Module code: MSM70
ECTS Credit Points: 30
Module Type: Compulsory
Year: 1st
Language: Greek
Module general description: The objective of the module MSM70 is to teach fundamental mathematical knowledge and techniques from Analysis, Linear Algebra, Probability Theory, and Statistics, in order for students to obtain the necessary foundations to successfully participate in the thematic units of the second term of studies. Lessons will focus on fundamental mathematical principles and familiarization with calculations.
Curriculum of the Module: In particular, the curriculum of MSM70 includes the following topics:
Metric spaces (topology of metric spaces, sequences, continuity). Complete metric spaces. Fixed point theorems. Completion of metric spaces. Normed vector spaces. Matrices. Eigenvalues – Eigenvectors. Linear Systems. Probability. Conditional probability and independence. Discrete random variables.
Learning Outcomes: Upon successful completion of the module MSM70 “Fundamental Theories and Methods in Mathematics”, students will have developed the following skills:
General Learning Outcomes: Upon successful completion of MSM70, students will have obtained the following:
Subjects Covered:
Prerequisites: There are no prerequisite courses.
Evaluation: Students are assigned to submit six (6) written assignments during the academic year. The average grade of the six (6) written assignments, weighted at 30%, is taken into consideration for the calculation of the final grade. The grade of written assignments is activated only with a score equal to or above the pass level (≥5) in the final or resit exams.
The grade of the final or the resit exams shall be weighted at 70 % for the calculation of the final grade.
Module code: MSM71
ECTS Credit Points: 30
Module Type: Compulsory
Year: 1st
Language: Greek
Module general description: The module MSM71 aims to provide knowledge on mathematical methods and tools for the analysis of mathematical models in Physical Sciences and Modern Technology.
Curriculum of the Module: In particular, the curriculum of THE MSM71 includes the following topics:
Ordinary Differential Equations (ODEs) and Systems of ODEs: Qualitative theory. Phase plane, equilibrium point characterization, Bifurcations, Lyapunov method for studying stability, Sturm theory, Sturm Liouville boundary value problems (BVP).
Partial Differential Equations (PDEs): First order PDEs: Methods for solving linear and non-linear PDEs, classical solutions, weak solutions, shock waves. Second order Linear PDEs: classification, methods of solution: method of separation of variables, Fourier and Laplace integral transforms, Poisson Integral, Green’s method. Properties of harmonic functions. Extreme values theorems. Green’s function for BVPs.
Linear Operators: Duality and conjugate operators. Green’s method for BVPs with linear operator. Existence of solution via alternative Fredholm theorem.
Integral Equations of Fredholm and Volterra type: existence of solution via alternative Fredholm theorem, or fixed point theorem, iterative methods (Successive Approximations, solvent kernel via iterative kernels or Fredholm determinants), Characteristic values and eigenfunctions. Separable kernels and symmetric kernels (Hilbert–Schmidt theory) Fourier and Laplace Integral transform methods. Transformation of integral equations to BVPs or to Initial Value Problems and vice versa.
Learning Outcomes: Upon successful completion of MSM70 “Mathematical Models in the Physical Sciences”, students will have developed the following skills:
General Learning Outcomes: Upon successful completion of MSM71, students will have obtained the following:
Prerequisites: There are no prerequisite courses.
Subjects Covered:
Evaluation: Students are assigned to submit six (6) written assignments during the academic year. The average grade of the six (6) written assignments, weighted at 30%, is taken into consideration for the calculation of the final grade. The grade of written assignments is activated only with a score equal to or above the pass level (≥5) in the final or resit exams.
The grade of the final or the resit exams shall be weighted at 70 % for the calculation of the final grade.
Module code: MSM80
ECTS Credit Points: 20
Module Type: Compulsory
Year: 2nd
Language: Greek
Module general description: The principal aim of the module MSM80 is to provide the student with substantial experience in using scientific software packages for symbolic and numerical computations for teaching and research, in order to solve problems stemming from natural sciences corroborating a variety of analytical and computational methods.
