Curricullum Mathematics - · Web viewMean approximation of a given function by a trigonometric polynomial. The convergence of a Fourier series in a given point. The Fourier series

Bachelor’s Degree Study Programme

Affirmed in the Councilof Faculty of Physics and Mathematics

on November 2, 1998

Bachelor’s Degree Programme in Mathematics

Table of Contents GOAL OF PROGRAMME..................................................................................................................3TASKS OF PROGRAMME................................................................................................................3ANNOTATION OF PROGRAMME.....................................................................................................3DESCRIPTION OF PROGRAMMES....................................................................................................3LENGTH OF STUDIES AND SCORE OF SUBJECT MATTERS.............................................................4IMMATRICULATION TERMS...........................................................................................................4REQUIREMENTS IN ASSESSMENT OF STUDIES AND ASSESSMENT ORDER....................................4SYLLABUS OF STUDIES.................................................................................................................5SUBJECT OF CURRICULUM STUDIES PROGRAMME........................................................................6STUDIES MODULE IN TECHNOMATHEMATICS................................................................................8MEANS OF FULFILMENT THE PROGRAMME.................................................................................10COSTS OF STUDIES PER BACHELOR’S DEGREE PROGRAMME STUDENT PER ANNUM (LS.).......12TERMS OF AWARDING ACADEMIC DEGREE’S..............................................................................13POSSIBILITIES OF ACQUIRING NATURAL SCIENCES, SOCIAL SCIENCES AND HUMANITIES AS WELL AS OTHER STUDYING POSSIBILITIES.................................................................................13

University of Latvia, Faculty of Physics and Mathematics 15


GOAL OF PROGRAMMEThe goal of the studies programme for baccalaureate in mathematics is to provide for

academic education in the science Mathematics, maintaining a historically established inheritance of the traditions of the science of Mathematics in Latvia and facilitating further development of a possibly greater number of directions in Mathematics.

TASKS OF PROGRAMMEThe tasks of the studies programme for a baccalaureate is:to offer the students of this programme extended knowledge in one or several

separate directions of Mathematics and their application;to provide the required basis of academic knowledge to prepare highly qualified

professionals for the application of mathematics in national economy (mathematical modelling and mathematical statistics) and to provide for the education of mathematics in all levels;

to prepare the specialists with an independent and creative approach in acquiring the latest achievements of Mathematics and putting them effectively into practice.

ANNOTATION OF PROGRAMMEWithin the baccalaureate’s studies programme, the students obtain basic knowledge

in the fundamental studies subjects of Mathematics and in the subjects which are topical for the practical applications of Mathematics as well as separate selected sub-programmes of Mathematics in the relevant subjects. Equally great attention has been paid to the theoretical aspects of these subjects and to ability of applying the acquired knowledge for the mathematical modelling of nature, technology and social processes. For the emphasis of the unity of theory and practice serve the course and the bachelor’s degree papers.

A significant role in the studies programme is played by the students acquiring proficient application of computing technologies, especially for the usage of solving different mathematical problems as well as by the courses in the Sciences. Within the programme, it is feasible for the students to acquire those courses of the social sciences and Humanities they take interest in.

DESCRIPTION OF PROGRAMMESThe studies for a baccalaureate at the University of Latvia were opened in 1990. The

studies programme has been developed on the basis of up to that time existing studies programmes of the speciality of Mathematics at the faculty of Physics and Mathematics and has been modified in compliance with the corresponding studies programmes of the leading European universities, taking into consideration the historical traditions characteristic of the University of Latvia and the peculiarities of the development of Mathematics in Latvia.

The baccalaureate’s programme in mathematics is the leading studies programme in Mathematics in Latvia’s higher educational establishments.

The studies programmes determinates the subject and the syllabus of the studies, specific requirements for the assessment of the studies and successful fulfilment of the



programmes, secured by academic and financial resources as well as by the required material and informative bases. The programme and their courses are added in the supplement.

Successful mastering of the baccalaureate’s programme makes it possible to continue studies for a master’s degree in Latvia and abroad.

Mastering of the baccalaureate’s programme makes it possible to compete successfully in the multinational labour market.

LENGTH OF STUDIES AND SCORE OF SUBJECT MATTERSThe total score of the baccalaureate’s programme in mathematics is 161 credit points

and the length of the studies for full time form studies is 4 years.The course of the programme are divided into three parts:part A – the compulsory courses (94 credit points, 58% of the common score of

studies programme);part B – the optional courses (52 credit points, 32% of the common score of studies

programme);part C – the free optional courses of studies (15 credit points, 10% common score of

studies programme);

IMMATRICULATION TERMSThe candidates for the baccalaureate’s studies programme in mathematics are subject to

the general immatriculation terms of the University of Latvia and to the orders of the UL Studies Vice–Rector.In addition to the requirements mentioned in the General Terms in 1998, there was

an entrance examination in mathematics (in a written form) at the bachelor’s studies programme in Mathematics. In case of an equal assessment, a candidate took a higher position in the competition with higher average mark in geometry and algebra from the secondary education document (certificate). Without the entrance examinations for the studies at the bachelor’s degree programme in mathematics could register the following candidates: the first three price winners of LR and international mathematics and computer science olympiads; students with an honours diploma of A.Liepa Correspondence Mathematics School; and, in the competition order, the students whose final examination mark of mathematics in the secondary education document (certificate) is not less than 8.

REQUIREMENTS IN ASSESSMENT OF STUDIES AND ASSESSMENT ORDERThe students of the bachelor’s studies programme in Mathematics are subject to the UL

assessment requirements and assessment order.



SYLLABUS OF STUDIES The planning of the bachelor’s studies programme in mathematics is shown in the

following tableA: Compulsory courses 1.sem 2.sem 3.sem 4.sem 5.sem 6.sem 7.sem 8.sem

Programming and Computer Science I 4Algebra I 4Analytical Geometry 4Mathematical Logic 2Calculus I 8Programming and Computer Science II 4Algebra II 8Calculus II 8Programming and Computer Science III 2Calculus III 8Differential Equations I 4Numerical Methods I 2Probability Theory 4Calculus IV 4Numerical Methods II 2Mathematical Statistics 4Numerical Methods III 4Function Theory of Complex Variable

4

Course Paper 4Bachelor’s Work 10

94 22 20 16 10 12 4 10

B: Optional Courses 1.sem 2.sem 3.sem 4.sem 5.sem 6.sem 7.sem 8.sem

Foreign Language 1 1Natural Sciences 4 2 3Special courses in Mathematics 2 2 4 4 15 8 6

52 1 3 2 4 8 17 11 6

C: Free Optional Courses 1.sem 2.sem 3.sem 4.sem 5.sem 6.sem 7.sem 8.sem

15 3 3 3 3 3



Total Score of Credit Points

1.sem 2.sem 3.sem 4.sem 5.sem 6.sem 7.sem 8.sem

161 23 23 21 17 23 20 18 16



SUBJECT OF CURRICULUM STUDIES PROGRAMMEThe subject of the bachelor’s degree programme is shown in the following table:

No. Name of course Credit points

Testing Method

Part A (94 credit points)Algebra I 4 ExamAlgebra II 8 ExamAnalytical Geometry 4 ExamDifferential Equations I 4 Exam Complex Variable Function Theory 4 Exam Calculus I 8 Exam Calculus II 8 Exam Calculus III 8 Exam Calculus IV 4 Exam Mathematical Logic 2 Test Mathematical Statistics 4 Exam Numerical Methods I 2 Test Numerical Methods II 2 Test Numerical Methods III 4 ExamProgramming and Computer Science I 4 Test Programming and Computer Science II 4 Test Programming and Computer Science III 2 Test Probability Theory 4 Test Course Paper 4 DefenceBachelor’s Work 10 Defence

Courses in Natural Sciences (9 credit points)Natural Sciences I 4 Test Natural Sciences II 2 Test Natural Sciences III 3 Exam

Optional Courses in Mathematics (41 credit points)Analytical Solutions 2 Exam Applications of Numerical Methods for Solution of Mathematical Physics and Hydrodynamics Problems

2 Exam

Calculus of Variations 4 Test Correctness of Problems 2 Exam Differential Equations II 4 Exam Differential Geometry 4 Exam Discrete Mathematics 2 Exam Elements of Combinatorics 3 Exam



Equations of Mathematical Physics 4 Exam Fixed Point Method 2 Exam Functional Analysis 4 Test Fundamentals of Actuarial Mathematics 4 Test Fundamentals of Geometry 2 Exam Fuzzy Sets and Systems I 4 ExamFuzzy Sets and Systems II 4 ExamGeneral Theory of Optimal Algorithms 4 Test Integral Equations 4 Exam Integral Splines and their Applications 2 Exam Introduction to Algorithm Theory 2 Exam Introduction to Number Theory 3 ExamLebesque Integrals 4 Exam Mathematical Models in Differential Equations 2 Test Mathematical Models of Chemical Reactors Theory 2 Exam Mathematical Models of Continuous Medium Mechanics

2 Test

Mathematical Principles of Economic Models 2 TestMathematical, Statistical and Special Software Products

4 Test

Methods of Mathematical Physics 4 Exam Methods of Optimisation 4 Exam Microeconomics of Insurance 4 TestNon-linear Boundary Value Problems in Applications

2 Exam

Numerical Methods IV 3 Exam Numerical Methods of Optimisation 4 Test Open - Key Criptography 2 Test Operational Research 4 Exam Optimal Control of Processes 4 ExamPortfolios of Securities and their Management 4 TestPractical Logic I 2 TestPractical Logic II 2 TestPrinciples of Mathematical Modelling 2 Exam Regression analysis 2 Exam Seminar on Program Packages 2 Test Seminar for Data Handling of Continuous Process

2 Test

Solution of Boundary Value Problems in Layered Media

2 Exam

Special Numerical Methods 2 ExamSupplementary Chapters of Mathematical Statistics

4 Exam



Survey Sampling 4 ExamTopology I 2 Exam Topology II 2 Exam

STUDIES MODULE IN TECHNOMATHEMATICSMotivation of Necessity of Studies module. The European Consortium for Mathematics in

Industry (ECMI) has worked out and henceforth, in most leading European universities, carried out an educational conception of mathematicians for the needs of industry and other branches of economy. The Latvian Academy of Sciences and the UL Institute of Mathematics are members of the above mentioned organisation, therefore since 1995 the higher education and science integration project supported financially by the RL Education and Science Ministry, Development of Studies Programme Orientated to Technomathematics, has been made (project director doc. J.Cepitis). As a result, for a baccalaureate’s studies programme in mathematics a supplementary module in technomathematics is opened, which after having mastered it, could serve the students as a basis being awarded a certificate issued by the ECMI Board for a performance of its requirements in the ECMI training centres.Description of Studies module. At the supplementary module of technomathematics

studies the main emphasis is laid upon the analytical and numerical methods of differential equations, the problems of non-linear analysis and optimisation as well as the analysis of regression and the elements of discrete mathematics. An essential studies component is seminars in mathematical modelling, which provide for the students’ active participation in the working out the process of mathematical modelling, in its analysis, in solving a mathematical problem, making a written report and giving its verbal presentation.

Subject of Supplementary module. A supplementary module in technomathematics studies records the optional courses of the bachelor’s programme in mathematics, part B, with the common score of 44 credit points, leaving a selection relevant to the specialisation with the score of 8 credit points. A course paper is substituted by seminars in mathematical modelling (e.g. Seminar in Programme Packages, Seminar in Continuous Data Processing) envisaged for performance.

Compulsory courses, part B (44 credit points)

Natural Sciences I 4 Test

Natural Sciences II 2 Test

Natural Sciences III 3 Exam

Differential Equations II 4 Exam

Discrete Mathematics 2 Exam

Functional Analysis 4 Test

Equations of Mathematical Physics 4 Exam

Principles of Mathematical Modelling 2 Exam



Operations Research 4 Exam

Methods of Optimisation 4 Exam

Regressions analysis 2 Exam

Numerical methods IV 3 Exam Applications of Numerical Methods for Solution of Mathematical Physics and Hydrodynamics Problems

2 Exam

Topology I 2 Exam

Optional courses, part B (8 credit points)

Analytical Solutions 2 Exam

Integral Splines and their Applications 2 Exam

Correctness of Problems 2 Exam Solution of Boundary Value Problems in Layered Media

2 Exam

Mathematical Models of Mechanics in Continuous Environment

2 Test

Non-linear boundary Value Problems in Applications

2 Exam

Note. The optional courses of the technomathematics supplementary module from part B can be substituted by the relevant master’s degree studies courses.



MEANS OF FULFILMENT THE PROGRAMME List of Teaching StaffN0. Name, surname Academic Title Scientific Degree

The compulsory (A) part of the programme is fulfilled by

Baiba ABOLTINA Lecturer M.Math.

Svetlana ASMUSS Docent Dr.Math.

Mihails BELOVS Docent Dr.Math.

Margarita BUIKE Lecturer M.Math.

Viktorija CARKOVA Docent Dr.Math.

Jānis CEPITIS Docent Dr.Math.

Andrejs CIBULIS Docent Dr.Math.

Janis CIRULIS Docent Dr.Math.

Teodors CIRULIS Professor Dr.Hab.Math.

Harijs KALIS Professor Dr.Hab.Math., Dr.Hab.Phys.

Halina LAPINA Lecturer M.Math.

Janis LAPINS Docent Dr.Math.

Janis MENCIS Docent Dr.Paed.

Rasma MILLERE Asistents M.Math.

Visvaldis NEIMANIS Lecturer M.Math.

Laila NIEDRITE Lecturer M.Comp.Sc.

Janis SMOTROVS Lecturer M.Math.

Aleksandrs SOSTAKS Professor Dr.Hab.Math.

Viesturs VEZIS Lecturer M.Comp.Sc.

Peteris ZARINS Docent Dr.Paed.

Arta ZODZINA Assistant M.Math.

The optional (B) part of the programme is fulfilled by

Agnis ANDZANS Professor Dr.Hab.Math.



Aivars BERZINS Docent Dr.Math.

Andris BUIKIS Professor Dr.Hab.Math.

Leonids BULIGINS Docent Dr.Phys.

Inese BULA Docent Dr.Math.

Janis BULS Docent Dr.Math.

Silvija CERANE Docent Dr.Math.

Sandris LACIS Lecturer Dr.Phys.

Andris LIEPINS Docent Dr.Math.

Ojars LIETUVIETIS Docent Dr.Math.

Visvaldis NEIMANIS Lecturer M.Math.

Uldis RAITUMS Professor Dr.Hab.Math.

Liga RAMANA Lecturer M.Math.

Janis VUCANS ASS.PROFESSOR Dr.Math.



COSTS OF STUDIES PER BACHELOR’S DEGREE PROGRAMME STUDENT PER ANNUM (LS.)No. Line No Calculation formula Calcula

ted value

A B CCalculation of individual earnings for one teacher per student annually.

Title Average pay of teaching

staff per month

Ratio of teaching staff to provide for studies programme

Professor 350 0,25 1 D1=A1*B1 99,3Associates professor 280 0,04 2 D2=A2*B2 0Docent 125 0,58 3 D3=A3*B3 72,57Lecturer 102 0,13 4 D4=A4*B4 11Assistant 80 0 5 D5=A5*B5 0Assistant 6 D6=A6*B6 0

D7=(D1+D2+D3+D4+D5+D6)*12 2194,44one teachers average pay per annum, Ls. 7average students’ number per teacher 8 4One teacher’s pay per student per annum, Ls 9 D9=D7/D8 548,61number of other employees (except teachers) and ratio 10 1,5

ratio of individual earnings fund of other employees stud.progr.

11 2,4Other employees pay per student per annum, Ls. 12 D12=D9*D10 342,88

N1 Individual earnings fund per student per annum, LLs.Ls.

13 D13=D9+D12 891,49

N2 Employer’s social payments per student per annum (28%) 14 D14=D13*0,28 249,62

N3 Costs of business and service trips per student per annum 15 15costs of postal and other services per student per annum 16 1other services (copying, typography, fax etc.) 17 24

N4 Payment of services – total, Ls. 18 D18=D16+D17 25supply of teaching aids and materials per student per annum 19 4,5stationery and low – price inventory 20 5,5

N5 Supply of materials and low – price inventory per student per annum, Ls 21 D21=D19+D20 10textbooks per student per annum 22 14length of wear and tear of textbooks (in years) 23 10price of 1 textbook 24 30costs of textbook supply per student per annum, Ls. 25 D25=D22*D24/D23 42costs of journal supply per student per annum, Ls. 26 3

N6 Costs of textbook and journal supply per student per annum, Ls. 27 D27=D25+D26 45sports per student per annum Ls. 32 8amateur activity per student per annum, Ls 33 8

N7 Students social welfare per student per annum, Ls. 34 D34=D32+D33 16supply of equipment per student per annum, Ls.

Ls. 35 45,5investments for modernisation of equipment – 20%of inventory 36 0,2costs for modernisation of equipment 37 D37=D35*D36 9,1

N8 Costs of supply and modernisation of equipment per student 38 D38=D35+D37 54,6

Total direct costs per student per annum, Ls. 39 D39=D13+D14+D15+D18+D21+D27+D334+D38

1306,71

N9 Expenses for UL library (3% of existing means of 40 D40=D39*0,0309 40,38structural unit), Ls.

N1 Allocated expenses for operation of UL per student per annum (26.7%), LsLs

41 D41=(D39+D40)*0,36 490,34



Gross total costs per student per annum 42 D42=D39+D40+D41 1837,43

TERMS OF AWARDING ACADEMIC DEGREE’STo obtain the academic baccalaureate in mathematics, the candidate is required to fulfil

the baccalaureate studies programme, pass the exam and defend the bachelor’s thesis.The bachelor’s examination and the bachelor’s thesis are accepted by the

Baccalaureate Examination Committee, which due to the proposal of the Council of the faculty of Physics and Mathematics, is affirmed by the order of the UL Rector. The Board of the Sub-programmes of the Mathematics Sciences determinate the bachelor’s examination papers, which are connected with the courses of the bachelor’s programme in mathematics from part A. The Common Score of the bachelor’s thesis is 10 credit points, and its themes to the students are offered by the Chairs. The themes are affirmed in the University of Latvia in the established order. The principle show to work out and design the bachelor’s thesis as well as how to assess the thesis, are described in the document Writing and Defence of Bachelor and Master’s Thesis in Mathematics.

POSSIBILITIES OF ACQUIRING NATURAL SCIENCES, SOCIAL SCIENCES AND HUMANITIES AS WELL AS OTHER STUDYING POSSIBILITIES

The bachelor’s studies programme in mathematics offers the students to acquire the courses in the social, natural and humanitarian sciences with the common score of 28 credit points; furthermore, from part B, it is possible for the students to select separate social or humanitarian curses from the professional mathematician – statistician or secondary school teacher studies programmes. The students are offered to add to their knowledge in foreign languages, if necessary – in the Latvian language, as well as to go in for physical education and sports. These possibilities are provided by the UL organisational activities and the relevant structural units.



Curricullum Mathematics

Course Abstracts

University of LatviaRīga, 2000



TABLE OF CONTENTSCOMPULSORY COURSES (PART A)..................................................................................................................17

ALGEBRA I..............................................................................................................................................18ALGEBRA II.............................................................................................................................................19ANALYTIC GEOMETRY........................................................................................................................20CALCULUS I............................................................................................................................................22CALCULUS II...........................................................................................................................................24CALCULUS III.........................................................................................................................................26CALCULUS IV.........................................................................................................................................28COMPLEX VARIABLE FUNCTION THEORY.....................................................................................30DIFFERENTIAL EQUATIONS - I...........................................................................................................31MATHEMATICAL LOGIC......................................................................................................................33MATHEMATICAL STATISTICS............................................................................................................34NUMERICAL METHODS I.....................................................................................................................36NUMERICAL METHODS II....................................................................................................................38NUMERICAL METHODS III..................................................................................................................40PROBABILITY THEORY........................................................................................................................42PROGRAMMING AND COMPUTERS I................................................................................................43PROGRAMMING AND COMPUTERS II...............................................................................................45PROGRAMMING AND COMPUTERS III.............................................................................................46

OPTIONAL COURSES IN MATHEMATICS (PART B).........................................................................................47ANALYTICAL SOLUTIONS..................................................................................................................48APPLICATIONS OF NUMERICAL METHODS FOR SOLUTIONS OF MATHEMATICAL PHYSICS AND HYDRODYNAMICS PROBLEMS...............................................................................50APPLIED PROBLEMS OF OPTIMISATION IN ECONOMICS AND MANAGEMENT SCIENCE. .51CALCULUS OF VARIATIONS...............................................................................................................53CORRECTNESS OF PROBLEMS...........................................................................................................54DIFFERENTIAL EQUATIONS - II.........................................................................................................55DIFFERENTIAL GEOMETRY................................................................................................................55ELEMENTS OF COMBINATORICS......................................................................................................56EQUATIONS OF MATHEMATICAL PHYSICS....................................................................................57FIXED POINT METHOD.........................................................................................................................58FUNCTIONAL ANALYSIS.....................................................................................................................60FUNDAMENTALS OF ACTUARIAL MATHEMATICS......................................................................62FUNDAMENTALS OF GEOMETRY.....................................................................................................64FUZZY SETS AND SYSTEMS I.............................................................................................................66FUZZY SETS AND SYSTEMS II............................................................................................................68GENERAL THEORY OF OPTIMAL ALGORITHMS...........................................................................70INTEGRAL EQUATIONS........................................................................................................................72INTEGRAL SPLINES AND THEIR APPLICATIONS...........................................................................74INTRODUCTION TO ALGORITHM THEORY.....................................................................................75INTRODUCTION TO NUMBER THEORY............................................................................................77LEBESGUE INTEGRAL..........................................................................................................................78MATHEMATICAL MODELS IN THE DIFFERENTIAL EQUATIONS..............................................79



MATHEMATICAL MODELS OF CHEMICAL REACTOR THEORY.................................................80MATHEMATICAL MODELS OF CONTINUOUS MEDIUM MECHANICS......................................81MATHEMATICAL PRINCIPLES OF ECONOMIC MODELS..............................................................82MATHEMATICAL, STATISTICAL AND SPECIAL SOFTWARE PRODUCTS................................84METHODS OF MATHEMATICAL PHYSICS.......................................................................................85METHODS OF OPTIMISATION.............................................................................................................86MICROECONOMICS OF INSURANCE.................................................................................................87NON-LINEAR BOUNDARY VALUE PROBLEMS IN APPLICATIONS............................................89NUMERICAL METHODS IV..................................................................................................................90NUMERICAL METHODS OF OPTIMISATION....................................................................................92OPEN–KEY CRYPTOGRAPHY..............................................................................................................94OPERATIONAL RESEARCH..................................................................................................................96OPTIMAL CONTROL OF PROCESSES.................................................................................................98PORTFOLIOS OF SECURITIES AND THEIR MANAGEMENT.......................................................100PRACTICAL LOGIC I............................................................................................................................102PRACTICAL LOGIC II..........................................................................................................................103PRINCIPLES OF MATHEMATICAL MODELLING...........................................................................104REGRESSION ANALYSIS....................................................................................................................105SEMINAR FOR DATA HANDLING OF CONTINUOUS PROCESS.................................................106SEMINAR OF PROGRAM PACKAGES..............................................................................................107SOLUTION OF BOUNDARY PROBLEMS IN LAYERED MEDIA...................................................108SPECIAL NUMERICAL METHODS....................................................................................................109SUPPLEMENTARY CHAPTERS OF MATHEMATICAL STATISTICS...........................................110SURVEY SAMPLING............................................................................................................................112TOPOLOGY I..........................................................................................................................................114TOPOLOGY II........................................................................................................................................116

COURSES IN NATURAL SCIENCES.................................................................................................................118NATURAL SCIENCES I (THEORETICAL MECHANICS)................................................................119NATURAL SCIENCES II (THEORY OF ELECTROMAGNETIC FIELD)........................................121NATURAL SCIENCES III (MATHEMATICAL MODELS OF PHYSICAL PROCESSES)..............122



COMPULSORY COURSES (PART A)



ALGEBRA I(Algebra)

Course code Mate - 1001Author Lecturer Baiba Aboltina, M.Math.,Credits 4 creditsRequired for grade examPrerequisites noneAnnotation. The objectives of the course are: Methods of solving of Dependent Linear Systems, methods of calculation of Determinant, Matrix Algebra, the Complex Numbers’ plane.Subjects:1. Introduction lecture. Determination of Dependent Linear Systems.2. The equivalent of Dependent Systems. Solving of Dependent Linear Systems by the

Gauss method.3. Second order determinant, its features and usage. Third order determinant, its usage.4. High order determinant, its features.5. Methods of calculation of “N” order determinant. The Kramer Formula general usage.6. The compound of Complex Numbers. Algebraic form of Complex Numbers.

