· 3 contents contents...

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UNIVERSITY OF MONTENEGRO

Faculty of Mathematics and Natural Sciences

T E M P U S JEP_19099_2004

PODGORICA, 2007.

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CONTENTS CONTENTS ...............................................................................................................3 INTRODUCTION......................................................................................................8 Academic Undergraduate Programme of Study BIOLOGY ....................................10 Mathematics .............................................................................................................13 Citology and Histology .............................................................................................14 Plant Anatomy..........................................................................................................15 General and Non-Organic Chemistry ......................................................................16 Invertebrates I ..........................................................................................................17 History of Biology ....................................................................................................18 Physics......................................................................................................................19 Histology of Organs with Embryology .....................................................................20 Organic Chemistry ...................................................................................................21 Invertebrates II.........................................................................................................22 Algae, fungi and lichens ..........................................................................................23 Anthropology............................................................................................................24 Biochemistry I ..........................................................................................................25 Physiology ................................................................................................................26 Comparative Anatomy and Sistematics of Vertebrates I.........................................27 Systematic of Cormophyta I .....................................................................................28 Biochemistry II.........................................................................................................29 Comparative Anatomy and Sistematics of Vertebrates II .......................................30 Systematic of Cormophyta II....................................................................................31 Microbiology ............................................................................................................32 Genetics ....................................................................................................................33 Plant Physiology.......................................................................................................34 Ecology of Animals I................................................................................................35 Ecology of Plants I ...................................................................................................36 Environmental Protection I....................................................................................37 Hidrobiology.............................................................................................................38 Molecular Biology II................................................................................................39 Comparative Physiology...........................................................................................40 Evolution ..................................................................................................................41 Ecology of Animals II ..............................................................................................42 Ecology of Plants II .................................................................................................43 Environmental Protection II ..................................................................................44 Human Ecology........................................................................................................45

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Academic Undergraduate Programme of Study PHYSICS .....................................47 Linear algebra and analytical geometry ..................................................................51 Analysis I..................................................................................................................52 Physical Mechanics..................................................................................................53 Introduction to Experimental Physics I ...................................................................54 Laboratory Physics I (Mechanics) ...........................................................................55 Introduction to Computing I ....................................................................................56 English for Physics I ................................................................................................57 Analysis II ................................................................................................................58 Molecular Physics with Thermodynamics ...............................................................59 Introduction to Experimental Physics II..................................................................60 Laboratory Physics I (Waves and thermodynamics)................................................61 Introduction to Computing II...................................................................................62 Analysis III...............................................................................................................63 Differential Equations..............................................................................................64 Electromagnetism.....................................................................................................65 Basics of Physics Measurement Techniques I ........................................................66 Physics Laboratory II (Electromagnetism) ..............................................................67 English for Physics II ..............................................................................................68 Complex Analysis .....................................................................................................69 Numerical Methods..................................................................................................70 Probability Theory and Statistics .............................................................................71 Basics of Physics Measurements Techniques II ....................................................72 Optics........................................................................................................................73 Physics Laboratory II (Optics) .................................................................................74 Physic of Atoms........................................................................................................75 English for Physics III .............................................................................................76 Mathematical Methods in Physics ...........................................................................77 Theoretical physics I ................................................................................................78 Quantum physics I ...................................................................................................79 Practicum III............................................................................................................81 Statistical Physics .....................................................................................................82 Theoretical physics II...............................................................................................83 Quantum physics II..................................................................................................84 Introduction to Nuclear Physics ..............................................................................85 History and Philosophy of Physics...........................................................................86 Laboratory Practicum III -Practicum in Nuclear Physics.......................................87

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Academic Undergraduate Programme of Study MATHEMATICS.........................88 Analysis I..................................................................................................................91 Linear Algebra I.......................................................................................................92 Computers and Programming..................................................................................93 Introduction to Logic ...............................................................................................94 English for Mathematics I .......................................................................................95 Principles of Programming......................................................................................96 Introduction to Combinatorics .................................................................................97 Linear Algebra II .....................................................................................................98 Analitical Geometry .................................................................................................99 Analysis II ..............................................................................................................100 English for Mathematics II....................................................................................101 Analysis 3 ...............................................................................................................102 Algebra I.................................................................................................................103 Introduction to Geometry .......................................................................................104 Programming 1 ......................................................................................................105 Discrete Mathematics.............................................................................................106 Complex analysis I .................................................................................................107 Analysis 4 ...............................................................................................................108 Algebra 2 ................................................................................................................109 Probability Theory..................................................................................................110 Programming 2 ......................................................................................................111 Differential Equations............................................................................................112 Algebra 3 ................................................................................................................113 Statistics..................................................................................................................114 Measure and Integral.............................................................................................115 Partial Differential Equations................................................................................116 Theoretical Mechanics ...........................................................................................117 English for Mathematics III ..................................................................................118 Complex Analysis II ...............................................................................................119 Introduction to Differential Geometry ...................................................................120 Functional Analysis ...............................................................................................121 Numerical Analysis ................................................................................................122 Algebraic Topology ................................................................................................123 History and Philosophy of Mathematics ................................................................124 English for Mathematics IV...................................................................................125

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Academic Undergraduate Programme of Study ......................................................... MATEMATICS AND COMPUTER SCIENCE ....................................................126 Analysis I................................................................................................................128 Linear Algebra I.....................................................................................................129 Computers and Programming................................................................................130 Introduction to Logic .............................................................................................131 English for Mathematics I .....................................................................................133 Principles of Programming....................................................................................134 Introduction to Combinatorics ...............................................................................135 Linear Algebra II ...................................................................................................136 Analitical Geometry ...............................................................................................137 Analysis II ..............................................................................................................138 English for Mathematics II....................................................................................139 Analysis 3 ...............................................................................................................140 Algebra I.................................................................................................................141 Operating Systems..................................................................................................142 Programming 1 ......................................................................................................143 Discrete Mathematics.............................................................................................144 Complex analysis I .................................................................................................145 Analysis 4 ...............................................................................................................146 Algebra 2 ................................................................................................................147 Probability Theory..................................................................................................148 Programming 2 ......................................................................................................149 Differential Equations............................................................................................150 Statistics..................................................................................................................151 Computer Networks................................................................................................152 Measure and Integral.............................................................................................153 Database Systems ...................................................................................................154 Object Oriented Programming...............................................................................155 English for Mathematics III ..................................................................................156 Introduction to Differential Geometry ...................................................................157 Functional Analysis ...............................................................................................158 Numerical Analysis ................................................................................................159 Compilers ...............................................................................................................160 Visualization and Computer Graphics...................................................................161 Internet Technologies ............................................................................................162 English for Mathematics IV...................................................................................163

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Academic Undergraduate Programme of Study COMPUTER SCIENCE............164 Introduction to Computer Science .........................................................................167 Computers and Programming................................................................................168 Analysis I................................................................................................................169 Analitical Geometry ...............................................................................................170 Introduction to Mathematical Logic ......................................................................171 English for Computer Sciences I ...........................................................................172 Principles of programming ....................................................................................173 Data Structures ......................................................................................................174 Analysis II ..............................................................................................................175 Linear Algebra .......................................................................................................176 Algebra ...................................................................................................................177 English for Computer Sciences II..........................................................................178 Programming 1 ......................................................................................................179 Discrete mathematics I...........................................................................................180 Computer Networks................................................................................................181 Operating Systems..................................................................................................182 Analysis III.............................................................................................................183 Object Oriented Programming...............................................................................184 Programming 2 ......................................................................................................185 Visualization and Computer Graphics...................................................................186 Peripheral Devices and Interfaces .........................................................................187 Probability Theory and Statistics ...........................................................................188 Differential Equations............................................................................................189 Discrete mathematics II .........................................................................................190 Computer System Architecture...............................................................................191 Database Systems ...................................................................................................192 Programming Languages.......................................................................................193 Numerical Analysis ...............................................................................................194 Computer Security..................................................................................................195 Artificial Intelligence .............................................................................................197 Introduction to Information Systems .....................................................................198 English for Computer Sciences III ........................................................................199 Software Engineering ............................................................................................200 Advanced Topics in Programming.........................................................................201 Compilers ...............................................................................................................202 Internet Technologies ............................................................................................203 Distributed Systems ................................................................................................204 Advanced Database Systems ..................................................................................205 English for Computer Sciences IV.........................................................................206

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I

INTRODUCTION

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II

Academic Undergraduate Programme of Study

BIOLOGY

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Programme of study Level of Studies Academic/Applied

BIOLOGY Undergraduate Academic I year

Course

Mandatory

Elective

Winter Semester

hours weekly

EC

TS

Ljetnji semestar

hours weekly

EC

TS

1. Mathematics X 2+0 3 2. Citology and Histology X 3+2 6 3. Plant Anatomy X 4+3 8 4. Opšta I neorg.hemija X 3+2 5 5. Invertebrates I X 3+2 5 6. History of Biology X 2+0 3 7. Physics X 3+1 4 8. Histology of Organs with

Embriology X 2+2 5

9. Organic Chemistry X 3+2 6 10. Invertebrates II X 3+2 7 11. Algae, fungi and lichens X 4+4 8

Total 11 17+9 30 15+11 30 II year 1 Anthropology X 3+2 5 2. Biochemistry I X 3+2 6 3. Physiology X 4+3 8 4. Comparative Anatomy and

Sistematics of Vertebrates I X 3+3 7

5. Systematic of Cormophyta I X 2+1 4 6. Biochemistry II X 2+2 5 7. Comparative Anatomy and

Sistematics of Vertebrates II X 2+2 5

8. Systematic of Cormophyta II X 2+3 5 9. Microbiology X 4+3 8 10. Genetics X 4+2 7

Total 10 15+11 30 14+12 30 III year 1. Plant Physiology X 4+3 9 2. Molecular Biology I X 3+2 6 3. Ecology of Animals I X 2+1 4 4. Ecology of Plants I X 2+2 4 5. Environmental Protection I X 2+0 2 6. Hidrobiology X 3+2 5 7. Molecular Biology II X 3+2 6 8. Comparative Physiology X 2+2 4 9. Evolution X 4+0 5 10. Ecology of Animals II X 2+2 5 11. Ecology of Plants II X 2+2 5 12. Environmental Protection II X 2+0 2 13. Human ecology x 3+0 3

Total 16+10 *30 18+8 30

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Name of the course: Mathematics Programmes of Studies:

Academic study program Biology

Level of the course:

Bachelor level, I year, I semester Number of ECTS credits: 3

Contact hours: 2 Lectures per week, 30 hours in semester for consultations = 60 contact hours in semester

Total hours: 3 x 30 = 90 hours in semester

Structure: 26 hours - lectures, 4 hours - exams, 30 hours - consultations, 20 hours – homework (individual solving of problems), 10 hours – individual study.

Language: Serbian or English Prerequisites: There is no prerequisites

Aim: This course is an introduction to basic techniques that can be used in biology. The special attention is being pointed to the statistical methods that have application in biology and ecology.

Contents:

Dates and representation of dates. Characteristics of one-dimensional dates. Characteristics of multy-dimensional dates. Definitions of probability. Gauss, Poisson and binomial distribution. Errors in statistics. Prošireni metod maksimalne vjerodostojnosti. Metod momenata. Stratifikovano biranje uzorka. Osnove metoda najmanjih kvadrata. Fitovanje pravom linijom. Fitovanje binomnih podataka. Linearni i nelinearni metod najmanjih kvadrata. Intervali povjerenja. Testiranje hipoteza. Interpretacija eksperimenta.

Main texts: Statistics, R.J.Barlow

Further readings: - capacity of understanding a problem.

- ability to understand the proofs of theorems, and to construct independently simple proofs of mathematical statement.

Competences to be developed:

- ability for analytical thinking and capacity to argu the own opinion and statements.

Methods of teaching:

Lectures and seminars with the active participation of students, individual home tasks, group and individual consultations.

Examination: Written exams (3 times in semester), problem solving - home tasks, estimation of individual activity on lectures and seminars

Methods of self-evaluation:

Students pools, results of exams, direct communications with the students.

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Name of the course: Citology and Histology Programmes of Studies:

Academic study programmes Biology


Bachelor level, 1st year, 1st semester Number of ECTS credits: 6

Contact hours: 3 Lectures + 2 laboratory per week, 15 hours in semester for consultations = 128 contact hours in semester


Structure: 48 hours - lectures, 32 hours - laboratory, 6 hours - exams, 15 hours - consultations, 79 hours – individual study.

Language: Serbian or English Prerequisites: No prerequisites.

Aim:

This course is aimed to introduce students with basic Citology, Basic Histology and Basic Histological Tehniques. Histologu is a branch of Anatomy that studiestissuesof animals and plants. This course discussesonly animals and more specifically human tissues.

Contents:

Cells. Cytoplasm. Cell membrane. Membrane transport. Ribosomes. Endoplasmic Reticulum.. Polyribosomes. Golgi Apparatus. Peroxisomes. Mitohondria. Cytoskeletion. Nucleus. Chromatin. Cell cycle.Extracellular Matrix. Epithelium and Glands. Connective Tissue. Cartilage and Bone. Musgle NervousTissue. Blood and Hemopoiesis. Zlatko Andjelkovic,LJ. Somer, M. Matavulj, V. Lackovic, D. LalosevicL. Nikolic, Z. Milosavljevic,V. Danilovic,.GIP*Bonafides*, Nis 2002. V. Lackovic at all. Histoloski atlas 2003.

Main texts:

Leslie P. Gartner, James L. Hiatt, Color Textbook of Histology,Philadelphia, W.B.Saunders Company 2001. Further readings: Understand Basic Gells and Tissue. LearnHistologigal Tehniques Learn microscopicstructure of the cell and tissues.



Lectures and laboratory with the active participation of students, group and individual consultations.

Examination: Written exams (3 times in semester), , estimation of individual activity on lectures and laboratory.



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Name of the course: Plant Anatomy Programmes of Studies:

Academic study programmes Biology , Faculty of Science and Mathematics, University of Montenegro


Bachelor level, III year, I semester Number of ECTS credits: 8

Contact hours: (4 Lectures +3 laboratory work) per week; 15 hours in semester for consultation = 120 contact hours in semester


Structure: Lectures: 52 hours; Exercises: 39 hours; Exam: 14 hours; Consultation: 15 hours; Seminars: 25 hours; Individual work: 95 hours

Language: Montenegrian Prerequisites: - Aim: Examination of structure and function plant cells, tissues and organs

Contents:

Plant cell anatomy. Cytosol. Cell membranes. Cell wall. Endoplasmic reticulum. Nucleus. Ribosomes. Mitochondrion. Chloroplast. Cell reproduction: mitosis, meiosis. Plant tissues. Embryonic tissue – meristem. Epidermal tissue. Ground tissue. Vascular tissue. Plant organs. Monocot versus dicot plants. Structure of monocot and dicot roots.How roots differ. Structure of monocot and dicot stems. How stems become woody. How stems differ. Structure of leaves. How leaves differ. Reproduction in plants. Flowering plants undergo alternation of generations. Flower-sepals, petals, stamens and a pistil. Polinations. Fertilization. Seeds. Fruits. Tati� B., Petkovi� B., Morfologija biljaka. Zavod za udžbenike i nastavna sredstva, Beograd 1998. (in Searbian)

Main texts:

Mader S., Biology. Mc Graw-Hill. Boston, Massachusetts, 1996. Further readings: - capacity to understanding plant cell anatomy

- ability to understand plant tissues anatomy - ability to understand structure of plant organs


- capacity to understanding reproduction in plants


Leactures, laboratory practice, seminars, group and individual consultation

Examination: Laboratory practice examination, estimation of individual activity on lectures and seminars



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Name of the course: General and Non-Organic Chemistry

Course: OPŠTA I NEORGANSKA HEMIJA

Šifra predmeta Status predmeta Semestar Broj ECTS kredita Fond �asova

ONH.01.01. Mandatory I 5 3P+2V

Studijski programi za koje se organizuj : Akademske Undergraduate PMF-a, studijski program Biologija,studije traju 6 semestara, 180 ECTS kredita Uslovljenost drugim predmetima: Nema

Ciljevi izu�avanja predmeta: Detaljnije upoznavanje osnova opšte hemije. Prou�avanje osobina hemijskih elemenata i njihovih najzna�ajnijih jedinjenja sa posebnim akcentom na biogene elemente i mikroelemente zna�ajne za živi svijet. Ime i prezime nastavnika i saradnika: Prof. dr Željko Ja�imovi�, Dr I. Boškovi� Metod nastave i savladanja gradiva: Predavanja, laboratorijske i ra�unske vježbe, samostalna izrada doma�ih zadataka. Konsultacije.

Sadržaj predmeta Pripremne nedjelje

I nedjelja II nedjelja III nedjelja IV nedjelja V nedjelja VI nedjelja VII nedjelja VIII nedjelja IX nedjelja X nedjelja XI nedjelja XII nedjelja XIII nedjelja XIV nedjelja XV nedjelja XVI nedjelja

Završna nedjelja XVIII-XXI nedjelja

Priprema i upis semestra Uvod, pojam materije, smješe i �iste supstance, elementi i jedinjenja. Osnovni hemijski zakoni, atomska i molekulska teorija. Struktura atoma i teorije o strukturi atoma. Energetski nivoi elektrona, elektronska konfiguracija i PSE. Hemijska veza, me�umolekulske sile, kristalni sistemi. Gasni zakoni. Stehiometrija.Izra�unavanja iz hemijskih jedna�ina. Slobodna nedjelja Rastvori, osobine razblaženih rastvora, Vant-Hoffov i Raulovi zakoni. Rastvori elektrolita. Jaki i slabi elektroliti, kiseline, baze i soli. Brzina hemijske reakcije, hemijska ravnoteža. Test Termohemija i hemijska termodinamika,kompleksna jedinjenja,koloidni rastvori. Pregled elemenata i njihovih jedinjenja: vodonik,kiseonik,halogeni elem.,alkalni metali, plem. gasovi. 15 i 16 grupa PSE, metali i prelazni elementi. Kolokvijum Stabilni i nestabilni izotopi, tipovi radioaktivnosti. Završni ispit Ovjera semestra i upis ocjena Dopunska nastava i poravni ispitni rok

OPTERE�ENJE STUDENTA

Nedjeljno 5 kredita x 40/30 = 6 sati i 40 minuta

Struktura: - 3 sata predavanja; - 2 sata laboratorijskih vježbi; - 1 sat i 40 minuta individualnog rada studenata uklju�uju�i i konsultacije.

U semestru Nastava i završni ispit:6sati i 40min. x 16= 106sati i 40min. Neophodne pripreme prije po�etka semestra(administracija, upis i ovjera): 2 x 6sati i 40 min.= 13 sati i 20 min. Ukupno optere�enje za predmet: 5x 30 = 150 sati Dopunski rad za pripremu ispita u popravnom ispitnom roku,uklju�uju�i i polaganje popravnog ispita od 0 do 41 sat i 50 minuta. Struktura optere�enja: 106sati i 30min.(nastava)+13sati i 20min.(priprema)+41sata i 50 minuta(dopunski rad)

Obaveze studenata u toku nastave: Studenti su Mandatory da poha�aju nastavu,rade laboratorijske vježbe, predaju doma�e zadatke, rade test i kolokvijum.

Literatura Literatura: (1) Arsenijevi�, , Opšta i neorganska hemija, Nau�na knjiga- Beograd, 1998 (2) D Poleti , Opšta hemija II dio- Hemija elemenata , TMF, Beograd 2003. (3) Filipovi�, S. Lipanovi�, Op�a i anorganska kemija, Školska knjiga, Zagreb, 1988.

(4) V. �ešljevi�, V. Leovac, E. Ivegeš, Praktikum neorganske hemije- prvi dio, PMF Novi Sad 1997 (5) Milan Sikirica, Stehiometrija, Školska knjiga, Zagreb, 1989., Zbirka zadataka. Oblici provjere znanja i ocjenjivanje: Aktivnost na vježbama i predati izvještaji – 4 poena, 3 doma�a zadatka po 2 poena - 6 poena, Test iz laboratorijskih vježbi- 10 poena, Kolokvijum- 30 poena, Završni ispit- 50 poena. Ispit je položen sa 51 poenom Posebnu naznaku za predmet: Laboratorijske vježbe se izvode u grupama sa najviše 13 studenata.

Ime i prezime nastavnika koji je pripremio podatke: Prof. dr Željko Ja�imovi� Napomena: Dodatne informacije o predmetu na sajtu

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Name of the course: Invertebrates I Programmes of Studies:




Contact hours: (3 Lectures + 2 Seminar) per week, 15 hours in semester for consultations = 80 contact hours in semester


Structure: 39 hours - lectures, 26 hours - seminars, 10 hours - exams, 15 hours - consultations, 30 hours – homework (individual solving of problems), 30 hours – individual study.

Language: Serbian or English Prerequisites: Without prerequistes

Aim:

This course is aimed to introduce students with basic notions of morphology, anatomy, ecology and philogetical position of different groups of lower invertebrates. The course include fundamental knowledge on different aspects of zoology of lower invertebrates .

Contents:

Clasification and Taxonomy. Protozoa. Orgin of Metazoa. Spongia. Cnidaria. Ctenophora. Plathemminthes. Turbelaria, Trematodes, Cestodes. Nemertina. Cycloneuralia: Kinorhyncha, Gastrotricha, Loricifera, Nematomorpha, Nematoda. Gnathifera, Gnathostomulida, Micrognathozoa. Mollusca (general introduction) Gastropoda, Bivalvia, Cephalopoda. M.Brajkovi�: Zoologija invertebrata I Dio. -Zavod za udžbenike Beograd 2003. Main texts: Further readings: - capacity of zoological understanding of different groups of invertebrates.

- ability to understand the importance of invertebrates, and to construct independently simple proofs of zoological statement. - capability to apply the knowledge of morphology, anatomy and ecology of invertebtaeres in different areas of biology;





Examination: exams (3 times in semester), problem solving - home tasks, estimation of individual activity on lectures and seminars



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Name of the course: History of Biology Program of Studies:

Academic study program: Biology (Faculty of Natural Science, Department of Biology) Course: History of Biology



Contact hours:

Weekly: 3 credits x 40/30 = 4 hours

Structure: 2 hours teaching 2 hours of individual work including consulatation

Total hours During semester:

90 hours in semester.

Structure:

Teaching and final examination: 4 hours 16 =64 hours Necessary preparation: administration, enrolment, notarization before start of semester, 2x4 hours =8 hours. Total: 3x30=90 Additional work: preparation of examination in makeup examination period, including makeup examination for students from 0 do 36 hours (all another time from the first two points are until to total load of 240 hours). Load structure: 64 hours (teaching) + 8 hours (preparation) + 18 hours (additional work).

Language: Native language or English Prerequisites: /

Aim:

This course is aimed to introduce students with chronological development of biology; contribution of scientists in development of biology science; technical inovation in the function of biology research; formulation of biology concepts etc.

Contents:

Historical review of biology science. Ancient philosophy, ancient Greeks philosophers who research in biology.Ancient Roman’s philospohers. Biology in middle century (Albert Magnus, Frederich II Hofenstaufen etc). Biological research in XVI century (Michael Servetus, Andres Vesalius, Fabricius d’Aquapendente, etc).Biological research in XVII century (Francis Bacon, Robert Hooke, Jan Svammerdam, etc).Biology in XVII century II part (Regnier de Graff, etc). Biology in XVIII century (Carl Line, Georges Luis Leclerc, Luigi Galvani, Lazzaro Spalanzani, etc).Biology in XVIII century II (Antoine Laurent Lavoisier, Rene Antoine Ferchault, Caspar Friedrich Wolf etc).Biology in XIX century I and II (Jean-Baptiste Lamarck, Mathias Jacob Schleiden, Theodore Schwan etc). Biology in XIX century (Robert Brown, Xavier Bichat, Jean Bapttiste Dumas, Heckel, Charles .Darvin etc).Biology in XX century I (Watson, Crick, Landstainer etc). Biology in XX century II (J.Dausset, Thomas Hunt Morgan, Hugo de Vries etc).

Main text: Teodorides, Ž. History of biology, 1-127, VII publishing: Ministry of culture, France. Printed in Belgrade, 1995.


to understand chronological development of biology science, technical inovation in the function of biology research; formulation of biology concepts etc.


Lectures and seminars with the active participation of students, individual home tasks, consultations.

Examination: Written exams (2 times per semester),seminar paper, final exam. Methods of self-evaluation:

Student’s pools, results of exams, direct communications with the students.

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Name of the course: Physics Programmers of Studies:

Academic study programs Biology


Bachelor level, I year, II semester Number of ECTS credits: 4

Contact hours: (2 Lectures + 1 laboratory hour) per week, 15 hours in semester for consultations = 60 contact hours in semester


Structure: 26 hours - lectures, 13 hours -laboratory, 6 hours - exams, 15 hours - consultations, 30 hours – homework (individual solving of problems), and 30 hours – individual study.

Language: Serbian or English

Prerequisites: This course belongs to the first bases of physics and essentially needs no foreknowledge in physics. However, notions of physics and mathematics on a secondary school level are recommended

Aim:

The aim is to learn the student the basic knowledge and principles of these parts of physics listed in the content. These items are important in the understanding of further courses in the biology curriculum. The aim of this course is to teach students some procedures necessary for a laboratory work too. This course is also an important training in scientific thinking and working.

Contents: Kinematics,Dynamics,Gravitation,Work and Energy, Linear momentum, Rotation, Hydrostatics, Hydrodynamics,Oscillations,Waves, Gases, Thermodynamics,Electromagnetism,Geometrical Optics,Nuclear physics Resnic, Halliday and Krane: Physics, volume 1 and 2 (fifth edition)

Janji�, Bikit i Cindro: Opšti kurs fizike I i I I.Savi�, A.Sre�kovi� Fizika za studente biologije (skripta) V.Vu�i� Osnovna mjerenja u fizici

Main texts:

Further readings: This course enables the student to acquire the fundamental skills of mechanics,thermodynamics,waves and oscillations and the behavior of fluids (liquids and gasses), electromagnetism.The mastering of a scientific way of working is here an end aim. A broader context is the understanding of modern society and its technological evolution. This course is a good help to do later scientific research.



Lectures and seminars with the active participation of students, individual home tasks, laboratory practice, group and individual consultations.

Estimation: Written exams, estimation of individual activity in laboratory practice Methods of self-evaluation:


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Name of the course: Histology of Organs with Embryology Programmes of Studies:

Academic study programmes BIology


Bachelor level, 1st, year, 2nd semester Number of ECTS credits: 5

Contact hours: (2 Lectures + 2 laboratory) per week, 16 hours in semester for consultations = 80 contact hours in semester


Structure: 32 hours - lectures, 32 hours - laboratory 6 hours - exams, 16 hours - consultations, 64 hours – individual study.

Language: Serbian or English Prerequisites: No prerequisites Aim: This course is aimed to introduce students with basic .

Contents:

Cyrcylatory System, Lymphoid System, EndocrineSystem, Integument,Respyratory System,Digestive System /Oral Caviity/,Digestive System/Alimentary Canal/, Digestive System/Glands/, Urinary System,FemaleReproductive System, Male Reproductive System, Special Senses/ specialized peripheral receptors,eye, ear/. Embryology organs and organ sustems. Z.Andjelkovic, Lj Somer, M.Matavullj. V.LackovicI.Nikolic, Z.Milosavljevic, V.DanilovicHistoloska gradja organa, GIP*Bonafides, Nis 2002. I.Nikolic, G.Rancic, G.Radenkovic, V.Lackovic, V.Todorovic. D.Mitic,Embriologija coveka,DATA STATUS, Beograd, 2006. V.Lackovic, S.Popovic, Histoloski atlas, Bonafides Nis, 2001.

Main texts:

Laslie P. Gartner, James L. Hiatt,Color Textbook of HistologyW.B.Saunders Company Philadelphia, 2001. Further readings: Understand microscopic structure organs and organ systems.

Learn basis and special of embryology organs and organ system.



Lectures and laboratory with the active participation of students, individual home tasks, group and individual consultations.

Examination: Written exams (3 times in semester), estimation of individual activity on lectures and laboratory.



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Name of the course: Organic Chemistry Programmes of Studies:

Academic Study Program Biology.


Bachelor level, 1st year, 2nd semester. Number of ECTS credits: 6

Contact hours: 3 lectures + 2 laboratory per week, 15 hours in semester for consultations = 128 contact hours in semester



Language: Serbian Prerequisites: No prerequisites.

Aim: Obtaining knowledge of modern understandings and successes of organic chemistry. Study of structure, characteristics of organic compounds and mechanisms of their reactions.

Contents:

Introduction, characteristic features of organic compounds, qualitative elementary analyses of organic compounds. Structure of organic compounds, reaction and reagents, classification of organic compounds. Carbohydrates: alkanes, alkenes, structure and isomery. Alkines, alkadienes, cyclic carbohydrates. Aromatic compounds. Halogen derivate of carbohydrates. Alcohols, phenols, ethers, tioalcohols, tioethers. Aldehydes and ketones. Carbonyl acids. Derivates of organic acids. Substituted organic acids-halogen-hydroxyl-keto-acids. Aromatic nitro compounds, amines. Amino acids, proteins. Carbohydrates – monosaccharides. Carbohydrates – disaccharides, polysaccharides. S. Arsenijevic, Organska hemija, Naucna knjiga, Beograd,1997. A. Taylor, Organska hemija, Naucna knjiga , Beograd,1995. D. Rondovic, M. Puric, Hemija, Univerzitet Crne Gore, Podgorica, 2003. R. Kastratovic, Praktikum organske hemije, Podgorica, 1997.

Main texts:

Further readings:


- ability to understand connections between atoms and atomic groups in organic molecules;

- understanding of characteristic transformation of functional groups; - understanding of structural characteristics and their influence on behavior

of organic compounds; - to cope with basic mechanisms of organic reactions and to understand

transformations in biological systems; - to cope with basic techniques of laboratory work and fulfillment of

experiments in organic chemistry laboratory Methods of teaching:


Examination: Written exams. Homeworks, tests, final examination. Methods of self-evaluation:

Student pools, results of exams, direct communications with the students.

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Name of the course: Invertebrates II Programmes of Studies:








Aim:

This course is aimed to introduce students with basic notions of morphology, anatomy, ecology and philogetical position of different groups of higher invertebrates. The course include fundamental knowledge on different aspects of zoology of higher invertebrates.

Contents:

Annelida (General Introduction). Polychaeta. Oligochaeta. Hirudinomorpha. Priapulida, Echiurida, Sipunculida, Tardigrada. Panarthropoda. Arthropoda (General Introduction). Trilobitomorpha, Trilobita, Chelicerata (Merostomata, Pycnogonida). Arachnida – General introduction; Diversity of Arachnida. Crustacea – General introduction; Diversity. of Crustacea: Malacostraca Myriapoda Hexapoda (General Introduction) Apterygota, Insecta. Diversity of Insecta. Tentaculata : General introduction and Diversity. Echinodermata (General Introduction), Echinozoa, Asterozoa, Crinoizoa. Chaetognatha. Filogeny of Animals M.Brajkovi�: Zoologija invertebrata II Dio. -Zavod za udžbenike Beograd 2003.

Main texts:

Further readings: - capacity of zoological understanding of different groups of invertebrates.

- ability to understand the importance of invertebrates, and to construct independently simple proofs of zoological statement. - capability to apply the knowledge of morphology, anatomy and ecology of invertebtaeres in different areas of biology;








23

Name of the course: Algae, fungi and lichens Programmes of Studies:

Academic - Biology.


Bachelor level, 1st year, 2nd semester. Number of ECTS credits: 9



Structure: 64 hours - lectures, 64 hours - exercises, 6 hours - exams, 24 hours - consultations.

Language: Serbian or English Prerequisites: No prerequisites.

Aim: Study of morphology, anatomy, systematic and taxonomy of algae, fungi and lichens

Contents:

Morphology, anatomy and taxonomy of algae from following divisions: Cyanophyta, Rhodophyta, Pyrrophyta, Euglenophyta, Xanthophyta, Chrysophyta, Bacillariophyta, Chlorophyta and Charrophyta. Ecological groups of algae. Morphology, anatomy and taxonomy of fungi from divisions: Myxomycota and Eumycota (Zygomycotina, Mastigomycotina, Ascomycotina, Basidiomycotina and Deuteromycotina). Morphology, anatomy and taxonomy of lichens (Ascolichenes). Blažen�ic, J. (2000): Sistematika algi. Nau�na knjiga, Beograd Cvijan, M. (1996): Praktikum iz algologije, Biološki fakultet Univerziteta u Beogradu; Rankovic, B. (2003): Sistematika gljiva. PMF, Kragujevac. Marinovic, R. (1988): Osnovi mikologije i lihenologije. Naucna knjiga, Beograd. Muntanola - Cvetkovic, M. (1987): Opšta mikologija, Književne novine, Beograd. Vukojevic, J. (2000): Praktikum iz mikologije i lihenologije. NNK Internacional, Beograd.

Main texts:

Further readings:

Treinor, F., R. (1978): Introductory Phycology. John Wiley & Sons, New York. Alexopolos, C.J., Mims, C.W., Blackwell, M. (1996): Introductory Mycology. John Wiley & Sons Inc., New York. Ozenda, P., Clauzade, G. (1970): Les lichens. Masson et Cle, Paris.


Familiarity with basic concepts in Systematic of Algae, Fungi and Lichens Capacity to understand morphology and anatomy of different groups of Algae, Fungi and Lichens Learning methods for sampling, preservation and preparation of different groups of algae, fungi and lichens Ability to recognize the most common representatives of algae, fungi and lichens in field Developing the oral and written communication


Lectures and laboratory exercises with the active participation of students, group and individual consultations.

Examination: 2 written exams (colloquiums), 2 practical exams (tests) Methods of self-evaluation:


24

Name of the course: Anthropology Programmes of Studies:

Academic study programme Biology.


Bachelor level, III year, VI semester Number of ECTS credits: 5

Contact hours: 3 Lectures + 2 laboratory per week, 20 hours in semester for consultations = 106 contact hours in semester


Structure: 48 hours - lectures, 32 hours - seminars, 6 hours - exams, 20 hours - consultations, 44 hours individual study.

Language: Serbian Prerequisites: No prerequisites

Aim:

This course is aimed to introduce students with basic notions of anthropological principles and basic anthropometrics techniques. Also, it gives basic knowledge in anthropogenesis and variability of human population.

Contents:

Position of the man in biology. Human body, prenatal development, postnatal ontogenesis, factors of growth and development. Body proportions, sexual and population differences. Acceleration in the human populations. Constitution, anthropological, medical and ecological aspects. Composition of the human body and organic systems. Evolution of hominids. Modern man end evolution. Geographic distribution anthropological types. B.M.Ivanovic� Anthropology I, Anthropomorphology. Unirex, Podgorica, 1996. P. Vlahovic. The man in time and spatial. Anthropology. Vuk Karadzic, Belgrade, 1996 R. Hadziselimovic. Anthropogenesis. Svjetlost, Sarajevo, 1988. �

Main texts:

E.N.Hrisanfonova, Perevozcikov. Antropologia. MGU, Moscow Further

readings: - capacity to understand anthropological problems. analytical thinking, search the literature, developing the oral and written communication. - ability for individual work in laboratory.







25

Name of the course: Biochemistry I Programmes of Studies:



Bachelor level, 2nd year, 3rd semester Number of ECTS credits: 6



Structure: 48hours - lectures, 32 hours - laboratory, 6 hours - exams, 20 hours - consultations, 74 hours – individual study.

Language: Prerequisites: No prerequisites.

Aim: This course is aimed to introduce students with basic notions of biochemical molecules along with an introduction to metabolism.

Contents:

The scope of biochemistry. Interactions of biomolecules. H2O. Amino acids, peptides and the three-dimensional strucure of proteins. Structure of nucleic acids (DNA and RNA), stability, and basic recombinant technology. Protein stability, folding, and dynamics related to function. Carbohydrates. Lipids and membranes. Transport trough membrane. Enzyme kinetics, regulation of enzyme activity, and mechanisms of enzyme catalysis. Introduction to metabolism. Lj. Topisirovi�, Dinami�ka biohemija, Beograd, 1999.

D.W. Martin, P.A. Mayes, V.W. Rodwell, D.K. Granner, Harperov pregled biohemije. Savremena administracija. Beograd, 1992.

Main texts:

J. Berq, J.L. Tymoczko, L. Stryer. Biochemistry. W.H. Freeman & Company. 6 edition,2006. Further readings: D. Woet, Biochemistry. Wiley; 3 Har/Com edition, 2004. - capacity to understand structure and function of biochemical molecules.

- ability to understand membrane structure and membrane transport

- Understand basic principles of metabolism


- ability for individual work in biochemistry lab



Examination: Written exams (2 times in semester), oral exam, problem solving - home tasks, estimation of individual activity on lectures and seminars



26

Name of the course: Physiology Programmes of Studies:

Academic study of Biology, Faculty of Science, University of Montenegro


Bachelor level, II year, IV semester Number of ECTS credits: 8

Contact hours:

(4 Lectures + 3 laboratory work) per week., 15 hours in semester for consultations = 120 contact hours in semester

Total hours:

8 x 30 = 240 hours in semester

Structure: 52 hours - lectures, 39 hours - laboratory exercises , 14 hours - exams, 15 hours - consultations, 30 hours – homework, 90 hours – individual study.

