Unified List of Graduate Courses and Thesis

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Remarks

1. The outlines are sorted based on their alphabetic order of their codes, rather than the order of their titles.
2. The outline of the Master's Thesis is located at the end of the listing.

Real Analysis (AN1)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	AN1
Semester	1
Course Title	Real Analysis
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	General Background
Prerequisite Courses	Introduction to Topology
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes

The plan of the course is the deeper study of the theory of metric spaces. The Stone - Weirstrass theorem is presented and also there are studied theorems that involve families of equicontinuous functions. Among others there are studied the following topics: the Cantor set, totally bounded and compact metric spaces, the Hausdorff metric and the Tietze theorem. Moreover, applications of the above theorems are given.

General Competences

The objective of the course is the graduate student’s ability achievement in analysis and synthesis of deeper knowledge of Real Analysis.

Syllabus

The Ascoli - Arzela and Stone - Weirstrass theorems and applications, the Cantor set, characterization of totally bounded metric spaces via subsets of Cantor set, extensions of continuous functions and the Tietze theorem, the space S(X) of closed and bounded subsets of a metric space and the metric Hausdorff on S(X), characterization of completeness of the metric space S(X) equipped with the metric Hausdorff and applications, the selection Blashke theorem, applications of the fixed point theorem of Banach, partitions of unity.

Teaching and Learning Methods - Evaluation

Delivery	Teaching with talks on the blackboard.
Use of Information and Communications Technology	-
Teaching Methods	Activity Semester Workload Lectures 39 Individual study 110 Exercises solving 38.5 Course total 187.5
Student Performance Evaluation	Final written exam or student's presentations on the blackboard. The student can choose either of the above ways of examination or both with final grade the higher one.

Attached Bibliography

• Charalambos D. Aliprantis, Owen Burkinshaw, Principles of Real Analysis, Academic Press.
• Michael O Searcoid, Metric Spaces, Springer Undergraduate Mathematics Series.

General Topology (AN2)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	AN2
Semester	1
Course Title	General Topology
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	General Background
Prerequisite Courses	-
Language of Instruction and Examinations	Language of Instruction (lectures): Greek Language of Instruction (activities other than lectures): Greek and English Language of Examinations: Greek and English
Is the Course Offered to Erasmus Students	Yes
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes

Using the Bloom Taxonomy. All the following sets are considered to be arbitrary subsets of an arbitrary Euclidean normed space of finite dimension. Remembering: Topological spaces, open and closed sets, interior and closure of sets. Continuous functions in topological spaces. Axioms of separation. Convergence in topological spaces. Metric spaces and metrizable spaces. Dimension of topological spaces. Dimension of metrizable space. Comprehension: Methods of generating topologies. Homeomorphisms. Frechet spaces. Operations on topological spaces. Functions spaces. Compact spaces, locally compact spaces, compactifications, countably compact spaces, pseudocompact spaces, sequentially compact spaces. Totally bounded and complete metric spaces. Paracompact spaces, countably paracompact spaces. Connected spaces, kinds of disconnectedness. Uniform spaces, totally bounded, complete and compact uniform spaces, proximity spaces. Applying: Thorough study of topological spaces. Thorough study of continuous functions in topological spaces. Evaluating: Teaching undergraduate courses.

General Competences

Production of free, analytic and inductive thinking. Required for the production of new ideas. Working independently. Team work. Decision making.

Syllabus

Topological spaces, methods of generating topologies, continuous mappings, axioms of separation, Frechet spaces, subspaces, Cartesian products, quotient spaces, function spaces, compact spaces, locally compact spaces, compactifications, countably compact spaces, pseudocompact spaces, sequentially compact spaces, totally bounded and complete metric spaces, paracompact spaces, countably paracompact spaces, connected spaces, kinds of disconnectedness, dimension of topological spaces and its basic properties, uniform spaces, totally bounded, complete and compact uniform spaces, proximity spaces.

Teaching and Learning Methods - Evaluation

Delivery	Lectures in class. Teaching is assisted by Learning Management System. Teaching is assisted by the use of online forums where students can participate in order to improve their problem solving skills, as well as their understanding of the theory they are taught. Teaching is assisted by the use of pre-recorded videos.
Use of Information and Communications Technology	Use of Learning Management System, combined with File Sharing Platform as well as Blog Management System for distributing teaching material, submission of assignments, course announcements, gradebook keeping for all students evaluation procedures, and communicating with students. Use of Appointment Scheduling System for organising appointments between students and the teacher. Use of Survey Web Application for submitting anonymous evaluations regarding the teacher. Use of Wiki Engine for publishing manuals regarding the regulations applied at the exams processes, the way teaching is organized, the grading methods, as well as the use of the online tools used within the course.
Teaching Methods	Activity Semester Workload Lectures 39 Study and analysis of bibliography 78 Preparation of assignments and interactive teaching 70.5 Course total 187.5
Student Performance Evaluation	Language of evaluation: Greek and English. Methods of evaluation: Weekly presentations - oral exams, combined with weekly written assignments. In any case, all students can participate in written exams at the end of the semester. The aforementioned information along with all the required details are available through the course’s website. The information is explained in detail at the beginning of the semester, as well as, throughout the semester, during the lectures. Reminders are also posted at the beginning of the semester and throughout the semester, through the course’s website. Upon request, all the information is provided using email or social networks.

Attached Bibliography

1. Ryszard Engelking - General Topology.
2. James Munkres - Topology.
3. John Kelley - General Topology.

Complex Analysis (AN3)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	AN3
Semester	2
Course Title	Complex Analysis
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	General Background
Prerequisite Courses	Introduction to Complex Analysis (undergraduate)
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes

The course aims, firstly, to provide a more complete picture of the subject matter of Complex Analysis, and, secondly, to highlight the impact of its results concerning the properties of various functions of real variables and its relation – mainly through the notion of a harmonic function - to other areas of Mathematics, such as Harmonic Analysis, Geometry and Partial Differential Equations, but also to present some applications of Complex Analysis within various fields of the Natural Sciences. Concerning the skills and competences which the students will acquire, the subject is especially suitable for highlighting the connections between various mathematical areas, the power of generalization of a notion in order to understand the properties of a certain subcase of it, and the usefulness of looking at a subject from different points of view.

General Competences

Search for, analysis and synthesis of data and information Adapting to new situations Decision-making Working independently Criticism and self-criticism Production of free, creative and inductive thinking

Syllabus

Holomorphic, entire, and meromorphic functions. Conformal mappings. Analytic continuation. Weierstrass’ Convergence Theorem. The Gamma function. Infinite Products. The Riemann Mapping Theorem. Harmonic functions and applications.

Teaching and Learning Methods - Evaluation

Delivery	Face-to-face
Use of Information and Communications Technology	-
Teaching Methods	Activity Semester Workload Lectures 39 Self-study 78 Homework 70.5 Course total 187.5
Student Performance Evaluation	The evaluation is carried out as a combination of Written exam. Assigned homework. Presentation and oral examination.

Attached Bibliography

• J. Bak and D. J. Newman, Complex Analysis (3rd ed.), Springer, 2010.
• S. Lang, Complex Analysis (4th ed.), Springer, 1999.
• I. Markushevich, Theory of Functions of a Complex Variable (2nd ed.), Vol. 1-3, AMS Chelsea, 2011.
• I. Markushevich, The Theory of Analytic Functions: A Brief Course, Mir Publishers, 1983.
• R. Remmert, Theory of Complex Functions, Springer, 1990.
• R. Remmert, Classical Topics in Complex Function Theory, Springer, 1998.
• K. Jänich, Funktionentheorie (6te Aufl.), Springer, 2011.

Functional Analysis (AN4)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	AN4
Semester	1
Course Title	Functional Analysis
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	General Background
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes	The objectives of the course are: The acquisition of background from the students on the basic structures and techniques of Functional Analysis, as independent knowledge as well as a tool for the other branches of Analysis, so that they will have the potential to apply the knowledge they get in applications.
General Competences	The course aims to give the postgraduate student the ability to analyse and synthesize advanced concepts of Functional Analysis. The goal is to acquire the skills for autonomous work and teamwork in an interdisciplinary environment and to able to produce new research ideas.

Syllabus

1. Normes spaces, Banach spaces and Hilbert spaces, classical examples (sequence spaces and function spaces). Basic theorems.
2. General theory of topological vector spaces, locally convex spaces, separation theorems.
3. Weak topologies, theorems of Mazur, Alaoglu and Goldstine, weak compactness.
4. Schauder bases and basic sequences.
5. Extreme points, Krein Milman theorem.
6. Riesz representation theorem, Lp spaces.
7. Fixed point theorems.

Teaching and Learning Methods - Evaluation

Delivery	Face to face
Use of Information and Communications Technology	-
Teaching Methods	Activity Semester Workload Lectures 39 Study and analysis of bibliography 78 Preparation of assignments and interactive teaching 70.5 Course total 187.5
Student Performance Evaluation	Written examination at the end of the semester.

Attached Bibliography

• J. Bak and D. J. Newman, Complex Analysis (3rd ed.), Springer, 2010.
• S. Lang, Complex Analysis (4th ed.), Springer, 1999.
• I. Markushevich, Theory of Functions of a Complex Variable (2nd ed.), Vol. 1-3, AMS Chelsea, 2011.
• I. Markushevich, The Theory of Analytic Functions: A Brief Course, Mir Publishers, 1983.
• R. Remmert, Theory of Complex Functions, Springer, 1990.
• R. Remmert, Classical Topics in Complex Function Theory, Springer, 1998.
• K. Jänich, Funktionentheorie (6te Aufl.), Springer, 2011.

Differential Equations (AN5)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	AN5
Semester	2
Course Title	Differential Equations
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Special Background
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes

This course is aiming at familiarizing the students with a variety of advanced subjects related to differential equations. Both classical and modern subjects are studied. After attending this course, the students should be able to: be familiar with a vast set of subjects related to differential equations, start researching on subjects regarding the qualitative theory of differential equations, and familiarize himself with the bibliography related to the subjects he was taught.

General Competences

Working independently. Team work. Production of new research ideas. Production of free, creative and inductive thinking. Search for, analysis and synthesis of data and information, with the use of the necessary technology. Develop critical thinking skills.

Syllabus

Second order linear ordinary differential equations: Sturm-type theorems, oscillation and non-oscillation theorems. Reducing differential equation problems to integral ones. Volterra integral equations: existence and uniqueness of solutions. Existence of solutions. The linear equation. The first order linear equation. Some problems on the semi-axis. Fredholm theory for linear integral equations: the resolvent kernel. The entire functions of Fredholm and their applications. Eigenvalues, eigenfunctions and conjugate functions. Some integral inequalities: Gronwall and Bihari Lemmas, and their applications. Delay differential equations: Introduction, Examples and the stepping method. Some remarkable examples and some "wrong" questions. Lipschitz condition and uniqueness for the basic initial problem. Notations and uniqueness for systems with bounded delay. Existence for systems with bounded delay. Linear delay differential systems: superposition. Fixed coefficients. Variation of parameters. Stability for delayed differential systems: Definitions and examples. Asymptotic stability. Linear and almost linear differential systems. Fractional differential equations: Definitions and basic calculus. Initial and boundary systems. Dynamical systems: definitions and calculus. Equations and problems. Various subjects.

Teaching and Learning Methods - Evaluation

Delivery	Lectures on class
Use of Information and Communications Technology	-
Teaching Methods	Activity Semester Workload Lectures 39 Exercises / Homework 52.5 Autonomous Study 96 Course total 187.5
Student Performance Evaluation	The students can choose one of the following options: Presentation in the class - Written homework - Exercises. Written final exam. If both methods are used, then the final grade is the maximum of the two. The criteria regarding the grading are publised in the "E-Course" platform.

Attached Bibliography

• C. Corduneanu, Principles of Differential and Integral Equations
• R. D. Driver, Ordinary and Delay Differential Equations
• T. A. Burton, Volterra Integral and Differential Equations
• R. K. Miller, Nonlinear Volterra Integral Equations
• P. Hartman, Ordinary Differential Equations
• Κ. Diethelm, The Analysis of Fractional Differential Equations
• Y. Zhou, Basic Theory of Fractional Differential Equations
• M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications.

Partial Differential Equations (AN6)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	AN6
Semester	2
Course Title	Partial Differential Equations
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	General Background
Prerequisite Courses	Vector Analysis (undergraduate), Real Analysis, Functional Analysis
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes

The course, apart from the instruction in the classical quartet of Partial Differential Equations (PDEs) (transport, Laplace, heat and wave) in several space variables, aims, first, to highlight the modern, analytic approach to the theory of PDEs and the reasons that suggest it, and, second, to provide an introduction to nonlinear PDEs, especially concerning first-order and hyperbolic equations. The skills and competences the students will acquire concern, on the one hand, the paradigmatic transition from the resolution of a problem to the theoretical analysis of its properties and the investigation of its structural foundation, and, on the other hand, the recognition of the essential difference between linear and nonlinear problems and the limitations of the method of approximation of a nonlinear problem by linear problems.

General Competences

Search for, analysis and synthesis of data and information Adapting to new situations Decision-making Working independently Working in an interdisciplinary environment Criticism and self-criticism Production of free, creative and inductive thinking

Syllabus

Transport, Laplace, heat and wave equations for several space variables. Nonlinear first-order equations (method of characteristics, introduction to Hamilton-Jacobi equations and to conservation laws, weak solutions). The Cauchy-Kovalevskaya Theorem. Sobolev Spaces and weak derivatives. Theory of second-order linear equations. Semigroup Theory. Nonlinear hyperbolic and dispersion equations.

Teaching and Learning Methods - Evaluation

Delivery	Face-to-face
Use of Information and Communications Technology	-
Teaching Methods	Activity Semester Workload Lectures 39 Self-study 78 Homework 70.5 Course total 187.5
Student Performance Evaluation	The evaluation is carried out as a combination of Written exam. Assigned homework. Presentation and oral examination.

Attached Bibliography

• H.Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, 2011
• L.C. Evans, Partial Differential Equations (2nd ed.). AMS, 2010
• G. Folland, Introduction to Partial Differential Equations, Princeton University Press, 1976
• L. Hörmander, The Analysis of Linear Partial Differential Operators, Vol. 1-4, Springer, 1983-85
• J. Jost, Partial Differential Equations (2nd ed.), Springer, 2007
• T. Tao, Nonlinear Dispersive Equations: Local and Global Analysis, CBMS, AMS, 2006
• M. Taylor, Partial Differential Equations, Vol. I-III, Springer, 1996
• G.B.Whitham, Linear and Nonlinear Waves, Wiley-Interscience, 1974.

Measure Theory (AN7)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	AN7
Semester	1
Course Title	Measure Theory
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	General Background
Prerequisite Courses	-
Language of Instruction and Examinations	Language of Instruction (lectures): Greek Language of Instruction (activities other than lectures): Greek and English Language of Examinations: Greek and English
Is the Course Offered to Erasmus Students	Yes
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes

Using the Bloom Taxonomy. All the following sets are considered to be arbitrary subsets of an arbitrary Euclidean normed space of finite dimension. Remembering: The notion of the rectangle and the notion of the volume of a rectangle. The notion of the outer measure. The notion of the Lebesgue measure. The notion of the σ-Algebra. The Borel set. The notion of the characteristic function, of the step function, of the simple function and of the measurable function. The “almost everywhere” validity of a property. The notion of the Lebesgue integral. Definition of the L1 space, of the integrable functions. The notion of the absolutely continuous function. The notion of the locally integrable function. The notion of the Lebesgue density. The Lebesgue set of a locally integrable function. Good kernels and approximations to the identity. The notion of the bounded variation function. The notion of abstract measurable spaces. The Caratheodory measurable sets. The metric outer measures. The notion of the pre-signed measure. Comprehension: The Cantor set. Properties of the outer measure. Properties of the Lebesgue measure. Translation invariance property of the Lebesgue measure. Conditions under which sets are measurable. Construction of non-measurable sets. Properties of measurable functions. Approximation of measurable functions by simple or step functions. The three principles of Littlewood. Brunn – Minkowskii inequality. Properties of the Lebesgue integral. Relation between the Lebesgue integral and the Riemann integral. Fatou Lemma. Uniform convergence theorem. Riesz – Fischer theorem. Translation invariance property of the Lebesgue integral. Fubini theorem. Relation between integrable and measurable function. The Hardy - Littlewood maximal function. Properties of bounded variation functions. Properties of absolutely continuous and differentiable functions. Properties of abstract measurable spaces. Integration in abstract measurable spaces. Absolute continuity of measures. Applying: Calculating the measure of a set. Finding examples of non-measurable sets. Calculating the Lebesgue integral. Finding the mean value of a function. Evaluating: Teaching undergraduate courses.

General Competences

Production of free, analytic and inductive thinking. Required for the production of new ideas. Working independently. Team work. Decision making.

Syllabus

Measure spaces, Lebesgue measure, measurable functions and Lebesgue integral, Monotone convergence Theorem and Dominated convergence Theorem, relation between Riemann and Lebesgue integral. Product measures, Fubini Theorem. L^p spaces. Signed measures, Hahn decomposition, Radon-Nikodym Theorem. Convergence of sequences of measurable functions.

Teaching and Learning Methods - Evaluation

Delivery	Lectures in class. Teaching is assisted by Learning Management System. Teaching is assisted by the use of online forums where students can participate in order to improve their problem solving skills, as well as their understanding of the theory they are taught. Teaching is assisted by the use of pre-recorded videos.
Use of Information and Communications Technology	Use of Learning Management System, combined with File Sharing Platform as well as Blog Management System for distributing teaching material, submission of assignments, course announcements, gradebook keeping for all students evaluation procedures, and communicating with students. Use of Appointment Scheduling System for organising appointments between students and the teacher. Use of Survey Web Application for submitting anonymous evaluations regarding the teacher. Use of Wiki Engine for publishing manuals regarding the regulations applied at the exams processes, the way teaching is organized, the grading methods, as well as the use of the online tools used within the course.
Teaching Methods	Activity Semester Workload Lectures 39 Study and analysis of bibliography 78 Preparation of assignments and interactive teaching 70.5 Course total 187.5
Student Performance Evaluation	Language of evaluation: Greek and English. Methods of evaluation: Weekly presentations - oral exams, combined with weekly written assignments. In any case, all students can participate in written exams at the end of the semester. The aforementioned information along with all the required details are available through the course’s website. The information is explained in detail at the beginning of the semester, as well as, throughout the semester, during the lectures. Reminders are also posted at the beginning of the semester and throughout the semester, through the course’s website. Upon request, all the information is provided using email or social networks.

Attached Bibliography

• H.Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, 2011
• L.C. Evans, Partial Differential Equations (2nd ed.). AMS, 2010
• G. Folland, Introduction to Partial Differential Equations, Princeton University Press, 1976
• L. Hörmander, The Analysis of Linear Partial Differential Operators, Vol. 1-4, Springer, 1983-85
• J. Jost, Partial Differential Equations (2nd ed.), Springer, 2007
• T. Tao, Nonlinear Dispersive Equations: Local and Global Analysis, CBMS, AMS, 2006
• M. Taylor, Partial Differential Equations, Vol. I-III, Springer, 1996
• G.B.Whitham, Linear and Nonlinear Waves, Wiley-Interscience, 1974.

Harmonic Analysis (AN8)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	AN8
Semester	2
Course Title	Harmonic Analysis
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Special Background
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes	The plan of the course is the achievement by the graduate student of the theoretical background in the theory of differentiation bases on Euclidean spaces, their connection with maximal operators and the study of fractal sets. Also the study of general theory of maximal operators will be considered.
General Competences	The objective of the course is the graduate student’s ability achievement in analysis and synthesis of the basic background in the area of Harmonic Analysis connected with differentiation bases on Euclidian spaces and geometric measure theory on the plane. Also connections with maximal operators will be given.

Syllabus

Busemman – Feller differentiation bases on Euclidian spaces and associated maximal operators, covering lemmas and applications to the behavior of maximal operators, connection of differentiation bases of certain spaces with respective properties of the related maximal operators, the basis B2 of intervals on Euclidian spaces and its differentiation properties, covering properties of the basis B2, the basis B3 of rectangles on Euclidian spaces: The Perron tree, Fefferman’s lemma, Besicovitch and Kakeya sets, the Nikodym set, subbases of B3 and differentiation properties, Hausdorff measure and dimension on the plane, fractal sets and densities, regular and irregular sets, tangency and projection properties, the theory of maximal operators from the general point of view.

Teaching and Learning Methods - Evaluation

Delivery	Teaching with talks on the blackboard.
Use of Information and Communications Technology	-
Teaching Methods	Activity Semester Workload Lectures 39 Independent study 110 Exercises solutions 38.5 Course total 187.5
Student Performance Evaluation	Combination of writing and oral examination at the end of the semester.

Attached Bibliography

• M. De Guzman, Real Variable Methods in Fourier Analysis, North Holland - Mathematical Studies.

Operator Theory (AN9)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	AN9
Semester	2
Course Title	Operator Theory
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Specialized general knowledge
Prerequisite Courses	Functional Analysis
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes	The aim of this course is for postgraduate students to acquire a special background in Operator Theory in general Banach spaces and in particular in Hilbert spaces
General Competences	The course aims to enable the graduate student to acquire the ability to analyze and synthesize advanced concepts of Operator Theory. The goal is to acquire the resources for independent and group work in an interdisciplinary environment.

Syllabus

Bounded linear operators on Banach spaces and Hilbert spaces. Spectrum of an operator, the spectrum of a self-adjoint operator. Functions of self-adjoint operators, spectral theorem. Topologies in operator spaces.

