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DEPARTMENT OF MATHEMATICS AND STATISTICS

UNDERGRADUATE PROGRAMME OF STUDY

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Undergraduate Programme of Study The famous Platonic inscription “let no one ignorant of geometry enter” has been adopted, directly or indirectly, by all universities in the world and, appropriately, the Department of Mathematics and Statistics was one of the departments with which the University of Cyprus commenced its operation in 1992. The primary aim of the Department is the promotion, through scientific research and teaching, of the Mathematical Sciences. The achievement of this aim is inextricably linked with the need to produce well-trained scientists who will contribute to the continuation of the cultural and economic progress of Cyprus. Because of the pivotal role of Mathematics and Statistics in Science, it is necessary to create a department of high calibre. Important steps in achieving this ambition are the development of links with corresponding institutions abroad and the creation of high-level programmes of study. The undergraduate programme started in September 1992 and has been designed so that the student can advance to graduate studies. The Department of Mathematics and Statistics offers the following undergraduate degree:

Bachelor in Mathematics and Statistics - Specialisation in Applied Mathematics - Specialisation in Pure Mathematics - Specialisation in Probability/Statistics

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Essential Programme Information The undergraduate studies programme is based on the European Credit Transfer System (ECTS). To obtain a degree a student must earn 240 ECTS. The expected time of completion of the required credits is 8 semesters. Under certain special circumstances the duration of the programme may be extended to a maximum of 12 semesters. Should a student obtain a postponement of studies, its duration will not be counted towards the 12 semesters. Class attendance is mandatory. The Department assigns to each undergraduate an Academic Advisor, who is a member of the Academic Faculty of the Department and who provides the student with guidance on academic affairs. Each semester students register for courses according to the requirements of their programme of study. A student is considered to have full-time status if she/he is registered for a total of 27-30 ECTS. A student may register for up to 38 ECTS per semester. After completing the 2nd year of studies a student who is repeating two courses may register for a maximum of 42 credits. The Vice-Rector of Academic Affairs may grant permission, in exceptional cases, for up to 45 ECTS, after a well-substantiated written recommendation from the Head of the Department. Any requests for more than 45 ECTS which will be recommended by the Head of the Department and the Vice-Rector of Academic Affairs, will be submitted to the Senate for final approval. An undergraduating student who is also pursuing a minor programme of studies, may be granted permission to register for up to 42 ECTS per semester. If a student fails to complete the requirements of her/his programme of study in 12 semesters, then hers/his studies are terminated automatically, without being awarded a degree. The studies of an undergraduate student will also be terminated automatically in the following cases:

(a) the student has not registered for any courses (by the 4th

week of a semester), (b) a first year student receives a grade of (0) in all of her/his courses, (c) after the second semester the student receives a grade of (0) in all of her/his courses for two

consecutive semesters, (d) the student, after a postponement of her/his studies, does not register for any courses (e) the student has withdrawn from all courses.

The Undergraduate Studies Office notifies the student in writing for her/his termination of studies. The student can file an appeal justifying her/his absence within a 15-day framework. Otherwise the student is automatically erased from the academic registry. Curriculum

The curriculum is divided into four levels and six groups. Level 101-199 corresponds mainly to courses of the first year of studies, level 201-299 corresponds mainly to courses of the second year of studies, level 301-399 corresponds mainly to courses of the third year of studies, and finally, level 401-499 corresponds mainly to courses of the fourth year of studies. Level 001-099 courses correspond to service courses (Table 3) and are not open to Mathematics or Statistics majors (except MAS 007; see Elective Courses).

The six groups into which the courses are divided correspond approximately to the following areas of Mathematics: Analysis, Algebra, Geometry, Probability/Statistics, Numerical Analysis and Applied Mathematics. The second digit of the course number determines the area of mathematics that the course belongs to. The characteristic digit (2nd digit of the course number) of the six areas is 0 & 1, 2, 3, 5 & 6, 7 and 8, respectively, and they appear in Table A.

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Each undergraduate course is usually a four-hour course (four hours per week, one of which may correspond to a recitation hour). Each course has a corresponding amount of ECTS as they appear in Table A.

Undergraduate Programme of Studies The undergraduate programme guide provides the programme’s courses, a description for the majority of the courses and the requirements for obtaining the specialisations offered.

TABLE A COURSES FOR STUDENTS OF MATHEMATICS AND STATISTICS:

Course Title ECTS Pure Mathematics

Applied Mathematics

Statistics

MAS101 - Calculus I 8 ▲ ▲ ▲

MAS102 - Calculus II 8 ▲ ▲ ▲

MAS121 - Linear Algebra Ι 8 ▲ ▲ ▲

MAS122 - Linear Algebra II 8 ▲ ▲ ▲

MAS131 - Basic Mathematics I 7 ▲ ▲ ▲

MAS132 - Basic Mathematics II 7 ▲ ▲ ▲

MAS133 - Sets and Algebraic Structures 7 ▲ ▲ ▲

MAS191 - Mathematics with computers 8 ▲ ▲ ▲

MAS201 - Multivariable Differential calculus 7 ▲ ▲ ▲

MAS202 - Multivariable Integral calculus 7 ▲ ▲ ▲

MAS203 - Ordinary Differential Equations 7 ▲ ▲ ▲

MAS222 - Number Theory 7

MAS261 - Probability I 7 ▲ ▲ ▲

MAS262 - Statistics I 7 ▲ ▲ ▲

MAS271 - Numerical Analysis I 7 ▲ ▲ ▲

MAS301 - Real Analysis 8 ▲ ▲ ▲

MAS302 - Complex variables I 7 ▲ ▲ ▲

MAS303 - Partial Differential Equations 7 ΜΑS(Pure)Α ΜΑS(Applied)Α or

ΜΑS(Pure)**

ΜΑS(Pure)**

MAS304 - Functional Analysis 7 ΜΑS(Pure)Α ΜΑS(Applied)Α or

ΜΑS(Pure)**

ΜΑS(Pure)**

MAS321 - Introduction to Algebra 7 ▲ ΜΑS(Pure)** ΜΑS(Pure)**

MAS331 - Classical Differential Geometry 7 ▲ ▲ ▲

MAS350 - Stochastic Processes 7 ΜΑS(Statistics) ΜΑS(Statistics) ▲

MAS361 - Probability II 7 ΜΑS(Statistics) ΜΑS(Statistics) ▲

MAS362 - Statistics II 7 ΜΑS(Statistics) ΜΑS(Statistics) ▲

MAS371 - Numerical Analysis II 7 ΜΑS(Applied)Α

MAS401 - Measure Theory and Integration 7 ΜΑS(Pure)Α ΜΑS(Pure)** ΜΑS(Pure)**

MAS402 - Complex Analysis II 7

MAS403 - Ordinary Differential Equations II 7 ΜΑS(Applied)Β

MAS418 - An Introduction to Fourier Analysis 7 ΜΑS(Pure)Α ΜΑS(Pure)** ΜΑS(Pure)**

ΜΑS419 - Topics in Analysis 7

ΜΑS420 - Approximation Theory 7 ΜΑS(Applied)Β

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ΜΑS422 - Introduction to Coding Theory 7

