{"id":18993,"date":"2008-07-15T09:22:23","date_gmt":"2008-07-15T09:22:23","guid":{"rendered":"https:\/\/www.upjs.sk\/prirodovedecka-fakulta\/studium\/dalsie-vzdelavanie\/predmety-rigoroznej-skusky-matematika\/"},"modified":"2022-10-26T08:12:39","modified_gmt":"2022-10-26T08:12:39","slug":"predmety-rigoroznej-skusky-matematika","status":"publish","type":"page","link":"https:\/\/www.upjs.sk\/prirodovedecka-fakulta\/studium\/dalsie-vzdelavanie\/predmety-rigoroznej-skusky-matematika\/","title":{"rendered":"Predmety rigor\u00f3znej sk\u00fa\u0161ky – Matematika"},"content":{"rendered":"

\"\" \"Zoznam <\/p>\n

<\/p>\n

\u0160tudijn\u00e9 predmety s\u00fa rozdelen\u00e9 do skupiny povinn\u00fdch a volite\u013en\u00fdch predmetov. Uch\u00e1dza\u010d si vyberie dva \u0161tudijn\u00e9 predmety, z ktor\u00fdch jeden mus\u00ed by\u0165 zo skupiny povinn\u00fdch predmetov. Obsah volite\u013en\u00e9ho predmetu s n\u00e1zvom Vybran\u00e9 partie … ur\u010d\u00ed predseda komisie pri rozhovore s uch\u00e1dza\u010dom pod\u013ea zamerania rigor\u00f3znej pr\u00e1ce.<\/p>\n

Povinn\u00e9 predmety<\/strong><\/p>\n

Analytick\u00e1 a diferenci\u00e1lna geometria<\/strong><\/p>\n

Metrick\u00e9 vlastnosti vektorov. Afinn\u00e9 a metrick\u00e9 vlastnosti line\u00e1rnych \u00fatvarov. Afinn\u00e9 a metrick\u00e9 vlastnosti kvadratick\u00fdch \u00fatvarov. Krivky euklidovsk\u00e9ho priestoru. Frenetove vzorce a ich d\u00f4sledky. Plochy euklidovsk\u00e9ho priestoru. Z\u00e1kladn\u00e9 formy pl\u00f4ch. Niektor\u00e9 krivky na ploche.
Predmetom sk\u00fa\u0161ky s\u00fa kapitoly, ktor\u00e9 si uch\u00e1dza\u010d vyberie (pod\u013ea svojho z\u00e1ujmu a t\u00e9my rigor\u00f3znej pr\u00e1ce) po dohode so sk\u00fa\u0161aj\u00facim.
Literat\u00fara: <\/strong>B. Budinsk\u00fd, Analytick\u00e1 a diferenci\u00e1ln\u00ed geometrie, SNTL Praha 1983.
H.S.M. Coxeter, Introduction\u00a0 to geometry, New York 1961.<\/p>\n

Diferenci\u00e1lne rovnice<\/strong><\/p>\n

Z\u00e1kladn\u00e9 pojmy. Element\u00e1rne met\u00f3dy rie\u0161enia diferenci\u00e1lnych rovn\u00edc. Existencia a jednozna\u010dnos\u0165 rie\u0161enia. Z\u00e1kladn\u00e9 vlastnosti rie\u0161en\u00ed line\u00e1rneho diferenci\u00e1lneho syst\u00e9mu a line\u00e1rnej diferenci\u00e1lnej rovnice. Line\u00e1rny diferenci\u00e1lny syst\u00e9m (line\u00e1rna diferenci\u00e1lna rovnica) s kon\u0161tantn\u00fdmi koeficientami. Porovn\u00e1vacie vety.
Literat\u00fara:<\/strong>
M. Gregu\u0161, M. \u0160vec, V. \u0160eda, Oby\u010dajn\u00e9 diferenci\u00e1lne rovnice, Alfa, Bratislava 1985.
I. Kluv\u00e1nek, L. Mi\u0161\u00edk, M. \u0160vec, Matematika II, Alfa, Bratislava 1961.<\/p>\n

Funkcie komplexnej premennej<\/strong><\/p>\n

Postupnosti a rady komplexn\u00fdch \u010d\u00edsel. Funkcia \u2013 jednozna\u010dn\u00e1, mnohozna\u010dn\u00e1, inverzn\u00e1. Analytick\u00e9 funkcie. Konformn\u00e9 zobrazenie. Poten\u010dn\u00e9 rady. Funkcie \u2013 exponenci\u00e1lna, trigonometrick\u00e9, logaritmick\u00e1, mocninn\u00e1. Integr\u00e1l funkcie komplexnej premennej. Cauchyho integr\u00e1lna veta. Cauchyho integr\u00e1lny vzorec. Taylorov rad. Laurentov rad.
Literat\u00fara:<\/strong>
I. Kluv\u00e1nek, L. Mi\u0161\u00edk, M. \u0160vec, Matematika II, Alfa, Bratislava 1961.
I. I. Privalov, Vvedenie v teoriju funkcii komplexnovo peremennovo, Nauka, Moskva 1977.<\/p>\n