Teaching and Learning Methods - Evaluation

Delivery	Teaching on the blackboard
Use of Information and Communications Technology	Communication with the students via e-mail
Teaching Methods	Activity Semester Workload Lectures 39 Individual study 78 Resolving exercises and assignments 70.5 Course total 187.5
Student Performance Evaluation	Written exam at the end of the semester (mandatory), delivery of assignments and exercises during the semester (mandatory), lecture-presentation on the board by the student (optional).

Attached Bibliography

• Y. Abramovic C. Aliprantis, An invitation to Operator Theory.
• J. Conway, A course in Functional Analysis.
• R. Douglas, Banach Algebra Techniques in Operator Theory.
• V. Sunder Functional Analysis, Spectral Theory.
• W. Rudin Functional Analysis.

Topological Methods in Differential Equations (AN10)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	AN10
Semester	2
Course Title	Topological Methods in Differential Equations
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Specialized general knowledge
Prerequisite Courses	Differential Equations, General Topology, Functional Analysis, Real Analysis
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes	Knowledge of topics in functional analysis with application in differential equations. Ability to start research in problems related to qualitative theory of differential equations. Become familiar with research bibliography concerning qualitative theory in a wide sector of differential equations.
General Competences	Search for, analysis and synthesis of data and information, with the use of the necessary technology. Production of new research ideas. Contact research bibliography concerning qualitative theory in a wide sector of differential equations.

Syllabus

Application of topological fixed point theorems in the theory of differential equations, contraction theorems, theorems of Schauder, Schaefer, degree theory, nonlinear alternative, fixed point theorems in cones, Krasnoselskii’s theorems, theorems of Leggett-Williams type. Applications in initial value and boundary value problems, in integro-differential equations and functional differential equations. Existence of solutions, of positive solutions, of upper and lower solutions.

Teaching and Learning Methods - Evaluation

Delivery	Face to face
Use of Information and Communications Technology	-
Teaching Methods	Activity Semester Workload Lectures, seminars 45 Exercises, Projects 52.5 Personal study 90 Course total 187.5
Student Performance Evaluation	Problem solving, written work, essay/report, oral or/and written examination, public presentation.

Attached Bibliography

• H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev., 18, No. 4 ,1976 (pages 620-709)
• K. Deimling, Nonlinear functional analysis, Springer-Verlag, New York,1985
• R. D. Driver, Ordinary and delay differential equations, Springer Verlag, New York, 1976
• D. Guo and V. Lakshmikantham, Nonlinear problems in abstract cones, Academic Press, San Diego,1988
• J. K. Hale and S. M. V. Lunel, Introduction to functional differential equations, Springer Verlag, New York, 1993.

Convex Analysis (AN11)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	AN11
Semester	2
Course Title	Convex Analysis
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Specialized general knowledge
Prerequisite Courses	Real Analysis, Calculus I and Calculus II
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes

The course aims to an introduction to convex analysis at graduate level. Material varies from classical topics on convex analysis to recent research problems. The students should: get knowledge on issues from a wide area of topics on convex analysis, aim ability to start research on problems on the theory of convex analysis, be introduced to the literature on problems in the area of convex analysis that he/she was taught.

General Competences

Working independently. Team work. Production of free, creative and inductive thinking. Production of analytic and synthetic thinking. Search for, analysis and synthesis of data and information, with the use of the necessary technology. Get in touch with specialized knowledge and evolve abilities for comparing, obtaining and evaluating results on the specific area of interest.

Syllabus

Basic notions. Convex functions and convex sets, convexity criteria. Normed spaces. Dual spaces and the Legendre transform. The Caratheodory Theorem and its applications to geometry. Radon’s and Helly’s theorems. Minkowski’s First theorem and its applications to Optimization Theory. The concentration of measure phenomenon on the sphere. Dvoretzky’s theorem and the Quotient of Subspace theorem. The Brunn-Minkowski inequality and its generalizations (Lp variants and functional forms). Mixed volumes and inequalities of Aleksandrov-Fenchel type. Inequalities of isoperimetric type (e.g. the classical isoperimetric inequality and the Blaschke-Santalo inequality). The Brascamp-Lieb inequality and reverse isoperimetric inequalities. Area measures of convex hypersurfaces. The Minkowski Existence and Uniqueness problem and its generalizations, applications to the Theory of Monge-Ampere equations. Classical open problems.

Teaching and Learning Methods - Evaluation

Delivery	Lectures/ Class presentations
Use of Information and Communications Technology	-
Teaching Methods	Activity Semester Workload Lectures/Presentations 45 Assignments/Essays 52.5 Individual study 90 Course total 187.5
Student Performance Evaluation	Students choose evaluation by one or both of the following: Class presentation - Essays - Assignments Final Written Examination In case that a student participates to both, the final grade is the maximum of the two grades. Evaluation criteria and all steps of the evaluation procedure will be accessible to students through the platform “E-course” of the University of Ioannina.

Attached Bibliography

• J. Bakelman, Convex Analysis And Nonlinear Geometric Elliptic Equations
• R. J. Gardner, Geometric tomography. Second edition.
• H. Groemer, Geometric Applications of Fourier Series and Spherical Harmonics.
• Koldobsky, Fourier Analysis in Convex Geometry.
• M. Ledoux, The Concentration of Measure Phenomenon.
• V.D. Milman and G. Schechtman, Asymptotic Theory of Finite Dimensional Normed Spaces
• R. Tyrel Rockafellar, Convex Analysis.
• R. Schneider, Convex bodies: the Brunn-Minkowski theory. Second expanded edition.
• R. Schneider and W. Weil, Stochastic and Integral Geometry.
• C. Thompson, Minkowski Geometry.

Independent Study in Analysis I (AN12)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	AN12
Semester	1
Course Title	Independent Study in Analysis I
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Special Background
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes

Th course aims at introducing the students at the advanced subjects of Mathematical Analysis, which are covered by the rest of the graduate courses. The syllabus is decided by the professors, who may be members of the faculty, or scientists from Greece or from abroad (i.e. visiting professors from other Greek or non-Greek academic institutions, professors emeritus, invited speakers etc), and may contain classical theoretical subjects or applied subjects from Mathematical Analysis or other modern research fields. After attending this course, the students should be able to: be familiar with a vast set of subjects related to differential equations, start researching on subjects regarding the qualitative theory of differential equations, and familiarize himself with the bibliography related to the subjects he was taught.

General Competences

Working independently. Team work. Production of new research ideas. Production of free, creative and inductive thinking. Search for, analysis and synthesis of data and information, with the use of the necessary technology. Develop critical thinking skills.

Syllabus

Will be decided by the teaching professor.

Teaching and Learning Methods - Evaluation

Delivery	Will be decided by the teaching professor.
Use of Information and Communications Technology	-
Teaching Methods	Activity Semester Workload Lectures 39 Exercises / Homework 52.5 Autonomous Study 96 Course total 187.5
Student Performance Evaluation	Will be decided by the teaching professor. The criteria will be published at the course's webpage.

Attached Bibliography

Θα καθορίζεται από τον διδάσκοντα / Will be determined by the teacher

Independent Study in Analysis II (AN13)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	AN13
Semester	2
Course Title	Independent Study in Analysis II
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Special Background
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes

Th course aims at introducing the students at the advanced subjects of Mathematical Analysis, which are covered by the rest of the graduate courses. The syllabus is decided by the professors, who may be members of the faculty, or scientists from Greece or from abroad (i.e. visiting professors from other Greek or non-Greek academic institutions, professors emeritus, invited speakers etc), and may contain classical theoretical subjects or applied subjects from Mathematical Analysis or other modern research fields. After attending this course, the students should be able to: be familiar with a vast set of subjects related to differential equations, start researching on subjects regarding the qualitative theory of differential equations, and familiarize himself with the bibliography related to the subjects he was taught.

General Competences

Working independently. Team work. Production of new research ideas. Production of free, creative and inductive thinking. Search for, analysis and synthesis of data and information, with the use of the necessary technology. Develop critical thinking skills.

Syllabus

Will be decided by the teaching professor.

Teaching and Learning Methods - Evaluation

Delivery	Will be decided by the teaching professor.
Use of Information and Communications Technology	-
Teaching Methods	Activity Semester Workload Lectures 39 Exercises / Homework 52.5 Autonomous Study 96 Course total 187.5
Student Performance Evaluation	Will be decided by the teaching professor. The criteria will be published at the course's webpage.

Attached Bibliography

Θα καθορίζεται από τον διδάσκοντα / Will be determined by the teacher

Algebra I (ΑΛ1)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	AΛ1
Semester	1
Course Title	Algebra I
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Special Background
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes	The objectives of the course are: The postgraduate student to reach a good level of theoretical background on topics related to the theory of group actions, the Sylow theorems and the general theory of modules over associative rings.
General Competences	The aim of the course is to enpower the postgraduate student to analyse and compose basic notions of advanced Algebra. This will allow him to work in an international interdisciplinary environment.

Syllabus

Group actions on a set, Sylow theorems and applications, Direct and semidirect products, Finitely generated abelian groups, Free groups, Amalgamated free product of groups, Jordan-Hoelder theorem, Modules and homomorphisms between modules, Free modules, Direct sum and product of modules, Exact sequences and functors, Noetherian rings and modules, Semisimple rings and modules, Elements of multilinear and tensor algebra.

Teaching and Learning Methods - Evaluation

Delivery	Face to face
Use of Information and Communications Technology	-
Teaching Methods	Activity Semester Workload Lectures 39 Study and analysis of bibliography 78 Preparation of assignments and interactive teaching 70.5 Course total 187.5
Student Performance Evaluation	Written exam at the end of semester (obligatory), problem solving or/and intermediate exams (optional).

Attached Bibliography

Θα καθορίζεται από τον διδάσκοντα / Will be determined by the teacher

Algebra II (ΑΛ2)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	ΑΛ2
Semester	2
Course Title	Algebra II
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	General Background
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes

The main purpose of the course is to introduce the student to the basic concepts, results, tools and methods of the Representation Theory of Finite Groups and its applications to other areas of Mathematics, mainly in Group Theory, and other related sciences, e.g. in Physics. At the end of the course, we expect the student to understand the basic concepts and the main theorems that are analysed in the course, to understand how these are applied to concrete examples arising from different thematic areas of Mathematics and related sciences, to be able to apply them to derive new elementary consequences in various fields, and finally to be able to perform some (not so obvious) calculations related to several problems arising in Group Theory.

General Competences

The course aims at enabling the graduate student to acquire the ability to analyse and synthesize basic knowledge of the basic Representation Theory of Finite Groups, which is an important part of modern Mathematics with numerous applications to other sciences, for instance in Physics. When the graduate student comes in for the first time in connection with the basic notions of representation theory and its applications to group theory, (s)he strengthens her/his creative, analytical and inductive thinking, and her/his ability to apply abstract knowledge in different areas of central interest in Mathematics and related sciences.

Syllabus

• Representations and characters of groups.
• Groups and homomorphisms.
• FG-modules και group-algebras.
• Schur’s Lemma and Maschke’s Theorem.
• Group-algebras and irreducible modules.
• Conjugacy classes and characters.
• Character tables and orthogonality relations.
• Normal subgroups and lifted characters.
• Elementary examples of characters tables.
• Tensor products. Restricting representations to subgroups.
• Applications.

Teaching and Learning Methods - Evaluation

Delivery	Face to face
Use of Information and Communications Technology	-
Teaching Methods	Activity Semester Workload Lectures 39 Student's study Hours 78 Exercises: Problem Solving 70.5 Course total 187.5
Student Performance Evaluation	The evaluation is based on the combined performance of the graduate student in: Weekly homeworks, Presentations during the semester, Major Homework at the end of the course, Written examination at the end of the courses in Greek with questions and problems of development of theoretical topics and problem solving.

Attached Bibliography

• J.P. Serre: “Linear Representations of Finite Groups”, Springer-Verlag, (1977).
• B. Steinberg: “Representation Theory of Finite Groups: An Introductory Approach”, Springer, (2012).
• C.W. Curtis and V. Reiner: “Methods of Representation Theory: With Applications to Finite Groups and Orders”, Wiley, (1981).
• P. Etingof et al: “Introduction to Representation Theory”, Student Mathematical Library 59, AMS, (2011).
• J.L. Alperin and R.B. Bell: “Groups and Representations”, Springer (1995).
• M. Burrow: “Representation Theory of Finite Groups”, Academic Press, (1965).
• M. Liebeck and G. James: “Representations and Characters of Groups”, CUP, (2001).

Applied Algebra (ΑΛ3)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	ΑΛ3
Semester	2
Course Title	Applied Algebra
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	General Background
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes

The main purpose of the course is to introduce the student to the basic concepts, results, tools and methods of the theory of polynomials over finite fields and its applications to algebraic cryptography and coding theory, using tools from the theory of algebraic curves. In addition, the elementary theory of elliptic curves is developed and several applications are given to various areas of Mathematics and other sciences. At the end of the course, we expect the student to understand the basic concepts and the main theorems that are analysed in the course, to understand how these are applied to concrete examples arising in Mathematics and related sciences, to be able to apply them to derive new elementary consequences in various thematic fields, and finally to be able to perform some (not so obvious) calculations related to the construction and analysis of algebraic codes and encrypted messages.

General Competences

The course aims at enabling the graduate to acquire the ability to analyse and synthesize basic knowledge of the theory of polynomials over finite fields in connection with the basic elements of algebraic curves, in particular of elliptic curves, which is an important part of modern Mathematics with numerous applications in other sciences, with a view to applications in coding theory and algebraic cryptography. In particular, in the course are analyzed: the basic theory of codes (linear and cyclic codes), the elementary theory of elliptic curves and their applications to cryptography. When the graduate comes in for the first time in connection with the basic notions of coding theory and the central concepts of elliptic curves and their applications to contemporary cryptography, (s)he strengthens her/his creative, analytical and inductive thinking, and her/his ability to apply abstract knowledge in different areas of central interest with numerous applications in everyday life.

Syllabus

• Finite Fields and Polynomials: (1) Rings, ideals, homomorphisms, polynomials, fields, algebraic extensions, (2) Finite fields, irreducible polynomials over finite fields, factorization of polynomials over finite fields, and (3) Reminders from elementary number theory.
• The null-space of a matrix. Linear and cyclic codes.
• Algebraic cryptography.
• Basic theory of algebraic curves.
• Elliptic curves.
• Applications of elliptic curves to algebraic cryptography.

Teaching and Learning Methods - Evaluation

Delivery	Face to face
Use of Information and Communications Technology	-
Teaching Methods	Activity Semester Workload Lectures 39 Student's study Hours 78 Exercises: Problem Solving 70.5 Course total 187.5
Student Performance Evaluation	The evaluation is based on the combined performance of the graduate student in: Weekly homeworks, Presentations during the semester, Major Homework at the end of the course, Written examination at the end of the courses in Greek with questions and problems of development of theoretical topics and problem solving.

Attached Bibliography

• N. Koblitz: “Algebraic aspects of cryptography”, Springer-Verlag, (1998).
• Δ. Πουλάκης: “Κρυπτογραφία”, Εκδόσεις Ζήτη, (2004).
• Δ. Πουλάκης: “Γεωμετρία των Αλγεβρικών Καμπυλών”, Εκδόσεις Ζήτη, (2006).
• Ι. Αντωνιάδης και Α. Κοντογεώργης: “Πεπερασμένα Σώματα και Κρυπτογραφία”, Σύνδεσμος Ελληνικών Ακαδημαϊκών Βιβλιοθηκών, (2015).
• I.F. Blake, G. Seroussi, and N. Smart: “Elliptic Curves in Cryptography”, Lecture Note Series. Cambridge University Press, (1999).
• N. Koblitz: “A Course in Number Theory and Cryptography”, Springer-Verlag, (1994).

Algebraic Number Theory (ΑΛ4)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	ΑΛ4
Semester	2
Course Title	Algebraic Number Theory
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Special Background
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes	The aim of the course is the postgraduate student to reach a good level of theoretical background on topics related to the algebraic number theory.
General Competences	The aim of the course is to empower the postgraduate student to analyse and compose advanced notions of Algebraic number theory.

Syllabus

Dedekind domains, norm, discriminant, finiteness of class number, Dirichlet unit theorem, quadratic and cyclotomic extensions, quadratic reciprocity, completions and local fields.

Teaching and Learning Methods - Evaluation

Delivery	Face-to-face
Use of Information and Communications Technology	-
Teaching Methods	Activity Semester Workload Lectures 39 Study of theory 78 Solving of exercises 70.5 Course total 187.5
Student Performance Evaluation	Written exam at the end of semester (obligatory) , problem solving or/and intermediate exams (optional).

Attached Bibliography

• Milne, James S., Algebraic Number Theory (v3.07), (2017). Available at www.jmilne.org/math/.
• Jarvis Frazer, Algebraic Number Theory, Springer, 2014.

Homological Algebra (ΑΛ5)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	ΑΛ5
Semester	2
Course Title	Homological Algebra
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	General Background
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes

The main purpose of the course is to introduce the main concepts, tools and methods of Homological Algebra, and to indicate some direct applications in various areas of Mathematics and other related sciences. At the end of the course, we expect the student to understand the basic concepts and the main theorems that are analysed in the course, to understand how these are applied to concrete examples arising from different thematic contexts, to be able to apply them to derive new elementary consequences in various areas, and finally to be able to perform some (not so obvious) calculations related to the thematic core of the course.

General Competences

The course aims at enabling the graduate to acquire the ability to analyse and synthesize basic knowledge of Homological Algebra, which is an important part of modern Mathematics with numerous applications in other sciences. When the graduate comes in for the first time in connection with the basic notions of Homological Algebra, (s)he strengthens her/his creative, analytical and inductive thinking, and her/his ability to apply abstract knowledge in different thematic fields.

Syllabus

• Basic concepts and results from Ring Theory.
• Introduction to Module Theory.  
• Fundamental constructions of modules.
• Introduction to the basic elements of Category Theory.
• Projective, injective and flat modules.
• Complexes and (Co)Homology.
• Projective and Injective Resolutions.
• Derived Functors.
• Ext and Tor. 
• Homological Dimension. 
• Applications of Homological Algebra.

Teaching and Learning Methods - Evaluation

Delivery	Face to face
Use of Information and Communications Technology	-
Teaching Methods	Activity Semester Workload Lectures 39 Student's study Hours 78 Exercises: Problem Solving 70.5 Course total 187.5
Student Performance Evaluation	The evaluation is based on the combined performance of the graduate student in: Weekly homeworks, Presentations during the semester, Major Homework at the end of the course, Written examination at the end of the courses in Greek with questions and problems of development of theoretical topics and problem solving.

Attached Bibliography

• P. Hilton and U. Stammbach: "A Course in Homological Algebra", Springer-Verlag, (1971).
• J. Rotman: "An Introduction to Homological Algebra", Springer, Second Edition, (2009).
• M. Scott Osborne: "Basic Homological Algebra", Springer, (2000).
• Ch. Weibel: "An Introduction to Homological Algebra", Cambridge University Press, (1994).
• S.I. Gelfand and Yu. Manin: "Methods of Homological Algebra", Springer, Second Edition, (2003).
• P. Bland: "Rings and their Modules", De Gruyter, (2011).

Specialized Topics in Algebra (ΑΛ6)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	ΑΛ6
Semester	2
Course Title	Specialized Topics in Algebra
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Special Background
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes	The aim of the course is for the postgraduate student to reach a good level of theoretical background on topics related to the theory of commutative rings.
General Competences	The aim of the course is to empower the postgraduate student to analyse and compose basic notions of Commutative Algebra.

Syllabus

Topics of Commutative and Combinatorial Algebra: Hilbert's Basis theorem, Primary Decomposition, Localization, Dimension, Hilbert Series, Groebner Bases, Simplicial complexes and homology, Stanley-Reisner ideals, Hilbert's Nullstellensatz theorem.

Teaching and Learning Methods - Evaluation

Delivery	Face to face
Use of Information and Communications Technology	-
Teaching Methods	Activity Semester Workload Lectures 39 Study of theory 78 Solving of Exercises 70.5 Course total 187.5
Student Performance Evaluation	Written exam at the end of semester (obligatory), problem solving or/and intermediate exams (optional)

Attached Bibliography

• Μαλιάκας Μιχάλης, Εισαγωγή στην Μεταθετική Άλεβρα, Εκδόσεις Σοφία, 2008
• Atiyah, M. F.; Macdonald, I. G., Introduction to commutative algebra. Addison-Wesley Publishing Co., 1969 ix+128 pp.

Classical Differential Geometry (ΓΕ1)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	ΓΕ1
Semester	1
Course Title	Classical Differential Geometry
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Special Background
Prerequisite Courses	Topology, Calculus of Several Variables, Complex Analysis.
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English).
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes	In this lecture we introduce basic notions of Classical Differential Geometry. More precisely, we introduce among others the notions of a manifold as a subset of the Euclidean space. Then, we present various local and global theorems concerning minimal submanifolds.
General Competences	Work autonomously. Work in teams. Develop critical thinking skills.

Syllabus

• Manifolds of the Euclidean space.
• Tangent and normal bundles.
• 1st and 2nd fundamental forms.
• Weingarten operator and Gauss map.
• Convex hypersurfaces.
• Hadamard’s Theorem.
• 1st and 2nd variation of area.
• Minimal submanifolds.
• Weierstrass representation.
• Bernstein’s Τheorem.

Teaching and Learning Methods - Evaluation

Delivery	Face-to-face.
Use of Information and Communications Technology	-
Teaching Methods	Activity Semester Workload Lectures 39 Autonomous Study 78 Solution of Exercises - Homeworks 70.5 Course total 187.5
Student Performance Evaluation	Weakly HomeWorks, presentations of the HomeWorks in the blackboard, written final examination.