MAS424 - Theory of Rings and Modules 7

MAS425 - Group Theory 7

MAS426 - Field Theory 7

MAS427 - Group Representation Theory 7

ΜΑS429 – Topics in Algebra 7

MAS431 - Introduction to Differentiable Manifolds 7 ΜΑS(Pure)Β ΜΑS(Pure)** ΜΑS(Pure)**

MAS432 - Introduction to Riemannian Geometry 7 ΜΑS(Pure)Β ΜΑS(Pure)** ΜΑS(Pure)**

MAS433 - Topology 7 ΜΑS(Pure)Β ΜΑS(Pure)** ΜΑS(Pure)**

MAS434 - Introduction to Algebraic Topology 7

MAS439 - Introduction to Algebraic Geometry 7

MAS451 - Linear Models I 8 ΜΑS(Statistics) ΜΑS(Statistics) ▲

MAS452 - Linear Models II 7

ΜΑS454 - Non-Parametric Statistics 7 ΜΑS(Statistics) ΜΑS(Statistics) ▲

MAS455 - Sampling Theory 7

MAS456 - Time Series 7

MAS458 - Statistical Data Analysis 7 ΜΑS(Statistics) ΜΑS(Statistics) ▲

MAS459 - Multivariable Analysis 7

MAS466 - Survival Analysis 7

ΜΑS468 - Topics in Probability – Statistics I 7

ΜΑS469 - Topics in Probability – Statistics II 7

MAS471 - Numerical Solution of Ordinary Differential Equations 7 ΜΑS(Applied)Β

MAS472 - Numerical Solution of Partial Differential Equations 7 ΜΑS(Applied)Β

MAS473 - Introduction to the Finite Element Method 7 ΜΑS(Applied)Β

MAS481 - Applied Mathematical Analysis 7 ΜΑS(Applied)Β

MAS482 - Classical Mechanics 7 ΜΑS(Applied)Β

MAS 483 - Fluid Dynamics 7 ΜΑS(Applied)Β

ΜΑΣS484 - Introduction to Mathematical Modeling 7

ΜΑS487 - Topics from Applied Mathematics 7

ΜΑS499 - Independent Study 7

MAS501, MAS502, MAS503 – Work Placement I, II, III ** 1

ΜΑS857*, ΜΑS858* - Mathematical Problem Solving Techniques 4

Notes:

* = The Course does not have a typical course code since students can registerforit at any time during their studies.

** = The students have the choice of registering in the work placement program during the summer months. The duration of the placement is 1-3 months, with 1 ECTS earned for each month of work. The ECTS earned from a work placement do not contribute towards the sum of 240 ECTS required for obtaining the degree. Students are allowed to do a work placement (1-3 ECTS) only once, in the summer term between either the 2nd and 3rd or the 3rd and 4th year of their studies.

▲ = Compulsory Course.

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PROGRAMME OF STUDY DEGREE REQUIREMENTS The degree of Mathematics and Statistics requires 240 ECTS obtained from the following courses:

(1) 17 Compulsory Courses for all students

1) MAS101 - Calculus I 2) MAS102 - Calculus II 3) MAS121 - Linear Algebra Ι 4) MAS122 - Linear Algebra II 5) MAS131 - Basic Mathematics I 6) MAS132 - Basic Mathematics II 7) MAS133 – Sets and Algebraic Structures 8) MAS191 - Mathematics with computers 9) MAS201 - Multivariable Differential calculus 10) MAS202 - Multivariable Integral calculus 11) MAS203 - Ordinary Differential Equations 12) MAS261 - Probability I 13) MAS262 - Statistics I 14) MAS271 - Numerical Analysis I 15) MAS301 - Real Analysis 16) MAS302 - Complex variables I 17) MAS331 - Classical Differential Geometry

(2) Compulsory Course from other Department

Course CS 031 Introduction to Programming (7 ECTS).

(3) Free electives from other departments:

Students are required to take 3-5 courses from other departments. These courses must belong to at least 3 different faculties. The Faculty of Pure and Applied Sciences may be one of the faculties (excluding however the courses offered by the Department MAS). Free electives from the Athletics Department and the Language Centre are considered courses of their respective independent faculties. Only one first-level foreign language course may be considered as a free elective, unless the student completes the second-level as well, in which case both levels count.

(4) Students are required to take two english language courses. (5) Restricted Elective courses within the Department (MASXXX)

Any course which does not carry a characteristic symbol (see Table A) is a restricted elective course (MASXXX). Any Departmental course within or outside a student’s specialisation may count as a restricted elective course, provided that the student’s specialisation requirements have been fulfilled. Moreover, a student may not use the same course to cover multiple requirements of her/his specialisation.

(6) Compulsory courses according to the areas of specialisation:

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Statistics Pure Mathematics Applied Mathematics

Compulsory Courses from other Department (a) PHYΧΧΧ

Compulsory Courses from other Department (a) PHYΧΧΧ

Compulsory Courses within the Department (a) MAS350 - Stochastic Processes (b) MAS361 - Probability II (c) MAS362 - Statistics II (d) MAS451 - Linear Models I (e) ΜΑS454 - Non-Parametric Statistics (f) MAS458 - Statistical Data Analysis (g) MAS321 - Introduction to Algebra or one course from the list ΜΑS (Pure) Α or one course from the list ΜΑS (Pure) B.

Compulsory Courses within the Department (a) MAS321 - Introduction to Algebra (b) One course from the list ΜΑS(Pure)Α (c) One course from the list ΜΑS(Pure)B (d) Two courses from the list ΜΑS(Statistics)

Compulsory Courses within the Department (a) Two courses from the list ΜΑS(Applied)Α (b) Two courses from the list ΜΑS(Applied)Β (c) Two courses from the list ΜΑS(Statistics) (d) MAS321 - Introduction to Algebra or one course from the list ΜΑS(Pure)Α or one course from the list ΜΑS(Pure)B.

Επεξηγήσεις:

ΜΑS(Applied)Α = (a) MAS303 – Partial Differential Equations (b) MAS304 – Functional Analysis (c) MAS371 - Numerical Analysis II

ΜΑS(Applied)Β = (a) MAS403 - Ordinary Differential Equations II (b) ΜΑS420 - Approximation Theory (c) MAS471 - Numerical Solution of Ordinary Differential Equations (d) MAS472 - Numerical Solution of Partial Differential Equations (e) MAS473 - Introduction to the Finite Element Method (f) MAS481 - Applied Mathematical Analysis (g) MAS482 - Classical Mechanics (h) MAS 483 - Fluid Dynamics

ΜΑS(Pure)Α = (a) MAS303 - Partial Differential Equations (b) MAS304 - Functional Analysis (c) MAS401 - Measure Theory and Integration (d) MAS418 - An Introduction to Fourier Analysis

ΜΑS(Pure)Β = (a) MAS426 - Field Theory (b) MAS431 - Introduction to Differentiable Manifolds (c) MAS432 - Introduction to Riemannian Geometry (d) MAS433 - Topology

ΜΑS(Statistics) = (a) MAS350 - Stochastic Processes (b) MAS361 - Probability II (c) MAS362 - Statistics II (d) MAS451 - Linear Models I (e) ΜΑS454 - Non-Parametric Statistics (f) MAS458 - Statistical Data Analysis

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REMARKS (a) Chosing a specialisation: students choose and state their specialisation during their 2

nd year of

studies (at the beginning of the spring semester). In order to change specialisation, a student needs to apply in writing to the Head of Department before the commencement of the semester during which the student desires the change to take effect.

