&)p Fm(x)1112 528 y Fk(2)1164 448 y Fq(u)p Fv(\()p Fq(t;)17 b(x)p Fv(\))22 b Fo(\000)56 b Fv(lim)1552 516 y Fm(x)p Fp(\045)p Fm(x)1717 528 y Fk(1)1770 448 y Fq(u)p Fv(\()p Fq(t;)17 b(x)p Fv(\))27 b Fh(6)2178 380 y Fq(x)2233 395 y Fn(2)2298 380 y Fo(\000)c Fq(x)2453 395 y Fn(1)p 2178 425 318 4 v 2319 516 a Fq(t)2522 448 y(:)-165 729 y Fi(Ak)m(o)34 b(je)f Fq(u)201 744 y Fn(0)272 729 y Fh(6)c Fq(M)44 b Fi(onda)34 b(je)f Fq(u)28 b Fh(6)i Fq(M)10 b Fi(.)1237 704 y(\024)1230 729 y(Sto)m(vi)-5 b(\024)-44 b(se,)34 b(uvjet)h(\(16\))d(tada)h(da)5 b(je)34 b(k)-5 b(arakterizaciju)32 b(k)m(o)5 b(ja)34 b(izdv)-5 b(a)5 b(ja)-165 848 y Fq(u)32 b Fi(med)p 68 796 30 2 v(u)h(svim)e(rje)-5 b(\024)-44 b(senjima)32 b(Burgerso)m(v)m(e)j (jedand)m(\024)-46 b(zb)s(e,)33 b(te)g Fq(u)f Fi(zo)m(v)m(emo)h(Hopfo)m (vim)f(rje)-5 b(\024)-44 b(senjem.)-165 991 y Fz(Dem.)100 b Fv(Sv)m(o)5 b(jstv)m(o)37 b(monotonosti)d(iz)g(leme)h(p)s(ok)-5 b(azuje)35 b(da)g(za)g(\014ksan)i Fq(t)e Fv(imamo)d(jednak)m(ost)37 b Fq(y)3280 1006 y Fp(\000)3343 991 y Fv(\()p Fq(t;)17 b(x)p Fv(\))33 b(=)-165 1130 y Fq(y)-117 1145 y Fn(+)-54 1130 y Fv(\()p Fq(t;)17 b(x)p Fv(\))38 b(osim)e(u)i(prebro)5 b(jiv)m(o)38 b(mnogo)e Fq(x)p Fv(,)k(i)c(da)i(tada)f(imamo)e(k)m(on)m (v)m(ergenciju)40 b Fq(u)2865 1147 y Fm(k)2910 1130 y Fv(\()p Fq(t;)17 b(x)p Fv(\))37 b Fo(!)3303 1080 y Fm(x)p Fp(\000)p Fm(y)3445 1090 y Fk(+)3499 1080 y Fn(\()p Fm(t;x)p Fn(\))p 3303 1107 349 4 v 3463 1164 a Fm(t)-165 1278 y Fv(p)s(o)44 b(\(14\).)81 b(De\014nira)5 b(jmo)44 b(da)h(je)g Fq(u)p Fv(\()p Fq(t;)17 b(x)p Fv(\))49 b(=)1534 1228 y Fm(x)p Fp(\000)p Fm(y)1676 1238 y Fk(+)1730 1228 y Fn(\()p Fm(t;x)p Fn(\))p 1534 1255 V 1695 1312 a Fm(t)1938 1278 y Fv(op)m(\023)-46 b(cenito.)81 b(Budu)m(\023)-46 b(ci)45 b(da)g(za)h(sv)-5 b(aki)45 b Fq(t)g Fv(imamo)-165 1397 y(k)m(on)m(v)m(ergenciju)38 b(za)f(sk)m(oro)h(sv)-5 b(aki)37 b Fq(x)p Fv(,)h(p)s(o)f(F)-8 b(ubinijev)m(om)35 b(teorem)m(u)i(zaklju)m(\024)-46 b(cujemo)36 b(da)h(imamo)d(k)m(on)m(v) m(er-)-165 1515 y(genciju)h(sk)m(oro)h(svuda)g(na)f Fw(R)933 1479 y Fn(+)1021 1515 y Fo(\002)24 b Fw(R)p Fv(.)52 b(Iz)36 b(\(14\))e(tak)m(od)p 1811 1464 30 2 v 1 w(er)i(imamo,)d(ak)m(o)j(je)f Fq(y)2651 1530 y Fp(\000)2714 1515 y Fv(\()p Fq(t;)17 b(x)p Fv(\))33 b(=)f Fq(y)3113 1530 y Fn(+)3177 1515 y Fv(\()p Fq(t;)17 b(x)p Fv(\),)36 b(da)f(je)-165 1634 y Fq(u)d Fv(neprekidna)h(u)g(\()p Fq(t;)17 b(x)p Fv(\),)33 b(a)f(neprekidnost)i(s)f(lijev)-5 b(a,)30 b(o)s(dnosno)j(zdesna,)h (funk)m(cija)e Fq(y)2933 1649 y Fp(\006)3029 1634 y Fv(nam)g(da)5 b(je)33 b(\(15\).)57 1753 y(Dok)-5 b(a)m(\024)-46 b(zimo)21 b(slabu)i(k)m(on)m(v)m(ergenciju)i(nizo)m(v)-5 b(a)23 b(\()p Fq(u)1731 1770 y Fm(k)1777 1753 y Fv(\))g(i)f(\()p Fq(u)1982 1717 y Fn(2)1982 1785 y Fm(k)2028 1753 y Fv(\))h(u)h(L)2228 1717 y Fp(1)2228 1785 y Fn(lo)r(c)2328 1753 y Fv(\()p Fw(R)2450 1717 y Fn(+)2536 1753 y Fo(\002)e Fw(R)p Fv(\))p Fo(\003)p Fv(.)40 b(Iz)24 b(\(14\))f(slijedi)e(da)i(za)-165 1872 y(sv)-5 b(aku)26 b(to)m(\024)-46 b(cku)26 b(\()p Fq(t;)17 b(x)p Fv(\))25 b(p)s(osto)5 b(je)26 b Fq(\026)989 1887 y Fn(0)1059 1872 y Fq(>)h Fv(0)e(i)f Fq(\016)32 b(>)27 b Fv(0)e(takvi)g(da)g(za)h(sv)-5 b(aki)25 b Fq(\026)i Fh(6)h Fq(\026)2515 1887 y Fn(0)2583 1872 y Fv(i)c(sv)-5 b(aki)25 b(\()p Fq(s;)17 b(z)t Fv(\))28 b Fo(2)g Fv(K\(\()p Fq(t;)17 b(x)p Fv(\))p Fq(;)g(\016)t Fv(\))-165 1990 y(vrijedi)754 2093 y Fq(x)23 b Fo(\000)f Fq(y)979 2108 y Fn(+)1043 2093 y Fv(\()p Fq(t;)17 b(x)p Fv(\))p 754 2137 499 4 v 986 2228 a Fq(t)1285 2160 y Fo(\000)23 b Fv(1)k Fh(6)h Fq(u)1622 2175 y Fm(\026)1672 2160 y Fv(\()p Fq(s;)17 b(z)t Fv(\))28 b Fh(6)2030 2093 y Fq(x)23 b Fo(\000)g Fq(y)2256 2108 y Fp(\000)2319 2093 y Fv(\()p Fq(t;)17 b(x)p Fv(\))p 2030 2137 V 2262 2228 a Fq(t)2561 2160 y Fv(+)22 b(1)17 b Fq(:)-165 2366 y Fv(Iz)49 b(izlaganja)d(k)m(o)5 b(je)49 b(je)g(pretho)s(dilo)e(dok)-5 b(azu)49 b(nejednak)m(osti)g (\(14\))f(slijedi)e(da)j(za)g(sv)-5 b(aki)48 b(k)m(ompaktni)-165 2485 y(skup)33 b Fq(K)39 b Fv(p)s(osto)5 b(ji)32 b(uniformna)e(o)s (cjena)i Fq(R)h Fv(na)f(vrijednosti)g(funk)m(cija)g Fq(y)2425 2500 y Fp(\000)2488 2485 y Fv(\()p Fq(t;)17 b(x)p Fv(\))33 b(i)e Fq(y)2838 2500 y Fn(+)2901 2485 y Fv(\()p Fq(t;)17 b(x)p Fv(\).)44 b(Na)32 b([)p Fo(\000)p Fq(R)q(;)17 b(R)q Fv(])-165 2604 y(funk)m(cija)32 b Fq(u)259 2619 y Fn(0)334 2604 y Fv(ima)e(uniformn)m(u)i(ogradu)g Fq(A)p Fv(,)h(a)f(budu)m(\023) -46 b(ci)33 b(da)f(je)909 2876 y Fq(F)14 b Fv(\()p Fq(t;)j(x;)g(y)t Fv(\))26 b(=)1432 2809 y(\()p Fq(x)c Fo(\000)h Fq(y)t Fv(\))1737 2772 y Fn(2)p 1432 2853 347 4 v 1563 2944 a Fv(2)p Fq(t)1811 2876 y Fv(+)1909 2740 y Fl(Z)2008 2767 y Fm(y)1964 2966 y(x)2070 2876 y Fq(u)2126 2891 y Fn(0)2168 2876 y Fv(\()p Fq(\030)5 b Fv(\))17 b Fq(d\030)30 b Fh(6)e Fv(0)-165 3142 y(k)-5 b(ad)47 b(je)h Fq(y)i Fv(to)m(\024)-46 b(ck)-5 b(a)48 b(minim)m(uma)c(za)j(funk)m(ciju)h Fq(F)14 b Fv(\()p Fq(t;)j(x;)g Fo(\001)p Fv(\))47 b(mo)m(\024)-46 b(zemo)46 b(o)s(cijeniti)f(da)i(je)h(u)g(tim)d(to)m(\024)-46 b(ck)-5 b(ama)-155 3231 y Fp(j)p Fm(x)p Fp(\000)p Fm(y)s Fp(j)p -155 3257 187 4 v -76 3315 a Fm(t)95 3280 y Fh(6)54 b Fv(2)p Fq(A)p Fv(.)89 b(P)m(osebno)49 b(dobiv)-5 b(amo)46 b Fo(j)1358 3231 y Fm(x)p Fp(\000)p Fm(y)1500 3241 y Fd(\007)1555 3231 y Fn(\()p Fm(t;x)p Fn(\))p 1358 3257 350 4 v 1519 3315 a Fm(t)1717 3280 y Fo(j)54 b Fh(6)g Fv(2)p Fq(A)p Fv(,)e(o)s(dnosno)c(dobiv)-5 b(amo)46 b(da)h(je)h Fq(u)g Fv(lok)-5 b(alno)-165 3399 y(ograni)m(\024)-46 b(cena.)42 b(Fiksira)5 b(jmo)31 b(funk)m(ciju)i Fq(')27 b Fo(2)h Fv(C)1482 3363 y Fp(1)1482 3424 y Fm(c)1564 3399 y Fv(\()p Fw(R)1686 3363 y Fn(+)1771 3399 y Fo(\002)23 b Fw(R)p Fv(\))o(.)44 b(T)-8 b(ada)32 b(familija)d(otv)m(orenih)j (kugala)1069 3611 y Fo(f)p Fv(K\(\()p Fq(t;)17 b(x)p Fv(\))p Fq(;)g(\016)1530 3626 y Fm(t;x)1627 3611 y Fv(\))27 b(:)45 b(\()p Fq(t;)17 b(x)p Fv(\))27 b Fo(2)i Fv(supp)17 b Fq(')p Fo(g)-168 3822 y Fv(\024)-46 b(cini)42 b(otv)m(oren)i(p)s (okriv)-5 b(a)m(\024)-46 b(c)43 b(k)m(ompaktnog)g(skupa)i Fq(K)54 b 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Fk(2)1297 4790 y Fq(u)1353 4808 y Fm(k)1399 4790 y Fq(')17 b(dtdx)28 b Fo(!)1827 4655 y Fl(Z)1882 4880 y Fg(R)1948 4859 y Fk(2)2006 4790 y Fq(u')17 b(dtdx)g(:)-165 5065 y Fv(Na)32 b(isti)f(na)m(\024)-46 b(cin)32 b(slijedi)f(da)h(su)i Fo(j)p Fq(u)1037 5029 y Fn(2)1037 5097 y Fm(k)1082 5065 y Fq(')p Fo(j)e Fv(omed)p 1400 5013 V(eni)g(s)h(\(2)p Fq(A)22 b Fv(+)g(1\))2029 5029 y Fn(2)2072 5065 y Fo(j)p Fq(')p Fo(j)p Fv(,)32 b(te)g(da)1097 5198 y Fl(Z)1152 5424 y Fg(R)1218 5402 y Fk(2)1276 5334 y Fq(u)1332 5292 y Fn(2)1332 5360 y Fm(k)1378 5334 y Fq(')17 b(dtdx)27 b Fo(!)1806 5198 y Fl(Z)1861 5424 y Fg(R)1927 5402 y Fk(2)1985 5334 y Fq(u)2041 5292 y Fn(2)2083 5334 y Fq(')17 b(dtdx)g(:)57 5594 y Fv(Znamo)31 b(da)h(je)h(ispunjeno)1183 5805 y Fq(@)1234 5820 y Fm(t)1266 5805 y Fq(u)1322 5820 y Fm(\026)1394 5805 y Fv(+)22 b Fq(@)1543 5820 y Fm(x)1591 5805 y Fv(\()1639 5766 y Fn(1)p 1639 5782 39 4 v 1639 5839 a(2)1687 5805 y Fq(u)1743 5764 y Fn(2)1743 5830 y Fm(\026)1793 5805 y Fv(\))28 b(=)g Fq(\026@)2073 5820 y Fm(xx)2164 5805 y Fq(u)2220 5820 y Fm(\026)2286 5805 y Fv(;)-165 6010 y Ft(22)p eop %%Page: 23 28 23 27 bop 3102 -116 a Fr(Burgerso)n(v)-5 b(a)25 b(jednad)n(\024)-39 b(zba)71 83 y Fv(p)s(omno)m(\024)-46 b(zimo)28 b(jednad)m(\024)-46 b(zbu)32 b(funk)m(cijom)e Fq(')e Fo(2)g Fv(C)1763 47 y Fp(1)1763 108 y Fm(c)1845 83 y Fv(\()p Fw(R)1967 47 y Fn(+)2052 83 y Fo(\002)23 b Fw(R)p Fv(\))30 b(i)g(in)m(tegrira)5 b(jmo.)41 b(P)m(arcijalnom)28 b(in)m(tegraci-)71 202 y(jom)j(i)h(primjenom)e(Gauss-Greeno)m(v)m(og)k(teorema)e(dobit)d(\023) -46 b(cemo)917 470 y Fo(\000)1011 334 y Fl(Z)1066 560 y Fg(R)1132 538 y Fk(2)1174 470 y Fv(\()p Fq(u)1268 485 y Fm(\026)1318 470 y Fq(@)1369 485 y Fm(t)1401 470 y Fq(')22 b Fv(+)g Fq(u)1641 429 y Fn(2)1641 494 y Fm(\026)1691 470 y Fq(@)1742 485 y Fm(x)1790 470 y Fq(')p Fv(\))17 b Fq(dtdx)27 b Fv(=)h Fq(\026)2308 334 y Fl(Z)2363 560 y Fg(R)2429 538 y Fk(2)2487 470 y Fq(u)2543 485 y Fm(\026)2593 470 y Fq(@)2644 485 y Fm(xx)2736 470 y Fq(')17 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v(eno)h(vrijedno)-5 b(\024)-44 b(s)m(\023)e(cu)44 b(inicijalnog)c(uvjeta)k Fq(u)3267 1830 y Fn(0)3352 1815 y Fv(na)g(segmen)m(tu)71 1934 y([)p Fo(\000)p Fq(R)q(;)17 b(R)q Fv(].)72 b(Aproksimiramo)39 b(inicijalni)f(uvjet)k Fq(u)1915 1949 y Fn(0)1999 1934 y Fv(nizom)f(\()p Fq(u)2390 1885 y Fm(j)2390 1963 y Fn(0)2432 1934 y Fv(\))h(funk)m(cija)f(klase)h(C)3210 1897 y Fn(1)3210 1958 y Fm(c)3253 1934 y Fv(\()p Fw(R)p Fv(\))g(takvim)f(da)71 2070 y Fq(u)127 2021 y Fm(j)127 2100 y Fn(0)197 2070 y Fo(!)27 b Fq(u)380 2085 y Fn(0)453 2070 y Fv(u)k(L)599 2034 y Fn(1)599 2102 y(lo)r(c)699 2070 y Fv(\()p Fw(R)p Fv(\))o(,)h(k)m(o)5 b(je)32 b(jednolik)m(o)d(zado)m(v)m(olja)m(v)-5 b(a)5 b(ju)31 b(o)s(cjen)m(u)h(rasta)f(iz)f(isk)-5 b(aza)30 b(teorema.)42 b(T)-8 b(ak)j(a)m(v)32 b(niz)71 2189 y(bi)g(se)h(na)g (primjer)e(mogao)g(dobiti)f(tak)m(o)j(da)g(za)f(sv)-5 b(aki)33 b Fq(j)38 b Fv(funk)m(ciju)1431 2443 y Fq(u)1487 2394 y Fm(j)t Fp(\003)1487 2472 y Fn(0)1565 2443 y Fv(\()p Fq(x)p Fv(\))28 b(=)1828 2332 y Fl(n)1911 2396 y Fq(u)1967 2411 y Fn(0)2009 2396 y Fv(\()p Fq(x)p Fv(\))98 b Fo(j)p Fq(x)p Fo(j)27 b Fh(6)h Fq(j)1911 2514 y Fv(0)278 b(ina)m(\024)-46 b(ce)71 2697 y(k)m(on)m(v)m(oluiramo)31 b(sa)i(standardnim)e(izglad)p 1581 2645 V -1 w(iv)-5 b(a)m(\024)-46 b(cem)32 b Fq(\032)1987 2712 y Fm(n)2070 2697 y Fv(o)s(dabranim)e(tak)m(o)j(da)g(je)1406 2852 y Fl(Z)1505 2878 y Fm(j)1461 3078 y Fp(\000)p Fm(j)1577 2988 y Fo(j)p Fq(u)1661 3003 y Fn(0)1703 2988 y Fv(\()p Fq(x)p Fv(\))22 b Fo(\000)h Fq(u)2012 2939 y Fm(j)2012 3017 y Fn(0)2054 2988 y Fv(\()p Fq(x)p Fv(\))p Fo(j)p Fq(dx)28 b(<)2460 2920 y Fv(1)p 2460 2965 49 4 v 2461 3056 a Fq(j)2536 2988 y(:)71 3303 y Fv(F)-8 b(unk)m(cije)38 b Fq(F)528 3318 y Fm(j)568 3303 y Fv(\()p Fq(t;)17 b(x;)g(y)t Fv(\))37 b(zadane)h(k)-5 b(ao)38 b(i)e Fq(F)14 b Fv(,)39 b(gdje)f(je)g Fq(u)2011 3318 y Fn(0)2091 3303 y Fv(supstituiran)f(s)i Fq(u)2772 3254 y Fm(j)2772 3333 y Fn(0)2851 3303 y Fv(na)f(sv)-5 b(ak)m(om)38 b(k)m(ompaktnom)71 3439 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Fq(@)1994 3977 y Fm(x)2043 3962 y Fq(')2107 3882 y Fl(\001)2185 3962 y Fq(dtdx)28 b Fv(=)2509 3827 y Fl(Z)2564 4052 y Fg(R)2651 3962 y Fq(u)2707 3914 y Fm(j)2707 3992 y Fn(0)2749 3962 y Fq(')17 b(dx)g(;)71 4235 y Fv(pa)32 b(nam)g(Lesb)s(egueso)m(v)j (teorem)d(o)h(dominirano)5 b(j)30 b(k)m(on)m(v)m(ergenciji)j(da)5 b(je)33 b(tvrdnju.)293 4355 y(Uzmimo)d(da)j(su)g Fq(x)996 4370 y Fn(1)1039 4355 y Fq(;)17 b(x)1138 4370 y Fn(2)1208 4355 y Fo(2)29 b Fw(R)j Fv(takvi)g(da)g(je)h Fq(x)1964 4370 y Fn(1)2035 4355 y Fh(6)28 b Fq(x)2195 4370 y Fn(2)2238 4355 y Fv(,)33 b(i)e(ra)m(\024)-46 b(cuna)5 b(jmo:)714 4630 y(lim)682 4698 y Fm(x)p Fp(&)p Fm(x)847 4710 y Fk(2)899 4630 y Fq(u)p Fv(\()p Fq(t;)17 b(x)p Fv(\))22 b Fo(\000)56 b Fv(lim)1287 4698 y Fm(x)p Fp(\045)p Fm(x)1452 4710 y Fk(1)1504 4630 y Fq(u)p Fv(\()p Fq(t;)17 b(x)p Fv(\))28 b(=)1911 4562 y Fq(x)1966 4577 y Fn(2)2032 4562 y Fo(\000)22 b Fq(y)2179 4577 y Fn(+)2243 4562 y Fv(\()p Fq(t;)17 b(x)2415 4577 y Fn(2)2457 4562 y Fv(\))p 1911 4607 584 4 v 2186 4698 a Fq(t)2527 4630 y Fo(\000)2637 4562 y Fq(x)2692 4577 y Fn(1)2757 4562 y Fo(\000)23 b Fq(y)2905 4577 y Fp(\000)2968 4562 y Fv(\()p Fq(t;)17 b(x)3140 4577 y Fn(1)3183 4562 y Fv(\))p 2637 4607 V 2911 4698 a Fq(t)1798 4898 y Fv(=)1911 4830 y Fq(x)1966 4845 y Fn(2)2032 4830 y Fo(\000)22 b Fq(y)2179 4845 y Fn(+)2243 4830 y Fv(\()p Fq(t;)17 b(x)2415 4845 y Fn(2)2457 4830 y Fv(\))23 b Fo(\000)f Fv(\()p Fq(x)2710 4845 y Fn(1)2775 4830 y Fo(\000)h Fq(y)2923 4845 y Fp(\000)2986 4830 y Fv(\()p Fq(t;)17 b(x)3158 4845 y Fn(1)3201 4830 y Fv(\)\))p 1911 4875 1366 4 v 2577 4966 a Fq(t)1798 5125 y Fh(6)1913 5057 y Fq(x)1968 5072 y Fn(2)2033 5057 y Fo(\000)23 b Fq(x)2188 5072 y Fn(1)p 1913 5102 318 4 v 2054 5193 a Fq(t)2257 5125 y(;)71 5375 y Fv(gdje)33 b(smo)f(primijenili)27 b(tvrdnju)34 b(Leme)e(2.)293 5496 y(Ak)m(o)h(je)g Fq(y)650 5511 y Fn(+)713 5496 y Fv(\()p Fq(t;)17 b(x)p Fv(\))28 b(=)g Fq(y)1103 5511 y Fp(\000)1166 5496 y Fv(\()p Fq(t;)17 b(x)p Fv(\),)33 b(onda)f(iz)g(druge)h(tvrdnje)h(Leme)e(2)h(slijedi)d (da)j(je)1463 5776 y Fq(u)27 b Fv(=)h Fq(u)p Fv(\()p Fq(t;)17 b(x)p Fv(\))27 b(=)2057 5708 y Fq(x)22 b Fo(\000)h Fq(y)t Fv(\()p Fq(t;)17 b(x)p Fv(\))p 2057 5753 439 4 v 2258 5844 a Fq(t)3770 6010 y Ft(23)p eop %%Page: 24 29 24 28 bop -165 -116 a Fr(Nelinearni)27 b(zak)n(oni)f(sa)n(\024)-39 b(cuv)-5 b(anja)-165 83 y Fv(k)m(onstan)m(tna)33 b(i)f(neprekidna)h(na) f(otv)m(orenom)h(djelu)f(segmen)m(ta)g(k)m(o)5 b(ji)33 b(spa)5 b(ja)33 b(\()p Fq(t;)17 b(x)p Fv(\))32 b(i)g(\(0)p Fq(;)17 b(y)t Fv(\()p Fq(t;)g(x)p Fv(\)\).)42 b(Kada)-165 202 y(je)34 b Fq(u)-2 217 y Fn(0)71 202 y Fo(2)c Fv(C)237 166 y Fn(1)237 226 y Fm(c)280 202 y Fv(\()p Fw(R)p Fv(\),)35 b(tada)e(otv)m(orene)j(linije)31 b(zadiru)j(u)h(p)s(o)s(dru)m(\024)-46 b(cje)34 b(na)g(k)m(o)5 b(jem)35 b(znamo)e(da)h(je)h Fo(j)p Fq(u)p Fo(j)29 b Fh(6)i Fq(M)10 b Fv(,)35 b(pa)-165 320 y(slijedi)25 b(da)j(je)g(tada)f Fo(j)p Fq(u)p Fo(j)f Fh(6)j Fq(M)38 b Fv(i)27 b(ina)m(\024)-46 b(ce.)41 b(Za)27 b(generalni)f Fq(u)1878 335 y Fn(0)1948 320 y Fv(tvrdnja)i(slijedi)e (iz)h(aproksimacije)f(k)-5 b(ao)27 b(prije.)57 439 y(Jedinstv)m(enost) 37 b(je)f(p)s(oslijedica)d(T)-8 b(eorema)36 b(1)f(iz)g(prv)m(og)h(o)s (djeljk)-5 b(a.)52 b(T)-8 b(o)m(\024)-46 b(cnije,)36 b(ak)m(o)g(p)s(osto)5 b(ji)35 b(drugo)-165 558 y(ograni)m(\024)-46 b(ceno)27 b(rje)-5 b(\024)-44 b(senje)29 b(Cauc)m(h)m(yjev)m(og)h (problema)c(za)i(Burgerso)m(vu)h(jedna)m(\024)-46 b(zbu)29 b(uz)f(iste)g(inicijalne)c(uvjete)-165 677 y Fq(v)t Fv(,)32 b(k)m(o)5 b(je)34 b(zado)m(v)m(olja)m(v)-5 b(a)32 b(\(16\))g(\(Hopfo)m (v)m(o)h(rje)-5 b(\024)-44 b(senje\),)34 b(tada)e(je)h Fq(w)d Fv(=)d Fq(u)22 b Fo(\000)g Fq(v)37 b Fv(slab)s(o)31 b(rje)-5 b(\024)-44 b(senje)34 b(jednad)m(\024)-46 b(zb)s(e)1349 850 y Fq(@)1400 865 y Fm(t)1432 850 y Fq(w)25 b Fv(+)d Fq(@)1676 865 y Fm(x)1724 850 y Fv(\()p Fq(aw)s Fv(\))27 b(=)h(0)17 b Fq(;)-165 1024 y Fv(s)33 b(n)m(ulom)e(k)-5 b(ao)32 b(inicijalnim)c(uvjetom,)k(i)g(\(2\))g(je)h(ispunjeno)f(s)h Fq(C)7 b Fv(\()p Fq(t)p Fv(\))28 b(=)2423 985 y Fn(1)p 2423 1001 39 4 v 2428 1058 a Fm(t)2471 1024 y Fv(.)3324 1143 y Fw(Q.E.D.)57 1301 y Fv(Kao)34 b(\024)-44 b(sto)41 b(smo)f(vidjeli)e(u)j(o)m(v)m(om)f(dok)-5 b(azu,)42 b(uz)f Fq(t)g(>)g Fv(0)f(sv)-5 b(aki)40 b(\()p Fq(t;)17 b(x)p Fv(\))41 b(mo)m(\024)-46 b(zemo)39 b(sh)m(v)-5 b(atiti)39 b(k)-5 b(ao)40 b(vrh)-165 1420 y(trokuta)33 b Fq(T)242 1435 y Fm(t;x)338 1420 y Fv(,)g(s)h(ostalim)c(vrho)m(vima)i(\(0)p Fq(;)17 b(y)1422 1435 y Fp(\006)1485 1420 y Fv(\()p Fq(t;)g(x)p Fv(\)\).)45 b(Za)32 b Fq(u)g Fv(imamo)e(da)j(je)g(neprekidna)g(na)g (otv)m(orenim)-165 1563 y(stranama)g(trokuta)h Fq(T)673 1578 y Fm(t;x)803 1563 y Fv(i)f(da)h(je)g(jednak)-5 b(a)34 b(brzini)1755 1514 y Fm(x)p Fp(\000)p Fm(y)1897 1524 y Fd(\006)1952 1514 y Fn(\()p Fm(t;x)p Fn(\))p 1755 1540 350 4 v 1916 1598 a Fm(t)2115 1563 y Fv(.)47 b(Tvrdimo)33 b(da)h(za)g(sv)-5 b(aki)34 b Fq(t)3138 1527 y Fp(0)3193 1563 y Fq(>)c(t)k Fv(p)s(osto)5 b(ji)-165 1682 y(jedinstv)m(eni)46 b Fq(x)394 1646 y Fp(0)466 1682 y Fv(tak)-5 b(a)m(v)46 b(da)g(je)g Fq(T)1067 1697 y Fm(t;x)1215 1682 y Fo(\022)51 b Fq(T)1400 1701 y Fm(t)1428 1680 y Fd(0)1451 1701 y Fm(;x)1516 1680 y Fd(0)1543 1682 y Fv(.)84 b(Da)45 b(bismo)g(to)h(dok) -5 b(azali,)48 b(p)s(ok)-5 b(azat)42 b(\023)-46 b(cemo)46 b(da)g(je)g(ili)-165 1801 y Fq(T)-108 1816 y Fm(t;x)16 1801 y Fo(\022)28 b Fq(T)178 1820 y Fm(t)206 1799 y Fd(0)230 1820 y Fm(;x)295 1799 y Fd(0)354 1801 y Fv(ili)i(su)j(trokuti)f (disjunktni.)57 1919 y(Otv)m(orene)37 b(stranice)g(se)h(mogu)d(sje)m (\023)-46 b(ci)37 b(samo)f(ak)m(o)h(ima)5 b(ju)35 b(iste)h(nagib)s(e,)h (a)f(tada)h(je)f(jedan)h(trokut)-165 2038 y(sadr)m(\024)-46 b(zan)36 b(u)g(drugom;)g(ak)m(o)g(\()p Fq(t;)17 b(x)p Fv(\))36 b(le)m(\024)-46 b(zi)34 b(na)i(otv)m(orenom)f(dijelu)f (stranice)i(trokuta)f Fq(T)2950 2058 y Fm(t)2978 2036 y Fd(0)3002 2058 y Fm(;x)3067 2036 y Fd(0)3094 2038 y Fv(,)h(onda)g(je)f Fq(T)3564 2053 y Fm(t;x)-165 2157 y Fv(sadr)m(\024)-46 b(zan)35 b(u)f(to)5 b(j)35 b(stranici.)48 b(Za)34 b Fq(x)1029 2121 y Fp(0)1089 2157 y Fv(uzimamo)e(in\014m)m(um)h (svih)i Fq(y)j Fv(takvih)c(da)g Fq(T)2683 2176 y Fm(t)2711 2155 y Fd(0)2735 2176 y Fm(;y)2835 2157 y Fv(le)m(\024)-46 b(zi)33 b(desno)j(o)s(d)e(\()p Fq(t;)17 b(x)p Fv(\).)-165 2276 y(T)-8 b(ada)39 b(desna)h(strana)f(trokuta)g Fq(T)1084 2295 y Fm(t)1112 2274 y Fd(0)1135 2295 y Fm(;x)1200 2274 y Fd(0)1266 2276 y Fv(le)m(\024)-46 b(zi)38 b(zdesna)j(o)s(d)d(\()p Fq(t;)17 b(x)p Fv(\))39 b(zb)s(og)g(neprekidnosti)g(zdesna)i(funk)m (cije)-165 2394 y Fq(y)-117 2409 y Fn(+)-54 2394 y Fv(\()p Fq(t)19 2358 y Fp(0)44 2394 y Fq(;)17 b Fo(\001)p Fv(\),)40 b(ali)d(lijev)-5 b(a)37 b(strana)i(strana)g(ne)h(mo)m(\024)-46 b(ze)38 b(le)m(\024)-46 b(zati)38 b(strogo)h(zdesna)h(zb)s(og)f (neprekidnosti)g(s)g(lijev)-5 b(a)-165 2513 y(funk)m(cije)44 b Fq(y)257 2528 y Fn(+)320 2513 y Fv(\()p Fq(t)393 2477 y Fp(0)418 2513 y Fq(;)17 b Fo(\001)p Fv(\))43 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b(a,)39 b(a)d(geometrijski)e(je)j(jasno)f(da)h(je)f(to)g(Lipshitzo)m(v) -5 b(a)-165 3107 y(funk)m(cija.)43 b(Nek)-5 b(a)33 b(je)g Fq(x)p Fv(\()p Fq(t)p Fv(\))g(takv)-5 b(a)32 b(krivulja)g(kroz)h(\()p Fq(t;)17 b(x)p Fv(\).)43 b(Tvrdimo)32 b(da)h(je)-165 3340 y(\(17\))679 b(lim)580 3410 y Fm(t)608 3389 y Fd(0)631 3410 y Fp(&)p Fm(t;t)785 3389 y Fd(00)827 3410 y Fp(&)p Fm(t)959 3273 y Fq(x)p Fv(\()p Fq(t)1087 3236 y Fp(00)1133 3273 y Fv(\))22 b Fo(\000)h Fq(x)p Fv(\()p Fq(t)1421 3236 y Fp(0)1446 3273 y Fv(\))p 959 3317 526 4 v 1090 3408 a Fq(t)1125 3379 y Fp(00)1193 3408 y Fo(\000)g Fq(t)1328 3379 y Fp(0)1522 3340 y Fv(=)1635 3273 y(1)p 1635 3317 49 4 v 1635 3408 a(2)1711 3199 y Fl(\022)1797 3340 y Fv(lim)1784 3408 y Fm(y)s Fp(\045)p Fm(x)1962 3340 y Fq(u)p Fv(\()p Fq(t;)17 b(y)t Fv(\))j(+)35 b(lim)2343 3408 y Fm(y)s Fp(&)p Fm(x)2521 3340 y Fq(u)p Fv(\()p Fq(t;)17 b(y)t Fv(\))2782 3199 y Fl(\023)2889 3340 y Fq(:)-165 3576 y Fv(Kak)m(o)41 b(bismo)f(to)g(dok)-5 b(azali,)42 b(uzmimo)d(da)i(su)g Fq(x)1640 3539 y Fp(0)1708 3576 y Fv(=)h Fq(x)p Fv(\()p Fq(t)1954 3539 y Fp(0)1980 3576 y Fv(\))f(i)f Fq(x)2182 3539 y Fp(00)2270 3576 y Fv(=)i Fq(x)p Fv(\()p Fq(t)2516 3539 y Fp(00)2563 3576 y Fv(\),)h(te)e(pretp)s(osta)m(vimo)f(da)h(je)-165 3694 y Fq(t)-130 3658 y Fp(00)-57 3694 y Fq(>)28 b(t)82 3658 y Fp(0)135 3694 y Fq(>)f(t)p Fv(.)44 b(T)-8 b(ada)33 b(imamo)287 3868 y Fq(y)335 3883 y Fp(\000)399 3868 y Fv(\()p Fq(t)472 3827 y Fp(00)518 3868 y Fq(;)17 b(x)617 3827 y Fp(00)663 3868 y Fv(\))27 b Fh(6)i Fq(y)882 3883 y Fp(\000)945 3868 y Fv(\()p Fq(t)1018 3827 y Fp(0)1043 3868 y Fq(;)17 b(x)1142 3827 y Fp(0)1167 3868 y Fv(\))28 b Fh(6)g Fq(y)1386 3883 y Fp(\000)1450 3868 y Fv(\()p Fq(t;)17 b(x)p Fv(\))27 b Fh(6)i Fq(y)1841 3883 y Fn(+)1904 3868 y Fv(\()p Fq(t;)17 b(x)p Fv(\))28 b Fh(6)g Fq(y)2295 3883 y Fn(+)2359 3868 y Fv(\()p Fq(t)2432 3827 y Fp(0)2457 3868 y Fq(;)17 b(x)2556 3827 y Fp(0)2581 3868 y Fv(\))28 b Fh(6)g Fq(y)2800 3883 y Fn(+)2863 3868 y Fv(\()p Fq(t)2936 3827 y Fp(00)2982 3868 y Fq(;)17 b(x)3081 3827 y Fp(00)3127 3868 y Fv(\))g Fq(;)-165 4042 y Fv(i)42 b(imamo)f(i)i(k)m(on)m(v)m (ergenciju)i Fq(y)978 4057 y Fp(\006)1042 4042 y Fv(\()p Fq(t)1115 4005 y Fp(00)1161 4042 y Fq(;)17 b(x)1260 4005 y Fp(00)1306 4042 y Fv(\))46 b Fo(!)h Fq(y)1585 4057 y Fp(\006)1648 4042 y Fv(\()p Fq(t;)17 b(x)p Fv(\))44 b(k)-5 b(ad)44 b Fq(t)2130 4005 y Fp(00)2222 4042 y Fo(!)j Fq(t)p Fv(,)41 b(\024)-44 b(sto)44 b(slijedi)e(iz)h(dok)-5 b(aza)43 b(zdesne)-165 4160 y(\(s)h(lijev)-5 b(a\))41 b(neprekidnosti)j(funk)m(cija)g Fq(y)1302 4175 y Fp(\006)1365 4160 y Fv(\()p Fq(t;)17 b Fo(\001)p Fv(\))43 b(u)h(lemi)d(2.)76 b(Ak)m(o)44 b(je)g Fq(y)2442 4175 y Fn(+)2506 4160 y Fv(\()p Fq(t)2579 4124 y Fp(0)2604 4160 y Fq(;)17 b(x)2703 4124 y Fp(0)2728 4160 y Fv(\))46 b(=)g Fq(y)2982 4175 y Fp(\000)3046 4160 y Fv(\()p Fq(t)3119 4124 y Fp(0)3144 4160 y Fq(;)17 b(x)3243 4124 y Fp(0)3268 4160 y Fv(\))44 b(za)f(neki)-165 4279 y Fq(t)-130 4243 y Fp(0)-69 4279 y Fq(>)36 b(t)p Fv(,)j(tada)e(je)h(\(17\))f(trivijalno)d(ispunjeno,)39 b(pa)c(\023)-46 b(cemo)37 b(stoga)g(dalje)g(razmatrati)e(samo)i(ta)5 b(j)37 b(slu)m(\024)-46 b(ca)5 b(j.)