Attached Bibliography

• M. do Carmo, Riemannian Geometry, Birkhaüser Boston, Inc., Boston, MA, 1992.
• J. Jost, Riemannian Geometry and Geometric Analysis, Universitext, Springer, 2017.
• J. Lee, Introduction to smooth manifolds, Second edition, Graduate Texts in Mathematics, 218, Springer, 2013.
• Δ. Κουτρουφιώτης, Διαφορική Γεωμετρία, Πανεπιστήμιο Ιωαννίνων, 1994.

Differential Geometry (ΓΕ2)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	ΓΕ2
Semester	1
Course Title	Differential Geometry
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Special Background
Prerequisite Courses	Linear Algebra, Topology, Calculus of Several Variables.
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes	In this lecture we introduce basic notions of Differential Geometry. More precisely, we introduce among others the notions of manifold, manifold with boundary, vector bundle, connection, parallel transport, submanifold, differential form and de Rham cohomology.
General Competences	Work autonomously Work in teams Develop critical thinking skills.

Syllabus

• Topological and smooth manifolds.
• Tangent and cotangent bundles.
• Vector fields and their flows.
• Submanifolds and Frobenius’ Theorem.
• Vector bundles.
• Connection and parallel transport.
• Differential forms.
• De Rham cohomology.
• Integration.
• Stokes’ Theorem.

Teaching and Learning Methods - Evaluation

Delivery	Face-to-face.
Use of Information and Communications Technology	-
Teaching Methods	Activity Semester Workload Lectures 39 Autonomous Study 78 Solution of Exercises - Homeworks 70.5 Course total 187.5
Student Performance Evaluation	Weakly HomeWorks, presentation in the blackboard of the HomeWorks, written final examination.

Attached Bibliography

• M. do Carmo, Riemannian Geometry, Birkhaüser Boston, Inc., Boston, MA, 1992.
• J. Jost, Riemannian Geometry and Geometric Analysis, Universitext, Springer, 2017.
• J. Lee, Introduction to smooth manifolds, Second edition, Graduate Texts in Mathematics, 218, Springer, 2013.
• Δ. Κουτρουφιώτης, Διαφορική Γεωμετρία, Πανεπιστήμιο Ιωαννίνων, 1994.

Riemannian Geometry (ΓΕ3)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	ΓΕ3
Semester	2
Course Title	Riemannian Geometry
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Special Background
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes	In this lecture we introduce basic notions of Riemannian Geometry. More precisely, we introduce among others the notions of Riemannian metric, Levi-Civita connection, holonomy, curvature operator, Ricci curvature, sectional curvature, scalar curvature and Jacobi field.
General Competences	Work autonomously Work in teams Develop critical thinking skills.

Syllabus

• Riemannian metrics, isometries, conformal maps.
• Geodesics and exponential maps.
• Parallel transport and holonomy.
• Hopf-Rinow’s Theorem.
• Curvature operator, Ricci curvature, scalar curvature.
• Riemannian submanifolds.
• Gauss-Codazzi-Ricci equations.
• 1st and 2nd variation of length.
• Jacobi fields.
• Comparison theorems.
• Homeomorphic sphere theorem.

Teaching and Learning Methods - Evaluation

Delivery	Face-to-face
Use of Information and Communications Technology	-
Teaching Methods	Activity Semester Workload Lectures 39 Autonomous Study 78 Solution of Exercises - Homeworks 70.5 Course total 187.5
Student Performance Evaluation	Written final examination, presentations of HomeWorks.

Attached Bibliography

• M. do Carmo, Riemannian Geometry, Birkhaüser Boston, Inc., Boston, MA, 1992.
• J.-Η. Eschenburg, Comparison theorems in Riemannian Geometry, Lecture Notes, Universität Augsburg, 1994.
• J. Jost, Riemannian Geometry and Geometric Analysis, Seventh edition, Universitext, Springer, 2017.
• J. Lee, Riemannian manifolds: An introduction to curvature, Graduate Texts in Mathematics, 176, Springer, 1997.
• P. Petersen, Riemannian Geometry, Third edition, Graduate Texts in Mathematics, 171, Springer, 2016.
• Δ. Κουτρουφιώτης, Διαφορική Γεωμετρία, Πανεπιστήμιο Ιωαννίνων, 1994.

Differential Topology (ΓΕ4)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	ΓΕ4
Semester	2
Course Title	Differential Topology
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Special Background
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in Greek)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes	In this lecture we present applications of Algebraic and Differential Topology in the study of topological invariants of smooth manifolds.
General Competences	Work autonomously Work in teams Develop critical thinking skills

Syllabus

• Manifolds.
• Immersions, embeddings and submersions.
• Milnor’s proof of the fundamental theorem of algebra.
• Sard’s theorem and Morse functions.
• Partition of unity and Whitney’s embedding theorem.
• Homotopy and isotopy.
• Brouwer’s degree.
• Whitney’s approximation theorem.
• Differential forms and integration.
• Hopf's invariant.
• Hopf's degree theorem.

Teaching and Learning Methods - Evaluation

Delivery	Face-to-face
Use of Information and Communications Technology	-
Teaching Methods	Activity Semester Workload Lectures 39 Autonomous Study 78 Solution of Exercises-Homeworks 70.5 Course total 187.5
Student Performance Evaluation	Weakly HomeWorks, presentations in the blackboard of HomeWorks, written final examination.

Attached Bibliography

• T. Bröcker, K. Jänich, Introduction to differential topology, Cambridge Univ. Press, 1982.
• V. Guillemin and A. Pollack, Differential Topology, Prentice Hall, 1974.
• J. Milnor, Morse Theory, Annals of Mathematical Studies, 51. Princeton University Press, Princeton, N.J. 1963.
• J. Milnor, Topology from a differentiable viewpoint, The University Press of Virginia, Charlottesville, Va. 1965.

Algebraic Topology I (ΓΕ5)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	ΓΕ5
Semester	2
Course Title	Algebraic Topology I
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Special background. Specialized general knowledge. Skills development.
Prerequisite Courses	General Topology, Algebraic Structures I
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes

Algebraic topology is a twentieth century field of mathematics that can trace its origins and connections back to the ancient beginnings of mathematics. One of the strengths of algebraic topology has always been its wide degree of applicability to other fields. Nowadays that includes fields like physics, differential geometry, algebraic geometry, and number theory, biology, financial sciences, computer sciences. We expect familiarity with basic notions from point set topology. We study the compact open topology and function spaces as an introduction to homotopy between maps. The main Learning Outcomes can be described as the application of Cell complexes and the category of CW spaces in connection between homotopy-homology and important problems in geometry. Moreover, how do we compute using homotopy? How can we distinguish between topological spaces? We compute the fundamental groups of basic topological spaces and classify covering spaces. Singular homology is introduced along with the main technics of computations.

General Competences

Search for analysis and synthesis of data and information related with topological and geometrical problems. Working independently and in a Team work. Working in an interdisciplinary environment aiming at production of new research ideas related to the syllabus of the course.

Syllabus

Compact open topology. Homotopy, fundamental group. Homotopy of the circle. Cell complexes. Real and Complex projective spaces. Covering spaces. Deformations. Classification of covering spaces. Applications. Scheifert-Van Kampen Theorem. Fundamental groups of surfaces.

Singular homology. Homotopic maps and homology. The long exact sequence of a pair. Homology of the sphere. Relative homology, excision. The degree of maps of spheres. Fixed point Theorems.

Teaching and Learning Methods - Evaluation

Delivery	Face-to-face
Use of Information and Communications Technology	-
Teaching Methods	Activity Semester Workload Lectures 39 Working hours in class 28 Project 30 Assignments 40.5 Final 50 Course total 187.5
Student Performance Evaluation	Written Examination, Oral Presentation, tests, written assignments.

Attached Bibliography

• A first course in topology, j. Muncres, Prentice Hall.
• Algebraic Topology, A. Hatcher, https://www.math.cornell.edu/~hatcher/AT/
• A Concise Course in Algebraic Topology, J. P. May, https://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf

Algebraic Topology II (ΓΕ6)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	ΓΕ6
Semester	1
Course Title	Algebraic Topology II
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Special background. Specialized general knowledge. Skills development in connections with topology geometry and algebra.
Prerequisite Courses	ΓΕ5 - Algebraic Topology I
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes

Algebraic Topology begins its creation by H. Poincare in 1900. In its first thirty years the field seemed limited in application in algebraic geometry, but this changed dramatically in 1930 with the creation of differential topology by G. De Rham and E. Cartan and of homotopy theory by W. Hurewicz and H. Hopf. Its influence began to spread to more and more branches as it gradually took on a central role in mathematics. This course is a continuation of the course Algebraic Topology I and aims in studying and managing advanced skills in order to calculate and solve difficult problems in topology-geometry. The key idea is to attach algebraic structures to topological spaces and their maps in such a away the algebra is both invariant under a variety of deformation of spaces and maps, and computable. Our aim is to transform difficult geometric problems to homotopic ones. We also study and develop homotopical tools. We calculate homological modules as well as cohomological rings for important spaces. Homotopical and cohomological sequences are concerned.

General Competences

Search for analysis and synthesis of data and information related with topological and geometrical problems. Working independently and in a Team work. Working in an interdisciplinary environment aiming at production of new research ideas related to the syllabus of the course.

Syllabus

Polyhedral, simplicial and singular homology theory, Lefschetz fixed-point theorem, cohomology and products, Künneth and universal coefficient theorems, Poincare and Alexander duality theorems. Cofibrations, cofiber homotopy equivalence, fibrations, fiber homotopy equivalence, cofiber-fiber sequences, the cellular approximation theorem. Hopf invariant problem, CW and cellular homology, subdivision and excision, the generalized Jordan curve theorem, Borsuk-Ulam. Classifying spaces Eilenberg-MacLane spaces, Meyer-Vietoris sequences, vector bundles characteristic classes.

Teaching and Learning Methods - Evaluation

Delivery	Face-to-face
Use of Information and Communications Technology	-
Teaching Methods	Activity Semester Workload Lectures 39 Working hours in class 28 Project 30 Assignments 40.5 Final 50 Course total 187.5
Student Performance Evaluation	Written Examination, Oral Presentation, tests, written assignments.

Attached Bibliography

• Elements of Homotopy Theory, George Whitehead. Springer Verlag .
• Algebraic Topology, A. Hatcher, https://www.math.cornell.edu/~hatcher/AT/
• A Concise Course in Algebraic Topology, J. P. May, https://math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf

Algebraic Geometry (ΓΕ7)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	ΓΕ7
Semester	2
Course Title	Algebraic Geometry
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Special Background
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes	The aim of the course is the postgraduate student to reach a good level of theoretical background on topics related to the algebraic geormetry.
General Competences	The aim of the course is to empower the postgraduate student to analyse and compose advanced notions of Algebraic Geometry.

Syllabus

Affine Varieties, Nullstellensatz, dimension, Regular and rational functions on Varieties, Projective varieties, birational geometry, tangent space and nonsingularity, divisors, differential forms, canonical class, Riemann-Roch theorem.

Teaching and Learning Methods - Evaluation

Delivery	Face-to-face
Use of Information and Communications Technology	-
Teaching Methods	Activity Semester Workload Lectures 39 Study of theory 78 Solving of exercises 70.5 Course total 187.5
Student Performance Evaluation	Written exam at the end of semester (obligatory), problem solving or/and intermediate exams (optional).

Attached Bibliography

• Shafarevich, Igor R. Basic algebraic geometry 1, Varieties in Projective Space, Springer, 2013.
• Shafarevich, Igor R. Basic algebraic geometry 2, Schemes and Complex Manifolds, Springer, 2013.

Specialized Topics in Geometry (ΓΕ8)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	ΓΕ8
Semester	2
Course Title	Special Topics in Geometry
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Special Background
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes	In this lecture we discuss several topics concerning contemporary topics in Differential Geometry, e.g. symplectic and Kähler manifolds, theory of isometric immersions, minimal surfaces and geometric evolution equations.
General Competences	Work autonomously Work in teams Develop critical thinking skills

Syllabus

• Complex manifolds.
• Kählerian manifolds.
• Riemannian submersions and projective spaces.
• Homogeneous and symmetric spaces.
• Holonomy groups.
• The Bochner technique.
• Harmonic maps and harmonic forms.
• Minimal submanifolds.
• Convergence of Riemannian manifolds.
• Comparison theorems.
• Geometric flows.

Teaching and Learning Methods - Evaluation

Delivery	Face-to-face
Use of Information and Communications Technology	-
Teaching Methods	Activity Semester Workload Lectures 39 Autonomous Study 78 Solution of Exercises - Homeworks 70.5 Course total 187.5
Student Performance Evaluation	Weakly homeworks, presentations.

Attached Bibliography

• B. Andrews and C. Hopper, The Ricci flow in Riemannian Geometry, Springer, 2011.
• T. Colding and W. Minicozzi, A course in minimal surfaces, Graduate Studies in Mathematics, Volume 121, 2011.
• M. Dajczer and R. Tojeiro, Submanifolds theory beyond an introduction, Springer, 2019.
• J. Jost, Riemannian Geometry and Geometric Analysis, 7th edition, Springer, 2017.
• P. Petersen, Riemannian Geometry, 3rd edition, Graduate Texts in Mathematics, 171, Springer, 2016.

Mathematical Statistics (ΣΕΕ1)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	ΣΣΕ1
Semester	1
Course Title	Mathematical Statistics
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Special Background
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English, reading Course)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes	The course aims to extend the knowledge which the students have obtained during their undergraduate studies on several themes of Mathematical Statistics and to present some special topics of Mathematical Statistics.
General Competences	Working independently Decision-making Production of free, creative and inductive thinking Criticism and self-criticism All the above will give to the stundetns the opportunity to work in an international multidisciplinary environment.

Syllabus

Extensions of the following subjects: Unbiasdness, Sufficient, Minimal Sufficient, Completeness, Consistency, Theorem of: Rao-Blackwell, Lehmann-Scheffé, Basu. Maximum Likelihood Estimators: Properties-Asymptotic Properties. Decision Theory: minimax, Bayes estimators. Modified Likelihood, EM algorithm, Numerical methods of finding estimators. Confidence intervals: pivotal quantity, Asymptotic method etc. Delta Method- Asymptotic statistics.

Teaching and Learning Methods - Evaluation

Delivery	Classroom (face-to-face)
Use of Information and Communications Technology	Use of ICT in communication with students
Teaching Methods	Activity Semester Workload Lectures 39 Study and analysis of bibliography 78 Preparation of assignments and interactive teaching 70.5 Course total 187.5
Student Performance Evaluation	Final written exam in Greek (in case of Erasmus students in English).

Attached Bibliography

• B. Andrews and C. Hopper, The Ricci flow in Riemannian Geometry, Springer, 2011.
• T. Colding and W. Minicozzi, A course in minimal surfaces, Graduate Studies in Mathematics, Volume 121, 2011.
• M. Dajczer and R. Tojeiro, Submanifolds theory beyond an introduction, Springer, 2019.
• J. Jost, Riemannian Geometry and Geometric Analysis, 7th edition, Springer, 2017.
• P. Petersen, Riemannian Geometry, 3rd edition, Graduate Texts in Mathematics, 171, Springer, 2016.

Linear Models (ΣΕΕ2)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	ΣΣΕ2
Semester	1
Course Title	Linear Models
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Specialized general knowledge
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English, reading Course)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning Outcomes

By the end of the course students are expected to demonstrate: A strong foundation in simple linear, multiple regression and in the one- and two-way analysis of variance as well as in extending these concepts, Deep knowledge of the main assumptions of the general linear model and their implications when violated, How to conduct diagnostics and correct for the violation of the assumptions of the general linear model, How to interpret various coefficients and in general how to analyze data with linear models, How to deal with multicollinearity effects, missing data e.t.c..

General Competences

Working independently Decision-making Adapting to new situations Production of free, creative and inductive thinking Synthesis of data and information, with the use of the necessary technology Working in an interdisciplinary environment

Syllabus

The General Linear Model of full Rank and its statistical properties. Multiple Regression Analysis. Hypothesis tests, diagnostic measures and residual analysis. Variable selection. Models of non full rank. Estimable functions, One and two-way analysis of variance with equal and unequal numbers per cell.

Teaching and Learning Methods - Evaluation

Delivery	Face-to-face
Use of Information and Communications Technology	Use of ICT in communication with students
Teaching Methods	Activity Semester Workload Lectures 39 Independent study 70 Study and analysis of bibliography, Fieldwork 78.5 Course total 187.5
Student Performance Evaluation	Final written exam in Greek (in case of Erasmus students in English).

Attached Bibliography

• Καρακώστας, Κ. (2002). Γραμμικά Μοντέλα: Παλινδρόμηση και Ανάλυση Διακύμανσης. Πανεπιστήμιο Ιωαννίνων.
• Λουκάς, Σ. (2014).  Γενικό Γραμμικό Μοντέλο. Πανεπιστήμιο Ιωαννίνων.
• Οικονόμου, Π. και Καρώνη, Χ. (2010). Στατιστικά Μοντέλα Παλινδρόμησης, Εκδόσεις Συμεών.
• Draper, N.R. and H. Smith, (1998). Applied Regression Analysis, Third Edition, Wiley,
• Searle, S.R., (1997). Linear Models, Wiley Classics Library, Wiley,
• Seber, G.A.F. and A.J. Lee, (2003). Linear Regression Analysis, 2nd Edition, Wiley.

Mathematical Programming (ΣΕΕ3)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	ΣΣΕ3
Semester	1
Course Title	Mathematical Programming
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Special Background
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes

The course learning outcomes are: the presentation of mathematical programming problems, the presentation of their solution techniques and their applications in several areas such as production, distribution, routing, etc. Upon successful completion of the course the student will be able to: model complex systems comprehend the mathematical foundation of the Simplex method and the dual theory select the appropriate algorithm for a particular optimization problem understand and apply the appropriate techniques required to solve linear optimization problems understand the principles of dynamic programming and apply dynamic programming solution techniques recognize and apply the appropriate inventory management policies (depending, each time, on underlying assumptions of the system)

General Competences

Working independently Decision-making Adapting to new situations Production of free, creative and inductive thinking Synthesis of data and information, with the use of the necessary technology Project planning and management

Syllabus

Linear programming problems formulation. The Simplex algorithm. Big M-method. Two-Phase method. Revised Simplex method. Duality theory. Dual Simplex algorithm. Sensitivity analysis. Parametric analysis. Transportation problem. Transhipment problem. Assignment problem. Dynamic programming: Bellman principle of optimality, finite and infinite horizon problems. Applications of dynamic programming. Inventory control.

Teaching and Learning Methods - Evaluation

Delivery	Face-to-face
Use of Information and Communications Technology	Lindo/Lingo Software, Mathematica, Email, Class Web
Teaching Methods	Activity Semester Workload Lectures 39 Study and analysis of bibliography 78 Preparation of assignments and interactive teaching 70.5 Course total 187.5
Student Performance Evaluation	LANGUAGE OF EVALUATION: Greek METHODS OF EVALUATION: Written work (30%), Final exam (70%).

Attached Bibliography

• Καρακώστας, Κ. (2002). Γραμμικά Μοντέλα: Παλινδρόμηση και Ανάλυση Διακύμανσης. Πανεπιστήμιο Ιωαννίνων.
• Λουκάς, Σ. (2014).  Γενικό Γραμμικό Μοντέλο. Πανεπιστήμιο Ιωαννίνων.
• Οικονόμου, Π. και Καρώνη, Χ. (2010). Στατιστικά Μοντέλα Παλινδρόμησης, Εκδόσεις Συμεών.
• Draper, N.R. and H. Smith, (1998). Applied Regression Analysis, Third Edition, Wiley,
• Searle, S.R., (1997). Linear Models, Wiley Classics Library, Wiley,
• Seber, G.A.F. and A.J. Lee, (2003). Linear Regression Analysis, 2nd Edition, Wiley.

Biostatistics (ΣΕΕ4)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	ΣΣΕ4
Semester	1
Course Title	Biostatistics
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Specialized general knowledge
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English, reading Course)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes	Students completing this course should be able to: Understand and apply statistical methods for the design of biomedical research and analysis of biomedical research data. Understand and use mathematical and statistical theory underlying the application of biostatistical methods; use and interpret results from specialized computer software for the statistical analysis of research data;
General Competences	Working independently Decision-making Production of free, creative and inductive thinking Criticism and self-criticism

Syllabus

Biomedical research design elements - Diagnostic tests - Percentages and standardization - Independence test and linear stress tests - Relative risk - Odds ratio - Relevance and agreement - McNemar, Cochran - Mantel - Haenszel - Logistic Regression - ROC Curves.

Teaching and Learning Methods - Evaluation

Delivery	Classroom (face-to-face)
Use of Information and Communications Technology	Statistical software Use of ICT in communication with students
Teaching Methods	Activity Semester Workload Lectures 39 Working independently 78 Exercises-Homework 70.5 Course total 187.5
Student Performance Evaluation	Final written exam in Greek (in case of Erasmus students in English).

