Curriculum - AIKNCaiknc.lv/zinojumi/en/LuMatMaE.doc · Web viewThe system of orthogonal functions in Hilbert space, Fourier series. Least squares approximation, constructing of best

TABLE OF CONTENTS

GOAL OF PROGRAMME................................................................................................................................2

TASKS OF PROGRAMME...............................................................................................................................2

DESCRIPTION OF PROGRAMMES..................................................................................................................2

SUBJECT OF PROGRAMME............................................................................................................................2

LENGTH OF STUDIES AND SCORE OF SUBJECT MATTERS...........................................................................3

IMMATRICULATION TERMS..........................................................................................................................3

REQUIREMENTS IN ASSESSMENT OF STUDIES AND ASSESSMENT ORDER...................................................4

SYLLABUS OF STUDIES PROGRAMME..........................................................................................................5

SUBJECT MATTERS OF STUDIES PROGRAMME............................................................................................5

MEANS OF FULFILMENT THE PROGRAMME. LIST OF TEACHING STAFF......................................................7

COSTS OF STUDIES PER MASTER’S DEGREE PROGRAMME STUDENT PER ANNUM (LS.)............................8

TERMS FOR AWARDING ACADEMIC DEGREE’S...........................................................................................9

POSSIBILITIES OF ACQUIRING NATURAL SCIENCES, SOCIAL SCIENCES AND HUMANITIES AS WELL AS OTHER STUDYING POSSIBILITIES.................................................................................................................9

STUDIES PROGRAMME DEVELOPMENT PLAN..............................................................................................9

GOAL OF PROGRAMMEThe goal of the studies programme for master’s degree 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 master’s degree 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.

DESCRIPTION OF PROGRAMMESThe studies for a master’s degree at the University of Latvia have taken place since 1992.

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. The programme is modified in compliance with the corresponding studies programmes of the leading European universities, taking into consideration the historical traditions of the University of Latvia and the peculiarities of the development of Mathematics in Latvia. The master’s degree programme is made in accordance with the bachelor’s degree programme.

The studies programmes in mathematics for a master’s degree are predominant in the Highest Level programmes in mathematics at higher educational establishments of Latvia.

The studies programmes determinate the subject and the syllabus of the studies, define the immatriculation terms and assessment order, specific requirements for the assessment of the studies, for academic and financial resources as well as for the required material and informative bases which are responsible for a successful fulfilment of the studies.

Having performed the programmes for a master’s degree, enable the students compete in the multinational labour market.

SUBJECT OF PROGRAMMEWithin the master’s degree programme, the students obtain extended knowledge in one or

several separate directions of mathematics.For those students who study at this programme it is possible to specialise in the following

directions: Probability theory and Mathematical Statistics; Mathematical Didactics; Modern Elementary Mathematics; Differential Equations and Mathematical Modelling;

Function Theory; Topology; Mathematical Structures in Business and Economy.

LENGTH OF STUDIES AND SCORE OF SUBJECT MATTERSThe total score of the master’s degree programme amounts to 80 credit points and the length

of the studies comprises 2 years.The course of the programme are divided into three parts: part A – the compulsory courses (52 credit points, 65% of the total score of studies)

including working out the master’s paper (30 credit points); part B – the optional courses (20 credit points, 25% of the total score of studies); part C – the free optional courses of studies (8 credit points, 10% of the total score of

studies).The division of the studies courses at the master’s degree programme into parts A and B is

relative. Each student, when starting the master’s degree studies programme, selects one of the above

mentioned 7 directions and, with the assistance of the tutor of this direction, works out the student’s individual studies programme in which the courses from part A are clarified in the following way:

the first group of part A contains 4 general courses in mathematics (8 credit points), which are common to all the master’s degree sub-programmes;

the second group of part A (14 credit points) contains those master’s degree programme courses in mathematics which are compulsory for the students who have chosen the relevant speciality.

The individual studies programmes of the master’s degree student are affirmed at the University of Latvia in the established order.

IMMATRICULATION TERMSImmatriculation for the master’s degree studies at the University of Latvia takes place in

compliance with the affirmed by UL Senate Terms of Immatriculation for the Master’s Degree Studies at the University of Latvia.

In the year 1999, to be enrolled for a master’s degree the following specific terms were laid down for the candidates.

The qualification on the previous education.Higher academic education at the bachelor’s level (a baccalaureate certifying diploma):the University of Latvia;an accredited higher school of Latvia ( for the fulfilment of at least a three-year course of the

bachelor’s programme, the score and the subject of which answer the requirements of the UL bachelor’s programme, part A);

a foreign country higher school acknowledged by the Republic of Latvia;in the transition period, the correspondence of the bachelor’s education to the UL

requirements at non-accredited private higher schools is being decided by the Entrance Examination Committee of the relevant master’s degree programme.

Higher (five–year) professional education, which also includes the subject of the relevant bachelor’s academic studies programme (a diploma for higher professional education).

Previously obtained higher education (a higher school diploma which affirms the performance of the studies programmes started up to 1991).

Immatriculation Articles.

To the obtainers of a baccalaureate and to the obtainers of higher professional education, who wish to continue their studies at the master’s degree programme for a branch of science, relevant to the master’s degree programme in mathematics, the diploma competition takes place. A list of those higher schools and bachelor’s degree obtainers relevant to the master’s degree, who are determined for the diploma competition, is affirmed by the Council of the faculty of Physics and Mathematics.

The other higher schools candidates who do not go into the category of previous Article and who have fulfilled the requirements on the previous education take one entrance exam in mathematics.

Entrance exams.The entrance exams take place in the scope of the requirements of the bachelor’s degree

programme in mathematics, part A.The requirements of the entrance exams are proposed by the Council of the faculty and

affirmed by Vice-Rector 5 months before the Entrance Examination Committee starts its activity.CompetitionOut of competition are immatriculated: the obtainers of the UL bachelor’s degree ‘with

honours’ in the following branches of the Science: Mathematics, Physics, Computer Science.The other candidates take part in the common competition.The results of the competition are assessed by the diploma competition or a candidate’s

assessment at the entrance exams. If necessary, for a more exact assessment of the master’s degree competition, for Entrance Examination Committee can organise the discourse with candidates.

In accordance with the resolution of 9 February, 1998, adopted by a Council of the faculty of Physics and Mathematics, at the studies for a master’s degree in mathematics without entrance exams are enrolled: the owners of the bachelor’s diploma in mathematics from all Latvia higher schools, the owners of a diploma which is made equal to the bachelor’s diploma in mathematics, as well as the obtainers of the UL baccalaureate in Computer Science and Physics.

REQUIREMENTS IN ASSESSMENT OF STUDIES AND ASSESSMENT ORDERThe students of the master’s degree programme in mathematics are subject to the UL

assessment requirement and assessment order.

SYLLABUS OF STUDIES PROGRAMMEThe syllabus of the master’s degree programme in mathematics is presented in the following

tableBlocks of programme Credit points Total

1.sem. 2.sem. 3.sem. 4.sem. credit pointsCommon score (part A) 4 4 - - 8Score of direction (part A) 0 – 4 0 – 4 0 – 4 0 – 4 14Selectives (part B) 4–8 4 4 4 16 - 20Scientific seminars (part B) 0 – 4 0 – 4 0 – 4 0 – 4 0 – 16Humanitarian selectives (part C) 0 – 4 0 – 4 0 – 4 0 – 2 8Master thesis 0 – 4 0 – 4 4 – 8 12 – 20 30

Total in semester 20 20 20 20 80

Notes1. Within each semester, the student selects a course in compliance with the offered lectures

at the time table, submitting it to tutor’s approval who is responsible for the master’s degree direction.

2. If the number of students is insufficient, separate courses of the direction are given once two years.

3. The average score of the obtained credit points calculated to one semester, by the end of each semester should not be less than 20.

SUBJECT MATTERS OF STUDIES PROGRAMMECommon compulsory courses To All Directions

1. Selected chapters in functional analysis and function theory

2 cr.p. exam

2. Selected chapters in discrete mathematics 2 cr.p exam

3. Selected chapters in ordinary and partial differential equations

2 cr.p exam

4. Selected chapters in probability theory and mathematical statistics

2 cr.p exam

OPTIONAL COURSESAFFINE, PROJECTIVE AND COMBINATORIAL GEOMETRY

ANALYTICAL METHODS OF ORDINARY DIFFERENTIAL EQUATIONSANALYTICAL METHODS OF PARTIAL DIFFERENTIAL EQUATIONSAPPLICATION BY SPLINES IN MATHEMATICAL PHYSICSAPPLIED ANALYSIS (NON-LINEAR)APPLIED ANALYSIS (OPTIMISATION)APPROXIMATIONS THEORYASYMPTOTIC METHODS IN MATHEMATICSCOMBINATORIAL ALGORITHMSCRYPTOGRAPHYDEVELOPMENT TECHNOLOGIES OF ELECTRONIC TEACHING AIDS.ELEMENTS OF SET AND CATEGORY THEORIESELEMENTS OF TOPOLOGYFACTOR ANALYSIS AND MATHEMATICAL STATISTICSGENERAL METHODS OF ELEMENTARY MATHEMATICSLINEAR SYSTEM THEORY AND REGRESSION ANALYSISMARKOVA PROCESSES WITH DISCRETE STATES SPACE.MATHEMATICAL MODELLINGMATHEMATICAL MODELS OF PROCESSES IN POROUS MEDIAMATHEMATICAL STRUCTURES IMATHEMATICAL STRUCTURES IIMATHEMATICAL THEORY OF MARKET EQUILIBRIUMMATHEMATICAL THINKING PSYCHOLOGYMODERN ELEMENTARY GEOMETRYMULTIDIMENSIONAL STATISTICAL ANALYSISNON-LINEAR BOUNDARY VALUE PROBLEMSNUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONSPERTURBATION ANALYSISPRACTICAL CLASSES OF MATHEMATICAL MODELLING WITH COMPUTER PROGRAM PACKAGES IPRACTICAL CLASSES OF MATHEMATICAL MODELLING WITH COMPUTER-PROGRAM PACKAGES IIPRACTICAL CLASSES OF MATHEMATICAL MODELLING WITH COMPUTER-PROGRAM PACKAGES IIIPRACTICE IN OLYMPIAD PROBLEMS SOLVINGRISK THEORYSELECTED CHAPTERS OF COMPUTER APPLICATION PROGRAMSSELECTED CHAPTERS OF NUMBER THEORYSOLVABILITY OF NONLINEAR EQUATIONSSPECIAL FINITE-DIFFERENCE METHODSSPECIAL SECTIONS OF CALCULUS.SPLINES AND THEIR APPLICATIONSSTATISTICAL MODELLINGSTOCHASTIC PROCESSES.SUPPLEMENTARY CHAPTERS OF MATHEMATICAL STATISTICS.THE HISTORY OF LATVIAN SCHOOLSTHEORY OF ALGORITHMS AND AUTOMATATOPOLOGY IIIVISUAL BASIC AS THE UNIVERSAL TOOL IN CREATING TEACHING-CONTROLLING PROGRAMS.

MEANS OF FULFILMENT THE PROGRAMME. LIST OF TEACHING STAFF.The master’s degree programme in mathematics is fulfilled by:

N. Name, surname Academic title Scientific degree

1. Agnis Andzans Professor Dr.Hab.Math.

2. Svetlana Asmuss Docent Dr.Math.

3. Mihails Belovs Docent Dr.Math.

4. Aivars Berzins Docent Dr.Math.

5. Andris Broks Docent Dr.fiz.

6. Andris Buikis Professor Dr.Hab.Math.

7. Inese Bula Docent Dr.Math.

8. Janis Buls Docent Dr.Math.

9. Viktorija Carkova Docent Dr.Math.

10. Janis Cepitis Docent Dr.Math.

11. Andrejs Cibulis Docent Dr.Math.

12. Teodors Cirulis Professor Dr.Hab.Math.

13. Silvija Cerane Docent Dr.Math.

14. Harijs Kalis Professor Dr.Hab.Math., Dr.Hab.Phys.

15. Janis Lapins Docent Dr.Math.

16. Andris Liepins Docent Dr.Math.

17. Ojars Lietuvietis Docent Dr.Math.

18. Uldis Raitums Professor Dr.Hab.Math.

19. Liga Ramana Lecturer M.Math.

20. Andrejs Reinfelds Lead. researcher Dr.Hab.Math.

21. Janis Smotrovs Lecturer M.Math.

22. Aleksandrs Sostaks Professor Dr.Hab.Math.

23. Viesturs Vezis Lecturer M.C.Sc.

24. Janis Vucans Ass. Professor Dr.Math.

COSTS OF STUDIES PER MASTER’S DEGREE PROGRAMME STUDENT PER ANNUM (LS.)

No. Line No Calculation formula Calculated

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 331 0,3 1 D1=A1*B1 99,3Associates professor 146 0 2 D2=A2*B2 0Docent 125 0,59 3 D3=A3*B3 72,57Lecturer 100 0,11 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)*0,36 490,34Gross total costs per student per annum 42 D42=D39+D40+D41 1837,43

TERMS FOR AWARDING ACADEMIC DEGREE’STo obtain the academic master’s degree in mathematics, the candidate is required to fulfil the

master’s degree studies programme and defend the master’s thesis.The Master’s Degree Candidate 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 accepts the defence of the master’s thesis. The common score of the master’s thesis is 30 credit points, and the Chairs offer its themes to the candidates. The themes are affirmed in the University of Latvia in the accepted order. The principles how to work out and design the master’s thesis as well as how to assess the thesis, are described in the document Writing and Defence of Bachelor and Master’s Theses in Mathematics.

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

Besides the courses from part B, included in the master’s studies programme in mathematics which are related to the natural sciences, social sciences and the Humanities, the candidates from the studies programme, part C, can select the courses from other master’s degree programmes, including the courses in the social sciences and Humanities.

STUDIES PROGRAMME DEVELOPMENT PLANIn the nearest future, cardinal changes are not planned in the subject matters of the master’s degree

studies programme in mathematics. Separate changes in the optional parts of the programmes are for certain continuously being brought in by the novelties of Mathematics. To improve the quality of the studies programme, a determinant factor is providing the financial resources for raising the individual earnings of the teaching staff and off the subsidiary employees as well as for the replenishment and the development of the material and informative basis.