Operations with Complex Numbers.7. Trigonometric form of Complex Numbers. Operations with Complex Numbers in

Trigonometric form. General form of complex numbers.8. Calculation of matrix and multiplication with another number. Multiplication of

matrix, its features.9. Identity and inverse matrix, its features.10.Calculation of inverse matrix by the Gauss method. Elementary matrix.11.Determination and features of Linear Space Vectors. Subspace system. Isomorphism.12.Linear association and vector systems. Linear co-ordinate system. The base of co-

ordinate systems and numbers of dimensions.13.Features of arithmetical spaces. Bond between space systems with different bases.14.Rank of matrix, its features. Minor of matrix’ base.15.The maximal number of independent row and column of matrix. Theorem of

Kronecker-Kapelli. Necessary and sufficient conditions of nullification of Determinant.16.The fundamental system of decision of Linear Homogeneous Dependent Systems.Requirement for receiving of the credits: 32 hours lectures, 32 hours practice work.

1. Personal work “Complex Numbers”2. Practical writing test “Determinants and Matrix Algebra”4. Practical writing test “Spaces of linear vectors and rank of matrix” (One hour)

Recommended literature:1. N.Eņģele. Ievads algebrā. LVU, R.1981.2. N.Eņģele. Ievads matemātisko struktūru teorijā. LVU, R.1984.3. Кострикин А.И. Введение в алгебру.4. Тальцев А.И. Основы линейной алгебры.5. Гельфанд И.М. лекции по линейной алгебре.6. Фаддеев Д.К., Соминский И.С. Сборник задач по высшей алгебре.7. Проскуряков И.В. Сборник задач по линей8. Сборник задач по алгебре под ред. А.И.Кострикина.



ALGEBRA II(Algebra II)

Course code Mate – 1002 Authors Lecturer Baiba Aboltina, M.Math.,Credits 2 creditsRequired for grade examPrerequisites Mate - 1001Annotation. Polynomials and theirs roots. Conception of general coherency, group, circle and field theory. Linear Algebra paragraphs: linear vector space, linear operators and Euklide’ space, polynomial matrix calculation and quadrant form.Subjects:1. Polynomial, its root. Euklide’ algorithm, Taylor’s formula.2. General theory of Algebra. Calculation and limits of real root of real polynomial.3. Polynomials with various arguments, symmetrical polynomials.4. Unity, sub-unity, its characteristics. Reflection, its characteristics.5. Conception of group. Sub-group, cyclic group.6. Close classes of groups. Factor-group.7. Homorphism and isomorphism of group. Symmetrical groups.8. Circle and area inside the circle. Factor-circle. Full area.9. Linear operators, proper value and proper vector.10. Euklid’ space, isomorphism. Ortonormalised base.11.Scalar and geometrical multiplication. Orthogonal transformation.12. Symmetrical and orthogonal operator.13. Quadrate form its canonical and normal perform.14. Positive determined form. Quadrate form of pair.15. Polynomial matrixes, their canonical perform.16. Jordan’ matrix. Normal form of Jordan’ matrix.Requirements for receiving of credits: 32 hours lectures, 32 hours practise1. One test works in theory “Definitors” — theory of mathematical structure.2. 3 practical test works (30 min.) — polynomials, mathematical; structure, linear operators.3. 4 individual home works — polynomials, mathematical structure, quadrate form, polynomial matrix.Recommended literature:1. N.Eņģele. Ievads matemātisko struktūru teorijā. LVU, R.1984.2. N.Eņģele. Lineārās algebras papildu nodaļas. -LVU,R.1985.3. Кострикин А.И. Введение в алгебру.4. Куликов Л.Я. Алгебра и теория чисел.5. Курош А.Г. Курс высшей алгебры.6. Сборник задач по алгебре – под ред. А.И.Кострикина.7. Фаддеев Д.К., Соминский И.Сю Сборник задач по высшсей алгеьре.8. Проскеряков И.В. Сборнил задач по линейной алгебре.9. Шрамов Щ.Д. Задачник по линейной алгебре.



ANALYTIC GEOMETRY(Analītiskā ģeometrija)

Course code Mate - 1011Author Docent D.Taimina, Dr.Math.,Credits 4 creditsRequired for grade examPrerequisites

Annotation.Course review main conceptions of analytic Geometry and its use on Flatness and

in Space.

Subjects:1. Vectors

1.1 Conception of Vector. Linear operation with vectors. Vector linear equability and non-equability. Base. Co-ordinates.1.2 Vector scalar multiplication.1.3 Vector multiplication of vectors.1.4 Vector mixing multiplication.1.5 Using of vector plane geometry exercises.

2. Various co-ordinate systems on flatness and in space.2.1 Polar co-ordinate system on flatness and in space, sphere and cylinder space coordinate system.2.2 Transformation of Decart’s co-ordinate system on flatness and in space.

3. Co-ordinate system on straight line and on flatness.3.1 Straight line on flatness. Similar straight lines.3.2 Straight line and flatness in space. Similar flatness and straight lines. Array of parallel flatness and straight lines in space.

4. Canonical form of the Curve of 2-nd order.4.1 Curve of 2-nd order as part of cone. Ellipse, canonical view, characteristics.4.2 Hyperbole, canonical view, characteristics.4.3 Parabola, canonical view, characteristics.4.4 Curve of 2-nd order in the Polar co-ordinate system. Hyperbole asymptotical approaching.

5. Canonical bodies with 2-nd order surfaces.5.1 Sphere, cylinder and canonical bodies. Rotation bodies.5.2 Ellipsoid, hyperboloid, paraboloid.5.3 Investigation of surfaces of 2-nd order bodies by method of parallel profile.

6. General concepts about 2-nd order bodies

Requirement for received of credits: 32 hours lecture, 32 hours of practical work. Oral exam. During semester period to pass 2 test works and 16 home works.



Recommended literature:1. Tomsons J., Trupins Š. Analītiskā ģeometrija: Lekciju kurss. I - VII d.- R. LVU

rotaprints 1966.- 1979.g.2. Александров П.С. Лекции по аналитической геометрии.- М., Наука, 19683. Цубербиллер О.Н. Задачи и упражнения по аналитической геометрии. - М., Наука,

19684. Моденов П.С., Пархоменко А.С. Сборник задач по аналитической геометрии. - М.,

Наука, 19765. Клетеник Д.В. Сборник задач по аналитической геометрии. - М., Наука, 1977



CALCULUS I (Matemātiskā analīze I)

Course code Mate-1062Authors Prof. A. Šostaks, Dr. habil. Math.;

docent S.Asmuss, Dr Math.Credits 4 creditsRequired for grade exam

Annotation. Elements of the Cantorian set theory. Real numbers. Functions of one variable: limits, continuity. Fundamentals of differential calculus. Investigation of a function by methods of differential calculus.

Subjects: 1. Elements of the Cantorian Set theoryConcept of a set (Intuitive approach). Subset, operations with sets (union, intersection,

and subtraction). Cartesian product, its properties. Euler-Venn diagrams. Families of sets. De Morgan formulae. Concept of a function. An image and a pre-image of a set. Special functions: injections, surjections, bijections. Composition of functions. Finite and infinite sets. Countable sets. Basic properties of countable sets. Uncountable sets. Continuum.

2. Real numbersBasic sets of numbers: natural numbers N, integers Z, rational numbers Q, irrational

numbers, real numbers R. Elementary properties of these sets. Bounded and unbounded sets of real numbers. The minimal and the maximal elements of a set. The supremum and the infimum of a set. A neighbourhood of a set. Extension R* of the set R.

3. Limit of a sequence of real numbers. Limit of a sequence of real numbers: definitions, examples. Uniqueness of the limit.

Elementary properties of limits of sequences. Convergent and divergent sequences. Infinitely small and infinitely large sequences. Arithmetic operations and limits. Limits and inequalities. Existence of limits for monotone bounded sequences. The Cauchy criterion. Weierstrass theorems. Number e. Concept of a number series. Convergent and divergent series - definitions, examples. Elementary properties of number series.

4. Limit of a function of one variable.Elementary functions, their basic properties. The limit of a function: Cauchy and Heine

definitions; their equivalence. One-side limits. Examples. Baisc properties of limits. Limits and arithmetic operations. Limits and inequalities. Existence of a limit: the Cauchy criterion. Infinitely small and infinitely large functions, their comparison. Symbols o and O. Important examples of limits.

5. Continuity of a function of one variableContinuity of a function at a point. Arithmetic operations with continuous functions.

Composition of continuous functions. Continuity of a monotone function. Inverse function and its continuity. Continuity of elementary functions. Points of



discontinuity and their classification. Functions continuous on a closed interval. Weierstrass theorems. Bolzano-Cauchy theorems. Uniform continuity of a function: Cantor theorem.

6. Differential calculus of a function of one variableIntroduction to the concept of a derivative. Derivative at a point, its geometric and

physical interpretation. Differential of a function, its interpretation. Examples. Existence of derivative and continuity of a function. Arithmetic operations and derivation of functions. Derivative of composition; derivative of an inverse function. Derivatives of elementary functions. Higher order derivatives and differentials.

7. Application of differential calculus for the search of functions. Fermat, Rolle, Cauchy and Lagrange theorems. L'Hopital theorems. (Indeterminate forms

.Taylor-Maclaurin formula and Taylor-Maclaurin series: definitions, basic properties, examples. Remainder in a Taylor-Maclaurin formula. Extremum of a function and its characterisation by methods of differential calculus. Local and global maximum and minimum of a function. Applied maximum and minimum problems. Monotonicity of a function; concavity of a function. Investigation of a function and sketching its graph by applying methods of differential calculus.

Requirements for received of credits: 64 hours lectures, 32 hours practical work, 32 hours laboratory work. Students are required to fulfil 12 independent home works, and to write 1 theoretical and 1 practical control work. The exam takes place in oral form. Students must show understanding of theoretical material considered at lectures and demonstrate the ability of solving practical problems corresponding to the course.

Recommended literature:1. S. Nikoļskis Matemātiskā analīze, 1.d. - Rīga, Zvaigzne, 19762. M. Zandere, Diferenciālrēķini - Rīga, Latvijas universitāte, 1991.3. I. Kārkliņš, I. Kārkliņa, Funkcijas nepārtrauktība, funkcijas robeža - Rīga, Latvijas

universitāte, 1975. 4. K. Šteiners, Matemātiskās analīzes elementi, Rīga, Zvaigzne, 1993. 5. Г.М. Фихтенгольц, Основы математического анализа, т. 1 - Москва, Наука, 1968. 6. В.И. Зорич, Математический анализ, т. 1 - Москва, Наука, 1981. 7. Л.Д. Кудрявцев, Краткий курс математического анализа, Москва, Наука, 1989. 8. Б.П. Демидович, Сборник задач и упражнений по математическому анализу,

Москва, Наука, 1972. 9. Л.Д. Кудрявцев, А.Д. Кутасов, В.И. Чехлов, М.И. Шабунин, Сборник задач по

математическому анализу: Предел, непрерывность, дифференцируемость. Москва, Наука, 1984.

10.И.И. Ляшко, А.К. Боярчук, Я.К. Гай, Г.П. Головач, Математический анализ в примерах и задачах, т.1. - Киев, Вища школа, 1974.

11.В.Ф. Бутузов, Н.Ч. Крутицкая, Г.Н. Медведев, А.А. Шишкин, Математический анализ в вопросах и задачах, т.1. - Москва, Высшая школа, 1984.



CALCULUS II(Matemātiskā analīze II)

Course code Mate-1063Authors Prof. A. Šostaks, Dr. habil. Math.;

docent S.Asmuss, Dr Math.Credits 4 creditsRequired for grade examPrerequisities: Mate-1062

Annotation. The objects of the course are: Antiderivative and indefinite integral. Definite integral. Applications of definite integral. Concept of a metric space. Euclidean space Rm. Functions of several variables: limits, continuity. Differential calculus for functions of many variables.

Subjects: 1. Antiderivative and indefinite integral.Concept of an antiderivative of a function and indefinite integral. Basic properties of

indefinite integrals. Methods of integration: substitution, integration by parts, integration of rational functions, integration of trigonometric and hyperbolic functions

2. Definite integral.Introduction to the concept of a definite integral. Darboux sums and integral sums.

Definition of a definite integral. Integration criteria. Basic properties of definite integrals. Important classes of functions for which the definite integral exists (continuous functions, monotone functions). Integral as the function of its upper limit. Newton - Liebniz formula.

3. Applications of definite integrals and methods of evaluation of integrals.Important applications of definite integrals: evaluation of areas. Evaluation of volumes by

slicing and by cylindrical shells. Length of curve. Area of a revolution. Work. Simplest methods for evaluation of definite integrals: method of rectangles, method of trapezoids, Simpson method.

4. Improper integrals. Two types of improper integrals: improper integrals from functions defined on

unbounded intervals and improper integrals from unbounded functions. Convergence of integrals. Cauchy criterion of convergence. Evaluation of improper integrals. Absolute and conventional convergence. Examples.

5. Euclidean space Rm

Space Rm as a real vector space. Rm as a metric space. Scalar product of vectors and norm of a vector in Rm . Open and closed balls, cubes and rectangles in Rm . Neighbourhood of a point. Sequences, convergence of a sequence fundamental sequences. Completeness of the space Rm. Open and closed sets in Rm. Compact sets in Rm .

6. Functions of several variables.The concept of a function of several variables. The concept of a vector-function. Limit of

a function at a point. Continuity of a function in a point. Arithmetic operations with continuous functions. Composition of continuous functions. Theorem about a



continuous function on a compact set. Weierstrass theorems. Bolcano-Cauchy theorems. Uniform continuity of a function: Cantor theorem.

7. Differential calculus of a function of several variablesPartial derivatives. Differentiability of a function and differential. Necessary conditions

and sufficient conditions for differentiability of a function. Basic properties of differentiable functions. Examples. Directional derivative, Gradient. Geometrical interpretation, tangent plane and normal vector.

8. Applications of differential calculusThe concept of extremum of a function of several variables: definition, methods of

evaluation, examples. Applied maximum and minimum problem for functions of several variables.


Recommended literature:1. S. Nikoļskis. Matemātiskā analīze, 1.d; 2.d. - Rīga, Zvaigzne, 19762. K. Šteiners, Matemātiskās analīzes elementi, Rīga, Zvaigzne, 1993. 3. Г.М. Фихтенгольц, Основы математического анализа, т.1, т.2 - Москва, Наука,

1968. 4. В.И. Зорич, Математический анализ, т.1, т.2 - Москва, Наука, 1981. 5. Л.Д. Кудрявцев, Краткий курс математического анализа, Москва, Наука, 1989. 6. Б.П. Демидович, Сборник задач и упражнений по математическому анализу,

Москва, Наука, 1972. 7. Л.Д. Кудрявцев, А.Д. Кутасов, В.И. Чехлов, М.И. Шабунин, Сборник задач по

математическому анализу: Интеграл. Москва, Наука, 1984. 8. И.И. Ляшко, А.К. Боярчук, Я.К. Гай, Г.П. Головач, Математический анализ в

примерах и задачах, т.1, т. 2 - Киев, Вища школа, 1974.



CALCULUS III (Matemātiskā analīze III)

Course code Mate-2064Authors Prof. A. Sostaks, Dr. Hab. Math.; Docent S.Asmuss, Dr.Math.Credits 4 creditsRequired for grade examPrerequisities: Mate-1063

Annotation. The objects of the course are: Implicit functions, Multiple integrals. Line integrals. Surface integrals. Fundamentals of vector calculus, Real number series. Functional series. Power series.

Subjects: 1. Implicit functionsThe concept of an implicit function. Existence and properties of an implicit function: the

case of a function of one variable. Existence and properties of an implicit function: the case of a function of several variables. Implicit functions determined by systems of functional equations. The concept of dependence of a system of functions. Functional matrices, its range. Functional determinants.

2. Multiple integrals: general theory.Concept of a measurable set. Measure of a set. Criteria of measurability of a set. Sets

whose Lebesque measure equals zero. Examples of measurable sets in R2 and R3. Upper and lower Darboux sums. Integral of a function. Criterion of existence of an integral and its corollaries.

Basic properties of multiple integrals.3. Double and triple integrals.Double and triple integrals. Examples. Evaluation of double and triple integrals by

reduction to iterated (repeated) integrals. Transformation of variables in double and triple integrals. Geometric interpretation of transformation of variables. Polar, cylindrical and spherical co-ordinate systems and corresponding transformations of variables. Jacobi determinants. Geometric and physical applications: evaluation of areas of surfaces and volumes, centre of gravity.

4. Line integrals.Concept of a smooth curve. Parametric description of a curve. Line integrals of the first

type. Line integrals of the second type. Connections between the two types of line integrals. Green formula and its applications. Integrals independent of the path (2 dimensional case). Full differential expressions.

5. Surface integralsConcept of a surface. Parametric description of a surface. Tangent plane and normal to a

surface. The problem of orientation of a surface. Evaluation of area of a surface. Surface integrals of the first type. Surface integrals of the second type. Connections between the two types of surface integrals. Stokes theorem and its applications. Integrals independent of the path (3 dimensional case) Gauss-Ostrogradsky theorem and its applications.



6. Fundamentals of vector calculus.Scalar and vector fields. Vector lines and vector surfaces. Gradient, its physical and

geometrical interpretation. Nabla-operator. Divergence. Solenoidal fields. Rotor. Potential fields. Gauss-Ostrogradsky theorem in vector form. Stokes theorem in vector form.

7. Real number series.Concept of a series of real numbers. Convergent and divergent series. The sum of a

series. Convergence of a series with positive terms: comparison tests, sufficient conditions. Cauchy, and D'Alamber convergence tests. Integral convergence test. Alternating series and Leibniz convergence test. Absolute and conditional convergence. Examples.

8. Functional series. Functional sequences and functional series: pointwise convergence versus uniform

convergence. Weierstrass test of uniform convergence. Cauchy criterion of uniform convergence. Dirichlet and Abel's tests. Examples. Limit of a functional sequence; its continuity and differentiability. Sum of a functional series, its continuity and differentiability.

9. Power series. Concept of a power series. Abel theorem on the convergence of power series. The radius

of convergence and the area of convergence. Examples. Uniform and absolute convergence. Cauchy-Hadamard theorem. Differentiation and integration and of power series. Power series of some elementary functions. Applications of functional series. Computations using series.


Recommended literature:1. S. Nikoļskis Matemātiskā analīze, 2.d. - Rīga, Zvaigzne, 19762. Г.М. Фихтенгольц, Основы математического анализа, т.2 - Москва, Наука, 1968. 3. В.И. Зорич, Математический анализ, т.2 - Москва, Наука, 1981. 4. Л.Д. Кудрявцев, Краткий курс математического анализа, Москва, Наука, 1989. 5. Б.П. Демидович, Сборник задач и упражнений по математическому анализу,

Москва, Наука, 1972.



CALCULUS IV(Matemātiskā analīze IV)

Course code Mate-2065Authors Prof. A. Šostaks, Dr. habil. Math.; Docent S.Asmuss, Dr.Math.Credits 4 creditsRequired for grade examPrerequisities: Mate-2064

Annotation. The objects of the course are: Integrals depending on a parameter (proper and improper). Applications of integrals depending on a parameter. Euler integrals: function and function. Fourier series. Fourier integrals.Subjects: 1. Integrals depending on parameters.Integral as a function of parameter; its continuity. Convergence of an integral with

respect to the parameter. Limit transition on the parameter in an integral. Differentiation and integration of an integral on a parameter.

2. Improper integrals depending on parameter.An improper integral as a function of parameter: the case of an unbounded interval; the

case of unbounded functions. Uniform convergence of an improper integral with respect to the parameter. Sufficient conditions for uniform convergence. Differentiation and integration of an improper integral on a parameter. Limit transition on the parameter in an improper integral. Euler integrals: -function and

function. Applications of improper integrals for evaluation of definite and indefinite integrals.

3. Fourier series: general theory. Concept of a Hilbert space. Orthogonal and orthonormal systems of vectors in a Hilbert

space. Fourier coefficients of a function. Expansion of a function into a Fourier series. Bessel inequality. Parseval equality. Closed orthonormal systems. Complete orthonormal systems.

4. Fourier series on trigonometric systems.Trigonometric system of functions, its closeness and completeness. Trigonometric series.

Expansion of a periodic function into a trigonometric series. Fourier series for even and odd functions. Expansion of a non-periodic function into a Fourier series. The Dirichlet integral. Mean approximation of a given function by a trigonometric polynomial. The convergence of a Fourier series in a given point. The Fourier series in a complex form.

5. Fourier integral.The Fourier integral. The Fourier integral in a complex form. Applications of a Fourier

integrals.

Requirements for receiving of credits: 32 hours lectures, 32 hours practical work. Students are required to fulfil 6 independent home works, and to write 1 practical control work. The exam takes place in oral form. Students must show understanding of theoretical material considered at lectures and demonstrate the ability of solving practical problems corresponding to the course.



Recommended literature:1. S. Nikoļskis. Matemātiskā analīze, 2.d. - Rīga, Zvaigzne, 19762. N. Piskunov. Differential and integral calculus, vol 2, MIR publ., Moscow, 1981.3. Г.М. Фихтенгольц. Основы математического анализа, т.2 - Москва, Наука, 1968. 4. В.И. Зорич. Математический анализ, т.2 - Москва, Наука, 1981. 5. Б.П. Демидович. Сборник задач и упражнений по математическому анализу,

Москва, Наука, 1972.



COMPLEX VARIABLE FUNCTION THEORY.(Kompleksā mainīgā funkciju teorija.)

Course code Mate-4025Authors Prof. T.Cīrulis, Dr.Hab.Math., lekt. J.Smotrovs, Mag.Math.Credits 4 creditsRequired for grade examPrerequisites Mate-2065, Mate-1002, Mate-1011, Mate-3015 Annotation. The objectives of course are: complex numbers, their geometrical interpretation, algebraic and transcendental operations with complex numbers, complex variable and properties of its mapping, complex integral, complex series, singular points, residue and its applications, analytical extension.Subjects:1. Complex numbers, their geometrical interpretation. Trigonometric and exponential

forms.2. Algebraic and transcendental operations with complex numbers.3. Lines and domains in the plane of complex numbers. Infinite point. Neighborhoods of

the points.4. Complex variable and its geometrical interpretation. Limit, continuity and derivative of

the function. 5. Geometric interpretation of derivative of the function. Conformal mapping and its

properties.6. Linear and linear-fractional functions.7. Other elementary functions.8. Concept of the multifunction and its one-valued branchs. Riemannian surface.9. Complex integral and its properties. Cauchy theorem. Newton-Leibniz formulae.10.Cauchy integral formulae and its consequences.11.Series of the numbers and functions in the plane of the complex numbers, their

common properties. Weierstrass theorem. 12.Power series. Taylor and Laurent series.13.Singular points of the complex function, their classification. Residues. Cauchy

residues theorem.14.Application of residues in calculus of the integrals.15.Other applications of the residues.16.Analytical extension, its properties. Principle of the permanence.Requirement for received of credits: 48 hours lectures, 16 hours practical work. Students are required to fulfil 2 independent home works. The examination takes place in an oral form. Ticket of the examination contain two questions of the theory and one problem.Recommended literature:1. H.Kalis. Matemātiskās fizikas vienādojumi, klasifikācija un izvedumi. Stīgas

svārstības vienādojums. Rīga, LU, 1992.2. H.Kalis. Puasona un siltuma vadīšanas vienādojums. Rīga, LU, 1992.3. E.Riekstiņš. Matemātiskās fizikas vienādojumi. Rīga. Zvaigzne, 1964.4. А.Н.Тихонов, А.А.Самарский. Уравнения математической физики. Москва.,

Наука, 1977.