-165 4398 y(Fiksira)g(jmo)30 b Fq(y)379 4362 y Fp(0)431 4398 y Fv(=)d Fq(y)582 4413 y Fp(\000)646 4398 y Fv(\()p Fq(t)719 4362 y Fp(0)744 4398 y Fq(;)17 b(x)843 4362 y Fp(0)868 4398 y Fv(\))33 b(i)e Fq(y)1050 4362 y Fp(00)1123 4398 y Fv(=)c Fq(y)1274 4413 y Fn(+)1338 4398 y Fv(\()p Fq(t)1411 4362 y Fp(00)1457 4398 y Fq(;)17 b(x)1556 4362 y Fp(00)1602 4398 y Fv(\),)33 b(te)f(de\014nira)5 b(jmo)32 b(p)s(omo)m(\023)-46 b(cne)32 b(funk)m(cije)1255 4631 y Fq(I)1306 4590 y Fp(0)1331 4631 y Fv(\()p Fq(\030)5 b Fv(\))27 b(=)h Fq(u)1642 4646 y Fn(0)1684 4631 y Fv(\()p Fq(\030)5 b Fv(\))21 b(+)1937 4564 y Fq(\030)27 b Fo(\000)22 b Fq(x)2161 4527 y Fp(0)p 1937 4608 250 4 v 2032 4699 a Fq(t)2067 4671 y Fp(0)1235 4878 y Fq(I)1286 4836 y Fp(00)1331 4878 y Fv(\()p Fq(\030)5 b Fv(\))27 b(=)h Fq(u)1642 4893 y Fn(0)1684 4878 y Fv(\()p Fq(\030)5 b Fv(\))21 b(+)1937 4810 y Fq(\030)27 b Fo(\000)22 b Fq(x)2161 4774 y Fp(00)p 1937 4855 271 4 v 2032 4946 a Fq(t)2067 4917 y Fp(00)2234 4878 y Fq(:)-165 5077 y Fv(Ra)m(\024)-46 b(cunamo:)606 5139 y Fl(Z)706 5166 y Fm(y)747 5134 y Fd(00)661 5365 y Fm(y)702 5343 y Fd(0)809 5275 y Fq(I)860 5234 y Fp(0)885 5275 y Fv(\()p Fq(\030)5 b Fv(\))17 b Fq(d\030)31 b Fv(=)1254 5139 y Fl(Z)1354 5166 y Fm(y)1395 5134 y Fd(00)1310 5365 y Fm(y)1351 5343 y Fd(0)1458 5275 y Fq(u)1514 5290 y Fn(0)1556 5275 y Fv(\()p Fq(\030)5 b Fv(\))17 b Fq(d\030)25 b Fv(+)1934 5208 y(\()p Fq(\030)h Fo(\000)d Fq(x)2196 5171 y Fp(0)2221 5208 y Fv(\))2259 5171 y Fn(2)p 1934 5252 369 4 v 2063 5343 a Fv(2)p Fq(t)2147 5315 y Fp(0)2312 5131 y Fl(\014)2312 5190 y(\014)2312 5250 y(\014)2312 5310 y(\014)2345 5150 y Fm(y)2386 5118 y Fd(00)2345 5374 y Fm(y)2386 5352 y Fd(0)1151 5595 y Fv(=)1254 5459 y Fl(Z)1354 5486 y Fm(y)1395 5454 y Fd(00)1310 5685 y Fm(y)1351 5663 y Fd(0)1458 5595 y Fq(u)1514 5610 y Fn(0)1556 5595 y Fv(\()p Fq(\030)5 b Fv(\))17 b Fq(d\030)25 b Fv(+)1924 5527 y(\()p Fq(y)2014 5491 y Fp(00)2081 5527 y Fo(\000)e Fq(x)2236 5491 y Fp(0)2261 5527 y Fv(\))2299 5491 y Fn(2)p 1924 5572 418 4 v 2078 5663 a Fv(2)p Fq(t)2162 5634 y Fp(0)2374 5595 y Fo(\000)2483 5527 y Fv(\()p Fq(y)2573 5491 y Fp(0)2620 5527 y Fo(\000)f Fq(x)2774 5491 y Fp(0)2800 5527 y Fv(\))2838 5491 y Fn(2)p 2483 5572 397 4 v 2627 5663 a Fv(2)p Fq(t)2711 5634 y Fp(0)1151 5815 y Fv(=)27 b Fq(F)14 b Fv(\()p Fq(t)1404 5774 y Fp(0)1429 5815 y Fq(;)j(x)1528 5774 y Fp(0)1553 5815 y Fq(;)g(y)1649 5774 y Fp(00)1694 5815 y Fv(\))22 b Fo(\000)h Fq(F)14 b Fv(\()p Fq(t)2004 5774 y Fp(0)2029 5815 y Fq(;)j(x)2128 5774 y Fp(0)2153 5815 y Fq(;)g(y)2249 5774 y Fp(0)2273 5815 y Fv(\))27 b Fh(>)i Fv(0)-165 6010 y Ft(24)p eop %%Page: 25 30 25 29 bop 3102 -116 a Fr(Burgerso)n(v)-5 b(a)25 b(jednad)n(\024)-39 b(zba)71 83 y Fv(zb)s(og)32 b(tog)27 b(\024)-44 b(sto)33 b Fq(y)675 98 y Fp(\000)771 83 y Fv(minimizira)28 b(funk)m(ciju)33 b Fq(F)14 b Fv(\()p Fq(t;)j(x;)g Fo(\001)p Fv(\).)42 b(Isto)33 b(tak)m(o)g(dobiv)-5 b(amo)31 b(i)g(nejednak)m(ost)k(za)d Fq(I)3682 47 y Fp(00)3728 83 y Fv(:)1612 248 y Fl(Z)1712 275 y Fm(y)1753 243 y Fd(0)1667 474 y Fm(y)1708 452 y Fd(00)1796 384 y Fq(I)1847 343 y Fp(00)1893 384 y Fv(\()p Fq(\030)5 b Fv(\))17 b Fq(d\030)31 b Fh(>)d Fv(0)17 b Fq(:)71 660 y Fv(Dalje)31 b(imamo:)730 820 y Fl(Z)830 846 y Fm(y)871 814 y Fd(0)786 1045 y Fm(y)827 1024 y Fd(00)898 955 y Fv(\()p Fq(I)987 914 y Fp(00)1033 955 y Fv(\()p Fq(\030)5 b Fv(\))21 b Fo(\000)i Fq(I)1329 914 y Fp(0)1354 955 y Fv(\()p Fq(\030)5 b Fv(\)\))17 b Fq(d\030)30 b Fv(=)e Fo(\000)1855 820 y Fl(Z)1954 846 y Fm(y)1995 814 y Fd(00)1910 1045 y Fm(y)1951 1024 y Fd(0)2058 955 y Fq(I)2109 914 y Fp(0)2134 955 y Fv(\()p Fq(\030)5 b Fv(\))17 b Fq(d\030)25 b Fo(\000)2494 820 y Fl(Z)2593 846 y Fm(y)2634 814 y Fd(0)2549 1045 y Fm(y)2590 1024 y Fd(00)2678 955 y Fq(I)2729 914 y Fp(00)2775 955 y Fv(\()p Fq(\030)5 b Fv(\))17 b Fq(d\030)30 b Fh(6)f Fv(0)17 b Fq(;)71 1232 y Fv(o)s(dnosno)32 b(da)h(vrijedi:)945 1505 y(\()p Fq(y)1035 1464 y Fp(00)1102 1505 y Fo(\000)22 b Fq(y)1253 1464 y Fp(0)1278 1505 y Fv(\))1333 1365 y Fl(\022)1415 1438 y Fq(x)1470 1402 y Fp(0)p 1415 1482 81 4 v 1425 1574 a Fq(t)1460 1545 y Fp(0)1528 1505 y Fo(\000)1637 1438 y Fq(x)1692 1402 y Fp(00)p 1637 1482 102 4 v 1647 1574 a Fq(t)1682 1545 y Fp(00)1749 1365 y Fl(\023)1844 1505 y Fv(+)1942 1365 y Fl(\022)2031 1438 y Fv(1)p 2026 1482 61 4 v 2026 1574 a Fq(t)2061 1545 y Fp(0)2118 1505 y Fv(+)2242 1438 y(1)p 2226 1482 82 4 v 2226 1574 a Fq(t)2261 1545 y Fp(00)2317 1365 y Fl(\023)2417 1438 y Fq(y)2469 1402 y Fp(00)p Fn(2)2574 1438 y Fo(\000)h Fq(y)2726 1402 y Fp(0)p Fn(2)p 2417 1482 372 4 v 2578 1574 a Fv(2)2826 1505 y Fh(6)28 b Fv(0)17 b Fq(;)66 1772 y Fv(\024)-44 b(sto)33 b(dalje)e(da)5 b(je:)1153 1877 y Fq(x)1208 1841 y Fp(0)p 1153 1922 81 4 v 1163 2013 a Fq(t)1198 1984 y Fp(0)1265 1945 y Fo(\000)1375 1877 y Fq(x)1430 1841 y Fp(00)p 1375 1922 102 4 v 1385 2013 a Fq(t)1420 1984 y Fp(00)1514 1945 y Fh(6)1619 1804 y Fl(\022)1708 1877 y Fv(1)p 1702 1922 61 4 v 1702 2013 a Fq(t)1737 1984 y Fp(0)1795 1945 y Fv(+)1919 1877 y(1)p 1903 1922 82 4 v 1903 2013 a Fq(t)1938 1984 y Fp(00)1994 1804 y Fl(\023)2094 1877 y Fq(y)2146 1841 y Fp(00)2213 1877 y Fv(+)22 b Fq(y)2363 1841 y Fp(0)p 2094 1922 294 4 v 2216 2013 a Fv(2)1041 2217 y Fq(x)1096 2176 y Fp(00)1143 2217 y Fq(t)1178 2176 y Fp(0)1225 2217 y Fo(\000)h Fq(x)1380 2176 y Fp(0)1405 2217 y Fq(t)1440 2176 y Fp(00)1514 2217 y Fh(6)28 b Fv(\()p Fq(t)1692 2176 y Fp(00)1760 2217 y Fo(\000)23 b Fq(t)1895 2176 y Fp(0)1920 2217 y Fv(\))1968 2150 y Fq(y)2016 2165 y Fn(+)2080 2150 y Fv(\()p Fq(t)2153 2114 y Fp(00)2199 2150 y Fq(;)17 b(x)2298 2114 y Fp(00)2344 2150 y Fv(\))22 b(+)g Fq(y)2550 2165 y Fp(\000)2613 2150 y Fv(\()p Fq(t)2686 2114 y Fp(0)2711 2150 y Fq(;)17 b(x)2810 2114 y Fp(0)2836 2150 y Fv(\))p 1968 2195 906 4 v 2396 2286 a(2)2900 2217 y Fq(:)71 2419 y Fv(Sli)m(\024)-46 b(cno)31 b(slijedi)g(i)1041 2592 y Fq(x)1096 2551 y Fp(0)1122 2592 y Fq(t)1157 2551 y Fp(00)1225 2592 y Fo(\000)23 b Fq(x)1380 2551 y Fp(00)1426 2592 y Fq(t)1461 2551 y Fp(0)1514 2592 y Fh(6)28 b Fv(\()p Fq(t)1692 2551 y Fp(0)1739 2592 y Fo(\000)23 b Fq(t)1874 2551 y Fp(00)1920 2592 y Fv(\))1968 2524 y Fq(y)2016 2539 y Fp(\000)2080 2524 y Fv(\()p Fq(t)2153 2488 y Fp(0)2178 2524 y Fq(;)17 b(x)2277 2488 y Fp(0)2302 2524 y Fv(\))22 b(+)g Fq(y)2508 2539 y Fn(+)2571 2524 y Fv(\()p Fq(t)2644 2488 y Fp(00)2690 2524 y Fq(;)17 b(x)2789 2488 y Fp(00)2836 2524 y Fv(\))p 1968 2569 V 2396 2660 a(2)2900 2592 y Fq(;)66 2793 y Fv(\024)-44 b(sto)33 b(skupa)g(s)g(pretho)s(dnom)f(nejednak)m(o)-5 b(\024)-44 b(s)m(\023)e(cu)35 b(da)5 b(je)1042 3060 y Fq(x)1097 3019 y Fp(00)1144 3060 y Fq(t)1179 3019 y Fp(0)1226 3060 y Fo(\000)23 b Fq(x)1381 3019 y Fp(0)1406 3060 y Fq(t)1441 3019 y Fp(00)1515 3060 y Fv(=)k(\()p Fq(t)1691 3019 y Fp(00)1760 3060 y Fo(\000)22 b Fq(t)1894 3019 y Fp(0)1919 3060 y Fv(\))1967 2993 y Fq(y)2015 3008 y Fn(+)2079 2993 y Fv(\()p Fq(t)2152 2957 y Fp(00)2198 2993 y Fq(;)17 b(x)2297 2957 y Fp(00)2343 2993 y Fv(\))22 b(+)g Fq(y)2549 3008 y Fp(\000)2612 2993 y Fv(\()p Fq(t)2685 2957 y Fp(0)2711 2993 y Fq(;)17 b(x)2810 2957 y Fp(0)2835 2993 y Fv(\))p 1967 3038 V 2396 3129 a(2)2899 3060 y Fq(:)71 3335 y Fv(Dalje,)364 3295 y Fm(x)408 3264 y Fd(00)450 3295 y Fm(t)478 3264 y Fd(0)501 3295 y Fp(\000)p Fm(x)605 3264 y Fd(0)628 3295 y Fm(t)656 3264 y Fd(00)p 364 3312 334 4 v 440 3369 a Fm(t)468 3348 y Fd(00)511 3369 y Fp(\000)p Fm(t)599 3348 y Fd(0)740 3335 y Fv(mo)m(\024)-46 b(zemo)32 b(zapisati)f(k)-5 b(ao)32 b Fq(x)1709 3298 y Fp(0)1756 3335 y Fo(\000)23 b Fq(t)1891 3298 y Fp(0)1926 3295 y Fm(x)1970 3264 y Fd(00)2012 3295 y Fp(\000)p Fm(x)2116 3264 y Fd(0)p 1926 3312 213 4 v 1942 3369 a Fm(t)1970 3348 y Fd(00)2012 3369 y Fp(\000)p Fm(t)2100 3348 y Fd(0)2149 3335 y Fv(,)32 b(o)s(dnosno)h(dobiv)-5 b(amo:)1119 3557 y Fq(x)1174 3520 y Fp(00)1242 3557 y Fo(\000)23 b Fq(x)1397 3520 y Fp(0)p 1119 3601 304 4 v 1139 3692 a Fq(t)1174 3664 y Fp(00)1242 3692 y Fo(\000)g Fq(t)1377 3664 y Fp(0)1460 3624 y Fv(=)1574 3557 y(2)p Fq(x)1678 3520 y Fp(0)1725 3557 y Fo(\000)g Fv(\()p Fq(y)1911 3572 y Fp(\000)1974 3557 y Fv(\()p Fq(t)2047 3520 y Fp(0)2072 3557 y Fq(;)17 b(x)2171 3520 y Fp(0)2196 3557 y Fv(\))22 b(+)g Fq(y)2402 3572 y Fn(+)2466 3557 y Fv(\()p Fq(t)2539 3520 y Fp(00)2585 3557 y Fq(;)17 b(x)2684 3520 y Fp(00)2730 3557 y Fv(\)\))p 1574 3601 1233 4 v 2135 3692 a(2)p Fq(t)2219 3664 y Fp(0)2832 3624 y Fq(;)71 3878 y Fv(iz)29 b(\024)-46 b(cega)33 b(slijedi)d (\(17\).)293 3997 y(Vidimo,)g(iz)i(\(17\))g(slijedi)f(da)h(je)h Fq(x)28 b Fv(=)g Fq(x)p Fv(\()p Fq(t)p Fv(\))33 b(in)m(tegralna)e (krivulja)g(diferencijalne)g(jednad)m(\024)-46 b(zb)s(e)1627 4170 y Fm(d)p 1613 4187 68 4 v 1613 4244 a(dt)1691 4210 y Fq(x)p Fv(\()p Fq(t)p Fv(\))28 b(=)f Fq(u)p Fv(\()p Fq(t;)17 b(x)p Fv(\()p Fq(t)p Fv(\)\))71 4423 y(u)32 b(smislu)f(T)-8 b(eorema)32 b(4)g(iz)f(prv)m(og)i(p)s(ogla)m(vlja.)41 b(Iz)32 b(T)-8 b(eorema)32 b(5)g(u)g(istom)f(p)s(ogla)m(vlju,)f(uzima)5 b(ju)m(\023)-46 b(ci)31 b(u)h(obzir)71 4541 y(\(16\),)c(zaklju)m(\024) -46 b(cujemo)27 b(da)g(je)h(sv)-5 b(ak)g(a)27 b(takv)-5 b(a)27 b(krivulja)g(jedinstv)m(eno)h(o)s(dred)p 2699 4490 30 2 v 1 w(ena)f(u)h(budu)m(\023)-46 b(cnosti)28 b(p)s(o)m(\024)-46 b(cetnom)71 4660 y(to)m(\024)g(ck)m(om.)86 b(Unatrag,)50 b(p)s(o)m(\024)-46 b(cev)-5 b(\024)-44 b(si)47 b(iz)f(\()p Fq(t;)17 b(x)p Fv(\),)51 b(vidimo)45 b(da)h(su)i(ob)5 b(je)47 b(stranice)g(trokuta)g Fq(T)3334 4675 y Fm(t;x)3478 4660 y Fv(in)m(tegralne)71 4779 y(krivulje,)32 b(pa)g(nema)h(go)m(v)m(ora)f(o)g(jedinstv)m(enosti.)293 4898 y(Ak)m(o)27 b(je)f Fq(u)645 4913 y Fn(0)715 4898 y Fo(2)i Fv(C)879 4861 y Fm(k)926 4898 y Fv(,)g(za)e(neki)h Fq(k)j Fh(>)f Fv(1,)e(i)e(ak)m(o)i Fq(F)14 b Fv(\()p Fq(t;)j(x;)g Fo(\001)p Fv(\))26 b(ima)e(jedinstv)m(en)m(u)k(to)m(\024) -46 b(cku)27 b(minim)m(uma)c Fq(y)t Fv(\()p Fq(t;)17 b(x)p Fv(\))71 5016 y(u)34 b(k)m(o)5 b(jo)g(j)34 b(je)g Fq(@)567 5031 y Fm(y)s(y)652 5016 y Fq(F)43 b(>)30 b Fv(0,)j(tada)g(je)h(to)f(to)m(\024)-46 b(cno)34 b(na)f(nek)m(o)5 b(j)35 b(ok)m(olini)d(to)m(\024)-46 b(ck)m(e)34 b(\()p Fq(t;)17 b(x)p Fv(\))34 b(i)f Fq(u)c Fo(2)g Fv(C)3216 4980 y Fm(k)3297 5016 y Fv(na)k(to)5 b(j)34 b(ok)m(olini.)71 5135 y(T)-8 b(o)m(\024)-46 b(cnije,)32 b(p)s(o)h(teorem)m(u)f(o)h (implicitno)28 b(zadanim)j(funk)m(cijama,)h(jednad)m(\024)-46 b(zba)1630 5348 y Fq(@)1681 5363 y Fm(y)1726 5348 y Fq(F)14 b Fv(\()p Fq(t)1876 5307 y Fp(0)1901 5348 y Fq(;)j(x)2000 5307 y Fp(0)2025 5348 y Fq(;)g(y)t Fv(\))27 b(=)g(0)71 5561 y(ima)k(jedinstv)m(eno)i(rje)-5 b(\024)-44 b(senje)34 b Fq(y)t Fv(\()p Fq(t)1251 5525 y Fp(0)1275 5561 y Fq(;)17 b(x)1374 5525 y Fp(0)1400 5561 y Fv(\))32 b(u)h(blizini)c Fq(y)t Fv(,)j(k)m(o)5 b(je)34 b(je)e(C)2344 5525 y Fm(k)2424 5561 y Fv(funk)m(cija)g(\()p Fq(t)2865 5525 y Fp(0)2890 5561 y Fq(;)17 b(x)2989 5525 y Fp(0)3015 5561 y Fv(\).)293 5680 y(Primijetimo)40 b(da,)47 b(ak)m(o)d(je)g Fq(u)1385 5695 y Fn(0)1471 5680 y Fv(neprekidna)g(funk)m(cija,)j(tada)2637 5634 y Fm(y)s Fp(\000)p Fm(x)p 2637 5657 144 4 v 2695 5714 a(t)2820 5680 y Fv(+)30 b Fq(u)2982 5695 y Fn(0)3024 5680 y Fv(\()p Fq(y)t Fv(\))46 b(=)h(0)d(k)-5 b(ad)44 b(je)g Fq(y)50 b Fv(=)71 5810 y Fq(y)119 5825 y Fp(\006)182 5810 y Fv(\()p Fq(t;)17 b(x)p Fv(\).)55 b(Dalje,)36 b(ak)m(o)h(je)f Fq(u)1110 5825 y Fn(0)1186 5810 y Fo(2)e Fv(C)1356 5774 y Fn(1)1399 5810 y Fv(\()p Fw(R)p Fv(\),)j(tada)f(vidimo)e(da)i Fq(u)2370 5774 y Fp(0)2370 5838 y Fn(0)2437 5810 y Fv(+)2548 5771 y Fn(1)p 2548 5787 39 4 v 2553 5844 a Fm(t)2632 5810 y Fv(ima)f(n)m(ulto)m(\024)-46 b(cku)36 b(izmad)p 3452 5758 30 2 v(u)g Fq(y)3624 5825 y Fp(\000)3687 5810 y Fv(\()p Fq(t;)17 b(x)p Fv(\))3770 6010 y Ft(25)p eop %%Page: 26 31 26 30 bop -165 -116 a Fr(Nelinearni)27 b(zak)n(oni)f(sa)n(\024)-39 b(cuv)-5 b(anja)-165 83 y Fv(i)38 b Fq(y)-51 98 y Fn(+)12 83 y Fv(\()p Fq(t;)17 b(x)p Fv(\))39 b(k)-5 b(ad)39 b(su)h(oni)e(razli) m(\024)-46 b(citi.)59 b(Ak)m(o)40 b(je)f(p)s(o)m(\024)-46 b(cetni)39 b(uvjet)h Fq(u)2144 98 y Fn(0)2225 83 y Fv(o)s(dabran)e(tak) m(o)h Fq(u)2892 47 y Fp(0)2892 111 y Fn(0)2973 83 y Fv(ne)h(p)s(oprima) d(istu)-165 202 y(vrijednost)42 b(razli)m(\024)-46 b(citu)40 b(o)s(d)i(n)m(ule)g(vi)-5 b(\024)-44 b(se)42 b(o)s(d)g(k)m(ona)m(\024) -46 b(cano)43 b(mnogo)e(puta,)j(slijedi)c(da)j(za)f(\014ksan)h Fq(t)f Fv(p)s(osto)5 b(ji)-165 320 y(samo)33 b(k)m(ona)m(\024)-46 b(cno)34 b(mnogo)f(disk)m(on)m(tin)m(uiteta.)47 b(Dakle,)33 b Fq(u)h Fv(je)f(tada)h(u)g(C)2375 284 y Fm(k)2455 320 y Fv(v)-5 b(an)34 b(nek)m(og)h(skupa)g(krivulja)d(na)-165 439 y(k)m(o)5 b(jima)31 b(zado)m(v)m(olja)m(v)-5 b(a)32 b(Rankine-Hugonioto)m(v)g(uvjet.)1394 636 y Fw(Uvjet)37 b(en)m(tropije)57 794 y Fv(Mo)m(\024)-46 b(zemo)43 b(p)s(osta)m(viti)e (uvjet)j(na)f(slab)s(o)e(rje)-5 b(\024)-44 b(senje)45 b(k)m(o)5 b(ji)42 b(je)h(ekviv)-5 b(alen)m(tan)43 b(\(16\),)i(ali)c(je) i(primjen-)-165 913 y(jiviji)31 b(u)i(op)m(\023)-46 b(cenitijim)31 b(situacijama.)44 b(Da)33 b(bismo)g(to)g(p)s(ostigli,)e(p)s(omno)m (\024)-46 b(zimo)31 b(visk)m(ozn)m(u)k(aproksimaciju)-165 1032 y(Burgerso)m(v)m(e)f(jednad)m(\024)-46 b(zb)s(e)1197 1150 y Fq(@)1248 1165 y Fm(t)1280 1150 y Fq(u)1336 1165 y Fm(\026)1408 1150 y Fv(+)22 b Fq(u)1562 1165 y Fm(\026)1612 1150 y Fq(@)1663 1165 y Fm(x)1711 1150 y Fq(u)1767 1165 y Fm(\026)1845 1150 y Fv(=)27 b Fq(\026@)2058 1165 y Fm(xx)2150 1150 y Fq(u)2206 1165 y Fm(\026)2272 1150 y Fq(;)-165 1306 y Fv(s)33 b(\010)-24 1270 y Fp(0)1 1306 y Fv(\()p Fq(u)95 1321 y Fm(\026)145 1306 y Fv(\),)g(gdje)f(je)h(\010) 28 b Fo(2)g Fv(C)819 1270 y Fn(2)862 1306 y Fv(,)33 b(te)g(rezultat)f (zapi)-5 b(\024)-44 b(simo)30 b(u)j(obliku)625 1485 y Fq(@)676 1500 y Fm(t)709 1485 y Fv(\010\()p Fq(u)873 1500 y Fm(\026)923 1485 y Fv(\))22 b(+)g Fq(@)1132 1500 y Fm(x)1180 1485 y Fq(Y)f Fv(\()p Fq(u)1352 1500 y Fm(\026)1402 1485 y Fv(\))28 b(=)f Fq(\026@)1681 1500 y Fm(xx)1773 1485 y Fv(\010\()p Fq(u)1937 1500 y Fm(\026)1987 1485 y Fv(\))22 b Fo(\000)h Fq(\026)p Fv(\010)2276 1444 y Fp(00)2322 1485 y Fv(\()p Fq(u)2416 1500 y Fm(\026)2466 1485 y Fv(\)\()p Fq(@)2593 1500 y Fm(x)2641 1485 y Fq(u)2697 1500 y Fm(\026)2747 1485 y Fv(\))2785 1444 y Fn(2)2844 1485 y Fq(:)-165 1655 y Fv(Pritom)31 b(je)h(funk)m(cija)h Fq(Y)54 b Fv(zadana)32 b(tak)m(o)h(da)f(vrijedi)1416 1826 y Fq(Y)1495 1785 y Fp(0)1520 1826 y Fv(\()p Fq(s)p Fv(\))27 b(=)h Fq(s)p Fv(\010)1889 1785 y Fp(0)1914 1826 y Fv(\()p Fq(s)p Fv(\))17 b Fq(:)-165 1996 y Fv(Ak)m(o)33 b(\010)g(o)s(dab)s(eremo)e(k)m(on)m(v)m(eksnom,)k(o)s(dnosno)e(zah)m (tijev)-5 b(amo)31 b(da)i(je)f(\010)2447 1960 y Fp(00)2521 1996 y Fh(>)d Fv(0,)j(slijedi)f(nejednak)m(ost)1025 2167 y Fq(@)1076 2182 y Fm(t)1109 2167 y Fv(\010\()p Fq(u)1273 2182 y Fm(\026)1323 2167 y Fv(\))22 b(+)g Fq(@)1532 2182 y Fm(x)1580 2167 y Fq(Y)g Fv(\()p Fq(u)1753 2182 y Fm(\026)1803 2167 y Fv(\))27 b Fh(6)h Fq(\026@)2083 2182 y Fm(xx)2175 2167 y Fv(\010\()p Fq(u)2339 2182 y Fm(\026)2389 2167 y Fv(\))17 b Fq(:)-165 2337 y Fv(T)-8 b(akvu)36 b(k)m(on)m(v)m(eksn)m (u)k(funk)m(ciju)c(\010)g(obi)m(\024)-46 b(cno)35 b(zo)m(v)m(emo)h Fz(entropija)p Fv(,)g(a)g Fq(Y)56 b Fz(tok)m(om)36 b(entropije)p Fv(.)54 b(Budu)m(\023)-46 b(ci)35 b(da)h(su)-165 2456 y Fq(u)-109 2471 y Fm(\026)-27 2456 y Fv(lok)-5 b(alno)30 b(ograni)m(\024)-46 b(cene)32 b(u)g(limesu)f(k)-5 b(ad)32 b Fq(\026)c Fo(!)f Fv(0,)32 b(to)g(imamo)e(da)i(je)g(u)h(smislu)e (teorije)g(distribucija)f(na)-165 2575 y Fw(R)-81 2539 y Fn(+)4 2575 y Fo(\002)23 b Fw(R)32 b Fv(ispunjeno)-165 2745 y(\(18\))1269 b Fq(@)1329 2760 y Fm(t)1362 2745 y Fv(\010\()p Fq(u)p Fv(\))22 b(+)g Fq(@)1735 2760 y Fm(x)1783 2745 y Fq(Y)f Fv(\()p Fq(u)p Fv(\))27 b Fh(6)h Fv(0)17 b Fq(:)-165 2916 y Fv(Uvjet)44 b(\(18\))g(p)s(onek)-5 b(ad)44 b(zo)m(v)m(emo)h Fz(uvjet)f(entropije)p Fv(.)79 b(Eksplicitno)43 b(to)h(zna)m(\024)-46 b(ci)44 b(da)g(za)g(sv)-5 b(aku)45 b(p)s(ozitivn)m(u)-165 3035 y(funk)m(ciju)33 b Fq(')27 b Fo(2)h Fv(C)464 2998 y Fp(1)464 3059 y Fm(c)546 3035 y Fv(\()p Fw(R)668 2998 y Fn(2)732 3035 y Fo(\002)23 b Fw(R)p Fv(\))32 b(vrijedi)-165 3255 y(\(19\))1006 3120 y Fl(Z)1122 3255 y Fv(\(\010\()p Fq(u)p Fv(\))p Fq(@)1413 3270 y Fm(t)1445 3255 y Fq(')22 b Fv(+)g Fq(Y)f Fv(\()p Fq(u)p Fv(\))p Fq(@)1890 3270 y Fm(x)1938 3255 y Fq(')p Fv(\))33 b Fq(dtdx)28 b Fh(>)g Fv(0)17 b Fq(:)-165 3485 y Fv(T)-8 b(o)32 b(je)h(ispunjeno)g(za)f(sv)-5 b(aku)34 b(k)m(on)m(v)m(eksn)m(u)i(C)1456 3449 y Fn(2)1532 3485 y Fv(funk)m(ciju.)44 b(Uzimamo)30 b(li)h(za)h(nek)m(e)i Fq(k)d Fo(2)d Fw(R)k Fv(i)g Fq(")27 b(>)h Fv(0)1240 3694 y(\010\()p Fq(s)p Fv(\))g(=)1564 3573 y Fl(q)p 1663 3573 549 4 v 1663 3694 a Fv(\()p Fq(s)22 b Fo(\000)h Fq(k)s Fv(\))1961 3665 y Fn(2)2026 3694 y Fv(+)f Fq(")2170 3665 y Fn(2)2229 3694 y Fq(;)-165 3877 y Fv(u)32 b(limesu)f(k)-5 b(ad)33 b Fq(")27 b Fo(!)h Fv(0)k(vidimo)e(da)j(mo)m(\024)-46 b(zemo)31 b(uzeti)1426 4048 y(\010\()p Fq(s)p Fv(\))d(=)f Fo(j)p Fq(s)22 b Fo(\000)h Fq(k)s Fo(j)17 b Fq(;)-165 4219 y Fv(i)31 b(tada)h(imamo)e(da)j(je)1120 4336 y Fq(Y)22 b Fv(\()p Fq(s)p Fv(\))27 b(=)1462 4296 y Fn(1)p 1462 4313 39 4 v 1462 4370 a(2)1510 4336 y Fv(\()p Fq(s)1594 4294 y Fn(2)1659 4336 y Fo(\000)22 b Fq(k)1812 4294 y Fn(2)1855 4336 y Fv(\)sign)16 b(\()p Fq(s)22 b Fo(\000)h Fq(k)s Fv(\))1348 4490 y(=)1462 4451 y Fn(1)p 1462 4468 V 1462 4525 a(2)1510 4490 y Fo(j)p Fq(s)f Fo(\000)g Fq(k)s Fo(j)p Fv(\()p Fq(s)g Fv(+)g Fq(k)s Fv(\))17 b Fq(:)-165 4645 y Fv(Stoga)32 b(\(19\))g(za)g(sv)-5 b(aki)33 b Fq(k)d Fo(2)e Fw(R)33 b Fv(da)5 b(je)-165 4865 y(\(20\))450 4730 y Fl(Z)566 4865 y Fo(j)p Fq(u)p Fv(\()p Fq(t;)17 b(x)p Fv(\))k Fo(\000)i Fq(k)s Fo(j)1080 4785 y Fl(\000)1125 4865 y Fq(@)1176 4880 y Fm(t)1208 4865 y Fq(')p Fv(\()p Fq(t;)17 b(x)p Fv(\))23 b(+)1613 4826 y Fn(1)p 1613 4842 V 1613 4900 a(2)1661 4865 y Fv(\()p Fq(u)p Fv(\()p Fq(t;)17 b(x)p Fv(\))22 b(+)g Fq(k)s Fv(\))p Fq(@)2228 4880 y Fm(x)2276 4865 y Fq(')p Fv(\()p Fq(t;)17 b(x)p Fv(\))2550 4785 y Fl(\001)2629 4865 y Fq(dtdx)28 b Fh(>)g Fv(0)17 b Fq(:)-165 5086 y Fv(Obrn)m(uto,)46 b(nejednak)m(ost)g(\(19\))c (slijedi)g(iz)h(\(20\),)i(jer)f(se)g(sv)-5 b(ak)g(a)44 b(k)m(on)m(v)m(eksna)i(funk)m(cija)e(na)f(k)m(ompaktu)-165 5205 y(mo)m(\024)-46 b(ze)31 b(prik)-5 b(azati)30 b(k)-5 b(ao)31 b(limes)f(sup)s(erp)s(ozicija)g(funk)m(cija)i(oblik)-5 b(a)29 b Fo(j)p Fq(x)20 b Fo(\000)h Fq(k)s Fo(j)31 b Fv(i)g(a\014ne)h(funk)m(cije.)43 b(Primijetimo)-165 5323 y(da)30 b(ak)m(o)h(uzmemo)e Fq(k)34 b Fv(do)m(v)m(oljno)c(v)m(elik)g(p) s(o)g(apsolutno)5 b(j)30 b(vrijednosti)g(prv)m(o)h(p)s(ozitiv)-5 b(an,)29 b(pa)h(negativ)-5 b(an,)30 b(iz)-165 5442 y(\(20\))i(slijedi) 1051 5611 y Fo(\007)1145 5475 y Fl(Z)1261 5530 y(\000)1307 5611 y Fq(u@)1414 5626 y Fm(t)1446 5611 y Fq(')22 b Fv(+)1640 5572 y Fn(1)p 1640 5588 V 1640 5645 a(2)1688 5611 y Fq(u)1744 5570 y Fn(2)1786 5611 y Fq(@)1837 5626 y Fm(x)1885 5611 y Fq(')1949 5530 y Fl(\001)2028 5611 y Fq(dtdx)28 b Fh(>)g Fv(0)17 b Fq(;)-171 5811 y Fv(\024)-44 b(sto)33 b(implicira)c(da)j(je)h Fq(u)f Fv(slab)s(o)g(rje)-5 b(\024)-44 b(senje)33 b(Burgerso)m(v)m(e)i (jednad)m(\024)-46 b(zb)s(e.)