Galoisova te\u00f3ria<\/strong><\/p>\n

Jednoduch\u00e9 roz\u0161\u00edrenia pol\u00ed. Algebraick\u00fd a transcendentn\u00fd prvok nad po\u013eom. Adjunkcia kore\u0148ov. Stupne a kone\u010dn\u00e9 roz\u0161\u00edrenia. Algebraick\u00e9 \u010d\u00edsla. Rozkladov\u00e9 polia polyn\u00f3mov. Kone\u010dn\u00e9 polia.
Literat\u00fara:<\/strong>
G. Birkhoff, S. Mac Lane, Preh\u013ead modernej algebry, Alfa, Bratislava 1979, kapitola 14 a 15.<\/p>\n

Kombinatorick\u00e1 optimaliz\u00e1cia<\/strong><\/p>\n

V\u00fdpo\u010dtov\u00e1\u00a0 zlo\u017eitos\u0165 algoritmov. NP-\u00faplnos\u0165 kombinatorick\u00fdch algoritmov. Preh\u013ead\u00e1vanie grafov. Stromy a kostry \u2013 vyh\u013eadanie v\u0161etk\u00fdch kostier, \u00faloha o minim\u00e1lnej kostre. \u00dalohy o cest\u00e1ch v grafoch. \u00dalohy o tokoch. P\u00e1rovacie a prira\u010fovacie\u00a0 a dopravn\u00e9 probl\u00e9my. Vyu\u017eitie met\u00f3d line\u00e1rneho programovania \u2013 prim\u00e1rno du\u00e1lny algoritmus. Pribli\u017en\u00e9 algoritmy.
Literat\u00fara:<\/strong>
N. Christofides: Graph Theory – An Algorithmic approach,\u00a0\u00a0 Academic Press, New York 1975 (rusk\u00fd preklad z r. 1978).
J. Plesn\u00edk: Grafov\u00e9 algoritmy, Veda Bratislava 1983.
G. Chartrand, O. R. Vellermann, Applied and Algorithmic Graph Theory, McGraw-Hill, Inc., New York 1993.<\/p>\n

Kombinatorika <\/strong><\/p>\n

Permut\u00e1cie, kombin\u00e1cie, kombin\u00e1cie s opakovan\u00edm. D\u00f4le\u017eit\u00e9 kombinatorick\u00e9 identity. Vytv\u00e1raj\u00face funkcie a ich vyu\u017eitie pri rie\u0161en\u00ed kombinatorick\u00fdch \u00faloh. Part\u00edcie. Blokov\u00e9 sch\u00e9my a ich inciden\u010dn\u00e9 matice. Z\u00e1kladn\u00e9 tvrdenia o blokov\u00fdch sch\u00e9mach. Princ\u00edp inkl\u00fazie a exkl\u00fazie. Syst\u00e9my r\u00f4znych reprezentantov.
Literat\u00fara: <\/strong>N.J. Vilenkin, Kombinatorika, Moskva – Firmatiz 1969 (ex. slovensk\u00fd preklad).
M. Hall, Combinatorial\u00a0 Theory, Blaisdell Publishing Company 1967 (rusk\u00fd preklad: Kombinatorika, Moskva 1970).<\/p>\n

Line\u00e1rna optimaliz\u00e1cia<\/strong><\/p>\n

Vlastnosti \u00falohy line\u00e1rneho programovania. Konvexnos\u0165 pr\u00edpustnej mno\u017einy. Existencia optim\u00e1lneho krajn\u00e9ho bodu. B\u00e1zick\u00e9 pr\u00edpustn\u00e9 rie\u0161enia a prechod medzi nimi. Simplexov\u00e1 met\u00f3da a jej \u0161tartovanie. Probl\u00e9m zacyklenia a jeho odstr\u00e1nenie. Dualita v line\u00e1rnom programovan\u00ed.\u00a0 Vz\u0165ah medzi rie\u0161eniami prim\u00e1rnej a du\u00e1lnej \u00falohy. Ekonomick\u00e1 interpret\u00e1cia duality. Du\u00e1lna a revidovan\u00e1 simplexov\u00e1 met\u00f3da. Anal\u00fdza senzitivity a postoptimaliza\u010dn\u00e1 anal\u00fdza.
Literat\u00fara:<\/strong>
Plesn\u00edk, Dupa\u010dov\u00e1, Vlach, Line\u00e1rne programovanie, Alfa\u00a0 Bratislava 1992.<\/p>\n

