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Faculty of Science and Mathematics / MATHEMATICS / INTRODUCTION TO MATHEMATICAL LOGIC

Course:INTRODUCTION TO MATHEMATICAL LOGIC/
Course IDCourse statusSemesterECTS creditsLessons (Lessons+Exercises+Laboratory)
3979Obavezan142+1+0
ProgramsMATHEMATICS
Prerequisites
Aims
Learning outcomes
Lecturer / Teaching assistant
Methodology
Plan and program of work
Preparing weekPreparation and registration of the semester
I week lectures
I week exercises
II week lectures
II week exercises
III week lectures
III week exercises
IV week lectures
IV week exercises
V week lectures
V week exercises
VI week lectures
VI week exercises
VII week lectures
VII week exercises
VIII week lectures
VIII week exercises
IX week lectures
IX week exercises
X week lectures
X week exercises
XI week lectures
XI week exercises
XII week lectures
XII week exercises
XIII week lectures
XIII week exercises
XIV week lectures
XIV week exercises
XV week lectures
XV week exercises
Student workload
Per weekPer semester
4 credits x 40/30=5 hours and 20 minuts
2 sat(a) theoretical classes
0 sat(a) practical classes
1 excercises
2 hour(s) i 20 minuts
of independent work, including consultations 		Classes and final exam:
5 hour(s) i 20 minuts x 16 =85 hour(s) i 20 minuts
Necessary preparation before the beginning of the semester (administration, registration, certification):
5 hour(s) i 20 minuts x 2 =10 hour(s) i 40 minuts
Total workload for the subject:
4 x 30=120 hour(s)
Additional work for exam preparation in the preparing exam period, including taking the remedial exam from 0 to 30 hours (remaining time from the first two items to the total load for the item)
24 hour(s) i 0 minuts
Workload structure: 85 hour(s) i 20 minuts (cources), 10 hour(s) i 40 minuts (preparation), 24 hour(s) i 0 minuts (additional work)
Student obligations
Consultations
Literature
Examination methods
Special remarks
Comment
Grade:FEDCBA
Number of pointsless than 50 pointsgreater than or equal to 50 points and less than 60 pointsgreater than or equal to 60 points and less than 70 pointsgreater than or equal to 70 points and less than 80 pointsgreater than or equal to 80 points and less than 90 pointsgreater than or equal to 90 points

Faculty of Science and Mathematics / MATHEMATICS / ENGLISH LANGUAGE 1

Course:ENGLISH LANGUAGE 1/
Course IDCourse statusSemesterECTS creditsLessons (Lessons+Exercises+Laboratory)
5545Obavezan142+1+0
ProgramsMATHEMATICS
Prerequisites There are no pre-requisites for the course. However, the students should command intermediate English in order to be able to follow the classes.
Aims To master the basic grammar structures and use the English language in everyday situations.
Learning outcomes After passing this exam, the students will be able to: - Understand the English discourse messages on topics commonly encountered (family, professions, hobbies, etiquette, customs), as well as the basic messages of the more complex English texts and audio recordings on various concrete and abstract topics (art, travel, media, school systems, weather), - Speak English relatively fluently on familiar topics using simple structures, exchange information and participate in conversation on familiar topics as well as those covered in classes, - Describe experience, events, plans, provide explanation and arguments in the English language, - Command the English grammar at the lower-intermediate level, - Write a short essay in English on a familiar topic, - Be aware of the connection between the foreign language and culture, and be familiar with some traditions in the English-speaking countries.
Lecturer / Teaching assistantMilica Vuković Stamatović, Savo Kostić
MethodologyA short introduction to the topics covered, with the focus on the participation of students in various types of exercises - conversation and writing, pairwork, groupwork, presentations, discussions etc.
Plan and program of work
Preparing weekPreparation and registration of the semester
I week lecturesIntroduction to the course; Present Simple vs Present Continuous
I week exercises Present Simple vs Present Continuous, exercises
II week lecturesPast Simple (regular/irregular verbs); Used to
II week exercises Past Simple (regular/irregular verbs); Used to, exercises
III week lecturesPast Continuous (Past Simple vs Past Continuous)
III week exercisesPast Continuous (Past Simple vs Past Continuous), exercises
IV week lecturesPresent Perfect Simple (Past Simple vs Present Perfect Simple)
IV week exercisesPresent Perfect Simple (Past Simple vs Present Perfect Simple), exercises
V week lecturesFuture (Future simple – Be going to – Present Continuous)
V week exercisesFuture (Future simple – Be going to – Present Continuous), exercises
VI week lecturesMid-term test
VI week exercisesMid-term test
VII week lecturesRevision, error correcting
VII week exercisesRevision, error correcting
VIII week lecturesPronouns; Infinitives
VIII week exercisesPronouns; Infinitives, exercises
IX week lecturesAdjectives
IX week exercisesAdjectives, exercises
X week lecturesModal Verbs
X week exercisesModal Verbs, exercises
XI week lecturesPast Perfect Simple; Past Perfect Continuous
XI week exercisesPast Perfect Simple; Past Perfect Continuous, exercises
XII week lecturesPassive Voice
XII week exercisesPassive Voice, exercises
XIII week lecturesReported Speech
XIII week exercisesReported Speech, exercises
XIV week lecturesConditionals - Wishes
XIV week exercisesConditionals - Wishes, exercises
XV week lecturesPreparation for the exam
XV week exercisesPreparation for the exam
Student workload2 hours 40 minutes
Per weekPer semester
4 credits x 40/30=5 hours and 20 minuts
2 sat(a) theoretical classes
0 sat(a) practical classes
1 excercises
2 hour(s) i 20 minuts
of independent work, including consultations 		Classes and final exam:
5 hour(s) i 20 minuts x 16 =85 hour(s) i 20 minuts
Necessary preparation before the beginning of the semester (administration, registration, certification):
5 hour(s) i 20 minuts x 2 =10 hour(s) i 40 minuts
Total workload for the subject:
4 x 30=120 hour(s)
Additional work for exam preparation in the preparing exam period, including taking the remedial exam from 0 to 30 hours (remaining time from the first two items to the total load for the item)
24 hour(s) i 0 minuts
Workload structure: 85 hour(s) i 20 minuts (cources), 10 hour(s) i 40 minuts (preparation), 24 hour(s) i 0 minuts (additional work)
Student obligations Attendance, doing homework, active participation in classes
Consultations
LiteratureLiteratura: Jenny Dooley and Virginia Evans, Grammarway 3, Express Publishing
Examination methods
Special remarksAdopted on 21-7-2016: http://senat.ucg.ac.me/data/1469020997-Akreditacija%20PMF%202017%20final.pdf
Comment/
Grade:FEDCBA
Number of pointsless than 50 pointsgreater than or equal to 50 points and less than 60 pointsgreater than or equal to 60 points and less than 70 pointsgreater than or equal to 70 points and less than 80 pointsgreater than or equal to 80 points and less than 90 pointsgreater than or equal to 90 points

Faculty of Science and Mathematics / MATHEMATICS / COMPUTERS AND PROGRAMMING

Course:COMPUTERS AND PROGRAMMING/
Course IDCourse statusSemesterECTS creditsLessons (Lessons+Exercises+Laboratory)
495Obavezan163+3+0
ProgramsMATHEMATICS
Prerequisites
Aims
Learning outcomes
Lecturer / Teaching assistant
Methodology
Plan and program of work
Preparing weekPreparation and registration of the semester
I week lectures
I week exercises
II week lectures
II week exercises
III week lectures
III week exercises
IV week lectures
IV week exercises
V week lectures
V week exercises
VI week lectures
VI week exercises
VII week lectures
VII week exercises
VIII week lectures
VIII week exercises
IX week lectures
IX week exercises
X week lectures
X week exercises
XI week lectures
XI week exercises
XII week lectures
XII week exercises
XIII week lectures
XIII week exercises
XIV week lectures
XIV week exercises
XV week lectures
XV week exercises
Student workload
Per weekPer semester
6 credits x 40/30=8 hours and 0 minuts
3 sat(a) theoretical classes
0 sat(a) practical classes
3 excercises
2 hour(s) i 0 minuts
of independent work, including consultations 		Classes and final exam:
8 hour(s) i 0 minuts x 16 =128 hour(s) i 0 minuts
Necessary preparation before the beginning of the semester (administration, registration, certification):
8 hour(s) i 0 minuts x 2 =16 hour(s) i 0 minuts
Total workload for the subject:
6 x 30=180 hour(s)
Additional work for exam preparation in the preparing exam period, including taking the remedial exam from 0 to 30 hours (remaining time from the first two items to the total load for the item)
36 hour(s) i 0 minuts
Workload structure: 128 hour(s) i 0 minuts (cources), 16 hour(s) i 0 minuts (preparation), 36 hour(s) i 0 minuts (additional work)
Student obligations
Consultations
Literature
Examination methods
Special remarks
Comment
Grade:FEDCBA
Number of pointsless than 50 pointsgreater than or equal to 50 points and less than 60 pointsgreater than or equal to 60 points and less than 70 pointsgreater than or equal to 70 points and less than 80 pointsgreater than or equal to 80 points and less than 90 pointsgreater than or equal to 90 points

Faculty of Science and Mathematics / MATHEMATICS / LINEAR ALGEBRA 1

Course:LINEAR ALGEBRA 1/
Course IDCourse statusSemesterECTS creditsLessons (Lessons+Exercises+Laboratory)
3967Obavezan184+3+0
ProgramsMATHEMATICS
Prerequisites no
Aims Standard course of Linear algebra for students of mathematics. Includes theory of finite-dimensional vector spaces, matrices, systems of linear equations and linear mappings in finite-dimensional vector spaces (including spectral theory).
Learning outcomes
Lecturer / Teaching assistantVladimir Jaćimović, Dušica Slović
Methodologylectures, seminars, consultations
Plan and program of work
Preparing weekPreparation and registration of the semester
I week lecturesGroups and fields. Vector spaces. Definition. Examples. Vector subspaces. Linear span.
I week exercisesGroups and fields. Fields of real and complex numbers. Geometric vectors in the plane.
II week lecturesLinearly dependent and independent vectors. Base and dimension of vector spaces. Isomorfism of vector spaces.
II week exercises Vector spaces. R^n and C^n. Vector subspaces. Linear span.
III week lecturesMatrices. Gauss method for solving linear systems of equations. Matrices of elementary transforms.
III week exercisesLinearly dependent and independent vectors. Base and dimension of vector spaces. Problems and examples in R^n. Subspaces in R^n. Systems of linear equations.
IV week lecturesDeterminants of square matrices. Rank of matrix.
IV week exercisesGauss method for solving systems of linear equations. Matrices. Matrices of elementary transforms.
V week lecturesInverse matrix. Regular and singular matrices. Matrices of change of bases. Equivalent matrices.
V week exercisesDeterminant and rank of matrix.
VI week lecturesSystems of linear equations. Existence and uniqueness of solution. General solution. Kronecker Capelli theorem. Cramers' rule.
VI week exercisesInverse matrix. Regular and singular matrices. Matrices of coordinate change.
VII week lectures1st test
VII week exercises1st test
VIII week lecturesEmpty week.
VIII week exercisesEmpty week.
IX week lecturesLinear mappings in vector spaces. Definition. Examples. Kernel and image of linear mapping.
IX week exercisesHomogeneous and nonhomogeneous systems of linear equations. Methods of solving. Existence and uniqueness of solution. Cramers' rule.
X week lecturesMatrix of linear mapping. Similar matrices. Inverse mapping. Rank of linear mapping.
X week exercisesLinear mappings in vector spaces. Kernel and image of linear mapping. Examples: operators of projection, rotation and differentiation of polynomials.
XI week lecturesInvariant subspaces of linear mapping. Eigenvalues and eigenvectors. Eigenspaces.
XI week exercisesMatrix of linear mapping. Inverse mapping. Rank of linear mapping.
XII week lecturesFundamental theorem of algebra. Characteristic polynomial of linear mapping. Polynomials of matrices/operators. Hamilton-Cayley theorem.
XII week exercisesEigenvalues and eigenvectors of linear mapping. Characteristic polynomial of linear mapping.
XIII week lecturesJordan form and cannonical base of nilpotent linear mapping.
XIII week exercisesMethod of calculation of eigenvectors. Eigenspaces.
XIV week lecturesJordan form of linear mapping. Examples.
XIV week exercisesJordan form of linear mapping. Similar matrices.
XV week lectures2nd test
XV week exercises2nd test
Student workload4 hours/week lectures + 3 hours/week seminars + 4 hours/week homework = 11 hours/week. Total: 11 hours/week x 16 weeks = 176 hours
Per weekPer semester
8 credits x 40/30=10 hours and 40 minuts
4 sat(a) theoretical classes
0 sat(a) practical classes
3 excercises
3 hour(s) i 40 minuts
of independent work, including consultations 		Classes and final exam:
10 hour(s) i 40 minuts x 16 =170 hour(s) i 40 minuts
Necessary preparation before the beginning of the semester (administration, registration, certification):
10 hour(s) i 40 minuts x 2 =21 hour(s) i 20 minuts
Total workload for the subject:
8 x 30=240 hour(s)
Additional work for exam preparation in the preparing exam period, including taking the remedial exam from 0 to 30 hours (remaining time from the first two items to the total load for the item)
48 hour(s) i 0 minuts
Workload structure: 170 hour(s) i 40 minuts (cources), 21 hour(s) i 20 minuts (preparation), 48 hour(s) i 0 minuts (additional work)
Student obligations
Consultations1 hour/week (lectures) + 1 hour/week (seminars)
LiteratureM. Jaćimović, I. Krnić „Linearna algebra, teoreme i zadaci“ (skripta) E. Shikin „Lineinie prostranstva i otobrazheniya“, Moskva 1987. S. Friedberg, A. Insel, L. Spence „Linear algebra, 4th edition“ Pearson, 2002.
Examination methodsattendance (5 points), homework (5x1 points), 2 tests (2x30 points), one corrective test, final exam (30 points), corrective final exam, 2 brief oral exams (optional – 2x5 points)
Special remarksThe language of instruction is Serbo-Croat. Lectures can be given in English or Russian language.
Comment
Grade:FEDCBA
Number of pointsless than 50 pointsgreater than or equal to 50 points and less than 60 pointsgreater than or equal to 60 points and less than 70 pointsgreater than or equal to 70 points and less than 80 pointsgreater than or equal to 80 points and less than 90 pointsgreater than or equal to 90 points

Faculty of Science and Mathematics / MATHEMATICS / ANALYSIS 1

Course:ANALYSIS 1/
Course IDCourse statusSemesterECTS creditsLessons (Lessons+Exercises+Laboratory)
3977Obavezan184+3+0
ProgramsMATHEMATICS
Prerequisites None.
Aims The aim of the course is for students to adopt and master the basics of mathematical analysis: limit theory, elements of differential and integral calculus and the theory of series.
Learning outcomes On successful completion of this course students will be able to: 1. Define the basic notions of Mathematical analysis 1: the set of real numbers, the limit of a sequence and function, differentiability of functions, derivatives and antiderivatives on segments. 2. Define the basic properties of the set of real numbers. 3. Derive basic propositions of limit theory and differential calculus, establish when a sequence or function has a limit or the property of continuity or differentiability. 4. Examine and relate properties of functions of one variable using differential calculus. 5. Apply the acquired knowledge to solving different tasks related to the stated content of mathematical analysis. 6. Apply the acquired knowledge to solving real tasks and problems.
Lecturer / Teaching assistantProf. dr Žarko Pavićević - lecturer, Nikola Konatar - teaching assistant
MethodologyLectures, exercises, homework assignments, consultations, exams.
Plan and program of work
Preparing weekPreparation and registration of the semester
I week lecturesIntroducing students to basic topics covered by the course.
I week exercisesIntroducing students to basic topics covered by the course.
II week lecturesThe set of real numbers - axiomatic construction.
II week exercises The set of real numbers - axiomatic construction.
III week lecturesCompleteness principles of the set of real numbers.
III week exercisesCompleteness principles of the set of real numbers.
IV week lecturesConvergent sequence theory.
IV week exercisesConvergent sequence theory.
V week lecturesBolzano’s and Cauchy’s theorem for sequences. Banach fixed-point theorem.
V week exercisesBolzano’s and Cauchy’s theorem for sequences. Banach fixed-point theorem.
VI week lecturesTopology on the set of real numbers.
VI week exercisesTopology on the set of real numbers.
VII week lecturesLimit of a function. Continuity of a function at a point.
VII week exercisesLimit of a function. Continuity of a function at a point.
VIII week lecturesGlobal properties of functions continuous on segments.
VIII week exercisesGlobal properties of functions continuous on segments.
IX week lecturesUniform continuity of functions.
IX week exercisesUniform continuity of functions.
X week lecturesReview. First midterm exam.
X week exercisesReview. First midterm exam.
XI week lecturesDifferentiability of functions at a point. Derivative of a function.
XI week exercisesDifferentiability of functions at a point. Derivative of a function.
XII week lecturesDerivatives of higher order.
XII week exercisesDerivatives of higher order.
XIII week lecturesMean value theorems of differential calculus. Bernouli-LHospital rule. Taylor formulas.
XIII week exercisesMean value theorems of differential calculus. Bernouli-LHospital rule. Taylor formulas.
XIV week lecturesMonotonicity and extrema of differentiable functions. Convexity of functions. Points of inflexion.
XIV week exercisesMonotonicity and extrema of differentiable functions. Convexity of functions. Points of inflexion.
XV week lecturesExamining properties and sketching graphs of functions. Second midterm exam.
XV week exercisesExamining properties and sketching graphs of functions. Second midterm exam.
Student workload
Per weekPer semester
8 credits x 40/30=10 hours and 40 minuts
4 sat(a) theoretical classes
0 sat(a) practical classes
3 excercises
3 hour(s) i 40 minuts
of independent work, including consultations 		Classes and final exam:
10 hour(s) i 40 minuts x 16 =170 hour(s) i 40 minuts
Necessary preparation before the beginning of the semester (administration, registration, certification):
10 hour(s) i 40 minuts x 2 =21 hour(s) i 20 minuts
Total workload for the subject:
8 x 30=240 hour(s)
Additional work for exam preparation in the preparing exam period, including taking the remedial exam from 0 to 30 hours (remaining time from the first two items to the total load for the item)
48 hour(s) i 0 minuts
Workload structure: 170 hour(s) i 40 minuts (cources), 21 hour(s) i 20 minuts (preparation), 48 hour(s) i 0 minuts (additional work)
Student obligations Students are required to attend classes, do the homework assignments and take all exams.
ConsultationsAs agreed with students.
LiteratureV. I. Gavrilov,,Ž. Pavićević, Matematička analiza I, I.M. Lavrentjev, R. Šćepanović, Zbirka zadataka iz mat. analize I, B.P. Demidovič: Zbirka zadataka iz matematičke analize (Prevod)
Examination methodsTwo homeworks or tests are graded with 8 points (4 points for each homework or test). 2 points are awarded for attendance to lectures and exercises. Two midterm exams are graded with 20 points each (40 points in total). Final exam - 50 points. A passing grade is awarded to students who accumulate at least 50 points.
Special remarks
Comment
Grade:FEDCBA
Number of pointsless than 50 pointsgreater than or equal to 50 points and less than 60 pointsgreater than or equal to 60 points and less than 70 pointsgreater than or equal to 70 points and less than 80 pointsgreater than or equal to 80 points and less than 90 pointsgreater than or equal to 90 points