(b) In exceptional cases, a student in their final year may take at most two postgraduate level courses as restricted-elective courses (only one of these two courses can be a Seminar), with the approval of both the courses’ instructor and the academic advisor. An exceptional case is defined to be a student with at least 8.5 grade point average in the Deparment’s courses. It is understood that the credit units of each course are counted only once.

(c) A student may complete her/his studies with more than 240 credits, depending on her/his

choice of elective courses inside and outside the Department. (d) Indicative programmes of study for the Bachelor in Mathematics and Statistics with

specialisation in Pure Mathematics, Applied Mathematics and Statistics/Probability are given in Tables C1, C2 and C3, respectively.

(e) Reading courses are not offered at the undergraduate level. Students may enroll in an Independent Study instead.

(f) The students have the choice of registering in the work placement program during the

summer months. The duration of the placement is 1-3 months, with 1 ECTS earned for each month of work. The ECTS earned from a work placement do not contribute towards the sum of 240 ECTS required for obtaining the degree.

Students are allowed to do a work placement (1-3 ECTS) only once, in the summer term between either the 2nd and 3rd or the 3rd and 4th year of their studies.

Internship program Codes: MAS501- Work Placement Ι (1ECTS) ΜΑΣ502- Work Placement ΙI (1ECTS) ΜΑΣ503- Work Placement ΙII (1ECTS)

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Independent Study (IS) Regulations (MAS499) General Students may register in MAS499 as a restricted elective course (MASXXX), during the last two semesters of their studies. Every IS must contain sufficient elements of initiative, independent study and originality, in the broader sense of the term. It may have a theoretical and/or applied character. The IS may be an extension of a known theory, methods or techniques and/or aplication of these to specific problems. The student registers for an IS after the approval of his/her academic advisor. The coordination of the preparation and the evaluation of every IS is performed by the Head of the Undergraduate Studies Committee (IS Coordinator). IS Topic Submission and Announcement Faculty members may compile a list of possible topics and submit it to the department. The submission of a topic for IS by the faculty is optional. Each topic has a title and a short description. The IS Coordinator, in cooperation with the Head of Department, prepares a list of the submitted IS topics which he announces to the faculty and the final-year students before the commencement of the registration period. Assumption of IS topic The students interested in conducting an IS, need to contact a member of the academic faculty. In order to secure a specific topic, the students need the approval of the respective faculty member who is the Supervisor of the IS. IS Supervision The supervision of the student conducting an IS is carried out by the faculty member (Supervisor). The monitoring of the IS’s progress takes place during regular meetings between the student and the Supervisor. IS Evaluation The IS is evaluated by the Supervisor and another faculty member, named the Second Evaluator. The list of evaluators is prepared by the IS Coordinator in collaboration with the Supervisors and the Head of Department. The Coordinator prepares the Schedule of the IS Presentations. The presentations take place during the exam period. At least one week before the IS presentation date, the student submits two copies of the IS to the Department (one for the Department’s archive and the other for the Supervisor). The student presents the project orally to the Supervisor and the Second Evaluator, according to the Schedule of the IS Presentations. The presentation is open to all faculty members and the students of the Department. After the presentation, the Supervisor, in agreement with the Second Evaluator, submits a well-documented grade for the IS using a special Evaluation Form is accordance with the procedure of submission of course grades. The Deparment’s Council resolves cases of disagreement on the IS grade. The primary evaluation criteria of an IS are: a) Project quality (e.g. accuracy and completeness of analysis, suitability of methodology, validity of

theoretical results, quality of software code, potential for extension, coherence). b) Degree to which the study is completed. c) Degree to which the student comprehends the broader area of the IS topic. d) Quality of the written project (e.g. organization and structure, clarity, readability and coherence). IS Requirements Every IS report should be a complete text which is divided into chapters. The text should contain an introduction to the topic, an analysis of the importance of the study, the study’s connection with similar studies, a literature review, a description of the methodology, a statement and evaluation of the results, and finally, conclusions and suggestions for potential extensions of the study.

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The format of the IS is dictated by the Supervisor. Independent studies which create and utilize software programmes, need to contain the relevant source codes and the instructions in special appendices. The source codes themselves may not consist the main text of the IS. The IS may be written in English, provided an extended summary in Greek is attached.

MINOR PROGRAMME OF STUDY

The minor programme in Mathematics is open to all University students, except for students from the Department of Mathematics and Statistics.

According to the University regulations, a minor program consists of courses totaling at least 60 ECTS credits. The earliest a student can enroll on a minor program is the 3rd semester of her/his studies. The requirements for the minor in Mathematics are the successful completion of eight (9) courses which must include: MAS101, MAS102, MAS121, MAS133, MAS261, MAS262, MAS271, and two (2) additional courses worth 7ects or 8ects from courses within the Department of Mathematics and Statistics.

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Courses offered to other Departments Courses that are offered exclusively to other Departments and their respective ECTS are listed in table below (Table B). TABLE Β

Courses offered to other Departments

Course Title Department ECTS

ΜΑS001 Mathematics I ECO, PBA, AFN, ΒΙΟ 6

ΜΑS002 Mathematics II PBA, AFN, ΒΙΟ 6

ΜΑS003 Elements of Complex Analysis PHY 7,5

ΜΑS007 History of Mathematics ΜΑS, «Ε» 5

ΜΑS012 Calculus for Computer Scientists Ι CS 5

ΜΑS013 Calculus for Computer Scientists II CS 5

ΜΑS018 Introductory Mathematics I PHY, CHE 5

ΜΑS019 Introductory Mathematics II PHY 5

MAS020 Introductory Mathematics II (CHE) CHE

ΜΑS025 Engineering Mathematics I ECE, CEE, ΜΜE 5

ΜΑS026 Engineering Mathematics II ECE, ΜΜE 5

ΜΑS027 Engineering Mathematics III ECE, CEE, ΜΜE 5

ΜΑS029 Elements of Linear Algebra ECE, CEE, ΜΜE, CS 5

ΜΑΣ030 Introduction to Probability and Statistics ECE, CEE 5

ΜΑS051 Statistical Methods I ΕDU, SPS, PSY 5

ΜΑS055 Introduction to Probability and Statistics CS 7

ΜΑS061 Statistical Analysis I ECO, PBA, AFN 6

ΜΑS062 Statistical Analysis ΙΙ PBA, AFN, PSY 6

Note:

«Ε» = Free Elective Course

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TABLE C1: PURE MATHEMATICS - INDICATIVE PROGRAMME OF STUDIES Semesters Code - Course Title ECTS