DIFFERENTIAL EQUATIONS - I (Diferenciālvienādojumi - I)

Course code Mate-2134Author Docent J.Cepītis, Dr.math.Credits 4 credits Required for grade examPrerequisites Mate-1002, Mate-1011, Mate-1063Annotation. The course considers basic concepts and interpretation of ordinary differential equations, analytical methods of their solving and introduction of the qualitative theory.Subjects:1. Ordinary differential equations of the first order, their geometrical and physical

interpretation, primary differential equations. Ordinary differential equations of higher order and their systems.

2. Cauchy’s problem: extensibility and non-extensibility of solutions, existence, local and global uniqueness of solutions.

3. Linear differential equations and systems. General properties of their solutions, solving in the case of constant coefficients. Euler’s equation.

4. Linear differential equations of the second order. The canonical form, theorems of comparison, oscillating and non-oscillating solutions.

5. Two point linear boundary value problems for the second order differential equations. Green’s function and its using.

6. Using of the series for the solving of second order linear homogenous ordinary differential equations. Bessel’s equation.

7. Systems of the two first order linear homogenous differential equations. Behaviour of the trajectories in the interior of stationary point.

8. Autonomous systems. Stability of the stationary solution.Requirements for receiving of credits: 32 hours lectures, 32 hours practical work.Students are required to fulfil 2 independent home works and 2 laboratory works. The exam takes place in an oral form. Students must answer 2 theoretical questions and show an ability of analytical solving of some ordinary differential equation. Recommended literature:1. J.Cepītis. Pirmās kārtas parastais diferenciālvienādojums. Rīga, LU, 1994.2. J.Cepītis. Košī problēma pirmās kārtas parastam diferenciālvienādojumam.

Rīga,LU,1992.3. S.Čerāne. Diferenciālvienādojumu kurss. Eksistences teorēma. Lineāri vienādojumi.

Rīga, LU, 1980.4. S.Čerāne. Diferenciālvienādojumu kurss. Speciāli jautājumi., Rīga, LU, 1981.5. И.Г.Петровский, Лекции по теории обыкновенных дифференциальных

уравнений, М.,Наука,1981.



DIFFERENTIAL GEOMETRY(Diferenciālģeometrija)

Course code Mate-3015Author lecturer V.Neimanis, M.Math.Credits 4 creditsRequired for grade examinationPrerequisites Mate – 2065, Mate – 1011, Mate-1002

Annotation.This course reviews classic questions of Differential Geometry, which linked with curves, with surfaces in Euklide Space.Subjects:1. Vector function conception in Calculus.2. Various definitions of line in Euklide Space. Examples.3. Local theory of lines in space E3: main base of lines.4. The Frene’ Formula. Line splitting and rotation.5. Natural definitions of line. Point’s area on line.6. Line on flatness: tangent line to circle, evolut and evolvent.7. Various definitions of surfaces in Evklide Geometry. Examples.8. Differential Geometry, surfaces in E3 space: normal tangent.9. Surface 1-st and 2-nd quadrature form, its Geometrical interpretation.10.Using of Surface 1-st and 2-nd quadrature form, beam’s length of surface’ line, circle

located between two lines of surface, surface’s part area, surface splitting with lines.11.Main directions of surfaces and main theirs splitters. Its calculation.12.Co-ordinate lines of surface, splitting lines, asymptote line, geodetic lines, their

location.13.Menie’ Theorem, Euler’s Formula. Depending characteristic curve.14.Surface points’ classification. Classification of surfaces.15.Conception of surface 3-d quadrature form and its use.16.Non-closed surfaces, its characteristics and definitions.17.N-dimensions spaces conception.

Requirements for received of credits: 1. Practical test work “Classical Linear Differential Geometry”.2. Practical test work “Classical Differential Geometry of Surfaces”.3. Test work in theory of main course’ conceptions, theirs definition and examples, use of main results.4. 2 theoretical questions exam with 2 exercises which show good knowledge of theoretical questions.

Recommended literature:1. T.Cīrulis, V.Neimanis. Diferenciālģeometrija, R.1990.2. Сборник задач и упражнений по дифференциальной геометрии, под. pед.

В.Т.Воднева, Минск 1970



MATHEMATICAL LOGIC(Matemātiskā loģika)

Course code Mate-1044Author Docent Janis Cirulis, Dr.Math.Credits 2 creditsRequired for grade testANNOTATIONAn introductory course. Main topics: Basic notions of propositional and predicate logics, representation of the logical structure of statements and arguments by means of mathematical logic, methods of verifying of formulas and checking of arguments. Its purpose is to aid the student to get skills in solving problems of this kind.SUBJECTS1. Propositional logic

1.1. Statements and their truth values. Simple and compound statements, logical operations.1.2. Formulas. Representation of logical structure of statements by formulas.1.3. Universally valid formulas, compatible formulas, equivalence, entailment.1.4. Algebraic properties of logical operations, transformations of formulas.1.5. Practical checking of arguments and of compatibility.

2. Predicate logic2.1. Limits of propositional logic. Subject-predicate structure of simple statements. Free and bounded variables, quantifiers.2.2. Representation of structure of statements by means of predicate logic2.3. Formulas and interpretations.2.4. Universal validity, compatibility and other semantic notions in predicate logic. Properties of quantifiers.2.5. Practical checking of arguments and of compatibility in predicate logic.

REQUIREMENTS FOR RECEIVED OF CREDITSDuring the term, the student has to pass two tests in problem solving): after each of the two main themes. Each problem is evaluated by a certain number of points. To get the credit, it is necessary and sufficient to gain at last a half of the possible total sum of points in both tests.

RECOMMENDED LITERATURE1. J. Cīrulis. Matemātiskā loģika nematemātiķiem. R., LVU, 1982. 2. V. Detlovs. Matemātiskās loģikas un kopu teorijas elementi. R., LVU, 1967, 1969.3. Логика, множества, аксиоматические теории, M., 1973.4. Цирулис Я.П. Лекции по математической логике. Р., 1973



MATHEMATICAL STATISTICS (Matemātiskā statistika)

Course code Mate-3030Author docent Viktorija Carkova, Dr. math.Credits 4 credits Required for grade examPrerequisites Mate-2032

AnnotationThe course includes descriptive sample statistics; empirical distribution function; quantities, the sample correlation coefficient; point and interval estimates; large sample techniques; chi-square test procedures; methods for linear regression and hypothesis testing.

Subjects1. Statistical method. Collecting data and statistical thinking.2. Set of order statistics. Histograms. The empirical distribution function.3. Point estimation.4. Method of maximum likelihood. Moments method.5. Confidence probability. Sample confidence interval.6. Confidence interval construction.7. Statistics’ based on the Normal distribution.8. Confidence Interval for the Mean of normal populations.9. Confidence Interval for the variance of normal populations.10.Hypothesis testing. Null hypothesis. Alternative hypothesis.11.Testing hypothesis about the population proportion p.12.First kind error, second kind error. Testing hypothesis about the population mean.13.Estimation and hypothesis testing with two independent samples: difference between

two means, retio of two variances.14.Chi-square test of independence for contingency tables.15.Random samples homogeneity.16.Statistical hypothesis testing about random variables distribution.17.Regression. Population regression curve. Sample regression curve.18.Linear regression. Method of least squares.19.A model for linear regression.20.Confidence intervals and hypotheses test for linear regression parameters.21.Correlation coefficient. Correlation ratio.Requirements for received of creditsThe examination for this course is a three-hour written-answer examinationRecommended literature1. V.Carkova. Mathematical statistics. LU, Riga, 1979, 79 p.2. O.Krastiņš. Varbūtību teorija un matemātiskā statistika. Rīga, Zvaigzne, 1978.



3. R.Iman, W.Conover. A Modern Approach to Statistics. Wiley, NY,1983.



NUMERICAL METHODS I (Skaitliskās metodes I)

Course code Mate-2132Authors Prof. H. Kalis Dr. hab. math., Dr. hab. phys. Lect. R. Millere, Mag. math.Credits 2 creditsRequired for grade testPrerequisites Mate-1063, Mate-1002, Mate-1025

Annotation. The objectives of the course are: sources of classical problems in numerical analysis; methods for solving of the algebraic and transcendental equations and systems of equations, approximate and exact methods for solving the systems of linear algebraic equations, algorithms for computation of eigenvalues and eigenvectors of a matrix.

Subjects:1. Mathematical modelling - the method of the reference the laws of real worlds. 2. Calculations with approximate numbers. Absolute, relative and inevitable errors.3. Methods for solving non-linear equations, numbers of solutions. The bisection method. 4. Simple iterative method for solving of equation, the convergence, estimation of error.5. Newton’s, secant and combined methods for solving of equation. 6. Solution of system of non-linear equations with simple iteration and Newton methods. 7. The programming packages (Mathematica, Maple) for solving equations and systems. 8. Gauss method and its versions (method of factorisation) for solution of simultaneous

algebraic equations.9. Square root and ortogonalisation method for solution of algebraic equations.10.Simple iterative method for solution of simultaneous algebraic equations. 11.Seidel method. Optimisation of method algorithm. 12.Methods of relaxation . 13.Methods for computation of coefficients of characteristic equation. Solutions of the

full problem of eigenvalues. 14.Fadejev-Leverje method for computation coefficients of a characteristic equation. 15.The programming packages (Mathematica, Maple, a.o.) for solution of simultaneous

algebraic equations and computation of eigenvalues. 16.Method of iteration for calculation of the part problem of eigenvalues.

Requirements for receiving of credits: 32 hours lectures, 32 hours practical work, 2 test works. Students are required to fulfil 4 laboratory works.



Recommended literature:1. В.И. Крылов, В.В.Бобков, П.И. Монастырский. Численные методы высшей

математики. Минск, 1972.2. Н.С. Бахвалов, Н.С. Жидков, Г.М. Кобельков. Численные методы М.: 1987. 3. I.Pagodkina, R.Millere. Algebras un matemātiskās analīzes uzdevumu skaitliskā

risināšana. Rīga, 1996.R.J. Goult, R.F.Hoskins, J.A.Milner, H.J.Pratt. Computational methods in Linear Algebra. Stanby Thornes (Publishers) Ltd., 1974.

4. A.Heck. Introduction to MAPLE. Springer-Verlag, New-York, Inc., 1966.5. H.Kalis, S.Lācis, O.Lietuvietis, I.Pagodkina. Programmu paketes “Mathematica”

lietošana mācību procesā. Mācību grāmata, Rīgā, 1997.



NUMERICAL METHODS II (Skaitliskās metodes II)

Course code Mate-2138Authors Prof. H. Kalis Dr. hab. math., Dr. hab. phys. Lect. R. Millere Mag. math. Credits 2 creditsRequired for grade testPrerequisites Mate-1002, Mate-2064, Mate-2026, Mate-2132

Annotation. The objectives of the course are: sources of classical problems in numerical analysis; continuous and discrete forms of functions approximation. Simple and multiple integral, numerical solution of integral equations.

Subjects:1. The problem of interpolation. The algebraic interpolating polynomial, existence and

uniqueness. Lagrange’s interpolation polynomial.2. Convergence of interpolation process, errors and Lagrangian interpolation.3. The divided and finite differences, the difference table, derivatives and differences.4. Newton interpolation polynomial for interpolation a begin and end of the table. 5. Interpolation in middle of tables. Interpolation with equal and unequal intervals, errors.6. Trigonometric interpolation polynomial for periodical functions.7. Spline approximation and interpolation: linear, quadratic and cubic splines.8. Least squares approximation algebraic and trigonometrically polynomials.9. Numerical integration for a definite integral. The rectangular quadrature rules, error. 10.Quadrature formulas of interpolation type. Newton-Cotes , trapezoidal and Simpson

quadrature rules, composite rules. The programming packages (Mathematica, Maple).

11.Gaussian quadrature rules, the error of integration formulae.12.Constructing of the Gauss quadrature rule.13.Multiple integrals, improper integrals and integration of oscillating function.14.Computing process for solving indefinite integrals. Difference equations with constant

coefficients. Stability of a numerical process.15.Quadrature rule method for solving integral equations of Fredholm and Volterra types.16.Method of iterations for solving integral equations.

Requirements for receiving of credits: 32 hours lectures, 32 hours practical work, 2 test works. Students are required to fulfil 3 laboratory works.



Recommended literature:1. Н.С. Бахвалов, Н.С. Жидков, Г.М. Кобельков. Численные методы М.: Наука,

1987.2. В.И. Крылов, В.В.Бобков, П.И. Монастырский. Численные методы высшей

математики. Минск, 1972.3. I.Pagodkina. Tuvinātās metodes. Skaitliskā integrēšana, Rīgā, LVU, 1982.4. I.Pagodkina, R.Millere. Algebras un matemātiskās analīzes uzdevumu skaitliskā

risināšana. Rīgā, 1996.5. S.O. Conte, C.Boor. Elementary Numerical Analysis an algorithmic approach.

Mc.Graw-Hill Book Company, 1972.6. G.M.Phillips, P.J.Taylor. Theory and Applications of Numerical Analysis. Acad. Press:

London and New York, 1973.7. A.Heck. Introduction to MAPLE. Springer-Verlag, New-York, Inc., 1966.8. H.Kalis, S.Lācis, O.Lietuvietis, I.Pagodkina. Programmu paketes “Mathematica”




NUMERICAL METHODS III(Skaitliskās metodes III )

Course code Mate-3139Authors Prof. H. Kalis, Dr. Hab. Math., Dr. Hab. Phys.Credits 4 creditsRequired for grade examPrerequisites Mate-2133, Mate-2135, Mate-2138

Annotation. The objectives of the course are: methods for numerical solutions of Cauchy initial-value problem and boundary-value problem for ordinary differential equations (ODE), eigenvalues and eigenfunctions for the second order differential operators, initial-boundary and boundary-value problems for partial differential equations (PDE).

Subjects:1. The programming packages (Mathematica, Maple) for solving Cauchy problem of ODE.2. Euler`s, Euler`s-Cauchy, predictor-corrector methods. Error estimates. Approximation, convergence and stability,3. Numerical integration by Taylor series, Runge-Kutta methods.4. Multi-step formulas. Adams methods. Schur`s criterion on the roots of a polynomial in the unit circle. Stability of numerical methods. 5. Reduction of the boundary-value problem for ODE to the Cauchy problem. Shooting methods. The programming packages.6. Solution of the boundary-value problem by finite-difference methods, different order of approximation.7. Derivation of finite-difference scheme, integration-interpolation method (method of finite volume). Discrete factorisation methods (Thomas algorithm).8. Parabolic equations of PDE. Numerical solutions of initial-boundary value problem for heat transfer equation, two-level finite-difference schemes. Approximation, stability and convergence.9. Some three-level finite-difference schemes: Richardson and Du Fort-Frankel’s schemes. Scheme with higher order of approximation.10.Line method for solving of the heat transfer equation, analytical solution. The alternating direction method (ADI).11.The initial-boundary value problem for the equation of string oscillations. The explicit finite-difference scheme, stability.12.Line method for solving the equation of string oscillations, analytical solution.13.PDE of elliptic type. Approximation of equations and boundary conditions.14.Dirichlet problem for the Poisson’s equation in a rectangle, finite-difference scheme of second and fourth orders.15.Dirichlet problem for the Poisson’s equation in a rectangle, the methods of iterations and relaxation, analytical solution.16.Line method for solving the Poisson’s equation in a rectangle, alternating direction methods (ADI).



Requirements for received of credits: 32 hours lectures, 32 hours practical work, 2 testworks. Students are required to fulfil 3 laboratory works. The exam takes place in an oral form.

Recommended literature:1. H. Kalis. Diferenciālvienādojumu tuvinātās risināšanas metodes. Rīgā, 1986.2. А.А. Самарский. Теория разностных схем. М., 1977.3. H. Kalis. Nepārtraukto un diskrēto matemātiskās fizikas problēmu analītiskie

atrisinājumi. LU, 1992.4. E. At. Kendall. An introduction to numerical analysis. University of Iowa, New York:

1989.5. G.M.Phillips, P.J.Taylor. Theory and Applications of Numerical Analysis. Acad Press:

London and New York, 1973.6. W.F.Ames. Numerical method for partial differential equations. Acad. Pres.: New-

York, 1977. 7. G.D.Smith. Numerical solution of partial differential equations. Finite difference

methods. Clar. Press. Oxford, 1978.8. A.Heck. Introduction to MAPLE. Springer-Verlag, New-York, Inc., 1996.9. H.Kalis, S.Lācis, O.Lietuvietis, I.Pagodkina. Programmu paketes “Mathematica”




PROBABILITY THEORY(Varbūtību teorija)

Course code Mate-2032Author Docent, Dr.Math. Viktorija CarkovaCredits 4credits/64 hoursRequired for grade examPrerequisites Mate-2064, Mate-1002

Annotation. The course includes probability space, Bernoulli trials, random variables and their characteristics, limit theorems.

Subjects1. Random trails, random events and their probability.2. Probability theory rules: additional rule, conditional probability total probability rule,

Bayes’ formula.3. Discrete spaces, examples.4. Discrete random variables and their distributions.5. Discrete random variables and their characteristics.6. Discrete random variables: the uniform, binomial, geometric, negative binomial and

Poisson distributions.7. Discrete bivariate random variable. The correlation coefficient.8. Independent trials. Bernoulli trails. Limit theorems: law of large numbers, Poisson

theorem, normal approximation to the binomial. Examples.9. Axioms of probability. Probability space.10.Probability measure. Properties.11.Sigma-algebra of events. Completion of a measure.12.Continuous random variables. The uniform, exponential, normal random variables.13.Properties of the random variable distribution function.14.Multivariate distribution. Joint distribution function.15.Function of random variables.16.Independent random variables.17.Definitions of random variables convergences.18.Strong law of large numbers. Chebyshev’s inequality. Chebyshev’s theorem.

Bernoulli theorem.19.Borel-Cantelli lemma. Borel theorem.20.Properties of the random variables characteristic functions.21. Central limit theorem, and. their application.Requirements for received of credits. The examination for this course is a three-hour written-answer examinationRecommended literature1. V.Carkova, M.Buikis. 25 lectures of Probability theory. LU, Riga, 1965, 175 p. 2. J Lapins.(editor). Exercises. LU, Riga, 1981.3. Sh.M.Ross. Introduction to Probability Models. Fifth Edition, Acad.Press, NY, 1995.4. A.Francis. Advanced Level Statistics. Stanley Thornes LTD, Great Britain, 1979.



PROGRAMMING AND COMPUTERS I(Programmēšana un datori)

Course code DatZ1024Authors Mg. Comp., lect. Viesturs Vezis, Mg. Comp., lect. Laila NiedriteCredits 3Required for grade test

Annotation:The whole study course consists of three parts what include training in basic computer using skills, to use Internet services, to work in programming language PASCAL, to use application program packages. The first part of the course is dedicated to gain basic skills of using computer, Internet services and application program packages. SUBJECTS1. Informatics and its basic concepts. Short history of computer technology development.

Computer technology generations and classification. 2. The role and meaning of computer in information processing. Main parts of computer

(blocks) and auxiliaries: their tasks, meaning and functions. 3. Information entering, saving and deleting. The concept of operational systems.

Computer viruses and protection against them. Archive programs, their usage.4. Microsoft Windows environment and principles of functioning. Microsoft Windows

applications: running and closing. Windows Accessories. Multimedia tools. Adjusting Windows environment to users’ needs (taste).

5. Computer networks: importance, principles of functioning and usage possibilities. Internet - worldwide global network: possibilities, development history and perspective. IP addresses and domain name system. Organization of information transmission. Internet access levels: comparison of speed and other parameters. Legal restrictions and etiquette. Potential of Internet services usage.

6. Introduction to word processing (formatting) systems. Possibilities of using Word processor Microsoft Word in teaching.

7. Spreadsheets. Usage of Microsoft Excel in solving mathematical problems. Introduction to databases, possibilities of database usage.

8. Program package Mathematica 2.2.3, using in simple calculations, equation and equation system solving, constructing function graphs and geometrical figures.

Requirements for received of credits:Completion of all practical exercises.Two projects have to be worked out:Creation and formatting of one lesson (topic) conspectus.Creation of electronic class journal.Credits are awarded after successful completing of all exercises and both projects.



Recommended literature:1. I.Murāne Ar INTERNET uz tu.2. I.Murāne INTERNET tas ir vienkārši 3. Эд Крол Все об интернет4. H.Kalis, S.Lācis, O.Lietuvietis, I.Pagodkina Programmu paketes “Mathematica” 5. Microsoft® Windows® 95 Step by Step Catapult, Inc6. Microsoft® Office 97 Professional 6-in-1 Step by Step Perspection, Inc.7. Microsoft® Word 97 Step by Step Catapult, Inc.8. Microsoft® Excel 97 Step by Step Catapult, Inc



PROGRAMMING AND COMPUTERS II(Programmēšana un datori II)

Course code DatZ-1025Authors Mg. Comp.Sc., lect. Viesturs Vezis, Mg. Comp.Sc., lect. Laila NiedriteCredits 4Required for grade testPrerequisites The first part of this course.Annotation The whole study course consists of three parts what include training in basic computer using skills, to use Internet services, to work in programming language PASCAL, to use application program packages. The second part of the course is dedicated to programming language PASCAL: operators and the simplest data structures, different algorithms necessary to mathematicians. SUBJECTS1. The concept of algorithm and its tracing formats. Programming language PASCAL as

one of the formats. Concept of command (operator). Constants and variables. Expressions. Data input and output operators. Simple program. Entering, editing and running a program. Informatics and its basic concepts. Short history of computer technology development. Computer technology generations and classification.

2. Built-in functions. 3. Conditional operator.4. Single and compound operators. The change of consecution between operators.5. Variant operator. 6. Loop. Loop operator with counter. 7. Loop operators with begin and end condition. 8. Functions and procedures, examples.9. Global, local variables. Scope.10.Concept of recursion. Examples.11.User defined data type. Scalar and restricted data types. One-dimensional array and its

processing.12.Sorting algorithms. Bubble and Hoare methods.13.Multi-dimensional arrays. 14.Methods of array processing.15.PASCAL graphics commands.16.Basic principles of good programming style.Requirements for received of credits:Completion of all practical exercises.One project: given mathematical problem have to be solved.Credits are awarded after successful completing of all exercises and the project. Recommended literature:1. Вирт Н., Язык прогаммирования Паскаль. М.1981.2. Грогоно П., Язык прогаммирования Паскаль. М.1982.3. Вирт Н., Алгоритмы+структуры данных=прогаммы. М.1985.