Attached Bibliography

• Rosner, B. (2010). Fundamentals of Biostatistics. 7th International edition, Brooks/Cole
• Armitage P., Berry G., Mathews JNS (2002). Statistical Methods in Medical Research. 4th Edition. Blackwell Science.
• Friedman L.M., Furberg C.D. and DeMets, D.L. (2010). Fundamentals of Clinical Trials. 4th edition, Springer.
• S. Piantadosi (2005). Clinical Trials: A Methodological Perspective Second Edition. Wiley.
• [Περιοδικό / Journal] Biostatistics
• [Περιοδικό / Journal] The International Journal of Biostatistics

Data Analysis and Statistical Packages (ΣΕΕ5)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	ΣΣΕ5
Semester	2
Course Title	Data Analysis and Statistical Packages
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Specialised general knowledge
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English, reading Course)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes	The students should be able to understand which statistical method is appropriate for each problem under study and to confirm that the relative assumptions hold. Moreover, the student should be able to present and interpret the results of the above analysis.
General Competences	Working independently Decision-making Production of free, creative and inductive thinking Criticism and self-criticism

Syllabus

Descriptive Statistics. Review of basic undergraduate concepts like t-test, Mann Whitney test etc. Multiple Regression and Diagnostics. One Way and Two Way Analysis of Variance. Introduction to Logistic Regression. Repeated Measures Analysis. Reliability and Factor Analysis. Introduction to Survival Analysis: Roc curves, Cox PH Regression Model.

Teaching and Learning Methods - Evaluation

Delivery	Classroom (face-to-face)
Use of Information and Communications Technology	Use of ICT in communication with students
Teaching Methods	Activity Semester Workload Lectures 39 Working independently 78 Exercises - Homework 70.5 Course total 187.5
Student Performance Evaluation	Final written exam in Greek (in case of Erasmus students in English) which includes analysis of real data sets.

Attached Bibliography

• Barnett, V. and Lewis, T. (1978). Outliers in statistical data. Wiley, New York.
• Belsley, David A, Kuh, Edwin and Welsch, Roy E. (1980). Regression diagnostics: identifying influential data and sources of collinearity. Wiley Series in Probability and Mathematical Statistics. John Wiley & Sons.
• Samprit Chatterjee, Ali S. Hadi (2012). Regression analysis by examples. John Wiley & Sons, Inc.
• Coakes, S. and Steed, L ( 1999). S.P.S.S. Analysis without Anguish. Wiley.
• Field, A. P. (2005). Discovering statistics using S.P.S.S. (Second Edition). London: Sage.
• Landau, S. and Everitt (2004). A Handbook of Statistical Analyses using S.P.S.S.. Chapman and Hall.
• Neter, J., Kutner, M., Nachtsheim, C. and Wassserman, W. (1996). Applied linear statistical models. 4th Edition, Irwin, Inc.
• Rawlings, J. O. (1988). Applied regression analysis: a research tool. Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA.
• Rencher, A. C. (2000). Linear Models in Statistics. Wiley.
• Searle, S. R. (1971). Linear models. John Wiley & Sons, Inc.
• Seber, G. A. F. (1977). Linear regression analysis. John Wiley & Sons.

Multivariate Analysis (ΣΕΕ6)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	ΣΣΕ6
Semester	2
Course Title	Multivariate Analysis
Independent Teaching Activities	Lectures-Laboratory (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Specialized general knowledge
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes

The aim of the course is to present techniques and methods of Multivariate Statistical Analysis. The interest is initially focused on the study of multivariate distributions and, in particular, the multivariate normal distribution that predominates in the classical multivariate analysis. Estimation techniques and statistical tests on the parameters of the multivariate normal distribution are presented and studied. Afterwards the following subjects are presented: Principal Components, Discriminant Analysis and Cluster Analysis. The above methods are presented and studied theoretically. Their implementation is provided by the use of appropriate statistical software. Upon completing the course students should be able to elaborate research issues on Multivariate Statistical Analysis. They also should be able to apply the aforementioned multivariate techniques in a real data set.

General Competences

Working independently Decision-making Production of free, creative and inductive thinking Criticism and self-criticism

Syllabus

The multivariate normal distribution. The non-central chi-square and F distributions. Quadratic forms: Independence, distributions. Spherical and Elliptical distributions. Maximum likelihood estimators (m.l.e) of the parameters of the multivariate normal distribution. Classical properties of m.l.e. The Wishart distribution. Tests of hypotheses of mean vectors. Likelihood ratio method - Union/Intersection method. Hotelling's T2 statistic and distribution. One-way MANOVA. Tests concerning variance-covariance matrices. Tests of independence. Principal Components. Discriminant Analysis. Cluster Analysis

Teaching and Learning Methods - Evaluation

Delivery	Classroom (face-to-face)
Use of Information and Communications Technology	Use of ICT in communication with students
Teaching Methods	Activity Semester Workload Lectures 39 Working independently 78 Exercises-Homework 70.5 Course total 187.5
Student Performance Evaluation	Final written exam in Greek (in case of Erasmus students in English).

Attached Bibliography

• Anderson, T. W. (2003). An Introduction to Multivariate Statistical Analysis. 3rd Edition. Wiley.
• Fang, K.T., and Zhang, Y.T.. (1990). Generalized Multivariate Analysis. Springer. Berlin.
• Flury, B. (1997). A first course in multivariate statistics. Springer.
• Johnson, R. A. and Wichern, D. W. (2006). Applied Multivariate Statistical Analysis. Prentice Hall.
• Mardia, K. V., Kent, J. T. and Bibby, J. M. (1979). Multivariate Analysis. Academic Press.
• Muirhead, R. J. (1982). Aspects of Multivariate Statistical Theory. Wiley.
• Rencher, A. C. (1995). Methods of Multivariate Analysis. Wiley.
• Srivastava, M. S. (2002). Methods of multivariate statistics. Wiley.

Non Linear Programming (ΣΕΕ7)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	ΣΣΕ7
Semester	2
Course Title	Non Linear Programing
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Special Background
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes	The course aims to introduce students to the fundamentals of non-linear optimization. Upon successful completion of the course the student will be able to: understand the basic principles of nonlinear optimization problems. use some of the commonly used algorithms for nonlinear optimization (unconstrained and constrained). select the appropriate algorithm for a particular optimization problem.
General Competences	Working independently Decision-making Adapting to new situations Production of free, creative and inductive thinking Synthesis of data and information, with the use of the necessary technology

Syllabus

Introduction to unconstrained and constrained optimization, Lagrange Multipliers, Karush-Kuhn-Tucker conditions, Line Search, Trust Region, Conjugate Gradient, Newton, Quasi-Newton methods, Quadratic Programming, Penalty Barrier and Augmented Lagrangian Methods.

Teaching and Learning Methods - Evaluation

Delivery	Face-to-face
Use of Information and Communications Technology	Lindo/Lingo Software, Mathematica, Email, class web
Teaching Methods	Activity Semester Workload Lectures 39 Study and analysis of bibliography 78 Preparation of assignments and interactive teaching 70.5 Course total 187.5
Student Performance Evaluation	LANGUAGE OF EVALUATION: Greek METHODS OF EVALUATION: Written work (30%), Final exam (70%).

Attached Bibliography

• Anderson, T. W. (2003). An Introduction to Multivariate Statistical Analysis. 3rd Edition. Wiley.
• Fang, K.T., and Zhang, Y.T.. (1990). Generalized Multivariate Analysis. Springer. Berlin.
• Flury, B. (1997). A first course in multivariate statistics. Springer.
• Johnson, R. A. and Wichern, D. W. (2006). Applied Multivariate Statistical Analysis. Prentice Hall.
• Mardia, K. V., Kent, J. T. and Bibby, J. M. (1979). Multivariate Analysis. Academic Press.
• Muirhead, R. J. (1982). Aspects of Multivariate Statistical Theory. Wiley.
• Rencher, A. C. (1995). Methods of Multivariate Analysis. Wiley.
• Srivastava, M. S. (2002). Methods of multivariate statistics. Wiley.

Sampling Theory (ΣΕΕ8)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	ΣΣΕ8
Semester	2
Course Title	Sampling Theory
Independent Teaching Activities	Lectures-Laboratory (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Specialized general knowledge
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes

Students completing this course should be able to: Basic methods of selecting a sample from a finite population. Application of basic methods of selecting a sample from a finite population. Select the most appropriate one sampling scheme from several alternatives. Compute estimators, standard errors and confidence intervals and to conduct the appropriate statistical analysis based on the sampling scheme used. Separate the sampling and non-sampling errors and ways of minimizing them. Elements of research methodology. Questionnaires.

General Competences

Working independently Decision-making Production of free, creative and inductive thinking Criticism and self-criticism

Syllabus

Sampling and non sampling errors, simple random sampling, stratified sampling, systematic sampling, cluster sampling, ratio estimators, determination of optimal sample size, bias in sampling theory. Some elements of missing data and basic methods of imputation.

Teaching and Learning Methods - Evaluation

Delivery	Classroom (face-to-face)
Use of Information and Communications Technology	Use of ICT in communication with students
Teaching Methods	Activity Semester Workload Lectures 39 Working independently 78 Exercises-Homework 70.5 Course total 187.5
Student Performance Evaluation	Final written exam in Greek (in case of Erasmus students in English).

Attached Bibliography

• Cochran, W.G. (1977). Sampling Techniques. Wiley.
• Δαμιανού, Χ. (2007). Μεθοδολογία Δειγματοληψίας, Τεχνικές και Εφαρμογές. Εκδόσεις Σοφία. (in Greek)
• Fuller, W. A. (2009). Sampling Statistics. Wiley.
• Lohr, S. L. (2010). Sampling: Design and Analysis. 2nd Edition. Brooks/Cole.
• Thompson, S. K. (2012). Sampling. Wiley.
• Παπαγεωργίου, Ι. (2015). Θεωρία Δειγματοληψίας. Ελληνικά Ακαδημαϊκά Ηλεκτρονικά Συγγράμματα και Βοηθήματα, ΣΕΑΒ 2015, www.kallipos.gr. (in Greek)
• Φαρμάκης Ν. (2015) Δειγματοληψία και Εφαρμογές. Ελληνικά Ακαδημαϊκά Ηλεκτρονικά Συγγράμματα και Βοηθήματα, ΣΕΑΒ 2015, www.kallipos.gr. (in Greek).
• Χριστοφίδης, Τ. Δειγματοληψία (Πρόχειρες Σημειώσεις). Παν/μιο Κύπρου. (in Greek).

Probability Theory (ΣΕΕ9)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	ΣΣΕ9
Semester	2
Course Title	Probability Theory
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Specialized general knowledge
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English, reading Course)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes

This course treats the fundamentals of probability theory with a focus on proofs and rigorous mathematical theory. Upon its completion the students will be able to: explain the foundations of probability in the language of measure theory, state the strong law of large numbers have a working knowledge of weak convergence, characteristic functions, and the central limit theorem, explain the concept of conditional expectation, its properties and applications give an introduction to discrete time martingales and the martingale convergence theorem be able to solve basic problems related to the theory

General Competences

Working independently Decision-making Production of free, creative and inductive thinking

Syllabus

Measure-theoretic foundations of probability theory (σ-algebras, measure and probability spaces, generated sigma-algebras. Caratheodory extension theorem, Lebesgue measure, Random variables and their distribution Lebesgue integral and expectation. Almost sure convergence. Convergence in probability and in Lp. Monotone convergence theorem, dominated convergence theorem, Change of variables. Independent random variables). Key limit theorems (Weak law of large numbers, Borel-Cantelli lemmas, Kolmogorov extension theorem, strong law of large numbers, Lindeberg central limit theorem ) Martingales (Martingale Convergence, Applications)

Teaching and Learning Methods - Evaluation

Delivery	Classroom (face-to-face)
Use of Information and Communications Technology	Use of ICT in communication with students
Teaching Methods	Activity Semester Workload Lectures 39 Working independently 78 Exercises - Homework 70.5 Course total 187.5
Student Performance Evaluation	Final written exam in Greek (in case of Erasmus students in English).

Attached Bibliography

• Billingsley P., Probability and Measure, 4th Edition, 1995, John Wiley and Sons
• M. Capinski and E. Kopp, Measure, integral and probability, Springer. (Springer-Verlag London, Ltd., second edition, 2004).
• R. Durrett, Probability: Theory and Examples, 4th Edition, Cambridge Series in Statistical and Probabilistic Mathematics, 2010.
• Kingman, J. F. C. and Taylor, S. J. An Introduction to Measure and Probability. Cambridge, England: Cambridge University Press, 1966.
• Rao, M. M. Measure Theory And Integration. New York: Wiley, 1987.
• D. Stroock, Probability: An Analytic View, 2nd Edition, Cambridge University Press, 2011
• [Περιοδικό / Journal] Advances in Applied Probability
• [Περιοδικό / Journal] Annals of Applied Probability
• [Περιοδικό / Journal] Annals of Probability
• [Περιοδικό / Journal] Journal of Applied Probability
• [Περιοδικό / Journal] Journal of Theoretical Probability
• [Περιοδικό / Journal] Probability Surveys
• [Περιοδικό / Journal] Theory of Probability and Its Applications

Applied Multivariate Analysis (ΣΕΕ10)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	ΣΣΕ10
Semester	2
Course Title	Applied Multivariate Analysis
Independent Teaching Activities	Lectures-Laboratory (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Specialized general knowledge
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English, reading Course)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes

Students completing this course should be able to: Apply basic multivariate methods of statistical analysis Choose the appropriate method of multivariate data analysis for a given multivariate data set, depending on the objectives of the study Implement methods of dimension reduction Interpret the results of multivariate data analyses. Carry out multivariate data analysis through a statistical software (SPSS, SAS, Matlab, R)

General Competences

Working independently Decision-making Production of free, creative and inductive thinking Project planning and management Criticism and self-criticism

Syllabus

This course covers the following topics with applications mainly with SPSS and R: Graphical display of multivariate data, Data reduction techniques, Principal component analysis, Factor analysis, Canonical correlation analysis, Cluster analysis, Discriminant analysis, MANOVA, Repeated measurement analysis, Neural Networks. Applications with SPSS and R.

Teaching and Learning Methods - Evaluation

Delivery	Classroom (face-to-face)
Use of Information and Communications Technology	Use of ICT in communication with students
Teaching Methods	Activity Semester Workload Lectures 39 Working independently 78 Exercises-Homework 70.5 Course total 187.5
Student Performance Evaluation	Final written exam in Greek (in case of Erasmus students in English).

Attached Bibliography

• Anderson, T.W. (2003). An Introduction to Multivariate Statistical Methods, 3nd ed., Wiley.
• Giri, N.J. (2004). Multivariate Statistical Analysis, 2nd edition, Marcel Dekker, New York.
• Johnson, R. A. and Wichern, D.W. (1998). Applied Multivariate Statistical Analysis, 4th ed. Prentice Hall.
• Timm, N. H. (2002). Applied Multivariate Analysis. Springer.
• Καρλής, Δ. (2005).Πολυμεταβλητή Στατιστική Ανάλυση. Εκδόσεις Σταμούλη
• [Περιοδικό / Journal] Journal of Multivariate Analysis

Time series (ΣΕΕ11)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	ΣΣΕ11
Semester	2
Course Title	Time series
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Specialized general knowledge
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes	Upon completion of this course, a student will: be familiar with properties of the major types of time series be able to identify appropriate models for time series. be able to diagnose model adequacy. construct time series models from data and verify model fits use statistical packages to construct time series models and conduct analysis
General Competences	Working independently Decision-making Adapting to new situations Production of free, creative and inductive thinking Synthesis of data and information, with the use of the necessary technology Working in an interdisciplinary environment

Syllabus

Introduction to stationary time series. Simple models for time series. Linear processes, general autoregressive-moving average models. Prediction of stationary time series. The families of ARMA, ARIMA and State space models. Seasonality in time series. Modelling stochastic volatility. Time series regression. Nonlinear non-Gaussian time series. Multivariate time series. Multivariate autoregressive model.

Teaching and Learning Methods - Evaluation

Delivery	Face-to-face
Use of Information and Communications Technology	Statistical Software Use of ICT in communication with students
Teaching Methods	Activity Semester Workload Lectures 39 Working independently 70 Study and analysis of bibliography, Fieldwork 78.5 Course total 187.5
Student Performance Evaluation	LANGUAGE OF EVALUATION: Greek METHODS OF EVALUATION: written work (20%), Final exam (80%)

Attached Bibliography

• Shumway, R.H. and Stoffer, D.S. (2017) Time Series Analysis and Its Applications with R Examples, 4rd edition, Springer-Verlag, New York.
• Brockwell, P.J. and R. A.. Davis (2016) Introduction to Time Series and Forecasting, 3nd edition, Springer-Verlag, New York.
• Cowpertwait, P.S.P. and A.V. Metcalfe (2009) Introductory Time Series with R, Spinger-Verlag.
• Cryer, J.D. and K-S Chan (2010) Time Series Analysis: with applications in R, 2nd Edition, Springer
• Δημέλη Σ. (2003, 3η Έκδοση): Σύγχρονες Μέθοδοι Ανάλυσης Χρονολογικών Σειρών, Εκδόσεις ΚΡΙΤΙΚΗ, Αθήνα.
• Θαλασσινός Λευτέρης Ι. Ανάλυση χρονολογικών σειρών Μεθοδολογία Box-Jenkins.
• [Περιοδικό / Journal] Journal of Time Series Analysis
• [Περιοδικό / Journal] Journal of Time Series Econometrics

Computational Statistics Analysis (ΣΕΕ12)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	ΣΣΕ12
Semester	2
Course Title	Computational Statistics Analysis
Independent Teaching Activities	Lectures-Laboratory (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Specialized general knowledge
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English, reading Course)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes

Students completing this course should be able to: Use R and other statistical software to implement computational statistics techniques. Be able to generate random numbers from a variety of distributions and asses their quality. Be able to apply the Jacknife, the Bootstrap and other computational statistics techniques under the appropriate settings and assumptions. Plan and implement a statistical simulation study in an efficient way. Interpret the results from a simulation study.

General Competences

Working independently Decision-making Production of free, creative and inductive thinking Criticism and self-criticism

Syllabus

This course covers the following topics and relies on heavy use of R: random number generation techniques. The jacknife, bootstrap and their theoretical properties. Cross validation, kernel density estimation, local regression. Monte Carlo simulation and its applications.

Teaching and Learning Methods - Evaluation

Delivery	Classroom (face-to-face)
Use of Information and Communications Technology	Use of ICT in communication with students
Teaching Methods	Activity Semester Workload Lectures 39 Working independently 78 Exercises-Homework 70.5 Course total 187.5
Student Performance Evaluation	Final written exam in Greek (in case of Erasmus students in English).

Attached Bibliography

• Davison, A. C., Hinkley, D. V., (1997). Bootstrap methods and their application. Cambridge University Press.
• Rizzo, M. L., (2007). Statistical computing with R. Chapman & Hall/CRC.
• Robert, C. P., Casella, G., (2009). Introducing Monte Carlo methods with R. Springer Verlag.
• Gentle, J. E., (2009). Computational Statistics, Springer.
• Givens, G.H. and Hoeting, J.A., (2012). Computational Statistics, Wiley.
• [Περιοδικό / Journal] Statistics and Computing
• [Περιοδικό / Journal] Computational Statistics.
• [Περιοδικό / Journal] Computational Statistics & Data Analysis.

Survival Analysis (ΣΕΕ13)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	ΣΣΕ13
Semester	1
Course Title	Survival Analysis
Independent Teaching Activities	Lectures-Laboratory (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Specialized general knowledge
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English, reading Course)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes

This course describes the various methods and underlying theory used for modeling, analyzing and interpreting survival data. Students taking this course will master their understanding of survival techniques and will be able to: understand different types of censoring, and learn to estimate and interpret survival characteristics. compare survival rates in different groups. assess the relationship of risk factors and survival times using the Cox regression model, and assess the appropriateness and adequacy of the model. understand issues in the design, analysis, and interpretation of studies involving time-dependent covariates.

General Competences

Working independently Decision-making Production of free, creative and inductive thinking Criticism and self-criticism

Syllabus

Types of censored data: right censoring, left censoring, interval censoring, e.t.c. Functions of interest in survival studies: The survival function, the hazard rate function, the cumulative hazard rate function and their connection with the probability density function and the cumulative distribution function. Estimation of the survival function: the Nelson-Aalen estimate, the Kaplan-Meier estimate. Parametric estimators. Comparison of two or more survival curves. Estimation of the hazard rate function: parametric estimates, kernel (nonparametric) estimation. Semiparametric estimation: Cox regression and its extensions. Model properties and limitations, variable selection, model validation measures (residual types and analysis, remedial measures).

Teaching and Learning Methods - Evaluation

Delivery	Classroom (face-to-face)
Use of Information and Communications Technology	Use of ICT in communication with students
Teaching Methods	Activity Semester Workload Lectures 39 Working independently 78 Exercises-Homework 70.5 Course total 187.5
Student Performance Evaluation	Final written exam in Greek (in case of Erasmus students in English).

Attached Bibliography

• Lawless, J.L. (2002), Statistical Models and Methods for Lifetime Data 2nd Edition, Wiley.
• Cox D.R. and D. Oakes (1994), Analysis of Survival Data, Chapman and Hall,
• Klein and Moschberger (2003). Survival Analysis: Techniques for Censored and Truncated Data, 2nd edition. Springer.
• Kleinbaum, D. G. and Klein M., (2005), Survival Analysis: A Self-Learning Text, 2nd Edition. New York: Springer
• Therneau T, and Grambsch, P. (2000). Modeling Survival Data: Extending the Cox Model. New York: Springer.
• Hosmer, Jr, DW. and Lemeshow, S. (2008). Applied Survival Analysis: Regression Modeling of Time to Event Data, 2nd Edition. Wiley,
• Kalbfleish, JD. and Prentice, RL. (2002). The Statistical Analysis of Failure Time Data. Wiley,
• Collet, D. (2003). Modeling Survival Data in Medical Research. London: Chapman and Hall.
• [Περιοδικό / Journal] Lifetime data analysis

Non Parametric Statistics (ΣΕΕ14)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	ΣΣΕ14
Semester	2
Course Title	Non Parametric Statistics
Independent Teaching Activities	Lectures-Laboratory (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Specialized general knowledge
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English, reading Course)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes

This course aims at introducing nonparametric techniques in statistical analysis and the use of these techniques in a variety of disciplines. The course will focus on the so-called smoothing procedures for curve estimation. Students taking this course will develop an appreciation of nonparametric statistics and will be able to: understand the concept and scope of nonparametric techniques, explain the fundamental principles of smoothing and nonparametric curve estimation, estimate functions of interest without strong parametric assumptions, test hypotheses about these functions and construct confidence regions, use in practice the modern nonparametric techniques to answer concrete questions about real data sets, use the R software to generate output in regard to the previous point and for computing intensive methods such as the bootstrap.