Curriculum

Mathematics

Course Abstracts

University of LatviaRīga, 2000

Master’s Degree

Study Program

Affirmed in the Council

University of Latvia Faculty of Physics and Mathematics

11

of Department of Physics and Mathematicson November 2, 1998

University of Latvia Faculty of Physics and Mathematics12

Master’s Degree Programme in Mathematics

PART A (COMPULSORY COURSES)

University of Latvia Faculty of Physics and Mathematics

13


SELECTED CHAPTERS IN PROBABILITY THEORY AND MATHEMATICAL STATISTICS

(Varbūtību teorijas un matemātiskās statistikas izvēlētas nodaļas)

Course code: Mate-5217Author Dr.hab. Math., Professor A.Lorencs, Dr.mat. N.SinenkoCredits 2 credits / 32 hoursRequired for grade test or examPrerequisites Mate-3030 or Mate-2036

AnnotationThe aim of this course is to introduce student’s methods of statistical data collecting, planning of simple statistical experiments and analysis main methods of obtained results.

Subjects1. Distribution functions and moments of random variable.2. Sample characteristics.3. Measurement data analysis.4. Basics of errors theory.5. Sample survey methods.6. Regression analysis.7. Dispersion analysis.8. Factor analysis.9. Outlier analysis.10. Marcov chains.

Requirements to received of credits1. Students should be able to construct the simplest experiment plans.2. Students should be able to construct an adequate mathematical model.3. Students should be able to obtain statistical estimations of the model parameters.4. Students should be able to test adequacy of the model using sample data.5. Students must know how to pick out main components from random vector and be able to

find factors.

Recommended literature1. Г.Крамер, Математические методы статистики, Изд. Мир, М.,1975.2. У.Кокрен, Методы выборочного исследования, Изд. Статистика, М., 19763. В.Феллер, Введение в теорию вероятностей и её приложения, Изд. Мир, М.,1984.



SELECTED CHAPTERS OF DISCRETE MATHEMATICS AND ALGEBRA(Diskrētās matemātikas un algebras izvēlētās nodaļas)

Course code Mate-5214Authors Professor A. Andžāns, Dr. habil.mat., Docent A. Bērziņš, Dr.mat.Credits 2 creditsRequired for grade examPrerequisites none

Annotation. The objectiveness of the course is selected problematic and methods of reasoning of discrete mathematics, theoretical computer science and algebra; some central results with general mathematical and educational value; connections between considered problems and school mathematics and informatics.

Subjects:1. General concept of algorithm2. Algorithmically unsolved problems3. Recursively innumerable sets. Universal sets4. Finite automata5. Special classes of automata6. Ramsey theory 7. General combinatorial methods8. The concept of the complexity of a combinatorial problems9. General algebraic structures10. Universal algebra11. Algebraic constructions12. The concept of category 13. Free objects14. Fields and their extensions15. Algebraic extensions16. Quadratic extensions and geometrical constructions

Requirements for received of credits: 64 hours lectures. Students are required to fulfil one independent homework. The exam takes place in an oval form. Students must show the ability of problem resolving and the understanding of basic concepts, results and the ideas of proof.

Recommended literature:1. Х. Роджерс. Теория рекурсивных функций и эффективная вычислимост. М.,Мир,1972-

624 с. 2. М.Холл. Комбинаторика. М., Мир,1970- 424 с.3. A.Andžāns, J.Čakste, T. Larfelds, L.Ramāna, M.Seile. vidējās vērtības metode. Aizkraukle,

krauklītis, 1996-231 lpp.4. Б. Трахтеньрот, Я. Барздинь. Конечные автоматы. М., Наука, 1970-400 с.5. А. Курош. Лекции по обшей алгеьре. М., Физматгиз, 1962-420 с.6. С. Ленг. Алгеьра. М., Мир,1968-564 с.



SELECTED CHAPTERS OF ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS (Parasto un parciālo diferenciālvienādojumu izvēlētas nodaļas)

Course code Mate - 5216

Authors Prof. A.Buiķis, Dr.Hab.Math., docent J.Cepītis, Dr.Math.

Credits 2 credits

Required for grade exam

Prerequisites

Annotation. The aim of the course is to give the sufficient knowledge of the basic methods for investigation of ordinary and partial differential equations.

Subjects:1. Cauchy’s problem for the system of the first order ordinary differential equations.2. Boundary value problems for the second order ordinary differential equations.3. Behaviour of solutions of non-linear ordinary differential equations, examples from

applications. 4. Partial differential equations as the models of natural process. Conservation principles.5. Cauchy’s- Kowalewska’s theorem. 6. Sturm’s- Liouville’s problem.7. Generalised solutions.8. Numerical methods for differential equations, differential fitting of difference schemes.

Requirements for received of credits: 32 hours lectures.The exam takes place in the oral form. Students should be familiarised with the problems considered in the program.Recommended literature:1. J.Cepītis. Košī problēma pirmās kārtas parastam diferenciālvienādojumam. Rīga, LU, 1992.2. Ph.Hartman. Ordinary Differential Equations. Moscow, 1970, etc.3. M.Braun. Differential Equations and their applications. An Introduction to Applied

Mathematics. Springer-Verlag, 1993.4. E.Riekstiņš. Matemātiskās fizikas vienādojumi, Rīga,1964.5. А.Н.Тихонов, А.А.Самарский,Уравнения математической физики, М., Наука,1977.6. K.W.Morton. Numerical Solution of Convection Diffusion Problems, Chapmann&Hall, 1996.7. A.J. Chorin, E.J.Marsden. A Mathematical Introduction to Fluid Mechanics, Springer, 1993.8. M.Renardy, R.C.Rogers. An Introduction to Partial Differential Equations, Springer, 1992.9. J.W.Thomas. Numerical Partial Differential Equations, Springer, 1995.



SPECIAL SECTIONS OF FUNCTION THEORY AND FUNCTIONAL ANALYSIS.(Funkcionālanalīzes un funkciju teorijas izvēlētas nodaļas.)

Course code Mate-5215Authors Prof. T.Cīrulis, Dr.h.math.,Credits 2 creditsRequired for grade examPrerequisites Mate-2065, Mate-1002, Mate-1011, Mate-3015

Annotation.The objectives of course are applications of functional analysis for problems of the function theory and for a solving of equations with functions.

Subjects:1. Functional analysis as mean conceptions ideas, methods and terminology of mathematics.2. A necessity to generalise the concept of functions for the solving problems of mathematical

physics. Regular and singular distributions.3. Theorem of the complection of normed spaces until Banach spaces. Lebesgue’s and Sobolev’s

spaces and its applications in mathematical physics.4. Properties of Hilbert space. Linear operators and functionals in Hilbert and Banach spaces.

Matrices as linear operators and their norms. 5. Theorems of the embedding. Examples about embedding spaces. Riesz theorem about the

representation of functionals in Hilbert space. Applications. 6. Conjugate spaces and conjugate operators. Self – conjugate operators in Hilbert space.7. Projectors. Ortoprojectors in Hilbert spaces and is applications.8. Non-linear maps and equations in Hilbert and Banach spaces. Frechet differential and its

applications. 9. Monotone operators in Hilbert and Banach spaces. Convex functionals.

Requirement for received of credits: 32 hours lectures. Students are required to fulfil one homework. The examination takes place in an oral form. Students must to show an understanding of methods of functional analysis for functions and equations.

Recommended literature:1. Л.В.Кантoрович, Г.П.Акилов. Функциональный анализ. Москва. 1977., 741 с.2. В.А.Треногин. Функциональный анализ. Москва “Наука”. 1980., 495 с.3. Л.А.Люстерник,В.И.Соболев. Элементы функционального анализа.4. А.Н.Колмогоров, С.В.Фомин. Элементы теории функций и функционального анализа.5. Ю.Рудин. Функциональный анализ. Москва.1975., 440 с.6. П.Халмош. Гилбертово пространство в задачах. Москва.1970., 351 с.7. В.С. Владимиров. Уравнения математической физики. Москва.1971., 512 с.



PART B (OPTIONAL COURSES)



AFFINE, PROJECTIVE AND COMBINATORIAL GEOMETRY(Afīnā, projektīvā un kombinatoriskā ģeometrija)

Course code Mate – 5220Authors Lecturer L. Ramāna, Mag.Math.Credits 4 creditsRequired for grade examPrerequisites Mate-5247

Annotation. The objectiveness of the course is main methods of affine, projective and combinatorial geometry and their applications to problem solving in elementary geometry.

Subjects:1. Various geometry’s as transformation groups. Erlangen programmes.2. The group of affine transformation. Affine invariant and their uses in elementary geometry.

The parallel projections as an example of affine transformation.3. Central projection and its uses in elementary geometry problem solving.4. The group of projective transformation. Interpretations of projective geometry.5. Classical problems of combinatorial geometry in Euclidean plane.6. Geometrical objects in a discrete plane.7. Elements of geometry fractals.

Requirements for receiving of credits: 64 hours lectures. Students are required to work at home on problem solving constantly. The exam is in a written form and consists of problem solving.

Recommended literature:1. http://archives.Math.utk.edu/topics/fractals.gtml 2. Математитеческий цветник. Москва.,1983.3. Д.Гилберт, С.Кон-Фоссен. Наглядная геометрия. Москва.,1981.


http://archives.math.utk.edu/topics/fractals.gtml


ANALYTICAL METHODS OF ORDINARY DIFFERENTIAL EQUATIONS (Parasto diferenciālvienādojumu analītiskās metodes)

Course code Mate - 5251Author Docent J.Cepītis, Dr.Math.Credits 2 credits Required for grade examPrerequisites

Annotation. The aim of the course is to give the sufficient knowledge of the basic methods for applying and investigation of ordinary differential equations.

Subjects:9. Ordinary differential equations of the first order in applications.10. Linear ordinary differential equations of the second order in applications.11. Systems of linear and non-linear ordinary differential equations, stability of the steady state. 12. Phase plane analysis.13. Theorems of the Poincare-Bendixon type.14. Bifurcation in the real applications.15. Integral funnels, antifunnels and enclosures.

Requirements for receiving of credits: 32 hours lectures.The exam takes place in the oral form. Students should be familiarised with the problems considered in the program.

Recommended literature:10. M.Braun. Differential Equations and Their applications. An Introduction to Applied

Mathematics. Springer-Verlag,1993.11. J.H.Hubbard, B.H.West. Differential Equations: A Dynamical Systems Approach., Springer-

Verlag,1991.



ANALYTICAL METHODS OF PARTIAL DIFFERENTIAL EQUATIONS(Parciālo diferenciālvienādojumu analītiskās metodes)

Course code Mate-5252Author Prof. A. Buiķis, Dr.Hab.Math.Credits 2 creditsRequired for grade examPrerequisites

Annotation. The aim of the course is to give the students sufficient knowledge of the basic methods for solving partial differential equations.

Subjects:1. Conservation laws.2. Theoretical problems of solution of 1-st order equations.3. Discontinuity of the scalar solutions of conservation laws, weak solutions, jumps (the

Rankine-Hugoniot condition).4. The systems of 1-st order equations (characteristics, hyperbolicity).5. Hyperbolical systems with constant coefficients.6. The main properties of elliptic equations, harmonic functions; the maximum principle7. Analytical methods for solving quasilinear systems.8. Notion of the Green function and its construction for specific regions.9. The method of boundary elements.

Requirements for receiving of credit: 24 hours lectures, 8 hours workshops.The exam takes place in the oral form. Students should be familiarized with the problems considered in the programme.

Recommended literature:1. G. Barenblatt. Similarity, Self-similarity, Interasymptoticy. Leningrad, 1978 (Russian).2. A. Berezovskij. Lectures about Nonlinear Boundary Problems Mathematical Physics. Part 2.

Kiev, 1974 (Russian).3. Nonlinear waves. Moscow, 1977 (Russian).4. M. Renardy, R. Rogers. An Introduction to Partial Differential Equations. Springer, 1993. 5. A. Samarskij, V. Galaktionov. Solutions with Sharpening for Quasilinear Parabolic

Equations. Moscow, 1987 (Russian).6. M. Taylor. Partial Differential Equations. Springer, !986.7. C.Brebbia, J.Telles, L.Wrobel. Boundary Element Techniques. Springer, 1984 (Russian,

1987).



APPLICATION BY SPLINES IN MATHEMATICAL PHYSICS(Splainu izmantošana matemātiskajā fizikā)

Course code Mate-5263Author: Prof. A. Buiķis, Dr.Hab.Math.Credits 2 creditsRequired for grade examPrerequisites

Annotation. The course familiarise the students with the main properties of classical and special splines and their applications to solving various problems of mathematical physics.

Subjects:1. The basic constructive and numerical properties of splines ( classical, Hermitean, exponential,

rational).2. Formulas of classical and special representation.3. Parabolic and integral splines and their applications in the numerical approximation of

differential equations. 4. Differential equations with discontinuous coefficients: the method of spline collocation.5. The Galerkin’s method and splines.6. Numerical solution of integral equations with the help of splines.7. Multidimensional splines and numerical methods.

Requirements for received of credits: 16 lecture hours, 16 lecture seminars. The exam takes place in oral form. Students should master the main properties of different splines and be able to apply them to solving specific real problems of differential equations.

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

Cambridge University Press, 1966.2. T. Rusakov. Methods of Spline-functions in Numerical Hydrodynamic. Perma, 1987

(Russian).3. H. Spaeth. Eindimensionale Spline – Interpolationsalgorithmen. Verlag Muenchen,

1990(German).4. S. Stetshkin, J. Subotin. Spline in Numerical Mathematics. M., 1976 (Russian ). 5. J. Zavjalov, B. Kvasov, V.Miroshnitshenko. Methods of Spline- Functions. M., 1980

(Russian ). 6. A.Buikis. Problems of Mathematical Physics with Discontinuous Coefficients and their

Applications. Riga, 1991. Manuscript (Russian).



APPLIED ANALYSIS (NON-LINEAR)(Pielietojamā analīze (nelineārā))

Course code: Mate-5254Author: Prof. U.Raitums, Dr.Hab.Math.Credits: 3 creditsRequired for grade: examPrerequisites:

Annotation. The course considers those elements of functional analysis, which are necessary to deal with various nonlinear equations and to substantiate numerical methods.

Subjects:1. Integration of real argument functions with values in Banach spaces.2. Derivatives of operators.3. Finite increment formulae.4. Implicit functions theorems.5. Monotone operators and Minty's theorem.6. Iteration methods for strongly monotone operators.7. Newton's method in Banach spaces.8. Galerkin's and Ritz's methods in Hilbert spaces.9. Schauder's principle.