Language: Montenegrian Prerequisites: Basic courses of zoology and cytology

Aim:

This course is aimed to introduce students with basic study of relationships between the cell structure and their functions. To give a general sketch of the structure of animal tissues and organs and emphasize how the organization of tissues and organs confirm their functions.

Contents:

Cellular physiology. Biological membranes, basic neuronal activity. Cellular connections, arrangements of tissues. Reflexive arch. Phenomenon of excitability on the cell level. Characteristic function of the blood. Circulation system, regulation of circulation. Heart activity and its humoral and neuronal regulation. Breathing and its regulation. Renal function. The digestive tract and its activity-regulation. The endocrine system. Energetic metabolism. The sensory system and perception. The motor system and movement regulation. Higher neuronal activity. A. Guyton and J. Hall, Medical Physiology, tenth edition, Savremena administracija, Belgrade, 2003. Main texts: M. Paši�, Physiology of Nervous System, Naucna Knjiga, Beograd, 1993. J. Pinel, Biopsychology, Fourth Edition, Allyn and Bacon, 2000. Further readings: R. Berne and M. Levy, Physiology, Fourth Edition, Mosby, 1998. - Course provides capacity to understand the complex mechanisms and function of the intact organism and its emphasis on the processes that regulate the important properties of living systems. Competences to

be developed: - Within the homework tasks students provide ability for individual work with different sources of scientific literature, performing independent literature research, preparing individual presentations about different scientific physiological problems and defending scientific view in group discussions.


-Lectures, seminars and practical laboratory exercises are organized with the active discussions and student’s participation, individual home tasks, group and individual consultations.




27

Name of the course:

Comparative Anatomy and Sistematics of Vertebrates I

Programmes of Studies:



Bachelor level, II year, III semester Number of ECTS credits: 7




Language: Serbian or English Prerequisites: Courses of Inverebrates I and Invertebrates II

Aim: This course is aimed to introduce students with basic notions of morphology, anatomy, function, adaptation pattern and phylogenetic relations between different groups of vertebrates.

Contents:

Chordata: general characteristics, systematic and phylogenetic position of Vertebrates. The main characteristics, origin and ontogeny. Morphological characteristics and evolutionary differentiation of organic systems of the subphylum Vertebrata ( Integumentary system, skeletal system, muscular system, nervous system, sense organs, endocrine system, digestive system, respiratory system, circulatory system, excretory system and reproductive system).

Main texts:

Kalezi� M. Osnovi morfologije ki�menjaka. Savremena administracija- Beograd, 1995.

Kalezi� M. Hordati (skripta). Biološki fakultet Univerziteta u Beogradu – Beograd, 1998.

Stankovi� S. Uporedna anatomija ki�menjaka. Nau�na knjiga – Beograd, 1950.

Šori� V. Morfologija i sistematika Hordata. Univerzitet u Kragujevcu, PMF – Kragujevac, 1997.

Further readings: Pough H, Janis CM & Heiser JB 2005. Vertebrate Life. 7th Ed.

Pearson, Prentice Hall - Capacity of zoological understanding of general characteristics, systematic and phylogenetic position of Vertebrates. - Ability to understand the evolutionary differentiation of organic systems of the subphylum Vertebrata. - Capability to apply the knowledge of morphology and anatomy of vertebrates in different areas of biology.


- Ability for analytical thinking and capacity to argue the own opinion and statements.


Lectures and seminars with the active participation of students, fieldwork, individual home tasks, group and individual consultations.

Examination: Quiz (2 times in semester), written and oral examination, problem solving - home tasks, estimation of individual activity on lectures and seminars.


Students’ pools, results of exams, direct communications with the students.

28

Name of the course: Systematic of Cormophyta I Programmes of Studies:

Academic study programme Biology



Contact hours: Per week 2h Lectures + 1h Practicum and 1h and 20min of individual work and consultations


Structure: 85h and 30 min (2h of lectures + 1h of practical work + 1h and 20min of individual work and consultations- per week; exams), 10h and 40 min administrative preparation and 23h and 50 min additional work

Language: Serbian or English Prerequisites: Basic courses of Plant Anatomy and Morphology

Aim: This course is aimed to introduce students with basic principals of taxonomy of plants and Takhtadjan’s systematical and phylogenetical system.

Contents:

Introduction in taxonomy and systematic of higher plants. Origin of vascular plants, evolution of plant organs, classification of higher plants. Division: Rhyniophyta & Zoosterophyllophyta. Division Bryophyta. Division Lycopodiophyta. Division Psilotophyta. Divion Equisetophyta.Division Polypodiophyta. Division Pinophyta International Code of Botanical Nomenclature (Tokyo Code) ��, �. ��. (1978): �� , ��i, ��, �� , ��i� �� , ��, �� Main texts: Domac, R. (1984): Mala flora Hrvatske i susjednih podru�ja, Škosla knjiga Zagreb Ble�i�, V. (1964): Sistematika Viših Biljaka, Beograd Javorka S., Csapody, V. (1975): Iconographia Florae Partis Austro-Orientalis Europae Centralis, Akademiai Kiado, Budapest

Lauber, K. & Wagner G. (2001): Flora Helvetica, Verlag Paul Haupt, Bern-Stuttgart-Wienn Rohlena, J. (1942): Conspectus florae Montenegrinae, Preslia 20/2 Pulevi�, V. (2005): Gradja za Vaskularnu Floru CG, Zavod za Zaštitu Prirode Crne Gore, Posebna Izdanja, knjiga 2

Further readings:

Sitte, P., Ziegler, H., Ehrendorfer, F. & Bresinsky, A. Eds. (1998): Strasburger - Lehrbuch der Botanik fuer Hochschule, Gustav Fischer Verlag, Stuttgart, Jena Capacity to understand the basic principals of plant taxonomy and to became familiar with codex of botanical nomenclature Capacity to understand basic principals of systematic of plants (Takhtajan’s system of classification) Capacity to identify the most common representatives from the flora of Montenegro and to recognize them in field


Capacity to search the literature and develop written and oral communication Methods of teaching:

Lectures, practical work, homeworks, seminar papers, consultations.

Examination: 2 written examinations, 1 practical examination, homework, seminar paper Methods of self-evaluation:

Students pools, results of examinations, direct communications with the students.

29

Name of the course: Biochemistry II Programmes of Studies:



Bachelor level, 2nd year, 4th semester Number of ECTS credits: 5




Language: Prerequisites: No prerequisites.

Aim: This course is aimed to introduce students in concept of metabolism (anabolism and catabolism).

Contents: Energy transduction (oxidative phosphorilation). Metabolism of carbohydrates. Metabolism of amino acid and proteins. Metabolism of lipids. Metabolism of nucleic acids. Energy metabolism. Lj. Topisirovi�, Dinami�ka biohemija, Beograd, 1999.

D.W. Martin, P.A. Mayes, V.W. Rodwell, D.K. Granner, Harperov pregled biohemije. Savremena administracija. Beograd, 1992.

Main texts:

J. Berq, J.L. Tymoczko, L. Stryer. Biochemistry. W.H. Freeman & Company. 6 edition,2006. Further readings: D. Woet, Biochemistry. Wiley; 3 Har/Com edition, 2004. - capacity to understand metabolism of carbochydrates, amino acids, lipids and nucleic acids. - analytical thinking, search the literature, developing the oral and written communication


- ability for individual work in biochemistry lab






30

Name of the course:

Comparative Anatomy and Sistematics of Vertebrates II








Language: Serbian or English Prerequisites: Course of Comparative Anatomy and Systematics of Vertebrates I

Aim: This course is aimed to introduce students with systematics of Chordata and Vertebrata.

Contents: Classification and Taxonomy. Chordata: Hemichordata, Tunicata and Cephalochordata. The general characteristics and classification of Agnatha, Chondrichthyes, Osteichthyes, Amphibia, Reptilia, Aves and Mammalia.

Main texts:

Radovanovi� M. Zoologija (drugi deo – sistematika životinja). Nau�na knjiga – Beograd, 1965.

Kalezi� M. Hordati (skripta). Biološki fakultet Univerziteta u Beogradu – Beograd, 1998.

Šori� V. Morfologija i sistematika Hordata. Univerzitet u Kragujevcu, PMF. 1997.

�orovi� A., Kalezi� M. Morfologija hordata (praktikum). Biološki fakultet Univerziteta u Beogradu – Beograd, 1996.

Further readings:

- Capacity of zoological understanding of classification and taxonomy of different groups of vertebrates. - Ability to understand systematic and phylogenetic position of different groups of the subphylum Vertebrata. - Capability to apply the knowledge of taxonomy of vertebrates in different areas of biology.


- Ability for analytical thinking and capacity to argue the own opinion and statements.


Lectures and seminars with the active participation of students, fieldwork, individual home tasks, group and individual consultations.




31

Name of the course: Systematic of Cormophyta II Programmes of Studies:

Academic study programme Biology



Contact hours: Per week 2h Lectures + 2h Practicum + 1h field work and 1h and 40min of individual work and consultations


Structure: 106h and 40 min (2h of lectures + 2h of practical work + 1h of foeld work + 1h and 40min of individual work and consultations- per week; exams), 13h and 20 min administrative preparation and 41h and 50 min additional work

Language: Serbian or English Prerequisites: Basic course in Systematic of Cormophyta I

Aim: This course is aimed to introduce students with basic principals of systematic of flowering plants (Takhtadjan’s systematical and phylogenetical system).

Contents:

Introduction in systematic of flowering plants, general features of division Magnoliophyta, subordinate classification. Subclass Magnoliidae & Ranunculidae. Subclass Hamamelidae. Subclass Caryophyllidae. Subclass Dillenidae. Subclass Rosiidae. Subclass Asteriidae. Subclass. Commelinidae & Arecidae Takhtajan, A. (1997): Diversity and Classification of Flowering Plants, Columbia University Press Main texts: Domac, R. (1984): Mala Flora Hrvatske i susjednih podru�ja, Školska Knjiga Zagreb Ble�i�, V. (1964): Sistematika Viših Biljaka, Beograd Javorka S., Csapody, V. (1975): Iconographia Florae Partis Austro-Orientalis Europae Centralis, Akademiai Kiado, Budapest Lauber, K. & Wagner G. (2001): Flora Helvetica, Verlag Paul Haupt, Bern-Stuttgart-Wienn Pulevi�, V. (2005): Gradja za Vaskularnu Floru CG, Zavod za Zaštitu Prirode Crne Gore, Posebna Izdanja, knjiga 2 Rohlena, J. (1942): Conspectus florae Montenegrinae, Preslia 20/2

Further readings:

Sitte, P., Ziegler, H., Ehrendorfer, F. & Bresinsky, A. Eds. (1998): Strasburger - Lehrbuch der Botanik fuer Hochschule, Gustav Fischer Verlag, Stuttgart, Jena Capacity to understand basic principals of Takhtajan’s system of classification of flowering plants and its phylogeny Ability to identify the most common representatives of flowering plants in the flora of Montenegro (use of keys for plant identification) Ability to recognize the most common representatives of flowering plants in field Herbarium preparation Search the literature


Developing the oral and written communication Methods of teaching:

Lectures, practical work, field excursions, homeworks, seminar papers, consultations.

Examination: 2 written examinations, 2 practical examination, seminar paper, herbarium Methods of self-evaluation:

Students pools, results of examinations, direct communications with the students.

32

Name of the course: Microbiology Programmes of Studies:

Academic study of Biology, Faculty of Science and Mathematics, University of Montenegro



Contact hours:

(4 Lectures + 3 laboratory work) per week; 15 hours in semester for consultations = 120 contact hours in semester.

Total hours:


Structure: Lectures: 52 hours; Exercises: 39 hours; Exams: 14 hours; Consultations: 15 hours; Seminars: 25 hours; Individual work: 95 hours.

Language: Montenegrian Prerequisites: Elementary knowledge of Biochemistry and Molecular biology

Aim:

This course will provide the student with understanding of general microbiological principles and basic microbiological techniques. Moreover, introduce to student in bioengineering (DNA technology), microbial ecology and medical microbiology.

Contents:

The topics of lectures concern an overview of microbial life, the structure of prokaryotic and eukaryotic cell/function, general systematic and diversity of bacteria, fungi and viruses as well as their biology and ecology, growth of microorganisms and its regulation, metabolic diversity of microorganisms and interactions of microorganisms with the host. Microbial genetics, genetic engineering and biotechnology. Waste water treatment, Water purification; Food preservation; Industrial microbiology. The practical work will include techniques for cultivation of microorganisms and isolation of different groups of microorganisms from natural environment, microscopy, sterilization and disinfection, enumeration of microbial populations, PCR technique and identification of microorganisms, analysis bacteria community and bacterial resistance. 1. Madigan M.T., Martinko J.M: Brock Biology of microorganisms, eleventh edition, 2006, Prentice Hall. 2. Jemcev V.E., Djukic D: Microbiology, Belgrade, 2000. 3. Vuk�evic K.J., Simic D: Methods in Microbiology, Faculty of Biology, Belgrade, 1997.

Main texts:

McKane L., Kandel J: Microbiology essentials and applications, 1996, McGraw-Hill Further readings: The student will have a general and broad basic knowledge of prokaryote microbiology with emphasis on microbiological topics and perspectives relevant to interdisciplinary work in environmental, biotechnology and biomedicine and in industry. Practical skills in microbiology.



Lectures, laboratory practice, student quiz and discussions, seminars, students' independent work

Examination: Laboratory practice examination, lecture written or oral examination.


Student questionnaire (opinion poll), results of exams and direct communications with the students.

33

Name of the course: Genetics Programmes of Studies:

Academic Study Program Biology






Language: Serbian or English Prerequisites: No prerequisites

Aim: This course is aimed to introduce students with basic notions of heredity and variation of different traits in living organisms.

Contents:

Cellular reproduction.Prokaryotic and Eukaryotic cells. Mitosis and meiosis.Cell cycle. The basic principles of inheritance. Mendel`s rules of heredity. Monohybrid crosses. Dihybrid crosses. Extension of Mendelian inheritance. Interaction between alleles. Interaction between genes. Multiple alleles and blood groups in humans. Genes and chromosomes. Inheritance of sex-linked genes. Mechanisms of sex determination. Linkage, crossing-over and genetic mapping in eukaryotes. Chromosome aberrations. Variation in chromosome structure. Variation in chromosome number. Cytological techniques. Molecular structure of DNA and RNA. DNA replication. Transcription and RNA processing. Translation and the genetic code. Regulation of gene expression in prokaryotes. Regulation of gene expression in eukaryotes. Genetic basis of cancer and other human diseases. Inheritance of quantitative traits. Polygenic inheritance. Heritability. Genes in populations. Estimating allele and genotypic frequencies. The Hardy-Weinberg equilibrium. Evolutionary genetics. Genetic variation in natural populations. Factors that change allele frequencies in populations. Molecular evolution. Marinkovic D., Tucic N., Kekic V. 1981. Genetika. Naucna knjiga, Beograd. Borojevic K. 1991. Geni i populacija. Forum,Novi Sad. Misic P. 1999. Genetika. Partenon, Beograd.

Main texts:

Lynch M. & Walsh B. 1998. Genetics and analysis of quantitative traits. Sinauer Associates,Inc., Sunderland, Massachusetts, USA. Further readings: Tamarin R.H. 2002. Principles of genetics. McGraw – Hill, New York, USA. - student`s ability to understand the basic genetic principles ( by presenting the important concepts of classical, molecular and population genetics). - student`s ability to analyze data, design experiments and/or appreciate the relevance of experimental techniques. - students can appreciate the relationship between experimentation and quantitative analysis.




Examination: Written exams. Methods of self-evaluation:


34

Name of the course: Plant Physiology Programmes of Studies:

Academic study programmes Biology, Faculty of Science and Mathematics, University of Montenegro


Bachelor level, III year, IV semester Number of ECTS credits: 9

Contact hours: (4 Lectures +4 laboratory work) per week; 15 hours in semester for consultation = 120 contact hours in semester


Structure: Lectures: 96 hours; Exercises: 96 hours; Exam: 14 hours; Consultation: 15 hours; Seminars: 25 hours; Individual work: 24 hours

Language: Montenegrian Prerequisites: Plant Anatomy, Biochemistry Aim: Examination process of life in plants

Contents:

Physiology of plant cell. Water and plant cells (structure and properties of water, water transport proces). Water balance of the plant (water absorption by the root, water is transported throught tracheides and xylem, water vapor from the leaf to the atmosphere by diffusion through stomata). Photosynthesis:light reaction (structure of the photosynthetic apparatus, organization of light-absorbing antenna system, mechanisms of electron and proton transport, regulation and repair of the photosynthetic apparatus). Photosynthesis: carbon reactions (the Calvin cycle, the photorespiratory carbon oxidation cycle, the C4

carbon cycle, synthesis of starch and sucrose). Respiration and lipid metabolism (glycolysis, Tricaroxylic acid cycle, electron transport and ATP synthesis, lipid metabolism). Mineral nutrients (assimilation of mineral nutrients, macro- and microelements). Growth, development and differentiation. Phytochrome. Auxins. Gibberellins. Cytokinins. Ethylene. Abscisic Acid. The control of flowering. Physiology of seeds and fruits. Movement of plants. Stress physiology. Stankovi� Ž., Petrovi� Krsti�, B., Eri�, Ž.,Plant physiology, Novi Sad, 2006. (in Serbian) Neškovi� M., Konjevi� R., �ulafi�, Lj., Plant physiology, NNK-International, Belgrade, 2003. (in Serbian) Kastori R., Plant physiology, Feljton, Novi Sad, 1998. (in Serbian)

Main texts:

Taiz L., Zeiger E., Plant Physiology. Sinauer Associates, Inc., Sunderland, Massachusetts, 1998. Further readings: - capacity to understand water balans of the plant

- capacity of understanding plants photosynthesis and respiration - ability to comprehend plants mineral nutriment


- capacity to understand plants growth, development and differentiation


Leactures, laboratory practice, seminars, group and individual consultation

Examination: Laboratory practice examination, estimation of individual activity on lectures and seminars



35

Name of the course: Ecology of Animals I Programmes of Studies:



Bachelor level, III year, V semester Number of ECTS credits: 4





Aim: This course is aimed to introduce students with basic notions of Ecology The course include fundamental knowledge on different aspects of animal ecology

Contents:

Intriduction in Ecology, Relationship betwwen organism and enviromental. Conditions. Resurs. Unitare and modular organism. Migration and Competition within species. Commpetition between the species. Characteristics of the predators. Dyinamisc of predator –plen relationship. Detritophags and reducents. .Mutualisam. Parasitism. Peši�, V. Principles of Ecology (scripta)

Main texts: Esa Ranta, Per Lundberg, Veijo Kaitala (2005) Ecology of populations, Science, 388pp. Further readings: - capacity of ecological understanding of different groups of animals.

- ability to understand the importance of Ecology, and to construct independently simple proofs of ecological statement. - capability to apply the knowledge of ecology of animals in different areas of biology;








36

Name of the course: Ecology of Plants I Programmes of Studies:








Aim: This course is aimed to introduce students with basic notions of Ecologic conditions and their influence on plants

Contents:

Intriduction in Ecology. Fitoecology. Basic notions of Biogeography. Ecological conditions: sunshine, temperature, soils, air, water. Ca like ecological condition. Plants which lives on the sands. Relationship between plants – competition, parasitism; relationship between plants and animals. Jankovi� M., Fitoekologija sa elementima fitocenologije i pregled vegetacije na zemlji. Nau�na knjiga, Beograd, 1990. Main texts:

Stevanovi� B., Jankovi� M., Ekologija biljaka. International, Beograd, 2001. Further readings: - capacity to understand the importance of Ecology

- ability to understand the role of plants in the nature and importance of plants for men - capability to apply the knowledge of ecology of plants in the others areas


- analytical thinking


Lectures, laboratory and fieldwork with the active participation of students, individual home tasks, group and individual consultations.




37

Name of the course: Environmental Protection I Programmes of Studies:







Language: Serbian or English Prerequisites: Courses of Animal Ecology and Plant Ecology

Aim:

This course is aimed to introduce students with basic notions of structure, functioning and protection of environment as well as possibility of interpretation of phenomena and processes taking place in terrestrial and aquatic ecosystems and to acquaint students with the degree of degradation and revalorization of human environment; with systematic approach to the protection of nature and environment with improving attitudes towards various activities.

Contents:

Basic ecological topics. Biosphere and technosphere. Types of pollution. Cycling of pollutants. Theory of environmental protection. Impact of energy production and its use on environment. Exploitation of mineral resources and pollution. Chemical and physical characteristics of natural water ecosystem (atmospheric, superficial and underground water). Hydrological cycle. Chemical, physical and biological indicators of water quality. Water pollution and its negative impact. Toxicology of waters. Trophy, saprobity and autopurification in waters. Eutrophication. General characteristics of atmosphere. Air circulation. Temperature regime and meteorological conditions. Temperature inversion. Water in atmosphere. Sources and cycling of the natural components of air.

Main texts:

D.S. Veselinovi�, I.A. Gržeti�, Š.A. �armati, D.A. Markovi�. 1995. Stanja i procesi u životnoj sredini-K. I, Fakultet za fizi�ku hemiju, Beograd. M. �ukanovi�, Ekološki izazov, Elit, Beograd. R. Kastori. 1995. Zaštita agroekosistema, Novi Sad. D. Tuhtar. 1990. Zaga�ivanje zraka i vode, Svjetlost, Sarajevo.

Further readings:

Grupa autora.Vodi� za dobro upravljanje u oblasti životne sredine. UNDP Anderson, J&D Shiers. The green guide to specification. Blackwell publishing. - Capacity to show basic and partly deepened knowledge in biology/ecology. - Ability to analyze the focus of modern environmental protection work. - Capability to apply the knowledge of the different threats on the environment Competences to

be developed: - Ability for analytical thinking and capacity to argue the own opinion and statements in the public debate on environmental problems.


Lectures with the active participation of students, fieldwork, individual home tasks, group and individual consultations.




38

Name of the course: Hidrobiology Programmes of Studies:

Academic - Biology.


Bachelor level, 3rd year, 5th semester. Number of ECTS credits: 5

Contact hours:

3 hours of lectures + 2 hours of exercises per week, 15 hours in semester for consultations = 95 contact hours in semester

Total hours:


Structure: 48 hours - lectures, 32 hours - exercises, 6 hours - exams, 15 hours – consultations + 48 hours - individual work.


Prerequisites: Comparative systematic and anatomy of vertebrates Algae, fungi and lichens

Aim: Study of main characteristics of freshwater and marine ecosystems

Contents:

Introduction – water as a substance. Didtribution of aquatic biotops. Abiotic environment and water chemistry. Lotic systems: origin, classification and general characteristics. Biocenoses of lotic systems. Adaptations of organisms. Lentic freshwater systems: origin, morphology and abiotic environment, adaptations of organisms. Vertical and horizontal zonation of continental waters. Energy input. Structure and dynamic of lake biocenoses in litoral and pelagial. Wetlands and estuars: general characteristics, abiotic environment and biocenoses. See – coastal zone, abiotic environment. Diversity of biotops and litoral biocenoses. Open ocean: abiotic environment. Diversity of biotops and pelagial biocenoses. Energy input in oceans – food chain, adaptations of organisms and succesions. Matonickin & Pavletic: Zivot naših rijeka-biologija teku�ih voda. Školska knjiga, Zagreb 1971. Karleskint G. Introduction to Marine Biology. Brooks Cole, 1997. Wetzel R. - Limnology, lake and river ecosystem. Academic press, 2000.

Main texts:

Stankovi�, S. Jezera svijeta. Further readings: Dobson M. & Frid Ch. – Ecology of aquatic systems. 1998


Familiarity with general characteristics (abiotic and biotic) of aquatic environment. Capacity to understand mutual relationship and interaction between abiotic and biotic environment in aquatic ecosystems. Ability to understand, explain and make conclusions about different appearances in aquatic environment (cause and consequence). Knowledge about analysing of basic abiotic characteristics of water and about sampling, preservation and examination of organisms of different aquatic biocenoses. Developing the oral and written communication


Lectures and laboratory exercises with the active participation of students, group and individual consultations.

Examination: 2 written exams (colloquiums), 2 practical exams (tests) Methods of self-evaluation:


39

Name of the course: Molecular Biology II Programmes of Studies:

Academic Study Program Biology.



Contact hours: 3 lectures + 2 laboratory per week, 15 hours in semester for consultations =128 contact hours in semester


Structure: 48 hours - lectures, 32 hours - seminars, 6 hours - exams, 15 hours - consultations, 79 hours – individual study.

Language: Prerequisites: No prerequisites. Aim: Study of biology structures and mechanisms at molecular level.

Contents:

Translation of eukaryotes, selection of correct AUG codon in translation initiation, endoplasmatic reticulum and signal hypothesis, overlapping genes, regulation of gene expression, lactose operon, levels of gene activity regulation, cloning and gene reprogramming, principles of cloning and significance, stem cells, types and application of stem cells, PCR (principles, application and parameters that influence reaction), mutations, reverse mutation, types and mechanisms of DNA repair, principles genetic recombination, differences in genetic organization of prokaryotes and eukaryotes, gene families, transposons, principles of transpositions, genetic engineering, molecular biology of malignant cell, factors that influence malignant transformation, mechanisms of genesis of tumors, principles of cell protection of malignant transformation, molecular immunology, antibody structure, immunogenetics, organization of immune system, clonal selection theory. Bruce Alberts et al.: Molecular Biology of the Cell, Garland Science – a member of the Taylor & Francis Group, New York, USA, 2002. George M. Malacinski, Essentials of Molecular Biology, Jones and Bartlett, Boston, USA, 2003. David Freifelder: Molecular Biology, Jones and Bartlett, Boston, USA, 1987. Danko Obradovi�: Svetlosni mikroskopi, Zavod za udžbenike i nastavna sredstva, Beograd, Srbija i Crna Gora, 2002.

Main texts:

Danko Obradovi�: Reakcija polimerizacije lanca; Kloniranje organizama; Manuscripts.

Further readings: Benjamin Lewin: Genes VIII, Prentice Hall, Lebanon , USA, 2004.


- Understand principles of molecular mechanisms. - See how molecular biology is used to answer a wide range of biological

questions. - Learn critical molecular biology methods. - Understand how the methods work.



Examination: Written exams. Methods of self-evaluation:


40

Name of the course: Comparative Physiology Programmes of Studies:

Academic study of Biology, Faculty of Science, University of Montenegro



Contact hours:

(2 Lectures + 2 laboratory work) per week., 15 hours in semester for consultations = 60 contact hours in semester

Total hours:


Structure: 26 hours - lectures, 26 hours - laboratory exercises , 8 hours - exams, 15 hours - consultations, 15 hours – homework, 30 hours – individual study.

Language: Montenegrian Prerequisites: Basic courses of zoology, cytology and Physiology.

Aim:

The course starts with a general introduction to physiological processes in the animal kingdom. Examples will be given on how evolution has led to the development of fundamentally different body- and organ constitutions, to solve similar physiological problems using different strategies. The course will highlight how knowledge of different physiological specializations in the animal kingdom can be studied.

Contents:

Adaptation to high hydrostatic pressure, Living in extreme environments. Circulation and respiration. Hydrothermal vents. Osmoregulation. Electroreception. Thermoregulation. Sleep: Behavioral, neurophysiology and evolutionary perspectives. Circadian rhythms and biological clocks. Biochemical and physiological adaptations. Sensory specialization in migrating animals. Evolutionary and comparative aspects of ageing. V.Petrovi� and R. Radoji�i�, Comparative Physiology, I and II part, Zavod za udžbenike i nastavna sredstva, Beograd, 1995. Main texts: V. Davidovi�. Comparative Physiology, Zavod za udžbenike i nastavna sredstva, Beograd, 2000. J. Pinel, Biopsychology, Fourth Edition, Allyn and Bacon, 2000. Further readings: R. Berne and M. Levy, Physiology, Fourth Edition, Mosby, 1998. - Course provides capacity to understand the complex mechanisms and function of the intact organism and its emphasis on the processes that regulate the important properties of living systems in evolutionary and comparative perspectives. Competences to

be developed: - Within the homework tasks students provide ability for individual work with different sources of scientific literature, performing independent literature research, preparing individual presentations about different scientific physiological problems and defending scientific view in group discussions.


-Lectures, seminars and practical laboratory exercises are organized with the active discussions and student’s participation, individual home tasks, group and individual consultations.




41

Name of the course: Evolution Programmes of Studies:

Academic - Biology.



Contact hours: 4 hours of lectures per week, 15 hours in semester for consultations = 63 contact hours in semester


Structure: 48 hours - lectures, 3 hours - exams, 15 hours – consultations + 84 hours - individual work.


Prerequisites: Comparative systematic and anatomy of vertebrates Genetic

Aim: Study of general principles of evolution process

Contents:

Evolution theory: deffinition, structure, history and connection with other sciences. Origin and history of life. Manifestations of organic evolution – evidences. Variability, organic diversity. Biological species, conceptions, reproductive isolation. Mechanisms of evolutional changes: Selection – Darvin's and modern cocept, genetical base. Adaptation. Genesis and origin of species – speciation. Theories: phyletic, divergent, apomictic, polyploid and aneuploid. Alopatric – geographical speciation: definition and manifestation, mechanisms and factors. Examples of geographical speciations. Parapatric and simpatric speciation. Genetical changes during speciation and genetical base of mechanisms of isolation. Genesis of higher taxonomical categories, adaptive type and adaptive zone. Trends of evolution process, speed and directions of evolution. Radoman Pavle: Teorija organske evolucije. Zavod za izdavanje udžbenika, Beograd, 1971. Tuci� Nikola: Uvod u teoriju evolucije. Zavod za udžbenike i nastavna sredstva. Beograd, 1987

Main texts:

Tuci� N., Cvetkovi� D. – Evoluciona biologija, Beograd, 2000 Further readings:


Familiarity with Darvin’s and modern concept of evolutional theory. Capacity to understand main mechanisms of evolution process. Ability to understand and interpret evidences of organic evolution. Developing the oral and written communication


Lectures with the active participation of students, group and individual consultations.

Examination: 2 written exams (colloquiums) Methods of self-evaluation:


42

Name of the course: Ecology of Animals II Programmes of Studies:








Aim: This course is aimed to introduce students with basic notions of Ecology The course include fundamental knowledge on different aspects of animal ecology

Contents:

Diversity of the life cycles. Abundance of the Populations. Ecology of the Communities. Island Communities. Diversity of the species. Conservation Biology- Introduction. Zoogeography: areal of the species. Orgin of Fauna. Zoogeographical elements. Zoogeography of land. Zoogeography of the water ecosystems. Zoogeography of Balkan Peninsula. Ecology and Anthropogenic influence. Peši�, V. Principles of Ecology (scripta)

Main texts:

Malcolm L. Hunter 2002. Fundamentals of Conservation Biology, Blackwwell Publishing, 496pp. Further readings: C Barry, Cox Peter, D Moore. Biogeography An Ecological and Evolutionary Approach Seventh Edition - capacity of ecological understanding of different groups of animals.

- ability to understand the importance of Ecology, and to construct independently simple proofs of ecological statement. - capability to apply the knowledge of ecology of animals in different areas of biology;








43

Name of the course: Ecology of Plants II Programmes of Studies:








Aim: This course is aimed to introduce students in basic concept of beginning of plants community , their structure and dynamic; concept of distribution vegetation on the Earth

Contents:

Intriduction in Fitocenology. Beginning of plants community. Features of plants community. Concept of distributions of vegetation on the Earth. Impact of ecologic conditions on distributions vegetation. Vertical distribution. Horizontal distribution. Lignosa, Herbosa, Deserta, Errantia.

Jankovi� M., Fitoekologija sa elementima fitocenologije i pregled vegetacije na zemlji. Nau�na knjiga, Beograd, 1990. Main texts:

Jankovi� M., Fitogeografija. Nau�na knjiga, Beograd, 2001. Further readings: - capacity to understand features of plants community and importance of their protection - ability to understand the impact of ecological conditions on distributions of vegetation - capability to apply the knowledge of ecology of plants in the others areas


- analytical thinking


Lectures, laboratory and fieldwork with the active participation of students, individual home tasks, group and individual consultations.




44

Name of the course: Environmental Protection II Programmes of Studies:







Language: Serbian or English Prerequisites: Course of Environmental Protection I

Aim:

This course is aimed to introduce students with basic notions of structure, functioning and protection of environment as well as possibility of interpretation of phenomena and processes taking place in terrestrial and aquatic ecosystems and to acquaint students with the degree of degradation and revalorization of human environment; with systematic approach to the protection of nature and environment with improving attitudes towards various activities.

Contents:

Local and global air pollution. Smog. Radioactive pollutants. Homogeneous and heterogeneous processes in atmosphere. Effects of air pollution. Biological consequences of global climate change. Biological consequences of ozone depletion. Soil as a complex system. Soil degradation. Waste water and dangerous substances treatment. Assessing polluted substances in water, soil and air. Pedosphere pollution. Soil erosion. The city as an ecosystem. Biodiversity (evolution and meaning of the term, definitions, endanger and protection). Sustainable development.

Main texts:

D.S. Veselinovi�, I.A. Gržeti�, Š.A. �armati, D.A. Markovi�. 1995. Stanja i procesi u životnoj sredini-K. I, Fakultet za fizi�ku hemiju, Beograd. M. �ukanovi�, Ekološki izazov, Elit, Beograd. R. Kastori. 1995. Zaštita agroekosistema, Novi Sad. D. Tuhtar. 1990. Zaga�ivanje zraka i vode, Svjetlost, Sarajevo.

Further readings:

Grupa autora. 2003. Vodi� za dobro upravljanje u oblasti životne sredine. UNDP. 334p.

Anderson, J. & D. Shiers. 2004. The green guide to specification. Blackwell publishing. 98p.

- Capacity to show basic and partly deepened knowledge in biology/ecology.

- Ability to analyze the focus of modern environmental protection work. - Capability to apply the knowledge of the different threats on the

environment. Competences to be developed:

- Ability for analytical thinking and capacity to argue the own opinion and statements in the public debate on environmental problems.


Lectures with the active participation of students, fieldwork, individual home tasks, group and individual consultations.




45

Name of the course: Human Ecology Program of Studies:

Academic study program Biology



Contact hours: Weekly3 credits x 40/30 = 4 hoursStructure:2 hours teaching 2 hours of individual work including consulatation

Total hours: 90 hours in semester.

Structure:

Teaching and final examination: 4 hours 16 =64 hours Necessary preparation: administration, enrolment, notarization before start of semester, 2x4 hours =8 hours. Total: 3x30=90 Additional work: preparation of examination in makeup examination period, including makeup examination for students from 0 do 36 hours (all another time from the first two points are until to the total load of 240 hours). Load structure: 64 hours (teaching) + 8 hours (preparation) + 18 hours (additional work).

Language: Native language or English. Prerequisites: /

Aim:

This course is aimed to introduce students with basic information in human ecology. Human population like all another population are parts of biocenosis i.e. they are part of global macro ecosystems – biosphere. That's way is understandable review of ecology of human popuation from ecosystems aspects, as well as from aspect of urban ecosystem environment.

Contents:

Human ecology: definition, basic information. Ecological aspects of human development. Urban evolution, urban agglomerations, urban ecosystems. Demography and biology, demography and ecology. Human population growth, demographic explosion. Human population structure, nutrition, food pollution. Working place as environment, working place as part of urban and industrial environment. Climate and microclimate as factor of thermal stress, acclimatisation. Human influence and air pollution. Human influence and soil pollution. Human influence and water pollution. Genetic consequences of environmental devastation –genotoxical agens etc. I. Hrestomatija tekstova, Reading book, Department of Biology, University of Belgrade. Klepac. R. Osnove ekologije (Basic Ecology ) 1- 180 ISBN 86-7111-024-9. Publishing: jugoslovenska medicinska naklada, 1988. Main text:

Steiner, F Huuman Ecology, 1-237.Island Press. Washington -Covelo-London, Washington D.C.USA, 2002 Kocijan�i�, R. Higijena (Hygiene), 1-609. ISBN 86-17-09084-7. Medicine Faculty, University of Belgrade, 2002. Further readings: Moore, Gary. S. Living with the Earth (1-596). Water pollution overview. Lewis publisher company. Library of Congress cataloguing. USA., 2002


- to understand human influence in biosphere (air pollution, soil pollution, water pollution, genetic consequences of pollution) and evident feedback, to learn about environmental protection, to understand specialty of urban ecosystems etc.


Lectures and seminars with the active participation of students, individual home tasks, consultations.