Curriculum of the Module: In particular, the curriculum of MSM80 includes the following topics:
Introduction to basic software commands and the Wolfram Language: Definition of functions, symbolic and numerical computation of series and integrals. Functions of several variables, derivatives and partial derivatives. Basic commands for graphical representations, plots of parametric curves in 2D and 3D. Basic commands for handling linear algebra problems (matrix calculus, determinants, eigenvalues and eigenvectors). Lists. Basic symbolic and numerical solving commands for ordinary differential equations. Introduction to the mathematical theory of dynamical systems – Study of non-linear phenomena using software (limit cycles and chaotic phenomena). Basic commands for the study of power series and Fourier series – Implementations of representations by using the software capabilities. Introduction to finite difference numerical methods for partial differential equations. Eigenvalue problems and studies with relevant software commands. Study of partial differential equations by using the software capabilities: Implementations of the representations of the solutions for elliptic, parabolic and hyperbolic equations – Nonlinear problems (nonlinear wave propagation phenomena, reaction-diffusion equations and relevant topics) – Calculus of variations and mathematical software. Integral transformations and mathematical software.
Learning Outcomes: The successful completion of the module MSM80 “Computational methods and software for Mathematics” provides the opportunity for the student to develop the following skills
General Learning Outcomes: Upon successful completion of MSMB80, students will have obtained the following:
By learning to use the software package, students will obtain a valuable asset for writing both their diploma theses and scientific papers in the fields of Mathematics and the Natural Sciences in general.
Subjects Covered:
Prerequisites: There are no prerequisite courses.
Evaluation: Students are assigned to submit six (6) written assignments during the academic year. The average grade of the six (6) written assignments, weighted at 30%, is taken into consideration for the calculation of the final grade. The grade of written assignments is activated only with a score equal to or above the pass level (≥5) in the final or resit exams.
The grade of the final or the resit exams shall be weighted at 70 % for the calculation of the final grade.
Module code: MSM81
ECTS Credit Points: 20
Module Type: Compulsory for division C
Division: Mathematics Education
Year:2nd
Language: Greek
Module general description: The module MSM 81 aims to present the historical evolution of fundamental mathematical concepts as well as diachronic and contemporary trends in Mathematics Education.
Learning Outcomes: Upon successful completion of MSMB81 “History and Teaching of Mathematics”, students will have developed the following skills:
More particularly, learning outcomes are the following:
Students are required to actively participate at two levels: a) in the search for reputable literature from free sources (open access journals) that supplements the arguments and critical assessment of their positions, and b) in the application of the knowledge they have obtained in practical teaching circumstances.
Students will obtain the ability to base their assertions not only on their existing experience, but also on references to reputable sources.
The unit offers students the opportunity to develop their ability to manage the knowledge they have obtained, edit scientific papers, and write critical, scientifically documented assertions.
Furthermore, students shall become acquainted with assessing scientific papers drawn from journals and drawing key positions and conclusions from them.
To better connect the aforementioned subjects, students will write “case studies”.
Finally, a separate part of the unit dedicated to the interaction between modern Mathematics and Cognitive Science.
Subjects Covered:
Evaluation: Submission of six (6) written assignments during the academic year, the weighted average grade of which constitutes a 33 percent of each student’s grade, if a pass grade (≥5) is obtained in the final or resit examination. The grade of the final or the resit exams is weighted at 67% for the calculation of the final grade.
Module code: MSM82
ECTS Credit Points: 20
Module Type: Optional Mandatory (Students choose between course modules MSM82 and MSM84)
Division: Applied Mathematics
Year: 2nd
Language: Greek
Module general description: The course aims to give students a detailed and practical introduction to basic concepts of Mathematical Modeling and specifically to the concepts of dimensional analysis, scaling, perturbation methods (regular perturbation methods, multiple scales methods, boundary layer method) with the use certain case studies. Additionally the process of mathematical modelling is applied to present the basic concepts of acoustic scattering and tumor growth, as well as the application of electroencephalography (EEG) and magnetoencephalography (MEG). Finally an introduction of the basic concepts of simulation and the programming package OCTAVE is presented.
Curriculum of the Module: The curriculum of MSMΒ82 includes the following topics:
Learning Outcomes: Upon successful completion of MSM82 “Applied Mathematical Standardization”, students will have developed the following skills:
General Learning Outcomes: Upon successful completion of MSM82, students will have:
Subjects Covered:
Prerequisite Courses: Successful completion of MSM71.