Matematick\u00e1 logika<\/strong><\/p>\n

Predik\u00e1tov\u00fd po\u010det: pojem termu, formuly, vo\u013en\u00e1 a viazan\u00e1 premenn\u00e1. Formalizovan\u00e1 te\u00f3ria a d\u00f4kaz. Met\u00f3dy d\u00f4kazu. Veta o kompaktnosti. Interpret\u00e1cia jazyka, model te\u00f3rie. G\u00f6delova veta o \u00faplnosti. Neprotire\u010divos\u0165 te\u00f3rie a existencia modelu.
Literat\u00fara:<\/strong>
Sochor A.,\u00a0 Klasick\u00e1 matematick\u00e1 logika, Praha 2001.
Mendelson E., Introduction to Mathematical Logic, van Nostrand 1964 (existuj\u00fa \u010fal\u0161ie vydania a rusk\u00fd preklad).<\/p>\n

Mno\u017einov\u00e1 topol\u00f3gia<\/strong><\/p>\n

Topologick\u00fd priestor, rozli\u010dn\u00e9 mo\u017enosti zadania topol\u00f3gie. Konvergencia. Axi\u00f3my odde\u013eovania a ich v\u00fdznam. Kompaktnos\u0165 a jej z\u00e1kladn\u00e9 vlastnosti. Kompaktifik\u00e1cia. Kompaktnos\u0165 metrick\u00fdch priestorov. S\u00favislos\u0165 priestoru. Vety typu Bairea.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0
Literat\u00fara:<\/strong>
Engelking R., Topologia og\u00f3lna, Warszawa 1975 (exist. aj anglick\u00e9 vydania).
Bourbaki N., Topologie g\u00e9n\u00e9ral, Chap. 1.\u20133., Paris 1956 (existuje viacero nov\u0161\u00edch vydan\u00ed a rusk\u00fd preklad).<\/p>\n

Syntetick\u00e1 geometria<\/strong><\/p>\n

Trojuholn\u00edky. Pravideln\u00e9 n-uholn\u00edky. Zhodnosti a podobnosti euklidovskej roviny. Zobrazenia euklidovsk\u00e9ho priestoru. Kru\u017enice a gu\u013eov\u00e9 plochy. Afinn\u00e1 a projekt\u00edvna geometria. Axiomatick\u00e1 v\u00fdstavba geometrie (Hilbertova s\u00fastava axi\u00f3m.)
Predmetom sk\u00fa\u0161ky s\u00fa kapitoly, ktor\u00e9 si uch\u00e1dza\u010d vyberie (pod\u013ea svojho z\u00e1ujmu a t\u00e9my rigor\u00f3znej pr\u00e1ce) po dohode so sk\u00fa\u0161aj\u00facim.
Literat\u00fara:<\/strong>
H.S.M. Coxeter, Introduction to geometry John Wiley 1961.
J. \u0160vr\u010dek, J. Van\u017eura, Geometrie troj\u00faheln\u00edka, SNTL, Praha 1988.
V. Svitek, Logick\u00e9 z\u00e1klady geometrie, SPN, Bratislava 1969.<\/p>\n

Te\u00f3ria \u010d\u00edsel<\/strong><\/p>\n

Pojmy, tvrdenia, met\u00f3dy a v\u00fdsledky element\u00e1rnej te\u00f3rie \u010d\u00edsel v rozsahu hociktorej
z ni\u017e\u0161ie citovan\u00fdch polo\u017eiek.
Literat\u00fara:<\/strong> \u00a0
M. Kolibiar, Algebra a pr\u00edbuzn\u00e9 discipl\u00edny, Alfa, Bratislava 1991 (\u010cas\u0165 A: 2-6).
T.\u0160al\u00e1t: Vybran\u00e9 kapitoly z element\u00e1rnej te\u00f3rie \u010d\u00edsel, SPN, Bratislava 1969
(Kapitoly: 1-4).
I. M. Vinogradov, Z\u00e1klady theorie \u010d\u00edsel, N\u010cSAV, Praha 1953 (Kapitoly: 1-5).<\/p>\n

Te\u00f3ria grafov<\/strong><\/p>\n

Stromy, Eulerovsk\u00e9 a Hamiltonovsk\u00e9 grafy. Farebnos\u0165 v grafoch, \u00davod do topologickej te\u00f3rie grafov. \u00davod do extrem\u00e1lnej te\u00f3rie grafov, Orientovan\u00e9 grafy. Siete,\u00a0 Ramseyova te\u00f3ria pre grafy.
Literat\u00fara:<\/strong>
J.A. Bondy and U.S.R. Murty, Graph Theory with applications, North\u00a0 Holland, New York 1976.\u00a0
R. Diestel, Graph Theory, Springer, New York, Berlin, 1996.
M. Kolibiar a kol., Algebra a pr\u00edbuzn\u00e9 discipl\u00edny. Alfa, Bratislava 1991, \u010das\u0165 B.<\/p>\n