General Competences

Working independently Decision-making Production of free, creative and inductive thinking Criticism and self-criticism

Syllabus

Presentation and Introduction to nonparametric methods. Nonparametric estimation of the probability density function (p.d.f.) by histogram and by kernel density estimation. Asymptotic properties of the derived estimates. Non parametric estimation of the cumulative distribution function (e.c.d.f.) with the empirical c.d.f., kernel smoothing and properties of the derived estimatres. Methods and techniques for bandwidth selection. Improvements of kernel estimates: elimination of boundary bias, variable bandwidth kernel estimates and transformation-based estimates. Nonparametric regression: the Nadaraya-Watson estimate and the local polynomial estimate. Multivariate kernel estimation and special topics.

Teaching and Learning Methods - Evaluation

Delivery	Classroom (face-to-face)
Use of Information and Communications Technology	Statistical software Use of ICT in communication with students
Teaching Methods	Activity Semester Workload Lectures 39 Working independently 78 Exercises-Homework 70.5 Course total 187.5
Student Performance Evaluation	Final written exam in Greek (in case of Erasmus students in English).

Attached Bibliography

• Silverman, B. (1986). Density Estimation for Statistics and Data Analysis, Chapman and Hall.
• Wand, M.P. and Jones, M.C. (1994). Kernel smoothing, First Edition, Chapman and Hall.
• Simonoff, J.S. (1996). Smoothing Methods in Statistics, Springer.
• Fan, J. and Gijbels, I. (1996). Local Polynomial Modelling and Its Applications, Chapman and Hall.
• Loader, C. (1999). Local Regression and Likelihood, Springer.
• Scott, D. (2015). Multivariate Density Estimation: Theory, Practice, and Visualization, Second edition, Wiley.
• Takezawa, K. (2006). Introduction to Nonparametric Regression, Wiley.
• Wasserman, L. (2006). All of Nonparametric Statistics, Springer.
• Klemela, J. (2009). Smoothing of Multivariate Data: Density Estimation and Visualization, Wiley.
• Tsybakov, A.B. (2009). Introduction to Nonparametric Estimation Springer.
• Chacón, J.E. and Duong, T. (2018). Multivariate Kernel Smoothing and its Applications, Taylor and Francis.
• [Περιοδικό / Journal] Journal of Nonparametric Statistics.

Stochastic analysis with applications (ΣΕΕ15)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	ΣΣΕ15
Semester	2
Course Title	Stochastic analysis with applications
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Specialized general knowledge
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English, reading Course)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes

The course learning outcomes are: the presentation of the theoretical and practical fundamental concepts within Ito calculus, martingale methods, stochastic differential equations and diffusion processes. The application of this theory within linear filtering, optimal stopping and stochastic control, financial derivative. Upon successful completion of the course the student will be able to: know the main results and basic applications of stochastic Ito calculus understand stochastic differential equations understand of martingales in continuous time use numerical methods for stochastic differential equations use methods of stochastic analysis for modeling in different application areas

General Competences

Working independently Decision-making Adapting to new situations Production of free, creative and inductive thinking Synthesis of data and information, with the use of the necessary technology Working in an interdisciplinary environment

Syllabus

Stochastic processes in continuous time, processes adapted to an information flow, processes predictable with respect to an information flow, Brownian motion. Ito stochastic calculus. Martingales and representation theorems. Stochastic differential equations: existence and uniqueness of the solution. Theory of diffusions: Markov processes, Dynkin formula, Girsanov theorem. Applications: linear filtering, optimal stopping and stochastic control theory, Financial Derivatives.

Teaching and Learning Methods - Evaluation

Delivery	Face-to-face
Use of Information and Communications Technology	Use of ICT in communication with students
Teaching Methods	Activity Semester Workload Lectures 39 Independent study 70 Study and analysis of bibliography, Fieldwork 78.5 Course total 187.5
Student Performance Evaluation	LANGUAGE OF EVALUATION: Greek METHODS OF EVALUATION: written work (20%), Final exam (80%)

Attached Bibliography

• Karatzas I. and S. Shreve. Brownian Motion and Stochastic Calculus. Springer. 1998
• Lamberton, D. & Lapeyre, B. (1996). Introduction to Stochastic Calculus Applied to Finance. Chapman & Hall.
• Oksendal B.: Stochastic Differential Equations, 6th edition. Springer 2007.
• Revuz D. and M. Yor. Continuous martingales and Brownian motion. Springer. 2001
• Rogers L.C. and D. Williams.Diffusions, Markov Processes and Martingales. Vol.1 and 2, Cambridge University Press. 2002
• Steele J. M., Stochastic Calculus and Financial Applications, 2001.
• [Περιοδικό / Journal] Stochastic Analysis and Applications.

Risk Management (ΣΕΕ16)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	ΣΣΕ16
Semester	1
Course Title	Risk Management
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Specialized general knowledge
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes	By the end of the course, successful students will be able to: have a broader view on risk analysis and management acquire state-of-the-art quantitative techniques for modelling risk factors and managing risk use common risk models, as well as develop customized versions for different risk management problems.
General Competences	Working independently Decision-making Adapting to new situations Production of free, creative and inductive thinking Synthesis of data and information, with the use of the necessary technology Project planning and management

Syllabus

• Structure of uncertainty and risk - Bayesian method, portfolio risk
• Measuring risk: Value at risk and conditional value at risk, generic risk measures, coherent risk measures, application of the risk measures to simple problems.
• Decision analysis: utility theory, stochastic dominance models, St. Petersburg paradox, Allais paradox
• Stochastic modelling and applications: portfolio optimization, production planning, robust approaches for stochastic optimization

Teaching and Learning Methods - Evaluation

Delivery	Face-to-face
Use of Information and Communications Technology	-
Teaching Methods	Activity Semester Workload Lectures 39 Independent study 70 Study and analysis of bibliography, Fieldwork 78.5 Course total 187.5
Student Performance Evaluation	LANGUAGE OF EVALUATION: Greek METHODS OF EVALUATION: written work (20%), Final exam (80%)

Attached Bibliography

• Anderson EJ (2014). Business Risk Management: Models and Analysis. Wiley
• Boudoukh A., J., A. Saunders (2009) Understanding Market, Credit, and Operational Risk: The Value at Risk Approach Linda, Wiley-Blackwell
• McNeil, A.J., R. Frey and P. Embrechts, (2005) Quantitative Risk Management, Princeton University Press, New Jersey
• [Περιοδικό / Journal] Risk Management

Game theory (ΣΕΕ17)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	ΣΣΕ17
Semester	2
Course Title	Game theory
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Specialized general knowledge
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes	This course aims to provide students with a foundation to: recognize and model strategic situations understand the language and concepts of game theory understand the theoretical models within the field apply game-theoretic analysis to negotiation and bargaining situations.
General Competences	Working independently Decision-making Adapting to new situations Production of free, creative and inductive thinking Synthesis of data and information, with the use of the necessary technology Working in an interdisciplinary environment

Syllabus

• Static Games of complete information (basic theory: Normal form Games, Nash equilibrium, Mixed strategies and existence of equilibrium, Applications)
• Dynamic Games of complete information (Dynamic games of complete and perfect information, two stage games of complete but imperfect information, repeated games)
• Static Games of Incomplete information (static Bayesian games and Bayesian Nash equilibrium)
• Dynamic Games of Incomplete information (perfect Bayesian equilibrium, Signalling games)
• Applications

Teaching and Learning Methods - Evaluation

Delivery	Face-to-face
Use of Information and Communications Technology	Use of ICT communication with students
Teaching Methods	Activity Semester Workload Lectures 39 Independent study 70 Study and analysis of bibliography, Fieldwork 78.5 Course total 187.5
Student Performance Evaluation	LANGUAGE OF EVALUATION: Greek METHODS OF EVALUATION: written work (20%), Final exam (80%)

Attached Bibliography

• Dixit A. and B. Nalebuff. (1991) Thinking Strategically, Norton.
• Watson J.. Strategy (2002). An Introduction to Game Theory, Norton
• Dutta P.K.. (1999) Strategies and Games: Theory And Practice, MIT
• Vega-Redondo F., (2003) Economics and the Theory of Games, Cambridge Univ. Press
• Owen G. (1995) Game Theory, Academic Press, 3rd ed.
• Myerson R. (1991) Game Theory, Harvard University Press.
• Fudenberg D. and J. Tirole (1991) Game Theory, MIT Press.
• Rubinstein M. & J. Osborne (1994) A Course in Game Theory, MIT Press.
• Ritzberger Kl. (2002) Foundations of Non-Cooperative Game Theory, Oxford University Press.
• Gibbons R. (2009) Εισαγωγή στη θεωρία παιγνίων, Εκδ. Gutenberg
• Μαγείρου Ευ. (2009), Παίγνια και Αποφάσεις, Μια Εισαγωγική Προσέγγιση, Εκδ. Κριτική, Αθήνα.
• Μηλολιδάκης, Κ. (2009) Θεωρία Παιγνίων, Μαθηματικά Μοντέλα Σύγκρουσης και Συνεργασίας. Εκδόσεις Σοφία, Θεσσαλονίκη.
• [Περιοδικό / Journal] International Journal of Game Theory.

Inventory theory (ΣΕΕ18)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	ΣΣΕ18
Semester	2
Course Title	Inventory theory
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Specialized general knowledge
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes	Upon completion of this course, a student will: understand the classical inventory models (like EOQ, base-stock models, periodic models) acquire the quantitative tools for analysing the costs and optimal solutions for such policies understand of the relationship among the classical models learn approaches to multi-echelon inventory systems that have been proposed in the literature
General Competences	Working independently Decision-making Adapting to new situations Production of free, creative and inductive thinking Synthesis of data and information, with the use of the necessary technology Working in an interdisciplinary environment

Syllabus

Inventory Systems and models, One item with constant demand rate, Several products and locations, Stochastic demand, Stochastic leadtimes, Time varying stochastic demand, Empirical Bayesian inventory models. Research frontiers.

Teaching and Learning Methods - Evaluation

Delivery	Face-to-face
Use of Information and Communications Technology	Mathematica / Matlab Use of ICT in communication with students
Teaching Methods	Activity Semester Workload Lectures 39 Independent study 70 Study and analysis of bibliography, Fieldwork 78.5 Course total 187.5
Student Performance Evaluation	LANGUAGE OF EVALUATION: Greek METHODS OF EVALUATION: written work (20%), Final exam (80%)

Attached Bibliography

• Axsater, S. Inventory Control. Norwell, MA: Kluwer, 2000.
• Hadley G. and T.M. Whitin. Analysis of inventory systems. Englewood Cliffs, N.J., Prentice-Hall, 1963.
• Porteus, E. L. Foundations of Stochastic Inventory Theory. Stanford, CA: Stanford University Press, 2002.
• Silver, E. A., D. F. Pike, and R. Peterson. Inventory Management and Production Planning and Scheduling, 3rd ed. Hoboken, NJ: Wiley, 1998.
• Zipkin P. Foundations of Inventory Management. Boston: McGraw-Hill, 2000.
• [Περιοδικό / Journal] International Journal of Production Economics
• [Περιοδικό / Journal] European Journal of Operational Research
• [Περιοδικό / Journal] Manufacturing and Service Operations Management
• [Περιοδικό / Journal] Management Science
• [Περιοδικό / Journal] Omega
• [Περιοδικό / Journal] Operations Research
• [Περιοδικό / Journal] Production and Operations Management
• [Περιοδικό / Journal] Production Planning and Control.

Advanced Topics in Statistics (ΣΕΕ19)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	ΣΣΕ19
Semester	2
Course Title	Advanced Topics in Statistics
Independent Teaching Activities	Lectures-Laboratory (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Specialized general knowledge
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English, reading Course)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes

The purpose of this advanced course is to enrich students' knowledge with current advanced topics of statistical methodology and theory that are not closely related to other courses of the subject. Moreover, this course would be characterized by a close relationship with other areas of mathematics and computing, among others, and the aim is to be characterized by an interdisciplinary character. In the learning outcomes is included the familiarization of the students, by means of this course, with an interdisciplinary type of thinking for solving problems of the real world.

General Competences

Working independently Decision-making Production of free, creative and inductive thinking Criticism and self-criticism

Syllabus

The precise content of this course may vary from time to time, but it will consist of selected, advanced topics of contemporary research interest in statistical methodology, depending on both demands from students and the availability of appropriate course leaders. Examples include parametric lifetime modeling, experimental design, extreme value statistics, advanced stochastic simulation, graphical modeling, statistics quality control etc. The course will be of interest to students who want to develop their basic knowledge of statistics methodology. See the specific semester page for a more detailed description of the course.

Teaching and Learning Methods - Evaluation

Delivery	Classroom (face-to-face)
Use of Information and Communications Technology	Use of ICT in communication with students
Teaching Methods	Activity Semester Workload Lectures 39 Working independently 78 Exercises-Homework 70.5 Course total 187.5
Student Performance Evaluation	Final written exam in Greek (in case of Erasmus students in English).

Attached Bibliography

Θα καθορίζεται από τον διδάσκοντα / Will be determined by the teacher

Advanced Topics in Operational Research (ΣΕΕ20)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	ΣΣΕ20
Semester	2
Course Title	Advanced Topics in Operational Research
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Specialized general knowledge
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes	This course will combine the technical OR and Optimization skills with the computing and modelling techniques to study several fields in which OR is applied on a daily basis.
General Competences	Working independently Decision-making Adapting to new situations Production of free, creative and inductive thinking Synthesis of data and information, with the use of the necessary technology Working in an interdisciplinary environment

Syllabus

The syllabus could be vary from year to year. Possible topics include:

• Operational Research in Production Planning
• Operational Research in Logistics
• Operational Research in Telecommunications
• Operational Research in Energy
• Operational Research in Healthcare
• Operational Research in Sport
• Operational Research in the Environment

Teaching and Learning Methods - Evaluation

Delivery	Face-to-face
Use of Information and Communications Technology	CPLEX, Mathematica/Matlab Use of ICT in communication with students
Teaching Methods	Activity Semester Workload Lectures 39 Independent study 70 Study and analysis of bibliography, Fieldwork 78.5 Course total 187.5
Student Performance Evaluation	LANGUAGE OF EVALUATION: Greek METHODS OF EVALUATION: written work (30%), Final exam (70%)

Attached Bibliography

• Allen, A. O. (1990). Probability, Statistics, and Queueing Theory - With Computer Science Applications (Second Edition). Academic Press, Orlando, Florida.
• Cohen, S. S. (1985). Operational Research. Edward Arnold, London.
• Hillier, F. S. and Lieberman, G. J. (2001). Introduction to Operations Research (Seventh Edition). McGraw-Hill, New York.
• Hopp, W. J. and Spearman, M. L. (2000). Factory Physics: Foundations of Manufacturing Management (Second Edition). Irwin/McGrawHill, New York.
• Larson, R. C. and Odoni, A. R. (1981). Urban Operations Research. PrenticeHall, Englewood Cliffs, New Jersey.
• Lewis, C. D. (1970). Scientific Inventory Control. Elsevier, New York.
• Ross, S. M. (1983). Stochastic Processes. Wiley, New York.
• Silver, E. A., Pyke, D, F. and Peterson, R. (1998). Inventory Management and Production Planning and Scheduling. (Third Edition). Wiley, New York.
• Winston W. L., Operations research (Applications and algorithms). Duxbury Press (International Thomson Publishing) 1994.
• [Περιοδικό / Journal] Operations Research
• [Περιοδικό / Journal] Management Science
• [Περιοδικό / Journal] Manufacturing and Service Operations Management
• [Περιοδικό / Journal] Production and Operations Management
• [Περιοδικό / Journal] Journal of Business Logistics
• [Περιοδικό / Journal] Sport Management Review
• [Περιοδικό / Journal] Journal of Scheduling
• [Περιοδικό / Journal] Interfaces
• [Περιοδικό / Journal] Public Transport
• [Περιοδικό / Journal] Transportation Research Part A: Policy and Practice.

Numerical Analysis (ΑΑ1)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	AA1
Semester	1
Course Title	Numerical Analysis
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Special background, skills development.
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes

Upon successful completion of this course, students will be able to: apply advanced theoretical techniques in the multidimensional space to prove and analyze the convergence and stability of numerical methods for the solution of a variety of problems. evaluate and compare numerical methods in terms of their accuracy, efficacy, and applicability. demonstrate independence in the use of research material to prove key results. implement numerical methods in Python or Octave and construct appropriate numerical experiments to verify the corresponding theoretical results. evaluate the correctness of numerical results by comparing them with both the theory of numerical methods and the theory of continuous problems.

General Competences

Search for, analysis and synthesis of data and information, with the use of the necessary technology. Adapting to new situations. Working independently. Decision-making. Production of free, creative, and inductive thinking. Promotion of analytical and synthetic thinking. Working in an interdisciplinary environment.

Syllabus

• Differentiation in n, Fréchet and Gateaux derivatives. Newton’s method for systems of nonlinear equations. Fixed-point and contraction theorems. Order of convergence of Newton’s method.
• Numerical solution of systems of ordinary differential equations. Single-step and multistep methods. Consistency, stability, and convergence. Stiff problems.
• Polynomial interpolation: Lagrange and Hermite interpolation. Linear and cubic splines. Error analysis of interpolation.

Teaching and Learning Methods - Evaluation

Delivery	Face-to-face.
Use of Information and Communications Technology	Use of a tablet device to deliver teaching. Lecture materials in pdf-format are made available to students, for later review, on Moodle learning platform. Provision of study materials in Moodle e-learning platform. Use of online quizzes in Moodle platform, which aim to enhance student engagement and motivation in learning. Provision of model solutions for some exercises in podcast format. Communication with students through e-mails, Moodle platform and Microsoft teams. Use of sophisticated software (python or Octave) to enhance students’ understanding and learning by demonstrating numerical examples in the classroom.
Teaching Methods	Activity Semester Workload Lectures 39 Study and analysis of bibliography 70 Worksheets 30 Project 30 Presentation 18.5 Course total 187.5
Student Performance Evaluation	Solution of worksheets (Weighting 35%, addressing learning outcomes 1-3) Project, produced with LaTeX (Weighting 40%, addressing learning outcomes 1-5) Presentation, produced with Beamer (Weighting 25%, addressing learning outcomes 1-5)

Attached Bibliography

• Αριθμητική Ανάλυση, Β. Δουγαλής, Πανεπιστημίου Αθηνών.

Approximation Theory (AA2)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	AA2
Semester	1
Course Title	Approximation Theory
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Special Background
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes	After successful end of this course, students will be able to: know the basic items of approximation from a linear space to a subspace, know the differences (advantages and disadvantages) of different kinds of approximations, know the basic numerical methods for the polynomial approximation, implement the algorithms of such methods on a computer.
General Competences	Search for, analysis and synthesis of data and information, with the use of the necessary technology Adapting to new situations Criticism and self-criticism Production of free, creative and inductive thinking

Syllabus

• General Theory of existence and uniqueness of approximation.
• Uniform Approximation: Weierstrass, Bernstein, Jackson theorems, approximation of continuous functions, approximation of discrete functions, Remez algorithm.
• Least Squares Polynomial Approximation: Systems of Normal Equations, Orthogonal Polynomials, approximation of continuous functions, approximation of discrete functions, connection with Uniform approximation.
• First Power Polynomial Approximation: Characterization, approximation of continuous functions, approximation of discrete functions,.
• Rational Approximation: Characterization, connection with Uniform approximation, Remez algorithm.
• Rational Interpolation.

Teaching and Learning Methods - Evaluation

Delivery	In the classroom
Use of Information and Communications Technology	-
Teaching Methods	Activity Semester Workload Lectures 39 Working Independently 78 Exercise - Homework 70.5 Course total 187.5
Student Performance Evaluation	Written examination

Attached Bibliography

• Theodor J. Rivlin: An Introduction to the Approximation of Functions. Dover Publications Inc. New York, 1969.

Numerical Linear Algebra I (ΑΑ3)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	AA3
Semester	1
Course Title	Numerical Linear Algebra I
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Special Background
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes

After successful end of this course, students will be able to: know and understand the Perron-Frobenius Theory, know the differences of Perron-Frobenius Theory as applied to different classes of matrices (irreducible, cyclic, primitive and reducible), know the efficiency of the Perron-Frobenius Theory in applications, know and understand the theory of Krylov subspace methods, know error analysis, know the preconditioned techniques and the necessity of preconditioning, implement the above methods with programs on a computer.