-165 6010 y Ft(26)p eop %%Page: 27 32 27 31 bop 3102 -116 a Fr(Burgerso)n(v)-5 b(a)25 b(jednad)n(\024)-39 b(zba)71 83 y Fw(T)-9 b(eorem)31 b(4.)98 b Fi(Nek)-5 b(a)28 b(su)g(funk)m(cije)h Fq(u)e Fi(i)g Fq(v)k Fi(apsolutno)d(omed)p 2223 31 30 2 v(ene)g(vrijedno)-5 b(\024)-44 b(s)m(\023)e(cu)29 b Fq(M)39 b Fi(na)27 b(pruzi)h Fo(h)p Fv(0)p Fq(;)17 b(T)d Fo(i)e(\002)g Fw(R)p Fi(,)71 202 y(i)31 b(nek)-5 b(a)33 b(za)g(sv)-5 b(aki)32 b Fq(k)f Fo(2)d Fw(R)k Fi(zado)m(v)m(olja) m(v)-5 b(a)5 b(ju)33 b(\(20\).)43 b(T)-8 b(ada)33 b(slijedi)d(da)j(je) 1409 362 y Fl(Z)1509 389 y Fm(b)p Fp(\000)p Fm(M)8 b(t)1465 588 y(a)p Fn(+)p Fm(M)g(t)1732 498 y Fo(j)p Fq(u)p Fv(\()p Fq(t;)17 b(x)p Fv(\))22 b Fo(\000)h Fq(v)t Fv(\()p Fq(t;)17 b(x)p Fv(\))p Fo(j)g Fq(dx)71 774 y Fi(opada)5 b(ju)m(\023)-46 b(ca)33 b(funk)m(cija)f(p)s(o)g Fq(t)p Fi(,)h(dok)g(je)g Fq(b)22 b Fo(\000)h Fq(a)28 b(>)g Fv(2)p Fq(M)10 b(t)33 b Fi(i)e Fq(t)d(<)g(T)14 b Fi(.)43 b(P)m(osebno,)1465 909 y Fl(Z)1564 936 y Fp(1)1520 1135 y(\0001)1677 1045 y Fo(j)p Fq(u)p Fv(\()p Fq(t;)17 b(x)p Fv(\))22 b Fo(\000)g Fq(v)t Fv(\()p Fq(t;)17 b(x)p Fv(\))p Fo(j)g Fq(dx)71 1321 y Fi(je)33 b(opada)5 b(ju)m(\023)-46 b(ca)32 b(funk)m(cija)h Fq(t)p Fi(.)293 1440 y(Ak)m(o)h(su)h Fq(u)e Fi(i)g Fq(v)38 b Fi(rje)-5 b(\024)-44 b(senja)34 b(Burgerso)m(v)m(e)i(jednad)m(\024) -46 b(zb)s(e)35 b(dana)f(T)-8 b(eoremom)33 b(3,)h(uz)g(p)s(o)m(\024)-46 b(cetne)35 b(uvjete)g Fq(u)3856 1455 y Fn(0)71 1559 y Fi(i)c Fq(v)177 1574 y Fn(0)220 1559 y Fi(,)i(tada)f(u)h(limesu)e(k)-5 b(ad)32 b Fq(t)c Fo(!)f Fv(0)33 b Fi(imamo)1511 1714 y Fl(Z)1610 1740 y Fm(b)1566 1940 y(a)1664 1850 y Fo(j)p Fq(u)1748 1865 y Fn(0)1790 1850 y Fv(\()p Fq(x)p Fv(\))22 b Fo(\000)h Fq(v)2090 1865 y Fn(0)2132 1850 y Fv(\()p Fq(x)p Fv(\))p Fo(j)17 b Fq(dx)g(:)71 2151 y Fz(Dem.)113 b Fv(Nek)-5 b(a)40 b(je)f Fq(')h Fo(2)f Fv(C)1043 2115 y Fp(1)1043 2175 y Fm(c)1125 2151 y Fv(\()p Fo(h)p Fv(0)p Fq(;)17 b(T)d Fo(i)20 b(\002)j Fw(R)p Fv(\))39 b(nenegativna)g(funk)m (cija.)64 b(Primijenimo)36 b(li)h(\(20\))i(na)g Fq(u)p Fv(,)i(uz)71 2270 y Fq(k)31 b Fv(=)c Fq(v)t Fv(\()p Fq(s;)17 b(y)t Fv(\))31 b(dobiv)-5 b(amo)473 2404 y Fl(Z)589 2540 y Fo(j)p Fq(u)p Fv(\()p Fq(t;)17 b(x)p Fv(\))k Fo(\000)i Fq(v)t Fv(\()p Fq(s;)17 b(y)t Fv(\))p Fo(j)1318 2459 y Fl(\000)1361 2540 y Fq(@)1412 2555 y Fm(t)1445 2540 y Fq(')p Fv(\()p Fq(t;)g(x)p Fv(\))22 b(+)1849 2501 y Fn(1)p 1849 2517 39 4 v 1849 2574 a(2)1897 2540 y Fv(\()p Fq(u)p Fv(\()p Fq(t;)17 b(x)p Fv(\))22 b(+)g Fq(v)t Fv(\()p Fq(s;)17 b(y)t Fv(\)\))p Fq(@)2679 2555 y Fm(x)2726 2540 y Fq(')p Fv(\()p Fq(t;)g(x)p Fv(\))3000 2459 y Fl(\001)3079 2540 y Fq(dtdx)27 b Fh(>)i Fv(0)17 b Fq(:)71 2811 y Fv(Uzmemo)31 b(li)g(da)h(je)h Fq(')27 b Fo(2)h Fv(C)1057 2774 y Fp(1)1057 2835 y Fm(c)1139 2811 y Fv(\()p Fo(h)p Fv(0)p Fq(;)17 b(T)d Fo(i)20 b(\002)j Fw(R)f Fo(\002)g(h)p Fv(0)p Fq(;)17 b(T)d Fo(i)21 b(\002)i Fw(R)p Fv(\))32 b(mo)m(\024)-46 b(zemo)31 b(u)h(gornjem)g(uvrstiti)f Fq(')p Fv(\()p Fo(\001)p Fq(;)17 b Fo(\001)p Fq(;)g(s;)g(y)t Fv(\),)71 2929 y(a)24 b(p)s(otom)e(proin)m(tegrirati)g(i)h(p)s(o)h(\()p Fq(s;)17 b(y)t Fv(\).)39 b(U)m(\024)-46 b(cinimo)22 b(li)g(isto)i(s)g Fq(u)g Fv(i)f Fq(v)28 b Fv(med)p 2544 2877 30 2 v(usobno)d (zamijenjenim)c(dobiv)-5 b(amo)259 3064 y Fl(Z)376 3200 y Fo(j)p Fq(u)p Fv(\()p Fq(t;)17 b(x)p Fv(\))k Fo(\000)i Fq(v)t Fv(\()p Fq(s;)17 b(y)t Fv(\))p Fo(j)1105 3119 y Fl(\000)1148 3200 y Fq(@)1199 3215 y Fm(t)1232 3200 y Fq(')22 b Fv(+)g Fq(@)1467 3215 y Fm(s)1507 3200 y Fq(')g Fv(+)1701 3161 y Fn(1)p 1701 3177 39 4 v 1701 3234 a(2)1749 3200 y Fv(\()p Fq(u)p Fv(\()p Fq(t;)17 b(x)p Fv(\))22 b(+)g Fq(v)t Fv(\()p Fq(s;)17 b(y)t Fv(\)\)\()p Fq(@)2569 3215 y Fm(x)2616 3200 y Fq(')22 b Fv(+)g Fq(@)2851 3215 y Fm(y)2896 3200 y Fq(')p Fv(\))2998 3119 y Fl(\001)3077 3200 y Fq(dtdx)17 b(dsdy)29 b Fh(>)g Fv(0)17 b Fq(:)71 3470 y Fv(Zamijenimo)27 b Fq(t)k Fv(s)g(\()p Fq(t)19 b Fv(+)g Fq(s)p Fv(\))31 b(i)f Fq(x)h Fv(s)g(\()p Fq(x)19 b Fv(+)g Fq(y)t Fv(\),)30 b(vidimo)f(da)i(uz)h(oznaku)f Fq( )t Fv(\()p Fq(t;)17 b(x;)g(s;)g(y)t Fv(\))27 b(=)g Fq(')p Fv(\()p Fq(t)19 b Fv(+)g Fq(s;)e(x)h Fv(+)h Fq(y)t(;)e(s;)g(y)t Fv(\))71 3589 y(imamo)232 3699 y Fl(Z)348 3835 y Fo(j)p Fq(u)p Fv(\()p Fq(t)22 b Fv(+)g Fq(s;)17 b(x)22 b Fv(+)g Fq(y)t Fv(\))f Fo(\000)i Fq(v)t Fv(\()p Fq(s;)17 b(y)t Fv(\))p Fo(j)1415 3754 y Fl(\000)1458 3835 y Fq(@)1509 3850 y Fm(s)1550 3835 y Fq( )26 b Fv(+)1747 3796 y Fn(1)p 1747 3812 V 1747 3869 a(2)1795 3835 y Fv(\()p Fq(u)p Fv(\()p Fq(t)c Fv(+)g Fq(s;)17 b(x)22 b Fv(+)g Fq(y)t Fv(\))f(+)i Fq(v)t Fv(\()p Fq(s;)17 b(y)t Fv(\)\))p Fq(@)2915 3850 y Fm(y)2958 3835 y Fq( )3025 3754 y Fl(\001)3104 3835 y Fq(dtdx)g(dsdy)30 b Fh(>)e Fv(0)17 b Fq(;)71 4115 y Fv(nara)m(vno,)38 b(ak)m(o)f(za)g Fq( )h Fo(2)d Fv(C)1061 4079 y Fp(1)1061 4140 y Fm(c)1143 4115 y Fv(\()p Fw(R)1265 4079 y Fn(4)1307 4115 y Fv(\))h(imamo)e(da)j(je)f(supp)18 b Fq( )41 b Fv(sadr)m(\024)-46 b(zan)37 b(na)f(skupu)j(gdje)d(je)h(0)d Fq(<)h(t)g(<)f(T)51 b Fv(i)71 4234 y(0)35 b Fq(<)f(s)25 b Fv(+)g Fq(t)36 b(<)e(T)14 b Fv(.)56 b(Odab)s(erimo)35 b(funk)m(ciju)j Fq( )i Fv(takvu)e(da)f(je)g Fq( )t Fv(\()p Fq(t;)17 b(x;)g(s;)g(y)t Fv(\))34 b(=)g Fq( )2966 4249 y Fn(1)3009 4234 y Fv(\()p Fq(s;)17 b(y)t Fv(\))p Fq( )3290 4249 y Fn(2)3332 4234 y Fv(\()p Fq(t;)g(x)p Fv(\))37 b(za)g(nek)m(e)71 4352 y(funk)m(cije)c Fq( )497 4367 y Fn(1)540 4352 y Fq(;)17 b( )647 4367 y Fn(2)717 4352 y Fo(2)28 b Fv(C)881 4316 y Fp(1)881 4377 y Fm(c)963 4352 y Fv(\()p Fo(h)p Fv(0)p Fq(;)17 b(T)d Fo(i)21 b(\002)h Fw(R)p Fv(\).)293 4472 y(Dalje,)31 b(primijetimo)d(da)33 b(je)g(za)f(proizv)m(oljn)m(u)g(funk)m(ciju)h Fq( )2411 4487 y Fn(1)2482 4472 y Fo(2)28 b Fv(C)2646 4436 y Fp(1)2646 4497 y Fm(c)2727 4472 y Fv(\()p Fo(h)p Fv(0)p Fq(;)17 b(T)d Fo(i)21 b(\002)i Fw(R)p Fv(\))410 4607 y Fl(Z)526 4742 y Fo(j)p Fq(u)p Fv(\()p Fq(t)f Fv(+)g Fq(s;)17 b(x)22 b Fv(+)g Fq(y)t Fv(\))f Fo(\000)i Fq(v)t Fv(\()p Fq(s;)17 b(y)t Fv(\))p Fo(j)1593 4662 y Fl(\000)1636 4742 y Fq(@)1687 4757 y Fm(s)1728 4742 y Fq( )1791 4757 y Fn(1)1856 4742 y Fv(+)1964 4703 y Fn(1)p 1964 4720 V 1964 4777 a(2)2012 4742 y Fv(\()p Fq(u)p Fv(\()p Fq(t)22 b Fv(+)g Fq(s;)17 b(x)22 b Fv(+)g Fq(y)t Fv(\))f(+)h Fq(v)t Fv(\()p Fq(s;)17 b(y)t Fv(\)\))p Fq(@)3131 4757 y Fm(y)3175 4742 y Fq( )3238 4757 y Fn(1)3281 4662 y Fl(\001)3360 4742 y Fq(dsdy)71 5013 y Fv(neprekidna)35 b(funk)m(cija)g(p)s(o)f(\()p Fq(t;)17 b(x)p Fv(\),)35 b(jer)g(su)g Fo(j)p Fq(u)23 b Fo(\000)h Fq(v)t Fo(j)34 b Fv(i)f Fq(u)p Fo(j)p Fq(u)23 b Fo(\000)h Fq(v)t Fo(j)34 b Fv(Lipsc)m(hitzo)m(v)m(e)h(funk)m(cije)g (u)g(v)-5 b(arijablama)71 5132 y(\()p Fq(u;)17 b(v)t Fv(\))46 b(na)i(k)m(ompaktu)g([)p Fo(\000)p Fq(M)5 b(;)17 b(M)10 b Fv(])1346 5096 y Fn(2)1437 5132 y Fv(dok)48 b(je)g(\()p Fq(s;)17 b(y)t Fv(\))52 b Fo(7!)h Fq(u)p Fv(\()p Fo(\001)31 b Fv(+)h Fq(s;)17 b Fo(\001)32 b Fv(+)g Fq(y)t Fv(\))47 b(neprekidno)h(preslik)-5 b(a)m(v)g(anje)71 5250 y Fw(R)155 5214 y Fn(2)228 5250 y Fo(7!)30 b Fv(L)419 5214 y Fn(1)419 5282 y(lo)r(c)519 5250 y Fv(\()p Fw(R)641 5214 y Fn(2)684 5250 y Fv(\))o(.)50 b(Iz)34 b(zadnje)i(nejednak)m(osti) f(mo)m(\024)-46 b(zemo)33 b(zaklju)m(\024)-46 b(citi)33 b(i)g(da)h(je)h(ta)f(funk)m(cija)h(nenegativna)71 5369 y(za)23 b(sv)-5 b(aki)23 b(\()p Fq(t;)17 b(x)p Fv(\),)26 b(p)s(osebno)d(uzmimo)e(\()p Fq(t;)c(x)p Fv(\))28 b(=)g(\(0)p Fq(;)17 b Fv(0\).)39 b(Zaklju)m(\024)-46 b(cujemo)22 b(da)h(za)h(sv)-5 b(aku)24 b(p)s(ozitivn)m(u)e(funk)m(ciju)71 5488 y Fq( )134 5503 y Fn(1)205 5488 y Fo(2)28 b Fv(C)369 5452 y Fp(1)369 5513 y Fm(c)450 5488 y Fv(\()p Fo(h)p Fv(0)p Fq(;)17 b(T)d Fo(i)21 b(\002)h Fw(R)p Fv(\))32 b(vrijedi)298 5623 y Fl(Z)414 5758 y Fo(j)p Fq(u)p Fv(\()p Fq(t)21 b Fv(+)h Fq(s;)17 b(x)23 b Fv(+)f Fq(y)t Fv(\))f Fo(\000)h Fq(v)t Fv(\()p Fq(s;)17 b(y)t Fv(\))p Fo(j)1480 5678 y Fl(\000)1524 5758 y Fq(@)1575 5773 y Fm(s)1615 5758 y Fq( )1678 5773 y Fn(1)1743 5758 y Fv(+)1851 5719 y Fn(1)p 1851 5736 V 1851 5793 a(2)1900 5758 y Fv(\()p Fq(u)p Fv(\()p Fq(t)k Fv(+)h Fq(s;)17 b(x)23 b Fv(+)f Fq(y)t Fv(\))f(+)h Fq(v)t Fv(\()p Fq(s;)17 b(y)t Fv(\)\))p Fq(@)3019 5773 y Fm(y)3062 5758 y Fq( )3125 5773 y Fn(1)3168 5678 y Fl(\001)3247 5758 y Fq(dsdy)30 b Fh(>)e Fv(0)17 b Fq(:)3770 6010 y Ft(27)p eop %%Page: 28 33 28 32 bop -165 -116 a Fr(Nelinearni)27 b(zak)n(oni)f(sa)n(\024)-39 b(cuv)-5 b(anja)57 83 y Fv(Sad)33 b(o)s(dab)s(erimo)d(funk)m(ciju)j Fq( )1166 98 y Fn(1)1242 83 y Fv(tak)m(o)g(da)g(je)g Fq( )1764 98 y Fn(1)1807 83 y Fv(\()p Fq(s;)17 b(y)t Fv(\))26 b(=)i Fq(')p Fv(\()p Fq(s)p Fv(\))p Fq(c)p Fv(\()p Fq(s;)17 b(y)t Fv(\),)32 b(za)h(nek)m(e)h Fq(')28 b Fo(2)g Fv(C)3263 47 y Fp(1)3263 108 y Fm(c)3345 83 y Fv(\()p Fo(h)p Fv(0)p Fq(;)17 b(T)d Fo(i)p Fv(\))-165 202 y(i)31 b Fq(c)d Fo(2)g Fv(C)128 166 y Fp(1)128 226 y Fm(c)209 202 y Fv(\([0)p Fq(;)17 b(T)d Fv(])22 b Fo(\002)h Fw(R)p Fv(\))32 b(p)s(ozitivne)f(funk)m(cije.)44 b(Ak)m(o)34 b(na)e Fq(c)g Fv(p)s(osta)m(vimo)g(do)s(datni)f(zah)m(tjev)j(da)e(je) -165 433 y(\(21\))1350 b Fq(@)1410 448 y Fm(s)1451 433 y Fq(c)22 b Fv(+)g Fq(M)10 b Fo(j)p Fq(@)1796 448 y Fm(y)1842 433 y Fq(c)p Fo(j)27 b Fh(6)h Fv(0)17 b Fq(;)-165 664 y Fv(o)s(dakle)32 b(dalje)f(slijedi)g(da)h(je)877 895 y Fq(@)928 910 y Fm(s)969 895 y Fq(c)22 b Fv(+)1141 856 y Fn(1)p 1141 872 39 4 v 1141 930 a(2)1189 895 y Fv(\()p Fq(u)p Fv(\()p Fq(t)g Fv(+)g Fq(s;)17 b(x)22 b Fv(+)g Fq(y)t Fv(\))f(+)h Fq(v)t Fv(\()p Fq(s;)17 b(y)t Fv(\)\))p Fq(@)2308 910 y Fm(y)2352 895 y Fq(c)27 b Fh(6)h Fv(0)17 b Fq(;)-165 1126 y Fv(o)s(dnosno)32 b(za)h(p)s(ozitivn)m(u)f Fq(')g Fv(imamo)424 1267 y Fl(Z)541 1402 y Fo(j)p Fq(u)p Fv(\()p Fq(s;)17 b(y)t Fv(\))j Fo(\000)i Fq(v)t Fv(\()p Fq(s;)17 b(y)t Fv(\))p Fo(j)p Fq(')1340 1321 y Fl(\000)1383 1402 y Fq(@)1434 1417 y Fm(s)1475 1402 y Fq(c)22 b Fv(+)1647 1363 y Fn(1)p 1647 1379 V 1647 1436 a(2)1695 1402 y Fv(\()p Fq(u)p Fv(\()p Fq(s;)17 b(y)t Fv(\))k(+)h Fq(v)t Fv(\()p Fq(s;)17 b(y)t Fv(\)\))p Fq(@)2484 1417 y Fm(y)2527 1402 y Fq(c)2569 1321 y Fl(\001)2648 1402 y Fq(dsdy)30 b Fh(6)e Fv(0)17 b Fq(:)-165 1678 y Fv(Zaklju)m(\024)-46 b(cujemo)31 b(da)h(je)h(za)g(takv)m(e)h(funk)m(cije)f Fq(')f Fv(i)g Fq(c)g Fv(ispunjeno)877 1818 y Fl(Z)993 1954 y Fo(j)p Fq(u)p Fv(\()p Fq(s;)17 b(y)t Fv(\))j Fo(\000)j Fq(v)t Fv(\()p Fq(s;)17 b(y)t Fv(\))p Fo(j)p Fq(c)p Fv(\()p Fq(s;)g(y)t Fv(\))p Fq(')2036 1912 y Fp(0)2057 1954 y Fv(\()p Fq(s)p Fv(\))g Fq(dsdy)30 b Fh(>)e Fv(0)17 b Fq(:)-165 2235 y Fv(Namjesto)27 b Fq(c)h Fv(u)g(gornju)g(nejednak)m (ost)i(uvrstimo)d(niz)g(\()p Fq(c)1885 2252 y Fm(k)1931 2235 y Fv(\))h(k)m(o)5 b(ji)28 b(k)m(on)m(v)m(ergira)h(k)-5 b(arakteristi)m(\024)-46 b(cno)5 b(j)27 b(funk)m(ciji)-165 2354 y(skupa)310 2485 y Fo(f)p Fv(\()p Fq(s;)17 b(y)t Fv(\))p Fo(j)g Fq(a)j Fv(+)i Fq(M)10 b(s)28 b(<)g(y)i(<)e(b)23 b Fo(\000)f Fq(M)10 b(s)p Fo(g)28 b Fv(=)g Fo(f)p Fq(s;)17 b(y)t Fv(\))p Fo(j)g(j)p Fq(y)23 b Fo(\000)f Fq(y)2272 2500 y Fn(0)2315 2485 y Fo(j)f Fv(+)h Fq(M)10 b(s)29 b(<)2754 2446 y Fn(1)p 2754 2462 V 2754 2519 a(2)2803 2485 y Fv(\()p Fq(b)22 b Fo(\000)h Fq(a)p Fv(\))p Fo(g)17 b Fq(;)-165 2686 y Fv(gdje)31 b(je)h Fq(y)195 2701 y Fn(0)264 2686 y Fv(=)378 2647 y Fn(1)p 378 2663 V 378 2720 a(2)426 2686 y Fv(\()p Fq(a)20 b Fv(+)f Fq(b)p Fv(\).)44 b(T)-8 b(ak)j(a)m(v)32 b(niz)f(se)h(mo)m(\024)-46 b(ze)31 b(dobiti)f(standardnom)h(regularizacijom,)d(a)j(budu)m(\023)-46 b(ci)31 b(da)-165 2805 y(sv)-5 b(ak)g(a)35 b(opada)5 b(ju)m(\023)-46 b(ca)35 b(funk)m(cija)h(ima)30 b(\024)-46 b(zeljeno)35 b(sv)m(o)5 b(jsto)36 b(\(21\),)g(to)f(sv)m(o)5 b(jstv)m(o)34 b(\023)-46 b(ce)36 b(ispunja)m(v)-5 b(ati)34 b(i)g(svi)f(\024)-46 b(clano)m(vi)-165 2923 y(niza)27 b(\()p Fq(c)116 2941 y Fm(k)162 2923 y Fv(\).)42 b(Stoga)27 b(mo)m(\024)-46 b(zemo)27 b(zaklju)m(\024)-46 b(citi)26 b(da)h(za)h Fq(b)12 b Fo(\000)g Fq(a)30 b Fh(>)e Fv(2)p Fq(M)10 b(T)42 b Fv(i)27 b(p)s(ozitivn)m(u)g(funk)m(ciju)h Fq(')f Fo(2)h Fv(C)3263 2887 y Fp(1)3263 2948 y Fm(c)3345 2923 y Fv(\()p Fo(h)p Fv(0)p Fq(;)17 b(T)d Fo(i)p Fv(\))-165 3042 y(vrijedi)737 3121 y Fl(Z)853 3257 y Fq(')917 3216 y Fp(0)942 3257 y Fv(\()p Fq(s)p Fv(\))1081 3087 y Fl( )1159 3121 y(Z)1259 3148 y Fm(b)p Fp(\000)p Fm(M)8 b(s)1215 3347 y(a)p Fn(+)p Fm(M)g(s)1490 3257 y Fo(j)p Fq(u)p Fv(\()p Fq(s;)17 b(y)t Fv(\))j Fo(\000)j Fq(v)t Fv(\()p Fq(s;)17 b(y)t Fv(\))p Fo(j)g Fq(dy)2329 3087 y Fl(!)2437 3257 y Fq(ds)28 b Fh(>)g Fv(0)17 b Fq(;)-165 3530 y Fv(iz)29 b(\024)-46 b(cega)32 b(slijedi)f(da)i(je)f(un)m(utra)-5 b(\024)-44 b(snji)33 b(in)m(tegral)e(opada)5 b(ju)m(\023)-46 b(ca)33 b(funk)m(cija)f(p)s(o)g(v)-5 b(arijabli)30 b Fq(s)p Fv(.)57 3653 y(Nek)-5 b(a)35 b(je)h Fq(u)f Fv(Hopfo)m(v)m(o)h (rje)-5 b(\024)-44 b(senje)37 b(Cauc)m(h)m(yjev)m(e)h(zada)m(\023)-46 b(ce)37 b(za)e(Burgerso)m(vu)i(jednad)m(\024)-46 b(zbu)37 b(s)f(p)s(o)m(\024)-46 b(cetnim)-165 3772 y(uvjetom)34 b Fq(u)272 3787 y Fn(0)346 3772 y Fo(2)e Fv(L)505 3736 y Fp(1)586 3772 y Fv(\()p Fw(R)p Fv(\).)50 b(Odab)s(erimo)34 b(niz)g Fq(u)1548 3787 y Fn(0)p Fm(j)1658 3772 y Fo(2)e Fv(C)1826 3736 y Fp(1)1826 3796 y Fm(c)1907 3772 y Fv(\()p Fw(R)p Fv(\))j(tak)-5 b(a)m(v)35 b(da)g Fq(u)2561 3787 y Fn(0)p Fm(j)2670 3772 y Fo(!)d Fq(u)2858 3787 y Fn(0)2935 3772 y Fv(u)j(L)3085 3736 y Fn(1)3085 3803 y(lo)r(c)3185 3772 y Fv(\()p Fw(R)p Fv(\))o(,)h(i)e(nek)-5 b(a)-165 3890 y(su)33 b Fq(u)16 3905 y Fm(j)88 3890 y Fv(pripadna)f(Hopfo)m(v)-5 b(a)32 b(rje)-5 b(\024)-44 b(senja)33 b(Burgerso)m(v)m(e)i(jednad)m (\024)-46 b(zb)s(e.)44 b(T)-8 b(ada)545 4066 y Fl(Z)644 4093 y Fm(b)p Fp(\000)p Fm(M)8 b(t)600 4292 y(a)p Fn(+)p Fm(M)g(t)868 4202 y Fo(j)p Fq(u)952 4217 y Fm(j)991 4202 y Fv(\()p Fq(t;)17 b(x)p Fv(\))22 b Fo(\000)h Fq(u)1379 4219 y Fm(k)1425 4202 y Fv(\()p Fq(t;)17 b(x)p Fv(\))p Fo(j)g Fq(dx)27 b Fh(6)1918 4066 y Fl(Z)2018 4093 y Fm(b)1973 4292 y(a)2071 4202 y Fo(j)p Fq(u)2155 4217 y Fn(0)p Fm(j)2233 4202 y Fv(\()p Fq(x)p Fv(\))22 b Fo(\000)h Fq(u)2542 4219 y Fn(0)p Fm(k)2626 4202 y Fv(\()p Fq(x)p Fv(\))p Fo(j)17 b Fq(dx)g(;)-165 4489 y Fv(u)32 b(limesu)f(k)-5 b(ad)33 b Fq(k)e Fo(!)c(1)32 b Fv(dobiv)-5 b(amo)589 4635 y Fl(Z)689 4661 y Fm(b)p Fp(\000)p Fm(M)8 b(t)644 4861 y(a)p Fn(+)p Fm(M)g(t)912 4771 y Fo(j)p Fq(u)996 4786 y Fm(j)1035 4771 y Fv(\()p Fq(t;)17 b(x)p Fv(\))22 b Fo(\000)h Fq(u)p Fv(\()p Fq(t;)17 b(x)p Fv(\))p Fo(j)g Fq(dx)27 b Fh(6)1916 4635 y Fl(Z)2016 4661 y Fm(b)1971 4861 y(a)2069 4771 y Fo(j)p Fq(u)2153 4786 y Fn(0)p Fm(j)2231 4771 y Fv(\()p Fq(x)p Fv(\))22 b Fo(\000)h Fq(u)2540 4786 y Fn(0)2582 4771 y Fv(\()p Fq(x)p Fv(\))p Fo(j)17 b Fq(dx)g(:)-165 5073 y Fv(Budu)m(\023)-46 b(ci)32 b(da)h Fq(u)361 5088 y Fm(j)400 5073 y Fv(\()p Fq(t;)17 b Fo(\001)p Fv(\))27 b Fo(!)h Fq(u)794 5088 y Fn(0)p Fm(j)904 5073 y Fv(u)33 b(L)1052 5036 y Fn(1)1052 5104 y(lo)r(c)1152 5073 y Fv(\()p Fw(R)p Fv(\))f(k)-5 b(ad)32 b Fq(t)c Fo(!)f Fv(0,)33 b(dobiv)-5 b(amo)437 5386 y(lim)17 b(sup)515 5470 y Fm(t)p Fp(!)p Fn(0)752 5251 y Fl(Z)852 5277 y Fm(b)p Fp(\000)p Fm(M)8 b(t)807 5476 y(a)p Fn(+)p Fm(M)g(t)1075 5386 y Fo(j)p Fq(u)1159 5401 y Fn(0)1201 5386 y Fv(\()p Fq(x)p Fv(\))22 b Fo(\000)h Fq(u)p Fv(\()p Fq(t;)17 b(x)p Fv(\))p Fo(j)g Fq(dx)27 b Fh(6)h Fv(2)2069 5251 y Fl(Z)2168 5277 y Fm(b)2124 5476 y(a)2222 5386 y Fo(j)p Fq(u)2306 5401 y Fn(0)p Fm(j)2383 5386 y Fv(\()p Fq(x)p Fv(\))22 b Fo(\000)h Fq(u)2692 5401 y Fn(0)2734 5386 y Fv(\()p Fq(x)p Fv(\))p Fo(j)17 b Fq(dx)g(:)-165 5688 y Fv(Dakle)31 b Fq(u)p Fv(\()p Fq(t;)17 b Fo(\001)p Fv(\))27 b Fo(!)g Fq(u)561 5703 y Fn(0)636 5688 y Fv(u)33 b(L)784 5652 y Fn(1)784 5720 y(lo)r(c)884 5688 y Fv(\()p Fw(R)p Fv(\))f(k)-5 b(ad)32 b Fq(t)c Fo(!)f Fv(0.)3324 5811 y Fw(Q.E.D.)-165 6010 y Ft(28)p eop %%Page: 29 34 29 33 bop 3102 -116 a Fr(Burgerso)n(v)-5 b(a)25 b(jednad)n(\024)-39 b(zba)293 83 y Fv(Sv)m(o)5 b(jstv)m(o)36 b(sa)m(\024)-46 b(zimanja)33 b(u)h(T)-8 b(eorem)m(u)35 b(4)f(je,)h(nara)m(vno,)h(mnogo) d(ja)m(\024)-46 b(ce)34 b(o)s(d)g(teorema)g(jedinstv)m(enosti.)71 202 y(Dakle)44 b(\(16\))h(i)f(\(20\))h(su)h(dv)-5 b(a)45 b(ekviv)-5 b(alen)m(tna)45 b(na)m(\024)-46 b(cina)45 b(zada)m(v)-5 b(anja)45 b(dopustivih)g(rje)-5 b(\024)-44 b(senja)46 b(Burgerso)m(v)m(e)71 320 y(jednad)m(\024)-46 b(zb)s(e.)45 b(Na)33 b(disk)m(on)m(tin)m(uitetima)e(prv)m(og)i(reda)g (jasno)g(je)g(da)f(se)i(p)s(o)s(dudara)5 b(ju,)33 b(\(16\))f(k)-5 b(a)m(\024)-46 b(ze)33 b(da)g(sk)m(ok)71 439 y(mora)e(biti)g(negativ)-5 b(an,)32 b(dok)h(\(20\))f(k)-5 b(a)m(\024)-46 b(ze)33 b(da)f(ak)m(o)h(je)g(brzina)27 b(\024)-44 b(sok)-5 b(a)32 b Fq(s)p Fv(,)h(tada)f(vrijedi)834 634 y Fo(j)p Fq(u)918 649 y Fn(+)1003 634 y Fo(\000)22 b Fq(k)s Fo(j)1201 553 y Fl(\000)1246 634 y Fq(s)h Fo(\000)1424 595 y Fn(1)p 1424 611 39 4 v 1424 668 a(2)1473 634 y Fv(\()p Fq(u)1567 649 y Fn(+)1652 634 y Fv(+)f Fq(k)s Fv(\))1842 553 y Fl(\001)1915 634 y Fh(>)29 b Fo(j)p Fq(u)2105 649 y Fp(\000)2190 634 y Fo(\000)22 b Fq(k)s Fo(j)2388 553 y Fl(\000)2433 634 y Fq(s)g Fo(\000)2611 595 y Fn(1)p 2611 611 V 2611 668 a(2)2659 634 y Fv(\()p Fq(u)2753 649 y Fp(\000)2839 634 y Fv(+)g Fq(k)s Fv(\))3029 553 y Fl(\001)3108 634 y Fq(:)71 828 y Fv(Uzmemo)32 b(li)e Fq(k)h Fv(=)d Fq(u)803 843 y Fp(\006)866 828 y Fv(,)33 b(ta)5 b(j)32 b(uvjet)h(da)5 b(je)1450 1023 y(0)28 b Fh(6)1642 984 y Fn(1)p 1642 1000 V 1642 1057 a(2)1690 1023 y Fv(\()p Fq(u)1784 1038 y Fn(+)1870 1023 y Fv(+)22 b Fq(u)2024 1038 y Fp(\000)2087 1023 y Fv(\))g Fo(\000)h Fq(s)28 b Fh(6)g Fv(0)17 b Fq(;)66 1218 y Fv(\024)-44 b(sto)33 b(je)f(Rankine-Hugonioto)m(v)g(uvjet.)44 b(Za)32 b Fq(k)j Fv(v)-5 b(an)33 b([)p Fq(u)2020 1233 y Fp(\000)2083 1218 y Fq(;)17 b(u)2183 1233 y Fn(+)2246 1218 y Fv(])33 b(imamo)595 1416 y Fo(j)p Fq(u)679 1431 y Fn(+)764 1416 y Fo(\000)23 b Fq(k)s Fo(j)p Fv(\()p Fq(u)1040 1431 y Fn(+)1125 1416 y Fv(+)f Fq(k)s Fv(\))g Fo(\000)h(j)p Fq(u)1521 1431 y Fp(\000)1606 1416 y Fo(\000)f Fq(k)s Fo(j)p Fv(\()p Fq(u)1881 1431 y Fp(\000)1967 1416 y Fv(+)g Fq(k)s Fv(\))27 b(=)h(sign)16 b(\()p Fq(u)2567 1431 y Fn(+)2653 1416 y Fo(\000)22 b Fq(k)s Fv(\)\()p Fq(u)2938 1375 y Fn(2)2938 1441 y(+)3024 1416 y Fv(+)g Fq(u)3178 1375 y Fn(2)3178 1441 y Fp(\000)3241 1416 y Fv(\))2184 1560 y(=)28 b(2)p Fq(s)p Fv(\()p Fq(u)2477 1575 y Fn(+)2562 1560 y Fv(+)22 b Fq(u)2716 1575 y Fp(\000)2780 1560 y Fv(\)sign)16 b(\()p Fq(u)3097 1575 y Fn(+)3182 1560 y Fo(\000)23 b Fq(k)s Fv(\))71 1739 y(i)989 1858 y Fo(j)p Fq(u)1073 1873 y Fn(+)1158 1858 y Fo(\000)g Fq(k)s Fo(j)f(\000)g(j)p Fq(u)1545 1873 y Fp(\000)1631 1858 y Fo(\000)g Fq(k)s Fo(j)28 b Fv(=)f(\()p Fq(u)2037 1873 y Fn(+)2123 1858 y Fo(\000)22 b Fq(u)2278 1873 y Fp(\000)2342 1858 y Fv(\)sign)16 b(\()p Fq(u)2659 1873 y Fn(+)2744 1858 y Fo(\000)23 b Fq(k)s Fv(\))17 b Fq(;)71 2024 y Fv(pa)32 b(tada)h(imamo)c(jednak)m(ost.)293 2143 y(Uzmimo)d Fq(k)31 b Fv(=)871 2103 y Fn(1)p 871 2120 V 871 2177 a(2)920 2143 y Fv(\()p Fq(u)1014 2158 y Fn(+)1090 2143 y Fo(\000)13 b Fq(u)1236 2158 y Fp(\000)1300 2143 y Fv(\))28 b(=)f Fq(s)p Fv(.)42 b(T)-8 b(ada)29 b(su)f(u)h(gornjo)5 b(j)27 b(nejednak)m(osti)i(dvije)f(strane)3451 2103 y Fn(1)p 3451 2120 V 3451 2177 a(2)3500 2143 y Fo(j)p Fq(u)3584 2158 y Fn(+)3660 2143 y Fo(\000)13 b Fq(u)3806 2158 y Fp(\000)3870 2143 y Fo(j)71 2261 y Fv(p)s(omno)m(\024)-46 b(zene)33 b(sa)1392 2380 y Fq(s)22 b Fo(\000)1569 2341 y Fn(1)p 1569 2357 V 1569 2414 a(2)1618 2380 y Fv(\()p Fq(u)1712 2395 y Fp(\006)1797 2380 y Fo(\000)h Fq(s)p Fv(\))k(=)2122 2341 y Fn(1)p 2122 2357 V 2122 2414 a(2)2170 2380 y Fv(\()p Fq(s)22 b Fo(\000)h Fq(u)2432 2395 y Fp(\006)2495 2380 y Fv(\))17 b Fq(;)71 2546 y Fv(te)44 b(uvjet)h(p)s(osta)5 b(je)45 b Fq(u)867 2561 y Fn(+)978 2546 y Fh(6)j Fq(u)1159 2561 y Fp(\000)1222 2546 y Fv(.)79 b(T)-8 b(ada)44 b(je)g(o)m(v)m(o)h (to)m(\024)-46 b(cno)44 b(na)h(cijelom)d Fo(h)p Fq(u)2756 2561 y Fp(\000)2819 2546 y Fq(;)17 b(u)2919 2561 y Fn(+)2982 2546 y Fo(i)p Fv(,)47 b(jer)d(su)h(dvije)f(strane)71 2665 y(kv)-5 b(adrati)m(\024)-46 b(cne)33 b(i)e(jednak)m(e)j(u)f (rubnim)e(to)m(\024)-46 b(ck)-5 b(ama.)293 2784 y(Sv)m(o)5 b(jstv)m(o)35 b(sa)m(\024)-46 b(zimanja)32 b(u)i(T)-8 b(eorem)m(u)34 b(4,)f(tak)m(od)p 1944 2732 30 2 v 1 w(er)h(p)s(o)m(vla) m(\024)-46 b(ci)32 b(da)i(za)f(Hopfo)m(v)-5 b(a)34 b(rje)-5 b(\024)-44 b(senja)34 b(Burgerso)m(v)m(e)71 2902 y(jedna)m(\024)-46 b(zb)s(e)33 b(imamo)d(nejednak)m(ost)813 3006 y Fl(Z)930 3142 y Fo(j)p Fq(u)p Fv(\()p Fq(t;)17 b(x)k Fv(+)h Fq(y)t Fv(\))g Fo(\000)g Fq(u)p Fv(\()p Fq(t;)17 b(x)p Fv(\))p Fo(j)g Fq(dx)27 b Fh(6)2065 3006 y Fl(Z)2181 3142 y Fo(j)p Fq(u)2265 3157 y Fn(0)2307 3142 y Fv(\()p Fq(x)c Fv(+)f Fq(y)t Fv(\))f Fo(\000)i Fq(u)2788 3157 y Fn(0)2830 3142 y Fv(\()p Fq(x)p Fv(\))p Fo(j)17 b Fq(dx)g(;)71 3381 y Fv(jer)44 b(je)g(prostor)g(rje)-5 b(\024)-44 b(senja)45 b(translatorno)e(in)m(v)-5 b(arijan)m(tan.)76 b(P)m(osebno,)49 b(o)m(v)m(o)44 b(p)s(o)m(vla)m(\024)-46 b(ci)43 b(da)h(je)g Fq(u)g Fv(funk)m(cija)71 3499 y(omed)p 265 3448 V(ene)30 b(v)-5 b(arijacije)27 b(s)j(obzirom)e(na)h Fq(x)p Fv(,)h(ak)m(o)g(je)f (to)g(i)f Fq(u)2069 3514 y Fn(0)2111 3499 y Fv(.)43 b(Da)28 b(bi)h(to)g(vidjeli,)f(dok)-5 b(a)m(\024)-46 b(zimo)27 b(sljede)m(\023)-46 b(cu)30 b(lem)m(u.)71 3663 y Fw(Lema)38 b(3.)99 b Fi(Ak)m(o)33 b(je)g Fq(f)43 b Fi(funk)m(cija)33 b(omed)p 1524 3611 V(ene)g(v)-5 b(arijacije)31 b(na)h Fw(R)p Fi(,)h(tada)f(vrijedi)f(form)m(ula:)71 3902 y Fv(\(22\))1342 3766 y Fl(Z)1458 3902 y Fo(j)p Fq(f)11 b Fv(\()p Fq(x)22 b Fv(+)g Fq(y)t Fv(\))f Fo(\000)i Fq(f)11 b Fv(\()p Fq(x)p Fv(\))p Fo(j)17 b Fq(dx)27 b Fh(6)h Fo(j)p Fq(y)t Fo(j)p Fq(V)71 4156 y Fi(gdje)33 b(je)f Fq(V)54 b Fi(totalna)31 b(v)-5 b(arijacija)30 b(funk)m(cije)j Fq(f)11 b Fi(.)44 b(Obrn)m(uto,)33 b(ak)m(o)f(je)h Fq(f)38 b Fo(2)29 b Fv(L)2691 4120 y Fn(1)2733 4156 y Fv(\()p Fw(R)p Fv(\))j Fi(i)g(vrijedi)71 4415 y Fv(\(23\))877 b(lim)17 b(inf)1179 4480 y Fm(y)s Fp(!)p Fn(0)1418 4265 y Fl(R)1501 4345 y Fo(j)p Fq(f)11 b Fv(\()p Fq(x)22 b Fv(+)g Fq(y)t Fv(\))g Fo(\000)g Fq(f)11 b Fv(\()p Fq(x)p Fv(\))p Fo(j)17 b Fq(dx)p 1418 4392 935 4 v 1832 4483 a Fo(j)p Fq(y)t Fo(j)2390 4415 y Fv(=)28 b Fq(V)49 b(<)28 b Fo(1)17 b Fq(;)71 4667 y Fi(tada)32 b(je)h Fq(f)43 b Fi(omed)p 684 4615 30 2 v(ene)33 b(v)-5 b(arijacije)31 b(s)i(totalnom)d(v)-5 b(arijacijom)30 b Fq(V)21 b Fi(,)33 b(i)e(u)i(\(23\))f(p)s(osto)5 b(ji)32 b(i)f(limes.)71 4808 y Fz(Dem.)93 b Fv(Ak)m(o)33 b(je)g Fq(f)43 b Fv(funk)m(cija)32 b(omed)p 1343 4756 V(ene)i(v)-5 b(arijacije)30 b(i)i Fq(y)f(>)c Fv(0)33 b(tada)f(imamo)1308 5047 y Fq(!)t Fv(\()p Fq(y)t Fv(\))26 b(=)1630 4911 y Fl(Z)1747 5047 y Fo(j)p Fq(f)11 b Fv(\()p Fq(x)22 b Fv(+)g Fq(y)t Fv(\))f Fo(\000)h Fq(f)11 b Fv(\()p Fq(x)p Fv(\))p Fo(j)17 b Fq(dx)1527 5320 y Fh(6)1632 5184 y Fl(Z)1748 5179 y(\022)1822 5184 y(Z)1921 5210 y Fm(x)p Fn(+)p Fm(y)1877 5410 y(x)2086 5320 y Fo(j)p Fq(d)-16 b(f)11 b Fv(\()p