TOTAL

1st Semester MAS121 - Linear Algebra Ι 8

MAS131 - Basic Mathematics I 7

MAS133 - Sets and Algebraic Structures 7

CS031 Introduction to Programming 7 29

2nd Semester MAS101 - Calculus I 8

MAS122 - Linear Algebra II 8

MAS132 - Basic Mathematics II 7

MAS191 - Mathematics with computers 8 31

3rd Semester MAS102 - Calculus II 8

MAS201 - Multivariable Differential calculus 7

MAS261 - Probability I 7

MAS271 - Numerical Analysis I 7 29

4th Semester MAS202 - Multivariable Integral calculus 7

MAS203 - Ordinary Differential Equations 7

MAS262 - Statistics I 7

Foreign Language Course I 5

Elective Course from other departments 5 31

5th Semester MAS301 - Real Analysis 8

MAS321 - Introduction to Algebra 7

MAS331 - Classical Differential Geometry 7

ΜΑS(PURE)Α 7 29

6th Semester MAS302 - Complex variables I 7

ΜΑS(PURE)Β 7

ΜΑS(STATISTCS) 7

Foreign Language Course II 5

PHYΧΧΧ 5 31

7th Semester ΜΑS(STATISTCS) 7

ΜASXXX Restricted elective course within the Department 7


Free elective course from other departments (2 courses) or ΜAS XXX Restricted elective course within the Department *

8 29

8th Semester ΜASXXX Restricted elective course within the Department 7



Free elective course from other departments 5

Free elective course from other departments 5 31

GRAND TOTAL 240 Εxplanations

ΜΑSΧΧX = Elective Course within the Department

* = Elective Course from other departments (2courses) or ΜAS XXX Elective course within the Department * The credit units must add up to a minimum of 8 ECTS.

ΜΑS(PURE)Α = (a) MAS303 - Partial Differential Equations (b) MAS304 - Functional Analysis (c) MAS401 - Measure Theory and Integration (d) MAS418 - An Introduction to Fourier Analysis

ΜΑS(PURE)Β = (a) MAS426 - Field Theory (b) MAS431 - Introduction to Differentiable Manifolds (c) MAS432 - Introduction to Riemannian Geometry (d) MAS433 - Topology

ΜΑS(STATISTCS) = (a) MAS350 - Stochastic Processes (b) MAS361 - Probability II (c) MAS362 - Statistics II (d) MAS451 - Linear Models I (e) ΜΑS454 - Non-Parametric Statistics (f) MAS458 - Statistical Data Analysis

Note:

The same course may not be used to fulfill multiple requirements of a specialization.

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TABLE C2: APPLIED MATHEMATICS – INDICATIVE PROGRAMME OF STUDIES

Semesters Code - Course Title ECTS

TOTAL









MAS102 - Calculus II 8

3rd Semester MAS201 - Multivariable Differential calculus 7










ΜΑS(APPLIED)Α 7

ΜΑS(PURE)** 7 29


ΜΑS(STATISTICS) 7

ΜΑS(APPLIED)Β 7

Foreign Language Course ΙΙ 5

PHY ΧΧΧ 5 31

7th Semester ΜΑS(STATISTICS) 7

ΜΑS(APPLIED)Α 7

ΜΑS(APPLIED)Β 7

Elective Course from other departments (2 courses) ή ΜAS XX Elective course within the Department *

8 29

8th Semester ΜASXXX Elective course within the Department 7

ΜASXXX Elective course within the Department 7

ΜASXXX Elective course within the Department 7

Elective Course from other departments 5


GRAND TOTAL 240

Εxplanations

ΜΑSΧΧ = Restricted elective Course within the Department

* = Restricted elective Course from other departments (2courses) or ΜAS XXX Restricted elective course within the Department * The credit units must add up to a minimum of 8 ECTS.

ΜΑS(PURE)** = MAS321 - Introduction to Algebra or ΜΑS(PURE)Α or ΜΑS(PURE)Β

ΜΑS(APPLIED)Α = (a) MAS303 - Partial Differential Equations (b) MAS304 - Functional Analysis (c) MAS371 - Numerical Analysis II

ΜΑS(APPIED)Β = (a) MAS403 - Ordinary Differential Equations II (b) ΜΑS420 - Approximation Theory (c) MAS471 - Numerical Solution of Ordinary Differential Equations (d) MAS472 - Numerical Solution of Partial Differential Equations (e) MAS473 - Introduction to the Finite Element Method (f) MAS481 - Applied Mathematical Analysis (g) MAS482 - Classical Mechanics (h) MAS 483 - Fluid Dynamics

ΜΑS(STATISTICS) = (a) MAS350 - Stochastic Processes (b) MAS361 - Probability II (c) MAS362 - Statistics II (d) MAS451 - Linear Models I (e) ΜΑS454 - Non-Parametric Statistics (f) MAS458 - Statistical Data Analysis

Note: The same course may not be used to fulfill multiple requirements of a specialization.

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TABLE C3: Statistics – INDICATIVE PROGRAMME OF STUDIES Semesters Code - Course Title ECTS

TOTAL









3rd Semester MAS102 - Calculus II 8

MAS201 - Multivariable Differential calculus 7










MAS350 - Stochastic Processes 7

MAS261 - Probability I 7 29


MAS362 - Statistics II 7

MAS458 - Statistical Data Analysis 7

Foreign Language Course II 5


7th Semester ΜΑS454 - Non-Parametric Statistics 7

MAS451 - Linear Models I 8

ΜΑS(PURE)** 7

ΜASXXX Restricted elective course within the Department

7 29

8th Semester ΜASXXX Restricted elective course within the Department 7



Free elective course from other departments 5

Free elective course from other departments 5 31

GRAND TOTAL 240 Εxplanations

ΜΑSΧΧX = Restricted elective Course within the Department

ΜΑS(PURE)** = MAS321 - Introduction to Algebra or ΜΑS(PURE)Α or ΜΑS(PURE)Β

Note: The same course may not be used to fulfill multiple requirements of a specialization.

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COURSE DESCRIPTIONS MAS101 - Calculus I Fundamental properties of real numbers. Sup and Inf of a set and its basic properties. Sequences, its limits, properties of converging sequences. Subsequences. Nested interval principle. Functions and their limits. Sequential approach to limits of functions. Continuity of functions. Intermediate value Theorem and Existence of Extreme values Theorem. Uniform Convergence. Derivatives, basic results. Mean value Theorem and its variations. Continuity and derivative of Inverse function. Graph of a function. L’Hôpital’s Rule. MAS102 - Calculus II Required essential knowledge: MAS 101 Partitions, upper and lower sums, Riemann integral on a closed interval. Basic existence theorems of integrals. Computation of volumes and areas. The Fundamental Theorems of Calculus, generalised integrals. Logarithmic and exponential functions. Basic methods of integration, integration by parts, substitution, induction formulas, integration of rational functions. Taylor’s formula, computation of Taylor’s formula for various basic functions. Approximation of smooth functions by polynomials, the irrationality of e. Series, comparison test, Cauchy’s criterion, ratio test, nth root test, integral test, absolutely and conditionally convergentseries, Leibniz’s Theorem for alternating series, Abel’s and Dirichlet’s criteria, products of series. MAS121 - Linear Algebra Ι The algebra of matrices, invertible matrices. Reduced echelon form of a matrix and linear systems of equations. Vector spaces, base, dimension. Linear maps, matrix of a linear map, change of basis matrix, rank of a matrix. Determinants. The set of solutions of a linear system. Eigenvalues, eigenvectors and eigenspaces. MAS122 - Linear Algebra II Polynomial Ring. Characteristic polynomial, diagonalization, applications. Cayley-Hamilton theorem, minimal polynomial. Invariant subspaces, generalized eigenspaces. Primary decomposition theorem. Nilpotent endomorphisms, Jordan canonical form. Inner-product spaces, Gram-Schmidt method. Special matrices and their properties. MAS131 - Basic Mathematics I Functions and limits. Differentiation. Applications of differentiation, graphs, optimization problems. Integration (indefinite, definite and improper integrals), techniques of integration. Applications of integration (areas of domains in the plane, volumes of solids, arc lengths of curves and areas of surfaces of revolution). Differential equations. Complex numbers. MAS132 - Basic Mathematics II Analytic Geometry in R2: Vectors, inner product, length, distance between points. Equation for a line, tangent, vertical line to a curve. Circles, ellipses, parabolas, hyperbolas. Analytic Geometry in R