Examination: Written exams (2 times per semester), seminar paper, final exam. Methods of self-evaluation:


47

III


PHYSICS

49


PHYSICS Undergraduate Academic I year

Course

Mandatory

Elective

Winter Semester

hours weekly

EC

TS

Ljetnji semestar

hours weekly

EC

TS

1. Linear algebra and analytical geometry x 2+2 5

2. Analysis I x 3+3 7 3. Physical Mechanics x 4+4 9 4. Introduction to Experimental

Physics I x 2+1 4

5. Laboratory Physics I (Mechanics) x 0+3 3 6. Introduction to Computing I x 0+2 2 7. Foreign language I x 2+0 2 8. Analysis II x 3+3 7 9. Molecular Physics with

Thermodynamics x 4+4 10

10 Introduction to Experimental Physics II x 2+2 6

11 Laboratory Physics I (Waves and thermodynamics) x 0+3 3

12 Introduction to Computing II x 0+2 2 Total 30 30

II year 1 Analysis III x 3+2 6 2. Differential Equations x 2+2 4 3. Electromagnetism x 4+4 10 4. Basics of Physics Measurements

Techniques I x 2+2 5

5. Physics Laboratory II (Electromagnetism) x 0+3 3

6. Foreign language II x 2+0 2 7. Complex Analysis x 2+1 4 8. Numerical Methods x 2+2 4 9. Probability Theory and Statistics x 2+2 4 10 Basics of Physics Measurements

Techniques II 2+2 5

11 Optics 4+4 10 12 Physics Laboratory II (Optics) 0+3 3

Total 30 30 III year 1. Physics of Atom x 4+2 6 2. Foreign language III x 2+0 2 3. Mathematical Methods in Physics x 3+2 6 4. Theoretical Physics I x 2+2 6 5. Quantum Physics I x 3+2 7 6. Laboratory Practicum III -

practicum in nuclear physics x 0+3 3

7. Statistical Physics x 4+2 6 8. Theoretical Physics II x 2+2 6 9. Quantum Physics II x 3+2 7 10 Introduction to Nuclear Physics x 4+2 6 11 History and Philosophy of Physics x 2+0 2

50

12 Practicum III (atomic physics) x 0+3 3 Total

51

Name of the course: Linear algebra and analytical geometry Programmes of Studies:

Academic study program Physics



Contact hours: 2h Lectures + 2h Exercices , 1h 20min hours for consultations, per week = 75 contact hours in semester


Structure: 106 h 40 min lectures and exams, 13 h 20 min administrative work, 30 h – consultations and individual study.

Language: Serbian or Rusian Prerequisites: Basic courses of mathematics from secondary school

Aim: This course is aimed to introduce students with basic notions of linear algebra and analytical geometry and its applications in mathematical and technical science.

Contents:

Basic mathematical concepts. Basic algebraic structures. Linear spaces and linear maps (matrices). Polylinear maps (determinants). Laplacian development of matrices. Inverce matrix. System of lineal equations. Polynomial factorisation. Eigen vectors and eigenvalues. Similar matrices. Jordan canonical form. Operators (conjugate, orthogonal, normal). Eucklidean linear spaces (scalar, vector and mixed product and basic properties). Linie, plane and relation. Surfaces of second order (cylinder,conic, sphere and rotary) and classification. V. Daši� , Linear algebra and analytical geometry , Podgorica, 1995. (in Serbian) M. Kosmajac, Collection of resolute tasks from Linear algebra and analytical geometry, Titograd, 1985. (in Serbian)

Main texts:

. . ��!��, "��# ��$�%�, ��, 1974. Further readings: - capacity of geometrical understanding

- ability to understand the proofs of theorems, and to construct independently simple proofs of mathematical statement. - capability to apply the methods of algebra and analytical geometry in different areas of mathematics;








52

Name of the course: Analysis I Programmes of Studies:

Academic study programmes Physics





Structure: 45 hours - lectures, 45hours - seminars, 6 hours - exams, 20 hours - consultations, 40 hours – homework (individual solving of problems), 54 hours – individual study.

Language: Serbian or English Prerequisites: None

Aim:

This course is aimed to introduce students with initial notions of mathematical analysis. It includes the notion of number sequences and their convergence, limits of sequences and functions, continious function, the notion of differential of function and its properties. The course is aimed to detailed discussion of these notions as well as to the formulation classical theorems related to them.

Contents:

Introductory remarks on rational numbers. Real numbers – axiomatic approach. Convergent sequences. Theorem on convergence of monotonous and bounded sequences. Number e. Number series. The limit of function and its properties. The symbols o and O. Continuous functions. Uniformly continuous functions. Elementary functions and their graphics. Differential of the function at the point. Derivative. Derivatives of elementary functions. Derivative of composition of function, function defined by parameters and implicitly defined function. Higher derivatives. Leibnitz formula. Theorems on medium value. Lopital rule. Teylor formula. Monotonity, convexity, the curvature points of differentiable function. Drawing of graphics. D. Adnadjevic, Z.Kadelburg, Analysis I, Nauka, Beograd, 1995. (in Serbian) P.Mili�i�, M.Uš�umli�: Exercises in higher mathematics I, «Nau�na knjiga» Beograd 2000. (in Serbian) Main texts:

V. A. Zorich, Mathematical Analysis I, Sprienger, 2006

Further readings: B. P. Demidovich, Problems in Mathematical Analysis, Beekman Books Inc, 1975 -ability to understand the basic ideas of mathematical constructions.

- capacity to understand the mathematical notions and their applications.

- ability of understanding the importance and necessity of mathematical analysis in physics


- ability for analytical thinking and capacity to for individual solving of mathematical problems



Examination: Two written exams , problem solving - home tasks, estimation of individual activity on lectures and seminars, oral exam



53

Name of the course: Physical Mechanics

Course: FIZI�KA MEHANIKA


Obavezan I 9 4P+4V

Studijski programi za koje se organizuje: Akademske Undergraduate Prirodno-matemati�kog fakulteta, studijski program fizika (studije traju 6 semestara, 180 ECTS kredita) Uslovljenost drugim predmetima: Predmet mogu slušati svi koji upišu studij fizike. Ciljevi izu�avanja predmeta: Pošto se fizi�ke veli�ine koje se definišu u mehanici koriste u �itavoj fizici, studenti treba da ovladaju fizi�kim veli�inama u mehanici i zakonima koji ih povezuju, kao i granicama primjene tih zakona. Tako�e rješavanjem zadataka treba da steknu operativno znanje u primjeni tih veli�ina i zakona te na taj na�in prodube poimanje fizi�kih pojava i procesa u prirodi. Ime i prezime nastavnika i saradnika: Prof. dr Borko Vuji�i�, doc. dr Mira Vu�elji� Metod nastave i savladanja gradiva: Predavanja, vježbe, konsultacije, 10 doma�ih zadaka, 2 kolokvijuma

Sadržaj predmeta: Pripremne nedjelje

I nedjelja II nedjelja III nedjelja IV nedjelja V nedjelja VI nedjelja VII nedjelja VIII nedjelja IX nedjelja X nedjelja XI nedjelja XII nedjelja XIII nedjelja XIV nedjelja XV nedjelja XVI nedjelja


Priprema i upis semestra Uvodno predavanje. Kinematika materijalne ta�ke Dinamika materijalne ta�ke. Njutnovi zakoni, inercijalni sistemi, neinercijalne sile, centrifugalna mašina Sistem materijalnih ta�aka. Zakon odražanja impulsa. Kretanje centra mase. Energija, rad i snaga Kineti�ka i potencijalna energija Zakon održanja energije u mehanici, Sudari. I kolokvijum Mehanika rotacionog kretanja. Rotacija apsolutno �vrstog tijela oko nepokretne ose. Osnovni pojmovi jedna�ine dinamike rotacionog kretanja.Rad i energija kod rotacionog kretanja Slobodna nedjelja. Zakon održanja momenta impulsa. Napon i relativna deformacija. Hookov zakon. Elasti�ne dužinske deformacije. Smicanje. Torzija. Energija elasti�ne deformacije. Keplerovi zakoni i Newtonov zakon gravitacije. Gravitaciono polje (rad, potencijalna energija i potencijal). Kretanje u gravitacionom polju. II kolokvijum Napon u te�nostima - pritisak. Pascalov zakon. Hidrostati�ki pritisak. Barometarska formula. Arhimedov zakon. Plivanje tijela. Bernoullijeva jedna�ina. Toriccellijeva teorema. Venturijeva cijev. Zakon o održanju impulsa za fluide. Viskoznost. Poisseuillov zakon. Laminarno i turbolentno strujanje viskoznih fluida. Kretanje tijela kroz fluid - dinami�ki potisak Oscilatorno kretanje Prigušeno oscilovanje Završni ispit. Ovjera semestra Dopunska nastava i poravni ispitni rok

OPTERE�ENJE STUDENATA

nedjeljno 9 kredita X 40/30=12 sati

Struktura: - 4 sata predavanja; - 4 sata vježbi; - 4 sata samostalnog rada uklju�uju�i i konsultacije.

u semestru Nastava i završni ispit: 12 x 16 = 192 sata Neophodne pripreme prije po�etka semestra (administracija, upis, ovjera) 2 x 12 = 24 sata Ukupno optere�enje za predmet 9x30 = 270sati Struktura optere�enja: 192 sata (Nastava) + 24sati (Priprema) + 54 sati (Dopunski rad)

Studenti su Mandatory da redovno poha�aju nastavu, urade sve doma�e zadatke i oba kolokvijuma Literatura: 1. S. Backovi�, Fizi�ka mehanika, Zavod za udžbenike i nastavna sredstva, Podgorica, 1999.; 2. I. Irodov, Zbirka zadataka iz opšte fizike, Zavod za udžbenike i nastavna sredstva, Podgorica, 2000. ; 3. D. Hallidey, R. Resnick, K. Krane, Physics, John Willey & Sons, NY, 2002.

Ime i prezime nastavnika koji je pripremio podatke: Prof. dr Borko Vuji�i� Napomena:

54

Name of the course: Introduction to Experimental Physics I Programme of Studies:

Physics - academic study program


Bachelor level, 1st year, I semester Number of ECTS credits: 4

Contact hours: Per week: 2 hours lectures, one hour exercises; 2 hours 20 min consultations and individual study

Total hours: 120 hours in semester

Structure: The lectures and exam (5 hours 20 min) x 16 = 85 hrs 20 min), preparation 10 hrs 20 min, 8 hours - consultations, 6 hours – seminars, 16 hours individual study

Language: Montenegrin or English Prerequisites: -

Aim: The main goal of the course is to introduce students to the error theory, and to train them for the data analysis.

Contents:

Physics in science history. Experimental research and nature. Space and time. Physical system. System of knowledge. Physical quantities. Physical law. Experiment and theory. Causality. Mathematics and Physics. Relativity. Infinity Quantum limits of the measurements. Energy and thermodynamics. Physics and other sciences. Physics and technology. Applied Physics. Modern scientific research. Methodology and organization. Phases of Experiment. Physical quantities and units. SI (System International). Tables and graphs. ORIGIN. Excell. Introduction to random and systematic errors, precision and accuracy. Characteristic properties - mean, variance, skewness. Standard deviation of the mean. Propagated errors for one and several variables ; variance, covariance, Schwarz inequality. Elements of theory of probability. Integral probability. Probability content. Normal distribution. Binomial distribution. Poisson’s distribution. J.R. Taylor, An introduction to Error analysis, University Science Books Mili Valley, California 1982; J. Slivka, M. Terzi�, Obrada rezultata fizi�kih eksperimenata, Stylos, Novi Sad (in Serbian) 1995;

Skripta: I. Ani�in: Obrada rezultata mjerenja, Univ. of Belgrade (in Serbian), 1989. Barry C. Robertson, Modern Physics for Applied Science, John Willey and Sons, New York 1985 Further readings: Mladjenovic M. Development of Physics, Gradj.knjiga, Belgrade, 1983. Capacity to learn;

Capacity for analysis and synthesis and generating new ideas; Oral and written communication;


Critical and self-critical abilities.


Lectures and seminars with active students’ participation, individual homework, group and individual consultations.

Estimation: Written exams (two brief and final), seminar, homeworks, estimation of individual activity on lectures and seminars.


Results of the exams, questionnaires and direct communications with the students.

55

Name of the course: Laboratory Physics I (Mechanics) Programmers of Studies:

Academic study programs Physics



Contact hours: (3 hours in laboratory) per week, 15 hours in semester for consultations=60 contact hours in semester

Total hours: 3 x 30 =90 hours in semester

Structure: 30 hours – lectures with experimental work, 10 hours - seminars, 5 hours - exams, 15 hours - consultations, 30 hours – individual study.


Prerequisites: This course belongs to the first bases of physics and essentially needs no foreknowledge in physics. However, notions of physics on a secondary school level with respect to mechanics are recommended

Aim:

The aim of this course is learning the necessary skills to perform independently experiments, to analyze data and to deduce physically meaningful results. Getting acquainted with reporting the principles and the results of the performed experiment, taking into account error analysis and the reliability of the results obtained.

Contents:

Introduction to physical experimenting - Measuring physical quantities and error estimation - Error calculations - Error and statistics - Data treatment - Reporting –Some of the experiments that students perform independently are: Determination of the free fall acceleration by simple pendulum; Determination of the rotational inertia of a body by torsion oscillator; determination of the surface tension of the water… V.Vucic: Osnovna mjerenja u fizici Main texts:

John R. Taylor : An Introduction to Error Analysis - The study of Uncertainties in Physical Measurements, Oxford University Press, ISBN 0-935702-10-5 G.L. Squires : Practical Physics, Cambridge University Press, ISBN 0-52127095-2

Further readings:


This training enables the student to develop skills and insights in experiments in physics. This should allow him to understand, to perform and to interpret more advance experiments, which come up in the following part.


Lectures and seminars with the active participation of students, individual performing of experiments by the student.

Estimation:

The ability for practical knowledge and skills can be tested via de interaction during the laboratory workshops. Permanent testing the preparative knowledge and experimental skills. Periodical evaluation on the ability of the application of error analyses and insight in the various physical experiments, estimation of individual activity on lectures and seminars



56

Name of the course: Introduction to Computing I Programmes of Studies:

Academic study programmes Physics,



Contact hours: (0 Lectures + 2 Seminar and examinations +0.66 independent work) per week = 60 contact hours in semester

Total hours: 2 x 30 = 60 hours in two semesters

Structure: 42.56 hours – seminars end examinations 5.32 hours – preparation + 12.12 hours consultations and individual work

Language: Serbian or English Prerequisites: none

Aim:

To achieve competency in basic computer skills, such as file management, virus scan, scanners, printers, digital camera, Email and Internet Browsers. To introduce students to packages such as Word, Excel, PowerPoint, FrontPage and Corel Draw.

Contents:

INTRODUCTION TO COMPUTER SKILLS· Microcomputer fundamentals - CPU, RAM, keyboard, disk drives and printers. Disk and data storage. Management and care of disks. Basic network principles. Operating Systems. SYSTEM SOFTWARE· Basic network operating system. Use of basic operating system commands in a Windows environment. Directories/sub-directories. APPLICATIONS SOFTWARE· The basic applications to be taught are word processing and spreadsheet fundamentals. A graphical package will also be covered as well as PowerPoint.

Main text Any basic course of MS Windows Further

readings:


Ability to use the packages listed above. Production of a Laboratory Report using Word and incorporating a graph, table and diagram. Preparation of a written report on the topic in Word.


Seminars with active participation of students, individual home tasks, group and individual consultations.

Estimation: Written exams (3 times in a semester), problem solving - home tasks, estimation of individual activity at seminars



57

Name of the course: English for Physics I Programmes of Studies: Academic study program Physics

Level of the course: Bachelor level, II year, I semester Number of

ECTS credits: 2



Structure: 26 h - lectures, 4 h - exams, 15 h - consultations, 5 h – homework (individual solving of problems), 10 h – individual study.

Language: English Prerequisites: None

Aim: The introduction into the basics of English for Physics. The understanding of the importance of spoken and written English in both everyday life and in physics.

Contents:

Energy; grammar: Past simple vs. Past continuous. Energy 1; grammar: -ing forms and infinitives. Gravity; grammar: modal verbs must and have to. Gravity 1; grammar: Present perfect passive. Electrostatics; grammar: conditional sentences. Centrifugal force; grammar: Time clauses. Lasers; grammar: prepositions. Pressure; grammar: Present simple vs. present continuous. Sound; grammar: Reported speech. Material science; grammar: clauses of contrast. Thermal conductivity; grammar: Making predictions. Magnetics; grammar: will and would. Kinetics ; grammar: certainty. English for Physics – reader, compiled by Savo Kostic

Main texts:

Further readings: - reading skills which include comprehension of a given text - writing skills, which include composition of short essays

- speaking skills with the emphasis on computer science Competences to be developed:

- general grammar skills


Lectures with the active participation of students, individual home tasks, oral project presentation, group and individual consultations.

Examination: Written exams (2 times in semester), project assessment , estimation of individual activity on lectures , oral final examination


Students feedback, results of exams, comparison to the students from other universities.

58

Name of the course: Analysis II Programmes of Studies:

Academic study programme Physics






Language: Serbian or English Prerequisites: Basic knowledge from Analysis I

Aim:

This course is aimed to introduce students with further basic notions of mathematical analysis. It includes the notions of indefinite and definite integrals and their application, the notions of functional series and their convergence, both point by point and uniform and their properties, the notion of power series and its radius of convergence and Fourie series.

Contents:

Indefinite integral and primitive function. Method of variable changing. Method of partial integration. Definite integral. Main properties. The medium value theorem. Newton-Leibntiz formula. Application of definite integrals. Number series. Absolute and conditional convergence. The main convergence criterions: D’Alamber’s , Cauchy’s, Abel’s. Functional sequences. Functional series. Their convergence point by point and uniform. The sum of functional series. The differential and integral of functional series. Power series. Radius of convergence of power series. Fourie series. Bessel’s and Parseval’s equality. Dirichlet theorem. D. Adnadjevic, Z.Kadelburg, Analysis I, Nauka, Beograd, 1995. (in Serbian) P.Mili�i�, M.Uš�umli�, Exercise from higher mathematics II, «Nau�na knjiga» Beograd 2000. (in Serbian) Main texts:

V. A. Zorich, Mathematical Analysis II, Sprienger, 2004

Further readings: B. P. Demidovich, Problems in Mathematical Analysis, Beekman Books Inc, 1975 -ability to understand the basic ideas of mathematical constructions

- capacity to understand the mathematical notions and their applications. - ability of understanding the importance and necessity of mathematical analysis in physics


- ability for analytical thinking and capacity to for individual solving of mathematical problems



Examination: Two written exams , problem solving - home tasks, estimation of individual activity on lectures and seminars, oral exam



59

Name of the course: Molecular Physics with Thermodynamics Study program: Physics - academic study program Level of the course:

Bachelor level, 1st year, 2nd semester Number of ECTS credits: 10

Contact hours: (4 Lectures + 3 Exercises+ 1 Seminar) per week, 30 hours in semester for consultations = 150 contact hours in semester


Structure: 56 hours - lectures, 45 hours - exercises, 15 hours – seminars, 30 hours - consultations, 8 hours – exams (two brief and final), 146 hours individual study including preparation, individual solving of problems and homework

Language: Serbian or English Prerequisites: -

Aim: This course is aimed to introduce students with mechanical waves and the concepts and important laws and principles of molecular – kinetic theory and thermodynamics.

Contents:

Mechanical waves – types (transverse and longitudinal), mathematical description, speeds, energy, intensity, superposition and interference, boundary conditions. Standing waves. Resonance. Sound - characteristics. Intensity and sound level. Doppler effect. Supersonic speeds, shock waves. Temperature, thermal equilibrium, Zeroth law. Thermometers, scales. Thermal expansion. Thermal stress. Quantity of heat. Molar heat capacity. Calorimetry and phase changes. Heat transfer mechanisms. Thermal properties of matter – state equations. Molecular properties of matter. Molecular-kinetic theory of an ideal gas. Speeds and energies of molecules - distributions. Mean free path. Real gasses. Heat capacities, heat capacities of solids. Phases of matter. Thermodynamic systems and states. Work, internal energy. I principle of thermodynamics. Processes, directions. The cycles. Heat engines. Refrigerators. II principle of thermodynamics. The Carnot cycle; Stirling engine. Efficiencies of real engines. Clausius inequation. Entropy. Entropy and disorder. Entropy in cyclic and irreversible processes. Microscopic interpretation of entropy. Nernst’s theorem. Thermodynamic potentials. Halliday D., Resnick R., Walker J. Fundamentals of Physics, John Wiley&Sons, Inc., 2005. Ch.: 16, 17, 18, 19, 20. Main texts: Young H.D., Freedman R.A. Sears and Zemansky’s University Physics with Modern Physics, Pearson, 2004. Ch.: 15, 16, 17, 18, 19, 20. Žizic, B. Thermodynamics and molecular physics, Scientific Book, Belgrade, 1999. (in Serbian) Further

readings: D. Kondepudi, I. Prigogine. Modern thermodynamics, Wiley, 1998. Capacity to learn;

Basic knowledge, recognition and understanding of mechanical waves, molecular-kinetic and thermodynamic phenomena; Ability to apply the molecular-kinetic and thermodynamic method;


Oral and written communication and knowledge of the second language.


Lectures, exercises and seminars with active students’ participation, individual homework, group and individual consultations.

Estimation: Written exams (two brief and final), problem solving – homework, estimation of individual activity on lectures, exercises and seminars.



60

Name of the course: Introduction to Experimental Physics II Programme of Studies:

Physics - academic study program


Bachelor level, 1st year, II semester Number of ECTS credits: 6

Contact hours: Per week: 2 hours lectures, 2 hours exercises; 4 hours 20 min consultations and individual study


Structure: The lectures and exam (8 hours) x 16 = 128 hrs), preparation 16 hrs, 8 hours - consultations, 6 hours – seminars, 16 hours individual study, additional work up to 30 hrs.

Language: Montenegrin or English Prerequisites: -

Aim: The main goal of the course is to prepare students for the highly demanded multidisciplinary approach, to elevate their practical skills and, to train them for the advanced research activities.

Contents:

Comparison of means; large samples, significance testing; small samples.. Test of hypothesis at a prescribed level of significance. Least-squares fitting - straight line, linear correlation. The goodness of fit using a χ2 test. Dissemination of the results of scientific research. Classification of measurements. Measurement instruments. Calibration. Static properties of instruments. Ampermeter & voltmeter. Current sources. Dynamic properties of instruments. Thermal fluctuations and noise. Spectral analysis. Interpolation and extrapolation. How to present the results? Digitalization of signal. Digital-analogue interfaces. Semiconductor based sensors. Measurements of pressure, acceleration, photo-resistance, position, temperature. RS232 port. Acquisition of data via the interfaces GPIB IEEE488. NI LabView. J.R. Taylor, An introduction to Error analysis, University Science Books Mili Valley, California 1982; Rober Bishop (National Instruments, LABView 7), Learning with Lab View, Student Edition , 2005.

Skripta: I. Ani�in: Obrada rezultata mjerenja, Univ. of Belgrade (in Serbian), 1989. Barry C. Robertson, Modern Physics for Applied Science, John Willey and Sons, New York 1985 Further readings: John Madox, What Remains to be Discovered, Papermac, London, 1998 Capacity to learn;






Estimation: Written exams (two brief and final), seminar, homeworks, estimation of individual activity on lectures and seminars.



61

Name of the course:

Laboratory Physics I (Waves and thermodynamics)





Contact hours: (3 hours in laboratory) per week, 15 hours in semester for consultations=60 contact hours in semester


Structure: 30 hours – lectures with experimental work, 10 hours - seminars, 5 hours - exams, 15 hours - consultations, 30 hours – individual study.


Prerequisites:

This course belongs to the first bases of physics and essentially needs no foreknowledge in physics. However, notions of physics on a secondary school level with respect to mechanical ways and thermodynamics are recommended

Aim:

The aim of this course is learning the necessary skills to perform independently experiments, to analyze data and to deduce physically meaningful results. Getting acquainted with reporting the principles and the results of the performed experiment, taking into account error analysis and the reliability of the results obtained.

Contents:

This course is continuing of the Laboratory Physics from first semester with the experiments that covering an area of mechanical ways and thermodynamics. Some of the experiments that students perform independently are: Determination of the speed of sound from the air column; Interference of sound waves; Experimental checking of the Gey-Lisac low;Determination of the specific heat capacity of lead… V.Vucic: Osnovna mjerenja u fizici Main texts: John R. Taylor : An Introduction to Error Analysis - The study of Uncertainties in Physical Measurements, Oxford University Press, ISBN 0-935702-10-5 Further readings: G.L. Squires : Practical Physics, Cambridge University Press, ISBN 0-52127095-2 This training enables the student to develop skills and insights in experiments in area of the mechanical waves and thermodynamics. This should allow him to understand, to perform and to interpret more advance experiments, which come up in the following part.



Lectures and seminars with the active participation of students, individual performing of experiments by the student.

Estimation:

The ability for practical knowledge and skills can be tested via de interaction during the laboratory workshops., Permanent testing the preparative knowledge and experimental skills. Periodical evaluation on the ability of the application of error analyses and insight in the various physical experiments, estimation of individual activity on lectures and seminars.



62

Name of the course: Introduction to Computing II Programmes of Studies:




Contact hours: (0 Lectures + 2 Seminar and examinations +0.66 independent work) per week = 60 contact hours in semester


Structure: 42.56 hours – seminars end examinations 5.32 hours – preparation + 12.12 hours consultations and individual work

Language: Serbian or English Prerequisites: none

Aim: To achieve competency in basic computer skills in using Linux. To achieve basic programming skills and basic knowledge of FORTRAN programming language

Contents:

SYSTEM SOFTWARE: Basic network operating system. Use of basic operating system commands in a Linux environment. PROGRAMMING: Basic FORTRAN syntax. Use of loops, arrays, conditional statements. Different input and output formants. Any basic course of Linux

Main text Any basic course of FORTRAN Further

readings:


Ability to use Linux operating system in a simple manner. Ability to solve simple computational problems using FORTRAN using different inputs and outputs


Seminars with active participation of students, individual home tasks, group and individual consultations.

Estimation: Written exams (3 times in a semester), problem solving - home tasks, estimation of individual activity at seminars



63

Name of the course: Analysis III Programmes of Studies:





Total hours: 6x 30 = 180 hours in semester


Language: Serbian or English Prerequisites: Calculus I, Calculus II and Linear algebra.

Aim:

This course is aimed to introduce students with basic concepts of calculus in the multidimensional vector spaces. The course includes rigorous definitions of main notions of multidimensional calculus. The course includes as well as some classical theorems: Stokes, Gauss-Ostrogradski and Green.

Contents:

Functions of two or more vaiables. Neighborhoods. Regions. Limits. Iterated limits. Continuity. Partial derivatives. Higher order partial derivatives. Differentials. Implicit functions. Jacobians. Gradient, divergence and curl. Tangent plane to a surface. Normal line to a surface. Tangent line to a curve. Normal plane to a curve. Multiple integrals. Transformations of multiple integrals. Line integrals. Green theorem in the plane. Surface integrals. The divergence theorem. Stokes' theorem. 1. D. Adnadjevic, Z. Kadelburg: Matematicka analiza II, Beograd, 1994. 2. S. Lang: Calculus of several variables, Addison-Wesley. Main texts: 3. M. Spiegel: Advanced calculus, Schaum's outline series.

Further readings: - capacity of understanding of calculus in the multidimensional vector spaces.

- ability to understand the proofs of theorems.

- capability to apply the methods of multidimensional calculus in various problems of physics.


- ability for solving examples and simple problems.






64

Name of the course: Differential Equations Programmes of Studies:







Language: Serbian or English Prerequisites: Basic courses of Analysis and Linear algebra

Aim: In this course students become acquainted with basic terms of differential equations, theorems about existence of solutions, methods of solving and application.

Contents:

Simple differential equations of first order: term of solution of Kosi’s theses;theorem about existence of solution. Equations with separate variables, uniform and linear equations. Equations with total differential, integration factor. Simple differential equations of higher degree(solution,Kosi’s theses,theorems about existence of solution).Linear equation of n-order.Method of constant variation. Normal systems of simple differential equations(solution, Kosi’s theses, theorems about existence of solution).Method of elimination. Systems of linear differential equations(method of constant variations ,Ojler’s method) Linear and quasilinear partial differential equations of first order .Systems of nonlinear partial differential equations of first order(complete integration Pfafof’s equation. Classification of partial differential equations of second order.Reduction to canonic form. Wave equation (wire vacillation). Equation of heat conducting (cooling of cane).Dirihl’s circle problem.

Main texts: R.Š�epanovi�, M. Martinovi�: Diferencijalne jedna�ine, Unirex + PMF, 1994. Podgorica. (in Serbian). I.G.Petrovskij, Lekcii po teorii obxknavennxm uravneniqm, MGU, Moskva 1982 Further readings: I.G.Petrovskij, Lekcii ob uravneniqh s &astnxh proiyvodnxh, Nauka, Moskva 1950



Examination:

Ways of checking knowledge and giving marks by points: Presence on lectures 5 points. Two tests 25 points each. Doing homework 5 points. Final exam 40 points (oral exam). Passing grade will be given if more then 50 points is scored.



65

Name of the course: Electromagnetism Programmes of Studies:

Academic study program – Physics.



Contact hours: (4 Lectures+4 Seminars) per week, 15 hours in semester for consultations = 135 contact hours in semester


Structure: 56 hours – lectures, 56 hours – seminars, 8 hours – exams, 15 hours - consultations, 60 hours – homework (individual solving of problems), 105 hours – individual study.


Prerequisites: Mechanics. Mathematical background: Gradient; divergence; curl; Laplacian operator; Divergence Theorem; Stokes’ Theorem.

Aim:

The course aims to provide understanding and insight into the fundamental phenomena and principles of electromagnetism. Covers the fundamentals in electrostatics, magnetostatics, polarization and magnetization, AC-circuits . electromagnetism and Maxwell’s equations in differential and integral form.

Contents:

Electric charge. Coulomb’s law. Electric field, E. Electric potential. Voltage. Electrostatic energy. Gauss’ law. Electric dipole field and potential. Dipol in electric field. Electric multipoles. Dielectrics. Polarization of a dielectric. Polarization vector. Electric field in dielectric. Electric displacement vector, D. Field vectors at the interface between two linear dielectrics. Conductors in the electric field. Capacitor. Capacitance. Energy and energy density of the electric field. Electric current, intensity and density. Electromotive force. Ohm’s law. Kirchoff’s laws. Power and energy in the electric current circuit. Lorentz force. Magnetic field vector B. Magnetic fields of various current distributions. Solenoidal property of the magnetic field. Magnetic vector potential. Conductors carrying current in magnetic field. Magnetic dipole moment. Magnetic fields in matter. Magnetisation vector. Bound currents. The auxiliary field, H. Field vectors at the interface between two magnetic media. Diamagnetism, paramagnetism and ferromagnetism. Electromagnetic induction. Selfinduction. Mutual induction. Energy and energy density of the magnetic field. Circuit elements in AC circuits. LC circuits. Resistance in LC circuits – damping. Driven RLC circuits. Power in AC circuits. Transformers. Displacement current. Maxwell’s equations in differential and integral form. D. Burzan, Electromagnetism, in print, available on CD (in Serbian) I. V. Savelyev, General Physics, part 2 – Electricity and Magnetism (in Russian or English) Main texts:

I. Irodov, Problems in General Physics (in Serbian, Russian or English)

Further readings: D. J. Griffiths, Introduction to Electrodynamics, 3rd edition, 1999, Prentice Hall


Capacity of understanding the basic concepts and solving problems in electromagnetism. Ability to apply the concepts to concrete situations. Familiarity with some of the mathematical methods used to solve problems in electromagnetism. Background for further learning of physics.


Lectures and seminars with active participation of students, individual home tasks, group and individual consultations.

Estimation: Written exams (3 times in semester), homeworks, estimation of individual activity during lectures and seminars.


Student response, results of exams, direct communications with students.

66

Name of the course:

Basics of Physics Measurement Techniques I

Program of Study Academic study program Physics, Faculty of Natural Science and Physics Level of the course:

Bachelor level, Year II, Semester III Number of ECTS credits: 5

Contact hours:

(2 Lectures + 2 exercises with lab ) per week, 14 hours per semester for consultations = 70 contact hours per semester

Total hours:

5 x 30 = 150 hours per semester

Structure: 28 hours - Lectures, 28 hours – Exercises + Lab, 8 hours - Exams, 14 hours - Consultations, 30 hours – homework (2*15 individual solving of problems), 42 hours – Individual study.

Language: Serbian or English Prerequisites: Not needed

Aim:

The course is aimed at a basic introduction to measurement and instrumentation principles. It is designed to provide a practical - hands on – introduction to electronics, focused on measurements of signals. Due to its practical approach student will learn to work independently.

Contents:

Basic physical principles, measurement errors and probability functions, calculation precision and accuracy. Optimization of counting experiments. Distribution of time intervals. DC and AC voltages, current. Kirchhoff’s laws and Thevenin’s and Norton’s theorems. Passive and active devices, RC- and RL- circuits, RLC- circuits, alternating current behavior of RC- and RL- circuit. Semiconductor circuits, transistor amplifiers and op-amps, filters and oscillators. Other topics covered are fundamentals of light detectors, photoconductors, photodiodes and solar cells. p-i-n detectors for visible light, Schottky type infrared detectors, charge couple devices (CCD) for imaging, semiconductor x-ray sensors. D. Stankovi�, Fizi�ko tehni�ka merenja, Nau�na knjiga, Beograd 1997. Owen Bishop, Understand Electronics

Main texts:

Further readings: - Ability to understand basic principles of electronic design.

- Capability to know how to choose and design simple measurement systems for a given application, based on different physical measurement principles, and error analysis. - Development of analytical thinking and capability to defend own opinion and results.



Lectures and laboratory work with the active participation of students, individual home tasks, group and individual consultation.

Estimation: Written exams (3 times in semester), written laboratory report, problem solving - home tasks, estimation of individual activity on lectures and seminars



67

Name of the course: Physics Laboratory II (Electromagnetism) Programme of Studies:

Academic study programme Physics.



Contact hours: ( 3 Laboratories) per week, 14 hours in semester for consultations = 70 contact hours in semester


Structure: 36 hours - laboratories, 6 hours - exams, 14 hours - consultations, 12 hours – homework (individual solving of problems), 22 hours – individual study.

Language: Serbian or English Prerequisites: Physics Laboratory I

Aim:

To provide the student with wide experience in physics experiments which are appropriate to the junior and senior cycle physics curricula and enhance their ability and familiarity with related equipment and techniques. To complement and reinforce the theory covered in lecture modules, and in so doing, demonstrating to the student the important synergy of classroom and laboratory teaching in science education. To provide the student with training in the following areas: (i) good laboratory practice. (ii) keeping a laboratory notebook. (iii) data analysis and presentation. (iv) technical report writing. (v) appropriate aspects of laboratory safety. To enhance the practical skills of the student in performing experiments involving a wide range of physical concepts.

Contents:

Introduction. A range of short experiments of Electromagneism, which illustrate concepts from lectures. Measuring methods, techniques and instruments. Parameters of measuring instruments. Ohm’s law in DC circuits. Internal resistance of a source of EMS. Resistors connected in parallel and series. Temperature coefficient of resistance. Thermoelectric cell couple. Faraday’s law of eleclrolysis, electrochemisal Cu equivalent. Kirchhoff 's curent laws.�Voltmeter and ammeter. V. Vucic: Osnovna merenja u fizici.

Main texts: Students will know good procedures for carrying out an experiment. Students will know how to keep a good laboratory notebook and how to present a written scientific report, and appreciate the importance of these activities in relation to teaching in a secondary school. Students will develop a more critical analytical ability in evaluating data. Students will become aware of the physics experiments used in the secondary school physics curricula, the equipment required, and appropriate aspects of its correct use and maintenance.


Students will develop an appreciation of the important complementary nature of classroom and laboratory teaching for science education.


Laboratory works with the active participation of students, individual home tasks, group and individual consultations.

Estimation: Laboratory exams (3 times in semester), problem solving - home tasks, and estimation of individual activity on lectures and seminars.



68

Name of the course: English for Physics II Programmes of Studies: Academic study program Physics

Level of the course: Bachelor level, II year, II semester Number of

ECTS credits: 2





Aim: Further and more detailed insight into English for Physics with more thorough language and lexical content.

Contents:

Tools for physics; grammar: Past simple vs. Past continuous. Problem solving in mechanics; grammar: -ing forms and infinitives. Conceptual physics; grammar: modal verbs must and have to. Physics for architects; grammar: Present perfect passive. Physics for the Life Sciences; grammar: Conditional Sentences. Mechanics; grammar: Time clauses. Thermodynamics; grammar: prepositions. Electricity; Present simple vs. present continuous. Magnetism; grammar: Reported speech. Optics and Modern Physics; grammar: clauses of contrast. Advanced principles I – Mechanics; grammar: Making predictions. Advanced Principles II - Electricity and Magnetism; grammar: will and would. Advanced Principles III; grammar: certainty. English for Physics – reader, compiled by Savo Kostic

Main texts:

Further readings: The further development of 4 main language skills

The development of presentation techniques

Composition writing Competences to be developed:






69

Name of the course: Complex Analysis Programs of Studies:








Aim:

This course is aimed to introduce students with basic notions of complex analysis and its application in physics. The course includes some classical theorems of Theory of analytical functions, Complex integration, Conformal mapping and Laplace transformation.