Fq(t)p Fv(\))p Fo(j)17 b Fq(dt)2450 5179 y Fl(\023)2554 5320 y Fq(dx)1527 5603 y Fv(=)1630 5467 y Fl(Z)1747 5462 y(\022)1820 5603 y Fo(j)p Fq(d)-16 b(f)11 b Fv(\()p Fq(t)p Fv(\))p Fo(j)2098 5467 y Fl(Z)2196 5493 y Fm(t)2152 5693 y(t)p Fp(\000)p Fm(y)2317 5603 y Fq(dx)2423 5462 y Fl(\023)2530 5603 y Fq(dt)1527 5815 y Fv(=)27 b Fo(j)p Fq(y)t Fo(j)p Fq(V)37 b(;)3770 6010 y Ft(29)p eop %%Page: 30 35 30 34 bop -165 -116 a Fr(Nelinearni)27 b(zak)n(oni)f(sa)n(\024)-39 b(cuv)-5 b(anja)-171 83 y Fv(\024)-44 b(sto,)33 b(k)-5 b(ak)m(o)33 b(je)g(o)m(\024)-46 b(cigledno)31 b Fq(!)36 b Fv(parna)c(funk)m(cija,)h(dok)-5 b(azuje)33 b(\(22\).)57 202 y(Za)f(proizv)m(oljn)m(u)g(funk)m(ciju)h Fq(')27 b Fo(2)h Fv(C)1331 166 y Fn(1)1331 226 y Fm(c)1374 202 y Fv(\()p Fw(R)p Fv(\))k(imamo:)530 335 y Fl(\014)530 394 y(\014)530 454 y(\014)530 514 y(\014)564 343 y(Z)680 479 y Fo(\000)781 440 y Fm(d)p 767 456 68 4 v 767 513 a(dt)845 479 y Fq(')p Fv(\()p Fq(x)p Fv(\))p Fq(f)11 b Fv(\()p Fq(x)p Fv(\))17 b Fq(dx)1353 335 y Fl(\014)1353 394 y(\014)1353 454 y(\014)1353 514 y(\014)1414 479 y Fv(=)37 b(lim)1517 544 y Fm(y)s Fp(!)p Fn(0)1689 335 y Fl(\014)1689 394 y(\014)1689 454 y(\014)1689 514 y(\014)1723 343 y(Z)1839 479 y Fo(\000)1926 412 y Fq(')p Fv(\()p Fq(x)23 b Fv(+)f Fq(y)t Fv(\))f Fo(\000)i Fq(')p Fv(\()p Fq(x)p Fv(\))p 1926 456 684 4 v 2242 547 a Fq(y)2620 479 y(f)11 b Fv(\()p Fq(x)p Fv(\))17 b Fq(dx)2933 335 y Fl(\014)2933 394 y(\014)2933 454 y(\014)2933 514 y(\014)1414 751 y Fv(=)37 b(lim)1517 816 y Fm(y)s Fp(!)p Fn(0)1689 607 y Fl(\014)1689 667 y(\014)1689 726 y(\014)1689 786 y(\014)1723 616 y(Z)1849 684 y Fq(f)11 b Fv(\()p Fq(x)p Fv(\))22 b Fo(\000)h Fq(f)11 b Fv(\()p Fq(x)22 b Fo(\000)g Fq(y)t Fv(\))p 1849 728 675 4 v 2161 820 a Fq(y)2534 751 y(')p Fv(\()p Fq(x)p Fv(\))17 b Fq(dx)2852 607 y Fl(\014)2852 667 y(\014)2852 726 y(\014)2852 786 y(\014)1414 954 y Fh(6)28 b Fq(V)1576 969 y Fn(0)1635 954 y Fv(sup)17 b Fo(j)p Fq(')p Fo(j)g Fq(;)-165 1197 y Fv(gdje)41 b(je)g Fq(V)223 1212 y Fn(0)307 1197 y Fv(=)h(lim)17 b(inf)695 1212 y Fm(y)s Fp(!)p Fn(0)881 1147 y Fm(!)s Fn(\()p Fm(y)s Fn(\))p 881 1174 151 4 v 915 1234 a Fp(j)p Fm(y)s Fp(j)1042 1197 y Fv(.)69 b(Ak)m(o)41 b(je)g Fq(V)1520 1212 y Fn(0)1605 1197 y Fq(<)g Fo(1)p Fv(,)i(tada)e(je)g Fq(f)2296 1160 y Fp(0)2361 1197 y Fv(Radono)m(v)-5 b(a)41 b(mjera)f(\(deriv)-5 b(acija)39 b(je)-165 1335 y(sh)m(v)-5 b(a)m(\023)-46 b(cena)36 b(u)g(smislu)e(teorije)g(distribucija\),)g(ukupne)j(mase)e (manje)g(o)s(d)g Fq(V)2605 1350 y Fn(0)2647 1335 y Fv(,)h(o)s(dnosno,)g Fq(f)46 b Fv(je)35 b(funk)m(cija)-165 1454 y(omed)p 29 1402 30 2 v(ene)e(v)-5 b(arijacije)31 b(s)i(totalnom)d(v)-5 b(arijacijom)29 b Fq(V)49 b Fh(6)29 b Fq(V)1909 1469 y Fn(0)1951 1454 y Fv(.)44 b(Budu)m(\023)-46 b(ci)32 b(da)h(je)1147 1723 y Fq(V)1204 1738 y Fn(0)1274 1723 y Fh(6)28 b Fv(lim)17 b(sup)1451 1808 y Fm(y)s Fp(!)p Fn(0)1704 1656 y Fq(!)t Fv(\()p Fq(y)t Fv(\))p 1704 1701 192 4 v 1747 1792 a Fo(j)p Fq(y)t Fo(j)1933 1723 y Fh(6)29 b Fq(V)49 b Fh(6)28 b Fq(V)2307 1738 y Fn(0)-165 2015 y Fv(iz)k(\(23\),)27 b(\024)-44 b(sto)32 b(nam)g(upra)m(v)m(o)i(da)5 b(je)33 b(zadnju)g(tvrdnju.)3324 2133 y Fw(Q.E.D.)-165 6010 y Ft(30)p eop %%Page: 31 36 31 35 bop 1619 5211 a Fu(I)t(I)t(I.)55 b(Komp)t(enzirana)f(k)l (ompaktnost)p eop %%Page: 32 37 32 36 bop -165 -116 a Fr(Nelinearni)27 b(zak)n(oni)f(sa)n(\024)-39 b(cuv)-5 b(anja)57 83 y Fv(U)25 b(pretho)s(dnom)g(p)s(ogla)m(vlju)f (prou)m(\024)-46 b(ca)m(v)-5 b(ali)24 b(smo)h(Burgeso)m(vu)i(jednad)m (\024)-46 b(zbu.)43 b(Dobili)22 b(smo)j(p)s(osto)5 b(janje)-165 202 y(slab)s(og)37 b(rje)-5 b(\024)-44 b(senja,)41 b(te)f(nek)m(e)g (uvjete)g(k)m(o)5 b(ji)39 b(da)5 b(ju)39 b(jedinstv)m(enost.)64 b(Dali)36 b(smo)g(\024)-46 b(cak)39 b(i)f(form)m(ulu)f(p)s(o)i(k)m(o)5 b(jo)g(j)-165 320 y(se)38 b(takv)m(o)h(rje)-5 b(\024)-44 b(senje)39 b(mo)m(\024)-46 b(ze)38 b(izra)m(\024)-46 b(cunati.)58 b(Ko)s(d)37 b(op)m(\023)-46 b(cenitijih)36 b(zak)m(ona)j(sa)m(\024)-46 b(cuv)-5 b(anja)38 b(\(op)m(\023)-46 b(cenitija)36 b(funk)m(cija)-165 439 y(\015uksa)27 b Fq(f)11 b Fv(\))26 b(takvi)g(se)h(rezultati)e(ne)i(da)5 b(ju)27 b(tak)m(o)f(jednosta)m(vno)i(dobiti)c(up)s(otrebljenim)h(meto)s (dama.)39 b(Zato)-168 558 y(\023)-46 b(cemo)36 b(u)g(o)m(v)m(om)g(p)s (ogla)m(vlju)f(dok)-5 b(azati)35 b(egzistenciju)i(slab)s(og)e(rje)-5 b(\024)-44 b(senja)37 b(za)f(op)m(\023)-46 b(cenitiju)36 b(funk)m(ciju)g(\015uksa)-165 677 y Fq(f)43 b Fv(mo)s(dernijim)29 b(tehnik)-5 b(ama.)57 795 y(U)32 b(teorem)m(u)e(\023)-46 b(cemo)32 b(s)h(\012)g(ozna)m(\024)-46 b(ca)m(v)-5 b(ati)32 b(gornju)g(p)s(olura)m(vnin)m(u)g Fw(R)2371 759 y Fn(+)2457 795 y Fo(\002)23 b Fw(R)p Fv(.)-165 953 y Fw(T)-9 b(eorem)36 b(1.)100 b Fi(Nek)-5 b(a)33 b(je)f Fq(u)830 968 y Fn(0)900 953 y Fo(2)c Fv(W)1095 917 y Fn(1)p Fm(;)p Fp(1)1236 953 y Fv(\()p Fw(R)p Fv(\))p Fi(.)43 b(T)-8 b(ada)33 b(Cauc)m(h)m(yjev)i(problem)1300 1044 y Fl(\032)1392 1111 y Fq(@)1443 1126 y Fm(t)1475 1111 y Fq(u)22 b Fv(+)g Fq(@)1702 1126 y Fm(x)1750 1111 y Fq(f)11 b Fv(\()p Fq(u)p Fv(\))27 b(=)g(0)1688 1254 y Fq(u)p Fv(\(0)p Fq(;)17 b Fo(\001)p Fv(\))27 b(=)g Fq(u)2127 1269 y Fn(0)-165 1430 y Fi(ima)j(slab)s(o)i(rje)-5 b(\024)-44 b(senje)34 b Fq(u)27 b Fo(2)h Fv(L)868 1394 y Fp(1)868 1461 y Fn(lo)r(c)968 1430 y Fv(\()p 1006 1350 71 4 v(\012\))p Fi(.)p 3632 1473 30 43 v 57 1587 a Fv(Da)i(bismo)g(smo)h(dok)-5 b(azali)29 b(o)m(v)-5 b(a)5 b(j)31 b(teorem)g(trebat)e(\023)-46 b(ce)31 b(nam)g(neki)g(no)m(vi)g(p)s(o)5 b(jmo)m(vi)30 b(iz)g(matemati)m(\024)-46 b(ck)m(e)-165 1706 y(analize.)1374 1902 y Fw(Y)-9 b(oungo)m(v)m(e)38 b(mjere)57 2060 y Fv(Pretp)s(osta)m (vljamo)31 b(da)h(je)h(\012)28 b Fo(\022)g Fw(R)1318 2024 y Fm(d)1394 2060 y Fv(otv)m(oren)33 b(i)e(omed)p 1999 2008 30 2 v(en)i(skup.)44 b(Nek)-5 b(a)33 b(su,)g(nadalje,)f Fq(u)3249 2024 y Fm(n)3331 2060 y Fv(funk)m(cije)-165 2179 y(de\014nirane)g(na)h(\012.)44 b(Pretp)s(osta)m(vimo)32 b(da)g(za)h Fq(u)1532 2142 y Fm(n)1614 2179 y Fv(imamo)d(da)1417 2347 y Fq(u)1473 2306 y Fm(n)1615 2282 y Fn(L)1663 2250 y Fd(1)1734 2282 y Fn(\(\012\))p Fp(\003)1550 2347 y Fo(\000)-16 b(\000)g(\000)g(\000)g(\000)g Fq(*)27 b(u)17 b(:)-165 2516 y Fv(Pitamo)30 b(se)k(mo)m(\024)-46 b(zemo)31 b(li)g(iz)h(toga)g(zaklju)m(\024)-46 b(citi)30 b(da)1254 2685 y Fq(f)j Fo(\016)22 b Fq(u)1463 2644 y Fm(n)1606 2619 y Fn(L)1654 2587 y Fd(1)1724 2619 y Fn(\(\012\))p Fp(\003)1541 2685 y Fo(\000)-16 b(\000)g(\000)g(\000)f(\000)h Fq(*)28 b(f)k Fo(\016)22 b Fq(u)17 b Fv(?)-165 2853 y(Drugim)35 b(rije)m(\024)-46 b(cima,)37 b(jesu)h(li)e(op)s(eratori)g(k)m(o)5 b(ji)38 b(djeluju)f(k)-5 b(ao)37 b(k)m(omp)s(ozicija)f(s)i(neprekin)m (uto)h(diferencija-)-165 2972 y(bilnim)29 b(funk)m(cijama)j(slab)s(o)f Fo(\003)i Fv(nizo)m(vno)f(neprekin)m(uti?)57 3091 y(Bez)45 b(do)s(datnih)e(zah)m(tjev)-5 b(a)45 b(na)f(\()p Fq(u)1318 3055 y Fm(n)1368 3091 y Fv(\))g(to)g(op)m(\023)-46 b(cenito)44 b(ne)g(vrijedi,)d(\024)-44 b(sto)45 b(se)g(lak)m(o)e(mo)m(\024)-46 b(ze)44 b(vidjeti)f(na)-165 3210 y(sljede)m(\023)-46 b(cem)42 b(primjeru.)70 b(Nek)-5 b(a)42 b(su)h Fq(a)f Fv(i)f Fq(b)h Fv(razli)m(\024)-46 b(citi)38 b(realni)j(bro)5 b(jevi,)44 b(te)e Fq(\025)i Fo(2)g(h)p Fv(0)p Fq(;)17 b Fv(1)p Fo(i)p Fv(.)70 b(Dalje,)43 b(nek)-5 b(a)42 b(su)-165 3328 y Fq(u)-109 3292 y Fm(n)-32 3328 y Fv(:)28 b([0)p Fq(;)17 b Fv(1])27 b Fo(!)h Fw(R)k Fv(funk)m(cije)h(zadane)g(form)m (ulom)1162 3556 y Fq(u)1218 3515 y Fm(n)1268 3556 y Fv(\()p Fq(x)p Fv(\))28 b(=)1530 3415 y Fl(\032)1622 3517 y Fq(a)1782 3478 y Fm(k)p 1780 3494 47 4 v 1780 3552 a(n)1864 3517 y Fh(6)g Fq(x)h Fh(6)2168 3478 y Fm(k)r Fn(+)p Fm(\025)p 2168 3494 147 4 v 2218 3552 a(n)1622 3636 y Fq(b)107 b Fv(ina)m(\024)-46 b(ce)33 b(.)-165 3778 y(Za)f(ta)5 b(j)32 b(niz)g(lak)m(o)g(se)h(vidi)f(da)1205 3897 y Fq(u)1261 3856 y Fm(n)1402 3841 y Fp(\003)1311 3897 y Fo(\000)-16 b(\000)g Fq(*u)27 b Fv(=)h Fq(\025a)22 b Fv(+)g(\(1)g Fo(\000)g Fq(\025)p Fv(\))p Fq(b)-165 4052 y Fv(i)971 4170 y Fq(f)33 b Fo(\016)22 b Fq(u)1180 4129 y Fm(n)1320 4114 y Fp(\003)1230 4170 y Fo(\000)-16 b(\000)g Fq(*)p 1449 4089 59 4 v -1 w(f)39 b Fv(=)27 b Fq(\025f)11 b Fv(\()p Fq(a)p Fv(\))22 b(+)g(\(1)g Fo(\000)g Fq(\025)p Fv(\))p Fq(f)11 b Fv(\()p Fq(b)p Fv(\))17 b Fq(;)-165 4325 y Fv(a)32 b(znamo)g(da)g(za)h(prizv)m(oljn)m(u)f(funk)m(ciju)h Fq(f)43 b Fv(op)m(\023)-46 b(cenito)33 b(ne)g(vrijedi)e(jednak)m(ost) 886 4494 y Fq(f)11 b Fv(\()p Fq(\025a)22 b Fv(+)g(\(1)f Fo(\000)i Fq(\025)p Fv(\))p Fq(b)p Fv(\))28 b(=)f Fq(\025f)11 b Fv(\()p Fq(a)p Fv(\))22 b(+)g(\(1)g Fo(\000)h Fq(\025)p Fv(\))p Fq(f)11 b Fv(\()p Fq(b)p Fv(\))17 b Fq(:)57 4662 y Fv(Uk)m(olik)m(o,)36 b(recimo,)g(zah)m(tijev)-5 b(amo)35 b(da)h(su)h(deriv)-5 b(acije)35 b(funk)m(cija)h Fq(u)2465 4626 y Fm(n)2551 4662 y Fv(ograni)m(\024)-46 b(cene)36 b(u)g(L)3188 4626 y Fn(2)3231 4662 y Fv(,)h(onda)f(za-)-165 4781 y(klju)m(\024)-46 b(cujemo)37 b(da)h(je)h(niz)f(sadr)m(\024)-46 b(zan)39 b(u)f(L)1269 4745 y Fp(1)1350 4781 y Fo(\\)p Fv(H)1489 4745 y Fn(1)1532 4781 y Fv(,)i(pa)e(Rellic)m(ho)m(v)f(teorem) h(k)m(ompaktnosti)g(da)5 b(je)39 b(nizo)m(vn)m(u)-165 4900 y(slabu)32 b Fo(\003)g Fv(neprekidnost)i(za)f(sv)-5 b(aku)33 b(neprekin)m(utu)h(funk)m(ciju)f Fq(f)11 b Fv(.)57 5019 y(T)-8 b(akv)m(e)42 b(op)m(\023)-46 b(cenite)42 b(rezultate)e(nam)h(da)5 b(je)41 b(teorija)f(Y)-8 b(oungo)m(vih)40 b(mjera.)68 b(U)41 b(to)5 b(j)41 b(teoriji)e(p)s(o)s(dnizu)-165 5137 y(niza)d(\()p Fq(u)139 5101 y Fm(n)188 5137 y Fv(\))h(pridru)m (\024)-46 b(zujemo)36 b(familiju)c(vjero)5 b(jatnosnih)37 b(mjera)f(\()p Fq(\027)2238 5152 y Fm(x)2286 5137 y Fv(\))2324 5154 y Fm(x)p Fp(2)p Fn(\012)2514 5137 y Fv(na)h(k)m(o)s(domeni)e Fq(K)44 b Fv(\()p Fz(Y)-8 b(oungovu)-165 5256 y(mjeru)p Fv(\))32 b(takvu)i(da)e(je)h(slabi)e(limes)g(niza)h(\()p Fq(f)h Fo(\016)22 b Fq(u)1583 5220 y Fm(n)1629 5235 y Fj(k)1674 5256 y Fv(\))32 b(dan)h(s)g(o)m(\024)-46 b(cekiv)-5 b(anjem:)1206 5475 y Fo(h)p Fq(\027)1293 5490 y Fm(x)1340 5475 y Fq(;)17 b(f)11 b Fo(i)27 b Fv(=)1613 5339 y Fl(Z)1668 5565 y Fm(K)1759 5475 y Fq(f)11 b Fv(\()p Fq(\025)p Fv(\))17 b Fq(d\027)2067 5490 y Fm(x)2114 5475 y Fv(\()p Fq(\025)p Fv(\))g Fq(;)-165 5692 y Fv(za)32 b(sv)-5 b(aku)34 b(neprekidn)m(u)g (funk)m(ciju)f Fq(f)38 b Fv(:)28 b Fq(K)35 b Fo(!)27 b Fw(R)p Fv(.)57 5811 y(V)-8 b(erzija)32 b(osno)m(vnog)h(teorema)f(o)g (Y)-8 b(oungo)m(v)m(om)32 b(mjerama)f(je)i(dok)-5 b(azana)32 b(u)h([Ba].)-165 6010 y Ft(32)p eop %%Page: 33 38 33 37 bop 2873 -116 a Fr(Komp)r(enzirana)26 b(k)n(ompaktnost)71 88 y Fw(T)-9 b(eorem)39 b(2.)107 b Fi(Nek)-5 b(a)35 b(je)g Fv(\012)d Fo(\022)g Fw(R)1320 52 y Fm(d)1398 88 y Fi(skup)k(izmjeriv)e (u)h(Lesb)s(esgueo)m(v)m(om)h(smislu,)e Fq(K)39 b Fo(\022)32 b Fw(R)3428 52 y Fm(m)3535 88 y Fi(zatv)m(oren)71 207 y(skup,)k(i)e(nek)-5 b(a)35 b(je)g Fq(a)779 171 y Fm(n)860 207 y Fv(:)d(\012)f Fo(!)g Fw(R)1235 171 y Fm(m)1341 207 y Fi(niz)j(izmjerivih)e(funk)m(cija,)j(za)g(k)m(o)5 b(ji)34 b(vrijedi)g(Im)o Fv(\()p Fq(a)3187 171 y Fm(n)3238 207 y Fv(\))d Fo(!)g Fq(K)41 b Fi(u)35 b(mjeri,)71 325 y(o)s(dnosno)d(da)h(za)g(sv)-5 b(aki)32 b(otv)m(oreni)g(skup)i Fq(U)43 b Fi(k)m(o)5 b(ji)33 b(sadr)m(\024)-46 b(zi)32 b Fq(K)40 b Fi(vrijedi)1283 537 y Fv(lim)1327 597 y Fm(n)1435 537 y Fz(vol)16 b Fo(f)p Fq(x)28 b Fo(2)g Fv(\012)44 b(:)h Fq(a)2032 496 y Fm(n)2083 537 y Fv(\()p Fq(x)p Fv(\))39 b Fq(=)-60 b Fo(2)28 b Fq(U)10 b Fo(g)28 b Fv(=)f(0)17 b Fq(:)71 784 y Fi(T)-8 b(ada)49 b(p)s(osto)5 b(ji)47 b(p)s(o)s(dniz)h Fv(\()p Fq(a)1100 747 y Fm(n)1146 762 y Fj(k)1192 784 y Fv(\))1230 801 y Fm(k)1325 784 y Fi(i)f(izmjeriv)-5 b(a)47 b(familija)d(nenegativnih)49 b(Radono)m(vih)f(mjera)g(na)g Fw(R)3826 747 y Fm(m)71 902 y Fv(\()p Fq(\027)157 917 y Fm(x)205 902 y Fq(;)33 b(x)28 b Fo(2)g Fv(\012\))33 b Fi(takv)-5 b(a)33 b(da)f(vrijedi:)120 1021 y(\(a\))48 b Fo(k)p Fq(\027)391 1036 y Fm(x)439 1021 y Fo(k)489 1038 y Fp(M)612 1021 y Fv(=)716 941 y Fl(R)763 1056 y Fg(R)829 1034 y Fj(m)929 1021 y Fq(d\027)1028 1036 y Fm(x)1104 1021 y Fh(6)28 b Fv(1)f(\(ss)34 b Fq(x)28 b Fo(2)g Fv(\012\))p Fi(,)114 1140 y(\(b\))49 b Fv(supp)18 b Fq(\027)559 1155 y Fm(x)634 1140 y Fo(\022)28 b Fq(K)35 b Fv(\(ss)f Fq(x)28 b Fo(2)g Fv(\012\))p Fi(,)125 1258 y(\(c\))49 b Fq(f)33 b Fo(\016)22 b Fq(a)497 1222 y Fm(n)543 1237 y Fj(k)679 1203 y Fp(\003)588 1258 y Fo(\000)-16 b(\000)g Fq(*)p Fo(h)p Fq(\027)895 1273 y Fp(\001)920 1258 y Fq(;)17 b(f)11 b Fo(i)27 b Fv(=)1192 1178 y Fl(R)1239 1293 y Fg(R)1305 1272 y Fj(m)1390 1258 y Fq(f)11 b Fv(\()p Fq(\025)p Fv(\))17 b Fq(d\027)1698 1273 y Fp(\001)1722 1258 y Fv(\()p Fq(\025)p Fv(\))32 b Fi(u)h Fv(L)2035 1222 y Fp(1)2116 1258 y Fv(\(\012\))g Fi(za)f(sv)-5 b(aku)34 b(funk)m(ciju)f Fq(f)38 b Fo(2)28 b Fv(C)3315 1273 y Fn(0)3358 1258 y Fv(\()p Fw(R)3480 1222 y Fm(m)3552 1258 y Fv(\))p Fi(.)78 1364 y(\024)71 1389 y(Sto)m(vi)-5 b(\024)-44 b(se,)33 b(ak)m(o)g(p)s(o)s(dniz)f Fv(\()p Fq(a)1017 1353 y Fm(n)1063 1368 y Fj(k)1108 1389 y Fv(\))1146 1407 y Fm(k)1225 1389 y Fi(zado)m(v)m(olja)m(v)-5 b(a)32 b(i)g(uvjet)h (ograni)m(\024)-46 b(cenosti)71 1601 y Fv(\(1\))499 b(\()p Fo(8)p Fq(R)29 b(>)f Fv(0\))47 b(lim)1099 1670 y Fm(h)p Fp(!1)1313 1601 y Fv(sup)1365 1689 y Fm(k)1476 1601 y Fz(vol)16 b Fo(f)p Fq(x)28 b Fo(2)g Fv(\012)23 b Fo(\\)f Fv(K\(0)p Fq(;)17 b(R)q Fv(\))44 b(:)g Fo(j)p Fq(a)2531 1560 y Fm(n)2577 1575 y Fj(k)2622 1601 y Fv(\()p Fq(x)p Fv(\))p Fo(j)28 b Fh(>)g Fq(m)p Fo(g)g Fv(=)g(0)17 b Fq(;)71 1875 y Fi(onda)33 b(je)g Fo(k)p Fq(\027)514 1890 y Fm(x)562 1875 y Fo(k)612 1892 y Fp(M)736 1875 y Fv(=)c(1)f(\(ss)34 b Fq(x)29 b Fo(2)g Fv(\012\))34 b Fi(\()p Fq(\027)1473 1890 y Fm(x)1553 1875 y Fi(je)g(vjero)5 b(jatnosna)34 b(mjera\))e(i)g(za)h(sv)-5 b(aki)33 b(izmjeriv)f(skup)i Fq(A)29 b Fo(\022)g Fv(\012)71 1994 y Fi(vrijedi:)1481 2113 y Fq(f)k Fo(\016)22 b Fq(a)1685 2072 y Fm(n)1731 2087 y Fj(k)1904 2047 y Fn(L)1952 2015 y Fk(1)1989 2047 y Fn(\()p Fm(A)p Fn(\))1804 2113 y Fo(\000)-16 b(\000)g(\000)g(\000)g (\000)g Fq(*)27 b Fo(h)p Fq(\027)2321 2128 y Fp(\001)2346 2113 y Fq(;)17 b(f)11 b Fo(i)71 2287 y Fi(za)44 b(sv)-5 b(aku)44 b(neprekin)m(utu)i(funk)m(ciju)e Fq(f)57 b Fv(:)47 b Fw(R)1691 2251 y Fm(m)1809 2287 y Fo(!)f Fw(R)p Fi(,)h(takvu)d(da)g (je)g(niz)f Fv(\()p Fq(f)e Fo(\016)29 b Fq(a)3089 2251 y Fm(n)3135 2266 y Fj(k)3181 2287 y Fv(\))3219 2304 y Fm(k)3309 2287 y Fi(nizo)m(vno)44 b(slab)s(o)71 2405 y(relativno)31 b(k)m(ompaktan)i(u)f Fv(L)1134 2369 y Fn(1)1177 2405 y Fv(\()p Fq(A)p Fv(\))p Fi(.)p 3868 2448 30 43 v 293 2593 a Fv(U)h(daljnjem)28 b(\023)-46 b(cemo)32 b(s)h Fq(a)1171 2557 y Fm(k)1250 2593 y Fv(ozna)m(\024)-46 b(ca)m(v)-5 b(ati)29 b(\024)-46 b(clano)m(v)m(e)33 b(p)s(o)s(dniza)f (\()p Fq(a)2529 2557 y Fm(n)2575 2572 y Fj(k)2620 2593 y Fv(\))2658 2611 y Fm(k)2704 2593 y Fv(.)71 2712 y(Uvjet)h(ograni)m (\024)-46 b(cenosti)31 b(\(1\))i(je)f(ekviv)-5 b(alen)m(tan)33 b(s:)71 3081 y(\(2\))703 2847 y Fl(8)703 2936 y(>)703 2966 y(<)703 3146 y(>)703 3176 y(:)808 2924 y Fv(\()p Fo(8)p Fq(R)c(>)e Fv(0\)\()p Fo(9)p Fq(g)k Fo(2)d Fv(C\()p Fw(R)1651 2883 y Fn(+)1651 2954 y(0)1715 2924 y Fv(\)\))p Fq(g)36 b Fv(je)d(neopada)5 b(ju)m(\023)-46 b(c)q(a)o Fq(;)73 b Fv(lim)2590 2986 y Fm(t)p Fp(!1)2788 2924 y Fq(g)t Fv(\()p Fq(t)p Fv(\))27 b(=)g Fo(1)32 b Fv(i)1211 3165 y(sup)1263 3253 y Fm(k)1374 3030 y Fl(Z)1429 3255 y Fn(\012)p Fp(\\)p Fn(K\(0)p Fm(;R)q Fn(\))1795 3165 y Fq(g)t Fv(\()p Fo(j)p Fq(a)1963 3124 y Fm(k)2008 3165 y Fv(\()p Fq(x)p Fv(\))p Fo(j)p Fv(\))17 b Fq(dx)27 b(<)h Fo(1)71 3450 y Fv(Zaista,)h(uzmimo)g(nek)-5 b(a)30 b(vrijedi)g(\(2\).) 42 b(Fiksira)5 b(jmo)28 b Fq(R)q Fv(,)j(i)e(nek)-5 b(a)31 b(je)f Fq(g)k Fv(pripadna)29 b(neopada)5 b(ju)m(\023)-46 b(ca)32 b(funk)m(cija.)71 3569 y(T)-8 b(ada)32 b(vidimo)f(da)h(je)411 3825 y(sup)463 3913 y Fm(k)575 3825 y Fz(vol)707 3715 y Fl(n)774 3825 y Fq(x)c Fo(2)g Fv(\012)23 b Fo(\\)f Fv(K\(0)p Fq(;)17 b(R)q Fv(\))44 b(:)g Fo(j)p Fq(a)1646 3784 y Fm(k)1692 3825 y Fv(\()p Fq(x)p Fv(\))p Fo(j)28 b Fh(>)g Fq(t)2019 3715 y Fl(o)2102 3825 y Fq(g)t Fv(\()p Fq(t)p Fv(\))f Fh(6)i Fv(sup)2449 3913 y Fm(k)2560 3690 y Fl(Z)2615 3915 y Fn(\012)p Fp(\\)p Fn(K\(0)p Fm(;R)q Fn(\))2981 3825 y Fq(g)t Fv(\()p Fo(j)p Fq(a)3149 3784 y Fm(k)3194 3825 y Fv(\()p Fq(x)p Fv(\))p Fo(j)p Fv(\))17 b Fq(dx)g(;)71 4110 y Fv(a)32 b(k)-5 b(ak)m(o)33 b Fq(g)e Fo(\045)c(1)32 b Fv(to)h(slijedi)d(\(1\).)293 4229 y(Obrn)m(uto,)j(nek) -5 b(a)33 b(vrijedi)e(\(1\).)44 b(Za)32 b(\014ksan)h Fq(R)h Fv(mo)m(\024)-46 b(zemo)31 b(izabrati)g(rastu)m(\023)-46 b(ci)32 b(niz)g(\()p Fq(t)3252 4244 y Fm(n)3303 4229 y Fv(\))g(u)h Fw(R)3544 4193 y Fn(+)3608 4229 y Fv(,)f(tak)-5 b(a)m(v)71 4348 y(da)32 b(vrijedi)953 4490 y(sup)1005 4578 y Fm(k)1117 4490 y Fz(vol)1250 4379 y Fl(n)1316 4490 y Fq(x)c Fo(2)g Fv(\012)23 b Fo(\\)f Fv(K\(0)p Fq(;)17 b(R)q Fv(\))44 b(:)g Fo(j)p Fq(a)2188 4449 y Fm(k)2234 4490 y Fv(\()p Fq(x)p Fv(\))p Fo(j)28 b Fh(>)g Fq(t)2561 4505 y Fm(n)2612 4379 y Fl(o)2706 4490 y Fh(6)g Fq(n)2869 4449 y Fp(\000)p Fn(3)2988 4490 y Fq(;)71 4718 y Fv(te)33 b(uzeti:)p 1393 4815 51 4 v 1393 4870 a Fq(g)s Fv(\()p Fq(t)p Fv(\))28 b(=)1685 4730 y Fl(\032)1777 4811 y Fv(0)p Fq(;)106 b(t)28 b Fo(2)g Fv([0)p Fq(;)17 b(t)2271 4826 y Fn(1)2314 4811 y Fo(i)1777 4930 y Fq(n;)97 b(t)28 b Fo(2)g Fv([)p Fq(t)2178 4945 y Fm(n)2229 4930 y Fq(;)17 b(t)2308 4945 y Fm(n)p Fn(+1)2457 4930 y Fo(i)p Fv(.)71 5098 y(T)-8 b(ada)32 b(je)247 5385 y(sup)299 5473 y Fm(k)410 5250 y Fl(Z)465 5475 y Fn(\012)p Fp(\\)p Fn(K\(0)p Fm(;R)q Fn(\))p 831 5330 V 831 5385 a Fq(g)s Fv(\()p Fo(j)p Fq(a)998 5344 y Fm(k)1044 5385 y Fv(\()p Fq(x)p Fv(\))p Fo(j)p Fv(\))17 b Fq(dx)27 b Fv(=)h(sup)1547 5473 y Fm(k)1692 5261 y Fp(1)1659 5291 y Fl(X)1658 5505 y Fm(n)p Fn(=1)1820 5385 y Fq(n)p Fz(vol)2011 5275 y Fl(n)2077 5385 y Fq(x)g Fo(2)g Fv(\012)23 b Fo(\\)f Fv(K\(0)p Fq(;)17 b(R)q Fv(\))44 b(:)g Fo(j)p Fq(a)2949 5344 y Fm(k)2996 5385 y Fv(\()p Fq(x)p Fv(\))p Fo(j)27 b(2)h Fv([)p Fq(t)3338 5400 y Fm(n)3389 5385 y Fq(;)17 b(t)3468 5400 y Fm(n)p Fn(+1)3617 5385 y Fo(i)3656 5275 y Fl(o)1391 5715 y Fh(6)1530 5590 y Fp(1)1497 5620 y Fl(X)1497 5834 y Fm(n)p Fn(=1)1658 5715 y Fq(n)1716 5674 y Fp(\000)p Fn(2)1846 5715 y Fq(<)27 b Fo(1)17 b Fq(:)3770 6010 y Ft(33)p eop %%Page: 34 39 34 38 bop -165 -116 a Fr(Nelinearni)27 b(zak)n(oni)f(sa)n(\024)-39 b(cuv)-5 b(anja)-165 83 y Fv(Biranjem)31 b(neprekidne)j(funk)m(cije)f Fq(g)i Fv(za)e(k)m(o)5 b(ju)34 b(vrijedi)p 1848 28 51 4 v 31 w Fq(g)26 b Fo(\000)c Fv(1)28 b Fh(6)g Fq(g)j Fh(6)p 2385 28 V 28 w Fq(g)36 b Fv(dobiv)-5 b(amo)30 b(\(2\).)57 202 y(Ak)m(o)j(\()p Fq(a)349 166 y Fm(n)400 202 y Fv(\))f(je)h(niz)f(ograni)m(\024)-46 b(cen)32 b(u)h(L)1320 166 y Fp(1)1400 202 y Fv(\(\012;)17 b Fw(R)1636 166 y Fm(m)1708 202 y Fv(\),)32 b(to)h(jest)1327 394 y Fo(k)p Fq(a)1428 352 y Fm(n)1479 394 y Fo(k)1529 415 y Fn(L)1577 394 y Fd(1)1647 415 y Fn(\(\012;)p Fg(R)1819 394 y Fj(m)1883 415 y Fn(\))1945 394 y Fq(<)27 b(C)d(;)-165 585 y Fv(tada)k(je)h(i)f (\()p Fq(f)d Fo(\016)15 b Fq(a)438 549 y Fm(n)488 585 y Fv(\))29 b(ograni)m(\024)-46 b(cen)28 b(niz)h(u)g(L)1288 549 y Fp(1)1369 585 y Fv(\(\012;)17 b Fw(R)1605 549 y Fm(m)1676 585 y Fv(\))29 b(za)g(sv)-5 b(aku)30 b(neprekidn)m(u)g(funk)m (ciju)f Fq(f)11 b Fv(.)42 b(Dalje,)28 b(u)h(tom)-165 704 y(slu)m(\024)-46 b(ca)5 b(ju)37 b(je)g(ispunjen)g(i)f(uvjet)i (\(2\))e(s)h Fq(g)t Fv(\()p Fq(t)p Fv(\))e(=)g Fq(t)p Fv(,)j(pa)f(imamo)d(familiju)f(vjero)5 b(jatnostnih)37 b(mjera)f(\()p Fq(\027)3512 719 y Fm(x)3560 704 y Fv(\))g(i)-165 823 y(p)s(o)s(dniz)31 b(\()p Fq(a)243 787 y Fm(k)290 823 y Fv(\))h(tak)-5 b(a)m(v)33 b(da)g(je)1386 942 y Fq(f)g Fo(\016)22 b Fq(a)1590 900 y Fm(k)1727 886 y Fp(\003)1637 942 y Fo(\000)-16 b(\000)g Fq(*)o Fo(h)p Fq(\027)1943 957 y Fp(\001)1968 942 y Fq(;)17 b(f)11 b Fo(i)-165 1107 y Fv(u)29 b(L)-21 1070 y Fp(1)60 1107 y Fv(\(\012\),)h(za)f(sv)-5 b(aku)30 b(neprekidn)m(u)g(funk)m(ciju)g Fq(f)11 b Fv(.)42 b(Stoga,)29 b(u)g(tom)f(p)s(osebnom)h(slu)m(\024)-46 b(ca)5 b(ju)29 b(imamo)e(slijede)m(\023)-46 b(ci)-165 1225 y(k)m(orolar.)