Faculty of Science and Mathematics / MATHEMATICS / ENGLISH LANGUAGE 2

Course:ENGLISH LANGUAGE 2/
Course IDCourse statusSemesterECTS creditsLessons (Lessons+Exercises+Laboratory)
5546Obavezan222+1+0
ProgramsMATHEMATICS
Prerequisites None
Aims To understand and be able to use ESP (English for Mathematics)
Learning outcomes After passing this exam, the students will be able to: - Differentiate, understand and use the most basic mathematical English terminology in the field of number theory, applied mathematics, combinatorics and discrete mathematics, - Read simple mathematical expressions in English, - Understand the basic messages of popular-professional English texts in the field of mathematics, - Communicate in English independently, both orally and in writing, at the intermediate level, - Orally present in English on the mathematical topic chosen, - Write a summary of a popular-professional text or audio recording in English.
Lecturer / Teaching assistantMilica Vuković Stamatović, Savo Kostić
MethodologyLectures and exercises. Preparation of a presentation on a topic related to the content covered in the course. Studying for the test and the exam. Consultations.
Plan and program of work
Preparing weekPreparation and registration of the semester
I week lecturesIntroduction to the course. Reading: My Future Profession; Basic mathematical terms
I week exercisesVocabulary and grammar exercises
II week lecturesMathematical terms – algebra and geometry
II week exercises Vocabulary and grammar exercises
III week lecturesReading: A Genius Explains; Conditionals
III week exercisesConditional, exercises
IV week lecturesReading: Number Theory; Active and Passive
IV week exercisesActive and Passive, exercises
V week lecturesRevision
V week exercisesRevision
VI week lecturesReading: Applied Mathematics; Articles; Transformations
VI week exercisesTransformations, exercises
VII week lecturesPreparation for the mid-term test
VII week exercisesPreparation for the mid-term test
VIII week lecturesMid-term test
VIII week exercisesMid-term test
IX week lecturesReading: Combinatorics; Modal verbs
IX week exercisesModal verbs, exercises
X week lecturesReading: Discrete Mathematics; The Language of Proof
X week exercisesVocabulary exercises
XI week lecturesReading: An Interview with Leonardo Fibonacci; Vocabulary Revision
XI week exercisesVocabulary revision
XII week lecturesGrammar Revision
XII week exercisesGrammar Revision
XIII week lecturesMid-term test (2nd term)
XIII week exercisesMid-term test (2nd term)
XIV week lecturesTranslation exercises
XIV week exercisesTranslation exercises
XV week lecturesPreparation for the final exam
XV week exercises
Student workload2 hours 40 minutes
Per weekPer semester
2 credits x 40/30=2 hours and 40 minuts
2 sat(a) theoretical classes
0 sat(a) practical classes
1 excercises
-1 hour(s) i 40 minuts
of independent work, including consultations 		Classes and final exam:
2 hour(s) i 40 minuts x 16 =42 hour(s) i 40 minuts
Necessary preparation before the beginning of the semester (administration, registration, certification):
2 hour(s) i 40 minuts x 2 =5 hour(s) i 20 minuts
Total workload for the subject:
2 x 30=60 hour(s)
Additional work for exam preparation in the preparing exam period, including taking the remedial exam from 0 to 30 hours (remaining time from the first two items to the total load for the item)
12 hour(s) i 0 minuts
Workload structure: 42 hour(s) i 40 minuts (cources), 5 hour(s) i 20 minuts (preparation), 12 hour(s) i 0 minuts (additional work)
Student obligations Students have to attend the classes, do a presentation on a given topic and take the mid-term test and the final exam.
Consultations
LiteratureTextbook: English 2 (ESP - English for students of theoretical and applied mathematics)
Examination methods
Special remarksAdopted on 21-7-2016: http://senat.ucg.ac.me/data/1469020997-Akreditacija%20PMF%202017%20final.pdf
Comment/
Grade:FEDCBA
Number of pointsless than 50 pointsgreater than or equal to 50 points and less than 60 pointsgreater than or equal to 60 points and less than 70 pointsgreater than or equal to 70 points and less than 80 pointsgreater than or equal to 80 points and less than 90 pointsgreater than or equal to 90 points

Faculty of Science and Mathematics / MATHEMATICS / INTRODUCTION TO COMBINATORICS

Course:INTRODUCTION TO COMBINATORICS/
Course IDCourse statusSemesterECTS creditsLessons (Lessons+Exercises+Laboratory)
3981Obavezan242+2+0
ProgramsMATHEMATICS
Prerequisites
Aims
Learning outcomes
Lecturer / Teaching assistant
Methodology
Plan and program of work
Preparing weekPreparation and registration of the semester
I week lectures
I week exercises
II week lectures
II week exercises
III week lectures
III week exercises
IV week lectures
IV week exercises
V week lectures
V week exercises
VI week lectures
VI week exercises
VII week lectures
VII week exercises
VIII week lectures
VIII week exercises
IX week lectures
IX week exercises
X week lectures
X week exercises
XI week lectures
XI week exercises
XII week lectures
XII week exercises
XIII week lectures
XIII week exercises
XIV week lectures
XIV week exercises
XV week lectures
XV week exercises
Student workload
Per weekPer semester
4 credits x 40/30=5 hours and 20 minuts
2 sat(a) theoretical classes
0 sat(a) practical classes
2 excercises
1 hour(s) i 20 minuts
of independent work, including consultations 		Classes and final exam:
5 hour(s) i 20 minuts x 16 =85 hour(s) i 20 minuts
Necessary preparation before the beginning of the semester (administration, registration, certification):
5 hour(s) i 20 minuts x 2 =10 hour(s) i 40 minuts
Total workload for the subject:
4 x 30=120 hour(s)
Additional work for exam preparation in the preparing exam period, including taking the remedial exam from 0 to 30 hours (remaining time from the first two items to the total load for the item)
24 hour(s) i 0 minuts
Workload structure: 85 hour(s) i 20 minuts (cources), 10 hour(s) i 40 minuts (preparation), 24 hour(s) i 0 minuts (additional work)
Student obligations
Consultations
Literature
Examination methods
Special remarks
Comment
Grade:FEDCBA
Number of pointsless than 50 pointsgreater than or equal to 50 points and less than 60 pointsgreater than or equal to 60 points and less than 70 pointsgreater than or equal to 70 points and less than 80 pointsgreater than or equal to 80 points and less than 90 pointsgreater than or equal to 90 points

Faculty of Science and Mathematics / MATHEMATICS / ANALYTIC GEOMETRY

Course:ANALYTIC GEOMETRY/
Course IDCourse statusSemesterECTS creditsLessons (Lessons+Exercises+Laboratory)
1341Obavezan242+2+0
ProgramsMATHEMATICS
Prerequisites Attending and taking this course is not conditioned by other courses.
Aims The aim of this course is to introduce students to elements of vector algebra and the method of coordinates for investigation of geometrical objects and for solving of geometrical problems.
Learning outcomes On successful completion of the course, students will be able to: 1. Describe Cartesian, polar and sphere coordinate system and explain how basic geometric objects: point, line, plane, circle, ellipse, parabola and hyperbola can be presented in these coordinate systems. 2. Explain how the equations of a geometric object can be used establish their relation and position in plane and space. 3. Study the properties of geometric objects by using the equations they are described with. 4. Using the method of coordinates, solve some geometric tasks. 5. Using the equation of the second order of two and three variables, classify curves and surfaces of the second order.
Lecturer / Teaching assistantProf. dr Milojica Jaćimović – lecturer, Mr. Dušica Slović, assistant
MethodologyLectures and exercises with active participation of students, individual homework assignments, group and individual consultations.
Plan and program of work
Preparing weekPreparation and registration of the semester
I week lecturesCartesian coordinate systems in plane and in space. Polar and spherical coordinate systems.
I week exercisesCartesian coordinate systems in plane and in space. Polar and spherical coordinate systems.
II week lecturesVectors in coordinate system. Linear operations. Scalar, vector and mixed products.
II week exercises Vectors in coordinate system. Linear operations. Scalar, vector and mixed products.
III week lecturesCurves and surfaces and their equations. Examples.
III week exercisesCurves and surfaces and their equations. Examples.
IV week lecturesLine in the plane, plane in the space, line in the space, different equations of the line and the plane.
IV week exercisesLine in the plane, plane in the space, line in the space, different equations of the line and the plane.
V week lecturesRelations of lines and planes in space. Examples. Distance from a point to a plane and line.
V week exercisesRelations of lines and planes in space. Examples. Distance from a point to a plane and line.
VI week lecturesPlane in the n-dimensional Eucledian space. Dimension of the plane. Parallel planes.
VI week exercisesPlane in the n-dimensional Eucledian space. Dimension of the plane. Parallel planes.
VII week lecturesStudy break.
VII week exercisesStudy break.
VIII week lecturesLine and hyperplane. Distance from a point to the hyperplane. Plane as a intersection of hyperplanes. I written exam
VIII week exercisesLine and hyperplane. Distance from a point to the hyperplane. Plane as a intersection of hyperplanes. I written exam
IX week lecturesConvex set in a n-dimensional space. Segment, ray, half-space. Linear programming. Conic section. Classification. Canonical equations.
IX week exercisesConvex set in a n-dimensional space. Segment, ray, half-space. Linear programming. Conic section. Classification. Canonical equations.
X week lecturesProperties of the ellipse, hyperbola, parabola.
X week exercisesProperties of the ellipse, hyperbola, parabola.
XI week lecturesIsometric transformations of the Euclidean space. The group of isometric transformations.
XI week exercisesIsometric transformations of the Euclidean space. The group of isometric transformations.
XII week lecturesQuadric surfaces. Reduction to canonical form. Theorem of inertia. II written exam
XII week exercisesQuadric surfaces. Reduction to canonical form. Theorem of inertia. II written exam
XIII week lecturesSecond-order curves. Invariants. Properties, classification.
XIII week exercisesSecond-order curves. Invariants. Properties, classification.
XIV week lecturesSecond-order surfaces. Canonical form.
XIV week exercisesSecond-order surfaces. Canonical form.
XV week lecturesInvariants and second order surfaces.
XV week exercisesInvariants and second order surfaces.
Student workload2 hours of lectures 2 hours of exercises 1 hour 20 minutes of individual activity, including consultations
Per weekPer semester
4 credits x 40/30=5 hours and 20 minuts
2 sat(a) theoretical classes
0 sat(a) practical classes
2 excercises
1 hour(s) i 20 minuts
of independent work, including consultations 		Classes and final exam:
5 hour(s) i 20 minuts x 16 =85 hour(s) i 20 minuts
Necessary preparation before the beginning of the semester (administration, registration, certification):
5 hour(s) i 20 minuts x 2 =10 hour(s) i 40 minuts
Total workload for the subject:
4 x 30=120 hour(s)
Additional work for exam preparation in the preparing exam period, including taking the remedial exam from 0 to 30 hours (remaining time from the first two items to the total load for the item)
24 hour(s) i 0 minuts
Workload structure: 85 hour(s) i 20 minuts (cources), 10 hour(s) i 40 minuts (preparation), 24 hour(s) i 0 minuts (additional work)
Student obligations Students are required to attend classes.
ConsultationsAs agreed with the professor or teaching assistant.
LiteratureN. Elezović, Linearna algebra, Element, Zagreb, 2001; P.S. Modenov: Analiticka geometrija, Moskovski univerzitet; M. Jaćimović, I. Krnić: Linearna algebra – teoreme i zadaci, skripta, Podgorica
Examination methodsTwo written exams,( up to 30 points each), and the final exam (up to 40 points). Grading: 51-60 points- E; 61-70 points- D; 71-80 points- C; 81-90 points- B; 91-100 points- A.
Special remarks
Comment
Grade:FEDCBA
Number of pointsless than 50 pointsgreater than or equal to 50 points and less than 60 pointsgreater than or equal to 60 points and less than 70 pointsgreater than or equal to 70 points and less than 80 pointsgreater than or equal to 80 points and less than 90 pointsgreater than or equal to 90 points

Faculty of Science and Mathematics / MATHEMATICS / GEOMETRY OF SPACE LEVELS

Course:GEOMETRY OF SPACE LEVELS/
Course IDCourse statusSemesterECTS creditsLessons (Lessons+Exercises+Laboratory)
10106Obavezan242+2+0
ProgramsMATHEMATICS
Prerequisites
Aims
Learning outcomes
Lecturer / Teaching assistant
Methodology
Plan and program of work
Preparing weekPreparation and registration of the semester
I week lectures
I week exercises
II week lectures
II week exercises
III week lectures
III week exercises
IV week lectures
IV week exercises
V week lectures
V week exercises
VI week lectures
VI week exercises
VII week lectures
VII week exercises
VIII week lectures
VIII week exercises
IX week lectures
IX week exercises
X week lectures
X week exercises
XI week lectures
XI week exercises
XII week lectures
XII week exercises
XIII week lectures
XIII week exercises
XIV week lectures
XIV week exercises
XV week lectures
XV week exercises
Student workload
Per weekPer semester
4 credits x 40/30=5 hours and 20 minuts
2 sat(a) theoretical classes
0 sat(a) practical classes
2 excercises
1 hour(s) i 20 minuts
of independent work, including consultations 		Classes and final exam:
5 hour(s) i 20 minuts x 16 =85 hour(s) i 20 minuts
Necessary preparation before the beginning of the semester (administration, registration, certification):
5 hour(s) i 20 minuts x 2 =10 hour(s) i 40 minuts
Total workload for the subject:
4 x 30=120 hour(s)
Additional work for exam preparation in the preparing exam period, including taking the remedial exam from 0 to 30 hours (remaining time from the first two items to the total load for the item)
24 hour(s) i 0 minuts
Workload structure: 85 hour(s) i 20 minuts (cources), 10 hour(s) i 40 minuts (preparation), 24 hour(s) i 0 minuts (additional work)
Student obligations
Consultations
Literature
Examination methods
Special remarks
Comment
Grade:FEDCBA
Number of pointsless than 50 pointsgreater than or equal to 50 points and less than 60 pointsgreater than or equal to 60 points and less than 70 pointsgreater than or equal to 70 points and less than 80 pointsgreater than or equal to 80 points and less than 90 pointsgreater than or equal to 90 points

Faculty of Science and Mathematics / MATHEMATICS / PRINCIPLES OF PROGRAMMING

Course:PRINCIPLES OF PROGRAMMING/
Course IDCourse statusSemesterECTS creditsLessons (Lessons+Exercises+Laboratory)
1335Obavezan263+2+0
ProgramsMATHEMATICS
Prerequisites
Aims
Learning outcomes
Lecturer / Teaching assistant
Methodology
Plan and program of work
Preparing weekPreparation and registration of the semester
I week lectures
I week exercises
II week lectures
II week exercises
III week lectures
III week exercises
IV week lectures
IV week exercises
V week lectures
V week exercises
VI week lectures
VI week exercises
VII week lectures
VII week exercises
VIII week lectures
VIII week exercises
IX week lectures
IX week exercises
X week lectures
X week exercises
XI week lectures
XI week exercises
XII week lectures
XII week exercises
XIII week lectures
XIII week exercises
XIV week lectures
XIV week exercises
XV week lectures
XV week exercises
Student workload
Per weekPer semester
6 credits x 40/30=8 hours and 0 minuts
3 sat(a) theoretical classes
0 sat(a) practical classes
2 excercises
3 hour(s) i 0 minuts
of independent work, including consultations 		Classes and final exam:
8 hour(s) i 0 minuts x 16 =128 hour(s) i 0 minuts
Necessary preparation before the beginning of the semester (administration, registration, certification):
8 hour(s) i 0 minuts x 2 =16 hour(s) i 0 minuts
Total workload for the subject:
6 x 30=180 hour(s)
Additional work for exam preparation in the preparing exam period, including taking the remedial exam from 0 to 30 hours (remaining time from the first two items to the total load for the item)
36 hour(s) i 0 minuts
Workload structure: 128 hour(s) i 0 minuts (cources), 16 hour(s) i 0 minuts (preparation), 36 hour(s) i 0 minuts (additional work)
Student obligations
Consultations
Literature
Examination methods
Special remarks
Comment
Grade:FEDCBA
Number of pointsless than 50 pointsgreater than or equal to 50 points and less than 60 pointsgreater than or equal to 60 points and less than 70 pointsgreater than or equal to 70 points and less than 80 pointsgreater than or equal to 80 points and less than 90 pointsgreater than or equal to 90 points