General Competences

Search for, analysis and synthesis of data and information, with the use of the necessary technology Adapting to new situations Criticism and self-criticism Production of free, creative and inductive thinking

Syllabus

Perron-Frobenius Theory of Nonnegative Matrices: Irreducible Matrices, Cyclic and Primitive Matrices, Reducible Matrices, Extension of the Perron-Frobenius Theory, M-matrices, Applications of the Perron-Frobenius Theory. Minimization methods for the Solution of Linear Systems: Conjugate Gradient Method, Convergence Theory, Error Analysis, Preconditioning Techniques, Preconditioned Conjugate Gradient Methods, Applications.

Teaching and Learning Methods - Evaluation

Delivery	In the class
Use of Information and Communications Technology	-
Teaching Methods	Activity Semester Workload Lectures 39 Study and analysis of bibliography 78 Preparation of assignments and interactive teaching 70.5 Course total 187.5
Student Performance Evaluation	Written examination - Oral Examination.

Attached Bibliography

• Theodor J. Rivlin: An Introduction to the Approximation of Functions. Dover Publications Inc. New York, 1969.

Numerical Linear Algebra II (AA4)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	AA4
Semester	1
Course Title	Numerical Linear Algebra II
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Special Background
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in Greek)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes

After successful end of this course, students will be able to: know and understand the theory of methods for computation of the eigenvalues and singular values, know from applications, the necessity of this theory, know and understand the theory of Krylov subspace methods, know error analysis, know the preconditioned techniques and the necessity of preconditioning, implement the above methods with programs on the computer.

General Competences

Search for, analysis and synthesis of data and information, with the use of the necessary technology Adapting to new situations Criticism and self-criticism Production of free, creative and inductive thinking

Syllabus

Numerical methods for the computation of Eigenvalues and Eigenvectors: Power Method, QR Method, Stable algorithms (Howsholder Reflections, Givens Rotations). Singular Values: Singular Value Decomposition. Krylov subspace Methods for the solution of Large Scale Linear Systems: Preconditioned Conjugate Gradient Method. Generalized Minimal Residual Method (GMRES): Theory of Orthogonalization of Krylov Subspaces, Arnoldi and Lanczos Algorithms. Applications of Iterative Methods to boundary value problems and to Signal and Image Processing.

Teaching and Learning Methods - Evaluation

Delivery	In the classroom
Use of Information and Communications Technology	-
Teaching Methods	Activity Semester Workload Lectures 39 Study and analysis of bibliography 78 Exercises - Homework 70.5 Course total 187.5
Student Performance Evaluation	Written examination - Oral Examination

Attached Bibliography

• “Αριθμητική Γραμμική Άλγεβρα”, Β. Δουγαλής, Δ. Νούτσος, Α. Χατζηδήμος, Τυπογραφείο Πανεπιστημίου Ιωαννίνων.
• “Matrix Computations”, G. H. Golub, C. F. Van Loan, The John Hopkings University Press, Baltimore and London, 1996.

Numerical Solution of Ordinary Differential Equations (AA5)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	AA5
Semester	1
Course Title	Numerical Solution of Ordinary Differential Equations
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Special Background
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes

Numerical methods for initial value problems for ordinary differential equations are studied in this course; an introduction to numerical methods for the two-point boundary value problem is also given. Learning Objectives: Understanding the basic facts for initial value problems and the two-point boundary value problem. Understanding the fundamental qualitative characteristics of numerical methods for initial value problems, like consistency, order of accuracy, stability and convergence. It is expected that after taking the course the student will have: Understand the basic facts for initial value problems as well as for the two-point boundary value problem. Know the basic numerical methods for initial value problems and are familiar with their advantages and drawbacks. Understand the role of consistency, order of accuracy and stability of numerical methods for initial value problems. Know the basic numerical methods for initial value problems. Know the basic properties of finite difference and finite element methods for the two-point boundary value problem.

General Competences

Production of free, creative and inductive thinking. Consolidation, deepening and application of mathematical knowledge. Familiarity with numerical methods for initial as well as for boundary value problems.

Syllabus

• Short introduction to the theory of initial value problems.
• Analysis of the Euler methods: order of accuracy; stability properties, A-stability and B-stability; error estimates under various Lipschitz conditions (global, local and one-sided); a posteriori error estimates.
• Runge-Kutta and collocation methods: stability properties, order of accuracy, embedded pairs of methods and adaptive time step selection.
• Multistep methods: elements of the theory of difference equations, the root condition and stability, order of accuracy, one-leg methods, and G-stability.
• Introduction to the theory of the two-point boundary value problem: energy method and elliptic regularity.
• Finite difference methods for the two-point boundary value problem.
• Finite element method: construction of finite element spaces for various boundary conditions, Galerkin and Ritz methods, the Nitsche trick. Error estimates in the case of indefinite operators.

Teaching and Learning Methods - Evaluation

Delivery	In the class
Use of Information and Communications Technology	-
Teaching Methods	Activity Semester Workload Lectures 39 Working Independently 78 Exercise - Homework 70.5 Course total 187.5
Student Performance Evaluation	Mid-term and final written examinations

Attached Bibliography

• Γ. Δ. Ακρίβης, Β. Α. Δουγαλής: Αριθμητικές Μέθοδοι για Συνήθεις Διαφορικές Εξισώσεις. Πανεπιστημιακές Εκδόσεις Κρήτης, Ηράκλειο. Δεύτερη έκδοση, 2013, πρώτη ανατύπωση, 2015.

Numerical Solution of Partial Differential Equations (ΑΑ6)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	AA6
Semester	2
Course Title	Numerical Solution of Partial Differential Equations
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Special background, skills development.
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes

Upon successful completion of this course, students will be able to: apply advanced numerical analysis techniques to prove error estimates for numerical approximations of elliptic and parabolic problems. demonstrate independence in the use of research materials to prove key results. write FEM code in FEniCS or Octave and construct appropriate numerical experiments to verify theoretical results. evaluate the correctness of numerical results by comparing them with both the theory of numerical methods and the theory of continuous problems.

General Competences

Search for, analysis and synthesis of data and information, with the use of the necessary technology. Adapting to new situations. Working independently. Decision-making. Production of free, creative, and inductive thinking. Promotion of analytical and synthetic thinking. Working in an interdisciplinary environment.

Syllabus

• Hilbert spaces, Riesz’s representation theorem, Lax-Milgram’s theorem, Cea’s theorem.
• Sobolev spaces, weak derivatives, Poincare-Friedrichs inequalities.
• Weak formulation and the Finite Element Method (FEM) for elliptic boundary value problems in 1D and 2D. A priori and a posteriori error estimates, adaptivity.
• Semi-discrete and fully-discrete schemes for parabolic equations. Temporal discretization with the Explicit and Implicit Euler methods, and the Crank-Nicolson method.
• Computer implementation of FEMs.

Teaching and Learning Methods - Evaluation

Delivery	Face-to-face.
Use of Information and Communications Technology	Use of a tablet device to deliver teaching. Lecture materials in pdf-format are made available to students, for later review, on Moodle learning platform. Provision of study materials in Moodle e-learning platform. Use of online quizzes in Moodle platform, which aim to enhance student engagement and motivation in learning. Provision of model solutions for some exercises in podcast format. Communication with students through e-mails, Moodle platform and Microsoft teams. Use of sophisticated software (Octave ή FEniCS) for the computer implementation of FEM.
Teaching Methods	Activity Semester Workload Lectures 39 Study and analysis of bibliography 70 Worksheets 25 Project 35 Presentation 18.5 Course total 187.5
Student Performance Evaluation	Solution of worksheets (Weighting 30%, addressing learning outcomes 1-2) Project (Weighting 45%, addressing learning outcomes 1-4) Presentation (Weighting 25%, addressing learning outcomes 1-5)

Attached Bibliography

• “Μέθοδοι πεπερασμένων στοιχείων”, Γ. Δ. Ακρίβης, Λευκωσία, 2005.
• “Αριθμητική λύση μερικών διαφορικών εξισώσεων”, Μ. Πλεξουσάκης, & Π. Χατζηπαντελίδης, Κάλλιππος, 2015. http://hdl.handle.net/11419/665
• “The Mathematical Theory of Finite Element Methods”, S.C. Brenner, & L.R. Scott (Third ed., Vol. 15), Springer, New York, 2008.
• “Galerkin Finite Element Methods for Parabolic Problems”, V. Thomee, Springer-Verlag, 1997.

Specialized Topics in Numerical Analysis (AA7)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	AA7
Semester	2
Course Title	Specialized Topics in Numerical Analysis
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Special Background
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes	After successful end of this course, students will be able to: Combine the theory with applications. Analyze and study problems from applied sciences, choose the appropriate methods and finally to solve them. Understand the usefulness of Numerical Analysis to solve problems from nature and society.
General Competences	Search for, analysis and synthesis of data and information, with the use of the necessary technology Adapting to new situations Criticism and self-criticism Production of free, creative and inductive thinking

Syllabus

Solution of Large Scale problems from Fluid Mechanics by Numerical Methods: Analysis of the problem - Selection of the appropriate method - Implementation - Comparison of the methods. Solution of Large Scale Optimization problems coming from Control Theory Data mining - searching, Signal and Image Processing by Numerical Methods: Analysis of the problem - Selection of the appropriate method - Implementation - Comparison of the methods.

Teaching and Learning Methods - Evaluation

Delivery	In the class
Use of Information and Communications Technology	-
Teaching Methods	Activity Semester Workload Lectures 39 Study and analysis of bibliography 78 Exercises - Homework 70.5 Course total 187.5
Student Performance Evaluation	Written examination - Oral Examination - Project

Attached Bibliography

• Σημειώσεις Διδάσκοντα / Lecture notes
• Υλικό από το διαδίκτυο / Material from the web

Parallel Algorithms (AA8)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	AA8
Semester	2
Course Title	Parallel Algorithms
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Special Background
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes	After successful end of this course, students will be able to: understand techniques of Parallel Computations, Understand the necessity of Parallel Computations to solve large scale problems, Implement methods with programs on systems of parallel computations.
General Competences	Search for, analysis and synthesis of data and information, with the use of the necessary technology Adapting to new situations Criticism and self-criticism Production of free, creative and inductive thinking

Syllabus

• Techniques of Parallel Computations.
• Parallel Algorithms for the Solution of Linear Systems by Iterative Methods.
• Parallel Algorithms based on Domain Decomposition Methods for the solution of Boundary Value Problems.
• Parallelization factors of efficiency.

Teaching and Learning Methods - Evaluation

Delivery	In the class
Use of Information and Communications Technology	-
Teaching Methods	Activity Semester Workload Lectures 39 Study and analysis of bibliography 78 Exercises - Homework 70.5 Course total 187.5
Student Performance Evaluation	Written examination - Oral Examination - Project

Attached Bibliography

• Σημειώσεις Διδάσκοντα / Lecture notes.
• Υλικό από το διαδίκτυο / Material from the web.

Methods of Applied Mathematics I (ΕΜ1)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	EM1
Semester	2
Course Title	Methods of Applied Mathematics I
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Special Background
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes

The course is an introduction to the basic analytical and numerical methods of applied mathematics. The objectives of the course are: Development of the theoretical background of the postgraduate student in matters relating to Applied Mathematics. Ability of the student to apply analytical, approximate and numerical methods in problems of Mathematics, Physics and Engineering. Upon completion of the course the graduate student will be able to solve problems with analytical, approximate or numerical methods and further deepen the understanding of such methods.

General Competences

Adapting to new situations Decision-making Working independently Team work

Syllabus

Dimensional analysis and normalization, Perturbation theory for algebraic equations, integral and differential equations, Physical models described by partial differential equations (PDEs), Wave phenomena in continuous media, The course includes training in computational methods in the computer laboratory (Mechanics lab).

Teaching and Learning Methods - Evaluation

Delivery	In the class
Use of Information and Communications Technology	Use of computer (Mechanics) lab
Teaching Methods	Activity Semester Workload Lectures 39 Study and analysis of bibliography 78 Preparation of assignments and interactive teaching 70.5 Course total 187.5
Student Performance Evaluation	Weekly assignments Final project Written examination at the end of the semester

Attached Bibliography

• Σημειώσεις Διδάσκοντα / Lecture notes.
• Υλικό από το διαδίκτυο / Material from the web.

Methods of Applied Mathematics ΙI (EM2)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	EM2
Semester	2
Course Title	Methods of Applied Mathematics ΙI
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Special Background
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes

The course is a first introduction to the basic methods of applied mathematics and particularly in perturbation theory. There are many situations in mathematics where one finds expressions that cannot be calculated with absolute precision, or where exact answers are too complicated to provide useful information. In many of these cases, it is possible to find a relatively simple expression which, in practice, is just as good as the complete, exact solution. The asymptotic analysis deals with methods for finding such approximations and has a wide range of applications, both in the fields of pure mathematics such as combinatorics, probability, number theory and applied mathematics and computer science. The goal of this course is to introduce some of the basic techniques and to apply these methods to a variety of problems. Upon completion of this course students will be able to: Recognize the practical value of small or large parameters for calculating mathematical expressions. Understand the concept of (divergent) asymptotic series, and distinguish between regular and singular perturbations. Find dominant behaviors in algebraic and differential equations with small and large parameters. Calculate dominant behavior of integrals with a small parameter. Find a (in particular cases) the full asymptotic behavior of integrals. Identify the boundary layers in solutions of differential equations, and apply appropriate expansions to calculate the dominant solutions.

General Competences

Adapting to new situations Decision-making Working independently Team work

Syllabus

Introduction and notation of perturbation theory, Regular and singular perturbations, Asymptotic expansions of integrals, Asymptotic solutions of linear and nonlinear differential equations, Laplace and Fourier transforms (if time permits).

Teaching and Learning Methods - Evaluation

Delivery	In class
Use of Information and Communications Technology	Use of computer (Mechanics) lab
Teaching Methods	Activity Semester Workload Lectures 39 Self study 78 Homework - Projects 70.50 Course total 187.5
Student Performance Evaluation	Weekly assignments Final project

Attached Bibliography

• Εφαρμοσμένα Μαθηματικά, Logan D.J. Πανεπιστημιακές Εκδόσεις Κρήτης, Ηράκλειο, 1η έκδοση, 2010.
• Perturbation Methods, A.H. Nayfeh, 1η έκδοση, Willey-VCH, 2000.

Partial Differential Equations and Applications (EM3)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	EM3
Semester	1
Course Title	Partial Differential Equations and Applications
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Special Background
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes

The student in this course will apply the mathematical tools obtained from previous courses to better understand concepts arising from natural (and not only) phenomena and the way these are transformed into mathematical problems. More specifically, by completing this course, students should be able to use the method of characteristics to solve partial differential equations classify partial differential equations of second order in elliptic, parabolic and hyperbolic type use Green’s functions to solve elliptic type equations have a basic understanding of diffusion equations use separation of variables to solve linear partial differential equations

General Competences

Adapting to new situations Decision-making Working independently Team work

Syllabus

Basic concepts. Linear, quasi-linear and semi-linear equations of the first order. The Cauchy problem and its solution by the method of characteristic. Linear equations of 2nd order: classification (hyperbolic, parabolic, elliptic), examples (wave equation, heat equation, Laplace equation). Problems of initial and boundary values for the wave and heat equations. Boundary value problems and the Laplace equation. The Cauchy problem for the wave and heat equations.

Teaching and Learning Methods - Evaluation

Delivery	In class
Use of Information and Communications Technology	-
Teaching Methods	Activity Semester Workload Lectures 39 Self study 78 Homework - Projects 70.5 Course total 187.5
Student Performance Evaluation	Weekly assignments Final project

Attached Bibliography

• Fluid Mechanics with Applications, M. Xenos and E. Tzirtzilakis, 2018 (in Greek)
• Fluid Mechanics, Volume 1, A. Papaioanou, 2nd Edition, 2001 (in Greek).
• Computational Fluid Mechanics, I. Soulis, 1st Edition, 2008 (in Greek).
• Numerical heat transfer and fluid flow, S.V. Patankar, McGraw-Hill, New York, 1980.
• The Finite Element Method, Vol. 1, The Basis, O.C. Zienkiewicz, R.L. Taylor, 5th Ed., Butterworth-Heinemann, Oxford, 2000.
• Computational Techniques for fluid Dynamics, C.A.J. Fletcher Volumes I and II, 2nd Ed. Springer-Verlag, Berlin, 1991.

Fluid Mechanics (ΕΜ4)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	EM4
Semester	2
Course Title	Fluid Mechanics
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Special Background
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes

The course is an introduction to the analytical and numerical methods of Fluid Mechanics. The objectives of the course are: Development of the theoretical background of the postgraduate student in matters relating to Fluid Mechanics and ability of the student to apply analytical, approximate and numerical methods in Fluid Mechanics problems. Upon completion of the course the graduate student will be able to solve problems with analytical, approximate or numerical methods and further deepen the understanding of such methods.

General Competences

The course aims to enable the postgraduate student to: Develop the ability to analyse and synthesize basic knowledge of Fluid Mechanics. Adapt to new situations Decision-making Working independently Team work All the above will give to the students the opportunity to work in an international multidisciplinary environment.

Syllabus

Kinematics of Fluids, Fluid flow analysis, Equation of continuity and stream function, motion equations for Ideal and real fluids, laminar and turbulent flow, Boundary layer flows with adverse pressure gradient, Numerical Methods in Fluid Mechanics, Classification of fluid dynamics problems and relevant equations that describe basic numerical schemes, method of finite differences, compatibility, stability and convergence of numerical schemes, finite volume method, Introduction to the method of weighted residues, finite element method. The course includes training in computational methods in the computer laboratory (Applied and Computational Mathematics Laboratory).

Teaching and Learning Methods - Evaluation

Delivery	In class
Use of Information and Communications Technology	Use of computer lab (Applied and Computational Mathematics Laboratory).
Teaching Methods	Activity Semester Workload Lectures 39 Study study 78 Homework - Projects 70.5 Course total 187.5
Student Performance Evaluation	Weekly assignments Final project Written examination at the end of the semester

Attached Bibliography

• Fluid Mechanics with Applications, M. Xenos and E. Tzirtzilakis, 2018 (in Greek)
• Fluid Mechanics, Volume 1, A. Papaioanou, 2nd Edition, 2001 (in Greek).
• Computational Fluid Mechanics, I. Soulis, 1st Edition, 2008 (in Greek).
• Numerical heat transfer and fluid flow, S.V. Patankar, McGraw-Hill, New York, 1980.
• The Finite Element Method, Vol. 1, The Basis, O.C. Zienkiewicz, R.L. Taylor, 5th Ed., Butterworth-Heinemann, Oxford, 2000.
• Computational Techniques for fluid Dynamics, C.A.J. Fletcher Volumes I and II, 2nd Ed. Springer-Verlag, Berlin, 1991.

Dynamical Systems and Chaos (ΕΜ5)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	EM5
Semester	1
Course Title	Dynamical Systems and Chaos
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Special Background
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes

The course is an introduction to continuous and discrete dynamical systems. Non linear systems of differential equations often lead to non-deterministic (stochastic) results and chaotic situations. The objectives of the course are: Obtaining the theoretical background from the postgraduate student on issues related to dynamical systems described by differential equations. Obtaining the background from the student in computational methods to solve problems of dynamical systems. Description of chaotic situations of dynamical systems Upon completion of the course the postgraduate student will be able to solve with analytical and numerical mathematical methods problems of the dynamical systems and to further deepen their understanding.

General Competences

The course aims to enable the postgraduate student to: Develop the ability to analyse and synthesize basic knowledge of Dynamical Systems. Adapt to new situations Decision-making Working independently Team work All the above will give to the students the opportunity to work in an international multidisciplinary environment.

Syllabus

Dynamical systems and differential equations of motion, Equilibrium points of the dynamical system, Period doubling of non-linear differential equations, Phase space of the dynamical system, Chaotic trajectory of the system, Poincare map, Applications of the dynamical systems, Henon map, Mandelbrot και Julia sets, Self-similarity under scale change and Fractals. The course includes training in computational methods in the computer laboratory (Applied and Computational Mathematics Laboratory).

Teaching and Learning Methods - Evaluation

Delivery	In class
Use of Information and Communications Technology	Use of computer lab (Applied and Computational Mathematics Laboratory).
Teaching Methods	Activity Semester Workload Lectures 39 Study study 78 Homework - Projects 70.5 Course total 187.5
Student Performance Evaluation	Weekly assignments Final project Written examination at the end of the semester

Attached Bibliography

• Δυναμικά Συστήματα και Χάος, Πρώτος Τόμος, Α. Μπούντης, 1995, Εκδότης: Α. ΠΑΠΑΣΩΤΗΡΙΟΥ & ΣΙΑ Ι.Κ.Ε.
• Δυναμικά Συστήματα και Χάος, Δεύτερος Τόμος, Α. Μπούντης, 2001, Εκδότης: Εταιρεία Αξιοποίησης και Διαχείρισης Περιουσίας Πανεπιστημίου Πατρών.
• An Introduction to Dynamical Systems and Chaos, G.C. Layek, 2015, Editor: Springer.

Integrable Systems (EM6)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	EM6
Semester	1
Course Title	Integrable Systems
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Special Background
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes

Integrable systems are nonlinear differential equations which, in principle, can be solved analytically. This means that the solution can be reduced to a finite number of algebraic operations and integrations. Such systems are very rare - most nonlinear differential equations admit chaotic behavior and no explicit solutions can be written down. Integrable systems nevertheless lead to a very interesting mathematics ranging from differential geometry and complex analysis to quantum field theory and fluid dynamics. The main topics treated in the course, and the expected skill obtained by the students, are: Integrability of ODEs: Hamiltonian formalism, the Arnold-Liouville theorem, Painleve analysis. Integrability of PDEs: Solitons, Inverse Scattering Transform.