-165 1388 y Fw(Korolar)37 b(1.)101 b Fi(Nek)-5 b(a)34 b(je)f Fv(\012)c Fo(\022)h Fw(R)1079 1352 y Fm(d)1155 1388 y Fi(ograni)m(\024)-46 b(cen)33 b(i)f(otv)m(oren)i(skup,)h(a)e Fq(K)j Fo(\022)29 b Fw(R)2658 1352 y Fm(m)2763 1388 y Fi(zatv)m(oren)34 b(p)s(o)s(dskup,)g(te)-165 1507 y(nek)-5 b(a)33 b(je)f Fv(\()p Fq(a)254 1471 y Fm(n)305 1507 y Fv(\))g Fi(ograni)m(\024)-46 b(cen)32 b(niz)g(u)h Fv(L)1119 1471 y Fp(1)1200 1507 y Fv(\(\012;)17 b Fw(R)1436 1471 y Fm(m)1508 1507 y Fv(\))32 b Fi(tak)-5 b(a)m(v)33 b(da)f(vrijedi)1025 1699 y Fv(\()p Fo(8)p Fq(n)c Fo(2)g Fw(N)p Fv(\))98 b Fq(a)1573 1657 y Fm(n)1623 1699 y Fv(\()p Fq(x)p Fv(\))28 b Fo(2)g Fq(K)35 b Fv(\(ss)f Fq(x)28 b Fo(2)g Fv(\012\))17 b Fq(:)-165 1890 y Fi(T)-8 b(ada)26 b(p)s(osto)5 b(ji)26 b(p)s(o)s(dniz)f(tog)h(niza)g(\(radi)f(jednosta)m (vnosti)i(ozna)m(\024)-46 b(cimo)25 b(ga)h(jednak)m(o\))h(i)f(izmjeriv) -5 b(a)24 b(familija)-165 2009 y(vjero)5 b(jatnostnih)30 b(Radonovih)g(mjera)f Fv(\()p Fq(\027)1360 2024 y Fm(x)1408 2009 y Fv(\))1446 2026 y Fm(x)p Fp(2)p Fn(\012)1631 2009 y Fi(\(Y)-8 b(oungo)m(v)j(a)30 b(mjera)f(pridru)m(\024)-46 b(zena)31 b(p)s(o)s(dnizu)e Fv(\()p Fq(a)3328 1973 y Fm(n)3374 1988 y Fj(k)3420 2009 y Fv(\))h Fi(niza)-165 2128 y Fv(\()p Fq(a)-76 2092 y Fm(n)-26 2128 y Fv(\))p Fi(\))j(takv)-5 b(a)32 b(da)h(vrijedi)-95 2247 y(\(i\))48 b Fv(supp)17 b Fq(\027)322 2262 y Fm(x)398 2247 y Fo(\022)28 b Fq(K)35 b Fv(\(ss)e Fq(x)28 b Fo(2)h Fv(\012\))-122 2365 y Fi(\(ii\))47 b Fq(f)33 b Fo(\016)22 b Fq(a)261 2329 y Fm(n)307 2344 y Fj(k)443 2309 y Fp(\003)352 2365 y Fo(\000)-16 b(\000)g Fq(*)o Fo(h)p Fq(\027)658 2380 y Fp(\001)684 2365 y Fq(;)17 b(f)11 b Fo(i)27 b Fv(=)956 2285 y Fl(R)1003 2400 y Fg(R)1069 2378 y Fj(m)1153 2365 y Fq(f)11 b Fv(\()p Fq(\025)p Fv(\))17 b Fq(d\027)1461 2380 y Fp(\001)1486 2365 y Fv(\()p Fq(\025)p Fv(\))32 b Fi(u)h Fv(L)1799 2329 y Fp(1)1880 2365 y Fv(\(\012\))g Fi(za)f(sv)-5 b(aku)34 b(funk)m(ciju)f Fq(f)38 b Fo(2)28 b Fv(C)3079 2380 y Fn(0)3122 2365 y Fv(\()p Fw(R)3244 2329 y Fm(m)3315 2365 y Fv(\))p Fi(.)-165 2484 y(Obrn)m(uto,)38 b(za)f(sv)-5 b(aku)38 b(izmjerivu)d(familiju)e(vjero)5 b(jatnostnih)37 b(Radono)m(vih)f(mjera)g Fv(\()p Fq(\027)3001 2499 y Fm(x)3049 2484 y Fv(\))3087 2501 y Fm(x)p Fp(2)p Fn(\012)3278 2484 y Fi(takvu)h(da)-165 2603 y(vrijedi)26 b(\(i\),)h(p)s(osto)5 b(ji)26 b(niz)h Fv(\()p Fq(u)857 2567 y Fm(n)907 2603 y Fv(\))g Fi(ograni)m(\024)-46 b(cen)27 b(u)g Fv(L)1548 2567 y Fp(1)1629 2603 y Fv(\(\012;)17 b Fw(R)1865 2567 y Fm(m)1937 2603 y Fv(\))27 b Fi(tak)-5 b(a)m(v)27 b(da)g(je)h Fv(\()p Fq(\027)2576 2618 y Fm(x)2624 2603 y Fv(\))2662 2619 y Fm(x)p Fp(2)p Fn(\012)2843 2603 y Fi(pripadna)f(Y)-8 b(oungo)m(v)j(a)-165 2721 y(mjera.)p 3632 2764 30 43 v -165 2884 a Fw(Lema)40 b(1.)105 b Fi(Nek)-5 b(a)35 b(je)g Fv(\()p Fq(u)781 2848 y Fm(n)831 2884 y Fv(\))869 2899 y Fm(n)953 2884 y Fi(niz)g(u)f Fv(L)1262 2848 y Fp(1)1343 2884 y Fv(\(\012\))h Fi(tak)-5 b(a)m(v)35 b(da)f Fq(u)1982 2899 y Fm(n)2123 2828 y Fp(\003)2032 2884 y Fo(\000)-16 b(\000)g Fq(*u)p Fi(.)48 b(T)-8 b(ada)35 b Fq(u)2688 2902 y Fm(k)2765 2884 y Fo(!)c Fq(u)j Fi(jak)m(o)g(u)h Fv(L)3346 2848 y Fm(p)3388 2884 y Fv(\(\012\))g Fi(za)-165 3003 y Fq(p)27 b(<)h Fo(1)p Fi(,)k(ak)m(o)h(i)f(samo)f(ak)m(o)i(je)g Fq(\027)995 3018 y Fm(x)1070 3003 y Fv(=)28 b Fq(\016)1217 3024 y Fm(u)p Fn(\()p Fm(x)p Fn(\))1369 3003 y Fi(.)-165 3143 y Fz(Dem.)90 b Fv(Nu)m(\024)-46 b(znost:)41 b(iz)24 b Fq(u)710 3158 y Fm(n)850 3088 y Fp(\003)760 3143 y Fo(\000)-16 b(\000)g Fq(*)o(u)24 b Fv(u)h(L)1199 3107 y Fp(1)1280 3143 y Fv(\(\012\))g(p)s(o)g(T)-8 b(eorem)m(u)25 b(2)g(\(i)e(iz)i(raspra)m(v)m(e)h(k)m(o)5 b(ja)25 b(je)g(slijedila\))d (proizlazi)-165 3262 y(da)32 b(p)s(osto)5 b(ji)32 b(p)s(o)s(dniz)g(\()p Fq(u)710 3226 y Fm(n)756 3241 y Fj(k)800 3262 y Fv(\))838 3280 y Fm(k)917 3262 y Fv(i)g(familija)c(vjero)5 b(jatnostnih)33 b(mjera)e(\()p Fq(\027)2369 3277 y Fm(x)2417 3262 y Fv(\))2455 3279 y Fm(x)p Fp(2)p Fn(\012)2642 3262 y Fv(takv)-5 b(a)33 b(da)1133 3504 y Fq(f)g Fo(\016)22 b Fq(a)1337 3463 y Fm(k)1474 3448 y Fp(\003)1384 3504 y Fo(\000)-16 b(\000)f Fq(*)p Fo(h)p Fq(\027)1690 3519 y Fp(\001)1715 3504 y Fq(;)17 b(f)11 b Fo(i)27 b Fv(=)1988 3368 y Fl(Z)2043 3594 y Fn(\012)2119 3504 y Fq(f)h(d\027)2294 3519 y Fp(\001)2336 3504 y Fq(;)-165 3750 y Fv(za)e(sv)-5 b(aku)28 b(neprekidn)m(u)g(funk)m (ciju)g Fq(f)11 b Fv(.)41 b(Budu)m(\023)-46 b(ci)27 b(da)f Fq(u)1727 3714 y Fm(n)1805 3750 y Fo(!)h Fq(u)g Fv(u)f(L)2156 3714 y Fm(p)2199 3750 y Fv(\(\012\),)i(to)f(p)s(osto)5 b(ji)26 b(p)s(o)s(dniz)g(\()p Fq(u)3242 3714 y Fm(n)3288 3727 y Fj(j)3327 3750 y Fv(\))3365 3765 y Fm(j)3431 3750 y Fv(tak)-5 b(a)m(v)-165 3869 y(da)28 b Fq(u)22 3833 y Fm(n)68 3846 y Fj(j)135 3869 y Fo(!)f Fq(u)g Fv(\(ss)34 b(\012\),)c(pa)e(i)g Fq(f)c Fo(\016)14 b Fq(u)1037 3833 y Fm(n)1083 3846 y Fj(j)1150 3869 y Fo(!)27 b Fq(f)e Fo(\016)14 b Fq(u)27 b Fv(\(ss)33 b(\012\).)43 b(Prijelask)m(om)27 b(na)i(za)5 b(jedni)m(\024)-46 b(cki)28 b(p)s(o)s(dniz)g(dobiv)-5 b(amo)1391 4111 y Fq(f)33 b Fo(\016)22 b Fq(u)27 b Fv(=)1730 3975 y Fl(Z)1786 4201 y Fn(\012)1862 4111 y Fq(f)g(d\027)2036 4126 y Fp(\001)2078 4111 y Fq(;)-165 4351 y Fv(o)s(dnosno)32 b Fq(\027)265 4366 y Fp(\001)319 4351 y Fv(=)27 b Fq(\016)465 4373 y Fm(u)p Fn(\()p Fp(\001)p Fn(\))595 4351 y Fv(.)57 4470 y(Do)m(v)m(oljnost:)43 b Fq(\027)630 4485 y Fp(\001)683 4470 y Fv(=)27 b Fq(\016)829 4492 y Fm(u)p Fn(\()p Fp(\001)p Fn(\))992 4470 y Fv(p)s(o)m(vla)m(\024)-46 b(ci)31 b(da)h(za)h(sv)-5 b(aki)32 b Fq(p)c Fo(2)g(h)p Fv(1)p Fq(;)17 b Fo(1i)31 b Fv(imamo)1439 4662 y(\()p Fq(u)1533 4621 y Fm(n)1583 4662 y Fv(\))1621 4621 y Fm(p)1754 4606 y Fp(\003)1664 4662 y Fo(\000)-17 b(\000)h Fq(*)p Fv(\()p Fq(u)p Fv(\))2015 4621 y Fm(p)-165 4854 y Fv(i)25 b Fq(u)-56 4818 y Fm(n)84 4798 y Fp(\003)-6 4854 y Fo(\000)-16 b(\000)g Fq(*)o(u)p Fv(.)41 b(Budu)m(\023)-46 b(ci)26 b(da)h(je)f(\012)h(skup)g(k)m(ona)m (\024)-46 b(cne)28 b(mjere,)f(\()p Fq(u)1967 4818 y Fm(n)2017 4854 y Fv(\))f(je)g(sadr)m(\024)-46 b(zan)27 b(u)g(L)2674 4818 y Fm(p)2716 4854 y Fv(\(\012\))g(i)e(omed)p 3136 4802 30 2 v(en)i(u)f(njem)m(u.)-165 4972 y(Dakle,)31 b(zb)s(og)h(re\015eksivnosti)i(prostora)e(L)1402 4936 y Fm(p)1444 4972 y Fv(\(\012\))q(,)g(\()p Fq(u)1744 4936 y Fm(n)1794 4972 y Fv(\))h(ima)d(slab)s(o)h(k)m(on)m(v)m(ergen)m(tan)k (\(u)e(L)3088 4936 y Fm(p)3130 4972 y Fv(\(\012\))q(\))f(p)s(o)s(dniz,) -165 5091 y(a)i(jedin)m(tv)m(enost)i(limesa)d(k)-5 b(a)m(\024)-46 b(ze)35 b(da)g(slab)s(o)f(k)m(on)m(v)m(egira)h(k)h Fq(u)p Fv(.)49 b(No,)36 b(uzmemo)e(li)f(za)h Fq(f)43 b Fv(=)31 b Fo(j)23 b(\001)h(j)3151 5055 y Fm(p)3193 5091 y Fv(,)35 b(ima)5 b(ju)m(\023)-46 b(ci)33 b(u)-165 5210 y(vidu)f(da)h(je)f(na)h (skupu)h(omed)p 910 5158 V(ene)f(mjere)g(k)m(onstan)m(ta)g(1)f(in)m (tegrabilna)f(funk)m(cija,)h(dobiv)-5 b(amo)1084 5311 y Fl(Z)1140 5536 y Fn(\012)1216 5446 y Fo(j)p Fq(u)1300 5405 y Fm(n)1350 5446 y Fv(\()p Fq(x)p Fv(\))p Fo(j)1509 5405 y Fm(p)1568 5446 y Fq(dx)28 b Fo(!)1829 5311 y Fl(Z)1884 5536 y Fn(\012)1960 5446 y Fo(j)p Fq(u)p Fv(\()p Fq(x)p Fv(\))p Fo(j)2203 5405 y Fm(p)2262 5446 y Fq(dx)17 b(;)-165 5692 y Fv(o)s(dnosno)32 b(i)g(norme)g(k)m(on)m(v)m(ergira)5 b(ju,)29 b(\024)-44 b(sto)32 b(p)s(o)m(vla)m(\024)-46 b(ci)32 b(da)g Fq(u)1848 5656 y Fm(n)1926 5692 y Fo(!)27 b Fq(u)32 b Fv(u)h(L)2289 5656 y Fm(p)2332 5692 y Fv(\(\012\).)3324 5811 y Fw(Q.E.D.)-165 6010 y Ft(34)p eop %%Page: 35 40 35 39 bop 2873 -116 a Fr(Komp)r(enzirana)26 b(k)n(ompaktnost)1112 83 y Fw(T)-9 b(eorem)36 b(k)m(omp)s(enzirane)i(k)m(ompaktnosti)293 249 y Fv(Da)g(bismo)f(dok)-5 b(azali)37 b(teorem)h(k)m(o)5 b(ji)35 b(\023)-46 b(ce)39 b(dati)f(zado)m(v)m(olja)m(v)-5 b(a)5 b(ju)m(\023)-46 b(ce)39 b(uvjete)h(na)e(niz)g(\()p Fq(u)3386 213 y Fm(n)3436 249 y Fv(\))g(trebat)e(\023)-46 b(ce)71 367 y(nam)32 b(sljede)m(\023)-46 b(ci)32 b(teorem)g([T)-8 b(a1].)71 533 y Fw(T)f(eorem)52 b(3.)141 b(\(div-rot)52 b(lema\))45 b Fi(Nek)-5 b(a)47 b(je)f Fv(\012)52 b Fo(\022)f Fw(R)2181 497 y Fm(d)2271 533 y Fi(ograni)m(\024)-46 b(cen)46 b(i)f(otv)m(oren)j(skup,)j(te)46 b(nek)-5 b(a)47 b(su)71 652 y Fv(\()p Fz(u)159 616 y Fm(n)209 652 y Fv(\))p Fq(;)17 b Fv(\()p Fz(v)375 616 y Fm(n)425 652 y Fv(\))33 b Fi(ograni)m(\024)-46 b(ceni)31 b(nizo)m(vi)h(u)g Fv(L)1391 616 y Fp(1)1472 652 y Fv(\(\012;)17 b Fw(R)1708 616 y Fm(d)1752 652 y Fv(\))32 b Fi(takvi)h(da)f(je)h(ispunjeno)120 771 y(\(a\))48 b Fv(\()p Fz(div)17 b(u)516 734 y Fm(n)567 771 y Fv(\))32 b Fi(le)m(\024)-46 b(zi)32 b(u)g(k)m(ompaktnom)g(skupu)i (u)f Fv(H)1930 734 y Fp(\000)p Fn(1)2032 771 y Fv(\(\012\))114 889 y Fi(\(b\))49 b Fv(\()p Fz(rot)16 b(u)514 853 y Fm(n)565 889 y Fv(\))32 b Fi(le)m(\024)-46 b(zi)32 b(u)g(k)m(ompaktnom)g(skupu)i (u)f Fv(H)1928 853 y Fp(\000)p Fn(1)2031 889 y Fv(\(\012\))p Fi(.)71 1008 y(Nek)-5 b(a,)33 b(nadalje,)f(vrijede)g(k)m(on)m(v)m (ergencije)j Fz(u)1679 972 y Fm(n)1820 952 y Fp(\003)1729 1008 y Fo(\000)-16 b(\000)g Fq(*)p Fz(u)32 b Fi(i)g Fz(v)2137 972 y Fm(n)2278 952 y Fp(\003)2187 1008 y Fo(\000)-16 b(\000)g Fq(*)p Fz(v)33 b Fi(u)g Fv(L)2633 972 y Fp(1)2714 1008 y Fv(\(\012\))p Fi(.)44 b(T)-8 b(ada)32 b(slijedi)f(da)1656 1214 y Fz(u)1706 1172 y Fm(n)1779 1214 y Fo(\001)22 b Fz(v)1875 1172 y Fm(n)2016 1158 y Fp(\003)1925 1214 y Fo(\000)-16 b(\000)g Fq(*)o Fz(u)23 b Fo(\001)e Fz(v)71 1419 y Fi(u)33 b(smislu)e(teorije)h(distribucija.)p 3868 1462 30 43 v 293 1585 a Fv(Dok)-5 b(a)m(\024)-46 b(zimo)28 b(sada)j(teorem)g(k)m(o)5 b(ji)28 b(\023)-46 b(ce)31 b(biti)e(klju)m(\024)-46 b(can)31 b(u)g(dok)-5 b(azu)31 b(T)-8 b(eorema)31 b(1.)42 b(T)-8 b(o)31 b(je)g(upra)m(v)m(o)h(teorem) 71 1704 y(p)s(o)g(k)m(o)5 b(jem)33 b(smo)f(nazv)-5 b(ali)31 b(i)h(cijelo)f(p)s(ogla)m(vlje.)71 1869 y Fw(T)-9 b(eorem)41 b(4.)111 b Fi(Nek)-5 b(a)37 b(je)g Fv(\012)d Fo(\022)h Fw(R)1335 1833 y Fn(2)1414 1869 y Fi(ograni)m(\024)-46 b(cen)36 b(i)f(otv)m(oren)j(skup,)h(te)d Fq(f)45 b Fo(2)35 b Fv(C)2921 1833 y Fn(1)2964 1869 y Fv(\()p Fw(R)p Fv(\))p Fi(.)55 b(Nadalje,)37 b(nek)-5 b(a)36 b(je)71 1988 y Fv(\()p Fq(u)165 1952 y Fm(n)215 1988 y Fv(\))c Fi(niz)g(funk)m(cija)h (u)g Fv(L)959 1952 y Fp(1)1040 1988 y Fv(\(\012\))f Fi(tak)-5 b(a)m(v)33 b(da)g(vrijedi)1772 2194 y Fq(u)1828 2153 y Fm(n)1968 2138 y Fp(\003)1878 2194 y Fo(\000)-16 b(\000)g Fq(*)o(u)17 b(;)71 2399 y Fi(te)33 b(nek)-5 b(a)33 b(za)f(sv)-5 b(aku)34 b(k)m(on)m(v)m(eksn)m(u)i(funk)m(ciju)d Fv(\010)28 b(:)g Fw(R)f Fo(!)g Fw(R)33 b Fi(vrijedi)e(da)i(su)71 2605 y Fv(\()p Fq(E)6 b Fv(\))1258 b Fq(@)1534 2620 y Fm(t)1566 2605 y Fv(\(\010)22 b Fo(\016)g Fq(u)1824 2564 y Fm(n)1874 2605 y Fv(\))h(+)f Fq(@)2084 2620 y Fm(x)2132 2605 y Fv(\()p Fq(Y)43 b Fo(\016)22 b Fq(u)2398 2564 y Fm(n)2448 2605 y Fv(\))71 2810 y Fi(sadr)m(\024)-46 b(zani)30 b(u)h(k)m(ompaktnom)f(skupu)i(u)e Fv(H)1565 2774 y Fp(\000)p Fn(1)1668 2810 y Fv(\(\012\))p Fi(,)h(pri)c(\024)-46 b(cem)m(u)31 b Fq(Y)51 b Fi(zado)m(v)m(olja)m(v)-5 b(a)31 b Fq(Y)2977 2774 y Fp(0)3030 2810 y Fv(=)c Fq(f)3192 2774 y Fp(0)3217 2810 y Fv(\010)3287 2774 y Fp(0)3312 2810 y Fi(.)43 b(T)-8 b(ada)31 b(vrijedi)71 3016 y Fv(\(3\))1171 b Fq(f)33 b Fo(\016)22 b Fq(u)1576 2975 y Fm(n)1716 2960 y Fp(\003)1626 3016 y Fo(\000)-16 b(\000)f Fq(*f)33 b Fo(\016)22 b Fq(u)97 b Fv(\(u)32 b(L)2336 2975 y Fp(1)2417 3016 y Fv(\(\012\))q(\))71 3222 y Fi(i)71 3427 y Fv(\(4\))1150 b Fq(f)1405 3386 y Fp(0)1452 3427 y Fo(\016)22 b Fq(u)1580 3386 y Fm(n)1630 3427 y Fo(\000)-16 b(\000)g(!)p Fq(f)1911 3386 y Fp(0)1957 3427 y Fo(\016)22 b Fq(u)97 b Fv(\(u)33 b(L)2368 3386 y Fm(p)2411 3427 y Fv(\(\012\)\))p Fq(;)71 3633 y Fi(za)g(sv)-5 b(aki)32 b Fq(p)c(<)f Fo(1)p Fi(.Uk)m(olik)m(o)32 b Fq(f)43 b Fi(ni)32 b(na)g(k)m(o)5 b(jem)33 b(in)m(terv)-5 b(alu)32 b(nije)g(a\014na,)g(onda)c(\024)-44 b(sto)m(vi)-5 b(\024)-44 b(se)33 b(vrijedi)f(i)71 3838 y Fv(\(5\))1328 b Fq(u)1580 3797 y Fm(n)1630 3838 y Fo(\000)-16 b(\000)g(!)p Fq(u)97 b Fv(\(u)32 b(L)2190 3797 y Fm(p)2233 3838 y Fv(\(\012\)\))p Fq(;)71 4044 y Fi(za)h(sv)-5 b(aki)32 b Fq(p)c(<)f Fo(1)p Fi(.)293 4163 y Fv(Dok)-5 b(az)28 b(\023)-46 b(cemo)31 b(pro)m(v)m(esti)h(u)g(par)f(k)m(orak)-5 b(a,)32 b(no)f(prije)g(nego)g(li)f(prijed)p 2678 4111 30 2 v(emo)g(na)i(njega)f(k)m(omen)m(tira)5 b(jmo)71 4282 y(uvjet)47 b(\()p Fq(E)6 b Fv(\).)86 b(Sjetimo)45 b(li)g(se)j(k)-5 b(ak)m(o)47 b(smo)f(zadali)f(v)m(ezu)k(izmed)p 2403 4230 V -1 w(u)e(\010)g(i)f Fq(Y)68 b Fv(u)47 b(preho)s(dnom)f(p)s (ogla)m(vlju,)71 4400 y(primje)m(\023)-46 b(cujemo)32 b(da)h(se)h Fq(f)989 4364 y Fp(0)1013 4400 y Fv(\010)1083 4364 y Fp(0)1142 4400 y Fv(k)m(o)s(d)g(Burgerso)m(v)m(e)h(jednad)m (\024)-46 b(zb)s(e)34 b(upra)m(v)m(o)g(sv)m(o)s(di)g(na)f Fq(s)p Fv(\010)3121 4364 y Fp(0)3146 4400 y Fv(.)46 b(Stoga)32 b(\010)i(p)s(ono)m(v)m(o)71 4519 y(zo)m(v)m(emo)f(en)m(tropija,)f(a)g Fq(Y)54 b Fv(tok)33 b(en)m(tropije.)43 b(Zb)s(og)32 b(toga)g(o)m(vdje)h (\()p Fq(E)6 b Fv(\))32 b(zo)m(v)m(emo)i(uvjet)f(en)m(tropije.)71 4661 y Fz(Dem.)93 b(Ko)m(rak)33 b(1:)43 b Fv(Dok)-5 b(azujemo)31 b(\(3\).)43 b(Fiksira)5 b(jmo)31 b(k)m(on)m(v)m(eksn)m(u)36 b(funk)m(ciju)d(\010.)44 b(Uv)m(edimo)32 b(oznak)m(e)1141 4867 y Fq(v)1192 4826 y Fm(n)1270 4867 y Fv(=)c Fq(f)33 b Fo(\016)22 b Fq(u)1583 4826 y Fm(n)1633 4867 y Fq(;)17 b(w)1750 4826 y Fm(n)1827 4867 y Fv(=)27 b(\010)c Fo(\016)f Fq(u)2151 4826 y Fm(n)2234 4867 y Fv(i)31 b Fq(z)2342 4826 y Fm(n)2421 4867 y Fv(=)d Fq(f)k Fo(\016)22 b Fq(u)2733 4826 y Fm(n)2800 4867 y Fq(;)71 5072 y Fv(i)31 b(promatra)5 b(jmo)31 b(nizo)m(v)m(e)1424 5245 y Fz(u)1474 5203 y Fm(n)1552 5245 y Fv(=)1656 5104 y Fl(\024)1719 5177 y Fo(\000)p Fq(v)1847 5141 y Fm(n)1755 5313 y Fq(u)1811 5284 y Fm(n)1907 5104 y Fl(\025)2009 5245 y Fv(i)h Fz(v)2115 5203 y Fm(n)2193 5245 y Fv(=)2296 5104 y Fl(\024)2359 5177 y Fq(w)2432 5141 y Fm(n)2370 5313 y Fq(z)2419 5284 y Fm(n)2492 5104 y Fl(\025)71 5484 y Fv(u)h(L)219 5448 y Fp(1)299 5484 y Fv(\(\012;)17 b Fw(R)535 5448 y Fn(2)578 5484 y Fv(\).)43 b(P)m(osto)5 b(je)34 b(p)s(o)s(dnizo)m(vi)d(takvi)i (da)1360 5750 y Fz(u)1410 5708 y Fm(n)1551 5694 y Fp(\003)1461 5750 y Fo(\000)-17 b(\000)h Fq(*)1696 5609 y Fl(\024)1759 5682 y Fo(\000)p Fq(v)1795 5818 y(u)1897 5609 y Fl(\025)1999 5750 y Fv(i)32 b Fz(v)2105 5708 y Fm(n)2245 5694 y Fp(\003)2155 5750 y Fo(\000)-16 b(\000)g Fq(*)2374 5609 y Fl(\024)2437 5682 y Fq(w)2467 5818 y(z)2556 5609 y Fl(\025)3770 6010 y Ft(35)p eop %%Page: 36 41 36 40 bop -165 -116 a Fr(Nelinearni)27 b(zak)n(oni)f(sa)n(\024)-39 b(cuv)-5 b(anja)-165 83 y Fv(u)32 b(L)-18 47 y Fp(1)63 83 y Fv(\(\012\))q(.)43 b(Prema)32 b(\()p Fq(E)6 b Fv(\))33 b(imamo)c(da)1304 291 y Fz(div)17 b(v)1485 250 y Fm(n)1563 291 y Fv(=)28 b Fq(@)1718 306 y Fm(t)1750 291 y Fq(w)1823 250 y Fm(n)1895 291 y Fv(+)22 b Fq(@)2044 306 y Fm(x)2092 291 y Fq(z)2141 250 y Fm(n)-165 498 y Fv(le)m(\024)-46 b(zi)31 b(u)i(k)m(ompaktnom)f(p)s(o)s(dskupu)h(H)1203 462 y Fp(\000)p Fn(1)1306 498 y Fv(\(\012\),)g(a)f(uzmemo)g(li)f (\010\()p Fq(x)p Fv(\))d(=)g Fq(x)k Fv(isto)g(slijedi)f(i)h(za)1289 706 y Fz(rot)16 b(u)1472 665 y Fm(n)1550 706 y Fv(=)28 b Fq(@)1705 721 y Fm(t)1737 706 y Fq(u)1793 665 y Fm(n)1865 706 y Fv(+)22 b Fq(@)2014 721 y Fm(x)2063 706 y Fq(v)2114 665 y Fm(n)2180 706 y Fq(:)-165 914 y Fv(Primjenom)30 b(div-rot)i(leme)f(zaklju)m(\024)-46 b(cujemo)32 b(da)1187 1122 y Fq(u)1243 1080 y Fm(n)1293 1122 y Fq(z)1342 1080 y Fm(n)1416 1122 y Fo(\000)22 b Fq(v)1566 1080 y Fm(n)1616 1122 y Fq(w)1689 1080 y Fm(n)1830 1066 y Fp(\003)1739 1122 y Fo(\000)-16 b(\000)g Fq(*uz)26 b Fo(\000)d Fq(v)t(w)-165 1329 y Fv(u)42 b(distribucijama.)71 b(S)43 b(druge)g(pak)g(strane)g Fz(u)1570 1293 y Fm(n)1649 1329 y Fo(\001)29 b Fz(v)1752 1293 y Fm(n)1845 1329 y Fv(je)42 b(ograni)m(\024)-46 b(cen)42 b(niz)h(u)f(L)2734 1293 y Fp(1)2815 1329 y Fv(\(\012\))h(pa)f (ima)f(slab)s(o)h Fo(\003)-165 1448 y Fv(k)m(on)m(v)m(ergen)m(tan)35 b(p)s(o)s(dniz,)d(o)s(dakle)g(zb)s(og)g(jedinstv)m(enosti)h(limesa)1165 1656 y Fq(u)1221 1614 y Fm(n)1271 1656 y Fq(z)1320 1614 y Fm(n)1394 1656 y Fo(\000)22 b Fq(v)1544 1614 y Fm(n)1595 1656 y Fq(w)1668 1614 y Fm(n)1808 1600 y Fp(\003)1717 1656 y Fo(\000)-16 b(\000)g Fq(*uz)26 b Fo(\000)d Fq(v)t(w)c(;)-165 1863 y Fv(o)s(dnosno)-165 2071 y(\(6\))991 b Fq(u)1007 2030 y Fm(n)1057 2071 y Fq(Y)21 b Fv(\()p Fq(u)1229 2030 y Fm(n)1279 2071 y Fv(\))h Fo(\000)h Fq(f)11 b Fv(\()p Fq(u)1592 2030 y Fm(n)1642 2071 y Fv(\)\010\()p Fq(u)1844 2030 y Fm(n)1894 2071 y Fv(\))2022 2015 y Fp(\003)1932 2071 y Fo(\000)-16 b(\000)g Fq(*)o(uz)27 b Fo(\000)22 b Fq(v)t(w)d(:)-165 2279 y Fv(Primjenom)30 b(teorema)i(o)h(Y)-8 b(oungo)m(vim)31 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b(jednosta)m(vnosti)i(ra)m(\024)-46 b(cuna,)33 b(pretp)s(osta)m(vimo)f(da)g(u)h(to)m(\024)-46 b(cki)32 b(\()p Fq(t;)17 b(x)p Fv(\))33 b(vrijedi)1238 5485 y Fq(u)p Fv(\()p Fq(t;)17 b(x)p Fv(\))27 b(=)g Fq(f)11 b Fv(\()p Fq(u)p Fv(\()p Fq(t;)17 b(x)p Fv(\)\))28 b(=)f(0)17 b Fq(:)-165 5692 y Fv(T)-8 b(ada)32 b(\(8\))g(prelazi)g(u)1327 5811 y Fo(h)p Fq(\027)q(;)17 b Fv(1)1508 5828 y Fg(R)1578 5811 y Fq(Y)43 b Fo(\000)23 b Fq(f)11 b Fv(\010)p Fo(i)27 b Fv(=)h(0)17 b Fq(;)-165 6010 y Ft(36)p eop %%Page: 37 42 37 41 bop 2873 -116 a Fr(Komp)r(enzirana)26 b(k)n(ompaktnost)71 83 y Fv(za)33 b(sv)-5 b(aku)33 b(k)m(on)m(v)m(eksn)m(u)j(funk)m(ciju)d (\010.)44 b(Koriste)m(\023)-46 b(ci)32 b(\(7\))g(mo)m(\024)-46 b(zemo)31 b(zapisati)71 308 y(\(9\))1728 234 y Fo(h)p Fq(\027)q(;)17 b Fv(1)1909 251 y Fg(R)1978 234 y Fo(i)28 b Fv(=)f(0)1788 378 y Fo(h)p Fq(\027)q(;)17 b(f)11 b Fo(i)27 b Fv(=)g(0)17 b Fq(:)71 533 y Fv(Odab)s(erimo)26 b Fq(\013)j Fv(i)f Fq(\014)34 b Fv(tak)m(o)28 b(da)h(je)p 1254 478 197 4 v 28 w(con)m(v)s(\(supp)17 b Fq(\027)6 b Fv(\))28 b(=)g([)p Fq(\013)q(;)17 b(\014)6 b Fv(].)41 b(T)-8 b(ada)29 b(\(9\))2588 548 y Fn(1)2658 533 y Fv(p)s(o)m(vla)m (\024)-46 b(ci)27 b(da)i(je)f Fq(\013)h Fh(6)f Fv(0)f Fh(6)h Fq(\014)6 b Fv(.)42 b(Ak)m(o)71 652 y(bi)31 b(jedna)h(o)s(d)g (vrijednosti)f Fq(\013)i Fv(ili)c Fq(\014)38 b Fv(bili)29 b(n)m(ula)j(tad)f(bi)h(i)f(druga)h(vrijednost)g(bila)e(n)m(ula.)43 b(T)-8 b(o)32 b(bi)f(p)s(o)m(vla)m(\024)-46 b(cilo)71 770 y(da)35 b(je)h Fq(\027)j Fv(=)33 b Fq(\016)t Fv(,)e(\024)-44 b(sto)35 b(je)h(trivijalan)d(slu)m(\024)-46 b(ca)5 b(j,)36 b(stoga)f(pretp)s(osta)m(vljamo)g(da)g(je)h Fq(\013)d(<)f Fv(0)h Fq(<)f(\014)6 b Fv(.)52 b(De\014nira)5 b(jmo)71 889 y(p)s(omo)m(\023)-46 b(cne)32 b(funk)m(cije)h Fq(g)j Fv(i)c Fq(h)g Fv(form)m(ulama:)71 1253 y(\(10\))1534 1121 y Fq(g)t Fv(\()p Fq(\025)p Fv(\))27 b(=)1848 986 y Fl(Z)1948 1012 y Fm(\025)1904 1211 y(\013)2013 1121 y Fq(\030)21 b(d\027)6 b Fv(\()p Fq(\030)f Fv(\))1528 1406 y Fq(h)p Fv(\()p 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b(rastu)m(\023)-46 b(cu)31 b(funk)m(ciju)f Fq(')p Fv(.)42 b(Stoga)29 b(zaklju)m(\024)-46 b(cujemo)30 b(da)f(to)h(vrijedi) f(i)f(za)i(razliku)f(neopada-)71 3824 y(ju)m(\023)-46 b(cih)32 b(funk)m(cija,)h(i)e(dalje,)h(za)h(sv)m(e)h(glatk)m(e)e(funk)m (cije)h Fq(')p Fv(.)44 b(T)-8 b(ak)m(o)33 b(zaklju)m(\024)-46 b(cujemo)32 b(da)g(je)71 3986 y(\(12\))1272 b Fq(h)p Fv(\()p Fq(\025)p Fv(\))22 b Fo(\000)h Fq(f)1887 3945 y Fp(0)1912 3986 y Fv(\()p Fq(\025)p Fv(\))p Fq(g)t Fv(\()p Fq(\025)p Fv(\))j(=)i(0)17 b Fq(;)71 4148 y Fv(a)32 b(\(11\))g(da)5 b(je)33 b(da)g(vrijedi)1476 4266 y Fq(f)11 b Fv(\()p Fq(\025)p Fv(\))p Fq(g)1719 4225 y Fp(0)1743 4266 y Fv(\()p Fq(\025)p Fv(\))22 b Fo(\000)h Fq(\025h)2111 4225 y Fp(0)2136 4266 y Fv(\()p Fq(\025)p Fv(\))k(=)h(0)17 b Fq(:)71 4418 y Fv(Jedno)33 b(i)f(drugo)g(p)s(o)m(vla)m(\024)-46 b(ci)1479 4537 y(\()p Fq(f)11 b Fv(\()p Fq(\025)p Fv(\))p Fq(g)t Fv(\()p Fq(\025)p Fv(\))21 b Fo(\000)i Fq(h)p Fv(\()p Fq(\025)p Fv(\)\))2241 4496 y Fp(0)2293 4537 y Fv(=)28 b(0)17 b Fq(;)66 4689 y Fv(\024)-44 b(sto)33 b(za)5 b(jedno)33 b(s)d(\024)-46 b(cinjenicom)31 b(da)h(je)h Fq(g)e Fv(=)d Fq(h)f Fv(=)h(0)k(izv)-5 b(an)32 b([)p Fq(\013)q(;)17 b(\014)6 b Fv(])32 b(da)5 b(je)1530 4850 y Fq(f)11 b Fv(\()p Fq(\025)p Fv(\))p Fq(g)t Fv(\()p Fq(\025)p Fv(\))20 b Fo(\000)j Fq(h)p Fv(\()p Fq(\025)p Fv(\))28 b(=)f(0)17 b Fq(:)71 5012 y Fv(P)m(osljednje,)29 b(za)5 b(jedno)29 b(s)f(\(12\))f(i)d(\024)-46 b(cinjenicom)26 b(da)i(je)g Fq(g)j(<)c Fv(0)h(na)f Fo(h)p Fq(\013)q(;)17 b(\014)6 b Fo(i)26 b Fv(da)5 b(je)29 b Fq(f)11 b Fv(\()p Fq(\025)p Fv(\))h Fo(\000)g Fq(\025f)3228 4976 y Fp(0)3253 5012 y Fv(\()p Fq(\025)p Fv(\))27 b(=)h(0,)g(o)s(dakle)71 5131 y(slijedi)j(da)h(na)g(istom)g(in)m(terv)-5 b(alu)31 b(vrijedi)1752 5293 y Fq(f)11 b Fv(\()p Fq(\025)p Fv(\))27 b(=)g Fq(c\025)17 b(;)71 5455 y Fv(pa)32 b(je)h(tvrdnja)g(\(4\))f (trivijalno)e(ispunjena.)293 5574 y(Ak)m(o)37 b(nema)g(in)m(terv)-5 b(ala)35 b(na)i(k)m(o)5 b(jem)37 b(je)g Fq(f)48 b Fv(a\014na)37 b(onda)f(je)h(supp)18 b Fq(\027)41 b Fo(\022)36 b(f)p Fq(\013)q Fo(g)p Fv(,)h(pa)g(tvrdnja)g(slijedi)e(iz)71 5692 y(Leme)d(1.)3561 5811 y Fw(Q.E.D.)3770 6010 y Ft(37)p eop %%Page: 38 43 38 42 bop -165 -116 a Fr(Nelinearni)27 b(zak)n(oni)f(sa)n(\024)-39 b(cuv)-5 b(anja)1196 83 y Fw(Dok)f(az)37 b(gla)m(vnog)h(teorema)57 264 y Fv(P)m(ono)m(vno)31 b(\023)-46 b(cemo)32 b(k)m(oristiti)e(meto)s (du)i(is)m(\024)-46 b(ceza)m(v)-5 b(a)5 b(ju)m(\023)-46 b(ce)35 b(visk)m(oznosti.)