Requirements for received of credits: 48 hours lectures. The students are required to do 4 independent home works, related to four theoretical questions. The exam takes place in an oral form. The student is required to demonstrate knowledge and an understanding of the given theoretical questions.

Recommended literature.1. Kantorovich, L.V., Akilov, G.P., Functional Analysis, Moscow, 1977 (in Russian).2. Gajewski, H., Gröger, K., Zacharias, K., Non-linear Operator Equations and Differential

Operator Equations, Mir, Moscow, 1978 (in Russian).



APPLIED ANALYSIS (OPTIMISATION)(Pielietojamā analīze (optimizācija))


Annotation. The course considers the basic questions of the theory: existence of solutions of optimization problems, necessary optimality conditions, and optimal control problems.

Subjects:1. Weak convergence in Banach spaces.2. Sequential compactness in weak topologies.3. Weakly lower semi-continuous functionals.4. Existence of solutions for standard problems from calculus of variations (scalar case).5. Necessary optimality conditions for functionals.6. Examples of optimal control problems.7. Sensitivity analysis for optimal control problems.8. Introduction in numerical methods for optimisation problems.9. Extensions of extremal problems.

Requirements for receiving of credits: 48 hours lectures. The students are required to do 4 independent home works, related to four different theoretical questions. The exam takes place in an oral form. The student is required to show knowledge and an understanding of the role of weak convergence and necessary optimality conditions in the theory of optimisation.

Recommended literature:1. Dunford, N., Schwartz, J.T., Linear Operators. General Theory, Interscience Publishers, New

York, 1958, or Russian translation, IL, Moscow, 1962.2. Cea, J., Optimisation Theory and Algorithms, Mir, Moscow, 1973 (in Russian).3. Kantorovitch, L.V., Akilov, G.P., Functional Analysis, Nauka, Moscow, 1977 (in Russian).



APPROXIMATION OF FUNCTIONS.(Funkciju aproksimācija)

Course code Authors Prof. T.Cīrulis, Dr.Hab.Math.,Credits 4 creditsRequired for grade examPrerequisites Mate-2065, Mate-1002, Mate-1011, Mate-3015

Annotation. The objectives of course are the most important methods of approximation of functions. Mainly global approximations of functions. Mainly global approximations are considered. There is only Pade approximation as a local approximation. Approximations educate by means of methods of a functional analysis or of contractible methods.

Subjects:10. Local and global approximations of functions. Application of methods of functional

analysis. Methods of constructive еру function theory. Saturated and non-saturated approximations.

11. Better Chebyshev approximations in a metric, normed and Hilbert spaces. Properties of better approximations Jackson unequalities.

12. Lagrange and Hermite inerpolations. Non-saturatedness of interpolations.13. Approximations by means of splines. Classification, properties, constructions and

applications of splines. 14. Approximation of Pade and their connection with continued fractions. Two points Pade

approximations. Applications.15. Approximations with special functions.Requirement for receiving of the credits: 48 hours lectures, 16 hours practical work. Students are required to fulfill 2 independent home works about of themes 2., 3.,4.,5. The examination takes place in an oral form. Students must know the most important methods of approximation of functions and understand to apply them for mathematical problems. Recommended literature:8. А.Коллатц, И.Крабс. Теория приближений. Чебышевские приближения, Москва, Наука,

1978.9. А.Ф.Тиман. Теория приближения функций действительного переменного. Москва,

Гос.изд.физ.мат. 1960.10. Н.С.Бахвалов, Н.П.Жидков, Г.М.Кобельков. Численные методы. Москва, Наука,

1987.11. Н.С.Бабенко. Основы численного анализа. Москва, Наука, 1987.



APPROXIMATIONS THEORY(Aproksimācijas teorija )

Course code Mate-5221Authors Prof. H. Kalis Dr. Hab. Math., Dr. Hab. Phys.Credits 2 creditsRequired for grade examPrerequisites Mate-2085

Annotation. The objectives of the course are: theory of approximations by using the methods of functional analysis: best approximations in Banah’s and Hilbert’s spaces, approximation in linear normed space with Fourier series, orthogonal polynomials, Chebyshev polynomials, minimax approximation, algebraic, trigonometrically and spline approximation (interpolation). This course in binded with course “Practical classes of mathematical modelling with computer program packages”.

Subjects:1. The best element of approximation in linear normed space.2. The best approximation with general polynomials in continuous function space, Chebyshev

theorem.3. The best uniform approximation with algebraic polynomials, Weierstras’s theorem.4. The best approximation with trigonometrically polynomials.5. The system of orthogonal functions in Hilbert space, Fourier series. 6. Least squares approximation, constructing of best element.7. Orthogonal polynomials and its properties.8. Chebyshev`s polynomials of first and second kind and approximations of functions.9. Least squares approximations for functions defined with tables.10. Functions interpolation, Chebyshev’s system.11. Lagrange and Newton interpolating polynomial, Chebyshev’s mesh points.12. Hermits interpolating polynomial and error estimating.13. Trigonometrically interpolation.14. Interpolating with spline approximation.15. Rational approximations (Pade approximation).

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

Recommended literature:1. И.С.Березин, Н.П.Жидков. Методы вычислений I. М.,1959.2. Н.И.Ахиезер, И.М.Глазман. Теория линейных операторов в гильбертовом

пространстве. М., 1966.3. G.Engeln-Mullges, F.Uhlig. Numerical algorithms with Fortran. Springer, 1966.4. S.D.Conte, C.Boor. Elementary Numerical analysis an algorithmic approach. Mc Graw-Hill

Book Company, 1972.



ASYMPTOTIC METHODS IN MATHEMATICS(Asimptotiskās metodes matemātikā)

Course code Mate-5218Authors Docent M.Belovs, Dr.Math.Credits 4 creditsRequired for grade examPrerequisites Mathematics, Physics or Computer Science program for bachelor

grade

Annotation. The objects of the course are various asymptotic methods for function approximation and their applications: integral asymptotic, asymptotic methods of the solution for differential equations, asymptotic solutions for transcendent equations and their various generalizations.Subjects:1. The definitions and methods of asymptotic expansion their value in Mathematics and their

applications in modeling of natural processes. The most important operations with asymptotic expansions.

2. Asymptotic expansions of integrals.3. Asymptotic for multidimensional integrals.4. Asymptotic expansions of the roots of Algebraic and transcendent equations. Regular and

singular asymptotic.5. Uniform asymptotic expansions in the theory of nonlinear oscillation. 6. Differential equations with boundary layer.7. Singular perturbation for linear partial differential equations.8. Differential equations with large parameters. Methods of VKB.9. Connection of asymptotic solutions of differential equations with asymptotyc expensions for

integrals.10. Methods of asymptotic averaging for mathematical modeling of surroundings with pereodic

structure.11. Review of asymptotyc methods for the solution of operator equations.12. Advantages and deficiencies of asymptotic methods. Connection between calculating and

asymptotic methods.Requirements for receiving of credits: 48 hours lectueres, 16 hours practical work. Students are required to fulfill 2 independent home works for themes 2, 4, 5, 6, 8, 9. And to write a referate for one of the 2 to 11 themes. To pass an exam. The exam take place in an oral form.Recommended literature:1. Ф.Олвер. Асимптотика и специальные функции. Москва, «Наука», 1978., 376 стр. 2. А.Эрдейи. Асимптотические разложения. Москва, «Наука», 1981., 127 стр.3. М.В.Федорюк. Асимптотика. Интегралы и ряды, Москва, «Наука», 1987., 544 стр.4. А.Найфе. Введение в методы возмущений. Москва, «Мир», 1976., 455 стр.5. Н.Н.Моисеев. Асимптотические методы в нелинейной механике. Москва, «Наука»,

1981., 400 стр.6. А.М.Ильин. Согласование асимптотических разложений решений краевых задач.

Москва, «Наука», 1989., 336 стр.7. Н.С.Бахвалов, Г.П.Панасенко. Осреднение процессов в периодических средах. Москва,

«Наука», 1988., 312 стр.8. В.П.Маслов. Асимптотические методы теории возмущений. Москва, «Наука», 1988., 313 стр.



COMBINATORIAL ALGORITHMS(Kombinatoriskie algoritmi)

Course code Mate – 5234Authors Professor A.Andžāns, Dr. Hab.Math.Credits 4 creditsRequired for grade examPrerequisites Mate-1002, Mate-2065, Mate-2032

Annotation. The objectiveness of the course is method of elaboration, analysis and optimisation of combinatorial algorithms, their uses in various branches of mathematics and theoretical computer science, connections of combinatorial algorithms with school mathematics curricula.

Subjects:1. Algorithms of inductive constructions. Their uses in the construction of the networks.2. “Divide and conquer” algorithms. Their uses in the analysis of social processes.3. Probabilistic algorithms and their examples.4. Algorithms with incomplete or unreliable information.5. The concept and examples of nondeterministic algorithms.6. P-NP problematic.7. Fast algorithms for numbers, polynomials and matrices.8. Fast algorithms for graphs.

Requirements for receiving of credits: 64 hours lectures. Students are required to read and to report on two research papers. The exam is in an oral form. Students must show the ability of problem solving and the understanding of basic concepts, results and the ideas of proofs.Recommended literature:1. Д.Кнут. Искуство программирования. Москва,1976-19782. А.Ахо, Дж.Хопкрофт, Дж.Ульман. Построение и анализ вычислительных алгоритмоов.

Москва., Мир,1979



CRYPTOGRAPHY(Kriptogrāfija)

Course code Mate-5231Authors Docent J. Buls, Dr. Math.Credits 4 creditsRequired for grade exam

Annotation. Until the 1970’s cryptography was used mainly for military and diplomatic purposes. However, with the increased computerization of economic life, new needs for cryptography arose. The course treats all the essential core areas of cryptography: classical ad public-key cryptosystems via advanced theoretic and practical schemes.

Subjects:1. Introduction.2. Classical cryptosystems.3. Shannon’s theory.4. The data encription standard.5. Public-key cryptosystems.6. Signature schemes.7. Hash functions.8. Identification schemes and authentication codes.9. Secret sharing.10. Pseudo-random number generators.11. Zero-knowledge proofs.

Requirements to receiving of credits.64 hours lectures and seminars. Students are required to fulfill independent work. The exam in an oral form. Candidate must orient oneself in concepts and problems of course. The skill must be demonstrated as solution of problems. Candidate must unconstrained expound appointed theme.

Recommended literature.1. Neal Koblitz. Algebraic Aspects of Cryptography. Springer-Verlag, Berlin, Heidelberg, New

York, 1998.2. Alfred J. Menezes, Paul C. van Oorschot, Scott A. Vanstone. Handbook of Applied

Cryptography. CRC Press, Boca Raton, New York, London, Tokyo, 1997.3. Arto Saomaa. Public-key Cryptography. Springer-Verlag, Berlin, Heidelberg, New York,

1990.4. Douglas R. Stinson. Cryptography: theory and practice. CRC Press, Boca Raton, New York,

London, Tokyo, 1995.5. Dominic Welsh. Codes and Cryptography. Clarendon Press, Oxford,1996.



DEVELOPMENT TECHNOLOGIES OF ELECTRONIC TEACHING AIDS.(Elektronisku mācību līdzekļu izstrādes tehnoloģijas)

Course code: Mate-0226Author Mg. Comp., lect. Viesturs VezisCredits 4Required for grade testPrerequisites high school courses in Mathematics and Computer Sciences, ability to

use INTERNET services.Annotation The course consists of three blocks. It starts with overview of in teaching aids development and application technologies using Microsoft PowerPoint. Further the course introduces the program package Mathematica from the aspect of preparing teaching materials and using them in teaching process. In the conclusion of the course trainees are introduced to WWW page creating technologies.Subjects1. Computers’ place and role in lesson. Training programs: types, didactic aspects of developing

and application possibilities.2. Microsoft PowerPoint as the tool in preparing teaching materials. Evaluation of Microsoft

PowerPoint possibilities from methodical viewpoint. Creating and editing static and dynamic presentation materials. Computer usage as the projector of PowerPoint slides.

3. Application of Active Matrix Liquid Crystal Display (LCD) in demonstration of presentation materials, LCD usage methodology.

4. Possibilities of program package Mathematica usage in school and in creating visual aids.5. HTML (HyperText Markup Language) – language of WWW (World Wide Web).

Characterization of HTML. Simple Web page creating. Overview of different Web page editors. Microsoft FrontPage (or other) as the universal tool in creating complex Web pages.

6. The global structure of Web page: head, body. Comments, headings, lines, paragraphs and phrases. Bulleted and numbered lists, lists of definitions. Table creating and formatting. Link (hyperlink) creating and applying. Object, picture and applet including. Dynamic image creating.

7. Style and frame usage in Web pages. Insight in creating interactive Web pages: Forms and Scripts.

8. Overview of usage of the newest technologies in developing teaching aids.Requirements to received of credits:1. Two projects have to be worked out:

1.1. Development of presentation material set for one lesson (one topic) using program package Microsoft PowerPoint.

1.2. Presentment of one lesson (topic) conspectus in Web pages.2. Two given problems have to be solved using program package Mathematica.Credits are awarded after successful completing of both projects and solving both problems. Recommended literature1. H.Kalis, S.Lācis, O.Lietuvietis, I.Pagodkina Programmu paketes “Mathematca” izmantošana

mācību procesā2. Stephen W. Sagman, Running Microsoft® PowerPoint® 973. Microsoft® FrontPage® 97 Step by Step, Catapult, Inc



ELEMENTS OF SET AND CATEGORY THEORIES (Kopu un kategoriju teorijas elementi)

Course code Mate-4114Authors Prof. A. Sostaks, Dr. habil. Math.Credits 2 creditsRequired for grade exam Prerequisities: Mate-1060, 1061, 2086.

Annotation. Foundation of Cantor set theory. Elements of axiomatic approach to set theory. Foundations of theory of categories. Examples of categories from algebra, topology, functional analysis, etc.