Contents:

Complex numbers. Algebra and geometry in the complex plane. Riemann sphere. Holomorfic function. Cauchy-Riemann conditions. Elemantary complex functions. Integration in the complex plane. Cauchy theorem. Cauchy integral formula. Power series. Taylor's theorem. Zeros of holomorphic functions. The Identity theorem and the Uniquness theorem. Laurent's seriers. Singularities. Cauchy residue theorem. Application of contour integration. Conforml mapping. Bilinear mapping. General priciple of conformal mappimg. Laplace's transform and inverese Laplace transform. Application of Laplace's transform. H. A. Priestley, Introduction to Complex Analysis, (sec.edition), Oxford UIniversity Press, 2005. D. Kaljaj, Problems in Complex Analysis, University of Montenegro. Podgorica 2006. (in Serbian)

Main texts:

B. V. Shabat. Introduction to Complex Analysis, I part, Moscow, 1988 (Russian) Further readings: M. Perovich, Principles of Mathematgical Analysis, Nikshich, 1991. - capacity of understanding of methods theory of function of complex variables. - ability to understand complex mathematical models in physics and to solve simple models using methods of the complex variables. - capability to apply the results and methods of complex analysis in different areas of mathematics and physics;


- ability for analytical thinking and capacity to argue the own opinion and statements.






70

Name of the course: Numerical Methods Programmes of Studies:






Structure:

26 hours - lectures, 13 hours - seminars, 6 hours - exams, 15 hours - consultations, 30 hours – homework (individual solving of problems), 30 hours – individual study.


Aim:

In this course students get acquainted with basic numerical methods concerning approximation of function, approximate differentiation and integration, approximate solving of equations (algebraic, ordinary and partial differentials) and systems of algebraic equations. For convergency of iterative processes, theory about contractive copying is used.

Contents:

Numbers and operations with numbers. Final and devided differences. Approximation of functions: interpolarity (Langragee and Newton’s interpolar polynoms ) The smallest quadrate discrete case method. Differentiation and integration (rectangle formula, trapeze rule and Simpson’s formula). Solving of algebraic equations(method of bisection,method of tangent and secant ,method of simple iteration).Solving of simple differential equations (methods like Runge Kuta). Solving of partial differential equations(method of intersection and system)

Main texts: M. Martinovi�, R.Š�epanovi�: Numeri�ke metode, Unirex i PMF, 1995. Podgorica

Further readings: N.S. '�(��, )islenxe metodx I, Nauka, Moskva 1975 Methods of teaching:


Examination:

Ways of checking knowledge and giving marks by points: Presence on lectures 5 points. Two tests 25 points each. Doing homework 5 points. Final exam 40 points (oral exam).Passing grade will be given if more then 50 points is scored.



71

Name of the course: Probability Theory and Statistics Programmes of Studies:








Aim:

This course is aimed to introduce students with basic concepts of probability thoery and statistics and their applications. The course includes definitions of basic notions of probability theory and statistics. Course includes as well as some classical theorems of probability theory and statistics. This course is aimed to introduce students with statistics softwares STATISTICS and SPSS.

Contents:

Random experiment. Events as set. Probability. Conditional probability. Independence. Random variables. Random vectors. Probabilty mass function. Binomial and Poisson ditribution. Distribution function and density function. Examples of continuous distribution. The joint distribution function. Marginal distributions. The conditional ditribution function. Expectation. Moments. Functions of random variables. Characteristics function. Laws of large numbers. Central limit theorem. Sampling and Statistics. Order statistics. Chi square distribution, t-distribution and F-distribution. Fisher theorem. Estimation. Maximum likelihood method. Confidence intervals. Introduction to hypothesis testing. Inferences about normal models. A regression problem. Statistical softwares STATISTICS and SPSS 1. S. Stamatovi�: Vjerovatno�a. Statistika, PMF 2000.

2. B. Stamatovi�, S. Stamatovi�: Zbirka zadataka iz Kombinatorike, Vjerovatno�e i Statistike, PMF 2005. 3. Z. Ivkovi�: Teorija vjerovatno�e sa matemati�kom statistikom, Gra�evinska knjiga, Beograd, 1992.

Main texts:

4. Hogg, McKean, Craig: Introduction to mathematical Statistics, Pearson Prentice Hall, 2005.

Further readings: W. Feller: An introduction to probability theory and its application, Wiley. - capacity of understanding of probability computing and concept of statistics experiment. Competences to

be developed: - capability to apply the methods of probability theory and statistics in physics. Abilty to make a statistical analysis using statistical softwares.






72

Name of the course:

Basics of Physics Measurements Techniques II

Program of Study Academic study program Physics, Faculty of Natural Science and Physics Level of the course:

Bachelor level, Year II, Semester IV Number of ECTS credits: 5

Contact hours:

(2 Lectures + 2 exercises with lab) per week, 14 hours per semester for consultations = 70 contact hours per semester

Total hours:


Structure: 28 hours - Lectures, 28 hours – Exercises + Lab, 8 hours - Exams, 14 hours - Consultations, 30 hours – homework (2*15 individual solving of problems), 42 hours – Individual study.

Language: Serbian or English Prerequisites: Not needed

Aim:

The course is aimed at a basic introduction to measurement and instrumentation principles. It is designed to provide a practical - hands on – introduction to electronics, focused on measurements of signals. Due to its practical approach student will learn to work independently.

Contents:

Various applications of sensors, interface to computers, data acquisition, principles of sensors. Temperature sensors, pressure sensors, motion and acceleration sensors. Noise correlation, transmission lines, control system and digital signal processing with basics of digital electronics. Flip-Flop, Register. Converter (ADC, TDC) basics/applications are reviewed. General properties of radiation detectors (pulse height spectra, energy resolution, detection efficiency). Gas-filled detectors, scintillation detectors, semiconductor detectors. Detector electronics and pulse processing (pulse counting systems, pulse height analysis systems, pulse shape discrimination). Multichannel pulse analysis. D. Stankovi�, Fizi�ko tehni�ka merenja, Nau�na knjiga, Beograd 1997. Owen Bishop Understand Electronics Radiation detection and Measurement, Glenn F. Knoll

Main texts:

Further readings:

- Ability to understand the basics of electronic design.

- Capability to know how to choose and design simple measurement systems for a given application, based on different physical measurement principles, and error analysis.


- Development of analytical thinking and capability to defend own opinion and results.


Lectures and laboratory work with the active participation of students, individual home tasks, group and individual consultations.

Estimation: Written exams (3 times in semester), written laboratory report, problem solving - home tasks, estimation of individual activity on lectures and seminars



73

Name of the course: Optics Programme of Studies:



Bachelor level, 2nd year, 4th semester Number of ECTS credits: 9

Contact hours:

(3 hours lectures + 3 hours exercises class) per week, 30 hours in semester for consultations=108 contact hours in a semester

Total hours:

9 x 30 = 210 hours in a semester

Structure: 39 hours - lectures, 39 hours – class exercises, 8 hours - exams, 30 hours - consultations, 30 hours – homework (individual solving of problems), 64 hours – individual study.

Language: Serbian or English Prerequisites: 40% of passed exams of the 1st year study

Aim: This course is aimed to introduce students with classical optical phenomena and physics background of light as an electromagnetic wave. Students will understand geometric and wave optics.

Contents:

Basic concepts in physics: particles and waves: Maxwelian equations; Light as an electromagnetic wave, the velocity of light

Quantities which describe optical phenomena- light sources, photo-metrics, optical path, velocity of light etc.

Geometric Optics-Basic Laws, Fermat’s principle; optical elements and systems: mirror, lens, microscopes and telescopes

Theory of secondary sources- Huygens’ Frenel’s principle

Interference: Constructive and destructive interference . Diffraction: Fraunhofer’s and Frenel’s diffraction gratings and spectral systems Polarized light: Malus’ Law; Birefringent Crystals, filters and Brewster’s angle

Main texts:

1. Optics Matveev A.N. (English). Hardcover. 448 pp 2. Physics: A General Course. V.II. Savelyev I.V. (English). Hardcover. 512 pp 3. I. Irodov, Problems in General Physics (Zbirka zadataka iz opšte fizike, Zavod za udžbenike i nastavna sredstva Podgorica, 2000)(in serbian)

Further readings: Optics, E. Hecht, 2002, 4th edition Pearson education - Capacity to learn;

- Basic knowledge and understanding of optical phenomena and nature of light; - Problem solving skills in classical optical tasks;


- Literature search.


Lectures and class exercises, individual home tasks, group and individual consultations.

Examination: Three colloquia, problem solving - home tasks, estimation of individual activity on lectures and seminars, midterm examination, final exam.



74

Name of the course: Physics Laboratory II (Optics) Programme of Studies:

Academic study programme Physics.



Contact hours: ( 3 Laboratories) per week, 14 hours in semester for consultations = 70 contact hours in semester


Structure: 36 hours - laboratories, 6 hours - exams, 14 hours - consultations, 12 hours – homework (individual solving of problems), 22 hours – individual study.

Language: Serbian or English Prerequisites: Physics Laboratory I

Aim:

To provide the student with wide experience in physics experiments which are appropriate to the junior and senior cycle physics curricula and enhance their ability and familiarity with related equipment and techniques. To complement and reinforce the theory covered in lecture modules, and in so doing, demonstrating to the student the important synergy of classroom and laboratory teaching in science education. To provide the student with training in the following areas: (i) good laboratory practice. (ii) keeping a laboratory notebook. (iii) data analysis and presentation. (iv) technical report writing. (v) appropriate aspects of laboratory safety. To enhance the practical skills of the student in performing experiments involving a wide range of physical concepts.

Contents:

Introduction.A range of short experiments of alternating current circuits and optics, which illustrate concepts from lectures. Ohm’s law in alternating current circuits. The single loop RLC circuit. Discharging a capacitor. Light waves. Focal point of lenses. Index of refraction. Angle of minimum deviation. Optical instruments. Gratings and spectra. Polarization. Verification of relationships in Optics. V. Vucic: Osnovna merenja u fizici.

Main texts: Students will know good procedures for carrying out an experiment. Students will know how to keep a good laboratory notebook and how to present a written scientific report, and appreciate the importance of these activities in relation to teaching in a secondary school. Students will develop a more critical analytical ability in evaluating data. Students will become aware of the physics experiments used in the secondary school physics curricula, the equipment required, and appropriate aspects of its correct use and maintenance.


Students will develop an appreciation of the important complementary nature of classroom and laboratory teaching for science education.


Laboratory works with the active participation of students, individual home tasks, group and individual consultations.

Estimation: Laboratory exams (3 times in semester), problem solving - home tasks, and estimation of individual activity on lectures and seminars.



75

Name of the course: Physic of Atoms Programmes of Studies:






Structure: 52 hours - lectures, 26 hours - seminars, 6 hours - exams, 12 hours - consultations, 44 hours – homework (individual solving of problems and preparing of seminars), 50 hours – individual study.

Language: Serbian or English Prerequisites: Basic courses of Physics and Mathematics from earlier years

Aim:

This module takes as its starting point a discussion of the experiments that led to the birth of Modern Physics. The introduction to physic of atoms takes as its starting point ideas of atomic structure, the optical and X-ray spectra of atoms, the interaction of atoms with electric and magnetic fields, the theory of simple molecules and some atomic scattering theory.

Contents

Electron, photons and atoms. The elements of quantum mechanics. One-electron atoms. Interaction of one-electron atoms with electromagnetic radiation. Two-electron atoms. Many-electron atoms. The introduction to molecules.

Main text A. *. Mat�ee�; + ��# ,��; -. . .��/��0; + �� ,��; Further

readings: B. H. Bransden & C. J. Joachain; Physics of Atoms and Molecules; After studying the material presented in this course the student should be able to:

- give a brief account of the experiments that led to the introduction of quantum mechanics; - apply the Bohr theory of the atom to hydrogen and hydrogen like atoms and use this theory to explain and interpret their spectra; - ability for analytical thinking and capacity to argue the own opinion and statements;


-solve numerical problems based on all aspects of this course.



Estimation: Written exams (3 times in semester), problem solving - home tasks, estimation of individual activity on lectures and seminars



76

SAVO NAPRAVI TI OVAJ

Name of the course: English for Physics III Programmes of Studies: Academic study program Physics

Level of the course: Bachelor level, III year, V semester Number of

ECTS credits: 2





Aim: Further and more detailed insight into English for Physics with more thorough language and lexical content.

Contents:

Batteries; grammar: Past simple vs Past continuous; Electricity; grammar: -ing forms and infinitives; Basics of electrostatics; grammar: modal verbs must and have to ; Energy; grammar: Present perfect passive; Windpower; grammar: conditional sentences Kolokvijum (1 �as); govorne vježbe Slobodna nedjelja Fuel cells; grammar: Time clauses Lasers; grammar: prepositions Optics; Present simple vs present continuous Colour; grammar: Reported speech Lenses; grammar: clauses of contrast Magnetism; grammar: Making predictions Mechanics; grammar: will and would Nuclear science; grammar: certainty Završni ispit English for Physics – reader, compiled by Savo Kostic

Main texts:

Further readings: The further development of 4 main language skills The development of presentation techniques Composition writing







77

Name of the course: Mathematical Methods in Physics Programmes of Studies: Academic study program Physics


ECTS credits: 6

Contact hours:

(3h: Tutorial lectures + 2h: Exercices -exemples of solving practical problems ) per week, 14 hours in semester for consultations = 85 contact hours in semester

Total hours:


Structure: 39 h - lectures, 26 h - seminars, 6 h - exams, 14 h - consultations, 15 h – homework (individual solving of problems), 30 h – individual study.

Language: Monenegrin (Serbian, Croatian, Bosnian) or English or Rusian Prerequisites: University basic courses of Physics and Mathematics

Aim:

To introduce students with basic notions of mathematical description of physical laws with emphasis on diversity point of views, possible unifications which leads to generalisations and to develop and utilize a number of topics in mathematical physics.

Contents:

Variational calculus and aplication in analytical mechanics. Metric space. Lebesgue measure and integral. Probability spaces. Central boudary probability theorem. Entrophy. Thermodynamics second law . Banach’s and Hilbert’s spaces. Operators. Special functions (orthogonal polonomials). Aplications in atomic and quantum physics. Tensors. Tensors algebra elements. Tensors analysis elements. Integral transformations. Integral equations. Integral operators eigenfunctions. Introduction to group theory. Examples in solid state physics and quantum chemistry. Group representations. Aplications in the quantum mechanics and theory of relativity. Infinity groups. Aplications in quantum mechanics and elementary particle physics. W.Feller, An introd. to probability and its aplications, John Willey, N.Y, 1970. '.1��!��&, �� &��. ��2 �� (��, "��3, �, 2005. Mušicki, Mili�, Matemat. osnove teorijske fizike, Nau�na knj, Beograd, 1975. Mathews and Walker, Mathematical methods of physics, Benjamin, N.Y. 1964 S. Aljan�i�, Uvod u realnu i funkcionalnu analizu,Gra�. Knjiga, Beograd, 1979

Main texts:

S. Kurepa, Kona�no dim. vektroski prostori i primjene, SNL, Zagreb, 1986. G. Arfken, Mathematical methods for physicists, Academic Press, N.Y. 1970. Further readings: Ch. K. Chui, An introduction to wavelets, Academic Press, New York, 1992. - understanding of analytical problems. - ability to understand different ways of mathematical constructions in order to exactly describe essence of physical principles and laws - capability to apply the methods of functional analysis, group theory, tensor algebra and analysis, variational method and probability theory in different areas of physics;





Examination: Written exams (3 times in semester), problem solving - home tasks, estimation of individual activity on lectures and seminars, oral final examination

Methods of self-evaluation: Students pools, results of exams, direct communications with the students.

78

Name of the course: Theoretical physics I Programs of Studies: Academic study program Physics


ECTS credits: 6

Contact hours:

(2h: Tutorial lectures + 2h: Exercises -examples of solving practical problems ) per week, 15 hours in semester for consultations = 75 contact hours in semester

Total hours:



Language: Montenegrin (Serbian, Croatian, Bosnian) or English or Russian Prerequisites: Undergraduate basic courses of general physics and mathematics

Aim:

To introduce students with basic notions and physical laws in classical and relativistic mechanics with emphasis on the derivation and generalization on the physical basis and mathematical description and to develop and utilize a number of topics in mathematical physics.

Contents:

Description of motions in mechanics. Generalised coordinates. Lagrange’s equation of motion. Kinetic energy. The variational principle in mechanics. Energy, momentum and angular momentum conservation laws: first integrals. Central force field: reduction to the one-body problem. Kepler’s problem, particle colisions, scattering and Ratherford’s formula. Free, damped and forced oscilations. System small oscilations. Free vibrations frequencies and normal coordinates. Rigid body independent coordinates: Euler’s angles. Angular velocity. Inertia tensor and angular momentum. Rigid body equation of motion. Euler’s equations. Stability of rotations. Hamilton and Hamilton-Jaccobi’s equations. Phase space. Poisson’s brackets. 4-dim space-time interval. Lorentz transformations. 4-vectors. Relativistic particle action. Energy-momentum tensor. Relativistic particle decay and colision. I.V. Savelyev, Fundamentals of Theoretical Physics, V. 1, Mir, Moscow, 1982. 4. . 5��3��, 6�� &�� ,��, T.1, *��, ��, 1991. ". "��!��, -. "�,7��, ��(��.-�� !��,*��, ��, 1969.

Main texts:

�. Mušicki, Uvod u teorijsku fiziku, I dio, Nau�na knjiga, Beograd, 1980. H. Goldstein, Classical Mechanics, Addison-Wesley, Cambridge, 1953. Further readings: 4.6�3(��, 8�� &�� (�� !�# ,��, ��9, �.1974. - fundamental understanding of physical laws in classical and relativistic mechanics - derivation and generalization of mechanical quantities and laws - application of the fundamental mechanical theory to obtain transparent general solutions for many important problems


- understanding of relativistic mechanics phenomena through basic notions and mathematical description




Methods of self-evaluation: Student’s pools, results of exams, direct communications with the students.

79

Name of the course: Quantum physics I Programmes of Studies:




Contact hours:

(3 Lectures + 2 Problem Solving) per week, 16 hours in semester for consultations = 81 contact hours in semester

Total hours:


Structure: 39 hours - lectures, 26 hours – problem solving, 9 hours - exams, 16 hours - consultations, 30 hours – homework (individual solving of problems), 90 hours – individual study.

Language: Serbo-croatian or English Prerequisites: Classical mechanics nad Electrodynamics

Aim: This is an introductory quantum physics course with the basic aim of teaching basic quantum mechanical skills

Contents:

Wave function: Schroedinger equation. Statistical interpretation. Probability. Normalization. Momentum. Principle of uncertainity. Time-independent Schroedinger equation: Stationary states. Infinite square well. Harmonic oscillator. Finite square well. Free particle. »delta«-function potential. Mathematical formalism: Linear algebra. Function space. Generalized statistical interpretation. Quantum mechanics in three dimensions: Schroedinger equation in spherical coordinates. Hidrogen atom. Orbital momentum.

Main texts: Introduction to quantum mechanics, D.J. Griffiths, Prentice Hall, New Jersey

Further readings: Introduction to quantum mechanics (I and II part), Clod Cohen-Tannoudji, B. Diu, F. Laloe, Wiley Interscience, 1992.

80


• quote the Time Dependent Schrödinger Equation and Time Independent Schrödinger Equation and the conditions leading from one to the other; • be familiar with the concept of a wavefunction and the Born interpretation of the wavefunction; • be able to sketch wavefunctions and probability densities for simple problems; • be familiar with eigenfunctions and energy eigenstates of simple systems; • normalise wavefunctions; • be familiar with the concept of operators and resulting eigenvalue equations (specifically those relating to the energy, position and momentum); • calculate the expectation value of an observable using its related operator and calculate the uncertainty of an observable; • be familiar with and use the Dirac notation for inner products and for denoting states by means of bras and kets in the context of elementary linear algebra; • be familiar with the Heisenberg uncertainty relation and understand how it can be derived in the general mathematical form; • realise that quantum mechanics is based on postulates and have seen/discussed these postulates; • describe one dimensional quantum harmonic oscillators and use ladder operator (algebraic) techniques to obtain eigenfunctions and associated eigenvalues; • recall the Schrödinger Equation in three dimensions; • recall the quantization rules associated with angular momentum; • apply these concepts in the solution of the angular part of the Schrödinger Equation;


Lectures with the active participation of students, individual home tasks, group and individual consultations.

Estimation: Written exams (3 times in semester), problem solving - home tasks, estimation of individual activity on lectures.



81

Name of the course: Practicum III Programme of Studies:



Bachelor level, 3rd year, 6th semester Number of ECTS credits: 2

Contact hours:

One hour introductory lesson (2 hours laboratory exercises) per week, +3hours consultations in a semester=30 hours in a semester

Total hours:

2 x 30 = 60 hours in a semester

Structure: 1 hour - lecture, 26 hours – lab exercises, 4 hours - exams, 3 hours - consultations, 26 hours – individual study.

Language: Serbian or English Prerequisites: 40% of passed exams of the 2nd year study

Aim:

This course is aimed to improve and gain knowledge and skills in experimental physics through laboratory exercises in the field of Atomic Physics and spectroscopy. Through the autonomous work in the laboratory, students acknowledge theoretical concepts of quantum physics phenomena, determination of atomic constants and practical procedures in spectroscopy.

Contents:

Determination of Rydberg constant; Determination of Plank constant; Qualitative spectro-chemical analysis, Relative ratio of intensities of spectral line, Determination of the temperature of a DC arc in air, Emission spectroscopy; Apsorption spectroscopy. S. Mijovi�, praktikum III-Atomska fizika, Univerzitet Crne Gore, skripta, 1993. Laboratorijski priru�nici Main texts:

Further readings: + �� , �� 9 1987


Capacity to applying knowledge in practice; Report written skills; Ability to work autonomously; Concern for the quality; Skills required to conduct of standard laboratory procedures involved and used instrumentation taking into account any specific hazards associated with their use Skills obtained in practical measurements and determination of atomic constants;

Skills obtained in techniques of emission and absorption spectroscopy;


Lecture and lab exercises, individual home study, group and individual consultations.

Examination: Entered colloquia, written reports, final exam. Methods of self-evaluation:


82

Name of the course: Statistical Physics

Course: STATISTI�KA FIZIKA


Obavezan VI 6 4P+2V

Studijski programi za koje se organizuje: Akademske Undergraduate Prirodno-matemati�kog fakulteta, studijski program fizika (studije traju 6 semestara, 180 ECTS) Uslovljenost drugim predmetima: Položeni ispit iz fizi�ke mehanike sa termodinamikom i elektromagnetizama.

Ciljevi izu�avanja predmeta:Razvijanje termodinami�kih metoda koji se koriste za opisivanje makroskopskih sistema u stanju termodinami�ke ravnoteže. Razrada klasi�nih i kvantnih metoda pomo�u kojih se izra�unavaju termodinam�ki parametri sistema na osnovu njegove mikroskopske strukture i interakcija na mikroskopskom nivou. Ime i prezime nastavnika i saradnika: Prof. dr Borko Vuji�i�, doc. dr Žarko Kova�evi� Metod nastave i savladanja gradiva: Predavanja, vježbe, konsultacije, 10 doma�ih zadaka, 2 kolokvijuma

Sadržaj predmeta: Pripremne nedjelje

I nedjelja II nedjelja III nedjelja IV nedjelja V nedjelja VI nedjelja VII nedjelja

VIII nedjelja IX nedjelja X nedjelja XI nedjelja XII nedjelja XIII nedjelja XIV nedjelja XV nedjelja XVI nedjelja


Priprema i upis semestra Osnovni pojmovi i postulati termodinamike. Prvi princip termodinamike. Unutrašnja energija i toplota. Drugi princip termodinamike. Entropija. Funkcije odziva. Termodinami�ki potencijali za termomehani�ke, magnetne i elektri�ne sisteme. Osnovne termodinami�ke nejednakosti. Sistemi s promjenljivim brojem �estica. Tre�i princip termodinamike. Fazni prelazi I i II reda. Klauzius-Klapejronova jedna�ina i Erenfestove jedna�ine. Kriti�ni indeksi. I kolokvijum Osnovni postulati statisti�ke mehanike. Mikrokanonski ansambl. Slobodna nedjelja. Teorema o ravnomernoj raspodjeli energije po stepenima slobode. Entropija idealnog gasa. Kanonski ansambl. Veliki kanonski ansambl. Postulati kvantne statisti�ke fizike. Ansambli u kvantnoj statisti�koj mehanici. Idealan kvantni gas u velikom kanonskom ansamblu. Bose-Einsteinova, Fermi-Diracova i Boltzmannova funkcija raspodjele. II kolokvijum Dobijanje termodinami�kih funkcija idealnog kvantnog gasa. Jedna�ina stanja za idealni gas fermiona i za idealni gas bozona. Idealan gas fermiona na niskim temperaturama. Bose kondenzacija. Kineti�ka jedna�ina za funkciju raspodjele. Princip detaljnog balansa. Boltzmannova jedna�ina i Boltzmannova H-teorema. Završni ispit Ovjera semestra Dopunska nastava i poravni ispitni rok

OPTERE�ENJE STUDENATA nedjeljno

6 kredita x 40/30 = 8 sati

Struktura:

- 6 sati predavanja i vježbi; - 2 sata samostalnog rada, uklju�uju�i konsultacije. 7 kredita X 40/30=9 sati i 20 minuta

u semestru Nastava i završni ispit: (6 sati) x 16 = 96 sati Neophodne pripreme prije po�etka semestra (administracija, upis, ovjera) 2 x (10 sati) = 20 sati Ukupno optere�enje za predmet 6x30 = 180 sati Dopunski rad za pripremu ispita u popravnom ispitnom roku, uklju�uju�i i polaganje popravnog ispita od 0 do 45 sati (preostalo vrijeme od prve dvije stavke do ukupnog optere�enja za predmet 180 sati) Struktura optere�enja: 96 sati. (Nastava)+20 sati (Priprema)+32 sati (Dopunski rad)

Studenti su Mandatory da redovno poha�aju nastavu, urade sve doma�e zadatke i oba kolokvijuma Posebne naznake za predmet:

Ime i prezime nastavnika koji je pripremio podatke: Prof. dr Borko Vuji�i� Napomena:

83

Name of the course: Theoretical physics II Programs of Studies: Academic study program Physics

Level of the course: Bachelor level, III year, VI semester Number of

ECTS credits: 6

Contact hours:

(2h: Tutorial lectures + 2h: Exercises -examples of solving practical problems ) per week, 15 hours in semester for consultations = 75 contact hours in semester

Total hours:



Language: Montenegrin (Serbian, Croatian, Bosnian) or English or Russian Prerequisites: Undergraduate basic courses of general physics and mathematics

Aim:

To introduce students with basic notions and physical laws in electromagnetism with emphasis on the unity of electric and magnetic phenomena on physical basis and mathematical description, to develop and utilize a number of topics in mathematical physics.

Contents:

Gaussian and SI (MKSA) system of units. Electrostatic field in vacuum. Poisson's equation. Multipoles. Dielectics. Magnetostatic field in vacuum. Poisson's equation for vector potential. Biot-Savart law. Magnetic moment. Magnetics. Faraday's law of electromagnetic (elm) induction. Displacement current. Maxwell's equations. Potentials of elm field. D'Alembert's equation. Elm field density and flux energy (Poynting-Umov's vector). Elm momentum (Maxwell's stress tensor). Four-potential. Elm field tensor. Field transform. formulas. Field invariants. 4-dim form Maxwell's equations. Charged particle equation of motion. Charged particle action. Elm field action. 4-dim principle of least action. Elm field energy-momentum tensor. Charged particle: moment, Lagrangian and Hamiltonian. Elm field wave equation. Homogeneous and isotropic plane elm wave. Monochromatic plane elm wave in dielectrics and conductors. Non-monochromatic waves. Retarded elm potentials. Uniformly moving charge elm field. Accelerated charge elm field. System charges elm field. Electric dipole radiation. Magnetic dipole and quadrupole radiation. I.V. Savelyev, Fundamentals of Theoretical Physics, V. 1, Mir, Moscow, 1982. 4. . 5��3��, 6�� &�� ,��, T.1, *��, ��, 1991. ". "��!��, -. "�,7��, ��(��.-�� !��,*��, ��, 1969.

Main texts:

�. Mušicki, Uvod u teorijsku fiziku, II dio, Nau�na knjiga, Beograd, 1980. J. D. Jackson, Classical Electrodynamics, John Wiley, New York, 1998. Further readings: "."��!��, -."�,7��, :�� !��. ��7�2( ��!, *��, ��, 1982. - fundamental understanding of physical laws in electromagnetism - generalization of physical quantities and laws in electrodynamics in 4-dim - derivation forms of fundamental physical laws and physical quantities from principle of least action in 4 dim


- understanding of elm wave phenomena through wave quantities in different media and origin and characteristics of elm wave emission process




Methods of self-evaluation: Student’s pools, results of exams, direct communications with the students.

84

Name of the course: Quantum physics II Programmes of Studies:




Contact hours:

(3 Lectures + 2 Problem Solving) per week, 16 hours in semester for consultations = 81 contact hours in semester

Total hours:


Structure: 39 hours - lectures, 26 hours – problem solving, 9 hours - exams, 16 hours - consultations, 30 hours – homework (individual solving of problems), 90 hours – individual study.

Language: Serbo-croatian or English Prerequisites: Classical mechanics, Electrodynamics, and Quantum physics I

Aim: This is an introductory quantum physics course with the basic aim of teaching basic quantum mechanical skills

Contents:

IDENTICAL PARTICLES. Two-particle systems. Atoms and crystals. STATIONARY PERTURBATION THEORY. Perturbation of nondegenerate level. Perturbation of degenerate level. Hidorgen atom fine structure. Zeeman’s effect. Variation principle. Ground state of helium. Hidrogen molecule ion. TIME-DEPENDENT PERTURBATION THEORY. Two-level system. Emition and absorption. Spontaneous emition. SCATTERING THEORY. Partial wave analysis. Born approximation.

Main texts: Introduction to quantum mechanics, D. J. Griffiths, Prentice Hall, New Jersey 2005.

Further readings: Introduction to quantum mechanics (I and II part), Clod Cohen-Tannoudji, B. Diu, F. Laloe, Wiley Interscience, 1992.


• understand a more generalized state representation by means of column vectors and apply this to spin angular momentum;

• use ladder operator techniques to extract the quantization rules for spin angular momentum of spin ½ systems;

• understand and model spin ½ particles moving in magnetic fields – the Stern-Gerlach experiment;

• obtain eigenstates and eigenvalues for spin ½ and spin 1 systems; • ability to apply approximate methods • theoretical understanding of quantum scattering theory • composition of angular momenta • theoretical understanding of identical particles and Pauli principle



Estimation: Written exams (3 times in semester), problem solving - home tasks, estimation of individual activity on lectures.



85

Name of the course: Introduction to Nuclear Physics Programme of Studies:




Contact hours: (4 hours lectures + 2 hours seminars) per week, 30 hours in semester for consultations = 120 contact hours in semester



Language: Serbian or English Prerequisites: Basic course of Quantum Physics

Aim:

This course is aimed to introduce students with basic concepts of low energy nuclear physics, i.e. general properties of nuclei, characteristics of the nuclear force, principle models of the nucleus, radioactivity, nuclear reactions and to develop problem-solving skills in all these areas.

Contents:

Properties of stable nuclei and nuclear forces: Mass number and electric charge of the atomic nucleus. Nuclear and nucleonic mass. Nuclear binding energy and nuclear stability. Weizsacker's semiempirical formula. Nuclear radius. Spin and magnetic moment of nucleons and nuclei. Quadrupole electric moment. Parity. Isotopic spin. Nucleon-nucleon interactions: forces and potentials (Yukava potential, fundamentals of meson theory). Models of atomic nucleus: Liquid drop model. Fermi gas model. Shell model – foundations, shemes, experimental consequences, drawbacks. Generalized model – single-particle states in a nonspherical well, rotational states, vibrational levels, applicability of the model. Radioactive nuclear transformations: Radioactivity - nuclear instabilities, laws of radioactive decay. General laws and types of nuclear reactions: classification of nuclear reactions, conserwation laws, nuclear fission, thermonuclear reactions. K.N. Mukhin: Experimental Nuclear Physics. Vol I, Mir Publishers, Moscow 1987. Main texts: W.E. Burcham: Nuclear and Particle Physics, Nau�na knjiga Publisher, Belgrade, 1974 (in Serbian).

Further readings: B.R.Martin: Nuclear and Particle Physics – an introduction, John Wiley & Sons Ltd, 2006. - Capacity to learn;

- Basic knowledge and understanding of nuclear phenomena;

- Problem solving skills in nuclear physics tasks; Competences to be developed:

- Literature search.



Examination: Three colloquia, problem solving - home tasks, estimation of individual activity on lectures and seminars, midterm examination, final exam.



86

Name of the course: History and Philosophy of Physics Study program: Physics - academic study program Level of the course:

Bachelor level, 3rd year, VI semester Number of ECTS credits: 2

Contact hours: 2 lectures per week, 14 hours in semester for consultations and seminars = 44 contact hours in semester


Structure: 26 hours - lectures, 4 hours – exams (two brief and final), 14 hours – consultations and seminars, 16 hours - individual study and homework


Aim: This course is aimed to introduce students with historical development of physics and main concepts of its philosophy.

Contents:

Physics and metaphysics. Antic philosophy and physics. Atomists; Heraclites; Plato and Aristotle. Basic characteristic of middle ages doctrine. Teologia naturalis, Creatio ex nihilo. Eriugene – synthesis of Christian and Hellenic cognition of nature. Renaissance – Leonardo, Copernicus, Kepler, Galilei, Descartes. Scientific basis of physics – Newton, Huygens, Leibniz. Brief summary of electromagnetism (before and after Maxwell) and optics (Young and Fresnel). Development of thermodynamics; Entropy. Atom science. Quantum phenomenology. Planck, Schrodinger and Heisenberg. Nuclear and particle physics. Development of the scientific method. Space–time and geometry. The unification of physical phenomena. Applied physics. Mladjenovic M. Development of Physics, Gradjevinska knjiga, Belgrade, 1983. (in Serbian) Pavlovic B. Philosophy of Nature, Naprijed, Zagreb, 1978. (in Croatian)

Main texts:

Reichenbach H. The Rise of Scientific Philosophy, University of California Press, 1961. Further readings: Russell B. A History of Western Philosophy, Routledge, 1991. Capacity to learn;






Estimation: Written exams (two brief and final), estimation of individual activity on lectures and seminars.



87

Name of the course:

Laboratory Practicum III -Practicum in Nuclear Physics

Programme of Studies:




Contact hours: 3 hours in the lab per week, 15 hours in semester for consultations = 60 contact hours in semester


Structure: 39 hours – laboratory exercises, 8 hours - exams, 15 hours - consultations, 12 hours – homework, 16 hours – individual study.

Language: Serbian or English Prerequisites: Laboratory Practicum II

Aim:

Introducing students with simple instruments and methods in nuclear physics (particularly in spectroscopy and dosimetry of nuclear radiation) and analysis of raw data, the laboratory program will stress the development of their skills in designing and conducting experiments as well as in undertaking radiation protection measures.

Contents:

- General theoretical introduction to the data analysis and nuclear instruments and methods that will be used in this practicum, as well as with interaction of radiation with matter. - Nine laboratory experiments: 1. Statistical fluctuation in nuclear processes. 2. Geiger-Muller counter and its characteristics. 3. Determination of of gamma-ray energy by absorption in Pb. 4. Determination of maximum energy of beta-rays by absorption in Al. 5. Determination of energies of alpha-particles with nuclear emulsion. 6. Measurements of natural background radiation with ionisation chamber. 7. Measurement of beta-activity of environmental samples. 8. Dosimetry, ALARA principle, decontamination of working table in the lab. 9. Angular distribution of radiation beam. - Writte seminar work: Application of nuclear radiation in industry and medicine. - Examination of the final report on laboratory experiments. P. Vukotic, S. Dapcevic: Practicum in Nuclear Physics. Faculty of Natural Sciences and Mathematics, Podgorica, 1998. Main texts: I. Anicin, J. Puzovic: Practicum in Nuclear Physics. Faculty of Physics, Belgrade.

Further readings: I. Draganic, ed. : Radioactive Isotopes and Radiations – Books I, II. III. University of Belgrade and Institute Vinca, Belgrade, 1981 (in Serbian). - Basic capacity to measure characteristics of some nuclear phenomena;

- Ability to apply principles of radiation protection; Competences to be developed:

- Usage of nuclear data bases.