-165 386 y(Promotrimo)29 b(parab)s(oli)m (\024)-46 b(cku)31 b(aproksimaciju)g(Cauc)m(h)m(yjev)m(e)36 b(zada)m(\023)-46 b(ce)1141 540 y Fl(\032)1233 613 y Fq(@)1284 628 y Fm(t)1316 613 y Fq(u)1372 572 y Fm(n)1444 613 y Fv(+)22 b Fq(@)1593 628 y Fm(x)1641 613 y Fq(f)11 b Fv(\()p Fq(u)1794 572 y Fm(n)1844 613 y Fv(\))28 b(=)2027 574 y Fn(1)p 2023 590 47 4 v 2023 648 a Fm(n)2079 613 y Fq(@)2130 628 y Fm(xx)2222 613 y Fq(u)2278 572 y Fm(n)1630 757 y Fq(u)p Fv(\(0)p Fq(;)17 b Fo(\001)p Fv(\))27 b(=)g Fq(u)2069 772 y Fn(0)2128 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1460 y Fo(\000)-16 b(\000)g Fq(*)o(u)p Fv(.)57 1582 y(Cilj)21 b(nam)i(je)h(p)s(ok)-5 b(azati)22 b(da)h(je)g(tada)g(gomili)-5 b(\024)-44 b(ste)20 b Fq(u)j Fv(tra)m(\024)-46 b(zeno)24 b(slab)s(o)e(rje)-5 b(\024)-44 b(senje.)42 b(Do)m(v)m(oljno)22 b(je)i(p)s(ok)-5 b(azati)-165 1701 y(da)32 b(vrijedi)g(\()p Fq(E)6 b Fv(\),)32 b(jer)h(tada)f(tvrdnja)h(izlazi)d(iz)i(T)-8 b(eorema)32 b(4.)57 1824 y(Nek)-5 b(a)41 b(je)g(\010)i(:)g Fw(R)f Fo(!)g Fw(R)e Fv(k)m(on)m(v)m(eksna)45 b(funk)m(cije,)e(a)e (funk)m(cija)h Fq(Y)62 b Fv(takv)-5 b(a)41 b(da)g(vrijedi)f Fq(Y)3270 1788 y Fp(0)3337 1824 y Fv(=)i Fq(f)3514 1788 y Fp(0)3539 1824 y Fv(\010)3609 1788 y Fp(0)3634 1824 y Fv(.)-165 1943 y(Ra)m(\024)-46 b(cunamo:)429 2145 y Fq(@)480 2160 y Fm(t)513 2145 y Fv(\(\010)22 b Fo(\016)g Fq(u)771 2104 y Fm(n)821 2145 y Fv(\))g(+)g Fq(@)1030 2160 y Fm(x)1078 2145 y Fv(\()p Fq(Y)44 b Fo(\016)22 b Fq(u)1345 2104 y Fm(n)1395 2145 y Fv(\))27 b(=)h(\(\010)1672 2104 y Fp(0)1719 2145 y Fo(\016)22 b Fq(u)1847 2104 y 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Fm(n)1708 4888 y Fq(@)1759 4903 y Fm(x)1807 4888 y Fq(u)1863 4852 y Fm(n)1958 4888 y Fv(u)h(prostoru)f(L)2522 4852 y Fn(2)2565 4888 y Fv(\(\012\),)k(stoga)c(imamo)e(gornju)-165 5025 y(jednoliku)31 b(o)s(cjen)m(u.)57 5147 y(Tvrdimo,)k(k)g(tome,)g(i)g(da) g(su)g(\(\010)1287 5111 y Fp(00)1358 5147 y Fo(\016)23 b Fq(u)1487 5111 y Fm(n)1537 5147 y Fv(\))35 b(sadr)m(\024)-46 b(zani)35 b(u)g(ograni)m(\024)-46 b(cenom)34 b(skupu)i(u)g Fo(M)p Fv(\(\012\).)51 b(Ov)m(o)35 b(je)-165 5266 y(o)m(\024)-46 b(cito)31 b(zb)s(og)h(k)m(on)m(v)m(eksnosti)k(o)s(d)c(\010)h(i)c(\024) -46 b(cinjenice)32 b(da)g(je)1828 5227 y Fn(1)p 1824 5243 V 1824 5300 a Fm(n)1880 5266 y Fv(\()p Fq(@)1969 5281 y Fm(x)2017 5266 y Fq(u)2073 5230 y Fm(n)2123 5266 y Fv(\))2161 5230 y Fn(2)2236 5266 y Fv(L)2297 5230 y Fn(1)2372 5266 y Fv(funk)m(cija.)57 5389 y(Tvrdnja)h(teorema)f(sada)h (slijedi)d(iz)-165 5570 y Fw(Lema)j(2.)98 b(\(Murat\))29 b Fi(Nek)-5 b(a)29 b(je)g Fv(\()p Fq(g)1179 5533 y Fm(n)1229 5570 y Fv(\))f Fi(niz)h(k)m(o)5 b(ji)28 b(je)h(sadr)m(\024)-46 b(zan)30 b(u)f Fv(\()p Fq(A)14 b Fv(+)g Fq(B)5 b Fv(\))14 b Fo(\\)g Fq(C)7 b Fi(,)32 b(gdje)d(je:)41 b Fq(A)29 b Fi(k)m(ompaktan)-165 5688 y(u)j Fv(H)-6 5652 y Fp(\000)p Fn(1)97 5688 y Fv(\(\012\))p Fi(,)h Fq(B)k Fi(ograni)m(\024)-46 b(cen)32 b(u)h Fo(M)1060 5706 y Fm(b)1097 5688 y Fv(\(\012\))g Fi(i)e Fq(C)40 b Fi(ograni)m(\024)-46 b(cen)32 b(u)g Fv(W)2071 5652 y Fp(\000)p Fn(1)p Fm(;)p Fp(1)2272 5688 y Fv(\(\012\))q Fi(.)57 5811 y(T)-8 b(ada)32 b(je)h Fv(\()p Fq(g)498 5775 y Fm(n)548 5811 y Fv(\))f Fi(sadr)m(\024)-46 b(zan)33 b(u)g(k)m(ompaktnom)f(p)s(o)s(dskupu)i Fv(H)2172 5775 y Fp(\000)p Fn(1)2274 5811 y Fv(\(\012\))p Fi(.)-165 6010 y Ft(38)p eop %%Page: 39 44 39 43 bop 2873 -116 a Fr(Komp)r(enzirana)26 b(k)n(ompaktnost)71 83 y Fz(Dem.)93 b Fv(Promotrimo)29 b(Diric)m(hleto)m(vu)j(zada)m(\023) -46 b(cu)71 339 y(\(13\))1536 198 y Fl(\032)1627 264 y Fo(4)p Fq(v)1767 223 y Fm(n)1845 264 y Fv(=)p Fq(g)1972 223 y Fm(n)2119 264 y Fv(u)33 b(\012)1716 408 y Fq(v)1767 367 y Fm(n)1845 408 y Fv(=0)129 b(na)33 b Fq(@)5 b Fv(\012)18 b Fq(:)71 642 y Fv(Za)24 b Fq(p)k(>)f(d)p Fv(\(=)g(2\))e(k)m (ompaktnost)g(ulaganja)f(W)1750 593 y Fn(1)p Fm(;p)1749 672 y Fn(0)1853 642 y Fv(\(\012\))k Fq(,)-17 b Fo(!)28 b Fv(C)2235 657 y Fn(0)2278 642 y Fv(\(\012\))d(\(v.a)g([Br,)h(Th)m (\023)-46 b(eor)m(\022)g(eme)26 b(IX.16]\))f(p)s(o)m(vla)m(\024)-46 b(ci)71 771 y(da)32 b(je)h(ulaganje)f Fo(M)820 789 y Fm(b)856 771 y Fv(\(\012\))c Fq(,)-17 b Fo(!)28 b Fv(W)1269 735 y Fp(\000)p Fn(1)p Fm(;q)1430 771 y Fv(\(\012\))33 b(za)g Fq(q)e(<)1971 732 y Fm(d)p 1922 748 138 4 v 1922 805 a(d)p Fp(\000)p Fn(1)2097 771 y Fh(6)d Fv(2)33 b(tak)m(od)p 2488 719 30 2 v 1 w(er)f(k)m(ompaktno.)293 890 y(P)m(o)h(pretp)s(osta)m (v)m(ci)g Fq(g)1042 854 y Fm(n)1125 890 y Fv(mo)m(\024)-46 b(zemo)31 b(zapisati)g(na)i(sljede)m(\023)-46 b(ci)32 b(na)m(\024)-46 b(cin)1682 1108 y Fq(n)1740 1067 y Fm(n)1818 1108 y Fv(=)28 b Fq(g)1973 1067 y Fm(n)1969 1133 y Fn(1)2044 1108 y Fv(+)22 b Fq(g)2193 1067 y Fm(n)2189 1133 y Fn(2)2260 1108 y Fq(;)71 1326 y Fv(pri)27 b(\024)-46 b(cem)m(u)30 b(je)h Fq(g)625 1290 y Fm(n)621 1354 y Fn(1)703 1326 y Fo(2)d Fq(A)i Fv(i)g Fq(g)1009 1290 y Fm(n)1005 1354 y Fn(2)1086 1326 y Fo(2)e Fq(B)5 b Fv(.)43 b(Nek)-5 b(a)31 b(su)g Fq(v)1745 1290 y Fm(n)1741 1354 y Fn(1)1826 1326 y Fv(i)e Fq(v)1934 1290 y Fm(n)1930 1354 y Fn(2)2015 1326 y Fv(rje)-5 b(\024)-44 b(senja)31 b(rubne)g(zada)m(\023)-46 b(ce)32 b(\(13\))e(s)h Fq(g)3282 1290 y Fm(n)3278 1354 y Fn(1)3331 1326 y Fv(,)g(o)s(dnosno)g Fq(g)3821 1290 y Fm(n)3817 1354 y Fn(2)3871 1326 y Fv(,)71 1445 y(na)38 b(desno)5 b(j)39 b(strani.)59 b(Zb)s(og)37 b(linearnosti)e(slijedi)h (da)i(za)g(rje)-5 b(\024)-44 b(senje)39 b(zada)m(\023)-46 b(ce)39 b(vrijedi)e Fq(v)3183 1409 y Fm(n)3270 1445 y Fv(=)f Fq(v)3433 1409 y Fm(n)3429 1473 y Fn(1)3509 1445 y Fv(+)26 b Fq(v)3662 1409 y Fm(n)3658 1473 y Fn(2)3712 1445 y Fv(,)39 b(pri)68 1564 y(\024)-46 b(cem)m(u)33 b(je,)g(zb)s(og)f(elipti)m(\024)-46 b(ck)m(e)32 b(regularnosti)f(\(v.) 44 b([GT]\):)293 1694 y Fq(v)344 1658 y Fm(n)340 1722 y Fn(1)427 1694 y Fv(sadr)m(\024)-46 b(zan)33 b(u)g(k)m(ompaktnom)f(p)s (o)s(dskupu)h(W)2008 1645 y Fn(1)p Fm(;)p Fn(2)2007 1724 y(0)2111 1694 y Fv(\(\012\))28 b(=)g(H)2462 1658 y Fn(1)2462 1722 y(0)2504 1694 y Fv(\(\012\))293 1829 y Fq(v)344 1793 y Fm(n)340 1857 y Fn(2)427 1829 y Fv(sadr)m(\024)-46 b(zan)33 b(u)g(k)m(ompaktnom)f(p)s(o)s(dskupu)h(W)2008 1780 y Fn(1)p Fm(;q)2007 1859 y Fn(0)2110 1829 y Fv(\(\012\))293 1948 y(Budu)m(\023)-46 b(ci)33 b(da)h(je)f(tak)m(od)p 1075 1896 V 1 w(er)g Fq(g)1275 1912 y Fm(n)1354 1948 y Fo(2)c Fv(W)1550 1912 y Fp(\000)p Fn(1)p Fm(;)p Fp(1)1751 1948 y Fv(\(\012\))q(,)k(to)g(zaklju)m(\024)-46 b(cujemo)33 b(da)g(je)h(i)e Fq(v)3021 1912 y Fm(n)3104 1948 y Fv(sadr)m(\024)-46 b(zano)34 b(u)g(ograni-)68 2078 y(\024)-46 b(cenom)34 b(p)s(o)s(dskupu)h(W)926 2029 y Fn(1)p Fm(;r)925 2108 y Fn(0)1027 2078 y Fv(\(\012\))f(za)g Fq(r)e Fh(6)f Fo(1)p Fv(.)47 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setlinewidth } def /PL { stroke gnulinewidth setlinewidth } def /LTb { BL [] 0 0 0 DL } def /LTa { AL [1 dl 2 dl] 0 setdash 0 0 0 setrgbcolor } def /LT0 { PL [] 0 1 0 DL } def /LT1 { PL [4 dl 2 dl] 0 0 1 DL } def /LT2 { PL [2 dl 3 dl] 1 0 0 DL } def /LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def /LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def /LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def /LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def /LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def /LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def /P { stroke [] 0 setdash currentlinewidth 2 div sub M 0 currentlinewidth V stroke } def /D { stroke [] 0 setdash 2 copy vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke P } def /A { stroke [] 0 setdash vpt sub M 0 vpt2 V currentpoint stroke M hpt neg vpt neg R hpt2 0 V stroke } def /B { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke 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246 -45 R (0.1) Lshow 5026 1004 M -61 11 V 246 -45 R (0.15) Lshow 5267 1138 M -61 11 V 246 -45 R (0.2) Lshow 5507 1272 M -61 11 V 246 -45 R (0.25) Lshow 5748 1406 M -61 11 V 246 -45 R (0.3) Lshow 5988 1540 M -61 11 V 246 -45 R (0.35) Lshow 6229 1674 M -61 11 V 246 -45 R (0.4) Lshow 971 1936 M 63 0 V -126 0 R (0) Rshow 971 2650 M 63 0 V -126 0 R (0.5) Rshow 971 3366 M 63 0 V -126 0 R (1) Rshow 971 1221 M 0 2145 V LTa stroke grestore end showpage %%EndDocument endTexFig 496 3891 a 21521665 15156103 3289088 3289088 26970521 19866091 startTexFig 496 3891 a %%BeginDocument: god1.eps /gnudict 40 dict def gnudict begin /Color false def /Solid false def /gnulinewidth 5.000 def /vshift -46 def /dl {10 mul} def /hpt 31.5 def /vpt 31.5 def /M {moveto} bind def /L {lineto} bind def /R {rmoveto} bind def /V {rlineto} bind def /vpt2 vpt 2 mul def /hpt2 hpt 2 mul def /Lshow { currentpoint stroke M 0 vshift R show } def /Rshow { currentpoint stroke M dup stringwidth pop neg vshift R show } def /Cshow { currentpoint stroke M dup stringwidth pop -2 div vshift R show } def /DL { Color {setrgbcolor Solid {pop []} if 0 setdash } {pop pop pop Solid {pop []} if 0 setdash} ifelse } def /BL { stroke gnulinewidth 2 mul setlinewidth } def /AL { stroke gnulinewidth 2 div setlinewidth } def /PL { stroke gnulinewidth setlinewidth } def /LTb { BL [] 0 0 0 DL } def /LTa { AL [1 dl 2 dl] 0 setdash 0 0 0 setrgbcolor } def /LT0 { PL [] 0 1 0 DL } def /LT1 { PL [4 dl 2 dl] 0 0 1 DL } def /LT2 { PL [2 dl 3 dl] 1 0 0 DL } def /LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def /LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def /LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def /LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def /LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def /LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def /P { stroke [] 0 setdash currentlinewidth 2 div sub M 0 currentlinewidth V stroke } def /D { stroke [] 0 setdash 2 copy vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke P } def /A { stroke [] 0 setdash vpt sub M 0 vpt2 V currentpoint stroke M hpt neg vpt neg R hpt2 0 V stroke } def /B { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke P } def /C { stroke [] 0 setdash exch hpt sub exch vpt add M hpt2 vpt2 neg V currentpoint stroke M hpt2 neg 0 R hpt2 vpt2 V stroke } def /T { stroke [] 0 setdash 2 copy vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke P } def /S { 2 copy A C} def end gnudict begin gsave 50 50 translate 0.050 0.050 scale 0 setgray /Helvetica findfont 140 scalefont setfont newpath LTb LT0 6486 4486 M ("Godunov") Rshow 6570 4486 M 252 0 V 2896 4439 M 33 -6 V 34 -7 V 33 -6 V 33 -6 V 34 -6 V 33 -6 V 33 -7 V 34 -6 V 33 -6 V 33 -6 V 34 -6 V 33 -7 V 33 -6 V 34 -6 V 33 -6 V 33 -6 V 34 -7 V 33 -6 V 33 -6 V 34 -6 V 33 -6 V 32 -7 V 34 -6 V 33 -6 V 33 -6 V 34 -6 V 33 -7 V 33 -6 V 34 -6 V 33 -6 V 34 -6 V 33 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V 33 -6 V 34 -7 V 33 -6 V 33 -7 V 34 -8 V 33 -9 V 34 -12 V 33 -18 V 33 -43 V 34 -141 V 33 -444 V 33 -624 V 34 -181 V 33 -10 V 33 -7 V 34 -6 V 33 -6 V 33 -6 V 34 -6 V 33 -7 V 33 -6 V 34 -6 V 33 -6 V 33 -6 V 34 -7 V 33 -6 V 33 -6 V 34 -6 V 33 -6 V 33 -7 V 34 -6 V 33 -6 V 33 -6 V 34 -6 V 33 -7 V 33 -6 V 34 -6 V 33 -6 V 33 -6 V 34 -6 V 33 -7 V 33 -6 V 34 -6 V 33 -6 V 34 -6 V 33 -7 V 1933 3902 M 34 -6 V 33 -6 V 33 -6 V 34 -7 V 33 -6 V 33 -6 V 34 -6 V 33 -6 V 33 -6 V 34 -7 V 33 -6 V 34 -6 V 33 -6 V 33 -6 V 34 -7 V 33 -6 V 33 -6 V 34 -6 V 33 -6 V 33 -7 V 34 -6 V 33 -6 V 33 -6 V 34 -6 V 33 -7 V 33 -6 V 34 -6 V 33 -6 V 33 -6 V 34 -7 V 33 -6 V 33 -6 V 34 -6 V 33 -6 V 33 -7 V 34 -6 V 33 -6 V 33 -6 V 34 -6 V 33 -7 V 33 -6 V 34 -6 V 33 -6 V 33 -6 V 34 -7 V 33 -6 V 33 -6 V 34 -6 V 33 -7 V 33 -6 V 33 -7 V 33 -8 V 34 -10 V 33 -14 V 33 -20 V 34 -30 V 33 -48 V 33 -80 V 34 -140 V 33 -261 V 33 -458 V 34 -373 V 33 -60 V 33 -7 V 34 -6 V 33 -6 V 33 -6 V 34 -7 V 33 -6 V 33 -6 V 34 -6 V 33 -6 V 33 -7 V 34 -6 V 33 -6 V 33 -6 V 34 -6 V 33 -7 V 33 -6 V 34 -6 V 33 -6 V 33 -6 V 34 -7 V 33 -6 V 33 -6 V 34 -6 V 33 -6 V 33 -7 V 34 -6 V 33 -6 V 34 -6 V 33 -6 V 33 -7 V 34 -6 V 33 -6 V 33 -6 V 34 -6 V 33 -7 V 33 -6 V 34 -6 V 1452 3634 M 33 -6 V 34 -6 V 33 -7 V 34 -6 V 33 -6 V 33 -6 V 34 -6 V 33 -7 V 33 -6 V 34 -6 V 33 -6 V 33 -6 V 34 -7 V 33 -6 V 33 -6 V 34 -6 V 33 -6 V 33 -7 V 34 -6 V 33 -6 V 33 -6 V 34 -6 V 33 -6 V 33 -7 V 34 -6 V 33 -6 V 33 -6 V 34 -6 V 33 -7 V 33 -6 V 34 -6 V 33 -6 V 33 -6 V 34 -7 V 33 -6 V 33 -6 V 34 -6 V 33 -6 V 33 -7 V 34 -6 V 33 -6 V 34 -6 V 33 -7 V 33 -7 V 34 -8 V 33 -12 V 33 -17 V 34 -28 V 33 -41 V 33 -60 V 34 -80 V 33 -101 V 33 -120 V 34 -134 V 33 -141 V 33 -145 V 34 -148 V 33 -148 V 33 -148 V 34 -148 V 33 -53 V 33 -8 V 34 -6 V 33 -6 V 32 -6 V 34 -7 V 33 -6 V 33 -6 V 34 -6 V 33 -6 V 33 -7 V 34 -6 V 33 -6 V 33 -6 V 34 -6 V 33 -6 V 33 -7 V 34 -6 V 33 -6 V 34 -6 V 33 -6 V 33 -7 V 34 -6 V 33 -6 V 33 -6 V 34 -6 V 33 -7 V 33 -6 V 34 -6 V 33 -6 V 33 -6 V 34 -7 V 33 -6 V 33 -6 V currentpoint stroke M 34 -6 V 33 -6 V 33 -7 V 34 -6 V 33 -6 V 33 -6 V 971 3366 M 33 -6 V 34 -7 V 33 -6 V 33 -6 V 34 -6 V 33 -6 V 33 -7 V 34 -6 V 33 -6 V 33 -6 V 34 -6 V 33 -7 V 33 -6 V 34 -6 V 33 -6 V 33 -6 V 34 -7 V 33 -6 V 33 -6 V 34 -6 V 33 -6 V 33 -7 V 34 -6 V 33 -6 V 33 -6 V 34 -6 V 33 -7 V 33 -6 V 34 -6 V 33 -6 V 34 -6 V 33 -6 V 33 -7 V 34 -6 V 33 -6 V 33 -6 V 34 -6 V 33 -7 V 33 -6 V 34 -6 V 33 -78 V 33 -78 V 34 -77 V 33 -78 V 33 -78 V 34 -77 V 33 -78 V 33 -77 V 34 -78 V 33 -77 V 33 -78 V 34 -78 V 33 -78 V 33 -77 V 34 -78 V 33 -78 V 33 -77 V 34 -78 V 33 -78 V 33 -78 V 34 -6 V 33 -6 V 33 -6 V 34 -6 V 33 -7 V 33 -6 V 34 -6 V 33 -6 V 33 -6 V 34 -7 V 33 -6 V 34 -6 V 33 -6 V 33 -6 V 34 -7 V 33 -6 V 33 -6 V 34 -6 V 32 -6 V 33 -7 V 34 -6 V 33 -6 V 33 -6 V 34 -6 V 33 -6 V 33 -7 V 34 -6 V 33 -6 V 33 -6 V 34 -6 V 33 -7 V 33 -6 V 34 -6 V 33 -6 V 33 -6 V 34 -7 V 33 -6 V 33 -6 V 34 -6 V 33 -6 V 6229 2390 M 5748 2121 L 5267 1853 L 4785 1585 L 4304 1317 L 6196 2396 M 5715 2128 L 5233 1859 L 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V 33 -6 V 34 -6 V 33 -6 V 33 -7 V 34 -6 V 33 -6 V 33 -6 V 34 -6 V 33 -7 V 33 -6 V 34 -6 V 33 -6 V 33 -6 V 34 -7 V 33 -6 V 33 -6 V 34 -6 V 33 -6 V 33 -7 V 34 -6 V 33 -6 V 34 -6 V 33 -6 V 33 -7 V 34 -6 V 33 -6 V 33 -6 V 34 -6 V 33 -7 V 33 -6 V 34 -6 V 1452 3634 M 33 -6 V 34 -6 V 33 -7 V 34 -6 V 33 -6 V 33 -6 V 34 -6 V 33 -7 V 33 -6 V 34 -6 V 33 -6 V 33 -6 V 34 -7 V 33 -6 V 33 -6 V 34 -6 V 33 -6 V 33 -7 V 34 -6 V 33 -6 V 33 -6 V 34 -6 V 33 -6 V 33 -7 V 34 -6 V 33 -6 V 33 -6 V 34 -6 V 33 -7 V 33 -6 V 34 -6 V 33 -6 V 33 -6 V 34 -7 V 33 -6 V 33 -6 V 34 -6 V 33 -6 V 33 -7 V 34 -6 V 33 -6 V 34 -6 V 33 -6 V 33 -7 V 34 -6 V 33 -6 V 33 -6 V 34 -6 V 33 -7 V 33 -6 V 34 -6 V 33 -8 V 33 -18 V 34 -73 V 33 -298 V 33 -658 V 34 -385 V 33 -33 V 33 -6 V 34 -6 V 33 -7 V 33 -6 V 34 -6 V 33 -6 V 32 -6 V 34 -7 V 33 -6 V 33 -6 V 34 -6 V 33 -6 V 33 -7 V 34 -6 V 33 -6 V 33 -6 V 34 -6 V 33 -6 V 33 -7 V 34 -6 V 33 -6 V 34 -6 V 33 -6 V 33 -7 V 34 -6 V 33 -6 V 33 -6 V 34 -6 V 33 -7 V 33 -6 V 34 -6 V 33 -6 V 33 -6 V 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1221 M 4304 601 L 6229 1674 L 2896 2294 L 971 1221 L 971 1221 M 55 30 V 806 1130 M (0) Rshow 2638 911 M 55 30 V 2473 820 M (0.5) Rshow 4304 601 M 55 30 V 4139 509 M (1) Rshow 4304 601 M -61 11 V 246 -45 R (0) Lshow 4545 735 M -61 11 V 246 -45 R (0.05) Lshow 4785 869 M -61 11 V 246 -45 R (0.1) Lshow 5026 1004 M -61 11 V 246 -45 R (0.15) Lshow 5267 1138 M -61 11 V 246 -45 R (0.2) Lshow 5507 1272 M -61 11 V 246 -45 R (0.25) Lshow 5748 1406 M -61 11 V 246 -45 R (0.3) Lshow 5988 1540 M -61 11 V 246 -45 R (0.35) Lshow 6229 1674 M -61 11 V 246 -45 R (0.4) Lshow 971 1936 M 63 0 V -126 0 R (0) Rshow 971 2650 M 63 0 V -126 0 R (0.5) Rshow 971 3366 M 63 0 V -126 0 R (1) Rshow 971 1221 M 0 2145 V LTa stroke grestore end showpage %%EndDocument endTexFig 3770 6010 a Ft(45)p eop %%Page: 46 51 46 50 bop -165 -116 a Fr(Nelinearni)27 b(zak)n(oni)f(sa)n(\024)-39 b(cuv)-5 b(anja)496 0 y 21521665 15156103 3289088 3289088 26970521 19866091 startTexFig 496 0 a %%BeginDocument: lf3.eps /gnudict 40 dict def gnudict begin /Color false def /Solid false def /gnulinewidth 5.000 def /vshift -46 def /dl {10 mul} def /hpt 31.5 def /vpt 31.5 def /M {moveto} bind def /L {lineto} bind def /R {rmoveto} bind def /V {rlineto} bind def /vpt2 vpt 2 mul def /hpt2 hpt 2 mul def /Lshow { currentpoint stroke M 0 vshift R show } def /Rshow { currentpoint stroke M dup stringwidth pop neg vshift R show } def /Cshow { currentpoint stroke M dup stringwidth pop -2 div vshift R show } def /DL { Color {setrgbcolor Solid {pop []} if 0 setdash } {pop pop pop Solid {pop []} if 0 setdash} ifelse } def /BL { stroke gnulinewidth 2 mul setlinewidth } def /AL { stroke gnulinewidth 2 div setlinewidth } def /PL { stroke gnulinewidth setlinewidth } def /LTb { BL [] 0 0 0 DL } def /LTa { AL [1 dl 2 dl] 0 setdash 0 0 0 setrgbcolor } def /LT0 { PL [] 0 1 0 DL } def /LT1 { PL [4 dl 2 dl] 0 0 1 DL } def /LT2 { PL [2 dl 3 dl] 1 0 0 DL } def /LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def /LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def /LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def /LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def /LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def /LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def /P { stroke [] 0 setdash currentlinewidth 2 div sub M 0 currentlinewidth V stroke } def /D { stroke [] 0 setdash 2 copy vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke P } def /A { stroke [] 0 setdash vpt sub M 0 vpt2 V currentpoint stroke M hpt neg vpt neg R hpt2 0 V stroke } def /B { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke P } def /C { stroke [] 0 setdash exch hpt sub exch vpt add M hpt2 vpt2 neg V currentpoint stroke M hpt2 neg 0 R hpt2 vpt2 V stroke } def /T { stroke [] 0 setdash 2 copy vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke P } def /S { 2 copy A C} def end gnudict begin gsave 50 50 translate 0.050 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34 11 V 33 10 V 33 12 V 34 11 V 33 11 V 33 11 V 34 12 V 33 12 V 33 12 V 34 12 V 33 12 V 33 13 V 34 13 V 33 13 V 33 14 V 34 13 V 33 14 V 34 14 V 33 15 V 33 15 V 34 15 V 33 15 V 33 17 V 34 15 V 33 17 V 33 17 V 34 17 V 33 17 V 33 18 V 34 18 V 32 18 V 33 19 V 34 19 V 33 19 V 33 20 V 34 19 V 33 20 V 33 21 V 34 20 V 33 21 V 33 21 V 34 21 V 33 22 V 33 22 V 34 22 V 33 22 V 33 22 V 34 23 V 33 22 V 33 23 V 34 23 V 33 23 V 33 22 V 34 23 V 33 24 V 34 23 V 33 24 V 33 23 V 34 24 V 33 23 V 33 24 V 34 23 V 33 24 V 33 23 V 34 24 V 33 23 V 33 24 V 34 23 V 33 23 V 33 23 V 34 23 V 33 23 V 33 22 V 34 23 V 33 22 V 33 23 V 34 22 V 33 22 V 33 21 V 34 22 V 33 21 V 33 22 V 34 21 V 33 21 V 33 20 V 34 21 V 33 20 V 33 21 V 34 20 V 33 20 V 33 20 V 34 20 V 33 19 V 34 20 V 33 20 V 1933 1853 M 34 9 V 33 8 V 33 9 V 34 9 V 33 8 V 33 9 V 34 9 V 33 10 V 33 9 V 34 10 V 33 9 V 34 11 V 33 10 V 33 11 V 34 10 V 33 12 V 33 11 V 34 13 V 33 12 V 33 13 V 34 13 V 33 14 V 33 14 V 34 15 V 33 14 V 33 16 V 34 16 V 33 17 V 33 17 V 34 18 V 33 18 V 33 19 V 34 19 V 33 20 V 33 20 V 34 21 V 33 21 V 33 22 V 34 22 V 33 23 V 33 23 V 34 24 V 33 23 V 33 25 V 34 24 V 33 25 V 33 26 V 34 26 V 33 26 V 33 27 V 33 27 V 33 27 V 34 27 V 33 28 V 33 27 V 34 28 V 33 28 V 33 28 V 34 28 V 33 28 V 33 29 V 34 28 V 33 28 V 33 28 V 34 28 V 33 27 V 33 28 V 34 27 V 33 27 V 33 27 V 34 27 V 33 26 V 33 26 V 34 26 V 33 25 V 33 25 V 34 25 V 33 24 V 33 24 V 34 23 V 33 23 V 33 22 V 34 22 V 33 21 V 33 22 V 34 20 V 33 20 V 33 20 V 34 19 V 33 18 V 34 19 V 33 18 V 33 17 V 34 17 V 33 17 V 33 17 V 34 16 V 33 16 V 33 16 V 34 16 V 1452 2121 M 33 7 V 34 6 V 33 7 V 34 7 V 33 6 V 33 7 V 34 7 V 33 6 V 33 7 V 34 8 V 33 7 V 33 7 V 34 8 V 33 8 V 33 8 V 34 9 V 33 9 V 33 10 V 34 9 V 33 11 V 33 10 V 34 12 V 33 12 V 33 13 V 34 14 V 33 14 V 33 15 V 34 17 V 33 16 V 33 19 V 34 18 V 33 21 V 33 21 V 34 22 V 33 22 V 33 25 V 34 25 V 33 27 V 33 27 V 34 29 V 33 29 V 34 32 V 33 31 V 33 33 V 34 33 V 33 34 V 33 35 V 34 36 V 33 35 V 33 37 V 34 37 V 33 38 V 33 37 V 34 38 V 33 38 V 33 38 V 34 38 V 33 37 V 33 38 V 34 37 V 33 37 V 33 35 V 34 36 V 33 35 V 32 34 V 34 33 V 33 33 V 33 31 V 34 32 V 33 29 V 33 29 V 34 27 V 33 27 V 33 25 V 34 25 V 33 22 V 33 23 V 34 21 V 33 20 V 34 19 V 33 18 V 33 17 V 34 17 V 33 14 V 33 15 V 34 13 V 33 14 V 33 11 V 34 12 V 33 11 V 33 10 V 34 10 V 33 10 V 33 8 V currentpoint stroke M 34 9 V 33 9 V 33 8 V 34 8 V 33 8 V 33 8 V 971 2390 M 33 6 V 34 6 V 33 6 V 33 6 V 34 7 V 33 6 V 33 6 V 34 6 V 33 6 V 33 7 V 34 6 V 33 6 V 33 6 V 34 6 V 33 7 V 33 6 V 34 6 V 33 6 V 33 6 V 34 7 V 33 6 V 33 5 V 34 6 V 33 6 V 33 7 V 34 6 V 33 6 V 33 6 V 34 6 V 33 7 V 34 6 V 33 6 V 33 6 V 34 6 V 33 7 V 33 6 V 34 6 V 33 6 V 33 6 V 34 7 V 33 77 V 33 78 V 34 78 V 33 77 V 33 78 V 34 78 V 33 78 V 33 77 V 34 78 V 33 78 V 33 77 V 34 78 V 33 78 V 33 78 V 34 77 V 33 78 V 33 78 V 34 78 V 33 77 V 33 78 V 34 6 V 33 6 V 33 7 V 34 6 V 33 6 V 33 6 V 34 6 V 33 7 V 33 6 V 34 6 V 33 6 V 34 6 V 33 6 V 33 7 V 34 6 V 33 6 V 33 6 V 34 6 V 32 7 V 33 6 V 34 6 V 33 6 V 33 6 V 34 7 V 33 6 V 33 6 V 34 6 V 33 6 V 33 7 V 34 6 V 33 6 V 33 6 V 34 6 V 33 7 V 33 6 V 34 