Subjects: 1. Origins of set theory. 2. Sets. Operations with sets. Functions.3. Cardinality of a set. Cardinal numbers.4. Inequalities between cardinal numbers. Cantor-Bernshtein Theorem.5. Operations with cardinal numbers: sum, product, exponent.6. Countable sets. Continuum.7. Completely ordered sets.8. Axiom of Choice and Zermello theorem. Principle of complete oderability.9. Axiom of Choice and Kuratowski-Zorn principle.10. Axiom of Choice and Hausdorff principle.11. Analysis of the Axiom of Choice. Applications of the Axiom of Choice.12. Transfinite induction and transfinite recursion.13. Cardinal arithmetic.14. Powers of cardinals. Generalised Continuum Hypothesis.15. The cofinal character of a cardinal. Regular and singular cardinals.16. Ordinal numbers. Countable ordinals. Arithmetic of ordinal numbers.17. Concept of a category. Basic examples of categories. Morphisms.18. Special morphisms: sections,retractions, monomorphisms, epimorphisms, isomorphisms.19. Products and copruducts in a category.20. Diagrams in a category. Limits and colimits.21. Complete categories.22. Functors.23. Natural transformation of functors. Isomorphism and equivalence of categories.

Requirements for receiving credits: 32 hours lectures. 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. J. Adamek, Theory of Mathematical Structures, Reidel Publ. Comp., Dodrecht, Boston,

Lancaster, 1994.



2. J. Adamek, H. Herrlich, E. Strecker, Abstract and Concrete Categories, John Wiley&Sons, New York, 1990.

3. G. Birkhoff, Lattice Theory, Amer. Math. Soc. Providence, RI, 1987.4. J. Barwise (ed ) Handbook of Mtahematical Logic, Vol. II – Set Theory, North Holland 1977.5. А.В. Архангельский, Канторовская теория множеств, Москва, 1988.6. H. Herrlich, G. E. Strecker, Category Theory, Sigma Series in Pure Mathematics, 1987.7. Mac Lane, Categories for the working mathematicians – Springer Verlag 1971.8. К. Куратовский, А. Мостовский, Теория множеств, Москва, 1977.9. Cohen P. Set theory and the continuum hypothesis, New York,1966.10. Jech T. Lectures in set theory with particular emphasis on the method of forcing, Springer

Verlag, 1971.



ELEMENTS OF TOPOLOGY (Topoloģijas elementi)

Course code Mate-0266Authors Prof. A. Šostaks, Dr. Habil. Math.Credits 4 creditsRequired for grade exam Prerequisites: Mate-2064, Mate-1061

Annotation. The course basically contains the topics of the courses Mate-2086 "Topology 1" and Mate-3183 "Topology 2" and is mean for those master students who did not study these courses at the undergraduate level. On the other hand, as different from Mate-2086 and Mate-3183 the knowledge that the students must have from the course Mate- 3084 is used.

Subjects: 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.)1. Topological spaces – basic definitions and examples.2. Closed sets in a topological space.3. Structure of a set in a topological space.(Closure and interior of a set. Border of a set.

Neighbourhood of a point. Accumulation points of a set. Dense, nowhere dense and scattered subsets of a topological space.)

4. Base of a topology. Sub-base of a topology (Definition and examples of bases and sub-bases.. Construction of a topology from a base. Construction of a topology from a sub-base. Weight of a topology. Local weight of a topology in a point. Examples of constructed topologies from bases and sub-bases: Sorgenfrey space. Niemicky space.)

5. Separable spaces. Density of a topological space.6. Continuous mappings of topological spaces. (Basic definitions and characterisations.

Properties of continuous mappings. Examples of continuous mappings.)7. Homeomorphism. Homeomorphic spaces. Homeomorphic sets in topological spaces.8. Lower separation axioms: T0-spaces, T1-spaces, T2-spaces. 9. Higher separation axioms: Regular and T3-spaces. Normal and T4-spaces.10. Higher separation axioms and continuous functions: Urysohn Lemma. Extension of

continuous functions: Tietze - Urysohn theorems.11. Subspaces of topological spaces. Induced topology. Hereditary topological properties.12. Products of topological spaces: Definitions and basic properties of products. Properties of

projections. Productivity of separation axioms. Products of separable spaces.13. The diagonal theorem and its applications.

13.1. The diagonal theorem (The theorem about embedding of a topological space into certain products.)

13.2. Important products: Tychonoff cubes. Cantor cubes. Hilbert space.13.3. Special properties of diagonal mappings.

14. Compactness: Compact topological spaces and compact sets: definitions and examples. Basic properties of compact spaces and compact sets. Compactness and separation axioms. Continuous mappings on compact spaces. Continuous images of compact sets.

15. Product of compact spaces: Tychonoff theorem and its corollaries.



16. Stone-Weierstrass theorem and its corollaries.17. Connectedness. Connected spaces and connected sets: definitions and examples. Locally

connected spaces. Linearly connected spaces.18. Compact sets in metric spaces.

18.1. Pre-compact metric spaces and pre-compact subsets in metric spaces.18.2. Complete metric spaces and complete subsets in metric spaces.18.3. Characterisation of compact subsets in metric spaces.

19. Compactifications. One-point compactification of a locally compact space. Stone-Čech compactification and its properties.

20. Metrization of topological spaces.20.1. Problem of metrizability of a topological space.20.2. Urysohn Theorem on metrizability of a second countable regular space.

10.3. Bing-Nagata-Smirnov metrization criterion. (without proof).21. Elements of homology theory (Chain complexes, Homologies of complexes, Exact

sequences. Some applications of homology theory.)22. Elements of homotopy theory (Homotopy, retractions and deformations, fundamental group

of a space.)

Requirements for received of credits: 32 hours lectures + 32 hours workshop (seminars) Students are required to prepare a report related to the course and to give a talk at a seminar. 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. R. Engelking, General Topology – PWN, Warszawa, 1979.2. J.L. Kelley, General Topology – Van Nostrand Co, Inc., New York 1957.3. П.С. Александров, Введение в теорию множеств и общую топологию, Москва, 1982.4. А. Борубаев, А. Шостак, 50 примеров и конструкций из общей топологии, Фрунзе,

1986.5. L.A. Steen, J.A. Seebach, Counterexamples in topology. Dover publ. Inc., New York, 1996.



FACTOR ANALYSIS AND MATHEMATICAL STATISTICS.(Faktoranalīze un matemātiskā statistika )

Course code Mate-5228Author lect. Jānis Smotrovs, Mag.Math.Credits 4 creditsRequired for grade examPrerequisites Mate-2032, Mate-3030

AnnotationThe objectives of the course are mastering and practice of the profound methods of

mathematical statistics (factor analysis, cluster analysis, i.c.) especially, the treatment of the statistical data of sociology, pedagogy and psychology.Subjects:1. Basic ideas of mathematical statistics.2. Some questions of the theory of multidimensional random variables.3. General population and samples. Estimates of the parameters distribution. Confidence interval.4. Statistical hypothesis and its test.5. Analysis of variance (one and two factor cases).6. Correlation and regression analysis. Design of experiments.7. Method of principal components.8. Cluster analysis. Ideas of the qualitative factor analysis.9. Discrimination analysis. Linear and non-linear cases. Ideas of the canonical analysis.10. Factor analysis and its methods.Requirement for receiving of credits: 64 hours lectures. Students are required to fulfil one independent homework. The examination takes place in oral form: student defences his homework and answer of the questions.Recommended literature:1. V.Carkova. Matemātiskā statistika. Rīga, LVU, 1979.2. O.Krastiņš. Varbūtību teorija un matemātiskā statistika. Rīga, 1980.3. V.Labejevs. Varbūtību teorija un matemātiskā statistika fiziķiem. Rīga, LVU, 1980.4. Гаральд Крамер. Математические методы статистики. Москва, 1975.5. Иванова В.М., Калилина В.Н., Нешумова Л.А., Решетникова П.О. Математическая

статистика. Москва, 1989.6. В.Е.Дерябин. Многомерная биометрия для антропологов. Изд.Моск.Унив., 1983.7. М.Дж.Лендалл, А.Стюарт. Многомерный статистический анализ и временные ряды.

Москва, 1975.(т.3)8. И.Д.Мандель. Кластерный анализ. Москва, 1988.9. Д. Лоули, А.Максвел. Факторный анализ как статистический метод. Москва, 1967.



GENERAL METHODS OF ELEMENTARY MATHEMATICS(Elementārās matemātikas vispārējās metodes)

Course code Mate-5227Authors Professor A.Andžāns, Dr.habil. Math.Credits 4 creditsRequired for grade examPrerequisites none

Annotation. The objectiveness of the course is general methods of reasoning in solving, composing and classification of problems in elementary mathematics, uses of these methods in discrete mathematics and theoretical computer science, demonstrating the links between various branches of mathematics.

Subjects:1. The general idea of the interpretation method and the examples of it’s applications.2. The mean value method.3. The method of invariant.4. The main classes of algorithmic problems and their solution schemes,5. The role of analogy in problem solving and composing.6. The classification of problems and solutions.

Requirements for receiving of credits: 48 hours lectures, 16 hours problem solving, students are required to do regular homework in problem solving. The examination is in a written form and consists of problem solving.

Recommended literature:1. A.Andžāns, A.Ločmele, I.Palma, L.Ramāna, T.Larfelds, M.Seile. Vidējās vērtības metode.

Aizkraukle, Krauklītis, 1996-231 lpp.2. A.Andžāns, L.Ramāna, A.Reihenova, B. Johenansons. Invariantu metode. Rīga. 1997.3. E.Riekstiņš, A.Andžāns. Atrisini pats! Rīga,1984.4. L.Ramāna. Maģistra darbs. Rīga,1996. (rokraksts)



LINEAR SYSTEM THEORY AND REGRESSION ANALYSIS (Lineāro sistēmu teorija un regresijas analīze)

Course code Mate-5236Author Docent J.Cepītis, Dr.Math.Credits 4 credits Required for grade examPrerequisites

Annotation. The course considers various elements of linear system theory and regression analysis.

Subjects:1. Qualitative results of ordinary linear differential equation theory.2. General theory of linear difference equations and systems.3. Linear difference equations and systems with constant coefficients.4. Applications of linear differential and difference equations.5. Concept of regression analysis.6. Orthogonal polynomials.7. Hypothesis to verificated with the model of full series.8. The models with not full series.9. Plan for simple experiment.10. Analysis of non-responsibility.11. General linear hypothesis.12. Remarkable functions.

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 possess the basic concepts of linear system theory and regression analysis.

Recommended literature:1. A.Sen, M.Srivastava. Regression Analysis, Springer, 1996.2. S.N.Elaydi. An Introduction to Difference Equations. Springer-Verlag, 19963. R.A.Holmgren, A First Course in Discrete Dynamical Systems, Springer-Verlag, 1994 4. И.Г.Петровский. Лекции по теории обыкновенных дифференциальных уравнений, М.,

Наука, 1981.



MARKOVA PROCESSES WITH DISCRETE STATES SPACE./Stochastic Processes 1./

(Markova procesi ar diskrētu stāvokļu telpu)

Course code Mate-5237Author Docent Viktorija Carkova, Dr. Math.Credits 2 credits Required for grade examPrerequisites Mate-2032

AnnotationThis course makes students familiar with the Markov chains, Markov chain’s classification, ergodic theorem and some application.

Subjects1. Stochastic process definition.

1.1. Stochastic process classification, properties.1.2. Examples.

2. Stochastic processes with discrete states.2.1. Markov chains.2.2. Chapman-Kolmogorov equations.2.3. Examples. A Random walk model. Erenfests model. Transforming a process into a Markov chain. Branching processes.2.4. Classification of states.2.5. Recurrent states.2.6. Solidarity theorem.2.7. Periodic Markov chain.2.8. A random walk.2.9. An ergodic theorem.2.10 Continuous-time Markov chain.2.11 Examples. A model of class mobility. The Hardy-Weinberg law in genetic. The gambler’s ruin problem.

3. The Poisson process.3.1. Definition of the Poisson process. Properties.3.2. Erlang’s formulae.3.3. Examples.

Requirements for received of creditsThe examination for this course is a three-hour written-answer examination.

Recommended literature1 .V.Carkova, D.Kalnina. Stochastic processes. LU, Riga, 1981, ( 98 p.).2. S.M.Ross. Introduction to Probability Models. Academic press, Inc. Fifth edition, NY, 1995



MATHEMATICAL MODELLING(Matemātiskā modelēšana)

Course code Mate-5240Author Prof. A. Buiķis, Dr.Hab.Math.Credits 2 credits Required for grade examPrerequisites

Annotation. The course considers theoretical and methodical aspects of the mathematical modelling and numerical experiment, as well as construction of mathematical models, describing their computation and analysis of modelling results.

Subjects:1. Mathematical modelling and numerical experiment. Principles and general scheme. A real

process and its mathematical model. A mathematical model and a secondary model.2. Analogy in construction of models and a hierarchic approach to creation of models.3. Fundamental laws of nature and construction of mathematical models. 4. Mathematical models, dimensionless variables, similarity theory. Self-similar solutions.5. Overview and analysis of different specific mathematical models.6. Completion of the modelling works, presentation of the results.

Requirements for receiving of credits: 16 hours lectures, 16 hours workshops.The exam takes place in oral form. Students should master the principles of mathematical modelling, its scheme and methodology. They must be able to construct mathematical models for typical technical and natural processes.

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).3. J. Grasman. Asymptotic Methods for Relaxation Oscillations and Applications. Springer,

1987.4. N. J. Higham. Handbook of Writing for the Mathematical Sciences. SIAM Philadelphia,

1993.5. H. Ockendon, J. R. Ockendon. Viscous Flow. Cambridge University Press, 1995.6. A. Samarskij, A. Mihailov. Mathematical Modelling. M., 1997(Russian).



MATHEMATICAL MODELS OF PROCESSES IN POROUS MEDIA(Procesu porainās vidēs matemātiskie modeļi)

Course code Mate-5256 Author Prof. A. Buiķis, Dr.Hab.Math.Credits 2 credits Required for grade examPrerequisites

Annotation. The course familiarises the students with different models of filtration processes in porous media, starting with the models that can be reduced to the main types of the 2-nd order equations considered in the mathematical physics course, and further with essentially more complicated models.

Subjects:1. Basic concepts and relationships of the filtration theory: equations of continuity, motion, and

state.2. Models and problems of homogeneous liquid filtration in homogeneous and non-

homogeneous media.3. Models and problems of multiphase liquid filtration, taking and not taking into account the

capillary effect.4. Models and problems of homogeneous liquid and gas filtration in layered and porously –

fractured media.5. Analytical, analytically – numerical and numerical methods in the modelling of filtration

processes.6. Models and problems of convective diffusion and convective heat transfer processes in

porous media.7. Models of incompletely saturated media.