Supervised laboratory exercises, colloquia, written seminar work, group and individual consultations.

Examination: Two colloquia, estimation of individua final report on laboratory experiments and of written seminar work.



88

IV


MATHEMATICS

90


Mathematics Undergraduate Academic I year

Course

Mandatory

Elective

Winter Semester

hours weekly

EC

TS

Ljetnji semestar

hours weekly

EC

TS

1. Analysis 1 X 4+4 10 2. Linear algebra 1 X 4+3 8 3. Computers and

Programming X 3+3 6

4. Introduction to Logic X 2+1 4 5. English Language 1 X 2+0 2 6. Principles of Programming X 3+2 6 7. Introduction to

Combinatorics X 2+2 4

8. Linear algebra 2 X 2+2 6 9. Analytical Geometry X 2+2 4 10. Analysis 2 X 4+3 8 11. English Language 2 X 2+0 2

Total 11 15+11 30 15+11 30 II year 1. Analisys 3 X 3+2 6 2. Algebra 1 X 2+2 6 3. Introduction to Geometry X 3+2 4 4. Programming 1 X 3+2 6 5. Discrete Mathematics X 2+1 4 6. Complex Analisys 1 X 2+2 4 7. Analisys 4 X 3+2 6 8. Algebra 2 X 2+2 4 9. Probability Theory X 3+2 6 10. Programming 2 X 3+2 6 11. Differential Equations X 4+3 8

Total 11 15+11 30 15+11 30 III year 1. Algebra 3 X 4+3 8 2. Statistics X 3+2 6 3. Measure and Integral X 2+1 4 4. Partial Differential

Equations X 3+3 6

5. Theoretical Mechanics X 3+0 4 6. Engleski 3 X 2+0 2 7. Complex Analysys 2 X 3+2 6 8. Introduction to Differential

Geometry X 2+1 4

9. Functional Analisys X 3+1 4 10. Numerical Analisys X 2+2 6 11. Algebraic Topology X 3+1 6 12. History and Philosophy of

Mathematics X 2+0 2

13. Engleski 4 X 2+0 2 Total 13 17+9 15+11 30

91


Academic study programmes Mathematics, Mathematics and Computer Sciences






Language: Serbian Prerequisites: It has not.

Aim:

This course is aimed to introduce students on basic notions of set of real numbers, limit of a sequence, limit and continuity of a function, differential calculus and application of differential calculus. The course includes some classical theorems of Real Analysis.

Contents:

Set of real numbers-axiomatic fondation. Theory of convergent sequences. Bolzano’s and Cauchy’s theorems for numerical sequences. Banach’s theorem on fixed point. Topology on the set of real number. Limit of a function. Continuity of a function at a point. Basis of a set. Convergence and continuity of a function on the basis of a set. Global properties of continuous functions in a segment. Uniform continuity of functions. Differentiable function at a point. Derivative. Derivatives of higher orders. Theorems of the average values in differential calculus. Bernoulli-L’Hospital’s rule. Taylor’s formula. Monotonicity and extrema of differentiable functions. Convexness of functions. Saddle points. Investigation and ploting of graphs of functions. V. I. Gavrilov and Ž. Pavi�evi�, Matemati~ka analiza I, PMF Podgorica, Unirex, Podgorica, 1994, pages 538 (Serbian).

Main texts: V. A. Zorich, Matematicheskiy analiz I, Nauka, Moscow, 1981, pages 544 (Russian). B. P. Demidovich, Sbornik zadach i uprazhneniy po matematicheskomu analizu, Nauka, Moscow, 1977, pages 528 (Russian).

Further readings: W. Rudin, Principles of Mathematical Analysis, McGraw-Hill Company, 1964.


To get and further apply knowledge in the area of real numbers, Theory limit values, Theory of continuous functions and Theory of differential calculus






92

Name of the course: Linear Algebra I Programmes of Studies: Academic study program Mathematics, Mathematics and Computer Science

Level of the course: Bachelor level, I year, I semester Number of

ECTS credits: 8

Contact hours: (4 Lectures + 3 Seminars) per week, 20 hours in semester for consultations = 125 contact hours in semester




Aim:

Introducing students to fundamental concepts of Linear algebra, such as matrices, linear operators and spaces. This theory is crucial for understanding of advanced theories in mathematics, physics and computer sciences. Linear algebra, along with Mathematical Analysis, is basic and the most fundamental course in Mathematics.

Contents:

Systems of linear equalities. Gauss method. Matrices and elementary transformations. Determinant and rank of matrix. Vector spaces. Linear independence. Basis of vector space. Singular and invertible matrices. Kronecker – Kapelli theorem. Fundamental solution of linear system. Linear operators. Kernel and image of linear operator. Rank and defect. Matrix of linear operator. Matrix of transformation from one basis to another. Invariant subspaces. Eigenvalues and eigenvectors. Cayley – Hamilton theorem. Similar matrices. Nilpotent operators. Jordan form of matrix. A. Lipkovski, Linear algebra and analytical geometry, Belgrade. (in serbian) I. Krnic, �. Ja�imovi�, Linear Algebra – theorems and problems, skript, Podgorica, 2001. (in serbian) E. Shikin, Linear spaces and mappings, Moscow (in russian) S. Friedberg, A. Insel, L. Spence, Linear Algebra, Prentice Hall, 2002.

Main texts:

G. Strang, Linear Algebra and its Applications, Brooks Cole, 2005. Further readings: - introducing to the fundamental concepts of higher mathematics. - preparation for more advanced courses and theories

- developing of algebraic way of thinking and mathematical approach to scientific methods;


- developing ability of clear and critical thinking, understanding such concepts as proof, implication, precise statement etc.





Students pools, results of exams, comparison to the students from other universities.

93

Name of the course: Computers and Programming Programmes of Studies:

Mathematics, Mathematics and Computer Sciences




Total hours: 6 × 30 = 180 hours in semester

Structure: 39 hours - lectures, 39 hours - seminars, 12 hours - exams, 15 hours - consultations, 15 hours - homework, 30 hours - practical work in the computer classroom, 30 hours - individual study


Aim:

This is a general introductory course about computers. We shall learn the arithmetical, logical and physical bases of computers and the digital gates of computer system. We shall give an introduction to the Pascal programming language. To learn about computer hardware and Pascal.

Contents:

History of computers, von Neumann's concept. Number systems. Representation of numbers: two's complement, floating point. Boolean functions: definition, main identities, disjunctive form. Combinatorial circuits: adder, decoder, mux. Flipflops. Sequential circuits: register, counter, shift, serial adder. ALU and memory notions. Full example: Manchester Mark I (SSEM). Introduction to Windows and Linux. Introduction to Pascal software tools. Introductory notions about the Pascal programming language. Overview of Pascal data types. Integer and real data types. Char and Boolean data types. Overview of Pascal instruction types. If, case and goto instructions. For, while and repeat instructions. Solving of problems (Pascal). Work with the arrays in Pascal. Solving of problems from different areas (Pascal). M. Martinovi�, P. Staniši�, Computers and Principles of Programming, Podgorica, 2004 (in Serbian) R. Š�epanovi�, M. Martinovi�, Introduction to Programming and Problems in Pascal, Podgorica, 2000 (in Serbian)

Main texts:

M. Mano, Computer System Architecture, Prentice Hall, 1982 Further readings: Pascal user guide (from Internet) - learning of data representation and hardware components,

- knowledge of Boolean functions theory and problem's solving,

- capacity of individual working on computer, to process Pascal programs, Competences to be developed:

- capability to write elementary Pascal programs


Lectures and seminars with the active participation of students, group and individual consultations

Examination: Written exams (3 times in semester), estimation of individual activity on lectures and seminars


Students pools, results of exams, direct communications with the students

94

Name of the course: Introduction to Logic Programs of Studies:

Academic study programs Mathematics, Mathematics and Computer Sciences






Language: Serbian or English Prerequisites: Familiarity with elementary mathematical concepts.

Aim:

This course is an introduction to logic as the explicit study of the formal language in which mathematics may be described. It concerns the relation between mathematical theories, described in a formal language, and mathematical structures realizing those theories.

Contents:

Propositional calculus, including the notions of a propositional language, a tautology, and a formal system of axioms and rules for generating all tautologies as theorems (completeness theorem for propositional calculus). Abstract concept of a first order structure and the language associated with it. Interpretations of language and denotations of its terms and formulae with respect to a valuation of a given structure and the definition of satisfaction relation. Axioms and rules for the predicate calculus, deduction theorem and some metatheory including the principle of duality and prenex normal form theorem. The proof that the notion of formal derivability is consistent with, and no stronger than, the notion of logical consequence. Formulation and some consequences of the completeness theorem of the predicate calculus. Elementary equivalence and examples of important first order theories. Lemmon, E.J. Beginning logic, London, 1965 Kovijanic, Z. Introduction to mathematical logic, Podgorica, 2002. (in Serbian). Main texts:

Mendelson, E. Introduction to mathematical logic, Princeton, 1964. Further readings: Vujosevic, S. Mathematical logic, Podgorica, 1996. (in Serbian). - capacity of understanding abstract mathematical ideas.

- ability to distinguish the syntactic and semantic concepts and their importance in the foundation of mathematic. - capability to understand the method of formalization and its limitations.








95

Name of the course: English for Mathematics I Programmes of Studies: Academic study program Theoretical Mathematics


ECTS credits: 2





Aim: The introduction into the basics of English Mathematics. The understanding of the importance of spoken and written English in both everyday life and in mathematics.

Contents:

Basics of algebra; grammar: Past simple vs. Past continuous. Algebraic geometry; grammar: -ing forms and infinitives. Number theory; grammar: modal verbs must and have to. Applied mathematics; grammar: Present perfect passive. Combinatorics; grammar: conditional sentences. Discrete mathematics; grammar: Time clauses. Complex numbers; grammar: prepositions. Differential equations; Present simple vs. present continuous. Nonlinearity; grammar: Reported speech. Geometry and topology; grammar: clauses of contrast. Mathematical modelling; grammar: Making predictions. Numerical analysis; grammar: will and would. Probability theory; grammar: certainty. English for Mathematics – reader, compiled by Savo Kostic

Main texts:









96

Name of the course: Principles of Programming Programmes of Studies:








Aim:

This is an introductory course on programming. We shall learn the principles of programming and the assembly language. Also, we shall learn the Pascal programming language. Our goal is the knowledge of the hardware and software principles and Pascal.

Contents:

Basic computer (the model from the Mano's book): architecture, control unit, machine cycles, interrupt facility. The overview of the microprocessors development. Processor Intel 8086: architecture and instructions. How to use the debug.exe program. Examples of assembly language programs. Intel 8086 interrupt system. Operating systems concepts. Introduction to data structures: list, stack, queue, graph, Euler's cycle, binary tree. Pascal data types: intervals and enumerations. Records and sets. Arrays and matrices. Examples of programs (Pascal programming language). Procedures, functions, local variables and recursion. Files in Pascal. Pointers, procedures "new" and "dispose". Examples of programs from various areas (Pascal). M. Martinovi�, P. Staniši�, Computers and Principles of Programming, Podgorica, 2004 (in Serbian) R. Š�epanovi�, M. Martinovi�, Introduction to Programming and Problems in Pascal, Podgorica, 2000 (in Serbian)

Main texts:

Randall Hyde, The Art of Assembly Language Programming Further readings: Samuel L. Marateck, Turbo Pascal, John Wiley & Sons, 1991 - the full knowledge of the computer organization, with the emphasis on the software side, - capacity to write and read the assembly language code,

- the overall learning of the Pascal programming language, Competences to be developed:

- capability to write freely Pascal programs






97

Name of the course: Introduction to Combinatorics Programmes of Studies:








Aim: This course is aimed to introduce the fundamental notions of enumerative combinatorial theory. The aim is not to cover it in depth. Rather, we discuss a number of selected results and methods.

Contents:

The basic principles of enumerations. Sets and multy-sets. The Binomial theorem and Pascal’s triangle. Polynomial theorem. The principle of inclusion and exclusion, a generalization. The pigeonhole principle. Ramsey’s theorem, applications. Recurrence relations. Catalan number. Formal power series, generating functions. Partitions of the numbers. Partitions of the sets. Stirling numbers. Bell number. Enumeration under group action. D. Veljan, Combinatorics with Graph Theory, Zagreb (in Croatian)

Main texts: P. Cameron, Combinatorics: topics, techniques, algorithms, Cambrige University Press, 1994 Further readings: L.Lovasz, J.Palikan K. Vesztergombi, Discrete Mathematics, Springer, 2003. - capacity of understanding a combinatorial problems.









98

Name of the course: Linear Algebra II Programmes of Studies: Academic study program Mathematics, Mathematics and Computer Science

Level of the course: Bachelor level, I year, II semester Number of

ECTS credits: 6





Aim: Further introducing into the concepts of Linear algebra, such as scalar product, symmetric matrices, quadratic forms. Some basic applications of linear algebra to geometry and physics. Completion of standard education in Linear Algebra.

Contents:

Scalar product. Euclidian space. Unitary space. Orthogonal vectors. Orthonormal basis. Grammian matrix. Symmetric matrices and their eigenvalues. Positive and negative matrices. Bilinear and quadratic forms. Rank and index of quadratic form. Silvester criteria. Classification of hypersurfaces of second degree. Conjugate operators. Normal operators and matrices. Unitary and orthogonal operators. Hermitian operators. Norm of vector. System of linear equalities. Fredholm alternative. Normal solution and pseudosolution. A. Lipkovski, Linear algebra and analytical geometry, Belgrade. (in serbian) I. Krni�, �. Ja�imovi�, Linear Algebra – theorems and problems, skript, Podgorica, 2001. (in serbian) E. Shikin, Linear spaces and mappings, Moscow (in russian) S. Friedberg, A. Insel, L. Spence, Linear Algebra, Prentice Hall, 2002.

Main texts:

G. Strang, Linear Algebra and its Applications, Brooks Cole, 2005. Further readings: - further introducing into fundamental concepts of Linear Algebra. - fulfilling international standards in mathematical educations by teaching all the concepts and statements, that mathematical students are supposed to be familiar with after completion of first academic year. - developing of algebraic way of thinking and mathematical approach to scientific methods.


- offering first ideas about methods and problems of modern science.






99

Name of the course: Analitical Geometry Programs of Studies:

Academic study programs: Mathematics, Mathematics and Computer Sciences


Bachelor level, II year, II semester Number of ECTS credits: 4




Language: Serbian or English Prerequisites: Linear algebra I

Aim:

This course is aimed to introduce students with elements of vector algebra and method of coordinates in investigation of geometrical object. The course is also a good introduction in courses of synthetic geometry, differential geometry and linear algebra.

Contents:

Vectors in n – dimensional space (n=1,2,3). Linear operations. Basis and system of coordinates. Scalar and vector products. The curves in plane and in 3-dimensional space. Classification. The plane in n-dimensional space - dimension of the plane; the line and the hyperplane in Euclidian space. Geometric interpretation of the system of linear equations. Convex set. Linear programming problem Isometric transformation of Euclid space. Group of isometric transformation. Hypersurface of the second degree. Canonical form. Classification of the surface of the second surface. A. Lipkovski, Linear algebra and analytical geometry, Nau�na knjiga, Belgrade, 1995. (in Serbian) A. M. Postnikov, Lecture in geometrz, semester I, Analztic Geometry, Mir, Moscow, 1982.

Main texts:

D. L. Vossler, Exploring Analitic Geometry with Matematica, Academic Press, 1999.

Further readings: K. Horvati�, Linear algebra, Golden marketing – Tehni�ka knjiga, Zagreb, 2003 (in creation). - capacity of analytical interpretation of geometrical problems.

- ability to solve some simple problems using method of coordinates. - capability to apply the methods of coordinates in different areas of mathematics and its applications;








100








Language: Serbian Prerequisites: Analysis I

Aim:

This course is aimed to introduce students on basic notions of integral calculus, application of integral calculus, numerical series, functional sequences and series and Taylor`s series. The course includes some classical theorems from Real Analysis.

Contents:

Indefinite integral. Primitive function of a given function on the inteval. Primitive function on the segment. Definition of the Definite (Riemann’s) integral. Properties of integrable functions. Criteria for integrability of functions. Integral and derivation. Average values theorems in the integral calculus. Some integral formulae. Functions of bounded variations. Application of the definite integral. Improper integral. Series. Convergences of series. Functional sequences and series. Uniform convergence. Power series.

V. I. Gavrilov and Ž Pavi�evi�, Matemati~ka analiza I, PMF Podgorica, Unirex, Podgorica, 1994, pages 538 (Serbian).




To get and further apply the knowledge from the area: Theory of integrals, Theory of numerical sequences, Theory of functional sequences and series and Theory of power series.






101

Name of the course: English for Mathematics II Programmes of Studies:

Academic study program Mathematics and Mathematics and Computer Science


ECTS credits: 2





Aim: Further and more detailed insight into English for Mathematics with more thorough language and lexical content.

Contents:

Mathematical Logic and Foundation; grammar: Past simple vs. Past continuous. Combinatorics; grammar: -ing forms and infinitives. Ordered algebraic structures; grammar: modal verbs must and have to. General algebraic systems; grammar: Present perfect passive. Field theory; grammar: conditional sentences. Polynomials; grammar: Time clauses. Number theory; grammar: prepositions. Commutative rings and algebras; grammar: Present simple vs. present continuous. Algebraic geometry; grammar: Reported speech. Linear and multilinear algebra; grammar: clauses of contrast. Associative rings and algebras; grammar: Making predictions. Nonasociative rings and algebras; grammar: will and would. Category theory; grammar: certainty. English for Mathematics – reader, compiled by Savo Kostic

Main texts:









102

Name of the course: Analysis 3 Programmes of Studies:

Academic study programmes: Mathematics, Mathematics and Computer Sciences



Contact hours:

3 Lectures + 2 Seminars per week, 16 hours per semester for consultations and homework discussions =86 contact hours in semester

Total hours/semester:

16 x 8 = 128

Structure: 42 hours - lectures, 28 hours - seminars, 42 hours individual study and homework ( solving of problems), 16 hours for disputations.

Language: Montenegrin or English Prerequisites: Analysis 1, Analysis 2, Linear algebra

Aim: Analysis 3 and Analysis 4 introduce students to calculus (analysis) in n-dimensional Euclidean space.

Contents:

Metric and normed spaces. Topology of metric spaces and continuous mappings. Euclidean topology in Rn=the product topology in Rn. Completeness. Power series. Exponent of a matrix (operator). Connectedness and Compactness in Rn. Weierstrass and Bolzano theorems. Partial derivatives and derivatives of functions and mappings. Rules of differentiations. Jacobi’s matrix of a differential mapping. J(x,gf)=J(f(x),g)J(x,f). Gradient of functions. Mean value theorems. Higher partial derivatives. Taylor’s formulas. Local maxima and minima. Fixed point theorem. The inverse mapping and implicit function theorems. Rank of J(x,f) and charts. Conditional maxima and minima. M. Perovic, Foundations of Mathematical Analysis I, University Press, Podgorica, 1990 (in Montnegrin) K. Konigsberg, Analysis 2, Springer-Verlag, 1993.

Main texts:

J. Dieudonne, Foundations of modern analysis, 1960. Further

readings: - geometric understandings of metric and topological concepts

- understanding of motivations for definitions and statements

- ability to give instructive proofs. Competences to be developed:

- capacity to differentiate functions of several variables, finding tangent lines on curves and (hyper)planes on graphs and equipotential surfaces, Taylor’s series and maxima and minima on open and compact domains, geometric meaning of inverse mapping and implicit function theorems.


Lectures and seminars with contributions of students, individual discussions of homeworks, group and individual consultations.

Examination: Written weekly quizzes, homework (problem solving), one colloquium in the middle of the term, final exam.


Students pools, individual discussions of homeworks.

103

Name of the course: Algebra I Programmes of Studies:




Contact hours:

2h Lectures + 2h Exercices , 1h 20min hours for consultations, per week = 90 contact hours in semester


Structure: 128 h lectures and exams, 16 h administrative work, 36 h – consultations and individual study.

Language: Serbian or English or Rusian Prerequisites: Basic mathematical conceps from secondary school

Aim: This course is aimed to introduce students with basic notions in algebra and its applications in mathematical and technical sciences

Contents:

Operations and properties. Algebraic structures. Subalgebra. Congruence. Factor–algebra. Groupoid. Homomorphism of groupoids. Homomorphism Theorem of groupoids. Semigroups. Some classes of semigroups. Algebra of natural numbers. Peano system of axioms. Algebra sets, relations and maps. Lattices. Boolean algebras. Groups. Basic properties and examples. Subgroups. Basic properties. Lagrangian Theorem. Normal subgroup. Factor–group. Homomorphism of groups. Homomorphism Theorem of groups. Isomorphism Theorems of groups. Inner automorphisms. Cyclical group. Derivation group. V. Daši� , Introduction in general algebra, Podgorica, 2003. (in Serbian)

G. Kalajdži�, Algebra, Beograd, 1998. (in Serbian) B. Zekovi�, V. A. Artamonov Collection of resolute tasks from algebra (I part), Podgorica, 2003. (in Serbian) Z. Stojakovi�, �. Pauni�, Tasks from algebra, Novi Sad, 1998. (in Serbian)

Main texts:

S. Burris, H. P. Sankappanavar, A Course in Universal Algebra, Berlin, 1978. Further

readings: - ability to understand the proofs of theorems, and to construct independently simple proofs of mathematical statement. - capability to apply the methods of algebra in different areas of mathematics;








104

Name of the course: Introduction to Geometry Programs of Studies:




Contact hours: (3 Lectures + 2 Seminar) pe week, 15 hours in semester for consultations = 90 contact hours in semester



Language: Serbian or English Prerequisites: Basic courses of Algebra, Linear algebra and Analitical Geometry

Aim:

This course is aimed to introduce students with basic geometry of Euclidian space. It includes the concept of affine space and affine transformation, convexity and barycentric coordinates, regular polygons and regular polyhedrons.

Contents:

Short survey of the history of geometry. Linear and affine spaces. Affine subspaces. Geometry of affine subspaces. Affine transformations. Classic theorems of affine geometry. Convexity. Barycentric coordinates. Polygons and polyhedrons.in n-dimensional space. Simplexes. Extreme points of the set and convex shell. Euler formula. Regular polygons and regular polyhedrons. Cauchy theorems about polyhedrons. Spherics geometry. Geometry of sphere in n-dimensional space.

M. Audin, Geometry, Sprimger, 2003 Z. Lu�i�, Euclid and hyperbolic geometry, Mathematical faculty, Belgrade 1997. (in Serbian)

V.V. Prasolov, Geometry, MCCME, Moscow, 2007. (in Russian) Further readings: J. Silvester, Geometry, ancient and modern, Oxford University Press, 2001 - capacity of understanding of different concepts of geometry

- ability to understand the proofs of theorems, and to construct independently simple proofs of mathematical statement. - capability to apply geometrical methods in others areas of mathematics;








105

Name of the course: Programming 1 Programmes of Studies:

Mathematics, Mathematics and Computer Sciences, Computer Sciences






Language: Serbian or English Prerequisites: Computers and Programming, or Principles of Programming

Aim:

The Turing machines and the other models of computer. To learn what is a computer (in the theoretical sense) and what computer can do. The C programming language, in detail, with the intermediate level examples. To learn the C programming language.

Contents:

The detailed description of the Turing machine (in accordance to the Ebbinghaus' book). Church thesis. Modeling through single-letter alphabet. The machine word (the word which encodes the machine). Halting problem. The RAM model (Random Access Machine). Time and space complexity of the algorithms. The universal program. The C programming language: data types and instructions. Arrays and pointers (in C). Characters and strings. Input-output operations. Files. M. Martinovi�, R. Š�epanovi�, Theory of Algorithms and Pascal Programming, Podgorica, 1998 (in Serbian) Herbert S. Wilf, Algorithms and Complexity, Internet Edition, 1994 Brian W. Kernighan, Dennis M. Ritchie, The C Programming Language, Prentice Hall, 1988

Main texts:

A.V. Aho, J.E. Hopcroft, J.D. Ullman, The Analysis and Design of Computer Algorithms, Addison-Wesley, 1974 Further readings: - to learn the mathematical notion of the computer and to understand the computer's limitations, - capacity to write the programs using the C programming language



Lectures and seminars with the active participation of students, individual home tasks, group and individual consultations




106

Name of the course: Discrete Mathematics Programmes of Studies:







Language: Serbian or English Prerequisities: There is no prerequisites

Aim:

Course is aimed to introduce students with basic notions of Graph theory and its application. We discuss a number of selected results concerning the coloring of the graphs, Euler’s polyhedral formula, the planar graphs, the famous theorems of Kuratowski, Vizing and Brooks.

Contents:

The basic notions of graph theory. Path, cycles and connectivity. Trees, equivalent definitions of trees. Planar graphs, Kuratowski‘s theorem. Euler’s formula. Coloring maps and graphs: coloring graphs with two colors, Brooks’s theorem, “Five Color Theorem”, the first “proof” of the “Four Color Theorem”. Euler walks and Hamiltonian cycles. Matching in graphs. The “Marriage Theorem”. How to find a perfect matching. Latin squares. S. Cvetkovi�, Graph theory with applications, Belgrade, (in Serbian)

Main texts: R. Wilson, Graphs, Colourings and the Four-Color Theorem, Oxford University Press 2002 L.Lovasz, J.Palikan K. Vesztergombi, Discrete Mathematics, Springer, 2003. Further readings: - capability to translate a mathematical problem to a graph-theoretical one.

- capability to apply the methods of graph theory in different areas of mathematics; - ability to understand the proofs of theorems, and to construct independently simple proofs of mathematical statement.








107

Name of the course: Complex analysis I Programmes of Studies:


Level of the course: Bachelor level, II year, III semester Number of

ECTS credits: 4

Contact hours: 2h Lectures + 2h Exercices, for consultations 1h 20 min per week = 60 contact hours in semester


Structure: 85 h 20 min – lectures and exams, 10 h 40 min - administrative work, 24 h –consultations and individual study.

Language: Serbian or English or Rusian Prerequisites: Basic courses of Analysis I and II

Aim: This course is aimed to introduce students with basic notions in complex analysis and ots applications in mathematical and technical sciences

Contents:

Complex numbers. Field of complex numbers. Complex numbers geometrical interpretation. Modulus and complex conjugations. Metrics on C. Complex number argument and trigonometric form. Broadened complex plane. Riemann’s sphere and metrix on it. Complex number sequence. Numerical series. Connected and compact set. Curve. Contour. Orientation. Complex functions. Elemntary functions. Analytical and coplete functions. Harmonic funcrions. Complex integrations. Complex function integral on the line. Caushy’s theorem. Caushy formula.

Main texts: '. . . �%� , ��!�� Further readings:

- capacity of geometrical understanding of analytical problems.

- ability to understand the proofs of theorems, and to construct independently simple proofs of mathematical statement. - capability to apply the methods of complex analysis in different areas of mathematics;





Examination: Written exams (3 times in semester), problem solving - home tasks, estimation of individual activity on lectures and seminars and oral final examination


108





Contact hours:



16 x 8 = 128

Structure: 42 hours - lectures, 28 hours - seminars, 42 hours individual study and homework ( solving of problems), 16 hours for discussions.

Language: Montenegrin or English Prerequisites: Analysis 3

Aim: Analysis 4 with previous Analysis 3 introduce students to calculus (analysis) in n-dimensional Euclidean space.

Contents:

Jordan measure and Riemann integral in Rn. Lebesgues’ criterion of Riemann integrability. Fubini’s theorem. Change of variables. Differentiable manifolds in Rn (Curves and surfaces in R3). Tangent space (tangent line and tangent plane). Jordan’s measure and Riemann integral on (sub)manifolds of Rn. Orientation, tangent and transferal orientation on hypersurfaces. Integral of a vector field along the oriented curve and through the oriented (hyper)surface. Rotor and divergence f vector fields. Theorems of Green, Stokes and Ostrogradskii. The language of differential forms (correspondence between vector fields in R3 and 1-forms and 2-forms in R3). Exterior differential of forms as a mean to unite the three theorems. Relationship between the conditions A=grad f, rot A=0, div A=0. (Idea of homotopy equivalencies.) M. Perovic, Foundations of Mathematical Analysis I, University Press, Podgorica, 1990 (in Montnegrin) K. Konigsberg, Analysis 2, Springer-Verlag, 1993.

Main texts:

.M. Spivac, Calculus on manifolds, Benjamin Inc. 1965. Further

readings: L. Schwartz, Analyse Mathematique I, Hermann, Paris, 1967. - building analytic formalism for geometric ideas of curves and surfaces,.


- capacity to count Riemann integrals and integrals along curves and surfaces.


- ability to give instructive proofs.






109

Name of the course: Algebra 2 Programmes of Studies:







Language: Serbian or English or Rusian Prerequisites: Basic course of Algebra 1


Contents:

Symmetrical group. Cayley Theorem. Group of symmetries and isometries. Direct product of groups. Some properties. Ring. Field. Basic properties. Ideal of ring. Factor-ring. Characteristic of ring. Homomorphism of rings. Homomorphism-theorem. Subdirect product of rings. Isomorphism-theorems of rings. Maximal and simple ideals. Field quotients. Polynomial ring. Ring of polynomial functions. Exetension of field (basic concepts). Euclidean ring. V. Daši� , Introduction in general algebra, Podgorica, 2003. (in Serbian)


Main texts:

S. Burris, H. P. Sankappanavar, A Course in Universal Algebra, Berlin, 1978. Further readings:

- ability to understand the proofs of theorems, and to construct independently simple proofs of mathematical statement. - capability to apply the methods of algebra in different areas of mathematics;








110

Name of the course: Probability Theory Programmes of Studies:








Aim:

This course is aimed to introduce students with basic concepts of probability and their applications. The course includes rigorous Kolmogorov's definition of probability space and Lebesgues integral concept of expectation. Course includes as well as some classical theorems of Probability Theory: Bayes' theorem, Borel Cantelli's theorems, main theorem of mathematical expectation, central limit theorem, Bernoulli law of large number, Borel law of large number, Kolmogorov law of large number.

Contents:

Random experiment. Events as set. Probability. Conditional probability. Independence. Random variables. Random vectors. Probabilty mass function. Binomial and Poisson ditribution. Distribution function and density function. Examples of continuous distribution: Uniform, exponential, normal, gamma, Cauchy, beta, Weibul, Student, chi square. The joint distribution function. Marginal distributions. The conditional ditribution function. Multivariate normal ditribution. Expectation. Moments. Bianemes equality. Chebyshov's inequality. Functions of random variables. Characteristics function. Laws of large numbers. Central limit theorem. 1. S. Stamatovi�: Vjerovatno�a. Statistika, PMF 2000.


Main texts:

4. G. Grimett, D. Stirzaker: Probability and Random Processes, Oxford University Press, 2001.

Further readings: W. Feller: An introduction to probability theory and its application, Wiley. - capacity of understanding of probability space model and concept of probability computing. - ability to understand the proofs of theorems. - capability to apply the methods of probability theory in various branches of science.








111









Aim:

We shall learn about the algorithms and their logical complexity. The main topics are: techniques of programming, search trees, modern cryptography and NP-complete problems. We shall compose the corresponding C programs. Our goal is the knowledge of the theory of algorithms as well as C programming language.

Contents:

Data structures (graphs, trees, etc) and techniques of programming (recursion, dynamical programming, etc). Sorting algorithms (merge, quick, heap, and the like) and the lower bound in terms of comparisons. The binary search tree and the optimal binary search tree. Graph algorithms: Kruskal, Warshall, Floyd, Dijkstra, depth first search. Backtracking. Number theory: Euclidean algorithm, primality testing. Cryptography: RSA, Rabin, digital signature. Non-deterministic models of computation, NP-completeness hypothesis, reduction, SAT (Cook's theorem). Everywhere in this course: theory and C program. M. Martinovi�, R. Š�epanovi�, Theory of Algorithms and Pascal Programming, Podgorica, 1998 (in Serbian) Herbert S. Wilf, Algorithms and Complexity, Internet Edition, 1994 Udi Manber, Introduction to Algorithms: A Creative Approach, Addison-Wesley, 1989

Main texts:

Brian W. Kernighan, Dennis M. Ritchie, The C Programming Language, Prentice Hall, 1988 Michael R. Garey, David S. Johnson, Computers and Intractability, A Guide to the Theory of NP-completeness, W. H. Freeman, 1979 Further readings: Michael T. Goodrich, Roberto Tamassia, Data Structures & Algorithms in Java, John Wiley & Sons, 4th Edition - to learn the main classes of computer algorithms,

- to learn the complexity concept, - to acquire the knowledge of the advanced topics in the theory of algorithms (cryptography and NP-completeness),


- the full knowledge of C language Methods of teaching:





112


Academic study programmes Mathematics, Mathematics and Computer Sciences (A+B)


Bachelor level, II year,IV semester Number of ECTS credits: 10

Contact hours: (4 Lectures +3 Seminar) per week, 15 hours in semester for consultations = 60 contact hours in semester




Aim: In this course students get acquainted with simple differential equations, theorems about existence of solutions and methods of solution. In second part of the course students get to know dynamic systems, phase paths, stability of solutions and position of equilibrium.

Contents:

Simple differential equations of first order: term of solution and Kosi’s thesis, theorem about existence of solution, singular solutions Equations with separate variables, uniform and linear equation. Equations with total differential, integration parameter.Simple differential equations of higher degree (solution, Kosi’s thesis theorems about existence of solution lowering of degree).Linear equation of n-order Method of constant variation. Sturm’s theorems. Normal systems of simple differential equations (solution, Kosi’s thesis ,theorems about existence of solution). Method of elimination. Systems of linear differential equations (method of constant variation , Ojler’s and matrix method). Dynamic systems (integral curves, phase paths, closed phase paths).Stability of solution and position of equilibrium. Linear and quasilinear partial differential equations of first order. Kosi’s thesis. Systems of nonlinear partial differential equations of first order(complete integration, Pfafofa equations, method Lagranza Sarpija). R Š�epanovi�...Diferencijalne jedna�ine, Matemati�ki fakultet, Beograd 2005. god. (in Serbian). A.S.Pontrzgin, Obxknavenie differncilvnxe uravneniq, Nauka, Moskva, 1984 Main texts:

A.F.Filipov, Sbornik zada~ po differncilvnxm uravneniqm, Nauka, Moskva, 2005

Further readings: A.A.Arnolvd, Obxknavenie differncilvnxe uravneniq, Nauka, Moskva, 1984



Examination:

Ways of checking knowledge and giving marks by points: Presence on lectures 5 points. Two tests 25 points each.Doing homework 5 points. Final exam 40 points (oral exam). Passing grade will be given if more then 50 points is scored.



113


Academic study program Mathematics






Language: Serbian or English or Rusian Prerequisites: Basic courses of Algebra I and Algebra II

Aim: This course is aimed to introduce students with basic abstract structures and their characteristics.

Contents:

Extensions. Algebraic and transcendental extensions. Separable field..Algebraic closed field. Relative automorphisms. Subfields of invariants. Finite fields. Solvable groups. Normal and separabile extensions. Basic theorem of Theory Galois. Symmetric polynomials. Solvability of equations..Moduls. Algebras. Quaternion algebra. Frobenius Theorem Nonassociative algebras. Lie algebras. Representation of groups and algebras. n-semigroups and n-groups. (m,n)-rings. Multioperator-groups. Multioperator-semigroups and algebras. Universal algebras and subalgebras. Lattices of subalgebras. Homomorphisms, congruences and factor-algebras. Direct and subdirect product of algebras. Free universal algebras. Variety of algebras. Category and subcategory. Diagrams and duality. Monomorphisms, epimorphisms, isomorphisms.. Products and coproducts. Functor. Equivalency of categories. V. Daši� , Algebra, Podgorica, (in Serbian)

Lj. Ko�inac, A. Mandak, Algebra II, Priština, 1996. (in Serbian) B. Zekovi�, V. A. Artamonov Collection of resolute tasks from algebra (II part, appear), Podgorica (in Serbian) Z. Stojakovi�, �. Pauni�, Tasks from algebra, Novi Sad, 1998. (in Serbian)

Main texts:










114

Name of the course: Statistics Programmes of Studies:







Language: Serbian or English Prerequisites: Basic courses of Analysis, Linear algebra and Probability Theory.

Aim:

This course is aimed to introduce students with basic concepts of statistics and their applications. The course includes rigorous definitions of statistics experiments and main statistics notions. Course includes as well as some classical theorems of Statistics: Glivenko Cantelli theorem, Neyman-Pearson theorem, Fischer theorem. This course is aimed to introduce students with statistics softwares STATISTICS and SPSS.

Contents:

Sampling and Statistics. Order statistics. Glivenko-Cantelli theorem. Chi square distribution, t-distribution and F-distribution. Fisher theorem. Measures of quality of estimators. A sufficient statistic for a parameter. Properties of sufficient statistic. Rao-Cramer lower bound of efficiency. Maximum likelihood methods. Confidence intervals. Introduction to hypothesis testing. Neyman-Pearson theorem. Most powerful tests. Inferences about normal models. The anlysis of variance. A regression problem. Nonparametric statistics. Statistical softwares STATISTICS and SPSS. 1. S. Stamatovi�: Vjerovatno�a. Statistika, PMF 2000.