6 V 33 6 V 33 6 V 34 7 V 33 6 V 6229 3081 M -481 384 V -481 378 V -482 325 V -481 271 V 6196 3061 M -481 384 V -482 382 V -481 333 V -481 273 V 6162 3041 M -481 384 V -481 386 V -481 341 V -482 274 V 6129 3021 M -481 385 V -481 389 V -482 349 V -481 276 V 6096 3001 M -482 385 V -481 393 V -481 357 V -481 278 V 6062 2981 M -481 385 V -481 396 V -481 365 V -482 281 V 6029 2961 M -481 385 V -481 399 V -482 373 V -481 284 V 5996 2940 M -482 386 V -481 402 V -481 382 V -481 285 V 5962 2920 M -481 385 V -481 406 V -481 389 V -482 289 V 5929 2900 M -481 385 V -481 408 V -482 397 V -481 293 V 5896 2880 M -482 384 V -481 410 V -481 406 V -481 297 V 5862 2859 M -481 385 V -481 412 V -481 413 V -482 302 V 5829 2839 M -481 384 V -482 414 V -481 420 V -481 307 V 5796 2819 M -482 383 V -481 415 V -481 429 V -481 312 V 5762 2798 M -481 382 V -481 417 V -481 435 V -482 320 V 5729 2778 M -481 381 V -482 418 V -481 442 V -481 327 V 5696 2757 M -482 380 V -481 418 V -481 449 V -481 336 V 5662 2737 M -481 379 V -481 418 V -481 456 V -482 343 V 5629 2716 M -481 378 V -482 418 V -481 461 V -481 354 V 5596 2695 M -482 377 V -481 418 V -481 466 V -481 365 V 5562 2674 M -481 375 V -481 418 V -481 471 V -482 377 V 5529 2654 M -481 373 V -482 417 V -481 475 V -481 390 V 5496 2633 M -482 371 V -481 416 V -481 479 V -480 403 V 5462 2612 M -481 370 V -481 414 V -482 482 V -480 418 V 5429 2591 M -481 368 V -482 412 V -481 484 V -480 435 V 5396 2570 M -482 366 V -481 410 V -481 487 V -480 451 V 5362 2549 M -481 364 V -481 408 V -482 487 V -480 470 V 5329 2529 M -481 361 V -482 405 V -481 488 V -480 488 V 5296 2509 M -482 358 V -481 402 V -481 487 V -480 509 V 5262 2488 M -481 355 V -481 400 V -482 486 V -480 530 V 5229 2467 M -481 353 V -482 396 V -481 484 V -480 553 V 5195 2446 M -481 350 V -481 393 V -481 482 V -481 576 V 5162 2425 M -481 348 V -481 389 V -482 477 V -480 602 V 5129 2404 M -481 345 V -482 386 V -481 473 V 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-481 316 V -481 384 V 4495 2023 M -481 291 V -480 296 V -481 307 V -482 341 V 4462 2004 M -481 288 V -481 292 V -481 297 V -481 300 V 4429 1985 M -482 286 V -480 287 V -481 288 V -481 257 V 4395 1966 M -481 284 V -480 283 V -481 279 V -482 213 V 4362 1948 M -481 281 V -481 280 V -481 270 V -481 168 V 4329 1930 M -482 279 V -480 275 V -481 262 V -481 124 V 4295 1912 M -481 276 V -480 273 V currentpoint stroke M -481 254 V -482 77 V 4262 1894 M -481 274 V -481 269 V -481 246 V -481 31 V 4229 1876 M -482 273 V -480 265 V -481 240 V -481 -17 V 4195 1859 M -481 270 V -480 262 V -482 234 V -481 5 V 4162 1842 M -481 268 V -481 259 V -481 229 V -481 26 V 4129 1824 M -482 267 V -480 256 V -481 224 V -481 47 V 4095 1808 M -481 264 V -480 254 V -482 220 V -481 66 V 4062 1791 M -480 263 V -482 251 V -481 216 V -481 85 V 4029 1774 M -481 262 V -481 249 V -481 214 V -481 100 V 3995 1758 M -480 260 V -481 247 V -482 212 V -481 116 V 3962 1742 M -480 259 V -482 245 V -481 210 V -481 131 V 3929 1726 M -481 258 V -481 243 V -481 208 V -481 146 V 3895 1710 M -480 257 V -481 242 V -482 208 V -481 158 V 3862 1695 M -480 255 V -482 241 V -481 207 V -481 170 V 3828 1680 M -480 255 V -481 239 V -481 208 V -482 180 V 3795 1664 M -480 254 V -481 239 V -482 208 V -481 191 V 3762 1650 M -480 253 V -482 238 V -481 209 V -481 200 V 3728 1635 M -480 253 V -481 237 V -481 211 V -482 208 V 3695 1620 M -480 253 V -481 238 V -482 211 V -481 215 V 3662 1606 M -480 252 V -482 238 V -481 213 V -481 222 V 3628 1592 M -480 252 V -481 238 V -481 215 V -482 228 V 3596 1578 M -481 252 V -481 238 V -482 217 V -481 235 V 3563 1564 M -482 253 V -481 238 V -481 220 V -481 239 V 3529 1551 M -481 252 V -481 239 V -481 222 V -482 243 V 3496 1537 M -481 253 V -481 240 V -482 225 V -481 246 V 3463 1524 M -482 253 V -481 240 V -481 228 V -481 250 V 3429 1511 M -481 253 V -481 242 V -481 230 V -482 253 V 3396 1498 M -481 254 V -481 242 V -482 233 V -481 256 V 3363 1485 M -482 255 V -481 244 V -481 235 V -481 257 V 3329 1473 M -481 255 V -481 245 V -481 238 V -482 259 V 3296 1460 M -481 256 V -481 247 V -482 240 V -481 261 V 3263 1448 M -482 256 V -481 248 V -481 244 V -481 262 V 3229 1436 M -481 257 V -481 250 V -481 246 V -482 263 V 3196 1423 M -481 259 V -482 251 V -481 248 V -481 264 V 3163 1411 M -482 260 V -481 253 V -481 250 V -481 265 V 3129 1399 M -481 260 V -481 255 V -481 254 V -482 265 V 3096 1387 M -481 262 V -482 256 V -481 256 V -481 266 V 3063 1375 M -482 263 V -481 258 V -481 258 V -481 267 V 3029 1364 M -481 263 V -481 261 V -481 260 V -482 266 V 2996 1352 M -481 265 V -482 262 V -481 262 V -481 267 V 2963 1340 M -482 266 V -481 264 V -481 264 V -481 268 V 2929 1328 M -481 267 V -481 267 V -482 266 V -481 268 V 2896 1317 M -481 268 V -482 268 V -481 268 V 971 2390 L LTb 971 1674 M 3333 620 V 6229 1221 L 2896 601 L 971 1674 L 2896 601 M -55 30 V 3061 509 M (0) Lshow 4562 911 M -55 30 V 4727 820 M (0.5) Lshow 6229 1221 M -55 30 V 220 -121 R (1) Lshow 971 1674 M 61 11 V 786 1640 M (0) Rshow 1212 1540 M 61 11 V -246 -45 R (0.05) Rshow 1452 1406 M 61 11 V -246 -45 R (0.1) Rshow 1693 1272 M 61 11 V -246 -45 R (0.15) Rshow 1933 1138 M 61 11 V -246 -45 R (0.2) Rshow 2174 1004 M 61 11 V 1989 970 M (0.25) Rshow 2415 869 M 61 11 V 2230 835 M (0.3) Rshow 2655 735 M 61 11 V 2470 701 M (0.35) Rshow 2896 601 M 61 11 V 2711 567 M (0.4) Rshow 971 1674 M 63 0 V -126 0 R (-0.5) Rshow 971 2390 M 63 0 V -126 0 R (0) Rshow 971 3104 M 63 0 V -126 0 R (0.5) Rshow 971 3819 M 63 0 V -126 0 R (1) Rshow 971 1674 M 0 2145 V LTa stroke grestore end showpage %%EndDocument endTexFig 496 1946 a 21521665 15156103 3289088 3289088 26970521 19866091 startTexFig 496 1946 a %%BeginDocument: upw3.eps /gnudict 40 dict def gnudict begin /Color false def /Solid false def /gnulinewidth 5.000 def /vshift -46 def /dl {10 mul} def /hpt 31.5 def /vpt 31.5 def /M {moveto} bind def /L {lineto} bind def /R {rmoveto} bind def /V {rlineto} bind def /vpt2 vpt 2 mul def /hpt2 hpt 2 mul def /Lshow { currentpoint stroke M 0 vshift R show } def /Rshow { currentpoint stroke M dup stringwidth pop neg vshift R show } def /Cshow { currentpoint stroke M dup stringwidth pop -2 div vshift R show } def /DL { Color {setrgbcolor Solid {pop []} if 0 setdash } {pop pop pop Solid {pop []} if 0 setdash} ifelse } def /BL { stroke gnulinewidth 2 mul setlinewidth } def /AL { stroke gnulinewidth 2 div setlinewidth } def /PL { stroke gnulinewidth setlinewidth } def /LTb { BL [] 0 0 0 DL } def /LTa { AL [1 dl 2 dl] 0 setdash 0 0 0 setrgbcolor } def /LT0 { PL [] 0 1 0 DL } def /LT1 { PL [4 dl 2 dl] 0 0 1 DL } def /LT2 { PL [2 dl 3 dl] 1 0 0 DL } def /LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def /LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def /LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def /LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def /LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def /LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def /P { stroke [] 0 setdash currentlinewidth 2 div sub M 0 currentlinewidth V stroke } def /D { stroke [] 0 setdash 2 copy vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke P } def /A { stroke [] 0 setdash vpt sub M 0 vpt2 V currentpoint stroke M hpt neg vpt neg R hpt2 0 V stroke } def /B { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke P } def /C { stroke [] 0 setdash exch hpt sub exch vpt add M hpt2 vpt2 neg V currentpoint stroke M hpt2 neg 0 R hpt2 vpt2 V stroke } def /T { stroke [] 0 setdash 2 copy vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke P } def /S { 2 copy A C} def end gnudict begin gsave 50 50 translate 0.050 0.050 scale 0 setgray /Helvetica findfont 140 scalefont setfont newpath LTb LT0 6486 4486 M ("UpWind") Rshow 6570 4486 M 252 0 V 2896 1317 M 33 6 V 34 6 V 33 6 V 33 6 V 34 7 V 33 6 V 33 6 V 34 6 V 33 6 V 33 7 V 34 6 V 33 6 V 33 6 V 34 6 V 33 7 V 33 6 V 34 6 V 33 6 V 33 6 V 34 6 V 33 7 V 32 6 V 34 6 V 33 6 V 33 6 V 34 7 V 33 6 V 33 6 V 34 6 V 33 6 V 34 7 V 33 6 V 33 6 V 34 6 V 33 6 V 33 7 V 34 6 V 33 6 V 33 6 V 34 6 V 33 42 V 33 31 V 34 30 V 33 31 V 33 30 V 34 30 V 33 30 V 33 30 V 34 30 V 33 30 V 33 30 V 34 30 V 33 30 V 33 30 V 34 30 V 33 30 V 33 30 V 34 30 V 33 30 V 33 30 V 34 30 V 33 30 V 33 30 V 34 30 V 33 30 V 33 30 V 34 31 V 33 30 V 33 30 V 34 30 V 33 30 V 34 29 V 33 30 V 33 30 V 34 30 V 33 30 V 33 30 V 34 30 V 33 30 V 33 29 V 34 30 V 33 30 V 33 29 V 34 30 V 33 29 V 33 29 V 34 29 V 33 28 V 33 28 V 34 27 V 33 27 V 33 26 V 34 26 V 33 25 V 33 24 V 34 23 V 33 21 V 33 21 V 34 20 V 33 19 V 2415 1585 M 33 6 V 33 6 V 34 6 V 33 7 V 33 6 V 34 6 V 33 6 V 33 6 V 34 7 V 33 6 V 33 6 V 34 6 V 33 6 V 33 7 V 34 6 V 33 6 V 33 6 V 34 6 V 33 7 V 33 6 V 34 6 V 33 6 V 34 6 V 33 7 V 33 6 V 34 6 V 33 6 V 33 6 V 34 6 V 33 7 V 33 6 V 34 6 V 33 6 V 33 6 V 34 7 V 32 6 V 33 6 V 34 6 V 33 6 V 33 7 V 34 47 V 33 35 V 33 35 V 34 35 V 33 35 V 33 35 V 34 34 V 33 35 V 33 35 V 34 35 V 33 35 V 33 34 V 34 35 V 33 35 V 33 35 V 34 35 V 33 35 V 33 34 V 34 35 V 33 34 V 34 35 V 33 35 V 33 34 V 34 35 V 33 35 V 33 35 V 34 35 V 33 34 V 33 35 V 34 35 V 33 35 V 33 35 V 34 34 V 33 35 V 33 34 V 34 35 V 33 34 V 33 33 V 34 34 V 33 32 V 33 33 V 34 31 V 33 31 V 33 30 V 34 28 V 33 27 V 33 26 V 34 24 V 33 23 V 33 21 V 34 19 V 33 17 V 33 16 V 34 14 V 33 13 V 33 12 V 34 10 V 33 9 V 34 9 V 33 7 V 1933 1853 M 34 6 V 33 6 V 33 7 V 34 6 V 33 6 V 33 6 V 34 6 V 33 7 V 33 6 V 34 6 V 33 6 V 34 6 V 33 7 V 33 6 V 34 6 V 33 6 V 33 6 V 34 7 V 33 6 V 33 6 V 34 6 V 33 6 V 33 7 V 34 6 V 33 6 V 33 6 V 34 6 V 33 7 V 33 6 V 34 6 V 33 6 V 33 6 V 34 7 V 33 6 V 33 6 V 34 6 V 33 6 V 33 6 V 34 7 V 33 6 V 33 54 V 34 42 V 33 42 V 33 42 V 34 42 V 33 42 V 33 42 V 34 42 V 33 42 V 33 41 V 33 42 V 33 42 V 34 42 V 33 42 V 33 42 V 34 42 V 33 42 V 33 42 V 34 42 V 33 42 V 33 42 V 34 42 V 33 42 V 33 42 V 34 42 V 33 42 V 33 41 V 34 42 V 33 42 V 33 41 V 34 41 V 33 40 V 33 39 V 34 37 V 33 37 V 33 34 V 34 32 V 33 30 V 33 27 V 34 25 V 33 22 V 33 19 V 34 16 V 33 15 V 33 12 V 34 11 V 33 9 V 33 8 V 34 8 V 33 7 V 34 7 V 33 6 V 33 7 V 34 6 V 33 6 V 33 6 V 34 7 V 33 6 V 33 6 V 34 6 V 1452 2121 M 33 7 V 34 6 V 33 6 V 34 6 V 33 6 V 33 7 V 34 6 V 33 6 V 33 6 V 34 6 V 33 6 V 33 7 V 34 6 V 33 6 V 33 6 V 34 6 V 33 7 V 33 6 V 34 6 V 33 6 V 33 6 V 34 7 V 33 6 V 33 6 V 34 6 V 33 6 V 33 7 V 34 6 V 33 6 V 33 6 V 34 6 V 33 7 V 33 6 V 34 6 V 33 6 V 33 6 V 34 7 V 33 6 V 33 6 V 34 6 V 33 64 V 34 54 V 33 53 V 33 54 V 34 53 V 33 54 V 33 54 V 34 54 V 33 54 V 33 54 V 34 54 V 33 54 V 33 54 V 34 54 V 33 54 V 33 54 V 34 54 V 33 54 V 33 53 V 34 54 V 33 54 V 33 54 V 34 54 V 33 53 V 32 51 V 34 49 V 33 46 V 33 42 V 34 36 V 33 31 V 33 25 V 34 21 V 33 16 V 33 13 V 34 10 V 33 8 V 33 8 V 34 7 V 33 6 V 34 6 V 33 7 V 33 6 V 34 6 V 33 6 V 33 7 V 34 6 V 33 6 V 33 6 V 34 6 V 33 7 V 33 6 V 34 6 V 33 6 V 33 6 V currentpoint stroke M 34 7 V 33 6 V 33 6 V 34 6 V 33 6 V 33 7 V 971 2390 M 33 6 V 34 6 V 33 6 V 33 6 V 34 7 V 33 6 V 33 6 V 34 6 V 33 6 V 33 7 V 34 6 V 33 6 V 33 6 V 34 6 V 33 7 V 33 6 V 34 6 V 33 6 V 33 6 V 34 6 V 33 7 V 33 5 V 34 6 V 33 6 V 33 6 V 34 7 V 33 6 V 33 6 V 34 6 V 33 6 V 34 7 V 33 6 V 33 6 V 34 6 V 33 6 V 33 7 V 34 6 V 33 6 V 33 6 V 34 6 V 33 78 V 33 78 V 34 78 V 33 77 V 33 78 V 34 78 V 33 77 V 33 78 V 34 78 V 33 78 V 33 77 V 34 78 V 33 78 V 33 78 V 34 77 V 33 78 V 33 78 V 34 78 V 33 77 V 33 78 V 34 6 V 33 6 V 33 7 V 34 6 V 33 6 V 33 6 V 34 6 V 33 7 V 33 6 V 34 6 V 33 6 V 34 6 V 33 6 V 33 7 V 34 6 V 33 6 V 33 6 V 34 6 V 32 7 V 33 6 V 34 6 V 33 6 V 33 6 V 34 7 V 33 6 V 33 6 V 34 6 V 33 6 V 33 7 V 34 6 V 33 6 V 33 6 V 34 6 V 33 7 V 33 6 V 34 6 V 33 6 V 33 6 V 34 7 V 33 6 V 6229 3298 M -481 332 V -481 272 V -482 269 V -481 268 V 6196 3279 M -481 344 V -482 273 V -481 268 V -481 269 V 6162 3259 M -481 355 V -481 276 V -481 268 V -482 268 V 6129 3238 M -481 367 V -481 279 V -482 268 V -481 268 V 6096 3217 M -482 378 V -481 282 V -481 269 V -481 268 V 6062 3194 M -481 389 V -481 288 V -481 269 V -482 268 V 6029 3170 M -481 400 V -481 295 V -482 268 V -481 269 V 5996 3145 M -482 411 V -481 303 V -481 268 V -481 268 V 5962 3119 M -481 421 V -481 312 V -481 269 V -482 268 V 5929 3093 M -481 430 V -481 323 V -482 269 V -481 268 V 5896 3066 M -482 438 V -481 335 V -481 270 V -481 268 V 5862 3039 M -481 444 V -481 349 V -481 270 V -482 269 V 5829 3011 M -481 449 V -482 364 V -481 272 V -481 268 V 5796 2983 M -482 453 V -481 380 V -481 274 V -481 268 V 5762 2954 M -481 456 V -481 397 V -481 277 V -482 268 V 5729 2925 M -481 458 V -482 413 V -481 282 V -481 268 V 5696 2896 M -482 459 V -481 429 V -481 287 V -481 269 V 5662 2866 M -481 459 V -481 444 V -481 296 V -482 268 V 5629 2837 M -481 457 V -482 459 V -481 306 V -481 268 V 5596 2807 M -482 456 V -481 471 V -481 319 V -481 268 V 5562 2777 M -481 453 V -481 482 V -481 334 V -482 269 V 5529 2748 M -481 450 V -482 489 V -481 353 V -481 269 V 5496 2718 M -482 446 V -481 496 V -481 374 V -480 268 V 5462 2688 M -481 443 V -481 499 V -482 397 V -480 269 V 5429 2658 M -481 439 V -482 501 V -481 421 V -480 271 V 5396 2628 M -482 434 V -481 502 V -481 447 V -480 273 V 5362 2598 M -481 430 V -481 499 V -482 474 V -480 277 V 5329 2568 M -481 425 V -482 497 V -481 498 V -480 283 V 5296 2538 M -482 421 V -481 492 V -481 521 V -480 293 V 5262 2509 M -481 415 V -481 487 V -482 540 V -480 308 V 5229 2479 M -481 410 V -482 481 V -481 556 V -480 327 V 5195 2449 M -481 405 V -481 475 V -481 566 V -481 352 V 5162 2419 M -481 400 V -481 468 V -482 572 V -480 382 V 5129 2389 M -481 396 V -482 460 V -481 572 V -480 417 V 5095 2358 M -481 392 V -481 454 V -481 567 V -481 457 V 5062 2328 M -481 387 V -481 447 V -482 560 V -480 500 V 5029 2298 M -481 382 V -482 440 V -480 551 V -481 545 V 4995 2268 M -481 377 V -481 433 V -480 540 V -482 592 V 4962 2238 M -481 373 V -481 425 V -481 528 V -481 639 V 4929 2208 M -481 368 V -482 418 V -480 516 V -481 687 V 4895 2178 M -481 363 V -481 411 V -480 504 V -482 735 V 4862 2148 M -481 359 V -481 403 V -481 492 V -481 711 V 4829 2118 M -482 354 V -481 396 V -480 481 V -481 687 V 4795 2088 M -481 350 V -481 388 V -480 469 V -482 663 V 4762 2058 M -481 345 V -481 381 V -481 457 V -481 639 V 4729 2028 M -482 340 V -481 374 V -480 445 V -481 615 V 4695 1998 M -481 335 V -481 367 V -480 433 V -482 592 V 4662 1968 M -481 330 V -481 360 V -481 421 V -481 568 V 4629 1938 M -482 325 V -481 353 V -480 409 V -481 544 V 4595 1908 M -481 321 V -481 345 V -480 397 V -482 520 V 4562 1878 M -481 316 V -481 338 V -481 385 V -481 497 V 4529 1848 M -482 311 V -480 332 V -481 372 V -481 473 V 4495 1818 M -481 306 V -480 325 V -481 360 V -482 449 V 4462 1788 M -481 301 V -481 318 V -481 348 V -481 425 V 4429 1758 M -482 297 V -480 310 V -481 336 V -481 402 V 4395 1728 M -481 292 V -480 303 V -481 324 V -482 378 V 4362 1698 M -481 287 V -481 296 V -481 313 V -481 353 V 4329 1667 M -482 283 V -480 289 V -481 301 V -481 330 V 4295 1637 M -481 278 V -480 282 V currentpoint stroke M -481 290 V -482 305 V 4262 1606 M -481 274 V -481 275 V -481 278 V -481 281 V 4229 1564 M -482 269 V -480 268 V -481 268 V -481 267 V 4195 1558 M -481 268 V -480 269 V -482 268 V -481 267 V 4162 1552 M -481 268 V -481 268 V -481 269 V -481 267 V 4129 1546 M -482 268 V -480 268 V -481 269 V -481 267 V 4095 1540 M -481 268 V -480 268 V -482 268 V -481 268 V 4062 1533 M -480 269 V -482 268 V -481 268 V -481 267 V 4029 1527 M -481 268 V -481 269 V -481 268 V -481 267 V 3995 1521 M -480 268 V -481 269 V -482 268 V -481 267 V 3962 1515 M -480 268 V -482 268 V -481 269 V -481 267 V 3929 1509 M -481 268 V -481 268 V -481 268 V -481 268 V 3895 1502 M -480 269 V -481 268 V -482 268 V -481 267 V 3862 1496 M -480 268 V -482 269 V -481 268 V -481 267 V 3828 1490 M -480 268 V -481 269 V -481 268 V -482 267 V 3795 1484 M -480 268 V -481 268 V -482 269 V -481 267 V 3762 1478 M -480 268 V -482 268 V -481 268 V -481 268 V 3728 1471 M -480 269 V -481 268 V -481 268 V -482 267 V 3695 1465 M -480 269 V -481 268 V -482 268 V -481 267 V 3662 1459 M 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1348 M -482 268 V -481 268 V -481 268 V -481 269 V 3029 1341 M -481 269 V -481 268 V -481 268 V -482 268 V 2996 1335 M -481 268 V -482 269 V -481 268 V -481 268 V 2963 1329 M -482 268 V -481 268 V -481 269 V -481 268 V 2929 1323 M -481 268 V -481 268 V -482 269 V -481 268 V 2896 1317 M -481 268 V -482 268 V -481 268 V 971 2390 L LTb 971 1674 M 3333 620 V 6229 1221 L 2896 601 L 971 1674 L 2896 601 M -55 30 V 3061 509 M (0) Lshow 4562 911 M -55 30 V 4727 820 M (0.5) Lshow 6229 1221 M -55 30 V 220 -121 R (1) Lshow 971 1674 M 61 11 V 786 1640 M (0) Rshow 1212 1540 M 61 11 V -246 -45 R (0.05) Rshow 1452 1406 M 61 11 V -246 -45 R (0.1) Rshow 1693 1272 M 61 11 V -246 -45 R (0.15) Rshow 1933 1138 M 61 11 V -246 -45 R (0.2) Rshow 2174 1004 M 61 11 V 1989 970 M (0.25) Rshow 2415 869 M 61 11 V 2230 835 M (0.3) Rshow 2655 735 M 61 11 V 2470 701 M (0.35) Rshow 2896 601 M 61 11 V 2711 567 M (0.4) Rshow 971 2390 M 63 0 V -126 0 R (0) Rshow 971 3104 M 63 0 V -126 0 R (0.5) Rshow 971 3819 M 63 0 V -126 0 R (1) Rshow 971 1674 M 0 2145 V LTa stroke grestore end showpage %%EndDocument endTexFig 496 3891 a 21521665 15156103 3289088 3289088 26970521 19866091 startTexFig 496 3891 a %%BeginDocument: god3.eps /gnudict 40 dict def gnudict begin /Color false def /Solid false def /gnulinewidth 5.000 def /vshift -46 def /dl {10 mul} def /hpt 31.5 def /vpt 31.5 def /M {moveto} bind def /L {lineto} bind def /R {rmoveto} bind def /V {rlineto} bind def /vpt2 vpt 2 mul def /hpt2 hpt 2 mul def /Lshow { currentpoint stroke M 0 vshift R show } def /Rshow { currentpoint stroke M dup stringwidth pop neg vshift R show } def /Cshow { currentpoint stroke M dup stringwidth pop -2 div vshift R show } def /DL { Color {setrgbcolor Solid {pop []} if 0 setdash } {pop pop pop Solid {pop []} if 0 setdash} ifelse } def /BL { stroke gnulinewidth 2 mul setlinewidth } def /AL { stroke gnulinewidth 2 div setlinewidth } def /PL { stroke gnulinewidth setlinewidth } def /LTb { BL [] 0 0 0 DL } def /LTa { AL [1 dl 2 dl] 0 setdash 0 0 0 setrgbcolor } def /LT0 { PL [] 0 1 0 DL } def /LT1 { PL [4 dl 2 dl] 0 0 1 DL } def /LT2 { PL [2 dl 3 dl] 1 0 0 DL } def /LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def /LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def /LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def /LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def /LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def /LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def /P { stroke [] 0 setdash currentlinewidth 2 div sub M 0 currentlinewidth V stroke } def /D { stroke [] 0 setdash 2 copy vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke P } def /A { stroke [] 0 setdash vpt sub M 0 vpt2 V currentpoint stroke M hpt neg vpt neg R hpt2 0 V stroke } def /B { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke P } def /C { stroke [] 0 setdash exch hpt sub exch vpt add M hpt2 vpt2 neg V currentpoint stroke M hpt2 neg 0 R hpt2 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220 -121 R (1) Lshow 971 1674 M 61 11 V 786 1640 M (0) Rshow 1212 1540 M 61 11 V -246 -45 R (0.05) Rshow 1452 1406 M 61 11 V -246 -45 R (0.1) Rshow 1693 1272 M 61 11 V -246 -45 R (0.15) Rshow 1933 1138 M 61 11 V -246 -45 R (0.2) Rshow 2174 1004 M 61 11 V 1989 970 M (0.25) Rshow 2415 869 M 61 11 V 2230 835 M (0.3) Rshow 2655 735 M 61 11 V 2470 701 M (0.35) Rshow 2896 601 M 61 11 V 2711 567 M (0.4) Rshow 971 2390 M 63 0 V -126 0 R (0) Rshow 971 3104 M 63 0 V -126 0 R (0.5) Rshow 971 3819 M 63 0 V -126 0 R (1) Rshow 971 1674 M 0 2145 V LTa stroke grestore end showpage %%EndDocument endTexFig -165 6010 a Ft(46)p eop %%Page: 47 52 47 51 bop 2306 -116 a Fr(Numeri)n(\024)-39 b(ck)n(o)25 b(rje)l(\024)-37 b(sa)n(v)-5 b(anje)26 b(Burgerso)n(v)n(e)e(jednad)n (\024)-39 b(zb)r(e)732 0 y 21521665 15156103 3289088 3289088 26970521 19866091 startTexFig 732 0 a %%BeginDocument: lf4.eps /gnudict 40 dict def gnudict begin /Color false def /Solid false def /gnulinewidth 5.000 def /vshift -46 def /dl {10 mul} def /hpt 31.5 def /vpt 31.5 def /M {moveto} bind def /L {lineto} bind def /R {rmoveto} bind def /V {rlineto} bind def /vpt2 vpt 2 mul def /hpt2 hpt 2 mul def /Lshow { currentpoint stroke M 0 vshift R show } def /Rshow { currentpoint stroke M dup stringwidth pop neg vshift R show } def /Cshow { currentpoint stroke M dup stringwidth pop -2 div vshift R show } def /DL { Color {setrgbcolor Solid {pop []} if 0 setdash } {pop pop pop Solid {pop []} if 0 setdash} ifelse } def /BL { stroke gnulinewidth 2 mul setlinewidth } def /AL { stroke gnulinewidth 2 