Requirements for received of credits: 32 lecture hours. Students should master the typical models of filtration theory as well, as be able to provide formulation of the main types of problems for these models.Recommended literature:1. G. Barenblatt, V. Jentov, V. Rizhik. Motion of Liquid and Gas in the Natural Layers. M.,

1984 (Russian).2. J. Bear, A. Verruijt. Modeling Groundwater Flow and Pollution. D. Reidel Publ. Co,

Dordrecht, 1987.3. A. Buikis. Models of Filtration Processes. 1998 (Latvian, handwriting).4. U. Hornung. Homogenization and Porous Media. Springer, 1997.5. A. Konovalov. Problems of Multiphase Uncompressible Liquid Filtration. Novosibirsk, 1988

(Russian).6. Ne-Zheng Sun. Mathematical Modelling of Groundwater Pollution. Springer, 1996.7. P. Polubarinova-Kotshina. Theory of Groundwater Motion. M., 1997 (Russian).8. A.Buikis. Problems of Mathematical Physics with Discontinuous Coefficients and their

Applications. Riga, 1991. Manuscript (Russian).



MATHEMATICAL STRUCTURES I(Matemātiskās struktūras I)

Course code Mate-5245Authors Docent J. Buls, Dr. Math.; Docent A. Liepins, Dr. Math.Credits 4 creditsRequired for grade exam

Annotation. The course is intended as inter-discipline exposition. The first part deals with algebraic structures as self-contained branch but the second one demonstrates how it works in other mathematics. There are emphasised the three main structures: groups, rings and modules. So, the first part designed as basic standard course of abstract algebra.

Subjects1. Algebraic systems.2. Groups. Cyclic groups. Centre of group, centralise of element, commutators. Lagrange’s

theorem. Conjugacy classes, class equation. Normal subgroups, Noether’s isomorphism theorems. Correspondence theorem. Inner and other authomorphisms. Permutation representations, Cayley’s and Chauchy’s theorems. Groups acting on sets, orbit of element, stabiliser. Sylow’s theorems. Direct and semi-direct products. Extensions. Sequences of homomorphisms. Groups of low order.

3. Rings. Integral domains, division rings, fields. Quadratic fields, quaternion algebra’s, matrix algebra’s, algebra’s of polynomials, group rings. Quotient fields and localisation. Principal ideal domains, Euclidean domains, unique factorisation domains.

4. Modules and vector spaces, Direct products and sums. Finitely generated and cyclic modules. The rank of module. An annihilator, torsion element and module. Exact sequences. Free and projective modules. Free modules over principal ideal domains, free rank. Finitely generated modules over principal ideal domains.

5. Group representations. Definitions and basic properties. Maschke’s theorem and Schur’s lemma. Representations of abelian groups.

Requirements to received of credits.64 hours lectures and seminars. The exam in an oral form.

Recommended literature.1. W. A. Adkins, S. H. Weintraub. Algebra. An Approach via Module Theory. Springer-Verlag,

New York, Berlin, Heidelberg, London, Paris, Tokyo, Hong Hong, Barcelona, Budapest. 1992.

2. S. Lang. Algebra. Addison-Wesley, Mass. 1965.3. B. L. van der Warden. Algebra I. Achte Auflange der Moderner Algebra. Springer-Verlag.

Berlin, Heidelberg, New York.1971.4. B. L. van der Warden. Algebra II. Fünfte Auflange. Springer-Verlag. Berlin, Heidelberg,

New York.1967.5. Л. А. Скорняков. Элементы алгебры. “Наука”, Москва. 1986.



MATHEMATICAL STRUCTURES II(Matemātiskās struktūras II)

Course code Mate-5246Authors Docent A. Liepins, Dr. Math.; Docent J. Buls, Dr. Math.; Credits 4 creditsRequired for grade exam

Annotation. The course has interdisciplinary character: connection between different directions in mathematics is underlined on higher level of abstraction in comparison with undergraduate student’s programs having features of mathematics as human investigatory science to be the main objective. The course continue “Mathematical Structures I” integrating algebraic and topological (probabilistic) structures.

Subjects1. Topological structures.2. Convergence structures.3. Metric structures.4. Hilbert spaces. Banach spaces. Locally convex topological vector spaces.5. Topologies on space of linear continuous operators.6. Completeness and compactness. Completions and compactifications.7. Measurable structures.8. Probabilistic structures.9. Linearity and non-linearity in mathematics.10. Convex analysis.

Requirements to receiving of credits.64 hours lectures and seminars. The exam in an oral form. Student must show understanding the main objectives.

Recommended literature.1. Дж.Келли. Общая топология. Москва, Наука, 1968.2. A. Робертсон, В.Робертсон. Топологические векторные пространства. Москва, Мир,

1967 3. Дж.Окстоби. Мера и категория. Москва, Мир, 1974 4. П.-Ж. Лоран. Оптимизация и апроксимизация. Москва, Мир, 1975 5. Ж.-П. Обен, И.Экланд. Прикладной нелинейный анализ. Москва, Мир, 1988



MATHEMATICAL THEORY OF MARKET EQUILIBRIUM(Ekonomiskā līdzsvara matemātiskā teorija)

Course code Mate-0224Authors Docents A. Liepiņš, Dr. Math. Credits 4 creditsRequired for grade examPrerequisites Mate-3084

Annotation. Conditions sufficient to establish market equilibrium (using most popular mathematical models of economy) are analysed.

Subjects:1. Correspondences and continuity.2. Characteristic functions of sets: properties, duality.3. Valras’ conditions.4. Ky Fan’s inequality and connected results.5. Debreu-Gale-Nikaido theorem.6. Tangential condition.7. The basic theorem of solvability for correspondences.8. The main mathematical models of economy (due to Valras, von Neuman etc.) and market

equilibrium.

Requirements for received of credits.1. Have all individual or team tasks performed positively assessed.2. Have positive assessment in the test.

Recommended literature.1. J.-P. Aubin. Lànalyse non linèare et ser motirations èconomiques. Masson, Paris, 1984.2. S. Fischer, R. Dornbusch, R. Schmalensee.Economics. W.W. Norton & Company, 1994.



MATHEMATICAL THEORY OF MARKET EQUILIBRIUM(Ekonomiskā līdzsvara matemātiskā teorija)

Course code Mate-0224Authors Docents A. Liepins, Dr. Math. Credits 4 creditsRequired for grade examPrerequisites Mate-3084

Annotation. Conditions sufficient to establish market equivalibrium (using most popular mathematical models of economy) are analysed.

Course contents:1. Correspondences and continuity.2. Characteristic functions of sets: properties, duality.3. Valras’ conditions.4. Ky Fan’s inequality and connected results.5. Debreu-Gale-Nikaido theorem.6. Tangencial condition.7. The basic theorem of solvability for correspondences.8. The main mathematical models of economy (due to Valras, von Neuman etc.) and market

equvalibrium.

Requirements to receive credits.1. Have all individual or team tasks performed positively assessed.2. Have positive assessment in the test.

Recommended literature.1. J.-P. Aubin. Lànalyse non linèare et ser motirations èconomiques. Masson, Paris, 1984.2. S. Fischer, R. Dornbusch, R. Schmalensee.Economics. W.W. Norton & Company, 1994.



MATHEMATICAL THINKING PSYCHOLOGY(Matemātiskās domāšanas psiholoģija)

Course code Mate-0239Authors docent Daina Taimina, Dr. mat.Credits 2 creditsRequired for grade testPrerequisites Mathematical education or practical work experience in mathematics

teaching and knowledge in English are desirable.

Annotation. This course contemplates the aspect of mathematical thinking psychology. The course is in seminar form, where each participant is active. Shortly contemplate the history of mathematical psychology, the psychological aspect of different strategies of solving mathematical exercises, mathematical thinking as a process, creation psychology of mathematics, the social and formal aspects of mathematical proof, games in mathematics teaching.

Subjects:1. The birth of mathematical thinking psychology.2. Deductive and inductive mathematical thinking.3. Creation of mathematics.4. Mathematical proof.5. The role of the definition in the mathematics teaching and learning.6. Difficulty in learning mathematics.7. Games in mathematics teaching.8. Research in mathematical thinking psychology.9. Work with Internet for acquiring information about mathematical education and questions on

mathematics psychology.

Requirements for receiving of credits: 32 hours lectures and 1 test work.Active participating in the seminar studies, handed in and read out the final report about the picked theme, where is shown as theoretical knowledge, as analyse of personal experience.

Recommended literature:1. G.Birkhofs, Matemātika un psiholoģija, Maskava, 1977.(krievu val.)2. J.Hadmar, The psychology of invention in the mathematical field.3. R.E.Mayer, Thinking, problem solving, cognition, W.H. Freeman & Co, New York, 1954.4. J.Mason, L.Burton, K.Stacey, Thinking mathematically, Addison-Wesely,1992.



MODERN ELEMENTARY GEOMETRY(Modernā elementārā ģeometrija)

Course code Mate-5247Authors Lecturer L.Ramāna, Mg.Math.Credits 4 creditsRequired for grade examPrerequisites none

Annotation. The objectiveness of the course is general methods of solving geometrical problems, formal tools suitable for it, some classical results of elementary geometry.

Subjects:1. Uses of geometrical transformations in problem solving: rotation, homothety, parallel shift,

symmetry, and inversion.2. Methods of proving of geometrical inequalities.3. Radical axis and radical centre, a degree of a point.4. The method of correspondence between qualitative and qualitative assertions.5. Incidence theorem.6. Vectors in elementary geometry.

Requirements for received of credits: 64 hours lectures. Students are required to work regularly on problem solving at home. The exam is in a written form and consists of problem solving.

Recommended literature:1. E.Fogels, E.Lejnieks “Trijstūra ģeometrija”, Rīga, 1993.2. Howard Eves, Fundamentals of Modern Elementary Geometry, Boston, London, 1992.3. LIIS materiāli matemātiskajā izglītībā.



MULTIDIMENSIONAL STATISTICAL ANALYSIS(Daudzdimensionālā statistiskā analīze)

Course code Mate-5219Authors Prof. A.Lorencs, Dr.Hab.Math.Credits 4 creditsRequired for grade test or examPrerequisities: Mate-3030, Mate-2036

Annotation: This course considers statistical models for random type phenomena and processes for representation of which multivariate property vectors are used. Descriptions of multivariate individuals and phenomena are often found in sociological and biological research, and medicine, economics are not exceptions either. Therefore it should be considered an important part of studies – to introduce students to multivariate distribution types and their properties and to respective methods of statistical analysis.

Subjects: 1. Multivariate discrete distributions:

1.1. multinomial distributions;1.2. multivariate hipergeometric distributions.

Covariance matrices and characteristic functions of the distributions. The distributions as modules of natural statistical experiments. Application of the distributions in random phenomena analysis.2. Absolutely continuos multivariate distributions:

2.1. multivariate exponential distributions;2.2. multivariate normal distributions.

Covariance matrices and characteristic functions of the distributions. Real phenomena, which can be described by these distributions.3. Transformations of random vectors, calculation of distributions of the transformed random

vectors; normally distributed random vectors and their linear transformations.4. Design of a non-normal distributed random vector, the components of which are normal

distributed. Design of a normally distributed random vector, the components of which are dependent, normally distributed and uncorrelated.

5. Notion of marginal distribution. Marginal distributions of multivariate distributions. Conditional distributions, regression, homoscedasticity of a regression in a case of normal distribution.

6. Point estimates and regional estimates of multivariate distributions parameters; maximum likelihood method.

Adequacy testing of multivariate distributions; 2 test application for multivariate distributions; testing of statistical hypothesis for the mean value of normally distributed vector:a) when the covariance matrix is known;b) when the covariance matrix is unknown.7. T2-statistics, homogeneity testing of two normal distributed populations.8. Classification of observations, criteria of classifier optimality (Bayesian and mini-max

classifiers), Bayesian classifier design for two normally distributed populations, the distance



of Mahalanobis, the wrong probability of Bayesian classifier.Requirements for receiving of credits: 32 hours lectures. Students have to be able: - to design appropriate statistical model using the given statistical data (observations of the

population under consideration), i.e. appropriate multivariate distribution for the given population;

- to test statistical model using data of new observations;- to obtain point or regional estimates for parameters of the model;- to test the hypothesis of two populations mean values equality, using the given

observations data, if the covariance matrices can be considered being equal;- to design Bayesian classifier for a pair of normally distributed categories with equal

covariance matrices;- to calculate probability of classifier error.

Recommended literature:1. Y. L. Tong, The multivariate normal distribution, Springer, Berlin, Heidelberg etc. 1990.2. H. T. Nguyen, G. S. Rodgers, Fundamentals of mathematical statistics, Part 2. Statistical

Interface, Springer, Berlin, Heidelberg etc. 1989.3. L. Fahrmeir, G. Tutz, Multivariate statistical modelling based on generalised linear models,

Springer, Berlin, Heidelberg etc. 1996.



NON-LINEAR BOUNDARY VALUE PROBLEMS (Nelineārās robežproblēmas)

Course code Mate - 5248Author Docent J.Cepītis, Dr.Math.Credits 2 credits Required for grade examPrerequisites

Annotation. The course considers investigation of non-linear boundary value problems using method of a priori estimates.

Subjects:1. Lower and upper solutions, a priori estimate of solution.2. Bernstein’s, Shrader’s and Nagumo’s conditions.3. Cutset of Shrader’s condition using diagonals.4. Existence implied by uniqueness.5. Uniqueness theorem for solution having a priori estimate.6. Lower and upper solutions with break of derivative. Generalised solvability.7. Generalised two-point boundary value problem.8. Boundary value problems for equations with non-summable singularities.

Requirements for received of credits: 32 hours lectures. 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. Н.И.Васильев, Ю.А.Клоков, Основы теории краевых задач, Рига, Зинатне, 1978.3. Ц.На, Методы численного решения прикладных краевых задач, Москва, 1982.



NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATION(Skaitliskās metodes parastajiem diferenciālvienādojumiem )

Course code Mate-5260Authors Prof. H. Kalis, Dr. Hab. Math., Dr. Hab. Phys.Credits 3 creditsRequired for grade examPrerequisites Mate-3139

Annotation. The objectives of the course are: different methods for solving ordinary differential equations (ODE) are considered for joint point of view finite-difference schemes: the foundation for numerical methods, the order of approximation, stability and convergence (precision); special numerical methods, monotone difference schemes and effective algorithm. This course is made in compliance with program.