PROGRAMMING AND COMPUTERS III(Programmēšana un datori III)

Course code DatZ-2026Authors Mg. Comp.Sc., lect. Viesturs Vezis, Mg. Comp.Sc., lect. Laila NiedriteCredits: 4Required for grade testPrerequisites The first part of this courseAnnotation:The whole study course consists of three parts what include training in basic computer using skills, to use Internet services, to work in programming language PASCAL, to use application program packages. The third part of the course is dedicated to gaining programming language PASCAL graphics and to the most complicated data structures (file, pointers) databases control systems and one database developing. SUBJECTS1. Sorting algorithms. Bubble, Shell and Hoare sort methods. Merging method.

Evaluation of algorithmic efficiency.2. Multi-dimensional arrays. Array processing algorithms.3. Strings and records.4. Files. Direct access files and sequential access files. Working with text files.5. Naive text encryption methods. Text encryption using public key.6. Direct access files: examples of usage.7. Dynamic variables, usage.8. Stack and queue: realization. Examples.9. Lists: realization. List processing algorithms.10.Trees and graphs: kinds, realization and basic operations.11.Concept of database. Principles and functions of database management system.

Principles of database development, information input and editing.12.Query language. Report creating using information stored in database. Information

and result printing.13.Screen form creating, usage in database processing.14.The basic database models. Simple database creating.Requirements for received of credits:Completion of all practical exercises.Two projects have to be worked out:One given mathematical problem have to be solved.Creation of a database.Credits are awarded after successful completing of all exercises and both projects. Recommended literature:1. Вирт Н., Язык прогаммирования Паскаль. М.1981.2. Грогоно П., Язык прогаммирования Паскаль. М.1982.3. Вирт Н., Алгоритмы+структуры данных=прогаммы. М.1985.



OPTIONAL COURSES IN MATHEMATICS (PART B)



ANALYTICAL SOLUTIONS(Analītiskie atrisinājumi )

Course code Mate-4174Authors Prof. H. Kalis, Dr. Hab. Math., Dr. Hab. phys.Credits 2 creditsRequired for grade examPrerequisites Mate-3143, Mate-2135

Annotation. The objectives of the course are: analytical solutions for continuous and discrete problems of mathematical physics for linear differential equation of the second order.

Subjects:1. The orthogonal and bioorthogonal systems of eigenvectors for solution of linear

system of algebraic equations2. Linear difference equations of the first and second order with the constant

coefficients.3. Problem of eigenvalues and eigenfunctions for adjoint 2-order differential and

difference operators.4. Problem of eigenvalues and eigenfunctions for non-adjoint 2-order differential and

difference operators.5. Matrix-functions, their properties and calculations.6. Cauchy initial-value problem for ordinary differential equation (ODE) and system of

the first order.7. Boundary-value problem for ODE of the second order (continuous and discrete

cases).8. Boundary-value problem for the ODE system of the second order.9. Dirichlet problem for the Poisson`s type equations in rectangle (continuous, discrete

and line method version).10. The initial-boundary value problem for heat transfer equation.11. The schemes of alternating direction methods.

Requirements for receiving of credits: 32 hours lectures and 1 test work.Students are required to fulfil 3 independent laboratory works. The exam takes place in an oral form.

Recommended literature:1. H.Kalis. Nepārtraukto un diskrēto matemātiskās fizikas problēmu analītiskie

atrisinājumi. Rīga, LU, 1992. 2. H.Kalis. Diferenciālvienādojumu tuvinātās risināšanas metodes. Rīga, Zvaigzne,

1986.3. J.W.Thomas. Numerical partial differential equations: finite difference methods.

Springer, 1995.



APPLICATIONS OF NUMERICAL METHODS FOR SOLUTIONS OF MATHEMATICAL PHYSICS AND HYDRODYNAMICS PROBLEMS

(Skaitlisko metožu pielietošana fizikas un hidrodinamikas problēmu risināšanā )Course code Mate-4279Authors Prof. H. Kalis Dr. Hab. Math., Dr. Hab. phys.Credits 2 creditsRequired for grade examPrerequisites Mate-3140, Mate-3143Annotation. The objectives of the course are: modern numerical methods for solving of mathematical physics and hydrodynamics problems with computers, efficient numerical algorithm and its foundations (stability and precision), exact finite-difference schemes for one dimensional problems. This course is made in compliance with ECMI program.

Subjects:1. Cauchy problem for ordinary differential equations (ODE) of the first order.2. Boundary-value problem of ODE of the second order.3. Finite-difference A.Ilh`yn scheme in nonuniform grid.4. Finite-difference N.Bahvalov scheme in nonuniform grid.5. Exact finite-difference schemes for solutions of boundary-value problems of ODE

with piece-wise constant coefficients.6. Special finite-difference schemes for solutions of boundary-value problems of elliptic

type equations.7. Initial-boundary value problem for parabolic type equations, numerical methods.8. One dimensional boundary-value problems for hydrodynamics, finite-difference

schemes.9. Special finite-boundary schemes for solving of linear problems of

magnetohydrodynamics.10. Approximation a vorticity function on boundary.11. Monotone finite-difference scheme for calculations of vorticity function,12. Methods for investigation of stability of finite-difference schemes.13. The finite-elements methods.14. The boundary-elements methods.15. The spectral methods.Comparison of numerical methods for solutions of mass transfer equations.

Requirements for received of credits: 32 hours lectures and 1 test work.Students are required to fulfil 3 independent laboratory works. The exam takes place in an oral form.Recommended literature:1. H.Kalis. Speciālu skaitlisko metožu izstrāde un lietošana matemātiskās fizikas,

hidrodinamikas un magnētiskās hidrodinamikas problēmu risināšanā. Rīga, LU, 1993.2. П.Роуч. Bычислительная гидродинамика. - Москва: Мир, 1980.3. W.F.Ames. Numerical methods for partial differential equations. Acad. Press, New-

York, 1977.



APPLIED PROBLEMS OF OPTIMISATION IN ECONOMICS AND MANAGEMENT SCIENCE

(Optimizācijas lietišķās problēmas)

Course Code Mate - Author Assoc. Prof. Jānis Vucāns, Dr. Math. Credits 4 creditsRequired for grade examPrerequisites: Mate – 2065, Mate-3082 or Mate-4081, Mate-2134, Mate-

(Math. Foundations of Microeconomics)

Annotation: The principal aim of this course is to acquaint students with applied situations from economics, management science and relevant spheres, which can be described in terms of the problems of dynamic optimisation; to familiarise students with the basic methods of modelling, solution, and post-solution analysis of such problems. The material of the lecture course, being mathematical by applied methods of investigation, relates, from the point of view of its content, to different spheres of the social activity (economics, finances, management science etc.). Different problems of dynamic optimisation, referring to Calculus of Variations and Optimal Control Theory, as well as the analytical methods of their solution are considered.

Subjects:0. Introduction. 1. From statistic to dynamic optimisation. Classical Calculus of Variations. 2. The simplest problem of classical Calculus of Variations. 3. Discount factor.4. Economic applications of the simplest problem of classical Calculus of Variations. 5. Generalisations of the simplest problem of classical Calculus of Variations. 6. Necessary conditions of extrema for an other problem of Calculus of Variations: for

problems with different types of boundary conditions; LaGrange's problem.7. Different applications of Calculus of Variations. [Optimal economic growth with a

non-fixed amount of final capital; Minimisation of the road construction expenses; Optimal waste disposal; Optimal education and balanced growth; Micro foundations of macro models; Optimal capital accumulation in multisectoral production; Optimisation of the extraction process of the natural resources; Optimal economic growth (neo-classical model)].

8. Introduction into Optimal Control Theory. Pontryagin's Maximum principle. Optimal phasing of deregulation for economic process.

9. Dynamic programming. Problem of optimal distribution of resources. Salesman's problem.



Requirements for received of credits: 48 hours lectures, 16 hours seminars. During the semester students must solve a number of problems, assigned by the teacher for independent solving; they must present these solutions to the teacher by the indicated deadline. Students must independently prepare the analysis of some dynamic optimisation problem. During the seminar they must explain the obtained results. At the written examination student must be able to demonstrate the knowledge of the basic concepts and properties of dynamic optimisation, and must be able to apply them for situations from economics, management, and relevant spheres.

Recommended literature:1. Jānis Vucāns, Optimizācijas lietišķās problēmas ekonomikā un vadības zinātnē

(Applied Problems of Optimisation in Economics and Management Science), Lecture Notes, University of Latvia and Ventspils College, 1999.

2. Atle Seierstad and Knut Sydsaeter, Optimal Control Theory with Economic Applications, North-Holland, Amsterdam - New-York - Oxford - Tokyo, 1987.

3. Pierre N.V.Tu, Introductory Optimisation Dynamics: Optimal Control Theory with Economic and Management Science Applications, Springer-Verlag, Berlin - Heidelberg - New-York - Tokyo, 1984.

4. Jean-Pierre Aubin, Optima and Equilibra: An Introduction to Nonlinear Analysis, Springer-Verlag, Berlin - Heidelberg - New-York - London - Paris - Tokyo - Hong Kong - Barcelona etc., 1993.



CALCULUS OF VARIATIONS (Variāciju rēķini)

Course code Mate - 3082Author Docent Andrejs Cibulis, Dr. Math.

associate professor Jānis Vucāns, Dr. Math.Credits 4 creditsRequired for grade testPrerequisites Mate – 2065, Mate – 1002, Mate – 2132

ANNOTATIONANALYTIC AND NUMERICAL METHODS FOR SOLVING OPTIMISATION PROBLEMS, ELEMENTS OF OPTIMAL CONTROL THEORY.

SubjectsIntroductory lecture. Examples from Nature where some optimality principle holds.

Construction of mathematical models and solving them by elementary methods.Types of extremes. Necessary and sufficient extremes condition for functions of one and

several variables.Directional derivative. Instructive examples. Necessary and sufficient optimality

conditions for non-smooth problems.Optimisation problems with constraints. Methods of elimination and Lagrange

multipliers. Existence theorems for implicit functions. Necessary and sufficient optimality conditions

in the presence of equality and inequality constraints.Numerical optimum seeking methods. Unimodal functions. Active and passive strategies.

Principle of minimax. The dichotomous, golden section methods and Fibonacci search. Some minimisation

methods (gradient, gradient projections, Newton‘s ) for functions of several variables.Convexity. Properties of convex sets and functions. Separation theorems for convex sets.

The Karush-Kuhn-Tucker theorem.Classic calculus of variations (CCV). Historical review. The brachistochrone’s problem.

Weak and strong local extremum. The fundamental lemma of CCV. Extremals. Euler-Lagrange equation. Hilbert’s , Weierstrass’s, Bolza and other instructive examples . Bolza problem, isoperimetric and others problems of CCV. Lavrentiev phenomenon. Role of convexity for the existence of a global minimum. Variational principles.

Introduction to optimal control theory. Pontryagins maximum principle: discussion and applications.

Insight in dynamic programming.Requirements for received of creditsCourse consists of lectures and practical work in the ratio 2:1. Students must solve the given individual problems. Only after solving them students can pass a test or examination to get their credit.Recommended literatureJ. Engelsons. Optimizācijas metodes. I un II daļa.



CORRECTNESS OF PROBLEMS(Uzdevumu korektība)

Course code: Mate-4196Author: Prof. U.Raitums, Dr.Hab.Math.Credits: 2 creditsRequired for grade: examPrerequisites: Mate-2065, Mate-2135

Annotation. The course considers questions of correctness for problems, which before were studied from other points of view. The main attention is devoted to the continuous dependence of solutions of various problems on parameters.

Subjects:1. The notion of the correctness.2. Dependence of solutions of linear algebraic systems on perturbations of the

corresponding matrix.3. Dependence of eigenvalues on coefficients of matrices.4. Dependence of eigenvectors on entries of matrices.5. Non-linear equations in Euclidean spaces. Monotone mappings. The role of the

Lipschitz's condition.6. The role of Theorem of implicit function.7. Relationships between the convexity of a function and the monotonicity of the

corresponding gradient.8. Continuous dependence of solutions of minimisation problems on small perturbations

of the involved function.9. The role of the Lipschitz's condition in theory of ordinary differential equations.10.Gronwall's lemma.11.Continuous dependence of solutions of the Cauchy's problem for ordinary differential equations upon functional perturbations.12.Stability of solutions of linear Cauchy's problems under small nonlinear perturbations.13.Dependence of solutions of variational problems on perturbations of the integrand.14.The role of the convexity properties of the integrand.Requirements for receiving of credits: 32 hours lectures.During the course the students must fulfill 3 independent home works. The exam takes place in an oral form. The student is required to show an understanding of the role of various conditions on the continuous dependence of solutions of various problems upon parameters.Recommended literature:1. Fadeev, D.K., Fadeeva, V.N., Numerical Methods in Linear Algebra, Fiz-mat,

Moscow, 1963 (in Russian).2. Hartman, P., Ordinary Differential Equations, John Wiley&Sons, New York, 1964; or

Russian translation, Mir, Moscow, 1970.3. Gabasov, R., Kirillova, F.M., Methods of Optimization, Minsk, 1975 (in Russian).



DIFFERENTIAL EQUATIONS - II (Diferenciālvienādojumi - II)

Course code Mate-2135Author Docent J.Cepītis, Dr.Math.Credits 4 credits Required for grade examPrerequisites Mate-2064, Mate-2134Annotation. The course considers elements of the qualitative theory for ordinary differential equations and introduces with the basic concepts of the equations of mathematical physics.Subjects:1. Cauchy's problem for the ordinary differential equation: continuity and continuous

differentiability of the solution with respect parameters in the differential equation or initial condition.

2. Kinematic interpretation of autonomous systems. Properties of trajectories, their kinds and behaviour in limiting case. Cluster sets, their general properties and specific properties in the plane.

3. Bohl's – Brauer's theorem and its application in the qualitative theory of ordinary differential equations. Using of the vectors field rotation for the investigation of autonomous systems.

4. Lyapunov's methods for the investigation of stability. Stability theorem for the linear fitting.

5. Primary integrals, their properties and applying.6. Linear homogenous and quasilinear first order partial differential equations. Integral

surfaces and characteristics.7. Classification and canonical forms of the second order almost linear partial

differential equations.8. D'Alemberts formulae for the equation of hyperbolic type.9. Mixed problem for the equation of hyperbolic type, the method of separation of

variables.10. Sturm's – Liouville's problem. Properties of the eigenvalues and eigenfunctions.Requirements for receiving of credits: 32 hours lectures, 32 hours practical work.Students are required to fulfil 2 independent home works and 2 laboratory works. The exam takes place in an oral form. Students must answer 2 theoretical questions and practically solve one example. Recommended literature:1. J.Cepītis. Košī problēma pirmās kārtas parastam diferenciālvienādojumam.

Rīga,LU,1992.2. S.Čerāne. Diferenciālvienādojumu kurss. Speciāli jautājumi., Rīga, LU, 1981.3. И.Г.Петровский, Лекции по теории обыкновенных дифференциальных

уравнений, М.,Наука,1981.4. А.Н.Тихонов, А.А.Самарский, Уравнения математической физики,

М.,Наука,1977.



DIFFERENTIAL GEOMETRY(Diferenciālģeometrija)

Course code Mate-3015Author lector V.Neimanis, M.math.Credits 4 creditsRequired for grade examinationPrerequisites Mate – 2065, Mate – 1011, Mate-1002

Annotation.This course reviews classic questions of Differential Geometry, which linked with

curves, with surfaces in Euklide Space.Subjects:1. Vector function conception in Mathematical analysis.2. Various definitions of line in Euklide Space. Examples.3. Local theory of lines in space E3: main base of lines.4. The Frene’ Formula. Line splitting and rotation.5. Natural definitors of line. Point’s area on line.6. Line on flatness: tangent line to circle, evolut and evolvent.7. Various definitions of surfaces in Evklide Geometry. Examples.8. Differential Geometry, surfaces in E3 space: normal tangent.9. Surface 1-st and 2-nd quadrature form, its Geometrical interpretation.10. Using of Surface 1-st and 2-nd quadrature form, beam’s length of surface’ line, circle

located between two lines of surface, surface’s part area, surface splitting with lines.11. Main directions of surfaces and main theirs splitters. Its calculation.12. Co-ordinate lines of surface, splitting lines, asymptote line, geodetic lines, their

location.13. Menie’ Theorem, Eiler’ Formula. Depending characteristic curve.14. Surface points’ classification. Classification of surfaces.15. Conception of surface 3-d quadrature form and its use.16. Non-closed surfaces, its characteristics and definitions.N-dimensions spaces conception.Requirement for receiving of the credits: 1. Practical test work “Classical Linear Differential Geometry”.2. Practical test work “Classical Differential Geometry of Surfaces”.3. Test work in theory of main course’ conceptions, theirs definition and examples, use of main results.4. 2 theoretical questions exam with 2 exercises which show good knowledge of theoretical questions.Recommended literature:1. T.Cīrulis, V.Neimanis. Diferenciālģeometrija, R.1990.2. Сборник задач и упражнений по дифференциальной геометрии, под. pед.

В.Т.Воднева, Минск 1970



ELEMENTS OF COMBINATORICS(Kombinatorikas elementi)

Course code Mate - 3043Author Professor A.Andzans, Dr.Hab.Math.Credits 3 creditsRequired for grade examPrerequisites Mate – 1002, Mate – 2065

Annotation. The objectives of course are: elements of enumerative combinatorics and combinatorial algorithms.

Subjects:1. Elements of enumerative combinatorics: general rules of combinatorics, method of

recursive enumeration, recursive sequences with constant coefficients, combinatorial number systems, method of generating functions.

2. Elements of combinatorial algorithms.2.1. Mathematical games without / with prehistory, probabilistic games. Epp –

Fergusson theorem.2.2. Algorithms of sorting and searching: partial sorting, full sorting, sorting with full

information, determination of the result of sorting process.2.3. Elements of algorithms of inductive constructions.

Requirement for received of credits: 30 hours lectures, 10 hours practical work. Students are required to work at home on problem solving constantly. The exam takes place in a written form and consists of problem solving.

Recommended literature:1. A.Andžāns, P.Zariņš. Matemātiskās indukcijas metode un varbūtību teorijas elementi.

Rīga, Zvaigzne, 1983. - 186 lpp.2. A.Gailītis, A.Andžāns. Kārtošanas un meklēšanas uzdevumi. Aizkraukle, “Krauklītis”,

1995. -102 lpp.3. I.Kudapa. Matemātiskās spēles. Maģistra darbs (rokraksts). Rīga, LU, 1995. - 62 lpp.4. И.Я. Виленкин, Комбинаторика. Москва, Наука, 1969 - 328 с.5. М.Холл. Комбинаторика. Москва, Мир, 1970.- 424 с.6. А.Кофман. Введение в прикладную комбинаторику. Москва, Наука, 1975. - 479 с.7. И.И. Ежов, А.В.Скороход, П.И.Ядренко. Элементы комбинаторики. Москва,

Наука, 1977. - 80 с.8. Д. Кнут. Исскуство программирования для ЭВМ. Т.3. Сортировка и поиск.

Москва, Мир, 1978.- 844 с.



EQUATIONS OF MATHEMATICAL PHYSICS (Matemātiskās fizikas vienādojumi)

Course code Mate-3142Authors Prof. H.Kalis, Dr.Hab.Math., Dr.Hab.Phys.;

Docent J.Cepītis, Dr.Math.; Prof. U.Raitums, Dr.Hab.Math.

Credits 4 credits Required for grade examPrerequisites Mate-2065, Mate-2135

Annotation. The objectives of the course are: sources of classical problems in mathematical physics; solvability and uniqueness for initial or boundary value problems for canonical equations of the hyperbolic, parabolic and elliptic type; the method of separation of variables.

Subjects:1. Natural problems, which lead to hyperbolic equations, oscillating string.

Interpretation of d'Alemberts's formulae.2. Mixed problems for hyperbolic equations and their correctness, energy integral.3. Method of separation of variables for an oscillating string. Duhamel's superposition

principle.4. Natural problems, which lead to parabolic equations, the heat transfer equation.5. Stationary processes, Laplace's and Poisson's equations.6. Boundary value problems for Laplace's operator, physical interpretation. Hadamard's

example. 7. Harmonic functions, maximum principle for elliptic equations.8. Poisson's formulae via the method of separation of variables.9. Green's function, its physical interpretation.10. Potentials, their physical interpretation and applications to boundary value problems.11. Maximum principle for parabolic equations.12. Cauchy's problem for the heat transfer equation, its correctness.13. Mixed problem for the heat transfer equation on the line, its correctness.14. Examples with self-similar solutions.Requirements for receiving of credits: 48 hours lectures, 16 hours practical work.Students are required to fulfil 2 independent home works and 2 laboratory works. The exam takes place in an oral form. Students must show an understanding of problems formulations for classical equations of mathematical physics and their correctness.Recommended literature:1. H.Kalis. Matemātiskās fizikas vienādojumi, klasifikācija un izvedumi. Stīgas svārstības vienādojums.Rīga,LU, 1992.2. H.Kalis. Puasona un siltuma vadīšanas vienādojums. Rīga,LU,1992.3. E.Riekstiņš. Matemātiskās fizikas vienādojumi. Rīga, Zvaigzne,1964.4. А.Н.Тихонов, А.А.Самарский,Уравнения математической физики,

М.,Наука,1977.



FIXED POINT METHOD(Nekustīgā punkta metode)

Course code: Mate – 3098Author Docent Andris Liepins, Dr. Math.Credits: 2 creditsRequired for grade testPrerequisites Mate – 1002, Mate – 1120

AnnotationProblems of the existence and uniqueness of fixed points for mappings on a metric space, Bohl - Brouwer fixed point principle, applications in equation solvability problems and mathematical economics.

SubjectsBanach fixed point principle, applications.Remetrization problem.The main generalisations of Banach fixed point principle.Banach fixed point principle for multi-valued mappings.Fixed point theorems in partially ordered sets.Birkhoff – Tarski principle, applications.Coincidence theorems, applications.Catarski – Kirk theorem, applications.Fuzzy versions of fixed-point theorems.Fixed point of strictly constractive mappings.Fixed point of non-expansive mappings.Equations of evolutions accretive operations and semi-group.Convergence of iterationsBohl – Brauwer principle.Schauder theorem, applications.Some important statements with are equivalent to Bohl – Brauwer principle.Knaster – Kuratowski – Mazurkewicz principle.Ky Fan inequality.Schauder – Tichonov theorem.Markow – Kakutani theorem.Topological degree.Measures of noncompactness and their applications in fixed point theory.Fixed point theorems and solvability of integral and differential equations.Orbital cersiens of fixed point theorems.Local versions of fixed point theorems.Fixed point theorems in algebraic topology.Periodical points of mappings.



Requirements for receiving of creditsStudent must be able to apply the main concepts and the main results.

Recommended literatureJ. Dugundji, A. Granas. Fixed point theory. – Warszawa, P. W. N., 1982V. Istratescu. Fixed point theory. – Reidel Publishing Company, 1981.А.Иванов. Неподвижные точки отображений метрических пространств.

Исследования по топологии. ИМ им. А.Стеклова, Наука, 1976 М.Красносельский, П.Забрейко. Геометрические методы нелинейного анализа,

Москва. 1975 Ф.Куфнер, С.Фучик. Нелинейные дифференциальные уравнения. Москва. 1988 A.Liepiņš. Nekustīgā punkta princips. – Lekciju konspekts.Ж.П.Обен. Нелинейный анализ и его применения в экономике. Москва, 1988 Ж.П.Обен, И.Экланд. прикладной нелинейный анализ. Москва, 1988 Б.Садовский и др. Меры некомпактности и уплотняющие операторы Новосибирск,

1986 M. Taskovič. Osnove teorije fiksne tačke. – Beograd, Zavod za udžbenike i nastavna

sredstva,1986.