Te\u00f3ria gr\u00fap<\/strong><\/p>\n

Grupy, podgrupy. Cyklick\u00e9 grupy. Grupy permut\u00e1ci\u00ed. Rozklady na grup\u00e1ch, Lagrangeova veta. Norm\u00e1lne podgrupy, kongruencie na grup\u00e1ch, faktorov\u00e9 grupy, vety o izomorfizmoch. Priame s\u00fa\u010diny\u00a0\u00a0 gr\u00fap.\u00a0 \u0160trukt\u00fara kone\u010dnogenerovan\u00fdch gr\u00fap.\u00a0
Literat\u00fara:<\/strong>
A. Leg\u00e9\u0148, Grupy, okruhy a zv\u00e4zy, Alfa, Bratislava 1980.<\/p>\n

Vybran\u00e9 partie z matematickej anal\u00fdzy<\/strong><\/p>\n

Integr\u00e1lny po\u010det funkci\u00ed viac premenn\u00fdch \u2013 defin\u00edcia integr\u00e1lu na intervale, niektor\u00e9 vlastnosti a v\u00fdpo\u010det integr\u00e1lu na intervale, integr\u00e1l na mno\u017eine, niektor\u00e9 vlastnosti a v\u00fdpo\u010det integr\u00e1lu na mno\u017eine, dvojn\u00e9 a trojn\u00e9 integr\u00e1ly. Nevlastn\u00e9 integr\u00e1ly \u2013 defin\u00edcia, vety o existencii a met\u00f3dy v\u00fdpo\u010dtu nevlastn\u00fdch integr\u00e1lov. Parametrick\u00e9 integr\u00e1ly \u2013 vlastnosti funkcie danej parametrick\u00fdm integr\u00e1lom, pr\u00edklady v\u00fdpo\u010dtu integr\u00e1lov, funkcie gama a beta, Stirlingov vzorec.
Literat\u00fara:<\/strong>
I. Kluv\u00e1nek, L. Mi\u0161\u00edk, M. \u0160vec, Matematika II, Alfa, Bratislava 1961.<\/p>\n

Volite\u013en\u00e9 predmety<\/strong><\/p>\n

Algoritmy a zlo\u017eitos\u0165<\/strong><\/p>\n

Turingov stroj ako formaliz\u00e1cia pojmu algoritmus. \u010ciasto\u010dne rekurz\u00edvne funkcie. Kleeneho veta o norm\u00e1lnom tvare. Vz\u0165ah funkci\u00ed vypo\u010d\u00edtate\u013en\u00fdch na Turingovom stroji a \u010diasto\u010dne rekurz\u00edvnych funkci\u00ed. Probl\u00e9m zastavenia sa Turingovho stroja. Vlastnosti rekurz\u00edvne o\u010d\u00edslovate\u013en\u00fdch mno\u017e\u00edn. Univerz\u00e1lny syst\u00e9m programov. Veta o preklade, Riceova veta. Dynamick\u00e1 miera zlo\u017eitosti, pr\u00edklady. Veta o medzere, vz\u0165ah hodnoty funkcie a dynamickej miery zlo\u017eitosti.
Literat\u00fara:<\/strong>
Machtey M., Young P., An Introduction to the General Theory of Algorithms, North Holland 1978.
Bridges S., Computability: A mathematical Sketchbook, Springer 1994.
Bukovsk\u00fd L., Te\u00f3ria algoritmov, Ko\u0161ice 2004.<\/p>\n

Automaty a form\u00e1lne jazyky<\/strong><\/p>\n

Chomsk\u00e9ho hierachia jazykov a gramat\u00edk. Kone\u010dnostavov\u00fd automat, a jeho varianty: deterministick\u00e9, nedeterministick\u00e9, alternuj\u00face, pravdepodobnostn\u00e9, kvantov\u00e9, at\u010f. Jednosmern\u00e9, dvojsmern\u00e9, obratovo ohrani\u010den\u00e9 kone\u010dnostavov\u00e9 automaty. Regul\u00e1rne v\u00fdrazy a gramatiky. Un\u00e1rne regul\u00e1rne jazyky a ich vlastnosti. Problematika kon\u0161trukcie efekt\u00edvnych automatov jednotliv\u00fdch typov, z h\u013eadiska po\u010dtu stavov a hr\u00e1n. S\u00favislosti medzi te\u00f3riou kone\u010dnostavov\u00fdch automatov a te\u00f3riou zlo\u017eitosti.
Literat\u00fara:<\/strong>
J. E. Hopcroft, R. Motwani, J. D. Ullman, Introduction to automata theory\u00a0 languages, and computation, Addison-Wesley, 2001.<\/p>\n