Faculty of Science and Mathematics / MATHEMATICS / LINEAR ALGEBRA 2

Course:LINEAR ALGEBRA 2/
Course IDCourse statusSemesterECTS creditsLessons (Lessons+Exercises+Laboratory)
3968Obavezan262+2+0
ProgramsMATHEMATICS
Prerequisites Students are expected to have listened course of Linear algebra I.
Aims Standard course of Linear algebra II for students of mathematics. Includes theory of linear mapping in vector spaces with inner product.
Learning outcomes
Lecturer / Teaching assistantVladimir Jaćimović, Dušica Slović
Methodologylectures, seminars, consultations
Plan and program of work
Preparing weekPreparation and registration of the semester
I week lecturesSpaces with inner product. Hilbert and unitary spaces. Cauchy-Schwarz inequality.
I week exercisesInner product. Axiomatic framework, examples. Inner product of gemetric vectors. Inner product in R^n and C^n.
II week lecturesOrthogonal vectors. Orthonormal vector system. Orthonormal base in vector space. Gramian matrix. Gram-Schmidt orthogonalization algorithm.
II week exercises Orthogonal vectors. Orthonormal vector system. Orthonormal base in vector space. Gramian matrix. Gram-Schmidt orthogonalization algorithm.
III week lecturesQuadratic forms in Hilbert spaces. Sign of the quadratic form. Sylvester's criterion.
III week exercisesQuadratic forms in Hilbert spaces. Reduction of quadratic form to sum of squares by coordinate change.
IV week lecturesReduction of quadratic form to the sum of squares. Lagrange and Jacobi methods. Index of quadratic form. Law of inertia for quadratic forms.
IV week exercisesIndex of quadratic form. Sign of quadratic form. Law of inertia, Sylvester's criterion.
V week lecturesLinear mappings in unitary spaces. Adjoint operator. Existence and uniqueness. Matrix of adjoint operator.
V week exercisesAdjoint operator. Matrix of adjoint operator.
VI week lecturesKernel and image of adjoint operators. Normal operator.
VI week exercisesNormal operator.
VII week lectures1st test
VII week exercises1st test
VIII week lecturesEmpty week.
VIII week exercisesEmpty week.
IX week lecturesUnitary operator. Hermitian operator.
IX week exercisesUnitary operator. Examples and problems.
X week lecturesPositive operators. Square root of operators. Decompositions of operators.
X week exercisesHermitian operators. Square root of operators. Positive operators.
XI week lecturesLinear operator in Hilbert spaces. Symmetric operator.
XI week exercisesSymmetric operator. Eigenvalues of symmetric operator.
XII week lecturesOrthogonal operator. Reduction of orthogonal operator to the composition of simple rotations and reflections.
XII week exercisesOrthogonal operator. Orthogonal matrix.
XIII week lecturesClassification of hypersurfaces of second order in Hilbert spaces.
XIII week exercisesReduction of equation of second order hypersurface to canonical form.
XIV week lecturesLinear operator equations in unitary spaces. Existence and uniqueness of solution. Fredholm alternative.
XIV week exercisesLinear operator equations in unitary spaces. Fredholm alternative.
XV week lectures2nd test
XV week exercises2nd test
Student workload2 hours/week (lectures) + 2 hours/week (seminars) + 3 hours/week (homework) = 7 hours/week. Total: 7 hour/week x 16 week = 112 hours.
Per weekPer semester
6 credits x 40/30=8 hours and 0 minuts
2 sat(a) theoretical classes
0 sat(a) practical classes
2 excercises
4 hour(s) i 0 minuts
of independent work, including consultations 		Classes and final exam:
8 hour(s) i 0 minuts x 16 =128 hour(s) i 0 minuts
Necessary preparation before the beginning of the semester (administration, registration, certification):
8 hour(s) i 0 minuts x 2 =16 hour(s) i 0 minuts
Total workload for the subject:
6 x 30=180 hour(s)
Additional work for exam preparation in the preparing exam period, including taking the remedial exam from 0 to 30 hours (remaining time from the first two items to the total load for the item)
36 hour(s) i 0 minuts
Workload structure: 128 hour(s) i 0 minuts (cources), 16 hour(s) i 0 minuts (preparation), 36 hour(s) i 0 minuts (additional work)
Student obligations
Consultations1 hour/week lectures + 1 hour/week seminars
LiteratureM. Jaćimović, I. Krnić „Linearna algebra, teoreme i zadaci“ (skripta) E. Shikin „Lineinie prostranstva i otobrazheniya“, Moskva 1987. S. Friedberg, A. Insel, L. Spence „Linear algebra, 4th edition“ Pearson, 2002.
Examination methodsattendance (5 points), homework (5x1 points), 2 tests (2x30 points), one corrective test, final exam (30 points), corrective final exam, 2 brief oral exams (optional – 2x5 points)
Special remarksThe language of instruction is Serbo-Croat. Lectures can be given in English or Russian language.
Comment
Grade:FEDCBA
Number of pointsless than 50 pointsgreater than or equal to 50 points and less than 60 pointsgreater than or equal to 60 points and less than 70 pointsgreater than or equal to 70 points and less than 80 pointsgreater than or equal to 80 points and less than 90 pointsgreater than or equal to 90 points

Faculty of Science and Mathematics / MATHEMATICS / ANALYSIS 2

Course:ANALYSIS 2/
Course IDCourse statusSemesterECTS creditsLessons (Lessons+Exercises+Laboratory)
3978Obavezan284+3+0
ProgramsMATHEMATICS
Prerequisites None.
Aims The aim of the course is for students to adopt and master the basics of mathematical analysis: limit theory, elements of differential and integral calculus and the theory of series.
Learning outcomes On successful completion of the course, students will be able to: 1. Define the basic notions of mathematical analysis 2: Riemann integral on a closed interval, area of a curvilinear trapezoid, curve and curve length, volume and area of a solid of revolution, improper integral, convergent series. 2. Derive basic propositions related to the Riemann and improper integral and convergent series. 3. Calculate the Riemann integral as a limit of the sequence of integral sums. 4. Examine and associate the properties of differentiability and integrability of functions of a real variable. 5. Apply some integral formulas. 6. Apply the acquired knowledge to solving different tasks related to the stated content of mathematical analysis. 7. Apply the acquired knowledge to solving real tasks and problems.
Lecturer / Teaching assistantProf. dr Žarko Pavićević - lecturer, Nikola Konatar - teaching assistant
MethodologyLectures, exercises, homework assignments, consultations, written exams.
Plan and program of work
Preparing weekPreparation and registration of the semester
I week lecturesAntiderivative on an open interval. Indefinite integral.
I week exercisesAntiderivative on an open interval. Indefinite integral.
II week lecturesAntiderivative on an interval. Indefinite integral on an interval.
II week exercises Antiderivative on an interval. Indefinite integral on an interval.
III week lecturesDefinition of the Riemann integral. Properties.
III week exercisesDefinition of the Riemann integral. Properties.
IV week lecturesCriteria for the integrability of functions.
IV week exercisesCriteria for the integrability of functions.
V week lecturesProperties of the definite integral and integrable functions.
V week exercisesProperties of the definite integral and integrable functions.
VI week lecturesIntegral and derivative. Some integral functions.
VI week exercisesIntegral and derivative. Some integral functions.
VII week lecturesReview. First midterm exam.
VII week exercisesReview. First midterm exam.
VIII week lecturesFunctions of bounded variation.
VIII week exercisesFunctions of bounded variation.
IX week lecturesApplications of the definite integral.
IX week exercisesApplications of the definite integral.
X week lecturesImproper integral.
X week exercisesImproper integral.
XI week lecturesSeries. Convergence of series.
XI week exercisesSeries. Convergence of series.
XII week lecturesCriteria for the convergence of series with positive terms.
XII week exercisesCriteria for the convergence of series with positive terms.
XIII week lecturesFunctional sequences and series. Uniform convergence.
XIII week exercisesFunctional sequences and series. Uniform convergence.
XIV week lecturesReview. Second midterm exam.
XIV week exercisesReview. Second midterm exam.
XV week lecturesSome applications of Mathematical analysis in natural sciences.
XV week exercisesSome applications of Mathematical analysis in natural sciences.
Student workload
Per weekPer semester
8 credits x 40/30=10 hours and 40 minuts
4 sat(a) theoretical classes
0 sat(a) practical classes
3 excercises
3 hour(s) i 40 minuts
of independent work, including consultations 		Classes and final exam:
10 hour(s) i 40 minuts x 16 =170 hour(s) i 40 minuts
Necessary preparation before the beginning of the semester (administration, registration, certification):
10 hour(s) i 40 minuts x 2 =21 hour(s) i 20 minuts
Total workload for the subject:
8 x 30=240 hour(s)
Additional work for exam preparation in the preparing exam period, including taking the remedial exam from 0 to 30 hours (remaining time from the first two items to the total load for the item)
48 hour(s) i 0 minuts
Workload structure: 170 hour(s) i 40 minuts (cources), 21 hour(s) i 20 minuts (preparation), 48 hour(s) i 0 minuts (additional work)
Student obligations Students are required to attend classes, do the homework assignments and take both midterm exams.
ConsultationsAs agreed with students.
LiteratureV. I. Gavrilov,Ž. Pavićević, Matematička analiza I, D. Adnađević, Z. Kadelburg, Matematička analiza 2, I.M. Lavrentjev, R. Šćepanović, Zbirka zadataka iz mat. analize I, B.P. Demidovič: Zbirka zadataka iz matematičke analize.
Examination methodsTwo homeworks or tests are graded with 8 points (4 points for each homework or test). 2 points are awarded for attendance to lectures and exercises. Two midterm exams are graded with 20 points each (40 points in total). Final exam - 50 points. A passing grade is awarded to students who accumulate at least 50 points.
Special remarks
Comment
Grade:FEDCBA
Number of pointsless than 50 pointsgreater than or equal to 50 points and less than 60 pointsgreater than or equal to 60 points and less than 70 pointsgreater than or equal to 70 points and less than 80 pointsgreater than or equal to 80 points and less than 90 pointsgreater than or equal to 90 points

Faculty of Science and Mathematics / MATHEMATICS / ENGLISH LANGUAGE 3

Course:ENGLISH LANGUAGE 3/
Course IDCourse statusSemesterECTS creditsLessons (Lessons+Exercises+Laboratory)
5547Obavezan332+1+0
ProgramsMATHEMATICS
Prerequisites There are no formal prerequisites; however, the B2.2 level of English is needed to follow the course material.
Aims Mastering basic grammar structures and mathematical terminology, and actively use English for Specific Purposes.
Learning outcomes After passing the exam the student will be able to: - differentiate, understand and use the basic mathematical terminology in English referring to numbers, mathematical operations, fractions, roots, powers, logarithms, equations, inequalities, matrices and functions; understand the messages of popular and expert mathematical texts, as well as general texts, written in English, at the B2.3 level; - independently communicate in an oral and written form in English, at the B2.3 level; - explain his/her ideas by integrating the basic grammar structures and speaking skills, at the B2.3 level.
Lecturer / Teaching assistantDoc. dr Milica Vuković Stamatović
MethodologyA short introduction to the topics covered, with the focus on the participation of students in various types of exercises - conversation and writing, pairwork, groupwork, presentations, discussions etc.
Plan and program of work
Preparing weekPreparation and registration of the semester
I week lecturesMathematical Logic and Foundation; grammar: Past simple vs Past continuous;
I week exercisesPast simple vs Past continuous, exercises
II week lecturesCombinatorics: -ing forms and infinitives;
II week exercises -ing forms and infinitives, exercises
III week lecturesOrdered algebraic structures; grammar: modal verbs must and have to ;
III week exercisesmodal verbs must and have to, exercises
IV week lecturesGeneral algebraic systems; grammar: Present perfect passive;
IV week exercisesPresent perfect passive, exercises
V week lecturesField theory; grammar: conditional sentences
V week exercisesconditional sentences, exercises
VI week lecturesMidterm test
VI week exercisesSpeaking exercises
VII week lecturesRevision, error correction
VII week exercisesRevision, error correction
VIII week lecturesPolynomials; grammar: Time clauses
VIII week exercisesTime clauses, exercises
IX week lecturesNumber theory; grammar: prepositions
IX week exercisesprepositions, exercises
X week lecturesommutative rings and algebras; Present simple vs present continuous
X week exercisesPresent simple vs present continuous, exercises
XI week lecturesAlgebraic geometry; grammar: Reported speech
XI week exercisesReported speech, exercises
XII week lecturesLinear and multilinear algebra; grammar: clauses of contrast
XII week exercisesclauses of contrast, exercises
XIII week lecturesAssociative rings and algebras; grammar: Making predictions
XIII week exercisesMaking predictions, exercises
XIV week lecturesonasociative rings and algebras; grammar: will and would
XIV week exerciseswill and would, exercises
XV week lecturesCategory theory; grammar: certainty
XV week exercisesCertainty, exercises
Student workload
Per weekPer semester
3 credits x 40/30=4 hours and 0 minuts
2 sat(a) theoretical classes
0 sat(a) practical classes
1 excercises
1 hour(s) i 0 minuts
of independent work, including consultations 		Classes and final exam:
4 hour(s) i 0 minuts x 16 =64 hour(s) i 0 minuts
Necessary preparation before the beginning of the semester (administration, registration, certification):
4 hour(s) i 0 minuts x 2 =8 hour(s) i 0 minuts
Total workload for the subject:
3 x 30=90 hour(s)
Additional work for exam preparation in the preparing exam period, including taking the remedial exam from 0 to 30 hours (remaining time from the first two items to the total load for the item)
18 hour(s) i 0 minuts
Workload structure: 64 hour(s) i 0 minuts (cources), 8 hour(s) i 0 minuts (preparation), 18 hour(s) i 0 minuts (additional work)
Student obligations Redovno pohađanje nastave, priprema prezentacije, polaganje kolokvijuma i završnog ispita.
Consultations
LiteratureEnglish for Mathematics. Krukiewicz-Gacek and Trzaska. AGH University of Science and Technology Press: Krakow. 2012. English for Students of Mathematics. Milica Vuković Stamatović - skripta + handouts
Examination methods
Special remarksAdopted on 21-7-2016: http://senat.ucg.ac.me/data/1469020997-Akreditacija%20PMF%202017%20final.pdf
CommentNone
Grade:FEDCBA
Number of pointsless than 50 pointsgreater than or equal to 50 points and less than 60 pointsgreater than or equal to 60 points and less than 70 pointsgreater than or equal to 70 points and less than 80 pointsgreater than or equal to 80 points and less than 90 pointsgreater than or equal to 90 points

Faculty of Science and Mathematics / MATHEMATICS / DISCRETE MATHEMATICS

Course:DISCRETE MATHEMATICS/
Course IDCourse statusSemesterECTS creditsLessons (Lessons+Exercises+Laboratory)
503Obavezan342+1+0
ProgramsMATHEMATICS
Prerequisites
Aims
Learning outcomes
Lecturer / Teaching assistant
Methodology
Plan and program of work
Preparing weekPreparation and registration of the semester
I week lectures
I week exercises
II week lectures
II week exercises
III week lectures
III week exercises
IV week lectures
IV week exercises
V week lectures
V week exercises
VI week lectures
VI week exercises
VII week lectures
VII week exercises
VIII week lectures
VIII week exercises
IX week lectures
IX week exercises
X week lectures
X week exercises
XI week lectures
XI week exercises
XII week lectures
XII week exercises
XIII week lectures
XIII week exercises
XIV week lectures
XIV week exercises
XV week lectures
XV week exercises
Student workload
Per weekPer semester
4 credits x 40/30=5 hours and 20 minuts
2 sat(a) theoretical classes
0 sat(a) practical classes
1 excercises
2 hour(s) i 20 minuts
of independent work, including consultations 		Classes and final exam:
5 hour(s) i 20 minuts x 16 =85 hour(s) i 20 minuts
Necessary preparation before the beginning of the semester (administration, registration, certification):
5 hour(s) i 20 minuts x 2 =10 hour(s) i 40 minuts
Total workload for the subject:
4 x 30=120 hour(s)
Additional work for exam preparation in the preparing exam period, including taking the remedial exam from 0 to 30 hours (remaining time from the first two items to the total load for the item)
24 hour(s) i 0 minuts
Workload structure: 85 hour(s) i 20 minuts (cources), 10 hour(s) i 40 minuts (preparation), 24 hour(s) i 0 minuts (additional work)
Student obligations
Consultations
Literature
Examination methods
Special remarks
Comment
Grade:FEDCBA
Number of pointsless than 50 pointsgreater than or equal to 50 points and less than 60 pointsgreater than or equal to 60 points and less than 70 pointsgreater than or equal to 70 points and less than 80 pointsgreater than or equal to 80 points and less than 90 pointsgreater than or equal to 90 points