Evaluation: Students are assigned to submit six (6) written assignments during the academic year. The average grade of the six (6) written assignments, weighted at 30%, is taken into consideration for the calculation of the final grade. The grade of written assignments is activated only with a score equal to or above the pass level (≥5) in the final or resit exams.
The grade of the final or the resit exams shall be weighted at 70 % for the calculation of the final grade.
Students have the right to participate in the final/resit exams if (a) at least 50% of the potentially excellent grade has been obtained when adding the total of the six (6) assignments and (b) at least four (4) of the six (6) written assignments have been submitted.
Module code: MSM83
ECTS Credit Points: 20
Module Type: Optional Mandatory (Students choose between course modules MSM83 and MSM85)
Division: Pure Mathematics
Year: 2nd
Language: Greek
Module general description: The Module MSM83 discusses advanced elements of Functional Analysis and Operator Theory.
Curriculum of the Module: In particular, the curriculum of MSM83 includes the following topics:
Metric spaces (topology of metric spaces, sequences, continuity). Complete metric spaces. Fixed point theorems, Cantor, Baire. Completion of metric spaces. Normed vector spaces. Banach spaces. Continuous linear operators and linear functionals in normed spaces, norm of a linear operator, dual space and examples. Inner product spaces. Hilbert spaces (orthogonality, Riesz representation theorem, orthonormal bases). Linear operators on Hilbert spaces (adjoints, orthogonal projections, normal, hermitian and compact operators). Hahn-Banach theorem. Reflexive spaces. The principle of uniform boundedness. Open mapping theorem and bounded inverse theorem. Closed graph theorem. Locally convex spaces. Separation theorems. Continuous functionals in Schwartz spaces. Bounded linear operators on Βanach spaces (dual and compact operators). Banach algebras. Spectrum in Banach algebras. Ideals. Spectrum of bounded linear operators in Banach spaces (e.g point, approximate). Spectral theory of compact operators in Banach (or Hilbert) spaces. Unbounded operators in Hilbert spaces (closed, closable, symmetric, hermitian).
Learning Outcomes: Upon successful completion of MSM83 “Analysis”, students will have developed the following skills:
General Learning Outcomes: Upon successful completion of MSM83, students will have:
Subjects Covered:
Prerequisites: There are no prerequisite courses.
Evaluation: Students are assigned to submit six (6) written assignments during the academic year. The average grade of the six (6) written assignments, weighted at 30%, is taken into consideration for the calculation of the final grade. The grade of written assignments is activated only with a score equal to or above the pass level (≥5) in the final or resit exams.
The grade of the final or the resit exams shall be weighted at 70 % for the calculation of the final grade.
Module code: MSM84
ECTS Credit Points: 20
Module Type: Optional Mandatory (Students choose between course modules MSM84 and MSM82)
Division: Applied Mathematics
Year: 2nd
Language: Greek
Module general description: This course includes basic Probability theory and theory of Stochastic processes and also Applications in Stochastic Modelling.
Curriculum of the Module: A detailed description of the course content follows
Learning Outcomes: Upon successful completion of MSM84 “Stochastic Mathematics”, students will have developed the following skills:
General learning outcomes: Upon successful completion of MSM84, students will have obtained the following:
Subjects Covered:
Prerequisites: There are no prerequisite courses.
Evaluation: Students are assigned to submit six (6) written assignments during the academic year. The average grade of the six (6) written assignments, weighted at 30%, is taken into consideration for the calculation of the final grade. The grade of written assignments is activated only with a score equal to or above the pass level (≥5) in the final or resit exams.
The grade of the final or the resit exams shall be weighted at 70 % for the calculation of the final grade.
Module code: MSM85
ECTS Credit Points: 20
Module Type: Optional Mandatory (Students choose between course modules MSM85 and MSM83)
Division: Pure Mathematics
Year: 2nd
Language: Greek
Module general description: Module MSΜB85 discusses the basic elements of Number Theory and its applications in Cryptography, the basic elements of Group Theory and the theory of Euclidean spaces and their isometry groups.