Funkcion\u00e1lna anal\u00fdza<\/strong><\/p>\n

Z\u00e1klady te\u00f3rie mno\u017e\u00edn. Metrick\u00e9 a topologick\u00e9 priestory. Princ\u00edp kontrakt\u00edvnych zobrazen\u00ed. Line\u00e1rne priestory a line\u00e1rne funkcion\u00e1ly. Konvexn\u00e9 mno\u017einy a konvexn\u00e9 funkcion\u00e1ly. Normovan\u00e9 priestory. Euklidove priestory. Topologick\u00e9 line\u00e1rne priestory.
Literat\u00fara: <\/strong>A. N. Kolmogorov, S. V. Fomin, Z\u00e1klady te\u00f3rie funkc\u00ed a funkcion\u00e1ln\u00ed anal\u00fdzy, SNTL, Praha 1975.<\/p>\n

Grafov\u00e9 algoritmy<\/strong><\/p>\n

\u00davod do te\u00f3rie grafov. \u00dalohy a algoritmy, Dosiahnute\u013enos\u0165. Optimaln\u00e9 sledy, Toky v sie\u0165ach, Stromy a kostry. P\u00e1renia. Poch\u00f4dzky. \u00daloha \u010d\u00ednskeho po\u0161t\u00e1ra. \u00daloha obchodn\u00e9ho cestuj\u00faceho. Sie\u0165ov\u00e1 anal\u00fdza.
Literat\u00fara: <\/strong>J. Plesn\u00edk, Grafov\u00e9 algoritmy. Veda, Bratislava 1983.<\/p>\n

Kombinatorick\u00e1 topol\u00f3gia<\/strong><\/p>\n

Homot\u00f3pia a jej vlastnopsti. Homotopick\u00e1 grupa. Veta Seifertova \u2013 van Kampenova. V\u00fdpo\u010det najjednoduch\u0161\u00edch homotopick\u00fdch gr\u00fap. Brouwerova veta o pevnom bode. Nakr\u00fdvaj\u00faci priestor a jeho homotopick\u00e1 grupa. Varieta, pr\u00edklady. Klasifik\u00e1cia dvojrozmern\u00fdch kompaktn\u00fdch uzavret\u00fdch vari\u00e9t.
Literat\u00fara:<\/strong>
Singer I. M. a Thorpe J. A., Lecture Notes on Elementary Topology and Geometry, Springer 1967.
A.Gramain, Topologie des surfaces, Paris 1971.<\/p>\n

Konvexn\u00e9 mnohosteny<\/strong><\/p>\n

Platonsk\u00e9 a archimedovsk\u00e9 teles\u00e1. Eulerova veta a jej d\u00f4sledky. Steinitzova veta. Eberherdova veta a jej anal\u00f3gia. Vp\u00edsate\u013enos\u0165. Kotzigova veta a jej roz\u0161\u00edrenia. Kombinatorick\u00e9 vlastnosti. Viacrozmern\u00e9 mnohosteny.
Literat\u00fara:<\/strong>
P. R. Cromwell, Polyhedra, Cambridge University Press Cambridge, 1997.
E. Jucovi\u010d, Konvexn\u00e9 mnohosteny, Veda, Bratislava 1981.
Ziegler:, Lectures on Polytopes. Springer-Verlag, Berlin, 1995.<\/p>\n

Line\u00e1rne a celo\u010d\u00edseln\u00e9 programovanie<\/strong><\/p>\n

Geometrick\u00e9 vlastnosti\u00a0 \u00falohy line\u00e1rneho programovania. Simplexov\u00e1 met\u00f3da. Dualita v line\u00e1rnom programovan\u00ed. \u00daloha celo\u010d\u00edseln\u00e9ho line\u00e1rneho programovania \u2013\u00a0 vz\u0165ah k jej relax\u00e1cii a met\u00f3dy rie\u0161enia. Prim\u00e1rno-du\u00e1lny algoritmus a jeho aplik\u00e1cie na \u00falohy kombinatorickej optimaliz\u00e1cie
Literat\u00fara:<\/strong>
Papadimitriou, Steiglitz: Combinatorial Optimization, Prentice Hall, 1984.
Plesn\u00edk, Dupa\u010dov\u00e1, Vlach: Line\u00e1rne programovanie, Alfa, Bratislava, 1992.<\/p>\n