Faculty of Science and Mathematics / MATHEMATICS / ALGEBRA 1

Course:ALGEBRA 1/
Course IDCourse statusSemesterECTS creditsLessons (Lessons+Exercises+Laboratory)
3973Obavezan342+2+0
ProgramsMATHEMATICS
Prerequisites Prerequisities do not exist
Aims Introduction to the basic algebraic structures.
Learning outcomes After successful completion of this course the student will be able to: 1. Define the basic algebraic structures: groupoid, semigroup, monoid,group, ring and the field. 2. Describe algebra of sets,the algebra of functions and the algebra of natural numbers. 3. Explain and transmit the notion of lattice and complemented lattice. 4. Explain and transmit the basic notions of group theory such as the notions of subgroup, normal subgroup, factor group, cyclic groups, derived subgroup, group homomorphism and inner automorphism. 5. Prove and apply in practice Lagrange”s theorem and the fundamental theorem of group homomorphisms.
Lecturer / Teaching assistant Sanja Jančić-Rašović
MethodologyLectures, exercises,consultations.
Plan and program of work
Preparing weekPreparation and registration of the semester
I week lecturesThe notion of operation. Properties of operations.The notion of algebraic structure (algebra).
I week exercisesThe notion of operation. Properties of operations.The notion of algebraic structure (algebra).
II week lecturesSubalgebra. Congruence relation. Factor algebra
II week exercises Subalgebra. Congruence relation. Factor algebra
III week lecturesGroupoid. Homomorphism of groupoids.Fundamental theorem of groupoid homomorphisms
III week exercisesGroupoid. Homomorphism of groupoids.Fundamental theorem of groupoid homomorphisms
IV week lecturesSemigroup. Some classes of semigroups
IV week exercisesSemigroup. Some classes of semigroups
V week lecturesAlgebra of natural numbers.Peano axioms.Algebra of sets, relation algebra and the algebra of functions.
V week exercisesAlgebra of natural numbers.Peano axioms.Algebra of sets, relation algebra and the algebra of functions.
VI week lecturesLattices. Boolean algebras.
VI week exercisesLattices. Boolean algebras.
VII week lecturesInterim exam.
VII week exercisesInterim exam.
VIII week lecturesGroups. The basic properties and examples
VIII week exercisesGroups. The basic properties and examples
IX week lecturesSubgroups. The basic properties of subgroups. Lagrange's theorem (group theory).
IX week exercisesSubgroups. The basic properties of subgroups. Lagrange's theorem (group theory).
X week lecturesNormal subgroups. Factor group
X week exercisesNormal subgroups. Factor group
XI week lecturesGroup homomorphism.Fundamental theorem of group homomorphisms.
XI week exercisesGroup homomorphism.Fundamental theorem of group homomorphisms.
XII week lecturesIsomorphism theorems for groups. Inner automorphisms.
XII week exercisesIsomorphism theorems for groups. Inner automorphisms.
XIII week lecturesCyclic groups. Commutator (derived) subgroup
XIII week exercisesCyclic groups. Commutator (derived) subgroup
XIV week lecturesCorrectional exam for interim exam.
XIV week exercisesCorrectional exam for interim exam.
XV week lecturesFree groups.
XV week exercisesFree groups.
Student workloadA week 2 hours of lectures 2 hours of exercise 2 hours and 40 minutes of student work, including consultations During the semester Teachig and the final exam: 16x(5h 20min)=85h i 20 min Necessery preparation (before semester administration, enrollment and verification): 2x5h 20min=10h 40min. Total hours for the course::4x30 =120 hours Additional work : 0 to 24 hours Structure:: 85h 40min(lecture)+10h40min(preparation)+24h (additional work)
Per weekPer semester
4 credits x 40/30=5 hours and 20 minuts
2 sat(a) theoretical classes
0 sat(a) practical classes
2 excercises
1 hour(s) i 20 minuts
of independent work, including consultations 		Classes and final exam:
5 hour(s) i 20 minuts x 16 =85 hour(s) i 20 minuts
Necessary preparation before the beginning of the semester (administration, registration, certification):
5 hour(s) i 20 minuts x 2 =10 hour(s) i 40 minuts
Total workload for the subject:
4 x 30=120 hour(s)
Additional work for exam preparation in the preparing exam period, including taking the remedial exam from 0 to 30 hours (remaining time from the first two items to the total load for the item)
24 hour(s) i 0 minuts
Workload structure: 85 hour(s) i 20 minuts (cources), 10 hour(s) i 40 minuts (preparation), 24 hour(s) i 0 minuts (additional work)
Student obligations Students have to attend lectures and exercises, take interim exam and final exam.
ConsultationsAfter the lectures
LiteratureIntroduction to Algebra ,A.I.Kostrikin, Uvod u opstu algebru,V. Dasic, Zbirka rijesenih zadataka iz Algebre,(I dio),B.Zekovic,V..A..Artimonov Zbirka zadataka iz Algebre, Z.Stojakovic,Z.Mijajlovic
Examination methods- Interim exam 50 points - Final exam 50 points Grade A B C D E 91-100 81-90 71-80 61-70 51-60
Special remarks
Comment
Grade:FEDCBA
Number of pointsless than 50 pointsgreater than or equal to 50 points and less than 60 pointsgreater than or equal to 60 points and less than 70 pointsgreater than or equal to 70 points and less than 80 pointsgreater than or equal to 80 points and less than 90 pointsgreater than or equal to 90 points

Faculty of Science and Mathematics / MATHEMATICS / DISCRETE MATHEMATICS 1

Course:DISCRETE MATHEMATICS 1/
Course IDCourse statusSemesterECTS creditsLessons (Lessons+Exercises+Laboratory)
6593Obavezan353+1+0
ProgramsMATHEMATICS
Prerequisites
Aims
Learning outcomes
Lecturer / Teaching assistant
Methodology
Plan and program of work
Preparing weekPreparation and registration of the semester
I week lectures
I week exercises
II week lectures
II week exercises
III week lectures
III week exercises
IV week lectures
IV week exercises
V week lectures
V week exercises
VI week lectures
VI week exercises
VII week lectures
VII week exercises
VIII week lectures
VIII week exercises
IX week lectures
IX week exercises
X week lectures
X week exercises
XI week lectures
XI week exercises
XII week lectures
XII week exercises
XIII week lectures
XIII week exercises
XIV week lectures
XIV week exercises
XV week lectures
XV week exercises
Student workload
Per weekPer semester
5 credits x 40/30=6 hours and 40 minuts
3 sat(a) theoretical classes
0 sat(a) practical classes
1 excercises
2 hour(s) i 40 minuts
of independent work, including consultations 		Classes and final exam:
6 hour(s) i 40 minuts x 16 =106 hour(s) i 40 minuts
Necessary preparation before the beginning of the semester (administration, registration, certification):
6 hour(s) i 40 minuts x 2 =13 hour(s) i 20 minuts
Total workload for the subject:
5 x 30=150 hour(s)
Additional work for exam preparation in the preparing exam period, including taking the remedial exam from 0 to 30 hours (remaining time from the first two items to the total load for the item)
30 hour(s) i 0 minuts
Workload structure: 106 hour(s) i 40 minuts (cources), 13 hour(s) i 20 minuts (preparation), 30 hour(s) i 0 minuts (additional work)
Student obligations
Consultations
Literature
Examination methods
Special remarks
Comment
Grade:FEDCBA
Number of pointsless than 50 pointsgreater than or equal to 50 points and less than 60 pointsgreater than or equal to 60 points and less than 70 pointsgreater than or equal to 70 points and less than 80 pointsgreater than or equal to 80 points and less than 90 pointsgreater than or equal to 90 points

Faculty of Science and Mathematics / MATHEMATICS / ALGEBRA 1

Course:ALGEBRA 1/
Course IDCourse statusSemesterECTS creditsLessons (Lessons+Exercises+Laboratory)
8574Obavezan352+2+0
ProgramsMATHEMATICS
Prerequisites
Aims
Learning outcomes
Lecturer / Teaching assistant
Methodology
Plan and program of work
Preparing weekPreparation and registration of the semester
I week lectures
I week exercises
II week lectures
II week exercises
III week lectures
III week exercises
IV week lectures
IV week exercises
V week lectures
V week exercises
VI week lectures
VI week exercises
VII week lectures
VII week exercises
VIII week lectures
VIII week exercises
IX week lectures
IX week exercises
X week lectures
X week exercises
XI week lectures
XI week exercises
XII week lectures
XII week exercises
XIII week lectures
XIII week exercises
XIV week lectures
XIV week exercises
XV week lectures
XV week exercises
Student workload
Per weekPer semester
5 credits x 40/30=6 hours and 40 minuts
2 sat(a) theoretical classes
0 sat(a) practical classes
2 excercises
2 hour(s) i 40 minuts
of independent work, including consultations 		Classes and final exam:
6 hour(s) i 40 minuts x 16 =106 hour(s) i 40 minuts
Necessary preparation before the beginning of the semester (administration, registration, certification):
6 hour(s) i 40 minuts x 2 =13 hour(s) i 20 minuts
Total workload for the subject:
5 x 30=150 hour(s)
Additional work for exam preparation in the preparing exam period, including taking the remedial exam from 0 to 30 hours (remaining time from the first two items to the total load for the item)
30 hour(s) i 0 minuts
Workload structure: 106 hour(s) i 40 minuts (cources), 13 hour(s) i 20 minuts (preparation), 30 hour(s) i 0 minuts (additional work)
Student obligations
Consultations
Literature
Examination methods
Special remarks
Comment
Grade:FEDCBA
Number of pointsless than 50 pointsgreater than or equal to 50 points and less than 60 pointsgreater than or equal to 60 points and less than 70 pointsgreater than or equal to 70 points and less than 80 pointsgreater than or equal to 80 points and less than 90 pointsgreater than or equal to 90 points

Faculty of Science and Mathematics / MATHEMATICS / MECHANICS

Course:MECHANICS/
Course IDCourse statusSemesterECTS creditsLessons (Lessons+Exercises+Laboratory)
10151Obavezan353+2+0
ProgramsMATHEMATICS
Prerequisites no
Aims Introducing students to the basic concepts, principles and laws of classical mechanics.
Learning outcomes - That the student understands the basic concepts, principles and laws of mechanics and the role of the mathematical apparatus in their formulation; - That the student develops a feeling for mathematical modeling of movement problems and gains basic experience in formulating and solving them.
Lecturer / Teaching assistantProf. dr Ranislav Bulatović
MethodologyLectures, consultations, independent study and creation of assignments.
Plan and program of work
Preparing weekPreparation and registration of the semester
I week lecturesIntroduction. Space, time, movement, speed and acceleration of a point. Natural components of point acceleration.
I week exercisesIntroduction. Space, time, movement, speed and acceleration of a point. Natural components of point acceleration.
II week lecturesVelocity and acceleration of a point in curvilinear coordinates. I homework.
II week exercises Velocity and acceleration of a point in curvilinear coordinates. I homework.
III week lecturesAxioms of dynamics. Differential equations of motion of a material point. General theorems and first integrals.
III week exercisesAxioms of dynamics. Differential equations of motion of a material point. General theorems and first integrals.
IV week lecturesBasic models of rectilinear motion of a point. Qualitative examination of movement in a conservative force field. II homework.
IV week exercisesBasic models of rectilinear motion of a point. Qualitative examination of movement in a conservative force field. II homework.
V week lecturesMotion in the central force field. Keplers problem.
V week exercisesMotion in the central force field. Keplers problem.
VI week lecturesDynamics of the system of free material points. The two-body problem.
VI week exercisesDynamics of the system of free material points. The two-body problem.
VII week lecturesRecapitulation of the material covered. Preparation for the colloquium.
VII week exercisesRecapitulation of the material covered. Preparation for the colloquium.
VIII week lecturesColloquium
VIII week exercisesColloquium
IX week lecturesKinematics of a rigid body. Angular velocity vector. Eulers theorem. Rivals formula.
IX week exercisesKinematics of a rigid body. Angular velocity vector. Eulers theorem. Rivals formula.
X week lecturesSpecial cases of motion of a rigid body. III homework.
X week exercisesSpecial cases of motion of a rigid body. III homework.
XI week lecturesKinematics and dynamics of complex motion of a point.
XI week exercisesKinematics and dynamics of complex motion of a point.
XII week lecturesDynamics of a non-free system of material points. Lagrangian equations of the first kind. Lagrange-Dalembert principle.
XII week exercisesDynamics of a non-free system of material points. Lagrangian equations of the first kind. Lagrange-Dalembert principle.
XIII week lecturesLagrangian equations of the second kind. IV homework.
XIII week exercisesLagrangian equations of the second kind. IV homework.
XIV week lecturesEquilibrium stability of conservative systems. Small oscillations.
XIV week exercisesEquilibrium stability of conservative systems. Small oscillations.
XV week lecturesHamiltons principle. Hamiltons equations.
XV week exercisesHamiltons principle. Hamiltons equations.
Student workload
Per weekPer semester
5 credits x 40/30=6 hours and 40 minuts
3 sat(a) theoretical classes
0 sat(a) practical classes
2 excercises
1 hour(s) i 40 minuts
of independent work, including consultations 		Classes and final exam:
6 hour(s) i 40 minuts x 16 =106 hour(s) i 40 minuts
Necessary preparation before the beginning of the semester (administration, registration, certification):
6 hour(s) i 40 minuts x 2 =13 hour(s) i 20 minuts
Total workload for the subject:
5 x 30=150 hour(s)
Additional work for exam preparation in the preparing exam period, including taking the remedial exam from 0 to 30 hours (remaining time from the first two items to the total load for the item)
30 hour(s) i 0 minuts
Workload structure: 106 hour(s) i 40 minuts (cources), 13 hour(s) i 20 minuts (preparation), 30 hour(s) i 0 minuts (additional work)
Student obligations Students are required to attend classes regularly, do and hand in homework and take a colloquium.
ConsultationsMondays and Tuesdays from 11 a.m. to 12 p.m
LiteratureWritten lectures; R.D. Gregory, Classical Mechanics, Cambridge, 2006; V. G. Vilke, Teorijska mehanika (na ruskom), MGU, 1998; S.V. Bolotin i dr., Teorijska mehanika (na ruskom), „Akademija“, Moskva, 2010.
Examination methodsAttendance to classes 4; Homeworks 16; Colloquium 35; Final exam 45
Special remarks
Comment
Grade:FEDCBA
Number of pointsless than 50 pointsgreater than or equal to 50 points and less than 60 pointsgreater than or equal to 60 points and less than 70 pointsgreater than or equal to 70 points and less than 80 pointsgreater than or equal to 80 points and less than 90 pointsgreater than or equal to 90 points

Faculty of Science and Mathematics / MATHEMATICS / INTRODUCTION TO GEOMETRY

Course:INTRODUCTION TO GEOMETRY/
Course IDCourse statusSemesterECTS creditsLessons (Lessons+Exercises+Laboratory)
5345Obavezan363+2+0
ProgramsMATHEMATICS
Prerequisites None
Aims This is one of the basic courses in the study of mathematics. Students become familiar with foundations and the main concepts of geometry.
Learning outcomes
Lecturer / Teaching assistantSvjetlana Terzić - teacher, Goran Popivoda - assistant
MethodologyLectures, exercises, doing homework, consultaions
Plan and program of work
Preparing weekPreparation and registration of the semester
I week lecturesIntroduction into absolute geometry. The incidence axioms and their corollaries.
I week exercises
II week lecturesThe ordering axioms and their corollaries.
II week exercises
III week lecturesHalf line, half plane, half space, an angle line and anlge.
III week exercises
IV week lecturesPolygons and polyhedrons.
IV week exercises
V week lecturesCongruence axioms and their corollaries.
V week exercises
VI week lecturesThe main statements of the incidence, ordering and congruence axioms.
VI week exercises
VII week lecturesFirst mid term written exam.
VII week exercises
VIII week lectures
VIII week exercises
IX week lectures
IX week exercises
X week lectures
X week exercises
XI week lectures
XI week exercises
XII week lectures
XII week exercises
XIII week lectures
XIII week exercises
XIV week lectures
XIV week exercises
XV week lectures
XV week exercises
Student workload
Per weekPer semester
6 credits x 40/30=8 hours and 0 minuts
3 sat(a) theoretical classes
0 sat(a) practical classes
2 excercises
3 hour(s) i 0 minuts
of independent work, including consultations 		Classes and final exam:
8 hour(s) i 0 minuts x 16 =128 hour(s) i 0 minuts
Necessary preparation before the beginning of the semester (administration, registration, certification):
8 hour(s) i 0 minuts x 2 =16 hour(s) i 0 minuts
Total workload for the subject:
6 x 30=180 hour(s)
Additional work for exam preparation in the preparing exam period, including taking the remedial exam from 0 to 30 hours (remaining time from the first two items to the total load for the item)
36 hour(s) i 0 minuts
Workload structure: 128 hour(s) i 0 minuts (cources), 16 hour(s) i 0 minuts (preparation), 36 hour(s) i 0 minuts (additional work)
Student obligations
Consultations
Literature
Examination methods
Special remarks
Comment
Grade:FEDCBA
Number of pointsless than 50 pointsgreater than or equal to 50 points and less than 60 pointsgreater than or equal to 60 points and less than 70 pointsgreater than or equal to 70 points and less than 80 pointsgreater than or equal to 80 points and less than 90 pointsgreater than or equal to 90 points

Faculty of Science and Mathematics / MATHEMATICS / ANALYSIS 3

Course:ANALYSIS 3/
Course IDCourse statusSemesterECTS creditsLessons (Lessons+Exercises+Laboratory)
3969Obavezan363+2+0
ProgramsMATHEMATICS
Prerequisites
Aims
Learning outcomes
Lecturer / Teaching assistant
Methodology
Plan and program of work
Preparing weekPreparation and registration of the semester
I week lectures
I week exercises
II week lectures
II week exercises
III week lectures
III week exercises
IV week lectures
IV week exercises
V week lectures
V week exercises
VI week lectures
VI week exercises
VII week lectures
VII week exercises
VIII week lectures
VIII week exercises
IX week lectures
IX week exercises
X week lectures
X week exercises
XI week lectures
XI week exercises
XII week lectures
XII week exercises
XIII week lectures
XIII week exercises
XIV week lectures
XIV week exercises
XV week lectures
XV week exercises
Student workload
Per weekPer semester
6 credits x 40/30=8 hours and 0 minuts
3 sat(a) theoretical classes
0 sat(a) practical classes
2 excercises
3 hour(s) i 0 minuts
of independent work, including consultations 		Classes and final exam:
8 hour(s) i 0 minuts x 16 =128 hour(s) i 0 minuts
Necessary preparation before the beginning of the semester (administration, registration, certification):
8 hour(s) i 0 minuts x 2 =16 hour(s) i 0 minuts
Total workload for the subject:
6 x 30=180 hour(s)
Additional work for exam preparation in the preparing exam period, including taking the remedial exam from 0 to 30 hours (remaining time from the first two items to the total load for the item)
36 hour(s) i 0 minuts
Workload structure: 128 hour(s) i 0 minuts (cources), 16 hour(s) i 0 minuts (preparation), 36 hour(s) i 0 minuts (additional work)
Student obligations
Consultations
Literature
Examination methods
Special remarks
Comment
Grade:FEDCBA
Number of pointsless than 50 pointsgreater than or equal to 50 points and less than 60 pointsgreater than or equal to 60 points and less than 70 pointsgreater than or equal to 70 points and less than 80 pointsgreater than or equal to 80 points and less than 90 pointsgreater than or equal to 90 points