Subjects:1. Continuous and discrete Cauchy problem.2. Approximation, convergence and stability of finite-difference scheme.3. Multilevel, finite-difference scheme, stability, Schur’s criterion.4. Linear difference equations with constant coefficients, analytic solution.5. Properties and calculation of matrix-functions.6. The foundation of simple and modified Euler’s methods.7. The foundation of Euler-Cauchy method. 8. Difference scheme for Adam’s method, the method of undefined coefficients.9. Algorithm of Runge-Kutta methods.10. Reduction of the boundary-value problem to the Cauchy problem.11. Approximation of boundary-value problem with finite-difference schemes.12. Monotone finite-difference schemes.13. Method of finite volume, exact difference scheme.14. Variational type finite-difference schemes.15. Eigenvalues and eigenfunctions problem (continuous and discrete analogues).

Requirements for receiving of credits: 32 hours lectures, 16 hours practical work.Students are required to fulfil 3 independent laboratory works. The exam takes place in an oral form.

Recommended literature:1. H.Kalis. Parasto diferenciālvienādojumu skaitliskās metodes. Rīga, LU, 1996.2. H.Kalis. Diferenciālvienādojumu tuvinātās risināšanas metodes. Rīga, “Zvaigzne”, 1986.3. E.Hairer, G.Wanner. Solving ordinary differential equations, I, II, Springer, 1997.4. G.Engeln-Mulges, F.Uhlig. Numerical algorithm with Fortran. Springer, 1996.



NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS(Skaitliskās metodes parciālajiem diferenciālvienādojumiem )

Course code Mate-5261Authors Prof. H. Kalis Dr. Hab. Math., Dr. Hab. Phys.Credits 3 creditsRequired for grade examPrerequisites Mate-3143, Mate-3139

Annotation. The objectives of the course are: different methods for solving partial differential equations (PDE) are considered for joint point of view finite-difference schemes: the order of approximation, stability and convergence (precision); special numerical methods, monotone difference schemes. This course is made in compliance with ECMI program.

Subjects:1. Approximation of the partial different operators with finite-difference, computational

molecule.2. Approximation of the boundary conditions.3. Discrete Dirichlet problem in rectangle, stability and approximation.4. Eigenvalues and eigenfunctions problem for non-adjoint differential operators of the second

order, method of lines.5. A.Ilhìn monotone difference scheme.6. Method for solutions of discrete boundary-value problem of elliptic type PDE.7. Monotone approximation for heat transfer equation with convective term.8. Solving of heat transfer equation in multilayered media.9. Approximation, stability and convergence of two finite-difference schemes.10. Stability of three level difference schemes.11. Comparison analysis for grid methods, method of lines.12. Initial-boundary value problem for hyperbolic type PDE.13. Stability for finite-difference schemes: Fourier method, matrix stability analysis,

maximum principle, fon Neiman criterion, Gerschgorin`s circles.14. Finite-elements method.15. Boundary-elements method.

Requirements for received of credits: 32 hours lectures, 16 hours practical work.Students are required to fulfil 3 independent laboratory works. The exam takes place in an oral form.Recommended literature:1. H.Kalis. Diferenciālvienādojumu tuvinātās risināšanas metodes. Rīga, “Zvaigzne”, 1986.2. H.Kalis. Nepārtraukto un diskrēto matemātiskās fizikas problēmu analītiskie atrisinājumi.

Rīga, LU, 1992.3. H.Kalis. Speciālu skaitlisko metožu izstrāde un lietošana matemātiskās fizikas,

hidrodinamikas un magnētiskās hidrodinamikas problēmu risināšanā. Rīga, LU, 1973.4. J.W.Thomas. Numerical partial differential equations: finite difference methods. Springer,

1995. 5. W.F.Ames. Numerical methods for partial differential equations. Acad. Press, New-York,



1977.



PERTURBATION ANALYSIS(Perturbāciju analīze)


Annotation. The course considers the continuous dependence of solutions of equations on parameters, including functional perturbations. The aim of the course is to give an insight to what small changes of the mathematical model do not cause serious changes in the behaviour of the corresponding solutions.

Subjects:1. Linear algebraic systems, eigen-values and eigen-vectors.2. Perturbations of non-linear equations in Euclidean spaces.3. Perturbations of the Cauchy's problem for ordinary differential equations.4. Asymptotically stability of solutions of ordinary differential equations.5. Stability of critical points of functions in Euclidean spaces.6. Perturbations of integrands in calculus of variations.7. Perturbations of partial differential operators.

Requirements for receiving of credits: 32 hours lectures. The students have to do four independent home works related to four different theoretical questions. The exam takes place in an oral form. The students have to show knowledge and an understanding of the impact of various perturbations of equations to the behaviour of the corresponding solutions.

Recommended literature:1. Hartman, P., Ordinary Differential Equations, John Wiley&Sons, New York, 1964; or Russian

translation, Mir, Moscow, 1970.2. Bahvalov, A.N., Numerical Methods 1, Nauka, Moscow, 1973 (in Russian)3. Tikhonov, A.N., Samarskii, A.A., Equations of Mathematical Physics, Nauka, Moscow, 1977

(in Russian).



PRACTICAL CLASSES OF MATHEMATICAL MODELLING WITH COMPUTER PROGRAM PACKAGES I

(Matemātikas modelēšanas praktikums ar datorprogrammu paketēm I )

Course code Mate-0241Authors Prof. H. Kalis Dr. Hab. Math., Dr. Hab. Phys.Credits 2 creditsRequired for grade testPrerequisites Mate-2065, Mate-3139

Annotation. “Mathematica” is a programming language of many paradigms. This means that the user can write “Mathematica” programs in a variety of programming styles, from the traditional procedural style to the less conventional (but usually more efficient and elegant) rule-based and functional programming styles. The program packages “Mathematica” elements, graphics, programming, algebraic equations, functions, series, integration and differentiation, Cauchy problems of differential equations and others.

Subjects:1. Elements of program package “Mathematica”: symbols and constants, fixed functions,

arithmetic operators, real and complex variables.2. List of operators for forming expressions. Functions and functions-procedure, operators with

functions. Name, strings, arrays and tables, sets and lists.3. Graphics forming. The commands “Plot”, “Plot3D”. “Parametric Plot3D”, “ListPlot”,

“ContourPlot”, “DEnsityPlot”. The graphics packages: “Graphics”, “Animation”, “ImplicitPlot”.

4. Graphics primitives for forming 2-D and 3-D graphics. The graphics packages: “Polyhedra”, “Legend”.

5. Programming elements, Basic programming constructs: If [cond, exp1, exp2]; Which [test1, func1, func2,...]; Do [expr{i, imax}]; For [start, test, step, expr], While [test, expr]; Nest [func, expr, n]; Fixed Point [func, expr].

6. Operators with polynomials and rational expressions. Expression manipulation: commands “Expand”, “Simplify”, “Factor”.

7. Solving equations and system of equations: symbolical and numerical computations. The commands “Solve’, “Find Root”, “Eliminate”, “NSolve”.

8. Linear algebra. Arrays, vectors and matrixes. The “Linear Solve” command.9. Summations and product of expressions.10. The elements of geometry and analytic geometry.11. Computing the limiting value of expressions (command “Limit”). Differential and

integration of functions (commands “Integrate” and “D”).12. Numerical integration with command “NIntegrate”, multiple and line integrals.13. Computations of series from functions (commands “Series” and “InverseSeries”.14. General solutions of ordinary differential equations “ODE); command “DSolve” and

package “Calculus `DSolve`”. The method of constant-variations for nohomogeneous ODE.15. Cauchy problem for ODE. Graphics of solutions.16. Solutions of ODE by using series and integral transformations. The packages



“Algebra`Trigonometry`” and “Calculus`LaplaceTransform`”.

Requirements for receiving of credits: 8 hours lectures, 24 hours practical work.Students are required to fulfil 3 independent laboratory works using modern PC .

Recommended literature:1. S.Wolfram. Mathematica. Add.- Wesley publ. comp., 1988.2. T.B.Bahder. Mathematica for scientists and engineers. Add. - Wesley publ. comp. , 1991.3. H.R.Varian. Computational economics and finance. Modeling and analysis with

“Mathematica”. Springer-Verlag, New York, Inc., 1996.4. H.Kalis, S.Lācis, O.Lietuvietis, I.Pagodkina. Programmu paketes “Mathematica” lietošana

mācību procesā. Mācību grāmata, Rīga, 1997.



PRACTICAL CLASSES OF MATHEMATICAL MODELLING WITH COMPUTER-PROGRAM PACKAGES II

(Matemātiskās modelēšanas praktikums ar datorprogrammu paketēm II )

Course code Mate-0242Author Prof. H. Kalis Dr. Hab. Math., Dr. Hab. Phys.Credits 2 creditsRequired for grade testPrerequisites Mate-2065, Mate-3139

Annotation. “Mathematica” is a programming language for symbolical and numerical calculations. In this course are considered: numerical methods with “Mathematica”, interpolation and approximation of functions, numerical solutions of ordinary differential equations (ODE), functions of complex variables, elements of mathematical physics and others.

Subjects:1. Numerical methods and compute program package “Mathematica”, operators with real and

complex variables.2. Numerical solutions of algebraic and transcendental equations. The commands “Nhoots”,

“NSolve”, “FindRoot”.3. Interpolation and approximation of funkctions. The commands “InterpolatingPolynomial”,

“Interpolation”, “Fit”.4. Numerical differentiating and integrating. The commands “NIntegrate” and “D”.5. Solutions of Cauchy problem for ODE and system of ODE, “NDSolve”.6. Reductions the boundary-value problem to the Cauchy problem. 7. Solutions of boundary value problem of ODE by using finite-difference method. The

function-procedure with command “Module”.8. Optimisation methods, calculations of extreme for function: “FindMinimum”, “Grad”,

“Hess”, “ContrainedMin”, “ContrainedMax”. “LinearProgramming”.9. Functions of complex variables. The commands “Re”, “Im”, “Conjugate”, “Abs”, “Arg”,

“ComplexExpand”, “Residue”. Special functions and integrals.10. Elements of mathematical physics, analyses of vector fields, tensors: “Div”, “Curl”,

“Laplacian”. The packages “PlotField” and “PlotField3D”.11. The equation of string oscillations, the initial-boundary value problem.12. The heat transfer equation, initial-boundary value problem.13. Methods of lines for solutions initial-boundary value problem.14. Furier method in mathematical physics.15. Modelling of Braun`s movement.16. The elements of statistic, different probability distributions.

Requirements for received of credits: 8 hours lectures, 24 hours practical work. Students are required to fulfil 3 independent laboratory works using modern PC. Recommended literature:1. S.Wolfram. Mathematica. Add.-Wesley publ. comp., 1988.2. T.B.Bahder. Mathematica for scientists and engineers. Add.-Wesley publ. comp., 1991.3. H.Kalis, S.Lācis, O.Lietuvietis, I.Pagodkina. Programmu paketes “Mathematica” lietošana



mācību procesā. Mācību grāmata, Rīga, 1997.



PRACTICAL CLASSES OF MATHEMATICAL MODELLING WITH COMPUTER-PROGRAM PACKAGES III

(Matemātikas modelēšanas praktikums ar datorprogrammu paketēm III )

Course code Mate-0243Author Prof. H. Kalis, Dr. Hab. Math., Dr. Hab. Phys.Credits 2 creditsRequired for grade testPrerequisites Mate-2065, Mate-3139

Annotation. “Maple” is a language for symbolic mathematical calculations. “Maple” is a computer environment for doing mathematics. Symbolical, numerical and graphical computations can all be done with “Maple’. In this “Maple” course are considered: programming and system commands, computer graphics, calculus, linear algebra, solving equations, polynomials an common transforms, geometry and others. Subjects:1. Elements of computer program package “Maple”. Forming of tables and lists.2. Plotting functions. Command listing. The graphical package “plots”, “plottools”. 3. Programming elements. Basic programming constructs. Procedures.4. Polynomials and common transforms. Standard manipulations. Command list. Expression

manipulation: commands “convert”, “expand”, simplify”, “subs”. 5. Solving equations and system of equations: symbolical and numerical computations. The

commands “solve”, “isolve”, “msolve”, “fsolve” and “assign”.6. Linear algebra. Arrays, vectors and matrixes, special types of arrays, manipulating elements

of arrays. The “linalg” package. Command listing.7. Summation and product of expressions. The “sumtools” package.8. The elements of geometry and analytic geometry. The “geometry” package. 9. Calculus. Limiting value of expressions (command “limit”), differentiating and integrating of

expressions (commands “diff” and “int”). The “student” package.10. Computations of the truncated generalised series of functions, Taylor series (commands

“series” and “taylor”). The “powseries” package.11. The analytical solutions of ordinary differential equation (ODE), “dsolve” command.

Cauchy problem. The “DEtools” package.12. Solutions of ODE with series and integral transformations, “inttrans” package.13. Interpolation and approximation of functions, the commands “interp”, “maximise”,

“minimise”, “spline”; the “numapprox” package.14. Numerical solutions of ODE with option “type=numeric”, numerical methods .15. Functions of complex variables, elements of statistics. The “statt” package, statistical lists

and plots. Random number. Different probability distributions.16. Partial differential equations, command “pdsolve”, the package “PDEplot”.

Requirements for received of credits: 8 hours lectures, 24 hours practical work. Students are required to fulfil 3 independent laboratory works using modern PC.

Recommended literature:1. D.Redfern. The Maple Handbook. Maple V Release 4. Springer, 1996.2. C.T.J.Dodson, E.A.Gonzalez. Experiments in mathematics using Maple.Springer, 1995. 3. H.Kalis, R.Millere. Datorprogrammas “Maple” lietošana mācību procesā. Internetā, LIIS-



projekts.



PRACTICE IN OLYMPIAD PROBLEMS SOLVING(Olimpiāžu uzdevumu risināšanas praktikums )

Course code Mate-0250Authors Professor A.Andžāns., Dr.habil.Math.Credits 4 creditsRequired for grade examPrerequisites none

Annotation. The objectiveness of the course: skills in high-level problem solving and composing recent advances in mathematical contents.

Subjects:1. Extreme problem solving methods.2. Methods of proofs of inequalities.3. Methods of solving equations and their systems.4. Uses of complex numbers in various areas of elementary mathematics.5. Olympiad problem sets and trends in Latvia in the world.