General Competences

Adapting to new situations Decision-making Working independently Team work

Syllabus

Integrability in classical mechanics, Painleve analysis, Fourier transforms, the Inverse Scattering Transform and Soliton theory.

Teaching and Learning Methods - Evaluation

Delivery	In class
Use of Information and Communications Technology	-
Teaching Methods	Activity Semester Workload Lectures 39 Self study 78 Homework - Projects 70.5 Course total 187.5
Student Performance Evaluation	Weekly assignments Final project

Attached Bibliography

• P. G. Drazin, R. S. Johnson, Solitons: An Introduction, Cambridge University Press, 1989.
• M. J. Ablowitz, H. Segur, Solitons and the Inverse Scattering Transform, SIAM 1981.
• Προσωπικές σημειώσεις του διδάσκοντα.

Fractal Sets and Applications (EM7)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	EM7
Semester	2
Course Title	Fractal Sets and Applications
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Special Background
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes

The course is an introduction to Fractals and structures that have self-similarity under scale change. The objectives of the course are: Acquiring the theoretical background from the postgraduate student on topics related to Fractals. Obtaining the background from the student in analytical and computational methods to solve problems related to the Fractals. Understanding basic concepts of Fractals and extending to applications and nature. Upon completion of the course the postgraduate student will be able to use analytical and computational techniques to study problems related to Fractals and to further deepen their understanding.

General Competences

The course aims to enable the postgraduate student to: Develop the ability to analyse and synthesise basic knowledge of Fractals. Adapt to new situations Decision-making Working independently Team work All the above will give the students the opportunity to work in an international multidisciplinary environment.

Syllabus

Self-similarity under scale change, Fractal sets, Hausdorff dimension, Mandelbrot and Julia sets, Affine transformations in Euclidean space, Transformations in metric spaces, Theorem of contraction of images, Fractal construction, Collage theorem, Applications of Fractal sets. The course includes training in computational methods in the computer laboratory (Applied and Computational Mathematics Lab).

Teaching and Learning Methods - Evaluation

Delivery	In class
Use of Information and Communications Technology	Use of computer (Mechanics) lab
Teaching Methods	Activity Semester Workload Lectures 39 Self study 78 Homework - Projects 70.5 Course total 187.5
Student Performance Evaluation	Weekly assignments Final project Written examination at the end of the semester

Attached Bibliography

• Ο Θαυμαστός Κόσμος των Fractal, 2004, Α. Μπούντης, Εκδότης: Liberal Books Μονοπρόσωπη ΕΠΕ.
• Fractals Everywhere, 2nd edition, 2000, M. F. Barnsley, Publisher: Morgan Kaufmann.

Calculus of Complex Functions and Applications (EM8)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	EM8
Semester	2
Course Title	Calculus of Complex Functions and Applications
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Special Background
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes

By the end of the course the student should be able to: give an account of the concepts of analytic function and harmonic function and to explain the role of the Cauchy-Riemann equations. explain the concept of conformal mapping, describe its relation to analytic functions, and know the mapping properties of the elementary functions. describe the mapping properties of Möbius transformations and know how to use them for conformal mappings. define and evaluate complex contour integrals. give an account of and use the Cauchy integral theorem, the Cauchy integral formula and some of their consequences. analyze simple sequences and series of functions with respect to uniform convergence, describe the convergence properties of a power series, and determine the Taylor series or the Laurent series of an analytic function in a given region. give an account of the basic properties of singularities of analytic functions and be able to determine the order of zeros and poles, to compute residues and to evaluate integrals using residue techniques. use the theory, methods and techniques of the course to solve mathematical problems.

General Competences

Adapting to new situations Decision-making Working independently Team work

Syllabus

Complex numbers, topology in ℂ. Functions of one complex variable, limits, continuity and differentiability. The Cauchy-Riemann equations. Analytic and harmonic functions. Conformal mappings. Elementary functions from ℂ to ℂ, in particular Möbius transformations and the exponential function. Solution of boundary value problems in the plane for the Laplace equation using conformal mappings. Complex integration. Cauchy's integral theorem. The maximum principle for analytic and harmonic functions. Poisson's integral formula. Uniform convergence and analyticity. Power series. Taylor and Laurent series with applications. Zeros and isolated singularities. Residue calculus with applications. Rouché's theorem. Briefly about connections with Fourier series and Fourier integrals. The Riemann-Hilbert problem.

Teaching and Learning Methods - Evaluation

Delivery	In class
Use of Information and Communications Technology	-
Teaching Methods	Activity Semester Workload Lectures 39 Self study 78 Homework - Projects 70.5 Course total 187.5
Student Performance Evaluation	Weekly assignments Final project

Attached Bibliography

• ΜΙΓΑΔΙΚΕΣ ΣΥΝΑΡΤΗΣΕΙΣ ΚΑΙ ΕΦΑΡΜΟΓΕΣ, Κωδικός Βιβλίου στον Εύδοξο: 226, Έκδοση: 1η/2005, Συγγραφείς: CHURCHILL R., BROWN J., ISBN: 960-7309-41-3, Τύπος: Σύγγραμμα, Διαθέτης (Εκδότης): ΙΔΡΥΜΑ ΤΕΧΝΟΛΟΓΙΑΣ & ΕΡΕΥΝΑΣ-ΠΑΝΕΠΙΣΤΗΜΙΑΚΕΣ ΕΚΔΟΣΕΙΣ ΚΡΗΤΗΣ
• ΜΙΓΑΔΙΚΕΣ ΜΕΤΑΒΛΗΤΕΣ, Κωδικός Βιβλίου στον Εύδοξο: 12404786, Έκδοση: 1η/2011, Συγγραφείς: ABLOWITZ MARK J., ΦΩΚΑΣ ΑΘΑΝΑΣΙΟΣ Σ., ISBN: 978-960-524-337-1, Τύπος: Σύγγραμμα, Διαθέτης (Εκδότης): ΙΔΡΥΜΑ ΤΕΧΝΟΛΟΓΙΑΣ & ΕΡΕΥΝΑΣ-ΠΑΝΕΠΙΣΤΗΜΙΑΚΕΣ ΕΚΔΟΣΕΙΣ ΚΡΗΤΗΣ
• Αναλυτικές συναρτήσεις και μερικές εφαρμογές τους, Κωδικός Βιβλίου στον Εύδοξο: 12166, Έκδοση: 2η έκδ./1998, Συγγραφείς: Τερσένοβ Σάββας, ISBN: 978-960-7140-66-1, Τύπος: Σύγγραμμα, Διαθέτης (Εκδότης): ΔΙΑΥΛΟΣ Α.Ε. ΕΚΔΟΣΕΙΣ ΒΙΒΛΙΩΝ
• Μιγαδικές συναρτήσεις, Κωδικός Βιβλίου στον Εύδοξο: 11116, Έκδοση: 1η έκδ./1996, Συγγραφείς: Παντελίδης Γεώργιος Ν., Κραββαρίτης Δημήτρης Χ., Νασόπουλος Β., ISBN: 960-431-358-4, Τύπος: Σύγγραμμα, Διαθέτης (Εκδότης): Ζήτη Πελαγία & Σια Ι.Κ.Ε.
• Μιγαδικές συναρτήσεις, Κωδικός Βιβλίου στον Εύδοξο: 11115, Έκδοση: 1η έκδ./2008, Συγγραφείς: Ξένος Θανάσης Π., ISBN: 978-960-456-092-9, Τύπος: Σύγγραμμα, Διαθέτης (Εκδότης): Ζήτη Πελαγία & Σια Ι.Κ.Ε.

Specialized Topics in Applied Mathematics (EM9)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	EM9
Semester	2
Course Title	Specialized Topics in Applied Mathematics
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Special Background
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes	Introduction to computational or theoretical research on acceptable applied mathematics problems and supervision of reading on topics not covered by regular courses of instruction.
General Competences	Adapting to new situations Decision-making Working independently Team work

Syllabus

Depending on the students interests and Instructor availability.

Teaching and Learning Methods - Evaluation

Delivery	In class
Use of Information and Communications Technology	Use of computer (Mechanics) lab
Teaching Methods	Activity Semester Workload Lectures 39 Self study 78 Homework - Projects 70.5 Course total 187.5
Student Performance Evaluation	Weekly assignments Final project

Attached Bibliography

• ---

Specialized Topics in Fluid Mechanics (EM10)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	EM10
Semester	2
Course Title	Specialized Topics in Fluid Mechanics
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Special Background
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes	Introduction to computational or theoretical research on advanced Fluid Mechanics problems and supervision of reading on topics not covered by regular courses of instruction.
General Competences	Adapting to new situations Decision-making Working independently Team work

Syllabus

Depending on the students interests and Instructor availability.

Teaching and Learning Methods - Evaluation

Delivery	In class
Use of Information and Communications Technology	Use of Applied and Computational Mathematics computer lab
Teaching Methods	Activity Semester Workload Lectures 39 Self study 78 Homework - Projects 70.5 Course total 187.5
Student Performance Evaluation	Weekly assignments Final project Written examination at the end of the semester

Attached Bibliography

• ---

Complexity Theory (ΠΛ1)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	ΠΛ1
Semester	1
Course Title	Complexity Theory
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Elective
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes	This course aims at introducing to students the concepts of time and space complexities for solving difficult problems. After successfully passing this course the students will be able to: Understand complexity classes. Push further techniques for solving difficult problems. Understand difficult problems by using reductions.
General Competences	Search for, analysis and synthesis of data and information, with the use of the necessary technology Working independently Team work Project planning and management

Syllabus

• ΝΡ and Computational Intractibility
• The class of PSPACE
• Extending the limits of tractability
• Approximation Algorithms
• Local search.
• Randomized algorithms

Teaching and Learning Methods - Evaluation

Delivery	Lectures
Use of Information and Communications Technology	Use of projector and interactive board during lectures.
Teaching Methods	Activity Semester Workload Lectures 39 Working independently 78 Exercises - Homework 70.5 Course total 187.5
Student Performance Evaluation	Written work (50%). Essay / report (20%). Public presentation (30%).

Attached Bibliography

• Computational Complexity, Christos Papadimitriou.
• Computers and Intractability, M. R. Garey and D. S. Johnson.
• J. Kleinberg and E. Tardos, Σχεδιασμός Αλγορίθμων, ελληνική έκδοση, Εκδόσεις Κλειδάριθμος, 2008
• T. Cormen, C. Leiserson, R. Rivest, and C. Stein, Εισαγωγή στους Αλγορίθμους, ελληνική έκδοση, Πανεπιστημιακές Εκδόσεις Κρήτης, 2012.

Theory of Computation (ΠΛ2)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	ΠΛ2
Semester	1
Course Title	Theory of Computation
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Specialization
Prerequisite Courses	Undergraduate courses in Automata Theory and Formal Languages, Data Structures and Algorithms
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes

The goal of this course is the deeper understanding of Automata Theory and Language, Computability Theory and Complexity Theory as well as the introduction of students to critical thinking and research process. During the course a detailed examination of the following topics are done: Finite Automata (Deterministic FA, Nondeterministic FA, FA with Epsilon-Transitions) and their applications, Regular Expressions and Languages, Properties of Regular Languages Context-Free Grammars and languages, Pushdown Automata (Deterministic PDA, Acceptance by Fina lState, Acceptance by Empty Stack) , Properties of Context-Free Languages Turing Machines (Standard TM, Multitrack TM, Two-Way Tape TM, Multitape TM, Nondeterministic TM) Decidability and Computability Computational Complexity After completing the course the student can handle: theoretical documentation of problems solving exercises tracking applications which related to Finite Automata, Pushdown Automata, and Turing Machines as well as to Decidability and Computability and to Computational Complexity.

General Competences

Independent work Bibliographic search Effective selection and Design of the required machine and language.

Syllabus

• Properties of the Computation Theory Mathematical Models
• Problems classification to solvable and unsolvable
• Solvable Problems Classification

Teaching and Learning Methods - Evaluation

Delivery	Face to face
Use of Information and Communications Technology	Yes
Teaching Methods	Activity Semester Workload Lectures 39 Self study 78 Exercises 70.5 Course total 187.5
Student Performance Evaluation	Final essays (40%). Exercises - questions requiring critical thinking (30%). Presentations of related issues (30%).

Attached Bibliography

• Sudkamp, Thomas A. Languages and machines : an introduction to the theory of computer science / Thomas A. Sudkamp. - 2nd ed. ISBN 0-201-82136-2
• Hopcroft, John E., Rajeev Motwani, Jeffrey . Ullman Introduction to automata theory, languages and computation -2nd ed. ISBN 0321210298
• Michael Sipser. Introduction to the Theory of Computation (3rd ed.). Cengage Learning. ISBN 978-1-133-18779-0.

Advanced Algorithmic Topics (ΠΛ3)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	ΠΛ3
Semester	1
Course Title	Advanced Algorithmic Topics
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Specialization
Prerequisite Courses	Undergraduate courses in Data structures and Algorithms (optionally a course in Discrete mathematics)
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes

The goal of this course is the deeper understanding of the design and analysis of algorithms and address specific classes of problems and algorithms to solve them as well as the introduction of students to critical thinking and research process. A detailed examination of advanced methods of analysis and design of algorithms is done during the course. The analysis of an algorithm studies ways of finding its complexity. For the design of an algorithm for a problem we discuss basic design methods such as: greedy methods, dynamic programming, backtracking, recursion, exhaustive search of solution space, and branch and bound. We examine algorithms for problem categories such as sorting, searching, selection, graphs processing, integers and polynomials arithmetic, algorithms in matrices, and string handling algorithms. Complexity classes such as P, NP, NP-complete are defined. Some specific topics are also presented. After completing the course the student: Can analyze an algorithm Can select the most effective algorithm between algorithms for solving a problem. Has a good understanding of design methods and can design efficient algorithms for solving a problem. Knows algorithms to solve basic categories of problems and can use them as a building block for the design of other algorithms.

General Competences

Independent work Bibliographic search Complexity analysis of an algorithm Effective selection of an algorithm to solve a problem Design of efficient algorithms for the solution of a problem

Syllabus

• Complexity of algorithms
• Asymptomatic complexity
• Complexity analysis of algorithms
• Methods of algorithm design (divide and conquer greedy method, dynamic programming, backtracking, recursion, exhaustive search, branch and bound, etc.)
• Problems categories and corresponding algorithms (sorting, searching, selection, graph algorithms, sorting networks, matrix algorithms, integers and polynomials arithmetic, string processing, computational geometry, etc.)
• Complexity classes P, NP, NP-complete, etc.
• Specific topics

Teaching and Learning Methods - Evaluation

Delivery	Face to face
Use of Information and Communications Technology	Yes
Teaching Methods	Activity Semester Workload Lectures 39 Self study 78 Exercises 70.5 Course total 187.5
Student Performance Evaluation	Final examination (40%) comprised of: questions about the design and analysis of algorithms and their properties questions requiring critical thinking Exercises: design, analysis, implementation, algorithm properties (30%). Presentations of related issues (30%).

Attached Bibliography

• Cormen, Leiserson and Rivest, Introduction to Algorithms, MIT Press, 1990. (επίσης μεταφρασμένο από τις Πανεπιστημιακές Εκδόσεις Κρήτης)
• Δομές δεδομένων, αλγόριθμοι και εφαρμογές c++, Sahnii Sartaj, Εκδόσεις α. Τζιόλα
• Αλγόριθμοι σε C++, μέρη 1-4: θεμελιώδεις έννοιες, δομές δεδομένων, ταξινόμηση, αναζήτηση, Robert Sedgewick, Εκδόσεις Κλειδάριθμος
• Αλγοριθμοι σε C, μέρη 1-4: θεμελιώδεις έννοιες, δομές δεδομένων, ταξινόμηση, αναζήτηση, Robert Sedgewick, Εκδόσεις Κλειδάριθμος
• Mark Allen Weiss, Data Structures & Algorithm Analysis in Java, Addison-Wesley
• Clifford A. Shaffer, Data Structures and Algorithm Analysis, ebook, http://people.cs.vt.edu/shaffer/Book/
• A. Aho, J. Hopcroft, J. Ullman, (1983). Data Structures and Algorithms, Addison-Wesley.
• Baase, S., Computer Algorithms - Introduction to Design and Analysis, Second Edition, , Addison Wesley, Reading, Massachusetts, 1988.
• Knuth, D., The Art of Computer Programming - Vol. 1 Fundamental Algorithm, Addison Wesley, Reading, Massachusetts, 1973.
• Knuth, D., The Art of Computer Programming - Vol. 2 SemiNumerical Algorithm, Addison Wesley, Reading, Massachusetts, 1973.
• Knuth, D., The Art of Computer Programming - Vol. 3 Sorting and Searching, Addison Wesley, Reading, Massachusetts, 1973.

Algorithmic Graph Theory (ΠΛ4)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	ΠΛ4
Semester	2
Course Title	Algorithmic Graph Theory
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Special Background
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes	This course aims at introducing to students fundamental algorithmic techniques for solving problems related and modeled by graphs. After successfully passing this course the students will be able to: Understand graph theory. Design and analyze algorithms for graph problems. Understand difficult problems on graph classes.
General Competences	Search for, analysis and synthesis of data and information, with the use of the necessary technology Working independently Team work Project planning and management All the above will give to the stundetns the opportunity to work in an international multidisciplinary environment.

Syllabus

1. Fundamental Graph Theory
2. Algorithmic and Combinatorial Graph Problems
3. Complexity Classes and Parameterized Algorithms
4. Chordal graphs, Comparability graphs, Split graphs
5. Permutation graphs, Interval graphs, Cographs, Threshold graphs
6. Algorithmic problems and width parameters

Teaching and Learning Methods - Evaluation

Delivery	In the class
Use of Information and Communications Technology	Use of projector and interactive board during lectures.
Teaching Methods	Activity Semester Workload Lectures 39 Study and analysis of bibliography 78 Preparation of assignments and interactive teaching 70.5 Course total 187.5
Student Performance Evaluation	Written work (50%) Essay / report (20%) Public presentation (30%)

Attached Bibliography

• Cormen, Leiserson and Rivest, Introduction to Algorithms, MIT Press, 1990. (επίσης μεταφρασμένο από τις Πανεπιστημιακές Εκδόσεις Κρήτης)
• Δομές δεδομένων, αλγόριθμοι και εφαρμογές c++, Sahnii Sartaj, Εκδόσεις α. Τζιόλα
• Αλγόριθμοι σε C++, μέρη 1-4: θεμελιώδεις έννοιες, δομές δεδομένων, ταξινόμηση, αναζήτηση, Robert Sedgewick, Εκδόσεις Κλειδάριθμος
• Αλγοριθμοι σε C, μέρη 1-4: θεμελιώδεις έννοιες, δομές δεδομένων, ταξινόμηση, αναζήτηση, Robert Sedgewick, Εκδόσεις Κλειδάριθμος
• Mark Allen Weiss, Data Structures & Algorithm Analysis in Java, Addison-Wesley
• Clifford A. Shaffer, Data Structures and Algorithm Analysis, ebook, http://people.cs.vt.edu/shaffer/Book/
• A. Aho, J. Hopcroft, J. Ullman, (1983). Data Structures and Algorithms, Addison-Wesley.
• Baase, S., Computer Algorithms - Introduction to Design and Analysis, Second Edition, , Addison Wesley, Reading, Massachusetts, 1988.
• Knuth, D., The Art of Computer Programming - Vol. 1 Fundamental Algorithm, Addison Wesley, Reading, Massachusetts, 1973.
• Knuth, D., The Art of Computer Programming - Vol. 2 SemiNumerical Algorithm, Addison Wesley, Reading, Massachusetts, 1973.
• Knuth, D., The Art of Computer Programming - Vol. 3 Sorting and Searching, Addison Wesley, Reading, Massachusetts, 1973.

Symbolic Computations (ΠΛ5)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	ΠΛ5
Semester	2
Course Title	Symbolic Computations
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Specialization
Prerequisite Courses	Undergraduate courses in Data structures, Design and Analysis of Algorithms, Algebraic Structures, (optionally a course in Discrete Mathematics).
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes

The purpose of the course is an in-depth study of computer algebra and the algorithms used for the symbolic processing of mathematical expressions. The goal is the understanding of the algorithms and the applications of computer algebra and the training of the students in critical thinking for problem solving as well as the research process. Many basic computer algebra algorithms as well as advanced ones are examined and analyzed. Application of these algorithms is also discussed. With the completion of the course the student: Knows how mathematical objects are represented Knows the basic algorithms for symbolic algebraic computations as well as some more advanced algorithms Can use specialize software packages for the symbolic processing of mathematical expressions Can apply the necessary symbolic algebra algorithms for the solution of mathematical problems

General Competences

Working Independently Competence in Bibliographic search Application of symbolic algebra procedures and algorithms for the solution of a mathematical problem Use specific software in the area of computer algebra

Syllabus

• Introduction to computer algebra
• Symbolic computations compared to numerical computations.
• Basic algebraic structures.
• Representation of numbers, polynomials (one or many variables), rational expressions, functions, series.
• Simplifications of symbolic mathematical expressions.
• Basic algorithms: Greatest common devisor, Chinese remainder algorithm.
• Basic operations and algorithms on integers and polynomials.
• Integer and polynomial factorization.
• Modular algorithms.
• Linear algebra algorithms, solution of equations and systems.
• Gröbner bases and applications.
• Algorithms for symbolic integration and summation.
• Symbolic solution of differential equations.
• Software systems for the symbolic manipulation of mathematical expressions.
• Special topics

Teaching and Learning Methods - Evaluation

Delivery	Face to face
Use of Information and Communications Technology	Yes
Teaching Methods	Activity Semester Workload Lectures 39 Independent study 78 Exercises 70.5 Course total 187.5
Student Performance Evaluation	Final exam (40%) comprised of: Questions on the representation of mathematical data and the use of algorithms for the symbolic processing of mathematical expressions Questions requiring critical thinking Exercises - problem solution, programming using computer algebra software (30%). Presentations of related topics (30%).