Faculty of Science and Mathematics / MATHEMATICS / PROGRAMMING 1

Course:PROGRAMMING 1/
Course IDCourse statusSemesterECTS creditsLessons (Lessons+Exercises+Laboratory)
3983Obavezan363+2+0
ProgramsMATHEMATICS
Prerequisites
Aims
Learning outcomes
Lecturer / Teaching assistant
Methodology
Plan and program of work
Preparing weekPreparation and registration of the semester
I week lectures
I week exercises
II week lectures
II week exercises
III week lectures
III week exercises
IV week lectures
IV week exercises
V week lectures
V week exercises
VI week lectures
VI week exercises
VII week lectures
VII week exercises
VIII week lectures
VIII week exercises
IX week lectures
IX week exercises
X week lectures
X week exercises
XI week lectures
XI week exercises
XII week lectures
XII week exercises
XIII week lectures
XIII week exercises
XIV week lectures
XIV week exercises
XV week lectures
XV week exercises
Student workload
Per weekPer semester
6 credits x 40/30=8 hours and 0 minuts
3 sat(a) theoretical classes
0 sat(a) practical classes
2 excercises
3 hour(s) i 0 minuts
of independent work, including consultations 		Classes and final exam:
8 hour(s) i 0 minuts x 16 =128 hour(s) i 0 minuts
Necessary preparation before the beginning of the semester (administration, registration, certification):
8 hour(s) i 0 minuts x 2 =16 hour(s) i 0 minuts
Total workload for the subject:
6 x 30=180 hour(s)
Additional work for exam preparation in the preparing exam period, including taking the remedial exam from 0 to 30 hours (remaining time from the first two items to the total load for the item)
36 hour(s) i 0 minuts
Workload structure: 128 hour(s) i 0 minuts (cources), 16 hour(s) i 0 minuts (preparation), 36 hour(s) i 0 minuts (additional work)
Student obligations
Consultations
Literature
Examination methods
Special remarks
Comment
Grade:FEDCBA
Number of pointsless than 50 pointsgreater than or equal to 50 points and less than 60 pointsgreater than or equal to 60 points and less than 70 pointsgreater than or equal to 70 points and less than 80 pointsgreater than or equal to 80 points and less than 90 pointsgreater than or equal to 90 points

Faculty of Science and Mathematics / MATHEMATICS / ENGLISH LANGUAGE 4

Course:ENGLISH LANGUAGE 4/
Course IDCourse statusSemesterECTS creditsLessons (Lessons+Exercises+Laboratory)
5548Obavezan422+1+0
ProgramsMATHEMATICS
Prerequisites No prerequsites
Aims Students need to regularly attend classes, make a presentation and take a mid term and a final exam.
Learning outcomes After students pass the exam they will be able to: -distinguish, understand and use complex mathematical terminology in English from the areas of differential geometry, topology, vector products, mathematical analysis, -read more complex mathematical expressions in English, -understand basic messages of popular and expert texts, -carry out oral and written conversation in English at an intermediate level -present orally a topic in English
Lecturer / Teaching assistantMilica Vuković Stamatović, Savo Kostić
MethodologyLectures and practice. Presentations in English on a topic studied. Studying for mid term and final exams. Consultations.
Plan and program of work
Preparing weekPreparation and registration of the semester
I week lecturesHomological algebra; grammar: Past simple vs Past continuous;
I week exercisesPast simple vs Past continuous;
II week lecturesGroup theory and generalizations: -ing forms and infinitives;
II week exercises ing forms and infinitives;
III week lecturesTopological groups; grammar: modal verbs
III week exercisesmodal verbs exercises
IV week lecturesReal functions; grammar: Present perfect passive;
IV week exercises Present perfect passive;
V week lecturesMeasure and integrations; grammar: conditional sentences
V week exercisesvocabulary exercises
VI week lecturesMidterm test
VI week exercisesMid-term test
VII week lecturesRevision and error correction
VII week exercisesRevision and error correction
VIII week lecturesFunctions of a complex variable;
VIII week exercisesgrammar: revision of clauses
IX week lecturesPotential theory; grammar: prepositions
IX week exercisesrevision of prepositions
X week lecturesCommutative rings and algebras; Present simple vs present continuous
X week exercisesrevision of present tenses
XI week lecturesComplex variables and analytic spaces; grammar: Reported speech
XI week exercisesrevision of indirect speech, advanced
XII week lecturesSpecial functions; grammar: expressing contrast
XII week exercisesexpressing contrast
XIII week lecturesOrdinary differential equations; grammar: Making predictions
XIII week exercisesvocabulary exercises
XIV week lecturesPartial differential equations;
XIV week exercisesrevision of all texts
XV week lecturesPreparation for the final exam
XV week exercisesPreparation for the final exam
Student workload
Per weekPer semester
2 credits x 40/30=2 hours and 40 minuts
2 sat(a) theoretical classes
0 sat(a) practical classes
1 excercises
-1 hour(s) i 40 minuts
of independent work, including consultations 		Classes and final exam:
2 hour(s) i 40 minuts x 16 =42 hour(s) i 40 minuts
Necessary preparation before the beginning of the semester (administration, registration, certification):
2 hour(s) i 40 minuts x 2 =5 hour(s) i 20 minuts
Total workload for the subject:
2 x 30=60 hour(s)
Additional work for exam preparation in the preparing exam period, including taking the remedial exam from 0 to 30 hours (remaining time from the first two items to the total load for the item)
12 hour(s) i 0 minuts
Workload structure: 42 hour(s) i 40 minuts (cources), 5 hour(s) i 20 minuts (preparation), 12 hour(s) i 0 minuts (additional work)
Student obligations Students need to regularly attend classes, make a presentation and take a mid term and a final exam.
Consultationsonce a week for 2 hours
Literature"English for Mathematics" reader
Examination methods
Special remarksAdopted on 21-7-2016: http://senat.ucg.ac.me/data/1469020997-Akreditacija%20PMF%202017%20final.pdf
Comment/
Grade:FEDCBA
Number of pointsless than 50 pointsgreater than or equal to 50 points and less than 60 pointsgreater than or equal to 60 points and less than 70 pointsgreater than or equal to 70 points and less than 80 pointsgreater than or equal to 80 points and less than 90 pointsgreater than or equal to 90 points

Faculty of Science and Mathematics / MATHEMATICS / ALGEBRA 2

Course:ALGEBRA 2/
Course IDCourse statusSemesterECTS creditsLessons (Lessons+Exercises+Laboratory)
3972Obavezan452+2+0
ProgramsMATHEMATICS
Prerequisites None
Aims This course is aimed to introduce students with basic notions in algebra and its applications in mathematical and technical sciences
Learning outcomes On successful completion of this course, students will be able to: - describe the group of symmetry and isometry, direct product of groups and the symmetric group with the proof of the Cayley theorem - examine the structure of a ring in detail and define subrings, ideals, maximal and prime, quotient rings and direct products of rings - prove the Fundamental theorem on homomorphisms of rings, the first and second theorem of isomorphisms of rings with applications - define the characteristic of a ring and prove basic theorems related to it - describe the fraction field - describe the ring of polynomials and polynomial functions and prove the basic theorems about the factorization of polynomials with applications - describe the construction of field extensions and Euclidean rings, especially the Euclid’s algorithm of dividing with residue with applications
Lecturer / Teaching assistantProf.dr Biljana Zeković - lecturer, Dragana Borović - teaching assistant
MethodologyLectures and exercises, consultations, doing homework asignments
Plan and program of work
Preparing weekPreparation and registration of the semester
I week lecturesSymmetrical group. Cayley Theorem
I week exercisesSymmetrical group. Cayley Theorem
II week lecturesGroup of symmetries and isometries
II week exercises Group of symmetries and isometries
III week lecturesDirect product of groups. Some properties
III week exercisesDirect product of groups. Some properties
IV week lecturesRing. Field. Basic properties. (first homework assignment)
IV week exercisesRing. Field. Basic properties. (first homework assignment)
V week lecturesIdeal of ring. Factor-ring.
V week exercisesIdeal of ring. Factor-ring.
VI week lecturesCharacteristic of ring. Homomorphism of rings
VI week exercisesCharacteristic of ring. Homomorphism of rings
VII week lecturesHomomorphism-theorem.
VII week exercisesHomomorphism-theorem.
VIII week lecturesI written exam
VIII week exercisesI written exam
IX week lecturesSubdirect product of rings. Isomorphism-theorems of rings.
IX week exercisesSubdirect product of rings. Isomorphism-theorems of rings.
X week lecturesMaximal and prime ideals. Quotient field. (second homework assignment)
X week exercisesMaximal and prime ideals. Quotient field. (second homework assignment)
XI week lecturesPolynomial ring.
XI week exercisesPolynomial ring.
XII week lecturesRing of polynomial functions.
XII week exercisesRing of polynomial functions.
XIII week lecturesII written exam
XIII week exercisesII written exam
XIV week lecturesExtension of a field (basic concepts).
XIV week exercisesExtension of a field (basic concepts).
XV week lecturesEuclidean ring. (third homework assignment)
XV week exercisesEuclidean ring. (third homework assignment)
Student workload2 hours of lectures, 2 hours of exercises, 1 hour 20 minutes of individual work
Per weekPer semester
5 credits x 40/30=6 hours and 40 minuts
2 sat(a) theoretical classes
0 sat(a) practical classes
2 excercises
2 hour(s) i 40 minuts
of independent work, including consultations 		Classes and final exam:
6 hour(s) i 40 minuts x 16 =106 hour(s) i 40 minuts
Necessary preparation before the beginning of the semester (administration, registration, certification):
6 hour(s) i 40 minuts x 2 =13 hour(s) i 20 minuts
Total workload for the subject:
5 x 30=150 hour(s)
Additional work for exam preparation in the preparing exam period, including taking the remedial exam from 0 to 30 hours (remaining time from the first two items to the total load for the item)
30 hour(s) i 0 minuts
Workload structure: 106 hour(s) i 40 minuts (cources), 13 hour(s) i 20 minuts (preparation), 30 hour(s) i 0 minuts (additional work)
Student obligations Attendance, doing homework assignments, taking two written and the final exam
Consultations1 hour weekly (lectures), 1 hour weekly (exercises)
LiteratureUVOD U OPŠTU ALGEBRU, V. Dašić, ALGEBRA, G. Kalajdžić ZBIRKA REŠENIH ZADATAKA IZ ALGEBRE ( I deo), B. Zeković, V. A. Artamonov ZBIRKA ZADATAKA IZ ALGEBRE, Z.Stojaković, Ž.Mijajlović
Examination methodsThree homework assignments ( 2 points each), two written exams (21 point each) and the final exam (50 points), regular attendance (2 points) Everything is in written form, with oral examination in case of any unclarity or doubt that cheating devices wer
Special remarks
Comment
Grade:FEDCBA
Number of pointsless than 50 pointsgreater than or equal to 50 points and less than 60 pointsgreater than or equal to 60 points and less than 70 pointsgreater than or equal to 70 points and less than 80 pointsgreater than or equal to 80 points and less than 90 pointsgreater than or equal to 90 points

Faculty of Science and Mathematics / MATHEMATICS / DISCRETE MATHEMATICS 2

Course:DISCRETE MATHEMATICS 2/
Course IDCourse statusSemesterECTS creditsLessons (Lessons+Exercises+Laboratory)
6592Obavezan453+1+0
ProgramsMATHEMATICS
Prerequisites
Aims
Learning outcomes
Lecturer / Teaching assistant
Methodology
Plan and program of work
Preparing weekPreparation and registration of the semester
I week lectures
I week exercises
II week lectures
II week exercises
III week lectures
III week exercises
IV week lectures
IV week exercises
V week lectures
V week exercises
VI week lectures
VI week exercises
VII week lectures
VII week exercises
VIII week lectures
VIII week exercises
IX week lectures
IX week exercises
X week lectures
X week exercises
XI week lectures
XI week exercises
XII week lectures
XII week exercises
XIII week lectures
XIII week exercises
XIV week lectures
XIV week exercises
XV week lectures
XV week exercises
Student workload
Per weekPer semester
5 credits x 40/30=6 hours and 40 minuts
3 sat(a) theoretical classes
0 sat(a) practical classes
1 excercises
2 hour(s) i 40 minuts
of independent work, including consultations 		Classes and final exam:
6 hour(s) i 40 minuts x 16 =106 hour(s) i 40 minuts
Necessary preparation before the beginning of the semester (administration, registration, certification):
6 hour(s) i 40 minuts x 2 =13 hour(s) i 20 minuts
Total workload for the subject:
5 x 30=150 hour(s)
Additional work for exam preparation in the preparing exam period, including taking the remedial exam from 0 to 30 hours (remaining time from the first two items to the total load for the item)
30 hour(s) i 0 minuts
Workload structure: 106 hour(s) i 40 minuts (cources), 13 hour(s) i 20 minuts (preparation), 30 hour(s) i 0 minuts (additional work)
Student obligations
Consultations
Literature
Examination methods
Special remarks
Comment
Grade:FEDCBA
Number of pointsless than 50 pointsgreater than or equal to 50 points and less than 60 pointsgreater than or equal to 60 points and less than 70 pointsgreater than or equal to 70 points and less than 80 pointsgreater than or equal to 80 points and less than 90 pointsgreater than or equal to 90 points

Faculty of Science and Mathematics / MATHEMATICS / DIFFERENTIAL EQUATIONS

Course:DIFFERENTIAL EQUATIONS/
Course IDCourse statusSemesterECTS creditsLessons (Lessons+Exercises+Laboratory)
497Obavezan463+2+0
ProgramsMATHEMATICS
Prerequisites None
Aims In this course students get acquainted with simple differential equations, theorems about existence of solutions and methods of solutions. In second part of the course students get to know dynamic systems, phase paths, stability of solutions and position of equilibrium.
Learning outcomes
Lecturer / Teaching assistantNevena Mijajlović
Methodology
Plan and program of work
Preparing weekPreparation and registration of the semester
I week lecturesDifferential equations (DE) in normal form.
I week exercisesDifferential equations in normal form.
II week lecturesDifferential equations in symmetric form.
II week exercises Differential equations in symmetric form.
III week lecturesDE of higher degree. Lowering of degree of DE. Homogenous linear DE of n-order with variable coefficients.
III week exercisesDE of higher degree. Lowering of degree of DE. Homogenous linear DE of n-order with variable coefficients.
IV week lecturesNon-homogenous linear DE of n-order with variable coefficients. Method of constant variation. Homogenous linear DE with constant coefficients.
IV week exercisesNon-homogenous linear DE of n-order with variable coefficients. Method of constant variation. Homogenous linear DE with constant coefficients.
V week lecturesNon-homogenous linear DE of n-order with constant coefficients. Particular solutions. Lowering of degree of LDE of n-order when m linear independent solutions are known.
V week exercisesNonhomogenous linear DE of n-order with constant coefficients. Particular solutions. Lowering of degree of LDE of n-order when m linear independent solutions are known.
VI week lecturesSturm's theorems. Systems of LDE. Method of elimination.
VI week exercisesSturm's theorems. Systems of LDE (SLDE). Method of elimination.
VII week lecturesHomogenous and non-homogenous SLDE with variable coefficients. Method of constant variation.
VII week exercisesHomogenous and non-homogenous SLDE with variable coefficients. Method of constant variation.
VIII week lecturesHomogenous SLDE with constant coefficients. Oiler's and matrix methods. Non-homogenous SLDE with constant coefficients. Particular solutions.
VIII week exercisesHomogenous SLDE with constant coefficients. Oiler's and matrix methods. Non-homogenous SLDE with constant coefficients. Particular solutions.
IX week lecturesTest
IX week exercisesTest
X week lecturesLowering of number of equations. Solving of DE using series.
X week exercisesLowering of number of equations. Solving of DE using series.
XI week lecturesBoundary problem for LDE and SLDE.
XI week exercisesBoundary problem for LDE and SLDE.
XII week lecturesProof of theorem about existence of solution of DE. Dependance of solutions of parameters and initial conditions.
XII week exercisesDependance of solutions of parameters and initial conditions. Examples.
XIII week lecturesDynamical systems. Phase portrait. Stability of solution. Lyapunov and Chataev theorems.
XIII week exercisesDynamical systems. Phase portrait. Stability of solution. Lyapunov and Chataev theorems.
XIV week lecturesPartial DE of first order.
XIV week exercisesPartial DE of first order.
XV week lecturesCorrectional test.
XV week exercisesCorrectional test.
Student workload
Per weekPer semester
6 credits x 40/30=8 hours and 0 minuts
3 sat(a) theoretical classes
0 sat(a) practical classes
2 excercises
3 hour(s) i 0 minuts
of independent work, including consultations 		Classes and final exam:
8 hour(s) i 0 minuts x 16 =128 hour(s) i 0 minuts
Necessary preparation before the beginning of the semester (administration, registration, certification):
8 hour(s) i 0 minuts x 2 =16 hour(s) i 0 minuts
Total workload for the subject:
6 x 30=180 hour(s)
Additional work for exam preparation in the preparing exam period, including taking the remedial exam from 0 to 30 hours (remaining time from the first two items to the total load for the item)
36 hour(s) i 0 minuts
Workload structure: 128 hour(s) i 0 minuts (cources), 16 hour(s) i 0 minuts (preparation), 36 hour(s) i 0 minuts (additional work)
Student obligations
Consultations
Literature
Examination methods
Special remarks
Comment
Grade:FEDCBA
Number of pointsless than 50 pointsgreater than or equal to 50 points and less than 60 pointsgreater than or equal to 60 points and less than 70 pointsgreater than or equal to 70 points and less than 80 pointsgreater than or equal to 80 points and less than 90 pointsgreater than or equal to 90 points

Faculty of Science and Mathematics / MATHEMATICS / ANALYSIS 4

Course:ANALYSIS 4/
Course IDCourse statusSemesterECTS creditsLessons (Lessons+Exercises+Laboratory)
3971Obavezan463+2+0
ProgramsMATHEMATICS
Prerequisites
Aims
Learning outcomes
Lecturer / Teaching assistant
Methodology
Plan and program of work
Preparing weekPreparation and registration of the semester
I week lectures
I week exercises
II week lectures
II week exercises
III week lectures
III week exercises
IV week lectures
IV week exercises
V week lectures
V week exercises
VI week lectures
VI week exercises
VII week lectures
VII week exercises
VIII week lectures
VIII week exercises
IX week lectures
IX week exercises
X week lectures
X week exercises
XI week lectures
XI week exercises
XII week lectures
XII week exercises
XIII week lectures
XIII week exercises
XIV week lectures
XIV week exercises
XV week lectures
XV week exercises
Student workload
Per weekPer semester
6 credits x 40/30=8 hours and 0 minuts
3 sat(a) theoretical classes
0 sat(a) practical classes
2 excercises
3 hour(s) i 0 minuts
of independent work, including consultations 		Classes and final exam:
8 hour(s) i 0 minuts x 16 =128 hour(s) i 0 minuts
Necessary preparation before the beginning of the semester (administration, registration, certification):
8 hour(s) i 0 minuts x 2 =16 hour(s) i 0 minuts
Total workload for the subject:
6 x 30=180 hour(s)
Additional work for exam preparation in the preparing exam period, including taking the remedial exam from 0 to 30 hours (remaining time from the first two items to the total load for the item)
36 hour(s) i 0 minuts
Workload structure: 128 hour(s) i 0 minuts (cources), 16 hour(s) i 0 minuts (preparation), 36 hour(s) i 0 minuts (additional work)
Student obligations
Consultations
Literature
Examination methods
Special remarks
Comment
Grade:FEDCBA
Number of pointsless than 50 pointsgreater than or equal to 50 points and less than 60 pointsgreater than or equal to 60 points and less than 70 pointsgreater than or equal to 70 points and less than 80 pointsgreater than or equal to 80 points and less than 90 pointsgreater than or equal to 90 points