FUNCTIONAL ANALYSIS(Funkcionālanalīze)

Course code Mate-3084Authors Docent O.I.Kārkliņš, Dr.Math.Credits 4Required for grade examPrereguisities Mate-2065, Mate-1002

Annotation: The following main topics will be considered in the course: Introduction into the theory of continuous operators in Bahach and Hilbert spaces (the open operator principle, the principle of equiboundedness, the principle of extension of a continuous linear functional; applications of these principles); Duality in functional analysis; Fundamentals of the theory of compact operators, its application for solving of Fredholm integral equations; Freche differencial and elements of differential calculus in Banach spaces.

Subjects:1. Topological spaces, metric spaces, normed spaces. Examples. Continuous mappings.

Topological vector spaces.2. Completeness in metric spaces. The contraction principle and its applications

(Fredholm and Volterra integral equations of the second type; Cauchy problem).3. Continuity of a linear operator; its different characterisations. The norm of an operator.

Normed vector space LC(X,Y) and normed algebra LC(X,X), where X and Y are normed spaces.

4. Convergence of a sequence of operators: convergence in respect of a norm and pointwise convergence.

5. The equiboundedness principle for a family of operators. Banach-Steinhaus theorems.6. The principle of an open operator and its applications; in particular, its applications in

the stability theory of a solution of a linear equation.7. The principle of extension of a linear continuous functional. 8. Hahn-Banach theorems: Geometrical and analytic forms of these theorems.9. Duality in functional analysis.10. Hilbert spaces. Theorem on an orthogonal projection.11. Generalised Fourier series. Fourier coefficients.12. Riesz-Fisher theorem about homeomorphism of Hilbert spaces.13. Riesz theorem on the general form of a continuous linear functional in a Hilbert

space. Duality.14. Compact and precompact sets. Compact operators in a normed space. Ascoli-Arzela

theorem. Compactness of the Fredholm integral operator.15. Elements of the Fredholm - Riesz theory of. Spectrum of a reflective compact

operator. 16. Hilbert Shmidt theorem. 17. Fredholm operators of the second type. Volterra integral equations.18. Freche differential and elements of the differential calculus in Banach spaces.



Requirements for received of credits: 64 hours lectures. Students are required to fulfil independent home works at the during of the course. The exam takes place in oral form. Students must show understanding of theoretical material considered at lectures and demonstrate the ability of solving practical problems corresponding to the course.Recommended literature:1. S.Lang, Real analysis, Addison-Wesley Publ. Comp., 1969.2. I.Kārkliņš. Lebega integrāļi. Rīga, LU, 1991.



FUNDAMENTALS OF ACTUARIAL MATHEMATICS(Akruārmatemātikas pamati)

Coursse code Mate – 3212Author Docent Viktorija Carkova, Dr. Math.Credits 4 creditsRequired for grade testPrerequisites Mate – 2036

AnnotationThe aim of course is to provide an introduction to the actuarial scientific method and to introduce concepts of insurance models.

SubjectsLife insurance.The purpose and the principle of Life insurance (2hours lecture + 1 hours seminar).Life insurance probabilities (2+1).The Future Lifetime probabilities (2+1).Life Tables (2+1).The Force of Mortality probability distribution (2+2).Analytical distribution of Mortality (2+2).Annuities. Pensions (2+2).Pure Endowments (1+1).Insurance models including expenses (2+1).Insurance payment. Reserves (2+1).Pension calculated scheme (2+1).The Future Lifetime of a life aged x (2+1).Short-term Life insurance models analysis.Short-term Life insurance (2+0).Short-term analytical Risk Models (2+1).Short-term the individual Risk analysis (2+1).Central limit theorem (2+1).Collective Risk models for a single period (2+1).Life insurance models analysis over an extended period.Life insurance over an extended period (2+1).The individual Risk analysis over an extended period (2+1).Collective Risk models over an extended period (2+1).Recursion stochastic procedures (2+1).Requirements for received of credits.The examination for this course is a three-hour written-answer examination. The course covers insurance probabilistic models.Recommended literatureN.Bowers, H.Gerber, J.Hickman, D.Jones, C.Nesbitt, Actuarial Mathematics, Society of

Actuaries, U.S.A., 1986, (625 p.).



H.U.Gerber, Life Insurance Mathematics, Springer-Verlag, 1990, (131 p.)



FUNDAMENTALS OF GEOMETRY(Ģeometrijas pamati)

Course code Mate-0009Authors lect. V.Neimanis, M.Math..Credits 4 creditsRequired for grade examPrerequisites Mate-1011, Mate-1048, Mate-1117.Annotation. The objectives of the course are: Course review next geometry axioms structures: Euclide axiom system, general information and main conclusions; Lobachevski axioms system, general information and main conclusions; main concepts about projection “affino”-geometry and Riman’ Geometry.Subjects:1. Precision axioms system in Geometry: general information, main and additional

axioms and conclusions.2. Consequences and attached axioms; theorems which main points are linked with

conception of Line, triangles, beam, angles.3. Conception of dragging, axioms of dragging and conclusions, congruential coherence,

its main characteristics, congruential of line’ section, angles, triangles e.t.c.4. Comprising of angles and triangles, its subs traction and addition. Main concepts of angles.5. Deductial continuation of axioms, right angles and curves.6. Length of line’ section and angles’ volume, coordinate system on line, on 2-D surface and in space.7. Euclide Geometry axioms system. Similar triangles.8. Conception of interpretation of axioms system. Decart interpretation. Non- conflicting

geometrical system of Euclide’ axioms completeness and independence of axioms.9. Historical review of axioms methods and geometrical structures: Euclide’ “The

elements” and equivalents of 5-th Euclide’ postulate.10.Lobachevski’ geometrical axioms system, parallel angles and parallel lines. Features

of attitude of parallel lines.11.Triangles and squares in Lobachevski’ Geometry.12.Lobachevski’ Function (X) and its features.13.Parallel and divergence linear correlations of lines and flatness in Lobachevski’ space.14.Curves, equidistanties, arcs theirs features.15.Beltrami-Klein interpretation of Lobachevski’ Geometry.16.Puancare’ interpretation of Lobachevski’ Geometry.17.Axioms system of Riman’ Geometry. Conception of projection and “affino”-

geometrical system of axioms.Requirements for receiving of credits: 64 hours lectures.Include 2 practice test works, one theory test work and exam passing with 2 questions in theory and 1 exercise.Recommended literature:1. T.Cīrulis, V.Neimanis. Ģeometrijas pamati un diferenciālģeometrija. Rīga, LU, 1980.2. Ефимов Н.В. Высшая геометрия. Москва, 1971.3. Погорелов А.В. Основания геометрии, Москва, 1968.4. Щербаков Р.Н., Пичурин Л.Ф., От проективной геометрии к неевклидовой, Москва, 1979.



FUZZY SETS AND SYSTEMS I(Fazi kopas un sistēmasI)

Course code Mate-4184Authors Prof. A. Šostaks, Dr. Habil. Math.Credits 2 creditsRequired for grade exam Prerequisites: Mate-1060, 1061, 1063, 2086.

Annotation: The concept of a fuzzy set was introduced in 1965 by L. Zadeh. This concept generalises the notion of a usual set. The goal of such generalisation was to make mathematical concepts, constructions and results of set-theoretic nature more applicable in other branches of science. Since 1965 the mathematical theory of fuzzy sets and structures intensely developed in different directions; at the same time fuzzy sets, fuzzy structures and fuzzy systems have found many important applications far outside theoretical mathematics.. The main aim of the course is to give the students basic knowledge of the fundamentals of the theory of fuzzy sets and fuzzy structures. Some applications of the theory in practice will be also discussed.

Subjects: 1. Introduction: Basic ideas; examples, historical comments; some applications.2. Foundations of the theory of fuzzy sets:

2.1. Basic definitions, examples.2.2. Operations with fuzzy sets.2.3. Images and preimages of fuzzy sets.2.4. Level decomposition of fuzzy sets and its basic properties.

3. Fuzzy relations:3.1. Basic definitions, examples.3.2. Operations with fuzzy relations.3.3. Reflexive, symmetric and transitive fuzzy relations.3.4. Transitive closure of a fuzzy relation.3.5. Similarity relations.3.6. Difference relations.3.7. Level decomposition of relations and its properties..

4. Some concepts and results from lattice theory.5. L-fuzzy sets:

5.1. Definitions, examples of L-fuzzy sets.5.2. Operations on L-fuzzy sets; images and preimages of L-fuzzy sets.5.3. Level decompositions of L-fuzzy sets.

6. Representation technique:6.1. Representation 1st theorem and its applications for the theory of L-fuzzy sets.6.2. Representation 2nd theorem and its applications for the theory of L-fuzzy sets.

7. The .-field of fuzzy sets and fuzzy measures.7.1. Probability measures on a .-field of fuzzy sets.7.2. Probability fuzzy measures.



7.3. Lowen-Klement theorem on fuzzy measures.8. Some categorical aspects of fuzzy set theory:

8.1. A category, a subcategory, a morphism, an isomorphism. Examples of categories.8.2. Functors. Examples of functors.

9. Basic categories of fuzzy sets. Operations in categories of fuzzy sets.10. Fuzzy real numbers. Arithmetic operations on fuzzy numbers. Fuzzy real line.

Hyperfield of fuzzy numbers.11. Zadeh's extension principle. 12. Fuzzy equalities and fuzzy orders.13. Discussion on some applications of fuzzy sets.

Requirements for receiving credits: 28 hours lecture; 4 hours workshops. A students is required to prepare individually one report and to give a talk on its subject. The exam takes place in oral form. Students must show understanding of theoretical material considered at lectures and demonstrate the ability of solving practical problems corresponding to the course.

Recommended literature:1. A. Kaufman, Fuzzy Set Theory, Masson – Paris, Barcelona, New-York 1977.2. R. Kruse, J. Gebhardt, F. Klawon, Foundations of Fuzzy Systems – John Wiley &

Sons, 1994.3. Fuzzy Sets and Possibility Theory: Recent Developments, Ed. Ronald R. Yager,

Pergamon Press – New York, Oxford, Toronto- 1982.4. S.E. Rodabaugh, Fuzzy addition and the L-fuzzy real line, Fuzzy Sets and Systems,

N 8, (1982), pp. 39-52.5. Fuzzy Sets (Current trends): In Tatra Mountains – Mathematical publications, Vol.

12, Bratislava, 1997.6. Fuzzy Structures (Current trends): In Tatra Mountains – Mathematical publications,

Vol 13, Bratislava, 1997.



FUZZY SETS AND SYSTEMS II(Fazi kopas un sistēmas II)

Course code Mate-4185Author Prof. A. Šostaks, Dr. habil. Math.Credits 2 creditsRequired for grade exam Prerequisites: Mate-4184.

Annotation: This course is the continuation of Fuzzy Sets and Structures I (Mate-4184). Principal attention is devoted to fuzzy logic's, problems of fuzzy analysis, fuzzy topological and fuzzy algebraic structures. Some applications will be also discussed.

Subjects: 1. T-norms and T-conorms.2. Elements of the theory of GL-monoids. 3. Elements of the theory of MV-algebras.4. Elements of many-valued logics and fuzzy logics.5. Fuzzy algebraic structures: Fuzzy groups, fuzzy rings, categories of fuzzy algebraic

structures.6. Fuzzy metrics. Fuzzy metric spaces. Elements of fuzzy analysis.7. Fuzzy differential equations. 8. Elements of the theory of fuzzy integral.9. Fuzzy topological structures.

9.1. Chang-Goguen L-topologies. Stratified L-topologies.9.2. Hutton fuzzy topologies.9.3. (L,K)-fuzzy topologies.9.4. Problem of changing the base: Rodabaugh's approach to fuzzy topology.

10. L-unifortmities and L-fuzzy uniformities. 11. L-proximities and L-fuzzy proximites.12. Categories of fuzzily structured sets: general theory.13. Elements of the theory of fuzzy categories. Fuzzification of classical mathematical

categories and theories. 14. Fuzzy control. Fundamental problems of fuzzy control.

Requirements for received of credits: 32 hours lectures; 8 hours seminars. A student is required to prepare individually one report and to give a talk on its subject. The exam takes place in oral form. Students must show understanding of theoretical material considered at lectures and demonstrate the ability of solving practical problems corresponding to the course.



Recommended literature:1. A. Kaufman, Fuzzy Set Theory, Masson – Paris, Barcelona, New-York 1977.2. R.Kruse, J.Gebhardt, F.Klawon, Foundations of Fuzzy Systems, John Wiley & Sons,

1994.3. Fuzzy Sets and Possibility Theory: Recent Developments, Ed. Ronald R. Yager,

Pergamon Press – New York, Oxford, Toronto- 1982.4. S.E. Rodabaugh, Fuzzy addition and the L-fuzzy real line, Fuzzy Sets and Systems,

N 8, (1982), pp. 39-52.5. Fuzzy Sets (Current trends): In Tatra Mountains – Mathematical publications, Vol.

12, Bratislava, 1997.6. Fuzzy Structures (Current trends): In Tatra Mountains – Mathematical publications,

Vol 13, Bratislava, 1997.



GENERAL THEORY OF OPTIMAL ALGORITHMS(Optimālo algoritmu vispārīgā teorija)

Course code Mate-3190Author Docent S.Asmuss, Dr.Math.Credits 4 creditsRequired for grade testPrerequisites Mate-1002, Mate-2065, Mate-3140

AnnotationThe basic notions of this course are error and complexity of an algorithm, optimal information and optimal algorithm. Special attention is paid to the general theory. On this basis the efficiency of some algorithms for solving such problems as root - finding, approximation, numerical differentiation and integration, minimisation is investigated.

Subjects1. Error of algorithms.

1.1. Algorithms. Solution and information operators.1.2. Error of algorithms. Error optimal algorithms.1.3. Interpolatory algorithms, their error bounds.1.4. Central algorithms, their optimality.

2. Complexity of algorithms.2.1. Information and combinatorial complexity.2.2. Solvable and -solvable problems. -complexity of problems.2.3. Complexity optimal algorithms.

3. Theory of linear information.3.1. Linear information. Cardinality of linear information.3.3. Index of linear problems.3.4. Optimal information. Sufficient conditions for the existence of it.3.5. Optimal information for the problem of numerical integration.

4. Analysis of linear algorithms.4.1. Linear algorithms. Error and complexity.4.2. Optimal algorithms in Sard’s sense.4.3. Optimal algorithms in Nikolsky’s sense.4.4. Optimal quadrature formula.4.5. Optimal algorithms for linear problems.

5. Spline algorithms.5.1. Splines and spline algorithms5.2. Sufficient conditions for the optimality of spline algorithms.

6. Analysis of certain numerical algorithms.6.1. Approximation of linear functionals. Interpolation.6.3. Numerical differentiation and integration.6.4. Numerical solution of algebraic equations.6.5. Numerical solution of differential equations.

Requirements for received of credits



Student must be able to apply the main concepts and the main results.

Recommended literature1. Traub J.F., Wozniakowski H. A general theory of optimal algorithms. New York,

Academic Press, 1980.2. Aho A.V.,Hopcroft I.E., Ulman I.P. Construction and analysis of memorial algorithms,

Moscow, Mir, 1979.3. Laurent P.J. Approximation and Optimization, Paris, 1972.



INTEGRAL EQUATIONS(Integrālvienādojumi)

Course code Mate-4192Author Docent O.I.Kārkliņš, Dr.Math.Credits 4 creditsRequired for grade exam or testPrerequisites Mate-2065, Mate-1002 Anotation. This course considers Volterra and Fredholm integral equations of the second kind on the class of continuous functions and on the class of square-integrable functions. The existence problem and basic properties of solutions of these integral equations according to parameter are studied. Analogies with linear algebraic equation systems and the geometry of the Euclid space are discussed. An insight in numerical methods of solving integral equations is given. Reduction of differential equations problem to integral equations is considered.Subjects:1. Volterra and Fredholm integral equations of the first and second kind, relation

between them. The non-stability of solutions of the Fredholm equations of the first kind. Integral operators and their linearity. Integral equations with real or complex parameter. Characteristic values and characteristic functions of the integral equation, eigenvalues and eigenvectors of the integraloperator. Examples of the non-linear integral equations.

2. Solving of integral equations with method of successive approximations. Contraction theorem and its expansion.

3. Normed vector space of the continuous functions. Pre-Hilbert and Hilbert spaces of the square-integrable functions.

4. The proof of unique existence of the solution of Volterra integral equation of the second kind with Banach theorem expansion.

5. Solution of the Fredholm integral equation of the second kind with kernel of finite rank. Solving Fredholm equations by approximating its kernel with kernels of finite rank.

6. Solving of Fredholm integral equation of the second kind with methods of successive approximations. The iterated kernels. The Neumann series for the resolvent.

7.-8. The Fredholm theorems. Duality.9.-10. Continuous linear operators, boundedness property, operator norm. The Fredholm integraloperator continuity.

Compact and relatively compact sets. Ascoli-Arzela theorem.11.-12. Compact operators. Compactness of Fredholm integral operators. Approximation

of compact operator by linear operators of finite rank. The insight in Riesz theory.13.-14.Duality in normed vector spaces and Hilbert spaces. The Fredholm theorems in

operators form. Spectrum of self-adjoint compact operator on a Hilbert space. The Hilbert-Schmidt theorem.

15. Reduction of Sturm-Liouville problem to a Fredholm integral equation of the second kind.



Requirements for receiving of credits: 32 hours lectures, 32 hours workshops. Students are required to fulfill 3 independent home works. The exam takes in oral form. Students must show understanding of theoretical material considered at lectures and demonstrate the ability of solving practical problems corresponding to the course.

Recommended literature:1. J.A.Cochran. The Analysis of Linear Integral equations. McGraw-Hill Book

Company, 1972.2. J.Dieudonne. Foundations of Modern Analysis. Academic Press, 1960.



INTEGRAL SPLINES AND THEIR APPLICATIONS(Integrālie splaini un to lietojumi)

Course code Mate-0268Author Prof. A. Buiķis, Dr.Hab.Math.Credits 2 credits Required for grade examPrerequisites Mate-3143, Mate-3140

Annotation. The course considers physical processes in layered media and their mathematical models. The integral splines are defined, their properties and application to describing, both analytically and numerically, the constructed mathematical models.

Subjects:1. Cubic C2 splines. Their representation in the mi and Mi variables. The volumes of

necessary work (the labour consumption for the work). 2. Representation of the cubic splines by a matrix with independent from function

coefficients. Representation of non-homogeneous boundary conditions.3. Parabolic splines and their representation.4. Existence and uniqueness of the integral parabolic spline (IPS) for a piecewise smooth

function.5. A matrix with independent coefficients for IPS representation. The volume of work

and parallelisation of algorithm.6. The generalised integral parabolic spline (VIPS).7. Models of stationary processes for a two-layer system and their averaging.8. Models for three-layer system with an intermediate layer. Estimation of errors of the

method. 9. Application of using IPSs and VIPSs to the equations of mathematical physics with

discontinuous coefficients. Advantages of using IPSs and VIPSs as compared with classical splines.

Requirements for received of credits: 24 hours lectures, 8 hours workshops. The exam takes place in the oral form. Students should be familiarised with the problems considered in the program.

Recommended literature:1. K.Eriksen, D.Estep, P.Hansbo, C. Johanson. Computational Differential Equations. Cambridge

University Press, 1966.2. A.Buikis. Problems of Mathematical Physics with Discontinuous Coefficients and their

Applications. Riga, 1991. Manuscript (Russian ).3. T. Rusakov. Methods of Spline-functions in Numerical Hydrodynamic. Perma, 1987 (Russian).4.H.Spaeth.EindimensionaleSpline–Interpolationsalgorithmen.Verlag Muenchen, 1990 (German).5. S. Stetshkin, J. Subotin. Spline in Numerical Mathematics. M., 1976 (Russian ). 6. J.Zavjalov, B.Kvasov, V.Miroshnitshenko. Methods of Spline- Functions. M., 1980 (Russian).



INTRODUCTION TO ALGORITHM THEORY(Ievads algoritmu teorijā)

Course code Mate – 1107Author Docent Janis Buls, Dr. Math.Credits 2 credits Required for grade testPrerequisites

AnnotationIt was about sixty years ago when a notion of recursive function appeared. A new branch of mathematics was born - the theory of algorithms. Initially there was design to create a strict mathematical concept for intuitive notion on computable function. Nowadays the inferences of algorithm theory are exploited in mathematical logic, algebra, theoretical programming, cryptography, formal language theory, biology and philosophy. An effective calculability problem can be posed as follows: what is ability of modern computer if we ignore time, memory and resource restrictions? Therefore the theory of algorithms indulge neither in real existing computers or specific programming languages. It is interested in computers theoretical possibilities and limits. This course from the most basic concepts proceeds through the following stages: random access machines, computable functions, primitive recursive functions, partial recursive functions, Turing machines, Markov normal algorithms.

SubjectsRandom access machines(RAM).Functions computable by RAM.Programme in standard form. Juxtaposition of programs.Computability of composite function.Decidable and undecidable predicates, characteristic functions of theirs.Primitive recursive functions; examples.Logic calculus with primitive recursive predicates.Bounded sums and products.Bonded mu operator.Coding of tuples of nonnegative integers.Mu operator.Partial recursive functions.Ackermann generalised exponential.Markov normal algorithms.Turing machines and computability.

Requirements for received of creditsCandidate must orient one-self in concepts and problems of course. The skill must be demonstrated as solution of problems. Candidate must unconstrained expound appointed theme.



Recommended literatureКатленд Н. Вычислимость. Введение в теорию рекурсивных функций. Москва,1983.Пападимитриу Х. Комбинаторная оптимизация. Алгоритмы и сложность. Москва, 1985.Мальцев А.И. Алгоритмы и рекурсивные функции. Москва, 1986.Марков А.А., Нагорный Н.М. Теория алгорифмов. Москва, 1984.Роджерс Х. Теория рекурсивных функций и эффективная вычислимость. Москва, 1972.Успенский В.А., Семенов А.Л. Теория алгоритмов: основные открытия и приложения.

Москва, 1987.Юббинхауз Г.Д., Якос К., Ман Ф.К., Хермес Г. Машины Тюринга и рекурсивные

функции. Москва, 1972.



INTRODUCTION TO NUMBER THEORY(Ievads skaitļu teorijā)

Course code Mate – 1018Author Docent JānisBuls, Dr. Math.Credits 2 credits Required for grade testPrerequisitesAnnotationIntegers have always been among the fundamental notions of mathematics. Already ancient Greeks knew in VI century BC the integer solutions of the equations x2+y2=z2. German mathematician Carl Friedrich Gauss (XIX century AD) originated the basic methods in the theory of congruencies. These methods are applied nowadays in algebra, cryptography and coding theory. Number theory serves as an instrument in other areas of mathematics to prove results involving effective algorithms and complexity theory. The course considers elementary aspects of Number theory, namely, division and factorisation, arithmetical functions, multiplicative functions, congruencies.SubjectsAxioms of arithmetic.Algorithm of Euclid.Common divisors and multiples.Prime numbers. Sieve of Eratosthenes. Standard form of a natural number.Continued fractions. Recurrent sequences. Euler algorithm.Equation x +y =z.Arithmetic functions. Möbius function. Euler function.Congruencies and rings.Complete residue systems modulo m and reduced residue systems.Euler and Fermat theorems.Equations with congruencies.Linear equations with congruencies and continued fractions.System of congruencies and Chinese Remainder Theorem.Quadratic residues and non-residues.n=w +x +y +z .Primitive roots.Indices.Residue class character.Criterions for divisibility of numbers.Requirements for received of creditsCandidate must orient one-self in concepts and problems of course. The skill must be demonstrated as solution of problems. Candidate must unconstrained expound appointed theme. Recommended literatureN. Eņģele. Skaitļu teorija. – Rīga, 1980S. Mihelovičs. Skaitļu teorija. – DPU izdevniecība "Saule" 1996.