div setlinewidth } def /PL { stroke gnulinewidth setlinewidth } def /LTb { BL [] 0 0 0 DL } def /LTa { AL [1 dl 2 dl] 0 setdash 0 0 0 setrgbcolor } def /LT0 { PL [] 0 1 0 DL } def /LT1 { PL [4 dl 2 dl] 0 0 1 DL } def /LT2 { PL [2 dl 3 dl] 1 0 0 DL } def /LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def /LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def /LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def /LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def /LT7 { 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Rshow 2655 735 M 61 11 V 2470 701 M (0.35) Rshow 2896 601 M 61 11 V 2711 567 M (0.4) Rshow 971 1674 M 63 0 V -126 0 R (-0.5) Rshow 971 2390 M 63 0 V -126 0 R (0) Rshow 971 3104 M 63 0 V -126 0 R (0.5) Rshow 971 3819 M 63 0 V -126 0 R (1) Rshow 971 1674 M 0 2145 V LTa stroke grestore end showpage %%EndDocument endTexFig 732 1946 a 21521665 15156103 3289088 3289088 26970521 19866091 startTexFig 732 1946 a %%BeginDocument: upw4.eps /gnudict 40 dict def gnudict begin /Color false def /Solid false def /gnulinewidth 5.000 def /vshift -46 def /dl {10 mul} def /hpt 31.5 def /vpt 31.5 def /M {moveto} bind def /L {lineto} bind def /R {rmoveto} bind def /V {rlineto} bind def /vpt2 vpt 2 mul def /hpt2 hpt 2 mul def /Lshow { currentpoint stroke M 0 vshift R show } def /Rshow { currentpoint stroke M dup stringwidth pop neg vshift R show } def /Cshow { currentpoint stroke M dup stringwidth pop -2 div vshift R show } def /DL { Color {setrgbcolor Solid {pop []} if 0 setdash } {pop pop pop Solid {pop []} if 0 setdash} ifelse } def /BL { stroke gnulinewidth 2 mul setlinewidth } def /AL { stroke gnulinewidth 2 div setlinewidth } def /PL { stroke gnulinewidth setlinewidth } def /LTb { BL [] 0 0 0 DL } def /LTa { AL [1 dl 2 dl] 0 setdash 0 0 0 setrgbcolor } def /LT0 { PL [] 0 1 0 DL } def /LT1 { PL [4 dl 2 dl] 0 0 1 DL } def /LT2 { PL [2 dl 3 dl] 1 0 0 DL } def /LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def /LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def /LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def /LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def /LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def /LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def /P { stroke [] 0 setdash currentlinewidth 2 div sub M 0 currentlinewidth V stroke } def /D { stroke [] 0 setdash 2 copy vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke P } def /A { stroke [] 0 setdash vpt sub M 0 vpt2 V currentpoint stroke M hpt neg vpt neg R hpt2 0 V stroke } def /B { 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M -482 268 V -481 268 V -481 269 V -481 268 V 3429 1416 M -481 268 V -481 268 V -481 268 V -482 269 V 3396 1410 M -481 268 V -481 268 V -482 268 V -481 269 V 3363 1403 M -482 269 V -481 268 V -481 268 V -481 268 V 3329 1397 M -481 268 V -481 269 V -481 268 V -482 268 V 3296 1391 M -481 268 V -481 268 V -482 269 V -481 268 V 3263 1385 M -482 268 V -481 268 V -481 268 V -481 269 V 3229 1379 M -481 268 V -481 268 V -481 268 V -482 269 V 3196 1372 M -481 269 V -482 268 V -481 268 V -481 268 V 3163 1366 M -482 268 V -481 269 V -481 268 V -481 268 V 3129 1360 M -481 268 V -481 268 V -481 269 V -482 268 V 3096 1354 M -481 268 V -482 268 V -481 269 V -481 268 V 3063 1348 M -482 268 V -481 268 V -481 268 V -481 269 V 3029 1341 M -481 269 V -481 268 V -481 268 V -482 268 V 2996 1335 M -481 268 V -482 269 V -481 268 V -481 268 V 2963 1329 M -482 268 V -481 268 V -481 269 V -481 268 V 2929 1323 M -481 268 V -481 268 V -482 269 V -481 268 V 2896 1317 M -481 268 V -482 268 V -481 268 V 971 2390 L LTb 971 1674 M 3333 620 V 6229 1221 L 2896 601 L 971 1674 L 2896 601 M -55 30 V 3061 509 M (0) Lshow 4562 911 M -55 30 V 4727 820 M (0.5) Lshow 6229 1221 M -55 30 V 220 -121 R (1) Lshow 971 1674 M 61 11 V 786 1640 M (0) Rshow 1212 1540 M 61 11 V -246 -45 R (0.05) Rshow 1452 1406 M 61 11 V -246 -45 R (0.1) Rshow 1693 1272 M 61 11 V -246 -45 R (0.15) Rshow 1933 1138 M 61 11 V -246 -45 R (0.2) Rshow 2174 1004 M 61 11 V 1989 970 M (0.25) Rshow 2415 869 M 61 11 V 2230 835 M (0.3) Rshow 2655 735 M 61 11 V 2470 701 M (0.35) Rshow 2896 601 M 61 11 V 2711 567 M (0.4) Rshow 971 2390 M 63 0 V -126 0 R (0) Rshow 971 3104 M 63 0 V -126 0 R (0.5) Rshow 971 3819 M 63 0 V -126 0 R (1) Rshow 971 1674 M 0 2145 V LTa stroke grestore end showpage %%EndDocument endTexFig 732 3891 a 21521665 15156103 3289088 3289088 26970521 19866091 startTexFig 732 3891 a %%BeginDocument: god4.eps /gnudict 40 dict def gnudict begin /Color false def /Solid false def /gnulinewidth 5.000 def /vshift -46 def /dl {10 mul} def /hpt 31.5 def /vpt 31.5 def /M {moveto} bind def /L {lineto} bind def /R {rmoveto} bind def /V {rlineto} bind def /vpt2 vpt 2 mul def /hpt2 hpt 2 mul def /Lshow { currentpoint stroke M 0 vshift R show } def /Rshow { currentpoint stroke M dup stringwidth pop neg vshift R show } def /Cshow { currentpoint stroke M dup stringwidth pop -2 div vshift R show } def /DL { Color {setrgbcolor Solid {pop []} if 0 setdash } {pop pop pop Solid {pop []} if 0 setdash} ifelse } def /BL { stroke gnulinewidth 2 mul setlinewidth } def /AL { stroke gnulinewidth 2 div setlinewidth } def /PL { stroke gnulinewidth setlinewidth } def /LTb { BL [] 0 0 0 DL } def /LTa { AL [1 dl 2 dl] 0 setdash 0 0 0 setrgbcolor } def /LT0 { PL [] 0 1 0 DL } def /LT1 { PL [4 dl 2 dl] 0 0 1 DL } def /LT2 { PL [2 dl 3 dl] 1 0 0 DL } def /LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def /LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def /LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def /LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def /LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def /LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def /P { stroke [] 0 setdash currentlinewidth 2 div sub M 0 currentlinewidth V stroke } def /D { stroke [] 0 setdash 2 copy vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke P } def /A { stroke [] 0 setdash vpt sub M 0 vpt2 V currentpoint stroke M hpt neg vpt neg R hpt2 0 V stroke } def /B { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke P } def /C { stroke [] 0 setdash exch hpt sub exch vpt add M hpt2 vpt2 neg V currentpoint stroke M hpt2 neg 0 R hpt2 vpt2 V stroke } def /T { stroke [] 0 setdash 2 copy vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke P } def /S { 2 copy A C} def end gnudict begin gsave 50 50 translate 0.050 0.050 scale 0 setgray /Helvetica findfont 140 scalefont setfont newpath LTb LT0 6486 4486 M ("Godunov") Rshow 6570 4486 M 252 0 V 2896 1317 M 33 6 V 34 6 V 33 6 V 33 6 V 34 7 V 33 6 V 33 6 V 34 6 V 33 6 V 33 7 V 34 6 V 33 6 V 33 6 V 34 6 V 33 7 V 33 6 V 34 6 V 33 6 V 33 6 V 34 6 V 33 7 V 32 6 V 34 6 V 33 6 V 33 6 V 34 7 V 33 6 V 33 6 V 34 6 V 33 6 V 34 7 V 33 6 V 33 6 V 34 6 V 33 6 V 33 7 V 34 6 V 33 6 V 33 6 V 34 6 V 33 7 V 33 6 V 34 6 V 33 6 V 33 6 V 34 7 V 33 6 V 33 6 V 34 6 V 33 6 V 33 74 V 34 49 V 33 45 V 33 44 V 34 43 V 33 42 V 33 42 V 34 42 V 33 41 V 33 41 V 34 41 V 33 41 V 33 40 V 34 41 V 33 40 V 33 40 V 34 39 V 33 40 V 33 39 V 34 39 V 33 38 V 34 38 V 33 38 V 33 38 V 34 37 V 33 38 V 33 36 V 34 36 V 33 36 V 33 35 V 34 35 V 33 34 V 33 33 V 34 32 V 33 32 V 33 30 V 34 30 V 33 28 V 33 27 V 34 26 V 33 25 V 33 23 V 34 22 V 33 20 V 33 19 V 34 17 V 33 16 V 33 14 V 34 13 V 33 12 V 2415 1585 M 33 6 V 33 6 V 34 6 V 33 7 V 33 6 V 34 6 V 33 6 V 33 6 V 34 7 V 33 6 V 33 6 V 34 6 V 33 6 V 33 7 V 34 6 V 33 6 V 33 6 V 34 6 V 33 7 V 33 6 V 34 6 V 33 6 V 34 6 V 33 7 V 33 6 V 34 6 V 33 6 V 33 6 V 34 6 V 33 7 V 33 6 V 34 6 V 33 6 V 33 6 V 34 7 V 32 6 V 33 6 V 34 6 V 33 6 V 33 7 V 34 6 V 33 6 V 33 6 V 34 6 V 33 7 V 33 6 V 34 6 V 33 6 V 33 6 V 34 7 V 33 95 V 33 61 V 34 57 V 33 56 V 33 54 V 34 53 V 33 53 V 33 52 V 34 52 V 33 51 V 34 50 V 33 51 V 33 50 V 34 49 V 33 49 V 33 48 V 34 48 V 33 47 V 33 47 V 34 45 V 33 45 V 33 44 V 34 43 V 33 42 V 33 40 V 34 39 V 33 38 V 33 36 V 34 34 V 33 32 V 33 30 V 34 28 V 33 25 V 33 23 V 34 22 V 33 18 V 33 17 V 34 15 V 33 13 V 33 11 V 34 11 V 33 9 V 33 8 V 34 8 V 33 7 V 33 7 V 34 7 V 33 6 V 34 7 V 33 6 V 1933 1853 M 34 6 V 33 6 V 33 7 V 34 6 V 33 6 V 33 6 V 34 6 V 33 7 V 33 6 V 34 6 V 33 6 V 34 6 V 33 7 V 33 6 V 34 6 V 33 6 V 33 6 V 34 7 V 33 6 V 33 6 V 34 6 V 33 6 V 33 7 V 34 6 V 33 6 V 33 6 V 34 6 V 33 7 V 33 6 V 34 6 V 33 6 V 33 6 V 34 7 V 33 6 V 33 6 V 34 6 V 33 6 V 33 6 V 34 7 V 33 6 V 33 6 V 34 6 V 33 6 V 33 7 V 34 6 V 33 6 V 33 6 V 34 6 V 33 7 V 33 6 V 33 136 V 33 86 V 34 79 V 33 76 V 33 75 V 34 74 V 33 72 V 33 70 V 34 70 V 33 68 V 33 66 V 34 65 V 33 63 V 33 61 V 34 59 V 33 57 V 33 54 V 34 51 V 33 47 V 33 44 V 34 40 V 33 36 V 33 32 V 34 28 V 33 25 V 33 20 V 34 17 V 33 15 V 33 12 V 34 10 V 33 9 V 33 8 V 34 7 V 33 7 V 33 7 V 34 6 V 33 7 V 33 6 V 34 6 V 33 6 V 34 6 V 33 7 V 33 6 V 34 6 V 33 6 V 33 6 V 34 7 V 33 6 V 33 6 V 34 6 V 1452 2121 M 33 7 V 34 6 V 33 6 V 34 6 V 33 6 V 33 7 V 34 6 V 33 6 V 33 6 V 34 6 V 33 6 V 33 7 V 34 6 V 33 6 V 33 6 V 34 6 V 33 7 V 33 6 V 34 6 V 33 6 V 33 6 V 34 7 V 33 6 V 33 6 V 34 6 V 33 6 V 33 7 V 34 6 V 33 6 V 33 6 V 34 6 V 33 7 V 33 6 V 34 6 V 33 6 V 33 6 V 34 7 V 33 6 V 33 6 V 34 6 V 33 6 V 34 7 V 33 6 V 33 6 V 34 6 V 33 6 V 33 6 V 34 7 V 33 6 V 33 6 V 34 243 V 33 150 V 33 136 V 34 128 V 33 120 V 33 114 V 34 106 V 33 97 V 33 88 V 34 78 V 33 67 V 33 56 V 34 45 V 33 35 V 32 27 V 34 20 V 33 15 V 33 11 V 34 9 V 33 7 V 33 7 V 34 7 V 33 6 V 33 6 V 34 7 V 33 6 V 33 6 V 34 6 V 33 6 V 34 7 V 33 6 V 33 6 V 34 6 V 33 6 V 33 7 V 34 6 V 33 6 V 33 6 V 34 6 V 33 7 V 33 6 V 34 6 V 33 6 V 33 6 V currentpoint stroke M 34 7 V 33 6 V 33 6 V 34 6 V 33 6 V 33 7 V 971 2390 M 33 6 V 34 6 V 33 6 V 33 6 V 34 7 V 33 6 V 33 6 V 34 6 V 33 6 V 33 7 V 34 6 V 33 6 V 33 6 V 34 6 V 33 7 V 33 6 V 34 6 V 33 6 V 33 6 V 34 6 V 33 7 V 33 5 V 34 6 V 33 6 V 33 6 V 34 7 V 33 6 V 33 6 V 34 6 V 33 6 V 34 7 V 33 6 V 33 6 V 34 6 V 33 6 V 33 7 V 34 6 V 33 6 V 33 6 V 34 6 V 33 7 V 33 6 V 34 6 V 33 6 V 33 6 V 34 7 V 33 6 V 33 6 V 34 6 V 33 6 V 33 1437 V 34 6 V 33 7 V 33 6 V 34 6 V 33 6 V 33 6 V 34 7 V 33 6 V 33 6 V 34 6 V 33 6 V 33 7 V 34 6 V 33 6 V 33 6 V 34 6 V 33 7 V 33 6 V 34 6 V 33 6 V 34 6 V 33 6 V 33 7 V 34 6 V 33 6 V 33 6 V 34 6 V 32 7 V 33 6 V 34 6 V 33 6 V 33 6 V 34 7 V 33 6 V 33 6 V 34 6 V 33 6 V 33 7 V 34 6 V 33 6 V 33 6 V 34 6 V 33 7 V 33 6 V 34 6 V 33 6 V 33 6 V 34 7 V 33 6 V 6229 3347 M -481 287 V -481 268 V -482 269 V -481 268 V 6196 3335 M -481 293 V -482 268 V -481 268 V -481 269 V 6162 3322 M -481 299 V -481 269 V -481 268 V -482 268 V 6129 3308 M -481 307 V -481 269 V -482 268 V -481 268 V 6096 3292 M -482 316 V -481 269 V -481 269 V -481 268 V 6062 3275 M -481 326 V -481 270 V -481 269 V -482 268 V 6029 3256 M -481 338 V -481 271 V -482 268 V -481 269 V 5996 3236 M -482 350 V -481 273 V -481 268 V -481 268 V 5962 3214 M -481 364 V -481 275 V -481 268 V -482 268 V 5929 3191 M -481 378 V -481 277 V -482 269 V -481 268 V 5896 3166 M -482 392 V -481 282 V -481 269 V -481 268 V 5862 3140 M -481 407 V -481 287 V -481 268 V -482 269 V 5829 3113 M -481 421 V -482 294 V -481 268 V -481 268 V 5796 3085 M -482 434 V -481 303 V -481 268 V -481 268 V 5762 3055 M -481 447 V -481 313 V -481 269 V -482 268 V 5729 3025 M -481 459 V -482 325 V -481 269 V -481 268 V 5696 2993 M -482 469 V -481 340 V -481 269 V -481 269 V 5662 2961 M -481 478 V -481 356 V -481 270 V -482 268 V 5629 2928 M -481 486 V -482 374 V -481 271 V -481 268 V 5596 2894 M -482 492 V -481 394 V -481 273 V -481 268 V 5562 2859 M -481 497 V -481 415 V -481 276 V -482 268 V 5529 2824 M -481 500 V -482 437 V -481 279 V -481 269 V 5496 2788 M -482 502 V -481 459 V -481 285 V -480 268 V 5462 2752 M -481 502 V -481 480 V -482 294 V -480 268 V 5429 2716 M -481 500 V -482 501 V -481 305 V -480 268 V 5396 2678 M -482 499 V -481 520 V -481 319 V -480 268 V 5362 2641 M -481 496 V -481 535 V -482 337 V -480 269 V 5329 2603 M -481 492 V -482 549 V -481 359 V -480 268 V 5296 2565 M -482 487 V -481 560 V -481 385 V -480 268 V 5262 2527 M -481 481 V -481 568 V -482 414 V -480 269 V 5229 2489 M -481 474 V -482 573 V -481 447 V -480 270 V 5195 2450 M -481 468 V -481 574 V -481 484 V -481 271 V 5162 2411 M -481 460 V -481 574 V -482 522 V -480 274 V 5129 2371 M -481 453 V -482 570 V -481 562 V -480 278 V 5095 2332 M -481 444 V -481 564 V -481 601 V -481 287 V 5062 2292 M -481 436 V -481 555 V -482 638 V -480 301 V 5029 2252 M -481 427 V -482 545 V -480 670 V -481 322 V 4995 2211 M -481 419 V -481 533 V -480 696 V -482 351 V 4962 2171 M -481 409 V -481 520 V -481 714 V -481 389 V 4929 2130 M -481 399 V -482 506 V -480 723 V -481 439 V 4895 2089 M -481 390 V -481 490 V -480 722 V -482 500 V 4862 2048 M -481 380 V -481 473 V -481 712 V -481 572 V 4829 2007 M -482 369 V -481 455 V -480 694 V -481 654 V 4795 1965 M -481 359 V -481 437 V -480 667 V -482 744 V 4762 1923 M -481 348 V -481 418 V -481 633 V -481 844 V 4729 1881 M -482 337 V -481 397 V -480 593 V -481 952 V 4695 1838 M -481 326 V -481 376 V -480 548 V 2771 4154 L 4662 1794 M -481 314 V -481 356 V -481 496 V 2738 4148 L 4629 1749 M -482 302 V -481 334 V -480 439 V 2705 4141 L 4595 1700 M -481 290 V -481 309 V -480 375 V 2671 4135 L 4562 1626 M -481 269 V -481 268 V -481 268 V -481 267 V 4529 1620 M -482 268 V -480 269 V -481 268 V -481 267 V 4495 1614 M -481 268 V -480 268 V -481 269 V -482 267 V 4462 1608 M -481 268 V -481 268 V -481 268 V -481 268 V 4429 1602 M -482 268 V -480 268 V -481 268 V -481 268 V 4395 1595 M -481 269 V -480 268 V -481 268 V -482 267 V 4362 1589 M -481 268 V -481 269 V -481 268 V -481 267 V 4329 1583 M -482 268 V -480 268 V -481 269 V -481 267 V 4295 1577 M -481 268 V -480 268 V currentpoint stroke M -481 269 V -482 267 V 4262 1571 M -481 268 V -481 268 V -481 268 V -481 268 V 4229 1564 M -482 269 V -480 268 V -481 268 V -481 267 V 4195 1558 M -481 268 V -480 269 V -482 268 V -481 267 V 4162 1552 M -481 268 V -481 268 V -481 269 V -481 267 V 4129 1546 M -482 268 V -480 268 V -481 269 V -481 267 V 4095 1540 M -481 268 V -480 268 V -482 268 V -481 268 V 4062 1533 M -480 269 V -482 268 V -481 268 V -481 267 V 4029 1527 M -481 268 V -481 269 V -481 268 V -481 267 V 3995 1521 M -480 268 V -481 269 V -482 268 V -481 267 V 3962 1515 M -480 268 V -482 268 V -481 269 V -481 267 V 3929 1509 M -481 268 V -481 268 V -481 268 V -481 268 V 3895 1502 M -480 269 V -481 268 V -482 268 V -481 267 V 3862 1496 M -480 268 V -482 269 V -481 268 V -481 267 V 3828 1490 M -480 268 V -481 269 V -481 268 V -482 267 V 3795 1484 M -480 268 V -481 268 V -482 269 V -481 267 V 3762 1478 M -480 268 V -482 268 V -481 268 V -481 268 V 3728 1471 M -480 269 V -481 268 V -481 268 V -482 267 V 3695 1465 M -480 269 V -481 268 V -482 268 V -481 267 V 3662 1459 M -480 268 V -482 269 V -481 268 V -481 267 V 3628 1453 M -480 268 V -481 268 V -481 269 V -482 267 V 3596 1447 M -481 268 V -481 268 V -482 268 V -481 269 V 3563 1440 M -482 269 V -481 268 V -481 268 V -481 268 V 3529 1434 M -481 269 V -481 268 V -481 268 V -482 268 V 3496 1428 M -481 268 V -481 269 V -482 268 V -481 268 V 3463 1422 M -482 268 V -481 268 V -481 269 V -481 268 V 3429 1416 M -481 268 V -481 268 V -481 268 V -482 269 V 3396 1410 M -481 268 V -481 268 V -482 268 V -481 269 V 3363 1403 M -482 269 V -481 268 V -481 268 V -481 268 V 3329 1397 M -481 268 V -481 269 V -481 268 V -482 268 V 3296 1391 M -481 268 V -481 268 V -482 269 V -481 268 V 3263 1385 M -482 268 V -481 268 V -481 268 V -481 269 V 3229 1379 M -481 268 V -481 268 V -481 268 V -482 269 V 3196 1372 M -481 269 V -482 268 V -481 268 V -481 268 V 3163 1366 M -482 268 V -481 269 V -481 268 V -481 268 V 3129 1360 M -481 268 V -481 268 V -481 269 V -482 268 V 3096 1354 M -481 268 V -482 268 V -481 269 V -481 268 V 3063 1348 M -482 268 V -481 268 V -481 268 V -481 269 V 3029 1341 M -481 269 V -481 268 V -481 268 V -482 268 V 2996 1335 M -481 268 V -482 269 V -481 268 V -481 268 V 2963 1329 M -482 268 V -481 268 V -481 269 V -481 268 V 2929 1323 M -481 268 V -481 268 V -482 269 V -481 268 V 2896 1317 M -481 268 V -482 268 V -481 268 V 971 2390 L LTb 971 1674 M 3333 620 V 6229 1221 L 2896 601 L 971 1674 L 2896 601 M -55 30 V 3061 509 M (0) Lshow 4562 911 M -55 30 V 4727 820 M (0.5) Lshow 6229 1221 M -55 30 V 220 -121 R (1) Lshow 971 1674 M 61 11 V 786 1640 M (0) Rshow 1212 1540 M 61 11 V -246 -45 R (0.05) Rshow 1452 1406 M 61 11 V -246 -45 R (0.1) Rshow 1693 1272 M 61 11 V -246 -45 R (0.15) Rshow 1933 1138 M 61 11 V -246 -45 R (0.2) Rshow 2174 1004 M 61 11 V 1989 970 M (0.25) Rshow 2415 869 M 61 11 V 2230 835 M (0.3) Rshow 2655 735 M 61 11 V 2470 701 M (0.35) Rshow 2896 601 M 61 11 V 2711 567 M (0.4) Rshow 971 2390 M 63 0 V -126 0 R (0) Rshow 971 3104 M 63 0 V -126 0 R (0.5) Rshow 971 3819 M 63 0 V -126 0 R (1) Rshow 971 1674 M 0 2145 V LTa stroke grestore end showpage %%EndDocument endTexFig 3751 6010 a Ft(47)p eop %%Page: 48 53 48 52 bop -165 -116 a Fr(Nelinearni)27 b(zak)n(oni)f(sa)n(\024)-39 b(cuv)-5 b(anja)1553 83 y Fw(Program)57 258 y Fb(#include)53 b()57 378 y(#include)g()57 499 y(#define)g(KOMADA)g(4) 57 620 y(#define)g(Ah)f("Lax-Friedrichs")57 861 y(double)h(f\(double)g (x\);)57 982 y(double)g(fbl\(double)h(x\);)57 1103 y(double)f (f00\(double)h(x\);)57 1223 y(double)f(f01\(double)h(x\);)57 1344 y(double)f(f10\(double)h(x\);)57 1465 y(double)f(f11\(double)h (x\);)57 1586 y(double)f(Flux\(double)h(x,double)f(y,double)h (\(*F\)\(double\)\);)57 1706 y(void)e(itrak\(double)j(*X,int)d (n,double)i(t,double)f(\(*F\)\(double\),double)j(*I\);)57 1827 y(void)c(itrak1\(double)j(*X,int)e(n,double)g(t,double)g (\(*F\)\(double\),double)k(*I\);)57 1948 y(void)52 b(godunov\(double)j (*X,int)e(n,double)g(t,double)h(\(*F\)\(double\),double)i(*I\);)57 2068 y(void)c(pocet\(double)j(*X,int)d(n,double)i(\(*F\)\(double\)\);) 57 2189 y(void)e(ispis\(double)j(*X,int)d(n,int)h(g\);)57 2551 y(main\(\))57 2672 y({)159 2793 y(int)f(n=1000,m,i,j;)159 2914 y(double)h(*X,*Y,*Z,x,t;)159 3034 y(double)g(\(*F0\)\(double\);) 159 3155 y(void)g(\(*ITER\)\(double)i(*X,int)d(n,double)i (t,double\(*F\)\(double\),doubl)q(e)j(*I\);)159 3276 y(system\(\024)-51 b(clear"\);)159 3396 y(printf\("Kakove)55 b(zelis)e(pocetne)g(uvjete:\\n")518 3517 y("\\t\\t1\)\\tPadajucu)j (neprekidnu\\n")518 3638 y("\\t\\t2\)\\tPadajucu)g(sa)c(skokom\\n")518 3759 y("\\t\\t3\)\\tRastucu)k(neprekidnu\\n")518 3879 y("\\t\\t4\)\\tRastucu)g(sa)51 b(skokom\\t\\t"\);)159 4000 y(scanf\("\045d",&m\);)159 4121 y(switch\(m\){)159 4242 y(case)i(1:)e(F0=f00;break;)159 4362 y(case)i(2:)e(F0=f01;break;) 159 4483 y(case)i(3:)e(F0=f10;break;)159 4604 y(case)i(4:)e (F0=f11;break;)159 4724 y(})159 4845 y(printf\("Kojom)k(metodom)e (zelis)g(rjesavati)g(problem:\\n")518 4966 y ("\\t\\t1\)\\tLax-Friedrichs)58 b(-ovom\\n")518 5087 y("\\t\\t2\)\\tUpwind\\n")518 5207 y("\\t\\t3\)\\tGodunov)e (-ljevom\\t\\t\\t"\);)159 5328 y(scanf\("\045d",&m\);)159 5449 y(switch\(m\){)159 5570 y(case)d(1:)e(ITER=itrak;break;)159 5690 y(case)i(2:)e(ITER=itrak1;break;)159 5811 y(case)i(3:)e (ITER=godunov;break;)-165 6010 y Ft(48)p eop %%Page: 49 54 49 53 bop 2306 -116 a Fr(Numeri)n(\024)-39 b(ck)n(o)25 b(rje)l(\024)-37 b(sa)n(v)-5 b(anje)26 b(Burgerso)n(v)n(e)e(jednad)n (\024)-39 b(zb)r(e)395 83 y Fb(})395 202 y(printf\("\\nUpisi)56 b(broj)c(koraka)h(potrebnih)h(za)d(0.1\\t\\tn="\);)395 322 y(scanf\("\045d",&m\);)395 441 y(t=.1/m;)395 560 y(n=m/2;)395 680 y(/*alokacija)k(me-ram-a*/)395 799 y(X=\(double)f (*\)malloc\(\(n+1\)*sizeof\(x\))q(\);)395 918 y(Y=\(double)g (*\)malloc\(\(n+1\)*sizeof\(x\))q(\);)395 1038 y(/*unos)f(pocetnih)h (uvjeta*/)395 1157 y(pocet\(X,n,F0\);)395 1276 y(ispis\(X,n,0\);)395 1396 y(for\(j=1;j<=)h(KOMADA;j++\){)498 1515 y(for\(i=0;i=.6\))g (return)g(\(0\);)395 4737 y(else)g(return)g(\(1.-5*\(x-.4\)\);)293 4856 y(})293 4976 y(double)g(f01\(double)h(x\))293 5095 y({)395 5214 y(if)e(\(x<=.5\))h(return)g(\(1\);)395 5334 y(else)g(return)g(\(0\);)293 5453 y(})293 5572 y(double)g(f10\(double)h (x\))293 5692 y({)395 5811 y(if)e(\(x<=.4\))h(return)g(\(0\);)3751 6010 y Ft(49)p eop %%Page: 50 55 50 54 bop -165 -116 a Fr(Nelinearni)27 b(zak)n(oni)f(sa)n(\024)-39 b(cuv)-5 b(anja)159 83 y Fb(else)53 b(if\(x>=.6\))g(return)g(\(1\);)159 202 y(else)g(return)f(\(5*\(x-.4\)\);)57 322 y(})57 441 y(double)h(f11\(double)h(x\))57 560 y({)159 680 y(if)e(\(x<=.5\))h (return)g(\(0\);)159 799 y(else)104 b(return)53 b(\(1\);)57 918 y(})57 1157 y(/*Lax-Friedricson)j(metoda*/)57 1276 y(void)c(itrak\(double)j(*X,int)d(n,double)i(t,double)f (\(*F\)\(double\),double)j(*I\))57 1396 y({)159 1515 y(int)c(j;)159 1634 y(double)h(ih;)159 1754 y(ih=1./n;/*)h (printf\("\045g\\n",ih\);)i(*/)159 1873 y(for\(j=1;j