Faculty of Science and Mathematics / MATHEMATICS / PROGRAMMING 2

Course:PROGRAMMING 2/
Course IDCourse statusSemesterECTS creditsLessons (Lessons+Exercises+Laboratory)
3976Obavezan463+2+0
ProgramsMATHEMATICS
Prerequisites
Aims
Learning outcomes
Lecturer / Teaching assistant
Methodology
Plan and program of work
Preparing weekPreparation and registration of the semester
I week lectures
I week exercises
II week lectures
II week exercises
III week lectures
III week exercises
IV week lectures
IV week exercises
V week lectures
V week exercises
VI week lectures
VI week exercises
VII week lectures
VII week exercises
VIII week lectures
VIII week exercises
IX week lectures
IX week exercises
X week lectures
X week exercises
XI week lectures
XI week exercises
XII week lectures
XII week exercises
XIII week lectures
XIII week exercises
XIV week lectures
XIV week exercises
XV week lectures
XV week exercises
Student workload
Per weekPer semester
6 credits x 40/30=8 hours and 0 minuts
3 sat(a) theoretical classes
0 sat(a) practical classes
2 excercises
3 hour(s) i 0 minuts
of independent work, including consultations 		Classes and final exam:
8 hour(s) i 0 minuts x 16 =128 hour(s) i 0 minuts
Necessary preparation before the beginning of the semester (administration, registration, certification):
8 hour(s) i 0 minuts x 2 =16 hour(s) i 0 minuts
Total workload for the subject:
6 x 30=180 hour(s)
Additional work for exam preparation in the preparing exam period, including taking the remedial exam from 0 to 30 hours (remaining time from the first two items to the total load for the item)
36 hour(s) i 0 minuts
Workload structure: 128 hour(s) i 0 minuts (cources), 16 hour(s) i 0 minuts (preparation), 36 hour(s) i 0 minuts (additional work)
Student obligations
Consultations
Literature
Examination methods
Special remarks
Comment
Grade:FEDCBA
Number of pointsless than 50 pointsgreater than or equal to 50 points and less than 60 pointsgreater than or equal to 60 points and less than 70 pointsgreater than or equal to 70 points and less than 80 pointsgreater than or equal to 80 points and less than 90 pointsgreater than or equal to 90 points

Faculty of Science and Mathematics / MATHEMATICS / MEASURE AND INTEGRAL

Course:MEASURE AND INTEGRAL/
Course IDCourse statusSemesterECTS creditsLessons (Lessons+Exercises+Laboratory)
4410Obavezan542+1+0
ProgramsMATHEMATICS
Prerequisites Elementary courses of Analysis and Linear algebra passed.
Aims This course expands the students’ knowledge of Analysis.
Learning outcomes On successful completion of the course, students will be able to: 1. Precisely formulate the differences between finite and infinite sets, provide examples of countable and denumerable sets. They will also be able understand different formulations of the axiom of choice. 2. Explain the concepts of measurable spaces, measurable functions and abstract measure space using illustrative examples. 3. Describe the construction of Lebesgue measure and explain the difference between Jordan measure and Lebesgue measure, and present some corresponding examples. 4. Explain the construction of Lebesgue integral, formulate and prove the basic theorem about Lebesgue integral, including the monotone convergence theorem and the Lebesgue dominated convergence theorem. 5. Present Vitali’s immeasurable sets and examples of non-integrable functions. 6. Explain the different possibilities of proving the existence of mathematical objects with certain properties.
Lecturer / Teaching assistantProf.dr Milojica Jaćimović, lecturer; Nikola Konatar, teaching assistant
MethodologyLectures, exercises, individual homework assignments, consultations
Plan and program of work
Preparing weekPreparation and registration of the semester
I week lecturesSets, cardinality, the axiom of choice – equivalent formulations.
I week exercisesSets, cardinality, the axiom of choice – equivalent formulations.
II week lecturesRing and σ-ring of sets. Borel sets.
II week exercises Ring and σ-ring of sets. Borel sets.
III week lecturesOuter measure. Jordan’s extension of measure.
III week exercisesOuter measure. Jordan’s extension of measure.
IV week lecturesLebesgue’s extension of measure.
IV week exercisesLebesgue’s extension of measure.
V week lecturesMeasures in R^n.
V week exercisesMeasures in R^n.
VI week lecturesMeasurable functions.
VI week exercisesMeasurable functions.
VII week lecturesTest
VII week exercisesTest
VIII week lecturesIntegral of a simple i integral of a positive function
VIII week exercisesIntegral of a simple i integral of a positive function
IX week lecturesElementary theorems of integration.
IX week exercisesElementary theorems of integration.
X week lecturesElementary theorems of integration – continuation.
X week exercisesElementary theorems of integration – continuation.
XI week lecturesIntegrable functions.
XI week exercisesIntegrable functions.
XII week lecturesLebesgue spaces.
XII week exercisesLebesgue spaces.
XIII week lecturesTheorems of measure decomposition. Absolute continuity. Singular measures.
XIII week exercisesTheorems of measure decomposition. Absolute continuity. Singular measures. Correctional test.
XIV week lecturesRadon-Nikodym theorem.
XIV week exercisesRadon-Nikodym theorem.
XV week lecturesImmeasurable sets.
XV week exercisesImmeasurable sets.
Student workload2 hours of lectures, 1 hour of exercises, 2 hours 20 minutes of individual activity, including consultations
Per weekPer semester
4 credits x 40/30=5 hours and 20 minuts
2 sat(a) theoretical classes
0 sat(a) practical classes
1 excercises
2 hour(s) i 20 minuts
of independent work, including consultations 		Classes and final exam:
5 hour(s) i 20 minuts x 16 =85 hour(s) i 20 minuts
Necessary preparation before the beginning of the semester (administration, registration, certification):
5 hour(s) i 20 minuts x 2 =10 hour(s) i 40 minuts
Total workload for the subject:
4 x 30=120 hour(s)
Additional work for exam preparation in the preparing exam period, including taking the remedial exam from 0 to 30 hours (remaining time from the first two items to the total load for the item)
24 hour(s) i 0 minuts
Workload structure: 85 hour(s) i 20 minuts (cources), 10 hour(s) i 40 minuts (preparation), 24 hour(s) i 0 minuts (additional work)
Student obligations Students are required to attend classes, do the homework assignments, and take written exams and the final exam.
ConsultationsAs agreed with the professor or teaching assistant.
LiteratureS. Aljančić: Uvod u realnu i funkcionalnuanalizu, Beograd, Građevinska knjiga; S. Kurepa: Funkcionalna analiza, Zagreb, Školska knjiga
Examination methodsTest 50 points, Final exam 50 points.
Special remarks
Comment
Grade:FEDCBA
Number of pointsless than 50 pointsgreater than or equal to 50 points and less than 60 pointsgreater than or equal to 60 points and less than 70 pointsgreater than or equal to 70 points and less than 80 pointsgreater than or equal to 80 points and less than 90 pointsgreater than or equal to 90 points

Faculty of Science and Mathematics / MATHEMATICS / COMPLEX ANALYSIS 1

Course:COMPLEX ANALYSIS 1/
Course IDCourse statusSemesterECTS creditsLessons (Lessons+Exercises+Laboratory)
3970Obavezan552+2+0
ProgramsMATHEMATICS
Prerequisites Passed Subjects Analysis 1 and Analysis 2.
Aims Through this course, the student is introduced to the Complex Analysis, a classical mathematical disipline which is applicative both in Mathematics and in technical sciences.
Learning outcomes After passing the exam from the subject Complex Analysis I it is expected that the students are able: 1. To define the complex number in the algebraic and in the trigonometric form, to define the operations over complex numbers, to prove their properties and to present a geometric interpretation. 2. To define the metric on the sets C and Ĉ. 3. To define a sequence of complex numbers, the convergence of a sequence and to prove the basic properties of convergent sequences. 4. To define the elementary functions (power, polynomial, rational, exponential, trigonometric, hyperbolic, logarithmic, root, inverse trigonometric and inverse hyperbolic functions), and to prove their properties. 5. To define the differentiability of functions of complex variables. To define the harmonic function. To prove the basic properties. 6. To define the integral of a complex function. To formulate and to prove Cauchy’s theorem and its consequences. 7. To define Laurent series, the isolated singularities, to expand a function in a Laurent series. To determine the type of the singularity of a given function. 8. To define the concept of the residue. To formulate and prove theorems that relate to its application to the calculation of integrals of complex functions. 9. To define the conformal mappings. To formulate and prove theorems that relate to the properties of conformal mappings. 10. To define the bilinear mapping. To indicate and prove their properties.
Lecturer / Teaching assistantProf. dr Jela Šušić
MethodologyLectures, exercises, solving tests and final exam.
Plan and program of work
Preparing weekPreparation and registration of the semester
I week lecturesTo familiarize students with the work plan.Complex numbers, the operations over complex numbers. Geometric interpretation of complex number. The trigonometric form of a complex number.
I week exercisesComplex numbers, the operations over complex numbers. Geometric interpretation of complex number. The trigonometric form of a complex number.
II week lecturesThe extended complex plane, Riemann sphere, the metric on this sphere.
II week exercises The extended complex plane, Riemann sphere, the metric on this sphere.
III week lecturesA sequence of complex numbers, a bounded sequence, the convergent sequence, the properties. The series of numbers. The infinite product.
III week exercisesA sequence of complex numbers, a bounded sequence, the convergent sequence, the properties. The series of numbers. The infinite product.
IV week lecturesThe open and closed sets in C and Ĉ. The compactness.
IV week exercisesThe open and closed sets in C and Ĉ. The compactness.
V week lecturesThe path and the curve in C. The connected set and the domain in C.
V week exercisesThe path and the curve in C. The connected set and the domain in C.
VI week lecturesComplex functions. The limit and the continuity of a function of complex variable
VI week exercisesComplex functions. The limit and the continuity of a function of complex variable
VII week lecturesThe elementary functions: power, polynomial, rational, exponential, trigonometric, hyperbolic, logarithmic, root, general power function. Inverse trigonometric and inverse hyperbolic functions.
VII week exercisesThe elementary functions: power, polynomial, rational, exponential, trigonometric, hyperbolic, logarithmic, root, general power function. Inverse trigonometric and inverse hyperbolic functions.
VIII week lecturesFirst test - written part.
VIII week exercisesAssignments from the first test.
IX week lecturesResults of the first test and analysis of the achieved results. Oral part of the first test.
IX week exercisesResults of the first test and analysis of the achieved results. Oral part of the first test.
X week lecturesThe differentiability of a function of complex variable. The harmonic functions.
X week exercisesThe differentiability of a function of complex variable. The harmonic functions.
XI week lecturesThe integral of a complex function. Cauchy's theorem and Cauchy's integral formula and consequences.
XI week exercisesThe integral of a complex function. Cauchy's theorem and Cauchy's integral formula and consequences.
XII week lecturesLaurent series. The isolated singularities. The residue. Conformal mappings. Bilinear mappings.
XII week exercisesLaurent series. The isolated singularities. The residue. Conformal mappings. Bilinear mappings.
XIII week lecturesSecond test - written part.
XIII week exercisesAssignments from the second test.
XIV week lecturesResults of the second test and analysis of the achieved results. Oral part of the second test.
XIV week exercisesResults of the second test and analysis of the achieved results. Oral part of the second test.
XV week lecturesCorrectional first or second test - written part.
XV week exercisesResults of the correctional tests and analysis of the achieved results. Oral part of the correctional test.
Student workload
Per weekPer semester
5 credits x 40/30=6 hours and 40 minuts
2 sat(a) theoretical classes
0 sat(a) practical classes
2 excercises
2 hour(s) i 40 minuts
of independent work, including consultations 		Classes and final exam:
6 hour(s) i 40 minuts x 16 =106 hour(s) i 40 minuts
Necessary preparation before the beginning of the semester (administration, registration, certification):
6 hour(s) i 40 minuts x 2 =13 hour(s) i 20 minuts
Total workload for the subject:
5 x 30=150 hour(s)
Additional work for exam preparation in the preparing exam period, including taking the remedial exam from 0 to 30 hours (remaining time from the first two items to the total load for the item)
30 hour(s) i 0 minuts
Workload structure: 106 hour(s) i 40 minuts (cources), 13 hour(s) i 20 minuts (preparation), 30 hour(s) i 0 minuts (additional work)
Student obligations Attendance at lectures and exercises, solving tests.
ConsultationsMonday from 18 to 19 hours, office 220.
Literature
Examination methods70 points - 2 tests ( 30+5). 30 points - final exam (25+5). 50 points - minimum requirements to pass the exam.
Special remarks
CommentThe test and final exam are divided in written and oral parts. The written part of tests covers tasks and carries 30 points and oral part covers theory and carries 5 points. The written part of final exam covers tasks and carries 25 points and oral part c
Grade:FEDCBA
Number of pointsless than 50 pointsgreater than or equal to 50 points and less than 60 pointsgreater than or equal to 60 points and less than 70 pointsgreater than or equal to 70 points and less than 80 pointsgreater than or equal to 80 points and less than 90 pointsgreater than or equal to 90 points

Faculty of Science and Mathematics / MATHEMATICS / FUNCTIONAL ANALYSIS

Course:FUNCTIONAL ANALYSIS/
Course IDCourse statusSemesterECTS creditsLessons (Lessons+Exercises+Laboratory)
4099Obavezan553+1+0
ProgramsMATHEMATICS
Prerequisites Elementary courses of Analysis and Linear algebra passed.
Aims This course expands the students’ knowledge of Analysis.
Learning outcomes On successful completion of the course, students will be able to: 1. Explain the concepts and present some examples of metric spaces, topological spaces, normed spaces. They will also be able to define the convergence in a topological space, and explain the concept of complete metric space and basic theorems of metric spaces (Banach fixed point theorem, Baire’s category theorem, Cantor’s intersection theorem). 2. Explain the concept of the linear operator and operator norm. Understand the possibility of application of these concepts and theorems to prove the convergence of numerical methods for solving systems of linear equations. 3. Formulate and prove Hahn-Banach theorem and geometric Hahn-Banach theorem (hyperplane separation theorem). 4. Understand why certain theorems of functional analysis are particularly important (fundamental). 5. Read scientific papers, monographs and reference literature that use concepts and methods of functional analysis.
Lecturer / Teaching assistantProf.dr Milojica Jaćimović, lecturer; Nikola Konatar, teaching assistant
MethodologyLectures, exercises, individual homework assignments, consultations
Plan and program of work
Preparing weekPreparation and registration of the semester
I week lecturesNormed spaces. Metric spaces. Examples.
I week exercisesNormed spaces. Metric spaces. Examples.
II week lecturesTopological spaces. Examples
II week exercises Topological spaces. Examples
III week lecturesConvergence.
III week exercisesConvergence.
IV week lecturesCompleteness. Completion of a metric space. Examples.
IV week exercisesCompleteness. Completion of a metric space. Examples.
V week lecturesSets of the first and second category. Baire theorem.
V week exercisesSets of the first and second category. Baire theorem.
VI week lecturesCompactness. Continuity. Fixed points. (Written exam)
VI week exercisesCompactness. Continuity. Fixed points. (Written exam)
VII week lecturesStudy break
VII week exercisesStudy break
VIII week lecturesContinuity and the theorems of maximum and minimum.
VIII week exercisesContinuity and the theorems of maximum and minimum.
IX week lecturesLinear operator. Norm of a linear operator. The space of bounded linear operators.
IX week exercisesLinear operator. Norm of a linear operator. The space of bounded linear operators.
X week lecturesLinear functional. The space of continuous linear operators. Examples.
X week exercisesLinear functional. The space of continuous linear operators. Examples.
XI week lecturesHahn-Banach theorem. Geometric consequences.
XI week exercisesHahn-Banach theorem. Geometric consequences.
XII week lecturesConvexity. Weak convergence.
XII week exercisesConvexity. Weak convergence.
XIII week lecturesSpecter of a linear operator. (Written exam)
XIII week exercisesSpecter of a linear operator. (Written exam)
XIV week lecturesThe open mapping theorem. The closed graph theorem.
XIV week exercisesThe open mapping theorem. The closed graph theorem.
XV week lecturesHilbert space. Basis in a Hilbert space. Fourier series. Examples.
XV week exercisesHilbert space. Basis in a Hilbert space. Fourier series. Examples.
Student workload3 hours of lectures, 1 hour of exercises, 1 hours 20 minutes of individual activity, including consultations
Per weekPer semester
5 credits x 40/30=6 hours and 40 minuts
3 sat(a) theoretical classes
0 sat(a) practical classes
1 excercises
2 hour(s) i 40 minuts
of independent work, including consultations 		Classes and final exam:
6 hour(s) i 40 minuts x 16 =106 hour(s) i 40 minuts
Necessary preparation before the beginning of the semester (administration, registration, certification):
6 hour(s) i 40 minuts x 2 =13 hour(s) i 20 minuts
Total workload for the subject:
5 x 30=150 hour(s)
Additional work for exam preparation in the preparing exam period, including taking the remedial exam from 0 to 30 hours (remaining time from the first two items to the total load for the item)
30 hour(s) i 0 minuts
Workload structure: 106 hour(s) i 40 minuts (cources), 13 hour(s) i 20 minuts (preparation), 30 hour(s) i 0 minuts (additional work)
Student obligations Students are required to attend classes, do their homework assignments and take written exams.
ConsultationsAs agreed with the professor or teaching assistant.
LiteratureS. Aljančić: Uvod u realnu i funkcionalnu analizu, Beograd, Građevinska knjiga; S. Kurepa: Funkcionalna analiza, Zagreb, Školska knjiga. M.Jaćimović. Skripta
Examination methodsTwo written exams, 30 points each, 60 points in total. Final exam, 40 points.
Special remarks
Comment
Grade:FEDCBA
Number of pointsless than 50 pointsgreater than or equal to 50 points and less than 60 pointsgreater than or equal to 60 points and less than 70 pointsgreater than or equal to 70 points and less than 80 pointsgreater than or equal to 80 points and less than 90 pointsgreater than or equal to 90 points