Matematick\u00e1 \u0161tatistika<\/strong><\/p>\n

Bodov\u00e9 a intervalov\u00e9 odhady, vlastnosti odhadov, met\u00f3da maxim\u00e1lnej vierohodnosti, Rao-Cram\u00e9rova nerovnost. Testovanie \u0161tatistick\u00fdch hypot\u00e9z, z\u00e1kladn\u00e9 parametrick\u00e9 a neparametrick\u00e9 testy. Testy dobrej zhody, z\u00e1kladn\u00e9 testy v kontingen\u010dn\u00fdch tabu\u013ek\u00e1ch. Z\u00e1kladn\u00e9 vlastnosti line\u00e1rneho modelu – odhadnute\u013enos\u0165, testy, konkr\u00e9tne modely.
Literat\u00fara: <\/strong>J. And\u011bl, Matematick\u00e1 statistika, SNTL\/Alfa, Praha 1985.
F. Lamo\u0161, R. Potock\u00fd, Pravdepodobnos\u0165 a matematick\u00e1 \u0161tatistika, Alfa, Bratislava 1989.<\/p>\n

Neline\u00e1rne programovanie<\/strong><\/p>\n

Konvexn\u00e9 mno\u017einy a ich vlastnosti. Konvexn\u00e9 funkcie a ich vlastnosti, krit\u00e9ria konvexnosti funkcie. Nutn\u00e9 a posta\u010duj\u00face podmienky optima, Kuhn-Tuckerova te\u00f3ria. Dualita v neline\u00e1rnom programovan\u00ed. Kvadratick\u00e9 programovanie. \u0160peci\u00e1lne met\u00f3dy: line\u00e1rne a viacrozmern\u00e9 preh\u013ead\u00e1vanie. Met\u00f3dy bari\u00e9rov\u00fdch a penaliza\u010dn\u00fdch funkci\u00ed.
Literat\u00fara:<\/strong>
M. Hamala: Neline\u00e1rne programovanie, Alfa, Bratislava 1976.
Bazaraa, Sherali, Shetty: Nonlinear programming, Wiley, New York 1993.<\/p>\n

Okruhy a polia<\/strong><\/p>\n

Okruh, obor integrity, pole. R\u00e1d prvku, charakteristika oboru integrity. Podielov\u00e9 pole. Faktorov\u00e9 okruhy, ide\u00e1ly a homomorfizmy. Delite\u013enos\u0165 v okruhoch. Euklidovsk\u00e9 okruhy, okruhy hlavn\u00fdch ide\u00e1lov, Gaussove okruhy.
Literat\u00fara:<\/strong>
T. Katri\u0148\u00e1k a kol., Algebra a teoretick\u00e1 aritmetika (1), Alfa, SNTL, 1985.<\/p>\n

Te\u00f3ria frakt\u00e1lov<\/strong><\/p>\n

Intuit\u00edvny pojem samopodobnosti a rozmeru. Pojem frakt\u00e1lu a jeho z\u00e1kladn\u00e9 vlastnosti. Pr\u00edklady frakt\u00e1lov. Frakt\u00e1l ako invariantn\u00e1 mno\u017eina syst\u00e9mu kontrakci\u00ed. Banachova veta o pevnom bode. Existencia invariantnej mno\u017einy a jej jednozna\u010dnos\u0165. D\u00f4sledky pre kon\u0161trukciu frakt\u00e1lov. Hausdorffov rozmer. Vz\u0165ah Hausdorffovho a podobnostn\u00e9ho rozmeru. Dynamick\u00fd syst\u00e9m. Atraktor ako frakt\u00e1l. Juliove mno\u017einy, Mandelbrotova mno\u017eina.
Literat\u00fara:<\/strong>
Edgar G. A., Measure, Topology, and Fractal Geometrie, Springer 1990.
Falconer K., Fractal Geometry, John Wiley&Sons 1990.<\/p>\n

Te\u00f3ria hier<\/strong><\/p>\n

Extenz\u00edvny tvar hry, veta o hodnote hry. Maticov\u00e9 hry a met\u00f3dy ich rie\u0161enia. Bimaticov\u00e9 hry. Te\u00f3ria vyjedn\u00e1vania. Hry n-hr\u00e1\u010dov \u2013 jadro, Shapleyho hodnota, p\u00e1rovacie hry. Aplik\u00e1cie te\u00f3rie hier v ekon\u00f3mii \u2013 modely oligopolu, modely aukci\u00ed, hry s nesymetrickou inform\u00e1ciou.
Literat\u00fara:\u00a0 <\/strong>G. Owen,\u00a0 Game Theory, Academic Press\u00a0\u00a0 (existuje rusk\u00fd preklad).
L.C. Thomas,\u00a0 Games, Theory and Applications,\u00a0 Wiley, New York.
H.S. Bierman, L.Fernandez, Game Theory with Economic Applications, Addison-Wesley, 1998.
E. Rasmusen, Games and Information, Basil Blackwell, 1989.
\u00a0
Te\u00f3ria k\u00f3dovania<\/strong><\/p>\n