LEBESGUE INTEGRAL(Lebega integrāļi)

Course code Mate-3189Author Docent O.I.Kārkliņš, Dr.Math.Credits 4 creditsRequired for grade exam or testPrerequisites Mate-2065

Anotation. The theory of measure and Lebesgue integral is an essential part in mathematical education. Those are the foundation of functional analysis, the Probability theory, calculus of variations. Lebesgue integral and it simplifies many theorems and their proofs.

Subjects:Measures. Lebesgue measure.1. Algebras, -algebras. The -algebra of Borel sets.2. Additive set functions. Countable additivity. Measures.3. Outer measures. The measure induced by an outer measure.4. Lebesgue measure and -algebra and their properties. Measurable functions.5. Function measurability with respect to the -algebra: equivalent definitions.6. Operation with measurable functions. Approximation by simple functions. General

integral.7.-9. Definition and basic properties.10.-11. Levi, Fatou and Lebesgue convergence theorems. Absolute continuity of an

integral. Lebesgue integral in Rp.12.-13. The geometric interpretation of an integral. Fubini theorem.14.-15. Substitution in the lebesgue integral with a C¹-diffeomorphism. Lebesgue measure transformation by an invertible linear map.

Requirements for received of credits: 32 hours lectures, 32 hours workshops. Students are required to fulfil 3 independent home works. The exam takes in oral form. Students must show understanding of theoretical material considered at lectures and demonstrate the ability of solving practical problems corresponding to the course.

Recommended literature:1. I.Kārkliņš. Ievads integrāļa teorijā. Rīga, LU, 1990.2. I.Kārkliņš. Lebega integrāļi. Rīga, LU, 1991.3. A.E.Taylor. General Theory of Functions and Integration. Blaisdell Publ. Comp., 1965.



MATHEMATICAL MODELS IN THE DIFFERENTIAL EQUATIONS(Matemātiskie modeļi diferenciālvienādojumos)

Course code Mate – 3136Author Docent Silvija Cerane, Dr. Math.Credits 2 credits Required for grade testPrerequisites Mate-2065, Mate-2134

AnnotationMathematical Models of Nature (Mechanics, Biology, Chemistry etc.)

SubjectsGeneral Principles of stating the Differential Equations.Basic Notions of the Theory of Dynamical Systems.Stability. Lyapunov’s function. Lyapunov’s Theorem. Linearization. Examples.Bistability. Examples of Biology, Electrotechnik etc.Attractors, limit cycles, Puancare-Bendixon Theorem.Mathematical Problems of Biology. Dynamic of population. Lotka - Volterra equations.

Predator - prey ecosystem. Kolmogorow’s Theorem.Dynamical Systems in discrete timePuancare section, Puancare map.Rayleygh’s equation.Economical cycles.Bifurcation’s.Feigenbaum number. Attractors for discrete time dynamical systems. Differential equations for solving some medical problems.Lorenz equation.

Requirements for received of credits:32 hours lectures. Colloquiums.

Recommended literatureJohn L. Casti. Alternate Realities. Mathematical Models of Nature and Man. Wiley. New

York. 1989Calculus. Mathematics and Modelling. Addison- Wesley. 1997.Gotfried Jetschke. Mathematic der Selbstorganisation. Berlin, 1989Д.Эрроусмит, К.Плейс. Обыкновенные дифференциальные

уравнения. Качественная теория с приложениями. М., Мир, 1986



А.Д.Базыкин. Математическая биофизика взаимодействующих популяций. М. Наука, 1985



MATHEMATICAL MODELS OF CHEMICAL REACTOR THEORY (Ķīmisko rektoru teorijas matemātiskie modeļi)

Course code Mate - 4155Author Docent J.Cepītis, Dr.Math.Credits 2 credits Required for grade examPrerequisites Mate-2135

Annotation. The course considers mathematical models of chemical reactor theory which are reduced to the boundary value problems of non-linear ordinary differential equations.

Subjects:1. Some examples of chemical reactor theory mathematical models.2. Phase plane analysis for the models interpretation.3. Method of a priori estimates for the solvability proving.4. Shooting method.5. Iteration methods for the numerical solving.6. Transformation method for the numerical solving.7. Another methods for the numerical solving.8. Estimation of the number of solutions.9. Estimation of the solution stability.10. Estimation of the bifurcation parameters.

Requirements for receiving of credits: 28 hours lectures, 4 hours workshops. Students must once stand out in the workshop. The exam takes place in the oral form and students should be familiarised with the problems considered in the program.

Recommended literature:1. J.Cepītis. Parasto diferenciālvienādojumu nelineāras robežproblēmas. Rīga, LU, 1987.2. V.Voleski, I.Votruba, Mathematical models of chemical processes, Toronto, 1991.3. Ц.На, Методы численного решения прикладных краевых задач, Москва, 1982.



MATHEMATICAL MODELS OF CONTINUOUS MEDIUM MECHANICS (Nepārtrauktās vides mehānikas matemātiskie modeļi)

Course code Mate-0273Authors Docent L.Bulīgins, Dr.Phys., Docent S.Lācis, Dr.Phys.,

Prof. H.Kalis, Dr.Hab.Math., Dr.Hab.phys., Docent J.Cepītis, Dr.Math.Credits 4 credits Required for grade examPrerequisites Mate-3140, Mate-3143, Fizi-3167

Annotation. The course considers a mathematical description of physical processes using models of the continuous medium mechanics.

Subjects:1. Transfer processes in surroundings and technology. Concept of the continuous medium. 2.Lagrange’s and Euler’s coordinates.Equation of continuity. Stress tensor of the continuous medium.Equation of continuous medium motion. Boundary conditions. Models of ideal and

viscous fluid.Navier-Stokes equation.Equation of continous medium energy.Equation of inner energy. Fourier’s law. Equation of temperature.Motion of the continuous medium in the cylindrical and spherical co - ordinates.Reynold’s number. Stokes’ and Osen’s fittings.Boussinesk’s fitting.Pranndtl’s, Reley’s and Grassgof’s numbers. Instability of fluid.Critical Reley’s number in layer with free surfaces.Poissel’s ans Qetta’s flows, their stability.Taylor’s instability for the flow between two rotating cylinders. Concept of the package ‘Fluent’.

Requirements for received of credits: 32 hours lectures, 32 hours practical work.Students are required to fulfil 2 independent home works and 2 laboratory works. The exam takes place in an oral form.

Recommended literature:1. J.Batchelor. Introduction in Fluid Dynamics. Moscow, 1973.(Russian)2. L.I.Siedow. Mechanics of Continuous Medium. Moscow, 1976 (Russian)



MATHEMATICAL PRINCIPLES OF ECONOMIC MODELS(Ekonomisko modeļu matemātiskie pamati)

Course code Mate – 4195Author Docent Inese Bula, Dr. Math.Credits 4 creditsRequired for grade test, examPrerequisites Mate–2065, Mate – 2135

AnnotationMuch of the economic literature is preoccupied with the idea that market economies tend naturally to a equilibrium. We shall give three economic models and proofs of the existence of equilibrium or quasi-equilibrium. The basic mathematical tool is the "fixed point theorem" of Bohl-Brouwer and its generalisations to correspondences or to w-continuous mapping. Models of linear matrixes are nearer to reality. We consider difference equations and their applications in economic ("discrete" economic models).

SubjectsEconomic basic-model by K.J.Arrow and F.H.Hahn:Description of model,Continuity,Fixed point theorem of Bohl-Brouwer,Application of Bohl-Brouwer theorem to proof of equilibrium existence.Economic model by w-continuity:w-continuity and their properties,Deviations of model from the basic-model,Analog of Bohl-Brouwer-Schauder theorem for w-continuous mapping,Application of the analog of Bohl-Brouwer-Schuder theorem to proof of equilibrium

existence.Method least squares for design of curve.Economic model by "correspondence":Correspondences,Continuity,Description of model,Theorem of Kakutani,Application of Kakutani theorem to proof of equilibrium existence.Models of linear matrix’s:Model of Leontief,Model of Kantorovich.Difference equations:Basic-conceptions,Linear first-order equations with constant coefficients,Linear second-order equations with constant coefficients,



Linear n-order equations with constant coefficients.Equilibrium and stability of difference equations.Difference equations in economic:Supply-demand cycles,Model of Hicks,Model of Metzler.

Requirements for received of creditsThere must be written two test-works during the semester.The final examination is passed as a test (questions about various concepts and problems) or oral response.

Recommended literatureK.J.Arrow, F.H.Hahn General Competitive Analysis, North-Holland Publishing Company

Amsterdam, New York, Oxford, 1980.I.Bula Economic model of market equilibrium for w-continuous excess-demand function

(to appear).Ž.Debrē Vērtības teorija, Latvijas Akadēmiskā bibliotēka, 1997.S.Goldberg Differenzengleichungen und ihre Anwendung in Wirtschaftswissenschaft,

Psychologie and Soziologie, R.Oldenbourg Verlag, München, 1968.H.Nikaido. Convex structures and economic theory, Academic Press, New York and

London, 1968 (krieviski: H.Rommelfanger Differenzengleichungen, Bibliographisches Institut, Zürich, 1986.



MATHEMATICAL, STATISTICAL AND SPECIAL SOFTWARE PRODUCTS(Matemātiskās, statistiskās un speciālās datorprogrammu paketes)

Course code Mate-3271Authors Assoc. Prof. J.Vucans, Dr.Math.; Prof. H.Kalis, Dr.Hab.Math.; Dr.Hab. Phys. Docent O.Lietuvietis, Dr.Math.; Docent J.Lapiņš, Dr.Math.Credits 4Required for grade testPrerequisites Mate-1062, Mate-1063, Mate-2134

Annotation: The aim of the course is to learn the students the main principles which one must know in order to use software packages "Mathematica", "Maple V", MATLAB, SPSS, MINITAB as well as some less popular packages in his scientific and research work as a mathematician, economists, etc A survey about the areas where this software can be used and about the methods how it can be used will be given.

Subjects:The following questions will be considered in the course:1. Universal software package "Mathematica"; its use for analytic transforms and

numerical calculations.2. Universal software package Maple V, its use for analytic transforms and numerical

calculations.3. Universal software package MATLAB, its use for analytic transforms and numerical

calculations.4. Statistical software package MINITAB, its use for analytic transforms and numerical

calculations.5. Statistical software package MINITAB, its use for analytic transforms and numerical

calculations.6. Software package for operational research (in particular, for programming) TORA.7. Software package STORM for calculations and planning of economical processes (in

particular, for solving problems of linear programming, analysis of investments, planning of production, statistical analysis, etc.)

8. Possibilities and methods to put the results obtained by software packages in the WORD text.

Requirements for receiving of credits: during of the course student must fulfil individual laboratory works by means of the concrete software packages. The test takes place in oral form. Students must show good orientation in the possibilities of the software packages considered in the course.

Recommended literature:1. Instructions for the use of packages.



METHODS OF MATHEMATICAL PHYSICS (Matemātiskās fizikas metodes)

Course code Fizi - 2069Authors Prof. H.Kalis, Dr.Hab.Math., Dr.Hab.Phys.;

Docent J.Cepītis, Dr.Math.Credits 4 credits Required for grade examPrerequisites Mate-2065, Mate-2134

Annotation. The objectives of the course are: sources of classical problems in mathematical physics; solvability and uniqueness for initial or boundary value problems for canonical equations of the hyperbolic, parabolic and elliptic type; the method of separation of variables. Note that for this course Mate-2135 is not a perquisite.

Subjects:1. Linear homogenous and quasi-linear first orders partial differential equations. Integral

surfaces and characteristics.2. Classification and canonical forms of the second order almost linear partial

differential equations.3. The main principles for the obtaining of the equations of mathematical physics.

Oscillating string, heat transfer and other examples of natural appearances. Steady-state equations.

4. D'Alemberts formulae and its interpretation.5. Mixed problem for the equation of hyperbolic type, the method of separation of

variables.6. Sturm's – Liouville's problem. Properties of the eigenvalues and eigenfunctions.7. Mixed problem for the heat transfer equation on the line, its correctness.8. Boundary value problems for Laplace's operator, physical interpretation. Hadamard's

example. Harmonic functions, maximum principle for elliptic equations.9. Method of separation of variables, Green's function and potentials for elliptic

equations.

Requirements for receiving of credits: 32 hours lectures, 32 hours practical work.Students are required to fulfil 3 independent home works and 2 laboratory works. The exam takes place in an oral form. Students must show an understanding of problems formulations for classical equations of mathematical physics and their correctness.Recommended literature:1. H.Kalis. Matemātiskās fizikas vienādojumi, klasifikācija un izvedumi. Stīgas

svārstības vienādojums.Rīga,LU, 1992.2. H.Kalis. Puasona un siltuma vadīšanas vienādojums. Rīga, LU, 1992.3. E.Riekstiņš. Matemātiskās fizikas vienādojumi. Rīga, Zvaigzne, 1964.4. А.Н.Тихонов, А.А.Самарский,Уравнения математической физики, М., Наука,

1977.



METHODS OF OPTIMISATION(Optimizācijas metodes)

Course code Mate-3274Authors Docent O.I.Kārkliņš, Dr.Math.Credits 4Required for grade exam or testPrerequisities Mate-2065, Mate-2032, Mate-1002Annotation: The principal aim of the course is to develop different analytical and numerical methods for solving extreme problems for functions of one and several variables and for functionals. Besides, certain elements of the theory of optimal control will be discussed.Subjects:1. The contents of the subject and its history in brief.2. Necessary and sufficient conditions for the existence of an extremum in case of a

function of one variable.3. Necessary and sufficient conditions for the existence of an extremum in case of a

function of several variables.4. A directional derivative of a function. Lagrange first variation. Necessary and

sufficient conditions for the existence of an extreme.5. Problem of conditional minimisation for a function of several variables: it solving by

smoothing method. 6. The Lagrange method for solving problems of conditional minimisation: the case

when the conditions are given in the form of equality; the case when the conditions are given in the form of an inequality. Necessary and sufficient conditions for a conditional extreme.

7. Numerical methods of minimisation. Classification of methods. Minimisation of an unimodal functions of one variable. Active and passive methods. Minimax principle. Methods of minimisation of functions of several variables: gradient method, Newton method, etc.

8. Convex sets and convex functions. Theorems on separation of convex sets. Convex programming (planning).

9. Variations calculus. Brahistohrone problem. Elementary problems of variations calculus. Survey of some other problems of variations calculus.

10. Survey of the fundamentals of the theory of optimal control.Requirements for receiving of credits: 32 hours lectures, 32 hours practical work. Students are required to fulfil 5 independent home woks at the during of the course. The exam or test takes place in oral form. Students must show understanding of theoretical material considered at lectures and demonstrate the ability of solving practical problems corresponding to the course.Recommended literature:1. J.Engelsons. Optimizācijas metodes. I un II daļa.2. Ф.П.Васильев, Численные методы решения экстремальных задач, Наука , 1988.3. В.Д.Ногин и др., Основы теории оптимизации, Высш. Шк. , 1986.4. Э.М.Галеев, В.М.Тихомиров, Краткий курс теории экстремальных задач, Изд-во

МГУ 1989.



MICROECONOMICS OF INSURANCE (Apdrošināšanas mikroekonomika)

Course Code Mate - 3209Author Assoc. Prof. Jānis Vucāns, Dr. Math. Credits 4 creditsRequired for grade examPrerequisites Mate–3030, Mate-0272 Annotation: The principal aim of this course is to acquaint students with the economic and mathematical basis of the insurance theory.

Subjects:1. Introduction. What is "Microeconomics of Insurance?"2. General principles of insurance

2.1. Fundamental mechanism of the individual's insurance 2.2. Other forms of insurance

3. Decision making theory 3.1. Restrictions in using the criteria of actual expected value 3.2. Paradox of St-Petersburg3.3. The Bernoulli principle and the Fon Neuman - Morgenshtern axiomatic 3.4. Attitude to risk: definitions, measures 3.5. The risk of the random variable 3.6. Application: individual's attitude to savings 3.7. Critical notes about the applicability of the Bernoulli principle

4. How is the demand of insurance formed?4.1. The Mossen model of unique risk 4.2. The optimal form of the insurance contract 4.3. Self-insurance and its influence on insurance 4.4. Non-insurable risks and their influence on the insurance model

5. Life insurance - capitalisation5.1. Two periods model, basic assumptions and results 5.2. The economic basis of lifetime rent 5.3. General model5.4. Borsch's paradox5.5. Specification of model by taking into account fiscal aspects

6. Analysis of tariffing 6.1. The role of the intermediaries in the insurance business 6.2. The composition of the insurance costs 6.3. The microeconomic investigations of tariffing

7. Insurance market 7.1. The market equilibrium model 7.2. The role of the Mutual Insurance Companies in the automobile insurance market 7.3. The problem of the information asymmetry in the insurance market

8. Problems on the automobile insurance



8.1. A priori and a posterior tariffing systems and corresponding models 8.2. The experience collected in some countries in the domain of automobile

insurance 9. Systems of social insurance; the role of the State in the regulation of insurance market

Requirements for received of credits: 48 hours lectures, 16 hours seminars. During the semester students must solve a number of problems, assigned by the teacher for independent solving; they must present these solutions to the teacher by the indicated deadline. Students must independently prepare the analysis of some insurance problem. During the seminar they must explain the obtained results. At the written examination student must be able to demonstrate the knowledge of the basic concepts and properties of insurance and its microeconomic theory, explained during the lectures, and the ability to use them to formulate and to solve different insurance problems.

Recommended literature:1. D.Henriet, J.-Ch. Rochet, Microeconomie de l'assurance, Economica, Paris, 1991.2. P.Petauton, Théorie et pratique de l'assurance-vie, Dunod, Paris, 1996.3. V.V.Shahov, Strahovanie (Insurance - in Russian), Ed. company JuNITI, 1997.



NON-LINEAR BOUNDARY VALUE PROBLEMS IN APPLICATIONS (Nelineāras robežproblēmas pielietojumos)

Course code Mate - 3173Author Docent J.Cepītis, Dr.Math.Credits 2 credits Required for grade examPrerequisites Mate-2135, Mate-3143

Annotation. The course considers various mathematical models arising in applications, which are reduced to the non-linear boundary value problems for ordinary differential equations. The aim of the course is to give the sufficient knowledge of the basic methods for numerical solving and investigation of these non-linear boundary value problems.

Subjects:1. Classification of boundary value problems.2. Linear two-point boundary value problem.3. Solvability theorem for the quasi-linear boundary value problem. 4. Method of a priori estimates for the solvability proving of non-linear boundary value

problems.5. Boundary value problems for the Emden-Fowler type equations. Phase plane analysis.6. Boundary value problems for the equations and systems of higher order.7. Numerical methods for solving of non-linear boundary value problems: shooting

method, iteration methods, and transformation method.8. Methods for the estimating of number of solutions and solutions stability.

Requirements for receiving of credits: 28 hours lectures, 4 hours workshops.Students must fulfilled one home works and once stand out in the workshop. The exam takes place in the oral form and students should be familiarised with the problems considered in the program.Recommended literature:1. J.Cepītis. Parasto diferenciālvienādojumu nelineāras robežproblēmas. Rīga, LU,

1987.2. Ц.На, Методы численного решения прикладных краевых задач,Москва,1982.



NUMERICAL METHODS IV (Skaitliskās metodes IV)

Course code Mate-3140Authors Prof. H. Kalis Dr. Hab. Math., Dr. Hab. phys.Credits 3 creditsRequired for grade examPrerequisites Mate-3139, Mate-3143Annotation. The objectives of the course are: sources of non-classical problems in numerical analysis; methods and algorithm for numerical solutions of differential equations, methods for investigation and derivation of finite-difference schemes, variation type difference scheme, the alternating direction method.Subjects:1. Modification of Euler method for higher precision.2. Adam’s methods, the method of undefined coefficients.3. Multilevel finite-difference schemes for solution of Cauchy initial-value problem,

investigation of stability by using Schur’s criterion.4. Integration-interpolation method (method of finite volume), exact finite-difference

schemes.5. Monotone difference scheme, maximum principle in boundary value problem.6. A.Ilh'in exact finite-difference scheme.7. N.Bahvalov`s exact finite-difference scheme.8. The exact scheme for boundary-value problem with discontinuous coefficients.9. Exact scheme for solution of the system of differential equations.10. Ritz method for boundary-value problem, schemes of variation types.11. Galerkin method for solving of boundary-value problem, difference schemes of

variation types.12. Finite-elements method.13. Boundary-elements method.14. The problem on eigenvalues and eigenfunctions for solving of boundary-value

problem (continuous and discrete analogous).15. Analytical solutions of boundary-value problem with constant coefficients.16. Stability for difference schemes of parabolic type equations.

Requirements for received of credits: 32 hours lectures, 16 hours practical work. Students are required to fulfil 2 independent home works and 2 laboratory works. The exam takes place in an oral form.

Recommended literature:1. H.Kalis. Diferenciālvienādojumu tuvinātās risināšanas metodes. Rīga, - 1986.2. H.Kalis. Nepārtraukto un diskrēto matemātiskās fizikas problēmu analītiskie

atrisinājumi. - Rīga, LU, 1992. 3. А.А.Самарский, Е.С.Николаев. Методы решения разностных схем. - М., 1978.4. Н.С.Бахвалов. Численные методы. - М., 1973.5. W.F.Ames. Numerical Methods for partial differential equations. Acad. Press., 1977.



Number Theory(Skaitļu teorija.)

Course code Mate - 2019Author Professor A.Andzans, Dr.Hab.Math.,Credits 3 creditsRequired for grade examPrerequisites Mate – 1002, Mate – 1063,

Annotation.The objectives of course are: main results of elementary number theory, applications to cryptography, evaluation of complexity of some algorithms in number theory, elements of combinatorial number theory.

Subjects:1. Divisibility: elementary properties, divisors, multipler.2. Dividing modn: existence and unity, Euclidean algorithm, residue classes modn.3. The Basic Theorem of Arithmetics, condusions. The method of infinite sescent.4. Prime distribution: Euclid, Euler, Fermat proofs of the infinity of primes, (n)=(n),

Chebishev theorem.5. Some classical number theory results: Fermat theorem, Euler’s function, Euler’s

theorem, Vilson theorem.6. Congruences and their uses: concept, elementary properties, applications to theorem

proving and problem solving.7. Elements of cryptography: the concept of open cryptosystem, RSA – implementation,

tests of primality.8.* Structural theorem on the set of natural numbers: Minkovsky, Van der Waerden.

Schinarelman, Gell-Mann theorems, Varing problem and its partial solutions.

Requirement for receiving of the credits: 48 hours lectures. Students are required to study one topic from 8.* above independently. During a semester a control paper on problem solving must be written. The exam takes place in an oral (theory) and written (problem solving) form. Recommended literature:1. E.Lejnieks. Skaitļu teorija. Rīga, MZDB, 1936.- 294 lpp.2. N.Eņģele. Skaitļu teorija. Rīga, LVU, 1980.- 63 lpp.3. Š.Mihelovičs. Skaitļu teorija. Daugavpils, “Saule”, 1996. - 239 lpp.4. A.Bērziņa, A.Bērziņš. Skaitļu teorija. Elektroniskais almanahs “Matemātika”, Nr. 1 -

4, 6 - 7. R., LU NMS, 1997.5. В.Серпинский. 250 задач по элементарной теории чисел. Москва, Просвещение,

1968.- 160 c.6. А.Я.Хинчин. Три жемчужины теории чисел. Москва, Наука, 1979.- 62?.И.М.Виноградов. Основы теории чисел. Москва, Наука, 1981.-176 ?.8.* M.R.Schroeder. Number Theory in Science and Communication. Springer, 1985. - 374 pp.