Faculty of Science and Mathematics / MATHEMATICS / NUMERICAL ANALYSIS

Course:NUMERICAL ANALYSIS/
Course IDCourse statusSemesterECTS creditsLessons (Lessons+Exercises+Laboratory)
502Obavezan562+2+0
ProgramsMATHEMATICS
Prerequisites
Aims
Learning outcomes
Lecturer / Teaching assistant
Methodology
Plan and program of work
Preparing weekPreparation and registration of the semester
I week lectures
I week exercises
II week lectures
II week exercises
III week lectures
III week exercises
IV week lectures
IV week exercises
V week lectures
V week exercises
VI week lectures
VI week exercises
VII week lectures
VII week exercises
VIII week lectures
VIII week exercises
IX week lectures
IX week exercises
X week lectures
X week exercises
XI week lectures
XI week exercises
XII week lectures
XII week exercises
XIII week lectures
XIII week exercises
XIV week lectures
XIV week exercises
XV week lectures
XV week exercises
Student workload
Per weekPer semester
6 credits x 40/30=8 hours and 0 minuts
2 sat(a) theoretical classes
0 sat(a) practical classes
2 excercises
4 hour(s) i 0 minuts
of independent work, including consultations 		Classes and final exam:
8 hour(s) i 0 minuts x 16 =128 hour(s) i 0 minuts
Necessary preparation before the beginning of the semester (administration, registration, certification):
8 hour(s) i 0 minuts x 2 =16 hour(s) i 0 minuts
Total workload for the subject:
6 x 30=180 hour(s)
Additional work for exam preparation in the preparing exam period, including taking the remedial exam from 0 to 30 hours (remaining time from the first two items to the total load for the item)
36 hour(s) i 0 minuts
Workload structure: 128 hour(s) i 0 minuts (cources), 16 hour(s) i 0 minuts (preparation), 36 hour(s) i 0 minuts (additional work)
Student obligations
Consultations
Literature
Examination methods
Special remarks
Comment
Grade:FEDCBA
Number of pointsless than 50 pointsgreater than or equal to 50 points and less than 60 pointsgreater than or equal to 60 points and less than 70 pointsgreater than or equal to 70 points and less than 80 pointsgreater than or equal to 80 points and less than 90 pointsgreater than or equal to 90 points

Faculty of Science and Mathematics / MATHEMATICS / PROBABILITY THEORY

Course:PROBABILITY THEORY/
Course IDCourse statusSemesterECTS creditsLessons (Lessons+Exercises+Laboratory)
3975Obavezan563+2+0
ProgramsMATHEMATICS
Prerequisites It is not conditioned.
Aims Adopt the basic concepts of probability and trained for solving probabilistic tasks.
Learning outcomes After passing this exam student will be able to: 1. Precisely define the basic probabilty notions. 2. Formulate basic theorems. 3. Modele random experiment. 4. Recognizes practical problems which can be solved by Probabilty methods. 5. Use the theoretical results and standard procedures for dealing probablity tasks of medium difficulty.
Lecturer / Teaching assistantGoran Popivoda and Anđela Mijanović
MethodologyLectures, consultations and homeworks.
Plan and program of work
Preparing weekPreparation and registration of the semester
I week lecturesIntroduction to the subject. The concept of random events. Operations with events.
I week exercises
II week lecturesProbability, properties. Borel-Cantelli lemma.
II week exercises
III week lecturesClassical definition of probability. Examples. Conditional probability and independent events.
III week exercises
IV week lecturesThe concept of random variables and probability distribution.
IV week exercises
V week lecturesProbability distribution function. Properties.
V week exercises
VI week lecturesTypes of random variables.
VI week exercises
VII week lectures Important distributions.
VII week exercises
VIII week lecturesRandom vectors, marginal distribution. Independence of random variables.
VIII week exercises
IX week lecturesRandom variables obtained by Borel mapping. Transformation of random vectors.
IX week exercisesColloquium.
X week lecturesExpectation, properties and basic theorems.
X week exercises
XI week lecturesDispersion and correlation. Conditional expectation.
XI week exercises
XII week lecturesCharacteristic functions.
XII week exercises
XIII week lecturesTypes of convergence in probability.
XIII week exercises
XIV week lecturesLaw of large numbers.
XIV week exercises
XV week lecturesSecond colloquium.
XV week exercises
Student workload
Per weekPer semester
6 credits x 40/30=8 hours and 0 minuts
3 sat(a) theoretical classes
0 sat(a) practical classes
2 excercises
3 hour(s) i 0 minuts
of independent work, including consultations 		Classes and final exam:
8 hour(s) i 0 minuts x 16 =128 hour(s) i 0 minuts
Necessary preparation before the beginning of the semester (administration, registration, certification):
8 hour(s) i 0 minuts x 2 =16 hour(s) i 0 minuts
Total workload for the subject:
6 x 30=180 hour(s)
Additional work for exam preparation in the preparing exam period, including taking the remedial exam from 0 to 30 hours (remaining time from the first two items to the total load for the item)
36 hour(s) i 0 minuts
Workload structure: 128 hour(s) i 0 minuts (cources), 16 hour(s) i 0 minuts (preparation), 36 hour(s) i 0 minuts (additional work)
Student obligations Class attendance, taking the colloquiums and last exam.
Consultations
Literature1. S. Stamatović: Vjerovatnoća. Statistika, PMF 2000. 2. G. Grimett and D. Stirzaker: Probability and Random Processes, Oxford University Press, 2012. 3. B. Stamatović S. Stamatović; Zbirka zadataka iz Kombinatorike, Vjerovatnoće i Statistike, PMF 2005.
Examination methodsTwo colloquiums, maximum points are 30, each. Final exam, maximum points are 40. Mark E: from 50 to 59 points, mark D: from 60 to 69 points, mark C: from 70 to 79 points, mark B: from 80 to 89 points, mark A: from 90 to 100 points.
Special remarks
Comment
Grade:FEDCBA
Number of pointsless than 50 pointsgreater than or equal to 50 points and less than 60 pointsgreater than or equal to 60 points and less than 70 pointsgreater than or equal to 70 points and less than 80 pointsgreater than or equal to 80 points and less than 90 pointsgreater than or equal to 90 points

Faculty of Science and Mathematics / MATHEMATICS / ALGEBRA 3

Course:ALGEBRA 3/
Course IDCourse statusSemesterECTS creditsLessons (Lessons+Exercises+Laboratory)
3984Obavezan584+3+0
ProgramsMATHEMATICS
Prerequisites Algebra 1 and Algebra 2
Aims This course is designed as mainly Galois theory course, which means that its ultimate goal is to, by using complex algebraic tools, respond to the classic problem - finding the class of polynomials over the field of rational numbers, which roots can be expressed in terms of its coefficients where only basic operations (addition, substraction, multiplication, division and rational powers) are used. Similarly, these methods provide an answer to the other, classical geometry problems such as trisection angle using compass and ruler. Moving towards this goal, we will cultivate more detailed groups of permutations, group actions and partly group reprezentations.
Learning outcomes After passing this exam, the student should be able to 1. Define the basic notions and theorems about group actions and permutation groups. 2. Prove that S_n and A_n are simple groups for n>4. 3. Consider field extension as a vector space, and to conduct the algorithm for field extension. 4. To prove the Kronecker's theorem on the existence of the splitting field and to know its consequences. 5. Understand group of automorphisms of a field, Galois group and the correspondence between lattice of subgroups and lattice of subfields. 6. Prove that the fifth degree polynomials are not solvable by radicals. 7. Understand how Galois theory solves the classic problem of "solvability of equations by radicals" over some fields.
Lecturer / Teaching assistantVladimir Božović and Dragana Borović
MethodologyLectures, exercises, independent work, and consultations.
Plan and program of work
Preparing weekPreparation and registration of the semester
I week lecturesReview - basic notion of groups, Lagrange's theorem, Group homomorphisms
I week exercisesReview - basic notion of groups, Lagrange's theorem, Group homomorphisms
II week lecturesReview - basic theorems about group isomorphisms, Quotient groups, Permutation groups
II week exercises Review - basic theorems about group isomorphisms, Quotient groups, Permutation groups
III week lecturesGroup actions, Orbits and stabilizers, Simple groups
III week exercisesGroup actions, Orbits and stabilizers, Simple groups
IV week lecturesProof of simplicity of groups S_n and A_n, for n>4, Cauchy-Burnside theorem and its applications in combinatorics
IV week exercisesProof of simplicity of groups S_n and A_n, for n>4, Cauchy-Burnside theorem and its applications in combinatorics
V week lecturesCommutative rings, The notion of Integral domain and field, Field of fractions, Polynomial ring, Irreducible polynomials
V week exercisesCommutative rings, The notion of Integral domain and field, Field of fractions, Polynomial ring, Irreducible polynomials
VI week lecturesProof that the multiplicative group of a finite filed is cyclic, Primitive element of a finite filed, GCD and Eucledian algorithm in a polynomial ring over a filed
VI week exercisesProof that the multiplicative group of a finite filed is cyclic, Primitive element of a finite filed, GCD and Eucledian algorithm in a polynomial ring over a filed
VII week lecturesPrincipal ideal domains and Eucledian domains, Review - vector spaces, Quotient rings, Field extensions
VII week exercisesPrincipal ideal domains and Eucledian domains, Review - vector spaces, Quotient rings, Field extensions
VIII week lecturesAlgebraic extensions, Splitting fields, Kronecker's theorem, Finite fields - Galois fields, Galois group
VIII week exercisesAlgebraic extensions, Splitting fields, Kronecker's theorem, Finite fields - Galois fields, Galois group
IX week lecturesMidterm exam
IX week exercisesMidterm exam
X week lecturesGalois group action on the roots of a polynomial, Separabile extensions, The order of Galois group of the splitting filed of a separabile polynomial, Galois group of polynomial X^m-1
X week exercisesGalois group action on the roots of a polynomial, Separabile extensions, The order of Galois group of the splitting filed of a separabile polynomial, Galois group of polynomial X^m-1
XI week lecturesGalois group as a permutation group, Pure extension and Radical extension, Solvability of polynomials by radicals, Classical formulas for solvability of polynomials of small degree
XI week exercisesGalois group as a permutation group, Pure extension and Radical extension, Solvability of polynomials by radicals, Classical formulas for solvability of polynomials of small degree
XII week lecturesNormal extension and normal closure of a field. Solvable groups - basic theorems, Connection between the notion of solvability of Galois group and solvability of corresponding polynomial
XII week exercisesNormal extension and normal closure of a field. Solvable groups - basic theorems, Connection between the notion of solvability of Galois group and solvability of corresponding polynomial
XIII week lecturesTheorem of nonsolability of a polynomial of fifth degree, Group characters, Galois extensions
XIII week exercisesTheorem of nonsolability of a polynomial of fifth degree, Group characters, Galois extensions
XIV week lecturesFundamenthal theorem of Galois theory - Correspondence theorem, Applications of Galois theory, Applications of Galois theory - using SAGE and related computer algebra systems to compute examples in Galois theory
XIV week exercisesFundamenthal theorem of Galois theory - Correspondence theorem, Applications of Galois theory, Applications of Galois theory - using SAGE and related computer algebra systems to compute examples in Galois theory
XV week lecturesMakeup exam
XV week exercisesMakeup exam
Student workload
Per weekPer semester
8 credits x 40/30=10 hours and 40 minuts
4 sat(a) theoretical classes
0 sat(a) practical classes
3 excercises
3 hour(s) i 40 minuts
of independent work, including consultations 		Classes and final exam:
10 hour(s) i 40 minuts x 16 =170 hour(s) i 40 minuts
Necessary preparation before the beginning of the semester (administration, registration, certification):
10 hour(s) i 40 minuts x 2 =21 hour(s) i 20 minuts
Total workload for the subject:
8 x 30=240 hour(s)
Additional work for exam preparation in the preparing exam period, including taking the remedial exam from 0 to 30 hours (remaining time from the first two items to the total load for the item)
48 hour(s) i 0 minuts
Workload structure: 170 hour(s) i 40 minuts (cources), 21 hour(s) i 20 minuts (preparation), 48 hour(s) i 0 minuts (additional work)
Student obligations Students are encouraged to attend classes regularly, although this is not mandatory. However, it is doubtful that one will do well in the course if you miss too many lectures.
ConsultationsAs agreed with the professor or teaching assistant.
Literature1. Advanced Modern Algebra, Joseph J. Rotman, 2002. ISBN: 0-13-087868-5. 2. Algebra II , Veselin Perić, 1989.
Examination methodsThe forms of testing and grading 1. Midterm exam (up to 45 points) and Final exam (up to 45 points). 2. The points awarded for special commitment (up to 10 points). Grading scale: F (below 50 points), E (50-59 points), D (60-69), C (70-79), B (80-8
Special remarks
CommentIf opportunity to take a makeup test, or correctional final exam is used, then the results achieved on them will be treated as definitive.
Grade:FEDCBA
Number of pointsless than 50 pointsgreater than or equal to 50 points and less than 60 pointsgreater than or equal to 60 points and less than 70 pointsgreater than or equal to 70 points and less than 80 pointsgreater than or equal to 80 points and less than 90 pointsgreater than or equal to 90 points

Faculty of Science and Mathematics / MATHEMATICS / INTRODUCTION TO DIFFERENTIAL GEOMETRY

Course:INTRODUCTION TO DIFFERENTIAL GEOMETRY/
Course IDCourse statusSemesterECTS creditsLessons (Lessons+Exercises+Laboratory)
4291Obavezan642+1+0
ProgramsMATHEMATICS
Prerequisites
Aims
Learning outcomes
Lecturer / Teaching assistant
Methodology
Plan and program of work
Preparing weekPreparation and registration of the semester
I week lectures
I week exercises
II week lectures
II week exercises
III week lectures
III week exercises
IV week lectures
IV week exercises
V week lectures
V week exercises
VI week lectures
VI week exercises
VII week lectures
VII week exercises
VIII week lectures
VIII week exercises
IX week lectures
IX week exercises
X week lectures
X week exercises
XI week lectures
XI week exercises
XII week lectures
XII week exercises
XIII week lectures
XIII week exercises
XIV week lectures
XIV week exercises
XV week lectures
XV week exercises
Student workload
Per weekPer semester
4 credits x 40/30=5 hours and 20 minuts
2 sat(a) theoretical classes
0 sat(a) practical classes
1 excercises
2 hour(s) i 20 minuts
of independent work, including consultations 		Classes and final exam:
5 hour(s) i 20 minuts x 16 =85 hour(s) i 20 minuts
Necessary preparation before the beginning of the semester (administration, registration, certification):
5 hour(s) i 20 minuts x 2 =10 hour(s) i 40 minuts
Total workload for the subject:
4 x 30=120 hour(s)
Additional work for exam preparation in the preparing exam period, including taking the remedial exam from 0 to 30 hours (remaining time from the first two items to the total load for the item)
24 hour(s) i 0 minuts
Workload structure: 85 hour(s) i 20 minuts (cources), 10 hour(s) i 40 minuts (preparation), 24 hour(s) i 0 minuts (additional work)
Student obligations
Consultations
Literature
Examination methods
Special remarks
Comment
Grade:FEDCBA
Number of pointsless than 50 pointsgreater than or equal to 50 points and less than 60 pointsgreater than or equal to 60 points and less than 70 pointsgreater than or equal to 70 points and less than 80 pointsgreater than or equal to 80 points and less than 90 pointsgreater than or equal to 90 points

Faculty of Science and Mathematics / MATHEMATICS / ALGEBRAIC TOPOLOGY

Course:ALGEBRAIC TOPOLOGY/
Course IDCourse statusSemesterECTS creditsLessons (Lessons+Exercises+Laboratory)
1304Obavezan663+1+0
ProgramsMATHEMATICS
Prerequisites
Aims
Learning outcomes
Lecturer / Teaching assistant
Methodology
Plan and program of work
Preparing weekPreparation and registration of the semester
I week lectures
I week exercises
II week lectures
II week exercises
III week lectures
III week exercises
IV week lectures
IV week exercises
V week lectures
V week exercises
VI week lectures
VI week exercises
VII week lectures
VII week exercises
VIII week lectures
VIII week exercises
IX week lectures
IX week exercises
X week lectures
X week exercises
XI week lectures
XI week exercises
XII week lectures
XII week exercises
XIII week lectures
XIII week exercises
XIV week lectures
XIV week exercises
XV week lectures
XV week exercises
Student workload
Per weekPer semester
6 credits x 40/30=8 hours and 0 minuts
3 sat(a) theoretical classes
0 sat(a) practical classes
1 excercises
4 hour(s) i 0 minuts
of independent work, including consultations 		Classes and final exam:
8 hour(s) i 0 minuts x 16 =128 hour(s) i 0 minuts
Necessary preparation before the beginning of the semester (administration, registration, certification):
8 hour(s) i 0 minuts x 2 =16 hour(s) i 0 minuts
Total workload for the subject:
6 x 30=180 hour(s)
Additional work for exam preparation in the preparing exam period, including taking the remedial exam from 0 to 30 hours (remaining time from the first two items to the total load for the item)
36 hour(s) i 0 minuts
Workload structure: 128 hour(s) i 0 minuts (cources), 16 hour(s) i 0 minuts (preparation), 36 hour(s) i 0 minuts (additional work)
Student obligations
Consultations
Literature
Examination methods
Special remarks
Comment
Grade:FEDCBA
Number of pointsless than 50 pointsgreater than or equal to 50 points and less than 60 pointsgreater than or equal to 60 points and less than 70 pointsgreater than or equal to 70 points and less than 80 pointsgreater than or equal to 80 points and less than 90 pointsgreater than or equal to 90 points