Main texts:


Further readings: W. Feller: An introduction to probability theory and its application, Wiley. - capacity of understanding of statistic experiment, concept of statistic estimation and concept of hypothesis testing. - ability to understand the proofs of theorems. - capability to apply the methods of statistics in various branches of science. Abilty to make a statistics analysis using statistical softwares.


- ability for solving examples and simple problems. Methods of teaching:





115

Name of the course: Measure and Integral Programmes of Studies:








Aim:

This course covers Lebesgue's integration theory with applications to analysis. It includes the concepts of abstract measures, measurable functions and integral on abstract measure and fundamental theorems related ti these problem.

Contents:

Cardinality of sets. Axiom of choice – equivalent formulation. Ring and σ-ring of sets. Borel sets. Outer measure. Jordan extension of measure. Lebesgue extension of measure. Measures on R^n. Measurable function. Integral of simple function and integral of positive function. The basic theorems on integrals. Integrable functions. Lebesge spaces. Theorem on decomposition of measure. Apsolute continuity. Singular measures. Radon-Nikodim theorem. Fundamental theorem of calculus for Lebesgue integral. Functions of total variation. S. Aljan�i�, Introduction in real and functional analysis, Belgrade, 1972. (in Serbian) A. Bartle, The elements of integration and Lebesgue measure, John Wilez and Sons, 1995.

Main texts:

W. Rudin, Real and complex analysis, McGraw-Hill, 1966. Further readings: D. L. Cohn, Measure Theory, Birkhauser, 1980 - capacity of understanding the construction of new object in mathematics by completing. - ability to understand the proofs of theorems, and to construct independently simple proofs of mathematical statement. - capability to apply the methods of measure theory in different areas of mathematics;








116

Name of the course: Partial Differential Equations Programs of Studies:

Academic study program Mathematics







Aim:

This course is aimed to introduce students with basic equations of mathematical physics (the wave and heat equations) and the methods of their solving. It also covers the Sturm-Liouville theory and eigenfunction expansions, as well as the Dirichlet problem for Laplace's operator and potential theory.

Contents:

Linear partial differential equation of the second order. Reduction to canonical form. Classification of the partial differential equation of the second order. Cauchy problem for the hyperbolic equations - existence and uniqueness of the solution. Wave equation in plane and in space. Fourrier’s method. Cauchy problem for the parabolic equations - existence and uniqueness of the solutions.Fourrier’s method. Boundary problems for the elliptic partial equations. Sturm-Liouville problem. Laplace’s operator, Laplace’s equation. Method of the separation of the variables for solving Laplace’s and Poisson’s equations.Green’s function of the Laplace’s operator. Method of the potentials. Applications of Green’s functions for solving Dirichlet problem for ball. Integral equations. Method of the succesive approximations for Fredholom’s and for Voltera’s integral equation. Solving of Freddholm’s equation. R. Š�epanovi�, M. Martinovi�, Partaial differential equations, Podgorica 2000. (in Serbian) Main texts: V.M.Uroev, Equations of mathematical physics, Moscow, 1998. (in Russian) G. B. Folland, Introduction to Partial Differential Equations. Springer-Verlag, 2000. Second Edition �Further readings: I.Aganovi� ,K. Veseli�:Equations of Mathematical Physics,Zagreb,1985. (In Croatian) - capacity of understanding of mathematical models of physical phenomenons - ability to understand the proofs of theorems about partial differential equations and ability to construct independently simple models using partial differential equation and abilitz to solve them. - capability to apply the analytical methods in solving different problems in physics;








117

Name of the course: Theoretical Mechanics Programs of Studies:

Academic study programs Mathematics







Aim:

This course is aimed to introduce students with basic concepts of theoretical mechanics. The course includes axiomatic approach to mechanics, Lagrange-D’Alembert’s principle, stability of equilibrium in potential field, integral principles of mechanics and Hamilton’s equations.

Contents:

Space, time, motion, velocity, acceleration. Axioms of dynamics. Differential equations of motions of material point.General theorems and firsts integrals. Basic models of linearity motion of point. Motion in the field of central force. Keppler's problem. Kinematics and dynamics of complex motion of point. Elements of kinematics of solid body. Lagrange-D'Alembert's principles. General theorems of the dynamics of infree system. Lagrange's equations of the second type. Stability of the equilibrium in potential field. The small oscilations of the system about equilibrium. Forced oscillations. Dynamics of the sferical motion of the solid body. Integral principles of Mechanics. Hamilton's equations. V. G. Vilke, Teoretical mechanics (in russian, MGU, 1998; I. Aganovi�, K. Veseli�, Uvod u analiti�ku mehaniku, Prirodoslovno-matemati�ki fakultet u Zagrebu, 1990. V. Tatarinov, Classical dynamics (in Russian), MGU,1984.

Main texts:

I. Aganovi�, K. Veseli�, Introduction to analitical mechanics, Faculty of Mathematics and Natural Sciences, (in Croation), 1990. Further readings: - capacity of understanding of the basic concepts of classical mechanics.

- ability to understand the proofs of the basic theorems in mechanics, and to construct and study independently simple mathematical models in mechanics. - capability to apply the mathematical methods in studying of mechanical problems;








118

SAVO!!!!

Name of the course: English for Mathematics III Programmes of Studies:



ECTS credits: 2






Contents: English for Mathematics – reader, compiled by Savo Kostic

Main texts:









119

Name of the course: Complex Analysis II Programmes of Studies:



Bachelor level, III year, VI semester

Number of ECTS credits: 6

Contact hours:

(3 Lectures + 2 Seminar) per week, 15 hours in semester for consultations = 86 contact hours in semester

Total hours:



Language: Serbian or English Prerequisites: Complex analysis I

Aim:

This course is aimed to introduce students with basic notions of complex analysis. This is continuation of the complex analysis I. The course includes the main theorems of complex analysis and the main principles. The complex analysis is a classical mathematical discipline which is applicable in the most mathematical and technical disciplines.

Contents:

Taylor formula and the zeros of analytic functions. Maximum principle and Schwartz lemma. Uniqueness theorem and analytic extension. Weierstrass and Runge theorem. Laurent development series. Residue. Application of residue in calculation of integrals. Logarithmic derivative. Rouche's theorem and argument principle. Conformal mappings. Geometric and topological characteristics of analytic functions. Open and inverse mappings theorems. Mebius and the other elementary transformations. Continuation principle, analytic extension principle and Riemann Schwartz principle. Riemann mapping theorem. Poisson integral for the unit circle and for the real line. B. V. Šabat: Introduction to the complex analysis, „Nauka“, Moscow, Part I, 1976 (Russian). Main texts: D. Kalaj: Collection of problems of complex analysis, University of Montenegro, 2006, first edition, (Serbian). W. Rudin, Real and complex analysis, McGraw-Hill, 1966.

Further readings: L. V. Ahlfors, Complex Analysis. An introduction to the theory of analytic functions of complex variables, McGraw-Hill Book Company, New York, Toronto, London, 1966. - capacity of geometrical understanding of analytical problems.

- ability to understand the proofs of theorems, and to construct independently simple proofs of mathematical statement. - capability to apply the methods of functional analysis in different areas of mathematics;








120

Name of the course: Introduction to Differential Geometry Programmes of Studies:








Aim:

This course is aimed to introduce students with basic notions of Differential geometry in Euclidean space. It relates to application of differential calculus in studying geometry of curves and surfaces in Euclidean space. The course includes some classical statements based on the notions of curvature and torsion for curves and those one for the surfaces based on the existence of the first and second fundamental form. All statements are followed with geometric interpretation and application through examples.

Contents:

Local theory of curves. Natural parameterization. Curvature and torsion. Frenet-Serett formulas. The fundamental theorem for curves. Elementary surfaces, tangent vectors and tangent space. The first and second fundamental form. Weingarten map. Curves on surfaces. Normal and main curvatures. Gauss curvature. Geometric interpretation. Gauss and Codazzi equations. Egregium theorem. Geodesic curvature and geodesics lines. Uniqueness and minimalism of geodesics lines. Parallel vector fields and parallel transportation. Lining and rotating surfaces. The fundamental theorem for surfaces. N. Blazic, N. Bokan, An Introduction to Differential Geometry , Belgrade, 1996. (in Serbian) V. Dragovic, D. Milinkovic, Analysis on manifolds, Belgrade, 2003. (in Serbian)

Main texts:

S. Mischenko, A. T. Fomenko, Course of differential geometry and topology, Moscow, 2004. (in Russian) Further

readings: A. Presley, Elementary differential geometry, Sprienger, 2002 - capacity of visualization curves and surfaces in Euclidean space resulting in individual and creative thinking - capacity for understanding more complex geometrical features - understanding the differential calculus as powerful geometric tool


- capacity for application the techniques of differential calculus and linear algebra in investigation the shapes of curves and surfaces



Examination: Two written exams, homework’s, activity on lectures and seminars, oral exam



121

Name of the course: Functional Analysis Programmes of Studies:








Aim:


Contents:


Main texts:









122

Name of the course: Numerical Analysis Programmes of Studies:






Structure: 26 hours - lectures, 26 hours - seminars, 8 hours - exams, 15 hours - consultations, 15 hours - homework (individual solving of problems), 30 hours - individual study.


Aim:

To give an introduction to commonly used numerical methods. To demonstrate the application of numerical methods to calculations encountered in lecture courses and projects. To enable students to use numerical techniques to tackle problems that are not analytically soluble.

Contents:

Interpolation and similar matters: Curve fitting - Least squares approximations, Lagrange polynomial interpolation, Finite differences, Newton interpolation polynomials, Hermite interpolation, Cubic spline interpolation, Numerical differentiation. Numerical integration: Three formulas - rectangle, trapezium and Simpson, Runge's rule for practical error estimation (Richardson), Integral with weight function, Gaussian quadrature. Numerical methods of algebra: Gaussian elimination with pivoting, Matrix condition number, Simple iteration method for solving systems of linear equations, Jacobi and Gauss-Seidel methods, Relaxation method, Power method for matrix eigenvalues. Systems of nonlinear equations: Simple iteration method, Newton's method, Bisection and Secant methods (n=1). Numerical methods for ordinary differential equations: Euler method, Runge-Kutta methods, Adams' predictor-corrector method, Milne's method, Finite difference method for solving boundary value problem. Boško Jovanovi�, Desanka Radunovi�, Numeri�ka analiza, Beograd, 1993 (in Serbian) M. Martinovi�, R. Š�epanovi�, Numeri�ke metode, Nikši�, 1995 (in Serbian) C. Gerald, P. Wheatley, Applied Numerical Analysis, Addison-Wesley, 1997

Main texts:

N.S. Bahvalov, Numerical methods, FML, 2001 (in Russian) Further readings: G.W. Collins, Fundamental Numerical Methods and Data Analysis, 2003 - to learn some (elementary) classes of numerical methods, - to understand approximate solution and error estimation concepts, - capacity to solve problems (exercises) concerning the topics covered,


- capacity to apply the numerical techniques by means of writing programs Methods of teaching:





123

Name of the course: Algebraic Topology Programmes of Studies:






Structure: 40- hours – lectures, 14 hours - seminars, 6 hours - exams, 20 hours - consultations, 30 hours – homework (individual solving of problems), 40 hours – individual study.

Language: Serbian or English Prerequisites: Basic courses of Analysis and Algebra

Aim:

This course is aimed to introduce students with basic notions of Algebraic Topology. It relates to construction of some algebraic structures with aim to study topological spaces. The course includes the notions of homotopy, fundamental group, higher homotopy groups, coverings, fibrations, simplicial and singular homology.

Contents:

Topology and some constructions of topological spaces. Absolute and relative homotopy. Continuous maps and homotopy. Paths and the product of paths and its properties. Fundamental group. Fundamental group of circle. Proof of Brauer theorem and fundamental theorem of algebra. Coverings. Lifting homotopy theorem. Classification of coverings. The groups action and its application in coverings. Higher homotopy groups. Relative homotopy groups and exact homotopy sequence. Fibrations. Hopf fibrations. Exact homotopy sequence for coverings and fibrations. Simplicial complexes. Simplicial homology. Sphere, torus and Klein bottle. CW complexes. CW approximation theorem. Homotopy groups of CW complexes. Singular homology and exact homology sequence. Singular homology of CW complexes. C. Kosniowski, A first course in algebraic topology, Cambridge University Press, 1980 J. R. Mankres, Elements of algebraic topology, Addison-Wesley Publishing Company, 1984.

Main texts:

V. A. Vasil'ev, An Introduction to Topology, Fazis, Moscow, 1997 Further readings: A. Hatcher, Algebraic Topology, Cambridge University Press, 2001 - capacity of to understand the notion of continuous deformation of spaces

- ability to understand the relations and connectedness between different mathematical disciplines such as algebra and topology. - capability to apply the methods of algebraic topology in different areas of mathematics;


- ability for topological thinking and capacity to argue the algebraic- topological construction



Examination: Two written exams, homework, individual activity on lectures and seminars, oral exam



124

Name of the course:

History and Philosophy of Mathematics Programmes of Studies:



Bachelor level, III year,VI semester Number of ECTS credits: 3

Contact hours:



16 x4 = 64

Structure: 32 hours - lectures, 0 hours - seminars, 24 hours individual study and homework, 8 hours for discussions.

Language: Montenegrin or EnglishPrerequisites:

Aim:Contextual history of Mathematics to enlarge the general culture of students and “ the art of discovery be promoted and its method known through illustrious examples” (Leibniz).

Contents:

Protomathematics in prehistory and in ancient Egypt and Mesopotamia. Natural philosophers in Greece. Tales - the first mathematicians, Pythagoras – the founder of mathematics. Geometrisations of mathematics. Greek mathematics before Euclid: Eleatics and atomists, Plato’s circle, Eudoxus proves the atomist’s discoveries. Aristotle: Logic and the foundation of human knowledge. Three classical problems. Alexandria (Hellenism). Euclid, Archimedes, Apollonius. Alexandria under the Roman domination. Ancient and Medieval China. and India. Mathematics of Islamic World. Medieval Europe. Renaissance. Scientific revolution. Calculus and the triumph of science. Mathematical Analysis. Evolution of Algebra. Evolution of Geometry. Foundations of Mathematics. Logicism, formalism and intuitionism. Truth, and existence in Mathematics. V. J. Katz, A History of Mathematics, brief edition, Pearson, Addison Weslay, 2004.J. Fauvel, J. Gray, eds., The History of Mathematics: A Reader, Macmillan, 1987

Main texts:

M. Kline, Mathematical Thougt from Ancient to Modern Times, Oxford Univ. Press, 1972Further

readings: Abrege d’histoire des mathematiques 1700-1900, I, II, sous la direction de J. Dieudonne, Hermann, Paris, 1978- understand the evolution of mathematical ideas- develop an appreciation of the contributions made by various cultures and individuals to the growth and development of mathematical ideas (MAA)


- impact of mathematics to the development of human culture Methods of teaching:

Lectures, group and individual consultations.

Examination: Written two - weekly quizzes, one colloquium in the middle of the term, final exam.


Students pools.

125

SAVO!!!!

Name of the course: English for Mathematics IV Programmes of Studies:



ECTS credits: 2







Main texts:









126

V


MATEMATICS AND COMPUTER SCIENCE

127


Mathematics and Computer Science Undergraduate Academic I year

Course

Mandatory

Elective

Winter Semester

hours weekly

EC

TS

Ljetnji semestar

hours weekly

EC

TS

1. Analysis 1 X 4+4 10 2. Linear algebra 1 X 4+3 8 3. Computers and Programming X 3+3 6 4. Introduction to Logic X 2+1 4 5. English Language 1 x 2+0 2 6. Principles of Programming X 3+2 6 7. Introduction to Combinatorics X 2+2 4 8. Linear algebra 2 X 2+2 6 9. Analytical Geometry X 2+2 4 10. Analysis 2 X 4+3 8 11. English Language 2 X 2+0 2

Total 11 15+11 30 15+11 30

II year

1. Analisys 3 X 3+2 6 2. Algebra 1 X 2+2 6 3. Operating systems X 3+2 4 4. Programming 1 X 3+2 6 5. Discrete Mathematics x 2+1 4 6. Complex Analisys 1 X 2+2 4 7. Analisys 4 X 3+2 6 8. Algebra 2 X 2+2 4 9. Probability Theory X 3+2 6 10. Programming 2 X 3+2 6 11. Differential Equations X 4+3 8

Total 11 15+11 30 15+11 30 III year 1. Elective predmet 1 x 3+0 4 2. Statistics X 3+2 6 3. Measure and Integral X 2+1 4 4. Computer Networks X 3+2 4 5. Database Systems X 3+2 6 6. Object Oriented Programming X 2+1 4 7. English Language 3 X 2+0 2 8. Introduction to Differential

Geometry X 2+1 4

9. Functional Analysis X 3+1 4 10. Numerical Analysis X 2+2 4 11. Compilers X 2+2 4 12. Visualization and Computer

Graphics X 2+1 4

13. Internet technologies X 2+1 4 14. English Language 4 X 2+0 2 15. Elective predmet 2 X 3+0 4

Total 13 2 18+8 30 18+8 30

128









Aim:


Contents:











129

Name of the course: Linear Algebra I Programmes of Studies: Academic study program Mathematics, Mathematics and Computer Science


ECTS credits: 8





Aim:

Introducing students to fundamental concepts of Linear algebra, such as matrices, linear operators and spaces. This theory is crucial for understanding of advanced theories in mathematics, physics and computer sciences. Linear algebra, along with Mathematical Analysis, is basic and the most fundamental course in Mathematics.

Contents:

Systems of linear equalities. Gauss method. Matrices and elementary transformations. Determinant and rank of matrix. Vector spaces. Linear independence. Basis of vector space. Singular and invertible matrices. Kronecker – Kapelli theorem. Fundamental solution of linear system. Linear operators. Kernel and image of linear operator. Rank and defect. Matrix of linear operator. Matrix of transformation from one basis to another. Invariant subspaces. Eigenvalues and eigenvectors. Cayley – Hamilton theorem. Similar matrices. Nilpotent operators. Jordan form of matrix. A. Lipkovski, Linear algebra and analytical geometry, Belgrade. (in serbian) I. Krnic, �. Ja�imovi�, Linear Algebra – theorems and problems, skript, Podgorica, 2001. (in serbian) E. Shikin, Linear spaces and mappings, Moscow (in russian) S. Friedberg, A. Insel, L. Spence, Linear Algebra, Prentice Hall, 2002.

Main texts:

G. Strang, Linear Algebra and its Applications, Brooks Cole, 2005. Further readings: - introducing to the fundamental concepts of higher mathematics. - preparation for more advanced courses and theories



- developing ability of clear and critical thinking, understanding such concepts as proof, implication, precise statement etc.






130









Aim:


Contents:


Main texts:

M. Mano, Computer System Architecture, Prentice Hall, 1982 Further

readings: Pascal user guide (from Internet) - learning of data representation and hardware components,

- knowledge of Boolean functions theory and problem's solving, - capacity of individual working on computer, to process Pascal programs,


- capability to write elementary Pascal programs Methods of teaching:





131

Name of the course: Introduction to Logic Programs of Studies:







Language: Serbian or English Prerequisites: Familiarity with elementary mathematical concepts.

Aim:

This course is an introduction to logic as the explicit study of the formal language in which mathematics may be described. It concerns the relation between mathematical theories, described in a formal language, and mathematical structures realizing those theories.

Contents:

Propositional calculus, including the notions of a propositional language, a tautology, and a formal system of axioms and rules for generating all tautologies as theorems (completeness theorem for propositional calculus). Abstract concept of a first order structure and the language associated with it. Interpretations of language and denotations of its terms and formulae with respect to a valuation of a given structure and the definition of satisfaction relation. Axioms and rules for the predicate calculus, deduction theorem and some metatheory including the principle of duality and prenex normal form theorem. The proof that the notion of formal derivability is consistent with, and no stronger than, the notion of logical consequence. Formulation and some consequences of the completeness theorem of the predicate calculus. Elementary equivalence and examples of important first order theories. Lemmon, E.J. Beginning logic, London, 1965 Kovijanic, Z. Introduction to mathematical logic, Podgorica, 2002. (in Serbian). Main texts:

Mendelson, E. Introduction to mathematical logic, Princeton, 1964. Further readings: Vujosevic, S. Mathematical logic, Podgorica, 1996. (in Serbian). - capacity of understanding abstract mathematical ideas.

- ability to distinguish the syntactic and semantic concepts and their importance in the foundation of mathematic. - capability to understand the method of formalization and its limitations.








133

Name of the course: English for Mathematics I Programmes of Studies: Academic study program Theoretical Mathematics


ECTS credits: 2





Aim: The introduction into the basics of English Mathematics. The understanding of the importance of spoken and written English in both everyday life and in mathematics.

Contents:

Basics of algebra; grammar: Past simple vs. Past continuous. Algebraic geometry; grammar: -ing forms and infinitives. Number theory; grammar: modal verbs must and have to. Applied mathematics; grammar: Present perfect passive. Combinatorics; grammar: conditional sentences. Discrete mathematics; grammar: Time clauses. Complex numbers; grammar: prepositions. Differential equations; Present simple vs. present continuous. Nonlinearity; grammar: Reported speech. Geometry and topology; grammar: clauses of contrast. Mathematical modelling; grammar: Making predictions. Numerical analysis; grammar: will and would. Probability theory; grammar: certainty. English for Mathematics – reader, compiled by Savo Kostic

Main texts:









134

Name of the course: Principles of Programming Programmes of Studies:








Aim:

This is an introductory course on programming. We shall learn the principles of programming and the assembly language. Also, we shall learn the Pascal programming language. Our goal is the knowledge of the hardware and software principles and Pascal.

Contents:

Basic computer (the model from the Mano's book): architecture, control unit, machine cycles, interrupt facility. The overview of the microprocessors development. Processor Intel 8086: architecture and instructions. How to use the debug.exe program. Examples of assembly language programs. Intel 8086 interrupt system. Operating systems concepts. Introduction to data structures: list, stack, queue, graph, Euler's cycle, binary tree. Pascal data types: intervals and enumerations. Records and sets. Arrays and matrices. Examples of programs (Pascal programming language). Procedures, functions, local variables and recursion. Files in Pascal. Pointers, procedures "new" and "dispose". Examples of programs from various areas (Pascal). M. Martinovi�, P. Staniši�, Computers and Principles of Programming, Podgorica, 2004 (in Serbian) R. Š�epanovi�, M. Martinovi�, Introduction to Programming and Problems in Pascal, Podgorica, 2000 (in Serbian)

Main texts:

Randall Hyde, The Art of Assembly Language Programming Further readings: Samuel L. Marateck, Turbo Pascal, John Wiley & Sons, 1991 - the full knowledge of the computer organization, with the emphasis on the software side, - capacity to write and read the assembly language code,

- the overall learning of the Pascal programming language,


- capability to write freely Pascal programs Methods of teaching:





135

Name of the course: Introduction to Combinatorics Programmes of Studies:








Aim: This course is aimed to introduce the fundamental notions of enumerative combinatorial theory. The aim is not to cover it in depth. Rather, we discuss a number of selected results and methods.

Contents:

The basic principles of enumerations. Sets and multy-sets. The Binomial theorem and Pascal’s triangle. Polynomial theorem. The principle of inclusion and exclusion, a generalization. The pigeonhole principle. Ramsey’s theorem, applications. Recurrence relations. Catalan number. Formal power series, generating functions. Partitions of the numbers. Partitions of the sets. Stirling numbers. Bell number. Enumeration under group action. D. Veljan, Combinatorics with Graph Theory, Zagreb (in Croatian)

Main texts: P. Cameron, Combinatorics: topics, techniques, algorithms, Cambrige University Press, 1994 Further readings: L.Lovasz, J.Palikan K. Vesztergombi, Discrete Mathematics, Springer, 2003. - capacity of understanding a combinatorial problems.









136

Name of the course: Linear Algebra II Programmes of Studies: Academic study program Mathematics, Mathematics and Computer Science


ECTS credits: 6





Aim: Further introducing into the concepts of Linear algebra, such as scalar product, symmetric matrices, quadratic forms. Some basic applications of linear algebra to geometry and physics. Completion of standard education in Linear Algebra.

Contents:

Scalar product. Euclidian space. Unitary space. Orthogonal vectors. Orthonormal basis. Grammian matrix. Symmetric matrices and their eigenvalues. Positive and negative matrices. Bilinear and quadratic forms. Rank and index of quadratic form. Silvester criteria. Classification of hypersurfaces of second degree. Conjugate operators. Normal operators and matrices. Unitary and orthogonal operators. Hermitian operators. Norm of vector. System of linear equalities. Fredholm alternative. Normal solution and pseudosolution. A. Lipkovski, Linear algebra and analytical geometry, Belgrade. (in serbian) I. Krni�, �. Ja�imovi�, Linear Algebra – theorems and problems, skript, Podgorica, 2001. (in serbian) E. Shikin, Linear spaces and mappings, Moscow (in russian) S. Friedberg, A. Insel, L. Spence, Linear Algebra, Prentice Hall, 2002.

Main texts:

G. Strang, Linear Algebra and its Applications, Brooks Cole, 2005. Further readings: - further introducing into fundamental concepts of Linear Algebra. - fulfilling international standards in mathematical educations by teaching all the concepts and statements, that mathematical students are supposed to be familiar with after completion of first academic year. - developing of algebraic way of thinking and mathematical approach to scientific methods.


- offering first ideas about methods and problems of modern science.






137


Academic study programs: Mathematics, Mathematics and Computer Sciences


Bachelor level, II year, II semester Number of ECTS credits: 4




Language: Serbian or English Prerequisites: Linear algebra I

Aim:

This course is aimed to introduce students with elements of vector algebra and method of coordinates in investigation of geometrical object. The course is also a good introduction in courses of synthetic geometry, differential geometry and linear algebra.

Contents:

Vectors in n – dimensional space (n=1,2,3). Linear operations. Basis and system of coordinates. Scalar and vector products. The curves in plane and in 3-dimensional space. Classification. The plane in n-dimensional space - dimension of the plane; the line and the hyperplane in Euclidian space. Geometric interpretation of the system of linear equations. Convex set. Linear programming problem Isometric transformation of Euclid space. Group of isometric transformation. Hypersurface of the second degree. Canonical form. Classification of the surface of the second surface. A. Lipkovski, Linear algebra and analytical geometry, Nau�na knjiga, Belgrade, 1995. (in Serbian) A. M. Postnikov, Lecture in geometrz, semester I, Analztic Geometry, Mir, Moscow, 1982.

Main texts:

D. L. Vossler, Exploring Analitic Geometry with Matematica, Academic Press, 1999.

Further readings: K. Horvati�, Linear algebra, Golden marketing – Tehni�ka knjiga, Zagreb, 2003 (in creation). - capacity of analytical interpretation of geometrical problems.

- ability to solve some simple problems using method of coordinates. - capability to apply the methods of coordinates in different areas of mathematics and its applications;








138









Aim:


Contents:












139

Name of the course: English for Mathematics II Programmes of Studies:



ECTS credits: 2






Contents:

Mathematical Logic and Foundation; grammar: Past simple vs. Past continuous. Combinatorics; grammar: -ing forms and infinitives. Ordered algebraic structures; grammar: modal verbs must and have to. General algebraic systems; grammar: Present perfect passive. Field theory; grammar: conditional sentences. Polynomials; grammar: Time clauses. Number theory; grammar: prepositions. Commutative rings and algebras; grammar: Present simple vs. present continuous. Algebraic geometry; grammar: Reported speech. Linear and multilinear algebra; grammar: clauses of contrast. Associative rings and algebras; grammar: Making predictions. Nonasociative rings and algebras; grammar: will and would. Category theory; grammar: certainty. English for Mathematics – reader, compiled by Savo Kostic

Main texts:









140





Contact hours:



16 x 8 = 128

Structure: 42 hours - lectures, 28 hours - seminars, 42 hours individual study and homework ( solving of problems), 16 hours for disputations.

Language: Montenegrin or English Prerequisites: Analysis 1, Analysis 2, Linear algebra

Aim: Analysis 3 and Analysis 4 introduce students to calculus (analysis) in n-dimensional Euclidean space.

Contents:

Metric and normed spaces. Topology of metric spaces and continuous mappings. Euclidean topology in Rn=the product topology in Rn. Completeness. Power series. Exponent of a matrix (operator). Connectedness and Compactness in Rn. Weierstrass and Bolzano theorems. Partial derivatives and derivatives of functions and mappings. Rules of differentiations. Jacobi’s matrix of a differential mapping. J(x,gf)=J(f(x),g)J(x,f). Gradient of functions. Mean value theorems. Higher partial derivatives. Taylor’s formulas. Local maxima and minima. Fixed point theorem. The inverse mapping and implicit function theorems. Rank of J(x,f) and charts. Conditional maxima and minima. M. Perovic, Foundations of Mathematical Analysis I, University Press, Podgorica, 1990 (in Montnegrin) K. Konigsberg, Analysis 2, Springer-Verlag, 1993.

Main texts:

J. Dieudonne, Foundations of modern analysis, 1960. Further

readings: - geometric understandings of metric and topological concepts


- ability to give instructive proofs. Competences to be developed:

- capacity to differentiate functions of several variables, finding tangent lines on curves and (hyper)planes on graphs and equipotential surfaces, Taylor’s series and maxima and minima on open and compact domains, geometric meaning of inverse mapping and implicit function theorems.






141

Name of the course: Algebra I Programmes of Studies:






Structure: 128 h lectures and exams, 16 h administrative work, 36 h – consultations and individual study.

Language: Serbian or English or Rusian Prerequisites: Basic mathematical conceps from secondary school


Contents:

Operations and properties. Algebraic structures. Subalgebra. Congruence. Factor–algebra. Groupoid. Homomorphism of groupoids. Homomorphism Theorem of groupoids. Semigroups. Some classes of semigroups. Algebra of natural numbers. Peano system of axioms. Algebra sets, relations and maps. Lattices. Boolean algebras. Groups. Basic properties and examples. Subgroups. Basic properties. Lagrangian Theorem. Normal subgroup. Factor–group. Homomorphism of groups. Homomorphism Theorem of groups. Isomorphism Theorems of groups. Inner automorphisms. Cyclical group. Derivation group. V. Daši� , Introduction in general algebra, Podgorica, 2003. (in Serbian)


Main texts:

S. Burris, H. P. Sankappanavar, A Course in Universal Algebra, Berlin, 1978. Further readings: - ability to understand the proofs of theorems, and to construct independently simple proofs of mathematical statement. - capability to apply the methods of algebra in different areas of mathematics; Competences to

be developed:







142

Name of the course: Operating Systems Programs of Studies:

Academic study programs: Mathematics and Computer Sciences, Computer Science


Bachelor level, II year, I semester Number of ECTS credits: 4




Language: Serbian or English Prerequisites: Computers and Programming, Principles of Programming

Aim:

This course is aimed to introduce students to basic concepts of operating systems, their internal structure, ways of realisation, principles and criteria in their design. Students should also learn to use and to programm in several modern operating systems, including system calls and shell programming.

Contents:

Operating systems overview. OS objectives and functions. OS classification. The evolution of OS. Major achievements and design paradigms. Coputer system overview. Process. Process states, description, control. Threads, SMP and microkernels. Scheduling. Mutual exclusion and sychronisation. Deadlock and starvation. Memory management. Paging and segmentation. Virtual memory. I/O management and disk scheduling. File management. File organisation and access. File directiores. File sharing. Security. User interface. Tanenbaum: Modern Operating Systems, Prentice Hall International Stallings: Operating Systems Internals and Design Principles Main texts: Deitel, Deitel, Chofnes: Operating systems Silberchatz, Galvin: Operating Systems Concepts, Willey Further readings: Bic, Shaw: Operating Systems Principles, Prentice Hall - capacity of understanding concepts, structure and mechanisms of operating systems. - ability to recognize fundamental principles in contemporary design issues and current direction in the developement of operating systems - ability to use modern operating systems on advanced level - capability to practice programming using systems calls and shell programming;





Examination: Written exams (3 times in semester), oral exam at the end, problem solving - home tasks, estimation of individual activity on lectures and seminars



143









Aim:


Contents:


Main texts:








144

Name of the course: Discrete Mathematics Programmes of Studies:








Aim:

Course is aimed to introduce students with basic notions of Graph theory and its application. We discuss a number of selected results concerning the coloring of the graphs, Euler’s polyhedral formula, the planar graphs, the famous theorems of Kuratowski, Vizing and Brooks.

Contents:

The basic notions of graph theory. Path, cycles and connectivity. Trees, equivalent definitions of trees. Planar graphs, Kuratowski‘s theorem. Euler’s formula. Coloring maps and graphs: coloring graphs with two colors, Brooks’s theorem, “Five Color Theorem”, the first “proof” of the “Four Color Theorem”. Euler walks and Hamiltonian cycles. Matching in graphs. The “Marriage Theorem”. How to find a perfect matching. Latin squares. S. Cvetkovi�, Graph theory with applications, Belgrade, (in Serbian)

Main texts: R. Wilson, Graphs, Colourings and the Four-Color Theorem, Oxford University Press 2002 L.Lovasz, J.Palikan K. Vesztergombi, Discrete Mathematics, Springer, 2003. Further readings: - capability to translate a mathematical problem to a graph-theoretical one.

- capability to apply the methods of graph theory in different areas of mathematics; - ability to understand the proofs of theorems, and to construct independently simple proofs of mathematical statement.








145

Name of the course: Complex analysis I Programmes of Studies:


Level of the course: Bachelor level, II year, III semester Number of

ECTS credits: 4

Contact hours: 2h Lectures + 2h Exercices, for consultations 1h 20 min per week = 60 contact hours in semester


Structure: 85 h 20 min – lectures and exams, 10 h 40 min - administrative work, 24 h –consultations and individual study.

Language: Serbian or English or Rusian Prerequisites: Basic courses of Analysis I and II

Aim: This course is aimed to introduce students with basic notions in complex analysis and ots applications in mathematical and technical sciences

Contents:

Complex numbers. Field of complex numbers. Complex numbers geometrical interpretation. Modulus and complex conjugations. Metrics on C. Complex number argument and trigonometric form. Broadened complex plane. Riemann’s sphere and metrix on it. Complex number sequence. Numerical series. Connected and compact set. Curve. Contour. Orientation. Complex functions. Elemntary functions. Analytical and coplete functions. Harmonic funcrions. Complex integrations. Complex function integral on the line. Caushy’s theorem. Caushy formula.

Main texts: '. . . �%� , ��!�� Further readings:

- capacity of geometrical understanding of analytical problems.

- ability to understand the proofs of theorems, and to construct independently simple proofs of mathematical statement. - capability to apply the methods of complex analysis in different areas of mathematics;





Examination: Written exams (3 times in semester), problem solving - home tasks, estimation of individual activity on lectures and seminars and oral final examination


146





Contact hours:



16 x 8 = 128

Structure: 42 hours - lectures, 28 hours - seminars, 42 hours individual study and homework ( solving of problems), 16 hours for discussions.

Language: Montenegrin or English Prerequisites: Analysis 3

Aim: Analysis 4 with previous Analysis 3 introduce students to calculus (analysis) in n-dimensional Euclidean space.

Contents:

Jordan measure and Riemann integral in Rn. Lebesgues’ criterion of Riemann integrability. Fubini’s theorem. Change of variables. Differentiable manifolds in Rn (Curves and surfaces in R3). Tangent space (tangent line and tangent plane). Jordan’s measure and Riemann integral on (sub)manifolds of Rn. Orientation, tangent and transferal orientation on hypersurfaces. Integral of a vector field along the oriented curve and through the oriented (hyper)surface. Rotor and divergence f vector fields. Theorems of Green, Stokes and Ostrogradskii. The language of differential forms (correspondence between vector fields in R3 and 1-forms and 2-forms in R3). Exterior differential of forms as a mean to unite the three theorems. Relationship between the conditions A=grad f, rot A=0, div A=0. (Idea of homotopy equivalencies.) M. Perovic, Foundations of Mathematical Analysis I, University Press, Podgorica, 1990 (in Montnegrin) K. Konigsberg, Analysis 2, Springer-Verlag, 1993.

Main texts:

.M. Spivac, Calculus on manifolds, Benjamin Inc. 1965. Further

readings: L. Schwartz, Analyse Mathematique I, Hermann, Paris, 1967. - building analytic formalism for geometric ideas of curves and surfaces,.


- capacity to count Riemann integrals and integrals along curves and surfaces.


- ability to give instructive proofs.