Requirements for receiving of credits: 32 hours lectures and 32 hours practical work (problem solving). Students are required to fulfil one independent homework and regularly work at home or problem solving. The exam is in a written form and consists of problem solving.

Recommended literature:1. A.Vasiļevska, L.Ramāna. Ekstrēmu uzdevumu risināšanas metodes. Rīga, 1997.2. A.Andžāns, A.Ločmele, I.Palma, L.Ramāna, T.Larfelds. nevienādību pierādīšanas metodes.3. A.Andrejeva-Andersone, L.Ramāna, A.Andžāns. Vienādojuma sistēmu risināšanas metodes.

Aizkraukle, 19974. И.М.Яглом.Комплексные числа. М.,19685. Raksti žurnālā “Математика” 1979-1997 (Bulgārija)6. Kārtējo matemātikas olimpiāžu materiāli.7. LU A.Liepas NMS uzdevumu kartotēka.8. A.M. Schoenfeld. Mathematical Problem Solving. AP, 1992.



RISK THEORY(Riska teorija)

Course Code Mate - 5258Author Assoc. Prof. Jānis Vucans, Dr math.;

Lecturer Kristine Lomanovska, Mag.math.Credits 4 creditsRequired for grade examPrerequisites Mate – 3030

Annotation: The principal aim of this course is to acquaint students with the foundations of Risk Theory and with its applications in the practical spheres of economics and finances – in the insurance and other fields.

Subjects:

1. Introduction. The goal of Risk Theory. Different approaches: theoretical and practical. 2. Repetition about the stochastic processes. Foundations of Markov processes; Poisson process

and compounded Poisson process.3. Basic elements of Risk Theory. The probability of destruction and methods of its calculation.

The approximation of probability of destruction (formulation of the problem). 4. Applications of Risk Theory in the insurance. Process related to accidents. Distribution to the

total number of accidents. Estimation of the distribution to the total number of accidents. Compounded Poisson process. Normal approximation with step-wise functions. Edgeworth series. Equilibrium reserve as the defensive mean against the risk. Methods for calculation of equilibrium reserve limits.

5. Simulation as the method of risk analysis. Monte-Carlo method. Generation of random numbers. The approximate formula of probability of destruction. Wilson-Hilferty simulation. General model of simulation. The simulation of total global loses.

Requirements for receiving credits: 32 hours lectures, 32 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 Risk Theory 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 Risk Theory, explained during the lectures, and the ability to use them to formulate and to solve different problems – for example, from the insurance practice.

Recommended literature: 1. Buhlmann, Mathematical Methods in Risk Theory, Springer-Verlag, 1970.2. Beard, Pentikainen, Pesonen, Risk Theory, II edition, 1987.3. Daykin, Pentikainen, Pesonen, Practical Risk theory for actuaries, Chapman & Hall, 1993. 4. Pentikainen et. al. Insurance solvency and financial strength, Helsinki, 1989.



SELECTED CHAPTERS OF COMPUTER APPLICATION PROGRAMS(Lietišķo programmu izvēlētas nodaļas)

Course code Mate-0232Author Mg. Comp., lect. Viesturs VezisCredits: 4Required for grade testPrerequisitesAnnotation:The goal of the course is to flatten the level of computer application usage skills of trainees with different educational background. The course provides training to gain the skills to be able to use INTERNET and the most popular office programs towards creating and developing teaching materials and aids. The content of the course provides possibility to be dynamically changed according to different background of trainees.Subjects1. Information entering, saving and deleting in computer. The concept of operational systems.

Computer viruses and protection against them. Archive programs, their usage.2. Microsoft Windows environment and principles of functioning. Window elements and types.

Microsoft Windows applications: running and closing. Windows Accessories (calculator, simple text and graphic editors etc.) Multimedia tools. Usage of program and printing managers. Adjusting Windows environment to users’ needs (taste).

3. Microsoft Windows environment and principles of functioning. Microsoft Windows applications: running and closing. Windows Accessories. Multimedia tools. Usage of program and printing managers. Windows environment adjusting to users’ needs (taste).

4. 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 accesses levels: comparison of speed and other parameters. Legal restrictions and etiquette. Potential of Internet services usage.

5. Introduction to word processing (formatting) systems. Possibilities of using word processor Microsoft Word in teaching. (in-depth study).

6. Spreadsheets. Usage of Microsoft Excel in solving mathematical problems. (in-depth study).7. Introduction to databases. Possibilities of database usage.Requirements to received of credits:1. Completion of all practical exercises.2. Two projects have to be worked out:

2.1. Creation and formatting of one lesson (topic) conspectus.2.2. 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. Microsoft® Windows® 95 Step by Step Catapult, Inc5. Microsoft® Office 97 Professional 6-in-1 Step by Step Perspection, Inc.6. Microsoft® Word 97 Step by Step Catapult, Inc.7. Microsoft® Excel 97 Step by Step Catapult, Inc



SELECTED CHAPTERS OF NUMBER THEORY(Skaitļu teorijas izvēlētās nodaļas)

Course code Mate-5267Authors Docent A.Bērziņš, Dr.Math.Credits 4 creditsRequired for grade testPrerequisites Algebra course from mathematics bachelor program

Annotation. The objectiveness of the course is continued fractions, Diofantine equations, consequences, prime distribution, and fields of algebraic numbers.Subjects:1. Euclidean algorithm and theory of continued fractions.2. Linear consequences and Chinese theorem.3. Second-order consequence. Lagrange and Jacobi symbols.4. Diofantine equations. Hilbert 10th problem.5. The concept of residue class. Primitive root and index.6. Prime distribution problem. Dirichlet theorem.7. Fields of algebraic numbers. Theorem on primitive element.

Requirements for receiving of credits: 64 hours lectures. Students are required to write two papers or problem solving. The exam is in a written form and consists of two theoretical questions.

Recommended literature:1. Š. Mihelovičs. Skaitļu teorija. Daugavpils. 1996.2. И.М.Виноградов. основы теории чисел. Мockвa, 1981.



SOLVABILITY OF NONLINEAR EQUATIONS(Nelineāru vienādojumu atrisināmība)


Annotation. The course considers basic methods for obtaining the existence of solutions of non-linear equations. The exposition is mainly in Euclidean spaces.

Subjects:1. Bohl-Brouwer's theorem.2. Method of monotone operators.3. Method of potential operators.4. Topological degree.5. Schauder's principle.

Requirements for received of credits: 32 hours lectures. The exam takes place in an oral form. The students have to demonstrate full proofs.

Recommended literature:1. Raitums, U., Solvability of Nonlinear Equations, Teaching material, University of Latvia,

Riga, 1993 (in Latvian).2. Gajewski, H., Gröger, K., Zacharias, K., Nonlinear Operator Equations and Differential

Operator Equations, Mir, Moscow, 1978 (in Russian).



SPECIAL FINITE-DIFFERENCE METHODS(Speciālās galīgo diferenču metodes )

Course code Mate-5262Authors Prof. H. Kalis, Dr. Hab. Math., Dr. Hab. Phys.Credits 2 creditsRequired for grade examPrerequisites Mate-3139

Annotation. The presence of large parameters at first order derivatives or small parameters at second order derivatives in the system of differential equations cause additional difficulties for the application of general methods which become ineffective (small speed of convergence, low precision). Thus a topical task is to work out special finite-difference methods of solution (see works of Doolan, Miller, Allen, Sauthwell, Ilh`yn and others).

Subjects:1. Problems of mathematical physics with large (small) parameters, 2. Necessity of application of special finite-difference methods.3. Problems of mathematical physics with discontinuous coefficients in multi-layered domains. 4. Special finite-difference schemes for solution of Cauchy problem of ordinary differential

equations (ODEs).5. Special finite-difference schemes for solution of boundary-value problem of ODEs. 6. Special finite-difference schemes for solution of boundary value problem of elliptic type

partial differential equation. 7. Special finite-difference schemes for solution of boundary-initial value problem of heat

transfer and string oscillator equations.8. The equations of hydrodynamics and magnetohydrodynamics.

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. Speciālās diferenču shēmas matemātiskās fizikas problēmu risināšanai. Rīga, LU,

1991.2. H.Kalis. Diferenciālvienādojumu tuvinātās risināšanas metodes. Rīga, “Zvaigzne”, 1986.3. H.Kalis. Speciālu skaitlisko metožu izstrāde un lietošana matemātiskās fizikas,

hidrodinamikas un magnētiskās hidrodinamikas problēmu risināšanā. Rīga, LU, 1993. 4. H.Kalis. Ausarbeitung und Anwendung der spezialen numerischen Methoden zur Losung der

Probleme der mathematischen Physik, Hydrodynamik und Magnetohydrodynamik. Rîga, LU, 1993.

5. J.W.Thomas. Numerical partial differential equations: finite difference methods. Springer, 1995.



SPECIAL SECTIONS OF CALCULUS.(Matemātiskās analīzes speciālas nodaļas.)

Course code Mate-5238Authors Prof. T.Cīrulis, Dr.H.Math.Credits 4 creditsRequired for grade examPrerequisites Mate-2065, Mate-1002, Mate-1011, Mate-3015

Annotation.The objectives of course are methods of the complex calculus, integral transformations,

the most important special functions and their applications.

Subjects:1. Methods of calculus for representation of functions. Asymptotically and numerical methods

for the calculation of functions.2. Methods with the complex variable for calculus: analytic continuations, singular points,

applying of residue theory, meromorphic and integer functions.3. Integral transformations and their applications. 4. Special functions of mathematical physics. 5. Comform mapping and there applications.

Requirements for received of credits: 48 hours lectures, 16 hours practical work. The examination takes place in an oral form. Students must to show an understanding of methods for the solving of problems in the calculus.

Recommended literature:1. T.Cīrulis, Dz.Damberga. Kompleksā mainīgā funkciju teorijas lietojumi. Rīga, LU, 1992., 130

lpp.2. E.Riekstiņš. Matemātiskās fizikas metodes. Rīga, “Zvaigzne”, 1969., 629 lpp.3. М.А.Лаврентьев, Б.В.Шабат. Методы функций комплексного переменного. Москва,

Наука, 1978.4. Ф.Олвер. Асимптотические методы и специальные функции. Москва, Наука, 1978.



SPLINES AND THEIR APPLICATIONS(Splaini un to lietojumi)

Course code Mate-5264Author Docent Svetlana Asmuss, Dr.Math.Credits: 4Required for grade testPrerequisites Mate-1002, Mate-2065

Subjects1. Splines

1.1. Historical survey. Advantage of splines compared with polynomials in the approximation field.

1.2. Polynomial splines. Degree and defect. Space of splines.2. Cubic splines.

2.1. Cubic splines with boundary conditions of the type I, II, III and IV. Hermite splines. 2.2. Cubic B-splines.2.3. Local approximation formulas.2.7. Numerical differentiation and integration by cubic splines.

3. High degree splines.3.1. Interpolating splines. Its construction.3.2. B-splines; basic properties and formulas. Basis of the space of splines.3.3. Numerical differentiation and integration by splines. 3.6. Local approximation formulas.

4. Natural splines.4.1. Interpolating natural splines.4.2. Extremal property of interpolating splines. Smoothing splines. 4.3. Quadrature formula based on natural splines. Its optimality.

5. Numerical solution of differential and integral equations by splines.5.1. Collocation method.5.2. Subinterval method.5.3. Finite element method.

6. Bivariate splines.6.1. Bivariate splines over a regular guid. 6.2. Bivariate splines over a chaotic guid.6.3. Bicubic splines. Methods of its construction.

Recommended literature1. De Boor C. A practical guide to splines. New-York, Springer, 1978.2. Schumaker L.L. Spline functions: basic theory. New-York, Wiley, 1981.3. Nurnberg G. Approximation by spline functions. Berlin, Springer, 1989.4. Завьялов Ю.С., Квасов Б.И., Мирошниченко В.Л. Методы сплайн - функций. Москва, Наука, 1980. 5. Малоземов В.Н. , Певный А.В., Полиномиальные сплайны. Ленинград, 1986.



STATISTICAL MODELLING(Statistiskā modelēšana)

Course code Mate-0168Authors Prof. A.Lorencs, Dr.Hab.Math.Credits 4 creditsRequired for grade testPrerequisites: Mate-2065, Mate-2132, Mate-1002, Mate-4192

Annotation: The objectives of the course are to introduce students to methods of reformulating analytical problems into statistical ones, i.e. how to transform complex deterministic computational tasks into statistical or quasi-statistical experimental procedures. At the same time it is planned to provide knowledge about random number generator design problems.

Subjects: 1. History of Monte-Carlo methods.2. Problems of uniformly distributed random number physical-technical generator design;

fundamentals of pseudo-random number generators theory.3. Transformation of uniform distribution into the chosen kind of distribution.4. Problems of pseudo-uniform distribution transformation.5. Monte-Carlo methods for calculating multiple integrals.6. Monte-Carlo methods for finding solutions of linear algebraic equations; Monte-Carlo

procedures for matrix inversion.7. Monte-Carlo methods for calculating Fredholm integral equations.8. Estimating of Quasi-Monte-Carlo methods precision.9. Pseudo-Monte-Carlo methods substantiation problems.

Requirements for receiving of credits: 32 hours lectures. Students have to be able: to transform analytical task into statistical experimental procedure, to transform a uniform distribution into distribution appropriate for resolution of the given problem. Students have to solve a concrete problem and describe the results of the experiment.

Recommended literature:1. M. H. Kalos, P. A .Whitelock, Monte-Carlo methods, Vol.1: Basics, Wiley, New York, 1986,

IXIV+186p.2. L. Kuipers, H. Neiderreiter, Uniform distribution of sequences, Wiley, New York, 1974,

XI+390p.3. H. Neiderreiter, Quasi-Monte-carlo methods and pseudo-random numbers, Bul. Amer. Math.

Soc., Vol.84, N6, 1987, 957-1041p.



STOCHASTIC PROCESSES.(Gadījuma procesi)

Course code Mate-0047Author Docent, Dr.Math. Viktorija CarkovaCredits credits/64 hoursRequired for grade examPrerequisites Mate-2032, Mate-5237.AnnotationThis course makes students familiar with the Markov chains, Poisson process, continuous-time Markov Chains, Queuing theory, Brownian motion. Stationary processes and their application.