NUMERICAL METHODS OF OPTIMISATION(Optimizācijas skaitliskās metodes)

Courses code Mate-3079Author: Assoc. Prof., Dr. Math. Jānis Vucāns Credits 4 Required for grade testPrerequisites: Mate-2065, Mate-1002

AnnotationAim of the Course: to give the students practical skills in solving economic problems of numerical optimisation Course Outline: This is the practical course in numerical optimisation. The content of the course has little changes from year to year according to the interests of students. Every week students have two academic hours of lectures and two hours of laboratory works in which they must solve (generally by computer) individual exercises. The most of them are taken from the practical life. The aim of the lectures is to familiarise students with the methods of solution of such optimisation problems.

SubjectsAt the 1999./2000 academic year the problems to solve at the laboratory hours will correspond to the following themes: 1. Textual exercises that are reducible to the extremal problem for the functions of a single or several variables. 2. Numerical finding of the global extreme for oscillating functions. 3. Numerical minimisation of the unimodal functions. 4. Numerical finding of the minimal value of the single variable function in the given finite interval by the methods of broken lines, tangents and parabolas. 5. Construction of the linear programming problems and their solution by the program package TORA or by some equivalent package.6. Construction of the transportation type problems and their solution by the program package TORA. 7. Matrix games with mixed strategies and their equivalence with the problems of linear programming. Solution of such problems by the program package TORA. 8. Finding of the extremal values of the two variable functions by the geometrical or some other method. 9. Numerical finding of the extremal values of the two variable functions by the gradient method. 10. Applications of dynamic programming principle.11. Optimisation problems in Portfolio Analysis.

Requirements for receiving of credits: 64 hours, of which 32 of lectures, and 32 hours of laboratory works1. During the semester students must solve by computer the optimisation problems of different types.



2. To pass the examination every student must solve and present to teacher the package of problems distributed for individual solution. These problems form the content of the approximately 10 laboratory works.

Recommended literature:1. Р.Габасов, Ф.М.Кириллова. Методы оптимизации. БГУ, Минск 1975.2. Ф.П.Васильев. Численные методы решения экстремальных задач. Наука, М.,

1980.3. Н.Н.Мойсеев, Ю.П.Иванилов, Е.М.Столярова. Методы оптимизации. Наука, М.,

1978.4. D.Kļaviņš. Lineārās programmēšanas uzdevumi piemēros, "Zvaigzne", 1998.5. W.F.Sharp, G.J.Alexander, J.V.Bailey, Investments, Englewood Cliffs: Prentice Hall

International Inc., 1995.



OPEN–KEY CRYPTOGRAPHY(Atklātā kriptogrāfija)

Course code Mate – 3191Author Docent Jānis Buls, Dr. Math.

Docent Jānis Lapiņš, Dr. Math.Credits 2 creditsRequired for grade examPrerequisites Mate - 2036

AnnotationSome examples of classical and open-key cryptosystems, their analysis from the point of view of a legal user and cryptanalyst. The decryption methods based on the statistical properties of natural languages. Periodical cryptosystems, Kasiski's method of determining the period of a cryptosystem. Autoclave cryptosystems. Knapsack cryptosystems. Shamir’s cryptanalytic approach to analysis of knapsack cryptosystems. Dense knapsack cryptosystems over finite Galois field. RSA (Rivest’s, Shamir’s and Adleman’s) cryptosystem. Probabilistic algorithm for testing the primality of a number. Pseydoprime numbers. Cryptanalysis of RSA cryptosystems.

Subjects1. Introduction:

1.1. Cryptosystems and cryptoanalysis;1.2. Main requirements that must satisfy a good cryptosystem.

2. Some classical cryptosystems and approaches to cryptoanalytic attacks.2.1. Caesar cryptosystem and Key-word Caesar cryptosystem;2.2. Cryptoanalytic analysis of the Key-word Caesar cryptosystem.2.3. Substition and permutation cryptosystems. Polybios cryptosystem;2.4. Affine cryptosystem;2.5. Cryptoanalytic analysis of monoalphabetic cryptosystems based on letters’ frequencies of a natural language.2.6. Vigenere and Beaufort cryptosystems;2.7. Cryptoanalytic analysis of periodic cryptosystems. Kasiski’s methode of determining the period of a cryptosystem;2.8. Autoclave cryptosystems. The cryptoanalysis of autoclave cryptosystems;2.9. Mathematical model of the cryptographic machine C-36.

3. Characteristic of open-key cryptosystems:3.1. The idea of open-key cryptosystems. Basic construction principles of an open-key cryptosystem;3.2. The complexity description of cryptoanalysis of open-key cryptosystems;3.3. Advantages and disadvantages of open-key cryptosystems;3.4. Protocols, their application for strengthening communication security in public communication networks.

4. Knapsack cryptosystems:



4.1. Knapsack cryptosystems based on transformation of a super-increasing vector;4.2. Shamir’s cryptanalytic approach;4.3. Examples of the usage of Shamir’s algorithm;4.4. Super-reachable and hyper- reachable vectors, their properties;4.5. Constructing of knapsack cryptosystems based on application of hyper- reachable vectors;4.6. Constructing of dense knapsack cryptosystems based on application of properties of finite Galois field.

5. RSA cryptosystems:5.1. RSA cryptosystem, its properties;5.2. Practical problems arising in construction and usage of RSA with large prime numbers;5.3. Cryptanalyses of RSA cryptosystem;5.4. Pseido-prime numbers. Effects arising from errors made in the construction process of RSA cryptosystem.

Requirements for received of creditsBasic knowledge on construction and usage of classical and open-key cryptosystems, their properties and methods of cryptoanalytic analysis. Skill of encryption, decryption as well as cryptanalyses based on the statistical properties of natural languages.

Recommended literature1. Salomaa, Arto. Public-key cryptography. Springer Verlag, 1990.2.Koblitz, Neal. A course in number theory and cryptography, Springer Verlag, 1987.



OPERATIONAL RESEARCH(Operāciju pētīšana)

Course code Mate – 4077Author Docent Jānis Lapiņš, Dr. Math.Credits 2 creditsReceiving for grade testPrerequisites Mate-2065, DatZ–1025, Mate–2032, Mate-3030

AnnotationThe principles of modelling and the choice of decision procedures. Multiple objective decision problem and multicriteria's decision process. Linear and integer programming. Matrix game theory. Maximum flow problem, the algorithm of Ford-Fulkerson. Calendar graphics and the optimal allocation of resources. Stochastic model of analysis of calendar graphics. Queuing theory, single and multiple exponential systems. Decision systems for inventory management. Simulation experiments and simulation modelling.

Subjects1. Some historical remarks.2. Methodological principles and main stages of construction of a model.3. Linear programming, its geometric interpretation.4. Case studies. Linear programming problem as a model in economics.5. Simplex method for linear programming.6. Dual problem of linear programming.7. Dependence of the solution of linear programming on the parameters of problem.8. Economical analysis of the solution of a linear programming problem.9. Stability of a problem of linear programming. Methods of solving unstable linear programming problems.10. The finiteness of simplex method.11. Integer programming.12. Different algorithms for integer programming, their advantages and disadvantages.13. Games, classification of games.14. Matrix games. Payoff matrix. Necessary and sufficient conditions of existence of optimal pure strategies.15. Mixed strategies. Payoff function, its properties.16. The minimax theorem.17. Algorithm of a practical determining of optimal mixed strategies.18. n-person game. Equilibrium point.19. Co-operative games. 20. Basic definitions of the graph theory.21. Transportation network, maximum flow problem.22. Ford and Fulkerson theorem.23. Ford and Fulkerson algorithm.24. Calendar graphic as an implementation model of a large project.



25. Determination of time parameters of a project based on analysis of calendar graphic (deterministic model).26. Compressing of calendar graphics.27. Analysis of calendar graphics (stochastic model).28. Optimal allocation of resources over calendar graphics.29. Queuing problem.30. Birth process. The role of Poisson process in analysis of queuing problems.31. Birth and death process.32. M/M/n model (stationer regime). Optimisation problems.33. Inventory problems, their classification.34. Single product deterministic inventory model without and with a selling discount.35. Multi-product deterministic inventory model.36. Single product stochastic inventory model.37. Main stages of analysis of a complex system based on simulation experiments. The role of random numbers within the implementation of simulation experiments. Generation of random numbers with a given distribution.38. Design and data analysis of simulation experiments.

Requirements for received of creditsKnowledge on construction of models, their properties and methods of analysis. Skill in applying basic algorithms and correct interpretation of results.

Recommended literature1. Г.Вагнер. Основы исследования операций.т.1. Москва. Мир. 1972.2. Г.Вагнер. Основы исследования операций.т.2. Москва. Мир. 1973.3. D.Kļaviņš, J.Zelčs. Operāciju pētīšanas matemātiskās metodes, Rīga, 1979.4. Х.Таха. Введение в исследование операций.т.1. Москва. Мир. 1985.5. Х.Таха. Введение в исследование операций.т.2. Москва. Мир. 19856. I.Akuļičs, M.Purgailis. Masu apkalpošanas teorijas elementi, Rīga, Zvaigzne, 19807. C.Ф.Фишманов. Линейное программирование. Москва. Наука. 19818. Исследование операций.т.1. Методологические основы и математические методы. Мир. 19819. Исследование операций.т.2. Модели и применения. Мир. 1981



OPTIMAL CONTROL OF PROCESSES(Procesu optimālā vadīšana)

Course code Mate -4103Author Assoc. Prof. Janis Vucans, Dr Math.Credits 2 credits Required for grade testPrerequisites: Mate - 1061, Mate - 3079, Mate - 3082 or Mate - 4081

Annotation: Various optimal control problems and methods of their solution are considered in this course. The following questions are stressed: Pontryagin’s maximum principle as necessary condition of extremes; the problem of optimal control synthesis; the Bellman principle of dynamic programming, schemes of Bellman and Moiseev; methods of solution for non-correct extremal problems; examples of the optimal control problems. Course includes a part of practical works in which students must solve individually concrete optimal control problems (on the students choose - with or without use of the computers).

Subjects:1. Formulation of the Optimal Control Problem; examples. [1, §6.1 or 4]2. Maximum principle in problems with fixed trajectory endpoints and with fixed time.

Maximum principle in problems with free trajectory endpoints and with fixed beginning time moment of the movement. Maximum principle in problems with free trajectory endpoints and with free beginning and final time moments of the movement. Examples. [1, §6.2 or 4]

3. Proof of the maximum principle in the problem with free right trajectory endpoint. [1, §6.3 or 4]

4. Boundary problems of the maximum principle and methods of their solution. [1, §6.4]5. Relation between the maximum principle and the Classical Calculus of Variations. [1,

§6.5]6. Scheme of Bellman. Method of Dynamic Programming, its auxiliary problems. The

problem of synthesis for discrete systems. [1, §7.1]7. Scheme of Moiseev, methods of wandering channels and local variations. [1, §7.2] 8. Non-correct extremal problems and methods of their solution. Stabilisation. Normal

solution. Regularization and its basic lemmas. Tikhonov’s method, method of quasi-solutions and other methods. [2,§2.1-2.7]

9. Students’ reports on the individually solved problems.

Requirements for received of credits: 30 hours lectures, 10 hours seminars.1. During the semester students must solve a number of problems, assigned by the teacher

for independent solving; they must present these solutions to the teacher by the indicated deadline. During the seminar they must explain the obtained results

2. At the test students must demonstrate the knowledge of the theoretical material from themes 1 to 8, they must know to illustrate the theory by the examples of optimal control problems.



Recommended literature: 1. Jānis Vucāns, Optimizācijas lietišķās problēmas ekonomikā un vadības zinātnē. Lecture Notes, University of Latvia and Ventspils College, 19992. Ф.П.Васильев. Численные методы решения экстремальных задач. Москва, Наука, 1980.Ф.П.Васильев.Методы решения экстремальных задач. Москва, Наука, Jean-Pierre Aubin, Optima and Equilibra: An Introduction to Nonlinear Analysis, Springer-Verlag, Berlin - Heidelberg - New-York - London - Paris - Tokyo - Hong Kong - Barcelona etc., 1993.



PORTFOLIOS OF SECURITIES AND THEIR MANAGEMENT(Vērtspapīru portfeļi un to vadīšana)

Course Code Mate – 3210Author Assoc. Prof. Jānis Vucāns, Dr. Math. Credits 4 creditsRequired for grade examPrerequisites: Mate – 3030, Mate-1063

Annotation: Course makes the students familiar with the main mathematically-statistical principles of the theory of construction and management for the portfolios of securities.

Subjects:1. The role of the financial market. Return, risk, attitude towards risk (riskophobes and riskophyles individuals). Measurement of return and risk. Effects of diversification. Effects of correlation. Choice of the portfolio in the “three securities space”. Model of Markowitz. Tobin’s Model. Market Model. Diagonal or Sharpe model. Capital Asset Pricing Model - CAPM. Capital Market Line - CML. Security Market Line - SML. 2. Performance evaluation. Performance indexes. Performance decomposition. Calculation of return via weighted rates of return (time weighted and $ weighted). Options theory. Terminology, limits of evaluation, parity relation. Put and Call options of European and American type. Closed portfolio (with warranty) and binomial model. Portfolio in continuous media and Black-Schole model. Evaluation methods for options based on common stocks with or without dividends. Options of American type and their premature realisation. 3. Standardised fixed duration contracts. General definitions. Evaluation via arbitrage, keeping expenditures, intermediate return. Futures. Correction of the value of securities depending on their quality. Creation of synthetic assets. Economic evaluation via CAPM.

Requirements for received of credits: 48 hours lectures, 16 hours seminars. 1. During the semester students must solve a number of problems, assigned by the teacher

for independent solving; they must present these solutions to the teacher by the indicated deadline. During the seminar they must explain the obtained results.

2. At the written examination student must be able to demonstrate the knowledge of the basic concepts and properties of the theory of construction and management of the portfolios of securities, explained during the lectures, and the ability to use them to formulate and to solve different investment problems.

Recommended literature:1. R.Cobbaut, Théorie Financiere, Economica, Paris, 1994.2. R.Gibson, Option valuation: Analyzing and pricing Standartized Option Contracts,

McGraw Hill, 1991.3. W.Sharpe, G.Alexander, J.Bailey, Investments, Englewood Cliffs: Prentice Hall, 1995.



4. E.J.Elton, M.J.Gruber, Modern Portfolio Theory and Investment Analysis, J.Wiley & Sons, Inc., New-York, 1995.



PRACTICAL LOGIC I(Praktiskā loģika I)

Course code Mate – 1048Author Docent Jānis Cīrulis, Dr. Math.Credits 2 creditsRequired for grade testPrerequisites

AnnotationFor the greatest part, it is a practical (i.e. not theoretical) course. Its purpose is to aid the student to gain skills in analysis of the logical structure of propositions and to avoid some frequent logical mistakes in their formulations. Therefore, the bulk of the time will be spent for practical training.

Subjects1. Conditionals

1.1. A conditional, the converse and the contraposition of it. The logical square;1.2. Several interpretations of the conjunction "if_then";1.3. Non-standard forms of conditionals;1.4. Individual, general, and existential propositions;1.5. A general proposition as a conditional: its explanatory part, condition and conclusion.

2. Attributes of objects2.1. Universal, contingent, and inconsistent attributes;2.2. Most fundamental relationships between attributes of objects of the same class;2.3. Independent of attributes.

3. Logical structure of compound propositions3.1. Names, sentences, functors. Syntactic categories;3.2. Free and bounded variables. Quantifiers;

4. Picturing of the logical structure of a proposition by means of symbolic logic.5. Negative propositions.

Requirements for received of creditsEach of the three parts of the course ends with a test (in writing), where the student has to solve a number of problems; every problem is evaluated by a certain number of points. To get the credit, it is necessary and sufficient to gain at last a half of the possible total sum of points in all tests.



PRACTICAL LOGIC II(Praktiskā loģika I)

Course code Mate – 1204Author Docent Jānis Cīrulis, Dr. Math.Credits 2 credits Required for grade testPrerequisites Mate – 1048

AnnotationFor the greatest part, it is a practical (i.e. not theoretical) course. Its purpose is to aid the student to gain skills in analysis of the logical structure of propositions and to avoid some frequent logical mistakes in their formulations. Therefore, the bulk of the time will be spent for practical training.

Subjects1. Definitions

1.1. Types of mathematical definitions;1.2. Rules for correct definitions of functions and predicates.

2. Equivalent propositions2.1. Language of prepositional logic and language of predicate logic: syntax and semantics;2.2. Logical equivalence of formulas in prepositional and in predicate logic;2.3. Equivalent transformations of formulas and of propositions;2.4. Transforming negative propositions into positive ones.

3. Compatible propositions. Analysis of arguments3.1. What is compatibility;3.2. Logical structure of an argument. Direct and indirect arguments;3.3. Correct and incorrect arguments. Logical consequence relation;3.4. Proving compatibility of propositions and correctness of arguments---various methods (logical equations, semantic tableaux etc.).

Requirements for received of creditsEach of the three parts of the course ends with a test (in writing), where the student has to solve a number of problems; every problem is evaluated by a certain number of points. To get the credit, it is necessary and sufficient to gain at last a half of the possible total sum of points in all tests.

Recommended literature 1. Suppes P. Introduction to logic2. Georgacarakos G.N., Smith R. Elementary formal logic. McGraw-Hill, N.Y. ea., 1979.



PRINCIPLES OF MATHEMATICAL MODELLING(Matemātiskās modelēšanas principi)

Course code Mate-0270Author Prof. A. Buiķis, Dr.Hab.Math.Credits 2 creditsRequired for grade examPrerequisites Mate-3143

Annotation. In the course, the stages of mathematical modelling are analysed. Mathematical models of different processes are considered as well as analytical and numerical methods for their solution. Interpretation of the results is given.

Subjects:1.The concept of mathematical modelling, its principles and general scheme.2.Comparison of the mathematical model and the real process. Motion in the gravitation field.Typical mathematical models of various processes, including those with ordinary and

partial differential equations.Different real processes described by similar mathematical models.Additional boundary conditions; methods of solution applied depending on the

complexity of a mathematical model.Verification of the mathematical model, analysis of usefulness, gauge of the model. The

temperature waves.Dimensionless parameters. Self-similar solutions.Testing of the models describing the evolution of the Earth. Verification of the

hypotheses.Mathematical models for porous media, their specificity and applicationsPeculiarities of the mathematical models for non-linear processes.Completion of the processes of mathematical modelling. Presentation of results.

Requirements for received of credits: 20 hours lectures, 12 hours workshops. One unassisted work must be represented in workshop. The exam takes place in oral form.

Recommended literature:1. D. Edwards, M. Hamson. Guide to Mathematical Modelling. MacMillan Press, 1989.2.H. Gould, J.Tobochnik. An Introduction to Computer Simulation Methods. Applications to Physical Systems. Part 1, 2, 1990 (translation into Russian).J. Grasman. Asymptotic Methods for Relaxation Oscillations and Applications. Springer,

1987.N. J. Higham. Handbook of Writing for the Mathematical Sciences. SIAM Philadelphia,

1993.H. Ockendon, J. R. Ockendon. Viscous Flow. Cambridge University Press, 1995.A.Samarskij, A. Mihailov. Mathematical Modelling. M., 1997(Russian).



REGRESSION ANALYSIS(Regresijas analīze )

Course code Mate-0275Authors Doc. J. Cepitis Dr. Math. Prof. H. Kalis Dr. Hab. Math., Dr. Hab. Phys.Credits 2 creditsRequired for grade examPrerequisites Mate-3030, Mate-3139

Annotation. The objectives of the course are: different models are consider, investigation of parameters and comparison of models are carried out. This course is coordinated with ECMI program.

Subjects:1. Least squares investigation.2. Hypothesis to verificated with the model of full series.3. Regression analysis.4. Orthogonal polynomials.5. The models with not full series.6. Plan for simple experiment.7. Analysis of non-responsibility.8. General linear hypothesis.9. Remarkable functions.

Requirements for received of credits: 32 hours lectures and 1 test work.Students are required to fulfil 3 independent laboratory works.

Recommended literature:1. A.Sen, M.Srivastava. Regression Analysis, Springer, 1996.2. G.Engeln-Mullges, F.Uhlig. Numerical algorithms with Fortran. Springer, 1966.



SEMINAR FOR DATA HANDLING OF CONTINUOUS PROCESS(Seminārs nepārtraukto procesu datu apstrādē )

Course code Mate-0277Authors Prof. H. Kalis Dr. Hab. Math., Dr. Hab. Phys. Doc. J. Cepitis Dr. Math. Lect. R. Millere Mag. Math. Lect. M. Buike Mag. Math.Credits 2 creditsRequired for grade testPrerequisites Mate-3139, Mate-2135

Annotation. The objectives of the course are: approximations methods for linear and non-linear data handling with polynomials, splines and rational functions, applications with modern PC and programming packages (Mathematica, Maple). This course is in compliance with ECMI program.

Subjects:1. Functions approximations in Hilbert`s space.2. Functions approximation in Banah`s space.3. Algebraic and trigonometric interpolation of functions.4. Least squares method.5. Spline interpolation.6. Approximation methods for functions with two arguments.7. Data handling by using discrete Furier transform.8. Approximation with rational functions (Pade-approximation).9. Programming packages in approximation.10. Data handling by using programming language "Pascal" or "FORTRAN".

Requirements for received of credits: 32 hours lectures and 2 test works.Students are required to fulfil 3 independent laboratory works and 1 seminar works.

Recommended literature:1. I.Pagodkina, R.Millere. Algebras un matemātiskās analīzes uzdevumu skaitliskā

risināšana. Rīga, LU, 1996.2. S.D.Conte, C.Boor. Elementary numerical analysis an algorithmic approach. Mc

Graw-Hill, Book Company, 1972.3. G.Engeln-Mullges, F.Uhlig. Numerical algorithms with FORTRAN. Springer, 1996.4. A.Sen, M.Srivastava. Regression analysis, Springer, 1996.



SEMINAR OF PROGRAM PACKAGES(Seminārs programmu paketēs )

Course code Mate-3278Authors Prof. H. Kalis Dr. Hab. Math., Dr. Hab. phys. Doc. J. Cepītis Dr. Math. Lect. R.Millere Mag. Math. Lect. M. Buike Mag. Math. Credits 2 creditsRequired for grade testPrerequisites Mate-2065, Mate-2135, Mate-2138

Annotation. The objectives of the course are: concrete physical and technical problems, mathematical models, algorithm, programs and program packages, numerical analysis with Mathematica, Maple, Matlab, Matcad and others computer program packages. This course is co-ordinated with ECMI program.

Subjects:1. The computer program package "Mathematica".2. The computer program package "Maple"3. The computer program package "Matcad".4. The computer program package "Matlab".5. The main operators of mathematical analyses. 6. The main operators of algebra.7. The main operators of differential equations.8. The main operators of numerical methods.9. The main operators of computer graphics.10. Programming with packages.11. Programming language "Pascal" or "Fortran" for solutions of concrete problems of

mathematical physics.

Requirements for receiving of credits: 16 hours lectures, 16 hours practical work.Students are required to fulfil 3 independent laboratory works and 1 seminar works.

Recommended literature:1. T.B.Bahder. Mathematica for scientists and Engineers. Addison-Wesley publ. comp., 1995.2. R.E.Maeder. Programming in Mathematica. Addison-Wesley publ. comp., 1991.3. D.Redfern. The Maple Handbook. Maple Y Release 4, Springer, 1996.4. H.Kalis, S.Lācis, O.Lietuvietis, I.Pagodkina. Programmu paketes "Mathematica"

lietošana mācību procesā. Māc. grāmata, 1997.5. В.Ф. Очков. Matcad Plus 6.0 для студентов и инженеров. Москва, 1996.