Attached Bibliography

• Joel S. Cohen, "Computer Algebra and Symbolic Computation: Elementary Algorithms" Publisher: A K Peters/CRC Press, 2002
• Joel S. Cohen, "Computer Algebra and Symbolic Computation: Mathematical Methods" Publisher: A K Peters/CRC Press, 2003
• Keith O. Geddes, Stephen R. Czapor, George Labahn, “Algorithms for Computer Algebra”, Springer, 1992
• Davenport, J.H. and Siret, Y. and Tournier, E., Copmuter Algebra: Systems and Algorithms for Algebraic Computation, Academic Press, 1988.
• Akritas, A., Elements of Computer Algebra with Applications, Jhon Wiley, 1989,
• Modern Computer Algebra, Second Edition Joachim Von Zur Gathen, Juergen Gerhard Cambridge University Press, Cambridge, 2003.
• Computer algebra handbook. Foundations. Applications. Systems. Edited by Johannes Grabmeier, Erich Kaltofen and Volker Weispfenning. Springer-Verlag, Berlin, 2003.
• http://www.journals.elsevier.com/journal-of-symbolic-computation/

Natural Language Processing (ΠΛ6)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	ΠΛ6
Semester	1
Course Title	Natural Language Processing
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Specialization
Prerequisite Courses	Undergraduate courses in Automata Theory and Formal Languages, Introduction to Natural language Processing.
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes

The goal of this course is the deeper understanding of Natural Language Processing which concern to: the NL linguistics data formalization the codification of the NL syntax, morphology and semantics structure rules the parsing and generation algorithms of NL sentences as well as the introduction of students to critical thinking and research process. During the course a detailed examination of the above topics is done. After completing the course the student can handle theoretical documentation of problems and solving exercises, which are related to: definition and design of syntactic structure or phrase structure grammars as well as algorithms and syntactic analysis technics. formalization of morphological rules, design data bases and expert systems as well as algorithms and morphological analysis technics. formalization of semantic rules, design data bases and expert systems as well as algorithms and semantic analysis technics.

General Competences

Independent work Bibliographic search Effective selection and Design of the required machine and language.

Syllabus

• Properties of the Computation Theory Mathematical Models
• Problems classification to solvable and unsolvable
• Solvable Problems Classification

Teaching and Learning Methods - Evaluation

Delivery	Face to face
Use of Information and Communications Technology	Yes
Teaching Methods	Activity Semester Workload Lectures 39 Self study 78 Exercises 70.5 Course total 187.5
Student Performance Evaluation	Final essays (40%) . Exercises -questions requiring critical thinking (30%). Presentations of related issues (30%).

Attached Bibliography

• Mitkov Ruslan, The Oxford Handbook of Computational Linguistics. ISBN 0-19-823882
• Jurafsky Daniel & Martin H. James Speech and Language Processing - An Introduction to Ntural Language Proocessing, Computational Linguistics and Speech Recognition. ISBN 0-13-095069-6
• ALLEN James Natural Language Understanding. ISBN 0-8053-0334-0
• Natural Language Generation ed. by Gerard Kempen. ISBN 90-247-3558-0.

Cryptography Algorithms and Security of Information Systems (ΠΛ7)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	ΠΛ7
Semester	2
Course Title	Cryptography Algorithms and Security of Information Systems
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Specialized general knowledge
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes

Upon completion of this course postgraduate students: They will understand the fundamental concepts in the security of information systems and networks They will be able to identify vulnerabilities, program-level and service-level threats, and system and network-level risk assessments, and apply methodologies for identifying and addressing such risks They will be able to describe basic access control models and policies and be able to develop an appropriate security policy and the necessary protection mechanisms that will support it over an information system They will be able to develop cryptographic algorithms and implement encryption methods over a networking environment, to develop secure service mechanisms and algorithms using programming languages such as C / C ++ and libraries such as Libgcrypt and Libmcrypt. They will know the basic security features of network and network applications, methods of attacking (local-remote) and distinguish their criticality. They will be able to distinguish the basic vulnerabilities of Web and application service systems, especially SQL-injection attacks and buffer overflows, and develop appropriate defensive mechanisms. They will understand security concepts and technologies. They will have the ability to analyze the risk and synthesis of policies and technologies within an integrated IT security plan.

General Competences

Development and explanation of theoretical modules, presentation of specific case studies, analysis and evaluation of representative security technologies, security methods and algorithms for authentication, integrity and data encryption Developing secure software applications and services Making decisions, addressing real security and privacy issues Strategic planning and implementation – Embellishment Autonomous Work

Syllabus

1. Basic Concepts and Definitions in the Security of Information Systems.
2. Cryptographic issues: Symmetric, non-symmetric encryption, fraction cipher, cryptographic modes and public key cryptography flow, cryptographic summary functions, cryptanalysis.
3. Authentication Protocols and Authentication technologies, One-way hash functions, digital certificates, digital signatures, infrastructure (PKI) and public key algorithms (RSA, DSA, Diffie-Hellman, Elliptic curve) and number theory underlies.
4. Development and implementation of C/C++ cryptographic algorithms: Symmetric (DES, AES, 3DES, Blowfish)
5. Implementation in C / C++ public key security mechanisms (RSA, Diffie-Hellman), and summary (MD5, SHA)
6. Secure elections, anonymous protocols- Anonymity, Transaction Protocols, NFC protocols and security mechanisms, RFID Crypto-1 algorithm and key exchange infrastructures
7. Malware models and categories, rootkits, viruses, exploits. Database Security: Basic concepts, models and policies for database access control and methodological framework for safe database and application services design, SQL injections, services monitoring
8. System security and management at OS level Windows and Linux. Security issues at user level and administrative management techniques.
9. Security of Services: Portscan attacks, Denial of Service attacks, MitM attacks, remote exploits, buffer overflows, Network Security: IP spoofing, ARP spoofing, hijacking, sniffing
10. Security Strategies, Network Security components and mechanisms. Perimeter Network defense and safe management: Firewalls, NIDS, designing Identification and Encryption Services
11. Presentation of SSL and x.509 certificates. Create and verify digital signature: DSA algorithm, key creation, signing, and verification. Presentation of the GPG tool for email. Creating Certificates.
12. Perimeter defense - Firewalls: Create a firewall security policy. Secure network management: Use appropriate SNMP communication software for secure network management. Introduction to IPSec, Virtual Networks, OpenVPN Tool. Presentation of NIDS tools.
13. Learn script programming language for Windows AutoIT for Administrative and security tasks and secure tasks-methods implementation

Teaching and Learning Methods - Evaluation

Delivery	Classroom
Use of Information and Communications Technology	Use of Micro-computers Laboratory
Teaching Methods	Activity Semester Workload Lectures 39 Working Independently 78 Exercises - Homework 70.5 Course total 187.5
Student Performance Evaluation	Semester work and written examination

Attached Bibliography

• Ασφάλεια Δικτύων Υπολογιστών, Σ. Γκρίτζαλης, Σ. Κάτσικας, Δ. Γκρίτζαλης, Κωδικός Ευδόξου 9675, Εκδόσεις Παπασωτηρίου-Πολιτεία, Κωδικός Ευδόξου ISBN:9789607530455, 2004.
• Ασφάλεια Δικτύων Υπολογιστών, Α. Πομπόρτσης, Γ. Παπαδημητρίου, ISBN 960-8050-88-X, Εκδόσεις Τζιόλα, 2003.
• Κρυπτογραφία για Ασφάλεια Δικτύων Αρχές και Εφαρμογές, W. Stallings, Κωδικός Ευδόξου 12777632, ΜΑΡΙΑ ΠΑΡΙΚΟΥ & ΣΙΑ ΕΠΕ, ISBN: 9789604117307, 2011
• Ασφάλεια Υπολογιστών: Αρχές και Πρακτικές 3η Έκδοση, W. Stallings, L. Brown, Κωδικός Ευδόξου 50656354, Εκδόσεις Κλειδάριθμος, ISBN: 978-960-461-668-8, 2016
• Practical Unix and Internet Security, S. Garfinkel and G. Spafford , O’Reilly, ISBN: 978-0596003234, 2003
• Cryptography and Network Security Principles and Practice, 7th Edition, W. Stallings, Pearson Education, ISBN: 978-0134444284, 2017
• Applied Cryptography 2nd Edition, B. Schneier, Wiley, ISBN: 978-0471117094, 1996
• Computer Security, D. Gollmann, J. Wiley & Sons, ISBN: 978-0470741153, 2011
• Computer Security, M. Bishop, Addison Wesley, ISBN: 978-0321247445, 2005
• Instant AutoIT scripting, E. Fez Lazo, PACKT, ISBN: 978-1-78216-578-1, 2013
• Building Internet Firewalls, 2nd Edition, E.D. Zwicky, S. Cooper and B. Chapman, O Reilly, ISBN: 1-56592-871-7, 2000
• Network Intrusion Detection, 3rd Edition, S. Northcutt and J. Novak, New Riders, ISBN: 978-0735712652, 2002
• The GNU LibGCrypt reference manual, https://www.gnupg.org/documentation/manuals/gcrypt.pdf
• The Mcrypt library, N. Mavroyanopoulos, http://mcrypt.hellug.gr/index.html
• Implementing a Secure Local Area Network Environment, S. Kontogiannis, http://spooky.math.uoi.gr/~skontog/diplo.pdf, 2003.

Distributed Computing Systems and Applications (ΠΛ8)

• Ελληνική Έκδοση
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• Department of Mathematics
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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	ΠΛ8
Semester	1
Course Title	Distributed Computing Systems and Applications
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Specialized general knowledge
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes

Within this course the graduate students will understand the basic concepts of computational systems, micro-computing systems and IoT, digital systems, basic concepts of automatic control systems and the operation of actuators and sensors. The student will extend his programming skills with distributed microcomputers programming, ARM microprocessor programming and ATMEL AVR microcontrollers, using high level programming languages such as Python, C / C ++ and Qt for the development of Graphical User Interfaces. The student will understand concepts of wired, wireless Networks-Interconnection and transport-application protocols used by grid and distributed systems and will deal with algorithms and application protocols design and implementation on Distributed computing infrastructures.

General Competences

Data search, analysis and synthesis using Information Technologies Decision making Project design and implementation Working independently

Syllabus

1. Internet of Things and its extensions to different aspects of everyday life: smart cities, houses, smart farming, tourism (Cultural IoT-Virtual Reality driven), smart wearable devices. Presentation of basic concepts of digital systems, binary logic, combinational and sequential logic
2. Computer numerical systems and architectures, Input-Output, memory management and access. Advanced SPI and I2C Microcomputer Input / Output protocols, Interrupts and Interrupt handling. Presentation of the basic parts of the ARM microcomputer and ATMega328P microcontroller and their input and output-GPIO interfaces
3. Introduction to automatic control systems, open and closed loop control, P / PI / PD / PID controllers
4. Presentation of IEEE 802.x wired and wireless protocols, basic wireless networking protocols suite: TCP / IP / UDP / ICMP. Basic Application Services Serving the IoT (HTTP / CoAP / MQTT / ReST / SOAP / SNMP) and transfer computational data
5. Presentation of the Arduino IDE and C++ programming of the ATMega328P computing system, Examples using laboratory equipment. Presentation of the Wi-Fi library, I2C and SPI library, programmable cash handling, analog inputs (A2D) and PWM outputs and triggered events. Interfacing with Arduino as well as implementation of data transmission application protocols
6. Presentation and programming of the GPIO microcomputer RPi (BCM2837), PWM outputs for actuators and interrupts, using Python and C ++. Practical applications using laboratory equipment
7. Programming TCP / UDP client-server services in Python and C++. Programming HTTP requests for CoAP and ReST services. Design and implementation of data transmission and control application protocols. FSM, encoders-decoders. Practical applications on BCM2837.
8. Design and development of applications and application protocols, computational microcomputer systems programming and client-server data transfers
9. Introduction to Graphical Interface Programming for microsystems and mobile devices in C++/Qt. Presentation of Qt and IDE development tool (QtCreator)
10. Presentation of QWidgets, signals-slots mechanism and events. Normalized Object Orientation method
11. Programming simple graphical user interfaces that receive data from distributed computational systems and sensors
12. Advanced GUI programming. Programming for ARM micro devices using Qt, Qt Containers, and implementation of application protocols included in the GUI interface

Teaching and Learning Methods - Evaluation

Delivery	Classroom
Use of Information and Communications Technology	Use of Micro-computers Laboratory
Teaching Methods	Activity Semester Workload Lectures 39 Working Independently 78 Exercises - Homework 70.5 Course total 187.5
Student Performance Evaluation	Semester work and written examination

Attached Bibliography

• Programming the Raspberry Pi, Second Edition: Getting Started with Python, S. Monk, Mc Graw Hill, ISBN: 978-1-25-958740-5, 2015.
• Aνάπτυξη Εφαρμογών με το Arduino, Π. Παπάζογλου, Σπ. Π. Λιώνης, Εκδόσεις Τζιόλα, ISBN: 978-960-418-459-0, Kωδικός Ευδόξου 41954966, 2014.
• C++ GUI programming with Qt 4, Second Edition, J. Blanchette and M. Summerfield, Prentice Hall, ISBN: 978-0132354165, 2008.
• Mastering Qt 5, G. Lazar and R. Penea, Packt, ISBN 978-1-78646-712-6, 2016.
• TCP/IP Illustrated Vol 1: The protocols, W. R. Stevens, Addison-Wesley, 1994, ISBN 0-201-63346-9.
• Unix Network Programming, Volume 1: The Sockets Networking API (3rd Edition), W.R. Stevens, B. Fenner and A.M. Rudoff, Addison-Wesley, ISBN: 978-0131411555, 2010
• Mastering Python Networking, E. Chou, Packt, ISBN:978-1-78439-700-5, 2017
• Σχεδίαση Λογικών Κυκλωμάτων και Υπολογιστών 5η Έκδοση, Μ. Mano, K. Charles and M. Tom, Εκδόσεις Τζιόλα, ISBN: 978-960-418-641-9, Κωδικός Ευδόξου 59384943, 2017
• Foundations of Python Network Programming, 2nd Edition, B. Rhodes and J. Goerzen, APress, ISBN: 978-1430230038, 2010

Special Topics in Theoretical Computer Science (ΠΛ9)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	ΠΛ9
Semester	1
Course Title	Special Topics in Theoretical Computer Science
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Elective
Prerequisite Courses	641 - Design and Analysis of Algorithms
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes

Theoretical Computer Science is the foundation of information science and its main objective is to analyze and solve computational problems that are considered to be the most difficult and most fascinating in the history of mathematics. In addition to the purely mathematical aspect, Theoretical Computer Science offers new and effective techniques for dealing with practical computational problems that arise in all areas of scientific activity. The aim of the course is to specialize in areas covered by Theoretical Computer Science, such as Cryptography, Parallel Algorithms, Advanced Scientific Calculations, Approximation Algorithms, Semantics of Programming Languages, Computational Geometry, etc. The students of the course are expected to have advanced theoretical and practical skills in a wide range of subjects of vital importance for the Theoretical Computer Science and Mathematics. It will provide students with the opportunity to gain a strong background while exploring applications of Theoretical Computer Science in other areas such as economics, physics and biology. The course includes individual exercises, summary writing and presentation of relevant research papers. The course material will be adapted and specialized according to the necessary developments and requirements.

General Competences

Search for, analysis and synthesis of data and information, with the use of the necessary technology Working independently Team work Project planning and management

Syllabus

The main objective of the course is the specialization in areas covered by Theoretical Computer Science such as:

• Cryptography
• Parallel Algorithms
• Advanced Scientific Calculations
• Approximation Algorithms
• Programming Languages Semantics
• Computational Geometry
• Algorithm Engineering

The course matterial will be adapted and specialized according to the necessary developments and requirements.

Teaching and Learning Methods - Evaluation

Delivery	Lectures
Use of Information and Communications Technology	Use of projector and interactive board during lectures.
Teaching Methods	Activity Semester Workload Lectures 39 Working independently 78 Exercises - Homework 70.5 Course total 187.5
Student Performance Evaluation	Written exercises (50%) Essay / report (20%) Public presentation (30%)

Attached Bibliography

• Ζάχος, Ε., Παγουρτζής, Α., Σούλιου, Θ., Θεμελίωση επιστήμης υπολογιστών. Αποθετήριο «Κάλλιπος», 2015.
• Christos Papadimitriou, Computational Complexity, 1998.
• J. Kleinberg and E. Tardos, Σχεδιασμός Αλγορίθμων, Εκδόσεις Κλειδάριθμος, 2008.
• T. Cormen, C. Leiserson, R. Rivest, and C. Stein, Εισαγωγή στους Αλγορίθμους, Πανεπιστημιακές Εκδόσεις Κρήτης, 2012.

Special Topics in Computer Science (ΠΛ10)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	ΠΛ10
Semester	2
Course Title	Special Topics in Computer Science
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Elective
Prerequisite Courses	641 - Design and Analysis of Algorithms
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes

The aim of the course is to specialize in areas covered by Computer Science in applied fields. It provides background in data and information management. The specialization covers cognitive domains such as Databases, Machine Learning, Artificial Intelligence, Data Mining, etc. It also addresses all issues related to the design and optimization of computer hardware and software. This includes cognitive areas such as Programming Languages and their Implementation, Compilers, Hardware Design, Computer Architecture, Operating Systems, Distributed Systems, and more. The students of the course are expected to deepen in modern data processing techniques both theoretically and practically, while also acquiring a multifaceted knowledge of the principles of computer system design and programming. The course includes individual exercises, summary writing projects and presentation of relevant research papers. The material will be adapted and specialized according to the necessary developments and requirements.

General Competences

Search for, analysis and synthesis of data and information, with the use of the necessary technology Working independently Team work Project planning and management

Syllabus

The main objective of the course is to specialize in areas covered by Computer Science in applied fields such as:

• Data Mining
• Artificial Intelligence
• Database Systems
• Security of Information Systems
• Distributed Systems
• Mobile and Wireless Networks
• Pattern Recognition
• Machine Learning
• Signal Processing

The specialized subject will be adapted and specialized according to the necessary developments and requirements.

Teaching and Learning Methods - Evaluation

Delivery	Lectures
Use of Information and Communications Technology	Use of projector and interactive board during lectures.
Teaching Methods	Activity Semester Workload Lectures 39 Working independently 78 Exercises - Homework 70.5 Course total 187.5
Student Performance Evaluation	Written exercises (50%) Essay / report (20%) Public presentation (30%)

Attached Bibliography

• Evans Alan, Martin Kendall, Poatsy Mary Anne, Εισαγωγή στην πληροφορική: Θεωρία και Πράξη, Κωδικός Βιβλίου στον Εύδοξο: 41955480, 2014
• Παπαδόπουλος, Α., Μανωλόπουλος, Ι., Τσίχλας, Κ. 2015. Εισαγωγή στην Ανάκτηση Πληροφορίας, Αποθετήριο «Κάλλιπος», 2015.
• Παρασκευάς, Μιχαήλ, Ειδικά θέματα εφαρμογών της Κοινωνίας της Πληροφορίας, Αποθετήριο «Κάλλιπος», 2015.
• Δημακόπουλος, Β. Εισαγωγή: Παράλληλα Συστήματα και Προγραμματισμός, Αποθετήριο «Κάλλιπος», 2015.

Master's Thesis

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	ΜΔ
Semester	3
Course Title	Master's Thesis
Independent Teaching Activities	Independent Study (Credits: 30)
Course Type	Special background, specialised general knowledge, skills development.
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes

The Master's Thesis is prepared individually and independently by the graduate student and constitutes an in-depth study and treatment of a possibly non-original topic that is on the frontier of the research of a specific scientific field of the science of Mathematics. This study is conducted under the supervision of the supervising professor and is based on existing literature/research. The postgraduate student utilizes the knowledge and skills acquired during the studies in order to process the subject in a synthetic and rigorous way. The goal of the Master's Thesis is for the graduate student, under the supervision of the supervising professor, to develop critical, synthetic thinking and in-depth analysis skills of the subject in a scientific manner and with due mathematical rigor. After successful completion of the Master's Thesis, the student should be able to: To develop critical and synthetic thinking. To search for topics and make use of the available literature. Design a work plan and develop methods of approach and development of the subject. To substantiate claims in a scientific and rigorous manner as befits Mathematical science. To successfully write a scientific essay. To successfully present a specific topic.

General Competences

Work autonomously, develop critical thinking skills.

Syllabus

The supervisor professor indicates bibliography and references related to the subject of the thesis.

Teaching and Learning Methods - Evaluation

Delivery	Face-to-face.
Use of Information and Communications Technology	Physical presence and use of T.P.E. for literature search.
Teaching Methods	Activity Semester Workload Autonomous Study 700 Writing a Master's Thesis 140 Course total 840
Student Performance Evaluation	The evaluation of the Master's Thesis is done by a three-member Examination Committee which is appointed by the Department's Assembly. The evaluation includes the examination of the submitted essay of the Master's Thesis and the evaluation through the public presentation by the student of the following points: Understanding and deepening the assigned topic and overview of the relevant literature. Organization, planning and strict documentation of the conclusions of the Master's Thesis. Presentation of the results. Analysis of the conclusions.

Attached Bibliography

The recommended bibliography depends on the subject under study and is provided by the supervising professor.