Faculty of Science and Mathematics / MATHEMATICS / STATISTICS

Course:STATISTICS/
Course IDCourse statusSemesterECTS creditsLessons (Lessons+Exercises+Laboratory)
1361Obavezan663+2+0
ProgramsMATHEMATICS
Prerequisites Student must pass exam Probability theory.
Aims Adopt statistical concepts and methods and trained for solving statistical tasks.
Learning outcomes After passing this exam, will be able to: 1. Carefully define basic terms. 2. Formulate basic theorems. 3. It is understood that statistical tasks are inverse in relation to probabilistic. 4. Recognize practical problems to be solved by statistical methods. It is able to carry out a statistical analysis of the data and draw conclusions. 5. Using theoretical results and standard procedures for dealing with statistical tasks of medium difficulty.
Lecturer / Teaching assistantSiniša Stamatović and Goran Popivoda.
MethodologyLectures, exercises, consultations, homeworks.
Plan and program of work
Preparing weekPreparation and registration of the semester
I week lecturesCentral limit theorem and applications.
I week exercises
II week lecturesIntroduction in Statistics. main notions.
II week exercises
III week lecturesStatistical inferences. Rao Cramer theorem.
III week exercises
IV week lecturesConfidence intervals.
IV week exercises
V week lecturesSufficient statistics.
V week exercises
VI week lecturesIntroduction to hypothesis testing.
VI week exercises
VII week lecturesFree.
VII week exercises
VIII week lecturesFirst collouquium.
VIII week exercises
IX week lecturesNeyman Pearson theorem and applications.
IX week exercises
X week lecturesInferences about normal model.
X week exercises
XI week lecturesNonparametric hypothesis tests.
XI week exercises
XII week lecturesNonparametric tests.
XII week exercises
XIII week lecturesLinear regression.
XIII week exercises
XIV week lecturesAnalysis of variance.
XIV week exercises
XV week lecturesSecond colloquium.
XV week exercises
Student workload
Per weekPer semester
6 credits x 40/30=8 hours and 0 minuts
3 sat(a) theoretical classes
0 sat(a) practical classes
2 excercises
3 hour(s) i 0 minuts
of independent work, including consultations 		Classes and final exam:
8 hour(s) i 0 minuts x 16 =128 hour(s) i 0 minuts
Necessary preparation before the beginning of the semester (administration, registration, certification):
8 hour(s) i 0 minuts x 2 =16 hour(s) i 0 minuts
Total workload for the subject:
6 x 30=180 hour(s)
Additional work for exam preparation in the preparing exam period, including taking the remedial exam from 0 to 30 hours (remaining time from the first two items to the total load for the item)
36 hour(s) i 0 minuts
Workload structure: 128 hour(s) i 0 minuts (cources), 16 hour(s) i 0 minuts (preparation), 36 hour(s) i 0 minuts (additional work)
Student obligations Class attendance, taking the colloquiums and last exam.
Consultations
LiteratureHogg, McKean, Craig: Introduction to Mathematical Statistics, Pearson.
Examination methodsTwo colloquiums, maximum points are 30, each. Final exam, maximum points are 40. Mark E: from 50 to 59 points, mark D: from 60 to 69 points, mark C: from 70 to 79 points, mark B: from 80 to 89 points, mark A: from 90 to 100 points.
Special remarks
Comment
Grade:FEDCBA
Number of pointsless than 50 pointsgreater than or equal to 50 points and less than 60 pointsgreater than or equal to 60 points and less than 70 pointsgreater than or equal to 70 points and less than 80 pointsgreater than or equal to 80 points and less than 90 pointsgreater than or equal to 90 points

Faculty of Science and Mathematics / MATHEMATICS / PARTIAL EQUATIONS

Course:PARTIAL EQUATIONS/
Course IDCourse statusSemesterECTS creditsLessons (Lessons+Exercises+Laboratory)
3986Obavezan662+2+0
ProgramsMATHEMATICS
Prerequisites Enrolling in this course is not conditioned by passing other courses.
Aims The aim of this course is introducing students to basic notions related to partial differential equations
Learning outcomes On successful completion of this course, students will be able to: 1. Solve linear and quasilinear first order partial differential equations 2. Classify second order partial differential equations 3. Know basic methods for solving all three types of second order partial differential equations 4. Understand the notions of uniqueness and continuous dependance on initial conditions 5. Understand the physical meaning of these equations
Lecturer / Teaching assistantProf. dr Oleg Obradović, mr Nikola Konatar
MethodologyLectures, exercises, consultations
Plan and program of work
Preparing weekPreparation and registration of the semester
I week lecturesLinear and quasilinear first order partial differential equations. Method of characteristics.
I week exercisesLinear and quasilinear first order partial differential equations. Method of characteristics.
II week lecturesSolving linear and quasilinear first order partial differential equations.
II week exercises Solving linear and quasilinear first order partial differential equations.
III week lecturesSecond order linear partial differential equations, general notions. Reducing second order linear partial differential equations to canonic form.
III week exercisesSecond order linear partial differential equations, general notions. Reducing second order linear partial differential equations to canonic form.
IV week lecturesClassifying two variable second order partial differential equations with variable coefficients
IV week exercisesClassifying two variable second order partial differential equations with variable coefficients
V week lecturesDeriving the string equation. Existence of solution to the Cauchy problem for infinite string. (DAlembert formula.)
V week exercisesDeriving the string equation. Existence of solution to the Cauchy problem for infinite string. (DAlembert formula.)
VI week lecturesUniqueness of solution to the Cauchy problem. Continuous dependence of the solution to initial conditions.
VI week exercisesUniqueness of solution to the Cauchy problem. Continuous dependence of the solution to initial conditions.
VII week lecturesVibrating of the half-infinite string. Wave equation in space and plane. (Kirchhoff and Poisson formula)
VII week exercisesVibrating of the half-infinite string. Wave equation in space and plane. (Kirchhoff and Poisson formula)
VIII week lecturesFirst midterm exam.
VIII week exercisesFirst midterm exam.
IX week lecturesParabolic equations, general notions. The maximum and minimum theorem. Uniqueness of solution and continuous dependance on initial conditions.
IX week exercisesParabolic equations, general notions. The maximum and minimum theorem. Uniqueness of solution and continuous dependance on initial conditions.
X week lecturesFourier method for parabolic equations. (First boundary value problem. Second boundary value problem.)
X week exercisesFourier method for parabolic equations. (First boundary value problem. Second boundary value problem.)
XI week lecturesSolving one hyperbolic problem using the Fourier method.
XI week exercisesSolving one hyperbolic problem using the Fourier method.
XII week lecturesElliptic equation, general notions.
XII week exercisesElliptic equation, general notions.
XIII week lecturesGreen function for the Dirichlet problem. (three-dimensional case)
XIII week exercisesGreen function for the Dirichlet problem. (three-dimensional case)
XIV week lecturesSolving the Dirichlet problem on a ball.
XIV week exercisesSolving the Dirichlet problem on a ball.
XV week lecturesFourier method for elliptic equations. Second midterm exam.
XV week exercisesFourier method for elliptic equations. Second midterm exam.
Student workload
Per weekPer semester
6 credits x 40/30=8 hours and 0 minuts
2 sat(a) theoretical classes
0 sat(a) practical classes
2 excercises
4 hour(s) i 0 minuts
of independent work, including consultations 		Classes and final exam:
8 hour(s) i 0 minuts x 16 =128 hour(s) i 0 minuts
Necessary preparation before the beginning of the semester (administration, registration, certification):
8 hour(s) i 0 minuts x 2 =16 hour(s) i 0 minuts
Total workload for the subject:
6 x 30=180 hour(s)
Additional work for exam preparation in the preparing exam period, including taking the remedial exam from 0 to 30 hours (remaining time from the first two items to the total load for the item)
36 hour(s) i 0 minuts
Workload structure: 128 hour(s) i 0 minuts (cources), 16 hour(s) i 0 minuts (preparation), 36 hour(s) i 0 minuts (additional work)
Student obligations Students must attend lectures, do two midterm exams and the final exam.
ConsultationsAs agreed with students.
LiteratureR. Šćepanović, Diferencijalne jednačine, L. Evans, Weak convergence methods in PDEs, E. Pap, A. Takači, Đ. Takači, D. Kovačević, Zbirka zadataka iz parcijalnih diferencijalnih jednačina
Examination methodsTwo midterm exams, graded with a maximum of 25 points each. Final exam is graded with a maximum of 50 points.
Special remarks
Comment
Grade:FEDCBA
Number of pointsless than 50 pointsgreater than or equal to 50 points and less than 60 pointsgreater than or equal to 60 points and less than 70 pointsgreater than or equal to 70 points and less than 80 pointsgreater than or equal to 80 points and less than 90 pointsgreater than or equal to 90 points

Faculty of Science and Mathematics / MATHEMATICS / COMPLEX ANALYSIS 2

Course:COMPLEX ANALYSIS 2/
Course IDCourse statusSemesterECTS creditsLessons (Lessons+Exercises+Laboratory)
4289Obavezan663+2+0
ProgramsMATHEMATICS
Prerequisites
Aims
Learning outcomes
Lecturer / Teaching assistant
Methodology
Plan and program of work
Preparing weekPreparation and registration of the semester
I week lectures
I week exercises
II week lectures
II week exercises
III week lectures
III week exercises
IV week lectures
IV week exercises
V week lectures
V week exercises
VI week lectures
VI week exercises
VII week lectures
VII week exercises
VIII week lectures
VIII week exercises
IX week lectures
IX week exercises
X week lectures
X week exercises
XI week lectures
XI week exercises
XII week lectures
XII week exercises
XIII week lectures
XIII week exercises
XIV week lectures
XIV week exercises
XV week lectures
XV week exercises
Student workload
Per weekPer semester
6 credits x 40/30=8 hours and 0 minuts
3 sat(a) theoretical classes
0 sat(a) practical classes
2 excercises
3 hour(s) i 0 minuts
of independent work, including consultations 		Classes and final exam:
8 hour(s) i 0 minuts x 16 =128 hour(s) i 0 minuts
Necessary preparation before the beginning of the semester (administration, registration, certification):
8 hour(s) i 0 minuts x 2 =16 hour(s) i 0 minuts
Total workload for the subject:
6 x 30=180 hour(s)
Additional work for exam preparation in the preparing exam period, including taking the remedial exam from 0 to 30 hours (remaining time from the first two items to the total load for the item)
36 hour(s) i 0 minuts
Workload structure: 128 hour(s) i 0 minuts (cources), 16 hour(s) i 0 minuts (preparation), 36 hour(s) i 0 minuts (additional work)
Student obligations
Consultations
Literature
Examination methods
Special remarks
Comment
Grade:FEDCBA
Number of pointsless than 50 pointsgreater than or equal to 50 points and less than 60 pointsgreater than or equal to 60 points and less than 70 pointsgreater than or equal to 70 points and less than 80 pointsgreater than or equal to 80 points and less than 90 pointsgreater than or equal to 90 points

Faculty of Science and Mathematics / MATHEMATICS / MATHEMATICAL MODELING

Course:MATHEMATICAL MODELING/
Course IDCourse statusSemesterECTS creditsLessons (Lessons+Exercises+Laboratory)
5762Obavezan663+1+0
ProgramsMATHEMATICS
Prerequisites
Aims
Learning outcomes
Lecturer / Teaching assistant
Methodology
Plan and program of work
Preparing weekPreparation and registration of the semester
I week lectures
I week exercises
II week lectures
II week exercises
III week lectures
III week exercises
IV week lectures
IV week exercises
V week lectures
V week exercises
VI week lectures
VI week exercises
VII week lectures
VII week exercises
VIII week lectures
VIII week exercises
IX week lectures
IX week exercises
X week lectures
X week exercises
XI week lectures
XI week exercises
XII week lectures
XII week exercises
XIII week lectures
XIII week exercises
XIV week lectures
XIV week exercises
XV week lectures
XV week exercises
Student workload
Per weekPer semester
6 credits x 40/30=8 hours and 0 minuts
3 sat(a) theoretical classes
0 sat(a) practical classes
1 excercises
4 hour(s) i 0 minuts
of independent work, including consultations 		Classes and final exam:
8 hour(s) i 0 minuts x 16 =128 hour(s) i 0 minuts
Necessary preparation before the beginning of the semester (administration, registration, certification):
8 hour(s) i 0 minuts x 2 =16 hour(s) i 0 minuts
Total workload for the subject:
6 x 30=180 hour(s)
Additional work for exam preparation in the preparing exam period, including taking the remedial exam from 0 to 30 hours (remaining time from the first two items to the total load for the item)
36 hour(s) i 0 minuts
Workload structure: 128 hour(s) i 0 minuts (cources), 16 hour(s) i 0 minuts (preparation), 36 hour(s) i 0 minuts (additional work)
Student obligations
Consultations
Literature
Examination methods
Special remarks
Comment
Grade:FEDCBA
Number of pointsless than 50 pointsgreater than or equal to 50 points and less than 60 pointsgreater than or equal to 60 points and less than 70 pointsgreater than or equal to 70 points and less than 80 pointsgreater than or equal to 80 points and less than 90 pointsgreater than or equal to 90 points

Faculty of Science and Mathematics / MATHEMATICS / THEORY OF MESAURE

Course:THEORY OF MESAURE/
Course IDCourse statusSemesterECTS creditsLessons (Lessons+Exercises+Laboratory)
5889Obavezan662+2+0
ProgramsMATHEMATICS
Prerequisites
Aims
Learning outcomes
Lecturer / Teaching assistant
Methodology
Plan and program of work
Preparing weekPreparation and registration of the semester
I week lectures
I week exercises
II week lectures
II week exercises
III week lectures
III week exercises
IV week lectures
IV week exercises
V week lectures
V week exercises
VI week lectures
VI week exercises
VII week lectures
VII week exercises
VIII week lectures
VIII week exercises
IX week lectures
IX week exercises
X week lectures
X week exercises
XI week lectures
XI week exercises
XII week lectures
XII week exercises
XIII week lectures
XIII week exercises
XIV week lectures
XIV week exercises
XV week lectures
XV week exercises
Student workload
Per weekPer semester
6 credits x 40/30=8 hours and 0 minuts
2 sat(a) theoretical classes
0 sat(a) practical classes
2 excercises
4 hour(s) i 0 minuts
of independent work, including consultations 		Classes and final exam:
8 hour(s) i 0 minuts x 16 =128 hour(s) i 0 minuts
Necessary preparation before the beginning of the semester (administration, registration, certification):
8 hour(s) i 0 minuts x 2 =16 hour(s) i 0 minuts
Total workload for the subject:
6 x 30=180 hour(s)
Additional work for exam preparation in the preparing exam period, including taking the remedial exam from 0 to 30 hours (remaining time from the first two items to the total load for the item)
36 hour(s) i 0 minuts
Workload structure: 128 hour(s) i 0 minuts (cources), 16 hour(s) i 0 minuts (preparation), 36 hour(s) i 0 minuts (additional work)
Student obligations
Consultations
Literature
Examination methods
Special remarks
Comment
Grade:FEDCBA
Number of pointsless than 50 pointsgreater than or equal to 50 points and less than 60 pointsgreater than or equal to 60 points and less than 70 pointsgreater than or equal to 70 points and less than 80 pointsgreater than or equal to 80 points and less than 90 pointsgreater than or equal to 90 points

Faculty of Science and Mathematics / MATHEMATICS / MARKOV CHAINS

Course:MARKOV CHAINS/
Course IDCourse statusSemesterECTS creditsLessons (Lessons+Exercises+Laboratory)
6909Obavezan663+1+0
ProgramsMATHEMATICS
Prerequisites
Aims
Learning outcomes
Lecturer / Teaching assistant
Methodology
Plan and program of work
Preparing weekPreparation and registration of the semester
I week lectures
I week exercises
II week lectures
II week exercises
III week lectures
III week exercises
IV week lectures
IV week exercises
V week lectures
V week exercises
VI week lectures
VI week exercises
VII week lectures
VII week exercises
VIII week lectures
VIII week exercises
IX week lectures
IX week exercises
X week lectures
X week exercises
XI week lectures
XI week exercises
XII week lectures
XII week exercises
XIII week lectures
XIII week exercises
XIV week lectures
XIV week exercises
XV week lectures
XV week exercises
Student workload
Per weekPer semester
6 credits x 40/30=8 hours and 0 minuts
3 sat(a) theoretical classes
0 sat(a) practical classes
1 excercises
4 hour(s) i 0 minuts
of independent work, including consultations 		Classes and final exam:
8 hour(s) i 0 minuts x 16 =128 hour(s) i 0 minuts
Necessary preparation before the beginning of the semester (administration, registration, certification):
8 hour(s) i 0 minuts x 2 =16 hour(s) i 0 minuts
Total workload for the subject:
6 x 30=180 hour(s)
Additional work for exam preparation in the preparing exam period, including taking the remedial exam from 0 to 30 hours (remaining time from the first two items to the total load for the item)
36 hour(s) i 0 minuts
Workload structure: 128 hour(s) i 0 minuts (cources), 16 hour(s) i 0 minuts (preparation), 36 hour(s) i 0 minuts (additional work)
Student obligations
Consultations
Literature
Examination methods
Special remarks
Comment
Grade:FEDCBA
Number of pointsless than 50 pointsgreater than or equal to 50 points and less than 60 pointsgreater than or equal to 60 points and less than 70 pointsgreater than or equal to 70 points and less than 80 pointsgreater than or equal to 80 points and less than 90 pointsgreater than or equal to 90 points
Mathematica Montisnigri
Mathematica Montisnigri
Ecologica Montenegrina
Ecologica Montenegrina
Digital Herbarium
Digital Herbarium
Publications
Publications
Informator
Informator
EUA
EUA
Bulletins
Bulletins
E-theses
E-theses
Europass
Europass
EURAXESS - network
EURAXESS - network
//