3: Vectors, algebraic, geometric properties. Inner product, length, distance between points. Equation for a line (parametric-vector, cartesian format), distance of a point to a line. Regions in Euclidean space. Functions: Curves in the plane, regions between curves, curve intersections. Graphs of functions in R3, analytically and implicitly defined. Solids bounded by surfaces and intersections of surfaces. Transformations: Linear transforms, linear independence and geometric interpretation of determinant. Geometric transforms (translation, rotation, reflection, orthogonal transforms). Polar, cylindrical and spherical coordinates and regions defined in these coordinates. Curves: Curve parametrization in R

2 and R

3. Velocity, acceleration and tangent line. Arc length. Differentiation: Partial derivatives of multivariable functions. Tangent plane and linear approximation. Gradient and directional derivative. Integration: Double integrals over rectangles and general regions of R2. MAS133 - Sets and Algebraic Structures Set Theory: Sets, subsets. Set operations, complement, De Morgan’s laws, power set. Cartesian product. Relations, equivalence relations (equivalence classes modulo m, projective space, rational numbers). Venn diagrams. Elements of propositional logic (quantifiers, negation, truth diagrams). Functions: Image of a set, inverse image. Inverse function. Composition of functions, graphs. Sets of functions. Countable sets, uncountable sets. Diagonal procedure. Reductio ad absurdum and Mathematical Induction. Well Ordering Principle and Principle of Mathematical Induction. Examples

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from Number Theory and other areas of mathematics for understanding the procedure for proving a statement using these methods. Number Theory: Divisibility. Greatest common factor and least common multiple. Euclidean algorithm. Fundamental Theorem of Arithmetic. Applications to polynomials. Introduction to Algebraic Structures: Binary operations. Closure of operations. Properties of closed operations. Examples (composition of functions, matrix multiplication, inverse, congruence classes). Subgroups, groups (examples from cyclic groups (complex unit roots), symmetric group). The group (Ζn , +) as a quotient. Rings, fields and solving first order equations ax = b. MAS191 - Mathematics with computers MATLAB’s environment. MATLAB functions. For, while and if loops. Graphics in two and three dimensions. Programming. Polynomials. Reading from and writing in files. Computer arithmetic and error propagation. Symbolic computing. Special topics and applications (solution of nonlinear algebraic equations and linear systems, eigenvalue problems, numerical integration, ordinary differential equations). MAS201 - Multivariable Differential calculus Normed spaces: examples, Rn, equivalent norms, Cauchy-Schwartz inequality. Open and closed sets. Compactness (Heine-Borel, Bolzano- Weierstrass Theorems). Scalar and vector valued functions. Limits and continuity. Partial derivatives. Differentiability Criterion. Multivariable functions: gradient, differential, directional derivative, vector fields, divergence, curl, Laplacian operator. Vector functions of one variable: derivatives, arc length, change of parameter. Differentiation rules, chain rule, etc. Mean value Theorem. Derivatives of integrals with respect to a parameter. Taylor’s Theorem. Local extrema, conditional extrema (Lagrange multipliers). Inverse and implicit function Theorems. MAS202 - Multivariable Integral calculus Integrable functions and sets, properties. Fubini’s Theorem. Iterated integrals for continuous functions over a compact set (scalar functions over regions of the type Q = I1 × I2 × I3 … × In). Change of variables Theorem for linear and C

1-invertible transformations. Computation of volumes, Cavalieri’s

principle, examples such as the sphere, cylinder and cone. Convergence theorems (interchanging limits and integrals). Transform Theorem (without proof), applications. Parametrized surfaces, partition of unity. Surface and line integrals, computing the area of a surface. Differential forms, Stokes’ Theorem (Green, Gauss, Stokes), applications. MAS203 - Ordinary Differential Equations Separable ODEs. First order ODEs and integrating factors. Picard-Lindelöf theorem. Second order ODEs with constant coefficients. The method of undetermined coefficients and the method of variation of parameters. Systems of first order ODEs. MAS222 - Number Theory Divisibility, Euclidean algorithm, linear Diophantine equations. Prime numbers and the fundamental theorem of arithmetic. Congruences and the Chinese remainder theorem. Fermat’s and Wilson’s theorems. Arithmetic functions and perfect numbers. Euler’s theorem. Quadratic Reciprocity. Pell’s equation and continued fractions. MAS261 - Probability I Counting methods, combinatorics, probability measure space though σ-algebras, independence of events, random variables, cumulative distribution function, discrete and continuous random variables, mean value, multivariable distributions, multivariable normal distribution, sums of random variables, distributions of functions of random variables, covariance function, independence of random variables through the cumulative distribution function, moment generating function, characteristic function, introduction to the law of large numbers, introduction to the central limit theorem. MAS262 - Statistics I Random samples, statistical experiments, statistics, estimation methods (e.g., method of moments, method of maximum likelihood), properties of estimators (e.g., unbiasedness, sufficiency, completeness), exponential families, Rao-Blackwell theorem, Lehmann-Scheffe theorem, Cramer-Rao variance lower bound, confidence intervals, minimum length confidence intervals, hypotheses testing, properties of tests. Statistics, sufficiency, completeness, exponential families, unbiasedness, uniformly minimum variance unbiased estimators, Cramer-Rao lower bound, moment estimators, maximum likelihood estimators, confidence intervals, hypothesis testing.