147








Language: Serbian or English or Rusian Prerequisites: Basic course of Algebra 1


Contents:

Symmetrical group. Cayley Theorem. Group of symmetries and isometries. Direct product of groups. Some properties. Ring. Field. Basic properties. Ideal of ring. Factor-ring. Characteristic of ring. Homomorphism of rings. Homomorphism-theorem. Subdirect product of rings. Isomorphism-theorems of rings. Maximal and simple ideals. Field quotients. Polynomial ring. Ring of polynomial functions. Exetension of field (basic concepts). Euclidean ring. V. Daši� , Introduction in general algebra, Podgorica, 2003. (in Serbian)


Main texts:










148

Name of the course: Probability Theory Programmes of Studies:








Aim:

This course is aimed to introduce students with basic concepts of probability and their applications. The course includes rigorous Kolmogorov's definition of probability space and Lebesgues integral concept of expectation. Course includes as well as some classical theorems of Probability Theory: Bayes' theorem, Borel Cantelli's theorems, main theorem of mathematical expectation, central limit theorem, Bernoulli law of large number, Borel law of large number, Kolmogorov law of large number.

Contents:

Random experiment. Events as set. Probability. Conditional probability. Independence. Random variables. Random vectors. Probabilty mass function. Binomial and Poisson ditribution. Distribution function and density function. Examples of continuous distribution: Uniform, exponential, normal, gamma, Cauchy, beta, Weibul, Student, chi square. The joint distribution function. Marginal distributions. The conditional ditribution function. Multivariate normal ditribution. Expectation. Moments. Bianemes equality. Chebyshov's inequality. Functions of random variables. Characteristics function. Laws of large numbers. Central limit theorem. 1. S. Stamatovi�: Vjerovatno�a. Statistika, PMF 2000. 2. B. Stamatovi�, S. Stamatovi�: Zbirka zadataka iz Kombinatorike, Vjerovatno�e i Statistike, PMF 2005. 3. Z. Ivkovi�: Teorija vjerovatno�e sa matemati�kom statistikom, Gra�evinska knjiga, Beograd, 1992.

Main texts:

4. G. Grimett, D. Stirzaker: Probability and Random Processes, Oxford University Press, 2001.

Further readings: W. Feller: An introduction to probability theory and its application, Wiley. - capacity of understanding of probability space model and concept of probability computing. - ability to understand the proofs of theorems. - capability to apply the methods of probability theory in various branches of science.


- ability for solving examples and simple problems. Methods of teaching:





149









Aim:


Contents:


Main texts:









150


Academic study programmes Mathematics, Mathematics and Computer Sciences (A+B)


Bachelor level, II year,IV semester Number of ECTS credits: 10

Contact hours: (4 Lectures +3 Seminar) per week, 15 hours in semester for consultations = 60 contact hours in semester




Aim: In this course students get acquainted with simple differential equations, theorems about existence of solutions and methods of solution. In second part of the course students get to know dynamic systems, phase paths, stability of solutions and position of equilibrium.

Contents:

Simple differential equations of first order: term of solution and Kosi’s thesis, theorem about existence of solution, singular solutions Equations with separate variables, uniform and linear equation. Equations with total differential, integration parameter.Simple differential equations of higher degree (solution, Kosi’s thesis theorems about existence of solution lowering of degree).Linear equation of n-order Method of constant variation. Sturm’s theorems. Normal systems of simple differential equations (solution, Kosi’s thesis ,theorems about existence of solution). Method of elimination. Systems of linear differential equations (method of constant variation , Ojler’s and matrix method). Dynamic systems (integral curves, phase paths, closed phase paths).Stability of solution and position of equilibrium. Linear and quasilinear partial differential equations of first order. Kosi’s thesis. Systems of nonlinear partial differential equations of first order(complete integration, Pfafofa equations, method Lagranza Sarpija). R Š�epanovi�...Diferencijalne jedna�ine, Matemati�ki fakultet, Beograd 2005. god. (in Serbian). A.S.Pontrzgin, Obxknavenie differncilvnxe uravneniq, Nauka, Moskva, 1984 Main texts:

A.F.Filipov, Sbornik zada~ po differncilvnxm uravneniqm, Nauka, Moskva, 2005

Further readings: A.A.Arnolvd, Obxknavenie differncilvnxe uravneniq, Nauka, Moskva, 1984



Examination:

Ways of checking knowledge and giving marks by points: Presence on lectures 5 points. Two tests 25 points each.Doing homework 5 points. Final exam 40 points (oral exam). Passing grade will be given if more then 50 points is scored.



151

Name of the course: Statistics Programmes of Studies:







Language: Serbian or English Prerequisites: Basic courses of Analysis, Linear algebra and Probability Theory.

Aim:

This course is aimed to introduce students with basic concepts of statistics and their applications. The course includes rigorous definitions of statistics experiments and main statistics notions. Course includes as well as some classical theorems of Statistics: Glivenko Cantelli theorem, Neyman-Pearson theorem, Fischer theorem. This course is aimed to introduce students with statistics softwares STATISTICS and SPSS.

Contents:

Sampling and Statistics. Order statistics. Glivenko-Cantelli theorem. Chi square distribution, t-distribution and F-distribution. Fisher theorem. Measures of quality of estimators. A sufficient statistic for a parameter. Properties of sufficient statistic. Rao-Cramer lower bound of efficiency. Maximum likelihood methods. Confidence intervals. Introduction to hypothesis testing. Neyman-Pearson theorem. Most powerful tests. Inferences about normal models. The anlysis of variance. A regression problem. Nonparametric statistics. Statistical softwares STATISTICS and SPSS. 1. S. Stamatovi�: Vjerovatno�a. Statistika, PMF 2000. 2. B. Stamatovi�, S. Stamatovi�: Zbirka zadataka iz Kombinatorike, Vjerovatno�e i Statistike, PMF 2005. 3. Z. Ivkovi�: Teorija vjerovatno�e sa matemati�kom statistikom, Gra�evinska knjiga, Beograd, 1992.

Main texts:


Further readings: W. Feller: An introduction to probability theory and its application, Wiley. - capacity of understanding of statistic experiment, concept of statistic estimation and concept of hypothesis testing. - ability to understand the proofs of theorems. - capability to apply the methods of statistics in various branches of science. Abilty to make a statistics analysis using statistical softwares.








152

Name of the course: Computer Networks Programmes of Studies:

Academic study programme Applied Mathematics and Computer Sciences




Total hours: 4x30 = 120 hours in semester



Aim: Upoznavanje sa hardverskom i softverskom strukturom i osnovnim karakteristikama ra�unarskih mreža i njihovom prakti�nom primjenom.

Contents:

Design and analysis of computer networks. Introduction to local, metropolitan, and wide area networks using the standard OSI reference model as a framework. Introduction to the Internet protocol suite and to network tools and programming. Discussion of various networking technologies. Traffic flow management and error control. Routing algorithms and protocols. Switch and router architectures. Selected issues in high-speed network design. - Alberto Leon-Garcia, Indra Widjaja, - “Communication Networks: Fundamental Concepts and Key Architectures”, McGraw-Hill Companies, Inc., New York, San Francisco, St. Louis, Lisabon, London, Madrid, 2004. - F. Halsall, - “Data Communications, Computer Networks and Open Systems”, Addison-Wesley Publishing Company, New York, Paris, Amsterdam, Sidney …, 1996. - Shay William A., “Savremene komunikacione tehnologije i mreže“, Kompjuter biblioteka, ;a�ak 2004.

Main texts:

- .�. 6��,�, *.+. 6��,�, - “8��3< ��2� �� “, �� , 5�� -�� %�$, 2004.

Further readings: - - Competences to

be developed: -






153

Name of the course: Measure and Integral Programmes of Studies:








Aim:


Contents:


Main texts:









154

PEDJA

Name of the course: Database Systems Programmes of Studies:







Language: Serbian or English Prerequisites: Aim: Contents:

Main texts:

Further readings:







155

Name of the course: Object Oriented Programming Programmes of Studies:

Academic study programme Computer Science



Contact hours: (2 Lectures + 2 Exercises) per week, 12 hours in semester for consultations = 70 contact hours in semester


Structure: 26 hours - lectures, 26 hours - exercises, 6 hours - exams, 12 hours - consultations, 25 hours – homework (individual solving of problems), 25 hours – individual study.

Language: Serbian or English Prerequisites: Basic programming courses

Aim:

After being exposed to structured programming technique and procedural programming languages in previous courses, the aim of this is aimed to introduce students to a new paradigm – object oriented approach and its basic concepts using C++ as a target language. To goal of the course is also to make the students able to successfully apply these concepts in practical solving of typical programming problems.

Contents:

Introduction. About object-oriented programming paradigm. About C++ language. Example - the big picture. Language elements inherited from C. Language elements specific to C++. Functions, references, overloading. Abstraction and encapsulation. Classes and objects. Static members and functions. Access control rights. Friend functions and classes. Constructors and destructors. Inheritance. Static and dynamic binding. Polymorphism and virtual functions. Abstract classes. Multiple inheritance. Operator overloading mechanism. Overloading of some special operators. Template mechanism. Exception handling. Input-output functions. Standard library. Use of name spaces. Practical programming in Dev C++ an d Visual C++ development environments. 1. D. Mili�ev: Objektno orijentisano programiranje na jeziku C++, Mikroknjiga, Beograd 1995. 2. D. Mili�ev: Praktikum iz objektno orijentisanog programiranja, Mikroknjiga, Beograd. 3. L. Kraus: Programski jezik C++ sa rešenim zadacima, Akademska misao, Beograd, 2000.

Main texts:

Further readings: B. Eckel: Thinking in C++, Prentice Hall, 1995.

- accepting and deeper understanding of the object-oriented programming paradigm. - knowledge of C++ and appropriate development tools/ Competences to

be developed: - capability to solve practical problems using object-oriented approach..


Lectures and exercises with the active participation of students, homework assignments, group and individual consultations.

Examination: Two midterm exams and the final exam, homework grading, estimation of individual activity on lectures and exercises


Student pools, results of exams, direct communication with the students.

156

SAVO!!!!

Name of the course: English for Mathematics III Programmes of Studies:



ECTS credits: 2







Main texts:









157

Name of the course: Introduction to Differential Geometry Programmes of Studies:








Aim:

This course is aimed to introduce students with basic notions of Differential geometry in Euclidean space. It relates to application of differential calculus in studying geometry of curves and surfaces in Euclidean space. The course includes some classical statements based on the notions of curvature and torsion for curves and those one for the surfaces based on the existence of the first and second fundamental form. All statements are followed with geometric interpretation and application through examples.

Contents:

Local theory of curves. Natural parameterization. Curvature and torsion. Frenet-Serett formulas. The fundamental theorem for curves. Elementary surfaces, tangent vectors and tangent space. The first and second fundamental form. Weingarten map. Curves on surfaces. Normal and main curvatures. Gauss curvature. Geometric interpretation. Gauss and Codazzi equations. Egregium theorem. Geodesic curvature and geodesics lines. Uniqueness and minimalism of geodesics lines. Parallel vector fields and parallel transportation. Lining and rotating surfaces. The fundamental theorem for surfaces. N. Blazic, N. Bokan, An Introduction to Differential Geometry , Belgrade, 1996. (in Serbian) V. Dragovic, D. Milinkovic, Analysis on manifolds, Belgrade, 2003. (in Serbian)

Main texts:

S. Mischenko, A. T. Fomenko, Course of differential geometry and topology, Moscow, 2004. (in Russian) Further readings: A. Presley, Elementary differential geometry, Sprienger, 2002 - capacity of visualization curves and surfaces in Euclidean space resulting in individual and creative thinking - capacity for understanding more complex geometrical features - understanding the differential calculus as powerful geometric tool


- capacity for application the techniques of differential calculus and linear algebra in investigation the shapes of curves and surfaces



Examination: Two written exams, homework’s, activity on lectures and seminars, oral exam Methods of self-evaluation:


158

Name of the course: Functional Analysis Programs of Studies:








Aim:

This course is aimed to introduce students with basic notions of normed spaces bounded linear operators and linear functional. The course include some classical theorems of Functional Analysis: The uniform boundness principle, Hanh-Banach theorem, Bair’s category theorem, open mapping and closed graph theorem.

Contents:

Metric and normed space. Completeness. Finite dimensional space. Linear continuous operators. Norm of the linear operator. Completely continuous operator. Linear functional. Examples. Hanh-Banach theorem – analytical formulation and proof. Geometric formulation and proof. of Hanh-Banach theorem. Applications of Hanh-Banach theorem. Representation of linear functional in some normed space. Dual space of normed space. Weak topology in normed space. Adjoint and self-adjoint operators. Uniform boundness principle. Bair’s theorem and consequence. Open mapping and closed graph theorem. Hilbert space. Riesz representation theorem. Geometry of Hilbert space. Orthonormal basis of Hilbert space. Fourier series. Spectra of linear operator. S. Aljan�i�, Introduction in real and functional analysis, Belgrade, 1972. (in Serbian) A. Conway, A Course in Functional Analysis, Springer Verlag. 1985. Main texts:

W. Rudin, Real and complex analysis, McGraw-Hill, 1966. Further readings: L.V. Kantorivich, G.P. Akilov, Functional analysis, Pergamon Press, 1982. - capacity of geometrical understanding of analytical problems.

- ability to understand the proofs of theorems, and to construct independently simple proofs of mathematical statement. - capability to apply the methods of functional analysis in different areas of mathematics;








159

Name of the course: Numerical Analysis Programmes of Studies:






Structure: 26 hours - lectures, 26 hours - seminars, 8 hours - exams, 15 hours - consultations, 15 hours - homework (individual solving of problems), 30 hours - individual study.


Aim:

To give an introduction to commonly used numerical methods. To demonstrate the application of numerical methods to calculations encountered in lecture courses and projects. To enable students to use numerical techniques to tackle problems that are not analytically soluble.

Contents:

Interpolation and similar matters: Curve fitting - Least squares approximations, Lagrange polynomial interpolation, Finite differences, Newton interpolation polynomials, Hermite interpolation, Cubic spline interpolation, Numerical differentiation. Numerical integration: Three formulas - rectangle, trapezium and Simpson, Runge's rule for practical error estimation (Richardson), Integral with weight function, Gaussian quadrature. Numerical methods of algebra: Gaussian elimination with pivoting, Matrix condition number, Simple iteration method for solving systems of linear equations, Jacobi and Gauss-Seidel methods, Relaxation method, Power method for matrix eigenvalues. Systems of nonlinear equations: Simple iteration method, Newton's method, Bisection and Secant methods (n=1). Numerical methods for ordinary differential equations: Euler method, Runge-Kutta methods, Adams' predictor-corrector method, Milne's method, Finite difference method for solving boundary value problem. Boško Jovanovi�, Desanka Radunovi�, Numeri�ka analiza, Beograd, 1993 (in Serbian) M. Martinovi�, R. Š�epanovi�, Numeri�ke metode, Nikši�, 1995 (in Serbian) Main texts:

C. Gerald, P. Wheatley, Applied Numerical Analysis, Addison-Wesley, 1997 N.S. Bahvalov, Numerical methods, FML, 2001 (in Russian) Further readings: G.W. Collins, Fundamental Numerical Methods and Data Analysis, 2003 - to learn some (elementary) classes of numerical methods,

- to understand approximate solution and error estimation concepts, - capacity to solve problems (exercises) concerning the topics covered,


- capacity to apply the numerical techniques by means of writing programs Methods of teaching:





160

Name of the course: Compilers Programs of Studies:

Academic study programs: Computer Sciences, Mathematics and Computer Sciences


Bachelor level, III year, II semester Number of ECTS credits: 5



Structure: 26 hours - lectures, 26 hours - exercises, 6 hours - exams, 15 hours - consultations, 25 hours – homeworks (individual solving of problems), 26 hours – individual project, 25 hours – individual study.

Language: Serbian or English or Russian Prerequisites: Programming I, Programming II, advisable: Computer Architecture

Aim:

This course is an introduction to concepts and methods in compiler construction. The aims of course are:

- to describe the main concepts of compiler construction - to develop students' skills in building relatively complex software

project - to deepen understanding of programming languages, their semantics

and their applications.

Contents:

Introduction to compilation. Compilers and interpreters. Lexical analysis. Automata. Regular expressions. Flex. Grammars and languages. Top-down parsing. Bottom-up parsing.. LR(0), LR(1) and SLR(1). LALR. Syntax-directed translations. Semantical analysis. Type checking. Object-oriented and functional languages. Runtime behaviour. TAC. Code genaration. Program analysis. Introduction to dataflow analysis Code optimization. Loop optimiyation. Register allocation. .

Main texts: Cooper, Torczon – Engineering a Compiler, Morgan Kaufmann, 2003. (in English) Appel – Modern Compiler Implementation in Java (2nd edition), Cambridge University Press, 2002. (in English) Further readings: Aho, Sethi, Ullman – Compilers: Principles, Techniques and Tools, 2nd Edition, Addison Wesley, 2007 (in English) - be able to evaluate the suitability of different programming languages and compilers AI for various kinds of applications. - understanding basic data structures used for representing learned concepts, and the associated processing algorithms - be able to use these structures and algorithms in constructing simple applications which will recognise and evolve existing concepts


- capability to apply the compiler construction methods and algorithms in real life applications


Lectures and exercises with the active participation of students, individual homeworks, individual and group projects, group and individual consultations.

Examination: Midterm and final written exam, problem solving –programming and written homeworks, project presentation, estimation of student parcitipation on lectures and exercises



161

Name of the course: Visualization and Computer Graphics Programmes of Studies:







Language: Serbian or English Prerequisites: Basic courses of Algorithms and Mathematics

Aim:

This course is aimed to introduce students with basic concepts of visualization and Computer Graphics. Students shall learn how to write computer graphics applications in OpenGL. Course will cover graphics processor unit (GPU) programming using the OpenGL Shading Language (GLSL). GPUs are widely used today to drive cutting-edge 3D game engines, virtual reality simulations, and film pre-production. GPUs have changed computer graphics and how we teach it. Students will learn how to program GPUs to perform a variety of tasks, such as 3D lighting, animation, image processing, and special effects.

Contents:

Fundamentals of Computer Graphics. Mathematcal background of Computer Graphics. Introduction to 2D graphics. 2D Transfromations. Anti-aliasing. Introductions to 3D Graphics. 3D Transformations. Fractals. Cliping. Shading. The z-Buffer. Texture Mapping. Environment mapping. Introduction to Ray Tracing. Ray Tracing. Methods. Advance topics in Computer Graphics: 3D Face Modeling. 1. E.V. Šikin, A. V. Boreskov – "Computer Graphic", Dialog MIFI,2000. 2. OpenGL Programming Guide. (official OpenGL red-book). 3. OpenGL SuperBible. Wright, R.S., Sweet M. Waite Group Press, 2000

Main texts:

4. DirectX 3D Programming Bible, J.Sanchez, M. Canton , IDG 2000. Further readings: W. Feller: An introduction to probability theory and its application, Wiley.

- capacity of understanding of Visualization and Computer Graphics. Competences to be developed: - capacity to program GPUs to perform a variety of tasks, such as 3D lighting,

animation, image processing, and special effects.. Methods of teaching:





162

Name of the course: Internet Technologies Programs of Studies:




Contact hours:

(2 Lectures + 1 tutorial exercise) per week, 15 hours in semester for consultations = 60 contact hours in semester

Total hours:


Structure: 30 hours - lectures, 15 hours - exercises, 15 hours - consultations, 15 hours – homeworks (individual solving of problems), 30 hours – individual project, 15 hours – individual study.

Language: Serbian or English or Russian Prerequisites: Programming I, Programming II

Aim:

This course is an introduction to client-side and server-side technologies for designing and building web-based applications. The aims of course are:

- to describe the main concepts of web-based applications - to develop students' skills in building relatively complex software

project - to deepen understanding of various programming techniques

especially in context of web-applications - to gain first-hand experience of the problems of user- centred design

and visual layout

Contents:

Introduction to computer netorks. TCP-IP protocol. Web browser. Proxies. Intro to XHTML. Block–level and text-level elements. Tables. Lists. Images. CSS. HTTP. XHTML forms. JavaScript variables. Statements. Functions. DOM. Installing PHP. PHP variables and statements. PHP functions. PHP objects. Working with files. Database access. Session management in PHP. PHP security. Introduction to XML. Transforming XML using XSLT. Content management systems. Introduction to AJAX.

Main text: David Sklar – Learning PHP5, O'Reilly, 2004 (in English)

Dave Taylor - Creating Cool Web Sites with HTML, XHTML and CSS, Wiley, 2004 (in English) Further readings: Ellie Quigley – JavaScript by Example Prentice Hall, 2003 (in English) - capability to evaluate the suitability of different programming techiniques and tools for various kinds of web applications. - appreciation of the underlying principles of human-computer interaction and ergonomics; - understanding basic data structures used for representing learned concepts, and the associated processing algorithms


- ability to use these structures and algorithms in constructing applications which will recognise and evolve existing concepts



Examination: Problem solving –programming homeworks, project presentation, quizes, estimation of student parcitipation on lectures and exercises



163

SAVO!!!!

Name of the course: English for Mathematics IV Programmes of Studies:



ECTS credits: 2







Main texts:









164

VI


COMPUTER SCIENCE

166

Programme of study Level of Studies Academic/Applied Computer Science Undergraduate Academic

I year

Course

Mandatory

Elective

Winter Semester

hours weekly

EC

TS

Ljetnji semestar

hours weekly

EC

TS

1. Introduction to Computer Science X 2+3 6 2. Computers and Programming X 3+3 6 3. Analysis 1 X 3+2 6 4. Analitical Geometry X 2+2 5 5. Introduction to Mathematical Logic X 2+2 5 6. English for Computer Science 1 X 2+0 2 7. Principles of Programming X 3+2 6 8. Data Structures X 3+3 6 9. Analysis 2 X 3+2 6 10. Linear algebra X 2+2 5 11. Algebra X 2+2 5 12. English for Computer Science 2 X 2+0 2

Total 11 14+12 30 15+11 30 II year 1. Programming 1 X 3+2 5 2. Discrete Mathematics 1 X 3+0 4 3. Computer networks X 3+2 6 4. Operating systems X 3+2 6 5. Analysis 3 X 3+2 5 6. Object oriented programming X 2+1 4 7. Programming 2 X 3+2 5 8. Visualization and Computer

Graphics X 2+1 4

9. Peripheral Devices and Interfaces X 2+1 4 10. Probability Theory and Statistics X 3+2 5 11. Differential Equations X 2+2 4 12. Discrete Mathematics 2 X 3+0 4 13. Computer System Architecture X 2+1 4

Total 13 17+9 30 17+9 30 III year 1. Database Systems X 3+2 6 2. Programing Languages X 2+2 5 3. Numerical Analisys X 2+2 4 4. Computer Security X 2+0 4 5. Artificial Inteligence X 3+2 5 6. Introduction to Information Systems X 2+2 4 7. English for Computer Science 3 X 2+0 2 8. Softver Engineering X 3+2 5 9. Advanced Programming Technics X 2+2 5 10. Compilers X 2+2 5 11. Internet tehnologies X 2+1 4 12. Distributed Systems X 2+1 4 13. Advanced Database Systems X 3+2 5 14. English for Computer Science 4 X 2+0 2

Total 14 2 16+10 30 16+10 30

167

Name of the course: Introduction to Computer Science Programs of Studies:

Academic study program: Computer Sciences






Language: Serbian or English or Russian Prerequisites: There is no prerequisites for this course.

Aim:

This course is an introduction to main concepts of computer science including computer architectures, data organization, programmig languages, operating systems, databases and computer networks. The aims of course are:

- to give overview of main computer science areas - to develop students' computer skills including using various operating

systems, text processing, using spreadsheets and databases, communicating using e-mail and Internet

Contents:

Introduction. to computer architecture. Software and hardware. Programming paradigms. History of programming languages. Traditional programming concepts. Procedural units. Object-oriented programming. Declarative programming. Language implementation.. Programming concurrent activities. History of operating systems. Operating system architecture. Coordinating the machine’s activities. Handling competition among processes. Security. Computer networks' fundamentals. Internet and WWW. Internet protocols. XHTML. XML. Database systems. Relational data model. Introduction to SQL. DBMS. Introduction to software engineering. The Software Life Cycle. Modularity. Tools. Testing. Documenation.

Main texts: J.G. Brookshear - Computer Science: An Overview, 8th Edition, Addison Wesley, 2006. (in English)

Further readings: M. Martinovi�, P. Staniši� - Principles of Programming, University of Montenegro, Podgorica, 2004 (in Serbian) - understanding the role of computer science in modern world .

- capability to use various operating systems and Internet services as well as hardware platforms - ability to use text processing, spreadsheet and presentation software


- capability to communicate using computer


Lectures and exercises with the active participation of students, individual homeworks, individual essays and presentations, group and individual consultations.

Examination: Midterm and final written exam, written tasks and homeworks, project presentation, estimation of student parcitipation on lectures and exercises



168









Aim:


Contents:


Main texts:

M. Mano, Computer System Architecture, Prentice Hall, 1982 Further readings: Pascal user guide (from Internet) - learning of data representation and hardware components, - knowledge of Boolean functions theory and problem's solving, - capacity of individual working on computer, to process Pascal programs,


- capability to write elementary Pascal programs






169









Aim:


Contents:











170


Academic study programs: Computer Sciences







Aim:

This course is aimed to introduce students with fundamental elements of theory of matrices , determinants and solving systems of linear equalities. Students are also introducing with vector calculus and equality of line , plain and theory of curves and surfaces of second order.This course is fundamental for understanding of Linear algebra , Mathematical Analysis and other modern Math courses.

Contents:

Matrices and operations on matrices.Determinant, definition and examples.Properties of determinants. The inverse matrix.Elementary transformations of matrices.Rank of matrix.Systems of linear equalities. Kronecker-Kapelli theorem.Operations with vectors.Vector coordinates,Dot coordinates,Scalar,vector and scalar triple product of vectors Equality of line and plain. Curves of second order.Surfaces,.basics terms and examples.Curves of second order. Lectures of professor (skript in serbian) I. Krnic, �. Ja�imovi�, Linear Algebra – theorems and problems, skript, Podgorica, 2001. (in serbian) A. Lipkovski, Linear algebra and analytical geometry, Nau�na knjiga, Belgrade, 1995. (in Serbian)

Main texts:

A. M. Postnikov, Lecture in geometrz, semester I, Analztic Geometry, Mir, Moscow, 1982. Further readings: - introducing to basic concepts in Analitic geometry - capacity of analytical interpretation of geometrical problems. - ability in routine solving of some simple problems from Analitic geometry. - capability to apply the methods of coordinates in different areas of mathematics and its applications;


- developing of ability for analytical thinking and capacity to precise formulating and proving of theorems.






171

Name of the course: Introduction to Mathematical Logic Program of Studies:

Academic study program Computer Sciences







Aim:

Course is aimed to introduce students with the foundations of logic and mathematics and answers questions: what is the proof, why are there no recipes to design proofs, where do numerous mathematical rules come from, to distinguish semantic reasoning ("what is true?") from syntactic reasoning ("what can be shown?"). The first leads to Model Theory, the second, to Proof Theory. The student gets used to mathematical formalism and abstraction, and learns the way mathematicians present it.

Contents:

Truth tables. Tautologies. An axiom system for the propositional calculus. Quantifiers. First-order languages and their interpretations. Satisfiability and truth. Models. First order theories. Completeness theorems. Prenex normal form. S. Vujoševi�, Ž. Kovijani� Vuki�evi�-lecturenotes Main texts: E.Mendelson, Introduction to Mathematical Logic, 1987. Further readings: - capability to apply the results of mathematical logic in different areas of mathematics; - ability to understand the proofs of theorems, and to construct independently simple proofs of mathematical statement. - ability for analytical thinking and capacity to argue the own opinion and statements.


-ability to present mathematical formalism and abstraction,






172

Name of the course: English for Computer Sciences I Programmes of Studies: Academic study program Computer Science


ECTS credits: 2





Aim: The introduction into the basics of English for Computer Sciences. The understanding of the importance of spoken and written English in both everyday life and in information technology.

Contents:

Personal computing; grammar: Present Tenses. Portable computers; grammar: Present vs. Past tenses. Operating systems; grammar: Present vs. Past tenses 2. The processor; grammar: Present Tenses 2. Online services; grammar: articles. Analogue transmission; grammar: adjectives. Digital transmission; grammar: comparison of adjectives. Programming and languages; grammar: Present simple vs. present continuous. Programming 2; grammar: Conditional sentences type 1. Computer software; grammar: Conditional sentences type 2. Computer software 2; grammar: Conditional sentences type 3. Word processors; grammar: Mixed conditionals. Word processors 2; grammar: revision. Eric H. Glendenning, John McEwan Basic English for Computing

Main texts:

Keith Boeckner, P. Charles Brown Computing Further readings: - reading skills which include comprehension of a given text - writing skills, which include composition of short essays





Examination: Written exams (2 times in semester), project assesment , estimation of individual activity on lectures , oral final examination



173

Name of the course: Principles of programming Programmes of Studies:

Academic study programme, Computer Science





Structure: 39 hours - lectures, 39 hours - seminars, 6 hours - exams, 36hours - consultations, 40 hours – homework (individual solving of problems), 50 hours – individual study.

Language: Serbian or English Prerequisites: Basic courses of Logic

Aim:

This course stresses the important concepts of computer organization. It covers aspects of computer organization seen as a hierarchy of levels. These include the digital logic level, assembly language level and the conventional machine level. Also, the course introduces students with basic data structures and principles of working with files in Pascal.

Contents:

Minimization of Boolean function. Karnaugh maps. Logic gates. Examples of circuits (adder, decoder, multiplexer, MS, F, JK, T - flip flops). Registers. Counters. Overview of computer system organization. Processor organization and performance. Instruction set design and addressing modes. Pentium assembly language. Interrupts. Arrays. Sorting algorithms. RECORD, SET, clause WITH. Complex data types. Type FILE. Dynamic types. Recursion. (Pascal) M. Martinovi�, P. Staniši�: Principi programiranja, Univerzitet Crne Gore.

Main texts: G. Schneider, S. Bruell - "Advanced Programming and Problem Solving with Pascal", John Wiley & Sons

Further readings: S. Dandamundi, Introduction to assembly language programming, Springer-Verlag

-programming -motivation to learn assembly language -understanding computer organization


- algorithmic thinking Methods of teaching:





174

Name of the course: Data Structures Programmes of Studies:




Contact hours: (3 Lectures + 3 Practice) per week, 18 hours in semester for consultations = 70 contact hours in semester


Structure: 39 hours - lectures, 39 hours - practice, 9 hours - exams, 18 hours - consultations, 32 hours – homework (individual solving of problems), 33 hours – individual study.

Language: Serbian Prerequisites: No prerequisities

Aim:

This course is aimed to introduce students with basic concepts of data structures which they will need for studying other subjects. Students will introduce memory representation of data structures, their implementation and their application in creating efficient algorithms.

Contents:

Mathematical fundamentals. Algorithm analysis – complexity. Definition, classification, memory representation of data structures. Linear data structures. Arrays. Linked lists. Stack. Queue. Recursive programmes. Graph definition and terminology. Trees. Binary search trees. AVL-trees. Priority queues. Classical methods of sorting. Heapsort. Quicksort. Mergesort. B-trees. Practices will contain implementation of mentioned data structures in programming language Pascal. 1. Milo V. Tomaševi�, STRUKTURE PODATAKA. Elektrotehni�ki fakultet Univerziteta u Beogradu. Beograd, 2000. 2. Nenad Miti�,Saša Malkov,Vladimir Niki�, Osnovi programiranja:zbirka zadataka. Matemati�ki fakultet. Beograd, 2000. 3. Samanta,”Classic Data Structures”, 1e, 2001,PHI.

Main texts:

Further readings: Trembley, Sorenson,” An introduction to Data Structure with Applications”,2e, TMH. - capacity of understanding and implementation of basic data structures

Competences to be developed: - capability to apply data structures in creating of more complex algorithms


Lectures and practices with the active participation of students, individual home tasks, group and individual consultations.

Examination: Written exams (3 times in semester), problem solving - home tasks, estimation of individual activity on lectures and practices



175









Aim:


Contents:












176

Name of the course: Linear Algebra Programs of Studies:






Structure: 26 hours - lectures, 26 hours - seminars, 6 hours – exams, 25 hours - consultations, 22 hours – homework (individual solving of problems),45 hours – individual study.


Aim: This course is aimed to introduce students with elements of linear algebra such as spaces , linear operators ,quadratic forms , curves and surfaces.Thesee terms ase basic for understanding of modern math ,physics and computer sciences.

Contents:

Vector space.Subspace.Linear indipendance of vectors.Base and dimension.Matrix of transformation from one base to another.Euclidean spaces.Orthogonal system of vectors.Orthogonal complement of space.Linear operators.Rank and defect of operator,Operations on operators.Matrix of linear operator in different bases. Eigenvalues and eigenvectors of linear operator.Conugate operator.Symetric operator.Quadratic forms.Canonical form.Identifications of surfaces of second order. Lectures of professor( script in serbian) A. Lipkovski, Linear algebra and analytical geometry, Nau�na knjiga, Belgrade, 1995. (in serbian) I. Krnic, �. Ja�imovi�, Linear Algebra – theorems and problems, skript, Podgorica, 2001. (in serbian)

Main texts:

E. Shikin, Linear spaces and mappings, Moscow (in russian) Further readings: G. Strang, Linear Algebra and its Applications, Brooks Cole, 2005. - introducing to the fundamental concepts of higher mathematics.

- preparation for modern math theories



- developing ability of routine solving problems in Linear algebra. - clear understanding of precise formulation,proof ,etc.






177

Name of the course: Algebra Programmes of Studies:

Academic study program Computer Sciences






Language: Serbian or English or Rusian Prerequisites: Basic mathematics concepts from secondary school


Contents:

Basic algebraic structures. Algebra of natural numbers, relations, sets. Lattices. Semigroups Groups. Subgroups. Cyclical group.. Normal subgroups and homomorphisms of groups. Factor–group. Direct products of groups. Symmetrical group. Rings. Ideals. Factor- rings. Homomorphisms of rings. Direct product of rings. Polynomial rings. Polynomial factorization. V. Daši� , Introduction in general algebra, Podgorica, 2003. (in Serbian)

B. Zekovi�, V. A. Artamonov Collection of resolute tasks from algebra (I part), Podgorica, 2003. (in Serbian) Z. Stojakovi�, �. Pauni�, Tasks from algebra, Novi Sad, 1998. (in Serbian)

Main texts:










178

Name of the course: English for Computer Sciences II Programmes of Studies: Academic study program Computer Science


ECTS credits: 2





Aim: Further and more detailed insight into English for Computer Sciences with more thorough language and lexical content.

Contents:

The Internet, email and newsgroups; grammar: Past simple vs Past continuous. The Internet 2: World Wide Web; grammar: -ing forms and infinitives. Website Designer; grammar: modal verbs must and have to. Word processing; grammar: Present perfect passive. Data bases and spreadsheets; grammar: conditional sentences. Graphics and multimedia; grammar: Time clauses. Programming; grammar: prepositions. Analyst / programmer; grammar: Present simple vs. present continuous. Languages; grammar: Reported speech. Low level systems; grammar: clauses of contrast. Future trends; grammar: Making predictions. Future trends 2; grammar: will and would. IT Manager; grammar: certainty. Eric H. Glendenning, John McEwan Basic English for Computing(units 15-30)

Main texts:

Keith Boeckner, P. Charles Brown Computing Further readings: The further development of 4 main language skills The development of presentation techniques Composition writing







179









Aim:


Contents:


Main texts:








180

Name of the course: Discrete mathematics I Programmes of Studies:

Academic study programme, Mathematics and Computer Sciences






Language: Serbian or English Prerequisites: No

Aim:

This course is aimed to introduce students an important problem-solving skill - ability to count or enumerate objects. The course includes some basic techniques of counting. The stress is on performing combinatorial analysis to solve counting problems, not on applying formulae.

Contents:

Sets. Set operations. Basic counting principles (product rule, sum rule). Combinations. Permutations. Combinations with repetition. Permutations with repetition. Pigeonhole principle. Ramsey’s theorem. Inclusion-exclusion principle. Recurrence relations. Generating function. B. Stamatovi�, S. Stamatovi�: Zbirka zadataka iz Kombinatorike, Vjerovatno�e i Statistike, Prirodno-matemati�ki fakultet, Podgorica Main texts: D. Veljan: Kombinatorika i Teorija grafova, Školska knjiga Zagreb

Further readings: Kenneth H. Rosen, Discrete mathematics and its applications, Mc Graw Hill

- mathematical foundations for students of computer science

- ability to solve problems on different ways and to think mathematically Competences to be developed:

- ability to recognize and solve real-world problems






181

Name of the course: Computer Networks Programmes of Studies:

Academic study programme Applied Mathematics and Computer Sciences







Aim: Upoznavanje sa hardverskom i softverskom strukturom i osnovnim karakteristikama ra�unarskih mreža i njihovom prakti�nom primjenom.