SOLUTION OF BOUNDARY PROBLEMS IN LAYERED MEDIA(Robežproblēmu risināšana slāņainās vidēs)

Course code Mate-4276Author Prof. A. Buiķis, Dr.Hab.Math.Credits 2 credits (32 lecture hours.) Required for grade examPrerequisites Mate-3143, Mate-3140

Annotation. The course considers basics principles of mathematical models for porous media.

Subjects:1. Mathematical modelling and mechanics of continuous media. Basic specific and

special laws for different discontinuous media.2. Typical mathematical models with partial differential equations for specific

continuous media.3. Media with piecewise – constant properties. Conditions on the discontinuity surfaces.4. The method of conservative averaging for separates layers. Non-classical boundary

conditions.5. A system with a plane boundary layer. Estimation of the errors of the method.6. Models for two-layer systems and two-layer systems with an intermediate layer.7. Classical interpolating and integral splines. Existence and uniqueness of the integral

spline.8. Representation of the splines by a matrix with independent coefficients.9. The conservative diffusion problem and its peculiarities. The problem as applied to

porous media. 10. Convective heat transfer in a layer.11. Application of classical and integral splines to solving the boundary problems.12. Non-polynomial integral splines and problems with boundary layers Requirements for receiving of credits: 16 hours lectures, 16 hours workshops.Students must fulfil 2 test works. The exam takes place in oral form.Recommended literature:1. J. S. Zavjalov, B. I. Kvasov, V. L. Miroshnitshenko. Methods of Spline-Functions.

Moscow, Nauka, 1980 (Russian ).2. A. Buikis. Problems of Mathematical Physics with Discontinuous Coefficients and

their Applications. Riga, 1991. Manuscript (Russian ).3. H. Spaeth. Eindimensionale Spline-Interpolationsalgorithmen. R. Oldenburg Verlag.

Muenchen Wien,1990 .4. A. Tihonov, A. Samarskij. Equations of Mathematical Physics. Moscow, Nauka, 1987

and further editions (Russian).5. J. J. Fried. Groundwater pollution. Theory, Methods, Modelling and Practical

Methods. Moscow, Nedra, 1981 (Russian).6. K. W. Morton. Numerical Solution of Convection - Diffusion Problems. Chap.

Krank&Hall, London, 1996.



SPECIAL NUMERICAL METHODS (Speciālās skaitliskās metodes)

Course code Mate – 4141Author professor Harijs Kalis, Dr.Hab.phys., Dr.Hab.math Credits 2 credits Required for grade testPrerequisites Mate – 2065,Mate – 1002,Mate – 2134

AnnotationSpecial numerical methods are developed for solving the problems of the second order differential equations with large parameters at the derivative of first order. The advantage of this method is shown with respect to the classical ones.

Subjects1. Cauchy initial-value problem for linear ordinary differential equation of first order [1].2. Cauchy problem for the linear ordinary differential equations and systems of second order [1],[3]. 3. Boundary-value problem for ordinary differential equations of second order (uniform grid) [1],[2].4. Boundary-value problem for ordinary differential equations of second order (nonuniform grid) [1],[2].5. Boundary-value problem for system of linear ordinary differential equations [1],[3].6. Boundary-value problem for equation of elliptic type [1],[4].7. Continuous and discrete eigenvalue problem in one dimensional case [1],[4].8. Continuous and discrete eigenvalue problem in two dimensional case [1],[4].9. Solving of two dimensional boundary-value problems [1],[4].10. Solving of initial-boundary-value problems [1],[4].

Requirements for received of creditsTwo test works have to be assigned along the course. Students have to confirm their ability to solve the variety of concrete problems in mathematical physics as well as to show a deep knowledge concerning the questions described above.

Recommended literature1. H.Kalis. Speciālas diferenču shēmas matemātiskās fizikas problēmu risināšanā. LU, 1991.2. H.Kalis. Diferenciālvienādojumu tuvinātās risināšanas metodes. Rīga, Zvaigzne, 1986.3. Ф.Гантмахер. Теория матриц. Москва, Наука, 1967.4. H.Kalis. Nepārtraukto un diskrēto matemātiskās fizikas problēmu analītiskie atrisinājumi. LU, 1992.



SUPPLEMENTARY CHAPTERS OF MATHEMATICAL STATISTICS(Matemātiskās statistikas papildnodaļas)

Course code Mate – 3040Author Docent Viktorija Carkova, Dr. Math.Credits 4 creditsRequired for grade examPrerequisites Mate – 2065. Mate – 2032, Mate – 3030

ANNOTATIONTHIS COURSE MAKES STUDENTS FAMILIAR WITH THE VERIFICATION METHODS OF STATISTICS HYPOTHESES. THE OPTIMAL CRITERIA ARE CONSTRUCTED BY USING THE NEYMAN-PEARSON FUNDAMENTAL LEMMA AND THE COCHREN THEOREM.

SubjectsFundamental of Mathematical Statistics (4hours lecture + 2 hours seminar).The hypothesis.The lost function.The risk function.Test.The power function.Neyman-Pearson fundamental lemma (2+2).Most powerful test (2+2). Baysian and Minimax procedures (2+4).The optimal tests constructed by Neyman-Pearson lemma (2+2).Graphical representation for the optimal tests (2+2).One-parametrical class with monotone likelihood ratio (2+2).One-parametrical exponential class (2+2).One-sided hypotheses. Confidence sets (2+2).Test of hypotheses for parameters of one-dimensional normal sample (2+2).Test of hypotheses for parameters of multidimensional normal sample (4+2).Invariance principle (2+2).Linear hypotheses (4+2).Test of statistical hypotheses and confidence interval construction examples (2+2).Requirements for received of credits

Recommended literatureE.L. Lehmann,Testing Statistical Hypotheses, NY. John Wesley, 1959.J.Carkovs, Alternative statistical method, RTU, Riga, 1984, (Latvian).J. Carkovs, Simples hypotheses on parameters of normal distribution, RTU, Riga, 1986,

(Latvian).D.R.Cox and D.V.Hinkley, Theoretical Statistics, Chapman&Hall, London, 1994.



5.V.Carkova, Mathematical Statistics, LU, Riga, 1979, (Latvian).



SURVEY SAMPLING(Izlases apsekojumi)

Course code Mate – 3188Author Docent Jānis Lapiņš, Dr. Math.Credits 4 creditsRequired for grade testPrerequisites Mate - 2036

AnnotationMethods of gathering information from a sample of individuals are considered. Organisational and methodological principles of sampling surveys are discussed. Basic sampling and corresponding estimation procedures and their properties are analysed. Main sources of error in sample surveys and methods of reducing bias of estimates are discussed. Theoretical results are illustrated using experience from sample surveys of both – businesses and households and/or persons – carried out during last years at the Central Statistical Bureau of Latvia. Household Survey data for training purposes students can find also in Internet (see [6]).

Subjects1. Introduction.

1.1. What is a sample survey? Advantages of the sampling method. 1.2. The principal steps in a sample survey.1.3. Classification of sampling procedures.

2. Simple random sampling.2.1. Simple random sampling, selection procedures of a simple random sample.2.2. Estimation of population mean and population total, properties of estimates.2.3. Variances of estimates of population mean and population total and their estimation.2.4. Construction of confidence intervals for population mean and population total.2.5. Estimation of a ratio.2.6. Estimation of domain mean and domain total.2.7. Comparisons between domain means.

3. Stratified random sampling.3.1. Stratified random sampling, properties of estimates.3.2. Variance estimation and construction of confidence intervals.3.3. Proportional allocation (self-weighting sample), its properties.3.4. Optimum allocation.3.5. Relative precision of stratified random sampling and simple random sampling. 3.6. Effects of deviations from the optimum allocation and errors in the stratum sizes.



3.7. Post-stratification.4. Ratio estimates.

4.1. The ratio estimate.4.2. Approximate variance of the ratio estimate.4.3. Estimation of the variance from a sample.4.4. The bias of ratio estimates.4.5. Construction of confidence intervals for a ratio.4.6. Ratio estimates in stratified random sampling.

5. Regression estimators.5.1. Linear regression estimates with pre-assigned coefficient of linearity.5.2. Linear regression estimates when coefficient of linearity is estimated from a sample.

6. Cluster sampling and multistage sampling.6.1. Single-stage cluster sampling with clusters of equal sizes.6.2. Single-stage cluster sampling with probabilities proportional to cluster sizes.6.3. Designing of multistage sampling.6.4. Estimation in multistage sampling.6.5. Estimation of the variance of estimates in multistage sampling.6.6. Optimisation in multistage sampling.

7. Two face (double) sampling and use of auxiliary information.7.1. Technique of two face sampling. Optimum allocation of a sample.7.2. Use of auxiliary information for stratification.7.3. Use of auxiliary information for improvement of estimates.7.4. Use of auxiliary information for reducing of survey costs.

8. Sources of error in surveys.8.1. Sources of error in sample surveys.8.2. Frame imperfections and their impact.8.3. Non-response, methods of adjustment for non-response.8.4. Measurement errors.

Requirements for received of creditsBasic knowledge of sampling and estimation procedures, their statistical properties. Understanding of possibilities and limitations of modern statistical methodology for complex sample surveys. The first practical skills in sampling design and estimation (including variance estimation).

Recommended literature1. Cochran, William G. Sampling techniques. Wiley and Sons, 1977, 428 p.2. Särndal, Carl-Erik, and Swensson, Bengt, and Wretman, Jan. Model assisted survey sampling. Springer Verlag, 1992, 694 p.3. Lessler, Judith T., and Kalsbeek, William D. Nonsampling error in surveys. J.Wiley and Sons, 1991, 401 p.4. Krastiņš O., Krūmiņa I.. Izlases metode. LU Rīga, 1993, 70 lpp.5. http://www.stat.ncsu.edu/info/jse/v5n2/schwarz.supp/index.html – Stat Village



TOPOLOGY I (Topoloģija I)

Course code Mate-2086Authors Prof. A. Sostaks, Dr. habil. Math.Credits 2 creditsRequired for grade exam or testPrerequisities: Mate-1061, 1062, 2063, 2064

Annotation. The course is mainly devoted to set-theoretical aspects of topology. Basic ideas and concepts of topology as well as some fundamental topological results are considered. Special attention is devoted to the possibility of the use of topological ideas, methods and results in mathematical analysis and functional analysis as well as to the discussion of some relations between topology, on one side, and mathematical analysis, functional analysis and geometry, on the other.

Subjects:1. Introduction (Basic ideas of topology, A concise history of the origins and

development of topology, Some examples illustrating applications of topological ideas and elementary results.)

2. Topological spaces – basic definitions and examples.3. Closed sets in a topological space.4. Structure of a set in a topological space.5. Closure and interior of a set. Border of a set.6. Neighbourhood of a point. Accumulation points of a set.7. Dense subsets of a topological space.8. Base of a topology. Subbase of a topology.

8.1. Definition and examples of bases and subbases.8.2. Construction of a topology from a base. Construction of a topology from a

subbase.8.3. Weight of a topology. Local weight of a topology in a point.8.4. Examples of constructed topologies from bases and subbases: Sorgenfrey space.

Niemicky space.8.5. Separable spaces. Density of a topological space.

9. Continuous mappings of topological spaces.9.1. Basic definitions and characterisations.9.2. Properties of continuous mappings.9.3. Examples of continuous mappings.9.4. Homeomorphism.9.5. Definitions and examples. Basic properties of homeomorphisms.9.6. Homeomorphic topological spaces.9.7. Homeomorphic sets.

10. Subspaces of topological spaces. Induced topology. Hereditary topological properties.11. Direct sums of topological spaces.



12. Products of topological spaces (The case of a finite number of factors).12.1. Product topology.12.2. Projections. Basic properties of projections.12.3. Subspaces in products of topological spaces.12.4. Productive topological properties.

13. Lower separation axioms.13.1. T0-spaces13.2. T1-spaces13.3. T2-spaces13.4. Lower separation axioms and continuous real-valued functions.

14. Higher separation axioms14.1. Regular and T3-spaces.14.2. Normal and T4-spaces.

15. Higher separation axioms and continuous functions.16. Urysohn Lemma

16.1. Extension of continuous functions: Tietze-Urysohn theorems.16.2. Completely regular spaces.

17. Compactness.17.1. Compact topological spaces and compact sets: definitions and examples.

Basic properties of compact spaces and compact sets. 17.2. Compactness and separation axioms.17.3. Continuous mappings on compact spaces. Continuous images of compact

sets.18. Connectedness.

18.1. Connected spaces and connected sets: definitions and examples.18.2. Locally connected spaces. 18.3. Linearly connected spaces.18.4. Continuous functions on connected sets.

Requirements for receiving of credits: 32 hours lectures, Students are required to fulfil 2 independent home works, and to write 1 control work consisting of theoretical questions and problems. The exam (test) takes place in oral form. Students must show understanding of theoretical material considered at lectures and demonstrate the ability of solving practical problems corresponding to the course.

Recommended literature:1. A. Šostaks, M. Zandere, Topoloģijas elementi I. – Rīga, LVU, 1978.2. A. Šostaks, M. Zandere, Topoloģijas elementi I.I – Rīga, LVU, 1979.3. R. Engelking, General Topology – PWN, Warszawa, 1979.4. J.L. Kelley, General Topology – Van Nostrand Co, Inc., New York 1957.5. A. Matuzevičius, Topologija (Lithuanian), Vilnius, 1982.6. П.С. Александров, Введение в теорию множеств и общую топологию, Москва,

1982.П.А. Александрян, Э.А. Мирзахонян, Общая топология, Москва, 1979.



TOPOLOGY II (Topoloģija II)

Course code Mate-3183Authors Prof. A. Sostaks, Dr. Hab. Math.Credits 2 creditsRequired for grade exam or testPrerequisities: Mate-2086

Annotation. The course is the continuation of Mate-2086 “Topology I”. Principal attention here is devoted to the construction of the topological product and related topics. And to the study of topological properties of compactness type. Besides the problem of metrizability of a topological space is investigated.

Subjects: 1. Basic concepts of set-theoretic topology (a short repetition of some notions and results

from the course Topology I.)2. Products of topological spaces: the case of an infinite number of factors.

2.1. Definitions and basic properties of products. Properties of projections.2.2. Productivity of separation axioms.2.3. Products of separable spaces.

3. The diagonal theorem and its applications.3.1. The diagonal theorem (The theorem about embedding of a topological space

into certain products.) 3.2. Important products: Tychonoff cubes. Cantor cubes. Hilbert space.3.3. Special properties of diagonal mappings.

4. Compact spaces. Basic properties of compact spaces. Countably compact spaces. Product of compact spaces: Tychonoff theorem and its corollaries.Perfect mappings

Stone-Weierstrass theorem and its corollaries.Compact sets in metric spaces.

Precompact metric spaces and precompact subsets in metric spaces.Complete metric spaces and complete subsets in metric spaces.Characterisation of compact subsets in metric spaces.

7. Locally compact spaces.Basic definitions, examples.Operations on locally compact spaces.Images of locally compact spaces.

8. Compactifications.8.1. One-point compactification of a locally compact space.8.2. General theory of compactifications.8.3. Stone-Čech compactification and its properties.

9. Lindelof spaces.



10. Metrization of topological spaces.Problem of metrizability of a topological space.Urysohn Theorem on metrizability of a second countable regular space.Bing-Nagata-Smirnov metrization criterion. (without proof).

Elements of homology theory (Chain complexes, Homologies of complexes, Exact sequences. Some applications of homology theory.)

Elements of homotopy theory (Homotopy, retractions and deformations, fundamental group of a space.)

Requirements for received of credits: 32 hours lectures, Students are required to fulfil 2 independent home works, and to write 1 control work consisting of theoretical questions and problems. The exam (test) takes place in oral form. Students must show understanding of theoretical material considered at lectures and demonstrate the ability of solving practical problems corresponding to the course.

Recommended literature:1. A. Šostaks, M. Zandere, Topoloģijas elementi I. – Rīga, LVU, 1978.2.A. Šostaks, M. Zandere, Topoloģijas elementi I.I – Rīga, LVU, 1979.3. R. Engelking, General Topology – PWN, Warszawa, 1979.4. J.L. Kelley, General Topology – Van Nostrand Co, Inc., New York 1957.5. A. Matuzevičius, Topologija (Lithuanian), Vilnius, 1982.6. П.С. Александров, Введение в теорию множеств и общую топологию, Москва,

1982.7. П.А. Александрян, Э.А. Мирзахонян, Общая топология, Москва, 1979.8. E. H. Spanier, Algebraic topology, McGraw-Hill Book Company, 1971.



COURSES IN NATURAL SCIENCES



NATURAL SCIENCES I (THEORETICAL MECHANICS)(Dabas zinātnes I)

Course code Fizi-3166Author Dr.Phys., Dozent.L.BuliginsCredits 4Required for grade testPrerequisites Mate-2065, Mate-2143, Fizi-2069 or Mate-3143

AnnotationThe course considers principles of mathematical modelling of the mechanical aspect of natural and technical processes, based on the models of classical Newton mechanics, including formulation of the dynamical tasks and integration in the fields of various forces.

Subjects1. The subject of Mechanics science, its main theoretical models. Applicability limits of

classical mechanics. Components of theoretical mechanics.2. Systems of co-ordinates. Orthogonal curved co-ordinate systems. Co-ordinate lines,

surfaces and orts. Derivative of vector function by a scalar argument.3. Point kinematics. Trajectory, velocity and acceleration. Projections of the velocity and

acceleration in orthogonal co-ordinate system on the axis of natural tried. Sectorial velocity.

4. Dynamics of a material point. The first and second fundamental tasks of dynamics. Integration of one-dimensional motion equations, when force is a time, co-ordinate or velocity function. Conceptions of moment, kinetic and potential energies.

5. System of free material points. Internal and external forces. Moments of the system, the law of its conservation. Total mechanical energy of the system, the law of its conservation.

6. Motion equation of mass centre.7. Motion of a material point in the field of central forces. Two-body problem. Reduced

mass. Reduction of the two-body problem to one-particle problem.8. Potential appearance of central forces. Effective potential energy. General character of

motion in the field of central forces.9. Analytical mechanics. System of coupled material points. Bonds and reaction forces of

bonds. Classification of bonds. Virtual and real displacements. Degrees of freedom of the system and independent co-ordinates. Virtual action.

10.The principle of virtual displacements in static. Generalised force. Field of potential forces. Lagrange function. The 2-nd type equation of Lagrange for the field of potential forces.

11.The principle of smallest action. Lagrange equations as a Euler equations for the variation principle.

12.Small oscillations. The balance state of the system. Expressions for kinetic and potential energies in the vicinity of stable balance state.



13.Oscillations with one degree of freedom. Free oscillations, oscillations with friction, forced oscillations, forced oscillations with friction. Resonance.

14.Oscillations of a system with s degrees of freedom. Equation of frequencies. Normal oscillations.

15.Parametric oscillations. Non-linear oscillations.16.Computer modelling of motion of particles in fields of various sources.

Requirements for received of creditsUnderstanding of the lecture material as well as of its practical applications.

Recommended literature1. L.Landau and E.Lifshitz. Theoretical Physics (in Russian), vol. 1, Mechanics.

Moscow, 19882. Fluent v.4.3 Users Guide



NATURAL SCIENCES II (THEORY OF ELECTROMAGNETIC FIELD)Dabas zinātnes II (elektromagnētiskā lauka teorija)

Code number Fizi-3167Author prof.Gunārs Sermons, Dr.hab. phys.Credits 2 creditsRequired for grade examPrerequisites Mate-2065, Mate-2134, Fizi-2069 or Mate-3143

Annotation

The course contains foundations of theory of electromagnetic field: equation of electromagnetic field; methods solutions fields equations and applications; principle of relativity; electromagnetic fields in a substance and its mathematical modelling and problems for solving in studies.

Subjects.

1. Foundation of the vector analysis: scalar, vector and tensor fields. Scalar and vector product.

2. Differentiation operators of scalar and vector fields and its operations.3. Notions of electromagnetic field: electric and magnetic field, its fluxes and sources.4. A posterior law’s of electromagnetic field: The Colomb’s law, Bio – Savart law,

Ampere law, Faradey’s law of electromagnetic induction, Maxwell’s hypothesis of a displacement current existents.

5. The energy and momentum of electromagnetic field. Conservation law’s of the energy and momentum.

6. Maxwell’s equations, its differential and integral forms.7. Potentials of the electromagnetic field, its properties and potentialequations.8. Galilean and Einstein’s relativity principles, Galilean and Lorenz transformations. A

notion of four - dimensional space.9. Four-dimensional vectors and relativistic mechanics. Relationship between energy and

mass.10.Constant electric and magnetic fields.11.The electromagnetic waves. Plane and spherical electromagnetic waves.12.Radiation of electromagnetic fields. A notion of a wave zones.13.The electromagnetic fields in a substance. The substance polarisation. 14.Substance properties in electromagnetic field.15.Electric fields in a conductors and dielectrics. Magnetic properties of a substance.16.Computer modelling for electric fields.Requirements for receiving of credits:32 hours lectures. The exam takes place in an oral form. Students must to show an understanding for lectures material and its practical knowledge.Recommended literatureE.Šilters, G.Sermons, J.Miķelsons. Elektrodināmika. “Zvaigzne”, Rīga, 1986



NATURAL SCIENCES III (MATHEMATICAL MODELS OF PHYSICAL PROCESSES)

Dabas zinātnes III (fizikālo procesu matemātiskie modeļi)

Code number Fizi-4168Author lecturer Sandris Lācis, Dr. Phys.Credits 3 creditsRequired for grade examPrerequisites Mate-2065, Mate-1002, Mate-2134, Fizi-2069 or Mate-3143

AnnotationPresent course is one of three natural science course parts. It is devoted to mathematical description of physical processes by applying continuos media approach. The course includes three general topics:heat transfer: theory and applications:theory of elasticity applied to studies of deformations and stress in different

constructions;hydrodynamics of uncompressible fluids: inviscid an viscose fluids, solution of simple

problems.

Subjects:

1. Fourier’s law in heat conduction. Thermal resistance.2. Heat transfer equation. Convection heat transfer. Relation of boundary conditions in

the convection heat transfer to Fourier law. 3. Anisotropy in heat transfer.4. Heat transfer by radiation. The Stefan-Boltzman’s law.5. Basics of the tensor analysis in continuum mechanics.6. Volume and surface forces in models of continuous media. Stress tensor. Force, acting

to the plane with slope.7. Normal and shear deformations. The Hooke’s law.8. Strain tensor. Stress-stain relationships.9. Stress equations of equilibrium. Strain compatibility relations in tree dimensions.10.Two-dimensional state of stress, the plane-strain approach, plane stress. Airy’s stress

function, solution of biharmonic equation.11.Idealised equation of motion for continuum model. Inviscid and viscous fluids,

turbulence.12.Internal friction in real fluids due to a shearing stress. The Navier-Stokes equation,

constants of viscosity.13.Description of inviscid fluid by the Euler’s equation. Hydrostatics.14.Two-dimensional flows.15.Energy dissipation in a viscous fluid flow.16.Equation of mass transfer.



Requirements for receiving of credits:48 academic hours. The exam takes place in oral form.Understanding of principal course material; ability to solve course problems both analytically and by computer program packages.

Recommended literature:1. F.P. Incropera, D.P.De Witt. Fundamentals of heat and mass transfer, John Wiley &

Sons, 19902. G.K. Batchelor. An Introduction to Fluid dynamics, Cambridge University Press, 19673. A.R.Paterson. A first course in fluid dynamics, Cambridge University Press, 19944. J.W.Dally, W.F.Riley. Experimental stress analysis, McGraw Hill, 19915. Ю.А.Амензаде. Теория упругости, Высшая школа, 19716. L.D.Landau. Fluid Mechanics, Butterworth-Heinemann, 1995



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