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MAS271 - Numerical Analysis I Sources and propagation of error. Numerical solution of non-linear equations. Numerical solution of linear systems of equations. Polynomial interpolation. Numerical quadrature. MAS301 - Real Analysis Metric spaces, Normed spaces. Examples. Open and closed sets, interior and closure of a set. Accumulation points and the derived set. The Bolzano-Weierstrass Theorem. Convergence of sequences in metric spaces. Cauchy sequences. Complete metric spaces. The fixed point theorem. Compact sets in metric spaces. The Heine-Borel Theorem. Compact metric spaces. Continuous functions. Continuous and uniformly continuous functions. Continuity and compactness. Sequences and series of functions. Uniform convergence. Uniform convergence and continuity, uniform convergence and integration, uniform convergence and differentiation. The metric of uniform convergence. Sufficient conditions for uniform convergence of a series of functions. MAS302 - Complex variables I Complex numbers, analytic functions, Cauchy-Riemann equations. Harmonic functions. Exponential, trigonometric and logarithmic functions. Integration, Cauchy’s theorem, Cauchy’s integral formulas and inequalities. Liouville theorem and the fundamental theorem of Algebra. Maximum modulus principle. Taylor and Laurent series, residues. The argument principle. Conformal mappings and Mobius transformations. MAS303 - Partial Differential Equations 1

st order PDEs, Non-linear 1

st order PDEs, Linear PDEs of 2

nd order, Elliptic, Parabolic, Hyperbolic PDEs,

Separation of variables, Fourier series. MAS304 - Functional Analysis Metric and normed linear spaces, examples, series, Schauder bases, bounded linear operators, linear functionals, dual spaces. Inner product spaces, orthogonality, orthonormal sets, Bessel’s inequality, Hilbert spaces, projections, orthogonal complements. Riesz Representation Theorem, orthonormal bases. Zorn’s Lemma, Hahn-Banach Theorem with applications, the Principal of Uniform Boundedness with applications, the Open Mapping Theorem with applications, the Closed Graph Theorem with applications. MAS321 - Introduction to Algebra Groups, permutations and symmetric groups, cyclic groups. Subgroups and the Theorem of Lagrange. Homomorphisms and Quotient groups. Rings, integral domains and fields. Homomorphisms, ideals and quotient rings. Polynomial rings, divisibility in polynomial rings, prime and maximal ideals. Finite fields and field extensions. MAS331 - Classical Differential Geometry The Euclidean space R

n: inner product, Cauchy-Schwarz inequality, isometries. Curves in Rn:

parametized cuves, length, periodic, closed cuvres. Curves in R2: curvature, Frenet equalities, winding number, isoperimetric inequality, Hopf Theorem. Curves in R3: curvature, torsion, Frenet equalities, Fundamental theorem. Surfaces in R

3: regular surfaces, local parametrization, examples. Differentiable maps between surfaces, tangent space, total differential. First fundamental form, orientation, Gauss map, second fundamental form, principal curvatures, curvature lines, normal curvature, Gauss curvature, mean curvature. Integration on surfaces. Ruled, minimal surfaces, surfaces of revolution. Isometric (locally isometric) surfaces, Christoffel symbols, Theorema Egregium (Gauss). Parallel vector fields, geodesics, geodesic curvature. Gauss-Bonnet Theorem.

MAS350 - Stochastic Processes Stochastic process, stationary processes, stopping times. Markov chains, Poisson processes, Brownian motion. MAS361 - Probability II Review of basic elements from MAS 261. Stochastic independence through σ-algebras, Borel-Cantelli lemmas, Kolmogorov 0-1 laws, mean value as Lebesgue integral, basic inequalities, convergence of sequences of random variables, convergence of series of random variables, laws of large numbers,

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central limit theorems, conditional probability, conditional mean value, introduction to martingales, central limit theorem for martingales. MAS362 - Statistics II Asymptotic properties of estimators, asymptotic efficiency, asymptotic normality, introduction to statistical decision theory (minimax estimators, Bayes estimators), asymptotic properties of tests, optimal tests, goodness-of-fit tests, tests of independence. U-statistics. MAS371 - Numerical Analysis II Brief revision of the theory of eigenvalues and eigenvectors. Positive definite matrices. Vector and matrix norms. Iterative methods for the solution of linear systems. Gershgorin bounds for eigenvalues. Numerical methods for eigenvalues and eigenvectors. Lagrange interpolation. Hermite interpolation. Divided differences at repeated points. The Newton form of the Hermite interpolation polynomial. Orthogonal polynomials. Gaussian quadrature. MAS401 - Measure Theory and Integration General revision: Sets, orderings, cardinality, metric spaces. Measures: Algebras and σ-algebras, additive and σ- additive measures, outer measures, Borel measures on the real line. Integration: measurable functions, integration of positive functions, integration of complex valued functions, Convergence Theorems, modes of convergence, product measures, the n – dimensional Lebesgue integral, integration in polar coordinates, signed measures, the Radon – Nikodym theorem, complex measures, differentiation on Euclidean space, functions of bounded variation. L

p Spaces: The basic theory, the dual of L

p, the useful inequalities, the distribution function, weak – Lp spaces,

interpolation. MAS402 - Complex Analysis II Compactness and convergence in the space of analytic functions. The space of meromorphic functions. Riemann mapping theorem. Weierstrass Theorem on entire functions, analytic continuation. Elliptic funtions. Riemann surfaces. MAS403 - Ordinary Differential Equations II

Boundary Values Problems. Sturm-Liouville type problems. Asymptotic behaviour of nonlinear systems

of ODEs: Stability. Perturbation Theory of systems of ODEs which possess periodic solutions.

Perturbations of two-dimensional autonomous systems. Poincaré-Bendixson theory.

MAS418 - An Introduction to Fourier Analysis

Periodic functions, trigonometric polynomials, trigonometric series. Fourier series. Convergence of

Fourier series. Bessel’s inequality. Completness, Parseval’s Theorem. The Riemann-Lebesgue Lemma.

Dirichlet’s Theorem. Gibbs phenomenon. Differentiation and Integration of Fourier series. Cesaro and

Abel summability of Fourier series. Fejer’s Theorem. Poisson’s Theorem. The Fourier transform and its

properties. The inversion theorem and Plancherel’s identity. The convolution and its properties.

Applications to partial differential equations.

ΜΑS419 - Topics in Analysis Topics in Real Analysis, Complex Analysis, Harmonic Analysis or Differential Equations. ΜΑS420 - Approximation Theory Normed linearspaces and inner productspaces. Bounded linear operators. Fixed Point methods. Iterative methods for linear systems. Best approximation in normed linear spaces and inner product spaces. Orthogonal polynomials. ΜΑS422 - Introduction to Coding Theory Introduction to finite fields.Vector spaces over finite fields. Linear codes. Encoding and decoding with a linear code. Syndrome decoding. Hamming codes. Cyclic codes. MAS424 - Theory of Rings and Modules Rings and ideals. Homomorphism Theorems. Unique factorisation domains and principal ideal domains. Factor rings. Prime and maximal ideals. R-modules and homomorphisms. Finitely generated R-modules. Noetherian rings.

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MAS425 - Group Theory Normal subgroups, homomorphism theorems. Direct and semidirect products. Group actions. Normalizers and centralizers. Sylow theorems and p-groups. Simple groups. Finitely generated Abelian groups. Composition series and Jordan – Hölder theorem. Soluble groups. MAS426 - Field Theory Rings, ideals, polynomial rings. Fields, field extensions, algebraically closed fields, finite fields. Normal extensions and Galois extensions. The fundamental theorem of Galois theory. Solutions of equations by radicals, ruler and compass constructions. MAS427 - Group Representation Theory Representations. FG - modules, FG - submodules, FG - homomorphisms. Maschke’s Theorem and Schur’s Lemma. Irreducible modules. The group algebra, the centre of the group algebra. Characters, relation between characters and representations. ΜΑS429 - Topics in Algebra Topics in Algebra MAS431 - Introduction to Differentiable Manifolds Differentiable manifolds. Tangent space. Partition of unity. Sards Theorem. Vector fileds, flows. Frobenius theorem. Differential forms. Theorem of Stokes. Theorem of de Rham. MAS432 - Introduction to Riemannian Geometry Introduction to manifolds, tangent spaces and vector fields. Riemannian manifolds. Connections, geodesics, exponential map, normal coordinates, Gauss’ Lemma. Hopf-Rinow Theorem. Curvature. Jacobi fields. Theorems of Bonnet-Myers, Synge-Weinstein and Hadamard-Cartan.