Contents:

Design and analysis of computer networks. Introduction to local, metropolitan, and wide area networks using the standard OSI reference model as a framework. Introduction to the Internet protocol suite and to network tools and programming. Discussion of various networking technologies. Traffic flow management and error control. Routing algorithms and protocols. Switch and router architectures. Selected issues in high-speed network design. - Alberto Leon-Garcia, Indra Widjaja, - “Communication Networks: Fundamental Concepts and Key Architectures”, McGraw-Hill Companies, Inc., New York, San Francisco, St. Louis, Lisabon, London, Madrid, 2004. - F. Halsall, - “Data Communications, Computer Networks and Open Systems”, Addison-Wesley Publishing Company, New York, Paris, Amsterdam, Sidney …, 1996. - Shay William A., “Savremene komunikacione tehnologije i mreže“, Kompjuter biblioteka, ;a�ak 2004.

Main texts:

- .�. 6��,�, *.+. 6��,�, - “8��3< ��2� �� “, �� , 5�� -�� %�$, 2004.


be developed: -






182

Name of the course: Operating Systems Programs of Studies:

Academic study programs: Mathematics and Computer Sciences, Computer Science


Bachelor level, II year, I semester Number of ECTS credits: 4




Language: Serbian or English Prerequisites: Computers and Programming, Principles of Programming

Aim:

This course is aimed to introduce students to basic concepts of operating systems, their internal structure, ways of realisation, principles and criteria in their design. Students should also learn to use and to programm in several modern operating systems, including system calls and shell programming.

Contents:

Operating systems overview. OS objectives and functions. OS classification. The evolution of OS. Major achievements and design paradigms. Coputer system overview. Process. Process states, description, control. Threads, SMP and microkernels. Scheduling. Mutual exclusion and sychronisation. Deadlock and starvation. Memory management. Paging and segmentation. Virtual memory. I/O management and disk scheduling. File management. File organisation and access. File directiores. File sharing. Security. User interface. Tanenbaum: Modern Operating Systems, Prentice Hall International Stallings: Operating Systems Internals and Design Principles Main texts: Deitel, Deitel, Chofnes: Operating systems Silberchatz, Galvin: Operating Systems Concepts, Willey Further readings: Bic, Shaw: Operating Systems Principles, Prentice Hall - capacity of understanding concepts, structure and mechanisms of operating systems. - ability to recognize fundamental principles in contemporary design issues and current direction in the developement of operating systems - ability to use modern operating systems on advanced level - capability to practice programming using systems calls and shell programming;





Examination: Written exams (3 times in semester), oral exam at the end, problem solving - home tasks, estimation of individual activity on lectures and seminars



183

Name of the course: Analysis III Programmes of Studies:



Bachelor level, II year, III semester

Number of ECTS credits: 5

Contact hours:

(3 Lectures + 2 Seminar) per week, 15 hours in semester for consultations = 86 contact hours in semester

Total hours:



Language: Serbian or English Prerequisites: Analysis II and Linear algebra

Aim:

This course is aimed to introduce students with basic notions of Euclidean space, continuation, differentiation, integration of the mappings defined on the Euclidean space and an introduction to the theory of analytic functions. The course include some basic notation of Jordan measure, of Riemann integral and of one and two dimensional manifolds

Contents:

Topology of Euclidean space. Continuation of mappings of two and more variables. Differentiation of mappings of two and more variables. Taylor formula, local minimum and local maximum. Conditional extremum. Jordan measure. Riemann integral. Fubini's theorem. Change of variables theorem. Line integrals. Surface integrals. Complex plane. Definition of analytic functions. Caushy theorem and Caushy formula. Taylor formula. �. Dolji�anin, M. Perovi�, J. Solovev, Matematics III (Part II), Zavod za udzbenike i nastavna sredstva: Kosova, Priština, (Serbian). Main texts: M. Perovi�, Basics of mathematical analysis-Part I and Part II Univerzitetska rije�, Nikši�, 1991. (Serbian) K. Hoffman, Analysis in Euclidean Space, Prentice-Hall, Englewood Cliffs, New Jersey, 1975. Further

readings: V. Zorich, Mathematical analysis, Part II, Nauka, Moskva, 1984 (Russian). - capacity of geometrical understanding of analytical problems.

- ability to understand the proofs of theorems, and to construct independently simple proofs of mathematical statement. Competences to

be developed: - Facility with abstraction including the logical development of formal theories and the relationships between them






184

Name of the course: Object Oriented Programming Programmes of Studies:







Language: Serbian or English Prerequisites: Basic programming courses

Aim:

After being exposed to structured programming technique and procedural programming languages in previous courses, the aim of this is aimed to introduce students to a new paradigm – object oriented approach and its basic concepts using C++ as a target language. To goal of the course is also to make the students able to successfully apply these concepts in practical solving of typical programming problems.

Contents:

Introduction. About object-oriented programming paradigm. About C++ language. Example - the big picture. Language elements inherited from C. Language elements specific to C++. Functions, references, overloading. Abstraction and encapsulation. Classes and objects. Static members and functions. Access control rights. Friend functions and classes. Constructors and destructors. Inheritance. Static and dynamic binding. Polymorphism and virtual functions. Abstract classes. Multiple inheritance. Operator overloading mechanism. Overloading of some special operators. Template mechanism. Exception handling. Input-output functions. Standard library. Use of name spaces. Practical programming in Dev C++ an d Visual C++ development environments. 1. D. Mili�ev: Objektno orijentisano programiranje na jeziku C++, Mikroknjiga, Beograd 1995. 2. D. Mili�ev: Praktikum iz objektno orijentisanog programiranja, Mikroknjiga, Beograd. 3. L. Kraus: Programski jezik C++ sa rešenim zadacima, Akademska misao, Beograd, 2000.

Main texts:

Further readings: B. Eckel: Thinking in C++, Prentice Hall, 1995.

- accepting and deeper understanding of the object-oriented programming paradigm. - knowledge of C++ and appropriate development tools/ Competences to

be developed: - capability to solve practical problems using object-oriented approach..



Examination: Two midterm exams and the final exam, homework grading, estimation of individual activity on lectures and exercises



185









Aim:


Contents:


Main texts:









186

Name of the course: Visualization and Computer Graphics Programmes of Studies:







Language: Serbian or English Prerequisites: Basic courses of Algorithms and Mathematics

Aim:

This course is aimed to introduce students with basic concepts of visualization and Computer Graphics. Students shall learn how to write computer graphics applications in OpenGL. Course will cover graphics processor unit (GPU) programming using the OpenGL Shading Language (GLSL). GPUs are widely used today to drive cutting-edge 3D game engines, virtual reality simulations, and film pre-production. GPUs have changed computer graphics and how we teach it. Students will learn how to program GPUs to perform a variety of tasks, such as 3D lighting, animation, image processing, and special effects.

Contents:

Fundamentals of Computer Graphics. Mathematcal background of Computer Graphics. Introduction to 2D graphics. 2D Transfromations. Anti-aliasing. Introductions to 3D Graphics. 3D Transformations. Fractals. Cliping. Shading. The z-Buffer. Texture Mapping. Environment mapping. Introduction to Ray Tracing. Ray Tracing. Methods. Advance topics in Computer Graphics: 3D Face Modeling. 1. E.V. Šikin, A. V. Boreskov – "Computer Graphic", Dialog MIFI,2000. 2. OpenGL Programming Guide. (official OpenGL red-book). 3. OpenGL SuperBible. Wright, R.S., Sweet M. Waite Group Press, 2000

Main texts:

4. DirectX 3D Programming Bible, J.Sanchez, M. Canton , IDG 2000. Further readings: W. Feller: An introduction to probability theory and its application, Wiley.

- capacity of understanding of Visualization and Computer Graphics. Competences to be developed: - capacity to program GPUs to perform a variety of tasks, such as 3D lighting,

animation, image processing, and special effects.. Methods of teaching:





187

Name of the course: Peripheral Devices and Interfaces Programmes of Studies:

Academic study programme Computer Sciences







Aim: Prakti�no upoznavanje sa hardverskom i softverskom strukturom klase personalnih ra�unara. Ovladavanje osnovama asemblerskog jezika za mikroprocesore Intelove familije x86.

Contents:

Uvod. Osnovni pojmovi o ra�unarskim periferijama i interfejsima. I-P-O model ra�unarskog sistema. Arhitektura PC ra�unara IBM kompatibilnih. Osnovne komponente personalnih ra�unara i njihove osnovne funkcije. Na�in funkcionisanja i princip rada personalnog ra�unara. Prekidi, obrada prekida i vektori prekida. Softversko upravljanje perifernim ure�ajima. Funkcije BIOS-a i operativnog sistema. Funkcionalne karakteristike mikroprocesora. Arhitektura mikroprocesora 8086/88 firme Intel. Karakteristike asemblerskog programa. Programski model mikroprocesora 32-bitne Intelove arhitekture. Osnove asemblerskog programiranja. Evolucija familije mikroprocesora firme Intel. Karakteristike Pentijum mikroprocesora. Organizacija memorije kod arhitekture IA-32. Na�ini adresiranja operanada u asembleru. Operatori u asembleru. Rad sa segmentima u asembleru ili direktive segmentacije. Tipovi podataka i komande razmjene podataka. Primjeri programa na asembleru i zadaci. Pentijum 4 mikroprocesori. Operativni sistemi personalnih ra�unara. Instalacija i održavanje personalnih ra�unara. Operativna memorija personalnih ra�unara. Magistrale i disk podsistemi personalnih ra�unara. Fajl sistemi: FAT, FAT32, HPFS, NTFS i UNIX fajl sistem. Video podsistemi. Ulazno/izlazni portovi. Štampa�i i skeneri. Dodatni periferni ure�aji i oprema. - Scott Mueller, - "Nadgradnja i popravka PC-ja", CET, Beograd, 2003. (prevod 14. izdanja). - Hans-Peter Messmer, - “PC hardver”, Kompjuter Biblioteka, ;a�ak, 2002. Main texts: - Kip R. Irvine, - “Assembly Language for Intel-Based Computers“ (4th Edition), Prentice Hall, 2002.


be developed: -






188

Name of the course: Probability Theory and Statistics Programmes of Studies:








Aim:

This course is aimed to introduce students with basic concepts of probability thoery and statistics and their applications. The course includes definitions of basic notions of probability theory and statistics. Course includes as well as some classical theorems of probability theory and statistics. Theory is developing without meassure theory and Lebesgue integral concept. This course is aimed to introduce students with statistics softwares STATISTICS and SPSS.

Contents:

Random experiment. Events as set. Probability. Conditional probability. Independence. Random variables. Random vectors. Probabilty mass function. Binomial and Poisson ditribution. Distribution function and density function. Examples of continuous distribution. The joint distribution function. Marginal distributions. The conditional ditribution function. Expectation. Moments. Functions of random variables. Characteristics function. Laws of large numbers. Central limit theorem. Sampling and Statistics. Order statistics. Glivenko-Cantelli theorem. Chi square distribution, t-distribution and F-distribution. Fisher theorem. Estimation. Maximum likelihood method. Confidence intervals. Introduction to hypothesis testing. Inferences about normal models. A regression problem. Nonparametric statistics. Statistical softwares STATISTICS and SPSS 1. S. Stamatovi�: Vjerovatno�a. Statistika, PMF 2000. 2. B. Stamatovi�, S. Stamatovi�: Zbirka zadataka iz Kombinatorike, Vjerovatno�e i Statistike, PMF 2005. 3. Z. Ivkovi�: Teorija vjerovatno�e sa matemati�kom statistikom, Gra�evinska knjiga, Beograd, 1992.

Main texts:


Further readings: W. Feller: An introduction to probability theory and its application, Wiley. - capacity of understanding of probability computing and concept of statistics experiment. Competences to

be developed: - capability to apply the methods of probability theory and statistics in various branches of science. Abilty to make a statistical analysis using statistical softwares.






189


Academic study programmes Mathematics, Mathematics and Computer Sciences (C)







Aim: In this course students become acquainted with basic terms of differential equations, theorems about existence of solutions, methods of solving and application.

Contents:

Simple differential equations of first order: term of solution of Kosi’s theses;theorem about existence of solution. Equations with separate variables, uniform and linear equations. Equations with total differential, integration factor. Simple differential equations of higher degree(solution,Kosi’s theses,theorems about existence of solution).Linear equation of n-order.Method of constant variation. Normal systems of simple differential equations(solution, Kosi’s theses, theorems about existence of solution).Method of elimination. Systems of linear differential equations(method of constant variations ,Ojler’s method) Linear and quasilinear partial differential equations of first order .Systems of nonlinear partial differential equations of first order(complete integration Pfafof’s equation. Classification of partial differential equations of second order.Reduction to canonic form. Wave equation (wire vacillation). Equation of heat conducting (cooling of cane).Dirihl’s circle problem.

Main texts: R.Š�epanovi�, M. Martinovi�: Diferencijalne jedna�ine, Unirex + PMF, 1994. Podgorica. (in Serbian). I.G.Petrovskij, Lekcii po teorii obxknavennxm uravneniqm, MGU, Moskva 1982 Further readings: I.G.Petrovskij, Lekcii ob uravneniqh s &astnxh proiyvodnxh, Nauka, Moskva 1950



Examination:

Ways of checking knowledge and giving marks by points: Presence on lectures 5 points. Two tests 25 points each. Doing homework 5 points. Final exam 40 points (oral exam). Passing grade will be given if more then 50 points is scored.



190

Name of the course: Discrete mathematics II Programmes of Studies:

Academic study programme, Mathematics and Computer Sciences






Language: Serbian or English Prerequisites: No

Aim: This course is aimed to introduce students an old subject with many modern applications – Graph theory.

Contents:

Introduction to graph. Graph terminology. Representing graphs and graph isomorphism. Connectivity. Tree. Spanning trees. Minimum spanning tree. Graph coloring. Planar graphs. Euler and Hamilton Paths. Shortest path problem. Max-flow min- cut theorem. Matching theory. Douglas B. West: Introduction to graph theory, University of Illinois, Prentice Hall Main texts: D. Veljan: Kombinatorika i Teorija grafova, Školska knjiga Zagreb

Further readings: Reinhard Diestel: Graph Theory, Graduate texts in mathematics, Springer

- mathematical foundations for students of computer science - capability to solve problems in a conceivable discipline using graph models Competences to

be developed: - algorithmic thinking






191

Name of the course: Computer System Architecture Programmes of Studies:

Academic study programme Computer Sciences.



Contact hours:

(2 hours for teaching, 1 hour for exercises, 2 hours for consultations with Professor and Assistant) per week, = 85 contact hours in semester

Total hours:

4 × 30 = 120 hours in semester

Structure: 34 hours - lectures, 17 hours - exercises, 4 hours - exams, 34 hours - consultations, 8.5 hours – homework (individual solving of problems), 22.5 hours – individual study.

Language: Montenegrian or English

Prerequisites: - Basics of Computers, - Basics of Logic Design.

Aim:

Introduction to an organization and a modern computer system design by means of the MIPS computer system design. By designing an instruction set which enables complete computer system functioning, student gains necessary knowledge in this area.

Contents:

Binary logical elements, clocking. Latch. Logic function, logic circuit diagram. Logic circuit minimization, karnaugh maps, basic digital systems. Computer system design methodology, sequential circuit design. Moore and Mealy machine. Instructions – the language of the computer system,operations and operands of the computer hardware. Instruction types: R-type, memory-reference instructions, (un)conditional branching instructions. Procedures and their realization in the computer hardware. Assembler programming. MIPS R2000 programming language. Design of ALU used for introduced MIPS R2000 language instruction set implementation. Datapath and control unit, design methodology, simple (single clock cycle) implementation. Control unit, ALU control and CPU design. Multiple clock cycle CPU implementation, instruction execution dividing into separate cycles. 1. D.A. Paterson, J.L. Hennessy, Computer organization & Design, The

hardware/Software interface, Morgan Kaufmann Publishers, San Mateo, California, 1994.

2. M. Morris Mano and Charles R. Kime, Logic and Computer Design Fundamentals. Pearson Prentice Hall, 2004.

Main texts:

3. V. Ivanovi�, Handouts

Further readings: 1. J.L. Hennessy and D.A. Patterson, Computer architecture, A

quantitative approach, Morgan Kaufmann Publishers, San Mateo, California, 2003.

- capability to developed programs in assembler lenguage. Competences to be developed: - capacity of understanding internal processor design, including knowledge

instruction execution.. Methods of teaching:





192

PEDJA

Name of the course: Database Systems Programmes of Studies:








Main texts:

Further readings:







193

Name of the course: Programming Languages Programmes of Studies:







Language: Serbian or English Prerequisites: Basic programming courses, Object oriented programming

Aim:

Since the modeling has recently recognized as one of the key steps in the process of successful software design, the aim of the course is to introduce students to this process and to learn them to use the UML as a standard modeling language. Since the UML is quite complex, the course concentrates on most frequently used concepts and its practical use in visualization, specification, construction and documentation. The course also introduces design patterns as a key element of reusable object oriented design.

Contents:

Introduction. About software development process. Basic concepts of object-oriented modeling. An overview of UML: basic building blocks, rules, and general mechanisms. Classes, attributes and operations. Class diagrams. Relations: association, dependency, generalization. Functional specifications. Use cases case diagrams, include and extend relations. Interaction diagrams. Sequence diagrams. Object diagrams. Links and messages. Communication diagrams. Activity diagrams. States and transitions, state diagrams. Composite states with sequential and concurrent substates. Active classes, communication and synchronization. Standard stereotypes. Interfaces and its realization. Component diagrams. Deployment diagrams. Packages and subsystems. Composite structure diagrams. Interaction overview diagrams. Timing diagrams. Collaborations. Design patterns. Practical modeling in Rose. 1. Booch, G., Rumbaugh, J., Jacobson., I. : UML User’s Guide, Addison

Wesley, 1999 2. Martin Fowler: UML Distilled, Addison Wesley, 2004. 3. Gamma, E., Helm, R., Johnson, R., Vlissides, J., Design Patterns: Elements

of Reusable Object Orienetd Software, Addison-Wesley, 1995

Main texts:

Further readings: 1. D. Mili�ev: Objektno orijentisano modelovanje na jeziku UML, Mikroknjiga, Beograd, 2001.

- accepting and deeper understanding of the object-oriented modeling. - knowledge of UML and appropriate development tools Competences to

be developed: - habits and capabilities to apply modeling in practical projects.



Examination: One midterm exam and the final exam, one individual project, estimation of individual activity on lectures and exercises



194

Name of the course: Numerical Analysis Programmes of Studies: Academic study program Mathematics, Computer Science


ECTS credits: 5




Language: Serbian or English Prerequisites: Basic courses of analysis and algebra

Aim:

This course provides an elementary introduction to methods of numerical analysis. It includes some fundamental results and algorithms for the solution of widely varying mathematical problems. It prepares the students to deal with related questions and to apply the principles included in the course to new problems.

Contents:

Interpolation Polynomials Lagrange and Newton forms, Hermite Interpolation, Numerical Differentiation, Error Estimates for the Numerical Integration with Error Estimates, Numerical methods for solving Nonlinear Equations, Numerical methods for the system of Linear Equations, Numerical Method for Ordinary Differential Equations, Discretization, Runge-Kutta Methods, Numerical Method for Partial Differential Equations. D. Herceg, R. Vulanovic, Numerical Analysis, Novi Sad, 1989. (in Serbian) D. Herceg, N. Krejic, Numerical Analysis I, II , Novi Sad, 1998. (in Serbian) Main texts: A. A. Samarskii, A. V. Goolin, Numerical Methods, Moskow 1989. Further readings: - Build understanding of the applicability of this subject. - Ability to apply the numerical methods in different areas of mathematics. - Develop analytical thinking and the students own opinions and statements.




Evaluation: Written exams (3 times in semester), problem solving - home based tasks, assessment of individual activity during lectures and seminars, oral final examination


Students feedback polls, results of exams, comparison to students from other universities.

195

Name of the course: Computer Security Programmes of Studies:








Aim: Upoznavanje studenata sa prijetnjama bezbjednosti u ra�unarskim sistemima i na�inima, oblicima i metodama zaštite ra�unarskih sistema.

Contents:

Uvod. Osnovni pojmovi o bezbjednosti u ra�unarskim sistemima. Prijetnje bezbjednosti u ra�unarskim sistemima i principi izgradnje bezbjednog ra�unarskog sistema. Degradacija sistema pomo�u virusa i drugih štetnih programa. Preventivna zaštita ra�unara od virusa i Antivirus programi. Neophodna zaštita ra�unarskih sistema, politika i mehanizmi zaštite. Osnovni pojmovi iz kriptografije i kriptoanalize. Klasifikacija kriptosistema. Simetri�no ili klasi�no šifriranje. Apsolutno sigurna šifra. Konfuzija i difuzija i osnovni principi šifriranja. Blokovske šifre. Šifrovanje premještanjem i zamjenom. Fajstelova šifra. DES standard šifriranja podataka. Trojno šifrovanje. Otvaranje DES šifri. Ostale simetri�ne šifre. AES - napredni standard šifriranja. Rijndael-ova šifra. Pouzdanost koriš�enja simetri�nih šifri. Lokacija i razmještaj funkcija i ure�aja za šifriranje. Algoritmi sa otvorenim klju�evima. Algoritam RSA. Protokoli za provjeru i principi izgradnje protokola autenti�nosti. Instalacija dijeljenog klju�a i Difi-Helmanov protokol za razmjenu klju�eva. Provjera originalnosti kroz centar za distribuciju klju�eva i Protokol Nidhema-Šredera za provjeru autenti�nosti. Utvr�ivanje originalnosti protokolom Kerber. Elektronski potpis sa tajnim klju�em i elektronski potpis sa otvorenim klju�em. Hash funkcije. Generacija Message Digest koriš�enjem SHA-1. Elektronska uvjerenja. Kontrola pristupa i autorizacija kao mehanizam zaštite. Zaštita elektronske pošte. Zaštita Web-. Zaštita elektronskih transakcija. Zaštita na mrežnom nivou i IP zaštita. Transportni i tunelski režim zaštite. Virtuelne privatne mreže i tunelovanje. Zaštitna barijera (firewall). - M. Strib, ;. Perkins - “Firewalls zaštita od hakera", Kompjuter biblioteka, “Svetlost”, ;a�ak, 2003. - S. McClure, J. Scambray, G. Kurtz - “Sigurnost na mreži”, Kompjuter biblioteka, “Svetlost”, ;a�ak, 2001. Main texts:

- W. Stallings, - “Cryptography and Network Security.", Prentice-Hall, Inc., New Jersey, 1999.


be developed: - Methods of teaching:



Methods of self- Students pools, results of exams, direct communications with the students.

196

evaluation:

197

Name of the course: Artificial Intelligence Programs of Studies:

Academic study program: Computer Sciences





Structure: 39 hours - lectures, 26 hours - seminars, 6 hours - exams, 20 hours - consultations, 25 hours – homeworks (individual solving of problems), 24 hours – individual project, 25 hours – individual study.

Language: Serbian or English or Russian Prerequisites: Introduction to Logic, Programming I, Programming II

Aim:

This course is an introduction to concepts and methods in artificial intelligence, including search, constraint propagation, knowledge representation, planning, reasoning under uncertainty, and inductive learning. The aims of course are to describe the main concepts of artificial intelligence and to develop students' skills in designing and building serious artificial intelligence programs

Contents:

Intoduction to artificial intelligence. History and background. Informed and uninformed searh. Local search.Simulated annealing. Genetic algorithms. Taboo search. Games. Alpha-Beta Pruning. Constraint satisfaction problems. First order logic. Resolution. Unification. Forward and backward chaining. Rule based systems. Unceratinty. Bayesian networks. Bayesian inference. Introduction to machine learning. Decison trees. Nearest neighbor. Naive Bayes. Regreesion trees. Neural networks. Perceptron. Backpropagation. Introduction to kernel methods.

Main texts: S. Russell and P. Norvig. Artificial Intelligence: A Modern Approach. 2nd Edition, Prentice Hall, 2003. (in English)

Tom Mitchell - Machine Learning, McGraw Hill, 1997 (in English) Further readings: Christopher M. Bishop - Pattern Recognition and Machine Learning, Springer,

2006 (in English) - be able to evaluate the suitability of different AI algorithms and techniques for various kinds of applications. - understanding basic data structures used for representing learned concepts, and the associated processing algorithms - be able to use these structures and algorithms in constructing simple applications which will recognise and evolve existing concepts


- capability to apply the AI methods and algorithms in real life applications


Lectures and seminars with the active participation of students, individual homeworks, individual and group projects, group and individual consultations.

Examination: Midterm and final written exam, problem solving –programming and written homeworks, project presentation, estimation of student parcitipation on lectures and seminars



198

Name of the course: Introduction to Information Systems Programs of Studies:




Contact hours: (2 Lectures + 2 exercises) per week, 15 hours in semester for consultations = 75 contact hours in semester


Structure: 26 hours - lectures, 13 hours - exercises, 4 hours – exams and testings, 15 hours - consultations, 42 hours – individual project, 20 hours – individual study.

Language: Serbian or English Prerequisites: Data structures

Aim:

to describe the main concepts of organization and development of information systems to give students a thorough grounding in the techniques of systems analysis and design. to develop the student’s ability to plan the design of information systems and to consider the technical and communication skills required by a systems analyst to effectively operate in a commercial environment.

Contents:

Systems life-cycle models. Introduction to the concept of a methodology as a management tool and a vehicle for problem solving. The idea of a system. . Social skills. BSP. Data-flow diagrams. Process decision tools. Entity-relationship diagrams. Entity-life histories. Structured Walkthroughs. State Transition Diagrams. Object-oriented design. Correspondence between different modelling techniques. Documentation, Understanding of a Data Dictionary. . Software tools for systems design. Alternative strategies. Case tools and code generators. Software crisis. Prototyping. Reverse engineering.

Main text: Hawryszkiewycz I, Introduction to System Analysis and Design, 5/E, Pearson Education, 2000 (in English) Bocij P, Chaffey D, Greasley A, Hickie S, - Business Information Systems: Technology, Development and Management for the E-business, 3/E, Pearson Education, 2005 (in English) Curtis G, Cobham D, Business Information Systems: Analysis, Design & Practice, 5/E, Pearson Education, 2004 (in English)

Further readings:

Sommerville I, Software Engineering, Addison-Wesley, Pearson Education, 7th Edition, 2004 - capability to describe and model the concepts of systems identification and its boundaries with the outside world - appreciation of social aspects of undertaking systems analysis and design in commercial and other institutional environments - ability to describe and apply various structured analysis and design techniques to case-work


- ability to critically evaluate the modelling process by establishing consistency between the output of the different modelling techniques.



Examination: Tests, project presentation, estimation of student parcitipation on lectures and exercises



199

Name of the course: English for Computer Sciences III Programmes of Studies: Academic study program Computer Science

Level of the course: Bachelor level, III year, I semester Number of

ECTS credits: 2






Contents:

Computer Users; grammar: Past simple vs. Present perfect. Computer Architecture; grammar: Prepositions of place. Computer Applications; grammar: Present passive. Peripherals; grammar: Comparison and contrast. Operating Systems; grammar: -ing form as noun and after prepositions. Graphical User interface; grammar: Verb patterns. Applications Programs; grammar: Complex instructions. Multimedia; grammar: -ing clauses. Networks; grammar: Relative clauses. The Internet; grammar: Warnings. The World Wide Web; grammar: Time clauses. Websites; grammar: Giving advice. Communication systems; grammar: Collocations. Eric H. Glendenning, John McEwan Information Technology

Main texts:

Keith Boeckner, P. Charles Brown Computing Further readings: The further development of 4 main language skills








200

Name of the course: Software Engineering Programs of Studies:




Contact hours: (3 Lectures + 2 exercises) per week, 21 hours in semester for consultations = 96 contact hours in semester


Structure: 39 hours - lectures, 26 hours - exercises, 6 hours – exams and testings, 21 hours - consultations, 38 hours – individual project, 20 hours – individual study.

Language: Serbian or English Prerequisites: Introduction to Information Systems

Aim:

- to introduce basic concepts of requirements capture, modeling, formalisation and software measurement

- to give students a thorough grounding in the techniques of systems analysis and design.

- to give experience in teamwork in software engineering, simulating the software lifecycle

Contents:

Software engineering's aims. Software project management. Approaches to software design: evolutionary, incremental, prototyping, extreme programming. Formal specifications and methods. Configuration management. CASE tools. Requirement analysis. Software system architecture. Using CASE tools for designing. 4GL development tools. Object-oriented system design. UML. Rational Unified Process. Design patterns. Software quality assurance. Software metrics. Intelligent support for software design.

Main text: Sommerville I, Software Engineering, Addison-Wesley, Pearson Education, 7th Edition, 2004 (in English) Galin D, Software Quality Assurance, Addison-Wesley, Pearson Education, 2004.(in English) Kendall K, Kendall J, System Analysis and Design, 6th Edition, Prentice Hall, 2004 Further readings:

Mogin P, Lukovic I, Govedarica M, Principles of database system design , FTN Publishing, 2004 (in Serbian) - capability to describe and model the concepts of systems identification and its boundaries - understanding and ability to apply the main methods for testing software components and systems - to be familiar with the problems of constructing large software systems and of assuring their quality - have examined in depth some of the techniques and tools for trying to solve these problems.


- understand the role of software measurement within the planning and monitoring of software development projects



Examination: Tests, project presentation, estimation of student parcitipation on lectures and exercises



201

Name of the course: Advanced Topics in Programming Programmes of Studies:







Language: Serbian or English Prerequisites: Basic courses of Programming and Internet Technologies

Aim:

This course is intended to provide the concepts and the practical experience required to contribute to the design, implementation and integration of the Web Services paradigm. The students will learn the Web Services as a collection of technologies, including XML, Simple Object Access Protocol, Web Services Description Language and Universal Description, Discover and Integration, which allow building programming solutions for specific messaging and application integration problems.

Contents:

Service Oriented Architectures (SOA). DCOM. CORBA. RMI. Web Services paradigm. XML. Simple Object Access Protocol (SOAP): SOAP Transport Binding Framework, SOAP Serialization Framework, SOAP RPC Representation. Web Services Description Language (WSDL): Types, Message, Operation, Port Type, Binding, Port, Service. Universal Description, Discover and Integration (UDDI): Business Entity, Business Service, Binding Templates, Service types. Case stydies. 1. Ethan Cerami: Web Services: Essentiasls, O'Reilly 2002.

Main texts: 2. Selected papers.

Further readings: Selected papers. - Understanding of the Web Services paradigm.


- Appling the Web Services technology in solving the problems of how to flexibly modify, increment, and connect heterogeneous applications to meet the requirements of business.






202

Name of the course: Compilers Programs of Studies:







Language: Serbian or English or Russian Prerequisites: Programming I, Programming II, advisable: Computer Architecture

Aim:

This course is an introduction to concepts and methods in compiler construction. The aims of course are:

- to describe the main concepts of compiler construction - to develop students' skills in building relatively complex software

project - to deepen understanding of programming languages, their semantics

and their applications.

Contents:

Introduction to compilation. Compilers and interpreters. Lexical analysis. Automata. Regular expressions. Flex. Grammars and languages. Top-down parsing. Bottom-up parsing.. LR(0), LR(1) and SLR(1). LALR. Syntax-directed translations. Semantical analysis. Type checking. Object-oriented and functional languages. Runtime behaviour. TAC. Code genaration. Program analysis. Introduction to dataflow analysis Code optimization. Loop optimiyation. Register allocation. .

Main texts: Cooper, Torczon – Engineering a Compiler, Morgan Kaufmann, 2003. (in English)

Appel – Modern Compiler Implementation in Java (2nd edition), Cambridge University Press, 2002. (in English) Further readings:

Aho, Sethi, Ullman – Compilers: Principles, Techniques and Tools, 2nd Edition, Addison Wesley, 2007 (in English) - be able to evaluate the suitability of different programming languages and compilers AI for various kinds of applications. - understanding basic data structures used for representing learned concepts, and the associated processing algorithms - be able to use these structures and algorithms in constructing simple applications which will recognise and evolve existing concepts


- capability to apply the compiler construction methods and algorithms in real life applications



Examination: Midterm and final written exam, problem solving –programming and written homeworks, project presentation, estimation of student parcitipation on lectures and exercises



203

Name of the course: Internet Technologies Programs of Studies:




Contact hours:

(2 Lectures + 1 tutorial exercise) per week, 15 hours in semester for consultations = 60 contact hours in semester

Total hours:


Structure: 30 hours - lectures, 15 hours - exercises, 15 hours - consultations, 15 hours – homeworks (individual solving of problems), 30 hours – individual project, 15 hours – individual study.

Language: Serbian or English or Russian Prerequisites: Programming I, Programming II

Aim:

This course is an introduction to client-side and server-side technologies for designing and building web-based applications. The aims of course are:

- to describe the main concepts of web-based applications - to develop students' skills in building relatively complex software

project - to deepen understanding of various programming techniques

especially in context of web-applications - to gain first-hand experience of the problems of user- centred design

and visual layout

Contents:

Introduction to computer netorks. TCP-IP protocol. Web browser. Proxies. Intro to XHTML. Block–level and text-level elements. Tables. Lists. Images. CSS. HTTP. XHTML forms. JavaScript variables. Statements. Functions. DOM. Installing PHP. PHP variables and statements. PHP functions. PHP objects. Working with files. Database access. Session management in PHP. PHP security. Introduction to XML. Transforming XML using XSLT. Content management systems. Introduction to AJAX.

Main text: David Sklar – Learning PHP5, O'Reilly, 2004 (in English)

Dave Taylor - Creating Cool Web Sites with HTML, XHTML and CSS, Wiley, 2004 (in English) Further readings: Ellie Quigley – JavaScript by Example Prentice Hall, 2003 (in English) - capability to evaluate the suitability of different programming techiniques and tools for various kinds of web applications. - appreciation of the underlying principles of human-computer interaction and ergonomics; - understanding basic data structures used for representing learned concepts, and the associated processing algorithms


- ability to use these structures and algorithms in constructing applications which will recognise and evolve existing concepts



Examination: Problem solving –programming homeworks, project presentation, quizes, estimation of student parcitipation on lectures and exercises



204

Name of the course: Distributed Systems Programmes of Studies:







Language: Serbian or English Prerequisites: - Computer Networks and Communications

Aim: - Upoznavanje sa hardverskom i softverskom strukturom distribuiranih i

paralelnih ra�unarskih sistema, osnovama paralelnog programiranja i algoritmima za izvršavanja konkurentnih programa.

Contents:

- Uvod. Osnovni pojmovi. Karakteristike ra�unara visokih performansi. Super ra�unari. Klasifikacija i istorijat paralelnih i distribuiranih sistema. Primjeri distribuiranih sistema. Softverski koncept distribuiranih sistema. Performanse paralelnih i distribuiranih ra�unarskih sistema. Osnovni principi izgradnje distribuiranih sistema. Pravci budu�eg razvoja super ra�unara. Osnove paralelnog programiranja. Paralelizam zadataka i paralelizam poda-taka. Tehnologija klijent/server. Troslojni P-A-D model obrade podataka. Procesi i niti. Komunikacija i sinhronizacija konkurentnih procesa. Sinhronizacija vremena u distribuiranim sistemima. Algoritmi za me�usobno isklju�enja kriti�nih intervala. Odre�ivanje stanja distribuiranog sistema. Koordinacija distribuiranih procesa. Distribuirana zajedni�ka memorija. Distribuirani fajl sistem. Dupliranje (razmnožavanje) datoteka.

- +. S. Tanenbaum, M. van Steen - “Distributed Systems – Principles and paradigms”, Prentice-Hall, Inc., New Jersey, 2002. - +. S. Tanenbaum, - “Distributed Operating Systems”, Prentice-Hall, Inc., New Jersey, 1995. - G. Coulouris, J. Dollimore, T. Kindberg – “ Distributed Systems” , Addison-Wesley, Publishers Limited. London, New York … 2005.

Main texts:

- �. =. :�!<�, - “6��2 ��$�� &��$�, ��3��$� � ��!��$� ��$��# ”, ��3#��, ��, 2003.


be developed: -






205

PEDJA

Name of the course: Advanced Database Systems Programmes of Studies:








Main texts:

Further readings:







206

Name of the course: English for Computer Sciences IV Programmes of Studies: Academic study program Computer Science

Level of the course: Bachelor level, III year, II semester Number of

ECTS credits: 2






Contents:

Computing support; grammar: Giving advice. Data Security 1; grammar: Cause and effect. Data Security 2; grammar: Cause and effect 2. Software Engineering; grammar: If clauses. People in Computing; grammar: Requirements. Recent Developments in IT; grammar: ability. The Future of IT; grammar; grammar: Predictions. Electronic Publishing; grammar: Prefixes -ise verbs. Computer viruses; grammar: Listing. Computers in the office; grammar: The passive. Computers in Education; grammar: Giving examples. Computers in Medicine; grammar: Explanations and Definitions. Robotics; grammar: Compound nouns. Virtual reality; grammar: Classifying. Eric H. Glendenning, John McEwan Information Technology (units 15-30)

Main texts:

Keith Boeckner, P. Charles Brown Computing Further readings: The further development of 4 main language skills The development of presentation techniques Composition writing







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