Monoidy. Roz\u0161\u00edrite\u013enos\u0165 k\u00f3du na maxim\u00e1lny k\u00f3d. Grupov\u00e9 k\u00f3dy. Test na rozozn\u00e1vanie k\u00f3dov.\u00a0 Bernoulliho distrib\u00facia. Dyckov k\u00f3d. Kompoz\u00edcia k\u00f3dov. Rozklad k\u00f3du na nerozlo\u017eite\u013en\u00e9 k\u00f3dy.
Literat\u00fara: <\/strong>J. Berstel a D. Perrin, Theory of Codes, Academic Press 1985.<\/p>\n

Te\u00f3ria pravdepodobnosti<\/strong><\/p>\n

N\u00e1hodn\u00e9 vektory, ich rozdelenie a charakteristiky. Niektor\u00e9 \u0161peci\u00e1lne typy rozdelen\u00ed pravdepodobnosti. Z\u00e1kony ve\u013ek\u00fdch \u010d\u00edsel a centr\u00e1lne limitn\u00e9 vety. N\u00e1hodn\u00e9 procesy \u2013 Poissonov a Markovov proces. Kolmogorovove diferenci\u00e1lne rovnice.
Literat\u00fara:<\/strong>
A. R\u00e9nyi, Teorie pravd\u011bpodobnosti, Academia, Praha, 1972.
P. Mandl, Pravd\u011bpodobnostn\u00ed dynamick\u00e9 modely, Academia, Praha, 1985.
F. Lamo\u0161, R. Potock\u00fd, Pravdepodobnos\u0165 a matematick\u00e1 \u0161tatistika, Elfa, Bratislava, 1989.<\/p>\n

Te\u00f3ria rozmeru<\/strong><\/p>\n

Pojem mal\u00e9ho indukt\u00edvneho rozmeru a jeho z\u00e1kladn\u00e9 vlastnosti. Horn\u00e9 odhady rozmeru Euklidov\u00fdch priestorov. Vety o rozklade na 0 \u2013 rozmern\u00e9 podmno\u017einy. Pokr\u00fdvaj\u00faci rozmer a z\u00e1kladn\u00e9 vlastnosti. Vz\u0165ah indukt\u00edvneho a pokr\u00fdvaj\u00faceho rozmeru. Rozmer Euklidov\u00fdch priestorov.
Literat\u00fara:<\/strong>
Engelking R., Teoria wymiaru, Warszawa 1981 (existuj\u00fa \u010fal\u0161ie vydania a\u00a0\u00a0 anglick\u00fd preklad).
Hurewicz W.,\u00a0 Wallman H., Dimension Theory, Princeton 1941.
Aleksandrov P. S. a Pasynkov B. A., \u0392\u0432\u0435\u0434\u0435\u043d\u0438\u0435 \u0432 \u0442\u0435\u043e\u0440\u0438\u044e \u0440\u0430\u0437\u043c\u0435\u0440\u043d\u043e\u0441\u0442\u0438, Moskva 1973.<\/p>\n

Te\u00f3ria vyu\u010dovania matematiky<\/strong><\/p>\n

Predstavy pojmu \u010d\u00edslo, vyu\u017eitie t\u00fdchto predst\u00e1v pri zaveden\u00ed pojmu zlomok. Didaktick\u00e9 postupy pri roz\u0161irovan\u00ed \u010d\u00edseln\u00fdch mno\u017e\u00edn. Fylogen\u00e9za a ontogen\u00e9za funk\u010dn\u00e9ho myslenia. Ontogen\u00e9za argument\u00e1cie. Zavedenie pojmu funkcia. Propedeutika deriv\u00e1cie a ur\u010dit\u00e9ho integr\u00e1lu. Pr\u00ednos Pytagorasa a Euklidesa k rozvoju matematiky. Metodika planimetrie. Obsah a metodika stereometrie. Propedeutika analytickej geometrie. Koncepcia vyu\u010dovania analytickej geometrie. Vyu\u010dovanie trigonometrie a goniometrie. Vyu\u010dovanie kombinatoriky, pravdepodobnosti a \u0161tatistiky.
Literat\u00fara: <\/strong>Hejn\u00fd a kol. ,Te\u00f3ria vyu\u010dovania matematiky 2, SPN 1989.<\/p>\n