MAS433 - Topology Topological spaces, continuous functions, connected and compact sets, product spaces, the Tychonoff theorem, separation axioms, metric spaces. Homotopy, the fundamental group, the Seifert Van Kampen Theorem, Covering spaces. MAS434 - Introduction to Algebraic Topology Homology, Cohomology. CW complexes. Homology and Cohomology of product spaces, the Eilenberg Zielber theorem and the Kunneth formula. The cohomology ring. Cup product. Poincare duality. Applications: Homology and cohomology of compact surfaces, the Jordan Brouwer separation theorem, invariance of domain. MAS439 – Special Topics in Geometry Depends on the special interests of the staff member teaching it. MAS451 - Linear Models I Simple linear regression model: estimation, confidence intervals, hypothesis testing. Multiple linear regression model: estimation, confidence intervals, hypothesis testing. Goodness of fit, residual analysis and model selection. One and two-way ANOVA. MAS452 - Linear Models II Analysis of variance with one or more fixed-effects, Analysis of variance with one or more random factors, analysis of covariance. Generalized linear models: estimation in some examples, logistic regression, asymptotic properties of estimators. ΜΑS454 - Non-Parametric Statistics Distribution function estimation, probability density function estimation, regression function estimation, applications in R.

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MAS455 - Sampling Theory Sampling scheme design. Simple random sampling, stratified, systematic, cluster sampling, multistage sampling. Mean and variance estimation, ratio estimators, linear regression estimators, optimal choice of sample size, bias in survey methodology. MAS456 - Time Series Stationary processes, autocovariance function, spectral density, linear processes, ARMA processes, non-linear processes, ARCH and GARCH processes. Estimation of the mean and of the autocovariance function. Moment estimators, least squares estimators and maximum likelihood estimators of parameters. Asymptotic properties. MAS458 - Statistical Data Analysis Introduction to R, diagnostic statistics, simulation methods, Markov chain Monte-Carlo, simulation, optimization, resampling. MAS459 - Multivariable Analysis Multivariable distributions. Mean vector and covariance matrix estimation. Wishart distribution. Principal components, canonical correlation, cluster and discriminant analysis. Testing hypothesis in many dimensions. MAS466 - Survival Analysis Censored data, Truncated data. Survival function and hazard function. Nonparametric estimation of the survival function and the hazard function. Parametric models for the hazard function. Counting processes and martingales. Semiparametric Cox model. Tests for one or more populations, tests of class-K. ΜΑS468 - Topics in Probability – Statistics I Topics in Probability – Statistics ΜΑS469 - Topics in Probability – Statistics II Topics in Probability – Statistics MAS471 - Numerical Solution of Ordinary Differential Equations Brief revision of initial and boundary value problems for ordinary differential equations. Stability of initial and boundary value problems for difference equations. One-step methods for initial value problems. Runge-Kutta methods. Multistep methods for initial value problems. Shooting methods and finite difference for boundary value problems. MAS472 - Numerical Solution of Partial Differential Equations Brief revision of initial/boundary and boundary value problems for partial differential equations. One-dimensional parabolic equations. Explicit method, implicit method and Crank-Nicolson method. Stability analysis. Two-dimensional parabolic equations. Hyperbolic equations. CFL condition. The Wendroff, Lax-Wendroff and leap frog methods. Explicit and implicit methods for the one dimensional wave equation. Elliptic equations. The five-point and nine-point methods for the Poisson equation. Non-rectangular regions. Robin boundary conditions. MAS473 - Introduction to the Finite Element Method Variational formulation of elliptic problems. Methods of Galerkin and Ritz. Basis functions and discretization. Error bounds. Applications and examples. Finite elements for parabolic equations. MAS481 - Applied Mathematical Analysis Calculus of variations. Laplace transforms. Fourier analysis. Special functions. Integral equations.

MAS482 - Classical Mechanics Newton’s laws. Central forces. Moving coordinate systems. Systems of particles. Plane motion of rigid bodies. Space motion of rigid bodies. Lagrange’s equations.

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MAS 483 - Fluid Dynamics Introduction to vector and tensor calculus. Continuity and momentum equations in various coordinate systems. Laminar incompressible flows amenable to analytical solution. Steady-state and transient flows and applications. Streamfunction and Stokes flow. Boundary layer. Non-Newtonian flows. ΜΑΣS484 - Introduction to Mathematical Modeling This course emphasises the role of mathematical modelling as a tool for learning and appreciating mathematical techniques. Applications are drawn from diverse areas such as discrete dynamical systems, graphs and networks, linear programming. Extensive use of computer software is made throughout the course. ΜΑΣ487 - Topics from Applied Mathematics Topics from Applied Mathematics ΜΑS499 - Independent Study An independent study with sufficient elements of initiative and novelty under the guidance of a faculty member. MAS501, MAS502, MAS503 – Work Placement I, II, III The goal of this course is to offer the students of the Department of Mathematics and Statistics the possibility to apply the competencies acquired during their studies at the University in real world problems, and by doing so to form a more complete view of their field. This may be achieved either through a specific project assigned to the students by the Hosting Organization at the start of the term of the internship, or through the day to day work in the area of activity of the Hosting Organization. Moreover, the course aims to provide the students with industrial and/or research experience and as a result a better understanding of the job market. Through this course, the students will be able to make informed decisions relating to their career after graduating from the University.

ΜΑS857*, ΜΑS858* - Mathematical Problem Solving Techniques This course is geared towards undergraduate studentsinterested in mathematical problem solving. It will also prepare students who plan to participate in mathematical Olympiads. Emphasis will be given on problem solving techniques, creative thought and exposition skills. A variety of solving techniques will be introduced followed by a number of examples and problems. The problems will cover various areas of mathematics such as Algebra, Analysis, Combinatorics, Number Theory, Geometry, etc. This is an elective course, but it will be taken into serious consideration (or will be mandatory) in the selection ofstudents representing the Department in international mathematical competitions.