Subjects4. Definition of the stochastic process.

4.1. Classification of the stochastic processes, properties.4.2. Examples.4.3. Separable stochastic process.4.4. Martingale property of independent variables’ sum.4.5. Process with independent increments.

5. Stochastic processes with discrete states.5.1. Markov chains.5.2. Classification of states5.3. Continuous-time Markov chain.5.4. The Kolmogorov differential equations.5.5. Ergodic theorem.. 5.6. Information and entropy. Definition, properties.

6. The Poisson process.6.1. Definition of the Poisson process. Properties.6.2. Erlang formulae.6.3. Simulation of the Poisson process.

7. Brownian Motion.7.1. Properties of the Brownian motion.7.2. Non-regular properties.7.3. Geometric Brownian motions.

8. Stationary random processes.8.1. Correlation function.8.2. Stochastic integral.8.3. Wide sense stationary process.

Requirements for received of credits. The examination for this course is a three-hour written-answer examination.

Recommended literature1.V.Carkova. D.Kalniņa. Stohastiskie procesi. LU, Rīga, 1981, (98 lpp.).2.S.M.Ross. Introduction to Probability Models. Fifth Edition, Acad..Press, NY, 1995.



SUPPLEMENTARY CHAPTERS OF MATHEMATICAL STATISTICS.(Matemātiskās statistikas papildnodaļas)

Course code Mate-5244Author Docent Viktorija Carkova, Dr. Math.Volume 4 credits Testing form examPrerequisites Mate-1060, Mate-1061, Mate-1119, Mate-1120, Mate-2036

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.

Subjects1. Fundamental of Mathematical Statistics (4hours lecture + 2 hours seminar).

1.1 The hypothesis.1.2 The lost function.1.3 The risk function.1.4 Test.1.5 The power function.

2. Neyman-Pearson fundamental lemma (2+2).3 . Most powerful test (2+2). 4. Baysian and Minimax procedures (2+4).5. The optimal tests constructed by Neyman-Pearson lemma (2+2).6. Graphical representation for the optimal tests (2+2).7. One-parametrical class with monotone likelihood ratio (2+2).8. One-parametrical exponential class (2+2).9. One-sided hypotheses. Confidence sets (2+2).10. Test of hypotheses for parameters of one-dimensional normal sample (2+2).11. Test of hypotheses for parameters of multidimensional normal sample (4+2).12. Invariance principle (2+2).13. Linear hypotheses (4+2).14. Test of statistical hypotheses and confidence interval construction examples (2+2).

Requirements for received of credits34 hours lectures+30 hours seminars. The exam takes place in oral form.

Recommended literature1. E.L. Lehmann, Testing Statistical Hypotheses, NY. John Wesley, 1959.2. J.Carkovs, Alternative statistical method, RTU, Riga, 1984, (Latvian).3. J.Carkovs, Simples hypotheses on parameters of normal distribution, RTU, Riga, 1986, (Latvian).4. D.RCox and D.V.Hinkley, Theoretical Statistics, Chapman&Hall, London, 1994. 5.V.Carkova, Mathematical Statistics, LU, Riga, 1979, (Latvian).



THE HISTORY OF LATVIAN SCHOOLS(Latvijas skolu vēsture)

Course code Mate-0235Authors Docent Daina Taimiņa, Dr. Math.Credits 2 creditsRequired for grade testPrerequisites none

Annotation. The objectives of the course are: This course contemplates questions of the history of Latvian schools. The course is in the seminary form, when each participant is active.

Subjects:1. Schools in Latvia during feudal period.2. Schools of Vidzeme in 17th and 18th century.3. Schools in Kurzeme in 19th century.4. The history of gymnasium in Latvia.5. “Jaunlatvieši” and education.6. Education in Latvia during 20th century first half.7. The Soviet period in Latvians education.8. Educational reforms after revival of Latvia’s independence in 1991.

Requirements for receiving of credits: 32 hours lectures.Active participating in the seminar studies, handed in and read out the final report about the

picked theme, where is shown as theoretical knowledge, as analyse of personal experience.

Recommended literature:1. A.Vias, Latviešu skolu vēsture, Vidzeme, R., 1992.2. A.Vias, Latviešu skolu vēsture, Kurzeme no 1800. - 1885.gadam, R., 1994.3. Pedagoģiskā doma Latvijā no 1890.g.līdz 1940.g., R., 1994.4. J. Greste “… kā dzeņa vēders.”R., 1977.



THEORY OF ALGORITHMS AND AUTOMATA(Algoritmu un automātu teorija)

Course code Mate - 0222Authors Professor A.Andžāns, Dr.Habil. Math.; Docent Ē.Ikaunieks, Dr.Math.Credits 4 creditsRequired for grade examPrerequisites none

Annotation. The objectiveness of the course is various formalisations of the concept of algorithm, their uses in the analysis and modelling of processes, mathematical analysis of gnoseological problems.

Subjects:1. The historical development of the concept of the algorithm.2. Various formalisations of the concept of algorithm and connections between them.3. Recursive and recursively innumerable sets, their examples. Rice theorem.4. Recursively innumerable sets. Post theorem. Universal sets and their examples.5. Various concept of reducibility. Mhyill theorem. 6. Post problem. Priority method, Fridbery-Muchink theorem.7. The concept of the finite automaton.8. Automata with reduced outer memory: counters, pushdowns.9. Automata in the analysis of plane mazes.10. Cellular structures and their uses.11. The concept of synthesis in the limit.12. Elements and basic results of mathematical learning theory.

Requirements for receiving of credits: 48 hours lectures, 16 hours problem solving. The exam is in an oral form. Students must be able to solve simple problems and to understand concepts and ideas of proofs.

Recommended literature:1.A. Lorencs. Automātu kontrole un dignostika. Rīga, Pētergailis, 1994.2.R.Vollmar. Algorithmen in Zellularautomaten. Stuttgart, 1979.3.Автоматы (сб.статей). Москва ИЛ, 1956.



TOPOLOGY III (Topoloģija III)

Course code Mate-5265Authors Prof. A. Šostaks, Dr. Habil. Math.Credits 4 creditsRequired for grade exam Prerequisities: Mate-2086, Mate-3183

Annotation. The course is continuation of Mate-2086 “Topology I" and Mate-3183 "Topology II". Basic topics considered in the course include Fundamentals of the theory of compactifications, important generalisations of compactness (countable compactness, pseudo-compactness, para-compactness), fundamentals of the theory of uniformity’s and proximity’s.

Subjects: 1. Compact spaces: basic properties and behaviour under operations and mappings. 2. Locally compact spaces. Alexandroff one point extension of a locally compact space..3. Compactifications: general theory. Comparison of compactifications. Minimal and maximal

compactifications.4. Important special compactifications: Čech-Stone compactification and its basic properties.

Wallman compactification. Freudental compactification.5. Compactness as an imbedding property. Absolutely closed spaces and H-closed spaces.6. Countable compact spaces. Spaces of ordinals. Real valued functions on ordinal spaces. 7. Pseudo-compactness. 8. Para-compact spaces. Metrication criterion of para-compact spaces. Stone's theorem. 9. Connected spaces. Product of connected spaces. Monotone mappings.10. Uniformity’s. General theory of uniform spaces. Topology induced by uniformity. Uniformly

continuous mappings.11. Criterion of metrizability of uniform spaces.12. Proximities. General theory of proximity spaces. Topology induced by a proximity.

Proximally continuous mappings. Proximity induced by a uniformity.

Requirements for received of credits: 32 hours lectures + 32 hours workshop (seminars) Students are required to prepare a report related to the course and to give a talk at a seminar. 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. R. Engelking. General Topology – PWN, Warszawa, 1979.2. J.L. Kelley. General Topology – Van Nostrand Co, Inc., New York 1957.3. П.С. Александров. Введение в теорию множеств и общую топологию, Москва, 1982.4. А. Борубаев, А. Шостак. 50 примеров и конструкций из общей топологии, Фрунзе,

1986.5. L.A. Steen, J.A. Seebach. Counterexamples in topology. Dover publ. Inc., New York, 1996.



VISUAL BASIC AS THE UNIVERSAL TOOL IN CREATING TEACHING-CONTROLLING PROGRAMS.

(Visual Basic kā universāls līdzeklis apmācoši kontrolējošu programmu izstrādē)

Course code Mate-0233Author Mg. Comp., lect. Viesturs VezisCredits: 4Required for grade testPrerequisites Mate-0226, Mate-0232Annotation:The aim of the course is to introduce the technologies of creating teaching-controlling programs, to learn programming language Visual Basic as the universal tool in creating teaching-controlling programs.Subjects1. Computers’ place and role in lesson. Teaching programs - types, didactic aspects and

application possibilities.2. Tests, principles of preparing and applying.3. MS Visual Basic as the universal tool in test creating.4. MS Visual Basic - event oriented programming language. Introduction to environment of MS

Visual Basic - Control, Project, Properties and other windows. Examples.5. Screen Form and object - button (Command Button), properties. Simple program, running

and debugging.6. Objects, properties and methods.7. Standard dialog boxes, menu creating, MDI form.8. Principles of creating optimal computer user interface.9. Constants and variables, types and the scope. Overview of the most often used operations and

operators.10. Operators of Visual Basic: condition, selection and loop.11. Functions and procedures, application and linking to objects.12. Designing program as commercial product.13. Introduction to different development of teaching course systems.14. Set of tests creating.15. Introduction to the most recent technologies in development of teaching - controlling

programs.

Requirements for received of creditsProject - the set of tests for one study topic. Credits are awarded after successful completing of the project.Recommended literature:1. Michael Halvorson, Microsoft® Visual Basic® 5 Step by Step2. Microsoft® Visual Basic® 5.0 Programmer's Guide, Microsoft Corporation3. Microsoft® Visual Basic® Deluxe Learning Edition Version 5.0 Int'l Version, Microsoft

Press



Table of ContentsPART A (COMPULSORY COURSES)..............................................................................................................................................3

SELECTED CHAPTERS IN PROBABILITY THEORY AND MATHEMATICAL STATISTICS............................................4SELECTED CHAPTERS OF DISCRETE MATHEMATICS AND ALGEBRA..........................................................................5SELECTED CHAPTERS OF ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS...................................................6SPECIAL SECTIONS OF FUNCTION THEORY AND FUNCTIONAL ANALYSIS................................................................7

PART B (OPTIONAL COURSES)......................................................................................................................................................8AFFINE, PROJECTIVE AND COMBINATORIAL GEOMETRY...............................................................................................9ANALYTICAL METHODS OF ORDINARY DIFFERENTIAL EQUATIONS........................................................................10ANALYTICAL METHODS OF PARTIAL DIFFERENTIAL EQUATIONS.............................................................................11APPLICATION BY SPLINES IN MATHEMATICAL PHYSICS..............................................................................................12APPLIED ANALYSIS (NON-LINEAR)......................................................................................................................................13APPLIED ANALYSIS (OPTIMISATION)..................................................................................................................................14APPROXIMATIONS THEORY...................................................................................................................................................16ASYMPTOTIC METHODS IN MATHEMATICS......................................................................................................................17COMBINATORIAL ALGORITHMS...........................................................................................................................................18CRYPTOGRAPHY........................................................................................................................................................................19DEVELOPMENT TECHNOLOGIES OF ELECTRONIC TEACHING AIDS...........................................................................20ELEMENTS OF SET AND CATEGORY THEORIES................................................................................................................21ELEMENTS OF TOPOLOGY......................................................................................................................................................23FACTOR ANALYSIS AND MATHEMATICAL STATISTICS.................................................................................................25GENERAL METHODS OF ELEMENTARY MATHEMATICS................................................................................................26LINEAR SYSTEM THEORY AND REGRESSION ANALYSIS...............................................................................................27MARKOVA PROCESSES WITH DISCRETE STATES SPACE...............................................................................................28MATHEMATICAL MODELLING...............................................................................................................................................29MATHEMATICAL MODELS OF PROCESSES IN POROUS MEDIA.....................................................................................30MATHEMATICAL STRUCTURES I...........................................................................................................................................31MATHEMATICAL STRUCTURES II.........................................................................................................................................32MATHEMATICAL THEORY OF MARKET EQUILIBRIUM...................................................................................................34MATHEMATICAL THINKING PSYCHOLOGY.......................................................................................................................35MODERN ELEMENTARY GEOMETRY...................................................................................................................................36MULTIDIMENSIONAL STATISTICAL ANALYSIS................................................................................................................37NON-LINEAR BOUNDARY VALUE PROBLEMS...................................................................................................................39NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATION..........................................................................40NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS............................................................................41PERTURBATION ANALYSIS.....................................................................................................................................................42PRACTICAL CLASSES OF MATHEMATICAL MODELLING WITH COMPUTER PROGRAM PACKAGES I................43PRACTICAL CLASSES OF MATHEMATICAL MODELLING WITH COMPUTER-PROGRAM PACKAGES II..............45PRACTICAL CLASSES OF MATHEMATICAL MODELLING WITH COMPUTER-PROGRAM PACKAGES III.............46PRACTICE IN OLYMPIAD PROBLEMS SOLVING................................................................................................................47RISK THEORY..............................................................................................................................................................................48SELECTED CHAPTERS OF COMPUTER APPLICATION PROGRAMS...............................................................................49SELECTED CHAPTERS OF NUMBER THEORY.....................................................................................................................50SOLVABILITY OF NONLINEAR EQUATIONS.......................................................................................................................51SPECIAL FINITE-DIFFERENCE METHODS............................................................................................................................52SPECIAL SECTIONS OF CALCULUS.......................................................................................................................................53SPLINES AND THEIR APPLICATIONS....................................................................................................................................54STATISTICAL MODELLING......................................................................................................................................................55STOCHASTIC PROCESSES........................................................................................................................................................56SUPPLEMENTARY CHAPTERS OF MATHEMATICAL STATISTICS.................................................................................57THE HISTORY OF LATVIAN SCHOOLS..................................................................................................................................58THEORY OF ALGORITHMS AND AUTOMATA.....................................................................................................................59TOPOLOGY III.............................................................................................................................................................................60VISUAL BASIC AS THE UNIVERSAL TOOL IN CREATING TEACHING-CONTROLLING PROGRAMS.....................61


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Curriculum - AIKNCaiknc.lv/zinojumi/en/LuMatMaE.doc · Web viewThe system of orthogonal functions in Hilbert space, Fourier series. Least squares approximation, constructing of best