Univerz\u00e1lne algebry<\/strong><\/p>\n

Oper\u00e1cie, algebry. Homomorfizmy a kongruencie. Priame a polopriame\u00a0 s\u00fa\u010diny algebier. Vety o izomorfizmoch. Priame a polopriame rozklady. Oper\u00e1tory na triedach algebier. Triedy algebier a identity, variety algebier. Vo\u013en\u00e9 algebry.
Literat\u00fara:<\/strong>
M. Kolibiar a kol., Algebra a pr\u00edbuzn\u00e9 discipl\u00edny, Alfa, Bratislava 1991, \u010das\u0165 C 1 a C 3.<\/p>\n

Usporiadan\u00e9 mno\u017einy a zv\u00e4zy<\/strong><\/p>\n

\u010ciasto\u010dne usporiadan\u00e9 mno\u017einy. Podmienky re\u0165azcov. Princ\u00edp duality. Zv\u00e4zy a \u00fapln\u00e9 zv\u00e4zy. Distribut\u00edvne a modul\u00e1rne zv\u00e4zy. Podmienky o pokr\u00fdvan\u00ed. Komplement\u00e1rne zv\u00e4zy, Booleove algebry. Algebraick\u00e9 zv\u00e4zy, nekone\u010dn\u00e1 distribut\u00edvnos\u0165. Vn\u00e1ranie do \u00fapln\u00fdch zv\u00e4zov. Kongruencie na zv\u00e4zoch.
Literat\u00fara:<\/strong>
M. Kolibiar a kol., Algebra a pr\u00edbuzn\u00e9 discipl\u00edny, Alfa, Bratislava 1991, \u010das\u0165 C2.<\/p>\n

Vybran\u00e9 partie z te\u00f3rie mno\u017e\u00edn<\/strong><\/p>\n

Viacej mo\u017enost\u00ed pod\u013ea dohovoru so sk\u00fa\u0161aj\u00facim. Uv\u00e1dzame dva pr\u00edklady ponuky.
Nekone\u010dn\u00e1 kombinatorika: Kardin\u00e1lne charakteristiky podmno\u017e\u00edn re\u00e1lnej osi. Pr\u00edbuzn\u00e9 kardin\u00e1lne charakteristiky. Vz\u0165ahy medzi nimi. Cicho\u0144ov diagram. D\u00f4sledky pre existenciu v\u00fdnimo\u010dn\u00fdch mno\u017e\u00edn.
Deskript\u00edvna te\u00f3ria mno\u017e\u00edn: Borelovsk\u00e1 hierarchia mno\u017e\u00edn, jej z\u00e1kladn\u00e9 vlastnosti. Baireova hierarchia re\u00e1lnych funkci\u00ed. Vz\u0165ah medzi hierarchiami. Projekt\u00edvne mno\u017einy, z\u00e1kladn\u00e9 vlastnosti.
Literat\u00fara:<\/strong>\u00a0 pod\u013ea dohovoru.<\/p>\n

Predmet:<\/strong> \u013dubovo\u013en\u00fd predmet odboru Informatika \u2013 ur\u010d\u00ed predseda komisie pod\u013ea z\u00e1ujmu uch\u00e1dza\u010da a zamerania rigor\u00f3znej pr\u00e1ce.
Obsah a literat\u00fara:<\/strong> pod\u013ea v\u00fdberu predmetu.<\/p>\n

\"vytla\u010di\u00bb\"<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"

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Afinn\u00e9 a … \u010c\u00edta\u0165 \u010falej<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":17187,"menu_order":5,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_acf_changed":false,"_editorskit_title_hidden":false,"_editorskit_reading_time":0,"_editorskit_is_block_options_detached":false,"_editorskit_block_options_position":"{}","footnotes":""},"categories":[],"tags":[],"class_list":["post-18993","page","type-page","status-publish","hentry"],"acf":{"acf_link_na_externy_obsah":""},"yoast_head":"\nPredmety rigor\u00f3znej sk\u00fa\u0161ky - Matematika - Pr\u00edrodovedeck\u00e1 fakulta | UPJ\u0160<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.upjs.sk\/prirodovedecka-fakulta\/studium\/dalsie-vzdelavanie\/predmety-rigoroznej-skusky-matematika\/\" \/>\n<meta property=\"og:locale\" content=\"sk_SK\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Predmety rigor\u00f3znej sk\u00fa\u0161ky - Matematika - Pr\u00edrodovedeck\u00e1 fakulta | UPJ\u0160\" \/>\n<meta property=\"og:description\" content=\"\u0160tudijn\u00e9 predmety s\u00fa rozdelen\u00e9 do skupiny povinn\u00fdch a volite\u013en\u00fdch predmetov. 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