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Courses offered to other Departments

MAS001 Mathematics I Functions, limits and continuity of functions. Derivative, applications of derivatives. Integrals, applications of integrals. Hyperbolic functions. MAS002 Mathematics II Techniques of integration, improper integral. Sequences. Series, power series. Differential equations. Partial derivatives. Linear Systems, matrices, determinants, eigenvalues and eigenvectors. Vector Spaces. MAS003 Elements of Complex Analysis Complex numbers, analytic functions, Cauchy-Riemann equations, harmonic functions, exponential, trigonometric, logarithmic functions. Integrals, Cauchy’s Theorem, Cauchy’s Integral Formula. Morera’s Theorem, Liouville’s Theorem, Maximum Principle, Fundamental Theorem of Algebra. Taylor series, Laurent series, calculus of residues. Conformal mappings, linear rational transformations. Physical applications. MAS007 History of Mathematics Topics from ancient Greek mathematics, the middle ages and the modern era. MAS012 Calculus for Computer Scientists II Real numbers – Absolute value – Cartesian and polar coordinates – Equation of line, circle and

parabola - Functions of one variable – Inverse functions – Trigonometric functions – Exponential and

logarithmic functions – Hyperbolic functions - Limits – Computing limits – Continuity – Limits and

continuity of basic functions - Tangent lines and rates of change – The derivative function – Techniques

of differentiation – Derivatives of basic functions – Chain rule – Implicit differentiation – Derivatives of

inverse functions - Related rates – Linear approximation – Differentials –L’Hopital rules – Increase,

decrease and concavity – Local and absolute maxima and minima – Graphs –Newton’s method -

Rolle’s theorem – Mean value theorem - Indefinite integral – Definite integral (Riemann’s integral) –

First fundamental theorem of integral calculus – Average value of a function – Second fundamental

theorem of integral calculus – Integrals of basic functions - Integration by parts – Trigonometric and

algebraic substitutions – Integrating rational functions by partial functions.

ΜAS013 Calculus for Computer Scientists ΙΙ Review of principles of integration - Area between two curves – Volumes – Length of a curve – Area of

a surface of revolution –Improper integrals - Sequences – Monotone sequences - Infinite series –

Convergence tests – Alternating series – Absolute and conditional convergence - Taylor and Maclaurin

polynomials - Taylor and Maclaurin series - Convergence of Taylor series – Differentiating and

Integrating power series - Differential equations of first order – Linear differential equations of second

order – Special forms of differential equations - Functions of two or more variables – Limits and

continuity – Partial derivatives– Maxima and minima of functions of two variables – Complex variables

– Operations of complex variables – Exponential forms of complex numbers – Applications – Relations

between trigonometric and hyperbolic functions.

MAS018 Introductory Mathematics I Real functions of one variable. Limits and continuity of functions. Derivatives. Applications of derivatives. Exponential and logarithmic function. Trigonometric functions and their inverse. Hyperbolic functions and their inverse. Integrals of functions of one variable. Indefinite integral. Integral of basic functions. Integration techniques. Applications of integrals. Generalized integrals. Sequences and series of real numbers. Power series. Taylor series. MAS019 Introductory Mathematics II Vectors. Vector valued functions. Vector fields. Real-valued multivariable functions. Partial derivatives. Total differential. Gradient, divergence, curl. Higher order partial derivatives. Chain rule. Elements of linear algebra. Linear functions. Matrices, determinants. Linear systems. Vector spaces. Spaces with

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inner product. Linear independence, basis and dimension of vector space. Orthogonal and orthonormal bases. Eigenvalues and eigenvectors. MAS020 – Introductory Mathematics II (Chemistry) Differential equations of first order – Linear differential equations of second order – Special forms of

differential equations - Complex numbers and their properties – Polar and exponential forms –

Applications of complex numbers – Relations between trigonometric and hyperbolic functions –

Functions of two variables – Limits and continuity – Partial derivatives – Maxima and minimum –

Double integrals – Introduction to Linear Algebra - Systems of linear equations – Matrices –

Determinants – Vectors – Vector spaces – Eigenvalues and Eigenvectors

MAS025 Mathematics for Engineers I The real number system. Complex numbers (definition, elementary operations). Sequences of real numbers and limits. Real functions of one variable, limits, continuity. Hyperbolic, trigonometric functions. Derivatives of functions of one variable, tangent to a curve. Applications of derivatives. Mean value theorem, monotonicity, extrema, asymptotes. L’Hôpital’s rule. Riemannian integral. Fundemental Theorem of Calculus. Indefinite integrals. Integration techniques (substitution, integration by parts, partial fractions, trigonometric substitution, etc). Applications of integrals, calculation of area, volume and length of a curve. Real number series. Convergence criteria. Power series. Series and Taylor’s thereom. MAS026 Mathematics for Engineers II Multivariable functions. Tangent plane, differential. Partial derivatives. Vector fields. Gradient, divergence, curl. Vector valued functions. Curves on the plane and in space. Length of curve. Double and triple integrals. Change of variables in multiple integrals. Transformations, Jacobian determinants. Polar, spherical and cylindrical coordinate systems. Applications of multiple integrals. Line-integrals. Green’s Theorem. Surface integrals. Surface area. Stokes’ theorem. Gauss’ divergence theorem. MAS027 Mathematics for Engineers III Ordinary differential equations. Separable equations. Exact equations. Integrating factors. Solutions of linear and nonlinear first order differential equations. Second order differential equations. Fundamental solutions of homogeneous equations. The inhomogeneous problem. The methods of undetermined coefficients and variation of parameters. Solutions using power series. Applications of ordinary differential equations. Systems of linear differential equations. Laplace transform. MAS029 Elements of Linear Algebra Vectors on the plane and in space. Inner and outer product. Vector spaces. Linear independence. Basis and dimension of a vector space. Matrices. Linear systems and Gauss method. Inverse of a matrix. Orthogonal matrices. Rank. Determinants. Eigenvalues, eigenvectors, and diagonalization. Inner product spaces. Gram-Schmidt orthonormalization.

MAS030 Introduction to Probability and Statistics Descirptive statistics. Measures of central tendency and dispersion. Probability. Exponential families of distributions. ESTIMATION: Point estimation, sufficiency and completeness. Confidence intervals for the mean value, the variance, the difference between the means of independent and paired samples, the ratio of the variances of independent samples, the proportions and comparison of proportions of independent and paired samples. HYPOTHESIS TESTING: null hypothesis – alternative hypothesis, type I and II errors. Test for the mean value in small and large samples. Test for the comparison of the mean values of independent and paired samples. Test for the variance and the comparison of the variances of two populations. Test for the proportion and the comparison of proportions. Simple linear regression, analysis of variance. MAS051 Statistical Methods Descriptive statistics, probability, binomial distribution, normal distribution, sampling, confidence intervals, hypothesis testing, correlation, regression analysis, introduction to analysis of variance. MAS055 Introduction to Probability and Statistics (7 ECTS) Probability, conditional probability, Bayes Theorem, classical problems in probability (balls and bins, the birthday problem). Random variables, distributions (discrete and continuous). Independence.

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Expected value, applications (coupon collector’s problem). Probability inequalities (Jensen’s inequality, Markov’s inequality, Chebyshev’s inequality, Chernoff’s bounds). Introduction to stochastic processes, Markov chains, applications, random walks, Poisson process. Statistics, point estimation, confidence intervals, hypothesis testing, applications. Correlation, linear regression. Remark: The use of a statistical package or a statistical programming language (e.g. R) is an integral part of the course. Students are asked to familiarize themselves with the basic concepts of the course via a series of laboratory projects and applications. MAS061 Statistical Analysis I Descriptive statistics, probability models. random variables, expected value, sampling, central limit theorem. estimation, confidence intervals, hypothesis testing. Introduction to Regression analysis. MAS062 Statistical Analysis II Regression Analysis. Analysis of qualitative data. Chi-squared Tests. Analysis of variance. Nonparametrics. Time Series. Decision Theory.

department of mathematics and statistics … · mas434 - introduction to algebraic topology 7...

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