Curriculum of the Module: In particular, the curriculum of the MSM85 includes the following topics:
Number Theory and Algebraic Structures
a). Euclidean Division – Arithmetic Algorithms – Fast Multiplication – Greatest Common Divisor – Least Common Multiple – Euclidean Algorithm – Primes– Primitive Analysis and Applications.
b). Monoids – Groups – Subgroups – Cyclic Groups – Group Morphisms – Rings – Polynomials – Greatest Common Divisor – Euclidean Algorithm – Polynomial over a Field – Irreducible Polynomials.
c). Congruences – Linear Congruences – The Euler φ function – Order of an Integer mod n – Finite Fields.
d) Integer Factorization Algorithms – Primality Tests – Αlgorithms for the Computation of Discrete Logarithm.
Cryptography and Codes
Fundamentals of Cryptology – RSA, Rabin and ElGamal cryptosystems – Digital Signatures RSA, Rabin and DSA – Diffie-Hellman key protocol– Error Correcting Codes – Linear Codes – Generator Matrices – Control Matrices – Decoding.
Affine Geometry
Affine Spaces – Barycenter – Affine Subspaces – Affine Frames – Affine Maps – Affine Groups – Multilinear Maps – Multiaffine Maps – Polynomial Curves – Berstein Polynomials – Bézier Form of a Polynomial Curve – De Casteljau Algorithm – Subdivision Algorithm – De Boor Algorithm – Derivatives of Polynomial Curves – Joining Polynomial Curves.
Learning Outcomes: Upon successful completion of MSM85 “Algebra and Geometry”, students will have developed the following skills:
General Learning Outcomes: Upon successful completion of MSM85, students will have obtained the following:
Subjects Covered:
Prerequisites: There are no prerequisite courses.
Evaluation: Students are assigned to submit six (6) written assignments during the academic year. The average grade of the six (6) written assignments, weighted at 30%, is taken into consideration for the calculation of the final grade. The grade of written assignments is activated only with a score equal to or above the pass level (≥5) in the final or resit exams.
The grade of the final or the resit exams shall be weighted at 70 % for the calculation of the final grade.
Module code: MSM86
ECTS Credit Points: 20
Module Type: Compulsory
Year: 2nd
Language: Greek
General Description: The purpose of the Module MSMDE is to synthesize knowledge acquired during the student’s studies, through the elaboration of the Postgraduate Diploma Thesis.
The topics of the Diploma Theses concern Mathematical Science and its Applications. They are related to the specialized cognitive areas of the five Modules of the postgraduate program MSM. Also, topics can be proposed by both the Coordinators and the Instructors of the Program and they are subject to the approval of the Academic Committee of the Program. The approved topics are posted on the University’s website, before the beginning of the period of submission of the relevant statement by the students.
The M.D.E. can be theoretical-synthetic, applied-experimental or a combination of the two.
A) In theoretical-synthetic theses, students should understand a scientific topic or problem, applying for its study scientific knowledge and experience gained from the Study Program, in combination with a review of proposed literature. They should also be able to write their results and present them in public. .
B) In the applied – experimental projects, students should understand an applied scientific problem, present the tools and methodology for dealing with it and elaborate the process of solving it. They should also be able to write their findings and present them with scientific competence in public.
Therefore, the Module MSMDE, through the elaboration of the Diploma Thesis, provides the opportunity for synthesis and utilization of the knowledge acquired during the studies.
The scientific responsibility for preparing the D.E. is assigned to a three-member Evaluation Committee (EC), one member of which is responsible for the supervision and support of the student (Supervisor), in accordance with the prevailing ethics and scientific practice and respecting the principles of Open and Distance Education and Adult Education.
Learning Outcomes: Upon completion of the Module MSMDE, students will be able to:
General Regulation for Preparing Graduate Dissertations in PC with an annual Module Correspondence
For more information regarding the Specifications – Useful Material for writing Master’s Theses and uploading a Thesis at the H.O.U. Repository, you can go to the Digital Training Area http://study.eap.gr and especially to the Program of Studies section.
Prerequisites: The presentation of the Postgraduate Thesis takes place after the successful completion of the program’s Course Modules.